Hadron Structures and Perturbative QCD

17 Pages·0.3 MB·English

by  Yuji Koike

HADRON STRUCTURES AND PERTURBATIVE QCDa
YUJIKOIKE
Department of Physics, Niigata University
Ikarashi, Niigata 950-2181, Japan
E-mail: [e-mail]
In the first part of this talk, I will summarize recent developments in the study of the chiral-odd spin-
dependent parton distributions h1(x,Q2) and hL(x,Q2) of the nucleon, in particular, (i) Next-to-leading
order Q2 evolution of h1(x,Q2) and (ii) Leading order Q2 evolution of the twist-3 distribution hL(x,Q2)
9 andtheuniversalsimplificationoftheQ2evolutionofallthetwist-3distributionsinthelargeNclimit. The
9 second part of this talk will be devoted to a systematic analysis on the light-cone distribution amplitudes
9 of vector mesons (ρ, ω, φ, K∗ etc) relevant for exclusive processes producing these mesons. In particular,
1 twist-3distributionamplitudesarediscussedindetail.

n
a
1 Introduction
J

3 Highenergyprocessescanbeclassifiedintotwocategories,inclusiveandexclusiveprocesses. Quark-
1
gluonsubstructuresofhadronsinvolvedintheseprocessesrevealthemselvesasaformofpartondis-

tributionfunctionsintheinclusiveprocesses,andlight-conedistributionamplitudesintheexclusive
1
v proceeses. Understanding on both quantities constitutes a crucial step for the QCD description of
4 thehighenergyprocesses. Inthistalk,Iwillsummarizeourrecentstudiesonthequarkdistribtution
8 functions in the nucleon and the light-cone distribution amplitudes for the light vector mesons.
2
Spin dependent parton distribution functions for the nucleon measured by the polarized beams
1
andtargetsrepresent“spindistributions”carriedbyquarksandgluonsinsidethenucleon. Theyare
0
9 functions ofBjorken’s x whichrepresentparton’smomentum fractionin the nucleonanda scale Q2
9 atwhichtheyaremeasured. Untillnow,mostdataonthenucleon’sdistributionfunctionshavebeen
/ obtained through the lepton-nucleon deep inelastic scattering (DIS). The chiral-odd distributions,
h
p h1,L(x,Q2), are the new type of distribution functions which have not been measured so far: Due
- to the chiral-odd nature, they decouple from the inclusive DIS. They can, however, be measured
p
by the nucleon-nucleon polarized Drell-Yan process and semi-inclusive DIS which detect particular
e
h hadronsin the finalstate. They will hopefully be measuredbyplannedexperiments using polarized
: accelerators at BNL, DESY, CERN and SLAC etc1. In particular, RHIC at BNL is expected to
v
i provide first data on these distributions.
X
Inthestudyofthesedistributionfunctions,perturbativeQCDplaysanimportantroleinpredict-
r ingtheirQ2-dependence: Givenadistributionfunction, sayh (x,Q2),atonescaleQ2,perturbative
a 1 0 0
QCD predicts the shape of h (x,Q2) at an arbitrary scale Q2. This Q2 evolution is necessary not
1
only in extracting low energy hadron properties from high energy experimental data but also in
testing the x-dependence predicted by a non-perturbative QCD technique or a model with the high
energydata. In the firstpartofthis talk, I willsummarizeourrecentstudies onthe Q2-dependence
of h (x,Q2).
1,L
Light-cone distribution amplitudes (wave functions) for the vector mesons (ρ,ω,φ, and K )
∗
appearinvariousexclusiveprocessesproducingthesevectormesonsinfinalstates,suchasB decay,
B ℓνV, B ℓ+ℓ V, B γV, and electro-production, e+N N +V. (Study on the wave
− ′
→ → → →
functions for pseudoscalr mesons is less involved, and has been done by many works.) Analysis on
the wavefunctions is indispensable to testapplicability ofperturbativeQCDto exclusiveproceeses.
In particular, test of the standardmodel through the rare B decay requires the knowledge on these
wavefunctions. Inthesecondpartofthistalk,wepresentacompleteclassificationofthetwo-particle
(quark-antiquark)wavefunctions for the vectormesons basedontwist, chiralityand spin. This can
be done in parallel with that for the nucleon’s parton distribution functions. In particular, for the
aInvited talk presented at “RCNP International School of Physics of Hadrons and QCD”, October 12-13, 1998,
Osaka,Japan. Tobepublishedintheproceedings.
