Elementary Scalar Fields Research Articles

Multipoint correlators on the supersymmetric Wilson line defect CFT. Part II. Unprotected operators

We continue our study of multipoint correlators of scalar fields on the 1d defect CFT generated by inserting operators along the Maldacena-Wilson line in mathcal{N} = 4 SYM. We present a weak-coupling recursion relation that captures correlators at next-to-leading order involving an arbitrary number of the elementary scalar fields ϕi and ϕ6, the latter being unprotected. We can then build correlators of composite operators by pinching the scalar fields together. As a demonstration of our method, we give explicit results for correlators containing up to six points. We also expand some selected correlators using recently obtained conformal blocks in the comb and snowflake channel, and check that the extracted low-lying CFT data is consistent with explicit computations.

BICEP/Keck data and quadratic gravity

The recent results of the BICEP and Keck collaborations have put stringent bounds on many inflationary models, including some well-motivated ones. This is certainly the case when gravity remains described by Einstein's theory up to the inflationary scale, but can be avoided by introducing quadratic-in-curvature terms that are effective at that scale. Recently it has also been shown that these terms can UV complete gravity respecting stability and unitarity. Here the predictions of such quadratic gravity are computed and compared with the BICEP/Keck constraints by focusing on some of the inflationary scenarios that are best-motivated from the particle physics point of view and are already ruled out in Einstein gravity: (critical) Higgs inflation and natural inflation. The first scenario can be considered as the most economical option as the inflaton is identified with the only known elementary scalar field in the Standard Model and the near criticality of the Standard Model is used to remain in the perturbative regime. In the second one a pseudo-Nambu-Goldstone boson contributes to the inflationary dynamics and its potential is naturally flat. It is shown that in both scenarios one can restore the agreement with the observational constraints in quadratic gravity.

A composite Higgs with a heavy composite axion

We consider the strong dynamics associated with a composite Higgs model that simultaneously produces dynamical axions and solves the strong CP problem. The strong dynamics arises from a new Sp or SU(4) hypercolor gauge group containing QCD colored hyperfermions that confines at a high scale. The hypercolor global symmetry is weakly gauged by the Standard Model electroweak gauge group and an enlarged color group, SU(N + 3) × SU(N)′. When hyperfermion condensates form, they not only lead to an SU(5)/SO(5) composite Higgs model but also spontaneously break the enlarged color group to SU(3)c× SU(N)D. At lower energies, the SU(N)D group confines, producing two dynamical axions that eliminates all CP violation. Furthermore, small instantons from the SU(N)′ group can enhance the axion mass, giving rise to TeV scale axion masses that can be detected at collider experiments. Our model provides a way to unify the composite Higgs with dynamical axions, without introducing new elementary scalar fields, while also extending the range of axion masses that addresses the strong CP problem.

Non-abelian Toda theory on AdS2 and {mathrm{AdS}}_2/{mathrm{CFT}}_2^{raisebox{1ex}{1}!left/ !raisebox{-1ex}{2}right.} duality

It was recently observed that boundary correlators of the elementary scalar field of the Liouville theory on AdS2 background are the same (up to a non-trivial proportionality coefficient) as the correlators of the chiral stress tensor of the Liouville CFT on the complex plane restricted to the real line. The same relation generalizes to the conformal abelian Toda theory: boundary correlators of Toda scalars on AdS2 are directly related to the correlation functions of the chiral mathcal{W} -symmetry generators in the Toda CFT and thus are essentially controlled by the underlying infinite-dimensional symmetry. These may be viewed as examples of AdS2/CFT1 duality where the CFT1 is the chiral half of a 2d CFT; we shall to this as {mathrm{AdS}}_2/{mathrm{CFT}}_2^{raisebox{1ex}{1}!left/ !raisebox{-1ex}{2}right.} . In this paper we demonstrate that this duality applies also to the non-abelian Toda theory containing a Liouville scalar coupled to a 2d σ-model originating from the SL(2, ℝ)/U(1) gauged WZW model. Here the Liouville scalar is again dual to the chiral stress tensor T while the other two scalars are dual to the parafermionic operators V± of the non-abelian Toda CFT. We explicitly check the duality at the next-to-leading order in the large central charge expansion by matching the chiral CFT correlators of (T, V+, V−) (computed using a free field representation) with the boundary correlators of the three Toda scalars given by the tree-level and one-loop Witten diagrams in AdS2.

