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Dilepton anisotropy from p + Be and Ca + Ca collisions at BEVALAC energies
23 May 1996
PHYSICS ELSEVIER
Physics Letters B 376 (1996) 12-
LETTERS 6
18
Dilepton anisotropy from p + Be and Ca + Ca collisions at BEVALAC energies * E.L. Bratkovskaya
‘, W. Cassing, U. Mosel
Insfitut fir Theoretische Physik, Oniversitiit Giessen, D-35392 Giessen, Germany Received
15 January
1996;
revised manuscript received 16 February 1996 Editor: C. Mahaux
Abstract
A full calculation of lepton-pair angular characteristics is carried out for e+e- pairs created in p + Be and Ca + Ca collisions from 1.O to 2.1 GeV/A. It is demonstrated that the dilepton decay anisotropy depends sensitively on the different sources and may be used for disentangling them. Due to the dominance of the q- and A-Dalitz decays and only a small anisotropy coefficient for ST’VT- annihilation, the expected anisotropy coefficients show a decrease with invariant mass of the dilepton pair and change only moderately when comparing p + Be and Ca + Ca reactions at the same bombarding energy. PACS: 24.10-i; 14.60.-z Keywords: Heavy-ion collisions;
Leptons
Dileptons are quite attractive electromagnetic signals since they provide almost direct information on the hot and dense nuclear phase in heavy-ion collisions at BEVALAC/SIS and SPS energies [ I-41. The information carried out by leptons may tell us not only about the interaction dynamics of colliding nuclei, but also on properties of hadrons in the nuclear environment or on a possible phase transition of hadrons into a quark-gluon plasma (cf. [ 51) . However, there are a lot of hadronic sources for dileptons because the electromagnetic field couples to all charges and magnetic moments. In particular, in hadron-hadron collisions, the e+e- pairs are created due to the electromagnetic
* Supported by BMIT and GSI Darmstadt. ’Permanent address: Bogoliubov Laboratory Physics, Joint Institute for Nuclear Research, Moscow Region, Russia.
of Theoretical 141980 Dubna,
0370-2693/96/$12.00 @ 1996 Elsevier Science B.V. All rights reserved PI1 SO370-2693(96)00302-4
decay of time-like virtual photons which can result from the bremsstrahlung process or from the decay of baryonic and mesonic resonances including the direct conversion of vector mesons into virtual photons in accordance with the vector dominance hypothesis. In the nuclear medium, the properties of these sources may be modified and it is thus very desirable to have experimental observables which allow to disentangle the various channels of dilepton production. Recently, we have proposed to use lepton pair angular distributions for a distinction between different sources [ 6.71, Indeed, the coupling of a virtual photon to hadrons induces a dynamical spin alignment of both the resonances and the virtual photons. One thus can expect that the angular distribution of a lepton will be anisotropic with respect to the direction of the dilepton (i.e. virtual photon) emission. It has been shown that due to the spin alignment of the virtual photon and
E.L. Bratkouskuya et al./Physics
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Lerters B 376 (1996) 12-18
,
,
,
,
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0.6
0.7
0.8
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M (GeV)
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Fig. I The dilepton invariant mass spectra dcr/dM from p + 9Be and “‘Ca + 40Ca collisions at bombarding energies from 1 to 2.1 GeV/A in comparison to the experimental data [ I-31. The “7” denotes the contribution of the v-channel, the “A” labels the contribution of the A Dalitz decay, “N*” the N* Dalitz decay, “o -+ ?re+e- ” is the w -+ r&eDalitz decay, ‘@I” the proton-neutron bremsstmhlung, “TN” the Z-N bremsstrahlunz. ?r+’ the nion annihilation channel, while “p, w, W’ is the direct decay of vector mesons. The solid curves (denoted by “all”) show thesum of all sources.
the spins of colliding or decaying hadrons, this @ton decay anisotropy turns out to be quite sensitive to the specific production channel (cf. also [ 8,9] ) . Whereas in our previous work [ 6,7,10] we have considered the dilepton anisotropy for elementary nucleon-nucleon or pd collisions, we now evaluate this new observable for the first time for proton-nucleus and nucleus-nucleus collisions from 1 to 2 GeV/A bombarding energy, where the calculated inclusive dilepton spectra can be controlled in comparison to the DLS data [ l-31. The dynamical evolution of the proton-nucleus or nucleus-nucleus collision is described by a transport equation of the Boltzmann-Uehling-Uhlenbeck type evolving phase-space distribution functions for nucleons, A’s, N* ( 144O)‘s, N* ( 1535) ‘s,pions and 7’s with their isospin degrees of freedom. The details of the model are discussed in Refs. [ 11,121; here we use the same prescriptions and include additionally the pro-
duction channels from the direct decay of the vector mesons p, w, and Cp as well as the Dalitz-decay of the o-meson [ 131 employing the formfactors from Landsberg [ 141. In Fig. 1 we present the calculated dilepton invariant mass spectra dc/dM for p + 9Be and 40Ca + 40Ca collisions at bombarding energies from 1 to 2.1 GeV/A and compare them with the experimental data of the DLS collaboration [ l-31 including the DLS acceptance filter as well a mass resolution of 50 MeV. In the “cocktail” plot the A Dalitz decay (labeled by “A”), N’ Dalitz decay (“N*“), proton-neutron bremsstrahlung (“@>, TN bremsstrahlung (“TN”) (all without VDM formfactor), o -+ n-e+e- Dalitz decay, 7 Dalitz decay (“r]“), the pion annihilation channel (‘%rr”) as well as the direct decay of the vector mesons (“p, o, a”> are displayed explicitely while the solid curves (denoted by “all”) represent the sum
14
E.L. Bratkouskuyu
et d/Physics
of all sources. As seen from Fig. 1 the dilepton cross sections agree reasonably well with the experimental data as in [ 121. We will use the same cross sections, however, without experimental DLS filter for the calculation of the anisotropy coefficients. In order to characterize the dilepton decay anisotropy we have introduced in [ 671 the anisotropy coefficient B which allows one to describe the angular distribution of dileptons created in a hadron-hadron (h + h) reaction via da!‘” dhfd
cis 6/,h
= A!“( 1 + B”h cos2 8,,,,).
