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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 
 

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Theoretical Study of the Photons Production Kinetic In Hot Quark-Gluon 
Plasma Matter 

 
 
 
 
 
 
 
 

Article history: Received 28 January 2020, Accepted 15 March, Published October 2020.   

  Doi: 10.30526/33.4.2523  
 
 
Abstract 
      In this paper, we study flow of photons rate production in a quark-gluon QG plasma. 
General theory of this study is based on the field theory for hard interaction. The kinetic of 
photons production from hard interaction in charm with anti-top to production photons with 
gluon 𝑐𝑡̅ → 𝛾𝑔due to plasma phase at high temperatures (150, 200,250,300 and 350 MeV) .It 
has been investigated and studied using the postulate of quantum chromodynamic theory 
QCD .The photons production rate of hard photons with(𝐸 1GeV) are insensitive to 
strength coupling and depend mainly on the temperature of system T . Despite the different 
critical temperature (150 and 190MeV) comes, we find that same order of flow rate photons 
magnitude in both cases. In both cases, the flow rate of photons production in the QG 
plasmais increased with increased temperature of system and photons energy and decreases 
with increases the strength coupling strength. 
 
Key Word: Photons Production, Kinetic, Hot Quark-Gluon Plasma  
 
1. Introduction  
     The photons are important valuable tools to probing hot matter in relativistic heavy-ion 
collisions RHIC at BNL and Larger Hadronic Collision LHC at CERN[1-2]. The photons 
have been emitted from the different mechanism such that prompt photons, thermal photons 
and jet-medium photons .Photons are emited during the quark gluon plasma phase QGP and 
hadron gas Phase (HG) due to jets, and by decay of long-lived resonances into real photons 
[3]. Recently, the interaction of quark-gluon plasma has been achieved to understand the 
equilibrium state. Hard photons emission in both Compton and annihilation processes with 
having an energy large the temperature T of system [4]. The production of photons is 

Hadi J.M. Al-agealy    Rawnaq Qays Ghadhban Mohsin A. Hassooni 

Department of Physics, College of Education for Pure Sciences Ibn Al- Haitham, University of 
Baghdad, Baghdad, Iraq. 

Ibn Al Haitham Journal for Pure and Applied Science 

Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index 

Rownq.qph@gmail.com Hadialagealy2016@gmail.com  mohsin.a@ihcoedu.uobaghdad.edu.iq 
 



 
 
   

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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 

observable in higher energy relativistic heavy-ion collisions. The interaction of quark–gluon 
plasma in hot media has radiated thermal photons due to transverse momenta [5]. Hard 
photons are a important tool to investigate and study the characteristic and properties of quark 
matter in ultrarelativistic heavy-ion collision. It could be probing the dense system during 
interaction of quark-gluon plasma for a short time[6]. In this paper,we can find the relation 

between the photons rate and properties in 𝑐𝑡̅ → 𝛾𝑔quark system in plasma media according 
to the quantum chromodynamic postulate theory using MATLAP program .  
2.Theory  
     The production of gamma photons from plasma quark gluon interaction has been related to 
the retarded polarization of the gamma photon using field theory the number of photons 
emitted from quarks gluon plasma interaction per unit time and per unit volume is [7-8]. 

𝐸 F E Im ∏ 𝐸, 𝑞                                                                                                    (1) 

where E is the energy of system ,q is momentum of quarks ,∏ 𝐸, 𝑞  is the retarded self-

energy for photons emitted at temperature T and F E  is the Bose-Einstein distribution 
function for gluon and given by [9].  

F E                                                                                                                           (2) 

Where 𝜆  is the fugacity of gluon .Substituting Eq.(2) in Eq.(1) to produce:  

𝐸 𝜆 𝑒 1 Im ∏ 𝐸, 𝑞                                                                                       (3) 

The retarded propagators self-energy term related to photons emission according to spectral 

density 𝜌 𝑤, 𝑘  by relation [10]. 

