Nonlinear σ-model and nucleon structure

The elastic scattering of protons on protons at CERN's Intersecting Storage Rings and at FNAL, after the famous diffraction dip, yields a high crosssection when compared to the trend of the diffraction peak before the dip. What do these experimental results tell us? We will be justifying these principal answers: (1) the inelastic overlap function (which controls elastic scattering) can be explained in terms of the characteristics of certain experimentally measured crosssections (more precisely, semi-inclusive cross-sections), (2) a precise mechanism exists which, by bringing into play the inner part of the proton, builds up after-the-dip scattering from before-the-dip scattering, with corresponding predictions, in particular, for CERN's SPS Collider, and (3) after-the-dip scattering is no direct signal of the proton's constituents.

In this manuscript, we discuss a remarkable phenomenon concerning nonlinear and nonintegrable field theories in ($3+1$) dimensions, living at finite density and possessing nontrivial topological charges and non-Abelian internal symmetries (both local and global). With suitable types of Ans\"atze, one can construct infinite-dimensional families of analytic solutions with nonvanishing topological charges (representing the baryonic number) labeled by both two integer numbers and by free scalar fields in ($1+1$) dimensions. These exact configurations represent ($3+1$)-dimensional topological solitons hosting ($1+1$)-dimensional chiral modes localized at the energy density peaks. First, we analyze the Yang-Mills-Higgs model, in which the fields depend on all the space-time coordinates (to keep alive the topological Chern-Simons charge), but in such a way to reduce the equations system to the field equations of two-dimensional free massless chiral scalar fields. Then, we move to the nonlinear sigma model, showing that a suitable Ansatz reduces the field equations to the one of a two-dimensional free massless scalar field. Then, we discuss the Skyrme model concluding that the inclusion of the Skyrme term gives rise to a chiral two-dimensional free massless scalar field (instead of a free massless field in two dimensions as in the nonlinear sigma model) describing analytically spatially modulated hadronic layers and tubes. The comparison of the present approach both with the instanton-dyon liquid approach and with lattice QCD is shortly outlined.

The 331 model with right-handed neutrinos is reassessed to investigate the CP violation in the quark sector. After the spontaneous symmetry breaking, the masses and physical fields of the particle content are obtained. The fermions content of the 331 model is enlarged to include exotic quarks with known electric charge and with masses defined at the TeV scale. The existence of these exotic quarks induces extra CP violations via couplings with quarks of the Standard Model mediated by charged gauge boson with mass fixed at the TeV scale. An extra discrete [Formula: see text] symmetry is introduced in the 331 model to get a stable scalar field that can be a candidate to the dark matter content. The new scalar field interacts at the tree level with the [Formula: see text] gauge boson that works as a dark matter portal. The relic density associated with the scalar field is calculated to yield the solution mass that satisfies the observed dark matter. The region allowed on the parameter space of the dark matter mass versus [Formula: see text] mass is obtained to include the bounds of PANDAX2017, XENON1T(2t.y) and LUX experiments.

We study the influence of boundary conditions on energy levels of interacting fields in a box and discuss some consequences when we hange the size of the box. In order to do this we calculate the energy levels of bound states of a scalar massive field χ nteracting with another scalar field φ through the Lagrangian \(\mathcal{L}_{\operatorname{int} } \) = \(\frac{3}{2}g\phi ^2 \mathcal{X}^2 \)> in a one-dimensional box on which we impose Dirichlet boundary conditions. We find that the gap between the bound states changes with the size of the box in a nontrivial way. For the case where the masses of the two fields are equal and for large box the energy levels of Dashen-Hasslacher-Neveu (DHN model) are recovered and we have a kind of boson condensate for the ground state. Below a critical box size \(L \sim 2.93\left( {2\sqrt 2 /M} \right)\) the ground-state level splits, which we interpret as particle-antiparticle production under small perturbations of box size. Below other critical sizes, \(L \sim \left( {6/10} \right)\left( {2\sqrt 2 /M} \right)\) and \(L \sim 1.17\left( {2\sqrt 2 /M} \right)\), of the box, the ground state and firstexcited state merge in the continuum part of the spectrum.

