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

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

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

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

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

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 2 2 𝑟 1 2 Running coupling (including quark loop) 𝒦 𝑅𝐶 𝑟, 𝑟 1 ,𝑥 = 𝛼 𝑠 2𝜋 𝑟 2 𝑟 2 2 𝑟 1 2 + 1 𝑟 1 2 𝛼 𝑠 ( 𝑟 1 2 ) 𝛼 𝑠 ( 𝑟 2 2 ) −1 + 1 𝑟 2 2 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

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

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

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

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

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

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

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

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

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