Living without supersymmetry—the conformal alternative and a dynamical Higgs boson

We show that the key results of supersymmetry can be achieved via conformal symmetry instead. We propose that the Higgs boson be a dynamical fermion–antifermion bound state rather than an elementary scalar field, so that there is then no quadratically divergent self-energy problem for it and thus no need to invoke supersymmetry to resolve the problem. To obtain such a dynamical Higgs boson we study a conformal invariant gauge theory of interacting fermions and gauge bosons. The conformal invariance of the theory is realized via scaling with anomalous dimensions in the ultraviolet, and by a dynamical symmetry breaking via fermion bilinear condensates in the infrared, a breaking in which the dynamical dimension of the composite operator is reduced from three to two. With this reduction in dimension we can augment the gauge theory with a four-fermion interaction made renormalizable by this reduction, and can reinterpret the theory as a renormalizable version of the Nambu–Jona-Lasinio (NJL) model, with the gauge theory sector with its now massive fermion being a mean-field theory and the four-fermion interaction being the residual interaction. It is this residual interaction and not the mean field that then generates dynamical Goldstone and Higgs states, states that, as noted by Baker and Johnson, the gauge theory sector itself does not possess. The Higgs boson is found to be a narrow resonance just above threshold, with its width potentially being a diagnostic that could distinguish a dynamical Higgs boson from an elementary one. We couple the theory to a gravity theory, conformal gravity, that is equally conformal invariant, with the interplay between conformal gravity and the four-fermion interaction taking care of the vacuum energy problem. With conformal gravity being a unitary and renormalizable quantum theory of gravity there is no need for string theory with its supersymmetric underpinnings. With the vacuum energy problem being resolved and with conformal gravity fits to phenomena such as galactic rotation curves and the accelerating universe not needing dark matter, there is no need to introduce supersymmetry for either the vacuum energy problem or to provide a potential dark matter candidate. We propose that it is conformal symmetry rather than supersymmetry that is fundamental, with the theory of nature being a locally conformal, locally gauge invariant, non-Abelian NJL theory.

Dynamical clockwork axions

The clockwork mechanism is a novel method for generating a large separation between the dynamical scale and interaction scale of a theory. We demonstrate how the mechanism can arise from a sequence of strongly-coupled sectors. This framework avoids elementary scalar fields as well as ad hoc continuous global symmetries, both of which are subject to serious stability issues. The clockwork factor, q, is determined by the consistency of the strong dynamics. The preserved global U(1) of the clockwork appears as an accidental symmetry, resulting from discrete or U(1) gauge symmetries, and it is spontaneously broken by the chiral condensates. We apply such a dynamical clockwork to construct models with an effectively invisible QCD axion from TeV-scale strong dynamics. The axion couplings are determined by the localisation of the Standard Model interactions along the clockwork sequence. The TeV spectrum includes either coloured hadrons or vector-like quarks. Dark matter can be accounted for by the axion or the lightest neutral baryons, which are accidentally stable.

Strong dynamics in a classically scale invariant extension of the standard model with a flat potential

We investigate the scenario where the standard model is extended with classical scale invariance, which is broken by chiral symmetry breaking and confinement in a new strongly coupled gauge theory that resembles QCD. The standard model Higgs field emerges as a result of the mixing of a scalar meson in the new strong dynamics and a massless elementary scalar field. The mass and scalar decay constant of that scalar meson, which are generated dynamically in the new gauge theory, give rise to the Higgs field mass term, automatically possessing the correct negative sign by the bosonic seesaw mechanism. Using analogy with QCD, we evaluate the dynamical scale of the new gauge theory and further make quantitative predictions for light pseudo-Nambu-Goldstone bosons associated with the spontaneous breaking of axial symmetry along chiral symmetry breaking in the new gauge theory. A prominent consequence of the scenario is that there should be a standard model gauge singlet pseudo-Nambu-Goldstone boson with mass below 220 GeV, which couples to two electroweak gauge bosons through the Wess-Zumino-Witten term, whose strength is thus determined by the dynamical scale of the new gauge theory. Other pseudo-Nambu-Goldstone bosons, charged under the electroweak gauge groups, also appear. Concerning the theoretical aspects, it is shown that the scalar quartic coupling can vanish at the Planck scale with the top quark pole mass as large as 172.5 GeV, realizing the flatland scenario without being in tension with the current experimental data.

