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Higher loop effects in M(atrix) orbifolds
Nuclear Physics B (Proc.
Higher Loop Effects In M(atrix)
Suppl.)
PROCEEDINGS SUPPLEMENTS
68 (1998) 268-273
Orbifolds
Ori J. Ganor, Rajesh Gopakumar and Sanjaye Ramgoolam a aDepartment of Physics, Jadwin Hall Princeton University, Princeton, NJ 08544, USA We calculate the tweloop correction to the metric in the quantum mechanics which is relevant for the conjectured M(atrix) description of scattering a graviton off the fixed point of R’/&. Although it has the right dependence on distance, the factors of N do not match.
1.
INTRODUCTION
The M(atrix) conjecture [l] has passed many tests for flat R”tl space and toroidal compactifications. The leading low-velocity and long distance behavior of graviton scattering in lO+lD supergravity was reproduced in [l], and higher loop corrections have been studied in [2,3]. In these notes we are going to study higherloop corrections to the scattering of gravitons off orbifold fixed points. Our motivation was: ??
??
to study quantum effects in M(atrix)-theory beyond the leading low-energy supergravity. to study effects near orbifolds and, perhaps, to collect some clues for curved backgrounds in M(atrix) theory.
the loop-expansion 0 to understand in M(atrix) theory and its relation to the low-energy expansion of lO+l D supergravity. Orbifolds in M-theory exhibit quantum effects which are beyond classical supergravity. The orbifold R5/Zi (where 24 means orientation reversal) has an effective -i of a 5-brane charge [4,5] which can be detected by a low-energy scattering experiment. Similarly, the orbifold R8/& has an effective -& of a 2-brane charge, which follows from the ~CSAIB(R) interaction [6,7]. Another low-energy effect near orbifolds is the appearance of localized variables (in the IR limit). For example, for R4/& there is an SU(2) gauge multiplet 0920-5632/98/$19.00 PI1 SO920-5632(98)00159-S
0
1998
Elsevier
Science
B.V.
All
rights
reserved
(which is free in the IR), for R6/23 there are interacting degrees of freedom and for Rg/Z4 there is a chiral CFT [8]. We wish to isolate a non-zero 2-100~ effect in M(atrix)-theory and compare it with something that we know. We will show that scattering of a graviton off the fixed points of Rs/Z2 has a nonzero 2-100~ contribution which should capture the leading order of a qualitative effect - the (-&) membrane charge. We will calculate the M(atrix) 2-100~ contribution and show that it has the correct dependence on the velocity v and distance b but the dependence on N is problematic. These notes are based on the paper [9]. 2. ORBIFOLDS
IN M(ATRIX)-THEORY
Orbifolding in M(atrix)-theory [l,lO-141 is performed by picking a large gauge transformation U E U(o0) with ]U - I] = co and U2 = *I and leaving only the degrees of freedom which satisfy u-‘xu
= 7r(X),
(1)
where r is the space-time ZZ action. Sometimes one is required to add extra degrees of freedom [11,12,15]. Let us start with the simpler example of R5/Zh and show how scattering of a graviton off the fixed point reproduces the -3 5-brane charge (this was also discussed in [ll]). The M(atrix)-model for R5/Zh is O+lD quantum mechanics with 8 supersymmetries. It contains two multiplets [ll]. There is a vector multiplet: (Ao, X,, %),
i=1,2
(2)
0.1
whose N(2N SO(5) index
Ganor et al. /Nuclear
Physics B (Proc. Suppl.) 68 (1998) 268-273
fields are in the adjoint of Sp(N) (i.e. + 1) fields). p = 1. . .5 is a vector index. of (rotations of R5) and a = 1. . .4 is a spinor of SO(5). The other multiplet contains: a=6...9
(@)a, @%Y),
(3)
