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LEP established the Standard Model as a renormalizable quantum field theory with unprecedented precision. I take a personal and incomplete look back to that time and the impact Tini Veltman had on this exciting endeavour.

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I will recall three results of Martinus J.G. Veltman, which had a substantial importance in my scientific activity and which gave me the occasion of meeting him and of appreciating the great human and scientific gifts of his unforgettable personality.

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Dedicated to the memory of Professor Martinus Veltman, one of the founding fathers of our discipline: his legacy lives on. Many times we have to turn back and follow his footprints to find the right path. After reviewing general aspects of high-energy physics where he gave a seminal contribution we will introduce recent developments in the Standard Model Effective Field Theory, showing how the whole movement from renormalization to predictions plays from Veltman to SMEFT.

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This paper describes my personal appreciation of some of Tini Veltman’s great research achievements and how my own research career has followed the pathways he opened. Among the topics where he has been the most influential have been the pursuit and study of the Higgs boson and the calculation of radiative corrections that enabled the masses of the top quark and the Higgs boson to be predicted ahead of their discoveries. The search for physics beyond the Standard Model may require a complementary approach, such as the search for non-renormalizable interactions via the Standard Model Effective Field Theory.

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I am considering Veltman’s “The Infrared–Ultraviolet Connection” addressing the issue of quadratic divergences and the related huge radiative correction predicted by the electroweak Standard Model (SM) in the relationship between the bare and the renormalized theory, commonly called “the hierarchy problem” which usually is claimed that this has to be cured. After the discovery of the Higgs particle at CERN, which essentially completed the SM, an amazing interrelation of the leading interaction strengths of the gauge bosons, the top quark, and the Higgs boson showed up amounting that the SM allows for a perturbative extrapolation of the running couplings up to the Planck scale. The central question concerns the stability of the electroweak vacuum, which requires that the running Higgs self-coupling stays positive. Although several evaluations seem to favor the meta-stability within the experimental and theoretical parameter-uncertainties, one should not exclude the possibility that other experiments and improved matching conditions will be able to establish the absolute stability of the SM vacuum in the future. I will discuss the stable vacuum scenario and its impact on early cosmology, revealing the Higgs boson as the inflaton. It turns out that the Standard Model’s presumed “hierarchy problem” and similarly the “cosmological constant problem” resolve themselves when we understand the SM as a low-energy effective tail that is emergent from a cutoff-medium at the Planck scale. “The Infrared–Ultraviolet Connection” conveyed by the Higgs boson mass renormalization appears in a new light when the energy dependence of the SM couplings is taken into account. The bare Higgs boson mass square then changes sign below the Planck scale where it is activating the Higgs mechanism. At the same time, it reveals that the SM towards the Planck scale is in the symmetric phase, where the Higgs potential provides a high dark energy density triggering inflation, and four heavy Higgs bosons which decay and thereby are reheating the inflated early universe.

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Martinus (“Tini”) Veltman’s early contributions to the Standard Model were essential for its success. After some nostalgic reminiscences, I turn to the Standard Model with a minimalistic attitude, the point of view that beyond the SM there is only the Planck scale. Known since long, the gravitational force can be obtained as the gauge theory of local Poincaré symmetry, called gauge gravity. This gauge theory of gravity embodies per se a Palatini formulation. This causes the potential of non-minimally coupled Higgs inflation to have an intriguing improved large field behaviour. Some of its effects are experimentally accessible or refutable. The question of quantum corrections is discussed.

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We describe the non-minimal Standard Model, consisting of minimalistic extensions of the Standard Model, which for all we know is the theory of the universe, able to describe all of the universe from the beginning of time. Extensions discussed are an extra neutrino and a new Higgs model. We introduce the principle of global relativity and discuss how the theory can be largely derived from this principle. One is led to the unification of forces into SU(5) and a form of dark matter. We discuss the limitations of the theory, showing that it is not the theory of everything. However, we argue that it is the only part that is within conceivable reach of physical experiment or astronomical observation. It is argued that at the Planck scale the universe is effectively three-dimensional.

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Martinus Veltman was the first to point out the inconsistency of the experimental value for the decay rate of \(\pi ^0\rightarrow \gamma \gamma \) and its calculation by J. Steinberger with the very successful concept of the pion as the (pseudo) Nambu–Goldstone boson of the spontaneously broken global axial symmetry of strong interactions. That inconsistency has been resolved by J. Bell and R. Jackiw in their famous paper on the chiral anomalies. We review the connection between the decay amplitudes of an axion into two gauge bosons in Abelian vector-like and chiral gauge theories. The axion is the Nambu–Goldstone boson of a spontaneously broken axial global symmetry of the theory. Similarly as for the vector-like gauge theory, also in the chiral one, the axion decay amplitude is uniquely determined by the anomaly of the current of that global symmetry. Certain subtlety in the calculation of the anomaly in chiral gauge theories is emphasised.

