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Ambidextrous theory of the weak interactions
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Path Integral Formulation of Mean-Field Perturbation Theory. CARL M. BENDER, Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139; FRED COOPER, Theoretical Division, Los Alamos Scientific Laboratory, University of California, Los Alamos, New Mexico 87455; AM, GERALD S. GURALNIK, Physics Department, Brown University, Providence, Rhode Island 02912. We develop a convenient functional integration method for performing mean-field approximations in quantum field theories. This method is illustrated by applying it to a self-interacting 4” scalar field theory and a JJ‘J,, four-Fermion field theory. To solve the 9” theory we introduce an auxiliary field x and rewrite the Lagrangian so that the interaction term has the form ~4”. The vacuum generating functional is then expressed as a path integral over the fields x and 4. Since the x field is introduced to make the action no more than quadratic in +, we do the 4 integral exactly. Then we use Laplace’s method to expand the remaining x integral in an asymptotic series about the mean field x0. We show that there is a simple diagrammatic interpretation of this expansion in terms of the mean-field propagator for the elementary field Q and the mean-field bound-state propagator for the composite field x. The 4 and x propagators appear in these diagrams with the same topological structure that would have been obtained by expanding in the same manner a # field theory in which x and Q are both elementary fields. We therefore argue that by renormalizing these theories so that the meanfield propagators are equivalent, the two theories are described by the same renormalized Green’s functions containing the same three parameters, pa, ma, and g. The quartic theory is completely specified by the renormalized masses pL2and m2 of the x and 4 fields. These two masses determine the coupling constant g = gGz, m*). The cubic theory depends on $ and m2 and a thiid parameter g, , g = g($, ma, g,,), where g,, is the bare coupling constant. We indicate that g($, m2, go) < g($, mz) with equality obtained only in the limit g, -+ co. When g, -+ co the wavefunction renormalization constant for the x field in the cubic theory vanishes, and the cubic theory becomes identical to the quartic theory. Our approach guarantees that all quartic theories have the same graphical topology in the mean-field approximation. To illustrate this we show that the mean-field expansion of the four-Fermion current-current interaction theory is renormalizable and reproduces the results of the usual vector meson theory. A coupling-constant eigenvalue condition is derived which could serve to distinguish current-current interactions from normal electrodynamics. Ambidextrous Theory of the Weak Interactions. A. DE R~~JULA,HOWARD Geo~c~, AND S. L. GLASHOW, Lyman Laboratory of Physics, Harvard University; Cambridge, Massachusetts 02138. We construct a renormalizable model of weak interactions based on the gauge group SU(2)r x SU(2)a x U(1) and incorporating a natural suppression of all flavor-changing neutral currents. We resolve the dilemma between the observations of the “high-y anomaly” and of parity-violating neutral currents in neutrino physics. Parity violation in atomic physics is predicted to be smaller than in the standard gauge model. The decay p -+ ey need not be absolutely forbidden. The theory requires two charged and two neutral intermediate vector bosons with masses between 50 and 200 GeV; and at least six quark flavors, three charged leptons, and three massive neutral leptons. A Theory ofFlavor Mixing. A. DE R~~JULA,HOWARD GEORGI, AND S. L. GLASHOW, Lyman Laboratory of Physics, Harvard University, Cambridge, Massachusetts 02138. We investigate the origin of the Cabibbo angle and other flavor-mixing angles in an ambidextrous SV(2)‘ X su(2)a X U(1) gauge theory of weak and electromagnetic interactions involving 2n quark flavors. We show how a discrete symmetry of the Lagrangian leads to flavor-diagonal weak currents where all flavor-mixing angles vanish. Each quark is assigned a position on the “clock of flavor” and the charged-current weak interactions are identified as nearest-flavor couplings. Soft breakdown of the discrete symmetry leads to a one-parameter family of flavor-mixing angles in the charged
