Higgs decay into a lepton pair and a photon: a roadmap to $\mathbf{H\to Z\gamma}$ discovery and probes of new physics

The decay $H\to \ell^+\ell^- \gamma$, $\ell=e,\mu$, receives contributions from $H\to Z[\to \ell^+\ell^-] \gamma$ and a non-resonant contribution, both of which are loop-induced. We describe how one can separate these sub-processes in a gauge-independent way, define the decay rate $\Gamma(H\to Z\gamma)$, and extract the latter from differential $H\to \ell^+\ell^- \gamma$ branching ratios. For $\ell=\mu$ also the tree decay rate, which is driven by the muon Yukawa coupling, is important. We propose kinematic cuts optimized to separate the three contributions, paving the way to the milestones (i) discovery of $H\to Z \gamma$, (ii) discovery of $H\to \mu^+\mu^- \left. \gamma\right|_{\rm tree}$, and (iii) quantification of new physics in both the effective $H$-$Z$-$\gamma$ and non-resonant $H$-$\ell^+$-$\ell^-$-$\gamma$ couplings.

constitute the desired discovery of this decay mode 1 . At several steps of this derivation (for instance by modifying the NWA) one could change the definition of Γ(H → Zγ) by terms of order Γ 2 Z /M 2 Z and arrive at equally valid, yet different results. This feature is intrinsic to any decay into an unstable particle detected only through its decay products. In view of the smallness of Γ 2 Z /M 2 Z ∼ 10 −3 , however, this ambiguity is phenomenologically irrelevant.
The differential decay rate dΓ(H → µ + µ − γ)/dm µµ peaks at the photon and Z poles at m µµ = 0 and m µµ M Z , respectively, and rises towards the end of the spectrum at m µµ = M H (see Fig. 2 (b)).
The latter effect is due to the tree-level contribution involving the small muon Yukawa coupling. ATLAS has already found evidence for H → + − γ in the low invariant mass region dominated by the photon pole [9]. To discover H → Zγ one must study the complementary region and in the H → µ + µ − γ data carefully separate the Z peak from H → µ + µ − γ| tree . A discovery of the latter contribution will constitute a manifestation of the Higgs Yukawa coupling to muons, independent of and complementary to the observation of H → µ + µ − . The loop contribution to the decay rate of H → e + e − γ is several orders of magnitude larger than the corresponding tree contribution, as the latter is suppressed by the square of the tiny electron Yukawa coupling. We do not consider the process H → τ + τ − γ which is dominated by the tree-level contribution. The light lepton masses are neglected in the loop contributions which are found infrared-finite in this limit.
With increasing statistics one will be able to quantify deviations from the SM predictions not only for the effective H-Z-γ vertex, but also for the effective non-resonant H-+ -− -γ couplings. To this end the data sample with = e and = µ should not be combined, as new-physics (NP) contributions are likely to be different. Through the Higgs vev H-µ + -µ − -γ couplings can contribute to the anomalous magnetic moment of the muon, whose measurement significantly deviates from the SM prediction [7]. The nonresonant region between photon and Z pole is best suited to probe those NP operators which are unrelated to the effective H-Z-γ vertex, because the SM contribution is small.

