In this paper we elaborate on the translation-invariant renormalizable Phi^4 theory in 4-dimensional non-commutative space which was recently introduced by the Orsay group. By explicitly performing Feynman graph calculations at one loop and higher orders we illustrate the mechanism which overcomes the UV/IR mixing problem and ultimately leads to a renormalizable model. The obtained results show that the IR divergences are also suppressed in the massless case, which is of importance for the gauge field theoretic generalization of the scalar field model.
Motivated by the recent work of Vilar et al. arXiv:0902.2956 we enhance our non-commutative translation invariant gauge model arXiv:0901.1681 by introducing auxiliary fields and ghosts forming a BRST doublet structure. In this way localization of the problematic 1/D^2 term can be achieved without the necessity for any additional degrees of freedom. The resulting theory is suspected to be renormalizable. A rigorous proof, however, has not been accomplished up to now.
In this paper we find non-trivial vacuum states for the renormalizable non-commutative $ϕ^4$ model. An associated linear sigma model is then considered. We further investigate the corresponding spontaneous symmetry breaking.
We suggest a novel view on non-renormalizable interactions. It is based on the usual BPHZ R-operation which is equally applicable to any local QFT independently whether it is renormalizable or not. As a playground we take the phi^4_D theory in D dimensions for D=4,6,8,10 and consider the four-point scattering amplitude on shell. We derive the generalized RG equation and find the solution valid for any D that sums up the leading logarithms in all orders of PT in full analogy with the renormalizable case. It is found that the scattering amplitude in the phi^4_D theory possesses the Landau pole a…
Renormalizable $ϕ^{\star 4}_4$ models on Moyal space have been obtained by modifying the commutative propagator. But these models have a divergent "naive" commutative limit. We explain here how to obtain a coherent such commutative limit for a recently proposed translation-invariant model. The mechanism relies on the analysis of the uv/ir mixing in general Feynman graphs.
We formulate conditions for almost-commutative (spacetime) manifolds under which the asymptotically expanded spectral action is renormalizable. These conditions are of a graph-theoretical nature, involving the Krajewski diagrams that classify such geometries. This applies in particular to the Standard Model of particle physics, giving a graph-theoretical argument for its renormalizability. A promising potential application is in the selection of physical (renormalizable) field theories described by almost-commutative geometries, thereby going beyond the Standard Model.
We examine the issue of renormalizability of asymptotically free field theories on non-commutative spaces. As an example, we solve the non-commutative O(N) invariant Gross-Neveu model at large N. On commutative space this is a renormalizable model with non-trivial interactions. On the noncommutative space, if we take the translation invariant ground state, we find that the model is non-renormalizable. Removing the ultraviolet cutoff yields a trivial non-interacting theory.
The non-commutative O(N) Gross-Neveu model is solved in the large N limit in two and three space-time dimensions. The commutative version of the two dimensional model is a renormalizable quantum field theory, both in a coupling constant expansion and an expansion in 1/N. The non-commutative version has a renormalizable coupling constant expansion where ultraviolet divergences can be removed by adjusting counterterms to each order. On the other hand, in a previous work, we showed that the non-commutative theory is not renormalizable in the large N expansion. This is argued to be due to a combin…
When considering quantum field theories on non-commutative spaces one inevitably encounters the infamous UV/IR mixing problem. So far, only very few renormalizable models exist and all of them describe non-commutative scalar field theories on four-dimensional Euclidean Groenewold-Moyal deformed space, also known as `theta-deformed space'. In this work we discuss some major obstacles of constructing a renormalizable non-commutative gauge field model and sketch some possible ways out.
In this paper we propose a translation-invariant scalar model on the Moyal space. We prove that this model does not suffer from the UV/IR mixing and we establish its renormalizability to all orders in perturbation theory.
The spontaneous breaking of of translational invariance in non-commutative self-interacting scalar field theory in two dimensions is investigated by effective action techniques. The analysis confirms the existence of the stripe phase, already observed in lattice simulations, due to the non-local nature of the non-commutative dynamics.
