Global Constraints on Yukawa Operators in the Standard Model Effective Theory

CP-violating contributions to Higgs--fermion couplings are absent in the standard model of particle physics (SM), but are motivated by models of electroweak baryogenesis. Here, we employ the framework of the SM effective theory (SMEFT) to parameterise deviations from SM Yukawa couplings. We present the leading contributions of the relevant operators to the fermionic electric dipole moments (EDMs). We obtain constraints on the SMEFT Wilson coefficients from the combination of LHC data and experimental bounds on the electron, neutron, and mercury EDMs, and for the first time, we perform a combined fit to LHC and EDM data allowing the presence of CP-violating contributions from several fermion species simultaneously. Among other results, we find non-trivial correlations between EDM and LHC constraints even in the multi-parameter scans, for instance, when floating the CP-even and CP-odd couplings to all third-generation fermions.


Introduction
Charge-Parity (CP) violating contributions to Higgs-fermion couplings are a well-motivated possibility of physics beyond the standard model (SM) that might help address the problem of baryogenesis with new dynamics at the electroweak scale. As is well known, any such contributions are strongly constrained by null measurements of electric dipole moments (EDMs), and less strongly by Higgs production and decay data from colliders. It is also well known that the presence of several CPviolating phases can lead to cancellations and thus weaker bounds [1]. e interplay of EDM and collider constraints, in the presence of several CP-violating Higgs couplings to fermions simultaneously, is less well-known. In Ref. [2] constraints in the presence of two CP-violating parameters have been studied in detail. In this work, we scan over up to six di erent parameters.
Frequently, model-independent bounds on such interactions have obtained in the so-called "κ framework" by rescaling the SM Yukawa coupling by an overall factor and a complex phase. In Ref. [3], bounds on these factors have been obtained by studying contributions to EDMs through Barr-Zee diagrams with internal fermion loops. In Refs. [4,5] it was shown by explicit calculation within the κ framework that bosonic two-loop diagrams have a large impact on the EDM bounds for light quarks. However, it was later pointed out that the way these bosonic contributions had been been computed lead to a gauge dependent result [6]. is is related to the fact the naive implementation of the κ framework, in which only the dimension-four Higgs couplings are modi ed, is not a consistent quantum eld theory. A consistent alternative that resembles the κ framework most closely would be to use Higgs e ective theory (HEFT) [7]. In HEFT, the dimension-four couplings are complemented by the necessary higher dimension interactions to facilitate gauge-independent results.
Another consistent way of obtaining model-independent bounds for CP-violating Higgs-fermion interactions is the use of the SM e ective eld theory (SMEFT) [8]; see Refs. [2,[9][10][11] for recent work in this direction. We take the same approach in this article.
For the purpose of this work, we assume that the contributions to the EDMs of the SM fermions arise from physics at a scale Λ that is su ciently higher than the electroweak scale. is permits us to parameterize deviations from the SM in terms of operators in SMEFT. We will assume that any of the operator of the form (H † H)F L f R H may have non-zero coe cients. Here, H is the complex Higgs doublet eld, F L a le -handed doublet fermion eld, f R a right-handed singlet fermion eld, and we have suppressed avour indices. e reason for focussing on this class of operators is that they are the only dimension-six operators that induce tree-level modi cations to Higgs-fermion couplings. See Sec. 2 for a detailed discussion. ese operators will contribute to leptonic and hadronic dipole moments via a series of matching and renormalization-group (RG) evolution. Our analysis takes into account the leading e ects that arise at two-loop order in the electroweak interactions, as well as leading-logarithmic QCD corrections below the electroweak scale. We neglect any e ects of CP-odd, avour o -diagonal operators. e study of these e ects is relegated to future work.
To understand the complex interplay between the operators we consider requires us to vary multiple Wilson coe cients simultaneously. To explore this computationally challenging multidimensional parameter space, we make use of the GAMBIT [12] global ing framework, to which we have added a new module that allows for calculations of EDMs in this EFT, as well as the corresponding experimental likelihoods. We have also expanded the existing ColliderBit [13] module to be able to determine constraints on the Wilson coe cients from measured properties of the Higgs boson at the LHC.
is work contains several new aspects and presents nontrivial results. For the rst time, we perform a multi-parameter t to both LHC and EDM data. is corresponds to a more realistic scenario than single-parameter ts, as in a UV-complete model several CP phases are expected to be present. Moreover, we complement and correct some of the analytic expressions in the literature. In addition to updating existing constraints on either individual CP-odd Yukawa couplings (Figs. 4 and 5), or two CP-odd Yukawa couplings as in Ref. [2] (Fig. 6), we scan over up to six CP-even / CP-odd Yukawa couplings simultaneously for the rst time . While this represents still only a subset of all Yukawa operators, we argue that with current experimental results, including more parameters will not lead to additional signi cant constraints. For instance, allowing for any CP-odd contribution to the Yukawa of a light fermion (electron, or up and down quark) would allow to fully cancel the corresponding EDM constraint, thus leaving only the LHC bounds in a trivial way. However, we nd several nontrivial e ects. Focusing on the heavier fermions, we nd an intricate interplay between LHC and precision constraints that weakens the EDM bounds without li ing them fully. Not even by allowing for six independent parameters can we cancel all EDM constraints among the third-generation fermions, and nontrivial EDM bounds remain. A detailed discussion of these issues is found in Sec. 7.
In our analysis, we do not take into account perturbative and hadronic uncertainties. e e ects of hadronic uncertainties on EDM bounds have been studied in detail in Ref. [2] (see also Ref. [14]), and will not be repeated here. ey are relevant mainly for bounds on CP-violating Higgs couplings to the light quarks that arise from hadronic EDMs, as collider bounds are nearly absent. When allowing for the presence of several Wilson coe cients at the same time, EDM bounds tend to get canceled and only collider bounds remain, so hadronic uncertainties are less relevant. Note also that bounds from arising from the electron EDM have no hadronic uncertainty. However, there is a large, previously overlooked perturbative uncertainty regarding the bo om and charm couplings, as discussed in Sec. 3. is uncertainty is analogous to the case discussed in Ref. [15], and will be studied in detail in a forthcoming publication [16]. As not all relevant higher-order corrections are currently known, they are neglected in this analysis.
is paper is organised as follows. In Section 2 we specify the operators that we want to constrain; namely, those that modify the SM Yukawa couplings at tree level. We then derive the modi ed Higgs-fermion couplings in the broken electroweak phase. In Sec. 3 we discuss the e ective theory valid below the electroweak scale. e analytic contributions of the SMEFT operators to the partonic EDM of leptons and quarks are collected in Sec. 4. Most of these results are taken from the literature, although a few results are presented here for the rst time. Furthermore, we correct several errors in the literature. In Sec. 5 we summarize the contributions of the partonic EDMs to the EDM of the physical systems that are actually constrained by experiment. Sec. 6 describes the collider constraints on the SMEFT operators, and contains a short discussion of the expected contribution of operator with mass dimension eight. Our main results are contained in Sec. 7. We conclude in Sec. 8. App. A contains the full generic R ξ -gauge Lagrangian in the broken phase. In App. B we discuss the alternative avour basis in the electroweak broken phase that has been used in Ref. [17].

