The quark model pion and the Goldstone pion

Nudear P1lysics A 374(1982) S lab lc ® North-Holland PublishingCo., AmstcrdemNot to be reproduced by photoprint or microfilm without written permissionfrom the publisher.

THE QUARK MODEL PION AND THE GOLDSTONE PION

K. JOHNSON

Center for Theoretical Physics and Laboratory of Nuclear ScienceMassachusetts Institute of Technology

Cambridge, MA 02139USA

Abstract : An approximate vacuum wave-function for massless quarks is proposedwhich describes a collective, Goldstone pion with the internal structureof a quark model pion .

1 . Introduction

It is presently believed that the fundamental theory of the strong interac-tions is Quantum Chromodynamics (QCD) . This is a theory based upon a singlescale, usually called n, whose value is presently known only crudely (100 McV~n<_500 MeV) . At distances which are asymptotically small in comparison to 1/n, thecoupling between the strongly interacting particles vanishes . At a distance ofthe order 1/n, the coupling is of order 1 . At longer distances it is no longerYery meaningful to parametrize the interaction with a pert~urbative coupling para-meter . Here the theory is only qualitatively understood . )

Instead, at present, the description of strong interactions at this andlonger scales is largely based upon three "models" . Here I would like to brieflyreview these models . Then I would like to discuss how one aspect of these modelsmight be related to the fundamental theory . This is the relationship between thequark model pion, and the Goldstone or PCAC pion .

Ultimately one hopes to be able to justify the use of all of the models byobtaining them as approximations to the fundamental theory . In this way, oneshall also determine, in a precise way, their limitations .

Here, the principal concern will be QCb at low energies where the relevantquarks are the u and d (marginally, also the s quark) whose masses are small inunits of n . In fact, for much of our discussion, we shall be most interested inthe limit where this parameter is zero . In this limit, QCD is invariant under the"global" chiral flavor group, SU(2) X SU(2) . By making the hypothesis that thissymmetry is spontaneously broken down to the isospin sub-group, SU(2), particlephysicists have constructed an extremely successful phenomenological theory ofthe interactions of "soft" pious with hadrons

).This phenomenological theory can say almost nothing about hadron structure

since in the model hadrons have point-like interactions . It also says almostnothing about the hadron spectrum, the exception being the existence of the massless Goldstone boson, the pion, whose properties motivated the speculation thatits presence in the family of hadrons was a consequence of a spontaneous breakingof an approximate chiral syrmletry of the underlying Hamiltonian of the stronginteracts pns .

Together with the assumption of spontaneous symmetry breaking oneassumes 31 there is an additional small explicit breaking of chiral symmetry asthe consequence of small "current" masses : that is, explicit, as opposed to spon-taneous chiral breaking is described by

H I = f d3x (mu ùu + md ad)

(1 .1)

with the "current" masses m and an d whose origin comes from physics at some muchshorter distance scale . Wi~h these two hypotheses, a very successful phenomenol-ogy emerged before one had any idea what the chirally "symmetric" part of thebasic Hamiltonian actually involved beyond the necessary hypothesis that it should

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K.JOHNSON

contain quark operators .The version of the phenomenological chiral model which i the simplest and

most widely applied is that based upon effective Lagrangians ~) which containfields for each observed hadron coupled together in the most general way which ischi rally symmetric, with the pion field realizing a non-linear representation forthe chiral transfottnation as a kind of "gauge" field associated with the chiralgroup . As long as this Lagrangian is applied to the domain of "soft" interactionsits predictions are uniformly successful . This domain is defined by the so-called"tree" graphs which are obtained from the Lagrangian . Like all models when pushedtoo far, it fails (these based on "hard" pion calculations or loops) . We nowunderstand why this model should fail (the microscopic theory is QCD)

but so farwe have not really "derived" it since we have not yet found how to compute the"observed" hadron spectrum from QCD. That is, the parameters of the phenomenologi-cal model (masses and coupling constants) have not been calculated from "first"principles . Indeed, even the basic assumption of spontaneous chiral symmetrybreaking is not proven . Although the phenomenological model is excellent for des-cribing soft pion interactions, it fails to describe high energy scattering .

