Towards a microscopic understanding of nucleon polarizabilities

1.2 MB·

by  Gernot Eichmann

Few-Body Systems manuscript No.
(will be inserted by the editor)
Gernot Eichmann
Towards a microscopic understanding of nucleon
polarizabilities
6
Received: date / Accepted: date
1
0
2
Abstract We outline a microscopic framework to calculate nucleon Compton scattering from the
n
level of quarks and gluons within the covariant Faddeev approach. We explain the connection with
a
hadronic expansions of the Compton scattering amplitude and discuss the obstacles in maintaining
J
electromagnetic gauge invariance. Finally we give preliminary results for the nucleon polarizabilities.
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1 Keywords Nucleon Compton scattering Faddeev equations Dyson-Schwinger approach
· · ·

]
h
p 1 Introduction
-
p
There is much ongoing interest in the precision determination of the nucleon’s polarizabilities; see [1]
e
h for a recent review. The electric polarizability α and magnetic polarizability β tell us how the nucleon
[ responds to an external electromagnetic field, with current PDG values α = 11.2(4) 10−4 fm3 and
β = 2.5(4) 10−4 fm3 for the proton [2]. The polarizabilities are proportional to ×the volume and
1 ×
their smallness indicates that the proton is a rigid object due to the strong binding of its constituents.
v
Whereas α+β is constrained by a sum rule, the small value for β is commonly believed to be due to
4
5 a cancellation between the nucleon ‘quark core’ and the interaction with its pion cloud.
1 The polarizabilities are encoded in the nucleon Compton scattering (CS) amplitude Nγ∗ Nγ∗
→
4 which has many applications also beyond polarizabilities. The integrated CS amplitude is relevant for
0 two-photoncorrectionstonucleonformfactors[3]andperhapsalsofortheprotonradiuspuzzle[1].So
1. far,ourknowledgeoftheCSamplitudeisrestrictedtoafewkinematiclimitsincludingthe(generalized)
0 polarizabilities in real and virtual CS [4], the nucleon structure functions in the forward limit, and
6 deeply virtual CS (DVCS) from where generalized parton distributions are extracted [5]. In addition,
1 the crossed process pp γγ will be measured by PANDA.
v: Whilelatticecalcula→tionsforpolarizabilitiesareunderway(see[1]forreferences),themaintheoret-
i icaltoolstoaddressCSare‘hadronic’descriptionssuchaschiralperturbationtheory,whichprovidesa
X
systematicexpansionoftheCSamplitudeatlowenergies[6],anddispersionrelationswithadirectlink
r to experimentaldata [7;8]. On theother hand,handbag dominance inDVCSis well establishedand a
a
key ingredient to factorization theorems. Is it then possible to connect these two facets by a common,
underlying approach at the level of quarks and gluons that reproduces all established features, from
hadronic poles to the handbag picture? In the following we will briefly outline such an approach and
present first calculated results for the proton polarizabilities α and β.
This work is supported by the German Science Foundation DFG under project number DFG TR-16.
G. Eichmann
Institut fu¨r Theoretische Physik, Justus-Liebig-Universita¨t Giessen
Heinrich-Buff-Ring 16, 35392 Giessen, Germany
Tel.: [phone]
Fax: [phone]
E-mail: [email protected]
2
= + + +
+ + +
(cid:1836)(cid:2078) (cid:945)
Fig. 1 Three-quark Faddeev equation (top) and electromagnetic current matrix element (bottom).
2 The covariant Faddeev approach
Our tool of choice is the covariant three-body Faddeev approach established in [9]. Its basic equations
areillustratedinFig.1.TheFaddeevequationdeterminesthenucleonmassandbound-stateamplitude
(its ‘wave function’) by summing up all possible two- and three-body interactions between dressed
quarks. The electromagnetic current matrix element couples the photon to all microscopic ingredients
and thereby satisfies electromagnetic gauge invariance.
To solve the Faddeev equation one needs to specify its
input. Three-body interactions have been neglected so far,
and most studies have employed a rainbow-ladder trunca-
1.9
tion where the two-body kernel is given by a dressed gluon
exchange. The dressed quark propagator is solved from its 1.8
Dyson-Schwingerequationandtheresultingquarkmassfunc-
1.7 [(cid:1833)(cid:1857)(cid:1848)]
tion becomes momentum-dependent; it describes the transi-
tion from the input current-quark mass at large momenta to 1.6
a nonperturbative, dressed ‘constituent quark’ mass of a few
1.5
hundred MeV in the infrared. In general, any truncation must (cid:2007)(cid:903)
preservechiralsymmetrytoensureamasslesspioninthechiral 1.4
limit via the analogous Bethe-Salpeter equation; see e.g. [11]
1.3
for recent advances in this area.
1.2
Whereas the applicability of rainbow-ladder in the light-
meson sector is mainly limited to pseudoscalar and vector (cid:1986) 1.1
mesons, baryons fare much better: the approach reproduces
1.0
theoctetanddecupletgroundstatemasseswithin5 10%[12].
−
Fig. 2 shows results for the ρ meson, nucleon and ∆ masses 0.9
as functions of m2 (which is a−lso calculated) compared to lat-
π (cid:1840) 0.8 (cid:2004)
tice data and experiment. The only input is the quark-gluon
interaction for the two-body kernel whose model dependence 0.7
0.0 0.1 0.2 0.3 0.4 0.5
is given by the bands. In particular, once the model scale is (cid:2025)
set to reproduce the pion decay constant, there are no further
parameters or approximations and all subsequent results are Fig. 2 ρ−meson(cid:1865)[1(cid:2140)0(cid:1165)] [,(cid:1833)n(cid:1857)u(cid:1848)c(cid:1165)l]eon and ∆
predictions. masses[9]calculatedfromtheirBethe-
Salpeter and Faddeev equations. Stars
Apartfrommassspectra,arangeofformfactorshavebeen
arePDGvaluesandsymbolswitherror
calculatedaswellwithinthissetup.Amongthemarenucleon, bars are lattice data (see [9] for refer-
∆ and hyperon electromagnetic form factors, the N ∆γ ences).
→
transition, and nucleon axial form factors [13]. All these cases
exhibitgoodoverallagreementwithexperimentaldata(where
available)andalsolatticeresultsatlargerpionmasses,withdiscrepanciesatlowQ2 wherepion-cloud
effectsbecomeimportant.Whilethethree-bodyFaddeevapproachdoesnotdependonexplicitdiquark
degrees of freedom, it is conceptually close to the quark-diquark framework which typically yields
similar results and thereby establishes the presence of strong diquark correlations inside baryons [14].
An advantage is that the approach is not limited to two- and three-body systems: using the very same
buildingblocks,ithasbeenrecently alsoappliedtotetraquarksandthemuon g-2problem[15].Given
thebodyofresultssofaritisdesirabletogoastepfurtherandask:whatcanwelearnaboutCompton
scattering from such a microscopic perspective?
