The complete result for the effective potential for two graviton exchange at
two loops in M(atrix) theory can be expressed in terms of a generalized
hypergeometric function.Comment: 4 pages, TeX, no figure

Becker, Katrin

Becker, Melanie

English

arXiv

The complete result for the effective potential for two graviton exchange at two loops in M(atrix) theory can be expressed in terms of a generalized hypergeometric function

Caltech Authors - Main

J
H
E
P
0
9
(
1
9
9
8
)
0
1
9
Received: August 6, 1998, Accepted: September 18, 1998
HYPER VERSION
Complete solution for M(atrix) theory at two
loops
Katrin Becker and Melanie Becker
California Institute of Technology 452-48, Pasadena, CA 91125
E-mail: beckerk@theory.caltech.edu, mbecker@theory.caltech.ed
Abstract: The complete result for the effective potential for two graviton exchange at
two loops in M(atrix) theory can be expressed in terms of a generalized hypergeometric
function.
Keywords: D-branes, M(atrix) Theories, M-Theory.
J
H
E
P
0
9
(
1
9
9
8
)
0
1
9
It has recently become clear that DLCQ supergravity and M(atrix) theory for finite
N [1] are in perfect agreement even at the level of three graviton exchange [2].1
At this point all the calculations done in M(atrix) theory at two loops are based
on a perturbative expansion of some complicated integrals and no closed expression
for the effective action is known. Now more than ever before it will be important to
have a systematic approach to calculations in M(atrix) theory. In this way it might be
possible to go beyond computations in general relativity.
In this note we will make one step in this direction. We will present the complete
result for the effective potential for two graviton exchange in M(atrix) theory at two
loops. Our notation and conventions are those of [6].
The integrals that have to be solved for all diagrams appearing in double graviton
exchange at two loops in M(atrix) theory are (after the appropriate substitution) of
the type
I =
∫ ∞
1
dxdy
(−1 + x)α(−1 + y)β
(−1 + xy)γ
x−ζ+µy−ζ+ν. (1)
Here ζ = b2/2v and the other free parameters are constants whose value depend on the
concrete integral. This expression can be evaluated exactly and its solution is
B (α + 1, β + 1)B (ζ − ν − β − 1 + γ, 2 + α + β − γ)×
× 3F2 (1 + α, 2 + α + β − γ, 1 + α + µ− ν; 2 + α + β, ζ + α− ν + 1) , (2)
where B(x, y) = Γ(x)Γ(y)/Γ(x + y) is the beta function and 3F2 is a generalized hy-
pergeometric series of unit argument. Using the Stirling series
log Γ(z) = (z − 1/2) log z − z + 1/2 log(2π) +
m∑
n=1
B2n
2n(2n− 1)
z2n−1 +O(z−2m−1) , (3)
as z becomes large and the asymptotic expansion
3F2(a, b, c; d, e) = 1 +
abc
de
+ . . .+
(a)n(b)n(c)n
(d)n(e)n
+O(e−n−1) , (4)
as e becomes large (while a, b, c and d are fixed numbers), we can obtain the double ex-
pansion of the effective potential to all orders in the velocity and the impact parameter
as ζ →∞. Here B2n are the Bernoulli numbers whose numerical values and recurrence
relations can be found in Ramanujan [7].
The complete solution for the effective potential coming from the fermions is
F = −
4
√
2π
v3/2
Γ(ζ + 1)
Γ(ζ + 3
2
)
3F2(
1
2
,
3
2
,
3
2
; 3, ζ +
3
2
)−
−
135
√
2π
512v3/2
Γ(ζ + 1)
Γ(ζ + 7
2
)
3F2(
5
2
,
5
2
,
7
2
; 5, ζ +
7
2
) +
32
√
2π
v3/2
(−Γ(1
2
+ ζ)2 + Γ(ζ)Γ(1 + ζ))
Γ(ζ)Γ(1
2
+ ζ)
. (5)
1Previous work indicated that such an agreement might be possible [3, 4, 5].
1
J
H
E
P
0
9
(
1
9
9
8
)
0
1
9
Incidentally, one of the hypergeometric series could be summed up using Dougall’s
theorem. The complete solution coming from bosons and ghosts is given by the odd
terms in v of the formula
B =
π
√
2
3v3/2
Γ(−1
2
+ ζ)
Γ(ζ)
[49
8
3F2(
1
2
,
1
2
,
1
2
; 1, ζ)− 3F2(
3
2
,
3
2
,
3
2
; 3, 1 + ζ)−
−
137
16
(2ζ − 1)
ζ
3F2(
1
2
,
1
2
,
1
2
; 1, 1 + ζ)
]
. (6)
If we expand these formulas using the Stirling series we obtain the effective potential
computed in [6] up to order v8. The leading order corresponds to the relativistic
correction of the v4-term of [8], while the higher order terms correspond to quantum
gravity corrections.
The complete result of the effective potential of M(atrix) theory at one loop was
found by DKPS [9]. It sounds plausible that the complete solution at one and two
loops may teach us how to solve the model for an arbitrary number of loops.
Acknowledgments
This work was supported by the U.S. Department of Energy under grant DE-FG03-92-
ER40701.
References
[1] L. Susskind, Another conjecture about M(atrix) theory, hep-th/9704080.
[2] Y. Okawa and T. Yoneya, Multibody interactions of D particles in supergravity and matrix
theory, preprint UT-Komaba-98-13, hep-th/9806108.
[3] M. Fabbrichesi, G. Ferretti and R. Iengo, JHEP 06 (1998) 002 [hep-th/9806018].
[4] W. Taylor IV and M. Van Raamsdonk, hep-th/9806066.
[5] K. Becker and M. Becker, unpublished.
[6] K. Becker and M. Becker, A two-loop test of M(atrix) theory, Nucl. Phys. B 506 (1997)
48 [hep-th/9705091];
K. Becker, M. Becker, J. Polchinski and A. Tseytlin, Higher order graviton scattering in
M(atrix) theory, Phys. Rev. D 56 (1997) 3174 [hep-th/9706072];
K. Becker and M. Becker, On graviton scattering amplitudes in M theory, Phys. Rev. D
57 (1998) 6464 [hep-th/9712238].
[7] Ramanujan, Some properties of Bernoulli’s numbers, Collected Papers (1927), Cam-
bridge.
2
J
H
E
P
0
9
(
1
9
9
8
)
0
1
9
[8] T. Banks, W. Fischler, S.H. Shenker and L. Susskind, M theory as a matrix model: a
conjecture, Phys. Rev. D 55 (1997) 5112 [hep-th/9610043].
[9] M.R. Douglas, D. Kabat, P. Pouliot and S.H. Shenker, D-branes and short distances in
string theory, Nucl. Phys. B 485 (1997) 85 [hep-th/9608024].
3


