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RIMS-787—SPhT-91/140 (corrected)

arXiv:hep-th/9110058v1 21 Oct 1991RIMS-787—SPhT-91/140 (corrected)Root Systems and Purely Elastic S-Matrices IIPatrick Dorey†Service de Physique Th´eorique de Saclay,191191 Gif-sur-Yvette cedex, FranceStarting from a recently-proposed general formula, various properties of the ADEseries of purely elastic S-matrices are rederived in a universal way.In particular, therelationship between the pole structure and the bootstrap equations is shown to followfrom properties of root systems. The discussion leads to a formula for the signs of thethree-point couplings in the simply-laced affine Toda theories, and a simple proof of a resultdue to Klassen and Melzer of relevance to Thermodynamic Bethe Ansatz calculations.August 1991† dorey@poseidon.saclay.cea.fr1 Laboratoire de la Direction des Sciences de la Mati`ere du Commissariat `a l’Energie Atomique

1. IntroductionOccasionally, an integrable perturbation of a conformal field theory results in a massivescattering theory which is purely elastic, in that the S-matrix is diagonal in a suitablebasis [1].

This observation has lead to some work on such S-matrices as interesting objectsin their own right. A series of examples connected with the ADE series of Lie algebrashas been uncovered, both directly in the context of perturbed conformal field theory [2],and also via the study of affine Toda field theories [3–9] (in fact, there are slight differencesbetween the S-matrices found in the two contexts; these will be mentioned where relevant).As is often the case in the study of integrable quantum field theories, the proposed S-matrices have not (at least so far) been derived from first principles, but rather deducedon the basis of certain assumptions and consistency requirements.

However, for a theorywith a diagonal S-matrix, the Yang-Baxter equation – often a very powerful tool in thestudy of the S-matrices of integrable theories [10] – is trivially satisfied, and so gives noinformation. There does remain the possibility that two particles in the theory may fuse toform a third as a bound state [11].

As emphasised by Zamolodchikov [1], for purely elasticscattering the resulting bootstrap equations are sufficiently simple to provide a useful setof consistency conditions, constraining both the conserved charges and the S-matrix.The nature of the bootstrap solutions for the ADE theories is in fact closely linkedto properties of the corresponding root systems, and in particular the action on theseroot systems of the Coxeter element of the Weyl group [12]. The aim of this paper is toexplore this connection a little further, with particular emphasis on the implications for thestructure of the S-matrix elements.

By refining the notations used in [12], it turns out tobe possible to streamline the discussion considerably. After a description of some necessaryformulae, section two outlines how this goes.

While this section contains no new results,it does give a proof of the S-matrix bootstrap equations which simplifies and clarifies thatgiven previously. Section three is devoted to a discussion of the pole structure of S-matrix,and shows how this is exactly in accordance with the bound-state structure predicted bythe fusings.

Various empirically observed features of the purely elastic S-matrices turn outto be simple consequences of the properties of root systems. An application for some ofthe calculations in section three is given in section four, giving a description of one setof signs for the three-point couplings of the simply-laced affine Toda theories, in terms ofroots and weights.

Finally, section five gives a universal proof of an elegant formula dueto Klassen and Melzer [9], and section six contains some concluding remarks.1

2. PreliminariesSince this paper is a direct sequel to [12], the reader is referred back to that paperfor details of the motivations and many of the original formulae.

As mentioned in theintroduction, elements of the discussion given there become rather clearer if the notation(in particular, the labelling of the orbits of the Coxeter element), is changed slightly. Forthis, some results to be found in a paper by Kostant [13] will be needed, and this sectionstarts with a brief review of the relevant material.First though, note that various of the definitions and results to be given in this sectionwork equally well for simply-laced and non simply-laced root systems.

For example, thediscussion in [13] makes no distinction between the two cases. It is also worth noting thatthe discussion in [12] of the conserved charge bootstrap goes through essentially unchangedfor the non simply-laced root systems.

However, the S-matrix formulae given in that paperseem to be hard to generalise beyond the ADE series, possibly reflecting the difficultiesthat were found in the (quantum) problem of finding S-matrices for the non simply-lacedaffine Toda theories [4,7]. For this paper, then, attention will be restricted to the already-known purely elastic scattering theories, that is to those associated with the simply-lacedLie algebras.

The discussion will be based on a simply-laced root system Φ, of rank r,with {αi} a set of simple roots. Letting wi denote the Weyl reflection corresponding tothe simple root αi (so wi(x) = x −2α2i (αi, x)αi), setw = w1w2 .

. .

wr,so that w is a Coxeter element. Also, let ⟨w⟩be the subgroup of W, the Weyl group,generated by w. Finally, for i = 1 .

. .

r define a root φi byφi = wrwr−1 . .

. wi+1(γi).

