Heavy flavour contributions to the spin structure function g_1(x,Q^2) and the Bjorken sum r

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Heavy ?avour contributions to the spin structure function g 1(x,Q 2)and the Bj?rken sum rule Author W.L.van Neerven 1Instituut-Lorentz,University of Leiden,P.O.Box 9506,2300RA Leiden,The Netherlands Abstract We discuss the order α2s corrections to the longitudinal spin structure function g 1(x,Q 2,m 2)which are due to heavy ?avour contributions.Here Q denotes the virtuality of the photon and m stands for the heavy ?avour mass.Since the exact heavy quark coe?cient functions are not known yet we have used the asymptotic forms which are strictly speaking only valid in the region Q 2?m 2.However an analysis of the exact and asymp-totic expressions for F NLO 2(x,Q 2,m 2)and g LO 1(x,Q 2,m 2)reveals that the asymptotic forms can be also used at smaller Q 2.It appears that for the region 0.01

sum rule turns out to be very small.

1Introduction

Heavy quark production in deep inelastic electron-proton scattering proceeds via the following reaction (see Fig.1)

e ?(l 1)+P (p )→e ?(l 2)+Q (p 1)(ˉQ (p 1))+′X ′.(1)Where ′X ′denotes any inclusive hadronic ?nal state and V stands for the neutral intermediate vector bosons given by γ,Z .In this paper we treat the case that Q 2?M 2Z so that the process in Fig.1is dominated by the one photon exchange

?

ˉ)′

X ′

Figure 1:Kinematics of charm production in deep inelastic electron-proton scat-tering

mechanism only.If we also integrate over the momentum p 1of the heavy (anti)quark Q the unpolarised cross section is given by

d 2σ

(Q 2)2S eP {1+(1?y )2}F 2(x,Q 2,m 2)?y 2F L (x,Q 2,m 2) .(2)

Here S eP denotes the centre of mass energy of the electron-proton system and the heavy quark component of the spin averaged structure functions are given by F i (x,Q 2,m 2)(i =2,L ).In the case the proton is polarised parallel (→)or anti-parallel (←)with respect to the spin of the incoming electron we obtain the cross section

d 2σ(→)

dx dy =8πα2

2pq y =pq

n f n f k =1e 2k z max x dz z

,μ2)?L S i,q (z,Q 2μ2)+f S g (x m 2,m 2z ,μ2)?

L NS i,q (z,

Q 2μ2) +e 2Q f S q (x m 2,m 2z

,μ2)?H S i,g (z,Q 2

μ2) ,(5)

2

with

z max=Q2

z

Q2≥4m2.(6)

Further n f denotes the number of light?avours andμstands for the mass factor-ization scale which is put to be equal to the renormalization scale.The charges of the light quark k and the heavy quark Q are given by e k and e Q respectively. Notice that n f is determined by all quarks which are lighter than the heavy quark Q and Eq.(5)has to be understood in the?xed?avour number scheme.In Eq.

(5)the gluon density is denoted by f S g(z,μ2)and the singlet(S)and non singlet (NS)combination of quark densities are de?ned by

f S q(z,μ2)=

n f

k=1

[f k(z,μ2)+fˉk(z,μ2)],(7)

f NS k(z,μ2)=f k(z,μ2)+fˉk(z,μ2)?

1

which leads to the coe?cient function H PS,(2)

i,q .Here PS means purely singlet since

this function originates from the partonic process where a gluon(?avour singlet) is exchanged in the t-channel between the light quark q and the heavy quark Q. The second one is the Compton process where the virtual photon is coupled to the

light quark from which one obtains L NS,(2)

i,q =L S,(2)

i,q

.The expressions for the order

αs contribution H S,(1)

i,g

can be found in[1](i=L,2;unpolarised)and[2](i=1;

polarised).The exact analytic expressions for L NS,(2)

i,q are calculated in[3](i=

L,2;unpolarised)and[4](i=1;polarised).The exact heavy quark coe?cient

functions H PS,(2)

i,q and H S,(2)

i,g

have been only calculated in the unpolarised case

(i=L,2)(see[5])but the exact expressions in polarised scattering(i=1)are not known yet.However in[3]and[4]we could derive analytic formulae for the asymptotic heavy quark coe?cient functions up to orderα2s for unpolarised and polarised electroproduction respectively.The asymptotic heavy quark coe?cient functions are de?ned by

H asymp

i,k (x,

Q2

μ2

)=lim

Q2?m2 H exact i,k(x,

Q2

μ2

) .(12)

These asymptotic expressions are very useful because 1They provide us with a check on the exact calculation.

2The exact calculation of H PS,(2)

i,q and H S,(2)

i,g

(i=2,L)is semi-analytic and

they are only available in the form of long computer programs(see[5]). The latter show numerical instabilties at Q2?m2which are remedied by using in this region H asymp

i,k

.

3Further if these asymptotic heavy quark coe?cient functions are substituted in Eq.(5)it turns out that(see[6],[7])

R2(x,Q2,m2)=F asymp

2,Q

(x,Q2,m2)

m2,

m2

m2

ln j Q2

with an analogous expression for L asymp

i,k .There are two ways to determine Eq.

