Bdecay anomalies in a composite leptoquark model
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Abstract
The collection of a few anomalies in semileptonic Bdecays, especially in \(b\rightarrow c \tau \bar{\nu }\), invites to speculate about the emergence of some striking new phenomena, perhaps interpretable in terms of a weakly broken \(U(2)^n\) flavor symmetry and of leptoquark mediators. Here we aim at a partial UV completion of this interpretation by generalizing the minimal composite Higgs model to include a composite vector leptoquark as well.
1 Introduction
A number of anomalies in the decays of B mesons continue to receive much attention. As recalled below, the statistically most significant among these anomalies is of special interest since, at the partonic level, \(b\rightarrow c \tau \bar{\nu }\), it involves three thirdgeneration particles. As such it is suggestive of an explanation in terms of a \(U(2)^n\) flavor symmetry that distinguishes between the third generation of fermions as singlets and the first two generations as doublets [1].
Within this context Ref. [2] looked at the ability of leptoquark models, in particular spinone leptoquarks, to explain some of these anomalies. However, a model with massive vector fields cries out for a UV completion (see e.g. [3]). This is particularly true since, as one can anticipate from the relatively large size of the putative deviation from the Standard Model (SM) tree level amplitude, a fairly large coupling of the leptoquark must be invoked. The aim of this paper is to investigate whether it is possible to make a composite model that can serve as a (more) UV complete explanation of these flavor anomalies. In particular, we are looking to generalize the simplified composite Higgs models (CHM) of [4] to a case which includes leptoquarks. To this end we extend the global symmetry group of the strong sector from \(SU(3) \times SO(5) \times U(1)~\)[5] to \(SU(4) \times SO(5) \times U(1)\), where SU(4) is the Pati–Salam group. The extension from SU(3) to SU(4) can be seen as natural if one thinks of composite leptons as necessary to give masses to the standard leptons by bilinear mixing, as in the quark case often discussed.
Reference [48] is especially of interest in light of the goal of this work as it challenges the idea that the B decay anomalies could be due to simple extensions of the SM, i.e. a single leptoquark field. Specifically, when only the minimal set of operators needed to explain \(R_{D^{(*)}}\) and \(R_K\) are generated at some scale \(\Lambda \gg v\), the RG evolution of these operators generates unacceptably large deviations from lepton flavor universality in Z and \(\tau \) decays as well as lepton flavor violating \(\tau \) decays. The particular operators are \(Q_{\ell q}^{(1)}\) and \(Q_{\ell q}^{(3)}\); see [53, 54] for notation and the explicit form of the RGE. While a full oneloop RGE analysis is beyond the scope of this work, we note there are at least two effects that distinguish the model under consideration in this work from that of Ref. [48]. The first is that there are more dimensionsix operators than the two listed above, which are generated at tree level that contribute to the relevant RGE, e.g. \(Q_{\ell \ell }\) and \(Q_{H \Box }\), as well operators that do not contribute to the RGE of interest. Some of the additional operators contribute to the RGE with the opposite sign of the contribution coming from the operators considered in [48]. Secondly there are direct contributions to the observables of interest that are generated at the scale \(\Lambda \) at the oneloop level. Though these contributions do not have a logenhancement as the RGE contributions do, they can still serve to partially cancel the effects of the RGE contributions.
The rest of this paper is organized as follows. Section 2 describes the field content of the model as well as the mass spectra and mixing angles associated with the fermions and vector bosons. The tree level amplitudes and viable parameter space are presented in Sect. 3. This is followed by a discussion of electroweak precision data in Sect. 4. Then in Sect. 5 a description is given of a number of features of this model, which distinguish it from the usual partial CHM. Finally, our conclusions are given in Sect. 6.
2 Particle content
2.1 Vector boson masses and mixings
2.2 Fermion masses and mixings
We want to extend the socalled bidoublet model^{3} commonly considered in the standard Composite Higgs picture to the case of SU(4). The triplet scenario can be dealt with in a similar way and is discussed in Appendix A. The composite fermions transform under \(SU(4)\times SU(2)_L \times SU(2)_R \times U(1)_X\) as \(\psi _{\pm }=(4,2,2)_{\pm 1/2}\) and \(\chi _{\pm }=(4,1,1)_{\pm 1/2}\).
