IHJPAS. 53 (4)2022 

37 

 This work is licensed under a Creative Commons Attribution 4.0 International License. 

 

  

 

 

Theoretical Calculation of Photon Emission from Quark-Antiquark 

Annihilation Using QCD Theory 

 

 

 

 

 
 

 

Abstract 

In this work, we calculate and analyze the photon emission from quark and anti-quark interaction 

during annihilation process using simple model depending on phenomenology of quantum 

chromodynamic theory (QCD). The parameters, which include the running strength coupling, 

temperature of the system and the critical temperature, carry information regarding photon 

emission and have a significant impact on the photons yield. The emission of photon from strange 

interaction with anti-strange is large sensitive to decreases or increases there running strength 

coupling. The photons emission increases with decreases running strength coupling and vice versa. 

We introduce the influence of critical temperature on the  photon emission  rate in order to facilitate 

its further applied in  photon emission spectrum. Photon emission was increased with large critical 

temperature 𝑇𝑐 = 143 MeV comparing with photons emission at critical temperature 𝑇𝑐 =

126 MeV. We analyze and discuss the sensitive of the emission of photon to photons energy Eγ =

1.5 GeV to 5 GeV. It increases with decreased photons energy and vice versa. However, the 

photons emission increases with increases thermal energy of system T = 170 MeV to 270 Mev. It 

is implied that strength coupling, critical temperature and photons energy  can be as important as 

thermal energy of  system   for emission of  photon. 

Key words: Photon emission, Quark-Antiquark, Annihilation, QCD Theory. 

 

 

Doi: 10.30526/35.4.2879 

Article history: Received 6 June 2022, Accepted 21 August 2022, Published in October 2022. 

 

 

 

Elaf Mohammed Ahmed 
Department of Physics, College of 

Education Mustansiriyah University, 

Baghdad, Iraq. 
 elaafmuhamed@uomustansiriyah.edu.iq 

 

 

 

 

Ibn Al-Haitham Journal for Pure and Applied Sciences 

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

 

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 

 

 

 

 

 

Nada Farhan Kadhim 
Department of Physics, College 

of Education Mustansiriyah 

University, Baghdad, Iraq. 
dr.nada@uomustansiriyah.edu.iq 

 

https://creativecommons.org/licenses/by/4.0/
mailto:elaafmuhamed@uomustansiriyah.edu.iq
mailto:elaafmuhamed@uomustansiriyah.edu.iq
mailto:elaafmuhamed@uomustansiriyah.edu.iq
mailto:elaafmuhamed@uomustansiriyah.edu.iq
mailto:hadi.j.m@ihcoedu.uobaghdad.edu.iq
mailto:dr.nada@uomustansiriyah.edu.iq


IHJPAS. 53 (4)2022 
 

38 

1.Introduction 

     The elementary particle is an important field of physics, it deals with the fundamental building 

block of matter and their interactions. It's part of the Standard Model, that's the most successful 

grand model of fundamental physics [1]. In the 1960s, a variety of strong interaction particles are 

observed in experiments on nucleons [2]. Latterly, both George Zweig and Murray Gell-Mann 

introduce and proposed independently the quark model to describe hadrons in 1964 [3]. It 

fundamentally classification hadrons into mesons and baryons. The mesons contain quark-

antiquark while the baryons have three quark in bound states [4]. The standard model is the best 

model to describe the collision at higher energies for quarks interaction due to fundamental 

interactions by which it’s influenced [5]. Recently, different theories are introduced to investigate 

and study the interaction of quark-gluon and the construction of the nature of nucleon structure 

[6]. The basic constituents of the nucleon are quarks and gluon. these quarks and gluons  play a 

key role to produce nucleon mass. The spin of quarks fermion is 1/2  and has a spin equal to one 

[7]. The quark-gluon interaction matter has been observed and produced in relativistic heavy-ion 

collisions at the BNL collider. However, the complementary measurements at the CERN had been 

performed together with the development  of the quantum kinetic theories [8]. Shortly after the 

quarks model, several models were developed to investigate the characteristic of hadrons in terms 

of the quarks mode. Greenberg introduces the color hypothesis that describes the quarks as 

fermions having three color degrees of freedom [9]. Recently,  the structure of hadrons was 

understood from focused on the separated roles of gluons and quarks from theoretical and 

experimental work and integral properties; the quark and gluon spin and angular momentum[10]. 