Eq. ( 1) is the result of averaging the general polynomial angular distribution for the decay of a vector particle (cf. [ 151) over the azimuthal angle. In Eq. (1) 8,,,, is the angle between the electron momentum Z? and the virtual photon momentum q”” in the limit of 4 ‘I’ ---f 0, i.e. in the rest frame of the massive virtual phz Mathematically, t9,*his defined by cos 8/,,, = (IT, vi”) with the electron momentum 1: measured in the dilepton center-of-mass system (q* E 2: -i-Z: = 0), while yi;h = qhh/qih is the velocity of the dilepton c.m.s. relative to the h + h cm. system. M is the invariant mass of a lepton pair (M* = qi - q2) ; the coefficient Bfh describes the anisotropy while AFh determines the magnitude of the cross section. The total differential cross section for h + h collisions now can be represented as a sum of the differential cross sections for all channels da!‘”
dr’lh
c dMd cos Bhh = i=channe, dMd c;s e/t/, = A”” ( M) ( 1 + B”” ( M) cos* 8,,,,) , which leads to the total anisotropy B”“(M) =
c
(2)
coefficient
Letters B 376 (1996) 12-18
“weighted” anisotropy coefficients ( ( Bfh) ) for each channel i obtained by means of the convolution of BFh with the corresponding invariant mass distribution (cf.
[lOI). For heavy-ion reactions the situation becomes more complicated due to the nuclear dynamics and the explicit time evolution of the interacting system. Here we start from the point that the form of the angular distribution for all “elementary” interactions a + b, that occur in the nucleus-nucleus reaction A + B, are known. For this aim we employ the results of our previous works [ 6,7, IO] where the anisotropy coefficients for thepp, and pn bremsstrahlung, NN + A + NNe+e- Dalitz decay, v Dali& decay and n-+7~- annihilation channels were calculated explicitly in the hadron-hadron center-of-mass system a + 6. The differential angular distribution in elementary a+b collisions - before averaging over the momentum 4 ab _ can be represented as daeb dM dqab ii!cos eu,,
= Ayb (1 + Bfrbcos2 &/,) ,
(4)
where the coefficient Bj” for the elementary process a + b is a function of the dilepton mass M, the masses m,, mb and the initial invariant energy fi of the hadrons involved in the reaction: By6 = Bpb (M, fi; m,, mb) . It is important to note that we know the “elementary” coefficient only in the a + b system. In nucleus-nucleus collisions, however, the direction vi” = gb/qtfb is changed in each elementary a + b collision. Thus in order to define an anisotropy coefficient in the latter situation we have to perform an angular transformation from the elementary c.m.s. (0,b) to the c.m.s. of the colliding nuclei ( BAB) : cos 6,b = cos eABcos 8,
+
(B!“(M)),
sin 8.4~ sin 8y cos( $OpAB - $C$) ,
(5)
r=channel
dal’h, (B;‘*(M))
=
dM
B!‘h 1 + $3fh (3)
where 0, is the angle between the dilepton c.m.s. vein the c.m.s. of a+b and the dileplocity v$ = fb/gb ton c.m.s. velocity v,“” = qAB/qtB in the c.m.s. of the colliding nuclei A+B: cos0, = v~~.v~~/( Iv:“I Ivi’I).
where the special weighting factors originate from the necessary angle-integrations. Thus, the anisotropy coefficient Bhh for h + h reactions is the sum of the
Here, the vector qab is obtained by a Lorentz transformation of qAB: q“b = L(Vab)qAB, where Vab = (pa +&,)/(%I +E b) is the velocity of the a + b system relative to the A + B system.
15
E.L. Brutkmskuya et al./Physics Letters B 376 (19%) 12-18
Substituting (5) into Eq. (4) and using dfiA~ = d&b we get the respective distribution in the observ-
able angle @Asby integrating over the azimuthal angle FJAB:
da;” dMd cos OAB /dqAB
N
AybAi
(1
+B;(~,)cos~~AB),
(6)
the same angular transformation from the c.m.s. of the pions a + b to the c.m.s. of the nuclei A + B as in Eq. (5) with the replacement tY4+ 8,. Here, 8, is the angle between the pion momentum pz in the c.m.s. of a + b and the vector Y? = qAB/qtB. Following the same angular integration as for the previous cases we end up with the expression for the anisotropy coefficient for rTT+r- annihilation: BAB
=
Sdcos& h%) B(en)w(coseT)
Im
Jdcose,
pb
A(8,) = 1 + +
= ----(3c0s2eq 2&e,)
- 1)
(7)
using Afb from (4). Obviously, the transformation (5) does not change the quadratic form in cos 6 of the angular distribution. From Eqs. ( 1) and (6) and the normalization condition we finally get
B!B I
w(cose,)
’
(9)
sin2 eq,
B@
B(e,)
Ace,)
dzdL:B dz;AB.
