Effects of Coherence at Large x Paul Hoyer · PRL 109, 112001 (2012) PHYSICAL REVIEW LETTERS week...

Paul Hoyer ECT* 8 November 2017

1

Dilepton Productions with Meson and Antiproton BeamsECT*, Trento 6 – 10 November 2017

Inclusive DY: fq × fq

N ! T +X p !

L + n

transition revealed by T !

L (Q2) ! µ+µ

Exclusive DY: ϕπ × GPD

fq

fq

xF = x1 – x2

Paul HoyerUniversity of Helsinki

Effects of Coherence at Large x

xF ≈ 1

p

γ*Lϕπ

n

π–

GPD

uQ2

d u


2

γ*(Q2) xF → 1

μ+

μ–

Dynamics of

k(1-x)k+,-ℓ⊥

xk+,ℓ⊥

qq state is long-lived compared to γ*

q

q

k

+k

qq = m

2

m

2q + `

2?

x(1 x)k+k = m2

Q2

Q2 → ∞:Q

2 &m

2q + `

2?

x(1 x)

! ! µ+µ

1- x → 0: γ* is coherent on the entire qq state Q

2 . 2

1 x

and carries the helicity of the pion: γ*LBerger and Brodsky (BB), PRL 42 (1979) 940

for xF → 1

The γ*T couples to a single quark (LT)


3Short-lived qq state: BB dynamics for xF → 1

r?(qq) 1/Q

d

d cos / '(z) sin

2

The value of Q2(1-x) where BB > LT depends on the endpoint (z → 1) behavior of the pion DA ϕπ.

Pion is compact:

BB limit: Q2(1-x) fixed for Q2 → ∞ Mass of X is fixed.

k

z

1-z 1-x → 0

xF → 1

μ+

μ–u

p

r?(qq)

γ*L(Q2)ϕπ ≈1-z

d

u

For MX >> mN:

PH, Järvinen and Kurki 0808.0626

p

γ*(xF, p⊥)

π–

u

uQ2

d

X

Q

2 . 2

1 x

GPD ! fq/N

MX >> mNGPD ! fq/N

d quark hadronizes independently

π–


4

p ! L + n

Exclusive DY: ϕπ × GPD

xF ≈ 1

n

γ*Lϕπ

p

π–

GPD

u

du

Q2

L + n ! + p

DVMP: ϕπ × GPD

Dynamics of ExDY is analogous to DVMP

↔ xF ≈ 1

Available data on γ* N allows to estimate onset of asymptotic behavior.


5

described by recent Regge models [1,2], the focus of thisLetter is on the handbag mechanism in terms of quark andgluon degrees of freedom. Within the handbag interpreta-tion, the data appear to confirm the expectation that pseu-doscalar and, in particular, !0 electroproduction is auniquely sensitive process to access the transversityGPDs !ET and HT . The measured unseparated cross sectionis much larger than expected from leading-twist handbagcalculations. This means that the contribution of the lon-gitudinal cross section "L is small in comparison with "T .The same conclusion can be made in an almost modelindependent way from the comparison of the cross sections"U, "TT , and "LT [17].

Detailed interpretations are model dependent and quitedynamic in that they are strongly influenced by new data asthey become available. In particular, calculations are inprogress to compare the theoretical models with the singlebeam spin asymmetries obtained earlier with CLAS [18]and longitudinal target spin asymmetries, which are cur-rently under analysis.

In the near future, new data on# production and ratios of# to !0 cross sections are expected to further constrain

GPD models. Extracting "L and "T with improved statis-tical accuracy and performing new measurements withtransversely and longitudinally polarized targets wouldalso be very useful.We thank the staff of the Accelerator and Physics

Divisions at Jefferson Lab for making the experiment pos-sible. We also thank G. Goldstein, S. Goloskokov, P. Kroll,J.M. Laget, and S. Liuti for many informative discussionsand clarifications of their work, and for making available theresults of their calculations. This work was supported in partby the U.S. Department of Energy and National ScienceFoundation, the French Centre National de la RechercheScientifique and Commissariat a l’Energie Atomique, theFrench-American Cultural Exchange (FACE), the ItalianIstituto Nazionale di Fisica Nucleare, the ChileanComision Nacional de Investigacion Cientıfica yTecnologica (CONICYT), the National ResearchFoundation of Korea, and the UK Science and TechnologyFacilities Council (STFC). The Jefferson Science Associates(JSA) operates the Thomas Jefferson National AcceleratorFacility for the United States Department of Energy underContract No. DE-AC05-06OR23177.

FIG. 2 (color online). The extracted structure functions vs t for the bins with the best kinematic coverage and for which there aretheoretical calculations. The solid curves are theoretical predictions produced with the models of Ref. [15] and the dashed are from[16]. The data and curves are as follows: the positive value circles (black) and curves are "Uð¼ "T þ $"LÞ, the negative valuetriangles (blue) and curves are "TT , and the squares (red) with accompanying curves are "LT . The shaded bands reflect theexperimental systematic uncertainties.

PRL 109, 112001 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

14 SEPTEMBER 2012

112001-5

CLAS 1206.6355

σTT

σTL

σT + ε σL

p ! 0p

γ*L required for GPD factorization

described by recent Regge models [1,2], the focus of thisLetter is on the handbag mechanism in terms of quark andgluon degrees of freedom. Within the handbag interpreta-tion, the data appear to confirm the expectation that pseu-doscalar and, in particular, !0 electroproduction is auniquely sensitive process to access the transversityGPDs !ET and HT . The measured unseparated cross sectionis much larger than expected from leading-twist handbagcalculations. This means that the contribution of the lon-gitudinal cross section "L is small in comparison with "T .The same conclusion can be made in an almost modelindependent way from the comparison of the cross sections"U, "TT , and "LT [17].

Detailed interpretations are model dependent and quitedynamic in that they are strongly influenced by new data asthey become available. In particular, calculations are inprogress to compare the theoretical models with the singlebeam spin asymmetries obtained earlier with CLAS [18]and longitudinal target spin asymmetries, which are cur-rently under analysis.

In the near future, new data on# production and ratios of# to !0 cross sections are expected to further constrain

GPD models. Extracting "L and "T with improved statis-tical accuracy and performing new measurements withtransversely and longitudinally polarized targets wouldalso be very useful.We thank the staff of the Accelerator and Physics

Divisions at Jefferson Lab for making the experiment pos-sible. We also thank G. Goldstein, S. Goloskokov, P. Kroll,J.M. Laget, and S. Liuti for many informative discussionsand clarifications of their work, and for making available theresults of their calculations. This work was supported in partby the U.S. Department of Energy and National ScienceFoundation, the French Centre National de la RechercheScientifique and Commissariat a l’Energie Atomique, theFrench-American Cultural Exchange (FACE), the ItalianIstituto Nazionale di Fisica Nucleare, the ChileanComision Nacional de Investigacion Cientıfica yTecnologica (CONICYT), the National ResearchFoundation of Korea, and the UK Science and TechnologyFacilities Council (STFC). The Jefferson Science Associates(JSA) operates the Thomas Jefferson National AcceleratorFacility for the United States Department of Energy underContract No. DE-AC05-06OR23177.

FIG. 2 (color online). The extracted structure functions vs t for the bins with the best kinematic coverage and for which there aretheoretical calculations. The solid curves are theoretical predictions produced with the models of Ref. [15] and the dashed are from[16]. The data and curves are as follows: the positive value circles (black) and curves are "Uð¼ "T þ $"LÞ, the negative valuetriangles (blue) and curves are "TT , and the squares (red) with accompanying curves are "LT . The shaded bands reflect theexperimental systematic uncertainties.


14 SEPTEMBER 2012

112001-5

29Old Dominion University, Norfolk, Virginia 23529, USA30Rensselaer Polytechnic Institute, Troy, New York 12180-3590, USA

31University of Richmond, Richmond, Virginia 23173, USA32Universita’ di Roma Tor Vergata, 00133 Rome Italy

33Skobeltsyn Nuclear Physics Institute, 119899 Moscow, Russia34University of South Carolina, Columbia, South Carolina 29208, USA

35Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA36Union College, Schenectady, New York 12308, USA

37Universidad Tecnica Federico Santa Marıa, Casilla 110-V Valparaıso, Chile38University of Glasgow, Glasgow G12 8QQ, United Kingdom39University of Virginia, Charlottesville, Virginia 22901, USA

40College of William and Mary, Williamsburg, Virginia 23187-8795, USA41Yerevan Physics Institute, 375036 Yerevan, Armenia(Received 13 June 2012; published 10 September 2012)

Exclusive !0 electroproduction at a beam energy of 5.75 GeV has been measured with the Jefferson

Lab CLAS spectrometer. Differential cross sections were measured at more than 1800 kinematic values in

Q2, xB, t, and "!, in the Q2 range from 1.0 to 4:6 GeV2, !t up to 2 GeV2, and xB from 0.1 to 0.58.

Structure functions #T þ $#L, #TT , and #LT were extracted as functions of t for each of 17 combinations

of Q2 and xB. The data were compared directly with two handbag-based calculations including both

longitudinal and transversity generalized parton distributions (GPDs). Inclusion of only longitudinal

GPDs very strongly underestimates #T þ $#L and fails to account for #TT and #LT , while inclusion of

transversity GPDs brings the calculations into substantially better agreement with the data. There is very

strong sensitivity to the relative contributions of nucleon helicity-flip and helicity nonflip processes. The

results confirm that exclusive !0 electroproduction offers direct experimental access to the transversity

GPDs.

