Chiral-odd GPDs in large-N_c QCD

0.11 MB·

by  P. Schweitzer

6
Chiral-odd GPDs in large–N QCD
1 c
0
2

n
a
J

2
1
P. Schweitzer
∗
]
DepartmentofPhysics,UniversityofConnecticut,Storrs,CT06269,USA
h
p E-mail: [email protected]
-
p
C. Weiss
e
h TheoryCenter,JeffersonLab,NewportNews,VA23606,USA
[ E-mail: [email protected]

1
v
6 We study the flavor structure of the nucleon’s chiral-odd generalized parton distributions
1 (transversity GPDs) in the large–N limit of QCD. It is found that in the distributions H and
c T
0
3 E˜T theflavor–nonsingletcomponentu disleadinginthe1/Ncexpansion,whileinET andH˜T it
−
0 istheflavor–singletcomponentu+d. Thispatternisconsistentwiththeflavorstructureextracted
.
1 fromhardexclusivep 0 andh electroproductiondata,assumingthattheprocessesaredominated
0
bythetwist–3mechanisminvolvingthechiral-oddpseudoscalarmesondistributionamplitudes.
6
1
:
v
i
X
r
a
QCDEvolution2015-QCDEV2015-
26-30May2015
JeffersonLab(JLAB),NewportNewsVirginia,USA
Speaker.
∗
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommons
Attribution-NonCommercial-NoDerivatives4.0InternationalLicense(CCBY-NC-ND4.0). http://pos.sissa.it/
Chiral-oddGPDsinlarge–N QCD P.Schweitzer
c
1. Introduction
Generalized parton distributions (or GPDs) unify the concepts of parton density and elas-
tic form factor and enable a comprehensive description of the nucleon’s quark and gluon single–
particlestructureinQCD;seeRefs.[1,2,3,4]forareview. Attheleading–twistlevelthenucleon’s
quark structure is described by 4 chiral-even (quark helicity–conserving) and 4 chiral-odd (quark
helicity–flipping) GPDs[5]. The chiral-even GPDsreduce to the usual unpolarized and polarized
quark parton distribution functions (PDFs) in the limit of zero momentum transfer. These GPDs
appear in the amplitudes of hard exclusive processes such as deeply virtual Compton scattering
and exclusive meson production with longitudinal photon polarization, for which QCD factoriza-
tion theorems have been derived, and can be accessed experimentally in this way[6, 7, 8, 9]. The
chiral-odd GPDs reduce to the quark transversity PDFs in the limit of zero momentum transfer.
Relating these GPDs to hard exclusive processes has proved to be more challenging, as they de-
couple from vector meson production at leading twist in all orders in perturbative QCD due to
the chirality requirements for massless fermions [10]. The chiral-odd GPDshave been associated
with the diffractive electroproduction of two vector mesons at leading twist [11], but the process
is difficult to measure [12, 13, 14], and no data are available at present. Recent work suggests
thatpseudoscalar mesonelectroproduction (p 0,h ,p +)maybedominatedbyatwist–3mechanism
involving thechiral-oddnucleonGPDsandthechiral-odddistribution amplitude,whichoriginates
from the dynamical breaking of chiral symmetry in QCD[15, 16, 17, 18]. While no factorization
theorem exists at this level, the pseudoscalar production amplitudes have been calculated in the
modifiedhardscatteringapproachofRef.[16,17],whichimplementssuppression oflarge–sizeqq¯
configurations in the meson through the QCD Sudakov form factor. The results agree well with
the p 0 and h electroproduction data from the JLab CLAS experiment at 6 GeV incident energy,
regardingboththeabsolutecrosssectionsandtheL/T ratioasinferredfromtheazimuthal–angle–
dependent response functions [19, 20]. This has opened the prospect of more detailed studies of
the chiral-odd GPDs in further measurements of pseudoscalar meson electroproduction with the
JLab12GeVUpgrade.
