Root systems and purely elastic S-matrices

Nuclear Physics B35S (1991) 654--676 North-Holland

T SYSTEMS AN

PURELY ELASTIC S-MATRICES Patrick D®REY"

Service de Physique Théorique de Saclcar*~ F-91191 Gif=sur-Yvelte Ceiléï. France

Received -1 December 1990 A "fusing rule" for the three-point couplings in the known purely elastic scattering theories is proposed, in terms of which various universal features of these models have a simple explanation . A hypothesised expression for the general S-matrix element is shown to satisfy the bootstrap relations following from these couplings . and used to relate each S-matrix to the matrix of a particular Coxeter element in the relevant basis of simple roots.

1. Introduction The purpose of this paper is to describe an alternative characterisation of the three-point couplings- o r fusings. in the known purely elastic scattering theories connected with the simply-laced Lie algebras. Such theories have arisen in two areas : in perturbations of certain coset conformal field theories of the form g' x g 1 /g 2 [1,21. and in the study of affine Toda theories [3-12]. The S-matrices found in these two contexts are not quite the same, the proposals for the Toda theories having unphysical poles not found in the minimal hypotheses for the perturbed conformal theories, but the fusing structures are identical, and the considerations below are relevant to both situations. For details of the background the earlier papers should be consulted ; sect. 2 summarises the most relevant results. After a brief review of some facts about Weyl groups in sect. 3, the proposed fusing rule is described in sect. 4. (The term "fusing rule" is used throughout, to distinguish this rule from the fusion rules of conformal field theory .) Sect . 5 uses some more detailed information about Weyl groups to justify many of the statements made in sect. 4, and demonstrates how, for fusings obeying the rule, the bootstrap equations for the conserved charges are automatically satisfied by the eigenvectors of the Cartan matrix. Sect. 6 contains a discussion of the implications of this picture for various previously puzzling "universal" features that have been observed in these theories, with a couple of examples, while sect . 7 deals with the S-matrix, giving a general expression for the x Email address : dorey(a frsacl Lbitnet x x Laboratoire de la Direction des Sciences de la Matière du Commissariat à l'Energie Atomique.

0550-3213/91/$03 .500 1991 - Elsevier Science Publishers B.V . (North-Holland)

655

S-matrix elements and demonstrating that it satisfies the S-matrix bootstrap equations . Finally, some concluding remarks are made in sect. 8. The motivation for the rule to be described below was piovided by the observation that the mass triangles for various of the affine Toda theories could be found inside various polygons [7,13,14], and the subsequent observation [151 that these same pictures could be found in papers by Coxeter [161, as projections of various polytopes. 2. Purely elastic S-matrices The main interest in the following will be in the structure of the bootstrap equations, which constrain both the S-matrix elements S",,(®) and the eigenvalues q,° of the conserved charg,- acting on single-particle states (defined by Q, A"(8)> = q;' e`" I A "(O )>, where A J e) denotes a particle of type a and rapidity 0). A bootstrap equation arises for every fusing ab - c in the theory, that is for every non-zero three-point coupling C"'"'. For the S-matrix elements, the equation is [1] SdA 8 ) = S,,,,( 8 - iQ;',-) S,,(e + tUha.) ,

(2.1)

where the U's are certain angles, defined for every triple of particles possessing a nonvanishing three-point coupling . For the conserved charges, the bootstrap equation has a very similar form (and is in fact satisfied by the logarithmic derivative of the S-matrix [5]) (2.2) For later use, it will be more convenient to write this in a slightly different form . The effect of charge conjugation on the conserved charges is given by (2 .3)

This, together with the fact that rewritten as q ~~ + h e "'u h q .s

+ Q. + + q" A

e 's( U«h+ Uh~

al:cws eq. (2.2) to be - p

(2 .4)

where the angles U = 7r - U are called the fusing angles. In this form it is clear that the condition amounts to the closing in the complex plane of a triangle with sides q,s', q,;' and q.;', as shown in fig. 1 . A feature of the known theories is that the fusing angles are always integer multiples of -r/h, h being the Coxeter number of the Lie algebra associated to the theory. Thus for spin s, the angles in the triangle are certain integer multiples of sr/h. Note that the lengths of the sides may be negative, though for spin one they are the (positive) masses and the triangles are

' / R«mi s" +.D1e77Qs

65 6

q

a

S

Fig. 1 . The conserved charge bootstrap.

just the mass triangles . Given a set of masses and three-point couplings, the mass triangles serve to define the fusing angles, the angles in the "higher spin" triangles then being fixed as multiplies of these angles. A priori it is something of a surprise that sets of masses and couplings can be chosen in such a way that some higher-spin triangles do also close; a virtue of the rule to be given below is that it goes some way towards explaining this fact. The mass triangles were studied in ref. [7] in connection with the tiling problem associated with the identification of higher poles in the S-matrix . For the theory associated with the rank-r Lie algebra G, it turns out that nontrivial solutions to the conserved charge bootstrap only occur if the spin s, modulo h, is equal to an exponent of G. Furthermore, each of the r particles in the theory may be assigned to a spot on the unextended Dynkin diagram of G, and hence to a particular simple root « ;, in such a way that the set of conserved charges at spin s, when assembled into a vector q, = (q" I, q.(r 2 , . . . , q.,", ), forms an eigenvector of the Cartan matrix of G, with eigenvalue 2 - 2 cos -rrs/h [ 12, 6]. Thus, Cq, = A,q,

A ., = 2 - 2 cos

7rs/lr ,

(2 .5)

where s is an exponent of G. The resulting assignments of particles to spots on Dynkin diagrams are shown in table 1, the numbering of the particles being in ascending order of mass, the spin-1 charge . Collected together these masses form the Perron-Frobenius eigenvector for the adjacency matrix 1= 2 - C of the Dynkin diagram of G. Eq . (2 .5) turns out to follow naturally from the fusing rule to be defined below .

