the pion and an improved static bag model

P H Y S I C A L R E V I E W D V O L U M E 2 1 , N U M B E R 7 1 A P R I L 1 9 8 0

The pion and an improved static bag model

John F. Donoghue and K. Johnson Center .for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology,

Cambridge, Massachusetts 02139 (Received 25 July 1979)

Quark-model calculations involve an extended static object localized in space. We introduce new methods, involving momentum-space wave packets, which account for this localization. These methods have little effect on heavy states, whose sizes are large compared to their Compton size l/m, but are very important for light particles such as the pion. In this treatment the pion's mass is naturally very small, and, in order to connect with a spontaneously broken chiral symmetry, we require that m, vanish when the light quarks are massless. Expanding about this limit (and also readjusting the fit to other hadrons), we obtain m, = (mu + m,)/2 = 33 MeV. We calculate F, -- 145 MeV (using a normalization such that F,l,,, = 93 MeV), F,/F,z 1, and various corrections to static properties of baryons. In addition we explore the relationship of our methods with chiral perturbation theory, deriving the formula m T 2 = (mu + m,)(~(p)lq(O)q(O) IT@)) in the appropriate approximation and commenting on the quark mass obtained from the nucleon's cr term. Finally we discuss the bag model's use of the scalar density gq as an order parameter describing the separation of the spontaneously broken vacuum phase from the perturbative vacuum of the bag's interior.

I. INTRODUCTION

The stat ic bag model applied to hadrons made up of low-mass quarks has had a considerable phenomenological success in i t s reproduction of hadron masses and other parameters in t e rms of a few fundamental constants.'" The most notable exception to this success i s the n meson. It is the purpose of this paper to develop a simple improve- ment to the static approximation which will play a relatively minor role for the more massive s ta tes , but will be of crucial importance for the pion. In particular, i t will enable us to show that it may be possible to resolve the apparent dicho- tomy between the quark-model pion and the PCAC (part ial conservation of axial-vector current) pion.5

The bag model is a formillation which incorpor- ates in a local and relativistic framework the following features which have been abstracted frorn the observed properties of hadrons:

(1) Hadrons a r e composed of quarks which move relatively independently within a single hadron. Consistent with this i s the following:

(2) The effective interaction between the quarks at short distances i s governed by quantum chromo dynamics (QCD) treated perturbatiuely . These features a r e included in a model which a l - lows the hadrons to consist only of color-singlet combinations of quarks.

In this model i t i s hypothesized that the region of space within a hadron i s the usual perturbation- theory vacuum where quark interactions a r e relatively weak. The true (and complex) vacuum out- side of hadrons is unspecified except that ft has an energy per unit volume I3 lower than that inside.

For light quarks the location of the boundary between these phases i s governed by the sca lar density gq, with the bag action derivable from the ~ a g r a n ~ i a n '

As a further approximation the static bag model assumes that the region of space which contains the quarks is a fixed spherical cavity. In this paper we shall develop a method to correc t for the effects of "nailingJ' the quark wave functions to a fixed origin in space, The corrections a r e of inte res t by themselves but a s we stated above, they will be of particular importance for the pion,

The quark-model description of the pion is that of a quark-antiquark bound state with properties not much different from other hadrons. When spin- dependent forces due to colored-gluon exchange a r e taken into account perturbatively, the pion emerges naturally a s the lightest state.3 Spin forces provide a rather large interaction energy which subtracts from the unperturbed energy. However, despite th is , it is extremely difficult to obtain a pion with m,= 140 MeV. For example, the bag fit of Ref. 3 quotes a pion mass of 280 MeV.

The pion of the quark model appears superficial- ly quite different from the pion of PCAC."* With vanishing up- and down-quark masses the fundamental QCD theory has a chiral SU(2) x SU(2) sym - metry. This symmetry is assumed to be spontaneously broken, with the pion a s the associated massless Goldstone boson. For finite but smal l quark masses , the pion deviates only slightly from this description, picking up a small mass but re - taining the couplings of a collective excitation.

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1976 J O H N F . D O N O G H U E A N D K . J O H N S O N - 2 1

The connection between the PCAC pion and the pion of the quark model remains a persistent question.

However, even within the quark model, the pion has a property which distinguishes it from other hadrons: It i s the only particle whose "size" is smal ler than i t s Compton s ize ( l /m ,). Quark- model techniques have evolved from the bound- state treatments of atomic and nuclear physics. In these lat ter situations one deals with a heavy system whose spatial extent i s much la rger than i t s Compton size. This property i s also t rue for most quark bound states. For example, the proton has a radius of about 1 fm while i t s Compton s ize is & fm. Iluwever, the pion's Compton s ize i s 1.4 fm and estimates of i t s radius a r e near $ fm. In this paper we use this a s a basis for a reconsider- ation of the pion in the quark model.

This s i ze distinction requires that the naive static-bag techniques be modified in order to prop- erly treat the pion. The type of correction we have in mind is known generally a s a "center-of-mass" correction, but i t has some novel aspects in a re l - ativistic theory. As expected, the new methods produce very little change in the properties of heavy states. However, the pion emerges lighter than before. In particular, i t is possible to make the plon massless. We will do this in the limit of vanishing quark mass in an attempt to establish a connection with the pion of PCAC. Some interesting results emerge. We study the pion mass for smal l quark masses , and make a connection with chiral perturbation theory. The pion decay constant J?, i s calculated, and our treatment is such that it remains finite in the limit m,-0, a s is r e - quired by the chiral-symmetric theory. While we a r e f a r from uniting the two treatments, our work indicates that they a r e not a s disjoint a s previously thought.