Why supersymmetry? Physics beyond the standard model

The Naturalness Principle as a requirement that the heavy mass scales decouple from the physics of light mass scales is reviewed. In quantum field theories containing elementary scalar fields, such as the Standard Model of electroweak interactions containing the Higgs particle, mass of the scalar field is not a natural parameter as it receives large radiative corrections. How supersymmetry solves this Naturalness Problem is outlined. There are also other contexts where the presence of elementary scalar fields generically spoils the high–low mass scales decoupling in the quantum theory. As an example of this, the non-decoupling of possible Planck scale violation of Lorentz invariance due to quantum gravity effects from the physics at low scales in theories with elementary scalar fields such as the Higgs field is described. Here again supersymmetry provides a mechanism for ensuring that the decoupling of heavy–light mass scales is maintained.

My Life as a Boson: The Story of "The Higgs"

The story begins in 1960, when Nambu, inspired by the BCS theory of superconductivity, formulated chirally invariant relativistic models of interacting massless fermions in which spontaneous symmetry breaking generates fermionic masses (the analogue of the BCS gap). Around the same time Jeffrey Goldstone discussed spontaneous symmetry breaking in models containing elementary scalar fields (as in Ginzburg-Landau theory). I became interested in the problem of how to avoid a feature of both kinds of model, which seemed to preclude their relevance to the real world, namely the existence in the spectrum of massless spin-zero bosons (Goldstone bosons). By 1962 this feature of relativistic field theories had become the subject of the Goldstone theorem.

Light scalars in strongly-coupled extra-dimensional theories

The low-energy dynamics of five-dimensional Yang-Mills theories compactified on S^1 can be described by a four-dimensional gauge theory coupled to a scalar field in the adjoint representation of the gauge group. Perturbative calculations suggest that the mass of this elementary scalar field is protected against power divergences, and is controlled by the size of the extra dimension R. As a first step in the study of this phenomenon beyond perturbation theory, we investigate the phase diagram of a SU(2) Yang-Mills theory in five dimensions regularized on anisotropic lattices and we determine the ratios of the relevant physical scales. The lattice system shows a dimensionally reduced phase where the four-dimensional correlation length is much larger than the size of the extra dimension, but still smaller than the four-dimensional volume. In this region of the bare parameter space, at energies below 1/R, the non-perturbative spectrum contains a \emph{light} scalar state. This state has a mass that is independent of the cut-off, and a small overlap with glueball operators. Our results suggest that light scalar fields can be introduced in a lattice theory using compactified extra dimensions, rather than fine tuning the bare mass parameter.

Low-energy phenomenology of scalarless standard-model extensions with high-energy Lorentz violation

We consider renormalizable standard model extensions that violate Lorentz symmetry at high energies, but preserve $CPT$, and do not contain elementary scalar fields. A Nambu--Jona-Lasinio mechanism gives masses to fermions and gauge bosons and generates composite Higgs fields at low energies. We study the effective potential at the leading order of the large-${N}_{c}$ expansion, prove that there exists a broken phase, and study the phase space. In general, the minimum may break invariance under boosts, rotations, and $CPT$, but we give evidence that there exists a Lorentz invariant phase. We study the spectrum of composite bosons and the low-energy theory in the Lorentz phase. Our approach predicts relations among the parameters of the low-energy theory. We find that such relations are compatible with the experimental data within theoretical errors. We also study the mixing among generations, the emergence of the CKM matrix, and neutrino oscillations.

DYNAMICAL ELECTROWEAK SYMMETRY BREAKING IN STRING MODELS WITH D-BRANES

The possibility of dynamical electroweak symmetry breaking by strong coupling gauge interaction in models with D-branes in String Theory is examined. Instead of elementary scalar Higgs doublet fields, the gauge symmetry with strong coupling (technicolor) is introduced. As the first step of this direction, a toy model, which is not fully realistic, is concretely analyzed in some detail. The model consists of D-branes and anti-D-branes at orbifold singularities in (T2 × T2 × T2)/Z3 which preserves supersymmetry. Supersymmetry is broken through the brane supersymmetry breaking. It is pointed out that the problem of large S parameter in dynamical electroweak symmetry breaking scenario may be solved by natural existence of kinetic term mixings between hypercharge U(1) gauge boson and massive anomalous U(1) gauge bosons. The problems to be solved toward constructing more realistic models are clarified in the analysis.

