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Gauge-invariant modification of Witten's open superstring
Volume 213, number 4
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3 November 1988
G A U G E - I N V A R I A N T M O D I F I C A T I O N OF W I T T E N ' S O P E N S U P E R S T R I N G Olaf L E C H T E N F E L D and Stuart S A M U E L Physics Department, City Collegeof New York, New York, NY 10031, USA Received 18 May 1988
A violation of gauge invariance in Witten's open superstring field theory has recently been discovered. We propose a modification which resolves this problem by avoiding the collision of picture-changing operators. Feynman rules for the modified, gaugeinvariant theory are presented, including a useful representation for the Ramond propagator. All four-point amplitudes are offshell finite, agree on-shell with the Polyakov approach and exhibit decoupling of spurious physical states.
1. Introduction A non-perturbative approach to string theory [ 1,2 ] might be necessary to make contact with the four-dimensional world [3-5 ]. One way of extracting nonperturbative physics is through field theory. A covariant field theory for the open bosonic string was achieved by Witten in ref. [ 6 ]. This theory has been the subject of much research [ 7 - 2 0 ] . An extension to the open superstring has also been proposed by Witten [ 21 ]. However, recent work by Wendt [22 ] has revealed an associativity anomaly in its superstar product. It leads to an order g2 violation of gauge invariance and an infinite four-vector amplitude. The difficulty is traced to colliding picturechanging operators [ 22 ]. The superstar product of two Neveu-Schwarz fields involves the picturechanging operator, X ( z ) = ~ { Q, ~(z) }, at the string midpoint ~l. When there are two or more star operators, two or more X are present. Since X ( z ) X ( w ) blows up like ( z - w ) - 2 as z--, w, a singularity is generated that renders undefined multiple applications of superstar products. In ref. [ 22 ], the theory was regularized and a four-point contact term was added to the action to cancel the singularity on-shell. The offshell counterterm and modification to the gauge transformation law was obtained in ref. [23 ]. While this repairs the gauge violations to order g2 and yields The xf8 factor is a convention used to assure that the Ramond kinetic operator has the standard normalization [ 20 ]. 0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
finite four-point amplitudes, gauge violations at order g3 and infinities in five- and higher-point amplitudes most likely require higher-point counterterms in the action as well as additional modifications to the gauge transformation laws. An order by order patching up of Witten's proposal is arduous. The first and main goal of this letter is to overcome the above-mentioned difficulties by modifying Witten's theory [ 21 ] so that colliding X are avoided. An action, bilinear and trilinear in string fields, is discovered that is gauge-invariant and non-singular. The second purpose is to present the F e y n m a n rules for the gauge-fixed theory. In particular, a useful representation of the R a m o n d propagator is obtained. Finally, off-shell finiteness, on-shell agreement with first-quantized results and the decoupling of spurious states from physical amplitudes is demonstrated for the four-point amplitudes. Our notation and conventions are given in refs. [23-26, 20 ] with one exception: the coupling g in this paper corresponds to g in ref. [23 ]. Relevant papers for the present work are refs. [ 6,7,21-23,25,26 ]. We also rely on conformal field theory methods [27-38 ] as applied to string field theory [8,39-53] and offshell computations [25,26,54-56 ].
2. The modified theory Open string operators, O(a, z), are expressible in terms of left- and right-moving fields, O ( z ) and O(g), 431
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where z = e x p ( r + i a ) and g = e x p ( z - i a ) . The string strip, - ~ < z < + ~ and 0 < a < n, in tr and z space is the upper half-plane in the variable z and the lower half-plane in the variable g. The string midpoint, z = 0 and a=n/2, is z = i and g = - i . The union of the upper and lower half-planes forms the double cover of the open string configuration. Although left- and rightmoving fields are related by the open string boundary conditions, we may work on the full complex plane by using analytic fields only. This allows us to put midpoint insertions at z = i as well as at z = - i . The superstring star product and integral are obtained by inserting picture-changing and inverse picture-changing operators, xf8 Y(z) = c0~ × e x p ( - 2 q ~ ) (z), at the string midpoint [21 ]. Starting with the Witten action and transformation law [ 21 ], we considered all possible insertions of X(i), X ( - i ) , Y(i) and Y ( - i ) consistent with ghost number assignments and found that the following action,
case. Eq. (1) is rigorously invariant under eq. (2) and hence requires no counterterms. Although eq. (1) differs only slightly from the Witten proposal [21 ], it is a different theory. As pointed out in ref. [ 57 ], the free Ramond action couples fields of different mass levels because of the presence of Y. Our modified theory has the same feature in the Neveu-Schwarz sector; all levels couple at g = 0. As a result, off-shell amplitudes are not equal to those obtained from the Witten proposal. In particular, section 5 shows the finiteness of the four-boson amplitude. Under the assumption that the action is at most trilinear in string fields and that the structure of the gauge transformation is that ofeq. (2), eq. ( 1 ) is the unique gauge-invariant action except for the trivial reflection i ~ - i. The modified action is invariant under the following supersymmetry transformation:
S= f ~PNs*X(i)Y(--i)Q~PNS+ ~ R * Y(--i)Q~R
5~ys = X ( - i )
+ ]g~X(i)~VNS* ~VNS*~VNS+ 2gJ ~VNS*~V., ~R, (1) is invariant under the gauge transformation ~ ~r-tNS= QANs
+gX(-i)
[ ( ~UNS*ANs --ANs * 7tNs)
+ Y(i) ( ~'/R*At( --AR * ~T'/R) ] ,
5 ~R =
(3)
where ea is an anticommuting parameter. [e. S(_ 1/2)1o is the integral fermion emission operator at zero momentum with ea as the spinor wave function: [e.S(_l/2)]o=Vf(_l/2)(ua=e&, k = 0 ) . There is at least one other version of the supersymmetry transformations which uses the fermion emission operator in the + 1/2 picture:
8 ~'gR= V-~+ 1/2)(uot =edt, k = 0 ) ~NS,
[ ( ~[/R*ANs --ANs * ~JR)
"JI-( ~JNS*AR --AR * ~UNS) ].
(2)
Here, • and f denote the ordinary star product and string integral [ 6 ]. The original Witten proposal [ 21 ] is obtained by replacing Y ( - i ) by Y(i) in eq. ( 1 ) and replacing X ( - i ) by X(i) in eq. (2). In verifying the gauge invariance, no two X n o r two Y are multiplied at the same point. Using the proofs of the axioms in refs. [9,13,43,45], invariance to O ( g °) and O ( g ~) is easily shown. O ( g 2) invariance depends on the associativity of • in the absence of and in the presence of X ( i ) X ( - i ) . The former has been proven in refs. [25,55]. Associativity with a single X(i) present was proven in section 3 of ref. [23] but the same proof applies to the X ( i ) X ( - i ) 432
~ ~'/R= X ( - - i ) [e'S(- t/2) ]o ~NS,
8~NS = Y(i) V(+ l / 2 ) J (Uot = er/, k = 0) ~-JR,
QAR
+gX(-i)
[e'S(_,/2)]o Y(i) ~JR,
(4)
where the integral vertex is given in eq. (IV. lg) of ref. [ 24 ].