In Sec. I1 we present the new tools and illustrate them by a calculation of F,. Section 111 is devoted to applying our new methods to compute corrections to the pion's energy and a discussion of the pion mass. The methods developed require some l e s s significant changes in the other hadron states a l so , s o in Sec. IV we redo the fit to hadron masses. We show the relationship between our description and chiral perturbation theory in Sec. V. Concluding remarks a r e made in Sec. VI.

11. BAG WAVE PACKETS, THE PION DECAY CONSTANT, AND OTHER CORRECTIONS TO THE NAIVE STATIC-BAG

CALCULATIONS

In this section we use the calculation of F, and some corrections to the static-bag approximation for nucleons to demonstrate the construction of a bag wave packet. The crucial observatior~ is that tile static bag state (which we will denote simply

by use of the particle symbol with the subscript B , i.e., In),) is not a momentum eigenstate Lin usual notation jn(p))]. However, we shall show that i t can be related to a superposition of such s ta tes , o r wave packet. We define, tor a covariantly normalized momentum eigenstate ( ~ ( p ) I T ( $ ' ) ) = ( 2 ~ ) ~ 6 ' ~ ' ( p -pJ)2wP,

Here, $(p) i s a wave packet which describes the localization of the particle around the position x . We shall assume that it i s possible to choose an appropriate wave packet $ s o that the momentum spread approximates that of the total momentum of the quarks in a stat ic bag. We shall describe how this wave packet i s determined below. For baryons, a more convenient convention is

where here we have the usual

(B(P),X IB(P'),kt) =(271)36'S'(~ -P')(E/m)6AAe . The bag states a r e normalized in the conventional way

B(vjn)B=l ,

B ~ B , M IB,Mt)e = ~ Y , M , (4)

which corresponds to the wave-packet normaliza- tions

For simplicity in this paper we shall assume that it is possible to take a s the wave packet for a spin-$ baryon, Xf(p), a form

where ~ ( p ) i s a sca lar function, and uM(p) i s the covariantly normalized Dirac spinor with the spin quantized along the s ame spatial axis a s that of the fixed cavity state. With this convention Eq. (5b) becomes

For particles whose extension is larger than their

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Compton s i z e , the momentum spread of the quarks bag-model methods for calculating mat r jx r l e - in the s tat ic bag is s m a l l in comparison with the rnents. We can i l lustrate this by considering the m a s s . The r e v e r s e holds fo r the pion. axial-vector coupling constant g, ==g,(O) , defined

F o r heavy s t a t e s th i s leads to the usual s ta t ic- by

(k ' ,P(P1) ju(x)? Y5d(x) jk,n(p)) = ~ , , ( p ' ) r ~ 5 7 ~ h(p)ga((p -pf)2)e'X"P-p'' 4 ~ , - . is1

Transforming t o a bag located a t X = 0 ,

u(lb'',p / L ( X ) ? ' ~ ~ ~ ( X ) [ M , = ~ ( j 3 p d d p 1 ~ E$ f'. g i x ' ( f ' - f l ' ) - u . ~ ' ~ ' ~ ~ " h , ~ ( p ) ~ * ( p ' ) 9 ~ ( ( ~ ' ' ' (9 )

o r

u(M',* 1 ( d ' x G ' ~ ~ ( l l M , v ) , = l d 3 D ( 2 )2(2,1r \ y ( p ) 2 , g A ( O ~ , * ( f i ) ) 1 t i l u ( f i ) i ' ' . . (10)

Now le t us a s s u m e that the wave-packet spread is s m a l l in comparison to the nucleon m a s s , ( p 2 ) <<hf2. Then to f i r s t o r d e r , i f we expand the right- hand s ide of Eq. (10) we find

where

( p 2 ) = l d 3 f i ~ ( 2 n ) 3 ~ ~ ( ~ ) ~ 2 ~ 2 ' 0 (12)

is the mean monlentum spread in the wave packet. In the limit of the naive static-bag model, we r e - cover the s tandard resul t (gA) =(gFtlcbag). The f i r s t correct ion is

gA = g p c bag (I ti*) .

In the c a s e of vec tor density, t h e r e is no c o r r e c - tioil to the static-bag resul t for the total charge. Clearly th i s is necessary for the consistency of our method. When we go beyond the integrated quantitiy to obtain, fo r example, the mean square radius o r magnetic moment , we obtain (p"/m2 correct ions. F o r example,

The charge radius is m o r e subt le and i.s t reated in the Appendix, yielding

where ( y 2 X t a t ' L = (16)

I

Taking the tirne component and t ransfor~rl lng to the bag s ta te

Using the static-bag-model wave functions, the left-hand s ide is given by

where u ( I ) is the upper (lower) component in the static-bag wave function. With thls identification we can solve f o r the bag wave packet Np). Thts IS our c ruc la l s tep, We iden i~fy the wave packet which descr lbcs the total mornenturn spread of the quarks in the s tat ic hag w ~ t h that wilicl~ comes from the static-bag wave fllnctlons applied to the total annrhilatlon of the part ic les . T!lus, we frnd

