A discussion of duality and analyticity

IL NUOVO CIMENTO VoL. 1 A, N. 2 21 GennMo 1971

A Discussion of Duality and Analyticity (*).

C. BOLDRIGHINI

I s t i t u t o d i E i s i c a de l l 'Univers i t& - R o m a

L. SEI~TORIO

C E R N - Geneva

(ricevuto 1'1 Luglio 1970)

Summary . - - In this paper we first discuss how the Veneziano representa t ion contains Regge poles. I t does so in a very complicated way and even if the unitarizat ion program were taken for granted extra prob- lems would be expected. Second, the couplings of the elastic channel are studied as functions of the energy and of the angular momentum 1. Appropr ia te average values of the angular momentum I seem to have a clear meaning and this is conserved also when the straight-l ine rela- t ionship among l and m S of the model is broken. Although the general coupling case is not studied here the authors think tha t the dual model may have a more promising meaning if taken from this stat ist ical point of view.

1. - I t is wel l k n o w n t h a t a v igorous effort is b e i n g m a d e these days in

o r d e r to i n v e n t a n d deve lop a p e r t u r b a t i v e t h e o r y which , s t a r t i n g f rom t h e

gene ra l i zed Venez iano r e p r e s e n t a t i o n , shou ld conve rge t o w a r d s a ful l com-

p u t a t i o n of t h e S - m a t r i x (1).

As th i s p r o g r a m is s t i l l in t h e d e v e l o p m e n t s t age no s t r i c t ru les have been

s t a t e d to spec i fy i t . 51evertheless i t seems v e r y r e a s o n a b l e t h a t , as a l r e a d y

t h e four- a n d f ive -po in t a m p l i t u d e s seem to r e p r e s e n t some of t h e exper i -

(*) This research has been sponsored in par t by the Air Force Office of Scientific Research, through the European Office of Aerospace Research, OAR, United States Air Force, under contract f 61 05627 C 0084. (1) C. LOVELACE: V e n e z i a n o theory, CERN preprint TH. 1123 (1970).

293

~ C. BOLDRIGHINI and L. S]~RTORIO

mental data satisfactorily, the theory should provide a procedure which slightly changes such a first-order approximation {let us call it V°), at the same time restoring the analyticity of unitarity, this last property not being exhibited by the representation V °.

Looking from a naive point of view, one could believe that the properties of V ° should only be a little different from those eventually corrected, and that the latter should agree with previous knowledge of analytic S-matrix theory. But it may not be so; actually, we have that, on the road from V o to the previous results of the analytic S-matrix via perturbation, we encounter situations and/or, namely, conceptual alternatives. As an example we mention a most interesting conceptual issue which originated in the comparison of the properties of V o with previous results. Analyticity matches well with the Lorentz-pole hypothesis; the latter and the idea of straight-line trajectories (with unit spacing) are also compatible concepts, which also fit with Bethe- Salpeter unitarity. The V ° representation instead shows in its asymptotic expansion a behavior which, in addition to the well-known violation of nni- rarity analyticity (there are 6-functions in the variable which goes to infinity, for instance s), also violates the single-Lorentz-pole hypothesis. The Lorentz- pole hypothesis is a simplicity assumption inspired by the extra four-dimen- sional symmetry that the amplitudes have at t--~ 0. If a simple pole is the singularity furthest to the right in the complex plane of the SL~o Casimir index 4, infinite Regge poles spaced by one unit are predicted. Now, a simple Veneziano amplitude (without satellites) has an asymptotic behaviour which corresponds to the existence of infinite Lorentz poles all spaced by one unit below the leading one. Correspondingly, there are also infinite Regge poles again spaced by one and obviously with a multiplicity entirely different from that predicted for the family originating in a single leading Lorentz pole (5).

Should the perturbation program, together with the resulting unitarity analyticity, restore also some simplicity in the Lorentz plane? A subsequent study of n-point amplitudes has pushed much further the research of ref. (~), and has shown, with an actual level counting at all energies, that the dis- agreement with Lorentz poles turns out to be in fact an agreement with the hadronic mass spectrum of the Hagedorn thcrmodynamical model (3).

We learn from the preceding discussion that the program of leaving the zero-order approximation can either be expected to modify V ° in line with

the previous understanding and achievements (such a possibility in our example being that of studying the satellite addition) or to break previous schemes in favour of new concepts. I t seems reasonable, for instance, to expect that the

(3) M. L. 1)ACIELLO, •. SERTORIO and B. TAGLIENTI: NUOVO Cimento, 6 2 A , 713 (1969), and several papers thereaf te r by var ious authors. (a) S. FUBINI and G. VENEZIANO : •UOVO Cimento, 64 A, 811 (1969) and subsequent works.

A D ' I S C U S S I O N OF D U A L I T Y A N D A N A L Y T I C I T Y 9~95

level mult ipl ic i ty contained in V ° shall remain, perhaps approximate ly , also af ter res torat ion of un i ta r i ty ana ly t ie i ty (4).

Analogously to the L-pole issue there are other critical observat ions which can be profi tably made on V o, the purpose again being to have a clearer

unders tanding of the va r i e ty of propert ies t ha t the uni tar iza t ion project is expected to restore or preserve or break. Now the t a rge t of our criticism is the concept of Regge pole itself. I t is considered a good feature of the V o representat ion t ha t i t contains the l~egge-pole behaviour. Bu t it mus t be clear f rom the beginning t ha t the V o representa t ion mus t contain some bad features

as well as good features with respect to ]~egge poles for the simple reason tha t V o does not represent really a scat tering ampl i tude bu t ra ther represents the collection of the bound states of the theory. :Nevertheless, the bad and the

good features are closely linked. W h a t we hope is tha t un i ta r i ty will keep the

good features, a t the same t ime cancelling the bad ones and moreover, because

the good propert ies seem well inserted, the correction should be small and quick.

:Now, which are the good features and which, instead, are the bad features? We will t ry to answe;' this question in the following. We will t rea t in what follows only a four-point ampli tude. We th ink in fact t ha t there is no point in using the general n-point funct ion for our purposes. I n fact our (~ classical )> knowledge on the elastic ampli tudes is b y far more complete and also the meaning of dual i ty in this case is closer to the physiscs t ha t originated it.