3. Gauge fixing and propagators To commence perturbation theory, the gauge must be fixed. For the bosonic string and the superstring field theories, a convenient gauge is the SiegelFeynman one [58]: bo ~u=0,
(5)
where bo is the zero mode of the antighost, b(z). This gauge is achieved in the linearized theory, i.e. g = 0, by setting A = - Lff ' bo ~. In the absence of a Gribov
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ambiguity [ 59 ], this gauge should be possible in the interacting case. There have been suggestions [60,21 ] that an additional gauge constraint, flo ~ = 0, can be imposed in the Ramond sector of the superstring. Actually, not enough freedom remains in A to do this. For residual gauge transformations, i.e. those preserving bo ~v= 0, boQA must be zero. This yields A = QLfflboA which implies ~A~V=0, hence, the gauge is already fixed completely. There are two other reasons for believing that eq. (5) suffices for the Ramond sector. Firstly, a propagator exists (see eq. (8) below). Secondly, the dynamics at one loop are likely to be incorrect if, in addition, flo ~v= 0. In the light-cone analysis, the odd spin structure [ 61 ] has eight ~ zero modes, which yields a fourth order zero in its partition function. In the covariant formalism, there are ten zero modes for Vu; however, the 7o-flo system provides a zero in the denominator. If the constraint flo ~v= 0 is imposed, the fifth zero in the odd spin structure 9,~ partition function is no longer cancelled by the one from the 7o-flo system. This would lead to deviations from light-cone results in higher-point functions. For bo ~v= 0, the quadratic part of the action in eq. ( 1 ) becomes Squadratic = j ~VNS* n *X(i ) Y( - i ) Q~ ~VNS +f~R*~ZtY(--i)Q~R
,
(6)
where n=boco projects onto the space b o ~ = 0 and ~* = cobo projects onto Co~ = 0. They satisfy me= 7r, ~t;~t = ~ t and n + x*= 1. The kinetic operators, A, in eq. (6) are
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sector and B - 1= X( - i) for the Ramond sector. The propagator in eq. (8) is, in some sense, not unique. This is not due to the gauge-fixing procedure but to the singular nature of X ( z ) and Y ( z ) in the first-quantized theory. Although Y ( z ) and X ( z ) are mutual inverses, i.e. Y ( z ) X ( z ) = 1, there exist O ( z ) and O' (z) such that O ( z ) X ( z ) = 0 and O' ( z ) Y ( z ) = 0 (there is no contradiction here; associativity of operator products at the same point need not hold #2 ). Therefore, X(Y) is an inverse of Y ( X ) only modulo shifts with O ( O ' ) . However, restriction to a certain class of operators fixes this ambiguity. Among those restrictions are ghost number, conformal behavior and dimension, BRST-invariance, locality etc. The ambiguity reflects itself in eq. (8). If B - 1 is shifted by an element in the kernel of B, P still inverts A of eq. (7). We believe that the same constraints needed to uniquely fix X ( z ) and Y ( z ) must also be imposed on B - 1. In this sense, the propagator is unique. One positive feature is that the propagator in the Ramond sector is built from statistically correct operators [24], that is, operators which map GSO accepted states into GSO accepted states. The Ramond propagator used in refs. [ 22,62 ] is the product of two statistically incorrect operators, one of which is Fo. The computation of an off-shell amplitude requires deforming Fo onto external states, thereby relating the amplitude to one involving GSO-rejected states. For this reason the off-shell two-vector two-spinor amplitude in Witten's theory was not calculated in ref. [ 23 ]. With the Ramond propagator in eq. (8), there is no difficulty with statistically incorrect operators.
4. Feynman rules A=coboBQboco,
(7)
where B = Y ( - i ) in the Ramond sector and B = X(i) Y( - i) in the Neveu-Schwarz sector, the order of X, Y and Q being immaterial since they mutually commute. A propagator, P, is required to invert A, i.e. PA=Tz andAP=TC. It is easily seen that bo bo bo B - 1 , P = -~o B - I a - - = Lo Lo bo - Q ~oo B - I -bo = B - I -~o where B - ~ = X ( - i ) Y ( i )
bo B - ' b o Q Lo Lo
Refs. [6-8,21,63] have explained how secondquantized Feynman graphs are related to first-quantized world-sheet computations. For the bosonic theory, external states and propagators are represented by rectangular strips of width n. External states are of semi-infinite length and internal propagators are of length T, where T is to be integrated from 0 to oo. In addition, the following line integral, bo= ¢(dco/2ni)ogb(og), is taken across the world sheet
(8) for the Neveu-Schwarz
,2 Limit.... s [limit:, .:O I(z t )02(z2)O3(z3) ] may differ from limit:, .:, [limit:: .: 0, ( z I )02( z2 )Os ( z3 ) ]. 433
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(and its double). These results follow because the propagator in the bosonic theory is boL ff i = bo f ~ d T e x p ( - TLo), and exp ( - TLo) creates a world-sheet strip of length T. External and internal strings are thrice-wise joined together via the Witten method: The ends of each of the three strips are divided in two at zt/2 and the first half of one end is joined to the second half of the next. Since the interactions in eq. ( 1 ) are the same as in Witten's theory, the vertices of the Feynman graphs are the same. The R - R - N S vertex contains no picture changing operators, and the N S - N S - N S vertex has an X at the midpoint. The propagators, however, are different. Since they involve two Lff I (see eq. ( 8 ) ) , two strips of length TI and /'2 are used. The lengths, TI and T2, are independently integrated from 0 to oo. The operators bo and Q=~(dm/2ni)jaasT(tO) in P are expressed as line integrals across one of the strips. Picture-changing operators are inserted at a = r t / 2 between the two strips. Fig. la displays the propagator. The terms ( bo/ Lo )B- l and B-~ ( bo/ Lo ) in eq. (8) involve a single strip with insertions at one end as shown in fig. lb. It is straightforward to obtain the off-shell fourpoint amplitudes using the Giddings map [7 ] for T=TI+T2 and the methods in refs. [23,26]. The change of variables from ( TL, T2) to ( T, Tl ) must be made. It is not clear whether amplitudes are expressible in terms of elliptic functions as is the case for the Witten proposal [23 ] because of the extra variable Tl.
