Multiple Scales in Pati-Salam Unification Models

We investigate models where gauge unification (GUT) proceeds in steps that include Pati-Salam symmetry. Beyond the Standard Model, we allow for a well-defined set of small representations of the GUT gauge group. We show that all possible chains of Pati-Salam symmetry breaking can be realized in accordance with gauge-coupling unification. We identify, in particular, models with unification near the Planck scale, with intermediate left-right or SU(4) quark-lepton symmetries that are relevant for flavor physics, with new colored particles at accessible energies, and with an enlarged electroweak Higgs sector. We look both at supersymmetric and non-supersymmetric scenarios.


Introduction
The Standard Model (SM) of particle physics organizes all known particles, including the Higgs boson, in a linear representation of the space-time and gauge symmetries. This representation is reducible and decomposes into several irreducible multiplets. Therefore, the SM Lagrangian contains a considerable number of free parameters. A unified theory (GUT) where the SM gauge symmetry is embedded in a larger gauge group can reduce the number of free parameters.
Furthermore, there is a general expectation that a unified model of matter and interaction could explain some of the puzzles of particle physics, such as quark and neutrino flavor mixing, the smallness of neutrino masses, or dark matter.
Particularly promising GUT models are left-right symmetric extensions [1,2], the Pati-Salam model [3,4], the SU (5) theory by Georgi and Glashow [5], or the combination of both in an SO(10) [6] or E 6 [7] gauge theory. In these GUTs, matter fields unify within each generation, separately. The SM Higgs sector is enlarged. Further fields, symmetries, or geometrical structure are required to explain the breaking of GUT symmetries and the emergence of the SM (or its minimal supersymmetric extension, MSSM) at energies directly accessible to us.
In this paper, we investigate various classes of Pati-Salam (PS) models. PS symmetry, with a discrete left-right symmetry included, is the minimal symmetry group which collects all matter fields of a single generation in one irreducible multiplet. As an extension of the SM, it is particularly interesting because it naturally accommodates multiple energy thresholds or scales that fill the large gap between the SM (TeV) scale and the scale of ultimate unification, presumably the Planck scale. Such models have been widely discussed in case of both supersymmetric [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] and non-supersymmetric models [24][25][26]. PS models do not just exhibit multiple scales associated with symmetry breaking, but they can generate additional mass scales due to see-saw type patterns in the spectrum [27,28].
We want to study the relation of symmetry breaking and intermediate mass scales in PS models in a suitably generic way while keeping the sector of extra (Higgs) fields below the GUT scale as compact as possible. To this end, we impose conditions on the spectrum of new fields, so we can describe the chain of symmetry breaking via a staged Higgs mechanism with only small representations of the GUT group. In this setup, we scan over the complete set of models and investigate the allowed ranges and relations of the various mass scales they can provide.
There are several specific questions that one can address by such a survey of models. First of all, some GUT models are challenged by the non-observation of proton decay, so we may ask whether complete unification, and thus the appearance of baryon-number violating interactions, can be delayed up to the Planck scale. Secondly, whether the origin of neutrino masses and flavor mixing could be associated with a distinct scale, decoupled from complete unification.
Thirdly, whether there can be traces of unification, such as exotic particles, a low-lying leftright symmetry, or an enlarged Higgs sector, that may be observable in collider experiments or indirectly affect precision flavor data.
By restricting our framework to economical spectra with only small representations (see Sec. 3), we focus the study on phenomenologically viable models that look particularly attractive. Nevertheless, they span a large range of intermediate-scale configurations between the TeV and Planck scales. We do not impose extra technical conditions on the gauge-coupling running and matching ( [9], [12], [25]), so we can cover also corner cases where scale relations are tightly constrained.
After a short review of PS models in Section 2, we lay out the detailed framework of our study in Section 3. In Section 4 we take a brief look at the staged Higgs mechanism, before we discuss the overall spectrum patterns in Section 5. In Section 6 we implement the conditions of gauge coupling unification. Finally, Section 7 and Section 8 present generic properties and interesting details for the complete survey of PS models, divided in the two classes of supersymmetric and non-supersymmetric models respectively, before we draw conclusions.

Pati-Salam Symmetry
By the label of Pati-Salam gauge group, we refer to the symmetry group The SU (4) C factor is an extension of QCD with lepton number as fourth color, and there is a right-handed SU (2) R symmetry analogous to the weak interactions. We impose an exact discrete symmetry Z LR which exchanges SU (2) L with SU (2) R , and left-handed with righthanded matter fields, respectively.
If we impose this symmetry on any generation of quarks and leptons, they combine to a single irreducible representation Ψ L/R . Regarding the continuous gauge group only, this is a direct sum Ψ L ⊕ Ψ R , but it is rendered irreducible by the left-right symmetry 1 . In particular, this representation necessarily includes a right-handed neutrino field. The explicit embedding of the matter fields in the PS multiplet is as follows: (2) Here the subscript (r,g,b) are explicit color indices. This notation is compatible with the usual conventions for supersymmetric GUT models, where we will denote by Ψ L/R the corresponding superfield.
The The PS group is a subgroup of SO(10), but does not contain SU (5). It is well known that the gauge couplings unify for a pure SU (5) GUT model, if supersymmetry is imposed.
However, the recent discoveries about neutrino masses and mixing (see review article in [29]) suggest additional new physics at high energies. Concrete estimates tend to place the neutrino mass-generation scale below the GUT scale (either as direct mass scale [30] or via a see-saw mechanism [31]). We should expect multiple thresholds where the pattern of multiplets and symmetries changes. This idea fits well in the context of a PS symmetry and its breaking down to the SM.
Therefore, we investigate scenarios where a PS gauge symmetry is valid above some high energy scale. Below the PS scale, we allow for further thresholds where only a subgroup of PS is realized. One such subgroup is the minimal left-right symmetry SU (3) C × SU (2) L × PS models provide a partial explanation for the observed stability of the proton. SU (4) C contains the U (1) B−L subgroup, so B − L is conserved. In analogy with QCD, lepton number as the fourth color -and thus baryon number -is conserved by all interactions that do not involve the totally antisymmetric tensor abcd with four color indices. Counting field dimensions, this excludes baryon-number violating interactions of fermions in the fundamental or adjoint representations of SU (4) C , at the renormalizable level. In a minimal PS extension of the SM, proton-decay operators are thus excluded or, at least, naturally suppressed. In particular, the terms of the MSSM which mediate proton decay via squark fields are inconsistent with PS symmetry. 3 A field condensate that spontaneously breaks SU (4) C to SU (3) C carries lepton number, but no baryon number. Hence, PS breaking does not induce proton decay either, at the level of renormalizable SM or MSSM interactions. On the other hand, breaking PS symmetry down to the SM gauge groups can generate and relate the operators that provide neutrino Majorana masses and mixing in the lepton sector, as well as Yukawa terms and mixing in the quark sector.