I n Sect. 2 we discuss a first point of view. This is t ha t the bad features of V ° mus t be el iminated by uui tar i ty . I f all the quasi-trajectories appear ing in V o have to be tu rned into t rue Regge trajectories, special relations among

s-channel residues and coefficients of the asympto t i c expansion mus t be established.

In Sect. 3 we take a second point of view. I t is to develop the idea tha t with or wi thout un i ta r i ty corrections, V ° and t rue Regge poles are twin, complementary , ways to express the same physical reali ty, t ha t is pcripherMism. In this sense it is proposed t ha t a good use of V o is to consider averaged quan- tities over the huge assemblies of s tates tha t such a model provides.

I n Sect. 4 this idea is pushed a little fur ther by showing tha t the idea of averaging could even be considered the root of a stat ist ical in terpre ta t ion of the concept of duali ty.

I n Sect. 5 the fact t h a t at least for the elastic ampl i tude the average values

correspond to nonaverage values in other approaches in which some sort of rough uni ta r i ty has been forced into dual i ty is discussed.

We th ink tha t the great p rogram of uni tar izat ion using the field-theoretical techniques shall ve ry possibly proceed in the direction of these e lementary reasonings.

(4) Independent reasonings moreover discourage the satellite approach. See (i).

~9~ c, BOLDRIGHINI a n d L. SF~RTORIO

2. - We have first to explain what we mean by Regge poles. We assume as classic definition of Regge poles the existence of the Regge asymptotic behaviour which states that the s-dependence comes only through s ~ct), namely we say tha t the 0~. 1 harmonic analysis is thought to be made on the lull amplitude. Evidently, a nonhnear way of treating Regge poles may very well be devised. (We recall, for instance, the approaches of Com~ TAI~iXOUDJI et al., I~RAV~SCm and MARGOLIS, etc., but in that way duality is spoiled and, for this reason, we do not consider such cases here.)

Having stated our definition we can introduce the concept of asymptotic discrepancy. I t is the following. If we start from the amplitude

(1)

p ( 1 - ~ ( s ) ) / ' ( 1 - ~( t ) ) v(s, t) = P ( 1 - - a(s ) - - a ( t ) )

ot(s)=as +b , a ( t )=a t + b ,

we may consider its asymptotic behaviour, coming from the Stirling formula~

7t sin zt(a(s) + a(t)) (as)~(,) (2) V(s, t)--->F(a(t) ) sin~a(t) sin ~a(s)

with

I m 8 > 8 ,

l~e s

where a(t) is the function appearing in (1). On the other hand, consider the asymptotic behaviour of (1) as coming

from its Froissart-Gribov partial-wave analysis. The latter comes from

(3)

c o

V(l, t) =fdisc, V(s, t)Q~(zt)dz, , za

which, as a(t)-* l, behaves like

fl(t) (3') v ( t , t ) , - , 1 - o c t ) '

where a(t) is the pole furthest to the right, so that the Sommerfeld-Watson transformation automatically gives

(4) V(s, t) ~ - - ~/~[2a(t) + 1] fl(t) 2z,)~(t) F(o~(t) -f- 1) sin ~a(t~) ( - F(o~(t) + 1)"

A D I S C U S S I O N OF D U A L I T Y A N D A N A L Y T I C I T Y 297

There is a discrepancy between (2) and (4). Equat ion (2) contains the

(5) D ( s , t) = - - sin ~((z(s) + ~z(t))

s i n ~ ( s )

ext ra dependence on s. In fact, expression (4) is the asymptot ic form of a funct ion of which the analytic s t ructure of the 0~.~ t ransform is known. I t cannot contain s other than in the power s "m dependence. Formula (2) instead, is the direct asymptot ic form which fails just on the real axis. Formula (4) comes from the divergence of the Froissart-Gribov integral evaluated start ing from the s-channel expansion

(6) v ( s , t) = ~ o ~ - ~ , ~'~"'P~(z~) ,

= - - n=O l=O 8 - - 8 n

I m W ( 8 , t ) : ~ i ~n 'p l<Zs)(~< 8 - ~ n ) , n =O l~O

where

R=(t) = -F-~) k-i

and having used

1 1 - P - - + i 6 ( s - - s . ) ,

8 - - 8 n 8 - - 8 n

so tha t the integral becomes the series

(7) V(1, t )=~_o ~=o~ ~(. ) O. l + t _ 4 # V t _ 4 / ~ , z. = 1 + s ~ _ 4 / ~ .

The series (6) diverges al, l = ~(t) because the limit of the residue R,~(t)

for n large is

1 R . ( t ) -+ n ~(t) - - _r(~(t))"

I t is therefore a relation between the number ordering the pole n and the magnitude of the residue which constructs the Regge behaviour V + s ~<t~.

The s tudy of the discrepant factor D ( s , t) (see (6)) introduces us at once to the core of the problem. I t is this factor which reminds us tha t the V o amplitude is not a t rue scattering amplitude, it is a factor which affects the model and this is the proper ty of the model which is requested to disappear with the unitarization cure. A common reasoning is tha t apar t from the factor

9'98 C, BOLDRIGHINI and L. S~,RTOR~O

D(s, t) the propert ies of V ° are good (in par t icular the Regge behaviour) and

should be preserved. Moreover, because the good propert ies are also assumed to be benign, the el imination of wha t is b a d should be a small correction.

We consider the following r emark to be instruct ive a t this point . Consider instead of the t ra jec tory

~ - - : a s + b

the t ra jec tory

:¢ = as ~ b ~- es v , e ~ 0 , 7--~0,

which is the smallest depar ture f rom the s traight line. (This is the famous

t ra jec to ry which gives rise to the old prob lem of ghosts, in which we are not

interested.) Consider the new asympto t i c behaviour. We leave to Appendix A the

non immedia te calculations. We have

V(8, t) s~--~ flo(t)$ a(*) + fll(t)s a($,-l+y + f12(t)8 a``)-I + 0(8 °t-2"[~) ,

Such a sympto t i c series shows t h a t there is an unbalance in the counting among

t > 0 resonances and t < 0 Regge contributions. W h a t does this tell us?

