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A note on multi-loop vertices in string theory
Volume 214, number 4
PHYSICS LETTERS B
1 December 1988
A NOTE ON MULTI-LOOP VERTICES IN STRING THEORY J.L. P E T E R S E N J, J.R. S I D E N I U S 2 The Niels Bohr Institute, Universityof Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen O, Denmark
and A.K. TOLLSTI~N 3 Nordita,Blegdamsvej 17, DK-2100 Copenhagen 0, Denmark
Received 6 July 1988
We carry out a comparison between vertices proposed by various authors by discussing the multi-loop vertices for the bosonic string and for the NS-string in terms of oscillator modes.
1. Introduction In the past few years there has been a great deal o f interest in finding the general g-loop, N-string amplitude, i.e. the multi-loop vertex (see the reference list and references quoted therein for references to recent work). For the matter part o f the bosonic string this was found in refs. [ 1,2 ] and the complete result including ghosts and the measure for moduli arising from ghost zero modes was given (to arbitrary loop order and for general external states) for the closed bosonic string as eq. ( 5.1 ) o f ref. [ 3 ] and for the NS-string as eqs. ( 12 ), ( 13 ) o f ref. [ 4 ], repeated in eqs. ( 1 ), (5) below. The N-string g-loop vertex with the complete measure over moduli was independently obtained for the bosonic string in ref. [ 5 ] in explicit oscillator representations. The agreement o f these various expressions was established for v a c u u m diagrams but left as plausible for arbitrary external states. The expressions of ref. [ 5 ] have been supersymmetrized in ref. [ 6 ] and it was proposed that the formula obtained should be valid for the open NS-string (corresponding to 2 g o f the spin structures), and the connection to super-prime forms [ 7 ] was emphasized. This supersymmetrization turned out to agree with the result derived in ref. [ 4 ] as far as the vacu u m amplitude was concerned, whereas the agreement for arbitrary external string states was not completely clear. In this note we point out how to complete the p r o o f (for external string states corresponding to physical ghostvacua) by converting eq. (5) below into its oscillator counterpart. As a result the proposal o f ref. [ 6 ] is justified. We emphasize that eq. (5) as it stands is completely well defined and indeed - we believe - in itself very useful for general considerations of symmetry, etc. Throughout we shall deal explicitly only with the NS-string, however, the restriction to the bosonic string may be obtained by a completely trivial transcription. Address after 1 September 1988: CERN, CH-1211 Geneva 23, Switzerland. 2 Address after 1 September 1988: Department of Physics, Boston University, Boston, MA 02215, USA. 3 Address after 1 September 1988: Institute of Theoretical Physics, Chalmers Institute of Technology,S-412 96 Grteborg, Sweden. 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
533
Volume 214, number 4
PHYSICS LETTERS B
1 December 1988
2. The Neveu-Schwarz string
We refer to refs. [3,8,4,9 ] for notation and conventions. In ref. [ 4 ] we gave the general g-loop amplitude of a closed string for which the right-moving (analytic) sector is Neveu-Schwarz. It is given by the product of eqs. (12) and ( 13 ) of ref. [ 4 ]. The left-moving (anti-analytic) sector may be either fermionic or purely bosonic - it is not necessary for us to specify. Here we limit ourselves to considering external string states for which the ghost part is given by [q= 1; q' = - 1 ). Clearly this includes all physical states. For this case we give the amplitude as a product of a measure factor and a vertex. The measure factor may be given in terms of the multipliers, K~, and fixed points, Z,, and Zp,, of the appropriate super Schottky group, if, as o
¢~,) ,,=2
1- K ~
~v) ,=1
~
j
( 1)
(det 27~)-D/edltgdltN,
with a corresponding factor for the anti-analytic sewing non-zero modes. Here K~ is a complicated function of Ki, Z~,, Za, the phase of which depends on the 2g possible phase choices (spin structures) for K]/2. We may use superprojective invariance to fix Z~.,, Za, and Z~, at (0, 0), (oo, 0), and ( 1, 0,~, =O,~ ). Then d/.tg = f i dKi ( 1 - K ] / Z ~ - 2 g
i= I
"~
- -1
g-'
dZ,~,dZe,dZ,~dO,~,,
d/.tN=
fi
dZk
,
(2)
where we are using the notation
V:Z= (z, O)v, V(Z) = (VB(Z),
VF(Z) ) ,
(3)
for a superprojective map, V. The superspace difference is given by
Z - Z ' =z-z'-00' .