Due to the color confinement, the quarks are bounded into the hadrons by a variety of 

configurations. The baryons are formed by three quarks in nucleons and mesons make by the pair 

of quark and antiquark, they observe in collision experiments [11]. Photons are produced in a 

variety of sources;  prompt photons, thermal photons and photons are produced by hard interaction 

. There create experimentally in heavy-ion collisions [12]. The quark-antiquark state produces in 

the laboratory. It uses to investigate the transition in the nuclear matter in which quarks and gluons 

are confined in hadron [13]. In this paper, we present a theoretical calculation of photon rate that's 

emission from the annihilation of quark and antiquark using the QCD model. 

2.Theory  

The rate of photons emission from annihilation reaction of quark anti quarks is[3]. 

𝑅𝑝ℎ =
4𝑁𝑠

2

(2𝜋)6
𝐹𝑞 (𝑝𝛾 )(

𝑒𝑞

𝑒
)2 ∫ 𝜎𝑎(𝑠)√𝑠(𝑠 − 4𝑚

2) 𝑑𝑠.
1

4𝐸𝛾
∫ 𝐹�̅� (𝑝�̅� )[1 + 𝐹𝑔(𝑝�̅� )𝑑𝐸�̅�

∞
𝑠

4𝐸𝛾

∫ 𝑑𝜙
2𝜋

0
…(1) 

Where 𝑁𝑠, 𝑒𝑞  and 𝑒 are number of spins, charges and electric charge of quarks  𝜎𝑎(𝑠) is cross 

section, 𝐹𝑞 (𝑝𝛾 ) and 𝐹�̅�(𝑝�̅� ) are Juttner distribution of quark and gluon, 𝐸�̅� and 𝐸𝛾 are quark and 

photons energy, 𝑑𝑠, 𝑑𝜙 and 𝑑𝐸�̅�  are  element of  momentum, solid angle and quarks energies. The 

Juttner function of quark distribution relative to fugacity of anti-quark𝜆𝑞  is [14]. 

Ϝ�̅� (𝐸�̅� ) =  𝜆�̅� (𝑒
𝐸�̅�

𝑇 + 1)−1                                                                       (2) 

And the distribution of gluon F𝐵 (E) relative tofugacity of gluon 𝜆𝑔  is [15]. 



IHJPAS. 53 (4)2022 
 

39 

𝐹𝑔 (𝐸𝑔) = 𝜆𝑔(𝑒
𝐸𝑔/𝑇 − 1)−1                                                      (3) 

Inserting Eqs.(2)  and (3) in Eq.(1) and integral  over  𝐸�̅� ≥
𝑠

4𝐸𝛾
 to obtaine[16]. 

𝑅𝑝ℎ =
4𝑁𝑠

2

(2𝜋)6
𝐹𝑞 (𝑝𝛾 )(

𝑒𝑞

𝑒
)2.

1

4𝐸𝛾
[∫ [𝜆�̅� (𝑒

𝐸�̅�

𝑇 + 1)−1 + 𝜆�̅� 𝜆𝑔(𝑒
𝐸�̅�

𝑇 + 1)−1(𝑒𝐸𝑔/𝑇 −
∞
𝑠

4𝐸𝛾

1)−1] 𝑑𝐸�̅� ∫ 𝑑𝜙
2𝜋

0
∫ 𝜎𝑎(𝑠)√𝑠(𝑠 − 4𝑚

2) 𝑑𝑠…(4) 

