FCFeynmanParametrize
FCFeynmanParametrize[int, {q1, q2, ...}]
introduces Feynman parameters for the multi-loop integral int.
The function returns {fpInt,pref,vars}
, where fpInt
is the integrand in Feynman parameters, pref
is the prefactor free of Feynman parameters and vars
is the list of integration variables.
If the chosen parametrization contains a Dirac delta multiplying the integrand, it will be omitted unless the option DiracDelta
is set to True.
By default FCFeynmanParametrize
uses normalization that is common in multi-loop calculations, i.e. 1 i π D / 2 \frac{1}{i \pi^{D/2}} i π D /2 1 or 1 π D / 2 \frac{1}{\pi^{D/2}} π D /2 1 per loop for Minkowski or Euclidean/Cartesian integrals respectively.
If you want to have the standard 1 ( 2 π ) D \frac{1}{(2 \pi)^D} ( 2 π ) D 1 normalization or yet another value, please set the option FeynmanIntegralPrefactor
accordingly. Following values are available
“MultiLoop1” - default value explained above
“MultiLoop2” - like the default value but with an extra e γ E 4 − D 2 e^{\gamma_E \frac{4-D}{2}} e γ E 2 4 − D per loop
“Textbook” - 1 ( 2 π ) D \frac{1}{(2 \pi)^D} ( 2 π ) D 1 per loop
“Unity” - no extra prefactor multiplying the integral measure
“LoopTools” - overall prefactor 1 i ( π ) D / 2 r Γ \frac{1}{i (\pi)^{D/2} r_{\Gamma}} i ( π ) D /2 r Γ 1 with r Γ = Γ ( 3 − D / 2 ) Γ 2 ( D / 2 − 1 ) Γ ( D − 3 ) r_{\Gamma} = \frac{\Gamma(3-D/2) \Gamma^2 (D/2-1)}{\Gamma(D-3)} r Γ = Γ ( D − 3 ) Γ ( 3 − D /2 ) Γ 2 ( D /2 − 1 ) at 1 loop. This matches the the normalization of 1-loop integrals in LoopTools. For 2 loops and above an extra 1 i π D / 2 \frac{1}{i \pi^{D/2}} i π D /2 1 is added per loop.
The calculation of D D D -dimensional Minkowski integrals and D − 1 D-1 D − 1 -dimensional Cartesian integrals is straightforward.
To calculate a D D D -dimensional Euclidean integral (i.e. an integral defined with the Euclidean metric signature ( 1 , 1 , 1 , 1 ) (1,1,1,1) ( 1 , 1 , 1 , 1 ) you need to write it in terms of FVD
, SPD
, FAD
, SFAD
etc. and set the option "Euclidean"
to True
.
The function can derive different representations of a loop integral. The choice of the representation is controlled by the option Method
. Following representations are available
“Feynman” - the standard Feynman representation (default value). Both tensor integrals and integrals with scalar products in the numerator are supported.
“Lee-Pomeransky” - this representation was first introduced in 1308.6676 by Roman Lee and Andrei Pomeransky. Currently, only scalar integrals without numerators are supported.
FCFeynmanParametrize
can also be employed in conjunction with FCFeynmanParameterJoin
, where one first joins suitable propagators using auxiliary Feynman parameters and then finally integrates out loop momenta.
For a proper analysis of a loop integral one usually needs the U
and F
polynomials separately. Since internally FCFeynmanParametrize
uses FCFeynmanPrepare
, the information available from the latter is also accessible to FCFeynmanParametrize
.
By setting the option FCFeynmanPrepare
to True
, the output of FCFeynmanPrepare
will be added the the output of FCFeynmanParametrize
as the 4th list element.
See also
Overview , FCFeynmanPrepare , FCFeynmanProjectivize , FCFeynmanParameterJoin , SplitSymbolicPowers .
Examples
Feynman representation
1-loop tadpole
FCFeynmanParametrize[ FAD[{ q , m }], { q }, Names -> x ]
{ 1 , − ( m 2 ) D 2 − 1 Γ ( 1 − D 2 ) , { } } \left\{1,-\left(m^2\right)^{\frac{D}{2}-1} \Gamma \left(1-\frac{D}{2}\right),\{\}\right\} { 1 , − ( m 2 ) 2 D − 1 Γ ( 1 − 2 D ) , { } }
FCFeynmanParametrize[ FAD[{ q , m }], { q }, Names -> x , EtaSign -> True ]
{ 1 , − Γ ( 1 − D 2 ) ( m 2 − i η ) D 2 − 1 , { } } \left\{1,-\Gamma \left(1-\frac{D}{2}\right) \left(m^2-i \eta \right)^{\frac{D}{2}-1},\{\}\right\} { 1 , − Γ ( 1 − 2 D ) ( m 2 − i η ) 2 D − 1 , { } }
Massless 1-loop 2-point function
FCFeynmanParametrize[ FAD[ q , q - p ], { q }, Names -> x ]
{ ( x ( 1 ) + x ( 2 ) ) 2 − D ( − p 2 x ( 1 ) x ( 2 ) ) D 2 − 2 , Γ ( 2 − D 2 ) , { x ( 1 ) , x ( 2 ) } } \left\{(x(1)+x(2))^{2-D} \left(-p^2 x(1) x(2)\right)^{\frac{D}{2}-2},\Gamma \left(2-\frac{D}{2}\right),\{x(1),x(2)\}\right\} { ( x ( 1 ) + x ( 2 ) ) 2 − D ( − p 2 x ( 1 ) x ( 2 ) ) 2 D − 2 , Γ ( 2 − 2 D ) , { x ( 1 ) , x ( 2 )} }
FCFeynmanParametrize[ FAD[ q , q - p ], { q }, Names -> x , EtaSign -> True ]
{ ( x ( 1 ) + x ( 2 ) ) 2 − D ( − p 2 x ( 1 ) x ( 2 ) − i η ) D 2 − 2 , Γ ( 2 − D 2 ) , { x ( 1 ) , x ( 2 ) } } \left\{(x(1)+x(2))^{2-D} \left(-p^2 x(1) x(2)-i \eta \right)^{\frac{D}{2}-2},\Gamma \left(2-\frac{D}{2}\right),\{x(1),x(2)\}\right\} { ( x ( 1 ) + x ( 2 ) ) 2 − D ( − p 2 x ( 1 ) x ( 2 ) − i η ) 2 D − 2 , Γ ( 2 − 2 D ) , { x ( 1 ) , x ( 2 )} }
With p 2 p^2 p 2 replaced by pp
and D
set to 4 - 2 Epsilon
FCFeynmanParametrize[ FAD[ q , q - p ], { q }, Names -> x , FinalSubstitutions -> SPD[ p ] -> pp,
FCReplaceD -> { D -> 4 - 2 Epsilon}]
{ ( x ( 1 ) + x ( 2 ) ) 2 ε − 2 ( − pp x ( 1 ) x ( 2 ) ) − ε , Γ ( ε ) , { x ( 1 ) , x ( 2 ) } } \left\{(x(1)+x(2))^{2 \varepsilon -2} (-\text{pp} x(1) x(2))^{-\varepsilon },\Gamma (\varepsilon ),\{x(1),x(2)\}\right\} { ( x ( 1 ) + x ( 2 ) ) 2 ε − 2 ( − pp x ( 1 ) x ( 2 ) ) − ε , Γ ( ε ) , { x ( 1 ) , x ( 2 )} }
Standard text-book prefactor of the loop integral measure
FCFeynmanParametrize[ FAD[ q , q - p ], { q }, Names -> x , FinalSubstitutions -> SPD[ p ] -> pp,
FCReplaceD -> { D -> 4 - 2 Epsilon}, FeynmanIntegralPrefactor -> "Textbook" ]
{ ( x ( 1 ) + x ( 2 ) ) 2 ε − 2 ( − pp x ( 1 ) x ( 2 ) ) − ε , i 2 2 ε − 4 π ε − 2 Γ ( ε ) , { x ( 1 ) , x ( 2 ) } } \left\{(x(1)+x(2))^{2 \varepsilon -2} (-\text{pp} x(1) x(2))^{-\varepsilon },i 2^{2 \varepsilon -4} \pi ^{\varepsilon -2} \Gamma (\varepsilon ),\{x(1),x(2)\}\right\} { ( x ( 1 ) + x ( 2 ) ) 2 ε − 2 ( − pp x ( 1 ) x ( 2 ) ) − ε , i 2 2 ε − 4 π ε − 2 Γ ( ε ) , { x ( 1 ) , x ( 2 )} }
Same integral but with the Euclidean metric signature
FCFeynmanParametrize[ FAD[ q , q - p ], { q }, Names -> x , FinalSubstitutions -> SPD[ p ] -> pp,
FCReplaceD -> { D -> 4 - 2 Epsilon}, FeynmanIntegralPrefactor -> "Textbook" , "Euclidean" -> True ]
