Veneziano model and the finite energy sum rules

KASHYAP V. VASAVADA Department of Physics, University of Connecticut, Storrs, Connecticut

Received January 8 , 1970

We sum explicitly the left-hand side of the finite energy sum rules for the Veneziano amplitude. This shows that, for sufficiently large cutoff, the finite energy sum rules are satisfied with the leading con- tribution only. The actual sum contains other non-leading contributions.

Canadian Journal of Physics, 48. 1879 (1970)

Some time ago, Veneziano (1968) proposed a very attractive model for scattering amplitudes of strongly interacting particles, incorporating analyticity, resonance pole structure, crossing symmetry, and Regge behavior in a somewhat natural way. He also showed by induction that the finite energy sum rules (FESR) are satisfied for sufficiently large cutoff in his model. (For FESR see, for example, Dolen, Horn, and Schmid 1968; Igi 1962; Igi and Matsuda 1967; Logunov, Soloviev, and Tavkhelidze 1967.) The purpose of this article is to demonstrate that the left-hand side of the FESR can be summed ex- actly. For a sufficiently large value of the cutoff this sum becomes the right-hand side of the usual FESR, while for low cutoffs the sum contains contributions of the non-leading terms.

For simplicity we consider the x-x system, although the following treatment can be readily generalized to other cases.

The most general Veneziano type function for the x-x system is of the form'

where a(s) and a(t) are the s- and the t-channel trajectory functions (a@) = a, + a's). n and 1 are arbitrary positive integers with the restriction that

n < l < 2 n

R,, are arbitrary constants. The condition n < 1 is necessary to assure the maximal allowed Regge

'A term of the type T(n - a(s))T(m - a( t ) ) (n # m) can also occur. However, in the present case, crossing symmetry requires that there should be another term with n and rn interchanged and with identical coefficients. Hence these terms can always be combined to give a series of terms of the given form, where only terms with n = m occur. It should be noted, however, that the present proof goes through completely even for the general case n # m.

behavior. The other condition, 1 < 2n guarantees the required polynomial character of the reso- nance-pole residues and absence of the ancestors.

Different isospin amplitudes in various chan- nels (s, t, u) can be obtained by taking linear combinations of functions of the form [I]. In particular, the t-channel amplitudes correspond- ing to It = 0, 1, and 2 are given by

First, consider the It = 1, t-channel amplitude

Here we have

S - U v= - , X = a(s), y = a(t), w = a(u)

The FESR for this amplitude is found to be

Since G,(v, t) is odd under crossing, p is an even integer. is is the cutoff energy above which the Regge representation can be assumed to be good. P(t) is the residue function for the Regge-pole contribution.

The s-channel resonance poles can be exhibited by using the expansion

" T ( j + l + n - Z + y ) 1 l ) r ( l + + - M ) ( ~ + n-x)

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1880 CANADIAN JOURNAL O F PHYSICS. VOL. 48, 1970

Using this residue formula for the s- and u- channel poles, one gets

where D = x + y + w = 3a0 + 4m,2af. This implies that

FESR when the non-leading terms are also included.

For higher moment sum rules, one can derive a recursion relation. Let

Then

Hence we have The finite series for p = 0 can be summed

readily (see, for example, Mangulis) and gives ' - ") K p [I31 Kp+l =I?, + 1 - y - 1 +--- n (- 1)Zn-l-l 2

c81 where E?, is obtained from K, by replacing 1 by , .

T(N + 2 - 1 + y) 1 - 1.

x -- As shown above, KO is given by T(N - n + l )T( l + y - n)

For large N, [8] becomes [14] KO = 1 T(N - 1 + y + 2)

y + n - 1 + 1 T ( N - n + 1 ) n (- l )2r1-1-1 [9] lo"---- (N)y+l l - l+ l a' y + n - 1 + 1 T ( l + y - n) This gives

2 n - I - 1 ( 2 a l V ) y + n - ~ + ~ 1 T(N - 1 + y + 3) n ( -1 )

N -

a f y + n - 1 + l r ( l + y - n) C151 IZO = y + n - 1 + 2 T(N - n + 1)

Now asymptotically, we find that Thus, starting from KO all K,'s can be obtained. Then it becomes clear that the leading term in K,

[lo] Im Gl(v, t ) z 2 sin n(n - 1 + y) will be given by

N _ ' ' N a' T ( l + y - n ) which for large N becomes

X(2af)y+iz-1+1 y + n - I (4 1 y + n - I + p + l (N)y+n- l+p+l

Thus the FESR [4] is evidently satisfied for large enough N, in the case of p = 0. Equation [8] Again, this shows that the higher moment gives the exact form of the right-hand side of the FESR are satisfied for sufficiently large N by the

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VASAVADA: VENEZIANO MODEL AND THE FINITE ENERGY SUM RULES 1881

leading term only and the correction terms can be obtained by using the recursion relations.

Next, we consider the I, = 2 amplitude

In writing the Veneziano representation we have assumed that there is no I, = 2 Regge trajectory. The amplitude G, has no r function involving t , which would give rise to t-channel poles. For the FESR this would imply that the right-hand side should be zero in the leading approximation. Expansion of [I61 in the s-channel poles is given by

The residue of the pole at x = j + n contains the factor

r ( j + l + n - 1 + w ) r ( l + w - n)

which for fixed t becomes

The last factor can be clearly seen to alternate in sign as j goes over even and odd integers. Hence the contributions from the successive terms to the amplitude G2(v, t ) will be of opposite signs. Then, the FESR will be satisfied on the average as a result of these cancellations. This is also verified numerically in a recent publication by the author (see Vasavada 1970).

Finally, we make some comments about the

I, = 0 amplitude. The Pomeranchuk trajectory will contribute to this amplitude. Inclusion of the Pomeranchuk trajectory into the narrow reso- nance Veneziano model is not clear at present. Indeed, it has been suggested by Freund (1968) and Harari (1968) that the Pomeranchuk con- tribution may be related to the non-resonant background. Hence, the present considerations would not apply to the case of I, = 0 sum rules.

Thus, barring the last case mentioned above, we have demonstrated by explicit summation that the FESR are satisfied by the most general term of the Veneziano form for sufficiently large cutoff. In this way we also obtained the correction terms. If the low-energy data for various systems of interest become more precise in future, it may be possible to look at the non-leading contribu- tions to the FESR. In this case, the above ex- pressions will be explicitly useful. Also, it is possible to include approximately the Regge cuts into the Veneziano formalism (see Vasavada 1969). For such a purpose the explicit summations discussed here become useful.

DOLEN, R., HORN, D., and SCHMID, C. 1968. Phys. Rev. 166, 1768.

FREUND, P. G. 0. 1968. Phys. Rev. Lett. 20, 235. HARARI, H. 1968. Phys. Rev. Lett. 20, 1395. IGI. K. 1962. Phvs. Rev. Lett. 9. 76. IGI; K. and M A ~ S U D A , S. 1967. phys. Rev. Lett. 18, 625. LOGUNOV, A. A,, SOLOVIEV, L. D., and TAVKHELIDZE,

A. N. 1967. Phys. Lett. B, 24, 181. MANGULIS, V. Handbook of series for scientists and

engineers (The Academic Press, Inc., New York), p. 60.

VASAVADA, K. V. 1969. Veneziano representation and optical model (to be published).

1970. Veneziano model with secondary terms for pion-pion scattering. Phys. Rev. D, 1, 88.

VENEZIANO, G. 1968. Nuovo Cirn. A, 57, 190.

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