By embedding the Standard Model effective field theory (SMEFT) in the more general Higgs effective field theory (HEFT), we expose correlations among the coefficients of the latter that, if found to be violated in future data, would lead to the experimental falsification of the SMEFT framework. These are derived from the necessary symmetric point of HEFT and analyticity of the SMEFT Lagrangian that allows the construction of the SMEFT expansion, as laid out by other groups, and properties at that point of the Higgs-flare function ${ \mathcal F }(h)$ coupling Goldstone and Higgs bosons, of the Higgs potential V(h) and of the Higgs-top quark coupling function ${ \mathcal G }(h)$.
Raquel Gómez-Ambrosio, Felipe J Llanes-Estrada, Alexandre Salas-Bernárdez, Juan J Sanz-Cillero. SMEFT is falsifiable through multi-Higgs measurements (even in the absence of new light particles)[J]. Communications in Theoretical Physics, 2023, 75(9): 095202. DOI: 10.1088/1572-9494/ace95e
1. Introduction
Discovering new particles would entail the Standard Model (SM) being falsified in Popper’s sense [1] and force us to extend it. Absent such discovery, the SM is still falsifiable upon finding new forces among the known particles. Because the SM has a characteristic energy scale of 100 GeV (the mass of the Higgs boson at mh = 125 GeV, as well as the vacuum constant v = 246 GeV exemplify it), but no new particles below 1000 GeV have been found, there is a scale separation that begs the use of effective field theory.
The popular Standard Model effective field theory (SMEFT) extension of the SM electroweak symmetry breaking sector (EWSBS) adds operators classified by their mass dimension,
$\begin{eqnarray}{{ \mathcal L }}_{\mathrm{SMEFT}}={{ \mathcal L }}_{\mathrm{SM}}+\displaystyle \sum _{n=5}^{\infty }\displaystyle \sum _{i}\displaystyle \frac{{c}_{i}^{(n)}}{{{\rm{\Lambda }}}^{n-4}}{{ \mathcal O }}_{i}^{(n)}(H).\end{eqnarray}$
These operators ${{ \mathcal O }}_{i}^{(n)}(H)$, whose intensity is controlled by the Wilson coefficients ${c}_{i}^{(n)}$ for a given reference scale Λ, are the potentials of those new forces being sought. A non-zero ${c}_{i}^{(n)}$ would signal a departure from the SM, that then would need to be extended, perhaps by new resonances [2].
It is easy to ask oneself how the whole framework of SMEFT can be tested. Effective theories include all the possible interactions that are compatible with known particle content and the symmetries believed to hold. Would it not be that any separation from the SM could be recast in SMEFT form? In that case, absent some new light particle, any phenomena could be described by adding an operator with a parameter to the SM. This is not so, as we will detail.
The particle content of the electroweak symmetry breaking sector is packaged in a Higgs doublet field in the SM as in the SMEFT
where the Cartesian coordinates φa can be rearranged to the polar decomposition in terms of the ωi Goldstone bosons (which set the orientation of H through the unitary matrix U(ω) and the radial coordinate hSMEFT (with $| H| =(v+{h}_{\mathrm{SMEFT}})/\sqrt{2}$).