The quantities A(&), B(B,) are defined by Eq. (7) using B& = - 1 while cos & can be computed by the scalar products of the four mOmenta of pions pa,pb and colliding nuclei PA, PB, cos 8,
=
SdqAB @e,)B(e,)
-
j-dqAB
Ace,)
(8)
Eq. ( 8) implies that the numerator and denominator of BfB can be calculated independently within the BUD runs by integrating the functions A( 6’,) and B( 6,) - resulting from the Lorentz transformations - with the differential cross sections dofb/ ( dMdqAB) for the different channels i, which are defined in the same way as described in [ 11,121. The respective “reduced” differential cross sections dafB/dM are shown in Fig. 1 for the systems to be studied below. The above definition of the anisotropy coefficient in nucleus-nucleus collisions is valid for all channels except for rr+t--annihilation because there are only two particles in the initial and final states. For this particular reaction we have to use the angle of the pion momentum with respect to the lepton momentum in the c.m.s. of the leptons (or pions, which is the same for this channel), i.e. gb = pz + p;I = ZS_+ Z? = 0. As was shown in Ref. [ 61, the elementary anisotropy coefficient in the 7~72c.m.s. then is B,+,- = - 1. In heavy-ion collisions we can use the same definition for the polar angle as for the other channels because we can reconstruct the direction of qAB in the c.m.s. of the colliding nuclei A +B. Furthermore, we performed
with M2 = sob = (p, + pb)2 following Teryaev [ 161. A closer look at Eq. (9) shows that Bzt is a constant; its absolute value follows from the 7~+7r- annihilation angular distribution W( cos 0,) with cos 8, determined by ( 10). In case of an isotropic angular distribution, W(cos&) = const., the anisotropy coefficient Btt = 0 according to Eq. (9). Moreover, in the BUU calculation the angular distribution W( cos 0,) is also a function of time t due to the dynamical evolution of the system. Technically we define the distribution W(t) at time t (i.e. in the time interval [ t - At/2; t + At/2] ) as the ratio of the number of pion annihilation events with fixed cos 0, produced in the above time interval to the total number of &rTT- events produced in the A + B reaction, i.e. w(t,cose,)
= N(t,cose,)p,,,.
In order to demonstrate the rfrTangular anisotropy we show in Fig. 2 (1.h.s.) the rTr+w- angular distribution W( t, cos 0,) for p + 9Be at 2.1 GeV and 40Ca+40Ca at 1.0 and 2.0 GeV/A at those times t when the maximal number of pion annihilation events N(t) occured during the interval At; the respective &7r--annihilation rate fi( t) /IV,, is illustrated in the r.h.s. of Fig. 2 for the same reactions. Due to the low number of pion annihilation events for p + 9Be the angular distribution suffers from low statistics and
16
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Letters B 376 (1996) 12-18
I
t=7fm/c,
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T
J
I
:
‘Be, 2.1 GeV
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A ~+~Bie, ::
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:: j: .i :: i !i
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TT 9 2
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/j
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Fig. 2. The QT+~- angular distribution W( t, cos 0,) for p + “Be at 2.1 GeV and ” Ca + “‘Ca at 1.0, 2.0 GeV and time I (1.h.s.) where the maximal number of pions annihilation events N(t) occured. The pion annihilation rate lir(t)/N,,, is shown in the r.h.s. of the figure
for the same reactions.
the error bars - which result from different runs with 8000 testparticles per nucleon - are large. For the further analysis we have used the dotted line in Fig. 2 which represents a fit to the angular distribution. On the other hand, for 40Ca + 40Ca collisions the statistics is good enough and we show only fits to the “numerical” data in terms of the solid and dash-dotted lines, respectively. According to Eq. (10) the n-+7~- angular distribution W(cos 19,) is related to the energy and angular distribution of the pions that annihilate. In the c.m.s. of the pions - where B?r is defined - the angular distributions displayed in Fig. 2 (1.h.s.) correspond to annihilating pions having larger transverse then longitudinal momentum relative to qAB. A closer analysis of the events, furthermore, shows that the anisotropy of W( cos 0,) is essentially due to the nonequilibrium phase of the reaction. At 2 GeV/A, the r+rangular distribution becomes more isotropic than at 1 GeV/A in line with a “pionic fireball” scenario. As a consequence the resulting angular distribution shows only a very modest anisotropy. Before going over to the calculation of the
anisotropy coefficient for nucleus-nucleus collisions one has to take into account that resonances (A, N* ) can be created in quite different elementary channels than in pN or pd reactions. For example, A production via the z-N + A channel becomes quite important; an elementary channel for which the dilepton anisotropy has not been computed so far. We thus have calculated the efeanisotropy using the same vertices, delta-propagator, coupling constants and formfactors as in Refs. [ 7,171. Since this evaluation is straight forward, we do not present the details here. The results of our microscopic calculation for the anisotropy coefficient for the n-+ N + A + e+e- +X channel are displayed in Fig. 3. The anisotropy B=N_+h is a function of the dilepton invariant mass M and the invariant energy of the interacting particles s = (p, + pi)’ s Mi. For small fi only deltas with M MAO = 1.232 GeV appear and the coefficient MA B TN-A + 1 as expected for the Dali& decay of a free delta [6]. With increasing fi more energetic deltas can be created and (for fixed M) the phase-space for the final nucleon and virtual photon increases leading to a decrease of BrN-,~. We note that we do not take
E.L. Brutkovskaya
et d/Physics
1.” nN
--A
-e+r-+x
U.8
5
U.6
7
2
“A
“2
“.”
