Goldstone Bosons Research Articles

The open effective field theory of inflation

Abstract In our quest to understand the generation of cosmological perturbations, we face two serious obstacles: we do not have direct information about the environment experienced by primordial perturbations during inflation, and our observables are practically limited to correlators of massless fields, heavier fields and derivatives decaying exponentially in the number of e-foldings. The flexible and general framework of open systems has been developed precisely to face similar challenges. Building on previous work, we develop a Schwinger-Keldysh path integral description for an open effective field theory of inflation, describing the possibly dissipative and non-unitary evolution of the Goldstone boson of time translations interacting with an unspecified environment, under the key assumption of locality in space and time. Working in the decoupling limit, we study the linear and interacting theory in de Sitter and derive predictions for the power spectrum and bispectrum that depend on a finite number of effective couplings organised in a derivative expansion. The smoking gun of interactions with the environment is an enhanced but finite bispectrum close to the folded kinematical limit. We demonstrate the generality of our approach by matching our open effective theory to an explicit model. Our construction provides a standard model to simultaneously study phenomenological predictions as well as quantum information aspects of the inflationary dynamics.

Low-virtuality splitting in the Standard Model

When the available collision energy is much above the mass of the particles involved, scattering amplitudes feature kinematic configurations that are enhanced by the much lower virtuality of some intermediate particle. Such configurations generally factorise in terms of a hard scattering amplitude with exactly on-shell intermediate particle, times universal factors. In the case of real radiation emission, such factors are splitting amplitudes that describe the creation or the annihilation — for initial or final state splittings — of the low-virtuality particle and the creation of the real radiation particles. We compute at tree-level the amplitudes describing all the splittings that take place in the Standard Model when the collision energy is much above the electroweak scale. Unlike previous results, our splitting amplitudes fully describe the low-virtuality kinematic regime, which includes the region of collinear splitting, of soft emission, and combinations thereof. The splitting amplitudes are compactly represented as little-group tensors in an improved bi-spinor formalism for massive spin-1 particles that automatically incorporates the Goldstone Boson Equivalence Theorem. Simple explicit expressions are obtained using a suitably defined infinite-momentum helicity basis representation of the spinor variables. Our results, combined with the known virtual contributions, could enable systematic predictions of the leading electroweak radiation effects in high-energy scattering processes, with particularly promising phenomenological applications to the physics of future colliders with very high energy such as a muon collider.

Phase transition and gravitational waves in maximally symmetric composite Higgs model

In this paper we study phase transitions in a maximally symmetric composite Higgs model with next-to-minimal coset, where a pseudoscalar singlet emerges alongside the Higgs doublet. The maximal symmetry guarantees the finiteness of the radiatively generated scalar potential. We explore the scenario involving an explicit source of CP violation in the strong sector, which induces a ℤ2 asymmetric scalar potential, and consequently leads to nonzero vacuum expectation value for the singlet. Current experimental bounds from the LHC are imposed on the masses of the composite resonances, while the CP violating interactions of the pseudo Nambu-Goldstone bosons are tightly constrained from the measurements of the electric dipole moment of the electron. We compute the finite temperature corrections to the potential, incorporating the momentum-dependent form factors in the loop integrals to capture the effect of the strong dynamics. The impact of the resonances from the strong sector on the finite temperature potential are exponentially suppressed. The presence of explicit CP violation leads to strong first-order phase transition from a false vacuum to the electroweak vacuum where the pseudoscalar singlet has a non-zero vacuum expectation value. We illustrate that, as a result of such phase transitions, the production of potentially observable gravitational waves at future detectors will offer a complementary avenue to probe the composite Higgs models, distinct from collider experiments.

Complete CP-eigen bases of meson-baryon chiral lagrangian up to p5-order

Chiral perturbation theory describes the low energy dynamics of mesons and baryons in terms of the nonlinear Goldstone boson and fermion degrees of freedom. Through the Young tensor technique, we construct the on-shell operator bases for the meson-baryon system up to p5-order, using the chiral dimension power counting and heavy baryon expansion. For the Lorentz structure, additional treatments on off-shell external sources and operators with higher derivatives are necessarily considered, while for the internal structure, the invariant tensor basis is converted into the trace basis equivalently, and Cayley-Hamilton relations are utilized to classify different CP eigen-operators. Finally we present the complete operator set of C+P+, C+P-, C-P+, and C-P- eigen-operators at the p5-order, and obtain the operator counting from the Hilbert series.

