Notation and generic Euclidean integrals
(i) Notation:
The Euclidean four-momentum is denoted as P = (ωn,p) for bosons, where ωn = 2nπT, and
as P = (νn,p) for fermions, where νn = (2n + 1)πT. In dimensional regularization, space
integration is performed in d ≡ 3− 2 dimensions. Shorthand notation for the combined Matsubara sum and space integration:
bosons:
Z X
P ≡T X ωn Z p , fermions: Z X
{P} ≡T
X νn Z p , where Z p ≡ eγµ2
4π Z
d3−2p
(2π)3−2.
(ii) Generic Euclidean integrals:
Z dnp
(2π)nln(p
2+m2) =−Γ −
n
2
(4π)n/2m
n,
Z dnp
(2π)n
1
(p2+m2)α =
1 (4π)n/2
Γ α− n
2
Γ(α) m
n−2α,
Z
dnp
(2π)n
(p2)β (p2+m2)α =
1 (4π)n/2
Γ α− n
2 −β
Γ n
2 +β
Γ(α)Γ n2 m
n−2α+2β.
Three-dimensional integrals
(i) One-loop integrals:Z
p
ln(p2+m2) = − e
γ
(4π)3/2
µ
m 2
Γ −d
2
m3
=−m 3
6π µ
2m 2
1 + 8 3+
52 9 + π2 4 2 + 320 27 + 2 3π
2− 7 3ζ(3)
3+O(4)
,
Z
p
1
(p2+m2)α = eγ
(4π)3/2
µ
m
2Γ α− d
2
Γ(α) m 3−2α,
Z
p
1
p2+m2 =−
m
4π µ
2m 2
1 + 2+
4 + π 2
4
2 +
8 + π 2
2 − 7 3ζ(3)
3+O(4)
,
Z
p
1
(p2+m2)2 = 1 8πm
µ
2m 2
1 + π 2
4 2−7
3ζ(3) 3
+O(4)
,
Z
p
1
(p2+m2)3 = 1 32πm3
µ
2m 2
1 + 2+π 2 4 2+ π2 2 − 7 3ζ(3)
3+O(4)
,
Z
p
1
(p2+m2)4 = 1 64πm5
µ
2m 2
1 + 8 3+
4 3 + π2 4 2+ 2 3π
2− 7 3ζ(3)
3+O(4)
,
Z
p
1
(p2)α[(p+k)2]β = eγ
(4π)3/2
µ
k
2 Γ d
2 −α
Γ d2 −βΓ α+β−d
2
Γ(d−α−β)Γ(α)Γ(β) (k
2)32−α−β,
Z
p
1 (p2+m2
1)(p2+m22)
= e
γ
(4π)3/2
µ2
m1m2
Γ(−1 2)
m1 m2m1
−m2 m1m2
m2 2−m21
= 1
4π(m1+m2)
µ2 4m1m2
1 +
2− m1 +m2
m1−m2 lnm1
m2
+O(2)
(ii) Two-loop integrals:
Z
pq
1
(p2+m2)α(q2+m2)β[(p+q)2]δ = e2γ
(4π)3
µ
m 4
(m2)3−α−β−δ
× Γ
d
2 −δ
Γ α+δ− d
2
Γ β+δ− d
2
Γ(α+β+δ−d) Γ d2Γ(α)Γ(β)Γ(α+β+ 2δ−d) .
(iii) Integrals evaluated in coordinate space:
Z
pq
1
p2+m2 1
q2+m2
1
(p+q)2+m2 = 1 (8π)2
µ
2m
41
+ 2 + 4 ln 2−4 ln 3+
+h4 + 11 6 π
2+ 8 ln 2−8 ln 3 + 8 ln22−
−4 ln 2 ln 3 + 2 ln23 + 12 Li2(−2)i+O(2)
,
Z
pq
1
p2+m2 1
q2+m2
1
(p+q+k)2+m2
k=im
= 1 (8π)2
µ
2m 41
+ 6−8 ln 2+
+36−π 2
6 −48 ln 2 + 8 ln
22+O(2)
,
Z
pqr
1
p2+m2 1
q2+m2 1
r2+m2
1
(p+q+r)2+m2 =−
m
(4π)3
µ
2m 61
+ 8−4 ln 2+
+
52 + 17 12π
2−
32 ln 2 + 4 ln22
+O(2)
.
