On the Exotic Higgs Decays in Effective Field Theory

Eur. Phys. J. C (2016) 76:514 DOI 10.1140/epjc/s10052-016-4362-8

Regular Article - Theoretical Physics

On the exotic Higgs decays in effective ﬁeld theory

Hermès Bélusca-Maïtoa, Adam Falkowskib Laboratoire de Physique Théorique, Université Paris-Sud, Bat. 210, 91405 Orsay, France

Received: 22 April 2016 / Accepted: 9 September 2016 / Published online: 22 September 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com

Abstract We discuss exotic Higgs decays in an effec- in powers of 1/. The leading effects are expected from tive field theory where the Standard Model is extended by operators of dimension 6, as their coefficients are suppressed dimension-6 operators. We review and update the status of by 1/2. The first classification of dimension-6 operators two-body lepton- and quark-flavor-violating decays involv- was performed in Ref. [1]. For one generation of fermions, ing the Higgs boson. We also comment on the possibility of a complete non-redundant set (henceforth referred to as a observing three-body flavor-violating Higgs decays in this basis) was identified and explicitly written down in Ref. [2]. context. Reference [3] extended this to three generations of fermions, in which case a basis is characterized by 2499 independent parameters. Contents In this paper we are interested in the subset of these operators that lead to exotic decays of the 125 GeV Higgs 1 Introduction ...... 1 boson. By “exotic” we mean decays that are forbidden in 2 Exotic Higgs couplings from dimension-6 Lagrangian 2 the SM or predicted to occur with an extremely suppressed 3 Two-body Higgs decays ...... 3 branching fraction. More specifically, we are interested in 3.1 Lepton-flavor-violating decays ...... 3 decays that violate the lepton flavor or quark flavor. Lepton- 3.2 Flavor-changing top quark decays ...... 5 flavor-violating (LFV) processes are completely forbidden 4 Three-body decays ...... 6 in the SM in the limit of zero neutrino masses. Quark-flavor- 4.1 h → Wbq ...... 6 violating (QFV) Higgs decays as a flavor-changing neutral 4.2 h → 12γ ...... 6 current process are forbidden in the SM at tree level. The- 4.3 h → 12 Z ...... 7 oretical studies of exotic Higgs decays have a long history; 4.4 t → hqV ...... 8 see e.g. Refs. [4–11] and [12] for a review. Most of these 5 Conclusions ...... 9 papers assume new light degrees of freedom, in which case References ...... 9 the EFT approach described here is not adequate. On the other hand, Refs. [13,14] recently studied the possibility of LFV and QFV two-body decays of the 125 GeV Higgs within 1 Introduction the EFT framework. Such decays can arise in the presence of Yukawa-type dimension-6 operators [15]. These papers If new particles beyond the standard model (SM) are much demonstrated that LFV Higgs decays to τ ±μ∓ and τ ±e∓ heavier than 100 GeV, physics at the weak scale can be with the branching fraction as large as 10 % are allowed by described by an effective field theory (EFT) with the SM current indirect constraints. At the same time, the LHC is Lagrangian perturbed by higher-dimensional operators. The currently sensitive to branching fractions of order 1 % [16]. latter encode, in a model-independent way, possible effects This corresponds to probing the scale suppressing the corre- of new heavy particles at energies well below the new physics sponding dimension-6 operators at the level of ∼ 10 TeV. scale . The EFT framework allows for a systematic expan- The goal for this paper is to extend this study to a full set of sion of these effects in operator dimensions or, equivalently, dimension-6 operators. Apart from the Yukawa-type operators, exotic Higgs decays can arise in the presence of vertex- † ¯ μ ¯ μν a e-mail: [email protected] type ∼ H Dμ Hψγ ψ and dipole-type ∼ Hψσ ψ Fμν b e-mail: [email protected] operators. We systematically discuss these operators and the 123 514 Page 2 of 10 Eur. Phys. J. C (2016) 76 :514 new Higgs decay channels that they imply. The