DOI: 10.1103/PhysRevLett.109.112001 PACS numbers: 13.60.Le, 14.20.Dh, 14.40.Be, 24.85.+p

A major goal of hadronic physics is to describe the threedimensional structure of the nucleon in terms of its quarkand gluon fields. Deep inelastic scattering experimentshave provided a large body of information about quarklongitudinal momentum distributions. Exclusive electronscattering experiments, in which all final-state particles aremeasured, have been rather successfully analyzed and in-terpreted by Regge models which are based on hadronicdegrees of freedom (see, for example, Refs. [1,2]).However, during the past decade the handbag mechanismhas become the leading theoretical approach for extractingthe nucleon quark and gluon structure from exclusive re-actions such as deeply virtual Compton scattering (DVCS)and deeply virtual meson electroproduction. In this ap-proach, the quark distributions are parametrized in termsof generalized parton distributions (GPDs). The GPDscontain information about the distributions of both thelongitudinal momentum and the transverse position ofpartons in the nucleon. In the handbag mechanism, thereaction amplitude factorizes into two parts. One partdescribes the basic hard electroproduction process witha parton within the nucleon, and the other—the GPD—contains the distribution of partons within the nucleonwhich are the result of soft processes. While the formeris reaction dependent, the latter is a universal property ofthe nucleon structure common to the various exclusivereactions. This is schematically illustrated in Fig. 1.While the handbag mechanism should be most applicable

at asymptotically large photon virtualityQ2, DVCS experi-ments at Q2 as low as 1:5 GeV2 appear to be describedrather well at a leading twist by the handbag mechanism,while the range of validity of leading order applicability ofdeeply virtual meson electroproduction is not as clearlydetermined.There are eight GPDs. Four correspond to parton helic-

ity conserving (chiral-even) processes, denoted byHq, ~Hq,Eq, and ~Eq. Four correspond to parton helicity-flip (chiral-odd) processes [3,4], Hq

T , ~HqT , E

qT , and ~Eq

T . The GPDsdepend on three kinematic variables: x, %, and t, where xis the average parton longitudinal momentum fraction and% (skewness) is half of the longitudinal momentum fraction

FIG. 1 (color online). Schematic diagram of the !0 electro-production amplitude in the framework of the handbag mecha-nism. The helicities of the initial and final nucleons are denotedby & and &0, the incident photon and produced meson by ' and'0, and the active initial and final quark by ( and (0. The arrowsin the figure represent the corresponding helicities.


14 SEPTEMBER 2012

112001-2

L

Ee = 5.75 GeV

γ*T dominates DVMP at Ee ≲ 6 GeV2


6Slow approach of pion form factor to asymptopia

G. M. HUBER et al. PHYSICAL REVIEW C 78, 045203 (2008)

FIG. 6. (Color online) Q2Fπ data from this work, compared topreviously published data. The solid Brauel et al. [14] point has beenreanalyzed as discussed in the text. The outer error bars for the JLabdata and the reanalyzed Brauel et al. data include all experimentaland model uncertainties, added in quadrature, whereas the inner errorbars reflect the experimental uncertainties only. Also shown is themonopole fit by Amendolia et al. [9] as well as a 85% monopole+15%dipole fit to our data.

that assume the validity of the monopole parametrization overa wide range of Q2.

To illustrate the departure from the monopole curve, as wellas to provide an empirical fit that describes the data over themeasured Q2 range, we also show in Fig. 6 a fit that includesa small dipole component,

Ffit = 85%Fmono + 15%Fdip, (9)

where

Fdip = 1!1 + r2

dipQ2

12h2c2

"2(10)

and r2dip = 0.411 fm2. This dipole parametrization has nearly

the same χ2 for the elastic-scattering data as the monopolecurve shown [9], but it drops much more rapidly with Q2.The combined monopole plus dipole fit is consistent with ourintermediate Q2 data, while maintaining the quality of fit to theelastic-scattering data. Because a monopole parametrizationdoes not converge to the pQCD asymptotic limit [Eq. (2)], itis expected to fail at some point. Similarly, we should expectthis empirical monopole+dipole parametrization to show itslimitations when additional high Q2 data become available[41].

IV. COMPARISON WITH MODEL CALCULATIONS

The pion form factor can be calculated relatively easily ina large number of theoretical approaches that help advance ofour knowledge of hadronic structure. In this sense, Fπ plays arole similar to that of the positronium atom in QED. Here, wecompare our extracted Fπ values to a variety of calculations,selected to provide a representative sample of the approachesused.

A. Perturbative QCD

The most firmly grounded approach for the calculation ofFπ is that of pQCD. The large Q2 behavior of the pion formfactor has already been given in Eq. (1). By making use ofmodel-independent dimensional arguments, the infinitely largeQ2 behavior of the pion’s quark wave function (distributionamplitude, or DA) is identified as

φπ (x,Q2 → ∞) → 6fπx(1 − x), (11)

whose normalization is fixed from the π+ → µ+νµ decayconstant. Equation (2) follows from this expression.

Neither of these equations is expected to describe the pionform factor in the kinematic regime of our data, and so mucheffort has been expended to extend the calculation of Fπ

to experimentally accessible Q2. In this case, the pion DA,φπ (x,Q2), must be determined at finite Q2. Additional effects,such as quark transverse momentum and Sudakov suppression(essentially a suppression of large quark-quark separationconfigurations in elastic-scattering processes), must be takeninto account. A number of authors [42–46] have performedleading-twist next-to-leading order (NLO) analyses of Fπ atfinite Q2. The hard contributions to Fπ expand as a leadingorder part of order αs and an NLO part of order α2

s .Bakulev et al. [47] have investigated the dependence of the

form of the DA on the form factors, using data from a variety ofexperiments. These were the πγ γ transition form-factor datafrom CLEO [48] and CELLO [49], as well as our Fπ data. Theirresults are insensitive to the shape of the DA near x = 1/2,whereas its behavior at x = 0, 1 is decisive. The resultinghard contribution to the pion form factor is only slightly largerthan that calculated with the asymptotic DA in all consideredschemes. The result of their study, shown as F hard

π in Fig. 7, isfar below our data. The drop at low Q2 is due to their choiceof infrared renormalization, which is not necessarily shared

FIG. 7. (Color online) The Fπ data of Fig. 6 are compared witha hard LO+NLO contribution by Bakulev et al. [47] based on ananalysis of the pion-photon transition form-factor data from CLEO[48] and CELLO [49]. A soft component, estimated from a localquark-hadron duality model, is added to bring the calculation intoagreement with the experimental data. The band around the sumreflects nonperturbative uncertainties from nonlocal QCD sum rulesand renormalization scheme and scale ambiguities at the NLO level.

045203-10

G. M. HUBER et al. PHYSICAL REVIEW C 78, 045203 (2008)

methods are not applicable in the confinement regime. Chiralperturbation theory can give valuable insights, but it is limitedto small values of the photon virtuality Q2. Hence, in theintermediate Q2 regime one has to resort to models like theconstituent quark model or methods employing light-cone(LC) dynamics or the Bethe-Salpeter (plus Dyson-Schwinger)equation or to other approaches such as the use of dispersionrelations or (QCD or LC) sum rules.

Transitions and (transition) form factors are crucial ele-ments for gauging the ideas underlying these QCD-basedmodels. For example, the constituent quark model gives afairly good description of the meson and baryon spectrumand some transitions, but quark effective form factors aretypically required when describing hadronic form factors inthe experimentally accessible Q2 region. In this framework,the study of hadronic form factors can thus be viewed asa study of the transition from constituent to current quarkdegrees of freedom. As exemplified by the many calculationsof it, the electric form factor of the pion, Fπ , is one of thebest observables for the investigation of the transition of QCDeffective degrees of freedom in the soft regime, governed by allkinds of quark-gluon correlations at low Q2, to the perturbative(including next-to-leading order and transverse corrections)regime at higher Q2.

In contrast to the nucleon, the asymptotic normalizationof the pion wave function is known from pion decay. Thehard part of the π+ form factor can be calculated within theframework of pQCD as the sum of logarithms and powers ofQ2 [1]

Fπ (Q2) = 4πCF αs(Q2)Q2

!!!!!#∞n=0an

"log

#Q2

$2

$%−γn!!!!!

2

× 1 + O[αs(Q2),m/Q2], (1)

which in the Q2 → ∞ limit becomes [1,2]

Fπ (Q2) −−−→Q2→∞

16παs(Q2)f 2π

Q2, (2)

where fπ = 93 MeV is the pion decay constant [3].Because the pion’s qq valence structure is relatively

simple, the transition from “soft” (nonperturbative) to “hard”(perturbative) QCD is expected to occur at significantly lowervalues of Q2 for Fπ than for the nucleon form factors [4]. Someestimates [5] suggest that pQCD contributions to the pionform factor are already significant at Q2 ! 5 GeV2. However, arecent analysis [6] indicates that nonperturbative contributionsdominate the pion form factor up to relatively large valuesof Q2, giving more than half of the pion form factor upto Q2 = 20 GeV2. Thus, there is an ongoing theoreticaldebate on the interplay of these hard and soft componentsat intermediate Q2, and high-quality experimental data areneeded to help guide this discussion.

