DGLAP equations

In particle physics , the DGLAP equations describe how the parton densities depend on the energy scale under consideration. They were developed independently by the physicists Yuri Dokshitzer , Wladimir Naumowitsch Gribow and Lew Nikolajewitsch Lipatow , as well as Guido Altarelli and Giorgio Parisi , after whose first letters the equations are named. After the last two, the equations were formerly known as Altarelli-Parisi equations .

background

Parton densities are distribution functions of components of strongly bound systems of strong interaction, such as protons, and depend on the momentum fraction of the parton and the energy scale under consideration . The momentum fraction is limited, the energy scale, on the other hand, is freely open to high energies. Since the coupling constant of the strong interaction in bound systems becomes large, these systems cannot be described perturbatively ; the parton densities must therefore be determined experimentally. The DGLAP equations make it possible to carry out these experimental measurements, instead of for all possible and with a fixed energy scale, and to deduce from this data the behavior of the parton densities on any energy scales (with a fixed fraction of momentum). ${\displaystyle x}$ ${\displaystyle Q^{2}}$ ${\displaystyle 0\leq x\leq 1}$ ${\displaystyle x}$ ${\displaystyle Q^{2}}$

Leading order

The DGLAP equations in the leading order of the perturbation series in the coupling constant of the strong interaction are:

${\displaystyle Q^{2}{\frac {\partial }{\partial Q^{2}}}{\begin{pmatrix}q_{i}(x,Q^{2})\\{\bar {q}}_{i}(x,Q^{2})\\g(x,Q^{2})\end{pmatrix}}={\frac {\alpha _{s}(Q^{2})}{2\pi }}\sum _{j}\int _{x}^{1}{\frac {\mathrm {d} \xi }{\xi }}{\begin{pmatrix}P_{q_{i}q_{j}}(x/\xi )&0&P_{q_{i}g}(x/\xi )\\0&P_{{\bar {q}}_{i}{\bar {q}}_{j}}(x/\xi )&P_{{\bar {q}}_{i}g}(x/\xi )\\P_{gq_{j}}(x/\xi )&P_{g{\bar {q}}_{j}}(x/\xi )&P_{gg}(x/\xi )\end{pmatrix}}{\begin{pmatrix}q_{j}(\xi ,Q^{2})\\{\bar {q}}_{j}(\xi ,Q^{2})\\g(\xi ,Q^{2})\end{pmatrix}}}$

where denotes the splitting functions. Here is the energy scale of the observed process, the momentum fraction of the observed particle in comparison to the mother particle and the parton density function for quarks or that for antiquarks with flavor and that of the gluons . ${\displaystyle P}$ ${\displaystyle Q^{2}}$ ${\displaystyle x}$ ${\displaystyle q_{i}(x,Q^{2})}$ ${\displaystyle {\bar {q}}_{i}(x,Q^{2})}$ ${\displaystyle i}$ ${\displaystyle g(x,Q^{2})}$

Splitting functions

The splitting functions take on four different forms for the possible cases: - A quark emits a quark, - A quark emits a gluon, - A gluon emits a quark and - A gluon emits a gluon. For the splitting functions it is irrelevant whether it is quarks or antiquarks. In addition, the flavor of the quarks is also irrelevant for the gluon-quark splitting functions, while for the quark-quark splitting function only quarks with identical flavors merge. The splitting functions therefore have the form: ${\displaystyle P_{qq}}$ ${\displaystyle P_{gq}}$ ${\displaystyle P_{qg}}$ ${\displaystyle P_{gg}}$

