Published for SISSA by Springer Received: December 19, 2018 Revised: February 25, 2019 Accepted: April 5, 2019 Published: April 11, 2019 EFT approach to the electron electric dipole moment JHEP04(2019)090 at the two-loop level Giuliano Panico,a Alex Pomarolb,c and Marc Riembaud a Deutsches Elektronen-Synchrotron (DESY), 22607 Hamburg, Germany b IFAE and BIST, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain c Departamento de Fı́sica, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain d Départment de Physique Théorique, Université de Genève, Genève, Switzerland E-mail: gpanico@ifae.es, pomarol@ifae.es, marc.riembau@unige.ch Abstract: The ACME collaboration has recently reported a new bound on the electric dipole moment (EDM) of the electron, |de | < 1.1 × 10−29 e · cm at 90% confidence level, reaching an unprecedented accuracy level. This can translate into new relevant constraints on theories beyond the SM laying at the TeV scale, even when they contribute to the electron EDM at the two-loop level. We use the EFT approach to classify these correc- tions, presenting the contributions to the anomalous dimension of the CP-violating dipole operators of the electron up to the two-loop level. Selection rules based on helicity and CP play an important role to simplify this analysis. We use this result to provide new bounds on BSM with leptoquarks, extra Higgs, or constraints in sectors of the MSSM and composite Higgs models. The new ACME bound pushes natural theories significantly more into fine-tune territory, unless they have a way to accidentally preserve CP. Keywords: Beyond Standard Model, CP violation, Effective Field Theories ArXiv ePrint: 1810.09413 Open Access, c The Authors. https://doi.org/10.1007/JHEP04(2019)090 Article funded by SCOAP3 . Contents 1 Introduction 1 2 EDM of the electron in the EFT approach 3 2.1 SM EFT basis 3 2.2 Tree-level contributions 4 2.3 One-loop effects 4 JHEP04(2019)090 2.4 Two-loop effects 7 2.5 Dimension-8 operators 11 2.6 Threshold effects at the EW scale: the impact of CP-violating Yukawa’s 11 2.7 EFT below the electroweak scale and relevant RGEs 12 3 Impact on BSM 14 3.1 Power counting of the Wilson coefficients 14 3.2 Leptoquarks and extra Higgs 17 3.2.1 Scalar leptoquarks 17 3.2.2 Vector leptoquarks 19 3.2.3 The heavy Higgs 20 3.3 The MSSM 21 3.4 Composite Higgs 23 4 Conclusions 26 A EDM contributions from Barr-Zee diagrams 26 1 Introduction Electric dipole moments (EDM) provide one of the best indirect probes for new-physics. Since a non-zero EDM requires a violation of the CP symmetry, and the Standard Model (SM) contributions are accidentally highly suppressed, the EDM is an exceptionally clean observable to uncover beyond the SM (BSM) physics. Indeed, if BSM physics lies at the TeV scale, we expect new interactions and therefore new sources of CP violation to be present, 1 inducing sizable EDM to be observed in the near future. For this reason, experimental bounds on the electron and neutron EDM have provided the most substantial constraints on the best motivated BSM scenarios, such as supersymmetry or composite Higgs models. 1 As in the SM, we can expect that any parameter of the BSM that can be complex will be complex, providing unavoidably large new sources of CP violation. –1– The ACME experiment has recently released a new bound on the electron EDM that improve by a factor ∼ 8.6 their previous bound [1]: |de | < 1.1 · 10−29 e · cm . (1.1) This unprecedented level of accuracy allows for a sensitivity to BSM effects even if they appear at the two-loop level. Indeed, using the rough estimate,  2 2 de g me ' 2 , (1.2) e 16π Λ2 JHEP04(2019)090 we get from eq. (1.1) a bound on the scale of new-physics Λ & 2.5 TeV, being competitive with direct LHC searches. It is therefore of crucial interest to understand how and which BSM sectors affect the electron EDM up to the two-loop level, and which constraints can be derived from the bound eq. (1.1). The purpose of the paper is to use the Effective Field Theory (EFT) approach to provide a classification of the leading BSM effects on the EDM of the electron up to the two-loop level. In the EFT approach BSM indirect effects are encoded in the Wilson coef- ficients of higher-dimensional SM operators. At the loop level these Wilson coefficients can enter, via operator mixing, into the renormalization of the CP-violating dipole operators responsible for the EDM. By calculating the anomalous dimensions of these operators, we can provide all log-enhanced contributions to the EDM coming from new physics. At the leading order in a m2W /Λ2 expansion, the electron EDM arises from two dimension-6 operators, OeB and OeW (see below). We will present here the relevant anoma- lous dimensions of the imaginary part of the corresponding Wilson coefficients, CeB and CeW , up to the two-loop level. In particular, we will provide the leading correction (ei- ther at the one-loop level or two-loop level) of the different Wilson coefficients Ci to the imaginary part of CeB and CeW . We will see that due to selection rules, only few Wilson coefficients enter into the renormalization of CeB and CeW at the one-loop or two-loop level. Calculating these leading corrections will allow to extract bounds on these Wilson coefficients from the recent EDM measurement. In addition, we will also provide the most relevant one-loop anomalous dimensions of the dimension-8 operators affecting the electron EDM. Although sub-leading in the m2W /Λ2 expansion, dimension-8 operators give contributions of order de g 2 me m2W ' , (1.3) e 16π 2 Λ4 that can also be relevant as eq. (1.1) leads to the bound Λ & 2 TeV, similar to those from eq. (1.2). Our results can be useful to derive from eq. (1.1) new bounds on BSM particle masses. As an example, we will provide bounds on BSM with leptoquarks or extra Higgs, showing that we can exclude masses below hundreds of TeV. We will also present constraints on new regions of the parameter space of the MSSM, as well as bounds on top-partners in composite Higgs models. These bounds can be better than those from present and future direct searches at the LHC, unless the BSM preserve CP. –2– 2 EDM of the electron in the EFT approach We are interested in calculating new physics contributions to the electron EDM following the EFT approach. This is a valid approximation whenever the new-physics scale Λ is larger than the electroweak (EW) scale, such that new-physics effects on the SM can be char- acterized by the Wilson coefficients Ci (µ) of higher-dimensional SM operators. Assuming lepton number conservation, the leading effects arise from dimension-6 operators X Ci (µ) ∆L = Oi , (2.1) Λ2 i JHEP04(2019)090 where the Wilson coefficients are induced at the new-physics scale, Ci (µ = Λ), and must be evolved via the renormalization group equations (RGEs) down to the relevant physical scale at which the measurement takes place. Since the Wilson coefficients mix via loop effects in the RGEs, precise measurements, such as the EDM, can be sensitive to different Wilson coefficients induced by different sectors of the BSM. The EDM of the electron is measured at low-energies µ  me , and can then be extracted in the EFT approach from the coefficient of the operator i − de (µ) ēσµν γ5 eF µν , (2.2) 2 evaluated at the electron mass, de = de (µ = me ) , (2.3) where Fµν is the field-strength of the photon. The RG evolution of de (µ) from Λ to me must be computed in the EFT made with the states lighter than µ. This means that from the new-physics scale Λ down to the EW scale we must use the SM EFT, while below the EW scale we must use the effective theory including only light SM fermions, gluons and photons. Let us start discussing the contributions to de (µ) in the SM EFT. 2.1 SM EFT basis We will work mainly within the Warsaw basis [2], as the loop operator mixing is simpler in this basis due to the presence of many non-renormalization results. Nevertheless, we will make two changes in the four-fermion operators of the Warsaw basis. In particular, we will make the replacement2 (3) Olequ = (L̄aL σµν eR )εab (Q̄bL σ µν uR ) → Oluqe = (L̄aL uR )εab (Q̄bL eR ) , (2.4) Ole = (L̄L γ µ L0L )(ē0R γµ eR ) → Oleē0 l̄0 = (L̄L eR )(ē0R L0L ) , (2.5) where a, b denote the SU(2)L doublet indices, and LL and eR denote only the first generation lepton multiplets, while L0L and e0R the second and third generation ones.3 We also relabel the operator Oledq = (L̄aL eR )(d¯R QL a ) ≡ Oledq̄ ¯ . (2.6) 2 (3) These operators are related to the ones in the Warsaw basis [2] by Fierz identities, namely Olequ = (1) −8Oluqe − 4Olequ and Ole = −2Oleē0 l̄0 . 