Fortschritte der Physik / Progress of Physics: Band 14, Heft 7 1966 [Reprint 2021 ed.] 9783112500347, 9783112500330

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FORTSCHRITTE DER PHYSIK HERAUSGEGEBEN IM AUFTRAGE D E R PHYSIKALISCHEN GESELLSCHAFT IN DER DEUTSCHEN DEMOKRATISCHEN REPUBLIK VON F. KASCHLUHN, A. LÖSCHE, R. RITSCHL UND R. ROMPE

B A N D 14 • H E F T 7 • 1966

A K A D E M I E

- V E R L A G

•

B E R L I N

I N II A L T Seile

WOLFGANG K U M M E R : Introduction to Regge Poles

429

Die „ F O R T S C H R I T T E D E R P H Y S I K " sind durch den Buchhandel zu beziehen. Falls keine Bezugsmöglichkeit durch eine Buchhandlung vorhanden ist. wenden Sic sich bitte in der Deutschen Demokratischen Republik an den A K A D E M I E - V E R L A G , G m b l l , 108 Berlin 8, Leipziger Straße 3 - 4 in der Deutschen Bundesrepublik an die Auslieferungssteile: K U N S T UND W I S S E N , Inhaber Erich Bieber, 7 Stuttgart 1, Wilhelmstraße 4—6 bei Wohnsitz im Ausland an den Deutschen Buch-Export und -Import, GmbH, 701 Leipzig 1, Postschließfach 276 oder direkt an den A K A D E M I E - V E R L A G , GmbH, 108 Berlin K. Leipziger Straße 3—4

Fortschritte der Physik 14, 4 2 9 - 4 8 1 (1966)

Introduction to Regge Poles W O L F G A N G KTJMMEB

Institut

für theoretische

Physik

der Technischen

Hochschule

Wien,

Vienna,

Austria

Contents 1. Introduction

429

2. Potential scattering 2.1. Scattering amplitude at unphysical angular momentum 2.2. Complex transformations 2.3. Discussion of pole term 2.4. Solvable examples for Regge poles 2.5. Application to elementary particles and resonances

431 431 433 437 441 444

3. High energy behaviour of scattering amplitudes 446 3.1. Relativistic S-matrix, kinematics 446 3.2. Naive extension of potential scattering results to high energies in the crossed channel 448 3.3. Regge poles in field theory 450 3.3.1. Bethe-Salpeter equation 450 3.3.2. Regge poles in the weak coupling limit of the ladder approximation . . . . 452 3.3.3. Multiperipheral model 455 3.3.4. Survey of other field theoretic methods and results 457 3.4. Regge Poles in dispersion theory 458 3.4.1. Analyticity in energy and momentum transfer, unitarity 458 3.4.2. Singularities in energy and angular momentum 463 3.4.3. Higher spins, elementary particles, Khuri-Jones transform 467 3.5. Experimental consequences at high energies 469 3.5.1. Regge poles in differential and total cross-sections 469 3.5.2. Experimental situation, outlook 472 4. References

478

1. Introduction I t is v e r y t e m p t i n g to s u r v e y a certain field of physics, it it has passed from being a fashionable subject of theoretical a n d experimental discussions i n t o t h e stage w h e n critical e x a m i n a t i o n has already revealed the force a n d t h e limitations of i t s application to describe physical p h e n o m e n a . T h e extraordinary speed of d e v e l o p m e n t , 32