In the previous chapter, the Nonlinear Sigma Model (NLSM) in d dimensions has emerged as the continuum approximation to the quantum Heisenberg antiferromagnet in d-1 dimensions with additional Berry phases. The partition function is $${{Z}_{{NLSM}}} = \int_{\Lambda } {\mathcal{D}\hat{n} \exp \left( { - \frac{{{{\Lambda }^{{d - 2}}}}}{{2f}}\int {{{d}^{d}}x\sum\limits_{{\mu = 1}}^{d} {{{\partial }_{\mu }}\hat{n} \cdot } {{\partial }_{\mu }}\hat{n}} } \right)} .$$ (13.1) \(\left| {\hat n(x)} \right| = 1\) is a unit vector and x ∈ R d. Λ is the momentum cutoff, i.e., the shortest wavelength included in the \(D\hat n\). The path integral (13.1) is not well defined until we specify a regularization procedure. We consider a classical Heisenberg model on a d-dimensional cubic lattice, with N sites and lattice constant a. Its partition function is $${{Z}_{{HM}}} = \int {\prod\limits_{{i = 1}}^{\mathcal{N}} {d{{{\hat{\Omega }}}_{i}} \exp \left( {\frac{{\bar{J}}}{T}\sum\limits_{{\langle ij\rangle }} {{{{\hat{\Omega }}}_{j}} \cdot } {{{\hat{\Omega }}}_{j}}} \right)} .}$$ (13.2) The continuum limit of Z HM is given by the substitutions $$ \hat \Omega _i \to \eta _i \hat n(x_i ), $$ (13.3) where ηi is 1 (\((e^{ix_i \bar \pi } )\)) for the ferromagnet (antiferromagnet). Sums and differences are substituted by $$\begin{gathered} \begin{array}{*{20}{c}} \hfill {\sum\limits_{i} {F({{x}_{i}}) \to {{a}^{{ - d}}}\int {{{d}^{d}}xF(x),} } } \\ \hfill {\hat{\Omega }({{x}_{i}} + a{{{\hat{x}}}_{\mu }}) - \hat{\Omega }({{x}_{i}}) \to a{{\partial }_{\mu }}\hat{n},} \\ \end{array} \hfill \\ \begin{array}{*{20}{c}} {\sum\limits_{i} {F({{x}_{i}}) \to {{a}^{{ - d}}}\int {{{d}^{d}}xF(x),} } } \\ {\hat{\Omega }({{x}_{i}} + a{{{\hat{x}}}_{\mu }}) - \hat{\Omega }({{x}_{i}}) \to a{{\partial }_{\mu }}\hat{n},} \\ \end{array} \hfill \\ \end{gathered}$$ (13.4) and a dimensionless coupling constant is given by $$\frac{T}{{\bar{J}}} \leftrightarrow f.$$ (13.5) The measure is replaced by $$\mathcal{D}\hat{\Omega } \to \prod\limits_{{|q| \leqslant \Lambda }} {d{{{\hat{n}}}_{q}},}$$ (13.6) where η is the radius of a spherical (sph) Brillouin zone that has the same number of degrees of freedom as the cubic zone (cube) of the lattice $${{\mathcal{N}}^{{ - 1}}}\sum\limits_{{|{{k}_{\mu }}| \leqslant \tfrac{\pi }{a}}}^{{cube}} { = {{\mathcal{N}}^{{ - 1}}}} \sum\limits_{{|k| \leqslant \Lambda }}^{{sph}} { = 1.}$$ (13.7) This determines the spherical Brillouin zone’s radius to be $$\Lambda = \left\{ {\begin{array}{*{20}{c}} {\pi /a} \hfill & {d = 1} \hfill \\ {2\sqrt {\pi } /a} \hfill & {d = 2.} \hfill \\ {{{{(6{{\pi }^{2}})}}^{{1/3}}}/a} \hfill & {d = 3} \hfill \\ \end{array} } \right.$$ (1) Expanding the Hamiltonian in (13.2) in terms of gradients of \(\hat n\) and keeping up to quadratic terms, the relation $$ Z_{HM} (T/\bar J,a) \approx Z_{NLSM} (f,\Lambda ^{ - 1} )e^{dN/f} $$ (13.9) is established at low temperatures (f << 1). The factor e dN/f is the Boltzmann weight of the classical ground state. The continuum approximation holds also for the generating functional (see Appendix B) and for the longwavelength spin correlations, as long as the spin correlation length ξ is much larger than η-1. In this regime, the NLSM properties are weakly dependent on the regularization procedure, and therefore it serves as a model for diverse Heisenberg models. The crucial features of these models are their O(3) symmetry of the ground state manifold and their short-range interactions. Other details will primarily affect the choice of f and η.KeywordsPartition FunctionWeak CouplingSpin WaveNonlinear Sigma ModelWeak Coupling ExpansionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We consider the quark sector of theories containing three scalar SU(2)_L doublets in the triplet representation of A4 (or of S4) and three generations of quarks in arbitrary A4 (or S4) representations. We show that, for all possible choices of quark field representations and for all possible alignments of the Higgs vacuum expectation values that can constitute global minima of the scalar potential, it is not possible to obtain simultaneously non-vanishing quark masses and a non-vanishing CP-violating phase in the CKM quark mixing matrix. As a result, in this minimal form, models with three scalar fields in the triplet representation of A4 or of S4 cannot be extended to the quark sector in a way consistent with experiment.