Complete solution for M(atrix) theory at two loops

J
H
E
P09(1998)019
Received: August 6, 1998, Accepted: September 18, 1998
HYPER VERSION
Complete solution for M(atrix) theory at two
loops
Katrin Becker and Melanie Becker
California Institute of Technology 452-48, Pasadena, CA 91125
E-mail: beckerk@theory.caltech.edu, mbecker@theory.caltech.ed
Abstract: The complete result for the eective potential for two graviton exchange at
two loops in M(atrix) theory can be expressed in terms of a generalized hypergeometric
function.
Keywords: D-branes, M(atrix) Theories, M-Theory.
J
H
E
P09(1998)019
It has recently become clear that DLCQ supergravity and M(atrix) theory for nite
N [1] are in perfect agreement even at the level of three graviton exchange [2].1
At this point all the calculations done in M(atrix) theory at two loops are based
on a perturbative expansion of some complicated integrals and no closed expression
for the eective action is known. Now more than ever before it will be important to
have a systematic approach to calculations in M(atrix) theory. In this way it might be
possible to go beyond computations in general relativity.
In this note we will make one step in this direction. We will present the complete
result for the eective potential for two graviton exchange in M(atrix) theory at two
loops. Our notation and conventions are those of [6].
The integrals that have to be solved for all diagrams appearing in double graviton
exchange at two loops in M(atrix) theory are (after the appropriate substitution) of
the type
I =
Z 1
1
dxdy
(−1 + x)(−1 + y)
(−1 + xy)γ
x−+y−+: (1)
Here  = b2=2v and the other free parameters are constants whose value depend on the
concrete integral. This expression can be evaluated exactly and its solution is
B ( + 1;  + 1)B ( −  −  − 1 + γ; 2 +  +  − γ)
 3F2 (1 + ; 2 +  +  − γ; 1 +  + − ; 2 +  + ;  + −  + 1) ; (2)
where B(x; y) = Γ(x)Γ(y)=Γ(x + y) is the beta function and 3F2 is a generalized hy-
pergeometric series of unit argument. Using the Stirling series
log Γ(z) = (z − 1=2) log z − z + 1=2 log(2) +
mX
n=1
B2n
2n(2n− 1)
z2n−1 +O(z−2m−1) ; (3)
as z becomes large and the asymptotic expansion
3F2(a; b; c; d; e) = 1 +
abc
de
+ : : :+
(a)n(b)n(c)n
(d)n(e)n
+O(e−n−1) ; (4)
as e becomes large (while a; b; c and d are xed numbers), we can obtain the double ex-
pansion of the eective potential to all orders in the velocity and the impact parameter
as  !1. Here B2n are the Bernoulli numbers whose numerical values and recurrence
relations can be found in Ramanujan [7].
The complete solution for the eective potential coming from the fermions is
F = −
4
p
2
v3=2
Γ( + 1)
Γ( + 3
2
)
3F2(
1
2
;
3
2
;
3
2
; 3;  +
3
2
)−
−
135
p
2
512v3=2
Γ( + 1)
Γ( + 7
2
)
3F2(
5
2
;
5
2
;
7
2
; 5;  +
7
2
) +
32
p
2
v3=2
(−Γ(1
2
+ )2 + Γ()Γ(1 + ))
Γ()Γ(1
2
+ )
: (5)
1Previous work indicated that such an agreement might be possible [3, 4, 5].
1
J
H
E
P09(1998)019
Incidentally, one of the hypergeometric series could be summed up using Dougall’s
theorem. The complete solution coming from bosons and ghosts is given by the odd
terms in v of the formula
B =

