PHYSICAL REVIE%' 0 VOLUME 33, NUMBER 10 15 MAY 1986

Conservation laws in the supersymmetric sine-Gordon model

Soumitra Sengupta and Parthasarathi MajumdarSaba Institute of Nuclear Physics, 92 A.P.C. Road, Calcutta?00009, India

(Received 1Q October 1985)

+e compute all possible four-particle and six-particle scattering amplitudes in the supersymmetric sine-

Gordon model. They are found to be nonvanishing only when the particle number and the initial momen-

tum set are conserved.

The classical S matrix for the (1+1)-dimensional sine-Gordon model has the following features: (1) There is nomultiple production, and (2) the momenta of the final andinitial particles are equal. These properties are conse-quences of the complete integrability of the sine-Gordonmodel at the classical level. Perturbative calculations showthat these properties are also preserved for the quantumversion of the theory. ' Finally, an exact expression for theS matrix of the two-dimensional sine-Gordon model wasobtained explicitly. '

Exact integrability of the supersymmetric (SUSY) versionof the sine-Gordon theory" in I+1 dimensions has alsobeen established at the classical level, in terms of the ex-istence of an infinite number of constants of motion, all ininvolution. 5 The status of these conservation laws at thequantum level has not yet been established. 6 The verifica-tion of the validity of these properties in low orders of per-turbation theory is an essential step towards a completesolution of the model at the quantum level. This hasmotivated us to complete all possible four-particle and six-particle amplitudes in different channels in the tree approxi-mation for the quantum SUSY sine-Gordon theory. Theresults reveal the validity of both the properties at thelowest-order quantum 1evel, as we now proceed to demon-strate.

The Lagrangian for the SUSY sine-Gordon theory is

given by

~= ~ (8„4)'+ (m /P') (cosP@—1)+ TtTI i & mcos P—

(I)where qh is a real scalar field (sine-Gordon field) and P is atwo-component Majorana fermion. On expansion, inpowers of p, this yields

r

4 6W= ' [(t)pg)' —m'@']+ m'p'~ —m'p'~ +0(p')

and

g—1/2 Ig 1/2

u(p) —= u(a) = (m/2)'i' ti2,. ti2 (4)

normalized such that

u(p)u(p) =u(p)t (p) =2m

and

u(p) u(p) =@+m, u(p)u(p) =p —m

(5)

(6)

Once again the conservation of the initial set of momenta isevident from the energy-momentum conservation.

To calculate the amplitude in the 2 particle 2 particlechannel we observe that the Lagrangian admits the follow-

ing distinct processes to occur: 2b 2b, 2b 2f, and1blf 1 bl f (where b denotes a boson and f denotes afermion). In the tree approximation we have (Fig. 1)

A2b-2b= —im p

Also, energy-momentum conservation in two space-timedimensions, expressed in terms of the light-cone variables,admits only two discrete solutions for the a' s, viz. ,

I I r I

a, =al, a, =a2, or a, =a2, a. =al

Next, from Fig. 2 we obtain

. mp2-fIb — t MM

8

Therefore, using Eq. (3), we get

Atftb ]ftb —i (m P /8) (at ' a,' + at' a, ' )

+ t p,+ )&+ mP' 04'4 mP' 04'0 + 0(P6)8 2t 32 4!

(2)Using the Majorana representation of the y matrices andthe light-conc variables p —defined as7

p+ =—po+p'= ma, p —=p —p'= ma

we choose the spinors to be of the form

g—1/2 + I'g l/2

u(p) = u(a) = (m/2)'i2 —a +ia

1P'

g'r i2.P

FIG. 1. Diagram contributing to the process 2b 2b. Brokenlines represent bosons (b). Arrows indicate momentum flour.

33 3138 1986 The American Physical Society

33 BREEF REPORTS 3139

FIG. 2. Diagr™contributing to the process 1fib 1fib. Bro-ken lines represent bosons (b) and unbroken lines represent fer-mions (j). Arrows indicate momentum flow.

F16. 3. Diagram contributing to the process 2b 2f. The sym-bols are the same as in Fig. 2.

Figure 3 refers to the process 2b 2f with amplitude

A21, 2f = —i (mP2/8) uu

Using Eqs. (3) and (4) one obtains

A2$ 2f —i (m p /8)(a1 a2' —a1' a2 ' ) . (9)

gator is given as

[(p1+p2 —p5)' —m'] '- P {[(p1+p2 —p5)' —m']

+ &n 5 ( (p1+p2 —p5) —m )

(10)

Energy-momentum conservation ensures the conservationof the set of initial momenta.