Im ∏ 𝐸, 𝑞 ∑ 𝑒 𝑒 1 𝑑𝑤 𝑑𝑤 ~ 𝛿 𝐸 𝑤

𝑤 ~  𝐹 𝑤 . 𝐹 𝑤 𝑇𝑟 𝜉 𝑘, 𝑝 𝜌∗ 𝑤, 𝑘 𝜌 𝑤 𝐸, 𝑘 𝑝 𝜉 𝑘, 𝑝                                 (4) 

Where 𝑒  is square charge of quark , 𝑘 is soft momentum of quark propagator, 𝑘 𝑃 is the 

hard momentum of quark propagator ,𝐸 is the energy of system ,𝜌 𝑤, 𝑘  and 𝜌∗ 𝑤, 𝑘  are the 
spectral function and conjugate spectral function of quark, and 𝜉 𝑘, 𝑘, 𝑝 𝜉 𝑘, 𝑘, 𝑝 are 
the propagation and conjugate function of quarks system  
On the other hand, the 𝐹 𝑤 and 𝐹 𝑤 are Fermi Dirac representation for quark and 

anti-quark respectively. The quark and anti-quark distribution are given by the Juttner 
distribution functions [8]. 

𝐹  and𝐹 ́
́

́
                                                                                                                (5) 

Since the trace of system can be reformation using [12]. 

𝑇𝑟 𝜉 𝑘, 𝑘, 𝑝 𝜌∗ 𝑤, 𝑘 𝜌 𝑤 𝐸, 𝑘 𝑝 𝜉 𝑘, 𝑘, 𝑝 2𝑇𝑟 𝜌∗ 𝑤, 𝑘 𝜌 𝑤 𝐸, 𝑘 𝑝         (6) 

Substituting Eq.(6) ,Eq.(5) in Eq.(4) to result: 

𝐼𝑚∏ , 𝑒 ∑ 𝑒 𝑒 1
~

𝛿 𝐸 𝑤

𝑤 ~ 𝜆 𝜆 ́ ~  2𝑇𝑟 𝜌
∗ 𝑤, 𝑘 𝜌 𝑤 𝐸, 𝑘 𝑝                                                                                           (7) 



 
 
   

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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 

The Eq.(6) can simply be using the integral Dirac function [11].  
~

𝛿 𝐸 𝑤 𝑤 ~ ~ 𝛿 𝐸 𝑤 𝑤
~ 𝑒 1 𝑒 1 𝑑𝑤 ~ 𝑒

1 𝑒
́

1 1                                                                                                                                                           (8) 

Then Eq.(7) with Eq.(8) may be written as  

𝐼𝑚∏ , 𝑒 ∑ 𝑒 𝑒 1 𝜆 𝜆 ́ 𝑒 1 𝑒
́

1 1

𝑇𝑟 𝜌∗ 𝑤, 𝑘 𝜌 𝑤 𝐸, 𝑘 𝑝                                                                                                              (9) 
Then Eq.(3-46)simply due to solve the trace using identity of spectral functions to obtained 
[12]. 

𝐼𝑚∏ , 4𝜋 𝑒 ∑ 𝑒 𝑒 1 𝜆 𝜆 ́ 𝑒 1 𝑒
́

1

𝛿 𝑐𝑜𝑠𝜃 𝜌∗ 𝑤, 𝑘  1 𝜌∗ 𝑤, 𝑘  1                                                       (10) 

Since the quark –gluon plasma system has interested with large photon energy case have, 
𝐸 𝐸 ́ 𝐸 ≫ 𝑇 , that’s can use an approximation to given. 

𝑒 1 𝑒
́

1 𝑒 𝑒
́

𝑒
́

𝑒                                                           (11) 
Then Eq.(10) simply to:  

𝐼𝑚∏ , 4𝜋 𝑒 ∑ 𝑒 𝑒 1 𝑑𝑘 𝜆 𝜆 ́ 𝑒 𝛿 𝑐𝑜𝑠𝜃

𝜌∗ 𝑤, 𝑘  1 𝜌∗ 𝑤, 𝑘  1                                                                               (12) 

The Eq.(12) reformulated using 𝑑𝑘 𝛿 𝑐𝑜𝑠𝜃 𝑘𝑑𝑘to 

𝐼𝑚∏ , 4𝜋 𝑒 ∑ 𝑒 𝑒 1 𝑘𝑑𝑘 𝜆 𝜆 ́ 𝑒 𝜌
∗ 𝑤, 𝑘  1 

𝜌∗ 𝑤, 𝑘  1                                                                                                                     (13) 

However the recursion relation of integral [11].  

