Azimuthal Asymmetries in DIS as a Probe of Intrinsic Charm Content of the Proton

0.41 MB·English

by  L. N. Ananikyan

Azimuthal Asymmetries in DIS as a Probe
of Intrinsic Charm Content of the Proton
L.N. Ananikyan∗ and N.Ya. Ivanov†
Yerevan Physics Institute, Alikhanian Br.2, 375036 Yerevan, Armenia
(Dated: February 2, 2008)
We calculate the azimuthal dependence of the heavy-quark-initiated O(αs) contributions to the
lepton-nucleondeepinelasticscattering(DIS).Itisshownthat,contrarytothephoton-gluonfusion
(GF) component, the photon-quark scattering (QS) mechanism is practically cos2ϕ-independent.
WeinvestigatethepossibilitytodiscriminateexperimentallybetweentheGFandQScontributions
using their strongly different azimuthal distributions. Our analysis shows that the GF and QS
predictions for the azimuthal cos2ϕ asymmetry are quantitatively well defined in the fixed flavor
number scheme: they are stable, both parametrically and perturbatively. We conclude that mea-
surementsoftheazimuthaldistributionsatlargeBjorkenxcoulddirectlyprobetheintrinsiccharm
content of the proton. As to the variable flavor numberschemes, the charm densities of the recent
7
CTEQ and MRSTsets ofparton distributionshaveadramatic impact on thecos2ϕasymmetry in
0
thewhole region of x and,for this reason, can easily bemeasured.
0
2
PACSnumbers: 12.38.-t,13.60.-r,13.88.+e

n Keywords: PerturbativeQCD,HeavyFlavorLeptoproduction, IntrinsicCharm,AzimuthalAsymmetries
a
J

0
1
I. INTRODUCTION

1
v
The notionofthe intrinsiccharm(IC)contentofthe protonhasbeenintroducedover25yearsagoinRefs [1,2]. It
6
wasshownthat,inthelight-coneFockspacepicture[3,4],itisnaturaltoexpectafive-quarkstatecontributiontothe
7
0 proton wave function. The probability to find in a nucleon the five-quark component uudcc¯ is of higher twist since
| i
1 it scales as 1/m2 where m is the c-quark mass [5]. This component can be generated by gg cc¯fluctuations inside
→
0 the proton where the gluons are coupled to different valence quarks. Since all of the quarks tend to travelcoherently
7 at same rapidity in the uudcc¯ bound state, the heaviest constituents carry the largestmomentum fraction. For this
0 | i
reason, one would expect that the intrinsic charm component to be dominate the c -quark production cross sections
/
h at sufficiently large Bjorken x. So, the originalconcept of the charm density in the proton [1, 2] has nonperturbative
p nature and will be referred to in the present paper as nonperturbative IC.
- A decade agoanother point of view on the charmcontentof the protonhas been proposedin the frameworkof the
p
variable flavor number scheme (VFNS) [6, 7]. The VFNS is an approach alternative to the traditional fixed flavor
e
h number scheme (FFNS) where only light degrees of freedom (u,d,s and g) are considered as active. It is well known
: that a heavy quark production cross section contains potentially large logarithms of the type α ln Q2/m2 whose
v s
i contribution dominates at high energies, Q2 . Within the VFNS, these mass logarithms are resummed through
X theallordersintoaheavyquarkdensitywhic→he∞volveswithQ2 accordingtothestandardDGLAP[8,(cid:0)9,10]ev(cid:1)olution
r equation. Hence the VFN schemes introduce the parton distribution functions (PDFs) for the heavy quarks and
a
change the number of active flavorsby one unit when a heavy quark thresholdis crossed. We cansay that the charm
density arises within the VFNS perturbatively via the g cc¯evolution and will call it the perturbative IC.
→
Presently,bothperturbativeandnonperturbativeICarewidelyusedforaphenomenologicaldescriptionofavailable
data. (A recent review of the theory and experimental constraints on the charm quark distribution can be found in
Refs.[11,12]. SeealsoAppendixCinthepresentpaper). Inparticular,practicallyalltherecentversionsoftheCTEQ
[13] andMRST [14]sets of PDFs are basedon the VFN schemes and containa charmdensity. At the same time, the
keyquestionremainsopen: Howtomeasurethe intrinsiccharmcontentofthe proton? The basictheoreticalproblem
is that radiative corrections to the fixed order predictions for the production cross sections are large. In particular,
the next-to-leading order (NLO) corrections increase the leading order (LO) results for both charm and bottom
productioncrosssectionsby approximatelya factoroftwo atenergiesofthe fixedtargetexperiments. Moreover,soft
gluon resummation of the threshold Sudakov logarithms indicates that higher-order contributions are also essential.