The successful model for describing high energy strrong interactions is theRegge exchange model based upon straight line Regge-trajectories a(t),

The natural domain for this model is high energy and modest t(<O) exchange reac-tions .

However, by analyticity and relativity the model can also be applied to theasymptotic mass spectrum (t>O) . Although this model when applied to the massspectrum is extremely successful (all those beautiful rotational bands) it hasnothing correct to say about the ground states (a ) . However, at one point theRegge model and the chiral model overlap - the piôn like all other hadrons lieson a Regge trajectory - although the properties of the pion trajectory are lesswell established since the trajectory has a low value for ao . (The phenomenologyestablished is tested best when ao is larger .) Again, when pushed beyond its nat-ural domain the failures of this model are numerous So far the model has also

been "derived" from QCD, although the "string model" 5) may provide a bridge be-tween the "Regge-Dual" models and QCD, the parameters a' nd ao remain uncomputed(some think we are close to a semiquantitative estimate 6~ of a' in units of n) .The hadron-Reggeon couplings are not close to being calculable .

The third phenomenological model is the one used to describe hadron structure.In this case there are really

o closely related models, the constituent quarkmodel 7) and the parton model ~~ . In contrast to the other models both are basedupon particles which carry the same quantum numbers as the fields which appearin the fundamental theory, and therefore these models are at present the closestto "derivation" from QCD. The constituent quark model was originally postulatedto merely classify the observed hadron spectrum . Then it was proposed as aliteral model : hadrons are constructed of quarks m$ ing non-relativistically .The present version of the non-relativistic model ~ was provided in 1974 by in-troducing the spin dependent forces which one would expect if non-relativisticquarks were to interact at short distances as they would in perturbative QCD.This model has had many successes, its outstanding failure as a literal model isin its basic inconsistency: for u and d quarks, <p>/m is not small . As a conse-quence there is a tendency to only compare its correct predictions with experimentand to ignore its failures (like the prediction GA/G,, = 5/3 vs .ti1 .2) or introducead-hoc elaborations to "correct" the deficiencies . So far this model has beenuniformly successful when applied to the heavy quarks c and b where <p>/m issmall .

For heavy quarks this model can be said to be derived from QCD at leastin the case of the short distance interaction between the quarks . The long rangespin independent confining interaction has not yet been derived from QCD althoughat present we may be getting closer to achievi9

Sthat .

The relativistic constituent quark model

) (the M.I .T . BAG MODEL) also in-cluding the spin dependent forces associated with QCD encompasses all the success-ful predictions of the non-relativistic version (where they have been worked out),


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removes the basic inconsistency of the non-relativistic version (<p>/m can be any-thing) and adds new information (for example, it finds that the "constituent quarkmass" should really be regarded as the momentum of a massless quark localizedwithin a hadron . Thus, it is of order til/R where R is the characteristic hadronsize 1/R ti 200 MeV) . It also allows one to incorporate the small "current" massesmu and and appearing in (1 .1) as true dynamical masses as opposed to "constituent"masses, and this in turn allows one to relate them to the masses used in the re-normalization group .

It has also allowed one to form a bridgell) to the other extremely success-ful model, the parton model whose domain is the description of short space andtime current density correlations within hadrons . The parton model is the closestmodel to being derived from QCD, indeed, QCD has been directly applied here withgreat success .

Since the quark models of hadron structure are the models which are closestto the fundamental theory, there has been a natural tendency to try to extendthese models in the attempt to encompass the other models, or at least some features of the other models .

This is a very dangerous game .

It is only QCD whichshould encompass all the models . Although the domains of the models are not mutu-ally exclusive and there are many consistency requirements which allow them tocoexist, the perils of double counting may be severe if we try to mix them up .

