Resonance life time in BUU: observable consequences. Abstract

Transcription

1 Resonance life time in BUU: observable consequences. A.B. Larionov,M.Effenberger,S.Leupold,U.Mosel Institut für Theoretische Physik, Universität Giessen, D Giessen, Germany Abstract Within the transport BUU theory we study the influence of the choice for the -resonance life time on pion-nucleus reactions and on heavy-ion collisions at 1 A GeV. A quite small effect of these modifications on the pion absorption on nuclei is found provided that the absorption probability of the -resonance is modified consistently. Observable effects in the case of heavy-ion collisions are demonstrated. PACS numbers: 24.1.Cn; 24.1.Jv; 25.8.Ls; q; f Keywords: -resonance life time, BUU, pion absorption, Au+Au at 1 A GeV, pion production, collective flow, pion-proton correlations It is usually assumed that the life time of a resonance is given by its inverse total width 1/Γ. This assumption is widely used in the transport simulations of nuclear collisions in order to describe the decays of various baryonic and mesonic resonances. However, the quantum mechanical calculation of a two-particle scattering with only one partial wave gives the time delay in the transmission of the scattered wave equal to the derivative of the phase shift with respect to the center-of-mass (c.m.) energy [1]: Supported by GSI Darmstadt On leave from RRC I.V. Kurchatov Institute, Moscow, Russia Present address: SAP AG, D-6919 Walldorf, Germany 1

2 τ = dδ(e c.m.) de c.m.. (1) In the case of scattering through an intermediate resonance, it is natural, therefore, to associate the life time of the resonance with the delay time of the scattered wave. In particular, the life time of the (1232) resonance is given by Eq.(1) with δ(e c.m. ) δ 33 (E c.m. ), where δ 33 is the phase shift of the πn scattering in the S=3/2, I=3/2 channel. The two definitions of the life time, as 1/Γ and as the phase shift derivative, have completely different dependence on the c.m. energy (see Fig. 1 and Eqs. (7),(1) below): the inverse width decreases monotonically with E c.m. while the phase shift derivative has a maximum near m =1.232 GeV. Especially, at the threshold E c.m. = m N + m π =1.76 GeV, the inverse width becomes infinitely large while the phase shift derivative is zero. It is interesting, therefore, to find observable signals sensitive to the life time difference [2,3]. In the present work using the BUU model we study the influence of the -resonance life time on the pion absorption on nuclei and on some pionic observables from heavy-ion collisions at 1 A GeV. Recently it has been shown in Refs. [4,5] that Eq.(1) follows also from the Kadanoff-Baym formalism. Therein a transport equation has been derived for states with an arbitrary (continuous) mass spectrum without involving the commonly used quasiparticle approximation. Hence the formalism appears to be suited for states with large decay and/or collisional width. It turned out to be advantageous to incorporate a part of the collision processes in the propagation of the states. In this way it can be achieved that states created in a hot and dense medium can continuously change their in-medium properties towards their vacuum properties while they leave the dense region. Without this mechanism it might e.g. happen that states leave the system with an in-medium mass instead of the free one. (See [4,5] for details. For related works see also [6 8].) To account for the fact that a part of the collision processes is already covered by the propagation it turned out that the cross sections of the explicitly considered collisional events also have to be changed. The effect is completely analogous to the change of the resonance 2

3 life time from 1/Γ toeq.(1)namely d σ N NN dω = dσ N NN dω (Γτ) 1, (2) where dσ N NN /dω is a usual differential cross section obtained either by applying the detailed balance prescription from the measured NN N cross section [9] or directly as dσ N NN dω = 1 64π 2 M 2 p NN p N s 4 C NN, (3) where p NN and p N are the c.m. momenta of incoming and outgoing particles respectively, s is the c.m. energy squared, M 2 is the spin-averaged matrix element squared, and C NN =2 (1) if the final nucleons are identical (different). The same modification has to be done for any other process, where the -resonance enters in the initial state. Thus, the modified decay width in nuclear medium will be Γ Nπ =Γ Nπ (Γτ) 1, (4) where Γ Nπ is the standard decay width in nuclear matter taking into account the Pauli blocking for the final nucleon. The three-body absorption width, whenever it is included in BUU, also has to be changed as Γ NN NNN =Γ NN NNN (Γτ) 1. (5) Note that Eqs.(2),(4),(5) imply that the ratio of the partial life times of the -resonance with respect to decay and absorption is still equal to the ratio of the corresponding inverse decay widths (see Ref. [5] for details). Before discussing BUU results we will specify the explicit forms of the -resonance width and life time. Nonrelativistically, the phase shift near the pole energy E R of the resonance is expressed as (c.f. [1]) δ = arctan Γ 2(E R E c.m. ), (6) where Γ is the total width of the resonance. The pole energy and width are related to the real and imaginary parts of the retarded self energy of the resonance in its rest frame: 3