{ ( x ( 1 ) + x ( 2 ) ) 2 ε − 2 ( pp x ( 1 ) x ( 2 ) ) − ε , 2 2 ε − 4 π ε − 2 Γ ( ε ) , { x ( 1 ) , x ( 2 ) } } \left\{(x(1)+x(2))^{2 \varepsilon -2} (\text{pp} x(1) x(2))^{-\varepsilon },2^{2 \varepsilon -4} \pi ^{\varepsilon -2} \Gamma (\varepsilon ),\{x(1),x(2)\}\right\} { ( x ( 1 ) + x ( 2 ) ) 2 ε − 2 ( pp x ( 1 ) x ( 2 ) ) − ε , 2 2 ε − 4 π ε − 2 Γ ( ε ) , { x ( 1 ) , x ( 2 )} }
A tensor integral
FCFeynmanParametrize[ FAD[{ q , m }] FAD[{ q - p , m2}] FVD[ q , mu] FVD[ q , nu], { q },
Names -> x , FCE -> True ]
{ ( x ( 1 ) + x ( 2 ) ) − D ( m 2 x ( 1 ) 2 + m 2 x ( 1 ) x ( 2 ) + m2 2 x ( 2 ) 2 + m2 2 x ( 1 ) x ( 2 ) − p 2 x ( 1 ) x ( 2 ) ) D 2 − 2 ( x ( 2 ) 2 Γ ( 2 − D 2 ) p mu p nu − 1 2 Γ ( 1 − D 2 ) g mu nu ( m 2 x ( 1 ) 2 + m 2 x ( 1 ) x ( 2 ) + m2 2 x ( 2 ) 2 + m2 2 x ( 1 ) x ( 2 ) − p 2 x ( 1 ) x ( 2 ) ) ) , 1 , { x ( 1 ) , x ( 2 ) } } \left\{(x(1)+x(2))^{-D} \left(m^2 x(1)^2+m^2 x(1) x(2)+\text{m2}^2 x(2)^2+\text{m2}^2 x(1) x(2)-p^2 x(1) x(2)\right)^{\frac{D}{2}-2} \left(x(2)^2 \Gamma \left(2-\frac{D}{2}\right) p^{\text{mu}} p^{\text{nu}}-\frac{1}{2} \Gamma \left(1-\frac{D}{2}\right) g^{\text{mu}\;\text{nu}} \left(m^2 x(1)^2+m^2 x(1) x(2)+\text{m2}^2 x(2)^2+\text{m2}^2 x(1) x(2)-p^2 x(1) x(2)\right)\right),1,\{x(1),x(2)\}\right\} { ( x ( 1 ) + x ( 2 ) ) − D ( m 2 x ( 1 ) 2 + m 2 x ( 1 ) x ( 2 ) + m2 2 x ( 2 ) 2 + m2 2 x ( 1 ) x ( 2 ) − p 2 x ( 1 ) x ( 2 ) ) 2 D − 2 ( x ( 2 ) 2 Γ ( 2 − 2 D ) p mu p nu − 2 1 Γ ( 1 − 2 D ) g mu nu ( m 2 x ( 1 ) 2 + m 2 x ( 1 ) x ( 2 ) + m2 2 x ( 2 ) 2 + m2 2 x ( 1 ) x ( 2 ) − p 2 x ( 1 ) x ( 2 ) ) ) , 1 , { x ( 1 ) , x ( 2 )} }
1-loop master formulas for Minkowski integrals (cf. Eq. 9.49b in Sterman’s An introduction to QFT)
SFAD[{{ k , 2 p . k }, M ^ 2 , s }]
FCFeynmanParametrize[ % , { k }, Names -> x , FCE -> True , FeynmanIntegralPrefactor -> 1 ,
FCReplaceD -> { D -> n }]
( k 2 + 2 ( k ⋅ p ) − M 2 + i η ) − s (k^2+2 (k\cdot p)-M^2+i \eta )^{-s} ( k 2 + 2 ( k ⋅ p ) − M 2 + i η ) − s
{ 1 , i π n / 2 ( − 1 ) s Γ ( s − n 2 ) ( M 2 + p 2 ) n 2 − s Γ ( s ) , { } } \left\{1,\frac{i \pi ^{n/2} (-1)^s \Gamma \left(s-\frac{n}{2}\right) \left(M^2+p^2\right)^{\frac{n}{2}-s}}{\Gamma (s)},\{\}\right\} { 1 , Γ ( s ) i π n /2 ( − 1 ) s Γ ( s − 2 n ) ( M 2 + p 2 ) 2 n − s , { } }
FVD[ k , \ [ Mu]] SFAD[{{ k , 2 p . k }, M ^ 2 , s }]
FCFeynmanParametrize[ % , { k }, Names -> x , FCE -> True , FeynmanIntegralPrefactor -> 1 ,
FCReplaceD -> { D -> n }]
k μ ( k 2 + 2 ( k ⋅ p ) − M 2 + i η ) − s k^{\mu } (k^2+2 (k\cdot p)-M^2+i \eta )^{-s} k μ ( k 2 + 2 ( k ⋅ p ) − M 2 + i η ) − s
{ 1 , − i π n / 2 ( − 1 ) s p μ Γ ( s − n 2 ) ( M 2 + p 2 ) n 2 − s Γ ( s ) , { } } \left\{1,-\frac{i \pi ^{n/2} (-1)^s p^{\mu } \Gamma \left(s-\frac{n}{2}\right) \left(M^2+p^2\right)^{\frac{n}{2}-s}}{\Gamma (s)},\{\}\right\} { 1 , − Γ ( s ) i π n /2 ( − 1 ) s p μ Γ ( s − 2 n ) ( M 2 + p 2 ) 2 n − s , { } }
FVD[ k , \ [ Mu]] FVD[ k , \ [ Nu]] SFAD[{{ k , 2 p . k }, M ^ 2 , s }]
FCFeynmanParametrize[ % , { k }, Names -> x , FCE -> True , FeynmanIntegralPrefactor -> 1 ,
FCReplaceD -> { D -> n }]
k μ k ν ( k 2 + 2 ( k ⋅ p ) − M 2 + i η ) − s k^{\mu } k^{\nu } (k^2+2 (k\cdot p)-M^2+i \eta )^{-s} k μ k ν ( k 2 + 2 ( k ⋅ p ) − M 2 + i η ) − s
{ 1 , i π n / 2 ( − 1 ) s ( M 2 + p 2 ) n 2 − s ( p μ p ν Γ ( s − n 2 ) − 1 2 ( M 2 + p 2 ) g μ ν Γ ( − n 2 + s − 1 ) ) Γ ( s ) , { } } \left\{1,\frac{i \pi ^{n/2} (-1)^s \left(M^2+p^2\right)^{\frac{n}{2}-s} \left(p^{\mu } p^{\nu } \Gamma \left(s-\frac{n}{2}\right)-\frac{1}{2} \left(M^2+p^2\right) g^{\mu \nu } \Gamma \left(-\frac{n}{2}+s-1\right)\right)}{\Gamma (s)},\{\}\right\} { 1 , Γ ( s ) i π n /2 ( − 1 ) s ( M 2 + p 2 ) 2 n − s ( p μ p ν Γ ( s − 2 n ) − 2 1 ( M 2 + p 2 ) g μν Γ ( − 2 n + s − 1 ) ) , { } }
1-loop master formulas for Euclidean integrals (cf. Eq. 9.49a in Sterman’s An introduction to QFT)
SFAD[{{ k , 2 p . k }, - M ^ 2 , s }]
FCFeynmanParametrize[ % , { k }, Names -> x , FCE -> True , "Euclidean" -> True ,
FeynmanIntegralPrefactor -> I ]
( k 2 + 2 ( k ⋅ p ) + M 2 + i η ) − s (k^2+2 (k\cdot p)+M^2+i \eta )^{-s} ( k 2 + 2 ( k ⋅ p ) + M 2 + i η ) − s
{ 1 , i π D / 2 Γ ( s − D 2 ) ( M 2 − p 2 ) D 2 − s Γ ( s ) , { } } \left\{1,\frac{i \pi ^{D/2} \Gamma \left(s-\frac{D}{2}\right) \left(M^2-p^2\right)^{\frac{D}{2}-s}}{\Gamma (s)},\{\}\right\} ⎩ ⎨ ⎧ 1 , Γ ( s ) i π D /2 Γ ( s − 2 D ) ( M 2 − p 2 ) 2 D − s , { } ⎭ ⎬ ⎫
FVD[ k , \ [ Mu]] SFAD[{{ k , 2 p . k }, - M ^ 2 , s }]
FCFeynmanParametrize[ % , { k }, Names -> x , FCE -> True , FeynmanIntegralPrefactor -> I ,
FCReplaceD -> { D -> n }, "Euclidean" -> True ]
k μ ( k 2 + 2 ( k ⋅ p ) + M 2 + i η ) − s k^{\mu } (k^2+2 (k\cdot p)+M^2+i \eta )^{-s} k μ ( k 2 + 2 ( k ⋅ p ) + M 2 + i η ) − s
{ 1 , − i π n / 2 p μ Γ ( s − n 2 ) ( M 2 − p 2 ) n 2 − s Γ ( s ) , { } } \left\{1,-\frac{i \pi ^{n/2} p^{\mu } \Gamma \left(s-\frac{n}{2}\right) \left(M^2-p^2\right)^{\frac{n}{2}-s}}{\Gamma (s)},\{\}\right\} { 1 , − Γ ( s ) i π n /2 p μ Γ ( s − 2 n ) ( M 2 − p 2 ) 2 n − s , { } }
FVD[ k , \ [ Mu]] FVD[ k , \ [ Nu]] SFAD[{{ k , 2 p . k }, - M ^ 2 , s }]
FCFeynmanParametrize[ % , { k }, Names -> x , FCE -> True , FeynmanIntegralPrefactor -> I ,
FCReplaceD -> { D -> n }, "Euclidean" -> True ]
k μ k ν ( k 2 + 2 ( k ⋅ p ) + M 2 + i η ) − s k^{\mu } k^{\nu } (k^2+2 (k\cdot p)+M^2+i \eta )^{-s} k μ k ν ( k 2 + 2 ( k ⋅ p ) + M 2 + i η ) − s
{ 1 , i π n / 2 ( M 2 − p 2 ) n 2 − s ( 1 2 ( M 2 − p 2 ) g μ ν Γ ( − n 2 + s − 1 ) + p μ p ν Γ ( s − n 2 ) ) Γ ( s ) , { } } \left\{1,\frac{i \pi ^{n/2} \left(M^2-p^2\right)^{\frac{n}{2}-s} \left(\frac{1}{2} \left(M^2-p^2\right) g^{\mu \nu } \Gamma \left(-\frac{n}{2}+s-1\right)+p^{\mu } p^{\nu } \Gamma \left(s-\frac{n}{2}\right)\right)}{\Gamma (s)},\{\}\right\} { 1 , Γ ( s ) i π n /2 ( M 2 − p 2 ) 2 n − s ( 2 1 ( M 2 − p 2 ) g μν Γ ( − 2 n + s − 1 ) + p μ p ν Γ ( s − 2 n ) ) , { } }
1-loop massless box
FAD[ p , p + q1, p + q1 + q2, p + q1 + q2 + q3]
FCFeynmanParametrize[ % , { p }, Names -> x , FCReplaceD -> { D -> 4 - 2 Epsilon}]
