Leptoquarks in Oblique Corrections and Higgs Signal Strength: Status and Prospects

Prepared for submission to JHEP PSI-PR-20-08 , ZU-TH 20/20

Andreas Crivellina,b Dario Müllera,b Francesco Saturninoc aPhysik-Institut, UniversitätZürich,Winterthurerstrasse 190, CH-8057 Zürich,Switzerland bPaul Scherrer Institut, CH–5232 Villigen PSI, Switzerland aPhysik-Institut, UniversitätZürich,Winterthurerstrasse 190, CH-8057 Zürich,Switzerland cAlbert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of Bern, CH-3012 Bern, Switzerland E-mail: andreas.crivellin@cern.ch, [email protected], [email protected]

Abstract: Leptoquarks (LQs) are predicted within Grand Unified Theories and are well motivated by the current flavor anomalies. In this article we investigate the impact of scalar LQs on Higgs decays and oblique corrections as complementary observables in the search for them. Taking into account all five LQ representations under the Standard Model gauge group and including the most general mixing among them, we calculate the effects in h → γγ, h → gg, h → Zγ and the Peskin-Takeuchi parameters S, T and U. We find that these observables depend on the same Lagrangian parameters, leading to interesting correlations among them. While the current experimental bounds only yield weak constraints on the model, these correlations can be used to distinguish different LQ representations at future colliders (ILC, CLIC, FCC-ee and FCC-hh), arXiv:2006.10758v4 [hep-ph] 6 May 2021 whose discovery potential we are going to discuss.

Keywords: Beyond Standard Model, Phenomenological Models, Electroweak Sym- metry Breaking, Leptoquarks, Higgs Decays Contents

1 Introduction1

2 Setup and Conventions2

3 Oblique Corrections6

4 Higgs Couplings to g, γ and Z 8

5 Phenomenological Analysis 12

6 Conclusions 14

A Appendix 17 A.1 Loop Functions 17 A.2 Expanded Matrices 18 A.3 Exact Results for the Vacuum Polarization Functions 22 A.4 Leading Order SM Amplitudes in Higgs Decays 23

1 Introduction

Leptoquarks (LQs) are particles which have a specific interaction vertex, connecting a lepton with a quark. They are predicted in Grand Unified Theories [1–4] and were systematically classified in Ref. [5] into ten possible representations under the Standard Model (SM) gauge group (five scalar and five vector particles). In recent years, LQs experienced a renaissance due to the emergence of the flavor anomalies. In short, (∗) + − hints for new physics (NP) in R(D )[6–11], b → s` ` [12–17] and aµ [18] emerged, with a significance of > 3 σ [19–23], > 5σ [24–31] and > 3 σ [32], respectively. It has been shown that LQs can explain b → s`+`− data [33–57], R(D(∗))[33, 34, 36–40, 42–

44, 46, 47, 51–53, 55–90] and/or aµ [55, 56, 62, 71, 74, 77, 86, 91–107]. This strong motivation for LQs makes it also interesting to search for their signatures in other observables. Complementary to direct LHC searches [108–121], oblique electroweak (EW) parameters (S and T parameters [122, 123]) and the corrections to (eﬀective on-shell) couplings of the SM Higgs to photons (hγγ), Z and photon (hZγ)

– 1 – and gluons (hgg) allow to test LQ interactions with the Higgs, independently of the LQ couplings to fermions. In this context, LQs were briefly discussed in Ref. [124] based on analogous MSSM calculations [125–127], simplified model analysis [128–131], vacuum stability [132], LQ production at hadron colliders [133] and Higgs pair production [134]. In addition, Ref. [135] recently studied LQs in Higgs production and Ref. [136] considered h → γγ, while Ref. [137] performed the matching in the singlet-triplet model [86]. However, none of these analyses considered more than a single LQ representation at a time. The situation is similar concerning the S and T parameter. This was also briefly discussed in Ref. [124], based on simplified model calculations [138] and an analysis discussing only the SU(2)L doublet LQs [139]. Most importantly, the unavoidable correlations between Higgs couplings to gauge bosons and the oblique parameters were not considered so far. Importantly, these observables can be measured much more pre- cisely at future colliders such as the ILC [140], CLIC [141], and the FCC [142, 143]. Therefore, it is interesting to examine their estimated constraining power and discovery potential. In this article we will calculate the one-loop effects of LQs in oblique corrections, hγγ, hZγ and hgg, taking into account all five scalar LQ representations and the complete set of their interactions with the Higgs. In the next section we will define our setup and conventions before we turn to the calculation of the S and T parameters in Sec.3 and to hγγ, hZγ and hgg in Sec.4. We then perform our phenomenological analysis, examining the current status and future prospects for these observables in Sec.5, before we conclude in Sec.6. An appendix provides useful analytic (perturbative) expressions for LQ couplings and results for the loop functions.

2 Setup and Conventions

There are ten possible representations of LQs under the SM gauge group [5]. While for vector LQs a Higgs mechanism is necessary to render the model renormalizable, scalar LQs can simply be added to the SM. Since we are interested in loop effects in this work, we will focus on the latter ones in the following. The five different scalar LQs transform under the SM gauge group

GSM = SU(3)c × SU(2)L × U(1)Y (2.1) as given in Table1. We deﬁned the hypercharge Y such that the electromagnetic charge is given by 1 Q = Y + T , (2.2) 2 3

– 2 – ˜ ˜ Φ1 Φ1 Φ2 Φ2 Φ3 2 8 7 1 2 G 3, 1, − 3, 1, − 3, 2, 3, 2, 3, 3, − SM 3 3 3 3 3

Table 1: LQ representations under the SM gauge group.

with T3 representing the third component of weak isospin, e.g. ±1/2 for SU(2)L dou- blets and 1, 0, −1 for the SU(2)L triplet. Therefore, we have the following eigenstates with respect to the electric charge

−1/3 ˜ ˜ −4/3 Φ1 ≡ Φ1 , Φ1 ≡ Φ1 , ! ! √ ! Φ5/3 Φ˜ 2/3 Φ−1/3 2Φ2/3 (2.3) Φ ≡ 2 , Φ˜ ≡ 2 , τ · Φ ≡ √ 3 3 , 2 2/3 2 ˜ −1/3 3 −4/3 −1/3 Φ2 Φ2 2Φ3 −Φ3 obtained from the ﬁve representations. Note that the upper index refers to the electric charge and the lower one to the SU(2)L representation from which the ﬁeld originates.

