A field theory Lagrangian for the MIT bag model

Volume 78B, number 2, 3 PHYSICS LETTERS 25 September 1978

A FIELD THEORY LAGRANGIAN FOR THE MIT BAG MODEL ~

K. JOHNSON Center for Theoretical Physics, Laboratory for Nuclear Science and Department o f Physics, Massachusetts Institute o f Technology, Cambridge, MA 02139, USA

Received 16 June 1978

A novel kind of local field theory is presented whose Lagrange equations correspond to the equations for particles which follow from the MIT Bag Model.

In this note a reformulation of the MIT Bag Model [1,2] will be discussed. The reformulated model is essentially the same as the original one with the ex- ception that certain states are not present which in the earlier theory were somewhat troublesome. These are the so-called "empty bags" and "glueballs". Fur- thermore, in the reformulated version the basis for a full quantum mechanical treatment will be much clearer because it is a local field theory, albeit with an unconventional sort of Lagrangian. Thus the Lag- rangian for the reformulated theory will make no explicit reference to bag surfaces or surface boundary conditions. These will emerge naturally in the solutions of the equations. In this way the reformulated version of the model is related to earlier attempts [3] at finding a Lagrangian for a field theory with bag- like solutions. These earlier efforts were however un- successful in making a bagged version of QCD with the confinement boundary condition, which is what is needed for a realistic application to problems of hadron structure [4,5]. These field theories also involved ad-hoc limiting procedures which could be implemented at the level of the classical theory but not necessarily in the context of a local relativistic quantum theory [6,7].

Let us first consider a "free" Dirac field which will be bagged. We propose

This work is supported in part through funds provided by the U.S. Department of Energy (DOE) under contract EY- 76-C-02-3069.

= - - • - - 7 " q - m q q - O ( q q ) ,

Ox u x Ox u (1)

where 1 x > 0

O(x) = { 0 x<~O

Variation of (1) yields the field equation

O(#q) ~" +m q+5(cTq) B + ~ I 1

× ~ (qq)] q = 0. gx ~ l

(2)

That is, on the domain

~q > 0 (3)

(3 ' " 1 ~) ) -:- + m q = 0 (4) 1 Bx u

so we have a free Dirac field. On the boundary, where ~q = 0 it is required that the coefficient of 6(c7q) vanish,

[+ 1"1,# ~ ( ~ q ) + B J q = 0 . (5) ax"

Because the Dirac equation is first order, q = 0 on the boundary is, in general, too restrictive. Hence by iter- ation of (5),

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(½(3/8x u) (cTq)) 2 = B 2 , (6)

on the boundary. Since 3/3xU(~q) is normal to the surface ~Tq = 0, (6) implies the surface is space-like. We may therefore put

½(O/3x u) (77q) = BnU (7)

or

n. } (0 /0x) (4q) = 8 , (8)

where nu is the unit space-like normal to the surface. Since B > 0 and ~q > 0 inside, nU is the unit "in- ward" normal to the surface. Eq. (5) then becomes,

(iT"n - 1)q = 0. (9)

Exterior to the domain cTq > 0, there is no system. That is classically space divides itself into a domain qq > 0 with a space-like boundary (or boundaries) containing a free Dirac field (the bag); outside there is nothing.

(9) is of course consistent with cTq = 0 on the surface. Eqs. (4), (8) and (9) are the bag theory equations for a Dirac field [1].

Thus, the field theory with the Lagrangian (1) and the bag theory share at the classical level the same equations. The field theory however enforces the ad- ditional requirement cTq > 0 to define the spatial domain occupied by the fields. Since in our earlier formulation which involved the explicit introduction of surfaces dividing space, this condition was absent, the reformulated bag theory will have fewer solutions than the earlier version. However solutions of the present theory also solve the earlier "version.