(4)
The Vk are certain superprojective transformations defined by the 3-string vertex and by the sewing order. They satisfy Vk(0) = Zk where Zk is the kth super Koba-Nielsen parameter. The vertex is given by ref. [4], 1 (27~)D~D(EPi)exp[-~i,j ( ; (O)i~)xaut'-[-DXaUt(£ti)(2~')ql f (o.)j[)xaut.-{-DXaUt(f)j))l 1 ×exp(5~DX:"tOX:Ut)exp( -1N
DXkf)Xk)
where
D=Oo+O0:
(6)
(satisfying D2= 0_-) is the superspace derivative in local coordinates and xaut(z)=
N
~xe(z)' 7e";¢
N
Xe(Z)= E x(k)(Z)= E VkXk(Z), k= 1 k= I
E
Xk(Z)=--ipklogz+i ~ a~k) Z-"+iO ~ ~J(k)z-r--l/2, n=l
/'/
(7)
r=l/2
with appropriate expansions for the left-moving parts. The last factor in eq. (5) is an external wave function (re)normalization. It removes all infinities arising from self interactions leaving only the anomalous piece [4,9,3,10] 534
Volume 214, number 4
PHYSICS LETTERSB
f d2z [)X ~*)(Z) DX (k) (Z) - f
1 December1988
d22 [)Xj,.(Z) D X , ( Z ) =2pZlog DVk,v(0) .
(8)
The superholomorphic half forms, to~, may be given as [ 11 ]
,,, ( o - ~¥( o,~,)
O- ~,~(Op,)'~dZ ,
(9)
where Y~~:~,, omits elements with a rightmost generator factor Pf, n # 0. Although ref. [ 4 ] dealt explicitly only with closed strings we extract the multi-loop vertex for the somewhat simpler case of the open string by restricting ourselves to the holomorphic part and by replacing T o f ref. [ 4 ] by Im T. It is now straightforward to introduce eq. (7) into (the analytic part of) eq. (5) and thus obtain the multi-loop vertex for the open NS-string in terms of oscillators. The result is given by (2,)08 °
(
~ a~k) )
)/z, ~-[ exp (oo ~--o p(k)A~)D2zmIogDVkF(Z') [m], ' =0
k=l
k=l
~ X e x p (12 y~E'ff n,m=O~
l= : N a ( [L/'I k) k,ElT-'~'~ 1'~2n'-[m]! '~A(t) z , z ,~2mIog[V~,(Z)-~'V~(Z')] '"
t= t ' " j t: "N, :A ~D k ) 2A"(mO °EO ElVZ
1 m= °° Xexp ( ",
2m
z=z'=o )
g2~(Vk(Z))(ZztXmT)fflg2j(V~(Z'))
)
Z=Z '=0
,
(10)
where Y 'r~~,;means that the identity element is absent from the sum whenever k = l, [ n ] means integer part of n and the sum over n, m extends over all positive half odd integers and over all non-negative integers and we use the notation [ 6 ] A ~x) = a~k) n integer, A (k) = q/~k) n half odd integer
( 11 )
and Pk= ao~k) for notational convenience. The first abelian super integrals, t2~, satisfy D£2~= co~and may be given by
U)logZ-y(Z'~) ~,~,,, Z-y(Z~,) "
.Qi(Z)= ~
(12)
The convergence of the exponent of eq. (10) rests on momentum conservation Y~k=lNa~k) = 0. The calculations involved in obtaining eq. (10) from eq. ( 5 ) are straightforward; the only trick is the use of the super Cauchy theorem [ 12 ]. We give an outline of the calculation: first we have that ~ x ~~ ( z ) = v ~ x k ( z )
=
=x~(v~-'(z)
) +...
oo
a (nk)
-ipklog[Z- vk(0)] +i L --n-- [ Vk-:l(z)]-"+i[ V~,v'(Z)] L ~'[ V;,~(Z)]-~-'/2. n=
r= 1/2
(13)
Let us now look at the super integral (for k ~ l)
f dEZ[)X(k)(Z)
DX~t)(Z)=- ~
dZXtk)(Z)
DX~O(Z).
