Appendix A: Calculation of the three-point loop integrals
Here we present the details of calculation of the three-point loop integrals defined by Eq. (16).
Using Feynman parametrization we find that
$$\begin{aligned} K_{IR}(a,b)=-\frac{4ab}{i\pi ^2}\int \limits _0^1dy \int \limits _0^1dx \int \frac{x(2\pi \mu )^{4-n}d^nl}{(l^2-x^2c_y^2(a,b))^3}\;\;\nonumber \\ \end{aligned}$$
(A.1)
with \(c_y(a,b)=a y+b(1-y)\).
The integration over l in n-dimensional space gives:
$$\begin{aligned} K_{IR}(a,b)= & {} 2ab\frac{\Gamma \left( 3-n/2\right) }{(2\sqrt{\pi }\mu )^{n-4}} \int \limits _0^1dy \int \limits _0^1dx x^{n-5}c^{n-6}_y(a,b). \nonumber \\ \end{aligned}$$
(A.2)
After integration over x, and the expansion of the obtained expressions into the Laurent series around \(n=4\) result in:
$$\begin{aligned} K_{IR}(a,b)= & {} ab \int \limits _0^1 \frac{dy}{c^2_y(a,b)}\Biggl [ \displaystyle 2P_{IR} + \log \frac{c^2_y}{\mu ^2} \Biggr ] \nonumber \\= & {} 2ab\int \limits _0^1 dy{{\mathcal {K}}}_y(a,b), \end{aligned}$$
(A.3)
where the term representing the infrared divergence in the dimensional regularization reads
$$\begin{aligned} P_{IR}=\frac{1}{n-4}+\frac{1}{2}\gamma _E+\log \frac{1}{2\sqrt{\pi }}. \end{aligned}$$
(A.4)
After substitution into (A.3) photon mass regularization \(P_{IR}=\log \mu /\lambda \) we immediately find that our definition of \(K_{IR}(a,b)\) is equal to \(K(p_i,p_j)\) defined by Eq. (I.5) of [23].
Taking into account that \(c^2_y(k_1,-p_1)=c^2_y(k_2,-p_2)=\zeta _d(y)\) and \(c^2_y(k_1,p_2)=c^2_y(k_2,p_1)=\zeta _x(y)\), where
$$\begin{aligned} \zeta _d(y)= & {} y (m^2 y-S (1-y))+M^2 (y-1)^2, \nonumber \\ \zeta _x(y)= & {} y (m^2 y+X (1-y))+M^2 (y-1)^2 \end{aligned}$$
(A.5)
we will consider only two integrals, namely \(K_d=K_{IR}(k_1,-p_1)=K_{IR}(k_2,-p_2)\) and \(K_x=K_{IR}(k_1,p_2)=K_{IR}(k_2,p_1)\), that defined in the following way:
$$\begin{aligned} K_d= & {} -\frac{S}{2} \int \limits _0^1 \frac{dy}{\zeta _d(y)}\Biggl [ \displaystyle 2P_{IR} + \log \frac{\zeta _d(y)}{\mu ^2} \Biggr ] =\int \limits _0^1 dy{{\mathcal {K}}}_d(y), \nonumber \\ K_x= & {} \frac{X}{2} \int \limits _0^1 \frac{dy}{\zeta _x(y)}\Biggl [ \displaystyle 2P_{IR} + \log \frac{\zeta _x(y)}{\mu ^2} \Biggr ] =\int \limits _0^1 dy{{\mathcal {K}}}_x(y). \nonumber \\ \end{aligned}$$
(A.6)
The expression \(\zeta _{d,x}(y)\) can be presented in the following way:
$$\begin{aligned} \zeta _{d,x}(y)=\frac{M^2}{y_1^{d,x}y_2^{d,x}}(y-y_1^{d,x})(y-y_2^{d,x}), \end{aligned}$$
(A.7)
where
$$\begin{aligned} y_1^d=\frac{2M^2}{2M^2+S+{\sqrt{\lambda _{S}}}},\; y_2^d=\frac{2M^2}{2M^2+S-{\sqrt{\lambda _{S}}}},\;\;\;\; \nonumber \\ y_1^x=\frac{2M^2}{2M^2-X+{\sqrt{\lambda _{X}}}},\; y_2^x=\frac{2M^2}{2M^2-X-{\sqrt{\lambda _{X}}}}.\;\;\;\; \end{aligned}$$
(A.8)
It should be noted that for all \(S>2mM\) the quantities \(y_{1,2}^d\) belong to the region of the integration: \(1>y_2^d>y_1^d>0\). As a result, according to Eq. (A.7) the function \(\zeta _{d}(y)\) is positive for the two segments of the integration, namely \(0<y<y_1^d\) and \(y_2^d<y<1\), and negative between them. Moreover at the points \(y=y_{1,2}^d\) the integral over y in \(K_d\) defined by Eq. (A.6) diverges.