The solution of the first and second integral in Eq.(4) for 𝐸�̅� ≅ 𝐸𝑔is 

𝑅𝑝ℎ =
4𝑁𝑠

2

(2𝜋)6
𝐹𝑞 (𝑝𝛾 )(

𝑒𝑞

𝑒
)2. ∫[𝑇𝜆�̅� ∑

(−1)𝑛+1(𝑒

−𝑠
4𝐸𝛾𝑇)

𝑛

𝑛
∞
𝑛=1 +

𝑇𝜆�̅� 𝜆𝑔 ∑
𝑒

−(2𝑛+1)𝑠
4𝐸𝛾𝑇

2𝑛+1
∞
𝑛=1 ]𝜎𝑎(𝑠)√𝑠(𝑠 − 4𝑚

2) 𝑑𝑠                             (5) 

The final term in Eq.(5) is [17]. 

√𝑠(𝑠 − 4𝑚2) 𝜎(𝑠) = 4𝜋𝛼0𝛼𝑠𝑚
2[ln (

𝑠

𝑚2
) − 1]                       (6) 

The  Eq.(5) together  Eq.(6) for S > 4m2 leads to  

𝑅𝑝ℎ =
4𝑁𝑠

2

(2𝜋)6
𝐹𝑞 (𝑝𝛾 )(

𝑒𝑞

𝑒
)2. 4𝜋𝛼0𝛼𝑠𝑚

2𝑇𝜆�̅� ∫[∑
(−1)𝑛+1(𝑒

−𝑠
4𝐸𝛾𝑇)

𝑛

𝑛
∞
𝑛=1 + 𝜆𝑔 ∑

𝑒

−(2𝑛+1)𝑠
4𝐸𝛾𝑇

2𝑛+1
∞
𝑛=1 ] ln[(

𝑠

𝑚2
) −

1]𝑑𝑠                                                                                                          (7) 

The solution integral in Eq.(7) with  assume 𝑠 = 4𝑚2z in and 𝑚2 ≪ 𝐸𝛾 𝑇is. 

𝑅𝑝ℎ =
4𝑁𝑠

2

(2𝜋)6

𝐹𝑞(𝑝𝛾)

4𝐸𝛾
(

𝑒𝑞

𝑒
)

2

. 4𝜋𝛼0𝛼𝑠𝑚
2𝑇𝜆�̅�  [𝜆�̅� ∑

(−1)𝑛+1

𝑛
∞
𝑛=1 (

𝐸𝛾𝑇

𝑛𝑚2
) [ln (

4𝐸𝛾𝑇

𝑚2
) − 𝐶 − 𝑙𝑛𝑛 − 1] +

 𝜆�̅� 𝜆𝑔 ∑
(−1)𝑛+1

2𝑛+1
∞
𝑛=1

𝐸𝛾𝑇

(2𝑛+1)𝑚2
[ln (

4𝐸𝛾𝑇

𝑚2
) − 𝐶 − 𝑙𝑛(2𝑛 + 1) − 1]        (8) 

The  series in Eq.(8)reduce to  [18].. 

∑
(−1)𝑛+1

(2𝑛+1)2
∞
𝑛=1 = (1 −

1

22
) 𝜁(2)~

𝜋2

6
                                                            (9) 

And  

∑
(−1)𝑛+1

𝑛2
∞
𝑛=1 =

𝜋2

6
                                                                                          (10) 

Then the rate in  Eq.(8)  together Eq.(2), Eq.(9) and Eq.(10) with 𝑁𝑠 = 2 and 𝜆𝑔~1to become. 

