Vector boson polarizations in the decay of the Standard Model Higgs

The kinematic distributions of the lepton pairs produced in the decay of the Standard Model Higgs to ZZ and WW are related to the polarization fractions of the virtual vector bosons. The full amplitude can be decomposed analytically into a sum of polarized terms. Several observables, in particular the invariant mass of two charged leptons, one from each of the bosons, and the lepton angular distribution in the vector boson center of mass are shown to be sensitive to the boson polarizations. ar X iv :2 00 7. 12 08 0v 1 [ he pph ] 2 3 Ju l 2 02 0


Introduction
Vector Boson (VB) polarizations have attracted a great deal of attention in recent times. On the one hand, for single boson inclusive production, they can be unambigously predicted in the Standard Model. On the other hand, Vector Boson Scattering of longitudinally polarized is a crucial probe of the ElectroWeak Symmetry Breaking mechanism.
Since experiments can only observe the boson decay products within a limited subset of the full phase space, extracting vector polarizations is not straightforward.
VB polarizations at the LHC have been studied in a number of papers [1][2][3][4][5][6]. Both CMS and ATLAS have measured the W polarization fractions in the W + jets [7,8] channel and in tt events [9, 10]. Z polarization fractions at the LHC have been measured in [11,12]. The first polarization measurement at 13 TeV has been performed by ATLAS in W Z production [13].
In [14][15][16] a simple and natural way to define cross sections corresponding to vector bosons of definite polarization has been proposed. This allows to use polarized templates in fitting the data. Recently MadGraph5 aMC@NLO has introduced the possibility of generating polarized amplitudes [17].
In this paper, I discuss VB polarizations in the decay of the Standard Model Higgs to ZZ and W W , where only one of the VB can be on mass shell. The process is so simple that the decomposition of the full amplitude can be performed analitically, yielding a compact and transparent expression. The polarization fractions in Higgs decay are completely determined as in the case of single boson inclusive production. Their measurement would provide a test of the SM. This is a new take on a process which has been studied [18][19][20][21][22][23] since long before the discovery of the Higgs [24,25]. Precise predictions, including QCD and ElectroWeak NLO corrections, have been been provided in refs. [26][27][28]. An analysis including dimension six EFT operators can be found in [29].
The Higgs decay into four fermions has been studied experimentally, albeit with limited statistics, in order to determine the spin and parity of the Higgs, to set limits on its coupling strength and anomalous couplings to vector bosons [30][31][32][33][34].
In sect. 2 I recall the main ingredients needed in the analysis. In sect. 3-4 I discuss a number of observables whose distributions depend on the VB polarizations. Finally, in sect. 5 I discuss how the distributions are modified in the presence of leptonic cuts in a simple LHC-like framework.
2 Vector boson polarizations and angular distribution of its decay products Let us consider an amplitude in which a weak vector boson decays to a final state fermion pair. In the Unitary Gauge, it can be expressed as M and Γ are the vector boson mass and width, respectively, while c R and c L are the right and left handed couplings of the fermions to the W + (Z), as shown in tab. 1, and h denotes the chirality of the fermion. The polarization tensor, even when k 2 = M 2 , can be expressed in terms of four polarization vectors [35]: In a frame in which the off shell vector boson propagates along the (θ V , φ V ) axis, with three momentum κ, energy E and invariant mass Q 2 = √ E 2 − κ 2 , the polarization vectors read: On shell, the auxiliary polarization is zero and the longitudinal polarization reduces to the usual expression.
Consider the decays H → Z * Z * → µ + µ − e + e − and H → W − * W + * →ν µ µ − e + ν e in the center of mass of the Higgs. At lowest order the decay is described by a single, double resonant, diagram. The corresponding amplitude is Notice that, since M H < 2 M V , the double pole approximation of refs. [36][37][38][39][40] is not applicable.
Defining the decay amplitudes of the Vector Bosons as and using ε * R/L = −ε L/R one obtains where we have introduced factors f 0 , f L , f R to keep track from which vector polarization each term in the final result originates. f 0 , f L , f R are equal to one in the Standard Model, and one can envisage to measure them experimentally as a test of the SM.
