IHJPAS. 36(3)2023 109 This work is licensed under a Creative Commons Attribution 4.0 International License Theoretical Analysis Of The Photon Production Rate in the Quark-Gluon Interaction According To The Quantum Cromodynamic QCD Theory sabamustafa@uomustansiriyah.edu.iq :uthorA Corresponding* Abstract In this work, we have used the QCD dynamic scenario of the quark gluon interaction to investigate and study photon emission theoretically based on quantum theory. The QCD theory is implemented by deriving the photon emission rate equation of the state of ideal QGP at a chemical potential. The photon rate of the quark-gluon interaction has to be calculated for the anti up-gluon interaction in the �̅�g →�̅� γ system at the temperature of system (180 ≤ 𝑇 ≤ 360) 𝑀𝑒𝑉 with critical temperature (𝑇𝑐 =132.38, 158.86, 178.72 and 198.57) MeV and photon energy (1 ≤ 𝐸 ≤ 10) GeV. We investigated a significant effect of critical temperature, strength coupling, and photon energy on the photon rate contribution. Here, the increased photon emission rate and decreased strength coupling of the quark-gluon reaction due to the decrease in temperature of the system from 360 MeV to 180 MeV are predicted. Photon energy in the range (1 to 10) GeV and the rate spectrum of four varieties of critical temperatures are presented. The interesting point in our results is the minimum value of photon rate, especially in the photon energy E=10 GeV of 𝑛𝑓 = 3 flavor which reflects the poor coupling between quark and gluon in the the �̅�g →�̅� γ system which was already expected. The features of QCD results are achieved in the case of 𝑛𝑓 = 3 flavors for the photon energy E=1 to 10 GeV, the strength coupling and the doi.org/10.30526/36.3.3084 Article history: Received 23 October 2022, Accepted 18 January 2023, Published in July 2023. Saba Mustafa Hussein* Department of Physics, College of Education for Pure Science Ibn-AL-Haitham, University of Baghdad, Baghdad, Iraq. qsabamustafa@uomustansiriyah.edu.i Ibn Al-Haitham Journal for Pure and Applied Sciences Journal homepage: jih.uobaghdad.edu.iq Hadi J. M. AL-Agealy Department of Physics, College of Education for Pure Science Ibn-AL-Haitham, University of Baghdad, Baghdad, Iraq. hadi.j.m@ihcoedu.uobaghdad.edu.iq Al-Rubaiee A. A. Department of Physics, College of Science, Mustansiriyah University Dr.rubaiee@uomustansiriyah.edu.iq https://creativecommons.org/licenses/by/4.0/ mailto:sabamustafa@uomustansiriyah.edu.iq mailto:sabamustafa@uomustansiriyah.edu.iq mailto:sabamustafa@uomustansiriyah.edu.iq mailto:hadi.j.m@ihcoedu.uobaghdad.edu.iq mailto:hadi.j.m@ihcoedu.uobaghdad.edu.iq mailto:Dr.rubaiee@uomustansiriyah.edu.iq mailto:Dr.rubaiee@uomustansiriyah.edu.iq IHJPAS. 36 (3) 2023 110 photo meason rate are calculated theoretically. We can notice that the asymptotic behavior, which was characterized by a hadronic phase limit, will be satisfied. Keywords: Photon Production Rate, Quark-Gluon Interaction, QCD Theory. 1.Introduction Photon production is an important tool for probing the strong interaction matter in the Large Hadronic Collision (LHC) at CERN and relativistic heavy-ion collisions (RHIC) at BNL [1]. All the photons are divided into decay photons, which come from hadron decays, and direct photons. Furthermore, the direct photons are divided into prompt photons and thermal photons [2]. Mostly the quark-gluon has been created at relativistic heavy-ion collision experiments at the Relativistic Heavy Ion Collider (BNL) and the LHC at CERN [3]. High temperatures are not an extreme feature of the quark-gluon interaction that's produced in heavy ion collisions [4]. The strong interaction of quark-gluon undergoes a transition from one phase to another at an approximated temperature of 150 MeV [5]. The medium below the