F , is then determined by the norinalization condition Eq. (5a). With nz,=O, this resu l t s in

An interesting feature af this calculation is that F , has a finite limit a s n; .- 0 , a s long a s the pior) radius R , remains finite. Th is is as would be hoped for iri theories where nz. is s m a l l , and in- deed i s necessary for o u r calculation to be con- s is tent with ch i ra l perturbation theory. In 1.at:er sect ions we wil l determine the pion's radius to be i n the range I"= 3.3-3.5 G ~ v - ' , which correspozrds to a value

F,-140-150 MeV (22)

t o be compared with the experimental F,= 93 MeV. Clear ly , a l l of these cor rec t ions imply a commit- T h e r e is one considera.tion that we have neglected ment to a specific wave packet, and we sha l l now

above which would lower the value of F ,. Wt? have tu rn to our proposal f o r that by studying the meson s t a t e , in part icular the pion. calculated the amplitude for the removal of two

quarks from a bag, This , however, leaves a n F o r the pion, we shal l begin with the vacuum- "empty bag, " not the t r u e vacuum. Thus , there is to-a mat r ix element , that i s , a calculation of E',,

defined by a n ex t ra factor in Eq. (19) which descr ibes the overlap of the inside, perturbative vacuum of !be

( 0 l ~ x ) y " y s d ( x ) j n ( p ) ) = i f l - ~ . p ' " e ' ~ " . (17) bag with the t r u e vacuum. One might ant icipate ,

1978 J O E I N F . D O N O G H U E 4 N D K . J O H N S O N 2 1 -

because of the spa t ia l homogeneity of the vacuum E ~ ~ = ~ ( P / H / P ) B = @ ) , (26) s t a t e , that this overlap would take the form

where (E) is the average of (p2+m2)1/2 using the e x p ( - ~ ~ / " ~ ~ ) , with V - the volume of the pionbag, and X a dimensionless parameter . However, with- appropriate wave packet. F o r heavy s tates , with

R >> l /m, ( E ) o m , i n ze ro th approximation yield- out a theory of the t r u e QCD vacuum we a r e unable a t present to calculate the overlap. Since B ~ / ~ v , ing the usual equality of bag energy and m a s s

~ 0 . 4 is not l a rge , one might hope that th i s factor Eb,=ruz . (27) is not the dominating one. We can a l so compute F,, where i f VK - V,, the vacuum overlap should In fact, even the f i r s t correct ion t o this has been

b e the s a m e , That is, the ra t io F,/F, should be accounted f o r in the e a r l i e r static-bag calculations .3 Expanding the energy, one obtains insensitive to this. W e sha l l re tu rn to this cal-

culation below. l (E)=m +- ( P 2 ) .

The wave-packet amplitude 4 ( p ) wil l play a con- 2rn siderable role in subsequent discussions. It has the desirable feature that i t is insensitive to the bag boundary sur face , s ince (ZL' - 1 2 ) vanishes there . Alternative choices of the wave packet ( fo r example, that gotten by considering the annihila- tion amplitude associated with the operator &, d ) often have a discontinuity a t r = R , which produce very-high-momentum components in flp). We a r e using in $ ( p ) the cavity wave functions fo r the s t a t e with no gluons. The gluons introduced per - turbatively produce a s izable energy shlft in the pion, and could produce wave-function cor rec t ions also. However, the boundary conditions a t the sur face of the bag a r e independent of the coupling. Since the ( P ( P ) is a smooth function whose main property is that associated with i t s sca le , we be- l ieve that the propert ies of + ( p ) derived above a r e quite reasonable, and we will use Eq. (20) throughout the paper.

We can a t this t ime apply the s a m e fornlulation to the kaon, by using s trange-part ic le wave functions fo r one of the quarks an $(p) . The s a m e procedure with m, =0.5 GeV and m,- 0.3 GeV yields

Since R, =El, th i s produces 6, %Fro As we have remarked already, we believe that this resu l t i s m o r e accura te than o u r absolute calculation of e i ther F, o r FK since the "empty-bag," t r u e vacuum amplitude should cancel in the rat io F,/F,. We reca l l that the experimental rat io PK/F,= 1.20.

111. MASS OF THE PION

The localization of the quarks described above a l so produces a modification in the calculation of m a s s e s in the quark model. In momentum space we have the definition (using the proton t o b e specific)

( P ( P ) J H J P ( P ) ) = E , (24)

where E = ( ~ ~ + r n ~ ) ~ / ~ , SO that f o r a proton a t r e s t ,

($'(a) I H I P (0)) = m . (25)

However, when we convert t o bag s ta tes ,

Thus, i n a s ta te with independent part ic les , (p2) mn/B2, where n is the number of quarks. Since the zeroth approximation to the m a s s i n the bag model is nz = &n (2.04/R) (with m a s s l e s s quarks) there is therefore a n approximate relatior1

with C independent of n . However, this correct ion has the s a m e f o r m a s

the "zero-point energy" t e r m (-2JR) which was included i n the hadron m a s s e s calculated in Fkf. 3. We now observe that p a r t of Z, is accounted f o r by the momentum fluctuation effect. The phenomenologically determined value of 2, is about 1.8. We es t imate that C would be 0.6-0.8. Hence we s t i l l a ssume that EbW contains a t e r m -2,,/R, but where we shal l now expect 2,- 1.

