Running of Fermion Observables in Non-Supersymmetric SO(10) Models

We investigate the complete renormalization group running of fermion observables in two different realistic non-supersymmetric models based on the gauge group $\textrm{SO}(10)$ with intermediate symmetry breaking for both normal and inverted neutrino mass orderings. Contrary to results of previous works, we find that the model with the more minimal Yukawa sector of the Lagrangian fails to reproduce the measured values of observables at the electroweak scale, whereas the model with the more extended Yukawa sector can do so if the neutrino masses have normal ordering. The difficulty in finding acceptable fits to measured data is a result of the added complexity from the effect of an intermediate symmetry breaking as well as tension in the value of the leptonic mixing angle $\theta^\ell_{23}$.


I. INTRODUCTION
Grand unified theories provide an intriguing framework for physics beyond the Standard Model (SM). The SO(10) gauge group is a popular version since it accommodates all SM fermions and the right-handed neutrino in one representation [1,2]. However, in order to be a viable candidate, it must be able to reproduce the experimentally measured fermion masses and mixing parameters. Therefore, it is relevant to analyze how well the parameter values of a particular model can be fitted to the measured observables.
The issue of fermion masses and mixing parameters in non-supersymmetric (non-SUSY) SO(10) frameworks has been extensively discussed previously in the literature, see for example Refs. [3][4][5]. The most minimal choice of scalar representations in the Yukawa sector of the Lagrangian that can reproduce the desired fermion data are the 10 H and 126 H representations, which has been demonstrated in a number of previous fits [6][7][8][9][10][11]. One can also choose to extend the Yukawa sector by adding a 120 H representation [7,9,12,13].
In order to compare the parameters of a high-energy theory to low-energy observables, one must take into account the renormalization group equations (RGEs) [14]. Most previous analyses of fermion observables in SO(10) models use solutions of the RGEs for the SM to compare the parameters at the SO(10) breaking scale M GUT to observables extrapolated from the experimental energy scale up to that scale [7,8] or solve the RGEs for the parameter values from M GUT down to the electroweak scale M Z [9], assuming an SM-like model in the whole energy range. However, non-supersymmetric SO(10) models require an intermediate symmetry breaking [15], so it is worthwhile to consider a more complete analysis that * Electronic address: tohlsson@kth.se † Electronic address: pernow@kth.se  [16,17]. A commonly considered intermediate symmetry is the Pati-Salam (PS) gauge group [18]. The derivation of the complete set of RGEs for the gauge, Yukawa, and scalar couplings [19][20][21][22] in such a breaking chain as well as their matching conditions at M I was first attempted in Ref. [23]. A numerical analysis based on the RGEs and matching conditions presented therein demonstrated the substantial effect that an intermediate gauge group in the symmetry breaking can have on the RG running and fits to fermion observables in a minimal SO(10) model [10]. This analysis was later refined by deriving correct RGEs and also considering an extended (or non-minimal) SO(10) model [13]. The present work aims to extend the analysis of the two models in Refs. [10,13] in several ways. Firstly, we fit to the two neutrino mass-squared differences separately, whereas the above mentioned works performed the fits to only their ratio. Secondly, we consider both normal ordering (NO) and inverted ordering (IO) of the neutrino masses. Thus we consider four different cases, namely two different models, each with both NO and IO. Lastly, we update the values of the observables at M Z to the best-known values to date. This paper is organized as follows. First, in Sec. II, we briefly describe the models including the breaking chain down to M Z . Then in Sec. III, we describe the procedure used to perform the analysis. Next, in Sec. IV, the results of the analysis are presented, discussed, and compared to previous results. Finally, in Sec. V, we summarize our findings and conclude.

II. DESCRIPTION OF THE MINIMAL AND EXTENDED MODELS
In this section, we briefly outline the two models to which fits will be performed. More details on these models can be found in Refs. [10,13]. The two models are both non-supersymmetric and based on the SO(10) gauge group. In what follows, they are referred to as the minimal model and the extended model due to their difference in scalar representations (whether or not the 120 H is included). We assume that the SO(10) symmetry breaking to the spontaneously broken SM in both models proceeds via the PS group, viz.
The electroweak symmetry breaking scale is M Z = 91.1876 GeV [24] and the energy scales of the other two symmetry breakings are computed to be [13] M I = 4.8 · 10 11 GeV and M GUT = 10 16 GeV, (2) respectively. These energy scales are derived from the requirement of gauge coupling unification at M GUT with the coupling constants as shown in Fig. 1. Note that we can perform this analysis independent of the RG running of the Yukawa couplings since, to one-loop order, the RGEs for the gauge couplings are independent of those of the Yukawa couplings [19,20].

