Table Of ContentUH-511-1048-04
Perturbative Heavy Quark Fragmentation Function through O(α2)
s
Kirill Melnikov and Alexander Mitov
Department of Physics and Astronomy,
University of Hawaii,
Honolulu, HI, 96822
5 We derive the initial condition for the perturbative fragmentation function of a heavy quark
0 through order O(α2s) in theMS scheme. This initial condition is useful for computing heavy quark
0 (orlepton,incaseofQED)energydistributionsfromcalculationswithmasslesspartons. Inaddition,
2 theinitial condition at O(α2s) can beusedtoresum collinear logarithms ln(Q2/m2) inheavyquark
n energyspectrumwithnext-to-next-to-leadinglogarithmicaccuracybysolvingtheDGLAPequation.
a
J
3 I. INTRODUCTION
1
Production of heavy flavors (charm and bottom) in high energy processes has become an important subject in the
2
lastdecade,duetoexperimentsate+e−andhadroncolliders. Sincethecenterofmassenergyofthesemachinesismuch
v
3 largerthanbottomandcharmquarkmasses,it is tempting to considersuchprocessesinthe masslessapproximation.
4 Unfortunately, this is only possible for sufficiently inclusive observables, for example, total cross-sections. However,
1 differentialdistributionsaremorerelevantforexperimentalanalysisand,inaddition,containmoreinformationabout
4 the underlyingphysics. Aninterestingexampleisthe energydistributionofheavyhadronsproducedinahigh-energy
0
collision, since it gives direct access to the hadronization process. In perturbation theory, however, the heavy quark
4
energy distribution diverges in the limit m → 0 and the sensitivity to the quark mass m remains at arbitrary high
0
energies.
/
h This sensitivity is disagreeable for two reasons. First, it implies that energy spectra must be computed retaining
p
quark masses which makes the calculations quite involved. Second, in higher orders of perturbation theory, energy
-
p spectra contain powers of logarithms ln(Q2/m2), that may invalidate fixed order perturbative calculations.
e Both of these problems are solved by introducing perturbative fragmentation function [1]. This function describes
h the probability that a massless parton of a certain type fragments into a heavy quark. The mass of the heavy quark
:
v provides a natural cut-off for the collinear radiation, thus making perturbative predictions finite. The perturbative
i fragmentation function satisfies the DGLAP evolution equation [2]. By solving this equation, we can sum up large
X
collinearlogarithmsln(Q2/m2)thatappearinperturbativefragmentationfunctionandimprovethequalityofpertur-
r bative calculations. To avoid confusion, we note that a meaningful prediction for heavy hadron energy spectrum can
a
onlybeobtainediftheheavyquarkenergydistributionisconvolutedwithanon-perturbativefragmentationfunction.
Such functions are traditionally extracted by fitting hadron energy spectra in e+e− collisions. We will not discuss
this issue here; for recent work on the subject see [3]. We also note that the concept of perturbative fragmentationis
useful outside QCD; for example, QED effects in the electron energy spectrum in muon decay, currently under study
by the TWIST collaboration [4], can be analyzed using the same technique [5].
BecausetheQ2evolutionofthefragmentationfunctioncanbeobtainedfromtheDGLAPequation,theperturbative
fragmentation function at arbitrary Q2 can be fully reconstructed once the initial condition Dini and the time-like
splitting kernels for the DGLAP evolution are known. The initial condition is usually computed at Q2 ∼ m2; this
ensures that no large logarithms appear and fixed order perturbation theory is reliable.
The purpose of this paper is to provide such initial condition for the perturbative fragmentation function through
O(α2). As we explained earlier, the knowledge of Dini through next-to-next-to-leading order (NNLO) enables the
s
calculation of energy spectra of heavy quarks through O(α2) from massless results and, simultaneously, allows to
s
resumlargelogarithmsln(Q2/m2)usingtheDGLAP equation. AllpowercorrectionsO(m2/Q2)areneglectedwithin
this approach, but for many practical purposes such precision is sufficient.
Technically, the use of perturbative fragmentation functions makes it possible to compute the heavy quark energy
spectrum in two steps – we first compute the energy spectrum of massless partons produced in a hardscattering and
thenconvolutethis spectrumwithperturbativefragmentationfunction. The possibilityto removeallthe dependence
on the heavy quark mass from the calculation of the hard scattering simplifies the computations considerably.
2
The remainder of the paper is organized as follows. In Section II we describe the process-independent derivation
of the initial condition for the fragmentation function. Then, we discuss collinear factorization when finite quark
masses are present and explain how to compute Dini from relevant Feynman diagrams. We also suggest a suitable
modification of the original proposal [6, 7] for the process-independent computation of the initial condition; such
modification significantly simplifies NNLO calculations. In Section IV the initial condition for the fragmentation
function for both quark- and gluon-initiated processes is computed through NLO. This computation allows us to
demonstrate the details of our approach and to derive the NLO perturbative fragmentation function through O(ǫ),
whereǫisthedimensionalregularizationparameter. InSectionVwedescribethecalculationoftheO(α2)contribution
s
to the fragmentation function and present the result for the initial condition. We conclude in Section VI.
