The decoupling limit in the Georgi-Machacek model
We study the most general scalar potential of the Georgi-Machacek model, which adds isospin-triplet scalars to the Standard Model (SM) in a way that preserves custodial SU(2) symmetry. We show that this model possesses a decoupling limit, in which the predominantly-triplet states become heavy and degenerate while the couplings of the remaining light neutral scalar approach those of the SM Higgs boson. We find that the SM-like Higgs boson couplings to fermion pairs and gauge boson pairs can deviate from their SM values by corrections as large as , where is the SM Higgs vacuum expectation value and is the mass scale of the predominantly-triplet states. In particular, the SM-like Higgs boson couplings to and boson pairs can decouple much more slowly than in two Higgs doublet models, in which they deviate from their SM values like . Furthermore, near the decoupling limit the SM-like Higgs boson couplings to and pairs are always larger than their SM values, which cannot occur in two Higgs doublet models. As such, a precision measurement of Higgs couplings to and pairs may provide an effective method of distinguishing the Georgi-Machacek model from two Higgs doublet models. Using numerical scans, we show that the coupling deviations can reach 10% for as large as 800 GeV.
The recent discovery Aad:2012tfa () of a Standard Model (SM)-like Higgs boson at the CERN Large Hadron Collider (LHC) has focused much theoretical and experimental attention on possible extensions of the SM Higgs sector. One such extension is the Georgi-Machacek (GM) model Georgi:1985nv (), which adds isospin triplets to the SM Higgs sector in a way that preserves the SM prediction at tree level. The GM model is less theoretically attractive than Higgs-sector extensions involving isospin doublets or singlets because, in the GM model, the custodial symmetry that ensures at tree level is violated by hypercharge interactions. This leads to divergent radiative corrections to at the one-loop level Gunion:1990dt (), implying a relatively low cutoff scale (which would be needed in any case to solve the hierarchy problem).
On the other hand, the GM model is phenomenologically attractive due to two features not present in Higgs sector extensions containing only isospin doublets or singlets. These are the possibility that the SM-like Higgs boson has couplings to and pairs that are larger than predicted in the SM and the presence of a doubly-charged scalar ; in fact, these features are linked insofar as the doubly-charged scalar plays an important role in the unitarization of longitudinal scattering amplitudes when the SM-like Higgs boson coupling to pairs is enhanced relative to that in the SM Falkowski:2012vh (). This makes the GM model a valuable benchmark for studies of Higgs properties and searches for additional scalars beyond the SM, and its phenomenology has been extensively studied in the literature Chanowitz:1985ug (); Gunion:1989ci (); HHG (); Haber:1999zh (); Aoki:2007ah (); Godfrey:2010qb (); Low:2010jp (); Logan:2010en (); Chang:2012gn (); Chiang:2012cn (); Chiang:2013rua (); Kanemura:2013mc (); Englert:2013zpa (); Killick:2013mya (); Englert:2013wga (); Efrati:2014uta (). The GM model has also been incorporated into little Higgs Chang:2003un (); Chang:2003zn () and supersymmetric Cort:2013foa () models, and extensions with additional isospin doublets Hedri:2013wea () have also been considered.
Our objective in this paper is to study the approach to the decoupling limit Haber:1994mt () in the GM model. In the decoupling limit, all additional particles beyond those present in the SM become heavy and the couplings of the SM-like Higgs boson approach their SM values. This limit is of interest in the scenario that future measurements of the couplings of the SM-like Higgs boson at the LHC do not reveal large deviations from the SM expectations.