transition temperature is characterized by hadrons as having primary degrees of freedom. On the other hand, the medium above the transition temperature is characterized by quarks and gluons, the element degrees of freedom of QCD [6]. The standard model is an important theory that establishes the study of the dynamics and characteristics of quarks in nuclear matter [7]. The QCD is known to undergo the transition temperature at which partial deconfinement is restored. The nature of the QCD transition is discussed because of its relevance to heavy ion collisions [8]. The standard model of elementary particles in physics is the mathematical framework that describes the interactions of elementary particles: electromagnetism, weak interactions, and strong interactions [9]. Hadi, J.M., Al-agealy et al. (2016) evaluated the rate of the photon for quark-gluon collisions produced at Compton scattering using quantum consideration with different photon energy spectra [10]. In 2018, Hadi J. M. Al-Agealy et al. discussed the behavior of photons emitted from quark-gluon systems with different fugacity coefficients at high energy collisions using color quantum theory [11]. In 2020, Ahmed M. Ashwiekh et al. studied the flow rate of hard photon emission from quark-antiquark interaction at high temperatures using the lowest-order approximation of QCD theory. Results show an increase in flow rate with an increase in temperature of the media, which indicates a logarithmically divergent thermal effect on the photon product [12]. Elaf Mohammed et al. in 2022 calculated and analyzed the photon rate produced from the interaction of the quark with the anti- quark during the annihilation process depending on the phenomenology of QCD [13]. This paper aims to analyze and study the photon production rate from the interaction of quark and gluon via QCD theory simulations and investigate the effect of critical temperature, strength coupling, and photon distribution energy on its rate. 2.Theory The hard photon rate produced by the quark-gluon interaction with momentum P and energy E is given via QCD theory [14]. R𝑞𝑔 𝐻 (𝐸, 𝑃) = − 1 (2𝜋)3 f𝐵 (E)Im ∏ 𝑞𝑔 𝐻 (𝐸, 𝑃) (1) Where f𝐵 (E) is the Bose-Einstein distribution function and the imaginary part Im∏𝑞𝑔 𝐻 (𝐸, 𝑃) of the retarded self-energy polarization of photons at finite temperature is given by [15]. IHJPAS. 36 (3) 2023 111 Im∏𝑞𝑔 𝐻 (𝐸, 𝑃) = (−1) 𝑁𝑐𝐶𝑎 𝜋4 𝑔𝐸 2𝑔𝐻 2 𝑇 𝐸2 |IT,𝐿 | ∫ (𝑓𝑞 (𝑃) − 𝑓𝑞 (𝐸 − 𝑃)) (𝑃 2 + (𝐸 − 𝑃)2) ∞ 0 𝑑𝑃 (2) Where 𝑁𝑐 is color number, 𝐶𝑎 is the Casimir operator, 𝑔𝐸 is the quantum electrodynamics coupling,, 𝑔𝐻 is the quantum chromodynamic coupling, 𝑓𝑞 (𝑃) is the Fermi distribution function of a quark, |IT,𝐿 | is the self-integral of the system equal IT,𝐿 = IT − IL where IT and IL are dimensionless constants [16]. The Eq. (3-6) together with Eq.(3-3 ) and for all electric charges ∑ 𝑒𝑞 2 for quarks leads to Im ∏ 𝑞𝑔 𝐻 (𝐸, 𝑃) = (−1) 𝑁𝑐𝐶𝑎 𝜋4 𝑔𝐸 2𝑔𝐻 2 𝑇 𝐸2 |IT − IL| ∑ 𝑒𝑞 2 ∫ (𝑓𝑞 (𝑃) − 𝑓𝑞 (𝐸 − 𝑃)) (𝑃 2 + (𝐸 − ∞ 0 𝑃)2) 𝑑𝑃 (3) The Juttner distribution function 𝑓𝑞 (𝑃) and 𝑓𝑞 (𝑝 − 𝐸) as function of the fugacity coefficient 𝜆𝑞 for quarks and writs [17] 𝑓𝑞 (𝑃) = 𝜆𝑞 𝑒 (𝑝+𝜇𝑞) 𝑇 +𝜆𝑞 (4) 𝑓𝑞 (𝑝 − 𝐸) = 𝜆𝑞 𝑒 (𝐸−𝑝−𝜇𝑞) 𝑇 +𝜆𝑞 (5) where 𝜇𝑞 is the chemical potential coefficient. Obviously, the chemical potential with fugacity of quarks and gluons isλq,g = e μq,g T [18]. Substituting both Eq.(4) and Eq.(5) in Eq.(3) and expanding (𝑃2 + (𝐸 − 𝑃)2) = 𝑃2 + 𝐸2 − 2𝐸𝑃 + 𝑃2 = 2𝑃2 + 𝐸2 − 2𝐸𝑃 to results . Im∏𝑞𝑔 𝐻 (𝐸, 𝑃) = (−1) 𝑁𝑐𝐶𝑎 𝜋4 𝑔𝐸 2𝑔𝐻 2 𝑇 𝐸2 |IT − IL| ∑ 𝑒𝑞 2 [∫ (2𝑃2+𝐸2−2𝐸𝑃) ( 𝑒 (𝑝+𝜇𝑞) 𝑇 𝜆𝑞 +1) 𝑑𝑃 ∞ 0 − ∫ (2𝑃2+𝐸2−2𝐸𝑃) ( 𝑒 (𝐸−𝑝−𝜇𝑞) 𝑇 𝜆𝑞 +1) 𝑑𝑃 ∞ 0 ] (6) We assume that 𝐴1 = ∫ (2𝑃2+𝐸2−2𝐸𝑃) ( 𝑒 (𝑝+𝜇𝑞) 𝑇 𝜆𝑞 +1) 𝑑𝑃 ∞ 0 = ∫ [ 𝑒 −(𝑝+𝜇𝑞) 𝑇 𝜆𝑞 −1 − 𝑒 −2(𝑝+𝜇𝑞) 𝑇 𝜆𝑞 −2 + 𝑒 −3(𝑝+𝜇𝑞) 𝑇 𝜆𝑞 −3 − 𝑒 −4(𝑝+𝜇𝑞) 𝑇 𝜆𝑞 −4 − ∞ 0 ⋯ . . 