Bosonization relations as bag boundary conditions

Nuclear Physics B253 (1985) 308-322 ~~ North-Holland Publishing Company

B O S O N I Z A T I O N RELATIONS AS BAG B O U N D A R Y C O N D I T I O N S Sudhir NADKARNI

The Niels Bohr Institute, Unwersi(v of Copenhagen, Blegdamsvej 1 7, DK-2100 Copenhagen ¢), Denmark H.B. NIELSEN

The Niels Bohr Institute, Universit.v of Copenhagen and NORDITA, Blegdamsvej 17, DK-2100 Copenhagen O, Denmark Ismail ZAHED

Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Phvsws, Massach~*etts Dt*titute of Technology, Cambridge, Massachusetts 02139, USA and The Niels Bohr Institute, Unit,ersi(r" of Copenhagen, Blegdamsvej l 7, DK-2100 Copenhagen O, Denmark

Received 11 October 1984

The more .sophisticated bag models of hadrons become, the less precisely they seem to determine the bag radius. Idealizing this situation leads to the concept of exact bag models - "Cheshire cat" models (CCM) - where the physics is completely insensitive to changes in the bag radius. CCM are constructed explicitly in 1 + 1 dimensions, where exact bosonization relations are known. In the formalism of bag models, these relations appear as boundary conditions which ensure that the shifting of the bag wall has no physical effect. Other notable features of (1 + 1)-dimensional CCM are: (i) fermion number, though classically confined, can escape the bag via a vector current anomaly at the surface; (ii) essentially the same boundary action works for a variety of models and its symmetries determine those of the external boson fields. Remarkably enough, this (1 + l)-dimensional boundary, action has precisely the same form as the one used in (3 - 1)-dimensional chiral bag models, lending support to the belief that the latter are indeed approximate CCM. These (1 - 1)-dimensional results are expected to provide useful guidelines in the attempt to, at least approximately, bosonize (3 -~ 1)-dimensional QCD.

1. Introduction Quantum

c h r o m o d y n a m i c s (QCD), the reigning theory of the strong interactions,

h a s y e t t o b e s a t i s f a c t o r i l y v e r i f i e d in t h e l o w - e n e r g y r e g i m e , w h e r e it h a s s o far p r o v e n i n t r a c t a b l e . M o s t l o w - e n e r g y p r e d i c t i o n s are d e r i v e d f r o m m o d e l s b e l i e v e d to 308

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approximate QCD in that limit. These range from conventional quark bag models [1] to effective chiral models [2]. Bag models (for a recent review see [3]) depict hadrons as bubbles of perturbative vacuum containing weakly coupled quarks and gluons, i.e., "bags", which are immersed in a nonperturbative environment. Much of the successful phenomenology assumes a bag to be a static sphere having a definite radius. The latter is fixed by minimizing the total bag energy (or, equivalently, by balancing pressure at the bag surface) with disregard to quantum fluctuations. In such a framework the bag radius of a hadron is considered to be as real as its mass, charge or electromagnetic radii and is consequently an experimentally measurable parameter. But then it would be impossible to reconcile the large bare bag of the MIT school [4] with the smaller chiral version of Brown and collaborators [5]: in this "physical bag" picture, nature would be forced to choose one or the other. Surprisingly, perhaps, the more sophisticated these bag models of hadrons become, the less precisely they seem to determine the bag radius. Indeed, by improving on the description of the vacuum outside - incorporating quantum effects, boundary conditions, etc. - the concept of a fixed bag radius becomes fuzzier and fuzzier. Consequently, a large set of data points is now accommodated, within the new improved bag models, by a broad range of bag radii. In an entirely different approach, baryons are represented as stable, extended configurations of finite energy within an effective field theory involving exclusively mesonic degrees of freedom. There is increasing evidence that the light baryons of QCD can be described by such "chiral solitons', probably of the Skyrme [6] type to leading order. Recent work by the Princeton group [7] shows that the static properties of hedgehog-type skyrmions are strikingly similar to those predicted by bag models. One is therefore tempted to think that low-energy hadronic phenomenoiogy is, in general, rather insensitive to the value of the bag radius. Idealizing this situation, one can imagine exact bag models where the physics is completely independent of changes in the size or shape of the bag. The radius could then be pushed to infinity, resulting in a purely fermionic description (i.e., QCD), or to zero, so that the description is entirely bosonic (i.e., some effective mesonic theory). We call such idealized bag models "Cheshire cat" models (CCM), after Lewis Carroll's [8] Cheshire cat (here, the bag wall) which tends to fade away when examined closely, leaving behind only its grin (here, the bag boundary conditions which translate the fermionic and bosonic descriptions into one another). In the Cheshire cat philosophy, the bag is an unphysical concept and its use is merely a matter of technical convenience, having largely to do with the fact that the effective mesonic theory may not be known to sufficient accuracy. In such a case there might well exist, for a given low-energy observable, an optimum radius that minimizes the inaccuracy. Thus, a large bag might be more efficient for describing hadronic spectra whereas a smaller one might be better suited for probing hadronic interactions. But if enough corrections are taken into account, both descriptions