with fields in the anti-symmetric representation of Sp(N) (i.e. N(2N - 1) fields). Scattering of a graviton off the fixed point is calculated according to the methods developed in [16] for scattering DO-branes. The result is extracted from the lloop determinant (y = iv): det -‘(-dT2
+ y2r2 + b2 - 27)
.det4
ORBIFOLD
(5)
yr - ib a, >
& 77-+ib
Rs/Z2
R8/Z2 is obtained [lO,ll] by starting U(2N) SYM, picking the U(2N) matrix:
with a
0 -1NxN
>
and leaving only those modes which satisfy
;irJ = u-lx& x1 = u-lzl u, XI = -u-‘ jT;dJ, 5
=
b = (27~)~N$,
+ y2~2 + b2)
and gives the phase shift - & . 3. THE
We will denote the two U(N) gauge fields by Ao, Ah and the two bosonic fields corresponding to the two U(N)-s by X1,X;. The quotienting prescription leaves 8 fermions in the 8, of SO(8) and in the adjoint of U(N) x U(N) and additional fermions $J’ in the 8, of SO(8) and in the (N, &I) of U(N) x U(N). Let us explain why we expect a 2-100~ contribution. This is because the expected supergravity result for the phase shift of the scattering of a graviton off the fixed point is:
while behavior of L-loops diagrams (not taking into account wave-function contributions) is:
. det -I( -aT2 + y2~2 + b2 + 27) .det -‘(--&’
269
There is also the stringy argument that goes as follows. The tree-level supergravity result corresponds in type-IIA to exchange of a closed string which can also be interpretted as a loop of DD open strings [17]. The anomalous membrane charge of the R8/Z2 fixed point can be traced back to a tadpole in the l-loop closed string diagram [6]. The tadpole diagram, i.e. a torus with one D-boundary, can be deformed into a 2-100~ diagram of open DD strings. Before we proceed with the 2-100~ calculation itself, let us discuss what is the small parameter of the loop expansion. The action is given by [l]:
I = 2...9,
IQ.. .pu-‘i$u.
We obtain a U(N) x U(N) gauge group, one bosonic field Xi in the adjoint, and 8 bosonic fields X2 . . . Xs in the (N, m). In terms of these, the original U(2N) fields are
defining the impact parameter we find the L-loop contribution
z and velocity i3 in the form
cSL= R3(L-1)fL(&Z).
(6)
One can get rid of the R dependence by resealing
X =
R-l@,
7=-
t R’
0 = R-3/29.
In terms of the new variables, the action is
s =
J [;(aXd2+$x1:XJ12 CJt
(7)
270
0.1 Ganoret al./NuclearPhysics
Let b and w be the impact parameter and velocity in the new units. By dimensional analysis we find SL = b-3’L-1)j$).
(8)
Thus, the loop expansion is an expansion in “b” _ a low-energy expansion. 4. A TWO-LOOP
CALCULATION
B (Proc.
Suppl.)
68 (1998)
268-273
The quadratic piece can be written as Lf = i
+ 2vtll,fo2$ + 2bq/&li
(14) where < and 77are defined in terms of the original spinors as follows. We decompose
where X and p are real. and then group them as:
For the actual calculation we use the methods of [16] to calculate the phase-shift of a DO-brane moving past the fixed point of R8/Z2. We set x,
= vt,
X3 = b
(9)
and integrate off-diagonal elements. The Lagrangian is a sum of bosonic, fermionic and ghost terms: L = Lb + Lf + L,.
(10)
where i is an SO(6) spinor index i = 1. . .6 corresponding to the manifest SO(6) rotation group in directions transverse to the graviton trajectory and we have decomposed all SO(8) spinors under this SO(6) c SO(S). In the DKPS method, the Hamiltonian is expanded around the “free” Hamiltonians:
The bosonic piece is a sum of quadratic, cubic and quartic terms: Lb zzz@)
+ @)
+ L(4) b
(11)
.
The quadratic bosonic term is Lc2) b
(dtReX,)2
+ (dtReX2)2
+
;[a,(&
+
(C3,1mX,)2 - 4(b2 + v2t2)(ImX,)2
+
(dJmXs)2
+
(~3,@+)~ - 4(b2 + v2t2 + .)+t
+
(&(a_)2 - 4(b2 + v2t2 - v)@
+
$&(Xl
-
(b2 + v2t2)(X1 - Xi)“.
