Amplitude of $H\to\gamma Z$ process via one W loop in unitary gauge
(I. Details of calculation with Dyson scheme)

There are totally 5 Feynman diagrams. Three are similar as those of the $H\to\gamma\gamma$ process, via the direct coupling of Higgs and W-boson loop. We will demonstrate the calculations in details. The other two are via the direct 3-point coupling of Higgs and ZZ, while one Z transiting to gamma via the W bubble (3-point vertices) and W tadpole (4-point vertex). However, these latter two diagrams sum to be zero Wu:2017rxt , and the details of the calculation will not be presented in this paper. In this calculation the same surface term of the quadratic tensor integral is encountered, which will be investigated in details for the former three diagrams in the following ⁴⁴4As a matter of fact, these diagrams are self-energy like diagrams, and by Lorentz and $U(1)_{EM}$ gauge invariant arguments should be proportional to $k_{1}^{2}g_{\mu\alpha}-k_{1\mu}k_{1\alpha}$ ($\mu$ the on shell photon index, $\alpha$ some dummy index), so are zero. But just the same as the QED photon self-energy diagram, direct calculation is non-trivial. The superficial quadratic latter also need the correct quadratic surface term to get the ’proper’ form, which was regarded only available via a gauge invariant regularization. With the same argument, and also can be shown explicitly by the help of the same quadratic surface term, the fermion bubble of $\gamma Z$ also zero since $k_{1}$ on shell, i.e., $k_{1}^{2}=0,k_{1}\cdot\epsilon=0$. The application to the fermion case in fact is even more useful since here we can freely work in 4 dimension Minkowski momentum space without the ambiguity of $\gamma^{5}$.. This zero result together with the finite result of the former three diagrams show in this 1-loop subprocess no contribution to the $Z\gamma$ mixing renormalization constant.

Figure 1 showes the three diagrams to be calculated in the following, $k_{1}$ is the 4-momentum of $\gamma$. $k_{2}$ is the 4-momentum of the Z particle, $k_{2}^{2}=M_{Z}^{2}$, with the corresponding polarization vectors $\epsilon_{Z\lambda}^{\nu}$, $\lambda=1,2,3$, since Z is massive. We still have $k_{2\nu}\epsilon_{Z\lambda}^{\nu}=0,~{}\forall\lambda$ (see Eq. 87). There is also an extra $\cot\theta_{W}$ factor for the WWZ vertex, where $\theta_{W}$ is the Weinberg angle.

As convention, The S-matrix and T-matrix have the relation $S=I+iT$, and the matrix element between initial and final states $iT_{fi}=i(2\pi)^{4}\delta(P_{f}-P_{i})\mathfrak{M}_{fi}$ for the space-time displacement invariant case. Here we keep all the momenta respectively corresponding to each propagator and hence all $\delta$ functions respectively corresponding to each vertex. The one corresponding to the initial-final state energy momentum conservation is contained in these $\delta$ functions. After integrating over them, one will get the above form of T-matrix element with the $\mathfrak{M}_{fi}$ is the integration of the independent loop momenta only, without the $\delta$ functions attached to the vertices. This is the standard procedure in developing the Dyson-Wick perturbation theory in the interaction picture Dyson:1949ha . The four-momentum conservation $\delta$ function attached to each of the vertices is the result of integration of space-time variables in the perturbative expansion of the S-matrix, and is the manifestation of space-time displacement invariance ⁵⁵5This requires not only the boundary of space-time at infinity trivial but also no point or structure singular to ’block’ the displacement. We conjecture this may be the condition to exchange the integration order of configuration space and momentum space, see footnote 1. We also would like to point out ”no point or structure singular to ’block’ the displacement” guarantees the investigation on the surface term at infinity in momentum space Li:2017hnv .. In the following, we do not integrate out the $\delta$ functions of each vertex until have to and is allowed to integrate out some of the momenta with corresponding $\delta$ functions (only for ’hidden’ case, see the following). So here we deal with the matrix elements $T_{fi}$ rather than $\mathfrak{M}_{fi}$:

	$\displaystyle T_{1}$	$\displaystyle=$	$\displaystyle\frac{-ie^{2}gM\cot\theta_{W}}{(2\pi)^{4}}\int d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$		(1)
		$\displaystyle\times$	$\displaystyle(g_{\alpha}~{}^{\beta}-\frac{q_{1\alpha}q_{1}^{\beta}}{M^{2}})(g^{\rho\sigma}-\frac{q_{3}^{\rho}q_{3}^{\sigma}}{M^{2}})(g^{\alpha\gamma}-\frac{q_{2}^{\alpha}q_{2}^{\gamma}}{M^{2}})\frac{V_{\beta\mu\rho}(q_{1},-k_{1},-q_{3})~{}V_{\sigma\nu\gamma}(q_{3},-k_{2},-q_{2})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})},$

	$\displaystyle T_{2}$	$\displaystyle=$	$\displaystyle\frac{ie^{2}gM\cot\theta_{W}}{(2\pi)^{4}}\int d^{4}q_{1}d^{4}q_{2}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-q_{2}-k_{1}-k_{2})$		(2)
		$\displaystyle\times$	$\displaystyle(g_{\alpha}~{}^{\beta}-\frac{q_{1\alpha}q_{1}^{\beta}}{M^{2}})(g^{\alpha\gamma}-\frac{q_{2}^{\alpha}q_{2}^{\gamma}}{M^{2}})\frac{2g_{\mu\nu}g_{\beta\gamma}-g_{\mu\beta}g_{\nu\gamma}-g_{\mu\gamma}g_{\nu\beta}}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})},$

	$\displaystyle T_{3}$	$\displaystyle=$	$\displaystyle\frac{-ie^{2}gM\cot\theta_{W}}{(2\pi)^{4}}\int d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{2}-q_{3})\delta(q_{3}-k_{1}-q_{2})$		(3)
		$\displaystyle\times$	$\displaystyle(g_{\alpha}~{}^{\beta}-\frac{q_{1\alpha}q_{1}^{\beta}}{M^{2}})(g^{\rho\sigma}-\frac{q_{3}^{\rho}q_{3}^{\sigma}}{M^{2}})(g^{\alpha\gamma}-\frac{q_{2}^{\alpha}q_{2}^{\gamma}}{M^{2}})\frac{V_{\beta\nu\rho}(q_{1},-k_{2},-q_{3})~{}V_{\sigma\mu\gamma}(q_{3},-k_{1},-q_{2})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})}.$

Here we do not explicitly write the matrix element subscript$~{}_{fi}$, and all the $\delta$ fuctions are understood as four-dimensional one, i.e., $\delta(P_{1}-P_{2}):=\delta^{4}(P_{1}-P_{2})$. As GWW, we also omit the polarization vector, so, e.g., $T_{1}$ should be understood as $T_{1\mu\nu}$. In all this paper, we use M to represent the W mass. The relation between $T_{1}$ and $T_{3}$, i.e., $\mu\leftrightarrow\nu$, and $k_{1}\leftrightarrow k_{2}$ is clear to be read out.

To better illustrate the calculation procedure, we list the formulae we use repeatedly in Appendix A. They correspond to Eqs. (2.5-2.12) of GWW Gastmans:2011wh and can go to GWW (2.5-2.12) by simply taking $M_{Z}=0$ since there all for photons. Similarly as the $H\to\gamma\gamma$ process, the property of 3-particle vertex, WW$\gamma$ or WWZ, which is named as Ward Identity (W.I.) in GWW, is the key role in the evaluation, because the extra terms $q^{\alpha}q^{\beta}/M^{2}$ in W propagator product with the vertex is typical of the unitary gauge. There is more subtle elements in considering these W.I.’s, since they demonstrate the special relations of the various propagator momenta provided by the concrete dynamics in the standard mode. Furthermore, the equations (A9), (A10), and first and third terms of (A7), (A8) are independent of the integrated momentum choice (the $(q_{3}^{2}-M^{2})$ term reduces with one from the denominator, so in fact corresponding to a $g_{\mu\nu}$ term).

