The S-matrix of a factorizable supersymmetric Z(N) model

V01ume 191, num6er 3

PHY51C5 LE77ER5 8

11 June 1987

7 H E 5-MA7R1X 0 F A F A C 7 0 R 1 2 A 8 L E 5UPER5YMME7R1C 2 ( N ) M 0 D E L R. K 0 8 E R L E and V. K U R A K 1n5t1tut0 de F151cae Qu/m1ca de 5~0 Car105, Un1ver51dadede 5~0 Pau10, C.P. 369, 13560 5~0 Car105, 5P, 8ra211

Rece1ved 8 Decem6er 1986

7he exact 5-matr1x0f a tw0-d1men510na15uper5ymmetr1c2(N) m0de1 15c0mputed. Ant1part1c1e5appear a5 60und 5tate5 0f part1c1e55ett1n9a r1ch 5pectrum 5tructure.

2 ( N ) 5ymmetry ha5 6een 5tud1ed 60th a5 an 1nvar1ance 0f 5tat15t1ca1 mechan1ca1 m0de15 [ 1 ] and a150 0f tw0-d1men510na1 c0nf0rma1 1nvar1ant 4uantum f1e1d the0r1e5 [2]. A c0nnect10n 6etween the tw0 area5 can 6e e5ta6115hed, tak1n9 the 5ca11n911m1t [ 3 ] 0f a 1att1ce m0de1 1n 0rder t0 06ta1n a f1e1d the0ry, wh05e 5h0rt-d15tance 11m1t 5h0u1d 6e c0nf0rma11y 1nvar1ant. 7h15 ha5 6een carr1ed 0ut exp11c1t1y f0r the ca5e N = 2, c0rre5p0nd1n9 t0 the 151n9 m0de1, where the 5ca11n9 11m1t ha5 6een 5h0wn t0 ex15t and a c0mp1ete 5et 0f ma551ve 6reen•5 funct10n5 ha5 6een 06ta1ned v1a a 600t5trap pr09ram 5tart1n9 fr0m fact0r12at10n c0nd1t10n5 [4], wh1ch 1n turn are a c0n50 4uence 0f the Yan9-8axter e4uat10n5. 7he 2er0-ma55 11m1t 0f the 151n9 m0de1 can 6e v1ewed a5 the ca5e N = 2 0f a fact0r12ed 5-matr1x hav1n9 2 ( N ) 5ymm e t r y [ 5 ]. 7he h0pe 15 that th15 pr09ram can a150 6e carr1ed thr0u9h f0r m0re c0mp11cated 51tuat10n5. 7h15 w0u1d 6e very de51ra61e, 51nce recent1y 9reat pr09re55 ha5 6een made 1n c1a551fy1n9 a11 c0nf0rma1 1nvar1ant 4uantum f1e1d the0r1e5 [6,7]. 51nce th15 w0rk 15 6e1n9 extended t0 5uper5ymmetr1c the0r1e5 [ 7 ], 1t 15 thu5 0f 1ntere5t t0 06ta1n exact 5uper5ymmetr1c 5-matr1ce5. 7h15 15 carr1ed 0ut 1n the pre5ent paper f0r the 2 ( N ) m0de1. 7he rea50n f0r th15 name c0me5 fr0m the fact that 1n th15 m0de1 there 15 n0 ref1ect10n; ant1part1c1e5 are 60und 5tate5 0f N - 1 part1c1e5 and 1t5 re1at10n t0 the u5ua1 2 ( N ) m0de1 15 the 5ame a5 the re1at10n 6etween the 5uper5ymmetr1c (5U5Y) n0n11near a-m0de1 and the u5ua1 n0n-11near a-m0de1. 1n 0rder t0 exp1a1n the 1atter p01nt 1et u5 6r1ef1y