We show that the simplest non commutative renormalizable field theory, the $ϕ^4$ model on four dimensional Moyal space with harmonic potential is asymptotically safe to all orders in perturbation theory
Constructing renormalizable models on non-commutative spaces constitutes a big challenge. Only few examples of renormalizable theories are known, such as the scalar Grosse-Wulkenhaar model. Gauge fields are even more difficult, since new renormalization techniques are required which are compatible with the inherently non-local setting on the one hand, and also allow to properly treat the gauge symmetry on the other hand. In this proceeding (which is based on my talk given at the ``Workshop on Noncommutative Field Theory and Gravity'' in Corfu/Greece, September 8 -- 15, 2013), I focus on this l…
We study the polynomial Abelian or U(1)^d Tensorial Group Field Theories equipped with a gauge invariance condition in any dimension d. From our analysis, we prove the just renormalizability at all orders of perturbation of the phi^4_6 and phi^6_5 random tensor models. We also deduce that the phi^4_5 tensor model is super-renormalizable.
Non(anti)commutative gauge theories are supersymmetric Yang-Mills and matter system defined on a deformed superspace whose coordinates obey non(anti)commutative algebra. We prove that these theories in four dimensions with N=1/2 supersymmetry are renormalizable to all orders in perturbation theory. Our proof is based on operator analysis and symmetry arguments. In a case when the Grassman-even coordinates are commutative, deformation induced by non(anti)commutativity of the Grassman-odd coordinates contains operators of dimension-four or higher. Nevertheless, they do not lead to power divergen…
In this paper we prove that the Grosse-Wulkenhaar type non-commutative orientable complex scalar $ϕ^6_3$ theory, with two non-commutative coordinates and the third one commuting with the other two, is renormalizable to all orders in perturbation theory. Our proof relies on a multiscale analysis in x space.
Motivated by the recent construction of a translation-invariant renormalizable non-commutative model for a scalar field (see arXiv:0802.0791 [math-ph]), we introduce models for non-commutative U(1) gauge fields along the same lines. More precisely, we include some extra terms into the action with the aim of getting rid of the UV/IR mixing.
The simplest non commutative renormalizable field theory, the $ϕ_4$ model on four dimensional Moyal space with harmonic potential is asymptotically safe up to three loops, as shown by H. Grosse and R. Wulkenhaar, M. Disertori and V. Rivasseau. We extend this result to all orders.
We prove that the non-commutative Gross-Neveu model on the two-dimensional Moyal plane is renormalizable to all orders. Despite a remaining UV/IR mixing, renormalizability can be achieved. However, in the massive case, this forces us to introduce an additional counterterm of the form "psibar i gamma^{0} gamma^{1} psi". The massless case is renormalizable without such an addition.
The simplest non commutative renormalizable field theory, the $ϕ_4^4$ model on four dimensional Moyal space with harmonic potential is asymptotically safe at one loop, as shown by H. Grosse and R. Wulkenhaar. We extend this result up to three loops. If this remains true at any loop, it should allow a full non perturbative construction of this model.
We present a numerical study of the λϕ^{4} model in three Euclidean dimensions, where the two spatial coordinates are non-commutative (NC). We first show the explicit phase diagram of this model on a lattice. The ordered regime splits into a phase of uniform order and a ``striped phase''. Then we discuss the dispersion relation, which allows us to introduce a dimensionful lattice spacing. Thus we can study a double scaling limit to zero lattice spacing and infinite volume, which keeps the non-commutativity parameter constant. The dispersion relation in the disordered phase stabilizes in this l…
In this paper we discuss one-loop results for the translation invariant non-commutative gauge field model we recently introduced in arXiv:0804.1914. This model relies on the addition of some carefully chosen extra terms in the action which mix long and short scales in order to circumvent the infamous UV/IR mixing, and were motivated by the renormalizable non-commutative scalar model of Gurau et al. (cf. arXiv:0802.0791).
