We compute the energy spectrum of photons which originate from the quark-annihilation process ss¯&#8594;&#947;g in quark-gluon plasma. The spectrum peaks at an energy E&#947;max&#8764;2ms&#8764;400 MeV in the rest frame of the plasma. We expect one photon from the above process in the energy range of 2ms±0.25ms per hundred quark-gluon plasmas of a size R=3 fm and a lifetime &#964;=6 fm/c formed in nuclear collisions

Greiner, Walter

Rafelski, Johann

Staadt, Günther

English

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PHYSICAL REVIEW D  VOLUME  33, NUMBER  1  1 JANUARY  1986 Photons from strange-quark annihilation in a quark-gluon plasma G. Staadt and W. Greiner Institut fur  Theoretische Physik der J. W.  Goethe-Universität, 0-6000  Frankfurt am Main 11, Postfach 11  1932, West Germany J. Rafelski Institute of Theoretical Physics and Astrophysics,  University of Cape Town, Rondebosch  7700, Republic of South Africa (Received 20 February 1985) We  compute the energy spectrum of  photons which originate from the quark-annihilation process ss-~yg  in quark-gluon plasma.  The spectrum peaks at an energy E?  -2ms -400  MeV  in the rest frame of  the  plasma.  We  expect  one  photon  from  the  above  process  in the  energy  range  of 2m, IO.  25m, per hundred quark-gluon plasmas of  a size R =  3 fm  and a lifetime T= 6 fm/c formed in nuclear collisions. I.  INTRODUCTION Given the present  knowledge about  strong interaction, one has to assume that in collisions of  heavy nuclei with sufficiently high kinetic energy, a high-energy-density re- gion  can  be  formed  in  which  the hadrons  dissolve into their constituents, i.e., into quark-gluon  Several estimates2 lead to 0.5-2  ~e~/fm~  for the critical energy density.  The quark-gluon plasma represents a new physi- cal  phase  and  its  physics  can  often  approximately  be described by  methods of  perturbation theory.'r2  In order to verify this hypothesis of the existence of a new phase of matter one has to consider all processes, which could give a hint  to the existence of  the plasma.  In  this paper  we consider certain processes leading to high-energy photons. These photons will, in general, leave the interaction region without further interactions. Numerous photon production processes which have lit- tle to do with the plasma state will normally dominate the photon  spectra.  However, one can  perform  an in-depth study of  various contributions3 and it is conceivable that even a small effect may become visible in certain kinemat- ic domains, such as small  X, large PI,  once a correlation with strangeness overabundance is established. In the plasma phase, numerous pairs of strange and an- tistrange quarks are created by  gluons,4 which occasional- ly  annihilate again.  One possible,  though  rare, annihila- tion  process is to create a gluon and a photon.  Naively one expects that  these photons possess a  relatively high energy.  The  energy  of  photons  is  given  by E,=Es  +E,-E,  and  it should be  recorded that strange quarks and antiquarks carry their rest mass, in addition to the thermal kinetic energy, into the reaction  while emis- sion of low-energy gluons is stimulated by  the existence of numerous low-energy gluons in plasma.  In this paper we calculate the spectrum of the photons which are produced in such processes (sS-+yg)  quantitatively to first order in the perturbative approach. 11.  TRANSITION PROBABILITY:  sF4yg First  we  compute  the  transition  probability  per  unit time  and  volume  (U)  of  the  process  ss-+yg. For  that purpose  we  use  the lowest-order perturbative QCD and QED.  