General Setup
In this paper, we consider both supersymmetric and non-supersymmetric versions of a PS gauge theory. We first investigate supersymmetric models, where the model space is more constrained.
For this purpose, we assume N = 1 supersymmetry to hold over the full energy range above the TeV scale. Later we will also discuss the running of the gauge couplings in non-supersymmetric versions of the models.
Following a top-down approach, we put the highest scale under consideration at the Planck scale (we use M Planck = 10 18.2 GeV). In that energy range, gravitation is strong and a perturbative quantum field theory in four space-time dimensions is unlikely to be an appropriate description. Below this scale, gravitation becomes weak, and an effective weakly coupled fourdimensional gauge theory can emerge, with a GUT symmetry that contains PS as a subgroup.
By M GUT we denote the energy scale where this larger symmetry is broken down to PS. We note that M GUT may coincide with M Planck . Throughout this paper, we will leave unspecified the field content above M GUT and the mechanism responsible for GUT breaking down to the PS group.
Below M GUT , we are thus dealing with a PS model which we specify in more detail. In the supersymmetric case, we list all the PS-symmetric supermultiplets that are contained in the spectrum. All such supermultiplets can interact and contribute to the running of the gauge couplings. The effective Lagrangian may contain all symmetric renormalizable interactions of these superfields. It will also contain, in general, non-renormalizable interactions of arbitrary dimension which are suppressed by powers of the relevant higher mass scales, i.e., GUT or Planck scales. While non-renormalizable terms, as it turns out, are not required for the symmetry-breaking chains that we discuss below, they can contribute significantly to flavor physics in the low-energy effective theory. We will not discuss non-renormalizable effects in this paper in any detail, so we do not attempt a complete description of flavor. The necessary inclusion of all kinds of subleading terms is beyond the scope of this paper.
For the supersymmetric PS model and the effective theories down to the SM, we state the following simplifying assumptions: 1. The model is effectively a perturbative quantum field theory in four dimension at all scales below M GUT .
2. The breaking of the PS symmetry and its subgroups is due to Higgs mechanisms that involve the superfields contained in the given PS spectrum.
3. SUSY is broken softly by terms in the TeV energy range. 4. The effective scale-dependent gauge couplings coincide at M GUT , consistent with a GUT symmetry that contains SO(10).
The last point is the most important restriction that we impose on our models. (For a recent survey with different assumptions, see [9], [25]). As we will discuss below, this minimal field content can realize all possible chains of PS sub-symmetry breaking and addresses the phenomenological questions that we raised in the Introduction above. Simultaneously, a model with only small representations can easily be matched to a more ambitious theory of Planck-scale symmetry breaking. For instance, all multiplets that we consider, occur in the fundamental 248 representation of E 8 [32].
We now state explicitly the field spectrum that is consistent with our assumptions and display all possible irreducible PS multiplets in Tab. 1. The matter fields of the MSSM are contained in Ψ L/R multiplets. There are three copies of this representation, each one including a right-handed neutrino superfield. The other supermultiplets are new. They contain the MSSM Higgs bidoublet, various new Higgs superfields, and extra "exotic" matter. Table 1: Chiral superfield multiplets of the PS models considered in this paper. The fields are classified by their gauge-group quantum numbers; the discrete Z LR symmetry renders the multiplets left-right symmetric and thus irreducible.
Let us take a closer look at these superfields: 1. h could directly qualify as the MSSM Higgs bi-doublet.
2. Σ and T L/R are chiral multiplets in the adjoint representation. They have the same quantum numbers as the gauge fields of SU (4) C and SU (2) L,R , respectively.
3. Φ L/R and Φ L/R are extra multiplets with matter quantum numbers and their chargeconjugated images, respectively. Both must coexist in equal number, otherwise additional chiral matter generations would be present at the TeV scale 4 .
4. E and F contain only colored fields. Under SU (3) C they decompose into triplets and anti-triplets, so they can be viewed as vector-like quarks and their scalar superpartners.
Depending on their couplings, they may also behave as leptoquarks or diquarks. The F multiplet may have both leptoquark and diquark couplings and thus can violate baryon number (cf. Sec. 5.4). For the E multiplet, this is not possible. Note that Σ also provides vector-like quark and antiquark superfields, together with a color-octet and a color-singlet superfield. The Σ couplings conserve baryon number.
5. The singlet fields S 78 and S 27 (and any further singlet fields that originate from SO (10) or E 6 singlets) can couple to any gauge-invariant quadratic polynomial.
Chiral superfields which are not copies of the above are excluded by our basic assumptions.
An explicit (i.e., PS invariant) mass term in the superpotential can be either present in the superpotential ab initio, or generated by a singlet condensate. Without further dynamical assumptions, mass terms and mass thresholds are thus constrained only by matching conditions on the coupling parameters, and may exhibit a hierarchical pattern.
Some of the above mentioned multiplets contain an electroweak singlet and thus qualify as Higgs superfields for various steps of PS symmetry breaking, if the corresponding scalar component acquires a vacuum expectation value (vev). The associated vevs may be denoted as T (1, 1, 3) , Σ(15, 1, 1) , Φ(4, 1, 2) , and h(1, 2, 2) , corresponding to the Higgs fields introduced below. They induce the following breaking patterns: If there is a hierarchy between the vevs, we observe a cascade of intermediate symmetries.
Thus, we have the possibility for multi-step symmetry breaking. In the following, we distinguish six classes of PS models by their hierarchy patterns. We denote them as follows: The symmetry breaking chains associated to these classes are shown in Fig 1 (see also [28]). The scale Φ is always the lowest symmetry-breaking scale (above the electroweak scale), since this vev breaks all symmetries. The other vevs break the PS symmetry only partially and are only relevant if they lie above Φ . Nevertheless they are needed because we require the symmetry breaking to result from a renormalizable potential, which does not exist for Φ alone.
Class A and D are limiting cases of the other and will not be discussed in greater detail.
The number of matter multiplets (Ψ L/R ) is fixed to three. All other multiplets may appear in an arbitrary number of copies. We will specifically study scenarios where the multiplicity is either zero, one, or three. This already provides a wide range of possibilities for spectra and hierarchies and appears most natural in view of the generation pattern that is established for SM matter.
Let us further categorize models, constructed along these lines, that we will study below.
Each model type may fit into any of the above classes. It contains the three generations of MSSM matter superfields Ψ L/R together with the following extra superfields:  h, F , and S 27 appear in three copies since they combine with the matter multiplets. For the other multiplets, we set the multiplicity to one.

