1. Introduction
One of the most challenging questions in heavy-ion physics arises from the attempt to understand how the deconfined quarks and gluons, i.e. the quark-gluon plasma (QGP), evolve into the confined hadrons. This evolution involves apparently two different phase transitions: one is the confinement-deconfinement phase transition and the other is the chiral phase transition related to the breaking and restoration of the chiral symmetry of QCD. When the strange quark is in its physical mass and the u and d quarks are massless, i.e. in the chiral limit, the chiral phase transition in the regime of small chemical potential in the QCD phase diagram, see e.g. figure 9, might be of second order, and belongs to the O(4) symmetry universality class [1, 2]. With the increase of the baryon chemical potential, the second-order phase transition might be changed into a first-order one at the tricritical point. When the u and d quarks are in their small, but nonvanishing physical masses, due to the explicit breaking of the chiral symmetry, the O(4) second-order chiral phase transition turns into a continuous crossover [3], which is also consistent with experimental measurements, see e.g. [4]. The tricritical point in the phase diagram evolves into a critical end point (CEP), which is the end point of the first-order phase transition line at high baryon chemical potential or densities.
Although the phase transition at the CEP is of second order and belongs to the Z(2) symmetry universality class, the location of CEP and the size of the critical region around the CEP are non-universal. From the paradigm of the QCD phase diagram described above, one can easily find that the CEP plays a pivotal role in understanding the whole QCD phase structure in terms of the temperature and the baryon chemical potential. As a consequence, it becomes a very important task to search for and pin down the location of the CEP in the QCD phase diagram. Lattice simulations provide us with a wealth of knowledge of QCD phase transitions at vanishing and small baryon chemical potential, see e.g. [5–16], but due to the sign problem at finite chemical potentials, lattice calculations are usually restricted in the region of μB/T ≲ 2–3, where no signal of CEP is observed [17]. Notably, recent estimates of the location of CEP from first-principle functional approaches, such as the functional renormalization group (fRG) and Dyson–Schwinger equations (DSE), which pass through lattice benchmark tests at small baryon chemical potentials, converge in a rather small region at baryon chemical potentials of about 600 MeV [18–20], and see also, e.g. [21] for related discussions.
Search for the CEP is currently underway or planned at many facilities, see, e.g. [4, 22–33]. Since the CEP is of second order, the correlation length increases significantly in the critical region in the vicinity of a CEP. Moreover, it is well known that fluctuation observables, e.g. the fluctuations of conserved charges, are very sensitive to the critical dynamics, and the increased correlation length would result in the increase of the fluctuations as well. Therefore, it has been proposed that a non-monotonic dependence of the conserved charge fluctuations on the beam collision energy can be used to search for the CEP in experimental measurements [34–36], see also [22]. In the first phase of the Beam Energy Scan (BES-I) program at the relativistic heavy ion collider (RHIC) in the last decade, cumulants of net-proton, net-charge and net-kaon multiplicity distributions of different orders, and their correlations have been measured [37–44]. Notably, recently a non-monotonic dependence of the kurtosis of the net-proton multiplicity distribution on the collision energy is observed with 3.1 σ significance for central gold-on-gold (Au+Au) collisions [42].
In this work, we would like to present an overview on recent progress in studies of QCD at finite temperature and densities within the fRG approach. The fRG is a nonperturbative continuum field approach, in which quantum, thermal and density fluctuations are integrated successively with the evolution of the renormalization group (RG) scale [45], see also [46, 47]. For QCD-related reviews, see, e.g. [48–55]. Remarkably, recent years have seen significant progress in first-principle fRG calculations, for example, the state-of-the-art quantitative fRG results for Yang–Mills theory in the vacuum [56] and at finite temperature [57], vacuum QCD results in the quenched approximation [58], unquenched QCD in the vacuum [59–61] and at finite temperature and densities [18, 62].
In this paper, we try to present a self-contained overview, which includes some fundamental derivations. Although new researchers in this field, e.g. students, may find these derivations useful, familiar readers could just skip over them. Furthermore, in this paper we focus on studies of fRG at finite temperature and densities, so we have to give up some topics, which in fact are very important for the developments and applications of the fRG approach, such as the quantitative fRG calculations to QCD in the vacuum [56, 58, 61].
This paper is organized as follows: in section 2 we introduce the formalism of the fRG approach, including the Wetterich equation, the flow equations of correlation functions, the technique of dynamical hadronization, etc. Moreover, we also give a brief discussion about Wilson's recursion formula and the Polchinski equation, which are closely related to the fRG approach. In section 3 we discuss the application of fRG in the low energy effective field theories (LEFTs), including the Nambu–Jona-Lasinio (NJL) model and the quark-meson model. The relevant results in LEFTs, e.g. the phase structure, the equation of state (EoS), baryon number fluctuations, and critical exponents, are presented. In section 4 we turn to the application of fRG to QCD at finite temperature and densities. After a discussion about the flows of the propagators, strong couplings, four-quark couplings and the Yukawa couplings, we present and discuss the relevant results, e.g. the natural emergence of LEFTs from QCD, several different chiral condensates, QCD phase diagram and QCD phase structure, the inhomogeneous instability at large baryon chemical potentials, the magnetic EoS, etc. In section 5 we discuss the real-time fRG. After the derivation of the fRG flows on the Schwinger–Keldysh closed time path, one formulates the real-time effective action in terms of the ‘classical' and ‘quantum' fields in the physical representation. The spectral functions and the dynamical critical exponent of the O(N) scalar theory are discussed. In section 6 a summary with conclusions is given. Moreover, an example for the flow equations of the gluon and ghost self-energies in Yang–Mills theory at finite temperature is given in appendix A . The Fierz-complete basis of four-quark interactions of Nf = 2 flavors is listed in appendix B , and some useful flow functions are collected in appendix C .
2. Formalism of the fRG approach
We begin the section with a derivation of the Wetterich equation, i.e. the flow equation of the effective action, which is followed by a brief introduction about Wilson's recursion formula and the Polchinski equation, since they are closely related to the fRG approach. Then, the fRG approach is applied to QCD. A simple method to obtain the flow equations of correlation functions, i.e. equation (61 ), is presented. An example for the flow equations of the gluon and ghost self-energies in Yang–Mills theory at finite temperature is given in appendix A . Finally, the technique of dynamical hadronization is discussed in section 2.5 .
2.1. Flow equation of the effective action
We begin with a generating functional for a classical action $S[\hat{{\rm{\Phi }}}]$ with an infrared (IR) regulator as follows$\hat{{\rm{\Phi }}}$ is a collective symbol for all fields relevant in a specific physical problem, and the hat on the field is used to distinguish from its expected value in the following. It can even include fields which do not appear in the original classical action, and we will come back to this topic in what follows. The suffix of $\hat{{\rm{\Phi }}}$ , a, denotes not only the discrete degrees of freedom, e.g. the species, inner components of fields, etc, but also the continuous spacetime coordinates or the energy and momenta. The external source Ja is conjugated to ${\hat{{\rm{\Phi }}}}_{a}$ . A summation or/and integral is assumed for a repeated index as shown in equation (1 ). The k-dependent regulator ${\rm{\Delta }} {S}_{k}[\hat{{\rm{\Phi }}}]$ in equation (1 ) is used to suppress quantum fluctuations of momenta q ≲ k, while leaving those of q > k untouched. Usually, a bilinear term, convenient in actual computations, is adopted for the regulator, which reads${R}_{k}^{{ab}}={R}_{k}^{{ba}}$ for bosonic indices and ${R}_{k}^{{ab}}=-{R}_{k}^{{ba}}$ for fermionic ones. See e.g. [49] for discussions about generic regulators. We will see in the following that the IR cutoff k here is essentially the RG scale.
$\begin{eqnarray} {Z}_{k}[J]=\int ({ \mathcal D }\hat{{\rm{\Phi }}})\exp \left\{-S[\hat{{\rm{\Phi }}}]-{\rm{\Delta }} {S}_{k}[\hat{{\rm{\Phi }}}]+{J}^{a} {\hat{{\rm{\Phi }}}}_{a}\right\},\end{eqnarray}$
where the field $\begin{eqnarray} {\rm{\Delta }} {S}_{k}[\hat{{\rm{\Phi }}}]=\displaystyle \frac{1} {2} {\hat{{\rm{\Phi }}}}_{a} {R}_{k}^{{ab}} {\hat{{\rm{\Phi }}}}_{b},\end{eqnarray}$
with We proceed to taking a single scalar field φ for example, and the relevant regulator reads8 ). Here, we present two classes of regulators that are frequently used in the literature: one is the exponential-like regulator as follows
$\begin{eqnarray} {\rm{\Delta }} {S}_{k}[\varphi ]=\displaystyle \frac{1} {2}\int {{\rm{d}}}^{4}x{{\rm{d}}}^{4}y\varphi (x){R}_{k}(x,y)\varphi (y).\end{eqnarray}$
It is more convenient to work in the momentum space. Employing the Fourier transformation as follows $\begin{eqnarray}\varphi (x)=\int \displaystyle \frac{{{\rm{d}}}^{4}q} {{\left(2\pi \right)}^{4}}\varphi (q){{\rm{e}}}^{{\rm{i}} {qx}},\end{eqnarray}$
$\begin{eqnarray} {R}_{k}(x,y)=\int \displaystyle \frac{{{\rm{d}}}^{4}q} {{\left(2\pi \right)}^{4}} {R}_{k}(q){{\rm{e}}}^{{\rm{i}}q(x-y)},\end{eqnarray}$
one is led to $\begin{eqnarray} {\rm{\Delta }} {S}_{k}[\varphi ]=\displaystyle \frac{1} {2}\int \displaystyle \frac{{{\rm{d}}}^{4}q} {{\left(2\pi \right)}^{4}}\varphi (-q){R}_{k}(q)\varphi (q).\end{eqnarray}$
Here, the regulator has the following asymptotic properties which read $\begin{eqnarray} {R}_{k\to \infty }(q)\to \infty ,\quad \mathrm{and}\quad {R}_{k\to 0}(q)\to 0,\end{eqnarray}$
with a fixed q. In order to fulfill the aforementioned Requirement, viz., only suppressing quantum fluctuations of momenta q ≲ k selectively, one could make the choice as follows $\begin{eqnarray} {\left.{R}_{k}(q)\right|}_{q\lt k}\sim {k}^{2},\qquad {\left.{R}_{k}(q)\right|}_{q\gt k}\sim 0.\end{eqnarray}$
One may have already noticed that there are infinite regulators fulfilling equation ( $\begin{eqnarray} {R}_{k}^{\exp ,n}(q)={q}^{2} {r}_{\exp ,n}({q}^{2}/{k}^{2}),\end{eqnarray}$
with $\begin{eqnarray} {r}_{\exp ,n}(x)=\displaystyle \frac{{x}^{n-1}} {{{\rm{e}}}^{{x}^{n}}-1}.\end{eqnarray}$
The sharpness of the regulator in the vicinity of q = k is determined by the parameter n, as shown in figure 1, where the exponential regulators with n = 1 and 2 are depicted as functions of q with a fixed k. In figure 1 we also plot another commonly used regulator, i.e. the flat or optimized one [63, 64], which reads $\begin{eqnarray} {R}_{k}^{\mathrm{opt}}(q)={q}^{2} {r}_{\mathrm{opt}}({q}^{2}/{k}^{2}),\end{eqnarray}$
with $\begin{eqnarray} {r}_{\mathrm{opt}}(x)=\left(\displaystyle \frac{1} {x}-1\right){\rm{\Theta }}(1-x),\end{eqnarray}$
where Θ(x) is the Heaviside step function. Moreover, the derivative of the regulator with respect to the RG scale k, to wit, $\begin{eqnarray} {\partial }_{t} {R}_{k}(q)\equiv k{\partial }_{k} {R}_{k}(q),\end{eqnarray}$
is also shown in figure 1, where t is usually called as the RG time. For more discussions about regulators, see, e.g. [65].
Figure 1. Comparison of several different regulators and their derivatives with respect to the RG scale k as functions of the momentum q with a fixed k. |
It is more convenient to use the generating functional for connected correlation functions, viz.,18 ) into (17 ) and differentiating both sides of equation (17 ) with respect to Φa, one is led to16 ) is readily written as the inverse of the second-order derivative of Γk[Φ] w.r.t. Φ, i.e.
$\begin{eqnarray} {W}_{k}[J]=\mathrm{ln} {Z}_{k}[J],\end{eqnarray}$
which is also known as the Schwinger function. Then, the expected value of a field is readily obtained as $\begin{eqnarray} {{\rm{\Phi }}}_{a}\equiv \langle {\hat{{\rm{\Phi }}}}_{a}\rangle =\displaystyle \frac{\delta {W}_{k}[J]} {\delta {J}^{a}},\end{eqnarray}$
and the propagator reads $\begin{eqnarray}\begin{array} {rcl} {G}_{k,{ab}} & \equiv & \langle {\hat{{\rm{\Phi }}}}_{a} {\hat{{\rm{\Phi }}}}_{b} {\rangle }_{c}=\langle {\hat{{\rm{\Phi }}}}_{a} {\hat{{\rm{\Phi }}}}_{b}\rangle -\langle {\hat{{\rm{\Phi }}}}_{a}\rangle \langle {\hat{{\rm{\Phi }}}}_{b}\rangle \\ & = & \displaystyle \frac{{\delta }^{2} {W}_{k}[J]} {\delta {J}^{a} {J}^{b}},\end{array}\end{eqnarray}$
where the subscript c stands for ‘connected'. Moreover, Legendre transformation to the Schwinger function leaves us immediately with the one-particle irreducible (1PI) effective action, which reads $\begin{eqnarray} {{\rm{\Gamma }}}_{k}[{\rm{\Phi }}]=-{W}_{k}[J]+{J}^{a} {{\rm{\Phi }}}_{a}-{\rm{\Delta }} {S}_{k}[{\rm{\Phi }}].\end{eqnarray}$
In order to take both bosonic and fermionic fields into account altogether, we adopt the notation introduced in [49], to wit, $\begin{eqnarray} {J}^{a} {{\rm{\Phi }}}_{a}={\gamma }_{\,b}^{a} {{\rm{\Phi }}}_{a} {J}^{b},\end{eqnarray}$
with $\begin{eqnarray} {\gamma }_{\,{b}}^{{a}}=\left\{\begin{array} {ll}-{\delta }_{\,{b}}^{{a}}, & {a}\ \mathrm{and}\ {b}\ \mathrm{are}\ \mathrm{fermionic},\\ {\delta }_{\,{b}}^{{a}}, & \mathrm{others}.\end{array}\right.\end{eqnarray}$
Inserting equation ( $\begin{eqnarray}\displaystyle \frac{\delta ({{\rm{\Gamma }}}_{k}[{\rm{\Phi }}]+{\rm{\Delta }} {S}_{k}[{\rm{\Phi }}])} {\delta {{\rm{\Phi }}}_{a}}={\gamma }_{\,b}^{a} {J}^{b},\end{eqnarray}$
whereby, the propagator in equation ( $\begin{eqnarray} {G}_{k,{ab}}={\gamma }_{\,a}^{c} {\left({{\rm{\Gamma }}}_{k}^{(2)}[{\rm{\Phi }}]+{\rm{\Delta }} {S}_{k}^{(2)}[{\rm{\Phi }}]\right)}_{{cb}}^{-1},\end{eqnarray}$
with $\begin{eqnarray} {\left({{\rm{\Gamma }}}_{k}^{(2)}[{\rm{\Phi }}]+{\rm{\Delta }} {S}_{k}^{(2)}[{\rm{\Phi }}]\right)}^{{ab}}\equiv \displaystyle \frac{{\delta }^{2}({{\rm{\Gamma }}}_{k}[{\rm{\Phi }}]+{\rm{\Delta }} {S}_{k}[{\rm{\Phi }}])} {\delta {{\rm{\Phi }}}_{a}\delta {{\rm{\Phi }}}_{b}}.\end{eqnarray}$
We proceed to consider the evolution of the Schwinger function with the RG scale, i.e. the flow equation of Wk[J], which is straightforwardly obtained from equations (1 ) and (14 ). The resulting flow reads23 ) we have used the relation in equation (16 ). Applying Legendre transformation in equation (17 ) to the flow equation of Schwinger function in equation (23 ) once more, one immediately arrives at the flow equation of the effective action, as follows25 ) would be modified when the dynamical hadronization is encoded, see equation (80 ). In section 2.3 we would like to give an example for the application of fRG, and postpone discussions of the dynamical hadronization in section 2.5 .
$\begin{eqnarray} {\partial }_{t} {W}_{k}[J]=-\displaystyle \frac{1} {2}\mathrm{STr}\left[\left({\partial }_{t} {R}_{k}\right){G}_{k}\right]-\displaystyle \frac{1} {2} {{\rm{\Phi }}}_{a} {\partial }_{t} {R}_{k}^{{ab}} {{\rm{\Phi }}}_{b},\end{eqnarray}$
where we have introduced a notation super trace for compactness, which can also be expressed as $\begin{eqnarray}\mathrm{STr}\left[\left({\partial }_{t} {R}_{k}\right){G}_{k}\right]=\left({\partial }_{t} {R}_{k}^{{ab}}\right){\gamma }_{b}^{c} {G}_{k,{ca}}.\end{eqnarray}$
The factor γ in the equation above indicates that the super trace provides an additional minus sign for fermionic degrees of freedom. Note that in deriving equation ( $\begin{eqnarray}\begin{array} {rcl} {\partial }_{t} {{\rm{\Gamma }}}_{k}[{\rm{\Phi }}] & = & -{\partial }_{t} {W}_{k}[J]-{\partial }_{t} {\rm{\Delta }} {S}_{k}[{\rm{\Phi }}]\\ & = & \displaystyle \frac{1} {2}\mathrm{STr}\left[\left({\partial }_{t} {R}_{k}\right){G}_{k}\right],\end{array}\end{eqnarray}$
which is the Wetterich equation [45]. Note that the flow equation of effective action in equation (2.2. From Wilson's RG to Polchinski equation
The idea encoded in the Wetterich equation in equation (25 ) is that integrating out high momentum modes leaves us with an RG rescaled theory, and this theory is invariant at a second-order phase transition. This idea is also reflected in Wilson's RG and Polchinski equation, to be discussed in this subsection.
2.2.1. Wilson's RG and recursion formula
Here we follow [66] and begin with the Ginzburg–Landau Hamiltonian ${ \mathcal K }\equiv H[\sigma ]/T$ , which reads28 ) can also be replaced by that in terms of localized wave packets Wz(x) , i.e. the Wannier functions for the band of plane waves Λ/2 < q < Λ, that is,$\left|{\boldsymbol{x}}-{\boldsymbol{z}}\right|\gg 2{{\rm{\Lambda }}}^{-1}$ , and one also has27 ) into (26 ) and performing a functional integral over $\tilde{\sigma }({\boldsymbol{x}})$ or ${\tilde{\sigma }}_{z}$ , one arrives at$\bar{U}^{\prime} (0)=0;$ the mean square wave vector of the packet reads31 ), one has neglected the overlap between wave packets, the variation of $\sigma ^{\prime} ({\boldsymbol{x}})$ and the absolute value of Wz(x) within a block, and see [66–68] for details. Note that equation (31 ) is just the first step of the Kadanoff transformation.
$\begin{eqnarray} { \mathcal K }=\int {{\rm{d}}}^{d}x\left[\displaystyle \frac{1} {2}c{\left({\rm{\nabla }}\sigma \right)}^{2}+U(\sigma )\right].\end{eqnarray}$
Then one separates the field into two parts as follows $\begin{eqnarray}\sigma =\sigma ^{\prime} +\tilde{\sigma },\end{eqnarray}$
with $\begin{eqnarray}\tilde{\sigma }({\boldsymbol{x}})={L}^{-d/2}\sum _{{\rm{\Lambda }}/2\lt q\lt {\rm{\Lambda }}} {\sigma }_{{\boldsymbol{q}}} {{\rm{e}}}^{{\rm{i}} {\boldsymbol{q}}\cdot {\boldsymbol{x}}},\end{eqnarray}$
where L denotes the size of the system, Λ is a UV cutoff scale and Λ−1 can be regarded as the lattice spacing. The plane-wave expansion in equation ( $\begin{eqnarray}\tilde{\sigma }({\boldsymbol{x}})=\sum _{z} {\tilde{\sigma }}_{z} {W}_{z}({\boldsymbol{x}}).\end{eqnarray}$
Obviously, the wave packets have the property Wz(x) ∼ 0, if $\begin{eqnarray}\int {{\rm{d}}}^{d} {{xW}}_{z}({\boldsymbol{x}}){W}_{z^{\prime} }({\boldsymbol{x}})={\delta }_{{zz}^{\prime} },\end{eqnarray}$
i.e. they are orthonormal. Substituting equation ( $\begin{eqnarray}\begin{array} {l}\int ({ \mathcal D }\tilde{\sigma }){{\rm{e}}}^{-{ \mathcal K }}=\exp \left[-\int {{\rm{d}}}^{d}x\displaystyle \frac{1} {2}c{\left({\rm{\nabla }}\sigma ^{\prime} \right)}^{2}\right]\prod _{z}I(\sigma ^{\prime} )\\ =\,\exp \left\{-\int {{\rm{d}}}^{d}x\left[\displaystyle \frac{1} {2}c{\left({\rm{\nabla }}\sigma ^{\prime} \right)}^{2}+\bar{U}^{\prime} (\sigma ^{\prime} )\right]-{{AL}}^{d}\right\},\end{array}\end{eqnarray}$
with $\begin{eqnarray}\begin{array} {rcl}I(\sigma ^{\prime} ) & \equiv & {{\rm{\Omega }}}^{\tfrac{1} {2}}\int {\rm{d}}y\exp \left\{-\displaystyle \frac{c} {2}\bar{{q}^{2}} {\rm{\Omega }} {y}^{2}-\displaystyle \frac{{\rm{\Omega }}} {2}\left[U(\sigma ^{\prime} +y)\right.\right.\\ & & \left.\left.+U(\sigma ^{\prime} -y)\right]\right\}\\ & \equiv & \exp \left[-{\rm{\Omega }}\bar{U}^{\prime} (\sigma ^{\prime} )-{\rm{\Omega }}A\right],\end{array}\end{eqnarray}$
where Ω is the volume of a block or the wave packet; the constant A is determined by the condition $\begin{eqnarray}\bar{{q}^{2}}=\int {{\rm{d}}}^{d}x{\left({\rm{\nabla }} {W}_{z}({\boldsymbol{x}})\right)}^{2}.\end{eqnarray}$
In deriving equation (The second step of the Kadanoff transformation is to make a replacement for the remaining field $\sigma ^{\prime} $ and the coordinate in equation (31 ) as follows$c^{\prime} =c$ satisfied, one adopts η = 0. Choosing an appropriate value of c, such that32 ) is irrelevant.
$\begin{eqnarray}\sigma ^{\prime} ({\boldsymbol{x}})\to {2}^{1-\tfrac{d} {2}-\tfrac{\eta } {2}}\sigma ({\boldsymbol{x}}^{\prime} ),\qquad {\boldsymbol{x}}^{\prime} =\displaystyle \frac{{\boldsymbol{x}}} {2},\end{eqnarray}$
and thus one is left with $\begin{eqnarray} { \mathcal K }^{\prime} =\int {{\rm{d}}}^{d}x^{\prime} \left[\displaystyle \frac{1} {2}c^{\prime} {\left({\rm{\nabla }}\sigma \right)}^{2}+U^{\prime} (\sigma )\right],\end{eqnarray}$
with $\begin{eqnarray}c^{\prime} ={2}^{-\eta }c,\end{eqnarray}$
and $\begin{eqnarray}U^{\prime} (\sigma )=-{2}^{d} {{\rm{\Omega }}}^{-1}\mathrm{ln}\displaystyle \frac{I({2}^{1-\tfrac{d} {2}-\tfrac{\eta } {2}}\sigma )} {I(0)}.\end{eqnarray}$
In order to make $\begin{eqnarray}\displaystyle \frac{c} {2}\bar{{q}^{2}} {\rm{\Omega }}=1,\end{eqnarray}$
and defining $\begin{eqnarray}Q(\sigma )\equiv {\rm{\Omega }}U(\sigma ),\end{eqnarray}$
one arrives at $\begin{eqnarray}Q^{\prime} (\sigma )=-{2}^{d}\mathrm{ln}\displaystyle \frac{I({2}^{1-\tfrac{d} {2}}\sigma )} {I(0)}\end{eqnarray}$
with $\begin{eqnarray}I(\sigma )=\int {\rm{d}}y\exp \left\{-{y}^{2}-\displaystyle \frac{1} {2}\left[Q(\sigma +y)+Q(\sigma -y)\right]\right\},\end{eqnarray}$
which is Wilson's recursion formula [67, 68]. Note that the constant prefactor Ω1/2 in equation (2.2.2. Polchinski equation
In section 2.2.1 we have discussed the viewpoint of Wilson's RG, that is, integrating out modes of high scales successively leaves us with a low energy theory that evolves with the RG scale. This idea is also applied to a generic quantum field theory within the formalism of the functional integral, due to Polchinski [69]. We begin with a generating a functional for a scalar field theory in four Euclidean dimensions with a momentum cutoff, i.e.42 ) is the interaction Lagrangian at the scale Λ0, that for example reads${\rho }_{a}^{0}$ 's stand for bare quantities.
$\begin{eqnarray}\begin{array} {rcl}Z[J] & = & \int ({ \mathcal D }\phi )\exp \left\{\int \displaystyle \frac{{{\rm{d}}}^{4}p} {{\left(2\pi \right)}^{4}}\left[-\displaystyle \frac{1} {2}\phi (p)\phi (-p)({p}^{2}+{m}^{2})\right.\right.\\ & & \left.\left.\times {K}^{-1}\left(\displaystyle \frac{{p}^{2}} {{{\rm{\Lambda }}}_{0}^{2}}\right)+J(p)\phi (-p)\right]+{L}_{\mathrm{int}}(\phi )\right\},\end{array}\end{eqnarray}$
with a cutoff function given by $\begin{eqnarray}K({p}^{2}/{{\rm{\Lambda }}}_{0}^{2})=\left\{\begin{array} {ll}1, & {p}^{2}\lt {{\rm{\Lambda }}}_{0}^{2},\\ 0, & {p}^{2}\gg {{\rm{\Lambda }}}_{0}^{2}.\end{array}\right.\end{eqnarray}$
Evidently, here Λ0 plays a role as a UV cutoff scale, and Lint in equation ( $\begin{eqnarray}\begin{array} {rcl} {L}_{\mathrm{int}}(\phi ) & = & \int {{\rm{d}}}^{4}x\left[-\displaystyle \frac{1} {2} {\rho }_{1}^{0} {\phi }^{2}(x)-\displaystyle \frac{1} {2} {\rho }_{2}^{0} {\left({\partial }_{\mu }\phi (x)\right)}^{2}\right.\\ & & \left.-\displaystyle \frac{1} {4!} {\rho }_{3}^{0} {\phi }^{4}(x)\right],\end{array}\end{eqnarray}$
where we have used the notation in [69], and One would like to integrate out the high momentum modes of φ and reduce the UV cutoff Λ0 to a lower value, say Λ. In the meantime, one chooses $\left|{m}^{2}\right|\lt {\rm{\Lambda }}$ and J(p) = 0 for p2 > Λ2. As we have discussed in section 2.2.1 , when high momentum modes are integrated out, new effective interactions, included in the potential $U^{\prime} $ in equation (35 ) or the interaction Lagrangian Lint in equation (42 ), are generated. Thus, one is led to the following functional integral45 ) has to be satisfied, to wit,
$\begin{eqnarray}\begin{array} {l}Z[J,L,{\rm{\Lambda }}]=\int ({ \mathcal D }\phi )\exp \left\{\int \displaystyle \frac{{{\rm{d}}}^{4}p} {{\left(2\pi \right)}^{4}}\left[-\displaystyle \frac{1} {2}\phi (p)\phi (-p)\right.\right.\\ \left.\left.\times \left({p}^{2}+{m}^{2}){K}^{-1}\left(\displaystyle \frac{{p}^{2}} {{{\rm{\Lambda }}}^{2}}\right)+J(p)\phi (-p\right)\right]+L(\phi ,{\rm{\Lambda }})\right\}.\end{array}\end{eqnarray}$
If one wishes to take the generating functional Z[J, L, Λ] on the l.h.s. of the equation above to be independent of the scale Λ, i.e. $\begin{eqnarray} {\rm{\Lambda }}\displaystyle \frac{{\rm{d}} {Z}[J,L,{\rm{\Lambda }}]} {{\rm{d}} {\rm{\Lambda }}}=0,\end{eqnarray}$
the following evolution equation for the interaction Lagrangian in equation ( $\begin{eqnarray}\begin{array} {rcl} {\rm{\Lambda }}\displaystyle \frac{\partial L(\phi ,{\rm{\Lambda }})} {\partial {\rm{\Lambda }}} & = & -\int \displaystyle \frac{{{\rm{d}}}^{4}p} {{\left(2\pi \right)}^{4}}\displaystyle \frac{1} {2}\displaystyle \frac{1} {{p}^{2}+{m}^{2}} {\rm{\Lambda }}\displaystyle \frac{\partial K\left(\tfrac{{p}^{2}} {{{\rm{\Lambda }}}^{2}}\right)} {\partial {\rm{\Lambda }}}\\ & & \times \,\left[\displaystyle \frac{{\delta }^{2}L} {\delta \phi (p)\delta \phi (-p)}+\displaystyle \frac{\delta L} {\delta \phi (p)}\displaystyle \frac{\delta L} {\delta \phi (-p)}\right],\end{array}\end{eqnarray}$
which is the Polchinski equation [69].2.3. Application to QCD
In this section, we would like to apply the formalism of fRG discussed above to an effective action of rebosonized QCD in [18]. The truncation for the Euclidean effective action reads${\int }_{x}={\int }_{0}^{1/T} {\rm{d}} {x}_{0}\int {{\rm{d}}}^{3}x$ , T being the temperature. One can see that field contents in equation (48 ) include not only the fundamental fields in QCD, i.e. the gluon, Faddeev–Popov ghost, and the quark, but also the composite fields φ = (σ, π), the scalar and pseudo-scalar mesons respectively. Note that here the mesonic fields are not added by hands, but rather dynamically generated and transferred from the fundamental degrees of freedom via the dynamical hadronization technique, described in detail in section 2.5 . Consequently, there is no double counting for the degrees of freedom. In short, one is left with ${\rm{\Phi }}=(A,c,\bar{c},q,\bar{q},\sigma ,\pi )$ . The hk and λq,k in equation (48 ) denote the Yukawa coupling and the four-quark coupling, respectively,
$\begin{eqnarray}\begin{array} {l} {{\rm{\Gamma }}}_{k}[{\rm{\Phi }}]\\ =\,{\int }_{x}\left\{\displaystyle \frac{1} {4} {F}_{\mu \nu }^{a} {F}_{\mu \nu }^{a}+{Z}_{c,k}\left({\partial }_{\mu } {\bar{c}}^{a}\right){D}_{\mu }^{{ab}} {c}^{b}+\displaystyle \frac{1} {2\xi } {\left({\partial }_{\mu } {A}_{\mu }^{a}\right)}^{2}\right.\\ +\,{Z}_{q,k}\bar{q}\left({\gamma }_{\mu } {D}_{\mu }-{\gamma }_{0}\hat{\mu }\right)q+{m}_{s}({\sigma }_{s}){\bar{q}}_{s} {q}_{s}-{\lambda }_{q,k}\left[{\left({\bar{q}}_{l}\,{T}^{0} {q}_{l}\right)}^{2}\right.\\ \left.+\,{\left({\bar{q}}_{l}\,{\rm{i}} {\gamma }_{5} {\boldsymbol{T}} {q}_{l}\right)}^{2}\right]+{h}_{k} {\bar{q}}_{l}\left({T}^{0}\sigma +{\rm{i}} {\gamma }_{5} {\boldsymbol{T}}\cdot {\boldsymbol{\pi }}\right){q}_{l}\\ \left.+\,\displaystyle \frac{1} {2} {Z}_{\phi ,k} {\left({\partial }_{\mu }\phi \right)}^{2}+{V}_{k}(\rho ,{A}_{0})-{c}_{\sigma }\,\sigma -\displaystyle \frac{1} {\sqrt{2}}\,{c}_{{\sigma }_{s}}\,{\sigma }_{s}\right\}\\ +\,{\rm{\Delta }} {{\rm{\Gamma }}}_{\mathrm{glue}},\end{array}\end{eqnarray}$
with The first line on the r.h.s. of equation (48 ) denotes the classical action for the glue sector, while its non-classical contributions are collected in ΔΓglue. The gauge parameter ξ = 0, i.e. the Landau gauge, is commonly adopted in the computation of functional approaches. The wave function renormalization ZΦ,k of field Φ is defined as$\bar{{\rm{\Phi }}}$ . The gluonic field strength tensor reads4.2 . Therefore, it is necessary to distinguish different strong couplings. The renormalized strong couplings in the glue sector read${\lambda }_{{A}^{3},k}$ , ${\lambda }_{{A}^{4},k}$ and ${\lambda }_{\bar{c} {cA},k}$ stand for the three-gluon, four-gluon, ghost-gluon dressing functions, respectively, as shown in equations (A42 ), (A48 ), (A36 ). In equation (50 ) the gluonic strong couplings are denoted collectively as ${\bar{g}}_{\mathrm{glue},k}$ . In the same way, the quark-gluon coupling reads${\lambda }_{\bar{q} {qA},k}$ . The covariant derivative in the fundamental and adjoint representations of the color SU(Nc) group reads$\mathrm{Tr} {t}^{a} {t}^{b}=(1/2){\delta }^{{ab}}$ .