structure of The gauge couplings of SU(3) × SU(2) × U(1) are denoted the dimension-6 Lagrangian then implies certain relations by gS, gL , gY , respectively; we also define the electromag- between these Higgs couplings, as well as relations between = / 2 + 2 netic coupling e gL gY gL gY , and the weak angle single-Higgs interactions and Lagrangian terms without a sθ = g / g2 + g2 . The Higgs doublet H acquires the Higgs that affect precision observables. We give the limits Y L Y√ on each of the couplings from precision tests of the SM. That VEV H=(0,v/ 2), where v ≈ 246.2GeV.Wealso ˜ = ∗ information can be explored to place limits on the allowed define Hi ijH j . After electroweak symmetry break- ± magnitude of the Higgs couplings. We will discuss the maxi- ing, the gauge√ mass eigenstates are defined as W = mum exotic Higgs branching fraction that these limits permit. (W 1 ∓ iW2)/ 2, Z = cθ W 3 −sθ B, A = sθ W 3 + cθ B, The paper is organized as follows. In Sect. 2 we define where cθ = 1 − s2 = g / g2 + g2 . The fermions our notation and introduce the dimension-6 Lagrangian with θ L L Y = ( , † ) = (ν , ) LFV and QFV interactions in the Higgs basis. In Sect. 3 we qL uL VCKMdL and L L eL are doublets of review and update the results of Refs. [13,14] concerning the SU(2) gauge group. All fermions are three-component two-body exotic Higgs decays. In Sect. 4 we study the possi- vectors in the generation space. We work in the basis where u,d, bility of LFV and QFV Higgs decays mediated by vertex- and the fermions are mass eigenstates, thus Y are 3×3 diago- f √v nal matrices such that [Y ]ij = m f δij. The Higgs boson dipole-type operators, respectively. Obviously, studying the 2 i full parameter space of the dimension-6 Lagrangian would interactions following from Eq. (2), be an extremely difficult task. To deal with the degeneracies 2 SM h h 2 + − μ 2 μ among the parameters, one simplifying assumption we make L = + 2m W W + m Zμ Z h v v2 W μ Z throughout this paper is that the flavor-diagonal Higgs cou- 2 2 2 plings are not significantly affected by higher-dimensional h ¯ mh 3 mh 4 − m f ff− h − h , (3) operators. Furthermore, we will assume that there are no large v 2v 8v2 f fine-tuned cancellations between different parameters so as to satisfy constraints from precision experiments. In such a do not contain any LFV nor QFV couplings. constrained framework, we discuss the limits on the LFV and We move to describe the effect of dimension-6 opera- QFV Higgs couplings from various precision measurements. tors. In Eq. (1) we choose to normalize them by the elec- Given these constraints, we discuss the implications for the troweak scale v, while the new physics scale is absorbed 2 2 rate of exotic Higgs decays at the LHC. into the coefficients ci ∼ v / of these operators in the Lagrangian. A complete non-redundant LD=6 for three generations of fermions was explicitly written down in Ref. [3]. Here we work at the level of Higgs boson couplings with 2 Exotic Higgs couplings from dimension-6 Lagrangian other SM mass eigenstates, as in [17,18]. In this language, the Lagrangian is defined by a set of couplings [δy ], [δgVf], and We consider an effective theory where the SM is extended f [d ], which are in general 3×3 matrices with non-diagonal by dimension-6 operators: Vf elements for all fermion species f . A subset of these interactions violates lepton flavor and introduces tree-level flavor 1 = L = LSM + LD 6. (1) changing neutral currents for quark-flavor violation. eff v2 The first group is related to corrections to the SM Higgs Yukawa couplings in Eq. (3): We assume the SM electroweak symmetry is linearly real- L ized. This implies eff contains local operators invariant D=6 h ¯ under the SU(3) × SU(2) × U(1) symmetry; in particu- L =− m m [δy ] f , f , + h.c. . hff v fi f j f ij R i L j lar, the Higgs boson h enters the Lagrangian only through f ∈u,d,e i = j gauge invariant interactions of the Higgs doublet H. The SM (4) Lagrangian in our notation takes the form

