Collider phenomenology of a unified leptoquark model

Collider phenomenology of a unified leptoquark model

T. Faber thomas.faber@physik.uni-wuerzburg.de    Y. Liu yang.liu@uni-wuerzburg.de    W. Porod porod@physik.uni-wuerzburg.de Institut für Theoretische Physik und Astrophysik, Universität Würzburg, Germany    M. Hudec hudec@ipnp.troja.mff.cuni.cz    M. Malinský malinsky@ipnp.troja.mff.cuni.cz Institute for Particle and Nuclear Physics, Charles University, Czech Republic    F. Staub florian.staub@kit.edu Institute for Nuclear Physics, Karlsruhe Institute of Technology, Germany    H. Kolešová kolesova.hel@gmail.com University of Stavanger, Norway
Abstract

We demonstrate that in a recently proposed unified leptoquark model based on the gauge group one can explain without the need of extra heavy fermions. Low energy data, in particular lepton flavour violating decays and , severely constrain the available parameter space. We show that in the allowed part of the parameter space (i) some of the lepton-flavour-violating tau decay branching ratios are predicted to be close to their current experimental limits. (ii) The underlying scalar leptoquarks can be probed at the LHC via their dominant decay modes into tau-leptons and electrons and the third generation quarks. (iii) The constraints from meson oscillations imply that the masses of scalar gluons, another pair of coloured multiplets around, have to be bigger than around 15 TeV and, thus, they can be probed only at a future 100 TeV collider. In both neutral and charged variants, these scalars decay predominantly into third generation quarks, with up to (10%) branching ratios into family-mixed final states. Moreover, we comment on the phenomenology of the scalar gluons in the current scenarios in case that the -decay anomalies eventually disappear.

morekeywords=cp,-r,make,cd, numbers=left, stepnumber=1, numberstyle=, numbersep=10pt, backgroundcolor=, basicstyle=, stringstyle=, keywordstyle=

KA-TP-38-2018

I Introduction

The latest results of the LHC clearly show that the Standard Model (SM) continues to be a remarkably successful description of nature. So far, only a handful of experimental observations show deviations from its predictions. At the moment, exciting direct hints of physics beyond the SM are the recently observed anomalies in -meson decays Matyja et al. (2007); Bozek et al. (2010); Aaij et al. (2014); Huschle et al. (2015); Aaij et al. (2015, 2017), which suggest lepton flavour universality violation (LFUV) in the ratios , and , . Even though the individual discrepancies are between 2 and 3 , they all point in the same direction. A combined fit amounts to more than 4.5- deviations Altmannshofer et al. (2017); Capdevila et al. (2018).

Assuming that these anomalies are not a result of experimental systematics, they can be accounted for by leptoquarks (LQs) of various kinds Bauer and Neubert (2016a, b); Chao (2016); Murphy (2016); Alonso et al. (2015); Calibbi et al. (2015); Fajfer and Košnik (2016); Hiller et al. (2016); Deppisch et al. (2016); Bhattacharya et al. (2017); Barbieri and Tesi (2018); Buttazzo et al. (2017); Kumar et al. (2018). However, building viable ultraviolet (UV) complete models involving those particles is challenging, especially in light of very stringent constraints on lepton flavor violation (LFV) from various experimental searches, see e.g. Crivellin et al. (2017a, b). Several attempts to build UV completions exist already in the literature Di Luzio et al. (2017); Calibbi et al. (2017); Bordone et al. (2018a); Barbieri and Tesi (2018); Doršner et al. (2017); Blanke and Crivellin (2018); Greljo and Stefanek (2018); Bordone et al. (2018b); Matsuzaki et al. (2018); Heeck and Teresi (2018); Balaji et al. (2018); Fornal et al. (2018). Most of them aim for getting the vector leptoquark , which has quantum numbers under the SM gauge group , sufficiently light as it is an excellent candidate to explain the anomalies. It emerges naturally from the breaking of to which fixes the properties of up to effects from generation mixing of the fermions to which it couples. In these attempts the precise way in which the group is broken is ignored as well as the pattern of the couplings in the scalar sector and its mass spectrum.

In Faber et al. (2018) we have presented a detailed analysis of a model based on , proposed in Smirnov (1995); Fileviez Perez and Wise (2013), putting particular attention to the scalar sector. Keeping the minimal fermionic particle content minimal one cannot reduce the mass of well below 1000 TeV due to the constraints stemming from decays. Nevertheless, this setup in principle allows for an explanation of and predicts to be close to the SM expectation if the scalar leptoquarks are taken into account. In Faber et al. (2018) we have used the inspired assumption that all Yukawa couplings are symmetric in the flavor indices. As a consequence we found that one cannot explain without violating the experimental bounds from . This demonstrates that the scalar sector of such models must not be ignored.