𝑒 −𝑛(𝑝+𝜇𝑞) 𝑇 𝜆𝑞 −𝑛 ](2𝑃 2 − 2𝐸𝑃 + 𝐸2)𝑑𝑃) (7) 𝐴2 = ∫ (2𝑃2+𝐸2−2𝐸𝑃) ( 𝑒 (𝐸−𝑝−𝜇𝑞) 𝑇 𝜆𝑞 +1) 𝑑𝑃 ∞ 0 = (8 ) To solve both integrals Eq.(7) and Eq.(8), we can simply write both integrals in six terms; 𝐽1 = ∫ [ 𝑒 −(𝑝+𝜇𝑞) 𝑇 𝜆𝑞 −1 − 𝑒 −2(𝑝+𝜇𝑞) 𝑇 𝜆𝑞 −2 + 𝑒 −3(𝑝+𝜇𝑞) 𝑇 𝜆𝑞 −3 − 𝑒 −4(𝑝+𝜇𝑞) 𝑇 𝜆𝑞 −4 + 𝑒 −5(𝑝+𝜇𝑞) 𝑇 𝜆𝑞 −5 − ⋯ . . 𝑒 −𝑛(𝑝+𝜇𝑞) 𝑇 𝜆𝑞 −𝑛 ] (2𝑃 2)𝑑𝑃 ∞ 0 (9 ) The second integral is 𝐽2 = ∫ [ 𝑒 −(𝑝+𝜇𝑞) 𝑇 𝜆𝑞 −1 − 𝑒 −2(𝑝+𝜇𝑞) 𝑇 𝜆𝑞 −2 + 𝑒 −3(𝑝+𝜇𝑞) 𝑇 𝜆𝑞 −3 − 𝑒 −4(𝑝+𝜇𝑞) 𝑇 𝜆𝑞 −4 + 𝑒 −5(𝑝+𝜇𝑞) 𝑇 𝜆𝑞 −5 − ⋯ . . 𝑒 −𝑛(𝑝+𝜇𝑞) 𝑇 𝜆𝑞 −𝑛 ](2𝑃𝐸)𝑑𝑃 ∞ 0 (10) The third integral is 𝐽3 = ∫ [ 𝑒 −(𝑝+𝜇𝑞) 𝑇 𝜆𝑞 −1 − 𝑒 −2(𝑝+𝜇𝑞) 𝑇 𝜆𝑞 −2 + 𝑒 −3(𝑝+𝜇𝑞) 𝑇 𝜆𝑞 −3 − 𝑒 −4(𝑝+𝜇𝑞) 𝑇 𝜆𝑞 −4 + 𝑒 −5(𝑝+𝜇𝑞) 𝑇 𝜆𝑞 −5 − ⋯ . . 𝑒 −𝑛(𝑝+𝜇𝑞) 𝑇 𝜆𝑞 −𝑛 ](𝐸 2)𝑑𝑃 ∞ 0 (11 ) The fourth term is IHJPAS. 36 (3) 2023 112 𝐽4 = ∫ [ 𝑒 −(𝐸−𝑝−𝜇𝑞) 𝑇 𝜆𝑞 −1 − 𝑒 −2(𝐸−𝑝−𝜇𝑞) 𝑇 𝜆𝑞 −2 + 𝑒 𝑒 −3(𝐸−𝑝−𝜇𝑞) 𝑇 𝜆𝑞 −3 − 𝑒 −4(𝐸−𝑝−𝜇𝑞) 𝑇 𝜆𝑞 −4 + 𝑒 −5(𝐸−𝑝−𝜇𝑞) 𝑇 𝜆𝑞 −5 − ∞ 0 ⋯ … . . 𝑒 −𝑛(𝐸−𝑝−𝜇𝑞) 𝑇 𝜆𝑞 −𝑛 ](2𝑃 2)𝑑𝑃 (12 ) The fifth integral term is 𝐽5 = ∫ [ 𝑒 −(𝐸−𝑝−𝜇𝑞) 𝑇 𝜆𝑞 −1 − 𝑒 −2(𝐸−𝑝−𝜇𝑞) 𝑇 𝜆𝑞 −2 + 𝑒 𝑒 −3(𝐸−𝑝−𝜇𝑞) 𝑇 𝜆𝑞 −3 − 𝑒 −4(𝐸−𝑝−𝜇𝑞) 𝑇 𝜆𝑞 −4 + 𝑒 −5(𝐸−𝑝−𝜇𝑞) 𝑇 𝜆𝑞 −5 − ∞ 0 ⋯ … . . 𝑒 −𝑛(𝐸−𝑝−𝜇𝑞) 𝑇 𝜆𝑞 −𝑛 ](2𝑃𝐸)𝑑𝑃 (3) The final integral term is 𝐽6 = ∫ [ 𝑒 −(𝐸−𝑝−𝜇𝑞) 𝑇 𝜆𝑞 −1 − 𝑒 −2(𝐸−𝑝−𝜇𝑞) 𝑇 𝜆𝑞 −2 + 𝑒 𝑒 −3(𝐸−𝑝−𝜇𝑞) 𝑇 𝜆𝑞 −3 − 𝑒 −4(𝐸−𝑝−𝜇𝑞) 𝑇 𝜆𝑞 −4 + 𝑒 −5(𝐸−𝑝−𝜇𝑞) 𝑇 𝜆𝑞 −5 − ∞ 0 ⋯ … . . 𝑒 −𝑛(𝐸−𝑝−𝜇𝑞) 𝑇 𝜆𝑞 −𝑛 ](𝐸 2)𝑑𝑃 (14) The solutions of the six term are given by . 𝐽1 = 2𝑇 3 [ 𝜆𝑞 1 𝑒 −𝜇𝑞 𝑇 13 − 𝜆𝑞 2 𝑒 −2𝜇𝑞 𝑇 23 + 𝜆𝑞 3 𝑒 −3𝜇𝑞 𝑇 33 − 𝜆𝑞 4 𝑒 −4𝜇𝑞 𝑇 43 + 𝜆𝑞 5 𝑒 −5𝜇𝑞 𝑇 53 − ⋯ 𝜆𝑞 𝑛 𝑛3 𝑒 −𝑛𝜇𝑞 𝑇 ] Γ ( 3) (15) The second term in term 𝐽2 is. 𝐽2 = 2E𝑇 2Γ( 2) [ 𝜆𝑞 1 𝑒 −𝜇𝑞 𝑇 12 − 𝜆𝑞 2 𝑒 −2𝜇𝑞 𝑇 22 + 𝜆𝑞 3 𝑒 −3𝜇𝑞 𝑇 32 − 𝜆𝑞 4𝑒 −4𝜇𝑞 𝑇 42 + 𝜆𝑞 5 𝑒 −5𝜇𝑞 𝑇 52 − ⋯ 𝜆𝑞 𝑛 𝑒 −𝑛𝜇𝑞 𝑇 𝑛2 ] (16 ) The third integral term in 𝐽3 is 𝐽3 = 𝐸 2𝑇[ 𝜆𝑞 1 𝑒 −𝜇𝑞 𝑇 1 − 𝜆𝑞 2 𝑒 −2𝜇𝑞 𝑇 2 + 𝜆𝑞 3 𝑒 −3𝜇𝑞 𝑇 3 − 𝜆𝑞 4𝑒 −4𝜇𝑞 𝑇 4 + 𝜆𝑞 5 𝑒 −5𝜇𝑞 𝑇 5 − ⋯ 𝜆𝑞 𝑛𝑒 −𝑛𝜇𝑞 𝑇 𝑛 )𝛤( 1 ) ( 17) On the other hand, the fourth term is 𝐽4 = = 2𝑇 3Γ(3) [ 𝜆𝑞 1 𝑒 −(𝐸−𝜇𝑞) 𝑇 13 − 𝜆𝑞 2 𝑒 −2(𝐸−𝜇𝑞) 𝑇 23 + 𝜆𝑞 3 𝑒 −3(𝐸−𝜇𝑞) 𝑇 33 − 𝜆𝑞 4 𝑒 −4(𝐸−𝜇𝑞) 𝑇 43 + 𝜆𝑞 5 𝑒 −5(𝐸−𝜇𝑞) 𝑇 53 + ⋯ 𝜆𝑞 𝑛 𝑛3 𝑒 −𝑛(𝐸−𝜇𝑞) 𝑇 ] (18) The fifth integral term is 𝐽5 = 2E𝑇 2Γ(2) [ 𝜆𝑞 1 𝑒 −(𝐸−𝜇𝑞) 𝑇 12 − 𝜆𝑞 2 𝑒 −2(𝐸−𝜇𝑞) 𝑇 22 + 𝜆𝑞 3 𝑒 −3(𝐸−𝜇𝑞) 𝑇 32 + 𝜆𝑞 4 𝑒 −4(𝐸−𝜇𝑞) 𝑇 42 − 𝜆𝑞 5 𝑒 −5(𝐸−𝜇𝑞) 𝑇 52 + ⋯ 𝜆𝑞 𝑛 𝑛2 𝑒 −𝑛(𝐸−𝜇𝑞) 𝑇 ] (19) The sixth integral term gives 𝐽6 = 𝐸 2𝑇Γ(1) ( 𝜆𝑞 1 𝑒 −(𝐸−𝜇𝑞) 𝑇 1 − 𝜆𝑞 2 𝑒 −2(𝐸−𝜇𝑞) 𝑇 2 + 𝜆𝑞 3 𝑒 −3(𝐸−𝜇𝑞) 𝑇 3 − 𝜆𝑞 4 𝑒 −4(𝐸−𝜇𝑞) 𝑇 4 + 𝜆𝑞 5 𝑒 −5(𝐸−𝜇𝑞) 𝑇 5 … … … + 𝜆𝑞 𝑛 𝑛 𝑒 −𝑛(𝐸−𝜇𝑞) 𝑇 ) .( 20) Inserting the results of 𝐽1 , 𝐽2 ,and 𝐽3 in Eq.(7) gives 𝐴1 . IHJPAS. 36 (3) 2023 113 𝐴1 = ∫ 𝜆𝑞(2𝑃 2+𝐸2−2𝐸𝑃) 𝑒 (𝑝+𝜇𝑞) 𝑇 +𝜆𝑞 𝑑𝑃 ∞ 0 = 2𝑇3 [ 𝜆𝑞 1 𝑒 −𝜇𝑞 𝑇 13 − 𝜆𝑞 2 𝑒 −2𝜇𝑞 𝑇 23 + 𝜆𝑞 3 𝑒 −3𝜇𝑞 𝑇 33 − 𝜆𝑞 4 𝑒 −4𝜇𝑞 𝑇 43 + 𝜆𝑞 5 𝑒 −5𝜇𝑞 𝑇 53 − ⋯ 𝜆𝑞 𝑛 𝑛3 𝑒 −𝑛𝜇𝑞 𝑇 ] Γ( 3) + 2E𝑇2Γ( 2) [[ 𝜆𝑞 1 𝑒 −𝜇𝑞 𝑇 12 − 𝜆𝑞 2 𝑒 −2𝜇𝑞 𝑇 22 + 𝜆𝑞 3 𝑒 −3𝜇𝑞 𝑇 32 − 𝜆𝑞 4 𝑒 −4𝜇𝑞 𝑇 42 + 𝜆𝑞 5 𝑒 −5𝜇𝑞 𝑇 52 − ⋯ 𝜆𝑞 𝑛 𝑛2 𝑒 −𝑛𝜇𝑞 𝑇 ]] + 𝐸2𝑇 [ [ 𝜆𝑞 1 𝑒 −𝜇𝑞 𝑇 1 − 𝜆𝑞 2 𝑒 −2𝜇𝑞 𝑇 2 + 𝜆𝑞 3 𝑒 −3𝜇𝑞 𝑇 3 − 𝜆𝑞 4 𝑒 −4𝜇𝑞 𝑇 4 + 𝜆𝑞 5 𝑒 −5𝜇𝑞 𝑇 5 − ⋯ 𝜆𝑞 𝑛 𝑛 𝑒 −𝑛𝜇𝑞 𝑇 ]) 𝛤( 1 ) (21) On the other hand, we insert the 𝐽4 , 𝐽5, and 𝐽6 in Eq.(8) to give the integral term 𝐴2 . 𝐴2 = ∫ 𝜆𝑞 (2𝑃 2 + 𝐸2 − 2𝐸𝑃) 𝑒 (𝐸−𝑝−𝜇𝑞) 𝑇 + 𝜆𝑞 𝑑𝑃 ∞ 0 = 2𝑇3Γ(3) [ 𝜆𝑞 1 𝑒 −(𝐸−𝜇𝑞) 𝑇 13 − 𝜆𝑞 2 𝑒 −2(𝐸−𝜇𝑞) 𝑇 23 + 𝜆𝑞 3 𝑒 −3(𝐸−𝜇𝑞) 𝑇 33 − 𝜆𝑞 4 𝑒 −4(𝐸−𝜇𝑞) 𝑇 43 + 𝜆𝑞 5 𝑒 −5(𝐸−𝜇𝑞) 𝑇 53 + ⋯ 𝜆𝑞 𝑛 𝑛3 𝑒 −𝑛(𝐸−𝜇𝑞) 𝑇 ] +2E𝑇2Γ(2) [ 𝜆𝑞 1 𝑒 −(𝐸−𝜇𝑞) 𝑇 12 − 𝜆𝑞 2 𝑒 −2(𝐸−𝜇𝑞) 𝑇 22 + 𝜆𝑞 3 𝑒 −3(𝐸−𝜇𝑞) 𝑇 32 − 𝜆𝑞 4 𝑒 −4(𝐸−𝜇𝑞) 𝑇 42 + 𝜆𝑞 5 𝑒 −5(𝐸−𝜇𝑞) 𝑇 52 + ⋯ 𝜆𝑞 𝑛 𝑛2 𝑒 −𝑛(𝐸−𝜇𝑞) 𝑇 ] + 𝐸2𝑇Γ(1)[ 𝜆𝑞 1 𝑒 −(𝐸−𝜇𝑞) 𝑇 1 − 𝜆𝑞 2 𝑒 −2(𝐸−𝜇𝑞) 𝑇 2 + 𝜆𝑞 3 𝑒 −3(𝐸−𝜇𝑞) 𝑇 3 − 𝜆𝑞 4 𝑒 −4(𝐸−𝜇𝑞) 𝑇 4 + 𝜆𝑞 5 𝑒 −5(𝐸−𝜇𝑞) 𝑇 5 + ⋯ 𝜆𝑞 𝑛 𝑛 𝑒 −𝑛(𝐸−𝜇𝑞) 𝑇 ] (22) We assume 𝐽 = 𝐴1 + 𝐴2, then with using Eq.(21) and Eq.