$\boldsymbol{C\!P}\!$ violation in Higgs--gauge interactions: from tabletop experiments to the LHC

We investigate $C\!P$-violating interactions involving gauge bosons and the Higgs boson within an effective field theory approach, focusing on the six couplings that define the $C\!P$-violating sector of so-called universal theories. We compute the contributions of all relevant dimension-6 operators to electric dipole moments and the $C\!P\!$ asymmetry in $B\to X_s\gamma$, and compare the resulting current and prospective constraints to the projected sensitivity of the LHC. Low-energy measurements are shown to generally have a far stronger constraining power, which results in highly correlated allowed regions in coupling space-a distinctive pattern that could be probed at the high-luminosity LHC.


INTRODUCTION
To generate the observed matter-antimatter asymmetry in the Universe the Sakharov conditions [1] have to be satisfied. One of them requires that charge-parity (CP ) symmetry be violated. CP symmetry is broken in the Standard Model (SM) of particle physics with three generations of quarks, but only by the phase of the Cabibbo-Kobayashi-Maskawa (CKM) matrix and, potentially, the QCD θ term. The resulting amount of CP violation is, however, far too small to explain the observed matterantimatter asymmetry [2][3][4][5][6][7]. Scenarios of electroweak (EW) baryogenesis [8][9][10][11] demand new sources of CP violation not too far above the EW scale.
To address the above question we adopt an effective field theory (EFT) framework. We consider dimension-6 CP -violating operators within the SMEFT [32,33] that involve only the Higgs and gauge fields. In the Warsaw basis [33], these operators are defined as where ϕ is the Higgs doublet with ϕ = v/ √ 2, v 246 GeV, g s , g, and g are the SU (3) c , SU (2) L , and U (1) Y couplings, respectively, and G µν , W µν , and B µν the corresponding field strength tensors. We definẽ X µν = µναβ X αβ /2, with 0123 = +1. The Wilson coefficients C ϕX,X encode contributions from beyond-the-SM (BSM) physics scaling as 1/Λ 2 , where Λ is the BSM scale.
The six operators in Eq. (1) parameterize the CPviolating sector of so-called universal theories, 1 in which BSM particles couple only to SM bosons and/or to SM fermions only through the gauge and Yukawa currents, and thus, modulo field redefinitions, induce only bosonic operators at the scale Λ [37]. The SMEFT setup for the CP -conserving sector of universal theories and the effect of non-universal operators generated by the renormalization group (RG) flow have been studied in Refs. [37,38]. Here we focus on the CP -violating sector of universal theories, as this scenario provides a minimal consistent framework to study CP -violating Higgs-gauge interactions in the SMEFT context. Besides this practical motivation, these models are appealing because they can live at a relatively low-scale Λ (they satisfy minimal flavor violation [39,40] and generate CP -violating fermionic dipoles only through RG flow), a welcome feature for the viability of weak-scale baryogenesis.
The operators in Eq. (1) affect the cross sections of processes such as Higgs production via gluon or vector-boson fusion, Higgs production in association with EW gauge bosons, and Higgs decays, through non-interfering contributions quadratic in C ϕX and are thus suppressed by (v/Λ) 4 . Such dimension-8 contributions however still lead to significant constraints [20,41]. The Higgs-gauge operators contribute at O(v 2 /Λ 2 ) to CP -odd observables, such as the CP asymmetry in pp → h + 2j [21], angular distributions in associated HW and HZ production [20,41,42], or in h → 4l [43], while CW and C ϕW B contribute to CP -odd observables in diboson production [20,44]. CG gives tree-level corrections to pp → h + 2j and to multijet production [45]. In addition to these tree-level effects in collider observables, C ϕG , C ϕW , C ϕW B , C ϕB , CG, and CW contribute to lowenergy CP -violating observables, such as EDMs and the CP asymmetry in B → X s γ, at the loop level. In this Letter we set up the framework to include low-energy CP -violating probes and demonstrate that they put severe constraints on the CP -violating sector of universal theories. To establish the connection to existing collider bounds [21,46], we first concentrate the phenomenological analysis on the operators that involve the Higgs coupling, and later discuss the low-and high-energy input necessary for an analysis of all six parameters simultaneously.

RENORMALIZATION GROUP EVOLUTION
When the Higgs field acquires its vacuum expectation value, the operators in Eq. (1) generate θ-like terms by means of ϕ † ϕ → v 2 /2 + . . . , ϕ † τ i ϕ → −δ i3 v 2 /2 + . . . , where the dots denote terms that contain the Higgs scalar boson h. The parts of the operators in Eq. (1) that do not involve h can be absorbed in the SM θ terms. The U (1) Y and SU (2) L θ terms are unphysical because they can be removed by field rotations [47][48][49]. The gluonic operator effectively shifts the QCD θ term θ → θ − 16π 2 v 2 C ϕG , which is strongly constrained by the neutron EDM [50,51]. However, we will assume the presence of a Peccei-Quinn (PQ) mechanism [52] under which the total θ term vanishes dynamically, relieving C ϕG from its direct bound.