To further explore this possibility, it is necessary to gain more insight into the properties of
the chiral-odd GPDs and their relation to other measures of nucleon structure. Contrary to the
chiral-even GPDs, in the chiral-odd case neither the zero-momentum transfer limit of the GPDs
(transversity PDFs) nor the local operator limit of the GPD (form factor of local tensor operator)
correspondtostructuresthatareeasilymeasurable,sothatlittleusefulinformationcanbeobtained
in this way. The transversity PDFs can be extracted from observables in semi-inclusive deep-
inelastic scattering; see Refs. [21, 22, 23] and references therein. Theform factors of local tensor
operators, which constrain the lowest x–moment of the chiral-odd GPDs, have been calculated in
lattice QCD [24] and in the chiral quark-soliton model [25, 26, 27]. The x–dependent chiral-odd
GPDs have been studied in quark bound–state models of nucleon structure [28, 29, 30]. Besides
theseestimatesnotmuchisknownabouttheproperties ofthechiral-odd GPDs.
Herewereportaboutastudyoftheflavorstructureofthechiral-oddGPDsinQCDinthelimit
ofalargenumberofcolors(large–N limit)[31]. Thelarge–N limitrepresentsarigorousapproach
c c
toQCDinthenon-perturbativedomainandleadstomodel-independent relationsgoverningmeson
andnucleonstructureathadronicenergies[32]. Inthelarge-N limitQCDbecomessemi-classical,
c
2
Chiral-oddGPDsinlarge–N QCD P.Schweitzer
c
and baryons can bedescribed by meanfieldsolutions in termsof meson fields [33]. Itisassumed
that the qualitative properties of QCD, such as the dynamical breaking of chiral symmetry in the
ground state, are preserved as the limit is taken [34]. While the dynamics at large N remains
c
complex and cannot be solved exactly, and the form of the mean field solution is not known,
important insights can be obtained by exploiting known symmetry properties of the mean field
[35, 36]. In this way one obtains relations for baryon mass splittings, meson-baryon coupling
constants, electromagnetic and axial form factors, and other observables, which are generally in
good agreement with observations, see Ref. [37] for a review. In the matrix elements of quark
bilinear operators (vector or axial vector currents, tensor operators) the large–N limit identifies
c
leading andsubleading spin–flavor components. Theapproach canbeextended topartondensities
[38, 39], where e.g. it suggests a large flavor asymmetry of the polarized antiquark distribution
D u¯ D d¯asseemstobesupported bytherecentRHICW production data[40,41].
∓
−
Our study generalizes previous results on large–N relations for chiral-even GPDs [1], quark
c
transversitydistributions[42,43,44],andmatrixelementsoflocalchiral-oddoperators[25,26,27]
and relies onthe formal apparatus developed inthese earlier works (see Sec.3of Ref.[1]). In this
notewereviewthebasicpropertiesofchiral-oddGPDs(Sec.2),describethe1/N expansionofthe
c
chiral-odd GPDs(Sec.3),andcomparethepatternwiththepseudoscalar mesonelectroproduction
data(Sec.4).
2. Chiral-odd GPDs
GPDs parametrize the non-forward nucleon matrix elements of QCD light–ray operators of
thegeneral form[1,2,3,4]
dz
M(G )=P+ 2p− eixP+z−hN(p′,l ′)|y (−2z)G y (2z)|N(p,l )i , (2.1)
Z (cid:12)z+=0,~zT=0
(cid:12)
(cid:12)
whereP 1(p +p)istheaveragenucleon4–momentum,zisthedisplacem(cid:12)entofthequarkfields,
≡ 2 ′
and the 4–vectors are described by their light-cone components z =(z0 z3)/√2,~z =(z1,z2),
± T
±
etc. G denotes a generic matrix in spinor and flavor indices and defines the spin–flavor quantum
numbers ofthe operator. Inthe chiral-odd case thespinor matrix isofthe form G =is +j,and the
matrixelementisparametrized as[5]
P+D j D +Pj
M(is +jTq) = u¯(p′,l ′) is +j HTq+ M−2 H˜Tq
(cid:20) N
g +D j D +g j g +Pj P+g j
+ − Eq+ − E˜q u(p,l ). (2.2)
2M T M T
N N
(cid:21)
The GPDs Hq =Hq(x,x ,t), etc., are functions of the quark plus momentum fraction x, the plus
T T
momentumtransferx = D +/(2P+)=(p p)+/(p+p)+,andtheinvariantmomentumtransfer
′ ′
− −
t =D 2 =(p p)2. (For brevity we do not indicate the dependence of the GPDs on the normal-
′
−
ization scale.) Here we consider GPDs corresponding to a given quark flavor q=(u,d), and Tq
denotes thecorresponding projectoronquarkflavorindices.