4 E R = m R +ReΣ+ (E = E c.m., p =), Γ= 2ImΣ + (E = E c.m., p =), where E and p are the resonance energy and momentum. Thus, generally, the resonance parameters E R and Γ are the functions of the c.m. energy E c.m. of the scattered particles. Substituting (6) into (1) one gets the expression for the resonance life time: τ(e c.m. )= 1 2 A(E c.m.)(1 K), (7) where A(E c.m. )= is the spectral function satisfying the normalization condition and K = Γ (E c.m. E R ) 2 +Γ 2 /4, (8) + dω A(ω) =1 2π de R + E c.m. E R dγ. (9) de c.m. Γ de c.m. For the total width of the -resonance we use a parameterization with Γ(E c.m. )=Γ ( q q ) 3 m E c.m. β 2 + q2 β 2 + q 2 (1) q = 1 (E 2 4Ec.m. 2 c.m. m2 π + m2 N )2 m 2 N (11) being the pion momentum in the rest frame of, q being the value of q at E c.m. = m, Γ =.118 GeV and β =.2 GeV. For simplicity we will neglect the dependence of ReΣ + on E c.m. putting E R = m in Eqs.(8),(9). We have to comment, that Eq.(1) represents, in fact, the vacuum decay width of the -resonance. In practice, however, the total width of the -resonance in nuclear matter is very close to the vacuum decay width [1]. 4

5 Fig. 1 shows the dependence of the -resonance life time on the πn c.m. energy as given by the derivative of the phase shift (solid line) and by the inverse width (dashed line). The shape of the function τ(e c.m. ) is basically dominated by the presence of the spectral function A(E c.m. ) in Eq.(7). This implies that resonances with mass near the pole mass m have the longest life time. In the following we will address the question whether the difference between the life time prescriptions could be visible in observable quantities. The numerical calculations have been performed on the basis of the BUU code in the version of Ref. [6] using the soft momentum-dependent mean field with the incompressibility K = 22 MeV (SM). The resonance production/absorption quenching [11] has been implemented in order to reproduce the experimentally measured pion multiplicity in central Au+Au collisions at 1 A GeV. First, we have performed a BUU calculation of the pion absorption on nuclei. To this aim we have selected the experimental data from Ref. [12] on reactions π + +C,Featthe pion beam energies E π = 85, 125, 165, 25, 245 and 315 MeV. Fig. 2 shows the calculated excitation function of the π + absorption cross section in comparison with the data. The standard BUU calculation (dashed lines) employing the life time 1/Γ underpredicts the data at the lower energies. Using the life time of Eq.(7) (dotted lines) improves the agreement amplifying the absorption peak at E π = 15 2 MeV. The peak corresponds to the beam energy E π =(m 2 m2 π m2 N )/2m N m π = 192 MeV at which the -resonance is excited at the pole mass. However, the peak position is slightly changed by the Fermi motion. The physical reason for the increased absorption with the life time of Eq.(7) lies in longer living -resonances near the pole mass (see Fig. 1, solid line). This increases the probability that a will be absorbed in the collision with a nucleon. Applying now Eq.(7) for the life time and, in addition, modifying the N NN cross section according to Eq.(2) results in the absorption cross section (solid lines) practically indistinguishable from the standard calculation, since the factor (Γτ) 1 in (2) is less than 1 near the pole mass of the. The discrepance between our calculations and the data at E π < 25 MeV can be explained by missing three-body absorption mechanism of the -resonance: NN NNN. 5

