Higgs-flavon mixing and LHC phenomenology
in a simplified model of broken flavor symmetry
The LHC phenomenology of a low-scale gauged flavor symmetry model with inverted hierarchy is studied, through introduction of a simplified model of broken flavor symmetry. A new scalar (a flavon) and a new neutral top-philic massive gauge boson emerge with mass in the TeV range along with a new heavy fermion associated with the standard model top quark. After checking constraints from electroweak precision observables, we investigate the influence of the model on Higgs boson physics, notably on its production cross section and decay branching fractions. Limits on the flavon from heavy Higgs boson searches at the LHC at 7 and 8 TeV are presented. The branching fractions of the flavon are computed as a function of the flavon mass and the Higgs-flavon mixing angle. We also explore possible discovery of the flavon at 14 TeV, particularly via the decay channel in the final state, and through standard model Higgs boson pair production in the final state. We conclude that the flavon mass range up to GeV could probed down to quite small values of the Higgs-flavon mixing angle with 100 fb of integrated luminosity at 14 TeV.
The standard model (SM) of particle physics describes physics at the electroweak symmetry breaking (EWSB) scale of the visible sector remarkably well. With the discovery of a Higgs boson behaving much like that of the SM at the Large Hadron Collider (LHC) Aad et al. (2012); Chatrchyan et al. (2012), all of the expected SM particles have been detected. Attention has turned to precise determination of the properties of the Higgs boson, notably its decay branching fractions, and to the search for possible new physics beyond the SM. Given current precision, the branching fractions allow some, if limited, deviations from SM predictions.
The SM poses several puzzles. These include the origin of the fermion mass hierarchy and the flavor structure parametrized in the well-known CKM matrix Cabibbo (1963); Kobayashi and Maskawa (1973). The dynamics of flavor mixing is well described within a framework of three generations of quarks. The fact that no significant deviations from SM predictions have appeared in any flavor-related physics processes indicates that that any TeV scale new physics (NP) does not introduce any important new source of flavor change or CP violation beyond the SM. This hints at flavor symmetry (horizontal symmetry) in a NP model. The idea that the NP interactions are invariant under a flavor symmetry group is known as minimal flavor violation (MFV) D’Ambrosio et al. (2002).
In the MFV scenario, the SM flavor symmetry is broken explicitly by the non-vanishing SM Yukawa coupling constants. This symmetry could nevertheless be a true symmetry of nature at some high energy scale but broken by non-zero vacuum expectation values (vev’s) of scalar fields which are usually called flavons. In such a case, the SM Yukawa coupling constants are related to the ratio between the vev’s of flavons and some cutoff scale.
If the full non-abelian flavor symmetry of the fermion kinetic terms and gauge couplings is a global symmetry broken by flavons, this yields Goldstone bosons subject to stringent constraints. This problem can be avoided if the non-abelian flavor symmetry is gauged, giving mass to the Goldstone modes.111Other scenarios include abelian flavor symmetries Leurer et al. (1993, 1994) and discrete flavor symmetriesZwicky and Fischbacher (2009); these are not considered in this work. Anomaly cancellation in such a theory requires introduction of exotic fermionsBerezhiani and Chkareuli (1983); Berezhiani (1983); Berezhiani and Khlopov (1990); Grinstein et al. (2010); Feldmann (2011); Krnjaic and Stolarski (2013) to cancel gauge anomalies. These mix with the SM fermions. The SM fermion masses are the smaller eigenvalues of the mass matrix and are proportional to the inverse of the flavon vev’s, corresponding to an inverted hierarchy. The masses of all NP particles, such as the extra fermions, flavons, and flavor gauge bosons, are controlled by the flavon vev’s and therefore are approximately proportional to the inverse of the SM Yukawa constants. Constraints from low energy precision observables and flavor physics are carefully considered in Grinstein et al. (2010); Buras et al. (2012).
The lightest new particles in such a flavor symmetry model with inverted hierarchy are the exotic fermions, the flavon which couples to the third generation of SM fermions, and a massive top-philic gauge boson. Their masses could be at the TeV scale, and it should be possible to search for them at the LHC. The LHC phenomenology of the flavon, the top-philic gauge boson and the heavy fermion partner of the top-quark might be interestingly rich.
