Evolution equations in high-energy particle physics

High-energy particle physics focuses on the study understanding the fundamental forces of nature and particles in our universe.^[1] One big area of focus is gluons, which bind quarks together, so understanding Evolution equations play a major role gluon behavior at different energy levels. Some of these equations in this area include the BFKL equation, the BK equation, and the CCFM equation and these equations explain how the parton distribution varies in terms of energy scale. These equations are crucial for advancing our understanding the quantum chromodynamics (QCD) in high-energy conditions.

History

The development of evolution equations began in the 1970s. when researchers sought better ways to describe the behavior of partons (quarks and gluons) at high energy scales. The (Balitsky-Fadin-Kuraev-Lipatov) BFKL equation was one of the first to be formulated, thanks to Soviet physicists in Ioffe, Gribov, Lipatov, and Fadin. They aimed to understand the Regge limit of QCD, where energy in collisions vastly increases compared to other factors.

The 1980s and 1990s saw further progress was made with the BK (Balitsky-Kovchegov) and CCFM (Catani-Ciafaloni-Fiorani-Marchesini) equations, These equations introduced concepts like gluon saturation effects. Building on this early research, several scientists made significant contributions, contributing to the advancement of the field.

Major contributors

Vladimir S. Fadin is a renowned Russian physicist who made significant contributions to the development of the BFKL equation.^[2] His work provided valuable insights into gluon behavior at small Bjorken x values.^[3]
Ian I. Balitsky played a crucial role in advancing high-energy QCD through his work on the BK equation.^[4] He developed nonlinear evolution equations, which have greatly enhanced our understanding of gluon saturation.^[5]
Yuri V. Kovchegov independently derived the BK equation using a color dipole model, contributing substantially to our understanding of saturation and its associated phenomena in QCD.^[6]
Roberto Fiore, Guido Marchesini, and Giampiero Altarelli: Roberto Fiore, Guido Marchesini, and Giampiero Altarelli, in collaboration with Catani and Ciafaloni, formulated the CCFM equation.^[7] Their work aimed to connect the DGLAP and BFKL equations, resulting in an improved structure for parton evolution.^[8]

Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP)

The evolution of parton distributions based on collinear approach has been explained by these Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equations. They are a system of integro-differential equations which measure when the partonic particles locations in space change with energy scale.^[9]

 ${\frac {dF(x,t)}{dt}}={\frac {\alpha _{s}}{2\pi }}\int _{x}^{1}{\frac {dz}{z}}P(z)F\left({\frac {x}{z}},t\right)$

Gives the rate of change in momentum dependent parton density with times. While the splitting function is P(z), the parton distribution is F(x,t), coupled with the very strong constant α.^[10]

Balitsky-Fadin-Kuraev-Lipatov (BFKL) Equation

The Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation describes the evolution of parton distributions in the high-energy limit. It is a nonlinear integro-differential equation that describes the change in parton distributions as a function of energy scale. Balitsky, Fadin, Kuraev, and Lipatov worked on this together back in the late 1970s.

${\frac {\partial f(x\,k^{2})}{\partial \ln(1/x)}}={\frac {\alpha _{s}N_{c}}{\pi }}\int {\frac {d^{2}q}{\pi q^{2}}}\left[f(x\,\|{\vec {k}}+{\vec {q}}\|^{2})-{\frac {k^{2}f(x\,k^{2})}{\|k^{2}-q^{2}\|+q^{2}}}\right]$

This equation shows how quickly gluon density grows at low (x). As energy rises, more gluons pop up! This fact affects the behavior of protons act those high-energy collisions and contributes to understanding them.

Scientists analyze phenomena observed in deep inelastic scattering (DIS) experiments using the BFKL equation. HERA results have provided important information to support BFKL principles, while there are still major differences between theory and observation.^[11]

Balitsky-Kovchegov (BK) Equation

The BK equation provides a nonlinear view of gluon density saturation as x decreases. It builds upon the BFKL equation by factoring in saturation effects, which stabilize gluon density at very small x. Balitsky created it through operator product expansion while Kovchegov looked at it from a color dipole model viewpoint.

${\frac {\partial N(x,r)}{\partial \ln(1/x)}}={\frac {\alpha _{s}N_{c}}{2\pi ^{2}}}\int d^{2}r_{1}{\frac {r^{2}}{r_{1}^{2}r_{2}^{2}}}\left[N(x,r_{1})+N(x,r_{2})-N(x,r)-N(x,r_{1})N(x,r_{2})\right]$

The BK equation builds on BFKL by factoring in things like gluon saturation—which is super important when there are lots of them around! It suggests that when (x) gets very tiny, gluon density stops growing rapidly and becomes more stable.

