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IM SB RAS NNA 7-96

0 MECHANISMS OF THE REACTION π ? p → a0 0 (980)n → π ηn AT HIGH ENERGIES

arXiv:hep-ph/9610409v1 18 Oct 1996

N.N. Achasov and G.N. Shestakov

Laboratory of Theoretical Physics, S.L. Sobolev Institute for Mathematics, 630090, Novosibirsk 90, Russia

Abstract

0 The main dynamical mechanisms of the reaction π ? p → a0 0 (980)n → π ηn at high energies, currently investigated at Serpukhov and Brookhaven, are considered in detail. It is shown that the observed forward peak in its di?erential cross section can be explained within the framework of the Regge pole model only by the conspiring ρ2 Regge pole 0 exchange. The tentative estimates of the absolute π ? p → a0 0 (980)n → π ηn reaction ? π cross section at Plab = 18 GeV/c are obtained: σ ≈ 200 nb and, in the forward direction, dσ/dt ≈ 940 nb/GeV2 . The contribution of the one pion exchange, which is forbidden by 0 G-parity and which can rise owing to the f0 (980) ? a0 0 (980) mixing, is also estimate. A role of the Regge cuts in the non-?ip helicity amplitude is brie?y examined and a conclusion is made that the contributions of the cuts have to be inessential in comparison with the conspiring ρ2 Regge pole exchange.

PACS number(s): 13.85.-t, 11.80.Cr, 12.40.Gg

1

1

Introduction

In the q q ?-model (q is a light quark), every rotational excitation with the orbital angular moment L consists of four nonent: states 2S +1 LJ = 3 LL?1 , 3 LL , and 3 LL+1 with chargeparity C = (?1)L+1 and state 1 LL with C = (?1)L . However, so far there is a white spot in the lower-lying family with L = 2 [1]. The non-strange members of the 3 D2 nonet with the quantum numbers I G (J P C ) = 1+ (2?? ) and 0? (2?? ), i.e. the ρ2 , ω2 , and φ2 mesons (the masses of which are expected near 1.7 and 1.9 GeV [2-4]), are not yet identi?ed as peaks in corresponding multi-body mass spectra [1]. The discussions of the possible reasons of this unusual situation are contained, for example, in Refs. [3,4]. However, the mass distributions are not unique keepers of the information on the resonances. The resonance spectrum is also re?ected in the Regge behavior of the reaction cross sections at high energies. At present the detailed investigations of the reaction π ? p → π 0 ηn at π? Plab ≈ 40 and 18 GeV/c are carried out respectively at Serpukhov [5,6] and Brookhaven 0 [7]. The π 0 η mass spectrum in this reaction is dominated by the a0 0 (980) and a2 (1320) mesons [5-7]. In this connection, we should like to draw a special attention to the reaction 0 π ? p → a0 0 (980)n → π ηn because its di?erential cross section near the forward direction can be dominated by the Regge pole exchange with the quantum numbers of the “lost” ρ2 meson. In general, this reaction is unique in that it involves only unnatural parity exchanges in the t-channel. The purpose of this paper is to describe in detail the main dynamical mechanisms of 0 the reaction π ? p → a0 0 (980)n → π ηn in the Regge region. The paper is organized as follows. In Sec. II, we present the reggeization of the s-channel helicity amplitudes of the reaction π ? p → a0 0 (980)n and show that, in the framework of the Regge pole model, the observed forward peak in its cross section [7] can be explained by a very interesting and ?ne phenomenon such as conspiracy between the ρ2 trajectory and its daughter one. For the ?rst time, the ρ2 Regge trajectory was introduced (at that time it was named Z ) for ? the explanation of the absence of a dip near the forward direction in ρH 00 dσ/dt(π p → ωn) [8-10]. However, the nontrivial reason why the ρ2 Regge pole contribution in the s-channel amplitudes without helicity ?ip in the nucleon vertex and with zero helicity of the ω meson do not vanish at t = 0, i.e. conspiracy of the Regge poles in π ? p → ωn, did not discuss at all in Refs. [8-10]. Notice that, for the similar cases, the necessary type of conspiracy was known [11-14] well before the works [8-10]. Here we make up this omission by the example of the reaction π ? p → a0 0 (980)n. We present also the tentative estimate of the ? 0 0 π? π p → a0 (980)n → π ηn reaction cross section 1 at Plab = 18 GeV/c: σ ≈ 200 nb and in the forward direction dσ/dt ≈ 940 nb/GeV2 . In Sec. III, we remind one more interesting 0 0 0 feature of the reaction π ? p → a0 0 (980)n → π ηn associated with the f0 (980) ? a0 (980) mixing [15] and estimate the contribution of the one pion exchange which is possible owing to this mixing. In Sec. IV, the role of the Regge cuts in the non-?ip helicity amplitude is brie?y discussed. In Appendix, the conspiracy phenomenon is explained by the example of the elementary ρ2 exchange.

Often the normalization of the reaction events turns out to be a complicated problem. Probably in this connection, the experimental information on the absolute cross section of the reaction π ? p → 0 a0 0 (980)n → π ηn is so far absent.

