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Anisotropy of dilepton emission from nucleon-nucleon interactions
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PHYSICSLETTERSB ELSEVIER
PhysicsLetters J3348 (1995) 325-530
Anisotropy of dilepton emissionfrom nucleon-nucleon interactions* E.L. Bratkovskaya b, M . Schtifer a, W . Casing a, U. Mosel ‘, 0.\1 Teryaev b, V.D. Toneev b ’ Bngolubov
’ lnslitut ,flir Tkoretische Physik. Universitiit Giesscn. D-3S392 Giessen, Germans of Theoretical Physics, Joint Institute for Nr/rrkar Research, 141980 Dubna, d140scow Region, Russiu Laboratory
Received5 December 1994; revised manuscriptreceived27 January 1995 Editor: C. Mabaux
Abstract We studythe angular characteristics of efe- pairs produced in nucleon-nucleon interactions at intermediate energieson the basis of a one-boson-exchange model fitted to elastic NN scattering. Due to spin and angular momentum constraints, the dilepton anisotropy is found to be sensitive to the contribution of different sources. The anisotropy from NN-bremsstrahlung anddeltaresonance is calculatedandcompared to the ‘ialbz decayfrompseudo-scalar andvectormesons.
During the past few years a lot of work has been cqncentratedon the study of dilepton production in NN, pd, pA and AA reactionsin order to learn about the hadron dynamicsin the nuclear interior [ l-121. Due to their weak final state interaction leptonic probes are quite attractive since they provide more direct information on hot and dense nuclear matter at different stagesof its evolution in heavy-ioncollisions at energiesof about a few GeV/u. On the other hand there are a lot of hadronic sourcesfor dileptons becausethe electromagneticfield couples to all chargesand magneticmoments.It is thus very desirableto haveadditionalinformationwhich allows to disentanglethe varioussourcesexperimentally. In hadron-hadroncollisions,the e+e- pairs are createddue to the electromagneticdecayof time-like virtual photons.In turn, thesevirtual photonscan result from the bremsstrahlungprncessor from the decayof baryonicand mesonicresonancesincludingthe direct *Supported by BMFI’ and GSI Darmstadt.
conversionof vectormesonsinto virtual photonsin accordancewith the vectordominancehypothesis,In its rest frame, the decayof an unpolarizedphoton gives an isotropic angulardistribution for a createdlepton pair sincethereis no preferentialdirection. However, the couplingof the virtual photon to hadronsinduces a dynamicalspin alignmentof both, resonancesand virtual photons.We thus can expectthat the angular distributionof a lepton (e.g. e-) will be anisotropic with respectto the directionof the dilepton (i.e. virtual photon) emission.This decay anisotropy(definedfor the given dilepton massN) is carrying some informationon the spin alignmentof the virtual photon as well as on the spinsof colliding or decayinghadrons and therebyshouldallow to disentangledifferent productionprocesses. In this letter we focus on the investigationof angular distributionsof dileptonsemittedfrom nucleonnucleon interactionswithin two particular channels: the delta resonanceand NN-bremsstrahlung.In a precedingstudy these anisotropyeffects for differ-
0370-2693/95/$09.50 @ 1995 Elsevier ScienceB.V. All rights reserved LWIO370-2693(95)00140-9
326
E. L. Brarkmrkqa et rd. /Plzy,sicsLetters B 348 (1995) 32.5-330
* = ye
=o”> _ ]
S(M,O = 900)
Fig. 1. Illustration of the kinematical situa!ion for &epair production in pN interacllons. Z’Lh, 1’;1’ are the lemon momenta m the nucleon center-of-mass system (p(, + p,, = 0) while P is half the total lepton momentum I$ + /‘;I I-, I+ are the lemon momenta in the lepton center-of-mass system (I- + 1+ = 0). Op is the angle between the beam axis, given by the initial proton momentum po, and the momentum P. (Y and p are the angles between P and the relative momentum between bcth nucleons and both leptons, respectively. y and cp are ‘the additional rotational degrees of freedom around the P direction while B is the polar angle of the iepton momentum I- and P (P = 4 ).
ent channels (bremsstrahlung, Dalitz decays of delta resonance, v, 7r”, 7r+71---annihilation, Drell-Yan process) were discussed in the framework of the soft photon approximation [ 131; here we reanalyse the angular anisotropy on the basis of the microscopic One-Boson-Exchange (OBE) model developed for dilepton production in [ 141. To characterize the decay anisotropy, we choose the polar 8 and azimuthal q angles of the momentum Iof a created electron with respect to the momentum q of a virtual photon, where I-, I+ are measured in the rest frame of this virtual photon, i.e. I- + 1,. = 0. The kinematical situation is illustrated in Fig. 1. In order to compare the shape of the angular distribution for different channels, it is convenient to represent the differential cross section for dilepton production in the following form:
S(M,8) E
da
dM’d cos 0
=A(l+Bcos*8),
(1)
where M is the invariant mass of a lepton pair (IV* = C$ - q*) . The anisotropy coefficient B then is defined
(2)
.