+
+ (dtReXs)2
;[&(A0
+ A;)12
2
d iu3- - 2vta’ + 2ba2, dt
H
E
-d” +4v2t2 +4b2. dt2
p2 = --$
+ 4v2t2 + 4b2 + 2w2
We also have the zero-mode operators
- 4(b2 + w2t2)(ImXs)2 The propagators are defined as G(z, Y) = (YIH-‘I+
- Xi)12
Sk, Y) = -i(!M%)03, (16)
and
where ImX
E
we find
=
+ X;)12
$9
G% = (~I(Hf4v)%,
Mf4?14, (17)
= @+-@Jz
Go@, Y) =
Ao-A;
= h(@++@_).
SO(T
’
(12) The fermionic piece is a sum of a quadratic term and a Yukawa coupling Lf =LY)+L:y).
(13)
Y) =
-i(ylP;11+3
(18)
It is convenient to represent them as (see also [2]): G&y)
= X
Jdse-4b2s
n--l’2
e-~w(z-y)2
J2 siE4sv
coth2sv-3v(z+~)~tanh2sv
0.J
Ganor et al. /Nuclear
Physics B (Proc. Suppl.) 68 (I 998) 268-273
which gives, together bution, atotal of --l,(,-y)‘coth2sv-~v(z+y)*
2ba’ cash 2sw + 2ibg3 sinh 2sw].
The zero-mode propagators are calculated Go(xc,y)
=
1 --lx -YI, 2
So(x,Y)
=
i(e(x
to be
- Y) - O(Y - x))I.
In the definition of Go we subtracted an infinite constant. In the actual 2-100~ diagrams this constant can be seen to multiply a sum of contributions of bosons, fermions and ghost l-loops which vanishes. The cubic and quartic interaction vertices give rise to the following kinds of diagrams: Two cubic vertices joined by three bosonic propagators. Two cubic vertices joined by one bosonic and two fermionic propagators. Two cubic vertices joined by one bosonic and two ghost propagators.
fermionic+ Abosonic = --.
In fact, all the 2-100~ diagrams of the first three types contain exactly one Go or SO and two massive propagators. The calculation itself is rather tedious. The resulting contributions to the phase shift are: bosonic
---8192b5. 642~
n-
AC41
bosonic
Ac3-3) ghosts
A fermionic
=
+bv
=
-L+-
377rv +
64bv
=
--
=
A~,-,3c
=
7r --4bv
512b5’ 177rv 8192b5 ’ 25rrv
7r +4bv 256b5’
The total bosonic contribution A bosonic
14157W
477T =
is
+ Avionic + A~~,3,), 403nv 4096b5 ’
37Tv 4096b5
(19)
The leading & term cancelled leaving only a correction to the metric (which in the phase shift is proportional to v). The expected supergravity result can be obtained by solving the trajectory of a graviton which passes at a distance b >> 1, from the fixed point. The low-energy fields that the graviton feels are the same as those that would be generated from -& of a membrane located at the fixed point. This is because the total C’s charge . . . measured at infinity is - &. The classical solution for a membrane in lO+lD is given by (see for example [18]): ds2
=
H-2’3(-dt2
C,
=
H-‘dtAdylAdyll,
H
=
I+!$
+ dyf + dyf,)
+ H1’3dxidxi,
where Q2 = 8n~(2n)~(Z,)~ and 122 = -$ is the (effective) number of membranes. Solving for the trajectory gives the expected phase shift
A “figure-of-eight” bosonic diagram with a single quartic vertex.
