2
PHYSICAL REVIEW VOLUME 184. NUMBER 5 25 AUGUST 1969 Comments and Addenda The Comments and Addenda section is for short communications which are not of such urgency as to justify publication in Physical Review Letters and are not appropriate for regular articles. It includes only the following types of communications: (i) comments on papers pre- viously published in The Physical Review or Physical Review Letters; (2) addenda to papers previously published in The Physical Review or Physical Review Letters, in which the additional information can be presented without the need for writing a complete article. Manuscripts intended for this section may be accompanied by a brief abstract for information retrieval purposes. Accepted manuscripts will follow the same publication schedule as articles in this journal^ and galleys will be sent to authors. Spin-Matrix Polynomials and the Veneziano Formula* T. J. NELSONt Lawrence Radiation Laboratory^ University of California, Berkeley, California 94720 (Received 7 April 1969) We consider a partial-wave expansion for inr -^ iroi in terms of the spin-matrix polynomials PFn(z) =r(z+}»+J)/r(2~ Jw+f) rather than the conventional Legendre polynomials. This leads naturally to Veneziano's formula when his supplementary condition a{s)-{-a.{t)-^a{u) = 2 is invoked. T HE appearance of the ^ function B{x,y) =^T{x)V{y)/V{x-\-y) in Veneziano's representa- tion of the TTTT —» TTO) scattering amplitude^ is interesting. A general representation of the scattering amplitude in which this functional form appears naturally could conceivably be of fundamental significance. We will start from a partial-wave expansion in the s channel, A{sM) = hi:{2l+l)ai{s)lPi{z)-^Pi{-z)'} I ^{s^t.u), (1) for the symmetric factor in Veneziano's amplitude. Instead of an expansion in terms of Legendre poly- nomials, we wish to consider one in the spin-matrix polynomials defined by Wn{z) = V{z+ln+l)/V{z-ln+\), which are polynomials of degree n in the variable z for integer n. Weber and Williams^ have shown that A W „ a ^ - i ) = w!6fe,, (2) if k and n are both even or both odd, where A is the difference operator ^^(2) = g(s)-g(2-l). The function Wn{z) is called a spin-matrix polynomial because of its advantages in expansions of functions of s=s-^, where Si are the three (2^+l)X(2.y+l) spin matrices for spin s, and ^ is a unit vector along the axis of rotation. In particular, W2{a+n)+i{z) vanishes for ^^ 0 as a consequence of the Cayley-Hamilton theorem. * Work done under the auspices of the U. S. Atomic Energy Commission. t U. S. Atomic Energy Commission Postdoctoral Fellow. 1 G. Veneziano, Nuovo Cimento 57A, 190 (1968). 2T. A. Weber and S. A. Williams, J. Math. Phys. 6, 1980 (1965). ' 184 The Legendre polynomials in Eq. (1) can be expressed as an expansion in the spin-matrix polynomials in the form Pl{z)-^Pl{-z)= E bnlWn(2pz)+Wn{-2pz)-], (3) n=0 where p can be an arbitrary function of s. We now apply Eq. (2) with the argument 2pz with k even to get ''="'Kv)+''(-17) 2k\. (4) The coefficients hn for odd n drop out in Eq. (3) be- cause of the symmetry property T^^(—2)== {—YWn{z), so there will be no damage done if Eq. (4) defines hn for all non-negative integer values of n. The partial- wave expansion then takes the form \i:{2l+\)ai{s)lPi{z)+Pi{-z)-] I = i:-Cn{s)lWn{2pz)^Wn{-2pz)-], (5) n=0 w! where Cn{s) is defined by Cn{s)=\J:{2l+\)al{s)^- 1=0 xKvH-v)] , (6) if it is safe to interchange the two sums. We have ex- tended the range of the sum over / downward to zero because A^ will reduce a polynomial of degree / to zero a n>l. We also take Eq. (6) as the definition of 1954

Spin-Matrix Polynomials and the Veneziano Formula

  • Upload
    t-j

  • View
    213

  • Download
    0

Embed Size (px)

Citation preview

Page 1: Spin-Matrix Polynomials and the Veneziano Formula

P H Y S I C A L R E V I E W V O L U M E 1 8 4 . N U M B E R 5 2 5 A U G U S T 1 9 6 9

Comments and Addenda

The Comments and Addenda section is for short communications which are not of such urgency as to justify publication in Physical Review Letters and are not appropriate for regular articles. It includes only the following types of communications: (i) comments on papers pre­viously published in The Physical Review or Physical Review Letters; (2) addenda to papers previously published in The Physical Review or Physical Review Letters, in which the additional information can be presented without the need for writing a complete article. Manuscripts intended for this section may be accompanied by a brief abstract for information retrieval purposes. Accepted manuscripts will follow the same publication schedule as articles in this journal^ and galleys will be sent to authors.

Spin-Matrix Polynomials and the Veneziano Formula*

T. J. NELSONt Lawrence Radiation Laboratory^ University of California, Berkeley, California 94720

(Received 7 April 1969)

We consider a partial-wave expansion for inr -^ iroi in terms of the spin-matrix polynomials PFn(z) =r(z+}»+J)/r(2~ Jw+f) rather than the conventional Legendre polynomials. This leads naturally to Veneziano's formula when his supplementary condition a{s)-{-a.{t)-^a{u) = 2 is invoked.