In this paper we show that when releasing the symmetry conditions on the Yukawa matrices while still staying within the well-known minimal model, we can accommodate the anomalies without violating any other experimental bound. The corresponding parameter space is quite restricted which implies that the properties of the additional scalars are fixed to a high degree. Consequently, this leads to rather specific predictions for LHC searches.

The paper is organized as follows: in Sec. II we summarize the main features of the model with a particular focus on aspects relevant for the -physics anomalies. In Sec. III we discuss various constraints stemming from low energy data and their consequences for the properties of the new scalars. This is followed by discussion on the resulting collider phenomenology in Sec. IV. A brief summary is given in Sec. V.

For our investigation the SARAH package Staub (2008, 2010, 2011, 2013, 2014) needed to be extended considerably. We present this extension in the appendix. For our numerical calculations we used the generated model files to produce a spectrum generator based on SPheno Porod (2003); Porod and Staub (2012). For the calculation of cross section at hadron colliders we have used the SARAH-generated interface to MadGraph_aMC@NLO Alwall et al. (2011, 2015).

Ii Model aspects

We will briefly summarize here the main features of the model that are important for the subsequent discussion. For further details we refer to refs. Smirnov (1995); Fileviez Perez and Wise (2013); Faber et al. (2018). The Model is based on the gauge group where the SM emerges as part of the breaking. In this class of models, the leptons (including the right-handed neutrino) are unified with the quarks in representations as summarized in Tab. 1. The sub-eV neutrino masses and the observed leptonic mixing pattern are accommodated via an inverse seesaw mechanism Mohapatra and Valle (1986) by adding extra 3 generations of a gauge-singlet fermion to the original model of ref. Smirnov (1995) as proposed in Fileviez Perez and Wise (2013). Even though the breaking implies potentially also and/or breaking, it turns out that only the lepton number gets eventually broken while B remains a good symmetry to all orders in perturbation theory Faber et al. (2018).

Fermions Scalars
Table 1: Fermion and scalar content of the model at the or level.

ii.1 Symmetry breaking and scalar sector

The scalar sector consists of three irreducible representations of as given in Tab. 1. At the level of , the colourless part of the scalar sector consists of a complex singlet and two Higgs doublets and . The gauge symmetry is broken by their vacuum expectation values (VEVs) in the two successive steps

(1)

We parametrize the VEVs as

(2)

where the square brackets denote the structure, GeV, and TeV. The later is chosen in such a way that the vector leptoquark mass is consistent with the stringent bound set by the non-observation of . 111This bound can be actually lowered by more than an order of magnitude if one exploits maximally the freedom in the unitary interaction matrix Smirnov (2018); however, we need to save this freedom for configuring the scalar leptoquark interactions.

As in the usual two-Higgs-doublet models (2HDM) it is convenient to rotate the doublets via

(3)

which takes one to the basis where accommodates the entire electroweak VEV and contains also the would-be Nambu-Goldstone-bosons to be eaten by and , whereas is a second Higgs doublet which does not participate in the electroweak symmetry breaking. One can follow the analogy with the 2HDMs one step further. In particular, the physical component of the field defined by transformation Eq. 3 corresponds almost exactly to the SM Higgs because the current setting may be viewed as the 2HDM in the decoupling regime as is expected to be pushed up to the breaking scale . Furthermore, the admixture of in the physical Higgs is also suppressed by . All this can be readily verified by the analysis of the most general renormalizable scalar potential

(4)

where , , with and denoting the indices; the matrix notation is used to capture the structure and the traces run only over indices.

The coloured scalar degrees of freedom are the field originating from which dominates the Goldstone mode associated with the vector leptoquark, an doublet of charged and neutral scalar gluons and two other leptoquark doublets and , all of which stem from .

Although we have chosen so large that the effects of the extra vector bosons (the and the vector leptoquark ) are completely negligible the model allows for cases where a certain part of the scalar spectrum is much lighter. This can be easily seen by neglecting for the moment the effects of the breaking VEVs in the masses of the different components of the -field: 222Needless to say, the weak isospin mass splitting for a heavy doublet is only of the order GeV.

