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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
Paul Hoyer ECT* 8 November 2017
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)
Paul Hoyer ECT* 8 November 2017
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–
Paul Hoyer ECT* 8 November 2017
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.
Paul Hoyer ECT* 8 November 2017
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.
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
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.
PRL 109, 112001 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending
14 SEPTEMBER 2012
112001-2
L
Ee = 5.75 GeV
γ*T dominates DVMP at Ee ≲ 6 GeV2
Paul Hoyer ECT* 8 November 2017
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–
Paul Hoyer ECT* 8 November 2017
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.
Paul Hoyer ECT* 8 November 2017
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
Paul Hoyer ECT* 8 November 2017
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
Paul Hoyer ECT* 8 November 2017
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|
Paul Hoyer ECT* 8 November 2017
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
Paul Hoyer ECT* 8 November 2017
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
Paul Hoyer ECT* 8 November 2017
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*
Paul Hoyer ECT* 8 November 2017
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.
Paul Hoyer ECT* 8 November 2017
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
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onF
rase
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IUM
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ent
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okyo
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itut
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ent
ofP
hysi
csan
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Am
ster
dam
,The
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herl
ands
31A
ndrz
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ltan
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itut
efo
rN
ucle
arSt
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saw
,Po
land
32Ye
reva
nP
hysi
csIn
stit
ute,
3750
36Ye
reva
n,A
rmen
ia(R
ecei
ved
30Se
ptem
ber
2002
;pu
blis
hed
6Fe
brua
ry20
03)
Exc
lusi
veco
here
ntan
din
cohe
rent
elec
trop
rodu
ctio
nof
the!0
mes
onfr
om1H
and
14N
targ
ets
has
been
stud
ied
atth
eH
ER
ME
Sex
peri
men
tas
afu
ncti
onof
cohe
renc
ele
ngth
(lc)
,cor
resp
ondi
ngto
the
life
tim
eof
hadr
onic
fluct
uati
ons
ofth
evi
rtua
lph
oton
,an
dsq
uare
dfo
ur-m
omen
tum
ofth
evi
rtua
lph
oton
(!Q
2).
The
rati
oof
14N
to1H
cros
sse
ctio
nspe
rnu
cleo
n,ca
lled
nucl
ear
tran
spar
ency
,w
asfo
und
toin
crea
se(d
ecre
ase)
wit
hin
crea
sing
l cfo
rcoh
eren
t(in
cohe
rent
)!0
elec
trop
rodu
ctio
n.Fo
rfixe
dl c
,ari
seof
nucl
ear
tran
spar
ency
wit
hQ
2is
obse
rved
for
both
cohe
rent
and
inco
here
nt!0
prod
ucti
on,
whi
chis
inag
reem
ent
wit
hth
eore
tica
lca
lcul
atio
nsof
colo
rtr
ansp
aren
cy.
DO
I:10
.110
3/Ph
ysR
evL
ett.9
0.05
2501
PAC
Snu
mbe
rs:
24.8
5.+p
,13
.60.
Le,
14.4
0.C
s,25
.30.
Rw
One
ofth
efu
ndam
enta
lpr
edic
tion
sof
QC
Dis
the
exis
tenc
eof
aph
enom
enon
call
edco
lor
tran
spar
ency
(CT
),w
hose
char
acte
rist
icfe
atur
eis
that
,at
suffi
cien
tly
high
squa
red
four
-mom
entu
mtr
ansf
er"!
Q2#
toa
had-
ron,
the
init
ial
stat
ean
dfin
alst
ate
inte
ract
ions
ofth
atha
dron
trav
ersi
nga
nucl
ear
med
ium
vani
sh[1
–7].
