Beyond the Standard Model: A Noncommutative Approach
Christoph A. Stephan
备注：
To be published in the Proceedings of the XLIVth Rencontres de Moriond: Electroweak Interactions and Unified Theories (La Thuile, Italy, 7-14 March 2009)
During the last two decades Alain Connes developed Noncommutative Geometry (NCG), which allows to unify two of the basic theories of modern physics: General Relativity (GR) and the Standard Model (SM) of Particle Physics as classical field theories. In the noncommutative framework the Higgs boson, which had previously to be put in by hand, and many of the ad hoc features of the standard model appear in a natural way.
The aim of this presentation is to motivate this unification from basic physical principles and to give a flavour of its derivation. One basic prediction of the noncommutative approach to the SM is that the mass of the Higgs Boson should be of the order of 170 GeV if one assumes the Big Desert. This mass range is with reasonable probability excluded by the Tevatron and therefore it is interesting to investigate models beyond the SM that are compatible with NCG. Going beyond the SM is highly non-trivial within the NCG approach but possible extensions have been found and provide for phenomenologically interesting models. We will present in this article a short introduction into the NCG framework and describe one of these extensions of the SM. This model contains new scalar bosons (and fermions) which constitute a second Higgs-like sector mixing with theordinary Higgs sector and thus considerably modifying the mass eigenvalues.
Noncommutative geometry has become popular mathematics for describing speculative physics beyond the Standard Model. Noncommutative QED has long been known to fit within the framework of the Standard-Model Extension (SME). We argue in this work that noncommutative gravity also fits within the SME framework.
In this paper we study the noncommutative effects to the lepton spectrum from the decay of a polarized top quark. It is shown that the lowest contribution comes from the quadratic terms of the noncommutative parameter. The deviations from the standard model are significant for small values of the noncommutative characteristic scale. However, the charged lepton spin correlation coefficient has a remarkable deviation from the standard model from very low values of the noncommutative characteristic scale to 1 TeV.
We study the new structures appearing due to noncommutative effects in the inclusive decay b -> s γ^*, in the standard model. We present the corresponding coefficients which carries the space-space and space-time noncommutativity.
In this contribution we discuss the Noncommutative Standard Model and the associated Standard Model-forbidden decays that can possibly serve as an experimental signature of space-time noncommutativity.
We derive the most general Seiberg-Witten maps for noncommutative gauge theories in second order of the noncommutative parameter theta. Our results reveal the existence of more ambiguities than previously known. In particular, we demonstrate that some of these ambiguities enter observables like scattering cross sections and enlarge the parameter space of the noncommutative standard model beyond O(theta).
The $Z\rightarrow e^{+}e^{-}$ decay is studied basing on the noncommutative standard model (NCSM) with the hybrid gauge transformation. It is shown that if the latter is not included, the noncommutative correction to the amplitude of the $Z\rightarrow e^{+}e^{-}$ appears only as a phase factor, so that there is no new physical effect on the decay width. However, when the hybrid gauge transformation is included, the noncommutative effect appears in the two-body decay process. The discrepancy between the experimental branch ratio and the standard model prediction allows us to set the bound on th…
We consider the Higgs and Yukawa parts of the Non-Commutative Standard Model (NCSM). We explore the NC-action to give all Feynman rules for couplings of the Higgs boson to electro-weak gauge fields and fermions.
We study the top quark decay to b quark and W boson in the NonCommutative Standard Model (NCSM). The lowest contribution to the decay comes from the terms quadratic in the matrix describing the noncommutative (NC) effects while the linear term is seen to identically vanish because of symmetry. The NC effects are found to be significant only for low values of the NC characteristic scale.
Extensions of the Standard Model of elementary particle physics to noncommutative geometries are proposed by string models. Independent of this motivation, one may consider such a model as an effective field theory with higher-dimensional operators containing an antisymmetric rank-2 background field. We study the signals of such a Noncommutative Standard Model (NCSM) and analyze the discovery potential of fermion pair production at a future photon collider.
In this article we aim to estimate the bounds on the noncommutative scale $Λ_{NC}$ and to extract the 95% exclusion contours for some $θ_{μν}$ components using the recent measurements of the top quark width and the $W$ boson polarization in top pair events from CDF experiment at Tevatron.