This  means  that  we  evaluate  the  Feynman  dia- grams shown in Fig. 1. The symbols indicated in this figure have the following meaning:  p„p,,q,k  are the four-momenta of the particles, s I ,s2 the spin  of  the quarks, i,j the quark color,  a  the gluon color, 8,eG  the polarization vectors. Using the nomenclature of Ref. 5 the transition proba- bility w per unit time and volume can be written in terms of the invariant amplitude .M as The two contributions shown in Fig. 1 are given explicitly by6 We find that the propagator of  the virtual quark is not dependent on the presence of  the other plasma  particles, because the  chemical  potential  of  the strange quarks  is zero.  The chemical potential  must be zero, since strange quarks are always produced in  This independence can be established as follows:  A  fermion propagator in momentum space can be written7 66  @ 1985 The Amencan Physical Society PHOTONS FROM STRANGE-QUARK ANNIHILATION IN A . . .  67 with p'"=pa+pua  (5) (p=chemical  potential).  For  Strange  quarks  we  have p, =O  and Eq. (5) leads to pta=pa. From kinematic con- siderations  we  know  that p2  must  always be  unequal to Zero in order to have a massless photon emitted.  There- fore the second term in Eq. (4) does not yield a contribu- tion and Eq. (4) reduces to the propagator used in Eqs. (3). Let us here further note that these considerations are not valid for the light  (U  and d)  quarks, since their chemical potentials do not vanish in general. We evaluate  I d  1  in the rest frame of the s quark us- ing the radiation gauge.  The result is 2 means the sum over the spin/color  quantum numbers of  all  particles.  Expression  (6) is valid  only  in the rest frarne of the s quark.  However, we can transform Eq. (6) into a  function  of  two  invariant  Mandelstam  variables. Then  the  resulting  expression is valid  in  every  Lorentz System.  The three Mandelstam  variables of  the process FIG. 1.  sS+  yg in first order in a,a. described by  Fig. 1 are 2  s =(p,+pp) , 2  t =(P,-k)  , 2 U  =(P,-q)  , and are constrained by s+t+u=2m2. We make use of  s and  t as independent variables and ob- tain the final formula: In Eq. (8)  the QCD coupling constant a, =g2/4r is also a slowly changing  function of  s  and  t.  However, we  will use a constant value of a,  in further calculations which is to be  seen as a weighted average over space- and timelike transferred momenta. 111.  PHOTON SPECTRUM The number of events (sF-yg)  per unit time and volume is given by8 p(p,), P(&),  and pB(k)  are the distribution functions of the corresponding particles.  In order to obtain these func- tions, we make the simplest possible assumption, i.e., that quarks and gluons in the plasma can be described as an (nearly) ideal gas.1.2 Thus for the emitted  gluons we  in- clude the Bose function pB(k)  and notice the appearance of  the stimulated process (-PB)  next to the spontaneous one ( -  1) in the square brackets.  The present considera- tion of the quantum statistics effect-the  induced reaction (-pB )-goes  beyond the benign neglect of  these types of corrections,  such  as  the  neglect  of  Pauli  blocking  of strange-quark formation in Ref. 4.  We shall see that even the very  much  greater gluon density occurring  here will not significantly affect the computed reaction rate. The quark densities p(p, )  and  follow the Fermi distribution functions; in the rest frame of  the plasma we obtain For both F and s we  use only the first term of the series, which  corresponds  to the  classical  Boltzmann  distribu- tion.  The  overestimation  made  by  this  approach  is greatest at the lowest possible energy (the mass) of  the s and  F quark, where we  have  for both  quarks  E/T= 1. Since Boltzmann approximation simplifies the calculation greatly we have used it also in view of the other uncertain- ties prevailing here. We  are interested in the photon  spectrum and we will First compute the Lorentz-invariant quantity ~,d  3~  /d  3q. Since 2  1  .AT  I  is a function of  s and  t, we introduce in- tegrations over these variables: G. STAADT, W. GREINER, AND J. RAFELSKI We now perform the following operations. (1)  