Treatment of heavy quarks in QCD

Abstract

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

We show that to correctly describe the effects of the heavy-quark mass, m ^sub h^, in DGLAP evolution, it is necessary to work in the so-called 'physical' scheme. In this way, we automatically obtain a smooth transition through the heavy-quark thresholds. Moreover, we show that to obtain NLO accuracy, it is sufficient to account for the heavy-quark mass, m ^sub h^, just in the LO (one-loop) splitting function. The use of the ... factorization scheme is not appropriate, since at NLO we deal with a mixture of quarks and gluon (that is, the mass of the heavy parton is not well-defined). The formulas for the explicit m ^sub h^ dependence of the splitting functions, and for [alpha] ^sub s^, are presented.

Eur. Phys. J. C (2013) 73:2616DOI 10.1140/epjc/s10052-013-2616-2

Regular Article - Theoretical Physics

Treatment of heavy quarks in QCD

E.G. de Oliveira1,2, A.D. Martin1,a, M.G. Ryskin1,3, A.G. Shuvaev3

1Institute for Particle Physics Phenomenology, University of Durham, Durham DH1 3LE, UK

2Instituto de Fsica, Universidade de So Paulo, C.P. 66318, 05315-970 So Paulo, Brazil

3Petersburg Nuclear Physics Institute, NRC Kurchatov Institute, Gatchina, St. Petersburg, 188300, Russia

Received: 26 August 2013 / Published online: 25 October 2013 Springer-Verlag Berlin Heidelberg and Societ Italiana di Fisica 2013

Abstract We show that to correctly describe the effects of the heavy-quark mass, mh, in DGLAP evolution, it is necessary to work in the so-called physical scheme. In this way, we automatically obtain a smooth transition through the heavy-quark thresholds. Moreover, we show that to obtain NLO accuracy, it is sufcient to account for the heavy-quark mass, mh, just in the LO (one-loop) splitting function.

The use of the MS factorization scheme is not appropriate, since at NLO we deal with a mixture of quarks and gluon (that is, the mass of the heavy parton is not well-dened). The formulas for the explicit mh dependence of the splitting functions, and for s, are presented.

1 Introduction

The correct treatment of heavy quarks in an analysis of parton distributions is essential for precision measurements at hadron colliders. The up, down, and strange quarks, with m2 2QCD, can be treated as massless partons. However,

for charm, bottom or top quarks we must allow for the effects of their mass, mh with h = c, b or t. The problem is

that we require a consistent description of the evolution of parton distribution functions (PDFs) over regions which include both the Q2 m2h domain and the region Q2 m2h

where the heavy quark, h, can be treated as an additional massless quark.

Let us briey summarize how heavy quarks are treated in PDF analyses at present.1These analyses are performed in the MS scheme, in which the splitting and coefcient functions have been calculated using dimensional regularization. We will call this the conventional approach. Starting the evolution at a low scale we need consider only the three light quarks, taken as massless. As we evolve upwards we reach

1A detailed review can be found in [1].

a e-mail: [email protected]

the charm quark threshold Q2 = m2c. We could choose to

keep just the three light avors as quark PDFs, and include all the effects of the charm quark and its mass mc in the coefcient functions. Historically, higher-order calculations of charm production were done in such a so-called Fixed-Flavor-Number-Scheme (FFNS) [2].

Unfortunately, a FFNS cannot be used far above the threshold. For ln Q2 ln m2c, the charm quark starts to par

ticipate in the evolution. Therefore in the FFNS coefcient functions, we have to sum up an innite number of diagrams in order to reproduce inside these functions all the missing effects in the DGLAP evolution. Indeed, higher-order contributions, ns lnn(Q2/m2c), do not decrease in comparison to lower-order terms, and perturbation theory breaks down.

Here, we call this a 3-avor scheme (3FS). For higher Q2 we should include the c-quark (which is taken as massless) in the evolution and so generate2 a 4FS giving reliable results for m2c [lessorsimilar] Q2 [lessorsimilar] m2b, and 5FS giving reliable results for m2b [lessorsimilar] Q2 [lessorsimilar] m2t and so on.