I t tells us t h a t the poles mus t s tay on the real axis if we want the resemblanco with Regge asympto t ic behaviour. Consequently, wi th the poles on the real axis, the discrepant phenomenon mus t s tay wi th us. I t is an essential prop- e r ty of V ° if we wan t the assembly oi poles a t s = n to contain also the twin p rope r ty of the Regge behaviour (see (21) again). Nevertheless a honest Regge-

pole behaviour does no t have D(s, t). A first insight into this factor D(s, t) comes f rom a s tudy of JACOB and

MA~'DELB~0J~ (5), who have been able to show tha t a singulari ty in the vari-

able s mus t appea r when an infinite number of a sympto t i c contr ibutions fin(t). • s ~-(t) are added; this is a first root to unders tand a factor singular in s. ~qever-

theless we can see t ha t the singulari ty being just

sin ~ra(s)

is a fac t which bears ra ther deep consequences. Consider in fact the poles a t s = n as they enter into formula (6). We need 5-functions in order to

construct the contr ibutions which fill the l~roissart-Gribov par t ia l wave (3).

(6) M. JACOB and J. MANDELBROJT: NUOVO Cimento, 63A, 279 (1969).

A DISCUSSION OF DUALITY AND ANALYTICITY ~ 9 9

Let us now insist on (6). The order of summat ion expressed by (6), if we consider the square bracke t as a whole, is cleurly not the only possible one.

We m a y in f~et define v~rious interest ing urrays of poles.

F i g . 1.

tower ~ string p

S

Consider for instance Fig. 1. The vert ical a r ray is called a tower and its contr ibut ion to V is defined by

n - o z = o 8 - - 8 n 8 - - 8 n

The oblique a r ray is called a string, and its contr ibut ion to V is defined by

(n) 1 S~

n - ~ , 8 - - 8 n

and

V : ~ S v . ~- -0

We use the word (( str ing >> instead of (( t ra jec tory ~) because we just want

to point out how ambiguous, as we said above, i t is to call ~(s) Regge trajec- tory, a l though this shor t -cut name is popular ly used.

The tower is an interest ing set of poles because the T . grouping is na tura l

under B 4 mathemat ics . Dua l i ty of towers is ~ clear concept, and, as a ma t -

ter of fact , people who say tha t V 0 has dual i ty refer to tower dual i ty .

The string is also a re levant set of poles, because the S~ grouping is na tura l

for Reggc language. I n fact , dual i ty of l~egge poles is thought to be also a possible concept. A generalized string is ~ finite sum of strings

*;2

300 C. B O L D R I G H I N I a n d L. S E R T O R I O

In spite of the discrepant factor

D(s, t) = sin ~(~(s) + ~(t))

sin ~a(s)

the residues ~(z"' can t ruly be analytically continued from t > 4# 3 to the region t < 0, where they appear as factors of the asymptot ic expansion, always if one neglects the factor D(s, t). This can be seen easily if one considers, for instance, the upper string ~(~"), and, correspondingly, just the leading term of the s -+ c~ expansion (see eq. {2)).

The pole residues along the upper string are in fact

1

~(..,_ 2n + l f R 2 .(t)P.(z) dz. --1

In Appendix B we give a general expression for ~(") We derive from it the analytic continuation of ~ ) , ~(t), which satisfies

~(t) =/:( .) for .(t) = n , ~ n

because we want to see how the residue at the particle poles enters, for t < 0, in the s-channel asymptotic expansion. We limit our discussion to the leading term of the asymptotic expansion. I t can easily be verified tha t the expression

( s ) ~(t ) = a ( t ) (~ ( t ) - - ~)'(" F(a(t) + 1) .F(2a(t) + 1) ' c = b -~- 4/t~a,

satisfies the condition

~:(t) = ~") for a(t) = n

(see Appendix A, eq. (A.15)), and moreover

}(t) ___ e x p [ - - v] exp [-- ~(t)(2 log 2 - - 1) -k l o g ~(t ) ] .

Consequently, by virtue of Carlson's theorem (6) the function defined by (8)

is unique.

(e) Bateman Manuscript Project, Higher transcendental ]unctions, Vol. 1, formula 3.9.(2) (1955).

A D I S C U S S I O N O F D U A L I T Y A N D A N A L Y T I C I T Y 301

For s -+ c~ and t fixed we have (~)

Q_~(._~(z,) ~ / ~ 1 #(t) _ ( - 2z , ) ~,.)

cos :n~(t) 2 sin 7ea(t) (~( t ) - e) ~`,, F ( ~ ( t ) + ~)

F{2~(t))

which, when we make use of the duplication formula for the X-function, becomes

1 1 1 2 ( - 2z,)~"'(c~(t) - c)~"' - -

siny~(t) 2~(*)-lF(~(t)) "

The asymptot ic l imit s -> c~ is

Q-~.)- l (z , ) 1 1 2s (9) ~(t) cos xt~(t) "-+g - - ( - as)~'" F(~( t ) ) sin a~(t) ' z~ = 1 + t --4/z ~'

which proves our assertion.

In spite of the fact tha t ~(~") has a correct analyt ic continuation in t (or s, of course), as a l~egge residue should, the V ° representat ion has a peculiar proper ty affecting the meaning of I~egge pole, which we shall now discuss.

In fact let us insert a string or a generalized string into the Froissart- Gribov definition (3). l~ecalling tha t the asymptot ic behaviour of the Le- gendre polynomial P~_~(1 + 2 t / (Sn - -4#2 ) ) for large n is (7)

P._~ (1 + - - s _2~t4#2)~--~2~ 1 ,

we obtain tha t

(10) S~(1, t) - - z(z~) S~ (s, t) dz, + ~.-~P~_~(z~)z, d(s - - s~) dz, =

S o ~V

The convergence is proved by considering again the expression for .~(") given

in Appendix 13. F rom it we can majorize asymptotical ly the p - th string residue ~("~ for n--> c~ as n---~

(11) ~(2~ < c exp [-- n(2 log 2 -- 1) + (}p + 1) log n] .

Inserting (11) into (10) we have the result at once.

(') Bateman .Manuscript Project, Higher transcendental ]unctions, Vol, l , formula 3.5.(10) (1955).

2 0 - I 1 N u o v o C i m e n t o A .