𝑅𝑝ℎ =
𝛼0𝛼𝑠

3(2𝜋)2
(

𝑒𝑞

𝑒
)

2

𝑇2
𝜆�̅�𝜆𝑞

𝑒
𝐸𝛾
𝑇 +1

[[ln (
4𝐸𝛾𝑇

𝑚2
) − 𝐶 + 1 + 𝑙𝑛𝑛 + 𝑙𝑛(2𝑛 + 1)] (11) 



IHJPAS. 53 (4)2022 
 

40 

Where 𝛼0 is electrodynamic constant 𝛼0 = (137)
−1, 𝛼𝑠 is QCD strength constant, and 𝑚 is the 

finite quark mass 𝑚 = 𝑔𝑇 = √4𝜋𝛼𝑠𝑇 [19]  with 
1

𝑒
𝐸𝛾
𝑇 +1

≈ 𝑒
−𝐸𝛾

𝑇 for 𝐸𝛾 ≫ 𝑇 and 𝐶𝑝ℎ = 𝐶 − 1 −

𝑙𝑛𝑛 − 𝑙𝑛(2𝑛 + 1) the Eq.(11) reduce to  

𝑅𝑝ℎ =
𝛼0𝛼𝑠

3(2𝜋)2
(

𝑒𝑞

𝑒
)

2

𝑇2𝜆�̅� 𝜆𝑞 𝑒
−𝐸𝛾

𝑇 [ln (
4𝐸𝛾𝑇

𝑚2
) − 𝐶𝑝ℎ ]                               (12) 

The running strength constant is given by [12]. 

𝛼𝑠 =
6𝜋

(33−2𝑛𝑓)𝑙𝑛
8𝑇

𝑇𝑐

                                                                                         (13) 

The critical temperature  is [21]. 

𝑇𝑐 = (
90𝐵

𝜋2𝑛𝑔𝑞
)

1

4                                                                                               (14) 

Where 𝐵 is the Bag constant with bag model and 𝑛𝑔𝑞 = 𝑛𝑔 +
7

8
 (𝑛𝑞 + 𝑛�̅� ) is the number of 

gluons 𝑛𝑔 and quarks 𝑛𝑞  and anti quarks 𝑛�̅�  degrees of freedom. 

3. Results  

In order to evaluate the emission of photon coefficient from the interaction strange with anti-

strange quarks, we calculate the running strength coupling with the critical temperature, flavor 

number and quarks electric charge. Firstly, the flavor number estimates using summation 

flavors ∑ 𝑛𝑓 = 8 in s�̅�→γg system. The critical temperatures for system is calculated using 
Eq.(14) by take the Bag constant 𝐵1/4 = 230  𝑎𝑛𝑑 260 MeV [21] and inserts the spin 𝑛𝑠 = 2 and  

color number 𝑛𝑐 = 8 for gluon and takes 𝑛𝑠 = 2, 𝑛𝑐 = 3 𝑎𝑛𝑑 𝑛𝑓 = 8 for quarks, results are  𝑇𝑐 =

126𝑀𝑒𝑉 to  143𝑀𝑒𝑉. The running strength coupling  is evaluated using Eq.(13) as a function of 

the critical temperature, favor number and thermal energy of system. We  insert the thermal energy 

T=170,190,210,230,250 and 270 MeV, the flavor number 𝑛𝑓 = 8 and the critical temperature  𝑇𝑐 =

126 𝑡𝑜143 𝑀𝑒𝑉 in Eq.(13)  to be results running strength coupling are shown  in Table (1) and 

Table (2) at 𝑇𝑐 = 126 to 143 𝑀𝑒𝑉 alternatively  

.�̅�strange-ntiwith a = 126, 143 Mev for strange S quark interaction cThe running strength coupling at T .1 Table 

𝑇(𝑀𝑒𝑉) The running strength coupling 𝛼𝑠 The running strength coupling 𝛼𝑠 

  

170 0.46608 0.49227 

190 0.44526 0.46910 

210 0.42806 0.45005 

230 0.41353 0.43402 

250 0.40106 0.42030 

270 0.39020 0.40839 

 

Furthermore, The tuned values of electric charge of system estimates using  summation 

∑ (
𝑒𝑞

𝑒
)