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Chiral-oddGPDsinlarge–N QCD P.Schweitzer
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(cid:239) q,x+x ,mæ (cid:239) q,x - x ,m¢æ Figure 1: Representation of GPDs in the region
x <x<1 as nucleon–quarkhelicity amplitudes. In
thenucleonandquarkstates(denotedasN,q)thesec-
ond label denotes the fraction of the light-cone plus
momentum P+ carried by the particle, and the third
(cid:239) N,1+x ,læ (cid:239) N,1 - x ,l¢æ labeldenotesthelight-conehelicity.
Astheirchiral-evencounterparts, thechiral-odd GPDssatisfycertainsymmetryrelationsinx
duetotimereversalinvariance,
+GPD(x,x ,t) for GPD=Hq, H˜q, Eq,
GPD(x, x ,t)= T T T (2.3)
− ( GPD(x,x ,t) for GPD=E˜q.
− T
Theirintegralsoverx(orfirstmoments)coincide withtheformfactorsofthelocaltensoroperator
y¯(0)is mn y (0),
dxHq(x,x ,t) = Fq(t),
T T
Z
dxH˜q(x,x ,t) = F˜q(t),
T T
Z
dxEq(x,x ,t) = Eq(t),
T T
Z
dxE˜q(x,x ,t) = 0. (2.4)
T
Z
The vanishing of the first moment of E˜q is a consequence of the antisymmetry in x , Eq. (2.3); its
T
higher moments are non-zero. More generally, the higher x–moments of the chiral-odd GPDsare
polynomials inx (generalized tensorformfactors).
q
Inthelimitofzeromomentum transfer (forward limit) thechiral-odd GPDH reduces tothe
T
q
transversity PDFh (x),
1
Hq(x,x =0,t =0) = hq(x). (2.5)
T 1
Its first moment is known as the nucleon’s tensor charge. Because the local tensor operator is not
a conserved current, the tensor charge is scale–dependent and cannot directly be related to low–
energy properties ofthenucleon.
The partonic interpretation of the GPDs is described in Refs. [1, 2, 3, 4]. Our study of the
large–N limit of the chiral-odd GPDsrelies essentially on the representation of the GPDsas par-
c
tonic helicity amplitudes [45]. Thisrepresentation mostnaturally appears intheregionx <x<1,
where the GPDs describe the amplitude for the “emission” by the nucleon of a quark with plus
momentum fraction x+x and subsequent “absorption” of a quark with x x (see Fig. 1). (In the
−
region 1 < x < x the GPDs describe the emission and absorption of an antiquark, while in
− −
x <x<x theydescribe theemission ofaquark–antiquark pairbythenucleon. Wedonot need
−
to consider these regions explicitly in the subsequent arguments, as the N –scaling of the GPDs
c
is uniform in x.) The initial and final nucleon are described by light-cone helicity spinors, which
are obtained from rest-frame spinors (polarized along the z–direction) through a longitudinal and
transverse boost; these spinors transform in a simple manner under Lorentz transformations and
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Chiral-oddGPDsinlarge–N QCD P.Schweitzer
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have a clear connection to polarization states in the rest frame. The helicity amplitudes are then
definedas
dz
Aql ′m ′,lm =Z 2p− eixP+z−hN(P′,l ′)|Omq′m |N(p,l )i(cid:12)z+=0,~zT=0, (2.6)
(cid:12)
where l (l ) are the light-front helicity of the initial (final) nucleon(cid:12)and m (m ) those of the initial
′ (cid:12) ′
(final)quark. Withoutlossofgenerality onemaychoose aframeinwhichthenucleon3-momenta
~p,~p are in the x-z-plane. The chiral-odd light–ray operator associated with quark helicity flip is
′
givenby[5]
i
Oq =y ( z) s +1(1 g )y (z). (2.7)
+− q −2 4 − 5 q 2
Thehelicityamplitudes arerelated tochiraloddGPDsas
1 x
Aq =d H˜q+ − (Eq+E˜q) , (2.8a)
++,+ k T 2 T T
−
(cid:18) (cid:19)
1+x
Aq =d H˜q+ (Eq E˜q) , (2.8b)
−+,−− k T 2 T − T
(cid:18) (cid:19)
x 2 x
Aq = 1 x 2 Hq+d 2H˜q Eq+ E˜q , (2.8c)
++,−− − T k T −1 x 2 T 1 x 2 T
(cid:18) − − (cid:19)
p
Aq = 1 x 2d 2H˜q, (2.8d)
−+,+− − k T
wherethe“kinematic” prefactopr d isdefinedas
k
√t t 4M2x 2
d =sign P+D 1 D +P1 0− , t = N , (2.9)
k − 2M −0 1 x 2
N
(cid:18) (cid:19) −
inwhich t istheminimalvalueof t forthegivenvalueofx .