5. On-shell agreement and decoupling of spurious states Since the trilinear action in eq. ( 1 ) is identical to the Witten case, the trilinear couplings [ 18-20 ] are the same. It follows that the three-point functions agree with the standard first-quantized theory for onshell physical states and that spurious states decouple. These results are shown for Witten's theory in ref. [23]. On-shell agreement and the decoupling of spurious states can be demonstrated for the four-point amplitudes via picture-changing methods [ 22,34 ]. We use the techniques in section 5 ofref. [23 ]. Let us discuss the four-fermion case in detail. 434
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T~
(
T2
)<
do~ j (o~) ~ 2~xi BRST
~dto ob(o) 2~i
i~
)
cob(0~)
a T
<
>
~dco rob(o) 2/tl
II
.........................................................................................
b Fig. I. (a) The configuration generated by the propagator. For the Neveu-Schwarz case, there is a Yat • and an X a t m. For the R a m o n d case, there is an X at m. (b) The configuration generated by the asymmetric representation of the propagator in eq.
(8). There are six Feynman graphs that contribute to the amplitude [25 ]. Two of them are shown in fig. 2. First consider on-shell agreement. The second term in the asymmetric representation of the propagator
\ / ............../\.....
a
b
Fig. 2. Feynman graphs for four-fermion scattering. The solid lines denote fermions and the dotted lines denote bosons.
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(eq. (8) ) may be dropped for BRST-invariant external states. The propagator is then represented as in fig. lb and the four-fermion amplitude from fig. 2a becomes
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A 4F = + 4 g 2 tr (,~1,~2,~,3~.4)
oo
x(fdTbig
Aa4F = - 4 g 2 tr (~,t)],2~.3,~.4) ~ d T < V~((nl ) V2R((n2)
0 )< V3R((n3)bo
V4 (o)4)X(o)~) Y((no) 7,
(9)
where (no is the midpoint at one of the two vertices in the string configuration (see ref. [23] ) and (n3 is the midpoint in its copy (we work on the double). In eq. (9), V ~ ((ni) represent any BRST-invariant Ramond vertex operators. To establish the on-shell agreement, use picturechanging manipulations to move X at (n3 to (no. Go to the bigger algebra by inserting ~((n), write X((n3) =x/~{Q, ~((n3)} and commute or anticommute Q past other operators until it annihilates the left and right vacua. Two contributions are generated, one from ~((n) and one from bo: A a4F = -- 4g 2 tr (21222324 )
-~'8< v~((nl) vU(n2) v~((n3) v~ ((n4) X~((n~)Y((no)~((n)>big) ,
(11)
where again the limit (n-'(n0 is to be taken. The first term in eq. ( 11 ) completes the Koba-Nielsen integration region from 0 to ~ [ 7 ]. The second term cancels the second term in eq. (10) via the methods in refs. [23,25,55]. Now consider the decoupling of a spurious state. Suppose the fourth state is created by a trivial BRSTinvariant operator, Vl~ = {Q, ~4 } and the other states are BRST-invariant. Pushing Q past the other operators in diagrams 2a and 2b yields
oo
X(ffdZ < VI((nl )V2((n2)~'r3((n3)boV4 ((n4) 0
A~F +A 4F = + 4 g 2 tr (,~1~,2,~,3,~,4)
X [ < VI(o)I ) V2 (0.)2)V3(0)3) V4(0)4)
× ~((n~)Y((no)X((n) >big
XX(co~) Y((no) > T=°
- < v~((n,) v~((n~) P~((n,) vh((n+)
)< ~((n~) Y((no)~((n) >big).
(10)
Taking the limit (n-'(no, Y((no) and X((n) cancel in the first term in eq. (10). The zero mode of ~ must be soaked up by ~((n]) and what remains agrees [7,22,23 ] with the first-quantized approach except that the Koba-Nielsen integration range is from ½to 1. The second term involves the "collapsed configuration", i.e. T = 0. Performing similar manipulations on fig. 2b yields
xx(~)Y((no) >~=°1,
(12)
which is zero because associativity holds in the presence of the two picture-changing operators. The two-boson-two-fermion amplitude proceeds similarly. Two of the six Feynman graphs are shown in figs. 3a and 3b. On-shell agreement and spurious state decoupling occur if the "collapsed" diagrams cancel. Since fig. 3a includes X((n~)Y((no) from the propagator and X((no) for N S - N S - N S vertex, the diagram contains X((n]). Fig. 3b has X((n3 ) from the Ramond propagator. Hence, the two diagrams contain the same picture-changing operator and cancellation occurs. The four-boson diagrams are shown in figs. 4a and 4b. The diagrams involve X((n3)Y((no) from the propagator and X((no) and X((n~) from the vertices 435
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(
/
a
without adding higher-point terms to the action. By changing the quadratic action and gauge transformation law, colliding picture-changing operators are avoided. The four-boson amplitude is finite both onshell and off-shell. Let us examine the axioms of the modified theory. A superstar product can be defined by
b
(A*B)Ns = X ( - i )
Fig. 3. Feynman graphs for two-fermion-two-bosonscattering.
.., ... •... "%.....
../Y'"
....y""
%'......
•.,... . ..-
a
....
..
. ..../"
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"'.......
b
[ANs *BNs + Y(i)AR *BR],
( A * B ) R = X ( - - i ) ( A N s * B R +A~*BNs),
(13)
so that the gauge transformation in eq. (2) is 5 ~ = QA +g( ~ * A - A * ~). Since associativity of the superstar product is still violated due to colliding picture-changing operators, the geometric interpretation of the action is unclear. Formally, it is a ChernSimons form if the superstring integral reads 55="fX(i) Y ( - i ) Y ( - i ) " , where the second Y ( - i ) is to be removed by the X( - i) in *; this makes sense because the string integral always appears together with a star product. Alternatively, replace the integral-star combination by a dot product [42,64 ]:
Fig. 4. Feynman graphs for four-bosonscattering.
A.B=fY(--i)[X(i)ANs.BNs+AR.BR]. in both cases. Here, o95 is the location of the other vertex. In total, X(og~)X(oJo) appears in both diagrams and cancellation takes place. In conclusion, we have demonstrated agreement with the first-quantized approach for on-shell physical states and decoupling of spurious states for all physical four-point amplitudes. The off-shell four-fermion and two-boson-two-fermion amplitudes are finite because no two X collide. For the four-boson amplitude, the two X at the two N S - N S - N S vertices approach each other only when T~ and T2, the lengths of the two propagator strips, both go to zero. The measure dTl and dT2 as well as the presence of Yand the two bo in the Ramond propagator are sufficient to render the integral convergent in this region. Along the same lines, higher-point amplitudes are better behaved in moduli space than for Witten's theory, and are expected to be finite.
