PHYSICAL REVIEW D 97, 014011 (2018)

Positivity of the real part of the forward scattering amplitude

Andr´e Martin* CERN, 1211 , Switzerland

Tai Tsun Wu Gordon McKay Laboratory, , 02138 Cambridge, Massachusetts, USA

(Received 5 September 2017; published 23 January 2018; corrected 26 January 2018)

We prove the general theorem that the real part of the crossing even forward two-body scattering amplitude is positive at sufficiently high energies if, above a certain energy, the total cross section increases monotonically to infinity at infinite energy.

DOI: 10.1103/PhysRevD.97.014011

I. HISTORICAL INTRODUCTION The perturbation character disappears in the summation. From the point of view of physics, the advantage of using In 1965, Khuri and Kinoshita published a series of is that the following three basic interesting papers on the real part of the forward scattering features are all satisfied: (1) relativistic kinematics, (2) uni- amplitude obtained via dispersion relations from the tarity, and (3) particle production [6]. The result found this imaginary part which is proportional to the total cross way in 1970 was a surprise: the total cross section must section [1]. Among their results was the prediction that, if increase at high energies, essentially saturating the Froissart- the Froissart bound is saturated, i.e., if the total cross 2 Martin bound, and the real part of the forward scattering section behaves at high energies like ðln sÞ where s is the amplitude does have the Khuri-Kinoshita behavior. At that square of the center-of-mass energy [2], then the real part is time, the measured proton-proton total cross section was still positive at high energies. Moreover, the ratio ρ of the real decreasing, while the real part of the forward scattering π part to the imaginary part would behave as ln s at high amplitude was still negative but increasing rapidly [7]. Three energies. At that time, nobody thought that the total cross years later, two experiments at the Intersecting Storage Ring section would increase at high energies. Instead, the general (ISR) at CERN showed that the proton-proton total cross belief was that the Froissart-Martin bound is an upper section did turn around and start rising [8]. At CERN, after a bound and that all total cross sections would approach talk by Amaldi on these experimental results, one of us (AM) constants or decrease to zero at high energies. Earlier, in “predicted” that the real part would become positive. 1960, Gribov showed that it is not possible to have at the Actually it was a guess at that time; a rigorous proof is to same time a total cross section approaching a finite nonzero be presented in the present paper. A few years later, in 1977, limit and a diffraction peak approaching a fixed shape as a an experiment at CERN showed that the real part was indeed function of the momentum transfer [3]. Instead, he pre- becoming positive [9]. Years later, the rise of the total cross ferred the total cross section to decrease to zero. In 1962, it section and the positively of the real part were both was discovered that the diffraction peak for proton-proton confirmed [10,11]. Using the shrinking of the width of the diffraction peak scattering was shrinking [4]. and the rise of the total cross section, one of us (AM) In view of this situation, Cheng and one of us (TTW) proved in 1997 that, if the differential cross section at fixed decided to learn about the high-energy behavior of total negative t, for −T

2470-0010=2018=97(1)=014011(4) 014011-1 Published by the American Physical Society ANDRE´ MARTIN and TAI TSUN WU PHYS. REV. D 97, 014011 (2018)

II. THEOREM ON THE REAL The proof begins with the dispersion relation for fðξÞ PART AND ITS PROOF Z 2ξ2 ∞ dξ0 lm fðξÞ fðξÞ − fð0Þ¼ ð Þ The measurement of the real part of the scattering Re π ξ0 ξ02 − ξ2 5 amplitude in the forward direction is of great importance. μ

