Φ3 Theory Research Articles

Study of asymptotic free scalar field theories from adaptive perturbation method

We focus on the behavior of (2+1)d λϕ4 and (5+1)d λϕ3 or λ|ϕ|3 theories in different regimes and compare the results obtained from the adaptive perturbation method with those obtained from lattice simulation. These theories are simple models that exhibit asymptotic freedom, which is a property that is also observed in more complex theories such as QCD, which describes the strong interaction between quarks and gluons. Asymptotic freedom is an important feature of these theories because it allows for a perturbative treatment of interactions at high energies. However, the standard perturbation scheme breaks down in the presence of strong interactions, and the adaptive perturbation method, which involves resuming the Feynman diagrams, is more suitable for studying these interactions. Our research involves comparing the perturbation result to lattice simulation. In the case of the ϕ3 theory, there is no stable vacuum, so we explore evidence from the |ϕ|3 theory instead. Our results appear to show that resummation improves the strong coupling result for both the λϕ4 and λ|ϕ|3 theories. Additionally, we improve the resummation method for the three-point coupling vertex and study the RG flow to analyze the resummation contribution and theoretical properties.

Hidden zeros for particle/string amplitudes and the unity of colored scalars, pions and gluons

Recent years have seen the emergence of a new understanding of scattering amplitudes in the simplest theory of colored scalar particles — the Tr(ϕ3) theory — based on combinatorial and geometric ideas in the kinematic space of scattering data. In this paper we report a surprise: far from the toy model it appears to be, the “stringy” Tr(ϕ3) amplitudes secretly contains the scattering amplitudes for pions, as well as non-supersymmetric gluons, in any number of dimensions. The amplitudes for the different theories are given by one and the same function, related by a simple shift of the kinematics. This discovery was spurred by another fundamental observation: the tree-level Tr(ϕ3) field theory amplitudes have a hidden pattern of zeros when a special set of non-planar Mandelstam invariants is set to zero. These zeros are not manifest in Feynman diagrams but are made obvious by the connection of these amplitudes to the new understanding of associahedra arising from “causal diamonds” in kinematic space. Furthermore, near these zeros, the amplitudes simplify, by factoring into a non-trivial product of smaller amplitudes. Remarkably the amplitudes for pions and gluons are observed to also vanish in the same kinematical locus. These properties for Tr(ϕ3) amplitudes hold and further generalize to the “stringy” Tr(ϕ3) amplitudes. The “kinematic causal diamond” picture suggests a unique shift of the kinematic data that preserves the zeros, and this shift is precisely the one that unifies colored scalars, pions, and gluons into a single object. We will focus in this paper on explaining the hidden zeros and factorization properties and the connection between all the colored theories, working for simplicity at tree level. Subsequent works will describe this new formulation for the Non-linear Sigma Model and non-supersymmetric Yang-Mills theory, at all loop orders.

Nonlinear Sigma model amplitudes to all loop orders are contained in the Tr(Φ3) theory

Scattering amplitudes for the simplest theory of colored scalar particles—the Tr(Φ3) theory—have recently been the subject of active investigations. In this work we describe an unanticipated wider implication of this work: the Tr(Φ3) theory secretly contains nonlinear sigma model (NLSM) amplitudes to all loop orders. The NLSM amplitudes are obtained from Tr(Φ3) amplitudes by a unique shift of kinematic variables. We show that this shifted kinematics produces amplitudes for a cubic theory with a linear term in the potential, with extrema spontaneously breaking U(N)→U(N−k)×U(k). The Goldstone amplitudes for this theory coincide with those of pions in the U(N)×U(N)→U(N) chiral Lagrangian to all orders in the planar limit. We also give a purely on-shell understanding of this correspondence, showing that integrands defined by the kinematic shifts have the correct residues on poles and appropriately produce the Adler zero. Finally, we discuss how similar kinematic shifts produce certain infinite classes of mixed amplitudes of pions and Tr(Φ3) scalars, most of which are not interpretable from the Lagrangian description. Published by the American Physical Society 2024

