Table Of Content

Hadron physics and dynamical chiral symmetry breaking∗ 2 1 0 2 n LeiChang a PhysicsDivision,ArgonneNationalLaboratory,Argonne,Illinois60439,USA J E-mail: [email protected] 8 1 Craig D. Roberts† ] PhysicsDivision,ArgonneNationalLaboratory,Argonne,Illinois60439,USA; h t DepartmentofPhysics,IllinoisInstituteofTechnology,Chicago,Illinois - l E-mail: [email protected] c u DavidJ.Wilson n [ PhysicsDivision,ArgonneNationalLaboratory,Argonne,Illinois60439,USA E-mail: [email protected] 1 v 8 Physics is an experimental science; and a constructive feedback between theory and extant 1 and forthcomingexperimentsis necessary if an understandingof nonperturbativeQCD is to be 9 3 achieved.TheDyson-Schwingerequationsconnectconfinementwithdynamicalchiralsymmetry . 1 breaking,bothwiththeobservablepropertiesofhadrons,andhencecanplausiblyprovideameans 0 ofelucidatingtheempiricalcontentofstrongQCD.Weillustratethesepointsviacommentson: 2 1 in-hadron condensates; dressed-quark anomalous chromo- and electro-magnetic moments; the : v self-limitingmagnitudesofsuchmomentsandpion-loopcontributionstothegapequation;deep i X inelasticscattering;thespectraofmesonsandbaryons;thecriticalroleplayedbyhadron-hadron r interactionsinproducingthesespectra;andnucleonelasticandtransitionformfactors. a InternationalWorkshoponQCDGreen’sFunctions,ConfinementandPhenomenology 5-9September2011 Trento,Italy ∗Wearegratefultotheorganisersfortheopportunitytobeinvolvedinthisworkshop. Partsoftheoriginalmaterial describedinthiscontributionweredrawnfromcollaborationsanddiscussionswithA.Bashir,S.J.Brodsky,C.Chen, H. Chen, I.C. Cloët, B. El-Bennich, X. Gutiérrez-Guerrero, R.J. Holt, Y.-x. Liu, V. Mokeev, T. Nguyen, S.-x. Qin, H.L.L.Roberts,R.ShrockandP.C.Tandy.ThisworkwassupportedbyU.S.DepartmentofEnergy,OfficeofNuclear Physics,contractno.DE-AC02-06CH11357. †Speaker. (cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ Hadronphysicsanddynamicalchiralsymmetrybreaking CraigD.Roberts 1. Universal Truths Tosetthescene,werecapitulatesomebasicfacts. Thehadronspectrum,andhadronelasticand transition formfactors provide uniqueinformation aboutthelong-range interaction betweenlight- quarks and the distribution of a hadron’s characterising properties amongst its QCD constituents. Dynamical chiral symmetry breaking (DCSB) is the most important mass generating mechanism for visible matter in the Universe, and this means the Higgs mechanism is (almost) irrelevant to light-quarks. Notably, this is acknowledged by the Higgs-hunters at CERN who, in announcing inconclusive evidence of a Higgs sighting, stated: “The Higgs field is often said to give mass to everything. That is wrong. The Higgs field only gives mass to some very simple particles. The field accounts for only one or two percent of the mass of more complex things like atoms, molecules and everyday objects, from your mobile phone to your pet llama. The vast majority of masscomesfromtheenergyneededtoholdquarkstogetherinsideatoms.” Therunningofthequark mass, illustrated inFig.1, entails thatcalculations atevenmodest Q2 require aPoincaré-covariant approach. Furthermore, covariance plus the momentum-dependence of the interaction in QCD together guarantee the existence of quark orbital angular in a hadron’s rest-frame wave function. Confinement isexpressed through aviolent change inthe analytic structure of thepropagators for coloured particles; its presence can be read from a plot of a states’ dressed-propagator; and it is intimately connected with DCSB [1]. We now illustrate how the complex of Dyson-Schwinger equations (DSEs) has been employed to elucidate aspects of confinement and DCSB, and their impactonhadronobservables. Fullerexplanations maybefoundinrecentreviews[2–5]. 2. Condensates areconfined withinhadrons Dynamicalchiralsymmetrybreakinganditsconnectionwiththegenerationofhadronmasses wasfirstconsidered inRef.