The other kernel $R^{\text{in}}_{\bar{\nu}_{e}}$ is obtained from the detailed
balance, Eq. (20).

###  A.4 Pair creations/annihilations

In the pair process, $l^{-}+l^{+}\leftrightharpoons\nu+\bar{\nu}$, $\nu$ is
any one of $\nu_{e},\nu_{\mu},\nu_{\tau}$ and $l$ is either $e$ or $\mu$, and
the 4-momenta of neutrino and anti-neutrino are denoted by $q_{1}^{\alpha}$
and $q_{2}^{\alpha}$, respectively, and those of lepton and anti-lepton by
$p_{1}^{\alpha}$ and $p_{2}^{\alpha}$, respectively. The spin-averaged matrix
element squared is given in (18) and the coefficients $\alpha$ and $\beta_{i}$
are listed in Table 4. Here we presents the detailed expression of
$R_{\nu_{1}}^{\text{out}}$ in Eq. (16), which is written as

$\displaystyle R^{\text{out}}_{\nu_{1}}\left(E_{1},E_{2},\cos\theta\right)$
$\displaystyle=\iint\dfrac{d^{3}\bm{p}_{1}}{(2\pi)^{3}}\dfrac{d^{3}\bm{p}_{2}}{(2\pi)^{3}}\dfrac{\langle|\mathcal{M}|^{2}\rangle_{\text{pair}}}{16E_{1}E_{2}p_{1}^{0}p_{2}^{0}}2f_{l^{-}}\left(p_{1}^{0}\right)2f_{l^{+}}\left(p_{2}^{0}\right)\left(2\pi\right)^{4}\delta^{(4)}\left(q_{1}^{\alpha}+q_{2}^{\alpha}-p_{1}^{\alpha}-p_{2}^{\alpha}\right)$
(137)
$\displaystyle=\dfrac{2G_{F}^{2}}{(2\pi)^{2}E_{1}E_{2}}\left[\beta_{1}I_{1}\left(E_{1},E_{2},\cos\theta\right)+\beta_{2}I_{2}\left(E_{1},E_{2},\cos\theta\right)+\beta_{3}I_{3}\left(E_{1},E_{2},\cos\theta\right)\right],$
(138)

where $I_{1}$ can be cast into the following form:

$\displaystyle I_{1}\left(E_{1},E_{2},\cos\theta\right)$ $\displaystyle=\iint
d^{3}\bm{p}_{1}d^{3}\bm{p}_{2}\dfrac{1}{p_{1}^{0}p_{2}^{0}}\delta^{(4)}\left(q_{1}^{\alpha}+q_{2}^{\alpha}-p_{1}^{\alpha}-p_{2}^{\alpha}\right)f_{l^{-}}\left(p_{1}^{0}\right)f_{l^{+}}\left(p_{2}^{0}\right)\left(q_{1}\cdot
p_{1}\right)\left(q_{2}\cdot p_{2}\right)$ (139) $\displaystyle=\dfrac{2\pi
E_{1}^{2}E_{2}^{2}}{\Delta^{5}}\left(1-\cos\theta\right)^{2}\Theta(D)\left(A_{1}J_{2}+B_{1}J_{1}+C_{1}J_{0}\right),$
(140) $\displaystyle A_{1}$
$\displaystyle=E_{1}^{2}+E_{2}^{2}-E_{1}E_{2}\left(3+\cos\theta\right),$ (141)
$\displaystyle B_{1}$
$\displaystyle=E_{1}\left[-2E_{1}^{2}+E_{1}E_{2}\left(3-\cos\theta\right)+E_{2}^{2}\left(1+3\cos\theta\right)\right],$
(142) $\displaystyle C_{1}$
$\displaystyle=E_{1}^{2}\left[E_{1}^{2}+2E_{1}E_{2}\cos\theta+\dfrac{1}{2}E_{2}^{2}\left(3\cos^{2}\theta-1\right)-\dfrac{m_{l}^{2}}{2}\dfrac{1+\cos\theta}{1-\cos\theta}\dfrac{\Delta^{2}}{E_{1}^{2}}\right],$
(143) $\displaystyle\Delta$
$\displaystyle=\sqrt{E_{1}^{2}+E_{2}^{2}+2E_{1}E_{2}\cos\theta},$ (144)
$\displaystyle D$
$\displaystyle=1-\dfrac{2m_{l}^{2}}{E_{1}E_{2}\left(1-\cos\theta\right)}.$
(145)

Here again $J_{i}$’s are written in terms of the Fermi-Dirac integrals:

$\displaystyle J_{0}$
$\displaystyle=\dfrac{T}{e^{\left(E_{1}+E_{2}\right)/T}-1}\left[G_{0}\left(\beta\epsilon_{\text{min}}\right)-G_{0}\left(\beta\epsilon_{\text{max}}\right)\right],$
(146) $\displaystyle J_{1}$
$\displaystyle=\dfrac{T^{2}}{e^{\left(E_{1}+E_{2}\right)/T}-1}\left\\{\left[G_{1}\left(\beta\epsilon_{\text{min}}\right)+\beta\epsilon_{\text{min}}G_{0}\left(\beta\epsilon_{\text{min}}\right)\right]-\left[G_{1}\left(\beta\epsilon_{\text{max}}\right)+\beta\epsilon_{\text{max}}G_{0}\left(\beta\epsilon_{\text{max}}\right)\right]\right\\},$
(147) $\displaystyle J_{2}$
$\displaystyle=\dfrac{T^{3}}{e^{\left(E_{1}+E_{2}\right)/T}-1}\left\\{\left[G_{2}\left(\beta\epsilon_{\text{min}}\right)+2\beta\epsilon_{\text{min}}G_{1}\left(\beta\epsilon_{\text{min}}\right)+\left(\beta\epsilon_{\text{min}}\right)^{2}G_{0}\left(\beta\epsilon_{\text{min}}\right)\right]\right.$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left.-\left[G_{2}\left(\beta\epsilon_{\text{max}}\right)+2\beta\epsilon_{\text{max}}G_{1}\left(\beta\epsilon_{\text{max}}\right)+\left(\beta\epsilon_{\text{max}}\right)^{2}G_{0}\left(\beta\epsilon_{\text{max}}\right)\right]\right\\},$
(148) $\displaystyle\epsilon_{\text{min}}$
$\displaystyle=\max\left\\{m_{l},\epsilon_{-}\right\\},$ (149)
$\displaystyle\epsilon_{\text{max}}$
$\displaystyle=\min\left\\{E_{1}+E_{2}-m_{l},\epsilon_{+}\right\\},$ (150)
$\displaystyle\epsilon_{\pm}$
$\displaystyle=\dfrac{E_{1}+E_{2}}{2}\pm\dfrac{\Delta}{2}\sqrt{1-\dfrac{2m_{l}^{2}}{E_{1}E_{2}(1-\cos\theta)}}=\dfrac{E_{1}+E_{2}}{2}\pm\dfrac{\Delta}{2}\sqrt{D},$
(151)

with

$G_{i}\left(\beta\epsilon\right)=F_{i}\left[\beta\left(\mu_{l^{-}}+E_{1}+E_{2}-\epsilon\right)\right]-F_{i}\left[\beta\left(\mu_{l^{-}}-\epsilon\right)\right];$
(152)

$I_{2}$ and $I_{3}$ are calculated in the similar way as

$\displaystyle I_{2}\left(E_{1},E_{2},\cos\theta\right)$
$\displaystyle=I_{1}\left(E_{2},E_{1},\cos\theta\right),$ (153) $\displaystyle
I_{3}\left(E_{1},E_{2},\cos\theta\right)$ $\displaystyle=\dfrac{2\pi
m_{l}^{2}E_{1}E_{2}}{\Delta}\left(1-\cos\theta\right)\Theta\left(D\right)J_{0}.$
(154)

The other kernel $R^{\text{in}}_{\nu_{1}}$ is obtained from the detailed
balance, Eq. (21).

