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Production of η Mesons Near Threshold

arXiv:hep-ph/9303246v1 11 Mar 1993UCL-HEP-9302Production of η Mesons Near ThresholdColin WilkinUniversity College London,Gower St., London WC1E 6BT, U.K.AbstractIt is argued that the strong energy dependence of the p d →3He η cross section nearthreshold is due to a final state interaction between the η meson and the 3He nucleus. Thelarge scattering length that this implies is in accord with optical potential predictions andis evidence for a nearby virtual ‘‘bound’’ state in the η3He system.

This model suggests thatsharp structures should also be seen close to production thresholds on other light nuclei.The p d →3He η and p d →3He π0 cross sections show striking but very different energy depen-dences for proton energies within a few MeV of their respective production thresholds. To illustratethis, define average squared amplitudes through|fη(π)|2 =pppη(π) dσdΩcm,(1)where the phase space ratio of outgoing to incident centre-of-mass momenta has been factored out.If the centre-of-mass angular distributions are parameterised as|fη(π)|2 ∝1 + αη(π) cos θpη(π) ,(2)then |αη| ≤0.05 for pη ≤0.38 fm−1 [1].

In contrast απ = 0.73 ± 0.02 already by pπ = 0.14 fm−1 [2].On the other hand |fη|2 decreases by over a factor of three between threshold and pη = 0.35 fm−1,i.e. ∆Tp = 10 MeV [1, 3], but the angular average of |fπ|2 has a rather weak energy dependence nearthreshold [2, 4].The strong angular dependence seen in p d →3He π0 arises from the large πN P-wave, associatedwith the ∆(1232), interfering with an S-wave which is only significant within a few MeV of threshold.Taking this into account, both the cross section and deuteron tensor analysing power are describedquantitatively at low energies by a model involving a spectator nucleon [5].The most prominent feature of the low energy η N interaction is an L=0 resonance, the S11 N∗(1535),and the low energy π−p →η n reaction shows only a weak angular dependence [6].

I want to convinceyou that this strong S-wave resonance is also responsible for the rapid energy variation of the near-threshold p d →3He η cross section.The spectator-nucleon model, which was successful in π0 production, underestimates significantlythe low energy p d →3He η cross section [7, 8]. By including scattering of η’s and π’s on up to threenucleons, Laget and Lecolley [7] could enhance the cross section but their renormalised results, shownin fig.

1, fall offmuch less steeply than the data [1]. It is important to note that such a model isperturbative, treating all interactions only to lowest order.

Though this might be justified for lowenergy pions, it is dangerous for S-wave η-nucleon scattering, where the interaction is so strong.The sharpness of the energy scale, combined with the isotropy of the angular distribution, suggeststhat an S-wave η–3He final state interaction (FSI) is responsible for the phenomena rather than thedetails of the reaction mechanism or nuclear form factors.A common approximation [9], in the case of a weak transition to a channel with a strong FSI, leadsto an enhancement of the S-wave amplitude of the formfη =f Bηpη aη cotδη −ipη aη. (3)0Talk given at the BNL workshop on future directions in particle and nuclear physics at multi-GeV hadron beamfacilities, March 4–6, 1993.1

The influence of the S11 resonance is felt through the S-wave phase shift δη and scattering length aη.A reaction model is needed to estimate the amplitude f Bη , but it should be slowly varying for pηR < 1.At low energies it is often sufficient to takefη ≈f Bη1 −ipη aη. (4)This corresponds to imposing unitarity with constant K-matrix elements, i.e.

neglecting effective rangeeffects. At present it is pointless trying to go further.One cannot unambiguously extract values of the real (aR) and imaginary (aI) parts of the η 3Hescattering lengths directly from the present data using eq.

(4). Taking the updated SPES2 points [1],the valley of χ2 lies roughly alonga 2R + 0.866a 2I + 2.615aI = 21.69 ,(5)which demonstrates that the modulus of the scattering length has to be bigger than the nucleus itself!Of course Laget and Lecolley already have some FSI in their model [7], but only a modest amountassociated with η’s scattering offindividual nucleons, leading to aη ≈(1.2 + i0.6) fm.The difficulty in determining both real and imaginary parts is clear from the two fits shown infig.