In addition to the gauge interactions of the LQs, determined by the respective representation under the SM gauge group, LQs can couple to the SM Higgs doublet H (with hypercharge +1) via the Lagrangian [144]1

˜ † ˜ † † ˜ LHΦ = −A21˜ Φ2H Φ1 + A32˜ Φ2 τ · Φ3 H + Y22˜ Φ2H Hiτ2Φ2 † ˜ † † + Y31˜ Hiτ2 (τ · Φ3) H Φ1 + Y31 H (τ · Φ3) H Φ1 + h.c. † ˜ ˜ † − Y22 Hiτ2Φ2 Hiτ2Φ2 − Y˜˜ Hiτ2Φ2 Hiτ2Φ2 22 (2.4) † † − iY33εIJK H τI HΦ3,J Φ3,K 3 2 X 2 † † X 2 † ˜ † ˜ − mk + YkH H ΦkΦk − m˜ k + Yk˜H H ΦkΦk . k=1 k=1

2 Here mΦ represent the usual (bare) mass terms of the LQs, present without EW symmetry breaking and εIJK is the three-dimensional Levi-Civita tensor with ε123 = 1.

Note that A21˜ and A32˜ have mass dimension one, while the Y couplings are dimensionless. The LQ-Higgs interactions lead to additional contributions to the mass matrices. The mixing among them is depicted in Figure1.

1 Y2˜2˜ and Y22 were studied in Ref. [139] while the Y33 term was considered in Ref. [145].

– 3 – h h h

−1/3 ˜ −1/3 −1/3 ˜ −1/3 2/3 ˜ 2/3 Φ1 Φ2 Φ3 Φ2 Φ3 Φ2 A21˜ A32˜ A32˜

hh hh

−1/3 −1/3 ˜ 2/3 2/3 Φ3 Φ1 Φ2 Φ2 Y31 Y22˜

Figure 1: Feynman diagrams depicting LQ-Higgs interactions. Here the physical Higgs h can be replaced by its vev, leading to mixing among the LQs.

Once the Higgs acquires a vacuum expectation value (vev) with v ≈ 174 GeV, this generates the following mass matrices in the interaction basis  2 2 ∗ 2  m1 + v Y1 vA21˜ v Y31 −1/3 2 2 M =  vA21˜ m˜ 2 + v Y2˜ vA32˜  , 2 ∗ ∗ 2 2 v Y31 vA32˜ m3 + v Y3 m2 + v2Y v2Y 0  2 2 22˜ √ M2/3 = v2Y ∗ m˜ 2 + v2Y + Y − 2vA ,  22˜ 2 √ 2˜2˜ 2˜ 32˜  (2.5) ∗ 2 2 0 − 2vA m + v Y3 + Y33 √ 32˜ 3 m˜ 2 + v2Y 2v2Y ∗ −4/3 √1 1˜ 31˜ M = 2 2 2 , 2v Y31˜ m3 + v (Y3 − Y33) 5/3 2 2 M = m2 + v Y22 + Y2 , such that

† Q − ΦQM ΦQ ⊂ LHΦ . (2.6) This now parametrizes the mass terms in the Lagrangian, where Q is the electric charge and we deﬁned  −1/3  2/3 Φ Φ ! 1 2 Φ˜ −4/3 Φ ≡ ˜ −1/3 Φ ≡ ˜ 2/3 Φ ≡ 1 Φ ≡ Φ5/3 . (2.7) −1/3 Φ2  2/3 Φ2  −4/3 −4/3 5/3 2 −1/3 2/3 Φ3 Φ3 Φ3 In order to arrive at the physical basis we need to diagonalize the mass matrices in Eq. (2.5). This can be achieved via Mˆ Q = W QMQW Q† (2.8)

– 4 – with unitary matrices W Q. Thus, the interaction eigenstates in Eq. (2.7) are rotated as Q ˆ Q W ΦQ ≡ Φ (2.9) to arrive at the mass eigenstates. The matrices W Q for Q = −1/3 and Q = 2/3 too lengthy to be given analytically in full generality, but can of course be computed numerically. However, in order to obtain the explicit dependence on the Lagrangian parameters A and Y , we diagonalize the mass matrices perturbatively up to O(v2), which then yields the following expressions

 2 2 vA∗ v2(Y (m2−m˜ 2)+A∗ A )  v |A21˜ | 21˜ 31 1 2 21˜ 32˜ 1− 2 2 2 2 2 2 2 2 2 2(m1−m˜ 2) m1−m˜ 2 (m1−m3)(m1−m˜ 2)  2 2 2  −1/3 −vA21˜ v |A21˜ | |A32˜| −vA32˜ W ≈  2 2 1− 2 2 2 + 2 2 2 2 2  ,  m1−m˜ 2 2 (m1−m˜ 2) (m3−m˜ 2) m3−m˜ 2   −v2(Y ∗ (m2−m˜ 2)+A A∗ ) vA∗ 2 2  31 3 2 21˜ 32˜ 32˜ v |A32˜| 2 2 2 2 2 2 1− 2 2 2 (m1−m3)(m3−m˜ 2) m3−m˜ 2 2(m3−m˜ 2)  2  v Y22˜ 1 m2−m˜ 2 0 2 2 √  −v2Y ∗ 2 2  2/3 22˜ v |A32˜| − 2vA32˜ W ≈  2 2 1− 2 2 2 2 2  , (2.10)  m2−m˜ 2 √(m3−m˜ 2) m˜ 2−m3   2vA∗ 2 2  32˜ v |A32˜| 0 2 2 1 − 2 2 2 m˜ 2−m3 (m3−m˜ 2) √  2v2Y ∗  31˜ 1 2 2 −4/3 √ m˜ 1−m3 W ≈  2  . − 2v Y31˜ 2 2 1 m˜ 1−m3 The physical LQ masses then read 2 2 2 −1/32 2 2 |A21˜ | 2 2 |A21˜ | |A32˜| Ma ≈ m1 +v Y1 − 2 2 , m˜ 2 +v Y2˜ + 2 2 + 2 2 , m˜ 2 − m1 m˜ 2 − m1 m˜ 2 − m3 2 2 2 |A32˜| m3 + v Y3 − 2 2 , m˜ 2 − m3 a 2 2 2|A | M 2/3 ≈ m2 +v2Y , m˜ 2 +v2 Y +Y + 32˜ , a 2 2 2 2˜2˜ 2˜ 2 2 (2.11) m˜ 2 − m3 2 2 2 2|A32˜| m3 +v Y3 + Y33 − 2 2 , m˜ 2 − m3 a −4/32 2 2 2 2 Ma ≈ m˜ 1 + v Y1˜, m3 + v (Y3 − Y33) a , 5/32 2 2 M ≈ m2 + v (Y22 + Y2) , valid up to order v2, where a runs from 1 to 3 for Q = −1/3 and Q = 2/3 and from 1 to 2 for Q = −4/3, respectively.2

2For the calculation of the T parameter, we even needed the expansion of the mixing matrices and masses up to order v4. However, these equations are too lengthy to be included in this work explicitly.