In this work, we concentrate exclusively on the spacelikeregion of the pion form factor. For recent measurementsin the timelike region see Ref. [7]. At low values of Q2,where it is governed by the charge radius of the pion, Fπ

has been determined up to Q2 = 0.253 GeV2 [8,9] from thescattering of high-energy pions by atomic electrons. For the

determination of the pion form factor at higher values of Q2,one has to use high-energy electroproduction of pions on anucleon, i.e., employ the 1H(e, e′π+)n reaction. For selectedkinematic conditions, the longitudinal cross section is verysensitive to the pion form factor. In this way, data for Fπ wereobtained for values of Q2 up to 10 GeV2 at Cornell [10–12].However, those data suffer from relatively large statistical andsystematic uncertainties. More precise data were obtained atthe Deutsches Elektronen-Synchrotron (DESY) [13,14]. Withthe availability of high-intensity electron beams, combinedwith accurate magnetic spectrometers at the Thomas JeffersonNational Accelerator Facility (JLab), it has been possible todetermine L/T separated cross sections with high precision.The measurement of these cross sections in the regimeof Q2 = 0.60–1.60 GeV2 (Experiment Fpi-1 [15,16]) andQ2 = 1.60–2.45 GeV2 (Experiment Fpi-2 [17]) are describedin detail in the preceding article [18]. In this article, it isdiscussed how to determine Fπ from measured longitudinalcross sections, the values determined from the JLab andDESY data are presented, and the results of various theoreticalcalculations are compared with the experimental data.

Because the pion in the proton is virtual (off its mass-shell),the extraction of Fπ from the measured electroproduction crosssections requires some model or procedure. In the next section,the methods that have been used to determine Fπ from thedata are discussed. Section III presents the adopted extractionmethod and the values of Fπ thus determined, including a fulldiscussion of the uncertainties resulting from the experimentaldata and those from the adopted extraction procedure. Variousmodel calculations of Fπ are discussed and compared to thedata in Sec. IV. In the final section, some conclusions aredrawn and an outlook for the future is given.

II. METHODS OF DETERMINING THE PION CHARGEFORM FACTOR FROM DATA

The measurement of the pion form factor is challenging.As stated in the introduction, at low Q2Fπ can be measuredin a model-independent manner via the elastic scattering ofπ+ from atomic electrons, such as has been done up to Q2 =0.253 GeV2 at Fermilab [8] and at the CERN SPS [9]. Itis not possible to access significantly higher values of Q2

with this technique because of limitations in the energy of thepion beam together with the unfavorable momentum transfer.Therefore, at higher values of Q2 Fπ must be determined frompion electroproduction on the proton. The dependence on Fπ

enters the cross section via the t-channel process, in whichthe incident electron scatters from a virtual pion, bringing iton-shell. This process dominates near the pion-pole at t =m2

π , where t is the Mandelstam variable. The physical regionfor t in pion electroproduction is negative, so measurementsshould be performed at the smallest attainable values of −t .To minimize background contributions, it is also necessary toseparate out the longitudinal cross section σL, via a RosenbluthL/T (/LT/T T ) separation [19].

The minimum physical value of −t,−tmin, is nonzero andincreases with increasing Q2 and decreasing value of theinvariant mass, W , of the produced pion-nucleon system.

045203-2

Asymptotic limit of Fπ(Q2):

Jlab Fπ Coll. 0809.3052

Q2F(Q2) (GeV2)

ϕπu

u

ϕππ+

γ*(Q2<0)

π–

http://arxiv.org/abs/arXiv:0809.3052


7Approach to asymptopia for timelike FK

The data of the LA analysis [8] are shown in Fig. 2(a). They represent the most precise data for mK+K− <2.6 GeV/c2. The data at large masses seem to have the correct dependence on the mass according to the asymptoticQCD prediction, but the obtained values are above the absolute QCD prediction by a large factor.

The mass range covered by the SA analysis [9] is 2.6 − 7.5 GeV/c2 and the results are in agreement with thoseof the LA analysis in the common mass range. The measured scaled kaon FF, M2

K+K−× FK , is shown in Fig. 2(b).

The QCD model predictions from several authors [27, 28] are also shown. All the curves are below the data, but theseseem to approach the predictions at high energies.

(GeV)s’1.5 2 2.5 3 3.5 4 4.5 5

2 | +K

|F

-510

-410

-310

-210

-110

1

10

210CLEOBABARFitasymptotic QCD prediction

(GeV)′s1.5 2 2.5 3 3.5 4 4.5 5

2 | +K

|F

-510

-410

-310

-210

-110

1

10

210

(a)

CLEOSeth et al.

BABAR (SA ISR)

MK+K- (GeV/c2)

M2 K

+ K- |F

K| (

GeV

2 /c4 )

CZ (NLO)

Asy (LO)

Asy (NLO)

0

0.5

1

1.5

3 4 5 6 7

0.7

0.8

0.9

3.5 4

(b)

FIGURE 2: (a) The charged kaon form factor versus√

s′ = mK+K− measured by BABAR [8] in the LA analysis.The green band is the result of a fit with a function that has the shape of the QCD predictions to BABAR data for√

s′ > 3 GeV. The blue line is the absolute perturbative-QCD prediction. CLEO data [14, 22] (red empty squares)close to the ψ(2S) and above are consistent with BABAR ones. (b) The scaled charged kaon form factor measured byBABAR [9] in the SA analysis. Different QCD based predictions are shown by the curves lying below data points. Theregion near the ψ(3770) is shown in the inset.

Study of J/ψ→ K+K− and ψ→ K+K− DecaysThe cross section of the processes e+e− → ψ → K+K−, where ψ is either a J/ψ or a ψ(2S), has been measuredby using the data collected around the peaks of those resonances with a slightly relaxed event selection in order toincrease the statistics. A total of 462 ± 28 J/ψ and 66 ± 13 ψ(2S) have been selected by fitting the mass spectrumaround the resonance peaks, with a relativistic Breit-Wigner for the signal convolved with a double Gaussian functionto account for the mass resolution, and a linear function describing the nonresonant K+K−component. The B(ψ →K+K−) branching fractions have been obtained from the measured cross sections. However, to correctly perform thisoperation, the interference between the resonant and nonresonant e+e− → K+K− processes must be taken into account.The interference causes a shift of the measured B(ψ→ K+K−) relative to its true value by

δB = 2!

σ0

σψAs sin ϕ, (7)

where σ0 is the nonresonant cross section [Eq. (4)], and σψ = (12π/m2)B(ψ → e+e−). The ψ → K+K−decays canproceed through a single-photon Aγ and a three-gluon As amplitudes, with ϕ being their relative phase.

080002-4

Babar 1507.04638

Q2FK(Q2)

Q =

Babar has measured the time-like kaon form factor for

Q2 ≤ 56 GeV2

suggesting an approach tothe asymptotic QCD value.


8

γ*(Q2<0)

p n

π+

π

ep ! e+n

“Clearly, the Fπ values determined are strictly within the context of the VGL Regge model, and other values may result if other, better models become available in the future.”

Jlab Fπ Collaboration :

Spacelike pion form factor is hard to measure

Fπ


9Check pion form factor in exclusive DY

γ*(Q2<0)

p n

γ*(Q2>0)π+ π–

p n

Extract π poleπ π

ep ! e+n p ! µ+µn

Extraction of π pole involves similar uncertainties for Q2 < 0 and Q2 > 0.

Result for DY can be compared with Fπ from

e+e ! +

|F|2

Q/GeV

Babar 1205.2228

d

d=

↵2

4Q2|F(Q

2)|2 sin2


10

γ*L(Q2<0)

p p

ρ0L

Compactness of γ*(Q2) → ρ

Brodsky et al, hep-ph/9402283

Longitudinal polarization in e p → e ρ p indicates that the ρis produced in a compact configuration at high photon virtuality.

r?(qq)

P|


11Photon polarization in γ* p → ρ pσL/σtot in ρ production indicates the Q2 at which the γ* – ρ vertex becomes transversally compact.

γ*(Q2 ≳ 2 GeV2) compactifies the ρ wave function (at high s)

p ! 0p

Figure 2: (a) The π+π− invariant mass distribution for the final sample of events; the curveis a maximum likelihood fit with a non-relativistic Breit-Wigner distribution plus a flat (4%)background (see text for details); (b) the cross section for γ∗p → ρ0p as a function of Q2 for0.0014 < x < 0.004. Also shown are data from the NMC experiment [7]; the errors shown arejust the statistical errors. The ZEUS (NMC) data have an additional 31% (20%) normalisationuncertainty (not shown); (c) the cosθh distribution for the decay π+, in the s-channel helicitysystem, corrected for acceptance, for π+π− pairs in the mass range 0.6-1.0 GeV. The curve is afit to the form of Eq. (10); (d) the ρ0 density matrix element, r04

00, compared with results fromfixed target experiments [2,3,7] as a function of Q2. The thick error is the statistical error andthe thin error is the systematic error added in quadrature.