1

2

2

Reaction π ? p → a0 0 (980)n at high energies in the Regge pole model

1 ?λn (p2 )γ5 ?A ? γ ? (q1 + q2 )? B uλp (p1 ) , Mλn λp = u 2

The s-channel helicity amplitudes of this reaction can be written as: (1)

where q1 , p1 , q2 , and p2 are four-momenta of π ? , p, a0 0 , and n respectively, λp and λn are the proton and neutron helicities, A and B are the invariant amplitudes depending on s = (p1 + q1 )2 and t = (q1 ? q2 )2 and free of kinematical singularities [16]. Using normalization u ?u = 2mN and taking the proton and the neutron as “second particles” [17] we obtain that, in the c.m. system, √ √ (2) M++ = ?M?? = cos(θ/2) A ?tmin ? B ?tmax s , where θ is scattering angle, tmin and tmax are the values of the variable t at θ = 0? and 180? respectively, sin(θ/2) = [?(t ? tmin )/4|q1 ||q2 |]1/2 , and dσ 1 = |M++ |2 + |M?? |2 2 dt 64πs|q1 | . (4) √ √ M+? = +M?+ = sin(θ/2) A ?tmax ? B ?tmin s , (3)

Eqs. (2) and (3) have the most simple form at high energies. Taking into account that A α α?1 and B at ?xed t and s ? m2 respectively (see below) and also N behave like s and s 2 2 2 2 2 tmin ≈ ?mN (ma0 ? mπ ) /s and tmax ≈ ?s, we get M++ ≈ ?sB , M+? ≈ ?(t ? tmin )A . (5)

The helicity amplitudes in the t-channel c.m. system Fλp ?λn corresponding to the ? 0 reaction π a0 → p ?n have the form F++ = ?F?? = √ tA +

2 mN (m2 a0 ? mπ ) √ B, t

(6)

Φ(s, t, u) B, (7) t In these equations, the a0 0 (980) meson and neutron are taken as “second particles”, θt is scattering angle, |qt | and |pt | are the absolute values of the momenta of the particles in the 2 2 initial and ?nal states respectively. cos θt = (s ? u)/4|qt||pt |, u = 2m2 N + ma0 + mπ ? s ? t; the equation Φ(s, t, u) = 0 gives the boundary of the physical region. It is obvious, from Eqs. (6) and (7), that the helicity amplitudes F+? = +F?+ = 2|qt ||pt | sin θt B ≡ G++ = √ tF++ and G+ ? Φ(s, t, u) ? =? t

? ??1

F+? = B

(8)

are free of kinematical singularities. Their reggeization can be performed by the usual way [18,19]. 3

Constructing the helicity amplitudes with de?nite parity [18]

λp ??λn +|λp ??λn | /2 , = Gλp Gλp ?λn 1 ± ηπ ηa0 (?1) ?λn (±)

(9)

we obtain that, because of the di?erence of the intrinsic parities of the π and a0 mesons, (+) (+) (?) ηπ and ηa0 , the amplitudes G++ and G+? identically vanish and thus G++ ≡ G++ and (?) G+? ≡ G+? . Consequently, both amplitudes G++ and G+? have unnatural parity as it must be since each state of the πa0 system with angular moment J has parity Pπa0 = L+1 (?1)J +1 . It follows from the parity conservation condition Pp = Pπa0 , where ?n = (?1) L is angular moment of the p ?n system, that L = J both for the singlet p ?n spin state and for the triplet one. The amplitudes G+? and G++ correspond to the triplet and singlet (because G++ = ?G?? ) p ?n con?gurations respectively. The G-parity conservation condition (?1)L+S +I = (?1)J +S +1 = +1, where S and I are spin and isospin of p ?n, gives that in the triplet (singlet) state only even (odd) values of J are possible. The partial wave expansions of G++ and G+? are [18]: G++ =

J (2J + 1)f++ PJ (cos θt ) , J =1,3,...

G+ ? =

J (2J + 1)f+ ? J =2,4,...

′ PJ (cos θt )

J (J + 1)

. (10)

Thus, the amplitude G++ has to contain the Regge pole exchanges with I = 1, G = +1, signature τ = ?1, and “naturality” τ P = ?1. The high-lying Regge trajectory with such quantum numbers is the b1 trajectory (the well-known b1 (1235) meson is its lower-lying representative). The second independent amplitude G+? has to contain the Regge pole exchanges with I = 1, G = +1, τ = +1, and τ P = ?1 and here the ρ2 Regge trajectory is a leading one. Taking into account Eqs (10), the contributions of the b1 and ρ2 Regge pole exchanges in the physical region of the s-channel can be written as

1 Gb ++ = βb1 (t)

s s0

αb1 (t)

ie?iπαb1 (t)/2 ,

2 Gρ +? = βρ2 (t)

s s0

αρ2 (t)?1

e?iπαρ2 (t)/2 ,

(11)

where β (t), α(t), and complex factors are residues, trajectories and signatures of the corresponding Regge poles, and s0 = 1 GeV2 . For compensation of the nonphysical branch points in G+? connected with 1/ J (J + 1) [see Eq. (10)], the factor J (J + 1) J has been extracted from f+ ? [12]. Let us return to Eqs. (6) – (8) and express the invariant amplitudes A and B in terms of G++ and G+? . 1 2 G++ ? mN m2 (12) A= a0 ? mπ G+? , t B = G+ ? . (13) To avoid the 1/t singularity in the invariant amplitude A [see Eq. (12)], it is necessary to complement the reggeization scheme by the conditions on the behavior of the various contributions to Gλp ?λn as t → 0. Let us attempt to satisfy the analyticity of A assuming ρ2 1 the b1 and ρ2 exchanges only and also that the amplitudes Gb ++ and G+? do not vanish as t → 0. Then, substituting Eq. (11) to (12) and going to the limit t = 0, we obtain two relations: 2 (14) αb1 (0) = αρ2 (0) ? 1 , βb1 (0) = mN m2 a0 ? mπ βρ2 (0) , the ?rst of which is rather silly because, at the usual values of αb1 (0) ≈ ?(0.05 ÷ 0.3) [8-10,20], it requires αρ2 (0) ≈ 0.95 ÷ 0.7 (also, for the linear ρ2 trajecrory with the slope 4

