Theoretical Basis of Higgs-Spin Analysis in $H \to \gamma\gamma$ and $Z\gamma$ Decays

We chart the theoretical basis of radiative decays of the Higgs boson, $H \to \gamma\gamma$ and $Z\gamma$, for measuring the spin of the Higgs particle. These decay channels are complementary to other rare modes such as real/virtual $Z$-boson pair-decays. In systematic helicity analyses the angular distribution for zero-spin is confronted with hypothetical spin-$2^\pm$ and higher assignments to quantify the sensitivity.


1.
After the discovery of the Higgs particle, the properties must be examined experimentally to identify the particle as the element proper of the Higgs mechanism for breaking the electroweak symmetries [1] [for recent general reviews see Refs. [2]]. The dynamical steps include the confirmation of the scalar spin-zero character of the particle 1 . Higgs-boson decays to pairs of Z-bosons 2 , cf. Refs. [3][4][5] and additional references listed there, are promising candidates for examining the Higgs spin at the LHC, supplemented by other measurements in the Higgs-strahlung processes at LHC [6] and e + e − linear colliders [7]. Exploiting the primary Higgs-boson search channels, γγ and W W decays provide another self-evident tool for spin measurements [8,9]. [For a variety of methods see the recent literature in Refs. [10]]. All these channels are particularly difficult to analyze for low masses of the Higgs boson where the b-pair decays are overwhelming. However, with about 126 GeV this is precisely the mass range, where a new boson has been discovered by the LHC experiments [11][12][13], consistent with analyses of electroweak precision data [14] and compatible with Higgs patterns within the framework of the Standard Model [15].
In this study we examine to which extent radiative decays [16][17][18] H → γγ and Zγ (1.1) can be exploited to determine the spin of the Higgs boson. Even though these decays are rare, with BR ≃ 2 · 10 −3 for masses of the order of 126 GeV [19][20][21], the large number of light Higgs bosons produced in gluon fusion [20][21][22] generates a sample of order 10 4 events in γγ decays for an LHC luminosity of ∼ 100 fb −1 , and a similar number of Zγ-decay events [though reduced finally by the branching ratio for leptonic e, µ decays of the Z-boson]. These channels nicely supplement the other rare ZZ * channels. In the present report we perform general helicity analyses of the radiative decays to quantify the sensitivity to zero-spin of the Higgs boson in angular distributions.