15

Zeus hep-ex/9507001

Q2/GeV2

⇒

PMC Physics A 2007, 1:6 http://www.physmathcentral.com/1754-0410/1/6

Page 27 of 47

(page number not for citation purposes)

The Q2 dependence of for W = 90 GeV, averaged over the range 40 <W < 140 GeV, is

shown in Fig. 17 and listed in Table 7 together with the corresponding R values. The figure

includes three data points at lower Q2 from previous studies [10,53]. An initial steep rise of

with Q2 is observed and above Q2 ! 10 GeV2, the rise with Q2 becomes milder. At Q2 = 40 GeV2,

σL constitutes about 90% of the total γ*p cross section.

The comparison of the H1 and ZEUS results is presented in Fig. 18 in terms of the ratio R. TheH1 measurements are at W = 75 GeV and those of ZEUS at W = 90 GeV. Given the fact that Rseems to be independent of W (see below), both data sets can be directly compared. The twomeasurements are in good agreement.

The dependence of R on Mπ π is presented in Fig. 19 for two Q2 intervals. The value of R falls

rapidly with Mπ π above the central ρ0 mass value. Although a change of R with Mππ was antici-

pated to be ~10% [55], the effect seen in the data is much stronger. The effect remains strong also

at higher Q2, contrary to expectations [55]. Once averaged over the ρ0 mass region, the main con-

tribution to R comes from the central ρ0 mass value.

r0004

r0004

The ratio as a function o f Q 2 for W = 90 G eVFigure 17

The ratio as a function o f Q 2 for W = 90 G eV. Al so inc luded are values of from prev ious measurements

at lower Q2 values [10, 53]. The inner error bars indicate the stat ist ical uncertainty, the outer error bars represent the stat ist ical and systematic uncertainty added in quadra ture.

ZEUS

Q2 (GeV2)

r04 00

= σ

L / σ

tot

ZEUS 1994ZEUS 1995

ZEUS 120 pb-1

0

0.2

0.4

0.6

0.8

0 10 20 30 40 50

r0004 r00

04

Zeus 0708.1478

p ! 0p

L

tot

L

tot

W = 90 GeV


12γ* polarization in inclusive DY

VOLUME 55, NUMBER 24 PHYSICAL REVIEW LETTERS 9 DECEMBER 1985

pairs with M ) 4 GeV/c were produced only fromantiquarks with fraction x ) 0.4 of the momentum ofthe pion. [These variables are related by x„=(x„—xz)/(I —r ); x xz = r, where xz is the momentumfraction carried by the annihilating quark in a nu-cleon. j The resolution in the various reconstructedparameters of the muon pairs was limited primarily bymultiple scattering in the tungsten target and thespread in incident pion momentum. Typical values forthe rms reconstruction errors were 0.14 GeV/c2 formass M, 0.04 for xF, 0.03 for x, 0.04 for x~, 0.13GeV/c for PT, and 0.06 for cosg, .We have fitted the angular distribution of the muons

in the muon-pair rest frame with the form 1+X cos 0.We find X= 0.64 + 0.13 for 8 measured with respect tothe r -channel axis and A. = 0.66 + 0.17 using theCollins-Soper axis, ' integrating over all high-masspairs. Of greater interest is the dependence of parame-ter X on x as shown in Fig. 1 for the t channel. Atlarge x the angular distribution approaches the formsin H„which is model-independent evidence of longi-tudinal polarization of the virtual photon, as previouslyreported. 8

%e have also compared the data to the most generalform of the angular distribution for lepton-pair pro-duction via a virtual photon':

do- 2 2dQ

~ 1 + X cos 8 + p sin28 cos@+ t0 sin 8 cos2@.

0.8—

0-

—0 8—

0.4 0.5 0.6 0.7 0.8 0.9

FIG. 1. The parameter A. as a function of x obtainedfrom fits to the angular distribution of the muons in themuon-pair rest frame, using the form d o./d coss~ l+ ~ cos29, where 8 measures the p, + direction with respectto the t-channel axis. The solid curve is based on the QCDmodel of Berger and Brodsky (Ref. 9) with the value(kr2) = 0.62 GeV'/c' deduced from the observed pion struc-ture function.

The statistical power of the fit is limited by the size of the data sample. We find no significant departures ofparameters p and ~ from zero in any region of x, but these parameters are also consistent with the slightlynonzero values suggested in model calculations.Additional evidence for modifications to muon-pair production at large longitudinal momentum comes from the

transverse momentum of the pairs. To minimize the effect of reduction of transverse phase space for high longitu-dinal momentum, we define Feynman x in the center-of-mass frame as xF= PL/PL '" where

M +M~ (M —M~)2p max '+2, s s2

&/2T

We take the mass of the recoiling system, M„, to be 0.94 GeV/c . Our definition of PL '" distributes uniformly inx„ the reduction of phase space which occurs at high PT. The results for (P„) and (PT) are displayed in Fig. 2 asfunctions of xF. The value of (PT) —1.0 GeV/c at moderate xF is in good agreement with the trend of (PT) vs sobserved in other muon-pair production experiments. ' The average transverse momentum drops dramatically atlarge xF, in roughly the same region that the angular distribution changes its character.Information as to the structure of the pion, and of the target nucleons, was extracted from the observed muon-

pair cross section according to the model of quark-antiquark annihilation:

=K, X, e x x~ [q (x ) q ~ (x~ ) + q „(x„)q~ (x~ ) ].dx~ dx~ 9sr

Here q (x) is the probability of finding a quark carrying momentum fraction x of its parent hadron, '6 and K is anoverall normalization factor representing the net effect of a large class of higher-order QCD corrections. ' Our ac-ceptance was restricted to the region x )0.4, where only the valence quark distributions of the pion are signifi-cant. Taking note of 6-parity invariance, we define the structure function of the pion as

F (x ) =x u" (x ) =x d" (x ),

Plab = 80 GeV/c

E615 PRL 55 (1985) 2649

α

Q2 > 16 GeV2

xF

αN ! X

Recall: γ* is coherent on the entire qq state ford

d cos 1 + cos2

The DY data indicates σL dominancefor xF ≳ 0.9 when Q2 ≈ 20 GeV2 i.e.,

Q

2(1 x) . 2 GeV2

The γ* polarization should turn overat fixed Q2(1-x), when Q2 ≳ 2 GeV2 .

The “soft scale” Λ2 ≈ 2 GeV2 is consistent with that in γ* p → ρ p

Q

2 . 2

1 x


13Target form factor as xF → 1 in DY

p

γ*(xF, p⊥)

π–

u

uQ2

d

X

As xF → 1 the d-quark fuses with the proton remnants: X → N*

For xF < 1 the pion d-quark is energetic and hadronizes independently.

The p → N* transition form factor causes a decrease in

39 EXPERIMENTAL STUDY OF MUON PAIRS PRODUCED BY. . . 113

a a s a I s s a a I a s s a

I

10—2OOP

10~CLob I

2Mp (GeY/c)

5.00

FIG. 23. pT spectrum corrected for acceptance. The back-ground, as estimated from random pairings, is negligible.

10-1~

10 4~ 10&OI

0.00

I

0a) 104O~ 10+JD

10~0.00

10

&0.50 I- 10

10

10

280 5.00p, (Gev/c)

a s a I s a s a

cx„&0.70:10

10Ie

-102M 5.00

)ss (Gev/c)s a s I s a a a

cx,&0.90 r- 10

f- 10

0.00

0.00

&0.60 .=

290 5.00)ss (Gev/c)

a . a a I a a a s

080-

Jr2M 5.00~ (Gev/c)

a s s I s a a a

0.90cx,&1.00:

25 together with the parametrized fit. As can be seen, thequality of the fit is excellent.Spectra integrated over the range 0 &xF & 1 have also

been obtained in various regions of m„„. Figures 26 and27 show (pT), and pz. spectra as a function of m„„.Note especially the drop in (pT ) in the mass region ofthe upsilon resonances. This indicates a clear change inthe production mechanism for this interval. The resultsfor (pT ) vs m„„ indicate a slow increase in pT as m„„grows. Since this parameter is integrated over all xF it isquite insensitive to the small-cross-section region at highxFeA comparison with other experiments is given in Fig.

28, where (pT ) in the region xF & 0 is plotted as a func-

10~~

0.00 2M)ss (GeV/c)

- 10~5.00 0.00 2M

)sa (GeV/c)5.00

tion of s for the same value of &x=0.28 (Ref. 16). Theresults from this experiment agree with the general in-crease with s. This trend is predicted by QCD calcula-tions but as noted by Malhotra, ' the QCD calculationsfail to explain the magnitude of the increase in (pT ) withs. The QCD prediction for (pT) falls 60% too low at

FIG. 25. do/dpT for 4.05&m» &8.55 GeV/c, in regions ofxF, curves are result of fits.

O00)(3 125—I l

OlCL

O

0)~ 2.00-A

CV t-CLV

0.000.00 OM

Xp1.00

1.004.00 9.00

m~(GeY/c QI.O

FIG. 24. Mean pT for 4.05 (m» & 8.55 GeV/c, as a func-tion of xF. The values are those obtained from the fitted form.

FIG. 26. Fit results for 0(xF & 1, in regions of m» showinginferred values for (pr) vs m„„. Region near Y shows lowervalue of (pz. ).