(14)
ZI
Using the super Cauchy formula, partial integrations and variable shifts we get (at first only for n, m ~ 0) ~ Manipulating"the false singularity"of the log (refs. [3,8,4,9] ). 535
Volume 214, number 4
PHYSICS LETTERS B
1 December 1988
dZ' ~0 ' - 0 DX(O(Z,) . , ' X(k)(z)= + ~dZ,,log(Z,,_Z)DX(k)(z,,),
DXU)(Z)=-
z!
Zk
- ~ dZ X(k) ( Z) DX(° ( Z) = - ~ dY ' DXk( Y' ) ~ dY" DXz( Y" )Iog[ Vk( Y' ) - Vt(Y") ] ZI
0
0
A(,k) D 2" A(o "''
=
. . . . . o [ - - ~ .~
n2"""rVk(Z) [m]!--z, ,,,~L
~(Z') -
]
(15)
. z=~,
=o
The derivation has to be changed a little for the momentum modes. The reason is that these do not individually transform as scalar fields [ 10,3 ]. Instead V[ - i p , l o g ( Z - Z , ) ] - - i p , log[ Z - V(Z,) ] = -ip,log [ V - ' ( Z ) -Zi] + ip, log[ D VyI(Z)DV¢'(V(Z~) ) ]. (16) Thus one finds that the log in eq. ( 15 ) has to be replaced by log{ [ V~(Z)- Vz(Z')]/[DVk.F(Z)DVt.F(Z')]},
(17)
where the extra terms anyway drop out except for 7=1 and n = 0 or m = 0 . Now eq. (10) is immediate (using again momentum conservation). It is trivial to rewrite eq. (10) using the super prime form, E(Z, Z') (cf. refs. [6,7] ), which introduces convergence factors (in a non-unique way) permitting some interchanges of summations 1 ,_,. { z - ~ , ( z ' ) log E(Z, Z ' ) = l o g ( Z - Z ' ) + -~ r~ log~ - ~ _ - ~
z'-~,(z)
Z ' - 7 ( Z ' ),]
(18)
as
( 27t)Dc~(~ a ~/") ) k~__l eXP(m~=OP ~ k
xox.( 1 ..... o~', [n]!
~
~, ~ [--~.v~'z ~ xexp ~1 n,m=O k.l=1
) D2~'nlogD Vk,F(Z' ) ) z, =o
Dz, log[ Vk(Z)- Vl(Z' )1 z-z'-o/_ _
D~';N(Vk(Z), Vt(Z')) z=z'=o '
(19)
where
N(Z, Z' ) = log E(Z, Z' ) - l o g ( Z - Z' ) + ½t2~(Vk(Z) ) (2n Im T) ff J£2j( ~ ( Z ' ) ) .
(20)
This completes the comparison between our calculation [ 4 ] and the proposal in ref. [ 6 ]. Our eq. ( 19 ) constitutes a proof of the latter. It differs by a slight simplification as far as the tree diagram part ref. [ 13 ] (7=1) is concerned ~2 We are grateful to P. Di Vecchia and K. Hornfeck for numerous consultations.
~2 We thank K. Hornfeck for helping us in verifying the agreement of the two tree diagram pieces and for reminding us of some subtleties concerning the m o m e n t u m modes, and also for discussions concerning some different relative signs in the preprint version of ref. [6 ].
536
Volume 214, number 4
PHYSICS LETTERS B
1 December 1988
References [ 1 ] v. Alessandrini, Nuovo Cimento 2A ( 1971 ) 321; V. Alessandrini and D. Amati, Nuovo Cimento 4A ( 1971 ) 793. [2]C. Montonen, Nuovo Cimento 19A (1974) 69. [3]J.L. Petersen and J. Sidenius, Nucl. Phys. B 301 (1988) 247. [4] J.L. Petersen, J.R. Sidenius and A. Tollst6n, NBI preprint NBI-HE-88-30. [5] P. Di Vecchia, M. Frau, A. Lerda and S. Sciuto, Phys. Lett. B 199 (1987) 49; P. Di Vecchia, K. Hornfeck, M. Frau, A. Lerda and S. Sciuto, Phys. Lett. B 206 (1988) 643; G. Cristofano, R. Musto, F. Nicodemi and R. Pettorino, Naples preprint ( 1988 ). [6] P. Di Vecchia, M. Frau, K. Hornfeck, A. Lerda and S. Sciuto, preprint NORDITA-88/19P. [7] L. Alvarez-Gaum6 and C. Gomez, Phys. Lett. B 190 (1987) 55; E. Verlinde and H. Verlinde, Phys. Lett. B 192 (1987) 95; S. Mandelstam, in: Unified string theories, eds. M.B. Green and D.J. Gross (World Scientific, Singapore, 1986 ). [ 8 ] J.R. Sidenius, in: Perspectives in string theory, eds. P. Di Vecchia and J.L. Petersen (World Scientific, Singapore, 1988 ). [9] J.L. Petersen, J.R. Sidenius and A. Tollst6n, NBI preprint NBI-HE-88-29. [ 10 ] P. Di Vecchia, R. Nakayama, J.L. Petersen and J.R. Sidenius, Nucl. Phys. B 287 ( 1987 ) 621. [ 11 ] C. Lovelace, Phys. Lett. B 32 (1970) 703. [ 12] P. Di Vecchia, V.G. Knizhnik, J.L. Petersen and P. Rossi, Nucl. Phys. B 253 (1985) 701; D. Friedan, in: Unified string theories, eds. M.B. Green and D.J. Gross (World Scientific, Singapore, 1985). [ 13 ] A. Clarizia and F. Pezzella, Nucl. Phys. B 298 (1986) 636; A. Neveu and P. West, Phys. Lett. B 200 (1988) 275.