2

charge of strange  quark 𝑒𝑠 = −
1

3⁄ e and charge of anti-strange 𝑒�̅� = +
1

3⁄ e to  results  



IHJPAS. 53 (4)2022 
 

41 

1
9⁄ .  The emission of photons yields to the strange quark  interaction  with anti-strange calculates 

using Eq.(12) by substituting  the running strength coupling from Tables (1), the photon 

energy 𝐸𝛾 = 1.5,2,2.5,3,3.5,4 ,4.5 𝑎𝑛𝑑  5 𝐺𝑒𝑉 from experimental data [21],critical temperature 

Tc = 126 MeV to 143 MeV with taken fugacity of strange  and anti-strange quarks are 𝜆𝑞= 0.06, 

𝜆�̅� = 0.06  and annihilation parameter is  𝐶𝑝ℎ =  1.415 [22]. The results are tabulated in tables 

(2) and (3) for 𝑇𝑐 = 126 𝑀𝑒𝑉 𝑡𝑜 143 𝑀𝑒𝑉.  

Table 3. The emission of photon rate production  at 𝑇𝑐 = 143 𝑀𝑒𝑉 with 𝜆𝑞 = 0.06, 𝜆�̅�  = 0.06 for strange S quark 

interaction with anti-strange �̅� 

𝑅𝑝ℎ(
1

𝐺𝑒𝑉 2𝑓𝑚4
) 

 

1T=270 MeV T=250 MeV T=230 MeV T=210 MeV T=190 MeV T=170 MeV 

𝛼𝑠 = 0.39020 𝛼𝑠 = 0.40106 𝛼𝑠 = 0.41353 𝛼𝑠 = 0.42806 𝛼𝑠 = 0.44526 𝛼𝑠 = 0.46608 𝐸𝛾 (𝐺𝑒𝑉) 

5.21399E-13 4.46265E-13 3.1486E-13 1.87523E-13 9.32084E-14 3.72618E-14 1.5 

3.26636E-13 1.79666E-13 8.77242E-14 3.69096E-14 1.28228E-14 3.45285E-15 2 

8.10662E-14 3.68356E-14 1.45568E-14 4.81635E-15 1.26413E-15 2.43157E-16 2.5 

1.65446E-14 6.36962E-15 2.08115E-15 5.5137E-16 1.11044E-16 1.54642E-17 3 

3.10373E-15 1.02045E-15 2.77615E-16 5.92707E-17 9.21246E-18 9.33739E-19 3.5 

5.56063E-16 1.56675E-16 3.56054E-17 6.14427E-18 7.39102E-19 5.46644E-20 4 

9.68163E-17 2.34206E-17 4.45391E-18 6.22265E-19 5.80217E-20 3.13611E-21 4.5 

1.65349E-17 3.43802E-18 5.47698E-19 6.2015E-20 4.48661E-21 1.77389E-22 5 

 

The emission of photon production was plotted in in Figure (1) at 𝑇𝑐 = 126𝑀𝑒𝑉 and Figure (2) at 𝑇𝑐 =

143 𝑀𝑒𝑉 as a function of the photons energy  𝐸𝛾 (𝐺𝑒𝑉)for strange S quark  interaction with anti-

strange �̅� at fugacity 𝜆𝑞 = 0.06  and 𝜆𝑞 ̅= 0.06 . 

 

Figure 1. The emission of photon production 𝑅𝑝ℎ  foror strange S quark  interaction with anti-strange �̅� at 𝑇𝑐 =

126 𝑀𝑒𝑉 and  𝜆𝑞 = 0.06 and 𝜆�̅� = 0.06 . 



IHJPAS. 53 (4)2022 
 

42 

 

Figure 2. The emission of photon production 𝑅𝑝ℎ foror strange S quark  interaction with anti-strange �̅� at 𝑇𝑐 =

143 𝑀𝑒𝑉 and  𝜆𝑞 = 0.06 and 𝜆�̅� =0.06  . 