$\begin{eqnarray}\bar{{\rm{\Phi }}}={Z}_{{\rm{\Phi }},k}^{1/2} {\rm{\Phi }},\end{eqnarray}$
with the renormalized field $\begin{eqnarray} {F}_{\mu \nu }^{a}={Z}_{A,k}^{1/2}\left({\partial }_{\mu } {A}_{\nu }^{a}-{\partial }_{\nu } {A}_{\mu }^{a}+{Z}_{A,k}^{1/2} {\bar{g}}_{\ \mathrm{glue},k} {f}^{{abc}} {A}_{\mu }^{b} {A}_{\nu }^{c}\right).\end{eqnarray}$
Note that although different strong couplings are identical in the perturbative region, they can deviate from each other in the nonperturbative or even semiperturbative regime [56–58, 61], and see also relevant discussions in section $\begin{eqnarray} {\bar{g}}_{{A}^{3},k}=\displaystyle \frac{{\lambda }_{{A}^{3},k}} {{Z}_{A,k}^{3/2}},\quad {\bar{g}}_{{A}^{4},k}=\displaystyle \frac{{\lambda }_{{A}^{4},k}^{1/2}} {{Z}_{A,k}},\quad {\bar{g}}_{\bar{c} {cA},k}=\displaystyle \frac{{\lambda }_{\bar{c} {cA},k}} {{Z}_{A,k}^{1/2}\,{Z}_{c,k}},\end{eqnarray}$
where $\begin{eqnarray} {\bar{g}}_{\bar{q} {qA},k}=\displaystyle \frac{{\lambda }_{\bar{q} {qA},k}} {{Z}_{A,k}^{1/2}\,{Z}_{q,k}},\end{eqnarray}$
with the quark-gluon dressing function $\begin{eqnarray} {D}_{\mu }={\partial }_{\mu }-{{\rm{i}} {Z}}_{A,k}^{1/2} {\bar{g}}_{\bar{q} {qA},k} {A}_{\mu }^{a} {t}^{a},\end{eqnarray}$
$\begin{eqnarray} {D}_{\mu }^{{ab}}={\partial }_{\mu } {\delta }^{{ab}}-{Z}_{A,k}^{1/2} {\bar{g}}_{\bar{c} {cA},k} {f}^{{abc}} {A}_{\mu }^{c},\end{eqnarray}$
respectively. Here fabc is the antisymmetric structure constant of the SU(Nc) group, determined from its Lie algebra [ta, tb] = ifabctc, where the generators have the normalization In equation (48 ) the formalism of Nf = 3 flavor quark is built upon that of Nf = 2 by means of the addition of a dynamical strange quark qs, whose constituent quark mass ms(σs) is determined self-consistently from an extended effective potential of SU(Nf = 2), and see [18] for more details. Therefore, we have the quark field q = (ql, qs), where the u and d light quarks are denoted by ql = (qu, qd). The light quarks interact with themself through the four-quark coupling in the σ–π channel, and they are also coupled with the σ and π mesons via the Yukawa coupling. Here Ti (i = 1, 2, 3) are the generators of the group SU(Nf = 2) in the flavor space with $\mathrm{Tr} {T}^{i} {T}^{j}=(1/2){\delta }^{{ij}}$ and ${T}^{0}=(1/\sqrt{2{N}_{f}}){{\mathbb{1}}}_{{N}_{f}\times {N}_{f}}$ with Nf = 2. In equation (48 ) $\hat{\mu }=\mathrm{diag}({\mu }_{u},{\mu }_{d},{\mu }_{s})$ stands for the matrix of quark chemical potentials.
The effective potential in equation (48 ) can be decomposed into two parts as follows48 ) cσ and ${c}_{{\sigma }_{s}}$ are the parameters of explicit chiral symmetry breaking for the light and strange scalars σ, σs, respectively.
$\begin{eqnarray} {V}_{k}(\rho ,{A}_{0})={V}_{\mathrm{glue},k}({A}_{0})+{V}_{\mathrm{mat},k}(\rho ,{A}_{0}).\end{eqnarray}$
The first term on the r.h.s. of the equation above is the glue potential, or the Polyakov loop potential. The temporal gluon field A0 is intimately related to the Polyakov loop L[A0], see e.g. [70]; the second term is the mesonic effective potential of Nf = 2 flavors, which is O(4)-invariant with ρ = φ2/2. Moreover, in the effective action in equation (Applying the fRG flow equation in equation (25 ) to the QCD effective action in equation (48 ), one immediately arrives at the QCD flow equation, which is depicted in figure 2. One can see that the flow of effective potential receives contributions from the gluon, ghost, quark, and composite fields separately, each of which is a one-loop structure. It should be emphasized that, although it is a one-loop structure, the flow is composed of full propagators which are in turn dependent on the second derivative of the effective action, see equation (21 ). Thus, the flow equation in figure 2 a functional self-consistent differential equation, which will be explored further in the following.
Figure 2. Diagrammatic representation of the flow equation for QCD effective action. The lines denote full propagators for the gluon, ghost, quark, and meson, respectively. The crossed circles stand for the infrared regulators. |
2.4. Flow equations of correlation functions
Combining equation (21 ) one can reformulate Wetterich equation in equation (25 ) slightly such that$\tilde{{\partial }_{t}}$ , hits the RG-scale dependence only through the regulator. Note that ${{\rm{\Gamma }}}_{k}^{(2)}[{\rm{\Phi }}]$ in equation (56 ) is a bit different from that in equation (22 ), and here the factor ${\gamma }_{\hspace{0.15cm}b}^{a}$ is absorbed in ${{\rm{\Gamma }}}_{k}^{(2)}[{\rm{\Phi }}]$ . A convenient way to take this into account is to use the definition as follows
$\begin{eqnarray} {\partial }_{t} {{\rm{\Gamma }}}_{k}[{\rm{\Phi }}]=\displaystyle \frac{1} {2}\mathrm{STr}\left[\tilde{{\partial }_{t}}\mathrm{ln}\left({{\rm{\Gamma }}}_{k}^{(2)}[{\rm{\Phi }}]+{R}_{k}\right)\right],\end{eqnarray}$
where the differential operator with a tilde, $\begin{eqnarray} {\left({{\rm{\Gamma }}}_{k}^{(2)}[{\rm{\Phi }}]\right)}^{{ab}}\equiv \displaystyle \frac{\overrightarrow{\delta }} {\delta {{\rm{\Phi }}}_{a}} {{\rm{\Gamma }}}_{k}[{\rm{\Phi }}]\displaystyle \frac{\overleftarrow{\delta }} {\delta {{\rm{\Phi }}}_{b}},\end{eqnarray}$
where the left and right derivatives have been adopted.In order to derive the flow equations for various correlation functions of different orders, it is useful to express ${{\rm{\Gamma }}}_{k}^{(2)}$ in equation (57 ) in terms of a matrix, and the indices of matrix correspond to different fields involved in the theory concerned, e.g. ${\rm{\Phi }}=(A,c,\bar{c},q,\bar{q},\sigma ,\pi )$ in section 2.3 . This matrix is also called as the fluctuation matrix [71]. Then, one can make the division as follows${ \mathcal P }$ is the matrix of two-point correlation functions including the regulators, and its inverse gives rise to propagators; ${ \mathcal F }$ is the matrix of interaction which includes n-point correlation functions with n > 2, and thus terms in ${ \mathcal F }$ have the field dependence. Substituting equation (58 ) into (56 ) and making the Taylor expansion in order of ${ \mathcal F }/{ \mathcal P }$ , one arrives at
$\begin{eqnarray} {{\rm{\Gamma }}}_{k}^{(2)}+{R}_{k}={ \mathcal P }+{ \mathcal F },\end{eqnarray}$
where $\begin{eqnarray}\begin{array} {rcl} {\partial }_{t} {{\rm{\Gamma }}}_{k} & = & \displaystyle \frac{1} {2}\mathrm{STr}\left[{\tilde{\partial }}_{t}\mathrm{ln}({ \mathcal P }+{ \mathcal F })\right]\\ & = & \displaystyle \frac{1} {2}\mathrm{STr} {\tilde{\partial }}_{t}\mathrm{ln} { \mathcal P }+\displaystyle \frac{1} {2}\mathrm{STr} {\tilde{\partial }}_{t}\left(\displaystyle \frac{1} {{ \mathcal P }} { \mathcal F }\right)-\displaystyle \frac{1} {4}\mathrm{STr} {\tilde{\partial }}_{t} {\left(\displaystyle \frac{1} {{ \mathcal P }} { \mathcal F }\right)}^{2}\\ & & +\displaystyle \frac{1} {6}\mathrm{STr} {\tilde{\partial }}_{t} {\left(\displaystyle \frac{1} {{ \mathcal P }} { \mathcal F }\right)}^{3}-\displaystyle \frac{1} {8}\mathrm{STr} {\tilde{\partial }}_{t} {\left(\displaystyle \frac{1} {{ \mathcal P }} { \mathcal F }\right)}^{4}+\cdots ,\end{array}\end{eqnarray}$
from which it is straightforward to obtain the flow equations for various inverse propagators and vertices at some appropriate orders.Moreover, one can also employ some well-developed Mathematica packages, e.g. DoFun [72, 73], QMeS-Derivation [74], to derive the flow equations for correlation functions. We refrain from elaborating on details about the usage of these Mathematica packages in this review, which interested readers can find in the references above, but rather would like to introduce another easy-to-use approach to derive the flow equations described in detail in section 2.4.1 .
2.4.1. A simple approach to derivation of flow equations of correlation functions
We proceed with defining a generic 1PI n-point correlation function or vertex, as follows${V}_{k}^{(n)}$ in equation (60 ) can be represented schematically as the equation as follows$\tilde{{\partial }_{t}}$ hits the RG-scale dependence only through the regulator, and thus its implementation on diagrams would give rise to the insertion of a regulator for each inner propagator.
$\begin{eqnarray} {V}_{k,{{\rm{\Phi }}}_{{a}_{1}}\cdots {{\rm{\Phi }}}_{{a}_{n}}}^{(n)}\equiv -{{\rm{\Gamma }}}_{k,{{\rm{\Phi }}}_{{a}_{1}}\cdots {{\rm{\Phi }}}_{{a}_{n}}}^{(n)}=-{\left.\left(\displaystyle \frac{{\delta }^{n} {{\rm{\Gamma }}}_{k}[{\rm{\Phi }}]} {\delta {{\rm{\Phi }}}_{{a}_{1}}\cdots \delta {{\rm{\Phi }}}_{{a}_{n}}}\right)\right|}_{{\rm{\Phi }}=\langle {\rm{\Phi }}\rangle },\end{eqnarray}$
where ⟨Φ⟩ denotes the value of Φ on its equation of motion (EoM). The flow equation of vertex $\begin{eqnarray} {\partial }_{t} {V}_{k,{{\rm{\Phi }}}_{{a}_{1}}\cdots {{\rm{\Phi }}}_{{a}_{n}}}^{(n)}={\tilde{\partial }}_{t}\left(\begin{array} {l}\mathrm{all}\ \mathrm{one}-\,\mathrm{loop}\,\,\mathrm{correction}\\ \mathrm{diagrams}\ \mathrm{of}\ {V}_{k,{{\rm{\Phi }}}_{{a}_{1}}\cdots {{\rm{\Phi }}}_{{a}_{n}}}^{(n)}\end{array}\right).\end{eqnarray}$
Note that the one-loop diagrams on the r.h.s. are comprised of full propagators and vertices. As we have mentioned above, the partial derivative with a tilde As an example, we present the flow equation of the gluon self-energy in Yang–Mills theory in figure 3. One can see that it receives contributions from the gluon loop, the tadpole of the gluon, the ghost loop, and the tadpole of the ghost. Note that the last diagram on the r.h.s. of the equation in figure 3 arises from the non-classical two-ghost–two-gluon vertex. Remarkably, the factors in front of each diagram are in agreement with those in the perturbation theory. It, however, should be emphasized once more that although these diagrams are very similar with the formalism of perturbation theory, they are essentially nonperturbative, since both propagators and vertices in these diagrams are the full ones.
Figure 3. Diagrammatic representation of the flow equation for the gluon self-energy in Yang–Mills theory, where the last diagram arises from the non-classical two-ghost–two-gluon vertex. |
In order to let readers be familiar with the computation in fRG, we present some details about the flow equations of the gluon and ghost self-energies in the Yang–Mills theory at finite temperature in appendix A .
2.5. Dynamical hadronization
In section 2.3 we have mentioned that the mesonic fields in equation (48 ) are not added by hands. On the contrary, these composite degrees of freedom are dynamically generated from fundamental ones with the evolution of the RG scale from the ultraviolet (UV) toward the infrared (IR) limit. This is done via a technique called dynamical hadronization, which was proposed in [71, 75], and subsequently the formalism was further developed in [49]. Notably, recently the explicit chiral symmetry breaking and its role within the dynamical hadronization have been investigated in detail in [18], and a flow of dynamical hadronization with manifest chiral symmetry is put forward therein. In this section, we follow [18] and present the derivation of flow equation of the dynamical hadronization.
We denote the original or fundamental degrees of freedom in QCD as $\hat{\varphi }=(\hat{A},\hat{c},\hat{\bar{c}},\hat{q},\hat{\bar{q}})$ with the expected value $\varphi =\langle \hat{\varphi }\rangle $ , whereas composite degrees of freedom are introduced via a RG scale k-dependent composite field ${\hat{\phi }}_{k}(\hat{\varphi })$ [49, 71, 75], which is a function of the fundamental field $\hat{\varphi }$ . Then the superfield reads1 ) is modified a bit such that${J}_{\varphi }\,=({J}_{A},{J}_{c},{J}_{\bar{c}},{J}_{q},{J}_{\bar{q}})$ is conjugated to the field Φ = (φ, φk), distinguished with different labels of indices, i.e.64 ) reads17 ). Note that the term of explicit chiral symmetry breaking in the effective action, e.g. −cσσ or $-{c}_{{\sigma }_{s}} {\sigma }_{s}/\sqrt{2}$ in equation (48 ), does not contribute to the flow of effective action, and thus the effective action can always be decomposed into that in the case of chiral limit plus an explicit chiral symmetry breaking term. We follow [18] and separate the explicit breaking out, such that${\bar{{\rm{\Gamma }}}}_{k}$ in equation (67 ) stands for the effective action without the explicit chiral symmetry breaking. From equation (67 ), one arrives at68 ) combined with equation (17 ) leaves us with the relation for Schwinger functions as follows${\bar{W}}_{k}$ denotes the Schwinger function in the absence of the explicit chiral symmetry breaking. Similar to equation (25 ), one arrives at${\bar{W}}_{k}[\bar{J}]=\mathrm{ln} {\bar{Z}}_{k}[\bar{J}]$ can be obtained in equation (64 ) with $\bar{J}$ in lieu of J and a chiral symmetric ${S}_{\mathrm{QCD}}[\hat{\varphi }]$ . Subsequently, one arrives at75 ) into (74 ) and then to (73 ), one is led to16 ) has been used. Employing68 ) and the fact that cσ is independent of the RG scale k, we finally obtain the flow equation of effective action with dynamical hadronization, to wit,65 ) have been replaced with the super trace and trace, respectively; the integral over the spacetime coordinate is recovered for the last term on the r.h.s. of equation (80 ). The propagator Gk,ia in equation (79 ) is relabeled with ${G}_{k,{\phi }_{i} {{\rm{\Phi }}}_{a}}$ that has a clearer physical meaning. One can see that in comparison to equation (25 ), there are two additional terms, i.e. the last two on the r.h.s. of equation (80 ), in the flow equation of the effective action. These additional terms arise from the RG scale-dependent composite field in equation (63 ), and they can be employed to implement the Hubbard–Stratonovich transformation for every value of the RG scale, which eventually transfers the degrees of freedom from quarks to bound states.
$\begin{eqnarray} {\rm{\Phi }}=(\varphi ,{\phi }_{k})=(A,c,\bar{c},q,\bar{q},{\phi }_{k}),\end{eqnarray}$
with $\begin{eqnarray} {\phi }_{k}=\langle {\hat{\phi }}_{k}(\hat{\varphi })\rangle .\end{eqnarray}$
The generating functional in equation ( $\begin{eqnarray}\begin{array} {rcl} {Z}_{k}[J] & = & \exp \left({W}_{k}[J]\right)\\ & = & \int ({ \mathcal D }\hat{\varphi })\exp \left\{-{S}_{\mathrm{QCD}}[\hat{\varphi }]-{\rm{\Delta }} {S}_{k}[\hat{\varphi },{\hat{\phi }}_{k}]+{J}_{\varphi }\cdot \hat{\varphi }\right.\\ & & \left.+{J}_{\phi }\cdot {\hat{\phi }}_{k}\right\},\end{array}\end{eqnarray}$
where the external source J = (Jφ, Jφ) with $\begin{eqnarray}J\cdot \hat{{\rm{\Phi }}}={J}^{a} {\hat{{\rm{\Phi }}}}_{a},\quad {J}_{\varphi }\cdot \hat{\varphi }={J}_{\varphi }^{\alpha } {\hat{\varphi }}_{\alpha },\quad {J}_{\phi }\cdot {\hat{\phi }}_{k}={J}_{\phi }^{i} {\hat{\phi }}_{k,i}.\end{eqnarray}$
The regulator of bilinear fields in equation ( $\begin{eqnarray} {\rm{\Delta }} {S}_{k}[\hat{\varphi },{\hat{\phi }}_{k}]={\rm{\Delta }} {S}_{k}[\hat{{\rm{\Phi }}}]=\displaystyle \frac{1} {2} {\hat{{\rm{\Phi }}}}_{a} {R}_{k}^{{ab}} {\hat{{\rm{\Phi }}}}_{b}.\end{eqnarray}$
The effective action is obtained via a Legendre transformation to the Schwinger function as shown in equation ( $\begin{eqnarray} {{\rm{\Gamma }}}_{k}[{\rm{\Phi }}]={\bar{{\rm{\Gamma }}}}_{k}[{\rm{\Phi }}]-{c}_{\sigma }\sigma ,\end{eqnarray}$
where we have concentrated on the case of Nf = 2, which can be easily extended to include the strange quark. $\begin{eqnarray}\begin{array} {rcl} {{\rm{\Gamma }}}_{k}[{\rm{\Phi }}]-J\cdot {\rm{\Phi }} & = & {\bar{{\rm{\Gamma }}}}_{k}[{\rm{\Phi }}]-{c}_{\sigma }\sigma -J\cdot {\rm{\Phi }}\\ & = & {\bar{{\rm{\Gamma }}}}_{k}[{\rm{\Phi }}]-\bar{J}\cdot {\rm{\Phi }},\end{array}\end{eqnarray}$
with $\begin{eqnarray}\bar{J}=({J}_{\varphi },{J}_{\sigma }+{c}_{\sigma },{J}_{\pi }),\end{eqnarray}$
i.e. $\begin{eqnarray} {\bar{J}}_{\sigma }={J}_{\sigma }+{c}_{\sigma },\quad {\bar{J}}_{\varphi }={J}_{\varphi },\quad {\bar{J}}_{\pi }={J}_{\pi }.\end{eqnarray}$
Equation ( $\begin{eqnarray} {W}_{k}[J]={\bar{W}}_{k}[\bar{J}],\end{eqnarray}$
where one has $\begin{eqnarray} {\bar{W}}_{k}[J]={\left.{W}_{k}[J]\right|}_{{c}_{\sigma }\to 0},\end{eqnarray}$
that is, $\begin{eqnarray} {\partial }_{t} {\bar{{\rm{\Gamma }}}}_{k}[{\rm{\Phi }}]=-{\partial }_{t} {\bar{W}}_{k}[\bar{J}]-{\partial }_{t} {\rm{\Delta }} {S}_{k}[{\rm{\Phi }}],\end{eqnarray}$
where $\begin{eqnarray}\begin{array} {l} {\partial }_{t} {\bar{W}}_{k}[\bar{J}]\\ =\,\displaystyle \frac{1} {{\bar{Z}}_{k}[\bar{J}]}\int ({ \mathcal D }\hat{\varphi })\left(-{\partial }_{t} {\rm{\Delta }} {S}_{k}[\hat{{\rm{\Phi }}}]+{\bar{J}}_{\phi }\cdot {\partial }_{t} {\hat{\phi }}_{k}\right)\\ \times \,\exp \left\{-{S}_{\mathrm{QCD}}[\hat{\varphi }]-{\rm{\Delta }} {S}_{k}[\hat{{\rm{\Phi }}}]+{\bar{J}}_{\varphi }\cdot \hat{\varphi }+{\bar{J}}_{\phi }\cdot {\hat{\phi }}_{k}\right\}.\end{array}\end{eqnarray}$
It is straightforward to obtain $\begin{eqnarray} {\partial }_{t} {\rm{\Delta }} {S}_{k}[\hat{{\rm{\Phi }}}]=\displaystyle \frac{1} {2} {\hat{{\rm{\Phi }}}}_{a}\left({\partial }_{t} {R}_{k}^{{ab}}\right){\hat{{\rm{\Phi }}}}_{b}+{\hat{\phi }}_{k,i} {R}_{k,\phi }^{{ij}}\left({\partial }_{t} {\hat{\phi }}_{k,j}\right).\end{eqnarray}$
Inserting equation ( $\begin{eqnarray}\begin{array} {rcl} {\partial }_{t} {\bar{{\rm{\Gamma }}}}_{k}[{\rm{\Phi }}] & = & \displaystyle \frac{1} {2} {G}_{k,{ab}} {\partial }_{t} {R}_{k}^{{ab}}+\langle {\hat{\phi }}_{k,i} {R}_{k,\phi }^{{ij}}\left({\partial }_{t} {\hat{\phi }}_{k,j}\right)\rangle \\ & & -{\bar{J}}_{\phi }^{i}\langle {\partial }_{t} {\hat{\phi }}_{k,i}\rangle ,\end{array}\end{eqnarray}$
where equation ( $\begin{eqnarray}\langle {\hat{\phi }}_{k,i} {R}_{k,\phi }^{{ij}}\left({\partial }_{t} {\hat{\phi }}_{k,j}\right)\rangle =\left({G}_{k,{ia}}\displaystyle \frac{\delta } {\delta {{\rm{\Phi }}}_{a}}+{\phi }_{k,i}\right){R}_{k,\phi }^{{ij}}\langle {\partial }_{t} {\hat{\phi }}_{k,j}\rangle ,\end{eqnarray}$
and $\begin{eqnarray} {\bar{J}}_{\phi }^{i}=\displaystyle \frac{\delta \left({\bar{{\rm{\Gamma }}}}_{k}[{\rm{\Phi }}]+{\rm{\Delta }} {S}_{k}[{\rm{\Phi }}]\right)} {\delta {\phi }_{k,i}},\end{eqnarray}$
one arrives at $\begin{eqnarray}\begin{array} {rcl} {\partial }_{t} {\bar{{\rm{\Gamma }}}}_{k}[{\rm{\Phi }}] & = & \displaystyle \frac{1} {2} {G}_{k,{ab}} {\partial }_{t} {R}_{k}^{{ab}}+{G}_{k,{ia}}\left(\displaystyle \frac{\delta } {\delta {{\rm{\Phi }}}_{a}}\langle {\partial }_{t} {\hat{\phi }}_{k,j}\rangle \right){R}_{k,\phi }^{{ij}}\\ & & -\langle {\partial }_{t} {\hat{\phi }}_{k,i}\rangle \displaystyle \frac{\delta {\bar{{\rm{\Gamma }}}}_{k}[{\rm{\Phi }}]} {\delta {\phi }_{k,i}}.\end{array}\end{eqnarray}$
Given the relation in equation ( $\begin{eqnarray}\begin{array} {l} {\partial }_{t} {{\rm{\Gamma }}}_{k}[{\rm{\Phi }}]\\ =\,\displaystyle \frac{1} {2}\mathrm{STr}\left({G}_{k}[{\rm{\Phi }}]\,{\partial }_{t} {R}_{k}\right)+\mathrm{Tr}\left({G}_{\phi {{\rm{\Phi }}}_{a}}[{\rm{\Phi }}]\displaystyle \frac{\delta \langle {\partial }_{t} {\hat{\phi }}_{k}\rangle } {\delta {{\rm{\Phi }}}_{a}}\,{R}_{\phi }\right)\\ -\,\int \langle {\partial }_{t} {\hat{\phi }}_{k,i}\rangle \left(\displaystyle \frac{\delta {{\rm{\Gamma }}}_{k}[{\rm{\Phi }}]} {\delta {\phi }_{i}}+{c}_{\sigma } {\delta }_{i\,\sigma }\right),\end{array}\end{eqnarray}$
where some summations for the indices {a} and {i} as shown in equation (3. Low energy effective field theories
Prior to discussing the properties of the QCD matter at finite temperature and densities in section 4 , in this section, we would like to apply the formalism of fRG in section 2 to low energy effective field theories first. We adopt two commonly used formalisms of LEFTs, i.e. the purely fermionic NJL model in section 3.1 and the quark-meson (QM) model in section 3.2 , respectively.
3.1. NJL model
One prominent feature characteristic of the nonperturbative QCD is the dynamical chiral symmetry breaking, which is regarded as being responsible for the origin of the ∼98% mass of visible matter in the Universe [76–79], in contradistinction to the ∼2% electroweak mass. Within the fRG approach, the dynamical breaking or restoration of the chiral symmetry is well encoded in the four-fermion flows, which is illustrated briefly in what follows. For more details, see, e.g. [80, 81–89] and a related review [53].