P. Dorcjr / Roos sysieins

657

For reference, the eigenvectors are shown in table 2, the labelling used being in accordance with a splitting of the roots into two subsets {a;} and {ß,} to be defined in the next section . The particular expressions given are taken from ref. [171; they were in fact obtained using the bootstrap equations for the affine Toda couplings and then checked against eq. (2.5). Their values coincide with those given in ref. [181, apart from avoiding a problem in the latter's expression in the E 7 case at s=9. One more piece of notation will be introduced at this stage. If the Dynkin diagram of G has an automorphism a, then there is a corresponding degeneracy in the mass spectrum, with in . , = in,,,,,, . For all but the D, theories, the two particles are conjugate to each other, CA ., = A,r,r,. Allowing the charge conjugation operation to act on the particle labels, so that CA ., =A ii ,, this implies that in all but the (self-conjugate) D, theories, â ; = Qa ;. 3. Weyl groups This section is mainly included to establish notation; a good reference for the theory behind it is the book by Carter [191. Let 0 be the root system for the simply-laced Lie algebra G, of rank r. For each root a E qb, there is a corresponding reflection of Rr : wjx) =x-2

(a, x) (a, a)

a.

The reflections all map 0 to itself, and the finite group W that they generate is called the Weyl group of 0 . In fact, a subset of the w,, is enough to generate W, namely the reflections associated with any set of simple roots. The Coxeter elements of the Weyl group will be important below . These are elements of the form .1W.I W . . . W,,,-,

where a,, . . . , a,. make up a system lI of simple roots for 0. The Coxeter elements form a conjugaty class in W, and it is convenient to focus on a particular representative . To define this, II must be split into two subsets, each of which contains only mutually orthogonal roots: Il =la,, .

. .5akl U{PI, . . .-)ßr-k} '>

ai .ai-ßl'ß», =0 .

(3.1)

This splitting is unique (up to swapping the two subsets) and corresponds to a two-colouring of the (connected) Dynkin diagram of G (see table 1). Now elements w and w,, of W are defined by Wa=W« 1

. . .W«,,

Wh =Wßi . . .Wß-R .

TAMT I Mass assignments and two-colourings for the simply-laced Dynkin diagrams

A2n+l

An :

D2n+l :

:

&-O-~- - - -

MI

M3

M2

0

My

Mi-

0

M2n-3

0

0

0

A

E7

MI

Eg :

P Dorey / Root systems

659

TAne_r 2 Cartan matrix eigenvectors with eigenvalue 2 - 2 cos ®,. ®, = (-. /h )s. (second vector for D is s = n - 1, B, _ ,, /2) sin ®,

a4 Ps ail

ai f c

E, P2

a4 P3

a-1

stn 2®, stn 3®,

1

r,

1

a®i - 4

2cos2®, 2 cos 3®,

1.

sin 48, stn 5®, sin 6®,

D

All - s

a,

sin 110, 1, sin 80, - sin 20, sin 30, sin 20, sin 1

ß; E7 a 7 106

a4 Je t

a, 06

04

ß3

ai

2 cos ®,

B

,

-

s

an

exponent

1 0

2 cos 4®,'

sin 6®, - sin 48, sin 7®, - sin 3®, (sin 9®, - sin 7®, + sin s®, sin 4®, sin 30, sin 2®,

-

sin e, )

sin ®,

sin 70, - sin 5®, sin 88, - sin 49, sin 148, - sin 80, sin 5e, sin 4e, sin 39, sin 20, sin e,

Then w = w;, = 1, and w = ww,, is a Coxeter element . The Coxeter elements have period h, and Coxeter number of G, and so (w), the group generated by w, is isomorphic to Z h .

4. The fusing rule The idea is to split the root system into r disjoint subsets, one for each particle type in the scattering theory. To this end, for any root a define R(a) to be the set of all images of a under powers of the chosen Coxeter element w, that is the orbit of a under (w) . The projections to be described in the next section clearly show that each R(a) contains h elements, so r- such subsets, if disjoint, will exhaust the full set 0 of r - h roots. To specify a disjoint collection of the subsets, the splitting (3.1) of the set H of Jsim11!ple roots can be used. The subsets (R(a i )) and (M-13)) are all disjoint (this will be justified in the next section), and so for all simply-laced algebras, (P =

P Dort y / Root systems

66()

U [R(a;) U W-13)] . This motivates the following definition, for any simple root y : RI/ =

if y E la , , . .-a k ) ; if yE {ßI , . . . . ßi--k}

R(Y) R( -y),

Given this definition, -

U

y E11

(4 .1)

(4 .2)

Rv .

Thus each particle is associated with a coset in (P/ W, and via eq. (4.1) with a particular simple root . (The negation of half of the simple roots in eq. (4.1) is necessary both to ensure that the R Y are disjoint in all cases, and to facilitate the conserved charge argument to be given in the next section .) The action of charge conjugation on particle types associated to simple roots was given at the end of sect . 2; on the cosets there are three equivalent ways of achieving the same effect : R y - -Ry=w Ry=w,,R .y . This will be demonstrated later, but note that since w = w;, = 1 and wwt, = w which acts on the cosets as the identity, all these definitions are consistent with 1. The proposed fusing rule can now be given. Letting i, j and k label any three of the particle cosets, the statement is C`'ti i .e .

0

U

iff 3 roots a E R ;,,8 E R i , y E R ti

with a +,8 + y = () ,

iff 0ER ;+Rj +R ti .

(4.3)

The nexi section fills in some details missing above, and explains why this rule might be expected to reproduce the known couplings.

