Azimuthal asymmetry of direct photons in intermediate energy heavy-ion collisions
G. H. Liu, Y. G. Ma, X. Z. Cai, D. Q. Fang, W. Q. Shen, W. D. Tian, K. Wang
aa r X i v : . [ nu c l - e x ] A p r Azimuthal asymmetry of direct photons inintermediate energy heavy-ion collisions
G. H. Liu a,b , Y. G. Ma a ,X. Z. Cai a , D. Q. Fang a , W. Q. Shen a , W. D. Tian a , K. Wang a a Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China b Graduate School of the Chinese Academy of Sciences, Beijing 100080, China
Abstract
Hard photon emitted from energetic heavy ion collisions is of very interesting since it doesnot experience the late-stage nuclear interaction, therefore it is useful to explore the early-stageinformation of matter phase. In this work, we have presented a first calculation of azimuthalasymmetry, characterized by directed transverse flow parameter F and elliptic asymmetry co-efficient v , for proton-neutron bremsstrahlung hard photons in intermediate energy heavy-ioncollisions. The positive F and negative v of direct photons are illustrated and they seem to beanti-correlated to the corresponding free proton’s flow. Key words: proton-neutron bremsstrahlung, hard photons, azimuthal asymmetry, BUU
PACS:
The properties of nuclear matter at differ-ent temperatures or densities, especiallythe derivation of the Equation-of-State(EOS) of nuclear matter, are one of theforemost challenges of modern heavy-ionphysics. Since heavy ion collisions provideup to now the unique means to form andinvestigate hot and dense nuclear matterin the laboratory, many experimental andtheoretical efforts are under way towardsthis direction. Because of their relativelyhigh emission rates, nucleons, mesons, lightions and intermediate mass fragments,produced and emitted in the reactions, areconveniently used to obtain information onthe reaction dynamics of energetic heavyion collisions. However, these probes in-teract strongly with the nuclear mediumsuch that the information they convey may
Email address: [email protected] author. ( Y. G. Ma a , ). bring a blurred image of their source. For-tunately, energetic photons offer an attrac-tive alternative to the hadronic probes [1].Photons interacting only weakly throughthe electromagnetic force with the nuclearmedium are not subjected to distortionsby the final state (neither Coulomb norstrong) interactions. They therefore de-liver an undistorted picture of the emittingsource. For hard photons, defined as γ -rays with energies above 30 M eV in thispaper, many experimental facts supportedby model calculations [1,2,3] indicate thatin intermediate energy heavy-ion colli-sions they are mainly emitted during thefirst instants of the reaction in incoherentproton-neutron bremsstrahlung collisions, p + n → p + n + γ , occurring within theparticipant zone. This part of hard pho-tons are called as direct photon. Directhard photons have thus been exploited toprobe the pre-equilibrium conditions pre- Preprint submitted to Elsevier Science 1 November 2018 ailing in the initial high-density phase ofthe reaction [4,5]. Aside from the dominantproduction of hard photons in first-chancep-n collisions, a significant hard-photonproduction in a later stage of heavy-ionreactions, called as thermal photons, arealso predicted by the
Boltzmann-Uehling-Uhlenbeck ( BUU ) theory [6,7]. These ther-mal photons are emitted from a nearlythermalized source and still originate frombremsstrahlung production by individualp-n collisions, which was also confirmed bythe experiments at last decade [8,9].In this work, we take the
BUU trans-port model improved by Bauer [10]. Theisospin dependence was incorporated intothe model through the initialization andthe nuclear mean field. The nuclear meanfield U including isospin symmetry termsis parameterized as U ( ρ, τ z ) = a ( ρρ )+ b ( ρρ ) σ + C sym ( ρ n − ρ p ) ρ τ z , (1)where ρ is the normal nuclear matter den-sity; ρ , ρ n , and ρ p are the nucleon, neutronand proton densities, respectively; τ z equals1 or -1 for neutrons and protons, respec-tively; The coefficients a , b and σ are pa-rameters for nuclear equation of state. C sym is the symmetry energy strength due to thedensity difference of neutrons and protonsin nuclear medium, which is important forasymmetry nuclear matter (here C sym = 32 M eV is used), but it is trivial for the sym-metric system studied in the present work.For the calculation of the elementarydouble-differential hard photon productioncross sections on the basis of individualproton-neutron bremsstrahlung, the hard-sphere collision was adopted from
