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Relations between disc diagrams
Volume 217, number 4
PHYSICS LETTERS B
2 February 1989
RELATIONS BETWEEN DISC DIAGRAMS David LANCASTER and Paul MANSFIELD Department of Theoretical Physics, Universityof Oxford, 1 Keble Road, Oxford OX1 3NP, UK Received 8 October 1988; revised manuscript received 18 November 1988
We show that the amplitude for N open and M closed strings on a disc may be written as a sum of amplitudes with N + 2M open strings, with appropriate orderings and phases. This form may be simplified using Plahte identities which relate open string amplitudes with different orderings. We demonstrate that these also hold for type I superstrings.
1. Relations between disc diagrams
for both open and closed strings). Consider the complex integral
The relationship between open and closed strings is of fundamental concern in string field theory. Until recently the only covariant string field theory which had been shown to successfully reproduce the results of first quantisation was Witten's theory of the open bosonic string [ 1 ]. Closed strings are phenomenologically more interesting, and furthermore appear as a subsector of open string theories. This was responsible for their discovery in the context & t h e dual resonance model and recently this has been made the basis of a closed string field theory [2] which, it is claimed, reproduces the results of first quanfisation for closed string scattering amplitudes. The proposal is based on an analysis of the interacton between open and closed strings. In this letter we show that such a mixed amplitude at tree level can be expressed as a superposition of amplitudes for open strings alone. This generalises to the superstring, and is not only of interest in understanding the results of ref. [ 2 ], but also of practical value in simplifying the computation of such amplitudes, as was the work of ref. [ 3 ], which expressed pure closed string amplitudes in terms of open string ones. We also show that the Plahte identities [4] lead to further simplifications on both of our results, and those of ref. [ 3 ], and give their supersymmetric generalisation. We start by reminding readers of the Plahte identities [4]. The simplest example is the scattering of four tachyons with p 2= 2 (we take a ' = ~-throughout, 416
i dz zP"P2( 1 - z ) p2p3 , with branch cuts along the real axis. The integral converges at large z for Pl"P2+P2"p3
Fig. 1. Contour in upper half-plane yielding the four-particle Plahte identity.
0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
Volume 217, number 4
PHYSICS LETTERS B
0
t,*
exp(i~zpj "P2)
J dz Izl""~-'l 1-zl p2p3 -~x.,
+ ~ d z JzIP'P211-z] p2'p3 0
+ e x p ( - i z t p z ' p 3 ) ~( dz [zlP"P'-[1 -z[I'2P3-- 0. 1
exp (iz:p~ .p2)A(2134)+A(1234) + exp ( - i2zp2 'P3 )A ( 1 3 2 4 ) = 0. The complex conjugate relation also follows from using a similar contour in the lower half-plane. Making use of kinematics we obtain the real relation A(2134) sin ~P2 "P3
the positions of N - 3 vertex operators on the boundary of the world-sheet that we take to be the upper half-plane. For any integration variable we can write a vanishing complex integral which is the sum of amplitudes differing only by the position of the corresponding tachyon, multiplied by phases determined by appropriately negotiating the branch cuts. For example for five particles we obtain the following identity which we denote (!2345): A ( ! 2345 ) + exp (izrp~ "P2)A (2_1345 )
This relation between Euler beta functions can be interpreted in terms of the Veneziano amplitude for different particle orderings [ 5 ] :
A(1234) sin ztpl 'P3
2 February 1989
A(1324) sin 7~pr"P2
This is conveniently pictured using fig. 2. The area of this triangle is quadratic in open string amplitudes and is equal to the closed string scattering amplitude for four tachyons of momenta 2pi. This follows from the work of Kawai, Lewellen and Tye [3] and provides a manifestly symmetric representation of their result which can, if desired, be expressed in terms of a single open string amplitude. The Plahte identities hold for scattering of any number of open string tachyons in bosonic string theory. They may be derived by a simple generalisation of the above argument. The Koba-Nielsen expression for the N-tachyon amplitude is an integral over