– 5 – We now write the interaction terms of the Higgs with the LQs in the form ˜−1/3 ˆ −1/3 † ˆ −1/3 ˜2/3 ˆ 2/3 † ˆ 2/3 ˜−4/3 ˆ −4/3† ˆ −4/3 LHΦ = − Γab hΦa Φb − Γab hΦa Φb − Γab hΦa Φb 5/3 ˆ 5/3 † ˆ 5/3 ˜ −1/3 2 ˆ −1/3 † ˆ −1/3 ˜ 2/3 2 ˆ 2/3 † ˆ 2/3 − Γ hΦ Φ − Λab h Φa Φb − Λab h Φa Φb (2.12) ˜ −4/3 2 ˆ −4/3 † ˆ −4/3 5/3 2 ˆ 5/3 † ˆ 5/3 − Λab h Φa Φb − Λ h Φ Φ , with h as the physical Higgs ﬁeld, Φˆ Q being the mass eigenstates of charge Q with a, b again running from 1 to 3 for Q = −1/3 and Q = 2/3 and from 1 to 2 for Q = −4/3. In particular we have Γ˜−1/3 = W −1/3Γ−1/3W −1/3 † , Λ˜ 1/3 = W −1/3Λ−1/3W −1/3 † , Γ˜2/3 = W 2/3Γ2/3W 2/3 † , Λ˜ 2/3 = W 2/3Λ2/3W 2/3 † , (2.13) Γ˜−4/3 = W −4/3Γ−4/3W −4/3 † , Λ˜ −4/3 = W −4/3Λ−4/3W −4/3 † , with  ∗    2vY1 A21˜ 2vY31 Y1 0 Y31 −1/3 1 −1/3 1 Γ = √  A21˜ 2vY2˜ A32˜  , Λ =  0 Y2˜ 0  , 2 ∗ ∗ 2 ∗ 2vY31 A32˜ 2vY3 Y31 0 Y3  2vY 2vY ∗ 0   Y Y ∗ 0  2 22˜ √ 2 22˜ 2/3 1 ∗ 2/3 1 Γ = √ 2vY22˜ 2v Y2˜ + Y2˜2˜ − 2A ˜  , Λ = Y22˜ Y2˜ +Y2˜2˜ 0  , 2 √ 32 2 0 − 2A ˜ 2v Y3 +Y33 0 0 Y3 +Y33 √ 32 √ 1 2vY 2 2vY ∗ 1 Y 2Y ∗ Γ−4/3 = √ √ 1˜ 31˜ , Λ−4/3 = √ 1˜ 31˜ . 2 2 2vY31˜ 2v(Y3 −Y33) 2 2Y31˜ Y3 −Y33 √ 1 Γ5/3 = 2vY + Y , Λ5/3 = Y + Y . 22 2 2 22 2 (2.14)

The expanded expressions for Γ˜Q and Λ˜ Q up to O(v2) are given in the appendix.

3 Oblique Corrections

Oblique Corrections, i.e. radiative corrections to the EW breaking sector of the SM, can be parametrized via the Peskin-Takeuchi parameters S, T and U [146]. These parameters are expressed and calculated in terms of the vacuum polarization functions 2 ΠVV (q ), with V = W, Z, γ. We use the convention

V µ V ν 2 µν 2 µ ν = iΠVV (q )g − i∆(q )q q . (3.1)

– 6 – ˆ Q ˆ Q Φa ˆ Q0 Φa Φa V µ V ν

ˆ Q V µ V ν V µ h V ν Φb

Figure 2: The three diﬀerent topologies of Feynman diagrams that contribute to 2 ΠVV (q ) with V = W, Z, γ. The last diagram only exists for V = W, Z and has no impact on the S, T and U parameters as it is momentum independent.

Taking into account that our NP scale is higher than the EW breaking scale, we can expand the gauge bosons self-energies in q2/M 2. As ∆(q2) has no physical eﬀect, the three oblique parameters can be written as

2 2 2 2 4swcw 2 2 cw − sw 2 S = − 2 ΠZZ (0) − ΠZZ (mZ ) + Πγγ(mZ ) + ΠZγ(mZ ) , αmZ cwsw ΠWW (0) ΠZZ (0) T = 2 − 2 , αmW αmZ 2 2 2 2 (3.2) 4swcw ΠWW (0) − ΠWW (mW ) ΠZZ (0) − ΠZZ (mZ ) U = − 2 2 − 2 α cwmW mZ 2 2 2 sw Πγγ(mZ ) sw ΠZγ(mZ ) + 2 2 + 2 2 , cw mZ cw mZ where we used renormalization conditions for the vector ﬁelds such that

2 2 Πγγ(0) = ΠZγ(0) = Re ΠZZ (mZ ) = Re ΠWW (mW ) = 0 . (3.3)

These conditions are fulﬁlled automatically for Πγγ and ΠZγ because of the Ward identities. S, T and U can be calculated with the bare (unrenormalized) two-point correlation functions, the corresponding diagrams in our model are shown in Fig.2. Therefore, we used the check that all divergences disappear in the physical observables S, T and U after having summed over all SU(2)L components in the loop. The complete expressions for these parameters are quite lengthy and therefore given in the appendix. Expanding in addition in q2/M 2 and in v/M, i.e. perturbatively diagonalizing the LQ mass matrices, we can however obtain relatively compact expressions. Up to leading

– 7 – Q ˆ Q Φˆ γ Φa a γ h ˆ Q h Φa

ˆ Q γ γ Φa ˆ Q Φa Figure 3: The two types of diagrams that induce NP effects in h → γγ. For h → gg the photons can simply be replaced by gluons, for h → Zγ one photon can be replaced by a Z boson. The additional diagrams with reversed charge flow are not depicted. order in v we find

2 2 2 2 2 Nc v 7Y22 Y2˜2˜ 8Y33 |A21˜ | m1 17|A32˜| m3 S ≈ − 2 + 2 − 2 − 4 K1 2 + 4 K2 2 , 36π m2 m˜ 2 m3 10m ˜ 2 m˜ 2 10m ˜ 2 m˜ 2 2 2 2 2 4 2 4 2 2 2 Ncv Y22 Y2˜2˜ 4Y33 |A21˜ | m1 |A32˜| m3 Y2˜2˜|A21˜ | m1 T ≈ 2 2 2 + 2 + 2 + 6 K3 2 + 6 K4 2 + 4 K5 2 24πg2sw m2 m˜ 2 m3 10m ˜ 2 m˜ 2 2m ˜ 2 m˜ 2 2m ˜ 2 m˜ 2 2 2 2 2 2 2 2 2 Y2˜2˜|A32˜| m3 Y33|A32˜| m3 4|Y31| m1 4|Y31˜| m˜ 1 − 4 K5 2 − 2 4 K6 2 + 2 K7 2 − 2 K7 2 2m ˜ 2 m˜ 2 m˜ 2 m˜ 2 m3 m3 m3 m3 2 2 ∗ 2 2 2 2 2 2 2|Y22˜| m2 2< Y31A21˜ A32˜ m1 m3 |A21˜ | |A32˜| m1 m3 − 2 K7 2 − 4 K8 2 , 2 + 6 K9 2 , 2 , m˜ 2 m˜ 2 m˜ 2 m˜ 2 m˜ 2 5m ˜ 2 m˜ 2 m˜ 2 (3.4) U ≈ 0 , where the loop functions, given in the appendix, are normalized to be unity in case of equal masses. These expressions agree with Refs. [138, 139] for the special cases studied there. Note that U is approximately zero since it only arises at dimension 8.

