Study of the gluonic quartic gauge couplings at muon colliders

Ji-Chong Yang; Yu-Chen Guo; Yi-Fei Dong

doi:10.1088/1572-9494/acfd14

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Communications in Theoretical Physics ›› 2023, Vol. 75 ›› Issue (11) : 115201. DOI: 10.1088/1572-9494/acfd14

Particle Physics and Quantum Field Theory

Study of the gluonic quartic gauge couplings at muon colliders

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Abstract

The potential of muon colliders opens up new possibilities for the exploration of new physics beyond the Standard Model. It is worthwhile to investigate whether muon colliders are suitable for studying gluonic quartic gauge couplings (gQGCs), which can be contributed by dimension-8 operators in the framework of the Standard Model effective field theory, and are intensively studied recently. In this paper, we study the sensitivity of the process $μ^{+} μ^{-} \to j j ν \bar{ν}$ to gQGCs. Our results indicate that the muon colliders with c.m. energies larger than 4 TeV can be more sensitive to gQGCs than the Large Hadron Collider.

Key words

gluonic quartic gauge coupling (gQGC) / effective field theory (EFT) / muon collider / Vector Boson Fusion (VBF)

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Ji-Chong Yang, Yu-Chen Guo, Yi-Fei Dong. Study of the gluonic quartic gauge couplings at muon colliders[J]. Communications in Theoretical Physics, 2023, 75(11): 115201 https://doi.org/10.1088/1572-9494/acfd14

1. Introduction

So far, most experimental results have demonstrated the success of the Standard Model (SM). However, the theoretical difficulties of the theory itself have led people to believe that the SM is merely an effective theory, with the expectation of new physics (NP) potentially emerging at higher energy scales [1]. Due to the lack of clear signs of new physics, the SM effective field theory (SMEFT) [2–5] has become a popular theoretical tool for studying experimental results in a model-independent way. In most cases, NP effects are expected to manifest in physical processes that are described by dimension-6 operators of SMEFT. However, in some special cases, the contribution of dimension-8 operators may be more important [6–11]. These cases include anomalous quartic gauge coupling (aQGC) and neutral triple gauge coupling (nTGC), which have been the subject of many theoretical studies and experimental measurements [12–42]. These couplings are fixed by electroweak gauge bosons only. Another type of anomalous gauge coupling that does not exist in the SM is the quartic coupling between gluons and electroweak (EW) vector bosons. They can be described by dimension-8 operators, which are operators contributing to gluonic quartic gauge couplings (gQGCs) [7]. The quartic couplings of gluons to EW gauge bosons arise in the Born–Infeld (BI) extension of the SM, which was originally proposed to set an upper limit on the strength of the electromagnetic field [6]. It has been shown that the BI also appears in the theories inspired by M theory [43–45]. Recently, the gQGCs have been studied at the Large Hadron Collider (LHC)[7, 46].

High-energy collisions can directly generate new particles, while precise measurements assist us in identifying the indirect effects of unknown new physics and understanding the dynamics of known particles. The muon collider capitalizes on the advantages of two strategic approaches to exploit the complementarity between energy and precision. The inception of concepts pertaining to muon colliders dates back to earlier time [47–51]. At present, the energy frontier attainable by muon colliders remains undetermined. Ongoing investigations are centered on a 10 TeV configuration, with a targeted integrated luminosity of 10 ab⁻¹ [52, 53]. With the high energy and high luminosity, the muon collider not only possesses a stronger capability for probing NP compared to the LHC but also has the ability for more precise measures. As a result, high-energy muon colliders have gained much attention in the community [38–41, 54–64]. The muon colliders are ideal places to study the dimension-8 operators, including those contributing to the gQGCs. In a high-energy muon collider, the initial state muons emit low-virtuality vector bosons. The weak-boson fusion process transforms the muon collider into a high-luminosity vector boson collider [52, 65]. The gQGCs can affect the process

μ^{+} μ^{-} \to j j ν \bar{ν}

via both the vector boson fusion (VBF) processes and the tri-boson productions. In this paper, we study the sensitivity of the process

μ^{+} μ^{-} \to j j ν \bar{ν}

to the gQGCs.

The rest of this paper is organized as follows. In section 2, the dimension-8 operators contributing to gQGCs are introduced. In section 3, we compare the VBF process with the tri-boson process and discuss the unitarity bound on the operator coefficients. We also present our event selection strategy at muon colliders with different energies and luminosities. The numerical results of the constraints on the coefficients are presented in section 4. In section 5, we summarize and draw our conclusions.

2. Dimension-8 gluonic QGC operators

Although there are many possibilities for new physics at higher energy scales above the EW one, the low energy effective field theory (EFT) should be subject to the SM SU(3)_c × SU(2)_L × U(1)_Y gauge symmetries. A convenient way to take these symmetries into account is the SMEFT [66], which includes systematically all the allowed interactions with mass dimension d > 4. The extra dimensions are compensated by inverse powers of a mass scale M that is associated with heavy new particles. The gQGCs appear at dimension-8 level with 1/M⁴ suppression [7],

\begin{array}{rcl} O_{g T, 0} \equiv \frac{1}{16 M_{0}^{4}} \sum_{a} G_{μ ν}^{a} G^{a, μ ν} \times \sum_{i} W_{α β}^{i} W^{i, α β}, \end{array}

