conditions for regge behavior of an absolutely convergent veneziano series

P H Y S I C A L R E V I E W D V O L U M E 2 , N U h I B E R 1 0 1 5 N O V E M B E R 1 9 7 0

Conditions for Regge Behavior of an Absolutely Convergent Veneziano Series*

P. H. FRAMPTON t Elzrico Fevnzi Instikite, Unic,evsity of Cl~icago, Chicago, Illinois 60637

AND

C. W. GARDINER Pliysics Department, Syvacuse Unicersity, Syracz~se, iVew York 13210 (Received 30 March 1970; revised manuscript received 8 July 1970)

We investigate the conditions for the validity of a Regge asymptotic expansion of a scattering amplitude, A (s,t), which can he written in the form of an absolutely convergent series as

where S=cu(s), a real linear Regge trajectory, and T is similarly defined. We find both necessary and sufficient conditions for Regge behavior when /SI -+ m a t fixed T with 1 arg(--S) 1 < x , and for Regge behavior when IS1 -+a with h e d LL and 0 < jarg(S) I <x , in terms of the analyticity of two functions which are defined in terms of the coefficients CZh. IVe show also that the absolute convergence of the Veneziano series is sufficient to guarantee that A (s,t) is a meromorphic function, the residues of whose poles are given by sum- ming the residues of the corresponding pole in each term of the series. Several specific examples of Veneziano series are investigated to illustrate the utility of the methods.

I. INTRODUCTION where S= CY (s), T= ct ( t ) , and the trajectories are

F OR the four-particle scattering amplitude, Venezi- anol has proposed a simple formula which displays

Regge behavior in all channels. An infinite number of variations of this formula can be made, all with Regge behavior in all channels. The zero-width resonance approximation is made, and the amplitudes are meromorphic functions of the variables s, t, and ZL.

The problem which is addressed in this paper is the consideration of the conditions under which there exists a Regge asymptotic expansion for an absolutely convergent infinite series of Veneziano-like terms. All Veneziano-like terms can be included in a double infinite summation of the form

linear. We require the series to be absolutely convergent for

two reasons: first, so that the sum is unique and does not depend on the method of summation, and second, for Inathenlatical convenience.

The expansion (1.1) is such that any finite number of residues can be made arbitrary by an appropriate choice of the coefficients Clh. I t is possible to write other Veneziano-like terms of the form

with a#b, and it would seem natural to symmetrize (in S and T) such terms and include them in the expansion (1.1). This is not necessary, however, because these new terms are not linearly independent. The following explicit identity has been found by Kreps and Milgram2 for b<a:

where

[ q ] =integer part of q . * Work supported by the U. S. Atomic Energy Commission. t Present address: CERN, Geneva, Switzerland. G. Veneziano, Nuovo Cinlento 57A, 190 (1968). R. E. Kreps and RI. S. Milgram, Phys. Rev. D 1, 2271 (1970).

2 2378

2 C O N D I T I O N S F O R R E G G E B E H A V I O R O F A N A B S O L . U T E L Y . . . 2379

This identity shows that the symmetrized part of such terms can always be reduced to an expression of the form (1.1), so such terms need not be considered separately. Equation (1.2) also shows that the anti- symmetric part of an asymmetric A (s,t), expressible as an asymmetric Veneziano series, can always be written as a factor (S- T) multiplied by a symnletric Veneziano series.

We should mention that the representation (1.1) has been shown to be the unique dynamical solution from superconvergence condition^,^ when the assunlptions of the zero-width resonance approximation, linear trajectories, absence of exotic mesons, and neglect of the Pomeranchuk trajectory and cuts are made. This representation has also been derived by J i h ~ r i , ~ who was able to relate the coefficients Clh of Eq. (1.1) to the residues of Regge poles.

We wish to investigate the conditions under which the amplitude A(s,t) of Eq. (1.1) displays Regge behavior as 1s [ --+m a t fixed t , and also a t fixed u. I t can be seen that a t fixed h and T the asymptotic form of each term is independent of 1, and is given by

(where the a,, are certain functions of T), but it is not clear, even if the sumn~ation over I and h converges, that the asymptotic form of the sum is of a form similar to (1.3). T o show that a real problem does indeed exist, consider the series

where n is a positive integer. This is of the forin of Ecl. (1.1). After writing m=l+n, it can be written

and the hypergeometric function on the right-hand side can be evaluated to give

3 S. Matsuda, Phys. Rev. 185, 1811 (1969). N. N. Khuri, Phys. Rev. 185, 1876 (1969).

The first part of Eq. (1.6) gives Regge asymptotic behavior, since it is a finite sum, but the second part has an asymptotic expansion consisting of negative integral powers of S, starting from -n and decreasing, which could only come from fixed poles in the complex angular momentunl plane. Such a behavior is not expected by studying just the individual terms in the series.

