Pion Distribution Amplitude and Quasi-Distributions

0.73 MB·

by  Anatoly V. Radyushkin

Pion Distribution Amplitude and Quasi-Distributions
A. V. Radyushkin1,2
1Physics Department, Old Dominion University, Norfolk, VA 23529, USA
2Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA
We extend our analysis of quasi-distributions onto the pion distribution amplitude. Using the
formalism of parton virtuality distribution amplitudes (VDAs), we establish a connection between
the pion transverse momentum dependent distribution amplitude (TMDA) Ψ(x,k2) and the pion
⊥
quasi-distributionamplitude(QDA)Q (y,p ). WebuildmodelsfortheQDAsfromtheVDA-based
π 3
models for soft TMDAs, and analyze the p dependence of the resulting QDAs. As there are many
3
modelsclaimedtodescribetheprimordialshapeofthepionDA,wepresentthep -evolutionpatterns
3
formodelsproducingsomepopularproposals: Chernyak-Zhitnitsky,flatandasymptoticDAs. Our
results may be used as a guide for future studies of the pion distribution amplitude on the lattice
using the quasi-distribution approach.
7
1 PACSnumbers: 11.10.-z,12.38.-t,13.60.Fz
0
2
I. INTRODUCTION to the investigation of the 3-dimensional structure of
n
a hadrons.
J Our goal in the present paper is to perform a sim-
The parton distribution functions (PDFs) f(x), and
8 ilar analysis of the pion quasi-distribution amplitude
two-bodydistributionamplitudes(DAs)ϕ(x)arerelated
1 (QDA)Q (y,p )thatproducesthepionDAϕ (y)inthe
to matrix elements of bilocal operators on the light cone π 3 π

] z2 = 0, which prevents a straightforward calculation of large-p3 limit. The basic ingredients of our analysis are
h these functions in a lattice gauge theory formulated in virtualdistributionamplitudesandtransversemomentum
p dependent amplitudes introduced in Refs. [15, 16].
the Euclidean space. The usual way out is to calculate
-
p their moments. In particular, high precision lattice cal- The paper is organized as follows. We start in Section
e culations of the second moment of the pion distribution 2 with an introductory overview of the basic concepts
h amplitude ϕ (x) were reported in Ref. [1]. However, re- involved. First, we remind a covariant definition of the
π
[ cently,X.Ji[2]suggestedamethodallowingtocalculate twist-2 pion distribution amplitude. After that, we dis-
2 PDFsandDAsasfunctionsofx. Tothisend,heproposes cussitsdefinitionwithinthelight-frontformalism. Then
v to use purely space-like separations z =(0,0,0,z ). we outline the basics of the VDA/TMDA approach. In
3
8 Section 3, we discuss the quasi-distribution amplitudes.
The matrix elements of equal-time bilocal operators
8 In particular, we show that QDAs are completely deter-
producedistributionsQ(y,p )inthemomentump com-
6 3 3 mined by TMDAs through a rather simple transforma-
2 ponent (quasi-distributions). The crucial point is that
tion. Since the basic relations between the parton distri-
0 they tend to the light-cone distributions f(y), ϕ(y) in
butions are rather insensitive to complications brought
. the p →∞ limit. In case of PDFs, the results of lat-
1 3 by spin, in Section 3 we refer to a simple scalar model.
ticecalculationsofthepartonquasi-distributions(PQDs)
0
In Section 4, we discuss modifications related to quark
7 were reported in Refs. [3–8]. It is expected [9] that
spin and gauge nature of gluons in quantum chromody-
1 PQDs Q(y,p ) should have a mild perturbative evolu-
3
namics (QCD). In Section 5 we discuss the VDA-based
: tion [10–13] with respect to p for large p . However,
v 3 3 models for soft TMDAs, and present our results for non-
i the values of p3 used in the cited lattice calculations are
X perturbativeevolutionofQDAsobtainedinthesemodels.
notverylarge,andtheobservedstrongvariationofPQDs
The large-p limit of perturbative evolution is discussed
r with p does not have a perturbative form. 3
a 3 in Section 6. Our conclusions are given in Section 7.
