Modification of Higgs Couplings in Minimal Composite Models
Da Liu, Ian Low, Carlos E. M. Wagner
TL;DR
This paper investigates how Higgs couplings are modified in minimal composite models, analyzing the effects of different fermionic embeddings and their implications for Higgs phenomenology at the LHC.
Contribution
It provides a detailed analysis of Higgs coupling modifications in SO(5)/SO(4) composite models, highlighting the impact of fermionic embeddings and the relationship between tth and ggh couplings.
Findings
tth and bbh couplings are suppressed in 5 and 10 embeddings
In 14 embedding, couplings can be enhanced or suppressed
The tth to ggh coupling ratio is typically less than the Standard Model ratio
Abstract
We present a comprehensive study of the modifications of Higgs couplings in the SO(5)/SO(4) minimal composite model. We focus on three couplings of central importance to Higgs phenomenology at the LHC: the couplings to top and bottom quarks and the coupling to two gluons. We consider three possible embeddings of the fermionic partners in 5, 10 and 14 of SO(5) and find tth and bbh couplings to be always suppressed in 5 and 10, while in 14 they can be either enhanced or suppressed. Assuming partial compositeness, we analyze the interplay between the tth coupling and the top sector contribution to the Coleman-Weinberg potential for the Higgs boson, and the correlation between tth and ggh couplings. In particular, if the electroweak symmetry breaking is triggered radiatively by the top sector, we demonstrate that the ratio of the tth coupling in composite Higgs models over the Standard…
Click any figure to enlarge with its caption.
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10
Figure 11
Figure 12
Figure 13
Figure 14| Eq. (147) | ||||
| no bound | no bound | |||
| no bound | no bound | |||
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Modification of Higgs Couplings in Minimal Composite Models
Da Liua, Ian Lowa,b and Carlos E. M. Wagnera,c,d
a High Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439
b Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208
c Enrico Fermi Institute, University of Chicago, Chicago, IL 60637
d Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637
Abstract
We present a comprehensive study of the modifications of Higgs couplings in the minimal composite model. We focus on three couplings of central importance to Higgs phenomenology at the LHC: the couplings to top and bottom quarks and the coupling to two gluons. We consider three possible embeddings of the fermionic partners in , and of and find and couplings to be always suppressed in and , while in they can be either enhanced or suppressed. Assuming partial compositeness, we analyze the interplay between the coupling and the top sector contribution to the Coleman-Weinberg potential for the Higgs boson, and the correlation between and couplings. In particular, if the electroweak symmetry breaking is triggered radiatively by the top sector, we demonstrate that the ratio of the coupling in composite Higgs models over the Standard Model expectation is preferred to be less than the corresponding ratio of the coupling.
I Introduction
The discovery of the Higgs boson at the LHC Aad:2012tfa ; Chatrchyan:2012xdj has led to a new era in particle physics. The Standard Model (SM) has been validated as the proper low energy effective theory description of the interactions between the known fundamental particles. The Higgs boson production and decay rates seem to be in good agreement with those predicted in the SM Khachatryan:2016vau , suggesting that the mass generation proceeds from the Higgs mechanism, with the recently discovered Higgs being its observable consequence. The current precision of the Higgs rate measurements, however, leaves some room for departures from the simple SM picture. In particular, data collected at the LHC have only started to probe Higgs couplings with the third generation quarks, which will play a central role in future Higgs measurements, while couplings to fermions in the first two generations remain a challenge. Therefore, it is interesting to study what would be the possible consequences of the deviations of the third generation quark couplings to the Higgs boson and, in particular, what kind of high energy models can accommodate such deviations in a natural way.
A departure from the SM description is to be expected in any model that leads to the breakdown of the electroweak symmetry in a natural way. This could be achieved in models in which the Higgs boson is an elementary or a composite particle. If it is an elementary particle, with renormalizable interactions that remain perturbative until scales of order of the Planck scale, the natural implementation of electroweak symmetry implies a supersymmetric extension that renders the Higgs mass parameter insensitive to the ultraviolet physics Martin:1997ns . Due to the top-quark contribution to the loop-induced Higgs couplings, any modification of the Higgs coupling to top-quarks Gunion:1984yn will induce a similar modification of its coupling to gluons as measured in terms of their SM values. These two contributions may be rendered independent in the presence of light superpartners of the top-quark (stops) which could contribute in a relevant way to the loop-induced Higgs couplings. Based on this observation, an analysis of the possible enhancement of the Higgs couplings to top-quarks within supersymmetric models was recently presented in Refs. Badziak:2016exn ; Badziak:2016tzl .
Alternatively, a natural electroweak symmetry breaking may be achieved by assuming that the Higgs is a composite particle Terazawa:1976xx ; Terazawa:1979pj ; Kaplan:1983fs ; Kaplan:1983sm . There have been renewed interests in composite Higgs models, following the works in Ref. ArkaniHamed:2001nc ; ArkaniHamed:2002qx ; ArkaniHamed:2002qy ; Contino:2003ve ; Agashe:2004rs , and their interpretations as duals of models of gauge-Higgs unification in warped extra dimensions Hosotani:1988bm . In these models, the Higgs appears as a pseudo-Golstone boson and the insensitivity to the ultraviolet scale is ensured by its composite nature, as manifest by its gauge origin in the gauge-Higgs unification picture. The pseudo-Goldstone nature of the Higgs scalar stems from the spontaneous breakdown of a global symmetry group that includes the weak interaction group as a subgroup of it. One of the simplest and most attractive realization is when the global symmetry group is Agashe:2004rs , which breaks spontaneously into , that contains both the gauge group , as well as the custodial group . The four Nambu-Goldstone bosons associated with the breaking of the global group are identified with the four components of the Higgs doublet. The properties of the Higgs boson are determined by explicit symmetry breaking terms associated with the Yukawa coupling of the third generation quarks, which depend strongly on the representation of employed in the fermion sector.
