Dispersion theoretic calculation of the $H\rightarrow Z+\gamma$ amplitude

We have calculated the $W$-loop contribution to the amplitude of the decay $H\rightarrow Z+\gamma$ in two different methods: 1) in the $R_\xi$-gauge using dimensional regularization (DimReg), and 2) in the unitary gauge through the dispersion method. Using the dispersion method we have followed two approaches: i) without subtraction and ii) with subtraction, the subtraction constant being determined adopting the Goldstone boson equivalence theorem (GBET) at the limit $M_W\to 0$. The results of the calculations in $R_\xi$-gauge with DimReg and the dispersion method with the GBET completely coincide, which shows that DimReg is compatible with the dispersion method obeying the GBET.


I. INTRODUCTION
The calculation of the Higgs decay rate into two photons through the W -loop has become the subject of a controversy. After extracting the transverse factor which takes current conservation into account, the invariant amplitude is finite. Since however this amplitude is the sum of individually divergent Feynman diagrams most authors use dimensional regularization (DimReg) for its evaluation. Surprisingly, the DimReg result [1,2] 1 differs (by a real additive constant) from the outcome of a direct computation that works with the physical unitary gauge [4,5].
Responding to a criticism, [6], which points out that the delicate cancellation of divergences is ambiguous and thus one needs a regularization, the result of [4] was confirmed in [7] by applying unsubtracted dispersion relations in a calculation that deals only with absolutely convergent integrals. Nevertheless, this calculation was also subsequently criticized in [8]. The origin of the controversy stems from the fact that perturbative amplitudes may be ambiguous even if the corresponding momentum space integrals are convergent: the Feynman rules need to be supplemented by conditions like gauge invariance, or the associated Ward identities, alongside with locality (or causality [9]) which yields the analytic properties in momentum space. The argument for an unsubtracted dispersion relation follows directly from the fact that the only constants that may appear in perturbative calculations should be the coupling constants and masses that are part of the full renormalizable Lagrangian. Thus, the absence of Hγγ-coupling in the SM Lagrangian, implies a zero subtraction in the dispersion integrals for the H → γ + γ amplitude. The same argument holds for the H → Z + γ amplitude, as well.
However, since the SM is a spontaneously broken theory and masses are generated through the Higgs mechanism, it was argued that the considered amplitude should obey the boundary condition defined by the Goldstone boson equivalence theorem (GBET) [10,11]. In [12] the amplitude of H → γ + γ was calculated in the unitary gauge staying strictly in four dimensions but fulfilling the Goldstone boson equivalence theorem. Their result is the same as in [1]. In [13] it was shown how the amplitude for the decay H → γ + γ, calculated in the R ξ gauge and in the unitary gauge, may lead to different results.
These controversial results in the calculations of the amplitude for H → γ + γ motivated us to consider the decay H → Z + γ. These two processes are similar in a sense that at tree level they are both zero and induced by loop corrections only, the W -loops giving the main contribution. At M Z = 0 the process H → Z + γ should reproduce the results for H → γ + γ.
In this paper we calculate the one-loop W -contributions to H → Z + γ in two approaches: First we calculate the amplitude using the dispersion relation approach. We consider two cases: 1) we assume the unsubtracted dispersion relation, and 2) we assume a non-zero subtraction constant adopting the GBET in the limit M W → 0. Next we calculate the same amplitude in the commonly used R ξ -gauge using the conventional dimensional regularization (DimReg).
The goal of these calculations is to compare the two results: from the dispersion-relation approach, in which we deal with finite quantities only -with and without subtraction, to the result in R ξ -gauge with DimReg. The dispersionrelation approach can, in fact, be viewed as a general tool for resolving the ambiguities in the regularization scheme in quantum field theory. We show that with the dispersion-relation approach, where no regularization is necessary, and with subtraction determined by the GBET, we get exactly the same result as in R ξ -gauge with DimReg.
Previously the decay H → Z + γ was calculated using DimReg and R ξ -gauge by Cahn et al. [14], and later a complete analytic expression was obtained by other authors [15,16]. Recently in [17] this calculation was done in the unitary gauge, with the help of dimensional regularization. We completely agree with their results.
Before we go into the details it shall be mentioned that in this study we have used a couple of helpful Mathematica packages, [18][19][20][21][22][23].