3 Notice that in our formulae we will suppress the fermion generation indices, except in the cases in which they cannot be straightforwardly reconstructed from the context. –3– 2-loop tree level (1) Olequ = (L̄L eR )(Q̄L uR ) OeW = (L̄L σ a σ µν eR ) HWµν a Oe0 W = (L̄0L σ a σ µν e0R ) HWµν a OeB = (L̄L σ µν eR ) HBµν Oe0 B = (L̄0L σ µν e0R ) HBµν 1-loop OuW = (Q̄L σ a σ µν uR ) HWe a µν OuB = (Q̄L σ µν u e R ) HBµν Oluqe = (L̄L uR )(Q̄L eR ) O f = |H|2 W a µν W fa OdW = (Q̄L σ a σ µν dR ) HWµν a WW µν eµν OB Be = |H|2 B µν B OdB = (Q̄L σ µν dR ) HBµν JHEP04(2019)090 eµν OW Be = (H † σ a H) W a µν B ¯ = (L̄L eR )(d¯R QL ) Oledq̄ OW f aν bρ cµ Oleē0 l̄0 = (L̄L eR )(ē0R L0L ) f = εabc Wµ Wν Wρ Oye = |H|2 L̄L eR H Table 1. Operators involved in our analysis. Top-left: operators contributing to the electron EDM at tree-level. Bottom-left: operators contributing to the electron EDM at the one-loop level via mixing. Right: operators contributing to the electron EDM at the two-loop level via mixing. In this table we denoted by LL and eR only the first generation lepton multiplets, while L0L and e0R denote the second and third generation ones. Our labeling is to make clear that there are two types of operators Oψψψψ and Oψψψ̄ψ̄ that in Weyl notation are respectively ψ 4 of total helicity 2 and ψ 2 ψ̄ 2 of total helicity zero. As we will see in the following, the helicity of the operator plays a crucial role in understanding the properties of the operator mixing at the loop level [3]. In Dirac notation these two type of operators could also be written (after Fierzing) respectively as operators of type (Ψ̄γµ Ψ)(Ψ̄γ µ Ψ) and of type (Ψ̄L ΨR )(Ψ̄L ΨR ). In this case, for example, we would have ¯ = −(L̄L γµ QL a )(d¯R γ eR )/2. Oledq̄ a µ 2.2 Tree-level contributions At tree-level there are only two dimension-6 operators that contribute to the electron EDM, namely the dipole operators OeW and OeB , given in table 1. We have √ 2v de (µ) = 2 Im [sθW CeW (µ) − cθW CeB (µ)] , (2.7) Λ where we defined v ' 246 GeV as the Higgs VEV, and sθW ≡ sin θW with θW the weak angle (similarly for the other trigonometric functions). Notice that contributions to the electron EDM arised only from the imaginary part of CeW,eB . For this reason, we will only be interested in loop contributions to the dipole operators that can generate nonzero Im[CeW,eB ]. 2.3 One-loop effects At the one-loop level, however, other dimension-6 operators can mix with the dipole oper- ators OeW and OeB , giving a contribution to de . Selection rules, mainly based on helicity –4– y y F3 → Hψ 2 F , ψ 4 , H 2 F 2 Table 2. Selection rules [3, 4] for the mixing at the one-loop level between the different types of dimension-6 operators (in Weyl notation). arguments, dictate that only few operators contribute at the one-loop order to the anoma- lous dimension of the Wilson coefficients CeW and/or CeB , as has been argued in refs. [3, 4] based on the analysis of ref. [5]. The relevant selection rules for our analysis are given in table 2,4 where we use Weyl notation, and denote with F any SM field strength, with ψ JHEP04(2019)090 any Weyl fermion and with H any Higgs insertion. The OeW and OeB operators are of type Hψ 2 F and can then only receive contributions from operators of type ψ 4 , F 3 and H 2 F 2 of total helicity ≥ 2. There are four dimension-6 operators of type ψ 4 , but only two contain two leptons, (1) Oluqe and Olequ given in table 1. The second one, however, after closing the quark loop, can only give rise to the Lorentz-singlet structure L̄L eR , and therefore cannot contribute to the electron dipoles. Hence only Oluqe contributes at the one-loop level to the anomalous dimension of OeW and OeB .5 This contribution is given by ! ! d CeB yu g − 12 tθW Nc (YQ + Yu ) = 1 Cluqe , (2.8) d ln µ CeW 16π 2 4 Nc where Yf refers to the hypercharge of the fermion f (YQ = 1/6, Yu = 2/3 and Yd = −1/3), and Nc = 3 is the number of QCD colors. Since we can work in a basis where the Yukawa matrix yu is diagonal, the renormalization of the imaginary part of CeW,eB from eq. (2.8) only arises from the imaginary part of Cluqe . A second type of operators, involving SM bosons, are H 2 Fµν Fe µν . There are three operators of this type in the SM, presented in the bottom-left of table 1. All of them contribute to the EDM at the one-loop level [6, 7]:   ! ! C 3 d CeB ye g 0 2tθW (YL + Ye )  WW  f Im =− 2  C e  . BB (2.9) d ln µ CeW 16π 2 1 0 tθW (YL + Ye ) CW Be It is instructive to write eq. (2.9) in a more physically oriented way, by relating the con- tributions to the EDM to those to the CP-violating Higgs couplings hγγ, hγZ, and to the anomalous triple gauge coupling δκ̃γ , defined as vh   e µν + 2κ̃γZ Fµν Zeµν + ieδκ̃γ W + W − Fe µν . κ̃ γγ F µν F µ ν (2.10) Λ2 4 There is an exemption to these selection rules when the pair of Yukawas yu ye or yu yd is involved in the loop, as this can induce a mixing from ψ 2 ψ̄ 2 to ψ 4 [3, 4]. This can only affect the EDM operators at the two-loop level and will be discussed later. 5 As explained in ref. [4], this is easily seen in Weyl notation where the dipole operator is ∝ Lα Eβ F αβ with Lα and Eβ being respectively the SU(2)L doublet and singlet Weyl electron. Therefore, only four- fermion operators containing Lα Eβ (antisymmetric under α ↔ β) can contribute at the loop level to the dipole. The only one is Lα Eβ U α Qβ that corresponds to Oluqe in Dirac notation. Notice that this argument applies also to one-loop finite parts. –5– We have κ̃γγ = c2θW CB B̃ + s2θW CW W f − c θW s θW C W B e,  1 2 2 κ̃γZ = cθW sθW CW W f − CB Be − (cθW − sθW )CW Be, 2 1 v2 δκ̃γ = C e, (2.11) tθ W Λ 2 W B that, using eq. (2.9), leads to     d e me 4 1 Λ2 JHEP04(2019)090 2 de (µ) = 2 2 4Qe κ̃γγ − + 2Qe sθW κ̃γZ + 2 δκ̃γ , (2.12) d ln µ 8π Λ s2θW 2 v where Qe ≡ −1/2+YL = Ye = −1 is the electric charge of the electron. Due to the approxi- mate accidental cancellation in the electron vector coupling to the Z, (1/2+2Qe s2θW ) ∼ 0.04, the main contribution to the EDM comes from κ̃γγ and δκ̃γ . In fact, this second contribu- tion is often found to be small in many BSM scenarios, such as the MSSM or composite Higgs models, as we will see later. In these models the contribution from κ̃γγ is the dom- inant one. This allows in many cases to give a direct relation between the electron EDM and the CP-violating Higgs coupling to photons. Another class of operators that can in principle mix at the one-loop level with the electron dipole operators are other type of dipole operators Hψ 2 F , for example, those involving other fermions in the SM. It is easy to see, however, that there are no possible Feynman diagrams from quark dipole operators contributing to the electron EDM at the one-loop level. For dipole operators involving other SM leptons, for example, HeµF , these contributions are also absent at the one-loop level. Indeed, since we can work in a basis where the SM lepton Yukawa matrix ye is diagonal, none of these operators can affect the electron EDM at the one-loop level. Below we will see that there can be, however, contributions at the two-loop level. Finally, there are operators of type F 3 that could potentially mix with the dipole operators. In the SM there are two of these operators, either with gluons (OGe ) or W a bosons (OW f ). The OG e operator obviously can not give corrections to the electron dipole operators at one-loop or two-loop level, as there are no possible Feynman diagrams at these orders. On the other hand, the OW f operator can contribute to the OeW dipole at one loop. It turns out, however, that this contribution is finite, so it does not induce a running for the dipole operator. The finite contribution can be readily computed in dimensional regularization, leading to the result [8]6 3 Im[CeW ] = ye g 2 C W f. (2.13) 64π 2 6 Notice that the result depends on the regularization procedure used to compute the one-loop integral [8]. It has been argued in ref. [9] that, for this computation, dimensional regularization provides the only sensible regularization procedure within the EFT framework. –6– 1-loop (1) Olequ Oledq̄ ¯ , Olel̄′ē′ 2-loop 1-loop Oluqe Oe′ W , OuW , OdW OeW , OeB Oe′ B , OuB , OdB OW W f , OB B e , OW B e Oye OW f 1-loop (finite) Figure 1. Corrections to the electron EDM (imaginary part of CeW,eB ) induced up to the 2-loop JHEP04(2019)090 level. The dashed and solid arrows denote mixing at 1-loop and 2-loop order respectively. On the right we list the operators that generate contributions enhanced by a double logarithm (showing the 1-loop mixing patterns that generate it), whereas on the left we list operators giving rise to a single logarithm. 