Form factors in the eight-vertex model at reectionless points Yas-Hiro Quano Department of Clinical Engineering Suzuka University of Medical Science 7 Sep 005 @ LAPTH, France RAQIS 005
Introduction In this talk we consider form factors in the eightvertex model at the reectionless points. R-matrix in principal regime: R(u) =R(u r) = 4 a(u) d(u) b(u) c(u) c(u) b(u) d(u) a(u) 3 7 5 () Reectionless points: r N =+ N (N = 3 ) At r = r N, S-matrix S(u) =;R(u r ; )becomes (anti-)diagonal. Thus, we expect that the form factor formulae will be simple at reectionless points. In what follows, let O = c z = X "= " ;" (u ; ) " (u):
8V model is dicult. (Due to d-term) Baxter: Introduction of 8VSOS model. Translation from 8VSOS into 8V. Lashkevich-Pugai: Correlation fns in terms of SOS type I bosonization. Lashkevich: Form factor in terms of SOS type II bosonization. Quano: Solving bootstrap equations. Shiraishi: Direct bosonization, and (hidden) deformed W algebra symmetry. In this talk, we will derive an explicit formula of form factors at reectionless point, using Lashkevich's bosonization scheme. 3
Basic denitions The Boltzmann wts of SOS model(regime III): W W W 4 k k 4 4 k k k k k k k k k k u u u 3 7 5 = (u) 3 7 5 = (u) 3 7 5 = ; (u) [][k u] [ ; u][k] [u][k ] [ ; u][k] : (x = e ; = x ;u z = >0 0 <u<.) Vertex-face correspondence: = X b R(u ; u )t d c (u ) t a d (u ) W 4 c d b a 3 u ; u 5 t a b (u ) t b c (u ): () u c <? u d a c < X = u b 4 b? u d a
3 Review of Lashkevich's constraction Introduce the following basic bosons: [ m n ]=m [m] x[(r ; )m] x [m] x [rm] x m+n 0 [Q P ]= p ;: Let F l k := C [ ; ; ]jl ki, where n jl ki =0(n>0) Pjl ki =( k + l)jl ki t ; 0 t ; =(t ; )(t ; ) 0 = p r(r ; ) < : Type I VO in 8V SOS model on F l k : k+ k (u) = r; 4r [k] +(u) = z [k] :exp(' (z)) : ' (z) := ( p ;Q + P log z) ; X m=0 m m z;m : X(u) = k; k (u) = [k] +(u)x(u) I C dw p ;w A(v)[u ; v + ; k] [u ; v ; ] A(v) =w r; r : exp(;' (xz) ; ' (x ; z)) : : 5
Type II VO in 8V SOS model on F l k : r l+ 4(r;) l (u) = + (u) =z :exp(' (z)) : ' (z) := ( p X m [rm] x ;Q+P log z)+ z ;m : m [(r ; )m] m=0 x X 0 (u) = I dw 0 C 0 p ;w B(v)[u ; v ; + l]0 (u) = + (u)x0 (u) [u] 0 =[u]j r!r; l; l [u ; v + ]0 B(v) =w r r; :exp(;' (xz) ; ' (x ; z)) : : Translation into 8V/XYZ model from 8VSOS:. Tail operator: l0 k lk (u 0)=T l0 k 0 (u 0 )T lk (u 0 ) (3) l 0 l 0 + l 0 k 0 3 k 0 k 0 k 0 = ^ ^ l l+ l. (i) (Prod of four CTMs): (i) = X kl+i () lk =[k]x 4H lk H lk = P ^ ^ k 3 k k? k ^ u 0 T lk (u 0 ) lk [l] 0 T lk (u 0 ) (4) +X m>0 [m] x [rm] x [m] x [(r ; )m] x ;m m :
k 00 k 0 ^ k k = k k00 implies 0 l0 k lk = l0 l,and l0 k 0 lk =0 if k>k 0, l<l 0 or k<k 0, l>l 0. In what follows we assume that k > k 0 (l > l 0 ). From vertex-face correspondence we have lk0 lk (u 0)=(;) s[k0 ] [k] Xs (u 0 ) k 0 = k ; s: (5) l0 k 0 lk (u 0)=D l0 k 0 lk X0t; (u 0 ; )W ;(u 0 )Y s; (u 0 ) () where k 0 = k ; s, l 0 = l ; t, and Y (u) = I C dw p ;w A(v)[u ; v + 3 ; K] [u ; v + ] W ; (u) =W (u ; r; r(r;) ) W(u) =z :exp('0 (z)) ' 0 (z) =; 0 ( p X m [m] x ;Q+P log z); z ;m : m [(r ; )m] m=0 x Form factors in 8V/XYZ model can be obtained from the trace of product of,, l0 k 0 lk and, by using vertex-face transformation. 7