1 p 2 . ( p + q1 ) 2 . ( p + q1 + q2 ) 2 . ( p + q1 + q2 + q3 ) 2 \frac{1}{p^2.(p+\text{q1})^2.(p+\text{q1}+\text{q2})^2.(p+\text{q1}+\text{q2}+\text{q3})^2} p 2 . ( p + q1 ) 2 . ( p + q1 + q2 ) 2 . ( p + q1 + q2 + q3 ) 2 1
{ ( x ( 1 ) + x ( 2 ) + x ( 3 ) + x ( 4 ) ) 2 ε ( − 2 x ( 1 ) x ( 3 ) ( q1 ⋅ q2 ) − 2 x ( 1 ) x ( 4 ) ( q1 ⋅ q2 ) − 2 x ( 1 ) x ( 4 ) ( q1 ⋅ q3 ) − q1 2 x ( 1 ) x ( 2 ) − q1 2 x ( 1 ) x ( 3 ) − q1 2 x ( 1 ) x ( 4 ) − 2 x ( 4 ) x ( 2 ) ( q2 ⋅ q3 ) − 2 x ( 1 ) x ( 4 ) ( q2 ⋅ q3 ) − q2 2 x ( 3 ) x ( 2 ) − q2 2 x ( 4 ) x ( 2 ) − q2 2 x ( 1 ) x ( 3 ) − q2 2 x ( 1 ) x ( 4 ) − q3 2 x ( 4 ) x ( 2 ) − q3 2 x ( 1 ) x ( 4 ) − q3 2 x ( 3 ) x ( 4 ) ) − ε − 2 , Γ ( ε + 2 ) , { x ( 1 ) , x ( 2 ) , x ( 3 ) , x ( 4 ) } } \left\{(x(1)+x(2)+x(3)+x(4))^{2 \varepsilon } \left(-2 x(1) x(3) (\text{q1}\cdot \;\text{q2})-2 x(1) x(4) (\text{q1}\cdot \;\text{q2})-2 x(1) x(4) (\text{q1}\cdot \;\text{q3})-\text{q1}^2 x(1) x(2)-\text{q1}^2 x(1) x(3)-\text{q1}^2 x(1) x(4)-2 x(4) x(2) (\text{q2}\cdot \;\text{q3})-2 x(1) x(4) (\text{q2}\cdot \;\text{q3})-\text{q2}^2 x(3) x(2)-\text{q2}^2 x(4) x(2)-\text{q2}^2 x(1) x(3)-\text{q2}^2 x(1) x(4)-\text{q3}^2 x(4) x(2)-\text{q3}^2 x(1) x(4)-\text{q3}^2 x(3) x(4)\right)^{-\varepsilon -2},\Gamma (\varepsilon +2),\{x(1),x(2),x(3),x(4)\}\right\} { ( x ( 1 ) + x ( 2 ) + x ( 3 ) + x ( 4 ) ) 2 ε ( − 2 x ( 1 ) x ( 3 ) ( q1 ⋅ q2 ) − 2 x ( 1 ) x ( 4 ) ( q1 ⋅ q2 ) − 2 x ( 1 ) x ( 4 ) ( q1 ⋅ q3 ) − q1 2 x ( 1 ) x ( 2 ) − q1 2 x ( 1 ) x ( 3 ) − q1 2 x ( 1 ) x ( 4 ) − 2 x ( 4 ) x ( 2 ) ( q2 ⋅ q3 ) − 2 x ( 1 ) x ( 4 ) ( q2 ⋅ q3 ) − q2 2 x ( 3 ) x ( 2 ) − q2 2 x ( 4 ) x ( 2 ) − q2 2 x ( 1 ) x ( 3 ) − q2 2 x ( 1 ) x ( 4 ) − q3 2 x ( 4 ) x ( 2 ) − q3 2 x ( 1 ) x ( 4 ) − q3 2 x ( 3 ) x ( 4 ) ) − ε − 2 , Γ ( ε + 2 ) , { x ( 1 ) , x ( 2 ) , x ( 3 ) , x ( 4 )} }
3-loop self-energy with two massive lines
SFAD[{{ p1, 0 }, { m ^ 2 , 1 }, 1 }, {{ p2, 0 }, { 0 , 1 }, 1 }, {{ p3, 0 }, { 0 , 1 }, 1 },
{{ p2 + p3, 0 }, { 0 , 1 }, 1 }, {{ p1 - Q , 0 }, { m ^ 2 , 1 }, 1 }, {{ p2 - Q , 0 }, { 0 , 1 }, 1 },
{{ p2 + p3 - Q , 0 }, { 0 , 1 }, 1 }, {{ p1 + p2 + p3 - Q , 0 }, { 0 , 1 }, 1 }]
FCFeynmanParametrize[ % , { p1, p2, p3}, Names -> x , FCReplaceD -> { D -> 4 - 2 Epsilon}]
1 ( p1 2 − m 2 + i η ) . ( p2 2 + i η ) . ( p3 2 + i η ) . ( ( p2 + p3 ) 2 + i η ) . ( ( p1 − Q ) 2 − m 2 + i η ) . ( ( p2 − Q ) 2 + i η ) . ( ( p2 + p3 − Q ) 2 + i η ) . ( ( p1 + p2 + p3 − Q ) 2 + i η ) \frac{1}{(\text{p1}^2-m^2+i \eta ).(\text{p2}^2+i \eta ).(\text{p3}^2+i \eta ).((\text{p2}+\text{p3})^2+i \eta ).((\text{p1}-Q)^2-m^2+i \eta ).((\text{p2}-Q)^2+i \eta ).((\text{p2}+\text{p3}-Q)^2+i \eta ).((\text{p1}+\text{p2}+\text{p3}-Q)^2+i \eta )} ( p1 2 − m 2 + i η ) . ( p2 2 + i η ) . ( p3 2 + i η ) . (( p2 + p3 ) 2 + i η ) . (( p1 − Q ) 2 − m 2 + i η ) . (( p2 − Q ) 2 + i η ) . (( p2 + p3 − Q ) 2 + i η ) . (( p1 + p2 + p3 − Q ) 2 + i η ) 1
{ ( x ( 1 ) x ( 2 ) x ( 3 ) + x ( 1 ) x ( 4 ) x ( 3 ) + x ( 2 ) x ( 5 ) x ( 3 ) + x ( 4 ) x ( 5 ) x ( 3 ) + x ( 1 ) x ( 6 ) x ( 3 ) + x ( 5 ) x ( 6 ) x ( 3 ) + x ( 1 ) x ( 7 ) x ( 3 ) + x ( 5 ) x ( 7 ) x ( 3 ) + x ( 1 ) x ( 8 ) x ( 3 ) + x ( 2 ) x ( 8 ) x ( 3 ) + x ( 4 ) x ( 8 ) x ( 3 ) + x ( 5 ) x ( 8 ) x ( 3 ) + x ( 6 ) x ( 8 ) x ( 3 ) + x ( 7 ) x ( 8 ) x ( 3 ) + x ( 1 ) x ( 2 ) x ( 4 ) + x ( 2 ) x ( 4 ) x ( 5 ) + x ( 1 ) x ( 4 ) x ( 6 ) + x ( 4 ) x ( 5 ) x ( 6 ) + x ( 1 ) x ( 2 ) x ( 7 ) + x ( 2 ) x ( 5 ) x ( 7 ) + x ( 1 ) x ( 6 ) x ( 7 ) + x ( 5 ) x ( 6 ) x ( 7 ) + x ( 1 ) x ( 2 ) x ( 8 ) + x ( 2 ) x ( 4 ) x ( 8 ) + x ( 2 ) x ( 5 ) x ( 8 ) + x ( 1 ) x ( 6 ) x ( 8 ) + x ( 4 ) x ( 6 ) x ( 8 ) + x ( 5 ) x ( 6 ) x ( 8 ) + x ( 2 ) x ( 7 ) x ( 8 ) + x ( 6 ) x ( 7 ) x ( 8 ) ) 4 ε ( x ( 2 ) x ( 3 ) x ( 5 ) 2 m 2 + x ( 2 ) x ( 4 ) x ( 5 ) 2 m 2 + x ( 3 ) x ( 4 ) x ( 5 ) 2 m 2 + x ( 1 ) 2 x ( 2 ) x ( 3 ) m 2 + x ( 1 ) 2 x ( 2 ) x ( 4 ) m 2 + x ( 1 ) 2 x ( 3 ) x ( 4 ) m 2 + 2 x ( 1 ) x ( 2 ) x ( 3 ) x ( 5 ) m 2 + 2 x ( 1 ) x ( 2 ) x ( 4 ) x ( 5 ) m 2 + 2 x ( 1 ) x ( 3 ) x ( 4 ) x ( 5 ) m 2 + x ( 3 ) x ( 5 ) 2 x ( 6 ) m 2 + x ( 4 ) x ( 5 ) 2 x ( 6 ) m 2 + x ( 1 ) 2 x ( 3 ) x ( 6 ) m 2 + x ( 1 ) 2 x ( 4 ) x ( 6 ) m 2 + 2 x ( 1 ) x ( 3 ) x ( 5 ) x ( 6 ) m 2 + 2 x ( 1 ) x ( 4 ) x ( 5 ) x ( 6 ) m 2 + x ( 2 ) x ( 5 ) 2 x ( 7 ) m 2 + x ( 3 ) x ( 5 ) 2 x ( 7 ) m 2 + x ( 1 ) 2 x ( 2 ) x ( 7 ) m 2 + x ( 1 ) 2 x ( 3 ) x ( 7 ) m 2 + 2 x ( 1 ) x ( 2 ) x ( 5 ) x ( 7 ) m 2 + 2 x ( 1 ) x ( 3 ) x ( 5 ) x ( 7 ) m 2 + x ( 1 ) 2 x ( 6 ) x ( 7 ) m 2 + x ( 5 ) 2 x ( 6 ) x ( 7 ) m 2 + 2 x ( 1 ) x ( 5 ) x ( 6 ) x ( 7 ) m 2 + x ( 2 ) x ( 5 ) 2 x ( 8 ) m 2 + x ( 3 ) x ( 5 ) 2 x ( 8 ) m 2 + x ( 1 ) 2 x ( 2 ) x ( 8 ) m 2 + x ( 1 ) 2 x ( 3 ) x ( 8 ) m 2 + x ( 1 ) x ( 2 ) x ( 3 ) x ( 8 ) m 2 + x ( 1 ) x ( 2 ) x ( 4 ) x ( 8 ) m 2 + x ( 1 ) x ( 3 ) x ( 4 ) x ( 8 ) m 2 + 2 x ( 1 ) x ( 2 ) x ( 5 ) x ( 8 ) m 2 + 2 x ( 1 ) x ( 3 ) x ( 5 ) x ( 8 ) m 2 + x ( 2 ) x ( 3 ) x ( 5 ) x ( 8 ) m 2 + x ( 2 ) x ( 4 ) x ( 5 ) x ( 8 ) m 2 + x ( 3 ) x ( 4 ) x ( 5 ) x ( 8 ) m 2 + x ( 1 ) 2 x ( 6 ) x ( 8 ) m 2 + x ( 5 ) 2 x ( 6 ) x ( 8 ) m 2 + x ( 1 ) x ( 3 ) x ( 6 ) x ( 8 ) m 2 + x ( 1 ) x ( 4 ) x ( 6 ) x ( 8 ) m 2 + 2 x ( 1 ) x ( 5 ) x ( 6 ) x ( 8 ) m 2 + x ( 3 ) x ( 5 ) x ( 6 ) x ( 8 ) m 2 + x ( 4 ) x ( 5 ) x ( 6 ) x ( 8 ) m 2 + x ( 1 ) x ( 2 ) x ( 7 ) x ( 8 ) m 2 + x ( 1 ) x ( 3 ) x ( 7 ) x ( 8 ) m 2 + x ( 2 ) x ( 5 ) x ( 7 ) x ( 8 ) m 2 + x ( 3 ) x ( 5 ) x ( 7 ) x ( 8 ) m 2 + x ( 1 ) x ( 6 ) x ( 7 ) x ( 8 ) m 2 + x ( 5 ) x ( 6 ) x ( 7 ) x ( 8 ) m 2 − Q 2 x ( 1 ) x ( 2 ) x ( 3 ) x ( 5 ) − Q 2 x ( 1 ) x ( 2 ) x ( 4 ) x ( 5 ) − Q 2 x ( 1 ) x ( 3 ) x ( 4 ) x ( 5 ) − Q 2 x ( 1 ) x ( 2 ) x ( 3 ) x ( 6 ) − Q 2 x ( 1 ) x ( 2 ) x ( 4 ) x ( 6 ) − Q 2 x ( 1 ) x ( 3 ) x ( 4 ) x ( 6 ) − Q 2 x ( 1 ) x ( 3 ) x ( 5 ) x ( 6 ) − Q 2 x ( 2 ) x ( 3 ) x ( 5 ) x ( 6 ) − Q 2 x ( 1 ) x ( 4 ) x ( 5 ) x ( 6 ) − Q 2 x ( 2 ) x ( 4 ) x ( 5 ) x ( 6 ) − Q 2 x ( 3 ) x ( 4 ) x ( 5 ) x ( 6 ) − Q 2 x ( 1 ) x ( 2 ) x ( 3 ) x ( 7 ) − Q 2 x ( 1 ) x ( 2 ) x ( 4 ) x ( 7 ) − Q 2 x ( 1 ) x ( 3 ) x ( 4 ) x ( 7 ) − Q 2 x ( 1 ) x ( 2 ) x ( 5 ) x ( 7 ) − Q 2 x ( 1 ) x ( 3 ) x ( 5 ) x ( 7 ) − Q 2 x ( 2 ) x ( 3 ) x ( 5 ) x ( 7 ) − Q 2 x ( 2 ) x ( 4 ) x ( 5 ) x ( 7 ) − Q 2 x ( 3 ) x ( 4 ) x ( 5 ) x ( 7 ) − Q 2 x ( 1 ) x ( 2 ) x ( 6 ) x ( 7 ) − Q 2 x ( 1 ) x ( 4 ) x ( 6 ) x ( 7 ) − Q 2 x ( 1 ) x ( 5 ) x ( 6 ) x ( 7 ) − Q 2 x ( 2 ) x ( 5 ) x ( 6 ) x ( 7 ) − Q 2 x ( 4 ) x ( 5 ) x ( 6 ) x ( 7 ) − Q 2 x ( 1 ) x ( 2 ) x ( 3 ) x ( 8 ) − Q 2 x ( 1 ) x ( 2 ) x ( 4 ) x ( 8 ) − Q 2 x ( 1 ) x ( 3 ) x ( 4 ) x ( 8 ) − Q 2 x ( 1 ) x ( 2 ) x ( 5 ) x ( 8 ) − Q 2 x ( 1 ) x ( 3 ) x ( 5 ) x ( 8 ) − Q 2 x ( 1 ) x ( 2 ) x ( 6 ) x ( 8 ) − Q 2 x ( 2 ) x ( 3 ) x ( 6 ) x ( 8 ) − Q 2 x ( 1 ) x ( 4 ) x ( 6 ) x ( 8 ) − Q 2 x ( 2 ) x ( 4 ) x ( 6 ) x ( 8 ) − Q 2 x ( 3 ) x ( 4 ) x ( 6 ) x ( 8 ) − Q 2 x ( 1 ) x ( 5 ) x ( 6 ) x ( 8 ) − Q 2 x ( 2 ) x ( 5 ) x ( 6 ) x ( 8 ) − Q 2 x ( 3 ) x ( 5 ) x ( 6 ) x ( 8 ) − Q 2 x ( 2 ) x ( 3 ) x ( 7 ) x ( 8 ) − Q 2 x ( 2 ) x ( 4 ) x ( 7 ) x ( 8 ) − Q 2 x ( 3 ) x ( 4 ) x ( 7 ) x ( 8 ) − Q 2 x ( 2 ) x ( 5 ) x ( 7 ) x ( 8 ) − Q 2 x ( 3 ) x ( 5 ) x ( 7 ) x ( 8 ) − Q 2 x ( 2 ) x ( 6 ) x ( 7 ) x ( 8 ) − Q 2 x ( 4 ) x ( 6 ) x ( 7 ) x ( 8 ) − Q 2 x ( 5 ) x ( 6 ) x ( 7 ) x ( 8 ) ) − 3 ε − 2 , Γ ( 3 ε + 2 ) , { x ( 1 ) , x ( 2 ) , x ( 3 ) , x ( 4 ) , x ( 5 ) , x ( 6 ) , x ( 7 ) , x ( 8 ) } } \left\{(x(1) x(2) x(3)+x(1) x(4) x(3)+x(2) x(5) x(3)+x(4) x(5) x(3)+x(1) x(6) x(3)+x(5) x(6) x(3)+x(1) x(7) x(3)+x(5) x(7) x(3)+x(1) x(8) x(3)+x(2) x(8) x(3)+x(4) x(8) x(3)+x(5) x(8) x(3)+x(6) x(8) x(3)+x(7) x(8) x(3)+x(1) x(2) x(4)+x(2) x(4) x(5)+x(1) x(4) x(6)+x(4) x(5) x(6)+x(1) x(2) x(7)+x(2) x(5) x(7)+x(1) x(6) x(7)+x(5) x(6) x(7)+x(1) x(2) x(8)+x(2) x(4) x(8)+x(2) x(5) x(8)+x(1) x(6) x(8)+x(4) x(6) x(8)+x(5) x(6) x(8)+x(2) x(7) x(8)+x(6) x(7) x(8))^{4 \varepsilon } \left(x(2) x(3) x(5)^2 m^2+x(2) x(4) x(5)^2 m^2+x(3) x(4) x(5)^2 m^2+x(1)^2 x(2) x(3) m^2+x(1)^2 x(2) x(4) m^2+x(1)^2 x(3) x(4) m^2+2 x(1) x(2) x(3) x(5) m^2+2 x(1) x(2) x(4) x(5) m^2+2 x(1) x(3) x(4) x(5) m^2+x(3) x(5)^2 x(6) m^2+x(4) x(5)^2 x(6) m^2+x(1)^2 x(3) x(6) m^2+x(1)^2 x(4) x(6) m^2+2 x(1) x(3) x(5) x(6) m^2+2 x(1) x(4) x(5) x(6) m^2+x(2) x(5)^2 x(7) m^2+x(3) x(5)^2 x(7) m^2+x(1)^2 x(2) x(7) m^2+x(1)^2 x(3) x(7) m^2+2 x(1) x(2) x(5) x(7) m^2+2 x(1) x(3) x(5) x(7) m^2+x(1)^2 x(6) x(7) m^2+x(5)^2 x(6) x(7) m^2+2 x(1) x(5) x(6) x(7) m^2+x(2) x(5)^2 x(8) m^2+x(3) x(5)^2 x(8) m^2+x(1)^2 x(2) x(8) m^2+x(1)^2 x(3) x(8) m^2+x(1) x(2) x(3) x(8) m^2+x(1) x(2) x(4) x(8) m^2+x(1) x(3) x(4) x(8) m^2+2 x(1) x(2) x(5) x(8) m^2+2 x(1) x(3) x(5) x(8) m^2+x(2) x(3) x(5) x(8) m^2+x(2) x(4) x(5) x(8) m^2+x(3) x(4) x(5) x(8) m^2+x(1)^2 x(6) x(8) m^2+x(5)^2 x(6) x(8) m^2+x(1) x(3) x(6) x(8) m^2+x(1) x(4) x(6) x(8) m^2+2 x(1) x(5) x(6) x(8) m^2+x(3) x(5) x(6) x(8) m^2+x(4) x(5) x(6) x(8) m^2+x(1) x(2) x(7) x(8) m^2+x(1) x(3) x(7) x(8) m^2+x(2) x(5) x(7) x(8) m^2+x(3) x(5) x(7) x(8) m^2+x(1) x(6) x(7) x(8) m^2+x(5) x(6) x(7) x(8) m^2-Q^2 x(1) x(2) x(3) x(5)-Q^2 x(1) x(2) x(4) x(5)-Q^2 x(1) x(3) x(4) x(5)-Q^2 x(1) x(2) x(3) x(6)-Q^2 x(1) x(2) x(4) x(6)-Q^2 x(1) x(3) x(4) x(6)-Q^2 x(1) x(3) x(5) x(6)-Q^2 x(2) x(3) x(5) x(6)-Q^2 x(1) x(4) x(5) x(6)-Q^2 x(2) x(4) x(5) x(6)-Q^2 x(3) x(4) x(5) x(6)-Q^2 x(1) x(2) x(3) x(7)-Q^2 x(1) x(2) x(4) x(7)-Q^2 x(1) x(3) x(4) x(7)-Q^2 x(1) x(2) x(5) x(7)-Q^2 x(1) x(3) x(5) x(7)-Q^2 x(2) x(3) x(5) x(7)-Q^2 x(2) x(4) x(5) x(7)-Q^2 x(3) x(4) x(5) x(7)-Q^2 x(1) x(2) x(6) x(7)-Q^2 x(1) x(4) x(6) x(7)-Q^2 x(1) x(5) x(6) x(7)-Q^2 x(2) x(5) x(6) x(7)-Q^2 x(4) x(5) x(6) x(7)-Q^2 x(1) x(2) x(3) x(8)-Q^2 x(1) x(2) x(4) x(8)-Q^2 x(1) x(3) x(4) x(8)-Q^2 x(1) x(2) x(5) x(8)-Q^2 x(1) x(3) x(5) x(8)-Q^2 x(1) x(2) x(6) x(8)-Q^2 x(2) x(3) x(6) x(8)-Q^2 x(1) x(4) x(6) x(8)-Q^2 x(2) x(4) x(6) x(8)-Q^2 x(3) x(4) x(6) x(8)-Q^2 x(1) x(5) x(6) x(8)-Q^2 x(2) x(5) x(6) x(8)-Q^2 x(3) x(5) x(6) x(8)-Q^2 x(2) x(3) x(7) x(8)-Q^2 x(2) x(4) x(7) x(8)-Q^2 x(3) x(4) x(7) x(8)-Q^2 x(2) x(5) x(7) x(8)-Q^2 x(3) x(5) x(7) x(8)-Q^2 x(2) x(6) x(7) x(8)-Q^2 x(4) x(6) x(7) x(8)-Q^2 x(5) x(6) x(7) x(8)\right)^{-3 \varepsilon -2},\Gamma (3 \varepsilon +2),\{x(1),x(2),x(3),x(4),x(5),x(6),x(7),x(8)\}\right\} { ( x ( 1 ) x ( 2 ) x ( 3 ) + x ( 1 ) x ( 4 ) x ( 3 ) + x ( 2 ) x ( 5 ) x ( 3 ) + x ( 4 ) x ( 5 ) x ( 3 ) + x ( 1 ) x ( 6 ) x ( 3 ) + x ( 5 ) x ( 6 ) x ( 3 ) + x ( 1 ) x ( 7 ) x ( 3 ) + x ( 5 ) x ( 7 ) x ( 3 ) + x ( 1 ) x ( 8 ) x ( 3 ) + x ( 2 ) x ( 8 ) x ( 3 ) + x ( 4 ) x ( 8 ) x ( 3 ) + x ( 5 ) x ( 8 ) x ( 3 ) + x ( 6 ) x ( 8 ) x ( 3 ) + x ( 7 ) x ( 8 ) x ( 3 ) + x ( 1 ) x ( 2 ) x ( 4 ) + x ( 2 ) x ( 4 ) x ( 5 ) + x ( 1 ) x ( 4 ) x ( 6 ) + x ( 4 ) x ( 5 ) x ( 6 ) + x ( 1 ) x ( 2 ) x ( 7 ) + x ( 2 ) x ( 5 ) x ( 7 ) + x ( 1 ) x ( 6 ) x ( 7 ) + x ( 5 ) x ( 6 ) x ( 7 ) + x ( 1 ) x ( 2 ) x ( 8 ) + x ( 2 ) x ( 4 ) x ( 8 ) + x ( 2 ) x ( 5 ) x ( 8 ) + x ( 1 ) x ( 6 ) x ( 8 ) + x ( 4 ) x ( 6 ) x ( 8 ) + x ( 5 ) x ( 6 ) x ( 8 ) + x ( 2 ) x ( 7 ) x ( 8 ) + x ( 6 ) x ( 7 ) x ( 8 ) ) 4 ε ( x ( 2 ) x ( 3 ) x ( 5 ) 2 m 2 + x ( 2 ) x ( 4 ) x ( 5 ) 2 m 2 + x ( 3 ) x ( 4 ) x ( 5 ) 2 m 2 + x ( 1 ) 2 x ( 2 ) x ( 3 ) m 2 + x ( 1 ) 2 x ( 2 ) x ( 4 ) m 2 + x ( 1 ) 2 x ( 3 ) x ( 4 ) m 2 + 2 x ( 1 ) x ( 2 ) x ( 3 ) x ( 5 ) m 2 + 2 x ( 1 ) x ( 2 ) x ( 4 ) x ( 5 ) m 2 + 2 x ( 1 ) x ( 3 ) x ( 4 ) x ( 5 ) m 2 + x ( 3 ) x ( 5 ) 2 x ( 6 ) m 2 + x ( 4 ) x ( 5 ) 2 x ( 6 ) m 2 + x ( 1 ) 2 x ( 3 ) x ( 6 ) m 2 + x ( 1 ) 2 x ( 4 ) x ( 6 ) m 2 + 2 x ( 1 ) x ( 3 ) x ( 5 ) x ( 6 ) m 2 + 2 x ( 1 ) x ( 4 ) x ( 5 ) x ( 6 ) m 2 + x ( 2 ) x ( 5 ) 2 x ( 7 ) m 2 + x ( 3 ) x ( 5 ) 2 x ( 7 ) m 2 + x ( 1 ) 2 x ( 2 ) x ( 7 ) m 2 + x ( 1 ) 2 x ( 3 ) x ( 7 ) m 2 + 2 x ( 1 ) x ( 2 ) x ( 5 ) x ( 7 ) m 2 + 2 x ( 1 ) x ( 3 ) x ( 5 ) x ( 7 ) m 2 + x ( 1 ) 2 x ( 6 ) x ( 7 ) m 2 + x ( 5 ) 2 x ( 6 ) x ( 7 ) m 2 + 2 x ( 1 ) x ( 5 ) x ( 6 ) x ( 7 ) m 2 + x ( 2 ) x ( 5 ) 2 x ( 8 ) m 2 + x ( 3 ) x ( 5 ) 2 x ( 8 ) m 2 + x ( 1 ) 2 x ( 2 ) x ( 8 ) m 2 + x ( 1 ) 2 x ( 3 ) x ( 8 ) m 2 + x ( 1 ) x ( 2 ) x ( 3 ) x ( 8 ) m 2 + x ( 1 ) x ( 2 ) x ( 4 ) x ( 8 ) m 2 + x ( 1 ) x ( 3 ) x ( 4 ) x ( 8 ) m 2 + 2 x ( 1 ) x ( 2 ) x ( 5 ) x ( 8 ) m 2 + 2 x ( 1 ) x ( 3 ) x ( 5 ) x ( 8 ) m 2 + x ( 2 ) x ( 3 ) x ( 5 ) x ( 8 ) m 2 + x ( 2 ) x ( 4 ) x ( 5 ) x ( 8 ) m 2 + x ( 3 ) x ( 4 ) x ( 5 ) x ( 8 ) m 2 + x ( 1 ) 2 x ( 6 ) x ( 8 ) m 2 + x ( 5 ) 2 x ( 6 ) x ( 8 ) m 2 + x ( 1 ) x ( 3 ) x ( 6 ) x ( 8 ) m 2 + x ( 1 ) x ( 4 ) x ( 6 ) x ( 8 ) m 2 + 2 x ( 