The symmetric expressions for the open bosonic string amplitudes and one-loop unitarity

Volume 240, number 1,2 PHYSICS LETTERS B 19 April 1990

THE SYMMETRIC EXPRESSIONS FOR THE OPEN BOSONIC STRING AMPLITUDES A N D O N E - L O O P UNITARITY

Tsunehiro KOBAYASHI and Takashi S U Z U K I Institute of Physics, University of Tsukuba, Ibaraki 305, Japan

Received 18 December 1989

We propose a unified propagator defined by the sum of the twisted and the untwisted propagators which are introduced in the BRST invariant operator formalism for the open bosonic string theory. We apply the propagator to evaluate N-point tree amplitudes and one-loop amplitudes. It is shown that the tree amplitudes are represented by the symmetric sum of all inequivalent amplitudes, and that the one-loop amplitudes which are uniquely derived as the sum of all inequivalent one-loop amplitudes satisfy unitarity.

In the open bosonic string theory based on the BRST invar iant opera tor formalism, four pr imi t ive operators, that is, the one-loop p lanar tadpole [ 1,2], the non-or ientable tadpole [2] , the one-loop non- planar self-energy [ 3,4 ] and the two-loop non-planar tadpole [4] , have been constructed. Konno and the authors have shown that all these pr imi t ive operators are constructed from the 3-string vertex ( V~jkl, the twisted (T) and the untwisted (U) propagators and the reflection opera tor IRij) in a manifest ly dual t reatment . They enable us to construct general mul- t i loop ampli tudes . Indeed Konno and one of the authors (T.S.) have comple ted the calculus for the general mul t i loop ampl i tudes in terms of the above pr imi t ive structure and the secondary structure of open bosonic string theory [5] . But it is still un- known how we sum up the different d iagrams in order to satisfy unitari ty. It is, in principle, decided by string field theory. Although we cannot determine the entire weights among the different loop order diagrams because of the lack of a satisfactory string field theory, we will be able to fix the weights in the same loop order by imposing per turbat ive unitari ty.

In this paper we present a s imple expression for N- point tree ampl i tudes and vertices which are writ ten in terms of the symmetr ic sum of all possible inequivalent ones. We will show that this symmetr ic N- point vertex is powerful for evaluat ing loop ampl i - tudes. Explicit evaluat ions and discussions about

uni tar i ty are per formed at the one-loop level. Throughout these discussions we use a unif ied propagator defined by the sum of the twisted and untwisted propagators .

In ref. [3] , the untwisted propagator U for non- or ientable diagrams has been constructed in order to satisfy off-shell duali ty and the correct result of the one-loop non-orientable ampl i tude has been ob- tained. The twisted propagator T for the orientable d iagram and the untwisted propagator U are given as follows:

1

f :~ (x) ( 1 ) dx

T=(bo-b , ) x ( 1 - x ~ 0

with

i~ (x) = x~-O( 1 - x ) L°-L~ , I

U = - ( b o - b l ) x ( 1 - x ~ 0

1

= - ( b o - b ~ ) x ( i ~ x ) " \ 2 x - l J ' (2) 0

with

~ = g 2 e x p ( - L l) ,

where the choice of an arbi t rary phase convent ion for U is different from that chosen in ref. [ 3 ]. When we

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use not only the twisted propagator but also the untwisted propagator in the valuation of N-point tree amplitudes, an attractive feature appears.

In order to show this feature, we make a change of the integration variable of U: x / 2 x - 1 --+y. According to this change the integrant of U is equal to that of 7", but the integration region changes as _ f 6 _ ~ j - o + fT- Taking account of these changes, we see that the sum of U and T is represented by a unified form

i ,# (x) (3) cLv

D = (bo - b , ) x( 1 -x--~ " oo

When we construct the Veneziano amplitude by us- ing D instead of T, we can easily obtain

B~4)( PI, Pc, P3, P4) = < V12E IDU ( VF34 IRuF )

= ctvlxl-P'P~I 1--Xl PIP4 (4)

We divide the integration region [ - oo, ~ ] into three regions (i) [ - o c , 01, (ii) [0, 11, (iii) [1, ool, and change the integration variables of (i) and (iii) so that each integration region may be the ordinary region [0, 1 ]. Then we get

(i) i ¢tv Ixl-f~V2ll -xl--PIP4 oo

1

= ~ d ) ~ 2 - P I P 2 ( 1 - - 2 ) -PIP3 ( 5 )

0

where 2 = x / ( x - 1 ).