The
idea
isth
atth
edo
min
ant
ampl
itud
esfo
rex
clus
ive
reac
-ti
ons
athi
ghQ
2in
volv
eha
dron
sof
redu
ced
tran
sver
sesi
ze,
and
that
thes
esm
all
colo
r-si
ngle
tob
ject
sor
smal
lsi
zeco
nfigu
rati
ons
(SSC
)ha
vere
duce
din
tera
ctio
nsw
ith
hadr
ons
inth
esu
rrou
ndin
gnu
clea
rm
ediu
m.M
oreo
ver,
itis
assu
med
that
thes
eSS
Cre
mai
nsm
all
long
enou
ghto
trav
erse
the
nucl
eus.
Seve
rale
xper
imen
tsin
sear
chof
CT
[8–1
1]ha
vebe
enca
rrie
dou
tove
rth
ela
st20
year
s.A
ltho
ugh
none
ofth
ese
expe
rim
ents
isin
confl
ictw
ith
CT,
only
afe
wha
vesh
own
evid
ence
for
it.T
hefir
stev
iden
cefo
rC
Tca
me
from
quas
ifre
ech
arge
exch
ange
scat
teri
ngda
taof
40GeV
=cne
gativ
epi
ons
onca
rbon
[12]
assu
gges
ted
inR
ef.
[13]
.Fu
rthe
rev
iden
cefo
rC
Tco
mes
from
Ferm
ilab
expe
ri-
men
tE
791
onth
eA
depe
nden
ceof
cohe
rent
diff
ract
ive
diss
ocia
tion
of500GeV
=cpi
ons
into
di-j
ets
[14]
.T
his
resu
ltsh
ows
apl
atin
umto
carb
oncr
oss
sect
ion
abou
t10
tim
esla
rger
than
expe
cted
ifso
ftpr
oces
ses
wou
lddo
min
ate,
whi
chis
qual
itat
ivel
yco
nsis
tent
wit
hth
eore
ti-
cal
calc
ulat
ions
ofC
Tef
fect
s[1
5,16
].A
lso
expe
rim
ent
E66
5on
excl
usiv
ein
cohe
rent
!0
muo
prod
ucti
onfr
omnu
clei
[17]
give
san
indi
cati
onof
CT.
How
ever
,tha
tsig
nal
isof
inde
cisi
vest
atis
tica
lsi
gnifi
canc
e.In
this
Let
ter
we
repo
rtne
wev
iden
ceof
CT
inex
clus
ive
cohe
rent
and
inco
here
nt!0
elec
trop
rodu
ctio
nus
ing
ano
vel
anal
ysis
tech
niqu
e.W
hen
sear
chin
gfo
rC
T,a
com
mon
lyus
edob
serv
able
isth
enu
clea
rtr
ansp
aren
cyT$
"A="A"
p#,
whi
chis
the
rati
oof
the
nucl
earc
ross
sect
ion
pern
ucle
onto
that
onth
epr
oton
.Fo
rdi
ffra
ctiv
ein
cohe
rent
reac
tion
son
anu
clea
rta
rget
,w
here
the
nucl
eus
isex
cite
dor
brea
ksup
,C
Tpr
edic
tsth
atas
Q2
beco
mes
larg
e,T
appr
oach
esun
ity,
inde
pend
ento
fA.F
orco
here
ntre
acti
ons,
whe
reth
ein
ter-
acti
onle
aves
the
who
lenu
cleu
sin
tact
init
sgr
ound
stat
e,th
esa
me
obse
rvab
leis
used
insp
iteof
the
fact
that
itca
n
nolo
nger
beas
soci
ated
dire
ctly
wit
hth
epr
obab
ilit
yof
esca
peof
the
hadr
onfr
omth
enu
cleu
s.In
orde
rto
stud
yC
Tfo
rex
clus
ive
elec
trop
rodu
ctio
nof
!0
mes
ons,
one
has
tose
lect
asa
mpl
eof
!0
mes
ons
prod
uced
byph
oton
sw
ith
larg
eQ
2.