5. Projections and the conserved charge bootstrap In order to understand the fusing rule, some more information on the actions of w , w,, and w on the set of roots is required . This can be obtained by using the following result, which is quoted directly from ref. [19] (pp. 158-161), though in a slightly simplified form as only simply-laced root systems are of interest here. Result . Let (â ;, ß;) be the dual basis to the simple roots (a;, ßi}, C the Cartan matrix, A ., = 2 - 2 cos 0,. an eigenvalue of C not equal to 2, and q,. the correspond-

P. Dorz r / Rout

s1ste/ns

661

bs

bs Fig. 2. An invariant subspace of

i1-.

ing eigenvector . Define two vectors a,, b, in R" by a., =

q.;Y'ar

b,

= E q.~ , fi; .

(5 .1)

Then (a,., a,) = (b,, b.,), and the angle between a, and b, is 8,. . The dihedral subgroup (w, w,,) acts in the subspace spanned by a, and b,, with w being the reflection in the line Ob,, w,, the reflection in the line Oa, (see fig. 2). Hence w = wtip',, acts in the subspace as a rotation by 2e,, from a, to b, . If e., = 7rs/h, then letting s run over the exponents of G gives all the eigenvalues of C, so the period of w is indeed h (since the exponents are all integers, and one of them is 1). The complete action of w in root space is thus a series of rotations in orthogonal planes by angles 2H,, where s is an exponent of G. (Actually this double-counts : spin s and h - s correspond to the same plane, viewed from opposite sides . This can be verified by examination of the explicit expressions for and q.~ ,, and their behaviour as s -> h - s.) Proof. See ref. [191. For the one case not covered by this result, when C has an eigenvalue 2 corresponding to an exponent s = h/2, with e., = 7r/2, it is simple to adapt Carter's proof to see that one of a, or b,. is nonzero and that w acts in the relevant one-dimensional subspace as a reflection about the origin . It is now possible to obtain a useful characterisation of all the roots in terms of the simple roots and w, via a study of their projections into the various invariant subspaces . For the simple roots, eq. (5 .1) implies (ai , aj = q~., r , , (a i ,b,) =U,

(ß .i, b.,) = q,~, . (ß., , a,) = o ,

662

P. Dort'l' / Root s)slC'!)1s

Hence Pj ai) = q.", a, where P,. is the projector into the two-dimensional subspace spanned by a, and b,, and Id, -b.1 ) are dual to {a .S , b.1 } in that subspace . (The modification when s = h/2 is straightforward .) Thus the simple roots project onto two lines in the invariant subspace, at relative angle 7r - 8, . The chosen coset representatives Jai, -ßß,i) hence project onto two lines at relative angle ®,: cr,

P,(ai) = N,

^

a., ,

P,( -,Bi)

= q.~,b

(5 .2)

Since d, and b, have equal magnitude, this implies that the projections of the simple roots have lengths proportional to the q , the components of the eigenvectors of the Cartan matrix. To treat the R y defined in the last section, consider the set _t~, of all roots which may be written wl'ai or w 0'(-8d for some p in the range 0, . . ., h - I (recall that w has period h), and their projections into the various invariant subspaces . The Coxeter element w acts in each subspace as described above, the two lines being rotated in steps of 2Trs/h . Hence the projection P,(w''ai ) lies on the rotation of d, by 27rps/h, and similarly for P,(w1'(-t3.; )). In particular, in the s = I subspace the roots in J;~' project onto the vertices of a number of h-gons inscribed in circles with radii given by the particle masses (recall that the spin-one conserved charge of a particle is just its mass). Each R, gives one such h-gon, and so does indeed contain h elements . If there are no mass degeneracies, then all the /1-gons are distinct and the elements of _9~57 project onto rh distinct points, and so are themselves all distinct . There being rh roots in total, this means that _9' contains all of the roots in (P. Thus the desired splitting of 0 into r disjoint cosets (Ry),/ E  , each containing h elements, has been achieved . In the more usual situation that there is some mass degeneracy, extra work is required as some of the roots in Y may have the same projection into the s = I subspace, and it is possible that the definition of _9' involves some duplication . There are two cases to be considered, according to whether the simple roots corresponding to the duplicated mass project onto the same or different "arms". This is the same as asking whether the degenerate spots on the Dynkin diagram have the same or different colours in the two-colourings shown in table 1 . Recalling that the degenerate spots are related by diagram automorphisms (the one exception to this, D 3 + ,, will get a separate mention below), it is clear from table I that the latter possibility only occurs for Ci = A .,,,. This case will be treated second . If the mass-degenerate simple roots project onto the same arm, then it is straightforward to see that the situation is essentially unchanged from that already

P. Doi-ey / Root systems

663

described . If the two degenerate roots are ai and aJ , with P,(a;) = P,(aj ) = then wt'(a ;) and w`'(a) have the same projection if and only if p =q, so the only possibility for duplication would be if wP(a ;) = wP(a,) for some p. But then acting on both sides with w h- t' gives a; = a,, a contradiction . The same reasoning holds for the negatives of two simple roots /3;, Oj projecting to b,, and, with small modifications, to the situation when more than two simple roots project to a single point on the s = 1 plane . This is relevant to the only mass degeneracy not accounted for by diagram automorphisms, namely that found in the D; + , theories where an "accidental" (for n > 1) degeneracy is found between the "spinor" masses in, and nn,, and in, the mass of the nth "scalar" particle (note, the relevant three spots on the D 3n +, Dynkin diagram have the same colour for any n). For A,  the position is a little different . In these cases the coset representatives for particle and antiparticle project onto different arms and the possibility of duplication occurs if P,(n, t'a ;) = P,(- ß;) for some p. Looking at fig. 2, the requirement is 2pe, = -0,, but this is impossible as p must be an integer. Hence for A -,,, the two "equal mass" polygons in the s = 1 subspace are offset, and the projection into this subspace is enough to distinguish all the roots. Furthermore, as h here is odd, the polygons have an odd number of sides, and negation of all the roots interchanges them and thus swaps between particle and antiparticle . The example of A, illustrates all this clearly, showing how the choice of -ß; instead of Iß ; manages to capture all the roots. Thus (4.2) has been justified in all cases . Furthermore a complete description of the projections of all the roots into the invariant subspaces of w has been achieved, as illustrated in fig. 3. To see that the definition of charge conjugation given for A-,,, in fact holds for the other cases as well, the higher-spin projections must be examined, as for these theories particle and antiparticle cannot be separated in the s = 1 subspace . Let y be a simple root, and consider the projection of the root K, h /-'( - y ) into each

Fig. 3. Projections of roots.