In our recent paper [14] we have studied nonperturba-
tive evolution of PQDs using the formalism of virtuality
distribution functions [15, 16]. We found that PQDs can
be obtained from the transverse momentum dependent II. PION DISTRIBUTION AMPLITUDE
distributions (TMDs) F(x,k2). Then we built models
⊥
for the nonperturbative evolution of PQDs using simple
A. Covariant Definition
models for TMDs. Our results are in qualitative agree-
ment with the p -evolution patterns obtained in lattice
3
calculations [3–8] and also in diquark spectator models The pion distribution amplitude (DA) ϕπ(x,µ2) was
[17–19]. originally introduced [20] as a function ϕπ(x,µ2) whose
xn moments
As emphasized in Ref. [14], because of the relation
between PQDs and TMDs, the nonperturbative evolu-
(cid:90) 1
tion of PQDs reflects the k⊥-dependence of the TMDs fn(µ2)= xnϕπ(x,µ2)dx (1)
F(x,k⊥2), and thus its study provides a new approach 0
2
aregivenbyreducedmatrixelementsoftwist-2localop- subtraction. As a result, φ (x,µ2) has a nonpertur-
π
erators bative evolution with µ2 even if the perturbative evo-
lution is absent. Take a simple example ψ(x,k ) ∼
in+1Rµ2(cid:10)0|d¯(0)γ5{γνDν1...Dνn}u(0)|π+,P(cid:11) φ(x)e−k⊥2/Λ2. Then the zeroth x-moment of φπ(⊥x,µ2)
={P P ...P } f (µ2) . (2) has the ∼[1−e−µ2/Λ2]-dependence, i.e. it is not con-
ν ν1 νn n
stant, reaching f in the µ2 →∞ limit only.
π
Asusual{...}denotesthetwist-2projectionofaLorentz
Of course, if one has in mind only the applications in
structure,i.e.,symmetrizationofindicesandsubtraction whichnonperturbativepartoftheµ2-dependencemaybe
of traces. Since matrix elements of local operators with ignored, then φ (x,µ2) of the LF definition is very sim-
π
n>0 diverge, one needs to supply them by a renormal- ilar to the covariantly defined ϕ (x,µ2), and the differ-
ization procedure denoted above by R , with µ2 being π
µ2 encebetweenthemmaybetreatedastheuseofdifferent
the renormalization scale. In QCD, the standard choice
renormalization schemes.
ofR isbasedonthedimensionalregularizationandthe
µ2 As a matter of fact, in actual LF calculations one en-
modified minimal subtraction scheme MS. As a result of
counters LFWFs integrated to some process-dependent
such a renormalization, the zeroth moment f (µ2) does
0 scale µ, i.e. the choice of the renormalization prescrip-
not have µ2-dependence since the anomalous dimension
tion and the scale µ is dictated by diagrams. Moreover,
of the axial current is zero. Hence, f (µ2) for all µ2 is
0 if the relevant µ2’s are not extremely large, the simple
equal to the pion decay constant f
π example above shows that one may need to take into ac-
(cid:90) 1 countthenonperturbativeµ2-dependenceofφπ(x,µ2)re-
f (µ2)= ϕ (x,µ2)dx=f (3) flectingthetransversemomentumbehavioroftheLFWF
0 π π
0 ψ(x,k⊥), i.e., the 3-dimensional structure of the pion,
which may be essential for some processes.
known experimentally, f ≈130MeV.
π
This definition of DA is oriented on the use of the op- In particular, the photon-pion transition form factor
eratorproductexpansionandadescriptionofthepionin involves φπ(x,µ2 =x2Q2)/[xQ2], i.e. LFWF ψ(x,k⊥)
termsofthetwist-2DAϕ (x,µ2)thatgivesthecollinear integrated over k⊥ till xQ [32, 33]. As a result, the re-
π
distribution of the pion momentum p among its two va- mainingx-integralintheLFformulahasafiniteQ2 →0
lenceconstituents. Thedependenceofϕ (x,µ2)onµ2 is limit: the infrared small-x divergence is eliminated by a
π
governed by perturbative evolution [21–23] and does not cut-offprovidedbyφ(x,µ2 =x2Q2). Ontheotherhand,
reflect the primordial (nonperturbative) pion’s structure a formula involving MS-based DA ϕ(x,µ2) with a fixed
in the direction transverse to p. scale µ2 is singular in the Q2 →0 limit. One may ques-
As is well-known, for very large µ2, the pion DA tends tion the applicability of the LF formula down to Q2 =0,
to the “asymptotic DA” ϕas(x)=6f x(1−x) [24]. In but at least it does not give an infinite result for a quan-
π π
general, ϕ (x,µ2) may differ from its asymptotic form. tity that is known to be finite. For this reason, the LF
π
Over the years, several forms were proposed for the formula looks as a more attractive tool for modeling the
pion DA “at low normalization point”, e.g., Chernyak- form factor behavior at moderate Q2 than the perturba-
Zhitnitsky DA ϕCZ(x)=30f x(1−x)(1−2x)2 [25], tive QCD 1/Q2 twist expansion.