(2.1)Then the following results are given in [13]:(i) With the definition of positive and negative roots implied by the given choice of simpleroots, φi > 0 and w(φi) < 0. (ii) If α is a root such that α > 0 and w(α) < 0, then α is one of the φi.

(iii) Let Γi be the orbit of φi under ⟨w⟩. Then the Γi’s are disjoint, each has h elements,and thus their union is all of Φ.2

The set {φi} possesses one further useful property, which can be found for example in [14].If λi is the fundamental weight corresponding to the simple root αi, thenφi = (1 −w−1)λi. (2.2)(To prove this result, note that wi(λj) = λj −δijαj, which follows since the fundamentalweights are dual to the simple co-roots α∨i ≡2α2i αi.

For the simply-laced cases of interesthere, α2i = 2 and the λi are dual to the simple roots themselves.) This relation can beinverted, w having no eigenvalue equal to one.

Writing(1 −w−1)−1 = R,(2.3)it is easily checked thatR = 1hhX1pwp = −1hh−1Xp=0pw−p. (2.4)This mapping is not orthogonal, but rather satisfies(Rα, β) + (α, Rβ) = (α, β).

(2.5)Other identities, such as (Rα, β) = −(wα, Rβ), can also be found but only (2.5) will beused below. Projectors onto the various eigenspaces of w are given byPs = 1hh−1Xp=0ω−spwp = 1hh−1Xp=0ωspw−p,(2.6)with corresponding eigenvalues ωs = e2πis/h.

If s is not an exponent of the algebra, thenthe spin s eigenspace is null, and Ps = 0.It is often useful to focus on a particular ordering of the simple roots, linked to a two-colouring of the Dynkin diagram. This ordering is such that {αi} splits into two subsets,each of which contains only mutually orthogonal roots:{α1, α2, .

. .αr} = {α1, .

. .αk} ∪{αk+1, .

. .αr}(2.7a)Defining subsets of the indices 1, .

. .r as• = {1, 2, .

. .k},◦= {k + 1, k + 2, .

. .r}(2.7b)3

(‘black’ and ‘white’), the internal orthogonality amounts to the requirement that for i ̸= j,(αi, αj) = 0 if i and j have the same colour. By a small abuse of notation, the symbols•, •′, ◦, ◦′ and so on will occasionally be used to denote arbitrary indices taken from thecorresponding (black or white) subset.

A slightly different notation was used in [12]: blacksimple roots were called ‘type alpha’, white ones ‘type beta’.In this ordering,w = w{•}w{◦}withw{•} =Yi∈•wi,w{◦} =Yj∈◦wj.The internal orthogonality of the two subsets of the simple roots implies that the reflectionsfor simple roots of the same colour commute, and that a reflection for a given simple rootwill leave invariant all other simple roots of the same colour. It follows thatφ• = w{◦}(α•),φ◦= α◦.

(2.8)In [12], the coset representatives were α• and −α◦; since both w{◦} and −1 induce chargeconjugation on the cosets, this means that, strictly speaking, the assignment of cosets tosimple roots induced by the φi differs by an overall charge conjugation from that used inthe earlier paper. However this is merely a matter of labelling convention – for example,one can interchange the values of the conserved charges on particle and antiparticle simplyby negating the normalisations of the even spin charges – and so can be ignored.The ADE S-matrices will be built as products of functional ‘building blocks’.

In [12],the blocks used were{x}+ =x −1+x + 1+(perturbed conformal)x −1+x + 1+x −1 + B+x + 1 −B+(affine Toda)(2.9)wherex+ = sinhθ2 + iπx2h(2.10)andB(β) = 12πβ21 + β2/4π . (2.11)4

The block for an affine Toda theory contains an extra coupling-constant dependent partover the ‘minimal’ version suitable for perturbed conformal theories. This turns out tohave no effect on the physical pole structure (β being real, and B(β) therefore between0 and 2), and so it is reasonable to use the same notation for both cases.

In fact, thediscussion of pole structure will be a little more transparent if this block is swapped foranother, namely{x}−=(−{−x}−1+(perturbed conformal){−x}−1+(affine Toda)(2.12)Of course, any formula involving {x}−can be immediately rewritten in terms of {x}+.One further piece of notation will be needed. For any pair of roots α, β ∈Φ, an integeru(α, β) can be defined modulo 2h via the following relations:u(α, β) = −u(β, α)u(wα, β) = u(α, β) + 2u(φ•, φ•′) = u(φ◦, φ◦′) = 0u(φ◦, φ•) = 1.

(2.13)For the coset representatives φi, the abbreviated notation uij ≡u(φi, φj) will often be used.The definition is natural in that πsh u(α, β) is the (signed) angle between the projections ofthe roots α and β into the ωs eigenspace of w. Now the three-point couplings are describedby the following fusing rule [12] (also relevant in other contexts [15]):Cijk ̸= 0 iff∃roots α(i) ∈Γi, α(j) ∈Γj, α(k) ∈Γk with α(i) + α(j) + α(k) = 0. (2.14)(Note the use here of a convention that will be adhered to for the rest of this paper: α(i),for example, is used for any root that lies in Gi, the orbit of the root φi.