In the cases of proton-proton and antiproton-proton scatter- where μ ¼ 2mamb.fð0Þ is real, since fðξÞ is real between ings, when the measured total cross sections were still the left hand cut and the right hand cut i.e. −μ < ξ < þμ. decreasing as function of energy, the first indication that Let ξo be the value of ξ such that this total cross section would turn around and increase came from the measurement of the real parts [5]. dσ ≥ 0 ξ > ξ : ð Þ Specifically, at that time the measured values were negative dξ for 0 6 but becoming less so when the center-of-mass energy increased, as expected. However, the increase (i.e., less The dispersion relation (5) can be rewritten as negative) was too fast and showed a tendency to overshoot 2 Z 2ξ ξ0 1 to become positive. Indeed, this was the first indication that fðξÞ − fð0Þ − dξ0σðξ0Þ Re 02 2 the proton-proton total cross section, among others, would π μ ξ − ξ Z increase. As already mentioned, the theoretical prediction 2ξ2 ∞ 1 of increasing total proton-proton cross section was first ¼ dξ0σðξ0Þ ð Þ π ξ02 − ξ2 7 made in 1970 and the experimental observation in 1973. ξ0 It is the purpose of the present paper to study in general In the left-hand side (LHS) of (7), for ξ → ∞ the sign of the real part of the forward scattering amplitude. 2 Z Z The specific problem is: under what general and realistic 2ξ ξ0 1 2 ξ0 dξ0σðξ0Þ ∼− dξ0σðξ0Þ; ð Þ conditions on the total cross section, the real part of the 02 2 8 π μ ξ − ξ π μ forward two-body scattering amplitude can be guaranteed to be positive at sufficiently high energies? and is therefore bounded in absolute value. Clearly, this problem should be studied through Integrating the right-hand side (RHS) of (7) by parts dispersion relations. Consider the elastic scattering process leads to Z a þ b → a þ b; ð1Þ ξ ∞ ∂ ξ0 − ξ ð7Þ¼ dξ0 σðξ0Þ RHS of π ∂ξ0 ln ξ0 þ ξ let s, t, u be the Mandelstam variables. Throughout this ξ0 paper, t ¼ 0 so that the scattering is in the forward ξ ξ0 þ ξ ¼ σðξ0Þ ln þ IðξÞ; ð9Þ direction. In this case, π ξ0 − ξ 2 2 s þ u ¼ 2ma þ 2mb ð2Þ where

Z 0 0 where ma and mb are the masses of the particles a and b. ξ ∞ dσðξ Þ ξ þ ξ IðξÞ¼ dξ0 : ð Þ Thus, when t ¼ 0, the s − u symmetric variable is π dξ0 ln ξ0 − ξ 10 ξ0 1 ξ ¼ ðs − uÞ¼s − m2 − m2: ð Þ Note that the first term on the right-hand side of (9) is 2 a b 3 again bounded. This scattering amplitudes for a þ b → a þ b and a þ b¯ → a þ b¯ fðξÞ dσðξÞ are in general different. Define to If > 0 for ξ > ξ0 ð11Þ be the average of the forward scattering amplitude for these dξ two processes and σðξÞ that of the total cross sections, then IðξÞ is positive. pffiffiffi k s It only remains to show that this IðξÞ increases without fðξÞ¼ σðξÞðÞ ξ → ∞ Im 8π 4 bound for when when k is the c.m. momentum i.e. for ξ large σðξÞ → ∞ ð12Þ

ξσðξÞ It follows from Im f ≃ 16π ξ0 þ ξ ξ þ ξ 2ξ The main result of the present paper is: 0 0 ln 0 ≥ ln ≥ ð13Þ Theorem: If, for sufficiently large values of ξ, σðξÞ is ξ − ξ ξ − ξ0 ξ nondecreasing and approaches infinity as ξ → ∞. Then Re fðξÞ is positive for all sufficiently large values of ξ. when ξ > ξ0 that

014011-2 POSITIVITY OF THE REAL PART OF THE FORWARD … PHYS. REV. D 97, 014011 (2018) Z ξ ξ dσðξ0Þ ξ0 þ ξ IðξÞ ≥ dξ0 The improved version of (15) is then π dξ0 ln ξ0 − ξ Z ξ0 2 ξ Z 2ξ 0 0 0 1 ξ 0 fðξÞ ≥ fð0Þþ dξ σðξ Þ ξ dσðξ Þ 2ξ0 Re 02 2 ≥ dξ0 π μ ξ − ξ π dξ0 ξ ξ0 ξ ξ þ ξ 2ξ − σðξ Þ 0 þmax 1 ½σðξÞ − σðξ Þ: 2ξ0 0 ln ξ0≤ξ1≤ξ 1 ¼ ½σðξÞ − σðξ Þ: ð Þ π ξ0 − ξ π π 0 14 ð19Þ This proves that, because of (12), IðξÞ increases without If, for example, σðξÞ saturates the Froissart-Martin bound, bound as ξ → ∞. then this maximum over ξ0 ≤ ξ1 ≤ ξ is reached when The result is therefore, with the conditions (11) and (12), ξ1 ∼ξ=e.