These couplings arise from dimension-6√ operators of the LSM =−1 a a μν − 1 i i μν − 1 μν form c |H|2 fHf¯ , with [c ] ∼ m m [δy ] . 4 Gμν G 4 Wμν W 4 Bμν B f f ij fi f j f ij μ The second group is related to the contact interactions +D H † Dμ H + μ2 H † H − λ(H † H)2 H between the Higgs boson, fermions, and the massive SU(2) + ¯ γ μ + ¯ γ μ i fL Dμ fL i f R Dμ f R vector bosons: f ∈q, f ∈u,d,e − ˜ † ¯ u + † ¯ d + † ¯ + . . . 2 H u RY qL H dRY VCKMqL H eRY L h c D=6 gL h + μ Wq L = √ + W u¯ , γ [δg ] d , hV f f 1 v μ L i L ij L j (2) 2 i = j 123 Eur. Phys. J. C (2016) 76 :514 Page 3 of 10 514

+¯ γ μ[δ Wq] +¯ν γ μ[δ W] + . . u R,i gR ijdR, j L,i gL ijeL, j h c ing these parameters in the low-energy EFT is expected to ⎡ exhibit some sort of chiral suppression. Exactly this pattern 2 2 2 h ⎣ ¯ μ will arise from models following the minimal flavor-violation + g + g 1 + Zμ fL,i γ L Y v paradigm, where all sources of flavor violation are propor- ij f ∈u,d,e,ν ⎤ tional to the SM Yukawa matrices. Although, more generally, Zf ¯ μ Zf ⎦ the chiral suppression does not have to be proportional to the [δg ] f , + f , γ [δg ] f , (5) L ij L j R i R ij R j fermion masses, isolating the mass factor leads to a more f ∈u,d,e transparent picture for natural values of these parameters. where [δgVf] are Hermitian matrices. These couplings arise For the Yukawa interactions, the off-diagonal couplings can ¯ μ from dimension-6 operators of the form H † Dμ H f γ f . be more readily compared to the diagonal ones which, in this The gauge symmetry of the dimension-6 Lagrangian implies normalization, are just equal to 1 in the SM limit. δ Wq = δ Zu − δ Zd δ W = δ Zν −δ Ze In the rest of this paper, we discuss LFV and QFV exotic gL gL VCKM VCKM gL and gL gL gL . Furthermore, it implies that the Higgs boson enters via Higgs decays induced by the operators in Eqs. (4), (5), and (1+h/v)2. Therefore, the strength of the Higgs contact inter- (6). As mentioned before, we assume that the flavor-diagonal actions of this form is correlated with vertex corrections to Higgs couplings are not significantly affected by higher- the W and Z boson interactions with fermions. dimensional operators,1 and that there are no large fine-tuned Finally, we also consider the dipole-type Higgs interac- cancellations between different parameters so as to satisfy tions: constraints from precision experiments. In such a constrained + /v framework, we discuss the limits on the LFV and QFV Higgs LD=6 =−1 h dipole v2 couplings from various precision measurements. With these ⎡ assumptions, we give the limits on the couplings from pre- × ⎣ ¯ σ μν a[ ] a cision experiments and discuss the maximum exotic Higgs gS m fi m f j f R,i T dGf ij fL, j Gμν i = j f ∈u,d branching fractions allowed. + ¯ σ μν [ ] e m fi m f j f R,i dAf ij fL, j Aμν f ∈u,d,e 3 Two-body Higgs decays + 2 + 2 ¯ σ μν [ ] gL gY m fi m f j f R,i dZf ij fL, j Zμν f ∈u,d,e √ μν + In this section we discuss two-body flavor-violating decays + g m m u¯ , σ [d ] d , W 2 L ui u j R i Wu ij L j μν involving the Higgs boson. Such processes are generated via

¯ μν − + m m d , σ [d ] u , W the Yukawa couplings in Eq. (4). The important point is that di d j R i Wd ij L j μν [δ ] √ the y f ij are free parameters from the EFT point of view, + ν¯ σ μν [ ] + + . ., 2gL mei me j L,i dWe ijeR, j Wμν h c and can take any value within the EFT validity range. (6) 3.1 Lepton-flavor-violating decays σ = ı [γ ,γ ] [ ] × where μν 2 μ ν , and dVf are general 3 3 matrices. These couplings are absent in the SM at the tree No experimental dedicated searches have been done so far level, but they arise from dimension-6 operators of the form for h → μe and h → τe.Forh → τμ, the 95 % CL upper ¯ μν H f σ fVμν. The gauge symmetry of the dimension-6 limit on the branching ratio was set by CMS [20] and ATLAS Lagrangian implies that the W boson dipole couplings are [21]: related to those of the Z boson and the photon: η f dWf = 2 Br(h → τμ) ≤ 1.51 % (CMS); dZf + sθ dAf , ηu = 1, ηd,e =−1. Again, it also dictates that the Higgs boson enters via (1 + h/v). Therefore, the Br(h → τμ) ≤ 1.85 % (ATLAS). (7) strength of this type of Higgs interactions is correlated with . σ the strength of dipole interactions of the SM fermions and The CMS search shows a 2 4 excess over the expected 2 ( → τμ) = . +0.39 gauge bosons. null background, Br h 0 84−0.37 %, while √ σ ( → τμ) = In Eqs. (4) and (6) we isolated the factor m fi m f j in the “excess” in ATLAS is only 1 ,Brh the Yukawa and dipole interactions. This is done for conve- . +0.62 0 77−0.62 %. A naive combination of the ATLASand CMS [δ ] nience, and we do not assume any particular pattern of y f ij results yields and [dVf]ij. The Yukawa and dipole interactions are distin- guished by the fact that they violate chirality (they allow for 1 See e.g. [19] for a discussion of D = 8 operators in this context. transitions of left-handed fermions into right-handed ones 2 This excess may possibly be related to another one observed in the and vice versa), much like the fermion mass terms in the same sign di-muon final state in the tth¯ searches in ATLAS and CMS SM. Any model addressing the flavor problem and generat- [22]. 123 514 Page 4 of 10 Eur. Phys. J. C (2016) 76 :514