A(3-3)
with the fermionic contri-
tanh2sv
A=A +
271
6 = 37WN 2b5 where N is the longitudinal graviton in M(atrix) theory.
momentum
of the
5. SOME
ON THE
CAL-
COMMENTS CULATION
The 2-100~ calculation was performed for the N = 1 sector of the M(atrix)-model. The graviton had a single unit of longitudinal momentum. In the l-loop calculation of (11, if one represents two gravitons by two bound states of the SU(Ni) and SU(N2) quantum mechanics and separates them by a large distance one can ignore the details of the bound states if the separation is large. In the 2-100~ calculation we have also ignored the contribution from the non-trivial wave-function of the bound state and we need to justify why this is allowed in this subleading order as well.
212
0.J
Ganor et al. /Nuclear
Physics B (Proc. Suppl,) 68 (1998) 268-273
+O( It’- tj2),
For simplicity, let us analyze the wave-function contribution in the original BFSS model for R1’ql. We separate two gravitons at x and x’ and write: x=
Y X’I + 2’ > ’
XI + R Y+
(
(20)
where 2 and 2’ are SU(N) matrices which are small for the bulk of the bound-state wavefunction. The Hamiltonian is of the form H = He(ri’)+Hc(X’)+v(
x,x’,Y)+U(~,~‘,Y,z,x’).
(21)
He is the XI(N) Hamiltonian with the marginal bound state. V is the interaction with the offdiangonal elements Y which was responsible for the l-loop result and the 2-100~ contribution of the previous section. U is the interaction with the “internal” variables 2 and 2’ which we ignored before. U contains terms of the form j-J=@$+...
(22)
where 1c, are the fermionic superpartners of Y. Such a term will give in two-loops an expression like --
bhwN~01~~71~0)
(plus a bosonic contribution) where IQ,-,) is the ground-state wave-function of SU(N). This will contribute
I
dXdY(S(X7 YMX,
+W(x,
y)&G(x,
+ . . .)G(x-
Y>
where
is the propagator “within” the bound-state. We claim that for the order in which we are interested, the contribution of G(x - y) is determined by the commutation relations. The reason is that S(x, y) in localizes the x - y variable to the vicinity of (x - y) N (l/b) < 1. Thus, we may expand
+ ;ilt! - tj + 0(lt’ - t12)),
C is some unknown constant whose total contribution to the phase shift can be shown to sum to zero. The remaining contribution of It’- tl has the same form as the contribution from the free x and x’ propagators except that the x and x’ propanators contribute with a factor of N while the x” and 2’ contribute with a factor of (N2 - l)N. Altogether we have an N3 contribution. 6. CONCLUSIONS The expected result for the leading low-energy contribution to the phase shift of a graviton scattered off the R8/Zz fixed point is 6 = 3lrvN b5 . The result from the M(atrix)-model used is 6 = 37rN3v
that we have
256b5’ This suggests that some degrees of freedom might be missing from the orbifold O+lD quantum mechanics.
REFERENCES 1.
Y)+ Y)
= 6&C
T. Banks, W. Fischler, S.H. Shenker and L. Susskind, M Theory As A Matrix Model: A Conjecture, hep-th/9610043 2. K. Becker and M. Becker, A Two-loop test of M(atrix)-theory, hepth/9705091 3. K. Becker and M. Becker J. Polchinski and A. Tseytlin, Higher order graviton scattering in M(atti)-theory, Phys. Rev. D56 (1997) 3174. 4. E. Witten, Five-Branes and M-Theory On an Orbifold, Nucl. Phys. B463 (1996) 383. 5. E. Witten, On Flux Quantization In MTheory And The Effective Action, hepth/9609122 6. C. Vafa and E. Witten, A Strong Coupling Test of S-Duality, Nucl. Phys. B431 (1994) 3.
Polx2%l~o)+ ~(QO/[~,%~Qo)It’ - tl
0.1
7. 8. 9.
10. 11.
12.
13. 14.
15.
16.
17.
18.
