Nonvanishing Vacuum Research Articles

Model of U(1) chiral-symmetry breaking

An SU(2) gauge-theory model of spontaneous chiral-symmetry breaking through the nonvanishing vacuum expectation value of spin-zero fields is examined. For the case of two fermion flavors, the model explicitly demonstrates the 't Hooft resolution of the U(1) problem through instanton effects. The absence of any conflict with the current-algebra relations of chiral perturbation theory point to a clearer understanding of the gauge-theory vacuum $\ensuremath{\theta}$ periodicity in the case of broken chiral symmetries.

Field theory of massive vector bosons in a model with spontaneously broken symmetry

We study the spectrum of particle states for a charged scalar field interacting with an electromagnetic field in the Lorentz gauge. The Lagrangian contains the quartic self-coupling and imaginary-mass terms that give the scalar field a nonvanishing vacuum expectation value, and the symmetry of the model is spontaneously broken. We identify the massive vector boson and demonstrate that our identification is consistent by showing that the transverse and zero-helicity modes transform correctly among themselves in a Lorentz transformation. We show that each of the helicity modes obeys the Gupta-Bleuler subsidiary condition. We identify the Proca field contained in the theory.

Massive supergravity from spontaneously breaking orthosymplectic gauge symmetry

We construct a Lagrangian density which is manifestly invariant under the orthosymplectic gauge group OSP(1; 4) and under general coordinate transformations. This is done by the use of two multiplets, the symmetric and antisymmetric representations of OSP(1; 4). We present the general features of OSP( m; 2 n) and, in particular, its irreducible representations. The absence of OSP(1; 4) symmetry from the ground state indicates that one of the scalar fields, which is an element of the symmetric multiplet, has a nonvanishing vacuum expectation value. A shift in the fields reveals the physical spectrum of our Lagrangian. Two Goldstone fields are present, a vector and a spinor, corresponding to the breakdown of OSP(1; 4) to the Lorentzian group. The full Lagrangian contains a graviton, a massive spin- 3 2 field, and two massive scalar fields. The generalization to OSP(2; 4) is immediate.

CPConservation in the Presence of Pseudoparticles

We give an explanation of the $\mathrm{CP}$ conservation of strong interactions which includes the effects of pseudoparticles. We find it is a natural result for any theory where at least one flavor of fermion acquires its mass through a Yukawa coupling to a scalar field which has nonvanishing vacuum expectation value.

Reggeon field with non-vanishing vacuum expectation value

It is argued that if the vacuum expectation value of the reggeon field does not vanish, the solution of the reggeon calculus corresponds to the leading singularity of intercept one. The physical interpretation of the reggeon field theory with unstable vacuum is discussed.

Dynamical symmetry breaking in asymptotically free field theories

Two-dimensional massless fermion field theories with quartic interactions are analyzed. These models are asymptotically free. The models are expanded in powers of $\frac{1}{N,}$ where $N$ is the number of components of the fermion field. In such an expansion one can explicitly sum to all orders in the coupling constants. It is found that dynamical symmetry breaking occurs in these models for any value of the coupling constant. The resulting theories produce a fermion mass dynamically, in addition to a scalar bound state and, if the broken symmetry is continuous, a Goldstone boson. The resulting theories contain no adjustable parameters. The search for symmetry breaking is performed using functional techniques, the new feature here being that a composite field, say $\overline{\ensuremath{\psi}}\ensuremath{\psi}$, develops a nonvanishing vacuum expectation value. The potential of composite fields is discussed and constructed. General results are derived for arbitrary theories in which all masses are generated dynamically. It is proved that in asymptotically free theories the dynamical masses must depend on the coupling constants in a nonanalytic fashion, vanishing exponentially when these vanish. It is argued that infrared-stable theories, such as massless-fermion quantum electrodynamics, cannot produce masses dynamically. Four-dimensional scalar field theories with quartic interactions are analyzed in the large-$N$ limit and are shown to yield unphysical results. The models are extended to include gauge fields. It is then found that the gauge vector mesons acquire a mass through a dynamical Higgs mechanism. The higher-order corrections, of order $\frac{1}{N}$, to the models are analyzed. Essential singularities, of the Borel-summable type, are discovered at zero coupling constant. The origin of the singularities is the ultraviolet behavior of the theory.

Broken conformal symmetry. I. The hadron spectrum

The breaking of the asymptotic conformal symmetry of strong interactions is investigated. It is assumed that the symmetry is broken by the nonvanishing vacuum expectation value of a scalar field. The first-order symmetry-breaking term in the two-point function is completely determined by the conformally invariant limit of the theory. The model predicts families of almost linear Regge trajectories. It is predicted that the ratio of total widths to masses of resonances is —approximately—a universal constant. Comparison with baryon resonances shows reasonably good agreement with the prediction.

