Froissart Bound on Inelastic Cross Section Without Unknown Constants

Assuming that axiomatic local field theory results hold for hadron scattering, Andr\'e Martin and S. M. Roy recently obtained absolute bounds on the D-wave below threshold for pion-pion scattering and thereby determined the scale of the logarithm in the Froissart bound on total cross sections in terms of pion mass only. Previously, Martin proved a rigorous upper bound on the inelastic cross-section $\sigma_{inel}$ which is one-fourth of the corresponding upper bound on $\sigma_{tot}$, and Wu, Martin,Roy and Singh improved the bound by adding the constraint of a given $\sigma_{tot}$. Here we use unitarity and analyticity to determine, without any high energy approximation, upper bounds on energy averaged inelastic cross sections in terms of low energy data in the crossed channel. These are Froissart-type bounds without any unknown coefficient or unknown scale factors and can be tested experimentally. Alternatively, their asymptotic forms,together with the Martin-Roy absolute bounds on pion-pion D-waves below threshold, yield absolute bounds on energy-averaged inelastic cross sections. E.g. for $\pi^0 \pi^0$ scattering, defining $\sigma_{inel}=\sigma_{tot} -\big (\sigma^{\pi^0 \pi^0 \rightarrow \pi^0 \pi^0} + \sigma^{\pi^0 \pi^0 \rightarrow \pi^+ \pi^-} \big )$,we show that for c.m. energy $\sqrt{s}\rightarrow \infty $, $\bar{\sigma}_{inel }(s,\infty)\equiv s\int_{s} ^{\infty } ds'\sigma_{inel }(s')/s'^2 \leq (\pi /4) (m_{\pi })^{-2} [\ln (s/s_1)+(1/2)\ln \ln (s/s_1) +1]^2$ where $1/s_1= 34\pi \sqrt{2\pi }\>m_{\pi }^{-2} $ . This bound is asymptotically one-fourth of the corresponding Martin-Roy bound on the total cross section, and the scale factor $s_1$ is one-fourth of the scale factor in the total cross section bound. The average over the interval (s,2s) of the inelastic $\pi^0 \pi^0 $cross section has a bound of the same form with $1/s_1$ replaced by $1/s_2=2/s_1 $.

Assuming that axiomatic local field theory results hold for hadron scattering, André Martin and S. M. Roy recently obtained absolute bounds on the D-wave below threshold for pion-pion scattering and thereby determined the scale of the logarithm in the Froissart bound on total cross sections in terms of pion mass only. Previously, Martin proved a rigorous upper bound on the inelastic crosssection σ inel which is one-fourth of the corresponding upper bound on σtot, and Wu, Martin,Roy and Singh improved the bound by adding the constraint of a given σtot .
Here we use unitarity and analyticity to determine, without any high energy approximation, upper bounds on energy-averaged inelastic cross sections in terms of low energy data in the crossed channel. These are Froissart-type bounds without any unknown coefficient or unknown scale factors and can be tested experimentally. Alternatively, their asymptotic forms ,together with the Martin-Roy absolute bounds on pion-pion D-waves below threshold, yield absolute bounds on energy-averaged inelastic cross sections. E.g. for π 0 π 0 scattering , defining σ inel = σtot − σ π 0 π 0 →π 0 π 0 + σ π 0 π 0 →π + π − ,we show that for c.m. energy √ s → ∞,σ inel (s, ∞) ≡ s ∞ s ds ′ σ inel (s ′ )/s ′2 ≤ (π/4)(mπ) −2 [ln(s/s1) + (1/2) ln ln(s/s1) + 1] 2 where 1/s1 = 34π √ 2π m −2 π . This bound is asymptotically one-fourth of the corresponding Martin-Roy bound on the total cross section , and the scale factor s1 is one-fourth of the scale factor in the total cross section bound. The average over the interval (s,2s) of the inelastic π 0 π 0 cross section has a bound of the same form with 1/s1 replaced by 1/s2 = 2/s1.  [1] have obtained bounds on energy averages of the total cross-section without any unknown constants such as an overall constant factor or the scale factor in the logarithm. The purpose of the present work is to obtain analogous bounds on the energy averaged inelastic cross section without any unknown constants.The background and the basic postulates are again summarized below to make this work self-contained.