At energies below the EW scale, the Lagrangian contains flavor-conserving operators that induce leptonic and hadronic EDMs (L EDM ) as well as ∆B = ∆S = 1 operators that contribute to B → X s γ (L b→s ). 2 The leading contributions to these observables are generated 2 C ϕW B and CW induce b → dγ and s → dγ dipoles via the same diagrams that contribute to L b→s . The constraints from the direct CP asymmetry in B → X d γ and B → X s+d γ are, respectively, weaker than and degenerate with those from B → Xsγ. Similarly, constraints from CP -violating observables in kaon physics, such as K L → π 0 e + e − are not competitive.
by the following operators with where F µν and Z µν are the photon and Z field strengths, t a are the generators of SU (3) c , and Q f and T 3 f represent the electric charge and third component of weak isospin. The flavor-changing gluonic dipole O 8 is further suppressed by α em /(4π) with respect to C 7 , and can be safely neglected. We use e = −gs w = −g c w and t w = s w /c w , with s w = sin θ w , c w = cos θ w , and the weak mixing angle θ w . The Wilson coefficients of the dipole operators in Eq. (2) are complex, we write C The Higgs-gauge operators induce the operators in Eq. (2) through the diagrams shown in Fig. 1, giving the following matching conditions at with x t = m 2 t /m 2 W and loop functions [53] f One-loop diagrams involving Higgs-gauge operators that contribute to (gluonic) dipole operators. The red circles denote insertions of the SMEFT operators. The diagram on the right side also generates threshold corrections to flavor-violating dipole operators, such as O7. In addition to the above contributions, we take into account the RG evolution 3 between the BSM scale, µ = Λ, and the EW scale, µ = µ t . Furthermore,c g , and CG renormalize under QCD [56][57][58][59]. We evolve the lowenergy operators to the scale where QCD becomes nonperturbative, µ = Λ χ = 2 GeV, and take into account the bottom, charm, and strange thresholds, where the Weinberg operator obtains contributions analogous to the one in Eq. (4). The resulting fermion EDMs, chromo EDMs (CEDMs), and the Weinberg operator at the scale µ = Λ χ are given in Table I, where we assumed the initial scale µ 0 = Λ = 1 TeV. We notice that the weak operators C ϕB , C ϕW , C ϕW B , and CW contribute to the fermion EDMc   [50,51], mercury [61,62], xenon [63,64], and radium [65,66] EDMs in units of e cm (90% C.L.). The result for the CP asymmetry, AB→X sγ = 0.015 (20), is taken from Refs. [67][68][69].

LOW-ENERGY OBSERVABLES
Next, we discuss the connection of the coefficients in Table I to the most sensitive low-energy observables, starting with EDMs. Here, the most stringent limits are set by the neutron and 199 Hg atom, and by measurements on the polar molecule ThO. For the operators in Eqs. (1), the ThO measurement [60,70] can be interpreted as a probe of the electron EDM, d e = e Q e m ec (e) γ (Λ χ ) with a small theoretical uncertainty of O(15%) [71,72]. In contrast, nucleon EDMs receive contributions from several operators, with varying levels of theoretical uncertainties. Up-and down-quark EDM contributions are known with few percent accuracy [73], while the contribution of the strange EDM is only known to be non-zero at the two-sigma level from lattice simulations [73,74]. Even less is known about the other operators; QCD sum-rule calculations determined the contributions from the upand down-quark CEDMs with roughly 50% uncertainty, while the strange CEDM is assumed to vanish in the PQ scenario [22,[75][76][77]. The Weinberg operator appears with the largest uncertainty, O(100%), based on a combination of QCD sum-rules [78] and naive dimensional analysis estimates [56].
Finally, 199 Hg (and other diamagnetic atoms such as 129 Xe and 225 Ra) receives contributions from nucleon EDMs [79], as well as from the CP -odd isoscalar and Xsγ, LEP, and future collider constraints discussed below.
We summarize the current experimental limits in Table II, which also shows the limits on systems that are not yet competitive, but could provide interesting constraints in the future. EDM experiments on 225 Ra and 129 Xe atoms have already provided limits [63,65] and are quickly improving. In addition, plans exist to measure the EDMs of charged nuclei in electromagnetic storage rings [85], such as those of the proton and the deuteron. The EDM measurements of such light nuclei can more reliably be interpreted in terms of BSM operators than is the case for d Hg as the needed calculations are under much better theoretical control [86,87].