Jd3k(2~,)-'-Jd4k  S(k2)8(ko). (2)  Jd4k  is carried out using the S4 function and we set k =P, +ps-q. 2  1/2  (3)  We intend to carry out first the integrals over ps  and pF. Then we obtain, where ti  =m2-  ts$r(s2/4-m  s) We evaluate ps,& integrals in the rest frame of the plasma where the statistical distribution functions take a simple form. We notice that (1) d  'p-  I p /  EdE  da  for p,  and pF, (2)  three angular integrals are canceled by  the 6 functions and the last angular integral yields a factor 4~,  and (3)  the cancellation of the three angular integrals by  the 6 functions leads to restrictions of the domains of integration for Es and E,.  The result of these operations is given by the following expres- sion: where the integration regions  (B),  {B']  and  constants a,b,c are defined in Appendix  A  [see Eqs.  (A9), (AlO),  and (A5)-(A7)].  Using the substitution x =Es +EF,  y =G  the two integrals over Es and Es can be reduced to The spontaneous emission (SE) [corresponds to 1  in the large parentheses in Eq. (14)]  can now be obtained analytically. The integrations over x  and t [Eqs. (13)  and (1411 are carried out analytically and the remaining function of  s can be ap- proximated by  a simplified function of  which analytical integrals exist.  We obtain the simplified function from the in- vestigation of the asymptotic behavior of the exact function, where we can replace, e.g., As it turns out we  can use these approximations in the total domain of  integration.  The error of  the final result intro- duced in this way is of the order of magnitude of  1% and thus negligible.  The result for the spontaneous emission is 33  -  PHOTONS FROM STRANGE-QUARK ANNIHILATION  IN A . . .  69 We now turn to the calculation of the reduced emission (IE) [corresponds  to  the  Bose  function  in  the  large parentheses in Eq. (14)].  For small values of  (X -  EY  )/T the Bose  function can be  expanded in  a power  series in (X  -Ey)/T: where Bi  are the Bernoulli numbers.  For large values of (X  -E, )/T  (i.e., large gluon energies) the induced radia- tion has a very small yield and can be neglected.  We thus can replace the Bose function by  the approximate expres- sion We have verified that for all values of  Ey  the error of the final result made by  this approximation is only of the or- der  of  magnitude of  1%.  We  now  can perform  the in- tegrations over X and t analytically.  We obtain FIG. 2.  Energy  spectrum (spontaneous emission) in the rest frame of the plasma.  m is the (strange-)  quark mass. X  ~~~~~~d~~~(~)~  s  ,  (18) with However, we have not found a useful approximant for the remaining  s integral which must be  evaluated numerical- ly.  The result assumes the form ~~d~~/d~~  is  a  Lorentz-invariant  quantity,  which takes the form of Eq. (15)  plus Eq. (21)  in the rest frame of  the  plasma.  In  this  reference  system  the  spectrum d3A/d3q  depends  only  on  I q 1  =E„  i.e., d  3q =da  E,*~E,.  Therefore we  obtain the energy spec- trum in the rest frame of the plasma as Substitution of g2=4ra, and and into Eq. (22)  yields G.  STAADT, W. GREINER, AND J. RAFELSKI FIG. 3.  Energy spectrum (induced emission) in the rest frame of  the plasma.  m is the (Strange-)  quark mass. FIG. 4.  Energy spectrum (spontaneous + induced emission) In Figs.  2,  37  arid  we shOw fa, fb  arid fa  +fb  as func-  in  the rest frame of  the plasma.  m is the (strange-1  quark mass. tions of E, /m  for different values of  T/m. IV.  DISCUSSION OF RESULTS The position of  photon peaks from induced and spon- taneous  processes shown  in  Figs.  2 and  3 confirms  the qualitative argument  made  in  the Introduction:  the in- duced peaks are shifted to higher energies by  roughly 0.5 m  in  the  range  of  temperatures  considered  here (0.8 <T  < 1.2 m).  