Type g (Generic Model):
Here we classify all models that do not qualify as either of the above three types. We will denote these models by their class and a unique number. The numbering scheme is described in Appendix C.

Higgs Mechanism and Supersymmetry
Some of the superfields that we introduce beyond the MSSM matter fields, should act as Higgs fields and break the gauge symmetry, step by step, down to the SM gauge group. For each breaking step, we have to verify that the scalar potential has a local minimum for a non vanishing field configuration that breaks the gauge symmetry in the appropriate way. The value of the superpotential at this field configuration should be zero, so supersymmetry is maintained in the ground state.
To achieve this, we construct a generic renormalizable superpotential for the multiplets that we include in a specific model. Non-renormalizable terms could be added, but we will see that the various routes of gauge symmetry breaking result from renormalizable terms alone, so they can be ignored in a first approach. The specific models derive from a generic superpotential that includes all allowed superfields and their renormalizable interactions.

Generic Superpotential
We start at the GUT scale with the most general renormalizable PS-invariant superpotential for the superfields shown in Tab. 1 and impose parity (Z L/R ) as a symmetry at this scale. The superpotential can be broken down in several parts which we organize in view of the model types m, s, and e as introduced above.
The terms within W Φ/Σ/T generate the Higgs potential for all steps in the staged Higgs mechanism. This superpotential consists only of those fields that are allowed to get a vev.
Depending on the desired vev structure (case A to F), we may set some terms to zero, to obtain a minimal superpotential.
The superpotential W h/F is present in the models that contain the fields h and F . Similarly, the field E comes with the terms W E . The potential W S contains all interactions of the singlets with the other Higgs fields, where in each term, S indicates an arbitrary linear combination of all PS singlets present in the model.
If h, F are present, there is the W Yukawa superpotential. This part, which is the only renormalizable superpotential involving (two) matter superfields 5 , implicitly contains generation indices. Analogously, generation indices are implied for all superfields that occur in more than one copy. We note that Y contains only symmetric Yukawa couplings that are universal across leptons, neutrinos, up-and down-type quarks.
Only Ψ L and Ψ R are considered as matter superfields, containing all fermions of the SM.

Higgs Mechanism and Supersymmetry
For our supersymmetric models, we want to keep SUSY unbroken down to the TeV (MSSM) scale. To satisfy this constraint, we have to verify that all F and D terms vanish [33,34] after inserting the vacuum expectation values of the fields 6 : We explicitly calculate the scalar potential for the generic superpotential shown in (4). This covers all models under consideration; for any specific model, we set the couplings to zero that are not involved.
The fields that transform under PS but contain a singlet under the SM gauge group, are Φ R , Φ R , Σ, T R , and the matter superfields Ψ R . In particular, the SU (4) C singlets in Φ R and Φ R simultaneously break SU (2) R and thus induce direct breaking PS → MSSM. However, there is no nontrivial renormalizable superpotential of Φ R , Φ R alone, so we include the Σ and T R multiplets in the discussion. The vevs Σ and T R are particularly relevant to the symmetrybreaking chain if they are significantly larger than v Φ , since they leave subgroups of PS intact (cf. Fig. 1). We introduce the following notation: Here the subscript denotes the field component (or generator) that acquires the vev. These are all vevs we allow for in this paper.
For flavor physics, the vevs v Φ and v T are of particular interest, because they break the discrete Z LR symmetry. As long as the left-right symmetry is unbroken, there are no terms that distinguish right-handed up-type from down-type quarks. Therefore, the energy scale below which flavor mixing appears in the renormalizable part of the effective theory, is given by The vacuum expectation values of the F terms also have to vanish. Inserting the vevs, we obtain the necessary and sufficient conditions Solving for the mass parameters of Φ, T and Σ as well as for one of the singlet couplings, we obtain We have verified that these vev configurations also minimize the scalar potential while maintaining supersymmetry. Thus, depending on their mutual hierarchy, they realize the symmetrybreaking chains of the model classes A to D.
For the models classes E and F, we assume a vanishing vev of T or Σ, respectively. Nevertheless, PS symmetry is broken completely down to the MSSM gauge symmetry. A vanishing vev results if the corresponding multiplet does not couple to Φ R /Φ R . This can be realized by either omitting the multiplet entirely, or by setting l ΣΦ (l T Φ ) to zero, respectively. Otherwise, SUSY would be broken.
As long as we stay in the regime of renormalizable superpotentials, it is not allowed to set v Φ = 0, because in this case the solution of the minimization condition would be v T ≡ 0, so SU (2) R would remain intact. The reason is the absence of trilinear terms for T R in the renormalizable superpotential: SU (2) has no cubic invariant. In addition there would be no

