Sigma Model Research Articles

Six-meson amplitude in QCD-like theories

We calculate the relativistic six-meson scattering amplitude at low energy within the framework of QCD-like theories with $n$ degenerate quark flavors at next-to-leading order in the chiral counting. We discuss the cases of complex, real and pseudoreal representations, i.e. with global symmetry and breaking patterns $\mathrm{SU}(n)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(n)/\mathrm{SU}(n)$ (extending the QCD case), $\mathrm{SU}(2n)/\mathrm{SO}(2n)$, and $\mathrm{SU}(2n)/\mathrm{Sp}(2n)$. In case of the one-particle-irreducible part, we obtain analytical expressions in terms of ten six-meson subamplitudes based on the flavor and group structures. We extend on our previous results obtained within the framework of the $\mathrm{O}(N+1)/\mathrm{O}(N)$ nonlinear sigma model, with $N$ being the number of meson flavors. This work allows for studying a number of properties of six-particle amplitudes at one-loop level.

Black holes and black strings in the Einstein SU(N) nonlinear sigma model

We construct analytical solutions describing black holes and black strings in the Einstein $SU(N)$-nonlinear sigma model in ($3+1$) dimensions. This construction is carried out using the maximal embedding ansatz of $SU(2)$ together with the Euler parametrization of the $SU(N)$ group, in such a way that the nonlinear sigma model equations are automatically satisfied for arbitrary values of the flavor number $N$ while the Einstein equations can be solved analytically. In particular, we construct black holes with spherical and flat horizons as well as black strings that present the geometry of a three-dimensional charged Ba\~nados-Teitelboim-Zanelli black hole on the transverse section of the string. These configurations are not trivial embeddings of $SU(2)$ into $SU(N)$, which allow us to explicitly show the role that the flavor number plays on the geometry and thermodynamics of the black holes and black strings. Finally, we perform a thermal comparison between these configurations.

Homotopy momentum sections on multisymplectic manifolds

We introduce a notion of a homotopy momentum section on a Lie algebroid over a pre-multisymplectic manifold. A homotopy momentum section is a generalization of the momentum map with a Lie group action and the momentum section on a pre-symplectic manifold, and is also a generalization of the homotopy momentum map on a multisymplectic manifold. We show that a gauged nonlinear sigma model with Wess-Zumino term with Lie algebroid gauging has the homotopy momentum section structure.

Improvement of a Conserved Current Density Versus Adding a Total Derivative to a Lagrangian Density

AbstractFor classical relativistic field theory in Minkowski space‐time, the addition of a superpotential term to a conserved current density is trivial in the sense that it does not modify the local conservation law nor change the conserved charge, though it may allow us to obtain a current density with some improved properties. The addition of a total derivative term to a Lagrangian density is also trivial in the sense that it does not modify the equations of motion of the theory. These facts suggest that both operations are related and possibly equivalent to each other for any global symmetry of an action functional. We address this question following the study of two quite different (and well known) instances: the Callan‐Coleman‐Jackiw improvement of the canonical energy‐momentum tensor for scalar and vector fields (providing an on‐shell traceless energy‐momentum tensor) and the construction of a current density satisfying a zero curvature condition for two‐dimensional sigma models on deformed spaces (notably the squashed three‐sphere and warped AdS spaces). These instances correspond to fairly different implementations of the general results. An appendix addresses the precise relationship between the approaches to local conservation laws based on active and passive symmetry transformations, respectively.

Linear sigma dark matter

We present a model of self-interacting dark matter based on QCD-like theories and inspired by the proximity of a0(980 ± 20) to the Koverline{K} (990) threshold. Dark matter is comprised of dark pions which self-scatter via the σ resonance close to the ππ threshold. While the linear sigma model serves as a qualitative guide, a fully unitary description of the scattering in the strongly coupled regime is given by effective range theory. The introduction of a kinetically mixed dark photon allows the dark pion to either freeze-out or -in. We study the viable parameter space which explains the observed relic abundance while evading all current constraints. Searches for dark matter self interactions at different scales, (in)direct detection signals, and (in)visibly-decaying dark photons will test this model in the near future.