α′ ≈ 0.8 ÷ 1 GeV2 , it predicts the ρ2 mass mρ2 ≈ 1.02 ÷ 1.27 GeV). For the ρ2 trajectory heaving unnatural parity, this is evidently ruled out. In fact, we conclude that there is no way to make so that the residue of the b1 exchange in Eq. (11) would be ?nite at t = 0. Of course, in order for the amplitude A to be regular for t → 0, one can accept ρ2 2 1 that the amplitudes Gb In this case, the ++ and G+? are separately proportional to t. amplitude B in Eq. (13) and amplitude M++ in Eq. (5) caused by the ρ2 exchange are also proportional to t. Then from Eqs. (5) and (4), it follows immediately that, for b1 and ρ2 Regge pole exchanges, dσ/dt ? |t| at small |t|. Thus this Regge pole model predicts a dip near the forward direction in the π ? p → a0 0 n reaction cross section. On the contrary, the experiment [7] shows a clear forward peak. This means that the amplitude M++ with the quantum numbers of the ρ2 exchange in the t-channel does not vanish as t → 0. In the framework of the Regge pole model, this can be attended only in the case of a conspiracy of the ρ2 Regge trajectory with its daughter one (d), which has to have the quantum numbers of the b1 exchange. Let us written down the contribution of such daughter trajectory near t = 0 in the form Gd ++ = βd (t) s s0

αd (t)

ie?iπαd (t)/2 .

(15)

Then, the amplitude A should be regular at t = 0 [see Eq. (12)] if the following relations for the ρ2 , d and b1 exchanges are valid: αd (0) = αρ2 (0) ? 1 ,

2 βd (0) = mN m2 a0 ? mπ βρ2 (0) , 3

(16) (17)

βρ2 (0) = 0,

βb1 (t) ? t .

Now neither the amplitude B [see Eqs. (11), (13), and (17)] nor the amplitude M++ in (5) vanish at t = 0. Moreover, asymptotically (at large s) M++ is dominated by the ρ2 trajectory [(see Eqs. (5), (11) and (13)] and M+? is dominated by the b1 trajectory [(see Eqs. (5), (11), (12), (15), and (16)]. As for the daughter trajectory contribution and the non-asymptotic contribution of the ρ2 trajectory (which behaves as ? sαρ2 (t)?1 ) to the amplitude A and consequently to M+? then they can be neglected at all [see Eqs. (5), (11), (12), (15), and (16)]. Thus, on the one hand, a role of the daughter trajectory, in practice, comes to only the fact that the residue of the leading ρ2 Regge pole βρ2 (t), owing to a conspiracy [see Eqs. (16) and (17)], does not vanish when t → 0 and can be parametrized, for example, by the simplest exponential form: βρ2 (t) = ?γρ2 exp(b0 ρ2 t)/s0 . At the same time, the residue of the b1 Regge pole in Eq. (11) has to be proportional √ to t [see Eq. (17)] and can be parametrized, for example, as: βb1 (t) = tγb1 exp(b0 b1 t)/ s0 . In our normalization, the constants γρ2 and γb1 are dimensionless. On the other hand, if the daughter trajectory is parallel to the ρ2 trajectory (as, for example, in the Veneziano model) then, near 1.7 GeV, it should be expected a state with the b1 meson quantum numbers, which can be searched for in the a0 π , ωπ , and A2 π channels.

M ? ?N) ? ? N interactions the e?ective Lagrangians L(b1 πa0 ) ? j? b1 and L(b1 N Using for b1 πa0 and b1 N M B where j? = (q1 ? q2 )? and j? = u ?(p2 )γ5 (p2 ? p1 )? v (p1 ), one can easily verify that the contribution ? (p1 ) + N (p2 ) of the elementary b1 exchange to the amplitude G++ for the reaction π (q1 ) + a0 (q2 ) → N turns out to be really proportional to t. 3 The detailed explanation of the conspiracy phenomenon by the example of the elementary ρ2 exchange is contained in Appendix. 2 B ? j? b1 ,

5

Thus, in the model with the b1 and conspiring ρ2 Regge poles, the s-channel helicity amplitudes given by Eq. (5) can be written in the following form convenient for ?tting to the data: s αρ2 (0) ?iπαρ2 (t)/2 M++ = γρ2 ebρ2 (s)t , (18) e s0 s αb1 (0) ?iπαb (t)/2 1 ie , (19) M+? = ?(t ? tmin )/s0 γb1 ebb1 (s)t s0 ′ ′ where αR (t) = αR (0) + αR t, bR (s) = b0 R + αR ln(s/s0 ), R designates a Reggeon. Let us ′ ′ point out, as a guide, that αb1 (0) ≈ ?0.22 and αρ2 (0) ≈ ?0.3 for αb ≈ αρ ≈ 0.8 GeV?2 , 1 2 mb1 ≈ 1.235 GeV, and mρ2 ≈ 1.7 GeV. Using Eqs. (4), (18), and (19), we get that, at large s, dσ s 1 2 2bρ2 (s)t = γρ e 2 2 dt 16πs s0

2αρ2 (0)

+

?

tmin ? t 2 2bb (s)t s γb1 e 1 s0 s0

2αb1 (0)

.