537
PHYSICS LETTERS B
1 December 1988
A NOTE ON MULTI-LOOP VERTICES IN STRING THEORY J.L. P E T E R S E N J, J.R. S I D E N I U S 2 The Niels Bohr Institute, Universityof Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen O, Denmark
and A.K. TOLLSTI~N 3 Nordita,Blegdamsvej 17, DK-2100 Copenhagen 0, Denmark
Received 6 July 1988
We carry out a comparison between vertices proposed by various authors by discussing the multi-loop vertices for the bosonic string and for the NS-string in terms of oscillator modes.
1. Introduction In the past few years there has been a great deal o f interest in finding the general g-loop, N-string amplitude, i.e. the multi-loop vertex (see the reference list and references quoted therein for references to recent work). For the matter part o f the bosonic string this was found in refs. [ 1,2 ] and the complete result including ghosts and the measure for moduli arising from ghost zero modes was given (to arbitrary loop order and for general external states) for the closed bosonic string as eq. ( 5.1 ) o f ref. [ 3 ] and for the NS-string as eqs. ( 12 ), ( 13 ) o f ref. [ 4 ], repeated in eqs. ( 1 ), (5) below. The N-string g-loop vertex with the complete measure over moduli was independently obtained for the bosonic string in ref. [ 5 ] in explicit oscillator representations. The agreement o f these various expressions was established for v a c u u m diagrams but left as plausible for arbitrary external states. The expressions of ref. [ 5 ] have been supersymmetrized in ref. [ 6 ] and it was proposed that the formula obtained should be valid for the open NS-string (corresponding to 2 g o f the spin structures), and the connection to super-prime forms [ 7 ] was emphasized. This supersymmetrization turned out to agree with the result derived in ref. [ 4 ] as far as the vacu u m amplitude was concerned, whereas the agreement for arbitrary external string states was not completely clear. In this note we point out how to complete the p r o o f (for external string states corresponding to physical ghostvacua) by converting eq. (5) below into its oscillator counterpart. As a result the proposal o f ref. [ 6 ] is justified. We emphasize that eq. (5) as it stands is completely well defined and indeed - we believe - in itself very useful for general considerations of symmetry, etc. Throughout we shall deal explicitly only with the NS-string, however, the restriction to the bosonic string may be obtained by a completely trivial transcription. Address after 1 September 1988: CERN, CH-1211 Geneva 23, Switzerland. 2 Address after 1 September 1988: Department of Physics, Boston University, Boston, MA 02215, USA. 3 Address after 1 September 1988: Institute of Theoretical Physics, Chalmers Institute of Technology,S-412 96 Grteborg, Sweden. 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
533
Volume 214, number 4
PHYSICS LETTERS B
1 December 1988
2. The Neveu-Schwarz string
We refer to refs. [3,8,4,9 ] for notation and conventions. In ref. [ 4 ] we gave the general g-loop amplitude of a closed string for which the right-moving (analytic) sector is Neveu-Schwarz. It is given by the product of eqs. (12) and ( 13 ) of ref. [ 4 ]. The left-moving (anti-analytic) sector may be either fermionic or purely bosonic - it is not necessary for us to specify. Here we limit ourselves to considering external string states for which the ghost part is given by [q= 1; q' = - 1 ). Clearly this includes all physical states. For this case we give the amplitude as a product of a measure factor and a vertex. The measure factor may be given in terms of the multipliers, K~, and fixed points, Z,, and Zp,, of the appropriate super Schottky group, if, as o
¢~,) ,,=2
1- K ~
~v) ,=1
~
j
( 1)
(det 27~)-D/edltgdltN,
with a corresponding factor for the anti-analytic sewing non-zero modes. Here K~ is a complicated function of Ki, Z~,, Za, the phase of which depends on the 2g possible phase choices (spin structures) for K]/2. We may use superprojective invariance to fix Z~.,, Za, and Z~, at (0, 0), (oo, 0), and ( 1, 0,~, =O,~ ). Then d/.tg = f i dKi ( 1 - K ] / Z ~ - 2 g
i= I
"~
- -1
g-'
dZ,~,dZe,dZ,~dO,~,,
d/.tN=
fi
dZk
,
(2)
where we are using the notation
V:Z= (z, O)v, V(Z) = (VB(Z),
VF(Z) ) ,
(3)
for a superprojective map, V. The superspace difference is given by
Z - Z ' =z-z'-00' .