π π
“flat DA” ϕflat(x) = f [26–30], “root DA” ϕroot(x) = Still, aproblemwiththeLFformalismisthatLFWFs
π π π
(cid:112)
8f x(1−x)/π [31], etc. are not directly connected with the usual objects of the
π
covariantfieldtheory,suchasmatrixelementsoflocalor
nonlocal operators.
B. Light-Front Formalism Definition Inourpapers[15,16],wehavedevelopedtheformalism
of virtuality distribution amplitudes (VDAs) that is fully
Adifferentdefinition[23]isusedinthelight-front(LF) basedonthecovariantfieldtheoryconcepts. IntheVDA
quantizationframework,wherethepiondistributionam- approach,thepionisdescribedbythetransversemomen-
plitude φ (x,µ2) is understood as the k -integral tumdependentdistributionamplitude(TMDA)whichhas
π ⊥
a direct connection with the objects of the covariant
√
6 (cid:90) QFT.Ontheotherhand,justliketheLFwavefunctions,
φ (x,µ2)= ψ(x,k )d2k (4)
π (2π)3 ⊥ ⊥ the TMDAs give a 3-dimensional description of the pion
k2≤µ2
⊥ structure.
ofthelight-frontwavefunction(LFWF)ψ(x,k ). Wein-
⊥
tentionally use here a different notation φ (x,µ2) to em-
π
phasizethefactthatψ(x,k )isanobjectoftheHamilto-
⊥
C. Pion TMDA
nian light-front framework, while the pion DA ϕ (x,µ2)
π
in Eq. (1) is defined within the covariant Lagrangian
formulation of the quantum field theory (QFT). To omit inessential complications related to spin, we
Another difference is the use of a straightforward cut- illustrate the ideas underlying TMDAs using a simple
off k2 ≤ µ2 rather than a more sophisticated MS-like example of a scalar theory. The key element of our ap-
⊥
3
proach [15] is the VDA representation The TMDA may be written in terms of the VDA as
(cid:90) ∞ (cid:90) 1 i (cid:90) ∞ dσ
(cid:104)0|ψ(0)ψ(z)|p(cid:105)= dσ dx Ψ(x,k⊥2)=π σ Φ(x,σ) e−i(k⊥2−i(cid:15))/σ . (9)
0 0 0
×Φ(x,σ) e−ix(pz)−iσ(z2−i(cid:15))/4 (5)
The integrated TMDA
that basically reflects the fact that the matrix element (cid:90) µ2
(cid:104)0|ψ(0)ψ(z)|p(cid:105) depends on z through (pz) and z2. It f(x,µ2)≡π dk2Ψ(x,k2) (10)
⊥ ⊥
may be treated as a double Fourier representation with 0
respect to these variables.
isanalogoustotheµ2-dependentpiondistributionampli-
The main non-trivial feature of this representation is
tude φ(x,µ2) of the LF formalism (but, of course, being
in its specific limits of integration over x and σ. They
anobjectofthecovariantQFT,f(x,µ2)doesnotcoincide
hold for any contributing Feynman diagram [16], so we
with it). In terms of the VDA,
assume that this property is true in general. Note that
starting with the first loop, the diagram contributions
(cid:90) ∞ (cid:104) (cid:105)
are non-analytic in z2 due to lnz2 factors, but the VDA f(x,µ2)= dσ 1−e−i(µ2−i(cid:15))/σ Φ(x,σ) . (11)
representation,unliketheTaylorexpansioninz2,isvalid 0
nevertheless.
While the VDA representation is a fully covariant ex- Since it is defined by a straightforward cut-off, f(x,µ2)
pression, it is convenient to use a frame in which the evolves with µ2 even if the limit µ2 →∞ is finite, e.g. in
pion momentum p is purely longitudinal p=(E,0 ,P). a super-renormalizable theory. The evolution equation
⊥
Choosingsomespecialcasesofz,onecangetrepresenta-
d
tionsforseveralpartonfunctions,allintermsofoneand µ2 f(x,µ2)=πµ2Ψ(x,µ2) (12)
the same universal VDA Φ(x,σ). In particular, choosing dµ2
z on the light front z =0 and with z =0 (i.e., taking
+ ⊥
follows from the definition (10). When the TMDA
z =z ) gives the twist-2 distribution amplitude ϕ(x)
−
Ψ(x,k2) vanishes faster than 1/k2 (such a TMDA will
⊥ ⊥
(cid:90) 1 be called “soft”), evolution essentially stops at large µ2.
