New explicit expressions are derived for the one-loop two-point Feynman integral with arbitrary external momentum and masses and in dimensions. The results are given in terms of Appell functions, manifestly symmetric with respect to the masses . Equating our expressions with previously known results in terms of Gauss hypergeometric functions yields reduction relations for the involved Appell functions that are apparently new mathematical results.
REFERENCES
1.
P.
Appell
and J.
Kampé de Fériet
, Fonctions Hypergéométriques et Hypersphériques. Polynomes d′Hermite
(Gauthier-Villars
, Paris
, 1926
).2.
F. A.
Berends
, A. I.
Davydychev
, and V. A.
Smirnov
, Nucl. Phys. B
478
, 59
(1996
).3.
R. M.
Hornreich
, M.
Luban
, and S.
Shtrikman
, Phys. Rev. Lett.
35
, 1678
(1975
).4.
5.
W.
Selke
, in Phase Transitions and Critical Phenomena
, edited by C.
Domb
and J. L.
Lebowitz
(Academic
, London
, 1992
), Vol. 15
, pp. 1
–72
.6.
7.
H. W.
Diehl
, Pramana, J. Phys.
64
, 803
(2005
).8.
M.
Shpot
and H. W.
Diehl
, Nucl. Phys. B
612
, 340
(2001
).9.
M. A.
Shpot
, Y. M.
Pis’mak
, and H. W.
Diehl
, J. Phys.: Condens. Matter
17
, S1947
(2005
).10.
E. E.
Boos
and A. I.
Davydychev
[Theor. Math. Phys.
89
, 1052
(1992
)].11.
A. I.
Davydychev
and R.
Delbourgo
, J. Math. Phys.
39
, 4299
(1998
).12.
J.
Fleischer
, F.
Jegerlehner
, and O. V.
Tarasov
, Nucl. Phys. B
672
, 303
(2003
).13.
A. T.
Suzuki
, E. S.
Santos
, and A. G. M.
Schmidt
, J. Phys. A
36
, 4465
(2003
).14.
A. T.
Suzuki
, E. S.
Santos
, and A. G. M.
Schmidt
, J. Phys. A
36
, 11859
(2003
).15.
H. M.
Srivastava
and H. L.
Manocha
, A Treatise on Generating Functions
(Halsted∕Wiley
, New York
, 1984
).16.
A.
Prudnikov
, Y. A.
Brychkov
, and O. I.
Marichev
, Integrals and Series. More Special Functions
(Gordon and Breach
, New York
, 1990
), Vol. 3
.17.
A.
Erdélyi
, W.
Magnus
, F.
Oberhettinger
, and F. G.
Tricomi
, Higher Transcendental Functions
(McGraw-Hill
, New York
, 1953
), Vol. 1
.18.
M.
Abramowitz
and I. A.
Stegun
, Handbook of Mathematical Functions
(National Bureau of Standards
, New York
, 1972
).19.
20.
L. J.
Slater
, Generalized Hypergeometric Functions
(Cambridge University Press
, Cambridge
, 1966
).21.
The same formula quoted in Ref. 15, p. 92 misses the last parameter of .
22.
A.
Gervois
and H.
Navelet
, J. Math. Phys.
27
, 682
(1986
).23.
The denominator of in Eq. (2.4b) of Ref. 22 should read .
24.
MATHEMATICA 4.0, a product of Wolfram Research.
25.
A. K.
Rajantie
, Nucl. Phys. B
480
, 729
(1996
).27.
C.
Bervillier
and M.
Shpot
, Phys. Rev. B
46
, 955
(1992
).28.
H.
Exton
, J. Phys. A
28
, 631
(1995
).29.
A. A.
Inayat-Hussain
, J. Phys. A
20
, 4109
(1987
).30.
A. A.
Inayat-Hussain
, J. Phys. A
20
, 4119
(1987
).31.
R. G.
Buschman
and H. M.
Srivastava
, J. Phys. A
23
, 4707
(1990
).32.
H.
Srivastava
, M. L.
Glasser
, and V. S.
Adamchik
, Z. Anal. ihre Anwend.
19
, 831
(2000
).33.
34.
V. F.
Tarasov
, J. Math. Phys.
44
, 1449
(2003
).35.
S. B.
Opps
, N.
Saad
, and H. M.
Srivastava
, J. Math. Anal. Appl.
302
, 180
(2005
).© 2007 American Institute of Physics.
2007
American Institute of Physics
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