(4)
The Vk are certain superprojective transformations defined by the 3-string vertex and by the sewing order. They satisfy Vk(0) = Zk where Zk is the kth super Koba-Nielsen parameter. The vertex is given by ref. [4], 1 (27~)D~D(EPi)exp[-~i,j ( ; (O)i~)xaut'-[-DXaUt(£ti)(2~')ql f (o.)j[)xaut.-{-DXaUt(f)j))l 1 ×exp(5~DX:"tOX:Ut)exp( -1N
DXkf)Xk)
where
D=Oo+O0:
(6)
(satisfying D2= 0_-) is the superspace derivative in local coordinates and xaut(z)=
N
~xe(z)' 7e";¢
N
Xe(Z)= E x(k)(Z)= E VkXk(Z), k= 1 k= I
E
Xk(Z)=--ipklogz+i ~ a~k) Z-"+iO ~ ~J(k)z-r--l/2, n=l
/'/
(7)
r=l/2
with appropriate expansions for the left-moving parts. The last factor in eq. (5) is an external wave function (re)normalization. It removes all infinities arising from self interactions leaving only the anomalous piece [4,9,3,10] 534
Volume 214, number 4
PHYSICS LETTERSB
f d2z [)X ~*)(Z) DX (k) (Z) - f
1 December1988
d22 [)Xj,.(Z) D X , ( Z ) =2pZlog DVk,v(0) .
(8)
The superholomorphic half forms, to~, may be given as [ 11 ]
,,, ( o - ~¥( o,~,)
O- ~,~(Op,)'~dZ ,
(9)
where Y~~:~,, omits elements with a rightmost generator factor Pf, n # 0. Although ref. [ 4 ] dealt explicitly only with closed strings we extract the multi-loop vertex for the somewhat simpler case of the open string by restricting ourselves to the holomorphic part and by replacing T o f ref. [ 4 ] by Im T. It is now straightforward to introduce eq. (7) into (the analytic part of) eq. (5) and thus obtain the multi-loop vertex for the open NS-string in terms of oscillators. The result is given by (2,)08 °
(
~ a~k) )
)/z, ~-[ exp (oo ~--o p(k)A~)D2zmIogDVkF(Z') [m], ' =0
k=l
k=l
~ X e x p (12 y~E'ff n,m=O~
l= : N a ( [L/'I k) k,ElT-'~'~ 1'~2n'-[m]! '~A(t) z , z ,~2mIog[V~,(Z)-~'V~(Z')] '"
t= t ' " j t: "N, :A ~D k ) 2A"(mO °EO ElVZ
1 m= °° Xexp ( ",
2m
z=z'=o )
g2~(Vk(Z))(ZztXmT)fflg2j(V~(Z'))
)
Z=Z '=0
,
(10)
where Y 'r~~,;means that the identity element is absent from the sum whenever k = l, [ n ] means integer part of n and the sum over n, m extends over all positive half odd integers and over all non-negative integers and we use the notation [ 6 ] A ~x) = a~k) n integer, A (k) = q/~k) n half odd integer
( 11 )
and Pk= ao~k) for notational convenience. The first abelian super integrals, t2~, satisfy D£2~= co~and may be given by
U)logZ-y(Z'~) ~,~,,, Z-y(Z~,) "
.Qi(Z)= ~
(12)
The convergence of the exponent of eq. (10) rests on momentum conservation Y~k=lNa~k) = 0. The calculations involved in obtaining eq. (10) from eq. ( 5 ) are straightforward; the only trick is the use of the super Cauchy theorem [ 12 ]. We give an outline of the calculation: first we have that ~ x ~~ ( z ) = v ~ x k ( z )
=
=x~(v~-'(z)
) +...