Using the method to derive the flow equation for a generic vertex as shown in equation (61 ), one is able to obtain the flow equation of four-quark coupling, depicted schematically in figure 4. Here we refrain from going into the details of a realistic calculation, but rather try to infer behaviors of the four-quark flow connected to the breaking or restoration of the chiral symmetry. It follows from figure 4 that the β function for the dimensionless four-quark coupling $\bar{\lambda }\equiv {k}^{2}\lambda $ reads81 ) arising from the dimension of λ, the remaining three terms correspond to the three classes of diagrams in figure 5 one by one, and their coefficients are denoted by a, b, c, respectively. Further computation indicates one has a > 0 and c > 0 [90].
$\begin{eqnarray}\beta \equiv {\partial }_{t}\bar{\lambda }=(d-2)\bar{\lambda }-a{\bar{\lambda }}^{2}-b\bar{\lambda } {g}^{2}-{{cg}}^{4},\end{eqnarray}$
with the dimension of spacetime d = 4 and the strong coupling g. Note that apart from the first term on the r.h.s. of equation (
Figure 4. Schematic representation of the flow equation for the four-quark coupling. Those on the r.h.s. of equation stand for three classes of diagrams contributing to the flow of four-quark interaction, that is, the two-gluon exchange, purely self-interacting four-quark coupling, and mixture of the quark-gluon and four-quark interactions, respectively. Here, diagrams of different channels of momenta are not distinguished and prefactors for each diagram are not shown. |
Figure 5. Left panel: Sketch of a typical β function for the dimensionless four-quark coupling |
In the left panel of figure 5, a typical β function is plotted as a function of the dimensionless four-quark coupling schematically in different cases. When the gauge coupling g is vanishing and at zero temperature, there are two fixed points: one is the Gaussian IR fixed point $\bar{\lambda }=0$ , the other the UV attractive fixed point at a nonvanishing $\bar{\lambda }$ , and they are shown in the plot by red and purple dots, respectively. The position of the UV fixed point determines a critical value ${\bar{\lambda }}_{c}$ , which is necessitated in order to break the chiral symmetry, since only when $\bar{\lambda }\gt {\bar{\lambda }}_{c}$ , the four-quark coupling grows large and eventually diverges with the decreasing RG scale. When the temperature is turned on, the IR fixed point remains at the origin while the UV fixed point move towards right, as shown by the red dashed line. As a consequence, the broken chiral symmetry in the vacuum is restored at a finite T, if one has a value of $\bar{\lambda }$ with ${\bar{\lambda }}_{c}(T=0)\lt \bar{\lambda }\lt {\bar{\lambda }}_{c}(T)$ . When the strong coupling is nonzero, the parabola of β function moves downwards globally as shown by the blue curve in the left panel of figure 5. There is a critical value of the strong coupling, say gc, once one has g > gc, the whole curve of the beta function is below the line β = 0, which implies that the chiral symmetry breaking is bound to occur, no matter how large the initial value of $\bar{\lambda }$ is.
The four-fermion flow is also well suited for an analysis of the chiral symmetry breaking in an external magnetic field. The inclusion of a magnetic field would modify the four-quark flows as well as the fixed-point structure [80, 83]. The plot in the right panel of figure 5 is obtained in [80], which demonstrates that in the case of a magnetic field the flow pattern of the four-quark coupling has been changed and the chiral symmetry is always broken. This is in fact due to the dimensional reduction under a finite external magnetic field. Moreover, it is found that once the in-medium effects of temperature and densities are implemented, long-range correlations are screened and the vanishing critical coupling shown by the red line in the right panel of figure 5 is not zero anymore [80].
Recently, a Fierz-complete four-fermion model is employed to investigate the phase structure at finite temperature and quark chemical potential, and it is found that the inclusion of four-quark channels other than the conventionally used scalar-pseudoscalar ones not only plays an important role in the phase diagram at large chemical potential, but also affects the dynamics at small chemical potential [85]. The related fixed-point structure is analyzed at the finite chemical potential in the Fierz-complete NJL model, and it is found the dynamics are dominated by diquarks at large chemical potential [86]. Resorting to the fRG flows of four-quark interactions, one is able to observe the natural emergence of the NJL model at intermediate and low energy scales from fundamental quark-gluon interactions [87]. EoS of nuclear matter at supranuclear densities is also studied in the Fierz-complete setting [88], and a maximal speed of sound is found at supranuclear densities, which is related to the emergence of color superconductivity in the regime of high densities, i.e. the formation of a diquark gap, see also [89] for details. Moreover, the UA(1) symmetry and its effect on the chiral phase transition are investigated in the Fierz-complete basis [91].
Up to now, we have only discussed the chiral symmetry and its breaking by means of the four-fermion flows. As mentioned in the beginning of section 3.1 , a direct consequence of the dynamical chiral symmetry breaking is the production of mass, which is our central concern in the following. Here, we discuss recent progress in understanding the quark mass generation and the emergence of bound states in terms of RG flows [92]. Considering only the quark degrees of freedom in equation (48 ) and extending the scalar-pseudoscalar four-quark interaction to a Fierz-complete set of Nf = 2 flavors, denoted by ${ \mathcal B }$ in the following, one arrives at a RG-scale dependent effective action, given by${\rm{\Phi }}=(q,\bar{q})$ and ∫x,⋯ ≡ ∫d4x∫ ⋯ . Here ${{ \mathcal O }}_{{ijlm}}^{\alpha }$ stands for the four-quark operator of channel α, where the indices i, j, l, m run over the Dirac, flavor and color (Nc = 3) spaces, and the related coupling strength is given by λα,k. In the same way, the summation is assumed for repeated indices. In appendix B ten independent Fierz-complete channels of four-quark interactions of Nf = 2 flavors are presented.
$\begin{eqnarray}\begin{array} {l} {{\rm{\Gamma }}}_{k}[{\rm{\Phi }}]\\ =\,{\int }_{x,y}\left[{Z}_{q,k}(x,y)\bar{q}(x){\gamma }_{\mu } {\partial }_{\mu }q(y)+{m}_{q,k}(x,y)\bar{q}(x)q(y)\right]\\ -\,{\int }_{x,y,w,z}\sum _{\alpha \in { \mathcal B }} {\lambda }_{\alpha ,k}(x,y,w,z){{ \mathcal O }}_{{ijlm}}^{\alpha } {\bar{q}}_{i}(x){q}_{j}(y){\bar{q}}_{l}(w){q}_{m}(z),\end{array}\end{eqnarray}$
with The two-quark correlation function reads 
with86 ), in comparison to the bosonic one in equations (9 ) and (11 ), is implemented in the vector channel rather than the scalar one, which guarantees that the chiral symmetry is not broken by the regulator. In the same way, one is allowed to make a choice for the specific formalism of the fermionic regulator, e.g. the optimized one,10 ),
$\begin{eqnarray}\begin{array} {l} {{\rm{\Gamma }}}_{k,{ij}}^{(2)\bar{q}q}({p}^{{\prime} },p)\equiv -{\left.\frac{{\delta }^{2} {{\rm{\Gamma }}}_{k}} {\delta {\bar{q}}_{i}({p}^{^{\prime} })\delta {q}_{j}(p)}\right|}_{{\rm{\Phi }}=0}\\ =\left[{Z}_{q,k}(p){\rm{i}} {\left(\gamma \cdot p\right)}_{{ij}}+{m}_{q,k}(p){\delta }_{{ij}}\right]{\left(2\pi \right)}^{4} {\delta }^{4}({p}^{{\prime} }+p).\end{array}\end{eqnarray}$
Then one arrives at the quark propagator with an IR regulator, viz., $\begin{eqnarray}\begin{array} {rcl} {G}_{k}^{q\bar{q}}(p,p^{\prime} ) & = & {\left[{{\rm{\Gamma }}}_{k}^{(2)\bar{q}q}+{R}_{k}^{\bar{q}q}\right]}^{-1}\\ & = & {G}_{k}^{q}(p){\left(2\pi \right)}^{4} {\delta }^{4}(p^{\prime} +p),\end{array}\end{eqnarray}$
with a fermionic regulator given by $\begin{eqnarray} {R}_{k}^{\bar{q}q}={Z}_{q,k} {r}_{F}({p}^{2}/{k}^{2}){\rm{i}}\gamma \cdot p,\end{eqnarray}$
and $\begin{eqnarray} {G}_{k}^{q}(p)=\displaystyle \frac{1} {{Z}_{q,k}(p){\rm{i}}\gamma \cdot p+{Z}_{q,k} {r}_{F}({p}^{2}/{k}^{2}){\rm{i}}\gamma \cdot p+{m}_{q,k}(p)}.\end{eqnarray}$
Note that the fermionic regulator in equation ( $\begin{eqnarray} {r}_{F,\mathrm{opt}}(x)=\left(\displaystyle \frac{1} {\sqrt{x}}-1\right){\rm{\Theta }}(1-x),\end{eqnarray}$
or the exponential one in equation ( $\begin{eqnarray} {r}_{F}(x)={r}_{\exp ,n}(x)=\displaystyle \frac{{x}^{n-1}} {{{\rm{e}}}^{{x}^{n}}-1},\end{eqnarray}$
and even the simplest exponential regulator as follows $\begin{eqnarray} {r}_{F,\exp }(x)=\displaystyle \frac{1} {x} {{\rm{e}}}^{-x}.\end{eqnarray}$
The four-quark correlation function reads 
with
$\begin{eqnarray}\begin{array} {l} {{\rm{\Gamma }}}_{k,{ijlm}}^{(4)\bar{q}q\bar{q}q}(p,q,r,s)\equiv {\left.\displaystyle \frac{{\delta }^{4} {{\rm{\Gamma }}}_{k}} {\delta {\bar{q}}_{i}(p)\delta {q}_{j}(q)\delta {\bar{q}}_{l}(r)\delta {q}_{m}(s)}\right|}_{{\rm{\Phi }}=0}\\ =\,2\sum _{\alpha \in { \mathcal B }}\left({\lambda }_{\alpha ,k}^{S}(p,q,r,s)({{ \mathcal O }}_{{ijlm}}^{\alpha }-{{ \mathcal O }}_{{ljim}}^{\alpha })\right.\\ \,\,\,\left.+{\lambda }_{\alpha ,k}^{A}(p,q,r,s)({{ \mathcal O }}_{{ijlm}}^{\alpha }+{{ \mathcal O }}_{{ljim}}^{\alpha })\right)\\ \,\,\,\times {\left(2\pi \right)}^{4} {\delta }^{4}(p+q+r+s),\end{array}\end{eqnarray}$
where the symmetric and antisymmetric four-quark couplings read $\begin{eqnarray} {\lambda }_{\alpha ,k}^{S}(p,q,r,s)\equiv \left({\lambda }_{\alpha ,k}(p,q,r,s)+{\lambda }_{\alpha ,k}(r,q,p,s)\right)/2,\end{eqnarray}$
$\begin{eqnarray} {\lambda }_{\alpha ,k}^{A}(p,q,r,s)\equiv \left({\lambda }_{\alpha ,k}(p,q,r,s)-{\lambda }_{\alpha ,k}(r,q,p,s)\right)/2.\end{eqnarray}$
Neglecting the diagrams including the quark-gluon interaction in figure 4 and showing explicitly different channels of momenta, one is able to obtain the flow equation of four-quark coupling within the purely fermionic effective action in equation (82 ), shown diagrammatically in the second line of figure 6. We also depict the flow equation of the two-quark correlation function, i.e. the quark self-energy, in figure 6.
Figure 6. Diagrammatic representation of the flow equations for the two- and four-quark correlation functions, where prefactors and signs for each diagram are also included. The three diagrams on the r.h.s. of the flow equation of four-quark coupling stand for the t-, u-, and s-channels, respectively. |
3.1.1. Quark mass production
Following [92] we assume that the antisymmetric four-quark couplings ${\lambda }_{\alpha ,k}^{A}$ 's in equation (94 ) are vanishing in order to simplify calculations. Then, equations (93 ) and (94 ) leave us with${ \mathcal F }$ 's indicate that the relevant terms in equation (96 ) arise from the corresponding loop diagrams in the flow of four-quark coupling in figure 6. The coefficient ${ \mathcal F }$ 's depend on the quark propagators and regulators and see [92] for their explicit expressions. The flow of quark mass is readily obtained by projecting the flow equation of the quark self-energy in the first line in figure 6 onto the scalar channel, which reads${Z}_{q,k}^{R}(q)={Z}_{q,k}(q)+{R}_{F,k}(q)$ with RF,k(q) = Zq,krF(q2/k2), and${\bar{\lambda }}_{\alpha ,k}={\lambda }_{\alpha ,k} {k}^{2}$ and quark mass ${\bar{m}}_{q,k}={m}_{q,k}/k$ are also very useful in the following.
$\begin{eqnarray} {\lambda }_{\alpha ,k}^{S}(p,q,r,s)={\lambda }_{\alpha ,k}(p,q,r,s)={\lambda }_{\alpha ,k}(r,q,p,s),\end{eqnarray}$
and its flow equation reads $\begin{eqnarray}\displaystyle \begin{array} {l} {\partial }_{t} {\lambda }_{\alpha ,k}({p}_{1},{p}_{2},{p}_{3},{p}_{4})\\ =\,\sum _{\alpha ^{\prime} ,\alpha ^{\prime\prime} \in { \mathcal B }}\int \displaystyle \frac{{{\rm{d}}}^{4}q} {{\left(2\pi \right)}^{4}}\left[{\lambda }_{\alpha ^{\prime} ,k}({p}_{1},{p}_{2},q+{p}_{2}-{p}_{1},q)\right.\\ \,\,\,\times \,{\lambda }_{\alpha ^{\prime\prime} ,k}({p}_{3},{p}_{4},q,q+{p}_{2}-{p}_{1}){{ \mathcal F }}_{\alpha ^{\prime} \alpha ^{\prime\prime} ,\alpha }^{t}\\ \,\,\,+\,{\lambda }_{\alpha ^{\prime} ,k}({p}_{3},{p}_{2},q+{p}_{2}-{p}_{3},q)\\ \,\,\,\times \,{\lambda }_{\alpha ^{\prime\prime} ,k}({p}_{1},{p}_{4},q,q+{p}_{2}-{p}_{3}){{ \mathcal F }}_{\alpha ^{\prime} \alpha ^{\prime\prime} ,\alpha }^{u}\\ \,\,\,+\,{\lambda }_{\alpha ^{\prime} ,k}({p}_{1},q,{p}_{3},-q+{p}_{1}+{p}_{3})\\ \,\,\,\left.\times \,{\lambda }_{\alpha ^{\prime\prime} ,k}(q,{p}_{2},-q+{p}_{1}+{p}_{3},{p}_{4}){{ \mathcal F }}_{\alpha ^{\prime} \alpha ^{\prime\prime} ,\alpha }^{s}\right],\end{array}\end{eqnarray}$
where the superscripts t, u, and s of coefficient $\begin{eqnarray}\begin{array} {l} {\partial }_{t} {m}_{q,k}(p)\\ =\,\int \displaystyle \frac{{{\rm{d}}}^{4}q} {{\left(2\pi \right)}^{4}}\left({\tilde{\partial }}_{t} {\bar{G}}_{k}^{q}(q)\right){m}_{q,k}(q)\left[\displaystyle \frac{3} {2} {\lambda }_{\pi ,k}(p,p,q,q)\right.\\ \,\,\,+\,\displaystyle \frac{23} {2} {\lambda }_{\sigma ,k}(p,p,q,q)-\displaystyle \frac{3} {2} {\lambda }_{a,k}(p,p,q,q)\\ \,\,\,+\,\displaystyle \frac{1} {2} {\lambda }_{\eta ,k}(p,p,q,q)+\displaystyle \frac{8} {3} {\lambda }_{{\left(S+P\right)}_{-}^{\mathrm{adj}},k}(p,p,q,q)\\ \,\,\,\left.-\displaystyle \frac{16} {3} {\lambda }_{{\left(S+P\right)}_{+}^{\mathrm{adj}},k}(p,p,q,q)-4{\lambda }_{(V+A),k}(p,p,q,q)\right],\end{array}\end{eqnarray}$
with $\begin{eqnarray} {\tilde{\partial }}_{t} {\bar{G}}_{k}^{q}(q)=-2{\left({\bar{G}}_{k}^{q}(q)\right)}^{2} {Z}_{q,k}^{R}(q){q}^{2} {\partial }_{t} {R}_{F,k}(q).\end{eqnarray}$
Here one has $\begin{eqnarray} {\bar{G}}_{k}^{q}(q)=\displaystyle \frac{1} {{\left({Z}_{q,k}^{R}(q)\right)}^{2} {q}^{2}+{m}_{q,k}^{2}(q)}.\end{eqnarray}$
For the moment, we assume Zq,k = 1 and use the truncation as follows $\begin{eqnarray} {\lambda }_{\alpha ,k}={\lambda }_{\alpha ,k}({p}_{i}=0),\qquad (i=1,\cdots ,4),\end{eqnarray}$
$\begin{eqnarray} {m}_{q,k}={m}_{q,k}(p=0),\end{eqnarray}$
that is, neglecting the momentum dependence of the four-quark coupling and quark mass. The dimensionless four-quark coupling In order to focus on the mechanism of quark mass production, we make a further approximation as follows96 ) and (97 ) are simplified as${\bar{\lambda }}_{\sigma -\pi }$ and ${\bar{m}}_{q}$ in equations (104 ) and (105 ) can be found in [92].
$\begin{eqnarray} {\lambda }_{\alpha ,k}=0,\qquad (\alpha \notin \{\sigma ,\pi \}),\end{eqnarray}$
$\begin{eqnarray} {\lambda }_{\sigma -\pi ,k}\equiv {\lambda }_{\sigma ,k}={\lambda }_{\pi ,k},\end{eqnarray}$
i.e. only keeping the scalar-pseudoscalar σ and π channel. Then the flow equations in equations ( $\begin{eqnarray}\begin{array} {rcl} {\partial }_{t} {\bar{\lambda }}_{\sigma -\pi } & = & 2{\bar{\lambda }}_{\sigma -\pi }+\displaystyle \frac{{\bar{\lambda }}_{\sigma -\pi }^{2}} {2{\pi }^{2}} {\int }_{0}^{\infty } {\rm{d}} {x}\,{x}^{3} {r}_{F}^{\prime} (x)\\ & & \times \,\left[-4{\bar{m}}_{q}^{2}+7x{\left(1+{r}_{F}(x)\right)}^{2}\right]\\ & & \times \,\displaystyle \frac{1+{r}_{F}(x)} {{\left[{\left(1+{r}_{F}(x)\right)}^{2}x+{\bar{m}}_{q}^{2}\right]}^{3}},\end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array} {rcl} {\partial }_{t} {\bar{m}}_{q} & = & -{\bar{m}}_{q}+{\bar{m}}_{q} {\bar{\lambda }}_{\sigma -\pi }\displaystyle \frac{13} {4{\pi }^{2}} {\int }_{0}^{\infty } {\rm{d}} {x}\,{x}^{3} {r}_{F}^{\prime} (x)\\ & & \times \displaystyle \frac{1+{r}_{F}(x)} {{\left[{\left(1+{r}_{F}(x)\right)}^{2}x+{\bar{m}}_{q}^{2}\right]}^{2}},\end{array}\end{eqnarray}$
respectively. Here we have used the dimensionless variables, which entails that the RG scale k-dependence for these equations is removed. Numerical results of the RG flows of 3.1.2. Natural emergence of bound states
Properties of bound states of quarks or antiquarks, e.g. the pions and nucleons, in principle can be inferred from the relevant four-point and six-point vertices of quarks, in some specific channels and regimes of momenta [93]. In figure 7 a sketch map shows how this happens in the example of mesons. The square denotes a four-point vertex of quark and antiquark. If the total momentum of a quark and an antiquark, denoted by P here, is in the Minkowski spacetime and in the vicinity of the on-shell pole mass of a meson in some channel, i.e. ${P}^{2}\sim -{m}_{\mathrm{meson}}^{2}$ , the full four-quark vertex can be well described by a resonance of the meson, as shown on the r.h.s. of figure 7, where two quark-meson vertices are connected with the propagator of meson. Therefore, one has to calculate the full four-quark vertex or quark-meson vertex in some specific regime of momentum, which is usually realized by resuming a four-quark kernel to the order of infinity in the formalism of Bethe–Salpeter equations [94, 95]. Note that the necessary resummation for the four-quark vertex is well included in the flow equation of four-quark couplings in equation (96 ), and it is, therefore, natural to expect that the RG flows are also well-suited for the description of bound states as the same as the quark mass production in section 3.1.1 . Moreover, the advantage of RG flows is evident, that is, the self-consistency between the bound states encoded in the flow of four-quark vertices in equation (96 ) and that of quark mass gap in equation (97 ) can be well guaranteed, once a truncation is made on the level of the effective action, such as that in equation (82 ).
Figure 7. Schematic diagram showing the emergence of resonance at the pole of relevant meson mass from the four-quark vertex, where the square and half-circles stand for the full four-quark and quark-meson vertices, respectively. The dashed line denotes the meson propagator. |
In order to investigate the resonance behavior of four-quark vertices in equation (96 ), one has to go beyond the truncation in equation (100 ) and include appropriate momentum dependence for the four-quark vertices. The external momenta of couplings in equation (96 ) are parameterized as follows
$\begin{eqnarray} {p}_{1}=p+\displaystyle \frac{P} {2},\qquad {p}_{2}=p-\displaystyle \frac{P} {2},\end{eqnarray}$
$\begin{eqnarray} {p}_{3}=p^{\prime} -\displaystyle \frac{P} {2},\qquad {p}_{4}=p^{\prime} +\displaystyle \frac{P} {2}.\end{eqnarray}$
Then, one is left with the relevant Mandelstam variables given by $\begin{eqnarray}s={\left({p}_{1}+{p}_{3}\right)}^{2}={\left(p+p^{\prime} \right)}^{2},\end{eqnarray}$
$\begin{eqnarray}t={\left({p}_{1}-{p}_{2}\right)}^{2}={P}^{2},\end{eqnarray}$
$\begin{eqnarray}u={\left({p}_{1}-{p}_{4}\right)}^{2}={\left(p-p^{\prime} \right)}^{2}.\end{eqnarray}$
In the following, we focus on the π meson and assume, that the total momentum of quark and antiquark in the t-channel is near the regime of on-shell pion mass, viz., one has the t-variable ${P}^{2}\sim -{m}_{\pi }^{2}$ in equation (109 ). Consequently, the four-quark coupling of the pion channel would be significantly larger than those of other channels, and its dependence on external momenta would be dominated by the t-variable. Thus, one is allowed to make the approximation as follows96 ) as115 ) are momentum-independent. If one adopts further $p=p^{\prime} =0$ in equations (106 ) and (107 ), two Mandelstam variables are vanishing, i.e. s = u = 0, and one arrives at
$\begin{eqnarray} {\lambda }_{\pi ,k}({p}_{1},{p}_{2},{p}_{3},{p}_{4})\simeq {\lambda }_{\pi ,k}({P}^{2}),\end{eqnarray}$
$\begin{eqnarray} {\lambda }_{\alpha ,k}({p}_{1},{p}_{2},{p}_{3},{p}_{4})\simeq {\lambda }_{\alpha ,k}(0),\qquad \alpha \ne \pi .\end{eqnarray}$
Furthermore, insofar as the four-quark vertices on the r.h.s. of the flow of coupling in the second line of figure 6, a simple analysis of relevant momenta for each vertex indicates, that the t-variable dependence is only required to be kept for the vertices in the diagram of t channel, i.e. the first diagram. One is thus allowed to simplify the flow equation of λπ,k in equation ( $\begin{eqnarray} {\partial }_{t} {\lambda }_{\pi ,k}({P}^{2})={{ \mathcal C }}_{k}({P}^{2}){\lambda }_{\pi ,k}^{2}({P}^{2})+{{ \mathcal A }}_{k}(t,u,s),\end{eqnarray}$
with two coefficients given by $\begin{eqnarray} {{ \mathcal C }}_{k}({P}^{2})=\int \displaystyle \frac{{{\rm{d}}}^{4}q} {{\left(2\pi \right)}^{4}} {{ \mathcal F }}_{\pi \pi ,\pi }^{t},\end{eqnarray}$
and $\begin{eqnarray}\displaystyle \begin{array} {rcl} {{ \mathcal A }}_{k}(t,u,s) & = & \int \displaystyle \frac{{{\rm{d}}}^{4}q} {{\left(2\pi \right)}^{4}}\left\{\sum _{\alpha ^{\prime} ,\alpha ^{\prime\prime} \in { \mathcal B }}\left[{\lambda }_{\alpha ^{\prime} ,k} {\lambda }_{\alpha ^{\prime\prime} ,k}\left({{ \mathcal F }}_{\alpha ^{\prime} \alpha ^{\prime\prime} ,\pi }^{t}\right.\right.\right.\\ & & \left.\left.\left.+{{ \mathcal F }}_{\alpha ^{\prime} \alpha ^{\prime\prime} ,\pi }^{u}+{{ \mathcal F }}_{\alpha ^{\prime} \alpha ^{\prime\prime} ,\pi }^{s}\right)\right]-{\lambda }_{\pi ,k}^{2} {{ \mathcal F }}_{\pi \pi ,\pi }^{t}\right\}.\end{array}\end{eqnarray}$
Note that all the four-quark couplings in equation ( $\begin{eqnarray} {{ \mathcal A }}_{k}(t,u,s)\to {{ \mathcal A }}_{k}({P}^{2}).\end{eqnarray}$
One is able to observe the natural emergence of a bound state arising from the resummation of the four-quark vertex from equation (113 ), whose solution is readily obtained once the last term on the r.h.s. is ignored. One has117 ) is vanishing, the four-quark coupling in the IR λπ,k=0 is divergent. As a consequence, one can employ this condition to determine the pole mass of the bound state, i.e. the pion mass, which reads${{ \mathcal A }}_{k}({P}^{2})$ in equation (113 ) is taken into account, there is no analytic solution anymore. However, as shown in [92], the direct numerical calculation of equation (113 ) indicates that the qualitative behavior of the pole displayed by equation (115 ) is not changed, and more related numerical results of bound states are presented in [92].
$\begin{eqnarray} {\lambda }_{\pi ,k=0}({P}^{2})=\displaystyle \frac{{\lambda }_{\pi ,k={\rm{\Lambda }}}} {1-{\lambda }_{\pi ,k={\rm{\Lambda }}} {\int }_{{\rm{\Lambda }}}^{0} {{ \mathcal C }}_{k}({P}^{2})\tfrac{{\rm{d}} {k}} {k}},\end{eqnarray}$
with λπ,k=Λ the four-quark coupling strength of the pion channel at the UV cutoff, which is independent of external momenta. Evidently, when a value of P2 is chosen appropriately, such that the denominator in equation ( $\begin{eqnarray}1-{\lambda }_{\pi ,k={\rm{\Lambda }}} {\int }_{{\rm{\Lambda }}}^{0} {{ \mathcal C }}_{k}({P}^{2}=-{m}_{\pi }^{2})\displaystyle \frac{{\rm{d}}k} {k}=0.\end{eqnarray}$
When the coefficient 3.2. Quark-meson model
In this section, we discuss another formalism of the low energy effective field theories, i.e. the quark-meson model, which in principle can be obtained from the NJL model via the bosonization with the Hubbard–Stratonovich transformation. The effective interactions between quarks and mesons in the rebosonized QCD effective action in equation (48 ), and their natural emergence resulting from the RG evolution of fundamental degrees of freedom will be discussed in detail in section 4.4 . There is a wealth of studies in the literature relevant to the QM model within the fRG approach, see, e.g. [50, 96–151].
For the moment, we only consider the degrees of freedom of quarks and mesons and their interactions via the Yukawa coupling in equation (48 ). The Nf = 2 flavor QM effective action reads119 ) can be found in section 2.3 below equation (48 ). The effective potential in equation (119 ) can be decomposed into a sum of the contribution from the glue sector and that from the matter sector, which corresponds to the first and last two loops of the flow in figure 2, respectively. Thus, one is led to
$\begin{eqnarray}\begin{array} {rcl} {{\rm{\Gamma }}}_{k} & = & {\int }_{x}\left\{{Z}_{q,k}\bar{q}\left[{\gamma }_{\mu } {\partial }_{\mu }-{\gamma }_{0}(\hat{\mu }+{{\rm{i}} {g} {A}}_{0})\right]q+\displaystyle \frac{1} {2} {Z}_{\phi ,k} {\left({\partial }_{\mu }\phi \right)}^{2}\right.\\ & & \left.+{h}_{k}\bar{q}\left({T}^{0}\sigma +{\rm{i}} {\gamma }_{5} {\boldsymbol{T}}\cdot {\boldsymbol{\pi }}\right)q+{V}_{k}(\rho ,{A}_{0})-c\sigma \right\}.\end{array}\end{eqnarray}$
Note that explanations for most notations in equation ( $\begin{eqnarray} {V}_{k}(\rho ,{A}_{0})={V}_{\mathrm{glue},k}({A}_{0})+{V}_{\mathrm{mat},k}(\rho ,{A}_{0}),\end{eqnarray}$
where the first term on the r.h.s. is the glue potential or the Polyakov loop potential, since it is usually reformulated by means of the Polyakov loop L(A0), and the latter can be calculated through the flow equation within the QM model. In the following, we still use Vk(ρ) to stand for Vmat,k for simplicity in a slight abuse of notation.3.2.1. Flow of the effective potential
The flow equation of the effective potential reads$x=k\sqrt{1+{\tilde{m}}_{q,k}^{2}}-\mu $ . Here, L and $\bar{L}$ are the Polyakov loop and its conjugate. The dimensionless, renormalized meson and quark masses squared in equation (121 ) read121 ) are given by4.1 .