N ! X

Q2 > 16 GeV2

Plab = 252 GeV/c

E615 PRD 39 (1989) 92

xF

hp2?i

hp2?i of γ*


14

PMC Physics A 2007, 1:6 http://www.physmathcentral.com/1754-0410/1/6

Page 20 of 47

(page number not for citation purposes)

in Table 4. As expected, a steep decrease of the

cross section with Q2 is observed. The photo-

production and the low-Q2 (< 1 GeV2) meas-urements are also shown in the figure. An

attempt to fit the Q2 dependence with a sim-ple propagator term

with the normalisation and n as free parame-ters, failed to produce results with an accepta-

ble χ2 . The data appear to favour an n value

which increases with Q2.

10.3 W dependence of σ(γ*p → ρ0p)

The values of the cross section σ(γ*p → ρ0p) as a function of W, for fixed values of Q2, are plottedin Fig. 13 and given in Table 5. The cross sections increase with increasing W, with the rate of

increase growing with increasing Q2.

σ γ ρ ρ( ) ~ ( ) ,∗ −→ +p p Q m n0 2 2

Table 3: The slope b resulting from a fit to the differential cross-section dσ/dt to an exponential form for the reaction γ*p → ρ0p, for different Q2 intervals. The first column gives the Q2 bin, whi le the second column gives the Q2 value at which the differential cross sections are quoted. The first uncertainty is statistical, the second systematic.

Q2 bin (GeV2) Q2 (GeV2) b (GeV-2)

2–4 2.7

4–6.5 5.0

6.5–10 7.8

10–15 11.9

15–30 19.7

30–80 41.0

6 6 0 1 0 20 2. . ..± −

+

6 3 0 2 0 20 2. . ..± −

+

5 9 0 2 0 20 2. . ..± −

+

5 5 0 2 0 20 2. . ..± −

+

5 5 0 3 0 30 2. . ..± −

+

4 9 0 6 0 50 8. . ..± −

+

A compilat ion of the value o f the slope b from a fit o f the form dσ/d |t| ∝ e-b|t| for exclusive vector-meson electro-production, a s a function of Q2 + M2Figure 11A compilat ion of the value o f the slope b from a fit o f the form dσ/d |t| ∝ e-b|t| for exclusive vector-meson electro-production, as a function o f Q 2 + M2 . Al so inc luded is the DVCS result. The inner error ba rs indicate the sta tistical uncerta inty, the outer error bars represent the sta tistical and systemat ic uncerta inty added in quadrature.

ZEUS

ρ ZEUS 120 pb-1

ρ ZEUS 94

ρ ZEUS 95

φ ZEUS 98-00

φ ZEUS 94

J/ψ ZEUS 98-00

J/ψ ZEUS 96-97

J/ψ H1 96-00

ρ H1 95-96

DVCS H1 96-00

Q2+M2(GeV2)

b(G

eV-2

)

0

2

4

6

8

10

12

14

0 5 10 15 20 25 30 35 40 45 50

Zeus 0708.1478

p⊥ dependence in exclusive γ* p → ρ p

d

d|t| (p ! p) / exp(b|t|)

γ*(Q2<0)

p p

ρ0

t

As Q2 → – ∞ the γ* → ρ vertexcompactifies, leaving only thetarget proton form factor:

The upper vertex shrinksrapidly with Q2.

b

Linked to γ*T → γ*L

Fourier transform wrt p⊥gives impact parameter distr.


15

Color transparency in γ* A → ρ A´

VOLUME 74, NUMBER 9 PHYSICAL REVIEW LETTERS 27 FEBRUARY 1995

~400

~3000200

~~ 300.a) 250:—C3200:-

O150:—

c 10050:—

o 00 0.5 1 1.5 2m„„(Gev/c*)

0 ~~ I ~ t % I ~I I I I I I I J I I.. .t I

-25 -20 -15 -10 -5

(z,—1)/ez,

i/,0 5

FIG. 1. The (z —1)/Bz distribution for incoherent [i t'i )0.1 (GeV/c)2] po candidates passing the cuts described in thetext from the data (points) and the simulated sample (solidhistogram). The Monte Carlo sample includes contributionsfrom exclusive po (dotted histogram) and inclusive background(dashed histogram) events only. The Monte Carlo events weregenerated by a specially written generator (exclusive sample)and a Lund-based generator (inclusive sample) with fulldetector simulation and processed through the same analysisprocedure as that used for the data. The region accepted isindicated by arrows. Shown in the inset is the invariant-massdistribution for the p candidates passing all the cuts.

results are m~ = 0.780 ~ 0.004 GeV/cz, I ~ = 0.188 ~0.010 GeV/cz and n = 3.18 ~ 0.18.Exclusive production of vector mesons from a nuclear

target can be coherent, corresponding to production fromthe nucleus as a whole, or incoherent, corresponding toproduction from individual nucleons in the nucleus. Sincethe rate of falloff of the t' distribution for a diffrac-tive scattering process measures the physical size of thescatterer, one expects to see t' fall steeply initially (co-herent production), followed by a region with a shallowslope (incoherent production). The observed t' distribu-tions of the p candidates for hydrogen and calcium areshown in Fig. 2. Two distinct processes are clearly iden-tifiable. The, line for hydrogen is a fit by N, e ".I"I uing only the points between 0.08 and 0.50 GeVz (b„=

6.29 ~ 0.37 GeV z ). The lines for calcium are fits byNze bAlt'I using only points between 0.00 and 0.02 GeV(bz = 100 ~ 6 GeV ) and N„e ""i' using only pointsbetween 0.08 and 0.50 GeVz (b = 6.20 ~ 0.53 GeV ).In the present analysis, events with it'i ) 0.1 GeV

were selected for the incoherent sample (the same cutis applied for all targets). Contributions from coherentevents were estimated by integrating the fitted coherentexponential functions from 0.1 GeV to infinity. The levelof coherent background (subtracted from the signal) wasless than 1% for all but the deuterium target for which thecontamination was about 8%.The main source of background comes from events in

which a p, or a pair of oppositely charged hadrons withm consistent with the p mass, are produced throughfragmentation with a sum of g values close to 1 and noother detected particles. A Lund-based Monte Carlo [15]program was used to generate a sample of deep-inelasticevents. These events were subjected to the same analysisprocedure as that used for the data. The surviving eventswere then normalized to the data by demanding that thenumber of total events integrated over (z —I)/6z (from—~ to —3) from the Monte Carlo sample be equal to thatof the data. The estimated background events were thensubtracted from the data.For each Q2 region considered, the exclusive po pro-

duction cross section from a nucleus, o-&, can be fit by apower law o.~ = 0.0A in which o-0 and u are parameters.00 can be interpreted as the best estimate of the A = 1cross section, and o. characterizes the A dependence. Thetransparencies T = tTA/Atro, for incoherent po productionoff hydrogen, deuterium, carbon, calcium, and lead ver-sus A for the three Qz regions are shown in Fig. 3(a).Also shown in Fig. 3(a) are the functions T = A '. In

b

-110

10 lI10

A

I I I

10

10C3~10'»0~LLJ

10

0.9—0.8—0.7—0.6—0.5

10 1g'(Gev')

10

0.1I

0.2 0.3-I'(GeV')

0.4 0.5

FIG. 2. The t' distributions (not corrected for the experimentalresolution and acceptance) for hydrogen (lower curve) andcalcium (upper curve).

FIG. 3. (a) The transparency T, defined in the text, as afunction of A for three Q2 regions. Note that the points havebeen multiplied by 2, l, and 0.5, respectively, for the threeQ2 points. (b) n as a function of Q'. n = 1 corresponds tocomplete transparency. The errors shown in these plots arestatistical only.

1527

VOLUME 74, NUMBER 9 PHYSICAL REVIEW LETTERS 27 FEBRUARY 1995

~400

~3000200

~~ 300.a) 250:—C3200:-

O150:—

c 10050:—

o 00 0.5 1 1.5 2m„„(Gev/c*)

0 ~~ I ~ t % I ~I I I I I I I J I I.. .t I

-25 -20 -15 -10 -5

(z,—1)/ez,

i/,0 5

FIG. 1. The (z —1)/Bz distribution for incoherent [i t'i )0.1 (GeV/c)2] po candidates passing the cuts described in thetext from the data (points) and the simulated sample (solidhistogram). The Monte Carlo sample includes contributionsfrom exclusive po (dotted histogram) and inclusive background(dashed histogram) events only. The Monte Carlo events weregenerated by a specially written generator (exclusive sample)and a Lund-based generator (inclusive sample) with fulldetector simulation and processed through the same analysisprocedure as that used for the data. The region accepted isindicated by arrows. Shown in the inset is the invariant-massdistribution for the p candidates passing all the cuts.

results are m~ = 0.780 ~ 0.004 GeV/cz, I ~ = 0.188 ~0.010 GeV/cz and n = 3.18 ~ 0.18.Exclusive production of vector mesons from a nuclear

target can be coherent, corresponding to production fromthe nucleus as a whole, or incoherent, corresponding toproduction from individual nucleons in the nucleus. Sincethe rate of falloff of the t' distribution for a diffrac-tive scattering process measures the physical size of thescatterer, one expects to see t' fall steeply initially (co-herent production), followed by a region with a shallowslope (incoherent production). The observed t' distribu-tions of the p candidates for hydrogen and calcium areshown in Fig. 2. Two distinct processes are clearly iden-tifiable. The, line for hydrogen is a fit by N, e ".I"I uing only the points between 0.08 and 0.50 GeVz (b„=