Comb channel lightcone bootstrap: triple-twist anomalous dimensions

We advance the multipoint lightcone bootstrap and compute anomalous dimensions of triple-twist operators at large spin. In contrast to the well-studied double-twist operators, triple-twist primaries are highly degenerate so that their anomalous dimension is encoded in a matrix. At large spin, the degeneracy becomes infinite and the matrix becomes an integral operator. We compute this integral operator by studying a particular non-planar crossing equation for six-point functions of scalar operators in a lightcone limit. The bootstrap analysis is based on new formulas for six-point lightcone blocks in the comb-channel. For a consistency check of our results, we compare them to perturbative computations in the epsilon expansion of ϕ3 and ϕ4 theory. In both cases, we find perfect agreement between perturbative results and bootstrap predictions. As a byproduct of our studies, we complement previous results on triple-twist anomalous dimensions in scalar ϕ3 and ϕ4 theory at first and second order in epsilon, respectively.

Four loop renormalization in six dimensions using forcer

We employ the algorithm to renormalize a variety of six dimensional field theories to four loops. In order to achieve this we construct the master integrals in six dimensions from their four dimensional counterparts by using the Tarasov method. The ε expansion of the six dimensional masters are determined up to weight 9 where d=6−2ε. By applying the routine the four loop MS¯ renormalization of ϕ3 theory is reproduced before gauge theories are considered. The renormalization of these theories is also determined in the MOM˜ scheme. For instance the absence of ζ4 and ζ6 is confirmed to five loops in the MOM˜ renormalization of ϕ3 theory. We also evaluate the three loop β-function of the gauge coupling in six dimensional QCD. Published by the American Physical Society 2024

Correlation functions of scalar field theories from homotopy algebras

We present expressions for correlation functions of scalar field theories in perturbation theory using quantum A∞ algebras. Our expressions are highly explicit and can be used for theories both in Euclidean space and in Minkowski space including quantum mechanics. Correlation functions at a given order of perturbation theory can be calculated algebraically without using canonical quantization or the path integral, and we demonstrate it explicitly for φ3 theory. We show that the Schwinger-Dyson equations are satisfied as an immediate consequence of the form of the expressions based on quantum A∞ algebras.

Multicollinear singularities in celestial CFT

The purpose of this paper is to study the holomorphic multicollinear limit of (celestial) amplitudes and use it to further investigate the double residue condition for (hard celestial) amplitudes and the celestial operator product expansion. We first set up the notion of holomorphic multicollinear limits of amplitudes and derive the 3-collinear splitting functions for Yang-Mills theory, Einstein gravity, and massless ϕ3 theory. In particular, we find that in ϕ3 theory the celestial 3-OPE contains a term with a branch cut. This explicit example confirms that branch cuts can obstruct the double residue condition for hard celestial amplitudes, which is the underlying cause of the celestial Jacobi identities not holding for certain theories. This addresses an ongoing debate in the literature about associativity of the celestial OPEs and concretely demonstrates a new (multi-particle) term in the celestial OPE coming from the multi-particle channel in the amplitudes.

One-loop amplitudes in Einstein-Yang-Mills from forward limits

We present a method to compute the integrands of one-loop Einstein-Yang-Mills amplitudes for any number of external gauge and gravity multiplets. Our construction relies on the double-copy structure of Einstein-Yang-Mills as (super-)Yang-Mills with the so-called YM+ϕ3 theory — pure Yang-Mills coupled to bi-adjoint scalars — which we implement via one-loop Cachazo-He-Yuan formulae. The YM+ϕ3 building blocks are obtained from forward limits of tree-level input in external gluons and scalars, and we give the composition rules for any number of traces and orders in the couplings g and κ. On the one hand, we spell out supersymmetry- and dimension-agnostic relations that reduce loop integrands of Einstein-Yang-Mills to those of pure gauge theories. On the other hand, we present four-point results for maximal and half-maximal supersymmetry where all supersymmetry cancellations are exposed. In the half-maximal case, we determine six-dimensional anomalies due to chiral hypermultiplets in the loop.