(ForareviewseeRefs.[15,16]). Ontheotherhand,perturbativeinstabilityleadstoahighsensitivityofthetheoretical
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
2
calculations to standarduncertainties in the input QCD parameters. For this reason,it is difficult to comparepQCD
results for spin-averaged cross sections with experimental data directly, without additional assumptions. The total
uncertainties associated with the unknown values of the heavy quark mass, m, the factorization and renormalization
scales, µ and µ , Λ andthe PDFs are so large that one can only estimate the order ofmagnitude of the pQCD
F R QCD
predictions for production cross sections [17, 18].
Sinceproductioncrosssectionsarenotperturbativelystable,itisofspecialinteresttostudythoseobservablesthat
are well-defined in pQCD. A nontrivial example of such an observable was proposed in Refs. [19, 20, 21, 22] where
the azimuthal cos2ϕ asymmetry in heavy quark photo- and leptoproduction has been analyzed 1. In particular, the
Born level results have been considered [19] and the NLO soft-gluon corrections to the basic mechanism, photon-
gluon fusion (GF), have been calculated [20, 21]. It was shown that, contrary to the production cross sections,
the azimuthal asymmetry in heavy flavor photo- and leptoproduction is quantitatively well defined in pQCD: the
contribution of the dominant GF mechanism to the asymmetry is stable, both parametrically and perturbatively.
Therefore, measurements of this asymmetry would provide an ideal test of pQCD. As was shown in Ref. [22], the
azimuthalasymmetryinopencharmphotoproductioncouldhavebeenmeasuredwithanaccuracyofabouttenpercent
in the approved E160/E161experiments at SLAC [23] using the inclusive spectra of secondary (decay) leptons.
In the present paper we study the IC contribution to the azimuthal asymmetries in heavy quark leptoproduction:
l(ℓ)+N(p) l(ℓ q)+Q(p )+X[Q](p ). (1)
Q X
→ −
Neglecting the contribution of Z boson as well as the target mass effects, the cross section of the reaction (1) for
−
unpolarized initial states may be written as
d3σ α 1 y2
lN = em σ (x,Q2)+εσ (x,Q2)+εσ (x,Q2)cos2ϕ+2 ε(1+ε)σ (x,Q2)cosϕ , (2)
dxdQ2dϕ (2π)2xQ21 ε T L A I
− h p i
The quantity ε measures the degree of the longitudinal polarization of the virtual photon in the Breit frame [24],
2(1 y)
ε= − , (3)
1+(1 y)2
−
and the kinematic variables are defined by
Q2
S¯=(ℓ+p)2, Q2 = q2, x= ,
− 2p q
·
p q 4m2
y = · , Q2 =xyS¯, ρ= . (4)
p ℓ S¯
·
The cross sections σ (i=T,L,A,I) in Eq. (2) are related to the structure functions F (x,Q2) as follows:
i i
Q2
F (x,Q2) = σ (x,Q2), (i=T,L,A,I)
i 8π2α x i
em
Q2
F (x,Q2) = σ (x,Q2), (5)
2 4π2α 2
em
whereF =2x(F +F )andσ =σ +σ . InEq.(2),σ (σ )istheusualγ∗N crosssectiondescribingheavyquark
2 T L 2 T L T L
production by a transverse(longitudinal) virtual photon. The third cross section, σ , comes about from interference
A
between transverse states and is responsible for the cos2ϕ asymmetry which occurs in real photoproduction using
linearlypolarizedphotons[19, 20, 22]. The fourthcrosssection,σ ,originatesfrominterferencebetween longitudinal
I
and transverse components [24]. In the nucleon rest frame, the azimuth ϕ is the angle between the lepton scattering
planeandthe heavyquarkproductionplane,definedbythe exchangedphotonandthe detectedquarkQ(see Fig.1).
The covariantdefinition of ϕ is
r n Q2 1/x2+4m2 /Q2
cosϕ = · , sinϕ= N n ℓ, (6)
√ r2√ n2 2√ r2√ n2 ·
p
− − − −
rµ = εµναβp q ℓ , nµ =εµναβq p p . (7)
ν α β ν α Qβ
1 The well-known examples are the shapes of differential cross sections of heavy flavor production which are sufficiently stable under
radiativecorrections.
3
’
l l
γ ∗
N
Q
’
l l
ϕ
FIG. 1: Definition of theazimuthal angle ϕ in thenucleon rest frame.