In this paper we do not focus on details of flavor physics per se. Rather, we address the implications of flavons and the heavy fermion partner of the top-quark for Higgs boson physics, and the LHC phenomenology of a simplified flavor symmetry model with inverted hierarchy. We begin in Sec. II with an explanation of the motivation and origin for the inverted hierarchy in a flavor symmetry model. The simplified Lagrangian and the mass eigenstates are shown in this section also. In Sec. III, we briefly review the constraints from electroweak precision observables (EWPO) and flavor violation experiments. We study the effects of the flavor symmetry model on the production and decay properties of the SM Higgs boson in Sec. IV. The inclusive Higgs boson production cross section is suppressed relative to the SM by a factor , where is the mixing angle of the scalar flavon and the Higgs boson. This suppression is allowed by the LHC data at 7 and 8 TeV, within limits. We show that most of the Higgs boson couplings to SM particles are just rescaled by a factor , including the loop induced and vertices in the heavy fermion limit. The vertex deviates from the simple rescaling, but the deviation is not huge. The Higgs boson decay branching ratios are nearly unchanged relative to the SM since every sizable partial width is changed by an overall factor . In Sec. V, we investigate limits on the flavon from LHC data at 7 and 8 TeV and possible signals of the flavon at 14 TeV. Flavon searches at the LHC can focus on the SM Higgs-like decay channels (, ) and on the Higgs boson pair decay channel . We compute and display the decay branching fractions of the flavon as a function of the flavon mass and mixing angle . Only the and channels are significant in flavon decay. We examine bounds on the parameter space of flavons from heavy Higgs boson searches at 7 and 8 TeV. Because the flavon can be produced singly, if it decays into the final state with an appreciable decay branching ratio, the Higgs pair cross section will be enhanced significantly by this resonance effect. We perform a detailed simulation of the signal and backgrounds for the channel at 14 TeV for an assumed integrated luminosity of , deriving both standard deviation exclusion limits and standard deviation observation bounds as a function of flavon mass. In some regions of parameter space the search for a signal will give a stronger constraint on the NP model than the channel. Our conclusions are summarized in Sec. VI.
Ii From gauged flavor symmetry to a simplified model of broken flavor symmetry
The flavor symmetry of the quark kinetic terms and gauge couplings is
If this symmetry is gauged with only SM fermions present, the theory is anomalous. A “minimal” model of new fermions that cancel the anomalies was described in Grinstein et al. (2010). This model has exotic fermion partners of the SM quarks, flavor gauge bosons, and two scalar flavon fields and for the up-like and down-like quarks. remains anomalous, but the rest of the flavor symmetry (1) is taken to be gauged. The most general renormalizable interaction Lagrangian between the flavon fields and the SM and exotic fermions takes the form
Here , , are the SM quark fields, , , , and are the partner fermion fields, is the SM Higgs doublet field, and where is the anti-symmetric tensor with . and are dimensionless parameters, is a parameter with the dimensions of mass, and is the scalar potential. The representations under the gauge groups to which these these fields belong is shown in TABLE 1. One can verify that both the SM gauge symmetry and the flavor gauge symmetry are anomaly free with the contributions from the exotic fermion fields. If flavor symmetry breaks via flavon vevs with , the masses of the SM fermions are inversely proportional to the vev of the corresponding flavon field component, resulting in the asserted inverted hierarchy.
The large hierarchy between the masses of the SM quarks thus corresponds to a large hierarchy between the vevs of the flavons which suggests that the flavor symmetry could be broken sequentially Feldmann et al. (2009). Guided by this realization, we assume that the breaking of the Lagrangian in Eq (2) occurs in a sequence of steps. From the effective field theory point of view, one successively integrates out heavy degrees of freedom.
If we integrate out the heavy degrees of freedom associated with the first and second generations, we are left with a simplified flavor symmetry model with a manageable number of BSM degrees of freedom (a flavon, an exotic fermion, and a massive vector boson) associated with the top-quark and bottom-quark sectors. However, because the vev of the flavon associated with the bottom-quark is nearly two orders of magnitude larger than the vev of the flavon associated with the top-quark, this suggests finally integrating out the flavon associated with the bottom-quark. Thus, at the TeV scale, we have the effective Lagrangian
where is a complex flavon associated with the top-quark. There is also a residual gauged flavor symmetry under which only the , and fields carry (the same) charge. Chiral phase rotations of the fermions allow us to take, without loss of generality, . We can consider the Lagrangian (3) separately from our discussion of the higher-scale flavor structure, as a simplified model extending the top and Higgs sectors. The usual SM Yukawa interactions for the remaining fermions are added to this Lagrangian.222There are other models with similar Lagrangians, although arising from different motivations. For example, see Xiao and Yu (2014); He and Xianyu (2014).