In the following, we investigate the terms according to their minus power of M. The $\frac{-ieg^{2}M\cos\theta_{W}}{(2\pi)^{4}}$ factor will not explicitly written, and all terms should multiply with this factor to get the proper terms in the corresponding T amplitude of Eqs. (1-3). So when we mention a term, we do not take into account the overall M factor coming from the HWW coupling except explicitly addressed. Since there is still a WW$\gamma$ vertex, i.e., $V_{\beta\mu\rho}(q_{1},-k_{1},-q_{3})$ of $T_{1}$, $V_{\sigma\mu\gamma}(q_{3},-k_{1},-q_{2})$ of $T_{3}$, respectively, the $M^{-6}$ terms in $T_{1}$ and $T_{3}$ are again zero as the $H\to\gamma\gamma$ process, according to Eq. (95).

For the $M^{-4}$ terms, in $T_{1}$ and $T_{3}$, respectively, there are 3 ways of the combination of two of three $q_{i}^{\alpha_{i}}q_{i}^{\beta_{i}}/M^{2}$ ($i=1,2,3$) from the W propagators. One combination is zero because of the W.I. of the WW$\gamma$ vertex, Eq. (95). Another gives a $M_{Z}^{2}/M^{4}$ terms because of the W.I. of the WWZ vertex, Eq. (96). The third corresponds to those of the $H\to\gamma\gamma$ process, gives likely terms besides extra $M_{Z}^{2}/M^{4}$ terms.

Those from $T_{1}$:

	$\displaystyle T^{\prime\prime}_{11}$	$\displaystyle=$	$\displaystyle\frac{1}{M^{4}}\int d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$		(4)
		$\displaystyle\times$	$\displaystyle q_{1\alpha}q_{1}^{\beta}q_{3}^{\rho}q_{3}^{\sigma}g^{\alpha\gamma}\frac{V_{\beta\mu\rho}(q_{1},-k_{1},-q_{3})~{}V_{\sigma\nu\gamma}(q_{3},-k_{2},-q_{2})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})}=0,$

	$\displaystyle T^{\prime}_{11}$	$\displaystyle=$	$\displaystyle\frac{1}{M^{4}}\int d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$		(5)
		$\displaystyle\times$	$\displaystyle g_{\alpha}~{}^{\beta}q_{3}^{\rho}q_{3}^{\sigma}q_{2}^{\alpha}q_{2}^{\gamma}\frac{V_{\beta\mu\rho}(q_{1},-k_{1},-q_{3})~{}V_{\sigma\nu\gamma}(q_{3},-k_{2},-q_{2})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})}$
		$\displaystyle=$	$\displaystyle\frac{M_{Z}^{2}}{M^{4}}\int d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
		$\displaystyle\times$	$\displaystyle\frac{(-q_{1}^{2}q_{2\mu}q_{3\nu}+q_{1}\cdot q_{2}q_{1\mu}q_{3\nu})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})},$

employing the corresponding W.I.’s of Appendix A. Obviously, $T^{\prime}_{11}=0$ if $M_{Z}=0$, which is consistent with the $H\to\gamma\gamma$ case.

	$\displaystyle T_{11}$	$\displaystyle=$	$\displaystyle\frac{1}{M^{4}}\int d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$		(6)
		$\displaystyle\times$	$\displaystyle q_{1\alpha}q_{1}^{\beta}g^{\rho\sigma}q_{2}^{\alpha}q_{2}^{\gamma}\frac{V_{\beta\mu\rho}(q_{1},-k_{1},-q_{3})~{}V_{\sigma\nu\gamma}(q_{3},-k_{2},-q_{2})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})}.$

Now since $q_{1}^{\beta}V_{\beta\mu\rho}(q_{1},-k_{1},-q_{3})=(q_{3}^{2}-M^{2})g_{\mu\rho}-q_{3\mu}q_{3\rho}+M^{2}g_{\mu\rho}$, according to (93), $T_{11}=T_{111}+T_{112}+T_{113}$. For this kind of step we employ the W.I. for WW$\gamma$ (A7) in priority than WWZ vertex (A8) because it straightforwardly gives the proper minus power of M for each term.