II. PROCESS-INDEPENDENT DERIVATION OF Dini
Consider production of a heavy quark Q with mass m and a definite value of energy E in a hard scattering
Q
process. According to the QCD factorizationtheorems [8, 9, 10], the heavy quark energy spectrum can be computed
as a convolution of the energy distribution of massless partons produced in the hard process, and the fragmentation
functionthatdescribestheprobabilitythatthemasslesspartonfragmentsintoamassivequarkwithadefiniteenergy.
If the energy fractionE /E of the heavy quark is denoted by z, then the energy distribution of that quark can
Q Q,max
be written as:
dσ 1 dxdσˆ z µ
Q a
(z,Q,m)= (x,Q,µ)D , . (1)
dz x dx a/Q x m
Xa Zz (cid:16) (cid:17)
Herethesumrunsoverallpartons(quarks,antiquarksandgluons)thatcanbeproducedinthehardprocessandµis
thefactorizationscale. Thecoefficientfunctiondσˆ /dxistheMSrenormalizeddifferentialcross-sectionforproducing
a
a massless parton a 1 . It is defined indirectly through the equation
dσ dσˆ
a b
(z,Q,ǫ)= (z,Q,µ)⊗Γ (z,µ,ǫ), (2)
ba
dz dz
b
X
where dσ /dz is the bare energy distribution for the parton of type a; the collinear divergences in this distribution
a
are regularizedby workingin d=4−2ǫ dimensions. Γ are universalcollinear subtraction terms, defined in the MS
ab
scheme:
α P(0)(z) α 2 1 1
Γ =δ δ(1−z)− s ab + s P(0)⊗P(0)(z)+β P(0)(z) − P(1)(z) , (3)
ba ab 2π ǫ 2π 2ǫ2 ac cb 0 ab 2ǫ ab
(cid:16) (cid:17) (cid:16) (cid:17) (cid:20) (cid:16) (cid:17) (cid:21)
where α = α (µ) is the MS strong coupling constant, renormalized at the scale µ. The relation between the bare
s s
and the renormalized couplings reads:
α0 α α β
sS = s 1− s 0 +O(α2) , (4)
2π ǫ 2π 2π ǫ s
(cid:18) (cid:19)
whereS =(4π)ǫ e−ǫγ andγ isthe Eulerconstant. Also,β =(11C −4T n )/6is theO(α2)coefficientofthe QCD
ǫ 0 A R f s
β-function, C = 3, T = 1/2 are the QCD color factors, n denotes the number of fermion flavors (including Q)
A R f
and P(0,1) are the time-like splitting functions [11]. Our notations for the splitting functions follow Ref.[10].
ab
The functions D (x,µ/m) in Eq.(1) are the perturbative fragmentation functions [1]. They satisfy the DGLAP
a/Q
evolution equation and can be fully reconstructed from it, if the initial condition at a scale µ = µ is known. We
0
denote
µ µ
D z, 0 =Dini z, 0 . (5)
a/Q m a m
(cid:16) (cid:17) (cid:16) (cid:17)
1 In the evaluation of the coefficient function dσˆa/dx the heavy quark Q is considered as massless; therefore, the sum over indexes in
Eq.(1)includestheflavorQ.
3
If µ ∼m is chosen, the initial condition Dini does not contain large logarithms and can be derived from fixed order
0 a
perturbative calculations. Setting µ=µ in Eq.(1), we obtain:
0
dσfo dσˆ µ
Q(z,Q,m)= a(z,Q,µ )⊗Dini z, 0 , (6)
dz dz 0 a m
Xa (cid:16) (cid:17)
where dσfo is the fixed order cross-section for producing a quark Q with mass m, with all power corrections
O((m2/Q2)n), n ≥ 1 systematically neglected. It follows from this equation that the initial condition for the frag-
mentationfunctioncanbe extracted,ifthe heavyquarkenergyspectrumis computedinaparticularprocessthrough
certain order in α . This, however, is technically inconvenient.
s
A more convenient, process-independent approach was suggested recently in Refs.[6, 7]. Since hard scattering
cross-sections are insensitive to long-distance dynamics, we may write
dσfo dσ µ
Q(z,Q,m)= a(z,Q,µ )⊗D z, 0 , (7)
dz dz 0 a/Q m
Xa e (cid:16) (cid:17)
dσb(z,Q,ǫ)= dσa(z,Q,µ )⊗DLe (z,µ ,ǫ). (8)
dz dz 0 a/b 0
a
X e
e
The functions D and DL describe collinear radiation from massive (massless) quarks; we will show below that they
can be considered as bare fragmentation functions. Combining Eqs.(7,8) with Eqs.(6,2), we obtain:
e e
µ µ
D z, 0 = DL (z,µ ,ǫ)⊗[Γ(z,µ ,ǫ)]−1⊗Dini z, 0 . (9)
a/Q m a/b 0 0 bc c m
(cid:16) (cid:17) Xb,c (cid:16) (cid:17)
e e
It follows from this equation that Dini can be derived once D and DL are known. We describe how to compute D
and DL in the next Section.