The s tat ic bag s ta tes have (3 = 0 , but (p2)+0. This correct ion to the energy can therefore a l so be described a s a "center-of-mass" correct ion since i t removes the effect of nonzero (p2) f r o m the static-bag calculation. F o r light s ta tes such as the pion the "fluctuation" momentum ( ( P ~ ) ) ~ / ~ is not s m a l l i n comparison t o the m a s s , s o equat- ing the m a s s and the bag energy is clear ly not cor rec t . Rather, to compute the m a s s one should use

Even m a s s l e s s s t a t e s have ((@2)1/2)* 0 SO that the bag energy need not vanish. The important gener - a1 feature of Eq. (30) is that, f o r a given Eb,, wz is s q a l l e r than would be obtained f rom Eb,=m, since some of the energy goes into localizing the quarks i n a bag a t a fixed point i n space. This makes a low-energy bag s t a t e correspond t o an even-lower-mass part ic le; a light pion becomes even lighter.

When using a wave packet to es t imate the contribution to the bag's energy of the total momentum of the localized quarks, severa l a l ternat ives can be considered. A close paral le l between our t reatment and chiral per turbat ion theory ex is t s if

2 1 - T H E P I O N A N D A N I M P R O V E D S T A T I C B A G M O D E L 1979

we base our calculation on a fixed wave packet with z e r o par t i c le m a s s i n the normalization condition and z e r o quark m a s s i n the wave function (see Sec. V). The discussion in the remainder of this section is a l so based on such a wave packet. An al ternate proposal, which does not lead to such a close parallel, is to use a wave packet whose form is self-consistently fitted to the m a s s of the part ic le and the m a s s e s of the quarks. This is the method used in Sec. IV to determine the light quark m a s s a s p a r t of the overal l fit to the had- ronic spectrum. It tu rns out that the numerical values of the quark m a s s obtained in Sec. IV and Sec. V do not differ greatly. This would not be the c a s e if the pion m a s s were much s m a l l e r than 1/R. We have not made comparisons of which of these proposals gives the m o r e accurate es t ima- tion of the center-of-mass correct ion by using exact models. When making a variational t rea t - ment of the m a s s spec t rum based upon the bag's s ize, the proposal of a fixed wave packet, with both quark m a s s e s and part ic le m a s s e s s e t equal to zero, leads to the s imples t discussion, which is why we adopt i t i n the remainder of this section and in Sec. V.

Before proceeding fur ther we should d i scuss the spin-dependent fo rce due to gluons s ince i t has an important role in making the pion a low-mass p a r - ticle. By considering single-gluon exchange between quarks there is a contribution to the pion energy of the fo rm

where Go is a pure number (Go= 0.7) which comes f rom an integration over the quark wave function^.^ We have written the coupling constant a s a function of R to indicate the dependence of the effective coupling on the sca le s ize in the problem. With m a s s l e s s quarks the m a s s sca le is s e t by a/R (with a -2), which provides a low-momentum cut- off on the gluon propagator. Combined with the idea of asymptotic freedom th i s suggests that the effective a,@) should get weaker a t s m a l l 8 and s t ronger a t l a rge R . The specific fo rm of a ,@) will not be needed until Sec. IV, but the general var iat ion of a,@) is important in what follows.

The p r e c i s e predictions of the previous bag model fit3 do not apply in the new framework, s ince we m u s t now adjust fo r the center-of-mass effect and the running coupling constant. However, when one makes such adjustments the pion m a s s calculated f rom ((p2 +m:)1/2) = Ebag naturally fal ls close to zero. Therefore, the opportunity a r i s e s to put in by hand a connection with the PCAC pion associated with a spontaneously broken ch i ra l symmetry. It is not difficult to constrain the bag

model t o yield m,= 0 when the quark m a s s e s vanish. To do this, expand

and evaluate (p) and (1/2p) with the wave packet @ ( p ) determined with m,= 0 and nzq= 0. This yields

(P)=A/R, (33)

(1/2p) = R / C , (34)

with

and

The bag pion's m a s s is then given by the following function of the radius:

Here E,, contains

(1) the quark kinetic energy

E,,, = 2x/R

(x = 2.04.. . f o r m a s s l e s s quarks) ; (2) the gluon-exchange energy

where G0"0.7 f o r m a s s l e s s quarks; (3) the volume and "zero-point" energy,

The usual procedure is to minimize the energy a s a function of R to find the actual s ta te . H e r e we shal l minimize the m a s s ,

We sha l l a l so demand 2 '

m n / R " R , , = ~ ,

which may be regarded a s a constraint on the pa- r a m e t e r "z,." We note f i r s t that if ff, were a con- s tant , no solution to both (41a) and (41b) would exis t with R,> 0. Without the constraint (41b), a solution i n general would exis t with Z o l e s s than some c r i t i ca l value, but a t the minimum we would find ryz;> 0. If we t r y to enforce (b), we should find that Z o would be a t the c r i t i ca l value, and there R,=O. A z e r o rad ius pion is not sat isfactory