(1a)

\begin{array}{rcl} O_{g T, 1} \equiv \frac{1}{16 M_{1}^{4}} \sum_{a} G_{α ν}^{a} G^{a, μ β} \times \sum_{i} W_{μ β}^{i} W^{i, α ν}, \end{array}

(1b)

\begin{array}{rcl} O_{g T, 2} \equiv \frac{1}{16 M_{2}^{4}} \sum_{a} G_{α μ}^{a} G^{a, μ β} \times \sum_{i} W_{ν β}^{i} W^{i, α ν}, \end{array}

(1c)

\begin{array}{rcl} O_{g T, 3} \equiv \frac{1}{16 M_{3}^{4}} \sum_{a} G_{α μ}^{a} G_{β ν}^{a} \times \sum_{i} W^{i, μ β} W^{i, ν α}, \end{array}

(1d)

\begin{array}{rcl} O_{g T, 4} \equiv \frac{1}{16 M_{4}^{4}} \sum_{a} G_{μ ν}^{a} G^{a, μ ν} \times B_{α β} B^{α β}, \end{array}

(1e)

\begin{array}{rcl} O_{g T, 5} \equiv \frac{1}{16 M_{5}^{4}} \sum_{a} G_{α ν}^{a} G^{a, μ β} \times B_{μ β} B^{α ν}, \end{array}

(1f)

\begin{array}{rcl} O_{g T, 6} \equiv \frac{1}{16 M_{6}^{4}} \sum_{a} G_{α μ}^{a} G^{a, μ β} \times B_{ν β} B^{α ν}, \end{array}

(1g)

\begin{array}{rcl} O_{g T, 7} \equiv \frac{1}{16 M_{7}^{4}} \sum_{a} G_{α μ}^{a} G_{β ν}^{a} \times B^{μ β} B^{ν α}, \end{array}

(1h)

where

G_{μ ν}^{a}

is gluon field strengths,

W_{μ ν}^{i}

and B_μν are electroweak field strengths. Since gluons carry QCD color, denoted by the a superscript of

G_{μ ν}^{a}

, the gQGC operators must contain an even number of gluon field strengths, such as

G_{μ ν}^{a} G^{a, α β}

, so as to be colorless. The same thing applies for the SU(2)_L × U(1)_Y gauge boson field strengths, for example

W_{μ ν}^{i} W^{i, α β}

and B_μνB^αβ. Another symmetry to be imposed is Lorentz invariance. There are four different Lorentz-invariant contractions, as shown above. Hence we must consider eight gQGC operators in total.

The total and differential cross-sections are largely determined by the Lorentz structure of the gQGC operators. The eight operators can be classified into four pairs {O_gT,(0,4), O_gT,(1,5), O_gT,(2,6), O_gT,(3,7)}, each with a same Lorentz structure.

The hierarchical structure of the cross-sections generated by the eight gQGC operators is manifest in the 95% C.L. lower bounds derived from the ATLAS data [7] are

\begin{array}{rcl} \begin{array}{rcl} M_{0} & ⩾ & 1040 G e V, M_{1} ⩾ 777 G e V, \\ M_{2} & ⩾ & 750 G e V, M_{3} ⩾ 709 G e V, \\ M_{4} & ⩾ & 1399 G e V, M_{5} ⩾ 1046 G e V, \\ M_{6} & ⩾ & 1010 G e V, M_{7} ⩾ 954 G e V . \end{array} \end{array}

(2)

3. Features of the signal

3.1. Compare of the tri-boson process and the VBF process

The gQGCs can contribute to the process

μ^{+} μ^{-} \to j j ν \bar{ν}

via both the VBF and tri-boson processes, the Feynman diagrams are shown in figure 1. When a new particle X presents in the final states, the relative scaling between the VBF and annihilation contribution is [67]

\begin{array}{rcl} \frac{σ_{V B F}^{N P}}{σ_{a n n}^{N P}} \propto α_{W}^{2} \frac{s}{m_{X}^{2}} {l o g}^{2} (\frac{s}{M_{V}^{2}}) l o g (\frac{s}{M_{X}^{2}}) \end{array}

(3)

where

σ_{V B F}^{N P}

and

σ_{a n n}^{N P}

are cross-sections of VBF and annihilation processes, respectively. M_X,V denotes the masses of the X particle and the SM vector bosons, respectively. At high energies, the VBF process is double-logarithmic enhanced. However, it has been pointed out that, at lower energies the tri-boson process can be more sensitive to the dimension-8 operators [40]. Therefore, it is necessary to compare the contribution of tri-boson and VBF for the case of gQGCs.

Figure 1. The Feynman diagrams of the gQGC contributions. The tri-boson contribution is shown in the left panel, while the VBF contribution is shown in the right panel.

