The mass spectrum and coupling in affine Toda theories

Physics Letters B 266 ( 1991 ) 82-86 North-Holland

PHYSICS LETTERS B

The mass spectrum and coupling in affine Toda theories A. Fring, H . C . L i a o a n d D.I. Olive The Blackett Laboratory, Imperial College, London S W 7 2BZ, UK

Received 31 May 1991

We provide a unified derivation of the mass spectrum and the three point coupling of the classical affine Toda field theories, using general Lie algebraic techniques. The masses are proportional to the components of the right Perron-Frobenius vector and the three point coupling is proportional to the area of the triangle formed by the masses of the fusing particles.

The affine Toda field theory is a relativistic field theory describing the coupling of r independent scalar fields. It has been known for about ten years that in two dimensions its classical equations exhibit remarkable integrability properties [ 1-3 ]. There is such a theory for each affine root system but we shall consider only those associated with the root systems of an untwisted affine K a c - M o o d y algebra ~,. Then r, the number of fields, equals the rank of the finite dimensional simple Lie algebra g. Interest in this theory was intensified by the observation of Hollowood and Mansfield [4] that affine Toda field theories were integrable deformations (both classically and q u a n t u m mechanically) of a conformally invariant theory, namely the ordinary Toda field theory associated with g. In accordance with general ideas of Zamolodchikov [5], the infinite number o f conservation laws associated with the integrability are relics o f the broken conformal symmetry. It was then noticed on the basis o f case-by-case computation [ 1 , 6 - 9 ] that the masses m ~ , m 2 . . . . . m r and the three point coupling Cijk of the fields occurring in the classical equations o f motion display remarkable patterns that are independent of the magnitude of the symmetry breaking and seem to be preserved in the q u a n t u m theory, at least if g is simply laced. The masses are proportional to the components o f the Perron-Frobenius vector and the coupling has modulus proportional to the area o f the triangle formed by the masses of the three coupling particles. 82

The object o f this letter is to provide a unified derivation of these remarkable results, extended to be valid for any simple Lie algebra, whether simply laced or not, using results in Lie algebra theory and developing further ideas of Freeman [ 10] concerning the explanation o f the mass pattern. The lagrangian of the affine Toda theory reads ~ = t r ( ½0u~ 0 u ~ - eP~E e - ~ E t) ,

( 1)

where • is an element of the Cartan subalgebra h. Let T3 be the generator in h of a maximal S U ( 2 ) subalgebra embedded in g. Then if h denotes the Coxeter number of g, that is one less than the quotient o f the dimension of g by its rank, or equivalently one more than the height o f its highest root ~,, the quantity S.'= e 2~ir3/h

(2)

furnishes a Zh grading of g: g=go~gl~

--- ~ g h - I ,

S g n S - l = e2nin/hg n = ~ ng n .

(3)

We have go=h,

g,={Ec,,,...,Ec~,,E_~},

(4)

where a j, ..., ar are the simple roots o f g with respect to h. Then E occurring in the lagrangian (1) is an element of g~ which has to commute with its hermitian conjugate E* in order that the term linear in qb in (1) disappears, and qb=0 correctly describes the ground state. It is known that E is a "regular element" of g so

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PHYSICS LETTERS B

that the set of elements commuting with it forms a unique second Cartan subalgebra, h' (containing E*). The interplay between these two Cartan subalgebras h and h', said by Kostant [ 11 ] to be in apposition, is crucial to what follows and is known to be relevant in the construction of the conserved currents [2], which is achieved by gauging the zero curvature potentials into h' so that their duals become conserved. We denote by h, and e,~ the generators o f h ' and the corresponding step operators in a Cartan-Weyl basis, respectively. It then follows that Sx.hS- l = a(x).h~h', Seo~S-t ~ e , ( , ) where a must be an element of the Weyl group of g (and so orthogonal) of order h, a h = 1, as S h commutes with g. It is possible to choose phases so that, for all a,

Seo,S-~=eo(o,) .