Presumably, all the valid models must be obtained as approximations to theunderlying theory with their own more or less exclusive domains of validity . Cer-tainly, we have already noted that the models so far discussed most often failwhen they are pushed across each other's more or less exclusive boundaries . Itwould seem that at present the most productive program would be to try to see howthe successful models can be related to the fundamental theory and in that way de-termine what théir inherent limitations may be .

Although this is a very difficultprogram to implement, it seems to me that it is the only program which is likelyto be successful .

Since the heavy quark model is the one which is simplest (there is an addition-al, small parameter n/Mp ), it is the one which probably will turn out to be thebasic testing ground foP the precise predictions of QCD .

However, the important practical domain for the dnderstanding of ordinarymatter is the low energy region . Here I would like to describe the beginning ofa program in which we try to derive the low mass quark model (the static BAS model)by relating it at least crudely to ideas obtained from our present very limitedunderstanding of QCD . In particular, I would like to try to explain how twoseeming conflicting aspects of the quark model and the chiral model may be recon-ciled . This apparent conflict is associated with the role of the pion as aGoldstone boson appearing as a consequence of the spontaneous breakdown of chiralsymmetry and its role as a quark state, just one member of the (0- , 1 - ) super-multiplet of mesons where none of the other members of this multiplet are associa-ted with spontaneous symmetry breaking .

It is here that QCD should have its greatest impact on providing some guideas to the limitations of other "successful" models, such as meson models of thenuclear force and so on .

2 . The spontaneous breaking of chiral symmetry

At present, it is believed that quarkless QCD (that is, the self-coupledgauge field theory in the absence of colored Dirac particles) is responsible forthe principal novel aspect of the strong interaction : color confinement . Thequarks are spectators which respond to the background of the ground state wavefunction of the gauge field theory . The gauge theory is a no-parameter theorysince without the quark masses, the ammeter A is merely an overall scale . In thecase of the low mass quarks (u and d~, the "current" masses (m and m ) have asmall effect on the spectrum .

The most important effect is toumake th~

pion-massfinite .

In the limit of zero quark mass a,P.2 other hadronic parameters shouldremain finite in units of A . This includes things like the mean square chargeradius of the pion, the pion decay constant, and all other dimensionful hadronic

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K. JOHNSON

parameters such as nucleon magnetic moments, and mean square charge radii, e massand width, p mass and width, etc ., etc .

Since in this limit QCD is a no parameter theory, we can see that any overlysimple minded idea about QCD is likely to be wrong . However, the first goal mustbe the understanding of the spontaneous breaking of chiral symmetry no matter howcrude, since many if not all of the finite scales mentioned above are probablyassociated with this .

As many have stressedl2), spontaneous chiral symmetry breaking is most likelya consequence of confinement. Hence, to have a detailed and quantitative picturewe probably have to understand pure QCb first. However, here I would like to makea simple proposal based on a variational principle approach which may be systema-tically improved . The proposal I will make is motivated by the M .I .T . Bag model .However, to be finally justified the proposal to be made here will have to find itsjustification in the context of a detailed understanding of the ground state wavefunction of quark-less QCD.

It is believed that in the QCD ground state color density correlations fallrapidly to zero over distances larger than 1/n . This is the only feature of thiswave function which we shall take for granted here . This in contrast to QEDwhere (at equal times, in some gauge)

in QCD

where 0 means faster than any power.An immediate consequence of the absence of long range correlations in a

gauge theory is confinement : colors can not be separated over distances longerthan of the order 1/n without a cost in energy proportional to the distance ofseparation* .

