On the use of the KMR unintegrated parton distribution functions

We discuss the unintegrated parton distribution functions (UPDFs) introduced by Kimber, Martin and Ryskin (KMR), which are frequently used in phenomenological analyses of hard processes with transverse momenta of partons taken into account. We demonstrate numerically that the commonly used differential definition of the UPDFs leads to erroneous results for large transverse momenta. We identify the reason for that, being the use of the ordinary PDFs instead of the cutoff dependent distribution functions. We show that in phenomenological applications, the integral definition of the UPDFs with the ordinary PDFs can be used.

is used where the UPDFs are obtained by taking the derivative of the integrated PDFs where T a is the Sudakov form factor and D a (x, k ⊥ ) is the integrated parton distribution. This prescription is widely used in phenomenological analyses presented in the literature. It turns out however, that such a prescription leads to some unphysical results for large values of transverse momenta, k ⊥ ≥ Q. For example, we find negative or discontinuous UPDFs in one of the two discussed approximations, when the differential formula (1.1) is used. We identified the reason for such a behaviour and show how to compute the UPDFs which are free of such problems. This paper is organized as follows. In Sec. 2 we recall the KMR construction leading to the differential and integral forms of the UPDFs. In Sec. 3 we discuss two choices of the cutoff, used in the literature. In Sec. 4 we perform the numerical analysis, and illustrate the specific problems with the differential formula for the UPDFs. In Sec. 5 we show the equivalence between the differential and integral forms of the UPDFs using cutoff dependent integrated PDFs. Finally, in Sec. 6 we state our conclusions.

Unintegrated parton distributions
The starting point for the derivation of the KMR UPDFs in [17,18] are the DGLAP evolution equations for the integrated parton distributions D a (x, µ) where a denotes quark flavour/antiflavour or gluon and P aa ′ are the Altarelli-Parisi splitting functions We will consider here LO approximation, but the analysis can be extended to higher orders. The two integrals in Eq. (2.1) are separately divergent for ∆ = 0 due to the singular splitting functions P qq and P gg at z = 1. The first term describes the real emissions in the region µ 2 < k 2 ⊥ < µ 2 + δµ 2 , where k ⊥ is the transverse momentum of the exchanged parton, whereas the second term is responsible for the virtual emissions. In the DGLAP equations these singularities, which are due to soft emissions, cancel when the two terms are combined, through the plus prescription. However, by introducing a parameter ∆, one is able to separate the positive real emission term from the negative virtual emission one, which allows further manipulations leading to the definition of the UPDFs. In particular the choice of the cutoff will be physically motivated, and it will reflect the ordering of the parton emissions.
Let us take for the factorization scale the exchanged parton transverse momentum, µ = |k ⊥ | ≡ k ⊥ , and rewrite Eq. (2.1) in the form Let us also introduce the Sudakov formfactor which has the interpretation of the probability that the parton with transverse momentum k ⊥ will survive (without splitting) up to the factorization scale Q. After multiplying both sides of Eq. (2.3) by the Sudakov form factor, the l.h.s. can be written as a full derivative, Integrating both sides of the above equation over k ⊥ in the interval [Q 0 , Q], where Q 0 is an initial scale for the DGLAP evolution, we find on the l.h.s.
since T a (Q, Q) = 1. Thus, Eq. (2.5) takes the following form This form of Eq. (2.1) may serve as a basis for Monte Carlo simulations of parton branching processes, see for example [21]. The expression in the curly brackets in Eq. (2.7) defines the unintegrated parton distribution functions, (2.9) Formula (2.9) is commonly used to construct the UPDFs from the integrated PDFs, and is referred to as the KMR prescription [17,18]. The discussion of its applicability is the main subject of this paper.
For k ⊥ < Q 0 we need modeling, for example the UPDFs can be defined as below Thus, we assume a constant behaviour of the distribution f a (x, k ⊥ , Q)/k 2 ⊥ as a function of the transverse momentum.

