proba -- Sage

reset() SK3ll=1 var("m_l,m_a,E_nu,E_a,m_nu") SK3ull=(m_l^2-m_a^2+m_nu^2-2*E_nu*(E_nu-E_a))/(2*m_l*sqrt(E_nu^2-m_nu^2)) SK3ulul=((E_a-E_nu)^2-m_l^2)/m_l^2 SK3ual=(m_l^2-m_a^2-m_nu^2+2*E_a*E_nu)/(2*m_a*sqrt(E_nu^2-m_nu^2)) SK3uaua=(E_a^2-m_a^2)/m_a^2 SK3ulul=((E_a-E_nu)^2-m_l^2)/m_l^2 G_a=E_a/m_a G_l=(E_a-E_nu)/m_l SK3uaul=(m_nu^2-m_a^2-m_l^2+2*E_a*(E_a-E_nu))/(2*m_a*m_l) SK4uaul=(m_a^2 +m_l^2-m_nu^2)/(2*m_a*m_l) # VEKuaul=sqrt(-1/(4*m_a^2*m_l^2)*(4*E_a^2*m_nu^2-4*E_a*E_nu*m_a^2+4*E_a*E_nu*m_l^2-4*E_a*E_nu*m_nu^2+4*E_nu^2*m_a^2+m_a^4 - 2*m_a^2*m_l^2-2*m_a^2*m_nu^2+m_l^4-2*m_l^2*m_nu^2+m_nu^4)) SK3papl=m_a*m_l*SK3uaul SK3papa=m_a^2*SK3uaua SK3plpl=m_l^2*SK3ulul VEKpapl=m_a*m_l*VEKuaul # var("sigma_a,sigma_l,sigma_2") modR_s=sigma_a^2*sigma_l^2*sigma_2^4/(m_a^2*m_l^2)*((E_a-E_nu)^2*SK3papa-2*(E_a-E_nu)*E_a*SK3papl+E_a^2*SK3plpl-VEKpapl^2)

correctirovkabezcoefficientov=(SK3ull^2 - SK3ulul)*sigma_2^2*sigma_l^4 + (SK3ual^2 - SK3uaua)*sigma_2^2*sigma_a^4 + (G_a^2*SK3ll*SK3ulul - 2*G_a*G_l*SK3ll*SK3uaul + G_l^2*SK3ll*SK3uaua - 3*G_a^2*SK3ulul + 6*G_a*G_l*SK3uaul - 3*G_l^2*SK3uaua - SK3ll*VEKuaul^2 + 2*SK3ual*SK3ull*SK4uaul - 2*SK3uaul*SK4uaul + 3*VEKuaul^2)*sigma_2^2*sigma_a^2*sigma_l^2

fullcorrectirovka=1/(2*modR_s)*correctirovkabezcoefficientov

result=fullcorrectirovka(m_nu=0).expand().simplify_rational().factor().collect(sigma_a^2).collect(sigma_l^2).collect(sigma_2^2).collect(sigma_2^4).collect(sigma_l^4).collect(sigma_a^4)

show(result)

\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{{\left(4 \, E_{a} E_{\nu} m_{a}^{2} m_{l}^{2} - 4 \, E_{a} E_{\nu} m_{l}^{4} - 4 \, E_{\nu}^{2} m_{a}^{2} m_{l}^{2} - m_{a}^{4} m_{l}^{2} + 2 \, m_{a}^{2} m_{l}^{4} - m_{l}^{6}\right)} \sigma_{a}^{2}}{2 \, {\left(m_{a} - m_{l}\right)}^{2} {\left(m_{a} + m_{l}\right)}^{2} E_{\nu}^{2} \sigma_{2}^{2} \sigma_{l}^{2}} - \frac{{\left(4 \, E_{a} E_{\nu} m_{a}^{4} - 4 \, E_{a} E_{\nu} m_{a}^{2} m_{l}^{2} - 4 \, E_{\nu}^{2} m_{a}^{4} - m_{a}^{6} + 2 \, m_{a}^{4} m_{l}^{2} - m_{a}^{2} m_{l}^{4}\right)} \sigma_{l}^{2}}{2 \, {\left(m_{a} - m_{l}\right)}^{2} {\left(m_{a} + m_{l}\right)}^{2} E_{\nu}^{2} \sigma_{2}^{2} \sigma_{a}^{2}} - \frac{4 \, E_{a} E_{\nu} m_{a}^{4} - 4 \, E_{a} E_{\nu} m_{l}^{4} - 2 \, E_{\nu}^{2} m_{a}^{4} - 8 \, E_{\nu}^{2} m_{a}^{2} m_{l}^{2} + 2 \, E_{\nu}^{2} m_{l}^{4} - m_{a}^{6} + m_{a}^{4} m_{l}^{2} + m_{a}^{2} m_{l}^{4} - m_{l}^{6}}{2 \, {\left(m_{a} - m_{l}\right)}^{2} {\left(m_{a} + m_{l}\right)}^{2} E_{\nu}^{2} \sigma_{2}^{2}}