(cid:104)0|ψ(0)ψ(z−)|p(cid:105)= dxϕ(x)e−ixp+z− . (6) In a renormalizable theory, it makes sense to treat
0 Φ(x,σ) as a sum of a soft part Φsoft(x,σ), generating a
nonperturbative evolution of f(x,µ2), and a ∼ 1/σ hard
ComparingthisrelationwiththeVDArepresentationwe
tail. Toavoidnonperturbativeevolution,onemaychoose
have
an MS-type construction, e.g. regularize the σ-integral
(cid:90) ∞ in Eq. (7) by a σ−(cid:15) factor and then subtract 1/(cid:15) poles.
ϕ(x)= Φ(x,σ)dσ , (7)
However,justlikeintheLFformalism,theobjectsthat
0
appear in actual calculations are exactly the integrated
provided that the z2 → 0 limit is finite, e.g. in the TMDAs rather than their MS-type sisters. In particu-
super-renormalizable ϕ3 theory. In the renormalizable lar,thephoton-piontransitionformfactorisgiveninthe
ϕ4 theory, the function Φ(x,σ) has a ∼ 1/σ hard part, VDA approach by the x-integral of f(x,µ2)/[xQ2] taken
and the integral (7) is logarithmically divergent, reflect- at µ2 = xQ2 [15], i.e., it involves TMDA Ψ(x,k2) inte-
⊥
ing the perturbative evolution of the DA in such a the- grated over k2 till xQ2. As a result, the TMDA formula
⊥
ory. Inthiscase,onemayarrangearegularizationofthe has a finite Q2 → 0 limit. Furthermore, using simple
σ-integral characterized by some parameter µ2. Then models for soft TMDAs one can get a very close descrip-
ϕ(x)→ϕ(x,µ2). tion of experimental data by the nonperturbative evolu-
Light-cone singularities are avoided if we choose a tion of the integrated TMDA [16].
spacelike z, e.g., take z that has z and z components For very large µ2, the perturbative evolution domi-
− ⊥
only. Then we can introduce the transverse momentum nates and eventually brings f(x,µ2) to its asymptotic
dependent distribution amplitude Ψ(x,k2) as a Fourier form6f x(1−x). Thequestion,however,iswhatkindof
⊥ π
transform shapef(x,µ2)hasatlowscalesµ∼1 GeV,andalsohow
this shape changes with µ2. As we have discussed, this
(cid:90) 1
(cid:104)0|ψ(0)ψ(z ,z )|p(cid:105)= dx e−ixp+z− nonperturbativeµ2-evolutionreflects thek⊥ dependence
− ⊥
of the soft part of the pion TMDA.
0
(cid:90) Below, we shall see that there is another function,
× d2k⊥Ψ(x,k⊥2)ei(k⊥z⊥) (8) the pion quasi-distribution amplitude Q (y,P) whose
π
P-dependence is also determined by the k -dependence
⊥
ofthematrixelementwithrespecttoz andz . Because ofthepionTMDA.Thequasi-distributionshavebeenin-
− ⊥
of the rotational invariance in z plane, TMDA depends troduced recently by X. Ji [2] to facilitate a calculation
⊥
on k2 only, the fact already reflected in the notation. of light-front functions (PDFs, DAs, etc.) on the lattice.
⊥
4
III. QUASI-DISTRIBUTION AMPLITUDE C. Relation to TMDA
A. Definition Comparing the VDA representation (16) for Q (y,P)
π
withthatfortheTMDAΨ(x,k2)(9)(notethattheyare
⊥
ThebasicproposalofRef. [2]istoconsiderequal-time valid both for soft and hard parts) we conclude that
bilocaloperatorscorrespondingtoz =(0,0,0,z3)[or,for (cid:90) ∞ (cid:90) 1
brevity, z =z ]. Incorporating the VDA representation, Q (y,P)= dk dxP Ψ(x,k2+(x−y)2P2) .
3 π 1 1
we have −∞ 0
(20)
(cid:90) ∞ (cid:90) 1
(cid:104)0|ψ(0)ψ(z3)|p(cid:105)= dσ dxΦ(x,σ)eixp3z3+iσz32/4. Thus, the quasi-distribution amplitude Q (y,P) is com-
π
0 −1 pletely determined by the form of the TMDA Ψ(x,k2).