We study phenomenological consequences of a noncommutative extension of the standard model in the theta-expanded approach at the ILC. We estimate the sensitivity of the ILC for the noncommutative scale Lambda_NC. Comparing with earlier estimates for the LHC, we demonstrate the complementarity of the experiments at the two colliders.
We study Z--> l^+ l^- and W--> nu_l l^+ decays in the standard model including the noncommutative effects. We observe that these effects appear in the flavor dependent part of the decay widths of the processes under consideration and therefore, they are more effective for the heavy lepton decays.
We study the Möller and Bhabha scattering in the noncommutative extension of the standard model(SM) using the Seiberg-Witten maps of this to first order of the noncommutative parameter $θ_{μν}$. We look at the angular distribution $dσ/dΩ$ to explore the noncommutativity of space-time at around $Λ_{NC} \sim$ TeV and find that the distribution deviates significantly from the one obtained from the commutative version of the standard model.
We study the dipole moments, electric dipole moment, weak electric dipole moment, anomalous magnetic moment, anomalous weak magnetic moment, of fermions in the noncommutative extension of the SM. We observe that the noncommutative effects are among the possible candidates to explain the electric and weak electric dipole moment of fermions. Furthermore, the upper bounds for the parameters which carry space-time and space-space noncommutativity can be obtained by using the theoretical and experimental results of the fermion dipole moments.
We study the form factors appearing in the inclusive decay b -> s g^*, in the framework of the noncommutative standard model. Here g^* denotes the virtual gluon. We get additional structures and the corresponding form factors in the noncommutative geometry. We analyse the dependencies of the form factors to the parameter pΘk where p (k) are the four momenta of incoming (outgoing) b quark (virtual gluon g^*, Θis a parameter which measures the noncommutativity of the geometry. We see that the form factors are weakly
Extensions of the Standard Model of elementary particle physics to noncommutative geometries have been proposed as a low energy limit of string models. Independent of this motivation, one may consider such a model as an effective field theory with higher-dimensional operators containing an antisymmetric rank-two background field. We study the signals of such a Noncommutative Standard Model (NCSM) and analyze the discovery potential of a future photon collider, considering angular distributions in fermion pair production.
We study the associated Higgs production with Z boson at future linear colliders in the framework of the minimal noncommutative standard model. Using the Seiberg-Witten map, we calculate the production cross-section considering all orders of the noncommutative parameter $Θ_{μν}$. We consider the effect of earth's rotation on the orientation of $Θ_{μν}$ with respect to the laboratory frame and thus on the total cross-section, it's azimuthal distribution and rapidity distribution for the machine energy ranging from 0.5 TeV to 3.0 TeV corresponding to the noncommutative scale $Λ\ge $ 0.5 TeV.
Top quark decays before hadronization and its information transfer to final products. The top quark decays via the weak interaction, hence the azimuthal polarized decay rate is zero in the leading order of SM. Therefore the study of top quark could be taken into consideration as useful tools to search for new physics. In this paper, we calculate the decay of polarized top quark in the framework of the noncommutative standard model. We investigate the contribution of noncommutative space-time on the azimuthal polarized decay rate and spin analyzing power. Also we depict the energy spectrum for …
We study the pair production of charged gauge bosons at the LHC in a noncommutative extension of the standard model. We use angular distributions in the decays of the gauge bosons to partially reconstruct polarized cross sections. We use this, together with CP considerations, to construct more sensitive observables that allow to separate space-time from space-space noncommutativities.
We examine W pair production in the Noncommutative Standard Model constructed with the Seiberg-Witten map. Consideration of partial wave unitarity in the reactions WW to WW and e+e- to WW shows that the latter process is more sensitive and that tree-level unitarity is violated when scattering energies are of order a TeV and the noncommutative scale is below about a TeV. We find that WW production at the LHC is not sensitive to scales above the unitarity bounds. WW production in e+e- annihilation, however, provides a good probe of such effects with noncommutative scales below 300-400 GeV being …
We study $γγ$ scattering in noncommutative QED (NCQED) where the gauge field has Yang-Mills type coupling, giving new contributions to the scattering process and making it possible for it to occur at tree level. The process takes place at one loop level in the Standard Model (SM) and could be an important signal for physics beyond SM. But it is found that the Standard Model contribution far exceeds the tree level contribution of the noncommutative case.