~ C. BOLDRIGHINI and L. SERTORIO

I f we write the ident i ty

V ( s t ) = ~(st) + V'(st) ,

we have tha t the result (10) can be expressed by

(12) V'-> fl '(t)s ~('~ as well as V--> fl '( t)s ~m

with

fl'(t) = sin x(a(s) + a(t)) z~ sin z~a(s) sin zig(t)/'(:¢(t)) "

We have muti la ted the s, t crossing-symmetric ampli tude (1) wi thout changing

the l(t) poles. The Regge behaviour V-->s ~"> comes from the deep bottomless sea of

j / n << 1 direct-channel particles. Assemblies of s-channel poles can be added to V' in order to restore the

s, t symmetry . If the string S ---- ~ S~ is sufficiently thick we ma y use for V'

its asymptot ic form and write

V ~ ~ resonances + s ~(*~ .

The discussion of (12) is what we seek in this Section. First of all let us assure the reader tha t we are not going to speak of interference models. Wha t we read in (12) is the following. The set of residues belonging to s (n) is Reggeizable, as we have shown, and no other array has this proper ty . The

analytic continuation of ~(."> is

7~ ~,- , + / '(t) sin ha(t) "

)%ow we ex t rac t a string of s-channel poles (see (12)), and the leading Regge residue is unchanged. We may ext rac t a second string of poles, and so on, and the leading Regge residue remains unchanged. This is funny and not obvious, actually i t is ra ther surprising. V ° contains local duali ty in tha t par- t icular way, with poles on the real axis, and brings to the result (12).)%o connection among the assembly of s-channel resonances and the fl(t) residues of the asymptot ic series exists. When now we switch on uni tar i ty we may require

tha t the following obvious requirements are satisfied:

a) Regge poles (the various strings mentioned above) give rise to genuine

asymptot ic expansions;

b) dispersion relations are val id with finite discon$inuities;

A DISCUSSION OF D U A L I T Y ANIY ANALYTICITY 3 0 3

c) the particles of the V ° theory are kept ; the poles are s imply slipped in proper sheets.

Then, global dua l i ty m u s t be satisfie4 (as a consequence of a), b), e)). Links among the leading fl(t) and the s-channel resonances m u s t be created. The number of these links is infinite. Because these links are not embedded in V °, i t follows tha t un i t a r i ty mus t create such an infinite set of equations.

I t seems to us t h a t we have the following situation:

i) classical Regge poles are recovered only as a very complicated out- pu t of uni tar i ty ;

ii) or, if we wan t to preserve the present simplici ty of the Regge-polo phenomenological language, ra ther drastic modifications or depar tures f rom V 0 are expected.

A third possibil i ty is to discard the idea of making V o the star~ing point

of a theory and consider i t as a phenomenologiea] language which introduces an infinity of poles t h a t in practice are washed out in the sense t ha t they serve to build averaged quanti t ies (see nex t Section).

3. - In the preceding Section we have discussed some of the complications which are expected if the V o model local dual i ty as sole input should be per- tu rbed unti l it resembles what we call global duality.

In this Section we pursue the following idea. The pure resonance description

and the pure ]~egge-pole description of scat tering ampli tudes are twin descrip- tions. They bo th represent the peripheral pa r t of the interact ions in the sense t ha t they express those singularities in the m o m e n t u m space corresponding to the propert ies of the s t rong interact ion far away f rom the origin in con- figuration space. In this sense they can be considered equivalent approaches and, this way, ins tead of connecting t hem via uni ta r i ty one m a y seek a direct in terpre ta t ion of the physical propert ies of V °.

More precisely, we keep our poles on the real axis as V ° provides t hem and s imply consider V 0 as a way of describing huge assemblies of resonances, which is a little b i t inconsistent for analyt ic S-matr ix , bu t has the advan tage of

establishing a connection between the S-mat r ix and the rmodynamics (see the reference to HAGE])0~" in Sect. 1).

Here we show how V 0, as a model, nicely embodies propert ies of general peripheral na ture which we know f rom previous studies using the impac t pa ramete r representa t ion.

We know t h a t (~), if an elastic ampl i tude has a beh~viour l imited b y a power in s for fixed t, this corresponds to a finite l imit for the impac t pa ramete r

(s) M. JACOB: Acta Phys. Austr., Suppl., 6, 215 (1969).

~ 0 4 C. BOLDRIGmNI a n d L. S~.RTORIO

(we have a sphere of radius fixed with energy). I f we now impose the amplitude to be represented b y resonances we get t ha t the highest spin dependence on the energy mus t be of the fo rm 1 ~ e- ]p], namely ~ ~ cV~. W h a t we are going to show in detail is t h a t the V ° model contains the ment ioned prop- e r ty in a very interest ing way, namely , as average value of the angular mo- m e n t u m a t a given energy s, the average being t aken over the s ta tes belonging

to all strings piling over the value s. more precisely we will discuss

<~> = ~/<[~(l + 1)]>.

Moreover, and more impor tan t , we will show in detai l t ha t the behaviour

~7, also characterizes the large coupling poles. Namely , if we tu rn our a t ten-

t ion to the a r ray of poles 1 ~ ~ we see t ha t the poles 1 >> ~/s, ~ << V/s- are

Fig. 2.

t

• : ~ L~<~v~jcj+~i~

S

of negligible impor tance with respect to the poles with couplings ~ ; . Such an a r ray can be also defined as the a r ray which gives the m a x i m u m con- t r ibut ion to the Froissar t -Gribov par t ia l wave, namely t ha t ~(n) such t h a t

~ t t n ) (n) ] I lira :(~) < 1 ,

n'-"~¢~ '~ l ( n )

where l(n) is any other array. Obviously we do not consider for l(n) oscil-

lat ing functions. Le t us consider

l

--1

The polynomial R~(t) has the following structure:

(14) R . ( t ) = F(~(t) -k n) r ( ( ( ~ - ~1/2)(~- ~/+ n + b)

r(..)r(~(O) r ( ( (~- ~)i2)(~- ~) + b) r(n)'

~. D I S C U S S I O N OF D U A L I T Y AND A N I L Y T I C I T Y ~ 0 ~

where

zt n - - b c = b + 4/u2a , z : l + s . _ _ 4 # ~ ' s~ - - a

Let us fix our a t ten t ion first of all on the mean value of Z (') a t fixed n

n

(15) ~,,~, =-1 ~ ~.>.

