A Lower Bound on Inelasticity in Pion-Pion Scattering

Assuming that the pion-pion scattering amplitude and its absorptive part are analytic inside an ellipse in $t$- plane with foci $t=0$, $u=0$ and right extremity $t=4 m_{\pi}^2 +\epsilon $, ($\epsilon>0$), except for cuts prescribed by Mandelstam representation for $t\geq 4 m_{\pi}^2$, $u\geq 4 m_{\pi}^2$ , and bounded by $s^N$ on the boundary of this domain, we prove that for $s\rightarrow \infty$, \sigma_{inel} (s)>\frac{Const}{s^{5/2} }\exp {[-\frac{\sqrt{s}}{4} (N+5/2) \ln {s} ]}.

Assuming that the pion-pion scattering amplitude and its absorptive part are analytic inside an ellipse in tplane with foci t = 0, u = 0 and right extremity t = 4m 2 π + ǫ, (ǫ > 0), except for cuts prescribed by Mandelstam representation for t ≥ 4m 2 π , u ≥ 4m 2 π , and bounded by s N on the boundary of this domain, we prove that for s → ∞, It is well known that if there is no inelasticity, the scattering amplitude must be zero.However, there is no quantitative estimate of the amount of inelasticity required. This is what we try to do.There are various proofs of the fact that the scattering amplitude must be zero if there is no inelasticity. A very appealing attempt has been made by Cheung and Toll [1].Their idea is to use repeatedly elastic unitarity at all energies to the point where they get an absurd analyticity domain much too large.However, even after the enlargement of the pionpion analyticity domain by one of us in 1966 [2], it is not obvious that they have really succeeded.Alexander Dragt [3] has a proof which is nice but not quite complete: it uses the fact that partial wave amplitudes for very large angular momenta are dominated by the nearest singularities in the crossed channel. He needs more analyticity than what has been proved from field theory [2]. For instance, the Mandelstam representation [4] with a finite number of subtractions is largely sufficient. In fact he needs much less than that. Since we shall also use the dominance of the nearest singularities for large angular momenta, we state at the same time the assumption he needs and our assumption. If we use the standard mandelstam variables s, t, u , and choose units such that the * Electronic address: martina@mail.cern.ch † Electronic address: smroy@hbcse.tifr.res.in pion mass m π = 1, we need fixed energy analyticity in an ellipse with foci at t = 0 and u = 0 and right extremity at t = 4 + ǫ, minus the obvious cuts t ≥ 4, u ≥ 4 for the amplitude , and t ≥ 4 + 64/(s − 16), u ≥ 4 + 64/(s − 16) for the absorptive part (see Fig. 1). From field theory we only get ,for the absorptive part, an ellipse with right extremity at t = 4 exactly, and for the amplitude a region containing |t| < 4 . In fact for |t| < 4 fixed-t dispersion relations are valid, and with our assumptions they are valid for |t| < 4 + ǫ. With these assumptions we can prove that there must be inelasticity at energies such that s > 16 + 64/ǫ. For instance, if ǫ = 12 (corresponding to the full t-channel elastic strip), we must have inelasticity for s > 22.
For simplicity, we look first at π 0 π 0 scattering amplitude F (s, t) where π 0 is a fictitious iso-spin zero neutral pseudoscalar particle. It has the partial wave expansion, with the unitarity constraint The optical theorem gives, where F s (s, t) denotes the s-channel absorptive part ImF (s, t). Similar unitarity conditions hold in the t and u channels. The normalization specified by the above choice of ρ(s) corresponds to F (4, 0) = S-wave scattering length a 0 . For the generalization to real pions of iso-spin 1, we shall use the same normalizations as above, with F (s, t), f l (s), a l (s), σ tot , A s (s, 0), F (4, 0), a 0 being replaced respectively by the corresponding quantities with superscript I, e.g. F I (s, t), ., a I 0 . Our strategy will be the following.we write the partialwave amplitudes as well as their imaginary parts as contour integrals along the ellipse mentioned above, and add the contribution of the cuts (see fig. 1).Then we try to get an upper bound on the partial wave amplitude f l for which we need an upper bound B(s) on the whole ellipse. We also seek a lower bound on it's imaginary part Imf l , for which we need a bound on the discontinuity of the absorptive part which is nothing but the Mandelstam double spectral function. In fact this is what is missing in the work of Dragt [3].This will be done in the next section.