4 Higgs Couplings to g, γ and Z

The Feynman diagrams involving scalar LQs contributing to h → γγ, h → gg and h → Zγ are shown in Fig.3. The amplitude, induced by them, reads

2 ˜Q αNc X Q Γaa 2 A h → γ(p1)γ(p2) = m ε(p1)·ε(p2) − 2(ε(p1)·p2)(ε(p2)·p1) , (4.1) 24π Q 2 h Q,a (Ma ) with p1 and p2 representing the photon momenta, εµ(pi) the corresponding polarization vectors and a running over the number of mass eigenstates with the same electric

– 8 – charge Q = {−1/3, 2/3, −4/3, 5/3}. Here we used on-shell kinematics and expanded 2 2 in mh/M . Similarly, for the decay into a pair of gluons, we obtain ˜Q A A αs X Γaa 2 A A A A A h → g (p1)g (p2) = m ε (p1)·ε (p2) − 2(ε (p1)·p2)(ε (p2)·p1) , 48π Q 2 h Q (Ma ) where A labels the 8 gluons (no sum implied). For the Higgs decaying into a Z and a photon we obtain ˜Q ˜Q ˜Q ! αNc 1 X Tab Γba Q 2 2 Γaa A[h → Z(pZ )γ(pγ)] = Q 2 K7 xab − swQ 2 24π swcw Q Q Q (Mb ) (Ma ) (4.2) 2 2 × (mh − mZ ) ε(pZ ) · ε(pγ) − 2(ε(pZ )·pγ)(ε(pγ)·pZ ) ,

2 2 2 2 with a simultaneous expansion in mh/M and mZ /M and (M Q)2 xQ = a . (4.3) ab Q 2 (Mb )

The relevant observables in this context are the eﬀective on-shell hγγ, hgg and hZγ couplings, normalized to their SM values s s s Γh→γγ Γh→gg Γh→Zγ κγ = SM , κg = SM , κZγ = SM . (4.4) Γh→γγ Γh→gg Γh→Zγ We then have ˜Q 1 αNc X 2 Γaa κγ = 1 + Q , ASM 24π Q 2 h→γγ Q (Ma ) ˜Q 1 αs X Γaa κg = 1 + , (4.5) ASM 48π Q 2 h→gg Q (Ma ) ˜Q ˜Q ˜Q 1 αNc 1 X Tab Γba Q 2 2 Γaa κZγ = 1 − SM Q 2 K6 xab − swQ 2 , A 24π swcw Q Q h→Zγ Q (Mb ) (Ma ) with the LO SM amplitudes (see e.g. Ref. [147] for an overview) given by [124, 148–153]

SM α 4 A = √ A1(xW ) + A1/2(xt) , h→γγ 4π 2v 3 SM αs A = √ A1/2(xt) , (4.6) h→gg 8π 2v SM α −1 2 8 2 −1 Ah→Zγ = √ cwC1(xW , yW ) + 1 − sw C1/2(xt , yt) . 4πsw 2v cw 3

– 9 – FCC-ee sensitivity

CEPC sensitivity 0.05 II II T&S(1σ) 0.04

T&S(2σ) II 0.03 Y22 T Y~ ~ 0.02 2 2

II II II II 0.01 Y33

II  A~ /(m m ) 0.00 2 1 1 2  A ~/(m m ) -0.01 3 2 3 2 -0.010 -0.005 0.000 0.005 0.010 S Y/m2 = ±1/TeV 2

A/m2 = ± 1.5/TeV

Figure 4: Correlations between S and T for four different Lagrangian parameters in Eq. (2.4), assuming that only one of them is non-zero at a time. For simplicity, we assumed all LQ masses to be equal. While Y22 and Y2˜2˜ can yield both positive and negative effects in S, the effect in the T parameter is positive definite. Since our prediction for S and T depends on a single combination of parameters (Y/m2 or A2/m4), we used one degree of freedom to obtain the preferred region in the S-T plane, such that the region within the ellipse labelled by 1 σ (2 σ) corresponds to 68% C.L. (95% C.L.).

We deﬁned

2 2 mh 4mi xi = 2 , yi = 2 , (4.7) 4mi mZ while the loop functions are given in the appendix.3

In addition to the expansion of the loop functions, we can also expand the ex- ˜Q 2 ˜Q ˜Q Q 2 2 2 3 pressions Γ /M and T Γ K6(xab)/M in v /M up to O(v ), using Eq. (2.11). We 3Note that we did not include the eﬀects of bottom quarks in the SM prediction which would lead to a 10% destructive interference.

– 10 – T(1σ) T(2σ)

CEPC sensitivity

1.03 CLIC sensitivity

1.02 FCC-ee sensitivity

1.01 II II

FCC-hh sensitivity II II

1.00 II κγ II Y22 0.99 Y~ ~ II 2 2 0.98 Y33

0.97  A~ /(m m ) 2 1 1 2 0.00 0.02 0.04 0.06  A ~/(m m ) T 3 2 3 2

Y/m2 = ±1/TeV 2

A/m2 = ± 1.5/TeV

Figure 5: Correlations between κγ and T for diﬀerent Lagrangian parameters, assuming that only one of them is non-zero at a time and assuming all LQ masses to be equal. obtain

3 ˜−1/3 2 2 X Γaa √ Y1 Y˜ Y3 |A˜ | |A ˜| ≈ 2v + 2 + − 21 − 32 , −1/3 2 m2 m˜ 2 m2 m2m˜ 2 m2m˜ 2 a=1 (Ma ) 1 2 3 1 2 3 2 3 ˜2/3 2 X Γaa √ Y2 Y˜˜ + Y˜ Y3 + Y33 2|A ˜| ≈ 2v + 22 2 + − 32 , 2/3 2 m2 m˜ 2 m2 m˜ 2m2 a=1 (Ma ) 2 2 3 2 3 2 ˜−4/3 X Γaa √ Y˜ Y3 − Y33 ≈ 2v 1 + , −4/3 2 m˜ 2 m2 a=1 (Ma ) 1 3 Γ5/3 √ Y + Y ≈ 2v 22 2 , (4.8) 5/3 2 2 (M ) m2