3
10
(cid:2015)(cid:917) 8
(cid:1858)(cid:1182)(cid:1181)
6 (cid:1858)(cid:1182)
(cid:1858)(cid:1182)(cid:1182)
(cid:1986) (cid:1858)(cid:1182)(cid:1185)
FW P VCS 4 (cid:1858)(cid:1182)(cid:1183)
D G (cid:1858)(cid:1182)(cid:1184)
2
(cid:2001)
(cid:1843)’ (cid:1843)
RCS 0
-2
-4
0.0 0.1 0.2 0.3 0.4 0.5
(cid:1868)
(cid:2033)
Fig. 3 Left: Kinematics and p(cid:2015)h(cid:918)ase space in Compton scattering. Right: Dominant Co(cid:2015)m(cid:917)pton form factors
corresponding to the residue of the nucleon Born terms after removing the common pole factor. The bands
contain the full kinematic dependence on all four variables inside the cone.
3 Compton scattering
The nucleon CS amplitude depends on three independent momenta (see Fig. 3): the average nucleon
momentump,theaveragephotonmomentumΣ =(Q+Q)/2,andthemomentumtransfer∆=Q Q.
(cid:48) (cid:48)
−
The process is described by four Lorentz-invariant kinematic variables which we define as
Q2+Q2 Q Q Q2 Q2 p Σ
(cid:48) (cid:48) (cid:48)
η = , η = · , ω = − , λ= · , (1)
+ 2m2 − m2 2m2 m2
where m is the nucleon mass. The kinematic phase space in the variables η ,η ,ω is illustrated in
+
Fig. 3. The spacelike region that is integrated over in nucleon-lepton scatt{ering−form}s the interior of
a cone around the η axis. Its apex is where the static polarizabilities are defined, with momentum-
+
dependent extensions to real CS (η = ω = 0), the doubly-virtual forward limit (η = η , ω = 0),
+ +
and virtual CS (η =ω) including the generalized polarizabilities at η =0. −
+
−
Hadronic vs. microscopic decomposition. At the hadronic level the CS amplitude is given by the sum
of Born terms, which are determined by the nucleon form factors, and a one-particle-irreducible (1PI)
structure part that carries the dynamics and encodes the polarizabilities, see Fig. 4. The latter con-
tains s/u channel nucleon resonances beyond the nucleon Born terms (including the ∆, Roper, etc.),
−
t channel meson exchanges (pion, scalar, axialvector, ...), and pion loops, with well-established low-
−
energy expansions in chiral effective field theory. This is usually viewed as the ‘correct’ description at
low energies, whereas the handbag picture is interpreted as the ‘correct’ one at large photon virtual-
ities. Hence again the question: is there a common underlying description at the quark level that is
valid in all kinematic regions and encompasses both approaches?
InanalogytotheformfactordiagramsinFig.1onecanderiveaclosedexpressionfortheCSampli-
tudeatthequarklevel[16;17].Thetopologiesthatsurviveinarainbow-laddertruncation(apartfrom
permutations and symmetrizations) are collected in the second row of Fig. 4. Ambiguities stemming
fromintermediateoffshellhadronsneverariseherebecausehadronicdegreesoffreedomdonotappear
explicitly. Instead, the diagrams reproduce the onshell hadron pole contributions:
– Diagram(a)dependsonthethree-quarkscatteringmatrixwhichcontainsallpossiblebaryonpoles,
so it reproduces the nucleon Born terms as well as all s/u channel resonances.
−
– Diagram (b) contains the quark two-photon (quark Compton) vertex, which has an analogous
decomposition into quark Born terms and a 1PI part. The Born terms provide the handbag con-
tributions. The 1PI part features a quark-antiquark scattering matrix that contains all possible
t channel meson poles and thereby reproduces the meson exchanges in the first row.
−
Neither the handbag nor the cat’s-ears contributions from diagram (c) have a direct analogue in the
hadronicexpansion wheretheyare ratherabsorbedinto counterterms. Viceversa, thediagramsin the
bottom do not contain the microscopic representation of pion loops because those only enter beyond
rainbow-ladder.Inanycase,thesumofallgraphsintheboxsatisfieselectromagneticgaugeinvariance
3
B. Kinematicsanddefinitions
The nucleon Compton amplitude Γµν(p,Q(cid:31),Q) de- (cid:2015)(cid:917)
pendsonthreeindependentmomenta. Wewillalterna-
tively use the two sets {p,Q,Q(cid:31)} and {p,Σ,∆} which (cid:2028)’ (cid:1872)
arerelatedvia
(cid:2028)
p = 1(p +p ), (cid:2026)
Σ = 212(Qi+Qf(cid:31)), ∆=Q −Q(cid:31)=pf−pi, (8)
F VCS
withtheinverserelations W P
D G
p =p ∆, Q =Σ+∆,
i − 2 2 (9)
pf =p+∆2 , Q(cid:31) =Σ−∆2 . R C S
Withtheconstraintsp2=p2 = m2 theComptonam-
i f −
plitude depends on four Lorentz invariants. We work
withthedimensionlessvariables 3
B. Kinematicsanddefinitionsη+= Q22+m2Q(cid:31)2, η−= Qm·2Q(cid:31), ω= Q22−m2Q(cid:31)2, (10)
(cid:2033)
peTndhseonnutchleroeneinCdoemppentodnentammpolmituendtea.ΓWµνe(pow,rQi,llv(cid:31),iaQclet)evrendraes--aλ, = pm·2Σ = pm·2Q = pm·Q2(cid:31), (cid:2015)(cid:917) 4 FIG.2: Co(cid:2015)m(cid:918)ptonscatteringphasespaceinthevariablesη+,
atirveelryelauΣtpseed==thvie1212a((tpQwio++sQpeft(cid:31)))s,,{p,∆Q,=QQ(cid:31)}−anQd(cid:31)={pp,fΣ−,∆p(cid:31)i},QQwQ(cid:31)h22(iQc8(cid:30)h)(cid:31)==ΣΣ22(cid:2028)+’∆∆422 ±=(cid:2026)Σm·2∆η =,mF2(η+±ω), (11)(cid:1872) aCtCaηhn−rooyedmmaVcontppofChdttneooteSωhnnVeis(ssC(cid:2028)accctSlaoahttnteetltreieemnsrrcpaiiinonattniggcvwtee(al(hlViiRykneC:eCrseSτSrte),t)hghτeoile(cid:30)oin,vfngoeηtersh−twnhe,oeaanorprtardlaltitishnzl,ieeeimσndη,τtit−ep(cid:30)ωgo=.ar)altaax0Ttrtie.ishzd=Taeabonhi0vindeleitt(rebviF.reoiiWrsuoRtrnaueDdroaae)-fll
withtheinveprse=replatio∆n,s Q =Σ+∆s,othatth·eCompt−onf4ormfacto−rsinWDEq.(3)arediGmPen- defined(GP,τ(cid:30)=0a n dη−=0). + . . .