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should lead to similar predictions for any observable quantity. Under the Cheshire cat viewpoint, therefore, fitting the bag radius in a particular model is tantamount to fitting the errors of that model: a model without errors leads to an undefined bag radius. In practice, of course, errors are unavoidable when one attempts to mock up the complicated nonperturbative effects of Q C D by simple effective chiral models. However, even in such a situation, adherence to the Cheshire cat philosophy, i.e., that physics should be invariant under changes in bag size or shape, should lead to a set of consistency conditions on the model in question. These would relate the phenomenological parameters of the effective bosonic theory to the parameters of the fermionic theory and the surface couplings. Such constraints could substantially increase the predictive power of the model. It should be stressed that the correctness or otherwise of the Cheshire cat approach has direct physical consequences, because in CCM oscillations of the bag wall should lead to no new hadronic excitation. [Such oscillations have been used, for example, to explain the Roper resonance in the context of both conventional bag models [9] as well as models involving long tube-like bags [10]. We do not wish to imply here that all such computations are doomed to failure. One the contrary, one may start with a CCM and choose to restrict one's attention to only those states, inside and outside the bag, which are believed to be relevant to some particular physical process. In so doing one necessarily introduces errors which convert the exact C C M into an inexact bag model. However, since these errors have been deliberately introduced, with full knowledge of their effects, one should rather regard this procedure as being, in a crude sense, "gauge-fixing" of the CCM.] To demonstrate the existence of Cheshire cat models and to elucidate several of their important features, we go down to 1 + 1 dimensions, where bosonization is known to be exact (for a recent paper dealing with bosonization, see [11]). In sect. 2 we briefly review the general formalism of bag models and illustrate the Cheshire cat principle with a trivial example where we put bosons both inside as well as outside the bag. Sect. 3 describes the simplest example of a true CCM: one in which we place a free massless fermion inside the bag and the equivalent free massless boson outside. This model already contains all the key features of Cheshire cat models. In particular, the action placed on the boundary surface, which is determined uniquely by considerations of locality and symmetry, has a form which we believe to work for any CCM. The resulting bag boundary conditions turn out to be the well-known 2-dimensional bosonization relations. Using these relations, we show that shifting the bag wall has no physical effect. In sect. 4 we generalize to interacting fermions. Only simple interactions are considered here - the massless Thirring and Schwinger models. The former produces only a trivial rescaling of the effective boson field; the presence of local gauge invariance in the latter model is an extra complication that necessitates a gauge condition at the bag surface. Our conclusions are presented in sect. 5.

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2. General considerations In this section we shall briefly review some of the formalism pertaining to bag models, and then illustrate the Cheshire cat principle with a trivial example. To begin with, we divide D-dimensional spacetime into two regions, which we shall call "inside" and "outside", separated by a boundary surface, the " b a g wall". The regions need not necessarily be connected. One particular description of nature is used on the inside, and another one on the outside, via the actions

Si" = fin d°x

fin(X)'