(5)

The action of a on the system ofhr roots splits them into s disjoint orbits, say, each composing at most h elements. Hence hs>lhr. But the quantities 52~= ~ eo,~) associated with each orbit are linearly independent and c o m m u t e with S and so, by (4) lie in h = go. Hence s ~
Ai:= 1 x/n

eo,,~,,), i=l,...,r,

(6)

n=l

where Yi is a representative of the ith orbit 12j, to be specified later (see ( 2 3 ) ) . -g2~ is also an orbit and so must coincide with 12r, say, where/-is a permutation of order 2 of the numbers 1.... , r. We then have

At,=Ar,

[A~,As]=O, tr(A,Aj)=~,f,

(7)

in view of our chosen normalisations. Since A ~ h we can expand the Lie algebra valued fields

rio= ~ OiA,, so q~*=0r.

(8)

/=1

22 August 1991

Given E~ h' we can write E = q. h and calculate [E ?, [E, Aj] l = I~/.yj I2Aj,

(10)

so that the above quadratic term reads

_½j~2 E Iq'Yil2lO,[2,

(11)

i

and we recognise the particle masses rn,=fllq'Til ,

i = 1 ..... r.

(12)

It is intriguing that this is the Higgs formula for the masses of gauge particles for a specific Higgs field in the complex adjoint representation ~/. To proceed further we need to know more about 1/ and y~. As E~g~ c~h', q is an eigenvector of o with eigenvalue ~o=e :"i/h. Kostant [ 11 ] has confirmed, what is strongly suggested by the properties of a already demonstrated, that a is a "Coxeter element" of the Weyl group of g, that is, it is conjugate to a product of reflections a, in the simple roots a~ (henceforth with respect to h' rather than h), each taken just once: O l j " Ol i

a,(aj) =as-2 ~.2 ot,=aj-Ks, a,, Of. t.

(13)

where KS~is the Cartan matrix ofg. The simple roots correspond to the points of the Dynkin diagram of g which can always be "bicoloured" so that each point is either "black" or "white" and the lines link no two points of the same colour. It is then possible to choose a system of simple roots with respect to h' so that

a=aBaw,

(14)

where aB is a product of the reflections in the simple black roots, whose ordering is irrelevant as they commute, and similarly aw. Then it is possible to verify the identity [ 12 ]

(aB +aw)°ti= i (2~ij-Ko)°tJ"

(15)

j=l

The significance of this expansion is that the q~i are the mass eigenstates as we now see, by considering the term in the lagrangian ( 1 ) quadratic in q~. After rearrangement it is -½fl2tr(qb[Et,

As (aB + aw) 2 = 2 + a + a - l this relates eigenvalues of a and the Cartan matrix K. Let x denote the left eigenvector of K, to the eigenvalue 2 - 2 cos 0, so

i xigiJ = 2 ( 1 - c o s O)xj.

[E, qbl])

(16)

i=1

=

lfl2 ~ 0~Ojtr(A,[E*, [E, A A I ) . I,J

(9)

Because of the bicolouration we also have 83

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PHYSICS LETTERS B

__r

c(i)x, Ka=2(1 + c o s O)c(j)xj,

(17)

i=l

where c (i) = + 1 as i is black or white. Then, defining q w : = ~ xioti,

qB:= ~ xiot,,

JeW

i~B

(18)

we have, trivially, aw(qw) = - q w ,

22 August 1991

Ca k =f13 tr( [ [Ai, E ] , [Aj, E*] ]Ak).