A. SO(10) Lagrangians
Above M GUT , the Yukawa sector of the Lagrangian for the minimal model is given by where 16 F is the spinor representation containing the fermions, whereas 10 H and 126 H contain the Higgs scalars. Note that we forbid the coupling to the conjugate 10 * H by imposing a Peccei-Quinn U(1) PQ symmetry [4,5]. In the extended model, we also include the 120 H Higgs representation. Therefore, the Yukawa sector of the Lagrangian for this model is The Yukawa couplings h, f , and g are 3 × 3 matrices in flavor space. For simplicity, one can choose a basis in which h is real and diagonal. The other two matrices f and g are then complex symmetric and complex antisymmetric, respectively.
The fields that contribute to the particle masses are Φ 10 ≡ (1, 2, 2) 10 , Σ 126 ≡ (15, 2, 2) 126 , Φ 120 ≡ (1, 2, 2) 120 , Σ 120 ≡ (15, 2, 2) 120 , where the subscripts indicate which representation they originate from. Since these are the only scalars involved in the breaking chains, we appeal to the extended survival hypothesis to assume that they are the only ones that are present at this scale [8,15,25]. The Lagrangian for the minimal model between M GUT and M I is chosen as whereas for the extended model, we choose where C is the charge-conjugation matrix. The Yukawa coupling matrices Y are related to the ones appearing in the SO(10) Lagrangians by a set of matching conditions [16,23,26], for which we refer the reader to Refs. [10,13]. In Ref. [13], the correct RGEs can also be found, which determine the evolution of the gauge and Yukawa couplings between M GUT and M I .

C. SM-like Lagrangian
Below M I (and above M Z ), we assume as an SM-like model a two-Higgs-doublet model (2HDM) with the following Yukawa sector for both the minimal and extended models. Here, q L and L are the quark and lepton SU(2) L doublets, respectively, and u R , d R , e R , and N R are the quark and lepton SU(2) L singlets, respectively. The coefficients Y u , Y d , Y e , and Y D are Yukawa matrices for the up-type quarks, down-type quarks, charged leptons, and neutrinos, respectively, and φ 1 and φ 2 are the two Higgs scalars. The vacuum expectation values (vevs) after the symmetry breaking at M I are denoted as which are involved in the matching conditions for the Yukawa matrices found in Ref. [13]. Note that we have the constraint k 2 u + k 2 d = 246 GeV [27]. The RGEs for the evolution of the gauge and Yukawa couplings have previously been presented in the literature [13,23]. For neutrino masses, we assume a type-I seesaw mechanism with the seesaw scale close to M I . Thus, we have an effective neutrino mass matrix where M D = (k u / √ 2)Y D is the Dirac neutrino mass matrix and M R is the right-handed Majorana neutrino mass matrix. For more details on its relation to the Yukawa couplings in the PS model as well as its RGEs, the reader is referred to Ref. [13]. As explained therein, we also need to include a Higgs self-coupling for each Higgs doublet, since they affect the RG running of the neutrino mass matrix [28,29].