In Sec. 11 we find necessary and sufficient conditions on the Cli, for A (s,t) to possess Regge behavior arising from the trajectory ( ~ ( t ) and its daughters as [ s 1 -+a a t fixed t . In Sec. 111 ,the asymptotic behavior as I sl -+m

at fixed ZL is studied, and we find both necessary and suflicient conditions for the Regge behavior of A (s,t) in this channel. [By Iiegge behavior in the ZL channel of '4 (s,t), me mean that, like each individual term, the function decreases faster than any pourer of s as I s I -+w with arg(s) fixed and not equal to zero or T.] In Sec. IV we show that the absolute convergence of the Veneziano series (1.1) is sufficient to guarantee that the only singularities of A (s,t) come by adding up those in each individual term. Section V is devoted to a number of specific examples to illustrate the utility of the methods. The Appendices contain certain theorems and deriva- tions of an intricate nature.

11. TREATMENT OF I S I-+ a AT FIXED T

We derive an integral representation for A(s,t), as given by (1.1), by writing

where

lye now investigate under what conditions the interchange in order of summation and integration made in passing from (2.2) to (2.3) is justified. We shou, in Appendix R that the absolute convergence of the Veneziano series for A (s,t) implies that the power series (2.4) for +(a,b,T) converges absolutely in the region

and converges absolutely almost everywhere on the curve b= a(1-a) (*<a< 1). Furthermore, the validity

2380 P . H . F R A M P T O N A N D C . 1%'. G A R D I N E R 2

of the integral representation (2.3) is also shown to follow from only the absolute convergence of (2.4).

A strict converse does not turn out to be true, but if [for Re(T) and Re(S) negative]

converges absolutely in the same region as described for +(a,b,T), and $(x, x(1-x), T)~-"l(l--x)-~-l is integrable on [0,1], then the Veneziano series is given by (2.3) and is absolutely convergent for all S and T.

We now derive a necessary and sufficient condition for A(s,t) to have a Regge asymptotic expansion. Defining

Fib, TI=+(%, x(l-x), T I , (2.7) mTe have

A(s,t)= c l ~ x - ~ - ~ ( l - x ) - ~ - ~ F l ( x , T ) , (2.8) I' and substituting x=e-u, we see that

A(s,~) = l* dy es~g(e-~) , (2.9)

with g(x) = (~ -x ) -~ - 'F~(x ,T) . (2.10)

The representation (2.9) is valid only for negative values of Re(S) and Re(T). Watson's lemma, as ex- pounded in Appendix A, states that A(s,t) can be analytically continued to the region 1 arg(-S) 1 <a provided (i) g (e-u) is analytic for Re (y) > 0 and (ii) for each value of arg(y) between -$a and $a there exists a positive M such that for ( y 1 >0, lg(e-u) / <M. If in addition, g(e-u) satisfies the condition that (iii) as y -+ 0 with Re(y) > 0, g(e-u) has the asymptotic expansion

m

g (e-u) - C b (2") y n-T-l , (2.11) n=O

then A (s,t) will possess the asymptotic expansion valid as IS1 -+m with larg(-S)I<a,

I t can be seen that if F1(x,T) is analytic for I x I < 1, and possesses an asymptotic expansion as x - + 1 inside 1x1 = 1 of the form

F ~ ( ~ , T ) - C cn(T)(l-x)"t (2.13) n= 0

conditions (i) and (iii) will be satisfied. Also, if I arg(y) I <%a, as lyj -+a, e-g-+O, so

g(e-9 --+ Fl(0,T) = Coo, (2.14)

and if g(e-u) is analytic for Re(y) > 0, as required by condition (i), then condition (ii) must be satisfied.

Thus, the validity of conditions (i)-(iii) gives a Regge asymptotic expansion, valid so far only for Re(T)<O.

We now give a continuation to Re(T)>O, valid if g(x) satisfies the condition of analyticity in 1 x 1 < 1 and possesses the asymptotic expansion (2.12). We define

where M is any integer such the Re(M-T)>l. \Ve may continue (2.15) to Re(T) < O (keeping M fixed) and the integrals of the terms of the summation inside the square brackets exist and cancel with the terms of the first summation, giving Eq. (2.9). Thus (2.15) gives an analytic continuation of (2.9) for all T such that Re(M- T)> 1. Equation (2.15) exists only for Re(S) <0, but in the statement of \\Tatson's lemma in Appen- dix A, it is shown that A(s,t) may still be continued to (arg(--S) I <a, and the asymptotic expansion (2.12) will be valid in that region.

