Generalized parton distributions of the pion in chiral quark models and their QCD evolution

0.74 MB·English

by  Wojciech Broniowski

Generalized parton distributions of the pion
in chiral quark models and their QCD evolution∗
Wojciech Broniowski,1,2, Enrique Ruiz Arriola,3, and Krzysztof Golec-Biernat1,4,
† ‡ §
1The H. Niewodniczan´ski Institute of Nuclear Physics, PL-31342 Krak´ow, Poland
2Institute of Physics, S´wie¸tokrzyska Academy, ul. S´wie¸tokrzyska 15, PL-25406 Kielce, Poland
3Departamento de F´ısica At´omica, Molecular y Nuclear,
Universidad de Granada, E-18071 Granada, Spain
4Institute of Physics, Rzesz`ow University, PL-35959 Rzesz´ow, Poland
(Dated: 6 December 2007)
WeevaluateGeneralizedPartonDistributionsofthepionintwochiralquarkmodels: theSpectral
Quark Model and the Nambu–Jona-Lasinio model with a Pauli-Villars regularization. We proceed
8 bytheevaluationofdoubledistributionsthroughtheuseofamanifestlycovariantcalculationbased
0 on the α representation of propagators. As a result polynomiality is incorporated automatically
0 and calculations become simple. In addition, positivity and normalization constraints, sum rules
2 and soft pion theorems are fulfilled. We obtain explicit formulas, holding at the low-energy quark-
model scale. The expressions exhibit no factorization in the t-dependence. The QCD evolution of
n
thosepartondistributionsiscarried out toexperimentallyorlattice accessible scales. Weargue for
a
J the need of evolution by comparing the Parton Distribution Function and the Parton Distribution
Amplitude of the pion to the available experimental and lattice data, and confirm that the quark-
7
model scale is low, about 320 MeV.
1

PACSnumbers: 12.38.Lg,11.30,12.38.-t
]
h Keywords: generalizedpartondistributions,doubledistributions,light-coneQCD,exclusiveprocesses,chiral
p quarkmodels
-
p
e I. INTRODUCTION the pion. Although there are little chances of mea-
h
suring them directly in experiment, the pion GPD’s
[
are amenable to indirect experimental determination as
2 Generalized Parton Distributions (GPD’s) encode de- well as studies both on transverse [12] as well as Eu-
v taileddynamicalinformationonthe internalstructureof clidean [13] lattices (for a combination of experimental
2 hadrons and have thus become in recent times a ma-
and lattice-based reconstruction in the proton case see,
1 jor theoretical and experimental endeavor (for exten-
e.g., Ref. [14]). In addition, there are many theoreti-
0 sive reviews see e.g. [1, 2, 3, 4, 5, 6, 7, 8] and refer-
cal advantages for studying this quark-antiquark bound
1
ences therein). They represent a natural interpolation
. state. Inthefirstplace,inthechirallimitwherethecur-
2 between form factors and quark distribution functions.
rent quark masses vanish, the spontaneous breakdown
1 Actually, while form factors and distribution functions
of chiral symmetry of the QCD vacuum generates pions
7
provide in a separate way the spatial and momentum
0 as the zero modes. Their properties are expected to be
quark distributions in a hadronrespectively, GPD’s pro-
: dominatedby the brokenchiralsymmetry while confine-
v videasimultaneousphase-spacedescriptionofthe quark
ment effects are expected not to be crucial. Actually,
i hadron content as far as the position-momentum uncer-
X finite mass corrections to GPD’s of the pion have been
taintyrelationsallow[5,9]. Experimentally,GPD’sshow
r treated in the standard [15] and partially-quenched [16]
a up in hard exclusive processes such as Deeply Virtual chiral perturbation theory, while the breakdown of the
Compton Scattering (DVCS) or hard electroproduction expansion for small x m2/(4πf)2 has been pointed
of mesons. Factorization for hard exclusive electropro- ∼ π
out in Ref. [17]. Besides, compared to the nucleon there
duction of mesons in QCD was proved in Ref. [10]. Ef-
areno spin complicationsfor the pioncase,andthus the
fects of the Regge exchanges to exclusive processes were
study reduces to two singlescalarGPDfunction, one for
investigated in Ref. [11].
each isospin combination. Finally, the pion provides a
In the present paper we are interested in GPD’s for useful framework to learn on the interplay between the
chiralsymmetryandthe lightconefeatures,sinceweare
studying the behavior of the would-be Goldstone boson
in the infinite momentum frame.
∗Supported by Polish Ministry of Science and Higher Education,
grants 2P03B 02828 and N202 034 32/0918, Spanish DGI and Despite the intrinsiccomplexityofthe GPD’s,thereis
FEDERfundswithgrantFIS2005-00810,JuntadeAndaluc´ıagrant a number of simple conditions which ultimately are con-
FQM225-05, and EU Integrated Infrastructure Initiative Hadron sequences of the Poincar´eand electromagnetic gauge in-
PhysicsProjectcontractRII3-CT-2004-506078.