(13) ⊥
This formula may be also obtained if one takes
Using again the frame in which p = (E,0⊥,P), and in- z =(0,z1,0,z3) in the VDA representation and intro-
troducing the pion quasi-distribution amplitude through duces the momentum k conjugate to z . Then
1 1
(cid:90) ∞ (cid:90) ∞
(cid:104)0|ψ(0)ψ(z3)|p(cid:105)= dyQπ(y,P)e−iyPz3, (14) dyeiyPz3(cid:104)0|ψ(0)ψ(z1,z3)|p(cid:105)
−∞ −∞
we get a relation between QDA and VDA, (cid:90) ∞ (cid:90) 1
Q (y,P)=(cid:114)iP2 (cid:90) ∞ √dσ (cid:90) 1dxΦ(x,σ)e−i(x−y)2P2/σ . = −∞dk1e−ik1z1 0 dxΨ(x,k12+(x−y)2P2) . (21)
π
π σ Taking z = 0 gives Eq. (20). Furthermore, introducing
0 0 1
(15) the variable k ≡(x−y)P, we have
3
It is easy to see that, for large P, we have (cid:90) ∞ (cid:90) (1−y)P
(cid:114)iP2 σ Qπ(y,P)= dk1 dk3Ψ(y+k3/P,k12+k32) .
e−i(x−y)2P2/σ =δ(x−y)+ δ(cid:48)(cid:48)(x−y)+... −∞ −yP
πσ 4P2 (22)
(16)
Thus, Q (y,P) is given by an integral over a stripe
π
and Q (y,P → ∞) tends to the integral (7) leading of width P in the 2-dimensional (k ,k ) plane. When
π 1 3
to ϕ (y). This observation suggests that one may be P →∞forafixednonzeroy,thestripecoversthewhole
π
able to extract the “light-cone” distribution amplitude (k ,k ) plane. Moreover, for a soft TMDA Ψ(x,k2) that
1 3
ϕ (y) from the studies of the purely “space-like” func- rapidly decreases outside a region k2 (cid:46)Λ2, only the val-
π
tion Q (y,P) for large P [2]. ues of k (cid:46)Λ are essential, and for large P one may ap-
π 3
proximate the first argument of the TMDA by y. Hence,
the P →∞ limit gives ϕsoft(y).
π
B. Evolution For comparison, the integrated TMDA f(y,µ2) is ob-
tained by integrating Ψ(y,k2) over a circle of radius µ
⊥
Again, to study the P-evolution of Q (y,P) it makes in the k plane. Again, the circle covers the whole plane
π ⊥
sense to split Φ(x,σ) into the soft part, for which the when µ→∞, and fsoft(y,µ2)→ϕsoft(y).
π
integral over σ is finite, and the hard tail that generates Thus,whilethepatternsofthenonperturbativeevolu-
perturbative evolution. tion of Qsoft(y,P) and fsoft(y,µ2) are different, they be-
π
The nonperturbative evolution of Qsoft(y,P) with re- come more and more close for large P and µ, eventually
spect to P has the area-preserving property. Namely, producing the same function ϕsoft(y).
π
since
(cid:90) ∞ (cid:114)πσ
dye−i(x−y)2P2/σ = (17) IV. QCD
iP2
−∞
we have A. Spinor quarks
(cid:90) ∞ (cid:90) 1
dyQsoft(y,P)= dxϕsoft(x)=f . (18)
π π π In spinor case, one deals with the matrix element
−∞ 0
In other words, Qsoft(y,P) for any P has the same area Bα(z,p)≡(cid:104)0|ψ¯(0)γ5γαψ(z)|p(cid:105) . (23)
π
normalization as ϕsπoft(x). In this respect, the pion QDA It may be decomposed into pα and zα parts: Bα(z,p)=
pleasantly differs from the integrated TMDA fsoft(x,µ2) pαB (z,p)+zαB (z,p), or in the VDA representation
whose zeroth moment is µ2-dependent. p z
(cid:90) ∞ (cid:90) 1
Similarly, we have the momentum sum rule
Bα(z,p)= dσ dx
(cid:90) ∞ (cid:90) 1 0 −1
dyyQsπoft(y,P)= dxxϕsπoft(x) . (19) ×(cid:2)pαΦ(x,σ)+zαZ(x,σ)(cid:3) e−ix(pz)−iσ(z2−i(cid:15))/4 . (24)
−∞ 0
5
If we take z = (z ,z ) in the α = + component of V. MODELS FOR SOFT PART
− ⊥
Oα, the purely higher-twist zα-part drops out and we
can introduce the TMDA Ψ(x,k⊥2) that is related to the A. Models
VDA Φ(x,σ) by the scalar formula (9).