pfi =p−+∆22 , Q(cid:31) =Σ−∆22ηs.ionaleressefvu(en9nc)tuionndsercip(hηo+t,oηn−,cωro,sλs)in.gTahnedvcahriaarbgleescoηn+juagnad- R C S = Born terms Ns* /rue-scohnaannnceels t-mcheasonnnsel pion loops
WtfstHΣp∆ηpwssmsetWodw(cid:29)owioaoipx−rrluttooηiher.ei,meoatT=tietcnnr+mt==ttetrhceujvTaeehhwrht(cid:31)h,relaeende,irΣdie=eacchetcwfcl>o4ηesiλteiah∆attte/nLeQsrQikm+cQthhiuv0cnekse√Q,ooeott=2vdehatef(cid:31),neMv22−whn·2nur2odσewrRrre2ΣaedecQve2esnrireo−,pii,t(cid:30)nnaie+weηo2mtanssamg-ctσe(cid:31)Crrfbat)−ZnprdskYhiti.tn2usazeλo==i,2Qtlisahooe>hσtaeλ-nidon[hnEt,nawnnm1ioaa(cid:27)bs=(cid:31)nrssdnΣΣ2=nto]udtnattailrss:pev0ηsooeh,tdec22nkdωiorwσatp,ts+nennslcaimimdnΣh+−2oirsnptile·pYta,d=iseen(scg2+ηae22ωΣrhapeηwfsdηo−∆∆cs−enopasof+a,o−a44eruηteηonuts2in1t=rcr22frrv,+b=w−2rooerenteieqηaa=t<mcvs±locrn=.ieZu+ha−nr−cpo2temLttQemiartcenidthZp,acΣdfo·moiiemh=ηhnaηavωren2fpbp2rQ·dn−lωteo+t2ce·<e2tli,2QshisoΣ‘=Tetsnz(cid:29)n∆ttiλsη‘t(cid:26)a,os1s=geoati(cid:31)Ttaptio)−iesr·d,inn,zoln−b.ato=esst’s1∆,ts(cid:29)apg,lchnosZfmeoieTsmi+lmeot(nsesng·b,sωpalaasuu−hm2vrQi=bsao2endnIk2ηcraEaee=1e,cnutod-he(cid:31)md+trr(Yqemhet(cid:27)Imv’eivη<,sE,a.IasiecQa−a+a2votofneqh=scrη(aoiblamYvC2nftn.i3al+a±η2n2rt.laresttWtfsHΣp∆mset)dwrp−eωor(h(cid:29)(cid:29)−emid.braouEwioap<gxae?F−ss−rumnttωeohae·rl.emimleieboatqT=?t2QeV.etcimnarnmt=ri=Σ)terCpgr(cid:28)Wc1ηeujl.sTs)eehew,rtth,recaeea,(cid:31)e.nt−Tpdo2,mΣti(io2eeoscchbesηnwfcl>oooea4ηdr12irlmλia,nu∆att,./nL+eeanernsikdrm+ct0i.ihiuv0cnjbks(vm√len,wnooesopte)=2uhoate,eg((((ne(MTea2−watjnnsrr2otσow1t111w1rRbr.greΣaveerdrmfengovhhesireor−,31042tniecaoso,tienai)eweη2tans.ndakgaoeee-rtt)))))σ------,.rrfbt)−ZprdskYhii.t2sazoitlahoe>hσtaeλ-i[hnE,nawnn1ioaa(cid:27)bsn=nto]udtnattlrs:adcCtCaAupηωFTfTv0ηsoehtdecnkdωrhon−rwσate,pls+eInoosolhhaimiydoeΣhfiG2ndirWasncppmmemYta,deita=isneerscs•••••g2+ca.e2oeω:rapentwoppdηeeo−oficserfnh2atiaddEas,tto−aernPRQFGQVQuηtτesηrchnts:enoo1srtrpcrfrres+qbaw−ωo2h(eeonneoen(cid:31)(cid:31)2teieaeCiq=teaVastG<2µ.ncvslreorcri.ileZaeun+hnans−(c=soxo2ste(cid:2015)tmsaCtttw(cPe=caa=−pclartcecemchnidthe41uZitrdtoS,liaieQlh=o(cid:918)alηrhnaaaaηhvrmQ1trenpCsiipapdtcn0ntτη−0eolennωtez+tor2oeea)<2tltitusthlr(cid:31)eis+(cid:30)eoiΣ2‘nTondzadn(cid:29)efelto2ti⇒mns⇒wl‘tsl(cid:26)a,1=grierCoatimboTeanaptcFipcoaz=iet<si=·d,inynieodllnorcbonaattoihteset’si1oi∆0stspηl(cid:29)naigagηngc,mlchnoarcsdvZfv4ieηom0webηs+tl+el.+eototasta(sesleagi5(−ab,sp+ly(sltaaoshziuiun−utl4Vmi⇒rrp◦iRi=pbeoyst=aroket=ednInekdηncriag4<e+atCe:1:ee,scnuaoCtrio-hsemdc+ththhrrnYieem:oηbωet(cid:27)ImS’lτηeωohn∆iSvr<ssEea,ωtηanI−asilne−)t+,nactt.η)ea2Rhvot,ovefe+q=gscrghµhηsaoτi{bea+laemosYv,e=nηvfn.iilel+ee=a,ηs2nrt(cid:30)σ.oi.lpnrtzei=at−f,vipeωcr(hf(cid:29)−nei=dnt,.rhnauEapg0o<raee?F−ssηTnateωet·aiτe)rm[ibs=lli0bh−qE?tt}V.otit1m.aawnrηnηihsΣ(cid:31)Cghr(cid:28)1ηeuli.s)u=t,eo]inqt+−e⇒eea,la.=a.aλ−Tpno2emωt(inoosrip.sebsrnsotanr1t2ralrrmog0du=sp2ld,k.=e(aensdri0altd.FdtTpsQtiQd(fttraatGisgcfiiiη4p.1atbieQ(vleinisnhnsn+ooospne)hzrahheeoil,oran+m(egpma(((percT2uωesitj0ehimsfre{eetgaa2µi(cid:31)R1tw11vrrmbns.eσveoeeenprr):nreognfu2d.pgoηcτhhc12fit=e342tmmmeocosoaiηne,u=τΠpeps)trsiitid.Cn−,2ao.eIp(eee=eghprt)))xnrPt---−ei,.pc(cid:30)ωnireeo=gτsnsaVsatlbogsηoeelSotoλa=ii.e<a4e-fvµt(cid:31)tteraIgirpend)−aftasl}xeChotail)siatc0aηνea,acax=ncauitr0,e:inaTtisηrrto+i;tnaoSliritttce.wHlemcAupωTfTΠtrilcisnωd+te2nthvnnzrixrllsdrho0=foli)vh=t4plsmpe−erToaaihhnetlzoeeeaeaa.:ddn.tvWroaeca=auapauurbµmkoseeieesnabhtohaiisdersinsda•••••0ωvrietnt:r.rp(cid:2033)amνderscpnobrl,e0flusfib=shes0srrfrsitirbsitηiareEmtos(tt(pNPRQFGQVQroieτob.sia,oora.ceeqh?vivalti−siFrtoapor.blsQuwrsenqeacooa7urnewniesorion(cid:31)(cid:31)2uiadeifa=eteWaseewrsmsba2µ.ntuoRl(rrorutTtalntniawernsna)rcedr=sgtxut1itrnηtnsatew(cauchl=xencDea=−h-phh:llseohe5erco+e41ysduhiraloraaihsiott=pQdlslraaaieeeetkees)r)mQ1neardti-fCs,lliivtaoincmh0tηu0eoencnzooroa)o(cch-2nf.ataiuuncertainties.lsw(cid:31)h+ea2nohdaddnfparcpll2ata⇒ie⇒.twllenlwgaidC)mmbooeneaciFcsshdTzhr=nt<= yedndlortwoexperiments,orresolvedbyadjustingthenormalizationofsomedatasetswithintheassumedtδqrccoaaroihmfitiaeoioiopCcrtepηleaegηantohcmµugqardlv4inηm0obthxηiojc+lr.+hptotcsno‘ifnlacrosssectiondatatodetermineiftheobserveddiscrepancycouldbeexplainedbyproblemswithoneSνi5nea−eoud+ytosntogovznieieuui4iru⇒arp◦tpreot=rrcohnn=e dto+nanisagpa4ir<N+aantra:klslmapoEtitsextractionsof.Thus,thefirststepwasacarefulexaminationofthesignificantlychangethehighGQnccdihdtthhursianoriue:aobωm2i-salcηωob*hn∆rsaaukobeapωiηaoilstibciln ttg+gncttn.