Hence, we are led to a more general Variable-Flavor-Number-Scheme (VFNS), which is a composite of a sequence of nf -avor schemes, each with its own region of validity. As we pass through each transition point, Qtrans

(usually taken as Q2 = m2h), the number of quarks active

in the evolution increases from nf to nf + 1. So at a tran

sition point we have two different sets of PDFs: the nf -FS set for Q2 m2h and the (nf + 1)-FS set for Q2 m2h. The

two sets have to be matched together in the transition region. The matching conditions are

an+1i[parenleftbig]Q2[parenrightbig] = [summationdisplay]

k

Aik[parenleftbig]Q2/m2h[parenrightbig] ank[parenleftbig]Q2[parenrightbig], (1)

where denotes the convolution A a =

A[parenleftbig]x [parenrightbig] a[parenleftbig]x/x [parenrightbig], (2)

2There are special processes where a 4-avor set of partons is still necessary, see, for example [3].

[integraldisplay]

1

dx x

x

Page 2 of 7 Eur. Phys. J. C (2013) 73:2616

where the PDF set ani = g, q, h, with 3 light quarks and

n 3 heavy quarks h. We have suppressed the x arguments

in (1). The perturbative matrix elements Aik(Q2/m2h) contain ln(Q2/m2h) terms known to O(2s).3 In summary, the

various nf schemes are related to each other by perturbatively calculable transformation matrices between the PDFs and the coefcient functions.4

Note that the matrix A in (1) is not a square matrix; neglecting the NNLO correction there is some freedom in performing the matching at the transition points, which is exploited phenomenologically to ensure that the matching is as smooth as possible. The ACOT [4] and RT [5, 6] prescriptions were early attempts to implement this matching.An important development was the use of the so-called the General Mass (GM)-VFNS, which allows an estimate of the suppression of the nal-state phase space when heavy avor is produced. A rescaling variable

= x

[parenleftbigg]1 +

([summationtext] Mf )2

Q2

[parenrightbigg] (3)

is introduced, where the sum is over the heavy particles produced in the nal state. (For example, neutral-current heavy-avor production has h h in the nal state, whereas

a charge-current process has a single h.) Then the convolution CPDF, with the corresponding coefcient function,

C, should be integrated over the momentum fraction range < < 1. Rescaling shifts the momentum fraction variable in the PDF, a(, 2), to a higher value than in the zero-mass case. This rescaling prescription [7, 8] is known as ACOT.

The GM-VFNS is adopted in the MSTW [9], CTEQ (CT10) [10] and NNPDF [11] global parton analyses, although each analysis uses its own variant. For example, MSTW use the formalism of [12, 13], while NNPDF use a prescription [14] based on the xed-order next-to-leadinglog (FONLL) method. For comparison the most recent FFNS analysis [15] nds both the value of s(M2Z) and the size of the gluon PDF at large x, signicantly smaller than those of the GM-VFNS analyses.

The VFNS is well justied at LO accuracy. Indeed, at LO, in each cell (loop) of the evolution diagram, the transverse momenta, kti, or the virtualities, k2i, are strongly ordered; k2i k2i1 and a large logarithmic integration [integraltext]

k2i+1 k2i1

dk2i/k2i

compensates the small value of the QCD coupling s(k2i). The contribution from a nite interval of ln(k2i) (say, k2i

k2i+1) is considered as a NLO correction to the Leading-Log

evolution since here (from this extra loop) we get a small s now unaccompanied by a large logarithm. In the same way,

3In general the matching may be performed at any Q2 = c m2h, where

the value of c is c [greaterorsimilar] 1. Recall, however, that actually the matching (1) is done at one, xed point Q2 = Q2trans, say Q2 = m2h.

4In analogous way the smooth behavior of the coupling s(Q2) is provided.

we have to treat the heavy-quark mass dependence, which comes only from the nite region of ln(k2i) (that is, from k2i m2h) as a NLO effect. Correspondingly, the effect of

the running mass is a NNLO contribution.

If we account for the NLO corrections within the VFNS, where at each threshold, Q2 = m2h we just increase the num

ber of light active quarks by 1 (but each type of quark is considered as massless in the evolution), then, as mentioned above, we get JUMPs in the splitting functions (and kinks in s) when the value of nf is changed. This behavior is compensated by the matching condition (1). The effect of the kink is calculated and added to the NLO PDFs in such a way as to provide the correct behavior for Q2 m2h, assum

ing that there is only one threshold in this interval of evolution. The remaining kink in the derivative may be considered as a NNLO effect (and, in its turn, it can be compensated for in the region Q2 m2h at the NNLO level, again assuming

that there is only one threshold in this interval of evolution).