1 with (aR, aI) equal to (3.10, 2.52) and (5.0, 0.17) fm. Though the former gives a better overallagreement with the data, the place where the predictions really deviate is for pη ≤0.05 fm−1 and hereit is the hardest to measure due to intrinsic beam spread and energy loss in the target.

The lowestSPES2 point is averaged over a wide range of energies and the predicted counting rate dR/dT shown infig. 2 has a very skewed peak arising from the product of rapidly varying acceptance and cross section.A different approach will be needed to do better than this careful SPES2 measurement [1] and the onethat springs to mind is the Celsius storage ring with its cooled proton beam and gas jet target [10].Is such a large scattering length consistent with our (limited) knowledge of η-nucleus dynamics?

Inimpulse approximation the η 3He scattering length is about four times that of ηN but there are majorcorrections to this simple ansatz due to the strength of the interaction. A more reliable starting pointis to consider the lowest order η 3He optical potential for which2mRη N Vopt(r) = −4π A ρ(r) a (η N) ,(6)where mRη N is the η-nucleon reduced mass and A (= 3) the mass number.Bhalerao and Liu [11] analysed the πN and ηN coupled channels near the η threshold within anisobar model and from the πN phase shifts extracted a value for the η-nucleon scattering length ofa (ηN) = (0.27+ i0.22) fm..

However it is more consistent to apply eq. (4) directly to π−p →η n data.Using detailed balance and the optical theorem, a lower bound on the imaginary part of a (ηN) isprovided by the threshold π−p →η n cross section:Im[a (ηN)] ≥38πp2πpησtot(π−p →η n) .

(7)The data of ref. [6] require Im[a (ηN)] ≥(0.28 ± 0.04) fm which, though larger than that extracted in[11], is compatible with it.

Other channels, such as Nππ, are also open at the η threshold and theseincrease the inelasticity. We take Im[a (ηN)] = 0.30 fm, though this might be an underestimate.The energy dependence of the π−p →η n amplitude shown in fig.

3 is then enough to determinethe real part of the scattering length within the model, leading to a (ηN) = (0.55 ± 0.20 + i0.30) fm.The sign of the real part is ambiguous and has been chosen to be attractive to agree with Bhaleraoand Liu [11]. It should be noted that the larger value obtained here is not due to P-wave contaminationin the cross section.

This would go the wrong way but a difference could arise from different effectiverange assumptions. The puzzle could perhaps be clarified by new and very precise measurements ofγp →ηp with a monochromatic photon beam at Mainz.

Preliminary data show that the cross sectionremains isotropic until quite high energies [12].2

Using this value of a (ηN) in the optical potential of eq. (6), with a Gaussian nuclear densitycorresponding to an rms radius of 1.9 fm, leads to a predicted scattering length ofa (η 3He) = (−2.31 + i2.57) fm .

(8)The energy dependence of the p d →3He η cross section resulting from the simplified FSI formulaof eq. (4) with this scattering length is shown as the dashed line in fig.

4 and is compared with thepioneering SPES4 data [3] and updated SPES2 values [1]. The overall normalisation (the value of f Bη )is arbitrary.

Though the good numerical agreement with the data is perhaps a little fortuitous in viewof possible corrections to the lowest order optical potential, it nevertheless indicates that the rapid fallof the amplitude with energy is likely to be a consequence of the strong η 3He final state interaction.Once we have the potential of eq. (6) then we can calculate the phase shift at all energies, whichenables us to use the more general formula of eq.

(2) rather than the constant scattering length versionof eq. (3).

This makes little difference, as can be seen from the solid line in fig. 4.The sign of the real part of the scattering length indicates the possible presence of a ‘‘bound’’ ηstate for a much lighter nucleus than suggested by Haider and Liu [13].

It can be thought of as adisplacement of the S11 pole to below the η threshold through the repeated scattering of the η onall the nucleons in 3He. Other decay channels (pionic or nucleonic) are however open and the largeimaginary component in the scattering length severely limits observation of effects from this pole.The S-wave FSI enhancement factor of eq.

(2) is independent of the entrance channel, thoughthe particular nuclear reaction would influence the amount of P and higher waves present. It shouldtherefore be applicable also to the π−3He →η 3H reaction.