𝜌∗ 𝑤, 𝑘  1 𝜌∗ 𝑤, 𝑘  1 𝜎 𝑘 1 𝜎 𝑘 1

𝛽 𝑤, 𝑘 𝜃 𝑘 𝑤                                                                                                             (14) 

Then Eq.(13 ) with Eq .(14) lead to:  

𝐼𝑚∏ , 4𝜋 𝑒 ∑ 𝑒 𝑒 1 𝑘𝑑𝑘 𝜆 𝜆 ́ 𝑒 𝜎 𝑘 1 

𝜎 𝑘 1 𝛽 𝑤, 𝑘 𝜃 𝑘 𝑤                                                                                   (15) 

The second term in integral (3-59) refers to the correction term and is given by [12] 
The integral term in Eq. (15) lead to 

𝑘𝑑𝑘𝛽 𝑤, 𝑘 𝜃 𝑘 𝑤 𝑘𝑑𝑘 𝑚 Ω                                                                                (16) 

Where 𝛽 𝑤, 𝑘  is irrelevant for the cut term from Landau damping and is given by[12]. 

𝜎 𝑘 1 𝜎 𝑘 1 𝑘𝑑𝑘 2𝑚 𝑑𝑘                                   (17) 

 
 



 
 
   

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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 

Then Eq.(3-59) with definition integral in Eq.(3-67) and (Eq.(3-60) must to:  

𝐼𝑚∏ , 4𝜋 4𝜋𝛼 ∑ 𝑒 𝑒 1 𝜆 𝜆 ́ 𝑒 2𝑚 𝑑𝑘 𝑚 Ω .    

(18)          
The mass of quark is induced by QCD temperature by formula [13]. 

𝑚 𝛼 𝑇                                                                                                                       (19) 

And the integral in Eq.(18) is reduce to  

2 𝑑𝑘 𝐿𝑛 (20) 

It may be inserting Eq.(20) and Eq.(19) in Eq.(18) to be reduced to. 

𝐼𝑚∏ , 4𝜋 4𝜋𝛼 ∑ 𝑒 𝑒 1 𝜆 𝜆 ́ 𝑒 𝛼 𝑇 𝐿𝑛 Ω     (21) 

However, it can mentioned here that: 

Ω 1                                                                                                                                             (22) 

Inserting the Eq. (22) in Eq. (21) to reduce:  

𝐼𝑚∏ , 4𝜋 4𝜋𝛼 ∑ 𝑒 𝑒 1 𝜆 𝜆 ́ 𝑒 𝛼 𝑇 𝐿𝑛 1      (23) 

The current rate of photons can be calculation by substituting Eq.(23) in Eq.(3) to results  

𝐸 𝛼𝛼 ∑ 𝑒 𝜆 𝜆 𝜆 ́ 𝑇 𝑒 𝐿𝑛 1                                              (24) 

Where 𝛼 is the electro strength constant ,𝛼  is the strength coupling ,𝑒 is the square of 

electric charge of quarks system ,𝜆  is the fugacity of gluons , 𝜆  is the fugacity of quark ,𝜆 ́  

is the fugacity of anti-quark ,𝑇  is the square of temperature of system and 𝐸  is the photons 

energy . The strength coupling𝛼  is 

𝛼 𝑃                                                                                                             (25) 

Where 𝑁  is the flavor quantum number,𝑃  is the momentum of photons system and the 

critical temperature 𝑇  . 
 
3.Results  
     Table (1) shows strength coupling spectra of interaction 𝑐𝑡̅ → 𝛾𝑔system in plasma that is 
calculated according to quantum flavor number, critical temperature and temperature of 
system. To obtain the strength coupling, the momentum spectra of system were taken from 
experimental date Pc =1,2,3,4 and 5 GeV  [14] and elected the critical temperature Tc =150 
MeV and Tc= 190MeV.It should be mentioned at this point that quantum flavor number was 

calculated using the summation ∑ 𝑁 6 and estimation the total electric charge of quark 

using ∑ 𝑒  . The strength coupling was calculated using Eq.(25) and 

inserting𝑁 6 , P =1,2,3,4 and 5 GeV and elected Tc =150 MeV and Tc= 190MeV. 