– 11 – 3 ˜−1/3 ˜−1/3 2 2 2 2 X Tab Γba −1/3 v |A21| m1 |A32| m3 Y2˜ K7 x ≈ √ F1 + F1 − , −1/3 2 ab 4 2 4 2 2 2 2m ˜ 2 m˜ 2 2m ˜ 2 m˜ 2 m˜ 2 a,b=1 (Mb ) 3 ˜2/3 ˜2/3 2 2 X Tab Γba 2/3 v Y2˜ Y2 2(Y3 + Y33) Y2˜2˜ 3|A32| m3 K7 x ≈ √ − + + − F2 , 2/3 2 ab 2 2 2 2 4 2 2 m˜ 2 m2 m3 m˜ 2 m˜ 2 m˜ 2 a,b=1 (Mb ) 2 ˜−4/3 ˜−4/3 √ X Tab Γba −4/3 Y3 − Y33 K7 x ≈ − 2v , −4/3 2 ab 2 m3 a,b=1 (Mb ) T˜5/3 Γ˜5/3 v Y + Y ≈ √ 22 2 . (4.9) 5/3 2 2 (M ) 2 m2

Therefore, we have directly expressed κγ, κg and κZγ in terms of the Lagrangian parameters. The loop functions F1 and F2, given in the appendix, are again normalized to be unity in case of equal masses.

5 Phenomenological Analysis

Before we illustrate the eﬀects of LQs in the observables of our interest, let us recall the current experimental situation and the prospects at future colliders. Concerning the oblique corrections, the global ﬁt to electroweak precision measurements (including LEP [154], Tevatron [155] and LHC [156]) of Ref. [157] constrains the S and T parameter to lie within

S = [−0.06, 0.07] ,T = [−0.02, 0.05] , (5.1) at 95% C.L. within the 2-dimensional S-T plane, with a correlation factor of 0.72. Here, we can optimistically expect a sensitivity of 0.008 in the future at the FCC-ee [143]. For on-shell Higgs couplings, we used the results of Refs. [158, 159] for the current status, which are

+0.051 +0.055 κg = 1.066−0.050 , κγ = 0.999−0.053 . (5.2)

Concerning future prospects we expect for κγ (κg) an accuracy of 7% (2.3%) at the ILC [140], 3.7% (1.5%) at CEPC [160], 2.3% (0.9%) at CLIC [141], 3% (1.4%) at the FCC-ee [143] and 1.45% at the FCC-hh [161]. Finally, concerning h → Zγ, an accuracy of up to 1.8% in h → Zµ+µ−/h → µ+µ− can be achieved at the FCC-ee [143]. Let us start by considering the oblique parameters. Here and in the following, we will for deﬁniteness assume a LQ mass of 1 TeV, which is compatible with current LHC

– 12 – ILC Run-I sensitivity 1.15 1.15

I CEPC sensitivity 1.10 1.10 I I FCC-ee sensitivity 1.05 1.05 I I I FCC-hh sensitivity κg κg 1.00 1.00 I I I I CLIC sensitivity Out[1010]= I I I I I I κγ &κ g (1σ) 0.95 0.95 I I I κγ &κ g (2σ)

0.90 0.90 I Y1

0.90 0.900.95 0.951.00 1.00 1.051.05 1.101.10 Y~ 1 κ κγ γ Y2

Y~ 1.15 1.15 2

Y3 1.10 1.10 Y/m 2 = ±5/TeV 2 1.05 1.05 I I I I Y22 κg κg 1.00 1.00 Out[850]= IIII Y~ ~

II 2 2 II I I I I 0.95 0.95 Y33

II II  A~ /(m1 m2) 0.90 0.90 2 1  0.90 0.95 1.00 1.05 1.10 A ~/(m3 m2) 0.90 0.95 1.00 1.05 1.10 3 2 κ κγ γ A/m2 = ± 2 TeV-1

Figure 6: Correlations between κγ and κg for the diﬀerent Lagrangian parameters. Here we assumed all bi-linear LQ mass terms to be equal. Here we used one degree of freedom in the χ2 ﬁt for the allowed regions and the future prospects such that the intersection with the LQ line indicates the 68% and 95% CL for the corresponding parameter Y/m2 or A2/m4. limits [162–164] for a broad range of couplings to fermions. In Fig.4 we show the correlations between S and T for the four cases which contribute to both parameters

– 13 – simultaneously. As one can see, the effect in T is positive definite, as slightly preferred by current data. Note that the A parameters are dimensionful couplings which are naturally expected to be of the same order as the LQ masses and that similarly the dimensionless couplings Y are expected to be of order 1. Therefore, T already now sets relevant limits on these couplings and its future experimental sensitivity allows for stringent constraints or even to discover deviations from the SM within LQ models. Turning to the effects in Higgs couplings to gauge bosons, we show the correlations between κγ and T in Fig.5 and between κγ and κg in Fig.6. The currently allowed regions (1 σ and 2 σ, corresponding to 68% and 95% C.L. for one degree of freedom) are shown in color while the future prospects are indicated by dashed and dotted boundaries of the corresponding ellipses. Assuming a value close to the current best

ﬁt point in the κγ-κg plane is conﬁrmed in the future, this would point towards the ˜ LQ representation Φ2. Similarly, one can correlate κγ to κZγ, see Fig.7, which clearly provides complementary distinguishing power, especially at the FCC-hh. E.g. an anti- correlations between κγ to κZγ is not favored by either (single) Lagrangian parameter of coupling LQs to the Higgs.

6 Conclusions

LQs are prime candidates to explain the flavor anomalies, i.e. the discrepancies between the SM predictions and experiment in b → cτν and b → s`+`− processes and in the anomalous magnetic moment of the muon. Therefore, it is interesting to study alternative observables which are sensitive to LQs and could therefore as well show deviations from the SM predictions. In this context, parameters sensitive to additional electroweak symmetry breaking effects provide a complementary window. In particular, LQ couplings to the SM Higgs generate loop effects, which contribute to the oblique parameters (S and T ) and to effective Higgs couplings, entering on-shell Higgs boson production (gg → h) and decays (h → γγ, h → Zγ). All these observables have in common that (at the one-loop level) they do not depend on the LQ couplings to fermions but rather only on LQ couplings to Higgses (tri-linear and quadratic ones). Therefore, one can test this sector of the Lagrangian independently of the fermion couplings entering flavor observables. Taking into account the most general set of Higgs-LQ interactions, including mixing among different LQ representations, we calculated the one-loop contributions to the oblique parameters S, T and U. Using a perturbative expansion of the mixing matrices we were able to provide simple, analytic expressions for them. Similarly, we calculated the contributions to effective on-shell hgg, hγγ and hZγ couplings, expressing the corrections as simple analytic functions of the Lagrangian parameters.