Orbifolds by 2-groups and decomposition

In this paper we study three-dimensional orbifolds by 2-groups with a trivially-acting one-form symmetry group BK. These orbifolds have a global two-form symmetry, and so one expects that they decompose into (are equivalent to) a disjoint union of other three-dimensional theories, which we demonstrate. These theories can be interpreted as sigma models on 2-gerbes, whose formal structures reflect properties of the orbifold construction.

Integrable supersymmetric deformations of AdS3× S3× T4

We construct a family of type IIB string backgrounds that are deformations of AdS3× S3× T4 with a “squashed” AdS3× S3 metric supported by a combination of NSNS and RR fluxes. They have global SU(1, 1) × SU(2) symmetry, regular curvature, constant dilaton and preserve 8 supercharges. Upon compactification to 4 dimensions they reduce to mathcal{N} = 2 supersymmetric AdS2× S2 solutions with electric and magnetic Maxwell fluxes. These type IIB supergravity solutions can be found from the undeformed AdS3× S3× T4 background by a combination of T-dualities and S-duality. In contrast to T-duality, S-duality transformations of a type IIB supergravity background do not generally preserve the classical integrability of the corresponding Green-Schwarz superstring sigma model. Nevertheless, we show that integrability is preserved in the present case. Indeed, we find that these backgrounds can be obtained, up to T-dualities, from an integrable inhomogeneous Yang-Baxter deformation (with unimodular Drinfel’d-Jimbo R-matrix) of the original AdS3× S3 supercoset model.

Running coupling and non-perturbative corrections for O(N) free energy and for disk capacitor

We reconsider the complete solution of the linear TBA equation describing the energy density of finite density states in the O(N) nonlinear sigma models by the Wiener-Hopf method. We keep all perturbative and non-perturbative contributions and introduce a running coupling in terms of which all asymptotic series appearing in the problem can be represented as pure power series without logs. We work out the first non-perturbative contribution in the O(3) case and show that (presumably because of the instanton corrections) resurgence theory fails in this example. Using the relation of the O(3) problem to the coaxial disks capacitor problem we work out the leading non-perturbative terms for the latter and show that (at least to this order) resurgence theory, in particular the median resummation prescription, gives the correct answer. We demonstrate this by comparing the Wiener-Hopf results to the high precision numerical solution of the original integral equation.

Localizing non-linear {{mathcal {N}}}=(2,2) sigma model on S^2

We present a systematic study of {{mathcal {N}}}=(2,2) supersymmetric non-linear sigma models on S^2 with the target being a Kähler manifold. We discuss their reformulation in terms of cohomological field theory. In the cohomological formulation we use a novel version of 2D self-duality which involves a U(1) action on S^2. In addition to the generic model we discuss the theory with target space equivariance corresponding to a supersymmetric sigma model coupled to a non-dynamical supersymmetric background gauge multiplet. We discuss the localization locus and perform a one-loop calculation around the constant maps. We argue that the theory can be reduced to some exotic model over the moduli space of holomorphic disks.

Sensitivity of (multi)fractal eigenstates to a perturbation of the Hamiltonian

We study the response of an isolated quantum system governed by the Hamiltonian drawn from the Gaussian Rosenzweig-Porter random matrix ensemble to a perturbation controlled by a small parameter. We focus on the density of states, local density of states and the eigenfunction amplitude overlap correlation functions which are calculated exactly using the mapping to the supersymmetric nonlinear sigma model. We show that the susceptibility of eigenfunction fidelity to the parameter of perturbation can be expressed in terms of these correlation functions and is strongly peaked at the localization transition: It is independent of the effective disorder strength in the ergodic phase, grows exponentially with increasing disorder in the fractal phase and decreases exponentially in the localized phase. As a function of the matrix size, the fidelity susceptibility remains constant in the ergodic phase and increases in the fractal and in the localized phases at modestly strong disorder. We show that there is a critical disorder strength inside the insulating phase such that for disorder stronger than the critical the fidelity susceptibility decreases with increasing the system size. The overall behavior is very similar to the one observed numerically in a recent work by Sels and Polkovnikov [Phys. Rev. E 104, 054105 (2021)] for the normalized fidelity susceptibility in a disordered XXZ spin chain.