To perform the integration over y the method suggested by Kahane [32] is used. For this purpose the integration region is broken up into five segments as it is shown in Fig. 7. The contours \(C_2\) and \(C_4\) are chosen such that \(\zeta _{d}(y)\) has negative imaginary parts.
The integration for the regions \(C_1\), \(C_3\) and \(C_5\) can be expressed via Spence’s dilogarithm
$$\begin{aligned} \mathrm{Li }_2(x)=-\int \limits _0^x\frac{\log |1-y|}{y}dy \end{aligned}$$
(A.9)
in a following way
$$\begin{aligned}&K_{d}^{C_1} = \lim _{\delta _1\rightarrow 0}\int \limits _0^{y_1^d-\delta _1} dy {{\mathcal {K}}}_d(y) =\frac{S}{4{\sqrt{\lambda _{S}}}}\nonumber \\&\quad \times \left[ 2\biggl (2P_{IR}+\log \frac{{\sqrt{\lambda _{S}}}}{\mu ^2}\biggr ) \log \frac{\delta _1(S+2M^2+{\sqrt{\lambda _{S}}})^2}{4M^2{\sqrt{\lambda _{S}}}}\right. \nonumber \\&\quad + \log ^2\delta _1 +2 \log \frac{S+2M^2+{\sqrt{\lambda _{S}}}}{2{\sqrt{\lambda _{S}}}} \log \frac{{\sqrt{\lambda _{S}}}(S+2M^2+{\sqrt{\lambda _{S}}})}{2(S+M^2+m^2)^2}\nonumber \\&\quad \left. -\log ^2\frac{M^2}{{\sqrt{\lambda _{S}}}}-4 \mathrm{Li}_2\frac{S+2M^2+{\sqrt{\lambda _{S}}}}{2{\sqrt{\lambda _{S}}}} +\frac{2}{3}\pi ^2 \right] , \nonumber \\&K_{d}^{C_3} = \lim _{\delta _{1,2}\rightarrow 0}\int \limits _{y_1^d+\delta _1}^{y_2^d-\delta _2} dy{{\mathcal {K}}}_d(y) =-\frac{S}{4{\sqrt{\lambda _{S}}}}\nonumber \\&\quad \times \left[ 2 \biggl (2P_{IR}+\log \frac{{\sqrt{\lambda _{S}}}}{\mu ^2}\biggr ) \log \frac{\delta _1\delta _2(S+M^2+m^2)^2}{\lambda _S}\right. \nonumber \\&\quad + \log ^2\delta _1 + \log ^2\delta _2 -2\log ^2 \frac{S+M^2+m^2}{{\sqrt{\lambda _{S}}}}\nonumber \\&\quad \left. +\frac{2}{3}\pi ^2 +2i\pi \log \frac{\delta _1\delta _2(S+M^2+m^2)^2}{\lambda _S} \right] , \nonumber \\ K_{d}^{C_5}&= \lim _{\delta _2\rightarrow 0}\int \limits ^1_{y_2^d+\delta _2} dy{{\mathcal {K}}}_d(y) =\frac{S}{4{\sqrt{\lambda _{S}}}}\nonumber \\&\quad \times \left[ 2\biggl (2P_{IR}+\log \frac{{\sqrt{\lambda _{S}}}}{\mu ^2}\biggr ) \log \frac{\delta _2(S+2m^2+{\sqrt{\lambda _{S}}})^2}{4m^2{\sqrt{\lambda _{S}}}} + \log ^2\delta _2 \right. \nonumber \\&\quad +2 \log \frac{S+2m^2+{\sqrt{\lambda _{S}}}}{2{\sqrt{\lambda _{S}}}} \log \frac{2m^4}{{\sqrt{\lambda _{S}}}(S+2m^2+{\sqrt{\lambda _{S}}})}\nonumber \\&\quad \left. -\log ^2\frac{m^2}{{\sqrt{\lambda _{S}}}} +4 \mathrm{Li}_2\frac{{\sqrt{\lambda _{S}}}-S-2m^2}{2{\sqrt{\lambda _{S}}}} \right] . \end{aligned}$$
