Quantum corrections and the minimal Yukawa sector of SU (5)

It is well-known that the SU (5) grand unified theory, with the standard model quarks and leptons unified in 5 and 10 and the electroweak Higgs doublet residing in 5 dimensional representations, leads to relation, Y d = Y Te , between the Yukawa couplings of the down-type quarks and the charged leptons. We show that this degeneracy can be lifted in a phenomenologically viable way when quantum corrections to the tree-level matching conditions are taken into account in the presence of one or more copies of gauge singlet fermions. The 1-loop threshold corrections arising from heavy leptoquark scalar and vector bosons, already present in the minimal model, and heavy singlet fermions can lead to realistic Yukawa couplings provided their masses differ by at least two orders of magnitude. The latter can also lead to a realistic light neutrino mass spectrum through the type I seesaw mechanism if the colour partner of the Higgs stays close to the Planck scale. Most importantly, our findings demonstrate the viability of the simplest Yukawa sector when quantum corrections are considered and sizeable threshold effects are present.


I. INTRODUCTION
After the remarkable realization of the potential unification of the standard model (SM) gauge symmetries into a single gauge symmetry nearly fifty years ago [1][2][3], it has since become well-established that the Yukawa sector of the SM plays a pivotal role in determining the minimal and viable configurations of grand unified theories (GUT).The latter's potential to partially or completely unite quarks and leptons, in conjunction with the simplest choice of the Higgs field(s) in the Yukawa sector, often results in correlations among the effective SM Yukawa couplings that are inconsistent with observations.
The most glaring and simplest example of the above is the SU (5) GUTs with only 5-dimensional (5 and 5) Lorentz scalar(s) in the Yukawa sector in their ordinary (supersymmetric) versions.Both lead to at the scale of the unified symmetry breaking, namely M GUT , for the down-type quark and charged-lepton Yukawa coupling matrices Y d and Y e , respectively.The degeneracy between the two sectors predicted by Eq. ( 1) is not supported by the GUT scale extrapolated values of the effective Yukawa couplings determined from the measured masses of the down-type quarks and the charged leptons [4,5].The largest mismatch arises in the case of non-supersymmetric theories in which the extrapolation of the SM data implies, y b /y τ ≈ 2/3, y s /y µ ≈ 1/5 and y d /y e ≈ 2, at M GUT = 10 16 GeV.Deviation from the degeneracy shown in Eq. ( 1) can be achieved through several means: (a) Expanding the scalar sector [6][7][8][9][10][11], for instance, by introducing a 45dimensional Higgs field, or (b) Incorporating higherdimensional non-renormalizable operators [12][13][14][15][16][17][18], or (c) Introducing vector-like fermions that mix with the charged leptons and/or down-type quarks residing in the chiral multiplets of SU (5) [19][20][21][22][23][24][25][26][27][28].Each of these approaches alters the tree-level matching condition, Eq. ( 1), and introduces new couplings.These new couplings can be harnessed to obtain effective Yukawa couplings compatible with the SM.
In this article, we present a rather simple approach to alleviate the degeneracy between charged leptons and down-type quarks.Our method involves incorporating higher-order corrections to the tree-level matching conditions for the Yukawa couplings.Non-trivial implications of such corrections in the context of supersymmetric versions of SO (10) GUTs have been pointed out in [29][30][31] 1 .In the context of SU (5), we show that the inclusion of such corrections does not necessitate the introduction of new fermions or scalars charged under the SU (5) for modifying the tree-level Yukawa relations.This sets the present proposal apart from the previous ones outlined as (a-c) above.Specifically, we demonstrate that by expanding the minimal non-supersymmetric SU (5) framework to include fermion singlets and accounting for threshold corrections to the Yukawa couplings originating from these singlets, along with the leptoquark scalar and vector components already present in the minimal setup, a fully realistic fermion spectrum can be achieved.

