Veneziano-Type Form Factors for the Pion

Abstract

We clearly also have the restriction \-M u which determines which of the helicity amplitudes of the PI,PQ system survives at t t -0. 13 If the sum over J 2 i n (9) is dominated by a single resonance of spin S 2 , thenMj ^m.ixi(JQ+j it S 2 +J2).The scalar, pseudoscalar, vector, and axial-vector form factors of the pion are derived from (i) the Veneziano amplitudes for 7r + 7r"" -* 7r + 7r"" and 7r + 7r~--*.A 1 + 7r~, (ii) the hypothesis of the partially conserved axial-vector current, and (iii) the requirement that a chirally conjugate pair of form factors should exhibit similar structures. The resulting form factors feature an infinite sequence of poles corresponding to daughters of specific spin and parity, and definite signature. Various predictions on the clashing-beam production of 27r and on p-meson coupling constants are given.Recently, the Veneziano-type form factors for the pion have been discussed by several authors, 1 but these fail to satisfy a natural requirement that a chirally conjugate pair of sources, like the vector current V u a and the axial-vector current A M a , should couple in more or less symmetric ways to an infinite number of particles with specific spin and parity, whose existence is one consequence of the Veneziano model. 2 To illustrate our point, let us take for example the following off-shell TITT scattering amplitude which is consistent with the condition of partially conserved axial-vector current (PCAC) and which reduces to the Veneziano-Lovelace 3 amplitude on the mass shell:(7 r -(g')l9 X A x ( -, (0)k + (/,) )1 r-(g)) = V2/,m" 2 (fe 2 -m ¥ 2 )-1^11 ' 1 (s,awhere B k iJ {s,t)^T{i-a s )T{j-a t )/T{k-a s -a t ).Here k is the momentum of the off-shell n + meson and s = (p + q) 2 , t = (k-p) 2 , and u = (q-k) 2 . a s = a(s) represents the degenerate p-/trajectory and is given by a(s) = | + 6(s-m 7r 2 ) with b~x = 2(m p 2 -m 1T 2 ). A is the pion decay constant and 0o = £>7r7r 2 « Now by continuing to />-0 by means of the standard soft-pion technique the left-hand side of (1) reduces to the pion matrix element of the so-called £ term, t -k 2 , and using a {m ff 2 ) = \ we obtain 4(2)in order to obtain the above normalization. Thus, in spite of our original assumption expressed in (1), namely, that the pseudoscalar source d x A x {mm) (^) couples only to the IT meson, we obtain the result that its chiral counterpart 2 couples to all 0 + daughters of the p-/trajectory. We consider this asym-U 0 <+, W,9^X ( _> (0)]x 0 -0 = 2i6 3

General Treatment ofππ→πA1in the Veneziano Model

“…35 (b) These subtraction constants lead to a pion form factor with undesirable asymptotic behavior unless modifications are introduced, such as an infinite number of poles in the axial-vector current. 19 (c) The widths T(X-^Anr) for X along the leading trajectory decrease exponentially with / as expected, but the first few are quite large. The over-all scale in the present model can only be reduced to reasonable values if an appreciable amount of gs-type coupling is allowed within the limits of the SLAC data, indicating a preference in sign for GT/GL.…”

“…The most appealing is to assume that A M couples to all 1+ daughters of the w-Ai trajectory. 19 In this case, Eq. (15) is replaced by…”

Connection BetweenFπ(t)and the Amplitudes forππ→ππandGeffen1 1969Phys. Rev. Lett.11010View full textAdd to dashboardCite

“…PURC-2137-9 (unpublished). 6 Monte Carlo program based on technique of J. Butcher and H. Messell, NucL Phys.…”

Pion electromagnetic form factor, perturbative QCD, and large-NcRegge models

“…Quite generally, pQCD predicts integer powers and logarithms of Q 2 , whereas the models of Refs. [44,45,46,47] are able to generate fractional powers.…”

“…We term the mechanism the powerto-log transmutation, which essentially corresponds to a suitable superposition of fractional twist operators in the Regge model of Refs. [44,45,46,47] . The present study follows our investigation of the two-point functions [51,52].…”