Table of Contents
uPDFevolv is an evolution code for TMD parton densities using the CCFM evolution equation.
At high energies collisions of hadrons are described by parton densities dependent on the longitudinal momentum fraction , the transverse momentum and the evolution scale (transverse momentum dependent (TMD) or unintegrated parton density functions (uPDF)). The evolution of the parton density with the scale valid at both small and moderate is given by the CCFM evolution equation
QCD calculations of multiple-scale processes and complex final-states require in general transverse-momentum dependent (TMD), or unintegrated, parton density and parton decay functions,,,,,,,,, . TMD factorization has been proven recently} for inclusive and semi-inclusive deep-inelastic scattering (DIS). For special processes in hadron-hadron scattering, like heavy flavor or heavy boson (including Higgs) production, TMD factorization holds in the high-energy limit (small ),, .
In the framework of high-energy factorization, the deep-inelastic scattering cross section can be written as a convolution in both longitudinal and transverse momenta of the TMD parton density function with off-shell partonic matrix elements, as follows
with the DIS cross sections ( ) related to the structure functions and by . The hard-scattering kernels of Eq.(1) are -dependent and the evolution of the transverse momentum dependent gluon density is obtained by combining the resummation of small- logarithmic contributions,, with medium- and large- contributions to parton splitting,, according to the CCFM evolution equation,.
The factorization formula (1) allows one to resum logarithmically enhanced contributions to all orders in perturbation theory, both in the hard scattering coefficients and in the parton evolution, taking fully into account the dependence on the factorization scale and on the factorization scheme,.
The CCFM evolution equation,, is an exclusive equation for final state partons and includes finite- contributions to parton splitting. It incorporates soft gluon coherence for any value of .
The evolution equation for the TMD gluon density , depending on , and the evolution variable , is
where is related to the angle and -function specifies the ordering condition of the evolution.
The first term in the right hand side of Eq.(2) is the contribution of the non-resolvable branchings between the starting scale and the evolution scale , and is given by
where is the Sudakov form factor, and is the starting distribution at scale . The integral term in the right hand side of Eq.(2) gives the k_t-dependent branchings in terms of the Sudakov form factor and unintegrated splitting function . The Sudakov form factor is given by
where the splitting variable is given by , , and is the azimuthal angle of .
For the evolution of the parton densities, however, a forward evolution approach, starting from the low scale towards the hard scale , is used.
The splitting function for branching is given by (set by
where is the non-Sudakov form factor defined by
In addition to the full splitting function, simplified versions are useful in applications and are made available. One uses only the singular parts of the splitting function (set by
Another uses also for the small part (set by
In general a four-momentum can be written in light-cone variables as with and being the light-cone components and being the transverse component. The CCFM (as well as the BFKL) evolution depends only on one of the light-cone components. Assuming that the other one can be neglected, this leads to the condition that the virtuality of the parton propagator should be dominated by the transverse component, while the contribution from the longitudinal components is required to be small. The condition that leads to the so-called consistency constraint, which has been implemented in different forms (set by
where is the evolution scale. The quark splitting function is given by
In Eqs.~(4,5) the non-Sudakov form factor is not included, unlike the CCFM kernel given in the appendix~B of, because we only associate this factor with terms. The term in Eq.~(4) is the contribution of the non-resolvable branchings between starting scale and evolution scale , given by
where is the Sudakov form factor.
For a complete description of the final states also the contribution from sea-quarks needs to be included. We include splitting functions according to
with , and .
The splitting has been calculated in a - factorized form in,
with , and being the transverse momentum of the quark (gluon).
The evolution equation for the TMD sea-quark density , depending on , and the evolution variable , is
where is the non-resolvable branching probability similar to Eqs.(3).
The evolution of the TMD gluon density including the contribution from quarks is given by
The evolution equations Eqs.(6,7) are integral equations of the Fredholm type
and can be solved by iteration as a Neumann series
using the kernel , with the solution
Applying this to the evolution equations Eqs.(6,7), we identify with the first term in eqs.(7), where we use for simplicity here and in the following :
The first iteration involves one branching:
The second iteration involves two branchings,
In a Monte Carlo (MC) solution, we evolve from to a value obtained from the Sudakov factor . Note that the Sudakov factor gives the probability for evolving from to without resolvable branching. The value is obtained from solving for :
for a random number in .
If then the scale is reached and the evolution is stopped, and we are left with just the first term without any resolvable branching. If then we generate a branching at according to the splitting function , as described below, and continue the evolution using the Sudakov factor . If the evolution is stopped and we are left with just one resolvable branching at . If we continue the evolution as described above. This procedure is repeated until we generate . By this procedure we sum all kinematically allowed contributions in the series and obtain an MC estimate of the parton distribution function.
With the Sudakov factor and using
we can write the first iteration of the evolution equation as
The integrals can be solved by a Monte Carlo method : is generated from
with being a random number in , and is generated from
solving for , using from above and another random number in [0,1].
This completes the calculation on the first splitting. This procedure is repeated until and the evolution is stopped.
With and selected according to the above the first iteration of the evolution equation yields
The valence quark densities are normalised so that they fulfil for every the flavor sum rule.
The gluon and sea quark densities are normalised so that for every
When using the CCFM evolution in a fit program to determine the starting distribution , a full MC solution, is no longer suitable, since it is time consuming and suffers from numerical fluctuations. Instead a convolution method introduced in is used. The kernel is determined once from the Monte Carlo solution of the CCFM evolution equation, and then folded with the non-perturbative starting distribution ,
The kernel incorporates all of the dynamics of the evolution, including Sudakov form factors and splitting functions. It is determined on a grid of bins in . The binning in the grid is logarithmic, except for the longitudinal variable where we use 40 bins in logarithmic spacing below 0.1, and 10 bins in linear spacing above 0.1.
Using this method, the complete coupled evolution of gluon and sea quarks is more complicated, since it is no longer a simple convolution of the kernel with the starting distribution. To simplify the approach, here we allow only for one species of partons at the starting scale, either gluons or sea-quarks. During evolution the other species will be generated. This approach, while convenient for QCD fits, has the feature that sea-quarks, in the case of gluons only at , are generated with perturbative transverse momenta ( ), without contribution from the soft (non-perturbative) region.
For the starting distribution , at the starting scale , the following form is used:
with and free parameters .
with . At every scale the flavor sum rule is fulfilled for valence quarks.
A saturation ansatz for the starting distribution at scale is available, following the parameterisation of the saturation model by Eq.(18) of,
with . The free parameters are , , and . In order to be able to use this type of parameterisation over the full range, an additional factor of (see) is applied.
A simple plot program is included in the package. For a graphical web interface use TMDplotter.
uPDFevolv follows the standard AUTOMAKE convention. To install the program, do the following
The executable is in bin
run it with:
plot the result with:
We are very grateful to Bryan Webber for careful reading of the manuscript and clarifying comments.
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