[6]. Theeffectwasrepresented asavacuum phenomenon. Twoessen- tially inequivalent classes of ground-state were identified in the mean-field treatment of a meson- nucleonfieldtheory: symmetrypreserving(Wignerphase);andsymmetrybreaking(Nambuphase). Notably, within the symmetry breaking class, each of an uncountable infinity of distinct configu- rations isrelated toevery other byachiral rotation. Thisisarguably theorigin oftheconcept that strongly-interacting quantum fieldtheoriespossess anontrivial vacuum. With the introduction of the parton model to describe deep inelastic scattering (DIS), this notion was challenged via an argument [10] that DCSB can be realised as an intrinsic property of hadrons, instead of through a nontrivial vacuum exterior to the observable degrees of freedom. Suchaviewistenablebecause theessential ingredient required fordynamical symmetrybreaking in a composite system is the existence of a divergent number of constituents and DIS provided evidencefortheexistencewithineveryhadronofaseaoflow-momentumpartons. Thisperspective has, however, received scant attention. Instead the introduction of QCDsum rules asatheoretical artifice to estimate nonperturbative strong-interaction matrix elements entrenched the belief that theQCDvacuumischaracterised bynumerous distinct, spacetime-independent condensates. Faithinempiricalvacuumcondensatesmaybecomparedwithanearliermisguidedconviction that the universe was filled with a luminiferous aether. Recall that physics theories of the late 19th century postulated that, just as water waves must have a medium to move across (water), 2 Hadronphysicsanddynamicalchiralsymmetrybreaking CraigD.Roberts Rapid acquisition of mass is 0.4 effect of gluon cloud 0.3 V] mm == 03 0( CMheirVal limit) Ge m = 70 MeV p) [0.2 M( 0.1 0 0 1 2 3 p [GeV] Figure 1: Leftpanel– Dressed-quarkmass function, M(p): solid curves – DSE results, [7, 8], “data” – lattice-QCDsimulations[9].(N.B.m=70MeVistheuppermostcurve.Current-quarkmassdecreasesfrom toptobottom.) Theconstituentmassarisesfromacloudoflow-momentumgluonsattachingthemselvesto thecurrent-quark:DCSBisatrulynonperturbativeeffectthatgeneratesaquarkmassfromnothing;namely, itoccurseveninthechirallimit,asevidencedbythem=0curve.Middlepanel–Anobservableparticleis associatedwithapoleattimelike-P2.Thisbecomesabranchpointif,e.g.,theparticleisdressedbyphotons. Rightpanel–Whenthedressinginteractionisconfining,thereal-axismass-polesplits,movingintopairsof complexconjugatepolesorbranchpoints.Nomass-shellcanbeassociatedwithaparticlewhosepropagator exhibitssuchsingularitystructure[1]. and audible sound waves require a medium to move through (such as air or water), so also light waves require a medium, the “luminiferous aether.” This was apparently unassailable logic until, ofcourse, oneofthemostfamousfailedexperimentsinthehistoryofscience [11]. Notwithstanding the prevalence of the belief in empirical vacuum condensates, it does lead to problems; e.g., entailing a cosmological constant that is 1046-times greater than that which is observed [12, 13]. This unwelcome consequence is partly responsible for reconsideration of the possibility that the so-called vacuum condensates are in fact an intrinsic property of hadrons. Namely,inaconfiningtheorycondensatesarenotconstant,physicalmass-scalesthatfillallspace- time;instead,theyaremerelymass-dimensioned parametersthatserveapracticalpurposeinsome theoreticaltruncationschemesbutotherwisedonothaveanexistenceindependentofhadrons[13– 17]. Confinement isessential tothis viewand noeffort todecry itismeaningful unless the model usedintheattemptpossesses averacious expression ofthisphenomenon. This factor of 1046 mismatch is part of what has been called [12] “the greatest embarrass- ment in theoretical physics.” However, it vanishes if one discards the notion that condensates haveaphysical existence, whichisindependent ofthehadrons thatexpress QCD’sasymptotically realisable degrees offreedom [13]; namely, ifone accepts thatsuch condensates