( → τμ) = . +0.33 ; • τ → eγ : Br h 0 82−0.32 % Br(h → τμ) ≤ 1.47 % (ATLAS + CMS). (8) √ 2 2 CL mτ memτ m H [δy]eτ ≈ e 3ln − 4 ∗ 2 2 2 [δ ] In terms of the parameters in Eq. (4), the branching ratio CR 3m v mτ y τe H can be written as −10 −1 [δy]eτ ≈ 2.2 × 10 GeV ∗ , (14) [δy]τ ( → τμ) e Br h mμ 2 2 = |[δy]μτ | +|[δy]τμ| , (9) Br(h → ττ) mτ • τ → μγ :

√ → ττ 2 2 [δ ] where we assumed the h decay is not signiﬁcantly CL mτ mμmτ m H y μτ = . = ≈ e 3ln − 4 ∗ affected by new physics. Using mμ 105 7MeV,mτ C 2 v2 2 [δy]τμ R 3m H mτ 1.78 GeV, Br(h → ττ) = 6.3 % from the SM value, we [δ ] obtain the best ﬁt value and the 95 % CL bound on the EFT ≈ . × −9 −1 y μτ . 3 2 10 GeV [δ ]∗ (15) parameters: y τμ

|[δ ] |2 +|[δ ] |2 = . +0.88, Above, we kept only the contributions from diagrams with y μτ y τμ 2 19−0.85 (10) the τ lepton in the internal fermion line. Other contribu- |[δ ] |2 +|[δ ] |2 ≤ . . y μτ y τμ 1 98 tions are suppressed by mμ/mτ or me/mτ and can be neglected, unless there is a huge hierarchy between differ- The strongest constraints on the LFV Higgs couplings ent off-diagonal elements of [δy f ]. Such hierarchy is not come from 2 → 1γ decays [13,14]. In the SM, such pro- expected for EFT arising as low-energy approximation of cesses are completely forbidden in the limit of zero neutrino specific models where the flavor problem is addressed. Our masses, but they can be generated in the presence of D=6 results agree with Refs. [13,14]. operators. In the EFT with LFV Yukawa couplings, they It was pointed out in the literature [13,14,23,24] that cer- occur at one-loop level. The amplitude for the process is tain two-loop corrections, the so-called Barr–Zee diagrams parametrized as with a W or a top loop, may give comparable contributions as the one-loop diagrams computed above. Their analytical μν M = u(1)F2σ kνu(2) μ(k) with: forms can be found in Appendix A.2 of [14], which were adapted from the μ → eγ formulas of Chang et al. [25] and 1 1 ∓ γ5 F = (CL PL + CR PR); PL/R = , (11) Leigh et al. [26]. It turns out that Barr–Zee contributions are 2 16π 2 2 √ proportional to mi m j δyijC, where C is common for all the processes. Numerically, one finds and the decay width is given by (in the approximation m1 m2 ): • μ → eγ :

m3 2 2 2 → γ ≈ |C | +|C | . (12) C − − [δy] μ 2 1 5 L R L ≈ . × 10 1 e . 4096π 2 3 10 GeV [δ ]∗ (16) CR y μe

Evaluating the one-loop diagrams we ﬁnd the following • τ → eγ : results: [δ ] CL ≈ . × −10 −1 y eτ . 9 6 10 GeV [δ ]∗ (17) • μ → eγ : CR y τe • τ → μγ : √ 2 2 C mτ mμme m L ≈ e 2ln H − 3 C 2 v2 m2 [δ ]μτ R 2m H τ CL ≈ . × −8 −1 y . 1 4 10 GeV [δ ]∗ (18) [δ ] [δ ] CR y τμ × y eτ y τμ [δ ]∗ [δ ]∗ y μτ y τe Indeed, the two-loop contributions turn out to be dominant, [δ ] [δ ] ≈ . × −11 −1 y eτ y τμ , for τ → μγ and τ → eγ by approximately a factor of 4. 5 4 10 GeV [δ ]∗ [δ ]∗ (13) y μτ y τe For μ → eγ the ratio of two- and one-loop contributions 123 Eur. Phys. J. C (2016) 76 :514 Page 5 of 10 514

