Lawrence Berkeley National Laboratory Recent Work Title GLOSSARY FOR RBGGE POLE THEORY Permalink https://escholarship.org/uc/item/0sv8g2r3 Author Jackson, J.D. Publication Date 1969-09-24 eScholarship.org Powered by the California Digital Library University of California UCRL-19351 GLOSSARY FOR REGGE POLE THEORY R~CEIV • EO L''~WRENCE RADIATION LABORATORY J. D. Jackson OCT 2 0 1969 LIBRARy AN OOCUM.ENTs S.ECDTION September 24, 1969 TWO-WEEK LOAN COPY This is a library Circulating Copy which may be borrowed for two weeks. .1 for a personal retention copy, call Tech. Info. Diuision, Ext. 5545 c::: () -- -- - -- - - - ' ~ ~ LAWRENCE RADIATION LABORATORY~ 3 \J1 - 1UNIVERSITY of CALIFORNIA BERKELEY ~ ',1 j DISCLAIMER This document was prepared as an account of work sponsored by the United States Government. 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UCRL-19351 UC-34 Physics TID -4500 (54th Ed. ) UNIVERSITY OF CALIFORNIA Lawrence Radiation Laboratory Berkeley, California • AEC Contract No. W -7405 -eng-48 GLOSSARY FOR REGGE POLE THEORY J. D. Jackson September 24, 1969 • • • • 11 Printed in the United States of America Available from Clearinghouse for Federal Scientific and Technical Information National Bureau of Standards, U. S. Department of Commerce Springfield, Virginia 22151 Price: Printed Copy $3.00; Microfiche $0.65 -iii- UCRT~-19351 GLOSSARY FOR REGGE POLE THEORY* • J, D. Jackson Lawrence Radiation Laboratory · Universi~y of California Berkeley, California · September 24, 1969 ABSTRACT Part A contains a glossary for Regge pole theory for the use of readers of papers in this area of high energy theoretical physics. Part B contains definitions of invariant amplitudes and t-channel helicity amplitudes for meson-baryon elastic scattering and photoproduction of pions, as well as the connection between crossing symmetry and signature and a list of Regge trajectories. • * This work was suppor"l!ed in part by the u. S.Atomic Energy Commission. -1- UCRL-19351 INTRODUCTORY NOTE • This short glossary and accompanying·notes were prepared as an appendix to the version of my Lund review paper, "Models for High Energy Processes ir (UCRL-19205), that is to appear in the January, 1970 issue of Reviews of Modern Physics. It seems worthwhile to make the material available separately. The text is unaltered, apart from the addition of a bibliography necessary to make the present document almost self-contained. Phrases such as "Chapters of this review", ''see Eq. (VI.l3)", .''in Fig. 24·", etc •. refer to the text of UCRL-19205. For a general introduction to Regge theory the authoritative reference is now the book by Collins and Squires (1968). Some of the more advanced technicalities are described in detail by Bertocchi (1967) in his review at the Heidelberg Conference . • -2- UCRL-19351 · !APPENDIX The appendix is designed to aid the reader in understanding • this and other. papers concerned with Regge pole theory. Part A is a modest glossary of some of the commoner terms used by the experts. If some of the definitions sound like gibberish worthy of Lewis Carroll or Edward Lear {for example, see Compensation), I apologize. My hope is that a careful use of the glossary will, by stages, lead to an explanation of any term in words that a mere mortal (or experimenter) can understand. Part B consists of a number of definitions and examples of invariant amplitudes, crossing, Regge trajectories, etc . that provide ·background material for several of the Chapters of this review. • ... -3- UCRL-19351 A. Glossary • Ancestor states. States with L > N that arise when th1~ Veneziano amplitude (VI.2) is mutilated by replacing the real trajectory function by a complex function in order to obtain resonances with finite widths. When Re a(s) = N, the numerator in Eq. (VI.2) is no longer a real polynomial of degree N in t, .and so the partial wave content is no longer restricted to L <; N. Chew mechanism. See ghost-killing. Choosing sense or nonsense. See Nonsense-choosing or Sense-choosing. Compensation (mechanism). The use of a compensating trajectory of opposite parity to cancel a pole occurring in a nonsense~nonsense amplitude at a right-signature nonsense point for a Regge pole that chooses nonsense. If a pole chooses nonsense, the leading terms . a in .the helicity amplitudes behave as (cos ~t) .times 1,1,1 for ·Ass' Asn' Ann' respectively, at a right-signature nonsense point. For Ann alone there is a lOi~er order term, (cos 9t)-a-l (a - J)-1, ·that has a pole at a= J (J ~0). The compensating trajectory is such that ac = -J - 1 when a = J and its residue is such as to . 