Probing the nature of the Higgs-like Boson via h ->VF decays

We give a general decomposition of the h ->VF amplitude where V={W,Z} and F is a generic leptonic or hadronic final state, in the standard model (SM), and in the context of a general effective field theory. The differential distributions for F=l^+l^-, l nu (l =e, mu) are reported, and we show how such distributions can be used to determine modified Higgs couplings that cannot be directly extracted from a global fit to Higgs signal strengths. We also demonstrate how rare h ->VP decays, where P is a single hadron, with SM rates in the 10^{-5} range, can be used to provide complementary information on the couplings of the newly discovered Higgs-like scalar and are an interesting probe of the vacuum structure of the theory.

I. Introduction: Characterizing the properties of the newly discovered scalar boson at the LHC [1] is of central importance to determine experimentally the nature of electroweak symmetry breaking, and to investigate the possibility of physics beyond the Standard Model (SM). It is particularly important to determine how the new boson couples to the SU (2) L × U (1) Y gauge fields, since these couplings are directly related to symmetry breaking and gauge boson mass generation. The 125 GeV boson cannot decay into an on-shell pair of massive gauge bosons, but it can decay via h → V V * , V * → F, where one of the bosons is off-shell.
The purpose of this paper is to show that the offshellness of V can be viewed as a virtue in Higgs studies. It allows one to measure decays to final states F that would not be accessible if on-shell decays were allowed, and the additional decay channels increase the sensitivity to new-physics (NP) effects, as they affect kinematic distributions of F in addition to the total rate. We demonstrate this conclusion using two examples: (a) F is a pair of light leptons + − or ν, with = e, µ, and (b) F = P is a hadronic state composed of a single pseudoscalar or vector meson.
In the two-lepton case, most of the interesting information is encoded in the two-dimensional kinematical distributions of the leptons in the Higgs rest frame. We analyze such distributions both in the SM, and in the context of a general effective field theory (EFT) approach, neglecting lepton masses. We show that these distributions, which will soon be accessible at the LHC with increasing statistics, contain information that cannot be directly extracted from a global fit to signal strengths. In the h → V P case we show how these rare processes, with SM rates in the 10 −5 range, can provide a complementary tool to extract Higgs properties not easily accessible from the purely leptonic modes.
II. Amplitude Decomposition: Consider the h → V F amplitude where V = {W ± , Z 0 } is an on-shell massive weak gauge boson while F is a final state generated at tree level by the electroweak charged or neutral cur-rents, can be decomposed in terms of four independent Lorentz structures, which we define as Here g V = {g 2 , g 2 / cos θ W }, g 2 = e/ sin θ W , C V = {1/ √ 2, 1} are the coupling and normalization factors for V = {W ± , Z 0 }. Throughout this paper p will denote the V four-momentum and q the total four-momentum of the final state F. In writing Eq. (2), we have used p · ε = 0 for physical V bosons but we have not made any assumption about the angular momentum of the F state. We will also use the dimensionless variables = 0 violate CP . In general, the f i (q 2 ) are four independent dimensionless form factors. At q 2 = m 2 V , two of them satisfy the relation dictated by the requirement that the pole of the amplitude when q 2 → m 2 V corresponds to the exchange of an on-shell V . In the SM, f SM The different form-factors can be probed by using different final states. The differential decay rate summing over V polarizations is where λ(q 2 , ρ) = (1 +q 2 − ρ) 2 − 4q 2 and denotes the phase space of the final state F. In general, the tensor structure M µν depends on the formfactors f V i (q 2 ), but it is simplified for specific choices of F. For the two-lepton final state the currrent is conserved, q µ J F µ = 0, when the lepton masses are neglected, and M µν becomes where In the case of a single pseudoscalar meson, the current assumes the form J F µ ∝ q µ and Some information on the h decay amplitude is lost when summing over V polarizations. This information is essential to determine the spin and parity of h (see e.g. Ref. [2]) but is less relevant once we assume h to be a 0 + state, as in most realistic NP models.
III. Modifications of the Spectra: In order to investigate how the f i (q 2 ) can vary in possible extensions of the SM, we consider a general EFT that contains the SM scalar sector in the low energy theory. The EFT includes explicitly the Goldstone bosons associated with the breaking of SU (2) × U (1) Y → U (1) Q , as well as the remaining SM field content with a nonlinear realization of the SU (2) × U (1) Y symmetry and a singlet scalar field h [3][4][5][6][7]. The Goldstone bosons eaten by the W ± , Z bosons are denoted by π a where a = 1, 2, 3, and are grouped as Σ(x) = exp[iτ a π a /v]. The operators that are of interest in this work are Here c i are unknown Wilson coefficients, with c SM i = (1, 0, 0). The complete operator basis to sub-leading order for theories of this form is given in Ref. [8,9].
The subleading operators in Eq. (8) are chosen as they do not violate custodial symmetry at g 1 = 0; this simplifying choice is made to demonstrate the utility of the differential spectra. For simplicity we have also neglected possible NP effects giving rise to local operators coupling h, V and the fermonic currents directly, or modifying the currents themselves. These effects could still be described by the general decomposition in Eq. (2), but with contributions to the amplitude that will not have a pole as q 2 → m 2 V , and can contain non-universal (F-depedent) form factors.
We find that the projection of the custodial symmetry preserving operator basis to the form factor basis is The three parity-conserving form factors correspond to three independent combinations of the parameters of the EFT Lagrangian, and only one combination is determined by the total decay rate. The dependence of the differential rate on f i (q 2 ) in Eqs. (6)-(7) offers the opportunity to determine the individual form-factors, and hence the independent operator coefficients with sufficient data.