(22), we get 𝐽 = 2𝑇3 [ 𝜆𝑞 1 𝑒 −𝜇𝑞 𝑇 13 − 𝜆𝑞 2 𝑒 −2𝜇𝑞 𝑇 23 + 𝜆𝑞 3 𝑒 −3𝜇𝑞 𝑇 33 − 𝜆𝑞 4 𝑒 −4𝜇𝑞 𝑇 43 + 𝜆𝑞 5 𝑒 −5𝜇𝑞 𝑇 53 − ⋯ 𝜆𝑞 𝑛 𝑛3 𝑒 −𝑛𝜇𝑞 𝑇 ] Γ ( 3) + 2E𝑇2Γ( 2) [ 𝜆𝑞 1 𝑒 −𝜇𝑞 𝑇 12 − 𝜆𝑞 2 𝑒 −2𝜇𝑞 𝑇 22 + 𝜆𝑞 3 𝑒 −3𝜇𝑞 𝑇 32 − 𝜆𝑞 4 𝑒 −4𝜇𝑞 𝑇 42 + 𝜆𝑞 5 𝑒 −5𝜇𝑞 𝑇 52 − ⋯ 𝜆𝑞 𝑛 𝑛2 𝑒 −𝑛𝜇𝑞 𝑇 ] + 𝐸2𝑇 [ 𝜆𝑞 1 𝑒 −𝜇𝑞 𝑇 1 − 𝜆𝑞 2 𝑒 −2𝜇𝑞 𝑇 2 + 𝜆𝑞 3 𝑒 −3𝜇𝑞 𝑇 3 − 𝜆𝑞 4 𝑒 −4𝜇𝑞 𝑇 4 + 𝜆𝑞 5 𝑒 −5𝜇𝑞 𝑇 5 − ⋯ 𝜆𝑞 𝑛 𝑛 𝑒 −𝑛𝜇𝑞 𝑇 ) 𝛤( 1 ) + 2𝑇3Γ(3) [ 𝜆𝑞 1 𝑒 −(𝐸−𝜇𝑞) 𝑇 13 − 𝜆𝑞 2 𝑒 −2(𝐸−𝜇𝑞) 𝑇 23 + 𝜆𝑞 3 𝑒 −3(𝐸−𝜇𝑞) 𝑇 33 − 𝜆𝑞 4 𝑒 −4(𝐸−𝜇𝑞) 𝑇 43 + 𝜆𝑞 5 𝑒 −5(𝐸−𝜇𝑞) 𝑇 53 + ⋯ 𝜆𝑞 𝑛 𝑛3 𝑒 −𝑛(𝐸−𝜇𝑞) 𝑇 ] +2E𝑇2Γ(2) [ 𝜆𝑞 1 𝑒 −(𝐸−𝜇𝑞) 𝑇 12 − 𝜆𝑞 2 𝑒 −2(𝐸−𝜇𝑞) 𝑇 22 + 𝜆𝑞 3 𝑒 −3(𝐸−𝜇𝑞) 𝑇 32 − 𝜆𝑞 4 𝑒 −4(𝐸−𝜇𝑞) 𝑇 42 + 𝜆𝑞 5 𝑒 −5(𝐸−𝜇𝑞) 𝑇 52 + ⋯ 𝜆𝑞 𝑛 𝑛2 𝑒 −𝑛(𝐸−𝜇𝑞) 𝑇 ] + 𝐸2𝑇Γ(1)[ 𝜆𝑞 1 𝑒 −(𝐸−𝜇𝑞) 𝑇 1 − 𝜆𝑞 2 𝑒 −2(𝐸−𝜇𝑞) 𝑇 2 + 𝜆𝑞 3 𝑒 −3(𝐸−𝜇𝑞) 𝑇 3 − 𝜆𝑞 4 𝑒 −4(𝐸−𝜇𝑞) 𝑇 4 + 𝜆𝑞 5 𝑒 −5(𝐸−𝜇𝑞) 𝑇 5 + ⋯ 𝜆𝑞 𝑛 𝑛 𝑒 −𝑛(𝐸−𝜇𝑞) 𝑇 ] (23) IHJPAS. 36 (3) 2023 114 Then, we have 𝐽 = 2𝑇3 [ 𝜆𝑞 1 13 (𝑒 −𝜇𝑞 𝑇 + 𝑒 −(𝐸−𝜇𝑞) 𝑇 ) − 𝜆𝑞 2 23 (𝑒 −2𝜇𝑞 𝑇 + 𝑒 −2(𝐸−𝜇𝑞) 𝑇 ) + 𝜆𝑞 3 33 (𝑒 −3𝜇𝑞 𝑇 + 𝑒 −3(𝐸−𝜇𝑞) 𝑇 ) − 𝜆𝑞 4 43 (𝑒 −4𝜇𝑞 𝑇 + 𝑒 −4(𝐸−𝜇𝑞) 𝑇 ) + 𝜆𝑞 5 53 (𝑒 −5𝜇𝑞 𝑇 + 𝑒 −5(𝐸−𝜇𝑞) 𝑇 ) − ⋯ 𝜆𝑞 𝑛 𝑛3 (𝑒 −𝑛𝜇𝑞 𝑇 + 𝑒 −𝑛(𝐸−𝜇𝑞) 𝑇 )] Γ ( 3) + 2E𝑇2Γ( 2) [ 𝜆𝑞 1 12 (𝑒 −𝜇𝑞 𝑇 + 𝑒 −(𝐸−𝜇𝑞) 𝑇 ) − 𝜆𝑞 2 22 (𝑒 −2𝜇𝑞 𝑇 + 𝑒 −2(𝐸−𝜇𝑞) 𝑇 ) + 𝜆𝑞 3 32 (𝑒 −3𝜇𝑞 𝑇 + 𝑒 −3(𝐸−𝜇𝑞) 𝑇 ) − 𝜆𝑞 4 42 (𝑒 −4𝜇𝑞 𝑇 + 𝑒 −4(𝐸−𝜇𝑞) 𝑇 ) + 𝜆𝑞 5 52 (𝑒 −5𝜇𝑞 𝑇 + 𝑒 −5(𝐸−𝜇𝑞) 𝑇 ) − ⋯ 𝜆𝑞 𝑛 𝑛2 (𝑒 −𝑛𝜇𝑞 𝑇 + 𝑒 −𝑛(𝐸−𝜇𝑞) 𝑇 )] + 𝐸2𝑇 [ 𝜆𝑞 1 1 (𝑒 −𝜇𝑞 𝑇 + 𝑒 −(𝐸−𝜇𝑞) 𝑇 ) − 𝜆𝑞 2 2 (𝑒 −2𝜇𝑞 𝑇 + 𝑒 −2(𝐸−𝜇𝑞) 𝑇 ) + 𝜆𝑞 3 3 (𝑒 −3𝜇𝑞 𝑇 + 𝑒 −3(𝐸−𝜇𝑞) 𝑇 ) − 𝜆𝑞 4 4 (𝑒 −4𝜇𝑞 𝑇 + 𝑒 −4(𝐸−𝜇𝑞) 𝑇 ) + 𝜆𝑞 5 5 (𝑒 −5𝜇𝑞 𝑇 + 𝑒 −5(𝐸−𝜇𝑞) 𝑇 ) … 𝜆𝑞 𝑛 𝑛 (𝑒 −𝑛𝜇𝑞 𝑇 + 𝑒 −𝑛(𝐸−𝜇𝑞) 𝑇 ) ) 𝛤( 1 ) (24) Inserting Eq.(24) in Eq.(6) to give Im∏𝑞𝑔 𝐻 (𝐸, 𝑃) = (−1) 𝑁𝑐𝐶𝑎 𝜋4 𝑔𝐸 2𝑔𝐻 2 𝑇 𝐸2 |IT − IL| ∑ 𝑒𝑞 2 × 2𝑇3 [ 𝜆𝑞 1 1 3 (𝑒 −𝜇𝑞 𝑇 + 𝑒 −(𝐸−𝜇𝑞) 𝑇 ) − 𝜆𝑞 2 2 3 (𝑒 −2𝜇𝑞 𝑇 + 𝑒 −2(𝐸−𝜇𝑞) 𝑇 ) + 𝜆𝑞 3 3 3 (𝑒 −3𝜇𝑞 𝑇 + 𝑒 −3(𝐸−𝜇𝑞) 𝑇 ) − 𝜆𝑞 4 4 3 (𝑒 −4𝜇𝑞 𝑇 + 𝑒 −4(𝐸−𝜇𝑞) 𝑇 ) + 𝜆𝑞 5 5 3 (𝑒 −5𝜇𝑞 𝑇 + 𝑒 −5(𝐸−𝜇𝑞) 𝑇 ) − ⋯ 𝜆𝑞 𝑛 𝑛 3 (𝑒 −𝑛𝜇𝑞 𝑇 + 𝑒 −𝑛(𝐸−𝜇𝑞) 𝑇 )] Γ ( 3) + 2E𝑇 2 Γ( 2) [ 𝜆𝑞 1 1 2 (𝑒 −𝜇𝑞 𝑇 + 𝑒 −(𝐸−𝜇𝑞) 𝑇 ) − 𝜆𝑞 2 2 2 (𝑒 −2𝜇𝑞 𝑇 + 𝑒 −2(𝐸−𝜇𝑞) 𝑇 ) + 𝜆𝑞 3 3 2 (𝑒 −3𝜇𝑞 𝑇 + 𝑒 −3(𝐸−𝜇𝑞) 𝑇 ) − 𝜆𝑞 4 4 2 (𝑒 −4𝜇𝑞 𝑇 + 𝑒 −4(𝐸−𝜇𝑞) 𝑇 ) + 𝜆𝑞 5 5 2 (𝑒 −5𝜇𝑞 𝑇 + 𝑒 −5(𝐸−𝜇𝑞) 𝑇 ) − ⋯ 𝜆𝑞 𝑛 𝑛 2 (𝑒 −𝑛𝜇𝑞 𝑇 + 𝑒 −𝑛(𝐸−𝜇𝑞) 𝑇 )] + 𝐸2𝑇 [ 𝜆𝑞 1 1 (𝑒 −𝜇𝑞 𝑇 + 𝑒 −(𝐸−𝜇𝑞) 𝑇 ) − 𝜆𝑞 2 2 (𝑒 −2𝜇𝑞 𝑇 + 𝑒 −2(𝐸−𝜇𝑞) 𝑇 ) + 𝜆𝑞 3 3 (𝑒 −3𝜇𝑞 𝑇 + 𝑒 −3(𝐸−𝜇𝑞) 𝑇 ) − 𝜆𝑞 4 4 (𝑒 −4𝜇𝑞 𝑇 + 𝑒 −4(𝐸−𝜇𝑞) 𝑇 ) + 𝜆𝑞 5 5 (𝑒 −5𝜇𝑞 𝑇 + 𝑒 −5(𝐸−𝜇𝑞) 𝑇 ) … 𝜆𝑞 𝑛 𝑛 (𝑒 −𝑛𝜇𝑞 𝑇 + 𝑒 −𝑛(𝐸−𝜇𝑞) 𝑇 ) ) 𝛤( 1 ) (25) We have Im∏𝑞𝑔 𝐻 (𝐸, 𝑃) = (−1) 𝑁𝑐𝐶𝑎 𝜋4 𝑔𝐸 2𝑔𝐻 2 𝑇 𝐸2 |IT − IL| ∑ 𝑒𝑞 2 × [2𝑇3Γ ( 3) ( 𝜆𝑞 1 1 3 − 𝜆𝑞 2 2 3 + 𝜆𝑞 3 3 3 − 𝜆𝑞 4 4 3 + 𝜆𝑞 5 5 3 − ⋯ 𝜆𝑞 𝑛 𝑛 3 ) + 2E𝑇 2 Γ( 2) ( 𝜆𝑞 1 1 2 − 𝜆𝑞 2 2 2 + 𝜆𝑞 3 3 2 − 𝜆𝑞 4 4 2 + 𝜆𝑞 5 5 2 − ⋯ 𝜆𝑞 𝑛 𝑛 2 ) + 𝐸2𝑇𝛤( 1 )( 𝜆𝑞 1 1 − 𝜆𝑞 2 2 + 𝜆𝑞 3 3 − 𝜆𝑞 4 4 + 𝜆𝑞 5 5 − ⋯ 𝜆𝑞 𝑛 𝑛 )] [ (𝑒 −𝜇𝑞 𝑇 + 𝑒 −(𝐸−𝜇𝑞) 𝑇 ) + (𝑒 −2𝜇𝑞 𝑇 + 𝑒 −2(𝐸−𝜇𝑞) 𝑇 ) + (𝑒 −3𝜇𝑞 𝑇 + 𝑒 −3(𝐸−𝜇𝑞) 𝑇 ) + (𝑒 −4𝜇𝑞 𝑇 + 𝑒 −4(𝐸−𝜇𝑞) 𝑇 ) + (𝑒 −5𝜇𝑞 𝑇 + 𝑒 −5(𝐸−𝜇𝑞) 𝑇 ) … + (𝑒 −𝑛𝜇𝑞 𝑇 + 𝑒 −𝑛(𝐸−𝜇𝑞) 𝑇 ) ) (26) We assume that 𝛾(𝐸, 𝑇, 𝜆𝑞 , 𝜇𝑞 ) = [2𝑇 3Γ ( 3) ( 𝜆𝑞 1 13 − 𝜆𝑞 2 23 + 𝜆𝑞 3 33 − 𝜆𝑞 4 43 + 𝜆𝑞 5 53 − ⋯ 𝜆𝑞 𝑛 𝑛3 ) + 2E𝑇2Γ( 2) ( 𝜆𝑞 1 12 − 𝜆𝑞 2 22 + 𝜆𝑞 3 32 − 𝜆𝑞 4 42 + 𝜆𝑞 5 52 − ⋯ 𝜆𝑞 𝑛 𝑛2 ) + 𝐸2𝑇𝛤( 1 )( 𝜆𝑞 1 1 − 𝜆𝑞 2 2 + 𝜆𝑞 3 3 − 𝜆𝑞 4 4 + 𝜆𝑞 5 5 − ⋯ 𝜆𝑞 𝑛 𝑛 )] [ (𝑒 −𝜇𝑞 𝑇 + 𝑒 −(𝐸−𝜇𝑞) 𝑇 ) + (𝑒 −2𝜇𝑞 𝑇 + 𝑒 −2(𝐸−𝜇𝑞) 𝑇 ) + (𝑒 −3𝜇𝑞 𝑇 + 𝑒 −3(𝐸−𝜇𝑞) 𝑇 ) + (𝑒 −4𝜇𝑞 𝑇 + 𝑒 −4(𝐸−𝜇𝑞) 𝑇 ) + (𝑒 −5𝜇𝑞 𝑇 + 𝑒 −5(𝐸−𝜇𝑞) 𝑇 ) … + (𝑒 −𝑛𝜇𝑞 𝑇 + 𝑒 −𝑛(𝐸−𝜇𝑞) 𝑇 ) ) (27) The Eq.(26) together Eq.(27) give Im∏𝑞𝑔 𝐻 (𝐸, 𝑃) = (−1) 𝑁𝑐𝐶𝑎 𝜋4 𝑔𝐸 2𝑔𝐻 2 𝑇 𝐸2 |IT − IL| ∑ 𝑒𝑞 2 𝛾(𝐸, 𝑇, 𝜆𝑞 , 𝜇𝑞) (28) Inserting the Casimiro operator𝐶𝑎 = (𝑁𝑐 2−1) 2𝑁𝑐 = 4 3 relative to the color number 𝑁𝑐 = 3 [19] and the effective strength coupling parameter for QCD theory is 𝛼𝑄𝐶𝐷 (𝜇 2) = 𝑔 𝐻 2 4π [20] and quantum electrodynamics QED coupling constant is 𝛼QED= 𝑔𝐸 2 4𝜋 [21] in Eq.(28) to become Im∏𝑞𝑔 𝐻 (𝐸, 𝑃) = (−1)4 𝑁 𝜋2 16 3 𝛼QED𝛼𝑄𝐶𝐷(𝜇 2) ∑ 𝑒𝑞 2 |JT − JL| 𝑇 𝐸2 𝛾(𝐸, 𝑇, 𝜆𝑞 , 𝜇𝑞) (29) Inserting Eq.(29) in Eq.(1) to result R𝑞𝑔 𝐻 (𝐸, 𝑃) = 8𝑁 3𝜋5 𝛼QED𝛼𝑄𝐶𝐷 (𝜇 2) ∑ 𝑒𝑞 2 |J T − J L | 𝑇 𝐸2 F𝐵(E) 𝑇 𝐸2 𝛾(𝐸, 𝑇, 𝜆 𝑞 , 𝜇 𝑞 )(30) The Bosonic function distribution for gluon f𝐵 (E) = 𝜆𝑔 𝑒 𝐸 𝑇 −𝜆𝑔 [22],then Eq.