The Higgs couplings in the minimal model have been previously studied in the literature Falkowski:2007hz ; Low:2009di ; Low:2010mr ; Montull:2013mla ; Carena:2014ria , and it is known that the simplest representation choices lead to a suppression of both the third generation quark and gluon Higgs couplings with respect to the SM ones. In this article, we provide an analytical study of the pattern of the top, bottom and gluon couplings with the Higgs within this minimal model, for different choices of the representations in which the top quark is included. These three couplings are of central importance to the Higgs phenomenology at the LHC. One of our goals is to provide an analytical understanding of the capabilities of this model to fit the future Higgs data. Moreover, we compute the Coleman-Weinberg potential for the Higgs field that is induced by the top quark sector Falkowski:2006vi ; Medina:2007hz and study the constraints coming from the requirement of obtaining a proper electroweak symmetry breaking (EWSB) with a Higgs mass consistent with the observed one.
The presentation of this article is as follows. In section II we introduce a general framework for computing the relevant Higgs couplings and the Higgs potential by integrating out heavy partners of the third generation quarks in composite Higgs models. In section III we focus on the minimal model and analyze the case of introducing composite fermions in the 5 and 10 representation of . In section IV we analyze the case of employing the 14 representation of . We reserve section V for our conclusions,. In the Appendices we present some technical overview and details associated with the study. We also briefly discuss the more complicated scenario of using 5 + 10 representations of the group in the Appendix.
II General analysis
In this section, we present a general analysis of the relation between the and couplings, under broad assumptions that can be applied to arbitrary coset in composite Higgs models. We proceed by integrating out the new TeV scale strong dynamics, which results in an effective Lagrangian containing only SM particles. Effects of the strong dynamics are encoded in terms of the form factors of the SM particles in momentum space, in analogy with the form factors of the nucleons in low-energy QCD. Focusing on the third generation quarks for now, the form factors are defined as:
[TABLE]
where the form factors are the functions of and the proto-Yukawa couplings.
We will also assume that SM fermions obtain their masses from linear mixing with the new strong sector according to the hypothesis of partial compositeness Kaplan:1991dc , which means in the UV, we have the mixing Lagrangian:
[TABLE]
where the operators from the strong sector furnish some linear representations of . Note that must break explicitly and the proto-Yukawa couplings can be viewed as spurions parameterizing the effects of the explicit breaking. Then it should be clear that, after integrating out the strong dynamics, the wave function renormalizations with are proportional to , while the mass terms are proportional to . A detailed spurion analysis could put further constraints on the form factors, as will be shown later when we discuss specific embedding of the fermionic partners.
Since we are mainly interested in composite Higgs models in which the Higgs boson is realized as a pseudo-Nambu-Goldstone boson (pNGB), we will further assume that the wave function normalization form factors can be expanded in series of , where is Goldstone boson decay constant:
[TABLE]
where . The expansion follows from the observation that the Higgs boson is a double under and that there is a shift symmetry acting on the doublet Low:2014oga . For the form factors in front of the mass term, if the fermion is embedded in a vectorial representation , while for a spinorial representation it is simply . Since the spinorial representation of the model is severely constrained by the precision electroweak measurements Agashe:2006at , we will focus on the vectorial representations and its direct product:
[TABLE]
where are proportional to the mixing parameters .
To compute the Higgs couplings in models where the Higgs is a pNGB, it is important to recall that is not the same as the SM Higgs vacuum expectation value of GeV. Instead, by matching to the boson mass in the SM one obtains Low:2014oga
[TABLE]
where the misalignment angle is defined as . For coset this is explicitly demonstrated in Eq. (128) of Appendix A. The misalignment angle is related to the fine-tuning parameter
[TABLE]
which is commonly employed in the literature.
The Higgs coupling to fermions can be computed by noting that the fermion masses is computed from the form factors at the zero momentum:
[TABLE]
from which we can calculate the coupling strength with respect to SM as a function of the form factors:
[TABLE]
It turns out that for representations considered in this work, the expansions of the form factors terminate at and the coupling strength is given by:
[TABLE]
where . Note that the first term is the universal suppression factor coming from the term in the expansion, which can be rewritten in terms of :
[TABLE]
Before computing the coupling strength, it is worth recalling the observations made in Refs. Azatov:2011qy ; Furlan:2011uq ; Montull:2013mla , which states that, under the assumption of partial compositeness, the determinant of the fermion mass matrix is proportional to the the mass term form factor at the zero momentum. This is due to the particular form of the mass matrix:
[TABLE]
where denotes SM fermions and () is the mixing vector in the flavor space between the left-handed (right-handed) SM fermion and its composite partners. Here is the -symmetry-preserving mass matrix of the fermionic partners and does not depend on the Higgs field, because in the limit of zero mixing between SM and the composite sector, the G-symmetry is exact and all Higgs interactions must be derivatively coupled. For simplicity, we will assume that all mixing parameters are real and have chosen a basis in the flavor space where is diagonal. It is then not difficult to see:
[TABLE]
By integrating out the fermion partners using the equation of motion at the zero momentum from the Lagrangian in Eq. (11), we obtain:
[TABLE]
which in turn implies:
[TABLE]
Note that the Higgs dependence of the determinant is fully contained in the mass form factor .