Since thecoefficient B is sensitive to the spin structure of the interacting hadrons, it is in general a function of M and the masses of the hadrons involved in the reaction. Before actually presenting the numerical results for B we briefly recall the basic concepts of the microscopic OBE-model adopted [ 14j. We investigate hadronic interactions in first order covariant perturbation theory in the nucleon-nucleon interaction, where particles are produced from the external nucleon lines and from charged (internal) meson lines. Particle emission between subsequent NN interaction vertices is omitted in our calculation, but we include the more important effects of excited nucleon states, i.e. in particular the A-resonances (cf. [ 141). For the interaction between nuclcons and mesons we use the Hamiltonian:
ga = &r/d% where u, w, p, IT and a denote the scalar, vector, isovector-vector, isovector-pseudoscalar and isovector-axialvector meson fields, respectively. We account for the finite size of the hadrons by introducing formfactors Fi( k*) = (A: - ~2’) (A: - k*)-’ at each strong-interaction vertex where k is the four-momentum and M the mass of the exchanged meson. As nucleon resonances we only include the A in this study which couples to the nucleon via isospin-l-mesons, i.e. the pion and the p-meson. The corresponding vertex functions are: rNArr
@
=
~NAV --k/J’ , mT
..fNAp yh’+ = --1w
(4) (kvy”g,,
-
kpx-t)
~$7
(5)
%J
where kl, denotes the momentum of the outgoing meson, T the isospin-operator. From the decay A ‘4
E.L. Bratkovskqa et al. /Physics Letters B 348 (I 99s) 325-33G
N + rr we have determined the NAT-coupling constant to f.v~~ = 2.13. The A propagator is adopted from Ref. [ 151 and reads:
1 - 3Mdo(yPP;’ - PfYV)
/)
*
The authors of Ref. [ 151 give extensive arguments why this is the proper A-propagator. From a more practical point of view we find that the alternative expression proposed by Williams [ 161 leads to divergencies in the five-fold differential cross section for the PP -+ pn& reaction for Pi = 0 whereas the form (6) leads to the proper spectrum in comparison to the experimental data (cf. Fig. 5 of Ref. [ 141) . The mass of the A in the denominator in (6) is modified by an imaginary width MhO --: MAO - S/2 due to the fact that A’s have a short lifetime due to the pionic decay which depends on the invariant energy of the delta or equivalently on the momentum k, of the decaying pion in the restframe of the A. The actual parametrization of I’( &!!A) is adopted from [ 171. As in the case of nucleon-nu&;on scattering we use formfactors for the nucleon-A vertex but now of dipole form [ 18 ] in order to keep the self-energy of the nucleon finite (cf. [ 141). The coupling parameters and cutoffs in our approach are fixed in comparison to elastic scattering data for pp and pn reactions from 800 MeV up to 3 GeV as well as to experimental data of the pp -+ nA++ reaction from 970 MeV to 2.02 GeV. The results in comparison to the experimental data are given in Ref. [ 141 as well as the resulting couplings and cutoffs. Having fixed the fundamental couplings of the OBE-model we now turn to the calculation of the dilepton anisotropy. For the dilepton emission via a delta resonance, formed in the NN -+ AN --+ NNe+e- process, the dilepton differential cross section can be written in the form: do dM2dcos8
=
J
pdfiNdP
1 (e!y +e”_b)&IpJ27r)7
where pa, ph and p:, p; are the four momenta of initial and outgoing nucleons, respectively, En, Eh and pO, &, are the energy and momentum of nucleons in the nucleon-nucleon center-of-mass system (p,+ph = 0) ; Eab op and Z’?, l:b are the energy and momentum of leDions in the same reference frame. P = $ (Z? + 14”) is half of the total lepton momentum, Q,* = P!’ - P, and QN = p,’ - P. LYand p arc the angles between P and Qv and P and Q,, respectively. The angle @ is related with the polar angle 6 via
cosp= a-iyos* 21QLI with IQ,1 = i M2 + 4)P~2cos26. The transiiion matrix element T is obtained by summing all Feynman diagrams over the exchanged mesons rr, p: lTl2 = I J’J(T$) + Tj’)) I?-, where the term ]7j2 tacitly already includes a sum over final spin and average over initial spin degrees of freedom. For example, the matrix eiement for the,t-channel including a-exchange is given by T(f) ?I
=
JA JN Ir
L’ (8)
where the lepton current L, is defined as L@ = fx~-)Y,U(~+),
(9)
and I-, l+ are the four momenta of the leptons. The hadron current for the NAN-line is J:
= ii
$#‘A)
k&z
dp;),
c 10)
where k = pu - PA is the four momentum of the exchanged pion, and G&(PA) is the delta propagator (6). The NAy vertex function’is taken in the form:
328
E.L. Bratkovskayu et al. /Physics Letters B 348 (1995) 32.5-330
~~-._.____.___^_..~--~
-0.5
0.0 -0.5 0.0 0.5 1.9 COST3 COW9 Fig. 2. The differential cross section for dileptons in pp and pn reactions as a function of cos@ for fixed M = 0.1 CieV at 2.1 GeV bombarding energy. The “A” labels the contribution of the delta resonance term, the “A(I, a)” denotes the contribution of the r-channel with a single 7r-meson exchange only; “pp”. “~n”denote the Born terms ror the JJ~, pn collisions, respectively, and “Y the sum of all channels including the interferences.