\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{{\left(4 \, E_{a} E_{\nu} m_{a}^{2} m_{l}^{2} - 4 \, E_{a} E_{\nu} m_{l}^{4} - 4 \, E_{\nu}^{2} m_{a}^{2} m_{l}^{2} - m_{a}^{4} m_{l}^{2} + 2 \, m_{a}^{2} m_{l}^{4} - m_{l}^{6}\right)} \sigma_{a}^{2}}{2 \, {\left(m_{a} - m_{l}\right)}^{2} {\left(m_{a} + m_{l}\right)}^{2} E_{\nu}^{2} \sigma_{2}^{2} \sigma_{l}^{2}} - \frac{{\left(4 \, E_{a} E_{\nu} m_{a}^{4} - 4 \, E_{a} E_{\nu} m_{a}^{2} m_{l}^{2} - 4 \, E_{\nu}^{2} m_{a}^{4} - m_{a}^{6} + 2 \, m_{a}^{4} m_{l}^{2} - m_{a}^{2} m_{l}^{4}\right)} \sigma_{l}^{2}}{2 \, {\left(m_{a} - m_{l}\right)}^{2} {\left(m_{a} + m_{l}\right)}^{2} E_{\nu}^{2} \sigma_{2}^{2} \sigma_{a}^{2}} - \frac{4 \, E_{a} E_{\nu} m_{a}^{4} - 4 \, E_{a} E_{\nu} m_{l}^{4} - 2 \, E_{\nu}^{2} m_{a}^{4} - 8 \, E_{\nu}^{2} m_{a}^{2} m_{l}^{2} + 2 \, E_{\nu}^{2} m_{l}^{4} - m_{a}^{6} + m_{a}^{4} m_{l}^{2} + m_{a}^{2} m_{l}^{4} - m_{l}^{6}}{2 \, {\left(m_{a} - m_{l}\right)}^{2} {\left(m_{a} + m_{l}\right)}^{2} E_{\nu}^{2} \sigma_{2}^{2}}

print(latex(result))

-\frac{{\left(4 \, E_{a} E_{\nu} m_{a}^{2} m_{l}^{2} - 4 \, E_{a}
E_{\nu} m_{l}^{4} - 4 \, E_{\nu}^{2} m_{a}^{2} m_{l}^{2} - m_{a}^{4}
m_{l}^{2} + 2 \, m_{a}^{2} m_{l}^{4} - m_{l}^{6}\right)}
\sigma_{a}^{2}}{2 \, {\left(m_{a} - m_{l}\right)}^{2} {\left(m_{a} +
m_{l}\right)}^{2} E_{\nu}^{2} \sigma_{2}^{2} \sigma_{l}^{2}} -
\frac{{\left(4 \, E_{a} E_{\nu} m_{a}^{4} - 4 \, E_{a} E_{\nu} m_{a}^{2}
m_{l}^{2} - 4 \, E_{\nu}^{2} m_{a}^{4} - m_{a}^{6} + 2 \, m_{a}^{4}
m_{l}^{2} - m_{a}^{2} m_{l}^{4}\right)} \sigma_{l}^{2}}{2 \,
{\left(m_{a} - m_{l}\right)}^{2} {\left(m_{a} + m_{l}\right)}^{2}
E_{\nu}^{2} \sigma_{2}^{2} \sigma_{a}^{2}} - \frac{4 \, E_{a} E_{\nu}
m_{a}^{4} - 4 \, E_{a} E_{\nu} m_{l}^{4} - 2 \, E_{\nu}^{2} m_{a}^{4} -
8 \, E_{\nu}^{2} m_{a}^{2} m_{l}^{2} + 2 \, E_{\nu}^{2} m_{l}^{4} -
m_{a}^{6} + m_{a}^{4} m_{l}^{2} + m_{a}^{2} m_{l}^{4} - m_{l}^{6}}{2 \,
{\left(m_{a} - m_{l}\right)}^{2} {\left(m_{a} + m_{l}\right)}^{2}
E_{\nu}^{2} \sigma_{2}^{2}}

