Perturbation theory in the dilation model

Volume

43B. number

3

PHYSICS

PERTURBATION

THEORY

LETTERS

5 February

IN THE DILATON

1973

MODEL

Marjan BACE* Institute for Theoretical Physics, Utrech t, The Netherlands Received

27 November

1972

It is shown that the In E factors, found by Li and Pagels in perturbative expansions in E around spontaneously broken chiral symmetry limits, become factors of E-’ in the dilaton model. The method also furnishes an improved derivation of the Li-Pagels result.

We consider here the strong interaction H in the usual break-up H=Ho+eHl,

Hamiltonian

(1)

where Ho is assumed SU3 g SU3 invariant and HI breaks this symmetry. The popular [l] assumption for Hlis that it belongs to the (3,3) $ (3,3) representation of chiral SU, @ SU,. In the limit when the symmetry breaking parameter E goes to zero the physical vacuum goes to an SU3 symmetrical vacuum; there correspondingly emerges an octet of massless pseudoscalar mesons which we will refer to generically as the Goldstone pions. If baryon matrix elements of Ho and HI are assumed to be of essentially the same magnitude then the baryon mass differences lead to E between 3 and f. It therefore appears reasonable to attempt to do perturbation theory with the Hamiltonian of eq. (1). As pointed out by Li and Pagels [2], however, perturbation theory in the strict sense is not applicable because of the NambuGoldstone type of symmetry limit Ho is assumed to obey. We will be concerned in this letter with basically similar but more serious difficulties that arise if one or more scalar Nambu-Goldstone bosons are assumed in this limit. With the E = 0 world containing only pseudoscalar Goldstone particles Li and Pagels have shown that a perturbation expansion of a given matrix element in E, Map = My

+ E M$ t O(2),

(2)

will typically involve an M(1) which, instead of being a constant in the E + 0 limit, has a In E type singulaity. Although this situation is not yet fully understood, ex* F.O.M. research 222

Fellow.

pansions as eq. (2) are not necessarily rendered useless; numerically the In E factor is close to 1 for physical values of E. The model of eq. (1) has been extended [3] by the introduction of broken scale invariance: H, is assumed chiral and scale invariant, while HI breaks both symmetries. It will not be necessary for us to specify in further detail the structure of this breaking term. Furthermore the vacuum spontaneously breaks scale invariance so that in addition to the Goldstone pions the E + 0 limit involves a massless scalar Goldstone boson referred to as the dilation. Although the evidence is inconclusive for this model [4] it gained in popularity when it was pointed out by Altarelli, Cabibbo and Maiani [5], by Crewther [6] and by Mathur [7] that it explains in a natural way the large o-term found phenomenologically [8] in 71- N scattering, even if HI is taken from the (3,3) @ (5,3) representation. We will see that the emergence of the scalar Goldstone particle in the e + 0 limit typically leads to an e-l behaviour for M(1) and makes the power counting implied in expansion (2) not feasible. Consider the connected matrix element Map = (0118)

(3)

to be written for covariantly normalized states in the full Heisenberg picture in which the time development is generated by H. It is convenient for our purposes to introduce a picture in which the time development is determined by Ho. We will call this the symmetrical picture. It should be noted that this is not the well known interaction picture of quantum electrodynamics since Ho is not a free Hamiltonian. However, it has been introduced in analogy to the interaction picture whose problems [e.g. 91 apply to it. Ignoring such prob-

letns, it can be checked Heisenberg

by verifying the appropriate of motion that the Heisenberg

equations

and symmetrical unitary operator

pictures

are formally

related

by the

A

J E ffl(x)d4x

-i

t2

>

(4)

1

where HI (x) is the density operator corresponding to in the symmetrical picture get hats. The matrix element of eq. (3) can now be written as

HI, and operators

Map (&I U(m, - m)lj),

(5)

from which it is simple to calculate ML;‘: M$)

lezo = -iJd4x(&I

H1 (x)1@,

= - i(27r)4 F4(p a -p&(&l

ti, (0) I/?).

(6)

Since it is written in the symmetrical picture, the matrix element (& Ifi, Ifi) should be independent of E. If this were true, perturbation theory would be applicable to the dilation model. We examine the behaviour of the full matrix element (al HI (0) Ip) in the e + 0 limit. Separating the vertex and propagator parts we write (alffl(0)lS)=A(42)G(s,...)

(7)

where s=(FJ,+@,

4 =(P,-P& A(q*)=

-isexp

(iqx)(T(H1

(8) (x)/f,

(O))J,

d4x,

(9)

and G depends, in addition to s, on whatever other independent variables there may be. We now substitute eq. (7) in eq. (6) and take the limit E + 0. In this limit the vertex function G has, in general, no reason to vanish. If, for example, a,/3 are baryon states, G is simply proportional to the coupling constant of a scalar dilaton to a baryon pair. A(O), however, is dominated in the E + 0 limit by the single particle contribution, the dilaton. Since the dilaton mass squared is assumed proportional to E A(O)-c-l, E-+0

terms in the expansion of eq. (2) indistinguishable. We also obtain in this way a criterion which indicates when perturbation theory may nevertheless be applied; if the vertex

t1

CJ(t,, tz) = Texp

5 February 1973

PHYSICS LETTERS

Volume 43B, number 3

(IO)

and we conclude that in general M(l)-+ e-l in the symmetry limit. This makes the first and second order

function

G vanishes

linearly

in E then M(l)

has a well defined finite limit. Such a situation arises, for example, if a and 0 are pseudoscalar mesons [lo]. It can also be seen at this point how one obtains In E factors in perturbative expansions when there are only Goldstone pions in the model. In that case eqs. (6) and (7) are still valid but the first intermediate state contributing to A(cJ*) is an even parity pion-pair state. Writing out the Lehman-Killen [ 111 representation for A(0) and setting pion masses to zero in the integrand (but retaining a nonzero mass in the integration threshold) one is easily led (for example by dimensional arguments) to A,,(O)

- In E.