(iv) Some auxiliary identities:
Z ∞
0
dx x12+K31 2−
(x) = π 3/2
8√2
1
−γ+ 3 ln 2−4 ln 3 +
25 12π
2 +γ
2
2 +γ(4 ln 3−3 ln 2)+ +9
2ln 2
2 + 2 ln23 + 12 Li2(−2)
+O(2)
,
Z ∞
0
dx x2K41 2−
(x) =− π 2
4
1
+ 2(3−γ−3 ln 2) +
36 + 5 3π
2−12γ+ 2γ2+
+ 12(γ−3) ln 2 + 14 ln22
+O(2)
,
Z ∞
0
dx x2K31 2−
(x)I1
2−(x) = π
8
1
+ 2(3−γ−5 ln 2) +
36− π 2
6 −12γ+ 2γ 2+
+ 20(γ−3) ln 2 + 26 ln22
+O(2)
(v) Angular averaging in n dimensions and related identities:
Ωn =
2πn/2
Γ n2,
eip·xΩn =px 2
1−n2
Γ n2Jn
2−1(px), Z ∞
0
dx x
α+1Jα(bx)
(x2 +a2)β+1 =
b
2a β
aα
Γ(β+ 1)Kα−β(ab),
Z ∞
0
dx xα+1Jα(bx)e−ax2 = b
α
(2a)α+1e
−b2/4a ,
Z ∞
0
dx xα+1Iα(bx)e−ax 2
= b
α
(2a)α+1e
b2/4a ,
Z ∞
0
dxdy(xy)α+1Iα(cxy)e−ax 2
e−by2 = (2c)
αΓ(α+ 1)
(4ab−c2)α+1.
Coordinate space propagator in an arbitrary dimension
(i) Derivation by solution of a differential equation:
Define the coordinate space propagator V(R) by a Fourier transform,
V(R) =
Z
p
eip·R p2+m2.
We are going to prove the following formula,
V(R) =
eγµ2
4π
1
(2π)32− m
R 12−
K1
2−(mR). (1)
First note that the propagator is a Green’s function of the (Euclidean) Klein–Gordon equation in d = 3−2 dimensions. Using its definition, it is easy to show that it solves the differential equation (−∆ +m2)V(R) = Ξδd(R), where Ξ ≡ eγµ2/4π. Using the fact that V(R) only depends on the radial coordinate R and introducing the dimensionless variable x = mR, this equation becomes
d2 dx2 +
d−1
x
d dx−1
V(x) = 0,
everywhere except at the origin. This can be transformed by the substitutionV(x) =x−12+f(x)
into
x2f00+xf0−hx2+ 12 −2if = 0,
which is the modified Bessel equation. Its general solution is a linear combination of the modified Bessel functions of the first and second kinds. We thus obtain
V(x) =x−12+ h
cII1
2−(x) +cKK 1 2−(x)
i
. (2)
However, sinceIα(x) goes asymptotically asex/
√
evaluate the propagator at the origin. Combining Eq. (2) with the (double) Laurent series for the modified Bessel function of the second kind,
Kα(x) =
Γ(−α) 2
x
2
α
1 + x 2
4(1 +α)+O(x 4
)
+Γ(α) 2
x
2
−α
1 + x 2
4(1−α) +O(x 4
)
, (3)
one obtains V(0) = cKΓ − 12
/232−. On the other hand one finds, by direct integration, V(0) = Ξm1−2Γ − 1
2
/(4π)32−. This fixes the value of cK, leading immediately to Eq. (1).
Using the Eq. (3), one in turn derives the corresponding Laurent expansion of the propagator,
V(R) =
eγµ2
4
Γ 12 −
Γ 12
R−1+2
4π
1 + (mR) 2
2(1 + 2)+O(m 4
R4)
−
− eγµ2 Γ −
1 2 +
Γ −1 2
m1−2
4π
1 + (mR) 2
2(3−2)+O(m 4
R4)
.