Causal diamonds, cluster polytopes and scattering amplitudes

The “amplituhedron” for tree-level scattering amplitudes in the bi-adjoint ϕ3 theory is given by the ABHY associahedron in kinematic space, which has been generalized to give a realization for all finite-type cluster algebra polytopes, labelled by Dynkin diagrams. In this letter we identify a simple physical origin for these polytopes, associated with an interesting (1 + 1)-dimensional causal structure in kinematic space, along with solutions to the wave equation in this kinematic “spacetime” with a natural positivity property. The notion of time evolution in this kinematic spacetime can be abstracted away to a certain “walk”, associated with any acyclic quiver, remarkably yielding a finite cluster polytope for the case of Dynkin quivers. The mathcal{A} n−3, mathcal{B} n−1/ mathcal{C} n−1 and mathcal{D} n polytopes are the amplituhedra for n-point tree amplitudes, one-loop tadpole diagrams, and full integrand of one-loop amplitudes. We also introduce a polytope overline{mathcal{D}} n, which chops the mathcal{D} n polytope in half along a symmetry plane, capturing one-loop amplitudes in a more efficient way.

Recursive computation of Feynman periods

Feynman periods are Feynman integrals that do not depend on external kinematics. Their computation, which is necessary for many applications of quantum field theory, is greatly facilitated by graphical functions or the equivalent conformal four-point integrals. We describe a set of transformation rules that act on such functions and allow their recursive computation in arbitrary even dimensions. As a concrete example we compute all subdivergence-free Feynman periods in ϕ3 theory up to six loops and 561 of 607 Feynman periods at seven loops analytically. Our results support the conjectured existence of a coaction structure in quantum field theory and suggest that ϕ3 and ϕ4 theory share the same number content.

Random field ϕ3 model and Parisi-Sourlas supersymmetry

We use the RG framework set up in [1] to explore the ϕ3 theory with a random field interaction. According to the Parisi-Sourlas conjecture this theory admits a fixed point with emergent supersymmetry which is related to the pure Lee-Yang CFT in two less dimensions. We study the model using replica trick and Cardy variables in d = 8 − ϵ where the RG flow is perturbative. Allowed perturbations are singlets under the Sn symmetry that permutes the n replicas. These are decomposed into operators with different scaling dimensions: the lowest dimensional part, ‘leader’, controls the RG flow in the IR; the other operators, ‘followers’, can be neglected. The leaders are classified into: susy-writable, susy-null and non-susy-writable according to their mixing properties. We construct low lying leaders and compute the anomalous dimensions of a number of them. We argue that there is no operator that can destabilize the SUSY RG flow in d ≤ 8. This agrees with the well known numerical result for critical exponents of Branched Polymers (which are in the same universality class as the random field ϕ3 model) that match the ones of the pure Lee-Yang fixed point according to dimensional reduction in all 2 ≤ d ≤ 8. Hence this is a second strong check of the RG framework that was previously shown to correctly predict loss of dimensional reduction in random field Ising model.

Resonant resurgent asymptotics from quantum field theory

We perform an all-order resurgence analysis of a quantum field theory renormalon that contributes to an anomalous dimension in six-dimensional scalar ϕ3 theory and is governed by a third-order nonlinear differential equation. We augment the factorially divergent perturbative expansion associated to the renormalon by asymptotic expansions to all instanton orders, in a conjectured and well-tested formula. A distinctive feature of this renormalon singularity is the appearance of logarithmic terms, starting at second-instanton order in the trans-series. To highlight this and to illustrate our methods, we also analyze the trans-series for a closely related second-order nonlinear differential equation that exhibits a similarly resonant structure but lacks logarithmic contributions.

A finite energy-momentum tensor for the ϕ3 theory in 6 dimensions

Following Brown [1], we construct composite operators for the scalar ϕ3 theory in six dimensions using renormalisation group methods with dimensional regularisation. We express bare scalar operators in terms of renormalised composite operators of lower dimension, then do this with traceless tensor operators. We then express the bare energy momentum tensor in terms of the renormalised composite operators, with some terms having divergent coefficients. We subtract these away and obtain a manifestly finite energy momentum tensor. The subtracted terms are transverse, so this does not affect the conservation of the energy momentum tensor. The trace of this finite improved energy momentum tensor vanishes at the fixed point indicating conformal invariance. Interestingly it is not RG invariant except at the fixed point, but can be made RG invariant everywhere by further addition of finite transverse terms, whose coefficients vanish at the fixed point.

