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Comments on Nuclear and Particle Physics

arXiv:hep-ph/9212225v1 5 Dec 1992Submitted to:DOE/ER/40322-175Comments on Nuclear and Particle PhysicsU. of MD PP #93-019The Pion Cloud In Quenched QCDThomas D. Cohen and Derek B. LeinweberDepartment of PhysicsUniversity of Maryland, College Park, MD 20742August, 1992Diagrammatic methods and Large Nc QCD are used to argue that the nucleon ascalculated in quenched QCD contains physics which can be ascribed to the pion cloudof the nucleon.

In particular, it is argued that the physics corresponding to one-pionloops in chiral perturbation theory are included in the quenched approximation.Typeset Using REVTEX1

I. INTRODUCTIONAt the present time, the most promising technique for eventually deriving the low en-ergy properties of hadrons directly from quantum chromodynamics (QCD) is via numericalMonte Carlo simulations of the functional integral in a lattice regularized version of thetheory. Unfortunately, given the computational power currently available, it is not possibleto calculate hadron properties in a completely realistic manner in which quarks of physicalmass move in a large lattice volume with lattice spacings fine enough to guarantee asymp-totic scaling.

A common approximation that significantly reduces the numerical demandsof the simulations is the so-called quenched approximation. The computational gains asso-ciated with this approximation allow one to greatly improve on systematic and statisticaluncertainties, and to probe the extremes of the parameter space.In the Euclidean space formulation of the QCD functional integral, the weighting functionhas the following continuum formW = det [̸D + mq] exp(−SY M),(1)where SY M is the Euclidean action for a pure Yang-Mills theory (i.e.

only gluons). Integra-tion over the quark Grassmann fields gives rise to the functional determinant.

The quenchedapproximation consists of setting the functional determinant to a constant, independent ofthe gluon field configuration. Thus, in the quenched approximation, the weighting functionis simplified toWq = exp(−SY M) .

(2)This form is obviously amiable to Monte Carlo integration techniques.From a diagrammatic point of view, the omission of the fermion determinant in theweighting function corresponds to the neglect of all diagrams containing closed quark loopswhich are not connected to external sources.Given the fact that quenched calculations will continue to be used for some time to come,it is an important practical problem to determine what, if any, essential physics is lost whenone makes the quenched approximation. This issue is particularly important if one wishesto use the results of quenched lattice QCD calculations to gain insight into what physicsshould be included in models of hadrons.The physics of the pion cloud is believed to play an essential role in hadronic structure.Several models of the nucleon, including the Skyrmion [1], the cloudy bag model [2], thehybrid or chiral bag model [3] and chiral quark-meson models [4] stress the role of the pioncloud in the nucleon structure.

Thus, the question of whether the quenched approximationto QCD contains the physics of the pion cloud becomes important. In this comment, we willargue that pion cloud effects are, in fact, included in quenched calculations of nucleon prop-erties.

This result may seem to be counterintuitive, but as shall be shown, this conclusionis reached both in a large Nc analysis and in an explicit study of diagrams. The argumentsare somewhat heuristic but we believe rather compelling.Apart from the role of the pion cloud in various models, there is also a systematic methodto estimate the role of pions in hadronic observables, namely chiral perturbation theory(χPT).

This approach is an expansion which is based on a separation of scales between2

the pion mass and other scales in hadronic physics [5–8]. This approach should rigorouslyreproduce the low energy properties of QCD if the quark mass (and hence the pion masssquared) is sufficiently light.

Chiral perturbation theory predicts that certain quantities arenonanalytic in the quark mass as one approaches the chiral limit due to the effects of theinfrared behavior of pion loops. For our present purpose, we note that one can use theexistence of this nonanalytic behavior as a signature of pion cloud effects.Before proceeding it is worth noting that the the need for quenched calculations is par-ticularly strong if one wishes to study the role of the pion cloud.

At present, completelyrealistic lattice simulations of pion cloud effects are not possible. The basic difficulty stemsfrom the use of unphysically large quark masses in the lattice simulations which leads topions which are heavy and consequently of short range.A number of challenges are presented as one attempts to decrease the quark mass.

Thelong range nature of the pion demands larger physical lattice volumes which ultimatelyleads to larger lattices if one wishes to maintain a reasonable lattice spacing. Furthermore,calculations of fermion propagators and Monte-Carlo estimates of the functional determinantbecome increasingly difficult as the quark mass drops and convergence of the algorithmsslows.For finite computer resources one can always investigate larger lattice volumes and lighterquark masses in the quenched approximation than in full QCD.

It is likely that there willbe a significant period of time when quenched calculations are able to probe the parameterspace in areas where pion cloud effects are expected to be significant whereas full QCDsimulations will remain restricted to a more limited parameter space. Thus, it is importantto establish whether or not quenched calculations are capable of describing the physics ofthe pion cloud.In this spirit, it is important to make a more restrictive definition of the quenchedapproximation.Within the usual definition there remain diagrams which are extremelycumbersome to calculate and as a result are generally omitted in quenched analyses.

Figure1 displays two such diagrams with quark loops connected to external sources. These quarkline disconnected diagrams require knowledge of the spatial diagonal elements of the inversefermion matrix whereas a standard quark propagator requires knowledge of a single column.Hence, they are computationally quite difficult and for the purposes of this investigation wewill broaden our definition of the quenched approximation to exclude diagrams of the typeillustrated in figure 1.To make contact with physical observables, it is necessary to extrapolate from the largevalues of quark mass currently used in lattice calculations to their physical masses.

As hasbeen stressed elsewhere [9], nonanalytic terms in χPT can lead to important correctionsto this extrapolation in calculations of charge radii and other observables. It is clearly ofimportance to determine whether these chiral corrections to the extrapolations are presentin the quenched approximation or only in full QCD.II.

PION PROPERTIESThe question of whether the physics of pion loops is present in the quenched approx-imation has been discussed previously [10,11]. This discussion has been centered on the3

properties of pions and specifically whether the pion mass squared contains a term propor-tional to m2q ln(mq) which is predicted in χPT. The argument is that a new form of χPTmust be developed to deal with quenched QCD.