The scalar potential for the GM model first written down in Ref. Chanowitz:1985ug () and used throughout most of the literature Gunion:1990dt (); Gunion:1989ci (); Chang:2012gn (); Englert:2013zpa (); Efrati:2014uta () imposed a symmetry on the scalar triplets in order to simplify the form of the potential. Under this constraint, the potential depends on only two dimensionful parameters which can be eliminated in favor of the vacuum expectation values (vevs) of the doublet and triplet scalars after electroweak symmetry breaking. This implies that all the scalar masses can be written in the form , where GeV is the SM Higgs vev and represents some linear combination of the scalar quartic couplings. Because the sizes of the scalar quartic couplings are bounded by the requirement of tree-level perturbative unitarity to be , the masses of all the scalars in the GM model are bounded to be less than about 700 GeV Aoki:2007ah (); in particular, the -symmetric version of the GM model does not possess a decoupling limit.
In this paper we study the most general gauge- and custodial SU(2)-invariant tree level scalar potential of the GM model without imposing a symmetry. This full potential was first written down in Ref. Aoki:2007ah () and to our knowledge has been used for phenomenology only in Refs. Chiang:2012cn (); Chiang:2013rua (). The full potential contains two additional dimension-3 operators beyond those present in the -symmetric version, providing two additional dimensionful parameters that can drive a decoupling limit. We show that such a decoupling limit indeed exists and explore its phenomenology. In particular, we show that in the decoupling limit , (i) the additional scalars beyond the light SM-like Higgs boson become heavy with masses and increasingly degenerate with mass splittings ; and (ii) the tree-level couplings of the light SM-like Higgs boson to fermion pairs and or boson pairs, as well as the loop-induced couplings of to photon pairs or which receive contributions from the new charged scalars in the loop, deviate from the corresponding SM Higgs couplings by a relative correction of at most .
Our most interesting result is that, depending on how the decoupling limit is taken, the deviation of the coupling to or boson pairs can decouple as , in contrast to the situation in two Higgs doublet models or the Minimal Supersymmetric Standard Model in which this deviation vanishes as Gunion:2002zf (). Furthermore, near the decoupling limit the and couplings are always larger than their SM values, a phenomenon which cannot be achieved at tree level in models containing only scalar doublets or singlets. A precision measurement of the Higgs coupling to or boson pairs is thus extremely interesting in the GM model, and may provide the first evidence for scalars transforming under SU(2) as representations larger than doublets.
This paper is organized as follows. In Sec. II we write down the most general scalar potential and the resulting scalar mass eigenstates. In Sec. III we summarize the theoretical constraints on the model parameters from perturbative unitarity, boundedness-from-below of the scalar potential, and the avoidance of custodial SU(2)-breaking vacua. In Sec. IV we examine the approach to the decoupling limit and discuss the decoupling behavior of the couplings of the SM-like Higgs boson. We also compare the decoupling behavior to that in the two-Higgs-doublet model and scan over the GM model parameter space in order to evaluate the allowed ranges of couplings of the SM-like Higgs boson as a function of the masses of the heavier scalars. We conclude in Sec. V. Feynman rules, formulas for Higgs decays to and , and a translation table for the alternative parameterizations of the scalar potential used in the literature are collected in the appendices.
Ii The model
The scalar sector of the Georgi-Machacek model consists of the usual complex doublet with hypercharge111We use . , a real triplet with , and a complex triplet with . The doublet is responsible for the fermion masses as in the SM. In order to make the global SU(2)SU(2) symmetry explicit, we write the doublet in the form of a bi-doublet and combine the triplets to form a bi-triplet :
The vevs are defined by and , where the and boson masses constrain
Note that the two triplet fields and must obtain the same vev in order to preserve custodial SU(2). Furthermore we will decompose the neutral fields into real and imaginary parts according to
where we note that is already a real field.
The most general gauge-invariant scalar potential involving these fields that conserves custodial SU(2) is given by222Several different parameterizations of the scalar potential of the Georgi-Machacek model exist in the literature. We give a translation table in Appendix C.
Here the SU(2) generators for the doublet representation are with being the Pauli matrices, the generators for the triplet representation are
and the matrix , which rotates into the Cartesian basis, is given by Aoki:2007ah ()
We note that all the operators in Eq. (5) are manifestly Hermitian, so that the parameters in the scalar potential must all be real. Explicit CP violation is thus not possible in the Georgi-Machacek model.
