# High-\(p_T\) dilepton tails and flavor physics

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## Abstract

We investigate the impact of flavor-conserving, non-universal quark-lepton contact interactions on the dilepton invariant mass distribution in \(p~p \rightarrow \ell ^+ \ell ^-\) processes at the LHC. After recasting the recent ATLAS search performed at 13 TeV with 36.1 fb\(^{-1}\) of data, we derive the best up-to-date limits on the full set of 36 chirality-conserving four-fermion operators contributing to the processes and estimate the sensitivity achievable at the HL-LHC. We discuss how these high-\(p_T\) measurements can provide complementary information to the low-\(p_T\) rare meson decays. In particular, we find that the recent hints on lepton-flavor universality violation in \(b \rightarrow s \mu ^+ \mu ^-\) transitions are already in mild tension with the dimuon spectrum at high-\(p_T\) if the flavor structure follows minimal flavor violation. Even if the mass scale of new physics is well beyond the kinematical reach for on-shell production, the signal in the high-\(p_T\) dilepton tail might still be observed, a fact that has been often overlooked in the present literature. In scenarios where new physics couples predominantly to third generation quarks, instead, the HL-LHC phase is necessary in order to provide valuable information.

## 1 Introduction

Searches for new physics in flavor-changing neutral currents (FCNC) at low energies set strong limits on flavor-violating semileptonic four-fermion operators (\(qq'\ell \ell \)), often pushing the new physics mass scale \(\Lambda \) beyond the kinematical reach of the LHC [1]. For example, if the recent hints for lepton-flavor non-universality in \(b \rightarrow s \ell ^+ \ell ^-\) transitions [2, 3, 4, 5] are confirmed, the relevant dynamics might easily be outside the LHC range for on-shell production.

In this situation, an effective field theory (EFT) approach is applicable in the entire spectrum of momentum transfers in proton collisions at the LHC, including the most energetic processes. Since the leading deviations from the SM scale like \(\mathcal {O}(p^{2}/\Lambda ^{2})\), where \(p^2\) is a typical momentum exchange, less precise measurements at high-\(p_T\) could offer similar (or even better) sensitivity to new physics with respect to high-precision measurements at low energies. Indeed, opposite-sign same-flavor charged lepton production, \(p~p \rightarrow \ell ^+ \ell ^-\) (\(\ell =e, \mu \)), sets competitive constraints on new physics when compared to some low-energy measurements [6, 7, 8] or electroweak precision tests performed at LEP [9].

At the same time, motivated new physics flavor structures can allow for large flavor-conserving but flavor non-universal interactions. In this work we study the impact of such contact interactions on the tails of dilepton invariant mass distribution in \(p~p \rightarrow \ell ^+ \ell ^-\) and use the limits obtained in this way to derive bounds on class of models which aim to solve the recent \(b \rightarrow s \ell \ell \) anomalies. With a similar spirit, in Ref. [10] it was shown that the LHC measurements of \(p p \rightarrow \tau ^+ \tau ^-\) already set stringent constraints on models aimed at solving the charged-current \(b \rightarrow c \tau \bar{\nu }_\tau \) anomalies. The paper is organized as follows. In Sect. 2 we present a general parameterization of new physics effects in \(p~p \rightarrow \ell ^+ \ell ^-\) and perform a recast of the recent ATLAS search at 13 TeV with 36.1 fb\(^{-1}\) of data [11] to derive present and future-projected limits on flavor non-universal contact interactions for all quark flavors accessible in the initial protons. In Sect. 3 we discuss the implications of these results on the rare FCNC *B* meson decay anomalies. The conclusions are found in Sect. 4.

## 2 New physics in the dilepton tails

### 2.1 General considerations

We start the discussion on new physics contributions to dilepton production via Drell–Yan by listing the gauge-invariant dimension-six operators which can contribute at tree-level to the process. We opt to work in the Warsaw basis [12]. Neglecting chirality-flipping interactions (e.g. scalar or tensor currents, expected to be suppressed by the light fermion Yukawa couplings), dimension-six operators can contribute to \(q~ \bar{q} \rightarrow \ell ^+ \ell ^-\) either by modifying the SM contributions due to the *Z* exchange or via local four-fermion interactions. The former class of deviations can be probed with high precision by on-shell *Z* production and decays at both LEP-1 and LHC (see e.g. Ref. [13]). Also, such effects are not enhanced at high energies, scaling like \({\sim }v^2 / \Lambda ^2\), where \(v \simeq 246\) GeV.