where q = I- + 1, is the four momentum of virtual photon. The coupling constant has been obtained from the experimental partial width of the A of 0.6% for the decay into a real photon. The hadron current for the NN-line is
~~= ii(pl,)rNN7i(P;,).
(12)
We stress that we take into account the interference between all diagrams with rr, p meson exchange. For the cross section of diiepton production in the mm.-resonance processes NRi -+ NNefe- we use the same expression (7) as in the delta case, but now we add the diagrams with the emission of a virtual photon from the other nucleon line for the pp-channel and the pn-channel, because we also include the coupling to the anomalous magnetic momert of the nucleon. As a consequence aiso the neutron lines can contribute to the radiation of dileptons. The amplitude T includes the sum over the four meson-exchange diagrams taking into account the inter.fc*rencebetween al1 diegrams with the different meson; ; 7~,CT,p. ‘ti, a), Ty = c
Jz”’
J,:,!“’ I;&“.
iM
i = diagram,
31 =. ?r, cr. ji. co,i.
The hadron current t’c:
ij :‘ I;rL,
#a”) I*
= Q%)Y,
x PNMlt(y;))
h,,Y” - qpYv f mN) ((pa .- 9j2 - mi> (14)
where rNNM is the nucleon-nucleon-meson vertex (M = 7r, u, p, o, a). The final cross section then is given by coherently summing up over the T-matrix (8) and (13). As discussed in detail in Section 3.1 of Ref. [ 141 our approach for the evaluation of the differential dilepton cross section is fully gauge invariant. The respective results for the differential e+e- cross section for pp and pn collisions at 1.0 and 2.1 GeV are presented in Figs. 7, 8 of Ref. [ 141. In Fig. 2 we show the results of out calculations for the differential cross section for dileptons in p,p and pn reactions as a function of cos IBfor fixed M = 0.1 GeV at 2.1 GeV bombarding energy. The “A” labels the contribution of the delta resonance term, the “A(r, 7r)” denotes the contribution of the r-channel with r-meson exchange only; “pp”, “pd denote the Born terms for the 173,pn collisions, respectively, and “5’ the sum of all channels including the interfersnccs. As seen from this figure, the absolute value of the cross section related to the mechanism with an intcrmediate delta is sensitive to the model. Taking into dc;;oamt all diagrams of delta production and their intcrferences changes the value of the cross section by about a factor 4. However, the form of the cross settion remains the same as can be seen from the curves
E.L. Brntkovskuya et al. /Physics Letters B 348 (1992) 325-330
1.0._.. -......._._ A ‘I.._ .I.. I-0.7 f - - _ _ ..
0.6
...
*.
PP---,
0.4
'Lx_
.._.
‘\ ‘.
pp
-m
0.6
I.._ -* 0.2
~\_ 1
E -... ..~.
- _
0.1
-0.2
319
collieione
et
1.0 dev
I
1.0
0.7 0.4 0.4
h\ . \,
,--
IR\ 0.0
;
0.1
‘\ ‘,P”
@A) -.
-0.4
-0.6
' 0.1
J 0.4
0.2 M
(c&j'
-0.8
0.1
. --_
-
pn colli~ionti . m ’ 0.3
- - - _ ,- - _*’
at 2.1 dev ’ ’ c , ( ’ 0.6 0.7
h4 (cm)
Fig. 3. The anisotropy coefficients B for pp and pn collisions as functions of M at 1.O and 3.1 GeV. The solid curves with label “I” correspond to the calculation in the “full” OBE-model accounting for all mesons exchanges and all diagrams. The “A” lobels the contribution of the delta resonance term, “pp”, “@denote the Born terms for the pp, pn collisions, respectively. For the case of the delta we also present the results of a calculation involving a single n-exchange in the r-channel in terms of the long dashed curve labeled “Act, rr)“at 2.1 GeV (upper right). The result for the bremsstrahlung contribution in the soft photon approximation from 1131 is given by the dashed lint-s labeled “pn (SPA)“.