	$\displaystyle T_{113}$	$\displaystyle=$	$\displaystyle\frac{M^{2}}{M^{4}}\int d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$		(7)
		$\displaystyle\times$	$\displaystyle q_{1}\cdot q_{2}\frac{V_{\mu\nu\gamma}(q_{3},-k_{2},-q_{2})q_{2}^{\gamma}}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})}$

is obviously $M^{-2}$ term and to be discussed with other $M^{-2}$ terms later.

	$\displaystyle T_{112}$	$\displaystyle=$	$\displaystyle\frac{1}{M^{4}}\int d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$		(8)
		$\displaystyle\times$	$\displaystyle q_{1}\cdot q_{2}q_{3\mu}\frac{q_{3}^{\sigma}V_{\sigma\nu\gamma}(q_{3},-k_{2},-q_{2})(-q_{2}^{\gamma})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})}$
		$\displaystyle=$	$\displaystyle\frac{M_{Z}^{2}}{M^{4}}\int d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
		$\displaystyle\times$	$\displaystyle\frac{q_{1}\cdot q_{2}q_{3\mu}q_{3\nu}}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})},$

	$\displaystyle T_{111}$	$\displaystyle=$	$\displaystyle\frac{1}{M^{4}}\int d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$		(9)
		$\displaystyle\times$	$\displaystyle q_{1}\cdot q_{2}\frac{V_{\mu\nu\gamma}(q_{3},-k_{2},-q_{2})q_{2}^{\gamma}}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})}$
		$\displaystyle=$	$\displaystyle\frac{1}{M^{4}}\int d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
		$\displaystyle\times$	$\displaystyle q_{1}\cdot q_{2}\frac{q_{1}\cdot q_{2}g_{\mu\nu}-q_{1\mu}q_{2\nu}+(q_{1}\cdot k_{2}-q_{2}\cdot k_{1}-k_{1}\cdot k_{2})g_{\mu\nu}-M_{Z}^{2}g_{\mu\nu}}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})}.$

Again to employ the corresponding W.I. of these two kinds of 3-particle vertices. In the last step of Eq. (9), we also take into account that, $\int dxf(x)\delta(x-a)=\int dxf(a)\delta(x-a)$, to use the relation $q_{3}=q_{1}-k_{1}=q_{2}+k_{2}$, to get

$$(q_{3}^{2}g_{\mu\nu}-q_{3\mu}q_{3\nu})=q_{1}\cdot q_{2}g_{\mu\nu}-q_{1\mu}q_{2\nu}+(q_{1}\cdot k_{2}-q_{2}\cdot k_{1}-k_{1}\cdot k_{2})g_{\mu\nu}.$$

In fact, all the Ward identities for the 3-boson vertex we use here also have employed the energy momentum conservation at the vertex.

We write

	$\displaystyle T_{11B}$	$\displaystyle=$	$\displaystyle T^{\prime}_{11}+T_{112}$
		$\displaystyle=$	$\displaystyle\frac{M_{Z}^{2}}{M^{4}}\int d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
		$\displaystyle\times$	$\displaystyle\frac{(-q_{1}^{2}q_{2\mu}q_{2\nu}+2q_{1}\cdot q_{2}q_{3\mu}q_{3\nu})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})},$

so all the $M^{-4}$ term from $T_{1}$ are now $T_{11B}$ and $T_{111}$, with the 4th and 3th order divergences only in $T_{111}$ (formally similar as those in $H\to\gamma\gamma$).