2 Z In some cases, (19) leads to a considerable improvement 2ξ ξ0 1 fðξÞ ≥ fð0Þþ dξ0σðξ0Þ over (15). As an example, if for large ξ Re 02−ξ2 π μ ξ σðξÞ ∼ cðln ξÞγ ð20Þ ξ ξ þ ξ 2ξ − σðξ Þ 0 þ 0 ½σðξÞ − σðξ Þ: ð Þ 0 ln 0 15 with 0 < γ ≤ 2, then (15) gives π ξ0 − ξ π 2 fðξÞ > þ cξ ð ξÞγ; ð Þ 2ξ0 Re const 0 ln 21 On the right-hand side of this (15), the term π σðξÞ π approaches infinity as ξ → ∞, while all the other terms are while (19) gives bounded. Therefore 2γ fðξÞ > þ cξð ξÞγ−1: ð Þ Re const πe ln 22 Re fðξÞ > 0 ð16Þ The lower bound (22) is much stronger than that of (21). for all sufficiently large values of ξ. In fact, this bound (22) differs from the exact asymptotic This proves the Theorem. formula 1 fðξÞ ∼ πcγξð ξÞγ−1 ð Þ Re 2 ln 23 III. DISCUSSION by only a factor of 4 . Without any additional work, the lower bound for πe Re fðξÞ as given by (15) can be improved as follows. Of course, in all the bounds derived here, by a slight ξ ξ ξ IðξÞ adjustment of the constants, such as those of (21) and (22), Let 1 be any value between 0 and , then the defined ξ0s s by (10) clearly satisfies all the can be replaced by the Mandelstam variable . Z ξ ξ dσðξ0Þ ξ0 þ ξ ACKNOWLEDGMENTS IðξÞ ≥ dξ0 : ð Þ π dξ0 ln ξ0 − ξ 17 One of us (A. M.) thanks Anderson Kondi Ramida ξ1 Kohara for stimulating discussions and Marek Tasevsky The above argument then gives, entirely similar to (14), for making his participation at the EDS 2017 Conference in Prague possible. The other one (T. T. W.) is grateful to the 2ξ hospitality at CERN, where this work has been carried out. IðξÞ ≥ 1 ½σðξÞ − σðξ Þ: ð Þ π 1 18 We thank Shaojun Sun for typing this paper and Tullio Basaglia for help in the submission process.

[1] N. N. Khuri and T. Kinoshita, Phys. Rev. 137, B720 (1965); [2] M. Froissart, Phys. Rev. 132, 1953 (1961)., Nuovo Cimento 140, B706 (1965); Phys. Rev. Lett. 14, 84 (1965). In these 42, 930 (1966). papers, there is a small confusion: the authors say that E is [3] V. N. Gribov, Proceedings of the 1960 Annual International the so called “laboratory energy” of one particle, but if Conference on High Energy Physics, edited by E. G. C. masses are different, it is not. It is in fact proportional to our Sudarshan, J. H. Tinlot, and A. C. Melissinos, (Rochester variable ξ. University, Rochester, NY, 1960).

014011-3 ANDRE´ MARTIN and TAI TSUN WU PHYS. REV. D 97, 014011 (2018)

[4] A. N. Diddens et al., Proceedings of the 1962 International [9] U. Amaldi et al., Phys. Lett. B 66, 390 (1977). Conference on High Energy Physics at CERN, edited by J. [10] R Battiston et al. (UA4 Collaboration), Phys. Lett. 117B, Prentki, (CERN, Geneva, 1962). 126 (1982). [5] H. Cheng and T. T. Wu, Phys. Rev. Lett. 24, 1456 (1970). [11] A. Amos et al., Nucl. Phys. B 262, 689 (1985);L.A. [6] T. T. Wu, Scattering and production processes at high Fajardo et al., Phys. Rev. D 24, 46 (1981); C. Augier et al. energies, in: Scattering–Scattering and Inverse Scattering (UA4. 2 Collaboration), Phys. Lett. B 316, 448 (1993). in Pure and Applied Science, edited by R. Pike and [12] A. Martin, Phys. Lett. B 404, 137 (1997). In this text a small P. Sabatier, (Academic Press, San Diego, CA, 2002). correction should be made. It is not sufficient to say that [7] K. J. Foley, R. S. Jones, S. J. Lindenbaum, W. A. Love, S. Fðs; tÞ=s goes to zero for negative t. Instead, the correct Ozaki, E. D. Platner, C. A. Quarles, and E. H. Willen, Phys. condition is ðln sÞ1þϵFðs; tÞ=s → 0. Rev. Lett. 19, 857 (1967); G. G. Beznogikh et al., Nucl. Phys. B54, 78 (1973). “ 3” [8] U. Amaldi et al. (CERN-Rome), Phys. Lett. B 44, 119 Corrections: The license statement Funded by SCOAP was (1973). missing on the first page of the article and has been inserted.

014011-4