+ exp [izrp~ • (P2 +P3) ]A (23145) + exp [izrp~. (P2 +P3 +P4) ]A (234! 5 ) = 0 . This can be represented by a quadrilateral with sides proportional to the amplitudes and angles determined by the phases. We can construct other identities and quadrilaterals but they will not all be independent. Unlike the case of four particles, it is impossible to express every five-particle amplitude in terms of a single ordering. Amplitudes involving excited states of the bosonic string obey the same identities. This is because the branch points are due to ( x , - x j ) p'~ (from (exp(ip,.X), e x p ( i p / . X ) ) ) as in the tachyonic case. Additional factors of ( x , - x j ) "~, with n, integers, arise from derivatives of the logarithmic Green's functions (i.e. from the parts of vertex operators such as (OX, 3X) ), but they do not affect the phases. Similar identities hold for the type I superstring. For a given mass level, amplitudes can be written as a common piece that does not depend on the Lorentz structure of the external states, and a kinematic factor that does. For example, the scattering of four massless particles (gauge or fermion) has amplitude
[6] A(1234) _.=- ~ ,g - oF(p,.p2)F(p2.p3) K(1234),
F( 1 +pj P3)
K has overall symmetry for boson exchange and antisymmetry for fermion exchange, so cyclic reorderings do not necessarily have the same sign. This can be accommodated in the Plahte identity if a sign occurs when fermions are exchanged. If particles 1, 2 are fermionic and 3, 4 are bosonic, then (1234) means Fig. 2. Triangle representing the four-particle Plahte identity, the
area is the closed string amplitude. 417
Volume 217, number 4
PHYSICS LETTERS B
A (1234) - e x p (i~pa "P2)A (2134)
2 February 1989
this form, because the branch points are symmetrically disposed, phases arise as follows:
- e x p [i~p~ .(p~ +p_~) ]A (2314) = 0 . In this case the kinematical factor K(1234) has opposite sign from the others, which fits in with the fourboson case where there are no signs and all the K's are identical.
ly'] > I x - l ] : > Ix[:
exp(i~p~ .p~) , exp(i~pl 'P2) .
Now it is possible to change variables to ~=x+y' and
q=x-y'. J d~d~/I~[P' "21~/I'''p2
2. Mixed disc diagrams We now proceed to discuss the relation between amplitudes describing scattering of open and closed strings and amplitudes describing only open strings. The simplest situation to consider is ~%](3,1) illustrated in fig. 3 with three open (p2 = 2) and one closed tachyon (k2= 8). The amplitude is most simply derived in Polyakov's formalism. The Green's function is obtained from the method of images and so includes a contribution In Iz-gl. If we fix the SL(2, ) invariance in the usual way by placing the three open string taehyons at z = 0 , 1, ~ , the amplitude takes the form
× [ ~ - 1 [P' P~lt/- 1 IP' "3(~-q) 2 ( p h a s e ) . The phase condition retains a simple form (~-l)(r/-1)<0:
exp(izrp,.p3 ) ,
(~) (q) < 0 :
exp (brpL "P2) •
The integral is proportional to the amplitude for five open string tachyon scattering, it may be split into ranges corresponding to different five-particle orderings. If we associate P2 ~ 0, P3 ~ 1, P 4 ~ O0 and p~ ~ and q, there is a phase attached to each ordering according to the rule above. d(3a)ocA (23114) + exp(ilrp~ .P3)A (21314)
'~(3'l)°c f d2zlzlPZ'k'lz--llm'/'tlZ--212. u.h.p
+ e x p [i~rp~. (P2 +/73) ] M ( 1 2 3 1 4 )
We wish to expose this in the form of a five-point open string amplitude, as is naively suggested by regarding z and g as the two vertex operator locations to be integrated over. To do this carefully, we follow the same steps as Kawai, Lewellen and Tye [ 3 ]. As a point of notation, we define p~ =k~/2, so it obeys the open string tachyon mass condition p 12= 2. The amplitude can be written unambiguously in terms of x and y. In the complex y plane, independently of the location of the branch points, the branch cuts may be positioned on the imaginary axis, always avoiding the real axis. The y contour can be rotated to y' = - iy. In
+ exp (iTrpf "p2)A(12134) +A(11234)+A(21134). This may be simplified using the Plahte identities. By subtracting the combination (231!4) + (11234) + (2! 134) one obtains ,~3'~)ocsin npl .p2A(12134) . Invariance under interchange of particles 2, 3, 4 follows because the Plahte identities also allow the amplitude to be written in alternative forms. d~3'l)ocsin ~Pl "p3A(1312 4 ) , ocsin npj .P4A(14123) . It is instructive to see that both sides of the relation factorise on physical poles. A study of the residues would fix the normalisation. This is most easily accomplished using the F function representation:
",
\ P~
Fig. 3. A mixed open/closed string amplitude ~e/(3'J~, and its relation to the five-point open string amplitude.