The Froissart [2] bound on the total cross-section σ tot (s) for two particles at c.m. energy √ s , (where C, s 0 are unknown constants ) was initially proved assuming the Mandelstam representation . This assumption might not be valid , for example , if there are rising when s → ∞. For many processes, for example for ππ, KK, KK, πK, πN, πΛ scattering it is known [10] that t 0 = 4m 2 π ,, m π being the pion-mass . (Except when especially necessary to show the dependence on pion-mass, we shall choose units m π = 1). Using unitarity and validity of dispersion relations with a finite number of subtractions for −T < t ≤ 0 , Jin and Martin [11]) proved twice subtracted fixed-t dispersion relations for |t| < t 0 . From this Lukaszuk and Martin [12] fixed the constant C in the Froissart bound to obtain, where, where ǫ is an arbitrarily small positive constant. The Froissart-Martin bounds have inspired much work on high energy theorems (see e.g. [13], [14]) and on models of high energy scattering [15]. Further, Martin proved a bound on the total inelastic cross section σ inel (s) at high energy [16] which is one-fourth of the above bound σ max (s) on the total cross-section; and, Wu, Martin, Roy and Singh [17] obtained a bound on σ inel (s) which improves that bound if σ tot (s) is known, The motivation for getting a bound on the inelastic cross section is the almost general belief [15] that at high energies hadron total cross sections cannot exceed twice the inelastic cross section. Hence, gaining a factor of 4 in the inelastic cross section gains a factor of 2 in the total cross section. One shortcoming of these bounds from the standpoint of rigour is that [18] they are deduced by assuming that the absorptive part A(s, t), 0 ≤ t < t 0 is bounded by Const.s 2 / ln(s/s 0 ) for s → ∞, whereas axiomatic field theory results only guarantee that where s th is the s-channel threshold. From the practical point of view a more serious shortcoming is that they involve the unknown scale factor s 0 in the argument of the logarithm and the unknown arbitrarily small but nonzero constant ǫ.
We prove here analogous bounds on energy averages of the inelastic cross section. We choose the same normalizations as in Martin-Roy [1]. F (s, t) denotes 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 dσ/dΩ(s, t) is given by 4F (s, t)/ √ s 2 , 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) has the convergent partial wave expansion, with the unitarity constraint, Correspondingly, the optical theorem gives, for nonidentical partcles a, b, For identical particles ,e.g. for π 0 π 0 scattering, or for pion-pion scattering with Iso-spin I, the partial waves a l (s) → 2a I l (s) in the partial wave expansion,i.e.
and we have the same formula for the differential crosssection in terms of F (s, t), and the same form of the unitarity constraint,Ima I l (s) ≥ |a I l (s)| 2 , s ≥ 4 , as for non-identical particles. At threshold, F I (4, 0) = a I 0 , the S-wave scattering length for Iso-spin I.
It will be seen that proofs of the bounds for inelastic cross sections are considerably more involved than those for total cross sections, but the basic principles are the same. We give detailed derivations for the case of nonidentical particles a = b, and also quote the identical particle results when needed.

II. Convexity Properties of Lower Bound on
Absorptive Part in terms of Total Inelastic Cross Section. We obtain a lower bound on the absorptive part of F (s, t) (for s ≥ s th and 0 ≤ t < t 0 ) in terms of the inelastic cross section σ inel (s). Following [20] we prove that the bound is a convex function of the inelastic cross section. The absorptive part has the partial wave expansion, The corresponding expansion of the inelastic cross section is, Actually we shall vary Ima l (s) subject to the positivity restrictions (due to unitarity), to minimise A(s, t), given , (17) The bound will be seen to be an increasing function of σ inel,im (s). Further, therefore the bound will still hold when we replace σ inel,im (s) by the experimentally accessible σ inel (s). We work at a fixed s; so, unless specially needed, we shall suppress the s-dependence of Ima l (s), σ inel,im (s) and σ inel (s). The Lagrange multiplier method with positivity constraints on partial waves gives the variational solution (Ima l ) 0 , (19) where the integer L and the non-negative fraction λ − L, are to be determined so as to reproduce the given σ inel,im ; here P λ (z) for non-integer λ and z ≥ 1 is defined by, If A 0 (s, t) denotes the absorptive part with partial waves (Ima l ) 0 and A(s, t) that with arbitrary positive partial waves with the given σ inel,im (s) ,we obtain by direct subtraction, The last inequality follows because for z ≥ 1, and λ ≥ 0 ,P λ (z) is an increasing function of λ. We then have, where, Note that Σ I (λ) and A(λ) are monotonically increasing continuous functions of λ; hence λ and A(λ) may be considered functions of Σ I , and which is always positive, and also continuous at integer λ although (dA/dλ) and (dΣ I /dλ) are discontinuous there. Hence, which is discontinuous at integer λ, but always positive. This completes the proof that A(λ(Σ I )) is a convex function of Σ I ; i.e. at a given s the lower bound on A(s, t) is an increasing and convex fuction of σ inel.im , and hence of σ inel . III. Explicit evaluation of the bound. Explicitly, where λ ′ , L ′ corresponds to the value Σ ′ we get L ′ = 0 and the corresponding part of the integral can be evaluated exactly. Hence, In the remaining integral L ′ ≥ 1 and we shall prove that for Σ ′ I ≥ Σ I (1), Proof. From the partial wave expansion for Σ I (λ ′ ) ≡ Σ ′ I , we obtain The integral representation for P λ (z) given before yields, for z > 1, where the last inequality uses exp(−x) ≥ 1 − x. At high energies ln(z + √ z 2 − 1 ) goes to zero, and we assume moderately high energies (k > 6m π ) such that, with t < 4m 2 π , Then, which completes the proof.