Domain of Positivity of the Double Spectral Function and a Lower Bound In the first part, we recall the results of Mahoux and one of us [5] on the domain of positivity of the double spectral function. For s > 20, the absorptive part in the s−channel has a cut beginning at From t = 4 to t = 4 + ǫ < 16, the discontinuity across the cut is given by the Mandelstam form of the t-channel elastic unitarity condition on one of the double spectral functions ρ st (s, t), where , and with , The domain of integration in the z 1 , z 2 plane is bounded by the three lines, If we define then, the region (9 ) becomes just a triangle in the θ 1 , θ 2 plane bounded by the lines,(see Fig.2) These inequalities imply that for i = 1, 2, They also imply that θ ≥ 2θ 0 which gives the boundary curve of the spectral region It will be crucial to recall the observation of Mahoux and Martin [5] that when θ ≤ 3θ 0 , the inequalities (11 ) imply that only values of θ i ≤ 2θ 0 for i = 1, 2 , i.e. only values of F s (s i , t) outside the spectral region for i = 1, 2 are needed to compute the double spectral function. In this region, the convergent partial wave expansion, the positivity of Imf l (s i ) and the inequalities P l (1 + 2t/(s i − 4)) > 1 imply that F s (s i , t) > 0 for i = 1, 2. Hence,the double spectral function ρ st (s, t) is positive when θ ≤ 3θ 0 , i.e. for Since ρ(s, t) is symmetrical in its arguments, it is also positive for, II. Lower bound on inelasticity .
We shall now obtain a lower bound on ρ(s, t) in the domain (15) in terms of total cross sections σ tot (s 1 ), σ tot (s 2 ), where s 1 , s 2 are such that Eqn. (12) holds for the corresponding z 1 , z 2 . We then deduce a lower bound on inelasticity . It will then follow that if there is no inelasticity at one (and only one) energy in the s-channel (s > 20), the double spectral function must vanish in the range t = 4 + 64/(s − 16) to t = 4 + 32/( √ s − 6) , and hence that there is an interval of energy given by (12) in which the total cross section vanishes. This is impossible and hence the scattering amplitude is zero.It must be realized that only a small fraction of Mandelstam representation is used. Now, the question which was asked to one of us (AM) by Miguel F. Paulos, during a conference organized by João Penedones at EPFL , Lausanne was, if the inelastic cross section could be arbitrarily small. We want to show that, with some assumptions much weaker than the Mandelstam representation, but slightly stronger than what has been proved from local field theory, there exists a lower bound to inelasticity , The strategy we shall use is based on the results of Mahoux and Martin [5] on positivity of double spectral functions, and on the research made by Dragt [3], viz. that the real and imaginary parts of the partial wave amplitudes are dominated by the contributions of the nearby cuts in the crossed channel: where, Estimates of f l (s) and Imf l (s) We shall use a truncated Froissart-Gribov representation for Ref l (s) and Imf l (s). It follows from analyticity of F (s, t) in t within an ellipse with right extremity t = t M (s) and foci t = 0 and u = 0, except for cuts 4 ≤ t ≤ t M (s) and 4 ≤ u ≤ t M (s) .For l even, where Γ is an ellipse with foci at t = 0 and u = 0, and right extremity at t = 4 + 32 √ s−6 (see figure 1) . Hence, where ρ(s, t) is given by the Mandelstam equation (5).
