{'fraction_non_alphanumeric': 0.03429052699125692, 'fraction_numerical': 0.01134996390470843, 'mean_word_length': 4.471801623875357, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 3, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
On the Consistency of Perturbativity and Gauge Coupling Unification

We investigate constraints that the requirements of perturbativity and gauge coupling unification impose on extensions of the Standard Model and of the MSSM. In particular, we discuss the renormalization group running in several SUSY left-right symmetric and Pati-Salam models and show how the various scales appearing in these models have to be chosen in order to achieve unification. We find that unification in the considered models occurs typically at scales below M^{min}_{B violation} = 10^16 GeV, implying potential conflicts with the non-observation of proton decay. We emphasize that extending the particle content of a model in order to push the GUT scale higher or to achieve unification in the first place will very often lead to non-perturbative evolution. We generalize this observation to arbitrary extensions of the Standard Model and of the MSSM and show that the requirement of perturbativity up to M^{min}_{B violation}, if considered a valid guideline for model building, severely limits the particle content of any such model, especially in the supersymmetric case. However, we also discuss several mechanisms to circumvent perturbativity and proton decay issues, for example in certain classes of extra dimensional models.


Introduction
Among the many interesting open problems in particle physics is the question whether the Standard Model (SM) gauge coupling constants unify at some high energy scale. Even though such Grand Unification does not occur in the SM, one of its best motivated extensions, namely the Minimal Supersymmetric Standard Model (MSSM), does predict Grand Unification at M GUT ≃ 10 16 GeV [1], opening up the possibility to embed the model into a Grand Unified Theory (GUT) based, for example, on the gauge group SO (10).
Grand Unification is also possible in more complex models: Amaldi et al. have identified several extensions of the SM or MSSM particle content that would lead to gauge coupling unification [1,2], and Lindner and Weiser have performed a similar study in left-right symmetric models [3]. Recently, Calibbi et al. have derived a set of "magic fields" that can be added to the MSSM without spoiling unification [4]. Model-dependent studies have been carried out for example by Shaban and Stirling [5] for a non-SUSY left-right symmetric model and by Perez-Lorenzana and Mohapatra for models with extra dimensions [6]. A model independent study of non-SUSY left-right symmetric models has also been carried out by Perez-Lorenzana et al. [7]. Recently Aranda et al. carried out a study of extended Higgs sectors in SUSY GUTs, paying attention to the constraints coming from the requirements of perturbativity and freedom from anomalies [8].
Our aim in this work is to emphasize that the often adopted requirements that (1) all gauge couplings remain perturbative up to the GUT scale M GUT and (2) M GUT is large enough to suppress proton decay beyond the experimental limit severely constrain extensions of the SM or the MSSM. Even though non-perturbativity is not a principle problem but only a practical one, and proton decay can be suppressed even for low M GUT in suitably constructed GUTs, we will in this paper accept both (1) and (2) as valid guidelines for model building, and investigate in detail the constraints that a model has to fulfil in order to be compatible with them.
In sec. 2, we will begin by introducing the formalism of RGEs in order to fix our notation. We will then proceed to a detailed investigation of Grand Unification in left-right symmetric and Pati-Salam models in secs. 3 and 4, respectively. There, we will discuss how the various energy scales appearing in these models are constrained by the requirement of successful perturbative unification compatible with proton decay bounds. In sec. 5, we will generalize our results, and discuss perturbativity issues in arbitrary extensions of the SM and the MSSM. Finally, in sec. 6 we will outline how perturbativity and proton decay constraints can be circumvented by more elaborate model building constructs such as extra dimension. We will summarize our results and conclude in sec. 7.

Renormalization group evolution of gauge coupling constants
The dependence of the gauge coupling constants g i on the energy scale µ for a theory with gauge group G = i G i is given at one-loop order by where t = ln(µ/µ 0 ) and µ 0 is an arbitrary renormalization scale. The coefficients b i , which are determined by the particle content of the model, have for non-supersymmetric models the form [9] b Here, the sum runs over all representations of the gauge group factor G i , counted according to their multiplicity in the model. For example, in the Standard Model (SM) with its six left-handed quarks and six right-handed quarks, the 3-dimensional representation of SU(3) c has a multiplicity of 12. The Dynkin index T i (R) of the representation R of G i is defined by is the charge corresponding to the representation R.
