PERTURBATIVE QCD AND NUCLEON STRUCTURE FUNCTIONS.

31 Pages·0.23 MB·English

by  J. Kodaira

HUPD- 9504
January 1995
Perturbative QCD and Nucleon Structure
Functions
5
9
9
1
Jiro KODAIRA∗
n
a Dept. of Physics, Hiroshima University
J
6 Higashi-Hiroshima 724, JAPAN
2

1
v
1
Abstract
8
3
We review the basic aspects of the perturbative QCD based on the
1
0 operator product expansion to analyze the nucleon structure functions
5 in a pedagogical way. We explain the non-trivial relation between the
9
QCD results and the parton model especially to understand the polar-
/
h
ized nucleon structure functions which deserve much attentions in recent
p
- years.
p
e
h
:
v
i
X
r
a Invited talk at the
YITP Workshop on “From Hadronic Matter to Quark Matter:
Evolving View of Hadronic Matter”
YITP, Kyoto Japan, October 1994
to be published in the proceedings
∗Electronic address: [e-mail]
1 Introduction
The quantum chromodynamics (QCD) is a theory of the strong interaction. So far
all experimental data are consistent with the predictions of QCD. Especially the
high energy behavior of QCD is believed to be described by the perturbation theory
thanks to the asymptotically free nature of QCD. Among many interesting and
important problems, the spin structure of nucleon has been one of the most exciting
subjects in recent years. The data on the polarized deep inelastic scattering by
the EMC collaboration [1] result in excitement of not only particle physicists but
also nuclear physicists since the data seem to indicate that the nucleon’s spin is not
carried by quarks (partons). This EMC experiment has incited many (particle and
nuclear) physicists to challenge the so-called “spin crisis” problem in QCD. After
a flood of theoretical papers as well as new experiments [2]-[5], our understanding
on this problem is now much more improved: the interpretation of the QCD results
in terms of the parton model is never obvious and simpleminded one may fail: the
axial anomaly plays an important role: etc.
The aim of this talk is to provide a pedagogical introduction to the perturbative
QCD to study the nucleon structure through the deep inelastic process for non-
experts of QCD who are interested in the deep structure of the nucleon. Those who
are familiar with QCD and interested in recent progress in this field are referred to
recent nice article [6] and reviews [7].
In Sect.2 we review the kinematics of the deep inelastic lepton nucleon scattering
process. In Sec.3 the basic approach of the perturbative QCD based on the operator
product expansion and the renormalization group equation will be explained. The
relation between the QCD results and the parton model is discussed. It will be
stressed that the parton density is a “conception” which depends on the renormal-
ization scheme. In Sec.4 we consider the polarized structure functions. Concluding
1
remarks including some subtleties and/or controversial aspects which deserve more
investigations to understand recent experimental data are given in Sec.5.
2 The structure functions
The cross section for the deep inelastic lepton (l(k)) nucleon (N (p)) scattering
l(k)+N (p) l(k′)+X (Fig.1) is given in terms of the leptonic and the hadronic
→
X
k’
p
x
q
p
k
l N
Figure 1: Kinematic variables in inelastic lepton-nucleon scattering.
tensors according to the standard procedure in the field theory.
dσ 1 e2 2
k′ = LµνW ,
0d3k′ k p 4πQ2! µν
·
where we consider only the QED interaction between the lepton and nucleon and
keep only the lowest order in α . q is the momentum transfer from the lepton to
QED
the nucleon and q2 Q2 = (k k′)2 . The leptonic (L ) and the hadronic (W )
µν µν
≡ − −
tensors are defined as follows;
1
L = k,s j (0) k′,s′ k′,s′ j (0) k,s ,
µν µ ν
2 h | | ih | | i
s′
X
1
W = p,S J (0) X X J (0) p,S (2π)4δ4(p p q)
µν µ ν X
2π h | | ih | | i − −
X
X
1
= d4xeiq·x p,S [J (x), J (0)] p,S . (1)
µ ν
2π h | | i
Z
2
with the lepton’s (hadron’s) electromagnetic current j (J ). s(S) is the spin of the
µ ν
lepton (nucleon).
In general, the spin 4-vector of the fermion with mass m and momentum k is
defined as ~s 2 = 1, s2 = 1, s k = 0. So, for the longitudinally polarized (helicity
− ·
) states, we get,
±
1
sµ = (k, 0, 0, k0) , k = ~k 2 .
±m | |
We can use the following approximation for leptons msµ kµ since m 0.
lepton
≃ ± ≃
Using this approximation, we get for the leptonic tensor,
q2
L± = k k′ +k k′ + g iε qλkσ .
µν µ ν ν µ 2 µν ∓ µνλσ
On the other hand, the hadronic tensor contains all the information of the strong
interaction (QCD). Taking into account the various symmetries, namely the Lorentz
invariance, current conservation of the QED current, T and P invariance, we can
write down the general form for W ,
µν
W WS +iWA ,
µν ≡ µν µν
with
q q p q p q W
WS = g µ ν W + p · q p · q 2 ,
µν − µν − q2 ! 1 µ − q2 µ! ν − q2 ν! M2
G
WA = ε qλ SσMG +(p qSσ q Spσ) 2 .