Table 1 Experimental 90 % CL upper limits on the branching fraction hand, Br(h → μe) is constrained to be small by the μ → eγ for lepton radiative flavor-violating processes constraint, so as to be unobservable in practice. Process Upper limits on Br Refs./Exp. There is also the question which explicit BSM models may generate the pattern of LFV Yukawa couplings required to μ → γ . × −13 e 5 7 10 [27](MEG) produce Br(h → τμ/e) at the level of a percent to per-mille. −8 τ → eγ 3.3 × 10 [28] (BaBar) This turns out to be difficult in concrete models. Typically, − τ → μγ 4.4 × 10 8 [28] (BaBar) satisfying all constraints is either completely impossible [30], or requires some fine-tuning and/or challenging model building [19,31–42]. depends on the ratios of the different off-diagonal Yukawa couplings. 3.2 Flavor-changing top quark decays The experimental limits on these processes obtained by the BaBar collaboration (τ → γ ), and the MEG experiment Dimension-6 operators may also violate flavor in the quark (μ → eγ ) are collected in Table 1. Using those, we find the sector. In the SM, quark flavor is not conserved due to off- following constraints on the lepton-flavor-violating Yukawa diagonal CKM matrix elements, but flavor-changing neutral couplings: currents are forbidden at tree level. Therefore, the quark- flavor-violating processes involving the Higgs boson are sup- • μ → eγ : pressed by a loop factor, and in addition suppressed by the GIM mechanism. On the other hand, the couplings in Eq. (4) 2 2 may lead to flavor-changing neutral currents at tree level. [δy] μ + 0.2[δy] τ [δy]τμ + [δy]μ + 0.2[δy]μτ [δy]τ e e e e From the experimental point of view, the most inter- ≤ 0.048. (19) esting of these processes are the ones involving the top quark. ATLAS and CMS have performed direct searches • τ → eγ : for Higgs-mediated flavor-changing neutral currents in top → = , quark decays: t hq, q c u. Due to loop and GIM suppression, the branching fractions for these decays in the SM |[δy] τ |2 + |[δy]τ |2 ≤ 109. (20) e e are prohibitively small. However, in models beyond the SM with new sources of flavor violation these decays are often • τ → μγ : enhanced to a level that may be observable at the LHC; see e.g. [43]. 2 2 In the limit of massless charm or up quarks, the tree-level [δy]μτ + [δy]τμ ≤ 8.7. (21) decay width is given by the formula:

Limits on the off-diagonal Yukawa couplings from their one- → 2 2 2 loop contribution to 2 3 1 decays are weaker [29]. m mq m (t → hq) = t 1 − h |[δy ] |2 +|[δy ] |2 . Finally, motivated by the constraints discussed above, we πv2 2 u qt u tq 32 mt write the LFV Higgs branching fractions as (23) |[δy]μτ |2 +|[δy]τμ|2 Br(h → τμ) ≈ × 1.5%, 22 This translates to the branching fractions: 2 2 |[δy]eτ | +|[δy]τe| (h → τe) ≈ × , −3 2 2 Br 2 18 % (22) Br(t → hc) = 1.1 × 10 |[δyu]ct | +|[δyu]tc| , 100 2 2 |[δy]eμ| +|[δy]μe| − −6 2 2 Br(h → μe) ≈ × 4 × 10 9. Br(t → hu) = 1.9 × 10 |[δyu]ut| +|[δyu]tu| , (24) 0.062 ≈ . We can immediately see that the indirect constraints allow where we used t 1 35 GeV. for a sizable branching fraction of h → τe, and h → τμ The current 95 % upper limits on the branching fractions decays. In particular, the percent-level branching fraction for for these decays are given in Table 2. Using these, we ﬁnd h → τμ, hinted at by the CMS excess, can be addressed in the following constraints on the off-diagonal Higgs Yukawa the EFT context without any tension with τ → μγ bounds. couplings: However, one should note that the μ → eγ constraint does 2 2 not allow Br(h → τμ) and Br(h → τe) to be simultane- |[δyu]ct | +|[δyu]tc| ≤ 2.1, ously large. Observing both of these decays at the LHC would |[δy ] |2 +|[δy ] |2 ≤ 49. (25) thus signify a breakdown of the EFT approach. On the other u ut u tu 123 514 Page 6 of 10 Eur. Phys. J. C (2016) 76 :514