A Goldstone boson equivalence for inflation

The effective field theory of single-field inflation characterizes the inflationary epoch in terms of a pattern of symmetry breaking. An operator acquires a time-dependent vacuum expectation value, defining a preferred spatial slicing. In the absence of dynamical gravity, the fluctuations around the time-dependent background are described by the Goldstone boson associated with this symmetry breaking process. With gravity, the Goldstone is eaten by the metric, becoming the scalar metric fluctuation. In this paper, we will show that in general single-field inflation, the statistics of scalar metric fluctuations are given by the statistics of this Goldstone boson decoupled from gravity up to corrections that are controlled as an expansion in slow-roll parameters. This even holds in the presence of additional parameters, like the speed of sound, that naively enhance the impact of the gravitational terms. In the process, we derive expressions for leading and sub-leading gravitational corrections to all-orders in the Goldstone boson.

Automatic generation of helicity amplitudes in the Feynman-diagram gauge

We develop a method to calculate helicity amplitudes of an arbitrary tree-level process in the Feynman-diagram (FD) gauge for an arbitrary gauge model with adraph5_a@. We start from the ’t Hooft–Feynman gauge Lagrangian in eynules and generate scattering amplitudes by identifying the Goldstone boson as the 5th component of each weak boson. All the vertices of the 5-component weak bosons are then created automatically by assembling the relevant weak boson and Goldstone boson vertices in the Feynman gauge. The 5-component weak boson vertices are then connected by the 5×5 matrix propagator in the FD gauge. As a demonstration of the method we calculate the cross section for the process μ−μ+→νμν¯μtt¯H with complex top Yukawa coupling, which can be obtained by adding a gauge invariant dimension-6 operator to the Standard Model (SM) Lagrangian. The FD gauge and the unitary (U) gauge amplitudes give exactly the same cross section, and subtle gauge theory cancellation among diagrams in the U gauge at high energies is absent in the FD gauge, as has been observed for various SM processes. In addition, we find that the total cross sections at high energies are dominated by a single, or a set of nonvanishing Feynman amplitudes with the higher dimensional vertices in the FD gauge. Published by the American Physical Society 2024

On universal splittings of tree-level particle and string scattering amplitudes

In this paper, we study the newly discovered universal splitting behavior for tree-level scattering amplitudes of particles and strings [1]: when a set of Mandelstam variables (and Lorentz products involving polarizations for gluons/gravitons) vanish, the n-point amplitude factorizes as the product of two lower-point currents with n+3 external legs in total. We refer to any such subspace of the kinematic space of n massless momenta as “2-split kinematics”, where the scattering potential for string amplitudes and the corresponding scattering equations for particle amplitudes nicely split into two parts. Based on these, we provide a systematic and detailed study of the splitting behavior for essentially all ingredients which appear as integrands for open- and closed-string amplitudes as well as Cachazo-He-Yuan (CHY) formulas, including Parke-Taylor factors, correlators in superstring and bosonic string theories, and CHY integrands for a variety of amplitudes of scalars, gluons and gravitons. These results then immediately lead to the splitting behavior of string and particle amplitudes in a wide range of theories, including bi-adjoint ϕ3 (with string extension known as Z and J integrals), non-linear sigma model, Dirac-Born-Infeld, the special Galileon, etc., as well as Yang-Mills and Einstein gravity (with bosonic and superstring extensions). Our results imply and extend some other factorization behavior of tree amplitudes considered recently, including smooth splittings [2] and factorizations near zeros [3], to all these theories. A special case of splitting also yields soft theorems for gluons/gravitons as well as analogous soft behavior for Goldstone particles near their Adler zeros.