664

P. Dover / Root systems

invariant subspace (note, for all these other cases h/2 is an integer). Looking at fig. 3, and recalling the charge conjugation rule (2.3), it should be clear that

Having a common projection onto every invariant subspace, wt' l'( - y) and y must be equal, and so R y = - R,, in all cases. From their definitions, it is clear that w negates each a;, and w t , negates each ßßj . Since their combined action on the cosets is the identity, this is sufficient to see that these two operations also have the effect of negating each coset . (Note, since n',,w n = w -"w,, and wt,w n = w-''wt,, w and wt, do indeed map the cosets onto themselves .) This concludes the explanation of the formalism given in the last section. It remains to see why this rule might correspond with the couplings in the already known purely elastic scattering theories. The basic reasoning will be given first, and then some more detailed information about the fusing angles, necessary for later sections, will be derived . If C''k =A 0 then by the fusing rule there must be (not necessarily simple) roots a E R ;, /3 E R j , y E R k with a +,6 + y = 0. Hence the roots a, (3 and y placed nose-to-tail form a closed triangle in R'-. This triangle must project onto a closed triangle in each invariant subspace of w, and the results described above are sufficient to see what these projections must be . In the s = 1 subspace the sides have lengths m ;, ffi i and frzk , and thus the projection is a mass triangle . The relative angles between the sides must be integer multiples of 7r/h and are exactly the fusing angles defined for that mass triangle. Projection into an s > I subspace simply multiplies these relative angles by s, and replaces the masses m; by the numbers g ;. Referring to fig. 1 it is clear that the closure of this triangle exactly corresponds to the solution of the bootstrap equation (2.4), so long as the spin-s conserved charges are chosen to be proportional to the relevant eigenvector of the Cartan matrix. Since the charge that this procedure assigns to particle i is independent of j and k, this construction simultaneously solves the conserved charge bootstrap equations for all non-vanishing couplings . Note that the values taken by the charges for each coset show that the assignation of particles to simple roots that this construction provides via (4 .1) is indeed consistent with the earlier one given by eq. (2.5). However it does not of itself prove that the fusing rule given above exactly reproduces the couplings of the known theories, even though it is very strong evidence in support of the claim. Further support comes from the consequences of the rule to be described below, which reproduce various features previously observed for these models. For later use, the values of the fusing angles for each coupling will now be calculated in terms of the particular roots making up the relevant mass triangle . This involves consideration of the various possibilities for the distribution of the chosen coset representatives between the two arms in the s = I subspace .

P. Doie)l / Root S1sl('inti

It is convenient to use a single complex coordinate in each invariant subspace. The action of wt', a rotation by 2pq,, then becomes multiplication by e''-p',, so that for any root a, P,(wt'a) = e i ,pU, pja) (Note the formal similarity between P, and the conserved charge QS, in terms of which w acts as a finite Lorentz boost, increasing the "rapidity" by 2i8, .) As only relative angles are of interest, b., can be taken to point along the positive real axis. The projections of the roots shown in fig. 3 then become pc( wp,a i )

-_

~,Sf, e i(2p,+ 1)0, ,

pV( W l),(

- RJ ))

- Rß,

ei(?p,)B, .

(5 .3)

Consider first the case that particles i, j and k have the same colouring on the Dynkin diagram, so that their coset representatives ai, aj and a k project onto the same arm. In terms of these roots, eq. (4.3) becomes (5 .4)

wt''ai + wt'~aj + wp4 ak = 0,

for three integers pi, pi and P0 which can be ordered so that Pi < Pi < Pk . On projection into the spin-s subspace and multiplication by a phase this becomes q',r, + 9',r, ei2 -'(",- ",)N , + q .'Yl

, el,s(/~A-p,)N = 0 .

5 .5

Comparison with eq. (2.4) shows that for this case the fusing angles, in multiples of 0 1 = 7r/h and taken modulo 2h, are just Uü =2(pl Pi) ,

= 2(Pk - Pi ) ,

U~i=2(Pi-Pk),

(5 .6)

and that all the bootstrap equations for this coupling are indeed satisfied. If the representatives project onto different arms, the 2pi' s corresponding to a projection onto â simply have to be increased by one unit. So if the first two project onto â, the third onto b, then the angles become U -2(P;-Pi)+

I1; -2 (Pk -P;

Uki=2(Pi-Pk)+1 . (5 .7)

(The other possibilities are straightforwardly calculated .) This data implies the following rule for fusing angles: two particles associated with spots of the same colour on the Dynkin diagrams shown in table 1 only fuse at even multiples of 7/h, and particles of opposite colours only at odd multiples of -rr/h (this holding independent of the type of the bound-state particle produced) . This can be checked against the affine Toda couplings. One consequence is that in a specific S-matrix element Sii (0 ), the (physical) simple poles, which correspond to the

66 6

P. Dorey / Root systems

. formation of bound states, should always be separated by even multiples of 7r/h This fits in with the "brick wall" observation in ref. [17], that in a given S-matrix element all physical poles are separated by even multiples of 7r/h, despite there being no obvious necessity for this to be so. (The observation was so named because of its implications for the way in which the building blocks to be defined in sect . 7 can be "stacked" to make up an S-matrix element.) 6. Universality in purely elastic S-matrices In addition to explaining the existence of non-trivial solutions to the conserved charge bootstrap, the picture outlined above offers the hope of explaining many of the "universal" features that have previously been observed in the ADE scattering theories. If a desired result can be formulated "upstairs" in R'*, it becomes a general statement about simply-laced root systems which with luck can be universally proved . Thus the feature observed at the end of the last section was established without reference to any particular theory . However, there are more elegant examples to be found in various results which are true independent of particle type. If the reformulated result in root space makes no mention of the choice of simple roots and Coxeter element, then the information on the particle type associated to each root is lost . Only a single statement about root systems needs to be proved, which on projection (i .e. after the choice of a particular set of simple roots) resolves itself into a set of results, one for each particle type . The two examples below illustrate this idea. 6.1 . FLIPPING