$\begin{eqnarray}\begin{array} {rcl} {\partial }_{t} {V}_{k}(\rho ) & = & \displaystyle \frac{{k}^{4}} {4{\pi }^{2}}\left[\left({N}_{f}^{2}-1\right){l}_{0}^{(B,4)}({\tilde{m}}_{\pi ,k}^{2},{\eta }_{\phi ,k};T)\right.\\ & & +{l}_{0}^{(B,4)}({\tilde{m}}_{\sigma ,k}^{2},{\eta }_{\phi ,k};T)\\ & & \left.-4{N}_{c} {N}_{f} {l}_{0}^{(F,4)}({\tilde{m}}_{q,k}^{2},{\eta }_{q,k};T,\mu )\right],\end{array}\end{eqnarray}$
with the threshold functions given by $\begin{eqnarray}\begin{array} {l} {l}_{0}^{(B,4)}({\tilde{m}}_{\phi ,k}^{2},{\eta }_{\phi ,k};T)\\ =\,\displaystyle \frac{2} {3}\left(1-\displaystyle \frac{{\eta }_{\phi ,k}} {5}\right)\displaystyle \frac{1} {\sqrt{1+{\tilde{m}}_{\phi ,k}^{2}}}\left(\displaystyle \frac{1} {2}+{n}_{B}({\tilde{m}}_{\phi ,k}^{2};T)\right),\end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array} {l} {l}_{0}^{(F,4)}({\tilde{m}}_{q,k}^{2},{\eta }_{q,k};T,\mu )\\ =\,\displaystyle \frac{2} {3}\left(1-\displaystyle \frac{{\eta }_{q,k}} {4}\right)\displaystyle \frac{1} {2\sqrt{1+{\tilde{m}}_{q,k}^{2}}}\\ \,\times \,\left(1-{n}_{F}({\tilde{m}}_{q,k}^{2};T,\mu ,L,\bar{L})-{n}_{F}({\tilde{m}}_{q,k}^{2};T,-\mu ,\bar{L},L)\right),\end{array}\end{eqnarray}$
where the bosonic distribution function reads $\begin{eqnarray} {n}_{B}({\tilde{m}}_{\phi ,k}^{2};T)=\displaystyle \frac{1} {\exp \left\{\tfrac{k} {T}\sqrt{1+{\tilde{m}}_{\phi ,k}^{2}}\right\}-1},\end{eqnarray}$
and the fermionic one $\begin{eqnarray} {n}_{F}({\tilde{m}}_{q,k}^{2};T,\mu ,L,\bar{L})=\displaystyle \frac{1+2\bar{L} {{\rm{e}}}^{x/T}+L{{\rm{e}}}^{2x/T}} {1+3\bar{L} {{\rm{e}}}^{x/T}+3L{{\rm{e}}}^{2x/T}+{{\rm{e}}}^{3x/T}},\end{eqnarray}$
with $\begin{eqnarray} {\tilde{m}}_{\pi ,k}^{2}=\displaystyle \frac{V{{\prime} }_{k}(\rho )} {{k}^{2} {Z}_{\phi ,k}},\qquad {\tilde{m}}_{\sigma ,k}^{2}=\displaystyle \frac{V{{\prime} }_{k}(\rho )+2\rho {V}_{k}^{{\prime\prime} }(\rho )} {{k}^{2} {Z}_{\phi ,k}},\end{eqnarray}$
$\begin{eqnarray} {\tilde{m}}_{q,k}^{2}=\displaystyle \frac{{h}_{k}^{2}\rho } {2{k}^{2} {Z}_{q,k}^{2}}.\end{eqnarray}$
The quark and meson anomalous dimensions in equation ( $\begin{eqnarray} {\eta }_{q,k}=-\displaystyle \frac{{\partial }_{t} {Z}_{q,k}} {{Z}_{q,k}},\qquad {\eta }_{\phi ,k}=-\displaystyle \frac{{\partial }_{t} {Z}_{\phi ,k}} {{Z}_{\phi ,k}},\end{eqnarray}$
computation of which can be found in section There are two classes of methods to solve the flow equation of the effective potential in equation (121 ). The methods of the first class capture local properties of the potential and are very convenient for numerical calculations, but fail to obtain global properties of the potential. A typical method in this class is the Taylor expansion of the effective potential. On the contrary, the methods in the other class are able to provide us with the global properties of the potential, but usually, their numerical implements are relatively more difficult. The second class includes the discretization of the potential on a grid [97], the pseudo-spectral methods [152–155], e.g. the Chebyshev expansion of the potential [151], the discontinuous Galerkin method [156, 157], etc. In what follows we give a brief discussion about the Taylor expansion and the Chebyshev expansion of the potential.
The Taylor expansion of the effective potential reads129 ) in terms of the renormalized variables, one is led to130 ) into the left hand side of equation (121 ), one arrives at133 ) is vanishing. One can see that in this case, the expansion coefficients of different orders in equation (130 ) decouple from each other in the flow equations, which results in an improved convergence for the Taylor expansion and better numerical stability [116]. Another commonly used choice is the running expansion, whereof one expands the potential around the field on the EoS for every value of k, that is,
$\begin{eqnarray} {V}_{k}(\rho )=\sum _{n=0}^{{N}_{v}}\displaystyle \frac{{\lambda }_{n,k}} {n!} {\left(\rho -{\kappa }_{k}\right)}^{n},\end{eqnarray}$
with the expansion coefficients λn,k and the expansion point κk that might be k-dependent, where Nv is the maximal expanding order in the calculations. Reformulating equation ( $\begin{eqnarray} {\bar{V}}_{k}(\bar{\rho })=\sum _{n=0}^{{N}_{v}}\displaystyle \frac{{\bar{\lambda }}_{n,k}} {n!} {\left(\bar{\rho }-{\bar{\kappa }}_{k}\right)}^{n},\end{eqnarray}$
with $\begin{eqnarray} {\bar{V}}_{k}(\bar{\rho })={V}_{\mathrm{mat},k}(\rho ),\qquad \ \bar{\rho }={Z}_{\phi ,k}\rho ,\end{eqnarray}$
$\begin{eqnarray} {\bar{\kappa }}_{k}={Z}_{\phi ,k} {\kappa }_{k},\qquad {\bar{\lambda }}_{n,k}=\displaystyle \frac{{\lambda }_{n,k}} {{\left({Z}_{\phi ,k}\right)}^{n}}.\end{eqnarray}$
Substituting equation ( $\begin{eqnarray}\begin{array} {l} {\left.{\partial }_{\bar{\rho }}^{n}\left({\left.{\partial }_{t}\right|}_{\rho } {\bar{V}}_{k}(\bar{\rho })\right)\right|}_{\bar{\rho }={\bar{\kappa }}_{k}}\\ =\,({\partial }_{t}-n{\eta }_{\phi ,k}){\bar{\lambda }}_{n,k}-({\partial }_{t} {\bar{\kappa }}_{k}+{\eta }_{\phi ,k} {\bar{\kappa }}_{k}){\bar{\lambda }}_{n+1,k}.\end{array}\end{eqnarray}$
If the expansion point κk is independent of k, i.e. ∂tκk = 0, which is usually called the fixed point expansion, the expression in the second bracket on the r.h.s. of equation ( $\begin{eqnarray} {\left.\displaystyle \frac{\partial } {\partial \bar{\rho }}\left({\bar{V}}_{k}(\bar{\rho })-{\bar{c}}_{k}\bar{\sigma }\right)\right|}_{\bar{\rho }={\bar{\kappa }}_{k}}=0,\end{eqnarray}$
with $\begin{eqnarray}\bar{\sigma }={Z}_{\phi ,k}^{1/2}\sigma ,\qquad {\bar{c}}_{k}={Z}_{\phi ,k}^{-1/2}c.\end{eqnarray}$
Here c is independent of k. Then one arrives at the flow of the expansion point, as follows $\begin{eqnarray}\begin{array} {rcl} {\partial }_{t} {\bar{\kappa }}_{k} & = & \,-\displaystyle \frac{{\bar{c}}_{k}^{2}} {{\bar{\lambda }}_{1,k}^{3}+{\bar{c}}_{k}^{2} {\bar{\lambda }}_{2,k}}\left[{\left.{\partial }_{\bar{\rho }}\left({\left.{\partial }_{t}\right|}_{\rho } {\bar{V}}_{k}(\bar{\rho })\right)\right|}_{\bar{\rho }={\bar{\kappa }}_{k}}\right.\\ & & \left.+{\eta }_{\phi ,k}\left(\displaystyle \frac{{\bar{\lambda }}_{1,k}} {2}+{\bar{\kappa }}_{k} {\bar{\lambda }}_{2,k}\right)\right].\end{array}\end{eqnarray}$
For more discussions about the fixed and running expansions, see, e.g. [59, 116, 118, 128, 148, 158]. In the same way, the field dependence of the Yukawa coupling, i.e. hk(ρ) can also be studied within a similar Taylor expansion [116, 148], which encodes higher-order quark-meson interactions. Moreover, the effects of the thermal splitting for the mesonic wave function renormalization in the heat bath are also investigated in [148].As for the Chebyshev expansion, the effective potential reads${T}_{n}(\bar{\rho })$ and the respective expansion coefficients cn,k. Then, one arrives at the flow of the effective potential as follows${\eta }_{\phi ,k}(\bar{\rho })$ is taken into account. Note that the coefficients dn,k are related to cn,k through a recursion relation as shown in [151]. The flow equations of the coefficients in equation (137 ) are given by${T}_{N+1}(\bar{\rho })$ .
$\begin{eqnarray} {\bar{V}}_{k}(\bar{\rho })=\sum _{n=1}^{{N}_{v}} {c}_{n,k} {T}_{n}(\bar{\rho })+\displaystyle \frac{1} {2} {c}_{0,k},\end{eqnarray}$
with the Chebyshev polynomials $\begin{eqnarray}\begin{array} {rcl} {\left.{\partial }_{t}\right|}_{\rho } {\bar{V}}_{k}(\bar{\rho }) & = & \sum _{n=1}^{{N}_{v}}\left({\partial }_{t} {c}_{n,k}-{d}_{n,k} {\eta }_{\phi ,k}(\bar{\rho })\bar{\rho }\right){T}_{n}(\bar{\rho })\\ & & +\displaystyle \frac{1} {2}\left({\partial }_{t} {c}_{0,k}-{d}_{0,k} {\eta }_{\phi ,k}(\bar{\rho })\bar{\rho }\right),\end{array}\end{eqnarray}$
where the field dependence of the meson wave function renormalization Zφ,k(ρ) as well as the meson anomalous dimension $\begin{eqnarray}\displaystyle \begin{array} {rcl} {\partial }_{t} {c}_{m,k} & = & \displaystyle \frac{2} {N+1}\sum _{i=0}^{N}\left({\left.{\partial }_{t}\right|}_{\rho } {\bar{V}}_{k}({\bar{\rho }}_{i})\right){T}_{m}({\bar{\rho }}_{i})\\ & & +\displaystyle \frac{2} {N+1}\sum _{n=1}^{{N}_{v}}\sum _{i=0}^{N} {d}_{n,k} {T}_{m}({\bar{\rho }}_{i}){T}_{n}({\bar{\rho }}_{i}){\eta }_{\phi ,k}({\bar{\rho }}_{i}){\bar{\rho }}_{i}\\ & & +\displaystyle \frac{1} {N+1} {d}_{0,k}\sum _{i=0}^{N} {T}_{m}({\bar{\rho }}_{i}){\eta }_{\phi ,k}({\bar{\rho }}_{i}){\bar{\rho }}_{i},\end{array}\end{eqnarray}$
where the summation for i is performed on N + 1 zeros of the polynomial 3.2.2. Quark-meson model of Nf = 2 + 1 flavors
The effective action for the Nf = 2 + 1 flavor QM model, see, e.g. [109, 110, 128, 140, 143, 147], reads${T}^{0}=1/\sqrt{2{N}_{f}} {{\mathbb{1}}}_{{N}_{f}\times {N}_{f}}$ and Ta = λa/2 for a = 1, …, 8. Here λa are the Gell-Mann matrices. The covariant derivative on the mesonic fields is given by$\left[\hat{\mu },{\rm{\Sigma }}\right]$ denotes the commutator between the matrix of chemical potentials and the meson matrix in equation (141 ). Note that although mesons do not carry the baryon chemical potential, they might have chemical potentials for the electric charge and the strangeness. The quark chemical potentials are related to those of conserved charges through the relations as follows140 ) is a function of two chiral invariants, to wit,141 ) through the relationship as follows140 ) preserves SUA(3) while breaking UA(1) with
$\begin{eqnarray}\begin{array} {rcl} {{\rm{\Gamma }}}_{k} & = & {\int }_{x}\left\{{Z}_{q,k}\bar{q}\left[{\gamma }_{\mu } {\partial }_{\mu }-{\gamma }_{0}(\hat{\mu }+{{\rm{i}} {g} {A}}_{0})\right]q+{h}_{k}\,\bar{q}\,{{\rm{\Sigma }}}_{5}q\right.\\ & & +{Z}_{\phi ,k}\mathrm{tr}({\bar{D}}_{\mu } {\rm{\Sigma }}\cdot {\bar{D}}_{\mu } {{\rm{\Sigma }}}^{\dagger })+{V}_{\mathrm{glue}}(L,\bar{L})+{V}_{k}({\rho }_{1},{\rho }_{2})\\ & & \left.-{c}_{A}\xi -{c}_{l} {\sigma }_{l}-\displaystyle \frac{1} {\sqrt{2}}\,{c}_{s} {\sigma }_{s}\right\},\end{array}\end{eqnarray}$
where the scalar and pseudoscalar mesonic fields of nonets are in the adjoint representation of the U(Nf = 3) group, i.e. $\begin{eqnarray} {\rm{\Sigma }}={T}^{a}({\sigma }^{a}+{\rm{i}} {\pi }^{a}),\quad a=0,1,\ldots ,8\end{eqnarray}$
with $\begin{eqnarray} {\bar{D}}_{\mu } {\rm{\Sigma }}={\partial }_{\mu }+{\delta }_{\mu 0}\left[\hat{\mu },{\rm{\Sigma }}\right],\end{eqnarray}$
where $\begin{eqnarray} {\mu }_{u}=\displaystyle \frac{1} {3} {\mu }_{B}+\displaystyle \frac{2} {3} {\mu }_{Q},\end{eqnarray}$
$\begin{eqnarray} {\mu }_{d}=\displaystyle \frac{1} {3} {\mu }_{B}-\displaystyle \frac{1} {3} {\mu }_{Q},\end{eqnarray}$
$\begin{eqnarray} {\mu }_{s}=\displaystyle \frac{1} {3} {\mu }_{B}-\displaystyle \frac{1} {3} {\mu }_{Q}-{\mu }_{S}.\end{eqnarray}$
For the Yukawa coupling, one has $\begin{eqnarray} {{\rm{\Sigma }}}_{5}={T}^{a}({\sigma }^{a}+{\rm{i}} {\gamma }_{5} {\pi }^{a}).\end{eqnarray}$
In contradistinction to the two-flavor case, the effective potential in equation ( $\begin{eqnarray} {\rho }_{1}=\mathrm{tr}({\rm{\Sigma }}\cdot {{\rm{\Sigma }}}^{\dagger }),\end{eqnarray}$
$\begin{eqnarray} {\rho }_{2}=\mathrm{tr} {\left({\rm{\Sigma }}\cdot {{\rm{\Sigma }}}^{\dagger }-\displaystyle \frac{1} {3}\,{\rho }_{1}\,{{\mathbb{1}}}_{3\times 3}\right)}^{2}\end{eqnarray}$
which are both invariant not only under SUA(3) but also UA(1). On the EoM these two invariants read $\begin{eqnarray} {\left.{\rho }_{1}\right|}_{\mathrm{EoM}}=\displaystyle \frac{1} {2}({\sigma }_{l}^{2}+{\sigma }_{s}^{2}),\end{eqnarray}$
$\begin{eqnarray} {\left.{\rho }_{2}\right|}_{\mathrm{EoM}}=\displaystyle \frac{1} {24} {\left({\sigma }_{l}^{2}-2{\sigma }_{s}^{2}\right)}^{2},\end{eqnarray}$
where the light and strange fields are related to the zeroth and eighth components in equation ( $\begin{eqnarray}\left(\begin{array} {l} {\phi }_{l}\\ {\phi }_{s}\end{array}\right)=\displaystyle \frac{1} {\sqrt{3}}\left(\begin{array} {ll}1 & \sqrt{2}\\ -\sqrt{2} & 1\end{array}\right)\left(\begin{array} {l} {\phi }_{8}\\ {\phi }_{0}\end{array}\right).\end{eqnarray}$
The term of Kobayashi–Maskawa-'t Hooft determinant in equation ( $\begin{eqnarray}\xi =\det ({\rm{\Sigma }})+\det ({{\rm{\Sigma }}}^{\dagger }),\end{eqnarray}$
and the breaking strength is described by the coefficient cA. The light and strange quark masses read $\begin{eqnarray} {m}_{l,k}=\displaystyle \frac{{h}_{k}} {2} {\sigma }_{l},\qquad {m}_{s,k}=\displaystyle \frac{{h}_{k}} {\sqrt{2}} {\sigma }_{s},\end{eqnarray}$
and the pion and kaon decay constants are given by $\begin{eqnarray} {f}_{\pi }={\sigma }_{l},\quad {f}_{K}=\displaystyle \frac{{\sigma }_{l}+\sqrt{2}\,{\sigma }_{s}} {2},\end{eqnarray}$
where σl and σs are on their respective equations of motion. For more discussions about the flow equations in the Nf = 2 + 1 flavor QM model, see the aforementioned references in this section.3.2.3. Phase structure
In figure 8 the phase diagram of the Nf = 2 flavor QM model in the chiral limit obtained in [97] is shown. In order to investigate the phase structure, especially in the regime of large chemical potential, the global information of the effective potential in equation (121 ) is indispensable. Thus, the potential is discretized on a grid to resolve the flow equation and see [97] for more details. One can see that in the region of small chemical potentials, there is a second-order chiral phase transition denoted by the blue dashed line. With the decrease of the temperature, the second-order phase transition is changed into the first-order one at the tricritical point denoted by a green asterisk. The red solid lines stand for the first-order phase transition. Moreover, with the further decrease of the temperature, one observes that the first-order phase transition splits into two phase transition lines, one of which even evolves again into the second-order phase transition at high chemical potentials. Note that the first-order phase transition line backbends at high chemical potentials [135].
In figure 9 the phase diagram of the Nf = 2 flavor QM model obtained with the Chebyshev expansion in [151] is shown. The black dashed line denotes the O(4) second-order phase transition in the chiral limit, and the black dot stands for the tricritical point. The back solid line is the first-order phase transition in the chiral limit, where the splitting at large baryon chemical potential as shown in figure 8 is not shown explicitly here, and only the left branch is depicted. The solid lines of different colors in figure 9 correspond to different strengths of the explicit chiral symmetry breaking, i.e. different pion masses, and their end points, that is, the CEPs comprise a Z(2) second-order phase transition line, which is denoted by the red dashed line.
As we have mentioned above, the first-order phase transition line backbends at a large chemical potential. It is conjectured that this artifact might arise from the defect that the discontinuity for the potential at high baryon chemical potential is not well dealt with within the grid or pseudo-spectral methods. Recently, a discontinuous Galerkin method has been developed to resolve the flow equation of the effective potential [156, 157], and the relevant result for the two-flavor phase diagram is shown in figure 10 [157]. Within this approach, the formation and propagation of shocks are allowed, and see [156, 157] for a detailed discussion.
However, very recently another research in [159] finds that the back-bending behavior of the chiral phase boundary at large chemical potential has another different origin. It is found that the appearance of the back-bending behavior depends on the employed regulator [159]. In figure 11 results of the chiral phase boundary and the entropy density obtained with two different regulators are compared. The left panel corresponds to the usually used 3d flat or optimized regulator, see equations (12 ) and (88 ), and the right is obtained with a 3d mass-like regulator, see [159] for more details. The same truncation, i.e. the local potential approximation (LPA), is used in both calculations. It is observed that the usually used flat (Litim) regulator results in a back-bending of the chiral phase line at large chemical potential, which is also accompanied by a region of negative entropy density in the chirally symmetric phase, as shown by the blue area in the left panel of figure 11. On the contrary, in the right panel one finds that the chiral phase line shows no back-bending behavior and the entropy density stays positive if a mass-like regulator is used. See [159] for a more detailed discussion.
Figure 11. Entropy density for the 3d flat regulator (left panel) and 3d mass-like regulator (right panel) close to the zero temperature chiral phase boundary obtained in the QM model in LPA [159], where the solid and dashed lines denote the first-order phase transition and the crossover, respectively, and the black dot stands for the CEP. The blue region shows a negative entropy density. The plots are adopted from [159]. |
In heavy-ion collisions since the incident nuclei do not carry the strangeness, the produced QCD matter after collisions is of strangeness neutrality, that is, the strangeness density nS is vanishing. Usually, the condition of strangeness neutrality requires that the chemical potential of strangeness μS in equation (145 ) is nonvanishing with μB ≠ 0 [140, 141]. The influence of the strangeness neutrality on the phase boundary is investigated in [140], which is presented in figure 12. One can see in comparison to the case of μS = 0, the phase boundary moves up slightly due to the strangeness neutrality. In other words, the curvature of the phase boundary is decreased a bit by the condition of strangeness neutrality, see section 4.6 for more discussions about the curvature of the phase boundary.
Figure 12. Phase diagram of the Nf = 2 + 1 flavor QM model in the T–μB plane obtained in [140], where the phase boundaries in the regime of crossover for the two cases: μS = 0 and the strangeness neutrality nS = 0 are compared. The color stands for the relative error of the reduced condensate (see section |
3.2.4. Equation of state
From the effective action in equation (119 ) or (140 ), one is able to obtain the thermodynamic potential density, as follows
$\begin{eqnarray} {\rm{\Omega }}[T,\mu ]=\displaystyle \frac{T} {V}\left({\left.{{\rm{\Gamma }}}_{k=0}[{{\rm{\Phi }}}_{\mathrm{EoM}}]\right|}_{T,\mu }-{\left.{{\rm{\Gamma }}}_{k=0}[{{\rm{\Phi }}}_{\mathrm{EoM}}]\right|}_{T=\mu =0}\right),\end{eqnarray}$
where ΦEoM denotes the fields on the equations of motion. Then the pressure and the entropy density read $\begin{eqnarray}p=-{\rm{\Omega }}[T,\mu ],\end{eqnarray}$
and $\begin{eqnarray}s=\displaystyle \frac{\partial p} {\partial T}.\end{eqnarray}$
Moreover, the trace anomaly that is also called the interaction measure is given by $\begin{eqnarray} {\rm{\Delta }}=\epsilon -3p,\end{eqnarray}$
where the energy density reads $\begin{eqnarray}\epsilon =-p+{Ts}+\sum _{f=u,d,s} {\mu }_{f} {n}_{f},\end{eqnarray}$
with the number density for quark of flavor f $\begin{eqnarray} {n}_{f}=\displaystyle \frac{\partial p} {\partial {\mu }_{f}}.\end{eqnarray}$
In figure 13 the reduced condensate, see equation (229 ) and the relevant discussions in section 4.5 , is shown as a function of the reduced temperature t = (T − Tpc)/Tpc obtained in [110], where Tpc denotes the pseudocritical temperature for the chiral crossover. The fRG calculations are in agreement with the lattice simulations [160]. Furthermore, the mean-field results are also presented for comparison, where those labelled with ‘eMF' and ‘MF' are obtained by taking into account or not the fermionic vacuum loop contribution in the calculations, respectively, and see, e.g. [104] for more discussions. In figure 14 the pressure and the trace anomaly in the Nf = 2 + 1 flavor QM model obtained in [110] are shown as functions of the reduced temperature. The fRG results are compared with the lattice results [161, 162], and it is observed that the fRG results are consistent with the lattice results from the Wuppertal–Budapest collaboration (WB), except that the trace anomaly from fRG is a bit larger than that from WB in the regime of high temperature with t ≳ 0.3. Moreover, the mean-field results as well as those including a contribution of the thermal pion gas, are also presented for comparison.
The equations of state in figure 14 are obtained in fRG within the local potential approximation, where the RG-scale dependence only enters the effective potential in equation (119 ). The equations of state are also calculated beyond the LPA approximation in [118]. The relevant results are presented in figure 15. In the calculations, the wave function renormalizations for the mesons and quarks, and the RG scale dependence of the Yukawa coupling are taken into account. In figure 15 the results beyond LPA are in comparison to the lattice and LPA results. One can see that the trace anomaly is decreased a bit beyond LPA in the regime of T ≳ 1.1Tc. In figure 16 the influence of the strangeness neutrality on the isentropes is presented obtained in [140]. It is found that, in comparison to the case of μS = 0, the isentropes with nS = 0 move towards the r.h.s. a bit, and their turning around in the regime of crossover becomes more abrupt. Moreover, isentropes obtained from the Taylor expansion and the grid method for the effective potential are compared, and consistent results are found [163].
Figure 15. Left panel: Trace anomaly of the Nf = 2 QM model as a function of the temperature. Here the LPA result is compared with that beyond LPA, denoted by ‘full', in which the wave function renormalizations for the mesons and quarks, the RG scale dependence of the Yukawa coupling are taken into account. Right panel: Trace anomaly of the Nf = 2 + 1 QM model obtained beyond LPA (labelled with ‘LPA after rescale') as a function of the temperature, in comparison to the LPA result [110] and the lattice result from Wuppertal–Budapest collaboration [162]. Plots are adopted from [118]. |
Figure 16. Isentropes for different ratios of s/nB in the phase diagram in the Nf = 2 + 1 flavor QM model were obtained in [140]. Two cases with μS = 0 and the strangeness neutrality nS = 0 are compared. The black and gray lines stand for the chiral and deconfinement crossover lines, respectively. The plot is adapted from [140]. |
3.2.5. Baryon number fluctuations
Differentiating the pressure in equation (156 ) with respect to the baryon chemical potential n times, one is led to the nth order generalized susceptibility of the baryon number, i.e.
