1. The energy dependence of saturation scale at next-to-leading order
Mengliang WANG
Guizhou University of Finance and Economic

2. Outline Motivation The BK equation and the corrections
BK equation in NLO
The BK equation and the corrections
The higher order correction
The initial conditions
The numerical calculation
Summary and outlook
现代金融理论 <2000年
QTP2017, Xi'an, July 21-23
Mengliang WANG

3. The BK equation
At high energy, dipole-hadron scattering perturbative QCD (pQCD) predict non-linear phenomena:
gluon saturation
multiple scattering
𝒓 2
The non-linear solutions:
Balitsky-JIMWLK* hierarchy
+ mean field approximation：
Balitsky-Kovchegov (BK) equation
* Jalilian-Marian–Iancu–McLerran–Weigert–Leonidov–Kovner
At high energy, dipole-hadron scattering perturbative QCD (pQCD) predicts a rapid growth of the gluon density with increasing energy, which leads to non-linear phenomena:
The BK evolve equation of leading order:
QTP2017, Xi'an, July 21-23
Mengliang WANG

4. The BK equation of NLO
Insufficient for leading logarithmic accuracy compare with data:
structure functions in DIS at HERA
particle production in heavy ion collisions at LHC
A further corrections are needed to explain data:
BK equation of Full NLO reads:
QTP2017, Xi'an, July 21-23
Mengliang WANG

5. The higher order corrections
Beside the long full NLO-BK equation, simpler corrections can be used as to neglect the contribution from the high order of 𝛼 𝑠
However, the NLO BK equation suffers from a severe lack of stability:
solution can be negative with increasing rapidity (no-physically meaningful evolution)
The origin of this instability can be traced back to the large double transverse logarithmic NLO correction:
Several way of corrections have been developed to overcome the problem.
QTP2017, Xi'an, July 21-23
Mengliang WANG

6. The higher order corrections (1)
𝜕𝒩(𝑟,𝑥) 𝜕𝑌 = 𝑑 2 𝑟 1 𝓚 𝒓, 𝒓 𝟏 ,𝒙 𝒩 𝑟 1 ,𝑥 +𝒩 𝑟 2 ,𝑥 −𝒩 𝑟,𝑥 −𝒩( 𝑟 1 ,𝑥)𝒩( 𝑟 2 ,𝑥)
Leading order
𝒦 𝐿𝑂 𝑟, 𝑟 1 ,𝑥 = 𝛼 𝑠 2𝜋 𝑟 2 𝑟 𝑟 1 2
Running coupling (including quark loop)
𝒦 𝑅𝐶 𝑟, 𝑟 1 ,𝑥 = 𝛼 𝑠 2𝜋 𝑟 2 𝑟 𝑟 𝑟 𝛼 𝑠 ( 𝑟 1 2 ) 𝛼 𝑠 ( 𝑟 2 2 ) − 𝑟 𝛼 𝑠 ( 𝑟 2 2 ) 𝛼 𝑠 ( 𝑟 1 2 ) −1
The propagator of the emitted gluon can be understand as a pair of quark-antiquark, contract to the fix–coupling.
The quark-antiquark pair which added to the evolved wave function, not only can modify the BK kernel, but also can change the interaction structure of the evolution equation.
QTP2017, Xi'an, July 21-23
Mengliang WANG

7. The higher order corrections (2)
𝜕𝒩(𝑟,𝑥) 𝜕𝑌 = 𝑑 2 𝑟 1 𝓚 𝒓, 𝒓 𝟏 ,𝒙 𝒩 𝑟 1 ,𝑥 +𝒩 𝑟 2 ,𝑥 −𝒩 𝑟,𝑥 −𝒩( 𝑟 1 ,𝑥)𝒩( 𝑟 2 ,𝑥)
NLL (next-to-leading logarithmic) correction:
Double Logarithmic Approbation (DLA):
Resummation of large double logarithmic term from light-cone perturbative theory
Next-to-leading logarithmic (NLL):
Promoting this method to NLO-BK equation:
To avoid the disaster of the double logarithmic term, a effective method of resummation of large double logarithmic correction (DLA), by promoting this method to NLO-BK equation, established from light-cone perturbation theory. s to all orders
QTP2017, Xi'an, July 21-23
Mengliang WANG

8. Initial Condition
To solve these BK equations, the initial condition (Y=0) is necessary. Typical methods are GBW and MV (the results of with MV are shown in these slides):
Golec-Biernat and Wusthoff (GBW) model
McLerran-Venugopalan (MV) model
QTP2017, Xi'an, July 21-23
Mengliang WANG

9. Solutions from the BK equation
Results of several Y (=8, 16, 24):
The initial condition (Y=0) is shown in black dash line.
LO are in orange
RC are in green
NLL are in violate
QTP2017, Xi'an, July 21-23
Mengliang WANG

10. Saturation scale: 𝑌, 𝑟 𝐿𝑂 𝑌, 𝑟 𝑁𝐿𝐿 𝑌, 𝑟 RC
In order to show the energy dependence of the saturation scale, we extract the values of 𝑄 𝑠 from the numerical solutions in previous figure, via define 𝑁(𝑟=1/ 𝑄 𝑠 , 𝑌) =𝜅, we used 𝜅=0.5 in this slides.
𝑌, 𝑟 𝑁𝐿𝐿
𝑌, 𝑟 RC
𝑌, 𝑟 𝐿𝑂
QTP2017, Xi'an, July 21-23
Mengliang WANG

11. The energy/rapidity dependence
For the saturation 𝒩 𝑟,𝑌 =0.5 region, the different saturation scale 𝑄 𝑠 (=1/𝑟) are abstracted for the corresponding rapidity Y(0 ~ 25)
LO are in orange
RC are in green
NLL are in violate
Reasonable results for different experiment:
𝑸 𝒔 =0.5 ~ 2 from HERA, RHIC and LHC, which agrees with NLL and RC results
QTP2017, Xi'an, July 21-23
Mengliang WANG

12. Thank you ! Summary and outlook
With the MV initial condition, different approaches (LO, RC, NLL) are used of solving the BK equation in coordinates space.
The corresponding energy (rapidity) dependence are calculated.
The study of other approaches (DLA-rc, STL+DLA) are on going, together with the GBW Initial conditions.
Thank you !
QTP2017, Xi'an, July 21-23
Mengliang WANG

13. backup
QTP2017, Xi'an, July 21-23
Mengliang WANG

14. The higher order corrections
Leading order:
Running coupling (including quark loop):
NLL (next-to-leading logarithmic) correction:
QTP2017, Xi'an, July 21-23
Mengliang WANG

15. 𝒓 2
QTP2017, Xi'an, July 21-23
Mengliang WANG