The concept of flipping was introduced in ref. [7], as an aid to the identification of possible on-shell diagrams relevant for the perturbative treatment of higher poles in the affine Toda S-matrix. If affine Toda field theory is indeed integrable and purely elastic, then non-diagonal scattering is forbidden and this property should be visible at tree level in the perturbation theory . It is certainly possible, using the available couplings, to draw Feynman diagrams which contribute to an off-diagonal process, so the expectation is that the total contribution of these should cancel . Clearly, for this to happen the poles of the relevant diagrams must cancel, and it is this requirement that forms the basis for the flipping idea . The pole of a tree-level Feynman diagram occurs when the intermediate particle is on-shell, and hence when the dual diagram is formed from a pair of mass triangles . An s-channel pole has a positive residue, while t and et channels have negative residues. Thus to cancel a diagram contributing an s-channel pole, it must be possible to find a (t- or ii-channel) diagram involving the same external particles, which goes on-shell at the same point. In terms of the dual diagrams, this means that for any "non-diagonal" quadrilateral constructed from a pair of mass trian-

P. Doreh / Root systems k

667

t A 1 i

Fig . 4. Flipping on-shell processes .

gles, it must be possible to find another pair of triangles which tile the same quadrilateral but in the opposite sense, with a deficit parallelogram in the case of a u-channel cancellation . The two possibilities are illustrated in fig. 4. The first (t-channel) option will be called a type-I flip, the second (ii-channel) a type-II flip. If non-diagonal processes are to have vanishing amplitudes at tree level, then a flip of type I or II should always be possible. More details of the idea and its uses in constructing the more elaborate tilings relevant for higher poles are given in ref. [7]; however there was no proof that the procedure was always possible, only an expectation based on the assumed integrability of the theory and the checking of numerous examples . However in any theory where the couplings obey the fusing rule stated above, it is possible to give a general proof of flipping, based on a known result for root systems. Assume that there is an s-channel process involving particles i and j scattering to k and l, with intermediate particle q, and that i 0 k, j T 1 so that flipping should be possible. Couplings C'!`' and C`J !' I are non-vanishing, so upstairs there must be roots a, ß, y, 5, E representing particles i, j, k, l, q respectively, satisfying a + (3 - E = 0, E - y - 5 = 0 (recall the charge-conjugation rule). Note that by applying w to one triple it can be arranged that the representative for particle q is the same root E in both triangles, and, applying - w if necessary to reverse the orientation of one projected triangle, the complete picture will look as illustrated in fig. 5, with the projection into the s = 1 subspace giving the s-channel dual diagram picture . Thus upstairs there are four roots satisfying a + R ± ( - y) + ( - 5) = 0, with no pair equal and opposite (this follows from the fact that the process under discussion is non-diagonal) . A standard result (see for example, p. 55 of ref. [ 19]) then states that if a + /3 is a root (which it is in this case, being equal to E), then either ß + ( - y) and a + ( - 8), or ß + ( - 5) and a + ( - -/), are roots (or both). Projecting the two options down gives a type-I flip in the first case, and a type-11 flip in the second. Thus flipping is always possible. Though motivated by calculations in affine Toda field theory, this result does of course also hold for the perturbed conformal field theories, as it only refers to the fusing structures of the models.

R Am" / MN4

068

I

Pl

Fig. 5. A prpJJection to an on-shell pair of mass triangles .

6.2 . BUBBLE COUNTING

In refs. [1121. a curious property of the three-point couplings in the AIDE scattering theories was noted. Namely, if a particle type i is fixed and the number of ordered pairs ( j, k) such that Ciik * 0 is counted, then the result is h - 2. independent of the chosen particle i . (In the lowest-order treatment of mass renormalization in affine Toda theory, this is equivalent to counting how many .) ; bubble diagrams contribute to the finite renormalization of in In root space, this is equivalent to fixing a root a E: R, and asking how many ways its negative (also a root) can be written as the sum of two others, so that a coupling triangle can be formed. This statement is not completely trivial, and some care has to be taken to ensure that the counting is happening correctly . For example, the root triangles a,,8, y and wa, iv.8, ivy should not both be counted as they represent the same coupling. This has been dealt with by fixing the ruot a . A little more subtle is why the root counting should reproduce the fact that the pairs j. k must be ordered . This is best seen by considering the projection into the s = I subspace . If j * k then there are two possible orientations for the projections of a pair of roots which sum to -a (note that it is possible to swap between the two orientations by acting with which acts as the identity on the cosets), and so two contributions are expected to the root sum . If j = k, these two are in fact just one can, and so the correct counting is rcprcduced . Even without a "universal" proof of the result for all simply-laced algebras at once, it is straightforward to check that for each of A, D and E the total number of ways of expressing one root as the sum of two others is indeed h - 2, and so the result is proved . Note that this example illustrates the comment made at the beginning of this section, the fact that a E R i being "lost" once the result is re-expressed in root space . Once this has happened, the equivalence of all the

P DonT / Root siSteins

roots of a simply laced Lie algebra means that only one root needs to examined to obtain the result for all particle types at once. It seems plausible that simoiar methods can be used to establish various other universal features of the ADE scattering theories. although the identities for root systems required will get progressively more complicated . 7. The S- at' The main content of this section is a proposed expression for the general S-matrix element in the ADE scattering theories. This will be given in terms of the "building blocks" introduced in refs. [4,51. For the affine T a theories, these were defined to be (x- 1)(x +1) t {}

(x- 1 +B)(x+ 1 -B) .