(A.10)
The integration along \(C_2\) and \(C_4\) is done by replacing \(y\rightarrow r_1=y_1^d-\delta _1 \exp (-i\theta )\) and \(y\rightarrow r_2=y_2^d-\delta _1 \exp (i\theta )\) respectively:
$$\begin{aligned} K_{d}^{C_2}&= \lim _{\delta _1\rightarrow 0}\int \limits _0^{\pi } d\theta \frac{dr_1}{d\theta }{{\mathcal {K}}}_d(r_1)\nonumber \\&= -\frac{S}{4{\sqrt{\lambda _{S}}}} \left[ \pi ^2+2i\pi \biggl (2P_{IR}+\log \frac{\delta _1{\sqrt{\lambda _{S}}}}{\mu ^2}\biggr ) \Biggr ],\right. \nonumber \\ K_{d}^{C_4}&= \lim _{\delta _2\rightarrow 0}\int \limits _0^{\pi } d\theta \frac{dr_2}{d\theta }{{\mathcal {K}}}_d(r_2)\nonumber \\&=-\frac{S}{4{\sqrt{\lambda _{S}}}} \Biggl [\pi ^2+2i\pi \biggl (2P_{IR}+\log \frac{\delta _2{\sqrt{\lambda _{S}}}}{\mu ^2}\biggr ) \Biggr ].\nonumber \\ \end{aligned}$$
(A.11)
Summing up the integral over all five segments we obtained that:
$$\begin{aligned} K_{d}&=\sum _{i=1}^5K_{d}^{C_i} =\frac{S}{4{\sqrt{\lambda _{S}}}}\Biggl [\biggl (4P_{IR}+4\log \frac{m}{\mu }\nonumber \\&\quad -\log \frac{S+{\sqrt{\lambda _{S}}}}{S-{\sqrt{\lambda _{S}}}}\biggr )\log \frac{S+{\sqrt{\lambda _{S}}}}{S-{\sqrt{\lambda _{S}}}} +4\mathrm{Li}_2\frac{{\sqrt{\lambda _{S}}}-S-2m^2}{2{\sqrt{\lambda _{S}}}} \nonumber \\&\quad -4 \mathrm{Li}_2\frac{S+2M^2+{\sqrt{\lambda _{S}}}}{2{\sqrt{\lambda _{S}}}}\nonumber \\&\quad +2\log \frac{S-{\sqrt{\lambda _{S}}}}{2M^2}\log \frac{(S-{\sqrt{\lambda _{S}}})(S+2M^2-{\sqrt{\lambda _{S}}})^2}{8M^2\lambda _S}\nonumber \\&\quad -2\pi ^2 -4i\pi \biggl (2P_{IR}\nonumber \\&\quad +\log \frac{\delta _1\delta _2(S+m^2+M^2)}{\mu ^2}\biggr ) \Biggr ]. \end{aligned}$$
(A.12)
As opposed to \(\zeta _{d}(y)\) in the region \(0<y<1\) the other function \(\zeta _{x}(y)\) is always positive since \(y_2^x>y_1^x>1\) for \(2mM<X<M^2+m^2\) and \(y_1^x>1>0>y_2^x\) for \(X>M^2+m^2\). For both these situations we have:
$$\begin{aligned} K_{x}&= \frac{X}{4{\sqrt{\lambda _{X}}}}\Biggl [ \biggl (4P_{IR}+4\log \frac{m}{\mu }\nonumber \\&\quad +\log \frac{X+{\sqrt{\lambda _{X}}}}{X-{\sqrt{\lambda _{X}}}}\biggr )\log \frac{X+{\sqrt{\lambda _{X}}}}{X-{\sqrt{\lambda _{X}}}}\nonumber \\&\quad +4 \mathrm{Li}_2\frac{2M^2+{\sqrt{\lambda _{X}}}-X}{2{\sqrt{\lambda _{X}}}} \nonumber \\&\quad -4\mathrm{Li}_2\frac{X+{\sqrt{\lambda _{X}}}-2m^2}{2{\sqrt{\lambda _{X}}}}\nonumber \\&\quad -2\log \frac{X+{\sqrt{\lambda _{X}}}}{2M^2}\log \frac{(X+{\sqrt{\lambda _{X}}})(X-2M^2+{\sqrt{\lambda _{X}}})^2}{8M^2\lambda _X} \Biggr ]. \nonumber \\ \end{aligned}$$
