Renormalization group evolution with scalar leptoquarks

Leptoquarks are theoretically well-motivated and have received increasing attention in recent years as they can explain several hints for physics beyond the Standard Model. In this article, we calculate the renormalisation group evolution of models with scalar leptoquarks. We compute the anomalous dimensions for all couplings (gauge, Yukawa, Higgs and leptoquarks interactions) of the most general Lagrangian at the two-loop level and the corresponding threshold corrections at one-loop. The most relevant analytic results are presented in the appendix, while the notebook containing the full expressions can be downloaded at https://github.com/SumitBanikGit/SLQ-RG. In our phenomenological analysis, we consider some exemplary cases with focus on gauge and Yukawa coupling unification.


Introduction
The Standard Model (SM) of particle physics describes the known fundamental constituents of matter as well as their interactions.While the Higgs particle [1][2][3][4], last missing puzzle piece of the SM, was discovered in 2012 at the Large Hadron Collider (LHC) at CERN [5][6][7], it is clear that the SM cannot be the ultimate theory of Nature: It does not account for the astrophysical observation of Dark Matter nor for the non-vanishing neutrino masses required by neutrino oscillations.
A plethora of possible SM extensions have been proposed in the last decade.In this context, leptoquarks (LQs), i.e. hypothetical new particles which directly couple a quark to a lepton, are very interesting.They were first proposed within Grand Unified Theories (GUTs) [8][9][10][11] and composite models [12][13][14][15], and squarks can act as LQs within the Rparity violating MSSM (see e.g.Ref. [16] for a review).They were first classified in Ref. [17] into ten possible representations under the SM gauge group, of which five are scalars, and five are vector particles.
As LQs are possible (light) remnants of a GUT, it is interesting to assess their impact on gauge and Yukawa coupling unification.In particular, scalar LQs (SLQs) could be light states of a GUT symmetry-breaking sector [50] and it might even be possible that they are the only new particles (in addition to the SM) up to the GUT scale.In this case, they must alter the (renormalization group evaluation) RGE sufficiently to lead to the required coupling unification [193].
In this article, we will analyze the RGE in models with SLQs.For this, we will calculate the two-loop anomalous dimensions as well as the one-loop threshold corrections (at the LQ scale) and apply these results to the study of gauge and Yukawa coupling unification.For this, the article is structured as follows: In Sec. 2, we present our setup and conventions.In Sec. 3, we derive the β-function, give their analytic expression for some simple cases and examine gauge and Yukawa coupling unification for some specific examples.In Sec. 4, we summarize and discuss the results of this paper.In Appendix A, we give the full SLQ lagrangian, including all five possible SLQs.In Appendix B and C, we give the anomalous dimensions for the gauge (two-loop) and Yukawa couplings (one-loop), respectively.In Appendix D, we collect the one-loop threshold corrections of SM parameters on matching SLQ models with the SM.A notebook with our full results, including Higgs and LQ (self-) interactions etc., is available at [194].  1, where j = 1, 2, 3 is a flavour index.The left-handed fermions Q j and L j are doublets under SU (2) L and decompose into their components as

SM fields
while the right-handed fields u j , d j and j are SU (2) L singlets which we write as The electric charge q is given by the Gell-Mann-Nishijima formula, q = Y + I 3 where, I 3 is the third-component of the weak isospin and Y is the hypercharge related to the gauge factor U (1) Y .The SM Lagrangian is then written as where Y d , Y and Y u are the Yukawa couplings and B µν and the field strength tensors defined as, The covariant derivative for any field φ is defined as where Y is the U (1) Y hypercharge of φ. g1 , g 2 and g 3 are the gauge couplings associated with U (1) Y , SU (2) L and SU (3) c gauge groups, respectively.τ I and T α are the generators of SU (2) L and SU (3) c depending on the representation of φ.In the fundamental representation of SU (2) L and SU (3) c we have τ I = σ I 2 and T α = λα 2 , where σ I are the Pauli matrices and λ α are the Gell-Mann matrices.We now add to the SM the five scalar SLQs which transform under the SM gauge groups as given in Table 1.For later convenience, we decompose the SLQ Lagrangian as where, the terms L ij... contain terms involving only the LQs {Φ i , Φ j , ...}.The explicit expressions, as already presented in Ref. [195], for each Lagrangian term are given in Appendix A. Note that for our purpose, it is imperative to have the complete Lagrangian, e.g.terms involving LQ-Higgs interactions, such that the full set of counterterms needed to cancel all generated divergences is available.

