Froissart Bound on Total Cross-section without Unknown Constants

We determine the scale of the logarithm in the Froissart bound on total cross-sections using absolute bounds on the D-wave below threshold for $\pi\pi$ scattering. E.g. for $\pi^0 \pi^0$ scattering we show that for c.m. energy $\sqrt{s}\rightarrow \infty $, $\bar{\sigma}_{tot}(s,\infty)\equiv s\int_{s} ^{\infty} ds'\sigma_{tot}(s')/s'^2 \leq \pi (m_{\pi})^{-2} [\ln (s/s_0)+(1/2)\ln \ln (s/s_0) +1]^2$ where $m_\pi^2/s_0= 17\pi \sqrt{\pi/2} $ .

boundedness at fixed momentum transfer, were obtained by Epstein, Glaser and Martin [8] in the more general framework of the theory of local observables of Haag, Kastler and Ruelle [9].
Recently, Azimov has revisited the Froissart bound in a paper [10], Sec. 2 of which is similar to the 1962 and 1963 works of Martin [11].These papers were a precursor to Martin's later paper [2] which proved the bound rigorously from axiomatic field theory. Azimov has raised doubts about "application of the ideas and methods of axiomatic local field theory to hadron properties". His main point is that, "hadrons, consisting of quarks and gluons, cannot be pointlike", and might not be associated to local fields. However, Zimmermann [12] has shown that local fields can be associated to composite particles (for instance deuterons). We postulate that this construction applies to hadrons made of quarks. This is not obvious because, in spite of the practical successes of QCD, nobody knows how to incorporate particles without asymptotic fields in a field theory. Anyway this is a much weaker assumption than that of the validity of Mandelstam representation. In particular, we do not use the Froissart-Gribov representation of physical region partial waves for fixed s.
The Froissart-Martin bound has triggered much work on high energy theorems (see e.g. [13], [14]) and on models of high energy scattering [15]. Recently, Martin proved a bound on the total inelastic cross section at high energy [16] which is one-fourth of the bound σ max (s) on the total cross-section. Wu, Martin, Roy and Singh [17] obtained a bound on σ inel (s) in terms of σ tot (s) which vanishes both when the total cross-section vanishes and when it equals the unitarity upper bound.
In spite of all this progress, these bounds share severe shortcomings [17]. (i)They are deduced assuming that the absorptive part A(s, t), 0 ≤ t < t 0 is bounded by Const.s 2 / ln(s/s 0 ) for s → ∞. In fact, the Jin-Martin theorem on twice subtracted dispersion relations only guarantees that where s th is the s-channel threshold. As stressed by Yndurain and Common [18], this does not imply that A(s, t) ≤ Const.s 2 / ln(s/s 0 ) for all sequences of s → ∞.
(ii)The bounds are expressed in terms of σ max (s) which still contains the unknown scale s 0 of the logarithm, and the unknown positive parameter ǫ which can be chosen arbitrarily small but = 0. If ǫ is not fixed s 0 cannot be fixed since the advantage of a larger s 0 can be offset by a larger ǫ. We now remove both these shortcomings. We report definitive bounds on energy averages of the total crosssection in which the scale s 0 is determined in terms of C(t) which is a low energy (in fact below threshold) property in the t−channel. In some cases , e.g. for pion-pion scattering, for t → 4, C(t) is proportional to the Dwave scattering length [19] which is known phenomenologically; hence we obtain bounds on energy averages in terms of that scattering length. Even more exciting is the fact that for π 0 π 0 scattering we are able to obtain absolute bounds (in terms of pion-mass alone) on C(t) below threshold without assuming finiteness of the Dwave scattering length; this yields absolute bounds on the asymptotic energy averages of the total cross-section.
Normalizations. Let F (s, t) be an ab → ab scattering amplitude at c.m. energy √ s and momentum transfer squared t normalized for non-identical partcles a, b such that the differential cross-section is given by with t being given in terms of the c.m. momentum k and the scattering angle θ by the relation, Then, for fixed s larger than the physical s−channel threshold, F (s; cos θ) ≡ F (s, t) is analytic in the complex cos θ -plane inside the Lehmann-Martin ellipse with foci -1 and +1 and semi-major axis cos θ 0 = 1 + t 0 /(2k 2 ). Within the ellipse ,in particular, for |t| < t 0 , F (s, t) and the s-channel absorptive part F s (s, t) = A(s, t) have the convergent partial wave expansions, (2l + 1)P l (z)Ima l (s), (7) with the unitarity constraint, Correspondingly, the optical theorem gives, for a = b, For identical particles a = b e.g. for π 0 π 0 scattering, or for pion-pion scattering with Iso-spin I, we have the same formula for the differential cross-section, and the same form of the unitarity constraint, but the partial waves a l (s) → 2a I l (s) in the partial wave expansion,i.e.