CPviolation due to compactification

We address the challenging issue of how CP violation is realized in higher dimensional gauge theories without higher dimensional elementary scalar fields. In such theories interactions are basically governed by a gauge principle and therefore to get CP violating phases is a non-trivial task. It is demonstrated that CP violation is achieved as the result of compactification of extra dimensions, which is incompatible with the 4-dimensional CP transformation. As a simple example we adopt a 6-dimensional U(1) model compactified on a 2-dimensional orbifold $T^{2}/Z_{4}$. We argue that the 4-dimensional CP transformation is related to the complex structure of the extra space and show how the $Z_{4}$ orbifolding leads to CP violation. We confirm by explicit calculation of the interaction vertices that CP violating phases remain even after the re-phasing of relevant fields. For completeness, we derive a re-phasing invariant CP violating quantity, following a similar argument in the Kobayashi-Maskawa model which led to the Jarlskog parameter. As an example of a CP violating observable we briefly comment on the electric dipole moment of the electron.

Lectures on Walking Technicolor, Holography, and Gauge/Gravity Dualities

Dynamical electroweak symmetry breaking is an appealing, strongly coupled alternative to the weakly coupled models based on an elementary scalar field developing a vacuum expectation value. In Sections 2 and 3 of this set of lectures, I summarize the arguments, based on low‐energy phenomenology, supporting walking technicolor as a realistic realization of this idea. This pedagogical introduction to walking technicolor, and more generally to the physics of extensions of the standard model, makes extensive use of effective field theory arguments, symmetries, and counting rules. The strongly coupled nature of the underlying interactions, and the peculiar quasiconformal behavior of the theory, requires to use nonperturbative methods in order to address many fundamental questions within this framework. The recent development of gauge/gravity dualities provides an ideal set of such nonperturbative instruments. Sections 4 and 5 illustrate the potential of these techniques with two technical examples, one within the bottom‐up phenomenological approach to holography in five dimensions, the other within a more systematic top‐down construction derived from ten‐dimensional type‐IIB supergravity.

Standard Model without elementary scalars and high energy Lorentz violation

If Lorentz symmetry is violated at high energies, interactions that are usually non-renormalizable can become renormalizable by weighted power counting. Recently, a CPT invariant, Lorentz violating extension of the Standard Model containing two scalar-two fermion interactions (which can explain neutrino masses) and four fermion interactions (which can explain proton decay) was proposed. In this paper we consider a variant of this model, obtained suppressing the elementary scalar fields, and argue that it can reproduce the known low-energy physics. In the Nambu–Jona-Lasinio spirit, we show, using a large N c expansion, that a dynamical symmetry breaking takes place. The effective potential has a Lorentz invariant minimum and the Lorentz violation does not reverberate down to low energies. The mechanism generates fermion masses, gauge-boson masses and scalar bound states, to be identified with composite Higgs bosons. Our approach is not plagued by the ambiguities of approaches based on non-renormalizable vertices. The low-energy effective action is uniquely determined and predicts relations among parameters of the Standard Model.

Gravity in two-time physics

The field theoretic action for gravitational interactions in $d+2$ dimensions is constructed in the formalism of two-time (2T) physics. General relativity in $d$ dimensions emerges as a shadow of this theory with one less time and one less space dimensions. The gravitational constant turns out to be a shadow of a dilaton field in $d+2$ dimensions that appears as a constant to observers stuck in $d$ dimensions. If elementary scalar fields play a role in the fundamental theory (such as Higgs fields in the standard model coupled to gravity), then their shadows in $d$ dimensions must necessarily be conformal scalars. This has the physical consequence that the gravitational constant changes at each phase transition (inflation, grand unification, electroweak, etc.), implying interesting new scenarios in cosmological applications. The fundamental action for pure gravity, which includes the spacetime metric ${G}_{MN}(X)$, the dilaton $\ensuremath{\Omega}(X)$, and an additional auxiliary scalar field $W(X)$, all in $d+2$ dimensions with two times, has a mix of gauge symmetries to produce appropriate constraints that remove all ghosts or redundant degrees of freedom. The action produces on-shell classical field equations of motion in $d+2$ dimensions, with enough constraints for the theory to be in agreement with classical general relativity in $d$ dimensions. Therefore this action describes the correct classical gravitational physics directly in $d+2$ dimensions. Taken together with previous similar work on the standard model of particles and forces, the present paper shows that 2T physics is a general consistent framework for a physical theory. Furthermore, the 2T-physics approach reveals more physical information for observers stuck in the shadow in $d$ dimensions in the form of hidden symmetries and dualities, that are largely concealed in the usual one-time formulation of physics.

