Renormalizable Field Theory Research Articles

Single-valued multiple polylogarithms and a proof of the zig–zag conjecture

A long-standing conjecture in quantum field theory due to Broadhurst and Kreimer states that the periods of the zig–zag graphs are a certain explicit rational multiple of the odd values of the Riemann zeta function. In this paper we prove this conjecture by constructing a certain family of single-valued multiple polylogarithms which correspond to multiple zeta values ζ(2,…,2,3,2,…2) and using the method of graphical functions. The zig–zag graphs are the only infinite family of primitive graphs in ϕ44 theory (in fact, in any renormalisable quantum field theory in four dimensions) whose periods are now known.

Stable bound states of asymmetric dark matter

The simplest renormalizable effective field theories with asymmetric dark matter bound states contain two additional gauge singlet fields one being the dark matter and the other a mediator particle that the dark matter annihilates into. We examine the physics of one such model with a Dirac fermion as the dark matter and a real scalar mediator. For a range of parameters the Yukawa coupling of the dark matter to the mediator gives rise to stable asymmetric dark matter bound states. We derive properties of the bound states including nuggets formed from $N\gg1$ dark matter particles. We also consider the formation of bound states in the early universe and direct detection of dark matter bound states. Many of our results also hold for symmetric dark matter.

Asymptotic states and renormalization in Lorentz-violating quantum field theory

Asymptotic single-particle states in quantum field theories with small departures from Lorentz symmetry are investigated perturbatively with focus on potential phenomenological ramifications. To this end, one-loop radiative corrections for a sample Lorentz-violating Lagrangian contained in the Standard-Model Extension (SME) are studied at linear order in Lorentz breakdown. It is found that the spinor kinetic operator, and thus the free-particle physics, is modified by Lorentz-violating operators absent from the original Lagrangian. As a consequence of this result, both the standard renormalization procedure as well as the Lehmann-Symanzik-Zimmermann reduction formalism need to be adapted. The necessary adaptations are worked out explicitly at first order in Lorentz-breaking coefficients.

Driven surface diffusion with detailed balance and elastic phase transitions.

Driven surface diffusion occurs, for example, in molecular beam epitaxy when particles are deposited under an oblique angle. Elastic phase transitions happen when normal modes in crystals become soft due to the vanishing of certain elastic constants. We show that these seemingly entirely disparate systems fall under appropriate conditions into the same universality class. We derive the field-theoretic Hamiltonian for this universality class, and we use renormalized field theory to calculate critical exponents and logarithmic corrections for several experimentally relevant quantities.

Quark flavor transitions inLμ−Lτmodels

We consider a new massive vector-boson Z' that couples to leptons through the L_mu - L_tau current, and to quarks through an arbitrary set of couplings. We show that such a model can be obtained from a renormalizable field theory involving new heavy fermions in an anomaly-free representation. The model is a candidate explanation for the discrepancy observed recently by the LHCb collaboration in angular distributions of the final state particles in the rare decay B \to K* mu^+ mu^-. Interestingly, the new vector-boson contribution to the decay tau \to mu nu_tau \bar nu_mu can also remove a small tension in the measurement of the corresponding branching ratio. Constraints from light flavor meson-mixing restrict the coupling to the up- and down-quarks to be very small and thus direct production of the vector-boson at hadron colliders is strongly suppressed. The most promising ways to test the model is through the measurement of the Z decay to four leptons and through its effect on neutrino trident production of muon pairs. This latter process is a powerful but little-known constraint, which surprisingly rules out explanations of (g-2)_mu based on Z' gauge bosons coupled to muon number, with mass of at least a few GeV.

Feshbach resonance in a synthetic non-Abelian gauge field

We study the Feshbach resonance of spin-1/2 particles in the presence of a uniform synthetic non-Abelian gauge field that produces spin orbit coupling along with constant spin potentials. We develop a renormalizable quantum field theory that includes the closed channel boson which engenders the Feshbach resonance, in the presence of the gauge field. By a study of the scattering of two particles in the presence of the gauge field, we show that the Feshbach magnetic field, where the apparent low energy scattering length diverges, depends on the conserved centre of mass momentum of the two particles. For high symmetry gauge fields, such as the one which produces an isotropic Rashba spin orbit coupling, we show that the system supports two bound states over a regime of magnetic fields for a negative background scattering length and resonance width comparable to the energy scale of the spin orbit coupling. We discuss the consequences of these findings for the many body setting, and point out that a broad resonance (width larger than spin orbit coupling energy scale) is most favourable for the realization of the rashbon condensate.

Counterterms in the context of the universal Hopf algebra of renormalization

The manuscript discovers a new interpretation of counterterms of renormalizable Quantum Field Theories in terms of formal expansions of decorated rooted trees.