Canor et al. /Nuclear
Ph,vsics B (Proc. Suppl.) 68 (I 998) 268-273
M. Duff, J. Liu, and R. Minasian, Nucl. Phys. B452 (1995) 261. K. Dasgupta and S. Mukhi, Orbifolds of MTheory, Nucl. Phys. B465 (1996) 399. O.J. Ganor, R. Gopakumar and S. Ramgoolam, Higher loop effects in M(atrix) orbifolds, hepth/9705188, to appear in Nucl. Phys. B M. Douglas and G. Moore, D-branes, Quivers And ALE Instantons, hep-th/9603167. N. Kim and S.-J. Rey, M(atrix) Theory on an Orbifold and Twisted Membrane, hepth/9701139 S. Kachru and E. Silverstein, On gauge bosons in the M(atrix)-model approach to M-theory, Phys. Lett. B396 (1997) 70. T. Banks and L. Motl, Heterotic Strings from Matrices, hep-th/9703218. The U.H. Danielsson and G. Ferretti, Heterotic Life OF The D-Particle, hepth/9610082. T. Banks, N. Seiberg, and E. Silverstein, Zero and One-dimensional Probes with N=8 Supersymmetry, hep-th/9703052. M.R. Douglas, D. Kabat, P. Pouliot and S. Shenker, D-branes and Short Distances in String Theory, Nucl. Phys. B485 (1997) 85. J. Polchinski, Dirichlet-Branes and RamondRamond Charges, Phys. Rev.Lett. 75 (1995) 4724. J.G. Russo, BPS Bound States, Supermembranes and T-Duality in M-Theory, hepth/9703118.
273
Higher Loop Effects In M(atrix)
Suppl.)
PROCEEDINGS SUPPLEMENTS
68 (1998) 268-273
Orbifolds
Ori J. Ganor, Rajesh Gopakumar and Sanjaye Ramgoolam a aDepartment of Physics, Jadwin Hall Princeton University, Princeton, NJ 08544, USA We calculate the tweloop correction to the metric in the quantum mechanics which is relevant for the conjectured M(atrix) description of scattering a graviton off the fixed point of R’/&. Although it has the right dependence on distance, the factors of N do not match.
1.
INTRODUCTION
The M(atrix) conjecture [l] has passed many tests for flat R”tl space and toroidal compactifications. The leading low-velocity and long distance behavior of graviton scattering in lO+lD supergravity was reproduced in [l], and higher loop corrections have been studied in [2,3]. In these notes we are going to study higherloop corrections to the scattering of gravitons off orbifold fixed points. Our motivation was: ??
??
to study quantum effects in M(atrix)-theory beyond the leading low-energy supergravity. to study effects near orbifolds and, perhaps, to collect some clues for curved backgrounds in M(atrix) theory.
the loop-expansion 0 to understand in M(atrix) theory and its relation to the low-energy expansion of lO+l D supergravity. Orbifolds in M-theory exhibit quantum effects which are beyond classical supergravity. The orbifold R5/Zi (where 24 means orientation reversal) has an effective -i of a 5-brane charge [4,5] which can be detected by a low-energy scattering experiment. Similarly, the orbifold R8/& has an effective -& of a 2-brane charge, which follows from the ~CSAIB(R) interaction [6,7]. Another low-energy effect near orbifolds is the appearance of localized variables (in the IR limit). For example, for R4/& there is an SU(2) gauge multiplet 0920-5632/98/$19.00 PI1 SO920-5632(98)00159-S
0
1998
Elsevier
Science
B.V.