(20)

The best ?t (with χ2 ≈ 15.9 for 22 degrees of freedom) is obtained with Λ = 4.7 GeV?2 and C = 129 events/GeV2 . It is shown in Fig. 1 by the solid curve. Unfortunately, the Serpukhov data on dσ/dt(π ? p → a0 0 (980)n) at 40 GeV/c are not yet presented. It is 0 known only that, in the π η invariant mass region 1 ≤ mπ0 η ≤ 1.2 GeV and for ?tmin < ?t < 0.5 GeV2 the di?erential cross section dσ/dt(π ? p → π 0 ηn) has a similar peak in the forward direction [5]. Obviously, the Brookhaven data are described formally by the single amplitude M++ with the ρ2 exchange [see Eqs. (18) and (21)]. However, within ±(10 ? 20)% experimental uncertainties in dN/dt [7], the form (21) can be e?ectively reproduced for ?tmin < ?t < 0.6 by means of Eq. (20) where the b1 contribution should be also di?erent from zero. The ?t to the data [7] to the form dN/dt = C1 exp(Λ1 t) + (tmin ? t)C2 exp(Λ2 t) with C1 = 131 events/GeV2 , Λ1 = 7.6 GeV?2 , C2 = 340 events/GeV2 , and Λ2 = 5.8 GeV?2 gives a χ2 ≈ 15.9 for 20 degrees of freedom, and the corresponding curve is practically the same as the solid curve in Fig. 1. The dashed and dotted curves in Fig. 1 show the ρ2 [C1 exp(Λ1 t)] and b1 [(tmin ? t)C2 exp(Λ2 t)] contributions separately, with the latter yields approximately 34% of the integrated cross section. In order to determine rather accurately the parameters of the simplest Regge pole model given by Eqs. (18) – (20), the good data on dσ/dt(π ? p → a0 0 (980)n) at several appreciably di?erent energies are needed. First of all, we have in mind the energies of the π ? beams at Serpukhov (≈ 40 GeV), Brookhaven (≈ 18 GeV) and KEK (≈ 10 GeV). Notice that, according to the estimate σ ? (s)2α?2 with α ≈ ?0.3, the a0 0 (980) production cross section at KEK should be approximately 36 times as large as one at Serpukhov. So far the experimental information on the absolute values of the π ? p → a0 0 (980)n → 0 π ηn cross section is absent. Nevertheless, in order to have an idea of this cross section, 0 π? we shall estimate σ (π ? p → a0 0 (980)n → π ηn) at Plab = 18 GeV/c using the data on 0 the reaction π ? p → a0 2 (1320)n and the Brookhaven data on the π η mass spectrum in ? 0 π p → π ηn. According to Refs. [21,5] σ (π ? p → a0 2 (1320)n) = 18.5 ± 3.7 , 12.3 ± 2.5 , 2.7 ± 1.0 and 0.395 ± 0.080 ?b 6 (22)

π According to the Brookhaven data at Plab ≈ 18 GeV/c [7] the t distribution for events ? 0 0 of the reaction π p → a0 (980)n → π ηn is strongly peaked in the forward direction (see Fig. 1). These data are ?tted very well for ?tmin < ?t < 0.6 GeV2 by the single exponential form: 0 Λt dN/dt(π ? p → a0 . (21) 0 (980)n → π ηn) = C e

π at Plab = 12, 15, 40 and 100 GeV/c respectively. These data are ?tted quite well by the exponential function: π ?1.8 σ (π ? p → a0 . 2 (1320)n) ≈ 1.62mb[Plab /(1GeV/c)]

?

?

(23)

0 Then at 18 GeV/c, σ (π ? p → a0 2 (1320)n → π ηn) ≈ 1.29 ?b (here we have taken into account that B (a2 (1320) → πη ) ≈ 0.145 [1]). Fig. 2 shows the Brookhaven data (corrected by the registration e?ciency) on the π 0 η mass spectrum in the reaction π ? p → π 0 ηn at 18 GeV/c [7]. According to our estimate the ratio N (a0 (980))/N (a2 (1320)) ≈ 1/6 ? 1/7, where N (a0 (980)) and N (a2 (1320)) are the numbers of events, respectively, in the a0 (980) π? and a2 (1320) peaks above background. Thus, one can expect that, at Plab = 18 GeV/c, 0 ? 0 σ (π ? p → a0 (980) n → π ηn ) ≈ 200 nb and also [ dσ/dt ( π p → a (980) n → π 0 ηn)]t≈0 ≈ 0 0 940 nb/GeV2 according Eq. (21) with Λ = 4.7 GeV?2 . We emphasize that these estimates are rather tentative.

3

0 One pion exchange in π ? p → a0 0 (980)n → π ηn

It is now interesting to estimate the contribution to this reaction of the reggeized one pion exchange (OPE), which is forbidden by G-parity. The corresponding cross section has the form: 1 g2 dσ (OP E ) = 2 πN N dtdm πs 4π ?te2bπ (s)(t?mπ ) m3 ρππ σ (π + π ? → π 0 η ) , 2 (t ? m2 ) π

2

(24)

2 0 2 2 1/2 where gπN , N /4π ≈ 14.6, m is the invariant mass of the π η system, ρππ = (1 ? 4mπ /m ) 0 ′ ′ 2 bπ (s) = bπ + απ ln(s/s0 ), απ ≈ 0.8 GeV . This contribution arises owing to the f0 (980) ? 0 a0 0 (980) mixing violating isotopic invariance. The f0 (980) ? a0 (980) mixing phenomenon and its possible manifestations in the various reactions (for example, in π ± N → π 0 ηN ) were considered in detail in the works [15]. Therefore, here we give only the numerical estimates the absolute value of the OPE contribution at the Brookhaven and Serpukhov energies. Recall that the cross section of the reaction forbidden by G-parity π + π ? → π 0 η [see ? 0 ) → a0 (980) Eq. (24)] is determined mainly by the transitions f0 (980) → (K + K ? + K 0 K 0 + ? 0 ?0 and, in the region between the K K and K K thresholds, which has a width of 8 MeV, it can be on the average from 0.4 to 1 mb [15]. Outside the region 2mK + ≤ m ≤ 2mK 0 σ (π + π ? → π 0 η ) drops sharply. The mentioned uncertainty in the estimate of σ (π + π ? → π 0 η ) re?ects the spectrum of the model assumptions which were made by many authors for the determination of the coupling constants of the f0 (980) and a0 (980) ? and πη channels (see details in Refs. [15,22]). Note that the resonances with the ππ , K K + ? 0 ? 0 thresholds is controlled mainly value of σ (π π → π η ) between the K + K ? and K 0 K 2 2 2 2 by the production of ratios (gf + ? /gf π + π ? )(ga0 K + K ? /ga πη ) [15], where the coupling 0 0K K 0 0 constants g determine the corresponding decay widths of the scalar mesons, for example, 2 mΓf0 π+ π? (m) = (gf + ? /16π )ρππ and so on. 0π π Taking these remarks into account and integrating Eq. (24) over m from 2mK + to 2mK 0 , we get 2 dσ (OP E ) ?teΛπ (t?mπ ) (25) ≈ (12 ? 30)nb 2 dt (t ? m2 π)

