Contour Integral over Heaviside Function
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I am stuck on how to do a problem where I am deriving the Green's function for a nuclear scattering system. I am currently starting with the expression
$$G^+(mathbf{0})=frac{2m}{(2pi)^2}int dmathbf{p} frac{Theta(Lambda^2-mathbf{p}^2)}{mathbf{p}^2-mathbf{k}^2-iepsilon}$$
Where the heaviside function is introduced as a cut-off parameter. I started by evaluating the contour where the poles are located at $mathbf{p}=pm(mathbf{k}-iepsilon)$. I factored the denominator, but I am stuck on how to evaluate the residue given the Heaviside function in the argument:
$$Res{G^+(mathbf{0})}=lim_{pto p^+}({mathbf{p}-mathbf{k}-iepsilon})frac{Theta(Lambda^2-mathbf{p}^2)}{(mathbf{p}-mathbf{k}-iepsilon)(mathbf{p}+mathbf{k}-iepsilon)}=frac{Theta(Lambda^2-mathbf{p}^2)}{2mathbf{k}-2iepsilon}$$
Where the $2iepsilon$ term will go to zero after taking the limit $epsilonrightarrow0$.
The answer involves a $log(frac{Lambda^2}{-mathbf{k}^2})$, which confuses me more as to how I should evaluate this integral. Any help is appreciated!
contour-integration greens-function step-function
asked Jan 2 at 23:13
SKBSKB
62
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add a comment |
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I am stuck on how to do a problem where I am deriving the Green's function for a nuclear scattering system. I am currently starting with the expression
$$G^+(mathbf{0})=frac{2m}{(2pi)^2}int dmathbf{p} frac{Theta(Lambda^2-mathbf{p}^2)}{mathbf{p}^2-mathbf{k}^2-iepsilon}$$
Where the heaviside function is introduced as a cut-off parameter. I started by evaluating the contour where the poles are located at $mathbf{p}=pm(mathbf{k}-iepsilon)$. I factored the denominator, but I am stuck on how to evaluate the residue given the Heaviside function in the argument:
$$Res{G^+(mathbf{0})}=lim_{pto p^+}({mathbf{p}-mathbf{k}-iepsilon})frac{Theta(Lambda^2-mathbf{p}^2)}{(mathbf{p}-mathbf{k}-iepsilon)(mathbf{p}+mathbf{k}-iepsilon)}=frac{Theta(Lambda^2-mathbf{p}^2)}{2mathbf{k}-2iepsilon}$$
Where the $2iepsilon$ term will go to zero after taking the limit $epsilonrightarrow0$.
The answer involves a $log(frac{Lambda^2}{-mathbf{k}^2})$, which confuses me more as to how I should evaluate this integral. Any help is appreciated!
contour-integration greens-function step-function
asked Jan 2 at 23:13
SKBSKB
62
$endgroup$
add a comment |
$begingroup$
I am stuck on how to do a problem where I am deriving the Green's function for a nuclear scattering system. I am currently starting with the expression
$$G^+(mathbf{0})=frac{2m}{(2pi)^2}int dmathbf{p} frac{Theta(Lambda^2-mathbf{p}^2)}{mathbf{p}^2-mathbf{k}^2-iepsilon}$$
Where the heaviside function is introduced as a cut-off parameter. I started by evaluating the contour where the poles are located at $mathbf{p}=pm(mathbf{k}-iepsilon)$. I factored the denominator, but I am stuck on how to evaluate the residue given the Heaviside function in the argument:
$$Res{G^+(mathbf{0})}=lim_{pto p^+}({mathbf{p}-mathbf{k}-iepsilon})frac{Theta(Lambda^2-mathbf{p}^2)}{(mathbf{p}-mathbf{k}-iepsilon)(mathbf{p}+mathbf{k}-iepsilon)}=frac{Theta(Lambda^2-mathbf{p}^2)}{2mathbf{k}-2iepsilon}$$
Where the $2iepsilon$ term will go to zero after taking the limit $epsilonrightarrow0$.
The answer involves a $log(frac{Lambda^2}{-mathbf{k}^2})$, which confuses me more as to how I should evaluate this integral. Any help is appreciated!
contour-integration greens-function step-function
asked Jan 2 at 23:13
SKBSKB
62
$endgroup$
I am stuck on how to do a problem where I am deriving the Green's function for a nuclear scattering system. I am currently starting with the expression
$$G^+(mathbf{0})=frac{2m}{(2pi)^2}int dmathbf{p} frac{Theta(Lambda^2-mathbf{p}^2)}{mathbf{p}^2-mathbf{k}^2-iepsilon}$$
Where the heaviside function is introduced as a cut-off parameter. I started by evaluating the contour where the poles are located at $mathbf{p}=pm(mathbf{k}-iepsilon)$. I factored the denominator, but I am stuck on how to evaluate the residue given the Heaviside function in the argument:
$$Res{G^+(mathbf{0})}=lim_{pto p^+}({mathbf{p}-mathbf{k}-iepsilon})frac{Theta(Lambda^2-mathbf{p}^2)}{(mathbf{p}-mathbf{k}-iepsilon)(mathbf{p}+mathbf{k}-iepsilon)}=frac{Theta(Lambda^2-mathbf{p}^2)}{2mathbf{k}-2iepsilon}$$
Where the $2iepsilon$ term will go to zero after taking the limit $epsilonrightarrow0$.
The answer involves a $log(frac{Lambda^2}{-mathbf{k}^2})$, which confuses me more as to how I should evaluate this integral. Any help is appreciated!
contour-integration greens-function step-function
contour-integration greens-function step-function
asked Jan 2 at 23:13
SKBSKB
62
asked Jan 2 at 23:13
SKBSKB
62
asked Jan 2 at 23:13
SKBSKB
62
asked Jan 2 at 23:13
SKBSKB
62
asked Jan 2 at 23:13
SKBSKB
62
62
add a comment |
add a comment |
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