Discussion of the cutoff
In Ref. [17] the cutoff ∆ was set in accordance with the strong ordering (SO) in transverse momenta of the real parton emission in the DGLAP evolution, In the Sudakov form factor (2.4), k ⊥ is replaced by the loop momentum p ⊥ , and where ∆(p ⊥ ) = p ⊥ /Q. Since the integration limits in the real emission term in Eq. (2.8) should obey the condition x < (1 − ∆), the UPDFs are nonzero only for the transverse momenta With such a prescription, we always have k ⊥ < Q and T a (Q, k ⊥ ) < 1.
The prescription for the cutoff ∆ was further modified in Ref. [18,22] to account for the angular ordering (AO) in parton emissions in the sprint of the CCFM evolution [11][12][13][14], In such a case, the nonzero values of the UDPFs are given for The upper limit for k ⊥ is now bigger than in the DGLAP scheme. This is particularly important for small values of x, when k ⊥ < Q/x, which allows for a smooth transition of transverse momenta into the region k ⊥ ≫ Q, see Ref. [18,22] for more details. In this region, we have to decide on the form of the Sudakov form factor (3.2) in which ∆(p ⊥ ) = p ⊥ /(p ⊥ + Q). For k ⊥ > Q, the integration gives a negative value and T a (Q, k ⊥ ) > 1, which contradicts the interpretation of the Sudakov form factor as a probability of no real emission. In the usual approach, the Sudakov form factor is frozen to one Notice that with such a prescription, T a has the first derivative discontinuous at k ⊥ = Q. This effect will be seen in our numerical analysis.

Numerical analysis
Let us discuss the problem of the equivalence of the definitions (2.8) and (2.9) of the UPDFs. For the illustration, we use the unintegrated gluon distribution which is computed in the complete approach with quarks. The integrated PDFs in our numerical analysis are computed using the MSTW08 parametrization [23] of the initial conditions for the DGLAP evolution equations.
In Fig. 1 we show the unintegrated gluon distribution xf g (x, k ⊥ , Q)/k ⊥ 2 as a function of k 2 ⊥ for Q 2 = 100 GeV 2 and x = 10 −3 , 10 −2 , 10 −1 (from the top to the bottom) in the strong ordering (SO) (left plot) and angular ordering (AO) (right plot) approximations for the cutoff ∆. The solid lines are obtained from the integral form (2.8) while the dashed ones are from the differential formula (2.9).
In the SO case, shown on the left, we see a sharp cutoff for the solid curves resulting from condition (3.3). Such a cutoff is not present for the dashed curves computed from Eq. (2.9), which go into the forbidden region, k ⊥ > Q. In this region (4.1) due to condition (3.6), and the integrated gluon distribution on the r.h.s. has no limitations on the maximal value of the hard scale k ⊥ . Clearly, such a behaviour contradicts the assumption on the SO approximation. In the AO case, shown on the right plot in Fig. 1, the distributions from the integral formula (2.8) (solid lines) extend far beyond the point k ⊥ = Q, due to relation (3.5). The unphysical discontinuity at k ⊥ = Q of the distributions from the differential formula (2.9) (dashed lines) is a result of the discontinuity of the first derivative of the Sudakov form factor at this point. Notice also that the lowest lying dashed curve, which corresponds to x = 10 −1 , drops abruptly at k ⊥ = Q. For such a value of x, the integrated gluon distribution D g (x, k ⊥ ) decreases with rising k ⊥ , and its derivative (4.1) becomes negative (∼ −10 −2 ) which leads to a sharp drop on the logarithmic plot. On the other hand, the curves obtained from the integral formula behave in a smooth way without any discontinuities.