Ref. [11],and modified as in
Ref. [12] to allow for en-ergy conservation. The double differentialprobability is given by d σ elem dE γ d Ω γ = α R π E γ (2 β f + 3 sin θ γ β i ) . (2)Here R is the radius of the sphere, α isthe fine structure constant, β i and β f arethe initial and final velocity of the proton ] - b ( f m / c ) m / d t [ s d r / r time (fm/c) (a) (b) r / max r r / ærÆ Fig. 1. (a) Time evolution of hard photon pro-duction rate for the reaction Ca + Ca col-lisions at 30 M eV /nucleon for semi-centralevents (40–60%). (b) Time evolution of re-duced maximum density ρ max /ρ (closed cir-cles) and reduced average density h ρ i /ρ (opencircles) of the whole reaction system in thesame reaction. The blue dashed line representsthe time when the system ends up till the firstexpansion stage, and in the panel (a) it sep-arates direct photons (on the left side) andthermal photons (on the right side). in the proton-neutron center of mass sys-tem, and θ γ is the angle between incidentproton direction and photon emitting di-rection. More details for the model can befound in Ref. [10].In this paper, we simulate the reaction of Ca + Ca collisions at 30 M eV /nucleon ,and use the
EOS with the compressibility K of 235 M eV ( a = -218 M eV , b = 164 M eV , σ = 4/3) for the nuclear mean field U . As a first attempt to extract the pho-ton’s azimuthal asymmetry, we only takethe semi-central events (40 − t = 80 f m/c in thisreaction), as direct photons (on the leftside of blue dashed line in Fig. 1(a)). It isalso coincident with the definition of directphotons above. And we call the residualhard photons produced in the later stageas thermal photons (on the right side ofblue dashed line in Fig. 1(a)). So in thesimulation, we can identify the producedphoton as direct or thermal photon by theemitting time. Because of the sensitivityto the density oscillations of colliding sys-tem, hard photon may be sensitive to thenuclear incompressibility [6,7].It is well known that collective flow isan important observable in heavy ioncollisions and it can bring some essen-tial information of the nuclear matter,such as the nuclear equation of state[14,15,16,17,18,19,20,21,22,23]. Anisotropicflow is defined as the different n − th har-monic coefficient v n of the Fourier expan-sion for the particle invariant azimuthaldistribution [15]: dNdφ ∝ ∞ X n =1 v n cos( nφ ) , (3)where φ is the azimuthal angle between thetransverse momentum of the particle andthe reaction plane. Note that the z -axis is defined as the direction along the beamand the impact parameter axis is labelledas x -axis. Anisotropic flows generally de-pend on both particle transverse momen-tum and rapidity, and for a given rapiditythe anisotropic flows at transverse momen-tum p t ( p t = q p x + p y ) can be evaluatedaccording to v n ( p t ) = h cos ( nφ ) i , (4)where h· · · i denotes average over the az-imuthal distribution of particles with trans-verse momentum p t , p x and p y are projec-tions of particle transverse momentum inand perpendicular to the reaction plane,respectively. The first harmonic coefficient v is called directed flow parameter. Thesecond harmonic coefficient v is called theelliptic flow parameter v , which measuresthe eccentricity of the particle distributionin the momentum space.In relativistic heavy-ion collisions az-imuthal asymmetry of hard photons havebeen recently reported in the experimentsand theoretical calculations [24,25,26,27].It shows a very useful tool to explore theproperties of hot dense matter. However,so far there is still neither experimentaldata nor theoretical prediction on the az-imuthal asymmetry of hard photons inintermediate energy