Bag confinement chiral breaking effect on a goldstone pion

Volume 100B, number 3 PHYSICS LETTERS 2 April 1981

BAG CONFINEMENT CHIRAL BREAKING EFFECT ON A GOLDSTONE PION

T.J. GOLDMAN 1 California Institute o f Technology, Pasadena, CA 91125, USA

and

Richard W. HAYMAKER Louisiana State University, Baton Rouge, LA 70803, USA and Theory Division, Los A lamos Scientific Laboratory, Los A lamos, NM 8 7545, USA

Received 22 December 1980

We present a model in which the pion is a bound-state Goldstone boson in QCD and then the quarks are confined by a MIT bag-like boundary condition. We depart from the MIT cavity approximation in that we dynamically break chiral symmetry prior to confinement and we confine only in the q~ relative coordinate, leaving translation invariance in the total coordinate. We find MTr = 120 MeV, and a bag diameter R~r = 7 GeV -1 .

The MIT bag model of quark confinement [1 ] with perturbative corrections has been quite success- ful in reproducing the observed hadron spectrum with the notable exception o f the pion [2]. This state tends to come out too massive [3] ,1 but more important ly, this approach does not treat the pion as a Goldstone boson. Recently Donoghue and Johnson [4] made some progress on the mass problem. A static cavity containing two quarks is not in a eigenstate of momentum. They noted that it is important to sub- tract off the contr ibut ion to the energy that goes into center-of-mass fluctuations in calculating the mass o f the pion. On the other hand, there are many compel- ling reasons to adopt the view described by Pagels [5] and Pagels and Stokar [5] , in which the pion is a con- sequence of dynamical symmetry breaking in QCD and in which the current algebra quanti ty f~ (pion decay constant) can be calculated with confidence.

In this paper we present a model of the pion in the spirit o f the MIT bag model but at the same time tak-

1 Permanent address: Theory Division, Los Alamos Scientif- ic Laboratory, Los Alamos, NM 87545, USA.

,1 Ref. [3] shows that small instanton effects which split zr and r~ can significantly reduce this mass.

ing into account its Goldstone character. We depart significantly from the static cavity approximation pro- ceeding in the following two steps: (i) We include QCD interactions prior to confinement in order to break chiral symmetry dynamically. (ii) We then impose confinement only in the relative q?:l coordinate giving states that are eigenstates of energy. To implement relative coordinate confinement we found new boundary conditions on the q?:t wave functions analogous to the corresponding MIT conditions on q. Step (i) gives a fi- nite consti tuent quark mass and a massless Goldstone pion. Although step (ii) introduces explicit chiral symmetry breaking, as in the MIT case, it is far less serious a problem in our model. This is because it simply gives a small change to the Goldstone pion wave function of step (i) and gives the pion a small mass. In the MIT case the chiral breaking boundary condit ion provides the primary force responsible for the existence o f the pion. In neither case does the boundary condit ion cor- rectly represent a chiral-invariant confinement. How- ever in the light of our calculation we note in our conclusion that this chiral breaking could be interpreted as the growth o f current quark masses with scale size.

Hadrons can be described in the MIT bag picture as an effective lagrangian density of the form:

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L = L K + L~C D + L B + LU(1) A. (1)

L K consists of quark and gluon kinetic terms and pos- sibly quark mass terms. L(~)C D contains the convential quark-gluon and gluon-gluon interactions which are usually treated in ordinary Feyman perturbation theory. LU(1) A describes small instanton effects which break the axial U(1) symmetry. L B describes the confining effect o f the bag volume energy density. (See ref. [6] for a discussion on the possible origin o f L B.) L B breaks chiral symmetry explicitly. The calculational approach of the MIT cavity approximation is to treat L K + L B first and then treat L(~CD as a perturbation:

L O = L K + LB, L I = L~)CD + LU(1) A. (2)

L B violates chiral symmetry and so L o is not chiral invariant even for zero quark mass. This explicit breaking in L O precludes a proper t reatment of spontane- ous symmetry breaking. (Lu(1) A effects were only added in ref. [3] .)