ηeopaR,uecvveimayhavebeenduetosomeexperimentalerrorinoneormoreofthecrosssectionmeasurementsthatrcn+grephµenseQep{niea+esaes,evrddbnηvei=teoi(cid:1846)oe=p,srhxσ.rtotziso=slda−tfieiasctinlaµ.ItwasthereforearguedthattheobserveddifferencewouldincreaseinimportancewithincreasingQlk=ntti,.aopnardpotornosqaeleTa tu2ωeltsupiτoQib[b=Bnlcc0snm−Eaml}ouorusgFdt1arnηηh/daEM(cid:31),whichdifferencebetweensmall-andlarge-anglemeasurementscouldyieldlargecorrectionsto/GGroplhisaus=to;o]rinq+−ae⇒redleutaν=e.aλb[elnbepωosintar.eaot1srhnatcm.dea−nrbog0aa’wasmall,angle-dependentcorrectiontothecrosssection,leadingtothepossibilitythatasystematic=sig2dcQkk=og(mgiki8docieη4nps .1dTeQyeisrtinne+ot(icararn+m](imsp2eωs2iesdC0Esdif{ne,yieldsonlyGsuggestingthattheproblemwasrelatedtothecrosssectionmeasurements.AthighQ,wtheg(cid:31)Roatteeevht):ons2dr..(ηc2τh2efi=ehmpoaep=motxpisuettChe−c,2tee(iicxvcimbetweentheresultsofdifferentRosenbluthextractions[11,31,32,33,34],asillustratedinFig.3,oneTirmoh=τlntauaVathe)xmosηieoStλsyec<uns(cid:31)ttc,rnemgd−rssh}cmhCuipseaii)ico)understandingandresolvingthisdiscrepancy.Itwasnotedearly[16]thattherewassignificantscatterhηmes,cm=cecirtswu,l:iqeηttioffin+aolSletpuirlapwalsiω+om2oByuixrelnreiffi0iogm)mentsoftheprotonelectromagneticformfactorsshowninFig.2ledtosignificantactivityaimedat4nol(mmde−loinvnsncneed.:be.toat=asusuebsesetcaehiacg,)padctiωntontirThestrikingdifferencebetweenRosenbluth[30]andtheearlypolarizationtransfer[16,18]measure-tvclr dtaoactibtssr0sioair(ahfηolaeneaoebuioneflu.m.ateeai−nprtooc.rrennnitcaepBnboepostui)fttdntnuia-(ettsrmecn3.1Verificationofthediscrepancyta:yhiS,caSo mut1tloemsichlvntttngao(hhee5anri-sdadwehiahreeditdnceeyeus)nandstnastuellnrehnepihcpes,y.gescsosafataduriteeooisfireieeal3Experimentalobservablesandmeasurementshfshgnagelntcenlotls;tnenreanuriaalratatsdhe mantemdiaishaehrntindrdtrhntteobxsaamlpibedhis iitnczaroim-neonnnihahppsebeaepawtstqdanqsagpseeCrahnastrsobnvu(,-u.liesoiponvariousobservables.ergodlo)/anbthiedoaotaaDtnn:hrGtmgue−r)ahnfntidrocdiscrepancy,andthendescribetheoreticaleffortstocomputeTPEcorrectionsandassesstheirimpactotlrusa rcayi−d kyadadpcxoeipaonlnnncsfihrseeuietnioaofthecrosssection.Inthefollowingsectionswediscussexperimentaleffortstobetterunderstandthelnc1uoataocmC-oitsralghgtvPstqrebntnlh)nnnnr[ryeoaassociatedwithtwo-photonexchange,whichcanleadtoadditionalangular(andthus)dependenceεoaabiusaI1a-eceiifnrogcmnibScteausmtet7eneinsuiyisurementshavemostlyfocusedonimprovedtreatmentsofradiativecorrections,particularlythosecarhinucnsnts]slotdbpossakt.uuhuerevtnmutnittosactivity,boththeoreticallyandexperimentally,overthepastdecade.Attemptstoreconcilethemea-umicwnmlpoalasotoiorwtsocecnfergcritinsnEMtThediscrepancybetweentheLTandPTmeasurementsofhasstimulatedconsiderableG/Gnsanribonrw-hr(cQiupaooiscltsogagnmeegnaavrnnht)fos2strongviolationofthescalingbehavior(seealsoRefs.[1,2,28,29]).Qmr eegewteFdesc2clasteewcrderaoraaaihdeastshipabwhichisinviaLTorRosenbluthseparations,showinganapproximatelylineardecreaseofwithRQtxntirdrntnsee2neaemi‘aecriifighsedcgdrzndqathigexrandJeffersonLab[27].Theresults,illustratedinFig.2,areinstrikingcontrasttotheratioobtained ivnhanpetaetnrasuarbmameett.slgatyawnkiInaddition,therehavebeencomplementarymeasurementsusingpolarizedtargetsatMIT-Bates[26]aivuahnacbinrltammtTgsboutseuiirutehcetalietEM upto=85GeV.transfer(PT)techniquehasbeenusedtoaccuratelydeterminetheratio/GQ.Gnsehsearusoddsmlyehsme22wfstueiesineolmox.apnawInaseriesofrecentexperimentsatJeffersonLab[16,17,18,19,20,21,22,23,24,25],thepolarizationet(edpi.bcr,ngatlratncTet[mliaaheThradhn1iartnrp-dtattlhocirnt7ohecahcgatuomcou)opers]hueusleen;dl.eeymssfrvriaiatleoiaipauebafitersnntlsnegalvifcrlmitrar,yuuFigureadaptedfromRef.[12].cunzrehgoseisgtliqt(oecacctrnanbdo.t+iai1u(LT)separation[11](squares)andpolarizationtransfermeasurements[16,18](circles).tpollnvtoylceole euerhW2ii.adee,rffrptot ooogo8a.reomolFigure2:RatioofprotonelectrictomagneticformfactorsasextractedusingRosenbluthaeyiws nnnyaakceerr.)ts-fl,. aηptpwssWowioirrlevoηiiie,Tttniten+tthhuvadlhrh(cid:31)hlyeirde=satetcesltehhteaQusQoQhΣeeetpQnsntvdef(cid:31)ev2eh·u2udiee2dentcQe2==cnrpit(cid:30)nh+ovmslmhvBcpee(cid:31)Carneptien1212uofeeλ==r,2Q.iaiso((nnedonstmtpQe=(cid:31)nrssd==wΣΣ2iiaiKsnpeCro,o22+i+orepdtppnnciomnl+−onetsila·pQm−+pen(sepeη2tΣhfηfst∆∆(cid:31)ipm−eops))ofs+o∆∆4422notu2it=n22,,av,o=d{ror,,sηat=mnp±en=i−rpLn,cQmi∆p,atcΣsfao·Qmmaωr2fbm2rQ·maoc,=·e2l,2Q=etnspn∆QλQoηQoss=Q(cid:31)dtlm)−iri,(cid:31)z(cid:31)n−.t=}s,duepg−m==iTminmedann·ωatfinQeh2vQaΣΣ2n2dnEae=.(cid:31)td(cid:31)rΓ(iq+−h=vti{η,Waµ.eciQap+oνnh∆∆r(p22,eC2n(ti3a±f2apΣss)r,.−owm.b,−g,mωaQlie2Qel∆r)plpWs(cid:31)e,c(cid:31),it}a2oQηo,edl,n+tnwi)ejmwuh((raao11ngde((imncr1908naaehdk))))----- adCtCaηFhn−reIoo(cid:2028)yefiGdmma’nc.ontppeofh2ddttn:eooteωh(nnCVGeis(os(cid:2026)(cid:2015)sCPaccmctS,lao(cid:918)ahtptntτeetlttre(cid:30)ieeomns=rrncpaiiinonatt0sniggccvwteaFae(al(WhntlViiRyktneDdCe:eCrsreηSτiSrtn−e),t)hgghτeo=ile(cid:30)oipn,vfnhg0oeηtae)rsh−t(cid:2015).wnshe,eoeaa(cid:917)norprstardplaltGiatishnzl,Pcieeeimeσndη,τtiint−ep(cid:30)ωgo=(cid:1872).atr)alhtaRax0eTtrtie.Cishvzd=TaaeSarbonhVii0vin(cid:2033)adeleCitbt(rebvliF.reSeoiiWsrsuoRtrnη(cid:2028)aueD+droaa3e)-f,ll