Two classes of dual models with spontaneous symmetry breaking

Two new classes of dual models are presented. One required technique is the introduction of spontaneous symmetry breaking by means of assigning a non-vanishing vacuum expectation value to the dual-model operator L 0. Another is the use of ‘non-canonical’ spaces-spaces that cannot be generated by use of ordinary a- and b-type oscillators. The first class of models is characterized by two non-negative integers ( M, N) and has a ground-state mass μ 0 given by α′μ 0 2 = 3 2 M + 2N − 1 . These models posses L-type (Virasoro) gauge operators. The (0, 0) model is the usual Veneziano model with intercept one, while the (0, 1) model is probably equivalent to the intercept-minus-one model recently discovered by Gervais and Neveu. The second class of models is characterized by a single non-negative integer M and has a ground-state mass μ 0 given by α′μ 0 2 = 3 2 M − 1 2 . These models possess G-type gauges and each of them can be extended to include fermions. The M = 0 case corresponds to the dual pion model. Even though the new models have large guage algebras, all of them contain ghost states. A loose end in the presentation is the absence of a general proof of cyclic symmetry for the n-point amplitudes.

Is a q-number δ required in (8,8) chiral symmetry breaking?

We investigate the assumption that δ (the chiral-invariant scale-breaking part of the Hamiltonian density) has a dimension in the (8,8) model of chiral symmetry breaking. Using a spectral function sum rule for the vacuum expectation value of the chiral symmetry breaking Hamiltonian density u( x), restrictions for the mass m ϵ and the width Γ ϵ of η 0+(700–1000) are derived from this assumption. They imply that a q-number δ with dimension l δ ⩾ 1 is incompatible with Γ ϵ ⩽ m ϵ and m ϵ ⩾ 600 MeV. On the other hand, if δ is a c-number ( l δ = 0) it is shown that Γ ϵ ⩽ m ϵ is incompatible with m ϵ ⩾ 690 MeV while if Γ ϵ is below m ϵ (e.g. Γ ϵ ⩽ 0.9 m ϵ ) only an ϵ with a lower mass ( m ϵ ⩽ 670 MeV) is allowed. The present phenomenological situation concerning ϵ makes impossible a definite conclusion on the dimensional content of δ in the (8,8) model. However, experiments favor values of m ϵ and Γ ϵ which imply that δ is a q-number consisting of at least two terms with different dimensions and nonvanishing vacuum expectation values. Quite generally, our results may be used to check if a particular theoretical prediction for m ϵ and for Γ ϵ is compatible with a c-number δ in the (8,8) model. We assume in the paper that u has a dimension and that conventional ϵ-saturation is valid. However, independent of the latter assumption it is shown here that in the (8,8) chiral symmetry breaking scheme the dimension of u is between 0 and 4 if δ is a c-number (this may also be seen in the ( 3,3) ⊗ (3, 3) model from the analogous sum rule).

On the dimension of chiral and conformal symmetry breaking

We investigate the general consequences of Gell-Mann's decomposition of the energy density\(\theta _{00} (x) = 0_{00}^* (x) + \sum\limits_n {\delta _n (x) + u(x)} \), whereu is theSU3×SU3 breaking term transforming according to a\((3\bar 3) + (\bar 33)\) representation and being a Lorentz scalar of dimensiond. We show that there must be at least two terms among θ00*, δn with nonvanishing vacuum expectation values. If we further assume that there are only two such terms with dimensionsd' andd″ we can obtain a new sum rule involving the spectral functions of the propagators of θμμ, ∂μAμπ and ∂μAμK, which reads $$\int {\rho \theta (\mu ^2 )\frac{{d\mu ^2 }}{{\mu ^2 }} = X\left[ {\frac{{r - 2}}{{2r}}} \right]} \int {\rho \pi (\mu ^2 )\frac{{d\mu ^2 }}{{\mu ^2 }}} + \frac{{r + 2}}{{r + 1}}\int {\rho _K (\mu ^2 )\frac{{d\mu ^2 }}{{\mu ^2 }}} $$ , where\(r = - \tfrac{2}{3}(c + \sqrt 2 )/c\) andX=(d−d')(d″−d). Saturating with ρ, π, K, one finds, using an earlier result ongρππ, $$\frac{{m_\sigma }}{{\Gamma _{\sigma \pi \pi } }} = \frac{{32\pi }}{{3m_\sigma ^4 }}X\left[ {\frac{{r - 2}}{{2r}}f_\pi ^2 m_\pi ^2 + \frac{{r + 2}}{{r + 1}}f_{\rm K}^2 m_{\rm K}^2 } \right]\left( {1 + [d - 2]\frac{{m_\pi ^2 }}{{m_\sigma ^2 }}} \right)^{ - 2} $$ . We use the experimental valuesmρ≈700 and Γσππ≈200 to estimateX in both the Gell-Mann-Oakes-Renner and the Brandt-PreparataSU3×SU3 symmetry-breaking schemes. Unfortunately, for reasonabl]Literature values offKfπ our sum rule does not distinguish between these two schemes,e.g. forfKfπ=1.25, both schemes giveX≈3. Nevertheless, this typical value allows us to conclude that σ dominance is consistent with many models not involving operators of anomalously high dimensions and at the same time it allows us to exclude many other models.