In the SM the largest contribution to the coupling comes from the top quark. A detailed discussion of the coupling is given in Appendix B. Here we merely collect the essential results. In the limit of infinite top mass and resonance mass, the charge 2/3 sector contribution to can be obtained:
[TABLE]
For the charge -1/3 sector, the SM bottom quark contributes negligibly to the coupling, which need to be subtracted from the fermion mass matrix of the bottom sector. To be specific, we have:
[TABLE]
Note that starts from the linear order in , as we have neglected the SM bottom contribution.
We study the correlation between the and couplings by computing
[TABLE]
Note that if there is no Higgs dependence for all the wave function normalization form factors, is exactly equal to . We will see this limit from the specific calculations for the different representations of . It turns out, at the leading order in , has a strong correlation with the fermionic contribution to the Coleman-Weinberg potential of the Higgs boson, which in the Euclidean space is given by Contino:2010rs
[TABLE]
We are only interested in the Higgs potential to the quartic order in :
[TABLE]
where
[TABLE]
while is not relevant for present discussion. In the above is the cutoff of the composite model. Electroweak symmetry breaking requires
[TABLE]
It turns out that, for the embedding of composite resonances studied in this work, the last contribution inside the parenthesis in Eq. (20) is numerically subleading to the first two terms, whose value at dictates , as can be seen in Eq. (17). As a result, if the integral, Eq. (20), would receive its dominant contribution from the infrared regime, there would be a strong preference for in order to trigger EWSB. The interrelation between and will be studied in details when we consider embeddings of the composite resonances in and of .
Notice that is quadratically divergent. Specifically, from Eq. (II) it is clear that asymptotically in the Euclidean space,
[TABLE]
Under the expansion assumed in Eqs. (3) and (4) we see that
[TABLE]
These considerations suggest the quadratic divergence in resides only in the first two terms in Eq. (20), while the last term is finite.
In a viable model of natural EWSB, such quadratic divergences must cancel in the Higgs potential. The cancellation of quadratic divergent contributions to makes the infrared contribution to Eq. (20) more relevant and therefore the correlation between and stronger. In what follows we will always impose the cancellation of quadratic divergences in .
III 5 and 10 of
In this section, we study the cases where the composite resonances are embedded in the and of and mix with the elementary fermions according to Eq. (2). We will see that in neither case can the coupling be enhanced over the SM expectation. However, before we begin, it is useful to set up some notation in the CCWZ formalism Coleman:1969sm ; Callan:1969sn , which will be used heavily in this work. A brief overview of CCWZ can be found in Appendix A.
The main objects of consideration are the Goldstone matrix and the Goldstone gauge field , defined in Eqs. (126) and (127), respectively. The matrix transforms under the non-linearly realized as:
[TABLE]
where . Therefore, formally speaking, we can view the matrix as carrying an index on the left and an index on the right:
[TABLE]
where we use the capital Roman letters to denote the indices and the lower case Roman letters to denote the indices. In addition, we choose a basis such that the unbroken generators reside in the upper block of the generators. In this basis , , can be viewed as an vector and an vector, while transforms like an vector and an singlet. It will be useful to define such that
[TABLE]
where we have defined and evaluated the Goldstone matrix in the unitary gauge. will play a major role in building invariants.
III.1 5 of
We first discuss the case of embedding the composite resonances in the of in the top sector. Notice that we assume is explicitly broken by the mixing of the composite resonances with the elementary fermions. The composite resonances transform as () and () under transformations. The is unbroken in the strong sector, but it is also explicitly broken by the elementary-composite mixing. In other words, the mixing explicitly break to SM group . In addition, decomposes into two doublets of hypercharge , which are denoted as and , where and the subscriptsdenote the hypercharge values. More explicitly,
[TABLE]
The Lagrangian involving the composite fermions is then:
[TABLE]
where and . In the above the first line contains the fermion kinetic and Dirac mass terms, while the second and the third lines contain the mixing terms for and , respectively, with the elementary fermions and , which are the explicit realization of the partial compositeness assumption in Eq. (2). In particular, we have “uplifted” the elementary fermions, which only carry quantum number, to the space:
[TABLE]
where we have defined the following projection operators:
[TABLE]
In Eq. (28) the and , , are dimensionless parameters that are of order unity. Since we are not going to discuss the CP-violating effects in this paper, we assume these parameters are real.