1 ) x ( 5 ) x ( 6 ) x ( 8 ) m 2 + x ( 3 ) x ( 5 ) x ( 6 ) x ( 8 ) m 2 + x ( 4 ) x ( 5 ) x ( 6 ) x ( 8 ) m 2 + x ( 1 ) x ( 2 ) x ( 7 ) x ( 8 ) m 2 + x ( 1 ) x ( 3 ) x ( 7 ) x ( 8 ) m 2 + x ( 2 ) x ( 5 ) x ( 7 ) x ( 8 ) m 2 + x ( 3 ) x ( 5 ) x ( 7 ) x ( 8 ) m 2 + x ( 1 ) x ( 6 ) x ( 7 ) x ( 8 ) m 2 + x ( 5 ) x ( 6 ) x ( 7 ) x ( 8 ) m 2 − Q 2 x ( 1 ) x ( 2 ) x ( 3 ) x ( 5 ) − Q 2 x ( 1 ) x ( 2 ) x ( 4 ) x ( 5 ) − Q 2 x ( 1 ) x ( 3 ) x ( 4 ) x ( 5 ) − Q 2 x ( 1 ) x ( 2 ) x ( 3 ) x ( 6 ) − Q 2 x ( 1 ) x ( 2 ) x ( 4 ) x ( 6 ) − Q 2 x ( 1 ) x ( 3 ) x ( 4 ) x ( 6 ) − Q 2 x ( 1 ) x ( 3 ) x ( 5 ) x ( 6 ) − Q 2 x ( 2 ) x ( 3 ) x ( 5 ) x ( 6 ) − Q 2 x ( 1 ) x ( 4 ) x ( 5 ) x ( 6 ) − Q 2 x ( 2 ) x ( 4 ) x ( 5 ) x ( 6 ) − Q 2 x ( 3 ) x ( 4 ) x ( 5 ) x ( 6 ) − Q 2 x ( 1 ) x ( 2 ) x ( 3 ) x ( 7 ) − Q 2 x ( 1 ) x ( 2 ) x ( 4 ) x ( 7 ) − Q 2 x ( 1 ) x ( 3 ) x ( 4 ) x ( 7 ) − Q 2 x ( 1 ) x ( 2 ) x ( 5 ) x ( 7 ) − Q 2 x ( 1 ) x ( 3 ) x ( 5 ) x ( 7 ) − Q 2 x ( 2 ) x ( 3 ) x ( 5 ) x ( 7 ) − Q 2 x ( 2 ) x ( 4 ) x ( 5 ) x ( 7 ) − Q 2 x ( 3 ) x ( 4 ) x ( 5 ) x ( 7 ) − Q 2 x ( 1 ) x ( 2 ) x ( 6 ) x ( 7 ) − Q 2 x ( 1 ) x ( 4 ) x ( 6 ) x ( 7 ) − Q 2 x ( 1 ) x ( 5 ) x ( 6 ) x ( 7 ) − Q 2 x ( 2 ) x ( 5 ) x ( 6 ) x ( 7 ) − Q 2 x ( 4 ) x ( 5 ) x ( 6 ) x ( 7 ) − Q 2 x ( 1 ) x ( 2 ) x ( 3 ) x ( 8 ) − Q 2 x ( 1 ) x ( 2 ) x ( 4 ) x ( 8 ) − Q 2 x ( 1 ) x ( 3 ) x ( 4 ) x ( 8 ) − Q 2 x ( 1 ) x ( 2 ) x ( 5 ) x ( 8 ) − Q 2 x ( 1 ) x ( 3 ) x ( 5 ) x ( 8 ) − Q 2 x ( 1 ) x ( 2 ) x ( 6 ) x ( 8 ) − Q 2 x ( 2 ) x ( 3 ) x ( 6 ) x ( 8 ) − Q 2 x ( 1 ) x ( 4 ) x ( 6 ) x ( 8 ) − Q 2 x ( 2 ) x ( 4 ) x ( 6 ) x ( 8 ) − Q 2 x ( 3 ) x ( 4 ) x ( 6 ) x ( 8 ) − Q 2 x ( 1 ) x ( 5 ) x ( 6 ) x ( 8 ) − Q 2 x ( 2 ) x ( 5 ) x ( 6 ) x ( 8 ) − Q 2 x ( 3 ) x ( 5 ) x ( 6 ) x ( 8 ) − Q 2 x ( 2 ) x ( 3 ) x ( 7 ) x ( 8 ) − Q 2 x ( 2 ) x ( 4 ) x ( 7 ) x ( 8 ) − Q 2 x ( 3 ) x ( 4 ) x ( 7 ) x ( 8 ) − Q 2 x ( 2 ) x ( 5 ) x ( 7 ) x ( 8 ) − Q 2 x ( 3 ) x ( 5 ) x ( 7 ) x ( 8 ) − Q 2 x ( 2 ) x ( 6 ) x ( 7 ) x ( 8 ) − Q 2 x ( 4 ) x ( 6 ) x ( 7 ) x ( 8 ) − Q 2 x ( 5 ) x ( 6 ) x ( 7 ) x ( 8 ) ) − 3 ε − 2 , Γ ( 3 ε + 2 ) , { x ( 1 ) , x ( 2 ) , x ( 3 ) , x ( 4 ) , x ( 5 ) , x ( 6 ) , x ( 7 ) , x ( 8 )} }
An example of using FCFeynmanParametrize
together with FCFeynmanParameterJoin
props = { SFAD[{ p1, m ^ 2 }], SFAD[{ p3, m ^ 2 }], SFAD[{{ 0 , 2 p1 . n }}],
SFAD[{{ 0 , 2 (p1 + p3) . n }}]}
{ 1 ( p1 2 − m 2 + i η ) , 1 ( p3 2 − m 2 + i η ) , 1 ( 2 ( n ⋅ p1 ) + i η ) , 1 ( 2 ( n ⋅ ( p1 + p3 ) ) + i η ) } \left\{\frac{1}{(\text{p1}^2-m^2+i \eta )},\frac{1}{(\text{p3}^2-m^2+i \eta )},\frac{1}{(2 (n\cdot \;\text{p1})+i \eta )},\frac{1}{(2 (n\cdot (\text{p1}+\text{p3}))+i \eta )}\right\} { ( p1 2 − m 2 + i η ) 1 , ( p3 2 − m 2 + i η ) 1 , ( 2 ( n ⋅ p1 ) + i η ) 1 , ( 2 ( n ⋅ ( p1 + p3 )) + i η ) 1 }
intT = FCFeynmanParameterJoin[{{ props[[ 1 ]] props[[ 2 ]], 1 , x },
props[[ 3 ]] props[[ 4 ]], y }, { p1, p3}]
{ 1 ( ( − x ( 1 ) m 2 − x ( 2 ) m 2 + p1 2 x ( 1 ) + p3 2 x ( 2 ) ) y ( 1 ) + 2 ( n ⋅ p1 ) y ( 2 ) + ( 2 ( n ⋅ p1 ) + 2 ( n ⋅ p3 ) ) y ( 3 ) + i η ) 4 , 6 y ( 1 ) , { x ( 1 ) , x ( 2 ) , y ( 1 ) , y ( 2 ) , y ( 3 ) } } \left\{\frac{1}{(\left(-x(1) m^2-x(2) m^2+\text{p1}^2 x(1)+\text{p3}^2 x(2)\right) y(1)+2 (n\cdot \;\text{p1}) y(2)+(2 (n\cdot \;\text{p1})+2 (n\cdot \;\text{p3})) y(3)+i \eta )^4},6 y(1),\{x(1),x(2),y(1),y(2),y(3)\}\right\} { ( ( − x ( 1 ) m 2 − x ( 2 ) m 2 + p1 2 x ( 1 ) + p3 2 x ( 2 ) ) y ( 1 ) + 2 ( n ⋅ p1 ) y ( 2 ) + ( 2 ( n ⋅ p1 ) + 2 ( n ⋅ p3 )) y ( 3 ) + i η ) 4 1 , 6 y ( 1 ) , { x ( 1 ) , x ( 2 ) , y ( 1 ) , y ( 2 ) , y ( 3 )} }
Here the Feynman parameter variables x i x_i x i and y i y_i y i are independent from each other, i.e. we have δ ( 1 − x 1 − x 2 − x 3 ) × δ ( 1 − y 1 − y 2 − y 3 ) \delta(1-x_1-x_2-x_3) \times \delta(1-y_1-y_2-y_3) δ ( 1 − x 1 − x 2 − x 3 ) × δ ( 1 − y 1 − y 2 − y 3 ) . This gives us much more freedom when exploiting the Cheng-Wu theorem.
FCFeynmanParametrize[ intT[[ 1 ]], intT[[ 2 ]], { p1, p3}, Indexed -> True ,
FCReplaceD -> { D -> 4 - 2 ep}, FinalSubstitutions -> { SPD[ n ] -> 1 , m -> 1 }, Variables -> intT[[ 3 ]]]
{ y ( 1 ) ( x ( 1 ) x ( 2 ) y ( 1 ) 2 ) 3 ep − 2 ( y ( 1 ) ( x ( 1 ) x ( 2 ) 2 y ( 1 ) 2 + x ( 1 ) 2 x ( 2 ) y ( 1 ) 2 + x ( 2 ) y ( 2 ) 2 + x ( 1 ) y ( 3 ) 2 + x ( 2 ) y ( 3 ) 2 + 2 x ( 2 ) y ( 2 ) y ( 3 ) ) ) − 2 ep , Γ ( 2 ep ) , { x ( 1 ) , x ( 2 ) , y ( 1 ) , y ( 2 ) , y ( 3 ) } } \left\{y(1) \left(x(1) x(2) y(1)^2\right)^{3 \;\text{ep}-2} \left(y(1) \left(x(1) x(2)^2 y(1)^2+x(1)^2 x(2) y(1)^2+x(2) y(2)^2+x(1) y(3)^2+x(2) y(3)^2+2 x(2) y(2) y(3)\right)\right)^{-2 \;\text{ep}},\Gamma (2 \;\text{ep}),\{x(1),x(2),y(1),y(2),y(3)\}\right\} { y ( 1 ) ( x ( 1 ) x ( 2 ) y ( 1 ) 2 ) 3 ep − 2 ( y ( 1 ) ( x ( 1 ) x ( 2 ) 2 y ( 1 ) 2 + x ( 1 ) 2 x ( 2 ) y ( 1 ) 2 + x ( 2 ) y ( 2 ) 2 + x ( 1 ) y ( 3 ) 2 + x ( 2 ) y ( 3 ) 2 + 2 x ( 2 ) y ( 2 ) y ( 3 ) ) ) − 2 ep , Γ ( 2 ep ) , { x ( 1 ) , x ( 2 ) , y ( 1 ) , y ( 2 ) , y ( 3 )} }
In the case that we need U
and F
polynomials in addition to the normal output (e.g. for HyperInt)
(SFAD[{{ 0 , 2 * k1 . n }}] * SFAD[{{ 0 , 2 * k2 . n }}] * SFAD[{ k1, m ^ 2 }] *
SFAD[{ k2, m ^ 2 }] * SFAD[{ k1 - k2, m ^ 2 }] )
out = FCFeynmanParametrize[ % , { k1, k2}, Names -> x , FCReplaceD -> { D -> 4 - 2 Epsilon},
FCFeynmanPrepare -> True ]
1 ( k1 2 − m 2 + i η ) ( k2 2 − m 2 + i η ) ( ( k1 − k2 ) 2 − m 2 + i η ) ( 2 ( k1 ⋅ n ) + i η ) ( 2 ( k2 ⋅ n ) + i η ) \frac{1}{(\text{k1}^2-m^2+i \eta ) (\text{k2}^2-m^2+i \eta ) ((\text{k1}-\text{k2})^2-m^2+i \eta ) (2 (\text{k1}\cdot n)+i \eta ) (2 (\text{k2}\cdot n)+i \eta )} ( k1 2 − m 2 + i η ) ( k2 2 − m 2 + i η ) (( k1 − k2 ) 2 − m 2 + i η ) ( 2 ( k1 ⋅ n ) + i η ) ( 2 ( k2 ⋅ n ) + i η ) 1