We have

11

(lO) Z ~?' = ~ ~ P , ( 1 ) = R~(o) - - - I~0 l~O

F(n + b)

['(n) F(b) "

Using Stirling's formula we get

e x p I - - b] (17) ~: ~F' n' l=o -+ F(b )

and, therefore,

(18) ~(.~---~ exp [ - - b] n_l+ b

This power behaviour is expected because we know tha t after all the tower

mus t sa tura te the Yroissar t-Gribov integral. I t shows tha t there is a region ~(n) of l in which -z ~(") goes to zero less s trongly t han the str ing =,_~

~:(n) ~_ ,< c exp [ - -n (2 log2 - - 1 ) + (~p ~- 1) l o g n ] , ~, --> ( : ~ .

The behaviour (18), however, does not give any h in t abou t the m a x i m u m

of ~('> as a function of l a t fixed n, nei ther can it produce evidence t ha t there ~Z

is one. To show this we consider direct ly the integral

1

21+ I ; R 2 n(t) Pz(z) dz.

- - 1

The polynomials Rn and P~ have respect ively n and 1 zeros in the in terva l

- - 1 < z < 1. We can assume wi thout much loss of general i ty t ha t n and l are

both in the asympto t i c region; this simplifies the discussion. We divide the in tegrat ion in terval as follows:

(19)

I --l+S 1--e 1

--I --I --l+e l--S

~ C. BOLDRIGHINI and L. S]~RTORIO

wi th e small . I n s e r t i n g the St i r l ing a p p r o x i m a t i o n in the in tegra l f r o m - - l + e to l - - e , i.e.

(20) .R.(t) -+ ~ ( 1 - - z)~-~(1 + z ) ~ + b - + 2 - ~ -

• exp [ (n - - e) { ~ log l - - z ~ - + - ~ - log

((20) is va l id in the i n t e rva l - - 1 + e < z < 1 - - s wi th s > 0), one sees t h a t i t g ives a con t r ibu t ion which van i shes a t l eas t as f a s t as

[ e~ (n-¢}e/l

) .

The m a i n con t r ibu t ion is the re fore concen t r a t ed a r o u n d the end-poin ts , and

b e t t e r still, a r o u n d z = 1, since at z - = - - 1 R,(t(z)) is

(21) R.(t(z = - - 1)) F(e + b) - - zt ( - - 1)" sin ~(e + b) 2"(1 + n - - e - - b)

a n d goes as n - + c~ l ike

/'(c + b) (22) R . ( t ( z = - - 1 ) ) . _ ~ - - ( - - 1)" sin g(c A- b) e x p [ v + b - - 1 ] n 1-'-a ,

whereas R,(t(z = 1)) goes, as we saw above , l ike

1 ] ' (b) exp [ - - b]n ~ .

W e consider~ therefore~ on ly t he las t in tegra l in eq. (19). W e choose e in such

a w a y as to include in the i n t eg ra t i on i n t e r v a l t he first two zeros of Pt(z), which are a t z = z o - 1 - 3/12 and a~

15 Z = Z l Z 1 - - - -

ref. (9). This is a s sured b y the choice e--~ 1/l. T h e first two zeros of

R . ( t ) - _r'(~(t) + ~) _r'(n) p(~(t))

(o) Batemau Manuscript Project, Higher transcendental/unctions, Vol. l , Sect. 3.8 (1955).

A DISCUSSION OF DUALITY AND ANALYTICITY

are at ~ ( t )= 0 and zt(t)= 1, i.e. at

307

and

!

Z ~ Z o = I - - - -

!

Z = Z l ~ I

2b

2(b -4- 1)

Since both polynomials are positive at z = 1, the maximum coherence of phase is reached when l ~ k~ /n for a suitable k, and for this value of 1 the integral will have a maximum. The fact tha t e gets a dependence on n of the form 1/kn ~ is not disturbing because the integral in the middle of eq. (19) remains vanishing at least as

const \ ~ ] • n -<ll*~>v" .

That the dominant states are those with l around the value ~/n can be checked by calculating the average value of l(l + 1) (which classically corresponds to the square of the angular momentum) in the tower. This can easily be done by noticing tha t (9)

d ) j(j + 1) L ( z ) 2 '

so tha t

(23) d R~(a(t(z))) = ~ l(l q- 1)~:~.)

On the other hand, the derivative of R~(a(t(z))) for z = 1 can be explicitly calculated, giving

(d ) 1 n - - c F ( n + b ) ~_~ (24) dzz R~(t) , ~ = F(n i 2 _F(b) { ~ ( n q- b)-- ~(b)} , ~(x) = log F(x) .

The average value of ~(1 Jr 1) over the tower is, therefore, by comparing (23) and (24)

(n) (n)

~ 0 8 C. B O L D R I G H I N I and L . 8 E R T O R I O

Asymptotically it tends (lo) to n logn , and we get the result

The results just obtained allow us also a brief discussion of the ghost problem. In fact, consider for instance the sum of all ~") for 1 even, which can easily be calculated by observing tha t

~'-',(-)=1 ' ~ ~',-'P,(- 1) = ~ . ( t ( z = - 1 ) ) Z-O l=O

Making use of eq. (22) we find

1 n

, . ° - nb+ (--1)" sin a(e+b) - -

F(e + b) } ~b 1 - ¢ - b .

I t is easy to see tha t when 1 - - e -- b > b (which is equivalent to 3b + 4#~a < J) the ghosts are certainly present because the sum of the ~(~") for I even has a sign (--1)" asymptotically. The same happens with the odd spins. In such a case, moreover, even the average value of l ( l + 1) over the even or odd spin states can take negative values: in fact one shows easily in the same way as was done above tha t

(26) R.(t(z) = ~ (--1),+ 1 l(1 + 1) ~ . ) =

2~ - - - ( _ 1 ) . + 1 / ' ( b + e ) /~(n + 1 - - b - - e) s i n ~(b + e) ,

F(n)

so that , recalling the results (23) and (24), we obtain

(27j ~.) l(l 4- 1)

I=D 2 even

1 {n- - c F(n + b) - 2 ~ i rm---~ [~(n + b)-- ~(b)] +

+ (-- 1)" ~ _ c F(b + c) F(n + 1 - - b - - c) F(n) sin g(b + e).

