FeynCalc manual (development version)

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 dDq1dDqnd^D q_1 \ldots d^D q_n 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:

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))2ε2(q2x(1)x(2))ε,Γ(ε),{x(1),x(2)}}\left\{(x(1)+x(2))^{2 \varepsilon -2} \left(-q^2 x(1) x(2)\right)^{-\varepsilon },\Gamma (\varepsilon ),\{x(1),x(2)\}\right\}

FCFeynmanParametrize[FAD[p, p - q], {p}, Names -> x, FCReplaceD -> {D -> 4 - 2 Epsilon}] 
 
Times @@ Most[%]

{(x(1)+x(2))2ε2(q2x(1)x(2))ε,Γ(ε),{x(1),x(2)}}\left\{(x(1)+x(2))^{2 \varepsilon -2} \left(-q^2 x(1) x(2)\right)^{-\varepsilon },\Gamma (\varepsilon ),\{x(1),x(2)\}\right\}

Γ(ε)(x(1)+x(2))2ε2(q2x(1)x(2))ε\Gamma (\varepsilon ) (x(1)+x(2))^{2 \varepsilon -2} \left(-q^2 x(1) x(2)\right)^{-\varepsilon }

FCFeynmanParametrize[FAD[p, p - q], {p}, Names -> x, FCReplaceD -> {D -> 4 - 2 Epsilon}, 
   FeynmanIntegralPrefactor -> "Multiloop1"] 
 
Times @@ Most[%]

{(x(1)+x(2))2ε2(q2x(1)x(2))ε,Γ(ε),{x(1),x(2)}}\left\{(x(1)+x(2))^{2 \varepsilon -2} \left(-q^2 x(1) x(2)\right)^{-\varepsilon },\Gamma (\varepsilon ),\{x(1),x(2)\}\right\}

Γ(ε)(x(1)+x(2))2ε2(q2x(1)x(2))ε\Gamma (\varepsilon ) (x(1)+x(2))^{2 \varepsilon -2} \left(-q^2 x(1) x(2)\right)^{-\varepsilon }

FCFeynmanParametrize[FAD[p, p - q], {p}, Names -> x, FCReplaceD -> {D -> 4 - 2 Epsilon}, 
   FeynmanIntegralPrefactor -> "Unity"] 
 
Times @@ Most[%]

{(x(1)+x(2))2ε2(q2x(1)x(2))ε,iπ2εΓ(ε),{x(1),x(2)}}\left\{(x(1)+x(2))^{2 \varepsilon -2} \left(-q^2 x(1) x(2)\right)^{-\varepsilon },i \pi ^{2-\varepsilon } \Gamma (\varepsilon ),\{x(1),x(2)\}\right\}

iπ2εΓ(ε)(x(1)+x(2))2ε2(q2x(1)x(2))εi \pi ^{2-\varepsilon } \Gamma (\varepsilon ) (x(1)+x(2))^{2 \varepsilon -2} \left(-q^2 x(1) x(2)\right)^{-\varepsilon }

FCFeynmanParametrize[FAD[p, p - q], {p}, Names -> x, FCReplaceD -> {D -> 4 - 2 Epsilon}, 
   FeynmanIntegralPrefactor -> "Textbook"] 
 
Times @@ Most[%]

{(x(1)+x(2))2ε2(q2x(1)x(2))ε,i22ε4πε2Γ(ε),{x(1),x(2)}}\left\{(x(1)+x(2))^{2 \varepsilon -2} \left(-q^2 x(1) x(2)\right)^{-\varepsilon },i 2^{2 \varepsilon -4} \pi ^{\varepsilon -2} \Gamma (\varepsilon ),\{x(1),x(2)\}\right\}

i22ε4πε2Γ(ε)(x(1)+x(2))2ε2(q2x(1)x(2))εi 2^{2 \varepsilon -4} \pi ^{\varepsilon -2} \Gamma (\varepsilon ) (x(1)+x(2))^{2 \varepsilon -2} \left(-q^2 x(1) x(2)\right)^{-\varepsilon }

FCFeynmanParametrize[FAD[p, p - q], {p}, Names -> x, FCReplaceD -> {D -> 4 - 2 Epsilon}, 
   FeynmanIntegralPrefactor -> "Multiloop2"] 
 
Times @@ Most[%]

{(x(1)+x(2))2ε2(q2x(1)x(2))ε,eγεΓ(ε),{x(1),x(2)}}\left\{(x(1)+x(2))^{2 \varepsilon -2} \left(-q^2 x(1) x(2)\right)^{-\varepsilon },e^{\gamma \varepsilon } \Gamma (\varepsilon ),\{x(1),x(2)\}\right\}

eγεΓ(ε)(x(1)+x(2))2ε2(q2x(1)x(2))εe^{\gamma \varepsilon } \Gamma (\varepsilon ) (x(1)+x(2))^{2 \varepsilon -2} \left(-q^2 x(1) x(2)\right)^{-\varepsilon }

FCFeynmanParametrize[FAD[{p, m}], {p}, Names -> x, FCReplaceD -> {D -> 4 - 2 Epsilon}, 
   FeynmanIntegralPrefactor -> "Multiloop2"] 
 
Times @@ Most[%] 
 
Series[%, {Epsilon, 0, 1}] // Normal // FunctionExpand 
  
 

{1,eγεΓ(ε1)(m2)1ε,{}}\left\{1,-e^{\gamma \varepsilon } \Gamma (\varepsilon -1) \left(m^2\right)^{1-\varepsilon },\{\}\right\}

eγεΓ(ε1)(m2)1ε-e^{\gamma \varepsilon } \Gamma (\varepsilon -1) \left(m^2\right)^{1-\varepsilon }

m2ε+112ε(π2m2+12m2+6m2log2(m2)12m2log(m2))+m2+m2(log(m2))\frac{m^2}{\varepsilon }+\frac{1}{12} \varepsilon \left(\pi ^2 m^2+12 m^2+6 m^2 \log ^2\left(m^2\right)-12 m^2 \log \left(m^2\right)\right)+m^2+m^2 \left(-\log \left(m^2\right)\right)