7

π π at Plab = 18 GeV/c and approximately ?ve times smaller at Plab = 40 GeV/c. For the reactions with the one pion exchange, a typical slope in the considered energy region is: Λπ (= 2bπ (s)) ≈ (5 ? 7) GeV?2 . Then, the integral of the function con?ned in brackets in Eq. (25) over t turns out to be approximately equal 1. Hence we have σ (OP E ) ≈ (12 ? 30) 0 nb at 18 GeV/c. This is (6 – 15)% of our estimate, σ (π ? p → a0 0 (980)n → π ηn) ≈ 200 nb, obtained at the end of Sec. II. Due to a smallness of the π meson mass, the dσ (OP E )/dt is enhanced for small |t| [about (85 – 90)% of the integrated cross section σ (OP E ) originate from the region 0 < ?t < 0.2 GeV2 ]. At the maximum situated near t ≈ ?m2 π,

?

?

≈ (122 ? 305)nb/GeV2 . (dσ (OP E )/dt)t≈?m2 π

(26)

0 It can make up from 13 to 32.5% of [dσ/dt(π ? p → a0 0 (980)n → π ηn)]t≈0 which has been roughly estimated to be 940 nb/GeV2 at 18 GeV/c (see the end of Sec. II). Thus, the violating G-parity OPE contribution is able to play a quite appreciable 0 role in the formation of the peak in dσ/dt(π ? p → a0 0 (980)n → π ηn) near the forward direction. Note that the features of the interference between the π and b1 exchanges in the amplitude M+? were discussed in some detail in Ref. [15]. To extract uniquely the amplitude M+? which can be dominated in the low |t| range by the “forbidden” π exchange, a polarized target and a measurement of the neutron polarization in the 0 reaction π ? p → a0 0 (980)n → π ηn are necessary. It is also desirable to measure the charge0 0 0 symmetric reaction π ? n → a0 0 (980)p → π ηp in which the f0 (980) ? a0 (980) interference has to have opposite sign [15].

4

Contributions of the Regge cuts

The Regge cuts, just like the conspiring ρ2 Regge pole, can give a nonvanishing contribution to the amplitude M++ for t → 0. Generally speaking, it is di?cult to distinguish the contributions of the conspiring poles and cuts. However, the standard numerical estimates (such as below) show that in the considered reaction the Regge cuts have to be insigni?cant. First of all, we carry out a classi?cation of the two-Reggeon cuts contributing to the amplitude M++ of the reaction π ? p → a0 0 (980)n. According to Ref. [23], the signature of the cut is given by τc = τ1 τ2 , where τ1 and τ2 are the signatures of the Regge poles associated with the cut. The signature of the amplitude M++ is positive and therefore the τ1 and τ2 must be equal. Then, it is found that the two-Reggeon cuts associated with the Regge poles having the equal and opposite “naturalities” (τ P ) have a principle di?erent behavior as t → 0. Parity conservation gives that the cuts with (τ1 P1 )(τ2 P2 ) = ?1 do not vanish as t → 0 [24]. Among these are the a2 π , ρb1 , and ωa1 cuts and also the Pρ2 cut, where P is the Pomeron. The cuts with (τ1 P1 )(τ2 P2 ) = +1 give vanishing contributions to M++ as t → 0 (they turn out to be proportional to t) [24]. In this group, the ρρ and a2 a2 cuts are leading at large s. The amplitude of the two-Reggeon cut associated with the R1 and R2 Regge pole exchanges can be calculated in the absorption model approximation by the formula [2527]: i R1 R2 R1 R2 Mab d2 k⊥ Mab (27) →cd (s, t) = →ef (s, q ? k⊥ ) Mef →cd (s, k⊥ ) , 8π 2 s e,f

8

that is, considering the R1 R2 cut contribution as a process of a double quasielastic rescattering. In Eq. (27), k⊥ and q are the momenta transferred from the particle e to the particle c and from a to c respectively, q 2 ≈ ?t, the intermediate states e and f represent stable particles or narrow resonances. The accumulated wide experience of the work with the two-Reggeon cuts shows that reasonable estimates can be obtained considering the contributions of the simplest (lowest-lying) intermediate states. The calculation methods of the two-Reggeon cuts are well known (see, for example, Refs. [25-27,19,24]). Therefore, omitting details, we go at once to the discussion of the ?nal results. All these are π? concerned with Plab = 18 GeV/c. Begin with the a2 π cut. Taking into account in Eq. (27) the low-lying ηn intermediate state, we get the following contributions of the a2 π cut to the (dσ/dt)t≈0 and integral cross 0 section σ of the reaction π ? p → a0 0 (980)n → π ηn: dσ a2 π dt =

t≈0 a2 π ? ? 1 dσhf I (m2 m4 π (ba2 + bπ )) π dσ t dt t≈0 t dt 4π |? ba2 + ? bπ |2 ≈ 25 (nb/GeV2 ) B (a0 0 (980) → πη ) , t≈0