All
rights
reserved
(which is free in the IR), for R6/23 there are interacting degrees of freedom and for Rg/Z4 there is a chiral CFT [8]. We wish to isolate a non-zero 2-100~ effect in M(atrix)-theory and compare it with something that we know. We will show that scattering of a graviton off the fixed points of Rs/Z2 has a nonzero 2-100~ contribution which should capture the leading order of a qualitative effect - the (-&) membrane charge. We will calculate the M(atrix) 2-100~ contribution and show that it has the correct dependence on the velocity v and distance b but the dependence on N is problematic. These notes are based on the paper [9]. 2. ORBIFOLDS
IN M(ATRIX)-THEORY
Orbifolding in M(atrix)-theory [l,lO-141 is performed by picking a large gauge transformation U E U(o0) with ]U - I] = co and U2 = *I and leaving only the degrees of freedom which satisfy u-‘xu
= 7r(X),
(1)
where r is the space-time ZZ action. Sometimes one is required to add extra degrees of freedom [11,12,15]. Let us start with the simpler example of R5/Zh and show how scattering of a graviton off the fixed point reproduces the -3 5-brane charge (this was also discussed in [ll]). The M(atrix)-model for R5/Zh is O+lD quantum mechanics with 8 supersymmetries. It contains two multiplets [ll]. There is a vector multiplet: (Ao, X,, %),
i=1,2
(2)
0.1
whose N(2N SO(5) index
Ganor et al. /Nuclear
Physics B (Proc. Suppl.) 68 (1998) 268-273
fields are in the adjoint of Sp(N) (i.e. + 1) fields). p = 1. . .5 is a vector index. of (rotations of R5) and a = 1. . .4 is a spinor of SO(5). The other multiplet contains: a=6...9
(@)a, @%Y),
(3)
with fields in the anti-symmetric representation of Sp(N) (i.e. N(2N - 1) fields). Scattering of a graviton off the fixed point is calculated according to the methods developed in [16] for scattering DO-branes. The result is extracted from the lloop determinant (y = iv): det -‘(-dT2
+ y2r2 + b2 - 27)
.det4
ORBIFOLD
(5)
yr - ib a, >
& 77-+ib
Rs/Z2
R8/Z2 is obtained [lO,ll] by starting U(2N) SYM, picking the U(2N) matrix:
with a
0 -1NxN
>
and leaving only those modes which satisfy
;irJ = u-lx& x1 = u-lzl u, XI = -u-‘ jT;dJ, 5
=
b = (27~)~N$,
+ y2~2 + b2)
and gives the phase shift - & . 3. THE
We will denote the two U(N) gauge fields by Ao, Ah and the two bosonic fields corresponding to the two U(N)-s by X1,X;. The quotienting prescription leaves 8 fermions in the 8, of SO(8) and in the adjoint of U(N) x U(N) and additional fermions $J’ in the 8, of SO(8) and in the (N, &I) of U(N) x U(N). Let us explain why we expect a 2-100~ contribution. This is because the expected supergravity result for the phase shift of the scattering of a graviton off the fixed point is:
while behavior of L-loops diagrams (not taking into account wave-function contributions) is:
. det -I( -aT2 + y2~2 + b2 + 27) .det -‘(--&’
269
There is also the stringy argument that goes as follows. The tree-level supergravity result corresponds in type-IIA to exchange of a closed string which can also be interpretted as a loop of DD open strings [17]. The anomalous membrane charge of the R8/Z2 fixed point can be traced back to a tadpole in the l-loop closed string diagram [6]. The tadpole diagram, i.e. a torus with one D-boundary, can be deformed into a 2-100~ diagram of open DD strings. Before we proceed with the 2-100~ calculation itself, let us discuss what is the small parameter of the loop expansion. The action is given by [l]:
I = 2...9,
IQ.. .pu-‘i$u.
We obtain a U(N) x U(N) gauge group, one bosonic field Xi in the adjoint, and 8 bosonic fields X2 . . . Xs in the (N, m). In terms of these, the original U(2N) fields are
defining the impact parameter we find the L-loop contribution
z and velocity i3 in the form
cSL= R3(L-1)fL(&Z).
(6)
One can get rid of the R dependence by resealing
X =
R-l@,
7=-
t R’
0 = R-3/29.
In terms of the new variables, the action is
s =
J [;(aXd2+$x1:XJ12 CJt
(7)
270
0.1 Ganoret al./NuclearPhysics
Let b and w be the impact parameter and velocity in the new units. By dimensional analysis we find SL = b-3’L-1)j$).
(8)
Thus, the loop expansion is an expansion in “b” _ a low-energy expansion. 4. A TWO-LOOP
CALCULATION
B (Proc.