W -boson pair production at lepton colliders in the Feynman-diagram gauge

We calculate helicity amplitudes for e−e+→W−W+ analytically in the Feynman-diagram (FD) gauge. We show that, unlike in the unitary gauge, there is no energy growth of the individual Feynman amplitudes for the longitudinally polarized bosons, and the contributions from the associated Goldstone bosons are manifest even without taking the high-energy limit. We also find that the physical distributions can be interpreted by the individual amplitudes in the FD gauge. Published by the American Physical Society 2024

Dynamical edge modes and entanglement in Maxwell theory

Previous work on black hole partition functions and entanglement entropy suggests the existence of “edge” degrees of freedom living on the (stretched) horizon. We identify a local and “shrinkable” boundary condition on the stretched horizon that gives rise to such degrees of freedom. They can be interpreted as the Goldstone bosons of gauge transformations supported on the boundary, with the electric field component normal to the boundary as their symplectic conjugate. Applying the covariant phase space formalism for manifolds with boundary, we show that both the symplectic form and Hamiltonian exhibit a bulk-edge split. We then show that the thermal edge partition function is that of a codimension-two ghost compact scalar living on the horizon. In the context of a de Sitter static patch, this agrees with the edge partition functions found by Anninos et al. in arbitrary dimensions. It also yields a 4D entanglement entropy consistent with the conformal anomaly. Generalizing to Proca theory, we find that the prescription of Donnelly and Wall reproduces existing results for its edge partition function, while its classical phase space does not exhibit a bulk-edge split.

Helicity-changing decays of cosmological relic neutrinos

In this paper, we examine the possibility that massive neutrinos are unstable due to their invisible decays ν i → ν j + ϕ, where ν i and ν j (for i, j = 1, 2, 3) are any two of neutrino mass eigenstates with masses m i > m i and ϕ is a massless Nambu-Goldstone boson, and explore the implications for the detection of cosmological relic neutrinos in the present Universe. First, we carry out a complete calculation of neutrino decay rates in the general case where the individual helicities of parent and daughter neutrinos are specified. Then, the invisible decays of cosmological relic neutrinos are studied and their impact on the capture rates on the beta-decaying nuclei (e.g., ν e + 3H → 3He + e -) is analyzed. The invisible decays of massive neutrinos could substantially change the capture rates in the PTOLEMY-like experiments when compared to the case of stable neutrinos. In particular, we find that the helicity-changing decays of Dirac neutrinos play an important role whereas those of Majorana neutrinos have no practical effects. However, if a substantial fraction of heavier neutrinos decay into the lightest one, the detection of relic neutrinos will require a much higher energy resolution and thus be even more challenging.

The modulated soliton fields in the Goldstone boson model

It is well known that massless Goldstone bosons have not yet been observed. In the Goldstone boson model, after the spontaneous symmetry breaking under [Formula: see text], two coupled nonlinear equations are obtained for which we present the exact solitonic solutions. These solutions completely localize the energy density of the model and show the existence of two boson fields, one massive and the other massless. Also, it is seen that the solitonic waves are modulated and the massless wave rides on the massive wave. Finally, we calculate the charge density of the model, which again confirms the neutrality of these boson fields. Since massive bosons can be observed in the laboratory, these solitonic waves may be useful in tracking and detecting Goldstone bosons.

Dilaton forbidden dark matter

Dilaton effective field theory (dEFT) describes the long distance behavior of certain confining, near-conformal gauge theories that have been studied via lattice computation. Pseudo-Nambu- Goldstone bosons (pNGBs), emerging from the breaking of approximate, continuous, internal symmetries, are coupled to an additional scalar particle, the dilaton, arising from the spontaneous breaking of approximate scale invariance. This effective theory has been employed to study possible extensions of the standard model. In this paper, we propose a complementary role for dEFT, as a description of the dark matter of the Universe, with the pNGBs identified as the dark matter particles. We show that this theory provides a natural implementation of the “forbidden” dark matter mechanism, and we identify regions of parameter space for which the thermal history of dEFT yields the measured dark matter relic density. Published by the American Physical Society 2024