2.4 Two-loop effects At the two-loop level, more dimension-6 operators can contribute to the electron EDM by mixing with the dipoles OeW and OeB . We remark that we are only interested in calculating two-loop effects for those Wilson coefficients that did not mix with the EDM operators at the loop level. We are interested in extracting bounds to their corresponding Wilson coefficients and therefore we are only interested in calculating the leading correction to the EDM. For Wilson coefficients affecting the EDM already at the one-loop level, such as Cluqe , the two-loop corrections would only provide a small correction to their bound. New dimension-6 operators can contribute to the electron EDM by mixing with the dipoles OeW and OeB in two different ways. Either by mixing at the one-loop level with the operators we discussed in the previous section, Oluqe and OV Ṽ (V = W, B), that contribute at the one-loop level to the dipoles, or by direct two-loop contribution to the anomalous dimension of OeW and OeB (see figure 1). The first case can potentially give larger corrections, as in the leading-log approxi- mation, they will contain two logarithms, i.e. ∝ ln2 (Λ2 /m2W ). From the selection rules of table 2, we see that only two classes of operators can contribute at this order. One is given by the ψ 4 operators that could not generate an electron dipole at the one-loop due to the (1) absence of Feynman diagrams, namely the Olequ operator. The second class is given by dipole operators involving the second and third lepton generations, Oe0 W and Oe0 B , or the quarks, OuW , OuB , OdW and OdB . Notice that, as we pointed out before, there is an exception to the selection rules of table 2, corresponding to a possible mixing of ψ̄ 2 ψ 2 operators into ψ 4 when the pair of Yukawas either yu ye or yu yd is involved in the loop [3, 4]. Nevertheless, by working in the basis in which the lepton and up-type quark Yukawa matrices are real and diagonal, one can easily find that there are not ψ̄ 2 ψ 2 operators contributing to the imaginary part of Oluqe at the one-loop level. Indeed, in this basis yu ye is real and diagonal, and the only ψ̄ 2 ψ 2 operators that could contribute to Oluqe are the ones involving two electron fields and two same-generation quarks. The Wilson coefficients of these operators are necessarily real and do not induce CP-violating effects. –7– Therefore the one-loop mixing pattern and RGEs are the following. The Olequ operator can mix with Oluqe at the one-loop level [6, 7]: d g2   (1) Cluqe = 2 4(YL + Ye )(YQ + Yu )t2θW − 3 Clequ . (2.14) d ln µ 16π The dipole operators, on the other hand, mix with the Oluqe operator, [6, 7]7 d g ye h i Cluqe = − 8t θ (Y L + Y e )C uB + 12C uW , (2.15) d ln µ 16π 2 W JHEP04(2019)090 as well as with OV Ve operators [10]: d 2g h i CW W = − Im y e 0 C eW0 + y u N c C uW + y d N c C dW , (2.16) d ln µ f 16π 2 d 4g 0 h i CB Be = − Im y e 0 (Y L + Y e ) C 0 eB + y N (Y u c Q + Y u ) C uB + y N (Y d c Q + Y d ) C dB , d ln µ 16π 2 (2.17) d 2g h C e=− Im 2tθW ye0 (YL + Ye ) Ce0 W − yu Nc (YQ + Yu ) CuW d ln µ W B 16π 2  i + yd Nc (YQ + Yd ) CdW + ye0 Ce0 B − yu Nc CuB + yd Nc CdB . (2.18) Flavor indices are easily understood as we can always work with diagonal Yukawa matrix ye and either yu or yd . The operator OW f also enters at two loops via renormalization of the OV Ṽ opera- tors [6, 7], d 1 d 1 C f=− 15g 3 CW f, C e=+ 6g 0 g 2 YH CW f. (2.19) d ln µ W W 16π 2 d ln µ W B 16π 2 This leads to a two loop, double log contribution to the electron EDM, to be compared with the finite contribution at one loop in eq. (2.13). The two loop contribution becomes comparable to the one-loop one already for Λ ∼ 10 TeV. Let us now discuss those dimension-6 operators that can directly contribute at the two- loop level to the anomalous dimension of the electron dipole operators. In fact, we are only interested in the EDM, i.e. the imaginary part of the the dipole operators, and therefore only complex Wilson coefficients can contribute, as the SM interactions preserve CP up to small Yukawa couplings that we neglect. This reduces the list of possible dimension-6 operators to those to the right of table 1. For example, operators of the type Ψ̄γµ ΨH † Dµ H are Hermitian (and then have real Wilson coefficients) unless the two fermions involved are different, meaning that they must involve different flavors. But since in the SM we can work in the basis where ye and either yu or yd are diagonal, we cannot draw any two-loop Feynman diagram contributing to the electron EDM operators. 7 The heavy lepton dipole operators induce a running for Oluqe at one loop. However in the basis with diagonal lepton Yukawa’s they contribute only to the Oluqe involving heavy leptons, which then does not contribute to the electron dipoles at one loop. –8– Similar conclusions can be obtained for four-fermion operators, except for those in table 1, namely Oledq̄ ¯ and Oleē0 l̄0 . There is however an important subtlety related to these operators, which results in an ambiguity in the determination of their contributions to the electron EDM. Within the basis we are using, in which Oledq̄ ¯ and Oleē0 l̄0 are written as the product of scalar currents, it is simple to check that the 1-loop contributions to the electron EDM trivially vanish due to the tensor structure. However, through a Fierz rearrangement, Oledq̄ µ ¯ and Oleē0 l̄0 can also be rewritten in the form (Ψ̄γµ Ψ)(Ψ̄γ Ψ), i.e. as a product of vector currents. With this choice, if dimensional regularization (in particular the MS JHEP04(2019)090 scheme) is used to compute the contributions to the electron EDM, a finite 1-loop effect is found. The origin of this contribution is related to the presence of additional four- fermion interactions involving multiple gamma matrices that are generated at intermediate steps of the calculation. These are known as “evanescent operators” (for a review, see for example [11]). The coefficients of these interactions carry an  = 4 − d factor, but they can give finite effects in the presence of 1/ poles. As an example, we report the contributions to the electron EDM induced by the O`e = C`e (L̄L γ µ L0L )(ē0R γµ eR ) operator (see for instance [12, 13]) de m`0 =2 Im C`e . (2.20) e 16π 2 A similar result is obtained for the Oledq̄ ¯ operator in vector-current form. Summarizing the above discussion, one finds that the contributions from the Oledq̄ ¯ and Oleē0 l̄0 4-fermion operators crucially depend on the choice of the operator basis and on the regularization procedure. This, in turn, can affect the matching from a UV model. In particular, finite 1-loop contributions can be shifted from the matching to the 4-fermion operators into the EDM operators OeB and OeW and vice versa.8 As already mentioned, for our analysis we choose the scalar-current form for the 4-fermion operators, ( L̄L eR )(Ψ̄Ψ). Therefore no finite one-loop contributions arise for the electron EDM from the Oledq̄ ¯ and Oleē0 l̄0 operators. As we will see in section 3.2.3, this choice is particularly convenient for studying UV models including heavy Higgs-like states, in which case only scalar-current 4-fermion operators are obtained from the matching. We have then only Oye , Oledq̄ ¯ and Oleē0 l̄0 giving a two-loop mixing with the electric dipole operators. From Oye (see left-hand side of figure 2), we obtain ! ! d CeB g3 3 tθW YH + 4t3θW YH2 (YL + Ye ) = 1 2 2 C ye , (2.21) d ln µ CeW (16π 2 )2 4 2 + 3 tθW YH (YL + Ye ) while from Oledq̄ ¯ and Oleē0 l̄0 (see right-hand side of figure 2), we get respectively ! ! d CeB yd g 3 N c 3tθW YQ + 4t3θW (YL + Ye )(YQ2 + Yd2 ) Im = 1 2 Cledq̄ ¯ (2.22) d ln µ CeW (16π 2 )2 4 2 + 2tθW (YL + Ye )YQ 8 Analogous ambiguities are present in the case of 4-fermion contributions to the magnetic dipole mo- ments. For instance see the analysis of the b → sγ transitions in ref. [14]. –9– Figure 2. Feynman contributions to the anomalous dimension of OeW and OeB at the two-loop level from Oye (left) and Oledq̄ ¯ ,Oleē0 l̄0 (right). JHEP04(2019)090 and ! ! d CeB ye 0 g 3 1 3tθW YL + 4t3θW (YL + Ye )(YL2 + Ye2 ) Im = 1 2 Cleē0 l̄0 . (2.23) d ln µ CeW (16π 2 )2 4 2 + 2tθW (YL + Ye )YL We summarize our results by schematically presenting in figure 1 the mixing patterns of the effective operators contributing to the electron dipoles at 2-loop order. For complete- ness we also include the 2-loop mixing of the OW f operator, which, at 1-loop order, only induces finite corrections to the electron dipoles. We also provide below the leading-log approximation to the electron EDM that can be good enough since the new physics scale is not constrained yet to be far away from the electroweak scale, and therefore we do not need to resum the logs by exactly solving the RGEs. We find that the one-loop corrections are √ " !