The GM-VFNS allows us to correctly reproduce the evolution in a large ln Q2 interval, but it cannot describe precisely the behavior in the regions around the heavy-quark thresholds Q2 m2h. Such an approach does not remove

the jumps in the splitting functions at the transition point, Q = Qtrans. The kinks in these domains are only compen

sated in some average sense.

In this paper, we propose a completely different, physically motivated, approach, which automatically results in a smooth behavior of the PDFs, the coefcient functions and of s as the scale 2 passes through each heavy-quark threshold. In this physical approach the partons which occur in the Feynman diagrams are the basic entities. However, before we describe our approach, a comment about the conventional MS scheme is needed.

It was shown in [16] that the NLO coefcient functions, CNLO, obtained within the conventional MS prescription using the dimensional (D = 4 + 2 ) regularization, are dif

ferent from the results calculated in the physical approach of working in normal D = 4 space where the infrared di

vergency is removed by an appropriate subtraction of the contribution, CLO P LO, generated by the iteration of LO

evolution.

The above difference, C, is due to an / contribution coming from very large (non-physical) distances. It can be written as the convolution

Cik = CLOik k k(z), (4)

where ik(z) denotes the part of the LO splitting functions that is proportional to

Pik(z) = P LOik(z) + ik(z). (5) This contribution can be absorbed in the redenition of the partons, a(x, 2) = xg, xq, xh

Eur. Phys. J. C (2013) 73:2616 Page 3 of 7

aphys[parenleftbig]x, 2[parenrightbig] = aMS[parenleftbig]x, 2[parenrightbig]

+

s 2

ab(z) bMS[parenleftbig]x/z, 2[parenrightbig]. (6)

Correspondingly, there is a difference between the NLO splitting functions in DGLAP evolution equation obtained in the conventional MS scheme and the physical approach, see [17]. As seen from (6), at NLO, the conventional MS partons are rotated with respect to the physical partons by some angle. In particular, the singlet-quark distribution gets an admixture of gluons.5 This mixture greatly complicates the calculations of heavy-quark mass effects, and any other Feynman graph calculations, beyond LO, in the MS scheme. By working in the physical approach, where we calculate the explicit mh dependence of the DGLAP splitting functions (and of s), we obtain a well-dened, and simplied treatment of heavy-quark mass effects to NLO accuracy.

In fact, in the present paper we show how to account for the heavy-quark mass already in the LO splitting function. To do this we calculate the explicit m2h/Q2 dependence of the derivatives of the PDFs, a(x, Q2)/ ln Q2; instead of using the conventional splitting functions, which only depend on Q2 via the running coupling s(Q2). In Sect. 2 the corresponding splitting functions are calculated from the one-loop (i.e. LO) Feynman diagrams. We discuss whether, accounting for the mass effect already at the LO level, we have to correct the usual NLO splitting and the coefcient functions. In Sect. 3 we obtain an analogous LO formula which gives the effects of the heavy-quark masses to the running of s at NLO. This provides a smooth behavior of s across the heavy-quark thresholds. All the calculations are done in the physical scheme; so in the Appendix we present the formulas to provide the rotation from MS to physical scheme, and vice versa. We present our conclusions in Sect. 4.

2 Heavy-quark mass effects already included at LO

Since the heavy-quark mass effects come only from a nite interval of the ln Q2 evolution, to reach the NLO accuracy it is sufcient to account for mh only in the LO diagrams.

We will see that keeping the mass in the NLO (two-loop) graphs leads to a NNLO correction. As usual we use the axial gauge, where only the ladder (real emission) and the

5This rotation is compensated by a corresponding rotation back arising from the difference in the splitting functions. For heavy quarks we do not have an infrared problem. However, in general, using the conventional MS scheme, we do not know the mass of the parton that we are dealing with. It is therefore difcult to account for mass effects in the MS scheme.