In this quenched version of χPT there areno nonanalytic effects arising from the pion cloud surrounding a pion. This does not meanthat nonanalytic behavior is absent from mesonic observables.

Nonanalytic chiral behaviorcan have its origin in the cloud associated with the isoscalar η′ meson. While in nature theη′ is heavy due to anomalies and topological effects [12], in the quenched approximation theη′ is degenerate with the pion and η′ loops can yield nonanalytic chiral behavior.This discussion suggests that quenched QCD calculations of pion properties will notcorrectly reproduce the meson cloud effects of full QCD.

For example, in full QCD the pioncharge radius diverges logarithmically as the quark mass goes to zero [13]; quenched χPTpredicts it will remain finite in quenched QCD. The key point is that the η′ cloud cannotcorrectly simulate the role of the pion cloud for electromagnetic properties since the η′ isneutral and does not couple directly to photons.III.

NUCLEON PROPERTIESA. Diagrammatic considerationsWe will focus our attention on the properties of the nucleon and will argue that pionloop effects do contribute to nucleon properties even in the quenched approximation.

Atfirst thought this may seem absurd since pion loops require an intermediate state with aminimum of one qq pair and the formation of quark-antiquark pairs is apparently forbiddenby the restrictions imposed by the quenched approximations. However, as noted in Ref.

[14–16] this restriction is only apparent. Although the quenched approximation limits quarklines to those which are connected to external currents, the quark propagators used are fullyrelativistic Dirac propagators.

Such propagators contain “Z-graphs” in which the quark isscattered into a negative energy state and back. With the conventional hole interpretationof the Dirac propagator such processes are the creation and annihilation of qq pairs.

Therestriction imposed by the quenched approximation is merely that once a virtual pair iscreated along one quark line it must be annihilated on the same line. Thus, it is at leastpossible that quenched QCD may contain qq pairs and such pairs might be pionic in nature.The question of whether pion-loop physics is included in quenched calculations of cor-relation functions ultimately comes down to the question of whether intermediate statesare reached which have overlap with physical states containing pions plus other hadrons.This suggests that one should study old-fashioned energy denominator type time-ordereddiagrams and ask whether one can reach states which can be expressed as the product ofmore than one color singlet operator operating on the vacuum with at least one of theseoperators having the quantum numbers of the pion.

It is clear that this is a necessary butnot sufficient condition to establish the presence of pion loop physics. In addition, it mustbe shown that the singularity structure corresponds to pion plus hadron states.At the hadronic level, some of the pion cloud physics which accounts for nonanalyticbehavior in χPT can be represented as a single pion loop.

The one-pion-loop physics isrepresented by the hadronic time-ordered diagram in figure 2a. The corresponding diagram4

at the quark level commonly thought to give rise to pionic dressings but not surviving inthe quenched approximation is illustrated in figure 2b.Compare this with the quark level diagram for a nucleon correlation function in figure2c which survives in the quenched approximation [14]. The intermediate state contains aqq structure (at the top of figure 2c) which has both color-singlet and color-octet pieces.The quenched nature of the calculation requires that the antiquark must be the same flavoras one of the original quarks in the nucleon interpolating field.

Since there are always twodistinct flavors of quark in a nucleon, the qq structure can be either isospin zero or one. Thethree quark structure below the qq structure has a piece which is color singlet and isospinone half.

It is highly plausible that the part of this graph which consists of a color singletisospin one qq piece along with a color singlet isospin one half qqq piece has some overlapwith the physical pion-nucleon scattering state. Hence it appears that figure 2c contains,among other things, the essential physics of figure 2a.

Of course, the diagram in figure2c is only one of an infinite class of diagrams which appear to have nonzero overlap withpion-nucleon scattering states. One can add to the diagram an arbitrary number of gluons.It is worth noting at this point that a similar diagrammatic analysis would not givepionic intermediate states for meson correlation functions.

For example in figure 3a we showa one loop hadronic diagram for the ρ meson channel. The imaginary part of this graphgives the ρ to two pion decay.

In analogy to figure 2c we construct figure 3b. This diagramcertainly contains two isovector structures.

However, unlike in the nucleon case in figure 2c,these structures do not have pion quantum numbers. They are color 8 or 3 diquarks withbaryon number ±2/3.

Alternatively, one could arrange the diagram as in figure 3c. Thereis a component in which both structures are color singlets but one sees that at least one ofthe structures is isoscalar and cannot represent a pion.

The isoscalar piece has overlap withthe η′ meson discussed in Ref. [10].

Thus, this simple diagrammatic analysis shows that thephysics of figure 3a cannot be reproduced in the quenched approximation. This result isconsistent with Ref.

[15,11] and Ref. [10].5

B. Large Nc AnalysisThe preceding diagrammatic analysis is suggestive but not conclusive since it gives noinformation about the analytic properties of the diagram and we do not know whether thereis any spectral strength corresponding to nucleon plus pion states.

The presence of suchspectral strength will lead to nonanalytic behavior in m2π around zero. How can one learnwhether there is any nonanalytic behavior with respect to m2π in quenched QCD given thefact that explicit simulations with light quark masses are currently impractical?

Large NcQCD [17,18] provides considerable insight. The key point is that QCD to leading order ina 1/Nc expansion is quenched.

As shown by ‘t Hooft [17], in any diagram there is a 1/Ncsuppression factor associated with each closed fermion loop in a large Nc expansion. Thusthe diagrams which contribute to the leading order expression for any correlation functionhave the minimum number of fermion loops.

This is precisely the condition imposed bythe quenched approximation. It is amusing to note that the nonquenched diagram of figure2b, commonly thought to give rise to pionic dressings of the nucleon are 1/Nc suppressedrelative to that of figure 2c.Of course, the leading order large Nc approximation is a more drastic approximation thanthe quenched approximation since some graphs which do not contain internal fermion loops(e.g.

some non-planar graphs) are also 1/Nc suppressed. Since the large Nc approximationis more severe than the quenched approximation and contains the quenched approximationit is clear that if there is pion loop physics (as evidenced by nonanalytic behavior in m2π)in large Nc QCD, then the same physics should be present in the less severe quenchedapproximation.There are two distinct arguments which suggest that pion loop physics is present in theleading order large Nc approximation.