(7 .1)

where ({)

sinh(8/2 + i-x/2h ) sinh(8/2 - i-x/2h) '

(7

.2)

and B(ß) is a function of the coupling constant, the form of which is ifclevant to the bootstrap equations . With the factors of h included in eq. (7.2). it turns out that x takes integer values. The conjectures for the perturbed conformal field theory S-matrices are identical in form, but lack the coupling-constant dependence and so can all be written in terms of (7.3) {x) =(x- 1)(x+ 1) . The distinction between these two types of blocks will be ignored below, as they give rise to identical physical pole and bootstrap structures. The bootstrap equations (2.1) can be rewritten in a way which hides the 8-dependence . Define a shift operator r,. acting in the space of functions on C by (J-J )(0)

= f e+ (

i -y

= )

f( ® +i®,y) .

(This is "half' of the ~W operator defined by Christe and Mussardo in ref. [9].) The bootstrap equations become Sly = (`~7-U"A/01Sli)(/01SJj)"

(7 .4)

(The indices have been changed to fit the- labelling for particles being used in this

670

P. Dorc~y / Root systems

section.) Since J,. does not map the functions ( x ) to other such functions, it is helpful to work with a slightly more primitive unit, (x) + = sinh

(

8

2+

i7rx 2h )

,

so that . y(x i = (x +Y) , . The concern in this section is almost entirely with the bootstrap equations, and so it is convenient to modify the notation at this stage, and to assume that the basic blocks for S-matrices, (7.1) and (7.3), have been defined in terms of the simpler unit ( x )+ rather than ( x ), so for example the "minimal" block relevant for perturbed conformal field theory is now (x) +--- (x- 1) + (x+ 1) + , and it should be remembered below that in this notation the block used in earlier publications was (x) +/( -x) + . The revised block satisfies

Indices i, j, . . . will be used to stand for particles, and particle i will be described as being of type a or ß according to whether the corresponding simple root y; belongs to the first or the second subset of the simple roots defined in eq. (3 .1). This distinction is important, as the detailed form of the proposed two-particle S-matrix elements depends on the types of the particles involved. There are three cases: (a) Particles i and j of type a :

p=(1

11 ('r( 21n +

~

(7.5a)

(b) Particle i of type a, j of type ß : /I-

(c)

(7 .5b)

Particles i and j of type ß : S

( 7 .5c)

The sign of ßi in the last two equations may seem surprising, but recall that by cq. (4.1), the coset representative for R ; in these cases is -t3j .

P. Dorey / Root systems

671

Strictly speaking there is a fourth case to mention, namely i of type ß, j of type a. The expression for this is h-

(7.5d)

{2p}c~-"'",t

However this should follow directly from eq. (7.5b), given the symmetry Sii = Sii. In fact, neither this property nor unitarity (the requirement S(8)S( - 8) = 1) are obvious consequences of the expressions given in eq. (7.5), but they follow from the relations : (ai,

(a i , -

wP

ai) = (ai, w"a i) = - (ai,w-n-tai),

Wt' pi) =

(ej ,w" ai) =

~i+ -wnßi) = (~

'

i,

-(ai, -w-IPI), -W_

h't'pi) p i)

= -(~i,

+'ßi)-

These can be proved by considering the contributions to the inner products from each invariant subspace in turn. The symmetry property will be used below at various points. The first check to be made is that this hypothesis does indeed satisfy the bootstrap . Note that the differences between the three expressions given are in fact forced by the bootstrap equations . Consider first the case Ciiti - 0 with particles i, j and k all of the same type. This will check the internal consistency the three proposals, although the bootstrap for the fusing i, j - k will never relate the different cases. As the mechanism is identical for i, j and k all of type a or all of type /3, only the first possibility will be discussed . By considering the fusing rule and projections into the s = 1 subspace, it is straightforward to see that in this case aT.

=W ~À

/-ai

+W

A,

12

ai .

(7 .6)

(The angle U is assumed to be given as a multiple of 8, here .) Note that w rotates by units of 28, in the s = 1 subspace, and that all the powers of w are integers by the observation at the end of sect. 5. The relevant projection is shown in fig. 6.

k Fig. 6. The fusing ii - ti ; i, .i and ti of the same type .

672

P. Dor(~r / Rtx)r systems

If particle 1 is also of type a, then, on substituting for S,; and S,, using eq. (7.5a), (7 .4) becomes Sl _

h-l ~~=tl

1-=ip +

= F1 12( p

n=O h-l

P=O

12p

_

''

1 - ~~ k +

+ I

i

c(ià .tr

1

h-l P=O

~~(Y,, tt' , lt,

_

{2p + 1 + Cjk }(+'~

12

p + Uj'/2) +

-(t, + tv "td . =(t ' )

h-l p=0

'}+

where eq. (7.6) was used in the last equality . Thus eq. (7.5a) is reproduced. It is clear that if 1 had been of type /3, eq. (7.5d) and hence (7 .5b) would have been confirmed . The same reasoning holds for a ißß -,8 fusing, providing checks for eqs. (7.5b) and (7.5c). The only subtlety comes with a fusing relating particles of different types, and it is this that fixes the relative forms of eqs. (7.5a-c). The point to note is that roots a; and -ß,r of different types project onto different arms in the s = I subspace . This modifies fig. 8 slightly, and hence changes the powers of w from those appearing in eq. (7.6). The various possibilities all behave similarly, and only the case k of type 8, i and j of type a will be treated here as this is sufficient to relate eqs. (7.5a-c). The modification to fig. 6 is shown in fig. 7, and (recalling that the representative for k is -ßx.) eq. (7 .6) becomes - w ( û;A - i)/2a . + w, - (UA,+ i)12a . . (7 .7)