0.0
0.1
’ 0.1
\, “2
0.4
,\I B.5
0.6
1 0.7
M ,GCV,
Fig. 3. Results of our calculation for the anisotropy coefficient of the T + N + A --t e+r- + X channel at Mb = 4 from 1.232
to 1.8 GeV. into account the Dalitz decay of the higher nucleon resonances in the further calculations because their statistical weight is too low. Fig. 4 finally shows the computed weighted anisotropy coefficients (Bi(M)) for p + 9Be and %a + 40Ca collisions at the bombarding energies from 1 to 2.1 GeV/A. The main contributions arise from the 7 and A Dalitz decays due to their large “elementary” anisotropy coefficients and cross sections (cf. Fig. l), respectively. The contribution from pn bremsstrahlung is practically zero at all energies due to a smaller “elementary” anisotropy coefficient and due to a lower cross section as well. The weighted coefficient from T+~T- annihilation is rather small (M 0.1) even for the Ca + Ca reactions and decreases for M 2 mp due to the threshold behaviour of the cross section. However, compared to (B,+,-(M)) for p + 9Be, where pion annihilation is very low, a clear (but moderate) enhancement can be extracted. The contributions of the further channels (N*, TN bremsstrahlung, w + ?r”e+e-) are also negligible due to their small cross sections (cf. Fig. 1) . The cross section from direct decays of the vector mesons becomes compatible with pion annihilation for M x mp for p + 9Be at 2.1 GeV, but the anisotropy coefficient for the “free” vector meson in the vacuum is zero [ 181. We do not discuss here a possible modification of the p-meson properties in the medium that might lead to non-isotropic angular distributions of dileptons because for p + 9Be at 2.1 GeV one cannot reach sufficiently high baryon or pion densities. On the other hand, for 40Ca + 40Ca collisions at
Letters B 376 (1996) 12-18
17
2.0 GeV and for p + ‘Be at 2.1 GeV we observe a clear anisotropy from the 71 Dalitz decay, while for p + 9Be at 1 GeV the A Dalitz decay gives the main contribution at small invariant mass due to a dominant A cross section; the 7 coefficient increases at M from 0.4 to 0.5 GeV for the same reason. For clarity we briefly discuss the steps to “extract” the anisotropy coefficient from experimental data. In dilepton experiments the four-momenta of leptons in the c.m.s. of the nuclei A + B (or in the laboratory frame, which is connected with the c.m.s. by a simple Lorentz transformation) are measured - (&AB,lA_B), (&“+“,Z$ ). The four momentum q = (qf , qAB) of the virtual photon in the A + B c.m.s. then is given by: qt* = dB -t ctB, qAB = 1”” + I”+“. The angle 6AB has to be computed via
where ey[ is the angle between
the qAB and I!‘,
i.e.
cos&# = I”_“. qAB/(IZABI /qABI). In order to extract the anisotropy coefficient one has to count the number of dilepton events N( M, cos BAB) with fixed cos 6)ABand invariant mass M, The anisotropy coefficient for the invariant mass M then is simply given by BAB(M) =
N(M,COSeAB = 1) _ 1 N( M, COSeAB = 0)
’
(12)
Thus, summarizing, the calculated anisotropy coefficients for p + Be and Ca + Ca collisions support our suggestion in Refs. [ 6,7,10] that the dilepton decay anisotropy may serve as an additional observable to decompose the dilepton spectra into the various sources. Since the anisotropy vanishes in a hadronic fireball scenario our present results provide valuable information about the nonequilibrium stage of the reactions. We gratefully acknowledge many helpful discussions with S.S. Shimanskij, S. Teis, O.V. Teryaev, A.I. Titov, V.D. Toneev and Gy. Wolf.
E.L. Bratkovskaya et al./Physics
18
Letters B 376 (1996) 12-18
P+~B~
0.8
2.1 GeV
0.6 L
I
I.
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1.
0.3
I.
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I.
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F
%i+%~
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1 0.6 -
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0.3
0.4
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Fig. 4. The weighted anisotropy coefficients The notation is the same as in Fig. 1.
0.5
0.6
M (GeV)
M (GeV)
(Bi( M)) for p +YBe and 40Ca+“aCa
References 1 II G. Roche et al., Phys. Rev. Lett. 61 (1988) 1069. 12 1 C. Naudet et al., Phys. Rev. Len. 62 ( 1989) 2652. [3 I G. Roche et al., Phys. Len. B 226 (1989) 228. (4 I G. Agakichiev et al., Phys. Rev. Lea. 75 (1995) 1272. I 5 I U. Mosel, Ann. Rev. Nucl. Part. Sci. 41 ( 199 I ) 29. 16 I E.L. Bratkovskaya, O.V. Teryaev and V.D. Toneev, Phys. Lett. B 348 ( 1995) 283. E.L. Bratkovskaya, M. Schafer, W. Cassing, U. Mosel, O.V. Teryaev and V.D. Toneev, Phys. Lett. B 348 ( 1995) 325. J. Zhang, R. Tabti, Ch. Gale and K. Haglin, Preprint McGill/94-56, MSUCL-961. ‘1 T.I. Gulamov, Al. Titov and B. Kampfer, Preprint JINR E2-95153.
collisions
at bombarding
[lo] E.L. Bratkovskaya, [ 111 12) 131 141 IS]
_ [ 161 [17]
1IS]
energies from 1 to 2.1 GeV/A.
W. Cassing, U. Mosel, O.V. Teryaev, A.I. Titov and V.D. Toneev, Phys. Lett. B 362 ( 1995) 17. Gy. Wolf, G. Batko, W. Gassing, U. Mosel, K. Niita and M. Schafer, Nucl. Phys. A 517 (1990) 615. Gy. Wolf, W. Cassing and U. Mosel, Nucl. Phys. A 552 (1993) 549. W. Cassing, W. Ehehah and C.M. Ko, Phys. Lett. B 363 (1995) 35. L.G. Landsberg, Phys. Rep. 128 (1985) 301. B.T. Feld, Models of Elementary Particles, Blaisdell Publishing Company ( 1969). O.V. Teryaev, private communication. M. Schafer, H.C. Donges, A. Engel and U. Mosel, Nucl. Phys. A 575 (1994) 429. C. Gale and J.I. Kapusta, Nucl. Phys. B 357 (1991) 65.