(A.13)
Appendix B: Explicit expression for \(\theta _{ijk}^l(Q^2,\tau )\)
The quantities \(\theta _{ijk}^l(Q^2,\tau )\) read:
$$\begin{aligned} \theta ^1_{111}= & {} 4(S^2+X^2-2 Q^2 (m^2+M^2))F_{IR}, \\ \theta ^2_{111}= & {} 4S_p((Q^2-2M^2)F+2m^2F_d-(Q^2+\tau S )F_{1+} )\\&+\tau Q^2(4(m^2+M^2)-Q^2)(\tau F_d+F_{1+}) \\&+2((1+\tau ) (5 Q^2-6 S)+2(4-\tau )m^2-2\tau M^2)F_{IR}\\&+2(2 S^2+(Q^2+\tau S+4m^2)Q^2)F_{1+} +\frac{4}{1+\tau }\\&\times (S_p((Q^2+2M^2)F-2m^2F_d)+(2m^2Q^2-S X)F_{1+}), \\ \theta ^3_{111}= & {} 2 \tau (2(S_p+X)F+(\tau ^2 (M^2+m^2)\\&-(1+\tau )(\tau Q^2+4m^2))F_d +(\tau (4S-Q^2+m^2+M^2) \\&+2S+3Q^2)F_{1+})+2(4+8\tau +5\tau ^2)F_{IR} \\&+4((Q^2+2 m^2)F_{1+}+(4M^2-Q^2)F_{z_1}-(S+Q^2)F) \\&+\frac{4}{1+\tau } (XF-Q^2F_{z_1}), \\ \theta ^4_{111}= & {} -\tau (4(1+\tau )F+4F_{z_1}+(2+4\tau +3\tau ^2)(\tau F_d+F_{1+})), \\ \theta ^1_{112}= & {} 4Q^2(Q^2-2 m^2)F_{IR}, \\ \theta ^2_{112}= & {} 2Q^2\tau \biggl [ \frac{S_pF-m^2F_{1+}}{1+\tau }\\&-\tau (Q^2-2 m^2)F_d-(Q^2-m^2 )F_{1+} \biggr ] +\tau (5 Q^2-8 m^2)F_{IR}, \\ \theta ^3_{112}= & {} 2\tau (S+S_p)F+2\tau ^2F_{IR}+\frac{\tau }{2}\biggl [\tau ^2 (8 m^2-5 Q^2)F_d\\&-(5\tau Q^2+4(1-\tau )m^2)F_{1+}\biggr ] -8Q^2F_{z_1} \\&-\frac{2\tau }{1+\tau }((S_p+X)F-Q^2F_{z_1}-m^2F_{1+}), \\ \theta ^4_{112}= & {} -6\tau F_{z_1}-\tau ^3(\tau F_d+F_{1+}), \\ \theta ^1_{121}= & {} 4Q^2 (Q^2-2 m^2)F_{IR}, \\ \theta ^2_{121}= & {} \tau Q^2\biggl [2\tau (2m^2-Q^2)F_d+3F_{IR}+2(3m^2-Q^2)F_{1+}\\&+\frac{2\tau }{1+\tau }(m^2F_{1+}-S_pF)\biggr ], \\ \theta ^3_{121}= & {} 2(1+\tau )Q^2F+\tau ^2(F_{IR}\\&-\frac{3}{2}Q^2(\tau F_d+F_{1+}))-\frac{2Q^2}{1+\tau }(F+2F_{z_1}), \end{aligned}$$
$$\begin{aligned} \theta ^4_{121}= & {} -\frac{\tau ^3}{2}(\tau F_d+F_{1+}), \\ \theta ^1_{122}= & {} \frac{2Q^2}{M^2} (SX+M^2(Q^2-4m^2))F_{IR}, \\ \theta ^2_{122}= & {} \frac{1}{4M^2}\Biggl [ Q^2\biggl (8Q^2S_pF+\tau ^2(Q^2(Q^2-4M^2)\\&+16m^2M^2)F_d +(16m^2(Q^2+\tau M^2) \\&+2Q^2((3\tau -2)S-2\tau M^2)+(6+\tau )Q^4-8 \tau S^2)F_{1+}\\&-\frac{4\tau }{1+\tau }(S_p((Q^2+2M^2)F-2m^2F_d) \\&+(2m^2Q^2-SX)F_{1+})\biggr ) +2(8m^2(Q^2-\tau M^2)+2Q^2(2\tau M^2\\&-(3+4\tau )S)+(5+3\tau )Q^4+2\tau S^2)F_{IR} \Biggr ], \\ \theta ^3_{122}= & {} \frac{1}{8M^2}\Biggl [ 4Q^2((6\tau S+(1+\tau )(2M^2-5Q^2)-2X)F\\&+4M^2F_{z_1}) +(4(Q^2+S)(2Q^2-S) +16m^2(M^2+3Q^2) \\&+2\tau (8m^2(Q^2-M^2)+9Q^4+2S^2) +\tau ^2(Q^4-8M^2Q^2\\&+22Q^2S-8S^2))F_{1+} +\tau (\tau Q^2(\tau (Q^2-8M^2)-4Q^2)\\&+16m^2(\tau ^2M^2-(1+\tau )Q^2))F_d +2((1+\tau )(16m^2\\&+\tau (11Q^2-6S)) +2\tau (4Q^2+\tau M^2)+6Q^2) F_{IR} +\frac{4}{1+\tau }\\&\times ((SX-4M^2m^2)F_{1+}-Q^2(S_p+2M^2)F-4M^2Q^2F_{z_1})\Biggr ], \\ \theta ^4_{122}= & {} \frac{1}{4M^2}\biggl [4(\tau ^2(S-2Q^2)-\tau (Q^2+S)+Q^2)F+4Q^2F_{z_1}\\&+\tau (8 m^2+(1+2\tau )Q^2)(F_{1+}-\tau (1+\tau )F_d) \\&+\tau ^2 (\tau (2S-M^2)+2(1+\tau )X)F_{1+}\\&-\tau ^3(Q^2+\tau M^2)F_d+4\tau (\tau +1)^2F_{IR}\biggr ], \\ \theta ^5_{122}= & {} -\frac{1+\tau }{4M^2} \biggl [\tau ^2 (2F+(1+\tau )(\tau F_d+F_{1+}))-4F_{z_1}\biggr ], \\ \theta ^3_{211}= & {} 2(Q^2+(\tau -1)S)F-2Q^2F_{z_1}+ \tau ^2F_{IR}\\&+\frac{2}{1+\tau }(XF-Q^2F_{z_1}), \\ \theta ^4_{211}= & {} -\tau (2 F_{z_1}+\frac{\tau ^2}{2} (F_d \tau +F_{1+})), \\ \theta ^2_{212}= & {} \frac{\tau S_p}{4 M^2}\biggl [ (2Q^2(Q^2+4 m^2)+(\tau -1)(\lambda _S+\lambda _X))F_d-Q^2S_pF_{1+}\\&-\frac{1}{1+\tau }(Q^2 S_pF_{1+}-2Q^2(Q^2+4 M^2)F -(\lambda _S+\lambda _X)F_d) \biggr ], \\ \theta ^3_{212}= & {} \frac{1}{4M^2}\biggl [ 2(4\tau X(Q^2+M^2)+Q^2(\tau Q^2-4S_p))F\\&+4Q^4F_{z_1}+(\tau ^3((S_p+8m^2)M^2-3Q^4+8Q^2S-6S^2)\\&+2\tau (Q^2+4 m^2)(2\tau X-Q^2)+\tau S_p((1-\tau )M^2-Q^2))F_d\\&+(8m^2Q^2+2\tau \lambda _S +\tau Q^2(6S-4Q^2-M^2) \\&-\tau ^2(Q^2 (M^2-Q^2)+4 X^2))F_{1+} \\&-\frac{\tau }{1+\tau }(Q^2 (M^2(8F_{z_1}-F_d) +2Q^2(F+2F_{z_1})) \\&+(2M^2S-Q^2S_p)(4F+F_d) +(\lambda _S+\lambda _X-M^2 Q^2)F_{1+} ) \biggr ], \\ \theta ^4_{212}= & {} \frac{1}{4M^2} \biggl [2(\tau -1)(\tau S_p-2(\tau +2)Q^2)F+8(Q^2-\tau M^2)F_{z_1} \end{aligned}$$
$$\begin{aligned}&+\tau ^2(4(1+\tau )(\tau S-2m^2) -\tau ^2M^2 \nonumber \\&-(1+5\tau +3\tau ^2 )Q^2)F_d+\tau (\tau (2(1+2\tau )S-\tau M^2)\nonumber \\&-(3+5\tau +3\tau ^2)Q^2+8 m^2)F_{1+} \biggr ], \nonumber \\ \theta ^5_{212}= & {} \frac{1+\tau }{4 M^2} (4F_{z_1}-\tau ^2(2 F+(1+\tau ) (\tau F_d+F_{1+}))), \nonumber \\ \theta ^2_{221}= & {} \frac{\tau S_p}{4M^2}\biggl [ Q^2S_pF_{1+}-(2Q^2(Q^2+4 m^2) {\!}+{\!