###  A.5 Leptonic annihilations

The two reactions, $e^{-}+\mu^{+}\leftrightarrows\nu_{e}+\bar{\nu}_{\mu}$ and
$e^{+}+\mu^{-}\leftrightarrows\bar{\nu}_{e}+\nu_{\mu}$, are collectively
denoted in this subsection by $e+\mu\leftrightarrows\nu_{e}+\nu_{\mu}$. The
calculation of $R^{\text{out}}_{{\nu_{e}}}$ in Eq. (16) proceeds similarly to
that for the muon decay. It is expressed as

$\displaystyle
R^{\text{out}}_{{\nu_{e}}}\left(E_{{\nu_{e}}},E_{\nu_{\mu}},\cos\theta\right)$
$\displaystyle=\iint\dfrac{d^{3}\bm{p}_{\mu}}{(2\pi)^{3}}\dfrac{d^{3}\bm{p}_{e}}{(2\pi)^{3}}\dfrac{\langle|\mathcal{M}|^{2}\rangle_{\text{lep.ann.}}}{16E_{{\nu_{e}}}E_{\nu_{\mu}}p_{\mu}^{0}p_{e}^{0}}2f_{e}\left(p_{e}^{0}\right)2f_{\mu}\left(p_{\mu}^{0}\right)\left(2\pi\right)^{4}\delta^{(4)}\left(p_{e}^{\alpha}+p_{\mu}^{\alpha}-q_{{\nu_{e}}}^{\alpha}-q_{\nu_{\mu}}^{\alpha}\right)$
(155)
$\displaystyle=\dfrac{8G_{F}^{2}}{(2\pi)^{2}E_{{\nu_{e}}}E_{\nu_{\mu}}}I_{1}\left(E_{{\nu_{e}}},E_{\nu_{\mu}},\cos\theta\right),$
(156)

where the spin-averaged matrix element squared is given in Eq. (19) and
$I_{1}$ can be cast into the following form:

$\displaystyle I_{1}\left(E_{{\nu_{e}}},E_{\nu_{\mu}},\cos\theta\right)$
$\displaystyle=\iint
d^{3}\bm{p}_{\mu}d^{3}\bm{p}_{e}\dfrac{1}{p_{\mu}^{0}p_{e}^{0}}\delta^{(4)}\left(p_{e}^{\alpha}+p_{\mu}^{\alpha}-q_{{\nu_{e}}}^{\alpha}-q_{\nu_{\mu}}^{\alpha}\right)2f_{e}\left(p_{e}^{0}\right)2f_{\mu}\left(p_{\mu}^{0}\right)\left(q_{{\nu_{e}}}\cdot
p_{e}\right)\left(q_{\nu_{\mu}}\cdot p_{\mu}\right)$ (157)
$\displaystyle=\dfrac{2\pi
E_{{\nu_{e}}}E_{\nu_{\mu}}}{\Delta^{5}}\Theta(D)\left(A_{1}J_{2}+B_{1}J_{1}+C_{1}J_{0}\right),$
(158)

where $\Theta$ is again the Heaviside function and other factors are given as
follows:

$\displaystyle A_{1}$
$\displaystyle=E_{{\nu_{e}}}E_{\nu_{\mu}}\left(1-\cos\theta\right)^{2}\left[E_{{\nu_{e}}}^{2}+E_{\nu_{\mu}}^{2}-E_{{\nu_{e}}}E_{\nu_{\mu}}\left(3+\cos\theta\right)\right],$
(159) $\displaystyle B_{1}$
$\displaystyle=E_{{\nu_{e}}}^{2}E_{\nu_{\mu}}\left(1-\cos\theta\right)^{2}\left[E_{\nu_{\mu}}^{2}\left(1+3\cos\theta\right)+E_{\nu_{\mu}}E_{{\nu_{e}}}\left(3-\cos\theta\right)-2E_{{\nu_{e}}}^{2}\right]$
$\displaystyle\ \ \
-Q\left(E_{{\nu_{e}}}+E_{\nu_{\mu}}\right)\left(1-\cos\theta\right)\left[E_{\nu_{\mu}}^{2}-E_{\nu_{\mu}}E_{{\nu_{e}}}\left(3+\cos\theta\right)+E_{{\nu_{e}}}^{2}\right],$
(160) $\displaystyle C_{1}$
$\displaystyle=-E_{\nu_{\mu}}E_{{\nu_{e}}}^{3}\left(1-\cos\theta\right)^{2}\left[\dfrac{1}{2}E_{\nu_{\mu}}^{2}\left(3\cos^{2}\theta-1\right)+2E_{\nu_{\mu}}E_{{\nu_{e}}}\cos\theta+E_{{\nu_{e}}}^{2}\right]-\dfrac{m_{\mu}^{2}}{2}E_{\nu_{\mu}}E_{{\nu_{e}}}\left(1-\cos^{2}\theta\right)\Delta^{2}$
$\displaystyle\ \ \
+QE_{{\nu_{e}}}\left(1-\cos\theta\right)\left[-E_{\nu_{\mu}}^{3}\cos\theta-
E_{\nu_{\mu}}^{2}E_{{\nu_{e}}}\left(2-\cos^{2}\theta\right)+E_{\nu_{\mu}}E_{{\nu_{e}}}^{2}\cos\theta+E_{{\nu_{e}}}^{3}\right]$
$\displaystyle\ \ \
-Q^{2}\left[E_{\nu_{\mu}}^{2}\cos\theta+\dfrac{1}{2}E_{\nu_{\mu}}E_{{\nu_{e}}}\left(3+\cos^{2}\theta\right)+E_{{\nu_{e}}}^{2}\cos\theta\right],$
(161) $\displaystyle\Delta$
$\displaystyle=\sqrt{E_{\nu_{\mu}}^{2}+E_{{\nu_{e}}}^{2}+2E_{\nu_{\mu}}E_{{\nu_{e}}}\cos\theta},$
(162) $\displaystyle Q$
$\displaystyle=\dfrac{1}{2}\left(m_{\mu}^{2}-m_{e}^{2}\right),$ (163)
$\displaystyle D$
$\displaystyle=\left(1+\kappa\right)^{2}-\dfrac{2m_{\mu}^{2}}{E_{\nu_{\mu}}E_{\nu_{e}}\left(1-\cos\theta\right)},$
(164) $\displaystyle\kappa$
$\displaystyle=\dfrac{Q}{E_{\nu_{\mu}}E_{\nu_{e}}\left(1-\cos\theta\right)};$
(165)