(iii) i ch" Ixl P'P~II-xI-P'P~ -c-~c

1

= t d ~ X - P I P 3 ( 1 - - X ) -PIP4 , (6)

0

where 2 = l / x . Summing ( i ) - ( i i i ) , we obtain the symmetric form of the 4-tachyon amplitude as

B ( 4 ) ( s , t , u ) = A ( s , u ) + A ( s , t ) + A ( t , u ) . (7)

This relation is depicted in fig. 1, where the doubled internal line stands for the propagator/) . In eq. (7), the A (...) are the usual Veneziano amplitudes and the variables are defined by s = (P~ +/92) 2, t = (/91 +/93 ) 2,

2 3 2. 32 & 3 2

1 1 1 3 1

Fig. 1. The symmetric expression for 4-tachyon diagrams, where the doubled line represents the unified propagator D.

b/.-~-(el +P4) 2. Since A(x, y) has a symmetry under the change of variables x and y, B (4) (S, [, b/) has complete symmetries under the changes of s, t, u variables, that is, it satisfies crossing symmetries due to the Bose statistics of tachyons. Then we call the amplitude (7) "the complete 4-tachyon amplitude". It is worthwhile to note that the essential point in ob- taining the complete 4-tachyon amplitude is the fact that we use -Q as the twist operator in U. If we use the ordinary twist operator.(2 instead of-Q, we obtain A (s, t) +A (s, u) only. It has no such complete symmetries.

We can immediately extend this complete form to the 5-point case. In this case, we have 12 independent terms.

B ( s ) ( P I , P2, P3, P4, P s ) = JJ dxdy [xl-P,e2ly[-P,P5 -,ze

× [1 -xl--P2P3 [ 1 -3'1 1'3P4 [ 1 -xy[-P2P4

= ( 1 2 3 4 5 ) + (12354) + (12435) + (12453)

+ ( 1 3 2 4 5 ) + ( 1 3 2 5 4 ) + ( 1 3 4 2 5 ) + ( 1 3 4 5 2 )

+ ( 1 4 2 3 5 ) + ( 1 4 2 5 3 ) + ( 1 4 3 2 5 ) + ( 1 4 3 4 5 2 ) , (8)

where we use the notation

1

(abcde)= f f d~ dff O(1-2f])fc-PaPb~2 -PaPe o

× ( 1 - 2 ) -pbrc( 1 - f f)-P'Pd( 1 - 2 f ) PbPd

for the ordinary 5-point tree amplitude. The changes of the variables (x, y) to (2, 9) are exactly deter- mined by the conformal transformations h i ( i = 1, ..., 12) which transform the Koba-Nielsen variables Zr ( r = 1 ..... 5 ) in the complete form to Zf = h i (Zr) in each ordering of the RHS of the last equality in (8). By means of the transformations h i the Lovelace maps V,. are also transformed into V~=h. Vr. Thus the relation is extended to the off-shell amplitude, which is called the complete vertex hereafter. The x - y plane is divided into the 12 regions shown in fig. 2, each of which represents the region for one 5-point tree am-

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12435 14352

13452

14253 ~

' ~ ~ 14235

\~ "~.,. 14325

123451~13425 1113245 ~ x

12354 13254

Fig. 2. The division of the full x - y plane, i.e. the integration region of the complete 5-point tree amplitude, into those of 12 spe- cific ordering 5-point amplitudes.

plitude corresponding to one of the 12 orderings given in (8).

This preferred feature is generalized to arbitrary N- point tree diagrams. Namely, the N-point tree diagram constructed from N - 2 3-point vertices and N - 3 unified propagators includes ( N - 1 )!/2 independent noncyclic N-point amplitudes in a symmetric sum. We can call this symmetric form of the N- point amplitude the complete N-point amplitude or the complete N-point vertex.

Let us go to the discussion of loop diagrams. We will present here two simple examples at one-loop level, that is, the self energy type and the 4-point amplitude with one loop.

In order to construct the one-loop self energy, we sew the second and the third legs of the complete 4- point vertex with a propagator/5 (fig. 3). /5 represents the sum of the ordinary twisted and untwisted propagators (in fig. 3,/5 is depicted by a wavy line) as

1

~ ( y ) (1 + (2 ) . (9) dy

/5= ( b o - b , ) y ( i -~y) o

It is important to note that we must not use the uni-

fled propagator D instead of/5 in sewing the legs for the construction of loops. This is understood as follows: in the process constructing loops from the vertex, we have to sew the legs by a propagator which has poles at the points where the square of the mo- mentum of the combined leg is on the mass-shell. As was shown in fig. 1, the sewing process by D produces not only A (s, t) +A (s, u) having poles in the s-channel but also the extra term A (t, u) with no poles in the s-channel. It is also noticed that the second term in fig. 3 corresponding to the region - ~ ~< x~< 0 has two contributions if the internal x-propagator is described by an ordinary untwisted propagator with the twist operator £2 as shown in fig. 4. Namely, the integration region - ~ 4 x ~< 0 is divided into - 1 ~< x ~< 0 and - ~ ~< x ~< - 1 corresponding to the first and the second terms in fig. 4, respectively. After sewing the second and third legs, the diagrams coming from the first and third terms of fig. 3 give the same contributions, and also two contributions from the right-hand side of fig. 4 are the same. Thus we get the full one- loop self energy as shown in fig. 5. The first line of fig. 5 is coming from the region 0~<x~< 1 and the last line is coming from the region - 1 ~< x~< 0.