(30) becomes R𝑞𝑔 𝐻 (𝐸, 𝑃) = ( 8𝑁 3𝜋5 ) 𝜆𝑔 𝑒 𝐸/𝑇−𝜆𝑔 𝛼QED𝛼𝑄𝐶𝐷 (𝜇 2) 𝑇 𝐸2 |IT − IL| ∑ 𝑒𝑞 2 𝛾(𝐸, 𝑇, 𝜆𝑞 , 𝜇𝑞 ) (31) the strong coupling constant is [23]. 𝛼𝑄𝐶𝐷 (𝜇 2) = 6𝜋 (33− 2 𝑁𝐹) 𝑙𝑛 8𝑇 𝑇𝑐 (32) IHJPAS. 36 (3) 2023 115 Where 𝑁𝐹 is the flavor number of quarks, 𝑇 is the temperature of system and 𝑇𝑐 is the critical temperature for the quark – gluon interaction; it is given by [24]. 𝑇𝑐 = ( 90𝐵 𝜋2𝑑𝑔𝑞 ) 1 4 (33) where 𝐵 is the bag coefficient and 𝑑𝑔𝑞 is the degeneracy factor for gluons and quarks. It can be given by expression [25 ]. 𝑑𝑔𝑞 = 𝑑𝑔 + 7 8 (𝑑𝑞 + 𝑑�̅� ) (34) where 𝑑𝑔 is the number of gluons degrees of freedom as a function of the gluons spin state 𝑛𝑠 and gluons color states 𝑛𝑐 and 𝑑𝑞 is the number of quark degrees of freedom as function of the number of quark colour 𝑛𝑐 , spin 𝑛𝑠 and flavour degrees of freedom 𝑛𝑓 . Inserting Eq.(34) in Eq.(33) to result 𝑇𝑐 = ( 90𝐵 𝜋2[(𝑛𝑠×𝑛𝑐)+ 7 4 (𝑛𝑐×𝑛𝑠×𝑛𝑓)] ) 1 4 (35) 3.Results An essential estimation of the critical temperature is predicted near the phase transition scale, which is called the hadronic phase. One of the most technical predictions of the calculation of the critical temperature depends on the bag coefficient B in Eq.(33) and the degeneracy factors for gluons and quarks 𝑑𝑔𝑞 in Eq.(34) relative to the spin state 𝑛𝑠, color states 𝑛𝑐 and flavour degrees 𝑛𝑓 . Inserting 𝑛𝑠 = 2 together with 𝑛𝑐 = 8 for gluons and 𝑛𝑐 = 3 , 𝑛𝑠 = 2 𝑎𝑛𝑑 𝑛𝑓 = 3 for quarks system and the Bag constant (200,240,270 and 300)MeV in Eq.(35) to give the results listed in Table (1) Table 1 . Result of critical temperature uses the Bag mode of the quark –gluon system for �̅�g →�̅� γ System. Critical temperature 𝑇𝑐 𝑀𝑒𝑉 Bag constant 𝐵 1 4⁄ 𝑀𝑒𝑉 132.38 200 158.86 240 178.72 270 198.57 300 To evaluate the strength coupling from Eq.(32), we inert the critical temperature of the �̅�g →�̅� γ system from Table (1), taking the temperature of �̅�g →�̅� γ system in range (T=180, 210, 240, 270, 300, 330 and 360) MeV and 𝑛𝑓 = 3, the results of strength coupling are shown in Table 2. IHJPAS. 36 (3) 2023 116 Table 2 . Strength coupling for �̅�g →�̅� γ system at different critical temperature with variety temperature of system .. For simplicity in calculating the rate of the photon produced from the interaction of the quark- gluon system, we estimate the total quark charge and total flavour number in the quark system. The quark charge is the summation charge of ∑ 𝑒𝑞 2 = 5 9⁄ of �̅�g →�̅� γ system where the charge of the up quark is + 2 3⁄ and the anti-down is − 1 3⁄ ,. while the net favor is 𝑛𝑓 = 3 for the interaction system. The photon rate produced from the interaction of the anti-up with the anti- down system is calculated using Eq. (31) by inserting the photon energy from experimental data in range 𝐸 = 1 𝑡𝑜 10𝐺𝑒𝑉 [26], critical temperature from Table 1, strength coupling 𝛼𝑄𝐶𝐷 (𝜇 2) from Table 2, and fugacity of 𝜆𝑞=0.02 for quark, 𝜆𝑔=0.06 for gluon [23], taking the self-integral constants IT = 4.45 and IL = −4.26 [15] and using the chemical potential 𝜇𝑞 = 500 𝑀𝑒𝑉 [27] and 𝛼QED = 1/137 and 𝑁 = 3. Results are shown in tables (3), (4), (5), (6) and figures (1), (2), (3) and (4) taking IT = 4.45 and IL = −4.26 with 𝜆𝑔=0.06 for gluon , 𝜆�̅�=0.02 for quark in �̅�g →�̅� γ system . Table 3. Rate of photon production at 𝑇𝑐 =132.38 𝑀𝑒𝑉 , IT = 4.45 and IL = −4.26 with 𝜆𝑔=0.06, 𝜆�̅�=0.02 in �̅�g →�̅� γ system . 