$J_{i}$’s are expressed in terms of the Fermi-Dirac integrals:

$\displaystyle J_{0}$
$\displaystyle=\dfrac{T}{1-e^{\left(E_{{\nu_{e}}}+E_{\nu_{\mu}}+\mu_{e}-\mu_{\mu}\right)/T}}\left[G_{0}\left(\beta\epsilon_{\text{min}}\right)-G_{0}\left(\beta\epsilon_{\text{max}}\right)\right],$
(166) $\displaystyle J_{1}$
$\displaystyle=\dfrac{T^{2}}{1-e^{\left(E_{{\nu_{e}}}+E_{\nu_{\mu}}+\mu_{e}-\mu_{\mu}\right)/T}}\left\\{\left[G_{1}\left(\beta\epsilon_{\text{min}}\right)+\beta\epsilon_{\text{min}}G_{0}\left(\beta\epsilon_{\text{min}}\right)\right]-\left[G_{1}\left(\beta\epsilon_{\text{max}}\right)+\beta\epsilon_{\text{max}}G_{0}\left(\beta\epsilon_{\text{max}}\right)\right]\right\\},$
(167) $\displaystyle J_{2}$
$\displaystyle=\dfrac{T^{3}}{1-e^{\left(E_{{\nu_{e}}}+E_{\nu_{\mu}}+\mu_{e}-\mu_{\mu}\right)/T}}\left\\{\left[G_{2}\left(\beta\epsilon_{\text{min}}\right)+2\beta\epsilon_{\text{min}}G_{1}\left(\beta\epsilon_{\text{min}}\right)+\left(\beta\epsilon_{\text{min}}\right)^{2}G_{0}\left(\beta\epsilon_{\text{min}}\right)\right]\right.$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \
\left.-\left[G_{2}\left(\beta\epsilon_{\text{max}}\right)+2\beta\epsilon_{\text{max}}G_{1}\left(\beta\epsilon_{\text{max}}\right)+\left(\beta\epsilon_{\text{max}}\right)^{2}G_{0}\left(\beta\epsilon_{\text{max}}\right)\right]\right\\},$
(168) $\displaystyle\epsilon_{\text{min}}$
$\displaystyle=\max\left\\{m_{\mu},\epsilon_{-}\right\\},$ (169)
$\displaystyle\epsilon_{\text{max}}$
$\displaystyle=\min\left\\{\epsilon_{+},E_{{\nu_{e}}}+E_{\nu_{\mu}}-m_{e}\right\\},$
(170) $\displaystyle\epsilon_{\pm}$
$\displaystyle=\dfrac{E_{{\nu_{e}}}+E_{\nu_{\mu}}}{2}\left(\kappa+1\right)\pm\dfrac{\Delta}{2}\sqrt{\left(1+\kappa\right)^{2}-\dfrac{2m_{\mu}^{2}}{E_{{\nu_{e}}}E_{\nu_{\mu}}(1-\cos\theta)}}=\dfrac{E_{{\nu_{e}}}+E_{\nu_{\mu}}}{2}\left(\kappa+1\right)\pm\dfrac{\Delta}{2}\sqrt{D},$
(171)

and

$G_{i}\left(\beta\epsilon\right)=F_{i}\left[\beta\left(E_{{\nu_{e}}}+E_{\nu_{\mu}}-\mu_{e}-\epsilon\right)\right]-F_{i}\left[\beta\left(\mu_{\mu}-\epsilon\right)\right].$
(172)

An appropriate limit to $\mu_{\mu}=E_{{\nu_{e}}}+E_{\nu_{\mu}}-\mu_{e}$
results in the following expressions of $J_{0}$ through $J_{2}$:

$\displaystyle J_{0}$
$\displaystyle=T\left\\{F_{-1}\left[\beta\left(\mu_{\mu}-\epsilon_{\text{min}}\right)\right]-F_{-1}\left[\beta\left(\mu_{\mu}-\epsilon_{\text{max}}\right)\right]\right\\},$
(173) $\displaystyle J_{1}$
$\displaystyle=T^{2}\,\bm{\big{(}}\left\\{F_{0}\left[\beta\left(\mu_{\mu}-\epsilon_{\text{min}}\right)\right]+\beta\epsilon_{\text{min}}F_{-1}\left[\beta\left(\mu_{\mu}-\epsilon_{\text{min}}\right)\right]\right\\}-\left\\{F_{0}\left[\beta\left(\mu_{\mu}-\epsilon_{\text{max}}\right)\right]+\beta\epsilon_{\text{max}}F_{-1}\left[\beta\left(\mu_{\mu}-\epsilon_{\text{max}}\right)\right]\right\\}\bm{\big{)}},$
(174) $\displaystyle J_{2}$
$\displaystyle=T^{3}\,\bm{\Big{(}}\left\\{2F_{1}\left[\beta\left(\mu_{\mu}-\epsilon_{\text{min}}\right)\right]+2\beta\epsilon_{\text{min}}F_{0}\left[\beta\left(\mu_{\mu}-\epsilon_{\text{min}}\right)\right]+\left(\beta\epsilon_{\text{min}}\right)^{2}F_{-1}\left[\beta\left(\mu_{\mu}-\epsilon_{\text{min}}\right)\right]\right\\}$
$\displaystyle\ \ \ \ \ \ \ \ \
-\left\\{2F_{1}\left[\beta\left(\mu_{\mu}-\epsilon_{\text{max}}\right)\right]+2\beta\epsilon_{\text{max}}F_{0}\left[\beta\left(\mu_{\mu}-\epsilon_{\text{max}}\right)\right]+\left(\beta\epsilon_{\text{max}}\right)^{2}F_{-1}\left[\beta\left(\mu_{\mu}-\epsilon_{\text{max}}\right)\right]\right\\}\bm{\Big{)}}.$
(175)

Finally, the other kernel $R^{\text{in}}_{\nu_{e}}$ is obtained from the
detailed balance, Eq. (21).

##  Appendix B Nucleon structure function

We give here the detailed calculations of the structure functions of nucleons,
which, under the current approximation, are generally written (Eq. (36)) as

$\displaystyle\mathcal{S}^{\mu\nu}\left(q^{0},q\right)=\iint\dfrac{d^{3}\bm{p}_{2}}{(2\pi)^{3}2E_{2}^{*}}\dfrac{d^{3}\bm{p}_{4}}{(2\pi)^{3}2E_{4}^{*}}f_{2}\left(E_{2}^{*}\right)\left[1-f_{4}\left(E_{4}^{*}\right)\right]\,\Lambda^{\mu\nu}\,(2\pi)^{4}\delta^{(4)}(p_{1}^{\mu}+p_{2}^{\mu}-p_{3}^{\mu}-p_{4}^{\mu}).$
(176)

We follow the procedure given in Roberts2017_muon . First, we decompose the
hadronic tensor, $\Lambda^{\mu\nu}$, as follows:

$\Lambda^{\mu\nu}=AP_{1}^{\mu\nu}+BP_{2}^{\mu\nu}+CP_{3}^{\mu\nu}+DP_{4}^{\mu\nu}+EP_{5}^{\mu\nu}+FP_{6}^{\mu\nu},$
(177)

where $P_{i}^{\mu\nu}\ (i=1,\dots,6)$ are defined with the transferred 4
momentum $q^{\alpha}=(q^{0},0,0,q)$ and another vector
$n^{\alpha}=(q,0,0,q^{0})$, orthogonal to $q^{\alpha}$:

$\displaystyle P_{1}^{\mu\nu}$
$\displaystyle=\eta^{\mu\nu}-\dfrac{1}{q_{\alpha}^{2}}q^{\mu}q^{\nu}-\dfrac{1}{n^{2}}n^{\mu}n^{\nu},$
(178) $\displaystyle P_{2}^{\mu\nu}$
$\displaystyle=\dfrac{1}{q_{\alpha}^{2}}q^{\mu}q^{\nu},$ (179) $\displaystyle
P_{3}^{\mu\nu}$ $\displaystyle=\dfrac{1}{n^{2}}n^{\mu}n^{\nu},$ (180)
$\displaystyle P_{4}^{\mu\nu}$
$\displaystyle=\dfrac{1}{q_{\alpha}^{2}}\left(q^{\mu}n^{\nu}+q^{\nu}n^{\mu}\right),$
(181) $\displaystyle P_{5}^{\mu\nu}$
$\displaystyle=\dfrac{1}{q_{\alpha}^{2}}q_{\alpha}q_{\beta}\epsilon^{\mu\nu\alpha\beta},$
(182) $\displaystyle P_{6}^{\mu\nu}$
$\displaystyle=\dfrac{1}{q_{\alpha}^{2}}\left(q^{\mu}n^{\nu}-q^{\nu}n^{\mu}\right),$
(183)

where $q_{\alpha}^{2}=q_{\alpha}q^{\alpha}$, $n^{2}=n_{\alpha}n^{\alpha}$ and
$\epsilon^{\mu\nu\alpha\beta}$ is the anti-symmetric tensor. These tensors are
orthogonal to one another and satisfy the following relations:

$\displaystyle P_{1\,\mu\nu}P_{1}^{\mu\nu}=2,\ P_{2\,\mu\nu}P_{2}^{\mu\nu}=1,\
P_{3\,\mu\nu}P_{3}^{\mu\nu}=1,\ P_{4\,\mu\nu}P_{4}^{\mu\nu}=-2,\
P_{5\,\mu\nu}P_{5}^{\mu\nu}=-2,\ P_{6\,\mu\nu}P_{6}^{\mu\nu}=2.$ (184)

Then the coefficients in the decomposition (Eq. (177)) are given as follows:

$\displaystyle A$
$\displaystyle=4m_{2}^{*}m_{4}^{*}\left(G_{V}^{2}-G_{A}^{2}\right)-4\left(G_{V}^{2}+G_{A}^{2}\right)\dfrac{1}{q_{\alpha}^{2}}\left[\left(q\cdot\tilde{p}_{2}\right)\left(q\cdot\tilde{p}_{4}\right)-\left(n\cdot\tilde{p}_{2}\right)\left(n\cdot\tilde{p}_{4}\right)\right]$
$\displaystyle\ \ \ \
+4\dfrac{F_{2}G_{V}}{M}\left[m_{2}^{*}\left(q\cdot\tilde{p}_{4}\right)-m_{4}^{*}\left(q\cdot\tilde{p}_{2}\right)\right]+\dfrac{F_{2}^{2}}{M^{2}}\left[-\left(n\cdot\tilde{p}_{2}\right)\left(n\cdot\tilde{p}_{4}\right)+m_{2}^{*}m_{4}^{*}q_{\alpha}^{2}\right],$
(185) $\displaystyle B$
$\displaystyle=4m_{2}^{*}m_{4}^{*}\left(G_{V}^{2}-G_{A}^{2}\right)+4\left(G_{V}^{2}+G_{A}^{2}\right)\dfrac{1}{q_{\alpha}^{2}}\left[2\left(q\cdot\tilde{p}_{2}\right)\left(q\cdot\tilde{p}_{4}\right)-q_{\alpha}^{2}\left(\tilde{p}_{2}\cdot\tilde{p}_{4}\right)\right]$
$\displaystyle\ \ \ \
-2\dfrac{F_{2}^{2}}{M^{2}}\left(q\cdot\tilde{p}_{2}\right)\left(q\cdot\tilde{p}_{4}\right)+8\dfrac{G_{A}G_{P}}{M}\left[m_{4}^{*}\left(q\cdot\tilde{p}_{2}\right)-m_{2}^{*}\left(q\cdot\tilde{p}_{4}\right)\right]+4\dfrac{G_{P}^{2}}{M^{2}}q_{\alpha}^{2}\left[\left(\tilde{p}_{2}\cdot\tilde{p}_{4}\right)-m_{2}^{*}m_{4}^{*}\right],$
(186) $\displaystyle C$
$\displaystyle=4m_{2}^{*}m_{4}^{*}\left(G_{V}^{2}-G_{A}^{2}\right)-4\left(G_{V}^{2}+G_{A}^{2}\right)\dfrac{1}{n^{2}}\left[\left(n\cdot\tilde{p}_{2}\right)\left(n\cdot\tilde{p}_{4}\right)-n^{2}\left(\tilde{p}_{2}\cdot\tilde{p}_{4}\right)\right]$
$\displaystyle\ \ \ \
-4\dfrac{F_{2}G_{V}}{M}\left[m_{4}^{*}\left(q\cdot\tilde{p}_{2}\right)-m_{2}^{*}\left(q\cdot\tilde{p}_{4}\right)\right]$
$\displaystyle\ \ \ \
+\dfrac{F_{2}^{2}}{M^{2}}\left[-n^{2}\left(\tilde{p}_{2}\cdot\tilde{p}_{4}\right)+2\left(n\cdot\tilde{p}_{2}\right)\left(n\cdot\tilde{p}_{4}\right)-2\left(q\cdot\tilde{p}_{2}\right)\left(q\cdot\tilde{p}_{4}\right)+m_{2}^{*}m_{4}^{*}q_{\alpha}^{2}\right],$
(187) $\displaystyle D$
$\displaystyle=-4\left(G_{V}^{2}+G_{A}^{2}\right)\dfrac{1}{q_{\alpha}^{2}}\left[\left(q\cdot\tilde{p}_{2}\right)\left(n\cdot\tilde{p}_{4}\right)+\left(q\cdot\tilde{p}_{4}\right)\left(n\cdot\tilde{p}_{2}\right)\right]+2\dfrac{F_{2}G_{V}}{M}\left[m_{2}^{*}\left(n\cdot\tilde{p}_{4}\right)-m_{4}^{*}\left(n\cdot\tilde{p}_{2}\right)\right]$
$\displaystyle\ \ \ \
+4\dfrac{G_{A}G_{P}}{M}\left[m_{2}^{*}\left(n\cdot\tilde{p}_{4}\right)-m_{4}^{*}\left(n\cdot\tilde{p}_{2}\right)\right],$
(188) $\displaystyle E$
$\displaystyle=8iG_{V}G_{A}\dfrac{1}{q_{\alpha}^{2}}\left[\left(q\cdot\tilde{p}_{4}\right)\left(n\cdot\tilde{p}_{2}\right)-\left(q\cdot\tilde{p}_{2}\right)\left(n\cdot\tilde{p}_{4}\right)\right]+4i\dfrac{F_{2}G_{A}}{M}\left[m_{2}^{*}\left(n\cdot\tilde{p}_{4}\right)+m_{4}^{*}\left(n\cdot\tilde{p}_{2}\right)\right],$
(189) $\displaystyle F$
$\displaystyle=-4iG_{V}G_{A}\dfrac{1}{q_{\alpha}^{2}}\left(q_{\mu}n_{\nu}-q_{\nu}n_{\mu}\right)\tilde{p}_{4\alpha}\tilde{p}_{2\beta}\epsilon^{\alpha\mu\beta\nu}-2i\dfrac{F_{2}G_{P}}{M^{2}}n_{\mu}\tilde{p}_{4\rho}\tilde{p}_{2\sigma}q_{\beta}\epsilon^{\rho\sigma\beta\mu}.$
(190)