aremerelymass- dimensioned parameters in one or another theoretical truncation scheme. Thisappears mandatory in a confining theory [14, 15], a perspective one may embed in a broader context by considering justwhatisobservableinquantumfieldtheory[18]: “...althoughindividualquantumfieldtheories haveofcourseagooddealofcontent,quantumfieldtheoryitselfhasnocontentbeyondanalyticity, unitarity, cluster decomposition andsymmetry.” IfQCDisaconfining theory, thencluster decomposition isonlyrealisedforcoloursingletstates[1]andallobservable consequences ofthetheory, including itsground state,canbeexpressed viaanhadronic basis. Thisisquark-hadron duality. Thearguments inRefs.[14, 15] follow from twokey realisations. Thefirst isconnected with 3 Hadronphysicsanddynamicalchiralsymmetrybreaking CraigD.Roberts thein-pseudoscalar-meson quarkcondensate,whichisaquantitywithanexactexpressioninQCD; viz. [19,20],k z =r z f ,where(k =k±P/2) Pf1f2 Pf1f2 Pf1f2 ± L ifPf1f2Km =h0|q¯f2g5gm qf1|Pi = Z2(z ,L )trCDZk ig5gm Sf1(k+)G Pf1f2(k;K)Sf2(k−), (2.1) L ir z =−h0|f¯ig f |Pi = Z (z ,L )tr g S (k )G (k;K)S (k ), (2.2) Pf1f2 2 5 1 4 CDZk 5 f1 + Pf1f2 f2 − f2 m2 = [mz +mz ]k z . (2.3) Pf1f2 Pf1f2 f1 f2 Pf1f2 Here L is a Poincaré-invariant regularization of the integral, with L the ultraviolet regulariza- k tion mRass-scale, Z are renormalisation constants, with z the renormalisation point, G is the 2,4 Pf1f2 meson’s Bethe-Salpeter amplitude, and S are the component dressed-quark propagators, with f1,f2 m theassociated current-quark masses. Equation(2.1)describesthepseudoscalar meson’slep- f1,f2 tonicdecayconstant;i.e.,thepseudovectorprojectionofthemeson’sBethe-Salpeterwave-function onto the origin in configuration space; Eq.(2.2) describes its pseudsocalar analogue; and m is Pf1f2 the meson’s mass. The initial step in extending the concept was a proof that the in-pseudoscalar- meson quark condensate, k z , can rigorously be represented through the pseudoscalar-meson’s Pf1f2 z z z scalarformfactoratzeromomentumtransfer, Q2=0;viz.,(m =m +m ) f1f2 f1 f2 ¶ m2 k z ¶ k z S :=−hP |1(q¯ q +q¯ q )|P i= Pf1f2 = Pf1f2 +mz  Pf1f2. (2.4) Pf1f2 f1f2 2 f1 f2 f1 f2 f1f2 ¶ mz f2 f1f2¶ mz f2 f1f2 Pf1f2 f1f2  Pf1f2  Thesecondstepemploysamassformulaforscalarmesons,exactinQCD;viz. [15], f m2 = −[mz −mz ]r z , (2.5) Sf1f2 Sf1f2 f1 f2 S12 L fSf1f2Km = Z2trCDZk igm Sf1(k+)G Sf1f2(k;K)Sf2(k−), (2.6) L r z = −Z tr S (k )G (k;K)S (k ), (2.7) Sf1f2 4 CDZk f1 + Sf1f2 f2 − whichwasusedtoprovethatthein-scalar-meson quarkcondensate is,analogously andrigorously, connected with the scalar-meson’s scalar form factor at Q2 =0. Moreover the following limiting caseswerealsoestablished: f2 Sz chira=llimitk 0=−hq¯qi0 and f2 Sz heavy=quark(s)2k z , (2.8) P,S P,S P,S P,S P,S wherehq¯qi0ispreciselythequantitythatiswidelyknownasthevacuumquarkcondensatethrough anyofitsdefinitions [21]. With appeal then to demonstrable results of heavy-quark symmetry in QCD, it was argued that the Q2 =0 values of vector- and pseudovector-meson scalar form factors also determine the in-hadron condensates in these cases, and that this expression for the concept of in-hadron quark condensates is readily extended to the case of baryons. Thus, through the Q2 = 0 value of any hadron’s scalar form factor, one can extract the magnitude of a quark condensate in that hadron whichisareasonable andrealistic measureofdynamicalchiralsymmetrybreaking. 