Box Integral

In[]:=
​
V (Q1, Q1 + L, m1)[a, b] = V31
In[]:=
V31=-ik((g[a,c]*g[b,d]+g[a,d]*g[b,c])P[Q1,c]*P[-Q1-L,d])​​
Out[]=
-ik(g[a,d]g[b,c]+g[a,c]g[b,d])P[-L-Q1,d]P[Q1,c]
In[]:=
​
S(Q1+L,m1)=S31
In[]:=
S31=
-i
((Q1+L)^2+(m1)^2)
Out[]=
-
i
2
m1
+
2
(L+Q1)
V(Q1+L,Q2,m1)[e,f]=V12
In[]:=
V32=-ik((g[e,g]*g[f,h]+g[e,h]*g[f,g])*P[Q1+L,g]*P[Q2,h])​​
Out[]=
-ik(g[e,h]g[f,g]+g[e,g]g[f,h])P[L+Q1,g]P[Q2,h]
G(L+W)[e,f][i,j]=G11
In[]:=
G31=
-i
(W+L)^2
1
2
(g[e,i]*g[f,j]+g[e,j]*g[f,i])-
1
(D-2)
(g[e,f]*g[i,j])
Out[]=
-
i
1
2
(g[e,j]g[f,i]+g[e,i]g[f,j])-
g[e,f]g[i,j]
-2+D
2
(L+W)
V(L-Q4,Q3)[i,j]=V33
In[]:=
V33=-ik((g[i,k]*g[j,n]+g[i,n]*g[j,k])P[-(L-Q4),k]*P[Q3,n])
Out[]=
-ik(g[i,n]g[j,k]+g[i,k]g[j,n])P[Q3,n]P[-L+Q4,k]
S(L-Q4,m2)=S32
In[]:=
S32=
-i
((-L+Q4)^2+(m2)^2)
Out[]=
-
i
2
m2
+
2
(-L+Q4)
V(Q4,L-Q4)[o,p]=V24
In[]:=
V34=(-ik((g[o,r]*g[p,s]+g[o,s]*g[p,r])P[Q4,r]*P[L-Q4,s]))
Out[]=
-ik(g[o,s]g[p,r]+g[o,r]g[p,s])P[L-Q4,s]P[Q4,r]
G(L)[a,b][o,p]=G32
In[]:=
G32=
-i
L^2
1
2
(g[o,a]*g[p,b]+g[o,b]*g[p,a])-
1
(D-2)
*g[o,p]*g[a,b]
Out[]=
-
i-
g[a,b]g[o,p]
-2+D
+
1
2
(g[o,b]g[p,a]+g[o,a]g[p,b])
2
L

Inti3

In[]:=
inti3=(V31)*(S31)*(V32)*(G31)*(V33)*(S32)*(V34)*(G32)
Out[]=
4
i
4
ik
(g[a,d]g[b,c]+g[a,c]g[b,d])(g[e,h]g[f,g]+g[e,g]g[f,h])
1
2
(g[e,j]g[f,i]+g[e,i]g[f,j])-
g[e,f]g[i,j]
-2+D
(g[i,n]g[j,k]+g[i,k]g[j,n])-
g[a,b]g[o,p]
-2+D
+
1
2
(g[o,b]g[p,a]+g[o,a]g[p,b])(g[o,s]g[p,r]+g[o,r]g[p,s])P[-L-Q1,d]P[Q1,c]P[L+Q1,g]P[Q2,h]P[Q3,n]P[L-Q4,s]P[Q4,r]P[-L+Q4,k](
2
L
(
2
m1
+
2
(L+Q1)
)(
2
m2
+
2
(-L+Q4)
)
2
(L+W)
)
In[]:=
rules={P[x1_,x3_]*P[x2_,x3_]Dotp[x1,x2],P[x1_,x3_]*g[x2___,x3_,x4___]P[x1,x2,x4],P[x1_,x2_]^2Dotp[x1,x1],g[x1___,x5_,x2___]*g[x3___,x5_,x4___]g[x1,x3,x2,x4],g[x1_,x2_]^2g[x1,x1],g[x1_,x1_]D};​​​​num3=Factor[Numerator[inti3]]
Out[]=
1
4
2
(-2+D)
4
i
4
ik
(g[a,d]g[b,c]+g[a,c]g[b,d])(g[e,h]g[f,g]+g[e,g]g[f,h])(-2g[e,j]g[f,i]+Dg[e,j]g[f,i]-2g[e,i]g[f,j]+Dg[e,i]g[f,j]-2g[e,f]g[i,j])(g[i,n]g[j,k]+g[i,k]g[j,n])(-2g[a,b]g[o,p]-2g[o,b]g[p,a]+Dg[o,b]g[p,a]-2g[o,a]g[p,b]+Dg[o,a]g[p,b])(g[o,s]g[p,r]+g[o,r]g[p,s])P[-L-Q1,d]P[Q1,c]P[L+Q1,g]P[Q2,h]P[Q3,n]P[L-Q4,s]P[Q4,r]P[-L+Q4,k]
In[]:=
Enum3=Expand[num3];​​FSInt3=FullSimplify[Enum3//.rules]
Out[]=
1
2
(-2+D)
4
4
i
4
ik
((-2+D)(Dotp[L+Q1,-L+Q4]Dotp[Q2,Q3]+Dotp[L+Q1,Q3]Dotp[Q2,-L+Q4])-2Dotp[L+Q1,Q2]Dotp[Q3,-L+Q4])((-2+D)(Dotp[-L-Q1,Q4]Dotp[Q1,L-Q4]+Dotp[-L-Q1,L-Q4]Dotp[Q1,Q4])-2Dotp[-L-Q1,Q1]Dotp[L-Q4,Q4])
In[]:=
​​rull01={W:>Q1+Q2};​​DI3=Denominator[inti3]//.rull01
Out[]=
2
L
(
2
m1
+
2
(L+Q1)
)
2
(L+Q1+Q2)
(
2
m2
+
2
(-L+Q4)
)