Fig. 1 Part of the parton evolution chain which contains the g h h transition

[integraldisplay] dz [summationdisplay]

b

self-energy (virtual-loop contribution) diagrams give Leading Logarithms. Actually, for real emission we need to consider only the gluon-to-heavy-quark splitting function. Indeed the heavy-quark mass effects can be identied in the following subset of integrations:

[integraldisplay] dk2i1 k2i1

[integraldisplay] dk2ik2i

(k2i + m2h)2

[integraldisplay] dk2i+1 k2i+1

(7)

corresponding to the part of the parton chain containing the g h h transition, as shown in Fig. 1. The k2s are the

virtualities of the t-channel partons, and the heavy-quark mass effects enter in the k2i integration that results from the g h h transition. The kinematics responsible for the

LO result are when the virtualities are strongly ordered ( k2i1 k2i k2i+1 ). If two of the partons have com

parable virtuality, k2j k2j+1, then we lose a ln Q2 and ob

tain a NLO contribution of the form s(s ln Q2)n1 for n emitted partons.

At rst sight it appears that m2h should also have been retained in the integration over the heavy-quark line with virtuality ki+1. However, the heavy quark was produced at

Q2 m2h via the g h splitting. Due to the strong ordering

k2i+1 k2i in the evolution chain, we have k2i+1 m2h, and

so we may neglect m2h in the k2i+1 integration; otherwise this

would be the NNLO effect.

Note that in our NLO calculations, described below, we use a xed number mh(mh) for the heavy-quark mass.6 All the effects of the running quark mass should be regarded as part of the NNLO corrections.

2.1 Quark mass effects in the LO splitting functions

We are now in a position to calculate the heavy-quark mass effects in real LO h ( h) production which determines the ex

plicit m2/h2 dependence of Phg. This, in turn, allows us to account for the mh dependence in the heavy-quark virtual-loop contribution (that is, in the gluon self-energy), which gives an additional term in Pgg, which is proportional to

6Strictly speaking we may choose any reasonable xed value for mh, say mc (1.4 GeV), so that the NNLO correction is not large.

Page 4 of 7 Eur. Phys. J. C (2013) 73:2616

(1 z). Recall that the full g g splitting function has

the form

Pgg(z) = P realgg(z) (1 z)

P realhh[parenleftbig]z, Q2[parenrightbig] = CF

1 + z2

1 z

Q2 m2h + Q2

z(1 3z)

+ 1 z

Q2m2h

(Q2 + m2h)2

[parenrightbigg] (12)

[integraldisplay] 1 [parenleftbigg]z P real[parenleftbig]z [parenrightbig] + [summationdisplay]

f

Pqg[parenleftbig]z [parenrightbig][parenrightbigg] dz , (8)

where the [summationtext]f includes the summation over different type of quarks. Correspondingly

Phh(z) = P realhh(z) (1 z)

and for the h g transition

Pgh[parenleftbig]z, Q2[parenrightbig] = CF

1 + (1 z)2

z

0

Q2 m2h + Q2

+

1

z2 + z 2 z

Q2m2h

(Q2 + m2h)2

[parenrightbigg]

[integraldisplay] 0 P realgh[parenleftbig]z [parenrightbig] dz . (9)

To determine the m2h/Q2 dependence of Phg we must calculate the one-loop ladder (heavy-quark box)7 diagram. We denote the virtuality of the t-channel heavy quark h by k2i, as in Fig. 1. Now strong ordering means that the virtuality of incoming gluon k2i1 k2i and that k2i k2i+1. We nd

that the mh dependence of the LO gluon-to-heavy-quark splitting function, Phg, is

Phg[parenleftbig]z, Q2[parenrightbig] = TR

[parenleftbigg][bracketleftbig]z2 + (1 z)2[bracketrightbig]

Q2 m2h + Q2

[parenrightbigg]. (13)

We may summarize the LO evolution equations in the symbolic form

g = Pgg g + [summationdisplay]

q

Pgq q + [summationdisplay]

h

[parenleftbigg]Q2

zm2h 1 z

[parenrightbigg] (10)

where TR = 1/2. The rst term is the usual LO splitting

function Phg modied by a factor, Q2/(Q2 + m2h), which

tends to 1 for Q2 m2h, while for low Q2, Q2 m2h, this

contribution becomes negligible. The second term, proportional to m2h, accounts for the possibility to ip the helicity in the heavy-quark loop. It dies out for Q2 m2h. Finally,

the function accounts for the correct kinematics of heavy-quark production. We need energy to put the heavy-quark on-mass-shell. This leads to a minimum value of the (longitudinal part of) Q2.