Now, the calculation of the structure function is reduced to integrations over
$\bm{p}_{2}$ and $\bm{p}_{4}$ of the following 10 scalar variables:

$\displaystyle
1,\left(\tilde{p}_{2}\cdot\tilde{p}_{4}\right),\left(q\cdot\tilde{p}_{2}\right),\left(q\cdot\tilde{p}_{4}\right),\left(n\cdot\tilde{p}_{2}\right),\left(n\cdot\tilde{p}_{4}\right),\left(q\cdot\tilde{p}_{2}\right)\left(q\cdot\tilde{p}_{4}\right),\left(n\cdot\tilde{p}_{2}\right)\left(n\cdot\tilde{p}_{4}\right),\left(q\cdot\tilde{p}_{2}\right)\left(n\cdot\tilde{p}_{4}\right),\left(n\cdot\tilde{p}_{2}\right)\left(q\cdot\tilde{p}_{4}\right).$
(191)

As an example, we show the calculation for 1:

$\displaystyle\mathcal{I}_{1}\left(q^{0},q\right)=\iint\dfrac{d^{3}\bm{p}_{2}}{(2\pi)^{3}2E_{2}^{*}}\dfrac{d^{3}\bm{p}_{4}}{(2\pi)^{3}2E_{4}^{*}}f_{2}\left(E_{2}^{*}\right)\left[1-f_{4}\left(E_{4}^{*}\right)\right]\cdot
1\cdot(2\pi)^{4}\delta^{(4)}\left(p_{1}^{\mu}+p_{2}^{\mu}-p_{3}^{\mu}-p_{4}^{\mu}\right).$
(192)

The integration over $\bm{p}_{4}$ is done with the delta function as

$\displaystyle\mathcal{I}_{1}\left(q^{0},q\right)$
$\displaystyle=\dfrac{1}{16\pi^{2}}\iint\dfrac{d^{3}\bm{p}_{2}}{E_{2}^{*}}d^{3}\bm{p}_{4}\int\dfrac{dE_{4}^{*}}{E_{4}^{*}}\,\delta\left(E_{4}^{*}-\sqrt{|\bm{p}_{4}|^{2}+{m_{4}^{*}}^{2}}\right)f_{2}\left(E_{2}^{*}\right)\left[1-f_{4}\left(E_{4}^{*}\right)\right]\cdot
1\cdot\delta^{(4)}(p_{1}^{\mu}+p_{2}^{\mu}-p_{3}^{\mu}-p_{4}^{\mu})$ (193)
$\displaystyle=\dfrac{1}{8\pi^{2}}\int p_{2}\,dE_{2}^{*}\,d\cos\alpha\,d\beta
f_{2}\left(E_{2}^{*}\right)(1-f_{4}\left(E_{4}^{*}\right))\cdot
1\cdot\delta\left(\tilde{p}_{4}^{2}-{m_{4}^{*}}^{2}\right)\Theta\left(E_{4}^{*}-m_{4}^{*}\right),$
(194)

where we used the following relations,
$\frac{1}{E_{4}^{*}}\delta\left(E_{4}^{*}-\sqrt{|\bm{p}_{4}|^{2}+{m_{4}^{*}}^{2}}\right)=\delta\left(\tilde{p}_{4}^{2}-{m_{4}^{*}}^{2}\right)$,
$p_{2}dp_{2}=E_{2}^{*}dE_{2}^{*}$, and $\alpha$ and $\beta$ are the zenith and
azimuth angles of $\bm{p}_{2}$, respectively, with $\bm{q}$ being the z-axis;
we also note

$\displaystyle E_{4}^{*}$
$\displaystyle=E_{2}^{*}+U_{2}-U_{4}+q^{0}=:E_{2}^{*}+\tilde{q}^{0},$ (195)
$\displaystyle\bm{p}_{4}$ $\displaystyle=\bm{p}_{2}+\bm{q},$ (196)

and

$\displaystyle\delta\left(\tilde{p}_{4}^{2}-{m_{4}^{*}}^{2}\right)$
$\displaystyle=\dfrac{1}{|2qp_{2}|}\delta\left(\cos\alpha-\cos\alpha_{0}\right),$
(197) $\displaystyle\cos\alpha_{0}$
$\displaystyle=\dfrac{\left(\tilde{q}^{0}\right)^{2}+2\tilde{q}^{0}E_{2}^{*}-q^{2}+{m_{2}^{*}}^{2}-{m_{4}^{*}}^{2}}{2qp_{2}}.$
(198)

The integrations over $\beta$ and $\cos\alpha$ can be done to give

$\displaystyle\mathcal{I}_{1}\left(q^{0},q\right)=\dfrac{1}{8\pi|q|}\int_{E_{\text{min}}}^{E_{\text{max}}}dE_{2}^{*}f_{2}\left(E_{2}^{*}\right)\left[1-f_{4}\left(E_{4}^{*}\right)\right],$
(199)

where $E_{\text{min}}$ and $E_{\text{max}}$ are given as

$\displaystyle\begin{cases}E_{\text{min}}=\max\left\\{m_{2}^{*},m_{4}^{*}-\tilde{q}^{0},E^{*}_{2,+}\right\\},\
E_{\text{max}}=\infty&\left(\Delta^{2}:=\left(\tilde{q}^{0}\right)^{2}-q^{2}<0\right)\\\
E_{\text{min}}=\max\left\\{m_{2}^{*},m_{4}^{*}-\tilde{q}^{0},E^{*}_{2,-}\right\\},\
E_{\text{max}}=E^{*}_{2,+}&\left(\Delta^{2}>0\ \text{and if}\
\Delta^{2}<({m_{2}^{*}}^{2}-{m_{4}^{*}}^{2})\ \text{or}\
\Delta^{2}>({m_{2}^{*}}^{2}+{m_{4}^{*}}^{2})\right)\end{cases}$ (200)

with

$\displaystyle
E^{*}_{2,\pm}=-\dfrac{\tilde{q}^{0}}{2}\kappa\pm\dfrac{|q|}{2}\sqrt{\kappa^{2}-\dfrac{4{m_{2}^{*}}^{2}}{\Delta^{2}}},\
\kappa=1+\dfrac{{m_{2}^{*}}^{2}-{m_{4}^{*}}^{2}}{\Delta^{2}};$ (201)

otherwise $E_{2,\pm}^{*}$ has no real solutions for
$({m_{2}^{*}}^{2}-{m_{4}^{*}}^{2})<\Delta^{2}<({m_{2}^{*}}^{2}+{m_{4}^{*}}^{2})$
and $\mathcal{I}_{1}=0$.

