Contents

UAB-FT-689

Composite GUTs: models and expectations at the LHC

Michele Frigerio***E-mail: frigerio@ifae.es, Javi SerraE-mail: jserra@ifae.es, Alvise VaragnoloE-mail: alvise@ifae.es

Institut de Física d’Altes Energies, Universitat Autònoma de Barcelona, E-08193 Bellaterra, SPAIN

CNRS, Laboratoire Charles Coulomb, UMR 5221, F-34095 Montpellier, FRANCE

Université Montpellier 2, Laboratoire Charles Coulomb, UMR 5221, F-34095 Montpellier, FRANCE

We investigate grand unified theories (GUTs) in scenarios where electroweak (EW) symmetry breaking is triggered by a light composite Higgs, arising as a Nambu-Goldstone boson from a strongly interacting sector. The evolution of the standard model (SM) gauge couplings can be predicted at leading order, if the global symmetry of the composite sector is a simple group that contains the SM gauge group. It was noticed that, if the right-handed top quark is also composite, precision gauge unification can be achieved. We build minimal consistent models for a composite sector with these properties, thus demonstrating how composite GUTs may represent an alternative to supersymmetric GUTs. Taking into account the new contributions to the EW precision parameters, we compute the Higgs effective potential and prove that it realizes consistently EW symmetry breaking with little fine-tuning. The group structure and the requirement of proton stability determine the nature of the light composite states accompanying the Higgs and the top quark: a coloured triplet scalar and several vector-like fermions with exotic quantum numbers. We analyse the signatures of these composite partners at hadron colliders: distinctive final states contain multiple top and bottom quarks, either alone or accompanied by a heavy stable charged particle, or by missing transverse energy.

1 Introduction

Two paradigms have been proposed to account for the stability of the electroweak (EW) scale against quantum corrections, the so-called gauge hierarchy problem. One is a weakly coupled theory where the mass of the elementary scalar responsible for electroweak symmetry breaking (EWSB), that is the Higgs boson, is protected from ultraviolet scales by supersymmetry. The other involves a new strongly coupled sector, which generates the EW scale dynamically, in analogy with the origin of the QCD scale. The latter idea has been realized in several ways. A particularly attractive possibility is to achieve EWSB thanks to a Higgs-like composite field, emerging as a Nambu-Goldstone boson (NGB) from the strongly interacting sector: this is known as the composite-Higgs scenario [1, 2, 3]. It is generically favoured by precision measurements with respect to the simplest form of technicolour [4] or Higgsless models [5], and it has a more economical particle content with respect to little-Higgs models [6].

Whatever solution of the hierarchy problem is adopted, there are several robust indications for the unification of the gauge interactions at a scale , close but definitely smaller than the Planck scale , where gravitational interactions become strong. As a matter of fact, grand unified theories (GUTs) [7] elegantly account for the quantization of the electric charge, the quantum numbers of quarks and leptons and the basic relations between their Yukawa couplings, the cancellation of the gauge anomalies, the evidence for non-zero neutrino masses. While GUTs have been intensively studied mostly in the supersymmetric framework [8], the above-listed virtues of GUTs do not rely on the existence of low-energy supersymmetry. It is therefore sensible to try to realize precise gauge coupling unification in other extensions of the standard model (SM) that provide a natural explanation of the EW-GUT hierarchy.

In comparison with supersymmetric scenarios, strongly coupled models suffer from the obstruction to perturbative computations. Moreover, at first sight they do not share the striking prediction of precise gauge coupling unification, which pertains to the minimal supersymmetric SM. This gap has been significantly reduced over the years, thanks to the modern understanding of strongly coupled systems via the AdS/CFT correspondence [9], and in particular through the study of extra-dimensional scenarios of the Randall-Sundrum type [10], which permitted to extract both qualitative and quantitative information on EWSB, electroweak precision tests (EWPTs) and phenomenology. Following such developments, the investigation of gauge coupling unification has been pursued [11, 12, 13]. Actually, it turned out that some key features of strongly coupled models can be studied with no need to specify a dual extra-dimensional construction and independently from the details of the strong dynamics. In particular, a proposal for precise gauge coupling unification in the composite-Higgs scenario was advanced [14].

In this paper we further investigate composite-Higgs scenarios which exhibit gauge coupling unification, from a purely four-dimensional perspective. We will make use of two basic properties of the strongly interacting sector. The first is an approximate conformal symmetry, spontaneously broken at low energies, around few TeV, thus generating a mass gap in the spectrum of composite resonances. For energies above and up to the unification scale , the composite sector alone is well described by a strongly interacting conformal field theory (CFT). The conformal symmetry fixes the behaviour of the correlators of the composite sector, in particular those that affect the propagators of weakly coupled external fields. In the case of interest, these are the SM gauge fields, with the corresponding gauge couplings. It can be shown that the contribution of such composite sector to their running is logarithmic [15].

The second crucial ingredient is the invariance of the composite sector under a set of global symmetry transformations, analogous to the approximate chiral symmetry of QCD. If the SM gauged group, , is embedded in a simple group , then gauge coupling unification becomes independent from the strong dynamics at leading order. This is because the one loop contribution of the composite sector to the gauge coupling beta functions is universal, that is to say for . Moreover, the universal coefficient is constant between and because of the conformal symmetry.111In principle, the composite sector could exit the strong coupling regime before , thus modifying . This requires a breaking of the conformal symmetry at the intermediate scale where the transition between the two regimes occurs. The universality of the beta function coefficients is maintained also in this case, as long as is preserved. However, this option would introduce an unnecessary model-dependence. Therefore we will not consider it in this paper. At this point one is in the position to study quantitatively gauge unification, and subsequently undertake the construction of composite GUTs.

