Table Of ContentReduction of a Class of Three-Loop Vacuum Diagrams to
Tetrahedron Topologies
J.-M. Chung∗
Asia Pacific Center for Theoretical Physics, Seoul 130-012
B. K. Chung†
Asia Pacific Center for Theoretical Physics, Seoul 130-012
1
and
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0 Research Institute for Basic Sciences and Department of Physics,
2
Kyung Hee University, Seoul 130-701
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J
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Abstract
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We obtain finite parts (as well as ε-pole parts) of massive three-loop vac-
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0 uum diagrams with three-point and/or four-point interaction vertices by re-
1
ducing them to tetrahedron diagrams with both massive and massless lines,
0
1 whose finite parts were given analytically in a recent paper by Broadhurst.
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In the procedure of reduction, the method of integration-by-parts recurrence
/
h
relationsisemployed. WeuseourresulttocomputetheMSeffective potential
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- of the massive φ4 theory.
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PACS number(s): 11.10.Gh
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Typeset using REVTEX
∗Electronic address: [email protected]
†Electronic address: [email protected]
1
The calculations of the effective potential for a single-component massive φ4 theory
[1] and for a massless O(N) φ4 theory [2,3] were achieved at the three-loop level in four-
dimensional spacetime. (In three dimensional spacetime, see the work of Rajantie [4] for
a single-component φ4 theory.) The calculations in Ref. 1 and Refs. 2 and 3 are done
in the dimensional regularization scheme with a specific set of renormalization conditions.
The same calculations at a lower-loop level, in the cutoff regularization, with the same
renormalization conditions can be found in Ref. 5 and Ref. 6 respectively. We see that the
results agree with each other. Therefore, in the mass-dependent scheme, we do not need to
calculate finite the parts of three-loop diagrams. Knowledge of the pole terms is sufficient.
However, in a mass-independent scheme, such as the MS or MS scheme, we have to
calculate three-loop diagrams to the finite parts, i.e., to the ε0 order in the ε expansion.
Without imposing renormalization conditions at a specific scale, we just leave an arbitrary
constant µ, which is introduced inevitably for dimensional reasons, unspecified as in Eq. (13)
below. This has the drawback that it does not involve true physical parameters measured
at a given scale. Though it normally takes some effort to express physically measurable
quantities in terms of the parameters of the expression, the renormalization group (RG)
equation is dealt with much easier,1 and the calculations in complicated theories are much
more convenient.
The purpose of this note is to reduce a class of massive three-loop vacuum diagrams to
(three-loop)tetrahedron diagramswithbothmassive andmassless lines byusing themethod
of recurrence relations. Once the reductions are completed, the finite parts (as well as the
ε-pole parts) of the diagrams in question can be readily obtained because the finite parts of
these tetrahedron diagrams were determined analytically by Broadhurst [8].
Let us define three-loop vacuum integrals J, K, and L, which are nonfactorizable into
lower-loop integrals:
1
J ,
≡ Z (p2 +σ2)[(p+k)2 +σ2](q2 +σ2)[(q +k)2 +σ2]
kpq
1
K ,
≡ Z (k2 +σ2)(p2 +σ2)[(p+k)2 +σ2](q2 +σ2)[(q +k)2 +σ2]
kpq
1
L . (1)
≡ Z (k2 +σ2)2(p2 +σ2)[(p+k)2 +σ2](q2 +σ2)[(q +k)2 +σ2]
kpq
The momenta in Eq. (1) are all (Wick-rotated) Euclidean, and the abbreviated integration
measure is defined as
dnk
= µ4−n , (2)
Z Z (2π)n
k
where n = 4 2εis thespace-time dimension intheframework of dimensional regularization,
−
and µ is an arbitrary constant with a dimension of mass. The pole parts of all the integrals
in Eq. (1) are known [1, 3]. Now, the problem is to the find the finite parts of these integrals
in terms of the known transcendental numbers.
1See Ref. 7 for two-loop RG improvement of the effective potential.