4 Vertex operators at reectionless points Let r =+. Then N =0 =0. Recall l; l (u) = I dw 0 C 0 p + ;w (u)b(v)[v ; u ; + l]0 [v ; u + ]0 r (x (u)b(v) r; w=z x r; ) =z; r; : (u)b(v) :: + (x ; w=z x r; + ) The contour C 0 is chosen such that the poles from the factor [v ; u + ]0 at w = x ;+n(r;) z (v = u ; + n(r ; )) are inside if n Z >0 (outside if n Z <0 ) and the poles from (x ; w=z x r; ) (v = u+ ;n0 (r;), n 0 =0 N) are outside. Thus, the pinching occurs when n+n 0 = N. Hence we have l; X m l (u) =[l] 0 c n : (u)b(u + + n=0 where c n is a number. ; n(r ; )) : 8
For O = c z = X "= " ;" (u;) " (u), tail operator k0 l 0 kl with k 0 = k k are needed. Recall that kl0 kl (u 0)= l0 l : For k 0 = k ;, l;tk; lk (u 0 )=D l;tk; lk X 0t; (u 0 +u 0 )W ; (u 0 ): Now wewishtoshowthat l;tk; lk From = 0 for t>. B(v)W ; (u) =w r; (x ; z=w x r; ) (x z=w x r; ) : B(v)W ; (u) : no pinching occurs so that l;tk; lk =0ift>. Thus, the only nontrivial tail operator is l;k; lk (u 0 ) / W ; (u 0 ). Hence, l+ l and l; l and l;k; lk is the form of exp(boson), is the sum of exp(boson). The form factors for O is therefore expressed in terms of the sum of theta function (with no integral!). 9
5 Form factors in the 8V/XYZ model The m-particle form factors ( j = x ;u j ): F m (i)( m ) m = (i)tr H i ( m ( m ) ( )O (i) ) X my = F (lk)() m l l m l l l m j= t 0l j l j+ (u j ; u 0 ; ) where l m+ = l. From generalized ice condition, (7) F (i) m () m =0 unless ]fjj j > 0g n (mod ). Note that nonzero terms in the sum on (7) comes from the case l = l l. When l = l (7) has an integral. On the other hand, (7) for l = l can be expressed in terms of the sum of theta function (without integrals), otherwize is equal to 0. Since m ; 0 B @ m m C A > m; we can obtain no-integral formulae for c z form factors, in principle. 0
Explicit expressions: Let m =. (Lashkevich's result for 8r > ) F (i) (u u ) ; = c G (u ; u ) +u (;) ;i[u ; u] [ u ;u + +] 0 f Y j= u +u! ; ug f u ;u + g 0 G (u ; u j ) (8) w/ c is a scalar, [u] =[u]j r=, fug = fugj r=, G (u) =g (z)g (x 4 z ; ) z = x u g (z) = (z x4 x 4 x r; ) (x r+ z x 4 x 4 x r; ) (x z x 4 x 4 x r; ) (x r z x 4 x 4 x r; ) G (u) = : (x ; z x 4 ) (x 5 z ; x 4 ) Let m =. A certan sum of the component of F (i) (u u u 3 u 4 ) is as follows: F (i) +;+; + F (i) +;;+ + F (i) ;++; + F (i) ;+;+ = c (i) (u u ju 3 u 4 ) Y i<j G (u i ; u j ) [u +u ; u] [ u ;u + ; u] ] 0 [ u 4;u 3 + [u3+u4 This is valid for only r =+ N 4Y j= G (u ; u j ) ] 0 : (9) (N = ).
Summary and discussion In this talk, we showed that the type II VO and type II part of tail operators can be bosonized without any integration at reectionless points. Consequently, the form factors for z c in the 8V model can be expressed in terms of the sum of theta functions. Comment: Here we seto = c z. For generic O, the tail operator k0 l 0 kl with k 0 = k s (s > ) are needed. Nevertheless, there is no pinching for l 0 = l t with t>. Thus, the type II part of k0 l 0 kl is always written without integration. Future problem: Connection with Shiraishi's scheme.