InEqs.(4)and(6),mandm arethe massesoftheheavyquarkandthe target,respectively. Usually,the azimuthal
N
asymmetry associated with the cos2ϕ distribution, A (ρ,x,Q2), is defined by
2ϕ
d3σ (ϕ=0)+d3σ (ϕ=π) 2d3σ (ϕ=π/2)
A (ρ,x,Q2) = 2 cos2ϕ (ρ,x,Q2)= lN lN − lN
2ϕ h i d3σ (ϕ=0)+d3σ (ϕ=π)+2d3σ (ϕ=π/2)
lN lN lN
εσ (x,Q2) ε+εR(x,Q2)
= A =A(x,Q2) , (8)
σ (x,Q2)+εσ (x,Q2) 1+εR(x,Q2)
T L
d3σ
where d3σ (ϕ) lN (ρ,x,Q2,ϕ) and the mean value of cosnϕ is
lN ≡ dxdQ2dϕ
2π d3σ
dϕcosnϕ lN (ρ,x,Q2,ϕ)
dxdQ2dϕ
cosnϕ (ρ,x,Q2)= 0 . (9)
h i R 2π d3σ
dϕ lN (ρ,x,Q2,ϕ)
dxdQ2dϕ
0
R
In Eq. (8), the quantities R(x,Q2) and A(x,Q2) are defined as
σ F
R(x,Q2) = L(x,Q2)= L(x,Q2), (10)
σ F
T T
σ F
A(x,Q2) = A(x,Q2)=2x A(x,Q2). (11)
σ F
2 2
Likewise, we can define the azimuthal asymmetry associated with the cosϕ distribution, A (ρ,x,Q2):
ϕ
2d3σ (ϕ=0) 2d3σ (ϕ=π)
A (ρ,x,Q2) = 2 cosϕ (ρ,x,Q2)= lN − lN
ϕ h i d3σ (ϕ=0)+d3σ (ϕ=π)+2d3σ (ϕ=π/2)
lN lN lN
2 ε(1+ε)σ (x,Q2) 1+R(x,Q2)
= I =A (x,Q2) ε(1+ε)/2 , (12)
σ (x,Q2)+εσ (x,Q2) I 1+εR(x,Q2)
Tp L
p
where
σ F
A (x,Q2)=2√2 I(x,Q2)=4√2x I(x,Q2). (13)
I
σ F
2 2
Remember that y 1 in most of the experimentally reachable kinematic range. Taking also into account that
ε=1+ (y2), we fi≪nd:
O
A (ρ,x,Q2)=A(x,Q2)+ (y2), A (ρ,x,Q2)=A (x,Q2)+ (y2). (14)
2ϕ ϕ I
O O
4
So, like the σ (x,Q2) cross section in the ϕ-independent case, it is the parameters A(x,Q2) and A (x,Q2) that can
2 I
effectively be measured in the azimuth-dependent production.
In this paper we concentrate on the azimuthal asymmetry A(x,Q2) associated with the cos2ϕ-distribution. We
have calculated the IC contribution to the asymmetry which is described at the parton level by the photon-quark
scattering (QS) mechanism given in Fig. 2. Our main result can be formulated as follows:
⋆ Contrary to the basic GF component, the IC mechanism is practically cos2ϕ-independent. This is due to the
fact that the QS contribution to the σ (x,Q2) cross section is absent (for the kinematic reason) at LO and is
A
negligibly small (of the order of 1%) at NLO.
Astotheϕ-independentcrosssections,ourpartonlevelcalculationshavebeencomparedwiththepreviousresultsfor
theICcontributiontoσ (x,Q2)andσ (x,Q2)presentedinRefs.[25,26]. Apartfromtwotrivialmisprintsuncovered
2 L
in [25] for σ (x,Q2), a complete agreement between all the considered results is found.
L
Since the GF and QS mechanisms have strongly different cos2ϕ-distributions, we investigate the possibility to
discriminate between their contributions using the azimuthal asymmetry A(x,Q2). We analyze separately the non-
perturbative IC in the framework of the FFNS and the perturbative IC within the VFNS.
The following properties of the nonperturbative IC contribution to the azimuthal asymmetry within the FFNS are
found:
The nonperturbative IC is practically invisible at low x, but affects essentially the GF predictions at large x.
•
The dominance of the cos2ϕ-independent IC component at large x leads to a more rapid (in comparison with
the GF predictions) decreasing of A(x,Q2) with growth of x.
Contraryto the productioncrosssections,the cos2ϕasymmetryin charmazimuthaldistributions ispractically
•
insensitive to radiative corrections at Q2 m2. Perturbative stability of the combined GF+QS result for
A(x,Q2) is mainly due to the cancellation o∼f large NLO corrections in Eq. (11).
pQCD predictions for the cos2ϕ asymmetry are parametrically stable; the GF+QS contribution to A(x,Q2) is
•
practically insensitive to most of the standard uncertainties in the QCD input parameters: µ , µ , Λ and
R F QCD
PDFs.
Nonperturbativecorrectionstothecharmazimuthalasymmetryduetothegluontransversemotioninthetarget
• are of the order of 20% at Q2 m2 and rapidly vanish at Q2 >m2.