PHYSICS ELSEVIER
Physics Letters B 376 (1996) 12-
LETTERS 6
18
Dilepton anisotropy from p + Be and Ca + Ca collisions at BEVALAC energies * E.L. Bratkovskaya
‘, W. Cassing, U. Mosel
Insfitut fir Theoretische Physik, Oniversitiit Giessen, D-35392 Giessen, Germany Received
15 January
1996;
revised manuscript received 16 February 1996 Editor: C. Mahaux
Abstract
A full calculation of lepton-pair angular characteristics is carried out for e+e- pairs created in p + Be and Ca + Ca collisions from 1.O to 2.1 GeV/A. It is demonstrated that the dilepton decay anisotropy depends sensitively on the different sources and may be used for disentangling them. Due to the dominance of the q- and A-Dalitz decays and only a small anisotropy coefficient for ST’VT- annihilation, the expected anisotropy coefficients show a decrease with invariant mass of the dilepton pair and change only moderately when comparing p + Be and Ca + Ca reactions at the same bombarding energy. PACS: 24.10-i; 14.60.-z Keywords: Heavy-ion collisions;
Leptons
Dileptons are quite attractive electromagnetic signals since they provide almost direct information on the hot and dense nuclear phase in heavy-ion collisions at BEVALAC/SIS and SPS energies [ I-41. The information carried out by leptons may tell us not only about the interaction dynamics of colliding nuclei, but also on properties of hadrons in the nuclear environment or on a possible phase transition of hadrons into a quark-gluon plasma (cf. [ 51) . However, there are a lot of hadronic sources for dileptons because the electromagnetic field couples to all charges and magnetic moments. In particular, in hadron-hadron collisions, the e+e- pairs are created due to the electromagnetic
* Supported by BMIT and GSI Darmstadt. ’Permanent address: Bogoliubov Laboratory Physics, Joint Institute for Nuclear Research, Moscow Region, Russia.
of Theoretical 141980 Dubna,
0370-2693/96/$12.00 @ 1996 Elsevier Science B.V. All rights reserved PI1 SO370-2693(96)00302-4
decay of time-like virtual photons which can result from the bremsstrahlung process or from the decay of baryonic and mesonic resonances including the direct conversion of vector mesons into virtual photons in accordance with the vector dominance hypothesis. In the nuclear medium, the properties of these sources may be modified and it is thus very desirable to have experimental observables which allow to disentangle the various channels of dilepton production. Recently, we have proposed to use lepton pair angular distributions for a distinction between different sources [ 6.71, Indeed, the coupling of a virtual photon to hadrons induces a dynamical spin alignment of both the resonances and the virtual photons. One thus can expect that the angular distribution of a lepton will be anisotropic with respect to the direction of the dilepton (i.e. virtual photon) emission. It has been shown that due to the spin alignment of the virtual photon and
E.L. Bratkouskuya et al./Physics
0.15
0.20
0.25
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,
,
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0.3
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0.8
0.9
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M (GeV)
M (GeV) loo,
13
Lerters B 376 (1996) 12-18
,
,
,
,
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0.6
0.7
0.8
0.9
1.0
M (GeV)
0.2
0.3
0.4
0.5
0.6
M (GeV)
Fig. I The dilepton invariant mass spectra dcr/dM from p + 9Be and “‘Ca + 40Ca collisions at bombarding energies from 1 to 2.1 GeV/A in comparison to the experimental data [ I-31. The “7” denotes the contribution of the v-channel, the “A” labels the contribution of the A Dalitz decay, “N*” the N* Dalitz decay, “o -+ ?re+e- ” is the w -+ r&eDalitz decay, ‘@I” the proton-neutron bremsstmhlung, “TN” the Z-N bremsstrahlunz. ?r+’ the nion annihilation channel, while “p, w, W’ is the direct decay of vector mesons. The solid curves (denoted by “all”) show thesum of all sources.
the spins of colliding or decaying hadrons, this @ton decay anisotropy turns out to be quite sensitive to the specific production channel (cf. also [ 8,9] ) . Whereas in our previous work [ 6,7,10] we have considered the dilepton anisotropy for elementary nucleon-nucleon or pd collisions, we now evaluate this new observable for the first time for proton-nucleus and nucleus-nucleus collisions from 1 to 2 GeV/A bombarding energy, where the calculated inclusive dilepton spectra can be controlled in comparison to the DLS data [ l-31. The dynamical evolution of the proton-nucleus or nucleus-nucleus collision is described by a transport equation of the Boltzmann-Uehling-Uhlenbeck type evolving phase-space distribution functions for nucleons, A’s, N* ( 144O)‘s, N* ( 1535) ‘s,pions and 7’s with their isospin degrees of freedom. The details of the model are discussed in Refs. [ 11,121; here we use the same prescriptions and include additionally the pro-
duction channels from the direct decay of the vector mesons p, w, and Cp as well as the Dalitz-decay of the o-meson [ 131 employing the formfactors from Landsberg [ 141. In Fig. 1 we present the calculated dilepton invariant mass spectra dc/dM for p + 9Be and 40Ca + 40Ca collisions at bombarding energies from 1 to 2.1 GeV/A and compare them with the experimental data of the DLS collaboration [ l-31 including the DLS acceptance filter as well a mass resolution of 50 MeV. In the “cocktail” plot the A Dalitz decay (labeled by “A”), N’ Dalitz decay (“N*“), proton-neutron bremsstrahlung (“@>, TN bremsstrahlung (“TN”) (all without VDM formfactor), o -+ n-e+e- Dalitz decay, 7 Dalitz decay (“r]“), the pion annihilation channel (‘%rr”) as well as the direct decay of the vector mesons (“p, o, a”> are displayed explicitely while the solid curves (denoted by “all”) represent the sum
14
E.L. Bratkouskuyu
et d/Physics
of all sources. As seen from Fig. 1 the dilepton cross sections agree reasonably well with the experimental data as in [ 121. We will use the same cross sections, however, without experimental DLS filter for the calculation of the anisotropy coefficients. In order to characterize the dilepton decay anisotropy we have introduced in [ 671 the anisotropy coefficient B which allows one to describe the angular distribution of dileptons created in a hadron-hadron (h + h) reaction via da!‘” dhfd
cis 6/,h
= A!“( 1 + B”h cos2 8,,,,).