6. Conclusion In this letter, we have modified the Witten superstring field theory so as to make it gauge invariant 436
(14)
The action in eq. ( 1 ) becomes ~U.Q~U+2g~U.~*~u. The axioms that guarantee the gauge invariance of the theory are (a) Q 2= 0 (nilpotency of Q), (b) ( Q A ) ' B = - ( - 1 ) a A ' Q B (integration by parts), (c) Q(A*B ) = ( QA )*B + ( - 1 )aA*QB (gradedderivation law), (d) A . B = ( - 1 )abn'.4 (graded commutivity of dot product), (e) A. ( B ' C ) = ( - 1 )a~t'+C)B" ( C*A) (cyclicity under integral), ( f ) ( A ' B ) . ( C ' D ) = ( - 1 )a~,+c+d)(B'C)" ( D ' A ) (cyclicity under integral). With the identification A.B=~A*B, ( a ) - ( d ) are the same as in Witten's theory. Associativity is replaced by the weaker axioms (e) and (f). Some work still remains. (i) When two successive gauge transformations are performed, two picturechanging Xare multiplied at the same point as can be seen from eq. (13). What the gauge algebra closes onto is unclear. This problem is a remnant of the nonassociativity of the * product. (ii) Although it is
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s t r a i g h t f o r w a r d to o b t a i n the off-shell f o u r - p o i n t a m p l i t u d e s using t h e m e t h o d s in ref. [23 ], an e x p l i c i t c o m p u t a t i o n is w o r t h doing. ( i i i ) T h e Y i n s e r t i o n in the p r o p a g a t o r e l i m i n a t e s the c o l l i s i o n o f t h e t w o X c o m i n g f r o m t h e t w o v e r t i c e s in the f o u r - b o s o n a m plitude. V i a this m e c h a n i s m , the b o s o n i c f o u r - p o i n t a m p l i t u d e s are finite. It is i n t e r e s t i n g to a n a l y z e h o w this w o r k s in h i g h e r - p o i n t a m p l i t u d e s a n d to settle t h e q u e s t i o n o f tree-level finiteness. ( i v ) Finally, the B R S T q u a n t i z a t i o n p r o c e d u r e to gauge fix t h e seco n d - q u a n t i z e d superstring n e e d s to be c a r r i e d out. T h e steps s h o u l d be s i m i l a r to t h o s e o f the b o s o n i c string field t h e o r y [ 6 2 , 6 5 ] .
Acknowledgement T h i s w o r k was s u p p o r t e d in part by the N a t i o n a l Science F o u n d a t i o n g r a n t N S F - P H Y - 8 2 - 1 5 3 6 4 a n d by the D e p a r t m e n t o f Energy g r a n t D E - A C 0 2 83ER40107.
References [ 1 ] J.H. Schwarz, Phys. Rep. 89 (1982) 223; J. Scherk, Rev. Mod. Phys. 47 (1975 ) 123. [ 2 ] J.H. Schwarz, ed., Superstrings, Vols. I and II (World Scientific, Singapore, 1985 ); M.B. Green and D.J. Gross, eds., Unified string theories (World Scientific, Singapore, 1986 ); M.B. Green, J.H. Schwarz and E. Witten, Superstring theory, Vols. I and II (Cambridge U.P., Cambridge, 1987). [3] M.B. Green and J.H. Schwarz, Phys. Lett. B 149 (1984) l17;B 151 (1985) 21;Nucl. Phys. B 225 (1985) 93. [4] D.J. Gross, J.A. Harvey, E.J. Martinec and R. Rohm, Nucl. Phys. B 256 (1985) 253; B 267 (1986) 75. [ 5 ] P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, Nucl. Phys. B 258 (1985) 46. [6] E. Witten, Nucl. Phys. B 268 (1986) 253. [ 7 ] S. Giddings, Nucl. Phys. B 278 ( 1986 ) 242. [8 ] S. Giddings and E. Martinec, Nucl. Phys. B 278 ( 1986 ) 91. [9] D. Gross and A. Jevicki, Nucl. Phys. B 283 (1987) 1. [ 10 ] E. Cremmer, A. Schwimmer and C. Thorn, Phys. Lett. B 179 (1986) 57. [ 11 ] S. Samuel, Phys. Lett. B 181 ( 1986 ) 255. [ 12] N. Ohta, Phys. Rev. D 34 (1986) 3785. [ 13 ] D. Gross and A. Jevicki, Nucl. Phys. B 287 ( 1987 ) 225. [ 14] C. Thorn, The oscillator representation of Witten's three open string vertex function, in: Proc. XXIII Intern. Conf. of High energy physics (Berkeley, CA, 1986). [ 15 ] E. Cremmer, Vertex function in Witten's formulation of string field theory, LPTENS preprint (December 1986), in: Proc. Paris-Meudon Colloq. (September 1986).
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[ 16 ] S. Samuel, Introduction to string field theory, in: Proc. XVIII Intern. GIFT Conf. on Strings and superstrings (El Escorial, Spain, June 1987). [ 17 ] S. Samuel, Phys. Lett. B 181 (1986) 249. [ 18 ] D. Gross and A. Jevicki, Operator formulation of interacting string field theory III: NSR superstring, Santa Barbara preprint NSF-ITP-86-119 (March 1987). [ 19 ] K. Suehiro, Operator expression of Witten's superstring vertex, Kyoto preprint KUNS 857 (March 1987). [20] S. Samuel, Nucl. Phys. B 296 (1988) 187. [21 ] E. Witten, Nucl. Phys. B 276 (1986) 291. [22 ] C. Wendt, Scattering amplitudes and contact interactions in Witten's superstring field theory, SLAC preprint SLACPUB-4442 (November 1987). [ 23 ] O. Lechtenfeld and S. Samuel, Off-shell conformal methods for the superstring, CCNY preprint CCNY-HEP-88/3 (April 1988). [24] V.A. Kosteleck~, O. Lechtenfeld, W. Lerche, S. Samuel and S. Watamura, Nucl. Phys. B 288 (1987) 173. [25] S. Samuel, Nucl. Phys. B 308 (1988) 285. [26] S. Samuel, Nucl. Phys. B 308 (1988) 317. [27] C.S. Hsue, B. Sakita and M.A. Virasoro, Phys. Rev. D 2 (1971) 2857; J.-L. Gervais and B. Sakita, Phys. Rev. D 4 (1971) 2291; Nucl. Phys. B 34 (1971) 477; Phys. Rev. Lett. 30 (1973) 716. [28] L. Brink, D.I. Olive and J. Scherk, Nucl. Phys. B 61 ( 1973 ) 173. [29] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B 241 (1984) 333. [30] V.S. Dotsenko and V. Fateev, Nucl. Phys. B 240 (1984) 312. [ 31 ] D. Friedan, Z. Qiu and S. Shenker, in: Vertex operators in mathematics and physics, eds. J. Lepowsky, S. Mandelstam and I.M. Singer (Springer, Berlin, 1984) p. 419. [ 32 ] V.G. Knizhnik and A.B. Zamolodchikov, Nucl. Phys. B 247 (1984) 83. [ 33 ] M.A. Bershadsky, V.G. Knizhnik and M.G. Teitelman, Phys. Lett. B 151 (1985) 31. [ 34 ] D. Friedan, Z. Qiu and S. Shenker, Phys. Lett. B 151 ( 1985 ) 37. [ 35 ] V.G. Knizhnik, Phys. Lett. B 160 ( 1985 ) 403. [36] D. Friedan, E. Martinec and S. Shenker, Phys. Lett. B 160 ( 1985 ) 55; Nucl. Phys. B 271 (1986) 93. [37] J. Cohen, D. Friedan, Z. Qiu and S. Shenker, Nucl. Phys. B 278 (1986) 577. [38] V.G. Knizhnik, Commun. Math. Phys. 112 (1987) 567. [39] S. Mandelstam, Nucl. Phys. B 64 (1973) 205; B 69 (1974) 77. [40 ] E. Cremmer and J.L. Gervais, Nucl. Phys. B 76 (1974) 209; B 90 (1975) 410. [41 ] M.B. Green and J.H. Schwarz, Nucl. Phys. B 218 (1983) 43;B 243 (1983) 475; L. Brink, M.B. Green and J.H. Schwarz, Nucl. Phys. B 219 (1983) 437. [42] H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Phys. Rev. D 34 (1986) 2360. 437