The projection operators in Eq. (30) can be viewed as spurions carrying an index:
[TABLE]
These properties allow one to construct invariants which formally respect the full symmetry of the strong sector. The elementary fermions have the following global symmetry associated with them:
[TABLE]
so that the Lagrangian for the composite Higgs has a large global symmetry where is non-linear realized.111The is required to give the correct hypercharges Agashe:2004rs . The projection operators in Eq. (31) also transform under . When we integrate out the composite resonances, the resulting effective Lagrangian preserves this large symmetry and, as a consequence, can be constrained by performing a spurion analysis. More specifically, the effective Lagrangian can be constructed out of the following invariants:
[TABLE]
where is defined in Eq. (26). This argument implies the wave function form factors in Eq. (II), , are invariant under , while is built out of and with similar transformation properties under . In particular, we see can only contain a constant term and a term proportional to , while there is no dependence on the Goldston bosons (i.e. the Higgs boson) in .
The mass eigenstates before EWSB can be obtained by rotating the left-handed fields and right-handed fields with angles :
[TABLE]
The mixing matrices will obtain corrections after the Higgs receives a VEV upon EWSB. It is straightforward to obtain the full mass matrices by plugging Eq. (27) and the expression of the Goldstone matrix in Eq. (123) into the Lagrangian in Eq. (28). The result for the charge-2/3 states reads:
[TABLE]
where the mixing vectors and the composite mass matrix are:
[TABLE]
Recall is the vacuum misalignment angle defined as . The dependence on the in can be understood by restoring to its full doublet .
To determine the effects of the composite resonances on the Higgs couplings and the cancellation of quadratic divergences, we need to compute:
[TABLE]
We can see that in the limit , the mass determinant is zero and the top quark remains massless, since the full symmetry is unbroken. On the other hand, in the case of , there is no Higgs dependence in the trace of , which means that the quadratic divergence is cancelled in this limit.
The coupling is obtained by plugging in Eq. (37) into Eq. (140):
[TABLE]
where . Note that there is no composite mass dependence in the coupling Low:2009di ; Low:2010mr . This can be understood from the fact that there is only one -invariant that can be constructed out of with Higgs dependence:
[TABLE]
The determinant of the mass matrix (i.e. the form factor ) is then proportional to and the dependence on the mass scales can only enter through the proportionality constant, which drops out in Eq. (39).
On the other hand, to compute we need to compute the form factors defined in Eq. (II), which can be done by following the procedure of integrating out the composite resonances outlined in Appendix C. In terms of the expansion defined in Eqs. (3) and (4), the form factors are
[TABLE]
while all other terms vanish. A few comments are in order. First we see the new strong sector contributions to the wave function normalization and are proportional to and , respectively, and that to the mass term are proportional to . In the case where there is no breaking effects, i.e. , all the form factors vanish except the kinetic terms of the SM fermions, which are elementary fermions. On the other hand, in the limit
[TABLE]
the form factors vanish asymptotically, which signals the quadratic divergences in the Higgs mass cancel; see the discussion at the end of Sect. II. We will assume this is always the case so that the top quadratic divergence in the Higgs mass is cancelled.
To compute the top mass and the coupling, we first evaluate the form factors at the zero momentum:
[TABLE]
where we have defined:
[TABLE]
given our choice of cancellation of quadratic divergences in Eq. (42). In addition, we set in the form factor , because the sign of the fermion mass term is not physical. Now using Eqs. (7) and (8) we obtain and coupling:
[TABLE]
where we have neglected terms that are higher order in in the top mass. Note that to reproduce the observed top mass for and , we need the mixing parameters .
After substituting the form factors in Eq. (43) into Eq. (46), the leading modification in the coupling is given by
[TABLE]
Recall that . We see that is always smaller than zero, implying that the coupling is always suppressed.222We also checked that in the limit is fully composite, even though the mass matrix in Eq. (11) has an non-zero first diagonal entry and = 0, coupling is still reduced. In fact, it is possible to strengthen the bound in the above and prove that
[TABLE]
To see this, we observe from Eqs. (39) and (47) that
[TABLE]
where we have used the identity . Since as in Eq. (39), the bound in Eq. (49) follows. It is also worth noting that, in the region , :
[TABLE]
In Fig. 1 we plot contours of and as functions of and with and GeV (taken as a representative value of the running top quark mass at the TeV scale), for different values of . In the figures we always use the full formulas in Eqs. (7) and (8), which captures the full dependence on . For the bound in Eq. (49) gives
[TABLE]
which is consistent with the values shown in Fig. 1. Notice that gives the overall mass scale of the top partners.
Up to now, we have not considered effects of the composite resonances in the Higgs potential. As discussed in Section II , may be correlated with the coefficient , defined in Eq. (19), in front of the quadratic term in the potential. Let’s define
[TABLE]
where and
[TABLE]
For the case of , we can obtain
[TABLE]
where and is given in Eq. (45). Note that in our parametrization, is the case of the maximally symmetric composite Higgs considered in Ref. Csaki:2017cep . In this limit, all terms in vanish except , which is coming from the mass form factor square term in Eq. (54). Note that in this special limit, the dependence on the Higgs field of the wave function normalization form factors disappear, which implies exactly according to Eq. (17).