The decay amplitudes of the Vector Bosons depend on their polarization and the fermion chirality, which we denote as +/−. In the rest frame of the f f pair, they are: where (θ, φ) are polar and azimuthal angles of the positively charged lepton (antineutrino in the W − case), relative to the boson direction in the laboratory frame. Notice that, if the boson propagates in the negative z direction φ → −φ. If Q 2 i (i = 1, 2) are the invariant masses squared of the two fermion pairs, E in eqs.(2.8-2.10) is equal Q 2 i /2. For massless leptons, the decay amplitude for the auxiliary polarization is zero because ε µ A is proportional to the four-momentum of the virtual boson. Eqs.(2.8-2.10) show that each polarization is uniquely associated with a specific angular distribution of the charged lepton, even when the V boson is off mass shell and the notion of a vector boson with definite polarization is ill-defined.
The squared amplitude, summed over fermion polarizations, becomes: K denotes the product of longitudinal polarization vectors: since, in the Higgs rest frame we have K is larger when the invariant masses of the virtual vector bosons are small and, therefore, longitudinal polarizations yield a larger fraction of soft fermion pairs.
The interference terms in eq.(2.11) cancel when the squared amplitude is integrated over the full range of the angle φ, or, equivalently, when the charged lepton can be observed for any value of φ.
Notice that the azimuthal modulation depends quite strongly on the polar angles of the two decays. The amplitude of the oscillation is maximal when both θ 1 and θ 2 are equal to π/2 and becomes zero when either angle is zero or π.  3 The H → ZZ → 4 channel (3.1) in good agreement with refs. [26,28]. The coefficient of the cos(2 φ) term is about 10% of the constant one and about four times larger than the coefficient of the cos φ term. The presence of a large contribution proportional to cos(2 φ) was pointed out in [26], while the smaller term proportional to cos φ went unnoticed. The longitudinal longitudinal component accounts for about 60% of the partial width while each of the left left and right right components contribute 20%.  One could wonder whether the distributions discussed in this note have any chance of being measured. We recall that CMS, with 35 f b −1 at 13 TeV, collected about 50 four lepton events on the Higgs peak [34,41]. For comparison, Run 2 has provided about 140 f b −1 to each large experiment; Run 3 is expected to accumulate about 200 f b −1 at 14 TeV and finally HL-LHC will deliver 3000 f b −1 [42,43]. Therefore we can expect of the order of 500 events by the end of Run 3 and thousands of additional events from HL-LHC. Fig. 1 shows the distribution of the invariant mass of the same flavour, opposite charged leptons. The curves in the right hand side plot are normalized. In addition to the expected peak at the Z mass, the curves display a wide increase at small invariant masses. The secondary peak is wider and extends to smaller value for the longitudinally polarized virtual Z's than for the transversely polarized ones. Fig. 2 presents the distribution of the angle between the positively charged lepton and the direction of flight of the Z boson, see eq.(2.11). The longitudinally polarized part is distributed as sin 2 θ. The right and left terms depend only mildly on the angle, showing a weak preference for the forward(backward) direction in the right(left) polarized case. These distributions coincide with the familiar ones for on shell Z decay even though in H → ZZ each Z is on mass shell only about 50% of the times.
In fig. 3 we study the invariant mass of the two positively charged leptons. This quantity has the interesting property of depending on all five independent variables which describe the decay of the Higgs boson to four fermions. On the right hand side we show the same curves normalized to unit integral.
The e + µ + invariant mass shows some dependence on the underlying vector boson polarizations: the longitudinal longitudinal result is more peaked that the LL and RR ones. It is harder than the LL distribution. The RR curve is the widest one, with a tail at larger invariant masses. The contribution of the interference terms in eq.(2.11) is not zero. However, it is about two orders of magnitude smaller than those in fig. 3 and, therefore, not plotted. Fig. 4 shows the azimuthal separation of the two positively charged leptons in the Higgs center of mass system, with the decay axis in the z direction. The result agrees with eq.(3.2). The RL interference term provides the bulk of the azimuthal dependence. The term proportional to cos φ is due to the inteference between the longitudinal component and the R and L ones. NLO Electroweak corrections for the ∆φ differential distribution have been computed in refs. [27,28]. They are about -1% for ∆φ = π and +4% for ∆φ = 0.

