15 March 2001

Physics Letters B 502 (2001) 59–62www.elsevier.nl/locate/npe

Photoproduction of quasi-boundω mesons in nuclei

E. Marco∗,1, W. WeisePhysik Department, Technische Universität München, D-85747 Garching, Germany

Received 16 December 2000; accepted 24 January 2001Editor: J.-P. Blaizot

Abstract

We propose the(γ,p) reaction as a means of producing possible quasi-bound states ofω mesons in nuclei. We use an effectiveLagrangian, based on chiralSU(3) symmetry and vector meson dominance, in order to construct theω-nuclear potential. Thecontribution of boundω states is observable in the missing energy spectra of protons emitted in forward direction for severalnuclei. 2001 Published by Elsevier Science B.V.

PACS: 24.85.+p; 25.20.Lj

The behaviour of vector mesons in the nuclearmedium is one of the topics which attracts much at-tention in current nuclear physics. At high tempera-tures (and possibly at extremely high densities), thechiral symmetry of QCD is expected to be restoredand vector and axial vector mesons become degener-ate. At moderate densities characteristic of nuclei, oneexpects to see a downward shift of the spectral dis-tributions of vector mesons. QCD sum rules togetherwith current algebra considerations [1,2] suggest thatthe first moment of vector meson mass spectra shouldbe linked to the “chiral gap”, 4πfπ ∼ 1 GeV, and oneexpects the pion decay constantfπ to decrease withincreasing baryon density, roughly like the square rootof the chiral (quark) condensate.

Several studies, mostly concerned with theρ me-son spectrum and its possible implications for dilepton

Work supported in part by DFG.* Corresponding author.E-mail address: eugenio_marco@physik.tu-muenchen.de

(E. Marco).1 Fellow of the A. von Humboldt Foundation.

spectra produced in heavy-ion collisions, hint at a de-crease of vector meson masses in the nuclear medium.The study of QCD sum rules in the medium [3] and theBrown–Rho scaling hypothesis [4] suggest a droppingof the vector meson masses by approximately 15% atnormal nuclear matter density. Dynamical studies ofthe behaviour of vector mesons in the medium predicta considerable increase of theρ width, but no signifi-cant decrease in its mass [5–7]. For theω meson [8],such models predict a decrease of its mass accompa-nied by a moderate increase of its width.

The (d , 3He) reaction with recoilless kinematics hasbeen proposed in order to search for possibleη andωmeson bound states in nuclei [8–10]. This reaction al-lows the study of the in-medium behaviour of mesonsunder well controlled conditions, complementary toheavy-ion collisions.

In this Letter we propose the use of the(γ,p)reaction in nuclei to explore the behaviour of theωmeson in the nucleus. The difference between the(d ,3He) and(γ,p) reactions is that in the latter case,distortion effects are restricted only to the proton inthe final state. The optimal energies needed to produce

0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V.PII: S0370-2693(01)00188-5

60 E. Marco, W. Weise / Physics Letters B 502 (2001) 59–62

theω at rest, around 2.75 GeV in the photoproductioncase, are available at several experimental facilitiessuch as ELSA in Bonn and Spring8 in Japan.

In our calculation we have used the model devel-oped previously in [7,11,12] and updated in Ref. [8]to generate the (complex) potential experienced by theω in the nucleus. It is derived using an effective La-grangian which combines chiralSU(3) and vector me-son dominance. One of the important features of themodel is that the self-energy of theω in the medium isstrongly energy dependent, in contrast to the calcula-tions of [9,10], where a static potential is used. At nu-clear matter density theω mass is reduced by approxi-mately 15% and its width is increased up to around 40MeV. Unlike theρ meson with its prohibitively largein-medium width, theω has chances of being observedas a quasiparticle state in the nucleus.

Our choice of the kinematic conditions is such thattheω is produced practically at rest in the nucleus. Itslongitudinal and transverse self-energies are thereforenearly the same [7], and it is justified to work with asingle scalar self-energyΠ(E, r ). In the local densityapproximation, the potential that theω experiences inthe nuclear medium can be expressed as

Π(E, r ) ≡ 2EU

(E, r )

= −ReTωN(E)ρ(r )

(1)− i[EΓ (0)

ω (E)+ ImTωN(E)ρ(r )]

,

where,Γ (0)ω (E) is the freeω decay width, andTωN(E)

is the energy-dependentω-nucleon amplitude in freespace, which we take from [7].