It is worth recalling that a sufficient (although not necessary) condition for EWSB is through out the integration region. Among the three contributions in Eq. (55), is positive-definite and constant in . If one chooses TeV and , and is numerically small. On the other hand, the first and the last terms in Eq. (55) both grows with and, therefore, should dominate the integration in . As can be seen, the first contribution is strongly correlated with . Thus if is larger than in any significant way, one would need a sizeable positive contribution from to obtain a positive and EWSB. It turns out that can be positive only in the region
[TABLE]
Even in this region, is numerically small,
[TABLE]
In Fig. 2 we plot normalized to for and 0.9, to demonstrate the bound in Eq. (59).We conclude that the first term in Eq. (55), , is the dominant contribution to generically, and that EWSB prefers
[TABLE]
We will see that this pattern persists considering embeddings in and of . In Fig. 3 we present numerical scans of versus for , confirming the correlation derived from the analytical understanding. In the tiny sliver of region where and , we see is very small, at the percent level. Note that because the SM gauge boson contribution to the factor is always negative, including it will make the preference for even stronger.
To complete our discussion of the case of , we next discuss the implementation of the bottom Yukawa coupling. In this case we will introduce composite resonances to mix with and , but not . Similar to the top partners, the bottom partners can be embedded in the of but with a different charge, . This has the effect of mixing with the doublet of the composite resonances, as opposed to the doublet in the case of the top partner. The projection operators in this case are given by:
[TABLE]
Because of the similarity of these projection operators to their counter parts in the top sectors, all the formulas for the form factors remain the same except that the mass scales are now for the bottom partners. The leading term for the bottom mass reads:
[TABLE]
Note that in order to reproduce the bottom mass for and , we need . This implies, unless we have a large hierarchy between the left-handed and the right-handed mixing parameters, the contributions to the Higgs potential from the bottom sector can be safely neglected.
Now, by using Eqs. (9) and (16), we obtain
[TABLE]
where we have used notations similar to those in the top sector and . Since we are neglecting the small SM bottom contribution to the coupling, the bottom partner contribution to coupling in starts at the linear order in . Notice that in Eq. (63) has the same expression as in Eq. (47), with all the parameters now refer to the bottom sector. Therefore,
[TABLE]
Because the total width of the 125 GeV Higgs boson is dominated by the partial width in the channel, a reduction in the bottom Yukawa could result an overall increase in the observed signal strength across a variety of channels at the LHC, without modifying the production cross-section in the channel. For example, one can choose so that
[TABLE]
which implies for .
III.2 10 of
In this subsection, we embed the composite resonances in the two-index anti-symmetric representation of , which can be decomposed to under the unbroken . Here we assume that the and have the same mass, which is combined into a under and denoted by , with , for . We start from the top sector and the effective Lagrangians for the top partner fields:
[TABLE]
where . In the limit in which the , and , the invariance is restored. We assign two upper indices to the sixplet composite fields and as a result will have two upper and two lower indices. The “uplifting” of the elementary fermions and to the space in this case is defined by
[TABLE]
where
[TABLE]
which carry two upper indices and will be treated as spurions. Observe that the projection tensors are just the anti-symmetrized version of the product of the embedding vectors in the case of .
Similar to the case of , the spurion analysis reveals only one invariant for each projection operator (keep in mind our contraction convention):
[TABLE]
where and . To see this, we observe that the invariant involving the can always be reduced to the above by using the unitary constraints:
[TABLE]
To be specific, we have:
[TABLE]
which implies the expansion for the form factors in Eq. (3) stops at .
Calculating the form factors as before, the form factors for the left-handed sector are
[TABLE]
Similarly, for the right-handed sector,
[TABLE]
and the mass term:
[TABLE]
From the above, we see clearly that vanishes in the limit , and , and there is no Higgs dependence in the form factors. This is expected because in this limit the full symmetry is restored and the Goldstone field can be rotated away by redefinition of the the composite fields.
Cancellation of quadratic divergence in the top sector requires
[TABLE]
but not , since the mass term only breaks ”softly.” We assume Eq. (77).
We also define the following useful parameters
[TABLE]
Now it is straightforward to obtain the top mass at the leading order in ,
[TABLE]
which is the same as in the case of except the factor in front. The modifications to the and couplings from the top sector are then:
[TABLE]
where we have used the same convention for the ’s as in the case of . We see immediately that 1) can be either enhanced or reduced and 2) the top Yukawa coupling is not only always suppressed, but the suppression is in general stronger than in the case of . On the other hand,
[TABLE]
As for the Higgs potential, we compute in Eq. (53):
[TABLE]
where
[TABLE]
Notice that has a similar structure to the case of , where is related to the top mass and expected to be subdominant for TeV and . As for , one can show that it is positive only in the region
[TABLE]
In this region it is possible to demonstrate that
[TABLE]
So again there is a strong preference for in order to trigger EWSB, which is confirmed in the numerical scan shown in Fig. 4.
For the bottom sector, we again introduce composite fermions that mix with but not , which is similar to the case of and the results for the modifications of coupling and coupling read:
[TABLE]
Again is similar to its corresponding expression of in Eq. (81). As a result, the same bound
[TABLE]
applies. Also similar to the case of , if we assume that the mixing parameters in the bottom sector are small, it is possible to modify the coupling without chaning the coupling, as done in Eq. (66).
IV of
is the two-index symmetric-traceless representation of . This scenario is distinct from the cases of and in that there are two non-trivial invariants one can construct in the spurion analysis. As a result, the coupling now has a non-trivial dependence on the mass of the composite resonance Azatov:2011qy and, moreover, the coupling can be enhanced. A qualitatively similar, but numerically more complicated, scenario of embedding the composite fermions in simultaneously is discussed in Appendix D.