*One simple way to show this qualitatively is to study the expectation of aWilson loop in the ground state :

L = < P(exP(iJ

dx uAu))>W

where W is a loop and P stands for a path ordered integral . As W becomes largeenough, the loop integral can be factored into many "independent" pieces becauseof the rapid decrease of the correlations in A at distinct space points and thisleads to

L ti exp(-An2 tonst .)

where A is the area enclosed by the loop . By analytic continuation we can letone side of the loop -. iT and the other side be a space interval r, so

L + exp(-iT(rA2)const .)

rA2 const. can then be regarded as the potential of energy spatially separatedheavy quark with color "charges" as/2 . For more details and insightful reason-ing about gauge fields and their ground state, seé R .P . Feynman . 3)

Since color fluctuations cannot be correlated over distances long comparedto 1/A in the QCD vacuum, the correlation function of all particles which are thesources of color fields must fall to zero rapidly as well, that is,

<O~Ak (x) A~(Y)~0> 1ti(x_Y)

2(2 .1)

(A has dimension 1/L)

< O~Ak(x) AR (Y)~0> ti 0 , for (x-Y)2n2 » 1 (2.2)


<DIP~(x)P ß (Y)10> ~ 0 for (x-Y) 2 n2»1

(z .3)

where p

=color density of quarks = q+aa 2

q.

(Questions about gauge will beanswered below) . On the other hand, in the vacuum of massless free quarks

<OI P (x) P (Y)10> ti

1a B

(x-Y)6

(dimension of p = 1/L3)

What is the vacuum state of free quarks? It is a Dirac sea, or Slater de-terminant of negative energy plane waves . It is the spatial overlapping of thelong wave length, and massless wave functions which produce the power law falloff

. If one supposed that the principal modification to the wave function ofthe v cuum was to give the particles an effective mass m, then since when(x-y )~ »1/m2

<Olpa(x)Pß(Y)10> ~ ~~

e-zmlx-YI

(x-Y)

if m z n, the correlations would be properly suppressed . An effective massm = A for the up and down quarks would break chiral symmetry, but as was shownlong ago,l~) an effective Dirac mass is also a way of describing a spontaneouslybroken chiral symmetry . This is the standard argument for a "constituent" quarkmass and for the origin of the breaking of chiral symmetry . However, does avacuum formed using a Slater determinant formed with Dirac particles with aneffective mass give a good approximation to the true ground state? From a prac-tical standpoint it does not seem to be a very good ansatz, since the nearby ex-cited states orthogonal to such a ground state would be quasi-free quarks withan effective mass m, and this is not a very good description of reality . Hencesimply giving an effective mass m to quasi-free quarks cannot be the correctanswer for the ground state wave function . The excited states must be orthogonalto the proper ground state and also must not allow color to be separated overdistances large compared to 1/n . Because we know that quark shell models work,in some sense a quasi-free quark description should work . We should then try toguess a set of trial wave functions which have the property that

a)

<OIPa(x)P ß (Y)10> - 0

for (x-y)zt

lzn

b) in the low lying excited states with vacuum quantum numbers color cannotbe separated over distances large in comparison to 1/n .

One possible way to achieve this is motivated by the sort of Dirac wavefunctions which have been found to be so successful in static bag model calcula-tions . These are free Dirac wave functions for a quark in a box, with a boundarycondition on the surface,

(iy "n + 1)q = 0

(z.4)

(z.5)

(z .6)

where n is the unit vector normal to the surface of the box . The boundary con-dition (z .7) has been widely criticized because it is not chirally symmetric butthat could be a virtue of (z .7) rather than a defeçt since chiral symmetry isbroken . The question is rather, can the boundary condition used in (z .7) be re-garded as the signal of a spontaneous breakdown of chiral symmetry?

Indeed it was recognized very early l5

that (2 .7) can be viewed in a waywhich is quite analogous to the way in which an effective mass can be used todescribe a spontaneous breakdown of chiral symmetry .

We can see this as fol-lows .