Simultaneously we have to include heavy-quark loops in the gluon self-energy, as was mentioned in (8). That is, we must add a term to the gluongluon splitting function, Pgg,

Pgg = (1 z) [summationdisplay]

h

Pgh h,

q = Pqg g + Pqq q, (14)

h = Phg g + Phh h

where q = u, d, s denotes the light quark density functions

and h = c, b, t are the heavy-quark densities. We have ab

breviated P LO by P , and a = (2/S)a/ ln Q2.

Note that there are evolution equations, (14), for all type of partons (including heavy quarks) just starting from Q0.

The input heavy-quark distribution h(x, Q20) should be treated as an intrinsic PDF introduced in [18]. Of course, at low Q2 m2h the corresponding splitting functions are

strongly suppressed by the small value of the ratio Q2/m2h. So, actually the evolution of the heavy quark will start somewhere in the region Q2 m2h.

2.2 Quark mass effects in NLO diagrams

It turns out that to include heavy-quark mass effects in NLO evolution we do not need to modify the usual NLO splitting functions. In the absence of intrinsic heavy quark, we only have to take mh into account in Phg and then only in the LO part P (0)hg. (Of course, as a consequence, we must adjust the virtual corrections to Pgg.) The argument is as follows.

The k2i integral of (7) written with NLO accuracy has the form

[integraldisplay] dk2i A(k2i, k2i+1, m2h, z)

(k2i + m2h)2

[integraldisplay] A1(z)d(k2i + m2h)

(k2i + m2h) +

2m2hQ2z(1 z)

+ (Q2 + m2h)2

[parenrightbigg][parenleftbigg]Q2

zm2h 1 z

[integraldisplay] 0 Phg[parenleftbig]z , Q2[parenrightbig] dz , (11)

where the upper limits of integration, zh = Q2/(Q2 + m2h),

are determined by the function in (10).

For completeness, and to provide the smooth behavior in all the LO splitting functions, we present the other two LO kernels which involve the heavy quark. An analogous calculation for the h h splitting gives

7Better to say heavy-quark triangle, since the upper line with the largest kt at LO is treated as a point-like operator.

zh

[integraldisplay] A2(z) m2hdk2i

(k2i + m2h)2

+

= [integraldisplay] A3(z) dk2i

k2i+1

. (15)

Eur. Phys. J. C (2013) 73:2616 Page 5 of 7

The rst term gives the leading-logarithm contribution. To be specic we have

[integraldisplay]

Q2

k2i1

dk2(k2 + m2h) =

ln Q2 + m2h

m2h

(16)

for k2i1 m2h. Both the second term in (15), which is con

centrated in the region k2i m2h, and the third term, which

is concentrated near the upper limit, at k2i k2i+1, give non-

logarithmic contributions.

In the axial gauge the two rst terms on the right-hand-side of (15) come only from the pure ladder (and the corresponding self-energy) diagrams, from the region of k2i

k2i+1. That is, these two terms are exactly the same as those

generated by LO LO evolution, in which we have already

accounted for the mh effects. To avoid double counting, we have to subtract these contributions from (15). Thus the true NLO contribution is given by the third term only, in which we can omit the mh dependence since: (a) k2i+1 m2h, and,

(b) these order of O(m2h/k2i+1) terms kill the large logarithm

in the further [integraltext] dk2i+1/k2i+1 integration. That is, at NLO ac

curacy we can use the old, well-known, NLO splitting functions P (1)ik(z). If we were to account for the mass effect in P (1)ik(z), then we would be calculating a NNLO correction.8

In summary, there are no heavy-quark mass effects in the NLO splitting functions. Only LO P (0)hg needs to be modied in order to reach the NLO accuracy in the absence of intrinsic heavy quark.

2.3 NLO coefcient functions

Recall that the NLO coefcient function, CNLO(z, Q2), is

calculated assuming that the incoming parton virtuality, k2n is

much less than the scale Q2, so that the integral [integraltext]

Q2 dk2n/k2n

has a logarithmic form. This means that at NLO accuracy we may neglect the virtuality of the incoming parton. Moreover, in our physical scheme there is no mixture of different types of partons (like those generated by (1) in conventional VFNS). As a consequence there is no change to the NLO coefcient functions.

Note also that in the physical scheme we consistently use the x variable as the light-cone momentum fraction and it not necessary to introduce rescaling described in (3), and the subsequent text. That is, in the physical scheme we deal with quantities which have a clear physical interpretation.