labeled “A”and “A ( t, n)“, respectively. The anisotrspy coefficients B for pp and pn collisions as functions of M at 1.0 and 2.1 GeV are plotted in Fig. 3. As was shown in Ref. [ 131 the B values range from +l (for pseudoscalar meson (?rO,v) Dalitz decay) to -1 (for ~T+P- annihilation or p decay) and depend on both, the invariant mass M and the process considered. The solid curves with label “8” correspond to the calculation in the “full” QBE-model accounting for all mesons exchanges and all diagrams including the interferences. The ‘*A” labels the contribution of the delta resonance term, ‘Pp”, “pn”denote the Born terms for the pp, pn collisions, respectively. For the case of the delta we also present the results of a calculation involving only the r-exchange in the t-channel given by the long dashed curve labeled “A(t,r)” for the pp reaction at 2.1 GeV. The dotted and the long dashed line in this figure are almost identical since the coefficient B,
which is defined as a ratio of differential cross sections (cf. Eq. (2)), is not sensitive to the absolute value of the cross section, whereas the angular distributions are quite similar (1.h.s. of Fig. 2 j . In Ref. [ 131 the dilepton cross secrion from the A channel was approximated by a product of the delta production cross section and the dii’ferelltial width of the A Dalitz decay. The B value in this approximation is 4-l as for ihe case of pseudoscalar meson (TO, 91) Dalitz decay. We note that thotigh this approximation reproduces the form of the inclusive dilepton spectrum calculated within the OBE-model, it should be considered as rather unrealistic for the study of more subtle microscopic effects, such as polarization phe.. nomena and spin effects. Only for nearly real photons (M -+ 0) the coupling to the hadronic part (or delta alignment) becomes negligible. Thus we obtaitl a rather complicated behavior for the coefficient B in the OBE-model instead of B = 4-l in [ 131.
330
E.L. Bratkovskaya et al. /Physics Letters B 348 (1995) 325-330
For the same reasons the functional dependence of B for the Born terms changes very strongly with M: the dashed curves in Fig. 3 labeled “pn” are the results in the OBE-model, the dashed curves labeled “pn (SPA)” are the result from [ I31 based on the soft photon approximation (SPA) for the dilepton brzmsstrahlung. Here again the results are close at the lower energy ( 1 GeV) and low dilepton mass M. Note that at small Al an increase of the virtual photon energy leads to a decrease of B in the soft-photon approximation while B increases with M in the OBE-calculation and remains positive for all M contrary to the SPA. In summary we can conclude that the angular anisotropy B of dilepton pairs is very sensitive to the microscopic details of the interaction and can be used to determine the relative weight of the hadronic production channels in pN reactions as a function of the invariant mass M. It remains to be seen whether this angular anisotropy can also be used to distinguish between the various reaction channels in heavy-ion induced dilepton production. The authors acknowledge stimulating discussions with S.S. Shimanskiy and A.I. Titov. E.L.B. is grateful to the Institute for Theoretical Physics of the University ‘of Giessen for the support and kind hospitality during her visit. Part of this research was stimulated by discussions during the INT-04-3 program on ‘Hot and Dense Nuclear Matter’ in Seattle. Furthermore, this research was partly performed in the framework of the Grants NO_MP8000 and RFEOOOfrom the Intcrnational Science Foundation.
