Threshold effects in SO(10) models with one intermediate breaking scale

Despite the successes of the Standard Model of particle physics, it is known to suffer from a number of deficiencies. Several of these can be addressed within non-supersymmetric theories of grand unification based on $\text{SO}(10)$. However, achieving gauge coupling unification in such theories is known to require additional physics below the unification scale, such as symmetry breaking in multiple steps. Many such models are disfavored due to bounds on the proton lifetime. Corrections arising from threshold effects can, however, modify these conclusions. We analyze all seven relevant breaking chains with one intermediate symmetry breaking scale. Two are allowed by proton lifetime and two are disfavored by a failure to unify the gauge couplings. The remaining three unify at a too low scale, but can be salvaged by various amounts of threshold corrections. We parametrize this and thereby rank the models by the size of the threshold corrections required to save them.


I. INTRODUCTION
Grand unified theories (GUTs) in general [1], and in particular models based on the SO (10) gauge symmetry [2], are popular extensions of the Standard Model (SM) of particle physics. They can provide solutions to a number of open questions in the SM, such as the nature of charge quantization, anomaly cancellation, and the existence of three separate gauge groups [3]. Of a more phenomenological nature, SO(10) models naturally account for the generation of small neutrino masses through the type I [4][5][6][7][8] or type II [9][10][11] seesaw mechanisms.
A prerequisite of grand unification is that the evolution of the SM gauge couplings with energy scale, governed by the renormalization group equations (RGEs), must be such that they converge to each other. It is well-known that this does not occur in non-supersymmetric (non-SUSY) models, but that invoking SUSY can allow for successful gauge coupling unification [12][13][14]. On the other hand, SO(10) models allow for a symmetry breaking chain in multiple steps, due to it having rank five, which is one larger than the SM gauge group [15][16][17][18][19][20][21][22][23][24][25][26][27][28]. This can allow gauge coupling unification even without invoking SUSY.
Currently, the only experimentally relevant prediction of GUTs is the instability of protons.
The additional leptoquark scalar and gauge bosons that reside at the scale of unification M GUT in general mediate proton decay. This, together with the non-observation of proton decay, places a lower bound on the value of M GUT . In turn, this can disfavor some of the possible intermediate gauge groups since they predict a value of M GUT that is too low [29].
Threshold corrections [30,31] are one-loop corrections arising from fields lying at and around the scale of symmetry breaking that modify the matching conditions of the gauge couplings of the models above and below the energy scale of symmetry breaking. This can in turn modify the value of M GUT and thereby save some of the models that were previously disfavored [27,[32][33][34][35][36][37].
Furthermore, since threshold corrections modify the matching conditions of the gauge couplings, they can allow for unification in models where the gauge couplings do not unify [38][39][40].
In this work, we consider the direct breaking of SO (10)  Thereby, we quantify how large threshold corrections are required in order to save the models that a priori predict a proton lifetime that is too short. The renormalization group (RG) running is performed at two-loop level, with the one-loop level result given for comparison.
In Sec. II, we discuss the models that are analyzed in this work and present the solutions to the RGEs. We comment on which models achieve gauge coupling unification and the prediction for the proton lifetime in each of the models. Then, in Sec. III, we describe the computation of threshold corrections. In Sec. IV, we present the results of the threshold corrections for the different models.
Finally, in Sec. V, we summarize our findings and conclude.