4 Hadronphysicsanddynamicalchiralsymmetrybreaking CraigD.Roberts Notethatinthepresenceofconfinementitisimpossibletowriteavalidnonperturbativedefini- tionofasinglequarkorgluonannihilationoperator;andthereforeimpossibletorigorouslydefinea secondquantisedgroundstateforQCDuponafoundationofgluonandquark(quasiparticle) operators. Todosowouldbetoanswerthequestion: Whatisthestatethatisannihilatedbyanoperator which is – as appears at present – unknowable? However, with the assumptions that confinement is absolute and that it entails quark-hadron duality, the question changes completely. Then the nonperturbative Hamiltonian of observable phenomena in QCD is diagonalised by colour-singlet statesalone. Thegroundstateofthisstrong-interaction Hamiltonianisthestatewithzerohadrons. One may picture the creation and annihilation operators for such states as rigorously defined via smeared sources on aspacetime lattice. The ground-state is defined with reference to such operators, employing, e.g., theGell-Mann -Lowtheorem [22], whichisapplicable inthis casebecause therearewell-definedasymptotic statesandassociated annihilation andcreationoperators. Inlearning thattheso-called vacuum quark condensate isactually thechiral-limit valueofan in-pion property, some respond as follows. The electromagnetic radius of any hadron, rem, which H couples to pseudoscalar mesons must diverge in the chiral limit. This long-known effect arises becausethepropagationofmasslesson-shellcolour-singletpseudoscalarmesonsisundamped[23]. Therefore,doesnoteachpiongrowtofilltheuniverse;sothat,inthislimit,thein-pioncondensate reproduces theconventional paradigm? Confinement,again, vitiatesthisobjection. BothDSE-andlattice-QCDindicate thatconfine- ment entails dynamical mass generation for both gluons and quarks, Fig.1. The dynamical gluon and quark masses remain large in the limit of vanishing current-quark mass. In fact, the dynamical massesarealmost independent ofcurrent-quark massintheneighbourhood ofthe chiral limit. It follows that for any hadron the quark-gluon containment-radius does not diverge in the chiral limit. Instead, it is almost insensitive to the magnitude of the current-quark mass because the dynamical masses of the hadron’s constituents are frozen at large values; viz., 2−3L . These QCD considerations show that the divergence of rem does not correspond to expansion of a condensate p fromwithinthepionbutrathertothecopiousproductionandsubsequentpropagationofcomposite pions, eachofwhichcontains acondensate whosevalueisessentially unchanged fromitsnonzero current-quark massvaluewithinacontainment-domain whosesizeissimilarlyunaffected. There is moreto be said in connection with the definition and consequences of a chiral limit. Plainly, the existence of strongly-interacting massless composites would have an enormous impact on the evolution of the universe; and it is naive to imagine that one can simply set the renormalisation-point-invariantcurrent-quarkmassestozeroandconsideracircumscribedrangeof manageable consequences whilstignoring thewiderimplications forhadrons, theStandardModel andbeyond. Forexample,withallelseheldconstant,BigBangNucleosynthesisisverysensitiveto thevalueofthepion-mass[24]. Wearefortunatethattheabsenceofquarkswithzerocurrent-quark masshasproduced auniverseinwhichweexistsothatwemaycarefully ponderthealternative. Withquarkcondensates beinganintrinsicpropertyofhadrons,onearrivesatanewparadigm, asobserved inthepopular science press [25]: “EMPTYspace mayreallybeempty. Though quantum theory suggests that a vacuum should be fizzing with particle activity, it turns out that this paradoxical picture of nothingness may not be needed. A calmer view of the vacuum would also help resolve a nagging inconsistency with dark energy, the elusive force thought to be speeding up the expansion of the universe.” In connection with the cosmological constant, putting QCD 5 Hadronphysicsanddynamicalchiralsymmetrybreaking CraigD.Roberts condensates backintohadronsreducesthemismatchbetweenexperimentandtheorybyafactorof 1046. If technicolour-like theories are the correct scheme for extending the Standard Model [26], thentheimpactofthenotionofin-hadron condensates isfargreaterstill. 