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Inconsistency of logarithmically growing cross sections with Regge behavior of non-vacuum-exchange amplitudes
Volume 59B, number 1
PHYSICS LETTERS
13 October 1975
INCONSISTENCY OF LOGARITHMICALLY GROWING CROSS SECTIONS WITH REGGE BEHAVIOR OF NON-VACUUM-EXCHANGE AMPLITUDES ~ C.E. JONES and M.N. MISHELOFF
Behlen Laboratoryof Physics, Universityof Nebraska,Lincoln, Nebraska 68508, USA Received 7 August 1975 Using general theoretical considerations, we show that if total cross sections are to increase asymptotically like (log s)3, then high-energy non-vacuum exchange processes cannot be governed by simple factorizeable Regge behavior. The asymptotic behavior of scattering amplitudes has been a central problem in hadronic physics for many years. Recent developments, both experimental and theoretical, have cast grave doubt upon the idea of Regge pole domination of amplitudes involving vacuum quantum number exchange. The Pomeranchuk decoupling theorems [1 ] combined with the limitations imposed by the Froissart bound imply that if vacuum quantum number exchange amplitudes are dominated by a Regge pole, total cross sections must decrease asymptotically. Such a behavior would be at variance with the observed increase of total cross sections [2]. Moreover, several recent model calculations [3] have yielded cross sections proportional to (log s)~, where s is the square of the center of mass energy and/3 is a constant between zero and two. In this note we show that (log s)~ growth and more general kinds of growth of cross sections are inconsistent with simple Regge pole domination of non-vacuum quantum number exchange amplitudes. Thus, either (log s)# behavior of cross sections must be ruled out or the idea of simple Regge pole behavior must be applied with extreme caution. We treat the scattering of two arbitrary particles in the center of mass frame. We consider the initial and final particles as being divided into two clusters according to the signs of their rapidities [4]. Particles with positive rapidity are in the right cluster whereas those with negative rapidity are in the left cluster. We denote the momentum transfer between the clusters by Q and denote its square by t. We let M 2 and ML2 be the subenergies of the final particles in the right and left clusters respectively. The total cross section can be written as o
o(s)=
f do/dt dM2Rdm2
do _(2~)4fd4Q6(Q2_t)8[M2 dt dM2R dM2L × N~R--lf NR i~=12Pot(2n) d3pi
(1)
where
(p+Q)2]8[M2_(K_Q)2 ]
o(Pzi)~4 ( ~i Pi - P-Q)N~L=lJ./~=12ko/(27r)3
\
l
.
(2) /
× IA(P, K-~p 1, ...PNa, kl .... kNL)l 2. In eq. (2), the momenta of particles in the right and left clusters are denoted by Pi or P and k~ or K respectively. All possible final particles in the clusters are included in the sum. Because any event can be assigned unique values ° f N R , NL, P l , "'" PNR, and k I .... kNL, eqs. (1) and (2) count each event only once. We now restrict the sum over final particles to a sum over those such that the momentum transfer Q carries a specified set of quantum numbers, e.g., those of the rho meson. We assume that for large enough s/M2RM2 L, the amplitudes can be approximated by the exchange of a simple Regge pole: Supported in part by NSF Grant GP-43907. 53
Volume 59B, number 1
PHYSICS LETTERS
13 October 1975
A - - f ( t ) A R L(s/M2LM2R)a(t)"
(3)
In eq. (3), the Reggeon-particle scattering amplitudes A'R and A'L depend upon the momenta Q, P, P x , ' " , PNR and Q, K, k 1 , ..., kNL, respectively. We restrict the integral over t to a finite region and the integrals over ML2 and MR2 to the region M R2 ' jr,t2 ""L ~< c V~, where c is small enough so that eq. (3) is valid throughout the restricted region. In the restricted kinematical regions, we can insert eq. (3) into eq. (2) to obtain do ~ (27r)4 If(t)l 2(s/M2M2)2'~(t)fd4Q 5(Q 2 - t)8 [M2R - (P+ Q)21 5 [M 2 - (K - Q)2] 2s dt dM 2 dM 2
X
x[
lfi~=l 2Poi(2rr)30(Pzi,64(~iPi-p-Q)l'hRI2 NL=I
fl-I] =1 2 k o l ( 2 n ) a O(-kzi)64(~ki-K " \ ] " J
(4)
J