So,t = L . t d O x Eout(X). Varying the fields inside and outside, while keeping them fixed on the bag wall, leads in the usual manner to the corresponding equations of motion in the two regions. We shall refer to these as the "volume equations". In order to construct CCM, we must allow for information to pass across the bag wall, and therefore need boundary equations connecting fields at the surface. This is achieved by writing down an additional "surface action" So, =fon d~?U t~ ( x ) , where d ~ ~ - d~?- n ~ is an element of surface area, directed from the inside to the outside. Fields on the bag wall are now allowed to vary without constraint, resulting in the additional "surface equations" connecting the inside with the outside. The total action is thus S=Sin+Son+Sout

,

and its variation can be written ~ S = ~,~.volume + ~.K' surface - - - i n + out V ~ i n + out + on

"

Note that there are surface contributions from the volume action, since for a given volume action S = fvolumedDXe ( * , Op,@), the variation contains both volume and surface terms:

(

8~

dZ~, ( ) 8~ or,aco

0~,8__~ ~ 8¢ +

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312

The surface term emerges, of course, from a total derivative, which is usually discarded in open-space applications, but is all-important for our present purposes. We now consider a trivial example of the Cheshire cat principle, namely, we shall bosonize a boson. Consider a bosonic field theory defined in terms of a canonical boson field ~ on all spacetime. Let us divide spacetime as above and relabel the field " X " in the inside region. To make the discussion more general, let us also rename the potential energy function on the inside. Thus we have the following inside and outside actions:

These lead to the following volume equations:

a2x + U'(x)=O

(inside),

32¢ + v'(¢) = 0

(outside),

and the surface contribution

8s~:~':~== fo. d Z. {( a.× )Sx -( a.~)8~ }. To connect ~ and X we can use either of the following surface actions:

fdZ"(O-x)a.x s°"= f d Z " ( 0 - x) a.0, for definiteness we choose the former. The variation of the boundary action is then

8S=~r'ac°= fo dZ~ ( ( * - X)8(a.X) - (~X) 8X +(a.X) 8. ), and the total surface variation is 8Sf~r~%~. o. = L d 2~ (( ¢ - X) 8(a~X) + ( a~X - a~¢) 80 }. Note that since 8X.

a~,X is dotted into the surface normal, its variation is independent of

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313

We thus obtain two independent boundary equations:

n" O,X

n" 8,,#

)

boundary.

Differentiating the first one along the boundary surface, one has also

t" O~X = t ~ O~C

(boundary),

where t is any surface tangent vector; together with the second boundary equation we therefore have O.X = 0,¢

(boundary).

To obtain a true CCM, we require that any change in the position of the bag wall must leave the energy spectrum unchanged. This is achieved by setting

v(x = c) = v(c),

vc.

Our trivial CCM can then be rewritten entirely in x-language or in C-language, i.e., the boundary surface can be moved around without affecting the physics.

3. Massless free fermion Our simplest nontrivial CCM is based on the fact that in 1 + 1 dimensions, a massless free fermion bosonizes into a massless free scalar field*. Since the connection between fermions and bosons is essentially nonperturbative - for example, an elementary excitation in the fermionic sector becomes a topological excitation in boson language [13] - we should not expect the discussion to be quite as simple as for the trivial model of sect. 2. Indeed, anomalous quantum effects play a crucial role in the success of the model.It is therefore a bit surprising that the simple model we are about to discuss appears to contain all the crucial features of the more interesting models of interacting fermions in 1 + 1 dimensions. We take as our inside action the one for a free massless fermion field ~b:

while on the outside we place the free massless boson field C: 1

2

* Strictly speaking, one needs to exercise great care in defining such objects (see, e.g., ref. [12]); we shall ignore such niceties as they are irrelevant for our present purposes.

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Variation of Sin and Sout leads to the following volume equations of motion: i~6 = 0

(inside),

02¢ = 0

(outside),

and the surface variation ~ssurfaoCtt L n d ~v~' { - ½i ~ y~,# + ½i~Y~8# - (3~0) ~¢ }. =

We need a boundary action compatible with the Cheshire cat criterion: unrestricted communication of information across the bag wall. For reasons of locality and renormalizability, let us agree to exclude ratios of dimensionful operators (such as ~tk/~756) from appearing in our boundary action. The requirements of invariance under parity and global chiral rotations are then sufficient to determine the following unique form for the boundary action: Son