(11)

Thus the derivation presented here gives an alternative way of obtaining the Li-Pagels result; a way which makes clearer that one is consistently performing a standard perturbation expansion in E. Roughly speaking, then the difference between models containing a scalar Goldstone boson and those containing only pseudoscalar Goldstone bosons, i.e., the c-l and In E behaviour of M(l), is due to the difference in parities which allows single particle intermediate states in the first, and only two and more particle intermediate states in the second. Returning to the dilaton model we illustrate the situation by considering a concrete example: take the operator part of tl, to be equal to the scalar, SU, singlet So, from the (3,5) @ (3,3) representation. One establishes in the standard way the Ward identity A(,; (0) &, = E>

(12)

where babAC,) (q*) = -isexp

(iqx)

(T(Fa(x)Pb(0)))O

d4x.

(13) Here the P’s are the pseudoscalar members of the same representation as S, and the 6ab factor (a, b = 0,. . .,8) is due to our assumption for HI. These are all simplicity assumptions and are not relevant for what follows*. In eq. (12) lo is the vacuum expectation for HI so long as it contains an even parity SU3 singlet term.

* In particular, one can chose an arbitrary representation

223

Volume

4 3B, number

3

PHYSICS

value of So. Now, since to tends by assumption to a finite nonzero limit one has Ar,f (0) - E. Taking derivatives of eq. (12) leads to

(14) and the single particle contribution to Atp) (0) the one that leads to A(P) (0) - e-l, cancels on the right hand side of eq. (14). The two particle contribution of a dilaton cum pion intermediate state will give rise, as above, to a In E behaviour for a&J& in the E + 0 limit. We will call this an “exact” result because it stems straightforwardly from the exact Ward identity, eq. (12). In contradiction to this exact conclusion it will now be seen that perturbation theory leads to an e-l behaviour for a<,/& in the E + 0 limit. The operators So (x) and i. (x) are related by

so(xl = UT (x0>-

O3)

so(xl w, , - -9,

(15)

from which one obtains* y

/A =- iJd4yb. (xh so Wlret.

Sandwiching we find

this relation between symmetrical

E =

0.

(16) vacua

(17)

Commuting the e-derivative in eq. (16) with the vacuum is, of course, allowed since all E’Sin this equation and in the vacuum state are first set to zero. Due to the dominance of A(,)(O) by the dilaton pole one finds

LETTERS

5 February

1973

equation relating H(x) and &x), and taking the e-derivative arrives at eq. (16) with /-II substituted for So there. Taking meson matrix elements of such an equation one arrives at

(~,aH(O) .. a ae lpb)e=O = -i

sd4y(ial[H1

C_,v>,~(0)lret

l@b;b,.

(19)

The left hand side of eq. (19) is if Pa = Pb = 0, exactly A’,lb’from the expansion

(P,IHJP$ =‘4(,O$+EA$ + O(S). This is seen by noting that

(&Pal1

HlPb=o)=O

in the E + 0 limit, since HlPb= 0)= 0 in this limit. The right hand side of eq. (19) involves a factor of (E, - Eb) and a b-function in (Pa - Pb). The retarded commutator furnishes a 0-function which, upon integration, gives (I?, - Eb)-’ thus cancelling the previous such factor. One is then left with the formula Mai

= (Pa=OIEffl(0)lP~=0)+O(~2). (20)

From our general result, then, we conclude that the mass formulea of eq. (20) is incorrect in the dilaton model. This observation was made some time ago by Ellis, Weisz, and Zumino [ 121. 1 am grateful to Professors V.S. Mathur and S. Okubo, and Dr. B. de Wit for helpful discussions.

(18)

[l]

in obvious disagreement with the “exact” result. We have already shown that a scalar Goldstone meson makes power counting impossible. This example shows in addition that perturbation theory in E may lead to singularities which in fact are not present. We wish to end by pointing out that the way perturbation theory is done here, while more systematic, is equivalent to methods used by other authors. For example, the standard meson mass formula (1) is easily obtained with the above method. One writes down the

[2] [3] [4]

[S] [6] [7] [8] [9]

* Note that any c-number part that HI may have disappears in the commutator of eq. (16).

[lo] [ll]

224

[12]

M. Gell-Mann, R. Oakes and B. Renner, Phys. Rev. 175 (1968) 2195. L. -F. Li and H. Pagels, Phys. Rev. Lett. 26 (1971) 1204. P. Carruthers, Phys. Rev. D2 (1970) 2265. C.G. Callan in: Proc. Amsterdam Intern. Conf. on Elementary particles, eds. A.G. Tenner and M.J. Veltman (North Holland Publishing, Amsterdam, 1972). G. Altarelli, N. Cabibbo and L. Maiani, Phys. Lett. 35B (1971) 415. R.J. Crewther, Phys. Rev. D3 (1971) 3152. VS. Mathur, Phys. Rev. Lett. 27 (1971) 452, 700 (E). T.P. Cheng and R.F. Dashen, Phys. Rev. Lett. 26 (1971) 594. R.F. Streater and AS. Wightman, PCT, Spin and statistics, and all that (Benjamin, New York, 1964). J. Ellis, Phys. Lett. 33B (1970) 591. H. Lehman,Nuovo Cim. 11 (1954) 342; G. Kalldn, Helv. Phys. Acta 25 (1952) 417. J. Ellis, P.H. Weisz and B. Zumino, Phys. Lett. 34B (1971) 91.