We now prove the converse result. The converse of \Vatson's lemma, stated in Appendix

A, says that provided A (s,t) is analytic for 1 arg(-S) I <a, has only a simple pole a t S= 0, and as 1s I -+a has the asymptotic expansion (2.12) [valid for 1 arg(-S) I <a], there exists a u?zique g(e-u) satisfying conditions (i)-(iii), such that A (s,t) is given by (2.15).

We show in Sec. IV that if (1.1) is absolutely convergent, A(s,t) is analytic everywhere apart from poles when S is equal to a non-negative integer. We also know that if A(s,t) is an absolutely convergent Veneziano series, g (x) is given by (2. lo), and that Fl(x,t) is analytic a t x=O. Thus the substitution of y=lnx into g(e-u) does not give rise to a logarithmic branch point a t x= 0.

Thus, if Fl(x,t) is given by (2.7), it can be written as

and it is then straightforward to show that Fl(x,t) is analytic in 1 x 1 < 1 and possesses the asymptotic expansion (2.13).

Summarizing the results of this section :

(a) A sufficient condition for the absolute convergence of the Veneziano series for A(s,t) is that $(a,b,T) as defined in (2.6) is absolutely convergent in the region (2.5), and almost everywhere on the curve b=a(l--a) (+jail).

(b) If A(s,t) converges absolutely, i t possesses the integral representation (2.8).

(c) A necessary and sufficient condition for A (s,t) to possess a Regge asymptotic expansion as IS1 -+m a t fixed t is that (i) Fl (x, T) is analytic in x for 1 x 1 < 1, and (ii) As x -+ 1 inside 1 x I < 1, Fl(x,T) has the asymptotic expansion (2.13).

2 C O N D I T I O N S F O R R E G G E B E H A V I O R O F A N A B S O L U T E L Y . . . 2381

Comments

(1) The asymptotic expansion (2.13) is most easily guaranteed if Fl(x,T) is analytic a t x= 1, but this is not necessary.

(2) The rather cumbersome integrability condition on $(x, x(1-x), T) is most easily satisfied by requiring $(a,h,T) to be analytic in a and 6 in some region which contains all points of the region (2.5) and its boundary as interior points, but this also is not necessary.

111. ASYMPTOTIC BEHAVIOR AS Is1 --+a AT FIXED u

The conditions for Regge behavior as s 1 -+m with the i\landelstain variable zl fixed can be treated in some ways more simply than with T fixed, although the conditions turn out to be somewhat more stringent. Each term of the Veneziano series decreases exponentially as I S / --+w a t fixed z6 with 0< 1 arg(s) / <T, and it is natural to require that the sum have the same property. How- ever, all that is physically necessary is that the sum shall do no more than add extra Regge poles, and it is certainly not allowable that A(s,t) shall grow exponentially. We shall see that these three possibilities are intimately connected.

Proceeding in a manner similar to that used in Sec. 11, we develop an integral representation for A(s,t) as follows :

where a = S + T , (3.4)

and m C Z ~ r(21+2h-a)

Fz(x,a) = x - d + h . (3.5) LA--o I! r(1+2h--a)

FIG. 1. Contours used in the proof of Sec. 111. The original contour, along the real axis, may be distorted to the contour C, where the angle e is arbitrarily small, and C is arbitrarily close to the leading singularity a t -i@.

NOW if Im (S) > 0, and if h (y) has singularities only when y is nonzero and pure imaginary, the contour (- a , ) may be distorted to the contour C in Fig. 1, where the angle E may be arbitrarily small. After this distortion, the integral (3.6) provides a representation for A(s,t) for all S such that ~ < a r g ( S ) < ~ - E. [These assertions may be proved by first noting that the factor Fz in (3.7), when Re(y) -+ f a , gives F2(0,a), so that their truth depends only on the remaining factors, which can readily be checked.]

Defining the nearest singularity to the real axis to be a t y = -i@, we can then see that the asymptotic behavior as IS I --+w with E < arg(S) <T- e is given by eiPS, which decreases exponentially. A similar argument for 15'1--+a with - ~ > a r g ( S ) > - T + C shows that A(s,t) will decay exponentially if the only singularities of h(y) are on the imaginary axis.