– 14 – 1.10 1.10 ILC Run-I sensitivity

CEPC sensitivity 1.051.05 I I FCC-ee sensitivity

I I I I I I I FCC-hh sensitivity κκZZγγ1.001.00 II I I I II I I CLIC sensitivity

0.950.95 I I κγ (1σ)

Y1 0.90 0.90 Y~ 0.900.90 0.950.95 1.001.00 1.051.05 1.101.10 1 κ Y γκγ 2

1.10 Y~ 1.10 2 II Y3

1.05 1.05 Y/m 2 = ±5/TeV 2 I I Y I 22 κ 1.00 1.00 III I Zγ κZγ I III Out[852]= I Y~ ~ II 2 2 I I 0.95 0.95 Y33

 II A~ /(m1 m2) 2 1 0.90 0.90  0.90 0.95 1.00 1.05 1.10 A ~/(m3 m2) 0.90 0.95 1.00 1.05 1.10 3 2 κ κγ γ A/m2 = ± 4 TeV-1

Figure 7: Correlations between κγ and κZγ for the diﬀerent Lagrangian parameters coupling LQs to the Higgs. The currently preferred regions are shown as red ellipses and the future sensitivity is indicated by the dashed and dotted lines.

In our phenomenological analysis we correlated the effects in the oblique corrections with each other, see Fig.4, finding that the contribution to T is positive definite and that T is clearly more sensitive to LQs than S. Similarly, we correlated hgg with hγγ in Fig.6 and hγγ to hZγ in Fig.7. In the future it would be very interesting to include the

– 15 – NLO QCD corrections, in the spirit of Refs. [126, 127], as these interesting correlations open the possibility of distinguishing diﬀerent LQ representations, independently of their couplings to fermions, providing strong motivation for future colliders.

Acknowledgements — We thank Michael Spira for useful discussions. The work of A.C. and D.M. is supported by a Professorship Grant (PP00P2 176884) of the Swiss National Science Foundation. The work of F.S. is supported by the Swiss National Foundation grant 200020 175449/1.

– 16 – A Appendix

A.1 Loop Functions In this appendix we ﬁrst present the loop functions, which are used in (4.1) to write the results for the S and T parameters in a compact form

y3 + 2y2 − 19y + 4 (4y3 − 12y2 − 6y + 2) log(y) K (y) = −10 − , 1 (y − 1)4 (y − 1)5 10−y4 + 10y3 − 45y2 − 8y + 8 18(3y − 1) log(y) K (y) = + , 2 17 y(y − 1)4 (y − 1)5 y2 + 10y + 1 6y(y + 1) log(y) K (y) = 10 − , 3 (y − 1)4 (y − 1)5 (y + 4)(y2 + 10y + 1) 6(y + 1)(y + 4) log(y) K (y) = 2 − , 4 y(y − 1)4 (y − 1)5 2y2 + 5y − 1 6y2 log(y) K (y) = 2 − , 5 (y − 1)3 (y − 1)4 y2 − 5y − 2 6 K (y) = 2 + 6 (y − 1)3y (y − 1)4 3x2 − 1 − 2x log(x) K (x) = , 7 (x − 1)3 4 8x 4 K (x, y) = −3 + − 8 (x − 1)2(y − 1) (x − 1)(y − x)2 (x − 1)2(y − x) + 4x log(x) K10(x, y) + 4y log(y) K10(y, x) , 12x 2x2 − 7x − 13 6(x + 1) K (x, y) = 10 − − 9 (1 − x)2(x − y)2 2(x − 1)3(y − 1) (x − 1)3(y − x) 9(x − 3) 3 − − 2(x − 1)2(y − 1)2 (x − 1)(y − 1)3 + 3x log(x) K11(x, y) + 3y log(y) K11(y, x) , and 2x x − 2 K (x, y) = + , 10 (x − 1)(y − x)3 (x − 1)2(y − x)2 4x 4 x K (x, y) = − + . 11 (x − 1)2(y − x)3 (x − 1)3(y − x)2 (x − 1)4(y − x)

– 17 – In h → Zγ we used the following loop functions for the amplitude

x3 − 6x2 + 3x + 6x log(x) + 2 F (x) = 2 , 1 (x − 1)4 2 x4 − 2x3 + 9x2 − 6x2 log(x) − 10x + 2 F (x) = . 2 3 x(x − 1)4

A.2 Expanded Matrices Next, we will give the expressions for the coupling matrices, expanded in terms of the vacuum expectation value v. We have the weak isospin matrices T Q, which read in case of no LQ mixing

0 0 0 − 1 0 0 2 0 0 1 T −1/3 = 0 − 1 0 ,T 2/3 = 0 1 0 ,T −4/3 = ,T 5/3 = , (A.1)  2   2  0 −1 2 0 0 0 0 0 1 using the basis defined in Eq. (2.7). A unitary redefinition of the LQ fields in order to diagonalize the mass matrices in Eq. (2.5) also affects the T Q matrices

T˜Q = W QT QW Q† . (A.2)

Note that the LQ ﬁeld redeﬁnition has no impact the electromagnetic interaction, since the coupling matrix is proportional to the unit matrix and the W Q then cancel due to unitarity. If we use the perturbative diagonalization ansatz, we obtain

 2 2 vA∗ v2A A∗  −v |A21˜ | 21˜ 32˜ 21˜ 2 2 2 2 2 2 2 2 2 2(m1−m˜ 2) 2(m ˜ 2−m1) 2(m1−m˜ 2)(m ˜ 2−m3)  vA 2 |A |2 |A |2 vA  ˜−1/3 21˜ 1 v 21˜ 32˜ 32˜ T ≈ 2 2 − + 2 2 2 + 2 2 2 2 2  ,  2(m ˜ 2−m1) 2 2 (m1−m˜ 2) (m ˜ 2−m3) 2(m ˜ 2−m3)   v2A A∗ vA∗ 2 2  21˜ 32˜ 32˜ −v |A32˜| 2 2 2 2 2 2 2 2 2 2(m1−m˜ 2)(m ˜ 2−m3) 2(m ˜ 2−m3) 2(m ˜ 2−m3)  2  1 v Y22˜ − 2 2 0 2 m2−m˜ 2  v2Y ∗ v2|A |2 vA  (A.3) ˜2/3  22˜ 1 32˜ √ 32˜  T ≈ 2 2 2 + 2 2 2 2 2 ,  m2−m˜ 2 (m ˜ 2−m3) 2(m3−m˜ 2)   vA∗ v2|A |2  √ 32˜ 32˜ 0 2 2 1− 2 2 2 2(m3−m˜ 2) (m ˜ 2−m3) √  2v2Y ∗  31˜ 0 2 2 ˜−4/3 √ m3−m˜ 1 T ≈ 2  , 2v Y31˜ 2 2 −1 m3−m˜ 1 valid up to O(v2). T 5/3 is not aﬀected, since the LQ with charge Q = 5/3 does not mix.