Inth
ese
proc
esse
s,th
eha
dron
icst
ruct
ure
ofa
high
ener
gyvi
rtua
lph
oton
[18]
inth
efo
rmof
aq! qq
pair
has
atr
ansv
erse
size
r ?%
1=Q
[3].
The
q! qq
fluct
uati
onof
the
virt
ual
phot
onca
npr
opag
ate
over
adi
stan
cel c
know
nas
the
cohe
renc
ele
ngth
.It
isgi
ven
[4,1
8]by
l c$
2#Q
2&M
2 q! qq,
whe
re#
isth
e
virt
ual
phot
onen
ergy
andM
q! qq
isth
ein
vari
ant
mas
sof
theq! qq
pair.
Thi
sSS
Cca
npr
opag
ate
thro
ugh
the
nucl
ear
med
ium
wit
hli
ttle
inte
ract
ion.
Aft
erth
eq! qq
pair
ispu
ton
-she
ll,i
tw
ille
volv
eto
ano
rmal
-siz
e!0
mes
onov
era
dist
ance
l fca
lled
the
form
atio
nle
ngth
.It
isa
gove
rnin
gsc
ale
for
the
CT
effe
ctan
dis
give
n[3
]by
l f$
2#m
2 V0!
m2 V
,
whe
rem
Vis
the
mas
sof
the!0
mes
onin
the
grou
ndst
ate
andm
V0
the
mas
sof
its
first
radi
alex
cita
tion
.T
heph
enom
ena
dete
rmin
ing
nucl
ear
tran
spar
ency
form
anin
tric
ate
mix
ture
ofco
here
nce
and
form
atio
nle
ngth
effe
cts.
The
yha
vea
diff
eren
tap
pear
ance
for
co-
here
ntan
din
cohe
rent
!0
prod
ucti
on.F
orin
cohe
rent
pro-
duct
ion,
the
prob
abil
ity
for
theq! qq
pair
toin
tera
ctw
ith
the
nucl
ear
med
ium
incr
ease
sw
ithl c
unti
llc
exce
eds
the
nucl
ear
size
[19,
20].
For
valu
esof
thel c
smal
ler
than
the
nucl
eus,
this
cohe
renc
ele
ngth
effe
ct[2
0]ca
nm
imic
theQ
2de
pend
ence
ofth
enu
clea
rtr
ansp
aren
cypr
edic
ted
byC
T.Fo
rco
here
nt!0
prod
ucti
on,
inco
ntra
st,
the
nu-
clea
rfo
rmfa
ctor
supp
ress
esth
eap
pare
ntnu
clea
rtr
ans-
pare
ncy.
Smal
ll c
corr
espo
nds
toa
larg
elo
ngit
udin
alm
omen
tum
tran
sfer
(qc%
1=l c
),w
here
the
form
fact
oris
smal
l.H
ence
,Tde
crea
ses
wit
hQ
2in
cohe
rent
prod
uc-
tion
.Thi
sbeh
avio
rcan
notm
imic
CT,
buta
lso
inth
isca
se,
the
cohe
renc
ele
ngth
effe
cts
can
sign
ifica
ntly
mod
ify
the
Q2
depe
nden
ce,t
hus
obsc
urin
gth
ecl
ean
obse
rvat
ion
ofa
CT
effe
ct.I
nor
der
todi
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 .
Paul Hoyer ECT* 8 November 2017
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*
Paul Hoyer ECT* 8 November 2017
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
Paul Hoyer ECT* 8 November 2017
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
Paul Hoyer ECT* 8 November 2017
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
Paul Hoyer ECT* 8 November 2017
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)
Paul Hoyer ECT* 8 November 2017
21
Some observations
on the tantalizing simplicities of hadrons
which may guide our search
Paul Hoyer ECT* 8 November 2017
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
Paul Hoyer ECT* 8 November 2017
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
Paul Hoyer ECT* 8 November 2017
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
Paul Hoyer ECT* 8 November 2017
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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