We present a systematic investigation of one-loop renormalizability for nonanticommutative N=1/2, U(N) SYM theory in superspace. We first discuss classical gauge invariance of the pure gauge theory and show that in contradistinction to the ordinary anticommutative case, different representations of supercovariant derivatives and field strengths do not lead to equivalent descriptions of the theory. Subsequently we develop background field methods which allow us to compute a manifestly covariant gauge effective action. One-loop evaluation of divergent contributions reveals that the theory simply…
We show that the noncommutative Wess-Zumino model is renormalizable to all orders of perturbation theory. The noncommutative scalar potential by itself is non-renormalizable but the Yukawa terms demanded by supersymmetry improve the situation turning the theory into a renormalizable one. As in the commutative case, there are neither quadratic nor linear divergences. Hence, the IR/UV mixing does not give rise to quadratic infrared poles.
Inspired by the renormalizability of the non-commutative Phi^4 model with added oscillator term, we formulate a non-commutative gauge theory, where the oscillator enters as a gauge fixing term in a BRST invariant manner. All propagators turn out to be essentially given by the Mehler kernel and the bilinear part of the action is invariant under the Langmann-Szabo duality. The model is a promising candidate for a renormalizable non-commutative U(1) gauge theory.
We formulate conditions under which the asymptotically expanded spectral action on an almost commutative manifold is renormalizable as a higher-derivative gauge theory. These conditions are of graph theoretical nature, involving the Krajewski diagrams that classify such manifolds. This generalizes our previous result on (super)renormalizability of the asymptotically expanded Yang-Mills spectral action to a more general class of particle physics models that can be described geometrically in terms of a noncommutative space. In particular, it shows that the asymptotically expanded spectral action…
Motivated by the success of the non-commutative scalar Grosse-Wulkenhaar model, a non-commutative U(1) gauge field theory including an oscillator-like term in the action has been put forward in arXiv:0705.4205. The aim of the current work is to analyze whether that action can lead to a fully renormalizable gauge model on non-commutative Euclidean space. In a first step, explicit one-loop graph computations are hence presented, and their results as well as necessary modifications of the action are successively discussed.
We study the behavior of the photon two point function, in non-commutative QED, in a general covariant gauge and in arbitrary space-time dimensions. We show, to all orders, that the photon self-energy is transverse. Using an appropriate extension of the dimensional regularization method, we evaluate the one-loop corrections, which show that the theory is renormalizable. We also prove, to all orders, that the poles of the photon propagator are gauge independent and briefly discuss some other related aspects.
We show that the noncommutative Wess-Zumino (NCWZ) Lagrangian with permutation terms in the interaction parts is renormalizable at one-loop level by only a wave function renormalization. When the non-commutativity vanishes, the logarithmic divergence of the wave function renormalization of the NCWZ theory is the same as that of the commutative one. Next the algebras of noncommutative field theories (NCFT's) are studied. From Neother currents, the field representation for the generators of NCFT's is extracted. Then based on this representation, the commutation relations between the generators a…
In this paper we provide a new proof that the Grosse-Wulkenhaar non-commutative scalar Phi^4_4 theory is renormalizable to all orders in perturbation theory, and extend it to more general models with covariant derivatives. Our proof relies solely on a multiscale analysis in x space. We think this proof is simpler and could be more adapted to the future study of these theories (in particular at the non-perturbative or constructive level).