E ective theory above the electroweak scale -SMEFT
In this work we consider the SM augmented with SMEFT operators that induce tree-level modi cations to the Yukawa couplings. In the unbroken phase, the relevant part of the Lagrangian reads Here, u R , d R , R denote the triplets (in generation space) of right-handed up-, down-, and chargedlepton elds, while Q L and L L are the corresponding le -handed triplets. In accordance, the Yukawa matrices Y u , Y d , Y and Wilson coe cients C uH , C dH , C H , are generic complex 3 × 3 matrices. e primes indicate that they are not necessarily couplings in the mass-eigenstate basis. Phases in the Yukawa couplings and Wilson coe cients can potentially induce beyond-the-SM (BSM) CP-violating contributions to Higgs-fermion couplings. As is well-known, not all these complex parameters are physical, due to the freedom of choosing the phases of the le -and right-handed fermion elds. To isolate the physical BSM parameters, we express as many parameters as possible in terms of observed quantities, e.g., the fermion masses and CKM matrix elements. To this end, we rewrite the Lagrangian in the broken phase using the linear decomposition of the Higgs eld as e terms that induce the fermion masses are In analogy to the SM, we diagonalise the complete matrix in parentheses by biunitary transformations, parameterised by the eld rede nitions with f = u, d, , and U f , W f complex 3 × 3 matrices chosen such that a er this rotation, the mass Lagrangian is Here, the y SM f are real and diagonal matrices with entries that correspond to the observed fermion masses, i.e., e kinetic terms of the fermions are also a ected by the transformation in Eq. (4), which leads to the appearance of the CKM matrix V CKM ≡ U † u W d in the charged gauge interactions of the le -handed quarks, in complete analogy to the SM. e interaction terms of the fermions with a single Higgs eld from Eq. (1) a er performing the transformation in Eq. (4) are given by where we de ned We see that, in the most general case, we can parametrise any deviation with respect to the SM by the rotated Wilson coe cients C f H . For later convenience, we split these coe cients into their real and imaginary parts, with C f H±,ij real. is implies that the operator combinations proportional to the diagonal terms C f H±,ii are self-adjoint.
In the current work we consider the e ect of the avour-diagonal contributions, C f H,ii , for the case that they contain additional sources of CP violation. is implies, as we shall see, that C f H−,ii is non-zero. Note, however, that in the absence of any type of avour alignment with the SM Yukawas these operators also induce avour violation beyond the SM, which we do not consider in this work. In a concrete model, we thus expect additional to constraints from such avour o -diagonal contributions of C f H (see Ref. [18] for a recent discussion). It is, however, possible to obtain both CP-even and CP-odd modi cations in the avour-diagonal Yukawas without any avour-violation beyond the SM.
is corresponds to a UV scenario in which the unitary transformations in Eqs. (6) and (4) that rotate to the mass eigenstates simultaneously diagonalise the coe cients C f H . In this setup C f H is diagonal but not necessarily real, meaning that CP violation beyond the SM is still possible.
In unitarity gauge, all BSM e ects in our setup are contained in the Yukawa Lagrangian in the mass-eigenstate basis Here, the sum runs over all charged fermion elds, f = u, d, s, c, b, t, e, µ, τ , and we have traded the generation indices on the Wilson coe cients in favour of the avour label, f , since we focus on the avour-diagonal case. We will use this more compact notation in the remainder of this work. In App. A we present the corresponding Lagrangian for generic R ξ gauge. A di erent basis has been used in Ref. [17] to present the bounds, see App. B. O en, the κ framework is employed to parametrise avour-diagonal CP-violation in the Yukawas [3,5,15]. e corresponding Lagrangian reads with κ f a real parameter controlling the absolute value of the modi cation and φ f ∈ [0, 2π) a CPviolating phase, such that κ SM f = 1 and φ SM f = 0 reproduces the SM. Eq. (10) should be thought of as the dimension-four part of the corresponding HEFT Lagrangian in unitarity gauge. As long dimension-six SMEFT is a good approximation, and the h 2 and h 3 interactions in Eq. (9) do not a ect the observables considered, the bounds on C f H± directly translate to bounds on the κ-framework parameters via is is actually the case for the observables and the precision that we consider. A possible exception is the top contribution to the LHC observables, see the discussion in Sec. 6.
Using the Lagrangian in Eq. (9) we will be able to set constraints on the C f H± couplings from LHC measurements. Low-energy probes of CP-violation, however, also place signi cant constraints on these couplings. Our aim is capture the numerically leading e ects. To this end we work with a tower of e ective theories. Apart from a few exceptions that we discuss below, the RG evolution from Λ to µ ew within SMFET can be neglected as (most) Yukawa operators do not mix into the CP-violating dipoles for the electron and light-quarks, relevant for EDMs. In this case the leading e ects are obtained at the electroweak scale by matching to the e ective theory in which the heavy degrees of the SM are integrated out, and by subsequently running to the hadronic scale. In contrast, the few contributions that are induced from (two-loop) mixing within SMEFT will come with a UV-sensitive logarithm, log M h /Λ.

E ective theory below the electroweak scale
We will extract constraints on the coe cients in Eq. (9) from the experimental bounds on the electron, neutron, and mercury EDMs. We will make the assumption that these EDMs are induced solely by the electron and partonic CP-violating electric and chromoelectric dipole operators, and the purely gluonic CP-violating Weinberg operator [19] with coe cients d e , d q ,d q , and w, respectively. erefore, we neglect (subleading) contributions from the matrix elements of four-fermion operators. Here, q collectively denotes light quarks, u, d, s. e charm-quark is typically not included since presently there are no la ice computations for the hadronic matrix elements of its operators. e above coe cients are traditionally de ned via the e ective, CP-odd Lagrangian valid at hadronic energies µ had 2 GeV [20], Here, σ µν = i 2 [γ µ , γ ν ], T a are the generators of SU(n c ) in the fundamental representation normalised as Tr[T a , T b ] = δ ab /2, n c = 3 is the number of colors, and we collect our conventions for the eld strength and its dual below. e contributions from the Weinberg operator turn out to be subdominant because of its small nuclear matrix elements [20,21], but we keep them for completeness. e partonic dipole moments d q andd q as well as the coe cient w are obtained from the SMEFT Wilson coe cients in Eq. (9) via a sequence of matching at the weak scale, and the bo om-and charmavour thresholds, as well as the RG evolution between each scale, as outlined in Refs. [3,15,[22][23][24]. e resummation of logarithms is phenomenologically relevant for the hadronic dipole operators that mix under QCD. In the current analysis, we work at the leading-logarithmic order for quark operators. QCD corrections are also large for the bo om-and charm-quark contributions to the electron EDM.
ey are currently unknown [16] and are therefore not included in our analysis.
To perform the sequence of matching and RG evolution connecting the electroweak-scale Lagrangian, Eq. (9), with the hadronic-scale one, Eq. (12), requires the full avour-conserving, CP-odd e ective Lagrangian below the electroweak scale up to mass dimension six. In the conventions and basis of Ref. [15], adapted from Ref. [23], it reads: where the sums run over all active quarks with masses below the weak scale, e.g., q, q = u, d, s, c, b in the ve-avour theory. e linearly independent operators are 1 operators in terms of the -tensor is convenient when going beyond leading-logarithmic approximation, as they remain selfadjoint also in d = 4 − 2 dimensions. It is thus convenient to also de ne O q 2 and the dipoles O q 3 and O q 4 in an analogous manner. To express the operators in the more conventional iσ µν γ 5 notation one can use the d = 4 relation iσ µν γ5 where 0123 = − 0123 = 1. For details concerning the next-to-leading-logarithmic analysis see Ref. [15].
e powers of e and g s in the operator de nitions above are xed such that all Wilson coe cients have the same dimension, thus making the calculation of operator mixing more transparent.