It is remarkable that both the conformal symmetry and the -symmetry of the composite sector are also instrumental to generate hierarchical Yukawa couplings, by an elegant mechanism known as “partial compositeness” [16, 17]. The light SM fermions are taken to be elementary particles external to the composite sector, weakly coupled to it by mixing with composite fermionic operators. The Yukawa couplings arise from this mixing at low energies, and the hierarchies between them are due to the different scaling dimensions of the various operators (at least one for each SM chiral fermion). The required large anomalous dimensions can be generated if the composite sector is strongly coupled over a large range of energies, which naturally happens when it is close to an infrared-attractive fixed point, indicating that it is approximately conformal. Since the composite sector is responsible for generating the EW scale, it must carry EW charges. Now, in order to generate quark masses, it must also contain operators charged under colour. Then, the composite operators transform non-trivially under the full , which of course must be a subgroup of the global symmetry of the composite sector. Therefore the only extra assumption to move from the ordinary composite-Higgs scenario to the composite GUT scenario is the requirement of being simple.

Besides the breaking of the conformal symmetry at , the strong dynamics does also lead to the spontaneous breaking of part of the global symmetry, . This is necessary in order to obtain the Higgs as a Nambu-Goldstone boson (NGB), in analogy to the light pseudo-scalar mesons of QCD, that are the approximate NGBs of . The unbroken subgroup does not need to be simple, since this breaking is an infrared effect, which does not modify the gauge coupling evolution at leading order over the large hierarchy. Part of the global symmetries of the composite sector will be eventually broken explicitly by the gauge and fermion couplings to the elementary fields. Since these couplings are perturbatively small, the global symmetry of the composite sector holds in good approximation over the whole hierarchy between and . Nonetheless, below the explicit breaking generates a non-trivial effective potential for the NGBs of the composite sector, leading to EWSB.

In composite GUTs, the Higgs will be generically accompanied by other pseudo Nambu-Goldstone bosons (pNGBs) that fill with it a complete multiplet of the global symmetry . These necessarily light extra scalar fields are a distinctive feature of composite GUTs, to be contrasted with the usual weakly coupled GUTs, where the Higgs partners live at the GUT scale. The set of light scalars typically includes a coloured triplet that can potentially mediate proton decay. In supersymmetric GUTs the proton decay issue is usually cured imposing -parity and making the triplet super-heavy (), which requires to implement a doublet-triplet splitting mechanism. In composite GUTs, instead, the colour triplet (more in general, any operator generated by the strong dynamics) that might mediate proton decay cannot be decoupled, since its mass scale is around or below few TeV. The remedy will be to forbid the triplet couplings to SM fermions, and more in general to suppress baryon number violating operators, by imposing an appropriate symmetry. Similarly, composite operators that mediate lepton number violation need to be suppressed, not to generate too large neutrino masses.

The last important feature of composite GUTs is the presence of extra vector-like fermions at the EW scale, that eventually are responsible for the correction to the SM gauge coupling evolution, such that unification is achieved at , with a precision comparable to that of the MSSM. It is quite remarkable [14] that these extra fermions are automatically predicted, once one implements in a straightforward way the attractive features of the composite-Higgs scenario described above. Partial compositeness implies that the larger the mass of a SM fermion, the stronger its coupling is to the composite sector, and in turn the modification of its elementary properties. The degree of compositeness of the light SM fermions is thus small, but that of the top quark has to be large. In fact, the well-motivated possibility exists that the right-handed top quark is an entirely composite chiral fermion.222 It will be clear in the following why the left-handed top should not be composite. In this case, it must be accompanied by a set of composite partners, filling a complete multiplet of the global symmetry . In order to make these chiral top partners massive and to cancel gauge anomalies, one is forced to introduce extra elementary fermions. We will analyze in detail the impact of these new exotic particles on precise unification, EWPTs, and in EWSB, as well as their manifestation at the large hadron collider (LHC). Their observation would constitute another crucial signature of composite GUTs.

In this paper we will not attempt to build an explicit ultraviolet completion for the composite GUT scenario, that is to say, we will not construct a specific model at the scale . This definitely remains a very important task, in a territory that is presently largely unexplored. At least one comment is in order to settle the ground and avoid confusions. The full GUT must possess a gauge symmetry , which is a simple group containing . One may conceive that is broken only in the elementary sector, which is promptly realized assuming that the GUT breaking fields do not couple to the composite sector.333 To be concrete, think of chiral fermions in a complex representation of , such as a 5 or a 10 of . One example is provided by the SM fermions, which form chiral multiplets that only feel GUT breaking effects through Yukawa and gauge couplings. The required composite sector can be generated if there exist another set of such chiral fermions (i) charged under an additional gauge interaction in the non-perturbative regime and (ii) sufficiently weakly coupled to the breaking sector. Then, the composite sector would retain a global symmetry . However, the identification of the two groups may appear minimal but it is not necessary nor natural: the symmetry of the GUT theory may be larger, and boil down to a low energy global symmetry or, if the initial global symmetry of the strong sector was larger than the gauged , one could have . The only requirement is that belongs to the intersection of and . We will see that the suppression of baryon and lepton number violation implies further constraints on the interplay between these two groups.

The paper is organized as follows. In the next subsection we recall the general setup for the study of composite-Higgs scenarios, for the non-practitioners. In section 2 we describe in detail the evolution of gauge couplings when the SM Higgs boson and the right-handed top quark are composite states, and what the conditions for precision unification are. In section 3 we discuss the global symmetries of the composite sector that are required in order to respect the EWPTs, to implement gauge coupling unification and to avoid proton decay. The model which emerges as the simplest viable possibility has a global symmetry , with unbroken subgroup . The exotic fermion quantum numbers are then specified, and their contribution to EW precision parameters is computed. In section 4 we compute the effective potential for the pNGBs in this model, and derive the constraints for a satisfactory EWSB. In section 5 we describe the collider phenomenology of the Higgs and top quark composite partners in three different variants of the model. We finally summarize the substantial features of our composite GUT models in section 6.