Holography for Sp(2Nc) gauge dynamics: from composite Higgs to technicolour

We study Sp(2Nc) gauge dynamics with two Dirac fermion flavours in the fundamental representation. These strongly coupled systems underlie some composite Higgs models with a global symmetry breaking pattern SU(4)→Sp(4), leading to a light quartet of pseudo-Goldstone bosons that can play the role of the Higgs. Including four-fermion interactions can rotate the vacuum to a technicolour breaking pattern. Using gauge/gravity duality, we study the underlying gauge dynamics. Our model incorporates the Nc-specific running of the fermion anomalous dimension and can thus distinguish between specific gauge groups. We determine the bound-state spectrum of the UV SU(4)→Sp(4) symmetry breaking model, also including fermion masses and mass splitting, and display the Nc dependence. The inclusion of a four-fermion interaction shows the emergence of three Goldstone bosons on the path to technicolour dynamics.

Pions from higher-dimensional gluons: general realizations and stringy models

In this paper we revisit the general phenomenon that scattering amplitudes of pions can be obtained from “dimensional reduction” of gluons in higher dimensions in a more general context. We show that such “dimensional reduction” operations universally turn gluons into pions regardless of details of interactions: under such operations any amplitude that is gauge invariant and contains only local simple poles becomes one that satisfies Adler zero in the soft limit. As two such examples, we show that starting from gluon amplitudes in both superstring and bosonic string theories, the operations produce “stringy” completion of pion scattering amplitudes to all orders in α′, with leading order given by non-linear sigma model amplitudes. Via Kawai-Lewellen-Tye relations, they give closed-stringy completion for Born-Infeld theory and the special Galileon theory, which are directly related to gravity amplitudes in closed-string theories. We also discuss how they naturally produce stringy models for mixed amplitudes of pions and colored scalars.

Hairy black holes by spontaneous symmetry breaking

We study hairy black hole solutions in Einstein(-Maxwell)-scalar-Gauss-Bonnet theory. The complex scalar coupling function includes quadratic and quartic terms, so the gravitational action has a U(1) symmetry. We argued that when the effective mass of the scalar field is at the critical value, the black holes without hairs transform into hairy black holes in a symmetry-broken vacuum via spontaneous symmetry breaking. These hairy black holes are stable under scalar perturbations, and the Goldstone bosons are trivial. Moreover, we found that the spontaneous symmetry breaking associated with local U(1) is unlikely to occur in this theory. Published by the American Physical Society 2024

Soft theorems for boostless amplitudes

We consider effective field theories (EFTs) of scalar fields with broken Lorentz boosts, which arise by taking the decoupling and flat-space limits of the EFT of inflation, and derive constraints that must be satisfied by the corresponding scattering amplitudes if there is an underlying non-linearly realised symmetry. We primarily concentrate on extended shift symmetries which depend on the space-time coordinates, and find that combinations of scattering amplitudes obey enhanced Adler zeros. That is, such combinations vanish as one external momentum is taken soft, with the rate at which they vanish dictated by the corresponding symmetry. In our soft theorem derivation, we pay particular care to the energy and momentum-conserving delta functions that arise due to space-time translations, and show that when acted upon by derivatives with respect to spatial momenta, they yield a tower of soft theorems which are ultimately required for closure of the underlying symmetry algebra. All of our soft theorems correspond to constraints that must be satisfied by on-shell amplitudes and, even for symmetries that depend on the time coordinate, our soft theorems only require derivatives to be taken with respect to spatial momenta. We perform a soft bootstrap procedure to find solutions to our soft theorems, and compare these solutions to what we find from an off-shell analysis using the coset construction.

