QCD strings with spinning quarks

A consistent description of spin within a QCD string theory has been sought for many years. The addition of dynamical spin to the bosonic string led to the development of supersymmetry and superstring theory [1]. Such theories are more realistic as unified theories of elementary particle physics than as phenomenological descriptions of hadronic states. A more realistic description of hadronic states involves the replacement of the free end of the dual resonance string by the addition of a massive point quark to the end of the string. In 1977 Ida [2] analyzed the motion of a spinless massive quark on the end of a bosonic string. The relativistic flux tube model [3], derived from different assumptions, is mathematically equivalent to a bosonic string with a spinless quark end and produces realistic meson spectra on average, but there is no place for quark spin in this model. In this paper we make a modification of the bosonic string plus bosonic quark model to introduce quark spin. Our clue to constructing a consistent action comes from the suggestion of Buchmüller [4] that the spin of the quark should undergo pure Thomas precession because the quark sees a purely chromoelectric field in its rest frame. This seems to be supported by experimental data [5, 6] and is in agreement with QCD [7, 8]. We begin in Sec. II by discussing the treatment of spin in pseudoclassical language. We show how to construct actions for a free fermion as well as a fermion with background scalar and vector potentials. We analyze the case of a scalar potential in detail and show how the Thomas precession manifests itself in this language. In Sec. III we show in detail that the Fermi-Walker transport of the spin vector, which is the equation of motion of the spin vector for a particle in a scalar potential, leads to Thomas precession of the spin in its rest frame. In Sec. IV we use the example of a spinless quark coupled to the end of a string to argue for the form of the action for a spinning particle coupled to a modified Polyakov string action. The key idea is to obtain the equations of motion of the spin of the quark from boundary conditions, just as the equations of motion of the quark’s position arise from boundary conditions. To this end, we introduce new Grassmann-valued fields on the string worldsheet. In Sec. V we use the consistency of the equations of motion of the quark and the requirement of Thomas precession to fix the parameters in the string action. The result is that the only modification of a free spinning quark plus free bosonic string action is the replacement of the bosonic string position variable by the string position variable plus a term bilinear in worldsheet fermionic variables. In Sec. VI we explore the fermionic gauge invariance of our string action. In the phenomenologically interesting case, we find that the worldsheet fermionic variables are pure gauge degrees of freedom. We find the momentum and angular momentum from Noether’s theorem in Sec. VII. These conserved quantities are the usual starting point for the numerical quantization of the relativistic flux tube model. Finally, we conclude in Sec. VIII.