Two-loop (β6) S-matrix for A1(1) Affine Toda Field Theory

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V/stas/nAaronomy,Vol.37, pp. 149-152, 1993 Printedin GreatBritain.Allrightsreserved.

TWO-LOOP (~6) S-MATRIX FOR A1(1) AFFINE TODA FIELD THEORY

FreddyP e r m ~ a

Zen

Uji Research Center, Yukawa Institute for Theoretical Physics, Kyoto University, Uji 611, Japan, Department of Physics, Hiroshima University, Higashi Hiroshima 724, Japan and Department of Physics, Institute of Technology Bandung, Gancsha 10 Bandung 40132, Indonesia

Abstract:

We present perturbative calculation for the Affine Toda Field Theory (ATFT)

S-matrix to the sixth order for A~1) theory. The dispersion approach is used to calculate the complete form of the perturbation amplitude in contrast to the pole residues in previous papers. The result agrees with those S-matrix obtained in the S-matrix approach namely those based on analyticity, unitarity, crossing and bootstrap equation. 1. I n t r o d u c t i o n . The Affine Toda Field Theory (ATFT) corresponding to a simply-laced Lie algebra g is a massive two-dimensional bosonic field theory which is described by the lagrangian

L = lo#¢aO#¢a- m2

(11 i=0

Here r is the rank of g and ¢ a, a = 1,..., r is an r component bosonic field describing r massive paxticles. The input from the Lie algebra theory is encoded in the set of r dimensional vectors cq which are the simple roots of the Lie algebra g augmented by the affine root oto = - ~"~fi=l nlo~i ( h i are a set of integers known for each algebra and no = 1) , which is the lowest root in all the cases considered in this paper. We restrict ourselves to the theories based on simply laced algebras, so we adopt the normalization condition o~ = 2. Classically, these theories are integrable in the sense that the above lagrangian admits an infinite number of conserved quantities labelled by the spin. In this paper, we report that the proposed S-matrix gives the identical contribution with the perturbative calculation at the two loop level (/36) for A~1) theory. It should be emphasized that we calculate the entire perturbation anaplitude, in contrast to the pole residues of the amplitudes in most cases reported so far. In other words, our main concern is to show by simple example how the perturbation series together with possible renormalization counter

150

F.P. Zen

terms conspire to give the very special forms of the proposed exact S-matrix. We note that the all result of this paper has been reported in ref. 1. The proposed S-matrices for ATFT ~ are simple trigonometric functions of 0 and their possible poles are located at integer multiples of i~r/h, where i is the imaginary unit and h is the Coxeter number of g. For A 0) case, the S-matrices are

a+b-l~

Sob where

(T, -- ])(x + 1) for

11 ( x - I + B ) ( x + I - B ) la-bl+l step 2 (x) = sinh ~ - ½(0 - + ~-z) ~x)' 1

B

1+

a, b = 1, 2, ..., n,

(a)

84 =

(2)

+

+ otzj,

and the general formulae for higher order coupling constants take the form3

Ca,...% = ( _ l ) ~ ( _ i ) p

( f12 ) ~-1 Hp ma, for E ak = 0 (mod h), (4) k----I

k=l

and vanish otherwise. By using (2) and (3), we compute the coefficient of the sixth order (f16) expansion of S-Matrix for A~1) case, which takes the form

1

i

1 1 )

_

sinhO(l--~ + 167rsinhO

{

(1

32sinhO ~r2

1 sinh~O")"

(5)