# de 1 2v 1 − 4s2θW Λ2 Λ2 '− 2 2 yu Im Cluqe − ye 2κ̃γγ + κ̃γZ − 2 δκ̃γ ln 2 . (2.24) e 16π Λ s θW c θW 2v mh The double-log 2-loop corrections are given by √    de 1 2v 1 2 2 2 ' Im gy e y e 0 (11 + 9t θ )c e 0 W + 15yu (3 + t θ )C uW + 3y d (3 + t θ )C dW e (16π 2 )2 Λ2 8 W W W g   −3 ye ye0 (1 + 7t2θW )ce0 B − 3yu (1 + 7t2θW )CuB + yd (3 + 5t2θW )CdB tθ W  2 2 (1) 3 2  2 Λ 2 + 2g yu (3 + 5tθW )Clequ + ye g 13 + 3 tan θW CW ln . (2.25) m2h f Finally, the single-log 2-loop corrections are √   de g2 2v 3 2 1 2 1 2  Λ2 '− Im t C y + y d t C ¯ + y e 0 2 + 9t C 0 0 ln . (2.26) e (16π 2 )2 Λ2 8 θW e 8 θW ledq̄ 8 θW leē l̄ m2h Notice that we did not include in the above formula finite corrections that can arise at the matching scales. In fact, few other dimension-6 operators can enter in this way into the renormalization of the electron EDM. As we saw in section 2.3, an example is provided by the OWf operator, whose dominant corrections to the electron EDM arise as finite one-loop contributions. An analogous result is valid for the H 3 f¯f operator (f = u, d, e0 ) that modify the quarks and heavy lepton Yukawa couplings. These operators start to contribute to the electron EDM at the two-loop level through finite Barr-Zee-type diagrams [15]. We will discuss these operators in section 2.6. – 10 – 2.5 Dimension-8 operators Since we are calculating corrections to the EDM at the two-loop level coming from dimension-6 operators, it is appropriate to ask whether dimension-8 operators can also give similar contributions. The effects of these operators are suppressed with respect to dimension-6 operators by an extra m2W /Λ2 , but this could be overcome if their contributions to the EDM arises at the one-loop level instead of two-loops. We will only be interested here in dimension-8 operators that can be generated from integrating new physics at tree-level and that have not been constrained from previous dimension-6 operators. For example, dimension-8 operators involving extra |H|2 or extra derivatives will not be relevant. We JHEP04(2019)090 only find two of these operators, Oldqe = (L̄L dR H)(Q̄L eR H) , Ole0 l0 e = (L̄L e0R H)(L̄0L eR H) , (2.27) that at the one-loop level can mix with dimension-8 operators of dipole type, Oh2 eW = |H|2 OeW , Oh2 eB = |H|2 OeB , (2.28) similarly to eq. (2.8): ! ! d Ch2 eB yd g − 12 tθW Nc (YQ + Yd ) = 1 Cldqe , (2.29) d ln µ Ch2 eW 16π 2 4 Nc and equivalently for Cle0 l0 e with the replacement yd → ye , YQ → YL and Yd → Ye . We get a contribution to EDM of order √ 3 de 2v yd Im[Cldqe ] Nc Λ2 '− 4 ln . (2.30) e Λ 16π 2 24 m2W 2.6 Threshold effects at the EW scale: the impact of CP-violating Yukawa’s The log-enhanced contributions we considered in the previous sections are expected to give the dominant corrections to the electron EDM. Nevertheless, being this logarithm not always so large, there are cases in which finite corrections can be more important. A noticeable example, that we will discuss here, is the case of two-loop corrections to the EDM generated when integrating out the top and the Higgs at the EW scale. These corrections come from CP-violating Yukawa couplings induced by the SM dimension-6 operator Oyf at the EW scale: v2 cf hf¯L fR , cf = √ C yf . (2.31) 2Λ2 For the particular case of Cye we obtain, from Barr-Zee diagrams, that these contributions are given by   de 16 e2 m2t Im Cye '− v 2 + ln 2 , (2.32) e 3 (16π 2 )2 mh Λ2 where we only kept the leading corrections due to diagrams with a virtual photon and a top loop. Contributions from a virtual Z-boson are highly suppressed by the small vector – 11 – Z coupling to the electron, whereas diagrams involving light quarks are suppressed by the light quark Yukawa’s. Diagrams involving a virtual W boson are expected to be subleading with respect to the photon contribution, although not negligible. For simplicity we however neglect these contribution as often done in the literature. Notice that in the above formula we only included the leading terms in an expansion for large m2t /m2h , which reproduce the full result with an accuracy ∼ 10%.9 The appearance of ln m2t /m2h in eq. (2.32) can be understood as a RG running of the electron EDM from the top mass, where the top is integrated out generating a hF µν Fµν term at the one-loop level, down to the Higgs mass. In particular, this arises from a loop diagram involving hF µν Fµν and the CP-violating Yukawa eq. (2.31). JHEP04(2019)090 Comparing the result in eq. (2.32) with the two-loop running in eq. (2.26), we find that the former is typically dominant and is a factor of few larger than the latter for a cut-off scale in the 10 TeV range. This is because the contributions of eq. (2.26) are slightly smaller than the naive estimate since they are proportional to g 0 and are suppressed by an accidental factor 3/8. The two contributions become comparable only for a very large cut-off scale ∼ 104 TeV. Similarly, Cyu,d and Cye0 , which can give CP-violating corrections to the quark and to the heavy lepton Yukawa’s, can lead to finite Barr-Zee contributions to the electron EDM. For the top case, we have   de e2 2 me m2t Im Cyt '− 4Nc Qt v 2 + ln 2 . (2.33) e (16π 2 )2 mt mh Λ2 Again, the ln m2t /m2h can be understood as a RG running of the electron EDM from the top mass, where a hF F̃ term is generated from the CP-violating top Yukawa after integrating out the top, down to the Higgs mass. 2.7 EFT below the electroweak scale and relevant RGEs Let us now discuss the RG running effects below the EW scale. At the scale µ ∼ mW , we must integrate out all the heavy SM particles, the W , Z, the Higgs and top, and work for µ < mW with the EFT built with the light fermions, the photon and the gluons. The EDM of the electron arises in this EFT from a dimension-5 operator Oeγ = ēL σ µν eR Fµν , (2.34) whose coefficient, from the tree-level matching with the SM EFT at the EW scale, is given by v Ceγ (mW ) = √ (sθW CeW (mW ) − cθW CeB (mW )) 2Λ v3 + √ (sθW Ch2 eW (mW ) − cθW Ch2 eB (mW )) . (2.35) 2 2Λ3 9 We report the full expression in appendix A. – 12 – Other relevant operators below the EW scale are four-fermion operators made with the light SM fermions. The matching at the EW scale is given by Cluqe (mW )Oluqe = Ceuue (mW )(ēL uR )(ūL eR ) + · · · , (1) (1) Clequ (mW )Olequ = Ceeuu (mW )(ēL eR )(ūL uR ) + · · · , C ¯ (mW )O ¯ = C ¯¯(mW )(ēL eR )(d¯R dL ) + · · · , ledq̄ ledq̄ eedd Cleēl̄ (mW )Oleēl̄ = Cleēē (mW )(ēL eR )(ē0R e0L ) + · · · , (2.36) and JHEP04(2019)090 v2 Cldqe (mW )Oldqe = Cedde (mW )(ēL dR )(d¯L eR ) + · · · , 2Λ2 v2 Cle0 l0 e (mW )Ole0 l0 e = Cee0 e0 e (mW )(ēL e0R )(ē0L eR ) + · · · , 2Λ2 v2 Cldqe (mW )Oleqd = Ceedd (mW )(ēL eR )(d¯L dR ) + · · · , 2Λ2 v2 Cle0 l0 e (mW )Olel0 e0 = Ceee0 e0 (mW )(ēL eR )(ē0L e0R ) + · · · , (2.37) 2Λ2 where we have also included the matching of the dimension-8 SM operators Oleqd = (L̄L eR H)(Q̄L dR H) , Olel0 e0 = (L̄L eR H)(L̄0L e0R H) . (2.38) The above four-fermion operators can enter into the anomalous dimension of Oeγ at the one or two loop level. Using our previous results, we can easily extract the RGE for Ceγ since the renormalization of the photon can be read from that of the U(1)Y -boson in the SM. At the one-loop level we only have, equivalently to eq. (2.8) and eq. (2.29), (see also [16])   d e 1 mu md v 2 me 0 v 2 Ceγ = − √ Nc Qu Ceuue + N c Q d C edde + Qe Cee0 e0 e , (2.39) d ln µ 16π 2 2 Λ 2Λ3 2Λ3 where Qf refers to the EM charge of the fermion f . At the two-loop level, we can have contribution to Ceγ either by a one-loop mixing of Oeef f with Oef f e (f = u, d, e0 ), similarly to eq. (2.14), d e2 Cef f e = 2 Qe Qf Ceef f , (2.40) d ln µ π or by a direct contribution to Ceγ : d e3  md 2 me 3  ImCeγ = 4Nc Qe Qd Ceed¯d¯ + 4Qe Cee ē ē . 