1980 J O H N F . D O N O G H U E A N D K . J O H N S O N

because consistency with chiral per turbat ion the- x x --A m +-A

o r y requi res that F, remain finite when m, = 0, + * . , x,= 2.04.. R - R 2(x,- 1)

m a = 0. Since F,-l/R,, this means R,* 0. If we t r y to enforce both (a) and (b), i t is required that aas/aR>O, i.e., asymptotic freedom is necessary f o r our t rea tment of the pion to be possible. The enforcement of a condition on 2, to make the pion m a s s l e s s is not a "natural" condition in the con- text of the bag model; nevertheless , we believe that it could be here where the bag model a s a phenomenological version of QCD with a spontaneously broken chiral symmetry m u s t b e con- s t rained. The total volume and "zero-point" t e r m i n E,, r e f e r s to the energy of a bubble of per tu r - bative vacuum embedded in the t r u e vacuum. If these s t a t e s differ by a spontaneous breakdown of ch i ra l symmetry , then i t would be natural that 2, would be such that the pion should be mass less . Since we do not have a microscopic theory of th i s spontaneous symmetry breakdown, i t is necessary f o r us to enforce this requirement by hand. Equa- tions (41a) and (41b) t rans la te into two relat ions, one of which de te rmines R , i n t e r m s of B, Go, x , and Z,, and the other is the simple relation

This equation clear ly e x p r e s s e s the fact noted above that without asymptotic freedom our t rea t - ment of the pion would not be possible, s ince m , = 0 would a l so requ i re R,= 0. The fact that both s ides of Eq. (42) a r e governed by the scale param- e t e r of the s t rong interaction means that such a relationship is not absurd. F r o m a phenomenolog- i ca l point of view we sha l l regard Eq. (42) a s a n equation f o r R , if we a r e given B and a fo rm of a,. It tu rns out that the f o r m s of used i n Sec. IV a l l yield R,=3.3-3.5 GeV-I.

The physical pion m a s s i s not ze ro , but m,= 0.14 GeV. This can be accommodated by giving the quarks a m a s s . The pion m a s s can be solved f o r by calculating

where the left-hand side is obtained with a wave packet that is independent of both m, and ma. At each value of the radius, f o r fixed ma, this equality de te rmines a value of m,. F o r smal l m a we can expand Eb,(ma) a s

where the dependence of the quark 's kinetic ener - gy,

and Fig. 3 of Ref. 3 have been used. The m o s t imior tan t contribution is the change

in kinetic energy, and this l essens the sensitivity to the f o r m of a s @ ) . The f i t s of Sec. N al l give the observed pion m a s s f o r a quark m a s s

m a = 3 3 i 2 MeV, (46)

where the i2 indicates the range of possible values.

IV. FIT

The new methods which we have introduced r e - qu i re a redeterminat ion of the phenomenological quark parameters . The purpose of this section is to provide such a n evaluation. With the exception of Z,, the bag p a r a m e t e r s determined below a r e quite s i m i l a r to those of a previous fit in Ref. 3. A s mentioned before, Z , changes because s o m e portion of the phenomenologically determined Z , i s now accounted f o r by the center-of-mass c o r - rection.

We do not have the wave packets + ( p ) o r ~ ( p ) fo r a l l the hadrons. Here we will employ the s a m e f o r m of @ ( p ) f o r a l l pa r t i c les , and u s e Eq. (20), and the mass-dependent normalization of Eq. (5a). We do this f o r simplicity and because this f o r m has severa l a t t ract ive fea tures described above. In addition, we feel that the p rec i se f o r m of @ ( p ) should be only important f o r the pion.g

The contributions t o the bag energy a r e d i s - cussed in detai l in Ref. 3. The only change which we make is the introduction of a running effective coupling cu,(R) i n place of the fixed one. We of course d o not know the prec i se f o r m of a,(R). When R is s m a l l , one would expect that as(R) would change logarithmically a s determined by the renormalization group. However, a naive identification of the lowest-order QCD resul t with the bag coupling, i.e.,

f o r th ree f lavors (R, is the scale parameter analogous t o h in momentum space) , has an unfortunate consequence. This formula diverges a t R =R,. There a r e s o m e s ta tes , such as the nucleon and the pion, which receive a negative contribution t o the i r energy of the fo rm -(u,(R)M,/R. The m a s s of these par t i c les exhibits an instability a s one approaches R,, becoming a rb i t ra r i ly s m a l l o r negative. This c lea r ly i s unphysical, except that i t in s o m e way corresponds t o the instability of the perturbation-theory vacuum. Since this instability in a corresponds to low-momentum fluctuations,

T H E P I O N A N D A N I M P R O V E D S T A T I C B A G M O D E L 1981

TABLE 11. A comparison of the bag parameters of the previous fit (Ref. 3) with those obtained here (usingA = I ) .

Ref. 3 This paper (A = 1)

~ 1 / 4 0.145 GeV 0.135 GeV z~ 1.84 1.01 as@,) 2.2 2.1 RN 5 G ~ V - ' 5.5 G ~ V - ' R A 5.5 G ~ v - ' 5.6 G ~ V - '

v ) 2.2 1.5 R , 3.3 G ~ V - ' 3.5 G ~ V - '

The values of the dimensionless quantity (E)R a s a function of mR a r e given in Fig. 1. For large mR,(E)R-mR, while for m-O,(E)R-2.3. The .... equality (E) = E,,, then determines m(R) whose

FIG. 1 . The average value of ( E R ) vs m R using the wave function (P(p) . minimum value fixes the particle mass .

The basic input to the fit will be m,, m,, and the

"bag" effects will make a ( R ) finite except a s R - a. For the pion one could ignore the problem because R, remains safely away from R,. How- eve r , the proton has a la rger radius, and for this form of a, it i s not possible to both fit the N-A mass difference and retain a stable nucleon in lowest order .

