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Mass Renormalization and Chiral Symmetry Breaking in Nonlinear Spinor Theory W Band 30 a Zeitschrift für Naturforschung Heft 10 Mass Renormalization and Chiral Symmetry Breaking in Nonlinear Spinor Theory W. Bauhoff and D. Englert Institut für Theoretische Physik der Universität Tübingen (Z. Naturforsch. 30 a, 1215-tl223 [1975] ; received July 9,11975) The chiral invariance of conventional nonlinear spinor theory is assumed to be broken. This results in a finite mass renormalization of the nucleon mass. The self-consistency of the assumption is demonstrated. The meson mass eigenvalue equation is solved in lowest approximation. The spec- trum which is no longer parity degenerate is in reasonable agreement with experiment. 1. Introduction A way out of this dilemma is offered by the as- sumption of a spontaneous breaking of chiral in- Nonlinear spinor theory is an attempt for a uni- variance by the vacuum. We will follow this ap- fied description of high energy phenomena It proach in the present paper. In fact, this idea is not claims to calculate self-consistently all particle mas- new. It was used in a similar model in 3, and we take ses and coupling constants. So no (bare) mass term a great deal of our philosophy from this work. But for the fundamental field *f>{x) is introduced into there are some distinctive features of nonlinear the field equation: all masses are assumed to arise spinor theory, e.g. the finiteness of all integrals purely dynamically. Therefore, the propagator of arising from the indefinite metric. Furthermore, the the y-field is supposed to have a pole at a finite mass conclusions are very different from those of 3 due to value corresponding to the nucleon. a slightly different form of the interaction. But there is one problem associated with this line Already in the past, nonlinear spinor theory has of reasoning: Due to the absence of a mass term, been compared with models taken from solid state the field equation has as an additional symmetry physics. We mention only 4 where the analogy to a property the chiral transformations, i.e. it is in- semi-conductor model has been stressed since in variant under the substitution both cases one is dealing with the dynamics of xp(x)-^exp{iays}rp(x). "dressed" particles or clusters. The nucleons were related to the polarons and the mesons to the exci- So the nucleon is forced to stay massless if the chiral tons i.e. to polaron-antipolaron bound states. invariance is retained. The situation is similar to the vanishing photon mass in quantum electrodynamics Following 3, we will now propose an analogy of which is related to the gauge invariance of the nonlinear spinor theory to superconductivity. The theory. nucleon is considered as a mixture of bare fermions A possibility to render the nucleon massive in non- with opposite chirality, but the same charge, just like linear spinor theory is to associate a particle with the quasi-particle in a superconductor which is a opposite parity to the nucleon. Then we have a mixture of bare electrons with opposite charge but couple of particles degenerate in mass, and chiral the same spin. In both cases we find a spontaneous invariance is restored. This recipe was used in non- breakdown of a symmetry group which manifests linear spinor theory in the past 2. The price one has itself in a finite nucleon mass x or a finite gap pa- to pay is not only the introduction of an experi- rameter cp. If these two quantities vanish, the respec- mentally unobserved parity partner of the nucleon, tive particles are eigenstates of chirality and electric but also the mesonic spectrum turns out to be parity charge. The mesons are compared to collective exci- degenerate. So the bound states usually identified tations of quasi-particle pairs. We will not pursue with the n- or ^-meson are, in reality, mixtures of this analogy further mainly because of lack of com- scalar and pseudoscalar states. This remedy seems petence. to us not too attractive either. As is well known, the spontaneous breakdown of a symmetry is combined with the occurrence of a Reprint requests to Dr. W. Bauhoff, Institut für Theoreti- Goldstone particle (for a review see e.g. 5). In 3 a sche Physik der Universität Tübingen, D-7400 Tübingen, Auf der Morgenstelle 14. massless (pseudoscalar) meson was found which 1216 W. Bauhoff and D. Englert • Mass Renormalization in Nonlinear Spinor Theory could be identified with the Goldstone particle. We action. The overall sign of the interaction e = ± 1 do not find such a particle. This failure might be will be fixed in Section 4. due to the indefinite metric invalidating the Gold- By the requirement of invariance against