Suppl.)
68 (1998)
268-273
The quadratic piece can be written as Lf = i
+ 2vtll,fo2$ + 2bq/&li
(14) where < and 77are defined in terms of the original spinors as follows. We decompose
where X and p are real. and then group them as:
For the actual calculation we use the methods of [16] to calculate the phase-shift of a DO-brane moving past the fixed point of R8/Z2. We set x,
= vt,
X3 = b
(9)
and integrate off-diagonal elements. The Lagrangian is a sum of bosonic, fermionic and ghost terms: L = Lb + Lf + L,.
(10)
where i is an SO(6) spinor index i = 1. . .6 corresponding to the manifest SO(6) rotation group in directions transverse to the graviton trajectory and we have decomposed all SO(8) spinors under this SO(6) c SO(S). In the DKPS method, the Hamiltonian is expanded around the “free” Hamiltonians:
The bosonic piece is a sum of quadratic, cubic and quartic terms: Lb zzz@)
+ @)
+ L(4) b
(11)
.
The quadratic bosonic term is Lc2) b
(dtReX,)2
+ (dtReX2)2
+
;[a,(&
+
(C3,1mX,)2 - 4(b2 + v2t2)(ImX,)2
+
(dJmXs)2
+
(~3,@+)~ - 4(b2 + v2t2 + .)+t
+
(&(a_)2 - 4(b2 + v2t2 - v)@
+
$&(Xl
-
(b2 + v2t2)(X1 - Xi)“.
+
+ (dtReXs)2
;[&(A0
+ A;)12
2
d iu3- - 2vta’ + 2ba2, dt
H
E
-d” +4v2t2 +4b2. dt2
p2 = --$
+ 4v2t2 + 4b2 + 2w2
We also have the zero-mode operators
- 4(b2 + w2t2)(ImXs)2 The propagators are defined as G(z, Y) = (YIH-‘I+
- Xi)12
Sk, Y) = -i(!M%)03, (16)
and
where ImX
E
we find
=
+ X;)12
$9
G% = (~I(Hf4v)%,
Mf4?14, (17)
= @+-@Jz
Go@, Y) =
Ao-A;
= h(@++@_).
SO(T
’
(12) The fermionic piece is a sum of a quadratic term and a Yukawa coupling Lf =LY)+L:y).
(13)
Y) =
-i(ylP;11+3
(18)
It is convenient to represent them as (see also [2]): G&y)
= X
Jdse-4b2s
n--l’2
e-~w(z-y)2
J2 siE4sv
coth2sv-3v(z+~)~tanh2sv
0.J
Ganor et al. /Nuclear
Physics B (Proc. Suppl.) 68 (I 998) 268-273
which gives, together bution, atotal of --l,(,-y)‘coth2sv-~v(z+y)*
2ba’ cash 2sw + 2ibg3 sinh 2sw].
The zero-mode propagators are calculated Go(xc,y)
=
1 --lx -YI, 2
So(x,Y)
=
i(e(x
to be
- Y) - O(Y - x))I.
In the definition of Go we subtracted an infinite constant. In the actual 2-100~ diagrams this constant can be seen to multiply a sum of contributions of bosons, fermions and ghost l-loops which vanishes. The cubic and quartic interaction vertices give rise to the following kinds of diagrams: Two cubic vertices joined by three bosonic propagators. Two cubic vertices joined by one bosonic and two fermionic propagators. Two cubic vertices joined by one bosonic and two ghost propagators.
fermionic+ Abosonic = --.