0
− −
Inthe1/N expansion itisconvenient toworkwithlinear combinations ofthehelicity ampli-
c
tudes Eqs. (2.8a-2.8d) that have a simple interpretation in terms of spin transitions in the nucleon
restframe. Wedefinethemas
1
Aq= Aq +Aq (cid:181) d , (2.10a)
0 2 ++,+− −+,−− ll ′
(cid:18) (cid:19)
1
Aq= Aq +Aq (cid:181) s 1 , (2.10b)
1 2 ++,−− −+,+− ll ′
(cid:18) (cid:19)
1
Aq= Aq Aq (cid:181) s 2 , (2.10c)
2 2 ++,−−− −+,+− ll ′
(cid:18) (cid:19)
1
Aq= Aq Aq (cid:181) s 3 , (2.10d)
3 2 ++,+−− −+,−− ll ′
(cid:18) (cid:19)
wheres a arethePaulispinmatrices. Inthenucleonrestframe,thesecombinationsdescribespin
ll
′
transitions in which the matrix element of the spin operator has components along the x,y and z
axes.
3. 1/N expansionofchiral-odd GPDs
c
In the large-N the nucleon mass scales as M N , while the nucleon size remains stable,
c N c
∼
R N0. The 1/N expansion of GPDs is performed in a class of frames where the initial and
c c
∼
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Chiral-oddGPDsinlarge–N QCD P.Schweitzer
c
final nucleon move with 3–momenta pk, pk N0 (k=1,2,3) and have energies p0, p0 =M +
′ c ′ N
∼
O(1/N ),whichimpliesanenergyandmomentumtransfer D 0 N 1,D k N0,andthus
c c− c
∼ ∼
D i N0 (i=1,2), x N 1, t N0. (3.1)
c c− c
∼ ∼ | | ∼
Inthepartonicvariable xoneconsiders theparametricregion
x N 1, (3.2)
c−
∼
corresponding to non-exceptional longitudinal momenta of the quarks and antiquarks relative to
theslowlymovingnucleon, xM R 1 N0. Likewise,itisassumedthatthenormalizationscale
N − c
∼ ∼
of the light-ray operator is N0, so that the typical quark transverse momenta are N0. Equa-
c c
∼ ∼
tion (3.2) corresponds to the intuitive picture of a nucleon consisting of N “valence” quarks and
c
a “sea” of O(N ) quark–antiquark pairs, with each quark/antiquark carrying on average a frac-
c
tion 1/N of the nucleon momentum. Altogether, the N –scaling relations for GPDs can then
c c
∼
expressed intheform
GPD(x,x ,t) Nk Function(N x,N x ,t), (3.3)
c c c
∼ ×
wherethescalingexponentkdependsonthespin–flavorstructureandcanbeestablishedongeneral
grounds,whilethescalingfunctionontheright-hand-sidedoesnotexplicitlydependonN andcan
c
onlybedetermined inspecificdynamicalmodels.