To evaluate theω bound states one must solve thewave equation with the potential (1),

(2)[E2 + ∇2 −m2

ω −Π(E, r )]

φ(r ) = 0,

self-consistently to obtain the (complex) quasi-boundstate energiesEλ. We use a two parameter Fermi dis-tribution to describe the nuclear density distributions.The quasi-boundω states found for different nuclei aretabulated in Table 1, where we have introduced

(3)ελ = ReEλ −mω,

and the total decay widths

(4)Γλ = −2 ImEλ.

One observes that theω is bound even in light nucleisuch as6He. The widths are around 35–45 MeV.

Table 1Complex energy eigenvalues (see Eqs. (3), (4) of anω meson boundto several nuclei

(εnl ,Γnl ) [MeV]

Nucleus n l = 0 l = 1 l = 2

6ωHe (1) (−49,36) (−18,33) −

11ω B (1) (−66,41) (−40,39) (−13,37)

(2) (−14,34) − −

39ω K (1) (−88,44) (−73,45) (−57,45)

(2) (−54,45) (−36,44) (−17,44)

(3) (−16,41) − −

Their origin is primarily the reactionωN → πN

in the nucleus, as calculated in Ref. [7]. Althoughthese widths prohibit identifying individual peaks foreach state, one should nevertheless be able to observestrength at energies below the threshold for quasifreeω production.

The reaction that we propose in order to detectboundω states is(γ,p) on nuclei. At an incomingphoton energy of around 2.75 GeV, and with the pro-ton emitted in forward direction, theω is producednearly at rest. One can scan the contributions of theboundω states by observing the missing energy spec-trum, i.e., by measuring the energy of the outgoingproton in the forward direction and plotting the differ-ential cross section as a function ofEω −mω +|Bp| =Eγ +mp −Ep −mω, whereEp =mp + Tp is the de-tected energy of the outgoing proton, andBp is thebinding energy of the bound initial proton.

In order to evaluate the cross section we use the dis-torted wave impulse approximation method (DWIA)[13–15]. The nuclearω meson photoproduction crosssection, with the proton emitted at zero angle, is ex-pressed as(d2σγ+A→p+ω(A−1)

dE dΩ

)lab

θp=0

(5)=(dσγ+p→p+ω

dΩ

)lab

θp=0S(E).

For the free cross section,dσγ+p→p+ω/dΩ , we takea value of 0.3µb/sr from Ref. [16].

E. Marco, W. Weise / Physics Letters B 502 (2001) 59–62 61

S(E) is the response function, which takes intoaccount the removal of the proton from the nucleus,the binding of theω meson in the nucleus and thedistortion of the outgoing proton wave. The followingexpression holds forS(E):

S(E)=∑jp,lp

∑l,L

Np2l + 1

4π(lp0l0|L0)2

× Im

∞∫0

dr ′ r ′2w∗L

(r ′ )ψ∗

jplp

(r ′ )

×∞∫

0

dr r2wL(r)ψjplp (r)gl(E −Bp, r

′, r).

Hereψjplp (r) is the radial wave function of the initialbound proton,

(6)gl(E −Bp, r

′, r) = 2iEul(k, r<)v∗

l (k, r>),

is the radial Green function of theω meson for a givenangular momentuml, expressed in terms of the regularand outgoing solutions of the wave equation (2), and

wL(r)=1∫

−1

dcos(Θ) ei(pγ−pp)r cosΘ

(7)×D(z(Θ), b(Θ)

)PL(cosΘ),

with Legendre polynomialsPL. The distortion factorD(r ) is evaluated using the eikonal approximation forthe wave function of the outgoing proton:

(8)ψ†f

( pp, r) = e−ippzD

(r ).

This approximation is justified, given that the protonhas a kinetic energy of 1–2 GeV in our cases ofinterest. In terms ofz = r cosΘ and the impactparameterb, the distortion factor has the form

(9)D(r ) = exp

[−σpN

2

∞∫z

dz′ ρ(z′, b

)].

For the proton–nucleon cross section we have taken avalueσpN = 40 mb. Fig. 1 showsD(r )ρ(r )/ρ(0) inorder to illustrate the “active” zone of the process. Oneobserves that the reaction takes place predominantlyin the rear hemisphere of the nucleus, since theprotons are distorted on their way out. In the case ofthe (d , 3He) reaction, only the edges of the nuclear

Fig. 1. Values ofD(z,b)ρ(r)/ρ(0) for the (γ,p) reaction on12C.The reaction takes place predominantly at the darker regions. Thecircle indicates the r.m.s. radius of12C.