It is importantnot to confuse this with the simple root αi – in particular, even though the labelling of theorbits ultimately derives from the choice of simple roots, via (2.1), there are cases wherethe simple root αi does not lie in the orbit Γi.) Since the fusing angles U kij are the relativeangles of projections into the ω1 eigenspace, they are related to the u(α(i), α(j)) byU kij = πh|u(α(i), α(j))|,where α(i) + α(j) ∈Γ¯k.

(2.15)Being signed angles, the u(α, β)’s also satisfyu(α, β) + u(β, γ) + u(γ, α) = 0mod 2h(2.16)for (any) three roots α, β and γ.5

Armed with these conventions, the expression given in [12] for the two-particle S-matrix can be rewritten in a compact way:Sij =h−1Yp=0{2p + 1 + uij}(λi,w−pφj)±. (2.17)Note, the expression is identical in form whether it is written in terms of {x}+ or {x}−.This is equivalent to unitarity (Sij(θ)Sij(−θ) = 1) and follows from(λi, w−pφj) = −(λi, wp+1+uijφj).

(2.18a)Symmetry (Sij = Sji) can also be checked, using(λi, w−pφj) = (λj, w−p−uijφi). (2.18b)(A ‘mixed’ equality, (λi, w−pφj) = −(λj, wp+1φi), follows directly from (2.2), indepen-dently of the special root ordering (2.7).) Equivalent formulae were given in [12], but in aless compact way.Equation (2.17) can be put into a perhaps more suggestive form by noting from (2.13)that 2p + uij is just u(φi, w−pφj).

HenceSij =Yα(j)∈Γj{u(φi, α(j)) + 1}(λi,α(j))±. (2.19)The unitarity and symmetry of this formula can be checked directly by rewriting (2.18)as(λi, α(j)) = −(λi, wu(φi,α(j))+1α(j)),(2.20a)(λi, α(j)) = (λj, w(u(φj,α(i))−u(φi,α(j)))/2α(i)).

(2.20b)Note, the exponent of w in (2.20b) is an integer, since u(φj, α(i)) and u(φi, α(j)) are alwayseither both even or both odd. The important case for the symmetry of (2.19) is (λi, α(j)) =(λj, α(i)) if u(φj, α(i)) = u(φi, α(j)).The S-matrix bootstrap equations [1] can be checked very simply from (2.19).

Firstthese equations are rewritten, for each particle species l and each nonvanishing three-pointcoupling Cijk, asSli(θ)Slj(θ + iU kij)Slk(θ −iU jik) = 1. (2.21)(This alternative bootstrap equation, obtained from the more usual one via the equationsof unitarity (Sij(θ)Sij(−θ) = 1) and crossing (Sij(θ) = Si¯(iπ −θ)), is analogous to the6

symmetrical version of the conserved charge bootstrap equation used in [12].) The structureof this equation is clarified if a shift operator Ty is introduced, defined byTyf(θ) = f(θ + iπyh )(2.22)and acting on the blocks asTy{x}± = {x ± y}±.

(2.23)Recalling from (2.14) that Cijk ̸= 0 implies the existence of a root triangle {α(i), α(j), α(k)},the relation (2.15) can be used to write equation (2.21) asSliTu(α(i),α(j))SljTu(α(i),α(k))Slk= 1. (2.24)Note how the fact that u(α, β) is a signed angle takes care of the relative minus sign betweenthe two shifts in the earlier equation, (2.21).

Of course, depending on the orientation ofthe projection of the root triangle into the s = 1 subspace, there could be an overallnegation of the shifts in (2.24) compared to (2.21). In fact, root triangles projecting toboth orientations always exist – this will be discussed in more detail in section four – butthis is not a problem since (2.21) also holds with the labels j and k exchanged, an operationwhich itself has the effect of negating the two shifts.Finally, acting on both sides with Tu(φl,α(i)) and using (2.16) givesTu(φl,α(i))Sli Tu(φl,α(j))Slj Tu(φl,α(k))Slk= 1(α(i) + α(j) + α(k) = 0).

(2.25)To verify that the bootstrap equations in this form are satisfied by (2.19) is almost imme-diate. The left hand side of (2.25) involves three roots running through the orbits Γi, Γjand Γk for the S-matrix elements Sli, Slj and Slk respectively.