≤
WeconcludethatthecontributionsofbothGFandICcomponentstothecos2ϕasymmetryincharmleptoproduction
are quantitatively well defined in the FFNS: they are stable, both parametrically and perturbatively, and insensitive
(at Q2 >m2) to the gluon transverse motion in the proton. At large Bjorken x, the A(x,Q2) asymmetry could be a
sensitive probe of the nonperturbative IC.
The perturbative IC has been considered within the VFNS proposed in Refs. [6, 7]. The following features of the
azimuthal asymmetry should be emphasized:
Contrary to the nonperturbative IC component, the perturbative one is significant practically at all values of
∗ Bjorken x and Q2 >m2.
The charm densities of the recent CTEQ and MRST sets of PDFs lead to a sizeable reduction (by about 1/3)
∗
of the GF predictions for the cos2ϕ asymmetry.
We conclude that impact ofthe perturbative IC onthe cos2ϕasymmetry is sizeable in the whole regionof x and, for
this reason, can easily be detected.
Concerning the experimental aspects, azimuthal asymmetries in charm leptoproduction can, in principle, be mea-
sured in the COMPASS experiment at CERN, as well as in future studies at the proposedeRHIC [27, 28] and LHeC
[29] colliders at BNL and CERN, correspondingly.
The paper is organized as follows. In Section II we analyze the QS and GF parton level predictions for the ϕ-
dependent charm leptoproduction in the single-particle inclusive kinematics. In particular, we discuss our results for
the NLO QS cross sections and compare them with available calculations. Hadron level predictions for A(x,Q2) are
giveninSectionIII. WeconsidertheICcontributionstotheasymmetrywithintheFFNSandVFNSinawideregion
of x and Q2. Some details of our calculations of the QS cross sections are presented in Appendix A. An overview of
the soft-gluonresummation for the photon-gluonfusion mechanism is givenin Appendix B. Some experimentalfacts
in favor of the nonperturbative IC are briefly listed in Appendix C.
5
γ∗ γ∗ γ∗
Q Q
Q Q g
g
Q Q
(a)
(b)
γ∗ γ∗ γ∗
Q Q Q Q Q Q
(c)
FIG. 2: TheLO (a) and NLO (b and c) photon-quarkscattering diagrams.
II. PARTONIC CROSS SECTIONS
A. Quark Scattering Mechanism
The momentum assignment of the deep inelastic lepton-quark scattering will be denoted as
l(ℓ)+Q(k ) l(ℓ q)+Q(p )+X(p ). (15)
Q Q X
→ −
Taking into account the target mass effects, the corresponding partonic cross section can be written as follows [30]
d3σˆ α y2 √1+4λz2
lQ em
= σˆ (z,λ) (1 εˆ)σˆ (z,λ)+εˆσˆ (z,λ)cos2ϕ+2 εˆ(1+εˆ)σˆ (z,λ)cosϕ .
dzdQ2dϕ (2π)2zQ2 1 εˆ 2,Q − − L,Q A,Q I,Q
− h p (1i6)
In Eq. (16), we use the following definition of partonic kinematic variables:
q k Q2 m2
Q
y = · , z = , λ= . (17)
ℓ k 2q k Q2
Q Q
· ·
In the massive case, the (virtual) photon polarization parameter, εˆ, has the form [30]
2(1 y λz2y2)
εˆ= − − . (18)
1+(1 y)2+2λz2y2
−
At leading order, (α ), the only quark scattering subprocess is
em
O
γ∗(q)+Q(k ) Q(p ). (19)
Q Q
→
The γ∗Q cross sections, σˆ(0) (k =2,L,A,I), corresponding to the Born diagram (see Fig. 2a) are:
k,Q
σˆ(0)(z,λ) = σˆ (z) 1+4λz2δ(1 z),
2,Q B −
σˆ(0) (z,λ) = σˆ (z)p 4λz2 δ(1 z), (20)
L,Q B √1+4λz2 −
σˆ(0) (z,λ) = σˆ(0)(z,λ)=0,
A,Q I,Q
6
with
(2π)2e2α
σˆ (z)= Q em z, (21)
B Q2
where e is the quark charge in units of electromagnetic coupling constant.
Q
To take into account the NLO (α α ) contributions, one needs to calculate the virtual corrections to the Born
em s
O
process (given in Fig. 2c) as well as the real gluon emission (see Fig. 2b):
γ∗(q)+Q(k ) Q(p )+g(p ). (22)
Q Q g
→
The NLO ϕ-dependent crosssections, σˆ(1) andσˆ(1), aredescribedby the realgluonemissiononly. Corresponding
A,Q I,Q
contributions are free of any type of singularities and the quantities σˆ(1) and σˆ(1) can be calculated directly in four
A,Q I,Q
dimensions.