Spectra and Phenomenology
The various classes of models that we introduced above, lead to a great variety in the observable spectra. In the current work, we aim at a qualitative understanding, so we refrain from detailed numerical estimates or setting up benchmark models. Nevertheless, we observe a few characteristic patterns which can have interesting consequences for collider and flavor physics.
We discuss these patterns and their phenomenological consequences below.

Mass Matrix
Having verified that SUSY remains unbroken down to the TeV scale where soft-breaking terms appear, we consider the mass matrix of the scalars. We do not intend to derive quantitative results here, but just assign mass scales to the individual multiplets which depend on the hierarchy of the vevs. In Tab. 2, we list the results for all fields and the interesting model classes B, C, E and F. The classes A and D are limiting cases. Therefore they are not listed separately.
We should keep in mind that this table refers only to the mass contributions that result from symmetry breaking. All superfields except for the matter multiplets, may carry an individual PS-symmetric bilinear superpotential term. These mass terms are completely arbitrary and, with supersymmetry, do not cause naturalness problems [35,36]. Furthermore, a vev in any PS singlet field may contribute a similar term, again unrestricted by symmetries. Hence, by including a multiplet in the spectrum of the effective theory in a particular energy range, we implicitly assume that the sum of all mass terms for this field is either negligible compared to the energy, or fixed by the conditions that determine the vacuum expection values.
On the other hand, in many models, not all of the fields E, Σ and T receive masses from PS and subsequent symmetry breaking, given only the renormalizable Lagrangian terms above.
For these fields, either the bilinear mass term or effective masses induced by higher-dimensional operators play an important role and must be included as independent parameters. From a phenomenological viewpoint, we are mainly interested in the lowest possible mass for each of those fields. We discuss this issue in Sec. 7.1.
While some multiplets acquire masses proportional to either one of the symmetry-breaking which can be significantly smaller than v Φ if there is a hierarchy between the vevs. In other words, there is a new see-saw effect [27], unrelated to the well-known neutrino see-saw. We denote this induced mass scale by M IND . It is located below the scale where PS is completely broken down to the MSSM symmetry group. A generic expression is Depending on the model class, some of the field multiplets F , h, or T L become associated with M IND 7 (cf. Tab. 2). We thus get "light" supermultiplets consisting of scalars and fermions, which may be colored, charged, or neutral, and acquire a mass that does not coincide with either of the symmetry-breaking scales. If the hierarchy between the vevs is strong, M IND may be sufficiently low to become relevant for collider phenomenology. In model classes B, C, and F, it provides a µ term for h and may thus be related to electroweak symmetry breaking. In any case, the threshold M IND must be taken into account in the renormalization-group running of the gauge couplings. 7 In class F, this applies also to the singlet part of T R .
The classes A and D are a limiting case of B and C. Class A can be reached if one sets all vevs equal to a single vev v and class D if one just sets v Σ = v T = v. The order is thus, that fields which mix are grouped together. Thus the mass eigenvalues are a linear combination of the listed fields. Massless components which are the Goldstone Bosons are explicitly not considered here.

Goldstone Bosons
Not all of the scalar fields are physical: since the broken symmetries are gauged, nine of the scalar fields are Goldstone bosons that provide the longitudinal modes of the PS gauge bosons that are integrated out in the breaking down to the MSSM. Six of them come from The last one comes from the breaking of the U (1) subgroups We identify these Goldstone bosons as

MSSM Higgs
Apart from the matter fields and Goldstone bosons, the chiral superfield spectrum must provide the Higgs bi-doublet of the MSSM that is responsible for electroweak symmetry breaking.
Above the TeV scale, this multiplet should appear as effectively massless; mass terms (which together set the EWSB scale) are provided by soft-SUSY breaking parameters and the µ term which mixes both doublets. We note that the µ term may result from the vev of one of the electroweak singlets in the model, i.e., the model may implement a NMSSM-type solution for the µ problem [37,38].
The obvious candidate for the MSSM Higgs bi-doublet is the superfield h. From Tab. 2 we read off that the PS-breaking contribution to the h mass is see-saw suppressed in models with v T = 0 (B, C, D and F). In these models, the electroweak hierarchy is generated, at least partly, by a high-energy hierarchy in the PS symmetry-breaking chain.
However, the Φ L multiplets provide further Higgs candidates. In particular, while the right-handed doublets in Φ R serve as Goldstone bosons for the SU (2) R breaking and are thus unphysical, the mirror images in Φ L are to be considered as physical scalars (above the EWSB scale). Above the L-R breaking scale, these fields are protected against mass terms, since all contributions to the mass are absorbed in the minimization conditions 8 . Therefore they can only acquire nonzero mass due to mixing effects proportional to l hΦ and l T Φ breaking LRsymmetry. With respect to the SM gauge symmetry, they have the same quantum numbers as h, and will be called h Φ in the following.
In the limit v Φ v T , the mixing effect which provides a mass for h Φ becomes negligible, while there may still be an large contribution to the h mass. For the h/h Φ system, we obtain a mass matrix with an approximate eigenvalue structure where we allow for m h as an independent µ term, not directly related to PS breaking. Both mass terms may be as low as the TeV scale where soft-breaking terms come into play. In other words, the MSSM Higgs bi-doublet (in particular, the Higgs boson that has been observed) may belong to either h or h Φ , or be a mixture of both.
If v T vanishes (class E), the situation becomes more complicated. Now, both masses get a contribution of the order l hΦ v Φ . For m h = 0, both mass eigenvalues are degenerate. Conversely, if the mixing between h and h Φ vanishes, they are maximally split (m h , 0). For small mixing (l hΦ m h v Φ ), we get an additional see-saw effect and a factor l 2 hΦ v 2 Φ /m h for the smaller eigenvalue, which generates the effective µ term.
In short, in various classes of PS models, the MSSM Higgs bi-doublet may be naturally light, and it could actually originate from the Φ L superfields.