3. Dressed-quark Anomalous Moments TheappearanceandbehaviourofM(p)inFig.1areessentiallyquantumfieldtheoreticeffects, unrealisableinquantummechanics. Therunningmassconnectstheinfraredandultravioletregimes of the theory, and establishes that the constituent-quark and current-quark masses are simply two connected pointsonasinglecurveseparated byalargemomentuminterval. QCD’sdressed-quark behaves as a constituent-quark, a current-quark, or something in between, depending on the mo- mentumoftheprobewhichexplores thebound-state containing thedressed-quark. Theseremarks should makeclearthatQCD’sdressed-quarks arenotsimplyDiracparticles. Since the two-point functions of elementary excitations are strongly modified inthe infrared, then the same is generally true for three-point functions; i.e., the vertices. The bare vertex will thus be a poor approximation to the complete result unless there are extenuating circumstances. Thisisreadilymadeapparent, forwithadressed-quark propagator characterised byamomentum- dependentmass-function,onefindsimmediatelythattheWard-Takahashiidentityisbreached;viz., Pm igm 6=S−1(k+P/2)−S−1(k−P/2), (3.1) and the violation is significant whenever and wherever the mass function in Fig.1 is large. The mostimportantfeatureoftheperturbativeorbarevertexisthatitcannotcausespin-fliptransitions; namely, it is an helicity conserving interaction. However, one must expect that DCSB introduces nonperturbatively generatedstructuresthatverystronglybreakhelicityconservation. Thesecontri- butions will be large when the dressed-quark mass-function is large. Conversely, they will vanish intheultraviolet;i.e.,ontheperturbativedomain. Theexactformofthevertexcontributionsisstill thesubject ofstudybuttheirexistence ismodel-independent. A spectacular consequence is highlighted by considering dressed-quark anomalous magnetic moments. In QCD the analogue of Schwinger’s one-loop calculation can be performed to find an anomalous chromo-magnetic moment (ACM) for the quark. There are two diagrams: one similar in form tothat in QED;and another owingtothe gluon self-interaction. Onereads from Ref.[27] thattheperturbativeresultvanishesinthechirallimit. Thisisconsonantwithhelicityconservation in perturbative massless-QCD. However, Fig.1demonstrates that chiral symmetry is dynamically broken stronglyinQCDandonemusttherefore askwhetherthisaffectstheACM. Of course, it does, and it is now known that the effect is modulated by the strong momentum dependence of the dressed-quark mass-function [28]. In fact, dressed-quarks possess a large, dynamically-generatedACM,whichproducesanequallylargeanomalouselectromagneticmoment that has amaterial impact onnucleon magnetic and transition form factors [29, 30]. Furthermore, giventhemagnitudeofthemuon“gm −2anomaly”anditsassumedimportanceasanharbingerof physics beyond the Standard Model [31], it might also be worthwhile to make a quantitative esti- mateofthecontribution togm −2fromthequark’sDCSB-inducedanomalousmomentsfollowing, e.g.,thecomputational patternindicated inRef.[32]. 6 Hadronphysicsanddynamicalchiralsymmetrybreaking CraigD.Roberts We now illustrate an interesting possibility, which has recently been identified; viz., that the dressed-quark’s ACM is self-limiting. Consider the gap equation obtained from a symmetry- preserving regularisation of a vector-vector contact interaction [30] with a quark-gluon vertex of theformG na =G n l a/2[k= pf −pi,ℓ=(pf +pi)/2], G m (pf,pi;k) = gm A+s mn kn t2+s mn ℓn ℓ·kt3+[ℓm g ·k+igm s nr ℓn kr ]t4 −iℓm t5+ℓm g ·kℓ·kt6−ℓm g ·ℓt7+ℓm s nr ℓn kr t8. (3.2) Thedressed-quark propagator hastheformS(p)=1/(iAg ·p+B)=(1/A)/(ig ·p+M),where 16pa d4q 3M 1 A = 1+ IR m (tˆ −2), (3.3a) 3m2 Z (2p )4 2 A q2+M2 Q 5 G 16pa d4q M 1 B = m+ IR (4A−7Mm ), (3.3b) 3m2 Z (2p )4 A q2+M2 Q G withm =0.8GeV,amass-scaletypicalofgluons[33–36];anda aparametercharacterisingthe G IR infrared strength of the interaction. A context for the value of a is easily provided. In rainbow- IR laddertruncation withnonperturbatively-massless gaugebosons,thecouplingbelowwhichDCSB breakingisimpossibleviathegapequationsinQEDandQCDisa c/p ≈1/3[37,38],whilstwith a symmetry-preserving regularisation of a contact-interaction, a c /p ≈0.4 and a description of IR hadron phenomena requires a /p ≈1 [30]. The simplicity of Eqs.