The coefficient s(R) has the value 2/3 if R is a multiplet of chiral fermions, while c(R) = 1/3 if R is a multiplet of complex scalars. Finally, C 2i is the quadratic Casimir operator of the adjoint representation of G i . For supersymmetric models, eq. (2) has to be replaced by The solution of the one-loop renormalization group equation (1) can be written in the form where An important observation and one of the central points of this paper is that adding new (non-singlet) matter particles to a given model will always increase at least one of the b i , and hence will lead to larger values for the corresponding α i (t) at t > t 0 . For sufficiently large particle content, α i (t) will reach the non-perturbative regime at relatively low scales.
All results presented in this paper will be based on the above one-loop RGEs. In the case of weak couplings this approximation is certainly justified, but when approaching the nonperturbative regime, α −1 i 1, higher order effects will become relevant. Nevertheless, in the context of our study it is sufficient to define the non-perturbativity scale as the scale at which the one-loop approximations breaks down or, more specifically, as the scale at which the one-loop value of at least one of the α −1 i becomes negative. (Of course, the physical α −1 i will always remain positive, and it is just the invalidity of the one-loop approximation that can lead to negative values.) Also, note that we neglect threshold corrections, which lead to an uncertainty of typically one order of magnitude for the non-perturbativity scale and the GUT scale.

Grand Unification in Left-Right Symmetric Models
We now illustrate the constraints that perturbativity and unification place on models with large particle content by considering three different left-right (LR) symmetric extensions of the Standard Model: (i) A non-supersymmetric model with Higgs triplets [10,11,12,13], (ii) the "minimal" SUSY LR model [14], and (iii) a slightly extended SUSY LR model [15,16]. All three models have in common that quarks and leptons reside in the following (i) Non-SUSY LR model with triplet Higgs [10,11,12,13]. In the non-supersymmetric case, the LR symmetry is broken down to the Standard Model at a scale M LR by Higgs triplets ∆(1, 3, 1, 2) and ∆ c (1, 1, 3, −2) .
The second of these acquires a vev of order M LR and thus breaks SU(2) R ×U(1) B−L down to U(1) Y , while the first one is required only to keep the particle content left-right symmetric. Fermion masses are generated by a Higgs bidoublet with a vev of the order of the electroweak scale. Even though we do not need to worry about the details of the symmetry breaking mechanism in order to study the renormalization group evolution of the model, it is crucial to know the mass scales of all particles. A detailed investigation [13] shows that all Higgs particles in the model have masses of the order of M LR , except for an SU(2) L doublet emerging from the bidoublet Φ and playing the role of the SM Higgs boson. Note that, even though the model is generically non-supersymmetric, a high scale supersymmetrization at a scale M SUSY > M LR is imaginable.
(ii) Minimal SUSY LR model [14]. The Higgs sector of this model is given by that of the non-SUSY model (i) (with all fields promoted to superfields), supplemented by two additional triplets∆ required for anomaly cancellation and a singlet to ensure charge and R-parity conservation. Moreover, in the SUSY case two Higgs bidoublets Φ 1 and Φ 2 are required to allow for non-vanishing quark and lepton mixing angles. Of these Higgs superfields, four doublets as well as the doubly charged components of ∆ c and ∆ c are light and have masses of O(M SUSY ). In this study, we will make the simplifying assumption that these fields all have the same mass, M SUSY , whereas in practice, for low values of M SUSY some of the fields are required to have slightly higher masses to satisfy constraints on, for example, flavour changing neutral currents. We have checked this simplification has a very minimal effect on our results.