• [~(n + 1 - - b -- e) - - ~f(b + e)] - - z cos ~(b + c)}

(10) y . L. Lux~: The Special Punctions and Their Approximations, Vol. 1, formula 2.11.(8).

A D I S C U S S I O N O F D U A L I T Y A N D A ~ A L Y T I C I T Y ~ 0 ~

which asymptotical ly becomes

(2s) 1 ~--C n - - c 1] nl_b_¢ } { F ( ~ e x p [ - - b ] n b logn+( . . . . 1) 2z e x p [ c + b - - l ogn .

If 1 - b - c > b the sign ( - -1)" will dominate. We can summarize the content of this Sect ion in the following way.

The Veneziano representat ion prediction for the couplings of the resonances along the string

~ z ~ b

to the ~:~ channel is

(n) ~ e x ~. p [-- n(2 log 2 - -1 ) -~- l o g n ] .

They are slightly larger than the couplings predicted (u) by pure centrifugal barrier

~b~,, ~ exp [-- an.log n]

(a being a positive constant larger than one). Although the centrifugal barrier effect is not sufficient to explain the Veneziano couplings, they are still peripheral in the sense tha t the couplings ordered along the line l---V/s are the largest (for the ~r: channel). Moreover the resonances 1 << ~ /n are negligible and this can be seen from the fact tha t the average of 1 taken on all the couplings at a given energy s is just l~--~/s.

This means tha t the concept of (< effective i> t ra jec tory is also stable with respect to absorption corrections, which ~ffects the ~/n,~ s region and consequently the irrelevant tail in the l(~ (")) distribution.

The fact tha t the privileged behaviour

~ V Y

mentioned at the beginning is reached as an averaged value seems to suggest tha t the interesting physical content of the Veneziano representat ion is of

statistical-mechanics nature. This interpretat ion changes the ~tt i tude towards

ghosts. In fact, ghosts are hard objects to fight if one wishes to integrate each of them into the society of the good poles. If instead they are considered as members of ~ statistical assembly, they are only ~ mathemat ical feature of the model like the unt rue proper ty assumed in the statistical mechanics of gases about the molecules, namely to be elastic, pointlike, etc.

(11) H. GOLDBERG: Phys. Rev. Lett., 21, 779 (1968).

310: c. BOLDRIGHINI and L. SERTORIO

4. - The first thing to do in order to explain fur ther the idea of the statistical content of V ° is to show tha t the averaged quantit ies evaluated over the crystal s t ructure of V ° are unchanged after random destruct ion of all the string system. We will do this by steps.

We have proved previously tha t the unbroken Veneziano ampli tude satisfies the peripheral requirement, namely~ tha t from the fixed-t behaviour follows the average value

If we now displace all the poles (we let all towers collapse in whatsoever number of pieces) keeping unchanged the residues and we are able to show tha t the fixed-t behaviour is conserved we will have tha t also the <~> ~ / s - rule is conserved.

In order to do tha t let us consider the ~roissart-Gribov integral

(29)

¢o

n - O ] -0 zo

and let now the tower label n run over s,--> s~.j places. The index j ma y either run from 0 to n or may also include a subindex j ' labelling any extra mult iplici ty (for instance the Hagedorn multiplicity). Assume also tha t

(30) s,.¢ = s, -4- b..j

with

(31) ~..j = e~(n) , e~(n)n- ' ~ 0 .

Formula (19) then gives

f " dzt < ,,-o " n - O J-O

Due to the condition (31) we may now write

(33) s -z-~..~ ---- (n ÷ ~ . . . ~ - - - ---- n - " 1 - - (1--k 1) ~ .... ÷ ( / - 4 - 1 ) ( / - ] - 2 ) ( ~ ) ~ q - ' " } = 2 !

=n-"2Fl(1,1,1;--~), when ~F1 means Gauss hypergeometric function.

A DISCUSSION OF D U A L I T Y AND ANALYTICITY 311

Formula (32) shows tha t the Regge behaviour is conserved with residues fl(t) per tu rbed by a well-defined amount . This is wha t we set out to prove.

Cleurly, in this way, the crossing s y m m e t r y is changed by a small amount

(see (33)). Bu t we are not seeking here an actual procedure for destroying V °, af ter the good luck of hav ing found it. I t is ra ther a gedankenexper iment of

separat ing (splitting) all poles in the energy in ~ r andom way and of observing tha t this dest ruct ion of the V ° s t ruc ture does not alter its good properties.

I t seems to us t h a t this reasoning helps the stat ist ical in terpre ta t ion of the physical meaning of duali ty. Although we have not succeeded unt i l now we continue to seek a way of unders tanding why the mass spec t rum similar i ty exists be tween dual i ty and the rmodynamics t ha t was found by F ~ I ~ I and V E : N E Z I A ~ ' O .

We think t ha t the reasonings out l ined in this paper m a y be in the right direction.

5. - Up to this point we have always considered poles on the real axis; we have observed t h a t the na tura l concept in this context is t ha t of ruined tower ra ther than ordered t ra jectory. We want here to add a discussion on other peculiar facts which happen when poles are pu t in proper second sheets. Elsewhere is given (12~.b) a full discussion of an ampl i tude A(s, t) which

1) has a sys tem of an infinite number of thresholds in s (and in t),

2) contains the daughter sys tem as in t he bare Veneziano ampli tude.

When such an ampl i tude is inserted into the Froissar t -Gribov par t ia l -wave

formula, we can separate the contr ibut ion of the poles near the real axis f rom the contr ibut ion of the discont inui ty along the cuts:

(34) P(~, t) = F(~, t ) .o , . + .F(dt)ou," .

We can show tha t F(l , t)po~o, ( remember t h a t the poles are in infinite number ; not only: there is an ex t ra infinite number of pole strings as in V °) does not have a Regge pole in ~, and instead the asympto t i c behaviour

A ( s t ) - ~ s ~(~'

is due to F(1, t)out .. I n fact, the expression for F(l,t)pole, turns out to be

(1~) a) M. L. PACIELLO, L. SERTORIO and B. TAGLIENTI: NUOVO Cimento, 65A, 167 (1969); b)M. L. PACI~LLO, L. S]~RTORIO and B. TAGLIENTI: Nora interna n. 520, Istituto di Fisica (( G. Marconi ,~, Universith di Roma (1970).