References [I] C. Gaie and j. Kapusta, Phys. Rev. C 35 (1985) 2107; C 40 (1988) 745. 121 L.H. Xia, CM. KQ, L. Xiong and J.Q. Wu, Nucl. Phys. A 485 (1988) 721. 131 M. Schaferet al., Phys. Lett. B 221 (1989) I. 141 K. Haglin,I. Kapustaand C. Gale, Phys.Lett. B 224 ( 1989) 433. [51 CL. Korpa and S. Pratt, Phys. Rev. Lett. 64 ( 1990) 1502; CL. Kotpa et al., Phys. Lett. B 246 (1990) 333. [6] K.L. Haglin, Ann. Phys.212 (1991) 84. [71 M. Herrmann,B. Friman and W. Nijrenbetg, Nucl. Phys. A 545 (1992) 267~; A 560 (1993) 411. 181 2. Xiong, Z.G. Wu, CM. Ko and J.Q. Wu, Nucl. Phys. A 512 (1990) 772. [9] Gy. Wolf, G. Batko, W. Cassing,et al., Nucl. Phys. A 517 (1990) 615. [ 10 1 V.D. Toneev,K.K. Gudima and A.T. Titov, PreprintGSI-9205, Darmstadt,1992; Sov. Jour. of Nucl. Phys. 55 (1992) 1715; K.K. Gudima,A.I. Titov and V.D. Toneev,t’hys. Let?. B 287 (1992) 302. [ 1 I I Gy. Wolf, W. Cassing.W. Ehehaltand U. Mosel, Progr. Part. Nucl. Phys. 30 (1993) 273; Gy. Wolf, W. Gassingand U. Mosel, Nucl. Phys. A 552 (1993) 549. I 121 K. Haglin and C. Gale, Phys. Rev. C 49 (1994) 401. 113] E.L. Bratkovskaya, O.V. Teryaev, V.D. Toneev, preprint of the Institute for Nuclear Theory, NK-12, University of Washington, DOE/ER/40:61-167-lNT4-lS-01, submitted to Phys. Len. B. 1141 M. Schafer, H.C. Diinges, A. Engel and U. Mosel, Nucl. Phys. A 575 (1994) 429. 1151 M. Benmetroucheet al., Phys. Rev. C 39 (1989) 2339. [ 161 HT. Williams, Phys Rev. C 31 (1985) 2297. [ 171 V. Dimitriev, 0. Sushkovand C. Gaarde,Nucl. Phys. A 459 (1986) 503. [18] B.J. VerWest,Phys. Lett. B 83 (1979) 161.
PHYSICSLETTERSB ELSEVIER
PhysicsLetters J3348 (1995) 325-530
Anisotropy of dilepton emissionfrom nucleon-nucleon interactions* E.L. Bratkovskaya b, M . Schtifer a, W . Casing a, U. Mosel ‘, 0.\1 Teryaev b, V.D. Toneev b ’ Bngolubov
’ lnslitut ,flir Tkoretische Physik. Universitiit Giesscn. D-3S392 Giessen, Germans of Theoretical Physics, Joint Institute for Nr/rrkar Research, 141980 Dubna, d140scow Region, Russiu Laboratory
Received5 December 1994; revised manuscriptreceived27 January 1995 Editor: C. Mabaux
Abstract We studythe angular characteristics of efe- pairs produced in nucleon-nucleon interactions at intermediate energieson the basis of a one-boson-exchange model fitted to elastic NN scattering. Due to spin and angular momentum constraints, the dilepton anisotropy is found to be sensitive to the contribution of different sources. The anisotropy from NN-bremsstrahlung anddeltaresonance is calculatedandcompared to the ‘ialbz decayfrompseudo-scalar andvectormesons.
During the past few years a lot of work has been cqncentratedon the study of dilepton production in NN, pd, pA and AA reactionsin order to learn about the hadron dynamicsin the nuclear interior [ l-121. Due to their weak final state interaction leptonic probes are quite attractive since they provide more direct information on hot and dense nuclear matter at different stagesof its evolution in heavy-ioncollisions at energiesof about a few GeV/u. On the other hand there are a lot of hadronic sourcesfor dileptons becausethe electromagneticfield couples to all chargesand magneticmoments.It is thus very desirableto haveadditionalinformationwhich allows to disentanglethe varioussourcesexperimentally. In hadron-hadroncollisions,the e+e- pairs are createddue to the electromagneticdecayof time-like virtual photons.In turn, thesevirtual photonscan result from the bremsstrahlungprncessor from the decayof baryonicand mesonicresonancesincludingthe direct *Supported by BMFI’ and GSI Darmstadt.
conversionof vectormesonsinto virtual photonsin accordancewith the vectordominancehypothesis,In its rest frame, the decayof an unpolarizedphoton gives an isotropic angulardistribution for a createdlepton pair sincethereis no preferentialdirection. However, the couplingof the virtual photon to hadronsinduces a dynamicalspin alignmentof both, resonancesand virtual photons.We thus can expectthat the angular distributionof a lepton (e.g. e-) will be anisotropic with respectto the directionof the dilepton (i.e. virtual photon) emission.This decay anisotropy(definedfor the given dilepton massN) is carrying some informationon the spin alignmentof the virtual photon as well as on the spinsof colliding or decayinghadrons and therebyshouldallow to disentangledifferent productionprocesses. In this letter we focus on the investigationof angular distributionsof dileptonsemittedfrom nucleonnucleon interactionswithin two particular channels: the delta resonanceand NN-bremsstrahlung.In a precedingstudy these anisotropyeffects for differ-
0370-2693/95/$09.50 @ 1995 Elsevier ScienceB.V. All rights reserved LWIO370-2693(95)00140-9
326
E. L. Brarkmrkqa et rd. /Plzy,sicsLetters B 348 (1995) 32.5-330
* = ye
=o”> _ ]
S(M,O = 900)
Fig. 1. Illustration of the kinematical situa!ion for &epair production in pN interacllons. Z’Lh, 1’;1’ are the lemon momenta m the nucleon center-of-mass system (p(, + p,, = 0) while P is half the total lepton momentum I$ + /‘;I I-, I+ are the lemon momenta in the lepton center-of-mass system (I- + 1+ = 0). Op is the angle between the beam axis, given by the initial proton momentum po, and the momentum P. (Y and p are the angles between P and the relative momentum between bcth nucleons and both leptons, respectively. y and cp are ‘the additional rotational degrees of freedom around the P direction while B is the polar angle of the iepton momentum I- and P (P = 4 ).
