# Numerical analysis of the Balitsky-Kovchegov equation with running coupling: Dependence of the saturation scale on nuclear size and rapidity

## Abstract

We study the effects of including a running coupling constant in high-density QCD evolution. For fixed coupling constant, QCD evolution preserves the initial dependence of the saturation momentum Q{sub s} on the nuclear size A and results in an exponential dependence on rapidity Y, Q{sub s}{sup 2}(Y)=Q{sub s}{sup 2}(Y{sub 0})exp[{alpha}{sub s}d(Y-Y{sub 0})]. For the running coupling case, we rederive analytical estimates for the A and Y dependences of the saturation scale and test them numerically. The A dependence of Q{sub s} vanishes {proportional_to}1/{radical}(Y) for large A and Y. The Y dependence is reduced to Q{sub s}{sup 2}(Y){proportional_to}exp({delta}{sup '}{radical}(Y+X)), where we find numerically {delta}{sup '}{approx_equal}3.2. We study the behavior of the gluon distribution at large transverse momentum, characterizing it by an anomalous dimension 1-{gamma}, which we define in a fixed region of small dipole sizes. In contrast to previous analytical work, we find a marked difference between the fixed coupling ({gamma}{approx_equal}0.65) and running coupling ({gamma}{approx}0.85) results. Our numerical findings show that both a scaling function depending only on the variable rQ{sub s} and the perturbative double-leading-logarithmic expression provide equally good descriptions of the numerical solutions for very small r values below the so-called scaling window.

- Authors:

- Departamento de Fisica, Modulo C2, Planta baja, Campus de Rabanales, Universidad de Cordoba, 14071 Cordoba (Spain)
- (Switzerland)
- Department of Physics, CERN, Theory Division, CH-1211 Geneva 23 (Switzerland)
- (IST), CENTRA, Avenida Rovisco Pais, P-1049-001 Lisbon (Portugal)

- Publication Date:

- OSTI Identifier:
- 20705700

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. D, Particles Fields; Journal Volume: 71; Journal Issue: 1; Other Information: DOI: 10.1103/PhysRevD.71.014003; (c) 2005 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; ANOMALOUS DIMENSION; COUPLING; COUPLING CONSTANTS; DIPOLES; DISTRIBUTION; EVOLUTION; FIELD EQUATIONS; GLUONS; NUCLEAR RADII; NUMERICAL ANALYSIS; NUMERICAL SOLUTION; PARTICLE RAPIDITY; QUANTUM CHROMODYNAMICS; TRANSVERSE MOMENTUM

### Citation Formats

```
Albacete, J.L., Department of Physics, CERN, Theory Division, CH-1211 Geneva 23, Armesto, N., Salgado, C.A., Wiedemann, U.A., Milhano, J.G., and Instituto Superior Tecnico.
```*Numerical analysis of the Balitsky-Kovchegov equation with running coupling: Dependence of the saturation scale on nuclear size and rapidity*. United States: N. p., 2005.
Web. doi:10.1103/PhysRevD.71.014003.

```
Albacete, J.L., Department of Physics, CERN, Theory Division, CH-1211 Geneva 23, Armesto, N., Salgado, C.A., Wiedemann, U.A., Milhano, J.G., & Instituto Superior Tecnico.
```*Numerical analysis of the Balitsky-Kovchegov equation with running coupling: Dependence of the saturation scale on nuclear size and rapidity*. United States. doi:10.1103/PhysRevD.71.014003.

```
Albacete, J.L., Department of Physics, CERN, Theory Division, CH-1211 Geneva 23, Armesto, N., Salgado, C.A., Wiedemann, U.A., Milhano, J.G., and Instituto Superior Tecnico. Sat .
"Numerical analysis of the Balitsky-Kovchegov equation with running coupling: Dependence of the saturation scale on nuclear size and rapidity". United States.
doi:10.1103/PhysRevD.71.014003.
```

```
@article{osti_20705700,
```

title = {Numerical analysis of the Balitsky-Kovchegov equation with running coupling: Dependence of the saturation scale on nuclear size and rapidity},

author = {Albacete, J.L. and Department of Physics, CERN, Theory Division, CH-1211 Geneva 23 and Armesto, N. and Salgado, C.A. and Wiedemann, U.A. and Milhano, J.G. and Instituto Superior Tecnico},

abstractNote = {We study the effects of including a running coupling constant in high-density QCD evolution. For fixed coupling constant, QCD evolution preserves the initial dependence of the saturation momentum Q{sub s} on the nuclear size A and results in an exponential dependence on rapidity Y, Q{sub s}{sup 2}(Y)=Q{sub s}{sup 2}(Y{sub 0})exp[{alpha}{sub s}d(Y-Y{sub 0})]. For the running coupling case, we rederive analytical estimates for the A and Y dependences of the saturation scale and test them numerically. The A dependence of Q{sub s} vanishes {proportional_to}1/{radical}(Y) for large A and Y. The Y dependence is reduced to Q{sub s}{sup 2}(Y){proportional_to}exp({delta}{sup '}{radical}(Y+X)), where we find numerically {delta}{sup '}{approx_equal}3.2. We study the behavior of the gluon distribution at large transverse momentum, characterizing it by an anomalous dimension 1-{gamma}, which we define in a fixed region of small dipole sizes. In contrast to previous analytical work, we find a marked difference between the fixed coupling ({gamma}{approx_equal}0.65) and running coupling ({gamma}{approx}0.85) results. Our numerical findings show that both a scaling function depending only on the variable rQ{sub s} and the perturbative double-leading-logarithmic expression provide equally good descriptions of the numerical solutions for very small r values below the so-called scaling window.},

doi = {10.1103/PhysRevD.71.014003},

journal = {Physical Review. D, Particles Fields},

number = 1,

volume = 71,

place = {United States},

year = {Sat Jan 01 00:00:00 EST 2005},

month = {Sat Jan 01 00:00:00 EST 2005}

}