oo
a (nk)
-ipklog[Z- vk(0)] +i L --n-- [ Vk-:l(z)]-"+i[ V~,v'(Z)] L ~'[ V;,~(Z)]-~-'/2. n=
r= 1/2
(13)
Let us now look at the super integral (for k ~ l)
f dEZ[)X(k)(Z)
DX~t)(Z)=- ~
dZXtk)(Z)
DX~O(Z).
(14)
ZI
Using the super Cauchy formula, partial integrations and variable shifts we get (at first only for n, m ~ 0) ~ Manipulating"the false singularity"of the log (refs. [3,8,4,9] ). 535
Volume 214, number 4
PHYSICS LETTERS B
1 December 1988
dZ' ~0 ' - 0 DX(O(Z,) . , ' X(k)(z)= + ~dZ,,log(Z,,_Z)DX(k)(z,,),
DXU)(Z)=-
z!
Zk
- ~ dZ X(k) ( Z) DX(° ( Z) = - ~ dY ' DXk( Y' ) ~ dY" DXz( Y" )Iog[ Vk( Y' ) - Vt(Y") ] ZI
0
0
A(,k) D 2" A(o "''
=
. . . . . o [ - - ~ .~
n2"""rVk(Z) [m]!--z, ,,,~L
~(Z') -
]
(15)
. z=~,
=o
The derivation has to be changed a little for the momentum modes. The reason is that these do not individually transform as scalar fields [ 10,3 ]. Instead V[ - i p , l o g ( Z - Z , ) ] - - i p , log[ Z - V(Z,) ] = -ip,log [ V - ' ( Z ) -Zi] + ip, log[ D VyI(Z)DV¢'(V(Z~) ) ]. (16) Thus one finds that the log in eq. ( 15 ) has to be replaced by log{ [ V~(Z)- Vz(Z')]/[DVk.F(Z)DVt.F(Z')]},
(17)
where the extra terms anyway drop out except for 7=1 and n = 0 or m = 0 . Now eq. (10) is immediate (using again momentum conservation). It is trivial to rewrite eq. (10) using the super prime form, E(Z, Z') (cf. refs. [6,7] ), which introduces convergence factors (in a non-unique way) permitting some interchanges of summations 1 ,_,. { z - ~ , ( z ' ) log E(Z, Z ' ) = l o g ( Z - Z ' ) + -~ r~ log~ - ~ _ - ~
z'-~,(z)
Z ' - 7 ( Z ' ),]
(18)
as
( 27t)Dc~(~ a ~/") ) k~__l eXP(m~=OP ~ k
xox.( 1 ..... o~', [n]!
~
~, ~ [--~.v~'z ~ xexp ~1 n,m=O k.l=1
) D2~'nlogD Vk,F(Z' ) ) z, =o
Dz, log[ Vk(Z)- Vl(Z' )1 z-z'-o/_ _
D~';N(Vk(Z), Vt(Z')) z=z'=o '
(19)
where
N(Z, Z' ) = log E(Z, Z' ) - l o g ( Z - Z' ) + ½t2~(Vk(Z) ) (2n Im T) ff J£2j( ~ ( Z ' ) ) .
(20)
This completes the comparison between our calculation [ 4 ] and the proposal in ref. [ 6 ]. Our eq. ( 19 ) constitutes a proof of the latter. It differs by a slight simplification as far as the tree diagram part ref. [ 13 ] (7=1) is concerned ~2 We are grateful to P. Di Vecchia and K. Hornfeck for numerous consultations.
~2 We thank K. Hornfeck for helping us in verifying the agreement of the two tree diagram pieces and for reminding us of some subtleties concerning the m o m e n t u m modes, and also for discussions concerning some different relative signs in the preprint version of ref. [6 ].
536
Volume 214, number 4
PHYSICS LETTERS B
1 December 1988
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