*i*,

*j*,

*k*,

*l*are flavor indices, \(Q_i = ({V_{ji}^*} u^j_{L},d^i_L)^T\) and \(L_i = (\nu ^i_L,\ell ^i_L)^T\) are the SM left-handed quark and lepton weak doublets and \(d_i\), \(u_i\), \(e_i\) are the right-handed singlets.

*V*is the CKM flavor mixing matrix and \(\sigma ^a\) are the Pauli matrices acting on \(SU(2)_L\) space.

*Z*boson propagators), leading to

*Z*boson: in the SM \(g_Z^{f} = \frac{2 m_Z}{v} (T^3_f - Q_f \sin ^2 \theta _W)\). The contact terms \(\epsilon _{ij}^{q\ell }\) are related to the EFT coefficients in Eq. (1) by simple relations \(\epsilon _x = \frac{v^2}{\Lambda ^2} c_x\). The only constraint on the contact terms imposed by \(SU(2)_L\) invariance are \(\epsilon ^{d_L e_R^k}_{ij} = \epsilon ^{u_L e_R^k}_{ij} = c_{Q_{ij} e_{kk}} v^2 / \Lambda ^2\).

*Z*pole), the universal higher-order radiative QCD corrections factorize to a large extent. Therefore, consistently including those corrections in the SM prediction is enough to achieve good theoretical accuracy. It is still useful to define the differential LFU ratio,

*B*meson decays. The pattern of observed deviations can be explained with a new physics contribution to a single four-fermion \(bs\mu \mu \) contact interaction. As discussed in more detail in Sect. 3, a good fit of the flavor anomalies can be obtained with a left-handed chirality structure. For this reason, when discussing the connection to flavor in Sect. 3, we limit our attention to the \((\bar{L} L)(\bar{L} L)\) operators with muons given in the first line of Eq. (1).

^{1}To this purpose it is useful to rearrange the terms relevant to \(p ~p \rightarrow \mu ^+ \mu ^-\) as

^{2}:

### 2.2 Present limits and HL-LHC projections

*L*) is constructed treating every bin as an independent Poisson variable, with the expected number of events,

Furthermore, we independently cross-check the results by implementing the subset of operators in Eqs. (6, 7) in a FeynRules [16] model, and generating \(p p \rightarrow \mu ^+ \mu ^-\) events at 13 TeV with the same acceptance cuts as in the ATLAS search [11] using MadGraph5_aMC@NLO [17]. We find good agreement between the fits performed in both ways.

Focusing only on the \((\bar{L} L)(\bar{L} L)\) operators (in the notation of Eq. (6)), the 2\(\sigma \) limits, both from the present ATLAS search (blue) and projected for 3000 fb\(^{-1}\) (red), are shown in Fig. 2. The solid lines show the 2\(\sigma \) bounds when operators are taken one at a time. The dashed ones show the limits when all the others are marginalized. The small difference between the two, especially with present accuracy, confirms what we commented above. Further constraints on the operators with \(SU(2)_L\) triplet structure can be derived from the charged-current \(p p \rightarrow \ell \nu \) processes [6, 7, 9].

## 3 Implications for *R*(*K*) and \(R(K^*)\)

### 3.1 Effective field theory discussion

Recent measurements in rare semileptonic \(b \rightarrow s\) transitions provide strong hints for a new physics contribution to \(bs \mu \mu \) local interactions (see for example the recent analyses in Refs. [18, 19, 20, 21]). In particular, a good fit of the anomaly in the differential observable \(P_5'\) [22], together with the hints on LFU violation in \(R_K\) and \(R_{K^*}\) [23, 24, 25], is obtained by considering a new physics contribution to the \(C_{bs\mu }\) coefficient in Eqs. (6, 7). In terms of the SMEFT operators at the electroweak scale, this corresponds to a contribution to (at least) one of the two operators in the first row of Eq. (1) (see for example [26]). Moreover, the triplet operator could at the same time solve the anomalies in the charged-currrent (\(R_{D^{(*)}}\)) , see e.g. Refs. [27, 28, 29].

We concentrate on UV models in which new particles are above the scale of threshold production at the LHC, such that the EFT approach is applicable in the most energetic dilepton events. We stress however, that even for models with light new physics these searches can be relevant.