µν µνλσ 1 · − · M
(cid:26) (cid:27)
where M is the mass of the nucleon. WS (WA) is the symmetric (antisymmetric)
µν µν
part in µν and relevant to the unpolarized (polarized) process as shown below. Usu-
ally we define the following dimensionless structure functions (scaling functions):
ν Mν ν2
F W , F W , g G , g G .
1 1 2 2 1 1 2 2
≡ ≡ 2M ≡ 2 ≡ 2
with Mν p q. These structure functions depend on the Q2 and ν or Q2 and x
≡ ·
(Bjorken variable) x 1 = Q2 .
≡ ω 2Mν
3
The explicit formula for the cross sections for this process in the Laboratory
frame corresponding to the configuration in Fig.2 is easily calculated to be [8],
x
k’
s φ
θ
z
k α
S
Figure 2: Momentum and spin configuration in Lab. frame.
dσ±,S dσ¯ dσA
= ,
dE′dΩ dE′dΩ ± dE′dΩ
with
dσ¯ 2α2E′2 θ θ
= 2W sin2 +W cos2 ,
dE′dΩ Q4M 1 2 2 2!
dσA α2E′
= cosα (E +E′cosθ)MG Q2G
1 2
dE′dΩ −MQ2E −
h n o
+sinαcosφ E′sinθ MG +2EG .
1 2
{ }
i
The superscript refers to the lepton’s helicity and E′ k′ and E k . From
± ≡ 0 ≡ 0
these formulae, we can derive the expressions , for example, for the longitudinal
asymmetry which is the difference between the cross section for the nucleon’s spin
being parallel to the lepton’s ( ) and the nucleon’s spin being anti-parallel to the
↑↑
lepton’s ( ) :
↑↓
dσ↑↓ dσ↑↑ 2α2E′
= MG (E +E′cosθ) Q2G .
dE′dΩ − dE′dΩ MQ2E 1 − 2
n o
It is traditional and sometimes convenient to express the structure functions in
terms of the virtual photoabsorption cross sections (in Lab. frame). The definition
4
of the virtual photoabsorption cross section is given by,
πe2
σ ǫµ∗(q)W ǫν(q),
λ ≡ 2M(ν Q2/2M) λ µν λ
−
with the photon’s polarization vector ǫµ. We have adopted the Hand-Berkelman’s
λ
convention. Taking the direction of photon’s momentum q to be z-axis ( note that
~ ~
q = k ), we have the following polarization vector for photons in the nucleon’s rest
6
frame:
qµ = (ν, 0, 0, ν2 +Q2) ,
q
1 1
ǫ = (0, i, 1, 0) , ǫ = ( ν2 +Q2, 0, 0, ν) .
R S
L √2 ∓ √Q2
q
For the unpolarized structure functions W and W , we get the relations:
1 2
πe2 ν2
σ = W 1+ W ,
S 2M(ν Q2/2M) " 2 Q2!− 1#
−
πe2
σ = W .
T 2M(ν Q2/2M) 1
−
where T = R and/or L. The unpolarized lepton-nucleon scattering cross section is
given in terms of σ and σ ,
S T
dσ¯
= Γ (σ +εσ ) ,
dE′dΩ T T S
where
ν2 θ α E′ν Q2/2M
ε−1 = 1+2 1+ tan2 , Γ = − .
Q2! 2 T 2π2 E Q2(1 ε)
−
Here we note that ε means the ratio of T- and S-photon present in the virtual
photon .
To get the expressions for the polarized structure functions G and G , let us
1 2
consider the three types of the photoabsorption processes: σ (ǫ with Sµ =
1/2 L
R
(0,0,0, 1)); σ (ǫ with Sµ = (0,0,0, 1)); σ (the interference between T- and
3/2 R TS
± L ±
5
S- photon with S = 1). It is easy to obtain,
y
±
πe2
σ = [W +MνG Q2G ] ,
1/2 2M(ν Q2/2M) 1 1 − 2
−
πe2
σ = [W MνG +Q2G ] ,
3/2 2M(ν Q2/2M) 1 − 1 2
−
πe2
σ = Q2[MG +νG ] .
TS 2M(ν Q2/2M) 1 2
− q
Note that, σ = 1(σ +σ ) . By defining the asymmetries A and A :
T 2 1/2 3/2 1 2
σ σ MνG Q2G σ MG +νG
A 1/2 − 3/2 = 1 − 2 , A TS = Q2 1 2,
1 2
≡ σ +σ W ≡ σ W
1/2 3/2 1 T q 1
we can write the longitudinal asymmetry A as;
dσ↑↓ dσ↑↑
A dE′dΩ − dE′dΩ = D(A +ηA ).
≡ dσ↑↓ + dσ↑↑ 1 2
dE′dΩ dE′dΩ
where
D = 1−(E′/E)ε , η = ε√Q2 , R σS = 1+ Qν22 W2 −W1.