One way is to study large Nc hadrodynamics [19–21](i.e. a dynamical model based on hadron degrees of freedom).

The other is via the study ofmodels such as the Skyrme model which capture the correct leading order Nc physics fromQCD.The basic idea of large Nc hadrodynamics is that if one produces an effective hadronicmodel which reproduces the underlying physics of QCD then all of the parameters of thishadrodynamical model must scale with Nc in the manner prescribed by large Nc QCD[17,18]:Γnm ∼N(1−n/2)c;Mm ∼1;MB ∼Nc;gmBB ∼N1/2c;ΛBff ∼1 ,(3)where Γnm is a meson n-point vertex, Mm a meson mass, MB the baryon mass, gmBB ameson baryon coupling and ΛBff is a baryon form factor mass.The study of loops inlarge Nc hadrodynamics goes back to Witten [18] who showed that meson loops always givecorrections to the tree level meson properties which are suppressed in 1/Nc.Consider the one meson loop contribution to the meson propagator. The propagatorsin the loop are all of order unity but from (3) the three-meson vertices are order N−1/2c.There are two such vertices so the net effect is order 1/Nc which is indeed suppressedcompared to the leading order propagator which is of order unity.Note however, thatthis argument does not exclude the possibility of mesonic loop dressings to mesons in thequenched approximation since the large Nc approximation is more severe than the quenchedapproximation.6

As studied elsewhere [19–21], the situation is more interesting when one studies baryonproperties in large Nc hadrodynamics. For present purposes, the important issue is the mesonloop contribution to the nucleon mass.

Consider the nucleon self-energy at the hadronic level.This receives a contribution from a one-meson-loop diagram as in figure 2a. It is easy to seethat this self-energy is order Nc.

There are two N1/2cfactors at the meson-baryon couplings.The meson propagator is order unity since the mass is order one and the loop integral iscutoffby the form factor whose falloffis also order unity. In the large Nc limit where nucleonrecoil can be neglected, the nucleon propagator (working in the rest frame of the nucleon)goes like the inverse of the meson three momentum, i.e.order 1 .

Thus, one sees that inlarge Nc hadrodynamics (which is assumed to correctly reflect large Nc QCD), one mesonloop contributions to the nucleon mass are of order Nc which is leading order in Nc countingaccording to (3). As discussed above, the leading order terms in the 1/Nc expansion arequenched and hence the one-meson-loop physics, including one-pion-loop physics, should bepresent in the quenched approximation.The preceding argument is consistent with the leading nonanalytic behavior for pionloops obtained in χPT.

Perhaps the easiest place to look for this nonanalyticity is to studyd2MN/d(m2π)2.The leading order nonanalyticity from χPT comes from a one-pion-loopdiagram as in figure 2a. One obtainsd2 MNd(m2π)2 = −9128πg2Amπ f 2π+ O(1)(4)It is well known that gA ∼Nc and fπ ∼N1/2cso thatd2 MNd(m2π)2 ∼Nc(5)Integrating twice with respect m2π gives this pion loop contribution to MN scaling like Ncwhich is the leading order.The preceding argument has a drawback in that it is based on large Nc hadrodynamicsand not directly on QCD.

While it is highly plausible that the Nc behavior of hadrodynamics(including baryons) reproduces that of QCD, it has not been proven rigorously. Ideally, oneshould work directly in large Nc QCD which unfortunately is not possible.

Instead, one canlook at models which are believed to correctly reproduce QCD’s large Nc behavior. Theanalysis here will be based on the Skyrme model although all chiral large Nc soliton modelsof the nucleon (e.g.

the chiral bag model, the chiral quark meson soliton model) behave thesame way. The issue of whether the pion cloud physics associated with loops is present inlarge Nc QCD or models of large Nc QCD (such as the Skyrme model) is complicated.

Forthe present purposes it is reasonable to assert that pion cloud physics is present in largeNc QCD if various quantities calculated to leading order in Nc depend on the pion massin the same way as pion loop calculations of the same quantities in a hadronic picture.The lore of χPT is that the leading nonanalytic behavior in m2π is given by one-pion-loopgraphs in a hadronic model.Thus, ultimately the issue is whether the large Nc modelcalculations produce the same nonanalytic behavior near the chiral limit as one-pion-loophadronic calculations.Does the Skyrme model correctly reproduce the leading nonanalytic behavior in m2πpredicted in χPT? The answer is a qualified yes [22].

As discussed in detail in Ref. [22],7

the Skyrme model reproduces the leading nonanalytic properties of χPT in the followingsense. If one confines attention to vector-isovector and scalar-isoscalar operators (i.e.

whoseexpectation values which do not vanish in the hedgehog intrinsic state), and chooses theSkyrme model parameters to give the correct value of gA, then the Skyrme model predictionfor nucleon matrix elements will precisely reproduce the leading nonanalytic behavior ofχPT up to an overall factor which depends only on the quantum numbers of the operator.For scalar-isoscalar operators, the Skyrme model will always give a coefficient for theleading nonanalytic term which is a factor of three larger than χPT. The origin of thisfactor of three lies in the fact that the Skyrme model result depends on the order in whichthe chiral and large Nc limits are taken.

In χPT, it is explicitly assumed that the pion mass issmall compared to all relevant hadronic scales in the problem. Thus, the only physical statesenergetically near the nucleon are states with nucleon plus one pion.

On the other hand, inthe large Nc limit for a hedgehog model the ∆is degenerate with the nucleon. The N-∆splitting goes as 1/Nc.

Therefore, in the large Nc limit, loops with π-∆intermediate statesshould be included along with π-N intermediate states when doing χPT. The inclusion ofπ-∆intermediate states in χPT, assuming degenerate N and ∆masses and using gπN∆ascalculated in the Skyrme model, precisely accounts for the factor of three [22].