2
While massless multi-loop diagrams can be dealt with by essentially algebraic meth-
ods [9], the situation is more complicated in the case of massive diagrams. In notation of
Avdeev [10], the above three-loop diagrams J, K, and L correspond to B (0,0,1,1,1,1),
N
D (1,1,1,1,1,0), and D (1,1,1,1,2,0), respectively. The method of integration-by-parts
5 5
recurrence relations, given in Avdeev’s paper [10], allows us to connect the integrals
B (0,0,1,1,1,1), D (1,1,1,1,1,0), and D (1,1,1,1,2,0) to the tetrahedron integrals
N 5 5
B (1,1,1,1,1,1), B (1,1,1,1,1,1), and D (1,1,1,1,1,1). The analytic expressions of the
N M 5
finite parts, i.e., ε0 order terms, for all tetrahedron vacuum diagrams with different combi-
nations of massless and massive lines of a single mass scale were given in a recent paper by
Broadhurst [8]. With the convention of integration measure used in Ref. 8,
dnk
[dk] = ,
Z Z σn−4πn/2Γ(1+ε)
we quote the results of Broadhurst which are relevant to our calculation:
2ζ(3)
B (1,1,1,1,1,1)= V = +6ζ(3) 14ζ(4) 16U ,
N 4N 3,1
ε − −
2ζ(3)
B (1,1,1,1,1,1)= V = +6ζ(3) 9ζ(4) ,
M 3T
ε −
2ζ(3) 469 8 π
D (1,1,1,1,1,1)= V = +6ζ(3) + Cl2 16V , (3)
5 5 ε − 27 3 2(cid:16)3(cid:17)− 3,1
where U and V are defined as
3,1 3,1
( 1)m+n
U − ,
3,1 ≡ m3n
m>Xn>0
( 1)mcos(2πn/3)
V − ,
3,1 ≡ m3n
m>Xn>0
and can be expressed in terms of known transcendental numbers as2
ζ(4) ζ(2) 1 1
U = + ln22 ln42 2Li ,
3,1 4
2 2 − 12 − (cid:16)2(cid:17)
1 π 1 2π 13 259 3 2π
V = Cl2 πLs + ζ(3)ln3 ζ(4)+ Ls(1) . (4)
3,1 3 2(cid:16)3(cid:17)− 4 3(cid:16) 3 (cid:17) 24 − 108 8 4 (cid:16) 3 (cid:17)
(1)
In the above equation, Li (x), Cl (x), Ls (x), and Ls (x) are the polylogarithm, Clausen’s
4 2 3 4
polylogarithm, the log-sine integral, and the generalized log-sine integral, respectively [12],
whose numerical values at the given arguments are
1
Li = 0.517 479 061... ,
4
(cid:16)2(cid:17)
2The combination V in Eq. (4) in terms of the known transcendental numbers was found by
3,1
Fleischer and Kalmykov [11].
3
π
Cl = 1.014 941 606... ,
2
(cid:16)3(cid:17)
2π
Ls = 2.144 762 212... ,
3
(cid:16) 3 (cid:17) −
2π
(1)
Ls = 0.497 675 551... . (5)
4 (cid:16) 3 (cid:17) −
Using the the method of recurrence relations [9, 10, 13], we can arrive at following
connections:
1 4 2 1
σ−4B (0,0,1,1,1,1)= 32B +
N 4
(cid:20)2(1 3ε) 2 3ε − 1 2ε − 2(1 ε)(cid:21)
− − − −
486 729 35 3 2 512 10
+ + + +
− 1 3ε − 2(2 3ε) − 2ε ε2 ε3 1 2ε 1 ε
− − − −
Γ(1 ε)Γ2(1+2ε)Γ(1+3ε) 14 35 378 189 896 14
+ − + + + ,