{ ( x ( 3 ) x ( 4 ) + x ( 5 ) x ( 4 ) + x ( 3 ) x ( 5 ) ) 3 ε − 1 ( m 2 x ( 3 ) x ( 4 ) 2 + m 2 x ( 3 ) x ( 5 ) 2 + m 2 x ( 4 ) x ( 5 ) 2 + m 2 x ( 3 ) 2 x ( 4 ) + m 2 x ( 3 ) 2 x ( 5 ) + m 2 x ( 4 ) 2 x ( 5 ) + 3 m 2 x ( 3 ) x ( 4 ) x ( 5 ) + n 2 x ( 2 ) 2 x ( 3 ) + n 2 x ( 1 ) 2 x ( 4 ) + n 2 x ( 2 ) 2 x ( 4 ) + 2 n 2 x ( 1 ) x ( 2 ) x ( 4 ) + n 2 x ( 1 ) 2 x ( 5 ) ) − 2 ε − 1 , − Γ ( 2 ε + 1 ) , { x ( 1 ) , x ( 2 ) , x ( 3 ) , x ( 4 ) , x ( 5 ) } , { x ( 3 ) x ( 4 ) + x ( 5 ) x ( 4 ) + x ( 3 ) x ( 5 ) , m 2 x ( 3 ) x ( 4 ) 2 + m 2 x ( 3 ) x ( 5 ) 2 + m 2 x ( 4 ) x ( 5 ) 2 + m 2 x ( 3 ) 2 x ( 4 ) + m 2 x ( 3 ) 2 x ( 5 ) + m 2 x ( 4 ) 2 x ( 5 ) + 3 m 2 x ( 3 ) x ( 4 ) x ( 5 ) + n 2 x ( 2 ) 2 x ( 3 ) + n 2 x ( 1 ) 2 x ( 4 ) + n 2 x ( 2 ) 2 x ( 4 ) + 2 n 2 x ( 1 ) x ( 2 ) x ( 4 ) + n 2 x ( 1 ) 2 x ( 5 ) , ( x ( 1 ) 1 ( 2 ( k1 ⋅ n ) + i η ) 1 x ( 2 ) 1 ( 2 ( k2 ⋅ n ) + i η ) 1 x ( 3 ) 1 ( k1 2 − m 2 + i η ) 1 x ( 4 ) 1 ( ( k1 − k2 ) 2 − m 2 + i η ) 1 x ( 5 ) 1 ( k2 2 − m 2 + i η ) 1 ) , ( x ( 3 ) + x ( 4 ) − x ( 4 ) − x ( 4 ) x ( 4 ) + x ( 5 ) ) , { x ( 1 ) ( − n FCGV ( mu ) ) , x ( 2 ) ( − n FCGV ( mu ) ) } , − m 2 ( x ( 3 ) + x ( 4 ) + x ( 5 ) ) , 1 , 0 } } \left\{(x(3) x(4)+x(5) x(4)+x(3) x(5))^{3 \varepsilon -1} \left(m^2 x(3) x(4)^2+m^2 x(3) x(5)^2+m^2 x(4) x(5)^2+m^2 x(3)^2 x(4)+m^2 x(3)^2 x(5)+m^2 x(4)^2 x(5)+3 m^2 x(3) x(4) x(5)+n^2 x(2)^2 x(3)+n^2 x(1)^2 x(4)+n^2 x(2)^2 x(4)+2 n^2 x(1) x(2) x(4)+n^2 x(1)^2 x(5)\right)^{-2 \varepsilon -1},-\Gamma (2 \varepsilon +1),\{x(1),x(2),x(3),x(4),x(5)\},\left\{x(3) x(4)+x(5) x(4)+x(3) x(5),m^2 x(3) x(4)^2+m^2 x(3) x(5)^2+m^2 x(4) x(5)^2+m^2 x(3)^2 x(4)+m^2 x(3)^2 x(5)+m^2 x(4)^2 x(5)+3 m^2 x(3) x(4) x(5)+n^2 x(2)^2 x(3)+n^2 x(1)^2 x(4)+n^2 x(2)^2 x(4)+2 n^2 x(1) x(2) x(4)+n^2 x(1)^2 x(5),\left(
\begin{array}{ccc}
x(1) & \frac{1}{(2 (\text{k1}\cdot n)+i \eta )} & 1 \\
x(2) & \frac{1}{(2 (\text{k2}\cdot n)+i \eta )} & 1 \\
x(3) & \frac{1}{(\text{k1}^2-m^2+i \eta )} & 1 \\
x(4) & \frac{1}{((\text{k1}-\text{k2})^2-m^2+i \eta )} & 1 \\
x(5) & \frac{1}{(\text{k2}^2-m^2+i \eta )} & 1 \\
\end{array}
\right),\left(
\begin{array}{cc}
x(3)+x(4) & -x(4) \\
-x(4) & x(4)+x(5) \\
\end{array}
\right),\left\{x(1) \left(-n^{\text{FCGV}(\text{mu})}\right),x(2) \left(-n^{\text{FCGV}(\text{mu})}\right)\right\},-m^2 (x(3)+x(4)+x(5)),1,0\right\}\right\} ⎩ ⎨ ⎧ ( x ( 3 ) x ( 4 ) + x ( 5 ) x ( 4 ) + x ( 3 ) x ( 5 ) ) 3 ε − 1 ( m 2 x ( 3 ) x ( 4 ) 2 + m 2 x ( 3 ) x ( 5 ) 2 + m 2 x ( 4 ) x ( 5 ) 2 + m 2 x ( 3 ) 2 x ( 4 ) + m 2 x ( 3 ) 2 x ( 5 ) + m 2 x ( 4 ) 2 x ( 5 ) + 3 m 2 x ( 3 ) x ( 4 ) x ( 5 ) + n 2 x ( 2 ) 2 x ( 3 ) + n 2 x ( 1 ) 2 x ( 4 ) + n 2 x ( 2 ) 2 x ( 4 ) + 2 n 2 x ( 1 ) x ( 2 ) x ( 4 ) + n 2 x ( 1 ) 2 x ( 5 ) ) − 2 ε − 1 , − Γ ( 2 ε + 1 ) , { x ( 1 ) , x ( 2 ) , x ( 3 ) , x ( 4 ) , x ( 5 )} , ⎩ ⎨ ⎧ x ( 3 ) x ( 4 ) + x ( 5 ) x ( 4 ) + x ( 3 ) x ( 5 ) , m 2 x ( 3 ) x ( 4 ) 2 + m 2 x ( 3 ) x ( 5 ) 2 + m 2 x ( 4 ) x ( 5 ) 2 + m 2 x ( 3 ) 2 x ( 4 ) + m 2 x ( 3 ) 2 x ( 5 ) + m 2 x ( 4 ) 2 x ( 5 ) + 3 m 2 x ( 3 ) x ( 4 ) x ( 5 ) + n 2 x ( 2 ) 2 x ( 3 ) + n 2 x ( 1 ) 2 x ( 4 ) + n 2 x ( 2 ) 2 x ( 4 ) + 2 n 2 x ( 1 ) x ( 2 ) x ( 4 ) + n 2 x ( 1 ) 2 x ( 5 ) , x ( 1 ) x ( 2 ) x ( 3 ) x ( 4 ) x ( 5 ) ( 2 ( k1 ⋅ n ) + i η ) 1 ( 2 ( k2 ⋅ n ) + i η ) 1 ( k1 2 − m 2 + i η ) 1 (( k1 − k2 ) 2 − m 2 + i η ) 1 ( k2 2 − m 2 + i η ) 1 1 1 1 1 1 , ( x ( 3 ) + x ( 4 ) − x ( 4 ) − x ( 4 ) x ( 4 ) + x ( 5 ) ) , { x ( 1 ) ( − n FCGV ( mu ) ) , x ( 2 ) ( − n FCGV ( mu ) ) } , − m 2 ( x ( 3 ) + x ( 4 ) + x ( 5 )) , 1 , 0 ⎭ ⎬ ⎫ ⎭ ⎬ ⎫
From this output we can easily extract the integrand, its x i x_i x i -independent prefactor and the two Symanzik polynomials
{ integrand, pref} = out [[ 1 ;; 2 ]]
{ uPoly, fPoly} = out [[ 4 ]][[ 1 ;; 2 ]]
{ ( x ( 3 ) x ( 4 ) + x ( 5 ) x ( 4 ) + x ( 3 ) x ( 5 ) ) 3 ε − 1 ( m 2 x ( 3 ) x ( 4 ) 2 + m 2 x ( 3 ) x ( 5 ) 2 + m 2 x ( 4 ) x ( 5 ) 2 + m 2 x ( 3 ) 2 x ( 4 ) + m 2 x ( 3 ) 2 x ( 5 ) + m 2 x ( 4 ) 2 x ( 5 ) + 3 m 2 x ( 3 ) x ( 4 ) x ( 5 ) + n 2 x ( 2 ) 2 x ( 3 ) + n 2 x ( 1 ) 2 x ( 4 ) + n 2 x ( 2 ) 2 x ( 4 ) + 2 n 2 x ( 1 ) x ( 2 ) x ( 4 ) + n 2 x ( 1 ) 2 x ( 5 ) ) − 2 ε − 1 , − Γ ( 2 ε + 1 ) } \left\{(x(3) x(4)+x(5) x(4)+x(3) x(5))^{3 \varepsilon -1} \left(m^2 x(3) x(4)^2+m^2 x(3) x(5)^2+m^2 x(4) x(5)^2+m^2 x(3)^2 x(4)+m^2 x(3)^2 x(5)+m^2 x(4)^2 x(5)+3 m^2 x(3) x(4) x(5)+n^2 x(2)^2 x(3)+n^2 x(1)^2 x(4)+n^2 x(2)^2 x(4)+2 n^2 x(1) x(2) x(4)+n^2 x(1)^2 x(5)\right)^{-2 \varepsilon -1},-\Gamma (2 \varepsilon +1)\right\} { ( x ( 3 ) x ( 4 ) + x ( 5 ) x ( 4 ) + x ( 3 ) x ( 5 ) ) 3 ε − 1 ( m 2 x ( 3 ) x ( 4 ) 2 + m 2 x ( 3 ) x ( 5 ) 2 + m 2 x ( 4 ) x ( 5 ) 2 + m 2 x ( 3 ) 2 x ( 4 ) + m 2 x ( 3 ) 2 x ( 5 ) + m 2 x ( 4 ) 2 x ( 5 ) + 3 m 2 x ( 3 ) x ( 4 ) x ( 5 ) + n 2 x ( 2 ) 2 x ( 3 ) + n 2 x ( 1 ) 2 x ( 4 ) + n 2 x ( 2 ) 2 x ( 4 ) + 2 n 2 x ( 1 ) x ( 2 ) x ( 4 ) + n 2 x ( 1 ) 2 x ( 5 ) ) − 2 ε − 1 , − Γ ( 2 ε + 1 ) }
{ x ( 3 ) x ( 4 ) + x ( 5 ) x ( 4 ) + x ( 3 ) x ( 5 ) , m 2 x ( 3 ) x ( 4 ) 2 + m 2 x ( 3 ) x ( 5 ) 2 + m 2 x ( 4 ) x ( 5 ) 2 + m 2 x ( 3 ) 2 x ( 4 ) + m 2 x ( 3 ) 2 x ( 5 ) + m 2 x ( 4 ) 2 x ( 5 ) + 3 m 2 x ( 3 ) x ( 4 ) x ( 5 ) + n 2 x ( 2 ) 2 x ( 3 ) + n 2 x ( 1 ) 2 x ( 4 ) + n 2 x ( 2 ) 2 x ( 4 ) + 2 n 2 x ( 1 ) x ( 2 ) x ( 4 ) + n 2 x ( 1 ) 2 x ( 5 ) } \left\{x(3) x(4)+x(5) x(4)+x(3) x(5),m^2 x(3) x(4)^2+m^2 x(3) x(5)^2+m^2 x(4) x(5)^2+m^2 x(3)^2 x(4)+m^2 x(3)^2 x(5)+m^2 x(4)^2 x(5)+3 m^2 x(3) x(4) x(5)+n^2 x(2)^2 x(3)+n^2 x(1)^2 x(4)+n^2 x(2)^2 x(4)+2 n^2 x(1) x(2) x(4)+n^2 x(1)^2 x(5)\right\} { x ( 3 ) x ( 4 ) + x ( 5 ) x ( 4 ) + x ( 3 ) x ( 5 ) , m 2 x ( 3 ) x ( 4 ) 2 + m 2 x ( 3 ) x ( 5 ) 2 + m 2 x ( 4 ) x ( 5 ) 2 + m 2 x ( 3 ) 2 x ( 4 ) + m 2 x ( 3 ) 2 x ( 5 ) + m 2 x ( 4 ) 2 x ( 5 ) + 3 m 2 x ( 3 ) x ( 4 ) x ( 5 ) + n 2 x ( 2 ) 2 x ( 3 ) + n 2 x ( 1 ) 2 x ( 4 ) + n 2 x ( 2 ) 2 x ( 4 ) + 2 n 2 x ( 1 ) x ( 2 ) x ( 4 ) + n 2 x ( 1 ) 2 x ( 5 ) }
Symbolic propagator powers are fully supported
SFAD[{ I k , 0 , - 1 / 2 + ep}, { I (k + p ), 0 , 1 }, EtaSign -> - 1 ]
v1 = FCFeynmanParametrize[ % , { k }, Names -> x , FCReplaceD -> { D -> 4 - 2 ep},
FinalSubstitutions -> { SPD[ p ] -> 1 }]
1 ( − k 2 − i η ) ep − 1 2 . ( − ( k + p ) 2 − i η ) \frac{1}{(-k^2-i \eta )^{\text{ep}-\frac{1}{2}}.(-(k+p)^2-i \eta )} ( − k 2 − i η ) ep − 2 1 . ( − ( k + p ) 2 − i η ) 1