Therefore,at least for this class of observables, the nonanalytic behavior associated with pion loops isin fact present.An explicit example may help clarify this point. Once again, consider d2 MN/d(m2π)2.The one pion loop contribution in χPT (including only nucleon intermediate states is givenin (4).

If we also include a diagram analogous to figure 2a with a ∆-π intermediate stateand assume M∆= MN with gπN∆given by the Skyrme model value we obtaind2 MNd(m2π)2 = −27128πg2Amπ f 2π+ O(1)(6)Here the coefficient is precisely three times the usual χPT result.One can also calculate d2 MN/d(m2π)2 directly from the Skyrme model. With the identityd MNd(m2π) =⟨N | 14 f 2π tr(U −1) | N ⟩,(7)along with the asymptotic properties of U = exp(i⃗τ · ⃗φ/fπ) where ⃗φ is the nonlinear realiza-tion of the pion field, and the standard Skyrme model expression for gA, one can computed2 MN/d(m2π)2 rather easily.

The result is found to be the expression of (6). Thus, we seeexplicitly that a 1/Nc model such as the Skyrme model does have nonanalytic behavior in1/m2π precisely as one anticipates from a calculation of pion dressings of the nucleon with a∆degenerate in mass.IV.

SUMMARYIn summary, we have given heuristic—but compelling—arguments as to why simulationsof nucleons in quenched QCD contain pion cloud physics which chiral perturbation theoryascribes to one pion loop contributions at the hadronic level. We have shown that diagramssurviving in the quenched approximation contain pieces which “look like” nucleon plus pion8

states. We have also argued that one can use the 1/Nc approximation as a surrogate forthe quenched approximation since the 1/Nc approximation is quenched.

Two distinct ar-guments, one based on large Nc hadrodynamics and one based on 1/Nc hedgehog modelssuch the Skyrme model, both suggest that pion loop physics is present in large Nc QCD andhence in quenched QCD. These results contrast the superficial perception that the quenchedapproximation is incapable of including the physics of pionic dressings of baryons.We thank Wojciech Broniowski and Manoj Banerjee for helpful conversations.

D.B.L.thanks Richard Woloshyn and Terry Draper for early discussions which stimulated his in-terest in these issues. This work is supported in part by the U.S. Department of Energyunder grant DE-FG05-87ER-40322.

T.D.C. acknowledges additional financial support fromthe National Science Foundation though grant PHY-9058487.Thomas D. Cohen and Derek B. LeinweberDepartment of Physics, University of MarylandCollege Park , MD 207429

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[4] M. K. Banerjee, W. Broniowski, and T. D. Cohen, in Chiral Solitons, edited by K. F.Liu, (World Scientific, Singapore, 1987). [5] J. Gasser, M. E. Sainio, and A. Suarc, Nucl.

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FIGURESFIG. 1.

Skeleton diagrams of disconnected quark loops connected to external sources whichare generally not included in quenched QCD analyses. Diagram (a) contributes to the three-pointcorrelation function of a baryon current matrix element.

Diagram (b) contributes to the two-pointcorrelation function of an iso-scalar meson. The diagrams may be dressed with an arbitrary numberof gluons.FIG.

2. Time ordered diagrams for one-pion-loop dressings of the nucleon.

Time flows fromleft to right. For illustrative purposes we have selected the proton with a π+-n intermediate statefor all three diagrams.

Figure (a) illustrates the hadronic level dressing and figure (b) describesa quark level diagram na¨ıvely thought to exclusively account for pionic dressings of the nucleon.This diagram is excluded in the quenched approximation.Figure (c) illustrates a quark leveldiagram whose quantum numbers overlap with that of figure (a) and which survives in the quenchedapproximation.FIG. 3.

Time ordered diagrams for two pion intermediate states of the ρ-meson. Figure (a)illustrates the hadronic level intermediate state π0 π+.

Figure (b) describes a quark level diagramanalogous to that in figure 1c but which has no overlap with that of figure 2a. Figure (c) illustratesa quark level diagram which has overlap with a two meson intermediate state.

However, one of themesons is an isoscalar and cannot be identified with a pion.11

Submittedto:DOE/ER/0-CommentsonNuclearandParticlePhysicsU.ofMDPP#-0ThePionCloudInQuenchedQCDThomasD.CohenandDerekB.LeinweberDepartmentofPhysicsUniversityofMaryland,CollegePark,MD0August,DiagrammaticmethodsandLargeNcQCDareusedtoarguethatthenucleonascalculatedinquenchedQCDcontainsphysicswhichcanbeascribedtothepioncloudofthenucleon.Inparticular,itisarguedthatthephysicscorrespondingtoone-pionloopsinchiralperturbationtheoryareincludedinthequenchedapproximation.TypesetUsingREVTEX

