orm factors $a$ and $d$
-------------------------------------------

We calculate the form factor $a$ from Eq.  as a=\[ u \]&=&- 2g\^2 C\_F \_[{K}]{} e\^[-]{}\
&=& -2g\^2 C\_F e\^[-]{}\
&=& -2g\^2 C\_F \_[-]{}\^\
&=&- \_[-]{}\^ dk\_3 ,\[a\] where T\_2&=& \_[{k\_0}]{} ,\
T\_4&=& \_[{k\_0}]{} = -. Here we also note that in LLL, $p_{\perp}=0$. Now $T_2$ can be calculated as T\_2&=& -\
&& -\
&=& -. \_[-]{}\^ dk\_3 T\_2 &=&\
&&-(+ )- (+)\
&& -\_0\^ (+ )\
&&-\_0\^ ( + )\
&=&-\_0\^ ( + )\
&&-\_0\^ ( + )\
&=& +\^2 ’\[-2\]\
&&+\^3 \^4 ’\[-4\]-( \^2 \^2 -\^4 \^4 ’\[-4\] )+\[\]\
&=& 2+ ’\[-2\]+ ’\[-4\]-( - ’\[-4\] ). Similarly we get, \_[-]{}\^ dk\_3 T\_4 &=&- \_[-]{}\^ T\_2\
&=&-\
&=&- -( ’\[-2\]+’\[-4\]). T\_2&=& \_[{k\_0}]{} ,\
&=&-\
We note that \_[-]{}\^ n\_F\^(k\_3)&=&\_[0]{}\^ (n\_F\^++n\_F\^-),\
\_[-]{}\^ n\_F\^(k\_3)&=&\_[0]{}\^ ( n\_F\^+- n\_F\^-). Hence, \_[-]{}\^ dk\_3 T\_2&&- \_0\^\
&=&\_0\^( 2n\_B(k\_3)+n\_F(k\_3+)+n\_F(k\_3-))\
&=& 2-( (3)- ’(-4) )+O\[\^5\]. and \_[-]{}\^ dk\_3 q\_fB T\_4&=& -\_[-]{}\^ T\_2\
&=& \_[-]{}\^\
&& \_[0]{}\^ ( n\_F(k\_3+)-n\_F(k\_3-))\
&=&-( ’(-2)+’(-4))+O\[\^5\]. So the form factor $a=-d$ upto $\mathcal O[\mu^4]$ can be written as a=-d&=&- . \[ad\_final\]

Calculation of quark form factor $b$ and $c$
--------------------------------------------

Similarly one can calculate $b$ from Eq.  as b=-\[ n \]&=& 2g\^2 C\_F \_[{K}]{} e\^[-]{}\
&=& 2g\^2 C\_F e\^[-]{}k\_3\
&=& 2g\^2 C\_F \_[-]{}\^ k\_3\
&=& \_[-]{}\^ dk\_3 k\_3 , \[b\] where T\_1&=& \_[{k\_0}]{} ,\[T1\_def\]\
T\_3&=& \_[{k\_0}]{} =- . After doing the Matsubara sum, Eq.  becomes T\_1&=&\
&&\
&=& .

\_[-]{}\^ dk\_3 k\_3 T\_1 &=& I\_1 I\_1&=& \_[-]{}\^ dk\_3 . Fermion part of $I_1$ can be written as, &&-\_[-]{}\^ dk\_3\
&& -\_[0]{}\^ dk\_3\
&=& -\_[0]{}\^ dk\_3\
&=& +\[\]\[fermion\_I1\]. Bosonic part of $I_1$ is given as, &&\_[-]{}\^ dk\_3(n\_B(k\_3)-p\_3 )(-)\
&&\
&=&\
&=& (-- +)-+\[\] . \[bosonic\_I1\] After combining and , $I_1$ can be written as I\_1&=&-, \_[-]{}\^ dk\_3 k\_3 T\_3 &=&- dk\_3T\_1\
&&