If we want to find a set of wave functions where particles are localizedwithin a spatial region with a boundary, then we wish that the current throughthe surface vanish,

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(z .7)

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K. JOHNSON

and

n .qa y qb = 0

(2.8)

Here it is color which we wish to localize So color labels a and b (a=(ab)) arepart of the wave function . (2 .8) is a chi rally invariant condition, so the con-dition of localization by itself does not violate chiral symmetry.

Next let us consider a condition on the wave function which will insure(2.8) . Further let us imagine that q is a set of wave functions which also carrythe chiral labels for SU(2) x SU(2) generated by 1/2(1 ± y5)ß/2 .

Then consider+ + i~.ry

iy .nga = _eia .Te

5qa

(2 .9)

where we keep the color label a explicit . On the left we have, -Uqa where U isan arbitrary element of the chiral group . (2 .9) implies,

-qaiY.n

-

-qaei ~ .TyS e-iâ.r

(2 .10)

If we multiply (2 .9) byqb

on the left and (2.10) by qa on the right we find

+ +

1~ÎYqb

iY .n qa= _qb eia .r

e

5qa

(2 .lla)

-qbiy .n

qa=-4b

ei~TY5 e-iâTqa

.

(2 .llb)

This means that if â = 0, and for any ß, the right sides are equal andhence the color current through the surface vanishes .

Thus, the boundary condition

i~.TY5iy.n qa = - e

qa

(2 .12)

for any ~ guarantees localization of color in the box. However (2 .12) is not in-variant under chiral transformations . It is covariant under non-linear chiraltransform tions if we let 9 transform.

It is also covariant under the SU(2) s b-group if ~ transforms . To achieve this, we must enlarge the system to include

among the dynamical variables .

This was recognized very early and is the ori-gin of the hybr~~ models of pions and quarks which have become so popular in thepast few years .

) In this way the failure of (2 .12) to be invariant under chiraltransformations is similar to an effective mass . However, we don't rally wantto enlarge QCD with new degrees of freedom . Is it necessary to think of ~ thisway?

A set of free Dirac wave functions which obeys (2 .12) on the surface of a boxdefines a complete set of wave functions for the space within the box. Consequent-ly if we fill all of space with boxes and within each box choose a set which obeys

(2 .12) we have defined a complete set for all space .Once we have a suitable complete set of functions qi(x) in terms of which

we may expand the quark field,

q(x) = E qi(x) ei

(2 .13)i

we may then define a "trial vacuum", by dividing the set q .(x) into two classesof modes, those which give positive (p~(x)) and those which give negative (n~(x))expectations for the free Dirac Hamiltbnian . We fill the negative energy modes

and leave empty the positive energy modes to define the trial vacuum state . Thatis, we write

where


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define the state,~> . If our wave functions also carry one or more parameters,a i,we shall fix these variationally with (a/aai)<H>=0 . We also can write

with H defined by the diagonal elements of H in the basis q .(x) (or some part ofthe diâgonal elements) . We then will have the basis for a di~gramnatic improve-ment of our ansatz vacuum . To be practical, in a gauge theory the asymptoticallyshort waves in the basis q .(x) should look enough like plane waves so that wecan implement the renormal~zation program perturbatively .

To be specific let us take a periodic lattice Q of identical boxes a, as-sociated with a lattice vector xa. In the box a , we choose the complete set

gna(x) - qn(x - xa) for x in box a .

(2 .17)

= 0

for x outside of box a,

where the standard bag model wave functions q (x) belong to the box with latticevector xa = 0 . These functions define a compléte set of orthogonal functions forall space,

d3x q~a (x) gmß(x) =6a8dnm'

(2 .1B)

Expansions of the ordinary functions in terms of q

do not converge uni-formly in the boundaries . The functions q (x) have disgC$ntinuities on bound-aries, however since the Hamiltonian is fR~st order we need not worry about this- improvements here can be made later .