ti−on,whereasλandωswitchsigns(seeEq.(??)below). ThefirstthreeconstraintsinEq.(12)entail
Πµν(Q)=Π(Q2)tµν +Π(Q2)δµν, tµν =A Bδµν BµAν. (2)
QQ AB · − WeworkwithEuclideanconventionsbutallrelationsbe-
ThetransversedressingfunctionΠ(Q2)isf(cid:101)reeofkinematicsingularitiesandzeros.Thegaugepartδµν tfowremenfaLcotroernsttzh-iantvwareiadnetriqvueainntiTtiaebs,lessuIc,hIIasanthdeVC,oamrepttohne −η+<η−<η+, ω2+η−2 <η+2 . (15)
sameinMinkowskispace. Thisisacircular45◦ coneinη+ direction,withη and
isthetensorthatweeliminatedinthefirstplace,soΠ(Q2)mustvanishduetogaugeinvariance.This The variables η , η and ω also admit a simple geo- ω asthexandy variables. Theoppositecorners−ofthe
+
is what happens in dimensional regularization, whereas a cutoff breaks gauge invariance and induces metricunderstandingo−fthephasespace,cf. Fig.2. The cone are spanned by the σ,t and τ,τ(cid:31) axes because
{ } { }
a quadratic divergence (only) in the gauge part. If(cid:101)we did not know about the decomposition (2) spacelikeregionthatweneedtointegrateoverinorderto fromEq.(11)wealsohave
and performed a transverse projection, Π(Q2) would pick up a 1/Q2 pole from the gauge part which extracttwo-photoncorrectionstoobservablesissubject
iinnvvaalriidaantceesitshbereoxkternacbtyiomnoorfeΠth(Qan2a=c0u)t.offT,hfeortriannsstvaenrcsee/bgyauagneinsecpoamraptleiotne cisalaclusloatcioonnv:eunliteimntaitfelgyauthgee tothte>c0o,nstσra>int0s, 1<Z<1, 1<Y <1 (12) τ = 4Qm22 = η+4+ω, τ(cid:31)= 4Qm(cid:31)22 = η+4−ω.
− −
sum of all gauge parts must vanish, but the partial result for Π(Q2) is still free of kinematic problems where t, σ, Z and Y are the ‘spacelike’ variables intro- Acrosssectionthroughtheplanesoffixedtleadstothe
and — ideally — not strongly affected by gauge artifacts. ducedinRef.[1]: upperpanelofFig.4inRef.[1].
∆2 Σ2 Wecanalsolocalizethevariouskinematiclimitsinthis
t= 4m2, σ= m2, Z=Σ·∆, Y =p·ΣT. (13) plot:
HΣe=re,Σa/√hΣat2)daennodttehseasunbosrcmriapltiz(cid:29)‘eTd’(cid:29)sfotaunr-dmsofomr(cid:29)eanttr(cid:28)uamns(vee.rgs.e, • RQ2ea=lQCo2m=p0tonηsca=ttωer=ing0.(RCS):
projectionwithrespecttothetotalmomentumtransfer (cid:31) ⇒ +
Virtual Compton scattering (VCS):
∆(cid:29). ThesevariablesarerelatedtotheonesinEq.(10)via • Q2=0 η =ω.
t= η+−η−, σ= η++η−, Z= ω , G(cid:31)eneral⇒ized+polarizabilities:
λ=2 Y ω2+η2 2 η2 1+ (cid:27)2η+2 −.η−2 (14) •• FQo(cid:31)µrw=a0rd⇒liηm+it=: ω∆,µη=−=0⇒λ=η+0.=η−,ω=0.
−2 (cid:27) −− +(cid:26) η+−η− • Polarizabilities: η+=η−=ω=λ=0.
3
B. Kinematicsanddefinitions
The nucleon Compton amplitude Γµν(p,Q(cid:31),Q) de- (cid:2015)(cid:917)
pendsonthreeindependentmomenta. Wewillalterna-
tively use the two sets {p,Q,Q(cid:31)} and {p,Σ,∆} which (cid:2028)’ (cid:1872)
arerelatedvia
(cid:2028)
p = 1(p +p ), (cid:2026)
Σ = 212(Qi+Qf(cid:31)), ∆=Q −Q(cid:31)=pf−pi, (8)
F VCS
withtheinverserelations W P
D G
p =p ∆, Q =Σ+∆,
i − 2 2 (9)
pf =p+∆2 , Q(cid:31) =Σ−∆2 . R C S
Withtheconstraintsp2=p2 = m2 theComptonam-
i f −
plitude depends on four Lorentz invariants. We work
withthedimensionlessvariables 3
B. Kinematicsanddefinitionsη+= Q22+m2Q(cid:31)2, η−= Qm·2Q(cid:31), ω= Q22−m2Q(cid:31)2, (10)
The nucleon Compton amplitude Γµν(p,Q(cid:31),Q) de-λ= pm·2Σ = pm·2Q = pm·Q2(cid:31), (cid:2015)(cid:917) (cid:2015)(cid:918) (cid:2033)
pendsonthreeindependentmomenta. Weowri,llviacletevrenras-a, FIG.2: Comptonscatteringphasespaceinthevariablesη+,
atirveelryelauΣtpseed==thvie1212a((tpQwio++sQpeft(cid:31)))s,,{p,∆Q,=QQ(cid:31)}−anQd(cid:31)={pp,fΣ−,∆p(cid:31)i},QQwQ(cid:31)h22(iQc8(cid:30)h)(cid:31)==ΣΣ22(cid:2028)+’∆∆422±=(cid:2026)Σm·2∆η =,mF2(η+±ω), (11)(cid:1872) aCtCaηhn−rooyedmmaVcontppofChdttneooteSωhnnVeis(ssC(cid:2028)accctSlaoahttnteetltreieemnsrrcpaiiinonattniggcvwtee(al(hlViiRykneC:eCrseSτSrte),t)hghτeoile(cid:30)oin,vfngoeηtersh−twnhe,oeaanorprtardlaltitishnzl,ieeeimσndη,τtit−ep(cid:30)ωgo=.ar)altaax0Ttrtie.ishzd=Taeabonhi0vindeleitt(rebviF.reoiiWrsuoRtrnaueDdroaae)-fll 5
withtheinveprse=replatio∆n,s Q =Σ+∆s,othatth·eCompt−onf4ormfacto−rsinWDEq.(3)arediGmPen- defined(GP,τ(cid:30)=0andη−=0).