heavy ion collisions.Does the direct photon also exist azimuthalasymmetry so that it leads to non-zero di-rected transverse flow or elliptic asymme-try parameters in the intermediate energyrange? Moreover we know that direct pho-tons mostly originate from bremsstrahlungproduced in individual proton-neutron col-lisions, and free nucleons are also emittedfrom nucleon-nucleon collisions. Does theazimuthal asymmetry of the direct photonscorrelate with the one of free nucleons? Toanswer the above question, we focus onthe azimuthal asymmetry analysis for bothphotons and protons in this Letter.Fig. 2 shows the time evolution of the di-rected flow parameter v and elliptic flowparameter v for hard photons and freeprotons. Before we take further calculationand explanation, people should be cau-3ious about the word of ”flow” for photons.Since flow is associated with collectivitycaused by multiple interactions, which areexhibited by the nucleons, but not by thephotons. The photon emission pattern isbasically a result of the nucleon flow, andnot a photon flow per se. However, How-ever, in order to compare the results be-tween photons and protons, we still called v as directed flow parameter and v ellip-tic flow parameters for photons somewherein texts. Considering the nearly symmet-ric behavior for directed flow parameter( v ) versus rapidity, here we calculate theaverage v over only the positive rapidityrange, which can be taken as a measureof the directed transverse flow parameter.For emitted protons (open circles in thefigure) which are experimental measurableand are identified in our BUU calculationas those with local densities less than ρ / t = 30 f m/c before that the system is mostly infusion stage and protons are seldom emit-ted. The negative directed flow parameter v of free protons essentially stems from theattractive mean field. Up till t ∼ f m/c when the system is in the freeze-out stage,the directed flows become saturate. Forthe elliptic asymmetry parameter v of freeprotons, the positive values indicate of thepreferential in-plane emission driven bythe rotational collective motion due to theattractive mean filed. Similarly, the ellipticasymmetry parameter becomes saturatein the freeze-out stage. However, thereare obvious difference for proton-neutronbremsstrahlung photons (solid circles inthe figure) in comparison to protons. Con-trary to the negative directed transverseflow and positive flow, directed photonsshows the positive v and the negative v before t = 80 f m/c , i.e. the azimuthalanisotropy is shifted by a phase of π /2. Thetimes corresponding to the peak or valleyvalues of flows roughly keep synchronizedwith the compression or expansion oscilla-tion of the system evolution. For the late-stage thermal photons after t = 170 f m/c the azimuthal asymmetry vanishes, i.e. v and v fades-out.From the above calculations, we learn thatthermal photons in the later stage of reac- ( % ) v ( % ) v time (fm/c) -10-505101520 (a) hard photonfree proton (b) hard photonfree proton Fig. 2. The time evolution of v (a) and v (b) for hard photons (closed circles) and freeprotons (open circles). tion are emitted from a more thermalizedsystem, they prefer more isotropic emission(i.e. the vanishing ”flow” parameters) thandirect ones produced in the pre-equilibriumstage. Thereafter we only consider directphotons to discuss the azimuthal asymme-try results. For protons, we take the valuesof flows when the system has been alreadyin the freeze-out time at 180 f m/c .The directed transverse flow parameter atmid-rapidity can be also defined by theslope: F = d h p x i d ( y ) c.m. (cid:12)(cid:12)(cid:12) ( y ) c.m. =0 , where ( y ) c.m. is the rapidity of particles in the center ofmass and h p x i is the mean in-plane trans-verse momentum of photons or protons ina given rapidity region. In Fig. 3(a) and(b), we show h p x i plotted versus the c.m. rapidity y c.m. for direct photons (a) as wellas h p x i plotted versus the reduced c.m. rapidity ( y/y beam ) c.m. for free protons (b).The errors shown are only statistical. Agood linearity was seen in the mid-rapidityregion ( − . , .