${\displaystyle {\begin{aligned}P_{q_{i}q_{j}}=P_{{\bar {q}}_{i}{\bar {q}}_{j}}\equiv \delta _{ij}P_{qq}&=\delta _{ij}C_{F}\left({\frac {1+x^{2}}{(1-x)_{+}}}+{\frac {3}{2}}\delta (1-x)\right)\\P_{gq_{i}}=P_{g{\bar {q}}_{i}}\equiv P_{gq}&=C_{F}\left({\frac {1+(1-x)^{2}}{x}}\right)\\P_{q_{i}g}=P_{{\bar {q}}_{i}g}\equiv P_{qg}&=T_{F}\left(x^{2}+(1-x)^{2}\right)\\P_{gg}&=2C_{A}\left({\frac {x}{(1-x)_{+}}}+(1-x)\left(x+{\frac {1}{x}}\right)\right)+{\frac {11C_{A}-4n_{f}T_{F}}{6}}\delta (1-x)\end{aligned}}}$

Here are the quadratic Casimir element of the fundamental representation of the Lie group theory, the Standard Model of , the Casimir element of the adjoint representation, the index of the fundamental representation and the number of quark flavors. Also, the plus distribution was used, which is via the equation ${\displaystyle C_{F}=4/3}$ ${\displaystyle SU(3)}$ ${\displaystyle C_{A}=3}$ ${\displaystyle T_{F}=1/2}$ ${\displaystyle n_{f}=3}$

${\displaystyle \int _{0}^{1}{\frac {f(x)}{(1-x)_{+}}}\mathrm {d} x=\int _{0}^{1}{\frac {f(x)-f(1)}{1-x}}\mathrm {d} x}$

is defined.

Alternative basis

Instead of the physical base, the base can be used to simplify the equations . The following applies ${\displaystyle (q_{i},{\bar {q}}_{i},g)}$ ${\displaystyle (q_{i}^{\text{NS}},q_{i}^{\text{S}},g)}$

${\displaystyle {\begin{pmatrix}q_{i}^{\text{NS}}\\q_{i}^{\text{S}}\\g\end{pmatrix}}={\begin{pmatrix}1&-1&0\\1&1&0\\0&0&1\end{pmatrix}}{\begin{pmatrix}q_{i}\\{\bar {q}}_{i}\\g\end{pmatrix}}}$

The superscript denotes the non-singlet density function, while the superscript denotes the singlet density function. In this case the term singlet does not refer to the multiplicity , but to the baryon number , which results in the case of the NS state and in the case of the S state . ${\displaystyle {\text{NS}}}$ ${\displaystyle {\text{S}}}$ ${\displaystyle b=2/3}$ ${\displaystyle b=0}$

Due to the basic transformation, the DGLAP equations decouple insofar as the gluon distribution function is not required to solve the NS distribution functions:

${\displaystyle Q^{2}{\frac {\partial }{\partial Q^{2}}}{\begin{pmatrix}q_{i}^{\text{NS}}(x,Q^{2})\\q_{i}^{\text{S}}(x,Q^{2})\\g(x,Q^{2})\end{pmatrix}}={\frac {\alpha _{s}(Q^{2})}{2\pi }}\sum _{j}\int _{x}^{1}{\frac {\mathrm {d} \xi }{\xi }}{\begin{pmatrix}\delta _{ij}P_{qq}(x/\xi )&0&0\\0&\delta _{ij}P_{qq}(x/\xi )&2P_{qg}(x/\xi )\\0&P_{gq}(x/\xi )&P_{gg}(x/\xi )\end{pmatrix}}{\begin{pmatrix}q_{j}^{\text{NS}}(\xi ,Q^{2})\\q_{j}^{\text{S}}(\xi ,Q^{2})\\g(\xi ,Q^{2})\end{pmatrix}}}$

DGLAP equations in the Mellin space

The DGLAP equations can be represented in a simplified way after a Mellin transformation , since the integral changes into a product in the Mellin space. They are then:

${\displaystyle Q^{2}{\frac {\partial }{\partial Q^{2}}}{\begin{pmatrix}q_{i}(N,Q^{2})\\{\bar {q}}_{i}(N,Q^{2})\\g(N,Q^{2})\end{pmatrix}}={\frac {\alpha _{s}(Q^{2})}{2\pi }}\sum _{j}{\begin{pmatrix}\delta _{ij}\gamma _{qq}(N)&0&\gamma _{qg}(N)\\0&\delta _{ij}\gamma _{qq}(N)&\gamma _{qg}(N)\\\gamma _{gq}(N)&\gamma _{gq}(N)&\gamma _{gg}(N)\end{pmatrix}}{\begin{pmatrix}q_{j}(N,Q^{2})\\{\bar {q}}_{j}(N,Q^{2})\\g(N,Q^{2})\end{pmatrix}}}$