Table 2 List of experimental 95 % CL upper limits on the branching been seen in LHC Run-1 in the h → ZZ → 4 channel. fraction Br for Higgs-mediated quark-ﬂavor-violating processes Thus, if t → hq decays are observed at the LHC close to Process Upper limits on Br Refs. the current limit, it should be possible to also observe the h → Wbq decays in the future (although the tt¯ background → . × −3 t ch 4 6 10 (ATLAS) [44] will be a challenge in this case). −3 t → uh 4.5 × 10 (ATLAS) [44] The same decay can also be mediated by dipole-type cou- − t → q(= c + u)h 7.9 × 10 3 (ATLAS) [45] plings in Eq. (6). Implementing the relevant vertices in Feyn- − t → ch 5.6 × 10 3 (CMS) [46] Rules [48,49] and calculating the decay width numerically in aMC@NLO [50] one ﬁnds

−4 2 −7 2 Much as for LFV Higgs decays to tau leptons, the current Br(h → Wbc) = 1.7 × 10 |[dWu]cb| + 3.3 × 10 |[dWd]st| , −7 2 −8 2 indirect constraints on δyqt and δytq do not forbid the t → hq Br(h → Wbu) = 3.0 × 10 |[dWu]ub| + 1.7 × 10 |[dWd]dt| , branching fraction to be close to the current LHC limits. (27) While the relative phase between δyqt and δytq is severely constrained by neutron electric dipole moment searches [47], where we sum over W ± decay modes. Barring ﬁne-tuning, → the absolute values (which enter into the t hq widths) are in the EFT approach the W-boson dipole couplings dWq allowed to be large. One should also mention that D-meson are related to the analogous dipole couplings of the photon, 2 oscillations place more severe constraints on the products dWf ∼ sθ dAf . Given that, the parameters dWd entering the δ δ δ δ yut ytc and ytu yct ;see[47]. Therefore, in the EFT con- expressions above are subject to stringent constraints from text, it is impossible for both t → hc and t → hu branch- B-physics, such as the measurement of the b → sγ branching fractions to be close to the current experimental lim- ing fraction, and thus their contribution to the Higgs decays its. is completely negligible. As for the dWu parameters, they are less severely constrained by the limits on Br(t → qγ)set by the CMS experiment [51], and we estimate |[dWu]cb| 0.06, 4 Three-body decays |[dWu]ub| 0.4. This implies that the maximum branching fraction for dipole-mediated h → Wbq decays at most In the previous section we discussed two-body exotic decays ∼ 10−6, which is lower than the maximum Yukawa-mediated induced by dimension-6 operators of the Yukawa type. rate, and probably too low to be observable. We concluded that indirect constraints on the LFV and

QFV Higgs Yukawa couplings to fermions are consistent 4.2 h → 12γ with the branching fractions of h → τμ and h → τe decays that are readily observable at the LHC. In fact, We move to the dipole-type operators in Eq. (6). In the lep- the best limits on the relevant couplings currently come ton sector, these may lead to h → 12γ decays, where the from the LHC. This agrees with the conclusions from presence of a hard photon in the decay would allow experi- previous literature [14]. In this section we extend this ments to distinguish it from h → mediated by Yukawa discussion to three-body exotic Higgs decays and other couplings. operators appearing at the dimension-6 level in the EFT As discussed before, the strength of Higgs dipole-type Lagrangian. interactions is ﬁxed by the strength of the corresponding dipole interaction between fermions and a gauge boson. 4.1 h → Wbq Therefore the constraint on the Higgs coupling will come from dipole-mediated 1 → 2γ decays. In the limit where ∗ We begin with the h → t q → Wbq decays. These decays the leptons are massless, the width of the latter is given by are mediated by the same Yukawa couplings as the ones leading to the t → hc/u decays, and they are constrained by 2 4 e m m ( → γ)= 1 2 [ ] 2 + [ ] 2 , ATLAS and CMS searches as in Eq. (25): 1 2 dAe 12 dAe 21 4πv4 −4 2 2 (28) Br(h → Wbc) = 1.3 × 10 |[δyu]ct | +|[δyu]tc| ,

( → ) = . × −7 |[δ ] |2 +|[δ ] |2 , +− −+ Br h Wbu 2 3 10 yu ut yu tu where we summed over the 1 2 and 1 2 decay modes. (26) Using the experimental results from Table 1, we get the following constraints on the dipole couplings: + − where we summed over the W and W modes. Note that −4 2 2 −6 Higgs decays with O(10 ) branching fractions have already [dAe]eμ + [dAe]μe ≤ 1.2 × 10 , 123 Eur. Phys. J. C (2016) 76 :514 Page 7 of 10 514 2 2 −3 Table 3 Experimental 95 % CL upper limits on the branching fraction |[dAe]eτ | + |[dAe]τe| ≤ 2.6 × 10 , Br for LFV Z boson decays 2 2 −4 [dAe]μτ + [dAe]τμ ≤ 2.1 × 10 . (29) Process Upper limits on Br Refs./Exp.