The critical properties of the real phi^4 scalar field theory are studied numerically on the fuzzy sphere. The fuzzy sphere is a matrix (non commutative) discretisation of the algebra of functions on the usual two dimensional sphere. It is also one of the simplest examples of a non commutative space to study field theory on. Aside from the usual disordered and uniform phases present in the commutative scalar field theory, we find and discuss in detail a new phase with spontaneously broken rotational invariance, called matrix phase because the geometry of the fuzzy sphere, as expressed by the k…
We have two aims. The main one is to expound the idea of renormalization in quantum field theory, with no technical prerequisites (Sections 2 and 3). Our motivation is that renormalization is undoubtedly one of the great ideas, and great successes, of twentieth-century physics. Also it has strongly influenced in diverse ways, how physicists conceive of physical theories. So it is of considerable philosophical interest. Second, we will briefly relate renormalization to Ernest Nagel's account of inter-theoretic relations, especially reduction (Section 4). One theme will be a contrast between two…
This paper investigates the non-commutative version of the Abelian Higgs model at the one loop level. We find that the BRST invariance of the theory is maintained at this order in perturbation theory, rendering the theory one-loop renormalizable. Upon removing the gauge field from the theory we also obtain a consistent continuum renormalization of the broken O(2) linear sigma model, contradicting results found in the literature. The beta functions for the various couplings of the gauged U(1) theory are presented, as are the divergent contributions to every one particle irreducible (1PI) functi…
In a recent work a modified BPHZ scheme has been introduced and applied to one-loop Feynman graphs in non-commutative phi^4-theory. In the present paper, we first review the BPHZ method and then we apply the modified BPHZ scheme as well as Zimmermann's forest formula to the sunrise graph, i.e. a typical higher-loop graph involving overlapping divergences. Furthermore, we show that the application of the modified BPHZ scheme to the IR-singularities appearing in non-planar graphs (UV/IR mixing problem) leads to the introduction of a 1/p^2 term and thereby to a renormalizable model. Finally, we a…
We present a non-perturbative study of the lambda phi**4 model on a non-commutative plane. The lattice regularised form can be mapped onto a Hermitian matrix model, which enables Monte Carlo simulations. Numerical data reveal the phase diagram; at large lambda it contains a "striped phase", which is absent in the commutative case. We explore the question whether or not this phenomenon persists in a Double Scaling Limit (DSL), which extrapolates simultaneously to the continuum and to infinite volume, at a fixed non-commutativity parameter. To this end, we introduce a dimensional lattice spacing…
In this article we define and quantize a truncated form of the nonassociative and noncommutative Snyder phi^4 field theory using the functional method in momentum space. More precisely, the action is approximated by expanding up to the linear order in the Snyder deformation parameter beta, producing an effective model on commutative spacetime for the computation of the two-, four- and six-point functions. The two- and four-point functions at one loop have the same structure as at the tree level, with UV divergences faster than in the commutative theory. The same behavior appears in the six-poi…
Recently, a new type of renormalizable $ϕ^{\star 4}_{4}$ scalar model on the Moyal space was proved to be perturbatively renormalizable. It is translation-invariant and introduces in the action a $a/(θ^2p^2)$ term. We calculate here the $β$ and $γ$ functions at one-loop level for this model. The coupling constant $β_λ$ function is proved to have the same behaviour as the one of the $ϕ^4$ model on the commutative $\mathbb{R}^4$. The $β_a$ function of the new parameter $a$ is also calculated. Some interpretation of these results are done.
In this paper we analyze the one-loop renormalization of the $θ$-expanded $\rm SU(N)$ Yang-Mills theory. We show that the {\it freedom parameter} $a$, key to renormalization, originates from higher order non-commutative gauge interaction, represented by a higher derivative term $ b h θ^{μν}\hat F_{μν}\star\hat F_{ρσ}\star\hat F^{ρσ}$. The renormalization condition fixes the allowed values of the parameter $a$ to one of the two solutions: $a=1$ or $a=3$, i.e. to $b=0$ or to $b=1/2$, respectively. When the higher order interaction is switched on, ($a=3$), pure non-commutative SU(N) gauge theory …
We review the Moyal and Wick-Voros products, and more in general the translation invariant non-commutative products, and apply them to classical and quantum field theory. We investigate phi^4 field theories calculating their Green's functions up to one-loop for the two- and four-point cases. We also review the connections of these theories with Drinfeld twists.
We study infrared divergences due to ultraviolet-infrared mixing in quantum field theory on Moyal space with Lorentzian signature in the Yang-Feldman formalism. Concretely, we are considering the phi^4 and the phi^3 model in arbitrary even dimension. It turns out that the situation is worse than in the Euclidean setting, in the sense that we find infrared divergences in graphs that are finite there. We briefly discuss the problems one faces when trying to adapt the nonlocal counterterms that render the Euclidean model renormalizable.
Quantum field theory has been shown recently renormalizable on flat Moyal space and better behaved than on ordinary space-time. Some models at least should be completely finite, even beyond perturbation theory. In this paper a first step is taken to extend such theories to non-flat backgrounds such as solvable symmetric spaces.
We construct and analyze a phase diagram of a self-interacting matrix field coupled to curvature of the non-commutative truncated Heisenberg space. The model reduces to the renormalizable Grosse-Wulkenhaar model in an infinite matrix size limit and exhibits a purely non-commutative non-uniformly ordered phase. Particular attention is given to scaling of model's parameters. We additionally provide the infinite matrix size limit for the disordered to ordered phase transition line.
