Two-loop massive Feynman integrals for ϕ4 field-theoretical model with long-range correlated disorder are considered. Massive integrals for the vertex function Γ(4) including two or three massless propagators for generic space dimension and for any value of the correlation parameter are evaluated analytically applying Mellin-Barnes method as well as familiar representation for one-loop integrals. Obtained expressions are presented in the form of hypergeometric functions.

1.
V. A.
Smirnov
, “
Evaluating Feynman integrals
,”
Springer Tracts  Mod. Phys.
(
Springer-Verlag
,
Berlin
,
2005
), Vol. 211;
V. A.
Smirnov
,
Feynman Integral Calculus
(
Springer-Verlag
,
Berlin
,
2006
).
2.
E.
Brézin
,
J. C.
Le Guillou
, and
J.
Zinn-Justin
, in
Phase Transitions and Critical Phenomena
, edited by
C.
Domb
and
M. S.
Green
(
Academic Press
,
London
,
1976
), Vol.
6
, p.
127
;
D. J.
Amit
,
Field Theory, the Renormalization Group, and Critical Phenomena
(
World Scientific
,
Singapore
,
1989
);
J.
Zinn-Justin
,
Quantum Field Theory and Critical Phenomena
(
Oxford University Press
,
1996
);
H.
Kleinert
and
V.
Schulte-Frohlinde
,
Critical Properties of ϕ4-Theories
(
World Scientific
,
Singapore
,
2001
).
3.
K. G.
Wilson
and
M. E.
Fisher
,
Phys. Rev. Lett.
28
,
240
(
1972
).
4.
R.
Guida
and
J.
Zinn-Justin
,
J. Phys. A: Math. Gen.
31
,
8103
(
1998
).
5.
G.
Parisi
, “
Field theory approach to second order phase transitions in three and two dimensional systems
,”
Lectures given at the Cargèse summer School
(1973), Columbia Uneversity, preprint;
G.
Parisi
,
J. Stat. Phys.
23
,
49
(
1980
).
6.
J. M.
Carmona
,
A.
Pelissetto
, and
E.
Vicari
,
Phys. Rev. B
61
,
15136
(
2000
).
7.
R.
Folk
,
Y.
Holovatch
, and
T.
Yavors’kii
,
Phys. Rev. B
62
,
12195
(
2000
).
8.
A.
Pelissetto
and
E.
Vicari
,
Phys. Rep.
368
,
549
(
2002
).
9.
G.
Grinstein
and
A.
Luther
,
Phys. Rev. B
13
,
1329
(
1976
).
10.
11.
I. O.
Mayer
,
A. I.
Sokolov
, and
B. N.
Shalaev
,
Ferroelectrics
95
,
93
(
1989
).
12.
D. V.
Pakhnin
and
A. I.
Sokolov
,
JETP Lett.
71
,
412
(
2000
), [
Pisma v ZhETF
71,
600
(
2000
) (in Russian)];
D. V.
Pakhnin
and
A. I.
Sokolov
,
Phys. Rev. B
61
,
15130
(
2000
).
13.
A.
Pelissetto
and
E.
Vicari
,
Phys. Rev. B
62
,
6393
(
2000
).
14.
Y.
Holovatch
,
M.
Dudka
, and
T.
Yavors’kii
,
J. Phys. Stud.
5
,
233
(
2001
).
15.
Y.
Holovatch
,
V.
Blavats’ka
,
M.
Dudka
,
C.
von Ferber
,
R.
Folk
, and
T.
Yavors’kii
,
Int. J. Mod. Phys. B
16
,
4027
(
2002
).
16.
Y.
Holovatch
,
R.
Folk
, and
T.
Yavors’kii
,
Physics-Uspiekhi
46
,
169
(
2003
).
17.
V. V.
Prudnikov
,
P. V.
Prudnikov
, and
A. A.
Fedorenko
,
JETP Lett.
68
,
950
(
1998
) [
Pisma Zh. Eksp. Teor. Fiz.
68,
900
(
1998
) (in Russian)].
18.
M.
Dudka
,
Y.
Holovatch
, and
R.
Folk
,
Condens. Matter Phys.
4
,
77
(
2001
).
19.
M.
Dudka
,
Y.
Holovatch
, and
R.
Folk
,
Condens. Matter Phys.
4
,
459
(
2001
).
20.
M.
Dudka
,
Y.
Holovatch
, and
R.
Folk
, in
Fluctuating Path and Fields
, edited by
W.
Janke
,
A.
Pelster
,
H.-J.
Schmidt
, and
V.
Bachmann
(
World Scientific
,
Singapore
,
2001
), p.
457
.
21.
P.
Calabrese
,
A.
Pelissetto
, and
E.