1
twist-3wavefunctions, weidentify the contributionfromthe three-particle(quark-gluon-antiquark)
twist-3distributionamplitudes,usingQCDequationofmotion. Therenormalizationandthemodel
building for the twist-3 wave functions can be/should be done starting from these exact relations,
which is discussed by Tanaka in the workshop.
2 Distribution Function of the Nucleon in Inclusive Processes
2.1 Chiral-Odd Distributions h (x,Q2)
1,L
Inclusive hard processes can be generally analyzed in the framework of the QCD factorization the-
orem2. This theorem generalizes the idea of the Bjorken-Feynman’s “parton model” and allows us
to include QCD correction in a systematic way. Here I restrict myself to the hard processes with
the nucleontarget,suchasdeep-inelastic lepton-nucleonscattering(DIS, l+p l +X),Drell-Yan
′
→
(p+p l+l +X), semi-inclusive DIS (l+p l +h+X). According to the above theorem, the
′ − ′
→ →
cross section (or the nucleon structure function) for these processes can be factorized into a “soft
part” and a “hard part”: The soft part represents the parton (quark or gluon) distribution in the
nucleon and the hard part describes the short distance cross section between the parton and the
external hard probe which is calculable within perturbation theory. For example, a nucleon struc-
ture function in DIS can be written as the imaginary part of the virtual photon-nucleon forward
Compton scattering amplitude. (Fig. 1 (b)) According to the above theorem, in the Bjorken limit,
i.e. Q2,ν =P q withx=Q2/2ν =finite,(Q2 = q2 isthevirtualityofthespace-likephoton,
· →∞ −
P is the nucleon’s four momentum), the structure function can be written as
1 dy x Q2
W(x,Q2)= Ha( , ,α (µ2))Φa(y,µ2), (1)
y y µ2 s
a Zx
X
where Φa represents a distribution of parton a in the nucleon and Ha describes the short distance
crosssectionofthepartonawiththevirtualphoton. µ2 isthefactorizationscale. InFig. 1(b),Φa is
identified by the dotted line. (Fig.1 (a)). Similarly to DIS, the crosssectionfor the nucleon-nucleon
Drell-Yan process can also be written in a factorized form at s=(P +P )2,Q2 with a fixed
A B
→∞
Q2/s (P are the momenta of the two nucleons, Q is the momentum of the virtual photon):
A,B
1 1 x x Q2
dσ dy dy Hab a, b,Q2; ,α (µ2) Φa(y ,µ2)Φb(y ,µ2), (2)
∼ a b y y µ2 s a b
a,b Zxa Zxb (cid:18) a b (cid:19)
X
where the two parton distributions, Φa and Φb, for the beam and the target appear as was shown
by dotted lines in Fig. 1(c).
(a)
(b) (c)
Figure1: (a) Quark distributionfunction. (b) Nucleon struction function inDIS. (c) Crosssection for the nucleon-
nucleonDrell-Yanprocess.
As is seen from Figs. 1(b),(c), the parton distribution can be regarded as a parton-nucleon
forward scattering amplitude shown in Fig. 1 (a) which appear in several different hard processes.
2
In particular, the quark distribution in the nucleon moving in the +eˆ direction can be written as
3
the light-cone Fourier transform of the quark correlationfunction in the nucleon:3
Φa(x,µ2)=P+ ∞ dz−eixP z PS ψ¯a(0)Γψa(z) PS , (3)
· µ
2π h | | | i
Z−∞
where PS denotesthenucleon(massM)statewithmomentumPµandspinSµ,andψaisthequark
| i
field with flavor a. In (3), we have suppressed for simplicity the gauge link operator which ensures
thegaugeinvarianceand indicatestheoperatorisrenormalizedatthescaleµ2. Afourvectoraµ is
µ
|
decomposedintotwolight-conecomponentsa = 1 (a0 a3)andthetransversecomponent~a . In
± √2 ± ⊥
(3), z+ =0,~z =~0, andz2 =0. Γ genericallyrepresentsγ-matrices,Γ=γ ,γ γ ,σ ,1. Φa(x,µ2)
µ µ 5 µν
measures the⊥distribution of the parton a to carry the momentum k+ =xP+ in the nucleon, which
is independent from particular hard proceeses.
