Table Of ContentThe Joint Physics Analysis Center Website
Vincent Mathieu1,a)
1CenterforExplorationofEnergyandMatter,IndianaUniversity,Bloomington,IN47403
a)[email protected]
6
1 URL:http://www.indiana.edu/∼jpac/index.html
0
2
Abstract.TheJointPhysicsAnalysisCenterisacollaborationbetweentheoristsandexperimentalistsworkinginhadronicphysics.
n Inordertofacilitatetheexchangeofinformationbetweenthedifferentactorsinhadronspectroscopy,wecreatedaninteractive
a website.Inthisnote,Isummarizethefirstprojectsavailableonthewebsite.
J
8
INTRODUCTION
]
h
p With the 12 GeV upgrade of the Thomas Jefferson National Laboratory (JLab), a new hall was created and a new
- detector, GlueX [1], was build. GlueX’s primary goal is the search of exotic mesons via real photo-production of a
p
fixedhydrogentarget.ThisprogramcomplementsthecurrenthadronspectroscopyprogramoftheCOMPASSdetector
e
h [2].Bothfacilitiesuseshighenergybeam(photonsat9GeVforGlueXandpionsat190GeVforCOMPASS)that
[ fragmentsintomultiplemesons.Revealingthespectrumofshort-livedresonancesdecayingintothesemesonsrequires
arobustamplitudeanalysis[3].Tothisend,parametrizationofthereactionshavetobeproposedandfittedtothedata.
1
Tofacilitatethiscollaborativeeffortsbetweentheoristsandexperimentalists,theJoinPhysicsAnalysisCenter(JPAC)
v
wascreated.Aftertwoyearsofactivities,severalpaperswerepublishedbytheJPACcollaborationanditsmembers.
1
5 Thetopicscoveredincludethephotoproductionofone[4]ortwomesons[5]athighenergies,elasticscatteringwith
7 pion[6]andkaon[7,8]andthethree-bodydecaysofheavy[9,10]andlight[11,12]mesons.Manyprojectshaveled
1 todeliverables(suchascodes)andaplatform,aninteractivewebpage[13],wascreatedtoexchangesmaterialamong
0 thecommunity.InthisproceedingIsummarizetheprojectsandtheirfeaturesavailableonlineviatheJPACwebsite.
.
1
0
6 NEUTRALPIONPHOTOPRODUCTIONATHIGHENERGIES
1
v: Thereactionγp→π0pathighenergiesiscontrolledbytheexchangeofReggeonswithnegativechargeconjugation.
i Thedominantexchanges,havingthelargerintercept,arethevector-likeReggeonsωandρhavingisospinI = 0and
X
I = 1 respectively. Sub-dominant exchanges are the axial-vector Reggeons h and b, whose influences show in the
r beamasymmetry.
a
The parametrization of the helicity amplitudes are efficiently done using a the Chew-Goldberger-Low-Nambu
(CGLN)scalaramplitudes[14].Theamplitudesforthereactionγp→π0pinitscenter-of-massframe(thes−channel
frame)isdecomposedinthefollowingway:
(cid:32) (cid:33)
1 i
As (s,t)=u¯(p ,µ )γ Fµν(k,λ ) γ γ A (s,t)+2q p A (s,t)+q γ A (s,t)+ ε γαqβ A (s,t) u(p ,µ ).
µ4,µ2µ1 4 4 5 1 2 µ ν 1 µ ν 2 µ ν 3 2 αβµν 4 2 2
(1)
ThephotonfieldstrengthisFµν(k,λ )=(cid:15)µ(k,λ )kν−kµ(cid:15)ν(k,λ )andtheCGLNamplitudesarescalarfunctionofthe
1 1 1
Mandelstamvariables=(k+p )2andt=(k−q)2.TheconventionforthemomentumaredisplayedonFigure1.
2
FromtheknowledgeofthefourscalarfunctionsA(s,t)allobservablescanbecomputed.Thecodeistherefore
i
organizedindifferentblocks.Themodelisspecifiedinaroutinecomputingthescalaramplitudes.Thehelicityampli-
tudesinthe s−channelarecomputedfromthescalaramplitudesandtheobservablesarecalculatedfromthehelicity
FIGURE1.Left:Conventionfortheparticlemomenta.Right:ModelcomparedtoAndersonetaldata[15].