 

4.Discussion  

     To understand the mechanism of emission of photons  yields, we analytically calculate the 

running strength coupling, critical temperature,thermal energy  and energy of photon. The running 

strength coupling as function to flavour number, critical temperature  and thermal energy of system  

forint reaction of strange quark and anti-strange quarkat annihilation process. Table (1) show that 

running strength coupling increases with decrease of thermal energy from 170 to 270 MeV of 

system. Also, the running strength coupling increases with increase critical temperature from  𝑇𝑐 =

126 to 143 MeV.  Emission of photon was increase with decreasing running strength coupling 

and vice versa. The emission of photon relative to photon energy are done in Figure (1) and figure 

(2). The emission of photon is maximum 𝑅𝑝ℎ = 5.21399 × 10
−13 1

𝐺𝑒𝑉 2𝑓𝑚4
 at critical temperature 𝑇𝑐 =

126 𝑀𝑒𝑉  with 𝐸𝛾 = 1.5 𝐺𝑒𝑉 and T = 270 MeV  and running strength coupling 𝛼𝑠 = 0.39020  while 

reach minimum 𝑅𝑝ℎ = 1.77389 × 10
−22 1

𝐺𝑒𝑉2𝑓𝑚
4 critical temperature 𝑇𝑐 = 126 𝑀𝑒𝑉 with 𝐸𝛾 =

 5 𝐺𝑒𝑉 and 𝛼𝑠 = 0.46608 for low thermal energy T=170Mev. However, the emission of photon reachs 

maximum 𝑅𝑝ℎ = 2.87147 × 10
−13

 at critical temperature𝑇𝑐 = 143 𝑀𝑒𝑉. With 𝐸𝛾 = 1.5 𝐺𝑒𝑉and 𝛼𝑠 =

0.40839 and T = 270 MeV comparing to minimum 𝑅𝑝ℎ = 1.80895 × 10
−22 𝑇𝑐 = 143 𝑀𝑒𝑉. With 𝐸𝛾 =

 5 𝐺𝑒𝑉and 𝛼𝑠 = 0.49227 for low thermal energy T=170 Mev. Figures (1) and (2) show the emission 

of photon yield as function of photon energy, it increases with decreases energy of photons and, 

the emission of photons have maximum for photon energy less than 2 GeV. Thus, emission of 

photon yields increase with increases thermal energy, and dramatically decrease the running 

strength coupling of interaction strange quark with anti-strange quark with larger critical  are more 

strongly enhanced  Generally, the emission of photon in Figure (1) and Table (3) larger than  in 

Figure (1)  and Table(2) because that photons are increased with increased critical temperature 

143MeV. We can extract the emission of photon increases with increasing thermal energy it has 



IHJPAS. 53 (4)2022 
 

43 

maximum at   𝑇 = 270 MeV  by comparing both Tables (2) and (3). However, the probability  of 

the emission of photon affected by running  strength coupling, critical temperature, thermal energy  

and photons energy and reach maximum at critical temperature TC = 143 MeV and temperature 𝑇 

= 270 MeV  of system. 

 

5. Conclusion  

     In conclusion, the emission of photon from strange quark interaction with anti-strange for 

annihilation processes depending on running strength coupling, critical temperature, thermal 

energy and photon energy. Therefore, one may demonstrated the emission of photon production 

be affected forcedly by running strength coupling, thermal and critical temperature. It is decreased 

with increasing the running strength coupling and decreased critical temperature and thermal 

energy of system for strange -anti strange annihilation process with flavor number 𝑛𝑓 = 8. The 

photons yield results are implied affected by photon energy. It each top emission for photon energy 

less than 2 GeV. We explore that emission of photon is   a unique feature and occurs at minimum 

running strength coupling and large thermal temperature and critical temperature. In conclusion, 

the emission of photon is produced forcedly at less strength coupling and large temperature T = 

270 MeV with critical temperature TC = 143MeV and photon energy less 2 GeV. 

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