varianceandprovidea priori testsonthe validityofthe-
†Electronicaddress: [email protected]
‡Electronicaddress: [email protected] oretical calculations. Proper support and polynomiality
§Electronicaddress: [email protected] restrictions on the GPD moments [1] are manifestations
2
of the Lorentz invariance. We note that polynomiality corporating chiral symmetry. Early calculations of pion
is not satisfied in light-front calculations. Double distri- GPD’s were done in an instanton-inspired model char-
butions do not suffer from this problem [18] (using the acterized by a momentum dependent mass function of
double distributions is a way of projecting the Lorentz- a dipole form [19, 36, 37], while PDF’s were evaluated
violating term onto the right space) although they re- in the same model in [38, 39]. The crossing-related
quiretheso-calledD-terms[19]tocomplywiththe most 2πGPA was also evaluated in Ref. [40] in the same in-
general polynomial allowed by the dimensional analysis. stanton model disregarding the momentum dependence
Normalizationconditionsandsum rulesarea manifesta- of the quark mass, a valid assumption in the limit of
tion ofthe gaugeinvariance,whichat the quarklevelre- small instantons. In that limit, end point discontinu-
quiresthe correctimplementationofthe electromagnetic ities arise. A full consideration of poles in the complex
Ward-Takahashiidentities. Thepositivitybound[20,21] plane in a non-local version of the NJL model was de-
underlines the Hilbert-space quantum-mechanical prob- scribed in Ref. [41]. Generally, the nonlocality of the
abilistic nature of pion light-cone wave functions, and quark mass function generated incorrect normalization,
may impose relevant constraints on admissible regular- since as noted later, PCAC should be properly incor-
izations based mainly on subtractions of ultraviolet di- porated [42], an issue also emphasized more recently in
vergences. Soft pion theorems based on PCAC relate Ref. [43]. A rather interesting feature of Ref. [42] is
GPD’s to Parton Distribution Amplitudes (PDA’s) [22]. thatend-pointdiscontinuitiesreappearafterPCACisin-
On a theoretical level, the amazing aspect of GPD’s is corporated, even for momentum-dependent quark mass
thattheconstraintsthatoughttobefulfilledaprioriare functions, against the widely spread prejudice that they
so demanding and intricate that it is extremely difficult onlyariseformomentum-independentmasses. TheOPE
to provide ansa¨tze fulfilling all of them simultaneously. and duality aspects of GPD’s have been discussed in
Thisiswhydynamicalcalculationsgoingbeyondreason- Ref. [44]. Light-front calculations have been undertaken
ablebutadmittedlyad hocparameterizationsarepartic- inRef.[45,46]forpointcouplingswithsubsequentinser-
ularly interesting and instructive. On the other hand, tion of Gaussian wave functions, however the approach
dynamical models providing GPD’s are also generating violates polynomiality. Power-law wave functions and
mutually consistent parton distribution functions, par- GPD’s of the pion proposed in Ref. [47] satisfy polyno-
ton distribution amplitudes, and form factors. Although miality but violate positivity (see [48]). Studies paying
this may appear a rather trivial statement, it imposes particular attention to polynomiality were first made by
demanding and tight constraints on details of the calcu- Tiburzi and Miller [48, 49] who proceeded via double
lation, and more specifically on the proper handling of distributions[50]. However,regularizationintheseworks
ultravioletdivergencesbasedonthe correctimplementa- wasdonebyintroducingmomentumdependentformfac-
tion of electromagnetic and chiral Ward-Takahashiiden- tors, which makes them difficult to reconcile with the
tities. Even in the case of a simple hadron such as the gaugeinvariance. ThemodelofRef.[51]basedonapseu-
pion in the chiral limit, the above mentioned necessary doscalarpion-quarkcoupling does not incorporatechiral
conditionsprovidepowerfullimitationsandinsomecases symmetry and does not fulfill the momentum sum rule.
clashwithwellestablishedprejudicesaboutthe meaning Noguera, Theußl, and Vento carried out a calculation
and realization of a relativistic bound-state wave func- in the NJL model based on the light-front coordinates,
tions in quantum field theory [23, 24, 25]. where the fulfillment of polynomiality for non-vanishing
momentumtransferisnotapparentanalytically[52],and
In the present work we determine GPD’s incorporat-
in fact numerical integration was required to establish
ing all the desirable properties required by the sym-
this property. Our NJL results agree with that work,
metries in two chiral quark models, the Nambu–Jona-
with the important methodological difference that the
Lasinio(NJL)(forreviewsseee.g. [26,27]andreferences
double distributions, where polynomiality is manifest,
therein) and the Spectral Quark model (SQM) [28, 29],
are used throughout. Moreover, our regularization is
which essentially is a way of introducing regularization
somewhat different than in the model of Ref. [52], we
in such a way that the vector-meson dominance is built
also use the non-linear rather than linear realization of
in. Thesefieldtheoreticalmodelsincorporatethesponta-
the chiral field.
neous breaking of chiral symmetry. Chiral quark models
use the large-N expansion at leading order, where the
c DespitethenumerousmodelcalculationsoftheGPD’s,
observables are obtained with one-loop quark diagrams.
itremainstodateunclearwhatistheirsignificanceorim-
The present calculation extends previous calculations of
pact on the interpretation of actual experiments and/or
PDF’s [23,28,30]andPDA’s[27,31]. DiagonalGPDin
lattice data. This is, perhaps, why most calculations of
impact parameterspace in these models wereconsidered
the GPD’s based on dynamical quark models and going
[32]. Our presentGPD resultreproduces consistently all
beyond just phenomenological parameterizations do not
theseparticularcases. Recently,theTransitionDistribu-
addressthisissue. However,whileexperimentalorlattice
tion Amplitude (TDA) [33, 34] has also been evaluated
results generate scale-dependent GPD’s, embodying the
in SQM [35].
well established logarithmic scaling violations in QCD,
TherehasbeenanumberofcalculationsofGPD’sand it is notorious that models generally produce scale inde-
related quantities of the pion within the framework in- pendent functions. Thus, quark models represent those
3
distributionsatagivenlowenergyscale. Itisnoteworthy thefinalresultsforthe GPD’scanbe writtenintermsof
thatscalingviolationscanonlybecomputedinthetwist rathersimplebutnon-trivialanalyticformulas,whichal-
expansionorderby order. Forinstance,for the structure lowsformoreinsightintotheir properties. We alsoshow
functions F(x,Q) with the Bjorken x and momentum Q thatourGPD’ssatisfythepositivitybounds. Inessence,
one has for the quark model all the knownconsistency conditions and constraints are
indeed satisfied in our calculation.