Full size|PPT slide

In the following, we consider the gQGC contribution to the process

μ^{+} μ^{-} \to ν \bar{ν} g g

. For simplicity, defining

f_{i} \equiv 1 / 16 M_{i}^{4}

, the triboson and the VBF contributions (denoting as σ_tri and σ_VBF) at the leading order of

M_{Z, W}^{2} / s

and at tree level are,

\begin{array}{rcl} \begin{array}{l} σ_{t r i} = B r (Z \to ν \bar{ν}) \times \frac{e^{2} s^{3}}{8847360 π^{3} c_{W}^{2} s_{W}^{2}} \\ \times {c_{W}^{4} [24 f_{0}^{2} + 4 f_{3} (f_{0} + 3 f_{1} + f_{2}) + 8 f_{0} f_{1} \\ + 12 f_{0} f_{2} + 14 f_{1}^{2} + 8 f_{1} f_{2} + 3 f_{2}^{2} + 4 f_{3}^{2}] \\ - 2 c_{W}^{2} s_{W}^{2} [2 f_{7} (f_{0} + 3 f_{1} + f_{2} + 2 f_{3}) + 24 f_{0} f_{4} + 4 f_{0} f_{5} \\ + 6 f_{0} f_{6} + 4 f_{1} f_{4} + 14 f_{1} f_{5} + 4 f_{1} f_{6} \\ + 6 f_{2} f_{4} + 4 f_{2} f_{5} + 3 f_{2} f_{6} + 2 f_{3} f_{4} + 6 f_{3} f_{5} + 2 f_{3} f_{6}] \\ + 5 s_{W}^{4} [24 f_{4}^{2} + 4 f_{7} (f_{4} + 3 f_{5} + f_{6}) + 8 f_{4} f_{5} + 12 f_{4} f_{6} \\ + 14 f_{5}^{2} + 8 f_{5} f_{6} + 3 f_{6}^{2} + 4 f_{7}^{2}]} \end{array} \end{array}

(4)

and

\begin{array}{rcl} \begin{array}{l} σ_{V B F} = \frac{e^{4} s^{3}}{4246732800000 π^{5} s_{W}^{4}} \\ \times {600 {l o g}^{2} (\frac{s}{16 M_{W}^{2}}) [800 ((6 f_{0}^{2} + f_{0} (2 f_{1} + 3 f_{2} + f_{3})) \\ + 1492 f_{1}^{2} + 52 f_{1} (31 f_{2} + 21 f_{3}) + 603 f_{2}^{2} \\ + 806 f_{2} f_{3} + 473 f_{3}^{2}] \\ - 40 l o g (\frac{s}{16 M_{W}^{2}}) \\ [160800 f_{0}^{2} + 26800 f_{0} (2 f_{1} + 3 f_{2} + f_{3}) \\ + 33944 f_{1}^{2} + 4 f_{1} (9491 f_{2} + 5136 f_{3}) \\ + 16191 f_{2}^{2} + 18982 f_{2} f_{3} + 11836 f_{3}^{2}] \\ + 4526400 f_{0}^{2} + 754400 f_{0} (2 f_{1} + 3 f_{2} + f_{3}) + 764788 f_{1}^{2} \\ + 877948 f_{1} f_{2} + 387588 f_{1} f_{3} \\ + 408087 f_{2}^{2} + 438974 f_{2} f_{3} + 285497 f_{3}^{2}} \end{array} \end{array}

(5)

where

\sqrt{s}

is the c.m. energies of the muon colliders

s_{W} = \sin (θ_{W})

and

c_{W} = \cos (θ_{W})

with the weak mixing angle denoted as θ_W,

B r (Z \to ν \bar{ν})

is taken as 20%. In equations (4) and (5), the interference between σ_tri and σ_VBF are ignored. The σ_VBF is obtained using the effective vector boson approximation [68–70],

\begin{array}{rcl} \begin{array}{l} σ_{V B F} (μ^{+} μ^{-} \to \bar{ν} ν g g) \\ = \sum_{λ_{1} λ_{2} λ_{3} λ_{4}} \int d ξ_{1} \\ \times \int d ξ_{2} f_{W_{λ_{1}}^{-} / μ^{-}} (ξ_{1}) f_{W_{λ_{2}}^{+} / μ^{+}} (ξ_{2}) σ_{W_{λ_{1}}^{+} W_{λ_{2}}^{-} \to g g} (\hat{s}), \\ f_{W_{+ 1}^{-} / μ_{L}^{-}} (ξ) = f_{W_{- 1}^{+} / μ_{L}^{+}} (ξ) = \frac{e^{2}}{8 π^{2} s_{W}^{2}} \frac{{(1 - ξ)}^{2}}{2 ξ} l o g \frac{μ_{f}^{2}}{M_{W}}, \\ f_{W_{- 1}^{-} / μ_{L}^{-}} (ξ) = f_{W_{+ 1}^{+} / μ_{L}^{+}} (ξ) = \frac{e^{2}}{8 π^{2} s_{W}^{2}} \frac{1}{2 ξ} l o g \frac{μ_{f}^{2}}{M_{W}}, \\ f_{W_{0}^{-} / μ_{L}^{-}} (ξ) = f_{W_{0}^{+} / μ_{L}^{+}} (ξ) = \frac{e^{2}}{8 π^{2} s_{W}^{2}} \frac{1 - ξ}{ξ}, \\ f_{W_{λ}^{\pm} / μ_{R}^{\pm}} = 0, f_{W_{λ}^{\pm} / μ^{\pm}} = \frac{f_{W_{λ}^{\pm} / μ_{L}^{\pm}} + f_{W_{λ}^{\pm} / μ_{R}^{\pm}}}{2}, \end{array} \end{array}

(6)

where

\sqrt{\hat{s}} = \sqrt{ξ_{1} ξ_{2} s}

is the c.m. energy of W⁺W⁻ → gg, and μ_f is the factorization scale set to be

\sqrt{\hat{s}} / 4

[70].