Since P λ ′ (z) is an increasing function of λ ′ , we obtain, Using the integral representation for P λ (z) given before and the analogous representation for the modified Bessel function , and the elementary inequality ln((z + cos φ z 2 − 1) ≥ (cos φ) ln((z + z 2 − 1), z > 1 we obtain [20], Substituting the above inequalities ,the integral over Σ ′ I in the expression for the lower bound can be evaluated exactly, and we have the exact result (without any high energy approximation), u ≡ ln((z + z 2 − 1)) k 2 σ inel /π; u 1 ≡ ln((z + z 2 − 1)) 1 − z −2 , (38) and the slightly weaker but simpler result, Note that at high energies z − 1 → 0, the last two terms give only a small positive contribution, Hence, at sufficiently high energies, but without any high energy approximation, we have the bound given by Eqs.
(37)-(38) and the slightly weaker but simpler bound, IV. Bound on energy averaged inelastic cross section.
Multiplying by s −3 and integrating over s, we obtain a lower bound on C (s1,s2) (t) which is the contribution from s 1 to s 2 to C(t), where s 2 > s 1 , and we used 2k ′ / √ s ′ > (1 − 4/s 1 ), and 2z 2 1 /(z 1 + 1) ≥ 2z 2 /(z + 1) ≥ 1 for s ′ in the interval (s 1 , s 2 ), and The lower bound on C(s 1 , s 2 )(t) is an average with the normalized weight function of an integrand which is a convex function of σ inel (s ′ ) .The convexity is readily proved; using (xI 1 (x)) ′ = xI 0 (x), and denoting, we have, Since the right-hand side is an increasing function of σ inel (s ′ ) ,we get the convexity property, Since the average of a convex function is greater than the convex function of the average [21] ,we have the bound where, To get bounds onσ inel (s, ∞) , andσ inel (s, 2s), we just choose the corresponding values for (s 1 , s 2 ).Thus we obtain, without any asymptotic approximations in s, and, where Asymptotic bounds Since we want asymptotic upper bounds on the energy averaged inelastic cross sections, we can assume without loss of generality that the arguments x 1 , x 2 of the modified Bessel functions tend to infinity , and obtain, and 32stC (s,2s) (t) √ 2π > ξ 2 exp ξ 2 (1 + O(1/ξ 2 )), (58) We now use the elementary lemma proved in [20], [1] , Lemma. If ξ > 1, and y ≥ √ ξ exp ξ, then, ξ < f (y) ≡ ln y − (1/2) ln ln y − 1 2 ln ln y .
With f (y) as defined above, we obtain the asymptotic bounds,σ where, and,σ inel (s, 2s) ≤ s→∞ Notice that the coefficients of (ln s) 2 in these bounds on the inelastic cross section are one-fourth of those in the corresponding bounds on the total cross section at high energies , and the scale factors in the inelastic case are also one-fourth of those in the corresponding total cross section bounds [1], In the case of pion-pion scattering, we may remove the unknown ǫ in t = 4m 2 π − ǫ rigorously by using absolute bounds on the D-wave below threshold derived in [1], or use phenological inputs on the D-wave scattering length and set ǫ = 0. The main qualitative difference from the case of non-identical particles is that only even partial waves occur in π 0 π 0 scattering, We first show that inspite of this difference, the bounds of this section at moderate energies as well as the asymptotic bounds on inelastic cross sections hold for π 0 π 0 scattering. V. Bounds on pion-pion inelastic cross sections. We shall exploit iso-spin invariance.