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[43] K. Itoh, K. Ogawa and K. Suehiro, BRS invariance of Witten's type vertex, Kyoto University preprint KUNS 846HE(TH) 86/06. [44] H. Kunitomo and K. Suehiro, A new method of constructing gauge string field theory, Kyoto University preprint, KUNS 855-HE(TH) 87/02 (January 1987). [45] T. Kugo, H. Kunitomo and K. Suehiro, BRS invariance of string vertex on general ghost vacuum, Kyoto University preprint, KUNS 855-HE(TH) 87/06 (May 1987). [46] E. Martinec, Nucl. Phys. B 281 (1986) 157. [47] G.T. Horowitz and S.P. Martin, Nucl. Phys. B 296 (1988) 220. [48] S.P. Martin, Conformal field theory representation and background independent action for superstring field theory, University of Santa Barbara preprint (December 1987 ). [49] S. Giddings, E. Martinec and E. Witten, Phys. Lett. B 176 (1986) 362. [50] A. Strominger, Phys. Lett. B 187 (1987) 295; Z. Qiu and A. Strominger, Phys. Rev. D 36 (1987) 1794. [51 ] A. LeClair, M.E. Peskin and C.R. Preitschopf, String field theory on the conformal plane. I. Kinematical principles, SLAC preprint SLAC-PUB-4306 (January 1988). [52] A. LeClair, M.E. Peskin and C.R. Preitschopf, String field theory on the conformal plane. II. Generalized gluing, SLAC preprint SLAC-PUB-4307 (January 1988).
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[53] A. Hosoya and H. Itoyama, The vertex as a Bogoliubovtransformed vacuum state in string field theory, Fermilab preprint FERMILAB-PUB-87/111-T OUHET-112 (January 1988). [ 54 ] J.H. Sloan, The scattering amplitude for four off-shell tachyons from functional integrals, Princeton University preprint PUPT-1075 (November 1987). [551 O. Lechtenfeld and S. Samuel, Nucl. Phys. B 308 (1988) 361. [56] V.A. Kosteleck~, and S. Samuel, Phys. Lett. B 207 (1988) 169. [571J.P. Yamron, Phys. Lett. B 187 (1987) 67. [ 58 ] W. Siegel, Phys. Lett. B 142 (1984) 276; B 151 ( 1985 ) 391, 396. [59] V.N. Gribov, Nucl. Phys. B 139 (1978) 1. [60] J.H. Schwarz, CalTech preprint CALT-68-1304 (October 1985). [ 61 ] N. Seiberg and E. Witten, Nucl. Phys. B 276 (1986) 272. [62] Y. Cai and A. He, Scattering amplitude for Witten's open superstring field theory, University of Texas preprint UTTG18-87 (1987). [63] C. Thorn, Nucl. Phys. B 287 (1987) 61. [64] T. Kugo, Gauge string field theory, Kyoto University preprint KUNS 865-HE(TH) 87/07 (1987). [ 65 ] M. Bochicchio, Princeton University preprint PUPT-1028 (1986).
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G A U G E - I N V A R I A N T M O D I F I C A T I O N OF W I T T E N ' S O P E N S U P E R S T R I N G Olaf L E C H T E N F E L D and Stuart S A M U E L Physics Department, City Collegeof New York, New York, NY 10031, USA Received 18 May 1988
A violation of gauge invariance in Witten's open superstring field theory has recently been discovered. We propose a modification which resolves this problem by avoiding the collision of picture-changing operators. Feynman rules for the modified, gaugeinvariant theory are presented, including a useful representation for the Ramond propagator. All four-point amplitudes are offshell finite, agree on-shell with the Polyakov approach and exhibit decoupling of spurious physical states.
1. Introduction A non-perturbative approach to string theory [ 1,2 ] might be necessary to make contact with the four-dimensional world [3-5 ]. One way of extracting nonperturbative physics is through field theory. A covariant field theory for the open bosonic string was achieved by Witten in ref. [ 6 ]. This theory has been the subject of much research [ 7 - 2 0 ] . An extension to the open superstring has also been proposed by Witten [ 21 ]. However, recent work by Wendt [22 ] has revealed an associativity anomaly in its superstar product. It leads to an order g2 violation of gauge invariance and an infinite four-vector amplitude. The difficulty is traced to colliding picturechanging operators [ 22 ]. The superstar product of two Neveu-Schwarz fields involves the picturechanging operator, X ( z ) = ~ { Q, ~(z) }, at the string midpoint ~l. When there are two or more star operators, two or more X are present. Since X ( z ) X ( w ) blows up like ( z - w ) - 2 as z--, w, a singularity is generated that renders undefined multiple applications of superstar products. In ref. [ 22 ], the theory was regularized and a four-point contact term was added to the action to cancel the singularity on-shell. The offshell counterterm and modification to the gauge transformation law was obtained in ref. [23 ]. While this repairs the gauge violations to order g2 and yields The xf8 factor is a convention used to assure that the Ramond kinetic operator has the standard normalization [ 20 ]. 0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
finite four-point amplitudes, gauge violations at order g3 and infinities in five- and higher-point amplitudes most likely require higher-point counterterms in the action as well as additional modifications to the gauge transformation laws. An order by order patching up of Witten's proposal is arduous. The first and main goal of this letter is to overcome the above-mentioned difficulties by modifying Witten's theory [ 21 ] so that colliding X are avoided. An action, bilinear and trilinear in string fields, is discovered that is gauge-invariant and non-singular. The second purpose is to present the F e y n m a n rules for the gauge-fixed theory. In particular, a useful representation of the R a m o n d propagator is obtained. Finally, off-shell finiteness, on-shell agreement with first-quantized results and the decoupling of spurious states from physical amplitudes is demonstrated for the four-point amplitudes. Our notation and conventions are given in refs. [23-26, 20 ] with one exception: the coupling g in this paper corresponds to g in ref. [23 ]. Relevant papers for the present work are refs. [ 6,7,21-23,25,26 ]. We also rely on conformal field theory methods [27-38 ] as applied to string field theory [8,39-53] and offshell computations [25,26,54-56 ].