The singularities of h(y), from (3.7), occur when y= ( 2 n + l ) ~ i , from the first factor, and when a singularity of the second factor is reached. Thus, we deduce the following :

For A(s,t) to go to zero faster than any power o j S as IS I -+w at fixed zl and with T> I arg (S) I > 0, it is sz~fii- ent that F2 (x,a) possess singzllarities only JOY $ < x < w . Conversely, let us define

and suppose that B(S,a) is analytic for all S apart from We have interchanged the order of summation and poles at S= 0, 1, 2, . . at s= a, a- 1, 2, . . . integration in going fro111 (3.2) to (3.3) which is justified (this follows fronl the absolute convergence of the in Appendix C , in ~ h i c h we can also deduce thatFz(x,u) ~~~~~i~~~ and is proved in set. ~ \ i ) . suppose is analytic inside r 1 <a. We now substitute q=ey to also that as I along any ray in the upper half- obtain plane, B @,a) decreases faster than eiPs, and as IS I -+a,

A (s,n = [ dy ecsuhb), (3.6) along any ray in the lower half-plane, B(S,a) decreases faster than e-iys. Then there exists a unique h(y) such

where that A (s,t) is given by (3.6), and h(y) is analytic in all h(y) = (l+eu)aF2(eu(l+e~)-2, a). (3.7) parts of the y plane, except where y is pure imaginary

2382 P . 13. F R A M P T O N A N D C . W . G A R D I N E R 2

and satisfies

We can prove this from the Fourier inversion theo- r e n ~ , ~ since (3.6) is a Fourier transform in which the transformed variable is is. Equation (3.6) converges for 0> Re(S)> Re(a), so that the inverse is given by

where c is a real number such that O>c>Re(a). Thus we define h(y) by (3.10) and continue to Re(a)<O by distorting the contour where necessary, but keeping the limits the same. Since H(S,a) is bounded by ei$s in the upper half-plane, and e-%rs in the lower half-plane, and since it is analytic apart from the poles mentioned above, we find, by rotating the contours where necessary, that h(y) does have the properties we set out to prove.

If, instead of requiring that B(S,a) be bounded by exponentials, we require simply that it shall go to zero faster than any power, we find that the region of analyticity of iz(y) is everywhere except for the imaginary axis. I t is clear, though, that such a result can also be proved by assuming that B(S,a) shall grow slower than any exponential which can include nonphysical behavior, such as a growth like exp[(iS)'I2].

Defining Fz by Eq. (3.7), and substituting r= lny, we deduce that ii h(y) is analytic everywhere apart from the imaginary axis, then Fz(x,a) must be analytic everywhere except for singularities on the line $<x< m, and a possible logarithmic branch point a t x = 0. This cannot exist, however, since x e have shown that Fz(x,a) is analytic a t x=O if the T'eneziano series converges absolutely.

The results of this section mav be summarized as follows : For A (s,f) to go to zero faster than any power of s as 1s 1 -+a a t fixed u, with n > I arg(S) / > 0, it is sufficient that Fz(x,a) possess singularities only when x is real and $<x< m . A necessary condition is that Fs(a,a) possess singularities only when x is real and '<x<m. 4-

IV. ANALYTICITY OF VENEZIANO SERIES

This section will show that an absolutely convergent Veneziano series converges uniformly in S for all finite S, and is therefore analytic where each term is analytic. We can write the modulus of a given term of the Venezi- ano series as

By use of the standard integral representations of the

E. C. Titchmarsh, Tlzeovy of Functions, 2nd ed. (Oxford U. P., New York, 1960).

beta and gamma functions, vie can deduce the inequality

r(h-T) I whenever

Re(T)<l+h, Re(S)<l+h, (4.3)

and where Y is any real number such that Re(S) < Y. Recause

r(h-Re(T)) iim 1 1, +w r(h-T) (4.4)

we can find a positive number M, such that the function whose limit is taken in (4.4) is less than M.

Thus we have the inequality that

I3owever, the right-hand side is a term of the series (1.1) evaluated a t S= Y, T = Re(T), and by hypothesis this se~.ies converges absolutely. Therefore, the 'lVeierstrass cornparison testG tells us that the Veneziano series converges uniformly in S for Re(S)< Y. Since Y is arbitrary, we derive uniform convergence in all the finite S plane (and similarly the T plane), and hence A (s,t) is analytic where its terms are. The only singularities in S of the terms are poles a t S = N , where N is a non- negative integer, which occur only in those terms for which I+h<N. Thus the poles in S are obtained by adding those in each term.

V. EXAMPLES

In this section we give some examples of infinite series which satisfy the requirements laid down.