In the QDA case, the easiest way to avoid the effects
To get an idea about patterns of the nonperturbative
of the zα admixture is to take the time component of
evolution of the QDAs, we need some explicit models of
Bα(z =z3,p) and define thek dependenceofsoftTMDAsΨ(x,k2). Wewilluse
⊥ ⊥
here the same models as in our papers [14, 16]. While
(cid:90) 1
B0(z ,p)=p0 dxQ (y,P) eiyPz3 . (25) TMDAsarefunctionsoftwoindependentvariablesxand
3 π
−1 k⊥2, we take, for simplicity, the case of factorized models
The connection between Q (y,P) and Φ(x,σ) is given Ψ(x,k2)=ϕ (x)ψ(k2) , (30)
π ⊥ π ⊥
thenbythesameformula(15)asinthescalarcase. Asa
result,wehavethesumrules(18)and(19)corresponding inwhichx-dependenceandk⊥-dependenceappearinsep-
to charge and momentum conservation. Furthermore, arate factors.
the quasi-distribution amplitude Qπ(y,P) is related to If we assume a Gaussian dependence on k⊥,
TMDA Ψ(x,k2) by the scalar conversion formula (20).
⊥ ϕ (x)
ΨG(x,k⊥2)= ππΛ2 e−k⊥2/Λ2 , (31)
B. Gauge fields the conversion formula (20) results in
In QCD, for π+ one should take the operator QG(y,P)= P√ (cid:90) 1dxϕ (x)e−(x−y)2P2/Λ2 . (32)
π Λ π π
0
Oα(0,z;A)≡d¯(0)γ γαEˆ(0,z;A)u(z) (26)
5
In the space of impact parameters z , the Gaussian
⊥
involving a straight-line path-ordered exponential model gives a e−z⊥2Λ2/4 fall-off that is too fast for large
z . Asanalternativeextremecase,wetakeamodelwith
⊥
(cid:20) (cid:90) 1 (cid:21) the 1/(1+z2Λ2/4) dependence on z , whose fall-off at
Eˆ(0,z;A)≡P exp igz dtAν(tz) (27) ⊥ ⊥
ν largez istoo slow. Itcorrespondstothe“slow”model
⊥
0
for the TMDA
in the quark (adjoint) representation. As is well-
K (2|k |/Λ)
known, its Taylor expansion has the same structure as ΨS(x,k2)=2ϕ (x) 0 ⊥ (33)
that for the original ψ¯(0)γ γαψ(z) operator, with the ⊥ π πΛ2
5
only change that one should use covariant derivatives that has a logarithmic singularity for small k reflecting
⊥
Dν =∂ν −igAν instead of the ordinary ∂ν ones. a too slow fall-off for large z . For the QDA, we have
⊥
Again, the zα admixture is avoided if the pion quasi-
distribution amplitude is defined through the time com- P (cid:90) 1
QS(y,P)= dxϕ (x)e−2|x−y|P/Λ . (34)
ponent of Oα. Then we have the same relation between π Λ π
0
the VDA and QDA as in the scalar case. Due to Eq.
(18), this results in the area preserving property for the Note that the Gaussian model and the “slow” model
soft part have the same ∼(1−z⊥2Λ2/4) behavior for small z⊥, i.e.
theycorrespondtothesamevalueofthe(cid:104)0|ϕ(0)∂2ϕ(0)|p(cid:105)
(cid:90) ∞
matrix element (in the scalar case), provided that one
dyQsoft(y,P)=f . (28)
π takes the same value of Λ in both models. For large z ,
−∞ ⊥
however, the Gaussian model has a fall-off that is too
Also, due to Eq. (19) we have the momentum sum rule fast, while the fall-off of the “slow” model is too slow.
Thus, they look like two extreme cases, and provide a
(cid:90) ∞
good illustration of the nonperturbative evolution of the
dy(y−1/2)Qsoft(y,P)=0 . (29)
q pion QDA, with expectation that other models would
−∞
produce results somewhere in between these two cases.