With this normalization, F I (4, 0) = a I 0 , the S-wave scattering length for Iso-spin I. and for pion-pion scattering the identical particle factors lead to, In the following, we shall consider non-identical particles a = b for detailed derivations and quote the identical particle results when needed. Convexity Properties of Lower Bound on Absorptive Part in terms of Total Cross-Section. Martin has proved unitarity lower bounds on A(s, t) for 0 < t < t 0 in terms of σ tot (s) [2],and in terms of σ inel (s) [16]. He has also proved [20] that these bounds are convex functions of σ tot (s), and σ inel (s) respectively. We recall first the convexity properties which will be crucial for our proofs of lower bounds on C(t) in terms of energy averages of total cross-sections. We work at a fixed-s , and suppress the s-dependence of Ima l (s), and σ tot (s) for simplicity of writing. Using 0 ≤ Ima l ≤ 1, the lower bound on A(s, t) for given σ tot is obtained by choosing, where, the fraction η, 0 ≤ η < 1, and the integer L are determined from the given σ tot . Thus, where, Hence, A(z) is a monotonically increasing function of Σ tot with piecewise constant positive derivative. Denoting Int(x) = integer part of x, which increases with L since z > 1, and hence with Σ tot when it crosses square of an integer. This proves that the lower bound A(z) is a convex function of Σ tot , and that, and, for Σ tot > 1 Using integral representations for P µ (z) and for the modified Bessel function I 0 we obtain for µ ≥ 0, z > 1, This yields the strict inequality ( without any high energy approximation ), At high energy, this gives, which is a convex function of σ tot (s).
Upper bound on energy-averaged total crosssection . Defining, and we obtain, since the average of a convex function must be greater than the convex function of the average [21]. At high energies ifσ tot (s, ∞) goes to ∞, the asymptotic expansion of I 1 (ξ) yields, where, To extract a bound on the cross-section, we need the following lemma [20]. If ξ > 1, and then, Proof. It is enough to prove this for y = √ ξ exp ξ, since the right-hand side is an increasing function of ξ.Taking logarithms ,and using ξ = ln y − (1/2) ln ξ ≡ ξ 1 repeatedly, For fixed y the derivative of the right-hand side with respect to ξ 1 is (4ξ 2 1 ) −1 which is positive, and ξ 1 < ln y for ξ > 1. Hence the stated upper bound on ξ follows.
Instead of the s-dependent C s (t) we shall use the simple s-independent upper bound , which follows by using A(s, t) > A(s, 0) for 4 > t > 0 and improves the value C(t) if low energy total crosssections are known. The integral of the weight function multiplying σ tot can be done. Thus, where,σ With f (y) as defined above, the upper bound on the average total cross-section in terms of C x (t)is, We may also find bounds on the average of the total cross-section in the interval (s, 2s), The lower bound on A(s, t) and its convexity yield, Asymptotically we obtain a bound of the same form as before, but with the scale factor in the logarithm being s 0 /2, Note that σ tot (s) <σ tot (s, 2s) if the cross-section increases with s in the interval (s, 2s);the above bound on energy averages therefore immediately yields a bound on σ tot (s) in that case.
For identical particles there are only even partial waves in the partial wave expansions, but the lower bound on the absorptive part is again a convex function of the total cross-section; the identical particle factors multiplying the partial waves ensure that inspite of only even partial waves contributing, the largest partial wave L in the variational bound which is of O( sσ tot (s)) has only O(1) corrections with respect to the non-identical particles case.The quoted asymptotic bounds on the absorptive part in terms of σ tot and on the energy averaged total cross-section in terms of C x (t) therefore remain unchanged.