If one puts Γ = γ ,γ γ , the chirality of ψ¯ and ψ becomes the same, namely it defines the
µ µ 5
chiral-even distributions. Likewise, putting Γ = σ ,1 defines the chiral-odd disributions. For the
µν
case of the deep-inelastic scattering (Fig. 1 (b)), the quark line emanating from the target nucleon
comesbacktotheoriginalnucleonafterpassingthroughthehardinteractions. Sincetheperturbative
interactioninthestandardmodelpreservesthechiralityexceptatinyquarkmasseffect,thechirality
of the two quark lines entering the nucleon in Fig. 1(b) is the same. Hence the DIS can probe only
the chiral-even quark distributions. On the other hand, in the Drell-Yan process (Fig. 1 (c)), there
isnocorrelationinchiralitybetweentwoquarklinesenteringeachnucleon. ThereforetheDrell-Yan
process probes both chiral even and odd distributions.
The chiral-odd distributions ha(x,µ2), ha(x,µ2) in our interest are defined by putting Γ =
1 L
σ iγ in (3):4
µν 5
dλ
eiλx PS ψ¯a(0)σ iγ ψa(λn) PS
µν 5 µ
2π h | | | i
Z
=2[ha(x,µ2)(S p S p )/M
1 ⊥µ ν − ⊥ν µ
+ha(x,µ2)M(p n p n )(S n)
L µ ν − ν µ ·
+ha(x,µ2)M(S n S n )] (4)
3 ⊥µ ν − ⊥ν µ
where we introduced two light-like vectors p, n (p2 = n2 = 0) by the relation Pµ = pµ + M2nµ,
2
p n = 1, p = n+ = 0. If we write P+ = , p = (1,0,0,1), n = 1 (1,0,0, 1). is a
· − P √P2 √2 − P
parameter which specifies the Lorentz frame of the system: corresPponds to the infinite
momentum frame, and M/√2 the rest frame of the nuclePon→. S∞µ is the transverse component
ofSµ definedby Sµ =(SP →n)pµ+(S p)nµ+Sµ. OnecanshowthatΦ⊥a definedin(3)hasasupport
1<x< 1. If one replac·es the quar·k field ψ⊥in (3) by its charge conjugation field Cψ¯T, it defines
−the anti-quark distribution Φ¯a. In particular ha (x,µ2) in (4) are related to their anti-quark
1,L,3
distribution by ha ( x,µ2)= h¯a (x,µ2).
1,L,3 − − 1,L,3
Φa appearsin a physicalcrosssectionin the formofthe convolutionwitha shortdistance cross
section in a partonlevel as is shown in (1) and (2). The crosssection can be expanded in powers of
1 as
√Q2
M M2
σ(Q2) A(lnQ2)+ B(lnQ2)+ C(lnQ2)+ , (5)
∼ Q2 Q2 ···
where each coefficient A, B, C receives lopgarithmic Q2-dependence due to the QCD radiative cor-
rection. In order to see how h can contribute in the expansion (5), it is convenient to move
1,L,3
into the infinite momentum frame ( Q ). In this limit the coefficient of h in (4) be-
1,L,3
P ∼ → ∞
haves, respectively, as O(Q), O(1), O(1/Q). Therefore if h contributes to the A term in (5), h
1 L
3
spin average longitudinal transverse
twist-2 f g h
1 1 1
twist-3 e h g
L T
Table 1: Clasification of the quark distributions based on spin, twist and chirality. Underlined distributions are
chiral-odd. Othersarechiral-even.
can contribute at most to the B-term, and h can contribute at most to the C-term. In general,
3
τ 2
−
when a distribution function contributes to hard processes at most in the order of 1 , the
√Q2
(cid:18) (cid:19)
distribution is called twist-τ. Therefore h , h , h in (4) is, respectively, twist-2, -3 and -4.
1 L 3
Twist-2 distribution h can be measured through the transversely polarized Drell-Yan5,6,4,7,
1
semi-inclusive deep inelastic scatterings which detect pion8, polarized baryons6,9,10, correlated two
pions11.
Fromthe discussionabove,one sees thatit is generallydifficult to isolateexperimentally higher
twist (τ 3) distributions in hard proceeses, since they are hidden by the leading twist-2 contribu-
≥
tion (A term in (5)). However, this is not the case for h and g . In particular spin asymmetries,
L T
they contribute to the B-term in the absence of A-term: g can be measured in the transversely
T
polarized DIS12, and h appears in the longitudinal versus transverse spin asymmetry in the po-
L
larized nucleon-nucleon Drell-Yan process4. Therefore the Q2-evolution of g and h can be a new
T L
test of perturbative QCD beyond the twist-2 level.