$\begin{eqnarray} {\chi }_{n}^{B}=\displaystyle \frac{{\partial }^{n}} {\partial {\left({\mu }_{B}/T\right)}^{n}}\displaystyle \frac{p} {{T}^{4}},\end{eqnarray}$
which is also related to the nth order cumulant of the net baryon number distributions, that is, $\begin{eqnarray}\langle {\left(\delta {N}_{B}\right)}^{n}\rangle =\sum _{{N}_{B}=-\infty }^{\infty } {\left(\delta {N}_{B}\right)}^{n}P({N}_{B})\end{eqnarray}$
with δNB = NB − ⟨NB⟩. Here P(NB) denotes the probability distribution of the net baryon number, which can be calculated theoretically from the canonical ensemble with an imaginary chemical potential, and see, e.g. [115, 142] for a detailed discussion. In experimental measurements, since neutrons are difficult to detect, the net proton number distribution is measured, as a proxy for the net baryon number distribution, and see, e.g. [22] for more details. Evidently, ⟨NB⟩ is the ensemble average of the net baryon number.The relations between the generalized susceptibilities in equation (161 ) and the cumulants in equation (162 ) for the lowest four orders read${\chi }_{n\gt 4}^{B}$ [104, 150, 166–168]. The relations between the hyper-order susceptibilities and the cumulants are given by [150]
$\begin{eqnarray} {\chi }_{1}^{B}=\displaystyle \frac{1} {{{VT}}^{3}}\langle {N}_{B}\rangle ,\end{eqnarray}$
$\begin{eqnarray} {\chi }_{2}^{B}=\displaystyle \frac{1} {{{VT}}^{3}}\langle {\left(\delta {N}_{B}\right)}^{2}\rangle ,\end{eqnarray}$
$\begin{eqnarray} {\chi }_{3}^{B}=\displaystyle \frac{1} {{{VT}}^{3}}\langle {\left(\delta {N}_{B}\right)}^{3}\rangle ,\end{eqnarray}$
$\begin{eqnarray} {\chi }_{4}^{B}=\displaystyle \frac{1} {{{VT}}^{3}}\left(\langle {\left(\delta {N}_{B}\right)}^{4}\rangle -3\langle {\left(\delta {N}_{B}\right)}^{2} {\rangle }^{2}\right),\end{eqnarray}$
which leaves us with the experimental observables as follows $\begin{eqnarray}\begin{array} {rcl}M & = & {{VT}}^{3} {\chi }_{1}^{{\rm{B}}},\qquad {\sigma }^{2}={{VT}}^{3} {\chi }_{2}^{{\rm{B}}},\\ S & = & \displaystyle \frac{{\chi }_{3}^{{\rm{B}}}} {{\chi }_{2}^{{\rm{B}}}\sigma },\qquad \kappa =\displaystyle \frac{{\chi }_{4}^{{\rm{B}}}} {{\chi }_{2}^{{\rm{B}}} {\sigma }^{2}}.\end{array}\end{eqnarray}$
Here M, σ2, S, κ stand for the mean value, the variance, the skewness, and the kurtosis of the net baryon or proton number distributions. In the same way, calculations can also be extended to the hyper-order baryon number fluctuations, i.e. $\begin{eqnarray} {\chi }_{5}^{B}=\displaystyle \frac{1} {{{VT}}^{3}}\left(\langle {\left(\delta {N}_{B}\right)}^{5}\rangle -10\langle {\left(\delta {N}_{B}\right)}^{2}\rangle \langle {\left(\delta {N}_{B}\right)}^{3}\rangle \right),\end{eqnarray}$
$\begin{eqnarray}\begin{array} {rcl} {\chi }_{6}^{B} & = & \displaystyle \frac{1} {{{VT}}^{3}}\left(\langle {\left(\delta {N}_{B}\right)}^{6}\rangle -15\langle {\left(\delta {N}_{B}\right)}^{4}\rangle \langle {\left(\delta {N}_{B}\right)}^{2}\rangle \right.\\ & & \left.-10\langle {\left(\delta {N}_{B}\right)}^{3} {\rangle }^{2}+30\langle {\left(\delta {N}_{B}\right)}^{2} {\rangle }^{3}\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array} {rcl} {\chi }_{7}^{B} & = & \displaystyle \frac{1} {{{VT}}^{3}}\left(\langle {\left(\delta {N}_{B}\right)}^{7}\rangle -21\langle {\left(\delta {N}_{B}\right)}^{5}\rangle \langle {\left(\delta {N}_{B}\right)}^{2}\rangle \right.\\ & & -35\langle {\left(\delta {N}_{B}\right)}^{4}\rangle \langle {\left(\delta {N}_{B}\right)}^{3}\rangle \\ & & \left.+210\langle {\left(\delta {N}_{B}\right)}^{3}\rangle \langle {\left(\delta {N}_{B}\right)}^{2} {\rangle }^{2}\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array} {rcl} {\chi }_{8}^{B} & = & \displaystyle \frac{1} {{{VT}}^{3}}\left(\langle {\left(\delta {N}_{B}\right)}^{8}\rangle -28\langle {\left(\delta {N}_{B}\right)}^{6}\rangle \langle {\left(\delta {N}_{B}\right)}^{2}\rangle \right.\\ & & -56\langle {\left(\delta {N}_{B}\right)}^{5}\rangle \langle {\left(\delta {N}_{B}\right)}^{3}\rangle -35\langle {\left(\delta {N}_{B}\right)}^{4} {\rangle }^{2}\\ & & +420\langle {\left(\delta {N}_{B}\right)}^{4}\rangle \langle {\left(\delta {N}_{B}\right)}^{2} {\rangle }^{2}\\ & & \left.+560\langle {\left(\delta {N}_{B}\right)}^{3} {\rangle }^{2}\langle {\left(\delta {N}_{B}\right)}^{2}\rangle -630\langle {\left(\delta {N}_{B}\right)}^{2} {\rangle }^{4}\right).\end{array}\end{eqnarray}$
It is more convenient to adopt the ratio between two susceptibilities of different orders, say $\begin{eqnarray} {R}_{{nm}}^{B}=\displaystyle \frac{{\chi }_{n}^{B}} {{\chi }_{m}^{B}},\end{eqnarray}$
where the explicit volume dependence is removed. The baryon number fluctuations as well as fluctuations of other conserved charges, e.g. the electric charge and the strangeness, and correlations among these conserved charges, have been widely studied in the literature. For lattice simulations, see, e.g. [5–7, 9–11, 15]. Investigations of fluctuations and correlations within the fRG approach to LEFTs can be found in, e.g. [99, 101, 118, 119, 127, 138, 140, 141, 143, 147, 150, 168, 169], and within the mean-field approximations in, e.g. [104, 167, 170–172]. For the relevant studies from DSEs, see, e.g. [173, 174]. Remarkably, recently QCD-assisted LEFTs with the fRG approach have been developed and used to study the skewness and kurtosis of the baryon number distributions [118, 119, 127], the baryon-strangeness correlations [140, 141], and the hyper-order baryon number fluctuations [150].In figure 17 the kurtosis of the baryon number distributions is shown as a function of the temperature at vanishing chemical potential. In the calculations the frequency dependence of the quark wave function renormalization is taken into account [127], whose contribution to the flow of the effective potential is shown schematically in figure 18. It is found that a summation for the quark external frequency is indispensable to the Silver Blaze property at finite chemical potential, see [127] for a more detailed discussion. From figure 17, one can see that the summation for the external frequency of the quark wave function renormalization improves the agreement between the fRG results and the lattice ones.
Figure 17. Kurtosis of the baryon number distributions as a function of the temperature obtained within the fRG approach to LEFTs [127]. In the calculations, the frequency dependence for the quark wave function renormalization is taken into account, and the relevant results (black solid line) are compared with those without the frequency dependence (pink dashed line) in [119]. The continuum-extrapolated lattice results from the Wuppertal–Budapest collaboration [6] are also presented for comparison. The plot is adapted from [127]. |
Figure 18. Schematic diagram of the frequency dependence of the quark anomalous dimension to the flow equation of the effective potential. The gray area stands for the contribution of the quark anomalous dimension, where a summation for the quark external frequency has been done. The plot is adapted from [127]. |
In figure 19 the fourth-, sixth-, and eighth-order baryon number fluctuations divided by the quadratic one are shown as functions of the temperature at vanishing baryon chemical potential. The fRG results [150] are compared with lattice results from the HotQCD collaboration [9, 10, 15] and the Wuppertal–Budapest collaboration (WB) [11]. It is observed that the fRG results are in quantitative, qualitative agreement with the WB and HotQCD results, respectively. In figure 20 the baryon number fluctuations of different orders, RB31, RB42, RB51, RB62, RB71 and RB82, are shown as functions of the temperature with several different values of the baryon chemical potential from 0 to 400 MeV. One finds that the dependence of fluctuations on the temperature oscillates more pronouncedly with the increasing order of cumulants. Moreover, with the increase of the baryon chemical potential the chiral crossover grows sharper, which leads to an increase in the magnitude of fluctuations significantly.
Figure 19. Baryon number fluctuations RB42 (left panel), RB62 (middle panel), and RB82 (right panel) as functions of the temperature at μB = 0, obtained within the fRG approach to a QCD-assisted LEFT in [150]. The fRG results are compared with lattice results from the HotQCD collaboration [9, 10, 15] and the Wuppertal–Budapest collaboration [11]. The inlay in the right panel shows the zoomed-out view of RB82. The plots are adopted from [150]. |
Figure 20. Baryon number fluctuations RB31 (top-left), RB42 (bottom-left), RB51 (top-middle), RB62 (bottom-middle), RB71 (top-right) and RB82 (bottom-right) as functions of the temperature with several different values of the baryon chemical potential, obtained within the fRG approach to a QCD-assisted LEFT in [150]. The inlays in each plot show the zoomed-out views. The plots are adopted from [150]. |
In the left panel of figure 21 the kurtosis of the baryon number distributions is depicted in the phase diagram [150]. The black dashed line denotes the chiral phase boundary in the crossover regime. The green dotted line stands for the chemical freeze-out curve obtained from the STAR freeze-out data [175], see [150] for a more detailed discussion. One can see that there is a narrow blue area near the phase boundary with μB ≳ 300 MeV, which indicates that the kurtosis becomes negative in this area. Moreover, it is found that with the increase of the baryon chemical potential, the freeze-out curve moves towards the blue area firstly, and then deviates from it a bit at large baryon chemical potential. The skewness of the baryon number distributions as a function of the collision energy calculated in the fRG approach is presented in the right panel of figure 21 [150], which is in comparison to the STAR data of the skewness of the net-proton number distributions Rp32 with centrality 0%–5% [42]. It is found that the fRG results are in good agreement with the experimental data except for the two lowest energy points, which is attributed to the fact that other effects, e.g. volume fluctuations [176–178], the global baryon number conservation [179–181], become important when the beam collision energy is low.
Figure 21. Left panel: Kurtosis of the baryon number distributions |
In figure 22 baryon number fluctuations RB42, RB62, RB82, RB31, RB51, and RB71 are shown as functions of the collision energy obtained within the fRG approach [150], where the chemical freeze-out curve obtained from STAR data as shown in the left panel of figure 21 is used. The fRG results are compared with experimental data of the kurtosis of the net-proton number distributions Rp42 with centrality 0%–5% [42], and the sixth-order cumulant Rp62 with centrality 0%–10% [43]. A non-monotonic dependence of the kurtosis RB42 on the collision energy is also observed in the fRG calculations, which is consistent with the experimental measurements of Rp42. Moreover, it is found that this non-monotonicity arises from the increasingly sharp crossover with the decrease of the collision energy, which is also reflected in the heat map of the kurtosis in the phase diagram in the left panel of figure 21. The fRG results of RB62 are also qualitatively consistent with the experimental data of Rp62.
Figure 22. Baryon number fluctuations RB42 (top-left), RB62 (middle-left), RB82 (bottom-left), RB31 (top-right), RB51 (middle-right), RB71 (bottom-right) as functions of the collision energy, obtained within the fRG approach to a QCD-assisted LEFT in [150]. Experimental data of the kurtosis of the net-proton number distributions Rp42 with centrality 0%–5% [42], and the sixth-order cumulant Rp62 with centrality 0%–10% [43] are presented for comparison. The plots are adopted from [150]. |
In figure 23 the full results of the baryon number fluctuations RB42 and RB62 are compared with those of Taylor expansion for the baryon chemical potential up to the eighth and tenth orders, which can be used to investigate the convergence properties of the Taylor expansion for the baryon chemical potential. For the two values of the temperature, T = 155 MeV and 160 MeV, it is found that the full results deviate from those of Taylor expansion significantly with μB/T ≳ 1.2–1.5. The oscillating behavior of the full results at large baryon chemical potential is not captured by the Taylor expansion, which hints that the convergence radius of the Taylor expansion for the baryon chemical potential might be restricted by some singularity, e.g. the Yang–Lee edge singularity, and see, e.g. [182–185] for more discussions.
Figure 23. Full results of the baryon number fluctuations RB42 (upper panels) and RB62 (lower panels) as functions of μB/T with two fixed values of the temperature, in comparison to the results of Taylor expansion up to order of |
We close this subsection with a discussion about the correlation between the baryon number and the strangeness, i.e.
$\begin{eqnarray} {C}_{{BS}}=-3\displaystyle \frac{{\chi }_{11}^{{BS}}} {{\chi }_{2}^{S}},\end{eqnarray}$
which is calculated within the fRG approach in [141]. There, the influence of the strangeness neutrality on CBS is investigated, and the relevant results are presented in figure 24. One can see that the correlation between the baryon number and the strangeness is suppressed by the condition of the strangeness neutrality.
Figure 24. Correlation between the baryon number and the strangeness CBS as a function of the temperature with several different values of the baryon chemical potential, obtained within the fRG approach in [141]. Two cases with μS = 0 (dashed lines) and the strangeness neutrality nS = 0 (solid lines) are compared. The plot is adapted from [141]. |
3.2.6. Critical exponents
According to the scaling argument, when a system is in the critical regime, its thermodynamic potential density can be decomposed into a sum of a singular and a regular part [66], viz.,174 ) and (175 ) t and h stand for the reduced temperature and magnetic field, respectively. They are given by119 ), respectively. In the following, the order parameter will be denoted as the magnetization density, and c as the magnetic field strength, i.e.175 ) leaves us with a set of scaling relations for various critical exponents, see [66, 186], such as,
$\begin{eqnarray} {\rm{\Omega }}\left(t,h\right)={f}_{s}(t,h)+{f}_{\mathrm{reg}}(t,h),\end{eqnarray}$
whereof, the singular part satisfies the scaling relation to the leading order as follows $\begin{eqnarray} {f}_{s}(t,h)={{\ell }}^{-d} {f}_{s}(t\,{{\ell }}^{{y}_{t}},h\,{{\ell }}^{{y}_{h}}),\end{eqnarray}$
with a dimensionless rescaling factor ℓ and the spatial dimension d. In equations ( $\begin{eqnarray}t=\displaystyle \frac{T-{T}_{c}} {{T}_{0}},\qquad h=\displaystyle \frac{c} {{c}_{0}},\end{eqnarray}$
where Tc is the critical temperature in the chiral limit, and T0 and c0 are some normalization values for the temperature T and the strength of the explicit chiral symmetry breaking c as shown in equation ( $\begin{eqnarray}M\equiv \sigma ,\qquad H\equiv c.\end{eqnarray}$
The scaling relation in equation ( $\begin{eqnarray}\begin{array} {l} {y}_{t}=\displaystyle \frac{1} {\nu },\quad {y}_{h}=\displaystyle \frac{\beta \delta } {\nu },\quad \beta =\displaystyle \frac{\nu } {2}(d-2+\eta ),\quad \gamma =\beta (\delta -1),\\ \gamma =(2-\eta )\nu ,\quad \delta =\displaystyle \frac{d+2-\eta } {d-2+\eta },\quad \nu d=\beta (1+\delta ).\end{array}\end{eqnarray}$
The critical behavior of the order parameter is described by the critical exponents β and δ, which read $\begin{eqnarray}M(t,h=0)\sim {\left(-t\right)}^{\beta },\end{eqnarray}$
with t < 0, and $\begin{eqnarray}M(t=0,h)\sim {h}^{1/\delta }.\end{eqnarray}$
Moreover, one has the susceptibility of the order parameter χ and the correlation length ξ, which scale as $\begin{eqnarray}\chi \sim {\left|t\right|}^{-\gamma },\qquad \mathrm{and}\qquad \xi \sim {\left|t\right|}^{-\nu }.\end{eqnarray}$
Recently, the pseudo-spectral method of the Chebyshev expansion for the effective potential has been used to calculate the critical exponents [151]. In the Nf = 2 QM model within the fRG approach, the critical exponents for the 3d O(4) and Z(2) symmetry universality classes, which correspond to the black dashed O(4) phase transition line and the red dashed Z(2) phase transition line as shown in figure 9 respectively, are calculated with the Chebyshev expansion of the effective potential. Both the LPA and LPA′ approximations are used in the calculations. In the LPA′ approximation a field-dependent mesonic wave function renormalization is taken into account. The relevant results are shown in table 1, which are also compared with results of critical exponents from other approaches, e.g. Taylor expansion of the effective potential [186–190] and the grid method [191] within the fRG approach, the derivative expansions (DE) up to orders of ${ \mathcal O }({\partial }^{4})$ and ${ \mathcal O }({\partial }^{6})$ with the fRG approach [192, 193], the conformal bootstrap for the 3d conformal field theories (CFTs) [194, 195], Monte Carlo simulations [196], and the d = 3 perturbation expansion [197]. In the meantime, the mean-field values of the critical exponents are also shown in the last line of table 1.
Table 1. Comparison of the critical exponents for the 3d O(4) and Z(2) symmetry universality classes from different approaches. The values with an asterisk are derived from the scaling relations in equation ( |
| Method | β | δ | γ | ν | νc | η | |
|---|---|---|---|---|---|---|---|
| O(4) QM LPA [151] | fRG Chebyshev | 0.3989(41) | 4.975(57) | 1.5458(68) | 0.7878(25) | 0.3982(17) | 0 |
| O(4) QM LPA′ [151] | fRG Chebyshev | 0.3832(31) | 4.859(37) | 1.4765(76) | 0.7475(27) | 0.4056(19) | 0.0252(91)* |
| Z(2) QM LPA [151] | fRG Chebyshev | 0.3352(12) | 4.941(22) | 1.3313(96) | 0.6635(17) | 0.4007(45) | 0 |
| Z(2) QM LPA′ [151] | fRG Chebyshev | 0.3259(01) | 4.808(14) | 1.2362(77) | 0.6305(23) | 0.4021(43) | 0.0337(38)* |
| O(4) scalar theories [187] | fRG Taylor | 0.409 | 4.80* | 1.556 | 0.791 | 0.034 | |
| O(4) KT phase transition [188] | fRG Taylor | 0.387* | 4.73* | 0.739 | 0.047 | ||
| Z(2) KT phase transition [188] | fRG Taylor | 0.6307 | 0.0467 | ||||
| O(4) scalar theories [189] | fRG Taylor | 0.4022* | 5.00* | 0.8043 | |||
| O(4) scalar theories LPA [186] | fRG Taylor | 0.4030(30) | 4.973(30) | 0.8053(60) | |||
| O(4) QM LPA [190] | fRG Taylor | 0.402 | 4.818 | 1.575 | 0.787 | 0.396 | |
| O(4) scalar theories [191] | fRG Grid | 0.40 | 4.79 | 0.78 | 0.037 | ||
| Z(2) scalar theories [191] | fRG Grid | 0.32 | 4.75 | 0.64 | 0.044 | ||
| O(4) scalar theories [192] | fRG DE | 0.7478(9) | 0.0360(12) | ||||
| Z(2) scalar theories [192, 193] | fRG DE | 0.63012(5) | 0.0361(3) | ||||
| O(4) CFTs [194] | Conformal bootstrap | 0.7472(87) | 0.0378(32) | ||||
| Z(2) CFTs [195] | Conformal bootstrap | 0.629971(4) | 0.0362978(20) | ||||
| O(4) spin model [196] | Monte Carlo | 0.3836(46) | 4.851(22) | 1.477(18) | 0.7479(90) | 0.4019(71)* | 0.025(24)* |
| Z(2) d = 3 expansion [197] | Summed perturbation | 0.3258(14) | 4.805(17)* | 1.2396(13) | 0.6304(13) | 0.4027(23) | 0.0335(25) |
| Mean field | 1/2 | 3 | 1 | 1/2 | 1/3 | 0 |
In figure 25 the critical exponent β for the 3d O(4) and Z(2) symmetry universality classes extracted from different ranges of temperature is shown, which is obtained with the Chebyshev expansion for the effective potential in the fRG [151]. It is found that only when the temperature is very close to the critical temperature, say ∣T − Tc∣ ≲ 0.01 MeV, the fRG results of both the O(4) and Z(2) symmetry universality classes are consistent with the conformal bootstrap results. This indicates that the size of the critical region is extremely small, far smaller than ∼1 MeV. Similar estimates for the size of the critical region are also found in, e.g. [98, 186, 198, 199].
Figure 25. Critical exponent β for the 3d O(4) (green lines) and Z(2) (red lines) symmetry universality classes extracted from different ranges of the temperature, obtained within the fRG approach in [151]. The conformal bootstrap results (gray dashed lines) [194, 195] are also presented for comparison. The plot is adapted from [151]. |
4. QCD at finite temperature and density
In this section, we would like to present a brief review on recent progress in studies of QCD at finite temperature and densities within the fRG approach to the first-principle QCD, which are mainly based on the work in [18, 62]. The relevant fRG flows at finite temperature and densities there have been developed from the counterparts in the vacuum in [59, 60]. Furthermore, these flows are also built upon the state-of-the-art quantitative fRG results for Yang–Mills theory in the vacuum [56] and at finite temperature [57], QCD in the vacuum [58, 61].
In the following, we focus on the chiral phase transition. As mentioned above, the QCD phase transitions involve both the chiral phase transition and the confinement-deconfinement phase transition. In fact, related subjects of the deconfinement phase transition, such as the Wilson loop, the Polyakov loop, the flows of the effective action of a background gauge field, the relation between the confinement and correlation functions, etc, have been widely studied within the fRG approach to Yang–Mills theory and QCD. Significant progress has been made in relevant studies, see [55] for a recent overview, and also e.g. [70, 108, 200–215].
4.1. Propagators and anomalous dimensions
The flow equations of inverse propagators are obtained by taking the second derivative of the Wetterich equation in equation (80 ) with respect to respective fields. The resulting flow equations of the inverse quark, gluon and meson propagators are shown diagrammatically in figure 26. Making appropriate projections for these flow equations, one is able to arrive at the flow of the wave function renormalization ZΦ,k as shown in equation (49 ) for a given field Φ, which includes nontrivial dispersion relation for the field, and is more conveniently reformulated as the anomalous dimension as follows84 ), see also equation (A24 ), reads62 ) on their respective equations of motion, that are vanishing except the σ field and the temporal gluon field A0. Note that the minus sign on the r.h.s. of equation (183 ), while not appearing in equation (A24 ), is due to the fact that the right derivative is used in equation (A24 ). Moreover, the quark chemical potentials in equation (48 ) have been assumed to be vanishing in equation (183 ). Apparently, the quark wave function renormalization and mass are readily obtained by projecting equation (183 ) onto the vector and scalar channels, respectively, which read185 ) is complex-valued at nonzero chemical potentials, are both related to constraints from the Silver-Blaze property for the correlation functions at finite chemical potentials, and see, e.g. [18, 119, 120, 127, 216] for more details. In equation (185 ) Re denotes that the real part of the expression in the square bracket is adopted. Note that in equation (185 ) the difference between the parts of the quark wave function renormalization longitudinal and transversal to the heat bath is ignored, where the projection is done on the spacial component, and thus the transversal anomalous dimension is used. The explicit expression of ηq,k is given in equation (C1 ). In the left panel of figure 27, the quark wave function renormalization Zq,k=0 (p0, p = 0) is depicted as a function of the temperature with different values of the baryon chemical potential.
$\begin{eqnarray} {\eta }_{{\rm{\Phi }},k}=-\displaystyle \frac{{\partial }_{t} {Z}_{{\rm{\Phi }},k}} {{Z}_{{\rm{\Phi }},k}}.\end{eqnarray}$
The quark two-point correlation function defined in equation ( $\begin{eqnarray}\begin{array} {rcl} {{\rm{\Gamma }}}_{k}^{(2)\bar{q}q}(p) & = & -{\left.\displaystyle \frac{{\delta }^{2} {{\rm{\Gamma }}}_{k}[{\rm{\Phi }}]} {\delta \bar{q}(-p)\delta q(p)}\right|}_{{\rm{\Phi }}={{\rm{\Phi }}}_{\mathrm{EoM}}}\\ & = & {Z}_{q,k}(p){\rm{i}}\gamma \cdot p+{m}_{q,k}(p),\end{array}\end{eqnarray}$
where ΦEoM denotes the fields in equation ( $\begin{eqnarray}\begin{array} {rcl} {Z}_{q,k}(p) & = & \displaystyle \frac{1} {4{\rm{i}}}\displaystyle \frac{1} {{p}^{2}}\mathrm{tr}\left(\gamma \cdot p\,{{\rm{\Gamma }}}_{k}^{(2)\bar{q}q}(p)\right)\\ {m}_{q,k}(p) & = & \displaystyle \frac{1} {4}\mathrm{tr}\left({{\rm{\Gamma }}}_{k}^{(2)\bar{q}q}(p)\right),\end{array}\end{eqnarray}$
where the trace runs only in the Dirac space. Neglecting the dependence of the quark wave function renormalization on the spacial momentum p, one arrives at the quark anomalous dimension as follows $\begin{eqnarray}\begin{array} {rcl} {\eta }_{q,k}({p}_{0}) & = & \displaystyle \frac{1} {4{Z}_{q,k}({p}_{0})}\\ & & \times \mathrm{Re} {\left[\displaystyle \frac{\partial } {\partial ({\left|{\boldsymbol{p}}\right|}^{2})}\mathrm{tr}\left({\rm{i}} {\boldsymbol{\gamma }}\cdot {\boldsymbol{p}}\left({\partial }_{t} {{\rm{\Gamma }}}_{k}^{(2)\bar{q}q}(p)\right)\right)\right]}_{{\boldsymbol{p}}=0},\end{array}\end{eqnarray}$
where the computation is done at p = 0, and p0 is a small, but nonvanishing frequency. The nontrivial choice of p0 and the fact that the expression in the square bracket in equation (
Figure 26. Diagrammatic representation of the flow equations for the inverse quark, gluon and meson propagators, respectively, where the dashed lines stand for the mesons and the dotted lines denote the ghost. |
Figure 27. Quark (left panel, Zq) and mesonic (right panel, |
The mesonic two-point correlation functions, i.e. those of the π and σ fields, are given by${\bar{Z}}_{\phi ,k=0}$ is depicted as a function of the temperature with different values of the baryon chemical potentials. The other is the case p = 0, and equation (189 ) is reduced the equation as follows${\bar{m}}_{\pi }$ and ${\bar{m}}_{\sigma }$ are approximately equal to their pole masses, respectively. The explicit expressions for equations (191 ) and (189 ) are presented in equations (C2 ) and (C3 ), respectively.
$\begin{eqnarray}\begin{array} {rcl} {{\rm{\Gamma }}}_{k}^{(2)\pi \pi }(p) & = & {\left.\displaystyle \frac{{\delta }^{2} {{\rm{\Gamma }}}_{k}[{\rm{\Phi }}]} {\delta {\pi }_{i}(-p)\delta {\pi }_{j}(p)}\right|}_{{\rm{\Phi }}={{\rm{\Phi }}}_{\mathrm{EoM}}}\\ & = & \left({Z}_{\pi ,k}(p){p}^{2}+{m}_{\pi ,k}^{2}\right){\delta }_{{ij}},\end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array} {rcl} {{\rm{\Gamma }}}_{k}^{(2)\sigma \sigma }(p) & = & {\left.\displaystyle \frac{{\delta }^{2} {{\rm{\Gamma }}}_{k}[{\rm{\Phi }}]} {\delta \sigma (-p)\delta \sigma (p)}\right|}_{{\rm{\Phi }}={{\rm{\Phi }}}_{\mathrm{EoM}}}\\ & = & {Z}_{\sigma ,k}(p){p}^{2}+{m}_{\sigma ,k}^{2},\end{array}\end{eqnarray}$
respectively, where the curvature masses of the mesons read $\begin{eqnarray} {m}_{\pi ,k}^{2}={V}_{k}^{\prime} (\rho ),\qquad {m}_{\sigma ,k}^{2}={V}_{k}^{\prime} (\rho )+2\rho {V}_{k}^{(2)}(\rho ).\end{eqnarray}$
As same as the quark wave function renormalization, the thermal splitting of the mesonic wave function renormalization in parts longitudinal and transversal to the heat bath is neglected. Moreover, one has assumed a unique renormalization for both the pion and sigma fields, viz., Zφ,k = Zπ,k = Zσ,k. The validity of these approximations has been verified seriously in [148]. The anomalous dimension of mesons at vanishing frequency p0 = 0 and a finite spacial momentum p is given by $\begin{eqnarray} {\eta }_{\phi }(0,{\boldsymbol{p}})=-\displaystyle \frac{{\delta }_{{ij}}} {3{Z}_{\phi }(0,{\boldsymbol{p}})}\displaystyle \frac{{\partial }_{t} {{\rm{\Gamma }}}_{{\pi }_{i} {\pi }_{j}}^{(2)}(0,{\boldsymbol{p}})-{\partial }_{t} {{\rm{\Gamma }}}_{{\pi }_{i} {\pi }_{j}}^{(2)}(0,0)} {{{\boldsymbol{p}}}^{2}}.\end{eqnarray}$
Two cases are of special interest: one is the choice p2 = k2, i.e. ηφ(0, k), which captures the most momentum dependence of the mesonic two-point correlation functions, and thus is usually used in the r.h.s. of flow equations involving mesonic degrees of freedom, for more detailed discussions see [18]. The anomalous dimension ηφ(0, k) leaves us with the renormalization constant defined as follows $\begin{eqnarray}\displaystyle \frac{1} {{\bar{Z}}_{\phi ,k}} {\partial }_{t} {\bar{Z}}_{\phi ,k}\equiv -{\eta }_{\phi }(0,k),\quad \mathrm{with}\quad {\bar{Z}}_{\phi ,k={\rm{\Lambda }}}=1.\end{eqnarray}$
In the right panel of figure 27, $\begin{eqnarray} {\eta }_{\phi }(0)=-\displaystyle \frac{1} {3{Z}_{\phi }(0)} {\delta }_{{ij}} {\left[\displaystyle \frac{\partial } {\partial {{\boldsymbol{p}}}^{2}} {\partial }_{t} {{\rm{\Gamma }}}_{{\pi }_{i} {\pi }_{j}}^{(2)}\right]}_{p=0}.\end{eqnarray}$
The related wave function renormalization Zφ,k(0) is used to extract the renormalized meson mass as $\begin{eqnarray} {\bar{m}}_{\pi }=\displaystyle \frac{{m}_{\pi ,k=0}} {\sqrt{{Z}_{\phi ,k=0}(0)}},\qquad {\bar{m}}_{\sigma }=\displaystyle \frac{{m}_{\sigma ,k=0}} {\sqrt{{Z}_{\phi ,k=0}(0)}},\end{eqnarray}$
where For the ghost anomalous dimension, one extracts it from the relation as follows${Z}_{c,k=0}^{\mathrm{QCD}}(p)$ denotes the momentum-dependent ghost wave function renormalization in QCD with Nf = 2 flavors in the vacuum obtained in [61]. Note that the ghost propagator is found to be very insensitive to the effects of finite temperature as well as the quark contributions for Nf = 2 and Nf = 2 + 1 flavors, see e.g. [57].
$\begin{eqnarray} {\eta }_{c}=-{\left.\displaystyle \frac{p{\partial }_{p} {Z}_{c,k=0}^{\mathrm{QCD}}(p)} {{Z}_{c,k=0}^{\mathrm{QCD}}(p)}\right|}_{p=k},\end{eqnarray}$
where The gluon anomalous dimension is decomposed into a sum of three parts, as follows194 ) is further expressed as${Z}_{A,k=0}^{\mathrm{QCD}}(p)$ for the Nf = 2 flavor QCD in [61], which reads${\rm{\Delta }} {\eta }_{A}^{q}$ in equation (194 ) and ${\eta }_{A,\mathrm{vac}}^{s}$ in equation (195 ) can be found in equations (C4 )–(C6 ). Moreover, the in-medium contribution to the gluon anomalous dimension resulting from the glue sector, i.e. the second term on the r.h.s. of equation (194 ), is taken into account in equation (C7 ).