=

fond~ v"{ ½n~,~e'2gvs'~b },

where ~ must be a pseudoscalar field that translates according to ~,--, ~ , - 8/v/~ under a chiral rotation ff --* eiV~°ff of the fermionic field. The form of Son is unique up to a trivial rescaling of ~ (necessitated, for example, by the Thirring interaction to be discussed in sect. 4) which would change the coefficient 2v/~- in the exponent. The bag normal n~ must be restricted to be spacelike (n 2= --1) or null (n 2= 0); timelike bag normals are incompatible with CCM, as will become clear very shortly. 3.1. CLASSICALBOUNDARY CONDITIONS Combining the variation of So, with the surface contributions from we get the following surface equations:

Sin and Sou"

(bag wall), where the first equation (which we shall call the @equation) has been used to rewrite the second one (the @equation). We can readily see, from the @equation, the necessity of excluding timelike bag normals: if n 2= + 1, the matrix (ici- n2e ~2#-Y5.) acting on ~b becomes invertible, implying that ~ vanishes on the boundary. This is clearly incompatible with CCM. The physical reason behind this stems from the representation of a fermion as a soliton in the boson picture [13]. Thus, a fermion leaving the bag causes a change in

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315

the vacuum well on one side of the fermion. For a timelike bag normal, i.e. a spacelike bag wall, information that the fermion has escaped the bag would have to travel at superluminal speed to produce the required change in vacua. The 0-equation merely states that the rate of flow of axial charge normal to the bag wall is conserved; it is just the normal component of the well-known bosonization relation

The q,-equation, on the other hand, is somewhat more complicated in its interpretation. Classically, it implies that the normal component of the vector current vanishes at the bag wall (this is readily seen by multiplying the if-equation on the left by ~, taking the hermitian conjugate equation, and adding the two together): ~$ff = 0

(bag wall).

Thus fermion number is classically confined inside the bag, and there can be no classical CCM. The hoped-for tangential component of the above bosonization relation, namely

must therefore obtain as a purely quantum effect. In fact, it is produced by a vector current anomaly operating at the bag surface via the tk-equation. The same quantum effects are responsible for making the pressure balance on both sides of the bag wall, independently of its position, thus making the wall into a Cheshire cat. We demonstrate these effects in the following subsection. 3.2. ANOMALOUS QUANTUM EFFECTS

We first show that although classically confined, fermions can escape the bag through quantum effects. For simplicity, we restrict the discussion to a single, static bag wall located at x = X, with bag normal ( n , ) = (0, 1). Inside this open-ended bag we have the continuity equation, = o,

x
which on integration from - o¢ to X yields the rate of change of fermion number N:

dN dt

= f_"xo¢d x 3°j°

=fx_o¢dxotj l =jl(x).

Classically, jl(X) vanishes as above. To demonstrate the anomalous quantum effects, it is most convenient to regularize the current by point-splitting in the time

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316

direction along the bag wall. Thus we define jx(t) = ~ ( t ) y ? / , ( t + e),

E ---) 0 .

Using the boundary condition ei2~r~q'(')~

(t) = iyl~k(t),

we may write yl~k(t + e) = ( - i)e'2¢';r'4't'+')4,(t + e), and for small e we may expand the exponential as follows: e n # ; ~ , . . . ~, = en#~,ou)[1 +

i2~/-~yse~( t)]e2~l*(,),*. -.)1

In computing the commutator of two O's on the border of a half-space, reflections from the surface must be taken into account. These change the free-space value [0(t, x ) , 0 ( 0 , 0 ) ] = - ~i[sgn(t + x) + s g n ( t - x)] by a factor of 2, resulting in [O(t),O(t+e)]=isgn(e)=

+i.

Thus the exponential of the commutator of two O's is unity along the bag wall, and we have

j,(t) = i f ( t ) ( - i)en~)'°(°[1 + i2¢'~-ys~+(t) ] ~ ( t + e). We use the boundary condition again, in its adjoint form ~, (t)e i2v~'r~q,(t)

= _

i~ (t))q,

to obtain

j,(t) = -j,(t) -i2qr~-eq)(t)~bt(t)~b(t + e). Finally, we replace the operator product by its leading short-distance singularity:

~*(t)~b(t + e) = /

+ [regular terms],

to obtain the desired result 1

•

jl(t) = ---~O(t), 71r

x = X.