If the theta-function factors could be neglected, the expressions appearing in brackets in eq. (4) could be replaced by expressions involving Reggeon-particle total cross sections. In fact, energy-momentum conservation implies that any p! is restricted to the region Pzl >~rn, - MR2/X/Tand k l to the region kzt ~ -rn~r + M2/x/~, quite apart from any restriction imposed by the theta functions. Therefore, if c is less than mTr, the theta functions are superfluous and may be omitted. To understand the restriction on Pzl, we note that kinematical relations at high s require that Qz = -(M2R + M2L)/2 V~ + O(1/V~) and ao = (M2R - M2)/2 V~. For any l, the z-component of the vector P + a - Pl is given by
(P+ a - Pl)z = V~-/2 - (MR2 + M2)/2X/s - Pzl + O(1/V~). Since the pion is the particle with the least mass, the time component o f P + Q - Pt satisfies the inequality
(P + Q - Pt)o <~V~/2 + (M 2 M2)/2 Vrf -- mrr. Since P + Q - Pt is the null vector or a positive time like vector, we have the inequality (P + Q - Pl)o >~ (P + Q - P l ) z which yields the above restriction on Pzt" A similar argument yields the restriction on kzt. Since the terms we are neglecting give a positive contribution to the cross section, we can combine eqs. (1) and (4) to obtain the inequality
oo)> ±
(5)
167r3
In eq. (5), ~ ( M 2; t) is the Reggeon-particle total cross section. We now assume that o(s) increases as (log s)~ for large s. We furthermore assume that the Reggeon-particle total cross section has the same asymptotic behavior, i.e., ~(M 2 ; t) ~ g(t)(log M2) ~. With these assumptions the right hand side of eq. (5) increases as (log s) 2~ as s increases; thus, the inequality is violated for B > 0. Before we discuss the possible resolutions of this inconsistency, we note that our assumptions may be weakened in several ways without resolving the inconsistency. First, we note that if we assume that the asymptotic behavior of non-vacuum exchange amplitudes is governed by an arbitrary factorizable J-plane singularity, not necessarily a Regge pole, eq. (5) is replaced by the inequality
o(s)>
54
l~ f dtfCVqdM~M~~(M~;t)f--
16rr3s 2
dM2LM2L ~d(M2L; t)f(t,s/M2R M2L),
(6)
Volume 59B, number 1
PHYSICS LETTERS
13 October 1975
where f is a positive function and ~(M2; t) is the appropriate total cross section for the factorized singularity. If ~(M2; t) behaves like (log s)~, the right hand side of eq. (6) increases at least as fast as (log s) 2~. Thus the inconsistency persists even for non-pole Regge behavior. Second, we note that cross section growths as mild as (In In s)~, (In In In s)~, etc. also lead to a violation of eq. (5). On the other hand, power behavior of total cross sections is not excluded by our considerations. If we assume that ~(M 2 ; t) ~ s a P - 1, the right hand side of eq. (5) behaves as s a p - 1 for Otp :/: 1. Power behavior is thus generally self reproducing (an exception to this occurs if Ctp = O~R(0) = 1, as shown in ref. [4]. There are several possible resolutions o f the violation of eq. (5): a) Reggeon-particle total cross sections do not have the same asymptotic behavior as particle-particle total cross sections. b) It may not be possible to choose a value of c so that all amplitudes appearing in eq. (2) are Regge behaved in the restricted integration region although it may be possible to do so for each individual amplitude. c) Total cross sections may not rise indefinitely. If this is the case, true asymptopia is not reached by present accelerator energies. d) Non-vacuum exchange amplitudes may not be Regge behaved, a possibility suggested to us by Patrascioiu [5]. Such a possibility should not be regarded as entirely unexpected in view of the fact that the iteration o f a Pomeranchuk singularity at J = 1 with a Reggeon to a singularity at least as high as the Reggeon (see ref. [6] for a calculation of this type within the framework of the Reggeon calculus). It is not clear how the mechanism in (d) resolves the violation of eq. (5). Corrections to simple Reggeon pole behavior o f the type discussed by Abarbanel and Sugar [6] will be in the form of eq. (6), if factorization holds, which, as mentioned, does not resolve the violation. Furthermore, the region o f integration responsible for the violation occurs when s / M 2 R M 2L is large but finite. Recent data [7] indicate that simple Regge pole exchange works extremely well in this region [7]. We are very grateful to P. Finkler, F.E. Low and A. Patrascioiu for helpful discussions.