Γ2(1+ε)Γ(1+4ε) (cid:20)3ε2 ε 1 3ε 2 3ε − 3(1 2ε) − 3(1 ε)(cid:21)
− − − −
2 1
D (1,1,1,1,1,1) = σ−2D (1,1,1,1,1,0) 1
5 5
−3 (cid:20) − 2ε(cid:21)
8 3 2 1 2
B + σ−2VL111(1,1,1)
− 3 4(cid:20)2(1 3ε) − 1 2ε 2(1 ε)(cid:21)− 3ε2
− − −
2 1 4 243 15 160 7
+ + + +
− 3ε4 − ε3 3ε2 2(1 3ε) ε − 3(1 2ε) 6(1 ε)
− − −
Γ(1 ε)Γ2(1+2ε)Γ(1+3ε) 7 14 175 189 224 7
− + + + + ,
− Γ2(1+ε)Γ(1+4ε) (cid:20)9ε3 3ε2 9ε 2(1 3ε) − 9(1 2ε) 18(1 ε)(cid:21)
− − −
1 32 1 1 1
D (1,1,1,1,2,0) = σ−2D (1,1,1,1,1,0)(1+ε) B +
5 5 4
3 − 3 (cid:20)2(1 3ε) − 1 2ε 2(1 ε)(cid:21)
− − −
4 162 44 4 256 10
σ−2VL111(1,1,1)+ + +
− 3ε 1 3ε 3ε − 3ε3 − 3(1 2ε) 3(1 ε)
− − −
Γ(1 ε)Γ2(1+2ε)Γ(1+3ε) 28 56 126 448 14
− + + + , (6)
− Γ2(1+ε)Γ(1+4ε) (cid:20)9ε2 3ε 1 3ε − 9(1 2ε) 9(1 ε)(cid:21)
− − −
where B is proportional to the difference between B (1,1,1,1,1,1) and B (1,1,1,1,1,1)
4 N M
[13], i.e.,
(1 2ε)(2 2ε)
B = − − B (1,1,1,1,1,1) B (1,1,1,1,1,1) .
4 M N
− 4 h − i
In Eq. (6), VL111(1,1,1) denotes a two-loop bubble integral, shown in Fig. 1 of the paper
by Fleischer and Kalmykov [11]. Its value up to ε3 order in the ε expansion can be found
in Eq. (7) of Ref. 11. For all higher-order terms, one may refer to Ref. 14. For our desired
accuracy, it is sufficient to take its value up to ε order:
3 1 4 π
σ−2VL111(1,1,1) = Cl
−2(1 ε)(1 2ε)(cid:20)ε2 − √3 2(cid:16)3(cid:17)
− −
4 π π3 2π
+ε Cl ln3 2√3Ls +O(ε2) . (7)
(cid:26)√3 2(cid:16)3(cid:17) − 3√3 − 3(cid:16) 3 (cid:17)(cid:27) (cid:21)
4
From Eqs. (3) — (7), we eventually obtain
2 23 35 275
B (0,0,1,1,1,1) = + + + +O(ε) ,
N
ε3 3ε2 2ε 12
1 17 1 67 12 π 229 60 π
D (1,1,1,1,1,0) = + + Cl + Cl
5 −ε3 − 3ε2 ε(cid:20)− 3 √3 2(cid:16)3(cid:17)(cid:21)− 3 √3 2(cid:16)3(cid:17)
12 π π3 2π
Cl ln3+ +6ζ(3)+6√3Ls +O(ε) ,
2 3
−√3 (cid:16)3(cid:17) √3 (cid:16) 3 (cid:17)
1 2 1 2 4 π
D (1,1,1,1,2,0) = + + Cl
5 3ε3 3ε2 ε(cid:20)3 − √3 2(cid:16)3(cid:17)(cid:21)
2 4 π π3 2π
+ Cl ln3+2ζ(3) 2√3Ls +O(ε) . (8)
2 3
−3 √3 (cid:16)3(cid:17) − 3√3 − (cid:16) 3 (cid:17)
We readily recover the values of J, K, and L in our original integration measure, Eq. (2),
by using the following relations:
σ4 σ2 −3ε 3ζ(2)
J = exp 3γε+ ε2 ζ(3)ε3+ B (0,0,1,1,1,1),
(4π)6(cid:18)4πµ2(cid:19) (cid:20)− 2 − ···(cid:21) N
σ2 σ2 −3ε 3ζ(2)
K = exp 3γε+ ε2 ζ(3)ε3+ D (1,1,1,1,1,0),
(4π)6(cid:18)4πµ2(cid:19) (cid:20)− 2 − ···(cid:21) 5
1 σ2 −3ε 3ζ(2)
L = exp 3γε+ ε2 ζ(3)ε3+ D (1,1,1,1,2,0),
(4π)6(cid:18)4πµ2(cid:19) (cid:20)− 2 − ···(cid:21) 5
where the exp[ ] factor comes from an expansion of Γ3(1+ε) and γ is the Euler constant.