To incorporate a local gauge theory of interac- tions is trivial once we have a globally gauge invariant free theory. Thus, since q(x)q(x) is locally gauge invariant, the Lagrangian of the complete theory is sim- ply,

./2 = (d?QC D - B)O(qq), (10)

where

"/2QCD=~{-Cl3"U(131~-~-gsAu) q

with

"~ gauge = 1 Tr [F/~VF r, ]

-- Tr[F~V(3zzA v - OvA # - igs[A,u, Au])] (11)

is the standard QCD field theory Lagrangian. It is an elementary exercise to verify that the bag theory boundary conditions [1] follow from (10). This in- cludes the boundary condition, nuFU v = 0 on the colored field tensor which insures color confinement. Thus, (10) gives the standard bag model except that bags are present only where qq > 0.

E m p t y bags are excluded since the present formulation is a field theory, in contrast to the earlier one which was a relativistic extended particle theory. The ground state is then not a particle (the "empty bag") but the vacuum. Bags containing colored glue excita- tions, but no quarks ("glueballs") [8] are excluded, at least at the semiclassical level since bags are present only where there are quarks (cTq > 0). Such states were present in our earlier formulation since the requirement qq > 0 was absent. There is of course no restriction on states which are mixtures of flavor singlet quark combinations and glue. (The r / i s most probably an example of such a state.)

The Lagrangian which we have presented may not be unique. An alternative possibility has in Particular been suggested by the Budapoest group [8]. In the above lagrangian, the quarks "confine" the colored vector fields in the sense that local gauge invariance was applied to a lagrangian which already describes a quark field that lumps into bags. Suppose we start with a Lagrangian which describes lumped color vector fields to which we attach quarks. One would then expect that the quarks will be confined since they are coupled to the vector fields. This latter theory would not permit us to consider the quark sector in the limit of vanishing coupling (just as the folwaula- tion based on [10] requires the presence of quarks to act as sources for the colored fields).

A Lagrangian which achieves this is

-~ = (d~Qc o - B)O [d2QC D - B ] . (12)

It is easy to find that this Lagrangian classically divides space into finite spatial regions (bags) where

-½Tr[F~VF l = +}Tr[E 2 - e 2] >/B (13)

in which there are colored fields. Because the quark system carries with it a destruction (or creation)

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operator for a point colored quark (or antiquark) wherever there is a finite quark amplitude, we expect a very large colored electric field so the inequality (13) will be satisfied. Hence we intuitively expect that a boundary (where 1Tr [E 2 - B 2 ] = B) can form only where q = 0. Indeed, if we write the Lagrange equations which follow from (12) we find that the boundary equation is

-½Tr [FUVFv] = B

(that is, the quark "pressure" term, n" l(a/ax) (q#) is absent). Further, as before, on this boundary, the vector field obeys nuFUV = 0 which is the signal for color confinement. In this case the Lagrange equations for the Dirac field can only be satisfied i fq = O, on the boundary. I fq = O, this is sufficient to make the quark "pressure", ½ n'(a/3x) (#q) vanish. We have earlier remarked that the condition q = 0 on a boundary in general is not compatible with the Dirac equation. Hence it is not clear that the Lagrange equations which follow from (12) are consistent. How- ever, when coupled to a gauge field which obeys the condition nuFUV = 0 on the same boundary, a sin- gularity may be present in the differential equation for the Dirac field on the boundary, and hence q = 0 could be consistent with a non-perturbative treatment of the equation [9]. In this case one could pos- sibly find non-perturbative solutions for the system (12) which correspond to classical bags filled with quarks in color singlet combinations [8] carrying colored fields. Thus, in this case the confinement of the gauge field leads to the confinement of quarks. The only difference between the Lagrange equations for (12) and (10) is that for (12), q = 0 on the surface while for (10), (iT"n - 1)q = 0. Hence, i ra sin- gularity in the differential equation forces q to vanish on the surface, then the equations for (10) and (12) would be the same. However, this equivalence would not be apparent without studying the field equations in a non-perturbative way. Hence, in the case of the model based upon (12), although many of the previous bag model results can be derived, it is not entirely clear how the results for the light quark hadrons [2] can be obtained.