418
.~/(3.1 )oc~rF(P~ "P2+ 2)F(p~ 'P3 + 2)F(p~ "P4 + 2 ) F( -Pl "p2)F(-P, 'p3)F(-P,
"P4)
Poles only occur for even mass level open string states
Volume 2 ! 7, number 4
PHYSICS LETTERSB
2 February 1989
that they imply that A (142343)=A (132434) so the symmetries are manifest) it becomes ag(2'2) oc - A ( 1 2 3 3 4 4 ) - A ( 1 2 4 4 3 3 ) +A(133244) -t- COS 7r (Pl "P4 + P 3 ' P a ) A ( 142343 ) .
Fig. 4. Factorisation channel for #d ~3.~1.
being exchanged in the channel of fig. 4. This is because the amplitude for the right-hand process in fig. 4 vanishes for odd mass levels. In order to proceed to the general situation we distinguish three cases where it is convenient to apply different SL(2, ~) fixing procedures. For three or more open strings and any number of closed strings, we fix the position of three open strings (conventionally at 0, 1, or) and then follow the same steps as for ~cJ(3"1). For one or two open strings SL(2, ~) is used to fix a closed string at La in the upper half-plane, and one open string is taken to oo [ 7 ], An analytic continuation of 2 to - i 2 ' is made before rotating the y contours. The simplest non-trivial example occurs for ~/{ 2,2):
~(2"2)cxT.Z3f d2z3 f dYl ,Z3--i~lk3'k4/2,Z3q-i~lk3"k4/2 X 123 --Xl [""k31Z3--273 [2IX1 --i~l pl'lt'4 oc/~'3 j d~3 dl/3 d x l ]~3 - J - '
]P3p41713-2' [p3.p,
x IG +;.' I"''~l ~3 + 2 ' I,'','4 x 1~3 - x , [p"p3[r/3 --Jr Ip'p~ × 1{3 --~3 121X~ --2' I~"'P"IX~+;t' WP"(phase), which is a six-point open string amplitude with SL(2, ~) fixed points at _+2' and Go. The phases depend on the ordering as ( ~ - ~ ' ) (~3 + 2 ' ) < 0 :
exp (inp3 'P4),
( ~ + 2 ' ) (tl3 - 2 ' ) < 0 :
exp (inp3 'p4 ) ,
( G - x ~ ) ( ~ - x ~ ) < 0 : exp(inp3-pj ) , Ix, 1<2' :
exp (irrp~ 'p4 ) •
After an expeditious use of Plahte identities (note
For the final case of a disc with only closed string vertices, it is only possible to make a formal argument because these amplitudes suffer a divergence due to the dilaton going to the vacuum. A suitable SL(2, N) choice is to fix one vertex at i2 and integrate another along the imaginary axis from 0 to i2. A very simple example is the two-tachyon amplitude, which gives rise to a higher order tree level mass shift. 2
d{o,2)< j dy 2 2 ( 2 2 - y 2) {y-i21-4]yq-i~{
~
4
o
× (2y)2(22) 2 , where the first term is the SL(2, ~) fixing determinant. This expression corresponds to the four-point open string amplitude A ( 1212 ) with the rather unusual SL(2, D) choice & = - 2 , & = - x > x4=2. This amplitude is sitting on a pole and is of course also divergent. For more complicated examples of this type one must analytically continue in 2 besides rotating the y contours. For certain three-point amplitudes, our results appear to fail because the restdting sum of open string amplitudes vanishes by Plahte identities. In these cases, ~¢{ 1,2} and ag (°'3), the kinematics are such that the branch points are in fact poles. This must be taken account of in performing the analytic continuations. We will now describe the general