}(\tau -1) (\lambda _S+\lambda _X))F_d \nonumber \\&+\frac{1}{1+\tau }(2 Q^2 (4 M^2+Q^2)F +(\lambda _S+\lambda _X)F_d-Q^2S_p F_{1+} )\biggr ], \nonumber \\ \theta ^3_{221}= & {} \frac{1}{4M^2}\biggl [ 4Q^2((\tau M^2 -(1+2\tau )S_p)F+4Q^4F_{z_1} \nonumber \\&+\tau (2 (Q^2+4 m^2) (Q^2-2\tau X) +2\tau ^2(XS_p+\lambda _X) \nonumber \\&+(1-\tau +\tau ^2)M^2S_p)F_d+(2 \tau ^2 (XS_p+\lambda _X) \nonumber \\&+Q^2 ((1+\tau )(8m^2-\tau M^2)-2 Q^2-4\tau S_p))F_{1+} \biggl ]\nonumber \\&+\frac{\tau }{4(1+\tau )}(Q^2(4 F+8 F_{z_1}+F_{1+})-S_pF_d),\nonumber \\ \theta ^4_{221}= & {} \frac{1}{4M^2}\biggl [ 2((1+\tau ) (2+5 \tau )Q^2-2\tau (2+3\tau )S)F\nonumber \\&+4(1+2 \tau )Q^2F_{z_1} +\tau ^2(2(1+\tau ) (Q^2-2\tau X+4m^2) \nonumber \\&-\tau ^2(S_p+M^2))F_d +\tau ((1+\tau )(8m^2-6\tau X)\nonumber \\&-\tau ^2(Q^2+M^2) )F_{1+} \biggr ],\nonumber \\ \theta ^5_{221}= & {} \frac{\tau (\tau +1)}{4 M^2}(4(1+\tau )F+4 F_{z_1}+\tau (1+2\tau )(\tau F_d+F_{1+})), \nonumber \\ \theta ^3_{222}= & {} \frac{1}{4M^2}\biggl [Q^2(2(Q^2+2M^2)(4F_{z_1}-\tau F) -S_p(\tau (1+\tau ^2)F_d\nonumber \\&+4(3+\tau )F)) +(\tau (1+\tau ) (\lambda _S+\lambda _X) \nonumber \\&-(2+\tau )Q^2(Q^2-8 m^2) )F_{1+} +\frac{\tau }{1+\tau } \bigl (Q^2 (S_p(4 F+F_d)\nonumber \\&-2(Q^2+2M^2)(F+2F_{z_1}))-(\lambda _S+\lambda _X)F_{1+}\bigr ) \biggr ],\nonumber \\ \theta ^4_{222}= & {} \frac{1}{4M^2} \biggl [4(3+3\tau +\tau ^2)Q^2F+4(3+2\tau )(Q^2F_{z_1}- \tau S F) \nonumber \\&+\tau ^2((1+\tau )Q^2-2 \tau ^2 X)F_d \nonumber \\&+\tau (2 (2+\tau ) (4 m^2-\tau S)-(3-\tau -2\tau ^2)Q^2)F_{1+}\biggr ], \nonumber \\ \theta ^5_{222}= & {} \frac{1+\tau }{4M^2}(2\tau (2+\tau )F+4(1+\tau )F_{z_1}+\tau ^3 (\tau F_d+F_{1+})). \end{aligned}$$
(B.1)
Here \(S_p=S+X=2S-Q^2\), and F and \(F_i\) (\(i=d,1+,z_1,IR\)) can be expressed through the invariant as
$$\begin{aligned} F= & {} \frac{1}{\sqrt{\lambda _q }}, \nonumber \\ F_d= & {} \frac{1}{2\pi \sqrt{\lambda _q }}\int \limits _0^{2\pi }\frac{d\phi _k}{z_1z_2} =\frac{1}{\tau }\Biggl (\frac{1}{C_2}-\frac{1}{C_1}\Biggr ), \nonumber \\ F_{1+}= & {} \frac{1}{2\pi \sqrt{\lambda _q }}\int \limits _0^{2\pi }d\phi _k \Biggl (\frac{1}{z_1}+\frac{1}{z_2}\Biggr ) =\frac{1}{C_1}+\frac{1}{C_2}, \nonumber \\ F_{z_1}= & {} \frac{1}{2\pi \sqrt{\lambda _q }}\int \limits _0^{2\pi }z_1d\phi _k \nonumber \\= & {} \frac{Q^2(S_p-v)+\tau (S(Q^2+v)+2M^2Q^2)}{\lambda _q^{3/2}}, \nonumber \\ F_{IR}= & {} \frac{R^2}{2\pi \sqrt{\lambda _q}}\int \limits _0^{2\pi }d\phi _k{\mathcal {F}}_{IR}=\frac{\tau ^2S_p F_d-(2+\tau )Q^2F_{1+}}{2(1+\tau ) }, \end{aligned}$$