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KUMMBR

especially in the field of elementary particles and high energy physics, makes the period of time very short which is required for this purpose. One typical example of this feature of modern physics is represented by the subject of this review. When the usefulness of T. REGGE'S [1] concept of complex angular momentum in potential theory was recognized in 1961 for high energy physics [2], it seemed to offer for the first time an ordering principle in the vast amount of experimental information, produced by the large accelerators in the GeV region. Till 1962 "everything" seemed to in be accordance with a simple model in which elementary particles and high energy scattering experiments were described by poles of the Scattering amplitude in the complex angular momentum plane, the so-called Regge poles. Very soon, however, the disagreement of the first, really most primitive, version of the theory with certain experimental facts, especially the nonshrinking diffraction peaks as observed in Some reactions between elementary particles [3], made such a deep impression that the exaggerated optimism made room for an equally unjustified disregard—mainly from the side of the experimental physicists. In the light of the experimental knowledge of today the situation of the Regge pole model for high energy processes is by no means bad—quite apart from the theoretical insight brought about by its concepts. I t seems that with polarization measurements at high energies being available in the near future, even some of the main predictions and applications of this model which could not be tested till, now, have good chances to become an essential part of the description of high energy phenomena. The theory of complex angular momentum has the pedagogical advantage of being understandable already using a comparatively very small amount of mathematics and starting at a very simple level. This consequence of its historical development from potential theory has led the author to present a pedestrian's approach to Regge poles [4] in a self-contained way in the sections 2.1 (till eq. (2.1/7)), 2.2 (till, eq (2.2/6)), 2.3, 2.4, 2.5, concerning low energy results in potential scattering, and the high energy part in sections 3.1, 3.2, and 3.5. The remaining chapters are devoted to field theory (3.3) and the dispersion theoretical discussion (3.4). The latter are not absolutely necessary in a first reading, but they may give a better unterstanding of the theoretical background. This report is intended to be a very simple introduction. Therefore the reader is already warned at this point that he will miss mathematical rigour nearly everywhere. For the mathematician this review will be disappointing, because the easiest approach (to the authors opinion), with the least amount of mathematics and the largest one of sloppiness was preferred wherever it was possible. From this guiding principle it is clear that the more involved problems could be mentioned in a descriptive way only; in this case always references to more detailed discussions can be found. Nevertheless the author did not intend to give an exhaustive list of references. The selection contained in this review can be supplemented easily by the extensive collection in the already existing monographs on REGGE poles [5, 6, 7]. Especially ref. [7] contains a complete account of the work done till 1964 which clearly includes the main "REGGE period". Some references to the relatively few important developments of the last two years are included in our own list. All formulae are written in units where c = h = 1 are dimensionless numbers, the metric of special relativity is gog = — git = 1, so that the momentum fourvector of a particle with mass M obeys j>2 = pi — p2 = M2.

431

Introduction to Regge Poles

Finally the author apologizes for maltreating the English language, to the reader who probably knows English better than be does. 2. Potential Scattering

2.1. S c a t t e r i n g A m p l i t u d e a t U n p h y s i c a l A n g u l a r M o m e n t u m We let the mathematical procedure start at something very familiar, so to speak at the very beginning of quantum theory, the S C H R O D I N G E R equation (-Zl/(2 M) + V - E ) y = 0

(2.1/1)

where M is the mass, E the energy of the particle under consideration. If it depends on a central symmetric potential V, we can split the wave function into factors which depend separately on the spherical co-ordinates ipf = r - 1 B (r, I, E) Pf pm eim

oo1). On the other hand Note e.g. the ambiguity, induced by an additional term g(l, E) x sin nl which would give no contributions at physical values of I. I t diverges however exponentially for (complex) I oo — a behaviour which is "unreasonable" within this context (cf. ref. [6], p. 140 for a more precise discussion). 32*

432

W . KUMMER

f(l, E) is given as part of a solution of the Schrddinger equation (2.1/2). We now insert the asymptotic behaviour of the incoming plane wave e

i

V

z

~

(

2

i

p r

) - i

j r

(21

+

1)

[e

i p r

(-1)'

-

P,(cos

&)

(2.1/6)

2=0

into eq. (2.1/3) in order to obtain the scattering matrix S(l, E) and its relation to the amplitude f ( l , E). R ( l , E )

S ( l , E )

G{e- :-" r

-

e™«- 1»

= 1 + 2i f

r—»oo =

8(1,

ejv r]

E )

(I, E ) .

(2.1/7)

I n a few cases only, the knowledge of the exact solution of eq. (2.1/2) permits a direct discussion for negative and complex values of I (see section 2.4). With a general potential the technique of Jost-functions and theorems from the theory of functions are needed to make general statements about the analytic properties of /. The reader must be referred for a detailed discussion especially to the work R E G G E [1, 5 ] , FROISSAJRT [ 9 ] , and N E W T O N [ 7 ] , We just note the important difference between potentials which increase at r — 0 as V ( r ) |

a r "

2

.