To conclude this Section, we comment on the analytic struecture ofeDini. The initial condition for the perturbativee
fragmeentation function can be expanded in series of αs:
n
µ α (µ ) µ
Dini z, 0 = s 0 d(n) z, 0 . (10)
a m 2π a m
(cid:16) (cid:17) nX=0(cid:18) (cid:19) (cid:16) (cid:17)
The results through O(α ) are known [1]:
s
d(0)(z)=δ δ(1−z),
a aQ
µ 1+z2 µ2
d(1) z, 0 =C ln 0 −1 ,
a=Q m F 1−z m2(1−z)2
(cid:16) (cid:17) (cid:20) (cid:18) (cid:18) (cid:19) (cid:19)(cid:21)+
µ µ2
d(1) z, 0 =T z2+(1−z)2 ln 0 ,
a=g m R m2
d(1) (cid:16) z,µ(cid:17)0 =0(cid:0), (cid:1) (cid:18) (cid:19) (11)
a6=Q,g m
(cid:16) (cid:17)
where C = 4/3 is the QCD color factor. In this paper, we are mostly concerned with the computation of the
F
coefficientsd(2) ,whereq standsforanymasslessquark. Thelogarithmicallyenhancedtermsind(2) followfrom
Q, Q, q, q¯
the DGLAP equation:
µ P(0)⊗P(0)(z) β µ2
d(2) z, 0 = ba Qb + 0P(0)(z) ln2 0
a m 2 2 Qa m2
(cid:16) (cid:17) " # (cid:18) (cid:19)
µ2
+ P(1)(z)+P(0)⊗d(1)(z,1)+β d(1)(z,1) ln 0 +d(2)(z,1). (12)
Qa ba b 0 a m2 a
h i (cid:18) (cid:19)
The mass-independentconstantofintegrationd(2)(z,1)cannotbe determinedfromthe DGLAP equationandhasto
a
be computed explicitly. This is the major goal of this paper. In the next Section we describe how this can be done.
4
III. THE COLLINEAR LIMIT
The idea that allows explicit computation of the functions DL and D and, therefore, of the initial condition for
the perturbative fragmentation function is as follows. Both D and DL are introduced to describe collinear radiation
withq⊥ ∼m≤µ0. Since wearenotinterestedinpower-suppresesedconteributionsto the cross-section,we canrestrict
ourselves to such terms in scattering amplitudes that, uponeintegraetion over the phase-space of final state particles,
produce O(lnn(m)), n ≥ 1 terms2. Such terms can be identified by applying power counting arguments [9] in the
collinear limit for scattering amplitudes. The collinear limit is defined as the kinematic limit where the relative
transverse momentum of two or more particles vanishes. When masses are introduced, the collinear limit is q →0,
⊥
m→0 and q /m=const [12].
⊥
Throughout the paper we work in the light-cone gauge n Aµ = 0, n2 = 0. This is necessary for the process-
µ
independent derivation of Dini because in such gauges, the O(lnn(m)), n ≥ 1 terms are produced only by diagrams
where collinearradiationis both emitted and absorbedby the same parton. The quantum interferenceeffects in such
gauges can be integrated over the phase-space of the final state particles in the massless approximation. Because of
that, Dini can be computed from self-energy type diagrams, integrated over the virtuality of the incoming parton.
Different cuts of such self-energy diagrams correspond to different contributions to the fragmentation function: at
O(α2), we have to deal with two-loop virtual corrections, one-loop virtual corrections to one-to-two splittings and,
s
finally, one-to-three splittings.
We beginby consideringthe one-to-threecollinearsplitting since the kinematics ofthe finalstate inthis caseis the
most general. We denote the four-momenta of the produced (massive or massless) particles by q and their sum
1,2,3
by p :
p=q +q +q . (13)
b 1 2 3
Weparameterizethecollineardirectionbypandusethegaugefixingvectornasthecomplimentarylight-conevector.
b
We then write:
n
q = z p+β +q . (14)
i i i i,⊥
(pn)
The components β are found from the on-shell conditions q2 =m2:
i i i
−q2 +m2−z2m2
i,⊥ i i p
β = , i=1,2,3 , (15)
i
2z
i
with p2 = m2. We are interested in the case when p,p and n belong to the same hyperplane in the d-dimensional
p
space3. Then:
b
z +z +z =1 , q +q +q =0. (16)
1 2 3 1,⊥ 2,⊥ 3,⊥
Using the above equations, we express the momentum p through p and n:
1 n
p=p+ 2b(p2−m2p)(pn), (17)
where: b b
2(pq )+2(pq )−2(q q )+m2−m2−m2−m2
p2−m2 = 2 3 2 3 1 2 3 p.
p z
Our notations are such that q always denotes the momentum of the heavy quark in the final state, whose energy is
b 1
measured. Forthatreasonweoftenwritez insteadofz ,todenoteitsenergyfraction. Eqs.(14,16)implytherelation:
1
(nq ) (nq )
2 3
z =1− − , (18)
(pn) (pn)
2 To be precise, we are interested in all contributions to the scattering amplitude that, upon integration over the phase space, scale as
m−nǫ, where n is some integer. Upon expanding in ǫ, those terms produce both ln(m)-enhanced and m-independent contributions to
theenergyspectrum.