II. THE FEYNMAN DIAGRAMS
We consider the contribution of the W -bosons loop-induced amplitude of the decay H → Z + γ. We work in the unitary gauge, when only the physical particles contribute. There are two types of diagrams. In Fig. 1 the three W -loop diagrams that contribute to the absorptive part of the amplitude are shown. These are the same diagrams as in the process H → γ + γ [4,7], in which one of the final photons is replaced by Z. In the same figure also the unitary cuts, needed for obtaining the absorptive parts of the amplitude are shown. In Fig. 2 the two additional diagrams that contribute to H → Z + γ are shown. These are H → Z + Z * with the subsequent transition Z * → γ with W + W − and W + in the loops. Clearly, kinematically their contribution to the absorptive part is zero and we don't consider them further. The amplitude for the process M is: where k 1 and k 2 are the momenta of the Z-boson and the photon, ζ 1 , ζ 2 are their polarizations, orthogonal to k 1 and k 2 , respectively: The contribution to M µν of the three diagrams on Fig. 1 is: Here, θ W is the Weinberg (weak mixing) angle and M = M W is the mass of the W-boson. The W W γ and W W Z vertices are denoted by V αβγ , the W W Zγ vertex is denoted by V αβµν , they are given in Appendix A, where all Feynman rules in the unitary gauge are recalled. We have also used the following brief notations: Taking into account the transformation properties under the reflection k → −k of the loop momentum, We relate M 3µν and M 1µν , thus simplifying our calculation:

III. ABSORPTIVE PART OF THE AMPLITUDE
We obtain the absorptive part through the Cutkosky rules which set the momenta of the W 's on-shell [24]: The imaginary part is obtained via the cut diagrams, M C iµν : Obviously, here we have taken into account Eq. (13). Further we define the invariant absorptive part A of the amplitude through the imaginary part of the amplitude: where P µν is the transverse-momentum (gauge invariant), given by Eq. (1) , Then A is obtained via the expression: where I µν is determined by the Feynman diagrams on Fig. 1. The two delta functions δ(D 1 ) and δ(D 3 ) in Eq. (18) reduce the one-loop integral to a phase-space integral. In the next section as the second step we will calculate from the absorptive part the real part of the amplitude by applying the dispersion integral technique. One can also inverse the step of computing the absorptive part. Instead of cutting the one-loop amplitude, one can sew appropriate treelevel amplitudes together to form the one-loop amplitude which turns the cutting step around, avoiding the explicit construction of one-loop Feynman diagrams. But then one can rely on evaluating Feynman integrals to do the second step [25]. These are the so-called unitarity cut methods based on [26], see also e. g. [27,28]. The tensor I µν is obtained via straightforward, but rather tedious calculations starting from the expressions (5)- (7). Also we make use of the following identities, that hold for both the W W γ and W W Z vertices: and After rather cumbersome calculations we end up with the following expression for I µν : Now we have to do the integration in (18). We perform it in the rest frame of the decaying Higgs boson, with the z-axis pointing along k 1 : The two δ-functions: δ( immediately determine k 0 and |k|: where Thus, we are left only with the 2-dimensional integral over the direction of k = |k|(sin θ cos φ, sin θ sin φ). For D 2 we obtain: The absorptive part of the amplitude is non-zero at τ > 1 and it reads: The details of the calculations are presented in Appendix B.