I n the first case the Z-dependent "centrifugal" term in eq. (2.1/2) is dominated completely by V So that near the origin the solution does not depend on I. Then the symmetry of eq. (2.1/2) appears in R too. This leads to f ( l , E ) = f ( - l - l , E ) ,

(2.1/8)

if use is made of eq. (2.1/7). Unfortunately in this case f can be shown to behave badly in the limit 111 oo which will be needed for the unambiguous extension to unphysical I as explained above. The second and the third case are subsumed in a potential oo

V

=

ar~ 2

+

r - i f o i o i ) e~ ar

doo

Til T ( - I

-

Va)

(—

I)

(— 2 z ) ~

u i

,

(2.2/10)

the new remaining integral is easily seen to decrease faster than before (cf. eq. (2.2/6)), namely as ( —z)~12/1. The first sum contributes powers of z up to z~ -*l* only, and the last sum gives terms proportional (— z) a sin -1 jix as did the corresponding ones in eq. (2.2/2). We note again, that in relativistic theory eq. (2.1/11) may not hold [12] which restricts the applicability of eq. (2.2/9) as compared to the one of eq. (2.2/2). Ih the case of such a symmetry one could try to push the path of integration in eq. (2.2/9) arbitrarily far to the left. Then the importance of the Second sum decreases further and in the limit only Regge pole terms survive. On the other hand each of the pole terms (3n(l — 72,

a monotonous connection between I and E near physical values. 5

) ^ and i2 are at first only the physical, integer values. ) The more exact calculation by means of eq. (2.1/2) gives

6

a 2 = J d r B 211 jdr ii 2 /r 2 j .

(2-3/9)

440

W.

KUMMER

Therefore a typical Regge trajectory of such a pole term in the Z-plane looks like the one in fig. 2/5. I t first progresses along the real, axis for increasing E < 0, passing through the real bound-states I — 0,1 a t E0 and Ev and turns away into the upper half-plane at E = 0, producing resonances in the amplitude in I = 2,3 at Er l and Er l. The latter resonance will be much less pronounced, because its distance from real values (where sin" 1 n I can become big) is much larger. Eq. (2.3/9) ceases to hold, if the pole has left the real values appreciably. Hence the path of the pole may eventually turn back and the real part of ex. decreases again (fig. 2/6).

There is a slightly different behaviour when we admit the appearance of a n exchange potential for the space coordinates in the Schrodinger-equation. The potential acts then differently depending on the symmetry or antisymmetry of the wave function in the co-ordinates. I n spherical co-ordinates this means a (anti-) symmetry in z = cos The factors Pt (z) in eq. (2.1/5) are even (odd) for I even (odd) under tins symmetry operation z —> —2. Hence the potential is then not the same for even and odd partial-waves in eq. (2.1/2) and consequently the Scattering amplitude differs in the two cases giving rise to different Regge poles in the symmetric case ("positive" signature) and antisymmetric case ("negative" signature) The corresponding amplitudes read

/ = /++/f± = ••• T P±{E) sin- 1

(E) [P„ ± ( B ) ( - z ) ± Px±iE)(2)].

(2.3/10)

For the plus (minus) sign the contribution of the square 'bracket vanishes if e r - 1 )-'/•].

(2.4/6) The amplitude exhibits cuts in the variable E starting from the thresholds for particle scattering and "hole" scattering E 2; M, E iS — M . The physical amplitude in the reactions has to be fixed again, [MjEf

- 1 ]-'/. = +» [1 -

(M/E) 2]- 1!',

for both cuts. A new feature emerges from the dependence on (cf. eq. (2.4/5)) I' = - 1 / . + 1/(i + V.)" — e«.