II. YUKAWA RELATIONS AT 1-LOOP
The Yukawa sector of the model is comprised of three generations of 10, 5 and N generations of the gauge singlet 1 Weyl fermions and a Lorentz scalar 5 H .The most general renormalizable interactions between these fields can be parametrized as with i, j = 1, 2, 3 and α = 1, ..., N denotes the generations and C is the usual charge-conjugation matrix.
We have suppressed the gauge and Lorentz indices for brevity.The symmetric nature of the first term implies Additionally, M N is the gauge invariant Majorana mass of the singlet fermions alias the right-handed (RH) neutrinos.
The SM quarks and leptons residing in the SU (5) multiplets are identified as a , 5 m = ϵ mn l n and 1 = ν C , where a, b, c denote the color while m, n are SU (2) indices.For the scalar, we define a colour triplet T a ≡ 5 a H and an electroweak doublet h m ≡ 5 m H [32]. Decompositions of Eq. ( 2) then lead to the following Yukawa interactions with the colour triplet and Higgs: where h = ϵh and we have suppressed the SU (3) and SU (2) contractions.Matching of with the SM Yukawa Lagrangian at tree level leads to Y u = Y 1 and Y d = Y T e = Y 2 at the renormalization scale µ = M GUT .For the matching at 1-loop, the Yukawa couplings receive two types of contributions.The first arises from the vertex corrections involving the colour triplet or the leptoquark gauge boson in the loop.The interaction of the latter with the SM fermions originates from the unified gauge interaction and it is given by [4,5] −L where X transforms as (3, 2, −5/6) under the SM gauge symmetry.The second type of contribution to the Yukawa threshold correction is due to wavefunction renormalization of fermions and scalar involving at least one of the heavy fields in the loop.The 1-loop corrected matching condition for the Yukawa couplings at a renormalization scale µ is given by where f = u, d, e, ν.The details of the derivation of the above expression are outlined in Appendix A. In Eq (6), δY f are the finite parts of 1-loop corrections to the Yukawa vertex Y f while K f,f C ,h are the finite parts of the wavefunction renormalization diagrams involving heavy particles in the loops evaluated in the MS scheme.Y 0 f denotes the tree-level Yukawa coupling matrix.As mentioned earlier, at µ = M GUT .
Next, we compute δY f using the interaction terms given in Eqs.(3,4,5) and assuming massive color triplet scalar T , vector leptoquark X and N generations of the RH neutrinos ν C α .We find, at the scale µ.Here, M Nα is the mass of ] is a loop integration factor and it is given in Eq. (B1) in the Appendix B. It can be noticed that other than the overall colour factor, δY d and δY e differ by the contribution from the heavy RH neutrinos.Because of the treelevel Yukawa couplings between d C i , ν C α and T in Eq. ( 3), the Y d gets threshold correction from the RH neutrinos and colour triplet scalar.It is noteworthy that the corrections δY f vanish in the supersymmetric version of the model [33], due to the perturbative non-renormalisation theorem for the supersymmetric field theories [34,35].
The computations of the finite parts of wavefunction renormalization for the light fermions and scalar at 1loop, involving at least one heavy fields in the loop, lead to: The above is the main result of this paper.It is noteworthy that Eq. ( 10) not only suggests Y d ̸ = Y T e but also implies that the difference between the two matrices is calculable in terms of the masses of the heavy scalar, gauge boson and RH neutrinos and their couplings.The latter also determines the masses of other fermions and hence can be severely constrained as we discuss in the next section.
Before assessing the viability of Eq. ( 10) in reproducing the complete and realistic fermion mass spectrum, we investigate its role for the third generation Yukawa couplings, namely y b and y τ , through a simplified analysis.Considering only one RH neutrino with M N1 = M N and only the third generation, one finds from Eq. ( 10): at the GUT scale.Here, y t is the top-quark Yukawa coupling and y ν = (Y 3 ) 31 .For some sample values of y t , y ν and µ = M X = 10 16 GeV, the contours corresponding to different values of the ratio y b /y τ on the M T -M N plane are displayed in Fig. 1.
The GUT scale extrapolation of the observed fermion mass data requires y b /y τ ≈ 2/3.As can be seen from Fig. 1, this can be achieved only if either M T or M N is larger than µ = M X by at least one to two orders of magnitude.Moreover, y ν is also required to be large.For g, y t < 1, it is the third term in Eq. (11) which is required to dominantly contribute to uplift the degeneracy between y b and y τ and hence the largest possible value of y ν is preferred.M T ≫ M GUT or M N ≫ M GUT along with large y ν are needed to overcome the loop suppression factor of 1/(16π) 2 .This simple picture provides a clear and qualitative understanding of the favourable mass scales of the colour triplet scalar and RH neutrino and it also holds more or less when the full three generation fermion spectrum is considered as we show in the next section.
It is noteworthy that the RH neutrino through its coupling with the lepton doublet generates a contribution to the light neutrino mass through the usual type I seesaw mechanism [36][37][38][39].It is obtained as Since M N cannot be much larger than M GUT in this case, phenomenologically viable y b /y τ can be achieved only if M T > M GUT .Conversely, when considering perturbative values of y ν and a situation where M N greatly surpasses M GUT , the RH neutrino's contribution to the light neutrino mass is rather negligible.This inadequacy to reproduce a viable atmospheric neutrino mass scale necessitates the inclusion of an additional source of neutrino masses.We also provide an example of this in the next section.