5) and the slope of a linearfit can be defined as the directed transverseflow parameter. The extracted value of thedirected transverse flow of direct photonsis about +3 . M eV /c , and that of free pro-tons is about − . M eV /c . Thus directphotons do exist the directed transverse4 ( M e V / c ) æ x p Æ ( nb ) F / d s d ( M e V / c ) æ x p ÆF d N / d c.m. (y) c.m. ) beam (y/y) (cid:176) ( F ) (cid:176) ( F -4 -3 -2 -1 0 1 2 3 4-3-2-10123 (a) direct hard photon -3 -2 -1 0 1 2 3-80-60-40-20020406080 -80-60-40-20020406080 (b) free proton -150 -100 -50 0 50 100 150580600620640660 (c) direct hard photon -150 -100 -50 0 50 100 150 (d) free proton Fig. 3. (a) Average in-plane transverse mo-mentum of direct photons as a function of c.m. rapidity for semi-central events (40 − − . ≤ y c.m. ≤ .
5. (b)Same as the panel (a) but for free protons. Thedashed line segment is a fit over the mid-ra-pidity region − . ≤ ( y/y beam ) c.m. ≤ .
5. (c)and (d) are the azimuthal distributions of di-rect photons and free protons, respectively,and both of them are fitted to the 4 th orderFourier expansion. asymmetry even though the absolute valueis smaller than the proton’s flow, and itssign is just opposite to that of free protons.As Eq. 3 shows, elliptic flow is defined asthe second harmonic coefficient v of anazimuthal Fourier expansion of the par-ticle invariant distribution. In order toextract the value of elliptic asymmetrycoefficient v and reduce the error of fits,we fit the azimuthal distribution to the4 th order Fourier expansion. Shown inFig. 3(c) and (d), direct photons demon-strate out-of-plane enhancement and the v is about − . v isabout +7 . v . Fig. 4shows v of direct photons (a) and freeprotons (b) as a function of transverse mo- ( % ) v ( % ) v (MeV/c) T p (MeV/c) T p (a) direct hard photon (b) free proton Fig. 4. v as a function of transverse momen-tum ( p T ) for direct photons (a) and free pro-tons (b). mentum p T . Similar to the directed trans-verse flow parameter, the values of ellipticasymmetry coefficient v of direct photonsand free protons also have the oppositesigns at this reaction energy, i.e. reflectinga different preferential transverse emissionin the direction of out-of-plane or in-plane,respectively. Meanwhile, the absolute val-ues of v for photons are smaller than theproton’s values as the behavior of trans-verse flow. Except the opposite sign, we seethat both v have similar tendency withthe increase of p T , i.e., their absolute valuesincrease at lower p T , and become graduallysaturated, especially for direct photons.To explain the above anti-correlation ofanisotropic emission between direct pho-tons and free protons, we should note thatdirect photons originate from the individ-ual proton-neutron collisions. As Eq. 2shows, we can roughly consider that in theindividual proton-neutron center of masssystem, in directions perpendicular to in-cident proton velocity, i.e. θ γ = π/
2, theprobability of hard photon production ismuch larger than that in the parallel di-rection, i.e. θ γ = 0, which is in agreementwith the theoretical calculations and theexperiments [29,30], that causes hard pho-ton preferential emission perpendicular tothe motion plane of corresponding nucle-ons. As a whole, the azimuthal anisotropyof hard photons is shifted by a phase of π/ F and v between them. Consequently, azimuthalanisotropic emission of hard photon and5ree nucleon are anti-correlated, presentingthe opposite behavior.In conclusion, we have presented a firstcalculation of azimuthal asymmetry, bothdirected and elliptic asymmetry, for