On the dependence of QCD splitting functions on the choice of the evolution variable

We show that already at the NLO level the DGLAP evolution kernel Pqq starts to depend on the choice of the evolution variable. We give an explicit example of such a variable, namely the maximum of transverse momenta of emitted partons and we identify a class of evolution variables that leave the NLO Pqq kernel unchanged with respect to the known standard MS-bar results. The kernels are calculated using a modified Curci-Furmanski-Petronzio method which is based on a direct Feynman-graphs calculation.


In tro d u ctio n
The choice of the evolution variable in the QCD evolution of the partonic densities is one of the key issues in the construction of any Monte Carlo parton shower [1]. The most popular choices are related to the virtuality, angle or transverse momentum of the emitted partons [2][3][4]. At the leading order (LO) level, commonly used for the simulations, the splitting functions are identical for all variables. In this note we investigate whether it is the case also beyond the LO. To calculate the evolution kernels we use slightly modified methodology of the Curci-Furmanski-Petronzio classical paper [5]. It is based on the direct calculation of the contributing Feynman graphs in the axial gauge, cf. [6]. The graphs are extracted by means of the projection operators which act by closing the fermionic or gluonic lines, putting the incoming partons on-shell and extracting pole parts of the expressions.
The distinct feature of this approach is the fact that the singularities are regularized by means of the dimensional regularization, except for the "spurious" ones which are regulated by the principal value (P V ) prescription. To this end, a dummy regulator 5 is introduced with the help of the replacement -In In ^ (ln)2 + 52(pn) 2 ' .
-1 -£ 6 0 (9 T 0 £) 80d3Hr F ig u re 1. Real graphs with double poles contributing to the NLO non-singlet P qq kernel. The solid lines represent quarks and the dotted lines stand for gluons.
The regulator 5 is directly linked to the definition of the P V operation and has a simple geometrical cut-off-like interpretation. This way some of the poles in e are replaced by the logarithms of 5. For more details we refer to the original paper [5] or to later calculations, for example [7][8][9]. The difference of our method with respect to the approach of [5] is the use of the New P V (N PV) prescription which we have introduced in [10,11]. N PV amounts to the extension of the geometrical regularization to all singularities in the light cone 1+ variable, not only to the "spurious" ones. This modification turns out to be essential, as it further reduces the number of higher-order poles in e by replacing them with the log 5 terms, and simplifies the contributions of the individual graphs.
There are three mechanisms which keep the kernel invariant under the change of the cut-off: (1) Invariance of a particular diagram. This applies to all diagrams with the single poles in e. (2) Pairwise cancellation between the matching real and virtual graphs, as in Vg and V f graphs of figure 1. (3) Cancellation between a graph and its counter-term. This is the case for ladder graphs. We will demonstrate that the mechanism (2) can fail already at the NLO level.
Our plan is the following. We will individually analyse the most singular diagrams con tributing to the Pqq kernel. There are three graphs with second-order poles in e contributing to the kernel; they are depicted in figure 1. We will calculate the difference between the kernel with the virtuality cut-off -q2 < Q2, as in the original paper [5], and with a set of different cut-offs. The cut-offs we consider are: the maximum and the scalar sum of the transverse momenta of the emitted partons, i.e. m a x jk^,k 2± } and k 1± + k 2±, as well as the maximum and the total rapidity of the emitted partons, i.e. m a x [k 1± / a 1, k 2± / a 2} and \k1± + k 2±\/(a1 + a 2) .1 The calculation will show th at three of these cut-offs leave the kernel unchanged with respect to the standard MS result, whereas, the one on the maxi mum of the transverse momenta leads to the change of the kernel. We will demonstrate in detail the mechanism of this change and we will formulate a general rule to identify cut-offs leading to it.
We will start with the diagram named Vg and its sibling Vf. Next, we will discuss the ladder graph B r and its counter term, Ct. Our analysis will demonstrate that only the Vg and V f diagrams depend on the chosen cut-off variable. In the case of the ladder graph the counter term cancels the dependence. Finally, we will comment on why the graphs with 1We define ki± = \ki±\.
-2-£ 6 0 (9 T 0 £) 80d3Hr F ig u re 2. The graph Yg contributing to the NLO non-singlet P qq kernel. The solid lines represent quarks and the dotted lines stand for gluons.
only single e poles do not contribute. This is also the reason why N PV is instrumental: it replaces 1/e3 poles of the diagram Y g (depicted in figure 2) by the single poles and logarithms of the regulator 5. As a consequence, this diagram does not contribute in N PV, whereas it would have a nontrivial contribution in the original P V prescription.