Insertion of other γ-matrices in (3) defines other distributions. In Table 1, we show the clas-
sification of the quark distributions up to twist-3.4 There f , g , e is defined, respectively, by
1 1,T
Γ = γ ,γ γ ,1 in (3). A similar classification can also be extended to the gluon distributions14.
µ µ 5
The distribution f contributes to the spin averagedstructurefunctions F (x,Q2) familiar inDIS.
1 1,2
The helicity distribution g contributes to the G (x,Q2) structure function measured in the longi-
1 1
tudinally polarizedDIS. By now there has been muchaccumulationof experimentaldata onf and
1
g , and the data on g triggered lots of theoretical discussion on the “origin of the nucleon spin”1.
1 1
The first nonzero data on g (=g g ) was also reported in Ref.13.
2 T 1
−
2.2 Next-to-leading order (NLO) Q2-evolution of h (x,Q2)
1
As we saw in the previous section, h is the third and the final twist-2 quark distribution. It has a
1
simple partonmodel interpretationascanbe seenby the Fourierexpansionofψ in (4). Itmeasures
theprobabilityinthetransverselypolarizednucleontofindaquarkpolarizedparalleltothenucleon
spin minus the probability to find it oppositely polarized. Here the transverse polarization refers
to the eigenstate of the transverse Pauli-Luban´ski operator γ S/ . If one replaces the transverse
5
⊥
polarization by the longitudinal one, it becomes the helicity distribution g . For nonrelativistic
1
quarks, h (x,µ2)=g (x,µ2). A model calculation suggests, h is the same order as g .4,15,16
1 1 1 1
The Q2-evolution of h is described by the usual DGLAP evolution equation17. Because of its
1
chiral-odd nature it does not mix with gluon distributions. Therefore the Q2-dependence of h is
1
described by the same equation both for singlet and nonsinglet distributions. For f and g , the
1 1
NLO Q2 evolutionwasderivedlong time ago18,19,20,21 andhasbeen frequentlyusedfor the analysis
of experiments22,23. The leading order (LO) Q2-evolution for h has been known for some time6.
1
In the recent literature, the next-to-leading order (NLO) Q2-evolution has also been completed by
two papers24,25: Vogelsang24 presented the light-cone gauge calculation for the two-loop splitting
function of h in the formalism originally used for f20. We 25 carried out the Feynman gauge
1 1
calculationofthe two-loopanomalousdimension followingthe method ofRef.18 for f . The results
1
of these calculations in the MS scheme agreed completely. In the following, I briefly discuss the
characteristic feature of the NLO Q2 evolution of h following Refs.25,26.
1
Analysis of (4) gives the connection between the n-th moment of h and a tower of twist-2
1
4
operators:
1 1
Mn[h1(µ2)] ≡Z−1dxxnh1(x,µ2)= 2−MhPS⊥|On⊥(µ2)|PS⊥i,
O =S ψ¯σναn iγ (in D)nψ, (6)
n⊥ ⊥ν α 5 ·
where S stands for the transversepolarizationandO (µ2) indicates the operatorO is renormal-
n⊥ n⊥
izaed at⊥the scale µ2. The contraction with nµ and Sµ (recall S n = 0, n2 = 0) in (6) projects
out the relevant twist-2 contribution from the compo⊥site operat⊥or·. (“Twist” for local composite
operators is defined as dimension minus spin.) By solving the renormalization group equation for
O , one gets the NLO Q2 dependence of [h (µ2)] as
n⊥ Mn 1
[h (Q2)] α (Q2) γn(0)/2β0 α (Q2) α (µ2)β γ(1) γ(0)
n 1 s s s 1 n n
M = 1+ − ,
Mn[h1(µ2)] (cid:18)αs(µ2)(cid:19) " 4π β0 2β1 − 2β0!#
(7)
where α (Q2) is the NLO QCD running coupling constant given by
s
α (Q2) 1 β lnln(Q2/Λ2)
s 1
= 1 , (8)
4π β ln(Q2/Λ2) − β2ln(Q2/Λ2)
0 (cid:20) 0 (cid:21)
withtheone-loopandtwo-loopcoefficientsoftheβ-functionβ =11 2/3N andβ =102 38/3N
0 f 1 f
− −
(N is the number of quark flavor) and the QCD scale parameter Λ. γ(0) and γ(1) are the one-loop
f n n
and two-loop coefficients of the anomalous dimension γ for OνS defined as
n n ν
⊥
α α 2
γ = sγ(0)+ s γ(1). (9)
n 4π n 4π n
(cid:16) (cid:17)
If one sets β 0 and γ(1) 0 in (7), the leading order (LO) Q2 evolution is obtained. γ(0) and
1 n n
→ →
γ(1) areobtained,respectively,bycalculatingtheone-loopandtwo-loopcorrectionstothetwo-point
n
GreenfunctionwhichimbedsOνS . Toobtainγ(1),calculationof18two-loopdiagramsisrequired
n ν n
in the Feynman gauge. Since the⊥expression for γ(1) is quite complicated, we refer the readers to
n
Refs.24,25 for them.
