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. 6 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.: +49 641 9933342 Fax: +49 641 9933309 E-mail: gernot.eichmann@theo.physik.uni-giessen.de 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 transition 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 contains 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 contributions. 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=toooea4ηdr12irlmλia,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=ietc0o,nstσra>int0s, 1o4ηesiλteiah∆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=toooea4ηders12irlmλia,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=iet0, 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. References 1. F. Hagelstein, R. Miskimen, and V. Pascalutsa, “Nucleon Polarizabilities: from Compton Scattering to Hydrogen Atom,” 1512.03765[nucl-th] 2. K. A. Olive et al., “Review of Particle Physics,” Chin. Phys. C38 (2014) 090001 3. J. Arrington, P. Blunden, and W. Melnitchouk, “Review of two-photon exchange in electron scattering,” Prog. Part. Nucl. Phys. 66 (2011) 782–833 4. E. J. Downie and H. Fonvieille, “Real and Virtual Compton Scattering: the nucleon polarisabilities,” Eur. Phys. J. ST 198 (2011) 287–306 5. M. Guidal, H. Moutarde, and M. Vanderhaeghen, “Generalized Parton Distributions in the valence region from Deeply Virtual Compton Scattering,” Rept. Prog. Phys. 76 (2013) 066202 6. H.W.Griesshammeretal.,“Usingeffectivefieldtheorytoanalyselow-energyComptonscatteringdatafrom protons and light nuclei,” Prog. Part. Nucl. Phys. 67 (2012) 841–897 7. D. Drechsel, B. Pasquini, and M. Vanderhaeghen, “Dispersion relations in real and virtual Compton scattering,” Phys. Rept. 378 (2003) 99–205 8. M.SchumacherandM.D.Scadron,“DispersiontheoryofnucleonComptonscatteringandpolarizabilities,” Fortsch. Phys. 61 (2013) 703–741 9. G. Eichmann et al., “Nucleon mass from a covariant three-quark Faddeev equation,” Phys. Rev. Lett. 104 (2010) 201601; H. Sanchis-Alepuz et al., “Delta and Omega masses in a three-quark covariant Faddeev approach,” Phys. Rev. D84 (2011) 096003; G. 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Towards a microscopic understanding of nucleon polarizabilities

Similar Towards a microscopic understanding of nucleon polarizabilities