If these orbits are labelledstarting at α(i), α(j) and α(k) (instead of the roots φi, φj and φk that would have been usedhad (2.17) been the starting point), then via (2.23) the left hand side ish−1Yp=0{u(φl, w−pα(i)) ± u(φl, α(i)) + 1}(λl,w−pα(i))±× {u(φl, w−pα(j)) ± u(φl, α(j)) + 1}(λl,w−pα(j))±× {u(φl, w−pα(k)) ± u(φl, α(k)) + 1}(λl,w−pα(k))±.That this is equal to one is apparent if the {x}−blocks have been used ({x}+ beingappropriate for the equivalent version of (2.25) with all shifts negated). From (2.13), for7

each value of p the three blocks involved are then equal to the same function, namely{2p + 1}−. The total power to which this is raised isλi, w−p(α(i) + α(j) + α(k)).

Butthis is zero, since α(i), α(j) and α(k) form a root triangle for the coupling Cijk. Hence thewhole expression is equal to one, as required.Section five will give a general proof of a result for the minimal scattering theories dueto Klassen and Melzer, for which it will be helpful to have an expression for the minimalS-matrix elements in terms of the unitary blocksx=x+/−x+.

(2.26)(This block, unitary in the sense that it individually satisfies the unitarity constraintmentioned above, was used in [4], along with the larger unitary block {x} = {x}+/{−x}+. )Via (2.9) and (2.26), the formulae (2.17) and (2.19) become, in the minimal cases,Sij =h−1Yp=02p + uij(λi,w−pφj) =Yα(j)∈Γju(φi, α(j))(λi,α(j)).(2.27)3.

Pole structureThis section will involve a detailed examination of the physical pole structure of S-matrices described by (2.17).The nature of the residues will be studied, for which alittle more information on the building block {x}−will be needed. This is a 2πi periodicfunction, real for imaginary θ.

It has simple poles at (x −1)πi/h and (x + 1)πi/h, withresidues positive multiples of −i, +i respectively. Outside the interval between these twopoles (or its repetition modulo 2πi), the function is positive on the imaginary axis.

Thesefacts hold equally well for the perturbed conformal or the affine Toda blocks.It is helpful to introduce a pictorial notation in which the S-matrix element is repre-sented by a ‘wall’ of rectangles, stacked along the imaginary axis. The building block {x}−is depicted by a single rectangle above the imaginary axis, stretching from (x −1)πi/h to(x + 1)πi/h:{x}−≡−i+i(x−1)πi/h(x+1)πi/h8

The residues of the poles at (x ± 1)πi/h, up to some real, positive constant, are shownabove the block. A product of blocks making up an S-matrix element is represented bystacking the rectangles to make a wall.

For example,{3}2−{5}−≡2πi/h4πi/h6πi/hPoles occur at the ends of the blocks, of higher order where the ends of blocks coincide.In this example there is a pole of order three at θ = 4πi/h, coming from the two blocks tothe left and one to the right. To represent a full S-matrix element, a wall of length 2π issufficient, since {x+2h}−= {x}−.

In fact, the unitarity constraint imposes that the heightfor {−x}−must be exactly the negative of that for {x}−, so a stretch of length π will do;it is convenient to let it straddle the physical strip, running from 0 to iπ. To give a coupleof examples, here are two of the S-matrix elements from the E8-related scattering theories:S35 :0•••••iπS88 :0•••••iπThe rest of each picture, a stretch of wall running from (say) iπ to 2iπ, can be obtainedby reflecting the piece shown about the line Im(θ) = iπ, and then negating all the heights.Note that the heights are positive inside the physical strip, and negative outside – a generalphenomenon that will be commented on at the end of this section.The positions ofexpected forward-channel poles (found from the three-point couplings and the masses) areshown by the symbols • below the axis — they occur precisely at the ‘downhill’ sectionsof wall (reading left to right).

This apparent coincidence exemplifies a well-establishedrelationship between the fusing structure and the nature of the pole residues, and will nowbe discussed.The nature of the residue of any pole is easy to find from the corresponding picture2(the term residue being used somewhat loosely to mean the coefficient of the most singular2 For the affine Toda S-matrices, this sort of block notation also allows the value of this residue,to leading order in β, to be identified; this was used in [5].9

part). Blocks not contributing directly to a pole simply multiply its residue by a positivereal number, and can be ignored (this is the reason for the minus sign in (2.12)).

Further-more, the contribution from two directly abutting blocks (one to the left and one to theright of the pole) is a positive multiple of (+i). (−i) = 1, so these can also be ignored.

Thusthe nature of the residue is determined solely by the difference in the number of blocksimmediately to the left and right of the pole, that is by the change in height of the wall ofblocks at the position of the pole. The residues for the three simplest possibilities are∼−i ,(3.1a)∼+1 ,(3.1b)∼+i .

(3.1c)In cases a and c, the total number of blocks to left and right of the pole is odd, and sothe pole itself is of odd order. These odd order poles always have an interpretation interms of the production of a bound state.

