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Dimensional regularization of Electron self-energy from Ryder's book

Physics Asked on May 25, 2021

I am Studying Electron self-energy using Ryder’s textbook, In page 334 we can see

Defining $k’=k-pz$ and avoiding the term linear in $k’$(because it integrates to zero) gives
begin{equation}
Sigma(p)=-ie^2mu^{4-d}int_0^1dzgamma_mu({not} p-{not}p z+m)gamma^muintfrac{d^dk’}{(2pi)^d}frac{1}{[k’^2-m^2z+p^2z(1-z)]^2}.label{r2.7}end{equation}

[…] This integral is performed with the help of equation (9A.5),
giving
begin{equation}
Sigma(p)=mu^{4-d}e^2frac{Gamma(2-frac{d}{2})}{(4pi)^{d/2}}int_0^1dzgamma_mu[{not}p(1-z)+m]gamma^nu[-m^2z+p^2z(1-z)]^{d/2-2}.
end{equation}

The equation 9A.5 is
begin{equation}
intfrac{d^dp}{(p^2+2pq-m^2)^{alpha}}=(-1)^{d/2}imathpi^{d/2}frac{Gammaleft(alpha-frac{d}{2}right)}{Gamma(alpha)}frac{1}{[-q^2-m^2]^{alpha-d/2}} .tag{9A.5}
end{equation}

I don’t understand how he applied this integral (9A.5) to obtain the result
begin{equation}
Sigma(p)=mu^{4-d}e^2frac{Gamma(2-frac{d}{2})}{(4pi)^{d/2}}int_0^1dzgamma_mu[{not}p(1-z)+m]gamma^nu[-m^2z+p^2z(1-z)]^{d/2-2}.
end{equation}
please help me to get an idea.

2 Answers

It's just a matter of applying the result (9A.5) to the integral in $d^d k^prime$. In fact call $M^2 = m^2z-p^2z(1-z)$ and put $q=0$ in the integral (9A.5) $$ intfrac{d^dk'}{(2pi)^d}frac{1}{[k'^2-m^2z+p^2z(1-z)]^2} = intfrac{d^dp}{(2pi)^d}frac{1}{[p^2-M^2]^2}=frac{1}{(2pi)^d}(-1)^{d/2}ipi^{d/2}frac{Gammaleft(2-dfrac{d}{2}right)}{Gamma(2)}frac{1}{[-M^2]^{2-d/2}}$$

where we just changed the integration variable from $k^prime$ to $p$ to make it clearer from the result 9A.5. Using the fact that $Gamma(2) = 1$, using the above definition of $M^2$ and simplifying a bit you get $$frac{(-1)^{d/2}}{2^d}ipi^{-d/2}Gammaleft(2-frac{d}{2}right)[-m^2z+p^2z(1-z)]^{d/2-2} = frac{i(-1)^{d/2}}{(4pi)^{d/2}}Gammaleft(2-dfrac{d}{2}right)[-m^2z+p^2z(1-z)]^{d/2-2}$$ where we used the fact that $2^d = 4^{d/2}$

Correct answer by Davide Morgante on May 25, 2021

Compare the second integrand in the first equation with ty he in the grand in 9A5. You see that $alpha rightarrow 2$, $q rightarrow 0$, $ -m^2 rightarrow etc.$ will transform one integrand into the other. Making the same substitutions in the rhs of 9A5 should give you the desired result.

Answered by my2cts on May 25, 2021

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