After EWSB and FSB,
in the unitary gauge, where GeV is the vev of the Higgs field, is the physical degree of freedom of the SM Higgs doublet field, and is the physical degree of freedom of the top flavon. The mass eigenstates are linear combinations of and which will be given below.
The mass matrix of the fermions is
It can be diagonalized by separate left and right rotations. This results in mass eigenvalues given by
which are positive and real for real . Here should be the running mass of the top quark which is 163 GeV Alekhin et al. (2012). One can find in terms of the other parameters, if is fixed to be the mass of the SM top quark. Reality of requires
The case and would correspond to the SM top quark being the heavier fermion; we do not treat this scenario because a light colored fermion which also couples to the electroweak gauge boson and the Higgs boson would be highly constrained by current data. Thus, will denote the heavy partner of the top quark.
When and , the other mass eigenvalue is
The right and left components of the two fermion mass eigenstates, the SM top quark and a heavy fermion , are
It is easy to derive
The Yukawa interactions are therefore
while the gauge interactions are
where () is the sine (cosine) of the weak angle.
In the scalar potential, the trilinear interaction between and is forbidden by the and flavor symmetries. However, the interaction
is still allowed. This term can be generated through a combined top-quark and heavy fermion loop in the one-loop effective potential even it is forbidden artificially at tree-level (FIG. 1).
The general renormalizable Lagrangian of the scalar fields of the complex gauge singlet extension of the SM can be written as
with scalar potential
the parameters and describe the self-interactions of the Higgs field and the flavon field. The gauge covariant derivative of the SM doublet Higgs field is the same as the one in the SM. The gauge covariant derivative of the flavon is
where is the gauge coupling constant of the residual flavor gauge symmetry, and is the gauge field of the residual flavor gauge symmetry.
The vev’s of neutral components are found to be
Avoidance of a flat direction of the vacuum requires
The vev of the doublet field is determined by the weak interaction coupling constant and the masses of the SM massive gauge bosons. Therefore there is a constraint
The physical degree of freedom which is dominated by should be the SM-like Higgs boson . Its mass should be GeV CMS (2013a); Aad et al. (2014a). Assuming the other scalar field mass eigenstate has mass , we can solve for the parameters and , and present the results in terms of and , where the first two parameters are determined by current experiments. We can also determine using these parameters as
We define the mass eigenstates of the scalar fields by
where the rotation angle is given by
The deviation of the Higgs field self-interaction strength from its value in the SM can be written as
The additional massive gauge boson , whose mass is , couples at tree-level only to the SM top-quark, the heavy fermion, the flavon, and the Higgs boson through
Searching for such a top-philic gauge boson is a challenging task at colliders when it does not mix with the SM at tree-level Chiang et al. (2007); Chen and Okada (2008); Jackson et al. (2010); Hsieh et al. (2010); Berger et al. (2011).
In summary, this section has presented a simplified model of spontaneous flavor symmetry breaking, which arises from the the gauged flavor symmetry model with inverted hierarchy in which only degrees of freedom related to the third generation are considered. There is a heavy fermion which mixes with the SM top-quark, a heavy scalar flavon which mixes with the SM-like Higgs boson, and a heavy top-philic vector boson . The basic couplings that we will need have been presented in this section.
Iii Constraints from the electroweak precision observables
The interactions between the fermions and the SM gauge bosons are different in the low-scale gauged flavor symmetry model from the SM interactions. The new scalar also couples to the SM gauge bosons such that the strength of these interactions should be constrained by SM electroweak precision observables. The modification of the charged current will also change the prediction of . Constraints from the EWPO and flavor physics were considered in the original paper Grinstein et al. (2010). In this work, we rexamine the EWPO constraints and include the contribution from the flavon in our calculation.