C w as well as some of the anomalous dimensions which control operator mixing depend on the conventions for the covariant derivatives for quarks and leptons, the eld strength tensor, and the dual tensors. Our conventions (same as in Ref. [15]) read with (not unimportantly) 0123 = − 0123 = 1. e comparison of Eqs. (12) and (13) gives the relation between d e , d q ,d q , w and the Wilson coe cients of the three-avour EFT evaluated at the hadronic scale µ had = 2 GeV: Note that C e 3 is scale independent with regard to the strong interaction, i.e., C e 3 (µ had ) C e 3 (µ ew ) with µ ew 100 GeV. e procedure to calculate C e 3 , C q 3 (µ had ), C q 4 (µ had ), and C w (µ had ) depends on the avour quantum numbers of the SMEFT operators. In the following section, we describe the di erent cases and present the corresponding results.

Electroweak matching and RG evolution to the hadronic scale
In this section we discuss how to obtain the Wilson coe cients of the EFT below the electroweak scale as a function of the Yukawa SMEFT Wilson coe cients. Since all operators in Eq. (13) are CP odd, all contributions are proportional to the CP-odd couplings C f H− . e nal results for the required initial conditions C qq 1 (µ ew ), C q 1 (µ ew ), C w (µ ew ), C e 3 (µ ew ), C q 3 (µ ew ), and C q 4 (µ ew ) are collected below in Eqs. (23), (24), (25), (26), and (27), respectively. ese are all the initial conditions required for the leading-log QCD analysis.
We begin the discussion with contributions that are UV sensitive, i.e., proportional to log(µ ew /Λ). ere are two equivalent ways of obtaining them. One way is to identify those SMEFT Yukawa operators O f H that mix into the SMEFT dipole operators O f W , O f B and O qG . e mixing induces dipole Wilson coe cient proportional to the leading UV logarithm log µ ew /Λ, and the subsequent tree-level matching onto the Lagrangian Eq. (13) potentially induces non-zero coe cients C e 3 (µ ew ), C q 3 (µ ew ), and C q 4 (µ ew ). In fact, only the following cases occur: the operator O eH mixes at the two-loop, electroweak level into the leptonic dipole operators Analogously, the Yukawa operators O qH with q = u, d, s mix into the quark dipole operators withĤ ≡ H for q = d, s andĤ ≡H for q = u. Here, σ a are the Pauli matrices acting in SU (2) L space. (Here, we considered only mixing into dipole operators with light fermions.) e two-loop mixing has been calculated for the electron case in Ref. [24]. An alternative way of obtaining these log µ ew /Λ terms is to directly perform the two-loop electroweak matching calculation to extract C f 3 (µ ew ), for the light-fermion avours f = e, u, d, s (see representative diagrams in the lower panels in Fig. 2). For these coe cients, the contributions to the matching proportional to the SMEFT coe cients C f H− are divergent and thus include a term proportional to the UV-sensitive logarithm log M h /Λ. We performed the matching calculation explicitly and nd that only the contributions to C f 3 (µ ew ) are UV sensitive, while the matching for C q 4 (µ ew ) is UV nite. To obtain gauge-independent results in this calculation, it was necessary [6] to include the dimension-ve vertices in the Lagrangian in the broken phase (see App. A) that were missed in Ref. [4]. e UV-divergent contributions are a direct consequence of including these dimension-ve vertices. e corresponding logarithmic terms in our results for C e 3 (µ ew ) are in agreement with the anomalous dimension presented in Ref. [24]. e UV-sensitive logarithms for the quark case are presented here for the rst time.
When electroweak UV logarithms are induced, the electroweak non-logarithmic terms in the matching at µ ew that accompany them are scheme dependent and formally part of the next-to-leadinglogarithmic (NLL) approximation. ey should not be included in the analysis and we thus disregard them.
For those contributions that are not UV sensitive, we directly perform the matching of the SMEFT Lagrangian Eq. (9) onto the e ective Lagrangian Eq. (13), integrating out the heavy degrees of the SM (top quark, Higgs, and the W and Z bosons). In all calculations we employ a general R ξ gauge for all gauge bosons and have veri ed the independence of our results of the gauge-xing parameters. All calculations were performed using self-wri en FORM [25] routines, implementing when necessary the two-loop recursion presented in Refs. [26,27]. e amplitudes were generated using QGRAF [28] and FeynArts [29].
At tree level and leading order in the 1/Λ expansion, integrating out the Higgs induces the following non-zero initial conditions for four-quark operators (see rst two diagrams in Fig. 1) with q, q = u, d, s, c, b. In principle, also the analogous operators with leptons are induced (qq¯ , ¯ ). However, since we will rely on a xed-order computation to predict d e ∝ C e 3 (see below), these operators do not enter our analysis. e matching at the electroweak scale also induces the Weinberg and dipole operators, but only at the two-loop level. e Weinberg operator obtains a two-loop initial condition at µ ew only from the top-quark operator (see third diagram in Fig. 1): with Φ(z) de ned in Eq. (39). is result agrees with Ref. [3], but is, as far as we are aware, presented here for the rst time in a non-parametric form. e electron photon dipole (O e 3 ), and light-quark photon and gluon dipoles (O q 3 , O q 4 with q = u, d, s) all receive two-loop initial conditions by integrating out the top quark and the Higgs, W , and Z bosons: with the precise decomposition of the coe cient functions A f , B q s , and B q ew discussed below. A few clari cations regarding Eqs. (25)- (27) are in order. We calculate the electroweak matching in xed-order perturbation theory whenever there are no QCD corrections that are enhanced by large logarithms related to light fermion masses. is is the case for all diagrams with top-quark loops, and the diagrams with lepton loops contributing to the electron dipole, i.e. the contributions C e 3 (µ ew ) that are proportional to C H− . (We also neglect the running of the electromagnetic coupling constant.) erefore, the coe cients C q 3/4 (µ ew ) do not receive threshold contributions from virtual quarks other than the top quark, since those are properly included via the RG-mixing of the four-quark operators (Eq. (23)) into C q 3/4 , see below. In principle, also the contributions to C e 3 (µ ew ) of all quarks other than the top should be obtained from the RG evolution; however, the corresponding mixed QED-QCD RG evolution is currently unknown 2 . us, we temporarily include the contribution of bo om and charm quarks to C e 3 (µ ew ) with a xed-order calculation, with the understanding that corrections to these results may be large. On the other hand, we do not include the corresponding xed-order results for the light-quark (u, d, s) contributions to C e 3 (µ ew ). In addition to the issue of the large unkown QCD corrections, these contributions are suppressed by the light-quark masses. erefore, it is not C e 3 but the C q 3/4 coe cients that provide the numerically leading constraints on the C qH− couplings for light quarks. Similarly, we do not include the contributions of the electron and muon to C q 3 (µ ew ). e appearance of their masses (that are much smaller than the lowest scale, µ had , where these operators can be meaningfully de ned) in the logarithms would spuriously enhance the corresponding bounds by large factors. ese contributions should be taken into account properly by using the RG. e tau, on the other hand, has a mass large enough to justify a xed-order calculation as an estimate and is thus included. Note, however, that numerically the by far strongest bounds on the coe cients of all three leptons arise from the electron EDM, and their contributions to the hadronic EDMs do not play 2 e QCD corrections to the bo om and charm contributions are naively estimated to be large (factor of a few in C e 3 ) but require a more complex calculation [16].  27) are the vector and axial-vector coupling of the Z boson to a fermion, respectively. In Fig. 2  . We have, however, kept the terms from top-quarks loops (proportional to L 1 and L 2 ) as they depend on the top-quark Yukawa and as such are independent of the scheme-dependent electroweak corrections. e loop functions (L i ) read where we de ned the mass ratios For the case of small fermion masses, i.e., when x 1, we have expanded for convenience the Φ(z) function. Our loop functions L 5 , L 6 , L 7 correct a global factor of √ 2 with respect to the corresponding ones in Ref. [5] and a typographical sign mistake in the x 3 hZ coe cient of Li 2 (1 − x hZ ) in L 5 in the same reference. e loop function L 2 implicitly corrects an error in Eq. (A.5) of Ref. [24], where two logarithms seem to have been incorrectly added. e function Φ(z) in the results above is given by [26] where λ ≡ 1 − 1/z, and the dilogarithm and Clausen function are de ned as respectively.