Z 0 → μe 2.5 × 10−6 [53] (DELPHI) → γ The dipole-mediated h 1 2 decay width is given by 1.7 × 10−6 [54] (OPAL) 7.5 × 10−7 [55] (ATLAS) 2 5 0 −5 e mhm1 m2 Z → τe 2.2 × 10 [53] (DELPHI) (h → 12γ) = π 3v6 −6 384 9.8 × 10 [54] (OPAL) × [ ] 2 + [ ] 2 , 0 → τμ . × −5 dAe 12 dAe 21 (30) Z 1 2 10 [53] (DELPHI) 1.7 × 10−5 [54] (OPAL) +− −+ where we summed over the 1 2 and 1 2 decay modes. Given Eq. (29), the branching fractions for dipole-mediated 2 2 3 → γ 3 (g + g )m m m h 1 2 decays are constrained as (Z → ) = L Y Z 1 2 1 2 πv4 6 2 2 −23 × [ ] + [ ] , Br(h → μeγ) ≤ 1.9 × 10 , dZe 12 dZe 21 (33) −15 Br(h → τeγ) ≤ 1.7 × 10 , (31) + − − + where we summed over the and decay modes. (h → τμγ) ≤ . × −15. 1 2 1 2 Br 2 3 10 This results in the following constraints on the dipole couplings: Unlike for Yukawa-mediated two-body decays, this time the decays with τ in the ﬁnal states are constrained to be 2 2 [dZe]eμ + [dZe]μe ≤ 76, extremely rare. As long as the EFT framework is adequate for describing Higgs decays, there is no prospect of observ- |[d ] τ |2 + |[d ]τ |2 ≤ 67, (34) Ze e Ze e ing the dipole-mediated LFV decays at the LHC or the future 2 2 100 TeV collider [52]. [dZe]μτ + [dZe]τμ ≤ 5.2. Stronger constraints on these couplings are obtained through

4.3 h → 12 Z their loop contributions to radiative lepton decays [56,57]. At one loop one ﬁnds Another process that can be generated by dipole-type inter- m4 m e2m2 (g2 + g2 ) → 1 2 Z L Y actions in Eq. (6)ish 1 2 Z. To calculate the width, (1 → 2γ) = ¯ μν 1024π 5v6 we have implemented the (1 + h/v)1σ 2Vμν vertex in 2 2 2 2 FeynRules, and calculated the decay width numerically in × 3 − 6cθ + 4cθ log cθ aMC@NLO. We obtain the branching fractions: 2 2 × [dZe] + [dZe] . (35) 1 2 2 1 −12 2 2 Br(h → μeZ) = 2.1 × 10 [dZe]eμ + [dZe]μe , Using the experimental results from Table 1, and assuming −11 2 2 no cancellations between the tree-level dAe and the one-loop Br(h → τeZ) = 3.5 × 10 |[dZe]eτ | + |[dZe]τe| , contribution from Z dipole, we get the following constraints −9 2 2 Br(h → τμZ) = 7.3 × 10 [dZe]μτ + [dZe]τμ , on dZe: (32) 2 2 −4 [d ] μ + [d ]μ ≤ 2.7 × 10 , Ze e Ze e + − − + where we summed over the and decay modes. |[d ] τ |2 + |[d ]τ |2 ≤ 0.63, (36) 1 2 1 2 Ze e Ze e Constraints on the parameters dZe come from experimen- 2 2 −2 tal limits on LFV Z boson decays summarized in Table 3. [dZe]μτ + [dZe]τμ ≤ 5.1 × 10 . The dipole mediated partial decay width is given by This translates to the constraints on the branching fractions:

− Br(h → μeZ) ≤ 1.5 × 10 19, 3 → γ − Of course, the process h 1 2 can occur with a larger branching Br(h → τeZ) ≤ 1.4 × 10 11, (37) fraction if it is mediated by off-diagonal Yukawa couplings and the − photon is emitted by one of the ﬁnal-state leptons. Br(h → τμZ) ≤ 1.9 × 10 11. 123 514 Page 8 of 10 Eur. Phys. J. C (2016) 76 :514