To overcome this , we introduce a se t of f o rms of a ,(R) which do not suffer f rom this problem. We require that they yield Eq. (47) a t smal l R , but be smoother at large R . This can be obtained by setting

. - requirement that m, vanish when m, = 0. The de - pendence of the parameters on the input i s of course coupled. However, to a large extent m,+ m, determines the bag constant B , m, - m, determines a, (o r in this case R,), and the condition on m, fixes 2, (and predicts R,). The resu l t s of such a procedure a r e given in Table I with a quark mass of 33 MeV. That mass was chosen by requiring that the pion be reproduced. Table I1 gives a brief comparison of the parameters from Ref. 3 with those for A = 1. The gluon coupling constant i s somewhat weaker, and radii a r e slightly changed, but otherwise the resu l t s a r e ra ther

(48) s imi la r .

The p meson differs from the pion in the sign

For A 3 1 there i s no singularity. For A 3 0.3 there remains a divergence in a ,(R), but it i s moved sufficiently f a r away from the region of inte res t that no problems ar i se . We will use 0 . 3 s A s 1 to obtain a feel for the dependence of our resu l t s on the form of a,(R).

We now proceed by setting

(E) = ((mZ+p2)1'2) = E , , ~ ~ . (49)

TABLE I. The bag parameters obtained for various forms of a, ( R ) = 2a/9 In@ + R o / R ) .

and magnitude of the gluon-induced spin-spin interaction. It it were not for this , the p would be degenerate with the pion a s in the SU(6) models. There a r e no ex t ra parameters to be determined, and the mass of the p i s a prediction of the model. It i s listed in Table 111. We see that gluon exchange does provide a large splitting of the p and pion, placing the p close to i ts experimental mass , although a little low. It i s not surpris ing that the

TABLE 111. The mass and radii of the p meson pre- dicted from our fit, for various values of A .

1982 J O H N F . D O N O G H U E A N D K . J O H N S O N 2 1 -

p mass should be sensitive to the form of as(R) since that i s the parameter which most directly governs the p-n splitting. The results favor A = 1.

Part icles containing strange quarks may be accommodated by determining the strange q u a r k mass . For strange baryons, rn,= 0.28 GeV (the same a s in Ref. 3) reproduces the A S = 1 mass differences quite well. This value then predicts a kaon mass m,= 0.44 GeV (A= 0.3)-0.46 GeV (A= I ) , which i s quite reasonable. Alternatively one could use the kaon mass to determine m,, yielding wz,= 0.33* 0.01 for the various forms of

1. The basic static properties of the nucleon with

the above parameters

(y2)~tatic bag = (0.77)2 fm2 . ( 5 0 ~ )

The momentum fluctuation inherent in the wave function which we a r e using corresponds to

The f irst corrections to the static limit, given in Sec. IT, can therefore be computed, with the result

The most serious discrepancy in the static proper- t ies calculated in Ref. 3 was the low value of the nucleons' gyromagnetic ratio (2mppp= 1.9). The combination of a slightly la rger proton radius and the fluctuation correction improves this consider- ably.

V. THE RELATIONSHIP WITH CHIRAL PERTURBA'ITIOW THE.ORY

A inassless pion i s an exact consequence of a spontaneously broken chiral symmetry. In theo- r i e s such a s QCD with small, explicit breaking of chiral symmetry in the form of quark masses, the pion i s an approximate Goldstone boson, and a chiral perturbation theory has been developed.'

In this scheme, one adds to the chirally invar- iant Lagrangian the mass t e rms

The pion mass in lowest-order perturbation theory i s

where the matrix elements a r e evaluated with m, -- 0. The mass parameters here a r e those of "current quarks."

Equation (56) also a r i s e s in an approximation to our treatment. To make this clear, we shall f i r s t establish the connection between the field-theory matrix element (n(p) 1 ?j(x)q(x) I ~ ( 0 ' ) ) and the bag- model matrix element (n ] q(x)q(x)\ n), using the fixed wave packet g5 with m,= 0, and zero quark masses a s discussed in Sec. 11. This i s

so in particular

Equation (58) will be basic in what follows. Now we can do perturbation theory in the bag model, again with the assumption that when the quarks a r e

If we assume that in the bag model we compute mass less rn, = 0. Then we have (as above) a/am(Ebu) in lowest order from the quark mass

Eb,=((p2 + m,2)11') = ( p ) + m r 2 ( 1 / 2 ~ ) , (59) term, we obtain

21 - T H E P I O N A N D A N I M P R O V E D S T A T I C B A G M O D E L 1983

where q, is the zero-quark-mass s tat ic bag wave function. With the bas ic relation (58), and Eqs. (61) and (62) we find the s a m e expression f o r mV2, i.e.,

In part icular , numerical ly using Eq. (4) and Eq. (60), .

and f o r l a t e r reference,

The quark m a s s e s in the bag model have always been light, m o r e like "current" quark m a s s e s than those of "constituent" q u a r k s which appear in non- relat ivis t ic t reatments . This derivation makes c l e a r the identification of the bag-model quark m a s s with that of the cur ren t quarks. Of course, absolute quark m a s s e s only make sense when the prescr ipt ion used t o renormalize them is spelled out. The use of the bag wave function in evaluating m, i s one reasonable method.