Unfortunately the lowest energy for whichthis has been measured [14] corresponds to pη = 0.41 fm−1, which is just offthe scale of fig. 1 !What happens for heavier nuclei?

Data exist for d d →4He η, but only away from threshold [15].Taking an rms radius of 1.63 fm, the η 4He potential is stronger but of shorter range than for η 3He.The predicted scattering length of (−2.00 + i0.97) fm corresponds to a somewhat less steep energydependence than for η 3He production. Including effective range effects through eq.

(2), the decreasein |f|2 between pη = 0.1 and 0.4 fm−1 is expected to be about 2.8 for 3He but only 1.9 for 4He.Coherent η production has been measured in p 6Li →η 7Be∗[16], though the resolution obtainedby detecting the η through its 2γ decay was insufficient to isolate individual states in the 7Be nucleus.Since the optical potential of eq. (6) predicts a scattering length of (−2.92 + i1.21) fm and the typicalη centre-of-mass momentum in this experiment was pη ∼0.5 fm−1, these data lie outside the FSI peak.Turning now to lighter systems, Ueda suggested [17] that, even for a system as diffuse as thedeuteron, the S11 pole should be moved towards the η threshold.

Using a separable coupled channelmodel, with π and η exchange, he predicted a scattering length aηd ≈(1.8 + i2.5) fm.In their analysis of old np →dη data, the authors of ref. [18] assumed that the cross section wasmodulated by an ad hoc function of the form1(1 + b2p2η)2 .

(9)In order to explain the shape of the spectrum, a large value of b was required (≈3 fm), but this wasnot a dedicated experiment and the available range in pη was rather small. However evidence for thesize of b can be obtained in another way.It has been shown that at 1.3 GeV the yield of η’s in pn collisions is about a factor of 10 largerthan in pp [19].

Furthermore there has recently been an independent measurement of the pp →ppηtotal cross section near threshold [20]. Using this as normalisation, the combined experiments suggestthat σT (np →η X) ≈0.03 mb at pη ≈0.6 fm−1.

If this is dominated by deuteron formation, then thecombination with the threshold normalisation claimed in ref. [18] implies that b is at least 3 fm.Theoretical predictions pp →ppη total cross section largely follow a phase space behaviour in termsof the energy Qcm above threshold [21].

To fit the energy dependence of the three experimental pointsof ref. [20], the authors used the functional form of eq.

(9) with the same value of b = 3 fm.Though none of these nucleon-nucleon determinations is yet compelling, they do seem to supportUeda’s contention [17] that the S11 pole is significantly displaced already in the two-nucleon sector forboth isospin-zero and one. Unfortunately neither of these states is coupled to the πd system.3

In summary, there is ample evidence that the low energy η interaction with the few-nucleon systemis very strong but much more experimental work is needed to pin it down in detail. In particular Iwould suggest the measurement of the following near-threshold reactions:• np →dη.

This might be done using pd →d ps η with the cooled proton beam and gas jet targetat Celsius. The spectator proton ps could then be picked up in a solid state counter.• More points are needed in the pp →ppη excitation function.• The separation of the real and imaginary parts of the scattering length in the η3He case couldbe done through measurements at Celsius at fractions of MeV above threshold.• The π−3He →η 3H of ref.

[14] should be measured closer to threshold. The results might not bequite as clear as for the proton beams since the pion reactions could be more peripheral.• d d →4He η looks like a strong possibility at Saturne.• p 3H →4He η has been foiled at LAMPF but n 3He →4He η might be easier.• In the p 6Li →η 7Be∗case [16], it might be advantageous to study the cross sections at say 1–2MeV above the threshold for the excitation of a particular level.What should theoreticians do while waiting for the results from some of the above list?• A study of more refined models of the S11 and other resonances which decay into ηN.Onedifficulty is that the S11 is strongly coupled to πN, ηN and ρN.• More refined models of the ηNN system where the ρ coupling has to be included.• Better estimates of f Bη in the p d →3He η case so that one can insert the FSI’s more reliably,without having to make the constant scattering length approximation in the Watson FSI factor.Progress has already been made on semi-empirical three-nucleon models [22, 23].However in the end theoreticians do what they want to and experimentalists do what they can getmoney for!Discussions with R.Kessler and A.Moalem on the results of ref.