Renormalization Group Evolution and Phenomenological Analysis
We calculate the complete two-loop β-functions and the one-loop threshold corrections of all couplings of our LQ Lagrangian in the MS-scheme.For the anomalous dimensions, we used the Python package PyR@TE [196]. 1 The PyR@TE model files and full analytic expressions for the β-functions, as well as the threshold corrections, are available online publicly on Github [194].The one-loop threshold corrections are presented in Appendix D, which were derived using Matchete [198] and cross-checked for some selected cases with matchmakereft [199].We also checked that including the threshold corrections reduce, as expected, the matching scale dependence.The QCD contribution to some LQ Yukawa couplings was cross-checked with Ref. [200].We use the following convention throughout this paper for the β-function for any parameter A where µ is the renormalization scale and β (n) (A) denotes the contribution at n-loop order and the dots are the higher-order terms.The β-functions form a system of coupled differential equations which we solve numerically, as explained below.
At the one-loop level, we have the following coefficients for the gauge couplings where, n 1 , n 1, n 2 , n 2 and n 3 denote the number of copies, i.e. generations, of Φ 1 , Φ 1, Φ 2 , Φ 2 and Φ 3 , respectively.The first term is the well know SM term, and we have used the SU (5) GUT normalization g 1 = 5 3 g1 w.r.t. the values given in Table 1.The one-loop expressions for the β-function for the Yukawa couplings and the corresponding threshold Figure 1: Renormalization group evolution of the gauge couplings α i = g2 i /4π in SM and its extensions with the different LQ representations.In order to better visualize the effect of adding LQ, we considered three generations of each LQ of mass 3 TeV.For simplicity, we assumed all Yukawa and Higgs couplings involving LQs to be zero (at the LQ scale).correction are provided in Appendixes C and D, respectively.The two-loop β-function coefficients for the gauge couplings are given in Appendix B.
With these results at hand, we can next study the phenomenology of some selected cases.Here, we focus primarily on the evolution of the gauge and Higgs Yukawa couplings from the electroweak scale to the GUT scale. 2 As the starting point for the evolution, we take the following initial values of the SM parameters at the top scale, above the top threshold (i.e. with 6 active flavours) in the MS-scheme We evolve the couplings using the β-functions of the SM up to the LQ scale m x (with Figure 2: Left: Two-loop renormalization group evolution of gauge couplings for SM+Φ 3 . We observe gauge unification at around 10 14 GeV when we set the LQ scale to m 3 ≈ 10 6 TeV.We vary the initial value of the three diagonal components of Y LL 3 between −0.6 to 0.6 (keeping the initial value of all other couplings to zero) to show that their effect on running is minimal.Right: Two-loop renormalization group running of gauge couplings for SM+Φ 2 + Φ 3 .We observe gauge unification for m 2 = 3 TeV and m 3 = 4 × 103 TeV.
After evolving the couplings to the LQ scale, we include the one-loop threshold corrections determined by comparing the theory with SLQs, to the one without them (i.e. the SM at the LQ scale).This gives a shift to the SM fermion Yukawa couplings depending mainly on the initial values (at the LQ scale) of the LQ Yukawa couplings. 3We then run all couplings, the SM ones as well as the ones including LQs, using the β-functions of the full model from the LQ scale to the high scale, for which we take as an upper limit the Planck scale (≈ 10 19 ) where gravitational effects would become important.
As a first step, we illustrate the running of the gauge coupling as a function of the scale Q in Fig. 1 for the five different LQ representations for a fixed LQ scale of 3 TeV, which is compatible with direct LHC searches [173][174][175][176][177][178][179][180][181][182][183][184][185][186][187][188][189][190][191][192].In order to highlight and enhance the impact of adding the LQs, we considered three generations of the same LQ representation.As the LQ couplings have a minimal effect on the running of gauge couplings, we take their initial value to be zero at the LQ scale.
We also observe that no single LQ representation at the TeV scale can lead to gauge coupling unification.Nonetheless, if we allow for higher LQ masses, it turns out that a single generation of Φ 3 can lead to gauge coupling unification at ≈ 10 14 GeV for an LQ mass of around 10 6 TeV, as shown in Fig. 2 (left).However, such a low unification scale would conflict with the limits from proton decay [236], at least for a standard SU (5) GUT.In Fig. 2 (left), we also show the impact of the LQ Yukawa couplings on SM fermions  (which are free parameters).One can see that for the gauge coupling evolution, their effect is suppressed since they only enter at the two-loop level.In fact, we scanned over the initial values of all the three diagonal elements of Y LL 3 between −0.6 and 0.6 (while keeping the initial value of other couplings to zero), leading to a slight thickening of the curves.However, for larger values of the LQ Yukawa couplings large non-perturbative values for some couplings are induced below the GUT scale, setting an upper limit on the Yukawa couplings if perturbativity is required.
We now illustrate the RGE of bottom and tau Yukawa couplings in Fig. 3.While in the SM, bottom-tau Yukawa coupling unification already happens around 10 6 GeV, in LQ models, all representations have a small impact on the RGE of bottom Yukawa.Furthermore, the same is true for the tau one, except for Φ 1 and Φ 2 .For them, the oneloop threshold corrections to the tau Yukawa coupling can be sizable and their running strongly changes (w.r.t. the SM). 4 The running of the tau Yukawa coupling of SM+Φ 1  reversed for the solid lines.The initial value of the remaining LQ couplings was set to zero.In fact, as can be seen in Fig. 3, the trajectories all lie within the quite narrow blue band except for, Φ 1 and Φ 2 .
Here a very strong impact on the evaluation of the tau Yukawa is possible due to the chiral enhancement.
We next consider the case, motivated by SU (5) GUTs [50], where SM is extended by Φ 2 and Φ 3 .Here, we find that for m 2 ≈ 3 TeV and m 3 ≈ 10 3.5 TeV, we can achieve gauge coupling unification at ≈ 10 13 TeV as shown in Fig. 2 (right) as well as Yukawa coupling unification around the same energy scale.However, this scale is again naively in conflict with proton decay.Finally, we consider the case where we extend the SM with (a single representation) all five possible SLQs (see Table 1).In this case, we obtain gauge coupling unification at around 10 13 TeV for a common LQ scale of 125 TeV as shown in Fig. 4 (top).We also show in Fig. 4 (bottom) that bottom-tau Yukawa coupling unification is possible nearly at the same energy scale for the following initial values of the (3, 3) component of the following LQ Yukawa couplings at the LQ scale while taking the initial value of other LQ couplings to be zero.