𝐸𝛾 𝐺𝑒𝑉 R𝑞𝑔 𝐻 (𝐸, 𝑃) 1 𝐺𝑒𝑉 2𝑓𝑚4 T=180MeV T=210MeV T=240MeV T=270MeV T=300MeV T=330MeV T=360MeV 𝛼𝑄𝐶𝐷 =0.29 25 𝛼𝑄𝐶𝐷 =0.27 48 𝛼𝑄𝐶𝐷 =0.26 10 𝛼𝑄𝐶𝐷 =0.25 00 𝛼𝑄𝐶𝐷 =0.24 09 𝛼𝑄𝐶𝐷 =0.23 33 𝛼𝑄𝐶𝐷 =0.22 67 1 8.0004E-12 3.7264E-11 1.2547E-10 3.3889E-10 7.8194E-10 1.6049E-09 3.0106E-09 2 1.2629E-14 1.2604E-13 7.4724E-13 3.1194E-12 1.0159E-11 2.7551E-11 6.5013E-11 3 4.5519E-17 9.8877E-16 1.0445E-14 6.7986E-14 3.1444E-13 1.1319E-12 3.3708E-12 4 1.7034E-19 8.1351E-18 1.5491E-16 1.5921E-15 1.0596E-14 5.1299E-14 1.9529E-13 5 6.4597E-22 6.7993E-20 2.3400E-18 3.8075E-17 3.6571E-16 2.3886E-15 1.1663E-14 6 2.4658E-24 5.7269E-22 3.5661E-20 9.1982E-19 1.2766E-17 1.1263E-16 7.0626E-16 7 9.4472E-27 4.8446E-24 5.4620E-22 2.2347E-20 4.4845E-19 5.3484E-18 4.3101E-17 8 3.6278E-29 4.1093E-26 8.3919E-24 5.4486E-22 1.5816E-20 2.5509E-19 2.6430E-18 9 1.3952E-31 3.4919E-28 1.2920E-25 1.3315E-23 5.5925E-22 1.2201E-20 1.6258E-19 10 5.3715E-34 2.9708E-30 1.9919E-27 3.2592E-25 1.9811E-23 5.8476E-22 1.0023E-20 𝑇𝐶 𝛼𝑄𝐶𝐷 T=180Me V T=210Me V T=240Me V T=270Me V T=300Me V T=330Me V T=360Me V 132.386068274 0.2925 0.2748 0.2610 0.2500 0.2409 0.2333 0.2267 158.863281928 8 0.3167 0.2960 0.2801 0.2675 0.2571 0.2484 0.2409 178.721192169 9 0.3346 0.3116 0.2940 0.2801 0.2688 0.2593 0.2512 198.579102411 0.3524 0.3269 0.3077 0.2925 0.2801 0.2698 0.2610 IHJPAS. 36 (3) 2023 117 Figure1. The rate of photon produces as afunction to gamma energy produced at 𝑇𝑐 =132.38𝑀𝑒𝑉 Table 4. Rate of photon production at 𝑇𝑐 = 158.86 𝑀𝑒𝑉 , IT = 4.45 and IL = −4.26 with 𝜆𝑔=0.06, 𝜆�̅� =0.02 in system �̅�g →�̅� γ system . 𝐸𝛾 𝐺𝑒𝑉 R𝑞𝑔 𝐻 (𝐸, 𝑃) 1 𝐺𝑒𝑉 2𝑓𝑚4 T=180MeV T=210MeV T=240MeV T=270MeV T=300MeV T=330MeV T=360MeV 𝛼𝑄𝐶𝐷 =0.31 67 𝛼𝑄𝐶𝐷 =0.29 60 𝛼𝑄𝐶𝐷 =0.28 01 𝛼𝑄𝐶𝐷 =0.26 75 𝛼𝑄𝐶𝐷 =0.25 71 𝛼𝑄𝐶𝐷 =0.24 84 𝛼𝑄𝐶𝐷 =0.24 09 1 8.6621E-12 4.0145E-11 1.3465E-10 3.6256E-10 8.3445E-10 1.7090E-09 3.2001E-09 2 1.3674E-14 1.3579E-13 8.0190E-13 3.3373E-12 1.0841E-11 2.9338E-11 6.9104E-11 3 4.9284E-17 1.0652E-15 1.1209E-14 7.2736E-14 3.3555E-13 1.2053E-12 3.5829E-12 4 1.8443E-19 8.7640E-18 1.6625E-16 1.7033E-15 1.1308E-14 5.4627E-14 2.0757E-13 5 6.9940E-22 7.3250E-20 2.5112E-18 4.0735E-17 3.9026E-16 2.5436E-15 1.2397E-14 6 2.6697E-24 6.1696E-22 3.8270E-20 9.8408E-19 1.3623E-17 1.1994E-16 7.5070E-16 7 1.0229E-26 5.2191E-24 5.8616E-22 2.3909E-20 4.7856E-19 5.6954E-18 4.5814E-17 8 3.9278E-29 4.4270E-26 9.0059E-24 5.8292E-22 1.6878E-20 2.7163E-19 2.8093E-18 9 1.5106E-31 3.7618E-28 1.3865E-25 1.4245E-23 5.9680E-22 1.2993E-20 1.7281E-19 10 5.8157E-34 3.2005E-30 2.1377E-27 3.4869E-25 2.1141E-23 6.2269E-22 1.0654E-20 1 2 3 4 5 6 7 8 9 10 10 -35 10 -30 10 -25 10 -20 10 -15 10 -10 10 -5 E (GeV) lo g (R H q g ) 1 /G e V 2 f m 4 T =180 Mev T =210 Mev T =240 Mev T =270 Mev T =300 Mev T =330 Mev T =360 Mev IHJPAS. 36 (3) 2023 118 Figure2. The rate of photon produces as afunction to gamma energy produced at 𝑇𝑐 = 158.86𝑀𝑒𝑉 . Table 5. Rate of photon production at 𝑇𝑐 = 178.72 𝑀𝑒𝑉 , IT = 4.45 and IL = −4.26 with 𝜆𝑔=006, 𝜆�̅� =0.02 in �̅�g →�̅� γ system. 𝐸𝛾 𝐺𝑒𝑉 R𝑞𝑔 𝐻 (𝐸, 𝑃) 1 𝐺𝑒𝑉 2𝑓𝑚4 T=180MeV T=210MeV T=240MeV T=270MeV T=300MeV T=330MeV T=360MeV 𝛼𝑄𝐶𝐷 =0.33 46 𝛼𝑄𝐶𝐷 =0.31 16 𝛼𝑄𝐶𝐷 =0.29 40 𝛼𝑄𝐶𝐷 =0.28 01 𝛼𝑄𝐶𝐷 =0.26 88 𝛼𝑄𝐶𝐷 =0.25 93 𝛼𝑄𝐶𝐷 =0.25 12 1 9.1511E-12 4.2255E-11 1.4133E-10 3.7970E-10 8.7229E-10 1.7837E-09 3.3357E-09 2 1.4446E-14 1.4292E-13 8.4169E-13 3.4951E-12 1.1332E-11 3.0621E-11 7.2032E-11 3 5.2066E-17 1.1212E-15 1.1765E-14 7.6174E-14 3.5077E-13 1.2580E-12 3.7347E-12 4 1.9484E-19 9.2247E-18 1.7450E-16 1.7838E-15 1.1820E-14 5.7016E-14 2.1637E-13 5 7.3888E-22 7.7100E-20 2.6357E-18 4.2661E-17 4.0796E-16 2.6549E-15 1.2922E-14 6 2.8204E-24 6.4939E-22 4.0168E-20 1.0306E-18 1.4240E-17 1.2518E-16 7.8251E-16 7 1.0806E-26 5.4935E-24 6.1524E-22 2.5039E-20 5.0026E-19 5.9445E-18 4.7755E-17 8 4.1495E-29 4.6597E-26 9.4526E-24 6.1047E-22 1.7643E-20 2.8352E-19 2.9283E-18 9 1.5959E-31 3.9595E-28 1.4553E-25 1.4919E-23 6.2387E-22 1.3561E-20 1.8013E-19 10 6.1440E-34 3.3687E-30 2.2437E-27 3.6517E-25 2.2100E-23 6.4993E-22 1.1105E-20 1 2 3 4 5 6 7 8 9 10 10 -35 10 -30 10 -25 10 -20 10 -15 10 -10 10 -5 E (GeV) lo g (R H q g ) 1 /G e V 2 f m 4 T =180 Mev T =210 Mev T =240 Mev T =270 Mev T =300 Mev T =330 Mev T =360 Mev IHJPAS. 