In terms of the vevs, the scalar potential is given by333We will discuss the conditions required to avoid alternative minima in Sec. III.3.
Minimizing this potential yields the following constraints:
Inserting [Eq. (3)] into Eq. (10) yields a cubic equation for in terms of , , , , , , , and . With (and hence ) in hand, Eq. (9) can be used to eliminate in terms of the parameters in the previous sentence together with . We illustrate below how can also be eliminated in favor of one of the custodial singlet Higgs masses or [see Eq. (22)].
The physical field content is as follows. When expanded around the minimum, the scalar potential gives rise to ten real physical fields together with three Goldstone bosons. The Goldstone bosons are given by
The physical fields can be organized by their transformation properties under the custodial SU(2) symmetry into a fiveplet, a triplet, and two singlets. The fiveplet and triplet states are given by
Note that the ratio is finite in the limit , as can be seen from Eq. (10) which yields
The two custodial SU(2) singlets are given in the gauge basis by
These states mix by an angle to form the two custodial-singlet mass eigenstates and , defined such that :
The mixing is controlled by the mass-squared matrix
The mixing angle is fixed by
with the masses given by
It is convenient to use the measured mass of the observed SM-like Higgs boson as an input parameter. The coupling can be eliminated in favor of this mass by inverting Eq. (21):
Note that in deriving this expression for , the distinction between and is lost. This means that, depending on the values of and the other parameters, this (unique) solution for will correspond to either the lighter or the heavier custodial singlet having a mass equal to the observed SM-like Higgs mass.
Iii Theoretical Constraints on Lagrangian Parameters
iii.1 Perturbative unitarity of scalar field scattering amplitudes
Perturbative unitarity of scalar field scattering amplitudes requires that the zeroth partial wave amplitude, , satisfy or . Because the scalar field scattering amplitudes are real at tree level, we adopt the second, more stringent, constraint. The partial wave amplitude is related to the matrix element of the process by
where is the (orbital) angular momentum and are the Legendre polynomials. We will use this to constrain the magnitudes of the scalar quartic couplings . These unitarity bounds for the scalar quartic couplings in the GM model were previously computed in Ref. Aoki:2007ah (). We have recomputed them independently and agree with the results of Ref. Aoki:2007ah ().
We work in the high energy limit, in which the only tree-level diagrams that contribute to scalar scattering are those involving the four-point scalar couplings; all diagrams involving scalar propagators are suppressed by the square of the collision energy. Thus the dimensionful couplings , , , and are not constrained directly by perturbative unitarity. In the high energy limit we can ignore electroweak symmetry breaking and include the Goldstone bosons as physical fields (this is equivalent to including scattering processes involving longitudinally polarized and bosons). We neglect scattering processes involving transversely polarized gauge bosons or fermions.
Under these conditions, only the zeroth partial wave amplitude contributes to , so that the constraint corresponds to . This condition must be applied to each of the eigenvalues of the coupled-channel scattering matrix including each possible combination of two scalar fields in the initial and final states. Because the scalar potential is invariant under SU(2)U(1), the scattering processes preserve electric charge and hypercharge and can be conveniently classified by the total electric charge and hypercharge of the incoming and outgoing states. We include a symmetry factor of for each pair of identical particles in the initial and final states. The basis states and resulting eigenvalues of are summarized in Table 1.
The eigenvalues of comprise the following independent combinations of (defined in the same way as in Ref. Aoki:2007ah ()):444Our notation for the is different from that of Ref. Aoki:2007ah (). This has been taken into account in the definitions of and in Eq. (24). A translation between our notation and that of Ref. Aoki:2007ah () is given in Appendix C.
Requiring imposes the conditions and , which must all be simultaneously satisfied.555The unitarity constraint imposed in Ref. Chiang:2012cn () corresponds to , which is obtained by requiring rather than our more stringent constraint .