Fig. 2. Missing energy spectra for the12C(γ,p)ω11B reaction atEγ = 2.75 GeV. Dotted lines represent the contributions from twoparticular combinations of boundω and proton-hole states.

surface are actively involved [8], because the incomingdeuteron and the outgoing3He are both stronglyabsorbed. As a consequence, the reduction of the (d ,3He) cross section forω production is about 10 timesstronger than in the photoproduction case.

In Fig. 2 we show the calculated proton missingenergy spectrum for the12C(γ,p)ω11B reaction withan incoming photon energy of 2.75 GeV. This is theenergy at which a freeω would be produced at rest.Pronounced structures coming from different boundω

states can be seen below the threshold for quasi-freeω

production. The figure also shows separately the con-

62 E. Marco, W. Weise / Physics Letters B 502 (2001) 59–62

tributions of two of the more prominent combinationsof ω and proton states. The prominent structure seen atEω −mω + |Bp| 13 MeV is characteristic ofω11Bwith theω and the initially bound proton inp-orbitals,and reflects a threshold effect.

The results for the(γ,p) reaction on 40Ca atEγ = 1.5 GeV and 2.75 GeV are shown in Figs. 3and 4, respectively. In both cases there are importantcontributions coming from the boundω mesons.For Eγ = 1.5 GeV, a freeω meson would have amomentum of around 130 MeV/c, comparable to thatin the suggested (d , 3He) experiments [8,9], atTd =4 GeV. The cross section atEγ = 2.75 GeV is slightlyhigher than atEγ = 1.5 GeV.

Fig. 3. Missing energy spectra for the40Ca(γ,p)ω39K reaction atEγ = 1.5 GeV.

Fig. 4. Missing energy spectra for the40Ca(γ,p)ω39K reaction atEγ = 2.75 GeV.

If only the missing energy of the recoiling protonis detected, the spectrum of produced quasiboundω

mesons is expected to sit on a background, the crosssection of which may be approximately five times aslarge as theω meson signal itself. This backgroundshould be flat, resulting primarily from the(γ,p)reactions leading toρ meson and continuumππproduction, with theρ meson width strongly increasedin the presence of the nucleus. Ideally, the backgroundwould be reduced by detecting a characteristic decaymode of theω meson together with the forward proton.

Acknowledgements

We thank Albrecht Gillitzer, Satoru Hirenzaki, PaulKienle, Eberhard Klempt, Berthold Schoch and Hi-roshi Toki for helpful discussions.

References

[1] E. Marco, W. Weise, Phys. Lett. B 482 (2000) 87;F. Klingl, W. Weise, Eur. Phys. J. A 4 (1999) 225.

[2] M. Golterman, S. Peris, Phys. Rev. D 61 (2000) 034018.[3] T. Hatsuda, S.H. Lee, Phys. Rev. C 46 (1992) 34.[4] G.E. Brown, M. Rho, Phys. Rev. Lett. 66 (1991) 2720.[5] G. Chanfray, R. Rapp, J. Wambach, Phys. Rev. Lett. 76 (1996)

368.[6] R. Rapp, J. Wambach, hep-ph/9909229, Adv. Nucl. Phys.

(2000), submitted for publication.[7] F. Klingl, N. Kaiser, W. Weise, Nucl. Phys. A 624 (1997) 527.[8] F. Klingl, T. Waas, W. Weise, Nucl. Phys. A 650 (1999) 299.[9] R.S. Hayano, S. Hirenzaki, A. Gillitzer, Eur. Phys. J. A (1999)

99.[10] K. Tsushima, D.H. Lu, A.W. Thomas, K. Saito, Phys. Lett.

B 443 (1998) 26;K. Saito, K. Tsushima, D.H. Lu, A.W. Thomas, Phys. Rev.C 59 (1999) 1203.

[11] F. Klingl, N. Kaiser, W. Weise, Z. Phys. A 356 (1996) 193.[12] N. Kaiser, T. Waas, W. Weise, Nucl. Phys. A 612 (1997) 297.[13] O. Morimatsu, K. Yazaki, Nucl. Phys. A 483 (1988) 493;

O. Morimatsu, K. Yazaki, Nucl. Phys. A 435 (1985) 727.[14] J. Hüfner, S.Y. Lee, H.A. Weidenmüller, Nucl. Phys. A 234

(1974) 42.[15] C.B. Dover, L. Ludeking, G.E. Walker, Phys. Rev. C 22 (1980)

2073.[16] E. Klempt, in: D. Menze, B. Metsch (Eds.), Proceedings

of International Conference “Baryons ’98”, World Scientific,Singapore, 1999, p. 25.