On the chiral transformation properties of the pseudoscalar-vector-vector interaction

I t has already been observed by many authors (1.3) that the partially conserved axial-current hypothesis (a), PCAC, and the current-field proportionality (a) lead to a vanishing ¢~p~ coupling in the effective-Lagrangian formalism (e). In order to eliminate this difficulty one has to introduce directly an (opT: coupling that violates PCAC. It is the scope of this work to discuss, in the framework of effective Lagrangiaus and the field-current proportionality (~), the transformation properties of this pseudoscalarvector-vector interaction under the chiral SU3Q SUn group. We consider the nonet of pseudosealar fields P and the nonet of dependent or independent scalar fields S to transform like the representations (3, g) and (3, 3) of SUaQ SUs. That is, the 3 × 3 matrix B = S + iP transforms like (3, $) and B + ~ S i P like (g, 3). The 8 vector mesons V# and their 8 ehiral partners At, transform like the representations (8, 1) and (1, 8) of SU3Q SUs. That is, the 3 × 3 traceless matrices /~+) = V#÷~¢# transform like (8, 1) and 2~ ' = V~--g~/# like (1, 8). The SUn scalar 1 transforms like the representation (1, 1) of SUaQ SUa. vector meson ¢% It is then at once evident that one cannot construct an SUsQSUsinvar ian t VVP or ~IVP coupling. Let us furthermore demonstrate that it is not possible to have an SUaQ SUsinvariant interaction of two vector, one pseudoscalar and one scalar meson when the vector mesons couple through their covariant curls in order to maintain the currentfield proportionality. That is, one cannot generate a vector-vector-pseudoscalar meson interaction through that four-meson interaction when the nonvanishing vacuum expectat ion value of the T = 0 scalar fields is considered.

Broken symmetries for any spin

It is shown that in the absence of interactions one can construct nonvanishing vacuum expectation values for the field operators which describe massless particles of an arbitrary spin. Since this necessarily implies a formal breakdown of Lorentz invariance, it might well be anticipated that the vacuum expectation value of the commutators of the canonical fields with the generators of pure Lorentz transformations could place nontrivial constraints upon the theory. However a proof is given of the fact that such commutators fail to impose any condition upon the theory and that the broken symmetry is consequently entirely self-consistent.

Renormalization of the σ-model

We consider the renormalization of the σ-model in the absence of the baryon fields. The model is characterized by three parameters. In order that the current algebra and the partially conserved axial vector current condition are valid in the renormalized σ-model, it is necessary that all divergences of the theory may be absorbed in two constants, the mass and the coupling constant, and no new ad-hoc parameters be introduced to remove divergences. A perturbation method is presented, in which the lowest order gives rise to the usual results of the current algebra (or chiral dynamics). It is shown that indeed all divergences can be absorbed into the redefinitions of the mass parameter and the one coupling constant. This is achieved by demonstrating the existence of an intermediate renormalization (wherein the counter terms are precisely those of a chiral symmetric theory) which makes the theory finite. As a consequence, the mass difference between the scalar and pseudoscalar particles is finite and all the three point and four point couplings are expressible in terms of one parameter. The vacuum expectation value of the scalar σ field is given by a non-linear equation. A consistent treatment of this equation is outlined in detail, and the equation is shown to be finite and well-defined after the intermediate renormalization. The possibility of the spontaneous breakdown of the chiral symmetry, through the non-vanishing vacuum expectation value of the σ-field even when the Lagrangian is chiral invariant, is described and the Goldstone theorem is verified up to the second order in perturbation expansion. The treatment of the axial vector current - pion vertex (such as appears in the pion decay), which contains a linear divergence not encountered in the S-matrix expansion, is detailed. We relate the pion decay constant to the vacuum expectation value of the renormalized σ field in all orders of perturbation theory.

Spontaneous Symmetry Breakdown in a Nonlinear Boson Field: a Model forCViolation

A boson field with quartic self-coupling is discussed in the BCS approximation. For certain values of the coupling constant the transition from simple products to normal ordered products by a Bogolyubov transformation implies a spontaneous breakdown of the original symmetries in the Lagrangian. In a real scalar field a discrete symmetry can be broken; in more complex fields mass splitting also occurs. If the boson is characterized by the assignment ${0}^{+\ensuremath{-}}$, the nonvanishing vacuum expectation value of its field operator may result in tadpole diagrams, which offer an explanation of the $C$-violating transitions. The experimental consequences of different coupling schemes between the ${0}^{+\ensuremath{-}}$ meson and other particles are discussed in detail. The role of the $\ensuremath{\rho}\ensuremath{\pi}\ensuremath{\eta}$ and $\ensuremath{\rho}\ensuremath{\pi}X$ matrix elements is emphasized in explaining different $C$-violating transitions in $\mathrm{SU}(2)$- and $\mathrm{SU}(3)$-symmetric ways, respectively.