Dynamical symmetry breaking and the Nambu–Goldstone theorem in the Gaussian wave functional approximation

We analyze the group-theoretical ramifications of the Nambu–Goldstone (NG) theorem in the self-consistent relativistic variational Gaussian wave functional approximation to spinless field theories. In an illustrative example we show how the Nambu–Goldstone theorem would work in the O(N) symmetric φ4 scalar field theory, if the residual symmetry of the vacuum were lesser than O(N−1), e.g., if the vacuum were O(N−2), or O(N−3),… symmetric. (This does not imply that any of the “lesser” vacua is actually the absolute energy minimum: stability analysis has not been done.) The requisite number of NG bosons would be (2N−3), or (3N−6),…, respectively, which may exceed N, the number of elementary fields in the Lagrangian. We show how the requisite new NG bosons would appear even in channels that do not carry the same quantum numbers as one of N “elementary particles” [scalar field quanta, or Castillejo–Dalitz–Dyson (CDD) poles] in the Lagrangian, i.e., in those “flavor” channels that have no CDD poles. The corresponding Nambu–Goldstone bosons are composites (bound states) of pairs of massive elementary (CDD) scalar fields excitations. As a nontrivial example of this method we apply it to the physically more interesting ’t Hooft σ model (an extended Nf=2 bosonic linear σ model with four scalar and four pseudoscalar fields), with spontaneously and explicitly broken chiral O(4)×O(2)≃SUR(2)×SUL(2)×UA(1) symmetry.

CHIRALLY EXTENDED QUANTUM CHROMODYNAMICS

We propose an extended Quantum Chromodynamics (XQCD) Lagrangian in which the fermions are coupled to elementary scalar fields through a Yukawa coupling which preserves chiral invariance. Our principle motivation is to find a new lattice formulation for QCD which avoids the source of critical slowing down usually encountered as the bare quark mass is tuned to the chiral limit. The phase diagram and the weak coupling limit for XQCD are studied. They suggest a conjecture that the continuum limit of XQCD is the same as the continuum limit of conventional lattice formulation of QCD. As examples of such universality, we present the large N solutions of two prototype models for XQCD, in which the mass of the spurious pion and sigma resonance go to infinity with the cut-off. Even if the universality conjecture turns out to be false, we believe that XQCD will still be useful as a low energy effective action for QCD phenomenology on the lattice. Numerical simulations are recommended to further investigate the possible benefits of XQCD in extracting QCD predictions.

Chirally extended quantum chromodynamics

We propose a chirally extended Quantum Chromodynamics (XQCD) Lagrangian in which the fermions are coupled to elementary scalar fields through a Yukawa coupling which preserves chiral invariance. Our principle motivation is to find a new lattice formulation for QCD which avoids the source of critical slowing down usually encountered as the bare quark mass is tuned to the chiral limit. The phase diagram and the weak coupling limit for XQCD support our conjecture that the continuum limit of XQCD is in the same universality class as conventional lattice QCD. The large N solution of a toy model for XQCD is studied and here it is shown that the mass of the spurious pion and σ resonance go to infinity with the cut-off. Numerical simulations are being pursued to investigate further the possible benefits of XQCD in extracting useful QCD predictions.

Phase Structure of the Gauged Yukawa Model

Based on the ladder Schwinger-Dyson equation, we investigate phase structure of the gauged Yukawa model possessing a global SU(2)L X SU(2)R symmetry and an unbroken (vector-like) gauge symmetry. We show that even when we tune the squared mass of the scalar boson in the Lagrangian to be positive, there still exists the dynamical chiral symmetry breaking due to fermion pair conden­ sate (VEV of the composite scalar) triggered by the strong Yukawa coupling larger than a certain critical value. We find a nontrivial ultraviolet fixed line and renormalized trajectories in the three-dimensional coupling space of the Yukawa coupling, the gauge coupling and the hopping parameter of the elementary scalar field. Presence of the gauge coupling is crucial to existence of the fixed line. Implications of the result for the lattice calculation and the top quark condensate model are also discussed.