IV. REAL PART OF THE AMPLITUDE
The full invariant amplitude F (τ, a) is defined by where P µν is the transverse-momentum factor (1). The vanishing of the absorptive part of the amplitude at τ < 1 tells us that the invariant amplitude F at τ < 1, which is the physically interested region, is only real. Following the analytic properties of the amplitude, we define the invariant unsubtracted amplitude F un (τ, a) in this region, τ < 1, through the convergent dispersion integral: From the explicit expression for A and its behaviour at τ → ∞ we obtain that this integral is absolutely convergent. This however does not imply that there are no subtractions in (33): the dispersion integral (33) determines the full amplitude F (τ, a) up to an additive constant C(a): In order to determine C(a) we need some additional information about the amplitude -some boundary condition or a physical measurable quantity at some fixed value of τ . In our calculations we fix C(a) through the Goldstone Boson equivalence theorem (GBET) [10], which fixes the behaviour of the amplitude at τ → ∞.
In accordance with this we calculate the amplitude F (τ, a) in two steps: 1. First we calculate F un (τ, a) using the dispersion relation Eq. (33). 2. We calculate C(a) using the GBET.
A. The unsubtracted amplitude F un (τ, a) The unsubtracted amplitude F un (τ, a) is determined by the convergent dispersion integral Eq. (33). The integrals in Eq. (33) are taken analytically -they are given in Appendix C, and we obtain: The result for τ > 1 in the above formula is obtained via analytic continuation. (The same result may be found if we had set τ > 1 in the integrand and taken the iǫ prescription in D 2 into account.) There are several important physical consequences for this amplitude.
1. The amplitude at threshold, τ = a, is finite. We have: The absence of singularities in the amplitude is in accordance with the required analytic properties of F un (τ, a), necessary for the validity of the dispersion relations.
2. In the asymptotic limit τ → ∞, which implies M 2 H ≫ M 2 at fixed a, we obtain: 3. In the limit of a → 0, we have to recover the corresponding invariant amplitude F γγ (τ ) for the H → γ + γ process: where P µν is the same transverse bilinear combination as Eq. (1) with the (on shell) photon momenta k 1 , k 2 . We obtain: which is exactly the result for F γγ un (τ ), obtained in the unitary gauge, both, with direct calculations without renormalization in [4], and using the dispersive relations approach without subtraction in [7]. 4. We calculated also the amplitude of the process in the commonly used R ξ -gauge using DimReg. The calculation was done with the help of the automatic tools FeynArts [18] and FormCalc [19]. There are 20 Feynman triangle vertex graphs, 6 Feynman vertex graphs with a four-point interaction and 10 graphs with selfenergies from Z * − γ transition. It is checked that the result is UV finite and independent of ξ and it coincides with the one, obtained earlier in [15]. However, the result for the amplitude F DimReg (τ ), obtained using dimensional regularization, differs by a real additive constant from our result for F un (τ ): 2πF DimReg (τ, a) = 2πF un (τ, a) which leads to a non-vanishing asymptotic behaviour at τ → ∞.

B. The charged ghost contribution and the constant C(a)
We determine the constant C(a) through the charged ghost contribution adopting the GBET, which implies that at M W → 0, i.e. at τ → ∞, the SU (2) × U (1) symmetry of the SM is restored and the longitudinal components of the physical W ± -bosons are replaced by the physical Goldstone bosons φ ± . In the following M φ µν denotes the amplitude of H → Z + γ in which the W ± are replaced by their Goldstone bosons φ ± . The GBET implies [10]: We calculate the charged ghost contribution in two different ways: through direct calculations and via the dispersion integral. Both calculations lead to the same result.
• There are 3 vertex graphs and 2 selfenergy graphs, shown in Figs. 3 and 4, that possibly can contribute. We denote the contribution from the vertex diagrams by M φ 1+2+3,µν . Following the Feynmann rules for the φ ± -vertices, given in Appendix A, with direct calculations using DimReg we learn that the selfenergy graphs do not contribute, the result is finite and gauge invariant, as expected: • However, as the goal of our approach with the dispersion integrals is to obtain the amplitude using only finite quantities, we shall obtain the Goldstone-boson contribution by using the dispersion method. Analogously to Eq.(32), we single out the coupling constants (see the Feynman rules in Appendix A) and define the invariant part F φ of the decay amplitude M φ µν in the Higgs-Goldstone boson scalar theory: We shall apply the dispersive approach (without subtraction) to the function F φ (τ, a). In order to obtain the form factor τ F φ (τ, a) that enters the amplitude M φ µν , Eq. (48), we must multiply the result for F φ (τ, a) by τ . (The same strategy was elaborated for the H → γ + γ process in [8].) In general, a constant term can, of course, be always added and in order to fix the subtraction constant some additional physical boundary conditions are required. In contrast to the SM, where the GBET is a boundary condition that fixes the subtraction constant, in the Higgs-Goldstone scalar theory there are no asymptotic theorems one could refer to.
However, the GBET allows to define a boundary condition for F φ (τ, a), as well. According to the GBET, the constant C(a) is obtained as the large-τ limit, Eq. (45), which in terms of the form factors reads: Since C(a) is a finite quantity, the structure of Eq. (48) and more precisely the presence of the factor M 2 H in the coupling, implies that the large-τ behavior of the function F φ (τ, a) is of the form F φ (τ, a) ∼ O(τ −x ), with x ≥ 1. Therefore, the value of the integral (1/π) ARC dyF φ (y, a)/(y − τ ) over the infinite arc in the complex τ -plane, is zero. This, and the fact that the dispersion integral (see Eq. (51) bellow) is convergent, implies that the dispersion relation applied for F φ (τ, a) does not need a subtraction.
The absorptive part A φ (τ, a) of the function F φ (τ, a) is obtained via the Cutkosky rules from the cut diagrams in Fig. 3. Evidently the selfenergy graphs, see Fig. 4, have no absorptive parts. We obtain (see Appendix D): The expression for the function F φ (τ, a), valid in the whole τ -interval, is obtained via the dispersion integral: where I 2 (τ, a) and J 2 (τ, a) are convergent and given in Appendix C.