IV. VIABILITY TEST AND RESULTS
To establish if the Y u , Y d and Y e are evaluated from Eqs. (6,8,9) can reproduce the realistic values of the SM Yukawa couplings and the quark mixing (CKM) matrix, we carry out the χ 2 optimization.Focusing on the minimal setup, we first consider only one RH neutrino with mass M N1 ≡ M N as mentioned in the previous section.The χ 2 function (see for example [40,41] for the definition and optimization procedure) includes 9 diagonal charged fermion Yukawa couplings and 4 CKM parameters.For the input values of these parameters at the GUT scale, we evolve the SM Yukawa couplings from µ = M t (M t being the top pole mass) to µ = M GUT = 10 16 GeV using the 2-loop renormalization group equations (RGEs) in the MS scheme following the procedure outlined in [41].The 2-loop SM RG equations have been computed using the PyR@TE 3 package [42].The values of the SM Yukawa and gauge couplings at µ = M t are taken from [43].The RGE extrapolated values at the GUT scale are listed as O exp in Table I.For the standard deviations, we use ±30% in the light quark Yukawa couplings (y u,d,s ) and ±10% in the rest of the observables as considered in the previous fits [41].
Using the freedom to choose a basis in Eq. ( 2), we set Y 1 diagonal and real.The RH neutrino mass matrix, M N , in general N flavour case can also be chosen real and diagonal simultaneously.Y 2,3 are complex in this basis.Using the Eqs.(6,8,9), we then compute the matrices Y u,d,e and diagonalize them to obtain the nine diagonal Yukawa couplings and quark mixing parameters.These quantities are fitted to the extrapolated data at µ = M GUT by minimizing the χ 2 function.We set M X = M GUT and g = 0.53 which is an approximate value of the RGE evolved SM gauge couplings at µ = 10 16  GeV.Fixing M T and M N to some values, we then minimize the χ 2 along with a constrain |(Y 1,2,3 ) ij | < √ 4π on all the input Yukawa couplings to ensure that they are within the perturbative limits [44].We repeat this procedure for several values of M T and M N .The obtained distribution of the minimized χ 2 (≡ χ 2 min ) is displayed in Fig. 2.
Note that without 1-loop corrections, i.e. with Y d = Y T e , the obtained value of χ 2 min is 53.Therefore, values of χ 2 min < 53 show improvements due to quantum corrected matching conditions in the model.In particular, for χ 2 min < 9, it is ensured that no observable is more than ±3σ away from its central value and, therefore, can be considered to lead to viable charged fermion mass spectrum and the quark mixing.
It can be seen from Fig. 2, a very good fit of the entire charged fermion mass spectrum and the quark mixing parameters can be obtained if M T or M N ≥ 10 17.2 GeV.These results are in a very good agreement with the limits on M T and M N obtained for y b /y τ ≲ 2/3 in a simplified case discussed earlier and shown in Fig. 1.
The three-generation χ 2 analysis also reveals that all the underlying 13 observables can be fitted within their ±1σ range (corresponding to χ 2 min ≤ 3) provided (i) M T ≤ 10 14.5 GeV and M N ≥ 10 17.2 GeV, or (ii) M T ≥ 10 18.2 GeV.While the second leads to M T alarmingly close to the Planck scale making the doublet-triplet splitting problem [45][46][47] more severe, the possibility (i) is conceptually allowed and technically a safe choice.Since M N is a scale independent of M GUT in the present framework, the large hierarchy between them is permitted.Also, M T can be significantly smaller than M GUT provided it satisfies the proton lifetime limit, M T ≳ 10 11 GeV [48].We list explicitly one benchmark solution from the region (i) which is displayed as Solution I in Table I.The fitted values of the corresponding input parameters are given in Appendix C.
Although, the RH neutrino is introduced to reproduce  the viable charged fermion mass spectrum, its mass and couplings are not constrained from the requirement of the light neutrino masses and mixing parameters.To account for both the solar and atmospheric neutrino mass scales, one needs at least two RH neutrinos in the minimal realization.The light neutrino masses are then generated through the usual type I seesaw mechanism: Here, M ν is 3 × 3 light neutrino mass matrix while M N is 2 × 2 heavy neutrino mass matrix.Y ν is 3 × 2 matrix which can be computed using Eqs.(6,8,9).The above leads to one massless light neutrino.We extend the χ 2 function to include the solar and atmospheric squared mass differences, three mixing angles and a Dirac CP phase to assess if Eq. ( 13) along with Eqs.(6,8,9) can provide a realistic spectrum of quarks and leptons.For the input values of neutrino observables, we use the results of the latest fit from [49] and set ±10% uncertainty as earlier.The RGE effects in neutrino data are neglected as they are known to be small [50][51][52][53] and within the set uncertainty for normal hierarchy in the neutrino masses which is the case considered here.
The result of χ 2 minimization for this case is shown in Table I as Solution II and the optimized values of parameters are listed in the Appendix C. As it can be seen, we find a very good agreement with all the fermion masses and mixing parameters with χ 2 min = 4.The resulting values of M N1 and M N2 are smaller than M GUT which requires M T > 10 17.2 GeV as anticipated from Fig. 2.
As a simple extension of the possibilities discussed above, it is straightforward to anticipate a case in which there are more than two RH neutrinos present.At least one of them is strongly coupled with the SM fermions and has a mass greater than M GUT .It leads to the required threshold corrections for a viable charged fermion spectrum, however its contribution to the neutrino masses is sub-dominant.The other RH neutrinos have sub-GUT scale masses and can lead to a realistic light neutrino spectrum without significantly altering the threshold corrections.This scenario is exemplified by Solution III in Table I.In this case, N 3 , with M N2 > M GUT , couples to the SM leptons with large couplings and gives the required threshold corrections to down-type quark sector.Notably, in this context, it is evident that the colour triplet scalar need not approach Planck-scale values to fulfil its role.