2. The modified theory Open string operators, O(a, z), are expressible in terms of left- and right-moving fields, O ( z ) and O(g), 431
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where z = e x p ( r + i a ) and g = e x p ( z - i a ) . The string strip, - ~ < z < + ~ and 0 < a < n, in tr and z space is the upper half-plane in the variable z and the lower half-plane in the variable g. The string midpoint, z = 0 and a=n/2, is z = i and g = - i . The union of the upper and lower half-planes forms the double cover of the open string configuration. Although left- and rightmoving fields are related by the open string boundary conditions, we may work on the full complex plane by using analytic fields only. This allows us to put midpoint insertions at z = i as well as at z = - i . The superstring star product and integral are obtained by inserting picture-changing and inverse picture-changing operators, xf8 Y(z) = c0~ × e x p ( - 2 q ~ ) (z), at the string midpoint [21 ]. Starting with the Witten action and transformation law [ 21 ], we considered all possible insertions of X(i), X ( - i ) , Y(i) and Y ( - i ) consistent with ghost number assignments and found that the following action,
case. Eq. (1) is rigorously invariant under eq. (2) and hence requires no counterterms. Although eq. (1) differs only slightly from the Witten proposal [21 ], it is a different theory. As pointed out in ref. [ 57 ], the free Ramond action couples fields of different mass levels because of the presence of Y. Our modified theory has the same feature in the Neveu-Schwarz sector; all levels couple at g = 0. As a result, off-shell amplitudes are not equal to those obtained from the Witten proposal. In particular, section 5 shows the finiteness of the four-boson amplitude. Under the assumption that the action is at most trilinear in string fields and that the structure of the gauge transformation is that ofeq. (2), eq. ( 1 ) is the unique gauge-invariant action except for the trivial reflection i ~ - i. The modified action is invariant under the following supersymmetry transformation:
S= f ~PNs*X(i)Y(--i)Q~PNS+ ~ R * Y(--i)Q~R
5~ys = X ( - i )
+ ]g~X(i)~VNS* ~VNS*~VNS+ 2gJ ~VNS*~V., ~R, (1) is invariant under the gauge transformation ~ ~r-tNS= QANs
+gX(-i)
[ ( ~UNS*ANs --ANs * 7tNs)
+ Y(i) ( ~'/R*At( --AR * ~T'/R) ] ,
5 ~R =
(3)
where ea is an anticommuting parameter. [e. S(_ 1/2)1o is the integral fermion emission operator at zero momentum with ea as the spinor wave function: [e.S(_l/2)]o=Vf(_l/2)(ua=e&, k = 0 ) . There is at least one other version of the supersymmetry transformations which uses the fermion emission operator in the + 1/2 picture:
8 ~'gR= V-~+ 1/2)(uot =edt, k = 0 ) ~NS,
[ ( ~[/R*ANs --ANs * ~JR)
"JI-( ~JNS*AR --AR * ~UNS) ].
(2)
Here, • and f denote the ordinary star product and string integral [ 6 ]. The original Witten proposal [ 21 ] is obtained by replacing Y ( - i ) by Y(i) in eq. ( 1 ) and replacing X ( - i ) by X(i) in eq. (2). In verifying the gauge invariance, no two X n o r two Y are multiplied at the same point. Using the proofs of the axioms in refs. [9,13,43,45], invariance to O ( g °) and O ( g ~) is easily shown. O ( g 2) invariance depends on the associativity of • in the absence of and in the presence of X ( i ) X ( - i ) . The former has been proven in refs. [25,55]. Associativity with a single X(i) present was proven in section 3 of ref. [23] but the same proof applies to the X ( i ) X ( - i ) 432
~ ~'/R= X ( - - i ) [e'S(- t/2) ]o ~NS,
8~NS = Y(i) V(+ l / 2 ) J (Uot = er/, k = 0) ~-JR,
QAR
+gX(-i)
[e'S(_,/2)]o Y(i) ~JR,
(4)
where the integral vertex is given in eq. (IV. lg) of ref. [ 24 ].
3. Gauge fixing and propagators To commence perturbation theory, the gauge must be fixed. For the bosonic string and the superstring field theories, a convenient gauge is the SiegelFeynman one [58]: bo ~u=0,
(5)
where bo is the zero mode of the antighost, b(z). This gauge is achieved in the linearized theory, i.e. g = 0, by setting A = - Lff ' bo ~. In the absence of a Gribov
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ambiguity [ 59 ], this gauge should be possible in the interacting case. There have been suggestions [60,21 ] that an additional gauge constraint, flo ~ = 0, can be imposed in the Ramond sector of the superstring. Actually, not enough freedom remains in A to do this. For residual gauge transformations, i.e. those preserving bo ~v= 0, boQA must be zero. This yields A = QLfflboA which implies ~A~V=0, hence, the gauge is already fixed completely. There are two other reasons for believing that eq. (5) suffices for the Ramond sector. Firstly, a propagator exists (see eq. (8) below). Secondly, the dynamics at one loop are likely to be incorrect if, in addition, flo ~v= 0. In the light-cone analysis, the odd spin structure [ 61 ] has eight ~ zero modes, which yields a fourth order zero in its partition function. In the covariant formalism, there are ten zero modes for Vu; however, the 7o-flo system provides a zero in the denominator. If the constraint flo ~v= 0 is imposed, the fifth zero in the odd spin structure 9,~ partition function is no longer cancelled by the one from the 7o-flo system. This would lead to deviations from light-cone results in higher-point functions. For bo ~v= 0, the quadratic part of the action in eq. ( 1 ) becomes Squadratic = j ~VNS* n *X(i ) Y( - i ) Q~ ~VNS +f~R*~ZtY(--i)Q~R
,
(6)
where n=boco projects onto the space b o ~ = 0 and ~* = cobo projects onto Co~ = 0. They satisfy me= 7r, ~t;~t = ~ t and n + x*= 1. The kinetic operators, A, in eq. (6) are
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sector and B - 1= X( - i) for the Ramond sector. The propagator in eq. (8) is, in some sense, not unique. This is not due to the gauge-fixing procedure but to the singular nature of X ( z ) and Y ( z ) in the first-quantized theory. Although Y ( z ) and X ( z ) are mutual inverses, i.e. Y ( z ) X ( z ) = 1, there exist O ( z ) and O' (z) such that O ( z ) X ( z ) = 0 and O' ( z ) Y ( z ) = 0 (there is no contradiction here; associativity of operator products at the same point need not hold #2 ). Therefore, X(Y) is an inverse of Y ( X ) only modulo shifts with O ( O ' ) . However, restriction to a certain class of operators fixes this ambiguity. Among those restrictions are ghost number, conformal behavior and dimension, BRST-invariance, locality etc. The ambiguity reflects itself in eq. (8). If B - 1 is shifted by an element in the kernel of B, P still inverts A of eq. (7). We believe that the same constraints needed to uniquely fix X ( z ) and Y ( z ) must also be imposed on B - 1. In this sense, the propagator is unique. One positive feature is that the propagator in the Ramond sector is built from statistically correct operators [24], that is, operators which map GSO accepted states into GSO accepted states. The Ramond propagator used in refs. [ 22,62 ] is the product of two statistically incorrect operators, one of which is Fo. The computation of an off-shell amplitude requires deforming Fo onto external states, thereby relating the amplitude to one involving GSO-rejected states. For this reason the off-shell two-vector two-spinor amplitude in Witten's theory was not calculated in ref. [ 23 ]. With the Ramond propagator in eq. (8), there is no difficulty with statistically incorrect operators.