(5)
(6)
(7)
(8)

where has been eliminated using the minimization conditions for the potential. This yields the approximate tree-level sum rule Faber et al. (2018)

(9)

It is well known that, unlike , the leptoquark has the potential to simultaneously accommodate and . From Eq. 9 one can see that can be in the TeV range even in case of a rather large if there is an appropriate interplay between , and .

Assuming for the moment being that is at least of the order , one sees from Eq. 9 that relatively light scalar gluons are possible in scenarios where is light and heavy. We will thus investigate such scenarios. In principle also could be smaller yielding somewhat lighter and states. However, the contribution of to lepton flavour violating observables imply that the masses should be in the multi-TeV range. For completeness, we note that the large number of parameters allows to obtain easily a SM-like Higgs boson with a mass GeV.

ii.2 Fermionic sector

The fermion masses are generated by the following Lagrangian:

(10)

where are Yukawa couplings and is a Majorana mass matrix. Without loss of generality, we work in a basis where the lepton mass matrix is flavor-diagonal. The up- and down-type quarks in the mass basis are obtained via and for , with the four arbitrary unitary matrices in the flavour space being constrained by .

Two of the Yukawa couplings are strongly related to the masses of down-type quarks and charged leptons,

(11)
(12)

where are diagonal matrices of the corresponding fermion masses.

The Yukawa interactions of the LQs and scalar gluons are encoded solely in and . Eq. 11 and Eq. 12 determine up to the two rotation matrices. On the other hand, due to the extended neutrino sector, the other important matrix , as well as , can be chosen essentially arbitrarily. Indeed, the measured up-type quark masses satisfying

(13)

can be always achieved by a suitable choice of . The light Majorana neutrino mass matrix, from which the neutrino masses and PMNS matrix follow, can be then obtained via a proper choice of the Majorana mass matrix .

While both and generally contribute to various lepton-flavour violating processes, only the interactions arising from are sufficient for a tree-level explanation of the anomalies. Hence, for simplicity we have assumed small entries in in order to fulfill the bounds on LFV violating muon decays Faber et al. (2018).

Iii Constraints from rare lepton and meson decays

The interactions of following from the term proportional to in Eq. 10 read

(14)

with the relevant Yukawa matrix parametrized as

(15)

As we have stated in the previous section, we neglect the other pair of the interaction terms arising from . Without referring to the specific pattern of the above matrix imposed by the extended symmetry of the model, a few simple but important observation are to be made.

First, the interactions in Eq. 14 involve the right-handed leptons. In view of , this implies that the corresponding tree-level contributions to and (entering at the scale where the leptoquarks are integrated out) have not only the same magnitude but also the same sign. Thus, there is only a very small interference between the NP and SM contributions in the amplitudes. Notice that there are ways to circumvent this feature by making the loop contributions dominant, see Bečirević and Sumensari (2017); Fajfer et al. (2018); however, we do not opt here for this scenario.

Second, interaction in Eq. 14 generally induces new sources of LFUV whenever two columns of Eq. 15 differ. In this respect, can be achieved if and only if the LQs couple more to the electrons than to the muons D’Amico et al. (2017), i.e., when .

The third point is that the interactions in Eq. 14 mediate lepton flavour violating processes (LFV) whenever there are nonzero entries of in two different columns. For example, a very stringent experimental bound arises from the limits on or from whose amplitudes are given by linear combinations of . It is clear that all the muon number violating process mediated by will be suppressed if

(16)

approximately holds.

As indicated earlier, cannot be chosen arbitrarily in our model as it is a subject of the extended symmetry constraints. In particular, applying the flavour rotations in Eq. 10 and using the relations from Eq. 11 and Eq. 12 one obtains the following pattern:

(17)

with and being arbitrary unitary matrices. The question is now whether this pattern is compatible with significant deviation of and supressed LFV.

In Ref. Faber et al. (2018), this model was studied under an extra inspired assumption and neglecting possible phases in a second step. In such a case, the interaction matrix in Eq. 17 simplifies to

(18)

where denotes the elements of the mixing matrix. Clearly, the requirements like Eq. 16 are in contradiction with the unitarity of and thus LFV is principally unavoidable. In Faber et al. (2018) it was found by scanning over the considered parameter space that the experimental bound BR Tanabashi et al. (2018) inevitably leads to , in contradiction with measurements.