F (x)
2
F(x,Q)=F0(x)+ Q2 +..., (1.1) Theoutlineofthepaperisasfollows: InSect.IIwelist
the definitions and properties of the GPD’s. We intro-
while for QCD duce both the asymmetric and symmetric kinematics, as
wellasdefinetheisospinprojections. Wegivethegeneric
F (x,α(Q2)) quark-model expressions in terms of the two- and three-
F(x,Q)=F (x,α(Q2))+ 2 +..., (1.2)
0 Q2 point functions. Sections III and IV contain the results
oftheSpectralQuarkModelandtheNJLmodel,respec-
whereF (x,α(Q2)) arelowenergymatrixelements with tively. We discuss the general need for the QCD evolu-
n
computable anomalous dimensions and depending log- tionofchiralquarkmodelsinSectionV,wherewedefine
arithmically on the scale through the running coupling the matching condition in the light of phenomenological
constant analyses, as well as Euclidean and transverse lattice cal-
culations. Inparticular,weshowthe evolvedforwarddi-
4π
α(Q2)= (1.3) agonalPDFofthepionandconfrontitwiththeresultsof
β0log(Q2/Λ2QCD) the E615experimentatFermilab[53]. Thisagreementis
11 2 quite remarkable, but sets the quark model momentum
β = N N . (1.4)
0 3 c− 3 f scale to very low values, Q0 320 MeV. Likewise we
≃
also discuss the evolved PDA as compared to the E791
In this work we take measurement [54] of the pion light-cone wave function
and to the lattice data. We show how the QCD evo-
Λ =226 MeV (1.5)
QCD lution leads to vanishing of the PDF at x = 1 and of
PDA at x = 0,1, which is the desired end-point behav-
and N = N = 3. The matching conditions are taking
c f ior. TheLOQCDevolutionofourgenuineGPD’s,based
ata givenscaleQ orderby orderin the twistexpansion
0 onthe standardERBL-DGLAPequations,iscarriedout
F (x) =F (x,α(Q2)) . (1.6) in Sect. VI, where the obtained quark-model initial con-
n |Model n 0 |QCD ditions are evolved to higher momentum scales. Finally,
Aquite differentissueis the operationaldefinitionofthe in Section VII we draw our main conclusions. The Ap-
low-energyreferencescaleQ . Herewewillusealongthe pendices contain the technique of analyzing the GPD’s
0
lines of previous works [23, 27, 30, 31] the momentum through the use of the α representation for the propa-
fraction carried by the valence quarks. It turns out that gators. This method, first introduced in the context of
in order to describe the available pion phenomenology structure functions in Refs. [38, 39], allows for a man-
the initial scale Q from the quark model must be very ifestly covariant calculation and leads to simple formal
0
low,around320MeV. Atsuchlowscaletheperturbative expressions for the basic two- and three-point functions
expansion parameter in the evolution equations is large, emerging in the analysis. The appearance of D-terms is
α(Q2)/2π = 0.34, which makes the evolution very fast manifest and natural in this treatment, based solely on
0
for scales close to the initial value Q . the Feynman diagrams in a conventional way. We also
0
Noneofthe previouschiral-quark-modelstudies ofthe list explicitly the basic two- and three-point functions of
genuineGPD’s(off-forwardnon-diagonal)carriedoutthe the two considered model. We proceed via the double
QCD evolution, starting from the initial condition pro- distributions, whichleads to a simple proofof polynomi-
vided by the models. The evolution is a major element ality [2].
of this work. As already mentioned, it is also a crucial
element if one wishes to compare the model prediction
to the data from experiments or lattice simulations. At
themomentthesedataareavailableonlyfortheforward
diagonalpartondistributionfunction ofthe pion(PDF), II. DEFINITIONS AND QUARK-MODEL
or the PDA. EXPRESSIONS
We have taken an effort to separate formal aspects
of the calculation from the model-dependent technical-
A. Formalism for pion GPD
ities of the regularization. That way we simply achieve
the desired features on generalgrounds,such as the sum
rules or the polynomiality conditions [1]. We stress this The kinematics of the process and the assignment of
isachievedwithoutafactorizedforminthetvariable. In momenta(inthe asymmetricway)isdisplayedinFigs.1
oneoftheconsideredmodels(theSpectralQuarkModel) and2. Forthe pionsonthemassshellwehave,adopting
4
the standard notation with the symmetry properties about the X =0 point,
p2 =m2π, q2 =−2p·q =t, HI=0(X,ξ,t) = HI=0( X,ξ,t),
n2 =0, p n=1, q n= ζ. (2.1) HI=1(X,ξ,t) = −HI=1( −X,ξ,t). (2.8)
· · −
−
Note the sign convention for t, positive in the physical
The following sum rules hold:
region.