Phenomenological Bounds in terms of D-wave Scattering Length. Rigorous results from axiomatic field theory do not guarantee finiteness of the D-wave scattering lengths. However if we use phenomenological values for them we can choose ǫ = 0 and evaluate C(t = 4) . We shall use, F π + π 0 →π + π 0 = 1/2(F 1 + F 2 ), F π 0 π 0 →π 0 π 0 = 1 3 F 0 + 2 3 F 2 , the crossing relation, and the total crossing symmetry of the π 0 π 0 → π 0 π 0 amplitude. If we denote F (s, t) = G(t, s) = G(t; z t ) where F (s, t) denotes the π + π 0 → π + π 0 or the π 0 π 0 → π 0 π 0 amplitude , then the corresponding G(t, s) has only even partial waves, For 0 < t < 4, with the absorptive part F s (s, t) defined by Eq. (7), the fixed-t dispersion relations with two subtractions imply the Froissart-Gribov formula rigorously for l ≥ 2, (39) where Q l denotes the Legendre function of the second kind.The positivity of the absorptive part then implies the positivity of g l (t) for 0 < t < 4; further, If the t-channel D-wave scattering lengths exist,the definitions of C(t) and of the D-wave scattering lengths a I 2 for iso-spin I yield, and Here we have defined the l-wave scattering lengths a I l as the q → 0 limits of the phase shifts δ I l (q) divided by q 2l+1 where q is the c.m. momentum . Then an S-wave scattering length is indeed a length, with dimension m −1 π , and the D-wave scattering lengths have dimension m −5 π . Then, phenomenologically [19] we have, and Roy [14] has obtained from low energy data, for x = 50,σ π 0 π 0 tot (x) = 8.2 ± 4 mb;σ π + π 0 tot (x) = 17 ± 3.5 mb. (44) With ǫ = 0 ,t = 4 and the values of C x (t = 4) given in terms of the scattering lengths, and the low energy total cross-sections, we have, from Eqs. (31)- (33), with x = 50, where we have indicated the separate contributions of the D-wave scattering lengths and low energy total crosssections to C x (4) but have not indicated the (substantial) errors on them which imply corresponding errors on the scale factors. Our bounds on average total cross-sections for π + π 0 and π 0 π 0 scattering therefore do not contain any unknown constants but the scale factor s 0 has large phenomenological errors. We cure this problem in the next section at the cost of getting poorer bounds.
Absolute bounds on the D-wave below threshold for π 0 π 0 scattering. Although threshold behaviour cannot be proved from first principles, it was shown long ago [22] that |f l (s)| < C(4 − s) l−1 must hold for 0 < s < 4. We derive an absolute bound of this form and use it to derive a rigorous asymptotic bound on energy averaged total cross-section for π 0 π 0 scattering without unknown constants.As noted already, for 0 < s < 4 and l ≥ 2, the Froissart-Gribov formula implies that f l (s) > 0. Hence, for 0 < s < 4, 4 − s < t < 4 the convergent partial wave expansion, is in fact a sum of positive terms and yields an upper bound for the l ≥ 2 partial waves if we can obtain a bound on F (s, t) − F (s, 0) using analyticity. The twice subtracted fixed-t dispersion relations in s can be rewritten in terms of the convenient variable z ≡ ( and the positivity of the absorptive part then yields, If s 1 < s < 4 and z 1 ≡ (s 1 − 2 + t/2) 2 , then and hence for z 1 < z < z 0 , Inserting this into the dispersion relation we have,for 4 > t > 4 − s − s 1 , and t ≥ 0, We now use absolute bounds on pion-pion amplitudes first discovered by Martin [23], and improved successively by [4], [24] and [25] in the improved final form, The partial wave expansion of F (s, 2)−F (s, 0) now yields for 3 < s < 4, f l (s) ≤ 6.25 + 33.99 which implies in particular, With s replaced by t in this formula, the Froissart-Gribov formula now yields, Absolute bound on energy averaged total crosssection for π 0 π 0 scattering at high energy. Inserting the bound on C(t) into the average cross-section bound, the optimum value of t turns out to be t = 4 − (1/8 ln(s/s 0 )) −1 , and the optimum bound, Forσ tot (s, 2s) we obtain the same form of the bound ,but with half the value of s 0 . Outlook and Acknowledgements. Our basic bound on the absorptive part, Eq. (20), is valid at all energies and its energy integral may be used for comparisons with experimental cross-section data which have a large non-asymptotic contribution at current energies. We have highlighted the simpler asymptotic upper bounds on average total cross-sections .
We believe that our result is important as a matter of principle. However, we also believe that the magnitude of the coefficient in front of the Froissart bound is not satisfactory, especially if one decides to beiieve that the Froissart term is universal and compares with p-p and ppbar cross-sections at the ISR [26],at the SppbarS [27], at the Tevatron [28] and at the LHC [29]. All these indicate the existence of a Froissart like contribution with a much smaller coefficient, and a much larger scale and are well reproduced by, for instance, the BSW model [15] which incorporates automatically the Froissart behaviour. Returning to ππ scattering, can the situation be improved? Yes, because one has to enforce crossing symmetry and unitarity. Kupsch [30] has constructed a crossing symmetric model satisfying Eq.(8), but never tried to get numerical results. Also, we believe that unitarity in the elastic strips could be important. This led to the discovery by Gribov [31] that the behaviour sF(t) for the total amplitude is impossible. If you remove the elastic unitarity constraint [32] the Gribov theorem disappears. To attack the problem one could use a variational approach taking as an input the inelastic double spectral function in the Mandelstam representation. All we need is to find someone courageous not looking for a job.
Similar bounds on inelastic cross-sections without any unknown constants will be reported separately [33].