Hence, the property (2) is verified for all possible 2 2

channels.%e now concentrate on six-particle amplitudes. Clearly

the following processes may take place: 4f 2 b, 3f—1f2b, 3b- 1b2f, and 4b- 2f, 3b 3b, 2b 4bFirst we compute the connected part of the S-matrix ele-

rnent for the process 4f 2b (Fig. 4). The bosonic propa-

where P([(p1+p2 —p5)2 —m2] '} is the principal part ofthe bosonic propagator.

Also, it is easy to verify that, in terms of the light-conevariables,

Q& + Q2 Q~—Q5 Q2 —

Q5(P1 +P2 P5) N1 Pl

Using (11), the amplitude turns out to be

~ 2 4

A4f 2$ = [ 6(1)u (2)u (3)u (4) [a1a2a5(a1+ a2) (a1 —a5) (a2 —a5)64

+ a1a2a6(a1+ a2) '(a1 —a6) '(a2 —a4) ']

—v(1)u(3)6(2) u(4) [a1a3a5(a1+ a3) '(a1 —a5) '(a3 —a5)

+ a1a3a6(a1+ a3) '(a1 —a4) '(a3 —a6) ']

+ V(1)u (4)V(2) u (3) [a1a4a5(a1+ a4) '(a1 —a5) '(a4 —a5)

+ a1a4a6(a1+ a4) '(a1 —a6) '(a4 —a6) ']} (12)

Now for a 3 particle —3 particle channel, the energy-momentum conservation can be written as' a ~ + a 2 + a 3= a4+ a5+ a6 and a~ ' + Q2' + a3 ' = a4 ' + a5 ' + Q6 '. These yield the identity

(a1 —a4) (a1 —a5) (a1 —a4) = a1a2 'a3 '(a2+ a3) (a4a5a6 —a1a2a3)

and similar identities with a1 replaced by a2, a3 in the left-hand side of Eq. (13).For 4 particle 2 particle processes identities similar to Eq. (13) can be obtained by replacing a4 by —a4, i.e.,

(a1+ a4) (a1 —a5) (a1 —a6) = —a1a2 'a3 (a2+ a3) (a4a5a4+ a1a2a3)

and similarly for the others. Using (14) Eq. (12) becomes

A4f 2b 64 ala2a3[(a1+a2)(a2+a3)(a3+al)(ala2a3+ 4a5 6)]im2p4 —1

(13)

X [ u(1) u(2)u(3) u(4)(a1a2a5+ a3a4a6+ a1a2a6+ a3a4a5) —v(1) u(3)u(2) u(4)

x (a1a3a5+ a2a4a6+ a1a3a4+ a2a4a5) + 11 (1)u (4)v(2) u (3) (a2a3a5+ a1a4a6+ a2a3a4+ a1a4a5) ]

Using Eqs. (3), (4), and (14), and after some algebraic manipulation, we obtain A4f 2y 0. Thus the connected part of the

3140 BRIEF REPORTS 33

S matrix for the process 4f —2b does indeed vanish.The 5-function term in the propagator, 8((pt+@2—/23)

—m2), which has hitherto been neglected, gives a contribu-tion when the sets of incoming and outgoing mornenta areequal. But that will lead, by the energy-momentum conser-vation equation, to

a6= a2+a3+a4 (identifying a1 with a3),

a6 =a2 +a3 +a4—1 —1

But the above two sets of equations are inconsistent (sinceall a's are positive). Therefore, the 5-function term cannotcontribute for this channel. Hence, A4f 2$ 0.

Next, we turn our attention to the process 3f 1f2b(Fig. 5). A similar calculation yields

I/ /

F16. 4. Diagram contributing to the process 4f—2b.

2 4

fl 3f lf2 b ala2a3(al + a2) ( at + a3) ( a2+ a3) (a1a2a3 a4a3a6)

x [ 6(1)u (2) u (4) u (3)(at a2a3 —a3a4a6+ ata2a6 —a3a4a3)

—V(1)u (3)u (4) u (2) ( a1a3a3 —a2a4a6+ at a3a6 —a2a4a3)

+ t (2) u (3)u(4) u (1)(a2a3a3 —a 1a4a6+ a2a3a6 —a 1a4a3) ]

Using Eqs. (3), (4), and (13) we get

A3f tf2s 64ata2a3(a3+ a6) (at + a2) (at+ a3) (a2+ a3) (ata2a3 —a4a3a6)

im2p4 —1 —1 —1

x [ (a —1/2a 1/2 a 1/2a —1/2) (a —1/2a 1/2+ a 1/2a —1/2) (a a a a )

(a —1/2a 1/2 a 1/2a —1/2) (a —1/2a 1/2+ a 1/2a —1/2) (a a a a )

+(a2 ' a3' —a2' a3 ' )(a4 ' a1' +aq' a1 ' )(a2a3 —ataq)]

This amplitude vanishes identically after a few tedious steps. But now it is easy to verify that the 8-function term gives acontribution when the sets of incoming and outgoing mornenta are equal.