4. Feynman rules A=coboBQboco,
(7)
where B = Y ( - i ) in the Ramond sector and B = X(i) Y( - i) in the Neveu-Schwarz sector, the order of X, Y and Q being immaterial since they mutually commute. A propagator, P, is required to invert A, i.e. PA=Tz andAP=TC. It is easily seen that bo bo bo B - 1 , P = -~o B - I a - - = Lo Lo bo - Q ~oo B - I -bo = B - I -~o where B - ~ = X ( - i ) Y ( i )
bo B - ' b o Q Lo Lo
Refs. [6-8,21,63] have explained how secondquantized Feynman graphs are related to first-quantized world-sheet computations. For the bosonic theory, external states and propagators are represented by rectangular strips of width n. External states are of semi-infinite length and internal propagators are of length T, where T is to be integrated from 0 to oo. In addition, the following line integral, bo= ¢(dco/2ni)ogb(og), is taken across the world sheet
(8) for the Neveu-Schwarz
,2 Limit.... s [limit:, .:O I(z t )02(z2)O3(z3) ] may differ from limit:, .:, [limit:: .: 0, ( z I )02( z2 )Os ( z3 ) ]. 433
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(and its double). These results follow because the propagator in the bosonic theory is boL ff i = bo f ~ d T e x p ( - TLo), and exp ( - TLo) creates a world-sheet strip of length T. External and internal strings are thrice-wise joined together via the Witten method: The ends of each of the three strips are divided in two at zt/2 and the first half of one end is joined to the second half of the next. Since the interactions in eq. ( 1 ) are the same as in Witten's theory, the vertices of the Feynman graphs are the same. The R - R - N S vertex contains no picture changing operators, and the N S - N S - N S vertex has an X at the midpoint. The propagators, however, are different. Since they involve two Lff I (see eq. ( 8 ) ) , two strips of length TI and /'2 are used. The lengths, TI and T2, are independently integrated from 0 to oo. The operators bo and Q=~(dm/2ni)jaasT(tO) in P are expressed as line integrals across one of the strips. Picture-changing operators are inserted at a = r t / 2 between the two strips. Fig. la displays the propagator. The terms ( bo/ Lo )B- l and B-~ ( bo/ Lo ) in eq. (8) involve a single strip with insertions at one end as shown in fig. lb. It is straightforward to obtain the off-shell fourpoint amplitudes using the Giddings map [7 ] for T=TI+T2 and the methods in refs. [23,26]. The change of variables from ( TL, T2) to ( T, Tl ) must be made. It is not clear whether amplitudes are expressible in terms of elliptic functions as is the case for the Witten proposal [23 ] because of the extra variable Tl.
5. On-shell agreement and decoupling of spurious states Since the trilinear action in eq. ( 1 ) is identical to the Witten case, the trilinear couplings [ 18-20 ] are the same. It follows that the three-point functions agree with the standard first-quantized theory for onshell physical states and that spurious states decouple. These results are shown for Witten's theory in ref. [23]. On-shell agreement and the decoupling of spurious states can be demonstrated for the four-point amplitudes via picture-changing methods [ 22,34 ]. We use the techniques in section 5 ofref. [23 ]. Let us discuss the four-fermion case in detail. 434
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T~
(
T2
)<
do~ j (o~) ~ 2~xi BRST
~dto ob(o) 2~i
i~
)
cob(0~)
a T
<
>
~dco rob(o) 2/tl
II
.........................................................................................
b Fig. I. (a) The configuration generated by the propagator. For the Neveu-Schwarz case, there is a Yat • and an X a t m. For the R a m o n d case, there is an X at m. (b) The configuration generated by the asymmetric representation of the propagator in eq.
(8). There are six Feynman graphs that contribute to the amplitude [25 ]. Two of them are shown in fig. 2. First consider on-shell agreement. The second term in the asymmetric representation of the propagator
\ / ............../\.....
a
b
Fig. 2. Feynman graphs for four-fermion scattering. The solid lines denote fermions and the dotted lines denote bosons.
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(eq. (8) ) may be dropped for BRST-invariant external states. The propagator is then represented as in fig. lb and the four-fermion amplitude from fig. 2a becomes
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A 4F = + 4 g 2 tr (,~1,~2,~,3~.4)
oo
x(fdT
Aa4F = - 4 g 2 tr (~,t)],2~.3,~.4) ~ d T < V~((nl ) V2R((n2)
0 )< V3R((n3)bo
V4 (o)4)X(o)~) Y((no) 7,
(9)
where (no is the midpoint at one of the two vertices in the string configuration (see ref. [23] ) and (n3 is the midpoint in its copy (we work on the double). In eq. (9), V ~ ((ni) represent any BRST-invariant Ramond vertex operators. To establish the on-shell agreement, use picturechanging manipulations to move X at (n3 to (no. Go to the bigger algebra by inserting ~((n), write X((n3) =x/~{Q, ~((n3)} and commute or anticommute Q past other operators until it annihilates the left and right vacua. Two contributions are generated, one from ~((n) and one from bo: A a4F = -- 4g 2 tr (21222324 )
-~'8< v~((nl) vU(n2) v~((n3) v~ ((n4) X~((n~)Y((no)~((n)>big) ,
(11)
where again the limit (n-'(n0 is to be taken. The first term in eq. ( 11 ) completes the Koba-Nielsen integration region from 0 to ~ [ 7 ]. The second term cancels the second term in eq. (10) via the methods in refs. [23,25,55]. Now consider the decoupling of a spurious state. Suppose the fourth state is created by a trivial BRSTinvariant operator, Vl~ = {Q, ~4 } and the other states are BRST-invariant. Pushing Q past the other operators in diagrams 2a and 2b yields
oo
X(ffdZ < VI((nl )V2((n2)~'r3((n3)boV4 ((n4) 0
A~F +A 4F = + 4 g 2 tr (,~1~,2,~,3,~,4)
X [ < VI(o)I ) V2 (0.)2)V3(0)3) V4(0)4)
× ~((n~)Y((no)X((n) >big
XX(co~) Y((no) > T=°
- < v~((n,) v~((n~) P~((n,) vh((n+)
)< ~((n~) Y((no)~((n) >big).
