INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

Transcription

1 IC/69/63 INTEENAL REPORT (Limited distribution) INTERNATIONAL ATOMIC ENERGY AGENCY INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS A RIGOROUS LOWER BOUND ' ON THE KSFR RELATION FROM FIELD THEORY: THE STATUS OF FIELD-CURRENT IDENTITY FOR INTERACTING FIELDS + R, ACHARYA * P. NARAYANASWAMY ** and T.S. SANTHANAM*** International Centre for Theoretical Physics, Trieste, Italy. ABSTRACT The possibility of retaining the axial-vector field-current identity for interacting JT and A, fields which are dynamically independent in the sense that [ a, JTJ =0 and [ a, v] =0 at equal times is shown to lead to an inoonsistenoy unless the mass of the pion is zero. A lower b )und on the parameter n?_ is established. MIRAMARE - TRIESTE 2 July To be Submitted for publication. * Address from September 1969: Institut filr theoretische Physik, Universitat Bern, Switzerland. ** Address from September 1969: Physics Department, Southern Illinois University, Edwardsville, 111., USA. *** On leave from MATSCIENCE, Madras, India.

2

3 2 2 2 It Is well known that the KSFR relation G /(2m ) =*F cannot be derived from current algebra alone. In this paper we have attempt- ed an objective study of the status of the KSFR relation 2) with a view to determining the exact relationship between G i m and F as dictated by the fodowing assumptions: a) field-current identity,. b) canonical commutation relations of field theory and c) Weinberg's 3) first spectral function sum rule. We combine these assumptions with the information on the pion wave function renormalization constant 4) provided by the Drell-Finn-Hearn bound, which is a rigorous result of field theory, to show that G 0.7F' 2m 2 P We also demonstrate that the field-current identity for the interacting * and a fields is inconsistent with the non-zero mass of the pion if the two fields are dynamically independent in the sense at equal times. [a jr] = [a, r] = 0, (1) Li fi We begin with the axial vector field-current identity A (x) = G A a (x) + F d * (2) where A (x) is the weak axial vector current and a (x) is the pure \X A* spin-one field. Let us now consider the vacuum expectation -value of the equaltime commutator of A Q (x) and A.(x), i = 1, 2, 3, following from eqs. (1) and (2). -2-

4 fi(x Q ) <[A 0 <x), Al * 6(x 0 ) <^ra 0 *(x), 3^ TO)]^. (3) Introducing the spectral representation for the two-point functions, we write eq. (3) as J90 ffi(u^) 2 r A dju + / p ( / n ) d j u - Z F ^ O (4) 0 " 0 where we have used the positivity of the spectral function of the a e\ A field ; p {p ) and p (/J ) are^respectively,the spin-one and spin- 3) -1 zero spectral functions introduced by Weinberg. Here Z is defined in terms of the pion spectral function Z l / /» * 3) Eq. (4), along with Weinberg's first sum rule ' yields (6) We now appeal to the well-known Drell-Finn-Hearn bound 4) Z^1 > 1.4 (7) which is a rigorous result of local field theory and hence arrive at the inequality 00 " 2. V / JLCjl dm 2 > 1.4, F. 0-3-

5 This rigorous bound can, in principle, be tested experimentally since V 2 the function p (ju ) is directly measurable in terms of the total cross section of the process e + e -> (T = 1). Eq. (8) can then be re- expressed in the form 7) " 6s s < + t e "-* (T = 1! {.)> 1.4 (16* 3 a 2 F 2 ) (9) / xox ^ 0 2 where a = e /4T, For the present we employ p-dominance to obtain an estimate on the bound.2 G 2 2 F > 0.7. (10) Eq. (10) provides a rigorous lower bound which is quite consistent with the approximate validity of the KSFR relation. It should be stressed that eq. (10) is in no way dependent on Weinberg's second sum rule 3) We shall now prove that the field-current identity for the interacting a and y fields is inconsistent with the requirement that these fields are dynamically independent in the sense of eq. (1) unless the mass of the pion is zero. From eqs, (1) and (2) we obtain [A.(x), sr(o)] 5(x Q ) = TA 0 (x), 7r(0)] 6(x Q ) = 0 (11) and U.(x) ;] 6(x Q )= -[A Q (x), 9. = IF Z' 1 9, 8 4 (x) From eq. (12) it follows that i = 1,2,3. (12) 2 ) d)u 2 = F Z" 1 (13)