TABLE1.γπ0→ pp¯(t−channel)quantum
numbersoftheCGLNscalaramplitudes.
JPC I =0 I =1
A (1,3,5,...)−− ω ρ
1
A (1,3,5,...)+− h b
2
A (2,4,6,...)−− − −
3
A (1,3,5,...)−− ω ρ
4
amplitudes.Thelattertwofunctionsarecommontoeveryparametrizationandareinherenttothereaction.Theuser
needsonlytospecifythescalaramplitudestochangethemodel.
SofarthemodelavailableonlineconcernsthehighenergydomainE > 4GeVwherethedominanceofRegge
γ
poles is manifest [4]. The scalar amplitudes have good quantum numbers in the t−channel, the center-of-mass of
crossed channel reaction γπ0 → pp¯, cf. Table 1. It is then easy to parametrized the scalar amplitudes at high en-
ergies. The energy dependence and the phase are automatically predicted by the Regge theory and the remaining
t−dependenceoftheresiduesaretakenfromone-particle-exchangemodel.
A Fortran code producing the differential cross section is available for download and for simulations on the
website. In addition the interested user will find aC code for the amplitudes and the AmpTools class associated. A
Mathematicapackage,usedtoperformedthefit,isalsoavailable.
THREE-BODYDECAYSOFLIGHTMESONS
η → 3π
From the theoretical point of view η → 3π decays are of interest because of isospin violation. These decays are
dominatedbytheintrinsicisospinbreakingeffectsinQuantumChromo-Dynamics(QCD)aselectromagneticeffects
areexpectedtobesmall.Consequently,thedecaywidthforη → 3πisexpectedtobeproportionaltothelightquark
massdifferenceandthedecayamplitudeisoftenexpressedintermsofthequantity,1/Q2definedby
1 m2−m2
= d u . (2)
Q2 m2−(m +m )2/4
s d u
IntheReference[11],wedeterminedQbymatchingaparametrizationoftheDalitzdistributionη→π+π−π0obtained
byWASA@COSY[16]withtheNext-to-Leading-Order(NLO)expansionofChiralPerturbationTheory(ChPT)[17].
Since the available phase space for this decay is small, the distribution will be described by only the lowest partial
FIGURE2.Dalitzplotsinreducedvariablesforthethreepionsdecaysofeta,omegaandphimesons.
wavesineachchannels
L(cid:88)max=12L+1(cid:32)2 (cid:33)
A(s,t,u)= P (z )[a (s)−a (s)]+P (z)[a (t)+a (t)]−P (z )[a (u)−a (u)] . (3)
L s 0L 2L L t 1L 2L L u 1L 2L
2 3
L=0
The first index of the amplitudes a is the isospin, and z , z, z are the cosine of the scattering angles in the s, t,
IL s t u
u channels respectively. Unitarity in the 3 channels imposes conditions on the amplitudes a written in the form a
IL
coupled channel integral equations for the functions a . Indeed one sees easily that the partial waves, obtained by
IL
projectionwithP (z )fortheLwaveinthes−channelforinstance,involvesalltheamplitudesa .Thisisbecauseof
L s IL
thenon-orthogonalitybetweenLegendrepolynomialswithdifferentarguments(P (z )andP (z)forinstance).The
L s L(cid:48) t
contributionfromcrossed-channelsarecalledthe“3bodyinteractions”.Themodelwithoutthe3bodyinteractionsis
denotedthe“2body”approximationandcorrespondstothestandardisobarmodel.Asinputweusedtwo-pionscatter-
ingamplitudesfromtheanalysisof[18].Theparametersofthefitarethesubtractionconstantsforeachcontributing
partialwave,seetheReference[11]forthedetails.TheDalitzplotdistributionfittedtotheWASA@COSYdata[16]
ispresentedonFigure2.TheDalitzplotisexpressedusingthedimensionlessreducedvariables(x,y)
√
3(t−u) 3(M2/3+µ2−s)
x= , y= , (4)
2M(M−3µ) 2M(M−3µ)
withMthemassofthedecayingparticles,theηmeson,andµthepionmass.
