PHYSICAL REVIEW D VOLUME 37, NUMBER 11 1 JUNE 1988 Explicit o meson, topology, and the large-N limit of the Skyrmion Thomas D. Cohen Departrnerlt of Physicv arjd Astronomy, University of Maryland, College Park, Maryland 20742 (Received 12 November 1987) A generalization of the Skyrme model based on the linear u model, i.e., with an explicit u-meson degree of freedom, is proposed. Although the winding number he., the baryon number) of such a nlodel is not strictly conserved at the semiclassical level, it is shown that in the large-N limit all pro- cesses which violate baryon conservation are exponentially suppressed. The possibility that higher- order 1/,qr effects will keep baryon conservation exact at finite N is discussed as are possible phe- nomenological advantages of this kind of model. Over the past several years there has been a revival of interest' in Skyrme's old idea2 that baryons are topologi- cal solitons of chiral fields in which one identifies the baryon number with the topological winding number. Much of this interest stems from 't Hooft's demonstra- tion that in the large-N limit QCD is equivalent to a theory of weakly interacting meson^.^ Witten's analysis of scaling properties of baryons for large N shows that baryons scale with 1 /N in the same way as a classical sol- iton scales with the coupling constant4 also added plausi- bility to the Skyrme model. Witten's demonstration that a Wess-Zumino term could be added to the Skyrme La- grangian to account for the anomaly structure of QCD and that this term had a topological number equal to the number of colors in QCD (Ref. 5) also served to justify the model. While these developments have certainly added plausi- bility to Skyrme's original idea one important ingredient has not really been justified-the use of a nonlinear a model. Let us consider for a moment why the Skyrmion is based on the nonlinear, as opposed to the linear, a model. The basic reason is topology. An unambiguous algebraically conserved density can be constructed from the fields in the nonlinear a model: where L, = Utd,U and U is the 2 X 2 matrix which con- tains the chiral fields. (For reasons of simplicity we will restrict our attention to the two-flavor case.) The non- t linear constraint is that U is unitary, i.e., UU = 1. In terms of u and rr fields U is ( a + i r . ~ ) / f , and the non- linear constraint is that a2+rr2= f t. Because the density in Eq. (1) is unambiguously conserved, the space integral of B0 is a constant of motion for any dynamics (i.e., any equations of motion) and is a topologically conserved ob- ject which one might plausibly associate with the con- served baryon number. Consider now a linear a model. Such a model can also be expressed in terms of a 2 x 2 matrix U =(a + ir.rr)/ f, but the constraint that USU= 1 is dropped. Thus such a model has an explicit degree of freedom associated with the a meson. As in the case of the nonlinear model Eq. (1) defines an algebraically conserved density with L, defined for the linear model by One justification for using Eqs. (1) and (2) to define a baryon current is the work of Goldstone and ~ i l c z e k . ~ They consider a u model coupled to fermions. By in- tegrating out the fermions and making an adiabatic ap- proximation they get a mesonic expression for the baryon number given by the integral of B0 of Eq. (1) with L, given in Eq. (2). This has been used in the context of the nonlinear u model [in which Lp in Eq. (2) becomes UtapU] as a justification of the Skyrme ~ u r r e n t . ~ Here we merely note that this justification is equally good for the linear version of the model. Although the density given by Eq. (1) is algebraically conserved wherever it is well defined in the linear model it is ambiguous. In partic- ular when UtU =o, L,,, and hence B,, is not well defined. Thus the possibility exists that for some dynam- ics UtU will pass through zero at some space-time point and at such a point the space-integrated zero component of the density could conceivably change discontinuously: the space integral of B, is not a topologically conserved object. This poses a potential problem for soliton models in which this current is identified with the everywhere- conserved baryon density. Note that in a linear a model there is a Mexican hat potential v ( a 2 + n 2 ) which makes it costly in energetic terms to reach "dangerous" configurations with u2+n2=0, and that the larger the a mass is the more energy such configurations cost. Thus the dynamics of the linear a model with a large a mass will tend to exclude the dangerous configurations. Nev- ertheless, no matter how large the u mass is, such configurations do exist and the conservation of B, cannot be guaranteed. Topology is in some sense the essence of the Skyrme model and perhaps is sufficient to justify the choice of the nonlinear a model. There are, however, consequences of this choice which should be discussed. The nonlinear a model can be thought of as the limit of the linear model in which the explicit a degree of freedom plays no role, i.e., where the u mass tends to infinity and the chiral 3344 @ 1988 The American Physical Society