Now we come to $T_{3}$, from external variables, its relation with $T_{1}$ is $\mu\leftrightarrow\nu$, and $k_{1}\leftrightarrow k_{2}$; from internal momenta, it is $q_{1}\leftrightarrow-q_{2}$ and $q_{3}\leftrightarrow-q_{3}$. For easy to investigate, we now deal with it separately, according to the same thread of $T_{1}$. Similarly for the three ways of combination, $T^{\prime\prime}_{31}=0$, and

	$\displaystyle T^{\prime}_{31}$	$\displaystyle=$	$\displaystyle\frac{M_{Z}^{2}}{M^{4}}\int d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{2}-q_{3})\delta(q_{3}-k_{1}-q_{2})$		(11)
		$\displaystyle\times$	$\displaystyle\frac{(-q_{2}^{2}q_{1\mu}q_{1\nu}+q_{1}\cdot q_{2}q_{3\mu}q_{3\nu})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})}.$

For $T_{31}$, we also have $T_{31}=T_{311}+T_{312}+T_{313}$, with

	$\displaystyle T_{313}$	$\displaystyle=$	$\displaystyle\frac{M^{2}}{M^{4}}\int d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{2}-q_{3})\delta(q_{3}-k_{1}-q_{2})$		(12)
		$\displaystyle\times$	$\displaystyle q_{1}\cdot q_{2}\frac{q_{1}^{\beta}V_{\beta\nu\mu}(q_{1},-k_{2},-q_{3})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})}$

is obviously $M^{-2}$ term and to be discussed with other $M^{-2}$ terms later.

	$\displaystyle T_{312}$	$\displaystyle=$	$\displaystyle\frac{M_{Z}^{2}}{M^{4}}\int d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{2}-q_{3})\delta(q_{3}-k_{1}-q_{2})$		(13)
		$\displaystyle\times$	$\displaystyle\frac{q_{1}\cdot q_{2}q_{3\mu}q_{3\nu}}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})},$

	$\displaystyle T_{311}$	$\displaystyle=$	$\displaystyle\frac{1}{M^{4}}\int d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{2}-q_{3})\delta(q_{3}-k_{1}-q_{2})$		(14)
		$\displaystyle\times$	$\displaystyle q_{1}\cdot q_{2}\frac{q_{1}\cdot q_{2}g_{\mu\nu}-q_{1\nu}q_{2\mu}+(q_{1}\cdot k_{1}-q_{2}\cdot k_{2}-k_{1}\cdot k_{2})g_{\mu\nu}-M_{Z}^{2}g_{\mu\nu}}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})}.$

We write

	$\displaystyle T_{31B}$	$\displaystyle=$	$\displaystyle T^{\prime}_{31}+T_{312}$
		$\displaystyle=$	$\displaystyle\frac{M_{Z}^{2}}{M^{4}}\int d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{2}-q_{3})\delta(q_{3}-k_{1}-q_{2})$
		$\displaystyle\times$	$\displaystyle\frac{(-q_{2}^{2}q_{1\mu}q_{1\nu}+2q_{1}\cdot q_{2}q_{3\mu}q_{3\nu})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})},$

again all the $M^{-4}$ term from $T_{3}$ are now $T_{31B}$ and $T_{311}$, with the 4th and 3th order divergences only in $T_{311}$. $T_{111}$ and $T_{311}$ can be combined, since for both of them $q_{3}$ only appear in the $\delta$ functions and can be integrated out. Then the 4th and 3th order divergences cancel after summing with those of $T_{2}$ and only quadratic one proportional to $M_{Z}^{2}/M^{4}$ left (zero for $M_{Z}=0$).

	$\displaystyle T_{111}+T_{311}$	$\displaystyle=$	$\displaystyle\frac{1}{M^{4}}\int d^{4}q_{1}d^{4}q_{2}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-q_{2}-k_{1}-k_{2})$		(16)
		$\displaystyle\times$	$\displaystyle q_{1}\cdot q_{2}~{}~{}\frac{2q_{1}\cdot q_{2}g_{\mu\nu}-q_{1\mu}q_{2\nu}-q_{2\mu}q_{1\nu}-M_{Z}^{2}g_{\mu\nu}}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})},$

since $(q_{1}\cdot k_{2}-q_{2}\cdot k_{1}-k_{1}\cdot k_{2})g_{\mu\nu}+(q_{1}\cdot k_{1}-q_{2}\cdot k_{2}-k_{1}\cdot k_{2})g_{\mu\nu}=k_{2}^{2}g_{\mu\nu}=M_{Z}^{2}g_{\mu\nu}$, takeing now $q_{1}-q_{2}=k_{1}+k_{2}$ from the $\delta$ function. The $M^{-4}$ term in $T_{2}$ is