Having computed the initial conditions at the electroweak scale, we perform the QCD RGE evolution from µ ew to the hadronic scale µ had = 2 GeV (see Refs. [3,15] for details). For operators with quarks the RGE evolution resums large QCD logarithms and accounts for mixing among di erent operators. In the case of light-quark SMEFT operators, the tree-level induced four-quark operators (O qq 1 and O q 1 ) mix at one-loop under QCD into the quark dipoles. Nevertheless, the contributions to C q 3 (µ ew ) and C q 4 (µ ew ) of two-loop diagrams with top-quark and Z-boson loops provide the numerically dominant e ect [19] as the four-fermion operators are additionally suppressed by an extra light-quark mass (see Eq. (23)). e situation is di erent for contributions from bo om-and charm-quark SMEFT operators. Here the leading contribution to partonic dipoles (d q ,d q ) and the Weinberg operator (w) are induced by mixing during the RG evolution.
e main reason is that the nuclear matrix elements of bo om and charm dipole operators are tiny. erefore, diagrams like the ones in Fig. 2 do not contribute, i.e., bo om and charm quarks only enter as virtual particles in loop diagrams. At the same time, their mass is signi cantly below the electroweak scale. For this reason, a tree-level matching at the electroweak scale and the subsequent one-loop RG evolution [23], which must include the mixing of four-fermion operators into dipole and Weinberg operators, is su cient (and necessary) to obtain the leading-logarithmic result. e two-loop calculation has been performed in Ref. [15] but is not used in this work, as perturbative uncertainties and higher-order corrections are generally neglected in our analysis (see the discussion in Sec. 7.3).

Electric dipole moments
Non-zero coe cients of the higher dimension operators in Eq. (9) induce in general electric dipole moments in nucleons, atoms, and molecular systems via contributions to the hadronic Lagrangian in Eq. (12). Here we summarize the status of the induced dipole moments for the systems used in our t, namely, the electron, neutron, and mercury EDMs.
In addition to these EDMs there are also constraints from the experimental measurement of the muon EDM [30], as well as other systems with a hadronic component. e direct constraint from the current muon EDM measurement leads to constraints that are about six orders of magnitude weaker than the one obtained via virtual muon contributions to the electron EDM, and will thus not be used. Concerning other hadronic EDMs, we have checked that within our se ing the experimental bounds on the radium [31] and xenon [32] EDMs are not competitive with the neutron and mercury bounds and we do not include them in the t either.
is value was obtained using O molecules, neglecting any CP-violating electron-nucleon couplings. Bounds on the electron EDM have also been obtained using YbF [34] and HfF + [35]. e resulting bounds are currently not competitive with the O bound and are thus not used in our t.

Neutron
e simplest hadronic system used in our t is the neutron. e most recent experimental bound on the neutron EDM is [36] |d n | < 1.8 × 10 −26 e cm @ 90% CL .
e future projections estimate an improvement of the limit to |d n | < 10 −27 e cm [36]. roughout this work, we assume that the θ QCD term has negligible e ect on any EDMs and all e ects arise from the Yukawa SMEFT operators. e dipole moments of the partons then contribute to the neutron EDM as e hadronic matrix elements of the chromoelectric dipole operators (d q ) and the Weinberg operator (w) are estimated using QCD sum rules and chiral techniques [20,21,37]. For the matrix elements of the electric dipole operators (d q ) we use the la ice results [38] g u T = −0.204 (15), g d T = 0.784 (30), g s T = −0.0027 (16). e sign of the hadronic matrix element of the Weinberg operator is not known; to be de nite, we choose the positive sign in our analysis. Note that in our scenario the contribution of the Weinberg operator is always subdominant (either suppressed by small quark masses, or numerically small compared to the electron EDM bound), such that the sign ambiguity does not a ect the results of our global t.

Mercury
Signi cant constraints in our t arise also from the mercury EDM. e experimental bound is [39] |d Hg | < 7.4 × 10 −30 e cm @ 95% CL .
e relation of the partonic EDMs to that of mercury is given by [20] d Hg e = κ S g πN N a 0 eḡ Here, κ S = −2.8 × 10 −4 fm −2 denotes the contribution of the Schi moment to the mercury EDM, with an error not exceeding 20% [40]. e expression in square brackets is the Schi moment for mercury. e CP-odd isoscalar and isovector pion-nucleon interactions are given by g πN N = (5±10)× (d u +d d fm −1 , g (1) πN N = 20 +40 −10 × (d u −d d fm −1 , obtained from QCD sum-rule estimates [41], while g πN N 13.5 [42,43] is the CP-even pion-nucleon coupling. e contribution of these interactions to the Schi moment is given by the parameters a 0 = 0.01 e fm 3 and a 1 = ±0.02 e fm 3 . We took these "best values" from Ref. [20]; they have an intrinsic uncertainty of about an order of magnitude. e contributions of unpaired proton and neutron to the Schi moment has been calculated in Ref. [44], with result s p = 0.20(2) fm 2 and s n = 1.895(35) fm 2 . Finally, the contribution of the electron EDM is subdominant; no explicit uncertainty is given for the prefactor a e = 10 −2 [20,41]; however, di erent evaluations lead to di erent signs [45]. See Ref. [20] for a detailed discussion of all contributions to the mercury EDM. Employing "central values" for all parameters, we nd numerically with d n /e given in Eq. (43).
We have neglected several contributions in Eq. (45). e isotensor pion-nucleon coupling contributes in principle, but is expected to be small in comparison to the scalar and vector coupling, as it arises at higher order in chiral perturbation theory [20]. Contributions of four-fermion operators are smaller than the chromo EDM contributions by two orders of magnitude, and are also neglected. Finally, in our analysis we neglected the proton EDM contribution that is one order of magnitude smaller than the neutron EDM contribution, as well as the small electron EDM contribution. 6 Collider observables e SMEFT couplings in the Lagrangian in Eq. (9) do not only induce electric dipole moments, but also a ect the production cross sections and decay branching ratios of the Higgs boson. In this section we give an overview of the most important e ects; the actual numerical implementation of the LHC constraints is discussed in Sec. 7. e main e ects are captured by considering modi cations of the gluon fusion production cross section (gg → h) and the h → γγ branching ratio (both are one-loop induced both in the SM and in our setup), as well as of the branching ratios to fermions observed by ATLAS and CMS (h → bb, h → cc, h → τ τ , h → µµ), and the total decay width of the Higgs. It is convenient to parameterise these modi cations in terms of the parameters Using Eq. (11) we can readily translate r f,± to the κ-framework parameters, i.e., κ f cos φ f = 1 − r f,+ and κ f sin φ f = −r f,− . e deviation from the SM gluon fusion cross section or the decay to gluon-induced light jets can be e ectively captured by (see Ref. [3] for details) where we have de ned τ q ≡ 4 m 2 q /M 2 h − iε, the sum runs over all quark avours q = u, d, s, c, b, t, and the fermionic one-loop functions are given by Similarly, the modi cation of the Higgs decays into two photons with respect to the SM reads where the sums run over all charged fermions f with n c (f ) = 3 for quarks and n c (f ) = 1 for leptons, and the bosonic contribution is given by We keep the modi cations of the Higgs coupling to W W and ZZ bosons unmodi ed as the contributions from Yukawa operators are loop induced compared to the tree-level contributions already present at the dimension-four level. Moreover, h → Zγ is omi ed as its decay width is suppressed by a smaller coupling compared to h → γγ. e signal strengths of searches for Higgs decays to fermions are also a ected by the corresponding modi cations of the partial widths, namely: As of now, the ATLAS and CMS collaborations have observed Higgs decays into f = b, c, τ, µ. However, the modi cations of the partial widths to any fermion a ect the total width of the Higgs via which in turn a ects all its branching ratios, i.e., all the signal strength for Higgs searches. We conclude this section with some remarks concerning the convergence of the SMEFT expansion and the potential impact of dimension-eight operators. e issue arises because the squared amplitude enters the expression (52), leading to terms proportional to 1/Λ 4 in R f f that we keep in our numerics. What would be the impact of including dimension-eight Yukawa operators (of the generic form L HC qH q R ) in the amplitude? While this question is hard to answer quantitatively without actually performing the analysis, the following arguments suggest that the impact would result only in minimal changes on the bounds of the dimension-six coe cients.