In the ϕ-independent case, σˆ(1) and σˆ(1) , we also work in four dimensions. The virtual contribution (Fig. 2c)
2,Q L,Q
containsultraviolet(UV) singularitythatis removedusing the on-mass-shellregularizationscheme. Inparticular,we
calculate the absorptive part of the Feynman diagram which has no UV divergences. The real part is then obtained
by using the appropriate dispersion relations. As to the infrared (IR) singularity, it is regularized with the help of
an infinitesimal gluon mass. This IR divergence is cancelled when we add the bremsstrahlung contribution (Fig. 2b).
Some details of our calculations are given in Appendix A.
The final (real+virtual) results for γ∗Q cross sections can be cast into the following form:
α 1+2λ
σˆ(1)(z,λ)= sC σˆ (1)√1+4λδ(1 z) 2+4lnλ √1+4λ lnr+ 2Li (r2)+4Li ( r)
2,Q 2π F B − − − √1+4λ 2 2 −
n h
+3ln2(r) 4lnr+4lnrln(1+4λ) 2lnrlnλ
− −
α 1 1 io
+ sC σˆ (z) 1 3z 4z2+6z3+8z4 8z5
4π F B (1+4λz2)3/2 [1 (1 λ)z]2 − − −
(cid:26) − − h
+6λz 3 18z+13z2+10z3 8z4
− −
+4λ2z(cid:0)2 8 77z+65z2 2z3 (cid:1) (23)
− −
+16λ3z3(cid:0) 1 21z+12z2 1(cid:1)28λ4z5
− −
2lnD(z,λ) (cid:0) (cid:1) i
+ 1+z+2z2+2z3 +2λz 2 11z 11z2 +8λ2z2(1 9z)
√1+4λz2 − − − −
h (cid:0) 8(1+4(cid:1)λ)2z4 (cid:0)4(1+2λ)(1+4(cid:1)λ)2z4lnD(z,λ) i
,
− (1 z) − √1+4λz2 (1 z)
− + − + (cid:27)
α 2λ 4λ 1+2λ
σˆ(1) (z,λ)= sC σˆ (1) δ(1 z) 2+4lnλ lnr+ 2Li (r2)+4Li ( r)
L,Q π F B √1+4λ − − − √1+4λ √1+4λ 2 2 −
n h
+3ln2(r) 4lnr+4lnrln(1+4λ) 2lnrlnλ
− −
α 1 z io
+ sC σˆ (z) (1 z)2 λz 13 19z 2z2+8z3
π F B (1+4λz2)3/2 [1 (1 λ)z]2 − − − −
(cid:26) − − h (cid:0) (cid:1)
2λ2z2 31 39z+8z2 (24)
− −
8λ3z3(cid:0)(10 7z) 32λ(cid:1)4z4
− − −
2λz2lnD(z,λ) i
[3+3z+16λz]
− √1+4λz2
8λ(1+4λ)z4 4λ(1+2λ)(1+4λ)z4lnD(z,λ)
,
− (1 z) − √1+4λz2 (1 z)
− + − + (cid:27)
α z(1 z) 1 2λlnD(z,λ)
σˆ(1) (z,λ)= sC σˆ (z) − 1+2λ(4 3z)+8λ2z + [2+z+4λz] , (25)
A,Q 2π F B (1+4λz2)3/2 [1 (1 λ)z] − √1+4λz2
(cid:26) − − (cid:27)
(cid:2) (cid:3)
7
α 1 √z
σˆ(1)(z,λ)= s C σˆ (z) (1 z)(1+2z) 4λz 10 10z z2+2z3
I,Q 8√2 F B (1+4λz2)2[1 (1 λ)z]3/2 − − − − −
(cid:26)
− − (cid:0) (cid:1)
8λ2z2 25 29z+8z2 96λ3z3(3 2z) 128λ4z4 (26)
− − − − −
+8 λz[1 (1 λ)z] 1 z(cid:0)2+λz(13 11z(cid:1))+4λ2z2(7 4z)+16λ3z3 .