6 We have performed the calculation including the three-body absorption as parameterized by Oset and Salcedo [13] (see dash-dotted lines in Fig. 2). In this case we show only results with the life time of Eq.(7) and the modified absorption probability in the processes N NN and NN NNN (Eqs.(2),(5)), since these modifications, when done all simultaneously, have only a very small effect on the absorption cross section (c.f. solid and dashed lines). We see now a good description at the lower energies, but at higher energies the data are overpredicted somewhat. The authors of Ref. [3] have observed an influence of the life time variations on the K + in-plane flow in central Ni+Ni collisions at 1.93 A GeV. We have studied the π + in-plane flow for the system Au+Au at 1 A GeV and b=6 fm. Fig. 3 shows our calculations in comparison with the data from Ref. [14]. The acceptance of the detector [14] for π + s is good only at positive c.m. rapidities. This causes the measured <p x > (Y () )-dependence to be asymmetric with respect to Y () =. AtY () > all calculations generally agree with data within errorbars. However, there is a difference in the flow ( d<p x >/dy () at Y () = ) between the calculations: the calculation with the modified life time (7) and the modified cross section (2) produces less (negative) flow than the standard calculation, while the results with only modified life time (7) are practically the same within statistics with standard ones. Indeed, π + s exhibit the antiflow due to the superposition of the shadowing and the Coulomb repulsion from protons. A smaller cross section σ N NN near the pole mass gives less shadowing and, therefore, less antiflow. Fig. 4 shows the invariant mass spectrum of the correlated proton-pion pairs from the central collision Au+Au at 1.6 A GeV in comparison with the data from Ref. [15]. We have extracted the correlated pairs by selecting the proton and pion which are emitted from the same resonance and did not rescatter afterwards (see Ref. [16] for the comparison of this method with the background subtraction technique). The shape of the calculated spectrum well agrees with data, but the peak position is overpredicted by about 5 MeV. We checked, that the peak position is not changed and the spectrum gets slightly wider when taking into account also those pairs, where the nucleon experienced rescattering on other nucleons 6

7 one or two times. The calculations with the modified life time (7) (dotted line) and with modifying both life time and cross section (2) (solid line) result in somewhat sharper peaks at the invariant mass of 1.2 GeV. This is due to longer living -resonances near the pole mass, which reach the late freeze-out stage and then decay to the proton-pion pairs. The - resonances propagate now in an expanding nuclear matter and, therefore, their absorption is not so effective as in the pion-nucleus reactions. Thus, increasing the life time of the -resonances near the pole mass does not lead to their increased absorption, in distinction to the pion-nucleus case. To describe the peak position, some other important physical contribution should be taken into account in our calculations. That could be the off-shell pion dynamics [17]: Pions get off-shell due to large πn cross section. Then we expect some softening of the invariant mass (p, π) spectrum due to decreasing of the threshold mass. In conclusion, we have performed a comparative study of the two choices of the - resonance life time, i.e. the standard one (1/Γ) and the modified one of Eq.(1) (or Eq.(7)), which is the delay time of the scattered wave. A consistent transport theory should include also the corresponding modifications of the -resonance absorption probabilities according to Eqs.(2),(5). We observed in our calculations of the pion absorption on nuclei at E π = MeV that when all modifications are done simultaneously, their total effect becomes negligible. This is caused by two mutually compensating effects: longer partial life time of the -resonance with respect to decay and smaller absorption cross section N NN.In the case of Au+Au collisions at 1 A GeV the transverse in-plane pion flow and the invariant mass spectrum of the correlated (p, π + ) pairs do show some sensitivity to the life time and absorption modifications. Thus, detailed comparison with more precise data on the pion flow and (p, π) correlations could provide the constraints on the life time in nuclear matter. Authors are grateful to W. Cassing for useful discussions. 7