In fact, all the 2-100~ diagrams of the first three types contain exactly one Go or SO and two massive propagators. The calculation itself is rather tedious. The resulting contributions to the phase shift are: bosonic
---8192b5. 642~
n-
AC41
bosonic
Ac3-3) ghosts
A fermionic
=
+bv
=
-L+-
377rv +
64bv
=
--
=
A~,-,3c
=
7r --4bv
512b5’ 177rv 8192b5 ’ 25rrv
7r +4bv 256b5’
The total bosonic contribution A bosonic
14157W
477T =
is
+ Avionic + A~~,3,), 403nv 4096b5 ’
37Tv 4096b5
(19)
The leading & term cancelled leaving only a correction to the metric (which in the phase shift is proportional to v). The expected supergravity result can be obtained by solving the trajectory of a graviton which passes at a distance b >> 1, from the fixed point. The low-energy fields that the graviton feels are the same as those that would be generated from -& of a membrane located at the fixed point. This is because the total C’s charge . . . measured at infinity is - &. The classical solution for a membrane in lO+lD is given by (see for example [18]): ds2
=
H-2’3(-dt2
C,
=
H-‘dtAdylAdyll,
H
=
I+!$
+ dyf + dyf,)
+ H1’3dxidxi,
where Q2 = 8n~(2n)~(Z,)~ and 122 = -$ is the (effective) number of membranes. Solving for the trajectory gives the expected phase shift
A “figure-of-eight” bosonic diagram with a single quartic vertex.
A(3-3)
with the fermionic contri-
tanh2sv
A=A +
271
6 = 37WN 2b5 where N is the longitudinal graviton in M(atrix) theory.
momentum
of the
5. SOME
ON THE
CAL-
COMMENTS CULATION
The 2-100~ calculation was performed for the N = 1 sector of the M(atrix)-model. The graviton had a single unit of longitudinal momentum. In the l-loop calculation of (11, if one represents two gravitons by two bound states of the SU(Ni) and SU(N2) quantum mechanics and separates them by a large distance one can ignore the details of the bound states if the separation is large. In the 2-100~ calculation we have also ignored the contribution from the non-trivial wave-function of the bound state and we need to justify why this is allowed in this subleading order as well.
212
0.J
Ganor et al. /Nuclear
Physics B (Proc. Suppl,) 68 (1998) 268-273
+O( It’- tj2),
For simplicity, let us analyze the wave-function contribution in the original BFSS model for R1’ql. We separate two gravitons at x and x’ and write: x=
Y X’I + 2’ > ’
XI + R Y+
(
(20)
where 2 and 2’ are SU(N) matrices which are small for the bulk of the bound-state wavefunction. The Hamiltonian is of the form H = He(ri’)+Hc(X’)+v(
x,x’,Y)+U(~,~‘,Y,z,x’).
(21)
He is the XI(N) Hamiltonian with the marginal bound state. V is the interaction with the offdiangonal elements Y which was responsible for the l-loop result and the 2-100~ contribution of the previous section. U is the interaction with the “internal” variables 2 and 2’ which we ignored before. U contains terms of the form j-J=@$+...
(22)
where 1c, are the fermionic superpartners of Y. Such a term will give in two-loops an expression like --
bhwN~01~~71~0)
(plus a bosonic contribution) where IQ,-,) is the ground-state wave-function of SU(N). This will contribute
I
dXdY(S(X7 YMX,
+W(x,
y)&G(x,
+ . . .)G(x-
Y>
where
is the propagator “within” the bound-state. We claim that for the order in which we are interested, the contribution of G(x - y) is determined by the commutation relations. The reason is that S(x, y) in localizes the x - y variable to the vicinity of (x - y) N (l/b) < 1. Thus, we may expand
+ ;ilt! - tj + 0(lt’ - t12)),
C is some unknown constant whose total contribution to the phase shift can be shown to sum to zero. The remaining contribution of It’- tl has the same form as the contribution from the free x and x’ propagators except that the x and x’ propanators contribute with a factor of N while the x” and 2’ contribute with a factor of (N2 - l)N. Altogether we have an N3 contribution. 6. CONCLUSIONS The expected result for the leading low-energy contribution to the phase shift of a graviton scattered off the R8/Zz fixed point is 6 = 3lrvN b5 . The result from the M(atrix)-model used is 6 = 37rN3v
that we have
256b5’ This suggests that some degrees of freedom might be missing from the orbifold O+lD quantum mechanics.
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Y)+ Y)
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