Examination of the ADE scattering theorieson a case-by-case basis has shown that case c, the downhill pole with a +i residue, isalways forward channel, while case a, uphill, is crossed channel.The pictures give asimple ‘uphill/downhill’ mnemonic by which to decide if an odd-order pole correspondsto a forward channel bound state, in agreement with what has already been observed inthe two E8 examples. (Note, though, that it remains unclear from the point of view ofperturbation theory why the +i residue should be forward channel for the higher odd-orderpoles.

The mechanism by which this rule is reproduced, even for the third-order poles, isquite complicated [5]. )Another ‘empirical’ observation can be made: apart from the occasional exception atthe very edge of the physical strip (for example, at 0 and iπ in S88 above) the wall heightnever changes by more than ±1.

In the exceptional cases, the height change is always from−1 to 1 or back. Wall segments of negative height have zeroes instead of poles, so there isa cancellation and S-matrix is analytic at these points (and in fact is, by unitarity, forcedto be equal to ±1).

Hence (3.1) turns out to cover all possibilities for S-matrix poles. Inparticular, even-order poles always have positive real residues.It might be expected that all the observations described above should have a universalexplanation in the context of root systems.

In fact, this can be achieved using only themost elementary properties of the simply-laced roots.10

Referring back to (2.19), consider the pole in Sij at relative rapidityπih u(φi, α(j))(with 0 < u(φi, α(j)) < h for the physical strip). The two blocks contributing to thispole involve α(j) and wα(j), that involving wα(j) being to the left (recall from (2.13) thatu(φi, wα(j)) = u(φi, α(j)) −2).

Thus the change δh in wall height at this pole is given byδh = (λi, α(j)) −(λi, wα(j)). (3.2)Using (2.2) for the second equality, this simplifies:δh =(1 −w−1)λi, α(j)= (φi, α(j)).

(3.3)Being the inner product of two roots (of a simply-laced algebra), it is now clear thatthe change in wall height, if not zero, can only be ±1 or ±2. These possibilities can beexamined in turn.A change of −1 should correspond to a bound state.

But (φi, α(j)) = −1 implies thatφi + α(j) is a root, −α(k) say, since it is just the Weyl reflection of φi with respect toα(j). This gives a root triangle {φi, α(j), α(k)} and from the fusing rule a non-zero three-point coupling Cijk.

The fusing angle for the ¯k bound state is, by (2.15), πhu(φi, α(j)) –exactly that corresponding to the pole under discussion. Conversely if there is a three-point coupling such that a bound state at the relevant fusing angle is a possibility, thenδh = −1 is forced (since φi + α(j) = −α(k) is then a root, and so δh = (φi, α(j)) =12(α2(k) −φ2i −α2(j)) = −1).The story for δh = 1 is similar, with the conclusion that δh = 1 if and only if there isa bound state in the crossed channel.These two cases have taken up both forward and crossed channels, so an even-orderpole can never have an associated single-particle bound state.

It only remains to remarkthat δh = ±2 implies φi = ±α(j), and a relative rapidity for the putative pole of 0 or iπ,the first to be found in ii scattering, the second in i¯ı. As already mentioned, such ‘poles’disappear by unitarity.This then provides a universal explanation for the interplay between the bootstrapequations and the pole structure of the purely elastic S-matrices.

In particular it elucidatesthe ‘internal’ consistency of these S-matrices, obeying as they do the very equations thatthey imply via their pole structure.To close this section, a comment on a slightly simpler feature of (2.17) and (2.19),which also has an interpretation in terms of root systems. Over the full range from 0 to11

2πi, the wall height can be both positive and negative, and indeed must be so to satisfythe unitarity constraint. But on physical grounds, the height had better not be negativefor blocks in the physical strip: in such an eventuality, a perturbed conformal theory S-matrix would have zeroes in the physical strip, while the affine Toda theory would gainphysical-strip poles with coupling-constant dependent positions.

Now this height is givenas the inner product of some root with a fundamental weight, and so is positive or negativeaccording to whether the relevant root is positive or negative with respect to the given setof simple roots (to say this in another way, the wall height around πihu(φi, α(j)) + 1isjust one ‘component’ of the height of the root α(j), namely that piece due to the simpleroot αi). Referring in particular back to (2.17), it is clear that the physical requirementreduces, traversing each orbit Γj of the Coxeter element starting at the special root φj,to a discussion of which roots are positive and which negative.

Although the details willbe omitted here, it is straightforward to use results (i), (ii) and (iii) from the beginning ofsection two, together with the implication from unitarity that approximately half3 of theroots in a given orbit are positive, to see that the desired property of the wall heights doesindeed follow from general theory.4. The signs of the three-point couplingsIn the perturbative treatment of the affine Toda theories, expansion of the potentialto order β results in a set of three-point couplings Cijk which, if non-vanishing, obey the‘area rule’:Cijk = σijk 4β√h∆ijk,where ∆ijk is the area of a triangle of sides mi, mj and mk (the particle masses) and σijkis a phase of unit modulus, which given the hermiticity of the original Lagrangian can betaken to be plus or minus one.