**1. Minimal flavor violation**

*B*meson decays has a \(V_{t s}\) suppression, while the dilepton signal at high-\(p_T\) receives an universal contribution dominated by the valence quarks in the proton. The flavor fit in Eq. (10) combined with this flavor structure would imply a value of \(|C_{D\mu }| \sim 1.4 \times 10^{-3}\) which, as can be seen from the limits in Fig. 3, is already probed by the ATLAS dimuon search [11] depending on the origin of the operator (i.e. from the SU(2) singlet or triplet structure) and will definitely be investigated at high luminosity.

^{3}Allowing for more freedom and setting \(C_{bs\mu } \equiv \lambda _{bs} C_{D\mu }\), we show in the top (central) panel of Fig. 4 the 95% CL limit in the \(C_{D\mu }\)–\(|\lambda _{bs}|\) plane, where \(C_{U\mu }\) is related to \(C_{D\mu }\) by assuming the triplet (singlet) structure. As discussed before, a direct upper limit on \(\lambda _{bs}\) via \(b-s\) fusion can be derived only for very large values. On the other hand, requiring \(C_{bs\mu }\) to fit the

*B*decay anomalies already probes interesting regions in parameter space, excluding the MFV scenario (\(\lambda _{bs} = V_{ts}\)) for both singlet and triplet cases.

**2.** \(U(2)_Q\) **flavor symmetry**

In the lower panel of Fig. 4 we show the present and projected limits in the \(C_{b\mu }\)–\(\lambda _{bs}\) plane (here we set \(C_{D\mu } = C_{U\mu } = 0\), after checking that no large correlation with them is present). As for the MFV case, the fit of the flavor anomalies in Eq. (10) combined with the upper limit on \(|C_{b\mu }|\), provides a lower bound on \(|\lambda _{bs}|\). In this case, while at present this limit is much lower than the natural value predicted from *U*(2) symmetry, \(\lambda _{bs} \sim V_{ts}\), with high luminosity an interesting region will be probed. For example, in the *U*(2) flavor models of Refs. [29, 33, 34, 57] a small value of \(\lambda _{bs}\) is necessary in order to pass the bounds from \(B-\bar{B}\) mixing.

**3. Single-operator benchmarks**

### 3.2 Model examples

We briefly speculate on the UV scenarios capable of explaining the observed pattern of deviations in the rare *B* meson decays. For our EFT approach to be valid, we focus on models with new resonances beyond the kinematical reach for threshold production at the LHC. In such models, the effective operators in Eq. (1) are presumably generated at the tree level.^{4} We focus here on the single mediator models in which the required effect is obtained by integrating out a single resonance. These include either an extra \(Z'\) bosons [29, 33, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52] or a leptoquark [28, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62] (for a recent review on leptoquarks see [63]).

We note that a full set of single mediator models with tree-level matching to the vector triplet (\(c^{(3)}_{Q_{ij} L_{kl}}\)) or singlet (\(c^{(1)}_{Q_{ij} L_{kl}}\)) operators consists of color-singlet vectors \(Z'_\mu \sim (\mathbf{1},\mathbf{1},0)\) and \(W'_\mu \sim (\mathbf{1},\mathbf{3},0)\), color-triplet scalar \(S_3 \sim (\bar{\mathbf{3}},\mathbf{3},1/3)\), and vectors \(U_1^\mu \sim (\mathbf{3},\mathbf{1},2/3)\), \(U_3^\mu \sim (\mathbf{3},\mathbf{3},2/3)\), in the notation of Ref. [63]. The quantum numbers in brackets indicate color, weak, and hypercharge representations, respectively.

*and*\(W'\)

*models*A color-singlet vector resonance gives rise to an

*s*-channel resonant contribution to the dilepton invariant mass distributions if \(M_{Z'}\) is kinematically accessible. Otherwise, the deviation in the tails is described well by the dimension-six operators in Eq. (1) with \(\Lambda = M_V\) and

^{5}and in the EFT approach. The results are shown in Fig. 5. The limits in the full model are shown with solid-blue and those in the EFT are shown with dashed-blue. We see that for a mass \(M_{Z'} \gtrsim 4-5\) TeV the limits in the two approaches agree well but for the lower masses the EFT still provide conservative bounds.

^{6}On top of this we show with green lines the best fit and 2\(\sigma \) interval that reproduces the \(b \rightarrow s \mu \mu \) flavor anomalies, showing how LHC dimuon searches already exclude such a scenario independently of the \(Z'\) mass. The red solid line indicates the naive bound obtained when interpreting the limits on the narrow-width resonance production \(\sigma (p p \rightarrow Z') \times \mathcal {B}(Z'\rightarrow \mu ^+ \mu ^-)\) from Fig. 6 of Ref. [11].