Actually, the derivation of the chiral per turba- tion formula Eq. (56) does not depend on the bag model. It is independent of the p rec i se f o r m of q5(p), provided the wave packet itself is independ- en t of m,. Note, however, that the wave packet used in Sec. 1V involved the hadron m a s s v ia Eq. (5a), and cannot b e used to obtain Eq. (63), since the expansion of Eq. (59) does not apply to such a wave packet. The important ingredients a r e (1) the equality of (E) with the total quark energy E,,, (2) the vanishing of m, when m,= 0, and in both cases , f i r s t -o rder perturbation theory.

Di rec t evaluation of Eq. (64) yields m a = 22 MeV. This corresponds to a ch i ra l parameter

2 -0.5). Current-algebra m a s s ratioslor" then suggest m,* = $(m:+ m,*) = 6 MeV. The renormalization constant Z is readily calculable in the bag ( see above). In fact, i t is only the product m,Z which e n t e r s the calculation of m, through Eq. (62). The resu l t to be compared with the chiral per tur- bation theory m a s s i s m,*= 12 MeV ( o r as in Sec. IV, more accurately, m,* = 17 MeV). The bag model 's values of the up- and down-quark m a s s e s a r e higher than those of ch i ra l SU(3) by a fac tor of 2 o r 3.

The quark m a s s e s determined in our method f r o m the pion m a s s a r e consistent with those found by a study of the nucleon's a t e r m defined from1'

[F; , a,A:(O)l= 6 , ~

o r (67)

a = m , ( ~ u + d d ) .

Taking mat r ix elements of a between nucleon s t a t e s yields m,* = m,Z directly'?

m,* = $~,,,(l- (p2)/2mb2). (68)

Es t imates of a,, range f r o m 51 to 70 MeV o r m,* = 14-19 MeV. This t rans la tes t o quark m a s s e s m,= 25-37 MeV, consis tent with the quark m a s s found in Sec. IV. Jaffee has recently discussed the a t e r m in h i s discussion of scale-dependent light-quark masses.14

The perturbation method developed in th i s section, Eq. (64), yields m,= 22 MeV, while the alternative method based on a wave packet satisfy- ing Eq. (5a) gave a value m q = 33 MeV in Sec, IV. The disagreement i s due to two effects. F i r s t , we have included gluon effects in Sec. IV, but omitted them f rom Eq. (62). Second, the m a s s dependence of $ ( p ) is included in Sec. IV (but not in Secs. 111 and V) and introduces t e r m s proportional to mn2 l n ( m p f f ) in the expansion analogous to Eq. (59). But they a r e not a major source of discrepancy in practice, since the f i t of Sec. IV gave a value m,R. -0.5. In chiral perturbation theory c o r r e c -

,, . ( O ~ ~ q ~ O ) = - F , 2 ( ? i l ~ q l ' ~ ) = - ( 0 . 1 5 GeV)3 tions to the lowest-order resul t a l so involve

m n 2 lnmff2, although the i r effect is numerical ly f o r R, = 3.3-3.5 GeV. The m a s s m, i s not meant to

different f rom ours. b e compared direct ly to m a s s e s f rom cur ren t al-

If we naively extended the bag-model vers ion of gebra. The current-algebra study of pseudoscalar

ch i ra l perturbation theory to include the kaon, we m a s s e s yields only m a s s ratios. Weinberg h a s proposed a m e a s u r e of absolute size." If would obtain the usual chiral-SU(3) X SU(3) current-

a lgebra ra t ios (neglecting isospin breaking) ( ~ I q q q ( ~ ) =ZN,, where Z i s a renormalization con- s tant and N, is the number of quarks in H, then - mff2 1

m,+ a,, 27nK2=26 *-(HlmvqqlH)=

m~ - mqZ ' (66) N v

m,/mSe &. can b e defined. AS = 1 m a s s differences can b e (70)

used to indicate m z - 1 5 0 MeV (this i s t rue in the Instead, the c o r r e c t bag procedure, using a s in- bag model a l so s ince there m, = 280-330 MeV, and put m, and m,, yields

1984 J O H N I? . 1 ) O N O C B U E A R D K . J O H N S O N 21 -

m q / m s ~ &. (71) symmetric phase and qq < 0, the spontaneously

In this application SU(3) i s no! a good symmetry, and significant differences in the treatment of pions and kaons a r e seen. This explains why we agree with Weinberg's value of m,*, but not nz:.

V I CONCLUSIONS

We have shown how the static bag model can be corrected to take into account in a simple way the effects associated with the localization of the particle a t a fixed point in space, Although the modifications a r e smal l for hadrons whose s ize is large in corriparison to the scale se t by the rnass, they have a very sul:stantial effect on the lightest hadron, the pion. By incorporating these corrections, we have s h o ~ v t ~ how it i s possible for the traditional quark-rnodel s tate wlth a substantial extended s ize to coincide with the PCAC pion, which is a "collective state" associated with the spontaneous breakdown of chiral symmetry, -41- though we have by no means g0n.e al l the way to reconcile these two s e e m i ~ g l y very different physical pictures of the pion, we h a w at least jndi- cated how they could be compatible with each other. Our methods have allowed us to estimate with some success the n and K decay constants, and to derive a connection with chiral perturbation theory,