It is necessary to check that this is totally symmetric in the indices. This follows remembering that the A~, Aj and E, E t mutually commute. In fact, as the commutator in (26) commutes with S, defined in eq. (2), we can write [ [E,A~], [Et, Aj]]=fl-3CakA~.

aB(q~) = - - q B ,

(19)

(26)

(27)

But

a n d b y (15) and (17) [ [E, A~], [E t, AA ] aw (q~) =qB + 2 cos Oqw, aH(qw) = q w + 2 cos 0 qB-

Thus both aB and aw act on the plane in rootspace spanned by qB and qw and we can readily find the eigenvector o f a in this plane. We have

a(q) =e2i°q

if q = e - i ° / Z q w +eiO/2qB ,

(22)

The possible values of 0 are nyc/h where n~.... , n, are the exponents o f g taking values between 1 and h - 1. The eigenvalue n~ = 1, i.e., 0 = tc/h yields the smallest eigenvalue of K. By the Perron-Frobenius theorem the corresponding components of x~ never vanish and can all taken to be positive. Taking into account these facts, one can deduce that the roots c(i)ai lie on distinct orbits of a. Hence we can choose as our representative o f the orbit f2,

yi=c(i)ai.

(23)

By (22) and (23) the masses (12) are now given by mi =fiX, Or2 sin n/ h ,

(q'7i)(q*'Yj) h y~ ~oq-P[e~,,(y,), eoq(~j)] ,

P,q

(28)

which we rearrange as

(tl'Yi)(q*'YJ) h

~

SP(~q °)q[ey,, eo,(yj) ] ) S -p .

(29)

(21)

and can then calculate q.aj = i c ( / ) sin Oxja 2 e -ic(j)°/2 .

-

(20)

(24)

and are thus proportional to the components of the right Perron-Frobenius vector. (As Kis not symmetric ifg is not simply laced the left and right eigenvectors to the same eigenvalue may differ.) We can now examine the three point coupling by expanding the lagrangian ( 1 ) to get the cubic term 1

The commutator of two step operators is non zero either when the roots concerned add to zero or add to a third root. In the first case we have [ey,, e~(yl) ] =Ti'h •

As the result of the calculation must c o m m u t e with S, and this does not, we expect these terms to sum up to zero. Indeed

~ SPTi.hS-P=(l-ah)(1-ff)-lyi.h=O

(31)

P

as a h = 1 and a # 1. In the second case the c o m m u t a t o r may give a step operator for a root in the orbit g2g and hence contribute to Cak. Defining

Q( i,j, ~):={q~771yi +aq(Tj)~g2~} ,

(32)

we have [ [E, A,], [E t, Aj] ] -

--

3~ CakoiOJOk

(30)

(~'y,) (~*'yj) h

(q7;)X/~ (~*'yA ~ ~ ogq~(i,j,q)A~,

(33)

k- q c Q

=~ f13 tr(q~[ [q~, E ] , [q~, E*] ] ) . Expanding q) by (8) we see 84

(25)

where e(i,j, q) is the structure constant defined by [ey,, e~(yj)]=~(i,j, q)e~,+~(~j) .

(34)

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This is the confirmation of Dorey's rule [ 13 ] for the coupling of particle Eto particle i andj. We shall now show that when the set Q (32) is not empty, it contains precisely two elements. The particle Ehas mass

fl2[[E,A,], [Et, AA ] _ # 2 (~'~') (~*'~'J)

x~~

~ e(i,j, q)(09q--ogq')AE E

2i = ~--~mimjsinOis(q)~(i,j,q)A~.

(43)

E

(35)

/31 rl.7~1 = fll ~" (~'i + a~ (~'j))1,

22 August 1991

Comparing with eq. (27) we finally obtain which can be reexpressed as

mE = I rn~ + mj e~°O(q) I

(36)

using ( 22 ) and the fact that r/. a q( orj) = 09- qq.otj where

O~fiq)'.={2q-½[c(i)-c(j)]}O,

O=n/h.

(37)

So the mass squared is 2 2 mE=m ~+mj2 + 2 m t m j cos Oo(q) ,

(38)

that is m~, ms-, m~ form a triangle with the angle between sides mi and rnj being n-Oo(q). Any other q'eQ(i,j, ~) must also couple y~, 7j to the orbit ~2g, and produce the same mass mE. Thus cos Oo(q)= cos Oo(q') and q' must satisfy one of the two equations

Oo(q)+_Oo(q')=2nn,

nE~_.