1.1 The setup of composite-Higgs models

The lagrangian for composite-Higgs models can be expressed, in the same spirit of Refs. [17, 18], as

(1)

There is a sector of elementary weakly coupled fields, whose dynamics is described by , invariant under the SM gauge symmetries, . The field content of this sector is the one of the SM, without the Higgs. In addition, there exists a new strongly interacting sector, described by , made of composite bound states. Such sector is characterized by a scale , associated to the mass of the lightest massive resonances (massless composites are also present), and by an inter-composite coupling . The latter is larger than the elementary weak couplings (generically denoted by ), although it can be significantly smaller than the naive dimensional analysis (NDA) estimate in fully strongly interacting theories, that is . The composite sector is invariant under a global symmetry , which contains as a subgroup. At a scale close to , is spontaneously broken to , giving rise to a set of NGBs parametrizing the coset space ; this set includes the Higgs doublet . The NGBs remain massless in the limit , and their dynamics is described by a non-linear -model with characteristic scale . This scale controls the interaction among the NGBs and it is related to the composite sector parameters as

(2)

in analogy with QCD, where Eq. (2) relates the pion decay constant to the mass of the QCD resonances. Also, is approximately conformal invariant at energies above , and it remains strongly coupled over the whole hierarchy between and .

The interaction between the elementary and composite sectors is described by , that in general respects only the SM gauge symmetries. We assume that has the form dictated by partial compositeness, which applies both to the coupling of the elementary gauge bosons as well as of the elementary fermions. Generically, the former can be expressed as

(3)

where is the elementary gauge boson, coupled with strength to the corresponding composite sector current . Analogously, there is a coupling for each elementary chiral fermion. In order to describe the Yukawa couplings, it is convenient to write these couplings as

(4)

where the chiral fermions are coupled to the composite operators with strength . The operators transform under in such a way as to generate a coupling between , and at low energies, below . Therefore, the low energy values of are constrained to reproduce the observed Yukawa couplings [16, 17]:444 Here we assume that each chiral elementary fermion couples dominantly to a unique composite operator, which is responsible for inducing the corresponding Yukawa coupling. Nonetheless, extra couplings to other operators could be present, if allowed by the gauge (and global) symmetries of the full lagrangian.

(5)

Of course both Eq. (3) and Eq. (4) must respect the SM gauge symmetries. In addition, extra global symmetries might be approximately preserved (in particular, consistency will require to impose baryon and lepton number conservation, as discussed later). However, Eq. (3) and Eq. (4) do not respect the symmetry, therefore introducing a (weak) explicit breaking of the global symmetries of the composite sector, in particular of the NGB symmetries. As a consequence, an effective potential for the NGBs will be generated by loops of elementary fields. For instance, the mass of the Higgs field will receive corrections that scale like , with the scale acting as the cut-off for the elementary loops. This is in analogy to QCD, where the charged pion mass receives divergent loop corrections from the photon that are cut at the meson mass scale.

The effective potential, which is an expansion in the small explicit breaking couplings, , and in the number of elementary loops, will induce a VEV for the Higgs, , breaking the EW symmetry. This introduces the last parameter of our framework, , which describes the departure from an elementary Higgs scenario, obtained in the limit , or from a so-called Higgsless scenario, in the limit (in this case the longitudinal gauge boson scattering amplitudes are unitarized as in technicolour). The deviations from the SM predictions introduced by the composite sector will then be proportional to , and thus the EWPTs set an upper bound on this parameter, as reviewed in section 3.1. Besides, constitutes a rough measure of the degree of fine-tuning in the model.555Composite models naturally tend to predict (or ), which is ruled out phenomenologically (see section 3.1). Achieving a separation of scales requires a tuning of the parameters of the model, specifically of those responsible for the generation of the Higgs effective potential: this is the incarnation, in composite-Higgs scenarios, of what is customarily dubbed as the little hierarchy problem. We shall aim, for concreteness, at , representing a 10% fine-tuning, which would be highly competitive with respect to alternative scenarios (e.g. supersymmetric constructions). Note that and imply , which we take as a reference value.

2 Gauge coupling evolution

In the spirit of GUTs, we assume that the SM gauge interactions have a common strength at high energies, which we observe at low energies as three different gauge couplings due to the GUT spontaneous symmetry breaking, occurring at the scale , and the consequent differential running to lower energies. Actually, an indication in favour of this assumption is provided by the SM particle content, since the evolution of the SM gauge couplings from the EW scale to high energies yields a rough convergence of their values, at the 20% level. The contribution of the SM fields to the renormalization of the gauge couplings can be parametrized at one-loop by the -function coefficients , where refer to the , , groups, respectively. While the contribution of the fermions is universal (i.e. independent from ), since they fill complete multiplets, the ones of the gauge bosons and of the Higgs are not.

The usual test for gauge coupling unification at one-loop consists in the comparison of the ratio of -function coefficients, , with the value determined by the measurements of the gauge couplings at the scale , . The SM prediction is . Although thresholds might arise at scales close to or at intermediate scales, there are no observational nor theoretical reasons why they should be large enough to achieve precision unification. On the other hand, the new physics associated with EWSB, which is required in particular to address the hierarchy problem, may significantly contribute to the -function coefficients and improve unification with respect to the SM. We explain in the following how the needed states can arise in the context of composite-Higgs models [14].