The F multiplet
The mass of the multiplet F is generically see-saw suppressed and thus comparatively light. This appears as a common feature of all model classes with more than one scale. It can couple to matter via W Yukawa . Since F is a SU (4) C antisymmetric tensor, the possible Yukawa couplings 8 The mass of Φ R is fixed since it is proportional to the vev which we keep fixed provide both diquark and leptoquark couplings, explicitly breaking baryon number in the lowenergy theory. In fact, F is the analog of the colored Higgs field which in SU (5) GUT models induces rapid proton decay unless it is very heavy.
However, in PS models the Yukawa matrices Y F and Y are not related, so there is no doublet-triplet splitting problem [21,22]. Hence, by omitting the coupling of F to Ψ, proton decay is excluded in the presence of all gauge symmetries. This can be achieved, for instance, by a flavor symmetry or by an appropriate discrete quantum number.
If the F multiplet is sufficiently light, it may provide detectable new particles at colliders.
Without the Y F Yukawa coupling, there is no immediate decay to MSSM matter fields, but other terms in the Lagrangian provide indirect decay channels. In this situation, the particles (color-triplet scalars and fermion superpartners) may become rather narrow as resonances.

SM Singlets
The most complicated mass matrix belongs to the electroweak singlets that are contained in the various PS multiplets. Even if we do not consider PS singlets, there are still five electroweak singlets which can mix non-trivially. In the general case, their masses cannot be calculated in closed analytical form. To get a handle on these particles, we computed the dependence on the different scales numerically. Given one of the scale hierarchy patterns introduced above, we find additional scales and new hierarchy patterns which may have interesting consequences for flavor and Higgs physics. However, singlets do not contribute to the running of the gauge couplings at leading-logarithmic level, so we do not attempt a detailed discussion of the singlet sector in the present work.

Matter Couplings
The renormalizable superpotential contains terms that couple h to matter Ψ, so matter fields can get masses via the h vev, after electroweak symmetry breaking. However, there is no reason for flavor physics to originate solely from the renormalizable superpotential. In particular, if h Φ turns out to be the MSSM Higgs, there are no contributions from this superpotential at all. Instead, we expect significant and non-symmetric contributions to masses and mixing from non-renormalizable terms that induce effective Yukawa couplings at one of the symmetrybreaking scales. The generated terms are proportional to powers of the various vevs in the model, and suppressed by masses of heavier particles, by M GUT , or M Planck . There is ample space for flavor hierarchy in the resulting coefficients. For instance, if we consider dimension-four terms in the superpotential, we identify the following interactions that can affect matter-Higgs Yukawa couplings in the low-energy effective theory, There are additional couplings to gauge singlets, which may also be flavored, so overall there is great freedom in assigning masses and generating hierarchies in the mass and mixing parameters.

Neutrino Mass
As a left-right-symmetric extension of the SM, PS models contain right-handed neutrinos and allow for a Dirac neutrino mass term. Furthermore, the extra fields that are present in our setup can induce any of the three neutrino see-saw mechanisms for mass generation [39,40].
The fields Φ R and T R provide singlets with a vev that can couple to right-handed neutrinos, generating a Majorana mass term proportional to v Φ and v T , respectively. Combined with the Dirac mass, this results in a type I see-saw. The field T L contains SU (2) L -triplet scalars and fermions and may thus induce a type-II or type-III see-saw mechanism.

Unification conditions
Within the framework of PS model classes that we have defined in the previous sections, we now impose unification conditions on the gauge couplings. As stated above, we require complete unification for all gauge groups to a GUT symmetry that contains SO (10). The unification scale M GUT where this should happen is not fixed but depends on the spectrum. At each threshold below this where the spectrum changes, i.e., particles are integrated out, we state the appropriate matching conditions.
We work only to leading-logarithmic level, where non-abelian running gauge couplings are continuous in energy, and matching conditions depend only on the spectra 9 . Abelian gauge couplings do exhibit discontinuous behavior as an artifact of differing normalization conventions in different effective theories.
The leading-logarithmic running of a gauge couplings between fixed scales µ 1 to µ 2 is given Here, the b i are group theoretical factors that can be calculated from the representations of the particles [41].
Inserting the intermediate mass scales (case B for definiteness) as discussed above, the complete running is a sum of multiple terms, where i . The calculation of the coefficients in this formula is straightforward and can be found in Appendix A. For simplicity, we always assume that the listed scales exhaust the available hierarchies, and no further hierarchies from couplings become relevant here. Thus, we assume all additional scalar fields to be integrated out at their "natural" mass scale, which is determined by the considerations in the previous section.
In passing, we note that the calculation is actually independent from a supersymmetry assumption. The supersymmetric and non-supersymmetric frameworks differ only in the form of the β-function (cf. (30), (31)).
Regarding U (1) couplings with their normalization ambiguity, we have to explicitly consider the unification condition for To define the hypercharge coupling strength, we explicitly calculate the unbroken direction and identify the charges of the larger groups. This results in a relation between the group generators and therefore between charges and couplings. We obtain 10 For the non-abelian symmetry breaking steps, the unification conditions just depend on the breaking pattern, i.e., where the GUT group contains SO(10). These three breaking patterns result in the matching respectively.
In addition to the unification and matching conditions, we have the additional constraint that the mass scales are properly ordered. For instance, for class B we have: Furthermore, the coupling strengths α i have to be sufficiently small and positive at all mass scales, so we do not leave the perturbative regime.
Counting the number of conditions and free parameters (scales), we observe that the models are still under-constrained. Hence, we can derive constraints for the mass scales and exclude particular models, but not fix all scales completely. Nevertheless, imposing unification does restrict the model parameter space significantly, as we can show in the following sections.