The leading-twist off-forward (t = 0) non-diagonal
6 1
(ζ = 0) generalized parton distribution (GPD) of the dXHI=1(X,ξ,t)=2F (t), (2.9)
6 V
pion is defined as Z−1
1
ab(x,ζ,t)= dz−eixp+z− (2.2) dXXHI=0(X,ξ,t)=θ2(t)−ξ2θ1(t), (2.10)
H Z 4π × Z−1
πb(p+q)ψ¯(0)γ nT ψ(z)πa(p) ,
h | · | i z+=0,z⊥=0 where FV(t) is the electromagnetic form factor, while
θ (t) and θ (t) are the gravitational form factors of the
where0≤ζ andthexvariable,−1+ζ ≤x(cid:12)(cid:12)≤1,isdefined p1ion(seeAp2pendixD)whichsatisfythelowenergytheo-
in the asymmetric notation (cf. Fig. 1), a and b are
rem θ (0)=θ (0) in the chirallimit [55]. Sum rule (2.9)
1 2
isospinindices forthe pion,T is the isospinmatrixequal
expressesthe electricchargeconservation,while(2.10)is
1 for the isoscalar and τ for the isovector cases, finally
3 responsiblefor the momentum sumrule indeepinelastic
ψ is the quark field and z is the light-cone coordinate.
scattering. Finally, for X 0
Explicitly, the two isospin projections are equal to ≥
δ I=0(x,ζ,t)= dz−eixp+z− (2.3) HI=0,1(X,0,0)=q(X),
ab
H 4π ×
Z relating the distributions to the the pion’s forwarddiag-
πb(p+q)ψ¯(0)γ nψ(z)πa(p) ,
h | · | i z+=0,z⊥=0 onal parton distribution function (PDF), q(X).
iǫ I=1(x,ζ,t)= dz−eixp+z(cid:12)(cid:12)− (2.4) The polynomiality conditions [1, 2] state that
3ab
H 4π ×
hπb(p+q)|ψ¯(0)γ·nZψ(z)τ3|πa(p)i z+=0,z⊥=0. d1XX2jHI=1(X,ξ,t)= j A(ij)(t)ξ2i,
One can form the combinations termed (cid:12)(cid:12)the quark and Z−1 Xi=0
antiquark GPD’s of the pion, 1 j+1
dXX2j+1HI=0(X,ξ,t)= B(j)(t)ξ2i, (2.11)
i
(x,ζ,t) = 1( (x,ζ,t)+ (x,ζ,t)), Z−1 Xi=0
q I=0 I=1
H 2 H H
1 where A(j)(t) and B(j)(t) are the coefficient functions
(x,ζ,t) = ( (x,ζ,t) (x,ζ,t)). (2.5) i i
Hq¯ 2 HI=0 −HI=1 (form factors) depending on j and i. The polynomiality
conditions follow from basic field-theoretic assumptions
From the general formulation it follows that (x,ζ,t)
Hq such as the Lorentz invariance, time reversal, and her-
has the support x [0,1], whereas (x,ζ,t) the sup-
∈ Hq¯ miticity, hence are automatically satisfied in approaches
port x [ 1+ζ,ζ]. The range x [0,ζ] is called the
∈ − ∈ that obey these requirements. Conditions (2.11) supply
ERBL region,while x [ 1+ζ,0] and x [ζ,1] are the
∈ − ∈ importanttestsofconsistency. Inourapproachthepoly-
DGLAP regions, where the nomenclature refers to the
nomiality will be demonstrated straightforwardly in an
QCD evolution, see Sect. VI.
analyticwaythroughthe use ofdouble distributions, see
Inthesymmetricnotation,somewhatmoreconvenient
Appendix A.
in certain applications1, one introduces
Another constraint for the GPD’s, the positivity
ζ x ζ/2 bound [21], is derived with the help of the Schwartz in-
ξ = , X = − , (2.6) equality and the momentum representation of the pion
2 ζ 1 ζ/2
− − light-cone wave functions. In the simplest form the con-
where 0 ξ 1 and 1 X 1. Then straint states that (for t 0)
≤ ≤ − ≤ ≤ ≤
ξ+X 2ξ
HI=0,1(X,ξ,t)= I=0,1 , ,t . (2.7) H (X,ξ,t) q(x )q(x ), ξ X 1. (2.12)
q in out
H ξ+1 ξ+1 | |≤ ≤ ≤
(cid:18) (cid:19)
p
where x =(x+ξ)/(1+ξ), x =(x ξ)/(1 ξ).
in out
− −
The off-forward (∆ = 0) diagonal (ξ = 0) GPD of
the pion (we take π+)⊥ca6n be written as
1 Inthispaperweswitchbackandforthbetweenthetwoconven-
tions, since explicit expressions are shorter in the asymmetric
notation, whilesomeformalfeatures are simplerto state in the H(x,ξ =0, ∆2)= d2bei∆⊥·bq(x,b). (2.13)
symmetricnotation. − ⊥
Z
5
We use here
q(x,b)= dz−eixp+z− (2.14)
4π
Z
z z
π+(p′)q¯(0, −,b)γ+q(0, −,b)π+(p) ,
× h | − 2 2 | i
where x is the Bjorken x, ∆ = p p lies in the trans-
′
verse plane, and b is an imp⊥act par−ameter. The model-
independent relationfound in Ref. [56] reads in the pion
case
1 d2q
Z0 dxq(x,b)=Z (2π)⊥2eiq⊥·bFV(−q2⊥) (2.15)
By crossing, the process related to the Deeply Virtual
Compton Scattering (DVCS) off the pion, i.e., two pion
production in γ γ collisions, can be measured at low in-
∗
variant masses [57]. The relevant matrix element reads FIG.1: Thedirect(a)andcrossed(b)Feynmandiagramsfor
thequark-modelevaluation of theGPD of thepion.
Φab(u,ζ,W2)= dz−eixp+z− (2.16)
4π ×
Z
hπa(p1)πb(p2)|ψ¯(0)γ·nTψ(z)|0i z+=0,z⊥=0, B. Formal results for chiral quark models
(cid:12)
whereW2 =(p +p )2,ζ =p n/P nand(cid:12)u=(p p )2.