However, the flow photonic rate 𝐸  has calculated using Eq. (24) at critical temperature 

TC=150MeV using fugacity of quark 𝜆 =0.85, anti-quark𝜆 ́ 0.9 and gluon 𝜆  0.08 with 
various temperature T=150,200,250.300 and 350MeV with MATLAB program .Results are 



 
 
   

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listed in table (2) with Tc =150 MeV and table (3) with Tc =190 MeV. Figures (1) and (2)  
show the flow of photons rate plotted verse photons energy.  

 
Table 1. Strength coupling quantum color at Tc =150 MeV and Tc= 190MeV. 

Critical 
Temperature  

The 𝒄�̅� → 𝜸𝒈system 

𝑷𝒄=1 GeV 𝑷𝒄=2GeV 𝑷𝒄=3 GeV 𝑷𝒄=4 GeV 𝑷𝒄=5 GeV 

150 MeV 0.7642 0.5597 0.4840 0.4416 0.4135 
190MeV 0.8730 0.6159 0.5254 0.4758 0.4433 

 
Table 2. Result of flow photonic rate𝐸  in𝑐𝑡̅ → 𝛾𝑔 at TC=150MeV with 𝜆 =0.85 𝜆 ́ 0.9 ,𝜆  0.08 . 

𝐸 Gev 
𝐸

𝑑𝑅
𝑑 𝑞

𝐺𝑒𝑉 𝑓𝑚  

T=150 MeV T=200 MeV T=250MeV T=300MeV T=350MeV 
𝜶𝑺𝒕=0.7642 𝜶𝑺𝒕=0.5597 𝜶𝑺𝒕=0.4840 𝜶𝑺𝒕=0.4416 𝜶𝑺𝒕=0.4135 

1.0 8.6587E-10 6.6690E-09 2.5694E-08 6.7689E-08 1.4174E-07 
1.5 3.5605E-11 6.2224E-10 3.9304E-09 1.4403E-08 3.8180E-08 
2.0 1.3896E-12 5.5434E-11 5.7542E-10 2.9372E-09 9.8661E-09 
2.5 5.2875E-14 4.8277E-12 8.2441E-11 5.8652E-10 2.4976E-09 
3.0 1.9825E-15 4.1489E-13 1.1662E-11 1.1568E-10 6.2461E-10 
3.5 7.3629E-17 3.5347E-14 1.6361E-12 2.2632E-11 1.5497E-10 
4.0 2.7164E-18 2.9933E-15 2.2819E-13 4.4028E-12 3.8236E-11 

 
 

Table 3. Result of photonic rate 𝐸  in 𝑐𝑡̅ → 𝛾𝑔atTC=190MeV with 𝜆 =0.85 𝜆 ́ 0.9 ,𝜆  0.08 

𝐸 Gev 
𝐸

𝑑𝑅
𝑑 𝑞

𝐺𝑒𝑉 𝑓𝑚  

T=150 MeV T=200 MeV T=250MeV T=300MeV T=350MeV 
𝜶𝑺𝒕=0.8730 𝜶𝑺𝒕=0.6159 𝜶𝑺𝒕=0.5254 𝜶𝑺𝒕=0.4758 𝜶𝑺𝒕=0.4433 

1.0 9.3956E-10 7.1020E-09 2.7156E-08 7.1233E-08 1.4872E-07 
1.5 3.8906E-11 6.6529E-10 4.1670E-09 1.5198E-08 4.0157E-08 
2.0 1.5243E-12 5.9405E-11 6.1116E-10 3.1041E-09 1.0391E-08 
2.5 5.8152E-14 5.1815E-12 8.7669E-11 6.2051E-10 2.6330E-09 
3.0 2.1845E-15 4.4580E-13 1.2413E-11 1.2248E-10 6.5896E-10 
3.5 8.1247E-17 3.8015E-14 1.7426E-12 2.3977E-11 1.6358E-10 
4.0 3.0009E-18 3.2215E-15 2.4319E-13 4.6667E-12 4.0379E-11 

 
 

4. Discussion  
     We discuss behavior of photons flow rateproduction in quark anti quark interaction 
processes from a non-equilibrated QG plasma at finite potential with effect of coupling 
strength constant. The results indicate  important information concerning photon produced as 
a signature of QG plasma. The photons flow rateof QG interaction with quantum quark flavor 
number 𝑁 10 are discussed as follows. Figure (1) indicate flow of photon production rate 
at temperature T = 0.150, 0.200, 0.250, 0.300 and 0.350 GeV cross annihilation process for 