As noted earlier, if s is in the Mahoux-Martin domain (15), ρ(s, t) is positive. Now we postulate that F (s, t) and F s (s, t) are bounded by B(s) in the ellipse Γ.The behaviour of B(s) for s → ∞ will be discussed later.Now we need some estimates on the Q l 's. We prove that, for z real and > 1, (see Appendix) and for z = cosh((θ 1 + iθ 2 )), (see Appendix), This means that on an ellipse with foci cosθ = ±1 the modulus of Q l cosθ) is maximum at the right extremity. We can get a bound on |f l | where L(s) is the perimeter of the ellipse with extremities at plus 4 times the length of the cuts t = 4 to t = 4 + 32 For s > 16, We need now a lower bound for Imf l (s). Imf l (s) is given by a contour integral including the contribution from the cuts and the ellipse. We use the fact that Q l () is a decreasing function for an argument > 1. We limit arbitrarily the integration on the cuts to where , which is certainly valid for sufficiently large s. A lower bound on Imf l is given by Notice that ρ(s, t) according to [5] is strictly positive, given by the double integral of Mandelstam in the strip 4 < t < 4 + 32/( √ s − 6). Now,given B(s), L(s) and ρ(s, t) it is possible to prove that |f l | 2 is strictly less than Imf l for l sufficiently large. We have and so where we define, ), It is convenient to denote, Note that , x 2 < x 1 , and for s sufficiently large, We now obtain bounds on the relevant Legendre functions. Using the results (90) and (96) in the Appendix, we have, Further Equations (96) and (81) in the Appendix yield, and Eqn (96) gives, We now have, without asymptotic approximations. For s → ∞, It is clear that since R 2 > 1, for l large enough, i.e. for l > L 0 (s) = Const.s ln s, s → ∞, the contribution of the first term on the right-hand side of Eqn. (37) involving a positive double spectral function is dominant, and that term implies that Imf l |f l | 2 → ∞, l > Const.s ln s .
Hence the inelastic cross section is dominant and nonzero for l > L 0 (s) . The fact that ρ(s, t) is different from zero is essential. We now evaluate the lower bound on Imf l , and hence on the inelastic cross section at high energies.
III. Lower bound on the double spectral function .
We must get a lower bound on ρ(s, t).This is relatively easy. We return to the Mandelstam equation (5 ) for 4 < t < 16 and restrict ourselves to the Mahoux-Martin domain (15) of positivity of ρ(s, t). To get a lower bound on ρ(s, t) we shall do rather wild majorizations.
1) We reduce the domain of intgration in the θ 1 , θ 2 plane (11)to the union of three regions A, B, C (see figure  2) Notice that under z 1 ↔ z 2 , the regions B ↔ C and A ↔ A.
2) Using Eqns. (7), (8), we shall replace H(z, z 1 , z 2 ) in the denominator by simple upper bounds on it in the three regions: It will be convenient to define, 3) Since we are in the Mahoux-Martin domain in which F s (s 1 , t) and F s (s 2 , t) have convergent partial wave expansions with positive partial waves, and t is positive, the absorptive parts obey the bounds, They also obey stronger bounds in terms of σ tot (s i ) , originally derived by Martin [2] or 0 < t < 4, but also valid for 4 < t < 4+ 32 √ s−6 under the present assumptions. At high energies they have the simple form, Using the majorizations 1), 2) and the weaker bound (42) in 3), we obtain where the first term in the braces on the right is the contribution of region A and the second term of regions B and C , and I(s 1M ), I(s 2M ) are defined similarly by replacing s M by s 1M and s 2M respectively. Note that s M ,s 1M and s 2M depend on s, t. E.g.
A simple bound is obtained by retaining only the region A. In addition to the above results for general P (s), we shall evaluate bounds on I(s M ), ρ(s, t) and the integral over t of ρ(s, t), for two simple choices of P (s).
(i) P(s) independent of s. Let P 1 < p < P 2 then we can get a lower bound on the integral over t of ρ(s, t) by restricting to the interval Then, For fixed s large enough, s M is an increasing function of t, and hence it's minimum value is at the lowest value Finally we have the bound, which is positive definite and > Const.s −3/2 unless the total cross section vanishes identically at all energies upto (s M )min.