The vanishing/non-vanishing of the coupling is describedby the rule (2.14). This, together with the normalisation of Cijk, has now been derivedin a general way [16].

However the signs σijk are a little more subtle. Clearly they can bechanged around by negating some of the fields, but this does not mean that they can allbe set to 1.

Indeed, cancellations necessary for perturbation theory to be compatible withintegrability often depend crucially on the presence of relative phases between different3 in fact exactly half, except for A2n for which h is odd and the situation is marginally morecomplicated.12

terms, which would not be present if all the σ’s were equal to 1 (for some examples of this,see [5,6]).Despite the apparent arbitrariness involved, there is a special choice of normalisationswhich connects with the root system data already described. Recall that any three-pointcoupling Cijk results in a bound-state pole in ij scattering of odd order 2m + 1, and notethat the field normalisations used in [4] can be altered so that in every case,σijk = (−1)m.(4.1)This is actually only a small increase in information over a formula given in [5], itself aconsequence of a formula found by Braden and Sasaki [6].

Their result reads:σijk = −σil ¯mσjm¯nσkn¯l,(4.2)holding in this form whenever the triangle ∆ijk is tiled internally by the three othermass/coupling triangles, ∆il ¯m, ∆jm¯n and ∆kn¯l. Note, (4.2) does not change with changesto the field normalisations.

The consequence of this, remarked in [5], is that if ∆ijk is tiledin a ‘nested’ fashion by 2q + 1 other triangles {∆A}, A = 1, . .

.2q + 1 (with phase factors{σA}), thenσijk = (−1)q YAσA. (4.3)(The tiling of a triangle is nested if it is tiled by three other triangles each of which iseither untiled, or is itself tiled in a nested way.

This allows (4.1) to be used inductively toderive (4.3). )Now higher poles are also associated with tilings by mass triangles[5].

For an odd-order pole of order 2m + 1, with ij producing a bound state ¯k, the triangle ∆ijk has a‘maximal’ nested tiling (in fact, many such) by 2m+1 triangles, maximal in the sense thateach constituent triangle cannot be further tiled. The number 2m+1 was called the depthof the coupling triangle ∆ijk in [5].

Applying (4.3) then gives the phases for all couplingtriangles as products of the phases for triangles of depth 1. It is then only necessary tocheck that all the unit depth phases can all be set to 1 to deduce (4.1) from (4.3).Now (4.1), together with the discussion of the last section, can be used to give anexpression for the signs in terms of the roots and weights.

The physical strip pole for the¯k bound state in ij scattering will be at πih u(φi, αo(j)) for some αo(j) ∈Γj (cf the discussion13

preceding (3.2)). Note, {φi, αo(j), αo(k)}, the corresponding root triangle for Cijk, is orientedsuch that 0 < u(φi, αo(j)) < h. The order of the pole is2m + 1 = (λi, αo(j)) + (λi, wαo(j)).

(4.4)Now the change δh in the wall height at this point is −1, as the pole is forward channel;so combining (4.4) with (3.2) givesm = (λi, αo(j))(4.5)and σijk = (−1)(λi,αo(j)). This expression for σijk involves the choice of one particular roottriangle, and furthermore its symmetry in i, j and k is not at all obvious.

To remedy thesedefects, the idea of the orientation of a root triangle will have to be made a little moreprecise. Let {α(i), α(j), α(k)} be any root triangle implying the nonvanishing of the couplingCijk; the orientation ǫ(α(i), α(j), α(k)) is then defined to be +1 if the projection into thes = 1 subspace has a clockwise sense (going from i to j to k), and −1 if anticlockwise.

Sinceǫ(α(i), α(j), α(k)) = +1 if and only if 0 < u(α(i), α(j)) < h, the root triangle {φi, αo(j), αo(k)}used above has orientation +1. It will be helpful to define another quantity for (general)root triangles, namelyf(α(i), α(j), α(k)) = 12 + (Rα(i), α(j)),(4.6)where R, given by (2.3), has the important property that Rφi = λi.

Finally, setm(α(i), α(j), α(k)) = ǫ(α(i), α(j), α(k))f(α(i), α(j), α(k)) −12. (4.7)For the original root triangle {φi, αo(j), αo(k)}, this coincides with (4.5) (this is the reason forthe −12).

But m(α(i), α(j), α(k)) is the same for any triangle of roots for Cijk, and further-more is symmetrical in i, j and k. To establish these properties, some more information onthe set of root triangles is needed. There are in fact 2h ordered triplets {α(i), α(j), α(k)} foreach non-zero three-point coupling (the ordering being needed to count correctly the caseswhen, say, i = j).