III. PARAMETER-FITTING PROCEDURE
In this section, we describe the procedure and numerical tools used to perform the parameter fits, which follows closely Refs. [10,13]. The general procedure consists of minimizing a χ 2 function, which is formed by comparing measured data at M Z with the RG running of parameter values from M GUT to M Z in a given SO(10) model. This RG running is performed by solving the relevant RGEs of the model parameters from M GUT to M Z , taking into account the change of parameters at M I . Due to the nature of the matching conditions at M I , it is not possible to extrapolate the observables from M Z to M GUT and we are forced to perform the RG running from the high-energy model down to the low-energy observables. In the minimal model, there are 22 parameters: three in h, twelve in f , four in the complex vevs v u and v d , one in the ratio of the real vevs k u /k d , one in the real vev v R , and one in the Higgs self-coupling λ (since the two are assumed to be equal above M I ). The extended model has a total of 34 parameters, which are the 22 of the minimal model and an extra twelve: six in g, four in the complex vevs t u and z u , and two in the real vevs t d and z d .   [30,31] and mean values of the quark mixing parameters are computed from values given in Ref. [24]. The neutrino masssquared differences and the leptonic mixing angles are taken from Refs. [32,33].
In order to determine the values of the abovementioned parameters that provide the best fit to measured data, we employ the following strategy: mixing parameters) at M Z and compare these to measured data by calculating the corresponding value of the χ 2 function. 4. Repeat the above steps to find the parameter values the provide the best fit and the corresponding value of the χ 2 function. The χ 2 function is defined as where X i is the measured value of the ith observable at M Z with corresponding error σ i and µ i is the corresponding predicted value of the given model for the current choice of parameter values. We also define the pulls p i as above for later convenience. For the sampling of the parameters, we interchangeably use the packages Multi-Nest [34][35][36], which is a nested sampling algorithm, and Diver [37], which is a differential evolution algorithm. Prior distributions are used to generate the next iteration of parameter values such that the elements of h, f , and g are sampled from logarithmic priors between 10 −20 and 10 −1 (and allowed to be negative), λ is sampled from a uniform prior between −1 and 1, and the vevs are sampled from uniform priors between −550 GeV and 550 GeV, except for v R which is sampled from a uniform prior between 10 12 GeV and 10 16 GeV and the ratio k u /k d which is sampled from a uniform prior between −550 and 550. The ranges of the above-mentioned priors are obtained from their expected orders of magnitude as well as preliminary numerical tests. After the sampling algorithm has converged on a set of parameter values, a Nelder-Mead simplex algorithm [38] is used to further evolve the parameter values to a set that provides an even better fit. However, note that one can never be sure that the global minimum is found. The best that one can do is to restart the minimization procedure several times with different starting parameter values.
In Table I, we list the measured values of the 18 observables that we fit to. Some comments regarding the choice of values and their corresponding errors are in order. Firstly, the values of the quark and charged-lepton masses are taken from an updated RG running analysis, using the same method as in Refs. [30,31]. The relative errors of the quark masses are set to values between 50 % (up and down quarks) and 2 % (top quark), motivated by large theoretical uncertainties in the quark masses, whereas the charged-lepton masses have relative errors set to 5 %, due to their almost negligible experimental errors, in order to facilitate the fitting procedure. Secondly, the values of the quark mixing angles are calculated from the elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix given in Ref. [24], whereas the Dirac CP-violating phase of the CKM matrix is computed from the Wolfenstein parameters of the same reference. The chosen relative errors between 1 % and 10 % reflect the relation among the uncertainties of the quark mixing parameters. Finally, the values of the leptonic mixing angles and the neutrino mass-squared differences are taken from Refs. [32,33], as are the associated errors of the leptonic mixing angles. For the mass-squared differences, we choose the relative errors so that their ratio has a relative error of 10 %, since the neutrino mass-squared differences have larger uncertainties than the charged-lepton masses. Note that we do not fit to the leptonic Dirac CP-violating phase, since knowledge of its value is limited to indications from global fits, see for example Refs. [32,33].