IV.1 The Top Sector
Under the unbroken , can be decomposed into . Therefore we introduce three mass scales, and for the 9-plet, 4-plet and the singlet, respectively. The uplifting of the elementary fermion to the space achieved through the following projection operators:
[TABLE]
[TABLE]
These projection operators carry two indices, which are just the symmeterized version of the tensor product of two projection operators in the case of .
The effective Lagrangians for the top partner fields:
[TABLE]
where and the numerical factors in front of the mixing terms are such that, in the limit in which , , and , the full symmetry is recovered. Notice that, in the above, the 9-plet fermion is a symmetric, traceless rank-2 tensor field with , which means that only the traceless part of mixing with it. The corresponding in the kinetic term therefore has two upper and two lower indices.
Similar to the case of , the 4-plet can be decomposed as two doublets which are denoted as and ; see Eq. (27). For the 9-plet, it can be decomposed as three degenerate triplets () with hypercharge respectively. From these quantum numbers we see, before EWSB, the triplet fermions and cannot mix with the elementary fermions, while and the singlet can mix with the and , respectively, which is similar to the case of . The mass spectrum before EWSB is then
[TABLE]
where are the masses of the triplets, the doublet , the doublet and the singlet respectively. After EWSB, the physical masses will be corrected at , except the exotic electric charge states. We then define the following mixing parameters:
[TABLE]
which is similar to the case of .
As in previous cases, we first explore the independent invariants involving the Goldstone matrix and the projection operators in Eq. (92). A useful observation in this regard is the fact that an invariant involving the can be rewritten by using the unitary constraints:
[TABLE]
As a result, the following decomposition applies
[TABLE]
In the end there are precisely two and only two invariants for the form
[TABLE]
where here generally denotes .
An important consequence of having two different invariants is that now we have two Higgs-dependent terms in the and, as a result, the coupling depends on the composite mass scales, unlike in the case of and . To be specific, we have:
[TABLE]
which implies the mass form factor now contains two different trigonometric combinations: and . In other words, is now non-vanishing in Eq. (4). A similar computation for the form factors and shows that they contain expansion up to , while the expansion in stops at because . In the end, the form factors in the case of have the following expansions,
[TABLE]
Similar to previous cases, cancellation of quadratic divergences in the top sector requires the following condition,
[TABLE]
which we impose in our analysis. Furthermore, we define
[TABLE]
Now it is straightforward to obtain the top mass from the form factors:
[TABLE]
where again we have neglected the higher order terms in . Notice the above formula is the same as the case of except the numerical factor . One relevant difference with respect to the and cases is that the top-quark mass dependence on include higher order terms which are proportional to and not to . This implies that for , the dominant dependent of the top-quark mass is not linear on the Higgs field, what makes it difficult to generate a realistic top mass and a light Higgs boson at the same time, and also causes an unacceptably large departure of the top-quark coupling to the Higgs with respect to its SM value. Hence, a phenomenologically viable model can only be obtained if the leading contribution proportional to , Eq. (105), is sizable compared to the higher order terms proportional to . The formulae below are derived under this assumption.
By following the same calculation as in before, we obtain the modifications to the and couplings:
[TABLE]
where and . The superscript in indicates this is the contribution from the top sector. Notice that, in Eqs. (106) and (107), there is a fictitious pole at , which is just a reflection of our previously stated approximation of keeping only the leading contribution in to the top quark mass, Eq. (105). Again, such a fictitious pole does not arise in the case of and because in these two cases and do not have explicit dependence on the composite scales.
Let’s analyze the and couplings without considering the Higgs potential, for now. The most important distinctions from the cases of and is that, in the current scenario, and can be either positive or negative. In other words, there are enough degrees of freedom in Eqs. (106) and (107) such that and could be either enhanced or reduced. It is easiest to demonstrate this numerically. In Fig. 6 we plot contours of and for the benchmark of and (left panel), (right panel), where it is clear that they can be enhanced or reduced over the SM expectations for the reasonable values of , which is the overall mass scale for the top partners.
Next let’s turn to the difference between the and couplings, which is given by:
[TABLE]
It is evident that:
[TABLE]
from which we can see that:
[TABLE]
Again this quantity enters into the Higgs potential through Eq. (53),
[TABLE]
where:
[TABLE]
Again through out the integration region, , is a sufficient condition to trigger EWSB. Notice that the second contribution to in Eq. (113) is identical to in the case of in Eq. (57)333Recall that this is the singlet fermion contribution to the in the Higgs potential., apart from the numerical coefficient of 5/2. It was shown there that this contribution is quite insignificant when it is positive in the region of ; see Eq. (59). The first term in is positive in the following region:
[TABLE]
and can become sizeable if is either much smaller than 1 or much larger than 1. Actually, it is possible to prove:
[TABLE]
In Fig. 6, we perform numerical scans over , with determined by the mass of for . In the upper plots of versus , we see that there is a strong preference for to have a positive to trigger the EWSB. Although there are regions where a significant positive can be obtained for , we show in the lower plot that, once we require that both the value of and the Higgs mass are reproduced by the Higgs potential, is always less than in the case of . Although, we find some points with in the case of , both and are very small and are not very phenomenologically interesting. Our results in Fig. 6 not only confirm the findings in Ref. Montull:2013mla , but also provide an analytic understanding of the strong correlation.