A. Summation over h Only

In this case we have a series of beta functions only, with Clh zero unless 1 is zero. Thus we find that Fl(x,T) and Fz(x, S+T) have a very simple form, namely,

so that we need to consider only one function. / e require Fz(z, S+ T) to have singularities only when x= q, where q is a real number satisfying $<q< 00. When this is the case, the singularities oi Fl(x,T) occur when x = v, where

z'-'L2= q 1 -- (5.3)

6 G. E. T. Whittaker and G. N. Watson, d4ode~n Analysis, 4th ed. (Cambridge U. P., New York, 1962).


so that the condition for Regge behavior a t fixed t re- quires that qrl. We can test also that when q= 1, v=;( l+ia) , which allows Regge behavior a t fixed t. Notice also that the function $(a,b,T) given in (2.4) is given by

which is required by the condition q2 1 to be analytic in I b I < 1, which includes the region (2.5), showing that the series for A (s,t) is absolutely convergent there. Thus we see that the function

displays Regge behavior in all channels, and is absolutely convergent if and only if Fs(x,T) has singularities only on the real line greater than or ecluztl to 1.

B. Mandelstam's Amplitude

This is a special case of the preceding example, which has the remarkable property that the trajectories are spaced by two units.' I t is given by setting

which satisfies the conditions put on the previous example.

C. Summation Only over 1

In this part we shall only sketch proofs; the reader niay fill in the details. With only an I summation,

For absolute convergence of the original series, x-S-'(l-~)-~-' multiplied by this must be integrable on [0,1]. Since i t is a simple power series, it must repre- sent therefore an analytic function in I x 1 < 1, and the only point left to check is the behavior as x-+ 1. K O precise condition can be given, but it can be checked that the integrability condition is satisfied if

for some positive e . [This is also verifiable directly to be sufficient for absolute convergence of (1.1).]

Noul defining a function

7 S. Mandelstam, Phys. Rev. Letters 21, 1724 (1968).

we can write

the contour enclosing the points x=O and x=z, but remaining inside the circle Izj < 1, which is the radius of convergence of (5.9). The necessary change of order of integration and summation is justified by distorting the contour to lie outside 1 z / = 1 x 1 , so that the integration takes place in the region of uniform convergence of both series involved. If p(z) is analytic a t z= 1, the representation (5.9) shows that Fl(x,T) is analytic a t x= 1, so Regge behavior a t fixed t will be guaranteed.

We now study Fz(x, S+T). I n this case writing again a=Sf T, we can show that

where the contour encloses the origin and the point z=4x, and includes none of the singularities of p(z). The coincidence of a singularity of p (z) a t z= y, say, with the singularity of the hypergeometric function a t z= 4x will give rise to a singularity in Fz(x,a) a t x=$y. Thus if we wish to have singularities of Fz(x,a) only on the interval $<x< 0 3 , it will be necessary that all singularities of p(z) be confined to the region of real z greater than 1. I n this case CZ-X1, for some positive A < 1, so that absolute convergence will also be guaranteed.

D. Special Case of Clh= wlvh

We study this case because it is explicitly soluble, and gives a nontrivial problem to solve. I t also shows that the conditions of Regge behavior a t fixed t, a t fixed u, and absolute convergence are quite independent. In this case, straightforward calculation shows that

and

2384 P . H . F R A M P T O N A N D C . W . G A R D I N E R 2

We now study first F2(x,a), since it is most restrictive. Analyzing all the various pinches and endpoint singularities that can occur, we find that singularities of Fz(x,a) can occur when

and r=o.

Ho\vever, the singularity a t x=O cannot occur on the first sheet of F2(x,a) since our analysis in Sec. 111 shows that it is always analytic a t x=O. Consequently, to restrict the singularities of F2(x,a) to the interval $<x< w , we shall require w and v to be real, and

Since we are now restricted to real za and v, we shall carry out the remainder of the analysis also with real w and v, which maltes the work simpler to understand.

We now check what range of w and v will guarantee an absolutely convergent Veneziano series. I t is readily checked that $(a,b,T) as given in (5.14) has as region of absolute convergence the region

This region contains every point of the region (2.5) and its boundary as an interior point provided that

We prove this as follows: The boundary of (5.18) does not intersect the boundary of (2.5) provided that the line

does not cut the parabola ( b l = la[--la12, for l a l < l . Substituting, we iind that the line meets the parabola when

, # i ~ ~ + l w l . t C ( l ~ l + I ~ I ) " - 4 I ~ I l ~ ~ ~

Thus, either the roots must be imaginary, yielding the second condition (5.19), or the smallest root must be greater than one, i.e.,

The root is implicitly positive, so that I w I > 1 v I ; squar- ing, we derive the first condition (5.19). Thus tl/(a,b,T) will then be analytic on the curve b = a(1-a) ( 0 5 ~ 5 1) and therefore integrable there, so that the Veneziano series will converge absolutely if (5.19) is satisfied.