TH E appearance of the ^ function B{x,y) =^T{x)V{y)/V{x-\-y) in Veneziano's representa­

tion of the TTTT —» TTO) scattering amplitude^ is interesting. A general representation of the scattering amplitude in which this functional form appears naturally could conceivably be of fundamental significance. We will start from a partial-wave expansion in the s channel,

A{sM) = hi:{2l+l)ai{s)lPi{z)-^Pi{-z)'} I

^{s^t.u), (1)

for the symmetric factor in Veneziano's amplitude. Instead of an expansion in terms of Legendre poly­nomials, we wish to consider one in the spin-matrix polynomials defined by

Wn{z) = V{z+ln+l)/V{z-ln+\),

which are polynomials of degree n in the variable z for integer n. Weber and Williams^ have shown that

A W „ a ^ - i ) = w!6fe,, (2)

if k and n are both even or both odd, where A is the difference operator

^^(2) = g ( s ) - g ( 2 - l ) .

The function Wn{z) is called a spin-matrix polynomial because of its advantages in expansions of functions of s = s - ^ , where Si are the three (2^+l )X(2 .y+l ) spin matrices for spin s, and ^ is a unit vector along the axis of rotation. In particular, W2{a+n)+i{z) vanishes for ^ ^ 0 as a consequence of the Cayley-Hamilton theorem.

* Work done under the auspices of the U. S. Atomic Energy Commission.

t U. S. Atomic Energy Commission Postdoctoral Fellow. 1 G. Veneziano, Nuovo Cimento 57A, 190 (1968). 2T. A. Weber and S. A. Williams, J. Math. Phys. 6, 1980

(1965). ' 184

The Legendre polynomials in Eq. (1) can be expressed as an expansion in the spin-matrix polynomials in the form

Pl{z)-^Pl{-z)= E bnlWn(2pz)+Wn{-2pz)-], ( 3 ) n=0

where p can be an arbitrary function of s. We now apply Eq. (2) with the argument 2pz with k even to get

''="'Kv)+''(-17) 2k\. (4)

The coefficients hn for odd n drop out in Eq. (3) be­cause of the symmetry property T^^(—2)== {—YWn{z), so there will be no damage done if Eq. (4) defines hn for all non-negative integer values of n. The partial-wave expansion then takes the form

\i:{2l+\)ai{s)lPi{z)+Pi{-z)-] I

= i:-Cn{s)lWn{2pz)^Wn{-2pz)-], (5 ) n=0 w !

where Cn{s) is defined by

Cn{s)=\J:{2l+\)al{s)^-1=0

xKvH-v)] , (6)

if it is safe to interchange the two sums. We have ex­tended the range of the sum over / downward to zero because A^ will reduce a polynomial of degree / to zero a n>l. We also take Eq. (6) as the definition of

1954

Page 2: Spin-Matrix Polynomials and the Veneziano Formula

184 S P I N - M A T R I X P O L Y N O M I A L S A N D T H E V E N E Z I A N O F O R M U L A 1955

Cn(s) for odd values of n, since they drop out of the amplitude anyway. The form of Eq. (6) is interesting in that it relates Cn(s) to discrete values of the entire Legendre series. We now consider the Sommerfeld-Watson transform of the right-hand side of Eq. (5). This is given by

00 1

E -Cn(s)[Wn(2pz) + Wn(-2pz)2 n=onl

cin,s) 1 r c{t — / dn ilTTJc T(n («+i)

r(2pz+i«+i) r(-2pz+iM+i)\ TT hn+k)/su ,r(2p2—5W+I) r(—2pz—5»+5)/sin7rw

where the contour C passes around the positive real axis of the complex n plane in the clockwise direction. We now distort the contour past the leading pole of c(n,s), which we suppose to be located a.tn{s) = a(s) — l, picking up the contribution

T(2pz+Ms)) r ( - 2 p z + | a ( 5 ) ) \ + ) .

KT(2pz-^a(s)+l) r ( - 2 p z - i a ( 5 ) + l ) / X

We can easily interpret this contribution to the ampli­tude A (s,t,u) if we assume linear trajectories. We have z=(t—u)/4kikf, so that taking p to be bkik/, where b is the slope, we get

2p2= -aiu)-ia(s)+Uia+b^)

=a(t)+ia{s)-i{3a+bX),

where a is the intercept. Our contribution from the pole is then

yS(5)r(l -ais)){

m<T-a(t)) -am \ a(s)-a(t)))' T(l+ia-a(s)-am

(7)

where a=Sa+b2 = a(s)-{-a(t)+a(u). Evidently, Jo-has to be an integer if we are to avoid poles at non-integral values of the trajectory function in the t and u channels. Experimentally, Jcr is close to unity. The supplementary condition J<r=l was discussed by Veneziano and also by Mandelstam.^ Taking fi(s) to be constant, say, P/lw, our symmetrized contribution to A (SjtjU) becomes

A(sAu)= WT)lB(l-a{s), l-a{t))

+B(l~a(s), l-a(u))+B(l~a(t), l - a ( # ) ) ] ,

which coincides with Veneziano's proposal. Evidently there are several steps in the deduction

here that will need to be filled in with arguments, rather than mere enthusiasm. Pending the outcome of these investigations, it seems fair to say that the use of the spin-matrix polynomials instead of the Legendre polynomials can probably reduce the dy­namics inherent in the Veneziano formula to just that of the pole in the complex n plane.

I t is a pleasure to thank Professor G. F. Chew for the hospitality of the Lawrence Radiation Laboratory.

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