First, note that this issue concerns only the real parts of the Wilson coe cients. As there is no interference with the SM in the CP-odd part of the amplitude, such dimension-eight terms would be of order 1/Λ 8 . e rst term on the right side of Eq. (52), on the other hand, would be changed to Here, C (6) and C (8) denote generic dimension-six and dimension-eight Wilson coe cients, respectively. We see that the additional last term in Eq. (54) is parametrically suppressed with respect to the linear dimension-six term by a factor v 2 /Λ 2 , and with respect to the quadratic dimension-six term by a factor m f /v. A potential problem arises if the term quadratic in C (6) dominates the t. A direct comparison between the last two terms in Eq. (54), under the "EFT assumption" C (6) ∼ C (8) , shows that the dimension-eight contribution will be subleading as long as C (6) > √ 2m f /v. We will see in Fig. 4 that this condition is clearly satis ed for all fermions apart from the top quark. While this is only a rough estimate, we interpret this as an indication that the impact of the dimension-eight terms on the t would be tiny. By contrast, Fig. 5 shows that for the top quark C tH+ 3, which is larger than √ 2m t /v ∼ 1, such that for values of C (8) ∼ 9 the dimension-eight term would contribute signi cantly to the t. While this required value of C (8) is close to the perturbativity limit, we conclude that the EFT expansion in the collider observables might not work as well for the top quark as it does for all other fermions. CP-odd contributions, on the other hand, are not expected to receive large corrections from dimension-eight operators for any of the fermions, as discussed above.

Global Analyses
Having obtained in the previous sections the expressions for the relevant observables, EDMs, Higgs production cross sections (σ), and branching ratios (BR) decay widths we now combine the constraints on the considered Wilson coe cients in a global analysis. For this purpose we use the GAMBIT global ing framework [12] to calculate a combined likelihood based on these two sets of constraints. e collider likelihoods are taken from the HiggsSignals 2.5.0 and HiggsBounds 5.8.0 codes [46,47] interfaced with the ColliderBit module [13] of GAMBIT.
HiggsSignals contains three modules to compute likelihoods from Higgs measurement at the LHC. Each module calls HiggsBounds to calculate the various production cross sections and branching ratios within our SMEFT model and to obtain the signal strengths implemented in each module. e rst module computes a likelihood of Run 1 Higgs measurements using a set of signal strengths provided by the ATLAS and CMS combination of Run 1 data [48]. In this module, the signal strengths are "pure channels" in the sense that one decay channel is combined with one speci c production channel, i.e., µ f i = (σ i BR f )/(σ i,SM BR f SM ) with i and f indicating di erent production modes (ggh, VBF, W h, Zh, tth) and decay channels (ZZ, W W , γγ τ τ , bb), respectively. Run 1 data are sensitive to 20 such signal strengths. e (Log)likelihood is then obtained from where µ are the vectors containing the 20 signal strengths computed as a function of the SMEFT Wilson coe cients, µ exp are the corresponding experimental combinations, and the superscript "T " denotes transposition. e matrix C cov is the signal-strength covariance matrix describing the uncertainties of and correlations among the signal strengths. For Run 2 measurements there are two HiggsSignals modules to compute a likelihood, one that uses the simpli ed template cross section measurements (STXS) [49] and another that uses the peak-centered method. In our analysis we mainly use the former since the peak-centered method increases the computing time by almost an order of magnitude and has also less constraining power, as we have checked explicitly. e only exception in which we do use the peak-centered-method module is to include the two h → µµ analyses containing in total 34 peak measurements [50,51]. Each signal strength is a combination of various production modes weighted by the corresponding experimental e ciency ( i ), which accounts for the detector performance in identifying signal events, i.e., In our analysis we include the signal strengths of 56 measurements originating from experimental searches from June 2021 [52][53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68][69][70][71]. e corresponding (Log)likelihood is then obtained in an analogous manner as for the Run 1 data in Eq. (55). e SM predictions for the cross sections are obtained through a t to the predictions from Yellow Report 4 [49].
To compute a likelihood from the EDM measurements, we assume their experimental uncertainties σ exp to be Gaussian distributed yielding the likelihoods (log L) with X = n, e, Hg. e uncertainties σ X,exp are obtained from the upper limits given in Eqs. (41), (42), and (44) by assuming zero central values for d exp X . e total likelihood of a parameter point is computed from the sum of the χ 2 values from the EDM and LHC likelihoods. To identify preferred regions in a subspace of the scanned parameters we must pro le over the remaining scanned parameters. We perform the pro ling using pippi [72], which computes the lowest possible χ 2 value of each parameter point in the subspace of interest by allowing all other parameters to oat simultaneously to minimize the χ 2 . When projecting onto a two-parameter subspace, the allowed regions at 68% and 95% CL correspond to the parameter space for which the di erence χ 2 − χ 2 best is less or equal than χ 2 68% ≈ 2.28 and χ 2 95% ≈ 5.99, respectively. χ 2 best is the χ 2 value of the best-t point, i.e., the lowest χ 2 value. In our case the χ 2 value of the SM (all SMEFT Wilson coe cients are zero) divided by the number of the included N d.o.f. = 79 observables is where we did not include the 34 peak observables from the h → µµ measurements; they only a ect the muon Yukawa. In the scan that we present the di erence between the χ 2 of the best-t point and the SM is always less than 0.1, so we do not display the best-t value in the plots.

One-flavour scans
To set the stage, we rst perform two-dimensional scans over the Wilson coe cients of each fermion individually, with the Wilson coe cients of all other fermions set to zero. e results are shown in Fig. 4. We allow the Wilson coe cients to oat within their perturbative values, |C f H± | ≤ 4π, and x Λ = 1 TeV in our scan. All numerical input parameters are taken from Ref. [73].
In this work, we parameterize the e ects beyond the SM by the SMEFT Wilson coe cients in Eq. (9). EDMs only constrain the imaginary part of the Wilson coe cients, C f H− , while LHC observables constrain also the real parts, C f H+ . In general, the electron EDM gives the strongest bounds on CPviolating parts of the lepton and heavy-quark (top, bo om, charm) coe cients, while the corresponding light-quark (up, down, strange) coe cients are mainly constrained by a combination of neutron and mercury EDM measurements.
• Top e constraints from the modi cation of Higgs production in gluon fusion have the parametric dependence µ gg = (1 − r t,+ ) 2 + r 2 t,− , where we used the asymptotic values A(∞) = B(∞) = 1, valid to good approximation for the top quark, and neglected the contribution of all other quark avours. In the same approximation, we have µ γγ = (1.0 + 0.27r t,+ ) 2 + 0.17r 2 t,− . ese are the dominant LHC constraints on the top couplings; note, however, that all production channels are included in our analysis. e corresponding constraints are shown in Fig. 4 (bo om le panel). e strongest individual bound on the CP-odd coe cient C tH− arises from the electron EDM, while the constraints arising from the hadronic systems are much weaker, see Fig. 4. A zoomed-in version of the combined t region is shown in Fig. 5.