Eq. ( 1) is the result of averaging the general polynomial angular distribution for the decay of a vector particle (cf. [ 151) over the azimuthal angle. In Eq. (1) 8,,,, is the angle between the electron momentum Z? and the virtual photon momentum q”” in the limit of 4 ‘I’ ---f 0, i.e. in the rest frame of the massive virtual phz Mathematically, t9,*his defined by cos 8/,,, = (IT, vi”) with the electron momentum 1: measured in the dilepton center-of-mass system (q* E 2: -i-Z: = 0), while yi;h = qhh/qih is the velocity of the dilepton c.m.s. relative to the h + h cm. system. M is the invariant mass of a lepton pair (M* = qi - q2) ; the coefficient Bfh describes the anisotropy while AFh determines the magnitude of the cross section. The total differential cross section for h + h collisions now can be represented as a sum of the differential cross sections for all channels da!‘”
dr’lh
c dMd cos Bhh = i=channe, dMd c;s e/t/, = A”” ( M) ( 1 + B”” ( M) cos* 8,,,,) , which leads to the total anisotropy B”“(M) =
c
(2)
coefficient
Letters B 376 (1996) 12-18
“weighted” anisotropy coefficients ( ( Bfh) ) for each channel i obtained by means of the convolution of BFh with the corresponding invariant mass distribution (cf.
[lOI). For heavy-ion reactions the situation becomes more complicated due to the nuclear dynamics and the explicit time evolution of the interacting system. Here we start from the point that the form of the angular distribution for all “elementary” interactions a + b, that occur in the nucleus-nucleus reaction A + B, are known. For this aim we employ the results of our previous works [ 6,7, IO] where the anisotropy coefficients for thepp, and pn bremsstrahlung, NN + A + NNe+e- Dalitz decay, v Dali& decay and n-+7~- annihilation channels were calculated explicitly in the hadron-hadron center-of-mass system a + 6. The differential angular distribution in elementary a+b collisions - before averaging over the momentum 4 ab _ can be represented as daeb dM dqab ii!cos eu,,
= Ayb (1 + Bfrbcos2 &/,) ,
(4)
where the coefficient Bj” for the elementary process a + b is a function of the dilepton mass M, the masses m,, mb and the initial invariant energy fi of the hadrons involved in the reaction: By6 = Bpb (M, fi; m,, mb) . It is important to note that we know the “elementary” coefficient only in the a + b system. In nucleus-nucleus collisions, however, the direction vi” = gb/qtfb is changed in each elementary a + b collision. Thus in order to define an anisotropy coefficient in the latter situation we have to perform an angular transformation from the elementary c.m.s. (0,b) to the c.m.s. of the colliding nuclei ( BAB) : cos 6,b = cos eABcos 8,
+
(B!“(M)),
sin 8.4~ sin 8y cos( $OpAB - $C$) ,
(5)
r=channel
dal’h, (B;‘*(M))
=
dM
B!‘h 1 + $3fh (3)
where 0, is the angle between the dilepton c.m.s. vein the c.m.s. of a+b and the dileplocity v$ = fb/gb ton c.m.s. velocity v,“” = qAB/qtB in the c.m.s. of the colliding nuclei A+B: cos0, = v~~.v~~/( Iv:“I Ivi’I).
where the special weighting factors originate from the necessary angle-integrations. Thus, the anisotropy coefficient Bhh for h + h reactions is the sum of the
Here, the vector qab is obtained by a Lorentz transformation of qAB: q“b = L(Vab)qAB, where Vab = (pa +&,)/(%I +E b) is the velocity of the a + b system relative to the A + B system.
15
E.L. Brutkmskuya et al./Physics Letters B 376 (19%) 12-18
Substituting (5) into Eq. (4) and using dfiA~ = d&b we get the respective distribution in the observ-
able angle @Asby integrating over the azimuthal angle FJAB:
da;” dMd cos OAB /dqAB
N
AybAi
(1
+B;(~,)cos~~AB),
(6)
the same angular transformation from the c.m.s. of the pions a + b to the c.m.s. of the nuclei A + B as in Eq. (5) with the replacement tY4+ 8,. Here, 8, is the angle between the pion momentum pz in the c.m.s. of a + b and the vector Y? = qAB/qtB. Following the same angular integration as for the previous cases we end up with the expression for the anisotropy coefficient for rTT+r- annihilation: BAB
=
Sdcos& h%) B(en)w(coseT)
Im
Jdcose,
pb
A(8,) = 1 + +
= ----(3c0s2eq 2&e,)
- 1)
(7)
using Afb from (4). Obviously, the transformation (5) does not change the quadratic form in cos 6 of the angular distribution. From Eqs. ( 1) and (6) and the normalization condition we finally get
B!B I
w(cose,)
’
(9)
sin2 eq,
B@
B(e,)
Ace,)
dzdL:B dz;AB.