These conditions allow us to determine the maximum range allowed by unitarity for each of the parameters , which will be useful for setting up numerical parameter scans. We first note that the conditions take the general form , which can be rewritten without loss of generality as . This equation describes the region bounded by a pair of cones with apices at that meet at a unit circle in the – plane. Clearly, then, the maximum allowed range of (i.e., or ) is obtained by setting , and the maximum allowed range in the – plane is obtained by setting .
The coupling is constrained by the unitarity conditions on and . The least stringent constraints come from setting and read from and from . We thus obtain the maximum range from unitarity,
Constraints on the couplings and come from the unitarity conditions on , , , and . These are shown in the left panel of Fig. 1, where we again take in and in for the least stringent constraints. The allowed region in the – plane is a six-sided region bounded by the constraints on , , and . The constraint on does not provide any additional information. Simultaneously satisfying all constraints, we obtain the maximum ranges from unitarity,
Constraints on the couplings and come from the unitarity constraints on , , , , and . These are shown in the right panel of Fig. 1, where we take in and for the least stringent constraints. (The constraint from yields , which corresponds to the left and right edges of the plot.) The allowed region in the – plane is a parallelogram bounded by the constraints on and . The constraints on and do not provide any additional information. Simultaneously satisfying all constraints, we obtain the maximum ranges from unitarity,
Within these maximum ranges the unitarity constraints must still be imposed. Discarding expressions that provide no additional information, we obtain the minimal set of unitarity conditions,666Imposing instead of would double the right-hand side of each of these expressions.
iii.2 Bounded-from-below requirement on the scalar potential
The constraints that must be satisfied at tree level for the scalar potential to be bounded from below can be determined by considering only the terms in the scalar potential [Eq. (5)] that are quartic in the fields, because these terms dominate at large field values. Following the approach of Ref. Arhrib:2011uy (), we parametrize the potential using the following definitions:
Scanning all possible field values yields the parameter ranges
The ranges of and will be discussed in more detail below.
The quartic terms in the potential are given in this parametrization by,
The potential will be bounded from below if the expression multiplying in Eq. (31) is always positive. The expression in the square brackets in Eq. (31) is a bi-quadratic in of the form . Such an expression is positive for all values of when
We thus obtain the bounded-from-below conditions,
These conditions must be satisfied for all allowed values of and .
The field combination is given explicitly by
To derive the allowed range of , we can work in a basis where the Hermitian matrix is diagonalized with positive real eigenvalues , and . In this basis,
from which it follows (using ) that .777Our desired vacuum, with , corresponds to .
To derive the allowed range of , we can start by choosing the SU(2) basis so that the field value of lies entirely in the real neutral component, . Then,
Inserting this into the expression for in Eq. (29) yields
Because is invariant under custodial SU(2), this expression can be rewritten in terms of custodial SU(2) eigenstates as follows. We first define the custodial singlet, triplet, and fiveplet contained in according to
In terms of the custodial symmetry eigenstates, we have
where and , and
From this form it can be easily seen that .888Our desired vacuum, with , corresponds to .
The region in the plane populated by taking all possible combinations of field values is shown in Fig. 2. For a given , the region encompasses , where999These bounds on are obtained by noting that the curved part of the boundary in Fig. 2 is traced out by field combinations in which only and are nonzero. Taking into account the normalization by , the formulas for and along this boundary can then be expressed as functions of a single variable, which can in turn be expressed in terms of .
Following Ref. Arhrib:2011uy (), the monotonic dependence on and in Eq. (33) can be used to obtain the following bounded-from-below constraints:101010Reference Chiang:2012cn () computed the bounded-from-below constraints taking into account all combinations of two nonzero scalar fields. Because our treatment allows any number of the scalar fields to be nonzero, our bounded-from-below constraints are more stringent than those of Ref. Chiang:2012cn ().