What about glueballs for the model based on (12)? It would seem that just as (1) permits quark bags without colored glue that (12) would permit pure gauge fields without quarks in the limit of weak cou-

pling, but this is not true. The Lagrangian which permits static vector field bags analogous to the static

quark bags given by (1) instead of (12) is,

= (~gauge -- B)O [B - .Cgauge], (14)

that is, the sign of the argument of the 0 function must be opposite to that in (12). The sign in (12) is dictated by the fact that 1(E2 -- B 2) is expected to be large and positive near a point quark. The sign in (14) is a necessary condition for solutions in the limit gs -+ 0 because for such solutions, a virial theorem applies [1 ] which implies

< f d3xl(E2-B2)>=O, BAG

where the brackets indicate a time average. Conse- quently the inequality ½(E 2 - B 2) > B inside the bag cannot be satisfied.

This also means that in the absence of coupling glueballs cannot exist for this Lagrangian. However, glueballs certainly could exist when gs 4= 0. Never- theless just as we are uncertain about the calculation of the quark bag masses for this model, we are also uncertain about the glueball masses.

It is possible that many of the successful phenom- enological results which follow from (10) can also be obtained for (12) since many of the equations are identical. However, it would be most interesting to study those cases where the results are distinct to find which model most accurately describes the physics ofhadrons, or indeed to find if they are equiv- alent. One difference is that in the absence of quark masses, (12) is formally chiral invariant while (10) is not.

It is suggestive that the second model where +½ Tr [E 2 - B 2 ] ~> B describes a vacuum structure where the existence of colored fields requires a min- imum "electric field strength", analogous to the type of ground state found in super-conductors with the replacement of electric by magnetic fields. The first model with the requirement that 6,(x)q(x) > 0 also carries this implication since in general the quarks act as sources of electric colored fields.

Although a formulation of the bag model as a local field theory has been found, one should not minimize the difficulties in constructing a full quantum theory even in the case of the "free" Dirac system with the classical Lagrangian (1). Nonlinear

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Lagrangians are notoriously plagued by infinities and (1) is certainly quite non-linear! [7] In spite of this, one could be somewhat optimistic because the function O(~q) involves no new scale, which more "ana- lytic" quantities would introduce because ~q carries the dimension 1/L 3 . Certainly the first step in this program will involve the technology which has been developed for the soliton solutions of more conven- tional but physically uninteresting Lagrangians [ 10].

References

[1] A. Chodos et al., Phys. Rev. D9 (1974) 3471. [2] For a review see K. Johnson, Acta Phys. Polonica B6

(1975) 865.

[3] M. Creutz, Phys. Rev. D10 (1974) 1749; W.A. Bardeen et al., Phys. Rev. Dl l (1975) 1094; R. Friedberg and T.D. Lee, Phys. Rev. D16 (1977) 1096:

[4] M. Creutz and K.S. Soh, Phys. Rev. DI2 (1975) 443. [5] See however, R. Friedberg and T.D. Lee, Columbia

University preprint CU-TP-118 (1978). [6] The present formulation shares features.in common

with A. Chodos, Phys. Rev. D12 (1975) 2397. [7] A. Chodos and A. Klein, Phys. Rev. D14 (1976) 1663. [8] R.L. Jaffe and K. Johnson, Phys. Lett. 60B (1976) 201. [9] P. Gn~dig, P. Hasenfratz, J. Kuti and A.S. Szalay, in:

Proc. Neutrino 1975 IUPAP Conf. BalatonfiJred, June 1975, Vol. II, p. 251; and Phys. Lett. 64B (1976) 62; P. Hazenfratz and J. Kuti, preprint; submitted to Phys. Reports C, 1977.

[10] See, e.g., Extended systems in field theory, Phys. Lett. 23C (1976).

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A field theory Lagrangian for the MIT bag model