3 1 2 c . BOLDRIGHINI a n d L. SERTORIO

(ref. (I~*))

(35a)

(35b)

f:'- F(t, 0,.,.. = a~ Z ~es V.(r ' . i ' t )Q,(z , ) ~(8 - r'. i ') dz , = ,t-O

zo

. ~ 4 . , t )O~(z , ) (5(s - - . j , +

zo

~f

The finite integral gives rise to no singularities. The infinite integral is

(36) - , " ' t) 1 + . - - ~ ' ~-o . . , r . , t ~ - ~ # ~ / t - - 4 # *

The residue Res V.(..:(t~, t) is (l~b)

(37) Res V r~ (i~ t) • n ~ - n , - -

We can majorize the sum

( =?) a~. F(i + 1) ~ 1~ 1 + n --

• /~(1 + ~ - ~ -F 1) s i n . ( ~ 3- ]) .

i n

Res . . . . . <J~ t) vn ('G, , t - 0

in expression (36) by taking j~ times its largest term. In this way, considering tha t j~ has to go to infinity like n 1-v if one wants constant widths, ref. (lsb)

1-o ---- COl]St ,

one finds tha t the series (36) always converges, because the ratio of two con- secutive terms ](A,+I)/(A~) ] goes always to zero as n approaches infinity.

The fact t ha t the saturat ion of the l~roissart-Gribov projection is due to the contribution of strong s-plane cuts is not surprising; after all this is what we were used to in the Bethe-Salpeter approximation. We think i t is worth recalling tha t in the model with complex poles we just mentioned, the behaviour a(s)--~v/s (see formulae (16) and ref. (l~a)) is the highest angular momentum allowed as a function of s (or t), and this behaviour comes solely from the dynamical assumption of the cascade decay of resonances with true finite life.

A D I S C U S S I O N OF D U A L I T ~ A N D A N A L Y T I C I T Y 313

What we are interested in here is not making remarks on the merits or defects of the ref. (l~.b) approach. We mention it here as a well-defined example of how the concepts outlined in the preceding Section can work. In fact, in ref. (l~a.b) the s-plane cuts are introduced with minor changes in the Veneziano-pole strengths. Nevertheless this had the effect of unveiling these new facts:

a) a precise dynamical assumption is introduced (cascade decay law) nstead of taking averaged values;

b) in spite of the minor modifications with respect to V 0 the model in ref. (1,a.b) has

1) not local dual i ty any more bu t 'global dual i ty; the asymptot ic Regge expansions are built by s-plane cut contributions;

2) Regge poles and Regge trajectories are not recovered as a simple feature.

This amounts to saying tha t there is equivalence between properties of cut-equipped ampli tude and average properties of the model ampli tude V o.

6 . - C o n c l u s i o n .

In this paper we s tar t f r om a critical discussion of what Regge poles are in the Veneziano elastic representation. This real pole representat ion natural ly contains Regge poles in an anomalous way. The recovery of s tandard analytic properties appears to need the satisfaction of such str ingent requirements tha t ei ther the concept of (dual) Regge poles might tu rn out infinitely complicated or the modifications of the Veneziano representat ion can be expected to change the actual approximation in a conceptually drastic way {Sect. 1 and 5).

This criticism is made more constructive by considering non-l~egge sets of poles (the effective array) and introducing the concept of average values as physically significant quantit ies in the Veneziano representat ion (Sect. 3).

Fur ther , a random displacement of all poles is studied and 'the stabil i ty

of ~he concept of (~ dual mean values ~) is proved (although in a gross way)

with respect to the number of poles and order of magni tude of the residues. The stabil i ty of average values under statistical disordering of the trajec-

tories (Sect. 4) seems to us a good conceptual h in t for a deeper understanding of what duali ty means.

I t is a pleasure to thank Profs. A. DI GIACOMO and M. TOLLER for m a n y discussions and Prof. M. JACOB for kind and constructive criticism.

314 e. BOLDRIOI-IINI and L. SERTORIO

A P P E N D I X A

The Veneziano four-point ampli tude does not accept complex traj ectorie for two reasons; the first is t ha t when ~!s) = as + b is replaced by a complex function the ghost ancestors appear. These ghosts have been extensively studied in the literature. We want to discuss here another still worse discrepancy appearing when a(s) is a complex function. We examine the case

(A.1) o~(s) = as + b + es v , y N O , s , - ,O,

namely we consider the smallest possible variat ion from ~ ( s ) - as + b. The ampli tude

_ r ( 1 - ~ ( s ) ) F ( 1 - ~ ( t ) ) = v ( s , t ) (A.2) F(1 - - oc(s) - - oc(t) )

corresponding to (A.1) has poles for a(s)-= n or :¢(t)= n. Consider now the full asymptot ic behaviour of (A.2), i t can be evaluated as follows:

sin ~(.(s) + .(t)) F(o~(s) + o~(t)) V(s, t) = / ' (1 --.(t)) sin =.(s) r(~(8))

Moreover, we have (13)

(A.3) F(~(s) + ~ 8 ( t ) ) > ( o d s ) ) ~ , ) [ l + o ~ ( t ) ( a ( t ) - - l ) ( ~ ) ] l'(~(s)) ~(" . . . . . . 2.(s) + 0

We can introduce into (A.3) the asymptot ic behaviour

(a(s))~(') = (as+esV-~b)~( ' )=(as )~" ) ( l+esvas -~b)~"L(as )~ ( t ) ( l+g( t ) e sva+sb+ ' ' ' )

and (14)

~(s) -- (ss~' + b) -~ as/(ss v + b) + 1 ~-'g (esv + b)-I ~ as \ ~ - s / + ) finally we get

( A A ) V(s t ) - - ~ ~") + f l l ( t )s ~")-1+~ O(s ~")-~+~) ~ fl(t)s + &(t)s ~"'-1 +

(13) E. T. CoI-sol~: Asymptotic Expansions (Cambridge, 1965), p. 61. (14) E. T. COl, SON: Asymptotic Expansions (Cambridgo, 1965), p. 7.