(iii) Non-minimal SUSY LR model [15,16]. The particle content of the non-minimal model is similar to that of the minimal SUSY-LR model, with the singlet S being replaced by two triplets and uncharged under U(1) B−L . Breaking of the left-right symmetry proceeds in two steps in this model: The set of light particles includes the usual MSSM Higgses at the electroweak scale, the neutral components of ∆ c and∆ c at M B−L , and the Ω field at max(M 2 B−L /M LR , M SUSY ) [16]. In fig. 1 we compare the one-loop renormalization group running of the three considered models and of the MSSM. The embedding of the SM or MSSM into the standard GUT groups SU(5) and SO(10) requires α 3 (M GUT ) = α 2 (M GUT ) = 20 3 α 1 (M GUT ), but for the graphical presentation we have absorbed the factor 20 3 into the definition of α 1 , so that at M GUT the curves for α 1 , α 2 , and α 3 meet at one point. In the left-right models, the GUT normalization factor is 8 3 instead of 20 3 . The matching condition for the GUT-normalized U(1) coupling constants at M LR reads for models (i) and (ii) where α 1,LR is the U(1) B−L coupling constant, while α 1,SM and α 2 correspond to U(1) Y and SU(2) L , respectively. In model (iii), a condition of the form (13) is imposed not at M LR but at M B−L , with α 2 replaced by the U(1) R coupling constant. Fig. 1 shows that in all three models, unification is possible, but, especially in case (ii), tends to occur at rather low scales, in possible conflict with bounds from proton decay. In fact, from dimensional analysis, we expect the proton lifetime to be The bound τ p > 2. can be embedded into a Grand Unified Theory only if special measures are taken to forbid or suppress proton decay operators beyond the estimate (14). Note also that the unified coupling constant α GUT has a much larger value in the supersymmetric models than in the non-SUSY case. The reason is that according to eqs. (2) and (3) the additional particle content of SUSY models always increases the beta function coefficients b i .
In general, in supersymmetric models, besides the (dimension-six) operators induced by X and Y gauge boson exchange, proton decay can be induced by additional dimensionfive operators arising from the exchange of coloured higgsinos. These are really dangerous operators, since they lead to extremely fast proton decay. Indeed, if they are present, the proton life-time τ p becomes proportional only to the second power in the GUT scale, instead of the fourth power, as reported in eq. (14).
The purely supersymmetric contributions to proton decay have already been used to set limits on SUSY-GUT models. For example, the minimal supersymmetric SU(5) GUT model has been tightly constrained by the Super-Kamiokande lower bound on p → K +ν decay channel [18], assuming that the gauge coupling unification is satisfied. Several works have dealt with the possibility of suppressing dimension-five operators. Some of these models invoke extra-dimensions, see e.g. [19,20] or a more complicated higgs sector [21,22].
However, in our work we decided to pursue a conservative approach and thus apply only the constraint on the unification scale derived through eq. (14). It is, however, possible that some models with a unification scale M GUT M min  [13], (ii) the minimal supersymmetric LR model [14], and (iii) a non-minimal SUSY-LR model [15,16]. The light curves in the background correspond to the renormalization group running in the MSSM. (or even at M Pl ) by extending the particle content of the LR model. In particular, the addition of extra coloured particles could postpone unification, but, as discussed above, any new particle will inevitably bring the model closer to non-perturbativity. The orange shaded regions in the upper panels of fig. 2 show for which values of M SUSY and M LR it is definitely impossible to reconcile unification, perturbativity and the proton decay bounds by adding extra matter because at least one of the α i becomes non-perturbative below M min / B , even without additional particles in the model. For model (iii) there is also a yellow region in which this happens only for some choices of M B−L . In these regions of parameter space, any attempt to increase M GUT by adding new scalar or fermionic particles would be in even greater conflict with perturbativity. We see that the problem is particularly severe in the minimal SUSY-LR model (ii). The reason is that this model has many particles with low masses around M SUSY . In particular, the doubly charged scalars δ c−− andδ c++ have a very strong impact on the running of α 1 .

Grand Unification in a SUSY Pati-Salam model
Let us now investigate Grand Unification in another well-motivated class of models, namely those of the Pati-Salam (PS) type [24] with the gauge group SU(2) L × SU(2) R × SU(4). In particular, we will study the minimal supersymmetric PS model discussed in [25]. The gauge symmetry breaking pattern of this model is and SUSY is broken at M SUSY < M LR . The matter particles reside in the representations ψ(2, 1, 4) and ψ c (1, 2, 4 * ) of the PS gauge group. They get masses from the vevs of the Higgs bidoublets Φ(2, 2, 1) and Φ (2,2,15) . The

Model-independent discussion of perturbativity and Grand Unification
Let us now generalize our findings from the previous sections to arbitrary extensions of the Standard Model. The observation that models with large particle content enter the non-perturbativity regime at relatively low scales is quite generic (for exceptions, see sec. 6) since it follows directly from the fact that additional matter particles always increase the coefficients b i (see equation (3)) Therefore, if perturbativity up to the GUT scale is demanded in such models, gauge coupling unification must also occur at relatively low scales, in tension with proton decay bounds that suggest M GUT M min / B ∼ 10 16 GeV. Thus, models with large particle content are disfavoured over more economic ones.