result. Any scattering amplitude involving Mclosed string vertex operators attached to a world-sheet with the topology of a disc and an arbitrary number of open string operators attached to the boundary, can be written as a sum of amplitudes with M pairs of additional open string vertex operators (each member of the pair carrying half the momentum of the original closed string) multiplied by phases depending on the momenta. The different orderings included in the sum depends on the choice ofSL(2, ~) gauge fixing, and so there are three cases, as above. If we work in the upper halfplane: (A) Three or more open strings. The three open strings used to effect the gauge fixing occur in a fixed 419
Volume 217, number 4
PHYSICS LETTERS B
order and we sum over all possible orderings of the other vertices. (B) One or two open strings. The closed string at i2 gives rise to a pair o f open strings at +,~', the open string used for gauge fixing is at infinity, and we sum over all orderings o f the other vertices. ( C ) No open strings. Again the closed string at i2 gives rise to a pair o f vertices at _+2', and the closed string on the imaginary axis leads to two open strings at +_y', y < 2', and we sum over all orderings o f the r e m a i n i n g vertices. The phases are assigned to the open string amplitudes as follows: F o r every open string vertex o f mom e n t u m p which did not originate from a closed string and appears sandwiched between a pair o f vertices which did, we multiply by exp (bzp.k) where k is the m o m e n t u m of each m e m b e r o f the pair. W h e n e v e r a pair of vertices, each of m o m e n t u m k~, originating from a closed string appears sandwiched between another pair, each o f m o m e n t u m k2, we multiply by exp (2i~zk~. k2 ), whereas if the two pairs are arranged so that their m e m b e r s alternate we multiply by exp(inkj.k2). The form o f these results is the same for all species o f particles, the pairs o f open string vertices include a p p r o p r i a t e derivatives o f the target space coordi-
420
2 February 1989
nares corresponding to the factorisation o f the closed string vertices into left and right m o v i n g modes, as in ref. [ 3 ]. Although the formulae o b t a i n e d d e p e n d on the details of the SL(2, R) fixing, the Plahte identities m a y be used to transform from one choice to another.
Acknowledgement We would like to thank Alessandro d ' A d d a for describing Plahte's results to us.
References [ 1] E. Witten, Nucl. Phys. B 268 (1986) 79. [ 2] P. Mansfield, Nucl. Phys. B 306 (1988) 630. [3 ] H. Kawai, D.C. Lewellen and S.-H.H. Tye, Nucl. Phys. B 269 (1986) 1. [4] E. Plahte, Nuovo Cimento 116 A (1970) 713. [5] D.B. Fairlie and K. Jones, Nucl. Phys. B 15 (1970) 323. [6 ] M.B. Green, J.H. Schwarz and E. Witten, Superstring theory (Cambridge U.P., Cambridge, 1987 ). [7] M. Ademollo, A.D'Adda, R.D'Auria, E. Napolitano, P. Di Vecchia, F. G[iozzi and S. Sciuto, Nucl, Phys. B 77 (1974) 189; L. Clavelli, Phys. Rev. D 9 (1974) 3449.