In order to get a rough idea about the NLO Q2 dependence of h , we plotted in Fig. 2 γh(1)
1 n
(1) fg(1) (1)
(γ for h ) in comparison with γ (γ for the nonsinglet f and g ) for N = 3,5. One sees
n 1 n n 1 1 f
from Fig. 2 γh(1) > γfg(1) especially at small n. This suggests that the NLO Q2 evolution of h is
n n 1
quite different fromthatoff andg inthe smallx region. The relationγh(1) >γfg(1) is inparallel
1 1 n n
with and even more conspicuous than the LO anomalous dimensions which read
n+1
1
γh(0) = 2C 1+4 ,
n F  j
j=2
X
 
n+1
2 1
γfg(0) = 2C 1 +4 . (10)
n F  − (n+1)(n+2) j
j=2
X
 
To illustrate the generic feature of the Q2 evolution, we have applied the obtained Q2 evolution to
a reference distribution for g and h . As a reference distribution, we take GRSV g distribution23
1 1 1
andassumeh (x,µ2)=g (x,µ2)atalowenergyinputscale(µ2 =0.23GeV2 for LOandµ2 =0.34
1 1
GeV2 for NLO evolution) as is suggested by a nucleon model4,15. We then evolve them to Q2 =20
GeV2 and see how much deviation is produced between them. The result is shown in Fig. 3. As is
5
expectedfromtheanomalousdimension,thedrasticdifferenceintheQ2evolutionbetweenh andg
1 1
isobservedinthesmallxregion,andthistendencyismoresignificantfortheNLOevolution.27,28,26
(Although g for u-quark mixes with the gluon distribution, the same tendency in the difference
1
from h is observed for the nonsinglet distribution.)
1
300
N = 3
f
200
1) N = 5
(n f
g
100
f , g
1 1
h
1
0
0 10 20 30
n
Figure2: TheNLOanomalous dimensionγnh(1) incomparisonwithγnfg(1). Thisfigureistaken fromRef.25.
3.5 3.5
D u LO evolution D u NLO evolution
3.0 3.0
Q2 = 20 GeV2 Q2 = 20 GeV2
2.5 m 2= 0.23 GeV2 2.5 m 2= 0.34 GeV2
2.0 2.0
LO input NLO input
1.5 1.5
d u d u
1.0 1.0
0.5 0.5
(a) (b)
0.0 0.0
0.1 1 0.1 1
x x
Figure3: (a)TheLOQ2 evolutionofh1 (denotedbyδu)andg1 (denotedby∆u)fortheu-quark. (b)TheNLOQ2
evolutionofh1 andg1 fortheu-quark. ThisfigureistakenfromRef.26
In Ref. 30, the Regge asymptotics of h was studied and the small-x behavior was predicted to
1
h(0) h(1)
be h (x) constant (x 0). On the other hand, the rightmost singularity of γ and γ are,
1 n n
∼ →
respectively,locatedatn= 2andn= 1inthecomplexnplane,which,respectively,corresponds
− −
to h (x) x and h (x) constant as x 0. Therefore inclusion of the NLO effect in the DGLAP
1 1
∼ ∼ →
asymptotics gives consistentbehaviorat x 0 as the Regge asymptotics. This is in contrastto the
→
(nonsinglet) f and g distributions, whose LO and NLO DGLAP asymptotics are the same.