(B.2)
where \({{\mathcal {F}}}_{IR}\) is defined by Eq. (35), while \(z_{1,2}=kk_{1,2}/kp_1\) depends on \(\phi _k\) as
$$\begin{aligned} z_1= & {} \frac{1}{\lambda _q}\Biggl [ Q^2(S_p-v) +\tau ((Q^2+v)S+2M^2Q^2)-2M\sqrt{\lambda _z}\cos \phi _k\Biggr ], \nonumber \\ z_2= & {} \frac{1}{\lambda _q}\Biggl [ Q^2(S_p-v) +\tau ((Q^2+v)(X-v)-2M^2Q^2) \nonumber \\&-2M\sqrt{\lambda _z}\cos \phi _k\Biggr ], \nonumber \\ \lambda _z= & {} (\tau ^q_{max}-\tau )(\tau -\tau ^q_{min})(Q^2(S(X-v)-M^2Q^2) -m^2\lambda _q), \end{aligned}$$
(B.3)
and \(\tau ^q_{max/min}\) defined by Eq. (40).
At last
$$\begin{aligned} C_1= & {} \sqrt{4m^2M^2(\tau ^q_{max}-\tau )(\tau -\tau ^q_{min}) +(Q^2+\tau S)^2}\,, \nonumber \\ C_2= & {} \sqrt{4m^2M^2(\tau ^q_{max}-\tau )(\tau -\tau ^q_{min})+(Q^2+\tau (v-X))^2}\,. \end{aligned}$$
(B.4)
Appendix C: Calculation of \(\delta _S\)
The real photon phase space in the dimensional regularization has a form
$$\begin{aligned} \frac{d^3k }{k_0}\rightarrow & {} \frac{d^{n-1}k^\prime }{(2\pi \mu )^{n-4}k_0^\prime } \nonumber \\= & {} \frac{2\pi ^{n/2-1}k_0^{\prime n-3}dk_0^\prime \sin ^{n-3}\theta _k d\theta _k }{(2\pi \mu )^{n-4}\Gamma (n/2-1)}, \end{aligned}$$
(C.1)
where \(\theta _k\) is defined as the spatial angle between the photon three-momentum and \(\mathbf{k}_{1,2}^\prime \) or \(\mathbf{k}_{s,x}^\prime \) that are introduced below, and \(\mu \) is an arbitrary parameter of the dimension of a mass. Here and later the upper prime index means that the energy or three-momentum is defined in the system \(\mathbf{p}_1\mathbf{+ q=0}\).
The Feynman parametrization of the propagators in \({{\mathcal {F}}}_{IR}\):
$$\begin{aligned} {{\mathcal {F}}}_{IR}= & {} \frac{1}{4k^{\prime 2}_0}\int \limits _0^1dy \Biggl [\frac{2}{1-x\beta _1}-\frac{2}{1-x\beta _2} -\frac{S}{k_{s0}^{\prime 2}(1-x\beta _s)^2} \nonumber \\&+\frac{X}{k_{x0}^{\prime 2}(1-x\beta _x)^2}\Biggr ] =\frac{1}{4k^{\prime 2}_0}\int \limits _0^1dy {{\mathcal {F}}}(x,y), \end{aligned}$$
(C.2)
where y is the Feynman parameter, \(x=\cos \theta _k \), \(\beta _i=|\mathbf{k}^{\prime }_i|/k_{i0}^{\prime }\) for \(i=1,2,s,x\), \(k_{s0}^{\prime }=yk_{10}^{\prime }+(1-y)p_{10}^{\prime }\) and \(k_{x0}^{\prime }=yk_{20}^{\prime }+(1-y)p_{10}^{\prime }\).