Because the new physics effects occur above the -pole, their influence on the SM EWPO can be described with the well known oblique parameters , , Peskin and Takeuchi (1990). When the SM reference values of and are chosen to be
the best fit values of the oblique parameters are Baak et al. (2012)
while the correlation coefficients matrix is Baak et al. (2012)
The contribution from the exotic real scalar boson to the oblique parameters Barger et al. (2008) is suppressed by the mixing angle . The contribution from the third generation is given in Grinstein et al. (2010). Using a check, we show the one standard deviation (1), 2 and 3 fit regions in FIG. 2. A detailed analysis of flavor physics observables and of in this model is presented in Ref. Buras et al. (2012). After inclusion of the contributions from the gauge bosons of the flavor group, this model could resolve the tension but result in a more serious tension from and than in the SM. Contributions from the flavons are not included. Interested readers can find the constraints in Ref. Buras et al. (2012).
Iv Higgs Boson Physics
In this section, we investigate Higgs boson physics in the simplified flavor symmetry model under consideration. The interactions between the SM-like Higgs boson and other SM particles are different from those in the pure SM. The differences have two origins. First, there is mixing between the doublet and the flavon. Second, there is sizable mixing between the SM top-quark and the heavy fermion .
Gluon fusion is the most important production channel of the SM Higgs boson at the LHC. In the NP model, the interaction between the SM-like Higgs boson and the gluon is mediated by both the SM top-quark and the heavy fermion . Denoting the SM and the NP interactions as
respectively, we have
where , and
where . The ratio is nearly independent of and in the 3 fit region from EWPO. In the limit of large fermion mass, we obtain
The interaction is also modified in the NP model. The contributions from the light fermions are highly suppressed by the fermion mass, so we consider only the contribution from , and . Denoting the SM and the new physics (NP) interaction as
, and are the color factor and charge of the top-quark. In the limit of large fermion mass,
is a very good numerical approximation for this model.
Because does not couple to the SM fermions except the top-quark, all of the other coupling strengths are rescaled by a factor of . Therefore the NP effects will not change the SM-like Higgs boson decay branching ratios, but they will change the production cross section. The gluon-gluon fusion channel, vector boson fusion (VBF) channel, and the vector boson associated production (VH) channel are all suppressed by a factor of .
For the interaction, the ratio between the coupling constant and the top-quark Yukawa coupling constant in the SM is
It will deviate from by a small amount, but the production channel has a much smaller cross section than the other three channels.
The results from a fit of the Higgs boson inclusive cross section by the CMS collaboration CMS (2013a) is
The result from the ATLAS collaboration
would exclude most of the parameter space of the NP model ATL (2014a). However, at the 3 C.L., the region is still allowed.
Although the and decay branching ratios are not changed in this NP model, owing to the universal rescaling factor of and ( is forbidden because ), it is worth checking the decay branching ratio. In the NP model, there are additional contributions from both the -loop and the -loop (FIG. 3).
The contributions from the -loop and the -loop can be read out from the rescaling of the SM amplitude. The additional contribution from the Feynman diagrams shown in FIG. 3 must be calculated independently. The effective operator can be written as , where and are the field strengths of the and the electromagnetic field, respectively. The partial decay width of the Higgs boson is
In the SM, there are contributions from the fermion loops and the loop. The contribution from the top-quark loop is
where , and are the standard Pasarino-Veltman functions. We use the coupling constant in the SM. The contributions from the pure top and loops in the NP case are
The new contributions from the mixing loops are
The analytic formulas of the Passarino-Veltman functions can be found in ’t Hooft and Veltman (1979); Passarino and Veltman (1979). The correction from NP is comparable to the contribution from the SM fermion loops. Because the partial width is not rescaled by just but has a more complicated behavior, the branching ratio is changed by the NP. We show the ratio between the in the NP and the SM in FIG. 4.
As seen in FIG. 4, the NP contribution increases . When the and parameters satisfy the SM EWPO at 3 C.L., the correction to is small.
Last but not least, an important question is how the SM Higgs pair-production cross section Shao et al. (2013) is changed in this model of NP. There are two sources of change. The first is from flavon decay. The flavon can be produced singly at the LHC. If it decays into the final state with a sizable decay branching ratio, the cross section will be enhanced significantly owing to the resonance. The second source comes from corrections to the and vertices. We leave flavon production and decay to the next section but discuss the modifications of the vertices here.