{ ( − x ( 1 ) − x ( 2 ) ) 3 ep − 7 2 x ( 2 ) ep − 3 2 ( − x ( 1 ) x ( 2 ) ) 3 2 − 2 ep , ( − 1 ) ep + 1 2 Γ ( 2 ep − 3 2 ) Γ ( ep − 1 2 ) , { x ( 1 ) , x ( 2 ) } } \left\{(-x(1)-x(2))^{3 \;\text{ep}-\frac{7}{2}} x(2)^{\text{ep}-\frac{3}{2}} (-x(1) x(2))^{\frac{3}{2}-2 \;\text{ep}},\frac{(-1)^{\text{ep}+\frac{1}{2}} \Gamma \left(2 \;\text{ep}-\frac{3}{2}\right)}{\Gamma \left(\text{ep}-\frac{1}{2}\right)},\{x(1),x(2)\}\right\} { ( − x ( 1 ) − x ( 2 ) ) 3 ep − 2 7 x ( 2 ) ep − 2 3 ( − x ( 1 ) x ( 2 ) ) 2 3 − 2 ep , Γ ( ep − 2 1 ) ( − 1 ) ep + 2 1 Γ ( 2 ep − 2 3 ) , { x ( 1 ) , x ( 2 )} }
An alternative representation for symbolic powers can be obtained using the option SplitSymbolicPowers
SFAD[{ I k , 0 , - 1 / 2 + ep}, { I (k + p ), 0 , 1 }, EtaSign -> - 1 ]
v2 = FCFeynmanParametrize[ % , { k }, Names -> x , FCReplaceD -> { D -> 4 - 2 ep},
FinalSubstitutions -> { SPD[ p ] -> 1 }, SplitSymbolicPowers -> True ]
1 ( − k 2 − i η ) ep − 1 2 . ( − ( k + p ) 2 − i η ) \frac{1}{(-k^2-i \eta )^{\text{ep}-\frac{1}{2}}.(-(k+p)^2-i \eta )} ( − k 2 − i η ) ep − 2 1 . ( − ( k + p ) 2 − i η ) 1
{ x ( 2 ) ep − 1 2 ( ( 1 2 ( 1 − 2 ep ) + 1 2 ( 4 − 2 ep ) − 1 ) x ( 1 ) ( − x ( 1 ) − x ( 2 ) ) 3 ep − 7 2 ( − x ( 1 ) x ( 2 ) ) 1 2 − 2 ep + ( 2 ep + 1 2 ( 2 ep − 1 ) − 3 ) ( − x ( 1 ) − x ( 2 ) ) 3 ep − 9 2 ( − x ( 1 ) x ( 2 ) ) 3 2 − 2 ep ) , ( − 1 ) ep + 1 2 Γ ( 2 ep − 3 2 ) Γ ( ep + 1 2 ) , { x ( 1 ) , x ( 2 ) } } \left\{x(2)^{\text{ep}-\frac{1}{2}} \left(\left(\frac{1}{2} (1-2 \;\text{ep})+\frac{1}{2} (4-2 \;\text{ep})-1\right) x(1) (-x(1)-x(2))^{3 \;\text{ep}-\frac{7}{2}} (-x(1) x(2))^{\frac{1}{2}-2 \;\text{ep}}+\left(2 \;\text{ep}+\frac{1}{2} (2 \;\text{ep}-1)-3\right) (-x(1)-x(2))^{3 \;\text{ep}-\frac{9}{2}} (-x(1) x(2))^{\frac{3}{2}-2 \;\text{ep}}\right),\frac{(-1)^{\text{ep}+\frac{1}{2}} \Gamma \left(2 \;\text{ep}-\frac{3}{2}\right)}{\Gamma \left(\text{ep}+\frac{1}{2}\right)},\{x(1),x(2)\}\right\} { x ( 2 ) ep − 2 1 ( ( 2 1 ( 1 − 2 ep ) + 2 1 ( 4 − 2 ep ) − 1 ) x ( 1 ) ( − x ( 1 ) − x ( 2 ) ) 3 ep − 2 7 ( − x ( 1 ) x ( 2 ) ) 2 1 − 2 ep + ( 2 ep + 2 1 ( 2 ep − 1 ) − 3 ) ( − x ( 1 ) − x ( 2 ) ) 3 ep − 2 9 ( − x ( 1 ) x ( 2 ) ) 2 3 − 2 ep ) , Γ ( ep + 2 1 ) ( − 1 ) ep + 2 1 Γ ( 2 ep − 2 3 ) , { x ( 1 ) , x ( 2 )} }
Even though the parametric integrals evaluate to different values, the product of the integral and its prefactor remains the same
Integrate [ Normal [ Series [ v1[[ 1 ]] /. x [ 1 ] -> 1 , { ep, 0 , 0 }]] /. x [ 1 ] -> 1 , { x [ 2 ], 0 , Infinity }]
Normal@Series[ v1[[ 2 ]] % , { ep, 0 , 0 }]
2 5 \frac{2}{5} 5 2
− 4 i 15 -\frac{4 i}{15} − 15 4 i
Integrate [ Normal [ Series [ v2[[ 1 ]] /. x [ 1 ] -> 1 , { ep, 0 , 0 }]] /. x [ 1 ] -> 1 , { x [ 2 ], 0 , Infinity }]
Normal@Series[ v2[[ 2 ]] % , { ep, 0 , 0 }]
− 1 5 -\frac{1}{5} − 5 1
− 4 i 15 -\frac{4 i}{15} − 15 4 i
Calculate the simplest divergent triangle integral from QCDLoop
FCClearScalarProducts[] ;
SPD[ r ] = 0 ;
SPD[ s ] = 0 ;
SPD[ r , s ] = - 1 / 2 ;
int = FAD[{ q , 0 }, { q - r , 0 }, { q - s , 0 }]
1 q 2 . ( q − r ) 2 . ( q − s ) 2 \frac{1}{q^2.(q-r)^2.(q-s)^2} q 2 . ( q − r ) 2 . ( q − s ) 2 1
i π 2 C 0 ( 0 , 0 , 1 , 0 , 0 , 0 ) i \pi ^2 \;\text{C}_0(0,0,1,0,0,0) i π 2 C 0 ( 0 , 0 , 1 , 0 , 0 , 0 )
fp = FCFeynmanParametrize[ int, { q }, Names -> x , FCReplaceD -> { D -> 4 - 2 ep}, FeynmanIntegralPrefactor -> "LoopTools" ]
{ ( − x ( 2 ) x ( 3 ) ) − ep − 1 ( x ( 1 ) + x ( 2 ) + x ( 3 ) ) 2 ep − 1 , − Γ ( 1 − 2 ep ) Γ ( 1 − ep ) 2 , { x ( 1 ) , x ( 2 ) , x ( 3 ) } } \left\{(-x(2) x(3))^{-\text{ep}-1} (x(1)+x(2)+x(3))^{2 \;\text{ep}-1},-\frac{\Gamma (1-2 \;\text{ep})}{\Gamma (1-\text{ep})^2},\{x(1),x(2),x(3)\}\right\} { ( − x ( 2 ) x ( 3 ) ) − ep − 1 ( x ( 1 ) + x ( 2 ) + x ( 3 ) ) 2 ep − 1 , − Γ ( 1 − ep ) 2 Γ ( 1 − 2 ep ) , { x ( 1 ) , x ( 2 ) , x ( 3 )} }
intRaw = Integrate [ fp[[ 1 ]] /. x [ 2 ] -> 1 , { x [ 1 ], 0 , Infinity }, Assumptions -> { ep < 0 , x [ 3 ] >= 0 }]
− ( − x ( 3 ) ) − ep − 1 ( x ( 3 ) + 1 ) 2 ep 2 ep -\frac{(-x(3))^{-\text{ep}-1} (x(3)+1)^{2 \;\text{ep}}}{2 \;\text{ep}} − 2 ep ( − x ( 3 ) ) − ep − 1 ( x ( 3 ) + 1 ) 2 ep
Reintroduce the correct i η i \eta i η -prescription to get the imaginary part right
intRes = Integrate [ intRaw, { x [ 3 ], 0 , Infinity }, Assumptions -> { ep < 0 }] /. (- 1 )^ (- ep) -> (- 1 - I eta)^ (- ep)
( − 1 − i eta ) − ep Γ ( − ep ) 2 2 ep Γ ( − 2 ep ) \frac{(-1-i \;\text{eta})^{-\text{ep}} \Gamma (-\text{ep})^2}{2 \;\text{ep} \Gamma (-2 \;\text{ep})} 2 ep Γ ( − 2 ep ) ( − 1 − i eta ) − ep Γ ( − ep ) 2
res = (Series [ fp[[ 2 ]] intRes, { ep, 0 , 0 }] // Normal ) /. Log [ - 1 - I eta] -> Log [ 1 ] - I Pi
1 ep 2 + i π ep − π 2 2 \frac{1}{\text{ep}^2}+\frac{i \pi }{\text{ep}}-\frac{\pi ^2}{2} ep 2 1 + ep iπ − 2 π 2
Compare to the known result
resLit = Series [ ScaleMu^ (2 ep)/ ep^ 2 1 / pp^ 2 (- pp - I eta)^ (- ep), { ep, 0 , 0 }] /. Log [ - pp - I eta] -> Log [ pp] - I Pi // Normal
1 ep 2 pp 2 + 2 log ( μ ) − log ( pp ) + i π ep pp 2 + 4 log 2 ( μ ) − 4 log ( μ ) ( log ( pp ) − i π ) + ( log ( pp ) − i π ) 2 2 pp 2 \frac{1}{\text{ep}^2 \;\text{pp}^2}+\frac{2 \log (\mu )-\log (\text{pp})+i \pi }{\text{ep} \;\text{pp}^2}+\frac{4 \log ^2(\mu )-4 \log (\mu ) (\log (\text{pp})-i \pi )+(\log (\text{pp})-i \pi )^2}{2 \;\text{pp}^2} ep 2 pp 2 1 + ep pp 2 2 log ( μ ) − log ( pp ) + iπ + 2 pp 2 4 log 2 ( μ ) − 4 log ( μ ) ( log ( pp ) − iπ ) + ( log ( pp ) − iπ ) 2
(res - resLit) /. pp | ScaleMu -> 1
0 0 0
Notice that one can also keep the i η i \eta i η -prescription explicit in the integrand by setting the option EtaSign
to True
. However, for integrating such representation using Mathematica’s Integrate
it is better to remove it
tmp = FCFeynmanParametrize[ int, { q }, Names -> x , FCReplaceD -> { D -> 4 - 2 ep}, FeynmanIntegralPrefactor -> "LoopTools" , EtaSign -> True ]
{ ( x ( 1 ) + x ( 2 ) + x ( 3 ) ) 2 ep − 1 ( − x ( 2 ) x ( 3 ) − i η ) − ep − 1 , − Γ ( 1 − 2 ep ) Γ ( 1 − ep ) 2 , { x ( 1 ) , x ( 2 ) , x ( 3 ) } } \left\{(x(1)+x(2)+x(3))^{2 \;\text{ep}-1} (-x(2) x(3)-i \eta )^{-\text{ep}-1},-\frac{\Gamma (1-2 \;\text{ep})}{\Gamma (1-\text{ep})^2},\{x(1),x(2),x(3)\}\right\} { ( x ( 1 ) + x ( 2 ) + x ( 3 ) ) 2 ep − 1 ( − x ( 2 ) x ( 3 ) − i η ) − ep − 1 , − Γ ( 1 − ep ) 2 Γ ( 1 − 2 ep ) , { x ( 1 ) , x ( 2 ) , x ( 3 )} }