A D I S C U S S I O N O F D U A L I T Y A N D A N A L Y T I C I T Y 315

with

fl(t) = / " ( 1 - - .(t)) (cos z~(t) + ctg ~a(s) sin u.( t))a ~"' ,

~ ( t ) = ~(t) a ~(t) ,

~(t) - - 1~. fl~(t) = fl(t)a(t)( b + 2a l

I t can a l ready be seen tha t , b y going to higher orders, one gets all the contr ibut ions of order s ~(t)-" as well as all those of order s ~(t)-"+~.

We therefore find the existence of contr ibut ions s ~")-"+v which do not have the coun te rpa r t of resonances in the t > 0 region. These contr ibut ions are of order s in the residue. They can be called (< asympto t ic -cont r ibu t ion ghosts )> and indicate a severe mi sma tch of singularit ies as we consider the poles direct ly or through the Fro issar t -Gr ibov definition which is governed b y the a sympto t i c expansion t e r m b y t e rm.

APPE~ND I X B

n Evaluation of ~ .

.C(n) The explicit eva lua t ion of ~ can be made as follows. We recall the definition

(B.I) 1

( .,;'"' ---- j + ~(7(t)) Pj(z) dz

--1

with R . as a funct ion of z, given b y (since ~ ( t )~ at + b = (n - -c ) ( ( z - -1 ) /2 ) + b; c ~ 4#2a + b)

(B.2) F ( ( n - - v)((z-- 1)/2) + n + b) ,,

R . ( ~ ( t ( z ) ) ) = r ( n ) r ( ( n - ~) ( ( z - 1 ) / 2 ) + b ) = ,-oX d~°'~ ' •

The coefficients d~ are known f rom the double expansion

n

Rn(~(g)) = ~ c~ (a:(t)) k - 1

, . " 21 \--2-I z',

316 c. BOLDRIGHINI and L. S~.RTORIO

so t h a t

re.a) = ~ ek b - - ~-~,~-o m ~ / ~ , -g- I

Z m .

Recal l ing now t h a t (15)

(B.4)

1

fzmp~(z)dz={ 0 i f m < j , I~.j if m > ) ,

wi th

2J'q-ltf~,!(½{m "~ :)) ! 1 --~ ( - - 1 ) m-' I~., : (½(m--:)) !(:-F m + l)! 2

we get, a f t e r some rea r rangement

(B.5) ~.Z~,o,,,,( n- -cy ( ( n - - c ) / 2 ) =

.,*'"' = , b - - 7 b - (~ :7)~)/2

(2i + 1)2'k!(1(~ + 1))! (½(~ - j))!(j + ~ + 1)!(k-- ~)!

1 --F (-- 1 ) " - '

I n (B.5) are separa te ly exhibi ted the t r a j ec to ry pa rame te r s a and b, and (n)

t he B-funct ion s t ructure which is represen ted b y the coefficients ck • These coefficients are known f rom the expansion

( B . 6 ) = 1 f l ( x - - n - ~ k ) _ 1 ~_ 1 x~ ~c c") R~(x) ~ ~_1 V(n) _ f:'(O) ~ -~ ~-~ ~ x~,

where ] ~ indicates t he k-th der iva t ive of ],(x) = R~(x)F(n) eva lua ted a t x = 0. Such der iva t ives are given b y

im~im- i~,~. • .~Q ~-I

f rom which we get

n--I n--1 n--1 ( n-1 )

ix-1 J l= l tm-i ~1 i=1 i:~Jl ..... jra-t

~-1 (n - - 1) ! : m ~ (n - - ~1) ( 3 - - ~m--1) 11. ja,...~m_l~l • • •

(15) F.. T. WHITTAKER and G. N. WATSON: A Course in Modern Analysis (Cambridge, 1952).

A DISCUSSION OF DUALITY A K D ANALYTICITY

and, findly,

so that by comparison of (B.4) and (B.5) we have

From (14) and (11) we have the full expression of [ y ' , which we do not write here at length.

One can show, starting from expression (B.5), that

Making use of the Stirling expansion one gets

The behaviour of 6;"' when n is large and n-1 is fixed, n-I = p , is similar to that of 6:'. To show this, consider the expression of Er!,"l,

I t is easy to show from expression (B.8) that

n - 1 1 ( n - l ) !

( (k - l ) ! )2 (n -k ) ! '

I n fact, cr' is the sum of terms, each of which is a number smaller

than or equal to l / ( k - l ) ! We therefore find

(2(n-p) + 1)2n-pk!(4(m + n-p ) ) ! --

($(m-n.+p)) ! (n-p + m + I)!@--m)! '

31 - I1 ~Vuovo Cimento A .

318

T h e n u m b e r of t e r m s in t h i s s u m is ~ ( k - - n + p q - 1 ) - - - - p ( p q - 3 ) / 2 . t e r m is s m a l l e r t h a n o r e q u a l t o ~ - ~

2-~(n - - 1 ) ! (n - - e - - 2b )"n !(n - - p /2 ) !

((n - p - 1) !)~(2(n - p ) ) !

T h e s u m (B.13) is t h e r e f o r e m a j o r i z e d as n goes t o i n f i n i t y b y

e. e x p [ - - n(2 l og 2 - - 1) -~ (~p + 1) l og n ] .

C. BOLDRIGaINI and L. SEl~rOmO

E a c h

• R I A S S U N T 0

In questo lavoro prim~ si diseute in ehe mode 1~ rappresentaz ione di Veneziano eontenga poli di Regge. ~ un mode ver~mente compl ica te e se ~nehe il p rogramm~ di uni tar izzazione fosse d~to per sconta to si mos t ra che altro non si incont re rebbe ehe un ' anco r pifi complie~ta situazione. Suceess ivamente ci si dedica allo s tudio dei residui dei poli, l i m i t a t a m e n t e al c~so elastico, in funzione d i s e di 1. Si mos t ra che cer t i valor i medi di l hanno un significnto fisico chinro ehe r imane ~nche se si t e m p e 1~ legge di l inear i t~ de]le t ra ie t tor ie impos ta dal modello. Questo mode di ragionare st~tist ieo sembra promet~ente nonost~nte lo s tudio dei residui non sia s t a te t r a t t a t o con la mass ima generalitY.

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A discussion of duality and analyticity