References [1] H.D.I. Abarbanel et al., Phys. Rev. Lett. 26 (1971) 937; C.E. DeTa.r, D.Z. Freedman and G. Veneziano, Phys. Rev. D4 (1971) 906; C.E. Jones et al., Phys. Rev. D6 (1972) 1033; R. Brower and J. Weis, Phys. Lett. 41B (1972) 631. [2] M. Holder et al., Phys. Lett. 35B (1971) 361; U. Amaldi et al., Phys. Lett. 43B (1973) 231 and Phys. Lett. 44B (1973) 112; S.R. Amendolia et al., Phys. Lett. 44B (1973) 119. [3] H.D.I. Abarbanel and J.B. Bronzan, Phys. Lett. 48B (1974) 345; H. Cheng and T.T. Wu, Phys. Lett. 36B (1971) 357; J. Ball and F. Zachariasen, Nucl. Phys. B78 (1974) 77. [4] A similar cluster decomposition was employed by R.C. Brower et al., Phys. Lett. 46B (1973) 105. [5] A. Patrascioiu (private communication). [6] H.D.I. Abarbanel and R.L. Sugar, Phys. Rev. D10 (1974) 721. [7] D.J. Mellema et al., Proc. XVII Intern, Conf. on High energy physics, 1-37, ed. J.R. Smith, London (1974) See also V. Barger, Plenary report on Reaction mechanisms at high energy, Proc. XVII Intern. Conf. on High energy physics, 1-37, ed. J.R. Smith, London (1974).
55
PHYSICS LETTERS
13 October 1975
INCONSISTENCY OF LOGARITHMICALLY GROWING CROSS SECTIONS WITH REGGE BEHAVIOR OF NON-VACUUM-EXCHANGE AMPLITUDES ~ C.E. JONES and M.N. MISHELOFF
Behlen Laboratoryof Physics, Universityof Nebraska,Lincoln, Nebraska 68508, USA Received 7 August 1975 Using general theoretical considerations, we show that if total cross sections are to increase asymptotically like (log s)3, then high-energy non-vacuum exchange processes cannot be governed by simple factorizeable Regge behavior. The asymptotic behavior of scattering amplitudes has been a central problem in hadronic physics for many years. Recent developments, both experimental and theoretical, have cast grave doubt upon the idea of Regge pole domination of amplitudes involving vacuum quantum number exchange. The Pomeranchuk decoupling theorems [1 ] combined with the limitations imposed by the Froissart bound imply that if vacuum quantum number exchange amplitudes are dominated by a Regge pole, total cross sections must decrease asymptotically. Such a behavior would be at variance with the observed increase of total cross sections [2]. Moreover, several recent model calculations [3] have yielded cross sections proportional to (log s)~, where s is the square of the center of mass energy and/3 is a constant between zero and two. In this note we show that (log s)~ growth and more general kinds of growth of cross sections are inconsistent with simple Regge pole domination of non-vacuum quantum number exchange amplitudes. Thus, either (log s)# behavior of cross sections must be ruled out or the idea of simple Regge pole behavior must be applied with extreme caution. We treat the scattering of two arbitrary particles in the center of mass frame. We consider the initial and final particles as being divided into two clusters according to the signs of their rapidities [4]. Particles with positive rapidity are in the right cluster whereas those with negative rapidity are in the left cluster. We denote the momentum transfer between the clusters by Q and denote its square by t. We let M 2 and ML2 be the subenergies of the final particles in the right and left clusters respectively. The total cross section can be written as o
o(s)=
f do/dt dM2Rdm2
do _(2~)4fd4Q6(Q2_t)8[M2 dt dM2R dM2L × N~R--lf NR i~=12Pot(2n) d3pi
(1)
where
(p+Q)2]8[M2_(K_Q)2 ]
o(Pzi)~4 ( ~i Pi - P-Q)N~L=lJ./~=12ko/(27r)3
\
l
.