CrosBox Integral

Rules 4

V(-Q2-L,Q2)[ab]=V41
In[]:=
V41=(-ik(g[a,d]g[b,c]+g[a,c]g[b,d])P[-(L+Q2),c]P[Q2,d])
Out[]=
-ik(g[a,d]g[b,c]+g[a,c]g[b,d])P[-L-Q2,c]P[Q2,d]
S(Q1+L,m1)=S41
In[]:=
S41=
-i
2
(L+Q2)
+
2
m1
Out[]=
-
i
2
m1
+
2
(L+Q2)
V(Q1,Q2+L)[e,f]=V42
In[]:=
V42=(-ik(g[e,h]g[f,g]+g[e,g]g[f,h])P[L+Q2,g]P[Q1,h])
Out[]=
-ik(g[e,h]g[f,g]+g[e,g]g[f,h])P[Q1,h]P[L+Q2,g]
G(L+W)[e,f][i,j]=G41
In[]:=
G41=
-i
(W+L)^2
1
2
(g[e,i]*g[f,j]+g[e,j]*g[f,i])-
1
(D-2)
(g[e,f]*g[i,j])​​
Out[]=
-
i
1
2
(g[e,j]g[f,i]+g[e,i]g[f,j])-
g[e,f]g[i,j]
-2+D
2
(L+W)
V(Q4,L+Q3)[i,j]=V23
In[]:=
​​V43=(-ik(g[i,k]*g[j,n]+g[i,n]*g[j,k])P[-(L-Q4),n]P[Q3,k])
Out[]=
-ik(g[i,n]g[j,k]+g[i,k]g[j,n])P[Q3,k]P[-L+Q4,n]
In[]:=
S42=
-i
(Q4-L)^2+m2^2
Out[]=
-
i
2
m2
+
2
(-L+Q4)
V(L-Q4,Q3)[o,p]=V44
In[]:=
V44=(-ik(g[o,s]g[p,r]+g[o,r]g[p,s])P[Q4,r]P[L-Q4,s])​​
Out[]=
-ik(g[o,s]g[p,r]+g[o,r]g[p,s])P[L-Q4,s]P[Q4,r]

G(L)[a,b][o,p ]=G32

In[]:=
G22=
-i
(L^2)
1
2
(g[o,a]*g[p,b]+g[o,b]*g[p,a])-
1
(D-2)
(g[o,p]*g[a,b])
Out[]=
-
i-
g[a,b]g[o,p]
-2+D
+
1
2
(g[o,b]g[p,a]+g[o,a]g[p,b])
2
L

Inti4

​
Linearity of the inner product of the Box and CrosBox

Development:

Defitons
Exampels:
Rull2n
Test:
Final rule

Development minus-rule

Defs:
Final rule

Rule 24 and 31 act on FSInt3 (Box).

Rule 24 and 31 act on FSInt2 (CrosBox)

Formulierung in Terms of Mandelstam variables and Decomposition of Integrand.

Defs Mandelstam variables:

​

Rules 41, 51 act on FSR33 (Box)

Rules 41, 51 act on FSR43 (CrosBox)

Development of the Decomposition rule :

Rull61 Sym of Dotp[Qn,L]
Rull71 mass wiki https://en.wikipedia.org/wiki/Energy%E2%80%93momentum_relation

Decomposition rule for the Box (FSR35).

Sorse: https://en.wikipedia.org/wiki/Mandelstam_variables

Application of the decomposition rull3171, from EQS3 to FSR35

Rules 41, 51 act on FSR43 (CrosBox)

Decomposition rule for the CrosBox (FSR45).

Sorse: https://en.wikipedia.org/wiki/Mandelstam_variables

Application of the decomposition rull4171, from EQS4 to FSR45

Excluding invers Propagator:

Rull divelupmend for IProp Exlusion:

rull27n:
Test:

Application of the IProp Exlusion rull271 to R323 Box:

Application of the IProp Exlusion rull271 to R426 CrosBox

Comperisen Whit Litutur:

Text Def test valus:
​
Box
CrosBox

Baikov representation

Nominator of Int:

Nominator of Box:
Nominator of CrosBox:

Denominator:

Baikov representation rull:
Denominator of Box and CrosBox:
Denominator in Baikov representation:

Mometum Gram matrix

Loop momentum Gram matrix

Mometum Gram matrix of Box

Baikov representation rull for IProp

Mometum Gram matrix of CrosBox.

TBD

Compersen withs H.

rull from H to T convetion.
​
H Box Expresion:
Numerical Comparison for Box and B-Const

CrosBox

H Mometum Gram matrix :
Numerical Comparison for CrosBox and B-Const
Seff Compersen:

Triangel

Rull
Tignale decomposition rull IProp[m1^2 + (L + Q1)^2]
Tignale decomposition rull IProp[(L + Q1 + Q2)^2]
Tignale decomposition rull IProp[(Q4 - L)^2 + m2^2]
Tignale decomposition rull IProp[L2]