3 Moregeneralconfigurationhasbeenconsideredin[13].
5
which is nothing but the energy conservation condition.
Although we have only discussed the kinematics of the one-to-three splitting, the kinematics of the one-to-two
splitting can be obtained as a particular case. For this, it is sufficient to set the momentum q and the mass m to
3 3
zero in the equations above.
In QCD, factorization properties of the cross-sections, as in Eqs.(7,8), can be traced back to the factorization
propertiesofboththephase-spaceandthe matrixelementsinthe collinearkinematics. We illustratethoseproperties
by considering the tree level amplitudes and then generalize these considerations to include virtual corrections.
Consider a hardscatteringprocess characterizedby some scale Q≫m. Suppose that n+2 partons with momenta
k ,...,k ,q ,q ,q are produced. We consider momenta k ,...,k as non-exceptional, while momenta q ,q ,q
1 n−1 1 2 3 1 n−1 1 2 3
are collinear, as described above. First, we discuss the phase-space factorization.
We denote the phase-space of n+2 partons as dPS(n+2)(k ,...,k ,q ,q ,q ). In the limit when q ,q and q
1 n−1 1 2 3 1 2 3
become collinear q +q +q =p+O(q ), the (n+2)-particle phase space factorizes:
1 2 3 ⊥
dPS(n+2)(k ,...,k ,q ,q ,q )=dPS(n)(k ,...,k ,p) dΦcoll(q ,q ). (19)
1 n−1 1 2 3 1 n−1 2 3
We use the following notations:
1
dΦcoll(q ,q )= [dq ;m ][dq ;m ], (20)
2 3 2 2 3 3
z
and [dq;m ] is the d-dimensional one-particle phase space:
q
ddq
[dq;m ]= δ+(q2−m2). (21)
q (2π)d−1 q
To derive Eq.(19), we neglect the O(q ,m) difference between pˆand p in dPS(n) and integrate overq using Eq.(13).
⊥ 1
Wenowconsiderfactorizationpropertiesofthetree-levelmatrixelement. Inthemasslesscase,theyweredescribed
in [13, 14]. Below we generalize those results to the case of non-zero mass m. We consider massive quarks produced
in the fragmentation of a quark-like parton of either the same or different flavor. The case of a gluon fragmentation
into a heavy quark can be dealt with in a similar way.
We already mentioned that, in physical gauges, the relevant contributions in the collinear limit come from di-
agrams where the same parton emits and absorbs collinear radiation. Taken in conjunction with power counting
arguments, this observation can be used to simplify tree-level matrix elements in the collinear limit and derive the
factorization formula. Consider a process with n+2 particles in the final state described by the matrix element
M(n+2)(k ,...,k ,q ,q ,q ). In physical gauges,the collinear splitting p→q +q +q decouples from the rest of
1 n−1 1 2 3 1 2 3
the process, so that the squared, spin-averagedamplitude |M(n+2)|2 can be written as [13]:
b
|M(n+2)(k ,...,k ,q ,q ,q )|2 =M(n)(k ,...,k ,p) Vcoll(p,q ,q ). (22)
1 n−1 1 2 3 1 n−1 β,α α,β 2 3
Here α,β are spinor indices. If, in the collinear limit, Vcoll can be written as
b b
Vcoll =6p W +O(q ), (23)
α,β α,β ⊥
we can rewrite Eq.(22) to make the factorization of the matrix element explicit:
|M(n+2)(k ,...,k ,q ,q ,q )|2 =|M(n)(k ,...,k ,p)|2 W(n,p,q ,q )+O(q ). (24)
1 n−1 1 2 3 1 n−1 2 3 ⊥
Here |M(n)(....,p)|2 is the amplitude squared for producing n on-shell particles with non-exceptional momenta; the
b
scalar function W contains all the information about the collinear splitting.
We now explain why Vcoll can be written as in Eq.(23). Because the collinear splitting p→q +q +q decouples
1 2 3
fromthe restofthe process,Vcoll isobtainedasthe sumsquaredofallpossiblediagramsthatdescribeatransitionof
anoff-shellpartonwith momentump into three particles with momenta q . The propagatorofthe off-shellparton
1,2,3 b
p is included in Vcoll. In the massless case the matrix Vcoll(n,p,q ,q ) can be written as [13]:
2 3
b
µ2ǫ 2 3 6n
b Vcoll = 0 A 6bq +B p2 . (25)
p2 i i (pn)
(cid:18) (cid:19) i=1 !