To formulate the perturbativity constraints on models of new physics more quantitatively, let us assume an arbitrary extensions of the SM or MSSM particle content at a scale In In the MSSM, there is slightly less freedom to extend the particle content: Perturbativity of α 2 is lost below 10 16 GeV if 10 additional SU(2) doublets or 3 triplets are added, and α 1 would run to ∞ below 10 16 GeV if 9 Y = 2 superfields are introduced. Many extensions of the SM focus especially on the electroweak sector, which has much more room for interesting new phenomena at high energy than the QCD sector. Therefore, we will now consider models with no new coloured particles, i.e. b new 3 = 0, but with arbitrary contributions to b 1 and b 2 from particles at the TeV scale. For models of this type, we plot in fig. 7 the non-perturbativity scales for α 1 and α 2 (shaded areas), and we indicate those values. Of course, even in those cases where no gauge coupling unification has been found in our plot, it can be forced by adding suitably chosen coloured representations to the model, provided that the points where α −1 1 and α −1 2 meet lies below the SM/MSSM curve for α −1 3 , but still in the perturbative regime. We can also read from fig. 7 that for large beta function coefficients the GUT scale is always rather low, in possible conflict with proton decay. Again, one might hope to increase M GUT by adding more particles, especially coloured ones, but this will inevitably lead to non-perturbative evolution if the beta function coefficients become too large.
Thus we conclude that, for models in which the SM or the MSSM is extended only by additional matter fields or Higgs particles at the electroweak scale, Grand Unification above M min To end this section, we summarize the implications of perturbativity, unification, and proton decay constraints for model building in the flow chart shown in fig. 8. Perturbativity is particularly problematic in non-minimal SUSY models (scenarios (A2) and (B2)) because these tend to have very large particle content. If SUSY does not exist up to M Pl (cases (C1) and (C2)), the perturbativity constraint can be fulfilled more easily, but we encounter other difficulties, in particular the hierarchy problem. Taking these considerations into account, the most attractive of the considered scenarios is SUSY in its minimal form -the MSSM (or, equivalently, the NMSSM, which differs from the MSSM only by the addition of one gauge singlet, see [26] and references therein). Since the MSSM does not provide gauge coupling unification and cannot solve the hierarchy problem if M SUSY ≫ M Z (case (B1)), we are left with case (A1), the MSSM with M SUSY around the LHC scale.

Circumventing the perturbativity constraints
In the analysis carried out in the previous sections, two main problems with extensions of the Standard Model have appeared.  Table 1: Contributions of hypothetical new particles to the β-function coefficients b i . Numbers are given for chiral superfields; in the non-SUSY case, they have to be multiplied by 1/3 for complex scalars and by 2/3 for chiral fermions. Note that the values for b 1 include the GUT normalization factor 3 20 .  The first one is represented by the non-perturbative running of the gauge couplings at high scales when the particle content is increased with respect to the Standard Model or the MSSM. In figs. 5 and 6 we have quantified the limits that the new contributions to the β-function coefficients have to fulfil to preserve perturbative coupling constants α i .
In principle, we could also accept the divergent evolution of the coupling constants at high scales. However this means that we lose the ability to make predications about physics above the non-perturbativity scale and, moreover, we lose the ability to verify one of the main theoretical justifications for physics beyond the SM, i.e. gauge coupling unification.