Finally perhaps it would be interesting if we re- mark on a possibly significant similarity between the c h i d field theory and the hag-model description of hadrons cornposed principally of light quarks. In field theory, the ratio

in the limit of ~ l l a s s l e s s pions. If we say / a s in the bag model) that the interior of extended hadrons i s accurately described by a perturbative vacuum (that is, a chirally symmetric state) , then at "edge" where the effects of valence quarks should be lowest one might expect that $q=O (as in the bag model). Inside, wit11 the presence of the valence quarks tc j '> 0. Consequently (~(0) / fq 1 d p ) ) :. 0- Hence, because of Eq. (72), (0 / (Iq 10) <O, that i s , g(x)q(~) changes dlscontinu- ously from zero to l e s s than zero at the boundary of the hadron, In this way it is s imi lar to the order parameter a t the boundary between two pha- s e s which differ by a f irst-order change. Viewed in this way, the bag lnodel for light quarks corresponds to a model of a "two-phase'Vacuuin with the order parameter <q> O defining the chirally

broken state where the chiral symmetry is spontaneously broken, it cannot itself be a chirally syinmetric model. It i s , however, natural that the bag model's chiral density finds i t s source only on the bag boundary.

We would like to thank our colleagues a t the Center for Theoretical Physics for valuable discussions. In particular, we wish to thank V. Bal- uni for some very helpful comments, This work was supported in part through funds provided by the U. S. Department of Energy (DOE) under Con- t rac t No. EY-76-C-02-3069,

APPENDIX: NONSTATIC MATRIX ELEMENTS

For static quantities such a s axial-vector coupling constants o r magnetic rrioments, the methods derived in the main text a r e sufficient, However, for nonstatic quantities, such a s charge radius contained in the form factor F(q) , a further irn- provement is needed. This appendix provides the necessary treatment.

When a photon couples to a particle it t ransfers to i t a finite momentum q. Therefore we must necessarily go beyond a purely static treatment, so rather than considering only fixed static bags, let us consider a moving bag a s a wave packet with a nonzero net momentum 3 . We do this by transforming each of the momentum eigenstates in the wave packet by G, i . e., for mesons

"* $(P) l f i l (p+ 0)) . (Al) Ub2 w p+p.)l I 2

The normalization factor i s such that

This superposition produces a normalized bag state, centered around the origin, with an overall

+ momentum q. This wave packet can be written in a more convenient fashion by redefining the integration variable.

T o calculate the form factor F ( q ) defined by

(111(p') IJ*(x) I I Z / I ( ~ ) ) = ( ~ + p ' ) ~ F ( p - ~ ' ) e ~ " ( * - ~ ) ,

(A41

we transfornl to a matrix element between one static-bag state and one moving state:

2 1 - T H E P I O N A N D A N I M P R O V E D S T A T I C B A G M O D E L 1985

Lf we now integrate this with a factor e-"'' we find

where, dropping small kinematic terms of order q2/yn2, we identify

r , y ( g ) = / d 3 x e - t 9 . x ( M , q I J O ( ~ IM, 0 ) ( A n

with the Fourier transform of the static bag charge density

~ ~ . , = f d ' x r-Lq'yp::: t ic(x) . (A81

Here we wish to compute corrections to the form factor of a heavy particle, the proton, so again we

I

shall neglect such small kinematic quantities of order q 2 / m 2 . In this case, the spin-dependent effects a re also absent and the formulas for mesons and baryons coincide. If we express the result in terms of the charge radius defined by

we find

(v2),,,= (1 + i (p z ) /m9 , (A101

which is also Eq. (15).

'A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn, and V. F. Weisskopf, Phys. Rev. D?, 3471 (1974).

2 ~ . Chodos, K. L. Jaffe, K. Johnson, and C. B. Thorn, Phys. Rev. D g, 2599 (1974).

%. &Grand, R. L. Jaffe, K. Johnson, and J. Kiskis, Phys. Rev. D 12, 2060 (1975).

4 ~ . Donoghue, E. Golowich, and B. R. Holstein, Phys. Rev. D 2, 2875 (1975).

5~ preliminary account of some of this work i s contained in K. Johnson, report given a t Orbis Scientiae, Coral Gables, 1979 (unpublished).

6 ~ . Johnson, Phys Lett. z, 259 (1978). 'Fee, for example, S. Adler and R. Dashen, Current Algebras and Applications to Par t ic le Physics (Ben- jamin, N. Y., 1968) and references therein.

8 ~ o r a review of chi ra l perturbation theory see H. Pagels, Phys. Rep. E C , 219 (1975).

%e have also performed a fit with the baryon wave

packet using the quark wave functions in the form (u2 -12)3/2 instead of (u2-12) in an attempt to simulate the effect of three quarks in a baryon. The parameters found a r e very similar to those of the fit quoted in the text.

' O S . We~nberg, in Festschrift f o r I. I. Rabi, edited by Lloyd Motz (New York Academy of Sciences, N. Y., 1977).

"P. Langacker and H. Pagels, Phys Rev D s , 2070 (1979).

or a review see J. F. Gunion, P. C. McNamee, and M. D. Scadron, Nucl. Phys. m, 445 (1978).

l 3 ~ h e IT t e r m has been studied in the bag model in Ref. 4 and by D. J. Broadhurst, Nucl. Phys. E 5 , 319 (1976).

' 4 ~ . L. Jaffe, Oxford Report No. Ref. 15/79 (unpublished).

the pion and an improved static bag model

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