• Bottom, charm, tau, muon None of the bo om, charm, tau, and muon EDMs themselves have been measured with high precision. Consequently, EDM bounds on CP-odd Higgs couplings of the fermions arise from their virtual contributions to the electron EDM (again, the bounds arising from hadronic systems are negligible in comparison). It is remarkable that, the presence of a large quadratic logarithm (see Eq. (33)), makes the electron EDM bound weaker than that on the top by only a factor 5 for the bo om, a factor 2 for the charm, a factor 6 for the muon, and of the same order for the tau, rather than being suppressed by the much larger factor Q 2 t m t /Q 2 f m f = O(50 − 500), expected naively [24] from the Yukawa suppression. For the bo om and charm quarks, this strong logarithmic enhancement indicates that QCD corrections are large in these cases and should be included [16], similar to the case of hadronic dipole moments [3,15] (see the discussion in Sec. 3).
Measurements at the LHC directly constrain the decay of the Higgs into bo om, charm, tau, and muon pairs. e main e ect comes from the modi cation of the partial h → ff widths (see Eq. (52)).
is is a simpli ed picture, as modifying any Yukawa also a ects the total Higgs decay width (see Eq. (53)), the h → γγ decay and, for quarks, also Higgs production via gluon-fusion. is e ect is largest for the bo om quark (that dominates the SM Higgs decay width); nevertheless, we include the e ect of all fermions on the total Higgs width. Note that a negative value for the bo om Yukawa (in the sense of Eq. (11)) is excluded at 68% CL (bo om middle panel in Fig. 4). Note also that for the muon Yukawa, the recent LHC measurements are more constraining than the electron EDM by an order of magnitude. Regarding tau couplings, the CMS analysis on the CP structure of the h → τ τ decay [74] disfavours large values of |C τ H− |. While this analysis is not included here, as it is not (yet) part of the HiggsSignals data set, it would split the 2σ LHC constraint (blue region in lower right panel of Fig. 4) in two distinct regions as illustrated in the CMS analysis [74].
• Electron, up, down, strange e main EDM bounds on the electron and light-quark (up, down, strange) SMEFT coe cients arise from their contributions to the electron EDM and the neutron and mercury EDMs, respectively, as discussed in Sec. 4. Note the di erent impact of the neutron and mercury EDMs on the up and down coe cients, due to the strong isospin dependence of the corresponding EDM predictions.
LHC bounds on the Wilson coe cients for the electron and the up and down quarks arise from modi cations of the total Higgs decay width Eq. (53). Indeed, we checked that the contributions to gluon fusion and h → γγ are subleading. (For modi cations of Higgs production induced by parton distribution functions, see Ref. [75].) However, the resulting constraints are very weak when compared to the SM Yukawas, as illustrated by the corresponding ratios |r e,+ | 4000, |r u,+ | 500, and |r d,+ | 225. In contrast, the bound on the CP-odd strange-quark coe cient from LHC is almost competitive with the hadronic EDM bounds (central panel in Fig. 4). Figure 6: Le : Constraints resulting from the combined 2D scan of CP-odd up and down Yukawas (C uH− , C dH− ). As the EDMs are not sensitive to the CP-even parts and the corresponding LHC constraints are weak we do not scan over those parameters. Right: Constraints resulting from the combined 4D scan of CP-even and CP-odd bo om and strange Yukawas (C bH± , C sH± ). Contours represent the allowed 68% and 95% con dence regions, the colour coding of individual constraints is given in the legend, and gray/black areas correspond to the combined regions. For more details see caption of Fig. 4.

Two-, three-flavour scans, and beyond
Next, we let the Wilson coe cients of more than one fermion avour oat simultaneously. is allows for the cancellation of the contributions of the considered Wilson coe cients to the constraining EDMs, thereby relaxing the bounds in certain regions of parameter space. ere are no such cancellations possible in the collider observables, because the main e ect comes from partial decay widths where no interference is possible. ( e small interference term between top-and bo om-quark contribution to Higgs production via gluon fusion and h → γγ is negligibly small.) However, the bounds on the di erent Wilson coe cients are still correlated. For instance, the Higgs decay into bo om quark dominates the total Higgs decay width, and thus the h → bb rate a ects all branching ratios signi cantly. e same is true, albeit to a lesser extent, for all other Higgs decays.
• Up and down In the le panel of Fig. 6 we show the results of a two-parameter scan over C uH− and C dH− . We do not scan over the corresponding CP-even Wilson coe cients as they do not enter the EDM predictions. e di erent dependence of the neutron and the mercury EDMs on the up-and down-quark coe cients is clearly visible. With the isoscalar pion-nucleon coupling (entering the mercury EDM prediction) being subdominant, the bands of the two observables are nearly orthogonal, thus allowing to set stringent constraints on both C uH− and C dH− .
• Bottom and strange In the right panel of Fig. 6 we show the results of a four-parameter scan over C bH± and C sH± a er pro ling over the CP-even couplings. As we neglect the tiny strange-quark contribution to the electron EDM, the only constraint on C sH− arises from the two hadronic EDMs. On the other hand, C bH− receives its dominant constraint from the electron EDM, while the contributions to the hadronic EDMs are also taken into account.
We include the CP-even Wilson coe cients in the scan as they are both bounded by LHC measurements. As there is no direct measurement of h → ss, the correlation between C sH− and C bH− results from the contributions to the total Higgs decay width. e analogous plots with CP-even Wilson coe cients do not contain any additional information compared to the one-avour scans and are thus not shown here.
• Top and bottom; top and tau; top and charm In Fig. 7 we show the results of three di erent four-parameter scans, oating C tH± simultaneously with C bH± ( rst column), with C τ H± (second column), and with C cH± (third column). e electron EDM bounds on C tH− could, in principle, be li ed by two orders of magnitude compared to the single-avour scan by choosing values close to the perturbativity limit for C bH− , C τ H− , and C cH− . However, given the LHC bounds on these parameters the bound on C tH− is weakened by only a factor of the order of ve. Note also that allowing C tH+ to oat signi cantly increases the allowed range of values for C bH+ at the 68% CL (upper le panel in Fig. 7). e relaxation of the bounds compared to the single-avour t is even more pronounced for the charm quark (upper right panel in Fig. 7). In the last row, we present the constraints on the parameters C tH+ and C tH− , with the other parameters pro led. is can be directly compared to Fig. 5, showing a signi cant relaxation of the bounds.
Interestingly, the combination of the electron EDM with LHC constraints has a profound impact on the constraint on the CP-even Wilson coe cients and not only on the CP-odd ones as one would naively expect. To understand this be er, we rst consider the second panel from above in the rst column of Fig. 7 showing the allowed C tH− -C bH− region. Here, the allowed combined region corresponds to the intersection of the electron EDM and the LHC constraint, which implies that for each allowed pair of (C tH− , C bH− ), it is possible to nd corresponding allowed values of C bH+ and C tH+ (which have been pro led out in the plot). is can easily be veri ed by looking at the bo om-le and bo om-center panels of Fig. 4. By contrast, the electron EDM on its own does not constrain the C tH− -C bH+ subspace at all, i.e., the whole parameter space of the third panel in the rst column of Fig. 7 (C tH− -C bH+ plot) is allowed with respect to the electron EDM. e reason being that one can always cancel the top against the bo om contributions to the EDM (see green band in the upper le panel). However, the combined EDM-LHC region is smaller than the one allowed by LHC alone, as not all values of C bH− required to cancel the contributions of C tH− are allowed by LHC bounds. In fact, the bo om single-avour analysis (bo om center panel of Fig. 4) indicates that roughly C bH− ≈ C bH+ , resulting in the much smaller allowed combined region in the 4D scan. Similar arguments apply to the plots that show the top-tau and top-charm coe cients. Regarding the case of the charm quark, note that its contribution to the electron EDM is larger than that of the bo om quark, while the LHC constraint is comparatively weaker, resulting in a larger allowed combined region. Finally, we remark that while the contribution to the electron EDM of the muon is similar to that of the bo om, the muon Yukawa is so strongly constrained by recent LHC measurement that no appreciable cancellation can occur, which is why we do not present this scan. Figure 7: Constraints resulting from a 4D scan of top and bo om coe cients (C tH± , C bH± ; rst column), top and tau coe cients (C tH± , C τ H± ; second column), and top and charm coe cients (C tH± , C cH± ; third column), assuming Λ = 1 TeV. In each plot only two parameters are shown, the remaining two are pro led over (see main text). Contours represent the allowed 68% and 95% con dence regions, the colour coding of individual constraints is given in the legend, and gray/black areas correspond to the combined regions. For details see main text and the caption of Fig. 4.