(2) /
× IA(P, K-~p 1, ...PNa, kl .... kNL)l 2. In eq. (2), the momenta of particles in the right and left clusters are denoted by Pi or P and k~ or K respectively. All possible final particles in the clusters are included in the sum. Because any event can be assigned unique values ° f N R , NL, P l , "'" PNR, and k I .... kNL, eqs. (1) and (2) count each event only once. We now restrict the sum over final particles to a sum over those such that the momentum transfer Q carries a specified set of quantum numbers, e.g., those of the rho meson. We assume that for large enough s/M2RM2 L, the amplitudes can be approximated by the exchange of a simple Regge pole: Supported in part by NSF Grant GP-43907. 53
Volume 59B, number 1
PHYSICS LETTERS
13 October 1975
A - - f ( t ) A R L(s/M2LM2R)a(t)"
(3)
In eq. (3), the Reggeon-particle scattering amplitudes A'R and A'L depend upon the momenta Q, P, P x , ' " , PNR and Q, K, k 1 , ..., kNL, respectively. We restrict the integral over t to a finite region and the integrals over ML2 and MR2 to the region M R2 ' jr,t2 ""L ~< c V~, where c is small enough so that eq. (3) is valid throughout the restricted region. In the restricted kinematical regions, we can insert eq. (3) into eq. (2) to obtain do ~ (27r)4 If(t)l 2(s/M2M2)2'~(t)fd4Q 5(Q 2 - t)8 [M2R - (P+ Q)21 5 [M 2 - (K - Q)2] 2s dt dM 2 dM 2
X
x[
lfi~=l 2Poi(2rr)30(Pzi,64(~iPi-p-Q)l'hRI2 NL=I
fl-I] =1 2 k o l ( 2 n ) a O(-kzi)64(~ki-K " \ ] " J
(4)
J
If the theta-function factors could be neglected, the expressions appearing in brackets in eq. (4) could be replaced by expressions involving Reggeon-particle total cross sections. In fact, energy-momentum conservation implies that any p! is restricted to the region Pzl >~rn, - MR2/X/Tand k l to the region kzt ~ -rn~r + M2/x/~, quite apart from any restriction imposed by the theta functions. Therefore, if c is less than mTr, the theta functions are superfluous and may be omitted. To understand the restriction on Pzl, we note that kinematical relations at high s require that Qz = -(M2R + M2L)/2 V~ + O(1/V~) and ao = (M2R - M2)/2 V~. For any l, the z-component of the vector P + a - Pl is given by
(P+ a - Pl)z = V~-/2 - (MR2 + M2)/2X/s - Pzl + O(1/V~). Since the pion is the particle with the least mass, the time component o f P + Q - Pt satisfies the inequality
(P + Q - Pt)o <~V~/2 + (M 2 M2)/2 Vrf -- mrr. Since P + Q - Pt is the null vector or a positive time like vector, we have the inequality (P + Q - Pl)o >~ (P + Q - P l ) z which yields the above restriction on Pzt" A similar argument yields the restriction on kzt. Since the terms we are neglecting give a positive contribution to the cross section, we can combine eqs. (1) and (4) to obtain the inequality
oo)> ±
(5)
167r3
In eq. (5), ~ ( M 2; t) is the Reggeon-particle total cross section. We now assume that o(s) increases as (log s)~ for large s. We furthermore assume that the Reggeon-particle total cross section has the same asymptotic behavior, i.e., ~(M 2 ; t) ~ g(t)(log M2) ~. With these assumptions the right hand side of eq. (5) increases as (log s) 2~ as s increases; thus, the inequality is violated for B > 0. Before we discuss the possible resolutions of this inconsistency, we note that our assumptions may be weakened in several ways without resolving the inconsistency. First, we note that if we assume that the asymptotic behavior of non-vacuum exchange amplitudes is governed by an arbitrary factorizable J-plane singularity, not necessarily a Regge pole, eq. (5) is replaced by the inequality