2.
The zero-spin character of the Higgs boson reflects itself in the isotropic decay distribution of the radiative decays in the rest frame, i.e. with Θ denoting the polar angle of the γγ and Zγ axis, defined, for example, in the Higgs-boson rest frame with regard to the LHC beam axis, Fig. 1(a), and with the normalization including the proper statistical factor of the two-photon state. The axis can be reconstructed experimentally in the exclusive γγ and Zγ final states. Since the Z-boson is produced in helicity ±1 states, the ℓℓ distribution in leptonic Z decays is of the familiar 1 + cos 2 θ ℓ form, θ ℓ being the polar lepton angle in the rest frame of the Z-boson. The general parity-invariant polar angular distribution can be cast into a compact form for any Higgs-spin J decays to γγ and Zγ in gluon fusion: the functional forms displayed in Fig. 1(c,d). The [non-negative] reduced production and decay helicity probabilities X for gluon fusion, Y for γγ and Y ′ for Zγ final states, are model-dependent parameters, obeying the sum rules [The formalism can easily be transferred to qq production with S z = 0 or ±1 by substituting X J 1 for X J 2 and D J 1k for D J 2k (k = 0, 1, 2) in the cross sections and in the X sum rule. Note however that the rate for signal Higgs production in qq collisions is negligibly small for light quark beams. The initial states gg and qq mix incoherently in the most general configurations.] The polar angle of the Z decay distribution can easily be integrated out, equivalent to the substitutions (1 + cos 2 θ ℓ ) → 8/3 and sin 2 θ ℓ → 4/3. Since the reduced helicity probabilities are non-negative, the coefficients of both the (1 + cos 2 θ ℓ ) term and the sin 2 θ ℓ term inevitably generate non-vanishing maximum/minimum ± cos 2J Θ terms. Thus, observing [in the experimentally ideal case] that the angular distribution is independent of cos Θ, proves unambiguously the spin-zero character of the Higgs particle. At the same time the sin 2 θ ℓ term in Zγ final states is predicted to be absent, providing an independent cross-check.  The observation of spin states in gg → H → γγ allows [partial] conclusions also on the parity of the states. From Bose symmetry and parity symmetry of the helicity amplitudes , referring separately to initial and final states [3], the selection rules presented in Tab. I can easily be derived [see also Ref. [24]], complementing global rules noted earlier in the literature. Scalar and pseudoscalar Higgs bosons cannot be discriminated in the γγ decay mode, neither even/odd parity by observing D J 00 , and spin correlation effects [7,25,26] must be exploited for discrimination. But observing any state with helicity difference ∆λ = 2 in initial or final state determines unambiguously the even-parity character of the Higgs boson. Analogous rules apply also to Zγ final states in gluon fusion and γγ final states in qq annihilation. Either the first or the second index ∆λ in the D functions coming with the production or decay amplitude, respectively, is restricted in parallel to the rules in Tab. I while the respective companion index is unrestricted apart from the standard spin constraints.
The γγ channel is described by two-by-two independent probabilities for production and decay, X J 0,2 and Y J 0,2 . Popular choices for experimental simulations are the complementary {J; 00} ′ scalar-type ′ and the {J; 22} ′ tensor-type assignments ′ : supplemented by the ′ mixed-type assignment ′ : for the signal J = 0 and the hypothetical alternatives J ≥ 2.
[The γγ coupling of the tensor-type assignment is equivalent to the KK graviton coupling in d = 5 scenarios [8].] The configurations can be exploited in two ways: (i) One of the two scalar-or tensor-type configurations for J ≥ 2 for instance, is sufficient to prove that the spin-zero test of the signal is non-trivial; (ii) However, to prove experimentally that the spin-2 assignment is not realized, the three configurations, which are mutually independent, must necessarily be shown absent. Not observing the double index {J; 00}, the state {J − } is ruled out, while neither observing distributions carrying at least one index 2, the state {J + } is ruled out, too. Thus any spin J ≥ 2 can be rejected for both parities ± by angular analyses. -Due to potential longitudinal Z polarization accounted for by Y ′ J 1 , the Zγ final state is described by three independent decay probabilities.
Disregarding J = 1, as forbidden by the Landau-Yang theorem in γγ decays [24,27], we will choose J = 2 for illustration. [Part of the distributions have also been noted in Refs. [8,28].] None of the possible helicity states would generate a flat distribution like spin-zero Higgs-boson decays. It is shown in Fig. 1 how the flat signal distribution contrasts with the distributions generated by hypothetical spin J = 2 assignments of both even and odd parity, and the angular distribution of the backgrounds as well [to be analyzed next]. In the final section we will compare the spin-0 distribution, assigned J = 0 and m = λ γ,Z − λ ′ γ = 0, specifically with the scalar-type {2; 00} and the tensor-type {2; 22} distributions, representing the state J = 2 with the two parities ±. The {2; 00} scalar-type distribution is pronounced in the center like spin-0 after angular cut, and rises in the forward/backward directions like the continuum backgrounds, thus providing a non-trivial analogue to be discriminated experimentally from the Higgs signal. The {2; 22} tensor-type distribution rises monotonically to the left and to the right of the center, providing also a valuable discriminant. Both types must be ruled out necessarily to reject experimentally the spin-2 assignment for even and odd parity.
A first global comparison between spin-0 and all the spin-2 distributions is offered by the moments of the polar-angle distributions of the γγ and Zγ event axes 3 noted in Tab. II. The first moments of the spin-2 distributions are characteristically different 4 from the spin-0 distribution and may provide early information on the Higgs spin.
3. The large continuum background generated in pp(qq) → γγ and Zγ processes [and, to a lesser extent, loop-induced gluon fusion] requires stringent cuts in order not to dwarf the signals. The angular characteristics allow to reduce  the continuum backgrounds considerably.
The angular distribution of the background events is strongly peaked in the forward and backward directions as a result of the t and u-channel exchange mechanisms, in contrast to the flat distribution of the signal. In the rest frame of the parton system, cutting out the singular forward and backward directions Θ → 0 and π [so long as the cross sections are not regularized properly]: for the invariant parton energy √ s = M H , electric and weak quark charges denoted by Q, v, a [29,30]; the helicity decomposition is familiar from electron-positron collisions, cf. appendix in Ref. [31]. The parton subprocesses are peaked at small angles, regularized by the strong interaction scale Λ QCD . For the given Higgs mass, the angular distribution of gg → γγ [32] is surprisingly close to the qq process, cf. Fig. 1(b). Induced by radiative return [33] in Zγ, and self-evident in γγ, the γ's and the Z-bosons are traveling primarily along the LHC beam axis. By restricting | cos Θ| to less than cos Θ cut ≤ 1/ √ 2, the signal is reduced only modestly, but the background strongly.