II. SEPARATING THE RESONANT CONTRIBUTION
We parametrize the loop-induced amplitude for the process h → γ as: where, using the notation of Ref. [2], we denote the four-momenta of photon, lepton and antilepton by k, p 1 , p 2 , respectively, while the chiral projectors are P L,R = (1 ∓ γ 5 )/2.
The loop-functions a 1,2 and b 1,2 depend on the Mandelstam variables where m and m H denote the masses of lepton and Higgs boson. The coefficients a 2 and b 2 are obtained by exchanging the variables t and u within a 1 and b 1 , respectively. Explicit one-loop expressions for the coefficients a 1 and b 1 can be found in Ref. [2] and corresponding ancillary files.
As mentioned in the Introduction, the off-shell amplitude for H → γZ * , which determines α(s) and β(s), depends on the unphysical gauge parameter ξ. However, the process H → γZ involving the on-shell Z boson does not depend on the gauge. Thus, we can isolate the ξ-independent part of the amplitude for H → γZ * [→ + − ] sub-process by setting s = m 2 Z in α(s), β(s), i.e. the residue of the Z-boson propagator is gauge-independent. In the following we denote this term the "resonant" contribution.
Separating the resonant and non-resonant terms in this way yields a 1 (s, t) = a nr 1 (s, t) + a res 1 (s) , We write where the tree contribution in the second term is to be dropped for = e. The loop contribution to the differential decay rate over the variables s and t is given by the formula: where we have neglected the light lepton masses in the phase space and u is to be substituted for the expression in Eq. (2). The non-zero value of the lepton mass impacts the value of the loop induced contribution to the decay rate only in the dilepton invariant-mass region close to the production threshold, m ∼ 2m , via the kinematic effect. We avoid this region by using the cut m ,min ≡ √s The square of the magnitude of a 1 in Eq. (6) contains three distinguishable pieces: and mutatis mutandis for a 2 and b 1,2 . Corresponding contributions to the one-loop decay rate are where the small interference term, denoted by Γ int , corresponds to the third term in Eq. (8) and can be safely neglected for the purposes of expected near-future measurements.
The differential decay rate for the tree contribution for H → µ + µ − γ reads: where For this distribution, we keep the nonvanishing muon mass in the formulas for physical kinematic limits given in Eq. (12). Note that the muon mass cannot be neglected in the phase space integral of the tree contribution, see Eq.(12) below. With data on d 2 Γ ds dt one can implement a very simple discovery strategy for H → Zγ: Just insert a res 1 from Eq. (6) into Eq. (5) and the resulting expression for a 1 into Eq. (8) (and treat a 2 and b 1,2 in the same way), then use these results in Eq. (7), and finally add d 2 Γtree ds dt . When using this formula to fit the three quantities α(m 2 The resulting resonant and non-resonant one-loop distributions are shown in the left plot in Fig. 2. Since FIG. 2: One-loop contributions to differential decay rate with respect to the invariant dilepton mass for = e (left) and = µ (right). The full one-loop, resonant and nonresonant contributions are denoted by black dashed, solid red and turquoise dot-dashed curves, respectively. For the case = e the full one-loop contribution represents the full rate, while for = µ, the additional, tree-level contribution needs to be accounted for.
the masses of electrons and muons can be safely neglected in the one-loop calculation, plot (a) represents the loop correction for both cases. Furthermore, since the tree contribution for H → e + e − γ is negligible, dΓ loop /dm also represents the total contribution for H → e + e − γ. The effect of the tree contribution is shown in the plot 2 (b). The only kinematic cut imposed for these plots is the one for the photon energy in the Higgs rest frame, E γ, min = 5 GeV, which only lowers the maximum value of m .
In Fig. 3 we display the interference contribution. As expected, this distribution changes sign at the value of m corresponding to the Z-pole and is approximately symmetric around the null-axis in this region. However, its magnitude turns out negligible within the full rate -this term is completely dropped in the following discussion.

III. KINEMATIC CUTS
In this section we study the impacts of the kinematic cuts on the minimal values of the variables t and u on the resonant-, nonresonant-and tree contributions.
We fix the kinematic range for the variable s all the way until the section III D as: where E γ,min the minimal photon energy in the rest frame of the Higgs.
The full physical range for the variable t is given in Eq. (12). We introduce the kinematic cuts on the below On the other hand, the tree contribution is peaking for the small values of t, as can be seen from the Dalitz plot boundary parallel to s-axis, and for the small values of u, as can be seen from the diagonal boundary of the plot in Fig. 1 (b).

A. Resonant contribution
The resonant distribution is given by: with the mass of the light lepton neglected in the evaluations of both the kinematics and the amplitude. With  Integrating over the variable t, while imposing the cutst min andũ min , we have: A further integration over the variable s can also be performed analytically, but results in a somewhat lengthy expression. In Fig. 4 we illustrate the variations of the resonant differential decay rate dΓ res /dm for different values of the cuts (t min ,ũ min ).
The effects of the cuts are more noticeable in the fully integrated decay rate. Integrating over s in the range given in Eq. (13) we have, e.g.

B. Nonresonant contribution
The analytic form of the non-resonant contribution turns out rather lengthy -its explicit form can be read off from the expressions given in Appendix A of Ref. [2]. As in the previous case, we integrate where Γ nr [t min = 0,ũ min = 0] = 0.043 keV .
Therefore, we find weak dependence on the t, u-cuts as long as the values of the latter are not such that they remove a significant amount of the phase space.
It is convenient to display the shapes of the distributions shown in Fig. 5 in an approximate numerical form. Since the dependence on the cuts is small, we represent the shape that does not involve any cuts on variables t, u as the following power series:  This is an acceptable approximation given that the non-resonant part is itself a small contribution to the full decay rate in the interesting region around Z-boson peak.