0 0 (2.41) d ln µ (16π 2 )2 Λ Λ The RGE running should be considered from mW down to the mass of the heaviest fermion in the four-fermion operator, where the state should be integrated out. 10 Therefore this running can be important for lighter fermions. 10 We have also to include here self-renormalization effects. These are however small and generically correct the bounds on the Wilson coefficients by roughly 10%. – 13 – We can compare our results with those obtained from Barr-Zee diagrams arising from CP-violating Yukawa interactions eq. (2.31). For light quarks and leptons e0 , one finds respectively ! de e2 2 me mq 2 m2q π 2 Im Cyq '− 4Nc Qq v ln + , (2.42) e (16π 2 )2 m2h m2h 3 Λ2 and   de e2 me me0 m2e0 π2 Im Cye0 '− 4Q2e v ln2 + . (2.43) e 2 (16π ) 2 m2h mh2 3 Λ2 The double logarithm in eqs. (2.42) and (2.43) can be understood from an EFT perspective JHEP04(2019)090 as the RG running from the Higgs mass, where integrating the Higgs generates a complex Ceef f from eq. (2.31), down to the light fermion masses. The relevant RGEs are indeed those in eq. (2.39) and eq. (2.40). For light quarks, however, a better bound on the corresponding Wilson coefficient can be obtained from constraints on CP-violating electron-nucleon interactions, that the ACME collaboration has also recently reported [1]: GF   Z − N (1) (0) (1) (0) − √ iēγ5 eN̄ CS + CS σ3 N , C S = CS + CS < 7.3 · 10−10 . (2.44) 2 Z + N Neglecting isospin breakings that in the ThO are small [1], we have for the down-quarks (0) Im[C ¯¯] ¯ CS ' CS = − √ eedd hN |dd|N i. (2.45) 2GF Λ2 We have checked that bounds from de are slightly better than those from CS for operators involving the bottom (using hN |b̄b|N i ' 74 MeV/mb [17]), while the bound from CS is better for ligher quarks. 3 Impact on BSM 3.1 Power counting of the Wilson coefficients So far we have presented the leading contributions of the dimension-6 operators to the anomalous dimension of the electron electric dipole operators up to two loops. All the possible contributing operators are given in table 1. Nevertheless, the importance of the different operators depends on the size of their Wilson coefficients, which crucially depends on the BSM dynamics. In the following we will be interested in BSM theories that can be described as weakly- coupled renormalizable theories. These includes all possible extensions of the SM with extra particles with renormalizable interactions. In these BSM theories we can classify the Wilson coefficients as those that can be generated at tree-level and those generated at most at the loop level. For example, among the operators of table 1, the only ones with Wilson coefficients that can be induced at tree-level by integrating out new heavy states are all the four-fermion operators and Oye ; the rest, involving always field-strengths, can only be generated by loops. Therefore, we expect g∗3 g g∗2 g 2 2 Cf V ∼ , CV Ve ∼ , ¯ , Cleē0 l̄0 , Cye ∼ g∗ , Cluqe , Clequ , Cledq̄ (3.1) 16π 2 16π 2 – 14 – Oluqe Scalar (3, 2, 7/6) Scalar (3̄, 1, 1/3) (1) Olequ Scalar (1, 2, 1/2) Scalar (3̄, 1, 1/3) Oy e Fermion (1, 2, −1/2) ⊕ (1, 1(3), −1) Oledq̄ ¯ Vector (3̄, 2, 5/6) Fermion (1, 2, −1/2) ⊕ (1, 1(3), 0) Vector (3, 1, 2/3) Fermion (1, 2, −3/2) ⊕ (1, 1(3), 0) Scalar (1, 2, 1/2) Scalar (1, 2, 1/2) Oleē0 l̄0 Vector (1, 1, 0) JHEP04(2019)090 Vector (1, 2, 1/2) Scalar (1, 2, 1/2) Table 3. States transforming under the SM group SU(3)c × SU(2)L × U(1)Y contributing at the tree-level to the operators of table 1. where g∗ refers to a generic coupling of the BSM dynamics to the SM, f = e, u, d and V = W, B. In this class of BSM theories the contributions to the electron EDM have the following loop expansion. The leading contributions are of order de /e ∼ 2 2 2 (g∗ /16π )(mu /Λ ) ln(Λ/mW ) and can arise from those particular BSM that contribute to Oluqe at tree-level. There are only two types of particles that can generate Oluqe at tree-level (see table 3), the leptoquarks R2 and S1 that will be discussed below. Next, we can have BSM dynamics contributing directly to the Wilson coefficients of the electron dipole operators, that can give corrections of one-loop order (but without a log-enhancement), de /e ∼ (g∗2 /16π 2 )(g∗ v/Λ2 ). This can happen in BSM theories that contain fermions and bosons coupled to the electron, with at least one of them charged, as for example, the selectron and wino in supersymmetric theories. Contributing at the two-loop level, we have those BSM theories induc- (1) ing Olequ at tree-level, which leads to a double-log enhanced EDM, de /e ∼ g 2 g∗2 /(16π 2 )2 (mu /Λ2 ) ln2 (Λ/mW ). As shown in table 3, this includes BSM theories with extra Higgses. Single-log two-loop contributions can come from BSM scenarios generating OV Ṽ at the loop level (those BSM containing extra charged fermions coupled to the Higgs), or generating Oledq̄ ¯ , Oleē0 l̄0 or Oye at tree-level (see table 3). Also BSM theories generating Oluqe at one-loop level can lead to a single-log two-loop contribution to the electron EDM, as we will see later for the case of the MSSM. Finally, BSM theories with extra SU(2)L fermions can generate OW f at the loop-level, leading to a contribution to the electron EDM at the two-loop level with no log enhance- ment. On the other hand, BSM theories contributing to the EDM of a fermion different from the electron are expected to give negligible effects to the electron EDM, as these arise at best at the three-loop level. Examples of these classes of BSM theories will be given below. We must also notice that there is a large class of low-energy effective descriptions of strongly-coupled theories – 15 – tree-level two-loops CeW 5.5 × 10−5 ye g Clequ 3.8 × 10−2 ye yt CeB 5.5 × 10−5 ye g 0 Cτ W 260 yτ g two-loops finite Cτ B 380 yτ g 0 one-loop C ye 14 ye λh CtW 6.9 × 10−3 yt g Cluqe 1.0 × 10−3 ye yt C yt 14 yt λh CtB 1.2 × 10−2 yt g 0 CW W 4.7 × 10−3 g 2 C yb 2.9 × 103 yb λh f CbW 64 yb g CB Be 5.2 × 10−3 g 0 2 C yτ 3.4 × 103 yτ λh CbB 47 yb g 0 CW Be 2.4 × 10−3 gg 0 Cledq̄ 10 ye yt (yt /yb ) JHEP04(2019)090 ¯ CW f 6.4 × 10−2 g 3 Cleē0 l̄0 0.63 ye yt (yt /yτ ) Table 4. Bounds on the Wilson coefficients coming from eq. (1.1) taking Λ = 10 TeV. For a better appreciation of the bound, we have extracted the Yukawa, gauge or Higgs coupling (λh = 0.1) that we naturally expect to carry these Wilson coefficients. For Cledq̄ ¯ and Cleē0 l̄0 we have further extracted a factor (yt /yb ) and (yt /yτ ) respectively to reflect the fact that these coefficients can be potentially larger consistently with their natural sizes eq. (3.2). that follow the same power counting described above. These are those theories that were assumed to follow the “minimal coupling” assumption [18], and correspond to holographic models as well as their deconstructed versions. It is also important to keep in mind that operators of table 1 containing the fields L̄L and eR can potentially give a contribution to the electron mass. This places a constraint on the natural size of their Wilson coefficients. In particular, we find   CeV v Cye v 3 Clequ mu Cluqe mu Cledq̄ ¯ md Cleēl̄ me0 , , , , , . me . (3.2) 16π 2 Λ2 16π 2 16π 2 16π 2 16π 2 In fact, in most of the UV-complete BSM theories (e.g. supersymmetry, composite Higgs or theories with flavor symmetries only broken by Yukawas) we expect operators with chirality flips to carry yukawa couplings, i.e., C f V ∝ yf , C ye ∝ y e , Clequ ∝ ye yu , Cluqe ∝ ye yu , ¯ ∝ ye yd , Cleē0 l̄0 ∝ ye ye0 , Cledq̄ (3.3) implying that we only expect sizable contributions to the electron EDM from four-fermion operators involving the third family. All these considerations can be useful for a proper interpretation of the recent ACME bound. In table 4 we list the bounds on individual Wilson coefficients that can be inferred from the new electron EDM measurement eq. (1.1). To derive the bounds we considered the various Wilson coefficients one-by-one. Although typical BSM theories give rise to simultaneous contributions to several Wilson coefficients, strong cancellations are typically not present. In such situation the bounds obtained on single Wilson coefficients remain approximately valid.11 11 Bounds on effective operators coming from measurements of the electron EDM were previously derived in the literature in refs. [13, 19–22]. – 16 – 3.2 Leptoquarks and extra Higgs As a first example of an application of the above EFT analysis, we focus here on new- physics models containing states of table 3. In particular, we focus on leptoquarks and heavy Higgs-like states.12 As can be seen from table 3, four leptoquark multiplets can give rise to electron EDM contributions up to two-loop order.13 Among scalar leptoquarks only the R2 and the S1 multiplets give rise to contributions to de . In both cases the contributions arise at one-loop level and include a logarithmically-enhanced term. On the other hand, vector leptoquarks, in particular the V2 and the U1 multiplets can contribute to the electron EDM at one-loop JHEP04(2019)090 order only with finite contributions. We analyze the various cases in the following, providing the matching with the EFT operators in the limit of heavy multiplet masses. For the scalar leptoquarks, we also com- pare the leading running contributions to the EDM with the full results, which are already known in the literature. For simplicity we only include couplings to third-generation quarks, since interactions with the light generations give rise to EDM contributions suppressed by the light-fermion Yukawa couplings. 