PHYSICS LETTERS B
2 February 1989
RELATIONS BETWEEN DISC DIAGRAMS David LANCASTER and Paul MANSFIELD Department of Theoretical Physics, Universityof Oxford, 1 Keble Road, Oxford OX1 3NP, UK Received 8 October 1988; revised manuscript received 18 November 1988
We show that the amplitude for N open and M closed strings on a disc may be written as a sum of amplitudes with N + 2M open strings, with appropriate orderings and phases. This form may be simplified using Plahte identities which relate open string amplitudes with different orderings. We demonstrate that these also hold for type I superstrings.
1. Relations between disc diagrams
for both open and closed strings). Consider the complex integral
The relationship between open and closed strings is of fundamental concern in string field theory. Until recently the only covariant string field theory which had been shown to successfully reproduce the results of first quantisation was Witten's theory of the open bosonic string [ 1 ]. Closed strings are phenomenologically more interesting, and furthermore appear as a subsector of open string theories. This was responsible for their discovery in the context & t h e dual resonance model and recently this has been made the basis of a closed string field theory [2] which, it is claimed, reproduces the results of first quanfisation for closed string scattering amplitudes. The proposal is based on an analysis of the interacton between open and closed strings. In this letter we show that such a mixed amplitude at tree level can be expressed as a superposition of amplitudes for open strings alone. This generalises to the superstring, and is not only of interest in understanding the results of ref. [ 2 ], but also of practical value in simplifying the computation of such amplitudes, as was the work of ref. [ 3 ], which expressed pure closed string amplitudes in terms of open string ones. We also show that the Plahte identities [4] lead to further simplifications on both of our results, and those of ref. [ 3 ], and give their supersymmetric generalisation. We start by reminding readers of the Plahte identities [4]. The simplest example is the scattering of four tachyons with p 2= 2 (we take a ' = ~-throughout, 416
i dz zP"P2( 1 - z ) p2p3 , with branch cuts along the real axis. The integral converges at large z for Pl"P2+P2"p3
Fig. 1. Contour in upper half-plane yielding the four-particle Plahte identity.
0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
Volume 217, number 4
PHYSICS LETTERS B
0
t,*
exp(i~zpj "P2)
J dz Izl""~-'l 1-zl p2p3 -~x.,
+ ~ d z JzIP'P211-z] p2'p3 0
+ e x p ( - i z t p z ' p 3 ) ~( dz [zlP"P'-[1 -z[I'2P3-- 0. 1
exp (iz:p~ .p2)A(2134)+A(1234) + exp ( - i2zp2 'P3 )A ( 1 3 2 4 ) = 0. The complex conjugate relation also follows from using a similar contour in the lower half-plane. Making use of kinematics we obtain the real relation A(2134) sin ~P2 "P3
the positions of N - 3 vertex operators on the boundary of the world-sheet that we take to be the upper half-plane. For any integration variable we can write a vanishing complex integral which is the sum of amplitudes differing only by the position of the corresponding tachyon, multiplied by phases determined by appropriately negotiating the branch cuts. For example for five particles we obtain the following identity which we denote (!2345): A ( ! 2345 ) + exp (izrp~ "P2)A (2_1345 )
This relation between Euler beta functions can be interpreted in terms of the Veneziano amplitude for different particle orderings [ 5 ] :
A(1234) sin ztpl 'P3
2 February 1989
A(1324) sin 7~pr"P2
This is conveniently pictured using fig. 2. The area of this triangle is quadratic in open string amplitudes and is equal to the closed string scattering amplitude for four tachyons of momenta 2pi. This follows from the work of Kawai, Lewellen and Tye [3] and provides a manifestly symmetric representation of their result which can, if desired, be expressed in terms of a single open string amplitude. The Plahte identities hold for scattering of any number of open string tachyons in bosonic string theory. They may be derived by a simple generalisation of the above argument. The Koba-Nielsen expression for the N-tachyon amplitude is an integral over