Critical exponents from five-loop scalar theory renormalization near six-dimensions

We present five-loop results for the renormalization of various models with a cubic interaction (in d=6−2ε dimensions). For the scalar model and its O(n)-symmetric extension we provide renormalization constants, anomalous dimensions and critical exponents. We discuss in detail the method of calculation, and provide all counterterms up to five loops. This allows one to consider generalizations of the φ3 theory to other symmetries.

One-loop Parke-Taylor factors for quadratic propagators from massless scattering equations

In this paper we reconsider the Cachazo-He-Yuan construction (CHY) of the so called scattering amplitudes at one-loop, in order to obtain quadratic propagators. In theories with colour ordering the key ingredient is the redefinition of the Parke-Taylor factors. After classifying all the possible one-loop CHY-integrands we conjecture a new one-loop amplitude for the massless Bi-adjoint Φ3 theory. The prescription directly reproduces the quadratic propagators of the traditional Feynman approach.

CHY loop integrands from holomorphic forms

Recently, the Cachazo-He-Yuan (CHY) approach for calculating scattering amplitudes has been extended beyond tree level. In this paper, we introduce a way of constructing CHY integrands for Φ3 theory up to two loops from holomorphic forms on Riemann surfaces. We give simple rules for translating Feynman diagrams into the corresponding CHY integrands. As a complementary result, we extend the Λ-algorithm, originally introduced in arXiv:1604.05373, to two loops. Using this approach, we are able to analytically verify our prescription for the CHY integrands up to seven external particles at two loops. In addition, it gives a natural way of extending to higher-loop orders.

Renormalization-group theory of first-order phase transition dynamics in field-driven scalar model

Through a detailed study of the mean-field approximation, the Gaussian approximation, the perturbation expansion, and the field-theoretic renormalization-group analysis of a φ3 theory, we show that the instability fixed points of the theory, together with their associated instability exponents, are quite probably relevant to the scaling and universality behavior exhibited by the first-order phase transitions in a field-driven scalar ϕ4 model, below its critical temperature and near the instability points. Finitetime scaling and leading corrections to the scaling are considered. We also show that the instability exponents of the first-order phase transitions are equivalent to those of the Yang–Lee edge singularity, and employ the latter to improve our estimates of the former. The outcomes agree well with existing numerical results.

Fragmentation in the ϕ3 theory and the LPHD hypothesis

We present analytic solution of the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) equation at leading order (LO) in the [Formula: see text] theory in 6 space-time dimensions. If the [Formula: see text] model was the theory of strong interactions, the obtained solution would describe the distribution of partons in a jet. We point out that the local parton-hadron duality (LPHD) conjecture does not work in this hypothetical situation. That is, treatment of hadronisation of shower partons is essential for the description of hadron distributions in jets stemming from proton-proton (pp) collisions at [Formula: see text] TeV and from electron-positron ([Formula: see text]) annihilations at various collision energies. We use a statistical model for the description of hadronisation.

CHY-construction of planar loop integrands of cubic scalar theory

In this paper, by treating massive loop momenta as massless momenta in higher dimensions, we are able to treat all-loop scattering equations as tree ones. As an application of the new perspective, we consider the CHY-construction of bi-adjoint ϕ 3 theory. We present the explicit formula for two-loop planar integrands. We discuss in details how to subtract various forward singularities in the construction. We count the number of terms obtained by our formula and by direct Feynman diagram calculation and find the perfect match, thus provide a strong support for our results.

ϵ-expansions near three dimensions from conformal field theory

We formally extend the CFT techniques introduced in arXiv: 1505.00963, to $$ \phi \frac{2{d}_0}{d_0-2} $$ theory in d = d 0 − ϵ dimensions and use it to compute anomalous dimensions near d 0 = 3, 4 in a unified manner. We also do a similar analysis of the O(N) model in three dimensions by developing a recursive combinatorial approach for OPE contractions. Our results match precisely with low loop perturbative computations. Finally, using 3-point correlators in the CFT, we comment on why the ϕ 3 theory in d 0 = 6 is qualitatively different.