ent channels (bremsstrahlung, Dalitz decays of delta resonance, v, 7r”, 7r+71---annihilation, Drell-Yan process) were discussed in the framework of the soft photon approximation [ 131; here we reanalyse the angular anisotropy on the basis of the microscopic One-Boson-Exchange (OBE) model developed for dilepton production in [ 141. To characterize the decay anisotropy, we choose the polar 8 and azimuthal q angles of the momentum Iof a created electron with respect to the momentum q of a virtual photon, where I-, I+ are measured in the rest frame of this virtual photon, i.e. I- + 1,. = 0. The kinematical situation is illustrated in Fig. 1. In order to compare the shape of the angular distribution for different channels, it is convenient to represent the differential cross section for dilepton production in the following form:
S(M,8) E
da
dM’d cos 0
=A(l+Bcos*8),
(1)
where M is the invariant mass of a lepton pair (IV* = C$ - q*) . The anisotropy coefficient B then is defined
(2)
.
Since thecoefficient B is sensitive to the spin structure of the interacting hadrons, it is in general a function of M and the masses of the hadrons involved in the reaction. Before actually presenting the numerical results for B we briefly recall the basic concepts of the microscopic OBE-model adopted [ 14j. We investigate hadronic interactions in first order covariant perturbation theory in the nucleon-nucleon interaction, where particles are produced from the external nucleon lines and from charged (internal) meson lines. Particle emission between subsequent NN interaction vertices is omitted in our calculation, but we include the more important effects of excited nucleon states, i.e. in particular the A-resonances (cf. [ 141). For the interaction between nuclcons and mesons we use the Hamiltonian:
ga = &r/d% where u, w, p, IT and a denote the scalar, vector, isovector-vector, isovector-pseudoscalar and isovector-axialvector meson fields, respectively. We account for the finite size of the hadrons by introducing formfactors Fi( k*) = (A: - ~2’) (A: - k*)-’ at each strong-interaction vertex where k is the four-momentum and M the mass of the exchanged meson. As nucleon resonances we only include the A in this study which couples to the nucleon via isospin-l-mesons, i.e. the pion and the p-meson. The corresponding vertex functions are: rNArr
@
=
~NAV --k/J’ , mT
..fNAp yh’+ = --1w
(4) (kvy”g,,
-
kpx-t)
~$7
(5)
%J
where kl, denotes the momentum of the outgoing meson, T the isospin-operator. From the decay A ‘4
E.L. Bratkovskqa et al. /Physics Letters B 348 (I 99s) 325-33G
N + rr we have determined the NAT-coupling constant to f.v~~ = 2.13. The A propagator is adopted from Ref. [ 151 and reads:
1 - 3Mdo(yPP;’ - PfYV)
/)
*
The authors of Ref. [ 151 give extensive arguments why this is the proper A-propagator. From a more practical point of view we find that the alternative expression proposed by Williams [ 161 leads to divergencies in the five-fold differential cross section for the PP -+ pn& reaction for Pi = 0 whereas the form (6) leads to the proper spectrum in comparison to the experimental data (cf. Fig. 5 of Ref. [ 141) . The mass of the A in the denominator in (6) is modified by an imaginary width MhO --: MAO - S/2 due to the fact that A’s have a short lifetime due to the pionic decay which depends on the invariant energy of the delta or equivalently on the momentum k, of the decaying pion in the restframe of the A. The actual parametrization of I’( &!!A) is adopted from [ 171. As in the case of nucleon-nu&;on scattering we use formfactors for the nucleon-A vertex but now of dipole form [ 18 ] in order to keep the self-energy of the nucleon finite (cf. [ 141). The coupling parameters and cutoffs in our approach are fixed in comparison to elastic scattering data for pp and pn reactions from 800 MeV up to 3 GeV as well as to experimental data of the pp -+ nA++ reaction from 970 MeV to 2.02 GeV. The results in comparison to the experimental data are given in Ref. [ 141 as well as the resulting couplings and cutoffs. Having fixed the fundamental couplings of the OBE-model we now turn to the calculation of the dilepton anisotropy. For the dilepton emission via a delta resonance, formed in the NN -+ AN --+ NNe+e- process, the dilepton differential cross section can be written in the form: do dM2dcos8
=
J
pdfiNdP
1 (e!y +e”_b)&IpJ27r)7