+ exp [izrp~ • (P2 +P3) ]A (23145) + exp [izrp~. (P2 +P3 +P4) ]A (234! 5 ) = 0 . This can be represented by a quadrilateral with sides proportional to the amplitudes and angles determined by the phases. We can construct other identities and quadrilaterals but they will not all be independent. Unlike the case of four particles, it is impossible to express every five-particle amplitude in terms of a single ordering. Amplitudes involving excited states of the bosonic string obey the same identities. This is because the branch points are due to ( x , - x j ) p'~ (from (exp(ip,.X), e x p ( i p / . X ) ) ) as in the tachyonic case. Additional factors of ( x , - x j ) "~, with n, integers, arise from derivatives of the logarithmic Green's functions (i.e. from the parts of vertex operators such as (OX, 3X) ), but they do not affect the phases. Similar identities hold for the type I superstring. For a given mass level, amplitudes can be written as a common piece that does not depend on the Lorentz structure of the external states, and a kinematic factor that does. For example, the scattering of four massless particles (gauge or fermion) has amplitude
[6] A(1234) _.=- ~ ,g - oF(p,.p2)F(p2.p3) K(1234),
F( 1 +pj P3)
K has overall symmetry for boson exchange and antisymmetry for fermion exchange, so cyclic reorderings do not necessarily have the same sign. This can be accommodated in the Plahte identity if a sign occurs when fermions are exchanged. If particles 1, 2 are fermionic and 3, 4 are bosonic, then (1234) means Fig. 2. Triangle representing the four-particle Plahte identity, the
area is the closed string amplitude. 417
Volume 217, number 4
PHYSICS LETTERS B
A (1234) - e x p (i~pa "P2)A (2134)
2 February 1989
this form, because the branch points are symmetrically disposed, phases arise as follows:
- e x p [i~p~ .(p~ +p_~) ]A (2314) = 0 . In this case the kinematical factor K(1234) has opposite sign from the others, which fits in with the fourboson case where there are no signs and all the K's are identical.
ly'] > I x - l ] : > Ix[:
exp(i~p~ .p~) , exp(i~pl 'P2) .
Now it is possible to change variables to ~=x+y' and
q=x-y'. J d~d~/I~[P' "21~/I'''p2
2. Mixed disc diagrams We now proceed to discuss the relation between amplitudes describing scattering of open and closed strings and amplitudes describing only open strings. The simplest situation to consider is ~%](3,1) illustrated in fig. 3 with three open (p2 = 2) and one closed tachyon (k2= 8). The amplitude is most simply derived in Polyakov's formalism. The Green's function is obtained from the method of images and so includes a contribution In Iz-gl. If we fix the SL(2, ) invariance in the usual way by placing the three open string taehyons at z = 0 , 1, ~ , the amplitude takes the form
× [ ~ - 1 [P' P~lt/- 1 IP' "3(~-q) 2 ( p h a s e ) . The phase condition retains a simple form (~-l)(r/-1)<0:
exp(izrp,.p3 ) ,
(~) (q) < 0 :
exp (brpL "P2) •
The integral is proportional to the amplitude for five open string tachyon scattering, it may be split into ranges corresponding to different five-particle orderings. If we associate P2 ~ 0, P3 ~ 1, P 4 ~ O0 and p~ ~ and q, there is a phase attached to each ordering according to the rule above. d(3a)ocA (23114) + exp(ilrp~ .P3)A (21314)
'~(3'l)°c f d2zlzlPZ'k'lz--llm'/'tlZ--212. u.h.p
+ e x p [i~rp~. (P2 +/73) ] M ( 1 2 3 1 4 )
We wish to expose this in the form of a five-point open string amplitude, as is naively suggested by regarding z and g as the two vertex operator locations to be integrated over. To do this carefully, we follow the same steps as Kawai, Lewellen and Tye [ 3 ]. As a point of notation, we define p~ =k~/2, so it obeys the open string tachyon mass condition p 12= 2. The amplitude can be written unambiguously in terms of x and y. In the complex y plane, independently of the location of the branch points, the branch cuts may be positioned on the imaginary axis, always avoiding the real axis. The y contour can be rotated to y' = - iy. In
+ exp (iTrpf "p2)A(12134) +A(11234)+A(21134). This may be simplified using the Plahte identities. By subtracting the combination (231!4) + (11234) + (2! 134) one obtains ,~3'~)ocsin npl .p2A(12134) . Invariance under interchange of particles 2, 3, 4 follows because the Plahte identities also allow the amplitude to be written in alternative forms. d~3'l)ocsin ~Pl "p3A(1312 4 ) , ocsin npj .P4A(14123) . It is instructive to see that both sides of the relation factorise on physical poles. A study of the residues would fix the normalisation. This is most easily accomplished using the F function representation:
",
\ P~
Fig. 3. A mixed open/closed string amplitude ~e/(3'J~, and its relation to the five-point open string amplitude.