(14).

1Evaluate the phase space integrals and the Feynman integrals in the usual way for Q2?m2.

2Use operator product expansion techniques and the theorem of mass fac-torization.

The last method was applied in[3]and[4]to compute the asymptotic forms of the heavy quark coe?cient functions for unpolarised and polarised scattering respectively.

2Orderα2s contributions to g1(x,Q2,m2)

In this section we will analyse the charm component of the longitudinal spin structure function g1.To compute the latter in NLO the following coe?cient

functions are available:H exact,(1)

1,g (Born),L exact,(2)

1,q

(Compton process),H asymp,(2)

1,g

(gluon bremsstrahlung),H asymp,(2)

1,q

(Bethe-Heitler).Like in unpolarised scattering the gluonic coe?cient functions indicated by H(l)1,g are the most important ones. Therefore deep inelastic electroproduction of charm quarks provide us with an excellent way to determine the gluon density in polarised scattering(see Eq.(5)). However as we will show below the charm component in the spin case is much smaller with respect to the light parton contribution to the structure function as

observed for F2in unpolarised scattering.This is revealed by H asymp,(2)

1,g in Fig.

2where we observe a strongly oscillatory behaviour which is in contrast to its unpolarised analogue shown in Fig.8of[9].Further it satis?es the sum rule

1 0dx H(l)1,g(x,

Q2

μ2

)=0,(15)

which holds in any scheme.Notice that the Born contribution H(1)1,g also satis?es the above relation and we may safely assume that Eq.(15)holds in all orders for general Q2and m2.Further it turns out that the coe?cient functions H(l)1,k (k=q,g)are much smaller than their unpolarised analogues.In particular the large logarithmic terms of the type ln i x

(1?4m2

x -0.04-0.020

0.02

0.04

0.06

0.08

0.1

0.12

x 10

-2

10-410-310-210-11

Figure 3:The charm component of xg 1(x,Q 2,m 2c )at Q 2=50(GeV /c)2.Dotted line:xg exact 1(Born);solid line:xg approx 1(NLO)

.

Choosing the parton density set in [10](Standard Scenario)with n f =3

and Λ=200MeV we have plotted the charm component of g exact 1(Born)and g approx 1(NLO)at Q 2=50(GeV /c)2in Fig.3.From this ?gure we infer that the

charm component becomes maximal in the region 0.01

7

x

10-6

10-5

10-4

10-3

10-2

10-410-310-210-11

Figure 4:The charm component and the light parton contribution to xg 1(x,Q 2,m 2c )at Q 2=50(GeV /c)2.Dotted line:xg approx 1(charm ,NLO);solid line:xg light 1(NLO)

.

3Order α2s contributions to the Bj?rken sum rule

Since we know the exact result for L NS ,(2)1,q [4]we can compute the order α2s cor-

rection to the Bj?rken sum rule due to heavy ?avour contributions.The latter is given by

1

0dx g P 1(x,Q 2,m 2)?g N 1(x,Q 2,m 2) =1

G V | L NS 1,q (1),(16)

where the superscript (1)denotes the ?rst moment (Mellin transform).In order α2s we have the following properties

Q 2?m 2→ L NS ,(2)1,q (1)～Q 2

4π

2 ?4ln(Q 2

6|

G A

where C NS1,q denotes the massless quark coe?cient function.The?rst moment of the latter quantity has been calculated up to orderα3s in[13]and the orderα4s term has been estimated too(see e.g.[14].It turns out that the charm contribution to the Bj?rken sum rule is even smaller than the the estimated orderα4s correction (see[15]).The contributions coming from the bottom and top quark are even smaller which can be attributed to the decoupling of heavy quarks shown in Eq.

17.Finally we have the relation

C NS1,q(n f,x,Q2

m2

,

m2

μ2

),(20)

so that for Q2?m2the large logarithm ln Q2

[8]M.A.G.Aivazis,J.C.Collins,F.I.Olness and W.K.Tung,Phys.Rev.D50

(1994)3102.

[9]W.L.van Neerven,”Charm Production in deep Inelastic Lepton-Hadron

Scattering”,Lectures presented at the XXXVIIth Cracow School of Theo-retical Physics,30th May-10th June,Zakopane,Poland.To be pub.in Acta Phys.Polon.B.

[10]M.Gl¨u ck,E.Reya,M.Stratmann and W.Vogelsang,Phys.Rev.D53(1996)

4775.

[11]C.Adlo?et al.(H1Collaboration),Z.Phys.C72(1996)593.

[12]M.Derrick et al.(ZEUS Collaboration),Phys.Lett.B349(1995)225;

M.Derrick et al.(ZEUS Collaboration),XXVII Int.Conf.on HEP’96, Warsaw(1996);

J.Breitweg et al.(ZEUS Collaboration),DESY-97-089.

[13]https://www.doczj.com/doc/f53987801.html,rin and J.A.M.Vermaseren,Phys.Lett.B259(1991)345.

[14]M.A.Samuel,J.Ellis,M.Karliner,Phys.Rev.Lett.74(1995)4380.

[15]J.Bl¨u mlein and W.L.van Neerven,to appear.

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