Conclusions
LQs are well-motivated extensions of the SM.They arise in composite or extra-dimensional setups and, most importantly, are predicted by GUTs.Furthermore, they have been under intensified investigation in the last years as they are potential candidates for describing several tensions between SM measurements and experiments.
In this article, we computed the two-loop renormalization group evolution as well as the one-loop threshold corrections, for all parameters within generic SLQ models.This includes gauge couplings, SM Yukawa couplings, LQ Yukawa couplings (to quarks and leptons), Higgs and LQ (self-)interactions.The appendices collect the full SLQ Lagrangian, two-loop β-functions of the gauge couplings, one-loop threshold corrections and one-loop β-functions for the SM Yukawa couplings.The full analytic expressions, together with the necessary model files, can be obtained from [194].
In our phenomenological analysis, we considered on the case in which one or more SLQs are light remnants of a symmetry-breaking sector of a GUT.In this setup, we focused on the renormalisation group evolution of the gauge and the Yukawa couplings, examining if unification can be achieved.Several simple scenarios were studied: 1) If one adds one of the 5 possible LQ representations to the SM, only Φ 3 (with a mass around 10 6 TeV) can lead to gauge coupling unification at ≈ 10 15 GeV.2) Extending the SM by all 5 possible LQ representations with a common mass scale ≈ 150 TeV, unification at ≈ 10 14 GeV is achieved.3) Adding the GUT-motivated LQs Φ 2 and Φ 3 to the SM, with masses of m 2 = 3 TeV and m 3 = 4 × 10 3 TeV, respectively, gauge coupling unification occurs at ≈ 10 13 GeV.4) Concerning the bottom-tau Yukawa unification, which happens at far too low scales in the SM, only the LQs Φ 1 and Φ 2 can lead to a sizable modification, due to possible chiral enhancement.Therefore, by choosing the initial value of LQ Yukawa couplings properly, one can always achieve bottom-tau unification at any desired (high) scale.quartic interaction with Higgs field and self-quartic interactions.
where we have omitted, at several places, color indices involving trivial contraction.For example, 3 and L 2 3 contain interactions involving precisely two LQs.