I.INTRODUCTIONAtthepresenttime,themostpromisingtechniqueforeventuallyderivingthelowenergypropertiesofhadronsdirectlyfromquantumchromodynamics(QCD)isvianumericalMonteCarlosimulationsofthefunctionalintegralinalatticeregularizedversionofthetheory.Unfortunately,giventhecomputationalpowercurrentlyavail-able,itisnotpossibletocalculatehadronpropertiesinacompletelyrealisticmannerinwhichquarksofphysicalmassmoveinalargelatticevolumewithlatticespacingsneenoughtoguaranteeasymptoticscaling.Acommonapproximationthatsignif-icantlyreducesthenumericaldemandsofthesimulationsistheso-calledquenchedapproximation.Thecomputationalgainsassociatedwiththisapproximationallowonetogreatlyimproveonsystematicandstatisticaluncertainties,andtoprobetheextremesoftheparameterspace.IntheEuclideanspaceformulationoftheQCDfunctionalintegral,theweightingfunctionhasthefollowingcontinuumformW=det[D+mq]exp(SYM);()whereSYMistheEuclideanactionforapureYang-Millstheory(i.e.onlygluons).IntegrationoverthequarkGrassmanneldsgivesrisetothefunctionaldetermi-nant.Thequenchedapproximationconsistsofsettingthefunctionaldeterminanttoaconstant,independentofthegluoneldconguration.Thus,inthequenchedapproximation,theweightingfunctionissimpliedtoWq=exp(SYM):()ThisformisobviouslyamiabletoMonteCarlointegrationtechniques.Fromadiagrammaticpointofview,theomissionofthefermiondeterminantintheweightingfunctioncorrespondstotheneglectofalldiagramscontainingclosedquarkloopswhicharenotconnectedtoexternalsources.Giventhefactthatquenchedcalculationswillcontinuetobeusedforsometimetocome,itisanimportantpracticalproblemtodeterminewhat,ifany,essentialphysicsislostwhenonemakesthequenchedapproximation.ThisissueisparticularlyimportantifonewishestousetheresultsofquenchedlatticeQCDcalculationstogaininsightintowhatphysicsshouldbeincludedinmodelsofhadrons.Thephysicsofthepioncloudisbelievedtoplayanessentialroleinhadronicstructure.Severalmodelsofthenucleon,includingtheSkyrmion[],thecloudybagmodel[],thehybridorchiralbagmodel[]andchiralquark-mesonmodels[]stresstheroleofthepioncloudinthenucleonstructure.Thus,thequestionofwhetherthequenchedapproximationtoQCDcontainsthephysicsofthepioncloudbecomesimportant.Inthiscomment,wewillarguethatpioncloudeectsare,infact,includedinquenchedcalculationsofnucleonproperties.Thisresultmayseemtobecounterintuitive,butasshallbeshown,thisconclusionisreachedbothinalargeNcanalysisandinanexplicitstudyofdiagrams.Theargumentsaresomewhatheuristicbutwebelieverathercompelling.

Apartfromtheroleofthepioncloudinvariousmodels,thereisalsoasystematicmethodtoestimatetheroleofpionsinhadronicobservables,namelychiralpertur-bationtheory(PT).Thisapproachisanexpansionwhichisbasedonaseparationofscalesbetweenthepionmassandotherscalesinhadronicphysics[{].Thisap-proachshouldrigorouslyreproducethelowenergypropertiesofQCDifthequarkmass(andhencethepionmasssquared)issucientlylight.Chiralperturbationtheorypredictsthatcertainquantitiesarenonanalyticinthequarkmassasoneap-proachesthechirallimitduetotheeectsoftheinfraredbehaviorofpionloops.Forourpresentpurpose,wenotethatonecanusetheexistenceofthisnonanalyticbehaviorasasignatureofpioncloudeects.Beforeproceedingitisworthnotingthatthetheneedforquenchedcalculationsisparticularlystrongifonewishestostudytheroleofthepioncloud.Atpresent,completelyrealisticlatticesimulationsofpioncloudeectsarenotpossible.Thebasicdicultystemsfromtheuseofunphysicallylargequarkmassesinthelatticesimulationswhichleadstopionswhichareheavyandconsequentlyofshortrange.Anumberofchallengesarepresentedasoneattemptstodecreasethequarkmass.Thelongrangenatureofthepiondemandslargerphysicallatticevolumeswhichultimatelyleadstolargerlatticesifonewishestomaintainareasonablelatticespacing.Furthermore,calculationsoffermionpropagatorsandMonte-Carloestimatesofthefunctionaldeterminantbecomeincreasinglydicultasthequarkmassdropsandconvergenceofthealgorithmsslows.FornitecomputerresourcesonecanalwaysinvestigatelargerlatticevolumesandlighterquarkmassesinthequenchedapproximationthaninfullQCD.ItislikelythattherewillbeasignicantperiodoftimewhenquenchedcalculationsareabletoprobetheparameterspaceinareaswherepioncloudeectsareexpectedtobesignicantwhereasfullQCDsimulationswillremainrestrictedtoamorelimitedparameterspace.Thus,itisimportanttoestablishwhetherornotquenchedcalculationsarecapableofdescribingthephysicsofthepioncloud.Inthisspirit,itisimportanttomakeamorerestrictivedenitionofthequenchedapproximation.Withintheusualdenitionthereremaindiagramswhichareex-tremelycumbersometocalculateandasaresultaregenerallyomittedinquenchedanalyses.Figuredisplaystwosuchdiagramswithquarkloopsconnectedtoexter-nalsources.Thesequarklinedisconnecteddiagramsrequireknowledgeofthespatialdiagonalelementsoftheinversefermionmatrixwhereasastandardquarkpropagatorrequiresknowledgeofasinglecolumn.Hence,theyarecomputationallyquitedi-cultandforthepurposesofthisinvestigationwewillbroadenourdenitionofthequenchedapproximationtoexcludediagramsofthetypeillustratedingure.Tomakecontactwithphysicalobservables,itisnecessarytoextrapolatefromthelargevaluesofquarkmasscurrentlyusedinlatticecalculationstotheirphysicalmasses.Ashasbeenstressedelsewhere[],nonanalytictermsinPTcanleadtoimportantcorrectionstothisextrapolationincalculationsofchargeradiiandotherobservables.Itisclearlyofimportancetodeterminewhetherthesechiralcorrections

FIG.SkeletondiagramsofdisconnectedquarkloopsconnectedtoexternalsourceswhicharegenerallynotincludedinquenchedQCDanalyses.Diagram(a)contributestothethree-pointcorrelationfunctionofabaryoncurrentmatrixelement.Diagram(b)contributestothetwo-pointcorrelationfunctionofaniso-scalarmeson.Thediagramsmaybedressedwithanarbitrarynumberofgluons.