-\frac{{\left(4 \, E_{a} E_{\nu} m_{a}^{2} m_{l}^{2} - 4 \, E_{a} E_{\nu} m_{l}^{4} - 4 \, E_{\nu}^{2} m_{a}^{2} m_{l}^{2} - m_{a}^{4} m_{l}^{2} + 2 \, m_{a}^{2} m_{l}^{4} - m_{l}^{6}\right)} \sigma_{a}^{2}}{2 \, {\left(m_{a} - m_{l}\right)}^{2} {\left(m_{a} + m_{l}\right)}^{2} E_{\nu}^{2} \sigma_{2}^{2} \sigma_{l}^{2}} - \frac{{\left(4 \, E_{a} E_{\nu} m_{a}^{4} - 4 \, E_{a} E_{\nu} m_{a}^{2} m_{l}^{2} - 4 \, E_{\nu}^{2} m_{a}^{4} - m_{a}^{6} + 2 \, m_{a}^{4} m_{l}^{2} - m_{a}^{2} m_{l}^{4}\right)} \sigma_{l}^{2}}{2 \, {\left(m_{a} - m_{l}\right)}^{2} {\left(m_{a} + m_{l}\right)}^{2} E_{\nu}^{2} \sigma_{2}^{2} \sigma_{a}^{2}} - \frac{4 \, E_{a} E_{\nu} m_{a}^{4} - 4 \, E_{a} E_{\nu} m_{l}^{4} - 2 \, E_{\nu}^{2} m_{a}^{4} - 8 \, E_{\nu}^{2} m_{a}^{2} m_{l}^{2} + 2 \, E_{\nu}^{2} m_{l}^{4} - m_{a}^{6} + m_{a}^{4} m_{l}^{2} + m_{a}^{2} m_{l}^{4} - m_{l}^{6}}{2 \, {\left(m_{a} - m_{l}\right)}^{2} {\left(m_{a} + m_{l}\right)}^{2} E_{\nu}^{2} \sigma_{2}^{2}}

reset() var("Ea,Ec,Eb_zv,ma,pa,pb,mb,Eb,mc") costheta=(Ea*Eb-Eb_zv*ma)/(pa*pb) sinthetavkv=(1-costheta^2) sinthetanewvkv=sinthetavkv(pa=sqrt(Ea^2-ma^2),pb=sqrt(Eb^2-mb^2))(Eb=Ea-Ec) sinthetavkv=sinthetanewvkv vektorproizvkv=((Ea^2-ma^2)*((Ea-Ec)^2-mb^2)/(ma^2*mb^2))*sinthetavkv latex(vektorproizvkv(Eb_zv=(ma^2+mb^2-mc^2)/(2*ma)).simplify_rational().factor())

-\frac{4 \, \mbox{Ea}^{2} \mbox{mc}^{2} - 4 \, \mbox{Ea} \mbox{Ec}
\mbox{ma}^{2} + 4 \, \mbox{Ea} \mbox{Ec} \mbox{mb}^{2} - 4 \, \mbox{Ea}
\mbox{Ec} \mbox{mc}^{2} + 4 \, \mbox{Ec}^{2} \mbox{ma}^{2} +
\mbox{ma}^{4} - 2 \, \mbox{ma}^{2} \mbox{mb}^{2} - 2 \, \mbox{ma}^{2}
\mbox{mc}^{2} + \mbox{mb}^{4} - 2 \, \mbox{mb}^{2} \mbox{mc}^{2} +
\mbox{mc}^{4}}{4 \, \mbox{ma}^{2} \mbox{mb}^{2}}

-\frac{4 \, \mbox{Ea}^{2} \mbox{mc}^{2} - 4 \, \mbox{Ea} \mbox{Ec} \mbox{ma}^{2} + 4 \, \mbox{Ea} \mbox{Ec} \mbox{mb}^{2} - 4 \, \mbox{Ea} \mbox{Ec} \mbox{mc}^{2} + 4 \, \mbox{Ec}^{2} \mbox{ma}^{2} + \mbox{ma}^{4} - 2 \, \mbox{ma}^{2} \mbox{mb}^{2} - 2 \, \mbox{ma}^{2} \mbox{mc}^{2} + \mbox{mb}^{4} - 2 \, \mbox{mb}^{2} \mbox{mc}^{2} + \mbox{mc}^{4}}{4 \, \mbox{ma}^{2} \mbox{mb}^{2}}

proba

452 days ago by akrobat