6 where the spectral function 2 %fo ) is defined by - i 8^ J. %(M 2 ) A (x ;,i 2 ) cfc 2. (14) 0 It is obvious that eq, (11) is consistent with the spectral representation eq.(14). When the mass of the pion is non-zeroj the equality F^ m 2 cr^2) = u 2 %(^). (15) which follows from PCAC,and eq. (14) enables eq. (13) to be rewritten as V* Fz Since F is non-zero, eq. (16) is manifestly incorrect unless crv) = «(A* 2 - m 2 ), (17) which implies that the pion is a free field (Z = 1) 7 contradicting our original assertion that it is not. When the pion mass is zero, eq. (16) is.no longer true and one has i»c&**jl p / %OI)M d/ - 0. (18) Eqs. (18) and (13) are compatible with each other since the spectral function %[n ) is not necessarily positive definite. Therefore, the consistency of eqs. (1) and (2) is restored in the presence of interaction provided the pion mass is zero. This would mean that one is dealing with a theory in which SU(2) x SU(2) symmetry is broken spontaneously. It appears rather remarkable that the vanishing of the pion mass is quite crucial to maintain the concept of field-current identity for dynamically independent interacting fields. -5-

7 Finally we observe that eq. ^6) yields an upper bound Z" 1 < 2. (19) where we have used p dominance and the approximate validity of the KSPR relation. In conjunction with the Drell-Finn-Hearn bound, eq. ( 7), we thus arrive at the conclusion that 0.5 Z Z 0.?'. (20) Eq, (20) has the implication that the pion is far from being a composite particle in the sense of field theory. ACKNOWLEDGMENTS The authors are grateful to Professors Abdus Salam and P. Budini and the International Atomic Energy Agency for their hospitality at the International Centre for Theoretical Physics, Trieste, -6-

8 REFERENCES 1) H. J. Schnitzer, Proceedings of the 1967 International Conference on Particles and Fields, p. 538, edited by C. R. Hagen, G. Guralnik and V. S. Mathur (Interscience Publishers, New York 1967); See also R. Arnowitt, M. H. Friedmann and P. Nath. Nucl. Phys. B5, 115 (1968). 2) K. Kawarabayashi and M. Suzuki, Phys, Rev. Letters 16_, 255 (1966);, '. Riazuddin and Fayyazuddin, Phys. Rev. 147_, 1071 (1966). 3) S. Weinberg, Phys. Rev. Lettersl8, 507 (1967); T. Das, V.S. Mathur and S. Okubo, Phys. Rev. Letters 18, 761 (1967). 4) S.D. DrelL A. C. Finn and A. C. Hearn, Phys. Rev. 136, B1439 (1964). 5) N. M. KrolL T. D. Lee and B. Zumino, Phys. Rev, 157, 1376 (1967); T.D. Lee, S. Weinberg and B. Zumino, Phys. Rev. Letters 18, 1029 (1967). The internal symmetry indices have been suppressed. 6) J. Schwinger, Phys. Rev. Letters _3, 296 (1959). 7) J.D. Bjorken, Phys. Rev. 148, 1467 (1966)* G.W. Barry, G.J. Gounaris and J. J. Sakurai, Phys. Rev. Letters 21, 941 (1968). 8) Abdus Salam, Nuovo Cimento 2$, 224 (1962); S. Weinberg, Phys. Rev. 130, 776 (1963). 1