In the standard MS scheme [15], the factors ln(4π) and γ are absorbed into the renor-
malization scale µ. Howerever, in the other widespread MS convention (see, e.g., Refs. 16
and 17), the factor ζ(2) is absorbed further into the scale µ. (This convention gives the same
result in the one-loop diagrams and is more convenient in higher-loop massive calculations.)
By introducing a new renormalization scale µ¯,
ζ(2)ε
µ¯2 = 4πµ2exp γ + , (9)
(cid:20)− 2 (cid:21)
the above three integrals J, K, and L are given as follows:
σ4 σ2 −3ε 2 23 35
J = + + +F ,
(4π)6(cid:18)µ¯2(cid:19) (cid:20)ε3 3ε2 2ε J(cid:21)
σ2 σ2 −3ε 1 17 1 67 12 π
K = + + Cl +F ,
(4π)6(cid:18)µ¯2(cid:19) (cid:20)−ǫ3 − 3ε2 ε(cid:26)− 3 √3 2(cid:16)3(cid:17)(cid:27) K(cid:21)
1 σ2 −3ε 1 2 1 2 4 π
L = + + Cl +F , (10)
(4π)6(cid:18)µ¯2(cid:19) (cid:20)3ε3 3ε2 ε(cid:26)3 − √3 2(cid:16)3(cid:17)(cid:27) L(cid:21)
where
275
F = 2ζ(3) ,
J
12 −
229 π3 60 π 12 π 2π
F = + +7ζ(3)+ Cl Cl ln3+6√3Ls ,
K 2 2 3
− 3 √3 √3 (cid:16)3(cid:17)− √3 (cid:16)3(cid:17) (cid:16) 3 (cid:17)
2 π3 5ζ(3) 4 π 2π
F = + + Cl ln3 2√3Ls . (11)
L 2 3
−3 − 3√3 3 √3 (cid:16)3(cid:17) − (cid:16) 3 (cid:17)
5
This completes the analytic evaluations of the three-loop vacuum integrals of Eq. (1) up to
the finite parts.
The purely numerical computation of the finite parts for some three-loop vacuum dia-
grams in a paper by Pelissetto and Vicari [18] enables us to extract the numerical values
for F , F , and F . In obtaining F and F , we assume first the unknown finite parts F
J K L J K J
and F for J and K in Eq. (10) since the pole parts are already known [1,3]. Then, we
K
differentiate J and K, twice and once, respectively, with respect to σ2. Meanwhile, we can
differentiate J and K in Eq. (1), twice and once, respectively, with respect to σ2 before the
momentum integrations, yielding the three-loop diagrams given in Appendix B of Ref. 18
whose finite parts have been numerically calculated. By equating the results of the differen-
tiations thus done, we determine the unknown values of F and F . The extracted results
J K
are
275 π
F = 6√3Cl 22ζ(3)+80S +20S 120S 30S 6S ,
J 2 1 2 5 6 7
12 − (cid:16)3(cid:17)− − − −
229 28 π
F = + Cl π2 +9ζ(3) 24S 6S +6S 12S ,
K 2 1 2 4 7
− 3 √3 (cid:16)3(cid:17)− − − −
2 π π2 5ζ(3)
F = 4√3Cl +8S +2S +2S , (12)
L 2 1 2 4
−3 − (cid:16)3(cid:17)− 3 − 3
where the quantities S , S , S , S , S , and S were calculated numerically in Appendix B
1 2 4 5 6 7
of Ref. 18. We see that the numerical values of F , F , and F given in Eq. (12) agree with
J K L
the analytical values in Eq. (11).3
Our results, Eqs. (10) and (11), together with the result for the all-massive-line tetrahe-
dron diagram in Ref. 8,
1
M
≡ Z (k2 +σ2)(p2 +σ2)(q2 +σ2)[(k p)2 +σ2][(p q)2 +σ2][(q k)2 +σ2]
kpq
− − −
1 σ2 −3ε 2ζ(3) 2π2 2 π 1
= +6ζ(3) 17ζ(4) ln22+ ln42 4Cl2 +16Li ,
(4π)6(cid:18)µ¯2(cid:19) (cid:20) ε − − 3 3 − 2(cid:16)3(cid:17) 4(cid:16)2(cid:17)(cid:21)
enable us to calculate the three-loop effective potential in the MS scheme for the single-
component massive φ4 theory. The renormalization of the three-loop effective potential is
straightforward, albeit long. Thus, we simply report the result:
V = V(0) +h¯V(1) +h¯2V(2) +h¯3V(3) +O(h¯4) ,
m2φ2 λφ4
V(0) = + ,
2 4!