(ii) P (s) → 0 for s → ∞. Then, we integrate over the region, 4 + (64 + p 1 (s)) (s − 16) < t = 4 + (64 + p(s)) (s − 16) where p 1 (s) and p 2 (s) → 0, for s → ∞, and we get s M − 4 ∼ p(s)/32 → 0. In the integral defining I(s M ) we can therefore replace where a 0 is the S− wave scattering length, and obtain Finally we obtain for s → ∞, p 1 (s) and p 2 (s) → 0, as slowly as we like, This bound is of interest as it shows that the asymptotic inelastic cross section cannot vanish if the S−wave scattering length is non-zero. However, the bound (50) is preferable as it does not need any asymptotic approximation.
IV. Asymptotic behaviour of the lower bound on inelastic cross section and discussion of the assumptions. Now we know that ,above a certain energy, the inelastic cross section cannot be zero.A lower bound can be obtained if we know something about B(s) and if we accept the postulated analyticity.If we believe in the validity of the Mandelstam representation with a finite number of subtractions, B(s) = s N . In fact we tend to believe that B(s) = s 2 /s 2 0 , because we postulate an ellipse (with cuts) which in the limit of high energy coincides with the ellipse with foci t = 0, u = 0 and extremities t = 4, u = 4.Inside this ellipse the absorptive part F s (s, t) is maximum for t real, 0 < t < 4, and the integral which means that F s (s, t) is almost everywhere less than s 2 . Concerning the dispersive part which is, modulo subtractions, the Hilbert transform of the absorptive part we have a rather tricky argument to show again that it is almost everywhere bounded by s 2+ǫ , ǫ arbitrarily small , for any t for which dispersion relations are valid.But we shall not use that result here. Using the lower bound on the integral of the double spectral function, and B(s) = s N ,we deduce that the ratio of the contributions of the cut term and the elliptical contour (Γ) term to Imf l goes to infinity if l > L 0 (s) = (N + 5/2) 16 s ln s.
The ratio of the contribution of the cut term to Imf l to the upper bound on |f l | 2 goes to infinity for a much smaller value, viz. if Hence, summing the contributions of partial waves with l > L 0 (s) we see that for s → ∞, V. Real Pions of Isotopic Spin 1. Let F (I) (s, t, u) denote the ππ → ππ amplitudes with total iso-spin I in the s-channel, I = 0, 1, 2, and F (I) (t, s, u) the amplitudes with iso-spin I in the t-channel. They are related by the crossing matrix C st , We do not assume the unsubtracted Mandelstam representation, However, we use the definitions  st (s, t), the superscripts I, I ′ denote iso-spins in the channel specified by the first argument, viz. t-channel and s-channel respectively. The Mandelstam unitarity equations for t-channel Iso-spin I, and 4 ≤ t ≤ 16 is given by Mahoux and Martin [5], s 1 , s 2 ), Crossing, Eq. (59) ,immediately yields where, which are identical to the values obtained in [5], and quoted again for ready reference. We now have, Mahoux and Martin [5] have noted that all the matrix elements of ζ 0 , ζ 0 − ζ 2 , ζ 0 + ζ 1 , ζ 0 − 2ζ 1 , and ζ 0 + 2ζ 2 (68) Hence, New results. The truncated Froissart-Gribov formula will enable us to obtain lower bounds on imaginary parts of s-channel partial waves of the following five amplitudes: 1 3 F (0) + F (1) + 5 3 F (2) (s, t) = F (0) (t, s); where the coefficients α I and β I can be read off the Equations (70) .E.g. α 0 = β 0 = 1/3, α 2 = β 2 = 2/3, α 1 = β 1 = 0 for the last amplitude which is just the π 0 π 0 → π 0 π 0 amplitude, F 00 ≡ 1 3 F (0) + 2F (2) .
The partial waves given by the truncated Froissart-Gribov formula are then, for even l + I, As before, Γ is an ellipse with foci at t = 0 and u = 0, and right extremity at t = 4+ 32 √ s−6 . As for pions without iso-spin, we prove, if we only use the region A in Fig. (2) ,that the combinations I β I ρ (I) (t, s) on the right-hand side are not ony positive, but also have a lower bound, provided that I β I ζ I I ′ ,I ′′ > 0 , for all I ′ , I ′′ , and tot (s 1 ) 16π .