### A not so trivial pursuit : explaining how elementary scalar particles can exist

**From past ...**

“I would like to offer a theoretical prediction at the 5% confidence level: within five years there will be a rigorous construction of the solutions ofλφ^{4}and spin- ~ quantum electrodynamics in four-dimensional space-time.”Arthur S. Wightman (1977)“When we try to pick out anything by itself, we find it hitched to everything else in the universe.”John Muir

Expectations often remain unfulfilled. Much of the early excitement over the standard model of the weak interactions was caused by the discovery that it was renormalizable, and hence could be considered a viable candidate for a fundamental theory of elementary particles. The standard model was a remarkable achievement, for it included the lucus a non lucendo of massive gauge bosons in a consistent fashion and so afforded a place for the mediators of the weak force, the W± and Z°. The standard model also posits the existence of an unobserved elementary scalar particle, the Higgs boson. The interactions of this Higgs particle are a necessary ingredient in the standard model.

As discussed in sections 2—4 of this review,strong evidence suggests that a theory which only includes Higgs particles is “trivial” or noninteracting. Should this triviality persist when the Higgs is “hitched to everything else” in the standard model, then the standard model scenario is in trouble. It would then be fair to ask if a real “Higgs particle” should exist, or whether this putative elementary scalar is instead just an invisible metaphor for a more complicated mechanism. The obvious question to ask—what this new mechanism may be—is not as obviously answerable.

A more informative set of possibilities does, however, exist if the Higgs sector of the standard model (or indeed of any realistic Higgs model) is nontrivial. In this case it is likely possible to calculate upper bounds upon the Higgs mass or even to predict its value. The requirement that the theory be nontrivial thus can imply phenomenological constraints on the theory (see section 5). By contrast, a naive semiclassical analysis of the scalar sector of the standard model does not yield any information on the Higgs mass.

It is evident that the question of the triviality of Higgs models is of more than philosophical interest. The inquiry as to whether an elementary scalar particle exists (and if its mass is predictable) is also given immediacy by the contemporary agenda for the construction of new accelerators. The intended purpose of these machines typically includes a search for the Higgs particle. Fortunately for these experimental efforts, progress is being made in the understanding of triviality and its implications for elementary particle phenomenology. Much remains to be done, however. Although techniques such as the analysis of lattice gauge theories by the Monte Carlo renormalization group may well hold the key to the riddle of triviality, the final answers are not yet known. This report therefore concludes with the same question as it started with: Can elementary scalar particles exist? The future holds the answer.

David J.E. CALLAWAY (Received February 1988)

**... to present**

Perturbatively renormalised quantum ﬁeld theory is an enormous phenomenological success, a success which lacks a mathematical understanding.The perturbation series is at best an asymptotic expansion which cannot converge at physical coupling constants. Some physical effects such as conﬁnement are out of reach for perturbation theory.

In two and partly three dimensions, methods of constructive physics [GJ87, Riv91], often combined with the Euclidean approach [Sch59, OS73, OS75], were used to rigorously establish quantum ﬁeld theory models. In four dimensions there was little success so far. It is generally believed that due to asymptotic freedom, non-Abelian gauge theory (i.e. Yang-Mills theory) has the chance of a rigorous construction. But this is a hard problem [JW00].What makes it so difficult is the fact that any simpler model such as quantum electrodynamics or the λφ^{4}-model cannot be constructed in four dimensions(Landau ghost problem [LAK54...] or triviality [Aiz81, Frö82]).

One of the main difficulties is the non-linearity of the models under consideration. Fixed point methods provide a standard approach to non-linear problems, but they are rarely used in quantum ﬁeld theory.In this contribution we review a sequence of papers [GW12b, GW13b, GW14] in which we successfully used symmetry and ﬁxed point methods to exactly solve a toy model for a quantum ﬁeld theory in four dimensions...

Taking the renormalisation group [WK74] serious, we would expect that General Relativity, because not renormalisable, is irrelevant and hence scaled away. The existence of gravity thus tells us that the scaling must stop at some length scale, and from the weakness of the gravitational coupling constant one deduces the value of that scale: the Planck length 10−35 m. There, the geometry of nature is expected to differ from the familiar structure of a differentiable manifold. One of many candidates for Planck scale physics is noncommutative geometry[Con94], a vast reformulation of geometry and topology in the language of operator algebras. The focus is shifted from manifolds to generalisations of the algebra of functions.This concept proved very successful in understanding the geometry of the Standard Model of particle physics as Riemannian geometry of a space which is the product of a manifold with a discrete space[Con96, CC96]. A large class of examples of noncommutative geometries comes from deformationsof the algebra of functions on manifolds. Schwartz functions on Euclideanspace R4 admit an R4-group action by translation. As shown by Rieffel [Rie93],this group action induces a noncommutative associative product on the space ofSchwartz functions, the Moyal product ... Whether or not the Moyal spaceis relevant for Planck scale physicsis pure speculation (although a refinement can be justified by uncertainty relationsfor position operators [DFR95]).In any case the Moyal space is a nice toymodel on which it is easy to formulate and to study (quantum) field theories...

We have shown that the φ.^{4}_{4}-model on noncommutative Moyal space, considered in the limit θ →∞ of extreme noncommutativity, is an exactly solvable and non-trivial matrix modelEuclidean symmetry is violated in the beginning, but we identiﬁed a limit which projects to diagonal matrices where Euclidean symmetry is restored. One would not expect that such a brutal projection can respect any quantum ﬁeld theory axioms. Surprisingly, the ﬁrst consistency checks, positivity of the lowest Widder criteria Lk,t[G••], are passed for the only interesting interval [λc,0] of the coupling constant!...

Harald Grosse (Vienna), Raimar Wulkenhaar (Münster)(Submitted on 5 Feb 2014)

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