Interaction effects on quantum Hall transitions: Dynamical scaling laws and superuniversality

We study the role of electron-electron interactions near integer and abelian fractional quantum Hall (QH) transitions using composite fermion (CF) representations. Interaction effects are encapsulated in CF theories as gauge fluctuations. Without gauge fluctuations, the CF system realizes a `dual' representation of the non-interacting QH transition. With gauge fluctuations, the system is governed by a gauged nonlinear sigma model (NLSM) with a $\theta-$term. While the transition is described by a strong-coupling fixed point of the NLSM, we are nevertheless able to deduce two of its properties. With $1/r$ interactions, 1) the transition has a dynamical exponent $z=1$, and 2) all transitions are `superuniversal': fractional and integer QH transitions are in the same universality class. With short-range interactions, $z=2$ and the fate of superuniversality remains unclear.

Geometry-kinematics duality

We propose a mapping between geometry and kinematics that implies the classical equivalence of any theory of massless bosons -- including spin and exhibiting arbitrary derivative or potential interactions -- to a nonlinear sigma model (NLSM) with a momentum-dependent metric in field space. From this kinematic metric we construct a corresponding kinematic connection, covariant derivative, and curvature, all of which transform appropriately under general field redefinitions, even including derivatives. We show explicitly how all tree-level on-shell scattering amplitudes of massless bosons are equal to those of the NLSM via the replacement of geometry with kinematics. Lastly, we describe how the recently introduced geometric soft theorem of the NLSM, which universally encodes all leading and subleading soft scalar theorems, also captures the soft photon theorems.

Fixed point annihilation for a spin in a fluctuating field

A quantum spin impurity coupled to a critical free field (the Bose-Kondo model) can be represented as a 0+1D field theory with long-range-in-time interactions that decay as $|t-t'|^{-(2-\delta)}$. This theory is a simpler analogue of nonlinear sigma models with topological Wess-Zumino-Witten terms in higher dimensions. In this note we show that the RG flows for the impurity problem exhibit an annihilation between two nontrivial RG fixed points at a critical value $\delta_c$ of the interaction exponent. The calculation is controlled at large spin $S$. This clarifies the phase diagram of the Bose-Kondo model and shows that it serves as a toy model for phenomena involving fixed-point annihilation and "quasiuniversality" in higher dimensions.

Double copy for massive scalar field theories

We explore extensions of the double copy to massive theories and find a new cubic theory with a local double copy. We consider the nonlinear sigma model and the special galileon theory, massless versions of which are known to be related through the double copy. We show that by performing a Kaluza-Klein reduction of these theories from five dimensions down to four, a double copy relation exists between the resulting massive four-dimensional scalar field theories. This requires the vanishing contribution of new galileon terms arising in high dimensions. We further explore if other interactions that do not arise from a dimensional reduction of the nonlinear sigma model could be double copied and find a new cubic interaction which satisfies the BCJ relations up to 5-point amplitudes.

The geometry of gauged (super)conformal mechanics

Motivated by recently explored examples, we undertake a systematic study of conformal invariance in one-dimensional sigma models where an isometry group has been gauged. Perhaps surprisingly, we uncover classes of sigma models which are only scale invariant in their ungauged form and become fully conformally invariant only after gauging. In these cases the target space of the gauged sigma model satisfies a deformation of the well-known conformal geometry constraints. We consider bosonic models as well as their mathcal{N} = 1, 2, 4 supersymmetric extensions. We solve the quantum ordering ambiguities in implementing (super-) conformal symmetry on the physical Hilbert space. Examples of our general results are furnished by the D(2, 1; 0)-invariant Coulomb branch quiver models relevant for black hole physics.