(2.4/7)

This shows that a cut exists in the ¿-plane too whose position does not change with energy, between the points l0 = — 1 / 2 ± e 2. The equation for the Regge poles is obtained by putting again the argument of the first /"-function in eq. (2.4/6) equal to a negative integer. The discussion [15] does not show any essential differences as compared to the unrelativistic case. The more interesting prdblem of Regge poles with a Yukawa potential which is more likely to serve as a model of strong interacting forces, V =

—ge-^jr

has been treated first by L O V E L A C E and M A S S O N ( [ / J ] p. 5 1 0 , [20]). Though no exact solution exists here, one can write down an eigenvalue equation of the type

444

W . RUMMER

(2.4/3) in the form of a series whose domain of convergence can be extended by the use of continued fractions. The resulting curves which are obtained by numerical computation for Re ex., look exactly like the one in fig. 2/6. In the case of the square-well potential other numerical computations \2T\ show trajectories which have rather peculiar properties. Those have their origin in the physically not sensible sharp edges of this potential. 2.5. A p p l i c a t i o n to E l e m e n t a r y P a r t i c l e s and R e s o n a n c e s The question arises, whether it is possible to carry over the concepts mentioned above to the theory of elementary particles. Up to this point we have based considerations on a wave equation of the type (2.1/1). If, however, bound-states are formed by interactions with large coupling constants and therefore possibly large binding energies, field theory becomes necessary. The potential treatment in general gives much too crude a picture of the complex phenomena which become extremely important in this region. Consequently the field theoretic generalization of the Schrodinger equation, the Bethe-Salpeter equation [22], is the appropriate tool for a description of scattering problems, bound-states, resonances, and eventually complex angular momentum and Regge poles. As we shall see in section 3.3 the approximations, performed to obtain solutions of this equation, amount to reducing the Bethe-Salpeter equation to a relation which can be understood as Schrodinger equation generalized in a certain sense. Some new features arise in the analyticity of the scattering amplitude. But regarding Regge poles in connection with bound-states and resonances, nothing essentially new appears as compared to simple potential theory. A different and more general source of information is the dispersion theoretic "pure (S'-Matrix" approach to Strong interaction physics (cf. section 3.4). Though important theorems on these very general grounds will be derived there, the actual application to the experimentally observed particles and resonances has to be done practically in the Same semiphenomenological way as within the frame work of potential theory. It is therefore not surprising that C H E W and F K A U T S C H I [2] obtained already in their first tentative application of the concept of Regge poles to the position of resonances, all essential points though they were relying on potential theory alone. They proposed to fit the strongly interacting particles (from now on we always include resonances in this term) into different Regge trajectories for different quantum states, belonging to one and the same global scheme (maximal analyticity and unitarity assumption of the ¿»-matrix). In field theoretic language (which these authors of course did not use) this means that all particles are bound-states and resonances of one fundamental field. Physically only positive, real integer or half-integer values of x(E) (spins) can be measured. Therefore Re a (E) is plotted as a function of t = (energy)2 as in the qualitative picture of fig. 2/6. In order to treat all particles and resonances on the same footing, C H E W and F R A U T S C H I assume that they all arise from poles in the complex angular momentum plane7). But among the known particles, only a few have the same "internal" quantum-numbers I (isotopic Spin), B (baryon number), S (strangeness) and can 7

) Of course all nucleus bound-states belong to this class too.