V. CONCLUSION
This article demonstrates that the seemingly unviable relationship, Y d = Y T e , predicted by the simplest and most minimal Yukawa sector of non-supersymmetric SU (5) GUT, can be rendered viable when accounting for 1-loop corrections to the tree-level matching conditions.This is accomplished by introducing one or more copies of fermion singlets.While they do not alter the tree-level matching conditions at the scale of unification, they can yield significant corrections at the 1-loop level through their direct Yukawa interactions with the down-type quarks and the colour triplet scalar.Sizeable non-degeneracy among the singlet fermions, colour triplet scalar, and leptoquark vector can thus impart large enough threshold corrections ensuring the compatibility of the minimal Yukawa sector with the effective SM description.
Our quantitative analysis reveals that achieving a realistic spectrum for the charged fermion Yukawa couplings and quark mixing necessitates either a significantly larger mass for the colour triplet scalar (M T ≫ M X ) or vastly higher masses for the RH neutrinos (M Nα ≫ M X ), under the assumption that the mass of the leptoquark gauge boson (M X ) defines the unification scale.The latter possibility is disfavoured if the same fermion singlets are expected to generate a viable light neutrino spectrum through the conventional type I seesaw mechanism.Nonetheless, the scenario of M Nα ≫ M X remains a plau-sible option if neutrinos acquire their masses through other means.This also includes type I seesaw mechanism with additional copies of RH neutrinos with sub-GUT scale masses and comparatively smaller couplings with the SM leptons.
It is noteworthy that the inclusion of quantum corrections can substantially alter the conclusions regarding the minimal Yukawa sector within the framework of an underlying grand unified theory.These findings provide motivation for conducting analogous investigations in the context of supersymmetric variants of SU (5)2 , as well as both the ordinary and supersymmetric versions of SO(10) GUTs, which feature more diverse particle spectra for threshold corrections and, simultaneously, more stringent symmetries that engage in intricate interplays.
In the case of Solution II, we get For all the solutions, we have used g = 0.53 as the value of unified coupling at the GUT scale and M X = 10 16 GeV.The optimized values of the masses of the colour triplet scalar and gauge singlet fermions are listed in Table I in the case of each solution.
at the scale µ.The loop integration factors are defined in Appendix B. Again, only K d C receives a contribution from the singlet fermions.As we show in the next sections, these contributions from singlet fermions are crucial for uplifting degeneracy between the charged lepton and down-type quarks.III.DEVIATION FROMY d = Y T eIt is seen from Eqs.(6,8,9) that the 1-loop corrections break the degeneracy between Y e and Y d .Explicitly, we obtain at the GUT scale:

√ 4π 2 0
this contribution is required to generate the atmospheric neutrino mass scale then one finds, M N = 7.6 × 10 16 GeV y ν .05eV m ν .

TABLE I .
The benchmark best-fit solutions obtained for three example cases as discussed in the text.Oexp denote the extrapolated values of the underlying observables at µ = 1016GeV.The reproduced values through χ 2 minimisation are listed under O th and corresponding pulls are given for each solution.The optimized values of the masses of leptoquark scalar and RH neutrinos are listed at the bottom of the table.