If we expand the field in terms of the set qa (x), each cell could in princi-ple have a distinct value of the parameter ~. But die shall see that this trialvacuum is not correct as follows . As outlined above the trial vacuum is a Diracsea given by filling the negative energy states in each box . Although this"vacuum" is correct in that it has short range color order,

<pa(x) o b (x)> = 0

for x and y distinct cells

(2 .19)

it is wrong because it is also only ehart range in chiral order (or any otherorder), for example

<Ai(x) A~(y)> = 0

for x and y in distinct cells (2 .20)

T "where A~ = q+~ y5 q

is the "chiral charge density" .

H = Ho + HI(2 .16)

If we had a reasonable ansatz for a ground state it should have short rangecolor order and at the same time, long range chiral order. In the exact vacuumwhich spontaneously breaks chiral symmetry,

<O~Ai(x) A~(y)~0> ti -F~ (xy (2.21)

q(x) = s +i

(Pi(x) a i ni (x) bi+ ) (2 .14)

a i ~> = 0 and bi ~> = 0 (2 .15)

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K.JOHNSON

as x-y -~ m. (2 .21) is a consequence of a massless pion Goldstone boson, for ingeneral

F~ in the pion decay constant,

and

a2and since W -"-~

i< O~T(Ai(x) A~(Y)~0>

_

- Fn a~2`~DF(xy))a ij +

. . .

(2 .22)

in the exact ground state ~0>, if chiral symmetry is spontaneously broken . Here

We should like our trial vacuum also to have the property (2 .21), so that ourapproximate treatment at each stage contains the collective Goldstone mode, eventhough the value of F may be approximate .

One difficulty which we may have is a conflict between the simultaneousrequirements of short range color order, and long range chiral order.

To obtain long range order the wave functions (2 .17) are clearly unsuitable .Let us define a new set as follows . We should like to be able to localize

the wave functions in space, so let us introduce the "Wannier factor"

W(x - xa ) =

J dyke

a

(2 .24)

In (2 .24), the integration takes place over the Brillouin zone associated withthe lattice, and V is the volume of the zone . W has the following properties :

E W(x - xa ) W(xa-Y) = W(x - Y)

(2 .25)a

W(xa- xß)

daB

(2.26)

2W = 1 in a cell and falls like ~

if we are at a distance d from a cell, whered » a = cell size .

dLet us now define the quasi-localized states

Qan (x) = W(x - xa) E gnY (x)

(2 .27)Y

where qn (;) as defined as before, except that in (2 .27), we choose the same chiralparameter ß in each cell . The states Qan (x) just as gnY (x) define a complete andorthogonal set of Dirac functions over all space. Since Q -~ 1/x2 as x -+ largedistances, the Q's are normalizable and in (?. .27) they are properly normalized .The funct~i7gns Q are an example of the Wannier wave functions used in solid statephysics1.

If we expand the field q(x) using the basis Q n (x) then the chiral order isestablished (with some spatial variation associate with the lattice structurewhich should be an artifact of the lattice basis - the vacuum should not be acrystal) . One can simply compute

<O~A~(x)A~(Y)~0>=W(x-Y)2 " ~~, nnY(x)(Y5 Z~)P~,(x)pmY~(Y)(Y5 ~)n~~ (Y)~

(x-Y)move the artificial variation associated with the lattice structure we shouldaverage x and y over the distant lattice cells in which they lie, so we thenfind if x-y = r with r»a,

(2 .28)

we have the proper correlations in chiral charge . To re-

3 3x-

cell ~ cell(nn(x)Y52 Pm (x)) " (Pm (Y)Y5-Z-nn (Y))7

(2 .29)

In (2 .29) we have o~dinary bag model wave functions . From (2 .29) we can obtainan expression for F~ .


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However, at the same time we seem to have also established long range co~Con on.de~c,which we certainly do not want to do . The color order does not appear (afteraveraging) in the color density correlation function, <p~pa(x)pb(y)~0> because

of the orthogonality of the baq model wave functions n(x) and p(x) . It wouldhowever appear in other densities, for example in the correlation function ofthe "colored" - chiral density,

However, we shall now show that the color order is not real, but a "gauge" arti-fact .