Related to the above analysis we comment on the model recently proposed in Ref. [52]. An anomaly-free horizontal gauge symmetry is introduced, with a corresponding gauge field (\(Z'_h\)) having MFV-like couplings in the quark sector. Figure 1 of Ref. [52] shows the preferred region from \(\Delta C_9^\mu \) in the mass versus coupling plane, as well as the constraint from the \(Z'\) resonance search (from the same experimental analysis used here [11]). While the limits from the resonance search are effective up to \(\sim 4\) TeV, we note that the limits from the tails go even beyond and are expected to probe all of the interesting parameter space of this model with the future-projected LHC data. Note that this statement is independent of the \(Z'\) mass (as long as the EFT is valid).

*Leptoquark models*A color-triplet resonance in the

*t*-channel gives rise to \(p p \rightarrow \ell ^+ \ell ^-\) at the LHC [64, 65, 66]. The relevant interaction Lagrangian for explaining

*B*decay anomalies is

## 4 Conclusions

In this work we discuss the contribution from flavor non-universal new physics to the high-\(p_T\) dilepton tails in \(p p \rightarrow \ell ^+ \ell ^-\), where \(\ell =e,\mu \). In particular, we set the best up-to-date limits on all 36 chirality-conserving four-fermion operators in the SMEFT which contribute to these processes by recasting ATLAS analysis at 13 TeV with 36.1 fb\(^{-1}\) of data, as well as estimate the final sensitivity for the high-luminosity phase at the LHC.

Recent results in rare semileptonic *B* meson decays show some intriguing hints for possible violation of lepton-flavor universality beyond the SM. It is particularly interesting to notice that several anomalies coherently point toward a new physics contribution in the left-handed \(b_L \rightarrow s_L \mu _L^+ \mu _L^-\) contact interaction. In most flavor models, the flavor-changing interactions are related (and usually suppressed with respect) to the flavor-diagonal ones. These in turn, are probed via the high-\(p_T\) dimuon tail, allowing us to already set relevant limits on the parameter space of some models.

In particular, our limits exclude or put in strong tension, scenarios which aim to describe the flavor anomalies using MFV structure that directly relates the \(bs\mu \mu \) contact interaction to the ones involving first generation quarks, tightly constrained from \(p p \rightarrow \mu ^+ \mu ^-\). On the other hand, scenarios with \(U(2)_Q\) flavor symmetry predominantly coupled to the third generation quarks lead to milder constraints. In order to further illustrate our point, we discuss a few explicit examples with heavy mediator states (colorless vectors and leptoquarks) and show a comparison of the limits obtained in the EFT with those obtained directly in the model.

If the flavor anomalies get confirmed with more data, correlated signals in high-\(p_T\) processes at the LHC will be crucial in order to decipher the responsible dynamics. We show how high-energy dilepton tails provide very valuable information in this direction.

## Footnotes

- 1.
Note that similar conclusions apply also for solutions of the flavor anomalies involving operators with different chirality structure.

- 2.
The down and up couplings are given by two orthogonal combinations of the triplet and singlet operators in the first line of Eq. (1): \(\mathbf{C}^{D(U)\mu }_{ij} = v^2 / \Lambda ^2 (c^{(1)}_{Q_{ij} L_{22}} \pm c^{(3)}_{Q_{ij} L_{22}} )\).

- 3.
It should also be noted that the triplet combination is bounded from the semileptonic hadron decays (CKM unitarity test) \(C_{U\mu }-C_{D\mu }=(0.46\pm 0.52)\times 10^{-3}\) [7], in the absence of other competing contributions.

- 4.
- 5.
The \(Z'\) decay width is determined by decays into the SM fermions \(u,d,s,c,b,t,\mu ,\nu _{\mu }\) via Eq. (18), i.e. \(\Gamma _{Z'} / M_{Z'} = 5 g_*^2 / (6 \pi )\).

- 6.
See Ref. [9] for a more detailed discussion on the EFT validity in high-\(p_T\) dilepton tails.

## Notes

### Acknowledgements

We would like to thank Martín González-Alonso and Gino Isidori for useful discussions. This work is supported in part by the Swiss National Science Foundation (SNF) under contract 200021-159720.

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