Consequently, this implies that the assumption is inconsistent with requirement of simultaneously explaining and respecting the bound from the decay. However, such a model assumption is only fully justified at the scale where one still has the left-right symmetry which, however, is broken well above the -breaking scale, see e.g. Bertolini et al. (2013) and refs. therein for explicit constructions. Renormalization group effects will lead to a breaking of . We also note that this model might not emerge from but from another framework.

In the general case of we have the freedom to choose 6 angles (apart from the phases). We impose the following constraints:

  1. To suppress the muon number violating processes we require the conditions from Eq. 16 to be satisfied.

  2. In order to maximize the LQ effect on , we need the product to become as large as possible; due to the smallness of electron mass, this condition affects only .

To achieve this, we take as starting point

(19)

Neglecting in Eq. 17 the mass of the first generation fermions and also the second generation if they appear together with one of the 3rd generation, we arrive at

(20)

In practice we see that this is not yet sufficient and that we also need closer to zero. Using now the freedom of the additional mixing angles of O() we can achieve the form

(21)

Note that all other choices of and satisfying the imposed conditions are, within this approximation, related to the presented ones up to the phases. In the numerical calculations presented below we have fulfilled Eq. 16 exactly, finding four separate closed curves in the parameter space when restricting to real and . For definiteness, we set in the following.

This construction allows to obtain the experimentally preferred values for and for

(22)

Hence, must apply in order to obey the bounds from direct leptoquark searches. As this and Eq. 21 define a rather special part of the parameter space, the question arises in which other observables such a setting can be tested. There are essentially two broad classes: low energy observables and LHC signals. We will focus here on the low energy part and discuss the collider aspects in the next section.

From the construction it is clear that there will be no additional constraints from any muon number violating decays such as . In fact, we can achieve any value of BR() between zero and the experimental bound by small deviations from the current limiting scenario Eq. 16 essentially without changing the findings below.

Numerical input values
 GeV
 GeV,  900 GeV
50
Table 2: Summary of the default input values used in this analysis except if stated otherwise. All other BSM scalars have masses of the order .
\includegraphics

[scale=0.35]tauepiVStau3e

Figure 1: Branching ratio BR) versus BR(). The experimental bound on BR() is given by the red line. We have used the input values given in Tab. 2 and the quark mixing matrices are chosen such that all other experimental constraints are fulfilled as discussed in the text.

In contrast, we do expect sizable effects in the sector. Some of the relevant experimental bounds are BR, BR and BR Tanabashi et al. (2018). The LQ contributes to the first two processes are loop level whereas to the last one also at the tree level. We have found that the bound on indeed starts to constrain the parameter space. This can be seen from Fig. 1 where we show BR) versus BR() scanning over the four lines in the allowed parameter space as described above. In addition we have found that also BR() is close to its experimental bound varying in the narrow range (2.2-2.7) , providing another test of the current scenario in upcoming experiments like Belle II. We note for completeness, that in the allowed parameter flavour violating decays into muons are strongly suppressed and, thus, an observation of would rule out our scenario.

We have also checked that the prediction for meson decays like are fully consistent with the current experimental data. In the context of leptoquarks a potentially constraining observable is the ratio BR()/BR(). However, due the required smallness of , all leptoquark effects on observables with neutrinos in the final state are suppressed and, thus, also this is consistent with data.

Staying in this part of the parameter space we have also checked whether the low energy data can constrain the masses of the other components of . Our construction implies that the scalar gluons, both the charged one and the neutral one, have flavour mixing couplings to quarks. This means in particular that the neutral one contributes at tree level to - and - () mixing. We find that within the experimental and theoretical uncertainties - requires  TeV whereas in case of the - mixing the bound is  TeV. It might be surprising that - mixing is only slightly more stringent than the -meson mixing but this is a consequence of the specific parameter space considered here. We note for completness, that in other parts of the parameter space this bound increases to  TeV.

Iv Collider phenomenology

iv.1 Collider phenomenology in the presence of flavour anomalies

In the previous section we have found a restricted region in parameter space where can be explained while being consistent with the constraints from other flavor observables. Here we discuss collider signatures testing this part of the parameter space. Eq. 9 allows for the cases where, apart from , also the scalar gluons , or even the whole scalar sector arising from , can be light enough to be tested either at the LHC or a prospective 100 TeV -collider.