TheChPTexpressionforthedecayη→π+π−π0isgivenby
1 m2(m2 −m2)(cid:32) 2 (cid:33)
A(s,t,u)=− K√ K π M (s)− M (s)+M (t)+M (u)+(s−u)M (t)+(s−t)M (u) . (5)
Q2 3 3m2F2 0 3 2 2 2 1 1
π π
The NNLO expressions of the isospin amplitudes M can be found in [17] and F = 92.3 MeV is the pion decay
I π
constant.WematchedEquation(3)and(5)attheAdlerzeros=4/3m2 todeterminetheQvalue.Weobtained
π
Q=21.4±0.4. (6)
Theneutraldecayη→π0π0π0canbeparametrizedsimilarly.Thecodesforthechargedandneutralmodesofthe
ηdecaysareavailablefordownloadandforsimulationsonline.Bothcodesproducestheamplitudesforthe2body
and3bodymodels(withorwithoutcrossed-channelre-scattering)atgiven(x,y)specifiedbytheuser.
ω,φ → 3π
Thenextlightmesondecayingintothreepionsarethevectormesons.Wecontinuedouranalysisofthreepionswith
the ω and φ decays into in Reference [12]. In this case the spin of the vector meson is factorized using the Lorentz
andisospindecomposition
(cid:18)−i (cid:19)
Aabc(s,t,u)=iε (cid:15)µ(p ,λ)pαpβpγ (cid:15)abc A(s,t,u). (7)
λ µαβγ V 1 2 3 2
Thescalaramplitudes A(s,t,u)isthenparametrizedwithatruncatedpartialwaveexpansioninallthreechannelsas
inEquations(3).ButwekeptonlytheP−wavesincethehigherwaves J = 3,5,...areexpectedtobeinsignifiantin
theωandφdecaysintothreepions:
A(s,t,u)= P(cid:48)(z )F(s)+P(cid:48)(z)F(t)+P(cid:48)(z )F(u). (8)
1 s 1 t 1 u
Theamplitudesatisfylinearintegralequationsasintheηdecay
1 [F(s+i(cid:15))−F(s−i(cid:15))]=(cid:16)1−4µ2/s(cid:17)1/2t∗(s)(cid:32)F(s)+ 3(cid:90) 1(1−z2)F(t[s,z ])dz (cid:33). (9)
2i 2 s s s
−1
Inthisunitarityequation,t(s)representtheP−waveelasticππscattering.Asintheηdecaywetooktheππphase-shift
parametrizationfromtheReference[18].ThestrategiestosolveEquation(9)aredetailedintheReference[12].We
found that the 3 body effects, the second term in Equation (9), are negligible. The model predictions for the Dalitz
distribution (normalized at the center of the Dalitz plot in the reduced variables) for the ω and φ are presented in
Figure2.
The Fortran code solving the integral equation and producing the amplitude for ω or φ decay in the 2 body
or3bodyapproximationareavailablefordownloadandforsimulationsonthededicatedwebpage.AMathematica
packageisalsoavailable.
COUPLEDCHANNELMODELFOR K¯N SCATTERING
0 πΣ ππΛKN πΣ* ηΛ K*N physical axis 0 πΛ πΣ KNππΣπΣ* πΛ*K∆ηΣ K*N physical axis
100 Λ(1520)Λ(1690)Λ(1710) Λ(1810) Λ(1890) Σ(1670) Σ(1915)
200 Λ(1600) Λ(1820) Λ(20Λ2(02)100) 100 Σ(1560)
Λ(1670) Σ(1770) Σ(1775) Σ(2000)
Λ(1405) Λ(1830)
V) 300 Λ(2050) V) 200 Σ(2030)
e Λ(2000) e
(MΓp765400000000 GDDSPPFF0000000011335577 Λ(2110) (MΓp430000 GDDSPPFF1111111111335577 Σ(2070)
800 500
1200 1400 1600 1800 2000 2200 2400 1200 1400 1600 1800 2000 2200 2400
Mp (MeV) Mp (MeV)
FIGURE3.SpectrumoftheΛ(I=0)andΣ(I=1)baryonsfromReference[7].