X
The functions A and B are dimensionless functions that depend on thebscalar products (q q ). To determine the
i b i j
behavior of the amplitude Vcoll in the collinear limit, we rescale the transverse momenta:
q →κq (26)
i,⊥ i,⊥
6
and consider the limit κ→0. Since (q q )∼O(q2), the functions A and B remain invariantunder that rescaling.
i j ⊥ 1−3
As follows from Eq.(14), the matrices 6q transform as: 6q →z 6p+O(κ). Therefore, after the rescaling Eq.(26), the
i i i
amplitude Vcoll becomes:
µ2ǫ 2 3
Vcoll =κ−4 0 A z 6p+O(κ−3). (27)
p2 i i
(cid:18) (cid:19) i=1 !
X
Because the collinear phase space Eq.(20) transforbms as:
dΦcoll(q ,q )→κ4−4ǫdΦcoll(q ,q ), (28)
2 3 2 3
under the rescaling Eq.(26), it follows that in the collinear limit κ → 0 only the first term in Eq.(27) gives non-
vanishing contribution after the phase-space integration. This proves that Vcoll has the form shown in Eq.(23). We
can extract the function W from Vcoll by applying the projection:
Tr 6nVcoll
W = . (29)
4(pn)
(cid:2) (cid:3)
We now generalize this resultto the case when atleast one of the finalstate particles is a quark with mass m. The
mass m is consideredto be of the order of the transverse momenta of the collinear particles. We follow the same line
of reasoning that leads to Eq.(25) allowing, however, for non-vanishing masses in the initial and final states of the
collinear splitting process. Eq.(25) generalizes to:
µ2ǫ 2 3 6n
Vcoll = 0 A(m) 6q +B(m) +Cm . (30)
p2−m2 i i (pn)
(cid:18) in(cid:19) i=1 !
X
In Eq.(30), m is the mass of the parbton that initiates the collinear splitting. The functions A(m), B(m) and C
in i
are scalar functions of dimension zero, two and zero respectively; they depend on (q q ) and m2. These functions
i j
possessaregularm→0 limit. Inthe collinearlimit, whenboth the transversemomentaandthe massm arerescaled
simultaneously,
q →κ q , m→κ m, (31)
i,⊥ i,⊥
the scaling properties of the functions A(m), B(m) and C coincide with their mass dimension. Therefore, under the
i
rescaling Eq.(31), the amplitude Vcoll behaves as:
µ2ǫ 2 3
Vcoll =κ−4 0 A(m)z 6p+O(κ−3). (32)
p2−m2 i i
(cid:18) in(cid:19) i=1 !
X
Thisresultimpliesthatalsoforthem6=0caseb,thefunctionW isgivenbyEq.(29). Finally,assemblingallthepieces,
we find that the function D(z) in Eq.(7), can be written as:
1 e Tr 6nVcoll(p,q2,q3,n;m) (nq2) (nq3)
D(z)= [dq ;m ][dq ;m ] δ 1−z− − . (33)
2 2 3 3
z 4(pn) (pn) (pn)
Z (cid:2) (cid:3) (cid:18) (cid:19)
b
Although we haeve derived Eq.(33) for the one-to-three splitting contribution to Dini(z,µ /m), similar expression is
a 0
validwhenvirtualcorrectionsareincluded. Inaddition,Eq.(33)canbeusedtoderivethe functionDL afterthe mass
m is set to zero everywhere.
We now discuss how to perform phase space integrations in Eq.(33). The approach proposedein [6, 7] requires
integration over the transverse momentum up to the scale µ = µ for both D and DL. While this allows a simple
⊥ 0
calculationofthefragmentationfunctionatO(α ),thisapproachbecomesimpracticalinhigherordersofperturbation
s
theory. It is possible to simplify the calculation considerably by realizing thatethe limeit µ⊥ →∞ can be taken when
the difference of massive and massless functions D is considered. The technical simplification associated with this
is twofold. First, all higher order QCD effects in the massless function DL vanish, because all the integrals that
contribute there become scaleless. We obtain: e
e
DL (z,µ ,ǫ)=δ δ(1−z). (34)
a/b 0 ab
e
7
Q Q
FIG. 1: Diagrams that contribute to perturbativefragmentation of a heavy quark Q→Q+X at O(αs). The dashed vertical
line indicates theintermediate state that has to be considered.
Second, in the massive case, only single-scale integrals have to be computed. Using Eq.(34), we obtain a simple
expression for the initial condition of the perturbative fragmentation function:
µ µ
Dini z, 0 = Γ (z,µ )⊗D z, 0 . (35)
a m ab 0 b/Q m
(cid:16) (cid:17) Xb (cid:16) (cid:17)
e
From this it follows, that D is the bare fragmentation function for the massive quark, whereas Dini represents its
b/Q
collinear renormalized version.