Supersymmetric Pati-Salam Models
In this section, we study the supersymmetric models that are consistent with our set of assumptions. We scan over all models by varying the number of superfield generations (0, 1,3) that are present in each effective theory (i.e., between the various symmetry breaking scales), independently for each gauge multiplet.
We force the masses of all superfields that are not part of the low-energy (MSSM) spectrum, to coincide with one of the thresholds M IND , v T , v Φ , etc., as explained in Sec. 5. In any case, order-one prefactors in the mass terms would only enter logarithmically in the gauge-coupling unification (16) and matching conditions (GCU) (18b, 20,21). This is a minor uncertainty. We should note, however, that additional coupling hierarchies, as they exist in the flavor sector of the known matter particles, are also possible and lead to a wider range of possibilities which we do not investigate further.
With these conventions understood, the model scan will be exhaustive, since we vary just discrete labels. In total, there are 1078 distinct configurations. Since we now are dealing with mass scales rather then vev structures or specific mass eigenvalues, we change our notation from vevs to mass scales (see Appendix B).
For each model, we analytically calculate the numerical values of mass scales for which the GCU conditions stated in the previous section can be fulfilled. Models where no solution is possible are not considered further. For the remaining models, we obtain model-specific relations between the mass scales. As a result, we can express those scales as functions of one or two independent mass parameters that we may choose as input. Varying those within the possible range, we obtain allowed ranges for all mass scales, within each model separately.
For all numerical results, we fix the common soft SUSY-breaking scale at M SUSY = 2.5 TeV. 12

Extra mass parameters
As discussed in the previous section, in some model classes the fields E, Σ and T do not necessarily obtain a mass term from symmetry breaking. Thus, their masses must be treated as extra free parameters. To get a handle on these scales, we considered all possibilities for assigning the mass scales of these superfields to the other mass scales in our framework, while keeping the GCU conditions. As a result, we can exclude the possibility that these extra scales are at the lower end of the spectrum.

General overview
Before we discuss the various classes and types of models in more detail, we summarize generic features and specific observations that we can extract from the study of all 828 supersymmetric models.
Roughly one half of all models are capable of GCU. Except for class E, all such configurations exhibit a unification scale M GUT > 10 16 GeV and are thus favored by the non-observation of proton decay. In class E this is true for half of them.
In contrast to the classes C, E and F, the allowed ranges in class B are rather constrained, so in this class, models can be fixed in a semi-quantitative way.
We now take a look at the low energy spectrum of the different classes. These can be extra One key feature of our configurations is that we allow for up to six light SU (2) L scalar bidoublets h and h Φ (three each). Most configurations with more than one fall in class E. But also in class C there is a handful of configurations. In class E most successful configurations have more than one light bidoublet. This is because h and h Φ are taken as degenerate in mass.
A more detailed discussion will follow when we look explicitly at class E.
In class E the LR-breaking scale is allowed to be rather light. We find 120 distinct models with M LR < 10 5 GeV. Also in class C we find some configurations.
As a generic observation, SU (2)  We illustrate these results in Fig. 2. The figure displays a considerable fraction of models where new matter is possible at the lowest scales (green squares), so we should be prepared to observe exotic particles, or at least their trace in precision observables, at collider experiments.  only 10% of the models result in a value in this range.
We also find that a neutrino mass scale in this range is associated with at most one generation of h fields. As an interesting non-standard scenario, hinting at E 6 grand unification, three light h generations are also possible, but accompanied by heavier right-handed neutrinos. In classes B and C, we find a few successful models with three h generations, and none in class F. However, there is quite some space for this scenario in class E.
So far, we treated M IND as a free parameter. To obtain more specific predictions, we imposed While this does not significantly reduce the number of allowed models, it drastically reduces the configurations with TeV-scale new particles. Most of these models still allowing TeV-scale new particles belong to class E. In this class, one quarter of all models allows for colored triplets below about 10 TeV.
In some models, the GCU constraints pin down all scales to a narrow range. The most obvious case is standard SO(10) coupling unification at the GUT scale, i.e., all vevs are of the same order of magnitude and located at M GUT . Clearly, this well-studied model is contained in our scan as a limiting case. We reproduce the observation that for this case, the only light multiplet is the MSSM Higgs h. However, we also find a few models where scales are essentially fixed, but the spectrum and unification pattern is different. Those are all classified as class E. Let us now consider in somewhat more detail, the three particular model types described in Sec. 3.