A possible way out could be provided by embedding the gauge group of the model into a larger group at an intermediate scale M I . This "partial unification" will change the matter contribution T i (R) as well as the gauge field contribution C 2i in eqs. (2) and (3) and could thus decrease b i , keeping the model perturbative up to the Planck scale. Moreover, the matching conditions at M I could be such that the coupling of the new gauge group above M I is smaller than the corresponding coupling constants of the SM, in the same way as the α 1 corresponding to U(1) B−L in the LR models can be smaller than the α 1 corresponding to U(1) Y in the SM by virtue of eq. (13), see fig. 1 (i). We extensively investigate different classes of left-right models in sec. 3 and Pati-Salam models in sec. 4. Our study shows that also in scenarios with partial unification, one can easily run into similar perturbativity problems as in models that preserve the SM gauge group up to high scales. Similar difficulties are encountered in other models, for example the SUSY Little Higgs model considered in ref. [27] which encounters non-perturbativity below the unification scale when extra matter  is added to generate the top quark mass. A realistic solution of the perturbativity problem is represented by extra-dimension models in which only the gauge fields propagate in the bulk, see e.g. [6]. In this case the number of gauge degrees of freedom increases thus also increasing the negative contributions to the β function. The direct consequence is that the gauge couplings are pulled away from the Landau pole and their perturbative evolution can be followed up to the unification scale.
Low energy unification scales, represent the second problem that we experience in extensions of the Standard Model. Using the naive dimensional estimate of eq. (14), it can be found that such low values for M GUT imply values of the proton lifetime τ p that do not fulfil the experimental constraints provided by the Super-Kamiokande detector [28,29]. However, the lower limit on the unification scale can be relaxed if proton decay is forbidden or sufficiently suppressed so that the proton lifetime τ p becomes much larger than the estimate of eq. (14).
One possibility to, in part, evade these experimental proton decay bounds is to construct a "contorted flavours" GUT model [30]. Indeed, there is no a-priori reason to couple the first quark family with the first lepton family in a GUT multiplet. For example we could have the (u, d) quarks in the same representation as the (ν τ , τ ) or (ν µ , µ). It has been shown in [30] that this pattern can lead to the correct fermion masses and can kinematically suppress the proton decay channels into charged leptons. However, the decay channels involving neutrinos are still present and the experimental constraints reported in [17,29] have to be taken into consideration. In this case, it is not possible to completely remove the constraints on the GUT scale arising from the experimental bounds on the proton lifetime.
An economical way to avoid the constraint on the GUT scale is achieved by extending the Higgs sector, see e.g. [31] for the case of an SU(5) GUT. In this way the baryon number violating mixing matrix is, in general, no longer related to the baryon conserving one and proton decay can be suppressed by correctly adjusting the mixing angle in the baryon number violating matrix. On the other hand, as we have shown, increasing the content of the Higgs sector of a theory can easily spoil or make difficult the perturbative unification of the gauge couplings, unless the the new Higgses lie at the GUT scale.
Finally, we want to stress that the gauge coupling unification scale could be different from the grand unification scale, as recently proposed in [32]. In this model, the proton can naturally become almost stable if the grand unification scale is big enough compared to the scale where the gauge couplings meet.

Conclusions
In this paper, we have argued that the particle content of any extension of the Standard Model or the MSSM is tightly constrained if gauge coupling unification, perturbativity, and a GUT scale above the generic scale of proton decay, M min / B ∼ 10 16 GeV, are demanded. For example, we have demonstrated that in many left-right symmetric and Pati-Salam models, it is impossible to fulfil all three requirements simultaneously (cf. fig. 2 : Either, one has to live with unification at lower scales and invent a mechanism to circumvent proton decay bounds, or one has to extend the particle content further in order to push M GUT higher, but at the price of losing perturbativity and thus predictivity. We have then generalized our observations to a wider class of SM or MSSM extensions, and have examined the constraints that perturbativity imposes on the scale of new physics and on its contribution to the βfunction coefficients (cf. figs. 5 and 6). Since extra matter particles will always increase the values of the running gauge coupling constants at high scales, our constraints favour extensions of the Standard Model that require not too many new particles at low scales. In other words, the problems of the SM should be solved in a rather economic way. In our opinion, this is another hint in favour of the MSSM with M SUSY 1 TeV.
We have finally discussed how the perturbativity constraints can be circumvented either by simply accepting non-perturbativity, by designing models with partial unification, by introducing extra dimensions, or by aiming for Grand Unification at low to intermediate scales. In the latter case, special measures have to be taken to forbid or suppress proton decay.
In cases where M 2 LR /M P S < M SUSY , we instead assume that the colour octet has a mass of M SUSY , and the following analytic expression for α −1 3 (M Z ) is found The expression for the unification scale however remains unchanged.