Here we are not concerned with neutrino masses and mixings, which can arise from a suitable Majorana mass matrix of the right handed neutrinos mixed with the composite \(\tilde{N}\) states. In any event, to an excellent level of approximation, we can study B anomalies in the basis of neutrino currenteigenstates, where the chargedcurrent leptonic weak interactions are flavor diagonal.
3 Tree level amplitudes for B anomalies
4 Electroweak precision constraints
Apart from flavor, composite Higgs models are constrained by electroweak precision data, which include oblique corrections, modified Zcouplings, and modified righthanded Wcouplings. Let us briefly comment on the new effects that are specific of the extension of SU(3) to SU(4) in the global symmetry group of the composite sector, related to the presence of the vector leptoquark.
5 LHC phenomenology

There are a number of \(Z^{\prime }\)like composite vector bosons, \(\tilde{V}_{\mu },\, W_{\mu }^{H3},\, X_{\mu },\, B^H_{\mu }\), all of which, except \(B^H_{\mu }\), have a large coupling to the lefthanded components of the thirdgeneration fermions. At LHC, by their exchange in the schannel, this results in a significant effect in \(b \bar{b} \rightarrow \tau ^+ \tau ^\).

There is a vector singlet composite leptoquark, \(V_\mu \), partly responsible for the anomalies in Bdecays, with branching ratios likely close to \(50\%\) for \(b\tau \) and \(t\nu _\tau \). \(V_\mu \) can be directly searched in QCD pair production. Its exchange in the tchannel also contributes to \(b \bar{b} \rightarrow \tau ^+ \tau ^\).

There are exotic composite leptons with a mass within a few \(\%\) degenerate with the exotic composite quarks that are normally discussed in the context of standard CHMs.
Summary of experimental results on searches for pair production of scalar leptoquarks. In the columns on the right a checkmark indicates that the corresponding decay is in principle possible for a spinone leptoquark that transforms as \(L_Y\) under \(SU(2)_L \times U(1)_Y\)
Reference  \(\sqrt{s}\) (TeV)  Decay mode  \(M_V\) range (GeV)  \(1_{2/3}\)  \(3_{2/3}\)  \(2_{5/6}\)  \(1_{5/3}\)  \(2_{1/6}\) 

[64]  8  \(V \rightarrow b \tau \)  \([200,\, 870]\)  \(\checkmark \)  \(\checkmark \)  \(\checkmark \)  
[65]  8  \(V \rightarrow t \tau \)  \([200,\, 800]\)  \(\checkmark \)  \(\checkmark \)  \(\checkmark \)  \(\checkmark \)  
[66]  8  \(V \rightarrow b \nu _{\tau }\)  \([200,\, 800]\)  \(\checkmark \)  \(\checkmark \)  
\(V \rightarrow t \nu _{\tau }\)  \(\checkmark \)  \(\checkmark \)  \(\checkmark \)  
[67]  13  \(V \rightarrow b \tau \)  \([600,\, 1000]\)  \(\checkmark \)  \(\checkmark \)  \(\checkmark \) 
5.1 Resonances in \(\tau ^ \tau ^+\)
Reference [44] was the first to point out that \(b \bar{b} \rightarrow \tau ^+ \tau ^\) can be a signal of models that attempt to explain \(R_{D^{(*)}}\). Subsequently Ref. [45] thoroughly investigated bounds on explanations of \(R_{D^{(*)}}\) coming from new physics searches involving pairs of tau leptons. We will compare our results to those of [45] at the end of this subsection.
The result of ATLAS at 8 TeV is the most constraining published experimental result on searches for new physics in \(\tau ^+ \tau ^\) [68]. CMS has released preliminary results at 13 TeV that are naïvely more constraining for \(M_{Z^{\prime }} \gtrsim 900\) GeV [67]. However, not enough information about cuts and efficiencies is provided to reinterpret this search in terms of the \(Z^{\prime }\)s of our model, which are not SMlike. We used the publicly available plot digitizer WebPlotDigitizer v3.10 [69] in performing this analysis.