– 18 – There are also interaction matrices for the ZZΦQΦQ vertex, which read in case of no LQ mixing

 2 2   2 2  sw 2sw 1 3 0 0 3 + 2 0 0  2   2  −1/3  s2  2/3  2s2  D =  0 w − 1 0  D =  0 w − 1 0   3 2   3 2  2 2 2 2  sw   2sw  0 0 3 0 0 3 − 1 2  4s2  w 0 5s2 12 D−4/3 = 3 D5/3 = w − .  2 2 4sw 3 2 0 3 − 1 (A.4)

If we include the LQ mixing, we have

4 2 2 2 (4s2 −3)vA∗ (4s2 −3)v2A∗ A  4sw (4sw−3)v |A21˜ | w 21˜ w 21˜ 32˜  − 2 2 2 2 2 2 2 2 2 3 (m1−m˜ 2) m˜ 2−m1 (m3−m˜ 2)(m ˜ 2−m1) 1 (4s2 −3)vA (4s2 −3)vA ˜ −1/3  w 21˜ ˜ w 32˜  D ≈  m˜ 2−m2 d22 m˜ 2−m2  , 12 2 1 2 3  (4s2 −3)v2A∗ A (4s2 −3)vA∗ 4 2 2 2  w 32˜ 21˜ w 32˜ 4sw (4sw−3)v |A32˜| 2 2 2 2 2 2 − 2 2 2 (m1−m˜ 2)(m ˜ 2−m3) m˜ 2−m3 3 (m3−m˜ 2) 2 2 2 2 2 2 2 2 ˜ (3 − 2sw) (4sw − 3)v |A21˜ | (4sw − 3)v |A32˜| with d22 = + 2 2 2 + 2 2 2 , 3 (m1 − m˜ 2) (m3 − m˜ 2)  2 2 2 2  (4sw+3) 16swv Y22˜ (A.5) 3 m˜ 2−m2 0 2 2 √ 1  16s2 v2Y ∗ 2 2 2(8s2 −9)v2|A |2 2(8s2 −9)vA  ˜ 2/3  w 22˜ (4sw−3) w 32˜ w 32˜  D ≈ m˜ 2−m2 3 − (m2−m˜ 2)2 m˜ 2−m2 , 12  2 2 √ 3 2 2 3   2(8s2 −9)vA∗ 2 2 2 2 2  w 32˜ 4(3−2sw) 2(8sw−9)v |A32˜| 0 2 2 + 2 2 2 m˜ 2−m3 3 (m3−m˜ 2) √  4 2 2 ∗  16sw 2(8sw−3)v Y31 1 2 2 ˜ −4/3 √ 3 m3−m˜ 1 D ≈  2 2 2 2  . 2(8sw−3)v Y31 (3−4sw) 3 2 2 m3−m˜ 1 3 Analogously to the Z boson, diﬀerent LQ generations mix under W interactions. With- out LQ mixing, the interactions with the W boson can be written in terms of the following matrices

0 0 0  0 0 0 B1 = √ ,B2 = 0 1 0 ,B3 = 1 0 0 , (A.6) 0 0 2  √  0 0 − 2 arranging the LQ in their charge eigenstates according to Eq. (2.7). B1 describes the interaction of LQs with electric charges Q = −4/3 and Q = −1/3, B2 the ones with Q = −1/3 and Q = 2/3, B3 with Q = 5/3 and Q = 2/3. If we include LQ mixing, the

– 19 – matrices expanded up to O(v2), then read

 2v2Y ∗  31˜ 0 0 2 1 √ m˜ 1−m3 B˜ ≈ √ ∗ ∗ ,  2 A˜ A 2vA √ v2|A |2  2v 21 32˜ ∗ 32˜ √ 32˜ 2 2 2 2 +Y31 2 2 2− 2 2 2 m1−m3 m1−m˜ 2 m˜ 2−m3 2(m ˜ 2−m3) √  vA∗ 2 A A∗  21˜ 2v 32˜ 21˜ 0 2 2 2 2 2 2 −Y31 m1−m˜ 2 m1−m3 m3−m˜ 2  v2Y ∗ 2 |A |2 |A |2  ˜2  22˜ v 21˜ 32˜  B ≈ 2 2 1− 2 2 2 − 2 2 2 0 ,  m2−m˜ 2 2 (m1−m˜ 2) (m ˜ 2−m3)   vA∗ √ v2|A |2  32˜ √ 32˜ 0 2 2 − 2− 2 2 2 m˜ 2−m3 2(m3−m˜ 2) −v2Y ∗ ˜3 22˜ B ≈ 1 2 2 0 . (A.7) m2−m˜ 2

We also have interaction matrices for the W +W −ΦQΦQ vertex. Without mixing, they read

0 0 0  1 0 0 2 0 0 1 F −1/3 = 0 1 0 F 2/3 = 0 1 0 F −4/3 = ,F 5/3 = . (A.8)  2   2  0 1 2 0 0 2 0 0 1

If we include mixing and expand up to order O(v2), we obtain

v2|A |2 vA∗  21˜ 21˜ ˜  2 2 2 2 2 f13 2(m1−m˜ 2) 2(m1−m˜ 2) vA 2 3|A |2 |A |2 3vA ˜−1/3  21˜ 1 v 32˜ 21˜ 32˜  F ≈ 2 2 + 2 2 2 − 2 2 2 2 2 ,  , 2(m1−m˜ 2) 2 2 (m ˜ 2−m3) (m ˜ 2−m1) 2(m ˜ 2−m3)  ∗  3vA 3v2|A |2 ˜∗ 32˜ 32˜ f13 2 2 2 − 2 2 2 , 2(m ˜ 2−m3) 2(m ˜ 2−m3) 2 v2A A∗ (m2 − 4m ˜ 2 + 3m2) ˜ 2v Y31 32˜ 21˜ 1 2 3 with f13 = 2 2 − 2 2 2 2 2 2 m1 − m3 2(m1 − m3)(m1 − m˜ 2)(m ˜ 2 − m3)  1  (A.9) 2 0 0 v2|A |2 vA 2/3  1 32˜ √ 32˜  F˜ ≈ 0 2 + 2 2 2 2 2 ,  (m ˜ 2−m3) 2(−m3m˜ 2)   vA∗ v2|A |2  √ 32˜ 32˜ 0 2 2 1 − 2 2 2 2(m3−m˜ 2) (m ˜ 2−m3) √  2v2Y ∗  31˜ 0 2 2 ˜−4/3 √ m1−m3 F ≈ 2  . 2v Y31˜ 2 2 1 m1−m3