Operator-product expansion with spontaneously broken symmetry

We have constructed self‐gravitating objects made of scalar fields in different excited states. We present numerical evidence of the stability of such objects. Within the Scalar Field Dark Matter model we find that such configurations could be a better explanation of galactic rotation curves than those obtained from self‐gravitating objects made of a single scalar field in the ground state.

We consider a scalar field theory in AdS d+1, and introduce a formalism on surfaces at equal values of the radial coordinate. In particular, we define the corresponding conjugate momentum. We compute the Noether currents for isometries in the bulk, and perform the asymptotic limit on the corresponding charges. We then introduce Poisson brackets at the border, and show that the asymptotic values of the bulk scalar field and the conjugate momentum transform as conformal fields of scaling dimensions Δ- and Δ+, respectively, where Δ± are the standard parameters giving the asymptotic behavior of the scalar field in AdS. Then we consider the case d = 2, where we obtain two copies of the Virasoro algebra, with vanishing central charge at the classical level. An AdS3/CFT2 prescription, giving the commutators of the boundary CFT in terms of the Poisson brackets at the border, arises in a natural way. We find that the boundary CFT is similar to a generalized ghost system. We introduce two different ground states, and then compute the normal ordering constants and quantum central charges, which depend on the mass of the scalar field and the AdS radius. We discuss certain implications of the results.

We define a distinguished "ground state" or "vacuum" for a free scalar quantum field in a globally hyperbolic region of an arbitrarily curved spacetime. Our prescription is motivated by the recent construction of a quantum field theory on a background causal set using only knowledge of the retarded Green's function. We generalize that construction to continuum spacetimes and find that it yields a distinguished vacuum or ground state for a non-interacting, massive or massless scalar field. This state is defined for all compact regions and for many noncompact ones. In a static spacetime we find that our vacuum coincides with the usual ground state. We determine it also for a radiation-filled, spatially homogeneous and isotropic cosmos, and show that the super-horizon correlations are approximately the same as those of a thermal state. Finally, we illustrate the inherent non-locality of our prescription with the example of a spacetime which sandwiches a region with curvature in-between flat initial and final regions.

In the warped compactification with a single Randall-Sundrum brane, a\npuzzling claim has been made that scalar fields can be bound to the brane but\ntheir Hodge dual higher-rank anti-symmetric tensors cannot. By explicitly\nrequiring the Hodge duality, a prescription to resolve this puzzle was recently\nproposed by Duff and Liu. In this note, we implement the Hodge duality via path\nintegral formulation in the presence of the background gravity fields of warped\ncompactifications. It is shown that the prescription of Duff and Liu can be\nnaturally understood within this framework.\n

We theoretically investigate the condensate state and collective excitations of a two-component Bose gas in a two-dimensional harmonic trap subject to isotropic Rashba spin-orbit coupling. In the weakly interacting regime when the interspecies interaction is larger than the intraspecies interaction (g↑↓&gt;g), we find that the condensate ground state has a half-quantum angular momentum vortex configuration with spatial rotational symmetry and skyrmion-type spin texture. Upon increasing the interatomic interaction beyond a threshold gc, the ground state starts to involve higher-order angular momentum components and thus breaks rotational symmetry. In the case of g↑↓gc and g↑↓

Instanton induced periodic potentials in non-linear sigma models

Measurements of pp and p-barp interactions at the CERN ISR and SPS colliders are reviewed. Results are given for cross sections, scattering amplitudes, multiplicity, and momentum transfer. (AIP)

Quantum field theory in the large N limit: a review