1 2 1· · 1− 2 The reduction formulas applied to the definition (2.4)
By comparing, we have
result in the amputated three-point Green function with
theconstrainedquarkmomentumintegration,k+ =xp+.
Φab(u,ζ,W2)= ab(x,ζ,t) (2.17) Large-N treatment leads to one-quark-loop diagrams,
H c
withmassivequarksduetospontaneouschiralsymmetry
One has the soft pion theorem [22], breaking. In nonlinear chiral quark models the quark-
pion interaction is described by the term ψ¯ωU5ψ in
ΦI=1(u,1,0)=HI=1(2u 1,1,0)=φ(u), (2.18) the effective action, where the pion field mat−rix is
−
where φ(u) represents the Pion Distribution Amplitude
(PDA) defined as U5 =exp(iγ5τ φ/f), (2.20)
·
1
hπa(p)|ψ¯(z)γµγ52τbψ(0)|0i|z+=0,z⊥=0 wheref denotes the pion decay constant. The resulting
Feynman rules and the definition (2.2) lead to the Feyn-
1
=ifpµδab dxeixpzφ(x). (2.19) man diagrams of Figs. 1 and 23. The presence of the
·
Z0 contact term is crucial for the preservation of the chi-
ral symmetry [19]. The evaluation of the diagrams is
Note that result(2.18) is basedon PCAC andhence is a straightforward, giving the following result for the isos-
consequence of the chiral symmetry. One of the reasons inglet and isovector parts:
to prefer GPD’s rather than 2πPDA is the absence of
finalstateinteractions,whicharesuppressedinthe large
N limit2. (x,ζ,t) = (x,ζ,t)+ (x,ζ,t)+ (x,ζ,t),
c I=0 a b c
H H H H
(x,ζ,t) = (x,ζ,t) (x,ζ,t). (2.21)
I=1 a b
H H −H
2 Thesimplestexampleillustratingthisfeatureisprovidedbythe
pionelectromagnetic formfactor. Theradiusreads The explicit contributions of the subsequent diagrams
hr2iπ = M6V2 "1− 4N1c log MmV2π2 !#,
the first contribution stemming from the quark loop and the 3 ThesimilarcalculationofRef.[52]usesthe linearrealizationof
secondcontributionanestimatefrompionloops[58]. thechiralsymmetry,withtheσ fieldpresent.
6
FIG.2: Thecontactcontribution(c)totheGPDofthepion,
responsible for the D-term.
FIG. 3: The contour C for evaluation of observables in the
to the GPD’s of Eq. (2.21) are meson dominance variant of SQM. MV denotes the ρ-meson
mass. The cross and hatched regions indicate theposition of
iN ω2 thepole and cutsof thespectral function Eq. (3.3).
(x,ζ,t)= c d4kδ(k n x)
Ha 4π2f2 · − ×
Z
ω2 k2 ζ ω2 k2+k p +x ω2 t k2+2k p k q
− − − · −2− · − · , The contribution of the diagram(c), having the support
(cid:0) DkD(cid:1)k+qD(cid:0)k p (cid:1) for x [0,ζ], is the D-term [19].
− ∈
iN ω2 Fromtheaboveformitisclearthatweneedtoconsider
(x,ζ,t)= c f2 d4kδ(k n x)
Hb 4π2 · − × two generic types of two- and three-point integrals:
Z
ω2+k2+ζk p+x ω2 t k2 2k p 2k q +k q
− i·N ω2D(cid:0)kD−k+2q−Dk+−p+q · −2x ·ζ(cid:1) · , I(x,l·n,l′·n,(l−l′)2)= −4iπN2cfω22 Z d4kδD(kk−·lnD−k−xl′),
Hc(x,ζ,t)= 4π2cf2 d4kδ(k·n−x)D D− . (2.22) J(x,l·n,l′·n,l2,l′2,l·l′)=
Z k k+q iN ω2 δ(k n x)
c d4k · − . (2.26)
The denominator of the proparator of quark of mass ω 4π2f2 Z DkDk−lDk−l′
and momentum l is denoted as
These are analyzed in detail Appendices A1 and A2.
D =l2 ω2+i0. (2.23) The two-point function I is logarithmically divergent,
l
−
hence the analysis needs regularization. This is where
The powers of the momentum k in the numerators may different quark models depart from one another. We
be eliminated with the following reduction formulas: mayseparatethe issuesofregularizationfromformalex-
pressions,whichisconvenientfortheoreticalaspectsand
k2 = Dk+ω2, the demonstration of the consistency conditions. Writ-
1 ten in terms of the basic two- and three-point functions
k q = (D t D ),
· 2 k+q − − k Eq. (2.25) become
1
k·p = −2(Dk−p−m2π −Dk), (x,ζ,t)= 1 I(x,0,1,m2)+(1 ζ)I(x,ζ,1,m2)
1 HI=0,1 2 π − π
k p = (D m2 D +t). (2.24)
· 2 k+p+q − π − k+q ∓I(x,−1+ζ,ζ,m(cid:2)2π)∓(1−ζ)I(x,−1+ζ,0,m2π)
t
Then the GPD’s become [(ζ 2x)m2 +t(x 1)]J[x,ζ,1,t,m2, ] (2.27)
− − π − π −2
HI=0,1(x,ζ,t)= −8iπN2cfω22 d4kδ(k·n−x)× (2.25) ±[(ζ−2x)m2π +t(x−ζ+1)]J[ζ−x,ζ,1,t,m2π,−2t] .