totheextrapolationsarepresentinthequenchedapproximationoronlyinfullQCD.II.PIONPROPERTIESThequestionofwhetherthephysicsofpionloopsispresentinthequenchedap-proximationhasbeendiscussedpreviously[0,].Thisdiscussionhasbeencenteredonthepropertiesofpionsandspecicallywhetherthepionmasssquaredcontainsatermproportionaltomqln(mq)whichispredictedinPT.TheargumentisthatanewformofPTmustbedevelopedtodealwithquenchedQCD.InthisquenchedversionofPTtherearenononanalyticeectsarisingfromthepioncloudsurround-ingapion.Thisdoesnotmeanthatnonanalyticbehaviorisabsentfrommesonicobservables.Nonanalyticchiralbehaviorcanhaveitsorigininthecloudassociatedwiththeisoscalar0meson.Whileinnaturethe0isheavyduetoanomaliesandtopologicaleects[],inthequenchedapproximationthe0isdegeneratewiththepionand0loopscanyieldnonanalyticchiralbehavior.ThisdiscussionsuggeststhatquenchedQCDcalculationsofpionpropertieswillnotcorrectlyreproducethemesoncloudeectsoffullQCD.Forexample,infullQCDthepionchargeradiusdivergeslogarithmicallyasthequarkmassgoestozero[];quenchedPTpredictsitwillremainniteinquenchedQCD.Thekeypointisthatthe0cloudcannotcorrectlysimulatetheroleofthepioncloudforelectromagneticpropertiessincethe0isneutralanddoesnotcoupledirectlytophotons.III.NUCLEONPROPERTIESA.DiagrammaticconsiderationsWewillfocusourattentiononthepropertiesofthenucleonandwillarguethatpionloopeectsdocontributetonucleonpropertieseveninthequenchedapproxima-tion.Atrstthoughtthismayseemabsurdsincepionloopsrequireanintermediatestatewithaminimumofoneqqpairandtheformationofquark-antiquarkpairsisapparentlyforbiddenbytherestrictionsimposedbythequenchedapproximations.However,asnotedinRef. [{]thisrestrictionisonlyapparent.Althoughthequenchedapproximationlimitsquarklinestothosewhichareconnectedtoexternalcurrents,thequarkpropagatorsusedarefullyrelativisticDiracpropagators.Suchpropagatorscontain\Z-graphs"inwhichthequarkisscatteredintoanegativeenergystateandback.WiththeconventionalholeinterpretationoftheDiracpropagatorsuchprocessesarethecreationandannihilationofqqpairs.Therestrictionimposedbythequenchedapproximationismerelythatonceavirtualpairiscreatedalongonequarklineitmustbeannihilatedonthesameline.Thus,itisatleastpossiblethatquenchedQCDmaycontainqqpairsandsuchpairsmightbepionicinnature.Thequestionofwhetherpion-loopphysicsisincludedinquenchedcalculationsofcorrelationfunctionsultimatelycomesdowntothequestionofwhetherintermediatestatesarereachedwhichhaveoverlapwithphysicalstatescontainingpionsplusother

hadrons.Thissuggeststhatoneshouldstudyold-fashionedenergydenominatortypetime-ordereddiagramsandaskwhetheronecanreachstateswhichcanbeexpressedastheproductofmorethanonecolorsingletoperatoroperatingonthevacuumwithatleastoneoftheseoperatorshavingthequantumnumbersofthepion.Itisclearthatthisisanecessarybutnotsucientconditiontoestablishthepresenceofpionloopphysics.Inaddition,itmustbeshownthatthesingularitystructurecorrespondstopionplushadronstates.Atthehadroniclevel,someofthepioncloudphysicswhichaccountsfornonan-alyticbehaviorinPTcanberepresentedasasinglepionloop.Theone-pion-loopphysicsisrepresentedbythehadronictime-ordereddiagramingurea.Thecorre-spondingdiagramatthequarklevelcommonlythoughttogiverisetopionicdressingsbutnotsurvivinginthequenchedapproximationisillustratedingureb.Comparethiswiththequarkleveldiagramforanucleoncorrelationfunctioningurecwhichsurvivesinthequenchedapproximation[].Theintermediatestatecontainsaqqstructure(atthetopofgurec)whichhasbothcolor-singletandcolor-octetpieces.Thequenchednatureofthecalculationrequiresthattheantiquarkmustbethesameavorasoneoftheoriginalquarksinthenucleoninterpolatingeld.Sincetherearealwaystwodistinctavorsofquarkinanucleon,theqqstructurecanbeeitherisospinzeroorone.Thethreequarkstructurebelowtheqqstructurehasapiecewhichiscolorsingletandisospinonehalf.Itishighlyplausiblethatthepartofthisgraphwhichconsistsofacolorsingletisospinoneqqpiecealongwithacolorsingletisospinonehalfqqqpiecehassomeoverlapwiththephysicalpion-nucleonscatteringstate.Henceitappearsthatgureccontains,amongotherthings,theessentialphysicsofgurea.Ofcourse,thediagramingurecisonlyoneofaninniteclassofdiagramswhichappeartohavenonzerooverlapwithpion-nucleonscatteringstates.Onecanaddtothediagramanarbitrarynumberofgluons.Itisworthnotingatthispointthatasimilardiagrammaticanalysiswouldnotgivepionicintermediatestatesformesoncorrelationfunctions.Forexampleingureaweshowaoneloophadronicdiagramforthemesonchannel.Theimaginarypartofthisgraphgivesthetotwopiondecay.Inanalogytogurecweconstructgureb.Thisdiagramcertainlycontainstwoisovectorstructures.However,unlikeinthenucleoncaseingurec,thesestructuresdonothavepionquantumnumbers.Theyarecolorordiquarkswithbaryonnumber=.Alternatively,onecouldarrangethediagramasingurec.Thereisacomponentinwhichbothstructuresarecolorsingletsbutoneseesthatatleastoneofthestructuresisisoscalarandcannotrepresentapion.Theisoscalarpiecehasoverlapwiththe0mesondiscussedinRef.[0].Thus,thissimplediagrammaticanalysisshowsthatthephysicsofgureacannotbereproducedinthequenchedapproximation.ThisresultisconsistentwithRef.[,]andRef. [0].

FIG.Timeordereddiagramsforone-pion-loopdressingsofthenucleon.Timeowsfromlefttoright.Forillustrativepurposeswehaveselectedtheprotonwitha+-nintermediatestateforallthreediagrams.Figure(a)illustratesthehadronicleveldressingandgure(b)describesaquarkleveldiagramnavelythoughttoexclusivelyaccountforpionicdressingsofthenucleon.Thisdiagramisexcludedinthequenchedapproximation.Figure(c)illustratesaquarkleveldiagramwhosequantumnumbersoverlapwiththatofgure(a)andwhichsurvivesinthequenchedapproximation.