The numerical results of equations (4) and (5) are shown in figure 2. The detector simulation is not included until the Monte Carlo (MC) simulation is applied. It can be seen that the tri-boson contribution is larger than the VBF at about

\sqrt{s} < 5 T e V

, at

\sqrt{s} = 30 T e V

, σ_VBF is about 5 times of σ_tri. Therefore the contribution of tri-boson is not negligible. In this paper, σ_VBF, σ_tri, and the interference between them are considered as the signal of the gQGCs.

Figure 2. σ_tri in equation (4) compared with σ_VBF in equation (5). When $\sqrt{s} < 5 T e V$ , σ_tri is larger than σ_VBF.

Full size|PPT slide

Apart from that, note that O_gT,4,5,6,7 operators do not contribute to the W-boson fusion processes. Since the energies considered are mainly above 5 TeV, in this paper, we only consider the contributions of the O_gT,0,1,2,3 operators. However, we shall emphasise that the process WW → gg provides a chance to study the O_gT,0,1,2,3 operators separately, compared with the pp → γγ and pp → Zγ processes where the contributions from O_gT,i and O_gT,i+4 are proportional to each other [46].

3.2. Unitarity bound

The SMEFT is not UV completed. As an EFT, the SMEFT is only valid under a certain energy scale. One of the signals when the SMEFT is no longer valid is the violation of unitarity. With dimension-8 operators, the amplitude of the process W⁺W⁻ → gg grows as

{\hat{s}}^{2}

which leads to the violation of unitarity [71–73] at large enough energy. The partial wave unitarity bound [74–76] is often used to check whether the energy scale considered is already invalid. For the subprocess

W_{λ_{1}}^{+} W_{λ_{2}}^{-} \to g_{λ_{3}} g_{λ_{4}}

, where λ_i correspond to the helicities, the amplitude can be expanded as [77]

\begin{array}{rcl} \begin{array}{l} M (W_{λ_{1}}^{+} W_{λ_{2}}^{-} \to g_{λ_{3}} g_{λ_{4}}) \\ = 8 π \sum_{J} (2 J + 1) \sqrt{1 + δ_{λ_{3}, λ_{4}}} e^{i (λ - λ^{'}) φ} d_{λ λ^{'}}^{J} (θ) T^{J} \end{array} \end{array}

(7)

where λ = λ₁ − λ₂,

λ^{'} = λ_{3} - λ_{4}

, θ and φ are zenith and azimuth angles of the one of the gluons in the final state, respectively, and

d_{λ λ^{'}}^{J} (θ)

are Wigner d-functions [77]. The partial wave unitarity bound is ∣T^J∣ ≤ 2 [74] which is widely used in previous works [33, 35–39, 78–81].

At leading order of

\hat{s}

, the relevant helicity amplitudes are

\begin{array}{rcl} \begin{array}{l} M (W_{+}^{+} W_{+}^{-} \to g_{+} g_{+}) = \frac{{\hat{s}}^{2}}{16} (4 f_{0} + f_{2} + f_{3}), \\ M (W_{+}^{+} W_{+}^{-} \to g_{-} g_{-}) \\ = \frac{{\hat{s}}^{2}}{64} (16 f_{0} + (2 f_{1} + f_{3}) \cos (2 θ) + 6 f_{1} + 4 f_{2} - f_{3}), \\ M (W_{+}^{+} W_{-}^{-} \to g_{+} g_{-}) = \frac{{\hat{s}}^{2}}{16} e^{2 i ϕ} \cos^{4} (\frac{θ}{2}) (2 f_{1} + f_{2} + f_{3}) . \end{array} \end{array}

(8)

There are other helicity amplitudes which lead to same unitarity bounds as those derived from equation (8), and therefore are not presented for simplicity. Assuming one operator at a time, the tightest bounds are

\begin{array}{rcl} \frac{{\hat{s}}^{2} | f_{0} |}{32 \sqrt{2} π} < 2, \frac{{\hat{s}}^{2} | f_{1} |}{96 \sqrt{2} π} < 2, \frac{{\hat{s}}^{2} | f_{2, 3} |}{128 \sqrt{2} π} < 2. \end{array}

(9)

As such, for the subprocess W⁺W⁻ → gg,

\hat{s} ⩽ s

, we consider the largest possible

\hat{s}

; the unitarity bounds on the coefficients at different energies are listed in table 1. The energies are chosen as the muon colliders [52, 53].

Table 1. The upper bounds of coefficients ∣f_i∣ and lower bounds of M_i in the sense of partial wave unitarity at different energies.

$\sqrt{\hat{s}} (T e V)$	3	10	14	30
∣f₀∣(TeV⁻⁴)	3.5	0.028	0.0074	0.00035
M₀(GeV)	365.56	1222.31	1704.76	3655.55
∣f₁∣(TeV⁻⁴)	10.5	0.085	0.022	0.001
M₁(GeV)	277.76	926.01	1298.27	2811.71
∣f_2,3∣(TeV⁻⁴)	14.0	0.114	0.030	0.004
M_2,3(GeV)	258.49	860.49	1201.41	1988.18

3.3. Event selection strategy

The background in this study is the SM contribution to

μ^{+} μ^{-} \to j j ν \bar{ν}

. The process μ⁺μ⁻ → jj is suppressed by the s-channel propagator, and can be further suppressed by the cuts on the missing energy, therefore is ignored. The process

μ^{+} μ^{-} \to j j ν ν \bar{ν} \bar{ν}

is suppressed by more EW couplings and therefore is also ignored.