II. MODELS
In this section, we discuss the eight models that we investigate in this work. Furthermore, we give some details on the particle content that is involved in each model and comment on the RG running.
The most minimal non-SUSY SO(10)-based model is one in which the gauge symmetry is broken directly to the SM. Following this logic, the next-to-minimal breaking chains are those with one intermediate gauge symmetry. The possible intermediate breaking chains may be seen, for example, in Refs. [26,41]. Here, we consider the direct breaking of SO (10) to the SM as well as all models with one intermediate symmetry breaking scale with at least two group factors in the intermediate symmetry. The reason that we require at least two group factors is that if there is only one, e.g.
SU (5), then this does not help enable gauge coupling unification.
In all models, the fermionic particle content consists of three generations of the spinorial 16 F .
In order to accommodate realistic fermion mass and mixing parameters, the scalar sector contains a complexified 10 H representation [42,43] and a 126 H representation. In order to retain some predictivity of the Yukawa sector of SO(10), we impose a Peccei-Quinn (PQ) symmetry [44,45], which forbids one of the two independent couplings between the fermions and the 10 H .
We assume that below the intermediate symmetry breaking scale M I , only the SM particle content survives and all other multiplets have masses around either M GUT or M I . This is in accordance with the "Survival Hypothesis", namely that scalars acquire the largest possible mass that is compatible with the symmetry breaking [16,46,47].
To one-loop order in perturbation theory, gauge couplings evolve from one scale M to another scale µ according to where the index i denotes the group to which the gauge coupling corresponds. The coefficient a i , known as the β coefficient, is determined by the particle content that exists in the relevant energy regime. It is given by [48,49] where C 2 (r) is the quadratic Casimir and S 2 (r) [sometimes also denoted C(r)] is the Dynkin index of the representation r, related to the quadratic Casimir by where G refers to the adjoint representation and d(r) denotes the dimension of representation r.
Furthermore, F i is the representation that the fermions belong to and S i is the representation of the scalars. The coefficient κ F is 1 for Dirac fermions and 1/2 for Weyl fermions and κ S is 1 for complex scalars and 1/2 for real scalars.
To two-loop order in perturbation theory, the gauge couplings obey the differential equation where the two-loop coefficients are given by [48,49] b There is also a contribution from the Yukawa coupling to the two-loop β function above. However, since the RG running of the Yukawa couplings is somewhat model-dependent, we neglect that term in Eq. (5). The β coefficients a i and b ij for the models considered are listed in Tab. II in App. A.
Given the values of the gauge couplings at the electroweak scale M Z 91.1876 GeV, the system of RGEs can be solved. Depending on the β functions, precise gauge coupling unification may be obtained. If it is possible, then that model is an allowed model for grand unification.
The relevant experimental prediction of grand unification related to the scale of unification is proton decay. From the scale of grand unification and the coupling strength g GUT at that scale, the proton lifetime in the most constraining channel can be computed as [50,51] where f π 139 MeV is the pion decay constant, A L 2.726 is a renormalization factor, α H 0.012 GeV 3 is the hadronic matrix element, and F q 7.6 is a quark-mixing factor. With these numerical factors, the proton lifetime in this channel can be estimated by Since proton decay has not been experimentally observed, there is a lower bound on the lifetime of the proton. The most constraining one is from Super-Kamiokande [52][53][54] with the bound τ (p → e + π 0 ) > 1.67 × 10 34 yr at 90 % confidence level. Any model must be able to accommodate a proton lifetime longer than the experimental bound.
In what follows, we employ the conventions that gauge couplings, β coefficients, and representations of fields appear in the order in which the gauge group is written. For example, in the first entry corresponds to SU(3) C , the second to SU(2) L , and the third to U(1) Y . For representations of fields, Abelian charges are always listed as subscripts.