We study collider signals for the noncommutative extension of the standard model using the Seiberg-Witten maps for SU(3)_C x SU(2)_L x U(1)_Y to first order in the noncommutativity parameters theta_munu. In particular, we investigate the ensitivity of Z-gamma-production at the Tevatron and the LHC to the components of theta_munu. We discuss the range of validity of this approximation and estimate exclusion limits from a Monte Carlo simulation.
Gauge unification is widely considered to be a desirable feature for extensions of the standard model. Unfortunately the standard model itself does not exhibit a unification of its running gauge couplings but it is required by grand unified theories as well as the noncommutative version of the standard model [2]. We will consider here the extension of the noncommutative standard model by vector doublets as proposed in [6]. Two consequences of this modification are: 1. the relations of the coupling constants at unification energy are altered with respect to the well known relation from grand un…
We study the Higgs boson pair production through $e^+e^-$ collision in the noncommutative(NC) extension of the standard model using the Seiberg-Witten maps of this to the first order of the noncommutative parameter $Θ_{μν}$. This process is forbidden in the standard model with background space-time being commutative. We find that the cross section of the pair production of Higgs boson (of intermediate and heavy mass) at the future Linear Collider(LC) can be quite significant for the NC scale $Λ$ lying in the range $0.5 - 1.0$ TeV. Finally, using the direct experimental(LEP II, Tevatron and glo…
We study muon pair production $ e^+ e^- \to μ^+ μ^-$ in the noncommutative(NC) extension of the standard model using the Seiberg-Witten maps of this to the second order of the noncommutative parameter $Θ_{μν}$. Using $\mathcal{O}(Θ^2)$ Feynman rules, we find the $\mathcal{O}(Θ^4)$ cross section(with all other lower order contributions simply cancelled) for the pair production. The momentum dependent $\mathcal{O}(Θ^2)$ NC interaction significantly modifies the cross section and angular distributions which are different from the commuting standard model. We study the collider signatures of the s…
The noncommutative standard model apparently violates the Lorentz symmetry. We compare the Lorentz violating terms in the Higgs sector of the noncommutative standard model with their counterparts in the standard model extension. We show that the Lorentz violating parameters in the Higgs sector can be expressed directly in terms of the noncommutative parameter without any background field. The absence of the background field enhances the obtained bounds on the noncommutative parameter from the standard model extension.
We study the muon pair production $ e^+ e^- \to μ^+ μ^-$ in the framework of the non-minimal noncommutative(NC) standard model to the second order of the NC parameter $Θ_{μν}$. The $\mathcal{O}(Θ^2)$ momentum dependent NC interaction significantly modifies the cross section and angular distributions which are different from the standard model. After including the effects of earth's rotation we analyse the time-averaged and time dependent observables in detail. The time-averaged azimuthal distribution of the cross section shows siginificant departure from the standard model. We find strong depe…
The triple neutral gauge boson and direct U(1)_Y-neutrino interactions, being absent in ordinary field theory, can arise quite naturally in noncommutative gauge field theories. Using non-perturbative methods and a Seiberg-Witten map based covariant approach to noncommutative gauge theory, we have found theta-exact expressions for both interactions, thereby eliminating previous restrictions to low-energy phenomena. In particular we obtain for the first time the covariant, theta-exact, triple neutral gauge boson interactions within the noncommutative Standard Model gauge sector including an addi…
The hybrid gauge transformation and its nontrivial phenomenological implications are investigated using the noncommutative gauge theory with the Seiberg-Witten map expanded scenario. Particularly, the $e^+e^- \toμ^+ μ^-$ process is studied with a generalized noncommutative standard model (NCSM) including massive neutrinos and neutrino-photon interaction. In this model, the hybrid gauge transformation in the lepton sector is naturally introduced through the requirement of gauge invariance of the seesaw neutrino mass term. It is shown that in the NCSM without hybrid gauge transformation the nonc…
It has been shown recently that the triple neutral gauge boson couplings are not uniquely determined in noncommutative extension of the Standard Model (NCSM). Depending on specific schemes used, the couplings are different and may even be zero. To distinguish different realizations of the NCSM, additional information either from theoretical or experimental considerations is needed. In this paper we show that these couplings can be uniquely determined from considerations of unification of electroweak and strong interactions. Using SU(5) as the underlying theory and integrating out the heavy deg…
Top-quark pair production in electron-positron collisions within the minimal noncommutative Standard Model
Université de Batna-1（巴特纳第一大学）；Laboratoire de Physique des Rayonnements et de leurs Interactions avec la Matière（射线及其与物质相互作用物理实验室）；Département de Physique, Faculté des Sciences de la Matière（物质科学学院物理系）
We consider Lorentz conserving noncommutative field theory to construct the Lorentz conserving noncommutative standard model based on the gauge group $SU(3)\times SU(2)\times U(1)$. We obtain the enveloping algebra-valued of Higgs field up to the second order of the noncommutativity parameter $þ_{μν}$. We derive the action at the leading order and find new vertices which are absent in the ordinary Standard Model as well as the minimal noncommutative standard model. We briefly study the phenomenological aspects of the model.