(3.3) wasattained as follows: IR weimposedtwoconstraintsfromperturbativeQCD[27],t +Mt =−m +t /2+Mt /2,m >0; 2 4 Q 5 7 Q wroteMt =t =(2+tˆ )m inordertosuppressaparameter;andused p2 =M2=p2,identifying 7 5 5 Q f i thefocusofintegration strength, inordertosimplifytheintegrands. One may now read that if t , t are large enough; viz., tˆ ≥2, then A≥1, in which case a 5 7 5 B>0solutionispossiblesolongasMm <(4A/7). Hence,thereisaninternallyconsistentupper Q bound on the dressed-quark ACM. Furthermore, including an ACM in the gap equation produces thesamesortofattraction asdoesintroducing resonant contributions viathevertex–theso-called meson-loop effects [39]. One should thus expect that their strength, too, is self-limiting. Finally, thissimpleanalysis confirmsalong-known resultthatdressing thevertex;i.e.,improvingsensibly upon the rainbow-ladder truncation, decreases the p2 =0 value of the mass-function: strength is shiftedoutto p&2L [40]. Thereforeoneshouldnotexpectvertexdressingtouniformlyboost QCD the magnitude of the dressed-quark mass function nor to materially reduce the critical coupling belowwhichDCSBisimpossible[38]. 4. Deep InelasticScattering Thepastfortyyearshaveseenatremendous efforttodeducethepartondistribution functions (PDFs) of the most accessible hadrons – the proton, neutron and pion. There are many reasons for this long sustained and thriving interest [3] but in large part it is motivated by the suspected process-independence oftheusualpartondistribution functionsandhenceanabilitytounifymany hadronic processes through their computation. In connection with uncovering the essence of the strong interaction, the behaviour of the valence-quark distribution functions at large Bjorken-x is most relevant. Furthermore, an accurate determination of thebehavior of distribution functions in 7 Hadronphysicsanddynamicalchiralsymmetrybreaking CraigD.Roberts 0.4 1.2 0.3 1 x) 0.8 u(v0.2 x uK/up 0.6 DSE-BSA, this work 27 GeV2 p -N Drell-Yan, <Q2> = 27 GeV2 0.1 E615 p N Drell-Yan 16.4 GeV2 0.4 Q2 = 27 GeV2, Full BSE Expt NLO analysis 27 GeV2 Q2 = 27 GeV2 (G = g , g g.P; q.P=0) DSE (Hecht et al.) 27 GeV2 0.2 5 5 Aicher et al. 27 GeV2 0.0 0 0.0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1 x x Figure 2: Leftpanel. Pion valence quark distribution function. Solid curve – full DSE calculation [43]; dot-dashedcurve–semi-phenomenologicalDSE-basedcalculationinRef.[44];filledcircles–experimental datafromRef.[45]; dashedcurve – NLO reanalysisofthe experimentaldata[46]; anddot-dot-dashed curve – NLO reanalysisofexperimentaldata withinclusionof soft-gluonresummation[47]. Rightpanel. DSE predictionfortheratioofu-quarkdistributionsinthekaonandpion[3,43]. ThefullBethe-Salpeter amplitudeproducesthesolid curve;areducedBSEvertexproducesthedashedcurve. Thereducedampli- tuderetainsonlytheinvariantsandamplitudesinvolvingpseudoscalarandaxialvectorDiracmatrices,and ignoresdependenceonthevariableq·P.Thesearepartofthereductionsthatoccurinapointliketreatment ofpseudoscalarmesons. Theexperimentaldataisfrom[48,49] the valence region is also important to high-energy physics. Particle discovery experiments and StandardModeltestswithcollidersareonlypossible iftheQCDbackground iscompletely under- stood. QCDevolution,apparentintheso-calledscalingviolationsbypartondistributionfunctions,1 entailsthatwithincreasingcenter-of-massenergy,thesupportatlarge-xinthedistributionsevolves tosmall-xandtherebycontributes materiallytothecollider background. OwingtothedichotomousnatureofGoldstonebosons,understandingthevalence-quarkdistri- butionfunctionsinthepionandkaonisofgreatimportance. Moreover,giventhelargevalueofthe ratio of s-to-u current-quark masses, a comparison between the pion and kaon structure functions offersthechancetocharteffectsofexplicitchiralsymmetrybreakingonthestructureofwould-be Goldstone modes. There is also the prediction [41, 42] that a theory in which the quarks interact via1/k2-vector-boson exchange willproducevalence-quark distribution functions forwhich q (x)(cid:181) (1−x)2+g , x&0.85, (4.1) v where g & 0 is an anomalous dimension that grows with increasing momentum transfer. (See Sec.VI.B.3ofRef.