8Before proceeding to NNLO, a phenomenological way to provide very smooth behavior of the NLO contribution would be to multiply the heavy-quark NLO terms (that is, those NLO terms which are proportional to nh) simply by the factor Q2/(Q2 + m2h).

3 Smooth s evolution across a heavy-quark threshold

In analogous way we account for the heavy-quark mass effect in the QCD coupling s(Q2). The running of the coupling to NLO is given by

dd ln Q2

[parenleftbigg] s 4

[parenrightbigg]

= 0

[parenleftbigg] s 4

2 1[parenleftbigg] s 4

3 , (17)

where the -function coefcients are

0(nf ) = 11

2

3nf , 1(nf ) = 102

38

3 nf . (18)

To determine the effect of a heavy-quark mass in the running of s at NLO, it is sufcient to calculate the gluon-to-heavy-quark loop insertion (that is, the gluon self-energy) to gluon propagator. This fermion loop insertion is responsible for the (2/3)nf term in the LO -function. In this

way we nd that, instead of changing nf from 3 to 4 (at Q2 = m2c), and from 4 to 5 (at Q2 = m2b), we must include

in nf a term

(r) =

[bracketleftbigg]1 6r + 12

r2 1 + 4r

ln 1 + 4r + 1

1 + 4r 1

[bracketrightbigg], (19)

for each heavy quark, where r m2h/Q2. In Fig. 2 we plot

as a function of Q2/m2h. As expected, 1 at large Q,

where the heavy quark acts as if it were massless, but even for Q2 10m2h we see that the effects of mh are very impor

tant. For Q2 m2h it vanishes as 1/5r = Q2/5m2h, so there

is only a small heavy-quark contribution to nf for Q < mh.

In Fig. 3 we compare the evolution of s in which the effects of the heavy-quark masses are included, with an evolution assuming all quarks are massless. In the latter case a prescription has been used to ensure that s is continuous across the heavy-quark thresholds. Different prescriptions are possible, but it is not possible to make the derivative also continuous, as can be seen from Fig. 3(b). In-

Fig. 2 The contribution of a heavy quark to the running of s, showing a smooth behavior across the heavy-quark threshold. If = 1, the

heavy quark acts as if it were massless

Page 6 of 7 Eur. Phys. J. C (2013) 73:2616

Fig. 3 (a) The running of s at NLO: the continuous curve is obtained with the effects of the heavy-quark masses mc, mb included, and the dashed curve is that used by MSTW. Both evolutions are normalized to s(M2Z) = 0.12. (b) The ratio of the above two evolutions of s

deed, with massless evolution, different reasonable prescriptions can lead to a difference of more than 0.5 % in going from Q2 20 GeV2 up to Q2 = M2Z, see the Appendix

in [3]. However, when the heavy-quark masses are properly accounted for, we see that the difference over this interval is about 4 %, and in fact up to 14 % starting from Q2 = 1 GeV2. The fact that the s curve, obtained with

mass effects included, lies consistently above that for massless evolution in Fig. 3(a) follows from the behavior of in Fig. 2 and that we have required both curves to have s(M2Z) = 0.12.

4 Conclusions

In order to account for the effects of the heavy-quark mass, mh, in DGLAP evolution, and to provide a smooth transition through the heavy-quark threshold regions, we include the mh dependence already in the LO (one-loop) splitting functions. We show that this modication of the LO splitting functions already provides NLO accuracy; there is no need to modify the known NLO splitting and coefcient

functions. The crucial difference of our approach with those of the conventional FFNS or VFNS, is the fact that the heavy-quark mass is included directly in the splitting functions; that is, the heavy-quark mass is retained throughout the evolution. The presence of the quark mass in the splitting function automatically suppresses the evolution of the heavy quark at low scales, Q2 m2h, while at large Q2 m2h the

massless limit is restored.

To express it another way, by explicitly calculating the appropriate Feynman diagrams, keeping the heavy-quark mass dependence, we obtained the corresponding expressions for the LO splitting functions and the running QCD coupling s. In this way, we have determined the full m2h/Q2 behavior of DGLAP evolution at NLO.