Effective field theories and cosmological scattering equations

We propose worldsheet formulae for wavefunction coefficients of the massive non-linear sigma model (NLSM), scalar Dirac-Born-Infeld (DBI), and special Galileon (sGal) theories in de Sitter momentum space in terms of the recently proposed cosmological scattering equations constructed from conformal generators in the future boundary. The four-point integrands are assembled from simple building blocks and we identify a double copy prescription mapping the NLSM wavefunction coefficient to the DBI and sGal wavefunction coefficients, including mass deformations and curvature corrections. Finally, we compute the soft limits of these wavefunction coefficients and find that they can be written in terms of boundary conformal generators acting on contact diagrams.

Erratum: Nonlinear sigma model with particle-hole asymmetry for the disordered two-dimensional electron gas [Phys. Rev. B 103, 125422 (2021)

Erratum: Nonlinear sigma model with particle-hole asymmetry for the disordered two-dimensional electron gas [Phys. Rev. B 103, 125422 (2021)

A stochastic PDE approach to large N problems in quantum field theory: A survey

In this Review, we review some recent rigorous results on large N problems in quantum field theory, stochastic quantization, and singular stochastic partial differential equations (SPDEs) and their mean field limit problems. In particular, we discuss the O(N) linear sigma model on a two- and three-dimensional torus. The stochastic quantization procedure leads to a coupled system of N interacting Φ4 equations. In d = 2, we show uniformity in N bounds for the dynamics and convergence to a mean-field singular SPDE. For large enough mass or small enough coupling, the invariant measures [i.e., the O(N) linear sigma model] converge to the massive Gaussian free field, the unique invariant measure of the mean-field dynamics, in a Wasserstein distance. We also obtain tightness for certain O(N) invariant observables as random fields in suitable Besov spaces as N → ∞, along with exact descriptions of the limiting correlations. In d = 3, the estimates become more involved since the equation is more singular. We discuss in this case how to prove convergence to the massive Gaussian free field. The proofs of these results build on the recent progress of singular SPDE theory and combine many new techniques, such as uniformity in N estimates and dynamical mean field theory. These are based on joint papers with Scott Smith, Rongchan Zhu, and Xiangchan Zhu.

The existence of solutions for the gravitational Maxwell gauged O(3) model on compact surfaces

We consider an elliptic equation induced from the Maxwell gauged O(3) sigma model coupled with gravity. In particular, we study the main equation as two cases: one is for only string and the other is for anti-string. On the compact surface, we prove the existence of ɛ-dependent solutions for each case by using the super-sub solutions method. Moreover, we find the second solution by using the Leray–Schauder degree theory. Furthermore, we estimate the asymptotic behavior of our solution as ɛ → 0.

Large charges on the Wilson loop in mathcal{N} = 4 SYM. Part II. Quantum fluctuations, OPE, and spectral curve

We continue our study of large charge limits of the defect CFT defined by the half-BPS Wilson loop in planar mathcal{N} = 4 supersymmetric Yang-Mills theory. In this paper, we compute 1/J corrections to the correlation function of two heavy insertions of charge J and two light insertions, in the double scaling limit where the charge J and the ’t Hooft coupling λ are sent to infinity with the ratio J/ sqrt{lambda } fixed. Holographically, they correspond to quantum fluctuations around a classical string solution with large angular momentum, and can be computed by evaluating Green’s functions on the worldsheet. We derive a representation of the Green’s functions in terms of a sum over residues in the complexified Fourier space, and show that it gives rise to the conformal block expansion in the heavy-light channel. This allows us to extract the scaling dimensions and structure constants for an infinite tower of non-protected dCFT operators. We also find a close connection between our results and the semi-classical integrability of the string sigma model. The series of poles of the Green’s functions in Fourier space corresponds to points on the spectral curve where the so-called quasi-momentum satisfies a quantization condition, and both the scaling dimensions and the structure constants in the heavy-light channel take simple forms when written in terms of the spectral curve. These observations suggest extensions of the results by Gromov, Schafer-Nameki and Vieira on the semiclassical energy of closed strings, and in particular hint at the possibility of determining the structure constants directly from the spectral curve.