≈ (28) (29)

′ Here ? bR = bR ? iπαR /2 (the argument s of the slope bR is omitted from this moment), a2 dσhf /dt is the part of the π ? p → ηn di?erential cross section caused by the a2 Regge pole exchange with a helicity-?ip in the nucleon vertex, dσ π /dt is the di?erential cross section 0 of the reaction ηn → a0 0 (980)n → π ηn caused by the π Regge pole exchange. According a2 the Fermilab data on π ? p → ηn [28], [(1/t)dσhf /dt]t≈0 ≈ 555 ?b/GeV4 , ba2 ≈ 4.18 ′ GeV?2 , and αa ≈ 0.8 GeV?2 (αa2 (0) ≈ 0.371). For the reaction with the π exchange, 2 4 π 2 2 2 0 2 [(mπ /t)dσ /dt]t≈0 = gπN N (ga0 πη /16π ) exp(?2bπ mπ )B (a0 (980) → πη ), where ga0 πη /16π = 2 2 1/2 Γa0 ηπ ma0 /ρηπ and ρηπ = [(1 ? (mη ? mπ )2 /m2 . According a0 )(1 ? (mη + mπ ) /ma0 )] the Particle Data Group [22], the width Γa0 ηπ can be from 50 to 300 MeV. We use its 2 ′ maximal value. Then, ga /16π ≈ 0.454 GeV2 . Also we assume that απ ≈ 0.8 GeV?2 and 0 ηπ 2 ? ? bπ ≈ 3.5 GeV?2 . The factor I (m2 π (ba2 + bπ )) in Eq. (28) has the form |1+ z exp(z )Ei(?z )| , 2 ? where z = mπ (ba2 + ? bπ ) and Ei(?z ) is the integral exponential function. Here we have 2 ? ? I (mπ (ba2 + bπ )) ≈ 0.55. Because B (a0 0 (980) → πη ) < 1, then Eqs. (28) and (29) give, respectively, less 0 than 2.7% and 1.7% of the expected values [dσ/dt(π ? p → a0 0 (980)n → π ηn)]t≈0 ≈ 940 0 4 nb/GeV2 and σ (π ? p → a0 . Even though we magnify these 0 (980)n → π ηn) ≈ 200 nb numbers by an order of magnitude (attributing the enhancement to the contributions of the other intermediate states), all the same they would be appreciably smaller of the expected values. Turn to the ρb1 cut. The contribution of the low-lying π ? p intermediate state is convenient- ly represented in the following form:

σ a2 π ≈ 3.4 (nb) B (a0 0 (980) → πη ) .

dσ ρb1 dt

=

t≈0

Note that the a2 π cut contribution to dσ/dt has a minimum around t ≈ ?0.4 GeV2 and decreases by approximately 54 times over the range of t from 0 to ?0.4 GeV2 . However, experimentally dσ/dt falls by a factor of 6.5 in this t-range and has not the minimum [see Eq. (21)].

4

1 1 dσ b1 1 dσhf = 4π |? bρ + ? bb1 |4 t dt t≈0 t dt t≈0 2 b2 4 ρ bb1 ρ σsf σ b1 < 0.5 nb/GeV2 , = π |? bρ + ? bb |4

1

ρ

(30)

9

ρ where σhf is the π ? p → π ? p cross section with the proton helicity-?ip caused by the ρ 0 Regge pole exchange, σ b1 is the cross section of the reaction π ? p → a0 0 (980)n → π ηn associated with the b1 Regge pole exchange. The limitation (30) has been obtained in ρ terms of the following inequalities: σhf < σ (π ? p → π 0 n)/2 ≈ 12.5 ?b [29], σ b1 < σ (π ? p → 0 2 2 ? ? 4 a0 0 (980)n → π ηn) ≈ 200 nb, bρ bb1 /|bρ + bb1 | < 1/16. Thus, the ρb1 cut contribution should be considered as a whole as very small. The ωa1 cut is more di?cult to estimate because there appear the amplitudes with the a1 Regge pole exchange which are directly unobservable by experiment. Consider the ? contributions of two simplest intermediate states ρ? p and b? 1 p. At the expense of the b1 p intermediate state, we have