The most intricate part of the problem involves the study of Fl(x,T). I t is clear that Fl(x,T) has a singularity a t x= 1 if and only if w= 1, so we shall immediately forbid w= 1, and investigate under what

conditions all singularities lie outside 1 x 1 < 1. The singularities occur a t the points

and we shall simply need to find in what region s of the wv plane these give rise to x with modulus greater than 1. With careful calculation, the region outside which

1 x 1 2 1 can be found to be the interior and the boundaries of the region given by the inequalities

Since it is only possible for x to be equal to 1 [in either of the expressions (5.20)] when w= 1, we can guarantee Regge behavior a t iixed t if we exclude the line w= 1 from the region (5.21).

We have drawn the regions (5.17), (5.19), and (5.21) in Fig. 2. Since the region of Regge behavior a t fixed t is included in the region for absolute convergence, we see that the series displays Regge behavior only if it is absolutely convergent; but notice that Regge behavior a t fixed zb is a quite independent concept.

Khuri4 has found a sufficient condition for absolute convergence of a general Veneziano series, ~irhich in our notation can be written

which is implied by this example, since this lneans w=v= $ lies in the region of absolute convergence. How- ever, such a bound is not sufficient to guarantee Regge behavior; for example, w= -3, v = -i does not lie in the region of Regge behavior a t fixed t or fixed z ~ .

E. Matsuda's Model

hZatsudas has given an interesting model in which the Veneziano terms can be summed explictly. The lnoclel is given by the integral

and may also be written in another form, by inaliing the substitution

with which it takes the form, for X20,

a =11 dZ[u1(~) - - c t ~ ( ~ ) / ~ ] u ( ~ ) - ~ ~ ~ ~ - ~ , (5.25)

where u ( z ) = - XZ+ [(A2- 1)z2+ 1]1/2. (5.26)

(In this form this model bears a close resemblance to those of Ref. 9.) From Eqs. (5.23) and (5.25), we see

S. Matsuda, Lawrence Radiation Laboratory Report No. UCRL-19266, 1969 (unpublished).

a C. W. Gardiner, Phys. Rev. D 1, 2888 (1970).


that

and

1+2q+q2

First, we see that the singularities of (5.28) occur a t the points

q=-1, and (5.29)

q= - A r t (A2- 1)1!2,

so that putting, as in Sec. 111, q=eu, the singularities in y will lie on the nonzero imaginary axis provided only that X is real and

- 1 < x 5 1 , (5.30)

which agrees with Matsuds's deduction. Turning now to Eq. (5.2) and keeping X real from now on, we now want to find the conditions on X for Fl(x,T) to be analytic a t x= 1 and inside 1 x 1 < 1. When AZ1, w(x) has a simple zero a t x= 1, so that the singularities of F1

will occur when o(x) is singular, and when w (x) = 0, for x f l . I t is readily checked that w(x) can only vanish when x= 1, but i t has a branch point a t x= 1/(1-X2)1'2, which can only lie outside the unit circle if O< X 2 5 2.

When X= 1, the branch point is a t x= 1, and is not allowed. Thus we have Regge behavior a t fixed t iflo

(a (b) (c) FIG. 2. Various regions described in Sec. V D. The shaded

regions in the wv plane are (a) region of absolute convergence of the series; (b) region of Regge behavior a t fixed t as Is1 -+m ; (c) region of Regge behavior a t fixed u as 1s 1 -+ m .

In the case of negative A, the substitution (5.24) is not one to one, and if care is taken, we get by the same substitution

where 0 and A are (ii) As t -+ 0 in

totic expansion

in which the contour starts a t z = 0, and passes around the singularity of w(z) a t z= I /(l-X2)1!2, and then ends a t z= 1 ; thus the endpoints of the integral are on differ- ent sheets. If / l/(l--X2)112j > 1, the asymptotic behavior as S -+ - cn will be exponential, i.e., i t will be

which is not Regge behavior. If / 1/(1-A2)1'2( < 1, the analysis of Sec. I1 can be modified to show that exponential growth will still exist in some direction. If X is negative, l / ( l - X2)1!2# 1, so we conclude that negative X are not allowed if Regge behavior a t fixed t is required.

VI. SUMMARY

In this paper a generalized Veneziano series expansion for a scattering amplitude has been studied. Tests have been derived to find whether a given series displays

10 Dr. Matsuda has confirmed that there was an error in his manuscript and this result is correct.

Regge behavior for / s / -+a a t fixed t (Sec. 11), and a t fixed u (Sec. 111). We have considered the analytic properties of A (s,t) (Sec. IV), and have given examples to illustrate the usefulness of the methods (Sec. V).