	$\displaystyle T_{21}$	$\displaystyle=$	$\displaystyle\frac{-1}{M^{4}}\int d^{4}q_{1}d^{4}q_{2}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-q_{2}-k_{1}-k_{2})$		(17)
		$\displaystyle\times$	$\displaystyle q_{1}\cdot q_{2}~{}~{}\frac{2q_{1}\cdot q_{2}g_{\mu\nu}-q_{1\mu}q_{2\nu}-q_{2\mu}q_{1\nu}}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})},$

so $T_{111}+T_{311}+T_{21}=T_{A}$:

	$\displaystyle T_{A}$	$\displaystyle=$	$\displaystyle\frac{1}{M^{4}}\int d^{4}q_{1}d^{4}q_{2}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-q_{2}-k_{1}-k_{2})\frac{-M_{Z}^{2}}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})}$		(18)
		$\displaystyle\times$	$\displaystyle q_{1}\cdot q_{2}g_{\mu\nu},$
		$\displaystyle=$	$\displaystyle\frac{M_{Z}^{2}}{M^{4}}\int d^{4}q_{1}d^{4}q_{2}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-q_{2}-k_{1}-k_{2})\frac{1}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})}$
		$\displaystyle\times$	$\displaystyle(\frac{(k_{1}+k_{2})^{2}}{2}+(-\frac{(q_{1}^{2}-M^{2})+(q_{2}^{2}-M^{2})}{2}-M^{2}))g_{\mu\nu}.$

All the uncancelled terms of $M^{-4}$ order, $T_{A}$, $T_{11B}$, $T_{31B}$, are proportional to $M_{Z}^{2}$, so are zero when Z mass goes to zero. Since $M_{Z}$ is a constant parameter in the dynamics, the result is not because of the in-proper choice of the integrated variables; as a matter of fact, all the above derivation is independent from any special choice of the integrated variable. In the integral $T_{A}$, the $q_{3}$ does not appear (or only appearing in the $\delta$ functions and is integrated out). This case, which has appeared above, is called ’hidden’ in the following.

We have

	$\displaystyle T_{11B}$	$\displaystyle=$	$\displaystyle\frac{M_{Z}^{2}}{M^{4}}\int d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$		(19)
		$\displaystyle\times$	$\displaystyle\frac{1}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})(q_{3}^{2}-M^{2})}$
		$\displaystyle\times$	$\displaystyle(-(k_{1}+k_{2})^{2}q_{3\mu}q_{3\nu}+q_{2}^{2}q_{3\mu}q_{3\nu}+(q_{3}^{2}-M^{2})k_{2\mu}q_{2\nu}+M^{2}k_{2\mu}q_{2\nu}+2k_{1}\cdot q_{3}k_{2\mu}q_{3\nu});$

	$\displaystyle T_{31B}$	$\displaystyle=$	$\displaystyle\frac{M_{Z}^{2}}{M^{4}}\int d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{2}-q_{3})\delta(q_{3}-k_{1}-q_{2})$		(20)
		$\displaystyle\times$	$\displaystyle\frac{1}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})(q_{3}^{2}-M^{2})}$
		$\displaystyle\times$	$\displaystyle(-(k_{1}+k_{2})^{2}q_{3\mu}q_{3\nu}+q_{1}^{2}q_{3\mu}q_{3\nu}-(q_{3}^{2}-M^{2})k_{2\mu}q_{1\nu}-M^{2}k_{2\mu}q_{1\nu}+2k_{1}\cdot q_{3}k_{2\mu}q_{3\nu}).$

In the above equations, $(q_{1}-q_{2})=(k_{1}+k_{2})$ is used. In the last line of each of Eqs. (18,19, 20), we have decomposed them into various terms with various orders of divergences.