IV. F = + − , ν: There are two kinematic variables needed to describe the final state, after averaging over lepton spins. Two convenient choices are either the two lepton energies in the h rest frame (E 1,2 ), or the lepton invariant mass q 2 and c θ ≡ cos θ, where θ is the angle between the lepton axis in the dilepton rest frame and the Higgs momentum. For these two cases we can write (y i = 2E i /m h , with i = 1 for the lepton and i = 2 for the antilepton) Neglecting lepton masses, the term between square brackets can be evaluated from Eq. (6) using where The general expression of the double differential energy distribution can be deduced from Eqs. (4)- (11).
In the SM, f 1 = 1, f 3,4 = 0, and the double differential rate is where The usefulness of these differential distribution is illustrated in Fig. 1, where we compare different spectra, with c i chosen to leave the total rate unchanged. 1 The dΓ/dq 2 spectrum exhibits large shape variations, which can be directly probed experimentally, and used to constrain the EFT. The variation in the q 2 spectrum is due to a modified weighting of the terms in Eq. (11), which have a different q 2 dependence.
On the other hand, the dependence of the lepton energy spectrum dΓ/dy 1 on c i is much weaker. Integrating over y 2 averages over a wide range ofq 2 . As a result the shape of the dΓ/dy 1 distribution is almost universal, even in presence of the NP effects in the EFT we consider. This stability of dΓ/dy 1 does not make it uninteresting -it provides a check for reconstruction errors or unaccounted for experimental systematics. The area under the curve of this spectrum is the total decay rate, 1 A detailed analysis of the present experimental constraints on the c i is beyond the scope of this work. However, we note that in an EFT with a linear realization of SU (2) L × U (1) Y in the scalar sector, it has been shown that current experimental bounds on the Wilson coefficients still allow a variation of the spectra on the order of ∼ 20% [11], similar to the variation shown in Fig. 1. and deviates from the SM value while maintaining this common shape. We have examined the possibility of using moments of the lepton energy and q 2 spectra to extract the Wilson coefficients of the operators in the EFT. However, these moments depend weakly on the c i . A more promising observable is the asymmetry A (either differential or integrated) given by weighting d 2 Γ/dy 1 dy 2 by sign(y 2 −ȳ 2 ) whereȳ 2 = (1 − ρ − y 1 )(1 − y 1 /2)/(1 − y 1 ) is the midpoint of the y 2 integration range. A is very sensitive to modifications of the form factors. In the SM, the integrated asymmetry is A = 15%, but it ranges from −9% to +27% for the illustrative EFT parameters adopted in Fig. 1.
V. Mesonic decays: The h → V P amplitude depends on the current matrix element where F P is the pseudoscalar meson decay constant. This current selects a single form-factor combination which has no pole as q 2 → m 2 V , by Eq. (3). In the SM, one has which is independent of g V (here v = 246 GeV). It is instructive to look at the structure of the amplitude for the h → V P process, This amplitude probes the ratio of two order parameters which both break SU (2) × U (1) in the SM, F P from the quark condensate and v from the Higgs sector. It provides a very interesting probe of the vacuum structure of the theory. The values of ci have been chosen for illustrative purposes, and are a bit larger than expected from EFT power counting. NDA [10] indicates that c2,3 ∼ O(v 2 /Λ 2 ) and c1 − 1 ∼ O(v 2 /Λ 2 ). The left plots are for mV = mZ , and the right plots are convoluted with a Breit-Wigner distribution of width ΓZ over the mass range mZ ± 10 GeV.
Compared to the leading decay mode of a light Higgs, h → bb, the h → V P decay amplitude is parametrically suppressed by the small ratio F P /m b . In the limit where stone component of the Yukawa interaction (as manifest from the gauge-less limit of the theory, see e.g. [12]). Computing the h → V P amplitude in the g → 0 limit treating V as an external field shows that the h → V P amplitude probes the trilinear h ∂µϕ V µ coupling, where ϕ is a (eaten) Goldstone boson.
we neglect the mass of the pseudoscalar meson, . This expression holds both for V = Z and for V = W ± , separately for each sign of charge.
Given the normalization of the currents in Eq. (1), the explicit expressions of F P in some of the most interesting modes are F π ± = V ud f π ,  [13][14][15]. From these values we deduce the SM rates reported in Table I. Despite the smallness of these rates, and the huge background at the LHC, we stress that some channels may have a relatively clean experimental signature, due to the displaced vertex of the subsequent P decay. Within the general EFT approach the h → V P rate assumes the following form with possible O(1) variations with respect to the SM. These variations are closely related to the possible variation of the dΓ(h → V )/dq 2 spectrum at q 2 = 0, which is quite difficult to access experimentally. As a further illustration of the complementarity of h → V and h → V P modes, we report here the dependence of the two total rates from the EFT parameters, adopting a common normalization for the leading term: Γ V ∝ c 2 1 + 0.9 c 1 c 2 + 1.3 c 1 c 3 + 0.6 c 2 c 3 + 0.2 c 2 2 + 0.5 c 2 3 , Γ V P ∝ c 2 1 + 0.9 c 1 c 2 + 0.9 c 1 c 3 + 0.4 c 2 c 3 + 0.2 c 2 2 + 0.2 c 2 3 .
In the limit where we neglect light hadron masses, Eqs. (19) and (20) continue to hold with P → P * , where P * is a vector meson, with decay constant defined by P * (q, )|J µ |0 = 1 2 F P * m P * µ .

VI. Conclusions:
We have shown the importance and utility of a decomposition of the h → V F amplitude into form-factors which can be probed by different final states, and how differential spectra can be used to disentangle the hV V * couplings in a general EFT approach. Complementary information is provided by leptonic and exclusive (semi-)hadronic final states, among which the h → V P decay is a simple and particularly interesting example. See [19] for a related analysis in the associated production process.