1 1
OneoftheinterestingapplicationsoftheobtainedNLOQ2 evolutionofh isthepreservationof
1
theSoffer’sinequality,312ha(x,Q2) fa(x,Q2)+ga(x,Q2). Althoughthevalidityofthisinequality
| 1 |≤ 1 1
hinges on schemes beyond LO32, the NLO Q2 evolution maintains the inequality at Q2 >Q2 if it is
0
6
satisfied at some (low) scale Q2 in suitably defined factorizationschemes such as MS and Drell-Yan
0
factorization schemes.33,24.
As wasdiscussedinSec. 2,a physicalcrosssectionisa convolutionofa partondistributionand
a short distance cross section. (See (1) and (2)) For the double transverse spin asymmetry (A )
TT
in the Drell-Yan process, the NLO short distance cross section has been calculated in Ref.34 in the
MS scheme. The analysis on A combined with the NLO tranversity distribution predicts modest
TT
but not negligible NLO effect.35
2.3 Q2-evolution of h (x,Q2) and its N limit
L c
→∞
Ingeneral,highertwist(τ 3)distributionsrepresentquark-gluoncorrelationinthenucleon. Using
≥
theQCDequationofmotion(see(33)later),oneobtainsfrom(4)the followingrelation(m =0)36:
q
1 dy
h (x,µ2)=2x h (y,µ2)+h (x,µ2), (11)
L y2 1 L
Zx
h (x,µ2)= iP+ ∞ dz−e 2ixPez 1udu u tdt
L − ·
M 2π
Z−∞ Z0 Z−u
e ×hPSk|ψ¯(uz)iγ5σµαgGνα(tz)zµzνψ(−uz)|PSki, (12)
where z2 = 0, z+ = 0 and S stands for the longitudinal polarization for the nucleon (Sµ =
Sµ =pµ M2nµ). This equatiokn means that h consists of the twist-2 contribution and h which
− 2 L L
rekpresents quark-gluon correlation in the nucleon. We call the latter contribution “purely twist-3”
contribution. (Expansion of (12) produces twist-3 local operators. See (15) below.) Equaetion (11)
reminds us of the Wandzura-Wilczek relation37 for g :
T
1 dy
g (x,µ2)= g (y,µ2)+g (x,µ2). (13)
T 1 T
y
Zx
For e and g , one can write down relations similar to (12).e
T
The Q2-evolution of the first and second terms in (11) is described separately. The evolution
of hL is queite complicated. A detailed analysis of (12) leads to the following relation for the n-th
moment of h 4:
L
e
[(n+1)/2]
e 2k 1
[h (µ2)]= 1 PS R (µ)PS , (14)
n L nk
M − n+2 2Mh k| | ki
k=2 (cid:18) (cid:19)
X
e 1
R = ψ¯σλαn iγ (in D)k 2igG nν(in D)n kψ (k n k+2) .
nk λ 5 − να −
2 · · − → −
(cid:2) (cid:3) (15)
We note that the number of independent operators R (k =2, ,[(n+1)/2]) increases with n.
nk
{ } ···
In the Q2-evolution, the mixing among R occurs and the renormalization is described by the
nk
{ }
anomalousdimensionmatrix[γ (g)] for R . Ifwe putthe LOanomalousdimensionmatrix for
n kl nk
{ }
R as [γ (g)] =(α /2π)[X ] corresponding to (9), the solution to the renormalizationgroup
nk n kl s n kl
{ }
equation for R takes the following matrix form:
nk
{ }
[(n+1)/2]
PS R (Q2)PS = LXn/β0 PS R (µ2)PS , (16)
nk nl
h k| | ki klh k| | ki
Xl=2 h i
whereL αs(Q2). X forh wasderivedinRef.38. TheQ2-evolutionforg andeisalsodescribed
≡ αs(µ2) n L T
by matrix equation similar to (16), and the solution was obtained in Refs.39,40 for g and in Ref.41
T
e e
7
for e.42 As is clear from (14) and (16) [h (Q2)] and [h (µ2)] are not connected by a simple
n L n L
M M
equationasinthecaseforthetwist-2distribution(see(7)).43Although(16)givescompleteprediction
for the Q2 evolution, it is generally difficulteto distinguish conetribution from many operators in the
analysis of experiments.
In orderto get a roughidea onthe Q2-evolutionof h , we plotted the eigenvaluesof X in Fig.
L n
4 (right). For comparison, we also showed in the same figure the LO anomalous dimension γ(0)/2
n
for h . (Note the differene in convention between (7) aned (16).) As is clear from this figure, the Q2
1
evolution of h is much faster than that of h . (See discussion below.)
