FeynmanIntegralPrefactor
FeynmanIntegralPrefactor
is an option for FCFeynmanParametrize
and other functions. It denotes an implicit prefactor that has to be understood in front of a loop integral in the usual FeynAmpDenominator
-notation. The prefactor is the quantity that multiplies the loop integral measure dDq1…dDqn and plays an important role e.g. when deriving the Feynman parameter representation of the given integral. Apart from specifying an explicit value, the user may also choose from the following predefined conventions:
- “Unity” - 1 for each loop
- “Textbook” - (2π)D1 for each loop.
- “Multiloop1” - iπD/21 for each loop if the integral is Minkowskian, iπD/21 or iπ(D−1)/21 for each loop if the integral is Euclidean or Cartesian respectively.
- “Multiloop2” - like “Multiloop1” but with an extra e2(4−D)γE for each loop
The standard value is “Multiloop1”.
See also
Overview, FCFeynmanParametrize.
Examples
FCFeynmanParametrize[FAD[p, p - q], {p}, Names -> x, FCReplaceD -> {D -> 4 - 2 Epsilon}]
{(x(1)+x(2))2ε−2(−q2x(1)x(2))−ε,Γ(ε),{x(1),x(2)}}
FCFeynmanParametrize[FAD[p, p - q], {p}, Names -> x, FCReplaceD -> {D -> 4 - 2 Epsilon}]
Times @@ Most[%]
{(x(1)+x(2))2ε−2(−q2x(1)x(2))−ε,Γ(ε),{x(1),x(2)}}
Γ(ε)(x(1)+x(2))2ε−2(−q2x(1)x(2))−ε
FCFeynmanParametrize[FAD[p, p - q], {p}, Names -> x, FCReplaceD -> {D -> 4 - 2 Epsilon},
FeynmanIntegralPrefactor -> "Multiloop1"]
Times @@ Most[%]
{(x(1)+x(2))2ε−2(−q2x(1)x(2))−ε,Γ(ε),{x(1),x(2)}}
Γ(ε)(x(1)+x(2))2ε−2(−q2x(1)x(2))−ε
FCFeynmanParametrize[FAD[p, p - q], {p}, Names -> x, FCReplaceD -> {D -> 4 - 2 Epsilon},
FeynmanIntegralPrefactor -> "Unity"]
Times @@ Most[%]
{(x(1)+x(2))2ε−2(−q2x(1)x(2))−ε,iπ2−εΓ(ε),{x(1),x(2)}}
iπ2−εΓ(ε)(x(1)+x(2))2ε−2(−q2x(1)x(2))−ε
FCFeynmanParametrize[FAD[p, p - q], {p}, Names -> x, FCReplaceD -> {D -> 4 - 2 Epsilon},
FeynmanIntegralPrefactor -> "Textbook"]
Times @@ Most[%]
{(x(1)+x(2))2ε−2(−q2x(1)x(2))−ε,i22ε−4πε−2Γ(ε),{x(1),x(2)}}
i22ε−4πε−2Γ(ε)(x(1)+x(2))2ε−2(−q2x(1)x(2))−ε
FCFeynmanParametrize[FAD[p, p - q], {p}, Names -> x, FCReplaceD -> {D -> 4 - 2 Epsilon},
FeynmanIntegralPrefactor -> "Multiloop2"]
Times @@ Most[%]
{(x(1)+x(2))2ε−2(−q2x(1)x(2))−ε,eγεΓ(ε),{x(1),x(2)}}
eγεΓ(ε)(x(1)+x(2))2ε−2(−q2x(1)x(2))−ε
FCFeynmanParametrize[FAD[{p, m}], {p}, Names -> x, FCReplaceD -> {D -> 4 - 2 Epsilon},
FeynmanIntegralPrefactor -> "Multiloop2"]
Times @@ Most[%]
Series[%, {Epsilon, 0, 1}] // Normal // FunctionExpand
{1,−eγεΓ(ε−1)(m2)1−ε,{}}
−eγεΓ(ε−1)(m2)1−ε
εm2+121ε(π2m2+12m2+6m2log2(m2)−12m2log(m2))+m2+m2(−log(m2))