p
2
3v3=2
Γ(−1
2
+ )
Γ()
h49
8
3F2(
1
2
;
1
2
;
1
2
; 1; )− 3F2(
3
2
;
3
2
;
3
2
; 3; 1 + )−
−
137
16
(2 − 1)

3F2(
1
2
;
1
2
;
1
2
; 1; 1 + )
i
: (6)
If we expand these formulas using the Stirling series we obtain the eective potential
computed in [6] up to order v8. The leading order corresponds to the relativistic
correction of the v4-term of [8], while the higher order terms correspond to quantum
gravity corrections.
The complete result of the eective potential of M(atrix) theory at one loop was
found by DKPS [9]. It sounds plausible that the complete solution at one and two
loops may teach us how to solve the model for an arbitrary number of loops.
Acknowledgments
This work was supported by the U.S. Department of Energy under grant DE-FG03-92-
ER40701.
References
[1] L. Susskind, Another conjecture about M(atrix) theory, hep-th/9704080.
[2] Y. Okawa and T. Yoneya, Multibody interactions of D particles in supergravity and matrix
theory, preprint UT-Komaba-98-13, hep-th/9806108.
[3] M. Fabbrichesi, G. Ferretti and R. Iengo, JHEP 06 (1998) 002 [hep-th/9806018].
[4] W. Taylor IV and M. Van Raamsdonk, hep-th/9806066.
[5] K. Becker and M. Becker, unpublished.
[6] K. Becker and M. Becker, A two-loop test of M(atrix) theory, Nucl. Phys. B 506 (1997)
48 [hep-th/9705091];
K. Becker, M. Becker, J. Polchinski and A. Tseytlin, Higher order graviton scattering in
M(atrix) theory, Phys. Rev. D 56 (1997) 3174 [hep-th/9706072];
K. Becker and M. Becker, On graviton scattering amplitudes in M theory, Phys. Rev. D
57 (1998) 6464 [hep-th/9712238].
[7] Ramanujan, Some properties of Bernoulli’s numbers, Collected Papers (1927), Cam-
bridge.
2
J
H
E
P09(1998)019
[8] T. Banks, W. Fischler, S.H. Shenker and L. Susskind, M theory as a matrix model: a
conjecture, Phys. Rev. D 55 (1997) 5112 [hep-th/9610043].
[9] M.R. Douglas, D. Kabat, P. Pouliot and S.H. Shenker, D-branes and short distances in
string theory, Nucl. Phys. B 485 (1997) 85 [hep-th/9608024].
3


Complete Solution for M(atrix) Theory at Two Loops

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