IV. RESULTS AND DISCUSSION
The χ 2 minimization procedure resulted in only one of the four cases having an acceptable fit, namely the extended model with NO, as shown in Table II  and the values of the remaining parameters are given in Table III   In Fig. 2, the pulls p i for each observable, defined in Eq. (13), are displayed. The sum of the squares of the pulls is the χ 2 function. It is evident that the largest contribution to the χ 2 function is due to the observable sin 2 θ 23 , for which the obtained prediction from the bestfit parameters is 0.287 (corresponding to θ 23 32.4 • in the lower octant), which is significantly lower than the measured value of 0.538 (corresponding θ 23 47.2 • in the higher octant). This tension due to the octant of θ 23 was also noted in a previous fit to observables at M GUT [8]. In fact, the measured value of sin 2 θ 23 used in the minimization procedure comes from a global fit [32,33] and although the 1σ range does not include our predicted value, it does allow for θ 23 in the lower octant. Furthermore, previous versions of the global fit [32,39] predicted a value of θ 23 in the lower octant, which was used in a fit similar to ours presented in Ref. [13]. Replacing the present measured value of sin 2 θ 23 by its previous value of 0.441, the χ 2 function for our current best-fit parameters takes the value 10.4, which is lower than the value 11.2 presented in Ref. [13]. Furthermore, we agree with their conclusion that significant tension in the fit is caused by the masses of the down and strange quarks.
Since the fit to the minimal model with NO is not totally unacceptable, it is worth to consider the significant contributions to its χ 2 function. The largest contribution comes from sin 2 θ 12 , followed by sin 2 θ 23 . In absolute terms, the best-fit value of sin 2 θ 12 is closer to the measured value than what is the case for sin 2 θ 23 , but since the relative error of the former is much smaller than that of the latter, it gives a larger contribution to the value of its χ 2 function. Similarly to the extended model, the best-fit parameter values of this model also predict a value of θ 23 in the lower octant.
In Fig. 3, the RG running of the fermion observables (except the quark mixing parameters since they exhibit small RG running) from M GUT down to M Z for the bestfit parameter values of extended model with NO are presented, with the dashed curves showing the RG running without the intermediate gauge group. That is, assuming the SM-like model discussed in Sec. II C for the whole energy range and with the same best-fit parameter values at M GUT . Note that since particle mass states are not well defined before electroweak symmetry breaking, the parameters above M Z are to be considered as effective parameters of the model. It is evident that the intermediate gauge group has a significant effect on the RG running. The quark masses display a diverging trend as the energy scale decreases from M GUT in the case of no intermediate gauge group. In fact, the parameters diverge so that the system of equations has no solution below a certain energy (which is why the dashed curves do not cover the full energy range even for the other observables). For the charged-lepton masses, the leptonic mixing angles, and the neutrino mass-squared differences, the difference in slope between the PS and 2HDM models is more pronounced. Particularly, the leptonic mixing angles exhibit large RG running in the PS model, but almost no RG running in the 2HDM model.
As a consistency check, one can observe that the RG running below M I in the case of an intermediate gauge group is of a similar form as that without an interme-  In order to describe the general behaviour of the RG running, particularly the quark and charged-lepton masses since these are linear in the Yukawa couplings, one can approximate the RGEs listed in Ref. [13], by their leading terms. For the energy region between M GUT and M I , the leading term is in all cases the one involving the gauge couplings. To an accuracy of a few percent, the RGEs can be approximated by where t = lnµ, the subscript i denotes the Yukawa coupling matrix in question and f i denotes a function of the three gauge couplings g j . The same applies to the RGEs in the 2HDM model for Y u and Y d , but for Y e the leading term is the one involving Y d such that the RGE may be approximated by In the energy region below M I , the approximation is not quite as good with the error of the charged-fermion masses at M Z between 10 % and 30 %. The fact that the RGEs for the quark and charged-lepton Yukawa matrices have different leading terms explains why the masses exhibit such different RG running. The mixing parameters and neutrino mass-squared differences cannot be well approximated by the leading terms, since these observables are related to the Yukawa couplings in a more complicated and non-linear way. The other two works that take into account the effect of the intermediate gauge group, Refs. [10,13], agree with our conclusion that it has a significant effect on the RG running of the parameters and thus on the fit itself. However, we find a considerably different behavior of the RG running of the parameters as well as increased difficulty in fitting the SO(10) models to the data. Comparing the effect of the intermediate gauge group on the RG running in our work with that of Ref. [13], we find a considerably closer similarity between the RG running behaviour of the parameters below M I with that in the absence of an intermediate gauge group. In comparison to Ref. [7], they, like us, concluded that the easiest model to fit to (out of the ones considered above) is the extended model with NO, in agreement also with Ref. [9]. The latter work also found that IO is more difficult to fit to than NO. However, they could find considerably better fits than we have found in all cases, since they do not take into account the intermediate gauge group, which increases the complexity of the problem considerably and complicates the fit. Of course, it must be noted that we cannot ensure that we have found the global minimum and cannot with complete certainty rule out the other three cases for which no acceptable fit was found. Furthermore, there may be other effects which may act to improve the ability to fit the models to the measured values of the observables, such as higher-order terms in the RGEs and threshold corrections to the RG running [11]. Such corrections may be particularly interesting for the minimal model with normal neutrino mass hierarchy, since this case is not too far from having a reasonable fit.

V. SUMMARY AND CONCLUSIONS
We have performed numerical fits to two different non-SUSY SO(10) models, namely the minimal model and the extended model, which differ by the inclusion of a 120 H representation in the Yukawa sector of the Lagrangian. The fits were performed with both NO and IO, and assuming a type-I seesaw mechanism for neutrino mass generation. The results of the fits show that out of the four cases considered, only the extended model with NO is viable with χ 2 18.6. One reason for the difficulty in finding acceptable fits of the models is the extra complexity introduced by the intermediate gauge group, which has been shown to have a considerable effect on the RG running as can be seen from the change in slope at M I in Fig. 3. Another reason for the difficulty in the fitting procedure is the fact that the best-known value of θ 23 is now in the higher octant, whereas a value in the lower octant (as previously predicted) would considerably improve the fit. However, before definitely ruling out the three cases that were unable to accommodate the measured values of the observables at M Z , one should investigate the effects of higher-order terms as well as threshold corrections.