rev1ew the 5uper5ymmetr1c 5tructure 0f fact0r12ed 5matr1ce5. Reca11 that the 5U5Y n0n-11near 0 ( N ) am0de1 can 6e v1ewed a5 e1ther the 5U5Y exten510n 0fthe 0 ( N ) n0n-11near a-m0de1 0r the 5U5Y exten510n 0f the 0 ( N ) 6r055-Neveu m0de1. 7he vect0r mu1t1p1et 5-matr1x 0f the 5U5Y n0n-11near a-m0de1 [ 8 ] 15 a pr0duct 0f tw0 fact0r5. 0 n e 0f them 15 prec15e1y the 5-matr1x 0f the 0 (N) 6r055-Neveu m0de1 [9]. 7he 0ther, 5ay Y, c0rre5p0nd5 t0 5uper5ymmetry. 7he re1at10n5 am0n9 the var10u5 amp11tude5 0f the 0 ( N ) vect0r mu1t1p1et 5catter1n9 can 6e 06ta1ned thr0u9h the 5upera19e6ra a5 1n ref. [ 8 ]. 8ut f0r the 1505p1n0r mu1t1p1et the 51tuat10n 15 n0t 4u1te 50. Even 1n the u5ua1 6r055-Neveu m0de1 the part1c1e5 0f the 1505p1n0r mu1t1p1et 06ey 1ntermed1ate 5tat15t1c5 [ 10 ] and 0ne 15 faced w1th the pr061em 0f c0n5truct1n9 the appr0pr1ate K1e1n fact0r5 [ 11 ]. 7he 51tuat10n 15 the 5ame 1n the 5U5Y C P N - 1 m0de1 [ 12]. 1t ha5 6een 5h0wn 1n ref. [ 13 ] that the na1Ve 5upera19e6ra 6reak5 d0wn 1n the 1 / N expan510n and 1t wa5 ar9ued [ 13 ] that the actua1 5upera19e6ra 5h0u1d c0nta1n centra1 term5 c0rre5p0nd1n9 t0 t0p01091ca1 4uantum num6er5. 0 n the 0ther hand, 1t wa5 065erved 1n ref. [14] that the Yan9-8axter e4uat10n5 can 6e 501ved w1th0ut 1mp051n9 the 5uper5ymmetry c0n5tra1nt5. (Actua11y, 1n ref. [14] the 065ervat10n wa5 f0r the 51ne-60rd0n 5uper5ymmetr1c m0de1, 6ut 1t can a150 6e 5h0wn f0r the 0 ( N ) ca5e 51nce the Yan9-8axter e4uat10n5 dec0up1e 1n 1nterna1 and 5uper5ymmetr1c 1nd1ce5. 7h15 15 the rea50n why the 5-matr1x 15 a pr0duct 0f tw0 fact0r5.) 50, 0ur 5trate9y 15 t0 c0n5truct a m0de1 6y 501v1n9 the

0370-2693/87/$ 03.50 • E15ev1er 5c1ence Pu6115her5 8.V. (N0rth-H011and Phy51c5 Pu6115h1n9 D1v1510n)

295

v01ume 191, num6er 3

PHY51C5 LE77ER5 8

Yan9-8axter e4Uat10n5 5UCh that tW0 fact0r5 C0me 0Ut f0r the 5-matr1x. 0ne, 5ay 17, C0rre5p0nd5 t0 5Uper5ymmetry. 7he reader W111Ver1fy that the fact0r 1706ta1ned 6e10w 15 determ1ned 6y the 5ame Un1tary e4Uat10n5 a5 Y ment10ned a60ve, the1r d1fference 6e1n9 ent1re1y C0nta1ned 1n the Cr0551n9re1at10n5. 7he 0ther fact0r C0rre5p0nd5 t0 the 1nterna1 5ymmetry, Wh1Ch we have Ch05en t0 6e 2 ( N ) . A5 a C0n5e4UenCe 0f the 5peCtrUm, 0Ur part1C1e5 06ey 1ntermed1ate 5tat15t1c5, 6ut we have n0t 06ta1ned up t0 n0w the K1e1n fact0r5 needed 1n 0rder t0 c0n5truct the 5uper a19e6ra (w1th centra1 term5). At any rate, 1n the 5e4ue1 we w111 name 0ur part1c1e5 6050n5 and ferm10n5, 6ear1n9 1n m1nd the a60ve 065ervat10n5. 7he m0de1 c0nta1n5 a5ympt0t1c 5tate5 c0n515t1n9 0f a de9enerate mu1t1p1et c0nta1n1n9 a ferm10n f, a 6050n 6 and the1r ant1part1c1e5 ]" and 6. 7he1r 5catter1n9 amp11tude5 f0r van15h1n9 ref1ect10n are (1a)

516•62 > =A(012)16261 > ,

(16)