L 1
It has been shown in Refs.44,36,41 that at largeN (the number of colors),a greatsimplification
c
occursintheeQ2-evolutionofthetwist-3distributions. RecallX in(16)isafunctionoftwoCasimir
n
operatorsCG =Nc andCF = N2c2N−c1. IfonetakesNc →∞,i.e. CF →Nc/2,(14)and(16)isreduced
to
[h (Q2)]= Lγnh/β0 [h (µ2)], (17)
n L n L
M M
n
e γh = 2Ne 1e 1 + 3 . (18)
n c j − 4 2(n+1)
j=1
X
e  
This evolution equation is just like those for the twist-2 distributions (see (7)). In Fig. 4 (left), we
showed the distribution of the eigenvalues of X obtained numerically at N . The solid line
n c
→ ∞
is the analytic solutionin (18), which shows (18)corresponds to the lowesteigenvalues at N .
c
→∞
Since (17) was obtained by a mere replacement C N /2 in (16), the correction to the result is
F c
→
of O(1/N2) 10 % level, which gives enough accuaracy for practical applications.
c ∼
Figure4: (Right) Complete spectrum ofthe eigenvalues ofthe anomalous dimensionmatrixforhL obtained inRef.
38. Thesymbol⋄denotes theone-loopanomalous dimensionfor h1. Thesolidlineistheanomalous dimension(21)
atlargen. (Left)Spectrum oftheeigenvalues oftheanomalousdimensionmatrixforhL atlargeeNc. Thesolidline
denotes theanalyticsolutiongivenin(18). ThisfigureistakenfromRef.36.
e
This large-N simplification is a consequence of the fact that the coefficients of R in (14)
c nk
constitutes the left eigenvector of X corresponding to the eigenvalue γh in this limit:
n n
[(n+1)/2]
1 2k [X ] = 1 2l eγh, (19)
− n+2 n kl − − n+2 n
k=2 (cid:18) (cid:19) (cid:18) (cid:19)
X
e
8
whichimpliesthatalltherighteigenvectorsofX excepttheonecorrespondingtoγh areorthogonal
n n
to the vector consisting of 1 2k . This leads to (17).
− n+2
e
This large-N simplifica(cid:16)tion of th(cid:17)e Q2 evolution was proved for the nonsinglet g in Ref.44 and
c T
for h and e in Ref.36. The corresponding anomalous dimensions for g and e are, respectively,
L T
e
e n e
1 1 1
γg =2N + ,
n c j − 4 2(n+1)
j=1
X
e  n 
1 1 1
γe =2N . (20)
n c j − 4 − 2(n+1)
j=1
X
e  
Corresponding to three twist-3 distributions in table 1, there are three independent twist-3
fragmentation functions.9 (Their number is doubled to 6 if one includes final state interactions.
See Ref.9) It has been shown in Ref.45 that at large N the Q2 evolution of all these nonsinglet
c
fragmentation functions is also described by a simple evolution equation similar to (17). Therefore
the simplification of the twist-3 evolution equationis universalto all twist-3 nonsinglet distribution
and fragmentation functions.
To illustrate the actual Q2 evolution of h , we have applied (17) to the bag model calculation
L
of h .46 (Fig. 5) Fig. 5(a) shows the bag calculation of h 4. At the bag scale, purely twist-3
L L
contributionh iscomparabletothe twist-2contribution. AftertheQ2 evolutiontoQ2 =10GeV2,
L
h is dominated by the twist-2 contribution (Fig. 5(b)). This can be ascribed to two facts: One
L
is the large aenomalous dimension (18) compared with the LO anomalous dimension of h ( in the
1
⋄
right figure of Fig. 4). The other is the presence of a node for h (x,Q2), which is taken as model
L
independent due to the constraint 1 dxh (x,Q2) = 0,47 which is an analogue of the Burkhardt-
0 L
Cottingham sum rule for g (x,Q2)48. A similar calculation wasedone for g in Ref.49. Using these
2 T
R
model calculations, the longitudinal-transeverse spin asymmetry, A , for the polarized Drell-Yan
LT
process was estimated in Ref.50.