We consider here an alternative calculational procedure:

L O = L K + L F L I = L B . (3) QCD + LU(1)A'

L O is now chiral invariant for vanishing quark mass terms. This separation in eq. (3) means the following: We now treat L(~CD as the driving term for the Nambu-Jona-Lasinio [7] mechanism (NJL). There are qT: 1 forces at medium short distances that may be strong enough to drive dynamical symmetry breaking. Caldi [8] has argued that small instantons give a strong enough force. We have investigated the short- range core of gluon exchange and found that it is also an important ingredient in the driving term [9]. We feel that this is strong evidence that L~C D should not be treated perturbatively but must be taken into account to redefine the vacuum prior to the introduc- tion of explicit chiral breaking.

Before we impose confinement we have a collec- tion of bound-state mesons and a non-zero constitu-

(exptl) ent quark mass provided a~ is greater than the cn (crit) "tical value a s at which the chiral phase transi- tion occurs. We then impose confinement to preclude the possibility of ionizing these states. This procedure has a consistency check. The NJL driving terms are the above stated interaction at length scale = A -1 . Since ^(exptl) decreases as A -1 decreases, consistency t t S

demands that Goldstone pions occur for A -1 suffi- ciently small so that confining effects can be neglected, but suffic.iently large such that a~ exptl) ~> a~ crit).

Technical details may be found in refs. [9] and [10] ; the purpose of this paper is to show that the up- shot of this view leads to modest changes in the calculational techniques of the standard MIT cavity approximation. We also wish to point out some simple and in- teresting consequences of this view that can be understood independently of the technical details. In refs. [9,10] (i) we discuss modifications of the NJL mechanism in adapting it to QCD, (ii) we calculate a~ crit), (iii) we discuss the electromagnetic and axial currents in our approximation and (iv) we give a derivation of the relative coordinate boundary conditions.

The k-space pion wave function of total momentum P in our model prior to confinement is the solu- tion of an inhomogeneous Bethe-Salpeter equation with a separable kernel. We simply state the result from ref. [9] ,2:

~Tr(k' P) = qS0(k' P) (4)

6 M 2 H 0 _~p) i75S(k _ 1 + S(k + 7P) , f27rQ) ~r- 1(p2 )

where ~ +r is the pion propagator (for p2 _+ 0, Q). -1/p2), frr is the pion decay constant ~93 MeV, M

is the quark mass, qS 0 is a superposition of on-shell plane waves

~u(P2)O(p 1) e x p ( - i P l "X 1) exp ( - iP2 "x2),

and

H 0 . d4k - 0 = 1 f ( 2 ~ tr[13,5¢ (k, P)] .

At the pion pole, p2 = 0, the second term in eq. (4) dominates and we are back to a purely bound-state wave function:

3sM 2 ~ f27rQ) ~-1 ($) - 4 7 4 J 2 s'd$-~-tg(s'-4M2)l/2 (5)"

.2 In ref. [9] we have a soft qq ~r matrix element with a chiral- invariant cut-off A but for simplicity we use a point vertex here with a dispersion cut-off ~2. Hence the Bethe-Salpeter kernel is just a point qq interaction going through a pseudo- scalar channel. Since this is so similar to NJL, eqs. (4), (5) and (6) can be obtained from ref. [7] with only minor QCD modifications.

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and

~2

- 7 • ( 6 ) 4rr 2 4M2

We now go to coordinate space and choose relative and total coordinates:

X 1 = X - x / 2 , x 2 = X + x / 2 .

We set ,3 the relative t ime x ° = 0 and relax the large- Ixl boundary condit ion on the wave function, replacing it with one on a confining sphere in the relative coordinate, i.e., (x 1 - x2)2 - R 2 = 0. There are no interactions in the to ta l coordinate and hence ¢ ~ exp ( - i P . X ) , P = (E, O), For E below threshold qS0(x, P) grows exponential ly for large Ixl. (We are free to use an inhomogeneous Bethe-Sa lpe te r equation since this growing behavior is no longer excluded by a boundary condit ion at infinity.) We also project out the jPC = 0 - + configuration. This gives ¢ = 40 + ¢1 where:

{ F'a E + 2M - I

q~°(x, P) - , (7) - ~ E - 2M F.a

3EM ¢1 (X, P) - rr2f2rrc/) ~-1 (S)

k 2 d k X - - [s - 4(k 2 + M 2 ) 1 - 1 (8)

0 ( k2 +M2)1/2

( EG.a [2(k 2 +M2)+EM] G) ×

\ [2(k 2 + M 2) - EM] G EG" a

where

F = Jo(qr)' G = lo(kr)' (9)