51f, f2 > =8(012)1f2f, > , 516,f2)=C(0,2)1f26,)+D(012)162f,)

,

(1c)

516,62) =A(1 --012)1626 • >

+D(1 --012 ) 1f2f~ ) ,

(1d)

51 f, f2 > =8(1 --0,2)11~2 f1 > (1e)

+ D ( 1 - 0 1 2 ) 1626, > ,

516,1~2 > =C(1-0,2)1}~26, >,

(1f)

where 0 , 2 = 0 ~ - 0 2 and 01 are the rap1d1ty var1a61e5 p0 = m c05h(620), p) = m 51nh(620) and where we have a1ready u5ed cr0551n9 5ymmetry t0 wr1te the 1a5t three amp11tude5. 7he un1tar1ty e4uat10n5 are A(0)A(-0)

= 1,

(2a)

8 ( ~ ) 8 ( - 0 ) = 1,

(26)

C ( 1 - 0 ) C ( 1 + 0 ) = 1,

(2c)

C( ~ ) C ( -- 0 ) + D( ¢ ) D ( - 0 ) = 1,

(2d)

C( ~ ) D ( - ~ )

296

+ D( ~ ) C ( - - ~ ) = 0

.

(2e)

11 June 1987

709ether w1th the Yan9-8axter e4uat10n5, wh1ch may 6e 06ta1ned 1n the u5ua1 way, they have the f0110w1n9 m1n1ma1 (w1th0ut p01e5) 501ut10n Am1n(0) = 51n[ 6 2 ( 0 +;t)] 51n(~90) Cm1n(0) ,

(3a)

~m1n(1~) = -- 51n[~1t(0--2)1 51n(~7t0) 6m1n(1~) ,

(36)

51n(~n2)

(3c)

D m 1 . ( 0 ) - 5 1 n ( ~ 0 ) Cm1.(0) , Crn1n(1~ 1) = ~r0( 1~ )

r(~0+~+0r(1-~0+0 = f=10F(1~0+~2+1)F(~0+1

X ,11 :,

)

F(•0-12+1)F(1-10+1) r(~-~0-~+1)r(~0+1)

"

(3d)

Rep1ac1n9 Cm1n(0) 6y 51n[~n(0+2)] Cm1.(0) C ( 0 ) = 51n[~rt(0-2)]

(4)

we 1ntr0duce a p01e at 0 = ~ . = 2 / N 1n the 6050n-6050n 5catter1n9 amp11tude A ( 0 ) [and c0n5e4uent1y 1nt0 the c0m61nat10n C ( 0 ) + D ( 0 ) ] 1n ana109y w1th the u5ua1 2 ( N ) m0de1 [5]. 7he ma55 5pectrum 15the 5ame a51n the u5ua1 2(N) m0de1. 1n part1cu1ar, there 15 an ( N - 1 ) part1c1e 60und 5tate at the 5ame ma55 a5 the fundamenta1 part1c1e5. 0 n e 0f the c0n5e4uence5 0f a 2 ( N ) 5ymmetry f0r part1c1e5 w1th0ut 1nterna1 de9ree5 0f freed0m 15 that ant1part1c1e5 are 60und 5tate5 0f ( N - 1 ) part1c1e5. We n0w 5h0w that 0ur amp11tude5 5upp0rt the f0110w1n9 1dent1f1cat10n: N--1

}•= 1-1 6 , , 1=1

(5a)

6 = f/6265...6N•1 +6/f263...6N•1 +... +616263...6N•2fN N 1

E 1~).

(56)

V01ume 191, num6er 3

PHY51C5 LE77ER5 8

7h15 mean5 that the ant1ferm10n 15 a 60und 5tate 0f ( N - 1 ) 6050n5 and the ant16050n 15 a 60und 5tate 0f ( N - 2 ) 6050n5 and 0ne ferm10n. 1n term5 0f0ur 5-matr1x amp11tude5 e45. (5) mean that, 1f 1n the N part1c1e 5catter1n9 amp11tude we pr0ject ( N - 1 ) part1c1e5 0nt0 the p01e 0f ma55 m, we have t0 repr0duce the c0rre5p0nd1n9 part1c1e amp11tude. U51n9 the n0tat10n ]~[6162...6N~] t0 1nd1cate that we have 90ne t0 the p01e 0fthe ( N - 1 ) part1c1e5 1n51de the 6racket t0 06ta1n an ant1ferm10n [and 0m1tt1n9 the 0ut51de 1etter f0r 6050n5, 51nce 0n1y the 11near c0m61nat10n e4. (56) 15 the ant16050n] we 06ta1n

and the ( N - 1 ) X ( N - 1 ) matr1ce5 M, N and R are 91ven 6y M1j = {CjN f0r 1 + j < N - 2 ;