C. Tree contribution
The definite integral over the variable t in Eq. (10) can be performed analytically. As before, for the lower limit we havet min , which is larger or equal to the the physical lower limit t min (s, m ), while the upper limit is t max (s, m ) −ũ min . Introducing the shorthand notation the resulting distribution with respect to s is: where we have temporarily suppressed an additional dependence of t min(max) on the lepton mass, for clarity of the notation. Note that the insertions of the Heaviside step function in the above equation confine the integration to the physically allowed region. The expression for I(t) is: The final formula for d 2 Γtree ds (s;t min ,ũ min ) is obtained by inserting the result of Eq. (25) into Eq. (24). We illustrate the dependence of the tree contribution on the cuts for several values oft min andũ min in Fig. 6.
Finally, integrating over the variable s in the limits given in Eq. (13), we have: where Γ tree [t min = t min (s, m µ )] = 0.104 keV .

D. Kinematic cuts and total rates
We now explore how each of the three contributions to integrated decay rate depends on the cuts on variables that also include s. We propose different cuts to optimize the sensitivity to the three milestones mentioned in the abstract. The results for several combinations of such cuts are shown in Table I.
Cuts 1 and 2 correspond to the choices of the three previous subsections 2 . For the cut 1 we find that the nonresonant contribution is around 20% of the resonant one, while the tree contribution is somewhat larger than about 10%. As noted before, the tree contribution receives a strong suppression with the increasing vales oft min andũ min . Cuts 3 and 4 isolate the resonant contribution stemming from H → Zγ, while cuts 5 and 6 probe the nonresonant contribution. The purpose of cut 7 is the isolation of the tree contribution.
Cut 8 simply illustrates an additional suppression of the tree contribution that results from tightening of cuts on t and u.

IV. RESONANT CONTRIBUTION AND THE NARROW-WIDTH APPROXIMATION
The resonant contribution is related to the decay rate of H → Zγ involving an on-shell Z boson that subsequently decays to a pair of light leptons.  We recall the amplitude for the process H → Zγ: where p Z , q, (p Z ), (q) denote momenta and polarizations of Z-boson and photon, respectively, while the loop functionÃ is given in Eq. (B1). The decay rate is: in agreement with the result in Ref. [14]. EvaluatingÃ in Eq. (B1) for the input values of Eq. (A1) gives the SM prediction Γ(H → Zγ) = 6.51 keV, again in agreement with the numerical result found from the analytic expression in Ref. [14]. This value is 3% larger than the central value quoted by the LHC Higgs Cross Section Working Group, Γ(H → Zγ) = 6.31 keV, in Table 177 on page 679 of Ref. [10], see also Eq. (III.1.18) on page 403. Ref. [10] finds an uncertainty of the theory prediction of order 5%, which could be reduced by a two-loop calculation.
Furthermore, the branching ratio of the process Z → at tree-level is Integration of the resonant distribution d 2 Γ res /(ds dt) over the variable t in the full range given in Eq. (12) results in We now apply the narrow-width approximation (NWA) for the Breit-Wigner distribution: where the limit is taken under the integral over s. Substituting this limit into Eq. (32), integrating this distribution over s, and using the relations (29) and (31) we find: provided that The latter relation can be explicitly confirmed using the functions α(s) and β(s), given in Eqs. A.1 and A.2 in Ref. [2]. Thus if α(m 2 Z ) 2 + β(m 2 Z ) 2 extracted from data, the desired decay width is calculated as obtained using the parameter inputs from Eq. (A1).

V. CONCLUSIONS
The decay rate dΓ(H→ + − γ) dm with = e or µ offers insights into different aspects of Higgs physics.
With increasing integrated luminosity it will be possible to (i) discover the decay H → Zγ and measure its branching ratio, (ii) discover the decay H → µ + µ − γ| tree driven by the muon Yukawa coupling, and (iii) ultimately quantify potential new physics contributions to both the loop-induced H → Zγ decay and the off-peak contributions to H → + − γ. The latter comprise the non-resonant loop contributions, best tested in the region between the photon and Z poles, and (for = µ) H → µ + µ − γ| tree which dominates dΓ(H→ + − γ) dm near the endpoint region with m > M Z .
In this paper we have proposed a gauge-independent, physical definition of the decay rate Γ(H → + − γ) and shown how it can be extracted from the measured decay spectrum dΓ(H→ + − γ) dm . To this end it is necessary to subtract the non-resonant contribution to dΓ(H→ + − γ) dm and we have derived easyto-use approximations for the cumbersome SM expression, see Eq. (21) above. We have further studied the dependence of dΓ(H→ + − γ) dm on kinematical cuts, which we only found to be a critical issue for H → µ + µ − γ| tree . In order to perform the three milestone measurements mentioned above we have proposed cuts to optimize the sensitivities to H → Zγ, H → µ + µ − γ| tree , and the non-resonant loop contribution, respectively, see Table I.