Of these, h can be found simply by acting ‘diagonally’ with powers of won an initial triangle. These all have the same orientation.

To see that there are exactlyh more, consider the action of ew ≡−w{•} (one could equally well set ew = w{◦}; all theidentities to be given below would be unchanged). As mentioned in [12], on the cosets ew14

has the effect of two successive charge conjugations, that is no effect at all: ewΓi = Γi. Butwhen acting on triangles, ew reverses the orientation:ǫ( ewα(i), ewα(j), ewα(k)) = −ǫ(α(i), α(j), α(k)).

(4.8)Applying powers of w to { ewα(i), ewα(j), ewα(k)} then gives h more possibilities, and thegeometry of the projection into the s = 1 subspace shows that there can be no more. (Tobe more precise, since U kij is fixed, (2.15) shows that once the root α(i) has been chosenthere are only two possible directions for the s = 1 projection of α(j).

But the roots in asingle orbit all project to different directions for s = 1, so there are at most two roots inΓj which can form a triangle involving α(i) and a root from Γk. Letting α(i) run throughits orbit then gives a maximum of 2h triangles.

)An important consequence of the above is that all triangles for a given coupling canbe obtained from an initial one by acting with w and ew. Now the effect of these twooperations on the orientation has already been given: w leaves ǫ invariant, while ew negatesit.

It is also clear that w leaves f unchanged, w being an orthogonal transformation whichcommutes with R. The action of ew is a little harder to see, but the identity ewR ew = 1 −Rtogether with the fact that the inner product of α(i) and α(j) must be −1 (their sum beinganother root) impliesf( ewα(i), ewα(j), ewα(k)) = −f(α(i), α(j), α(k)). (4.9)Comparing with (4.8) establishes the invariance of m(α(i), α(j), α(k)) under the diagonalaction of both w and ew, and hence its insensitivity to the choice of root triangle.There remains the symmetry of (4.7) between i, j and k. The orientation is completelyantisymmetric in its arguments, so it will be enough to demonstrate that the same is trueof f. First consider swapping α(i) and α(j) in (4.6).

From (2.5), and using again thatthe inner product of α(i) and α(j) must be −1, this sends f to −f.Now consider acyclic permutation of α(i), α(j) and α(k). The three roots sum to zero, so (Rα(j), α(k)) =(Rα(j), −α(i) −α(j)) = (Rα(i), α(j)), using (2.5) and also the fact that (Rα(j), α(j)) = 1.Thus f is unchanged by a cyclic permutation of its arguments, and this is enough toestablish antisymmetry.This completes the proof of the claims following equation (4.7).

The expression cannow be used in (4.1), it already having been mentioned that for one particular triangle(and hence for all) (4.7) coincides with the previously-derived (4.5). This can be used15

in a complete specification of the three-point couplings for the simply-laced affine Todatheories. With a simple rewriting of the resulting expression for (−1)m, the three-pointcoupling data can be summarised as follows:• Cijk ̸= 0 iff∃α(i) + α(j) + α(k) = 0 (where α(i) ∈Γi, α(j) ∈Γj, α(k) ∈Γk).• There is a normalisation of the fields such that in these casesCijk = ǫ(α(i), α(j), α(k))(−1)(Rα(i),α(j)) 4β√h∆ijk.5.

A formula due to Klassen and MelzerIn their investigations of the Thermodynamic Bethe Ansatz, Klassen and Melzer [9]observed an interesting universal feature of the minimal purely elastic S-matrices. If thematrix Nij is defined byNij = −12πihlnSij(θ)iθ=∞θ=−∞(5.1)then for the minimal (perturbed conformal field theory) S-matrices,N = 2C−1 −I,(5.2)where C is the relevant (non-affine) Cartan matrix, and I the unit matrix.

Using thisresult, Klassen and Melzer were able to calculate the central charges of the ultra-violetlimits of these theories, finding agreement with the idea that these S-matrices do indeeddescribe perturbations of certain conformal field theories.Such issues will not be theconcern here; the purpose of this section is merely to give a simple and universal proof of(5.2) starting from the general S-matrix expression of [12].Klassen and Melzer used unitary blocks fα(θ) ≡hα(θ), in terms of which the S-matrix element Sij was writtenSij =Yα∈Aijfα(θ).The parameter α was taken to satisfy −1 < α ≤1, and (5.1) reduced toNij =Xα∈Aij(1 −|α|)sgn(α),(5.3)with the convention that sgn(0) = 0 (since f0 ≡1). It will be more convenient belowto choose a different range for α, namely 0 ≤α < 2.