Z (cid:21)
1 1 ζ 1 1 ζ
+ − − This equation may be considered as the generic non-
D D D D ∓ D D ∓ D D
(cid:18) k k−p k+q k−p k+q k+p+q k k+p+q linear local chiral quark-model result for the isospin-
(ζ 2x)m2 +t(x 1) (ζ 2x)m2 +t(x ζ+1) projectedGPD’s of the pion. Model details, suchas reg-
+ − π − − π − ,
D D D ∓ D D D ularization,affectthespecificformofthetwo-andthree-
k k+q k−p k k+q k+p+q (cid:19) point functions, but leave the structure ofEq.(2.27) un-
with the upper (lower) signs corresponding to the case changed. The nontrivial features of the regularization
of I = 0 (I = 1). Note that the piece with 1/(D D ) will utterly be responsible for the fulfillment of the gen-
k k+q
cancels out due to the presence of contact diagram (c). eral properties of GPD’s described in Section IIA.
7
III. RESULTS OF THE SPECTRAL QUARK M =m =770MeV.ThecontourC fortheintegration
V ρ
MODEL in (3.2) is shown in Fig. 3. Despite the rather unusual
appearance of the spectral function, the model leads to
NowwecometotheevaluationofGPDinspecificmod- conventionalphenomenology[29,59]. Importantly,itim-
els. From now on we work for simplicity in the chiral plements the vector-mesondominance, yielding the pion
limit, electromagnetic form factor of the monopole form
m =0. (3.1)
π M2
FSQM(t)= V . (3.4)
The firstmodelweconsideris the SpectralQuarkModel V M2 t
V −
(SQM) ofRef.[29], where allthe necessarydetails ofthe
model can be found. The one-quark-loop action of this
For the gravitationalform factor we find
model is
Γ=−iNcZCdωρ(ω)Trlog(cid:0)i∂/−ωU5(cid:1), (3.2) θ1SQM(t)=θ2SQM(t)= MtV2 log(cid:18)MMV2V2−t(cid:19)≡FSSQM(t).
whereρ(ω)isthequarkgeneralizedspectralfunction,and (3.5)
U5isgiveninEq.(2.20). Inthecalculationsofthispaper
we only need the vector part of the spectral function,
Both the electromagnetic and gravitational form factors
which in the meson-dominance SQM [29] has the form
for SQM are plotted in Fig. 4 with solid lines.
1 1 1 With the help of Eq. (B1,B8) it is straightforward to
ρ (ω) = , (3.3)
V 2πiω(1 4ω2/M2)5/2 obtain the formulas for the GPD’s in SQM. The ex-
− V pressions are simple in the chiral limit, and shortest in
exhibiting the pole at the origin and cuts starting at the asymmetric notation. For the quark and antiquark
M /2, where M is the mass of the vector meson, GPD’s we obtain
V V
±
2(x 1) t(x 1)2+3(ζ 1)M2 θ(1 x)θ(x ζ)
2 (x,ζ,t)=θ((1 x)x)+θ((1 x)(x ζ))+t(1 x) − − − V − −
Hq − − − − " (cid:0) (t(x−1)2+(ζ−1)M(cid:1)V2)2
(x 1) t(x 1)2+3(ζ 1)M2 (x(ζ 2)+ζ) 3(ζ 1)ζ2M2 +t ζ2+8ζ 8 x2+2(4 5ζ)ζx+ζ2
+ − − − V + − − V − −
 (t(x(cid:0)−1)2+(ζ−1)MV2)2 (cid:1) (cid:0)ζ2+ 4txM(xV2−ζ) 3/2((cid:0)t(cid:0)(x−1)2+(ζ(cid:1)−1)MV2)2 (cid:1)(cid:1)
 θ(x)θ(ζ x)], (cid:16) (cid:17) 
× −
(x,ζ,t)= (ζ x,ζ,t). (3.6)
q¯ q
H H −
From these, the isospin combinations are trivial to ERBL and DGLAP regions:
get. The formulas satisfy the consistency relations
(2.8,2.9,2.10). In particular, upon passing to the sym-
metric notation and using the above formulas we verify
1
dXHI=1(X,ξ,t)=2FSQM(t),
V
Z−1 ζ 2 ζ M2 ζ M2 +t(1 ζ)
0 dXXHI=0(X,ξ,t)=(1 ξ2)FSQM(t).(3.7) dxHq(x,ζ,t)= −2 M2V t(2 ζ)(VM2 t−(1 ζ)),
Z−1 − S Z01 2 ζ MV 2− −2((cid:0)1 ζ)VM−2 −t(cid:1)
dx (x,ζ,t)= − V − V − .
Hq 2 M2 t(2 ζ)(M2 t(1 ζ))
Zζ V − − V(cid:0)− −(cid:1)
Theisovectornormisdecomposedasfollowsbetweenthe (3.8)
8
derivationthroughthedoubledistributionsshowninAp-
pendix A. Also note that the obtained formulas are cer-
F HtL
V tainly not of the form where the t-dependence is factor-
ized, i.e.
1.4
1.2 HI=0,1(X,ξ,t)=F(t)G(X,ξ). (3.11)
6
1
0.8 For the case oft=0 the formulas (3.6) simplify to the
well-know [19, 52] step-function results
0.6
0.4 (x,ζ,0)=θ[(1 x)(x ζ)] θ[ x(x+1 ζ)],
I=0
H − − − − −
0.2
(x,ζ,0)=θ[(1 x)(x+1 ζ)]. (3.12)
I=1
-t @GeV2D H − −
2 4 6 8 10
Another simple case is for ζ =0 and any value of t,
F HtL
S
1.4 (x,0,t)= MV2 MV2 +t(x−1)2 , (3.13)
1.2 Hq (M(cid:0)V2 −t(x−1)2)2 (cid:1)
1
which agrees with the result reported in [32]. The cor-
0.8
responding impact parameter representation obtained
0.6 there is given by the formula4
0.4
M2
0.2 q(x,b) = V (3.14)
2π(1 x)2 ×
-t @GeV2D −
2 4 6 8 10 bM bM bM
V V V
K K .