Understanding phenomena like black disk limits in QCD and heavy-ion collisions at RHIC and LHC depends on this equation.

Catani-Ciafaloni-Fiorani-Marchesini (CCFM) Equation

By including angular ordering, the CCFM equation connects the BFKL and DGLAP equations, that provides a more deep structure for parton density evolution. It can also be used in both high and low x regions.

$\Delta _{s}(x,k_{T}^{2})=\Delta _{NS}(x)+\int dz{\frac {d^{2}q}{\pi q^{2}}}\Theta (q^{2}-q_{0}^{2})P(z,q)\Delta _{s}(x/z,q^{2})$

This equation combines different types of radiation to accurately represent high-energy physics, helping to tell a complete picture about how gluons evolve when (x) is low or high.

This formula is widely used in parton shower Monte Carlo analyses and performs effectively when compared to experimental data obtained from the LHC.

References

^ Perkins, D. (2008). Introduction to High Energy Physics. Cambridge University Press. ISBN 978-0201057850.
^ Forshaw, J. R.; Ross, D. A. (2023). Quantum Chromodynamics and the Pomeron. Cambridge University Press. doi:10.1017/9781009290111.
^ Kovchegov, Y. V. (1999). "Unitarization of the BFKL Pomeron on a Nucleus". arXiv:hep-ph/9802290.
^ Balitsky, I. I. (2001). "High-energy QCD and Wilson lines". arXiv:hep-ph/0101042.
^ Kovchegov, Y. V.; Levin, E. (2003). Quantum Chromodynamics at High Energy. Cambridge University Press. doi:10.1017/9781009291446.
^ Kovchegov, Y. V. (1999). "Unitarization of the BFKL Pomeron on a Nucleus". Physical Review D. 60: 054025. arXiv:hep-ph/9903246. doi:10.1103/PhysRevD.60.054025.
^ Catani, S.; Ciafaloni, M.; Fiorani, F.; Marchesini, G. (1990). "Small-x Behavior of Initial State Radiation in Perturbative QCD". Nuclear Physics B. 336: 18–40. doi:10.1016/0550-3213(90)90342-B.
^ Marchesini, G.; Webber, B. R. (1984). "Monte Carlo simulation of general hard processes with coherent QCD radiation". Nuclear Physics B. 238: 1–29. doi:10.1016/0550-3213(88)90089-2.
^ "The Evolution of Parton Distributions" by W. T. Giele and S. A. Kulagin (2002)
^ High-Energy Particle Physics" by R. K. Ellis, W. J. Stirling, and B. R. Webber (2003)
^ Bartels J., Forshaw J., & Watt G., 2004 — ''The BFKL Pomeron''.

External links

[Perkins2008-1] Perkins, D. (2008). Introduction to High Energy Physics. Cambridge University Press. ISBN 978-0201057850.

[Forshaw2023-2] Forshaw, J. R.; Ross, D. A. (2023). Quantum Chromodynamics and the Pomeron. Cambridge University Press. doi:10.1017/9781009290111.

[Kovchegov1999-3] Kovchegov, Y. V. (1999). "Unitarization of the BFKL Pomeron on a Nucleus". arXiv:hep-ph/9802290.

[Balitsky2001-4] Balitsky, I. I. (2001). "High-energy QCD and Wilson lines". arXiv:hep-ph/0101042.

[Kovchegov2003-5] Kovchegov, Y. V.; Levin, E. (2003). Quantum Chromodynamics at High Energy. Cambridge University Press. doi:10.1017/9781009291446.

[Kovchegov12-6] Kovchegov, Y. V. (1999). "Unitarization of the BFKL Pomeron on a Nucleus". Physical Review D. 60: 054025. arXiv:hep-ph/9903246. doi:10.1103/PhysRevD.60.054025.

[Catani1990-7] Catani, S.; Ciafaloni, M.; Fiorani, F.; Marchesini, G. (1990). "Small-x Behavior of Initial State Radiation in Perturbative QCD". Nuclear Physics B. 336: 18–40. doi:10.1016/0550-3213(90)90342-B.

[Marchesini1984-8] Marchesini, G.; Webber, B. R. (1984). "Monte Carlo simulation of general hard processes with coherent QCD radiation". Nuclear Physics B. 238: 1–29. doi:10.1016/0550-3213(88)90089-2.

[9] "The Evolution of Parton Distributions" by W. T. Giele and S. A. Kulagin (2002)

[10] High-Energy Particle Physics" by R. K. Ellis, W. J. Stirling, and B. R. Webber (2003)

[11] Bartels J., Forshaw J., & Watt G., 2004 — ''The BFKL Pomeron''.

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