2.1 Composite sector contribution to the -functions

In general, a new strongly coupled sector may modify completely the gauge coupling evolution with respect to the SM. If such a sector is responsible for EWSB, it necessarily affects at leading order the evolution of the and couplings above the compositeness scale . While in the SM the contribution of the Higgs doublet to the evolution is relatively small, a larger number of degrees of freedom seems required to break the EW symmetry dynamically, so that no study of gauge coupling unification is feasible if the contribution of the composite sector cannot be computed.

Besides the obstruction to perturbative computations around the scale , that introduces a possibly large threshold, we do not know the structure of the EWSB sector between and the unification scale , making the analysis of unification highly model-dependent. Still, some deal of information may be extracted on the basis of symmetry considerations and of the consistency with electroweak data, as we now describe.

As a first general consideration, recall that the composite sector is supposed to stabilize the hierarchy between the electroweak scale and the unification (Planck) scale. This is achieved thanks to its approximate conformal symmetry, broken only at low energies . Then, if the composite sector is nearly conformal and weakly coupled to a set of external gauge fields, its main contribution to the running of the external gauge couplings will be logarithmic [15].666 The fact that the sector is conformal does not mean that it cannot contribute to the scale dependence of external fields coupled to it. What does not run is the intra-composite coupling. Technically, the logarithmic running follows from the fact that the correction to the gauge boson propagators is given by insertions, where is the CFT current coupled to the gauge bosons, and conformal invariance implies that [15, 19]. This is precisely what happens in theories with no intermediate scales, like the SM between the electroweak and the Planck scale, or above , where QCD is described by quarks and gluons.

Therefore, the contribution of the composite sector to the running of the gauge couplings , as a function of the renormalization scale , can be written as

(6)

and can be visualized diagrammatically as in Fig. 1. In general, the relative values of the coefficients cannot be computed perturbatively, nor the absolute size can be estimated in a model-independent way. Still, it was shown by Polyakov that [15], and recent studies aim to put lower bounds on these coefficients, as a function of the dimension of the scalar operators of a generic CFT [20]. These bounds could be of particular relevance for unification. Here we will assume that is small enough for the SM gauge couplings not to hit a Landau pole before .777 A warped extra-dimensional scenario yields , where is the AdS curvature radius, are the five-dimensional gauge couplings, and is the number of colours of the dual conformal theory [19]. However, the calculability in the warped extra-dimension requires a small ratio between the number of flavours and the number of colours, , since this is the expansion parameter of the theory. Unfortunately, in the scenario discussed in this paper, the number of flavours has to be large, due to the large global symmetry group , while the absence of a Landau pole requires , posing an upper bound on . Therefore we will not rely on warped extra-dimension estimates nor on large- arguments in this work.

Figure 1: Diagrammatic representation of the leading-order contribution from the strong sector to the SM gauge coupling running, parametrized in Eq. (6).

The differential running, that is, the dependence on the scale of the quantities , is affected at leading order by incomplete representations, e.g., in the case of the SM, the gauge bosons and the Higgs doublet. One knows, therefore, the amount of “ breaking” that should be introduced with respect to the SM in order to achieve precision unification. Then, the question is whether there are symmetries of the EWSB sector that allow to compute its contribution to the differential running, independently from the strong dynamics.

A straightforward (perhaps, the only) possibility [14] is to assume that the EWSB sector has a global symmetry , which is a simple group containing (therefore can be or a larger simple group). In this case the EWSB sector does not contribute to at the one-loop level, because for . Besides, since the Higgs doublet arises as a light composite state from the -symmetric sector, it does not contribute to the running. At most, it gives a small contribution below , the scale where is broken spontaneously to , that may be non-simple. Similarly, all low energy composite states may contribute to the differential running only below , as a sub-leading threshold effect.

In particular, if some of the SM fields are composite, they do not contribute to the differential running above , therefore it is convenient to denote with the -function coefficients of the elementary SM fields only. Specifically, when is part of the composite sector, the SM prediction is modified by the subtraction of , giving . The extra required correction to achieve precise unification will be provided by the interactions between the elementary and the composite fermions, as we now discuss.

The interactions of the elementary fields with the composite sector break explicitly and thus their effect on the differential running must be quantified. These are the SM gauge interactions of composite operators, Eq. (3), as well as the fermion mixing terms, Eq. (4). The contribution of these interactions to the running can be parametrized as [14]

(7)

where is summed over SM gauge bosons, and over fermions. These are formally two-loop contributions, as shown in Fig. 2, but with unknown coefficients. Since they are not universal, and not calculable a priori, they constitute an intrinsic theoretical uncertainty on unification in this scenario.888 Note that these two-loop contributions can be interpreted as threshold corrections associated with the ultraviolet brane in the warped extra-dimension picture. They can be explicitly computed by integrating over the bulk, and they are enhanced by the logarithm of the ultraviolet-infrared hierarchy. These non-leading corrections can be as large as the leading ones if the mixing with the composite sector is large, as it is the case for the top quark.

2.2 Top compositeness and precision unification

Figure 2: Example of sub-leading diagrams contributing to the differential running of the SM gauge couplings, on the left with a loop of elementary gauge bosons, and on the right with a loop of elementary fermions.

Since the values of the SM gauge couplings are fixed by experiment, the only couplings between the elementary and composite sectors that could modify significantly the running are the ’s which, in the framework of partial-compositeness, are related to the Yukawa couplings as explained in section 1.1. Explicitly, below the couplings in Eq. (4) generate, e.g. for a right-handed fermion , the lagrangian

(8)

where is a vector-like composite fermion (with the gauge quantum numbers of ) that arises as an excitation of the operator . By diagonalizing the associated mass matrix, the massless SM fermion can be written as , with . The fact that then leads to Eq. (5).