tWtfsHΣp∆ηpwssmsetWodw(cid:29)owioaoipx−rrluttooηiher.ei,meoatT=tietcnnr+mt==ttetrhceujvTaeehhwrht(cid:31)h,relaeende,irΣdie=eacchetcwfcl>o4ηesiλteiah∆attte/nLeQsrQikm+cQthhiuv0cnekse√Q,ooeott=2vdehatef(cid:31),neMv22−whn·2nur2odσewrRrre2ΣaedecQve2esnrireo,pii,t(cid:30)nnaie+weηo2mtanssamg-ctσpe(cid:31)Crrfbat)−ZnprdskYhiti.tnusafzeλo==i,2Qtliisahooe>hσtaeλ-nidon[hnEt,nawnnm1ioaabs=(cid:31)nrssdn=ΣΣ2=nto]udtnattailrss:pev0ηsooeh,tdec22nkdωiorwσatp,tsp+nennslcaimimdnΣh+−2oirsnptile·pYta,d−+=iseen(scg2+ηae22ωΣrhapeηwfsdηo−∆∆cs−enopasof+a∆,o−a44eruη22teηonuts2in1t=rcr22frrv,+b=w2rooerenteie,qηaa=t<mcvs±locrn=.ieZu+ha−nrcpo2temLttQemiartcenidthZp,acΣdfo·moiiemh=ηhnaηavωren2fpbp2rQ·dn−lωteot2ce·<e2tli,2QshisoΣ‘=Tetsnz(cid:29)n∆ttiλsQη‘ta,os1s=geoati(cid:31)Ttaptio)−iesr·d,inn,zo(cid:31)ln−b.ato=esst’s1∆,ts(cid:29)apg,lchnosZfmeo=ieTsmi+lmeot(nsesng·b,sωpalaasuu−hm2vrQi=bsaΣo2endnIk2craEaee=1e,cnutod-he(cid:31)mdtrr(Yqe−mhet(cid:27)Imv’eivη<,sE,a.IasiecQaa+a2votofneqh=sc∆rη(aoib22lamYvC2nftn.i3al+a±2n2rt.laresttWtfsHΣp∆ηsmset)wdr.p−eωor(h(cid:29)(cid:29)emid.braouEwioaip<gxae?F−ss−rumnttωeoohae·rl.emimleieboatqT=?t2QeV.etcimnnarnmt=ri=Σ)terCpgr(cid:28)Wc1ηeujl.sTsa)eehew,rtth,relcaeea,(cid:31)e.nt−Tpdeo2,rmΣti(io2eeoscchbesηnwfcl>oooea4ηders12irlmλia,nu∆att,./nL+eeasnernsikdrm+ct0i.ihiuv0cnjbks(vem√len,wnooesopte)=2uhoatef,eg((((ne(MvTea2−watjnnsrur2otσow1t111w1rRbr.greΣavee(redrmfengovhhesnireor−,319042tniecaoso,tnienai)eweη2tans.ndakgaoeee-crtt))))))σ------,.rrfbt)−ZprdskYhiti.t2usazoitliahoe>hσtaeλ-ion[hnE,nawnn1ioaa(cid:27)bsndn=nto]udtnattlrss:adcCtCaAupηωFTfTev0ηsoehtdecnkdωrhon−rrwσate,pls+eInoosolchhaimiydoeΣhfiG2ndirWasncppmmiempYta,deita=isne(erscs••••g2+ca.e2oeω:rhapentηwoppdηeeo−oficserfnh2atiado+dEas,tto−aernRQFGQVQuηtτesηrchnts:enoo1srttrpcrfrres,+qbaw−ωo2h(eeoonneen(cid:31)(cid:31)2teieaeCiq=teaηVastG<2µ.ncvslreorcrni.ieZaeun+hna−ns−(c=soxo2ste(cid:2015)tmsCtttw(cPe=caa=−pclartcecemchnidthe41uZ,ictdtoS,liaieQlh=o(cid:918)alηrhnaaaaηhvrmQ1tωrrenpCsipapdtcn0ntτη−0eolennωteo+tor2oeea)<2tltit,usthlr(cid:31)eis+(cid:30)eoiΣ2‘nTosndzdn(cid:29)efelto2ti⇒λmns⇒wl‘tsl(cid:26)a,s1=grierCoatimoTeanaptcFipcoaz)i=iet<si=·d,inynieodllnnorcbon.aattohteset’si1oi∆0stspη(cid:29)naigagηgngc,mlchnoarcsdvZfv4eηom0webηTs+tl+el.+eotoasta(sesleagi5(−ab,sp+ly(asltaaoshzuiun−utlh4Vmi⇒rrp◦iRi=pboyst=arokent=ednInekdηncriag4e<e+atCe:1:ee,cnuaoCtriod-hsemdc+ththhrrnYieemoηbωet(cid:27)ImSv’lτηeωohn∆iSvr<ssEea,ωtηanI−asiclane−)t+,nactt.)ea2Rhvot,ovefe+qh=gscrghµhrηsaoτi{bealaemosYv,e=nηvfn.iiileal+ee=a,ηs2nrt(cid:30)σ.oi.lapnrtzei=at−f,vripeωcr(hf(cid:29)−neidnt,.brhnauEapg0o<rgaee?F−ssηTFwωaiabbssffaostnateωet·aiτel)rm[ibs=fllei0aobh−qhE?titit}V.oetit1m.npfaawnrηnihsΣoo(cid:31)C,ginthr(cid:28)1iηeuli.ss)ru=crt,seo]einqt+cea⇒egea,lat.dp=ia.aλtt−Tpno2emωtu(iλnooosucritop.sebosηrnshotalnrhh1t.2TTtralrrmo∆g0rdonusp2eld,lk.ff+c=e(aenbsdrin0altd.hiiiiη4p.oa1atjbieQ(v5leinirsnhnt+osnpnhhe)tzusl,orn+m(sdtceg(((pcT2itesaiatnjs0eusfre{µAeetgic(cid:31)R1tw11hmb.gesσihveveeeiirr):ohonf2dni.goηcτhhc2nfi=ss342tdomνecaosoriηe,PτataBs)teeieit.Cndn−,2aoe(eeeehcrt)))xn-t---−eei,.tnp(cid:30)ωi=gτlnnrVatcsrogtηaemooSsol=.etB<ait(cid:31)ttoiirrndda)r−sacMl}.Csharftai)mnaRaηat,cixcufno0,oeitee:i112Ttηrtn-Ib+tootaoSlibwd05m5050e.wcAupωTfTCirlsfirωn+tr2phvzriexrldtnour)µ=a4ua0plmbiee−TnoaaihhnehSoneasr:l.nm.taW(cid:2009)asic=a0pνrbftmntsoeieesunhtehtsmpiiersiesetd•••••0ωgviiietn:r(cid:2033)a( daeaolseionofibe+ses0rsordgitbwteηpriauEnst(vnPRQFGQVQerneτb.srf.nceerhnvildl−siFraatpxo.,rsaenqe aancdeooe7eideotoi(cid:31)(cid:31)2.ia(cid:2010)eif=teWassmursesaQFna2µ.rnuoRn(crrrrltdailtaenn=sxut1Ttrrenηtlavi tw(aaucgshmte=enc1Da=−0aiphh lezh5e+se4(cid:48)1sduirmoraae Eoiga.t(cid:4670)kteQcdlhlr2,aaaeeepPeas)lr)mQ1,r-rfCs,llii(cid:883)apce0ηro.l0eoennzQnoroa)ebsq2hauo(cid:882)uncertainties.ul(cid:31)it+eIs2nodadnfslnl2t⇒n4o⇒wlil.(cid:903)nvhneioCedsm)boealoFczp=(cid:896)et<=wvyidedlortwoexperiments,orresolvedbyadjustingthenormalizationofsomedatasetswithintheassumeduair(cstedoioh(cid:3)etioioaa.=n(cid:136)pηalaagη2onmerar(cid:143)cidtnv4iηm0bη+frlld.+ettogtrsrT0yl)nsgsacrosssectiondatatodetermineiftheobserveddiscrepancycouldbeexplainedbyproblemswithonei5l−so+yestozi.ir(cid:895)ust4ie⇒4.rt,p◦ptoeoat=rm=era(cid:4671)hawnaisgt14or<+at:hcs(cid:3)aaoEstWiextractionsof.Thus,thefirststepwasacarefulexaminationofthesignificantlychangethehighGQiuchhnthhnnolie:nonbηrpeωo2lsηωomhan∆(cid:2015)sr egaωηaeycuoDDDPlnsrt+n+ctc.