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317

Thus fermions can enter or leave the bag if the value of ~ on the bag surface fluctuates in time. Next, we show that shifting the bag wall has no physical effect. To do this we must show that the so-called "pressure balance" condition, i.e., equality of the stress-energy-momentum tensors on both sides of the wall, is satisfied regardless of the position of the wall. The canonical stress-energy-momentum tensor for a free massless fermion field has the following components:

,~ Ol ~ ~ I0 ~ ~

--

½qJti~11~.

In 1 + 1 dimensions, quantum fluctuations give rise to the following operator equivalence [14]:

q / ~ al,k = - ~i~{ Jl + r~J0, + }, where the current is defined in a properly regularized manner, i.e. by point-splitting, and use is made of the leading singularity in the short-distance expansion of the product of q~ and ~t. The fermionic stress-energy-momentum tensor may thereby be rewritten in the so-called Sugawara form [14,15] involving only current operators:

.ff~=½~r(j~j~+j~j~_~l~jxjx),

~/oo= -1/" = 1.

At the bag surface, the bosonization relations 1

show that the fermionic stress-energy-momentum tensor is equal to the canonical stress-energy-momentum tensor for the equivalent free massless scalar field:

5 . ~ = {( a . , a~+ + a~, a~+ - 7"" a~, a~+). (Note: strictly speaking, the canonical and current forms of the fermionic f f ~ are equal only up to an infinite c-number which reflects the differences in regularizing the zero-point vacuum fluctuations in the two formulations; this c-number is calculable and must be inserted into the pressure balance equation in a real calculation.)

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4. Simple interactions When the basic CCM of the preceding section is extended to include simple interactions, of the Thirring and Schwinger type, we find that the boundary action remains essentially unchanged. 4.1. MASSLESSTHIRRING MODEL The Thirring interaction [16,12] is a four-fermion point interaction, corresponding to the inside action t

--

2

where g is a dimensionless coupling constant. Both kinetic and interaction terms in the action bosonize into the kinetic term for a massless scalar field. Clearly the latter requires rescaling in order to get the outside action in the canonical form

Sout= Lutd2x(12(8~,ck)2}; this rescaling then shows up in the bosonization relation

The above relation can be obtained as a bag boundary condition by appropriately rescaling qb in the exponent of the free fermion boundary action, which therefore takes the form

The rest of the discussion for the massless Thirring model goes along pretty much the same lines as for the free fermion case. The only differences are that various quantities, such as the stress-energy-momentum tensor, must be rescaled by factors of i/1 + these rescalings are well described in the literature [11,12] and need not be repeated here.

gilt;

4.2. MASSLESS SCHWlNGER MODEL

Of greater interest is the effect of adding a gauge interaction to the CCM of the preceding section. The simplest gauge model is the massless Schwinger model [17, 18], i.e., quantum electrodynamics of a massless fermion field in 1 + 1 dimensions. The model is exactly soluble, and bosonizes into a free scalar field of mass m = e / v ~ - , where e is the fermionic charge. In constructing the CCM for this

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319

system, we shall find it necessary to restrict gauge transformations on the bag surface; otherwise things work in much the same manner as for free fermions. The inside and outside actions are

Soo = fo.

2(e.,) _ {m2¢)

,

which lead to the following volume equations of motion:

( iO- eA)tp=O } a~,E = e~'y~,'ysg/, E =-e¢,/~a"d/~ ( oq2 4- m 2 ) , ~ = O,

m 2

~-

e 2

~r

(inside)

(outside),

and the surface variation

The electric field E in the Schwinger model is in a sense an auxiliary field; it is completely determined by the fermion field via the auxiliary equation

81E -- e~bt~b. Thus the bosonization relations for the free fermion theory are sufficient also for the massless Schwinger model. Note also that the equation of motion for E can be rewritten in the form

whence we identify, at the bag surface, E = m ~ + const, the additive constant being nothing more than the "background electric field" (in fermion language) or the "chiral vacuum angle" (in boson language) [18]. Since the electric field provides no new dynamical degrees of freedom, we may take as our surface action the same one as for the noninteracting case, namely

So.= fondU"

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S. Nadkarni et al. / Bag boundao' conditions

provided that we restrict our gauges so that no A, appears in ---i,~ ' A"~urfa~' out" e+ This means that we must impose the gauge condition ~ n ~ t A v[ bag wall = 0 .