3.2.1 Scalar leptoquarks We start our discussion with the case of scalar leptoquarks. The R2 leptoquark. The first case we consider is the R2 multiplet, whose SU(3)c × SU(2)L × U(1)Y quantum numbers are (3, 2, 7/6). The Lagrangian describing the relevant leptoquark interactions with the SM fermions is L = −y2RL tR Ra εab LbL1 + y2LR eR Ra∗ QaL3 + h.c. , (3.4) where LLi and QLi labels the i-generation lepton and quark respectively. In the limit of large mass, the R2 leptoquark gives rise to a contribution to the Oluqe effective operator, namely y2LR∗ y2RL∗ LR 2 eff = Oluqe + h.c. . (3.5) m2R2 Using eq. (2.24), we can obtain the log-enhanced one-loop contribution to the elec- tron EDM: de 1 mt  m2 ' 2 2 Im y2LR y2RL ln R22 . (3.6) e 8π mR2 mt The full one-loop contribution to the electron EDM is also known in the literature [27] de 3 mt LR RL  2 2  2 2  = 2 Im y 2 y 2 Q t I 2 mt /m R + Q LQ J 2 mt /m R , (3.7) e 32π 2 mR2 2 2 12 Notice that, in addition to the electron EDM, leptoquarks and heavy Higgs-like states, as well as supersymmetric scenarios, can also be constrained by the EDM of 199 Hg atom through the CP-odd electron- nucleon interaction [23, 24]. 13 See ref. [25] for a review of leptoquark properties and for the nomenclature. See also ref. [26] for a recent reappraisal of the contributions of the scalar leptoquarks to electron and light quark EDMs. – 17 – where Qt = 2/3 and QLQ = 5/3 are the electric charges of the top quark and R2 leptoquark, while the I2 and J2 functions are given by 1 1 I2 (x) = 3 (−3 + 4x − x2 − 2 ln x) , J2 (x) = (1 − x2 + 2x ln x) . (3.8) (1 − x) (1 − x)3 One can easily check that the leading logarithmic term in eq. (3.8) agrees with the result of the EFT calculation in eq. (3.6). In fact, the leading-log contribution in eq. (3.6) provides a quite good approximation of the full result even for relatively small leptoquark masses. The discrepancy is below 25% for mR2 > 300 GeV and below 10% for mR2 > 360 GeV. The recent bound on the electron EDM eq. (1.1) translates into the constraint14 s JHEP04(2019)090   |Im(y2LR y2RL )| |Im(y2LR y2RL )| mR2 & 420 TeV 1 + 0.075 ln , (3.9) ye yt ye yt where we have normalized y2LR y2RL to the electron and top Yukawa coupling, following the estimates presented in eq. (3.3). The S1 leptoquark. The second leptoquark state that can give rise to one-loop contri- butions to the electron EDM is the S1 multiplet, which has (3, 1, 1/3) quantum numbers. Its interactions with the SM fermions can be parametrized by Ca L = y1LL QL3 S1 εab LbL1 + y1RR tR S1 eR + h.c. , (3.10) T where the C superscript denotes the charge conjugation operation, namely ψ C ≡ Cψ with (1) C = iγ 2 γ 0 . Integrating out S1 gives rise to a contribution to Oluqe and O`equ , namely y LL∗ y RR h (1) i LSeff1 = 1 2 1 Oluqe + Olequ + h.c. . (3.11) mS 1 Therefore, from eq. (2.24), we obtain the following log-enhanced one-loop contribution to the electron EDM de 1 mt  m2 ' 2 2 Im y1LL y1RR∗ ln S21 . (3.12) e 8π mS1 mt The full one-loop result reads [25] de 1 mt   = 2 2 Im y1LL y1RR∗ G m2t /m2S1 , (3.13) e 32π mS1 where the G function is defined by 1 2  G(x) = 5 − 8x + 3x + 2(2 − x) ln x . (3.14) (1 − x)3 We find that the leading-log approximation in eq. (3.12) is in fair agreement with the full result, the difference being . 30% for mS1 & 220 GeV. Eq. (1.1) translates into the bound s   |Im(y1LL y1RR∗ )| |Im(y1LL y1RR∗ )| mS1 & 400 TeV 1 + 0.081 ln . (3.15) ye yt ye yt 14 Notice that eq. (3.9), as well as eqs. (3.15), (3.21), (3.24), (3.31), (3.32), (3.33) and (3.34), are valid as far as the mass bound is & 1 TeV. Below this value the leading-log approximation is not accurate since threshold effects can become relevant. – 18 – 3.2.2 Vector leptoquarks We now consider the case of vector leptoquarks. Before specializing the discussion to the V2 and U1 cases, we discuss some generic features of these models. As we already mentioned, vector leptoquarks can give rise to a finite contribution to the electron EDM at the 1-loop order. In order to compute the EDM effects, one needs first of all to embed the vector leptoquark into a well-behaved (i.e. renormalizable) UV theory. 15 For this purpose one can consider GUT-like extensions of the SM, as done in refs. [28, 29], which always lead to the following Lagrangian for the couplings of the Leptoquark Vµ to the photon Aµ JHEP04(2019)090 h i L ⊃ −i e QV (∂µ Vν† − ∂ν Vµ† )Aµ V ν − (∂µ Vν − ∂ν Vµ )Aµ V ν† + (Vµ Vν† − Vν Vµ† )∂ µ Aν , (3.16) where QV is the leptoquark electric charge. The couplings of a vector leptoquark to the electron and a quark q can be parametrized as L ⊃ gL ēL γ µ qL Vµ + gR ēR γ µ qR Vµ + h.c. . (3.17) The 1-loop contribution to the electron EDM is given by de Nc (QV − Qq ) mq ? = Im(gL gR ), (3.18) e 8π 2 m2V where mq and Qq are the mass and the electric charge of the quark, while mV is the leptoquark mass. Notice that contributions come from two type of diagrams, one in which the photon is attached to the quark line and one in which it is attached to the leptoquark line. The two contributions are therefore proportional to Qq ad QV respectively. Within our EFT description, these contributions must be matched directly into the Wilson coefficients CeW and CeB at the scale mV . Notice, however, that the leptoquark gives also rise to effective 4-fermion interactions of the form (L̄aL γµ QL a )(b̄R γ µ eR ) that after Fierzing leads to Oledq̄¯ . As we discussed, the Oledq̄ ¯ operator however do not contribute at 1-loop to the electron EDM but at the 2-loop order.16 The V2 leptoquark. Let us now consider V2 multiplet with quantum numbers (3, 2, 5/6). The relevant interactions with the SM fermions read C Ca L = xRL µ a ab b LR µ ab b 2 bR γ V2,µ ε LL1 + x2 QL3 γ ε V2,µ eR + h.c. . (3.19) Using the above formulae we find the following 1-loop contribution to the electron EDM de 5 mb = − 2 2 Im(xLR RL? 2 x2 ). (3.20) e 8π mV2 15 If the vector leptoquarks are described through the non-renormalizable Proca Lagrangian, divergent contributions are obtained for the EDM [29]. 16 If the 4-fermion operators are written in vector-current form and MS regularization is used, the 1- loop contribution to the electron EDM proportional to the quark charge would be matched onto the Wilson coefficient of (L̄aL γµ QL a )(b̄R γ µ eR ). We checked that the finite 1-loop contribution coming from this operator indeed matches the Qq term in eq. (3.18). – 19 – The new bound from the ACME collaboration leads to s Im(xLR 2 x2 RL? ) mV2 & 5.5 TeV . (3.21) ye yb The U1 leptoquark. The second possible vector leptoquark that gives a contributions to the electron EDM is the U1 state, with quantum numbers (3, 1, 2/3). Its Lagrangian reads a L = xLL µ a RR µ 1 QL3 γ U1,µ LL1 + x1 bR γ U1,µ eR + h.c. . (3.22) The 1-loop contributions to the electron EDM are given by . JHEP04(2019)090 de 1 mb = − 2 2 Im(xRR LL? 2 x2 ). (3.23) e 8π mU1 The bound in eq. (1.1) leads to s Im(xRR 2 x2 LL? ) mV2 & 2.5 TeV . (3.24) ye yb 3.2.3 The heavy Higgs The last type of massive multiplets we consider are Higgs-like states H with quantum numbers (1, 2, 1/2).17 Depending on their allowed couplings to the SM fermions they can give rise to different sets of contributions to the electron EDM. In general a Higgs-like multiplet can have couplings to all SM fermion species. For sim- plicity we consider only couplings to the electron family and to third generation fermions, discarding flavor-violating couplings. We parametrize the relevant heavy Higgs interac- tions as18 a e a QaL tR + κb H a QaL bR + κτ H a LaL τR + η(H † h)|h|2 + h.c. , (3.25) L = κe H a L L 1 e R + κt H 3 3 3 where He a ≡ εab H b∗ and h denotes here the SM Higgs doublet. In the limit of heavy mass, integrating out the Higgs-like state gives rise to three effective operators that contribute to the electron EDM, namely κe κt (1) κe κ∗b κe κ∗τ κe η LH eff = − 2 O lequ + 2 O ¯ ledq̄ + 2 Oleτ̄ l̄ + 2 Oye , (3.26) mH mH mH mH a where Oleτ l = (LL1 eR )(τ R LaL3 ). As we saw in the previous sections all the effective opera- tors in eq. (3.26) give rise to a contribution to the electron EDM at two-loop order. There (1) are however some important differences. The Olequ operator can potentially lead to the largest contribution, since its effects are enhanced by a double logarithm. The remaining two operators lead instead to the single logarithmic. 