D iag ram V g
In order to establish our notation and conventions, we give explicitly the starting formula for the contribution of the diagram Vg, corresponding to figure 1: f l -^m-2 The benefit of these variables is the diagonal form of the variables k 2 and q2 in which our formula is singular: The trace Wg is of the form (9 is the angle between Ki and K2) 8 I k 2 k 2 k 2 . Wg = ---------"2 ~2Tgc2 cos2 9 + * / -2 Tgc cos 9 + -2 T g k + T :n ) , (2.12) (2.15) a2 2 ) x 2 This allows us to rewrite formula (2.1) as -4 -£ 60(9 T 0 £) 80d3Hr

Cut-off on m a x (fc n ,fc 2 i } < Q
In eq. (2.19) we have shown only the singular parts of the integrand. The singularities of the integral are located at k2 = K2 = 0, i.e. at k2 = 0 and at -q2 = c2k2 + c2«2 = 0 i.e. at k \ = k2 = 0. As we can see from (2.19) , the q2 = 0 area is excluded due to the subtraction of the r G( -q2 < Q2) which is available in the literature [5 , 8]. The external integrals over d a cannot contribute additional 1/e poles as they are regulated by the N PV prescription.
This is one of the two key ingredients of the calculation. Since we are interested in the pole part of A r , we can expand the dn2 integrand in a standard way: The -q2 > Q 2 translates into (see eq. (2.11)) Comments are in order regarding the integration limits for both of the angular integrals.
One of the angles is trivial and covers the entire range ( 0 ,2n), as the system has rotational symmetry. The other angle, 0, between K1 and K2, has a non-trivial integration range, which depends on the kappas and alphas. However, there is a subspace where this angle is also unlimited. It is given by the conditions It ju st happens that in the limit k2 = 0 eq. (2.25) coincides with the entire range of K1.
This way we find (c0 = a 2/ a 1) Going back to eq. (2.22) we obtain (2.28) Performing the a -integrals we find where the symbol I0 denotes the IR-divergent integral regularized by means of the P V prescription with the geometrical 5 parameter: The result (2.29) differs from the shift in virtual corrections shown later in section 4 . We have obtained a net change of the kernel.

Cut-off on + k2± < Q
We have demonstrated in the previous section th at the change of real and virtual Vg-type diagrams do not compensate each other. L e t's consider the virtual correction Vg, figure 3 .
F ig u re 3. Real-virtual graph Vg contributing to NLO non-singlet P qq kernel. The solid lines represent quarks and the dotted lines stand for gluons.
The graph has one real gluon, labelled k, and the cut-off is unique and trivial: k± < Q.
However, if we look inside the graph we find two virtual momenta, k 1 and k2, such that k1 + k2 = k. Therefore, our k^-cut-off at the unintegrated level is \k1± + k 2±\ < Q.
This cut-off can be problem atic for the real gluons because it does not close the phase space. We will get back to this issue in the next paragraph. For now, let us note that, as argued in section 2.1, we calculate only the difference between the q2 and k± cut-offs.
This way we reproduced result (2.29) , but without the additional constant terms. It is identical to the change in the virtual corrections and there is no modification of the kernel.

Cut-off on |ki^ + k2±_\ < Q
Let us come back to the cut-off on the vector variable \k1± + k2±\ < Q. It indeed allows for the arbitrarily big values of \ki±\. The question is however whether it leads to well-defined and meaningful kernels? We will argue th at it does. 1 -, a 1x Translated into the K-variables of eq. (2.8) , the cut-off is simply k 1 < a i /(1x)Q, identical to the one of section 2.2. The K2 = Ki -k i. variable is unbounded because so is k i . (the k2. can always be adjusted to fulfill the cut-off) and the angle is also unlimited, 0 < d < 2n. Keeping in mind the discussion on the origin of the poles given around eq. (2.21) , we conclude that the upper limit on k2 does not m atter at all, and we can set it to infinity as well. Repeating all the steps of section 2.2 we recover the result (2.34) . In other words, we have ju st shown that the cut-off |ki. + k2.| < Q leads to a proper kernel.
One may be worried weather the higher order terms of the e-expansion of eq. (2.21) are finite. To answer this question let us inspect the original equations (2.1) and (2.12) . In the limit k2 ^ ro we have -q2 ~ (1 -x )x / (a i a 2) «2 and we find the integrals of the type / dKH ( K p , ( K i F 5 , ( K p ; , (2 '35) which are integrable at the infinity. We conclude th at the e expansion of eq. (2.21) is legit im ate and the cut-off |ki. + k2.| < Q is self consistent. The open question is though how will this cut-off perform with other graphs. Another question concerns its generalization to more than two real partons.

Cut-off on rapidity
Let us briefly comment on the cut-off on rapidity. B y rapidity we understand the quan tity a = |k.|/a (massless) or a = \J|k.|2 + k2/a (massive). For the case of two emis sions the analogy to virtual graph leads to a = |ki. + k2.| / (a i + a 2) < Q or a = \J|ki . + k2.| 2 + (k i + k2) 2/( a i + a 2) < Q. In the subspace k2 ~ k2 = 0 both formulas coincide and both are identical to the k .-ty p e formula with the cut-off Q shifted to Q(1 -x) in the k .-ty p e formula. This is ju st the result we have obtained for the virtual corrections.
Another option is m a x {a i , a 2} < Q. One has a i = (Ki -K2)/ a i and a 2 = Ki / a i + K2/ a2. At k2 = 0 this leads to Ki / a i < Q or equivalently |ki . + k2.| / (a i + a 2) < Q. This is identical to the previous case, so we expect the result to be in agreement with the virtual correction as well.

