By Abdus Salam (auth.), Paul Urban (eds.)
We summarize the placement concerning non-polynomial Lagrangians: I should still make the qualification that an enor mous quantity of verification is required prior to the issues of renormalizability are all looked after out, yet one may possibly ten tatively kingdom: 1) All matrix components are finite for theories the place the Dyson index D is below . 2) For the situations whilst D=2 or three, counter-terms were explicitly written which soak up all infinities and the theories are renormalizable. three) combined theories of polynomial and non-polynomial fields seem to be renormalizable supplied the Dyson in dices individually and together satisfy renormalizability standards. We think that vulnerable interactions, chiral los angeles grangians and Yang-Mills idea fall into this category notwithstanding precise proofs haven't but been built. four) it kind of feels most probably that to every order within the significant coupling (and to all orders within the minor coupling}the S matrix parts, as computed by means of tools defined, fulfill the required unitarity and analyticity requirements.
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Additional info for Developments in High Energy Physics: Proceedings of the IX. Internationale Universitätswochen für Kernphysik 1970 der Karl-Franzens-Universität Graz, at Schladming (Steiermark, Austria), 23rd February – 7th March 1970
28) v(s) ? 29) and for s>16 • It follows from eq. 26), and the theorem embodied in eqs. 30) where C1 is a quantity that depends only on the Holder index,~ • I leave you to show that there is no difficulty in handling the second piece of eq. 8), which only in- volves a vulgar non-singular integral. From eq. 10), rewritten with a prime on the left- hand side, one sees that 111m f I (s) II s [ C ~ + 1 ] b 2 +B Horeover, because of the phase-space factor in eq. 33) C2. 26) into itself. 35) is indeed observed, so that the set has been mapped into itself.
9) that is, when ~ t Hence, for any for t~16 s~4 . e. 12) §~2~~lY satisfied for s~16 , as should be the case. Above s=16, there is an inelastic contribution, Bpel(t,s) • In fact, one is free to add any crossing symmetric contribution that vanishes for s~16, so as to preserve elastic unitarity. We will use this freedom to ensure that the inelastic unitarity constraints are not violated. 6), and write them, not for p (s,t) directly, but rather for p(s,t) Instead of eq. ting function, which is constrained to vanish for s~16 and t~16 , and which will be chosen in such a way that the inelastic inequalities are observed for s>16.
S. Fradkin, Nucl. Phys. 49, 624 (1963); G. V. Efimov, Soviet Phys. l. 1417 (1963); G. Feinberg and A. Pais, Phys. Rev. ll, 2724 (1963), w. Gtittinger, Fortschr. Phys. i, 483 (1966); M. K. Volkov, Ann. Phys. ) ~, 202 (1968), H. M. Fried, Nuovo Cimento 52A, 1333 (1967); B. W. Lee and B. 10S3 (1969); D. I. Fivel and P. K. Mitter, "A theory of weak interactions without divergences", Univ. of Maryland preprint UMD-70-029 (1969); R. Delbourgo, Abdus Salam and J. Strathdee, ICTP, Trieste, preprint IC/69/17, to appear in Phys.
Developments in High Energy Physics: Proceedings of the IX. Internationale Universitätswochen für Kernphysik 1970 der Karl-Franzens-Universität Graz, at Schladming (Steiermark, Austria), 23rd February – 7th March 1970 by Abdus Salam (auth.), Paul Urban (eds.)