di-rect photons produced by proton-neutronbremsstrahlung from intermediate en-ergy heavy-ion collisions. It was, for thefirst time, presented that in the inter-mediate energy heavy-ion collisions theproton-neutron bremsstrahlung hard pho-ton shows non-zero directed transverse flowparameter and elliptic asymmetry coeffi-cient which have opposite sign to the corre-sponding free proton flow parameters. Thetime evolutions of azimuthal parameters v and v of hard photons exhibit rich struc-tures as the density oscillation of the sys-tem during the pre-equilibrium and ther-malization stage of reaction system. There-fore direct photons can server for a goodprobe to nuclear matter properties. Con-sidering that hard photons are dominantlyproduced by individual neutron-protonbremsstrahlung, so they are sensitive to thein-medium neutron-proton cross section,but not to the in-medium proton-proton orneutron-neutron cross section, that can beadvantaged in the isospin dependent studyof in-medium nucleon-nucleon cross sec-tion by direct photons. Of course, system-atic studies of the influences from equationof state, in-medium nucleon-nucleon crosssection, impact parameter and incidentenergy etc on the azimuthal asymmetry ofdirect photon should be carried out. Theprogress along this line is underway.This work was partially supported bythe National Basic Research Program ofChina (973 Program) under Contract No.2007CB815004, Shanghai DevelopmentFoundation from Science and Technol-ogy under Grant Numbers 06JC14082 and06QA14062, the National Natural Sci-ence Foundation of China under Grant No10535010 and 10775167. References [1] Y. Schutz et al., Nucl. Phys. A , 404(1997).[2] W. Cassing et al., Phys. Rep. , 363(1990).[3] H. Nifenecker et al., Annu. Rev. Nucl.Part. Sci. , 113 (1990) .[4] R. Wada et al., Phys. Rev. C , 497(1989).[5] M. Schmidt et al., Phys. Rev. Lett. ,20340 (2001).[6] Y. Schutz et al., Nucl. Phys. A , 97(1996).[7] Y. Schutz et al., Nucl. Phys. A , 126(1998).[8] G. Martinez et al., Phys. Lett. B , 23(1995).[9] D. G. d’Enterria et al., Phys. Rev. Lett. , 22701 (2001).[10] W. Bauer et al., Phys. Rev. C , 2127(1986).[11] J. D. Jackson, Classical Electrodynamics(Wiley, New York, 1962), p. 733.[12] W. Cassing et al., Phys. Lett. B , 21(1986).[13] F. M. Marqu´es et al., Phys. Lett. B ,30 (1995).[14] J. Y. Ollitrault, Phys. Rev. D , 229(1992).[15] S. Voloshin, Y. Zhang, Z. Phys. C , 665(1996).[16] H. Sorge, Phys. Lett. B , 251 (1997);Phys. Rev. Lett. (1997) 2309; , 2048(1999).[17] P. Danielewicz, R. A. Lacey, P. B.Gossiaux et al., Phys. Rev. Lett. , 2438(1998).[18] Y. G. Ma et al., Phys. Rev. C , R1492(1993); Z. Phys. A , 469 (1993); Phys.Rev. C , 1029 (1995); Phys. Rev. C , 3256 (1995); Nucl Phys. A , 611c(2007).
19] Y. M. Zheng, C. M. Ko, B. A. Li, and B.Zhang, Phys. Rev. Lett. , 253 (1999).[20] D. Persram and C. Gale, Phys. Rev. C ,064611 (2002).[21] J. Lukasik et al. (INDRA-ALDAINCollaboration), Phys. Lett. B , 223(2004).[22] T. Z. Yan, Y. G. Ma et al., Phys. Lett. B , 50 (2006).[23] J. H. Chen, Y. G. Ma, G. L. Ma et al.,Phys. Rev. C , 064902 (2006).[24] M. M. Aggarwal et al., Phys. Rev. Lett. , 022301 (2004); Nucl. Phys. A , 129(2005).[25] S. S. Adler et al., Phys. Rev. Lett. ,032302 (2006).[26] S. Turbide et al., Phys. Rev. Lett. ,032303 (2006).[27] R. Chatterjee, E. S. Frodermann, U.Heinz, and D. K. Srivastava, Phys. Rev.Lett. , 202302 (2006).[28] P. Danielwicz et al., Phys. Lett. , 146(1985).[29] V. Herrmann and J. Speth, Phys. Rev. C , 394 (1991).[30] Y. Safkan et al., Phys. Rev. C , 031001(2007), 031001(2007)