After substituting it into definition of \(\delta _S\) in (37) and, using \(\delta \) function, integration of the obtained result over the photon energy we found that
$$\begin{aligned} \delta _S= & {} \frac{1}{2(4\mu \sqrt{\pi })^{n-4}\Gamma (n/2-1)}\int \limits _{-1}^1dx(1-x^2)^{n/2-2} \nonumber \\&\times \int \limits _0^1dy {{\mathcal {F}}}(x,y) \int \limits _0^{{\bar{v}}}\frac{dv}{v}\left( \frac{v}{M}\right) ^{n-4}. \end{aligned}$$
(C.3)
Then, the integration over v, and expansion of the obtained expression into the Laurent series around \(n=4\) result in
$$\begin{aligned} \delta _S= & {} \delta _S^{IR}+\delta _S^1, \end{aligned}$$
(C.4)
where
$$\begin{aligned} \delta _S^{IR}= & {} \frac{1}{2}\biggl [P_{IR}+\log \frac{{{\bar{v}}}}{\mu M }\biggr ] \int \limits _0^1dy \int \limits _{-1}^1dx {{\mathcal {F}}}(x,y) \end{aligned}$$
(C.5)
and
$$\begin{aligned} \delta _S^1= & {} \frac{1}{4} \int \limits _0^1dy \int \limits _{-1}^1dx\log \biggl (\frac{1-x^2}{4}\biggr ) {\mathcal F}(x,y). \end{aligned}$$
(C.6)
Here \(P_{IR}\) is the infrared divergent term defined by Eq. (A.4). Since \(k_{s0}^{\prime 2}-|\mathbf{k}^\prime _s|^2=m_y^2(S)\) and \(k_{x0}^{\prime 2}-|\mathbf{k}^\prime _x|^2=m_y^2(X)\) where
$$\begin{aligned} m_y^2(S)=y(1-y)S+y^2m^2+(1-y)^2M^2, \end{aligned}$$
(C.7)
the integration over x and y variables in \(\delta _S^{IR}\) is performed explicitly:
$$\begin{aligned} \delta _S^{IR}= & {} 2(XL_X-SL_S)\biggl [P_{IR}+\log \frac{{{\bar{v}}}}{\mu M }\biggr ], \end{aligned}$$
(C.8)
where the quantities \(L_S\), and \(L_X\) are defined by Eqs. (20).
For the covariant analytical integration in \(\delta _S^1\) we express the initial and final lepton energies through the invariants:
$$\begin{aligned} k_{10}^\prime =\frac{X }{2M},\qquad k_{20}^\prime =\frac{S }{2M}. \end{aligned}$$
(C.9)
As a result,
$$\begin{aligned} \delta _S^1= & {} \frac{1}{4} S\sqrt{\lambda _S}L_S^2 -\frac{1}{4} X\sqrt{\lambda _X}L_X^2 \nonumber \\&+\frac{S}{\sqrt{\lambda _S}}\mathrm{Li}_2\biggl (\frac{2\sqrt{\lambda _S}}{S+\sqrt{\lambda _S}}\biggr ) -\frac{X}{\sqrt{\lambda _X}}\mathrm{Li}_2\biggl (\frac{2\sqrt{\lambda _X}}{X+\sqrt{\lambda _X}}\biggr ) \nonumber \\&+S_\phi (k_2,p_1,p_2) -S_\phi (k_1,p_1,p_2) . \end{aligned}$$
(C.10)
The function \(S_\phi \) can be expressed through the integration over x and y as
$$\begin{aligned} S_\phi (k_{1,2},p_1,p_2)= & {} \frac{k_{1,2}p_1}{2}\int \limits _{-1}^1dx\int \limits _0^1dy \frac{\log [(1-x^2)/4]}{k^{\prime 2}_{s0,x0}(1-x \beta _{s,x})^2}. \nonumber \\ \end{aligned}$$
(C.11)
The explicit expression for \(S_\phi \) can be found in Appendix B of [22].