The idea to account for the heavy-quark mass already in the splitting function was proposed in [19].9 However, there, the conventional MS factorization scheme was used and the splitting function still has some irregularity, since the heavy-quark part was included only at large enough Q2 above the heavy-quark threshold. It was shown in [22] that, in this form, the resulting physical cross sections are not different from those obtained in the conventional VFNS approach. On the other hand, in the present paper, we work in the physical scheme including the effect of the heavy-quark mass consistently starting from the input scale Q0 of the DGLAP evolution. Therefore, there are no irregularities at the heavy-quark thresholds. All the formulas at NLO level are quite simple. The generalization to NNLO is straightforwardwe need to account for mh in the two-loop (NLO) diagrams.

We emphasize the advantage of using the physical scheme where we deal with the true physical quantities: there is no mixture of the partons of different types, and no rescaling of the x variable as in (3). Our x is just the light-cone momentum fraction. Thus the mass of each parton is well-dened. In contrast in the MS factorization scheme, at NLO level, we deal with some mixture of partonsfor example the singlet-quark distribution has an admixture of gluons, and so on. Recall that also in NLO Monte Carlos, where the quantum numbers of each parton must be correctly dened, an alternative scheme to the MS scheme is used [23, 24]. Our approach should also be useful to compare Monte Carlo event generators, in which parton radiation is similarly performed including heavy-quark masses, with the analytical results.

To summarize, the treatment of heavy-quark mass effects is perturbatively calculable in QCD with no ambiguity,10 with the heavy-quark masses as free parameters. It is

9A similar splitting function which depends on quark mass was presented in earlier work by [20]. An analogous result for QED may be found in [21].

10There is the possibility of a small O(1/m2h) non-perturbative intrin

sic heavy-quark component in the starting heavy-quark distributions

to the DGLAP evolution.

Eur. Phys. J. C (2013) 73:2616 Page 7 of 7

not necessary to adopt one of the GM-VFN schemes (or a FFNS). In the physical scheme that we introduce, there is a smooth behavior of all quantities across the heavy-quark thresholds. Clearly, Fig. 3(b), for example, shows that a new global analysis of data is essential to determine the PDFs of the proton. However, rst, we must complete the calculation of all the splitting and coefcient functions in the physical scheme; this is under way.

Acknowledgements We thank Robert Thorne for valuable discussions. EGdO and MGR thank the IPPP at the University of Durham for hospitality. This work was supported by the grant RFBR 11-02-00120-a and by the Federal Program of the Russian State RSGSS-4801.2012.2; and by FAPESP (Brazil) under contract 2012/05469-4.

P realgg(z) = 2CA

[bracketleftbigg][parenleftbigg] z

1 z +

1 z

z + z(1 z)

[parenrightbigg]

[bracketrightbigg]. (25)

To be complete, recall also the relation between the MS NLO coefcient functions and those in the physical scheme. As was mentioned already in Sect. 1,

CNLOa(phys) = CNLOa(MS) [summationdisplay]

i

CLOi ia, (26)

see (4). References

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2534 (2013)

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[parenleftbig]1 + ln(1 z)[parenrightbig]

Appendix

At NLO accuracy the relation between the MS and the physical parton distributions are given by (6), where the long-distances part of the NLO coefcient function C originates from the term proportional to in the LO splitting functions

Pab(z) = P LOab(z) + ab(z). (20) The -dependent term, ab, is known (see for example [25]). However, in comparison with the results listed in [25], we have to add a contribution of pure kinematical origin. Indeed, in D = 4 + 2 space the logarithmic integration

[integraltext] dk2t/k2t is replaced by [integraltext] d2+2 kt/k2t (1/ )(k2t) . If ex

pressed in terms of the virtuality variable, this phase-space factor (k2t) reads

k2t[parenrightbig] = [parenleftbig]k2(1 z)[parenrightbig] = 1 + ln k2 + ln(1 z). (21) The last term in this expansion leads to an additional contribution to ab(z) of (20) of the form P LOab(z) ln(1 z). Thus

we obtain

P realqq(z) = CF

1 + z2

1 z

1 + ln(1 z)[parenrightbig] + (1 z)

[bracketrightbigg], (22)

Pqg(z) = TR[bracketleftbig][parenleftbig]z2 + (1 z)2[parenrightbig][parenleftbig]1 + ln(1 z)[parenrightbig]

+ 2z(1 z)[bracketrightbig], (23) Pgq(z) = CF

1 + (1 z)2

z [parenleftbig]1 + ln(1 z)[parenrightbig] + z

[bracketrightbigg], (24)