Very high energy physics needs a coherent description of the four fundamental forces. Non-commutative geometry is a promising mathematical framework which already allowed to unify the general relativity and the standard model, at the classical level, thanks to the spectral action principle. Quantum field theories on non-commutative spaces is a first step towards the quantification of such a model. These theories can't be obtained simply by writing usual field theory on non-commutative spaces. Such attempts exhibit indeed a new type of divergencies, called ultraviolet/infrared mixing, which pre…
We investigate the phase structure of non-commutative scalar field theories and find evidence for ordered phases which break translation invariance. A self-consistent one-loop analysis indicates that the transition into these ordered phases is first order. The phase structure and the existence of scaling limits provides an alternative to the structure of counter-terms in determining the renormalizability of non-commutative field theories. On the basis of the existence of a critical point in the closely related planar theory, we argue that there are renormalizable interacting non-commutative sc…
The $Ω$-background is defined for supergravity, and a very general class of such backgrounds is constructed for 11-dimensional supergravity. 11-dimensional supergravity in a certain $Ω$-background is shown to be equivalent to a 5-dimensional non-commutative gauge theory of Chern-Simons type. M2 and M5 branes are expressed as 1 and 2-dimensional extended objects in the 5-dimensional gauge theory. This 5-dimensional gauge theory is shown to admit a consistent quantization with two coupling constants, despite being formally non-renormalizable. A check of a twisted version of AdS/CFT is performed …
In this paper we investigate the Schwinger parametric representation for the Feynman amplitudes of the recently discovered renormalizable $ϕ^4_4$ quantum field theory on the Moyal non commutative ${\mathbb R^4}$ space. This representation involves new {\it hyperbolic} polynomials which are the non-commutative analogs of the usual "Kirchoff" or "Symanzik" polynomials of commutative field theory, but contain richer topological information.
We find transformation matrices allowing to express non-commutative three dimensional harmonic oscillator in terms of an isotropic commutative oscillator, following ``philosophy of simplicity'' approach. Non-commutative parameters have physical interpretation in terms of an external magnetic field. Furthermore, we show that for a particular choice of noncommutative parameters there is an equivalent anisotropic representation, whose transformation matrices are far more complicated. We indicate a way to obtain the more complex solutions from the simple ones.
We study the conditions for spontaneous symmetry breaking of the (2+1)-dimensional noncommutative phi^6 model in the small-theta limit. In this regime, considering the model as a cutoff theory, it is reasonable to assume translational invariance as a property of the vacuum state and study the conditions for spontaneous symmetry breaking by an effective potential analysis. An investigation of up to the two loop level reveals that noncommutative effects can modify drastically the shape of the effective potential. Under reasonable conditions, the nonplanar sector of the theory can become dominant…
We review the solutions of O(N) and U(N) quantum field theories in the large $N$ limit and as 1/N expansions, in the case of vector representations. Since invariant composite fields have small fluctuations for large $N$, the method relies on constructing effective field theories for composite fields after integration over the original degrees of freedom. We first solve a general scalar $U(\phib^2)$ field theory for $N$ large and discuss various non-perturbative physical issues such as critical behaviour. We show how large $N$ results can also be obtained from variational calculations.We illust…
The non-commutative Wess-Zumino model is used as a prototype for studying the low energy behaviour of a renormalizable non-commutative field theory. We start by deriving the potential mediating the fermion-fermion and boson-boson interactions in the non-relativistic regime. The quantum counterparts of these potentials are afflicted by irdering ambiguities but we show that there exists an ordering prescription which makes them hermitean. For space/space noncommutativity it turns out that Majorana fermions may be pictured as rods oriented perpendicularly to the direction of motion showing a lack…
We make here a short overview of the recent developments regarding translation-invariant models on the noncommutative Moyal space. A scalar model was first proposed and proved renormalizable. Its one-loop renormalization group flow and parametric representation were calculated. Furthermore, a mechanism to take its commutative limit was recently given. Finally, a proposition for a renormalizable, translation-invariant gauge model was made.