The typical Feynman diagrams for the background at tree level are shown in figure 3. An important feature is that the two jets are both from quarks, which is different from the case of gQGCs where the two jets are both from the gluons. As a consequence, there is no interference term between the SM and the gQGCs. Once the efficiencies of the event selection strategy for both the background and the signal are obtained, the cross-section can be read out.

Figure 3. Typical Feynman diagrams of the SM contribution.

Full size|PPT slide

To investigate the event selection strategy, the MC simulation is carried out with the help of the MadGraph5_aMC@NLO [82, 83] toolkit including a parton shower using Pythia82 [84]. The standard cuts are used as the default. The parton distribution function is NNPDF2.3 [85]. A fast detector simulation is then applied using Delphes [86] with the muon collider card. In the event generation, the complete syntax

μ^{+} μ^{-} \to j j ν_{l} {\bar{ν}}_{l}

is used, and the standard cuts are set as the default of ‘MadGraph5_aMC@NLO', the relevant cuts for jets are transverse momentum

p_{T}^{j} > 20 G e V

and pseudo-rapidity η_j < 5.0, and ΔR_jj > 0.4, where

Δ R = \sqrt{Δ ϕ_{j j}^{2} + Δ η_{j j}^{2}}

and where Δφ_jj and Δη_jj are the difference between the azimuth angles and pseudo-rapidities of any two jets.

At the energies of the muon colliders, the SM cross-sections (denoted as σ_SM) are obtained and listed in table 2. In the SM, the cross-section grows slowly with the energy.

Table 2. The cross-sections of the SM contribution and the upper bounds of coefficients ∣f_i∣ used in the phenomenological study. The cross-sections of the gQGCs are also shown.

$\sqrt{s} (T e V)$	3	10	14	30
σ_SM(pb)	0.8688	1.4548	1.6087	1.8988
∣f₀∣(TeV⁻⁴)	1	0.012	0.004	0.00035
σ_gT,0(pb)	0.00321	0.00115	0.00112	0.00114
∣f₁∣(TeV⁻⁴)	1.5	0.02	0.007	0.0006
σ_gT,1(pb)	0.00355	0.00138	0.00144	0.00134
∣f_2,3∣(TeV⁻⁴)	3	0.03	0.012	0.0012
σ_gT,2(pb)	0.00383	0.000956	0.00134	0.00178
σ_gT,3(pb)	0.00412	0.000924	0.00127	0.00161

The signal events are generated by assuming one operator at a time. The sensitivity of the process

μ^{+} μ^{-} \to j j ν \bar{ν}

to the gQGCs can be estimated with respect to the significance defined as [87, 88]

\begin{array}{rcl} S_{s t a t} = \sqrt{2 [(N_{b g} + N_{s}) l n (1 + N_{s} / N_{b g}) - N_{s}]}, \end{array}

(10)

where N_s = N_aQGC − N_SM and N_bg = N_SM. The N_bg can be obtained by σ_SM and the luminosities of the muon colliders, where the N_s can be provisionally estimated using σ_tri + σ_VBF where σ_tri and σ_VBF are given in equations (4) and (5). The coefficients are chosen so that the signal significance of the VBF contributions are about S_stat = 2 ∼ 3 before cuts at luminosity L = 1 ab⁻¹, L = 10 ab⁻¹ for

\sqrt{s} = 3 T e V

and

\sqrt{s} ⩾ 10 T e V

, respectively [52, 53]. The coefficients at different energies are listed in table 2. Note that they are also chosen to be within the partial wave unitarity bounds in table 1. The contributions of the gQGCs (denoted as σ_gT,i) obtained by MC are also shown in table 2.

To investigate the kinematic features, we require that the final states contain at least two jets, which is denoted as the N_j cut. To suppress the events from the process μ⁺μ⁻ → jj, we also require a minimal

{/ p}_{T}

for the events, where

{/ p}_{T}

is the transverse missing energy.

{/ p}_{T}

can also be used to suppress the events from the SM contribution to

μ^{+} μ^{-} \to j j ν \bar{ν}

. This is because the signal grows with the energy so that for the tri-boson signal events, the neutrinos are typically from a highly boosted Z boson. Since neutrinos play a role similar to that of the residual jets of the VBF processes at a hadron collider, similar to the standard VBF cut [89], the neutrinos are expected to be back-to-back. For the VBF signal events, although the neutrinos are back-to-back, the net transpose momentum is still typically larger than the case of the SM. The normalized distributions of

{/ p}_{T}

for the backgrounds and signals are shown in figure 4. We require the events to have a large

{/ p}_{T}

, which is denoted as the

{/ p}_{T}

cut.

Figure 4. The normalized distributions of ${/ p}_{T}$ for the SM and gQGCs at different energies. The top left panel corresponds to $\sqrt{s} = 3 T e V$ , the top right panel corresponds to $\sqrt{s} = 10 T e V$ , the bottom left panel corresponds to $\sqrt{s} = 14 T e V$ , and the bottom right panel corresponds to $\sqrt{s} = 30 T e V$ .

Full size|PPT slide

Due to the same reason, for both the tri-boson and VBF signal events, the jets are energetic. As a result, the invariant mass of the hardest two jets (denoted as m_jj) should be large for the signal events. The normalized distributions of m_jj for the background and signal are shown in figure 5. We require the events to have a large m_jj, which is denoted as the m_jj cut.

Figure 5. Same as figure 4 but for m_jj.