In the slice of the parameter space, where one explains while simultaneously respecting other low energy constraints such as and , the leptoquarks have rather special properties. The pattern of their Yukawa couplings Eq. 21 is reflected in their decays. For the charge particle one finds, regardless on which point in the allowed regions is chosen,

(23)

where is calculated at the scale . All other channels are negligible. Due to the hierarchical structure of the CKM matrix, the similar pattern appears for the charge particle, where the non-negligible decay channels satisfy

(24)

These particles are searched for by the ATLAS Aaboud et al. (2016) and CMS Sirunyan et al. (2018a) experiments. Assuming branching ratios of 100 % into a specific channel such as bounds up to 1.1 TeV have been set. However, as we have various combinations of different decay channels involving also leptons which are experimentally more difficult to measure, the actual bounds are expected to be somewhat weaker. However, recasting these analyses is beyond the scope of this article and will be left for future work.

We now turn to the next component of which can be potentially light, namely the doublet of charged and neutral scalar gluons. In the following we will consider the complex field even though it is split into its scalar and pseudoscalar component. However, this splitting is at most of GeV and thus can be neglected for the discussion here. The scalar gluon interactions arising from generally read

(25)

where the relevant Yukawa matrix satisfies

(26)
\includegraphics

[scale=0.4]FD.png

Figure 2: Exemplary Feynman graphs for the dominant production cross sections () and at the hadron colliders.

Note that the interactions of the scalar gluons with right-handed up-type quarks origin from which, as mentioned earlier, is suppressed in our model. For this reason our findings differ significantly from the ones of refs. Popov et al. (2005); Frolov et al. (2016); Martynov and Smirnov (2017). Due to the enhancement in Eq. 26, the neutral scalar gluons are generally predicted to prefer decays to the -quarks. In the slice discussed so far, combining Eq. 19 and Eq. 26 leads to

(27)

and one finds the following ranges for the various branching ratios

(28)
(29)
(30)

The neutral states have also loop induced couplings to the gluons Gresham and Wise (2007). Denoting the scalar (pseudoscalar) component of by () we find BR and BR. It has been noted already in ref. Gresham and Wise (2007) that the scalar contributions in the loop induced couplings are negligible even for and, thus, the parametric uncertainties due to the unknown are tiny. The remaining decays are into two quarks of the first two generations. We found in the previous section that the mass of the scalar gluon should be above  TeV due to the constraints on the - mixing. This is clearly too heavy for the LHC and, thus, one needs a 100 TeV -collider Arkani-Hamed et al. (2016); Contino et al. (2017) to probe these particles. In Fig. 2 we give some of the dominant Feynman diagrams for the processes () and . The cross sections for a 100 TeV collider are shown in Fig. 3 where we have included all QCD contributions as well as all couplings of scalar gluons to quarks. The relevant Yukawa coupling is choosen to be in the slice consistent with low energy data discussed in Sec. III. The cross sections include also the contributions from the production of a scalar gluon pair with the subsequent decay of one of the scalar gluons into the corresponding quark final state. For large scalar masses the production cross sections get a significant contribution from the quark initial states or are even dominated by those. Due to this, for example varies by about 20 per-cent because of its dependence on in the considered regions of parameter space. Note that the cross sections shown here are calculated at tree-level and we expect sizable QCD corrections. Combining the cross sections with branching ratio information, we have found that the dominant signals will be in the 4 -jet and 2+2-jets channels which are experimentally challenging.

\includegraphics

g0p_mass_cross_section

Figure 3: Various production cross sections at a prospective -collider with  TeV as a function of the corresponding mass. In addition also the channel and neglecting the electroweak contributions to the cross section one finds . Here we have used the parameters given in Tab. 2 except for the masses of the scalar gluons. The vertical line indicates the bound on obtained from meson mixing in the minimal model which excludes the lower mass values.

iv.2 Scalar gluons at colliders without flavour anomalies

\includegraphics

[scale=0.7]gptb_mass_cross_section \includegraphics[scale=0.7]g0_mass_cross_section

Figure 4: Production cross sections at the LHC with  TeV. On the left side the cross sections (purple line), (green line) and (blue line) are shown as a function of . The yellow line gives the current bound on this final state obtained by the ATLAS experiment Aaboud et al. (2018). On the right side the cross sections and are shown as a function of .

Since the measurements of the B anomalies still admit the case of being pure statistical fluctuations, in the following we focus on the situation when both leptoquarks are too heavy to contribute significantly to the low energy observables and when the lightest BSM fields are the scalar gluons.

These particles are interesting by their own, and, thus we study here the limiting case, where all flavour violating couplings of the neutral scalar gluons are absent and, thus they can have masses in range of the LHC. We still assume here that the elements of are smaller at least by an order of magnitude compared to those in . Consequently, Eq. 26 simplifies to

implying that BR is close to one and that the neutral states decay dominantly into . The latter can also decay into two gluons.