Table 4 Experimental 90 % CL upper limits on the branching fraction This translates to the following bounds on the branching frac- Br for four-lepton flavor-violating processes tions: Process Upper limits on Br Refs./Exp. − Br(h → μeZ) ≤ 8.1 × 10 17, − − μ → 3e 1.0 × 10 12 [58] (SINDRUM) Br(h → τeZ) ≤ 2.4 × 10 11, (42) τ → . × −8 − 3e 2 7 10 [59] (Belle) Br(h → τμZ) ≤ 1.2 × 10 11. τ → 3μ 2.1 × 10−8 [59] (Belle) The bounds are somewhat weaker than for the dipole- mediated Higgs decays, however, the suppression is still too much for any realistic prospects of experimental detection. The suppression is slightly smaller than for the decays with a photon in the final state, however, observing decays with 4.4 t → hqV this low branching fraction is impossible at the LHC or at the 100 TeV collider. Finally, we consider flavor-violating three-body decays of The same process (though with a different helicity struc- the top quark mediated by dipole-type operators: t → hqV, ture for the final-state fermions) can also be generated by where V is a photon or a gluon. We have implemented the μν μν vertex-type couplings in Eq. (5). Implementing the relevant t¯σ cVμν and ht¯σ cVμν vertices in FeynRules, and calcu- vertices in FeynRules and calculating the decay width numer- lated the decay width numerically in aMC@NLO. We find ically in aMC@NLO one finds Br(t → qVh) − ≈ 4.4 × 10 8. (43) Br(t → qV) Br(h → Z) 1 2 2 2 The current best constraints on Br(t → qγ) come from −5 Ze Ze ≈ . × [δ ] + [δ ] . 7 0 10 times gL 1 2 gR 1 2 searches for anomalous top production at the LHC. For the (38) dipole couplings to photons the strongest limits come from the CMS experiment [51]. They translate to the following limits on the branching fractions: Again, the off-diagonal vertex corrections are constrained by − LFV Z boson decays. The decay width is Br(t → uγ)≤ 1.3 × 10 4, − (44) Br(t → cγ)≤ 1.7 × 10 3. 2 2 (g + g )m Z (Z → ) = L Y For t → uγ , even stronger limits can be placed due 1 2 12π to the dipole contributions to the neutron electric dipole 2 2 × [δ Ze] + [δ Ze] . gL 12 gR 12 (39) moment [60], though these constraints do no apply when the dipole couplings are parity conserving. For the dipole couplings to the gluon, the strongest limits come from the Then the experimental constraints in Table 3 imply: ATLAS experiment [61]: Br(t → ug) ≤ 4.0 × 10−5, [δ Ze] 2 + [δ Ze] 2 ≤ . × −3, (45) gL eμ gR eμ 1 2 10 ( → ) ≤ . × −4. Br t cg 1 7 10 [δ Ze] 2 + [δ Ze] 2 ≤ . × −3, gL eτ gR eτ 4 3 10 (40) Limits on the flavor-violating dipole top couplings from tγ 2 2 −3 production at the LHC [62] are currently weaker. The experi- [δgZe]μτ + [δgZe]μτ ≤ 4.8 × 10 . L R mental bounds in Eqs. (44) and (45) translate to the following constraints on dipole-mediated top decays with the Higgs: Again stronger constraints arise through one-loop con- ( → γ ) ≤ . × −12, tributions to LFV lepton decays, for which the experimen- Br t u h 5 7 10 tal limits are collected in Table 4. Assuming no cancella- Br(t → cγ h) ≤ 7.5 × 10−11, (46) tions with tree-level contributions of 4-fermion operators, Br(t → ugh) ≤ 1.8 × 10−12, one obtains the bounds [29] Br(t → cgh) ≤ 7.5 × 10−12. + − 2 2 −6 As in the case of h → γ decays, the branching fraction [δgZe] μ + [δgZe] μ ≤ 1.1 × 10 , 1 2 L e R e → γ/ for t q gh may be larger than the limits in Eq. (46)if [δ Ze] 2 + [δ Ze] 2 ≤ . × −4, the process is mediated by off-diagonal Yukawa couplings gL eτ gR eτ 5 9 10 (41) and the photon or gluon is emitted from the final state quark. [δ Ze] 2 + [δ Ze] 2 ≤ . × −4. gL μτ gR μτ 4 1 10 Although the top production cross section is larger than that 123 Eur. Phys. J. 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