36 (3) 2023 119 Figure 3.The rate of photon produces as afunction to gamma energy produced at 𝑇𝑐 = 178.72 𝑀𝑒𝑉. Table 6. Rate of photon production at 𝑇𝑐 = 198.57 𝑀𝑒𝑉 , IT = 4.45 and IL = −4.26 with 𝜆𝑔=0.06, 𝜆�̅� =0.02 in �̅�g →�̅� γ system . 𝐸𝛾 𝐺𝑒𝑉 R𝑞𝑔 𝐻 (𝐸, 𝑃) 1 𝐺𝑒𝑉 2𝑓𝑚4 T=180MeV T=210MeV T=240MeV T=270MeV T=300MeV T=330MeV T=360MeV 𝛼𝑄𝐶𝐷 =0.35 24 𝛼𝑄𝐶𝐷 =0.32 69 𝛼𝑄𝐶𝐷 =0.30 77 𝛼𝑄𝐶𝐷 =0.29 25 𝛼𝑄𝐶𝐷 =0.28 01 𝛼𝑄𝐶𝐷 =0.26 98 𝛼𝑄𝐶𝐷 =0.26 10 1 9.6378E-12 4.4340E-11 1.4789E-10 3.9646E-10 9.0917E-10 1.8564E-09 3.4671E-09 2 1.5214E-14 1.4998E-13 8.8077E-13 3.6494E-12 1.1811E-11 3.1868E-11 7.4870E-11 3 5.4834E-17 1.1765E-15 1.2312E-14 7.9536E-14 3.6560E-13 1.3093E-12 3.8818E-12 4 2.0520E-19 9.6798E-18 1.8260E-16 1.8626E-15 1.2320E-14 5.9338E-14 2.2489E-13 5 7.7817E-22 8.0904E-20 2.7581E-18 4.4544E-17 4.2521E-16 2.7630E-15 1.3431E-14 6 2.9704E-24 6.8143E-22 4.2034E-20 1.0761E-18 1.4843E-17 1.3028E-16 8.1334E-16 7 1.1381E-26 5.7645E-24 6.4381E-22 2.6144E-20 5.2141E-19 6.1866E-18 4.9636E-17 8 4.3702E-29 4.8896E-26 9.8916E-24 6.3742E-22 1.8389E-20 2.9506E-19 3.0437E-18 9 1.6807E-31 4.1549E-28 1.5229E-25 1.5577E-23 6.5024E-22 1.4113E-20 1.8723E-19 10 6.4707E-34 3.5349E-30 2.3479E-27 3.8129E-25 2.3034E-23 6.7640E-22 1.1543E-20 1 2 3 4 5 6 7 8 9 10 10 -35 10 -30 10 -25 10 -20 10 -15 10 -10 10 -5 E (GeV) lo g (R H q g ) 1 /G e V 2 f m 4 T =180 Mev T =210 Mev T =240 Mev T =270 Mev T =300 Mev T =330 Mev T =360 Mev IHJPAS. 36 (3) 2023 120 Figure4. The rate of photon produces as afunction to gamma energy produced at 𝑇𝑐 = 198.57𝑀𝑒𝑉 . 4.Discussion Due to the photon rate expression in Eq. (31), we can find that the rate relates to many parameters such as strength coupling, temperature of the system, photon energy, fugacity of quark and gluon, quark charge, flavour number of the system, critical temperature, and strong coupling 𝛾(𝐸, 𝑇, 𝜆𝑞 , 𝜇𝑞 ). The critical temperature is influenced by the Bag constant and flavour number. The critical temperature 𝑇𝑐 in Table 1 increases with increasing the bag constant.It reaches a minimum of 132.38 at minimum bag constant of 200MeV and a maximum of 198.57 at a maximum bag constant 300MeV. This indicates that we consider the bag constant dependent on the density of quark-gluon matter in the �̅�g →�̅� γ system and this agrees with the results of the CERN SPS on the formation of a quark-gluon [28]. On the other hand, one of the main motivations for performing the calculation with different critical temperatures is to know the effect of the critical temperature on the photons that are produced. However, the critical temperature is the main factor affecting the rate of photons through its effect on strength coupling. The strength coupling result in Table 2 shows that it increases with an increase in the critical temperature. Moreover, the strength of coupling decreased with the increased in temperature of the system from 180 MeV to 360 MeV because the coupling between quarks and gluons decreases with increasing the temperature from 180 MeV to 360 MeV in �̅�g →�̅� γ system. However, the strength coupling is related to the critical temperature and the temperature of the �̅�g →�̅� γ system due to Eq. (34). It can be seen from Table 2 that the strength coupling increases with increasing the critical temperature from 132.38 MeV to 198.57 MeV and decreases with increasing the temperature of the system from 180 MeV to 360 MeV. Due to Eq.(31), the calculation of photon rate produced from interaction in �̅�g →�̅� γ system at the temperature system in range 180 MeV ≤ T and 𝑇 ≤ 1 2 3 4 5 6 7 8 9 10 10 -35 10 -30 10 -25 10 -20 10 -15 10 -10 10 -5 E (GeV) lo g (R H q g ) 1 /G e V 2 f m 4 T =180 Mev T =210 Mev T =240 Mev T =270 Mev T =300 Mev T =330 Mev T =360 Mev IHJPAS. 