where pa, ph and p:, p; are the four momenta of initial and outgoing nucleons, respectively, En, Eh and pO, &, are the energy and momentum of nucleons in the nucleon-nucleon center-of-mass system (p,+ph = 0) ; Eab op and Z’?, l:b are the energy and momentum of leDions in the same reference frame. P = $ (Z? + 14”) is half of the total lepton momentum, Q,* = P!’ - P, and QN = p,’ - P. LYand p arc the angles between P and Qv and P and Q,, respectively. The angle @ is related with the polar angle 6 via
cosp= a-iyos* 21QLI with IQ,1 = i M2 + 4)P~2cos26. The transiiion matrix element T is obtained by summing all Feynman diagrams over the exchanged mesons rr, p: lTl2 = I J’J(T$) + Tj’)) I?-, where the term ]7j2 tacitly already includes a sum over final spin and average over initial spin degrees of freedom. For example, the matrix eiement for the,t-channel including a-exchange is given by T(f) ?I
=
JA JN Ir
L’ (8)
where the lepton current L, is defined as L@ = fx~-)Y,U(~+),
(9)
and I-, l+ are the four momenta of the leptons. The hadron current for the NAN-line is J:
= ii
$#‘A)
k&z
dp;),
c 10)
where k = pu - PA is the four momentum of the exchanged pion, and G&(PA) is the delta propagator (6). The NAy vertex function’is taken in the form:
328
E.L. Bratkovskayu et al. /Physics Letters B 348 (1995) 32.5-330
~~-._.____.___^_..~--~
-0.5
0.0 -0.5 0.0 0.5 1.9 COST3 COW9 Fig. 2. The differential cross section for dileptons in pp and pn reactions as a function of cos@ for fixed M = 0.1 CieV at 2.1 GeV bombarding energy. The “A” labels the contribution of the delta resonance term, the “A(I, a)” denotes the contribution of the r-channel with a single 7r-meson exchange only; “pp”. “~n”denote the Born terms ror the JJ~, pn collisions, respectively, and “Y the sum of all channels including the interferences.
where q = I- + 1, is the four momentum of virtual photon. The coupling constant has been obtained from the experimental partial width of the A of 0.6% for the decay into a real photon. The hadron current for the NN-line is
~~= ii(pl,)rNN7i(P;,).
(12)
We stress that we take into account the interference between all diagrams with rr, p meson exchange. For the cross section of diiepton production in the mm.-resonance processes NRi -+ NNefe- we use the same expression (7) as in the delta case, but now we add the diagrams with the emission of a virtual photon from the other nucleon line for the pp-channel and the pn-channel, because we also include the coupling to the anomalous magnetic momert of the nucleon. As a consequence aiso the neutron lines can contribute to the radiation of dileptons. The amplitude T includes the sum over the four meson-exchange diagrams taking into account the inter.fc*rencebetween al1 diegrams with the different meson; ; 7~,CT,p. ‘ti, a), Ty = c
Jz”’
J,:,!“’ I;&“.
iM
i = diagram,
31 =. ?r, cr. ji. co,i.
The hadron current t’c:
ij :‘ I;rL,
#a”) I*
= Q%)Y,
x PNMlt(y;))
h,,Y” - qpYv f mN) ((pa .- 9j2 - mi> (14)
where rNNM is the nucleon-nucleon-meson vertex (M = 7r, u, p, o, a). The final cross section then is given by coherently summing up over the T-matrix (8) and (13). As discussed in detail in Section 3.1 of Ref. [ 141 our approach for the evaluation of the differential dilepton cross section is fully gauge invariant. The respective results for the differential e+e- cross section for pp and pn collisions at 1.0 and 2.1 GeV are presented in Figs. 7, 8 of Ref. [ 141. In Fig. 2 we show the results of out calculations for the differential cross section for dileptons in p,p and pn reactions as a function of cos IBfor fixed M = 0.1 GeV at 2.1 GeV bombarding energy. The “A” labels the contribution of the delta resonance term, the “A(r, 7r)” denotes the contribution of the r-channel with r-meson exchange only; “pp”, “pd denote the Born terms for the 173,pn collisions, respectively, and “5’ the sum of all channels including the interfersnccs. As seen from this figure, the absolute value of the cross section related to the mechanism with an intcrmediate delta is sensitive to the model. Taking into dc;;oamt all diagrams of delta production and their intcrferences changes the value of the cross section by about a factor 4. However, the form of the cross settion remains the same as can be seen from the curves
E.L. Brntkovskuya et al. /Physics Letters B 348 (1992) 325-330
1.0._.. -......._._ A ‘I.._ .I.. I-0.7 f - - _ _ ..