17 We are working in the basis at which the SM Higgs has no mass-mixing with the heavy Higgs-like states before EW symmetry breaking. This can always be achieved by an SU(2)L rotation. 18 For simplicity we neglect a possible coupling (H † h)2 + h.c., which could give a finite 1-loop contribution to the operators OV Ve that, in turn, give rise to a running for the electron EDM. – 20 – The leading contributions to the electron EDM from the four effective operators are given respectively by de 3g 2 + 5g 02 mt 2 mH 2 ' Im(κ e κ t ) ln , (3.27) e 2(16π 2 )2 m2H m2t  2  de 4 mb ∗ e m2H Z Z m2H g2 m2H '− Im (κe κb ) ln 2 + ge gb ln 2 − ln , (3.28) e (16π 2 )2 m2H 3 mb mZ 16 m2W   de 4 mτ ∗ 2 m2H Z 2 m2H g2 m2H '− Im (κe κτ ) e ln 2 + (ge ) ln 2 + ln , (3.29) e (16π 2 )2 m2H mτ mZ 16 m2W JHEP04(2019)090 and √ de 3g 2 t2θW 2v m2H '− Im (κ e η) ln . (3.30) e 8(16π 2 )2 m2H m2h These two-loop results agree with the Barr-Zee results [15, 30]. The new bound on the electron EDM leads to the constraints s   |Im(κe κt )| |Im(κe κt )| mH & 66 TeV 1 + 0.11 ln , (3.31) ye yt ye yt s   |Im(κe κb )| |Im(κe κb )| mH & 0.88 TeV 1 + 0.036 ln , (3.32) ye yt ye yt and s   |Im(κe κτ )| |Im(κe κτ )| mH & 1.4 TeV 1 + 0.092 ln , (3.33) ye yt ye yt for a Higgs coupling to top, bottom and tau respectively. In the presence of a non-vanishing η, the contribution to the CP-violating electron Yukawa leads to the additional bound s   |Im(κe η)| |Im(κe η)| mH & 4.9 TeV 1 + 0.18 ln . (3.34) ye ye Notice that this bound, for η ∼ 1, is significantly stronger than the ones derived in the presence of couplings to the bottom or τ , but is much weaker than the one expected in the presence of a sizeable coupling to the top quark. 3.3 The MSSM We will work within the MSSM assuming that the superpartner masses are larger than the EW scale. CP-violating phases can appear in several terms of the MSSM. Either in the supersymmetric parameter µ (that corresponds to the Higgsino mass), or in soft super- symmetry breaking terms: Bino and Wino masses, M1 and M2 respectively, Higgs mixing e L ũR . Nevertheless, only mass term, m212 Hu Hd , and the scalar trilinears, e.g. yu Au Hu Q those combinations of MSSM parameters whose phase cannot be removed by redefinitions of fields can lead to physical CP-violating effects. A recent analysis of the impact of the new ACME bound on the MSSM can be found in ref. [31]. – 21 – The main contribution from the MSSM arise from one-loop contributions to CeW and CeB , that generate an electron EDM calculated long ago -see for example [32]. From Winos and left-handed selectrons (L̃L ), we have " ! !# de g 2 me Im(M2 µ) M22 µ2 '− tan β I2 − I2 , (3.35) e 16π 2 m2 |M2 |2 − |µ|2 m2L̃ m2L̃ L̃L L L where I2 is defined in eq. (3.8). These effects are generated at Λ ∼ mass of the superpart- ners. Taking tan β sin(Arg[µM2 ]) ∼ 1, we get from the new bound eq. (1.1) JHEP04(2019)090 mL̃L & 25 (50) TeV , (3.36) for mL̃L = M2 = µ (mL̃L  µ = M2 ). At the two-loop order, we can get contributions from other regions of the parameter space of the MSSM. For example, Wino-Higgsino loops can induce the Wilson coefficients CW Wf , CB B e (the Bino contribution is much smaller for M1 ∼ M2 ), that, contrary to e , CW B the one-loop eq. (3.35), do not involve a L̃L . These are give by −8 + 27ρ − 24ρ2 + 5ρ3 + 6ρ2 ln ρ CW W f = Cloop , 16(ρ − 1)3 ρ(11 − 16ρ + 5ρ2 − 2(ρ − 4) ln ρ) CB Be = t2θW Cloop , 16(ρ − 1)3 ρ(7 − 8ρ + ρ2 + 2(ρ + 2) ln ρ) CW Be = tθW Cloop , (3.37) 8(ρ − 1)3 where ρ ≡ |M2 /µ|2 and g 4 sin 2β sin ϕ Cloop ≡ , ϕ = Arg[m212 µ∗ M2∗ ] . (3.38) 16π 2 |M2 µ| Using eq. (3.37) and eq. (2.12), we can obtain the contribution to the electron EDM in agreement with ref. [33] in the large log-approximation. From the ACME bound eq. (1.1), we get a limit on the Wino and Higgsino masses that can be approximately written as p |M2 µ| & 4 TeV , (3.39) where we have taken sin 2β ∼ sin ϕ ∼ 1. Another type of two-loop contributions to the electron EDM can arise from one-loop contributions to Cluqe . From loops involving selectrons, squarks, Winos and Higgsinos (see figure 3), we have 3g 2 Im[µM2 ] Im Cluqe = −ye yu F (m2i ) , (3.40) 16π 2 sin 2β where X m2i ln m2i F (m2i ) = − , (3.41) i Πi6=j (m2i − m2j ) – 22 – Figure 3. Feynman diagram of the MSSM contribution to Oluqe . JHEP04(2019)090 with i running over the mass of the Higgsino, Wino, ũR and L̃L . These results are valid for any quark generation u → u, c, t. For equal superpartner masses, we have F (m2i ) = 1/(12m4i ), and eq. (2.24) and the ACME bound lead to mi & 7.5 TeV , (3.42) for sin(Arg[µM2 ])/ sin 2β ∼ 1. Notice that, contrary to eq. (3.35) and eq. (3.38), the sin 2β appears in eq. (3.40) in the denominator and therefore becomes larger for small values of tan β. 3.4 Composite Higgs As a last example we consider the class of composite Higgs models. For definiteness we focus on minimal scenarios based on the SO(5) → SO(4) symmetry breaking pattern, which gives rise to a single Higgs doublet [34]. Depending on the implementation of the flavor structure different contributions to the electron EDM can arise. In models based on the anarchic partial compositeness paradigm naively extended to both quark and lepton sectors [35–38], large contributions arise at the one-loop level due to the presence of partners of the SM leptons and/or composite vector resonances. These contributions are generated at the mass scale of the composite states and can be estimated as [39, 40] de 1 me ∼ 2 2 , (3.43) e 8π f where f denotes the Goldstone Higgs decay constant (or equivalently the scale of sponta- neous SO(5) → SO(4) symmetry breaking). The new ACME result implies a severe bound on the compositeness scale f & 107 TeV , (3.44) which pushes these scenarios into highly fine-tuned territory. The one-loop contributions to the electron EDM can be efficiently suppressed by either introducing flavor symmetries (in particular a U(2) family symmetry involving the light fermion generations [41, 42]) or generating the light fermion masses by a bilinear f¯f mixing with the strong sector [43]. In both cases the leading corrections to the electron EDM arise at the two-loop level due to the presence of relatively light fermionic partners of the top quark [44–47]. – 23 – In a large set of minimal models,19 only derivative interactions involving the Higgs and the top partners give rise to CP-violating effects. Using a CCWZ notation (see ref. [40] for a review of the CCWZ formalism), a typical representative of such operators is given by ct diµ Ψ1 γ µ Ψi4 + h.c. , (3.45) where diµ denotes the CCWZ d-symbol, while Ψ1,4 are composite fermions in the singlet and fourplet SO(4) representation respectively. The ct coefficient is in general complex, thus containing a CP-violating phase. The two-loop corrections to the electron EDM arises from Barr-Zee-type diagrams and JHEP04(2019)090 contain a leading, log-enhanced contribution. The origin of the latter can be traced back to a two-step evolution. At the energy scale of the top partners a finite contribution to the OW Wf , OB B e and OW B e operators is generated, which then according to eq. (2.9) induces a running for the electron EDM [49]. As an explicit example we report the results for the 14 + 1 model with a light SO(4) fourplet and a fully composite right-handed top.20 By integrating out the heavy top- partners, we obtain, at leading order in the v/f expansion, Nc g 2 3 2 c T CW W f = (T ) , 16π 2 u v Nc g 02 2 cT (3.46) CB Be = Y , 16π 2 Q v Nc gg 0 cT CW Be = (−2Tu3 YQ ) , 16π 2 v while at the tree-level, that will be relevant later, we get √ Cyt = i 2yt cT , (3.47) where we have defined √ 2vyL4 yLt cT ≡ Im ct , (3.48) m2T with mT being the mass of the charged-2/3 top partner T , and yL4 , yLt the mixing of the QL doublet with the composite states as defined in ref. [49]. Notice that the contribution to Cyt is purely imaginary, i.e. CP-violating. Using the RGE eq. (2.12), we obtain an electron EDM given by " # de N c ye 2 2 g 2 3 2 2 1 2 3 m2T =− √ c T e Q u + Q u (T u − Q u s θW )(1 − 4s θW ) + g T Y u Q ln . (3.49) e 64π 4 2 4c2θW 2 m2t 19 These models are the ones in which only one SO(4)-invariant effective operator exists which gives rise to the Yukawa couplings. This happens, for instance, in the original holographic MCHM theories [34, 48], as well as in “minimally tuned” scenarios with a fully-composite right-handed top quark [47]. 