DjNf0r1+j=N-1; AjN f0r 1+j> N - 2 } ,

(1> 1 , j > 0 0 ,

=8jN

N--1

(1~1,j=0t) ,

(6a)

1=1

=QN N--1 V1 C1N1fNf~[61 . . . 6 N ~ ] 1=1

+ ~ u,~16N[6~...f,~...6N~1]),

QN f 0 r j - 1 > 0 } , (66)

0t=1

(86)

(8c)

and R~.)u 15 a ( N - a ) X ( N - 1 ) matr1x 91ven 6y

f0r (1=1,j=a)

R(90

=0jN

5 6 N-1 2 (61...fa...6N~)6N tX=1

f0r ( 2 < 1 < N - 1 , j = a )

(J~> c~,j-1= 1)

=Am

N-1

(8d)

f0r 0•>a,j-1>2)

Ua16N[6,...fa...6N--1] )

~=1

= CjN 0therw15e.

N-1N-1

H P1j1fN~[61~6N-21),

1=1 j = 1

(6c)

5 ~)N-1 ) 2 (61...fa...6N-,)fN N--1

w~1fu[ 6, ...f~...6N~ ,1) ,

(6d)

f0r N 0 d d , = N - 2 , N - 4 .... , 2 , 0 , - 2 .... , - ( N - 2 )

(9)

f0r N even.

where

N--1 Ua H MaJ• j=1

1n e45. ( 6 ) - ( 8 ) the ar9ument 0f the amp11tude5 w1th 1nd1ce5 (jN) 15

n j = N - 2 , N - 4 .... , + 1 ..... - ( N - 2 )

a=1

0t=1

0therw15e,

P1j= {AjN f0r 1-j> 0 ;DjN f0r 1=j;

)

N--1

+ 2

f0r (1= 1 , j = a ) ,

=DjN f0r (1=j, 2 < j < a )

= H A1016N~[61~6N-~] ) ,

511~[61...6N--,]fU=

(8a)

N[ff ) =A)N f0r (1= 1,j#ce)

51~[61...6N~1]6N

E

11 June 1987

U51n9 the 1dent1ty

N--1N--1 Va = 1=1 2 jH N}f~) , =1

N--1

H Y~(0j)=1~0(1-0),

j=1

(10)

N--1 N--1

W0t=

E H R(.) U

1=1 j = 1

(7)

297

V01ume 191, num6er 3

PHY51C5 LE77ER5 8

where Y6 (4~) --

51n[ ~n(4~ + 2 ) ] 1~,0( 4~) 51n(1n4~ )

(11)

0 n e 5ee5 that n 0 extra term5 a p p e a r 0 n the R H 5 0 f e4. ( 6 d ) a n d that N--1

1~ A,N(4~)=C(1-~),

(12a)

1=1 N-1

C1N(~1)=9(1--~) ,

(126)

t=1

U1 = / / 2

.....

v1 =v2 . . . . .

UN• 1

= 0 ( 1 -¢D)

vN-1 = A ( 1 - ~ )

(12c)

,

(12d)

,

N--1 N--1

U P0=D(1-~),

(12e)

1=1 ) = 1

W 1 =W 2 .....

WN•

1

=C(1 - ~ )

.

(12f)