F = iFqjl(qr), G = iFk]l(kr ),

and s = E 2 = p2 = 4(q2 + M2), and H 0 was evaluated

to give -E/(2rr)4M. Our boundary condit ion for equal-mass non-inter-

, a Setting the relative time equal to zero and confining in the relative space coordinate is an approximation to full confinement. In ref. [9] we discuss this further.

acting q and ?t in the center-of-momentum system in a spherical bag is the following:

i.7¢,,/0 _ 7 0 ¢ / . 7 + i ( ¢ - 7°i.7~i'77°)I surface = O. (10)

We refer the reader to ref. [10] for a discussion o f the virtues and limitations of the condit ion. We only note here that it guarantees zero current across the relative bag surface, tz.J = 0, where J = tr (~70~b¥ - ~rlfq~'),0 ) ,4 . We also give in ref. [10] a quadratic boundary condi- t ion needed for classical stability. Further motivation for this condit ion, eq. (10), can be seen by noting that if we write ~ in terms of 2 × 2 blocks q~ = (A B), then eq. (10) reduces to a single 2 X 2 condition:

F "nC + B i 'n + iA + i i 'oDi 'a = 0. (11)

Thus we get one rotationaUy invariant condit ion that does not overdetermine the wave function. Imposing this leads finally to the eigenvalue condit ion on the pion wave function:

- [or f~r~ (s)/M 21 [(E/2M)Jo(qR ) (q/M)Jl(qR)] 2 2 -1

3E (2 (°° kdk (k 2 +M2) 1/2 sin kR (12) + 4 - M \ R d k 2 _ q2

- - o o

o o

+ E R _~ (k 2 +M2)l/2(k 2 _ q2)

where R is the bag diameter, i.e. the qrq separation. First note that in the absence of interactions (inte-

gral terms absent), the condit ion is reminiscent but not identical to the MIT cavity condit ion for quarks in an l = 0 state. The lowest eigenvalue of the full equa- t ion is plot ted in fig. 1. The most significant feature of these curves is the exponential fall-off of ETrl BC with R. This can be easily understood by a potential theory example. Consider a bound state in a short-range attractive potential giving a wave function ~e - ~ r for

,4 We are tempted to call this the relative coordinate current since in terms of product wave functions ar = P2J1 - p 1./'2. To see this consider two non-interacting particle currents J~l' J~" Current conservation for J~ and J~ leads to the equation

~(plp2)/~T+ V X. (p2fl + PlJ2)/2 + Vx.(plY 2 - p2J1) = 0 in terms of total and relative coordinates. The second term is zero in the center-of-mass system and the third term in- volves our J.

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I 0 0 0 - -

I00

nrl

[33

I I [ I I I

,CL = 1.44 GeV

~ f = = 9 1 . 6 Mev

\ \

! ~ , \ (7500 e-2M~Mev M= 550 Mev / Y ~ , ~ R

I I I I I I 5 6 7 8 9 I0 I

R (GeV)-I l

Fig. 1. E~rl BC versus bag diameter R. ~-R is the radius of the relative bag. ETrl BC is the contribution to the pion mass aris- ing from relaxing the large-r boundary condition on the pion wave function and replacing it with the bag boundary condition. Volume energy effects are not included. An approxi- mate fit for this range of R is indicated on each curve.

large r, where K2/2p is the binding energy and # is the reduced mass. If we subsequently put this whole system into a deep well at radius R (much larger than the range of the attractive potential), the mass shift will be proportional to the wave function at R, i.e., e -2KR . However, if the state is very weakly bound or un- bound, the confining eigenvalue condition is of the form qR = constant. In the relativistic system, K is re- placed by the binding energy of the pion prior to con- fmement (2M) and so we can understand the exponential fall-off shown in the figure. Since the pion is a GoMstone boson and is the only deeply bound state it is the only state with the exponentially small confinement eigenenergy. Even though all mesons start on the same footing dynamics dictates this doubly spe-

cial status of the pion. Pursuing the bag calculation further let us add a

volume energy term * s. We write

4 1 3 % ( R ) = A e-2MRTr/N/r~ + "~rB(TR ) (13)

¢S For footnote see next column

( ~ R n is the radius of the sphere). Taking A from fig. 1, M = 300 MeV, B from the MIT bag value, B 1/4 = 0.14 GeV and choosing ~2 to give a good value of f~r (= 92 MeV) we minimize the energy to obtain bag