(Note that the powers of w are again integers.)

k Fig. 7. The fusing ii -> k, i, j of type a, 1, of type . ß

P. Durcy / Runt systems

673

For particle 1 of type a, eq. (7.5a) can be used in eq. (7.4) to find: SIX-

_

h-1

1-1 {2(p

- (U~ _

+

na ' )

2(p + (Ùj' +

P =()

1)/2)

t él . t

P=O h-)

F1 12p)

P=O

using eq. (7.7) in the last line. Thus eq. (7.5b) has been deduced from (7 .5a) . The effect of the "mixed" fusing is to reduce by one the numbers appearing inside the blocks, and so it is clear that taking 1 to be of type ß, eq. (7.5c) follows from (7.5b), via (7.5d) and the symmetry property . The other fusings go through in an analogous way, and so eq. (7.5) does indeed provide a set of functions which obey the S-matrix bootstrap - in fact, many such functions as the only property of the blocks Ix)-, used above was that {x},= {x + 2h} + (this was necessary to re-express the various products when verifying the bootstrap). However it is not very clear, apart from checking some simple examples, why choosing the minimal or full definitions of the blocks given above should reproduce the correct physical pole structure . Rather than going into this question here, the rest of this section will describe one non-trivial check that can be made on the proposal with only a small amount of work. The relevant observation is that the powers to which the blocks are raised, (â;, wPaj ) and the rest, are precisely the matrix elements for the various powers of the chosen Coxeter element w in a basis provided by the simple roots. If it is assumed that the affine Toda S-matrix elements are given by eq . (7.5), then the logic can be reversed, and the matrix form of w read from the S-matrix (the full set of S-matrix elements for all theories can be found in ref. [5]). All that is needed is to identify the power to which certain blocks are raised, and to take account of the negation of the roots of type ß . From eq. (7.5), the relevant blocks to examine are as follows : S.,../ = (3)(+,. :.«, . . . , {2i+(n"ß,) . . . , S,,,ß, Sß~

, =

I2)(0,",) .

+

..

I }-+

(f3, . :,~,), . . .

(7 .8)

(The dots stand for other blocks not relevant here.) Using this, an expression for w = ww,, in the basis of the relevant set of simple roots can be obtained for each simply-laced root system . The results are shown in table 3 . The labelling of the rows is shown to the left of each matrix, with roots of type a corresponding to the white dots in the two-colourings shown in table 1, type (3 to the black dots. The form of the general A or D matrix should be clear from figure, but there are slight modifications to the bottom right entry for A, and D, + , . For the A theories_ this

P. Dorey / Root systems

674

TABLE-_ 3

Matrices for u' = tis'~rit'r,, deduced from the affine Tuda i-matiices ßl a, ß;

a4 a,

a,, -, ; D ßnarr-4 ßn- ;

1-1 0 0 0 0 . . . 1 1 1 1 0 0... 0-1-1-1 0 0 . . . 0 1 1 1 1 1 ... 0 0 0-1-1-1 . . . 0 0 0 1 1 1 ...

W ß33 ß, E, a-t ß;

ai

-1 0-1 0 0 0 . . . 0-1-1 0 0 0 . . . 1 1 2 1 1 0... 0 0-1-1-1 0 . . . 0 0 1 1 1 1 ... 0 0 0 0-1-1 . . .

a,

ßc, a,, ß7

a;

ß;

ai

0 1 0 1 0 0 -1-1 0-1 0 0 0 0-1-1 0 0 1 1 1 2 1 1 0 0 0-1-1-1 0 0 0 1 fl 0

a, ß;

1

103 1

E7

0 1 () 1 1-1 0-1 0 0-1-1 1 1 1 2

() 0 0 1

0 0 ()

1 1 0-1 0 1

0 0 ()

0 0 0

1 () 0

0 0 0 1

0 1 0 1 0 0 O) -1-1 0-1 0 0 0 0 0-1-1 0 0 0 1 1 1 2 1 1 0 0 0 0-1-1-1 0 0 0 0 1 1 1 1 0 0 0 0 0-1-1)

0 0) 0 0 0 0 0 0 0 0 1 1 1-1 1 0)

is because S T-1 = S  _ (1), and lacks the factor {3} found in S,r,, for other values of a . Thus if particle 1 is of type a, the bottom right entry should be 0 rather than ± 1 . This occurs for A, . Similarly, for D,  + ,, particle 1 is of type « and S 11 _ (1)(h - 11, so the bottom right matrix entry should again be U. Various checks can be made on these matrices . The first arises from the one degree of freedom in the splitting of the simple roots, namely that the two sets a and ß can be interchanged, so that type a now corresponds to the black dots in table 1, type ß to the white dots. Noting that eq. (7.8) is not symmetrical between a and ß, this might seem to be a problem. However, recall that the particular Coxeter element w was defined to be wrr w,,, so swapping between the two sets, which interchanges w, and w,,, should send w to w - ' . Thus another matrix can be found inside the S-matrix by swapping the labellings on the two-colouring of the Dynkin diagram, and this matrix should be the inverse of the first. This was checked in all cases. The second check is simpler to state, if a little harder to verify . Since w" = 1, the same should hold for the matrices deduced from the S-matrices . This was checked for E 6 , E7 and Ex, and various A and D theories, and found to be true in all cases.