6.29 ~ 0.37 GeV z ). The lines for calcium are fits byNze bAlt'I using only points between 0.00 and 0.02 GeV(bz = 100 ~ 6 GeV ) and N„e ""i' using only pointsbetween 0.08 and 0.50 GeVz (b = 6.20 ~ 0.53 GeV ).In the present analysis, events with it'i ) 0.1 GeV

were selected for the incoherent sample (the same cutis applied for all targets). Contributions from coherentevents were estimated by integrating the fitted coherentexponential functions from 0.1 GeV to infinity. The levelof coherent background (subtracted from the signal) wasless than 1% for all but the deuterium target for which thecontamination was about 8%.The main source of background comes from events in

which a p, or a pair of oppositely charged hadrons withm consistent with the p mass, are produced throughfragmentation with a sum of g values close to 1 and noother detected particles. A Lund-based Monte Carlo [15]program was used to generate a sample of deep-inelasticevents. These events were subjected to the same analysisprocedure as that used for the data. The surviving eventswere then normalized to the data by demanding that thenumber of total events integrated over (z —I)/6z (from—~ to —3) from the Monte Carlo sample be equal to thatof the data. The estimated background events were thensubtracted from the data.For each Q2 region considered, the exclusive po pro-

duction cross section from a nucleus, o-&, can be fit by apower law o.~ = 0.0A in which o-0 and u are parameters.00 can be interpreted as the best estimate of the A = 1cross section, and o. characterizes the A dependence. Thetransparencies T = tTA/Atro, for incoherent po productionoff hydrogen, deuterium, carbon, calcium, and lead ver-sus A for the three Qz regions are shown in Fig. 3(a).Also shown in Fig. 3(a) are the functions T = A '. In

b

-110

10 lI10

A

I I I

10

10C3~10'»0~LLJ

10

0.9—0.8—0.7—0.6—0.5

10 1g'(Gev')

10

0.1I

0.2 0.3-I'(GeV')

0.4 0.5

FIG. 2. The t' distributions (not corrected for the experimentalresolution and acceptance) for hydrogen (lower curve) andcalcium (upper curve).

FIG. 3. (a) The transparency T, defined in the text, as afunction of A for three Q2 regions. Note that the points havebeen multiplied by 2, l, and 0.5, respectively, for the threeQ2 points. (b) n as a function of Q'. n = 1 corresponds tocomplete transparency. The errors shown in these plots arestatistical only.

1527

E665 PRL 74 (1995) 1525

Incoherent470 GeV μ

The systematic uncertainties are separated into Q2-and lc-dependent and kinematics-independent contribu-tions. The ratio of the integrated luminosities representsthe largest source of kinematics-independent uncertain-ties. An additional contribution comes from double-diffractive dissociation. The total estimated systematicuncertainty from all normalization factors is 11%. Thekinematics-dependent systematic uncertainties have beenstudied as a function of lc and Q2 on a bin-by-bin basis.The main contributions come from DIS background sub-traction, acceptance corrections, the efficiency of the !Eexclusivity cut, the corrections due to ‘‘Pauli blocking,’’and the application of radiative corrections. None of thekinematics-dependent systematic uncertainties cancel inthe coherent nuclear transparency because of the differ-ent t0 cuts that are applied, and they increase at small andat large coherence length values. At small lc, and corre-spondingly large Q2, the uncertainties in the coherent toincoherent separation via the t0 slope parameters bp andb14N, and the background subtraction dominate. At largelc, the uncertainty in the acceptance correction factorbecomes large. Thus, the contribution of the kinematics-dependent systematic uncertainty varies between 8% and14%. This results in a combined systematic uncertainty of14% to 18% for the nuclear transparency measurementspresented in Fig. 1.

A two-dimensional analysis of the nuclear transpar-ency as a function of coherence length and Q2 has beenperformed, which represents a new approach in the searchfor CT. It is constrained by the phase space boundaries

displayed in Fig. 2. Since the combination of statisticalsignificance and Q2 coverage is largest near lc ’ 2:0 fm,the region 1:3< lc < 2:5 fm has been chosen for thistwo-dimensional analysis. To deconvolute the CT andcoherence length effects, coherence length bins of 0.1 fmwere used. These finite bins introduce an additional sys-tematic uncertainty in the Q2 slope of 0.008 and

FIG. 2. Distribution of Q2 versus coherence length for exclu-sive !0 production on hydrogen and nitrogen. The regionsurrounded by the rectangle represents the subset thatwas used for the two-dimensional analysis of the nucleartransparency.

0.5

1Tc

0.5

1

1 2 3 4 5lc [fm]

Tin

c

FIG. 1 (color online). Nuclear transparency as a function ofcoherence length for coherent (top panel) and incoherent (bot-tom panel) !0 production on nitrogen, compared to predictionswith CT effects included (curves) [21]. The inner error barsinclude only statistical uncertainties, while the outer error barspresent the statistical and systematic uncertainties added inquadrature.

0

0.5

1<lc> = 1.35 fm

Tc(

l c,Q2 )

1.45 fm 1.55 fm 1.65 fm

0

0.5

1 1.75 fm 1.85 fm 1.95 fm 2.05 fm

0

0.5

1

0 2 4

2.15 fm

0 2 4

2.25 fm

0 2 4

2.35 fm

0 2 4

2.45 fm

Q2 [GeV2]

0

0.5

1

0

0.5

1

0

0.5

1

0 2 4 0 2 4 0 2 4

<lc> = 1.35 fm

Tc(

l c,Q2 )

1.45 fm 1.55 fm 1.65 fm

1.75 fm 1.85 fm 1.95 fm 2.05 fm

2.15 fm 2.25 fm 2.35 fm 2.45 fm

Q2 [GeV2]0 2 4

FIG. 3 (color online). Nuclear transparency as a function ofQ2 in specific coherence length bins (as indicated in eachpanel) for coherent !0 production on nitrogen. The straightline is the result of the common fit of the Q2 dependence. Theerror bars include only statistical uncertainties.

P H Y S I C A L R E V I E W L E T T E R S week ending7 FEBRUARY 2003VOLUME 90, NUMBER 5

052501-4 052501-4

2

The systematic uncertainties are separated into Q2-and lc-dependent and kinematics-independent contribu-tions. The ratio of the integrated luminosities representsthe largest source of kinematics-independent uncertain-ties. An additional contribution comes from double-diffractive dissociation. The total estimated systematicuncertainty from all normalization factors is 11%. Thekinematics-dependent systematic uncertainties have beenstudied as a function of lc and Q2 on a bin-by-bin basis.The main contributions come from DIS background sub-traction, acceptance corrections, the efficiency of the !Eexclusivity cut, the corrections due to ‘‘Pauli blocking,’’and the application of radiative corrections. None of thekinematics-dependent systematic uncertainties cancel inthe coherent nuclear transparency because of the differ-ent t0 cuts that are applied, and they increase at small andat large coherence length values. At small lc, and corre-spondingly large Q2, the uncertainties in the coherent toincoherent separation via the t0 slope parameters bp andb14N, and the background subtraction dominate. At largelc, the uncertainty in the acceptance correction factorbecomes large. Thus, the contribution of the kinematics-dependent systematic uncertainty varies between 8% and14%. This results in a combined systematic uncertainty of14% to 18% for the nuclear transparency measurementspresented in Fig. 1.

A two-dimensional analysis of the nuclear transpar-ency as a function of coherence length and Q2 has beenperformed, which represents a new approach in the searchfor CT. It is constrained by the phase space boundaries

displayed in Fig. 2. Since the combination of statisticalsignificance and Q2 coverage is largest near lc ’ 2:0 fm,the region 1:3< lc < 2:5 fm has been chosen for thistwo-dimensional analysis. To deconvolute the CT andcoherence length effects, coherence length bins of 0.1 fmwere used. These finite bins introduce an additional sys-tematic uncertainty in the Q2 slope of 0.008 and

FIG. 2. Distribution of Q2 versus coherence length for exclu-sive !0 production on hydrogen and nitrogen. The regionsurrounded by the rectangle represents the subset thatwas used for the two-dimensional analysis of the nucleartransparency.

0.5

1Tc

0.5

1

1 2 3 4 5lc [fm]

Tin

c

FIG. 1 (color online). Nuclear transparency as a function ofcoherence length for coherent (top panel) and incoherent (bot-tom panel) !0 production on nitrogen, compared to predictionswith CT effects included (curves) [21]. The inner error barsinclude only statistical uncertainties, while the outer error barspresent the statistical and systematic uncertainties added inquadrature.

0

0.5

1<lc> = 1.35 fm

Tc(

l c,Q2 )

1.45 fm 1.55 fm 1.65 fm

0

0.5

1 1.75 fm 1.85 fm 1.95 fm 2.05 fm

0

0.5

1

0 2 4

2.15 fm

0 2 4

2.25 fm

0 2 4

2.35 fm

0 2 4

2.45 fm

Q2 [GeV2]

0

0.5

1

0

0.5

1

0

0.5

1

0 2 4 0 2 4 0 2 4

<lc> = 1.35 fm

Tc(

l c,Q2 )

1.45 fm 1.55 fm 1.65 fm

1.75 fm 1.85 fm 1.95 fm 2.05 fm

2.15 fm 2.25 fm 2.35 fm 2.45 fm

Q2 [GeV2]0 2 4

FIG. 3 (color online). Nuclear transparency as a function ofQ2 in specific coherence length bins (as indicated in eachpanel) for coherent !0 production on nitrogen. The straightline is the result of the common fit of the Q2 dependence. Theerror bars include only statistical uncertainties.