Ageneralmethodforthe1/N expansionofnucleonmatrixelementsofquarkbilinearopera-
c
torshasbeengiveninRef.[1]. Itusesthefactthatthelarge–N nucleonisdescribed byaclassical
c
fieldwithacertainspin–isospin symmetry,andthatthenucleonstatesofdefinitespin,isospin,and
momentum areobtained through quantization ofthe(iso–)rotational andtranslational zeromodes
[35]. Inthisclassical picture thenucleon matrixelement ofthequark bilinear operator isobtained
by taking the expectation value of the operator in the static classical field, allowing for collective
(iso–) rotations, and computing the transition matrix element between collective wave functions
corresponding to the desired nucleon spin–isospin and momentum states. While the expectation
value of the operator in the classical field cannot be calculated from first principles and remains
unknown, its symmetry properties are unambiguously determined by the spin–isospin symmetry
of the classical field and the quantum numbers of the quark bilinear operator. Altogether, this al-
lowsonetoestablishtheN –scalingofthedifferentspin–isospincomponentsofthematrixelement
c
withoutcalculating thecoefficients accompanying thepowersof1/N .
c
We use the method of Ref. [1] to establish the N –scaling of the nucleon–quark helicity am-
c
plitudes andthe chiral-odd GPDs. Itisnatural towork withthecombinations Eqs.(2.10a–2.10d),
which correspond to definite spin transitions in the frame where the nucleons are moving slowly,
andwhichexhibithomogeneousscalingin1/N . Whenexpressingthehelicityamplitudesinterms
c
of the GPDs, we use that in the large–N limit the kinematic factor d N 1 in (2.9) and the
c k c−
∼
variable x become
D 1 D 3
d = , x = . (3.4)
k
2M − 2M
N N
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Chiral-oddGPDsinlarge–N QCD P.Schweitzer
c
Inthiswayweobtaininleading orderofthe1/N expansion therelations
c
d
N2 Au+d = k Eu+d+2H˜u+d , (3.5a)
c ∼ 0 2 T T
(cid:18) (cid:19)
1
N2 Au d = Hu d+x E˜u d , (3.5b)
c ∼ 1− 2 T− T−
(cid:18) (cid:19)
1
N2 Au d = Hu d+x E˜u d , (3.5c)
c ∼ 2− 2 T− T−
(cid:18) (cid:19)
d
N2 Au d = k E˜u d , (3.5d)
c ∼ 3− 2 T−
(cid:18) (cid:19)
which are to be understood in the sense of Eq. (3.3), and in which we indicate only the scaling
exponent. Insubleading orderofthe1/N expansion,
c
d
N Au d = k Eu d x E˜u d+2H˜u d , (3.6a)
c∼ 0− 2 T− − T− T−
(cid:18) (cid:19)
1
N Au+d = Hu+d x 2Eu+d+x E˜u+d+2d 2H˜u+d , (3.6b)
c∼ 1 2 T − T T k T
(cid:18) (cid:19)
1
N Au+d = Hu+d x 2Eu+d+x E˜u+d , (3.6c)
c∼ 2 2 T − T T
(cid:18) (cid:19)
d
N Au+d = k E˜u+d x Eu+d . (3.6d)
c∼ 3 2 T − T
(cid:18) (cid:19)
FromEqs.(3.5a—3.6d)wecanreadoffthelarge-N flavorstructureoftheGPDs,namely
c
Hu d N2, Hu+d N , (3.7)
T− ∼ c T ∼ c
Eu+d N3, Eu d N2, (3.8)
T ∼ c T− ∼ c
H˜u+d N3, H˜u d N2, (3.9)
T ∼ c T− ∼ c
E˜u d N3, E˜u+d N2. (3.10)
T− ∼ c T ∼ c
These relations generalize previous results for the N –scaling of the nucleon’s quark transversity
c
q q
PDF h [obtained as the forward limit of H , Eq. (2.5)] [42, 43, 44] and the tensor form factors
1 T
[obtained asthefirstx–moment, Eq.(2.4)][26,27].
The scaling relations Eq. (3.7) reveal several interesting properties of the chiral-odd GPDs.
First,oneseesthatinH andE˜ theflavor–nonsinglet componentisleadingandtheflavor–singlet
T T
onesubleading, whileinE andH˜ theorderisopposite.