In addition to EDMs, the operators OW and O ϕW B contribute to the CP asymmetry in B → X s γ and to CP -odd triple-gauge couplings that were probed at LEP. Concerning the B → X s γ asymmetry, we employ the expressions derived in Ref. [88] and take the required SM Wilson coefficients, as well as the hadronic parameters, from the same work. The triple-gauge vertices induced by OW and O ϕW B are of the form W + W − γ and W + W − Z, which were constrained using angular distributions in e + e − → W + W − [89,90]. In the notation of Ref. [91] we have,λ Z =λ γ = −2m 2 W CW and κ Z = −t 2 wκγ = 4t 2 w m 2 W C ϕW B , which leads to [92] v 2 C ϕW B = −0.93 +0.47 −0.31 , v 2 CW = 0.42 (33) . (6) ANALYSIS To constrain the Higgs-gauge operators, we use the limits on the EDMs of the electron, neutron, and mercury, and the CP asymmetry in B → X s γ as listed in Table II, as well as the LEP constraints on triple gauge couplings given in Eq. (6). Nuclear and hadronic EDMs as well as the CP asymmetry are affected by significant theoretical uncertainties. Here we follow Ref. [29] and present limits in a variety of cases: (i) the "central" scenario, in which we neglect all hadronic and nuclear uncertainties, (ii) the "Rfit" strategy, in which all hadronic and nuclear matrix elements are varied within their allowed ranges to minimize the χ 2 value, and (iii) the "Gaussian" strategy, in which the theoretical errors are treated in the same way as statistical errors are. This last strategy provides a realistic estimate of the impact of the theoretical errors when these are under control. We start by discussing the limits derived in the central case, which reflects the maximal constraining power of the low-energy measurements, assuming a single operator is present at the scale µ = Λ. We subsequently consider the impact of the theoretical uncertainties in the Rfit scenario, as well as a scenario in which multiple Higgs-gauge operators appear at the scale Λ.
Turning on a single operator at the scale Λ, we see from Table III that the low-energy limits are very stringent. The bounds on the operators with EW gauge bosons are dominated by the electron EDM, which constrains v 2 C ϕW ,ϕB,ϕW B,W to be O(10 −6 ), corresponding to a BSM scale of ∼ 100 TeV, assuming C i = 1/Λ 2 , or 10 TeV, including a loop factor, C i = 1/(4πΛ) 2 . The constraints from the neutron and 199 Hg EDMs are weaker, at the permille level for v 2 C ϕW and v 2 C ϕW B and at the percent level for v 2 C ϕB,W . The bounds on C ϕG and CG are dominated by the mercury EDM in the central case. For both operators, the large uncertainties on the matrix element of the Weinberg operator imply that the constraints weaken by an order of magnitude and become dominated by the neutron EDM when moving from the central to the Rfit strategy. In contrast, the limits on the EW operators are very similar when using the Rfit strategy, as they are dominated by the ThO measurement, which is not subject to large theoretical uncertainties. The fourth column of Table III shows the collider limits from Refs. [21] and [44] for comparison. 6 These highenergy probes are less sensitive by four to six orders of magnitude for most of the couplings, while they are com-   [21], while the lowenergy limits assume improved matrix elements and future EDM measurements as described in the text. All four couplings were turned on at the scale Λ = 1 TeV, and the lowenergy limits were obtained using the Gaussian strategy for the theoretical uncertainties.
petitive with the EDM constraints on v 2 C ϕG in the Rfit approach.
To see the effects of turning on multiple operators at the scale Λ, we investigate a scenario in which all Higgs-gauge couplings are present at µ = Λ, while keeping CG ,W (Λ) = 0. This allows us to directly compare the low-energy limits to those of Ref. [21], where the constraints from pp → h + 2j were analyzed in the same scenario. In this case there is one free direction left unconstrained by EDM measurements, even when neglecting theoretical uncertainties. For our choice of µ 0 = 1 TeV, this combination of couplings is given by ∼ 0.17 C ϕB + 0.86 C ϕW + 0.48 C ϕW B . As a result, EDM measurements alone are not sufficient to constrain all four dimension-6 operators simultaneously and the CP asymmetry in B → X s γ and LEP observables are needed to close the free direction. In addition, when treating the theoretical uncertainties in the Rfit or Gaussian approach, the constraints from d Hg and d n are degenerate, leading to another free direction. These free directions can be closed by reducing the errors on the theoretical predictions of matrix elements, or by considering improved constraints on the EDMs in Table II and bounds on the EDMs of additional systems, such as the proton or deuteron. Improvements on these three fronts are expected on the same timescale as the LHC Run III and the high-luminosity LHC, for which the limits in [21] were derived.