The remaining integration (199) can be written with the Fermi-Dirac integral,
which can be easily evaluated numerically in the following way:

$\mathcal{I}_{1}\left(q^{0},q\right)=\dfrac{1}{8\pi|q|}J_{0},$ (202)

where $J_{0}$ as well as $J_{1}$ and $J_{2}$, which will be needed later, are
expressed as

$\displaystyle J_{0}$
$\displaystyle=\dfrac{T}{1-\exp\left[\beta\left(-\tilde{q}^{0}-\mu_{2}+\mu_{4}\right)\right]}\left[G_{0}\left(\beta
E_{\text{min}}\right)-G_{0}\left(\beta E_{\text{max}}\right)\right],$ (203)
$\displaystyle J_{1}$
$\displaystyle=\dfrac{T^{2}}{1-\exp\left[\beta\left(-\tilde{q}^{0}-\mu_{2}+\mu_{4}\right)\right]}\left\\{\left[G_{1}\left(\beta
E_{\text{min}}\right)+\beta E_{\text{min}}G_{0}\left(\beta
E_{\text{min}}\right)\right]-\left[G_{1}\left(\beta
E_{\text{max}}\right)+\beta E_{\text{max}}G_{0}\left(\beta
E_{\text{max}}\right)\right]\right\\},$ (204) $\displaystyle J_{2}$
$\displaystyle=\dfrac{T^{3}}{1-\exp\left[\beta\left(-\tilde{q}^{0}-\mu_{2}+\mu_{4}\right)\right]}\left\\{\left[G_{2}\left(\beta
E_{\text{min}}\right)+2\beta E_{\text{min}}G_{1}\left(\beta
E_{\text{min}}\right)+\left(\beta E_{\text{min}}\right)^{2}G_{0}\left(\beta
E_{\text{min}}\right)\right]\right.$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left.-\left[G_{2}\left(\beta E_{\text{max}}\right)+2\beta
E_{\text{max}}G_{1}\left(\beta E_{\text{max}}\right)+\left(\beta
E_{\text{max}}\right)^{2}G_{0}\left(\beta
E_{\text{max}}\right)\right]\right\\},$ (205)

with

$G_{i}\left(\beta
E\right)=F_{i}\left[\beta\left(\mu_{2}-E\right)\right]-F_{i}\left[\beta\left(\mu_{4}-\tilde{q}^{0}-E\right)\right].$
(206)

Taking appropriate limits, we obtain for $\tilde{q}^{0}=\mu_{4}-\mu_{2}$

$\displaystyle J_{0}$
$\displaystyle=T\left\\{F_{-1}\left[\beta\left(\mu_{2}-E_{\text{min}}\right)\right]-F_{-1}\left[\beta\left(\mu_{2}-E_{\text{max}}\right)\right]\right\\},$
(207) $\displaystyle J_{1}$
$\displaystyle=T^{2}\,\bm{\big{(}}\left\\{F_{0}\left[\beta\left(\mu_{2}-E_{\text{min}}\right)\right]+\beta
E_{\text{min}}F_{-1}\left[\beta\left(\mu_{2}-E_{\text{min}}\right)\right]\right\\}$
$\displaystyle\ \ \ \ \ \ \ \ \
-\left\\{F_{0}\left[\beta\left(\mu_{2}-E_{\text{max}}\right)\right]+\beta
E_{\text{max}}F_{-1}\left[\beta\left(\mu_{2}-E_{\text{max}}\right)\right]\right\\}\bm{\big{)}},$
(208) $\displaystyle J_{2}$
$\displaystyle=T^{3}\,\bm{\Big{(}}\left\\{2F_{1}\left[\beta\left(\mu_{2}-E_{\text{min}}\right)\right]+2\beta
E_{\text{min}}F_{0}\left[\beta\left(\mu_{2}-E_{\text{min}}\right)\right]+\left(\beta
E_{\text{min}}\right)^{2}F_{-1}\left[\beta\left(\mu_{2}-E_{\text{min}}\right)\right]\right\\}$
$\displaystyle\ \ \ \ \ \ \ \ \
-\left\\{2F_{1}\left[\beta\left(\mu_{2}-E_{\text{max}}\right)\right]+2\beta
E_{\text{max}}F_{0}\left[\beta\left(\mu_{2}-E_{\text{max}}\right)\right]+\left(\beta
E_{\text{max}}\right)^{2}F_{-1}\left[\beta\left(\mu_{2}-E_{\text{max}}\right)\right]\right\\}\bm{\Big{)}}.$
(209)

In a similar way, other integrals for the remaining 9 scalars in Eq. (191) can
be accomplished to get the followings:

$\displaystyle\mathcal{I}_{\left(\tilde{p}_{2}\cdot\tilde{p}_{4}\right)}\left(q^{0},q\right)$
$\displaystyle=\dfrac{1}{8\pi|q|}\left[{m_{2}^{*}}^{2}-\dfrac{1}{2}\left(\Delta^{2}+{m_{2}^{*}}^{2}-{m_{4}^{*}}^{2}\right)\right]J_{0},$
(210)
$\displaystyle\mathcal{I}_{\left(q\cdot\tilde{p}_{2}\right)}\left(q^{0},q\right)$
$\displaystyle=\dfrac{1}{8\pi|q|}\left(b_{\left(q\cdot\tilde{p}_{2}\right)}J_{1}+c_{\left(q\cdot\tilde{p}_{2}\right)}J_{0}\right),$
(211) $\displaystyle b_{\left(q\cdot\tilde{p}_{2}\right)}$
$\displaystyle=U_{4}-U_{2}=:\Delta U,$ (212) $\displaystyle
c_{\left(q\cdot\tilde{p}_{2}\right)}$
$\displaystyle=-\dfrac{1}{2}\left(\Delta^{2}+{m_{2}^{*}}^{2}-{m_{4}^{*}}^{2}\right),$
(213)
$\displaystyle\mathcal{I}_{\left(q\cdot\tilde{p}_{4}\right)}\left(q^{0},q\right)$
$\displaystyle=\dfrac{1}{8\pi|q|}\left(b_{\left(q\cdot\tilde{p}_{4}\right)}J_{1}+c_{\left(q\cdot\tilde{p}_{4}\right)}J_{0}\right),$
(214) $\displaystyle b_{\left(q\cdot\tilde{p}_{4}\right)}$
$\displaystyle=\Delta U,$ (215) $\displaystyle
c_{\left(q\cdot\tilde{p}_{4}\right)}$
$\displaystyle=-\dfrac{1}{2}\left(\Delta^{2}+{m_{2}^{*}}^{2}-{m_{4}^{*}}^{2}\right)+q^{0}\tilde{q}^{0}-q^{2},$
(216)
$\displaystyle\mathcal{I}_{\left(n\cdot\tilde{p}_{2}\right)}\left(q^{0},q\right)$
$\displaystyle=\dfrac{1}{8\pi|q|}\left(b_{\left(n\cdot\tilde{p}_{2}\right)}J_{1}+c_{\left(n\cdot\tilde{p}_{2}\right)}J_{0}\right),$
(217) $\displaystyle b_{\left(n\cdot\tilde{p}_{2}\right)}$
$\displaystyle=q-\dfrac{q^{0}\tilde{q}^{0}}{q},$ (218) $\displaystyle
c_{\left(n\cdot\tilde{p}_{2}\right)}$
$\displaystyle=-\dfrac{q^{0}}{2q}\left(\Delta^{2}+{m_{2}^{*}}^{2}-{m_{4}^{*}}^{2}\right),$
(219)
$\displaystyle\mathcal{I}_{\left(n\cdot\tilde{p}_{4}\right)}\left(q^{0},q\right)$
$\displaystyle=\dfrac{1}{8\pi|q|}\left(b_{\left(n\cdot\tilde{p}_{4}\right)}J_{1}+c_{\left(n\cdot\tilde{p}_{4}\right)}J_{0}\right),$
(220) $\displaystyle b_{\left(n\cdot\tilde{p}_{4}\right)}$
$\displaystyle=q-\dfrac{q^{0}\tilde{q}^{0}}{q},$ (221) $\displaystyle
c_{\left(n\cdot\tilde{p}_{4}\right)}$
$\displaystyle=-\dfrac{q^{0}}{2q}\left(\Delta^{2}+{m_{2}^{*}}^{2}-{m_{4}^{*}}^{2}\right)+q\left(\tilde{q}^{0}-q^{0}\right),$
(222)
$\displaystyle\mathcal{I}_{\left(q\cdot\tilde{p}_{2}\right)\left(q\cdot\tilde{p}_{4}\right)}\left(q^{0},q\right)$
$\displaystyle=\dfrac{1}{8\pi|q|}\left(a_{\left(q\cdot\tilde{p}_{2}\right)\left(q\cdot\tilde{p}_{4}\right)}J_{2}+b_{\left(q\cdot\tilde{p}_{2}\right)\left(q\cdot\tilde{p}_{4}\right)}J_{1}+c_{\left(q\cdot\tilde{p}_{2}\right)\left(q\cdot\tilde{p}_{4}\right)}J_{0}\right),$
(223) $\displaystyle
a_{\left(q\cdot\tilde{p}_{2}\right)\left(q\cdot\tilde{p}_{4}\right)}$
$\displaystyle=(\Delta U)^{2},$ (224) $\displaystyle
b_{\left(q\cdot\tilde{p}_{2}\right)\left(q\cdot\tilde{p}_{4}\right)}$
$\displaystyle=\Delta
U\left[\left(q^{0}\tilde{q}^{0}-q^{2}\right)-\left(\Delta^{2}+{m_{2}^{*}}^{2}-{m_{4}^{*}}^{2}\right)\right],$
(225) $\displaystyle
c_{\left(q\cdot\tilde{p}_{2}\right)\left(q\cdot\tilde{p}_{4}\right)}$
$\displaystyle=\dfrac{1}{4}\left(\Delta^{2}+{m_{2}^{*}}^{2}-{m_{4}^{*}}^{2}\right)^{2}-\dfrac{1}{2}\left(\Delta^{2}+{m_{2}^{*}}^{2}-{m_{4}^{*}}^{2}\right)\left(q^{0}\tilde{q}^{0}-q^{2}\right),$
(226)
$\displaystyle\mathcal{I}_{\left(n\cdot\tilde{p}_{2}\right)\left(n\cdot\tilde{p}_{4}\right)}\left(q^{0},q\right)$
$\displaystyle=\dfrac{1}{8\pi|q|}\left(a_{\left(n\cdot\tilde{p}_{2}\right)\left(n\cdot\tilde{p}_{4}\right)}J_{2}+b_{\left(n\cdot\tilde{p}_{2}\right)\left(n\cdot\tilde{p}_{4}\right)}J_{1}+c_{\left(n\cdot\tilde{p}_{2}\right)\left(n\cdot\tilde{p}_{4}\right)}J_{0}\right),$
(227) $\displaystyle
a_{\left(n\cdot\tilde{p}_{2}\right)\left(n\cdot\tilde{p}_{4}\right)}$
$\displaystyle=\left(q-\dfrac{q^{0}\tilde{q}^{0}}{q}\right)^{2},$ (228)
$\displaystyle
b_{\left(n\cdot\tilde{p}_{2}\right)\left(n\cdot\tilde{p}_{4}\right)}$
$\displaystyle=q\left(\tilde{q}^{0}-q^{0}\right)\left(q-\dfrac{q^{0}\tilde{q}^{0}}{q}\right)-\dfrac{q^{0}}{q}\left(\Delta^{2}+{m_{2}^{*}}^{2}-{m_{4}^{*}}^{2}\right)\left(q-\dfrac{q^{0}\tilde{q}^{0}}{q}\right),$
(229) $\displaystyle
c_{\left(n\cdot\tilde{p}_{2}\right)\left(n\cdot\tilde{p}_{4}\right)}$
$\displaystyle=\dfrac{1}{4}\left(\dfrac{q^{0}}{q}\right)^{2}\left(\Delta^{2}+{m_{2}^{*}}^{2}-{m_{4}^{*}}^{2}\right)^{2}-\dfrac{1}{2}q^{0}\left(\tilde{q}^{0}-q^{0}\right)\left(\Delta^{2}+{m_{2}^{*}}^{2}-{m_{4}^{*}}^{2}\right),$
(230)
$\displaystyle\mathcal{I}_{\left(q\cdot\tilde{p}_{2}\right)\left(n\cdot\tilde{p}_{4}\right)}\left(q^{0},q\right)$
$\displaystyle=\dfrac{1}{8\pi|q|}\left(a_{\left(q\cdot\tilde{p}_{2}\right)\left(n\cdot\tilde{p}_{4}\right)}J_{2}+b_{\left(q\cdot\tilde{p}_{2}\right)\left(n\cdot\tilde{p}_{4}\right)}J_{1}+c_{\left(q\cdot\tilde{p}_{2}\right)\left(n\cdot\tilde{p}_{4}\right)}J_{0}\right),$
(231) $\displaystyle
a_{\left(q\cdot\tilde{p}_{2}\right)\left(n\cdot\tilde{p}_{4}\right)}$
$\displaystyle=\Delta U\left(q-\dfrac{q^{0}\tilde{q}^{0}}{q}\right),$ (232)
$\displaystyle
b_{\left(q\cdot\tilde{p}_{2}\right)\left(n\cdot\tilde{p}_{4}\right)}$
$\displaystyle=-\dfrac{1}{2}\left(\Delta^{2}+{m_{2}^{*}}^{2}-{m_{4}^{*}}^{2}\right)\left(q-\dfrac{q^{0}\tilde{q}^{0}}{q}\right)+\Delta
U\left[-\dfrac{q^{0}}{2q}\left(\Delta^{2}+{m_{2}^{*}}^{2}-{m_{4}^{*}}^{2}\right)+q\left(\tilde{q}^{0}-q^{0}\right)\right],$
(233) $\displaystyle
c_{\left(q\cdot\tilde{p}_{2}\right)\left(n\cdot\tilde{p}_{4}\right)}$
$\displaystyle=-\dfrac{1}{2}\left(\Delta^{2}+{m_{2}^{*}}^{2}-{m_{4}^{*}}^{2}\right)\left[-\dfrac{q^{0}}{2q}\left(\Delta^{2}+{m_{2}^{*}}^{2}-{m_{4}^{*}}^{2}\right)+q\left(\tilde{q}^{0}-q^{0}\right)\right],$
(234)
$\displaystyle\mathcal{I}_{\left(n\cdot\tilde{p}_{2}\right)\left(q\cdot\tilde{p}_{4}\right)}\left(q^{0},q\right)$
$\displaystyle=\dfrac{1}{8\pi|q|}\left(a_{\left(n\cdot\tilde{p}_{2}\right)\left(q\cdot\tilde{p}_{4}\right)}J_{2}+b_{\left(n\cdot\tilde{p}_{2}\right)\left(q\cdot\tilde{p}_{4}\right)}J_{1}+c_{\left(n\cdot\tilde{p}_{2}\right)\left(q\cdot\tilde{p}_{4}\right)}J_{0}\right),$
(235) $\displaystyle
a_{\left(n\cdot\tilde{p}_{2}\right)\left(q\cdot\tilde{p}_{4}\right)}$
$\displaystyle=\Delta U\left(q-\dfrac{q^{0}\tilde{q}^{0}}{q}\right),$ (236)
$\displaystyle
b_{\left(n\cdot\tilde{p}_{2}\right)\left(q\cdot\tilde{p}_{4}\right)}$
$\displaystyle=-\dfrac{{q}^{0}}{2q}\left(\Delta^{2}+{m_{2}^{*}}^{2}-{m_{4}^{*}}^{2}\right)\Delta
U+\left(q-\dfrac{q^{0}\tilde{q}^{0}}{q}\right)\left[-\dfrac{1}{2}\left(\Delta^{2}+{m_{2}^{*}}^{2}-{m_{4}^{*}}^{2}\right)+q^{0}\tilde{q}^{0}-q^{2}\right],$
(237) $\displaystyle
c_{\left(n\cdot\tilde{p}_{2}\right)\left(q\cdot\tilde{p}_{4}\right)}$
$\displaystyle=-\dfrac{q^{0}}{2q}\left(\Delta^{2}+{m_{2}^{*}}^{2}-{m_{4}^{*}}^{2}\right)\left[-\dfrac{1}{2}\left(\Delta^{2}+{m_{2}^{*}}^{2}-{m_{4}^{*}}^{2}\right)+q^{0}\tilde{q}^{0}-q^{2}\right].$
(238)