ηeRer(cid:917),etvemayhavebeenduetosomeexperimentalerrorinoneormoreofthecrosssectionmeasurementsthat+fBrheµteesviD{seas+uiiio(lsu,eaηvasaiceshe=,suσo.setzepi=Gaovp−→fggsndiepcono0.ItwasthereforearguedthattheobserveddifferencewouldincreaseinimportancewithincreasingQ=t,i.tnaapnrmdrefrrtoT.rxain2ωec6tmyfhaaaiτ[bs=l0rtdE}onlet1)ffartηtηhsmmEMg(cid:31)eb,whichdifferencebetweensmall-andlarge-anglemeasurementscouldyieldlargecorrectionsto/GGrypiiu=t0t]irnqie+di−⇒eamlas(cid:20)=so.aλtnoeω hsaie.let.s((Canh.nlstatroogn0bbasmall,angle-dependentcorrectiontothecrosssection,leadingtothepossibilitythatasystematic=s2dcek=(leoip d(cid:124)(cid:18)Ainη4p.ne))iyt1reQHitisrnF+oft eCnrnn+m(rp2hω+lsedr0BmEf{h,yieldsonlyGsuggestingthattheproblemwasrelatedtothecrosssectionmeasurements.AthighQegs0(cid:31)lRloeomF)o:aoodt d2e.η.c2τfe2ofi=∆8me=nfstohptwCn−,t2erec(sii1xsmibetweentheresultsofdifferentRosenbluthextractions[11,31,32,33,34],asillustratedinFig.3,oi=4τvnfuVathηterhoeηeeshSλnree<(cid:31)ttaoennsqd+i−hcter}paoCtai)vadunderstandingandresolvingthisdiscrepancy.Itwasnotedearly[16]thattherewassignificantscatterηtmtr,cr=fctlu,µQ:im.erηtta+rediaoeSletuyhwlω+gαl2lkionTaixrl(cid:48)rpe0o)mentsoftheprotonelectromagneticformfactorsshowninFig.2ledtosignificantactivityaimedat4rllrm1:ea−apinesen,is.:nr.t.Ro=nael0hvnDelslyhpiamttdωldsfentrThestrikingdifferencebetweenRosenbluth[30]andtheearlypolarizationtransfer[16,18]measure-eceefeαpyhib-ss0stevprivηoo,eifoQm...reseνbiec−c.teoe1olrnnecgatffoidfarmna[lf(ahfi+jal3.1Verificationofthediscrepancyctapg4ssTenut1toeuhtn(cid:123)ehhuehae]aa5sdricviidghfferee(cid:122)ds)fg--ecmua0123uasaget21tfawtel,engo0elccar2lhwbst(cid:2010)..2103Experimentalobservablesandmeasurementsednrtentruiiieyn8eeoon loTinste gl(cid:4670)rgtunprnroµQg-(cid:883)tCvethhehsed.tcbνa(cid:882)r.uah(cid:48)neeStolQe(cid:903)Nprr(rlwTeesb(cid:896)bafiteCiso0dapo(cid:3)aaoa(cid:136)+i.ttahrrnn2(cid:143)retnFamonvariousobservables.aatsntsiketuhefitheniir-(cid:895)tF.snisdppsc(cid:4671)ndiscrepancy,andthendescribetheoreticaleffortstocomputeTPEcorrectionsandassesstheirimpacti,egee.Q(cid:3)tsesllhngolisaa.she)pidtofthecrosssection.Inthefollowingsectionswediscussexperimentaleffortstobetterunderstandthe2teoaof(cid:19)ruoiaira(cid:125)0fune-xnoftungR.et4dnrtioassociatedwithtwo-photonexchange,whichcanleadtoadditionalangular(andthus)dependenceεdanogdet’o(hFedeansche+eηmipef(cid:2015)oplsurementshavemostlyfocusedonimprovedtreatmentsofradiativecorrections,particularlythosetl,cie+selozg(cid:917)vlfeita.wegyldnene.,Cie1nactivity,boththeoreticallyandexperimentally,overthepastdecade.Attemptstoreconcilethemea-erpaxo[rxivη4rgh0tc3d1uazFo((cid:124)pno.tpa−6eEMtegThediscrepancybetweentheLTandPTmeasurementsofhasstimulatedconsiderableG/Get9g.esemla(cid:123).Fresrererrr.a,n;xnio(cid:122)hnreaois.ωarupstrongviolationofthescalingbehavior(seealsoRefs.[1,2,28,29]).Qti7svsmos(cid:125)mm.2)iaocnsa],aaziPwrrnaTtowhichisinviaLTorRosenbluthseparations,showinganapproximatelylineardecreaseofwithRQcaλaenrtsb2(ac0nens,eengb)bhi.buo8ucandJeffersonLab[27].Theresults,illustratedinFig.2,areinstrikingcontrasttotheratioobtainedstn(cid:21)ite)ebwaht(mehallhei3irsywlauInaddition,therehavebeencomplementarymeasurementsusingpolarizedtargetsatMIT-Bates[26]eieizttepes)amnvsivl(ieiaerllcntEMpseaupto=85GeV.transfer(PT)techniquehasbeenusedtoaccuratelydeterminetheratio/GQ.Gdrlm-eerpshbee22odbli1ityctum.al)rmuu0eInaseriesofrecentexperimentsatJeffersonLab[16,17,18,19,20,21,22,23,24,25],thepolarizationeeaaeuteaDd,nstahnmiosredntnidooIavasatsftsidarrnnnurtawfnncserdi[iaeieearvahs2srtngascwdcntwoicnnoaeiive]TIoihltvvoLcrngttmd.ctsanosetJdieCeoilThhfgeaimatdrtF(F,epldneafearieyohLntlcniffiotrFotanvFigureadaptedfromRef.[12].iiat,:lsbonhrggTpsenoiesu(nitieirf(doneroFouuodbrssuoo3auP)scomolsrsi(LT)separation[11](squares)andpolarizationtransfermeasurements[16,18](circles).snlrrndhoysRayisassrr)tttoT-feeoweh,cst,tnosbrseha)iFigure2:RatioofprotonelectrictomagneticformfactorsasextractedusingRosenbluthita2aeespotsvohdL:netpchetnaefeentahdraarrRrcivbptoenaboemehwatlotaftcncunhbeih[rotoyet.o2tideeilsn-ahqohen7ntptIvtbeusif]lnhecr..yoseeed[Qoaoef1txteTbmwlhpt1fhp2slpooeyaah]ceeerenscsrreeRronc(uaaifnitbbsaomeertnsoqcllxfeeeiedilto.nueosedcnohntnumn[hawge1heetslraoxt2lips,euebennnspL]oslsasc,.egge)Teerhtthmeieirdromls,JiatailmewaeciuvetnwpccnnoffsiiedattrotntdhenioolaarrogirtvaprccensPa(myehctaffoossledTunlnleoadmyacwrerg,raLatatmeeniiratnspoanzaeeelevdabpsstatllaFetyuiooriet[ssciror1amiocudgRcxe6duntfro.emeioh,esemtnm2rtfs1emteestmo,7arrnp.sepeam,tatuxa[neans1fr1otpdilatses8,nfeysduetcf,ee2irtisrT1nldrtio,oiiatm9innPefrh2odsm,gcseeE8nteiG2aaanr,eaapd0rritEtaclsk2a,ioeaosv/di9t2.allenureGaei]1nxor)grecrA,e.gMtefficrGzcr2mtuceooaett2oalrErihedcea,nrsotmtn/aree2stnetGtcsr3pdsa(sotta,aMotrisfsao2sgnut[tnR1nb4iedsumt6dtt,sieopnos,,2twtuagth5rltat1piesaeu]toht8srR,ahtcsee]retQoMuosthdQεnrs(sni2e)aIccecd2Ttciu=tniprdhleioblw-ecooraeenB8llslphiruetsaot.ralshi5eitarybcdt)nihienhetm.Gzedasdtarmehpeiiatsn[nVtoaib2eohecscai6l2dnneeeet-.]