1 . cancel the pole. The situation is symmetrical around J = -2 W1th poles occurring when either J or -(J + 1) is a nonsense point. (See Gell-Mann et al., 1964, Appendix B). Conspiracy (conspiratorial solution). Relations at t = 0 among the trajectories and residues of a finite number of different Regge poles such that certain conspiracy relations are satisfied. -4- UCRL-19351 Con$piracy relation. An equation, us~ally holding at t = o, relating t-channel helicity amplitudes which receive contributions from • different parity sequences (different Regge poles). For an example, see Part B(e) of this Append,ix. For details and references, see Collins and Squires (1968), p.l09 ff and Rite (1969). Crossing (symmetry). The replacement of 4-momenta b~ their negatives (line reversal) such that s ~ u is. called s-u crossing. An amplitude that is even or odd.under this transformation is said to have a definite crossing srrometry (crossing symmetric or crossing antisymmetric, or just even or odd under crossing). The transforma- ·tion s ~~u generates v ~ -v. Other crossings for the 4-particle amplitude are s +----) t and t+---) u. Cut (in the J-plane). See Regge cut. Daughter trajectory. See Trajectory, daughter. Evasion (Evasive solution). The placing of restrictions on .the residues, but not on the trajectories, of a number of different Regge P?les in ·order to satisfy conspiracy relations. Exchange degeneracy. If two Regge poles, coupling to the same particles, and of opposite signature, have the same trajectory function, they are said to be exchange degenerate. This equality of trajectories is sometimes called "weak" exchange degeneracy. "Strong" exchange degeneracy requires equality of corresponding residue functions as well as trajectories. -5- UCRL-19351 Factorization. An almost intuitive property of Regge residues that follows from the definition·of a simple pole of definite quantum numbers and unitadty. If t3ba(t) is the residue of such a pole .. ·in an amplitude describing a transition from state a to state b (a,b specify particles and their helicities), then fac~orization ' 2 of residues requires that P. - P. P. for all values 6f t. ~-'ba - ~-'bb ~-'aa Fixed-pole. A pole in the complex angular momentum plane that does not move with the energy. In relativistic Regge theory there are fixed poles in the angular momentum plane at nonsense, wrong- signature points ifthe third double spectral function is nonvanishing. (see, for example, MandelstamandWang, 1967, and Oehme, 1968);· Froissart bound. A bound on the rate of increase of the total cross section, first proved by M. Froissart (see Eden, 1967, p.l68), as s ~co. This has the consequence in Regge pole theory that no.Regge trajectory can have an intercept greater than unity at t = 0. Gell-Mann mechanism. Same as nonsense-choosing mechanism. Ghost-killing (factor, mechanism). The standard Regge pole amplitude for spihless particles has its singularity structure determined by . ( ±1 + e -ina) r(a + l) sin-na. times a reduced residue: For odd signature this set of factors gives the desired poles at a= 1,3,5,··· and no other singular! ties. For even signature·, however, there are poles at a.== 01 2,4, · · · _and the point a = 0 may lie at negative t, corresponding to imagiriary mass and often in the physical region of ' -6- UCRL-19351 some pro.cess where poles are obviously not allowed. To eliminate this "ghost" pole it is necessary to make the reduced residue proportional to a(t). This factor of o:(t) is called the ghost­ killing factor. If the amplitude describes spin-flip and the residue "chooses sense" there will already be one or two factors of a: present in the residue. Then it becomes a matter of taste (at this state of.the art) whether a~y additional factor:of a: should be postulated, provided the 1.'equirements of factori~ation are satisfied. Chew's ghost-killing mechanism inserts i factor of a: in all even signature Regge amplitudes, independently of whether or not they need them to prevent a pole at a: = 0. Kinematic singularity. A square-root or power of a square-root singularity, generally in a helicity amplitude, at a threshold or pseudothreshold determined by the masses of the external particles, , ·or at s, t, or u equal zero .