Aswementionedearlier,etherearethreecontributionstoDb/Q(z,µ0,m),thathavetobeconsidered: double-virtual,
real-virtual and double-real. In all these cases the amplitude Vcoll can be constructed by applying conventional
Feynman rules to describe the transition of a parton withemomentum p to the final state of up to three particles
with momenta q . Virtual corrections should also be included, when appropriate. For each of these three cases,
1,2,3
the function W is obtained using the same projector as in Eq.(35). The only unusual feature of the amplitudes that
b
contribute to Vcoll is that the spinor, that describes the initial state parton, has to be replaced with the propagator
of the same parton with the off-shell momentum p. Vcoll is then given by the square of the corresponding matrix
element. Standard symmetry factors apply in this case as well. In particular, to evaluate D one has to sum over
spins and colors of the final state and to average obver colors in the initial state. The ultraviolet renormalization is
performed in a standard fashion. One has to renormalize the QCD coupling constant, the queark mass and external
quark and gluon fields. Since we work in the light cone gauge, we find it convenient to compute the quark and gluon
field renormalizationconstants by computing diagrams with self-energy insertions to external quark and gluon lines.
Finally,wecommentonthedoublevirtualcontributionstothefragmentationfunction. Suchcontributionsdescribe
theone-to-onesplittingandareproportionaltoδ(1−z). Unfortunately,computingthesecontributionsrequiresdealing
with well-known complications of light-cone gauges. For this reason, we decided to fix the (ǫ-dependent) coefficient
of δ(1−z) using the fermion number conservation sum rule:
1
dz Dini (z)−Dini =1. (36)
Q/Q Q/Q
Z0 (cid:16) (cid:17)
We do not encounter, however, any spurious light-cone singularities in the diagrams with one virtual loop. All
singularities originating from those diagrams are consistently regulated with dimensional regularization.
To perform integrations in phase-space and loop integrals we proceed as in [15, 16, 17], where further details of
the method can be found. The idea is to map all non-trivial phase-space integrals to loop integrals and use standard
multiloop methods [18], such as integration-by-partsand recurrence relations, to reduce all the phase-space and loop
integrals that have to be evaluated, to a few master integrals. For the reduction to master integrals we use the
algorithm[19]implementedin[20]. AllalgebraicmanipulationshavebeenperformedusingMaple [21]andForm[22].
We now discuss the calculation of the initial condition Dini through O(α ) which allows us to illustrate the details of
s
the method by considering a simple example.
IV. Dini AT NLO
In this Section we compute the initial condition for the fragmentation function at NLO through O(ǫ). Such terms
are needed for the computation of Dini at order O(α2). The function Vcoll for Q → Q+g splitting is obtained by
s
consideringthe diagramsshowninFig.1. Tofacilitate thecomparisonwiththe literature,weexpressthe functionW,
Eq.(29), through the so-called splitting function P [13]:
8πα µ2ǫ
W = s 0 P. (37)
p2−m2
(cid:18) (cid:19)
Introducing auxiliary vector n:
b
n
=(1−z)n, (38)
b (pn)
b
8
and using the projector Eq.(29), we obtain the splitting function:
(pn) (qn) (pn)−(qn)
P(1) =C 2 +(1−ǫ) −2−m2 . (39)
F
(qn) (pn) (pq)(pn)
(cid:18) (cid:19)
Combining Eqs.(18,38), we derive the real emission contribution to the fragmentation function D:
D(r)(z) = α0s 8π2µ20ǫ [dq;0]P(1)δ(1−(nq)). e (40)
2π 1−z (pq)
(cid:18) (cid:19) Z
Because of the constraint δ(1−(enq)) in Eq.(40), the splitting function simplifibes:
2 m2
bP(1) =C −1−z−ǫ(1−z)−z . (41)
F
1−z (pq)
(cid:18) (cid:19)
The result for P(1) in Eq.(41) agrees with Ref.[12]. When P(1) is used in Eq.(40) we observe that two integrals have
tobeevaluated. However,thoseintegralsarenotindependent; analgebraicrelationbetweenthemcanbefoundusing