In (2.27) the color label a is implicit, let us make it explicit and intro-duce another complete set of quasi-localized states,

Qan(x) = W(x-xa)E gnY (x) UY . x~

(2 .30)Y

v

where the color wave functions are x~, E x~x+~ = 1, and (x+aXa ) = say . Foreach cell, UY is an arbitrary color unitÄry transformation . The set Qan is alsocomplete and orthonormal, but the functions are not true Wannier functions be-cause the coefficient of W is not a periodic function on the lattice . A vacuumbased on the functions Q' has the same long range chiral properties as the onejust discussed, but no longer has any long range color order if we choose U tovary randomly from cell to cell . However, QCD is a local gauge theory and thefactor UY can be removed from the wave functions by a gauge transformation . Hencethe long range order which we found above is a "gauge artifact" - that is it canhave no physical consequences . It can be removed by a quasi-local gauge trans-formation .

We now have constructed a (very crude) model for the vacuum wave functionof quarks to be used in conjunction with a confining background wave functionfor gluons . It however has one desirable feature. It has the long range orderwhich is characteristic of a spontaneous breakdown of chiral symmetry, and ithas only a gauge artifact long range color order .

The parameter ~ may be removed by defining a "standard" set of wave functionsby letting

qn = exP(i 2 "tY5 ) qn "

(2.31)

The wave functions q~ obey the standard boundary condition,

iy.n q~ _ -q~

(2 .32)

on the cell surfaces . Since the Hamiltonian is invariant under the chiral groupthe "vacuun" defined with q' has the same energy as that defined by q', that iswe have a degenerate set of trial "vacua",

~~> = exp (ild3x q+(x)Y5 ~ " ~q(x))~~ =0>

(2 .33)

with a long range chiral order . The states J~> and ~~=0 > are of course ortho-gonal in the limit of m

volume but, if we allow ~ -. ~(x~ where ~(x) varies e.P..owLyon the scale defined by a, the cell size, then the states ~â(x)> are all low-lying and have a finite overlap on the state I~ = 0> . In this wav, we have the

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K. JOHNSON

basis for the treatment of a set of collective states, which are the pions . Theparameter 9(x) is not a new degree of freedom but rather the basis for a longwave length phenomenology . If we study the states ~ß(x)> locally, they will beparticle-hole excitations in each cell : that is, they are quark model pions . Thus,it is in the framework of the collective states defined by (2 .33) with ~ y ~(x)that the "Goldstone" pion and the quark model pion find their common meetingground . (2 .33) is of course general, it is also the collective state if the exactvacuum ~0> (instead of ~ß = 0 >) appears . However to be useful, we need to under-stand the quark structure of the exact vacuum . The state ~ß = 0> defined heremay be at least a crude beginning for understanding the quark basis of the "spon-taneous" breakdown of chiral symmetry .

In this vacuum a single scale, the cell size, appears . The cell size shouldbe fixed in terms of the QCD scale n by computing <H> and minimizing with respectto a . This is really a detail . The important thing about this model is ttiat it hasthe correct qualitative feature of the true QCD vacuum, namel it contains asingle scale, and it leads to all other dimensionful hadronic parameters beingdetermined in terms of it . There are no such things as "small" hadrons and "big"hadrons or different regions of hadronic phenomena based on adjustable parameters .It is therefore quite an interesting problem simply to compute hadronic phenomenausing this very crude guess for the ground state wave function .

3. Conclusions

The wave function proposed here has many deficiencies, for example, the longrange spatial order associated with the lattice which certainly must be absent inreality . Nevertheless, it still provides a description which may be locally correct in the true QCD vacuum . Its principle virtue is that it has the chiral andquark structure which most believe are the kind expected in the true QCD groundstate .*

* This work is supported in part through funds provided by the U .S .Department ofEnergy (D0E) under contract DE-AC02-76ER03069 .