A. No Intermediate Symmetry
The direct breaking of the SO(10) symmetry to the SM gauge group G 321 = SU(3) C × SU(2) L × U(1) Y can be achieved with a 144 H taking a vacuum expectation value (vev) in the appropriate direction [55]. We then assume that all multiplets from within the 144 H have masses at M GUT .
Assigning a non-zero PQ charge to the 144 H allows it to also break the PQ symmetry at M GUT .
Further, we assume that from the 10 H and the 126 H , only one combination of the SU(2) L doublets survives below M GUT . This is the SM Higgs doublet [43], such that the SM particle content is recovered below M GUT . The other fields that are not part of the SM field content reside at M GUT .
These are listed in Tab. III in App. B.
From the particle content described above, the β coefficients a i and b ij may be computed. They are listed in Tab. II in App. A. The resulting evolution of the SM gauge couplings is shown in literature [42,[57][58][59][60][61][62]. In this work, we follow the model described in Ref. [63], in which the SO (10) symmetry is broken by a vev in the 210 H . To break the PS symmetry as well as the PQ symmetry down to G 321 , we employ a vev in the 126 H together with a complex 45 H with a non-zero PQ charge.
The reason that two separate vevs are needed even though the 126 H has a non-zero PQ charge is to break the linear combination of PQ, B −L, and T 3,R which is otherwise left invariant [43,64,65].
Although the minimal choice of an extra representation for the breaking of the PQ symmetry could be considered to be a singlet, we will not use this. The reason is that singlets have mass terms The resulting RG evolution is shown in Fig on the fields such that (r 4 , r L , r R ) → (r 4 , r R , r L ) [66][67][68]. For a previous analysis of a similar model see e.g. Ref. [35].
In order to preserve D parity when breaking the SO (10)   The gauge group G 421 = SU(4) C × SU(2) L × U(1) R is a subgroup of the PS gauge group, but it may be reached directly by breaking the SO(10) symmetry. This is possible by assigning a vev to the appropriate direction of the 45 H . Models based on this gauge group have been previously analyzed in e.g. Refs. [33,69]. The breaking of G 421 down to G 321 can then be done using a vev of (10, 1) 1 from the 126 H . Since now the 45 H , which carries a PQ charge, is used to break the symmetry at M GUT , the PQ symmetry will also be broken at that scale. Contrary to the previously discussed models, we do not need to include two separate vevs to break the remaining linear combination of charges, since now both B − L and R remain unbroken at M GUT .
The matching conditions between G 321 and G 421 are identical to those given in Eqs. (10)- (11), with The fields that lie at M GUT and M I are given in Tab. VI in App. B.
At two-loop level, the resulting scales are M I ≈ 1.57 × 10 11 GeV and M GUT ≈ 2.69 × 10 14 GeV, giving a proton lifetime of τ p ≈ 5.8 × 10 29 yr. 3 Thus, as noted previously in the literature [29], this model is ruled out by proton decay bounds.
as studied in e.g. Refs. [70][71][72]. This may be reached by direct breaking of the SO (10)  Computing the β-functions between M I and M GUT , we first note that the fermions are embedded In this model, the matching condition at M I is more involved than in the above-discussed models due to the fact that the B − L needs to be appropriately normalized. Before normalization, the hypercharge Y may be expressed as In order to normalize these charges, the hypercharge Y is multiplied by the GUT normalization factor of 3/5 and the B −L charge is multiplied by 3/8. From this, one can derive the matching conditions of the appropriately normalized gauge couplings, namely In order to invert this relation, we face the issue that we are matching three gauge couplings to four. Thus, we introduce the parameter x such that α −1 This parameter is then solved for together with the scales M GUT and M I such that gauge coupling unification is achieved. The resulting matching conditions are The RG running is shown in Fig. 1e.
Solving for the scales that result in unification, we obtain at two-loop level M I ≈ 1.57×10 10 GeV and M GUT ≈ 5.18 × 10 15 GeV with the parameter x ≈ 1.38, resulting in a proton lifetime of τ p ≈ 9.4 × 10 34 yr. 4 Therefore, this model is allowed by proton lifetime considerations.
A similar model to the one in Sec. II E but with a surviving D parity may be constructed.
In this case, the SO(10) symmetry is broken down to the group From these fields, the β coefficients may be calculated and are given in App. A. Although this model has a similar gauge structure to the one in Sec. II E, the matching conditions become somewhat simpler due to the requirement that α −1 . This removes the extra freedom introduced by the parameter x above and the matching conditions simply read The resulting gauge coupling running is shown in Fig. 1f.
The scales that result in gauge coupling unification with at two-loop level are M I ≈ 3.13 × 10 11 GeV and M GUT ≈ 6.31 × 10 14 GeV, resulting in a proton lifetime of τ p ≈ 1.9 × 10 31 yr. 5 This model is therefore disfavored due to its prediction of the proton lifetime. in the 126 H . In order to also break the remaining combination of the Abelian charges and the PQ symmetry, a vev must be taken by one of the singlets in the 45 H .
The fermions are embedded as (3, From this, one can compute the β coefficients, given in App. A, as well as the RG running using the same matching conditions as in Sec. II E, replacing α −1 2R by α −1 1R . The result is shown in Fig. 1g, from which it is clear that unification is not achieved in this model. The reason is that the slopes of the two lines corresponding to the Abelian gauge couplings are too similar, meaning that they do not converge. Therefore, this model is disfavored on that ground and no prediction of the proton lifetime can be made. The final model considered is the breaking of SO(10) to a model of the SU(5) type. The reason that SU (5) is not considered on its own is that an intermediate symmetry that is a simple group does not help in achieving gauge coupling unification and instead changes the problem to requiring unification of the three gauge couplings at M I . We therefore consider the flipped SU(5) model, i.e. G 51 = SU(5) × U(1) X [74][75][76][77][78], in which the mixing between the external U(1) X and the Abelian charge from inside SU(5) to produce the hypercharge has the potential to help achieve gauge coupling unification.
The model which we construct, motivated by minimality, is one in which the SO (10)  To compute the RG running, the Abelian charge must be normalized by a factor of 1/ √ 40.
Normalizing also the hypercharge by its usual GUT factor, the matching conditions for the gauge couplings at M I can be derived. One must also take into account that since the SU(5) group contains the SU(3) C × SU(2) L part of G 321 , these must match at M I . The unification of α −1 2 and α −1 3 therefore determines M I . Hence, the matching conditions read Using these matching conditions, the RG running may be computed and is displayed in Fig. 1h.
As is shown, gauge coupling unification is not achieved in this model due to the diverging lines of α −1 5 and α −1 1X . To rectify this, the model would need to be made more complicated in order to either significantly change the RG running between M I and M GUT or to change the RG running in the SM region so as to change M I .