IV.2 The Bottom Sector
For the bottom sector, as similar to the case of and , we introduce composite fermions that mix with but not . The form factors are almost identical to the top except that the mass scales and the mixing parameters are now in the bottom sector. The elementary doublet are ”uplifted” to the space via
[TABLE]
[TABLE]
The invariants in this case are quite similar to those in the top sector,
[TABLE]
where the expansions in are evident and terminate at the order. The effective Lagrangian is similar to Eqs. (93) to (95), with replaced by .
Mixing parameters are defined similar to Eqs. (97) and (104), with all the parameters now referring to the bottom partners now. As such we still need in order to reproduce the small bottom quark mass for and TeV. Therefore their contribution to the Higgs potential can be ignored.
The modification to the bottom Yukawa coupling is identical to in Eq. (107), with all the parameters referring to their counterpart in the bottom sector. On the other hand, the coupling from the bottom sector is given by
[TABLE]
which is quite different from Eq. (106) because we have to subtract the SM bottom contribution. In Fig. 7, we plot the contours of with neglecting all the mixing parameters in the bottom sector. Due to the dependence on ratios of the mass scales, can be suppressed significantly. A phenomenological fit to current Higgs data at the LHC is left for future work.
V Conclusions
In this work we studied patterns of modifications in the Higgs couplings to third generation quarks and to two gluons in the minimal composite Higgs model. These three couplings play crucial roles in determining phenomenology of the Higgs boson at the LHC. We first presented a general framework for computing the aforementioned couplings by integrating out, at the leading order, partners of the third generation quarks. Then we applied the computation to three scenarios where the composite fermions are embedded in , and of . We also computed the contribution to the Coleman-Weinberg potential of the Higgs boson from the top sector and demonstrated a strong correlation between the and the couplings regions of parameter space where the electroweak symmetry breaking is triggered radiatively by the top sector.
Our findings are summarized in Table 1. The interesting patterns are
- •
the and couplings are always reduced relative to their SM expectations if the composite fermions are embedded in or , while these couplings can be either enhanced or suppressed in or .
- •
the coupling is always suppressed and independent of the mass scale of the top partner in . Such a pattern does not hold for embeddings in other representations.
- •
There exists strong correlations between and , assuming the top sector gives the dominant contribution to the Coleman-Weinberg potential of the Higgs boson. In regions of parameter space where the electroweak symmetry is triggered, is strongly preferred. Since the SM gauge boson contributions to the Higgs potential will always tend to preserve the electroweak symmetry, including them will make the preference even stronger.
These patterns could serve as diagnostic tools should a significant deviation appear in future Higgs measurements. In the absence of deviations, they can be used to potentially constrain the size of in the composite Higgs models.
It should be emphasized that we have only considered the minimal coset structure of . Obviously minimality is not always the best guideline when it comes to Nature. In particular, we have assumed through out this work that the dominant contribution to the Coleman-Weinberg potential arises from the top sector. It is conceivable that, if one introduces an additional contribution to the Higgs potential to trigger the EWSB, one could then weakened the strong correlation between and . One such possibility is to enlarge the coset structure to include an gauge boson, like in the original Georgi-Kaplan model in Refs. Kaplan:1983fs ; Kaplan:1983sm . We hope to return to such a scenario in the future.
Acknowledgements.
Work at University of Chicago is supported in part by U.S. Department of Energy grant number DE-FG02-13ER41958. Work at ANL is sup- ported in part by the U.S. Department of Energy under Contract No. DE-AC02-06CH11357. I.L. is supported in part by the U.S. Department of Energy under Contract No. DE-SC0010143.
Appendix A CCWZ for SO(5)/SO(4)
We present here our basis for the generators of the and generators, and review the CCWZ formalism for the coset. The generators are defined as:
[TABLE]
where and ,5. In Eq. (121) the () generators take the plus (minus) sign. The generators satisfy as well as the following commutation relations:
[TABLE]
The generators and correspond to the unbroken generators.
The Goldstone matrix is defined as , where the Goldstone fields are given by:
[TABLE]
where the factor is just a convention and can be absorbed by the redefinition of decay constant . The fourplet can be related with the doublet notation as follows:
[TABLE]
from which, we can see that the fourth component of will be our physical Higgs boson in the unitary gauge. The matrix transforms under the non-linearly realized as follows:
[TABLE]
where . The CCWZ covariant objects are defined as:
[TABLE]
which transform under the non-linearly realized :
[TABLE]
Note that transforms like a gauge field under . The leading two-derivative effective Lagrangian for the Goldstone bosons is
[TABLE]
where . By computing the boson mass we see 246 GeV. In addition the and couplings are given by
[TABLE]
where . For convenience we use a non-canonical basis for the gauge bosons, which are not relevant in our analysis.