On noncommutative spacetime, the Standard Model allows new, usually Standard Model forbidden, triple gauge boson interactions. In this letter we propose the Standard Model strictly forbidden Z -> gamma gamma and Z -> gg decay modes coming from the gauge sector of the Noncommutative Standard Model as a place where noncommutativity could be experimentally discovered.
I try to assess the weak and strong points of the standard model of electro-magnetic, weak and strong forces, how it can be derived from general relativity by generalizing Riemannian to noncommutative geometry and what post- and predictions this unification of all four forces entails in particle physics.
Restrictions imposed by gauge invariance in noncommutative spaces together with the effects of ultraviolet/infrared mixing lead to strong constraints on possible candidates for a noncommutative extension of the Standard Model. We study a general class of noncommutative models consistent with these restrictions. Specifically we consider models based upon a gauge theory with the gauge group U(N_1)\times U(N_2)\times ...\times U(N_m) coupled to matter fields transforming in the (anti)-fundamental, bi-fundamental and adjoint representations. We pay particular attention to overall trace-U(1) factor…
We present a noncommutative gauge theory that has the ordinary Standard Model as its low-energy limit. The model is based on the gauge group U(4) x U(3) x U(2) and is constructed to satisfy the key requirements imposed by noncommutativity: the UV/IR mixing effects, restrictions on representations and charges of matter fields, and the cancellation of noncommutative gauge anomalies. At energies well below the noncommutative mass scale our model flows to the commutative Standard Model plus additional free U(1) degrees of freedom which are decoupled due to the UV/IR mixing. Our model also predicts…
We examine the electroweak gauge sector of noncommutative standard model and in particular, obtain the $\mathcal{O} \,(θ)$ Feynman rules for all quadrilinear gauge boson couplings. Surprisingly, an electroweak-chromodynamics mixing appears in the gauge sector of noncommutative standard model, where photon as well as the neutral weak boson are coupled directly to three gluons. The phenomenological perspectives of the model in $W^-W^+ \rightarrow Z Z$ scattering are studied and it is shown that there is a characteristic oscillatory behaviour in azimuthal distribution of scattering cross sections…
In noncommutative geometry (NCG) the spectral action principle predicts the standard model (SM) particle masses by constraining the scalar and Yukawa couplings at some heavy scale, but gives an inconsistent value for the Higgs mass. Nevertheless, the scalar sector in the NCG approach to the standard model, is in general composed of two Higgs doublets and its phenomenology remains unexplored. In this work, we present a type-II two-Higgs-doublet model in NCG, with a SM-like Higgs mass compatible with the 125~GeV experimental value and extra scalars within the alignment limit without decoupling w…
We study the neutral Higgs boson pair production through $e^{+} e^{-}$ collision in the noncommutative(NC) extension of the standard model using the Seiberg-Witten maps of this to the first order of the noncommutative parameter $Θ_{μν}$. This process is forbidden in the standard model at the tree level with background space-time being commutative. After including the effects of earth's rotation we analyse the time-averaged cross section of the pair production of Higgs boson (in the light of LEP II and LHC data) at the future Linear Collider which can be quite significant for the NC scale $Λ$ l…
A noncommutative version of the usual electro-weak theory is constructed. We discuss how to overcome the two major problems: 1) although we can have noncommutative U(n) (which we denote by $U_{\star}(n)$) gauge theory we cannot have noncommutative SU(n) and 2) the charges in noncommutative QED are quantized to just $0, \pm 1$. We show how the problem with charge quantization, as well as with the gauge group, can be resolved by taking $U_{\star}(3)\times U_{\star}(2)\times U_{\star}(1)$ gauge group and reducing the extra U(1) factors in an appropriate way. Then we proceed with building the nonc…
In this work we propose the Z -> gamma gamma decay as a process strictly forbidden in the standard model, suitable for the search of noncommutativity of coordinates at very short distances. We computed the Z -> gamma gamma partial width in the framework of the recently proposed one-loop renormalizable gauge sector of the noncommutative standard model. New experimental possibilities at LHC are analyzed and a firm bound to the scale of noncommutativity parameter is set around 1 TeV.