2. Two loop S - m a t r i x (/~6) for A~1) case. It has been shown in ref. 3, that dispersion approach 4 is an alternative method of perturbative calculation for the ATFT. This approach has a merit that the nice properties of the on-shell lower order (sub) diagrams are available because of the on-shell intermediate states. Another merit is that the dispersion integral expressed in terms of the rapidity instead of the vaz'iable s enjoys a high degree of universality. In order to exploit the dispersion approach for evaluating a given amplitude, we first calculate the imaginary (absorptive) part by using the (generalized) unitarity relation 5. Here the intermediate states are physical, i.e. on-shell. Next we calculate the real (dispersive) part by the dispersion integral, which is essentially Cauchy's integration formula. All the diagrams contribution are shown in Figure 1. The chain lines in the fourth and fifth diagrams indicate where the intermediate states are inserted whereas the last diagram comes from two loop renormalization of the propagator. Let us only take the results which has been reported in ref. 1 chapter 4. The contribution of the cliagrams (a), (b) and (c) are

l'wo-Loop (f)6) S.Matrix i/36 (

02

1

diagram (a) : - --~-,~r2sinhaO diagram (b) : diagram (c) :

151

2i0

sinhaO

?rsinha6),

i/36 02 32 r 2 sinh3 0' i/36 1 321r2 sinh 0

(6)

The contribution of the diagrams (d) and (e) are computed by using dispersion relation 1 f ds' I(s) = -~ J s-7--L~_s (Im

(7)

I(st))unitary.

The results are i/36 (O - i~r)2 i/36 0 - i~r diagram (d) : 327r2 sinha 0 + 16r 2 sinh 2 0'

i/36

02

diagram (,.e) : 327r2 sinh3 0

i/36

0

(s)

16Ir2 sinh 2 0"

It is elementary to evaluate the contribution of diagram (f), which gives a constant term, i.e.

i/36 [Tr2 diagram (f) : ~

1

1

~-~ - 8]s-~nh0'

(9)

In order to calculate the contribution from diagram (g), we must consider two loop renormalization of the propagator. After somework, we get diagram (g)

• : ~/3~[3~4

1_1_]__~1 64r sinhS"

(10)

Summing up all the contributions, we get the two loop (/3o) S-matrix of A~1) theory from the field theory, which agrees with the result from the proposed exact S-Matrix in equation (5). 3. S u m m a r y . In this paper we have presented perturbative verification for the proposed exact S-matrices in ATFT, namely to the two loop S-matrix (/36) for A~1) theory by using dispersion approach. We get complete agreement between the S-matrix theory and the field theory provided the function B(/3) has the conjectured form (3). These simple examples show clearly that the integrability is encoded in the very special forms of the three-, four- and mnlti-point couplings (see equation (4)). They also indicate that the exact S-matrices of ATFT "know" the renormalization and that a possible exact solution of quantum field theory should contain the ordinary renorlnalized field theory as its weak coupling limit. We believe that the high degree of universality of the perturbation integrals encountered in the dispersion approach would be an interesting clue for finding an exact solution of the integrable quantum field theory.

152

F.P. Zen

Acknowledgements: The author would like to thank Ryu Sasaki for his guidance, discussion and encouragement. The author wishes to thank M. Ninomiya for his encouragement and S. Sawada for his help with TEX, as well as other members of the Uji Research Center, YITP, Kyoto University for the warmest hospitality. This work was support by the Ministry of Education, Science and Culture, Japan. REFERENCES.

1. R. Sasaki and F.P. Zen, The Affine Toda S-Matrices vs Perturbation Theory, Preprint YITP/U-92-08, Int. J. Mod. Phys. A (in press) and references therein. 2. H.W. Braden, E. Corrigan, P. Dorey and R. Sasaki, Phys. Left. B227 (1989), 44!; Nucl. Phys. B338 (1990), 689. 3. H.W. Braden and R. Sasaki, Preprint YITP/U-91-41, Nud. Phys. B, in press. 4. K. Nishijima, Fidds and Particles (Benyamin, New York, 1969). R.J. Eden, P.V. Landshoff, D.I. Olive and J.C. Polkinghorne, The Analityc S-Matrix (Cambridge University Press, 1966). 5. R.E. Cutkosky, J. Math. Phys. 1 (1960), 429. s-channel

t-channel

1

u-channel

I !

(a)

1

!

(b)

s-channel

(c)

t-channel

u-channel I x2

I

!

I

(d)

1 (e)

I

(f)

I

(g)

Figure I

I

I