An additional non-linear redefinition of hSMEFT allows us to rearrange the SMEFT Lagrangian in the form of a more general theory, the Higgs effective field theory (HEFT):
Our current focus therein is the flare function [3, 4]:
$\begin{eqnarray}{ \mathcal F }({h}_{\mathrm{HEFT}})=1+\displaystyle \sum _{n=1}^{\infty }{a}_{n}{\left(\displaystyle \frac{{h}_{\mathrm{HEFT}}}{v}\right)}^{n},\end{eqnarray}$
which amounts to a radial ‘scale’ (think of a(t) in a Friedmann−Robertson−Walker cosmology) in the field space of the (h, ωi) electroweak bosons (with ωi analogous to the spatial coordinates). What we call attention here is that the Taylor-series coefficients of ${ \mathcal F }$ as defined in equation (4) must satisfy experimental correlations or constraints as given in table 1 similar to the ones in [5] if the SMEFT is a valid description.3(3 Our advance since our previous publication in [5] consists of the new correlations among the HEFT t quark-Yukawa sector coefficients, see equation (19) and the improvements to the correlations among the Higgs self-interaction presented in equation (18) due to the cH□ operator. The results presented here have been reported to the Quark Confinement and the Hadron Spectrum conference resulting in a preliminary publication [6].) It is clear that an experimental program aimed at these correlations via the key process to access ${ \mathcal F }$, ωω → nh as sketched in figure 1, mh → nh to access V(hHEFT) (the Higgs potential), or $\bar{t}t\to {nh}$ to access the tree-level function modifying the Yukawa couplings, see equation (7), ${ \mathcal G }(h)$, can test the validity of the SMEFT itself and not only its parameters. As an example, the SMEFT correlation among a1 and a2 is shown in figure 2.
Figure 1. The ωω → nh processes can be the key to disentangling the nature of the EWSBS. They give direct access to the ai coefficients of the flare function ${ \mathcal F }$ and hence to their correlations, as listed in table 1. Current experimental constraints do not really extend beyond n = 2 and it will be challenging to examine higher coefficients. Better reconstruction techniques at the high-luminosity LHC run but especially a future high-energy collider (either hadronic or muonic) will hopefully improve the situation.
Figure 2. The correlation a2 = 2a1 − 3 that the SMEFT predicts at order 1/Λ2 is plotted against the current 95% confidence intervals for these two HEFT parameters [21, 22]. The dashed blue lines correspond to the direct experimental bounds prior to this work, while the solid green lines represent the bounds in table 1 coming from our 1/Λ2 correlation analysis.
Table 1. Correlations between the ai HEFT coefficients necessary for the SMEFT to exist, at order Λ−2 (first and second columns, with the numbers in the second consistent with 95% confidence level experimental bounds a1/2 ∈ [0.97, 1.09] [21]). The right column provides the corresponding numerical values at the next order [5]. They are quoted in terms of Δa1:=a1 − 2 and Δa2:=a2 − 1, so that all objects in the table vanish in the SM, with all the equalities becoming 0 = 0.
${ \mathcal O }(1/{{\rm{\Lambda }}}^{2})$
${ \mathcal O }(1/{{\rm{\Lambda }}}^{2})$
${ \mathcal O }(1/{{\rm{\Lambda }}}^{4})$
Δa2 = 2Δa1
Δa2 ∈ [−0.12, 0.36]
${a}_{3}=\tfrac{4}{3}{\rm{\Delta }}{a}_{1}$
a3 ∈ [−0.08, 0.24]
a3 ∈ [−3.1, 1.7]
${a}_{4}=\tfrac{1}{3}{\rm{\Delta }}{a}_{1}$
a4 ∈ [−0.02, 0.06]
a4 ∈ [−3.3, 1.5]
a5 = 0
a5 ∈ [−1.5, 0.6]
a6 = 0
a6 = a5
There are several reasons why the experimental tests of those correlations need large energies and statistics, at the limit of what is possible today at the Large Hadron Collider (LHC) and beyond. First, the ${ \mathcal F }$ function multiplies terms with derivatives of the Goldstone bosons ∂μωi → qμωi that yield couplings proportional to their four momenta and become more relevant at higher energies. Second, the equivalence theorem [7, 8] tells us that the scattering of longitudinal gauge bosons is related to scattering of Goldstone bosons (${\omega }_{i}\sim {W}_{L}^{\pm },{Z}_{L}$): the EW gauging of the HEFT Lagrangian (3) leads to the WW → nh interaction
$\begin{eqnarray*}{\rm{\Delta }}{{ \mathcal L }}_{\mathrm{HEFT}}^{{WW}\to {nh}}=\left(\displaystyle \frac{1}{2}{m}_{Z}^{2}{Z}_{\mu }{Z}^{\mu }+{m}_{W}^{2}{W}_{\mu }^{+}{W}_{\mu }^{-}\right){ \mathcal F }({h}_{\mathrm{HEFT}}),\end{eqnarray*}$
which for longitudinal gauge bosons clearly dominates over the non-derivative interactions from V only at high energies [9, 10]. Third, an increasing number of Higgs bosons (necessary to access each hn order of ${ \mathcal F }$, the Higgs-flare function) requires an ample phase space and, thus, high energy.