APPENDIX A : WATSON'S LEMMA

We give here a version of Watson's lemma suitable for our applications." Let g(t) be a function of t satisfying the following three properties :

(i) g(t) is analytic in the wedge

positive and less than ~ / 2 . the wedge (Al), g(t) has the asymp-

(iii) For every value of arg(t) satisfying the inequality (Al), there exists a positive M such that for 1 t I > 0,

Define a function J(x) by

where m is the smallest integer such that Re(af m)> 0. Then f(x) may analytically continued to the wedge

X++a>arg(x)> -8-+T, (A5

'1 G. N. Watson, Theory of Bessel Ft~nctions, 2nd ed. (Cab- bridge U. P., New York, 1966), p. 236.

2386 P . H . F R A X I P T O N A N D C . 1%'. G A K D I N E R 2

and inside this wedge it has the asymptotic expansion

Conversely, let J(x) be analytic in the wedge (-45) (with positive 0 and X) and possess inside it the asymptotic expansion (A6). Suppose also that f(x) is either integrable a t x= 0, or possesses a sirnplepole there. Then there exists a unique g ( t ) such that f(x) is given by (A4), and g( t ) satisfies the conditions (i)-(iii).

Proof

From (A4), if we prove the theorem for a > O , we may extend the result to all a . 1I7e therefore assume cu>O in all that follows.

Fronl the conditions (ii) and (iii) we can write, for real positive t,

The function g ( t ) is unique from the standard thcorj of the Laplace transform.

We may distort the contour as close as we like to the boundaries of the wedge (A5), and by a method similar to that used in the direct proof, show that J(x) is analytic in the wedge (Al).

If J(x) is integrable a t x=O, we may choose the contour so that no part of it lies in the region Re(.c)>O. It is then straightforward to show that g(t) satisfies condition (iii). Similarly, if there is only a simple pole a t z=O, we may pick up the residue of x=O (which is a constant), plus a tern1 \\-hich also satisfies condition (iii).

To prove condition (ii) we choose the contour K for the integration to be given by the wedge

N Let iz(x) be a function which goes to zero faster than I , f ( t) - C ant"?"/ <M't"r\'tl, ~ , - ( " + ' ~ ~ l ) , but is analytic in the wedge ('45). Define n=O

for some positive M', so that

N l f ( ~ ) - C anr(a+~~+1)x-(e*7'-t1)j

provided Re(x)>O, so that the integral over positive real t is a valid representation.

Thus, since the difference between !(a) and the first #-+I terms of the asymptotic expansion (A6) goes to zero faster than any of these, the expansion is asymptotic in the sense of Poincar6.

We now extend the proof to the full range of values of arg(x). We may rotate contour of integration in the t plane through an angle 4 provided that B > + > -A. The integral now converges in the range Re (ae*) > 0. Taking into account the range of $J, we find that j (z ) is analytic in the wedge (A5) and, b j similar reasoning to the above, we show that the asymptotic expansion (A6) is valid inside that wedge.

Proof of Converse

Assuming, as before, that a>O, we construct g ( t ) by the Laplace inverse transfornlation

(where c is a real positive number). Then,

and

[where the upper limit may be anywhere inside the wedge (A5)], it is true that Jz, (x) goes to zero faster than z-("+"+'-'). Thus we may integrate (A12) by parts A + 1 times to obtain

and hx+v+l(x) goes to zero faster than a-". This fact ma)- be used to show that as t -+ 0 in the wedge (Al),

Using then the asymptotic expansion (A6), we see that

where K y ( t ) goes to zero no slower than 1 t 1 n+e+l. This proves that g(u) satisfies condition (ii).

APPENDIX B

We prove here the result quoted in Sec. 11, that if the Veneziano series (1.1) is absolutely convergent, then the following result:


(i) For Re(S) and Re(T) negative, the series most all x in [a,b]. Hence the following statements are true :

m ClhI'(lfh-T) (i) The series (B4) converges for almost all r in [0,1], @(a, b, T) = - --------

I , ~ = o I! r(h-T) (") (ii) $(x, ( I - r ) , B)rA-l(l-.~)-B-l is integrable on

[0,11, and converges absolutely in the region (iii)

and converges absolutely almost everywhere on that part of the boundary of ( ~ 2 ) given by b = a ( l - a) IVe now prove the main result. Consider the formula

(+<a< 1).