Renormalization and redundancy in 2d quantum field theories

We analyze renormalization group (RG) flows in two-dimensional quantum field theories in the presence of redundant directions. We use the operator picture in which redundant operators are total derivatives. Our analysis has three levels of generality. We introduce a redundancy anomaly equation which is analyzed together with the RG anomaly equation previously considered by H.Osborn [8] and D.Friedan and A.Konechny [7]. The Wess-Zumino consistency conditions between these anomalies yield a number of general relations which should hold to all orders in perturbation theory. We further use conformal perturbation theory to study field theories in the vicinity of a fixed point when some of the symmetries of the fixed point are broken by the perturbation. We relate various anomaly coefficients to OPE coefficients at the fixed point and analyze which operators become redundant and how they participate in the RG flow. Finally, we illustrate our findings by three explicit models constructed as current-current perturbations of SU(2)_k WZW model. At each generality level we discuss the geometric picture behind redundancy and how one can reduce the number of couplings by taking a quotient with respect to the redundant directions. We point to the special role of polar representations for the redundancy groups.

Relations between the Hopf algebra and the B-series with new concepts

In this paper, the relationships between the Hopf algebra and the B-series approach are illustraded in details. The Hopf Algebra was proposed to solve problems in wider fields: non-commutative geometry, combinatorics of renormalization in quantum field theory. These two approaches generate the recursive composition rules based on the dual structure. We proposed to reformulate the composition rule on the tree space. It is found to be much simpler and easier without using the dual structure. In this paper, the B-series approach is represented in a way that applications are based on the understanding of the meaning hiding behind the formulae. This also enables us to describe the Hopf algebra in B-series language without using the dual structure. In order to show that the B-series approach could be applied to solve problems which is not restricted to ordinary differential systems, we use an example which is solved using the Hopf Algebra approach. Through the demonstration of applying the B-series approach, we shows that the key of appying the B-series approach is to construct the elementary differential of the system. It turns out that the results we derived are generated in an interesting way.

ISSUES OF EFFECTIVE FIELD THEORIES WITH RESONANCES

We address some issues of renormalization and symmetries of effective field theories with unstable particles - resonances. We also calculate anomalous contributions in the divergence of the singlet axial current in an effective field theory of massive SU(N) Yang-Mills fields interacting with fermions and discuss their possible relevance to the strong CP problem.

Implications of Neutrino Oscillations on the Dark-Matter World

According to my own belief that “The God wouldnʼt create a world that is so boring that a particle knows only the very feeble weak interaction.”, maybe we underestimate the roles of neutrinos. We note that right-handed neutrinos play no roles, or donʼt exist, in the minimal Standard Model. We discuss the language to write down an extended Standard Model - using renormalizable quantum field theory as the language; to start with a certain set of basic units under a certain gauge group; in fact, to use the three right-handed neutrinos to initiate the family gauge group SUf(3). Specifically we use the left-handed and right-handed spinors to form the basic units together with SUc(3)×SUL(2)×U(1)×SUf(3) as the gauge group. The dark-matter SUf(3) world couples with the lepton world, but not with the quark world. Amazingly enough, the space of the Standard-Model Higgs Φ(1,2), the family Higgs triplet Φ(3, 1), and the neutral part of the mixed family Higgs Φ0(3,2) undergoes the spontaneous symmetry breaking, i.e. the Standard-Model Higgs mechanism and the “project-out” family Higgs mechanism, to give rise to the weak bosons W± and Z0, one Standard-Model Higgs, the eight massive family gauge bosons, and the remaining four massive neutral family Higgs particles, and nothing more. Thus, the roles of neutrinos in this extended Standard Model are extremely interesting in connection with the dark-matter world.

ON THE PROBLEM OF VACUUM ENERGY IN FLRW UNIVERSES AND DARK ENERGY

We present a (hopefully) novel calculation of the vacuum energy in expanding FLRW spacetimes based on the renormalization of quantum field theory in nonzero backgrounds. We compute the renormalized effective action up to the two-point function and then apply the formalism to the cosmological backgrounds of interest. As an example we calculate for quasi de Sitter spacetimes the leading correction to the vacuum energy given by the tadpole diagram and show that it behaves as [Formula: see text] where H0 is the Hubble constant and Λ P is the Planck constant. This is of the same order of magnitude as the observed dark energy density in the universe.

Spontaneously broken conformal symmetry: dealing with the trace anomaly

The majority of renormalizable field theories possessing the scale invariance at the classical level exhibits the trace anomaly once quantum corrections are taken into account. This leads to the breaking of scale and conformal invariance. At the same time any realistic theory must contain gravity and is thus non-renormalizable. We show that discarding the renormalizability it is possible to construct viable models allowing to preserve the scale invariance at the quantum level. We present explicit one-loop computations for two toy models to demonstrate the main idea of the approach. Constructing the renormalized energy momentum tensor we show that it is traceless, meaning that the conformal invariance is also preserved.