o(s)>
54
l~ f dtfCVqdM~M~~(M~;t)f--
16rr3s 2
dM2LM2L ~d(M2L; t)f(t,s/M2R M2L),
(6)
Volume 59B, number 1
PHYSICS LETTERS
13 October 1975
where f is a positive function and ~(M2; t) is the appropriate total cross section for the factorized singularity. If ~(M2; t) behaves like (log s)~, the right hand side of eq. (6) increases at least as fast as (log s) 2~. Thus the inconsistency persists even for non-pole Regge behavior. Second, we note that cross section growths as mild as (In In s)~, (In In In s)~, etc. also lead to a violation of eq. (5). On the other hand, power behavior of total cross sections is not excluded by our considerations. If we assume that ~(M 2 ; t) ~ s a P - 1, the right hand side of eq. (5) behaves as s a p - 1 for Otp :/: 1. Power behavior is thus generally self reproducing (an exception to this occurs if Ctp = O~R(0) = 1, as shown in ref. [4]. There are several possible resolutions o f the violation of eq. (5): a) Reggeon-particle total cross sections do not have the same asymptotic behavior as particle-particle total cross sections. b) It may not be possible to choose a value of c so that all amplitudes appearing in eq. (2) are Regge behaved in the restricted integration region although it may be possible to do so for each individual amplitude. c) Total cross sections may not rise indefinitely. If this is the case, true asymptopia is not reached by present accelerator energies. d) Non-vacuum exchange amplitudes may not be Regge behaved, a possibility suggested to us by Patrascioiu [5]. Such a possibility should not be regarded as entirely unexpected in view of the fact that the iteration o f a Pomeranchuk singularity at J = 1 with a Reggeon to a singularity at least as high as the Reggeon (see ref. [6] for a calculation of this type within the framework of the Reggeon calculus). It is not clear how the mechanism in (d) resolves the violation of eq. (5). Corrections to simple Reggeon pole behavior o f the type discussed by Abarbanel and Sugar [6] will be in the form of eq. (6), if factorization holds, which, as mentioned, does not resolve the violation. Furthermore, the region o f integration responsible for the violation occurs when s / M 2 R M 2L is large but finite. Recent data [7] indicate that simple Regge pole exchange works extremely well in this region [7]. We are very grateful to P. Finkler, F.E. Low and A. Patrascioiu for helpful discussions.
References [1] H.D.I. Abarbanel et al., Phys. Rev. Lett. 26 (1971) 937; C.E. DeTa.r, D.Z. Freedman and G. Veneziano, Phys. Rev. D4 (1971) 906; C.E. Jones et al., Phys. Rev. D6 (1972) 1033; R. Brower and J. Weis, Phys. Lett. 41B (1972) 631. [2] M. Holder et al., Phys. Lett. 35B (1971) 361; U. Amaldi et al., Phys. Lett. 43B (1973) 231 and Phys. Lett. 44B (1973) 112; S.R. Amendolia et al., Phys. Lett. 44B (1973) 119. [3] H.D.I. Abarbanel and J.B. Bronzan, Phys. Lett. 48B (1974) 345; H. Cheng and T.T. Wu, Phys. Lett. 36B (1971) 357; J. Ball and F. Zachariasen, Nucl. Phys. B78 (1974) 77. [4] A similar cluster decomposition was employed by R.C. Brower et al., Phys. Lett. 46B (1973) 105. [5] A. Patrascioiu (private communication). [6] H.D.I. Abarbanel and R.L. Sugar, Phys. Rev. D10 (1974) 721. [7] D.J. Mellema et al., Proc. XVII Intern, Conf. on High energy physics, 1-37, ed. J.R. Smith, London (1974) See also V. Barger, Plenary report on Reaction mechanisms at high energy, Proc. XVII Intern. Conf. on High energy physics, 1-37, ed. J.R. Smith, London (1974).
55