The structure function is obtained in the following form:

$\displaystyle\mathcal{S}^{\mu\nu}\left(q^{0},q\right)=\bar{A}P_{1}^{\mu\nu}+\bar{B}P_{2}^{\mu\nu}+\bar{C}P_{3}^{\mu\nu}+\bar{D}P_{4}^{\mu\nu}+\bar{E}P_{5}^{\mu\nu},$
(239)

where the coefficients are written as

$\bar{X}=\iint\dfrac{d^{3}\bm{p}_{2}}{(2\pi)^{3}2E_{2}^{*}}\dfrac{d^{3}\bm{p}_{4}}{(2\pi)^{3}2E_{4}^{*}}\,f_{2}\left(E_{2}^{*}\right)\left[1-f_{4}\left(E_{4}^{*}\right)\right]\,X\,(2\pi)^{4}\delta^{(4)}(p_{1}^{\mu}+p_{2}^{\mu}-p_{3}^{\mu}-p_{4}^{\mu})\
\ \ \left(X=A,B,C,D,E\right),$ (240)

and are actually some linear combinations of $\mathcal{I}_{i}$ with
$i=1,\left(\tilde{p}_{2}\cdot\tilde{p}_{4}\right),\dots,\left(n\cdot\tilde{p}_{2}\right)\left(q\cdot\tilde{p}_{4}\right)$.
Note that the case with $X=F$ (see Eq. (190)) is omitted because it vanishes
as a result of energy momentum conservation,

$\displaystyle
q_{\mu}n_{\nu}\tilde{p}_{2\alpha}\tilde{p}_{4\beta}\epsilon^{\mu\nu\alpha\beta}$
$\displaystyle=q_{\mu}n_{\nu}\tilde{p}_{2\alpha}\left(\tilde{p}_{2\beta}+q_{\beta}+U_{\beta}\right)\epsilon^{\mu\nu\alpha\beta}$
$\displaystyle=(U_{2}-U_{4})q_{\mu}n_{\nu}\tilde{p}_{2\alpha}\epsilon^{\mu\nu\alpha
0}\ \ \ \ \left(U^{\beta}=(U_{2}-U_{4},0,0,0)\right)$ $\displaystyle=\Delta
Uq_{i}n_{j}\tilde{p}_{2k}\epsilon^{ijk}$ $\displaystyle=0.\ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \left(\because\bm{q}=(0,0,q),\bm{n}=(0,0,q^{0})\right)$ (241)

Finally the contraction of the projection tensors with the lepton tensor
yields the following results:

$\displaystyle L_{\mu\nu}P_{1}^{\mu\nu}$
$\displaystyle=-16\left[\dfrac{1}{q_{\alpha}^{2}}\left(p_{1}\cdot
q\right)\left(p_{3}\cdot q\right)+\dfrac{1}{n^{2}}\left(p_{1}\cdot
n\right)\left(p_{3}\cdot n\right)\right],$ (242) $\displaystyle
L_{\mu\nu}P_{2}^{\mu\nu}$
$\displaystyle=8\left[\dfrac{2}{q_{\alpha}^{2}}\left(p_{1}\cdot
q\right)\left(p_{3}\cdot q\right)-\left(p_{1}\cdot p_{3}\right)\right],$ (243)
$\displaystyle L_{\mu\nu}P_{3}^{\mu\nu}$
$\displaystyle=8\left[\dfrac{2}{n^{2}}\left(p_{1}\cdot
n\right)\left(p_{3}\cdot n\right)-\left(p_{1}\cdot p_{3}\right)\right],$ (244)
$\displaystyle L_{\mu\nu}P_{4}^{\mu\nu}$
$\displaystyle=\dfrac{16}{q_{\alpha}^{2}}\left[\left(p_{1}\cdot
q\right)\left(p_{3}\cdot n\right)+\left(p_{1}\cdot n\right)\left(p_{3}\cdot
q\right)\right],$ (245) $\displaystyle L_{\mu\nu}P_{5}^{\mu\nu}$
$\displaystyle=\pm i\dfrac{16}{q_{\alpha}^{2}}\left[\left(p_{1}\cdot
n\right)\left(p_{3}\cdot q\right)-\left(p_{1}\cdot q\right)\left(p_{3}\cdot
n\right)\right]\ \ \ \left(+:\text{neutrino},-:\text{antineutrino}\right).$
(246)