In this model the neutral scalar gluons have suppressed couplings to the top-quark compared to the models discussed for example in refs. Smirnov (1995); Gresham and Wise (2007); Hayreter and Valencia (2018) and, thus also the loop induced coupling is significantly smaller. Firstly this implies, that the decays into two gluons have at most a branching ratio of 5 per-cent. Secondly, this also implies that the bounds from processes like

(31)

obtained by the CMS experiment Sirunyan et al. (2018b, c) do not constrain our model even when taking QCD corrections via a K-factor of 1.7 Goncalves-Netto et al. (2012) into account. Here can be either a quark or a gluon jet. Instead we have found that the strongest constraints come from an ATLAS search for the production. Aaboud et al. (2018). We can see from Fig. 4 that this excludes scenarios with  TeV. This is actually a conservative bound in the sense as we assume here that BR( which maximizes the power of the experimental analysis. We want to stress here, that we have also included here the pair production combined with the subsequent decay . Due the steep decrease of the cross sections with the mass this plot indicates that the reach of the LHC will not be above 1.5 TeV. We therefore show in Fig. 3 the various cross sections at a 100 TeV collider starting from masses in the TeV range which clearly shows that the cross sections in the low mass range is so large that these particles should be found within the first data sets.

V Conclusions

In this paper we have studied a model based on the extended gauge symmetry which is arguably the most minimal UV-complete gauge framework including vector and scalar leptoquark fields. It has been shown recently Faber et al. (2018) that, among other features, this setup has the potential to accommodate the measured values of the and observables in semileptonic -decays. It is well known that, in this context, the strongest constraints stem from the non-observation of and . In order for these to be satisfied along with and a rather specific flavour pattern of leptoquark interactions with matter is required; for instance, all couplings of to muons need to be strongly suppressed. We have shown that there exists a narrow region in the parameter space where a fully consistent picture can be achieved. This, in turn, leads to a very predictive scenario in which several other phenomenological conclusions can be drawn.

First, there is a sharp prediction for the branching ratios of and which turn out to be close to their current experimental limits and, thus, should be observable in the next round of experiments such as Belle II.

Second, the charge- and scalar leptoquarks, whose masses should not be much above 1 TeV in order to address the -anomalies, turn out to have rather specific decay properties which can be tested either at LHC or at a future 100 TeV collider. In particular, we find that and . As such, a clear indication, if not a discovery, should be expected in the next LHC run (with the possible exception of scenarios with ).

Third, there is enough room in the allowed parameter space for relatively light scalar gluons (with electric charges 0 and 1) whose masses are constrained from meson mixing to be above some 15 TeV. Again, the branching ratios of their decays (including those into flavour violating channels) are fixed within narrow ranges which would facilitate their searches at future colliders.

Remarkably enough, the phenomenology of such relatively light scalar gluons in the model under consideration is interesting even if the -anomalies eventually disappear. It turns out that in such a case the stringent limits from the meson mixing can be alleviated and the bounds on their masses can be lowered into the LHC domain.

In this scenario the most stringent limits stems from the process where we get a bound TeV recasting an ATLAS search for . The usual bounds on do not apply in this model. In that situation the branching ratios into the third generation quarks, namely, BR() and BR(), amount to almost 100 %.

Acknowledgments

F.S. is supported by the ERC Recognition Award ERC-RA-0008 of the Helmholtz Association. T.F., Y.L. and W.P. have been supported by the DFG, project nr. PO-1337/7-1. M.H. and M.M. acknowledge the support from the Grant agency of the Czech Republic, project no. 17-04902S and from the Charles University Research Center UNCE/SCI/013. H.K. has been supported by the grant no. PR-10614 within the ToppForsk-UiS program of the University of Stavanger and the University Fund.

Appendix A Implementation in Sarah

a.1 Changes in Sarah

In the context of this project, we have extended the functionality of SARAH to work with unbroken subgroups in order to implement the Pati-Salam model. We summarize the main parts of the SARAH model file and explain the new commands. For all details of the standard commands we refer to Refs. Staub (2008, 2015). The following changes in SARAH have happened:

  1. The algebra was implemented to express the generators and structure constant of in terms of generators and structure constants of and Kronecker deltas.

