To local saturation of field commutators

Volume 55B, number 5

PHYSICS LETrERS

17 March 1975

TO LOCAL SATURATION OF FIELD COMMUTATORS

I.F. SKIRKO, B.L. VORONOV Lebedev Physical Institute, Moscow, USSR Received 29 January 1975 Any problem of local saturation of field commutator matrix elements for fields with asymptotic/n-limit is reduced to solvingthe appropriately truncated set of equations for the r-functions. A new version of local saturation is proposed. In applications of the field or current algebras, e.g. when deriving the sum rules, it is often suggested that only a finite number of intermediate particle states essentially contributes into the matrix elements of the field or current commutators. In order to get frame independent results, this approximation to commutators, known as saturation, should retain locality. In [1 ] the problem of local saturation was formulated in the general form as the finite-number-of-contractions approximation to commutators, the additional constraints on the matrix elements involved of fields (or currents) under consideration, like TCP-invariance, analyticity and unitarity properties, being imposed from the very beginning. (See also the previous works [2--4] on one and two particle local saturations of the simplest commutator matrix elements). When dealing with the saturation problem local operators are often taken as fields and currents which have the asymptotic ("in" or "out") limit, this limit being proportional to some definite free field. For brevity sake such local operators will be referred to as fields, and only the in-limit will be considered. We would like to draw attention to the fact that for fields which have the in-limit the construction of the locaUy saturated matrix elements of field commutators (f.c.m.e.) is reduced, in general, to solving the set of equations for matrix elements o f the Heisenberg currents (sources) (the so called r-functions). These equations are some approximation in the intermediate particles number to the known axiomatic equations for the r-functions [7, 8]. To illustrate, we restrict ourselves to one neutral scalar field. Let ~o(x), a neutral scalar field, have the asymptotic limit ~Pin(X)= (2rt)-3/2fd4k exp ( - ikx)'~in(k ) which describes free particles, the m a ~ r t i c l e s being m. Let us denote the n-particle states Ik 1 ... k n) as In), (k Ik') = 26o(k) 5(k - k ' ) , o~(k) = x/k 2 + m 2. ~o(x) is presented in the Yang-Feldman form: ~o(x) = ~Oin(X) - f d 4 x ' A R ( x x ' , m ) / ( x ' ) , the Heisenberg current (the source)/(x) = (O - m 2) ~0(x) satisfies the conditions (0 I/(x) 10) = (01](x)l 1)=0. ](x) may be expanded in terms of normal products of the in-field, -

](x)=(2~r)-a/2

fd4kexp(-ikx) n=2 ~ n-~.v/=IJld4k'~5'= k -

k

rn(k 1 . . . k n ) : ~ i n ( k l ) . . . ~ i n ( k n ) :

(1)

The r-functions r n (k 1 ... kn) are defined only on the mass shell: V i, ki E F+ O p - ,

F ± = {k = (kO, k) : kO = +w(k)) and have the evident symrnetry properties:

rn(k 1 ... kn) = rn(rt(k 1 ... kn) ) = I;n(-k 1 ... - k n ) = rn(Lk 1 ... L k n ) ,

(2)

rt is an arbitrary permutation among (1 ... n), L E Lt+. Up to the factor (2 ~)3/2, r-functions are the disconnected ("non-diagonal") matrix elements of the Heisenberg current](0). The problem of the local saturation of f.c.m.e, like (ll [~o(0), ~(x)] Im) = Cl+m(x) is evidently a question on an 477

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admissible form of the relevant rn(k 1 ... kn). This question may be partly answered if some known results of the axiomatic field theory are used. Namely, following [5] one can show that (ll [~0(0), ~0(x)] Im), Jl) = Iql ... ql ), Im) = IPl .-- Pm ), vanishes for x 2 < 0 if and only if


Im) = 0, x 2 < 0;

(3a)

rl+m +1((- q}, k, (p)) = - i f d 4 x exp ( - i k x ) O ( - x °) (/I [](0),/'(x)] Im) + At+ m ( ( - g ) , k(p}) (3b)

= Rl+ m ( ( - q), k, {p}) + Al+ m ( ( - q}, k, (p}), k E r + U F - ; A t + m ( ( - q } , k , ( p } ) - Al+m((-q}, Q -

9 - k,{p}) = [m 2 - ( Q -

9 - k ) 2 ] k t + m ( { - q } , k,{p}), k E P + U P - . (3c)