418
.~/(3.1 )oc~rF(P~ "P2+ 2)F(p~ 'P3 + 2)F(p~ "P4 + 2 ) F( -Pl "p2)F(-P, 'p3)F(-P,
"P4)
Poles only occur for even mass level open string states
Volume 2 ! 7, number 4
PHYSICS LETTERSB
2 February 1989
that they imply that A (142343)=A (132434) so the symmetries are manifest) it becomes ag(2'2) oc - A ( 1 2 3 3 4 4 ) - A ( 1 2 4 4 3 3 ) +A(133244) -t- COS 7r (Pl "P4 + P 3 ' P a ) A ( 142343 ) .
Fig. 4. Factorisation channel for #d ~3.~1.
being exchanged in the channel of fig. 4. This is because the amplitude for the right-hand process in fig. 4 vanishes for odd mass levels. In order to proceed to the general situation we distinguish three cases where it is convenient to apply different SL(2, ~) fixing procedures. For three or more open strings and any number of closed strings, we fix the position of three open strings (conventionally at 0, 1, or) and then follow the same steps as for ~cJ(3"1). For one or two open strings SL(2, ~) is used to fix a closed string at La in the upper half-plane, and one open string is taken to oo [ 7 ], An analytic continuation of 2 to - i 2 ' is made before rotating the y contours. The simplest non-trivial example occurs for ~/{ 2,2):
~(2"2)cxT.Z3f d2z3 f dYl ,Z3--i~lk3'k4/2,Z3q-i~lk3"k4/2 X 123 --Xl [""k31Z3--273 [2IX1 --i~l pl'lt'4 oc/~'3 j d~3 dl/3 d x l ]~3 - J - '
]P3p41713-2' [p3.p,
x IG +;.' I"''~l ~3 + 2 ' I,'','4 x 1~3 - x , [p"p3[r/3 --Jr Ip'p~ × 1{3 --~3 121X~ --2' I~"'P"IX~+;t' WP"(phase), which is a six-point open string amplitude with SL(2, ~) fixed points at _+2' and Go. The phases depend on the ordering as ( ~ - ~ ' ) (~3 + 2 ' ) < 0 :
exp (inp3 'P4),
( ~ + 2 ' ) (tl3 - 2 ' ) < 0 :
exp (inp3 'p4 ) ,
( G - x ~ ) ( ~ - x ~ ) < 0 : exp(inp3-pj ) , Ix, 1<2' :
exp (irrp~ 'p4 ) •
After an expeditious use of Plahte identities (note
For the final case of a disc with only closed string vertices, it is only possible to make a formal argument because these amplitudes suffer a divergence due to the dilaton going to the vacuum. A suitable SL(2, N) choice is to fix one vertex at i2 and integrate another along the imaginary axis from 0 to i2. A very simple example is the two-tachyon amplitude, which gives rise to a higher order tree level mass shift. 2
d{o,2)< j dy 2 2 ( 2 2 - y 2) {y-i21-4]yq-i~{
~
4
o
× (2y)2(22) 2 , where the first term is the SL(2, ~) fixing determinant. This expression corresponds to the four-point open string amplitude A ( 1212 ) with the rather unusual SL(2, D) choice & = - 2 , & = - x > x4=2. This amplitude is sitting on a pole and is of course also divergent. For more complicated examples of this type one must analytically continue in 2 besides rotating the y contours. For certain three-point amplitudes, our results appear to fail because the restdting sum of open string amplitudes vanishes by Plahte identities. In these cases, ~¢{ 1,2} and ag (°'3), the kinematics are such that the branch points are in fact poles. This must be taken account of in performing the analytic continuations. We will now describe the general result. Any