3.5 3.5
(a) (b)
3.0 3.0
Bag model Q2 = 10 GeV2
2.5 2.5 m 2 = 0.25 GeV2
bag
2) 2.0 2) 2.0
Q Q
x, 1.5 hL x, 1.5 hL
( (
hL 1.0 hL 1.0
0.5 0.5
0.0 0.0
~ ~
-0.5 hL -0.5 hL
-0.5 0.0 0.5 1.0 -0.5 0.0 0.5 1.0
x x
Figure5: (a)BagmodelpredictionforhL. Thedashedlinerepresentsthetwist-2contributiontohL. (b)Bagmodel
predictionforhL evolvedtoQ2=10GeV2 assumingthebagscaleisµ2 =0.25GeV2. Thesefigures aretaken from
Ref.46
Anothersimplificationofthetwist-3evolutionoccursatn .44,36 Inthislimit,allthetwist-3
→∞
distributions obey a simple DGLAP equation (17) with a common anomalous dimension which is
9
slightly shifted from (18) and (20):
n
1 3
γ =4C +N . (21)
n F c
 j − 4
j=1
X
 
This evolution equation satisfies the complete evolution equation to the O(ln(n)/n) accuracy44. In
the right figure of Fig. 4, (21) is shown by the solid line. One sees that it is close to the lowest
eigenvalues except for small n. Combined with this n result, the large-N evolution equation
c
→∞
in (17) with (18) and (20) for each distribution is valid to O((1/N2)ln(n)/n) accuracy.
c
3 Light-cone Distribution Amplitudes of Vector Mesons in QCD
In this section we present a systematic analysis on the light-cone distribution amplitudes (wave
functions)51 of the vector mesons (ρ, ω, φ, K etc) following our recent work53. These amplitudes
∗
arerelevantforthepreasymptoticcorrectiontovariousexclusiveprocessesproducingvectormesons
in the final states, such as B meson decay, B ℓνV (semi-leptonic), B γV (radiative), and
→ →
the electroproduction, γ +N N +V, etc. In particular, we show that the classification and
∗ ′
→
analysisofthelight-conedistributionamplitudesforvectormesonscanbedoneinparallelwiththat
of the distribution functions of the nucleon. (Analysis on the light-cone distribution amplitudes for
pseudo-scalarmesons is simpler. See e.g. Refs.51,52.) For definitness, we discuss the ρ meson wave
−
functions. Extention to other vector mesons is straightforward.
3.1 Definition and Classification
For the ρ -mesonmoving inthe positive+eˆ direction,the light-conewavefunctions aredefined as
− 3
φ(u,µ2)=P+ ∞ dz−eiuP z 0u¯(0)Γd(z) ρ (P,λ) , (22)
· µ −
2π h | | | i
Z−∞
where ρ (Pλ) standsfortheρ -meson(massm )statewiththemomentumP andthepolarization
− − ρ
| i
vector e(λ); P2 = m2, e(λ)2 = 1, P e(λ) = 0. Γ denotes generic γ matrices and z is the only
µ ρ − · −
nonzero componet of the space time cordinate z. The variable u in φ(u) represents a fraction of
“+”-momentum P+ carried by d quark and φ has a support on 0 < u < 1. Here and below the
gauge link operator [0,z] Pexp ig 0 dtzµA (tz) which retores gauge invariance is suppressed
≡ { 1 µ }
for simplicity. The only difference between the wave function (22) and the distribution functions
R
(3) is that the latter is a forwardmatrix elements while the former is a vacuum-to-mesontransition
amplitude. In order to classify the wave functions (22), it is convenient to introduce two light-like
vectorspandnaswasdoneinsection2.1in(4). Theysatisfytherelationsp n=1,P =p +1m2n
· µ µ 2 ρ µ
and e(λ) =(e(λ) n)p +(e(λ) p)n +e(λ). We introduce two coupling constants f and fT by the
µ · µ · µ µ ρ ρ
relation ⊥
h0|u¯(0)γµd(0)|ρ−(Pλ)i=fρmρeµ(λ), (23)
and
h0|u¯(0)σµνd(0)|ρ−(Pλ)i=fρT eµ(λ)Pν −eν(λ)Pµ . (24)
(cid:16) (cid:17)
With these definitions the classification of (22) can be done based on spin, chirality and twist, as
was the case for the distibution functions in the nucleon. The only difference is (i) e(λ) is a vector,
µ
while S for the nucleon is an axial vector, and (ii) the wave function (22) should be linear in e(λ),
µ µ
since it is a matrix element between the vacuum and the ρ meson state.
10