Here {-q} and {p} stands for - q 1""- ql, and p 1"" Pro, respectively, qi, t9]E I"+, Q = Z~=1 qi, ~ = Zm 1 P]" The so called quasilocal term At+ m ( ( - q ) , k, (p}) is a polynomial in k (therefore, ?~l+rn( ( - q ) , k, (p}) is alsb a polynomial in k) and represents the ambiguity in the definition of the retarded commutator. Rl+ m may be correctly, i.e. in a finite and relativistically invariant manner, defined [6] due to the Jost-LehmannDyson representation, then At+ m is also a finite and invariant function of all the momenta involved. With this definition some arbitrary 3-dimensional hypersurface Z enters eq. (3b) essentially. However, eq. (3a) garantees Rt+ m is also a finite and invariant function of all the momenta involved. With this definition some arbitrary 3-dimensional hypersurface Y~enters eq. (3b) essentially. However, eq. (3a) garantees Rt+rn to be independent on the choice of Z. And inversely, Z-independent solutions of eq. (3b) obey eq. (3a) automatically [9]. It is evident that eqs. (3) are the equations for the r-functions. However, when written in terms of the r-functions, eq. (3b) is a typical equation out of the known set of axiomatic equations for (nondiagonal) matrix elements of the Heisenberg currents [7, 8]. Due to [8], the indefinite quasilocal terms Al+ m may be excluded from eqs. (3b) if some boundary conditions on the r-functions, like (q I](0)Ip) ~ g, P ~ 0, etc. are imposed. Thus, the construction of local Ct+m(X ) is equivalent to solving eqs. (3). As it was noted, if ~-independent solutions are considered, it is sufficient to solve eq. (3b) only. Eqs. (3) enable us to treat an exact local theory, as well as any approximation to local field which preserves locality. It was shown [9] that ~-independent solutions of the entire set of eqs. (3b) (all II>, Ira) are considered) reconstruct local TCP-invariant quantum field theory with the unitary S-matrix. What concerns possible approximations to local field (in particular, various saturation schemes) its formulation in terms of eqs. (3)has practical advantages if one would be able to guarantee the Z-independence of solutions, or equivalently, to solve effectively eqs. (3a). Within some approximations eq. (3a) are easily solved. Namely, the perturbative solution of the entire set of eqs. (3b) for an exact local theory reduces these equations to the recurrent algebraic formulas which determine the higher rn'S in terms of lower ones [10]. Eqs. (3a) are fulfilled automatically and renormalized series expansion arises. One contraction approximation [ 1] is also easily treated in the framework of eqs. (3) since it corresponds to the lowest order approximation of perturbation theory. The pole represeiatation for all rn's, n > 2, follows in this case from eqs. (3b) immediately. Higher order contraction approximations for all Cl+m are hard to treat because of the inf'mite number of coupled equations and the nontriviality of eqs. (3a). Therefore we propose the different approach to the saturation problem. We consider this problem as the approximation to local field by finite functional polynomials in in-fields. I.e. in eq. (1) we put r n = O, n > N for some N < oo. The requirement of locality is preserved in some sector, i.e. for CI+m (x) with l + rn ~
Volume 55B, number 5

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approximation itself allows us to define consistently the S-matrix in some sector under certain constraints on the particle energies, the S-matrix satisfies the unitarity condition of the type SS += 1. What concerns the TCP-properties, we expect them to appear together with the analytical properties of the r-functions, as it will be demonstrated below. The local saturation problem so formulated is equivalent to solving the set of finite number of coupled equations (3), corresponding to the approximation adopted (N, M given, l +m ~ 3. Eq. (3a) holds identically and eqs. (3b) (3c) take the form

r2(k,p) = Al(k, p), k ~ r + u r - , p ~ r + ;

(4)

Al(k, p) - A l ( - p - k, p) = [m 2 - (p +k)2lXl (k, p ) .

(5)

Together with eqs. (2) this gives

r2(kl,k2)=P((kl +k2)2),

kl,k2~r+UP"

,

(6)

P(s) is an arbitrary real polynomial of s. We see that the solution given in [ 1,2] follows immediately. Now let us take into account also two-particle intermediate states [4] : r n = O, n >/4. It is important that starting from eqs. (3a), (3b) (1l), Im) = 10), Ip)) and eqs. (2), one can prove in a general case (irrespective of the number of terms in the expansion (1)) the dispersion relations, as well as TCP-condition (cf. eq. (7) with eq. (9) of ref. [4]) for r2(k 1, k2) : r2(k 1 , k2) = r(k 1 + k 2 + ie)2), (k + ie) 2 = k 2 + iek O, e ~ O, r(z) is a real analytical function in the whole plane cutted along the positive real axis from 4rn 2 up to ~. The discontinuity along the cut 4m 2 ~
~[r(s+ie)-r(s-ie)]=Imr(s+ie)=-½

fd4xexp(-iqx)(OI](O)](x)lp),

s=(q+p)2; q , p ~ P + .

(7)

Note that in [4] both the dispersion relations and TCP-conditions for r 2 were assumed. In the case considered eq. (3a) takes the form

(Ol[/(O),/(x)]lp)=fd4qexpOqx){f(q,p)-f(-p-q,p}=O,

x2<0 ;

(8)

f(q, p) = e(q O) r((q - i e ) 2) t0(q2, p .q, e(qO)) ;

(9)

1 3 fd4kld4k26(k2_m2)6(k2_m2)6(q_k1_k2 to(q2, P "q, e(qO)) - 2(2rr)

) (10)

X [®(k °) O(k °) + O(-kO)®(-KO)] r3(kl, k2, p),

pC P+.