1+εR E E′ε ≡ σ (cid:16) W(cid:17)
T 1
−
D is called as the depolarization factor and its physical meaning is obvious in
Fig.3. It is easily verified that,
k’
k θ
Z
ψ
q
N
Figure 3: Momentum configuration in inelastic lepton-nucleon scattering.
D = D D ,
1 2
×
6
with
1 (E′/E)ε
D cosψ = − ,
1
≡ √1 ε2
−
√1 ε2 σ
T
D probability to have T photon = − .
2
≡ − σ +εσ
T S
The structure functions g and g can be expressed in terms of A and A .
1 2 1 2
F √Q2 F ν
2 2
g = A + A , g = A A ,
1 1 2 2 2 1
2x(1+R) ν ! 2x(1+R) √Q2 − !
where the following relation has been used,
F Q2
2
F = W = 1+ .
1 1 x(1+R) ν2 !
Here it is to be noted that we have the inequalities from the unitarity arguments ;
A < 1 , A < √R. Furthermore if we consider the scaling region (Bjorken limit),
1 2
| | | |
Q2 = 4M2x2 1 , g is given by,
ν2 Q2 ≪ 1
A F
1 2
g = .
1 ∼
2x(1+R)
This is the basic formula on which the experimental determination of g is based.
1
3 The perturbative QCD
In this section, we review the fundamental aspects of QCD to analyze the structure
functions introduced above. At first, we discuss the general strategy of QCD based
on the operator product expansion (OPE) and the renormalization group equation
(RGE). Next, we consider the relation between the QCD results and the parton
model interpretations of the process we are considering.
7
3.1 Formal approach in the perturbative QCD
The kinematical region which we are interested in is the Bjorken limit, namely
Q2, ν with x = Q2 fixed. In this limit, we can easily recognize that the
→ ∞ 2Mν
hadronic tensor Eq.(1) is governed by the behavior of the current products near
the light-cone. So the light-cone expansion, which is a variant of the OPE, of two
currents might be applied tothis precess. However theOPE can not beused directly
here by the following reason. The OPE makes sense in the short distance limit and
this limit corresponds to the region where all component of q become infinity.
µ
Therefore, Q2 2p q. On the other hand, the physical region for the deep-inelastic
≥ ·
process is Q2 2p q. Fortunately we can overcome this dilemma by using the
≤ ·
dispersion relation [9] which relates the short distance limit to the Bjorken limit for
the deep-inelastic process.
Consider the time-ordered product of two currents which corresponds to the
forward Compton scattering of the virtual photon with “mass q2 ” , (the Lorentz
and the spin structure being neglected),
T = i dxeiq·x p TJ(x)J(0) p ,
h | | i
Z
the physical region of which is 2p q/ q2 = 2Mν/Q2 1. For this process, we can
· | | ≤
use the OPE. In general, the product of two operators (currents) can be expanded
as follows;
TJ(x)J(0) Cn(x2 iε)xµ1xµ2 xµnOi (0), (2)
∼ i − ··· µ1···µn
i,n
X
where Cn is a c-number function called Wilson’s coefficient function and Oi
i µ1µ2···µn
are local composite operators labeled by the index i. The dimensional argument
tells us that Cn(x2) behaves like,
i
Cn(x2) (x2)(dnO−n−dJ−dJ)/2,
i ∼
8
in the concerned limit of x2 0 with xµ small but = 0, where d is the dimension
→ 6
of the corresponding operator. So the operators with the lower twist (τ ):
N
τ dim. spin = dn n
N ≡ − O −
dominate. The Fourier transform of Eq.(2) assumes the following form with an
appropriate normalization to define Cn(Q2),
i
n
2
i d4xeiq·xTJ(x)J(0) Cn(Q2) qµ1 qµnOi .
∼ i Q2! ··· µ1···µn
Z i,n
X
By defining the matrix element of Oi ,
p Oi (0) p = 2Aip p trace terms,
h | µ1···µn | i n µ1 ··· µn −
(notethattheoperatorsshouldhavedefinitetwists, somustbetraceless) theforward
Compton scattering amplitude T is written as,
2Mν 1
T(ν,Q2) = 2 ωnAiCn(Q2) , ω = = .
n i Q2 x
i,n
X
To get information for the structure functions in the Bjorken limit, we rely on
the analytic structure of T in the complex ω-plane. There are cuts going out to
infinity from ω = 1 and the discontinuity of T is related to the structure function
±
W: W = 1 ImT. The next step is just to use the Cauchy’s theorem. The contour
π
integral of T around the origin in ω-plane picks up its n-th coefficient:
1
dωω−n−1T = 2 AiCn(Q2).
2πi n i
I i
X
Deforming the contour to pick up the discontinuity of T, the left hand side becomes,
2 ∞ 1
LHS = dωω−n−1ImT(ω,Q2) = 2 dxxn−1W(x,Q2),
π
Z1 Z0
where we have used the crossing symmetry,
T(q,p) = T( q,p) , W(q,p) = W( q,p).
− − −
9