The quantities A(&), B(B,) are defined by Eq. (7) using B& = - 1 while cos & can be computed by the scalar products of the four mOmenta of pions pa,pb and colliding nuclei PA, PB, cos 8,
=
SdqAB @e,)B(e,)
-
j-dqAB
Ace,)
(8)
Eq. ( 8) implies that the numerator and denominator of BfB can be calculated independently within the BUD runs by integrating the functions A( 6’,) and B( 6,) - resulting from the Lorentz transformations - with the differential cross sections dofb/ ( dMdqAB) for the different channels i, which are defined in the same way as described in [ 11,121. The respective “reduced” differential cross sections dafB/dM are shown in Fig. 1 for the systems to be studied below. The above definition of the anisotropy coefficient in nucleus-nucleus collisions is valid for all channels except for rr+t--annihilation because there are only two particles in the initial and final states. For this particular reaction we have to use the angle of the pion momentum with respect to the lepton momentum in the c.m.s. of the leptons (or pions, which is the same for this channel), i.e. gb = pz + p;I = ZS_+ Z? = 0. As was shown in Ref. [ 61, the elementary anisotropy coefficient in the 7~72c.m.s. then is B,+,- = - 1. In heavy-ion collisions we can use the same definition for the polar angle as for the other channels because we can reconstruct the direction of qAB in the c.m.s. of the colliding nuclei A +B. Furthermore, we performed
with M2 = sob = (p, + pb)2 following Teryaev [ 161. A closer look at Eq. (9) shows that Bzt is a constant; its absolute value follows from the 7~+7r- annihilation angular distribution W( cos 0,) with cos 8, determined by ( 10). In case of an isotropic angular distribution, W(cos&) = const., the anisotropy coefficient Btt = 0 according to Eq. (9). Moreover, in the BUU calculation the angular distribution W( cos 0,) is also a function of time t due to the dynamical evolution of the system. Technically we define the distribution W(t) at time t (i.e. in the time interval [ t - At/2; t + At/2] ) as the ratio of the number of pion annihilation events with fixed cos 0, produced in the above time interval to the total number of &rTT- events produced in the A + B reaction, i.e. w(t,cose,)
= N(t,cose,)p,,,.
In order to demonstrate the rfrTangular anisotropy we show in Fig. 2 (1.h.s.) the rTr+w- angular distribution W( t, cos 0,) for p + 9Be at 2.1 GeV and 40Ca+40Ca at 1.0 and 2.0 GeV/A at those times t when the maximal number of pion annihilation events N(t) occured during the interval At; the respective &7r--annihilation rate fi( t) /IV,, is illustrated in the r.h.s. of Fig. 2 for the same reactions. Due to the low number of pion annihilation events for p + 9Be the angular distribution suffers from low statistics and
16
E.L. Brutkooskaya et d/Physics
Letters B 376 (1996) 12-18
I
t=7fm/c,
T
T
J
I
:
‘Be, 2.1 GeV
T
A ~+~Bie, ::
2.1 Ge\
:: j: .i :: i !i
0.06
:,!i 4”Ca+40Ca, :g ; :: *.
TT 9 2
0.04
3 -E
0.02
i
I’
0.000
L
-1.0
:
I i
/j
1
-0.5
0.0 cos on
0.5
1.0
0.00 0
10
20
30 t
(fmk)
Fig. 2. The QT+~- angular distribution W( t, cos 0,) for p + “Be at 2.1 GeV and ” Ca + “‘Ca at 1.0, 2.0 GeV and time I (1.h.s.) where the maximal number of pions annihilation events N(t) occured. The pion annihilation rate lir(t)/N,,, is shown in the r.h.s. of the figure
for the same reactions.
the error bars - which result from different runs with 8000 testparticles per nucleon - are large. For the further analysis we have used the dotted line in Fig. 2 which represents a fit to the angular distribution. On the other hand, for 40Ca + 40Ca collisions the statistics is good enough and we show only fits to the “numerical” data in terms of the solid and dash-dotted lines, respectively. According to Eq. (10) the n-+7~- angular distribution W(cos 19,) is related to the energy and angular distribution of the pions that annihilate. In the c.m.s. of the pions - where B?r is defined - the angular distributions displayed in Fig. 2 (1.h.s.) correspond to annihilating pions having larger transverse then longitudinal momentum relative to qAB. A closer analysis of the events, furthermore, shows that the anisotropy of W( cos 0,) is essentially due to the nonequilibrium phase of the reaction. At 2 GeV/A, the r+rangular distribution becomes more isotropic than at 1 GeV/A in line with a “pionic fireball” scenario. As a consequence the resulting angular distribution shows only a very modest anisotropy. Before going over to the calculation of the
anisotropy coefficient for nucleus-nucleus collisions one has to take into account that resonances (A, N* ) can be created in quite different elementary channels than in pN or pd reactions. For example, A production via the z-N + A channel becomes quite important; an elementary channel for which the dilepton anisotropy has not been computed so far. We thus have calculated the efeanisotropy using the same vertices, delta-propagator, coupling constants and formfactors as in Refs. [ 7,171. Since this evaluation is straight forward, we do not present the details here. The results of our microscopic calculation for the anisotropy coefficient for the n-+ N + A + e+e- +X channel are displayed in Fig. 3. The anisotropy B=N_+h is a function of the dilepton invariant mass M and the invariant energy of the interacting particles s = (p, + pi)’ s Mi. For small fi only deltas with M MAO = 1.232 GeV appear and the coefficient MA B TN-A + 1 as expected for the Dali& decay of a free delta [6]. With increasing fi more energetic deltas can be created and (for fixed M) the phase-space for the final nucleon and virtual photon increases leading to a decrease of BrN-,~. We note that we do not take
E.L. Brutkovskaya
et d/Physics
1.” nN
--A
-e+r-+x
U.8
5
U.6
7
2
“A
“2
“.”