The composite component becomes large when , which requires a strongly coupled elementary field, . Then, the last term in Eq. (7) may become as large as a one-loop contribution:

(9)

where in the last step we used the rough strong coupling estimates and .999 The large- estimates would be and , where is the number of colours of the strongly coupled theory (thought of as a QCD-like theory).

Motivated by the large mass of the top quark, a natural possibility is to take the coupling of the right-handed top to be large, thus making the SM top quark mostly composite.101010 In general, the cannot be mostly composite, since gauge invariance would imply that is also mostly composite, which is strongly disfavoured by measurements of the coupling. However, this could be cured, as we will briefly review in section 3.1, if the theory respects an extra parity symmetry [21]. The alternative possibilities of mostly composite or [22] may also be interesting. In this case the distinction between composite and elementary fields becomes ambiguous: the large -violating coupling introduces a large uncertainty in the prediction for unification. The composite sector dynamics is significantly modified by , that cannot be treated as a small perturbation any longer. To overcome this ambiguity, one is led to consider the possibility of full compositeness of as proposed in [14], that is a scenario with no elementary state with the quantum numbers of in the low energy theory. The role of the right-handed top is then played by a composite state, denoted for simplicity , belonging to a chiral -multiplet , which is assumed to be massless before EWSB (no partner exists). Then, due to the unified -symmetry of the composite sector above , the contribution of must be subtracted from the -function coefficients .

If the low energy content of the elementary sector were just given by the SM without and , the ratio of -function coefficients would be , which is closer to but still far from the experimental value. However, a closer look at the composite-top scenario reveals that other chiral elementary fermions are needed to make the theory consistent. We give below two independent arguments leading to this conclusion. In particular, under well-motivated assumptions, we will show that a state with SM charges should also be subtracted from the differential running, so that the correct expectation for unification turns out to be , which is remarkably close to .111111 The MSSM predicts , but this sharp agreement with experiment at one-loop level is deteriorated when higher order corrections are included. In this scenario one may argue that precision unification is realized.

The first argument goes as follows. The elementary fermions of the complete GUT theory must belong to full -multiplets. The symmetry breaking at the GUT scale may well split the -multiplets containing the elementary SM fermions, giving a mass to the elementary right-handed top, , but not to the other SM species. However, the components which acquire a mass must form vector-like pairs (or be Majorana fermions, e.g. sterile neutrinos). Therefore, the remaining massless states should form full -multiplets up to vector-like pairs of states. The decoupling of each vector-like pair amounts to the subtraction of its contribution from the gauge coupling evolution. In particular, may decouple only if it pairs with an exotic fermion , and thus one achieves precision unification as described in the previous paragraph. Note that unification is enforced only by (i) the Higgs and right-handed top compositeness, and (ii) the -symmetry of the composite sector. In addition, exotic chiral fermions are predicted, in order to complete the -multiplet of .

We remark that this first argument holds only when the GUT symmetry is fully realized in four dimensions. On the contrary, in extra-dimensional scenarios where the GUT symmetry is realized in the bulk and it is broken explicitly on our brane by boundary conditions, the 4-dim chiral fermion zero-modes do not need to fill -multiplets (see e.g. Ref. [35]). In fact, in this case there is no unified gauge symmetry on our 4-dim brane. In these scenarios it may still be sensible to study precision unification of the 4-dim gauge couplings at , since GUT scale thresholds can be kept small; then, one cannot appeal to the above argument to subtract from the running.

The second argument for the existence of exotic fermions [14] applies when the -multiplet containing the right-handed top also contains other states, . Then, the extra chiral fermions necessarily require conjugate partners, in order to acquire a mass large enough to satisfy the experimental bounds. Also, such partners are needed to cancel the gauge anomalies, that were absent with the SM fermion content, but would be generated by the chiral fermions alone. Therefore, one must introduce exotic elementary fermions , with the same charges of (see also Ref. [23]). It is equivalent, and perhaps more elegant, to introduce a set of elementary exotic fermions that have the quantum numbers of a full -multiplet, with a lagrangian

(10)

The elementary pairs with making it super-heavy, with , the elementary pairs with the composite acquiring a mass , and the composite remains massless (neglecting the tiny mixing ).

The exotic fermions contribute of course to the gauge coupling evolution, in a way that depends on the choice of and of the -representation containing , i.e. . However, when is simple, the prediction for unification is univocal: the composite fermions form full representations, and so does the set . As a consequence, the addition of to the differential running is equivalent to the subtraction of , realizing precise unification as already discussed. Note that the case of simple tallies with the first argument for exotic fermions. When is not simple, the prediction for unification becomes model-dependent. Precise unification can still be obtained, if the set of corrects appropriately the -function coefficients, that is, if . We will come back briefly to this possibility at the end of section 3.2.

A comment is in order on the field content of the elementary sector at the EW scale. In general, composite-Higgs scenarios might have the potential to avoid the doublet-triplet splitting problem of supersymmetric GUTs, since emerges from the composite sector at the EW scale, independently from the GUT symmetry breaking sector. However, we have shown that to achieve unification one needs to introduce light elementary fermions in split representations, contrary to the supersymmetric case. We will see in section 3.3 that such splitting is needed also to prevent proton decay.

Finally, one may wonder if gauge anomalies constrain the emergence of chiral fermions from the composite sector and/or the set of chiral fermions of the elementary sector. In fact, if a -symmetric sector that undergoes condensation were anomalous under the SM, one would predict that the composite spectrum contains chiral fermions, because they must reproduce the anomaly (see e.g. Ref. [23]). Of course such anomaly should be compensated by the elementary sector: in particular, a composite must be compensated by the absence of , to recover the usual SM anomaly cancellation. The composite sector may well be, instead, anomaly-free. In fact, this is automatically the case when its global symmetry contains a subgroup . Then, either the composite sector contains no chiral fermions at all, or it contains a set that is anomaly free (e.g. a full representation). In this case also the elementary sector should be anomaly free (e.g. the SM fermions plus a full representation of exotic fermions). In all cases, the composite-GUT scenario under consideration is consistent by construction, since it has the SM chiral fermion content plus a set of vector-like fermions (both partially composite).