− − − − −
(cid:27)
p (cid:2) (cid:3)
In Eqs. (23-26), C =(N2 1)/(2N ), where N is number of colors, while
F c − c c
1+2λz √1+4λz2 √1+4λ 1
D(z,λ)= − , r = D(z =1,λ)= − . (27)
1+2λz+√1+4λz2 √1+4λ+1
p
The so-called ”plus” distributions are defined by
1
[g(z)] =g(z) δ(1 z) dζg(ζ). (28)
+ − −
Z
0
For any sufficiently regular test function h(z), Eq. (28) gives
1 1
lnk(1 z) lnk(1 z) lnk+1(1 a)
dzh(z) − = dz − [h(z) h(1)]+h(1) − . (29)
" 1 z # 1 z − k+1
Za − + Za −
To performa numericalinvestigationof the inclusive partonic crosssections, σˆ (k =T,L,A,I), it is convenient
k,Q
to introduce the dimensionless coefficient functions c(n,l),
k,Q
σˆ (η,λ,µ2)= e2Qαemαs(µ2) ∞ 4πα (µ2) n n c(n,l)(η,λ)lnl µ2 , (30)
k,Q m2 s k,Q m2
n=0 l=0 (cid:18) (cid:19)
X(cid:0) (cid:1) X
where µ is a factorization scale (we use µ = µ = µ ) and the variable η measures the distance to the partonic
F R
threshold:
s 1 z
η = 1= − , s=(q+k )2. (31)
m2 − λz Q
Our analysis of the quantity c(0,0)(η,λ) is given in Fig. 3. One can see that c(0,0) is negative at low Q2 (λ−1 < 1)
A,Q A,Q
and positive at high Q2 (λ−1 >20). For the intermediate values of Q2, c(0,0)(η,λ) is an alternating function of η∼.
A,Q
Our resultsfor the coefficientfunctionc(0,0)(η,λ) atseveralvalues ofλ arepresentedinFig.3. It is seenthatc(0,0)
I,Q I,Q
is negative at all values of η and λ. Note also the threshold behavior of the coefficient function:
√λ
c(0,0)(η 0,λ)= √2π2C + (η). (32)
I,Q → − F1+4λ O
This quantity takes its minimum value at λ =1/4: c(0,0)(η =0,λ )= π2C / 2√2 .
m I,Q m − F
(cid:0) (cid:1)
B. Comparison with Available Results
For the first time, the NLO (α α ) corrections to the ϕ-independent IC contribution have been calculated a
em s
O
long time ago by Hoffmann and Moore (HM) [25]. However, authors of Ref. [25] don’t give explicitly their definition
of the partonic cross sections that leads to a confusion in interpretation of the original HM results. To clarify the
situation,weneedfirsttoderivetherelationbetweenthelepton-quarkDIScrosssection,dσˆ ,andthepartoniccross
lQ
sections, σ(2) and σ(L), used in [25]. Using Eqs. (C.1) and (C.5) in Ref. [25], one can express the HM tensor σµν in
R
terms of”our”crosssectionsσˆ andσˆ definedby Eq.(16) inthe presentpaper. Comparingthe obtainedresults
2,Q L,Q
8
0
0.2
−1
0
λ) λ) −2
(0,0)η(,cA,Q −0.2 DSoottleidd:: λλ−−11==14 (0,0)η(,cI,Q −3 DSoottleidd:: λλ−−11==41
Dashed: λ−1=10 Dashed: λ−1=10
−0.4 Dash−dotted: λ−1=20 −4 Dash−dotted: λ−1=20
Long−dashed: λ−1=100 Long−dashed: λ−1=100
−5
0.001 0.01 0.1 1 10 100 1000 0.001 0.01 0.1 1 10 100 1000
η= s/m2−1 η= s/m2−1
FIG. 3: c(0,0)(η,λ) and c(0,0)(η,λ) coefficient functions at several values of λ.
A,Q I,Q
with the corresponding definition of σµν via the HM cross sections σ(2) and σ(L) (given by Eqs. (C.16) and (C.17) in
R
Ref. [25]), we find that
σˆ (z,λ) σˆ (z) 1+4λz2σ(2)(z,λ), (33)
2,Q B
≡
σˆ (z,λ) 2σˆBp(z) σ(L)(z,λ)+2λz2σ(2)(z,λ) . (34)
L,Q
≡ √1+4λz2
h i
Now we are able to compare our results with original HM ones. It is easy to see that the LO cross sections (defined
byEqs.(37)in[25]andEqs.(20)inourpaper)obeybothaboveidentities. Comparingwitheachotherthe quantities
σ(2) andσˆ(1) (givenby Eq.(51)in[25]andEq.(23)inthis paper,respectively),we findthatidentity (33)is satisfied
1 2,Q
atNLO too. The situation with longitudinalcrosssections is more complicated. We have uncoveredtwo misprints in
the NLO expression for σ(L) given by Eq. (52) in [25]. First, the r.h.s. of this Eq. must be multiplied by z. Second,
the sign in front of the last term (proportional to δ(1 z)) in Eq. (52) in Ref. [25] must be changed 2. Taking into
−
accountthesetypos,wefindthatrelation(34)holdsatNLOaswell. So,ourcalculationsofσˆ andσˆ agreewith
2,Q L,Q
the HM results.