We did not directly investigate the bounds on the leptoquark of this work coming from its sole contribution to \(\tau ^ \tau ^+\) searches. However, it is straightforward to translate the results of [45] into the parameters of our model. The composite leptoquarks have the same couplings structure as the leptoquarks in the socalled minimal model. The difference in terms on bounds is that in the composite model, \(R_{D^{(*)}}\) receives approximately equal contributions from the leptoquark and the \(W^{H3}\) boson. Thus in bounding the leptoquark parameter one should rescale the parameter \(g_U\) of [45] by a factor of \(1 / \sqrt{2}\). In doing so, the bounds on vector leptoquark from \(\tau ^ \tau ^+\) searches in the upper panel of figure 6 of [45] are relaxed.
6 Conclusions
Measurements in flavor physics in the years to come can compete with direct searches at the LHC in the attempt to discover deviations from the SM. In our view this is especially the case if a weakly broken \(U(2)^n\) symmetry plays some role in determining the structure of flavor. Given this premise it is natural to give consideration to a number of anomalies emerging in the decays of B mesons. On one side there is the fact that the statistically most significant among these anomalies, \(b\rightarrow c \tau \bar{\nu }\), involves three thirdgeneration particles. This matches with a \(U(2)^n\), which brings in a basic distinction between the third generation and the first two lighter families. On the other side there is the relatively large size of the putative deviation from a SM tree level amplitude, making it difficult to conceive a purely perturbative interpretation.
Building on these considerations and based on Ref. [2], in this work we have attempted to construct a partially UV complete explanation of the putative anomalies by extending the minimal CHM from an \(SU(3)\times SO(5)\times U(1)\) to an \(SU(4)\times SO(5)\times U(1)\) global symmetry group. With a suitable choice of the representation of the composite fermions under the unbroken global symmetry group we have shown that such construction can be performed without manifest contradiction with current experiments. To explain the anomalies requires large mixings of the lefthanded thirdgeneration quarks and leptons, \( s_{Lu3}s_{L\nu 3}\approx 0.7\div 0.8\) for \(\xi =0.1\) and, to be consistent with \(b \bar{b} \rightarrow \tau ^+ \tau ^\) searches at LHC, relatively large couplings of the composite vectors, \(g_{\rho }, g_{\rho R} \gtrsim 3\div 4\), always for \(\xi =0.1\).
In the case the anomalies will persist and perhaps be reinforced in experiments to come, several more detailed investigations can be performed of the model described here to prove its full compatibility with the various constraints, present and future. They include three main chapters: (i) electroweak corrections, both of oblique and nonoblique nature, extending and completing Sect. 4; (ii) flavor physics, as partly already discussed in [2], both in the quark and in the lepton sector; (iii) LHC searches, as outlined in Sect. 5.
Footnotes
 1.
 2.
This is a feature of the simplifying choice \(g_{\rho R}= g_X\).
 3.
We adopt the nomenclature of Ref. [55].
 4.
At tree level, contributions to \(b\rightarrow s \nu \bar{\nu }\) and \(s\rightarrow d \nu \bar{\nu }\) are also present, mediated in the schannel by the composite electroweak vectors \(W^{H3}\), \(\tilde{V}\), and X. They exhibit a CKM suppression proportional to \( r_d V_{tb}V^*_{ts}\) and \( r_d^2 V_{ts}V^*_{td}\) respectively. The constraint from \(\Delta B_s=2\) makes the corresponding experimental bounds irrelevant in the range of parameters considered in Fig. 1.
 5.
Note that we use \(v \approx 175\) GeV, whereas in [45] one uses \(v \approx 250\) GeV.
 6.
It is only necessary to impose the rapidity cut when \(p_T^2 < s / (4 \cosh ^2(Y_{\text {cut}}))\). For larger values of the transverse momentum, the rapidity is instead bounded by \(Y < \text {arccosh}(\sqrt{s / (4 p_T^2)})\). Computationally it is somewhat faster to integrate over one region as in Eq. (C.27) as opposed to two regions. This is made possible because in our implementation the PDFs evaluate to zero whenever x (or y / x) is greater than one.
Notes
Acknowledgements
We thank Oleksii Matsedonskyi for useful discussions, David Straub for useful discussions and comments on the manuscript, and the Institute of Theoretical Studies at ETH Zürich for its hospitality while part of this work was completed. The work of RB was supported in part by the Swiss National Science Foundation under contract 200021159720. The work of CM was supported in part by the Italian Ministry of Education, University and Research’s Fund for Investment in Basic Research under grant RBFR12H1MW, and by the United States Department of Energy under grant contract desc0012704.
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