P. Dorey / Root s)'stenis

675

This is in fact implied by the final check, that the characteristic polynomials of these matrices are in agreement with those previously deduced for the relevant Coxeter elements in ref. [20]. This gives a very non-trivial test of the conjecture made in eq. (7.5). I;. Conclusions There are various areas to be investigated in order to clarify the proposals made in this paper. The physical pole structure implied by eq. (7.5) should be elucidated, as should the relationship between the fusing rule (4.3) and the previously observed selection rule for couplings based on the Clebsch-Gordan series of the associated fundamental representations [6,12]. If the reformulation of the bootstrap equations presented above could somehow be worked in reverse, then perhaps some progress might be made on the general classification question for purely elastic scattering theories. A direct proof of the fusing rule (or even of the eigenvalue result (2.5)) starting from the affine Toda field theory lagrangian is still lacking, as is an understanding of its connections with the conformal theories which, when suitably perturbed, are expected to share the same fusing structure . In fact this might be an interesting aspect to pursue, as the relevant conformal theories are all related to W-algebras [2], and so it is possible that the fusing rule given here might have some implications for these models. Finally, it is possible that the rule described here may have uses beyond the purely elastic scattering theories. Various models with multiplet structure share the same mass spectrum as the affine Toda theories, for example the principal chiral models discussed in ref. [211. Such theories possess additional fractional spin charges [22] over those discussed in this paper, but it could be hoped that their bootstrap properties would mirror those found in the theories without multiplets . This seems to hold in the cases that have been studied so far [23,24], and if true in general would mean that the rule would also apply to some of the more complicated fusings of interest in studies of the Yang-Baxter equation [25] . I would like to thank Harry Braden, Ryu Sasaki and especially Ed Corrigan for the many discussions on affine Toda theory which were the motivation for the work described in this paper, C. Itzykson for the suggestion to check the characteristic polynomials for the Coxeter elements, and Ed Corrigan and J.-B. Zuber for helpful remarks on the manuscript. The observation of Ed Corrigan and Wolfgang Lerche on the possible relevance of polytopes, mentioned in sect. 1, was very important . I understand that Nick Warner and Wolfgang Lerche were also aware that root triangles project to mass triangles ; details of their work can be found in ref. [26]. 1 am grateful to the Royal Society for a fellowship under the European Science Exchange Programme .

67 6

P. Dorey / Root Systems

Note added in proof Since this paper was submitted, a paper by Freeman has come to our attention [27]. This independent work contains a general explanation for the connection between the affine Toda masses and the Perrone-Frobenius eigenvectors of Cartan matrices, and is also based on the properties of the Coxeter element. eferences [1] [21 [3] [4] [5] [6]

[7] [8] [9] [101

[I 1 ] [121 [13] [141 [15] [161 [17] [18] [19] [20] [211 [22] [23] [241 [251

[26] [27]

A.B . Zamolodchikov, Integrable Field Theory from Conformal Field Theory. Proc . Taniguchi Symposium, Kyoto (1988) ; A .B. Zamolodchikov, Int . J . Mod . Phys . A 4 (1989) 4235 V .A . Fateev and A .B . Zamolodchikov, Int . J . Mod . Phys . A 5 (199()) 1025 A .E . Arinshtein, V .A . Fateev and A .B. Zamolodchikov, Phys. Lett . B87 (1979) 389 H .W . Braden, E. Corrigan, P.E . Dorey and R . Sasaki, in Proc . NATO Conf . on Differential Geometric Methods in Theoretical Physics, Lake Tahoe, to be published H .W . Braden, E . Corrigan, P.E . Dorey and R . Sasaki, Nucl . Phys . B338 (1990) 689 H .W . Braden, E . Corrigan, P .E . Dorey and R . Sasaki, in Proc . 10th Winter School on Geometry and Physics, Srni, Czechoslovakia, to be published ; Integrable Systems and Quantum Groups, Pavia . Italy ; Spring Workshop on Quantum Groups, ANL, USA H .W. Braden, E . Corrigan, P .E . Dorey and R . Sasaki, Nucl . Phys . B356 (1991) 469 H .W. Braden and R . Sasaki, Phys . Lett . B255 (1991) 343 P. Christe and G . Mussardo, Nucl . Phys . B330 (199()) 465 P. Christe, in Proc . NATO Conf . on Differential Geometric Methods in Theoretical Physics, Lake Tahoe, 1989, to be published ; G . Mussardo, in Proc . NATO Conf. on Differential Geometric Methods in Theoretical Physics, Lake Tahoe, to be published ; P . Christe and G . Mussardo, Int . J . Mod . Phys . A 5 (1990) 4581 C . Destri and H .J . de Vega, Phys . Lett . B233 (1989) 336 T.R . Klassen and E. Melzer, Nucl . Phys . B338 (1991)) 485 C .J . Goebel, Prog . Theor. Phys . Suppl . 8 6 (1986) 261 P . Fendley, W . Lerche, S . Mathur and N . Warner, preprint CTP-1865, CALT-68-1631, HUTP9f1/A036 E . Corrigan and W . Lerche, private communication H . Coxeter, Am . J . Math . 62 (1941)) 457, H . Coxeter, Regular polytopes (Methuen, London, 1948) P . Dorey, Durham Ph .D . thesis, unpublished V. Pasquier, Nucl . Phys. B285 (1987) 162 R . Carter, Simple groups of Lie type (Wiley, New York, 1972) J . Frame, Duke Math . J . 18 (1951) 783 E . Ogievetsky and P . Wiegmann, Phys . Lett . B168 (1986) 360 D . Bernard and A . LeClair, preprint CLNS-9()/1()27, SPhT-9()/144 T .J . Hollowood, Oxford preprint OUTP-9t1-15P N .J . MacKay, Nucl . Phys . B356 (1991) 729 P . Kulish, E . Sklyanin and N . Reshetikhin . Lett . Math . Phys . 5 (1981) 393 ; M . Jimbo, Int . J . Mod . Phys. A 4 (1989) 3759 ; M . Jimbo, in Braid group . knot theory and statistical mechanics (World Scientific, Singapore, 1989) W . Lerche and N .P . Warner, Polytopes, Landau-Ginzburg solitons and supersymmetric Toda theories, preprint CALT-68-171)3 M .D . Freeman, On the mass spectrum of affïne Toda field theory, preprint KCL/91/Fl, Phys . Lett . B261, to be published