1 0
1 x 1 x − 1 x
FIG.4: (Coloronline)Thepionelectromagnetic,FV(t),(top) (cid:20) − (cid:18) − (cid:19) (cid:18) − (cid:19)(cid:21)
and gravitational, θ1(t) = θ2(t) ≡ FS(t), (bottom) form fac- From this expression one obtains
tors in SQM (solid line, Eq. (3.4,3.5)) and in NJL model
(dashed line, Eq.(4.3)). 1 M2K (bM )
dxq(x,b)= V 0 V . (3.15)
2π
Z0
Some special values of the GPD’s in SQM are:
This complies to the model-independent relation (2.15)
I=1(1,ζ,t)= I=0(1,ξ,t)=1, (3.9) when the vector-dominance form factor (3.4) is used,
H H since, explicitly,
M2 M2 +t(1 ζ)
I=1(ζ,ζ,t)= V V − ,
Hlim I=0(x,ζ,t)(M=(cid:0)V2M−V2t(M1−V2ζ+))t2(1(cid:1)−ζ) , Z d22πqMeV2iq+·bq2 =K0(bMV). (3.16)
x→ζ+H (M(cid:0)V2 −t(1−ζ))2 (cid:1) The case of Eq. (3.6) for ζ = 0.5 and several values
t 3M2 t(1 ζ) (1 ζ) of t is shown in Fig. 5. Results for other values of ζ are
lim I=0(x,ζ,t)= V − − − .
x→ζ−H (cid:0) (MV2 −t(1−ζ(cid:1)))2 qcoumalbitiantaitvieolnys,siHmIil=a0r,.1.FWiguerneo6teshthoawtstthheeIis=osp1iGnP0Danadnd1
The values at x = 1+ζ and x = 0 follow from the itsfirstderivativewithrespecttoxiscontinuousatx=0
−
symmetryrelations I=1,0(x,ζ,t)= I=1,0(ζ x,ζ,t). and x=ζ, while the I =0 combination is discontinuous
H ±H −
We note that the discontinuities at the end points x = at these points.
1 and for the I = 0 part at x = 0 and x = ζ are a At t = 0 we have the above-mentioned step-function
±
typical feature of quark-model calculations. The QCD results
evolution immediately washes out these discontinuities,
see Sect.VI. The derivativeof I=1(x,ζ,t) with respect HI=1 =θ(1 X2), (3.17)
H −
to x is continuous at the point x=ζ, where H =θ((1 X)(X ξ)) θ((1+X)( X ξ)).
I=0
− − − − −
d 2M2t 3M2 +t(1 ζ)
I=1(x,ζ,t) = V V − . As tincreases,thestrengthmovestothevicinityofthe
dxH |x=ζ − (M(cid:0)V2 −t(1−ζ))3 (cid:1) X =−±1points. Thelimitof−t(1−x)→∞inEqs.(3.6)
(3.10)
On general grounds, Eqs. (3.6) also satisfy the poly-
nomiality conditions (2.11), which can be seen from the 4 NoteanoverallsignmissinginRef.[32].
9
H Hx,Ζ,tL
q HI=1HX,Ξ,tL
2.5
1.5
2
1
1.5
0.5
1
x
0.2 0.4 0.6 0.8 1
0.5
-0.5
X
-1 -0.5 0.5 1
FIG. 5: (Color online) The SQM results for the quark GPD
of the pion, Hq of Eq. (2.5), plotted as a function of x for HI=0HX,Ξ,tL
ζ = 1/2 and t=0.2,0,−0.2,−1,−10,−100 GeV2, from top
to bottom (at x=0.9). Asymmetricnotation.
1.5
1
0.5
yield the asymptotic forms
X
-1 -0.5 0.5 1
I=1,0(1,ζ,t)=1, (3.18) -0.5
H
M2(1 ζ) -1
I=1(x,ζ,t) V − , x [0,1),
H ≃ t(1 x)2 ∈
-1.5
−
M2(1 ζ)
I=0(x,ζ,t) V − , x (ζ,1),
H ≃ t(1 x)2 ∈
−
lim I=0(x,ζ,t)= 1+ MV2 , FIG. 6: (Color online) Same as Fig. 5 for HI=1 (top) and
x ζ−H − (1 ζ)t HI=0 (bottom) in the symmetric notation, ξ=1/3.
→ −
M2(2x ζ) ζ2 3ζ+2
I=0(x,ζ,t) V − − , x (0,ζ).