the method of [15, 16, 17]. As a result, D(r)(z) can be expressed through a single “master” integral:
ddq
I(1) = e δ q2 δ[1−(nq)]=(π)1−ǫΓ(ǫ)m−2ǫ(1−z)1−2ǫ. (42)
(pq)
Z
(cid:2) (cid:3)
We finally obtain the real emission contribution to thebfragmentation function:
µ α0 4πµ2 ǫ (1−ǫ)Γ(ǫ)(1+z2)
D(r) z, 0 = s C 0 , (43)
m 2π F m2 (1−z)1+2ǫ
(cid:16) (cid:17) (cid:18) (cid:19) (cid:18) (cid:19)
valid to all orders in ǫ. To expaend this result in ǫ, we use
δ(1−z) (aǫ)n lnn(1−z)
(1−z)−1+aǫ = + . (44)
aǫ n! 1−z
n≥0 (cid:20) (cid:21)+
X
Virtual corrections are derived by considering self-energy diagrams in the light-cone gauge. The result reads:
µ α0 1 1 µ2 1 1 µ2 1 µ2 π2
D(v) z, 0 = s S C δ(1−z) + ln 0 + + ln2 0 + ln 0 + +2
m 2π ǫ F ǫ2 ǫ m2 2 2 m2 2 m2 12
(cid:16) (cid:17) (cid:18) (cid:19) ( (cid:20) (cid:18) (cid:19) (cid:21) (cid:18) (cid:19) (cid:18) (cid:19)
e 1 µ2 1 µ2 π2 µ2 π2 ζ(3)
+ǫ ln3 0 + ln2 0 + +2 ln 0 + − +4 +O(ǫ2). (45)
6 m2 4 m2 12 m2 24 3
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:21))
Combining Eqs.(43-45), we arrive at the bare perturbative fragmentation function through O(α ):
s
µ α0 1 1+z2 1+z2 µ2
D z, 0 =δ(1−z)+ s S C + ln 0 −2ln(1−z)−1
Q/Q m 2π ǫ F ǫ 1−z 1−z m2
(cid:16) (cid:17) (cid:18) (cid:19) ( (cid:20) (cid:21)+ (cid:20) (cid:18) (cid:18) (cid:19) (cid:19)(cid:21)+
e 1+z2 1 µ2 µ2 π2
+ǫ ln2 0 −ln 0 (2ln(1−z)+1)+2ln2(1−z)+2ln(1−z)+ +O(ǫ2) . (46)
1−z 2 m2 m2 12
(cid:20) (cid:18) (cid:18) (cid:19) (cid:18) (cid:19) (cid:19)(cid:21)+ )
As expected from the fermion number conservation condition, Eq.(36), the integral of D (z,µ /m) over z equals
Q/Q 0
1 and does not receive any corrections at order O(α ). Applying collinear renormalization Eq.(35) and substituting
s
α0sSǫ → αs in Eq.(46), we observe that the 1/ǫ term in Eq.(46) cancels and the remaineing terms are not modified.
Therefore Dini can be read off from Eq.(46) by simply omitting the 1/ǫ term. The O(ǫ0) part of Dini coincides with
the result of Ref.[1], while the O(ǫ) part is new.
The gluon fragmentation g →Q+Q can be treated in a similar way. The real emission contribution reads:
µ α04(2π)−1+2ǫµ2ǫ P
D z, 0 = s 0 [dq;m] g/Qδ(1−(nq)). (47)
g/Q m 2π z(1−z) p2
(cid:16) (cid:17) Z
e b
b
Q,q
9
Q Q Q
FIG. 2: Examples of diagrams that contribute to perturbative fragmentation of a heavy quark Q → Q+X at O(α2s). The
dashed vertical line indicates theintermediate state that has to be considered.
Evaluation of the splitting function P is described in [6]. We derive:
g/Q
1−z z m2
P =T 1−2z + , (48)
g/Q R 1−ǫ 1−ǫ(pq)
(cid:18) (cid:19)
which agrees with the result in Ref.[12]. There are two integrals of the type:
ddq
J(a) = δ q2−m2 δ[1−(nq)] , a=1,2 , (49)
(pq)a
Z
(cid:2) (cid:3)
that have to be computed, but only one of them is independent. Wbe obtain:
J(1) =(π)1−ǫΓ(ǫ)m−2ǫ(1−z). (50)
This allows us to derive the gluon-initiated contribution to the initial condition for the perturbative fragmentation
function at O(α ):
s
µ α0 1 µ2 1 µ2 π2
D z, 0 = s S T z2+(1−z)2 +ln 0 +ǫ ln2 0 + +O(ǫ2). (51)
g/Q m 2π ǫ R ǫ m2 2 m2 12
(cid:16) (cid:17) (cid:18) (cid:19) (cid:0) (cid:1)( (cid:18) (cid:19) (cid:20) (cid:18) (cid:19) (cid:21))
e
The 1/ǫpole cancelsafter collinearrenormalizationandthe knownresultfor D [1] is recoveredinthe limit ǫ→0.
g/Q
Having explained out method by considering a simple example, we now present the results for Dini through O(α2).
s
e
V. Dini AT NNLO
AtNNLO,largenumberofprocessescontribute,yet,allofthemcanbetreatedusingthemethodsdescribedabove.
Before we presentthe results for Dini throughO(α2), we briefly mentionsome technicaldetails that we find peculiar.
s
Severalsub-processescontributeatO(α2). Itisconvenienttosplitthemaccordingtotheircontributionstovarious
s
components of Dini. The real emission contributions to Dini are Q → Qgg,Qqq¯,QQQ, while for Dini and Dini the
a Q Q q,q¯
real emission sub-processes are Q →QQQ and q(q¯)→Qq(q¯)Q, respectively. Some of the contributing diagrams are
shown on Fig.2.