3 Global symmetries of the composite sector

In this section we identify the global symmetries of the strongly interacting EWSB sector. We begin with a review of the constraints coming from the electroweak precision tests (EWPTs), that will restrict the choice of the global symmetry group. The reader already familiar with EWPTs can move directly to section 3.2, where the minimal groups compatible with precision unification are classified. We then confront with the constraints coming from baryon and lepton number conservation. Finally, having determined all the needed global symmetries, we specify the quantum numbers of the exotic fermions associated with the composite top quark, and we estimate their contributions to EWPTs.

3.1 Constraints from the electroweak precision tests

In this section we shall briefly review the constraints on the EWSB sector coming from EWPTs, specializing to the case of composite-Higgs models. As customary, when evaluating extensions of the SM, we shall resort to an effective field theory description, following closely the analysis given in Ref. [18], where the relevant higher dimensional operators have been studied (we will use the same conventions). These are generated by the strong dynamics, and in particular they generically arise through tree-level exchange of heavy resonances. Therefore, such operators will be suppressed by powers of . However, this naive estimate has to be refined with some further considerations: (i) The Higgs belongs entirely to the composite sector, and therefore it couples to it with strength , so that higher dimensional operators modifying the EW vacuum or the properties shall be suppressed by powers of . (ii) The SM particles couple to the composite sector with strength dictated by partial-compositeness: each gauge boson has coupling , while each chiral fermion couples with strength . This is particularly relevant for the composite right-handed top quark , which couples with strength as the Higgs; the analysis of the viability and the consequences of a fully composite top quark has been presented in [25]. (iii) The low energy particle content of the composite GUT models, below the scale , is not that of the SM, since new light particles arise as the -partners of both and . The effects that these might have on precision observables are model dependent, and will be studied in section 3.5, after the relevant features of our scenario will have been settled.

We begin by recalling that electroweak data strongly favour the presence of a light Higgs-like particle in the spectrum [29]. The attempts to break the EW symmetry without a Higgs doublet, such as technicolour or Higgsless models, are generically difficult to reconcile with EWPTs, in particular due to large deviations in the Peskin-Takeuchi and parameters [30], with respect to the SM prediction. This motivates a preference for composite-Higgs models, in particular those where arises as a NGB from the strong dynamics, since in this case a hierarchy between the EW scale and naturally arises [18, 31].

Next, let us motivate the requirement that the new physics should be custodially symmetric, that is, it should (at least approximately) respect the custodial symmetry under which the three would-be NGBs, eventually eaten by the EW gauge bosons, transform as a triplet [32]. In our scenario, where EWSB is driven by the Higgs boson, this requirement translates into the requisite that the unbroken global symmetries of the composite sector, i.e. the group , should contain a subgroup , with transforming in the representation . In this case, when the EW symmetry is broken by , the diagonal subgroup of remains unbroken. The custodial symmetry is necessary to avoid large corrections to the tree level relation , that may arise from the operator

(11)

with , as prescribed by NDA, and . This operator can be generated purely by the strong dynamics, in the sense that does not vanish in the limit . It is extremely constrained by present data: , where we projected on the -axis the 95% C.L. ellipse in the plane [28]. In order to minimize the fine-tuning in our scenario, i.e. to allow for a higher value of compatible with the experimental constraints, the estimate given above implies that custodial symmetry must be imposed, because it automatically leads to . Similar conclusions can be drawn by analyzing the corrections to due to -violating operators involving a composite , so that custodial symmetry emerges as a generic requirement for the whole strong sector.121212 Here we are neglecting model dependent contributions to , due to couplings between the strong sector and the SM fields, which depend on and will be estimated later.

The other major source of concern, especially in strong dynamics scenarios, comes from the parameter.131313 Other parameters associated to the EW gauge boson properties are higher order in the number of derivatives, and they typically do not pose strong constraints [29]. The composite sector will generically generate the operators

(12)

with the NDA estimate .141414 This estimate is actually confirmed in holographic composite-Higgs models [2, 26] or other strongly interacting EWSB models with hidden local symmetries, see [27] for a review. The projection of the 95% C.L. ellipse in the plane [28] gives , that leads to the constraint . For a benchmark value , this gives , that lies within the window between and , the perturbativity limit for the coupling between resonances. In other words, the bound on pushes to larger values of for a fixed , which in turn should not be much larger than to avoid fine-tuning.

In composite-Higgs models, both and receive an additional contribution, arising at one-loop level because of the modified couplings of the Higgs to the gauge bosons. These couplings, which in the SM are such that scattering is unitarized, are suppressed due to the NGB nature of the Higgs boson. This leads to a mild sensitivity of the EW precision observables to the ultraviolet cut-off of the effective lagrangian for the NGBs, (as well as to the requirement of scattering unitarization by massive vector resonances). The leading effect can be accounted for by taking the SM expressions for and in the heavy Higgs approximation, and replacing the Higgs mass with an effective mass [31]:

(13)

The exponent accounts for the modification of the Higgs coupling to gauge bosons, that in the present scenario is given by , so that we have . Then, one has

(14)

where is the Fermi constant, the Weinberg angle, and is the reference Higgs mass that we adopted to define the allowed experimental ranges for and .