Full size|PPT slide

At different energies, we use different cuts. The event selection strategies are summarized in table 3. The cross-sections after cuts are listed in table 4. The efficiencies of the cuts (denoted as ε) are also shown in table 4. It can be seen that the event selection strategies can suppress the background significantly.

Table 3. The event selection strategies at different energies.

$\sqrt{s}$	${/ p}_{T}$	m_jj
(TeV)
3	>50 GeV	>1 TeV
10	>100 GeV	>3 TeV
14	>100 GeV	>5 TeV
30	>200 GeV	>10 TeV

Table 4. The contributions of the SM and gQGCs after cuts. The efficiencies of the cuts are shown in the last row.

$\sqrt{s}$	cut	SM	O_gT,0	O_gT,1	O_gT,2	O_gT,3
(TeV)		(fb)	(fb)	(fb)	(fb)	(fb)
	N_j	722.9	3.21	3.52	3.82	4.10
3	$⧸ P_{T}$	536.7	3.15	3.47	3.74	4.04
	m_jj	0.885	2.49	2.27	2.85	2.74
	ε	0.102%	77.6%	63.9%	74.4%	66.5%
$\sqrt{s}$	cut	SM	O_gT,0	O_gT,1	O_gT,2	O_gT,3
(TeV)		(fb)	(fb)	(fb)	(fb)	(fb)
	N_j	1191.9	1.15	1.37	0.955	0.921
10	$⧸ P_{T}$	508.2	1.12	1.34	0.930	0.902
	m_jj	0.256	0.961	1.06	0.799	0.728
	ε	0.0176%	83.6%	76.8%	83.6%	78.8%
$\sqrt{s}$	cut	SM	O_gT,0	O_gT,1	O_gT,2	O_gT,3
(TeV)		(fb)	(fb)	(fb)	(fb)	(fb)
	N_j	1315.5	1.12	1.43	1.34	1.26
14	$⧸ P_{T}$	531.9	1.10	1.42	1.31	1.24
	m_jj	0.105	0.846	1.02	1.03	0.916
	ε	0.00653%	75.5%	70.8%	76.9%	72.1%
$\sqrt{s}$	cut	SM	O_gT,0	O_gT,1	O_gT,2	O_gT,3
(TeV)		(fb)	(fb)	(fb)	(fb)	(fb)
	N_j	1542.4	1.14	1.33	1.78	1.61
30	$⧸ P_{T}$	256.4	1.11	1.30	1.73	1.57
	m_jj	0.0532	0.914	1.04	1.46	1.27
	ε	0.00280%	80.2%	77.6%	82.0%	78.9%

4. Constraints on the coefficients

Assuming one operator at a time, without the interference between the SM and the gQGCs, the cross-section after cuts can be expressed as,

\begin{array}{rcl} σ = ϵ_{S M} σ_{S M} + ϵ_{g T, i} \frac{f_{i}^{2}}{{\tilde{f}}_{i}^{2}} σ_{g T, i} ({\tilde{f}}_{i}) \end{array}

(11)

where σ_SM and

σ_{g T, i} ({\tilde{f}}_{i})

are the cross-sections of the SM and the gQGCs at

{\tilde{f}}_{i}

, and ε_SM and ε_gT,i are the cut efficiencies of the SM and gQGC events, respectively. Taking

{\tilde{f}}_{i}

as the ones in table 2, numerical results of σ_SM and σ_gT,i are listed in table 2, and ε_SM and ε_gT,i are listed in table 4. The expected constraints at muon colliders can be estimated by the signal significance defined in equation (10). The integrated luminosities for both the ‘conservative' and ‘optimistic' cases are considered [52, 53]. As a result, we can obtain the projected sensitivities on f_i and M_i by taking 2σ, 3σ or 5σ significance. The results are shown in tables 5 and 6. The expected constraints on f₂ and f₃ are close to each other, which is indicated by the fact that the leading helicity amplitudes for O_2,3 are the same.

Table 5. The projected sensitivities on the aQGC coefficients at the muon colliders with different c.m. energies and integrated luminosities for the ‘conservative' case.

	$S_{s t a t}$	3 TeV	10 TeV	14 TeV	30 TeV
		1 ab⁻¹	10 ab⁻¹	10 ab⁻¹	10 ab⁻¹
	2	<155	<1.24	<0.351	<0.025
∣f₀∣	3	<191	<1.52	<0.432	<0.03
(10⁻³TeV⁻⁴)	5	<248	<1.96	<0.561	<0.04
	2	>0.796	>2.67	>3.65	>7.07
M₀	3	>0.756	>2.53	>3.47	>6.71
(TeV)	5	>0.709	>2.38	>3.25	>6.29
	2	<244	<1.96	<0.561	<0.040
∣f₁∣	3	<300	<2.41	<0.689	<0.049
(10⁻³TeV⁻⁴)	5	<389	<3.11	<0.893	<0.064
	2	>0.711	>2.38	>3.25	>6.28
M₁	3	>0.675	>2.26	>3.09	>5.96
(TeV)	5	>0.633	>2.12	>2.89	>5.58
	2	<436	<3.39	<0.957	<0.068
∣f₂∣	3	<535	<4.16	<1.17	<0.083
(10⁻³TeV⁻⁴)	5	<695	<5.38	<1.52	<0.109
	2	>0.615	>2.07	>2.84	>5.07
M₂	3	>0.585	>1.97	>2.70	>5.23
(TeV)	5	>0.548	>1.85	>2.53	>4.90
	2	<445	<3.55	<1.01	<0.073
∣f₃∣	3	<546	<4.35	<1.25	<0.090
(10⁻³TeV⁻⁴)	5	<709	<5.64	<1.62	<0.116
	2	>0.612	>2.05	>2.80	>5.41
M₃	3	>0.582	>1.95	>2.66	>5.14
(TeV)	5	>0.545	>1.82	>2.49	>4.81

Table 6. The projected sensitivities on the aQGC coefficients at the muon colliders with different c.m. energies and integrated luminosities for the ‘optimistic' case.