0.6
...
*.
PP---,
0.4
'Lx_
.._.
‘\ ‘.
pp
-m
0.6
I.._ -* 0.2
~\_ 1
E -... ..~.
- _
0.1
-0.2
319
collieione
et
1.0 dev
I
1.0
0.7 0.4 0.4
h\ . \,
,--
IR\ 0.0
;
0.1
‘\ ‘,P”
@A) -.
-0.4
-0.6
' 0.1
J 0.4
0.2 M
(c&j'
-0.8
0.1
. --_
-
pn colli~ionti . m ’ 0.3
- - - _ ,- - _*’
at 2.1 dev ’ ’ c , ( ’ 0.6 0.7
h4 (cm)
Fig. 3. The anisotropy coefficients B for pp and pn collisions as functions of M at 1.O and 3.1 GeV. The solid curves with label “I” correspond to the calculation in the “full” OBE-model accounting for all mesons exchanges and all diagrams. The “A” lobels the contribution of the delta resonance term, “pp”, “@denote the Born terms for the pp, pn collisions, respectively. For the case of the delta we also present the results of a calculation involving a single n-exchange in the r-channel in terms of the long dashed curve labeled “Act, rr)“at 2.1 GeV (upper right). The result for the bremsstrahlung contribution in the soft photon approximation from 1131 is given by the dashed lint-s labeled “pn (SPA)“.
labeled “A”and “A ( t, n)“, respectively. The anisotrspy coefficients B for pp and pn collisions as functions of M at 1.0 and 2.1 GeV are plotted in Fig. 3. As was shown in Ref. [ 131 the B values range from +l (for pseudoscalar meson (?rO,v) Dalitz decay) to -1 (for ~T+P- annihilation or p decay) and depend on both, the invariant mass M and the process considered. The solid curves with label “8” correspond to the calculation in the “full” QBE-model accounting for all mesons exchanges and all diagrams including the interferences. The ‘*A” labels the contribution of the delta resonance term, ‘Pp”, “pn”denote the Born terms for the pp, pn collisions, respectively. For the case of the delta we also present the results of a calculation involving only the r-exchange in the t-channel given by the long dashed curve labeled “A(t,r)” for the pp reaction at 2.1 GeV. The dotted and the long dashed line in this figure are almost identical since the coefficient B,
which is defined as a ratio of differential cross sections (cf. Eq. (2)), is not sensitive to the absolute value of the cross section, whereas the angular distributions are quite similar (1.h.s. of Fig. 2 j . In Ref. [ 131 the dilepton cross secrion from the A channel was approximated by a product of the delta production cross section and the dii’ferelltial width of the A Dalitz decay. The B value in this approximation is 4-l as for ihe case of pseudoscalar meson (TO, 91) Dalitz decay. We note that thotigh this approximation reproduces the form of the inclusive dilepton spectrum calculated within the OBE-model, it should be considered as rather unrealistic for the study of more subtle microscopic effects, such as polarization phe.. nomena and spin effects. Only for nearly real photons (M -+ 0) the coupling to the hadronic part (or delta alignment) becomes negligible. Thus we obtaitl a rather complicated behavior for the coefficient B in the OBE-model instead of B = 4-l in [ 131.
330
E.L. Bratkovskaya et al. /Physics Letters B 348 (1995) 325-330
For the same reasons the functional dependence of B for the Born terms changes very strongly with M: the dashed curves in Fig. 3 labeled “pn” are the results in the OBE-model, the dashed curves labeled “pn (SPA)” are the result from [ I31 based on the soft photon approximation (SPA) for the dilepton brzmsstrahlung. Here again the results are close at the lower energy ( 1 GeV) and low dilepton mass M. Note that at small Al an increase of the virtual photon energy leads to a decrease of B in the soft-photon approximation while B increases with M in the OBE-calculation and remains positive for all M contrary to the SPA. In summary we can conclude that the angular anisotropy B of dilepton pairs is very sensitive to the microscopic details of the interaction and can be used to determine the relative weight of the hadronic production channels in pN reactions as a function of the invariant mass M. It remains to be seen whether this angular anisotropy can also be used to distinguish between the various reaction channels in heavy-ion induced dilepton production. The authors acknowledge stimulating discussions with S.S. Shimanskiy and A.I. Titov. E.L.B. is grateful to the Institute for Theoretical Physics of the University ‘of Giessen for the support and kind hospitality during her visit. Part of this research was stimulated by discussions during the INT-04-3 program on ‘Hot and Dense Nuclear Matter’ in Seattle. Furthermore, this research was partly performed in the framework of the Grants NO_MP8000 and RFEOOOfrom the Intcrnational Science Foundation.
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