2. Correlations in HEFT parameters induced by assuming the SMEFT’s validity
As we show after equation (12), the correlations mentioned above arise from the need for consistency of the SMEFT formulation when a change of variable hHEFT → hSMEFT is performed. This change affects any other piece of the Lagrangian involving the Higgs bosons, such as the Yukawa couplings to fermions, saliently the top quark, or the interactions among Higgs bosons themselves (both of which we examine here), as well as couplings to transversal gauge bosons (that we leave for future works).
The much-discussed V(H) Higgs potential, experimentally accessible at ‘low’ $\sqrt{s}$ because it contains no derivative couplings,
with v3 = 1, v4 = 1/4 and vn≥5 = 0 in the SM. Its coefficients also need to satisfy constraints that are exposed in table 2 and figure 3 if and when the SMEFT applies.
Figure 3. The correlation ${v}_{4}=\tfrac{3}{2}{v}_{3}-\tfrac{5}{4}-\tfrac{1}{6}{\rm{\Delta }}{a}_{1}$ that the SMEFT predicts at ${ \mathcal O }(1/{{\rm{\Lambda }}}^{2})$ is plotted making use of the current 95% confidence interval for v3 ∈[−2.5, 5.7] [23]. The experimental a1 uncertainty [21, 22], a1/2 ∈ [0.97, 1.09], is numerically negligible and allows to predict an SMEFT band given by the solid black line. An experimental measurement for v4 is still missing.
Table 2. Correlations among the coefficients Δv3:=v3 −1, Δv4:=v4 −1/4, v5 and v6 of the HEFT Higgs potential expansion in equation (6) that need to hold, at ${ \mathcal O }(1/{{\rm{\Lambda }}}^{2})$, if the SMEFT is a valid description of the electroweak sector. Based on the current bound Δv3 ∈[−2.5, 5.7] in Ref. [23], ${ \mathcal O }(1/{{\rm{\Lambda }}}^{2})$, the SMEFT predicts the coefficient intervals in the last column, testable in few-Higgs final states. A coupling cH□ ≠ 0 induces the correction Δa1 ∝ cH□, nevertheless numerically negligible since v3 experimental uncertainties much exceed those of a1. Likewise, we include the leading correlations for the Yukawa ${ \mathcal G }(h)$ function of equation (8), constraining c2 and c3 by c1 and a1 (from the correction to the value of the symmetric point h*). We make use of current 95% confidence interval for the top Yukawa coupling c1 ∈[0.84, 1.22] [24].
Similarly, the SM piece coupling the top quark to the Higgs boson is extended in the HEFT [11] by a multiplicative function ${ \mathcal G }(h)$
$\begin{eqnarray}{{ \mathcal L }}_{Y}=-{ \mathcal G }(h){M}_{t}\bar{t}t\sqrt{1-\displaystyle \frac{{\omega }^{2}}{{v}^{2}}},\end{eqnarray}$
with a Taylor expansion around the physical h = 0 vacuum given by
$\begin{eqnarray}{ \mathcal G }({h}_{\mathrm{HEFT}})=1+{c}_{1}\displaystyle \frac{{h}_{\mathrm{HEFT}}}{v}+{c}_{2}{\left(\displaystyle \frac{{h}_{\mathrm{HEFT}}}{v}\right)}^{2}+...\end{eqnarray}$
(with c1 = 1, ci≥2 = 0 in the SM). The correlations among these coefficients induced by the SMEFT at order 1/Λ2 are then again given in table 2 and figure 4.