4 . References

l . For a review, see : W. Marciano, H . Pagels, Phys . Rep . C 36, 137 (1978)2. For a review, see : H. Pagels, Phys . Rep . C 16, 219 (1975]3. M. Gell-Mann, R . Oakes, B . Renner, Phys . Rev. 175, 2195 (1968)

S . Glashow, S . Weinberg, Phys . Rev . Lett . 20, ~(1968)4 . S . Weinberg, Phys . Rev. Lett . 16, 169 (196bT

Phys . Rev. 166, 1568 (1967)R. Dashen, Phys . Rev. 1381245 (1969)R. Dashen, M . Weinstein,~hys . Rev . 183, 1291 (1969)

5 .

For reviews, see : "Dual Theory", ed.~ Jacob (North-Holland, Amsterdam, 1974)J. Scherk, Rev. Mod . Phys . ¢~, 123 (1975)

6 . M . Creutz, Phys . Rev . D 21, 2308 (1980)G . Bhanot, C. Rebbi, NucT. Phys . B 180 (FS2) 369 (1981)E . Pietarinen, Helsinki Preprint, HUTFT/80/49

7. G. Zweig~CERNP8182/TH4018 ' 2I4 (1364)

CERN 8419/TH412 (1964)8 . R . P . Feynmann, in "High-Energy Collisions", ed . C .N . Yang, J .A . Cole,

M .G .R . Hwa, J . Lee-Franzini, p. 237,(Gordon and Breach , New York)9 . A . DeRujula, H. Georgi, S .L . Glashow, Phys . Rev . D12, 147 (1975)

10 . A . Chodos, et . al ., Phys . Rev. D9, 3471 (1974)For reviews, see : K. Johnson, Act=a Physics Polonica B6, 865 (1975)P . Hasenfratz, J . Kuti, Phys . Rep . 40C, 75 (1978)

11 . R .L . Jaffe, G .C . Ross, Phys . Lett . ~J3S, 313 (1980)R .L . Jaffe, M . Soldate, MIT CTP 931981)

12 . See, for example: A. Casher, Ph.vs . Lett . 83B . 395 (19791


61c

13 . R.P . Feynmann, Cal . Tech . Preprint (1981)14 . Y. Nambu, J . Jona-Sasinio, Phys . Rev . 122, 345 (1961)15 . A . Chodos, C .B . Thorn, Phys . Rev . D12,272733 (1975)

T . Inoue, T . Maskawa, Prog . Theorel:- Phys . 54, 1833 (1975)

16 . C . Callen, R . Dashen, D. Gross, Phys . Rev . D19, 1826 (1979)G .E . Brown, M.Rho, Phys . Lett . _82B, 177(1979TPhys . Lett . 84B, 383 (1979)F . Myhrer, G~ Brown, Z .Xu, Nordita, 48 (1981)R .L . Jaffe, 1979 "Et tore Majorana School" 31/7-12/8 (1979)G .A . Miller, A .W . Thomas, S . The'berge, Phys . Lett . 91B, 192 (1980)S . The'berge, A.W . Thomas, G.A . Miller, Phys . Rev . D~ 2838 (1980)M.V . Barnhill III, A . Halprin, Phys . Rev . _D21, 1916~T980)C. DeTar, Phys . Rev . D24, 752, 762 (1981)

17 . See, for example : C . Ri~tel, "Quantum Theory of Solids" (J . Wiley, New York)

Discussion

d. Levinger (R.P.I., Troy) : We have ryood direct and indirect evidence for CouZomb'sLam . Horn good is the evidence for quantwn chromodynamics ?

K . Johnson : It is not quantitatively very good but it is impressive nevertheless .It predicts all signs correctly and all orders of magnitude correctly . Also, theresimply is no alternative which comes even close to achieving this .

Documents

The quark model pion and the Goldstone pion