• Bottom and tau In Fig. 8 we show the results of a scan of the four parameters C bH± and C τ H± . As in the previous fourparameter scans, there is an interesting interplay between EDM and LHC bounds. When considering EDM bounds only, we can always cancel the constraint if either C bH− or C τ H− is pro led. Hence, there are no pure EDM constraints in any but the upper center panel where the CP-odd coe cients C bH− and C τ H− are displayed. In contrast, the EDM constraints cannot always be satis ed if also LHC constraints are included. In the upper right plot with the two tau coe cients displayed, one can see that the allowed, combined region is enlarged compared to the single-avour scan. However, for extreme values for C τ H− , still allowed by LHC bounds, C bH− cannot be pro led such that it compensates the large tau contribution to the electron EDM and simultaneously still be within the 2σ-level bo om LHC bounds. In contrast, C τ H− can be pro led such that the C bH− coe cient in the upper le plot is only bounded by LHC constraints. e reason for this is apparent from Eq. (25): the contribution of the tau to the electron and quark EDMs are larger than those of the bo om by about a factor 3m τ log 2 (m τ /M h )/m b log 2 (m b /M h ) ≈ 2, meaning that the bo om contribution can always be fully canceled by τ contributions, but not vice versa. is implies that, given current data, the combined bounds (apart from the combination (C bH+ , C bH− )) are more stringent than either the LHC bounds or the EDM bounds alone. e e ect of the electron EDM, further restricting the parameter regions allowed by LHC data, is also clearly visible in the combined region in the lower center panel that shows the bounds on the two CP-even coe cients C bH+ and C τ H+ .
• Charm and tau In Fig. 9 we present the results of a scan of the four parameters C cH± and C τ H± . e results are analogous to the case of bo om and tau discussed above. Note that the contribution to the electron EDM of the charm quark is larger than that of the bo om quark by a factor of roughly , while the LHC bounds on the charm quark are considerably weaker that those on the bo om quark.
• Top, bottom, and tau (third generation) In Fig. 10 we present a scan of all six third-generation parameters C bH± , C τ H± , and C tH± . It is interesting to compare the results to the case in which only two out of the three avours were included in the t (Fig. 7).
First, we focus on the set of the four panels (upper and middle row, center and right) that show the same parameter combinations as the panels at the same positions in Fig. 7. We see that a er pro ling the remaining third-generation couplings, an "indirect" electron EDM constraint on C tH− remains, although the allowed region is now signi cantly larger. By contrast, the "indirect" electron EDM constraint can be li ed completely by C tH− (that is only weakly constrained from LHC measurements) in the panels in the bo om row and center le of Fig. 10, leaving only the LHC constraints. is should be compared to the corresponding much smaller combined regions in Fig. 8. e top le panel of Fig. 10 shows the constraints on both top Wilson coe cients, C tH± . Notice that the allowed values increased by about a factor of two for C tH+ and by one order of magnitude for C tH− compared to the single-avour scan (Fig. 5). e large computational resources required for scans with more than six parameters prevent us from scanning over more than three avours. Nevertheless, we can infer some results in this direction from the one-, two-, and three-avour scans above. Speci cally, we will consider how the constraints on the third-generation Wilson coe cients change when including more avours in the scans. Figure 8: Constraints resulting from a 4D scan of bo om and τ Wilson coe cients (C bH± , C τ H± ) assuming Λ = 1 TeV. In each plot only two parameters are shown, the remaining two are pro led over (see main text). Contours represent the allowed 68% and 95% con dence regions, the colour coding of individual constraints is given in the legend, and gray/black areas correspond to the combined regions.
For details see main text and the caption of Fig. 4. Figure 9: Constraints resulting from a 4D scan of charm and τ Wilson coe cients (C cH± , C τ H± ) assuming Λ = 1 TeV. In each plot only two parameters are shown, the remaining two are pro led over (see main text). Contours represent the allowed 68% and 95% con dence regions, the colour coding of individual constraints is given in the legend, and gray/black areas correspond to the combined regions.
For details see main text and the caption of Fig. 4.
• Top, bottom, (electron, light quark) As an example, we oat C tH± and C bH± under the presumption that any contributions to EDMs can be compensated by appropriate values of the coe cients C eH− and C qH− , q = u, d, which we, however, do not scan over. In this way only LHC bounds remain, which are one to two orders of magnitude weaker for C tH− with respect to the electron EDM. e results are shown in Fig. 11. We nd that the LHC constraints on C tH+ and C bH+ are somewhat li ed compared to the single-avour scans due to the large contribution of the bo om to the total Higgs width, while the absence of EDM bounds allows for much larger allowed ranges of C tH− and C bH− . e analogous results for the charm instead of the bo om couplings are shown in Fig. 12. More generally, the CP-violating up, down, and electron Wilson coe cients are merely and severely constrained by EDM measurements. Hence, including C eH− , C uH− , and C dH− in a scan over the Wilson coe cients of the second or third generation would allow to completely cancel any EDM constraints and again only LHC constraints would remain. Figure 10: Constraints resulting from a 6D scan of top, bo om, and τ Wilson coe cients (C tH± , C bH± , C τ H± ) assuming Λ = 1 TeV. In each plot only two parameters are shown, the remaining ones are pro led over (see main text). Contours represent the allowed 68% and 95% con dence regions, the colour coding of individual constraints is given in the legend, and gray/black areas correspond to the combined regions. For details see main text and the caption of Fig. 4.

Theory uncertainties
In this work, we did not study the impact of theoretical uncertainties on the bounds on the Wilson coe cients. Hence, a short discussion of these e ects is in order. e relevant uncertainties are: (i) uncertainties in the hadronic matrix elements; and (ii) perturbative uncertainties.
(i) e uncertainties on the hadronic matrix elements have been shown in Sec. 5. Note that, in addition to the ranges given for the parameters, also some of the relative signs are not determined. In our numerical analysis, we have taken the central values for the hadronic matrix elements, and (somewhat arbitrarily) chosen the positive signs where they were not determined. We did not include these uncertainties in our likelihood function. In fact, no bounds at the 68% CL would result from the mercury EDM, while the neutron EDM bounds would get weaker by about a factor 2. At 95% CL, there would also be no bound from the neutron EDM. (In Ref. [2], much smaller e ects of the hadronic uncertainties have been found. is may be related to the statistical treatment of the uncertainties; in our computation we take uncertainty on the EDMs from hadronic input, σ 2 had , to scale with the Wilson Figure 12: LHC constraints resulting from a 4D scan of top and charm Wilson coe cients (C tH± , C cH± ) assuming Λ = 1 TeV. In each plot only two parameters are shown, the remaining ones are pro led over (see main text). Contours represent the allowed 68% and 95% con dence regions, the colour coding of individual constraints is given in the legend, and gray/black areas correspond to the combined regions.