Type Em: Minimal Model
In class E, the minimal model is the standard MSSM without Higgs 13 , supplemented only by the additional fields Φ and Σ above their respective thresholds. Looking at Tab. 2, we see that this setup does not provide a see-saw scale. Hence, the sub-unification scales depend only on one free parameter, which we take to be M GUT . Fig. 3(a)   Another important difference is that the scales approach each other when M GUT gets larger.
In Fig.4      The intermediate scales tend to be higher than in class E above.
Out of the 216 class-F models only 29 configurations are consistent with GCU. In all cases, M GUT can be as large as the Planck scale.
As discussed in the overview, light degrees of freedom are not possible in this class. As above, we break down the set of configurations according to their content of light fields. There are now 6 categories. In addition to the ones mentioned above (zero, one, or three generations of F ), we have to distinguish cases of one or three generations of SU (2) triplets T L .
The minimal M IND value is strongly dependent on the number of SU (2) triplets. In the case of three triplets, it is strictly larger than 10 16 GeV, essentially independent of the number of F fields. Thus, let us look at the configurations with only one T L/R generation.
In these configurations, M IND is bound to be larger than M IND 10 6 GeV. It is realized for three generations of F and rises the less are included.
We conclude that in class F, the extra fields may play a role for flavor physics in an intermediate energy range, but are unlikely to be observable in collider experiments.

Type Fm: Minimal Model
The minimal model of class F contains the superfields Φ and T L/R in addition to the MSSM spectrum.
In models of this type, the lowest possible see-saw mass value is M IND ≈ 10 12 and LR-scale is M LR 10 15 . Thus these are ruled out, since the mass of the EWSB Higgs is associated to the LR-scale. A next to minimal setup explicitly including one generation of h is not able to produce GCU.

Type Fs/Fe
In type Fs, there is no model consistent with GCU. This is because α −1 3 grows to fast and overshoots α −1 2 before the condition (18b) for a possible QL-scale can be fulfilled. A model of type Fe consistent with GCU is also not possible.

Type Ff: Flavor-symmetry inspired Model
In the absence of the previous types, we take a look at a configuration which might be viewed as E 6 -inspired, but with the additional condition that two of three Higgs bidoublets get heavy In these classes, we effectively combine model classes E and F. There are five different scales, two of which are fixed by requiring GCU. For concreteness, we also fix M GUT = 10 18.2 GeV, i.e, we assume complete unification at the Planck scale. Still, we can choose two parameters independently, so we will obtain allowed and forbidden regions, but no one-to-one correspondences.
More specifically, we distinguish the cases v T ≤ v Σ (class B) and v T > v Σ (class C), where A and D appear as limits. The ordering of scales in the two scenarios is Here If we include one generation of h in addition, GCU is no longer possible.

Types Bs/Cs and Be/Ce
These setups do not allow GCU.

Class-B/C Models with M IND < 10 TeV
We may ask for models where the see-saw scale is sufficiently low (say, M IND < 10 TeV) that the new particles can have an impact on collider phenomenology. We find 8 (34) models where this is possible within class B (C), respectively. One configuration with normal hierarchy is model B199 14 , where we have three copies of h, F and Σ and no E. In the inverted case there is a similar model C211, which has the same spectrum, but three copies of Φ. The plots are shown in Fig. 6, 7.      Since we found that in the non-SUSY case the resulting mass thresholds tend to be lower, we fixed the GUT scale for the classes B and C to M GUT ≡ 10 16 GeV. For the value of M GUT = 10 18.2 GeV which we used in the SUSY case, we would have only one configuration (class C) for both classes that satisfy GCU. Lowering this scale even more, the model space would be less constrained, but this is disfavored by the proton decay limits.
We considered the same set of 828 models, with the SUSY partners removed, as in the previous section. Again, more than half of the models allow for GCU, given the modification mentioned in the previous paragraph. Still it turns out that without SUSY, classes A-D are disfavored but not excluded. In contrast to the supersymmetric case, a considerable set of the successful models fall in class F. Again, the most belong to class E. In any case, lower GUT scales tend to be favored.
As in the SUSY models discussed above, we observe a LR symmetry breaking scale roughly around 10 13 GeV in a large fraction of the successful models. Nevertheless, there are models where this scale can be much lower, down to below 100 TeV in some cases.
In class E, there are again many models with the possibility for light color triplets F in the TeV regime. In the non-SUSY setup, these are scalar particles and obviously cannot mix with quarks. We have to assume that there are couplings of either leptoquark or diquark type  here. Again we see, that the multiplicity of these bidoublets lowers the maximal unification scale.
As in the SUSY case, we find that three generations of light SU (2) triplets T are excluded.
In particular, in class C the lower bound for those is M T L 10 8 GeV. In class C, the bound becomes 10 11 GeV and in class B there is no GCU at all.
In the non-supersymetric case we again find in all classes a set of configurations fixing all mass scales "exactly". This usually corresponds to degenerate mass scales. A common scale for such classes is M GUT ≈ 10 14 GeV.
Again we find that fixing the induced scale M IND does not change the results very much.
Thus we see our assumptions justified to include this scale in our scans.
A general feature of non-SUSY spectra is the fact that the high-energy effective values of the gauge couplings are larger than in the SUSY case. This is due to the lower number of fields that contribute to the gauge-coupling running.
In the following, we do not repeat the detailed discussion of SUSY models but pick a few selected models and model types with particular features. More generic statistics can be read off Tab. 4.