Since the VDA Φ(x,σ) is defined through the ma-
trix element of a gauge-invariant operator, it is gauge-
invariant also. For this reason, TMDA Ψ(x,k2) is a B. Numerical Results
⊥
gauge-invariantobjectaswell. Itshouldnotbeconfused
withthek -dependent(andgauge-dependent)“underin- To compare evolution patterns induced by the Gaus-
T
tegrated distributions” that appear in perturbative loop sian and “slow” models, we take the Ansatz (30) with
calculations based on Sudakov decomposition of the in- ϕ (x)havingadrasticshapeoftheChernyak-Zhitnitsky
π
tegration momentum k. DA ϕCZ(x)=30f x(1−x)(1−2x)2. As one can see
π π
6
QC⇡Z(y,P)/f⇡ QCZ(y,P)/f
10 ⇡ ⇡ 10
1.5
1.5
5
5
1.0
1.0 3
3
0.5 0.5 1
1
y y
-1.0 -0.5 0.5 1.0 1.5 2.0 -1.0 -0.5 0.5 1.0 1.5 2.0
FIG. 1. Quasi-distribution amplitude QCZ(y,P) for P/Λ=1,3,5,10 in the Gaussian (left) and “slow” models (right).
π
from Fig. 1, for P/Λ = 1 the Gaussian model shows no flat limiting DA we have
indication of humps visible for higher P/Λ ratios. In the
“slow”model,smallhumpsarepresentevenforP/Λ=1. 1 P (cid:90) 1
QS,flat(y,P)= dxe−2|x−y|P/Λ . (35)
ForhighratiosP/Λ=5and10,thetwomodelsgiveclose f π Λ
π 0
results, with strong humps.
This integral can be calculated analytically. Writing
Assuming Λ ∼ 0.6 GeV suggested by the VDA-based
y =(1+η)/2 in terms of a symmetric variable η, we get
fitsofthephoton-piontransitionformfactorinRef. [16],
we expect that P ∼3 GeV would be required to support 1 (cid:16) (cid:17)
(or rule out) the CZ-type shape of the pion DA. f QπS,flat(y,P)= 1−e−P/Λcosh(Pη/Λ) θ(|η|≤1)
π
It is also interesting to note that the nonperturbative +sinh(P/Λ)e−P|η|/Λθ(|η|≥1) . (36)
evolution pattern here is exactly opposite to the pertur-
bative one. In the latter case, the humps of the initially
Similar, but more lengthy expressions may be obtained
CZ-shaped DA become less pronounced as the normal-
for two other models. As one can see from Fig. 2, for
izationscaleincreasesandeventuallydisappear,withthe
smallP =Λwehaveveryclosecurves. ForlargerP =3Λ
DA tending to the asymptotic ∼x(1−x) shape.
the difference becomes visible, and for large P =5Λ and
To compare patterns of the QDA’s nonperturbative P = 10Λ the curves shown in Fig. 3 are distinctly dif-
evolutionfordifferentshapesofthelimitingDA,wetake ferent. In fact, the P = 10Λ curves are very close to
three models for ϕ (x): Chernyak-Zhitnitsky ϕCZ(x), their limiting forms. Again, the nonperturbative evolu-
π π
flat ϕflat(x)=f and asymptotic ϕas(x)=6f x(1−x). tionpatternincaseoftheflatDAisoppositetotheper-
π π π π
The results in the Gaussian and the “slow” models are turbative one: as P increases, Qflat(y,P) broadens from
π
rather similar. To avoid plotting too many graphs, we a rather narrow function for P =Λ and becomes almost
take, for definiteness, the “slow” model. Then, for the constant for P =10Λ.
VI. LEADING-ORDER HARD TAIL has a ∼ 1/k2 behavior, but its explicit functional form
⊥
is much more complicated. In particular, it is finite in
the k →0 limit [16]. The infrared cut-off for the naive
⊥
The nonperturbative evolution of Q (y,P) essentially
π 1/k2 extrapolation is provided by the finite size of the
stops for P/Λ(cid:38)20, and for larger values of P the domi- ⊥
pion encoded in the parameters, like Λ, present in the
nantroleisplayedbytheperturbativeevolutioninduced
soft TMDA. Postponing the analysis of the interplay be-
bythehardpart. Inourpapers[15,16],itwassuggested
tweenthenonperturbativeandperturbativeevolutionfor
to take a purely soft TMDA (or VDA) as a starting ap-
future studies, we just outline below the VDA treatment
proximation, and then “generate” hard tail by adding
of the hard tail.
one-gluon exchanges. The only new parameter is the
For large σ, the lowest-order (in α ) hard tail has the
overall factor α , while the k -dependence of the hard s
s ⊥
tail of the TMDA Ψ(x,k2) is completely determined by form
⊥
the soft part.