{ ( − x ( 2 ) x ( 3 ) ) − ep − 1 ( x ( 1 ) + x ( 2 ) + x ( 3 ) ) 2 ep − 1 , − Γ ( 1 − 2 ep ) Γ ( 1 − ep ) 2 , { x ( 1 ) , x ( 2 ) , x ( 3 ) } } \left\{(-x(2) x(3))^{-\text{ep}-1} (x(1)+x(2)+x(3))^{2 \;\text{ep}-1},-\frac{\Gamma (1-2 \;\text{ep})}{\Gamma (1-\text{ep})^2},\{x(1),x(2),x(3)\}\right\} { ( − x ( 2 ) x ( 3 ) ) − ep − 1 ( x ( 1 ) + x ( 2 ) + x ( 3 ) ) 2 ep − 1 , − Γ ( 1 − ep ) 2 Γ ( 1 − 2 ep ) , { x ( 1 ) , x ( 2 ) , x ( 3 )} }
int = SFAD[{{ k , - m ^ 2 / Q k . n - k . nb Q }, { - m ^ 2 , 1 }}, {{ k , - m ^ 2 / Q k . nb - k . n Q }, { - m ^ 2 , 1 }}, { k , m ^ 2 }]
1 ( k 2 + − ( k ⋅ n ) m 2 Q − Q ( k ⋅ nb ) + m 2 + i η ) . ( k 2 + − ( k ⋅ nb ) m 2 Q − Q ( k ⋅ n ) + m 2 + i η ) . ( k 2 − m 2 + i η ) \frac{1}{(k^2+-\frac{(k\cdot n) m^2}{Q}-Q (k\cdot \;\text{nb})+m^2+i \eta ).(k^2+-\frac{(k\cdot \;\text{nb}) m^2}{Q}-Q (k\cdot n)+m^2+i \eta ).(k^2-m^2+i \eta )} ( k 2 + − Q ( k ⋅ n ) m 2 − Q ( k ⋅ nb ) + m 2 + i η ) . ( k 2 + − Q ( k ⋅ nb ) m 2 − Q ( k ⋅ n ) + m 2 + i η ) . ( k 2 − m 2 + i η ) 1
Sometimes loop integrals may require additional regulators beyond dimensional regularization (e.g. in SCET). For such cases we may add extra propagators acting as regulators via the option ExtraPropagators
FCFeynmanParametrize[ int, { k }, Names -> x , FCReplaceD -> { D -> 4 - 2 ep}, FinalSubstitutions -> { SPD[ nb] -> 0 , SPD[ n ] -> 0 , SPD[ nb, n ] -> 2 , Q -> 1 },
ExtraPropagators -> { SFAD[{{ 0 , n . k }, { 0 , + 1 }, al}]}]
{ x ( 4 ) al − 1 ( x ( 1 ) + x ( 2 ) + x ( 3 ) ) al + 2 ep − 1 ( m 4 x ( 2 ) x ( 3 ) + m 2 x ( 1 ) 2 − 2 m 2 x ( 2 ) x ( 3 ) − m 2 x ( 2 ) x ( 4 ) + x ( 2 ) x ( 3 ) − x ( 3 ) x ( 4 ) ) − al − ep − 1 , ( − 1 ) al + 3 Γ ( al + ep + 1 ) Γ ( al ) , { x ( 1 ) , x ( 2 ) , x ( 3 ) , x ( 4 ) } } \left\{x(4)^{\text{al}-1} (x(1)+x(2)+x(3))^{\text{al}+2 \;\text{ep}-1} \left(m^4 x(2) x(3)+m^2 x(1)^2-2 m^2 x(2) x(3)-m^2 x(2) x(4)+x(2) x(3)-x(3) x(4)\right)^{-\text{al}-\text{ep}-1},\frac{(-1)^{\text{al}+3} \Gamma (\text{al}+\text{ep}+1)}{\Gamma (\text{al})},\{x(1),x(2),x(3),x(4)\}\right\} { x ( 4 ) al − 1 ( x ( 1 ) + x ( 2 ) + x ( 3 ) ) al + 2 ep − 1 ( m 4 x ( 2 ) x ( 3 ) + m 2 x ( 1 ) 2 − 2 m 2 x ( 2 ) x ( 3 ) − m 2 x ( 2 ) x ( 4 ) + x ( 2 ) x ( 3 ) − x ( 3 ) x ( 4 ) ) − al − ep − 1 , Γ ( al ) ( − 1 ) al + 3 Γ ( al + ep + 1 ) , { x ( 1 ) , x ( 2 ) , x ( 3 ) , x ( 4 )} }
The option FCReplaceMomenta
is useful when we want to replace external momenta by linear combinations of other momenta. If the coefficients are symbolic, please keep in mind that you need to declare them as being of type FCVariable
.
DataType[ m , FCVariable] = True ;
DataType[ Q , FCVariable] = True ;
FCFeynmanParametrize[ SFAD[ k - pb, k + p , { k , m ^ 2 }], { k }, Names -> x , FCReplaceD -> { D -> 4 - 2 ep}, FinalSubstitutions -> { SPD[ nb] -> 0 , SPD[ n ] -> 0 , SPD[ nb, n ] -> 2 , Q -> 1 },
ExtraPropagators -> { SFAD[{{ 0 , n . k }, { 0 , + 1 }, al}]}, FCReplaceMomenta -> { p -> (Q n / 2 + m ^ 2 / Q nb/ 2 ), pb -> (Q nb/ 2 + m ^ 2 / Q n / 2 )}]
{ x ( 4 ) al − 1 ( x ( 1 ) + x ( 2 ) + x ( 3 ) ) al + 2 ep − 1 ( m 4 ( − x ( 2 ) ) x ( 3 ) + m 2 x ( 1 ) 2 − 2 m 2 x ( 2 ) x ( 3 ) + m 2 x ( 2 ) x ( 4 ) − x ( 2 ) x ( 3 ) − x ( 3 ) x ( 4 ) ) − al − ep − 1 , ( − 1 ) al + 3 Γ ( al + ep + 1 ) Γ ( al ) , { x ( 1 ) , x ( 2 ) , x ( 3 ) , x ( 4 ) } } \left\{x(4)^{\text{al}-1} (x(1)+x(2)+x(3))^{\text{al}+2 \;\text{ep}-1} \left(m^4 (-x(2)) x(3)+m^2 x(1)^2-2 m^2 x(2) x(3)+m^2 x(2) x(4)-x(2) x(3)-x(3) x(4)\right)^{-\text{al}-\text{ep}-1},\frac{(-1)^{\text{al}+3} \Gamma (\text{al}+\text{ep}+1)}{\Gamma (\text{al})},\{x(1),x(2),x(3),x(4)\}\right\} { x ( 4 ) al − 1 ( x ( 1 ) + x ( 2 ) + x ( 3 ) ) al + 2 ep − 1 ( m 4 ( − x ( 2 )) x ( 3 ) + m 2 x ( 1 ) 2 − 2 m 2 x ( 2 ) x ( 3 ) + m 2 x ( 2 ) x ( 4 ) − x ( 2 ) x ( 3 ) − x ( 3 ) x ( 4 ) ) − al − ep − 1 , Γ ( al ) ( − 1 ) al + 3 Γ ( al + ep + 1 ) , { x ( 1 ) , x ( 2 ) , x ( 3 ) , x ( 4 )} }
Lee-Pomeransky representation
1-loop tadpole
FCFeynmanParametrize[ FAD[{ q , m }], { q }, Names -> x , Method -> "Lee-Pomeransky" ]
{ ( m 2 x ( 1 ) 2 + x ( 1 ) ) − D / 2 , − Γ ( D 2 ) Γ ( D − 1 ) , { x ( 1 ) } } \left\{\left(m^2 x(1)^2+x(1)\right)^{-D/2},-\frac{\Gamma \left(\frac{D}{2}\right)}{\Gamma (D-1)},\{x(1)\}\right\} { ( m 2 x ( 1 ) 2 + x ( 1 ) ) − D /2 , − Γ ( D − 1 ) Γ ( 2 D ) , { x ( 1 )} }
Massless 1-loop 2-point function
FCFeynmanParametrize[ FAD[ q , q - p ], { q }, Names -> x , Method -> "Lee-Pomeransky" ]
{ ( − p 2 x ( 2 ) x ( 1 ) + x ( 1 ) + x ( 2 ) ) − D / 2 , Γ ( D 2 ) Γ ( D − 2 ) , { x ( 1 ) , x ( 2 ) } } \left\{\left(-p^2 x(2) x(1)+x(1)+x(2)\right)^{-D/2},\frac{\Gamma \left(\frac{D}{2}\right)}{\Gamma (D-2)},\{x(1),x(2)\}\right\} { ( − p 2 x ( 2 ) x ( 1 ) + x ( 1 ) + x ( 2 ) ) − D /2 , Γ ( D − 2 ) Γ ( 2 D ) , { x ( 1 ) , x ( 2 )} }
2-loop self-energy with 3 massive lines and two eikonal propagators
FCFeynmanParametrize[{ SFAD[{ p1, m ^ 2 }], SFAD[{ p3, m ^ 2 }],
SFAD[{ (p3 - p1), m ^ 2 }], SFAD[{{ 0 , 2 p1 . n }}], SFAD[{{ 0 , 2 p3 . n }}]}, { p1, p3},
Names -> x , Method -> "Lee-Pomeransky" , FCReplaceD -> { D -> 4 - 2 ep},
FinalSubstitutions -> { SPD[ n ] -> 1 , m -> 1 }]
{ ( x ( 2 ) x ( 1 ) 2 + x ( 3 ) x ( 1 ) 2 + x ( 2 ) 2 x ( 1 ) + x ( 3 ) 2 x ( 1 ) + x ( 5 ) 2 x ( 1 ) + x ( 2 ) x ( 1 ) + 3 x ( 2 ) x ( 3 ) x ( 1 ) + x ( 3 ) x ( 1 ) + x ( 2 ) x ( 3 ) 2 + x ( 2 ) x ( 4 ) 2 + x ( 3 ) x ( 4 ) 2 + x ( 3 ) x ( 5 ) 2 + x ( 2 ) 2 x ( 3 ) + x ( 2 ) x ( 3 ) + 2 x ( 3 ) x ( 4 ) x ( 5 ) ) ep − 2 , − Γ ( 2 − ep ) Γ ( 1 − 3 ep ) , { x ( 1 ) , x ( 2 ) , x ( 3 ) , x ( 4 ) , x ( 5 ) } } \left\{\left(x(2) x(1)^2+x(3) x(1)^2+x(2)^2 x(1)+x(3)^2 x(1)+x(5)^2 x(1)+x(2) x(1)+3 x(2) x(3) x(1)+x(3) x(1)+x(2) x(3)^2+x(2) x(4)^2+x(3) x(4)^2+x(3) x(5)^2+x(2)^2 x(3)+x(2) x(3)+2 x(3) x(4) x(5)\right)^{\text{ep}-2},-\frac{\Gamma (2-\text{ep})}{\Gamma (1-3 \;\text{ep})},\{x(1),x(2),x(3),x(4),x(5)\}\right\} { ( x ( 2 ) x ( 1 ) 2 + x ( 3 ) x ( 1 ) 2 + x ( 2 ) 2 x ( 1 ) + x ( 3 ) 2 x ( 1 ) + x ( 5 ) 2 x ( 1 ) + x ( 2 ) x ( 1 ) + 3 x ( 2 ) x ( 3 ) x ( 1 ) + x ( 3 ) x ( 1 ) + x ( 2 ) x ( 3 ) 2 + x ( 2 ) x ( 4 ) 2 + x ( 3 ) x ( 4 ) 2 + x ( 3 ) x ( 5 ) 2 + x ( 2 ) 2 x ( 3 ) + x ( 2 ) x ( 3 ) + 2 x ( 3 ) x ( 4 ) x ( 5 ) ) ep − 2 , − Γ ( 1 − 3 ep ) Γ ( 2 − ep ) , { x ( 1 ) , x ( 2 ) , x ( 3 ) , x ( 4 ) , x ( 5 )} }