Figure 4. The correlation, ${c}_{2}=\tfrac{3}{2}({c}_{1}-1)-\tfrac{1}{4}{\rm{\Delta }}{a}_{1}$, that the SMEFT predicts at ${ \mathcal O }(1/{{\rm{\Lambda }}}^{2})$ is plotted making use of the current 95% confidence interval for the top Yukawa coupling c1 ∈ [0.84, 1.22][24] (dashed red lines). The experimental a1 uncertainty [21, 22], a1/2 ∈ [0.97, 1.09] (at 95% CL), is reflected in the width of the gray band with the SMEFT correlation. An experimental determination of the $t\bar{t}\to {hh}$ coupling c2 is still missing. However, should it be measured. A test of the SMEFT is possible by comparing to c2 ∈ [−0.27, 0.35] (from linearly adding uncertainties with the existing data and analyses).
Let us then see, very briefly, how the various correlations come about. Instead of relying on the powerful geometric methods of [3, 12–15], we use the more pedestrian coordinate-dependent approach, more familiar to phenomenologists working on LHC physics. The goal is to see when is it possible to cast equation (3) into the specific SMEFT one, equation (1). This we write as
where the non-derivative and derivative terms, respectively given by V and B, collect typical SMEFT operators (think of them as expressed in the Warsaw basis). Note that we have only kept the partial derivative part of the SM Higgs doublet kinetic term ∣DH∣2 in the right-hand side of equation (9), as we are considering the equivalence theorem and focused on the scalar sector of the theory. At the lowest order correction, 1/Λ2, the relevant dimension-six SMEFT operators for our analysis are
$\begin{eqnarray}{{ \mathcal O }}_{H}\,:=\,{\left({H}^{\dagger }H\right)}^{3},\ \ {{ \mathcal O }}_{H\square }\,:=\,({H}^{\dagger }H)\square ({H}^{\dagger }H).\end{eqnarray}$
There are also other operators, such as, e.g. ${{ \mathcal O }}_{{HD}}\,={({H}^{\dagger }{D}_{\mu }H)}^{* }({H}^{\dagger }{D}^{\mu }H)$, but they break custodial symmetry, and Large Electron–Positron Collider studies suggest that the SU(2) × SU(2) → SU(2) electroweak symmetry breaking mechanism is the appropriate pattern, leaving the residual custodial SU(2) as a good approximate global symmetry of the scalar sector. The additional A(H) structure pointed out in [16] for Lagrangian (9) can be eliminated through partial integration and the use of the equations of motion [5].
To proceed, we need to perform the following conversion to pass from the SMEFT to the HEFT and vice versa:
The change from the SMEFT to the HEFT is straightforward and always possible, with the canonical, non-linear change of variables given in differential form as
where the flare function is provided by the relation
$\begin{eqnarray}{ \mathcal F }({h}_{\mathrm{HEFT}})={\left(1+{h}_{\mathrm{SMEFT}}/v\right)}^{2}.\end{eqnarray}$
However, the reverse conversion from the HEFT to the SMEFT,
$\begin{eqnarray}{h}_{\mathrm{HEFT}}={{ \mathcal F }}^{-1}\left({\left(1+{h}_{\mathrm{SMEFT}}/v\right)}^{2}\right),\end{eqnarray}$
runs into difficulty. This is because of the need to reconstruct squared operators of the Higgs doublet field H that is the basis of the SMEFT, such as
The extra ∣H∣2 on the right-hand side of the second equation ends in a denominator
$\begin{eqnarray}\begin{array}{l}{{ \mathcal L }}_{\mathrm{SMEFT}}=\mathop{\underbrace{| \partial H{| }^{2}}}\limits_{={{ \mathcal L }}_{\mathrm{SM}}}\\ +\,\mathop{\underbrace{\displaystyle \frac{1}{2}\left[\displaystyle \frac{8| H{| }^{2}}{{v}^{2}}{\left(({{ \mathcal F }}^{-1})^{\prime} \left(2| H{| }^{2}/{v}^{2}\right)\right)}^{2}-1\right]\,\displaystyle \frac{{\left(\partial | H{| }^{2}\right)}^{2}}{2| H{| }^{2}}}}\limits_{={\rm{\Delta }}{{ \mathcal L }}_{\mathrm{BSM}}}.\end{array}\end{eqnarray}$
As the SMEFT is assumed to have the analytical power expansion in equation (1), such singularity precludes its existence and needs to be cancelled by the preceding bracket in the second line of equation (16).