Becausethenumberofcontributingprocessesislarge,therearemanypossibilitiesofmassassignmentsforparticles
in the final state, relevant for the calculation of Dini. We have to consider all of those cases separately. As we
explained in Section III, we require the tree-level splitting amplitudes in the collinear limit when massive quarks are
present. Those results are not available in the literature; yet, a useful cross-check of our results is obtained once the
limit m→0 is taken. In that limit the splitting amplitudes derived in this paper coincide with those in Ref.[13].
We now present the final results for the fermion initiated contributions to the initial condition of the perturbative
fragmentationfunction at order α2. We give the results for the coefficients d(2)(z,µ /m), introduced in Eq.(10). Our
s a 0
results contain polylogarithmic functions up to rank three. These functions are defined through:
z Li (x) 1 z ln2(1−x)
n−1
Li (z)= dx, Li (z)=−ln(1−z), S (z)= dx. (52)
n 1 1,2
x 2 x
Z0 Z0
We begin with the simplest case of a light (anti)quark-initiated fragmentation process. Through O(α2), the two
s
contributions coincide:
µ µ
d(2) z, 0 =d(2) z, 0 . (53)
q m q m
(cid:16) (cid:17) (cid:16) (cid:17)
10
We write:
d(2) =C T F(CFTR), (54)
q F R q
where
(1−z)(4z2+7z+4) 8z2+27z+15
F(CFTR) = (1+z)ln(z)+ L2+ (1+z)ln2(z)− ln(z)
q 6z 3
(cid:26) (cid:27) (cid:26)
4(1−z)(14z2+23z+5) (1−z)(4z2+7z+4)
− L+ 2(1+z)ln(z)+ (Li (z)+ln(1−z)ln(z))
2
9z 3z
(cid:27) (cid:18) (cid:19)
1 1 15+27z+8z2
+ 4(1+z) Li (z)−ln(z)Li (z)+ ln3(z)− ln(1−z)ln2(z)−ζ(3) − ln2(z)
3 2
24 2 12
(cid:20) (cid:21)
16 56 56 7(1−z)(16+133z+88z2)
+ + z+ z2 ln(z)+ , (55)
3 3 9 54z
(cid:18) (cid:19)
where L=ln(µ2/m2).
0
Next, we present the heavy antiquark-initiated contribution d(2). We write:
Q
C
d(2) =C C − A F(CA,CF)+C T F(CFTR). (56)
F F F R
Q 2 Q Q
(cid:18) (cid:19)
The functions F(CA,CF) and F(CFTR) read:
Q Q
1+z2 π2
F(CA,CF) = −4Li (−z)+ln2(z)−4ln(1+z)ln(z)− +2(1+z)ln(z)+4(1−z) L
2
Q 1+z 3
(cid:26) (cid:18) (cid:19) (cid:27)
1+z2 ln(z)3
+ 4Li (z)−2Li (−z)+12S (−z)+ −2ln(z)Li (z)−2ln(z)Li (−z)
3 3 1,2 2 2
1+z 2
(cid:18)
2π2
+ 12ln(1+z)Li (−z)−3ln(1+z)ln(z)2+6ln2(1+z)ln(z)− ln(z)−7ζ(3)+π2ln(1+z)
2
3
(cid:19)
4(12z2+3z4+8z3+1+12z) 2(3z2−1−10z) z(−3z+3+z2+3z3)
− Li (z)+ Li (−z)− ln2(z)
3(1+z)3 2 3(1+z) 2 3(1+z)3
4(12z2+3z4+8z3+1+12z) 2(3z2−1−10z)
− ln(1−z)ln(z)+ ln(1+z)ln(z)
3(1+z)3 3(1+z)
−30z2+15z4−9−28z+36z3 30z2+28z3+15z4+36z+3 45z3−29z+29z2−45
+ ln(z)+ π2+ ,
6(1+z)3 18(1+z)3 6(1+z)2
(57)
and
4z2+3z−3 2 8z2+27z+15
F(CFTR) = (1+z)ln(z)− + L2+ (1+z)ln2(z)− ln(z)
Q 6 3z 3
(cid:26) (cid:27) (cid:26)
56z2+36z−72 20 1
+ − L+(1+z) 4ln(z)Li (z)+8ln(z)Li (−z)−8Li (z)−16Li (−z)+ ln3(z)−4ζ(3)
2 2 3 3
9 9z 6
(cid:27) (cid:18) (cid:19)
4(z−1)(z2+4z+1) 8z2+27z+15
+ (Li (z)+2Li (−z)+ln(1−z)ln(z)+2ln(1+z)ln(z))− ln2(z)
2 2
3z 12
20z5+351z+48+147z4+489z3+713z2 400z4−963z+1331z3−819+499z2 56
+ ln(z)− + . (58)
9(1+z)3 54(1+z)2 27z