$\begin{eqnarray} {\eta }_{A}={\eta }_{A,\mathrm{vac}}^{\mathrm{QCD}}+{\rm{\Delta }} {\eta }_{A}^{\mathrm{glue}}+{\rm{\Delta }} {\eta }_{A}^{q},\end{eqnarray}$
where the first term on the r.h.s. denotes the gluon anomalous dimension in the vacuum, and the other two terms stand for the medium contributions to the gluon anomalous dimension from the glue (gluons and ghosts) and quark loops, respectively. The vacuum contribution in equation ( $\begin{eqnarray} {\eta }_{A,\mathrm{vac}}^{\mathrm{QCD}}={\left.{\eta }_{A,\mathrm{vac}}^{\mathrm{QCD}}\right|}_{{N}_{f}=2}+{\eta }_{A,\mathrm{vac}}^{s},\end{eqnarray}$
where the two terms on the r.h.s. denote the contributions from the light quarks of Nf = 2 flavors and the strange quark. The former one is inferred from the gluon dressing function $\begin{eqnarray} {\left.{\eta }_{A,\mathrm{vac}}^{\mathrm{QCD}}\right|}_{{N}_{f}=2}=-{\left.\displaystyle \frac{p{\partial }_{p} {Z}_{A,k=0}^{\mathrm{QCD}}(p)} {{Z}_{A,k=0}^{\mathrm{QCD}}(p)}\right|}_{p=k}.\end{eqnarray}$
The explicit expressions of In figure 28 the gluon dressing functions 1/ZA(p) of Nf = 2 and Nf = 2 + 1 flavor QCD in the vacuum are shown. The fRG and lattice results are presented. Here the fRG gluon dressing of Nf = 2 flavors is inputted from [61], as shown in equation (196 ), which is also in quantitative agreement with the lattice result in [217]. Note that the gluon dressing of Nf = 2 + 1 flavors here is a genuine prediction, which is in good agreement with the respective lattice results [218, 219]. In figure 29 both the gluon dressing function and the gluon propagator of Nf = 2 + 1 flavors are presented. The calculated gluon dressing functions at finite temperature and baryon chemical potential in fRG are shown in figure 30. In the left panel of figure 30 several different values of temperature are chosen with μB = 0, and it is found that the gluon dressing function 1/ZA decreases with the increase in temperature. In the right panel, several different values of μB are adopted while with T = 150 MeV fixed. It is observed that the dependence of the gluon dressing function on the baryon chemical potential is very small. The gluon dressing functions at finite temperature obtained in fRG are compared with the relevant lattice results from [220] in figure 31, where several different values of temperature are chosen. One can see that the gluon dressing at finite temperature in fRG is comparable to that of lattice QCD.
Figure 29. Gluon dressing function 1/ZA(p) and the gluon propagator GA = 1/[ZA(p)p2] (inlay) as functions of momenta for QCD of Nf = 2 + 1 flavors in the vacuum, where the fRG results denoted by the black lines are compared with the continuum extrapolated lattice results by the RBC/UKQCD collaboration, see e.g. [218, 219, 221]. For the fRG results, the momentum dependence is given by p2 = k2. The plot is adapted from [18]. |
Figure 30. Gluon dressing function 1/ZA of the Nf = 2 + 1 QCD as a function of spatial momenta ∣p∣ with several different values of temperature (left panel) and baryon chemical potential (right panel), where the Matsubara frequency p0 is chosen to be vanishing. The identification p2 = k2 is used. The plot is adapted from [18]. |
4.2. Strong couplings
The quark-gluon coupling in equation (53 ), the ghost-gluon coupling in equation (54 ), and the three-gluon or four-gluon coupling in equation (50 ), etc, are consistent with one another in the perturbative regime of high energy, due to the Slavnov–Taylor identities (STIs) resulting from the gauge symmetry. However, when the momenta or the RG scale are decreased to k ≲ 1–3 GeV, the gluon mass gap begins to affect the running of strong couplings, and also the transversal strong couplings can not be identified with the longitudinal ones via the modified STIs, see, e.g. [56, 215, 222, 223] for more details. Consequently, different strong couplings deviate from one another in the low energy regime of k ≲ 1–3 GeV [56–58, 61], and thus it is necessary to distinguish them by adding appropriate suffixes. The couplings of the purely gluonic sector read${\lambda }_{{A}^{3},k}$ , ${\lambda }_{{A}^{4},k}$ and ${\lambda }_{\bar{c} {cA},k}$ denote the three-gluon, four-gluon, ghost-gluon dressing functions, respectively, as shown in equations (A42 ), (A48 ), (A36 ). The couplings of the matter sector read
$\begin{eqnarray} {\alpha }_{{A}^{3},k}=\displaystyle \frac{1} {4\pi }\displaystyle \frac{{\lambda }_{{A}^{3},k}^{2}} {{Z}_{A,k}^{3}},\qquad {\alpha }_{{A}^{4},k}=\displaystyle \frac{1} {4\pi }\displaystyle \frac{{\lambda }_{{A}^{4},k}} {{Z}_{A,k}^{2}},\end{eqnarray}$
$\begin{eqnarray} {\alpha }_{\bar{c} {cA},k}=\displaystyle \frac{1} {4\pi }\displaystyle \frac{{\lambda }_{\bar{c} {cA},k}^{2}} {{Z}_{A,k}\,{Z}_{c,k}^{2}},\end{eqnarray}$
where $\begin{eqnarray} {\alpha }_{\bar{l} {lA},k}=\displaystyle \frac{1} {4\pi }\displaystyle \frac{{\lambda }_{\bar{l} {lA},k}^{2}} {{Z}_{A,k}\,{Z}_{q,k}^{2}},\qquad {\alpha }_{\bar{s} {sA},k}=\displaystyle \frac{1} {4\pi }\displaystyle \frac{{\lambda }_{\bar{s} {sA},k}^{2}} {{Z}_{A,k}\,{Z}_{q,k}^{2}},\end{eqnarray}$
where the light-quark-gluon coupling and the strange-quark-gluon coupling have been distinguished. From the calculated results in, e.g. [59, 61], it is reasonable to adopt the approximation as follows $\begin{eqnarray} {\alpha }_{{A}^{4},k}={\alpha }_{{A}^{3},k},\qquad {\alpha }_{\bar{c} {cA},k}\simeq {\alpha }_{\bar{l} {lA},k}.\end{eqnarray}$
The flow equation of the quark-gluon vertex is presented in the first line of figure 32. Projecting onto the classical tensor structure of the quark-gluon vertex, denoted by${g}_{\bar{q} {qA},k}\equiv {\lambda }_{\bar{q} {qA},k}$ is used. In equation (202 ) the flow of quark-gluon vertex, i.e. the r.h.s. of the first line in figure 32, is denoted by201 ), some nonclassical tensor structures also play a sizable role in the dynamical breaking of the chiral symmetry [58, 61, 224], see, e.g. [18] for more detailed discussions. From equation (202 ) one formulates the flow of the light-quark–gluon coupling as205 ) and (206 ) can be found in equations (C9 ) and (C10 ).
$\begin{eqnarray} {\left({S}_{\bar{q} {qA}}^{(3)}\right)}_{\mu }^{a}\equiv -{\rm{i}} {\gamma }_{\mu } {t}^{a},\end{eqnarray}$
one is led to the flow of the quark-gluon coupling as follows $\begin{eqnarray}\begin{array} {rcl} {\partial }_{t} {\bar{g}}_{\bar{q} {qA},k} & = & \left(\displaystyle \frac{1} {2} {\eta }_{A}+{\eta }_{q}\right){\bar{g}}_{\bar{q} {qA},k}+\displaystyle \frac{1} {8({N}_{c}^{2}-1)}\\ & & \times \mathrm{tr}\left[{\left({\overline{\mathrm{Flow}}}_{\bar{q} {qA}}^{(3)}\right)}_{\mu }^{a} {\left({S}_{\bar{q} {qA}}^{(3)}\right)}_{\mu }^{a}\right]\left(\{p\}\right),\end{array}\end{eqnarray}$
where the trace sums over the Dirac and color spaces, and {p} stands for the set of external momenta for the vertex. Here $\begin{eqnarray} {\left({\overline{\mathrm{Flow}}}_{\bar{q} {qA}}^{(3)}\right)}_{\mu }^{a}=-\displaystyle \frac{1} {{Z}_{A,k}^{1/2} {Z}_{q,k}} {\left({\partial }_{t} {{\rm{\Gamma }}}_{k}^{(3)\bar{q} {qA}}\right)}_{\mu }^{a},\end{eqnarray}$
with $\begin{eqnarray} {\left({{\rm{\Gamma }}}_{k}^{(3)\bar{q} {qA}}\right)}_{\mu }^{a}\equiv \displaystyle \frac{\delta } {\delta {A}_{\mu }^{c}}\displaystyle \frac{\overrightarrow{\delta }} {\delta \bar{q}} {{\rm{\Gamma }}}_{k}\displaystyle \frac{\overleftarrow{\delta }} {\delta q}.\end{eqnarray}$
Note that besides the classical tensor structure of the quark-gluon vertex in equation ( $\begin{eqnarray} {\partial }_{t} {\bar{g}}_{\bar{l} {lA},k}=\left(\displaystyle \frac{1} {2} {\eta }_{A}+{\eta }_{q}\right){\bar{g}}_{\bar{l} {lA},k}+{\overline{\mathrm{Flow}}}_{(\bar{l} {lA})}^{(3),A}+{\overline{\mathrm{Flow}}}_{(\bar{l} {lA})}^{(3),\phi },\end{eqnarray}$
where the second term on the r.h.s. corresponds to the two diagrams on the r.h.s. of the flow equation for the quark-gluon vertex as shown in the first line of figure 32, and the last term to the last diagram, which arises from the quark-gluon, quark-meson fluctuations, respectively. For the strange-quark–gluon coupling, one has $\begin{eqnarray} {\partial }_{t} {\bar{g}}_{\bar{s} {sA},k}=\left(\displaystyle \frac{1} {2} {\eta }_{A}+{\eta }_{q}\right){\bar{g}}_{\bar{s} {sA},k}+{\overline{\mathrm{Flow}}}_{(\bar{s} {sA})}^{(3),A},\end{eqnarray}$
where the contribution from the strange-quark–meson interactions is neglected, that is reasonable due to relatively larger masses of the strange quark and strange mesons. The explicit expressions in equations (
Figure 32. Diagrammatic representation of the flow equations for the quark-gluon, three-gluon, and the quark-meson vertices, respectively. |
In the same way, the flow equation of the three-gluon coupling, ${g}_{{A}^{3},k}\equiv {\lambda }_{{A}^{3},k}$ , is readily obtained from the flow of the three-gluon vertex in the second line of figure 32. The flow of the three-gluon coupling is decomposed into a sum of the vacuum part and the in-medium contribution, as follows
$\begin{eqnarray} {\partial }_{t} {\bar{g}}_{{A}^{3},k}={\partial }_{t} {\bar{g}}_{{A}^{3},k}^{\mathrm{vac}}+{\partial }_{t} {\rm{\Delta }} {\bar{g}}_{{A}^{3},k},\end{eqnarray}$
where the vacuum part, i.e. the first term on the r.h.s. is computed in [59], and the second term is identified with the in-medium contribution of the quark-gluon coupling, to wit, $\begin{eqnarray} {\partial }_{t} {\rm{\Delta }} {\bar{g}}_{{A}^{3},k}={\partial }_{t} {\rm{\Delta }} {\bar{g}}_{\bar{l} {lA},k}.\end{eqnarray}$
In figure 33 the light-quark–gluon coupling, the strange-quark–gluon coupling, and the three-gluon coupling are depicted as functions of the RG scale for several different values of the temperature. It is observed that different couplings are consistent with one another in the regime of k ≳ 3 GeV, while deviations develop in the nonperturbative or even semiperturbative scale. Moreover, one can see that the strong couplings decrease with the increasing temperature.
Figure 33. Quark-gluon couplings for light quarks ( |
4.3. Dynamical hadronization, four-quark couplings and Yukawa couplings
Following section 2.5 , one performs the dynamical hadronization for the σ–π channel, and use48 ), where ${\dot{A}}_{k}$ is called the hadronization function. It is more convenient to adopt the dimensionless, renormalized hadronization function and four-quark coupling
$\begin{eqnarray}\langle {\partial }_{t} {\hat{\phi }}_{k}\rangle ={\dot{A}}_{k}\,\bar{q}\tau q,\end{eqnarray}$
with τ = (T0, iγ5T) as shown in equation ( $\begin{eqnarray}\dot{\tilde{A}}=\displaystyle \frac{{Z}_{\phi ,k}^{1/2}} {{Z}_{q,k}} {k}^{2} {\dot{A}}_{k},\qquad {\tilde{\lambda }}_{q,k}=\displaystyle \frac{{k}^{2} {\lambda }_{q,k}} {{Z}_{q,k}^{2}},\end{eqnarray}$
and the renormalized Yukawa coupling, $\begin{eqnarray} {\bar{h}}_{k}=\displaystyle \frac{{h}_{k}} {{Z}_{\phi ,k}^{1/2} {Z}_{q,k}}.\end{eqnarray}$
Inserting the effective action in equation (48 ) into equation (80 ) and performing a projection on the four-quark interaction in the σ–π channel, one is left with213 ) can be further written asC11 ) and (C12 ). In figure 35 the two terms on the r.h.s. of equation (214 ) and their ratios with respect to the total flow of the four-quark coupling are depicted as functions of the RG scale with T = 0 and μB = 0. It is observed that the flow resulting from the gluon exchange is dominant in the region of high energy, while that from the meson exchange plays a significant role in lower energy.
$\begin{eqnarray} {\partial }_{t} {\tilde{\lambda }}_{q,k}=2(1+{\eta }_{q,k}){\tilde{\lambda }}_{q,k}+{\overline{\mathrm{Flow}}}_{{\left(\bar{q}\tau q\right)}^{2}}^{(4)}+\dot{\tilde{A}}\,{\bar{h}}_{k},\end{eqnarray}$
where $\begin{eqnarray} {\overline{\mathrm{Flow}}}_{{\left(\bar{q}\tau q\right)}^{2}}^{(4)}\equiv -\displaystyle \frac{{k}^{2}} {{Z}_{q,k}^{2}} {\left({\partial }_{t} {{\rm{\Gamma }}}_{k,{\left(\bar{q}\tau q\right)}^{2}}^{(4)}\right)}_{{\dot{A}}_{k}=0},\end{eqnarray}$
is the flow of the four-quark coupling in the σ–π channel, whose contributions have been depicted in figure 34. One can see that the contributed diagrams can be classified into three sets, which correspond to the three lines on the r.h.s. of the flow equation in figure 34. The first line is comprised of diagrams with two gluon exchanges and the second line with two meson exchanges. The last line denotes the contributions from the mixed diagrams with one gluon exchange and one meson exchange. The mixed diagrams are negligible, since the dynamics of gluons and mesons dominate in different regimes of the RG scale, as shown in figure 35. Therefore, the four-quark flow in equation ( $\begin{eqnarray} {\overline{\mathrm{Flow}}}_{{\left(\bar{q}\tau q\right)}^{2}}^{(4)}={\overline{\mathrm{Flow}}}_{{\left(\bar{q}\tau q\right)}^{2}}^{(4),A}+{\overline{\mathrm{Flow}}}_{{\left(\bar{q}\tau q\right)}^{2}}^{(4),\phi },\end{eqnarray}$
where the two terms on the r.h.s. correspond to the contributions from the two gluon exchanges and two meson exchanges, respectively. Their explicit expressions are presented in equations (
Figure 34. Diagrammatic representation of the flow equation for the four-quark vertex. The first line on the r.h.s. denotes the contributions from two gluon exchanges and the second line those from two meson exchanges. The last line stands for the contributions from the mixed diagrams with one gluon exchange and one meson exchange. |
Figure 35. Left panel: Comparison between the flow of the four-quark coupling in the σ–π channel from two gluon exchanges, |
The dynamical hadronization is done by demanding212 ) one arrives at the hadronization function as follows
$\begin{eqnarray} {\tilde{\lambda }}_{q,k}=0,\end{eqnarray}$
for every value of the RG scale k, which is equivalent to the fact that the Hubbard–Stratonovich transformation is performed for every value of k. Thus from equation ( $\begin{eqnarray}\dot{\tilde{A}}=-\displaystyle \frac{1} {{\bar{h}}_{k}} {\overline{\mathrm{Flow}}}_{{\left(\bar{q}\tau q\right)}^{2}}^{(4)}.\end{eqnarray}$
Inserting the effective action in equation (48 ) into (80 ) and performing a projection on the Yukawa interactions between quarks and σ–π mesons, i.e. $\bar{q}\tau \cdot \phi q$ , one is led to217 )C13 ). Therefore, by substituting the hadronization function in equation (216 ) into (217 ), one obtains the total flow of the Yukawa coupling finally. Evidently, the dynamics of resonance of quarks are stored in the interactions between quarks and mesons through dynamic hadronization.
$\begin{eqnarray} {\partial }_{t} {\bar{h}}_{k}=\left(\displaystyle \frac{1} {2} {\eta }_{\phi ,k}+{\eta }_{q,k}\right){\bar{h}}_{k}+{\overline{\mathrm{Flow}}}_{(\bar{q} {\boldsymbol{\tau }}q){\boldsymbol{\pi }}}^{(3)}-{\tilde{m}}_{\pi ,k}^{2}\dot{\tilde{A}},\end{eqnarray}$
with the dimensionless and renormalized pion mass as follows $\begin{eqnarray} {\tilde{m}}_{\pi ,k}^{2}=\displaystyle \frac{{m}_{\pi ,k}^{2}} {{Z}_{\phi ,k} {k}^{2}}.\end{eqnarray}$
Here in equation ( $\begin{eqnarray} {\overline{\mathrm{Flow}}}_{(\bar{q} {\boldsymbol{\tau }}q){\boldsymbol{\pi }}}^{(3)}\equiv \displaystyle \frac{1} {{Z}_{\phi ,k}^{1/2} {Z}_{q,k}} {\left({\partial }_{t} {{\rm{\Gamma }}}_{k,(\bar{q} {\boldsymbol{\tau }}q){\boldsymbol{\pi }}}^{(3)}\right)}_{{\dot{A}}_{k}=0},\end{eqnarray}$
is the flow of the Yukawa coupling between the pion and quarks, which is shown in the third line of figure 32. Its explicit expression is presented in equation (In the left panel of figure 36, the renormalized Yukawa coupling with k = 0 is shown as a function of the temperature with several different values of baryon chemical potential. It is observed that with the increase of T or μB, the Yukawa coupling decays rapidly due to the restoration of the chiral symmetry. In the right panel of figure 36, the dependence of the renormalized Yukawa coupling on the RG scale for different values of temperature is investigated. One can see that the Yukawa coupling is stable in the region of k ≳ 1 GeV, and the effects of temperature play a role approximately in k ≲ 2πT. The effective four-quark coupling in the σ–π channel can be described by the ratio ${\bar{h}}_{k}^{2}/(2{\bar{m}}_{\pi ,k}^{2})$ [18, 59], which is shown as a function of the RG scale with several different values of temperature and μB = 0 in figure 37. It is observed that with the decrease of k and entering the regime of the chiral symmetry breaking, a resonance occurs in the scalar-pseudoscalar channel, which results in a rapid increase of the effective four-quark coupling. Moreover, the dependence of the effective four-quark coupling on the baryon chemical potential is shown in figure 38. One finds that the effective four-quark coupling decrease with the increasing temperature or baryon chemical potential.
Figure 36. Left panel: Renormalized Yukawa coupling of Nf = 2 + 1 flavor QCD at vanishing RG scale k = 0 as a function of the temperature with several different values of μB obtained in fRG. Right panel: Renormalized Yukawa coupling of Nf = 2 + 1 flavor QCD as a function of the RG scale with several different values of T and μB = 0. The plots are adopted from [18]. |
Figure 37. Effective four-quark coupling of Nf = 2 + 1 flavor QCD in the pseudoscalar channel |
Figure 38. Effective four-quark coupling of Nf = 2 + 1 flavor QCD in the pseudoscalar channel at vanishing RG scale |
4.4. Natural emergence of LEFTs from QCD
The fRG approach to QCD with the dynamical hadronization discussed above, provide us with a method to study the transition of degrees of freedom from fundamental to composite ones. One is also able to observe a natural emergence of LEFTs from the original QCD. To that end, figure 39 shows the four-quark single gluon exchange coupling for light quarks ${\bar{g}}_{\bar{l} {lA},k}^{2}$ and strange quarks ${\bar{g}}_{\bar{s} {sA},k}^{2}$ , dimensionless four-quark single meson exchange coupling ${\bar{h}}_{k}^{2}/(1+{\tilde{m}}_{\pi ,k}^{2})$ and ${\bar{h}}_{k}^{2}/(1+{\tilde{m}}_{\sigma ,k}^{2})$ as functions of the RG scale in the vacuum. Evidently, it is found that the gluonic exchange couplings are dominant in the perturbative regime of k ≳ 1 GeV. However, with the decrease of the RG scale the active dynamic is taken over gradually by the mesonic degrees of freedom, and one can see that the gluonic couplings and the Yukawa couplings are comparable to each other at k ≈ 600. In figure 40 dimensionless propagator gappings $1/(1+{\tilde{m}}_{{{\rm{\Phi }}}_{i},k}^{2})$ for Φi = l, s, σ, π are shown as functions of the RG scale. The gluon dressing function is also shown there for comparison. In figure 40 one observes the same information on decouplings as in figure 39, that is, as the RG scale evolves from the UV towards IR, the gluons decouple from the matter at first, then the quarks, and the mesons finally. For more related discussions see [18].
Figure 39. Four-quark single gluon exchange coupling for light quarks |
Figure 40. Dimensionless propagator gappings |
4.5. Chiral condensate
The chiral condensate of quark qi = u, d, s reads${m}_{{q}_{i}}^{0}$ is the current quark mass. Obviously, the light quark condensate is given by Δl = Δu = Δd with ${m}_{l}^{0}={m}_{u}^{0}={m}_{d}^{0}$ . The renormalized condensate is as follows220 ) to be subtracted, where the dimensionless ${{\rm{\Delta }}}_{{q}_{i},R}$ is normalized by a constant ${{ \mathcal N }}_{R}$ , e.g. ${{ \mathcal N }}_{R}\sim {f}_{\pi }^{4}$ . In the present fRG approach to QCD, the light quark condensate is given by48 ), one is led to
$\begin{eqnarray} {{\rm{\Delta }}}_{{q}_{i}}=-{m}_{{q}_{i}}^{0}T\sum _{n\in {\mathbb{Z}}}\int \displaystyle \frac{{{\rm{d}}}^{3}q} {{\left(2\pi \right)}^{3}}\mathrm{tr}\,{G}_{{q}_{i} {\bar{q}}_{i}}(q),\end{eqnarray}$
up to a renormalization term, where no summation for the index i is assumed and $\begin{eqnarray} {{\rm{\Delta }}}_{{q}_{i},R}=\displaystyle \frac{1} {{{ \mathcal N }}_{R}}\left[{{\rm{\Delta }}}_{{q}_{i}}(T,{\mu }_{q})-{{\rm{\Delta }}}_{{q}_{i}}(0,0)\right],\end{eqnarray}$
renders the vacuum part of the chiral condensate in equation ( $\begin{eqnarray} {{\rm{\Delta }}}_{l}=\displaystyle \frac{1} {2} {m}_{l}^{0}\displaystyle \frac{\partial {\rm{\Omega }}[{{\rm{\Phi }}}_{\mathrm{EoM}};T,{\mu }_{q}]} {\partial {m}_{l}^{0}}=\displaystyle \frac{1} {2} {c}_{\sigma }\displaystyle \frac{\partial {\rm{\Omega }}[{{\rm{\Phi }}}_{\mathrm{EoM}};T,{\mu }_{q}]} {\partial {c}_{\sigma }},\end{eqnarray}$
where Ω is the thermodynamic potential with the field ΦEoM being on is the equation of motion. From equation ( $\begin{eqnarray} {{\rm{\Delta }}}_{l}(T,{\mu }_{q})=-\displaystyle \frac{1} {2} {c}_{\sigma }\,{\sigma }_{\mathrm{EoM}}(T,{\mu }_{q}),\end{eqnarray}$
and $\begin{eqnarray} {{\rm{\Delta }}}_{l,R}(T,{\mu }_{q})=-\displaystyle \frac{{c}_{\sigma }} {2\,{{ \mathcal N }}_{R}}\left[{\sigma }_{\mathrm{EoM}}(T,{\mu }_{q})-{\sigma }_{\mathrm{EoM}}(0,0)\right].\end{eqnarray}$
The reduced condensate Δl,s is defined as the weighted difference between the light and strange quark condensates as follows,222 ), one arrives at
$\begin{eqnarray} {{\rm{\Delta }}}_{l,s}(T,{\mu }_{q})=\displaystyle \frac{1} {{{ \mathcal N }}_{l,s}}\left[{{\rm{\Delta }}}_{l}(T,{\mu }_{q})-{\left(\displaystyle \frac{{m}_{l}^{0}} {{m}_{s}^{0}}\right)}^{2} {{\rm{\Delta }}}_{s}(T,{\mu }_{q})\right],\end{eqnarray}$
which is usually normalized with its value in the vacuum, i.e. $\begin{eqnarray} {{\rm{\Delta }}}_{l,s}(T,{\mu }_{q})=\displaystyle \frac{{{\rm{\Delta }}}_{l}(T,{\mu }_{q})-{\left(\tfrac{{m}_{l}^{0}} {{m}_{s}^{0}}\right)}^{2} {{\rm{\Delta }}}_{s}(T,{\mu }_{q})} {{{\rm{\Delta }}}_{l}(0,0)-{\left(\tfrac{{m}_{l}^{0}} {{m}_{s}^{0}}\right)}^{2} {{\rm{\Delta }}}_{s}(0,0)}.\end{eqnarray}$
Similar to equation ( $\begin{eqnarray} {{\rm{\Delta }}}_{s}={m}_{s}^{0}\displaystyle \frac{\partial {\rm{\Omega }}[{{\rm{\Phi }}}_{\mathrm{EoM}};T,{\mu }_{q}]} {\partial {m}_{s}^{0}}={c}_{{\sigma }_{s}}\displaystyle \frac{\partial {\rm{\Omega }}[{{\rm{\Phi }}}_{\mathrm{EoM}};T,{\mu }_{q}]} {\partial {c}_{{\sigma }_{s}}},\end{eqnarray}$
and thus $\begin{eqnarray} {{\rm{\Delta }}}_{s}(T,{\mu }_{q})=-\displaystyle \frac{1} {\sqrt{2}} {c}_{{\sigma }_{s}}\,{\sigma }_{s,\mathrm{EoM}}(T,{\mu }_{q}).\end{eqnarray}$
Finally, one is led to $\begin{eqnarray} {{\rm{\Delta }}}_{l,s}(T,{\mu }_{q})=\displaystyle \frac{{\left(\sigma -\sqrt{2}\tfrac{{c}_{\sigma }} {{c}_{{\sigma }_{s}}} {\sigma }_{s}\right)}_{T,{\mu }_{q}}} {{\left(\sigma -\sqrt{2}\tfrac{{c}_{\sigma }} {{c}_{{\sigma }_{s}}} {\sigma }_{s}\right)}_{0,0}},\end{eqnarray}$
where one has used $\begin{eqnarray}\displaystyle \frac{{m}_{l}^{0}} {{m}_{s}^{0}}=\displaystyle \frac{{c}_{\sigma }} {{c}_{{\sigma }_{s}}}.\end{eqnarray}$
In the left panel figure 41, the renormalized light quark condensate is shown as a function of the temperature with several different values of baryon chemical potential. In the case of vanishing baryon chemical potential, the fRG result is compared with the continuum extrapolated result of lattice QCD [160], and excellent agreement is found. In the right panel of figure 41, the derivative of Δl,R with respect to the temperature, i.e. the thermal susceptibility of the renormalized light quark condensate, is shown. The peak position of the thermal susceptibility can be used to define the pseudocritical temperature, which is found to be Tc = 156 MeV for μB = 0, in good agreement with the lattice result. Figure 42 shows the reduced condensate as a function of the temperature with μB = 0, in comparison to the lattice simulations. One finds that for the physical current quark mass ratio, i.e. ${m}_{s}^{0}/{m}_{l}^{0}={c}_{{\sigma }_{s}}/{c}_{\sigma }\approx 27$ [225], the fRG results are in quantitative agreement with the lattice ones.
