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

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 $h\to\gamma\gamma$, $h\to gg$, $h \to Z \gamma$ 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), whose discovery potential we are going to discuss.

This strong motivation for LQs makes it also interesting to search for their signatures in other observables. Complementary to direct LHC searches [108][109][110][111][112][113][114][115][116][117][118][119][120][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) 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][126][127], simplified model analysis [128][129][130][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 precisely 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.

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 as given in Table 1.
We defined the hypercharge Y such that the electromagnetic charge is given by 2  3  3, 1, −  8  3  3, 2,  7  3  3, 2,  1  3 3, 3, − 2 3 with T 3 representing the third component of weak isospin, e.g. ±1/2 for SU (2) L doublets and 1, 0, −1 for the SU (2) L triplet. Therefore, we have the following eigenstates with respect to the electric charge We now write the interaction terms of the Higgs with the LQs in the form with h as the physical Higgs field,Φ 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 havẽ (2.14) The expanded expressions forΓ Q andΛ Q up to O(v 2 ) are given in the appendix.

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 Π V V (q 2 ), with V = W, Z, γ. We use the convention Figure 2: The three different topologies of Feynman diagrams that contribute to Π V V (q 2 ) 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 q 2 /M 2 . As ∆(q 2 ) has no physical effect, the three oblique parameters can be written as where we used renormalization conditions for the vector fields such that These conditions are fulfilled automatically for Π γγ and Π Zγ 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 q 2 /M 2 and in v/M , i.e. perturbatively diagonalizing the LQ mass matrices, we can however obtain relatively compact expressions. Up to leading 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.

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 Here we used on-shell kinematics and expanded in m 2 h /M 2 . Similarly, for the decay into a pair of gluons, we obtain where A labels the 8 gluons (no sum implied). For the Higgs decaying into a Z and a photon we obtain with a simultaneous expansion in m 2 h /M 2 and m 2 Z /M 2 and The relevant observables in this context are the effective on-shell hγγ, hgg and hZγ couplings, normalized to their SM values We then have with the LO SM amplitudes (see e.g. Ref. [147] for an overview) given by [124,[148][149][150][151][152][153]]  , assuming that only one of them is non-zero at a time. For simplicity, we assumed all LQ masses to be equal. While Y 22 and Y22 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 /m 2 or A 2 /m 4 ), 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 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- Y/m 2 = ± 1/TeV 2 A/m 2 = ± 1.5/TeV Figure 5: Correlations between κ γ 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.
Therefore, we have directly expressed κ γ , κ g and κ Zγ in terms of the Lagrangian parameters. The loop functions F 1 and F 2 , given in the appendix, are again normalized to be unity in case of equal masses.

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 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 A/m 2 = ± 2 TeV -1 Figure 6: Correlations between κ γ and κ g 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 χ 2 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 /m 2 or A 2 /m 4 .
limits [162][163][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 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 κ γ and T in Fig. 5 and between κ γ and κ g 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 κ γ -κ g plane is confirmed in the future, this would point towards the LQ representationΦ 2 . Similarly, one can correlate κ γ to κ Zγ , see Fig. 7, which clearly provides complementary distinguishing power, especially at the FCC-hh. E.g. an anticorrelations between κ γ to κ Zγ is not favored by either (single) Lagrangian parameter of coupling LQs to the Higgs.

Conclusions
LQs are prime candidates to explain the flavor anomalies, i.e. the discrepancies between the SM predictions and experiment in b → cτ ν 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.
A/m 2 = ± 4 TeV -1 Figure 7: Correlations between κ γ and κ Zγ 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 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.
In h → Zγ we used the following loop functions for the amplitude

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 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 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 obtaiñ valid up to O(v 2 ). T 5/3 is not affected, since the LQ with charge Q = 5/3 does not mix.
There are also interaction matrices for the ZZΦ Q Φ Q vertex, which read in case of no LQ mixing If we include the LQ mixing, we havẽ We also have interaction matrices for the W + W − Φ Q Φ Q vertex. Without mixing, they read If we include mixing and expand up to order O(v 2 ), we obtaiñ Finally we show the Higgs coupling matrices in (2.13) up to O(v) and

A.3 Exact Results for the Vacuum Polarization Functions
In this section we give the q 2 -expanded results for the vacuum polarization functions, with the LQ masses and couplings kept unexpanded 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 where again

A.4 Leading Order SM Amplitudes in Higgs Decays
The SM amplitudes for the hγγ, hgg and hZγ couplings in Eq. (4.6) read (A. 16)