Appendix B Partial Compositeness and The coupling
In this appendix, we review the general formula for the coupling induced by the composite resonances under the assumption of partial compositeness. We are working in the fermion mass eigenstates, whose interactions with Higgs are parametrized as (see Ref. Furlan:2011uq ; Azatov:2011qy ):
[TABLE]
where is the vacuum expectation value of the neutral Higgs boson and is a Hermitian matrix assumed to be real and symmetric. The partonic cross section for can be obtained by direct calculation Ellis:1975ap :
[TABLE]
where and fermion loop function is defined as
[TABLE]
and goes to 1 in the limit of :
[TABLE]
We define the coupling as:
[TABLE]
It is a very good approximation to include only heavy quarks with masses heavier than the Higgs boson in the loop function. In other words, the effective coupling is dominated by the sum over with Falkowski:2007hz :
[TABLE]
where we have used . Note that in our convention, the SM quark Yukawa coupling satisfies , where is the SM Higgs VEV. Since we work in the large top mass limit and neglect all the lighter quark contributions in the SM, we have . Throughout this work we define the ratio
[TABLE]
For the charge 2/3 particles including the top and assuming the mass matrix is real the result simply reads
[TABLE]
where and . For the charge -1/3 particles, we need to to extract the contribution from the SM bottom quark,
[TABLE]
Appendix C Effective Lagrangian After Integrating Out The Top Partners
In this appendix we present the formula for the effective Lagrangian obtained from integrating out the composite fields, for the SM quark (). The Lagrangian in the momentum space for the composite partners and the mixing terms can be generally written as:
[TABLE]
where we have neglected all gauge interactions. Here generally denotes the indices of the irreducible representations of the unbroken and is constructed with the Goldstone matrix and the elementary quark fields (). The effective Lagrangian after integrating out the top partners by using the equation of motion is simply:
[TABLE]
Note that for two-indices tensor representation of , the indices should be contracted as , which only have effects for anti-symmetric representation.
Appendix D
In this appendix, we discuss succinctly the case where the elementary quark fields are mixing with both and at the same time. We find there is a new invariant and, as a consequence, the and the quark Yukawa couplings are both dependent on the ratios of the composite scales as in the case of .
In the top sector, we will consider the scenario where mixes with composite resonances from both the and the , while only mixes with those in the . There are other possibilities for the mixing pattern which will not be pursued here. The effective Lagrangian is
[TABLE]
where the relevant projection operators are given in Eqs. (30) and (70) for and , respectively. Furthermore, is mixing angle between the 4-plet from the and the 4-plet from the . In addition to the invariants presented in Eq. (33) and Eq. (73), we have the following invariants constructed with embedding vectors :
[TABLE]
Now it is straightforward to calculate the form factors as before. After imposing the following condition for the cancellation of the quadratic divergence,
[TABLE]
the leading modifications to the and coupling strengths are given by:
[TABLE]
where the two sign in the are not correlated and we have also defined:
[TABLE]
Similar to the case of , there is enough degree of freedom such that both and can be positive or negative. It is not difficult to see that :
[TABLE]
In the bottom sector, we consider the possibilities that all elementary fields are ”uplifted” to multiplets with , which means should be uplifted to an vector with :
[TABLE]
and the embedding vectors for the left-handed fields are the same as the top sector. We consider the case that the left-handed quark doublet is mixing with both and . The invariants involving the right-handed bottom quarks are listed as follows:
[TABLE]
For simplicity we impose a similar condition to Eq. (146) to cancel the quadratic divergence in the bottom sector. The modification to the bottom Yukawa coupling is given by
[TABLE]
where the two signs are not correlated with each other and we have neglected the mixing angles in the bottom sector. Similar to the case of , there is enough degree of freedom to suppress bottom Yukawa coupling as needed.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1(1) ATLAS Collaboration, G. Aad et al. , “Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC,” Phys. Lett. B 716 (2012) 1–29 , ar Xiv:1207.7214 [hep-ex] . · doi ↗
- 2(2) CMS Collaboration, S. Chatrchyan et al. , “Observation of a new boson at a mass of 125 Ge V with the CMS experiment at the LHC,” Phys. Lett. B 716 (2012) 30–61 , ar Xiv:1207.7235 [hep-ex] . · doi ↗
- 3(3) ATLAS, CMS Collaboration, G. Aad et al. , “Measurements of the Higgs boson production and decay rates and constraints on its couplings from a combined ATLAS and CMS analysis of the LHC pp collision data at s = 7 𝑠 7 \sqrt{s}=7 and 8 Te V,” JHEP 08 (2016) 045 , ar Xiv:1606.02266 [hep-ex] . · doi ↗
- 4(4) S. P. Martin, “A Supersymmetry primer,” ar Xiv:hep-ph/9709356 [hep-ph] . [Adv. Ser. Direct. High Energy Phys.18,1(1998)].
- 5(5) J. F. Gunion and H. E. Haber, “Higgs Bosons in Supersymmetric Models. 1.,” Nucl. Phys. B 272 (1986) 1 . [Erratum: Nucl. Phys.B 402,567(1993)]. · doi ↗
- 6(6) M. Badziak and C. E. M. Wagner, “Enhancing the Higgs associated production with a top quark pair,” JHEP 05 (2016) 123 , ar Xiv:1602.06198 [hep-ph] . · doi ↗
- 7(7) M. Badziak and C. E. M. Wagner, “Enhanced Higgs associated production with a top quark pair in the NMSSM with light singlets,” ar Xiv:1611.02353 [hep-ph] .
- 8(8) H. Terazawa, K. Akama, and Y. Chikashige, “Unified Model of the Nambu-Jona-Lasinio Type for All Elementary Particle Forces,” Phys. Rev. D 15 (1977) 480 . · doi ↗