General rule
We can now generalize the analysis of the previous sections and formulate a more universal rule for identifying the variables that do or do not change the NLO kernel.

D iag ram V f
Let us now perform the analysis of the V f graph. It will heavily rely on the analysis done for the Vg graph. Let us begin with the m a x {k 2y ,k 2y } calculation. Our starting point is -9 -£60(9 T 0 £)80d3Hr the diagram depicted in figure 1. The analytical formula is analogous to eq. (2.1) : v2 = 342 P n n 1 ,2 ( 4 T fc2 cos2 0 + ,/ 4 T fc cos 0

V irtu a l d iag ram s
The shift in the virtual corrections due to the change of the cut-off can be found in ref. [9].
The ^-dependence of each diagram is given there. One finds th at there is no ^-dependence for the C F-type graphs and the only ones th at do depend on a are Vg and Vf, see eqs.
(4.25) and (4.31) in ref. [9]. Here we quote the change with respect to the virtuality case: Let us combine also the a -type cut-offs for the real V f+ V g graphs (5.3)

A d ded real and v irtu a l d iag ram s
We can now add changes of the real and the virtual V f+ V g graphs. For the a -type cut-offs we observe th at the contributions cancel each other and there is no net effect, as expected.

B r (lad d er) g rap h and co u n te r te rm
We now turn to the ladder graph and the counter term associated with it, shown in figure 1.
B o th of them have double e poles and therefore can be modified once the evolution variable changes. However, we will demonstrate that their difference remains unchanged. As before, we will calculate only the difference w .r.t. the result with cut-off on the virtuality, -q2 < Q 2. Therefore, the pole coming from the 1/q2 integrand is eliminated and we are forced to keep only terms th at generate the e pole from the dk2± integral. This means that we keep only T2, set to zero all other e-terms and expand dk^-in tegral, i.e. (7.8) This way we obtain (7.9) -12 -£ 6 0 (9 T 0 £) 80d3Hr a 1 a 1 Pqq(max{k1± ,k2± } < Q) -Pqq( -q2 < Q 2) = ( a s j 2 1 + x 2 \^ ( 2n2 rB q r =^4p4 / § d% x / f kirTi (e= 0 ) . L -q2>Q2 J The matching counter term r B . differs only by the "split" of the trace Wffr and an additional projection operator. The projection operator performs two actions: picks the epoles and sets on-shell the incoming quark (q1 in our case). These are minor modifications to (7.1) , ( 7.3) : (7.10) where and thanks to the condition qj2 = 0: We obtain (7.13) (7.14) It is easy to verify now th at these two quantities, r Br and r B . , are identical under the conditions ( 7.8) and the net change of the kernel is zero.
In appendix A we evaluate the change of the ladder graph alone caused by the change of the cut-off. This quantity is of interest, for example, in the construction of Monte Carlo algorithms.

C onclusions
In this paper we have discussed the change of the D G LA P kernel P qq due to the change of the evolution variable within the C F P scheme. We have demonstrated th at at the NLO level m ajority of the choices of the evolution variables lead to the same kernel, but there are ones, like the maximal transverse momentum, th at correspond to the modified kernel. We have explained the mechanism responsible for the change and we have formulated a simple rule to identify classes of variables th at leave the kernel unchanged at the NLO level.
There is an im portant open question related to our analysis: is the kernel dependence specific to the C F P method and specifically to the presence of the geometrical cut-off 5?
If all the singularities, including the "spurious" ones, were regulated by the dimensional -13 -£ 6 0 (9 T 0 £) 80d3Hr r Ct = C 2 g4 x P P I"(2n) 26 (1 + x2 + e(1 -x i ) 2), regulator has physical consequences. The same question holds for the modification of the original P V prescription of [5] to the N PV one used in this note.
Of course, this question can be addressed also from the perspective of different methods which employ calculation of the total cross sections for physical processes to obtain splitting functions. Such a viewpoint would allow us to interpret our result in terms of a finite scheme transform ation. This however, goes beyond the scope of the current work and we leave it for a future study. Our current results are valid within the C F P method.

A C han ge of ladder g rap h w ith cu t-o ff
In the appendix we calculate the change of the r Br for various cut-offs as it can be useful in constructing Monte Carlo algorithms. Let us continue with eq. (7.1) and let us implement the conditions (7.8): The upper limit depends on the chosen evolution variable. We will examine a few cases. eq. (7.1) transforms now into A r B -"2 = C F ( ) 2 /^lnl / y 1 2e5l-' -" i -"2x?(1+ x1)(x2+ x2) .

<A -5>
Let us continue with each case separately.