The result is the same as that obtained by geometric methods [16]; there must be a double zero of ${ \mathcal F }$ and a symmetric point with respect to the global SU(2) × SU(2) group so that the SMEFT expansion can be performed. Furthermore, analyticity requires that all its odd derivatives vanish at the symmetric point.
The particular case of the SM is given by ${ \mathcal F }={(1+{h}_{\mathrm{SMEFT}}/v)}^{2}$. As already pointed out, at higher orders in h/v, the existence of the SMEFT requires that the odd derivatives of ${ \mathcal F }$ at the symmetric point h* vanish.
The correlations from table 1 can then be obtained by matching the Taylor expansion of ${ \mathcal F }$ around such symmetric point hHEFT = h* with the expansion around our physical vacuum hHEFT = 0. Instead of that matching, one can also obtain the correlations by eliminating the SMEFT Wilson coefficients order-by-order. For example, at ${ \mathcal O }(1/{{\rm{\Lambda }}}^{2})$, there is only one operator, ${{ \mathcal O }}_{H\square }$, in equation (10), that controls all the HEFT coefficients of ${ \mathcal F }$:
The elimination, by substitution, of this cH□ coefficient from the HEFT ais yields the 1/Λ2 correlations of the second column of table 1. Proceeding to the next 1/Λ4 order in the SMEFT expansion brings in the Wilson coefficient ${c}_{H\square }^{(8)}$. Hence, one can likewise extract the weaker 1/Λ4 correlations among the HEFT parameters. The potential V(hHEFT) is in turn also affected by ${{ \mathcal O }}_{H}$,
Also, the ci in ${ \mathcal G }({h}_{\mathrm{HEFT}})$ modifying the Yukawa coupling receive analogous contributions from both SMEFT coefficients cH□ and cuH in standard notation, the second alternatively named ${c}_{{{tH}}^{+}}$ in [17]. The correction
can be carried on to the higher coefficients using the relations in table 2 (with ${\rm{\Delta }}{a}_{1}=2{c}_{H\square }{v}^{2}/{{\rm{\Lambda }}}^{2}+{ \mathcal O }(1/{{\rm{\Lambda }}}^{4})$).
3. Conclusions
Various authors, see e.g. [18], have pointed out differences between the SMEFT and HEFT formulations [19]. For example, in the SMEFT the Goldstone ωi and Higgs hSMEFT bosons are arranged in a left-SU(2) doublet while in the HEFT hHEFT is an SU(2) ⨂ SU(2) singlet, independent of the Goldstone triplet ωi. In addition, in the SMEFT, the Higgs field always appears in the combination (hSMEFT + v) and, thus, the HEFT deploys more independent higher-dimension effective operators (in exchange, it is less model-dependent). This means that the SMEFT is natural when hSMEFT is a fundamental field while the HEFT is typical for composite models of the EWSBS (such as those with hHEFT as a Goldstone boson). Finally, the counting of the SMEFT is based in a cutoff Λ expansion taking the canonical operator dimensions, ${ \mathcal O }(d)/{{\rm{\Lambda }}}^{d-4}$ (independently of Nloops), whereas the HEFT is a derivative expansion (independently of Nparticles) like the older electroweak chiral Lagrangian, with ${ \mathcal F }(h)$ inserted in the derivative Goldstone term.
Nevertheless, a lot of this is cosmetic and can be reorganized by changing variables hSMEFT ↔ hHEFT. What is key is the San Diego criterion [3, 12]: ${ \mathcal F }({h}_{\mathrm{HEFT}})$ must have a point h* symmetric under the global SU(2) × SU(2) group and, due to its existence and convergence in the h field space, the SMEFT is deployable if and only if (which is a statement about the HEFT Lagrangian)
•
$\exists {h}_{* }\in {\mathbb{R}}$ where ${ \mathcal F }({h}_{* })=0$, and
•
because of the need for ${{ \mathcal L }}_{\mathrm{SMEFT}}$ analyticity, ${ \mathcal F }$ is analytic between our vacuum h = 0 and h*, particularly around h*. Moreover its odd derivatives vanish.