The higgsstrahlung process $e^+e^-\to Z H$ and pair production process $e^+e^- \to H H$ are studied in the framework of the minimal noncommutative (NC) standard model. In particular, the Feynman rules involving all orders of the noncommutative parameter $θ$ are derived using reclusive formation of Seiberg-Witten map. It is shown that the total cross section and angular distribution can be significantly affected because of spacetime noncommutativity when the collision energy exceeds to 1 \tev. It is found that in each process, there is an optimal collision energy ($E_{oc}$) for achieving the gr…
Noncommutative geometry allows to unify the basic building blocks of particle physics, Yang-Mills-Higgs theory and General relativity, into a single geometrical framework. The resulting effective theory constrains the couplings of the particle model and reduces the number of degrees of freedom. After briefly introducing the basic ideas of noncommutative geometry, I will present its predictions for the Standard Model and the few known models beyond the Standard Model. Most of these models, including the Standard Model, are now ruled out by LHC data. But interesting extensions of the Standard Mo…
We study the Drell-Yan process at the Large Hadron Collider in presence of the noncommutative extension of standard model. Using the Seiberg-Witten map, we calculate the production cross-section to the first order in the noncommutative parameter $Θ_{μν}$. Although this idea is evolving for long, limited amount of phenomenological analysis was done so far and dominantly in the context of linear collider. Some outstanding feature from this non-minimal noncommutative standard model not only modify the couplings over SM production channel, also allow additional nonstandard vertices which can play …
Noncommutative (NC) space-time leads to some strong constraints on the possible choices of gauge groups and allowed representations of matter and gauge fields. The standard model based on $U(3)\times U(2)\times U(1)$ can be transcribed to NC space-time consistently with these constraints \cite{nNCSM}. In fact, through two new symmetry reduction mechanisms, this gauge group is reduced to the usual standard model one. In this paper, we show that, on the contrary to the usual standard model, the Goldestone boson equivalence theorem is violated due to the incompatibility of the new symmetry reduct…
We review the approach to the standard model of particle interactions based on spectral noncommutative geometry. The paper is (nearly) self-contained and presents both the mathematical and phenomenological aspects. In particular the bosonic spectral action and the fermionic action are discussed in detail, and how they lead to phenomenology. We also discuss the Euclidean vs. Lorentz issues and how to go beyond the standard model in this framework.
We consider the noncommutative Standard Model that contains Lorentz symmetry violation as a subset of the Standard Model extension. We introduce a constant electromagnetic field as a background to derive mutual relations between the free parameters of both theories. As the Lorentz violation parameters of the Standard Model extension are extensively explored in different experiments and many stringent bounds on these parameters are available, we can find new bounds on the scale of noncommutativity of the order of a few to tens of teraelectron volts.
We study cosmological consequences of the noncommutative approach to the standard model. Neglecting the nonminimal coupling of the Higgs field to the curvature, noncommutative corrections to Einstein's equations are present only for inhomogeneous and anisotropic space-times. Considering the nominimal coupling however, we obtain corrections even for background cosmologies. A link with dilatonic gravity as well as chameleon cosmology are briefly discussed, and potential experimental consequences are mentioned.