Unfortunately, we do not known the general solution of eqs. (8-10) in terms of r2, r 3. In [4] it is suggested that the approximation used together with TCP-conditions for r 3 necessitates f(q, p) to be independent ofp. Here we do not impose TCP-conditions on r3, these conditions appears naturally (to be true, only on the mass shell) if one satisfies locality of C2. However, following [4], let us consider the solution for which f(q, p) =f(q). Then eq. (8) together with the Lorentz invariance o f f ( q ) implies fd4q exp (iqx)f(q) = O, x 2 < O. Hence f(q) = e(qO) p(q2). Eq. (3b) immediately takes the form of the dispersion relations. From eq. (7) one has, with r(q 2) = r(q 2 + ie) = Ir(q 2) I exp (i8(q2)), (11)

to(q2, e(qO)) - ie(q0) [exp (2ie (q.0) 8(q2)) _ 1] O(q 2 - 4rn 2) (2704 479

Volume 55B, number 5

PHYSICS LETTERS

17 March 1975

so, the elastic unitarity of the s-wave scattering amplitude follows as the consequence of the approximation considered (again cf. [4]). Just as in [4] for r(z) satisfying eqs. ( 7 ) - ( 1 1 ) we get the Hilbert (MuschelishwiU-Omnes) problem [11] the solution of which under rather natural assumptions on 6 (s) is

r(z) = rr (z)exp I (z---~)Tr ' ~ 2 q6(s)(s)(s-dS-z)_]'l

(12)

n(z), g(z) are appropriate real polyonomials. 6(s) cannot be arbitrary due to the symmetry properties (2) for r 3, which are basic ones. Namely, it is yet not trivial to find symmetric r3(k 1 , k 2 , k3), k 1 , k 2, k 3 E P+, so that

f(q, p) does not depend on p. For example, if r 3 (k 1, k2, k3) = qb(kl "k2) + qb( k l . k 3) + ~ ( k 2 . k 3), then f(q, p) = f(q) implies ~ = C = const, but together with eq. (11) this gives C = 0 (i.e. ~ (q2) = 0) and we return to the previous case. The same is rather likely to hold if for r3(k 1 , k 2, k3) the Mandelstam type dispersion representation is valid. In any case the problem of to what extent the assumption f(q, p) = f ( q ) is compatible with symmetry properties (2) for r 3 together with unitarity condition (11) remains open. In conclusion we give here an example of the solution when f(q, p) depends essentially on p. Namely let r3(kl, k2, k3) = t((k I + k2+ k 3 + ie)2). Eqs. (3a) (3b) will be satisfied if r2(kl, k2) = P((k 1 + k2)2);

t(z) = C.P(z) ( z - m 2)

ds (s - m 2) (s - z) + Q(z

;

(13)

'-4m 2 P and Q are real polynomials, C = C. Although nontrivial, this solution gives no scattering. The more detailed treatment of the approach considered together with more general solutions of the particular problems will be presented elsewhere. The authors are indebted to Prof. V.Ya. Fainberg for useful discussions.

References [ 1] B. Stech, Prec. Symp. on Basic questions in elementary particle physics, Munich, 1971, p. 227. [2] B. Stech, Zeit. Phys. 239 (1970) 387. [3] H.D. Dahmen, K.D. Rothe, B. Stech, Phys. Lett. 34B (1971) 83; H.D. Dahmen, H.J. Rothe, K.D. Rothe, Nuovo Cim. 8A (1972) 416; D.G. Fakitov, JINR-prepfint P2-7660, Dubna (1974) (in Russian). [4] H.D. Dahmen, D. Gromes, K.D. Rothe, B. Stech, Phys. Lett. 35B (1971) 335. [5] V. Ya. Falnberg, Zh. Eksp. Teor. Fiz. 40 (1961) 1759. [6] B.L. Voronov, report to Kiev Conf. on Hight energy physics (1970). [7] B.V. Medvedev, M.K. Polivanov, Intern. Winter School on Theoretical Physics, vol. I, p. 77, Dubna (1964) (in Russian). [8] V.Ya. Fainberg, Intern. Winter School on Theoretical Physics, vol. I, p. 144, Dubna (1964) (in Russian). [9] l.F. Skirko, B.L. Voronov, P.N. Lebedev Physical Institute, preprint N143, Moscow, 1971, p. 51. [10] I.F. Skirko, B.L. Voronov, P.N. Lebedev, Physical Institute preprint N138, Moscow, 1970. [11] N.I. Muschelishwili, Singuljarnije integralnije uravnenija, Moskva (1962).

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