IntheReference[7]wepresentedaunitarymultichannelmodelfor K¯N scatteringintheresonanceregionthat
fulfills unitarity. Several coupled channels, indicated in the publication, were considered in the fitting procedure. In
theJPACwebpage,theobservablesandpartialwavesforthefollowingchannelscanbecomputed
K−p → K−p, K¯0n,
→ π−Σ+, π+Σ−, π0Σ0,
→ π0Λ. (10)
Allobservables,differentialcrosssectiondσ/dz ,polarizationobservable Pandtotalcrosssectionσ,areexpressed
s
intermsofthespin-non-flip f(s,z )andthespin-flipg(s,z )amplitudeswiththerelations
s s
dσ(s,z )= 1 (cid:104)|f(s,z )|2+|g(s,z )|2(cid:105), Pdσ(s,z )= 2 Im(cid:2)f(s,z )g∗(s,z )(cid:3), σ(s)=(cid:90) 1 dσ(s,z )dz (11)
dz s q2 s s dz s q2 s s dz s s
s s −1 s
Foragivenchannel(thechannelindexisomitted)theamplitudesadmitapartialwaveexpansion
(cid:88)∞
f(s,zs) = [((cid:96)+1)R(cid:96)+(s)+(cid:96)R(cid:96)−(s)]P(cid:96)(zs), (12)
(cid:96)=0
(cid:88)∞ (cid:113)
g(s,zs) = [R(cid:96)+(s)−R(cid:96)−(s)] 1−z2sP(cid:48)(cid:96)(zs). (13)
(cid:96)=1
Inagivenmeson-baryonchannel(cid:96)labelstherelativeorbitalangularmomentumandthetotalangularmomentumis
givenbyJ =(cid:96)±1/2.Foradetailedrelation,inallchannels,betweentheorbitalmomentumandthepartialwaveswe
referthereadertotheReference[7].
For a given orbital angular momentum (cid:96), we first remove the phase space factor with the introduction of a
diagonalmatrixC (s)
(cid:96)
R (s)=[C (s)]1/2T (s)[C (s)]1/2. (14)
(cid:96) (cid:96) (cid:96) (cid:96)
Thentheinverseofthereducedamplitudessatisfiesasimpleunitarityequations,ImT−1(s) = −iρ(s,(cid:96)),withρ(s,(cid:96))
(cid:96)
beingtheChew-Mandelstamfunction.Wecanthereforeuseareal K−matrixtoparametrizethereducedamplitudes
as
T (s)=(cid:104)K−1(s)−iρ(s,(cid:96))(cid:105)−1. (15)
(cid:96)
TheK−matrixisthesumoftheresonancecontributionsandaempiricalbackgroundterm.Eachwaveisparametrized
and fitted independently. The detailed procedure is described in the Reference [7]. Finally the partial waves are an-
alyticallycontinueontheunphysicalsheetandthepolepositionsareextracted.TheresultingspectrumforΛandΣ
baryonsisdisplayedonFigure3.
Thepartialwaves,binnedinenergysuppliedbytheuser,canbedowloadedonline.TheFortrancodeyielding
thepartialisalsoavailable.Thedifferentialcrosssection(togetherwiththepolarization)andthetotalcrosssection
havealsotheirdedicatedpages.Inthesamespiritofallpages,thecodesforproducingtheobservablescanbeboth
simulatedonlineanddowloaded.
CONCLUSIONS
After two years of activities, the JPAC has produced several papers concerning different hadronic reactions. The
emphasis is given to the appropriate constraints related to the physics of the reaction. In the resonance region, one
is interested extracting the properties of resonances (masses, widths and couplings) lying in the unphysical sheet.
Thereforeaparticularcareisgiventounitarity,whichcontrolstheanalyticcontinuation.Inathreebodydecaysofa
lightmeson,resonancesindifferentchannelsoverlap.Inthatcase,inadditiontounitarity,crossingsymmetryplays
an important role and leads to integral equation between the amplitudes. In a high energy scattering, the number of
relevantpartialwavesgrowsdrasticallyandpreventstheuseofthepartialwaveexpansion.Howeverbyananalytically
continuation of the partial waves in the complex angular momenta plane, one can trade the partial expansion by an
expansioninsingularitiesinangularmomentum,theReggepoles(andReggecuts).Theenergydependenceandthe
phasearepredictedandleadtosimpleparametrizationofhighenergydata.