36 (3) 2023 121 360 has been done in Tables (3), (4), (5) and (6) and is plotted in Figures (1), (2), (3) and (4), respectively. In the theoretical calculation of photon rate in order of 1 𝐺𝑒𝑉2𝑓𝑚4 for anti-up gluon–anti down photon interaction with different critical temperatures and strength couplings 𝛼𝑄𝐶𝐷 =0.2925, 0.2748, 0.2610, 0.2500, 0.2409, 0.2333 and 0.2267 in range of temperature of the system from 180 to 360 MeV. The maximum photon rate in all tables (3), (4), (5) and (6) is a maximum 𝑜𝑓 3.0106E − 09 1 𝐺𝑒𝑉2𝑓𝑚4 for 𝑇𝑐 =132.38 𝑀𝑒𝑉, 𝛼𝑄𝐶𝐷 = 3.2001E − 09 1 𝐺𝑒𝑉2𝑓𝑚4 , for 𝑇𝑐 = 158.86 𝑀𝑒𝑉, 3.3357E − 09 1 𝐺𝑒𝑉2𝑓𝑚4 for 𝑇𝑐 = 178.72 𝑀𝑒𝑉 and 3.4671E − 09 1 𝐺𝑒𝑉2𝑓𝑚4 for 𝑇𝑐 = 198.57 𝑀𝑒𝑉 at 𝐸 = 1 𝐺𝑒𝑉 and T=360 MeV with a minimum strength coupling of 𝛼𝑄𝐶𝐷 =0.2610, while the photon rates are minimum in tables (3),(4),(5) and (6) for a minimum of 5.3715E − 34 1 𝐺𝑒𝑉2𝑓𝑚4 for 𝑇𝑐 =132.38 𝑀𝑒𝑉, 5.8157E − 34 1 𝐺𝑒𝑉2𝑓𝑚4 , for𝑇𝑐 = 158.86 𝑀𝑒𝑉, 6.1440E − 34 1 𝐺𝑒𝑉2𝑓𝑚4 for 𝑇𝑐 = 178.72 𝑀𝑒𝑉 and 6.4707E − 34 1 𝐺𝑒𝑉2𝑓𝑚4 for 𝑇𝑐 = 198.57 𝑀𝑒𝑉 at 𝐸 = 10 𝐺𝑒𝑉 and T=180 MeV with a maximum strength coupling of 𝛼𝑄𝐶𝐷 = 0.2925. Figures 1, 2, 3 and 4 indicate the behavior of photon emission rate. There is a decrease with increased photons energy E(GeV) at a critical temperature of 132.38, 158.86, 178.72 and 198.57 𝑀𝑒𝑉 and variety temperatures of the system with flavors number 𝑛𝑓 = 3. The photon emission rate increases with the increase in temperature of the system and decreases with the strength of coupling at the critical temperature of �̅�g →�̅� γ system. Moreover, the rate of photon emission in Table 6 and Figure 4 with the critical temperature (𝑇𝑐 = 198.57 )𝑀𝑒𝑉 is larger than the photon rate in other Tables (3), (4), and (5). As we can see, the contribution of rate produced from interaction �̅�g →�̅� γ system in all Tables (3), (4), (5), and (6) and four Figures (1), (2), (3), (4), and (5) reach the maximum at the photons energy 𝐸 ≤ 2𝐺𝑒𝑉 comparing the minimum at 𝐸 ≥ 2𝐺𝑒𝑉 and reach the minimum at E=10 GeV. In fact, the theoretical calculation of the photon rate increases in the high and is affected by increasing the temperature of the system and decreasing the coupling between quark and gluon for all critical temperatures in Tables (3), (4), (5) and (6) for all four critical temperatures in the �̅�g →�̅� γ system with 𝑛𝑓 = 3. 5.Conclusion In the present work, the photon rate is calculated. It also studies the effect of the critical temperature and strong coupling of massless quark flavors at chemical potential 𝜇𝑞 = 500 𝑀𝑒𝑉 at the interaction of �̅�g →�̅� γ system. The equations of photon rate, critical temperatures and strength coupling are derived for the quark-gluon state consisting of an anti-up quark interacting with a gluon to produce an anti-down with photons gamma. Also, the total flavour number of the quark gluon state is 𝑛𝑓 = 3. We can conclude that there is a significant influence on the strength coupling and critical temperature term in the case of the study of the photon rate properties of the quark-gluon interaction. The strength coupling and critical temperature are the main pure features of the QCD effect on the photon rate of the anti up -gluon interaction at the range of temperature system 180-360 MeV. The QCD features of the critical temperature, strength coupling, and photon energy distribution for the photon emission state were quantitatively achieved in the system for 𝑛𝑓 = 2 + 1. This is a IHJPAS. 36 (3) 2023 122 unique feature of the photon rate spectrum. The interesting point in our results is the minimum value of photon rate, especially in the photon energy E=10 GeV of 𝑛𝑓 = 3 flavor, which reflects the weak correlation between quarks and gluons already expected. We conclude that the photon emission rate production at high energy is a good tool to study nucleon structure. References 1. 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