	$S_{s t a t}$	14 TeV	30 TeV
		20 ab⁻¹	90 ab⁻¹
	2	<2.95	<0.144
∣f₀∣	3	<3.63	<0.176
(10⁻⁴TeV⁻⁴)	5	<4.70	<0.228
	2	>3.81	>8.12
M₀	3	>3.62	>7.71
(TeV)	5	>3.40	>7.23
	2	<4.71	<0.231
∣f₁∣	3	<5.78	<0.284
(10⁻⁴TeV⁻⁴)	5	<7.49	<0.367
	2	>3.39	>7.21
M₁	3	>3.23	>6.85
(TeV)	5	>3.02	>6.42
	2	<8.03	<0.390
∣f₂∣	3	<9, 86	<0.479
(10⁻⁴TeV⁻⁴)	5	<12.8	<0.619
	2	>2.97	>6.33
M₂	3	>2.82	>6.01
(TeV)	5	>2.65	>5.64
	2	<8.52	<0.419
∣f₃∣	3	<10.5	<0.513
(10⁻⁴TeV⁻⁴)	5	<13.5	<0.664
	2	>2.93	>6.22
M₃	3	>2.78	>5.91
(TeV)	5	>2.61	>5.54

Compared with the expected constraints from the LHC in equation (2), the muon collider at

\sqrt{s} ⩾ 10 T e V

has tighter constraints. Since the operators contributing to the gQGCs are dimension-8, not surprisingly, higher energy muon colliders are more advantageous in detecting the gQGCs. Taking O_gT,0 as an example, at

\sqrt{s} = 3 T e V

, the sensitivity will exceed that of the LHC when the luminosity is about 70 ab⁻¹. However, for the luminosities in the ‘optimistic' case, i.e., L = 1, 10, 20 and 90 ab⁻¹ for

\sqrt{s} = 3, 10, 14

and 30 TeV, respectively, the 2σlower bounds on M₀ is about 27% of the c.m. energies of the muon collider, which indicates that the constraint on M₀ will exceed that of the LHC at a

\sqrt{s} = 4 T e V

muon collider. Meanwhile, in the same case, the 2σlower bounds on M_1,2,3 are about 24%, 21% and 20 ∼ 21% of

\sqrt{s}

, which are consistent with equation (2) in order of sensitivity. The above result can also be compared with the case of future hadron colliders such as the HL-LHC. In Ref. [46], taking O_gT,0 as an example, the 2σ constraint on M₀ by the pp → γγ channel is about 15% of

\sqrt{s}

, and the combined (combined of pp → γγ, pp → ℓ⁺ℓ⁻γ,

p p \to ν \bar{ν} γ

and

p p \to q \bar{q} γ

channels) constraint on M₀ is about 22% of

\sqrt{s}

. Thus, the sensitivities of the muon colliders to the gQGCs are competitive with future hadron colliders, and are even better at the same c.m. energies.

5. Summary

The muon colliders which are also called vector boson colliders are ideal places to study the quartic gauge couplings. In this paper, we investigate the capability of future muon colliders to detect the gQGC in the framework of SMEFT. The cross-sections of the process

μ^{+} μ^{-} \to j j ν \bar{ν}

with gQGCs present are calculated.

The gQGCs can affect the process

μ^{+} μ^{-} \to j j ν \bar{ν}

via both tri-boson and VBF contributions. The tri-boson contribution is larger than the VBF when

\sqrt{s} < 5 T e V

, and is about 1/5 of the VBF at

\sqrt{s} = 30 T e V

. Besides, the VBF process can be affected by O_gT,0,1,2,3. Therefore, we focus on O_gT,0,1,2,3 in this paper.

With partial wave unitarity bounds considered, the event selection strategy as well as the expected constraints on the operator coefficients are studied using MC simulation. For the luminosities in the ‘optimistic' case, 2σ lower bounds on M_0,1,2,3 are about 27%, 24%, 21% and 20 – 21% of the c.m. energies of the muon colliders, respectively. Our results indicate that the muon colliders with

\sqrt{s} > 4 T e V

can be more sensitive to the gQGCs than the LHC.

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Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grants Nos. 11905093 and 12147214, and the Natural Science Foundation of the Liaoning Scientific Committee No. LJKZ0978.