The H → W W → eµνν channel
Using M W = 80.38 GeV, Γ W = 2.10 GeV, the differential decay width with respect to φ in H → W W is +0.651 f L f R cos(2 φ)) × 10 −5 MeV degree . Eq.(4.2) shows that, in the W case, the coefficient of the cos φ term is comparable in magnitude with the constant term and about four times larger than the coefficient of the cos(2 φ) one, in general agreement with ref. [28] which, however, shows a different though related variable, the difference in azimuth in the laboratory transverse plane, which is easier to measure. Notice that for the negatively charged W, the − has opposite three momentum in the W rest frame compared to the antiparticle, whose distribution is described in eq.(2.11).
This implies that for the negatively charged lepton φ → π + φ and cos θ → − cos θ. The longitudinal longitudinal component accounts for about 60% of the partial width while the two left left and right right components contribute 20%. Fig. 5 presents the distribution of the angle between the positively charged lepton and the direction of flight of the W boson in the reference frame of the latter. The longitudinal polarized part is distributed as sin 2 θ, while the right and left terms are proportional to (1 ± cos θ) 2 respectively, as in on shell W decays.  In fig. 6 we study the invariant mass of the two charged leptons. On the right hand side we show the same curves normalized to unit integral. In the W case this variable has the additional advantage of not requiring the identification of the rest frame of the W pair which is notoriously extremely difficult to determine because of the two neutrinos in the final state.
The − + invariant mass again shows some dependence on the underlying vector boson polarizations: the RR and LL curves are identical, as expected, and softer than the longitudinal longitudinal result. The contribution of the interference terms in eq.(2.11) are not zero, however, they are about two orders of magnitude smaller than those in fig. 6. Fig. 7 shows the azimuthal separation of the two charged leptons in the laboratory frame. Eq.(4.2) shows that in the Higgs center of mass system all diagonal terms are independent of the angular separation in the plane orthogonal to the decay axis. In the lab, however, all polarization combinations depend non trivially on the difference in azimuth between the leptons. The full distribution favors small separations. At large values of ∆φ, there is a partial cancellation between the longitudinal longitudinal and transverse transverse components, and the longitudinal transverse interferences. Large interferences in ∆φ have also been reported in polarized W + W − production at the LHC [6]. NLO Electroweak corrections for the ∆φ differential distribution have been computed in ref. [28]. They are about 2.5% for ∆φ = π and 3.5% for ∆φ = 0.

A preliminary assessment of the effect of cuts in the LHC environment
In this section I investigate whether the differences of the kinematical distributions in the decay of polarized vector bosons survive in the LHC environment, where acceptance cuts and additional requirements, to improve the separation of signal from background, are necessary. Starting from a sample of gg → H events at leading order, the Higgs boson has been subsequently decayed to four leptons according to eq.(2.11), with a uniform angular distribution of the decay axis. In this simplified setup, all affects due to the transverse momentum of the Higgs boson are neglected.
The set of cuts for the ZZ case have been extracted from ref. [32] by CMS.
• pT > 7 GeV, |η | < 2.5 (acceptance) • 12 GeV < m + − < 120 GeV, m 4 > 70 GeV • ∆R , > 0.02, m + − > 4 GeV (veto on soft, collinear pairs) • N (pT > 20 GeV ) > 0, N (pT > 10 GeV ) > 1 (high pT leptons)  The transverse momentum distribution of the e + for ZZ events is shown in fig. 9 for the RR, LL and longitudinal longitudinal cases. The interference contributions are negligible. The three distribution show small differences. As expected, they exhibit two broad peaks at about half the value of the preferred lepton pair masses in fig. 1.     The set of cuts for the W W case have been taken from ref. [44] by ATLAS.
• pT > 15 GeV, |η | < 2.5 (acceptance) • 10 GeV < m + − < 55 GeV, N (pT > 22 GeV ) > 0 • pT > 30 GeV, pT miss > 20 GeV • ∆φ < 1.8, ∆φ ( )pT miss > π/2   The transverse momentum distribution of the e + for W W events is shown in fig. 13 for each of the six combinations of W polarizations, for the sum of the RR, LL and longitudinal longitudinal contributions and the sum of all contributions. There is a clear discontinuity at pT e + = 22 GeV related to the requirement of at least one charged lepton with such transverse momentum. The interference terms involving one longitudinal and one transverse W are large. The LR contribution is very small.