Φhard(x,σ) =∆(x)/σ , (37)
For large k , the generated hard part of the TMDA
⊥
7
Q (y,P)/f P = ⇤ Q (y,P)/f P = 3⇤
⇡ ⇡ ⇡ ⇡
asymptotic
asymptotic
0.7 1.2
0.6 flat 1.0
0.5 CZ flat
0.8
0.4
0.6
0.3
CZ
0.4
0.2
0.1 0.2
y y
-1.0 -0.5 0.5 1.0 1.5 2.0 -0.5 0.5 1.0 1.5
FIG. 2. Quasi-distribution amplitudes Q (y,P) in the “slow” model for P =Λ (left) and P =3Λ (right) evolving to CZ, flat
π
and asymptotic DAs.
Q⇡(y,P)/f⇡ P = 5⇤ Q⇡(y,P)/f⇡ P = 10⇤
asymptotic
1.4
asymptotic
1.2 1.5
1.0
flat
0.8 1.0
flat
0.6
0.4 0.5
CZ
0.2
CZ
y y
-0.5 0.5 1.0 1.5 -0.2 0.2 0.4 0.6 0.8 1.0 1.2
FIG.3. Quasi-distributionamplitudesQ (y,P)inthe“slow”modelforP =5Λ(left)andP =10Λ(right)evolvingtoCZ,flat
π
and asymptotic DAs.
with ∆(x) given by integral over σ
(cid:90) 1 I(x,y,P)= (cid:90) ∞ √dσ P e−(x−y)2P2/σ−m2/σ
∆(x)= dzV(x,z)ϕsoft(z) , (38) πσ σ
π 0
0 1
= . (40)
(cid:112)
(x−y)2+m2/P2
where V(x,z) is the perturbative evolution kernel
[21–23]. The asymptotic form (37) corresponds to a
∼1/k2 TMDA, which is singular for k = 0. As ex- This gives the hard part of the quasi-distribution ampli-
⊥ ⊥ tude
plained above, this singularity is absent in the exact
(rather complicated) expression for the hard tail. For il-
lustrationpurposes,wetakenowthesimplestregulariza- (cid:90) 1 ∆(x)
Qhard(y,P)= dx . (41)
tion 1/k⊥2 →1/(k⊥2 +m2). It corresponds to the change π 0 (cid:112)(x−y)2+m2/P2
1/σ →e−im2/σ/σ in the hard part of VDA,
It generates evolution with respect to P2 in the form
∆(x)
Φhard(x,σ) → e−im2/σ . (39)
σ d m2 (cid:90) 1 ∆(x)
P2 Qhard(y,P)= dx .
dP2 π 2P2 [(x−y)2+m2/P2]3/2
0
To proceed with the conversion formula, one needs the (42)
8
Taking the m/P →0 limit we have Just like in Ref. [14], we have established a simple
relationbetweenQDAsandTMDAsthatallowstoderive
m2 (cid:90) 1 V(x,z) models for QDAs from the models for TMDAs. Unlike
dx thePDFcase,therearemanydrasticallydifferentmodels
2P2 [(x−y)2+m2/P2]3/2
0 claimedtodescribetheprimordialshapeofthepionDA.
= V(y,z)+O(m2/P2) , (43) We have presented the p -evolution patterns for models
3
producingsomepopularproposals: Chernyak-Zhitnitsky,
i.e. forlargeP2 thequasi-distributionamplitudeevolves flat and asymptotic DAs. Our results may be used as a
accordingtotheperturbativeevolutionequationwithre- guideforfuturestudiesofthepiondistributionamplitude
spect to P2. on the lattice using the quasi-distribution approach.
As our estimates show, one would need P of the order
of a few GeV for the nonperturbative evolution to settle.
Itisnaturaltoexpectthatperturbativeevolutionwillbe
VII. CONCLUSIONS ratherimportantatsuchscales. Thus,aninterestingand
technically challenging question for future studies is the
In this paper, we extended the approach of Ref. [14], interplay between the nonperturbutive and perturbative
where we have been dealing with the parton distribu- evolution of the pion quasi-distribution amplitude.
tionfunctions,thebasicingredientsofperturbativeQCD
analysis of hard inclusive processes. Now we have dealt
with the pion distribution amplitude, the basic ingredi- ACKNOWLEDGEMENTS
ent of hard exclusive processes involving the pion. We
applied the formalism of virtuality distribution ampli- ThisworkissupportedbyJeffersonScienceAssociates,
tudes to study the p -dependence of quasi-distribution LLC under U.S. DOE Contract #DE-AC05-06OR23177
3
amplitudes Q (y,p ). and by U.S. DOE Grant #DE-FG02-97ER41028.
π 3
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