Theexamplespresentedinthisnotesinvolvetheextractionofphysicalquantities(Qvalue,Λ,Σspectrum)and
are buildings blocks for more complicated reactions. We hope that the codes available will help other physicists to
describe other reactions such as the photoproduction of a pion, a η or a ω meson. These processes are the basic
reactions to be soon studies at the JLab facilities. We note also our K¯N amplitudes can be easily embedded in the
photoproductionofakaonpair(inwhichbaryonresonancesactsasabackgroundformesonproductions)butalsoin
threebodydecayssuchasΛ → J/ψK−p.
b
The JPAC website is a part of broad collaborative project in hadron spectroscopy. We hope that the material
available online will help the practitioners to quickly develop new parametrization for other reactions and update
easily the current models. As far as possible, we separated, in the codes and in the publication, the parametrization
relatedtothekinematics(fixedforagivenreaction)andthemodel-dependence.
The website will grow as new projects are published. Previous project will be updated as new models or new
codesarereadyforsharing.Weinvitethecommunitytobrowsethewebsiteregularlyandsendtheircommentstothe
JPACmembers.
ACKNOWLEDGMENTS
This material is based upon work supported in part by the U.S. Department of Energy, Office of Science, Office of
NuclearPhysicsundercontractDE-AC05-06OR23177.ThisworkwasalsosupportedinpartbytheU.S.Department
ofEnergyunderGrantNo.DE-FG0287ER40365,NationalScienceFoundationunderGrantPHY-1415459.
REFERENCES
[1] J.Dudeketal.,Eur.Phys.J.A48,187(2012)[arXiv:1208.1244[hep-ex]].
[2] P.Abbonetal.[COMPASSCollaboration],Nucl.Instrum.Meth.A577,455(2007)[hep-ex/0703049].
[3] M.Battaglierietal.,ActaPhys.Polon.B46,257(2015)[arXiv:1412.6393[hep-ph]].
[4] V.Mathieu,G.FoxandA.P.Szczepaniak,Phys.Rev.D92,no.7,074013(2015)[arXiv:1505.02321[hep-
ph]].
[5] M.Shi,I.V.Danilkin,C.Ferna´ndez-Ram´ırez,V.Mathieu,M.R.Pennington,D.SchottandA.P.Szczepa-
niak,Phys.Rev.D91,no.3,034007(2015)[arXiv:1411.6237[hep-ph]].
[6] V. Mathieu, I. V. Danilkin, C. Ferna´ndez-Ram´ırez, M. R. Pennington, D. Schott, A. P. Szczepaniak and
G.Fox,Phys.Rev.D92,no.7,074004(2015)[arXiv:1506.01764[hep-ph]].
[7] C.Ferna´ndez-Ram´ırez,I.V.Danilkin,D.M.Manley,V.MathieuandA.P.Szczepaniak,arXiv:1510.07065
[hep-ph].
[8] C.Ferna´ndez-Ram´ırez,I.V.Danilkin,V.MathieuandA.P.Szczepaniak,arXiv:1512.03136[hep-ph].
[9] A.P.SzczepaniakandM.R.Pennington,Phys.Lett.B737,283(2014).
[10] A.P.Szczepaniak,Phys.Lett.B747,410(2015)[arXiv:1501.01691[hep-ph]].
[11] P.Guo,I.V.Danilkin,D.Schott,C.Ferna´ndez-Ram´ırez,V.MathieuandA.P.Szczepaniak,Phys.Rev.D92,
no.5,054016(2015)[arXiv:1505.01715[hep-ph]].
[12] I.V.Danilkin,C.Ferna´ndez-Ram´ırez,P.Guo,V.Mathieu,D.Schott,M.ShiandA.P.Szczepaniak,Phys.
Rev.D91,no.9,094029(2015)[arXiv:1409.7708[hep-ph]].
[13] http://www.indiana.edu/~jpac/index.html
[14] G.F.Chew,M.L.Goldberger,F.E.LowandY.Nambu,Phys.Rev.106,1345(1957).
[15] R. L. Anderson, D. Gustavson, J. R. Johnson, I. Overman, D. Ritson, B. H. Wiik and D. Worcester, Phys.
Rev.D4,1937(1971).
[16] M.Bashkanovetal.,Phys.Rev.C76,048201(2007)[arXiv:0708.2014[nucl-ex]].
[17] J.BijnensandK.Ghorbani,JHEP0711,030(2007)[arXiv:0709.0230[hep-ph]].
[18] R.Garcia-Martin,R.Kaminski,J.R.Pelaez,J.RuizdeElviraandF.J.Yndurain,Phys.Rev.D83,074004
(2011)[arXiv:1102.2183[hep-ph]].