To get a feeling for this condition, consider, for simplicity, the bag wall given by x = const, so that ( n ~ ) = (0,1). To achieve the gauge condition we start out by quantizing in the A 0 = 0 gauge. Then, we allow only those gauge transformations A, ---,A, + a,A for which the gauge function A(x, t) is constant along the boundary surface. Away from the surface A is arbitrary, so this is hardly such a severe restriction.

5. Conclusion

We have investigated the possibility of idealized bag models of hadrons, where the bag surface merely differentiates between equivalent fermionic and bosonic descriptions of the same underlying physical reality. We have shown that such "Cheshire cat" models can be implemented in 1 + 1 dimensions, for bag surfaces moving not faster than the speed of light, via a local boundary action, the form of which follows uniquely from symmetry arguments. It is remarkable that our boundary action looks precisely the same as the one used in (3 + 1)-dimensional chiral bag models, if one makes the identification (see appendix) f~a, l)= 1/2f~-, where f f is the pion decay constant in D dimensions. Since the connection between fermion and boson representations is nonperturbative, CCM cannot exist at the classical level. We have demonstrated the crucial role played by quantum effects, as manifested in the surface anomaly of the fermionic current as well as in the possibility of writing the fermionic stress-energy-momentum tensor solely in terms of fermionic currents. These were important steps in showing that the pressure balances independently of the position of the bag wall, thus producing a true CCM. For the sake of simplicity, we restricted our arguments to massless fermions which are governed by the particularly simple Sugawara form of the stress-energy-momentum tensor. With massive fermions the details get more complicated though the physics is still much the same. We shall take up these and other related matters in a forthcoming paper [20], which will deal mainly with the formalism of Cheshire cat models without reference to any specific interaction. Our programme is to extend this work to non-abelian gauge models in 1 + 1 dimensions and finally to examine the situation in the real (3 + 1)-dimensional world. While several of our results are clearly peculiar to 1 + 1 dimensions, we believe that much of this work can nevertheless be carried over to 3 + 1 dimensions, at least in an approximate way. In particular, the boundary action determines the symmetry properties of the external boson field, and guessing the correct boundary

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action, with the appropriate degrees of freedom, would go a long way towards establishing the effective bosonic theory. In this way CCM could be constructed in 3 + 1 dimensions, to some degree of approximation. The similarity of the boundary actions in 1 + 1 and 3 + 1 dimensions is, in this respect, very encouraging, and strengthens our belief that realistic (3 + 1)-dimensional bag models are approximate Cheshire cat models. The idea of unphysical bag walls and the name "Cheshire cat models" originated with K. Johnson, who is gratefully acknowledged. We wish to thank our colleagues at N B I / N O R D I T A for numerous useful discussions, and C. Mayuga for literary assistance. Research support for one of us (S.N.) was provided by the Danish Research Council and for another one of us (I.Z.) by the US National Science Foundation.

Appendix We show that the factor 2f~- in the exponent of our (1 + 1)-dimensional boundary action can be interpreted as the inverse of the (1 + l)-dimensional pion decay constant, f~ ~1. To make such an identification, recall that the lagrangian density for the nonlinear sigma model:

can be rewritten in terms of an angular variable 8, defined by o -f~cosS,

~r --- ~f~sin 8,

in the form ½fd(8~8) 2 + ½f,2sin28 ( 3 . # ) 2In terms of O, the surface term 1

b

2f. assumes the form

Since we are interested only in the scaling properties of 0, we can make the approximation # = const, or, alternatively, let 0 become very small. 0 is then related to a canonical field ~ by the scaling transformation 1 0=~¢,

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whence the surface coupling becomes

For the simple theories we have been considering, the factor ,r. ,~ can be disregarded and the "pion decay constant" f,, can then be read off from the exponent in the boundary action. The above arguments hold in any number of dimensions; in 1 + 1 dimensions we get 1

This is consistent with the result obtained in ref. [19].

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