  2. The possibility to define unbroken subgroups of a bigger gauge group was added

  3. All necessary routines to write the matter and gauge fields, which are defined for the bigger group, in terms of the unbroken subgroup were developed

We tried to keep the changes in SARAH as generic as possible. I.e. the new functionality is not restricted to the considered model or to Pati-Salam groups. However, we have only tested the function thoroughly for the model discussed in this paper. Therefore, one should be careful when using it with other models.

a.2 The Sarah model files

  1. The fundamental gauge groups () are defined as usually via the array Gauge:

    Gauge[[2]]={WL,   SU[2], left,  gL, True};
    Gauge[[3]]={PS,   SU[4], pati, g3,True};
    \endmdframed
  2. In order to define that get broken to an unbroken group , the following three steps are necessary:

    1. The name of the group which shall be broken as well as the name of the unbroken subgroups are defined via UnbrokenSubgroups

      \endmdframed\endmdframed

      Here, the first part of the rule must correspond to an entry in Gauge.

    2. The features of the unbroken gauge groups in the new array AuxGauge are defined. This is completely analogue to the definition of a group in Gauge.

      \endmdframed\endmdframed\endmdframed

      The third entry must be identical to the chosen name in UnbrokenSubgroups.

    3. Names for the new gauge bosons must be introduced. The mapping between the fundamental gauge bosons () to a set of new gauge bosons with dimensions under the unbroken subgroup is done as

      (32)

      This relation is defined in the model file using the new array RepGaugeBosons. For each unbroken subgroup a list must be given which consists of pairs of the name of a gauge boson and its dimension.

      {{VG,8}, {VX,3}, {VY,3}, {VS,1}}
      };
      \endmdframed\endmdframed\endmdframed\endmdframed

      Note, the names for the gauge bosons must always start with V. From this definition, also the mapping of the ghost is derived. The names of the ghost fields are those of the vector boson with V replaced by g.

  3. After the definition of the gauge groups, the matter fields are defined. This is done for non-supersymmetric fields using the arrays FermionField and ScalarField. For fields, which transform non-trivially under the broken gauge groups, the tensor notation is used. Thus, the fundamental representation is a vector of dimension . If the unbroken subgroup has dimension , the relation between the components of the fields are

    (33)

    The number of fields with a prime is .
    For the adjoint representation, an matrix is used. This matrix is then decomposed as

    (34)

    Here, is in the adjoint representation of the unbroken subgroup and all primed fields and are vectors under the unbroken subgroup. The fields to are singlets under the unbroken group.

    1. In the given model, the fermion fields are either singlets or transform in the (anti-) fundamental representation. This is defined via

        {FQL,3,{{uL[color,3], vL},{dL[color,3],eL}}, 0, 2, 4};
      FermionFields[[2]] =
        {FU, 3,{uR[color,-3], vR},                -1/2, 1,-4};
      FermionFields[[3]] =
        {FD, 3,{dR[color,-3], eR},                 1/2, 1,-4};
      FermionFields[[4]] =
        {Si, 3, Sing,                                0, 1, 1};
      \endmdframed\endmdframed\endmdframed\endmdframed\endmdframed

      Note, here the last three entries define the representation with respect to the gauge groups defined in Gauge. The representation with respect to the unbroken subgroup are defined for each component field in squared brackets, i.e. uL[color,3] means that the field uL is a colour triplet.

    2. In the scalar sector, the adjoint representation is needed in addition. All scalars are defined via

        {Chi,   1,      {Chiu[color, 3], Chi0},     1/2, 1, 4};
      ScalarFields[[2]] =
        {H,     1,      {Hp, H0},     1/2, 2, 1};
      ScalarFields[[3]]  =
        {Phi15, 1,
      {{{HGp15[color,colorb,8]+HSp15*dAB/Sqrt[12],HXp15[color,3]},
           {HYp15[colorb,3], -3*HSp15/Sqrt[12] }}
      ,{{HG015[color,colorb,8]+HS015*dAB/Sqrt[12],HX015[color,3]},
           {HY015[colorb,3], -3*HS015/Sqrt[12]  }}},
                                   1/2,  2, 15};
      dAB=Delta[color,colorb]
      \endmdframed\endmdframed\endmdframed\endmdframed\endmdframed\endmdframed

      Here, we have introduced the abbreviation dAB only for better readability. Note, that for the tensor representation the name of the second colour index is extended by b (i.e. colorb to prevent any ambiguity.
      There is one additional subtlety: in SARAH and other codes like MadGraph, CalcHep or WHIZARD the higher dimensional representations of unbroken gauge groups, i.e. the colour group, are not written as tensors but vectors. Therefore, it is necessary to re-write the neutral and charged octets. The necessary definitions are given in the list TensorRepToVector which reads in our case:

      {HG015,
        {color,HGV015,HG015[{p_,a_,b_}] :>
         sum[color/.subGC[gNN[p]],1,8] Lam[color/.subGC[gNN[p]],a,b]
             HGV015[{p,color/.subGC[gNN[p]]/Sqrt[2]}],
        {sum[col6,1,3] sum[col6b,1,3] Lam[col1,col6,col6b],
                                     {col1->col6,col1b->col6n} }
        }
      },
      {HGp15,
        {color,HGVp15,HGp15[{p_,a_,b_}] :>
         sum[color/.subGC[gNN[p]],1,8] Lam[color/.subGC[gNN[p]],a,b]
             HGVp15[{p,color/.subGC[gNN[p]]/Sqrt[2]}],
        {sum[col6,1,3] sum[col6b,1,3] Lam[col1,col6,col6b],
                                     {col1->col6,col1b->col6n} }
        }
      }
      };
      gNN[g_]:=5+ToExpression[StringTake[ToString[g],{-1}]];
      \endmdframed\endmdframed\endmdframed\endmdframed\endmdframed\endmdframed\endmdframed

      Each entry consists of the following pieces:

      • The name of tensor field (HG015, HGp15)

      • The name of the gauge group for which the re-writing shall take place (color)

      • The name which should be used for the vector representation (HGV015, HGVp15)

      • The substitution rule:

         HG015[{p_,a_,b_}] :>
           sum[color/.subGC[gNN[p]],1,8] Lam[color/.subGC[gNN[p]],a,b]}
                  HGV015[{p,color/.subGC[gNN[p]]/Sqrt[2]}]
        

        Here, p is an unique index (gen1, gen2, gen3, gen4) counting the fields in each interaction term and gNN is a function to shift this index by 5. Moreoever, a, b are the colour indices. Therefore, the above line is interpreted as

        (35)

        with a function to rename the indices.

      • Finally, one needs to define also the reverse operation, i.e. the relation to re-write the vector into the tensor representation. This is needed to derive the ghost interactions.

  4. Once the gauge sector and relation for the fields before EWSB are fixed, the rest of the model file is straightforward and follows the standard SARAH conventions:

    1. Lagrangian: the Lagrangian consists of two parts:

              {LagHC,  {AddHC->True}},
              {LagNoHC,{AddHC->False}}
      };
      \endmdframed\endmdframed\endmdframed\endmdframed\endmdframed\endmdframed\endmdframed\endmdframed

      For the first part, the hermitian conjugate needs to be added (AddHC->True). This part involves the fermion interactions as well as :

       + Y3 conj[H].FQL.FD +Y4 conj[Phi15].FQL.FD +Y5 FU.Chi.Si
       + \[Mu]/2 Si.Si  +lambda4 conj[H].conj[Chi].Phi15.Chi);
      \endmdframed\endmdframed\endmdframed\endmdframed\endmdframed\endmdframed\endmdframed\endmdframed\endmdframed

      All other parts of the Lagrangian are already hermitian and are defined via:

       + mPhi2 conj[Phi15].Phi15
       + lambda1 conj[H].H.conj[Chi].Chi
       + lambda2 conj[H].H.conj[Phi15].Phi15
       + lambda3 conj[Chi].Chi.conj[Phi15].Phi15
       + lambda5 conj[H].conj[Phi15].Phi15.H
       + lambda6 conj[Chi].Phi15.conj[Phi15].Chi
       + lambda7 conj[H].H.conj[H].H
       + lambda8 conj[Chi].Chi.conj[Chi].Chi
       +    Delta[pat1,pat2] Delta[pat2b,pat3b]*
            Delta[pat3,pat4] Delta[pat4b,pat1b]*
            Delta[lef1,lef2] Delta[lef3,lef4]*
        lambda9 conj[Phi15].Phi15.conj[Phi15].Phi15
       +    Delta[pat1,pat2] Delta[pat2b,pat1b]*
            Delta[pat3,pat4]Delta[pat4b,pat3b]*
            Delta[lef1,lef2] Delta[lef3,lef4]
        lambda10 conj[Phi15].Phi15.conj[Phi15].Phi15);
      \endmdframed\endmdframed\endmdframed\endmdframed\endmdframed\endmdframed\endmdframed\endmdframed\endmdframed\endmdframed

      For all terms but and