Class F
In class F, there are plenty of configurations leading to successful GCU. In general, we find that there is not much scope for scale variation. The GUT scale can be as large as M GUT ≈ 10 17 GeV. The scales are close to each other, since the lightest scale is fixed to be larger than M IND 10 13 GeV. In this class there are also models with GCU where all scales are essentially fixed, and not far from the GUT scale. Those lead to M GUT ≈ 2 × 10 14 GeV, which is rather low. The LR-scale emerges between 10 13 GeV ≤ M LR ≤ 2 × 10 15 GeV.
One exemplary configuration leading to GCU above 10 16 GeV is model F213. Here, we have three generations of F and one of T . These scalar particles can be rather light, potentially as low as the SUSY scale. In addition, this model contains three generations of the fields Φ and Σ, and one generation of E. We show the possible scale variation and a sample unification plot for this model in Fig. 9.

Class A to D
In class C, the QL-scale emerges typically close the PS scale. Likewise, the PS scale can become as large as the GUT-scale, such that the energy range with pure PS symmetry may vanish.
Looking at the possibility of three Higgs (h) generations, we do not find any configurations in class B, and a few in class C. On the other hand, these model classes favor three as the number of Φ generations.
In Fig. 10 and 11, we display the scale relations and gauge-coupling unification for two distinct models B53 and C45 15 . The former model contains three generations of F which can be light. In addition it contains three generations of E and all other fields ones. We found quite some range for scale variation. For the largest value of the PS scale (M PS ≈ 10 15 ) we find a degeneracy of all lower scales. The induced scale can be as light as some TeV.   Model C45 also features new particles at the TeV scale. Here we include three generations of Φ and Σ and all other fields ones. We again see quite some space to vary the scales.

Conclusions
We have presented a survey of models with gauge-coupling unification along a path that contains several intermediate scales, corresponding to left-right symmetry, quark-lepton unification, Pati-Salam symmetry, and SO(10) or larger GUT symmetry. We studied both supersymmetric 15 For the meaning of numerical model indices, see Appendix C.   and non-supersymmetric models, where the latter are derived from the former by omitting all superpartners.
We have restricted the allowed new chiral superfields (or scalar fields, in the non-SUSY case) below the GUT scale to a small well-defined set of multiplets, all of which fit in small representations of SO(10) or E 6 . A large subset of the resulting models is consistent with gauge-coupling unification, proceeding in several steps. In supersymmetric models, there is slightly more freedom in varying the scales than in non-supersymmetric models.
The assumptions and calculations do not constrain the model space in such a way that we can get precise numerical predictions, but we can deduce characteristic patterns in the scale hierarchies that correlate with specific choices for the spectrum.
In a wide range of models, GUT unification can be pushed up to the Planck scale. This fact, together with the properties of Pati-Salam symmetry, significantly reduces the strain that the non-observation of proton decay can put on GUT model building.
Additional thresholds in the intermediate range between observable energies and the GUT scale are likely. Being associated with left-right or quark-lepton symmetry breaking, they decouple flavor physics issues from the requirements of complete unification. We have not considered flavor physics in any more detail, but expect that it can generically be accommodated if non-renormalizable terms are properly included. Depending on the particular model and on the chosen set of flavor-dependent interactions, we expect specific hierarchies, relations and predictions for the flavor sector.
As an extra feature, the (super)fields Σ and T (adjoint of SU (4) C and SU (2) R ) and Φ (fundamental of SU (2) L × SU (2) R ) can cooperate to generate small mass terms for certain particles, including SM-like Higgs doublets and new exotic (s)quarks, such that they can be accessible at colliders. On the other hand, SU (2) L/R triplets as possible Pati-Salam breaking Higgs fields tend to raise the LR symmetry scale above the scale of quark-lepton unification and thus may enforce a direct relation between quark and lepton flavor physics.
Another generic property of the models under consideration is the scale of left-right symmetry breaking, naturally associated with neutrino mass generation, in an intermediate mass range. A neutrino mass scale significantly below the GUT scale is favored by numerical estimates of see-saw mechanisms that can explain the small observable neutrino masses.
We also encounter models where gauge-coupling unification at high energies implies a multi-Higgs doublet model at low scale, possibly with flavor quantum numbers. Furthermore, the observable Higgs doublet need not be a member of a (1, 2, 2) representation, as often assumed, but can also originate from a (4, 2, 1) + c.c. representation, i.e., behave as a scalar lepton. The effective µ term which sets the scale for low-energy Higgs physics, can have a see-saw like form and thus be naturally suppressed with respect to the higher symmetry-breaking scales.
It is remarkable that the most interesting phenomenology, which we may identify as Planckscale GUT unification, intermediate PS and LR scales, and new particles at TeV energies, can be simultaneously realized in a number of distinct models. This is not a generic feature. On the other hand, a large fraction of models is not consistent with unification at all.
In summary, Pati-Salam models can easily accommodate unification in a multi-scale framework. They provide a rich phenomenology and a promising background for new approaches to the lepton and quark flavor problem.

A Beta-Function Coefficients
As stated in the paper (cf. Sec. 6) the running of the gauge couplings can be described by The coefficient b i of the running coupling can be calculated by means of the representation of the fields contributing at the given mass scale alone [41]. For each set of gauge groups SU (N ) with N ≥ 2 the contribution of an field with representation (I 1 , . . . , I n ) to the running coefficientb i is given asb were d(I i ) is the dimension and the normalization of the representation T (I i ). These can be calculated using the representing matrices R a tr R a R b = T (R) δ ab , with T (N ) = 1 2 . For a U (1) the contribution is up to an consistent rescaling: The complete running coefficient depends now on whether we work in a supersymmetric theory or not. For the non-supersymetric case one has to divide the fields in scalar and fermionic contributions: where C 2 (G i ) = dim(G i ) is the quadratic Casimir operator.
Since in the supersymmetric case there is a superpartner for each scalar/fermionic field, there is no need to divide the fields in such a way. Thus, the relation simplifies to Tab. 5 displays the contribution of each field and its complete decomposition w.r.t. the subgroups of PS symmetry.