 
 
   

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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 

quarks quantum flavor number 𝑁 10. It is found the flow photons rate increased function 

of photons energy at critical temperature𝑇 150 MeV. It has been taken the T = 0.350 GeV 
to find large effect of flow photons rate contribution due to increase the photons energy (1, 
1.5, 2, 2.5, 3, 3.5 and 4 GeV). The strength coupling value also effect on contribute of the 
flow production rate and the rate is increased due to decreased the strength coupling and it has 
a larger flow rate produced at very higher temperature T=0.350 GeV and low strength 
coupling. 
 

 
Figure 1.Photon flow rate production throughct̅ → γgannihilation process due to Eγ at 𝑇 = 150 MeV, 𝑁 10, 

for with 𝜆 =0.85 𝜆 ́ 0.9 ,𝜆  0.08 

 
Figure 2. Photon flow rate production through ct̅ → γg annihilation process due to Eγ at T = 190 MeV, N 10, 

with fugacity of quark λ =0.85 anti-quark λ ́ 0.9 and gluon λ  0.08 

Similarly, in figure (2) it can be shown that the flow rate of photon emission is at same 
temperatures through the interaction of quark gluon at another critical temperature𝑇
190 MeV. The flow rate of photons yield is less increased with photons energy compare with 

1 1.5 2 2.5 3 3.5 4
10

-18

10
-16

10
-14

10
-12

10
-10

10
-8

10
-6

E  (GeV)

lo
g
( 

) 
  
1
/G

e
V
2
 f
m

4
 

 

 

T =150 Mev

T =200 Mev

T =250Mev
T =300 Mev

T =350 Mev

1 1.5 2 2.5 3 3.5 4
10

-18

10
-16

10
-14

10
-12

10
-10

10
-8

10
-6

E  (GeV)

lo
g
( 
) 
  
1
/G

e
V
2
 f
m

4
 

 

 

T =150 Mev

T =200 Mev

T =250Mev
T =300 Mev

T =350 Mev



 
 
   

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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 

figure (1) in same interaction process. In comparison with figure (1), the flow photons rate 
shows better outcome. This implies that critical temperature has more effect beside the 
strength coupling .In figure (2) ,we can show the less increases in contributions to the flow 
rate of photon at the same order of quantum flavor number as results of different in the 
strength coupling of QG plasma process. The flow rate increases with the effect of decreases 
strength coupling but decreases as temperature decreases. It is found that low temperature 
T=150 MeV produces less flow rate of photons in comparison to the large temperature T=350 
MeV during the QG plasma. This shows that the flow rate produced by QG plasma process is 
low in high strength coupling in comparison to the low strength coupling in same processes. 
The results in both tables (1) and (2) and figures (1) and (2) do not show any huge of flow 
rate of photons in QG plasma process due to contributed by fugacities of quarks and gluon . In 
𝑐𝑡̅ → 𝛾𝑔system, the flow rate contribution from Tc=150MeV is less than flow rate 
contribution from Tc=190MeV over QG plasma process. Overall results are less effective in 
the case of critical temperature whith compared with photons energy and strength coupling 
.Finally, in both figures (1) and (2) , we consider the QG plasma interaction has large effect  
with photons energy and strength coupling and less effect  with critical temperature . It is 
found that the flow rate of photons due to Tc=190 MeV is large in comparison to the Tc=150 
MeV. Also, it is found  a large flow photons rate due to same fugacity. 

 
5.Conclusion 
     In conclusion the flow rate of photons yield is evaluated by integrated the flow rate over 
plasma phase. The calculation of flow rate of photons production as a function of photons 
energy corporating with strength coupling and critical temperature .So, the flow rate of 
photons emission from a QG plasma system play an important role in the investigation of 
photon emission from higher energy collision .  
It is found that large increase of flow photons production rate with decreased strength 
coupling and with increasing the temperature T and critical temperature at QG plasma system. 
Therefore, the flow photon rate is  important for giving the information to create  quark-gluon 
plasma media and supply more knowledge for the study and understanding of QGP 
interaction. 
 
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