0.0
0.1
’ 0.1
\, “2
0.4
,\I B.5
0.6
1 0.7
M ,GCV,
Fig. 3. Results of our calculation for the anisotropy coefficient of the T + N + A --t e+r- + X channel at Mb = 4 from 1.232
to 1.8 GeV. into account the Dalitz decay of the higher nucleon resonances in the further calculations because their statistical weight is too low. Fig. 4 finally shows the computed weighted anisotropy coefficients (Bi(M)) for p + 9Be and %a + 40Ca collisions at the bombarding energies from 1 to 2.1 GeV/A. The main contributions arise from the 7 and A Dalitz decays due to their large “elementary” anisotropy coefficients and cross sections (cf. Fig. l), respectively. The contribution from pn bremsstrahlung is practically zero at all energies due to a smaller “elementary” anisotropy coefficient and due to a lower cross section as well. The weighted coefficient from T+~T- annihilation is rather small (M 0.1) even for the Ca + Ca reactions and decreases for M 2 mp due to the threshold behaviour of the cross section. However, compared to (B,+,-(M)) for p + 9Be, where pion annihilation is very low, a clear (but moderate) enhancement can be extracted. The contributions of the further channels (N*, TN bremsstrahlung, w + ?r”e+e-) are also negligible due to their small cross sections (cf. Fig. 1) . The cross section from direct decays of the vector mesons becomes compatible with pion annihilation for M x mp for p + 9Be at 2.1 GeV, but the anisotropy coefficient for the “free” vector meson in the vacuum is zero [ 181. We do not discuss here a possible modification of the p-meson properties in the medium that might lead to non-isotropic angular distributions of dileptons because for p + 9Be at 2.1 GeV one cannot reach sufficiently high baryon or pion densities. On the other hand, for 40Ca + 40Ca collisions at
Letters B 376 (1996) 12-18
17
2.0 GeV and for p + ‘Be at 2.1 GeV we observe a clear anisotropy from the 71 Dalitz decay, while for p + 9Be at 1 GeV the A Dalitz decay gives the main contribution at small invariant mass due to a dominant A cross section; the 7 coefficient increases at M from 0.4 to 0.5 GeV for the same reason. For clarity we briefly discuss the steps to “extract” the anisotropy coefficient from experimental data. In dilepton experiments the four-momenta of leptons in the c.m.s. of the nuclei A + B (or in the laboratory frame, which is connected with the c.m.s. by a simple Lorentz transformation) are measured - (&AB,lA_B), (&“+“,Z$ ). The four momentum q = (qf , qAB) of the virtual photon in the A + B c.m.s. then is given by: qt* = dB -t ctB, qAB = 1”” + I”+“. The angle 6AB has to be computed via
where ey[ is the angle between
the qAB and I!‘,
i.e.
cos&# = I”_“. qAB/(IZABI /qABI). In order to extract the anisotropy coefficient one has to count the number of dilepton events N( M, cos BAB) with fixed cos 6)ABand invariant mass M, The anisotropy coefficient for the invariant mass M then is simply given by BAB(M) =
N(M,COSeAB = 1) _ 1 N( M, COSeAB = 0)
’
(12)
Thus, summarizing, the calculated anisotropy coefficients for p + Be and Ca + Ca collisions support our suggestion in Refs. [ 6,7,10] that the dilepton decay anisotropy may serve as an additional observable to decompose the dilepton spectra into the various sources. Since the anisotropy vanishes in a hadronic fireball scenario our present results provide valuable information about the nonequilibrium stage of the reactions. We gratefully acknowledge many helpful discussions with S.S. Shimanskij, S. Teis, O.V. Teryaev, A.I. Titov, V.D. Toneev and Gy. Wolf.
E.L. Bratkovskaya et al./Physics
18
Letters B 376 (1996) 12-18
P+~B~
0.8
2.1 GeV
0.6 L
I
I.
0.2
1.
0.3
I.
0.4
I.
I.
0.5
0.6
I.
0.7
1.
0.8
I
0.9
.I 1.0
0.2
0.3
0.4
0.5
M (GeV)
0.6
0.7
0.8
0.9
1. 0
0.7
0.8
0.9
1.0
M (GeV)
F
%i+%~
0.8
1.0 GeV
1 0.6 -
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.2
0.3
0.4
Fig. 4. The weighted anisotropy coefficients The notation is the same as in Fig. 1.
0.5
0.6
M (GeV)
M (GeV)
(Bi( M)) for p +YBe and 40Ca+“aCa
References 1 II G. Roche et al., Phys. Rev. Lett. 61 (1988) 1069. 12 1 C. Naudet et al., Phys. Rev. Len. 62 ( 1989) 2652. [3 I G. Roche et al., Phys. Len. B 226 (1989) 228. (4 I G. Agakichiev et al., Phys. Rev. Lea. 75 (1995) 1272. I 5 I U. Mosel, Ann. Rev. Nucl. Part. Sci. 41 ( 199 I ) 29. 16 I E.L. Bratkovskaya, O.V. Teryaev and V.D. Toneev, Phys. Lett. B 348 ( 1995) 283. E.L. Bratkovskaya, M. Schafer, W. Cassing, U. Mosel, O.V. Teryaev and V.D. Toneev, Phys. Lett. B 348 ( 1995) 325. J. Zhang, R. Tabti, Ch. Gale and K. Haglin, Preprint McGill/94-56, MSUCL-961. ‘1 T.I. Gulamov, Al. Titov and B. Kampfer, Preprint JINR E2-95153.
collisions
at bombarding
[lo] E.L. Bratkovskaya, [ 111 12) 131 141 IS]
_ [ 161 [17]
1IS]
energies from 1 to 2.1 GeV/A.
W. Cassing, U. Mosel, O.V. Teryaev, A.I. Titov and V.D. Toneev, Phys. Lett. B 362 ( 1995) 17. Gy. Wolf, G. Batko, W. Gassing, U. Mosel, K. Niita and M. Schafer, Nucl. Phys. A 517 (1990) 615. Gy. Wolf, W. Cassing and U. Mosel, Nucl. Phys. A 552 (1993) 549. W. Cassing, W. Ehehah and C.M. Ko, Phys. Lett. B 363 (1995) 35. L.G. Landsberg, Phys. Rep. 128 (1985) 301. B.T. Feld, Models of Elementary Particles, Blaisdell Publishing Company ( 1969). O.V. Teryaev, private communication. M. Schafer, H.C. Donges, A. Engel and U. Mosel, Nucl. Phys. A 575 (1994) 429. C. Gale and J.I. Kapusta, Nucl. Phys. B 357 (1991) 65.