H ≃ 2(x 1)2(x ζ+1)2t ∈
− (cid:0)− (cid:1) IV. RESULTS OF THE
NAMBU–JONA-LASINIO MODEL
In the DGLAP region the absolute value of the I = 0,1
functions are bounded by unity. Note that at large t
− We use the non-linear NJL model with Pauli-Villars
the GPD’s continue to be equal to 1 at x = 1, however
(PV) regularization in the twice-subtracted version of
very quickly drop to zero in the DGLAP region. In the
Ref.[27]. Theprescriptionforregularizinganobservable
ERBL region the I =1 part drops, while the I =0 part
in this model is
tends to 1 as x ζ−, and drops to 0 elsewhere. O
− →
Sinceinourquarkmodelthepartondistributionfunc- d (Λ2)
tionis unity, q(x)=1,the positivity bound (2.12) states Oreg =O(0)−O(Λ2)+Λ2 OdΛ2 , (4.1)
that (for t 0)
≤
where Λ are the PV regulator. Note that Eq. (4.1) is
HSQM(X,ξ,t) 1, ξ X 1. (3.19) differentfromthe prescriptionused in[52], wherea vari-
| q |≤ ≤ ≤ ant of the PV regularizationwith two distinct cut-offs is
applied. We also use the non-linear rather than linear
It is a priori not obvious that the bound should hold in realization of the chiral field. In what follows we take
chiral quark models where finiteness of observables re- M = 280 MeV for the quark mass and Λ = 871 MeV,
sults from regularization involving subtractions. Never- whichyieldsf =93.3MeV[27]accordingthetheformula
theless, we have checked with Eqs. (3.6) that condition
(3.19) is actually satisfied in the DGLAP region for all
3M2
values of ξ and all negative t. This is also manifest in f2 = log(Λ2+M2) . (4.2)
− 4π2 reg
Fig. 5, as well as in Eq. (3.18). The bound is saturated
at the end points X = 1. Thus the positivity bound is (cid:0) (cid:1)
±
satisfied in SQM. ThepionelectromagneticformfactorintheNJLmodel
10
hence we do not give them here. More details may be
HI=1HX,Ξ,tL foundinRef.[52]. Forthespecialcaseoft=0Eq.(3.12)
2.5 holds. As in SQM, the conditions (2.9,2.10,2.8) are sat-
isfied in the considered NJL model with the PV regular-
2 ization.
ThenumericalresultsfortheNJLmodelaredisplayed
1.5 inFig.7. WhencomparingFigs.6and7wenoteastrik-
ing similarity between the two considered quark models.
1 The slight differences stem mainly from different form
factors in the two considered models, cf. Fig. 4. Our re-
0.5 sultsarealsoqualitativelysimilartothecaseofthechiral
limitinFig.6ofRef.[52]. AspointedoutinAppendixC,
x polynomiality followsfrom the derivationproceeding via
-1 -0.5 0.5 1
the doubledistributions. Wehavecheckedthatthesimi-
larlytoSQM,thepositivitybound(2.12)isalsosatisfied
HI=0HX,Ξ,tL in the NJL model at any ξ and all negative values of t.
1.5
V. QCD EVOLUTION OF QUARK MODELS
1
0.5 A. The need for evolution
x
-1 -0.5 0.5 1
Akeyquestion,notonlyforourmodelbutforanynon-
-0.5
perturbativecalculation,iswhatisthescaleatwhichour
-1 model result for the GPD’s holds. Ultimately, this boils
down to the issue on how the model predictions for the
-1.5
GPD’s might be confronted to experimental or lattice
data. In QCD, the GPD’s are scale dependent, while in
models they correspond to functions defined at a given
FIG. 7: (Color online) Same as Fig. 6 for for theNJL model scale. This is so because low energy models hold at a
with thePV regularization, M =280 MeV, Λ=871 MeV. scale above which scaling should set in. Perturbative
QCD and the corresponding scaling violations bring in
the issue of evolution equations for GPD’s which will be
is treatedindetailinSec.VI. Inthissectionwediscussand
N M2 updatetheprocedurealreadyusedinpreviousworks[23,
FNJL(t)=1+ c (4.3) 27,30,31]fortheevolutionofPDFandPDAandextract
V 8π2f2 ×
its consequences as compared to available experimental
√4(M2+Λ2) t √ t data or lattice results.
2 4(M2+Λ2) tlog − − −
− √4(M2+Λ2) t+√ t From the point of view of perturbative QCD where
 (cid:18) − − (cid:19) ,
p √ t both quarks and gluons contribute as explicit degrees of
−
  freedom,theroleofthelow-energychiralquarkmodelsis
 reg
  toprovideinitialconditionsfortheQCDevolutionequa-
The property lim F (t) = 0 follows from tions order by order in the twist expansion. Clearly, chi-
t NJL
Eq. (4.2). The isov→e−ct∞or form factors arising in both ralquarkmodelscontainnon-perturbativeQCDfeatures,
considered models are compared in Fig. 4. The formula particularly the spontaneous chiral symmetry breaking.
for the gravitational form factor in the NJL is lengthy, On the other hand, chiral quark models do not contain
henceweonlygivethenumericalresultsinFig.4. Inthis theQCDdegreesoffreedom,i.e., thecurrentquarksand
model also the two gravitational form factors are equal explicit gluons. So one expects typical high-energy per-
to eachother, θNJL(t)=θNJL(t). We note that although turbativeQCDfeatures,suchasradiativecorrections,to
1 2
the form factors in both models are qualitatively simi- be absent in the model. This is precisely the pattern of
lar, they are quantitatively somewhat different, which is logarithmic scaling violations which the models lack but
partlyduetothechoiceofparametersintheNJLmodel, whichhavetraditionallybeencomputedintheperturba-
as well as follows from different analytic structure of the tion theory in QCD.
corresponding formulas, in particular at large values of The procedure applied in this paper takes the quark-
t. model distributions at some low quark-model scale Q
0
−
TheapplicationoftheformulasderivedinAppendixC and evolves them to higher scales, where (for some ob-
leads to expressionssimilar to those of [52]. The analogs servables) the experimental or lattice data are available.
ofEq.(3.9,3.18)intheNJLmodelaremorecomplicated, Inthe followingweusethe leading-orderERBL-DGLAP