The flavor of a minimal composite S1 leptoquark and the b→cτν¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ b\ o c\ au \\overline{\ u} $$\\end{document} anomaly

Several experiments have measured a deviation in B → D(∗) semileptonic decays, that point to new physics at the TeV scale violating lepton flavor universality. A scalar leptoquark S1, with a suitable structure of couplings in flavor space, is known to be able to solve this anomaly modifying b→cτν¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ b\ o c\ au \\overline{\ u} $$\\end{document}. In the context of composite Higgs models, we consider a theory containing H and S1 as Nambu-Goldstone bosons (NGBs) of a new strongly interacting sector, with ordinary resonances at a scale O10\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O}(10) $$\\end{document} TeV. Assuming anarchic partial compositeness of the Standard Model (SM) fermions we calculate the potential of the NGBs that is dominated by the fermions of the third generation, we compute RD∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {R}_{D^{\\left(\\ast \\right)}} $$\\end{document} and estimate the corrections to flavor observables by the presence of S1. We find that the SM spectrum and mS1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {m}_{S_1} $$\\end{document} ∼ TeV can be obtained with a NGB decay constant of order ∼ 5 TeV. We obtain a robust correlation between the main corrections to RD∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {R}_{D^{\\left(\\ast \\right)}} $$\\end{document}, BK∗νν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {B}_{K^{\\left(\\ast \\right)}\ u \ u} $$\\end{document} and gτ/gμ, that leads to a sever bound on RD∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {R}_{D^{\\left(\\ast \\right)}} $$\\end{document}, roughly 2σ below the experimental value. Besides the bounds on the flavor observables gτW\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {g}_{\ au}^W $$\\end{document}, BR(τ → μγ) and ∆mBs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\Delta {m}_{B_s} $$\\end{document} are saturated, with the first one requiring a coupling between resonances g* ≲ 2, whereas the second one demands mS1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {m}_{S_1} $$\\end{document} ≳ 1.7 TeV, up to corrections of O1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O}(1) $$\\end{document}.

Scotogenic gauge mechanism for neutrino mass and dark matter

Scotogenic is a scheme for neutrino mass generation through the one-loop contribution of an inert scalar doublet and three sterile neutrinos. This work argues that such inert scalar doublet is a Goldstone boson mode associated with a gauge symmetry breaking. Hence, the resultant scotogenic gauge mechanism is very predictive, generating neutrino mass as contributed by a new gauge boson doublet that eats such Goldstone bosons. The dark matter stability is manifestly ensured by a matter parity as residual gauge symmetry for which a vector dark matter candidate is hinted.

Holography of broken U(1) symmetry

We examine the Abelian Higgs model in (d + 1)-dimensional anti-de Sitter space with an ultraviolet brane. The gauge symmetry is broken by a bulk Higgs vacuum expectation value triggered on the brane. We propose two separate Goldstone boson equivalence theorems for the boundary and bulk degrees of freedom. We compute the holographic self-energy of the gauge field and show that its spectrum is either a continuum, gapped continuum, or a discretuum as a function of the Higgs bulk mass. When the Higgs has no bulk mass, the AdS isometries are unbroken. We find in that case that the dual CFT has a non-conserved U(1) current whose anomalous dimension is proportional to the square of the Higgs vacuum expectation value. When the Higgs background weakly breaks the AdS isometries, we present an adapted WKB method to solve the gauge field equations. We show that the U(1) current dimension runs logarithmically with the energy scale in accordance with a nearly-marginal U(1)-breaking deformation of the CFT.

Effects of electroweak symmetry breaking on axionlike particles as dark matter

Axionlike particles (ALPs), the pseudo Nambu-Goldstone bosons associated to the spontaneous breaking of global symmetry, have emerged as promising dark matter candidates. Conventionally, in the context of misalignment mechanism, the nonthermally produced ALPs happen to stay frozen due to Hubble friction initially, and at a later stage, they begin to oscillate (before matter-radiation equality) at characteristic frequencies defined by their masses and behaving like cold dark matter. In this work, we study the influence of electroweak symmetry breaking (EWSB), through a higher order Higgs portal interaction, on the evolution of ALPs. Such an interaction is found to contribute partially to the ALP’s mass during EWSB, thereby modifying oscillation frequencies during EWSB as well as impacting upon the existing correlation between the scale of symmetry breaking and their masses. The novelty of the work lies in broadening the relic satisfied parameter space so as to probe it in near future via a wide range of experiments. Published by the American Physical Society 2024