20 For more details on the model see refs. [47, 50]. – 24 – ������ �� �� (���) ��� ��� �� ��� �� ��[�� ] ��� �� JHEP04(2019)090 ��� � � ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� Figure 4. Bounds on the mass of the top partner derived from the recent ACME results. The bounds are derived by setting f = 800 GeV, although the dependence on f is quite mild. The labels on the solid lines show the top partner mass mT in TeV. The three terms in square brackets come from diagrams containing a virtual photon, a virtual Z-boson and a virtual W -boson respectively. Since, as we said, the Z-boson vector coupling to the electron is quite suppressed, the main contribution is coming from the photon loop, whereas the W -boson term gives a ∼ 40% correction. In eq. (3.49) the RG running of the EDM starts at mT and stops at the top mass. At that scale, indeed, we have to integrate the top, inducing an additional finite contribution to CF Fe due to eq. (3.47). Surprisingly,21 this contribution exactly cancels the one coming from top partners loops, eq. (3.46), so that no net contribution to OF Fe is left below mt . In figure 4 we give the constraints on the mass of the T top partner in the (yL4 , Im[ct ]) plane. To derive these bounds we assumed that the result in eq. (3.49) provides the main correction to the electron EDM and no additional contributions (or at least no strong cancellations) are present. We can see that for natural values of the parameters of the theory, yL4 ∼ 1 and Im[ct ] ∼ 1, the bounds from the electron EDM measurements can exclude top partner masses up to ∼ 15 TeV. This bound is significantly stronger than the current direct LHC exclusions and cannot be matched even in the high-luminosity LHC runs [51]. 21 As noticed in ref. [49], the cancellation of the contributions to the OF Fe operators at low energy is a direct consequence of the fact that the derivative Higgs operator in eq. (3.45) induces purely off-diagonal couplings with the composite fermions. For this reason the trace of the coupling matrix vanishes and the top loop exactly cancels the contributions from the top partners. The cancellation is rather generic and happens in a large class of models. Indeed, since the dµ CCWZ symbol transform non-trivially under SO(4) (it is in the representation 4), it can only give rise to couplings involving fermions in two different SO(4) representations, which are therefore purely off-diagonal. – 25 – 4 Conclusions We performed a two-loop analysis of the EDM of the electron using the EFT approach. In particular, we calculated the RGEs of the dimension-6 CP-violating dipole operators at the two-loop order,22 as well as the one-loop RGEs of most relevant dimension-8 operators. We have shown that, due to selection rules, few operators can mix with EDM operators, as appreciated in figure 1 where we present a summary of which and how dimension-6 opera- tors enter into the EDM. We also commented on the RG running of the Wilson coefficients below the electroweak scale, and when CP-violating electron-nucleon interactions can be competitive with bounds on the electron EDM. JHEP04(2019)090 These results are important to provide a proper interpretation of the new ACME bound on the electron EDM in terms of constraints on BSM particles. The recent improvement on the bound allows to constrain TeV new-physics even when it only contributes at the two-loop level. We have shown this with some examples. In particular, we considered theories with leptoquarks or extra Higgs, obtaining bounds ranging 1 − 100 TeV. We also considered two of the most motivated BSM scenarios for TeV new-physics, supersymmetry and composite Higgs. We first reinterpreted previous calculations in the EFT language. Then, we used our RGE two-loop results to understand which sectors or which new regions of the parameter space of these BSM are now constrained by the recent ACME result. In the MSSM case, for example, we showed how our two-loop results can provide new constraints on the small tan β region in the s-electron and wino sector. For the composite Higgs, after reinterpreting calculations on top-partners in the EFT language, we showed that bounds on these particles put them out of the reach of the LHC, unless they have CP-conserving couplings. Therefore, we conclude that, unless we find a reason of why, contrary to the SM, the interactions in these BSM do respect CP, the ACME result makes these theories much less natural. More importantly, future improvement on the the electron EDM bound (see for example [53–55]) could constrain BSM beyond the reach of future colliders. Acknowledgments We thank G. M. Pruna and L. Vecchi for useful discussions. A.P. has been partly supported by the Catalan ICREA Academia Program, and grants FPA2014-55613-P, FPA2017-88915- P, 2014-SGR-1450 and Severo Ochoa excellence program SEV-2016-0588. A EDM contributions from Barr-Zee diagrams In this appendix we report the full expressions for the contributions to the electron EDM coming from CP-violating Higgs Yukawa’s originating from H 3 f¯f operators. 22 We have only calculated the leading effect to the EDM for each Wilson coefficient Ci up to the two-loop order. This means that we have not included, for example, self-renormalization effects, neither two-loop effects from Wilson coefficients entering in the renormalization of the EDM at the one-loop level. These effects are only expected to correct the derived bounds by less than O(1), and could be easily incorporated if needed (see for example [52] for the case of the magnetic dipole moment of the muon). – 26 – Before giving the formulae for the EDM contributions, it is useful to introduce a few definitions. We define the j(r, s) function as [30]   1 r ln r s ln s j(r, s) ≡ − . (A.1) r−s r−1 s−1 The Barr-Zee results with a CP-violating quark Yukawa can be rewritten in terms of the following integrals [56] Z 1   1 a F1 (a, 0) ≡ dx j 0, 0 x(1 − x) x(1 − x)   √   √  2 1 − 1 − 4a 1 + 1 − 4a JHEP04(2019)090 = √ Li2 1 − − Li2 1 − , (A.2) 1 − 4a 2a 2a and Z 1   1 a 1 F1 (a, b) ≡ dx j b, = [F1 (a, 0) − F1 (a/b, 0)] . (A.3) 0 x(1 − x) x(1 − x) 1−b Notice that the result in eq. (A.3) follows immediately from the relation j(r, s) = [j(0, s) − j(0, s/r)]/(1 − r). For the diagrams involving a CP-violating electron Yukawa we need the following integrals Z 1   (1 − x)2 + x2 a F2 (a, 0) ≡ dx j 0, = (1 − 2a)F1 (a, 0) + 2(ln a + 2) , (A.4) 0 x(1 − x) x(1 − x) and Z 1   (1 − x)2 + x2 a F2 (a, b) ≡ dx j b, 0 x(1 − x) x(1 − x)    1 a a  a = (1 − 2a)F1 (a, 0) − 1 − 2 F1 , 0 + 2 ln . (A.5) 1−b b b b The expansion of the F1 (a, 0) integral for large a is given by 1 F1 (a, 0) ' (2 + ln a) , (A.6) a while for small a one finds π2 F1 (a, 0) ' ln2 a + . (A.7) 3 We can now give the results for the Barr-Zee contributions to the electron EDM. The contribution from a Oyf operator for a generic quark or a heavy lepton is given by ! de Nc mf me Im Cyq X V V m2f m2V = Qf v ge gf F1 , , (A.8) e 64π 4 m2h Λ2 m2h m2h V =γ,Z V denote the vector couplings of the gauge boson V , namely g γ = eQ for the where ge,f f f photon and gfZ = g/(2cθW )(Tf3 − 2Qf s2θW ). The contribution from the Oye operator is instead  2  de Nc m2t Im Cye X V V mt m2V = Qt v ge gt F2 , , (A.9) e 64π 4 m2h Λ2 m2h m2h V =γ,Z where we only included the contributions from top loops, since the ones from the other fermions are suppressed by the small Yukawa’s. – 27 – Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. 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