However, the induced photons turn out to be about four times less numerous and hence the pho- ton spectrum is dominated by  the spontaneous process as shown in Fig. 4. Considering  the slopes of  the spectra we  find  that  in general they obey  an e-E'(KT' law, where K  ranges from 1.15 to 1.3.  Hence a naive approach in which slopes are used  as a measure of  the temperature would result in an overestimation by  a factor K of the prevailing plasma tem- perature.  Furthermore, the K'S  of the induced spectra are about 5% larger than those of the spontaneous radiation. We recall now that A is the rate per unit of  volume and time.  To find out the number of photons per unit of ener- gy  coming from a quark-gluon plasma, we have to multi- ply Eq. (23) with the volume  V of  the plasma and its es- timated lifetime T: For  V-4~/3 (3 fm13 and  r=6  fm/c  and  assuming rn  =  200 MeV, a, =  the normalized spectra (Figs. 2-41 have to be multiplied with a factor  1.8 (M~v)-'  in order to obtain the proper values.  We see that for T =  200 MeV in  100 quark-gluon plasma events one photon from sF an- nihilation  will  emerge in  an energy  range  of  AE, =  100 MeV around EraX  =  2ms. Integrating our spectra we  find one photon from sS  an- nihilation per ten plasma events living 6 fm/c and having a radius of 3 fm.  Thus we conclude that high-energy pho- tons from ss annihilation in quark-gluon plasma are rare. Finally, one may use our results, with all the due reser- vations,  also  as  an  order-of-magnitude estimate  for  the rate  of  high-energy  photon  radiation  from  light  u,d quarks.  Setting T =200 MeV, m x 10-20  MeV, which is appropriate to light  quarks, we  find  that the number  of emitted photons is enhanced roughly by  a factor 10.  The peaks are located at an energy of  Eyx  ,235-244  -  MeV. However, one should remember that light-quark processes are modified due to modifications of  the quark propaga- tor and quark distribution functions at finite baryon den- sity.  Furthermore, the up quarks have a four times larger coupling to the photon field. Of  Course  Compton-type processes  must  also be con- sidered  for  photon  production  in  quark-gluon  plasma. However, our expectation has been  that high-energy pho- tons  are dominated  by  annihilation diagrams studied  in this Paper. PHOTONS FROM STRANGE-QUARK ANNIHILATION IN A . . APPENDIX A Three angular integrals in Eq. ( 12) are carried out using  the 6  functions.  These operations lead  to the following restrictions  of  the domains  of  in- tegration for E, and &: where (Al) Equation (12)  further satisfies the kinematic constraint A study of the conditions Eqs. (Al)-(A4)  and (A8)  shows that it is sufficient to require  Eqs. (Al),  (A3), and (A4). Then Eqs. (A2) and (A8)  are always fulfilled.  We intend to carry out first the E, integration and after that the in- tegration  over  E,.  Hence  we  obtain  the  following domains of integration: 1  /2 J  d~?:  from - (S +t -m2)  m 'E,  J  d~,:  from  +  to w={B) 4EY  (s  +t -m2) 'S.  A.  Chin, Phys.  Lett.  78B, 552  (1978);  J. Rafelski and R.  3P. V. Chliapnikov et al., Phys. Lett. 141B, 276 (1984). Hagedorn, in  Thermodynamics of Quarks and Hadrons, edit-  4J. Rafelski and B. Müller, Phys. Rev. Lett. 48, 1066 (1982). ed  by  H.  Satz  (North-Holland,  Amsterdam,  1981);  G.  5J. D. Bjorken and S. D. Drell, Relativistic  Quantum Mechanics Domokos and J. I. Goldman, Phys. Rev. D 23, 203 (1981).  (McGraw-HiU,  New York, 1964). ?See, e.g., J. Rafelski and M. Danos, in Lecture Notes in Physics  6R. Cutter and D. Sivers, Phys. Rev. D 17, 196 (1978)  and Ref. 231,  edited by  W.  D.  Heiss (Springer, New  York,  19851, p.  5. 361;  B.  Müller,  Universität  Frankfurt  Report  No.  UFTP  'See,  e.g., H. A. Weldon, Phys. Rev. D 26, 1394 (1982). 125/83 (unpublished),  and references therein.  %ee, e.g., Ref. 5. 

Photons from strange-quark annihilation in a quark-gluon plasma

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