(10)
Taking the limit (n-'(no, Y((no) and X((n) cancel in the first term in eq. (10). The zero mode of ~ must be soaked up by ~((n]) and what remains agrees [7,22,23 ] with the first-quantized approach except that the Koba-Nielsen integration range is from ½to 1. The second term involves the "collapsed configuration", i.e. T = 0. Performing similar manipulations on fig. 2b yields
xx(~)Y((no) >~=°1,
(12)
which is zero because associativity holds in the presence of the two picture-changing operators. The two-boson-two-fermion amplitude proceeds similarly. Two of the six Feynman graphs are shown in figs. 3a and 3b. On-shell agreement and spurious state decoupling occur if the "collapsed" diagrams cancel. Since fig. 3a includes X((n~)Y((no) from the propagator and X((no) for N S - N S - N S vertex, the diagram contains X((n]). Fig. 3b has X((n3 ) from the Ramond propagator. Hence, the two diagrams contain the same picture-changing operator and cancellation occurs. The four-boson diagrams are shown in figs. 4a and 4b. The diagrams involve X((n3)Y((no) from the propagator and X((no) and X((n~) from the vertices 435
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''"-...
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(
/
a
without adding higher-point terms to the action. By changing the quadratic action and gauge transformation law, colliding picture-changing operators are avoided. The four-boson amplitude is finite both onshell and off-shell. Let us examine the axioms of the modified theory. A superstar product can be defined by
b
(A*B)Ns = X ( - i )
Fig. 3. Feynman graphs for two-fermion-two-bosonscattering.
.., ... •... "%.....
../Y'"
....y""
%'......
•.,... . ..-
a
....
..
. ..../"
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"'.......
b
[ANs *BNs + Y(i)AR *BR],
( A * B ) R = X ( - - i ) ( A N s * B R +A~*BNs),
(13)
so that the gauge transformation in eq. (2) is 5 ~ = QA +g( ~ * A - A * ~). Since associativity of the superstar product is still violated due to colliding picture-changing operators, the geometric interpretation of the action is unclear. Formally, it is a ChernSimons form if the superstring integral reads 55="fX(i) Y ( - i ) Y ( - i ) " , where the second Y ( - i ) is to be removed by the X( - i) in *; this makes sense because the string integral always appears together with a star product. Alternatively, replace the integral-star combination by a dot product [42,64 ]:
Fig. 4. Feynman graphs for four-bosonscattering.
A.B=fY(--i)[X(i)ANs.BNs+AR.BR]. in both cases. Here, o95 is the location of the other vertex. In total, X(og~)X(oJo) appears in both diagrams and cancellation takes place. In conclusion, we have demonstrated agreement with the first-quantized approach for on-shell physical states and decoupling of spurious states for all physical four-point amplitudes. The off-shell four-fermion and two-boson-two-fermion amplitudes are finite because no two X collide. For the four-boson amplitude, the two X at the two N S - N S - N S vertices approach each other only when T~ and T2, the lengths of the two propagator strips, both go to zero. The measure dTl and dT2 as well as the presence of Yand the two bo in the Ramond propagator are sufficient to render the integral convergent in this region. Along the same lines, higher-point amplitudes are better behaved in moduli space than for Witten's theory, and are expected to be finite.
6. Conclusion In this letter, we have modified the Witten superstring field theory so as to make it gauge invariant 436
(14)
The action in eq. ( 1 ) becomes ~U.Q~U+2g~U.~*~u. The axioms that guarantee the gauge invariance of the theory are (a) Q 2= 0 (nilpotency of Q), (b) ( Q A ) ' B = - ( - 1 ) a A ' Q B (integration by parts), (c) Q(A*B ) = ( QA )*B + ( - 1 )aA*QB (gradedderivation law), (d) A . B = ( - 1 )abn'.4 (graded commutivity of dot product), (e) A. ( B ' C ) = ( - 1 )a~t'+C)B" ( C*A) (cyclicity under integral), ( f ) ( A ' B ) . ( C ' D ) = ( - 1 )a~,+c+d)(B'C)" ( D ' A ) (cyclicity under integral). With the identification A.B=~A*B, ( a ) - ( d ) are the same as in Witten's theory. Associativity is replaced by the weaker axioms (e) and (f). Some work still remains. (i) When two successive gauge transformations are performed, two picturechanging Xare multiplied at the same point as can be seen from eq. (13). What the gauge algebra closes onto is unclear. This problem is a remnant of the nonassociativity of the * product. (ii) Although it is
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s t r a i g h t f o r w a r d to o b t a i n the off-shell f o u r - p o i n t a m p l i t u d e s using t h e m e t h o d s in ref. [23 ], an e x p l i c i t c o m p u t a t i o n is w o r t h doing. ( i i i ) T h e Y i n s e r t i o n in the p r o p a g a t o r e l i m i n a t e s the c o l l i s i o n o f t h e t w o X c o m i n g f r o m t h e t w o v e r t i c e s in the f o u r - b o s o n a m plitude. V i a this m e c h a n i s m , the b o s o n i c f o u r - p o i n t a m p l i t u d e s are finite. It is i n t e r e s t i n g to a n a l y z e h o w this w o r k s in h i g h e r - p o i n t a m p l i t u d e s a n d to settle t h e q u e s t i o n o f tree-level finiteness. ( i v ) Finally, the B R S T q u a n t i z a t i o n p r o c e d u r e to gauge fix t h e seco n d - q u a n t i z e d superstring n e e d s to be c a r r i e d out. T h e steps s h o u l d be s i m i l a r to t h o s e o f the b o s o n i c string field t h e o r y [ 6 2 , 6 5 ] .
Acknowledgement T h i s w o r k was s u p p o r t e d in part by the N a t i o n a l Science F o u n d a t i o n g r a n t N S F - P H Y - 8 2 - 1 5 3 6 4 a n d by the D e p a r t m e n t o f Energy g r a n t D E - A C 0 2 83ER40107.
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