ABSTRACTS
OF
PAPERS
TO
APPEAR
IN
FUTURE
ISSUES
Path Integral Formulation of Mean-Field Perturbation Theory. CARL M. BENDER, Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139; FRED COOPER, Theoretical Division, Los Alamos Scientific Laboratory, University of California, Los Alamos, New Mexico 87455; AM, GERALD S. GURALNIK, Physics Department, Brown University, Providence, Rhode Island 02912. We develop a convenient functional integration method for performing mean-field approximations in quantum field theories. This method is illustrated by applying it to a self-interacting 4” scalar field theory and a JJ‘J,, four-Fermion field theory. To solve the 9” theory we introduce an auxiliary field x and rewrite the Lagrangian so that the interaction term has the form ~4”. The vacuum generating functional is then expressed as a path integral over the fields x and 4. Since the x field is introduced to make the action no more than quadratic in +, we do the 4 integral exactly. Then we use Laplace’s method to expand the remaining x integral in an asymptotic series about the mean field x0. We show that there is a simple diagrammatic interpretation of this expansion in terms of the mean-field propagator for the elementary field Q and the mean-field bound-state propagator for the composite field x. The 4 and x propagators appear in these diagrams with the same topological structure that would have been obtained by expanding in the same manner a # field theory in which x and Q are both elementary fields. We therefore argue that by renormalizing these theories so that the meanfield propagators are equivalent, the two theories are described by the same renormalized Green’s functions containing the same three parameters, pa, ma, and g. The quartic theory is completely specified by the renormalized masses pL2and m2 of the x and 4 fields. These two masses determine the coupling constant g = gGz, m*). The cubic theory depends on $ and m2 and a thiid parameter g, , g = g($, ma, g,,), where g,, is the bare coupling constant. We indicate that g($, m2, go) < g($, mz) with equality obtained only in the limit g, -+ co. When g, -+ co the wavefunction renormalization constant for the x field in the cubic theory vanishes, and the cubic theory becomes identical to the quartic theory. Our approach guarantees that all quartic theories have the same graphical topology in the mean-field approximation. To illustrate this we show that the mean-field expansion of the four-Fermion current-current interaction theory is renormalizable and reproduces the results of the usual vector meson theory. A coupling-constant eigenvalue condition is derived which could serve to distinguish current-current interactions from normal electrodynamics. Ambidextrous Theory of the Weak Interactions. A. DE R~~JULA,HOWARD Geo~c~, AND S. L. GLASHOW, Lyman Laboratory of Physics, Harvard University; Cambridge, Massachusetts 02138. We construct a renormalizable model of weak interactions based on the gauge group SU(2)r x SU(2)a x U(1) and incorporating a natural suppression of all flavor-changing neutral currents. We resolve the dilemma between the observations of the “high-y anomaly” and of parity-violating neutral currents in neutrino physics. Parity violation in atomic physics is predicted to be smaller than in the standard gauge model. The decay p -+ ey need not be absolutely forbidden. The theory requires two charged and two neutral intermediate vector bosons with masses between 50 and 200 GeV; and at least six quark flavors, three charged leptons, and three massive neutral leptons. A Theory ofFlavor Mixing. A. DE R~~JULA,HOWARD GEORGI, AND S. L. GLASHOW, Lyman Laboratory of Physics, Harvard University, Cambridge, Massachusetts 02138. We investigate the origin of the Cabibbo angle and other flavor-mixing angles in an ambidextrous SV(2)‘ X su(2)a X U(1) gauge theory of weak and electromagnetic interactions involving 2n quark flavors. We show how a discrete symmetry of the Lagrangian leads to flavor-diagonal weak currents where all flavor-mixing angles vanish. Each quark is assigned a position on the “clock of flavor” and the charged-current weak interactions are identified as nearest-flavor couplings. Soft breakdown of the discrete symmetry leads to a one-parameter family of flavor-mixing angles in the charged