Lorentz stone theorem. But the reason might be also the un- transformations including reflections the propagator symmetrical treatment of the nucleon and the meson of the field if'{x) is restricted to the Lehmann spec- eigenvalue equation (second order in the coupling tral representation: constant for the nucleon, first order for the meson). oo Future work must be devoted to this question. Q1(m2) g.z{m2) F(p) = Jdm2 . (2.2) Before we start, we have to say something on the v p — m +1 £ y p + m + is o functional formulation of nonlinear spinor theory 6. It is needed for performing dynamical calculations, Since y{x) is somehow associated to the nucleon especially for the scattering problem. Since we are field, F{p) will have a pole at the (physical) nucleon mainly dealing with symmetry properties we may mass x. So the spectral function will contain a con- avoid the use of the complicated formalism of func- tribution ~d(m2 — x2) and it is essential that this tional quantum theory. For our purposes we need pole occurs in only one of the two spectral func- only the dynamical equations which were derived tions, since there is no particle with the same mass in'. The resulting equations are of conventional as the nucleon but with opposite parity. Bethe-Salpeter form and do not require the under- The bare mass x0 is related to the physical mass y. standing of the whole apparatus of functional quan- by tum theory. y. — XQ -f- by. (2.3) The paper is organized as follows: In Section 2 where the mass renormalization by is caused by the we introduce the symmetry breaking and discuss the interaction. In fact, the requirement by = 0 will lead relation to mass-renormalization. In Section 3 we immediately to the free theory /2 = 0. In nonlinear solve the nucleon eigenvalue equation and demon- spinor theory, by. will be finite due to the indefinite strate the consistency of the assumption of broken metric of the state space: we do not impose the symmetry. The boson mass eigenvalue equation is usual positivity conditions on @i(m2). solved in lowest approximation in Section 4. Finally, If we want to do a dynamical calculation in non- we give some concluding remarks. linear spinor theory we have to resort to Tamm- Dancoff-like approximations, since the canonical 2. Mass Renormalization and Symmetry Bethe-Salpeter methods do not work for indefinite Breaking metric8. So we will encounter not only the exact We will now start with the discussion of chiral propagator F in the Feynman graphs, but also the symmetry breaking and its relation to mass renor- Green's function G of the Dirac operator with the malization. In order to keep the algebra as simple bare mass x0 . It is the hope that finally G will be as possible, we shall use the non-Hermitean represen- dressed to F as it is done in9, but nobody knows tation and disregard the isospin. These degrees of how to do this in practice in nonlinear spinor theory. freedom will be introduced later on. Of course we This short-coming may be remedied, however, by will employ the four-component Dirac representation performing the mass-renormalization in (2.1). So which is parity-symmetric. The field equation is then: we add bxy(x) on both sides which leads to the equation: Z2 1 iyf dßyj(x) + *0v(x) = e — [y' y^(x)xp(x)ylixp(x) i yf dM y>(x) + x xp(x) = dxtp(x) (2.4) I2 + ysyflxp{x)y)(x)y5y^{x)]. (2.1) + € [yv(x)y{x)y„xp{x) We have added a (bare) mass term x0ip(x) as we i want to separate clearly the effects of mass renor- + VhY' V (*) y> (*) 75 7ß V (*) ] • malization and of symmetry breaking. Later on, we If we now invert the left-hand side Dirac operator the will return to the usual theory with x0 = 0 and Green's function G will have the pole at the physical exhibit the special problems related to this question. mass x as the exact propagator F. Of course, G will Our field equation differs from the model of 3 in the still differ from F (it does not contain continuum relative sign of the vector and axial-vector inter- contributions nor regularizing pole terms). We will 1217 W. Bauhoff and D. Englert • Mass Renormalization in Nonlinear Spinor Theory find additional graphs arising from the <5xy(x)- simply along the lines indicated above: We add the term, on which we will comment below. mass-renormalization term on both sides and invert Up to now, we have only performed a conven- the massive Dirac operator. Together with the chiral tional mass renormalization. Now we go over to the non-invariant F this will lead to a finite nucleon mass limit of the vanishing bare mass x0 = 0. The physical of only one parity and to a parity non-degenerate mass arises now only from the interaction: x — dx.
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