[3]foradetaileddiscussion.) Sincepseudoscalar mesontargetsaredifficulttoproduce, experimental knowledgeofthepar- tonstructure ofthepionandkaonarisesprimarily frompionicorkaonic Drell-Yanprocesses; and suchanexperiment [45]produced verydisturbing results: avalence-quark distribution function in striking conflictwithQCD.InsteadofEq.(4.1),thedataindicatedu (x)∼(1−x)1. Theexponent V 1DGLAPevolutionisdescribedinSec.IIDof Ref.[3]. TheevolutionequationsarederivedinperturbativeQCD anddeterminetherateofchangeofpartondensitieswhentheenergy-scalechosenfortheirdefinitionisvaried. 8 Hadronphysicsanddynamicalchiralsymmetrybreaking CraigD.Roberts “1” is that predicted by a theory without gluons; i.e., constituent-quark-like or contact-interaction models. Manyauthors employed such models toreproduce the dataand argued therefrom that in- ferencesfromQCDwerewrong. Thiscriticaldisagreementwasre-emphasizedbynonperturbative DSEpredictions [43,44]whichconfirmedtheexponent “2”anddemonstrated itsconnection with thedressed-quark mass-function thatisthesmokinggunfordynamical chiralsymmetrybreaking. Not until very recently was a resolution of the conflict between data and well-constrained theory presented [47]: itconfirmsthe DSEpredictions; and thereby emphasises thepredictive powerand strength ofusing asingle internally-consistent, well-constrained framework tocorrelate and unify thedescription ofhadron observables. WithRef.[43]asignificantmilestonewasachieved: thefirstunificationofthecomputationof distribution functionsthatariseinanalysesofdeepinelasticscattering withthatofnumerousother properties ofpseudoscalar mesons,including meson-mesonscattering [50]andthesuccessful prediction of electromagnetic elastic and transition form factors [51, 52]. The results confirmed the large-x behavior ofdistribution functions predicted bythe QCDparton model; provide agood ac- countofthep N Drell-Yandataforup (x);andaparameter-freepredictionfortheratiouK(x)/up (x) V V V thatagreeswithextantdata,showingastrongenvironment-dependence oftheu-quark distribution (right panel in Fig.2). The new Drell-Yan experiment running at FNAL is capable of validating thiscomparison, asistheCOMPASSIIexperiment atCERN.Suchanexperiment should bedone sothatcompleteunderstanding ofQCD’sGoldstonemodescanbeclaimed. Thesesuccessesshouldbecontrastedwithattemptstocomputepartondistributionfunctionsin constituent-quark-like models. Such quantum mechanical models cannot incorporate the momentum-dependent dressed-quark and dressed-gluon mass-functions; they have no connection with quantum field theory and therefore not with QCD; and they are not “symmetry-preserving” and hencecannotveraciously connectmesonandbaryonproperties. Amongsttheirdefects, suchmod- elstypicallyproduceveryharddistributions. Inorderthentomakecontactwiththerealworld’ssoft distributions, their adherents are forced to choose an unrealistically small value for the so-called hadronic scale,Q ,fromwhichDGLAPevolution mustbegin;viz.,Q ∼1.25L . Thereareno 0 0 QCD sound reasons to support an assertion that perturbative evolution can be used at such low scales. Indeed, it is plain from inspection of a plot of the dressed-quark mass function, Fig.1, that the domainQ <2L isessentiallynonperturbativebecausethereuponM(p2)exhibitstheinflexion 0 QCD point that ischaracteristic ofconfinement. Thusany useofDGLAPfrom such low mass-scales is p misguided. As made plain by the experience with u (x), it will only express model artefacts and V yieldnoreliableinformation aboutQCD. Thehardness ofthedistributions insuchmodelscanbetracedtothecompletelackofgluons, which is represented via enforcement of what may be called the “constituent-quark momentum sumrule;”viz, 1 N (cid:229) hxi = dxx q (x;Q )=1 (4.2) Q0 Z V 0 0 i=1 where N is the number of valence-quarks in the hadron. As elucidated in Sec.VI.B.3 of Ref.