Non-commutative (NC) field theories can be mapped onto twisted matrix models. This mapping enables their Monte Carlo simulation, where the large N limit of the matrix models describes the continuum limit of NC field theory. First we present numeric results for 2d NC gauge theory of rank 1, which turns out to be renormalizable. The area law for the Wilson loop holds at small area, but at large area we observe a rotating phase, which corresponds to an Aharonov-Bohm effect. Next we investigate the NC phi^4 model in d=3 and explore its phase diagram. Our results agree with a conjecture by Gubser a…
We study the IR/UV connection of the four-dimensional non-commutative phi^4 theory by using the Wilsonian Renormalization Group equation. Extending the usual formulation to the non-commutative case we are able to prove UV renormalizability to all orders in perturbation theory. The full RG equations are finite in the IR, but perturbative approximations of them are plagued by IR divergences. The latter can be systematically resummed in a way analogous to what is done in finite temperature field theory. As an application, next-to-leading order corrections to the two-point function are explicitly …
In this contribution to the proceedings of the Corfu Summer Institute 2015, I give an overview over quantum field theories on non-commutative Moyal space and renormalization. In particular, I review the new features and challenges one faces when constructing various scalar, fermionic and gauge field theories on Moyal space, and especially how the UV/IR mixing problem was solved for certain models. Finally, I outline more recent progress in constructing a renormalizable gauge field model on non-commutative space, and how one might attempt to prove renormalizability of such a model using a gener…
We consider the boundary states which describe D-branes in a constant B-field background. We show that the two-form field Phi, which interpolates commutative and noncommutative descriptions of D-branes, can be interpreted as the invariant field strength in the T-dual picture. We also show that the extended algebra parametrized by theta and Phi naturally appears as the commutation relations of the original and the T-dual coordinates.
We elaborate in this paper a translation-invariant model for fermions in 4-dimensional noncommutative Euclidean space. The construction is done on the basis of the renormalizable noncommutative translation-invariant Phi4 theory introduced by R. Gurau et al. We combine our model with the scalar model, in order to study the noncommutative pseudo-scalar Yukawa theory. After we derive the Feynman rules of the theory, we perform an explicit calculation of the quantum corrections at one loop level to the propagators and vertices.
We reconsider aspects of non-commutative dipole deformations of field theories. Among our findings there are hints to new phases with spontaneous breaking of translation invariance (stripe phases), similar to what happens in Moyal-deformed field theories. Furthermore, using zeta-function regularization, we calculate quantum corrections to KK-state masses. The corrections coming from non-planar diagrams show interesting but non-universal behaviour. Depending on the type of interaction the corrections can make the KK-states very heavy but also very light or even tachyonic. Finally we point out t…
We consider quantum corrections to a kink of noncommutative supersymmetric phi^4 theory in 1+1 dimensions. Despite the presence of an infinite number of time derivatives in the action, we are able to define supercharges and a Hamiltonian by using an unconventional canonical formalism. We calculate the quantum energy E of the kink (defined as a half-sum of the eigenfrequencies of fluctuations) which coincides with its' value in corresponding commutative theory independently of the noncommutativity parameter. The renormalization also proceeds precisely as in the commutative case. The vacuum expe…
We compactify the spaces $K(m,n)$ introduced by Maxim Kontsevich. The initial idea was to construct an $L_\infty$ algebra governing the deformations of a (co)associative bialgebra. However, this compactification leads not to a resolution of the PROP of (co)associative bialgebras, but to a new algebraic structure we call here a CROC. It turns out that these constructions are related to the non-commutative deformations of (co)associative bialgebras. We construct an associative dg algebra conjecturally governing the non-commutative deformations of a bialgebra. Then, using the Quillen duality, we …
It has been conjectured in the literature that renormalizability of the $θ$-expanded noncommutative gauge theories improves when one takes into account full nonuniqueness of the Seiberg-Witten expansion, which relates noncommutative (`high-energy') with commutative (`low-energy') fields. In order to check this conjecture we analyze renormalizability of the noncommutative chiral electrodynamics: we quantize the action which contains all possible terms implied by the SW map. After renormalization we arrive at a different theory in which the relation between the coupling constants is changed. Thi…