The last of the primary EWPTs that our scenario faces is the correction to the vertex. Experimentally, the coupling of the boson to the left-handed bottom quark current is known at the per mil level, at the level [33], if one allows for variations in (the coupling of the to the right-handed bottom).151515 Actually, the measured value of does not agree well with the SM prediction: the data on the forward-backward asymmetry and the branching fraction of into ’s suggest that should be larger than the SM value, , by roughly : the best fit is given by , with a range [33]. This is the interval we adopted to determine the allowed range for . If instead were enforced, the bound would become at the level. This suggests the introduction of a symmetry in the EWSB sector, and in its couplings to the elementary SM particles, in order to prevent large corrections to . Such a symmetry has been identified in [21]: the composite operator coupled to must transform under as an eigenstate of the parity exchanging with , that is to say, it has . This possibility is realized in one of the models we shall consider later. In this case the corrections to due to the composite sector are absent, more precisely at tree-level and for zero transferred momentum [21].

If one cannot enforce such symmetry protection mechanism, strong constraints on the parameters of the model come from the limits on . This effect can be parametrized by the higher dimensional operators

(15)

where is the top-bottom quark doublet. The coefficients depend on the coupling of to the composite sector, so that NDA gives . Assuming no cancellations are present, the estimate puts a strong bound on the degree of compositeness of and/or on . Actually, since is fully composite in our scenario, is determined by the requirement to reproduce the observed top Yukawa. This implies a bound , depending on the sign of the correction, which we cannot predict. This bound therefore can be even stronger than the one from . Nevertheless, the absence of a protection mechanism remains an open possibility. As a matter of fact, the data on suggests that beyond the SM physics might affect significantly the SM fit for (see for instance Ref. [33]).

When the symmetry introduced before is adopted, the tree-level contribution to vanishes (), and we only expect loop corrections to give a new physics contribution of the order , where is the SM top-loop contribution. This correction is safely below the experimental precision. Further loop corrections associated to compositeness are under control [25].

In addition to the deviations from the SM predictions for , and the coupling, also the couplings of will be significantly modified, due to its composite nature. However, there are no stronger constraints from present data. Such deviations could be observed (and eventually top-compositeness could be discovered) in the near future [25], by inspection of early LHC data.

3.2 Minimal global symmetry breaking patterns

The discussion of the previous sections leads to the following requirements on the global symmetry breaking pattern of the composite sector:

i) , in order to avoid leading order contributions to the differential running of the SM gauge couplings, that would spoil the calculability of unification.

ii) . The factor is needed to maintain a residual custodial symmetry after EWSB, while the extra abelian factor is necessary to properly embed the hypercharge gauged group . In fact, the simplest embedding turns out to be incompatible with the required hypercharges of composite fermions (which mix with the SM elementary ones).

iii) The broken generators in must include a multiplet of , which corresponds to the NGBs with the quantum numbers of the Higgs doublet.

With these requirements, the rank of should be equal or larger than . We find that there are only three possibilities with rank 5, listed in Table 1.

(a) 10
(b)
(c)
or
Table 1: Symmetry breaking patterns satisfying the requirements i)-iii), with the minimal restriction rank. The NGB representations under and are reported in the third and fourth column, respectively.

We indicated with the -representation of the broken generators in , i.e. of the NGBs of the composite sector. In the last column we provided the decomposition of under , with an arbitrary normalization of the charges (corresponding to the standard embedding, for the case ). Note that in option , may or may not be identified with the factor external to .

The group indicated in Table 1 is the maximal subgroup of satisfying the requirements i)-iii), but in all the three options the unbroken subgroup could actually be as small as . Of course, if a non-maximal is chosen, further NGBs appear, besides those listed in Table 1. We expect the possibilities with rank to be generalizations of these three cases with no qualitatively new features.161616 Note that when the rank is larger than 5, the possibility appears of semi-simple groups of the kind or , where unification is enforced by a permutation symmetry.

The hypercharge of composite states depends on the embedding of into , which is given in general by . To enforce custodial symmetry in the simplest possible way, we asked for the Higgs doublet to transform as under , so that one needs to obtain , while can be determined only by an additional requirement. In particular, the hypercharge of the colour triplet NGBs, , that are present for all three breaking patterns (a)-(c), is not fixed in general.

Consider for definiteness the case with the normalization , that is, the customary generator within the global SO(10) symmetry, not to be confused with the symmetry of the SM. If one requires the SM fermion quantum numbers to fit into a spinorial 16 representation of , one finds two solutions: (standard embedding into ) or (flipped ). Then the colour triplet NGBs have . In the following, we will also consider a different possibility, that is, to embed into a 10 representation of . In this case the two available solutions are and the colour triplet NGBs have .

Besides the light NGB scalar resonances, listed in Table 1, in the composite- limit one expects in general light fermionic resonances, corresponding to the partners of filling a -multiplet, and to the exotic elementary fermions that pair with them to form vector-like massive states. In principle, these fermion states can be absent all together, if the SM state forms a full -multiplet by itself. This requires , and . In all other cases exotic fermions are needed, with quantum numbers determined by the choice of the symmetry breaking pattern and of the -multiplet containing .

As discussed in section 2.2, the set of exotic fermions determines the fate of gauge coupling unification. A sufficient condition to realize it accurately is to take simple. This is the case only for the symmetry breaking pattern , with the largest possible unbroken subgroup, . This is the model whose phenomenology we will study in detail, motivated by unification.171717 Besides, the coset has the advantage of being the smallest coset of our list, with . This is a desirable feature, since the ultraviolet cut-off of our effective lagrangian for the NGBs is .

