Threshold Pion Production at Large Momentum Transfers

0.1 MB·English

by  V. M. Braun

9 Threshold Pion Production at Large Momentum
0
0 Transfers
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n V. M. Brauna, D. Yu. Ivanova,b, A. Lenza, A. Petersa
a
J aInstitut fu¨r Theoretische Physik, Universit¨atRegensburg, D-93040 Regensburg, Germany
bSobolev Institute of Mathematics, 630090 Novosibirsk, Russia
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2

]
h
p Abstract
-
p We consider threshold pion electroproduction on a proton target for photon virtualities in the
e region 1−10 GeV2. The S-wave multipoles at threshold, E0+ and L0+, are calculated using
h
light-cone sum rules.
[

1
v Key words: thresholdpionproduction, chiralsymmetry
4 PACS: 13.60.Le,12.38.Lg
8
1
3 Pionelectroproductionatthresholdfromaprotontarget,e(l)+p(P) e(l′)+π+(k)+
1. n(P′)ande(l)+p(P) e(l′)+π0(k)+p(P′),canbedescribedintermso→ftwogeneralised
0 →
form factors [1,2]
9
0
v: hN(P′)π(k)|jµem(0)|p(P)i= (1)
Xi =−fi N¯(P′)γ5(cid:26) γµq2−qµ6q m12Gπ1N(Q2)− iσ2µmνqνGπ2N(Q2)(cid:27)N(P),
r π (cid:0) (cid:1)
a
which can be related to the S-wave transverse E0+ and longitudinal L0+ multipoles (at
threshold). Here and below m=939 MeV is the nucleon mass.
The celebrated low-energy theorem (LET) [3,4,5] relates the S-wave multipoles or,
equivalently, the form factors G1,G2 at threshold, to the nucleon electromagnetic and
axial form factors in the chiral limit m =0
π
Q2Gπ0p= gA Q2 Gp , Gπ0p = 2gAm2 Gp , (2)
m2 1 2 (Q2+2m2) M 2 (Q2+2m2) E
Q2Gπ+n= gA Q2 Gn + 1 G , Gπ+n = 2√2gAm2 Gn .
m2 1 √2(Q2+2m2) M √2 A 2 (Q2+2m2) E
PreprintsubmittedtoElsevier 22January2009
Here the terms inG aredue to pionemissionoff the initial protonstate, whereasfor
M,E
charged pion in addition there is a contribution corresponding to the chiral rotation of
the electromagnetic current [3].
The subsequent discussion concentrated mainly on the corrections to (2) due to finite
pion mass [6,7]. More recently,the threshold pion productionfor small Q2 wasreconsid-
eredandthe low-energytheoremsre-derivedinthe frameworkofthe chiralperturbation
theory(CHPT), see [8] fora review.The new insightgainedfromCHPT calculations[9]
is that the expansion at small Q2 has to be done with care as the limits m 0 and
π
Q2 0donotcommute,ingeneral.The LETpredictionsseemtobe ingoodag→reement
→
withexperimentaldataonpionphotoproduction[10],However,itappears[11,8]thatthe
S-wave electroproduction cross section for already Q2 0.1 GeV2 cannot be explained
∼
without taking into account chiral loops.
Forlargermomentumtransfersthesituationismuchlessstudiedasthepowercounting
ofCHPTcannotbe applied.The traditionalderivationofLETusingPCACandcurrent
algebra does not seem to be affected as long as the emitted pion is ’soft’ with respect
to the initial and final state nucleons simultaneously. The corresponding condition is,
parametrically, Q2 Λ3/m (see, e.g. [6]) where Λ is some hadronic scale, and might
π
be satisfied for Q2 ≪1 GeV2 or evenhigher. We are not awareofany dedicatedanalysis
of the threshold pr∼oduction in the Q2 1 GeV2 region, however.
∼
It was suggested[12] that in the limit of very large momentum transfers the standard
pQCD collinear factorisation approach becomes applicable and the helicity-conserving
GπN form factor can be calculated for m = 0 in terms of chirally rotated nucleon
1 π
distribution amplitudes. In practice one expects that the onset of the pQCD regime
is postponed to very large momentum transfers because the factorisable contribution
involvesasmallfactorα2(Q)/π2andhastowinovernonperturbative“soft”contributions
s
that are suppressed by an extra power of Q2 but do not involve small coefficients.
The purpose of this study is to suggest a realistic QCD-motivated model for the Q2
dependence of the G1,2 form factors alias S-wave multipoles at threshold in the region
Q2 1 10 GeV2 that can be accessible in current and future experiments in Jefferson
∼ −
Laboratoryandelsewhere(HERMES,MAMI).InRef.[13]wehavedevelopedatechnique
to calculate baryon form factors for moderately large Q2 using light-cone sum rules
(LCSR). This approach is attractive because in LCSRs “soft” contributions to the form
factors are calculated in terms of the same nucleon distribution amplitudes (DAs) that
enterthepQCDcalculationandthereisnodoublecounting.Thus,theLCSRsprovideone
withthemostdirectrelationofthehadronformfactorsanddistributionamplitudesthat
is available at present, with no other nonperturbative parameters. The same technique
can be applied to pion electroproduction, taking into account the semi-disconnected
pion-nucleon contributions in the intermediate state. In Refs. [1,2] the G1 and G2 form
factors are estimated in the LCSR approachfor the range of momentum transfers Q2
1 10GeV2.WedemonstratethattheLETresultsin(2)areindeedreproducedatQ2 ∼
1−GeV2 to the required accuracy (m ), whereas the pQCD contribution considere∼d
π
in [12] formally corresponds to theOleading (at large Q2) part of the NNLO radiative
correction (α2) to the sum rules. Hence our approach describes both high-Q2 and
low-Q2 lim∼itsOcorrsectly and presents an extrapolation in between that makes maximal
use of quark-hadronduality and dispersion relations.
Accurate quantitative predictions are difficult for severalreasons,e,g,because the nu-
cleon distribution amplitudes are poorly known.In order to minimise the dependence of
2
0.5 0.6
0.4
0.4
0.3
0.2 0.2
0.1 E0π+0p/GD 0 E0π++n/GD
0
−0.1 −0.2
0 2 4 6 8 10 0 2 4 6 8 10
Q2 Q2
0.05 0.4
0
0.2
−0.05
−0.1
0
−0.15
−−00..225 L0π+0p/GD −0.2 Lπ0++n/GD
−0.3 −0.4
0 2 4 6 8 10 0 2 4 6 8 10
Q2 Q2
Fig.1.Thetree-levelLCSRresults(curves)comparedtoMAID07[14](points)fortheQ2dependenceof
theelectricandlongitudinal partialwaves atthreshold, E0+ andL0+ (inunits ofGeV−1),normalised
tothedipoleformula.
various parameters one may use the LCSRs to predict certain form factor ratios only,
and then normalise to the electromagnetic nucleon form factors as measured in the ex-
periment, see [2] for the details. The results are shown by the solid curves in Fig. 1,
where the four partial waves at threshold are plotted as a function of Q2, normalised to
the dipole formula G (Q2) = 1/(1+Q2/µ2)2 where µ2 = 0.71 GeV2. To give a rough
D 0 0
idea about possible uncertainties, the “pure” LCSR predictions (all form factors and
other input taken from the sum rules) are shown by dashed curves for comparison. The
accuracy can be improved in future and requires calculation of radiative corrections to
theLCSRs,especiallyifsufficientlypreciselatticecalculationsofthemomentsofnucleon
distribution amplitudes become available.
References
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