Figure 41. Left panel: Renormalized light quark chiral condensate Δl,R of Nf = 2 + 1 flavor QCD as a function of the temperature with several different values of baryon chemical potential obtained in fRG, in comparison to the lattice results at vanishing μB [160]. Right panel: Derivative of Δl,R with respect to the temperature, i.e. the thermal susceptibility of the renormalized light quark condensate, as a function of the temperature with several different values of baryon chemical potential. The plots are adopted from [18]. |
Figure 42. Reduced condensate of Nf = 2 + 1 flavor QCD as a function of the temperature with μB = 0 obtained in fRG, where the constituent quark mass differences are |
4.6. Phase structure
The QCD phase diagram in the plane of T and μB is shown in figure 43. The first-principle fRG results in [18] are in comparison to state-of-the-art calculations of other functional approaches, e.g. the DSE [19, 20], and lattice QCD [8, 12]. The phase boundary in the regime of continuous crossover at small or medium μB shows the dependence of the pseudocritical temperature on the value of the baryon chemical potential. In the calculations of fRG, this pseudocritical temperature is determined by the thermal susceptibility of the renormalized light quark chiral condensate, ∂Δl,R/∂T, as discussed in section 4.5 . Expanding the pseudocritical temperature around μB = 0, one arrives at
$\begin{eqnarray}\displaystyle \frac{{T}_{c}({\mu }_{B})} {{T}_{c}}=1-\kappa {\left(\displaystyle \frac{{\mu }_{B}} {{T}_{c}}\right)}^{2}+\lambda {\left(\displaystyle \frac{{\mu }_{B}} {{T}_{c}}\right)}^{4}+\cdots ,\end{eqnarray}$
with Tc = Tc(μB = 0), where the quadratic expansion coefficient κ is usually called the curvature of the phase boundary. It is a sensitive measure for the QCD dynamics at finite temperature and densities. Therefore, it provides a benchmark test for functional approaches to make a comparison of the curvature at small baryon chemical potential with lattice simulations. The curvature values of the phase boundary obtained from fRG, DSE, and lattice QCD are summarized in the second line of table 2. It is found recent results of the curvature from state-of-the-art functional approaches, e.g. fRG [18], DSE [19, 20], have already been compared to the lattice results. By contrast, those obtained from relatively former calculations of functional methods, e.g. [21, 226, 227], are significantly larger than the values of the curvature obtained from lattice simulations.
Figure 43. Phase diagram of Nf = 2 + 1 flavor QCD in the plane of the temperature and the baryon chemical potential. The fRG results [18] are compared with those from Dyson–Schwinger equations [19, 20], lattice QCD [8, 12]. The hatched red area denotes the region of inhomogeneous instability for the chiral condensate found in the calculations of fRG. Some freeze-out data are also shown in the phase diagram: [175] (STAR), [228] (Alba et al), [4] (Andronic et al), [229] (Becattini et al), [230] (Vovchenko et al), and [231] (Sagun et al). |
Besides the curvature of the phase boundary, another key ingredient of the QCD phase structure is the CEP of the first-order phase transition line at large baryon chemical potential, which is currently searched for with lots of effort in experiments [22, 37–43]. Lattice simulations, however, are restricted in the region of μB/T ≲ 2–3 because of the sign problem at finite chemical potentials, and in this region, no signal of CEP is observed [17]. Passing lattice benchmark tests at small baryon chemical potentials, functional approaches are able to provide relatively reliable estimates for the location of the CEP. The recent results of CEP after benchmark testing are presented in the third line of table 2, and also depicted in the phase diagram in figure 43. Remarkably, estimates of CEP from fRG and DSE converge in a rather small region at baryon chemical potentials of about 600 MeV. Note that results of [20] in table 2 are obtained without the dynamics of sigma and pion, and the relevant results are κ = 0.0167 and ${\left(T,{\mu }_{B}\right)}_{{}_{\mathrm{CEP}}}=(117,600)$ MeV when they are included, and see also, e.g. [21, 226, 233] for related discussions. It should be reminded that errors of functional approaches increase significantly when μB/T ≳ 4, for a detailed discussion see [18, 234], and thus one arrives at a more reasonable estimation for the location of CEP as $450\,\mathrm{MeV}\lesssim {{\mu }_{B}}_{\mathrm{CEP}}\lesssim 650\,\mathrm{MeV}$ .
4.6.1. Region of inhomogeneous instability at large baryon chemical potential
At large baryon chemical potentials, it is found within the fRG approach to QCD, that the mesonic wave function renormalization in equations (186 ) and (187 ) develops negative values at small momenta [18]. As shown in figure 44 the mesonic wave function renormalization at vanishing external momenta Zφ,k=0(0) is depicted as a function of the temperature with several different values of μB. One can see a negative Zφ,k=0(0) begins to appear for a temperature region when μB ≳ 420 MeV. The negative regime is clearly shown in the right plot of $\left|1/{Z}_{\phi ,k=0}(0)\right|$ between the two spikes, and it widens with the increase of the baryon chemical potential. From the two-point correlation functions of mesons, as shown in equations (186 ) and (187 ), the negative Zφ,k=0(0) implies that, for the dispersion relations of mesons, there is a minimum at a finite p2 ≠ 0. This nontrivial behavior that the minimum of dispersion is located at a finite momentum is indicative of an inhomogeneous instability, for instance, the formation of a spatially modulated chiral condensate. However, it should be reminded that a negative Zφ,k=0(0) is not bound to the formation of an inhomogeneous phase, it can also serve as a precursor for the inhomogeneous phase. See, e.g. [135, 137, 235–244] for more related discussions. Most notably, very recently consequence of the inhomogeneous instability indicated by the negative wave function renormalization on the phenomenology of heavy-ion collisions has been studied. It is found that this inhomogeneous instability would result in a moat regime in both the particle pT spectrum and the two-particle correlation, where the peaks are located at nonzero momenta [243, 244]. The region of negative Zφ,k=0(0) is shown in figure 45 by the blue area, while the red hatched area shows where this region overlaps with a sizable chiral condensate.
Figure 44. Left panel: Mesonic wave function renormalization at vanishing external momenta Zφ,k=0(0) as a function of the temperature with several different values of the baryon chemical potential, obtained in the fRG approach to Nf = 2 + 1 flavor QCD. Right panel: |
Figure 45. Phase diagram of Nf = 2 + 1 flavor QCD obtained in the fRG approach to QCD in comparison to freeze-out data. See also the caption of figure 43. The blue area denotes the region of negative mesonic wave function renormalization at vanishing external momenta Zφ,k=0(0), which is an indicator of inhomogeneous instability. The red hatched area stands for the regime with negative Zφ,k=0(0) and also sizable chiral condensate. The plot is adapted from [18]. |
4.7. Magnetic EoS
From equations (222 ), (224 ), and (225 ), it is convenient to define the corresponding susceptibilities for the light quark condensate, the renormalized light quark condensate, and the reduced condensate, i.e.
$\begin{eqnarray} {\chi }_{M}^{(i)}(T)=-\displaystyle \frac{\partial } {\partial {m}_{l}^{0}}\left(\displaystyle \frac{{{\rm{\Delta }}}_{i}(T)} {{m}_{l}^{0}}\right),\end{eqnarray}$
with (i) = (l), (l, R), (l, s) [62]. Similar to the difference between the light quark condensate and the renormalized light quark condensate, the susceptibilities for them differ by the vacuum contribution. Among these three susceptibilities, the reduced susceptibility for the reduced condensate defined in fRG and lattice QCD, e.g. [13], matches better.In figure 46 the susceptibility obtained from the reduced condensate as a function of the temperature for different pion masses in the fRG approach [62], is compared with the lattice results [13]. One can see that the fRG results are in good agreement with the lattice ones for pion masses mπ ≳ 100 MeV. There are some slight deviations from the two approaches for small pion masses. From the dependence of the reduced susceptibility on the temperature for a fixed pion mass in figure 46, one is able to define the pseudocritical temperature for the chiral crossover at such a value of pion mass, via the peak position of the curves, denoted by Tpc.
Figure 47 shows the dependence of the pseudocritical temperature on the pion mass, obtained both from the fRG approach and the lattice QCD. The values of pseudocritical temperature at finite pion masses are also extrapolated to the chiral limit, i.e. mπ → 0, which leaves us with the critical temperature Tc in the chiral limit. One obtains ${T}_{c}^{\mathrm{fRG}}\approx 142\,\mathrm{MeV}$ from the fRG approach [62], and ${T}_{c}^{\mathrm{lattice}}={132}_{-6}^{+3}$ MeV from the lattice QCD [13], that is, the critical temperature obtained in fRG is a bit larger than the lattice result. Recently, the critical temperature in the chiral limit from DSE is found to be ${T}_{c}^{\mathrm{DSE}}\approx 141\,\mathrm{MeV}$ [246].
Figure 47. Pseudocritical temperature as a function of the pion mass obtained in fRG (fQCD) in comparison to the lattice results [13]. The critical temperature in the chiral limit Tc has been obtained from an extrapolation of the pseudocritical temperature Tpc to mπ → 0. The plot is adapted from [62]. |
From the general scaling hypothesis, see e.g. [66], in the critical regime the pseudocritical temperature scales with the pion mass as$p\approx {0.91}_{-0.03}^{+0.03}$ [62]. This discrepancy indicates that the pion masses under investigation in [62], i.e. mπ ≳ 30 MeV are beyond the critical regime. In fact, the critical regime is found to be extremely small, mπ ≲ 1 MeV, in fRG studies of LEFTs, and see, e.g. [98, 151, 186, 198] for a more detailed discussion.
$\begin{eqnarray} {T}_{\mathrm{pc}}({m}_{\pi })\approx {T}_{{\rm{c}}}+c\,{m}_{\pi }^{p},\end{eqnarray}$
where c is a non-universal coefficient, while the exponent p is related to the critical exponents β and δ through p = 2/(βδ) [62]. Inputting the values of β and δ for the 3-d O(4) universality class [196, 247, 248], one arrives at p ≈ 1.08. On the contrary, fitting the fRG data in figure 47 leads to 5. Real-time fRG
In this section, we would like to give a brief introduction to the real-time fRG, based on a combination of the fRG approach and the formalism of Schwinger–Keldysh path integral, where the flow equations are formulated on the closed time path. The Schwinger–Keldysh path integral is devised to study real-time quantum dynamics [249, 250], which has been proved to be very powerful to cope with problems of both equilibrium and nonequilibrium many-body systems, see, e.g. [251–254] for relevant reviews.
Within the fRG approach on the closed time path, nonthermal fixed points of the O(N) scalar theory are investigated [255, 256]. The transition from unitary to dissipative dynamics is studied [257]. Spectral functions in a scalar field theory with d = 0 + 1 dimensions are computed within the real-time fRG approach [258]. Moreover, it has also been employed to study the nonequilibrium transport, dynamical critical behavior, etc in open quantum systems [259, 260], see also [55, 254] for related reviews. Recently, the real-time fRG is compared with other real-time methods [261].
Furthermore, it should be mentioned that another conceptually new fRG with the Keldysh functional integral is put forward in [262–265], and the regulation of the RG scale there is replaced with that of the time, which is also called the temporal RG. Noteworthily, recently the Källén–Lehmann spectral representation of correlation functions has been used in the approach of DSE, to study the spectral functions in the φ4-theory and Yang–Mills theory [266–268]. The spectral functions provide important information on the time-like properties of correlation functions, see, e.g. [269–272]. Besides the real-time fRG, another commonly adopted approach is the analytically continued fRG, where the Euclidean flow equations are analytically continued into the Minkowski spacetime on the level of analytic expressions with some specific truncations, and see, e.g. [112, 129, 273–275] for more details.
In this section, we would like to discuss the fRG formulated on the Keldysh path at the example of the O(N) scalar theory [276].
5.1. fRG with the Keldysh functional integral
On the closed time path the classical action in equation (1 ) for a closed system reads${\int }_{x}\equiv {\int }_{-\infty }^{\infty } {\rm{d}}t\int {{\rm{d}}}^{3}x$ , where the subscripts ± denote variables on the forward and backward time branches, respectively [251]. Now the Keldysh generating functional reads235 ) reads2 ) can be chosen as1 ) with the Keldysh functional integral reads14 ) reads239 ) now reading
$\begin{eqnarray}S[\hat{{\rm{\Phi }}}]={\int }_{x}\left({ \mathcal L }[{\hat{{\rm{\Phi }}}}_{+}]-{ \mathcal L }[{\hat{{\rm{\Phi }}}}_{-}]\right),\end{eqnarray}$
with $\begin{eqnarray}\begin{array} {l}Z[{J}_{+},{J}_{-}]\\ =\,\int \left({ \mathcal D } {\hat{{\rm{\Phi }}}}_{+} { \mathcal D } {\hat{{\rm{\Phi }}}}_{-}\right)\exp \left\{{\rm{i}}\left(S[\hat{{\rm{\Phi }}}]+\left({J}_{+}^{i} {\hat{{\rm{\Phi }}}}_{i,+}-{J}_{-}^{i} {\hat{{\rm{\Phi }}}}_{i,-}\right)\right)\right\}.\end{array}\end{eqnarray}$
It is more convenient to adopt the physical representation in terms of the ‘classical' and ‘quantum' variables, denoted by subscripts c and q, respectively, which are related to variables on the two-time branches by a Keldysh rotation, that is, $\begin{eqnarray}\left\{\begin{array} {l} {\hat{{\rm{\Phi }}}}_{i,+}=\displaystyle \frac{1} {\sqrt{2}}({\hat{{\rm{\Phi }}}}_{i,c}+{\hat{{\rm{\Phi }}}}_{i,q}),\\ {\hat{{\rm{\Phi }}}}_{i,-}=\displaystyle \frac{1} {\sqrt{2}}({\hat{{\rm{\Phi }}}}_{i,c}-{\hat{{\rm{\Phi }}}}_{i,q}),\end{array}\right.\end{eqnarray}$
and $\begin{eqnarray}\left\{\begin{array} {l} {J}_{+}^{i}=\displaystyle \frac{1} {\sqrt{2}}({J}_{c}^{i}+{J}_{q}^{i}),\\ {J}_{-}^{i}=\displaystyle \frac{1} {\sqrt{2}}({J}_{c}^{i}-{J}_{q}^{i})\end{array}\right..\end{eqnarray}$
Then, equation ( $\begin{eqnarray}\begin{array} {l}Z[{J}_{c},{J}_{q}]\\ =\,\int \left({ \mathcal D } {\hat{{\rm{\Phi }}}}_{c} { \mathcal D } {\hat{{\rm{\Phi }}}}_{q}\right)\exp \left\{{\rm{i}}\left(S[\hat{{\rm{\Phi }}}]+\left({J}_{q}^{i} {\hat{{\rm{\Phi }}}}_{i,c}+{J}_{c}^{i} {\hat{{\rm{\Phi }}}}_{i,q}\right)\right)\right\}.\end{array}\end{eqnarray}$
The bilinear regulator term in equation ( $\begin{eqnarray}\begin{array} {rcl} {\rm{\Delta }} {S}_{k}[\hat{{\rm{\Phi }}}] & = & \displaystyle \frac{1} {2}({\hat{{\rm{\Phi }}}}_{i,c},{\hat{{\rm{\Phi }}}}_{i,q})\left(\begin{array} {ll}0 & {R}_{k}^{{ij}}\\ {\left({R}_{k}^{{ij}}\right)}^{* } & 0\end{array}\right)\left(\begin{array} {l} {\hat{{\rm{\Phi }}}}_{j,c}\\ {\hat{{\rm{\Phi }}}}_{j,q}\end{array}\right)\\ & = & \displaystyle \frac{1} {2}\left({\hat{{\rm{\Phi }}}}_{i,c} {R}_{k}^{{ij}} {\hat{{\rm{\Phi }}}}_{j,q}+{\hat{{\rm{\Phi }}}}_{i,q} {\left({R}_{k}^{{ij}}\right)}^{* } {\hat{{\rm{\Phi }}}}_{j,c}\right).\end{array}\end{eqnarray}$
Then the RG scale k-dependent generating function equation ( $\begin{eqnarray}\begin{array} {rcl} {Z}_{k}[{J}_{c},{J}_{q}] & = & \int \left({ \mathcal D } {\hat{{\rm{\Phi }}}}_{c} { \mathcal D } {\hat{{\rm{\Phi }}}}_{q}\right)\exp \left\{{\rm{i}}\left(S[\hat{{\rm{\Phi }}}]+{\rm{\Delta }} {S}_{k}[\hat{{\rm{\Phi }}}]\right.\right.\\ & & \left.\left.+({J}_{q}^{i} {\hat{{\rm{\Phi }}}}_{i,c}+{J}_{c}^{i} {\hat{{\rm{\Phi }}}}_{i,q})\right)\right\}.\end{array}\end{eqnarray}$
The Schwinger functional in equation ( $\begin{eqnarray} {W}_{k}[{J}_{c},{J}_{q}]=-{\rm{i}}\mathrm{ln} {Z}_{k}[{J}_{c},{J}_{q}].\end{eqnarray}$
Combining the indices c, q and i for other degrees of freedom into one single label, say a, i.e. $\begin{eqnarray}\{{\hat{{\rm{\Phi }}}}_{a}\}=\left\{\{{\hat{{\rm{\Phi }}}}_{i,c}\},\{{\hat{{\rm{\Phi }}}}_{i,q}\}\right\},\end{eqnarray}$
$\begin{eqnarray}\{{J}^{a}\}=\left\{\{{J}_{q}^{i}\},\{{J}_{c}^{i}\}\right\}.\end{eqnarray}$
One is able to simplify notations significantly, for instance, the regulator term in equation ( $\begin{eqnarray} {\rm{\Delta }} {S}_{k}[\varphi ]=\displaystyle \frac{1} {2} {\hat{{\rm{\Phi }}}}_{a} {R}_{k}^{{ab}} {\hat{{\rm{\Phi }}}}_{b},\end{eqnarray}$
with $\begin{eqnarray} {R}_{k}^{{ab}}\equiv \left(\begin{array} {ll}0 & {R}_{k}^{{ij}}\\ {\left({R}_{k}^{{ij}}\right)}^{* } & 0\end{array}\right).\end{eqnarray}$
In the same way, the expectation value of the field reads $\begin{eqnarray} {{\rm{\Phi }}}_{a}\equiv \langle {\hat{{\rm{\Phi }}}}_{a}\rangle =\displaystyle \frac{\delta {W}_{k}[J]} {\delta {J}^{a}},\end{eqnarray}$
and the propagator is given by $\begin{eqnarray}\begin{array} {rcl} {G}_{k,{ab}} & \equiv & -{\rm{i}}\langle {\hat{{\rm{\Phi }}}}_{a} {\hat{{\rm{\Phi }}}}_{b} {\rangle }_{c}=-{\rm{i}}\left[\langle {\hat{{\rm{\Phi }}}}_{a} {\hat{{\rm{\Phi }}}}_{b}\rangle -\langle {\hat{{\rm{\Phi }}}}_{a}\rangle \langle {\hat{{\rm{\Phi }}}}_{b}\rangle \right]\\ & = & -\displaystyle \frac{{\delta }^{2} {W}_{k}[J]} {\delta {J}^{a} {J}^{b}}.\end{array}\end{eqnarray}$
The effective action is obtained from the Schwinger functional upon a Legendre transformation, viz.,20 ) and (21 ), one has$\tau \equiv \mathrm{ln}(k/{\rm{\Lambda }})$ , and ${R}_{k}^{{\rm{T}}}$ stands for the transpose of the regulator only in the c–q space as shown in equation (245 ). Finally, the flow equation of the effective action reads
$\begin{eqnarray} {{\rm{\Gamma }}}_{k}[{\rm{\Phi }}]={W}_{k}[J]-{\rm{\Delta }} {S}_{k}[{\rm{\Phi }}]-{J}^{a} {{\rm{\Phi }}}_{a}.\end{eqnarray}$
Similar to equations ( $\begin{eqnarray}\displaystyle \frac{\delta \left({{\rm{\Gamma }}}_{k}[{\rm{\Phi }}]+{\rm{\Delta }} {S}_{k}[{\rm{\Phi }}]\right)} {\delta {{\rm{\Phi }}}_{a}}=-{{\gamma }^{a}}_{b} {J}^{b},\end{eqnarray}$
and $\begin{eqnarray} {G}_{k,{ab}}={{\gamma }^{c}}_{a} {\left[{\left({{\rm{\Gamma }}}_{k}^{(2)}[{\rm{\Phi }}]+{\rm{\Delta }} {S}_{k}^{(2)}[{\rm{\Phi }}]\right)}^{-1}\right]}_{{cb}}\end{eqnarray}$
with $\begin{eqnarray} {\left({{\rm{\Gamma }}}_{k}^{(2)}[{\rm{\Phi }}]+{\rm{\Delta }} {S}_{k}^{(2)}[{\rm{\Phi }}]\right)}^{{ab}}\equiv \displaystyle \frac{{\delta }^{2}({{\rm{\Gamma }}}_{k}[{\rm{\Phi }}]+{\rm{\Delta }} {S}_{k}[{\rm{\Phi }}])} {\delta {{\rm{\Phi }}}_{a}\delta {{\rm{\Phi }}}_{b}}.\end{eqnarray}$
Then it is left to specify the flow equation of Schwinger functional with the Keldysh functional integral, which reads $\begin{eqnarray}\begin{array} {rcl} {\partial }_{\tau } {W}_{k}[J] & = & \displaystyle \frac{1} {2} {{\rm{i}} {G}}_{k,{ab}} {\partial }_{\tau } {R}_{k}^{{ab}}+\displaystyle \frac{1} {2} {{\rm{\Phi }}}_{a}\left({\partial }_{\tau } {R}_{k}^{{ab}}\right){{\rm{\Phi }}}_{b}\\ & = & \displaystyle \frac{{\rm{i}}} {2}\mathrm{STr}\left[\left({\partial }_{\tau } {R}_{k}^{{\rm{T}}}\right){G}_{k}\right]+\displaystyle \frac{1} {2} {{\rm{\Phi }}}_{a}\left({\partial }_{\tau } {R}_{k}^{{ab}}\right){{\rm{\Phi }}}_{b},\end{array}\end{eqnarray}$
where the RG time is denoted by $\begin{eqnarray}\begin{array} {rcl} {\partial }_{\tau } {{\rm{\Gamma }}}_{k}[{\rm{\Phi }}] & = & {\partial }_{\tau } {W}_{k}[J]-{\partial }_{\tau } {\rm{\Delta }} {S}_{k}[{\rm{\Phi }}]\\ & = & \displaystyle \frac{{\rm{i}}} {2}\mathrm{STr}\left[\left({\partial }_{\tau } {R}_{k}^{{\rm{T}}}\right){G}_{k}\right].\end{array}\end{eqnarray}$
5.2. Real-time O(N) scalar theory
The Keldysh effective action in equation (248 ) for the O(N) scalar theory reads${\rho }_{\pm }={\phi }_{\pm }^{2}/2$ . Here φi,± and φi,c/q (i = 0, 1, …N − 1) denote the fields of N components on the closed time branches or in the physical representation, respectively, and their relations are given in equation (236 ). Note that in equation (254 ) a local potential approximation with a k-dependent wave function renormalization Zφ,k has been adopted, which is usually called the LPA' approximation.
$\begin{eqnarray} {{\rm{\Gamma }}}_{k}[{\phi }_{c},{\phi }_{q}]={\int }_{x}\left[{Z}_{\phi ,k}({\partial }_{\mu } {\phi }_{q})\cdot ({\partial }^{\mu } {\phi }_{c})-{U}_{k}({\phi }_{c},{\phi }_{q})\right],\end{eqnarray}$
where the potential is given by $\begin{eqnarray} {U}_{k}({\phi }_{c},{\phi }_{q})={V}_{k}({\rho }_{+})-{V}_{k}({\rho }_{-}),\end{eqnarray}$
with When the O(N) symmetry is broken into the O(N − 1) one in the direction of component i = 0, the expectation values of the fields read${\bar{\rho }}_{c}\equiv {\bar{\phi }}_{c}^{2}/4$ .
$\begin{eqnarray} {\bar{\phi }}_{i,c}={\bar{\phi }}_{c} {\delta }_{i0},\qquad {\bar{\phi }}_{i,q}=0.\end{eqnarray}$
Then, the sigma and pion fields are given by $\begin{eqnarray} {\sigma }_{c}={\phi }_{0,c}-{\bar{\phi }}_{c},\qquad {\sigma }_{q}={\phi }_{0,q},\end{eqnarray}$
and $\begin{eqnarray} {\pi }_{i,c}={\phi }_{i,c},\qquad {\pi }_{i,q}={\phi }_{i,q},\qquad (i\ne 0).\end{eqnarray}$
The sigma and pion masses read $\begin{eqnarray} {m}_{\sigma ,k}^{2}\equiv V{{\prime} }_{k}({\bar{\rho }}_{c})+2{\bar{\rho }}_{c} {V}_{k}^{(2)}({\bar{\rho }}_{c}),\end{eqnarray}$
$\begin{eqnarray} {m}_{\pi ,k}^{2}\equiv V{{\prime} }_{k}({\bar{\rho }}_{c}),\end{eqnarray}$
with The regulator in equation (245 ) in the case of the O(N) scalar theory reads
$\begin{eqnarray} {R}_{k}(q)=\left(\begin{array} {ll}0 & {R}_{k}^{A}(q)\\ {R}_{k}^{R}(q) & 0\end{array}\right),\end{eqnarray}$
with $\begin{eqnarray} {R}_{k}^{R}(q)={R}_{k}^{A}(q)=\left(\begin{array} {ll} {R}_{\sigma ,k}(q) & 0\\ 0 & {R}_{\pi ,k}(q)\end{array}\right),\end{eqnarray}$
where one has $\begin{eqnarray} {R}_{\sigma ,k}(q)={R}_{\phi ,k}(q),\qquad {\left({R}_{\pi ,k}\right)}_{{ij}}(q)={R}_{\phi ,k}(q){\delta }_{{ij}},\end{eqnarray}$
with $\begin{eqnarray} {R}_{\phi ,k}(q)={Z}_{\phi ,k}\left(-{{\boldsymbol{q}}}^{2} {r}_{B}\left(\displaystyle \frac{{{\boldsymbol{q}}}^{2}} {{k}^{2}}\right)\right).\end{eqnarray}$
Here, the 3d flat regulator is used.The flow equation of effective action in equation (253 ) with the regulator in equation (261 ) can be reformulated as${{ \mathcal P }}_{k}$ and the interaction ${{ \mathcal F }}_{k}$ as shown in equation (58 ). In thermal equilibrium at a temperature T, the ${{ \mathcal P }}_{k}$ matrix reads267 ), i.e. ${{ \mathcal P }}_{k}^{K}$ , reads
$\begin{eqnarray} {\partial }_{\tau } {{\rm{\Gamma }}}_{k}[{\rm{\Phi }}]=\displaystyle \frac{{\rm{i}}} {2}\mathrm{STr}\left[{\tilde{\partial }}_{\tau }\mathrm{ln}\left({{\rm{\Gamma }}}_{k}^{(2)}[{\rm{\Phi }}]+{R}_{k}\right)\right],\end{eqnarray}$
with $\begin{eqnarray} {\left({{\rm{\Gamma }}}_{k}^{(2)}[{\rm{\Phi }}]\right)}_{{ab}}\equiv \displaystyle \frac{{\delta }^{2} {{\rm{\Gamma }}}_{k}[{\rm{\Phi }}]} {\delta {{\rm{\Phi }}}_{a}\delta {{\rm{\Phi }}}_{b}},\end{eqnarray}$
where the fluctuation matrix can be decomposed into a sum of the inverse propagator $\begin{eqnarray} {{ \mathcal P }}_{k}=\left(\begin{array} {ll}0 & {{ \mathcal P }}_{k}^{A}\\ {{ \mathcal P }}_{k}^{R} & {{ \mathcal P }}_{k}^{K}\end{array}\right),\end{eqnarray}$
where the inverse retarded propagator is given by $\begin{eqnarray} {{ \mathcal P }}_{k}^{R}=\left(\begin{array} {ll} {{ \mathcal P }}_{\sigma ,k}^{R} & 0\\ 0 & {{ \mathcal P }}_{\pi ,k}^{R}\end{array}\right),\end{eqnarray}$
with $\begin{eqnarray}\begin{array} {rcl} {{ \mathcal P }}_{\sigma ,k}^{R} & = & {Z}_{\phi ,k}\left[{q}_{0}^{2}-{{\boldsymbol{q}}}^{2}\left(1+{r}_{B}\left(\displaystyle \frac{{{\boldsymbol{q}}}^{2}} {{k}^{2}}\right)\right)\right]-{m}_{\sigma ,k}^{2}\\ & & +\mathrm{sgn}({q}_{0}){\rm{i}}\epsilon ,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array} {rcl} {\left({{ \mathcal P }}_{\pi ,k}^{R}\right)}_{{ij}} & = & \left\{{Z}_{\phi ,k}\left[{q}_{0}^{2}-{{\boldsymbol{q}}}^{2}\left(1+{r}_{B}\left(\frac{{{\boldsymbol{q}}}^{2}} {{k}^{2}}\right)\right)\right]-{m}_{\pi ,k}^{2}\right.\\ & & \left.+\mathrm{sgn}({q}_{0}){\rm{i}}\epsilon \right\} {\delta }_{{ij}}.\end{array}\end{eqnarray}$
The inverse advanced propagator is related to the retarded one through a complex conjugate, to wit, $\begin{eqnarray} {{ \mathcal P }}_{k}^{A}={\left({{ \mathcal P }}_{k}^{R}\right)}^{* }.\end{eqnarray}$
The qq component of the inverse propagator in equation ( $\begin{eqnarray} {{ \mathcal P }}_{k}^{K}=\left(\begin{array} {ll} {{ \mathcal P }}_{\sigma ,k}^{K} & 0\\ 0 & {{ \mathcal P }}_{\pi ,k}^{K}\end{array}\right),\end{eqnarray}$
with $\begin{eqnarray} {{ \mathcal P }}_{\sigma ,k}^{K}=2{\rm{i}}\epsilon \,\mathrm{sgn}({q}_{0})\coth \left(\displaystyle \frac{{q}_{0}} {2T}\right),\end{eqnarray}$
$\begin{eqnarray} {\left({{ \mathcal P }}_{\pi ,k}^{K}\right)}_{{ij}}=\left[2{\rm{i}}\epsilon \,\mathrm{sgn}({q}_{0})\coth \left(\frac{{q}_{0}} {2T}\right)\right]{\delta }_{{ij}}.\end{eqnarray}$
Then, one can obtain the propagator matrix as follows $\begin{eqnarray} {G}_{k}={\left({{ \mathcal P }}_{k}\right)}^{-1}=\left(\begin{array} {ll} {G}_{k}^{K} & {G}_{k}^{R}\\ {G}_{k}^{A} & 0\end{array}\right),\end{eqnarray}$
with the retarded and advanced propagator being $\begin{eqnarray} {G}_{k}^{R}={\left({{ \mathcal P }}_{k}^{R}\right)}^{-1},\qquad {G}_{k}^{A}={\left({{ \mathcal P }}_{k}^{A}\right)}^{-1},\end{eqnarray}$
and the correlation function or Keldysh propagator being $\begin{eqnarray} {G}_{k}^{K}=-{\left({{ \mathcal P }}_{k}^{R}\right)}^{-1} {{ \mathcal P }}_{k}^{K} {\left({{ \mathcal P }}_{k}^{A}\right)}^{-1}=-{G}_{k}^{R} {{ \mathcal P }}_{k}^{K} {G}_{k}^{A}.\end{eqnarray}$
Finally, one can verify the fluctuation-dissipation relation in thermal equilibrium, as follows $\begin{eqnarray} {G}_{k}^{K}=\left({G}_{k}^{R}-{G}_{k}^{A}\right)\coth \left(\displaystyle \frac{{q}_{0}} {2T}\right).\end{eqnarray}$
From equation (275 ) one arrives at the three different propagators which read280 ) follows directly from equation (277 ). The diagrammatic representation of these propagators is shown in figure 48.