We compute the beta-functions of the standard model formulated on a noncommutative spacetime. If we assume that the scale for spacetime noncommutativity is of the order of 2.2 \times 10^{15} GeV we find that the three gauge couplings of the standard model merge at a scale of 2.3 \times 10^{17} GeV. The proton lifetime is thus much longer than in conventional unification models.
The recent introduction of a deformed non-minimal version of the noncommutative Standard Model in the enveloping-algebra approach, having a one-loop renormalisable gauge sector involving a higher order gauge term, motivates us to consider the possibility of extending the fermion sector with additional deformations, i.e. higher order fermionic terms. Since the renormalisability properties of the fermion sector of the model are not yet fully known, we work with an effective fermion lagrangian which includes noncommutative higher order terms involving a contraction with the noncommutative $θ$ ten…
Restrictions imposed by gauge invariance in noncommutative spaces together with the effects of ultraviolet/infrared mixing lead to strong constraints on possible candidates for a noncommutative extension of the Standard Model. In this paper, we study a general class of 4-dimensional noncommutative models consistent with these restrictions. Specifically we consider models based upon a gauge theory with the gauge group U(N_1)\times U(N_2) \times ... \times U(N_m) coupled to matter fields transforming in the (anti)-fundamental, bi-fundamental and adjoint representations. Noncommutativity is intro…
We consider a complex singlet scalar in the spectral action approach to the standard model. It is shown that there is a range of initial values at the unification scale which is able to produce Higgs and top quark masses at low energies. The stability of the vacuum and the deviation of gauge couplings from experimental values are discussed and compared at the two-loop level with a real scalar singlet and the pure standard model.
In this paper, we derive the standard model with classical conformal invariance from the Yang--Mills--Higgs model in noncommutative geometry (NCG). In the ordinary context of the NCG, the {\it distance matrix} $M_{nm}$ which corresponds to the vacuum expectation value of Higgs fields is taken to be finite. However, since $M_{nm}$ is arbitrary in this formulation, we can take all $M_{nm}$ to be zero. In the original composite scheme, the Higgs field itself vanishes with the condition $M_{nm} = 0$. Then, we adopt the elemental scheme, in which the gauge and the Higgs bosons are regarded as eleme…
We explore the 1-loop renormalization group flow of two models coming from a generalization of the Connes-Lott version of Noncommutative Geometry in Lorentzian signature: the Noncommutative Standard Model and its B-L extension. Both make predictions on coupling constants at high energy, but only the latter is found to be compatible with the top quark and Higgs boson masses at the electroweak scale. We took into account corrections introduced by threshold effects and the relative positions of the Dirac and Majorana neutrino mass matrices and found them to be important. Some effects of 2-loop co…
In the standard model production of on-shell Z boson at a photon collider (or Z decays into $γγ$) is strictly forbidden by angular momentum conservation and Bose statistics (the Yang's Theorem). In the standard model with noncommutative space-time this process can occur. Therefore this process provides an important probe for the noncommutative space-time. The $γγ$ collision at the ILC by laser backscattering of the electron and positron beams offers an ideal place to carry out such a study. Assuming an integrated luminosity of 500 fb$^{-1}$, we show that the constraint which can be achieved on…
In this publication we present an extension of the Standard Model within the framework of Connes' noncommutative geometry [1]. The model presented here is based on a minimal spectral triple [7] which contains the Standard Model particles, new vectorlike fermions and a new U(1) gauge subgroup. Additionally a new complex scalar field appears that couples to the right-handed neutrino, the new fermions and the standard Higgs particle. The bosonic part of the action is given by the spectral action [1] which also determines relations among the gauge couplings, the quartic scalar couplings and the Yu…
We consider the noncommutative standard model based on $SU(3)\times SU(2)\times U(1)$. We study the gauge transformation of right handed neutrino and its direct interaction with photon in the noncommutative space-time. We show that the massive Dirac neutrinos, through the Higgs mechanism, can not accommodate this extension of the standard model; while the massive Majorana neutrinos are consistent with the gauge symmetry of the model. The electromagnetic properties and the dispersion relations for the neutrino in the noncommutative standard model is examined. We also compare the results with th…