Vicari
,
Phys. Rev. E
70
,
036104
(
2004
).
22.
A.
Pelissetto
,
P.
Rossi
, and
E.
Vicari
,
Phys. Rev. B
63
,
140414
(
2001
).
23.
B.
Delamotte
,
M.
Dudka
,
Y.
Holovatch
, and
D.
Mouhanna
,
Phys. Rev. B
82
,
104432
(
2010
).
24.
B.
Delamotte
,
M.
Dudka
,
Y.
Holovatch
, and
D.
Mouhanna
,
Condens. Matter Phys.
13
,
43703
(
2010
).
25.
Y.
Holovatch
and
M.
Shpot
,
J. Stat. Phys.
66
,
867
(
1992
).
26.
Y.
Holovatch
and
T.
Krokhmal’s’kii
,
J. Math. Phys.
35
,
3866
(
1994
).
27.
Y.
Holovatch
and
T.
Yavorskii
,
J. Stat. Phys.
92
,
785
(
1998
);
Y.
Holovatch
and
T.
Yavorskii
,
Condens. Matter Phys.
11
,
87
(
1997
).
28.
M. A.
Shpot
,
Cond. Matter Phys.
13
,
13101
(
2010
).
29.
R. M.
Hornreich
,
M.
Luban
, and
S.
Shtrikman
,
Phys. Rev. Lett.
35
,
1678
(
1975
).
30.
M. A.
Shpot
,
J. Math. Phys.
48
,
123512
(
2007
).
31.
S. M.
Dorogovtsev
,
Sov. Phys. Solid State
22
,
188
(1980) [
Fiz. Tverd. Tela (Leningrad)
22,
321
(
1980
) (in Russian)];
Sov. Phys. Solid State
22,
2141
(
1980
) [
Fiz. Tverd. Tela (Leningrad)
22,
3659
(
1980
) (in Russian)].
32.
D.
Boyanovsky
and
J. L.
Cardy
,
Phys. Rev. B
26
,
154
(
1982
);
D.
Boyanovsky
and
J. L.
Cardy
,
Phys. Rev. B
27
,
6971
(
2005
).
33.
I. D.
Lawrie
and
V. V.
Prudnikov
,
J. Phys. C: Solid State Phys.
17
,
1655
(
1984
).
34.
V.
Blavatska
,
C.
von Ferber
, and
Y.
Holovatch
,
Acta Phys. Slovaca
52
,
317
(
2002
).
35.
V.
Blavatska
,
C.
von Ferber
, and
Y.
Holovatch
,
Phys. Rev. B
67
,
094404
(
2003
).
36.
V.
Blavatska
,
M.
Dudka
,
R.
Folk
, and
Y.
Holovatch
,
Phys. Rev. B
72
,
064417
(
2005
).
37.
A.
Weinrib
and
B. I.
Halperin
,
Phys. Rev. B
27
,
413
(
1983
).
38.
V. V.
Prudnikov
,
P. V.
Prudnikov
, and
A. A.
Fedorenko
,
J. Phys. A: Math. Gen.
32
,
8587
(
1999
).
39.
V. V.
Prudnikov
,
P. V.
Prudnikov
, and
A. A.
Fedorenko
,
Phys. Rev. B
62
,
8777
(
2000
).
40.
E. R.
Korucheva
and
D. I.
Uzunov
,
Phys. Status Solidi B
126
,
K19
(
1984
).
41.
V. V.
Prudnikov
and
A. A.
Fedorenko
,
J. Phys. A: Math. Gen.
32
,
L405
(
1999
).
42.
E.
Korucheva
and
F. J.
de la Rubia
,
Phys. Rev. B
58
,
5153
(
1998
).
43.
Y.
Chen
,
S. H.
Guo
, and
Z. B.
Li
,
Phys. Status Solidi B
223
,
599
(
2001
).
44.
Z. B.
Li
,
Y.
Chen
, and
S. H.
Guo
,
Mod. Phys. Lett. B
15
,
43
(
2001
).
45.
Y.
Chen
and
Z. B.
Li
,
Phys. Rev. B
71
,
174433
(
2005
).
46.
P. V.
Prudnikov
and
V. V.
Prudnikov
,
Phys. Rev. B
80
,
024115
(
2009
).
47.
E. R.
Korucheva
and
Y. T.
Millev
,
J. Phys. A: Math. Gen.
17
,
L511
(
1984
).
48.
A. A.
Fedorenko
,
P.
Le Doussal
, and
K. J.
Wiese
,
Phys. Rev. E
74
,
061109
(
2006
).
49.
A. A.
Fedorenko
,
Phys. Rev. B
77
,
094203
(
2008
).
50.
V.
Blavatska
,
C.
von Ferber
, and
Y.