V. THE DECAY WIDTH OF H → Z + γ
A good approximation for the total width of the Higgs decay into Z +γ is given by the contributions of the W -boson and the top-quark loops (cf. [15]): where F t (τ t ) stands for the sum of the t-quark one-loop diagrams: and F W (τ ) stands for the sum of the W -boson one-loop diagrams.
Further, we identify F W (τ ) with the amplitude obtained with the dispersion integral, Eq. (34), in which the unsubtracted part is given in (35) and C(a) in (47): F W (τ ) = F (τ, a). This implies that at the measured value for the Higgs mass M H = 125.09 GeV, using m t = 172.44 GeV for the mass of the top-quark, we obtain the following value for the expected decay width: If, however, F W (τ ) was identified to the unsubtracted amplitude F W (τ ) = F un (τ, a), Eq. (35), the value for the decay width of H → Z + γ would be about 20 % smaller which, as we showed, seems not to be the correct result.

VI. CONCLUDING REMARKS
We have calculated the W -boson induced corrections to the decay H → Z + γ in the Standard Model in the unitary gauge using the dispersion-relation approach. This approach is very attractive as it deals only with finite quantities and thus does not involve any uncertainties related to regularization. However, the problem with the dispersion method is that it determines the amplitude merely up to an additive subtraction constant.
In accordance with this arbitrariness, we calculate the amplitude in two approaches: 1) without subtraction and 2) with subtraction. We use the the zero-mass limit at M W → 0 as determined by the GBET, to determine the subtraction constant. In this latter case we perform the calculations in two ways: i) through direct calculations of the amplitude determined by the GBET, using DimReg, and ii) via the dispersion method, starting from the absorptive part of the amplitude, and thus using only finite quantities. The two completely different calculations lead us to the same subtraction constant! Furthermore, we also calculated the amplitude in the commonly used R ξ -gauge class using dimensional regularization as regularization scheme and compared the result to the one obtained via the dispersion method. The R ξ -gauge result completely coincides with the dispersion method together with the subtraction term determined by the GBET.
Thus, we have shown that the dispersion-relation approach, with a subtraction determined by the GBET, presents an alternative method for calculating the H → Z + γ amplitude (and for H → γ + γ as also shown in [3]) to the commonly used R ξ -gauge technique. However, the dispersion method has two important advantages: 1) it deals only with finite quantities and thus is free of uncertainties related to the choice of regularization and 2) it's much simpler -working in the unitary gauge effectively we deal with only 3 Feynman diagrams, while in the R ξ -gauge one has to consider more than 20 graphs.
igM g αβ 2. Feynman vertex rule for the triple Higgs -Z-boson interaction Feynman rules involving the charged Higgs ghost in the R ξ gauge

2M
Appendix B: Integrals for the absorptive part A(τ ) Here we give the integrals involved in computation of the absorptive part A(τ ) of the amplitude.
The calculations are done in the rest frame of the Higgs boson, with z-axis taken along k 1 , the kinematics as given in Sec. III. We have used also the following relations: The evaluation of A(τ ) is reduced to the following integrals:
For the other integrals we have: They are expressed in terms of the integral J 0 (x), or equivalently of the elementary function f (x): where f (x) ≡ arcsin 2 √ x. (C10) We have: When |τ − a| < 1 for the functions F (a, τ ) and G(a, τ ), that enter the amplitude (35), we have: (C18)