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Abstract
Key words
Cite this article
1. Introduction
2. Dimension-8 gluonic QGC operators
3. Features of the signal
3.1. Compare of the tri-boson process and the VBF process
Figure 1. The Feynman diagrams of the gQGC contributions. The tri-boson contribution is shown in the left panel, while the VBF contribution is shown in the right panel.
Figure 2. σtri in equation (4) compared with σVBF in equation (5). When $\sqrt{s} < 5 T e V$ , σtri is larger than σVBF.
3.2. Unitarity bound
Table 1. The upper bounds of coefficients ∣fi∣ and lower bounds of Mi in the sense of partial wave unitarity at different energies.
3.3. Event selection strategy
Figure 3. Typical Feynman diagrams of the SM contribution.
Table 2. The cross-sections of the SM contribution and the upper bounds of coefficients ∣fi∣ used in the phenomenological study. The cross-sections of the gQGCs are also shown.
Figure 4. The normalized distributions of ${/ p}_{T}$ for the SM and gQGCs at different energies. The top left panel corresponds to $\sqrt{s} = 3 T e V$ , the top right panel corresponds to $\sqrt{s} = 10 T e V$ , the bottom left panel corresponds to $\sqrt{s} = 14 T e V$ , and the bottom right panel corresponds to $\sqrt{s} = 30 T e V$ .
Figure 5. Same as figure 4 but for mjj.
Table 3. The event selection strategies at different energies.
Table 4. The contributions of the SM and gQGCs after cuts. The efficiencies of the cuts are shown in the last row.
4. Constraints on the coefficients
Table 5. The projected sensitivities on the aQGC coefficients at the muon colliders with different c.m. energies and integrated luminosities for the ‘conservative' case.
Table 6. The projected sensitivities on the aQGC coefficients at the muon colliders with different c.m. energies and integrated luminosities for the ‘optimistic' case.
5. Summary
References
Acknowledgments
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Received	Revised	Accepted	Published
2023-08-21	2023-09-15	2023-09-26	2023-11-10
Issue Date
2023-11-10

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Abstract

Key words

Cite this article

1. Introduction

2. Dimension-8 gluonic QGC operators

3. Features of the signal

3.1. Compare of the tri-boson process and the VBF process

Figure 1. The Feynman diagrams of the gQGC contributions. The tri-boson contribution is shown in the left panel, while the VBF contribution is shown in the right panel.

Figure 2. σ_tri in equation (4) compared with σ_VBF in equation (5). When $\sqrt{s} < 5 T e V$ , σ_tri is larger than σ_VBF.

3.2. Unitarity bound

Table 1. The upper bounds of coefficients ∣f_i∣ and lower bounds of M_i in the sense of partial wave unitarity at different energies.

3.3. Event selection strategy

Figure 3. Typical Feynman diagrams of the SM contribution.

Table 2. The cross-sections of the SM contribution and the upper bounds of coefficients ∣f_i∣ used in the phenomenological study. The cross-sections of the gQGCs are also shown.

Figure 5. Same as figure 4 but for m_jj.

Table 3. The event selection strategies at different energies.

Table 4. The contributions of the SM and gQGCs after cuts. The efficiencies of the cuts are shown in the last row.

4. Constraints on the coefficients

Table 5. The projected sensitivities on the aQGC coefficients at the muon colliders with different c.m. energies and integrated luminosities for the ‘conservative' case.

Table 6. The projected sensitivities on the aQGC coefficients at the muon colliders with different c.m. energies and integrated luminosities for the ‘optimistic' case.

5. Summary

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References

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Footnotes

Acknowledgments

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Content to export

Abstract

Key words

Cite this article

1. Introduction

2. Dimension-8 gluonic QGC operators

3. Features of the signal

3.1. Compare of the tri-boson process and the VBF process

Figure 1. The Feynman diagrams of the gQGC contributions. The tri-boson contribution is shown in the left panel, while the VBF contribution is shown in the right panel.

Figure 2. σtri in equation (4) compared with σVBF in equation (5). When s<5TeV, σtri is larger than σVBF.

3.2. Unitarity bound

Table 1. The upper bounds of coefficients ∣fi∣ and lower bounds of Mi in the sense of partial wave unitarity at different energies.

3.3. Event selection strategy

Figure 3. Typical Feynman diagrams of the SM contribution.

Table 2. The cross-sections of the SM contribution and the upper bounds of coefficients ∣fi∣ used in the phenomenological study. The cross-sections of the gQGCs are also shown.

Figure 4. The normalized distributions of /pT for the SM and gQGCs at different energies. The top left panel corresponds to s=3TeV, the top right panel corresponds to s=10TeV, the bottom left panel corresponds to s=14TeV, and the bottom right panel corresponds to s=30TeV.

Figure 5. Same as figure 4 but for mjj.

Table 3. The event selection strategies at different energies.

Table 4. The contributions of the SM and gQGCs after cuts. The efficiencies of the cuts are shown in the last row.

4. Constraints on the coefficients

Table 5. The projected sensitivities on the aQGC coefficients at the muon colliders with different c.m. energies and integrated luminosities for the ‘conservative' case.

Table 6. The projected sensitivities on the aQGC coefficients at the muon colliders with different c.m. energies and integrated luminosities for the ‘optimistic' case.

5. Summary

{{custom_sec.title}}

{{custom_sec.title}}

References

{{custom_fnGroup.title_en}}

Footnotes

Acknowledgments

{{custom_ack.title_en}}

RIGHTS & PERMISSIONS

Figure 2. σ_tri in equation (4) compared with σ_VBF in equation (5). When $\sqrt{s} < 5 T e V$ , σ_tri is larger than σ_VBF.

Table 1. The upper bounds of coefficients ∣f_i∣ and lower bounds of M_i in the sense of partial wave unitarity at different energies.

Table 2. The cross-sections of the SM contribution and the upper bounds of coefficients ∣f_i∣ used in the phenomenological study. The cross-sections of the gQGCs are also shown.

Figure 5. Same as figure 4 but for m_jj.