B Two-loop β-functions of gauge couplings
In this Appendix, we collect the two-loop β-functions of gauge couplings of some of the SLQ models considered in this article.
C One-Loop β-functions of SM Yukawa couplings This Appendix presents the one-loop β-functions of the SM Yukawa couplings.We use n 1 , n 1, n 2 , n 2 and n 3 to denote the number of generations of Φ 1 , Φ 1, Φ 2 , Φ 2 and Φ 3 , respectively.

D One-Loop threshold corrections
In this Appendix, we collect the one-loop threshold corrections of the SM gauge, Yukawa, and quartic couplings on matching the SM with SM+Φ 1 + • • • + Φ 3 .To simplify the expressions, we assumed all the LQs to have the same mass m (for unequal mass, see Ref. [194]) and set the dimensionful trilinear couplings to zero.The superscript 0 denotes the original parameters in the SM lagrangian and we use the shorthand notation {1, .., 8} , corresponding to the gauge groups U (1) Y , SU (2) L and SU (3) c , respectively.The structure constants of SU (2) L and SU (3) c are denoted by f IJK and f αβγ .For better readability, SU (2) L indices inside parenthesis (• • • ) are contracted such that they form gauge singlets.

Figure 3 :
Figure 3: Two-loop renormalization group running of bottom and tau Yukawa couplings in SM+Φ 1 and SM+Φ 2 for a single-generation SLQ of mass 3 TeV.The dotted line denotes the tau-Yukawa running when we set the initial value of the (3, 3) components of Y RR 1 , Y LL 1 , Y RL

2 and Y LR 2 to 0 . 1 .
For solid lines, we reverse the sign of the(3,3) components of Y LL 1 and Y LR 2 .The blue bands show the variation of running in SM+Φ i , for i = 1, 1, 2, 2, 3 for bottom Yukawa and i = 1, 2, 3 for tau Yukawa, while varying the (3, 3) components of LQ Yukawa couplings.

Figure 4 :
Figure 4: Two-loop renormalization group running of gauge couplings (left), and bottom and tau Yukawa couplings (right) for SM+Φ 1 +Φ 1+Φ 2 +Φ 2+Φ 3 .We observe gauge coupling unification and bottom-tau unification at around the same energy scale ≈ 10 13 GeV when we set the mass of all LQs to ≈ 125 TeV and take the non-zero initial values of LQ couplings given in Eq. (3.6).

Table 1 :
Representations of the SM and LQ fields under the SU (3) c × SU (2) L × U (1) Y gauge group.The SM fields transform under the SU (3) c × SU (2) L × U (1) Y gauge group as given in Table 1)(Y e ) = 3 2 Y e Y † e Y e + 3Tr Y † u Y u Y e + 3Tr Y † d Y d Y e + Tr Y † e Y e Y e −