[1] and ref. [20] respectively weremuch appreciated.References[1] M. Gar¸con et al., in Spin and Symmetry in the Standard Model, Lake Louise, 1992, edited by B.A.

Campbell (World Scientific, Singapore 1992) p 337; R. Kessler, Ph.D. thesis (UCLA) 1992 andprivate communication (1993). [2] M. Pickar, A. D. Bacher, H. O. Meyer, R. E. Pollock & G. T. Emery, Phys.Rev.

C46, 397 (1992). [3] J. Berger et al., Phys.Rev.Lett.

61, 919 (1988). [4] C. Kerboul et al., Phys.Lett.

181B, 28 (1986); A. Boudard et al., Phys.Lett. 214B, 6 (1988).

[5] J. F. Germond & C. Wilkin, J.Phys. G16, 381 (1990).

[6] D. M. Binnie et al., Phys.Rev. D8, 2789 (1973); R. M. Brown et al., Nucl.

Phys. B153, 89 (1979);F. Bulos et al., Phys.Rev.Lett.

13, 486 (1964); W. Deinet et al., Nucl.Phys. B11, 495 (1969).

[7] J. M. Laget & J. F. Lecolley, Phys.Rev.Lett. 61, 2069 (1988).4

[8] J.-F. Germond & C. Wilkin, J.Phys. G15, 437 (1989).

[9] M. Goldberger & K. M. Watson, Collision Theory (New York, Wiley, 1964). [10] B. H¨oistad & T. Johansson, Celsius proposal C23, 1992 (unpublished).

[11] R. S. Bhalerao & L. C. Liu, Phys.Rev.Lett. 54, 865 (1985).

[12] H. Str¨oher (private communication of the results of the TAPS collaboration). [13] Q. Haider & L. C. Liu, Phys.Lett.

B172, 257 (1986). [14] J. C. Peng et al., Phys.Rev.Lett.

63, 2353 (1989). [15] J. Banaigs et al., Nucl.Phys.

B105, 52 (1976); J. Banaigs et al., Phys.Rev. C32, 1448 (1985).

[16] E. Scomparin et al., J.Phys. G19 , L51 (1993).

[17] T. Ueda, Phys.Rev.Lett. 66, 297 (1991).

[18] F. Plouin, P. Fleury & C. Wilkin, Phys.Rev.Lett. B65, 690 (1990).

[19] G. Dellacasa, contribution to this meeting. [20] A. M. Bergdolt et al, submitted for publication.

[21] J.-F. Germond & C. Wilkin, Nucl.Phys. A518, 308 (1990); J. M. Laget, F. Wellers & J. F. Lecolley,Phys.Lett.

B257, 254 (1991); T. Vetter et al., Phys.Lett. B263 153 (1991).

[22] K. Kilian & H. Nann in Meson Production near Threshold, edited by H. Nann & E. J. Stephenson,AIP Conf. Proc.

No. 221 (AIP, New York, 1990) p.

185. [23] C. Wilkin, in 6th Journ´ees d’Etudes Saturne, Mont Ste Odile, 1992, edited by M.Bordry (Labo-ratoire National Saturne, Paris 1992) p 121.Figure CaptionsFig.

1: Fits to the p d →3He η data of ref. [1] with complex scattering lengths a = (3.1 +i2.5) fm (solid curve) and (5.0 + i0.2) fm (dashed curve).

The Laget and Lecolley predictions(LL) [7] fall offmuch too slowly with pη.Fig. 2: The counting rate (solid curve) at the lowest energy point of ref.

[1] as a functionof ∆Tp, the energy above threshold in the p d →3He η reaction. This is a product of thecross section and experimental acceptance.Fig.

3: The square of the π−p →η n amplitude extracted from the total cross section dataof ref. [6], as a function of pη.

The solid line is a fit using eq. (4) with the imaginary part ofthe ηN scattering length constrained by unitarity, leading to (0.55 + i0.30) fm.

The dashedline is the best fit with Im[a (ηN)] = 0.Fig. 4: Predictions for the square of the p d →3He η amplitude using the optical potentialand eq.

(3) (solid curve) and its approximation by eq. (4) (dashed).

The pioneering data ofSPES4 [3] are shown as crosses and those of SPES2 [1] as circles.5

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