Cutoff dependent PDFs
An important question arises here, why the formulae (2.8) and (2.9) for the UPDFs give different results despite their seemingly mathematical equivalence. To answer this question, we have to realize that the equivalence crucially depends on the existence of the cutoff ∆. To compute the UPDFs, we have to solve first Eq. (2.1) (or its equivalent form (2.5)) which gives the cutoff dependent integrated PDFs, D a (x, k ⊥ , ∆). With such distributions, the UPDFs from Eqs. (2.8) and (2.9) will be the same. However, in the numerical analysis in the previous section, we follow the standard approach with the PDFs obtained from the DGLAP evolution equations with ∆ = 0, in which the singularity at z = 1 is regularized by the plus prescription. This is why we find different results for the UPDFs from the two prescriptions.
In order to demonstrate this effect, we solve Eq. (2.5) with the cutoffs in the SO and AO cases. We also use prescription (3.6) for the values of the Sudakov form factor for k ⊥ > Q. In Fig. 2 we show, as an example, the cutoff dependent integrated gluon distribution, xD g (x, k ⊥ , ∆), as a function of the factorization scale k 2 ⊥ for Q 2 = 100 GeV 2 (dashed lines). The ordinary gluon distribution obtained from the DGLAP equations with ∆ = 0 is shown as the solid lines. In the SO case (left plot), we plot the cutoff dependent distribution in the forbidden region, k ⊥ > Q, which is equal to a constant since the r.h.s of Eq. (2.5) vanishes there. Thus, the unintegrated gluon distribution equals zero in this region, which is clearly seen on the left plot in Fig. 3 where we plot the UPDFs obtained from the cutoff dependent PDFs in the SO approximation. Now, we can check that the integral and differential prescriptions for the unintegrated gluon distributions are exactly equivalent, provided the cutoff dependent integrated parton densities are used. This is seen in Fig. 3, where we demonstrate the equality of the results on the unintegrated gluon distribution, xf g (x, k ⊥ , Q)/k 2 ⊥ , obtained from the integral and differential prescriptions of the UPDFs.
Since the parametrizations of the integrated PDFs are only available for the cutoff independent case, it is important to check how numerically big is the effect of the cutoff on the unintegrated distributions. In Fig. 4, we show the comparison of the unintegrated gluon distributions computed from the integral formula (2.8) in the SO and AO cases. The solid curves show the results obtained with the ordinary integrated PDFs while the dashed curves are found with the cutoff dependent parton distributions. As we see, the difference is marginal. Therefore, the standard procedure to compute the UPDFs from the ordinary PDFs is acceptable as long as the integral definition (2.8) is used. The differential form (2.9), however, causes problems for large values of transverse momenta, k ⊥ ∼ Q and should be avoided.

Conclusions
We critically re-examined the derivation and hidden assumptions leading to the UPDFs proposed by Kimber, Martin and Ryskin [17,18], which are commonly used in the phenomenological analyses with parton distributions which additionally depend on parton transverse momementum, k ⊥ . We found that in the standard approach, when the ordinary PDFs found from the global fits to data are used, the definitions (2.8) and (2.9) of the UPDFs give different results in the large transverse momentum region, k ⊥ ∼ Q. In particular, the UPDFs from the differential formula (2.9) extends in the SO approximation into the forbidden region, k ⊥ ≥ Q, and are discontinuous or negative in this region in the AO approximation.
We identified the reason for such a pathological behaviour, being the use of the ordinary PDFs instead of the the cutoff dependent PDFs which guarantee the mathematical equivalence of the two definitions of UPDFs. We demonstrated such an equivalence nu-merically, using the equation (2.1) with the cutoff ∆ in the SO and AO approximations. With the cutoff dependent PDFs, the UPDFs no longer suffer from the described above pathological behaviour.
However, the use of the cutoff dependent PDFs is cumbersome and might spoil the effectiveness of the phenomenological analyses with the KMR UPDFs. The good news is that the UPDFs computed from the formula (2.8) are practically the same, regardless of the choice of the ordinary or cutoff dependent PDFs in the calculations. Thus, as a final conclusion, the KMR UDPFs should only be computed from the integral formula (2.8) in which the PDFs from the global fits can used.