Phase transitions in two dimensions and multiloop renormalization group expansions

We discuss using the field theory renormalization group (RG) to study the critical behavior of twodimensional (2D) models. We write the RG functions of the 2D λϕ4 Euclidean n-vector theory up to five-loop terms, give numerical estimates obtained from these series by Pade-Borel-Leroy resummation, and compare them with their exact counterparts known for n = 1, 0,−1. From the RG series, we then derive pseudo-e-expansions for the Wilson fixed point location g*, critical exponents, and the universal ratio R6 = g6/g2, where g6 is the effective sextic coupling constant. We show that the obtained expansions are “friendler” than the original RG series: the higher-order coefficients of the pseudo-e-expansions for g*, R6, and γ−1 turn out to be considerably smaller than their RG analogues. This allows resumming the pseudo-e-expansions using simple Pade approximants without the Borel-Leroy transformation. Moreover, we find that the numerical estimates obtained using the pseudo-e-expansions for g* and γ−1 are closer to the known exact values than those obtained from the five-loop RG series using the Pade-Borel-Leroy resummation.

Recipe theorem for the Tutte polynomial for matroids, renormalization group-like approach

Using a quantum field theory renormalization group-like differential equation, we give a new proof of the recipe theorem for the Tutte polynomial for matroids. The solution of such an equation is in fact given by some appropriate characters of the Hopf algebra of isomorphic classes of matroids, characters which are then related to the Tutte polynomial for matroids. This Hopf algebraic approach also allows to prove, in a new way, a matroid Tutte polynomial convolution formula appearing in [W. Kook, V. Reiner, D. Stanton, A convolution formula for the Tutte polynomial, J. Combin. Theory Ser. B 76 (1999) 297–300] and [G. Etienne, M. Las Vergnas, External and internal elements of a matroid basis, Discrete Math. 179 (1998) 111–119].

Addendum to: A Renormalizable 4-Dimensional Tensor Field Theory

This note fills a gap in the article with title above [1]. We provide the proof of Equation (82) of Lemma 5 in [1] and thereby complete its power counting analysis with a more precise next-to-leading-order estimate.

Supercurrent, supervirial, and superimprovement

Supersymmetric field theories possess a rich structure in their supercurrent supermultiplets. Some symmetries are manifest in one supercurrent supermultiplet but not in the others; for instance, $R$ symmetry is manifest in the $R$ multiplet but not in the Ferrara-Zumino multiplet. Similarly, we argue that dilatation symmetry is manifest in the virial multiplet (also known as a variant minimal supercurrent supermultiplet in the literature). It reveals that $R$ symmetry and dilatation symmetry are conceptually independent without further assumptions even though the superconformal symmetry connects the two. We show the structure of the virial multiplet for general renormalizable supersymmetric field theories in ($1+3$) dimensions to all orders in perturbation theory, and discuss the condition for the dilatation invariance (but not necessarily $R$ symmetric nor superconformal). We present novel scale invariant trajectories with a nilpotent structure in coupling constants for nonunitary Wess-Zumino models with nonvanishing beta functions, which are, therefore, not superconformal.

Renormalization of noncommutative quantum field theories

We report on a comprehensive analysis of the renormalization of noncommutative \phi^4 scalar field theories on the Groenewold-Moyal (GM) plane. These scalar field theories are twisted Poincar\'e invariant. Our main results are that these scalar field theories are renormalizable, free of UV/IR mixing, possess the same fixed points and \beta-functions for the couplings as their commutative counterparts. We also argue that similar results hold true for any generic noncommutative field theory with polynomial interactions and involving only pure matter fields. A secondary aim of this work is to provide a comprehensive review of different approaches for the computation of the noncommutative S-matrix: noncommutative interaction picture and noncommutative LSZ formalism.

BPS domain walls in $ \mathcal{N}=4 $ supergravity and dual flows

We establish the conditions for supersymmetric domain wall solutions to N=4 gauged supergravity in five dimensions. These read as BPS first-order equations for the warp factor and the scalar fields, driven by a superpotential and supplemented by a set of constraints that we specify in detail. Then we apply our results to certain consistent truncations of IIB supergravity, thus exploring their dual field theory renormalization group flows. We find a universal flow deforming superconformal theories on D3-branes at Calabi-Yau cones. Moreover, we obtain a superpotential for the solution corresponding to the baryonic branch of the Klebanov-Strassler theory, as well as the superpotential for the flow describing D3 and wrapped D5-branes on the resolved conifold.

On the renormalization of non-commutative field theories

This paper addresses three topics concerning the quantization of non-commutative field theories (as defined in terms of the Moyal star product involving a constant tensor describing the non-commutativity of coordinates in Euclidean space). To start with, we discuss the Quantum Action Principle and provide evidence for its validity for non-commutative quantum field theories by showing that the equation of motion considered as insertion in the generating functional Zc[j] of connected Green functions makes sense. Second, we consider the generalization of the BPHZ renormalization scheme to non-commutative field theories and apply it to the case of a self-interacting real scalar field. Finally, we discuss the renormalizability of various models for a self-interacting complex scalar field by using the approach of algebraic renormalization.