(39)

Since 0~
q+q'-- ½[c(i) - c O ' ) ] =nh=h.

(40)

The integer n cannot be zero otherwise i a n d j would be of different colour and either q or q' is zero. It means that ~,i+Tj = c ( i ) (a~ - a s ) is a root and or,, as. are simple, which is of course impossible. Therefore we can have at most two elements in Q related by eq. (40). On the other hand, the fact that A,, Aj commute implies that

Cijk =

4ifle(i,j, q) /~

dijk ,

(44)

where Aijkdenotes the area of the triangle bounded by the sides of length mi, rnj and m k = m;. The structure constants e (i, j, q) are most easily specified by saying that, if ~/is the highest root, ~ e(i,j, q) takes values + 1 unless i, j and k all correspond to short roots. Then the value is + 1/x/~ for B, C and F4 and + 2 / x / ~ for G2. (These facts are important for the vertex operator construction of non-simply laced Kac-Moody algebras [ 14 ]. ) An easier way to see the coupling must be proportional to Aok is to observe that in eq. (28), interchanging i a n d j has the effect ogq-P~ - o : -q. Since the coupling is symmetric in i and j, we are allowed to add such a term and produce a sine function. The "colour term" in 0o(q) also comes out correctly if we take care of the factor (r/'Yi)(r/*'9's). However, the above argument is still necessary to give a correct factor in eq. (44). Hiu Chung Liao would like to thank the Croucher Foundation for financial support under which this work is done. Andreas Fring would like to thank the Science and Engineering Research Council and the University of London for financial support and D. Ellwood for discussions.

References qeQ

~(i,j, q) = 0 ,

(41)

and Q has to contain more than one element. As a result, when Q is not empty,

~(i,j,q)+~(i,j,q')=O, and

(42)

[ 1 ] A.V. Mikhailov, M.A. Olshanetzky and A.M. Perelomov, Commun. Math. Phys. 79 ( 1981 ) 473. [ 2 ] D.I. Olive an d N. Turok, Nucl. Phys. B 257 [ FS 14 ] ( 1985 ) 277. [3] G. Wilson, Ergod. Theor. Dynam. Syst. 1 ( 1981 ) 361. [ 4 ] T.J. Hollowood and P. Mansfield, Phys. Lett. B 226 ( 1989 ) 73. [5]A.B. Zamolodchikov, Intern. J. Mod. Phys. A 1 (1989) 4235.

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[6] H.W. Braden, E. Corrigan, P.E. Dorey and R. Sasaki, Phys. Lett. B 227 (1989) 411. [ 7 ] H.W. Braden, E. Corrigan, P.E. Dorey and R. Sasaki, Aspects of perturbed conformal field theory and exact S-matrices, Proc. XVIII Intern. Conf. on Differential geometric methods in theoretical physics: physics and geometry (Lake Tahoe, 1989). [8] H.W. Braden, E. Corrigan, P.E. Dorey and R. Sasaki, Nucl. Phys. B 338 (1990) 689. [9] P. Christe and G. Mussardo, Nucl. Phys. B 330 (1990) 465. [ 10 ] M.D. Freeman, On the mass spectrum of affine Toda field theory, Kings College London preprint KCL/91/F1 ( 1991 ).

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[ 11 ] B. Kostant, Amer. J. Math. 81 (1959) 973. [ 12] S. Berman, Y.S. Lee, R.V. Moody, J. of Algebra 121 (1989) 339; see also B. Kostant, Proc. Nat. Acad. Sci. USA 16 (1984) 5275; R. Steinberg, Pacific J. Math. 118 (1985) 587; R. Carter, Simple groups of Lie type (Wiley, New York, 1972, 1989). [ 13 ] P. Dorey, Root systems and purely elastic S-matrices, Saclay preprint SPhT/90-169 (1990). [ 14] P. Goddard, W. Nahm, D. Olive and A. Schwimmer, Commun. Math. Phys. 107 (1986) 179.