– 20 – Finally we show the Higgs coupling matrices in (2.13) up to O(v)

2 A A∗ (m2+m2−2m ˜ 2)  |A21˜ | ∗ 32˜ 21˜ 1 3 2  2v Y1 + m2−m˜ 2 A˜ v 2Y31 − (m2−m˜ 2)(m ˜ 2−m2) 1 1 2 21 1 2 2 3 ˜−1/3  ˜−1/3  Γ ≈ √  A21˜ Γ22 A32˜  , 2  A A∗ (m2+m2−2m ˜ 2) 2  ∗ 21˜ 32˜ 1 3 2 ∗ |A32˜| v 2Y31 − 2 2 2 2 A ˜ 2v Y3 + 2 (m1−m˜ 2)(m ˜ 2−m3) 32 m3−m˜ 2 2 2 ˜−1/3 |A21˜ | |A32˜| with Γ22 = 2v Y2˜ − 2 2 − 2 2 , m1 − m˜ 2 m3 − m˜ 2   2vY2 2vY22˜ 0 2|A |2 √ 2/3 1  ∗ 32˜  ˜ √ 2vY˜ 2v Y2˜ + Y2˜2˜ − 2 2 − 2A32˜ Γ ≈  22 m3−m˜ 2  , 2  √ 2  ∗ 2|A32˜| 0 − 2A ˜ 2v Y3 + 2 2 32 m3−m˜ 2 Γ˜−4/3 ≈ Γ−4/3 . (A.10) and

 Y A∗ (Y −Y )A∗  31 32˜ 2˜ 1 21˜ Y1 v 2 2 + 2 2 Y31 m˜ 2−m3 m1−m˜ 2 1  Y ∗ A (Y −Y )A Y A (Y −Y )A  ˜ −1/3  31 32˜ 2˜ 1 21˜ 31 21˜ 2˜ 3 32˜  Λ ≈ v m˜ 2−m2 + m2−m˜ 2 Y2˜ v m˜ 2−m2 + m2−m˜ 2 , 2  2 3 1 2 2 1 3 2   Y ∗ A∗ (Y −Y )A∗  ∗ 31 21˜ 2˜ 3 32˜ Y31 v 2 2 + 2 2 Y3 m˜ 2−m1 m3−m˜ 2  √  2vY22˜ A32˜ Y2 Y22˜ 2 2 √ m˜ 2−m3 1  2v(Y −Y −Y )A  ˜ 2/3  ∗ 3 2˜ 2˜2˜ 32˜  Λ ≈ Y˜ Y2˜ + Y2˜2˜ m2−m˜ 2 , 2  √ 22 √ 3 2   2vY ∗ A∗ 2v(Y −Y −Y )A∗  22˜ 32˜ 3 2˜ 2˜2˜ 32˜ 2 2 2 2 Y3 m˜ 2−m3 m3−m˜ 2 Λ˜ −4/3 ≈ Λ−4/3 . (A.11)

– 21 – A.3 Exact Results for the Vacuum Polarization Functions In this section we give the q2-expanded results for the vacuum polarization functions, with the LQ masses and couplings kept unexpanded

2 2 2 ! 2 ! 2 X Nc e Q 2 1 µ q Πγγ(q ) = − q + log + , 48π2 ε Q 2 Q 2 Q (Ma ) 10(Ma ) ˜Q 2 ! ! Nc g2e Taa − swQ Q 2 2 2 X 2 1 µ q ΠZγ(q ) = − 2 q + log 2 + 2 , 48π cw ε Q Q Q (Ma ) 10(Ma ) Q 2 ˜ Q ! 2 ! ! 2 X Nc g2Γaa mZ 2g2Daa Q 2 1 µ ΠZZ (q ) = 2 2 − 2 (Ma ) + log 2 + 1 16π cw m c ε Q Q h w (Ma ) N g2 T˜Q − s2 Qδ T˜Q − s2 Qδ X c 2 ab w ab ba w ba 2 2 − × F (M Q) , (M Q) , 32π2c2 a b Q w 2 ! ! 2 X Nc Q mW 2 ˜Q Q 2 1 µ ΠWW (q ) = g2Γ − g F (M ) + log + 1 16π2 aa m2 2 aa a ε Q 2 Q h (Ma ) 2 2(3) 2 ˜1 Nc g B X 2 ab 2 2 −1/3 2 − (M −4/3) F (M −4/3) , (M ) 64π2 a a b a(b)=1 2 3 2 ˜2 Nc g B X 2 ab 2 2 2/3 2 − (M −1/3) F (M −1/3) , (M ) 64π2 a a b a,b=1 2 3 2 ˜3 Nc g B X 2 b 2 2 2/3 2 − (M 5/3) F (M 5/3) , (M ) , 64π2 b b=1 (A.12) where Q = {−1/3, 2/3, −4/3, 5/3} with a and b running from 1 to 3, 3, 2, and 1, respectively. Here we deﬁned

 2  2 2 ! Q 2 Q 2 Q 2 q q F (M ) , (M ) = (M ) f0 + f1 + f2 , (A.13) a b a  Q 2 Q 2  (Ma ) (Ma )

– 22 – with

1 µ2 3 2(xQ )2 log(xQ ) f = −2(xQ + 1) + log + + ba ba , 0 ba ε M 2 2 Q a xba − 1 2 Q Q 2 Q 3 Q Q 2 Q 2 1 µ 5 − 27xba + 27(xba) − 5(xba) − 6(3 − xba)(xba) log(xba) f1 = + log − , 3 ε M 2 Q 3 a 9(xba − 1) Q Q 3 Q 4 Q 2 Q −1 + 8xba − 8(xba) + (xba) + 12(xba) log(xba) f2 = , Q 5 6(xba − 1) (A.14) where again 2 (M Q) xQ = b . ba Q 2 (Ma ) A.4 Leading Order SM Amplitudes in Higgs Decays The SM amplitudes for the hγγ, hgg and hZγ couplings in Eq. (4.6) read

−(2x2 + 3x + 3(2x − 1)f(x)) A (x) = , 1 x2 2(x + (x − 1)f(x)) A1/2(x) = x2 (A.15) 2 2 sw 2 sw 2 C1(x, y) = 4 3 − 2 I2(x, y) + 1 + 2 − 5 + I1(x, y) , cw x cw x

C1/2(x, y) = I1(x, y) − I2(x, y) , with √ f(x) = arcsin2( x) , r 1 √ g(x) = − 1 arcsin( x) , x xy x2y2f(x−1) − f(y−1) x2yg(x−1) − g(y−1) (A.16) I (x, y) = + + , 1 2(x − y) 2(x − y)2 (x − y)2 −xyf(x−1) − f(y−1) I (x, y) = . 2 2(x − y)

– 23 – References

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