−2 (cid:27) −− +(cid:26) η+−η− • Polarizabilities: η+=reηa−s=onωa=blλe=ra0n. ge of model parametrizations the transverse CFFs remain almost unchanged.
AnotherremarkablefeatureisvisibleinFig.3:thebandscontainthefull kinematicdependenceon
3 Experimental observables and measurements
allfourvariablesη ,η ,ω andλinsidethecone,buteffectivelytheyonlydependonη .Theresidues
+ +
of the nucleon Born te−rms therefore scale with η , which reflects the symmetric makeup of the phase
+
3.1 Verification of the discrepancy
space.Thehadronicpolesformplanesinthephasespacethatwillgenerallycounteractthissymmetry
property: the nucleon Born poles appear at η = λ = 0; the nucleon resonance poles formThveesrttriikcianlg difference between Rosenbluth [30] and the early polarization transfer [16, 18] measure-
planes at fixed η < 0, where the value of λ d−epends on the width of the resonance; and tmecnhtasnofntehleproton electromagnetic form factorsshown in Fig.2 led tosignificant activity aimed at
meson poles appe−ar at fixed ∆2 = m2 η =η +m2/(2m2). −understandingandresolvingthisdiscrepancy. Itwasnotedearly[16]thattherewassignificantscatter
− i ⇒ − + i between the results of different Rosenbluth extractions [11, 31, 32, 33, 34], as illustrated in Fig. 3,
Polarizabilities. The nucleon polarizabilities α and β are related to the CFFs f1 and f2 instuhggeesltiimngitthattheproblemwasrelatedtothecrosssectionmeasurements. AthighQ2,GEyieldsonly
whereallkinematicvariablesarezero: α+β,β = f ,f α /m3.InFig.5weshowprealismmailnl,aarnygle-dependent correction to the cross section, leading to the possibility that a systematic
1 2 QED
resultsfromthequark-levelcalculation{extracte}dfro{mthe}b×asisinEq.3.Sofartheyareonlydibffaerlelpncaerbketweensmall-andlarge-anglemeasurementscouldyieldlargecorrectionstoGE/GM,which
wouldincreaseinimportancewithincreasingQ2. Itwasthereforearguedthattheobserveddifference
estimates: the quark Compton vertex that enters in the calculation depends on 6 Lorentz invariants
mayhavebeenduetosomeexperimentalerrorinoneormoreofthecrosssectionmeasurementsthat
and 128 tensor structures and its transverse/gauge separation is extremely sensitive to the nsiugnmifiecrainctsly.changethehighQ2extractionsofG . Thus,thefirststepwasacarefulexaminationofthe
E
We extracted the momentum dependence of α+β from f1 only whereas the standard deficnriotsisosnect[i7o]n datatodetermine iftheobserved discrepancy couldbeexplained byproblems with one
contains admixtures from higher CFFs at η >0, but for those our results are still too noisyo.rtwoexperiments, orresolved byadjustingthenormalizationofsomedatasetswithin theassumed
+
uncertainties.
1 ForarbitraryoffshellformfactorstheBorntermsmustbecombinedwith(non-diagrammatic)partsofthe
1PI contribution to arrive again at a gauge-invariant expression, see [17] for a discussion. 7
6
ThehatchedbandsinFig.5aretheoutcomeofdiagram(b)insidethecone.Forthetotalresultwe
addedthe∆resonance,thedominanthadroniccontributiontodiagram(a),usingaparametrizationfor
the experimental N ∆γ form factors. For comparison we plot the dispersion relation results for the
→
generalizedpolarizabilitiesfromRefs.[4;7].Thefiguremakesclearthatthesumα+β isdominatedby
diagram (b) and, as it turns out, especially by the handbag contributions. The magnetic polarizability
β is dominated by the ∆ pole from diagram (a) whereas (b) contributes little due to cancellations.
The discrepancy at low η is presumably due to missing pion loops — β is subject to cancellations
+
between the quark core (which then mainly comes from the ∆ pole) and pion cloud effects.
Tosummarize,wedemonstratedhowtoextractmicroscopicinformationonnucleonpolarizabilities
from the decomposition in Fig. 4. It will be further interesting to investigate spin polarizabilities and
gatherknowledgeonthespacelikemomentumdependenceoftheCSamplitude,whichwillimproveour
understanding of two-photon corrections to form factors as well as the proton radius puzzle. Finally,
the same framework can be adapted to other processes such as pion electroproduction, which has
contributed much to our knowledge of nucleon resonances and transition form factors. For their clean
extraction one needs to know the non-resonant ‘QCD background’ beyond hadronic exchanges, which
is information that a microscopic approach can provide.
Acknowledgements I am grateful to C. S. Fischer and G. Ramalho for valuable discussions, and I would
like to thank the organizers of the Light Cone 2015 conference for their support.
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