8 REFERENCES [1] A. Messiah, Quantum Mechanics, V. I, North-Holland Publishing Company, Amsterdam, [2] P. Danielewicz and S. Pratt, Phys. Rev. C 53, 249 (1996). [3] C. David, C. Hartnack and J. Aichelin, Nucl. Phys. A 65, 358 (1999). [4]S.Leupold,Nucl.Phys.A672, 475 (2). [5] S. Leupold, nucl-th/836, to appear in Nucl. Phys. A. [6] M. Effenberger, E.L. Bratkovskaya, and U. Mosel, Phys. Rev. C 6, (1999). [7] M. Effenberger and U. Mosel, Phys. Rev. C 6, 5191 (1999). [8] W. Cassing and S. Juchem, Nucl. Phys. A 665, 377 (2); Nucl. Phys. A 672, 417 (2); Nucl. Phys. A 677, 445 (2). [9]Gy.Wolf,W.CassingandU.Mosel,Nucl.Phys.A552, 549 (1993). [1] W. Ehehalt, W. Cassing, A. Engel, U. Mosel, and Gy. Wolf, Phys. Rev. C 47, R2467 (1993). [11] A.B. Larionov, W. Cassing, S. Leupold, U. Mosel, nucl-th/1319, to appear in Nucl. Phys. A. [12] D. Ashery et al., Phys. Rev. C 23, 2173 (1981). [13] E. Oset and L.L. Salcedo, Nucl. Phys. A 468, 631 (1987). [14] J.C. Kintner et al., Phys. Rev. Lett. 78, 4165 (1997). [15] M. Eskef et al., Eur. Phys. J. A 3, 335 (1998). [16] A.B. Larionov, W. Cassing, M. Effenberger, U. Mosel, Eur. Phys. J. A 7, 57 (2). [17] A.B. Larionov, W. Cassing, S. Leupold, U. Mosel, work in progress. 8

9 FIGURE CAPTIONS Fig. 1 The inverse width 1/Γ (dashed line), where Γ is given by Eq.(1), and the life time (solid line, Eq.(7)) of the -resonance as functions of the total c.m. energy of the pion and nucleon. Fig. 2 The beam energy dependence of the π + absorption cross section on carbon (upper panel) and on iron (lower panel). Standard BUU calculations (with 1/Γ life time) are represented by dashed line, while dotted and solid lines show respectively the results with the modified life time of Eq.(7) and with both modified life time and cross section N NN of Eq.(2). Dash-dotted line shows the calculations including the three-body absorption contribution [13] (see text for details). Experimental data are from Ref. [12]. Fig. 3 The π + average transverse momentum in the reaction plane vs. normalized c.m. rapidity Y () (y/y proj ) c.m. for the collision Au+Au at 1 A GeV and b=6 fm. Calculated curves are denoted as in Fig. 2. Data are from Ref. [14]. Fig. 4 Invariant mass spectrum of the correlated (p, π + ) pairs for the system Au+Au at 1 A GeV. Calculated curves are denoted as in Fig. 2. Data are from Ref. [15]. 9

10 6 5 1/Γ τ, 1/Γ (fm/c) τ E c.m. (GeV) FIG. 1.

11 σ abs (mb) data standard -life -life + cr. sect. Oset, -life + cr. sect. π + + C σ abs (mb) π + + Fe E π (MeV) FIG. 2.

12 <p x > (MeV/c) data, π + standard -life -life + cr. sect. Au+Au, 1 A GeV b=6 fm Y () FIG. 3.

13 dp/dm (GeV -1 ) data, (p,π + ) standard -life -life + cr. sect. Au+Au, 1 A GeV b= fm M (GeV) FIG. 4.

14 6 5 1/Γ τ, 1/Γ (fm/c) τ E c.m. (GeV)

15 σ abs (mb) data standard -life -life + cr. sect. Oset, -life + cr. sect. π + + C σ abs (mb) π + + Fe E π (MeV)

16 <p x > (MeV/c) data, π + standard -life -life + cr. sect. Au+Au, 1 A GeV b=6 fm Y ()

17 dp/dm (GeV -1 ) data, (p,π + ) standard -life -life + cr. sect. Au+Au, 1 A GeV b= fm M (GeV)