(ii) X-~- ' (~ -X) -~- '~ (X, x(1-x), T) is integrable on A(s,f)= dx ~ - ~ - ' ( l - x ) - ~ - ~ the interval [O,l] and

(iii) Conversely, let valid if Re(S) and Re(T) are negative. Notice also that

#(x, 4 1 -XI, R) r (12 - T) lirn ------------

c= h'" I'(lz--Re(T)) (B8) I clh1 r ( z + h - ~ )

= C x"f"(l -.)"- z,h=o I! r(h-B)

(B4) from which it follows that there exist nun~bers 0 and p

such that 0<0< 1 < p and converge absolutely when B is real and negative, for almostBl1 x in [O,l], and let

(1 - x)-"-'x-*-'$ (r, x (1 - x) , B)

be integrable on [O,ll for negativeA and 3 . The state- whenever /t>O. hToting that Ir>0 implies 1+11>0, we inents (i) and (ii) are true, and the VenezianO series see that the modulus of each integrand of (B7) is less (1.1) is absolutely convergent for all S and T. than or equal to

Proof dx x-Re(S)-l(l -%)-Re(T)-l We need first a preliminary lemma. Lemma. Let A and B be negative real numbers, and / Clh / r(1i-l~ -Re(T))

X --- ----xl+h consider (B10) I ! r(h-Re(T))

which is an absolutely convergent series, since it is of the form (1.1) with S = A and T = B , apart from the replacement of Clh by IClh(, which does not affect absolute convergence. I n Ref. 5 , Sec. 10.83, it is shown that if u,(x)>O for all t z and x, then

The terms of (B10) are now of the same form as the terms of (B4) (apart from the factor ,u/0). This then shows us two things:

(1) Since we have shown that (B5) is absolutely convergent almost everywhere on [0,1], @(x, x(1-x), T) is absolutely convergent almost everywhere on [O,l].

(2) The terms of the sequence of partial sums of the terins under the integral sign in (B7) are integrable and bounded by $(x, x(1-x), Re(T))x-Re(S)-l(l-x)-Re(T)-l, which is integrable on [0,1]. The general convergence theorem of Lebesgue (Ref. 5, Sec. 10.8) shows that we may interchange the order of suinmation and integration in (B7), and derive the second result which we wished to prove.

provided either side converges, and that the series The proof of the first result is straightforward. The Cn,om un(x) converges to an integrable function for al- function +is a double power series, so that if it converges

2388 P . H . F R A M P T O N A N D C . N T . G A R D I N E R

a t the point (a,P) it converges12 for 1 a / < I cr 1 , 1 b / < I P / . We have shown that the series converges absolutely almost everywhere on the curve

so i t must converge absolutely in the region (B2). The union of the regions (B 11) and (B 2) is the region stated in the first result.

Proof of Converse

The converse is proved by proving the converse of the lemma; i.e., if #(x,B) satisfies the conditions (i) and (ii) of the lemma, then H(A,B) can be written in the form (B4), which is a series of positive terms. This converse follows directly from theorem 10.83 of Titch- marsh.6 By techniques similar to those used in the direct proof, we then show that the Veneziano series is domi- nated by a multiple of a series of the form (B4), and therefore the Veneziano series (1.1) converges absolutely. This only holds for Re(S) and Re(T) negative. However, replacing T by T + l in (1.1) multiplies each term by a factor l+(h-S)/(l+h-T-1), which is bounded for 1, h>O. Thus the series (1.1) converges absolutely for Re(T) < 1. We can thus extend the region of absolute convergence to all S and T.

APPENDIX C

In this appendix me prove the results used in Sec. 111. These state that if

'2 B. A. Fuks, Introduction to the Theory of Analytic Functions oj Several Complex Variables (American Mathematical Society, Providence, R. I., 1963).

converges absolutely, then the follo~viilg obtain:

(i) The integral may be taken outside the summation; (ii)

is analytic in 1 xl <+. [We assume throughout this appendix that Re(S)>O, and Re(a)<-1, so that all the integrals exist.]

Proof

The proof is very similar to that in Appendix B. First ~nalre the substitution, in (Cl), of

where A and B are real, and also substitute lC1hl for Clh. Then, as in Appendix B, we can show that the integral inay be taken outside the summation, that the series

converges for almost all .2: in [O,$], and that

is integrable for 0 5 q< m . I n the same inanner as in Appendix B, we then show

that the modulus of each integrand in (Cl) is less than or equal to some positive multiple of

Hence i t follo~vs that the series for Fp(x,a) converges absolutely for lx 1 <$, proving result (ii). Result (i) follo\vs by similar reasoning to that in Appendix B, since each integrand is bounded by a positive multiple of (C6), which sums to an integrable function.

conditions for regge behavior of an absolutely convergent veneziano series

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