(i) First, the basic lower bound (from unitarity alone) on the absorptive part A(s, t) in terms of the inelastic cross section, given by Eq. (41), or in terms of the total cross section (Eq. (21) of [1] ) can be compared directly with phenomenological estimates of the absorptive part at energies where such estimates are available [26], [27]. This can be done for the amplitudes F π + π 0 →π + π 0 (s,t) = 1/2(F 1 + F 2 )(s, t), which have positive absorptive parts for s ≥ 4, 0 < t < 4. A violation of the bounds will indicate that the input absorptive part violates unitarity.
(iii) Thirdly, explicit asymptotic bounds on the averages of the inelastic cross section in the intervals (s, ∞) and (s, 2s) are given by Eqs. (61) and (63) , and in terms of the corresponding averages of the total cross section , in terms of a scale parameter s 0 ; s 0 is given by Eq. (62) in terms of C (s,∞) , an integral over absorptive parts in the interval (s, ∞) . Substituting the values of the D-wave scattering lengths given by [19], we have, choosing for s a value up to which absorptive parts can be reliably estimated, These equations give much stronger bounds than the absolute bounds. E.g. using only positivity of C π 0 π 0 →π 0 π 0 (4,s) (t = 4) we get, which is 800 times the absolute bound s 0 ≥ .015m 2 π . As the absorptive part integrals in the D-wave scattering length sum rules are rapidly convergent, even for the moderate value of √ s = 1, 6GeV , Colangelo et al [28] obtained a further big improvement in the values of the scale factor, when phenomenological values of absorptive parts upto √ s = 1, 6GeV are utilised , which are not very far from the scale factors used in phenomenological fits [27]. We should remember that the phenomenological values may be dependent on the particular parametrisation used to fit experimental cross sections. The implicit bounds (51)-(55) discussed in (i) and (ii) are without asymptotic approximations, and therefore can be compared directly with experiment. VIII. Concluding Remarks.
In this paper on inelastic cross sections and the previous one on total cross sections [1] we believe to have put the Froissart Bound on a solid ground, by using the notion of average cross sections which avoids completely the problem of the scale in the Froissart bound. These averages can be chosen rather arbitrarily but once you have chosen one you must stick to it. The simplest averages that we use are the ones from s to infinity and from s to 2s. The averaging interval must be sufficiently large if one wants to preserve the coefficients appearing in the Lukaszuk-Martin bound. The only unknown is the value of a certain integral on the absoptive part for some positive t. In the special case of pion-pion scattering all unknown constants are eliminated. The advantage of introducing the bound on the inelastic cross section is that, asymptotically, it is 4 times smaller than the one on the total cross section. So if you accept to believe that the elastic cross-section cannot be larger than the inelastic cross section, the limiting case being an expanding black disk you gain a factor 2 on the bound on the total cross section. However, not everybody agrees with this, for instance Troshin and Tyurin [29] believe that at high enrergy the scattering amplitude is dominantly elastic. It is tempting to make a rather daring and nonrigorous suggestion: if the amplitude is essentially elastic ( a small inelastic part is unavoidable according to well known theorems) then the effective large Lehmann ellipse has a right extremity at t = 16m 2 π , and the Froissart bound is divided by 4.
Anyway a factor of 2 or 4 is not sufficient to bring the absolute bounds near the experimental values [30], [31], [32] including the most recent experiments at LHC [33] , which indicate a definite increase of the crosssections compatible with a (ln(s)) 2 behaviour. There is little doubt that this trend will continue when LHC reaches higher energies.Towards quantitative improve-ment we may find unitarity bounds on the energy averages of the inelastic cross section given the total cross section as an input , in addition to absorptive part integrals at positive t [34].
However , as explained in Sec. VII above,if we are prepared to make phenomenological inputs such as the Dwave scattering lengths and low energy absorptive parts , the situation with respect to experimental comparisons improves dramatically [27].
What can we do on the theoretical side? In the case of pion pion scattering, Kupsch [35] has constructed an amplitude , crossing-symmetric satisfying "inelastic" unitarity and saturating the Froissart bound [35] ,but he does not give numbers. The result of Gribov [36] shows the importance of satisfying elastic unitarity in the "elastic strips" , [37]. This might help, but we dont know how, and there is the problem of finding people interested in working on this.