7h15 pr0ve5 0 u r 1dent1f1cat10n exh161ted 1n e4. (5). 7h15 1dent1f1cat10n ra15e5 the 5ame 4ue5t10n 0 f 1ntermed1ate 5tat15t1c5 [ 15 ] f0r 6 a n d f a5 1n the ch1ra1 6 r 0 5 5 - N e v e u [16] m0de1 a n d 51nce at the pre5ent t1me we are una61e t0 hand1e f1e1d5 w1th 1ntermed1ate 5tat15t1c5 d1rect1y, we a d 0 p t the 5ame p r 0 c e d u r e a5 e x p 0 u n d e d 1n ref. [16]. A5 a f1na1 r e m a r k we n 0 t e that 6y 1nc1ud1n9 a C . D . D . - t y p e p01e 1n 0 u r 501ut10n (a5 1n ref. [ 17] f0r the 7 0 d a cha1n) we 5h0u1d 6e a61e t0 9et the 5-matr1x 0 f the 5uper5ymmetr1e 7 0 d a cha1n [ 18 ]. 7h15 pr061em 15 u n d e r c u r r e n t 1nve5t19at10n. 7 h e he1p 0 f the referee 1n 1mpr0v1n9 the d15cu5510n 0 f a5pect5 re1at1n9 t0 5 u p e r 5 y m m e t r y 15 9ratefu11y ackn0w1ed9ed.

Reference5 [1] F.Y. Wu, Rev. M0d. Phy5. 54 (1982) 235; F.C. A1cara2 and R.K66er1e, J. Phy5. A 13 (1980) L153;

298

11 June 1987

V.A. Fateev and A.8. 2am010dch1k0v, Phy5.Lett. A 92 (1982) 37; 5.V.P0kr0v5ky and Yu.A.8a5h110v, C0mmun. Math. Phy5. 84 (1982) 103; 7.7. 7ru0n9, J. 5tat. Phy5.42 (1986) 349; F.C. A1cara2, Nuc1. Phy5. 8, t0 6e pu6115hed. [2] V.A. Fateev and A.8. 2am010dch1k0v, Landau 1n5t1tute prepr1nt (1985). [ 3 ] 8. McC0y and 7.7. Wu, 7he tw0-d1men510na1151n9m0de1, (Harvard U.P., Cam6r1d9e, MA, 1973); M.L.0•Carr01 and R.5. 5ch0r, Rev. 8ra5. F15., t0 6e pu6115hed. [ 4 ] 8.8er9, M. Kar0w5k1and P. We152,Phy5. Rev. D 19 (1979) 2477. [5] R. K66er1e and J.A. 5w1eca, Phy5. Lett. 8 86 (1979) 2. [6] A.A. 8e1av1n, A.M. P01yak0v and A.8. 2am010dch1k0v, Nuc1. Phy5.8 241 (1984) 333; D. Fr1edan, 2. Q1u and 5.H. 5henker, Phy5. Rev. Lett. 52 (1984) 1575; V.5. D0t5enk0 and V.A. Fateev, Nuc1. Phy5. 8 240 (F512) (1984) 312;825 (F513) (1985) 691. [ 7 ] M. 8er5had5ky, V. Kn12hn1kand M. 7e1te1man, Phy5. Len. 8 152 (1985) 85; D. Fr1edan, 2. Q1u and 5.H. 5henker, Phy5. Lett. 8 151 (1985) 37; Y.Y. 601d5chm1dt, Phy5. Rev. Lett. 56 (1986) 1627; 2. Qu1, Nuc1. Phy5. 8 270 (F516) (1986) 205. [8] R. 5hankar and E. W1tten, Phy5. Rev. D 17 (1978) 2134. [ 9 ] A.8.2am010dch1k0v and A1.8.2am010dch1k0v, Ann. Phy5. (NY) 120 (1979) 253. [ 10] R. 5hankar and E. W1tten, Nuc1. Phy5. 8 141 (1978) 349. [ 11 ] M. Kar0w5k1 and H.J. 7hun, Nue1. Phy5. 8 190 (1981) 61. [12] A. D•Adda, P. d1 Vecch1a and M. Lt15cher, Nuc1. Phy5. 8 152 (1979) 125. [ 13] E. W1tten, Nuc1. Phy5.8 149 (1979) 285. [ 14] M. D0r1a and R. 5hankar, J. 5tat. Phy5. 33 (1983) 695. [15] D.J. 7h0u1e55 and Y.-5. Wu, Phy5. Rev. 8 31 (1985) 1191, and reference5 there1n. [ 16] R. K66er1e, V. Kurak and J.A. 5w1eca, Phy5. Rev. D 20 (1979) 897. [ 17 ] A.E. Ar1n5hte1n,V.A. Fateev and A.8.2am010dch1k0v, Phy5. Lett. 8 87 (1979) 389. [ 18] M.A. 015hanet5ky, C0mmun. Math. Phy5. 88 (1983) 63.