The complete form of 4-point amplitudes with one loop is constructed from the 6-point complete vertex including 60 independent nonclyclic ordering vertices by sewing the a and b legs with/5. In the 6-point case, these 60 vertices are divided into the following 3 types; 24 of them are the planar type of fig. 6a. 24 other vertices are represented by the form of the left- hand side of fig. 6b and each of them is written by four twisted vertices given on the right-hand side of

x<0 1 .l~XSO 4 1 x~-I 4

Fig. 4. Diagrammatic description for the separation of the con- tribution coming from the integration region - ~ ~< x ~ 0 into two contributions corresponding to two different twisted diagrams.

1 4 1 L 1 3 1 4

Fig. 3. The construction of the one-loop self-energy from the complete 4-point vertex, where the wavy line represents the propagator D.

-

Fig. 5. The one-loop self energy diagrams.

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(a) i~..~ a b

i j k b i i k I [ i } k i i I k i I j k

~'l o b a b o b o b

i j b k i j [ k I k i j [ i k j

IClo _<,: ) , . . , " / . . . . . ~b b i I j k I i ) k i [ k j

b a

Fig. 6. The three types of 6-point vertices. (a) diagrams which have no twist; (b) twisted diagrams described by 4 different diagrams; (c) twisted diagrams described by 6 different diagrams.

fig. 6b. The last 12 vertices are in the form of the left- hand side of fig. 6c, which include six twisted vertices shown on the right-hand side of fig. 6c. The situa- tions pointed out in figs. 6b and 6c are the same one shown in fig. 4 for the self energy. Thus we have 1 9 2 = 2 4 + 2 4 × 4 + 12×6 vertices. Sewing the a and b legs with/5, we have 384 = 192 × 2 diagrams. They are reduced to 3 × 24= 48 diagrams, each of which has equal weights 8 (fig. 7 ). The meaning of 48 is written down as follows; the factor 3 corresponds to the number of non-cyclic permutations of four external particles, and 2 4 corresponds to the freedom for the four internal propagators to be either twisted or untwisted.

At this stage, we can discuss the one-loop unitarity, which was discussed in ref. [6]. Let us consider the two-body cut of the complete 4-point one-loop amplitude corresponding to the incoming particles, 1 and 2, and the outgoing ones, 3 and 4, as shown in fig. 8. Note that the last term on the right-hand side of fig. 7 does not contribute to the imaginary part of this two-body cut. Such a cut is represented by the multi- ple of two 4-point tree amplitudes as shown in the second equality of fig. 8 where the factor ½ arise from

2 3 2 32 43 2

1 4 1 41 31 4

Fig. 7. The 4-point diagrams with one loop constructed from the complete 6-point vertex.

2 3

{- 1 4

2 3

2 {P'qJ1 q 4

Fig. 8. The two-body cut for the one-loop 4-point diagram.

the change of the propagator/5 in the second diagram to the unified propagator D in the third diagram. Since both 4-point tree amplitudes are written by the complete 4-point amplitudes, one loop unitarity is ob- vious. Of course, the extra factor ~ suggested by Weinberg [ 7 ] should be taken into account.

For more than two-loops, we may expect to get all g-loop topological inequivalent diagrams by sewing g-pairs of neighbouring legs of the complete vertex with/5. It is very interesting to study the explicit form of the sum in higher loop amplitudes, but the diffi- culty of the evaluations lies in counting the multiplic- ity of each diagram in our complete amplitude. The explicit evaluations and also the discussions about unitarity for more than two-loop diagrams will be discussed in a forthcoming paper.

We thank Dr. H. Konno for valuable discussions in the early stage of this work.

References

[ 1 ] A. LeClair, Nucl. Phys. B 297 (1988) 603; G. Cristofano, F. Nicodemi and R. Pettorino, Phys. Lett. B 200 (1988) 292.

[2] T. Kobayashi, H. Konno and T. Suzuki, Phys. Rev. D 38 (1988) 1150.

[3] T. Kobayashi, H. Konno and T. Suzuki, Phys. Lett. B 211 (1988)86.

[4] T. Kobayashi, H. Konno and T. Suzuki, Tsukuba university preprint UTHEP-192 (1989).

[5] H. Konno and T. Suzuki, Tsukuba university preprint UTHEP-194 (1989).

[ 6 ] D.J. Gross, A. Neveu, J. Scherk and J.H.Schwarz, Phys. Rev. D 2 (1970) 697.

[7] S. Weinberg, Phys. Lett. B 187 (1987) 278.

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The symmetric expressions for the open bosonic string amplitudes and one-loop unitarity