P H Y S I C A L R E V I E W L E T T E R S week ending7 FEBRUARY 2003VOLUME 90, NUMBER 5

052501-4 052501-4

4

Hermes hep-ex/0209072

Coherent

27.5 GeV e+

Tran

spar

ency

α

The shrinking transverse size of γ* → qqwith increasing Q2 manifests itself through“Color transparency” on nuclear targets

A high energy qq state hadronizes outside the nucleus: Clearer CT signal

27D

epar

tmen

tof

Phy

sics

,Sim

onF

rase

rU

nive

rsit

y,B

urna

by,

Bri

tish

Col

umbi

aV

5A1S

6,C

anad

a28

TR

IUM

F,Va

ncou

ver,

Bri

tish

Col

umbi

aV6

T2A

3,C

anad

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uare

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(!Q

2).

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rved

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tly

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ean

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ate

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ract

ions

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ear

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ium

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sh[1

–7].

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atth

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ant

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itud

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ons

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ons

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med

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Cre

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[12]

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gges

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inR

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[13]

.Fu

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rev

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Tco

mes

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].A

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muo

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omnu

clei

[17]

give

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indi

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onof

CT.

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ever

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nal

isof

inde

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gnifi

canc

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this

Let

ter

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tech

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ucle

onto

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ener

gyvi

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[18]

inth

efo

rmof

aq! qq

pair

has

atr

ansv

erse

size

r ?%

1=Q

[3].

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q! qq

fluct

uati

onof

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virt

ual

phot

onca

npr

opag

ate

over

adi

stan

cel c

know

nas

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8]by

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2 q! qq,

whe

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virt

ual

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ergy

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ant

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pair.

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ate

thro

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ear

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ium

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]by

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,

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ndst

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the

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and

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atio

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tap

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ance

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orin

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abil

ity

for

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tera

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ith

the

nucl

ear

med

ium

incr

ease

sw

ithl c

unti

llc

exce

eds

the

nucl

ear

size

[19,

20].

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valu

esof

thel c

smal

ler

than

the

nucl

eus,

this

cohe

renc

ele

ngth

effe

ct[2

0]ca

nm

imic

theQ

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pend

ence

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enu

clea

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aren

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edic

ted

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rco

here

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ucti

on,

inco

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st,

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rmfa

ctor

supp

ress

esth

eap

pare

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ans-

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Smal

ll c

corr

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larg

elo

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udin

alm

omen

tum

tran

sfer

(qc%

1=l c

),w

here

the

form

fact

oris

smal

l.H

ence

,Tde

crea

ses

wit

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2in

cohe

rent

prod

uc-

tion

.Thi

sbeh

avio

rcan

notm

imic

CT,

buta

lso

inth

isca

se,

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cohe

renc

ele

ngth

effe

cts

can

sign

ifica

ntly

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ify

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Q2

depe

nden

ce,t

hus

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urin

gth

ecl

ean

obse

rvat

ion

ofa

CT

effe

ct.I

nor

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sent

angl

eth

eef

fect

sof

cohe

renc

ele

ngth

from

thos

eof

CT,

itis

impo

rtan

tto

stud

yth

eva

riat

ion

ofT

wit

hQ

2,

whi

leke

epin

gl c

fixed

[21]

.In

this

way

,ach

ange

ofT

wit

hQ

2ca

nbe

asso

ciat

edw

ith

the

onse

tof

CT.

PH

YS

ICA

LR

EV

IEW

LE

TT

ER

Sw

eek

endi

ng7

FE

BR

UA

RY

2003

VO

LU

ME

90,N

UM

BE

R5

0525

01-2

0525

01-2

Need to consider fixed qq life-time ℓc .


16

γ*(Q2)

pX → N*x2 → 1

p N*

γ*(Q2)

≈

Fp!N(Q2)

!

Bloom-Gilman duality in DIS

Fq/p(x2)

M

2X = M

2p +

(1 x2)Q2

x2

At high Q2, the electron scatters from a single quark in e p → e X

At the same value of x2 but low Q2, the process is coherent: e p → e N*


17Scaling F2(xB) distribution averages resonances

W. Melnitchouk (2010)

JLab Hall C

*

*

xB

F2(ep ! eX)

High Q2

TMC = Target Mass Correction

BG duality in e p → e N* works at low Q2 and xB << 1


18Resonances slide on the scaling curve as Q2 varies

ξ≈xB

Jlab Hall C

Q2 = 0.07 0.20 0.45 0.85 1.4 2.4 3.1 GeV2

1.2 < W2 < 1.9 GeV2

1.9 < W2 < 2.5 GeV2

W

2 = M

2N = M

2N +

(1 xB)Q2

xB C.S. Armstrong et al, hep-ph/0104055

Solid curve: Large Q2

“Δ”

“S11”

=2xB

1 +q1 + 4M2

px2B/Q

2


19

Fq/p(x2)

γ*T,L

π–

p

x1 → 1

x2=?

N*

Fq/(x1)

π–

p

γ*T,L(q)k

x2

x1

Compare Fq/π(x1) measured at high s, thus large MX, with the valueimplied by data at the same x1 at low energy, where MX = O(MN*).

This may be done at several (fixed) values of x2 = q+/p+

The system X should be fully inclusive, and q⊥ may be integrated over.

M

2X = (1 x2)

s(1 x1) + x1M

2p

q

2?

Is duality independent of the γ* polarization?

Bloom-Gilman duality in DY?Oleg Teryaev


20Duality is the key to hadron dynamics

Paul Hoyer IU 2015

3

s channel r esonancesR s t channel poles α j t

RA R s , t ≈

jA j s , t

R(s)j=

j(t)

Rs =

t

= =

Igi (1962), Dolen, Horn, Schmidt (1968)

“finite energy sum rules”

σπ+ p − σπ

− p

s-channel resonances

t-channel “Regge” poles

Duality in hadron-hadron scattering

4

W. Melnitchouk (2010)https://www.jlab.org/conferences/HiX2010/program.html

Lectures byD. Horn,…

Paul Hoyer IU 2015

11Duality in e+e– hadrons

ECM GeV Ezhela et al, hep-ph/0312114

10-1

1

10

10 2

0.5 1 1.5 2 2.5 3

3-loop pQCDNaive quark model

Sum of exclusivemeasurements

Inclusivemeasurements

R

ρ

ω

φ

ρ

u, d, s

2

3

4

5

6

7

3 3.5 4 4.5 5

Mark IMark I + LGWMark IIPLUTODASPCrystal BallBES

J/ψψ(2S)

ψ3770

ψ4040

ψ4160

ψ4415

c

2

3

4

5

6

7

8

9.5 10 10.5 11

ARGUSCrystal BallCLEO

CLEO IICUSBDASP

DHHMLENAMD-1

(1S)(2S)

(3S)

(4S)b

S GeV

Figure 2: (continued) Threshold regions in the e+ e− hadroproduction: u, d, s -, c- and b-flavour onset.Note the consistency between the exclusive and inclusive da ta at

√2 2 GeV.

6Paul Hoyer IU 2015

14Lecture II (5/43)

Parton CascadesHump-backed plateau

First confronted withtheory in e+e− → h+X .

CDF (Tevatron)

pp → 2 jets

Charged hadron yield asa function of ln(1/x) fordifferent values of jethardness, versus (MLLA)

QCD prediction.

One free parameter –overall normalization(the number of final π’sper extra gluon)

ξ = ln(1/x)

dN/dξ

CDF

MJJ=390 GeV

Dokshitzer (Les Houches 2008)

x = Eh/EJ

pp → h(x) + X

Paul Hoyer IU 2015

28

Duality and the EMC Effect

Red = resonance region data

Blue, purple, green = deep inelastic data from SLAC, EMC

Medium modifications to the structure functions are the same in the resonance region as in the DIS

Cross-over can be studied with new data

C/D

Fe/D

Au/D

J. Arrington, et al., nucl-ex/0307012 Cynthia Keppel (2005)


21

Some observations

on the tantalizing simplicities of hadrons

which may guide our search


22Hadrons as qq and qqq states?

Why are the sea quark dof’s not manifest in hadron spectrum?

e– bound states, defined by the Dirac wave function, have Fockcomponents with e+e– pairs, yet only e– quantum numbers.

|i = N0 exp

h b†p

B1

pm

Dmqd†q

i|0i

|M 0i =Z

dp

22E

Zdx

hb

†p

u

†(p)eipx + d

p

v

†(p)eipxi

'(x)(x)

|i

Dirac wf.

E.g., in D=1+1 dimensions, the eigenstates of the Dirac Hamiltonian are:

The Dirac equation demonstrates how this is possible:

PH 1605.01532

23

V=2m

2 4 6 8 10 12 14

-0.5

-0.25

0.25

0.5

0.75

1

m/e = 2.5

ex

30 32 34 361 2 3 4

0.20.40.60.8

1

x

ex

Wf

m/e = 4.0

Dirac φ(x)Schrödinger ρ(x) Φ

1(x) f f

ρ(x) Schrödinger

(a) (b)

_

Wf NR region: e–

e+

V (x) = 12 |x|Dirac vs. Schrödinger wave functions:

The Schrödinger ρ(x) decreases exponentially at large |x| (Airy fn.)