When is not simple, one can still hope to realize unification, if , where is the set of exotic fermions. To realize the latter condition, one should carefully choose and the -multiplet . First, notice that in the SM and meet at GeV, which is too early to unify with as well as to prevent gauge-mediated proton-decay (see section 3.3). To delay unification one needs a correction . If contains as an isolated factor, must be a singlet of in order to contain , and thus exotic fermions do not contribute to . In order to increase , one should resort to models where is part of a larger simple group, , and contains -non-singlet components. This is possible only for the symmetry breaking pattern , with the maximal unbroken subgroup and with . However, inspecting small -representations, we did not find one that contains and leads to reasonably good unification: the required contribution to is compensated by large contributions to and , that go in the wrong direction. We will not study further the case with non-simple in the following.

3.3 Constraints from proton stability

As usual in GUTs, the stability of the proton is endangered by baryon number violating interactions. These manifest in the SM effective lagrangian as higher dimensional operators suppressed by the scale of baryon number violation. In the elementary sector, this scale is the mass of GUT gauge bosons, . The interactions with the composite sector, however, may violate baryon number (and lepton number) at much smaller scales , thus invalidating the whole program of composite GUTs. Let us briefly comment, first, on the usual gauge-mediated proton decay at , and next move to the requirements to be imposed on the composite sector.

We explained how the differences of gauge coupling -function coefficients, , which control the differential running at leading order, can be determined thanks to the global symmetry of the composite sector. This not only allows the prediction for from the experimental values of , but also fixes the value of the scale , where and meet. With our recipe for precision unification, that is, with an elementary field content given by , one finds GeV, which is a factor of smaller than the lower bound on GUT gauge boson masses in the minimal model, GeV.181818 In the MSSM GeV, however supersymmetry enhances Higgs-mediated p-decay (since it is induced by dimension 5 operators), thus requiring an additional suppression. This gap shall be cured either by two-loops corrections, that may be enhanced by the strong dynamics, or by GUT thresholds, or by special structures of the Yukawa couplings leading to cancellations in the proton-decay operators (these can relax the lower bound on by more than one order of magnitude [34]). Also, if GUT breaking is realized in extra-dimensions by orbifolding, proton-decay operators can be forbidden or, more in general, their structure can be significantly different [35]. We do not elaborate more on these issues, since we do not control the strong dynamics at the two-loop level and we do not specify the theory at the GUT scale in this paper.

Let us now discuss possible low-energy sources of baryon and lepton number violation. While in the SM and are accidental symmetries of the renormalizable lagrangian (due to the gauge symmetry and the SM field content), in general the composite states may mediate - and -violating processes. This is particularly worrisome in view of the unified -symmetry of the composite sector, that will contain states with the charges of the SM ones together with their -partners. Therefore, one a priori expects e.g. resonances at the scale with the same quantum numbers of the gauge bosons . The couplings of these resonances to the SM elementary fermions (in particular those contained in the proton), as given in Eq. (4), will induce -violating operators of the form

(16)

where we used the rough relation , in order to illustrate that these operators cannot be sufficiently suppressed, if one wants to properly reproduce the SM Yukawa couplings.191919 Even if composite resonances coupling directly to the four elementary fermions are absent, -violating operators can be generated through non-perturbative effects by the strong sector, suppressed generically by the scale . This suppression is again far too small. Similarly, the dimension-5 operator could be induced, that would violate , generating Majorana neutrino masses far too large with respect to the observed ones.

One is then forced to postulate additional symmetries, to prevent too large and violations. Note that additional symmetries are also required in supersymmetric models: in particular, with the particle content of the MSSM, one needs to impose -parity in order to forbid and violating dim-4 interactions; this is sufficient in such weakly coupled theories, because one can assume that higher dimensional operators are generated only at scales much larger than the EW scale. In the composite GUT scenario, the extra symmetry should forbid also, at the very least, the dimension-5 and -6 operators. This extra “matter” symmetry, denoted by , shall be part of the global symmetries of the composite sector, either within the simple unified group, , or factored out, . Besides, it should be extended consistently to the elementary sector, that is, it must be (to very good approximation) a symmetry of the whole effective lagrangian in Eq. (1), and it should be preserved up to scales . In other words, while and are accidental symmetries of the SM alone at low energy, the symmetry must be imposed by hand on the couplings of the SM elementary fields to the composite sector.

To see concretely how proton decay arises and how it can be forbidden, let us consider the simple case of , left unbroken at . If the Higgs doublet belongs to a composite multiplet , the coloured triplet can mediate proton-decay, as it is well-known. Clearly, there is no generator internal to that can prevent the couplings of to SM fermions, without preventing at the same time the required couplings of . Therefore, one is forced to introduce an extra global symmetry external to . The most obvious option is with (i) the usual -assignments of elementary fermions, (ii) , as required since and belong to the same -multiplet; (iii) , in order to allow for the Yukawa coupling of .202020 Alternative choices could be e.g. , which may be inspired by Pati-Salam unification and allows for proton-decays into three leptons (these require operators with dimension larger than 6 and may be sufficiently suppressed [36]). In principle, even a discrete symmetry could be enough to guarantee the proton stability.

The same reasoning can be applied to the realistic case (we saw that is too small to accommodate the custodial symmetry), imposing a symmetry external to (and therefore also to SO(11)). However, in the case there is a second, slightly more economical, possibility: to identify the action of on the composite states with the action of one of the generators. In fact, once the embedding of the SM subgroup into is chosen, one is left with one independent linear combination of the generators in the Cartan subalgebra of , that is an independent symmetry, and one can take for any composite field . Since acts on elementary fermions as the usual baryon number, one needs . Recalling that the NGBs form a 10 representation of SO(10), a viable choice is . In particular .

As already mentioned, one also needs a symmetry to prevent neutrino masses larger than eV, as required by experiments, that is, the composite sector should respect in good approximation lepton number. We do not repeat a detailed discussion analogous to the one of baryon number, and we just adopt the most straightforward possibility: a