λ 3m4 3m2φ2 3λφ4 m4 m2φ2 λφ4 m2
V(1) = + + + ln φ ,
(4π)2(cid:20)− 8λ − 8 − 32 (cid:26)4λ 4 16 (cid:27) (cid:18) µ¯2 (cid:19)(cid:21)
3Also, in a recent paper by Anderson et al. [19], F was calculated numerically. The numerical
J
value of C in Eq. (18) of Ref. 19 agrees with our analytic value F in Eq. (11): our F +23π2/12
0 J J
means C of Ref. 19.
0
6
λ2 m4 3 1 π 11 1 π
V(2) = +m2φ2 Cl +λφ4 Cl
(4π)4(cid:20)8λ (cid:18)4 − 2√3 2(cid:16)3(cid:17)(cid:19) (cid:18)32 − 4√3 2(cid:16)3(cid:17)(cid:19)
m4 3m2φ2 5λφ4 m2 m4 m2φ2 3λφ4 m2
+ + ln φ + + + ln2 φ ,
−(cid:26)4λ 4 16 (cid:27) (cid:18) µ¯2 (cid:19) (cid:26)8λ 4 32 (cid:27) (cid:18) µ¯2 (cid:19)(cid:21)
λ3 1 m4 2363 13 π 3ζ(3)
V(3) = +m2φ2 + Cl +
2
(4π)6(cid:20)576 λ (cid:18)− 576 4√3 (cid:16)3(cid:17) 4 (cid:19)
4487 11 π 1 π 2 1 17ζ(4)
+λφ4 + Cl + Cl2 Li +
(cid:18)−2304 8√3 2(cid:16)3(cid:17) 6 2(cid:16)3(cid:17)− 3 4(cid:16)2(cid:17) 24
π2ln22 ln42 41m4 371 7 π
+ + +m2φ2 Cl
2
36 − 36 (cid:19) (cid:26) 96λ (cid:18) 96 − 4√3 (cid:16)3(cid:17)(cid:19)
701 3√3 π ζ(3) m2 17m4 37m2φ2
+λφ4 Cl + ln φ +
(cid:18)384 − 4 2(cid:16)3(cid:17) 4 (cid:19)(cid:27) (cid:18) µ¯2 (cid:19)−(cid:26) 48λ 24
143λφ4 m2 5m4 7m2φ2 9λφ4 m2
+ ln2 φ + + + ln3 φ , (13)
192 (cid:27) (cid:18) µ¯2 (cid:19) (cid:26)48λ 24 64 (cid:27) (cid:18) µ¯2 (cid:19)(cid:21)
where m2 is defined as m2 m2 + λφ2.
φ φ ≡ 2
In summary, using the method of integration-by-parts recurrence relations, we
have obtained the exact relations between the non-tetrahedron three-loop integrals
B (0,0,1,1,1,1),D (1,1,1,1,1,0),andD (1,1,1,1,2,0)andthetetrahedronthree-loopin-
N 5 5
tegrals B (1,1,1,1,1,1), B (1,1,1,1,1,1), and D (1,1,1,1,1,1), whose values are known
N M 5
to the finite parts. As an application of our loop calculations, the analytic evaluation of
three-loop effective potential in the MS scheme for the single-component massive φ4 theory
was obtained.
ACKNOWLEDGEMENT
This work was supported by a Korea Research Foundation grant (KRF-2000-015-
DP0066).
7
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