Using fα = fα+2 to relabel the16

negatively-indexed blocks, together with the fact that (1 −|α|)sgn(α) = 1 −(α + 2) for−1 < α < 0, equation (5.3) becomesNij =Xα∈Aij(1 −α) −|Aij ∩{0}|,(5.4)where now 0 ≤α < 2, and the second term undoes the overcounting in the first wheneverα = 0.To evaluate (5.4), it is simplest to use the first expression of (2.27). Each unitaryblock2p+uijcorresponds to an α ∈Aij equal to (2p+uij)/h, while |Aij ∩{0}| is simplythe number of blocks0.

Such blocks will only be found if uij = 0, and so |Aij ∩{0}| = 0if i and j have different colours. Otherwise,0is raised to the power (λi, φj).

For the theordering (2.7) of the simple roots, the full set of these inner products is(λ•, φ•′) = δ••′(λ◦, φ•′) = −C◦•′(λ•, φ◦′) = 0(λ◦, φ◦′) = δ◦◦′(5.5)Hence the general result is |Aij ∩{0}| = δij, and Nij is given byNij =h−1Xp=0(1 −(2p + uij)/h)(λi, w−pφj) −δij. (5.6)Strictly speaking, the fact that u•◦= −1 means that if i is of type • and j of type ◦, thefirst block counted by (5.6) corresponds to α = −1/h, outside the desired range.

This canbe ignored as the power to which this block is raised, (λi, φj), is zero here by (5.5).Making use of equations (2.4) and (2.6),Nij =λi,(h −uij)P0 + 2Rφj−δij = 2(λi, λj) −δij,the results P0 = 0 (0 is never an exponent) and Rφj = λj giving the second equality. Since(λi, λj) is exactly the inverse of the Cartan matrix, equation (5.2) now follows immediately.6.

ConclusionsVarious previously-observed features of the ADE purely elastic scattering theories arenow known to follow from general principles, especially given recent work on affine Todaperturbation theory [16] and the Clebsch-Gordan rule for fusings [17]. With regards to theS-matrices, a notable feature of the treatment given above, as compared to that in [12], is17

that the splitting of the roots according to type (colour) has become much less important.Essentially, once the notations (2.13) have been set up, this distinction can be forgotten.All that is needed is equation (2.15), relating the quantities u(α, β) to the fusing angles.However, it should not be forgotten that the splitting of the particles into two sets doesexpress a geometrical property of the projections of their orbits, reflected in the fact [12]that two particles of the same type always fuse at an even multiple of π/h, two particlesof opposite type at an odd multiple of π/h. Hence the split still has physical implications,even if these are best left hidden for most calculations.As regards future work, there are two obvious questions to ask.

Within the contextof theories with diagonal S-matrices, are there any other physically reasonable possibilitiesbeyond those associated with the ADE series?To formulate this question properly, itmust be decided exactly what is meant by ‘physically reasonable’. For example, whenevera subalgebra of the fusing algebra can be formed (such as emerges when a twisted fold-ing is considered [4][7]), a self-consistent set of S-matrix elements, obeying the bootstrapequations that they imply via their odd-order poles, can be obtained simply by takingthe corresponding submatrix of the ‘parent’ S-matrix.

However, in such cases there arealways higher-order physical poles which are inexplicable without the full set of particles ofthe parent theory, forcing their re-inclusion and returning the theory to the ADE set. (Asimilar phenomenon allows the solitons of the sine-Gordon theory to be inferred from theS-matrix elements of the breathers alone [18].) The only other purely elastic S-matricesthat seem to have been discussed so far [19,20] all obey rather different self-consistencyrequirements, in that the prescription for assigning forward and crossed channels to theodd-order poles is changed.

Such conditions appear to be appropriate for theories withnon-hermitian lagrangians [19], and so these S-matrices should in any case fall outside aninitial (perhaps ADE) classification of unitary purely elastic scattering theories. Neverthe-less, it would be interesting if they could also be given an interpretation in terms of rootsystems.The second question concerns the relevance of any of the above to theories withmultiplets and non-diagonal S-matrices.

This would require some similarity in structurebetween the bootstraps for the purely elastic scattering theories and those for at leasta subset of the more complicated models. At least at the level of the fusings, the samestructure has emerged in the study of perturbations of N=2 supersymmetric conformaltheories [15] (although in fact not for all the ADE series).Other possible candidatesfor inclusion in the subset include the principal chiral models [21].

S-matrices with the18

same scalar (CDD) part have been proposed for perturbations of certain conformal fieldtheories [22]. There are many coincidences (for example, of mass spectra) and even someexplicit calculations [23] suggesting that a connection may indeed be found, but the greatly-increased complexity of the bootstrap equations makes a complete analysis difficult.AcknowledgementsI would like to thank Ed Corrigan for interesting discussions and comments on themanuscript, and the Research Institute for Mathematical Sciences, Kyoto University fortheir kind hospitality while this work was being completed.

The work was supported inpart by a grant of the British Council. I am grateful to the Royal Society for a Fellowshipunder the European Science Exchange Programme.19

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