Introduction to Regge Poles

445

therefore originate from the same Regge trajectory. The others could lie on trajectories which turn down before reaching a second physical value (cf. fig. 2/6). Since exchange potentials should play an important role, particles which belong to the same trajectory differ in their spin values by 2 (cf. eq. (2.3/10)). In fig. 2/8 particles which are likely to belong to families are plotted in this way into apparent Regge trajectories [23]. We have included especially baryons, where it now seems possible to obtain altogether three straight lines with about the same slope. If eq. (2.3/9) is generalized formally to the relativistic case (the Square of the energy of two equal particles in the center of mass is then t = 4 (p2 + M2)), the effective range of the "potential" is of the order of the pion mass —a very satisfactory result. The three different baryon trajectories are so close together that exchange potentials obviously don't seem to be too important. This is remarkable with respect to higher symmetries which include spin (SUQ and its relativistic generalizations [24]), where this assumption is made implicitly. I t is certainly for this reason that an experimentally correct mass formula for the super-super-multiplet, which contains the «-octet and the ¿-decuplet of S U 3 [24], can be derived. In this formula the splitting between spins depends essentially on an expression which corresponds to our 1(1 + 1)- Hence—as far as this part of the mass formula is concerned —these higher symmetries don't give results which could not be understood in terms of Regge poles alone. On the Fig. 2/8 Chew-Frautschi plot for some baryons and mesons other hand one could say that the Regge pole theory for baryons, together with the experimental informations about the masses, shows the unimportance of exchange forces. Unfortunately for mesons no Regge trajectory can be drawn immediately, since till now no mesons have been found which could belong to one and the same Regge family. We have drawn nevertheless hypothetical trajectories for the /, A 3 and o 8 ) with roughly the same slopes as the ones for baryons, since the range of the interaction in eq. (2.3/9) should be of the same order of magnitude. We anticipate from section 3.5 that some information can be obtained concerning these trajectories, if certain high energy experiments can be described by the Regge pole model. I t is seen from fig. 2/8 that there is no trajectory which crosses t = 0 with the common slope at 1. The theoretical meaning of this fact will be discussed, together with the high energy amplitudes, in chapter three. The trajectory which belongs to the /-particle may have «(0) = 1 as assumed in the sketch (for a reason see again section 3.5). I t is called the vacuum trajectory, because the inter8

) The co lies so close to o that no separate curve is drawn.

33

Zeitschrift „Fortschritte der Physik", Heft 7

446

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nal quantum numbers of the /-particle are those of the vacuum (B = I = S = 0, P = G = + 1 , where P(0) is the (iso-)parity. Inspection of fig. 2/8 shows that trajectories with more complicated internal quantum numbers lie below the ones, where these numbers are simple. The vacuum trajectory is the highest one. This is plausible, because one expects (at one and the same angular momentum) states with higher symmetry (low quantum numbers) to be more tightly, bound than less symmetric ones. This can be explained from the fact that the total wave function consists of one part which Stems from the space-time properties (and angular momentum), and another one from the internal symmetries. It can be expected that the bound-state is rather stable as long as its symmetry properties as a whole are rather simple. For Small angular momentum of the, say, nucleons which form all bound-states also the space part will be simple. It follows that the Same should be true for the part depending on internal symmetries. Also if a second trajectory with the same quantum numbers as the first one appears, it should lie at higher energies and hence below the first one. Of course, after having found one resonance, one can never predict conclusively others with higher angular momentum belonging to the Same trajectory since Re x(t) may turn down as in fig. 2/6 before the next physical value is reached. For a physical particle the possibility exists that the trajectory turns down in the immediate neighbourhood of the respective physical value, but just below. Then as compared to our discussion of pole terms (eqs. (2.3/4), 2.3/5)), also a small real difference Hetx(E) — I may occur. Nevertheless just the /-particle was predicted as a Regge particle from the (more or less) known parameters of the vacuum trajectory [2], 3. High Energy Behaviour of Scattering Amplitudes 3.1. R e l a t i v i s t i c (Si-Matrix, K i n e m a t i c s Let us start with that part of the theoretical development which does not depend on the specific dynamical theory of the system. We will concentrate on elastic differential cross-sections. It will be an easy matter to use afterwards the optical theorem in order to calculate the total cross-section from the elastic forward amplitude. At high energies besides the particle itself, the antiparticle has to be treated on a similar footing. Hence the elastic scattering of two particles (fig. 3/1 —a) can be

a)

b) Fig. 3/1 The three channels of a general scattering problem

c)

Introduction to Regge Poles

447

looked upon in two other, different ways (fig. 3/1—b, 3/1—c), if we only change ingoing particles into outgoing antipartides. The invariant scattering amplitudes Fs (for simplicity of scalar particles with equal masses M), is defined in a relativistically invariant way by the ^-matrix element between the two-particle initial and final states i and / (f\8\i)

= ( f \ i ) + (2jt)- 2 i(16p 0>1 p0_2