Recently, the heavy quark initiated contributions to the ϕ-independent DIS structure functions, F and F , have
2 L
been calculated by Kretzer and Schienbein (KS) [26]. The final KS results are expressed in terms of the parton level
structure functions Hˆq and Hˆq. Using the definition of Hˆq and Hˆq given by Eqs. (7, 8) in Ref. [26], we obtain that
1 2 1 2
α σˆ (z) Hˆq(ξ′,λ) α 1+4λ
σˆ (z,λ) s B 1 , σˆ (z,λ) sσˆ (z) Hˆq(ξ′,λ), (35)
T,Q ≡ 2π√1+4λ√1+4λz2 2,Q ≡ 2π B 1+4λz2 2
r
whereσˆ =σˆ σˆ andσˆ aredefinedbyEq.(16)inourpaperandξ′ =z 1+√1+4λ 1+√1+4λz2 .
T,Q 2,Q L,Q L,Q
−
Totestidentities (35), oneneeds onlyto rewritethe NLOexpressionsfor the functions Hˆq(ξ′,λ) andHˆq(ξ′,λ) (given
(cid:0) 1 (cid:1)(cid:14)(cid:0) 2 (cid:1)
in Appendix C in Ref. [26]) in terms of variables z and λ. Our analysis shows that relations (35) hold at both LO
and NLO. Hence we coincide with the KS predictions for the γ∗Q cross sections.
However, we disagree with the conclusion of Ref. [26] that there are errors in the NLO expression for σ(2) given in
Ref.[25]3. Asexplainedabove,acorrectinterpretationofthequantitiesσ(2) andσ(L) usedin[25]leadstoacomplete
agreement between the HM, KS and our results for ϕ-independent cross sections.
As to the ϕ-dependent DIS, pQCD predictions for the γ∗Q cross sections σˆ (z,λ) and σˆ (z,λ) in the case
A,Q I,Q
of arbitrary values of m2 and Q2 are not, to our knowledge, available in the literature. For this reason, we have
performed several cross checks of our results against well known calculations in two limits: m2 0 and Q2 0.
→ →
In particular, in the chiral limit, we reproduce the original results of Georgi and Politzer [32] and M´endez [33] for
2 NotethatthistermoriginatesfromvirtualcorrectionsandthevirtualpartofthelongitudinalcrosssectiongivenbyEq.(39)inRef.[25]
alsohaswrongsign. SeeAppendixAformoredetails.
3 Indetail,theKSpointofviewontheHMresultsispresentedinPhDthesis[31],pp.158-160.
9
γ∗ γ∗
Q Q¯
Q¯ Q
g g
FIG. 4: The LO photon-gluon fusion diagrams.
σˆ (z,λ 0) and σˆ (z,λ 0). In the case of Q2 0, our predictions for σˆ (s,Q2 0) and σˆ (s,Q2 0)
I,Q A,Q 2,Q A,Q
→ → → → →
given by Eqs. (23,25) reduce to the QED textbook results for the Compton scattering of polarized photons [34].
C. Photon-Gluon Fusion
The gluon fusion component of the semi-inclusive DIS is the following parton level interaction:
l(ℓ)+g(k ) l(ℓ q)+Q(p )+X[Q](p ). (36)
g Q X
→ −
Corresponding lepton-gluon cross section, dσˆ , has the following decomposition in terms of the helicity γ∗g cross
lg
sections:
d3σˆ α 1 y2
lg em
= σˆ (z,λ) (1 ε)σˆ (z,λ)+εσˆ (z,λ)cos2ϕ+2 ε(1+ε)σˆ (z,λ)cosϕ , (37)
dzdQ2dϕ (2π)2zQ21 ε 2,g − − L,g A,g I,g
− h p i
where the quantity ε is defined by Eq. (3) with y = (q k )/(ℓ k ).
g g
· ·
At LO, (α α ), the only gluon fusion subprocess responsible for heavy flavor production is
em s
O
γ∗(q)+g(k ) Q(p )+Q(p ). (38)
g Q
→ Q
Theγ∗g crosssections,σˆ(0) (k =2,L,A,I),correspondingtotheBorndiagramsgiveninFig.4havetheform[35,36]:
k,g
α 1+β
σˆ(0)(z,λ) = sσˆ (z) (1 z)2+z2+4λz(1 3z) 8λ2z2 ln z [1+4z(1 z)(λ 2)]β ,
2,g 2π B − − − 1 β − − − z
z
2α n(cid:2) 1+β (cid:3) − o
σˆ(0)(z,λ) = sσˆ (z)z 2λzln z +(1 z)β ,
L,g π B − 1 β − z
z
n − o
α 1+β
σˆ(0)(z,λ) = sσˆ (z)z 2λ[1 2z(1+λ)]ln z +(1 2λ)(1 z)β , (39)
A,g π B − 1 β − − z
z
n − o
σˆ(0)(z,λ) = 0,
I,g
where σˆ (z) is defined by Eq. (21) and the following notations are used:
B
Q2 m2 4λz
z = , λ= , β = 1 . (40)
2q k Q2 z − 1 z
· g r −
Note that the cosϕ dependence vanishes in the GF mechanism due to the Q Q symmetry which, at leading order,
↔
requires invariance under ϕ ϕ+π [37].