(ii) Generalization and derivation by a direct integration:
Z
p
eip·R
(p2+m2)α =
eγµ2 4π
21−α
(2π)32−Γ(α) m
R
32−−α K3
Four-dimensional sum-integrals (bosons)
(i) One-loop sum-integrals:
Z X
P lnP
2 =−(eγµ2) 8π2
Γ −d
2
ζ(−d)
Γ 12 (2πT) 1+d
=−π 2T4
45
µ
4πT 2
1 +
8 3 + 2
ζ0(−3)
ζ(−3)
+
52 9 +
π2 4 +
16 3
ζ0(−3)
ζ(−3) + 2
ζ00(−3)
ζ(−3)
2+O(3)
,
Z X
P
1 (P2)α =
(eγµ2)
8π2
Γ α− d
2
ζ(2α−d) Γ 12Γ(α) (2πT)
1+d−2α,
Z X
P
1
P2 =
T2 12
µ
4πT 2
1 + 2
1 + ζ
0(−1)
ζ(−1)
+
4 + π 2
4 + 4
ζ0(−1)
ζ(−1) + 2
ζ00(−1)
ζ(−1)
2+O(3)
,
Z X
P
1 (P2)2 =
1 (4π)2
µ
4πT 21
+ 2γ+
π2 4 −4γ1
+
1 2γπ
2
+ 4γ2− 7 3ζ(3)
2+O(3)
,
Z X
P
1 (P2)3 =
2ζ(3) (4π)4T2
µ
4πT 2
1 + 2
1 + ζ
0(3)
ζ(3)
+
π2
4 + 4
ζ0(3)
ζ(3) + 2
ζ00(3)
ζ(3)
2 +O(3)
,
Z X
P
(P2 0)β (P2)α =
(eγµ2)
8π2
Γ α− d
2
ζ(2α−2β−d) Γ 12Γ(α) (2πT)
1+d−2α+2β,
Z X
P
(P2
0)β(p2)δ
(P2)α = (−1)
δ(eγµ2)
8π2
Γ α− d
2
Γ δ+d2Γ 1−α+d2ζ(2α−2β−2δ−d) Γ 12Γ(α)Γ d2Γ 1−α+δ+ d2 (2πT)
1+d−2α+2β+2δ,
Z X
P PµPν
(P2)2 = 1
2[δµiδνjδij−(1−2)δµ0δν0]
Z X
P
1
P2,
where the numbers γn are defined by
ζ(1 +) = 1
+
∞
X
n=0
(−1)n n! γn
n, γ
n= lim m→∞
" m X
k=1 lnnk
k −
lnn+1m n+ 1
#
Four-dimensional sum-integrals (fermions)
(i) One-loop sum-integrals:
Z X
{P}lnP
2 =− 1−2−d Z X
PlnP
2
= 1−2−d(e
γµ2)
8π2
Γ −d
2
ζ(−d) Γ 12 (2πT)
1+d
= 7π 2T4
360
µ
4πT 2
1 +
8 3 −
2
7ln 2 + 2
ζ0(−3)
ζ(−3)
+O(2)
,
Z X
{P}
1
(P2)α = 2
2α−d−1 Z X
P
1 (P2)α
= 22α−d−1(e γµ2)
8π2
Γ α− d
2
ζ(2α−d) Γ 12Γ(α) (2πT)
1+d−2α,
Z X
{P}
1
P2 =−
T2 24
µ
4πT 2
1 + 2
1−ln 2 + ζ
0(−1)
ζ(−1)
+O(2)
,
Z X
{P}
1 (P2)2 =
1 (4π)2
µ
4πT
21
+ 2γ+ 4 ln 2 +
π2
4 −4γ1+ 8γln 2 + 4 ln 2
2
+O(2)
,
Z X
{P}
1 (P2)3 =
14ζ(3) (4π)4T2
µ
4πT 2
1 + 2
1 + ζ
0
(3)
ζ(3) + 8 7ln 2
+O(2)
,
Z X
{P}
(P2 0)β (P2)α = 2
2α−2β−d−1 Z X
P
(P2 0)β (P2)α
Z X
{P}
PµPν
(P2)2 = 1
2[δµiδνjδij −(1−2)δµ0δν0]
Z X
{P}
1