1 1 dσ a1 bω ba1 dσ ω ≈ σ ω σ a1 < 2 2 ? ? ? ? dt dt π 4 π | b + b | | b + b | ω a1 ω a1 t≈0 t≈0 t≈0 1 ω a1 2 σ σ ≈ 10 (nb/GeV ) B (a0 (31) < 0 (980) → πη ) , 4π ? 0 where σ ω and σ a1 are cross sections of the reactions π ? p → b? 1 p and b1 p → a0 (980)n → π 0 ηn with the ω and a1 Regge pole exchanges respectively (there are not helicity-?ip in the nucleon vertices and the intermediate b? 1 meson has in the main helicity zero [30]). + ? ω + ? σ ≈ [σ (π p → b1 p) + σ (π p → b1 p) ? σ (π ? p → b0 1 n)]/2 ≈ 20 ?b [29-31]. To estimate a1 σ one can virtually be guided by only the data on the reaction π ? p → ρ0 n at 17.2 GeV/c [32]. The exchanges with the a1 quantum numbers make up approximately 4% of this reaction cross section (≈ 20% in the amplitude), i.e., ≈ 2.5 ?b [33]. To obtain the estimate 0 a1 ? 0 (31), we have use a rough assumption: σ a1 (b? 1 p → a0 n) ≈ σ (π p → ρ n). Similarly, one can obtain for the contribution of the ρ? p intermediate state with the transverse polarized ρ? meson that (dσ ωa1 /dt)t≈0 < (σ ω σ a1 /4π ) ≈ 7.5 (nb/GeV2 )B (a0 0 (980) → πη ), where σ ω ≈ [σ (π + p → ρ+ p) + σ (π ? p → ρ? p) ? σ (π ? p → ρ0 n)]/2 ≈ 15 ?b [29,34] and, for the ρ? p → a0 0 (980)n reaction cross section with the a1 exchange, we take simply the same 2.5 ?b as just above. A relative sign of the ρ? p and b? 1 p intermediate state contributions is unknown. As a result for the ωa1 cut, we have a very rough limitation: dσ ωa1 dt ≈ (dσ ωa1 /dt)t≈0 < 35 (nb/GeV2 ) B (a0 0 (980) → πη ). (32) As mentioned above, the contributions of the ρρ and a2 a2 cuts to M++ vanish as t → 0. However, the absorption corrections to these contributions, i.e., the ρρP and a2 a2 P cuts, are ?nite as t → 0 5 . The estimates of the ρρP and a2 a2 P cut contributions 0 to [dσ/dt(π ? p → a0 0 (980)n → π ηn)]t≈0 , quite similar done above, show that each of ones does not exceed by itself 2 nb/GeV2 , that is, very small. Moreover, a strong compensation between the ρρ and a2 a2 cuts (and analogously between ρρP and a2 a2 P cuts) takes place within the framework of the ρ ? a2 exchange degeneracy hypothesis because the productions of the Regge pole signature factors for these cuts are opposite in sign. As for the Pρ2 cut, the absorption correction of this type accompany any Regge pole exchange. In the cases that the pole contributions do not vanish as t → 0 (beyond general kinematic and factorization requirements), these corrections are e?ectively unimportant at least for the description of the di?erential cross sections. The reactions π ? p → π 0 n and π ? p → ηn give classical examples of this situation. At small |t| and in a wide energy region, the di?erential cross sections of these reactions are described remarkably well by a simple Regge pole model with the linear ρ and a2 trajectories [28,35].

5

Formulae for the such type cuts were obtained in Ref. [24].

10

5

Conclusion

We have considered the main dynamical mechanisms of the reaction π ? p → a0 0 (980)n → 0 π ηn at high energies and shown that the observed peak in its di?erential cross section in the forward direction can be explained within the framework of the Regge pole model only by a conspiracy of the ρ2 trajectory with its daughter one. Notice that there realizes another type of conspiracy in the well known cases of the reactions γp → π + n and pn → np (in which the corresponding peaks in the forward direction are usually described in terms of the Regge cuts [19,25,27]) than in the reaction π ? p → a0 0 (980)n [12]. We have also ? 0 0 π? obtained the estimates of the π p → a0 (980)n → π ηn reaction cross section at Plab = 18 0 GeV/c and of the OPE contribution which can be caused by the f0 (980) ? a0 (980) mixing. 0 Examining the Regge cut contributions to the non-?ip helicity amplitude, we have come to conclusion that they have to be inessential in comparison with the conspiring ρ2 Regge pole exchange. Certainly, it would be very interesting to ?nd some signs of the ρ2 state and its daughter state with the b1 meson quantum numbers, for example, in the a0 π , ωπ , and A2 π mass ? N annihilation. spectra around 1.7 GeV in the reactions induced by π ? mesons or in N

Acknowledgements

N.N. Achasov thanks A.R. Dzierba for many discussions. This work was partly supported by the Russian Foundation for Fundamental Researches Grant No. 94-02-05 188, the Grant No. 96-02-00548, and the INTAS Grant No. 943986.

Appendix

Let us explain a conspiracy phenomena by example of the elementary ρ2 exchange in ? (p1 ) + N (p2 ). The e?ective Lagrangians for the the reaction π (q1 ) + a0 (q2 ) → ρ2 (q ) → N ? ρ2 πa0 and ρ2 N N interactions can be written as (we omit the coupling constants)

M ?ν L(ρ2 πa0 ) = j?ν ρ2 , M j?ν = Q? Qν

(33)

1 B (34) j?ν =u ? (p 2 )γ 5 (γ ? P ν + γ ν P ? )v (p 1 ) , 4 2 where P = p2 ? p1 , Q = q1 ? q2 , q = q1 + q2 = p1 + p2 (P Q = s ? u, P q = 0, Qq = m2 π ? ma0 , ? q 2 = t). The helicity amplitudes of the process π ? a0 ?n are then 0 → ρ2 → p ? ) = j B ρ?ν L(ρ2 NN ?ν 2 ,

?ν Fλp ?λn = Vλp ?λn

Π?ν?′ ν ′ ′ ′ Q? Qν , 2 2 q ? mρ2

(35)

where

Π?ν?′ ν ′

1 ?ν ?λn (p2 )γ5 (γ ? P ν + γ ν P ? )vλp Vλ ? (p 1 ) , p ?λn = u 4 1 1 1 = π??′ πνν ′ + π?ν ′ πν?′ ? π?ν π?′ ν ′ , π?ν = g?ν ? q? qν /m2 ρ2 . 2 2 3

(36) (37)

11

O? mass shall (q 2 = m2 ρ2 ), the tensor Π?ν?′ ν ′ is not the spin-2 projection operator but contains the contributions of the lower (daughter) spins. In the case of coupling to nonconserved tensor currents, these daughter contributions appear in the physical amplitudes. In the case under consideration, the elementary ρ2 exchange with spin 2 is accompanied ? system by the spin-1 contribution only (the spin-0 contribution is uncoupled to the NN because it has the exotic quantum numbers I G (J P C ) = 1+ (0?? )). Using the relation B ?ν B ? ν j?ν g = 0 and j?ν q q = 0, Eq. (35) can be rewritten in the form Fλp ?λn = where the tensors 1 1 1 (2) P?ν?′ ν ′ = θ??′ θνν ′ + θ?ν ′ θν?′ ? θ?ν θ?′ ν ′ 2 2 3 and P?ν?′ ν ′ =