We therefore consider improved determinations of the matrix elements that were set as targets for the future in Ref. [27]. We assign 25% uncertainties to the nucleon EDM induced byc (u,d) g , and 50% uncertainties on the nucleon EDM from CG, the πN couplings fromc (u,d) g , and the nuclear structure matrix elements in d Hg , d Xe , and d Ra fromḡ 0,1 . Note that these uncertainty goals are by no means unrealistic considering recent lattice and nuclear-theory efforts [93][94][95][96], and in some cases have already been attained [97]. On the experimental side, we assume |d n | < 1.0 · 10 −27 e cm, which will be probed at the PSI and LANL neutron EDM experiments [98,99], and |d Ra | < 10 −27 e cm, well within reach of the ANL radium EDM experiment [66]. On a longer time scale, storage ring searches of the EDMs of light ions have the potential to compete with the neutron EDM [85], and we assume d p , d d < 1.0 · 10 −27 e cm. For the CP asymmetry in B → X s γ, Belle II will be sensitive to sub-percent values, |A B→Xsγ | < 4 · 10 −3 [100].
A comparison of the projected limits of Ref. [21] to the combination of future EDM and B → X s γ limits in the C ϕW -C ϕG and C ϕW -C ϕB planes are shown in Fig. 2 and in Table IV. Given our assumptions on the matrix elements, a Gaussian treatment of the theoretical uncertainties becomes justified. Marginalizing over the couplings for which the explicit dependence is not shown in the figure, the non-zero central values for the low-energy curves are driven by the LEP bound (6) on C ϕW B , which deviates from zero by ∼ 2σ. The gray, orange, and purple bands assume the proposed differential measurements in pp → h+2j have been performed on 36, 300, and 3000 fb −1 of integrated luminosity, respectively, while the red band shows the limits from low-energy experiments. The figure shows that the collider observables could in principle probe the C ϕW and C ϕB couplings at a comparable level as the low-energy limits with 36 fb −1 and 3000 fb −1 of data, respectively, but become relevant only when delicate cancellations between different couplings occur. In addition, the low-energy constraints on the gluonic operator C ϕG , are expected to be more stringent than the projected limits from the high-luminosity LHC by roughly two orders of magnitude, see Table IV.
The strong constraints that EDM experiments put on the parameter space will manifest themselves in correlations between observables at the LHC. For example, the electron EDM bound establishes correlations between C ϕW B , C ϕW , and C ϕB , as can be seen from the right panel of Fig. 2. An observation of large CP violation in the Higgs-gauge sector, of the size of the right column of Table IV, would then require a non-zero value for C ϕW B . In such a scenario one would therefore expect large effects in diboson production, induced by C ϕW B , to be consistent with EDM experiments.
We can finally relax the assumption CW ,G (Λ) = 0, and consider all the CP -violating operators expected in the framework of universal theories. As argued above, the dominant EDM constraints are only sensitive to two linear combinations of the weak couplings C ϕB , C ϕW , C ϕW B , and CW , so that EDM experiments could, in total, provide four independent constraints on the six operators in Eq. (1). As shown in this Letter, one possible strategy to close the open directions in parameter space relies on the CP asymmetry in B → X s γ and/or LEP observables, but of course complementary LHC measurements would provide the remaining two constraints as well. In either case, just as for the four-dimensional set of operators, one again expects strong correlations between CP -violating observables in the Higgs and weak boson  [21] in the C ϕW -C ϕG and C ϕW -C ϕB planes. The remaining couplings are marginalized over and the Gaussian strategy for the matrix elements is used.
sectors, which illustrates the enormous potential of the low-energy probes in constraining the CP -odd sector of universal theories.

CONCLUSIONS
In this Letter, we have analyzed the ability of lowenergy CP -violating probes to constrain Higgs-gauge CP -violating interactions. We have illustrated the impact of EDMs, and compared to current and future LHC sensitivities. Our work shows that EDMs cannot be neglected-in fact in a single-operator analysis they leave very little room for observing CP violation in the Higgs sector at the LHC. In a global analysis, flat or weakly bound directions from low-energy constraints are possible, and therefore, in order to fully probe CP -violating Higgs-gauge interactions in the context of universal theories, the inclusion of both EDMs and LHC measurements is mandatory.
Finally, we note that several lessons from our analysis extend beyond universal theories, where more CPviolating effective couplings appear. In this case EDMs enforce strong correlations among Higgs-gauge and other CP -violating couplings, which barring intricate cancellations lead to strong bounds on the individual couplings.