Dirac is constant at large |x| (1F1 fn.): Describes e+ density|'(x)|2 + |(x)|2


24

Lattice lessons

S. Aoki, et al., Phys. Rev. Lett. 84 (2000) 238

VOLUME 84, NUMBER 2 P HY S I CA L R EV I EW LE T T ER S 10 JANUARY 2000

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

m (G

eV)

K inputφ inputexperimentK

K*

φ

NΛ

Σ

Ξ

∆

Σ*

Ξ*

Ω

FIG. 4. Final spectrum results compared to experiment.

while our results are smaller by a similar magnitude. Howthese differences arise is shown in Fig. 3, where the resultsof Ref. [2] are plotted by open triangles. For the nucleon,both the results from Butler et al. [2] and the MILC col-laboration [3] are consistent with experiment; our value issmaller by 7% !2.5s".We also calculate PS decay constants and quark masses

using tadpole-improved one-loop values for renormaliza-tion constants. For the PS decay constant we find fp !120.0!5.7" MeV and fK ! 138.8!4.4" MeV with mK asinput, which are smaller than experiment by 9% !2s" and13% !5s", respectively. Quark masses are determined bya combined linear continuum extrapolation of mVWI

q andmAWI

q , since the large difference of values from the twodefinitions at finite lattice spacings [13,15] vanishes towardthe continuum limit [5]. We obtain mu,d ! 4.57!18" MeVand ms ! 115.6!2.3" MeV (mK input) or 143.7(5.8) MeV(mf input) in the modified minimal subtraction !MS "scheme at m ! 2 GeV. A 20% disagreement between thetwo values for ms originates from the small meson hyper-fine splitting, and hence represents a quenching effect.In conclusion, we have found that the light hadron spec-

trum in quenched QCD systematically deviates from theexperimental spectrum when examined with an accuracybetter than the 10% level. In the course of our analyses wehave observed strong support for the presence of quenchedchiral singularities for pseudoscalar mesons. Whether vec-tor mesons and baryons also have such singularities, how-ever, remains as a problem for future investigations.We thank all of the members of the CP-PACS Project

with whom the CP-PACS computer has been developed.Valuable discussions with M. Golterman and S. Sharpe

TABLE II. Spectrum results. Deviation from experiment withits statistical significance is also given.

mK input mf inputExpt. Mass (GeV) Deviation Mass (GeV) Deviation

K 0.4977 · · · · · · 0.553(10) 11.2% 5.6sK! 0.8961 0.858(09) 24.2% 4.3s 0.889(03) 20.8% 2.3sf 1.0194 0.957(13) 26.1% 4.8s · · · · · ·N 0.9396 0.878(25) 26.6% 2.5s 0.878(25) 26.6% 2.5sL 1.1157 1.019(20) 28.6% 4.7s 1.060(13) 25.0% 4.1sS 1.1926 1.117(19) 26.4% 4.1s 1.176(11) 21.4% 1.5sJ 1.3149 1.201(17) 28.7% 6.8s 1.288(08) 22.0% 3.5sD 1.2320 1.257(35) 2.0% 0.7s 1.257(35) 2.0% 0.7sS! 1.3837 1.359(29) 21.8% 0.9s 1.388(24) 0.3% 0.2sJ! 1.5318 1.459(26) 24.7% 2.8s 1.517(16) 21.0% 0.9sV 1.6725 1.561(24) 26.7% 4.7s 1.647(10) 21.5% 2.6s

are gratefully acknowledged. This work is supportedin part by the Grants-in-Aid of Ministry of Education(No. 08NP0101 and No. 09304029). G. B., S. E., andK.N. are supported by JSPS. H. P. S. is supported byJSPS Research for Future Program.

[1] For recent reviews, see R. Kenway, Nucl. Phys. B (Proc.Suppl.) 73, 16 (1999); T. Yoshié, ibid. 63, 3 (1998).

[2] F. Butler et al., Nucl. Phys. B430, 179 (1994).[3] MILC Collaboration, C. Bernard et al., Phys. Rev. Lett.

81, 3087 (1998).[4] Y. Iwasaki, Nucl. Phys. B (Proc. Suppl.) 53, 1007 (1997);

T. Boku, K. Itakura, H. Nakamura, and K. Nakazawa,in Proceedings of ACM International Conference on Su-percomputing ’97 (Association for Computing Machinery,New York, 1997), p. 108.

[5] CP-PACS Collaboration, S. Aoki et al., Nucl. Phys. B(Proc. Suppl.) 60A, 14 (1998); ibid. 63, 161 (1998); ibid.73, 189 (1999); R. Burkhalter, ibid. 73, 3 (1999).

[6] MILC Collaboration, C. Bernard et al., Nucl. Phys. B(Proc. Suppl.) 60A, 3 (1998).

[7] S. R. Sharpe, Phys. Rev. D 46, 3146 (1992).[8] C.W. Bernard and M. F. L. Golterman, Phys. Rev. D 46,

853 (1992).[9] C.W. Bernard and M. F. L. Golterman, Nucl. Phys. B (Proc.

Suppl.) 30, 217 (1993).[10] M. Booth et al., Phys. Rev. D 55, 3092 (1997).[11] J. N. Labrenz and S. R. Sharpe, Phys. Rev. D 54, 4595

(1996).[12] M. Bochicchio et al., Nucl. Phys. B262, 331 (1985).[13] S. Itoh et al., Nucl. Phys. B274, 33 (1986).[14] M. Golterman and S. Sharpe (private communication).[15] T. Bhattacharya, R. Gupta, G. Kilcup, and S. Sharpe, Phys.

Rev. D 53, 6486 (1996).

241

Light hadron spectrum in quenched approximation

even a pure Coulomb potential, σ = 0, implies a non-vanishing σeff at finite t ≪ r.Of course, the symmetry of the Wilson loop under interchange of r and t also impliesthat no plateau in V (r, t) can be found, unless t ≫ r. For smeared Wilson loops, onewould still expect a similar 1/t2 approach (with a different coefficient) of σeff towardsthe asymptotic limit, while effective masses, V (r, t), will approach V (r) exponentiallyfast at any r.

4.7.2 The quenched potential

-4

-3

-2

-1

0

1

2

3

0.5 1 1.5 2 2.5 3

[V(r)

-V(r 0

)] r 0

r/r0

β = 6.0β = 6.2β = 6.4Cornell

Figure 4.2: The quenched Wilson action SU(3) potential, normalised to V (r0) = 0.

In Figure 4.2, we display the quenched potential, obtained at three different β valuesin units of r0 ≈ 0.5 fm from the data of Refs. [173, 29]. The lattice spacings, determinedfrom r0, correspond to a ≈ 0.094 fm, 0.069 fm and 0.051 fm, respectively. The curverepresents the Cornell parametrisation with e = 0.295. At small distances the datapoints lie somewhat above the curve, indicating a weakening of the effective couplingand, therefore, asymptotic freedom. We will discuss this observation later. All datapoints for r > 4a collapse onto a universal curve, indicating that for β ≥ 6.0 the scalingregion is effectively reached for the static potential. Moreover, continuum rotationalsymmetry is restored: in addition to on-axis separations, many off-axis distances of thesources have been realised and the corresponding data points are well parameterised bythe Cornell fit for r > 0.6 r0. Prior to comparison between the potential at various β,the additive self-energy contribution, associated with the static sources, that divergesin the continuum limit has been removed. This is achieved by the parametrisation-independent normalisation of the data to V (r0) = 0.

42

Gunnar S. Bali, Phys.Rept. 343 (2001) 1

Confinement and CSBwithout quark loops.

Heavy quark potential agreeswith quarkonium phenomenology


25

Similarity of atomic and hadronic spectra

V (r) =

rV (r) = c r 4

3

s

rPQED: PQCD?

Adapted from presentation by J. Ritman (2005)

3D2

2900

3100

3300

3500

3700

3900

4100

η c(3590)

ηc(2980)

hc(3525)

ψ (3097)

ψ (3686)

ψ (3770)

ψ (4040)

χ 0(3415)

χ 1(3510)χ 2(3556)

3D1

3D3

1D2

3P2(~ 3940)

3P1(~ 3880)

3P0(~ 3800)

(~ 3800)

1 fmC C

~ 600 meV

-1000

-3000

-5000

-7000

11S0

13S1

21S0

23S1

21P1

23P2

23P1

23P0

031S

0 31D2

33D2

33D1

33D2

Ionization energy33S

1

e+ e-0.1 nm

Binding energy

[meV]

Mass [MeV]

DD Threshold

8!10-4 eV

10-4 eV

´´´

´´´

´

117 MeV

Positronium Charmonium

“The J/ψ is the Hydrogen atom of QCD”

V (r) = c r 4

3

s

rV (r) = ↵

r

QED QCD


26

The Schrödinger equation from QED

h r2

2me ↵

|x|

i(x) = Eb (x)

QM I: The Schrödinger equation for atoms is postulated:

QED: Schrödinger wf is O(α∞) and not unique starting point. It is distinguished by being of lowest order in ℏ

In the Born approximation QED is expanded around the classical photonfield generated by the electric charges of the electron and proton.

Higher order perturbative corrections are calculated using NRQED.

The Born approximation of QCD: Analytic insights into hadrons and duality.

PH 1605.01532

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