FIG.Timeordereddiagramsfortwopionintermediatestatesofthe-meson.Figure(a)illustratesthehadroniclevelintermediatestate0+.Figure(b)describesaquarkleveldiagramanalogoustothatingurecbutwhichhasnooverlapwiththatofgurea.Figure(c)illustratesaquarkleveldiagramwhichhasoverlapwithatwomesonintermediatestate.However,oneofthemesonsisanisoscalarandcannotbeidentiedwithapion.

B.LargeNcAnalysisTheprecedingdiagrammaticanalysisissuggestivebutnotconclusivesinceitgivesnoinformationabouttheanalyticpropertiesofthediagramandwedonotknowwhetherthereisanyspectralstrengthcorrespondingtonucleonpluspionstates.Thepresenceofsuchspectralstrengthwillleadtononanalyticbehaviorinmaroundzero.HowcanonelearnwhetherthereisanynonanalyticbehaviorwithrespecttominquenchedQCDgiventhefactthatexplicitsimulationswithlightquarkmassesarecurrentlyimpractical?LargeNcQCD[,]providesconsiderableinsight.ThekeypointisthatQCDtoleadingorderina=Ncexpansionisquenched.Asshownby`tHooft[],inanydiagramthereisa=NcsuppressionfactorassociatedwitheachclosedfermionloopinalargeNcexpansion.Thusthediagramswhichcontributetotheleadingorderexpressionforanycorrelationfunctionhavetheminimumnumberoffermionloops.Thisispreciselytheconditionimposedbythequenchedapproxi-mation.Itisamusingtonotethatthenonquencheddiagramofgureb,commonlythoughttogiverisetopionicdressingsofthenucleonare=Ncsuppressedrelativetothatofgurec.Ofcourse,theleadingorderlargeNcapproximationisamoredrasticapprox-imationthanthequenchedapproximationsincesomegraphswhichdonotcontaininternalfermionloops(e.g.somenon-planargraphs)arealso=Ncsuppressed.SincethelargeNcapproximationismoreseverethanthequenchedapproximationandcontainsthequenchedapproximationitisclearthatifthereispionloopphysics(asevidencedbynonanalyticbehaviorinm)inlargeNcQCD,thenthesamephysicsshouldbepresentinthelessseverequenchedapproximation.TherearetwodistinctargumentswhichsuggestthatpionloopphysicsispresentintheleadingorderlargeNcapproximation.OnewayistostudylargeNchadrody-namics[{](i.e.adynamicalmodelbasedonhadrondegreesoffreedom).TheotherisviathestudyofmodelssuchastheSkyrmemodelwhichcapturethecorrectleadingorderNcphysicsfromQCD.ThebasicideaoflargeNchadrodynamicsisthatifoneproducesaneectivehadronicmodelwhichreproducestheunderlyingphysicsofQCDthenalloftheparametersofthishadrodynamicalmodelmustscalewithNcinthemannerprescribedbylargeNcQCD[,]:nmN(n=)c;Mm;MBNc;gmBBN=c;Bff;()wherenmisamesonn-pointvertex,Mmamesonmass,MBthebaryonmass,gmBBamesonbaryoncouplingandBffisabaryonformfactormass.ThestudyofloopsinlargeNchadrodynamicsgoesbacktoWitten[]whoshowedthatmesonloopsalwaysgivecorrectionstothetreelevelmesonpropertieswhicharesuppressedin=Nc.Considertheonemesonloopcontributiontothemesonpropagator.Theprop-agatorsinthelooparealloforderunitybutfrom()thethree-mesonverticesareorderN=c.Therearetwosuchverticessotheneteectisorder=Ncwhichis

indeedsuppressedcomparedtotheleadingorderpropagatorwhichisoforderunity.Notehowever,thatthisargumentdoesnotexcludethepossibilityofmesonicloopdressingstomesonsinthequenchedapproximationsincethelargeNcapproximationismoreseverethanthequenchedapproximation.Asstudiedelsewhere[{],thesituationismoreinterestingwhenonestudiesbaryonpropertiesinlargeNchadrodynamics.Forpresentpurposes,theimportantissueisthemesonloopcontributiontothenucleonmass.Considerthenucleonself-energyatthehadroniclevel.Thisreceivesacontributionfromaone-meson-loopdiagramasingurea.Itiseasytoseethatthisself-energyisorderNc.TherearetwoN=cfactorsatthemeson-baryoncouplings.Themesonpropagatorisorderunitysincethemassisorderoneandtheloopintegraliscutobytheformfactorwhosefalloisalsoorderunity.InthelargeNclimitwherenucleonrecoilcanbeneglected,thenucleonpropagator(workingintherestframeofthenucleon)goesliketheinverseofthemesonthreemomentum,i.e.order.Thus,oneseesthatinlargeNchadrodynamics(whichisassumedtocorrectlyreectlargeNcQCD),onemesonloopcontributionstothenucleonmassareoforderNcwhichisleadingorderinNccountingaccordingto().Asdiscussedabove,theleadingordertermsinthe=Ncexpansionarequenchedandhencetheone-meson-loopphysics,includingone-pion-loopphysics,shouldbepresentinthequenchedapproximation.TheprecedingargumentisconsistentwiththeleadingnonanalyticbehaviorforpionloopsobtainedinPT.PerhapstheeasiestplacetolookforthisnonanalyticityistostudydMN=d(m).TheleadingordernonanalyticityfromPTcomesfromaone-pion-loopdiagramasingurea.OneobtainsdMNd(m)=gAmf+O()()ItiswellknownthatgANcandfN=csothatdMNd(m)Nc()IntegratingtwicewithrespectmgivesthispionloopcontributiontoMNscalinglikeNcwhichistheleadingorder.TheprecedingargumenthasadrawbackinthatitisbasedonlargeNchadrody-namicsandnotdirectlyonQCD.WhileitishighlyplausiblethattheNcbehaviorofhadrodynamics(includingbaryons)reproducesthatofQCD,ithasnotbeenprovenrigorously.Ideally,oneshouldworkdirectlyinlargeNcQCDwhichunfortunatelyisnotpossible.Instead,onecanlookatmodelswhicharebelievedtocorrectlyrepro-duceQCD'slargeNcbehavior.TheanalysisherewillbebasedontheSkyrmemodelalthoughallchirallargeNcsolitonmodelsofthenucleon(e.g.thechiralbagmodel,thechiralquarkmesonsolitonmodel)behavethesameway.TheissueofwhetherthepioncloudphysicsassociatedwithloopsispresentinlargeNcQCDormodelsoflargeNcQCD(suchastheSkyrmemodel)iscomplicated.Forthepresentpurposes0