Next we proceed to compute the amplitude 331, 1$2f (Figs. 6, 7, and 8). From Fig. 6, using (11) we get3

Bt = ' u(6)u(5) [ —ata2a3(at+ a2) '(a1+ a3) '(a2+ a3) '+ ata2a4(at+ a2) '(at —a4) '(a2 —a4)32

+ ata3a4(a1+ a3) '(a1 —a4) '(a3 —a4) '+ a2a3a4(a2+ a3) '(a2 —a4) '(a3 —a4) '] . (15)

Using Eqs. (5), (6), and (11) one obtains the contribution from Fig. 7:

//

/+3

//

/

FIG. 5. Diagram contributing to the process 3f 1f2b. FIG. 6. One-boson-exchange diagram for the process 3b I b2f.

33 BRIEF REPORTS 3141

I

~ /

lf

FIG. 7. One-fermion-exchange diagram for the process3b 1 b2f.

FIG. 8. Six-particle single-vertex diagram for the process 3b1 b2f.

8 = ' u(6)u(5)[ata2a6(al+a2) '(al —a6) '(a2 —a6) '+aia3a6(al+a3) '(ai —a6) '(a3 a6)2=32

+a2a3a6(a2+a3) '(a2 —a6) '(a3 —a6) '+ala2a5(at+a2) '(at —a5) '(a2 —a5)

+ ata3a5(ai+ a3) '(ai —a5) '(a3 —a5) '+ a2a3a5(a2+ a3) '(a2 —a5) '(a3 —a5) '] . (16)

The contribution from Fig. 8 can easily be calculated as

im4p43=32

(17)

—im'p41 2 (18)

Therefore, from Eqs. (17) and (18) the amplitude can beobtained as

~3b 1b2f B1+B2+ B3 0 ~

Once again, as in the last case, the 5-function term contri-butes only when the initial set of momenta is identical tothe final set of momenta. The amplitude for the process4b 2f can be obtained easily by replacing a4 by —a4 inthe expression of A3b ]b2f. But the connected part of thematrix element will obviously still vanish after this substitu-tion. But once again in this case, as discussed earlier, the5-function term does not contribute since the identificationof the initial set of momenta with the final set of momentaleads to an inconsistency in the energy-momentum conser-

Adding Eqs. (15) and (16), after some algebra, and usingthe identities (13), we get

I

vation equation.For the processes 3b 3b and 2b 4b, the contributing

terms in the Lagrangian are m2p2@4/4! and —m'p4$6/6!.This is the same as the ordinary sine-Gordon model. It hasalready been shown in the context of the ordinary sine-Gordon model that A3b 3b is nonvanishing only when theinitial set of the momenta is equal to the final set and

4b vanishes identically. '

Thus the results of all six-particle amplitudes a1so revealthe validity of the properties (1) and (2).

It should be observed that the equality of boson and fer-mion masses, characteristic of a SUSY theory, is crucial forthe above results. Furthermore, it is interesting to note thatthe total number of particles is to be conserved. Thenumber of bosons and fermions need not be conserved in-dependently.

It is our conjecture that the validity of properties (1) and(2), demonstrated above, should go through higher ordersof perturbation theory as well.

One of us (P.M.) thanks V. E. Korepin for an extremelyilluminating discussion. %e also thank G. Bhattacharya,A. Chatterjee, and P. Mitra for helpful discussions.

'I. Ya. Aref'eva and V. E. Korepin, Pis'ma Zh. Eksp. Teor. Fiz. 20,680 (1974) [JETP Lett. 20, 312 (1974)I.

2A. B. Zamolodchikov and A. B. Zamolodchikov, Ann. Phys.(N.Y.) 120, 253 (1979),

3J. Hruby, Nucl. Phys. 8131, 275 (1977); P. DiVecchia and S. Fer-rara, ibid. B130, 93 (1977).

4R. Shankar and E. %'itten, Phys. Rev. D 17, 2134 (1978).5P. P. Kulish and S. A. Tsyplyaev, Teor. Mat. Fiz. 46, 172 (1981)

[Theor. Math. Phys. 46, 114 (1981)].

The 5 matrix for the SUSY sine-Gordon theory proposed in Ref. 4is based on the assumption of the existence of an infinite set ofconserved quantities in the theory. Our purpose here, on the oth-er hand, is to prove two of these conservation laws fproperties (1)and (2) above] albeit in lowest-order perturbation theory.

78. Berg, M. Karowski, and H. J. Thun, Phys. Lett. 62B, 187(1976).