I. Gauge Coupling Running
The RG running of the gauge couplings for all models discussed in Sec. II are displayed in Fig. 1.

III. THRESHOLD CORRECTIONS
The results presented in Sec. II assume that the matching of two models occurs at tree-level, meaning that the gauge couplings of the subgroup are equal to a linear combination of the gauge couplings of the group from which it originates. At higher-loop orders, the matching conditions are modified by threshold corrections.
For the symmetry breaking of a group G m to another group G n at a scale M m→n , the matching condition with threshold corrections reads where the one-loop threshold corrections λ m n are given by [30,31] Here, the κ S i are 1 or 2 for real or complex representations, while k V i and k S

IV. RESULTS
Given the threshold corrections given in App. C, we randomly sample the masses of the scalars around the symmetry breaking scale and thereby find how large the deviations from the symmetry breaking scale are required to be in order to save the models.
In the case of no intermediate symmetry, the impact of the threshold effects on the matching conditions is such that they can compensate for the difference between the gauge couplings and therefore allow gauge coupling unification [39,40]. That is, since threshold corrections are meant to account for the failure of gauge coupling unification, we can compare the difference of the gauge couplings with the size of the threshold corrections. To this end, we define From the three gauge couplings in the SM, the failure of gauge coupling unification can be demonstrated by the two differences ∆λ 32 and ∆λ 21 . For each energy scale, we can plot the correlation of these two quantities, as is shown by the red lines in Fig. 2  which leads to a proton lifetime of τ p ≈ 5 × 10 34 yr, which is allowed but close to the current bound.
For the models with an intermediate symmetry that achieve gauge coupling unification without threshold corrections, the effect of threshold corrections on the scales M I and M GUT were found.
This was performed by solving the matching conditions to determine the two scales as functions of the η i parameters. Then, the η i were individually sampled in the same way as described above for To investigate if the models which were originally disfavored due to a too short proton lifetime (a) G422 (c) G421   We have solved the RG running in these models and studied whether or not gauge coupling unification is achieved and, if so, whether or not the prediction for the proton lifetime is above the experimental lower bound. Gauge coupling unification was achieved in models with G 422 , G 422D , G 421 , G 3221 , or G 3221D as intermediate symmetries. Of these, only G 422 and G 3221 predicted a proton lifetime that is above the experimental lower bound, with τ p ≈ 1.2 × 10 38 yr and τ p ≈ 9.4 × 10 34 yr, respectively. These two models are the well-studied PS and left-right models. Table I. Comparison of the viability of the models considered. The second column ("GCU") denotes whether or not gauge coupling unification is achieved without threshold corrections. The next four columns show if the predicted proton lifetime is above the lower bound for various sizes of threshold corrections. A checkmark ("") denotes an allowed model whereas a cross ("") denotes a disfavored model. The dashes in the last two rows signify that we did not investigate threshold corrections in those models. The results given are with RG running at two-loop level. Results at one-loop level may be found in the main text and the figures. The results in this work assume the specific model details as described in Sec. II. It should be noted that it is possible to modify some of these details while still achieving the same symmetry breaking chain. However, the results reported in this work can be seen as representative of these models. Furthermore, in the construction of the models, we have not taken into account any constraints from the scalar potential. It would be interesting to investigate this since there can be correlations between the masses of the scalars that could impact the results.
Additionally, we have not taken into account any bounds on physics related to M I . This may in some of the models be related to neutrino masses and/or leptogenesis. Furthermore, we have neglected the effect that perturbing the scalar fields around M I has on the RG running, as investigated for example in Ref. [35]. For larger perturbations, this effect may become substantial and have an effect on the conclusions. The above mentioned points, together with the investigation of the two models that were not considered here, namely those based on G 3211 and G 51 , would make for an interesting future study.

Appendix A: Beta Coefficients
The β coefficients for the models discussed in Sec. II are listed in Tab. II. The second column lists the β functions at one-loop level and the third column lists them at two-loop level.

Appendix C: Threshold Effects
In this appendix, we list the threshold corrections for the six models considered. They have been computed using Eq. (25) and the table of fields at each scale in App. C. We employ the notation that η i = ln(M i /M ), where M i is the mass of each scalar and M is the symmetry breaking scale at which the thresholds apply.
At M I , they are