For details see main text and the caption of Fig. 4.
coe cients.) On the other hand, the bounds on the heavy quarks (top, bo om, charm) are dominated by the electron EDM that has no hadronic uncertainties. Recall also, as discussed above, that EDM constraints become less relevant upon including more Wilson coe cients in the t, implying that the hadronic uncertainties do not (currently) play a signi cant role in the multi-parameter ts.
(ii) It is important to recognize that, in many cases, the perturbative uncertainties are as large as the nonperturbative uncertainties. As an example, consider the bounds on the bo om and charm Wilson coe cients, studied (within the κ framework) in Ref. [15]. ere it was shown that the QCD corrections are large; a er inclusion of the two-loop leading-logarithmic QCD corrections, the uncertainties on the electric and chomoelectric Wilson coe cients are reduced to order of 30%. We do not include the NLO corrections here, as la ice results are not available for all required matrix elements (see Refs. [76][77][78] for preliminary results). Maybe somewhat surprisingly, also the electron EDM bound on the CP-violating bo om and charm Yukawas receives large QCD corrections. e calculation of these e ects is ongoing [16] and also not included in this work. No theory uncertainties are included in the LHC constraints. Note that they are partially contained in the uncertainties quoted by the experiments.

Discussion and conclusions
In this work we have presented the rst high-dimensional ts of the coe cients of Yukawa-type SMEFT operators to multiple EDMs and LHC data. As expected, upon inclusion of a su cient number of Wilson coe cients, all EDM constraints can be evaded, and only LHC bounds remain. However, when considering the heavy fermions only, a nontrivial interplay between contributions remains.
ere are several ways to further extend our analysis in the future. As discussed in Sec. 7.3, we have neglect the impact of hadronic and perturbative uncertainties on our t results. is impact can be profound in the lower-dimensional scans, and further motivates the ongoing e orts to decrease the uncertainties.
Improved experimental bounds (or discoveries), as well as the inclusion of additional EDMs such as those of proton and deuterium, once they become available, are expected to have a signi cant impact on the global t. In particular, the di erent isospin dependence of the various hadronic systems will help to disentangle CP-odd Higgs couplings of the rst-generation quarks. e CP structure of the heavy-fermion Yukawas can be tested more directly at present and future colliders by studying observables that are designed speci cally to test the CP-odd couplings and typically require a vast amount of (expected) future data (see, e.g., Refs. [79][80][81][82][83][84][85]).
Finally, as explained in Sec. 2 where we introduced the theoretical framework for avour-diagonal CP-violation in Higgs Yukawas, we do in general expect new avour violating sources to accompany beyond-the-SM CP-violation. erefore, allowing for avour-changing contributions within SMEFT, and including the corresponding observables would be a further extension of the current analysis. While such contributions are generically expected in realistic UV models and are expected to lead to much stronger bounds, the question is whether, given the proliferation of parameters and observables, it is not be er to study selected UV models directly.
In the nal stages of this work, Ref. [86] was published, presenting analyses with a similar scope as ours. We brie y comment on the di erences between the two papers. In Ref. [86], the κ framework (discussed in this paper at the end of Sec. 2) is employed to perform multi-parameter ts to LHC data, including, in particular, the CMS CP analysis of the h → τ τ decay [74]. e interplay of LHC data and the electron EDM bound is discussed, but no combined t to both LHC and EDM data is performed. By contrast, constraints that arise from considerations of electroweak baryogenesis are discussed extensively in Ref. [86]. e expression for the electron EDM (Eq. (13) in Ref. [86]) is given in numerical form and thus hard to verify. However, it seems that the gauge-boson contribution must be incorrect (either gauge-dependent or containing an implicit logarithmic dependence on the UV cuto that is not made explicit; the cited references contain contradictory results). See our discussion in Sec. 4.
rotating to the mass-eigenstate basis for the fermions. We split the Lagrangian into L Yukawa = f =u,d,e (L dim-4,f + L dim-5,f + L dim-6,f ) , where Here, u = (u, c, t), d = (d, s, b), m u = (m u , m c , m t ), and the C uH± can also contain avour odiagonal pieces. e corresponding Lagrangian for down-type quarks is obtained by the obvious interchanges u ↔ d, G + ↔ G − , V † ↔ V , and by a reversal of sign in all terms that are odd in the Goldstone elds. Analogously, the corresponding Lagrangian for leptons is obtained from the up-type Lagrangian above by the obvious interchanges u → e, d → ν, G + ↔ G − , V † → 1, and again by reversing the sign in all terms that are odd in the Goldstone elds.
B An alternative flavour basis: real and diagonal Yukawas ere is a certain freedom of choosing the avour basis for the SMEFT Lagrangian to use for presenting the phenomenological constraints. is freedom stems from the fact that both the dimension-four Yukawas (Y f ) and the dimension-six SMEFT coe cients (C f H ) contribute to the observed masses (and mixings) of fermions. As long as this is guaranteed any basis is equivalent. However, since a UV extension of the SM matched to SMEFT would in general induce contributions both to Y f and C f H there is no clear notion of a "be er" or a more "physical" basis for Y f . We have thus opted to present the bounds in the mass-eigenstate basis as discussed in Sec. 2 (see also Ref. [18]). e advantage of this basis is that it is the one required for computations and is also directly related to the κ-framework basis (see discussion around Eq. (11)). Another approach would be to a empt to provide the constraints in terms of basis-independent, i.e., Jarlskog-type, invariants (see recent Ref. [87]).
In this appendix, we discuss a di erent choice of basis that has been used in the literature [17]. We comment on the di erences and point out a consistency condition on the Wilson coe cients in this basis that has been missed in the literature. As before, our starting point is the fermion mass term in Eq. (3): with Y f and C f H generic, complex 3 × 3 matrices. Now, instead of diagonalising the full matrix in parentheses, as we did, one may choose to rotate the fermion elds by a biunitary transformation that diagonalises only the dimension-four Yukawa matrices Y f . In other words, we rotate with the transformation matrices for f = u, d, , such that a er the rotation where now the matricesŶ f ≡Û † f Y fŴf are diagonal and real with entries that, however, do not correspond to the observed fermion masses. e kinetic terms of quarks is also a ected by the transformation in Eq. (63). At this stage this means that there is a matrixV ≡ U † u U d in the kinetic terms. Moreover, we have de nedĈ f H ≡Û † f C f HŴf . In general, the matricesĈ f H are not diagonal. However, similarly to the discussion of Sec. 2, one can make the UV assumption that they turn out to be complex but diagonal (or ignore o -diagonal entries). is is possible if the matrices U f , W f simultaneously diagonalise both Y f and C f H . Below we assume that this is the case. e analysis of Ref. [17] uses this basis, i.e.,Ĉ f H± or rescalings of them, to present the phenomenological constraints on the SMEFT parameter space.
Since in general, the elements ofĈ f H contain phases, we still have not rotated to the mass-eigenstate basis. To this end we perform an additional ( avour-diagonal) rotation on the right-handed elds as in Ref. [17]: Note that this chiral rotation leaves the kinetic term invariant. e phases θ f,i are xed such that they absorb any phase in the correspondingĈ f H . erefore a er the rotation Here, the y SM f are diagonal and real matrices with entries that correspond to the observed fermion masses, i.e., we have m f = v √ 2 y SM f , obtained from e phases θ f are thus xed and can be computed as functions of the physical masses and theĈ f H entries via As we have the restriction | sin θ f,i | ≤ 1, this gives a consistency bound onĈ f H− or equivalently on