scattering amplitude involving Mclosed string vertex operators attached to a world-sheet with the topology of a disc and an arbitrary number of open string operators attached to the boundary, can be written as a sum of amplitudes with M pairs of additional open string vertex operators (each member of the pair carrying half the momentum of the original closed string) multiplied by phases depending on the momenta. The different orderings included in the sum depends on the choice ofSL(2, ~) gauge fixing, and so there are three cases, as above. If we work in the upper halfplane: (A) Three or more open strings. The three open strings used to effect the gauge fixing occur in a fixed 419
Volume 217, number 4
PHYSICS LETTERS B
order and we sum over all possible orderings of the other vertices. (B) One or two open strings. The closed string at i2 gives rise to a pair o f open strings at +,~', the open string used for gauge fixing is at infinity, and we sum over all orderings o f the other vertices. ( C ) No open strings. Again the closed string at i2 gives rise to a pair o f vertices at _+2', and the closed string on the imaginary axis leads to two open strings at +_y', y < 2', and we sum over all orderings o f the r e m a i n i n g vertices. The phases are assigned to the open string amplitudes as follows: F o r every open string vertex o f mom e n t u m p which did not originate from a closed string and appears sandwiched between a pair o f vertices which did, we multiply by exp (bzp.k) where k is the m o m e n t u m of each m e m b e r o f the pair. W h e n e v e r a pair of vertices, each of m o m e n t u m k~, originating from a closed string appears sandwiched between another pair, each o f m o m e n t u m k2, we multiply by exp (2i~zk~. k2 ), whereas if the two pairs are arranged so that their m e m b e r s alternate we multiply by exp(inkj.k2). The form o f these results is the same for all species o f particles, the pairs o f open string vertices include a p p r o p r i a t e derivatives o f the target space coordi-
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nares corresponding to the factorisation o f the closed string vertices into left and right m o v i n g modes, as in ref. [ 3 ]. Although the formulae o b t a i n e d d e p e n d on the details of the SL(2, R) fixing, the Plahte identities m a y be used to transform from one choice to another.
Acknowledgement We would like to thank Alessandro d ' A d d a for describing Plahte's results to us.
References [ 1] E. Witten, Nucl. Phys. B 268 (1986) 79. [ 2] P. Mansfield, Nucl. Phys. B 306 (1988) 630. [3 ] H. Kawai, D.C. Lewellen and S.-H.H. Tye, Nucl. Phys. B 269 (1986) 1. [4] E. Plahte, Nuovo Cimento 116 A (1970) 713. [5] D.B. Fairlie and K. Jones, Nucl. Phys. B 15 (1970) 323. [6 ] M.B. Green, J.H. Schwarz and E. Witten, Superstring theory (Cambridge U.P., Cambridge, 1987 ). [7] M. Ademollo, A.D'Adda, R.D'Auria, E. Napolitano, P. Di Vecchia, F. G[iozzi and S. Sciuto, Nucl, Phys. B 77 (1974) 189; L. Clavelli, Phys. Rev. D 9 (1974) 3449.