(1)

Vλ?ν p ?λn

2 P?ν?′ ν ′ + [(q 2 ? m2 ′ ′ ρ2 )/mρ2 ]P?ν?′ ν ′ Q? Qν , 2 2 q ? mρ2

(2)

(1)

(38)

(θ?ν = g?ν ? q? qν /q 2 ) ,

(39)

1 [θ??′ qν qν ′ + θνν ′ q? q?′ + θ?ν ′ qν q?′ + θν?′ q? qν ′ ] 2q 2

(40)

are the spin-2 and spin-1 projection operators respectively [36] [(P (2) )2 = P (2) , (P (1) )2 = ?ν ?ν P (1) , P (2) = 5, P (1) = 3]. The spin-2 and spin-1 parts of the elementary ρ2 exchanges ?ν ?ν give the contributions to the amplitudes F+? and F++ respectively. This immediately follows from an explicit form of the angular and threshold behaviors of these amplitudes. F+? = ?8|qt |2 |pt |2 cos θt sin θt F++ = 4|qt ||pt | cos θt 1 Φ(s, t, u) s ? u ≡? , 2 t ? mρ2 t t ? m2 ρ2 (41)

2 (s ? u)mN (m2 mN (m2 ? m2 a0 ? mπ ) π) √ a0 √ ≡ . (42) tm2 tm2 ρ2 ρ2 √ Both amplitudes are singular as t → 0 as 1/ t. Now we go from the amplitudes F+? and F++ given by Eqs. (41) and (42) to the amplitudes G++ and G+? [see Eq. (8)]. Substituting G++ and G+? to Eq. (12), we see that, owing to the compensation between these helicity amplitudes having di?erent quantum numbers, the 1/t singularity in the invariant amplitude A is cancelled out:

1 2 G++ ? mN m2 a0 ? mπ G+? = t 2 2 mN (m2 s ? u mN (m2 1 1 a0 ? mπ ) a0 ? mπ ) = 4|qt ||pt | cos θt = + . t m2 t ? m2 t ? m2 m2 ρ2 ρ2 ρ2 ρ2 A= It takes place automatically since the conspiracy condition,

2 G++ = mN (m2 a0 ? mπ )G+?

(43)

at t = 0 ,

(44)

for the elementary ρ2 exchange is exactly ful?lled. The froward peak in dσ/dt is provided by the second invariant amplitude B = G+? = ?(s ? u)/(t ? m2 ρ2 ), which, as seen, does not vanish at t = 0 [see Eqs. (4) and (5)]. As for the contribution of the amplitude A to dσ/dt then it is an s times smaller at large s and therefore it can be neglected [see Eqs. (43), (5) and (4)]. 12

References

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13

[20] O.I. Dahl et al., Phys. Rev. Lett. 38, 54 (1977). [21] M.J. Corden et al., Nucl. Phys. B 138, 235 (1978). [22] Particle Data Group, L. Montanet et al., Phys. Rev. D 50, 1197, 1465, 1478 (1994). [23] V.N. Gribov, Zh. Eksp. Teor. Fiz. 53, 654 (1967). [24] N.N. Achasov, A.A. Kozhevnikov, G.N. Shestakov, Yad. Fiz. 31, 468 (1980); Yad. Fiz. 25, 1058 (1978). [25] A.B. Kaidalov, B.M. Karnakov, Phys. Lett. B 29, 372, 376 (1969); Yad. Fiz. 11, 216 (1970). [26] C. Michael, Phys. Lett. B 29, 230 (1969); Nucl. Phys. B 13, 644 (1969). [27] F. Henyey et al., Phys. Rev. 182, 1579 (1969); G.L. Kane et al., Phys. Rev. Lett. 25, 1519 (1970); R.A. Miller, Phys. Rev. D 2, 598 (1970). [28] O.I. Dahl et al., Phys. Rev. Lett. 37, 80 (1976). [29] V. Flamino et al., Compilation of Cross-Section, CERN-HERA 83-01, 1983. [30] A.C. Irving, V. Chaloupka, Nucl. Phys. B 89, 345 (1975). [31] D. Alde et al., Z. Phys. C – Particles and Fields 54, 553 (1992). [32] G. Grauer et al., Nucl. Phys. B 75, 189 (1974); H. Becker et al., Nucl. Phys. B 150, 301 (1979); Nucl. Phys. B 151, 46 (1979). [33] J.D. Kimel, J.F. Owens, Nucl. Phys. B 122, 464 (1977). [34] H.A. Gordon et al., Phys. Rev. D 8, 779 (1973). [35] A.V. Barnes et al., Phys. Rev. Lett. 37, 76 (1976). [36] K.J. Barnes, Journal of Math. Phys. 6, 788. (1965).

Figure captions

0 Fig. 1. The t distribution for the reaction π ? p → a0 0 (980)n → π ηn → 4γn at 18 Gev/c measured at Brookhaven [7]. The ?ts are described in the text.

Fig. 2. The π 0 η mass spectrum for the reaction π ? p →→ π 0 ηn → 4γn at 18 Gev/c measured at Brookhaven [7].

14

140

V e G 5 2 0. 0 / st n e v e , t d/ N d

120

100

80

60

40

20

0 0,0 0,2 0,4 0,6 0,8

-t , GeV

Fig. 1.

600

V e M 0 2 / st n e v E

a (1320) 500

400

a (980) 300

200

100

0 0,5 1,0 1,5 2,0 2,5

m(π

0

η)

, GeV

Fig. 2.

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