Holovatch
,
Phys. Rev. E
64
,
041102
(
2001
).
52.
A.
Weinrib
,
Phys. Rev. B
29
,
387
(
1984
).
53.
V.
Marinov
, “
Long range correlated percolation
,” e-print arXiv:cond-mat/0611533: cond-mat.stat-mech.
54.
A. A.
Fedorenko
,
D.
Carpentier
, and
E.
Orignac
,
Phys. Rev. B
85
,
125437
(
2012
).
55.
E. E.
Boos
and
A. I.
Davydychev
,
Theor. Math. Phys.
89
,
1052
(
1991
).
56.
A. I.
Davydychev
,
J. Math. Phys.
32
,
1052
(
1991
).
57.
A. I.
Davydychev
,
J. Math. Phys.
33
,
358
(
1992
).
58.
Higher Transcendental Functions
, edited by
A.
Erdelei
(
McGrawHill
,
New York
,
1953
), Vol.
I
.
59.
C.
Anastasiou
,
E. W. N.
Glover
, and
C.
Oleari
,
Nucl. Phys. B
572
,
307
(
2000
).
60.
V. V.
Bytev
,
M. Y.
Kalmykov
, and
B. A.
Kniel
,
Nucl. Phys. B
836
,
129
(
2010
).
61.
H.
Exton
,
Handbook of Hypergeometric Integrals: Theory, Applications, Tables, Computer Programs
(
Ellis Horwood
,
Chichester, England
,
1978
).
62.
I. S.
Gradshteyn
and
I. M.
Ryzhik
, in
Table of Integrals, Series, and Products
, 7th ed., edited by
A.
Jeffrey
and
D.
Zwillinger
(
Academic Press
,
2007
).
63.
L. J.
Slater
,
Generalised Hypergeometric Functions
(
University Press
,
Cambridge
,
1966
).
64.
A. P.
Prudnikov
,
Y. A.
Brychkov
, and
O. I.
Marychev
,
Integrals and Series. Special Functions
(
Gordon and Breach
,
New York
,
1986
), Vol.
3
.
65.
A. I.
Davydchev
,
Phys. Rev. D
61
,
087701
(
2000
).
66.
A. I.
Davydchev
and
M. Y.
Kalmykov
,
Nucl. Phys. B, Proc. Suppl.
89
,
283
(
2000
).
67.
A. I.
Davydchev
and
M. Y.
Kalmykov
,
Nucl. Phys. B
605
,
266
(
2001
).
68.
M. Y.
Kalmykov
,
J. High Energy Phys.
04
,
056
(
2006
).
69.
M. Y.
Kalmykov
,
B. F. L.
Ward
, and
S.
Yost
,
J. High Energy Phys.
02
,
040
(
2007
).
70.
M. Y.
Kalmykov
,
B. F. L.
Ward
, and
S.
Yost
,
J. High Energy Phys.
11
,
009
(
2007
).
71.
M. Y.
Kalmykov
and
B. A.
Kniehl
,
Phys. Part. Nucl.
41
,
942
(
2010
).
72.
S. A.
Yost
,
V. V.
Bytev
,
M. Y.
Kalmykov
,
B. A.
Kniehl
, and
B. F. L.
Ward
, in
Proceeding DPF-2011 Conference
,
Providence, RI
,
8-13 August 2011
;
e-print arXiv:1110.0210 [math-ph].
73.
V. V.
Bytev
,
M. Y.
Kalmykov
, and
B. A.
Kniehl
, in
Proceeding of 11th DESY Workshop on Elementary Particle Physics
,
Wernigerode, Germany
,
15-20 April 2012
;
74.
D.
Greynat
and
J.
Sesma
,
Comput. Phys. Commun.
185
,
472
(
2014
).
75.
S.
Moch
,
P.
Uwer
, and
S.
Weinzierl
,
J. Math. Phys.
43
,
3363
(
2002
).
76.
S.
Weinzierl
,
J. Math. Phys.
45
,
2656
(
2004
).
77.
S.
Weinzierl
,
Comput. Phys. Commun.
145
,
357
(
2002
).
78.
S.
Moch
and
P.
Uwer
,
Comput. Phys. Commun.
174
,
759
(
2006
).
79.
T.
Huber
and
D.
Maître
,
Comput. Phys. Commun.
175
,
122
(
2006
).
80.
T.
Huber
and
D.
Maître
,
Comput. Phys. Commun.
178
,
755
(
2008
).
81.
V. V.
Bytev
,
M. Y.
Kalmykov
, and
B. A.
Kniehl
,
Comput. Phys. Commun.
184
,
2332
(
2013
).
82.
B. A.
Kniehl
and
O. V.
Tarasov
,
Nucl. Phys. B
854
,
841
(
2012
).
You do not currently have access to this content.