itisreasonabletoassertthatpioncloudphysicsispresentinlargeNcQCDifvariousquantitiescalculatedtoleadingorderinNcdependonthepionmassinthesamewayaspionloopcalculationsofthesamequantitiesinahadronicpicture.TheloreofPTisthattheleadingnonanalyticbehaviorinmisgivenbyone-pion-loopgraphsinahadronicmodel.Thus,ultimatelytheissueiswhetherthelargeNcmodelcalcu-lationsproducethesamenonanalyticbehaviornearthechirallimitasone-pion-loophadroniccalculations.DoestheSkyrmemodelcorrectlyreproducetheleadingnonanalyticbehaviorinmpredictedinPT?Theanswerisaqualiedyes[].AsdiscussedindetailinRef. [],theSkyrmemodelreproducestheleadingnonanalyticpropertiesofPTinthefollowingsense.Ifoneconnesattentiontovector-isovectorandscalar-isoscalaroperators(i.e.whoseexpectationvalueswhichdonotvanishinthehedgehogintrinsicstate),andchoosestheSkyrmemodelparameterstogivethecorrectvalueofgA,thentheSkyrmemodelpredictionfornucleonmatrixelementswillpreciselyreproducetheleadingnonanalyticbehaviorofPTuptoanoverallfactorwhichdependsonlyonthequantumnumbersoftheoperator.Forscalar-isoscalaroperators,theSkyrmemodelwillalwaysgiveacoecientfortheleadingnonanalytictermwhichisafactorofthreelargerthanPT.TheoriginofthisfactorofthreeliesinthefactthattheSkyrmemodelresultdependsontheorderinwhichthechiralandlargeNclimitsaretaken.InPT,itisexplicitlyassumedthatthepionmassissmallcomparedtoallrelevanthadronicscalesintheproblem.Thus,theonlyphysicalstatesenergeticallynearthenucleonarestateswithnucleonplusonepion.Ontheotherhand,inthelargeNclimitforahedgehogmodeltheisdegeneratewiththenucleon.TheN-splittinggoesas=Nc.Therefore,inthelargeNclimit,loopswith-intermediatestatesshouldbeincludedalongwith-NintermediatestateswhendoingPT.Theinclusionof-intermediatestatesinPT,assumingdegenerateNandmassesandusinggNascalculatedintheSkyrmemodel,preciselyaccountsforthefactorofthree[].Therefore,atleastforthisclassofobservables,thenonanalyticbehaviorassociatedwithpionloopsisinfactpresent.Anexplicitexamplemayhelpclarifythispoint.Onceagain,considerdMN=d(m).TheonepionloopcontributioninPT(includingonlynucleonin-termediatestatesisgivenin().Ifwealsoincludeadiagramanalogoustogureawitha-intermediatestateandassumeM=MNwithgNgivenbytheSkyrmemodelvalueweobtaindMNd(m)=gAmf+O()()HerethecoecientispreciselythreetimestheusualPTresult.OnecanalsocalculatedMN=d(m)directlyfromtheSkyrmemodel.WiththeidentitydMNd(m)=hNjftr(U)jNi;()

alongwiththeasymptoticpropertiesofU=exp(i~~=f)where~isthenonlinearrealizationofthepioneld,andthestandardSkyrmemodelexpressionforgA,onecancomputedMN=d(m)rathereasily.Theresultisfoundtobetheexpressionof().Thus,weseeexplicitlythata=NcmodelsuchastheSkyrmemodeldoeshavenonanalyticbehaviorin=mpreciselyasoneanticipatesfromacalculationofpiondressingsofthenucleonwithadegenerateinmass.IV.SUMMARYInsummary,wehavegivenheuristic|butcompelling|argumentsastowhysim-ulationsofnucleonsinquenchedQCDcontainpioncloudphysicswhichchiralpertur-bationtheoryascribestoonepionloopcontributionsatthehadroniclevel.Wehaveshownthatdiagramssurvivinginthequenchedapproximationcontainpieceswhich\looklike"nucleonpluspionstates.Wehavealsoarguedthatonecanusethe=Ncapproximationasasurrogateforthequenchedapproximationsincethe=Ncapprox-imationisquenched.Twodistinctarguments,onebasedonlargeNchadrodynamicsandonebasedon=NchedgehogmodelssuchtheSkyrmemodel,bothsuggestthatpionloopphysicsispresentinlargeNcQCDandhenceinquenchedQCD.Thesere-sultscontrastthesupercialperceptionthatthequenchedapproximationisincapableofincludingthephysicsofpionicdressingsofbaryons.WethankWojciechBroniowskiandManojBanerjeeforhelpfulconversations.D.B.L.thanksRichardWoloshynandTerryDraperforearlydiscussionswhichstim-ulatedhisinterestintheseissues.ThisworkissupportedinpartbytheU.S.Depart-mentofEnergyundergrantDE-FG0-ER-0.T.D.C.acknowledgesadditionalnancialsupportfromtheNationalScienceFoundationthoughgrantPHY-0.ThomasD.CohenandDerekB.LeinweberDepartmentofPhysics,UniversityofMarylandCollegePark,MD0

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출처: arXiv:9212.225 • 원문 보기