1 radius ~TR~ ~ 3.5 GeV -1 , m~ ~ 120 MeV. (For these parameters, the first radial excitation occurs safely

high - at EJBC ~ 1.4 GeV. Since this is in the neigh- borhood of the cut-off it is beyond the range of valid- ity of the calculation. The volume energy would further add to this value.) Quark mass terms would fur-

ther raise mTr. We find these results encouraging. This bag radius

of 3.5 GeV -1 is close to the bag radii of other mesons which range from 3.0 to 5 GeV -1 in ref. [2]. Let us translate our g2 cut-off into a range R a = 1/K, where ~2 2 = 4(K 2 +M2). We obtain R = 1.5 GeV -1 which'is

well within the bag size R~ = 7 GeV -1 . This is a con-

sistency check on our lagrangian grouping procedure, eq. (3), since it tells us that the scale for dynamical

symmetry breaking, Rrz, is in the regime where confinement is not important.

In conclusion we find that the effect of the bag is similar to the effect of a small current quark mass term of order Mq/A ~ 1% in that it gives (MTr/A) 2

1% and leads to a small explicit chiral breaking of the spontaneously broken pion wave function. The difference is that the effect of the bag is not that of a local operator of mass dimension 3, and so one ex-

pects violations of the low energy pion theorems of the order of the explicit chiral breaking in the wave

function. However, it should be noted that quark mass terms are scale-dependent quantities and al-

,s Although we take the volume energy term from the MIT bag model we do not include the "zero-point energy", Zo/R, which is fitted to the spectrum in ref. [2] with an inexplicable negative value for Z o. Inexplicable that is, un- less it actually includes the color-electric ("coulombic") part of gluon exchanges which apparently vanish in ref. [2] because the quarks in the bag are uncorrelated. Fur- ther, Donoghue and Johnson have argued in ref. [4] that Z 0 includes center-of-momentum fluctuation effects. The physics of our relative coordinate bag is quite different from the static cavity approximation; we have reduced out the center-of-momentum and have implicitly included the "coulombic" part of gluon exchange since this provides part of the short-range driving term for dynamical symmetry breaking. The question of whether or not here is any residual zero-point energy which should be included is dis- cussed in ref. [9].

279


though they might be small at the dynamical symmetry breaking distance, R a , it is conceivable that they could trigger large effects at the confining distance R~. If for example the quark mass term grows rapidly with distance in the region of R~, then there would be a chiral breaking component to the confining force which should properly be represented by the bag rath- er than a small mass term perturbation. This does not preclude the existence o f other confining mechanisms active on this or larger scales.

We are happy to acknowledge useful conversation with C.G. Callan, L.H. Chan, F. Cooper, J. Donoghue, S. Frautschi, M. Gell-Mann, E. Golowich, K. Johnson, J. Kiskis, H. Pagels, H.D. Politzer, G. Preparata, S. Raby, A.K. Rajagopal, P. Ramond and F. Zachariasen. This work was supported in part by the US Department of Energy under Contract DE-AC-03-79ER0068. R.W.H. was supported in part by the US Department of Energy under Grant No. Eg-77-S-05.5490.

References

[1 ] A. Chodos, R.L. Jaffe, K. Johnson, C.B. Thorn and V.F. Weisskopf, Phys. Rev. 9 (1974) 3471.

[2] T. De Grand, R.L. Jaffe, K. Johnson and J. Kiskis, Phys. Rev. 12 (1975) 2060.

[3] D. Horn and S. Yankielowicz, Phys. Lett. 76B (1978) 343.

[4] J.F. Donoghue and K. Johnson, Phys. Rev. 21 (1980) 1975.

[5] H. Pagels, Phys. Rev. D19 (1979) 3080; H. Pagels and S. Stokar, Phys. Rev. D20 (1979) 2947.

[6] C.G. Callan, R. Dashen and D. Gross, Phys. Rev. D17 (1978) 2717.

[7] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345.

[8] D. Caldi, Phys. Rev. Lett. 39 (1977) 121. [9] T. Goldman and R.W. Haymaker, Dynamically broken

chiral symmetry with bag conf'mement, CalTech pre- print CALT-68-791.

[10] T. Goldman and R.W. Haymaker, Bag boundary conditions for conf'mement in the qc~ relative coordinate, in preparation.

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Documents

Bag confinement chiral breaking effect on a goldstone pion