The Mellin transform is given by: ${\displaystyle f(N)}$

${\displaystyle f(N)=\int _{0}^{\infty }\mathrm {d} xx^{N-1}f(x)}$

The functions that occur are called anomalous dimensions and are the Mellin transforms of the splitting functions. ${\displaystyle \gamma }$

Singlet / non-singlet base in Mellin space

The DGLAP equations in the singlet / non-singlet basis read accordingly in the Mellin space

${\displaystyle Q^{2}{\frac {\partial }{\partial Q^{2}}}{\begin{pmatrix}q_{i}^{\text{NS}}(N,Q^{2})\\q_{i}^{\text{S}}(N,Q^{2})\\g(N,Q^{2})\end{pmatrix}}={\frac {\alpha _{s}(Q^{2})}{2\pi }}\sum _{j}{\begin{pmatrix}\delta _{ij}\gamma _{qq}(N)&0&0\\0&\delta _{ij}\gamma _{qq}(N)&2\gamma _{qg}(N)\\0&\gamma _{gq}(N)&\gamma _{gg}(N)\end{pmatrix}}{\begin{pmatrix}q_{j}^{\text{NS}}(N,Q^{2})\\q_{j}^{\text{S}}(N,Q^{2})\\g(N,Q^{2})\end{pmatrix}}}$

solution

With this representation, a compact solution for the DGLAP equations for the non-singlet distribution functions can be given, since the energy scale dependence of the coupling constants is determined by the Callan-Symanzik equation . In leading order applies

${\displaystyle \alpha _{s}(Q^{2})={\frac {\alpha _{s}(\mu ^{2})}{1+b_{0}\alpha _{s}(\mu ^{2})\ln {\frac {Q^{2}}{\mu ^{2}}}}}}$

with a reference scale and a theory-dependent constant ${\displaystyle \mu ^{2}}$ ${\displaystyle b_{0}={\frac {33-2n_{f}}{12\pi }}}$

Then the solution is for the non-singlet distribution function

${\displaystyle q_{i}^{\text{NS}}(N,Q^{2})=q_{i}^{\text{NS}}(N,\mu ^{2})\left(1+\alpha _{s}b_{0}\ln {\frac {Q^{2}}{\mu ^{2}}}\right)^{\frac {\gamma _{qq}}{2\pi b_{0}}}}$

Further information

• ME Peskin, DV Schroeder: An Introduction to Quantum Field Theory . Westview Press, Boulder 1995, ISBN 0-201-50397-2 , pp. 590 ff .

Individual evidence

1. ^ ^{A b} Guido Altarelli: QCD evolution equations for parton densities . In: Scholarpedia . tape 4 , no. 1 , 2009, p. 7124 , doi : 10.4249 / scholarpedia.7124 .
2. ↑ Yuri L. Dokshitzer: Calculation of the Structure Functions for Deep Inelastic Scattering and e ⁺ e ^- Annihilation by Perturbation Theory in Quantum Chromodynamics . In: Sov. Phys. JETP . tape 46 , no. 4 , 1977, pp. 641–653 ( PDF [accessed March 9, 2014]).
3. ↑ V. Gribov, L. Lipatov: Deep inelastic ep scattering in perturbation theory . In: Sov. J. Nucl. Phys. tape 15 , 1972, p. 438-450 .
4. ^ G. Altarelli, G. Parisi: Asymptotic freedom in parton language . In: Nuclear Physics B . tape 126 , no. 2 , 1977, p. 298-318 , doi : 10.1016 / 0550-3213 (77) 90384-4 .
5. ↑ CTEQ Handbook .