$\begin{eqnarray} {{\rm{i}} {G}}_{\phi ,k}^{R}=\langle {T}_{p} {\phi }_{c}(x){\phi }_{q}(y)\rangle ,\quad {{\rm{i}} {G}}_{\phi ,k}^{A}=\langle {T}_{p} {\phi }_{q}(x){\phi }_{c}(y)\rangle ,\end{eqnarray}$
$\begin{eqnarray} {{\rm{i}} {G}}_{\phi ,k}^{K}=\langle {T}_{p} {\phi }_{c}(x){\phi }_{c}(y)\rangle =\left({{\rm{i}} {G}}_{\phi ,k}^{R}\right)\left(i{{ \mathcal P }}_{\phi ,k}^{K}\right)\left({{\rm{i}} {G}}_{\phi ,k}^{A}\right),\end{eqnarray}$
where Tp denotes the time ordering from the positive to negative time branch in the Keldysh closed time path, and equation (
Figure 48. Diagrammatic representation of the three different propagators for the sigma and pion mesons. A line associated with two end points labelled with ‘c, q' denotes the retarded propagator, while that with ‘q, c' denotes the advanced propagator. A line with an empty circle inserted in the middle is used to denote the Keldysh propagator, which results from equation ( |
Projecting the flow equation of the effective action in equation (265 ) onto a derivative with respect to the quantum sigma field σq, i.e.282 ) over ${\bar{\rho }}_{c}$ , one arrives at${\bar{\rho }}_{c}$ . Inserting equations (269 ) and (270 ), one has284 ) is just the flow equation of the effective potential in the LPA approximation obtained in the conventional Euclidean formalism of fRG.
$\begin{eqnarray} {\left.{\partial }_{\tau }\left(\displaystyle \frac{{\rm{i}}\delta {{\rm{\Gamma }}}_{k}[{\rm{\Phi }}]} {\delta {\sigma }_{q}}\right)\right|}_{{\rm{\Phi }}=0},\end{eqnarray}$
with Φ = (σc, {πi,c}, σq, {πi,q}), one is led to $\begin{array} {l} {\partial }_{\tau }V^{\prime}_{k}({\bar{\rho }}_{c})\\ =\,\frac{{\rm{i}}} {4}\int \frac{{{\rm{d}}}^{4}q} {{\left(2\pi \right)}^{4}} {\tilde{\partial }}_{\tau }\left[\frac{\partial {m}_{\sigma ,k}^{2}} {\partial {\bar{\rho }}_{c}} {G}_{\sigma ,k}^{K}(q)+\frac{\partial {m}_{\pi ,k}^{2}} {\partial {\bar{\rho }}_{c}} {\left({G}_{\pi ,k}^{K}\right)}_{{ii}}(q)\right]\\ =\,\frac{\partial } {\partial {\bar{\rho }}_{c}}\left\{-\frac{{\rm{i}}} {4}\int \frac{{{\rm{d}}}^{4}q} {{\left(2\pi \right)}^{4}}\left({\partial }_{\tau } {R}_{\phi ,k}(q)\right)\left[{G}_{\sigma ,k}^{K}(q)\right.\right.\\ \,\left.\left.+\,{\left({G}_{\pi ,k}^{K}\right)}_{{ii}}(q)\right]\right\},\end{array}$
which is diagrammatically shown in figure 49. Integrating both sides of equation ( $\begin{eqnarray}\displaystyle \begin{array} {l} {\partial }_{\tau } {V}_{k}({\bar{\rho }}_{c})\\ =\,-\frac{{\rm{i}}} {4}\int \frac{{{\rm{d}}}^{4}q} {{\left(2\pi \right)}^{4}}\left({\partial }_{\tau } {R}_{\phi ,k}(q)\right)\left[{G}_{\sigma ,k}^{K}(q)+{\left({G}_{\pi ,k}^{K}\right)}_{{ii}}(q)\right],\end{array}\end{eqnarray}$
up to a term independent of $\begin{eqnarray}\begin{array} {rcl} {\partial }_{\tau } {V}_{k}({\bar{\rho }}_{c}) & = & \displaystyle \frac{{k}^{4}} {4{\pi }^{2}}\left[{l}_{0}^{(B,4)}({\tilde{m}}_{\sigma ,k}^{2},{\eta }_{\phi ,k};T)\right.\\ & & \left.+\left(N-1\right){l}_{0}^{(B,4)}({\tilde{m}}_{\pi ,k}^{2},{\eta }_{\phi ,k};T)\right],\end{array}\end{eqnarray}$
with the threshold function given by $\begin{eqnarray}\begin{array} {l} {l}_{0}^{(B,4)}({\tilde{m}}_{\phi ,k}^{2},{\eta }_{\phi ,k};T)\\ =\,\displaystyle \frac{2} {3}\left(1-\displaystyle \frac{{\eta }_{\phi ,k}} {5}\right)\displaystyle \frac{1} {\sqrt{1+{\tilde{m}}_{\phi ,k}^{2}}}\left(\displaystyle \frac{1} {2}+{n}_{B}({\tilde{m}}_{\phi ,k}^{2};T)\right),\end{array}\end{eqnarray}$
and the renormalized dimensionless meson masses reading $\begin{eqnarray} {\tilde{m}}_{\sigma ,k}^{2}=\displaystyle \frac{{m}_{\sigma ,k}^{2}} {{k}^{2} {Z}_{\phi ,k}},\qquad {\tilde{m}}_{\pi ,k}^{2}=\displaystyle \frac{{m}_{\pi ,k}^{2}} {{k}^{2} {Z}_{\phi ,k}},\end{eqnarray}$
where the bosonic distribution function reads $\begin{eqnarray} {n}_{B}({\tilde{m}}_{\phi ,k}^{2};T)=\displaystyle \frac{1} {\exp \left\{\tfrac{k} {T}\sqrt{1+{\tilde{m}}_{\phi ,k}^{2}}\right\}-1}.\end{eqnarray}$
Note that equation (
Figure 49. Diagrammatic representation of the flow equation of the effective potential in the real-time O(N) scalar theory. The vertices are denoted by gray blobs, and their legs are distinguished for the ‘q' and ‘c' fields. the crossed circles stand for the regulator insertion. The plot is adapted from [276]. |
5.3. Flows of the two- and four-point correlation functions
The 1PI n-point correlation function or vertex in the Euclidean field theory is given in equation (60 ), and the counterpart in the Keldysh field theory reads${\bar{\phi }}_{c}=0$ in equation (256 ). In this case, the pion and sigma fields are degenerate, and they are collectively denoted by φi (i = 0, 1, …N − 1). The diagrammatic representation of the four-point vertex $i{{\rm{\Gamma }}}_{{\phi }_{q} {\phi }_{c} {\phi }_{c} {\phi }_{c}}^{(4)}$ reads 
whose flow equation is given in figure 50. According to the generic interchange symmetry of the external legs, the four-point vertex can be parameterized as${\lambda }_{4\pi ,k}^{\mathrm{eff}}$ is introduced. In the following, one adopts the truncation that the momentum dependence of the four-point vertex on the r.h.s. of the flow equation in figure 50 is neglected, and the four-point vertex there is identified as${\tilde{\partial }}_{\tau } {I}_{k}(p)$ in equation (292 ). The explicit expression of Ik(p) can be found in [276].
$\begin{eqnarray} {\rm{i}} {{\rm{\Gamma }}}_{k,{{\rm{\Phi }}}_{{a}_{1}}\cdots {{\rm{\Phi }}}_{{a}_{n}}}^{(n)}={\left.\left(\displaystyle \frac{{\rm{i}} {\delta }^{n} {{\rm{\Gamma }}}_{k}[{\rm{\Phi }}]} {\delta {{\rm{\Phi }}}_{{a}_{1}}\cdots \delta {{\rm{\Phi }}}_{{a}_{n}}}\right)\right|}_{{\rm{\Phi }}=\langle {\rm{\Phi }}\rangle }.\end{eqnarray}$
Note that a label ‘c' or ‘q' is associated with an external leg of the vertex to distinguish the classical or quantum field, as shown in figure 49. In the following, we consider the symmetric phase with $\begin{eqnarray}\begin{array} {l} {\mathrm{i\Gamma }}_{k,{\phi }_{i,q} {\phi }_{j,c} {\phi }_{k,c} {\phi }_{l,c}}^{(4)}({p}_{i},{p}_{j},{p}_{k},{p}_{l})\\ =\,-\frac{{\rm{i}}} {3}\left[{\lambda }_{4\pi ,k}^{\mathrm{eff}}({p}_{i},{p}_{j},{p}_{k},{p}_{l}){\delta }_{{il}} {\delta }_{{jk}}+{\lambda }_{4\pi ,k}^{\mathrm{eff}}({p}_{i},{p}_{k},{p}_{l},{p}_{j}){\delta }_{{ij}} {\delta }_{{kl}}\right.\\ \,\left.+\,{\lambda }_{4\pi ,k}^{\mathrm{eff}}({p}_{i},{p}_{l},{p}_{j},{p}_{k}){\delta }_{{ik}} {\delta }_{{jl}}\right],\end{array}\end{eqnarray}$
where an effective four-point coupling $\begin{eqnarray} {\lambda }_{4\pi ,k}={\lambda }_{4\pi ,k}^{\mathrm{eff}}(0).\end{eqnarray}$
Then, from the flow of the four-point vertex in figure 50, one arrives at $\begin{eqnarray}\begin{array} {l} {\partial }_{\tau } {\lambda }_{4\pi ,k}^{\mathrm{eff}}({p}_{i},{p}_{j},{p}_{k},{p}_{l})\\ =\,\displaystyle \frac{{\lambda }_{4\pi ,k}^{2}} {3}\left[(N+4){\tilde{\partial }}_{\tau } {I}_{k}(-{p}_{i}-{p}_{l})+2\,{\tilde{\partial }}_{\tau } {I}_{k}(-{p}_{i}-{p}_{k})\right.\\ \,\left.+\,2\,{\tilde{\partial }}_{\tau } {I}_{k}(-{p}_{i}-{p}_{j})\right],\end{array}\end{eqnarray}$
with $\begin{eqnarray} {I}_{k}(p)\equiv {\rm{i}}\int \displaystyle \frac{{{\rm{d}}}^{4}q} {{\left(2\pi \right)}^{4}} {G}_{\pi ,k}^{K}(q){G}_{\pi ,k}^{A}(q-p).\end{eqnarray}$
Note that the wave function renormalization Zφ,k = 1 is adopted to simplify the calculation of 
Figure 50. Diagrammatic representation of the flow equation of the four-point vertex in the symmetric phase. The plot is adapted from [276]. |
With the four-point vertex in equation (290 ) one is able to obtain the self-energy as follows 
which reads${\bar{\lambda }}_{4\pi ,k}^{\mathrm{eff}}$ is given by277 ), one arrives at299 ) reads${m}_{\pi ,k}^{2}$ can be extracted from the two-point correlation function at vanishing momentum, viz.,301 ) is negligible if the momentum dependence of the vertex is mild.
$\begin{eqnarray}\begin{array} {rcl}-{\mathrm{i\Sigma }}_{k,{ij}}(p) & = & {\delta }_{{ij}}(-\frac{{\rm{i}}} {6})(N+2)\\ & & \times \,\int \frac{{{\rm{d}}}^{4}q} {{\left(2\pi \right)}^{4}} {\rm{i}} {G}_{\pi ,k}^{K}(q){\bar{\lambda }}_{4\pi ,k}^{\mathrm{eff}}({p}_{0},\left|{\boldsymbol{p}}\right|,{q}_{0},| {\boldsymbol{q}}| ,\cos \theta ).\end{array}\end{eqnarray}$
Here, the function $\begin{eqnarray}\begin{array} {l} {\bar{\lambda }}_{4\pi ,k}^{\mathrm{eff}}({p}_{0},\left|{\boldsymbol{p}}\right|,{q}_{0},| {\boldsymbol{q}}| ,\cos \theta )\\ =\,\displaystyle \frac{1} {N+2}\left[N{\lambda }_{4\pi ,k}^{\mathrm{eff}}(-p,-q,q,p)+{\lambda }_{4\pi ,k}^{\mathrm{eff}}(-p,p,-q,q)\right.\\ \,\left.+\,{\lambda }_{4\pi ,k}^{\mathrm{eff}}(-p,q,p,-q)\right],\end{array}\end{eqnarray}$
with θ being the angle between the two momenta p and q. The inverse retarded propagator, i.e. the two-point correlation function with one q field and one c field, is given by $\begin{eqnarray}\begin{array} {rcl} {\mathrm{i\Gamma }}_{k,{\phi }_{i,q} {\phi }_{j,c}}^{(2)}(p) & \equiv & {\rm{i}}\frac{{\delta }^{2} {{\rm{\Gamma }}}_{k}[\phi ]} {\delta {\phi }_{i,q}(p)\delta {\phi }_{j,c}(-p)}\\ & & ={\rm{i}} {\delta }_{{ij}}\left({Z}_{\phi ,k}({p}^{2}){p}^{2}-{m}_{\pi ,k}^{2}\right),\end{array}\end{eqnarray}$
and its flow equation reads $\begin{eqnarray}\begin{array} {l} {\partial }_{\tau } {{\rm{\Gamma }}}_{k,{\phi }_{q} {\phi }_{c}}^{(2)}(p)=-{\tilde{\partial }}_{\tau } {{\rm{\Sigma }}}_{k}(p)\\ =\,(-\displaystyle \frac{{\rm{i}}} {6})(N+2)\int \displaystyle \frac{{{\rm{d}}}^{4}q} {{\left(2\pi \right)}^{4}} {\tilde{\partial }}_{\tau }\left({G}_{\pi ,k}^{K}(q)\right)\\ \times \,{\bar{\lambda }}_{4\pi ,k}^{\mathrm{eff}}({p}_{0},\left|{\boldsymbol{p}}\right|,{q}_{0},| {\boldsymbol{q}}| ,\cos \theta ).\end{array}\end{eqnarray}$
Substituting the Keldysh propagator in equation ( $\begin{eqnarray}\begin{array} {l} {\partial }_{\tau } {{\rm{\Gamma }}}_{k,{\phi }_{q} {\phi }_{c}}^{(2)}({p}_{0},\left|{\boldsymbol{p}}\right|)\\ =\,{\partial }_{\tau } {{\rm{\Gamma }}}_{k,{\phi }_{q} {\phi }_{c}}^{(2){\rm{I}}}({p}_{0},\left|{\boldsymbol{p}}\right|)+{\partial }_{\tau } {{\rm{\Gamma }}}_{k,{\phi }_{q} {\phi }_{c}}^{(2)\mathrm{II}}({p}_{0},\left|{\boldsymbol{p}}\right|).\end{array}\end{eqnarray}$
The first part of the r.h.s. of equation ( $\begin{eqnarray}\begin{array} {l} {\partial }_{\tau } {{\rm{\Gamma }}}_{k,{\phi }_{q} {\phi }_{c}}^{(2){\rm{I}}}\left.\left({p}_{0},\left|{\boldsymbol{p}}\right|\right)\right)\\ =\,-\displaystyle \frac{1} {24}\displaystyle \frac{(N+2)} {{\left(2\pi \right)}^{2}}\left[-\displaystyle \frac{\coth \left(\tfrac{{E}_{\pi ,k}(k)} {2T}\right)} {{\left({E}_{\pi ,k}(k)\right)}^{3}}-\displaystyle \frac{{\mathrm{csch}}^{2}\left(\tfrac{{E}_{\pi ,k}(k)} {2T}\right)} {2T{\left({E}_{\pi ,k}(k)\right)}^{2}}\right]\\ \times (2{k}^{2}){\int }_{0}^{k} {\rm{d}}\left|{\boldsymbol{q}}\right|{\left|{\boldsymbol{q}}\right|}^{2} {\int }_{-1}^{1}\mathrm{dcos}\theta \left[{\left.{\bar{\lambda }}_{4\pi ,k}^{\mathrm{eff}}\right|}_{{q}_{0}={E}_{\pi ,k}(k)}\right.\\ +\left.{\left.{\bar{\lambda }}_{4\pi ,k}^{\mathrm{eff}}\right|}_{{q}_{0}=-{E}_{\pi ,k}(k)}\right],\end{array}\end{eqnarray}$
and the second part is given by $\begin{eqnarray}\begin{array} {l} {\partial }_{\tau } {{\rm{\Gamma }}}_{k,{\phi }_{q} {\phi }_{c}}^{(2)\mathrm{II}}({p}_{0},\left|{\boldsymbol{p}}\right|)\\ =\,-\displaystyle \frac{1} {24}\displaystyle \frac{(N+2)} {{\left(2\pi \right)}^{2}}\displaystyle \frac{\coth \left(\tfrac{{E}_{\pi ,k}(k)} {2T}\right)} {{\left({E}_{\pi ,k}(k)\right)}^{2}}(2{k}^{2}){\int }_{0}^{k} {\rm{d}}\left|{\boldsymbol{q}}\right|{\left|{\boldsymbol{q}}\right|}^{2}\\ \times {\int }_{-1}^{1}\mathrm{dcos}\theta \left[\displaystyle \frac{\partial } {\partial {q}_{0}} {\left.{\bar{\lambda }}_{4\pi ,k}^{\mathrm{eff}}\right|}_{{q}_{0}={E}_{\pi ,k}(k)}\right.\\ \left.-\displaystyle \frac{\partial } {\partial {q}_{0}} {\left.{\bar{\lambda }}_{4\pi ,k}^{\mathrm{eff}}\right|}_{{q}_{0}=-{E}_{\pi ,k}(k)}\right]\end{array}\end{eqnarray}$
with $\begin{eqnarray} {E}_{\pi ,k}(k)={\left({k}^{2}+{m}_{\pi ,k}^{2}\right)}^{1/2},\end{eqnarray}$
where $\begin{eqnarray} {m}_{\pi ,k}^{2}=-{{\rm{\Gamma }}}_{k,{\phi }_{q} {\phi }_{c}}^{(2)}(0).\end{eqnarray}$
Note that the second part in equation (5.4. Spectral functions and dynamical critical exponent
The retarded propagator in the Källén–Lehmann spectral representation is related to the spectral function ρ via a relation as follows
$\begin{eqnarray} {G}_{R}({p}_{0},\left|{\boldsymbol{p}}\right|)=-{\int }_{-\infty }^{\infty }\displaystyle \frac{{\rm{d}} {p}_{0}^{\prime} } {2\pi }\displaystyle \frac{\rho ({p}_{0}^{\prime} ,\left|{\boldsymbol{p}}\right|)} {{p}_{0}^{\prime} -({p}_{0}+{\rm{i}}\epsilon )}.\end{eqnarray}$
Thus, the spectral function is proportional to the imaginary part of the retarded propagator, i.e. $\begin{eqnarray}\rho ({p}_{0},\left|{\boldsymbol{p}}\right|)=-2{\mathfrak{I}} {G}_{R}({p}_{0},\left|{\boldsymbol{p}}\right|).\end{eqnarray}$
Notice that here the IR limit k → 0 is tacitly assumed. The retarded propagator is just the inverse of the two-point correlation function, to wit, $\begin{eqnarray} {G}_{R}({p}_{0},\left|{\boldsymbol{p}}\right|)={\left[{{\rm{\Gamma }}}_{{\phi }_{q} {\phi }_{c}}^{(2)}({p}_{0},\left|{\boldsymbol{p}}\right|)\right]}^{-1}.\end{eqnarray}$
Consequently, the spectral function can be expressed in terms of the real and imaginary parts of the two-point correlation function, that is, $\begin{eqnarray}\rho ({p}_{0},\left|{\boldsymbol{p}}\right|)=\displaystyle \frac{2{\mathfrak{I}}\,{{\rm{\Gamma }}}_{{\phi }_{q} {\phi }_{c}}^{(2)}({p}_{0},\left|{\boldsymbol{p}}\right|)} {{\left[{\mathfrak{R}}\,{{\rm{\Gamma }}}_{{\phi }_{q} {\phi }_{c}}^{(2)}({p}_{0},\left|{\boldsymbol{p}}\right|)\right]}^{2}+{\left[{\mathfrak{I}}\,{{\rm{\Gamma }}}_{{\phi }_{q} {\phi }_{c}}^{(2)}({p}_{0},\left|{\boldsymbol{p}}\right|)\right]}^{2}}.\end{eqnarray}$
Evidently, one has $\begin{eqnarray}\rho (-{p}_{0},\left|{\boldsymbol{p}}\right|)=-\rho ({p}_{0},\left|{\boldsymbol{p}}\right|).\end{eqnarray}$
In figure 51, the spectral function $\rho ({p}_{0},\left|{\boldsymbol{p}}\right|=0)$ is shown as a function of p0 with different values of the temperature. In the left panel, the temperatures are above but close to the critical temperature Tc = 20.4 MeV for the phase transition, where the symmetry is broken from O(N) to O(N − 1) when T < Tc, while in the right panel, the temperatures are far larger than Tc. One can see that when temperature is large, one has a negative spectral function in the regime of small p0, and there is a minus peak structure around the pole mass. It is found that the negative spectral function results from the contributions of the Landau damping, and see [276] for a more detailed discussion. Nonetheless, when the temperature is decreased below about 60 MeV, the Landau damping is dominated by the process of creation and annihilation of particles, and the spectral function is positive. Moreover, when the temperature is closer and closer to the critical temperature, the peak structure on the spectral function becomes wider and wider, and finally disappears at the critical temperature. The 3D plots of the spectral function as a function of p0 and $\left|{\boldsymbol{p}}\right|$ are shown in figure 52 with two different values of the temperature.
Figure 51. Spectral function |
Figure 52. 3D plots of the spectral function as a function of p0 and |
We proceed to the discussion of the dynamical critical exponent, and begin with the kinetic coefficient ${\rm{\Gamma }}(\left|{\boldsymbol{p}}\right|)$ that is defined as$\left|{\boldsymbol{p}}\right|\gt {\xi }^{-1}$ , with $\xi \sim {m}_{\phi }^{-1}$ , the relaxation rate scales as
$\begin{eqnarray}\begin{array} {rcl}\displaystyle \frac{1} {{\rm{\Gamma }}(\left|{\boldsymbol{p}}\right|)} & = & -{\rm{i}} {\left.\displaystyle \frac{\partial {{\rm{\Gamma }}}_{{\phi }_{q} {\phi }_{c}}^{(2)}({p}_{0},\left|{\boldsymbol{p}}\right|)} {\partial {p}_{0}}\right|}_{{p}_{0}=0}\\ & = & {\left.\displaystyle \frac{\partial {\mathfrak{I}}\,{{\rm{\Gamma }}}_{{\phi }_{q} {\phi }_{c}}^{(2)}\left({p}_{0},\left|{\boldsymbol{p}}\right|\right)} {\partial {p}_{0}}\right|}_{{p}_{0}=0},\end{array}\end{eqnarray}$
where the parity properties of the real and imaginary parts of the two-point correlation function have been used, i.e. $\begin{eqnarray} {\mathfrak{R}}\,{{\rm{\Gamma }}}_{{\phi }_{q} {\phi }_{c}}^{(2)}(-{p}_{0},\left|{\boldsymbol{p}}\right|)={\mathfrak{R}}\,{{\rm{\Gamma }}}_{{\phi }_{q} {\phi }_{c}}^{(2)}({p}_{0},\left|{\boldsymbol{p}}\right|),\end{eqnarray}$
$\begin{eqnarray} {\mathfrak{I}}\,{{\rm{\Gamma }}}_{{\phi }_{q} {\phi }_{c}}^{(2)}(-{p}_{0},\left|{\boldsymbol{p}}\right|)=-{\mathfrak{I}}\,{{\rm{\Gamma }}}_{{\phi }_{q} {\phi }_{c}}^{(2)}({p}_{0},\left|{\boldsymbol{p}}\right|).\end{eqnarray}$
The relaxation rate, or dissipative characteristic frequency, is given by $\begin{eqnarray}\begin{array} {rcl}\omega (\left|{\boldsymbol{p}}\right|) & = & {\rm{\Gamma }}(\left|{\boldsymbol{p}}\right|)\left(-{{\rm{\Gamma }}}_{{\phi }_{q} {\phi }_{c}}^{(2)}({p}_{0}=0,\left|{\boldsymbol{p}}\right|)\right)\\ & = & -{\rm{\Gamma }}(\left|{\boldsymbol{p}}\right|){\mathfrak{R}}\,{{\rm{\Gamma }}}_{{\phi }_{q} {\phi }_{c}}^{(2)}({p}_{0}=0,\left|{\boldsymbol{p}}\right|).\end{array}\end{eqnarray}$
When the momentum is larger than the correlation length $\begin{eqnarray}\omega (\left|{\boldsymbol{p}}\right|)\propto {\left|{\boldsymbol{p}}\right|}^{z},\end{eqnarray}$
through which one can extract the dynamical critical exponent z [277]. From figure 53 the value of the dynamical critical exponent in the real-time 3d O(4) scalar theory is extracted, and one arrives at z = 2.02284(6), where the numerical error is shown in the bracket.
Figure 53. Double logarithm plot of the relaxation rate |
According to the standard classification for universalities of the critical dynamics [277], it is argued that the critical dynamics of the relativistic O(4) scalar theory belongs to the universality of Model G [278], see also [279], which leaves us with z = 3/2 in three dimensions. The dynamic critical exponent for the O(4) model is also calculated in real-time classical-statistical lattice simulations, and it is found that z is in favor of 2, but there is still a sizable numerical error [279]. The dynamic critical exponent in a relativistic O(N) vector model is also found to be close to 2 [257]. A similar result is found in a O(3) model in [280]. Moreover, z = 1.92(11) is found for Model A in three spatial dimensions from real-time classical-statistical lattice simulations [281]. In short, the dynamic critical exponent is far from clear and conclusive in comparison to the static critical exponent, and more studies of the critical dynamics from different approaches, including the real-time fRG, are necessary and desirable in the future.
6. Conclusions
In this paper, we present an overview on recent progress in studies of QCD at finite temperature and densities within the fRG approach. After a brief introduction of the formalism, the fRG approach is applied in low energy effective field theories and the first-principle QCD. The mechanism of quark mass production and natural emergence of bound states is well illustrated within the fRG approach, and a set of self-consistent flow equations of different correlation functions provide the necessary resummations, and plays the same role as the quark gap equation and Bethe–Salpeter equation of bound states.
We present results for the QCD phase structure and the location of the CEP, the QCD EoS, the magnetic EoS, baryon number fluctuations confronted with recent experimental measurements, various critical exponents, etc. It is found that the non-monotonic dependence of the kurtosis of the net-baryon or net-proton (proxy for net-baryon in experiments) number distributions could arise from the increasingly sharp crossover with the decrease of the beam collision energy, which in turn indicates that the non-monotonicity observed in experiments is highly non-trivial. Furthermore, recent estimates of the location of the CEP from first-principle QCD calculations within fRG and DSEs, which pass through lattice benchmark tests at small baryon chemical potentials, converge in a rather small region at baryon chemical potentials of about 600 MeV. But it should be reminded that errors of functional approaches increase significantly in the regime of μB/T ≳ 4, and thus one arrives at a more reasonable estimation for the location of CEP as $450\,\mathrm{MeV}\lesssim {{\mu }_{B}}_{\mathrm{CEP}}\lesssim 650\,\mathrm{MeV}$ . Moreover, a region of inhomogeneous instability indicated by a negative wave function renormalization is found with μB ≳ 420 MeV, and its consequence on the phenomenology of heavy-ion collisions has been investigated very recently [243, 244]. It is found that this inhomogeneous instability would result in a moat regime in both the particle pT spectrum and the two-particle correlation. By investigating the critical behavior and extracting critical exponents in the vicinity of the CEP, it is found that the size of the critical region is extremely small, quite smaller than ∼1 MeV.
In this review, we also discuss the real-time fRG, in which the flow equations are formulated on the Schwinger–Keldysh closed time path. By organizing the effective action and the flow equations in terms of the ‘classical' and ‘quantum' fields, one is able to give a concise diagrammatic representation for the flow equations of propagators and vertices in the real-time fRG. The spectral functions of the O(N) scalar theory in the critical regime in the proximity of the critical temperature are obtained. The dynamical critical exponent in the O(4) scalar theory in 3+1 dimensions is found to be z ≃ 2.023.