[3], in QCDthere isnoscale atwhichthe valence quarks cancarry all ofahadron’s momentum. This is readily explained. All hadrons are bound by gluons, which are invisible to the electromagnetic probe. Thus, some fraction of the hadron’s momentum is carried by gluons at all resolving scales 9 Hadronphysicsanddynamicalchiralsymmetrybreaking CraigD.Roberts unlessthehadronisapointparticle. Indeed,itisasimplealgebraicexercisetodemonstratethatthe only non-increasing, convex function which can produce hx0i=N and hxi=1, is the distribution q (x)=1,whichisuniquely connected withapointlike hadron; viz.,ahadron whosebound-state V amplitude is momentum-independent. The real consequence of the momentum sum-rule in QCD isthatatanyresolving scale,Q, 1 N (cid:229) dxx q (x;Q)<1. (4.3) Z V 0 i=1 Asemphasised above, PDFscanbeanexcellent probeoftherunning coupling butonlyifthe model input isrigorously connected with QCD,aswith the DSEprediction of the pion’s valence- quarkdistribution function [44],whichwasverifiedbythenewNLOanalysis ofextantdata[47]. 5. GrandUnification OwingtotheimportanceofDCSB,fullcapitalisationontheresultsofforthcomingexperimen- tal programmes is only possible if the properties of meson and baryon ground- and excited-states can be correlated within a single, symmetry-preserving framework, where symmetry-preserving means that all relevant Ward-Takahashi identities are satisfied. Constituent-quark-like models, whichcannotincorporate themomentum-dependent dressed-quark mass-function, failthistest. An alternative is being pursued within quantum field theory via the Faddeev equation. This analogueoftheBethe-Salpeterequationsumsallpossibleinteractionsthatcanoccurbetweenthree dressed-quarks. Asimplifiedequation[53]isfoundedontheobservation thataninteraction which describes color-singlet mesons also generates nonpointlike quark-quark (diquark) correlations in the color-antitriplet channel [54]. The dominant correlations for ground state octet and decuplet baryons are scalar and axial-vector diquarks because, e.g., the associated mass-scales are smaller than the baryons’ masses and their parity matches that of these baryons. (Recent studies confirm thefidelityofthenonpointlike diquarkapproximation withinthenucleonthree-body problem;e.g. Ref.[55].) On the other hand, pseudoscalar and vector diquarks dominate in the parity-partners of those ground states [56]. Thisapproach treats mesons and baryons on the same footing and, in particular, enablestheimpactofDCSBtobeexpressedintheprediction ofbaryonproperties. Building on lessons from meson studies [4], a unified spectrum of u,d-quark hadrons was obtained using a symmetry-preserving regularisation of a vector×vector contact interaction [56]. That study correlates the masses of meson and baryon ground- and excited-states within a single framework. In comparison with relevant quantities, the computation produces rms=13%, where rms is the root-mean-square-relative-error/degree-of freedom. The predictions uniformly overes- timate the PDG values of meson and baryon masses [57]. Given that the employed truncation deliberately omitted meson-cloud effects in the Faddeev kernel, this is a good outcome, since in- clusion ofsuchcontributions actstoreducethecomputedmassesbyjusttherequiredamount. Following this reasoning, a striking result is agreement between the DSE-computed baryon masses[30,56]andthebaremassesemployedinmoderncoupled-channelsmodelsofpion-nucleon reactions [65, 66]. The Roper resonance is very interesting. The DSE study [30] produces an excitation of the nucleon at 1.72±0.07GeV. This state is predominantly a radial excitation of the quark-diquark system, with both the scalar- and axial-vector diquark correlations in their ground 10