4.
For illustration of the J P sensitivity, we present a set of rough theoretical estimates, rescaled from available experimental data in γγ [11] or based on simulations in Zγ [34]. One photon with transverse energy ǫ perp in excess of 25 GeV was required for Zγ decays, and photons in excess of 40 GeV in γγ decays. Numerically, using for the corresponding polar angles in the Higgs rest frame, this is approximately equivalent to the restrictions | cos Θ| < 0.77 and 0.55, close to the theoretical cut 1/ √ 2 used in the previous section. The Z-boson was assumed to decay leptonically to e, µ pairs. The error estimates for the γγ final states were performed by rescaling the background event numbers of Ref. [11] in energy, using the theoretical energy dependence of qq → γγ [ignoring mis-identification from jets in this simplified theoretical estimate], and by raising the luminosity to 100 fb −1 , leaving us with 4.6k [146k] events after cuts for signal [background], in rough agreement, within a factor two with Ref. [35], after inserting the proper K-factor. The theoretical angular distribution of the photons in the center range left by the cuts was used as well according to Fig. 1. We have adopted the experimental efficiency of 40% and a resolution of ±2 GeV. Experimental refinements such as smearing effects etc. have not been considered in this coarse theoretical picture.
In the same way as Ref. [34] we determined the background event number from the cross section of the qq → Zγ process which dominates the background compared with gg collisions and the Drell-Yan process including final-state photon radiation. Similarly to the parameters in Ref. [34] we have adopted the values of 3 GeV for the mass resolution These theoretical estimates of rates and errors, not including detailed experimental refinements, should serve only as a rough illustration of theoretical expectations for Higgs spin analyses in radiative decays.
In the framework defined above, the results for the signals and the size of the roughly expected backgrounds are shown in Fig. 2 for H → γγ and H → Zγ in the theoretically cut cos Θ range. The solid lines are the [ideal] signals with very high statistics, the error bars, based on the event numbers defined above, represent the theoretical estimates of background fluctuations √ B, contaminating the angular distributions of the signals S [ ≪ B]. These distributions are compared with the expected decay characteristics of a hypothetical spin-2 particle, even/odd parity, derived for the representative angular tensor-type and scalar-type distributions in Eqs. (1.5). Note that the two distributions are mutually complementary to each other. The first | cos Θ| moments [normalized to zeroth moments] are in the ratio (0.52 ± 0.06)/(0.27 ± 0.05) for the tensor/scalar assignment in γγ final states, and (0.36 ± 0.10)/(0.18 ± 0.08) in Zγ final states. Even if added up, the resulting flattish behavior is fractured by the non-zero D 20 = D 02 contributions.
The small leptonic Z branching ratio together with the increased background cross section render experimental Zγ analyses much more demanding than γγ analyses, and a large increase of luminosity is required.
Studying experimentally the γγ and Zγ processes outside the Higgs mass window will yield a good understanding of the background shapes and normalizations. At the expense of a √ 2 increase of errors one could define a control region with a lower Higgs-type mass window to determine the shapes and use Monte Carlos to extrapolate from the control region to the signal region. 5 These parameters were extracted by means of PYTHIA [36] and ACERDET [37], with support by D.Zerwas gratefully acknowledged.
The transverse momentum of the photon was chosen in excess of 25 GeV. In Z decays to leptons, electrons or muons were reconstructed with transverse energy/momentum of more than 25 GeV, separated from the photon by ∆R ≥ 0.7, where ∆R is the square-root of the sum of the squares of the azimuthal angle and pseudo-rapidity differences. The invariant mass of the lepton pair was chosen compatible with the mass of the Z boson within 5 GeV. The reconstructed mass of the Higgs boson was allowed in a window of 3 GeV centered on its nominal mass. The second window can be chosen more restrictive as the Higgs boson width is negligible compared to the experimental resolution, whereas for the Z boson this is not the case. The resulting efficiency of 0.13 is close to the value reported in Ref. [34] when interpolating slightly different parameters.

5.
Summary: Dynamical characteristics of the Higgs boson in the Standard Model are particularly difficult to analyze experimentally in the mass region around 126 GeV since the overwhelming decays are b decays. In this report we have analyzed the theoretical potential of two decay modes, H → γγ decays and H → Zγ decays [the latter statistically more remote], to measure the spin of the Higgs boson. General helicity analyses prove the sensitivity of both decay modes to zero-spin of the Higgs boson, demonstrated by confronting J = 0 zero-spin for illustration to J = 2 ± , i.e. spin = 2 and even/odd parity.