T T
Second, onenotices thatthe amplitudes Au d and Au d aredegenerate inleading order ofthe
1− 2−
1/N expansion;seeEqs.(3.5b,3.5c). Thishappensbecausethemean–fieldpictureimpliedbythe
c
large–N limitmakesnodistinction between different transverse polarization directions inleading
c
order of the 1/N expansion. The equality Au d = Au d implies that the nucleon–quark double
c 1− 2−
helicity–flip amplitude vanishes, Au d =0, i.e., the amplitude in which the quark has helicity
−+,+
− −
oppositetothenucleonintheinitialstateandbothhelicitiesareflippedinthefinalstate. Ifweview
the nucleon as a system composed of the active quark and a “remnant,” the double–flip transition
involvesachangeoftherelativeorbitalangularmomentumbytwounits,D L=2[5]. Amongallthe
nucleon–quark helicity amplitudes (including also the chiral-even ones) this isthe only amplitude
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Chiral-oddGPDsinlarge–N QCD P.Schweitzer
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that is(i)double–flip, and (ii) vanishes exactly inthe leading order ofthe large-N limit. Itmerits
c
further study whether this observation could beexplained moredirectly inlight ofthemechanical
picture impliedbythelarge-N limit.
c
Third,onenoticesinEqs.(3.5a—3.6d)thatthecombinationEq+2H˜q E¯q emergesnaturally
T T ≡ T
fromthelarge-N expansionofthechiral-oddGPDs. ThedifferencesbetweentheindividualGPDs
c
Eq andH˜q aresuppressedin1/N . Interestingly,thiscombinationappearsalsointheamplitudefor
T T c
pseudoscalarmesonproduction[17],meaningthatthevirtualphotoncannot“distinguish”between
Eq andH˜q.
T T
4. Comparisonwithpseudoscalarmesonproduction data and outlook
It is interesting to compare our results with preliminary data from the JLab CLAS exclusive
pseudoscalar meson production experiments [19, 20] (cf. comments in Sec. 1). Analysis of the
azimuthal–angle dependent response functions shows that s s , which indicates domi-
LT TT
| |≪| |
nance of the twist-3 amplitudes, involving the chiral-odd GPDsHq and E¯ =Eq+2H˜q, over the
T T T T
twist–2 amplitudes involving the chiral-even GPD E˜q. A preliminary flavor decomposition was
performed assuming dominance of the twist–3 amplitudes and combining the data on p 0 and h
production, in which the u and d quark GPDs enter with different relative weight. Results show
opposite sign oftheexclusive amplitudes Hu and Hd , whichisconsistent withtheleading ap-
h Ti h Ti
pearanceoftheflavor-nonsinglet Hu d inthe1/N expansion. (Here ... denotestheintegralover
T− c h i
xoftheGPD,weightedwiththemesonwavefunction, hardprocessamplitude, andSudakovform
factor [17].) The results also suggest same sign of E¯u and E¯d , which is again consistent with
h Ti h Ti
theleadingappearanceoftheflavor-singlets Eu+d andH˜u+d inthe1/N expansion. Thesefindings
T T c
q
shouldbeinterpretedwithseveralcaveats: (a)theerrorsintheexperimentalextractionof H and
h Ti
E¯q are substantial; (b) the 1/N expansion predicts only the scaling behavior, not the absolute
h Ti c
magnitude oftheindividual flavorcombinations, cf.Eq.(3.3).
Itisencouraging that theflavor structure ofthe amplitudes extracted from the p 0 andh elec-
troproduction data is consistent with the pattern predicted by the 1/N expansion. Our findings
c
further support the idea that pseudoscalar meson production at x & 0.1 and Q2 fewGeV2 is
B
∼
governed bythetwist-3mechanism involving thechiral-odd GPDs. Itwouldbeinteresting tocal-
culate thechiral-odd GPDsindynamical modelsthatconsistently implement theN –scaling, such
c
asthechiral quark–soliton model. Suchcalculations would allow onetocalculate alsothe scaling
functions in the large–N relations, Eq. (3.3), and supplement the scaling studies with dynamical
c
information.
Acknowledgments
Wethank V. Kubarovsky for discussions on the preliminary JLab CLASpseudoscalar meson
production data. In our study we greatly benefited from discussions with D. Diakonov, V. Petrov,
P. Pobylitsa, and M. Polyakov during earlier joint work. P.S.is supported by NSF under Contract
No. 1406298. Notice: Authored by Jefferson Science Associates, LLCunder U.S. DOEContract
No. DE-AC05-06OR23177. The U.S. Government retains a non-exclusive, paid-up, irrevocable,
world-widelicense topublishorreproduce thismanuscript forU.S.Governmentpurposes.
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Chiral-oddGPDsinlarge–N QCD P.Schweitzer
c
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