We have presented new relations that implement this criterion at ${ \mathcal O }(1/{{\rm{\Lambda }}}^{2})$ and ${ \mathcal O }(1/{{\rm{\Lambda }}}^{4})$ in the 1/Λ counting; more precision is unnecessary until (if) separations from the SM are found. Then only with the scale Λ at hand out of separations of effective field theory coefficients from the SM can we decide how relevant the corrections due to the higher orders are expected to be and whether further work is warranted.
Among the three types of correlations that we have presented in tables 1 and 2, those for the coefficients of ${ \mathcal F }$ are more interesting for large values of the energy $\sqrt{s}\gg {m}_{h}\sim {m}_{W}\sim {m}_{Z}$, whereas those for V and ${ \mathcal G }$, that do not involve Goldstone bosons, are therefore of greater interest at low energies, when the potential competes with the derivative operators on equal ground, as $\sqrt{s}\sim {m}_{i}$.
In conclusion, we have newly translated these conditions into correlations among HEFT coefficients whose violation falsifies the SMEFT. Moreover, since many extensions of the SM incorporating supersymmetry, supergravity or other possibilities can be cast as an SMEFT, they can be likewise simultaneously falsified.
For the time being, no separations from the SM have been found [20] and one can only infer direct experimental bounds on the first terms, a1 and, perhaps, a2, so we have to wait for data with a larger number of Higgs bosons before assessing them. However, when this is done, the correlations will allow us to falsify the SMEFT in experiments even without new particles. We believe that this possibility improves the standing of the SMEFT as a scientific theory.
Supported by Spanish MICINN PID2019-108655GB-I00/AEI/10.13039/501100011033 grant, and Universidad Complutense de Madrid under research group 910309 and the IPARCOS institute; ERC Starting Grant REINVENT-714788; UCM CT42/18-CT43/18; the Fondazione Cariplo and Regione Lombardia, grant 2017-2070: and by Grant DataSMEFT23 (EUNextGeneration—PNRR—DM 247 08/22).
DobadoAPeláezJ R1994 On the equivalence theorem in the chiral perturbation theory description of the symmetry breaking sector of the Standard Model Nucl. Phys. B425 110 136 [Erratum: Nucl.Phys.B 434, 475–475 (1995)]
CastilloADelgadoR LDobadoALlanes-EstradaF J2017 Top–antitop production from ${W}_{L}^{+}{W}_{L}^{-}$ and ZLZL scattering under a strongly interacting symmetry-breaking sector Eur. Phys. J. C77 436
AlonsoRJenkinsE EManoharA V2016 A geometric formulation of Higgs effective field theory: measuring the curvature of scalar field space Phys. Lett. B754 335 342
A combination of measurements of Higgs boson production and decay using up to 139 fb−1 of proton–proton collision data at $\sqrt{s}=$ 13 TeV collected with the ATLAS experiment, ATLAS-CONF-2020-027, communication by the ATLAS coll. to 40th ICHEP, Prague (2020) https://cds.cern.ch/record/2725733/files/ATLAS-CONF-2020-027.pdf
22
TumasyanA2022 Search for Higgs boson pair production in the four b quark final state in proton-proton Collisions at $\sqrt{s}$=13 TeV Phys. Rev. Lett.129 081802
Search for Higgs boson pair production in the two bottom quarks plus two photons final state in pp collisions at $\sqrt{s}=13$ TeV with the ATLAS detector, ATLAS-CONF-2021-016, contribution of the ATLAS coll. to the 55th Rencontres de Moriond (2021) https://cds.cern.ch/record/2759683/files/ATLAS-CONF-2021-016.pdf
24
de BlasJEberhardtOKrauseC2018 Current and future constraints on Higgs couplings in the nonlinear effective theory JHEP JHEP07(2018)048