→
As to the NLO results, presently, only ϕ-independent quantities σˆ(1) and σˆ(1) are known exactly [38]. For this
2,g L,g
reason,wewilluseinouranalysistheso-calledsoft-gluonapproximationfortheNLOγ∗gcrosssections(seeAppendix
B). As shown in Refs. [20, 21, 39], at energies not so far from the production threshold, the soft-gluon radiation is
the dominant perturbative mechanism in the γ∗g interactions.
10
III. HADRON LEVEL RESULTS
A. Fixed Flavor Number Scheme and Nonperturbative Intrinsic Charm
In the fixed flavor number scheme 4, the wave function of the proton consists of light quarks u,d,s and gluons
g. Heavy flavor production in DIS is dominated by the gluon fusion mechanism. Corresponding hadron level cross
sections, σ (x,λ), have the form
k,GF
1
σ (x,λ) = dzg(z,µ )σˆ (x/z,λ,µ ), (k =2,L,A,I), (41)
k,GF F k,g F
Z
χ
χ = x(1+4λ), (42)
where g(z,µ ) describes gluon density in the proton evaluated at a factorization scale µ . The lowest order GF
F F
cross sections, σˆ(0) (k =2,L,A,I), are given by Eqs. (39). The NLO results, σˆ(1), to the next-to-leading logarithmic
k,g k,g
accuracy are presented in Appendix B.
We neglect the γ∗q(q¯) fusion subprocesses. This is justified as their contributions to heavy quark leptoproduction
vanish at LO and are small at NLO [38].
IntheFFNS,theintrinsicheavyflavorcomponentoftheprotonwavefunctionisgeneratedbygg QQ¯ fluctuations
→
where the gluons are coupled to different valence quarks. In the present paper, this component is referred to as the
nonperturbative intrinsic charm (bottom). The probability of the corresponding five-quark Fock state, uudQQ¯ , is
of higher twist since it scales as Λ2 m2 [5]. However, since all of the quarks tend to travel coherently at same
QCD (cid:12) (cid:11)
rapidityinthe uudQQ¯ boundstate,theheaviestconstituentscarrythelargestmomentumfraction. For(cid:12)thisreason,
(cid:14)
the heavy flavor distribution function has a more ”hard” z-behavior than the light parton densities. Since all of the
(cid:12) (cid:11)
densities vanish(cid:12)at z 1, the hardest PDF becomes dominant at sufficiently largez independently of normalization.
→
ConvolutionofPDFswithpartoniccrosssectionsdoesnotviolatethisobservation. Inparticular,assumingagluon
density g(z) (1 z)n (where n = 3 5), we obtain that the LO GF contribution to F scales as (1 χ)n+3/2
2
∼ − − −
at χ 1, where χ is defined by Eq. (42). In the case of Hoffman and Moore charm density (see below), the LO
→
IC contribution is proportional to (1 χ) at χ 1. It is easy to see that, independently of normalizations, the IC
− →
contribution to be dominate over the more ”soft” GF component at large enough x.
Forthefirsttime,theintrinsiccharmmomentumdistributioninthefive-quarkstate uudcc¯ wasderivedbyBrodsky,
| i
Hoyer,PetersonandSakai(BHPS) inthe frameworkofa light-cone model[1, 2]. Neglecting the transversemotionof
constituents, they have obtained in the heavy quark limit that
N
c(z)= 5z2 6z(1+z)lnz+(1 z)(1+10z+z2) , (43)
6 −
(cid:2) (cid:3)
1
where N =36 corresponds to a 1% probability for IC in the nucleon: c(z)dz =0.01.
5 0
Hoffmann and Moore (HM) [25] incorporated mass effects in the BHPS approach. They first introduced a mass
R
scaling variable ξ,
2ax 1+√1+4λ
ξ = , a= , (44)
1+√1+4λ x2 2
N
where λ = m2 Q2. To provide correct threshold behavior of the charm density, the constraint ξ γ < 1 was
N N ≤
imposed where
(cid:14)
2axˆ 1
γ = , xˆ= . (45)
1+√1+4λNxˆ2 1+4λ−λN
Resulting charm distribution function, c(ξ,γ), has the following form in the HM approach:
ξ
c(ξ) c(γ), ξ γ
c(ξ,γ)= − γ ≤ (46)

0, ξ >γ


4 Thisapproachissometimesreferredtoasthefixed-orderperturbationtheory(FOPT).