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First, we review Veltman’s suggestion to attack the naturalness problem in the Standard Model by requiring the absence of quadratic divergences and the resulting mass formula. Then, we emphasise the influence of Veltman’s suggestion in strengthening the belief that supersymmetry is the natural playground for solving the problem of quadratic divergences. Going further, we recall few sporadic suggestions concerning the cancellation of the logarithmic divergences too, which in the framework of supersymmetry has led to the construction of all-loop Finite Theories with the use of the idea of reduction of couplings. Eventually, we concentrate on a specific Finite Unified Theory and its successful predictions for the top and Higgs mass, among others, and the prospects of its final justification in future collider searches.

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Continuum spectra can be a way out to alleviate the tension generated by the elusiveness of narrow resonances of new physics in direct experimental searches. Motivated by the latter, we consider the linear dilaton model with a continuum spectrum of KK modes. It is provided by a critical exponential bulk potential for the scalar field stabilizing the distance, between the UV boundary at \(y=0\) and a naked (good) singularity at \(y=y_{\mathrm {s}}\), in proper coordinates, which corresponds in conformal coordinates to \(z_{\mathrm {s}}\to \infty \). The cutoff \(M_{\mathrm {s}}\) in this theory is an intermediate scale \(M_{\mathrm {s}}\simeq 10^{-5}M_{\mathrm {Pl}}\) and the warped factor solves the hierarchy between \(M_{\mathrm {s}}\) and the TeV, while the hierarchy between \(M_{\mathrm {Pl}}\) and \(M_{\mathrm {s}}\) has to be solved by a (Little) String Theory with coupling \(g_{\mathrm {s}}\simeq 10^{-5}\). The Standard Model is localized on a 4D IR brane. The graviton and radion Green’s and spectral functions have a continuum of states with a TeV mass gap, and isolated poles consisting of the 4D graviton and the light radion/dilaton. We construct the effective field theory below the mass gap where the continua of KK modes are integrated out, generating a set of dimension eight operators which contribute to low-energy electroweak precision observables, and high-energy violation of unitarity in vector boson scattering processes. The radion mass depends on the stabilizing UV brane potential and its wave function is localized toward the IR which enhances its coupling with the SM fields.

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Gauge theories have become the universal language of fundamental interactions. To this discovery, Martinus J.G. Veltman played a major role. In this short note, dedicated to his memory, we try to understand some of their geometrical properties. We show that a \(d\)-dimensional SU\((N)\) Yang–Mills theory can be formulated on a (\(d+2\))-dimensional space, with the extra two dimensions forming a surface with non-commutative geometry. The non-commutativity parameter is proportional to \(1/N\) and the equivalence is valid to any order in \(1/N\). We study explicitly the case of the sphere and the torus.

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In Veltman’s original view, the Higgs particle mass \(M_h\) was naturally large thus marking a second threshold in weak interactions, as with the \(W\) mass and the non-renormalizable 4-fermion \(\mathrm {V} - \mathrm {A}\) theory. Besides, with a large \(M_h\), Spontaneous Symmetry Breaking (SSB) would essentially be determined by the pure scalar sector regardless of the other sectors of the theory. Surprisingly, this picture is not completely ruled out. In fact, if SSB in \({\mit \Phi }^4\) theories is a weak first-order phase transition, as indicated by most recent lattice simulations, in addition to the known resonance with mass \(m_h\sim 125\) GeV, there might be a heavier excitation with mass \(M^2_h\sim m^2_h \ln ({\mit \Lambda }_{\mathrm {s}}/M_h)\), where \({\mit \Lambda }_{\mathrm {s}}\) is the ultraviolet cutoff of the scalar sector. The larger \(M_h\) controls vacuum stability and, differently from \(m_h\), would remain finite in units of the weak scale \(\langle {\mit \Phi } \rangle \sim 246.2\) GeV when \({\mit \Lambda }_{\mathrm {s}} \to \infty \). Lattice simulations of the propagator performed in the 4D Ising limit of the theory are consistent with this two-mass structure and lead to the estimate \(M_h\sim 700\) GeV. In spite of its large mass, however, the heavier state would couple to longitudinal vector bosons with the same typical strength of the low-mass state and would thus represent a relatively narrow resonance. In this way, such hypothetical resonance would naturally fit with some excess of 4-lepton events observed by ATLAS around 680 GeV. Analogous data from CMS are needed to confirm or disprove this interpretation. Finally, the effect of a two-mass structure on radiative corrections is discussed in connection with the value of \(\alpha _{\mathrm {s}}(M_Z)\) from \(e^+e^- \to {\mathrm {hadrons}}\).

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Based on a view of current flavour physics and motivated by the hierarchy problem and by the pattern of quark masses and mixings, I describe a picture of flavour physics that should give rise in a not too distant future to observable deviations from the SM in Higgs compositeness and/or in \(B\)-decays with violations of lepton flavour universality, as hinted by current data, or perhaps even in supersymmetry, depending on the specific realisation.

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We calculate the probability amplitude for tree-level elastic electron–muon scattering in Minkowski spacetime with carefully prepared initial and final wave packets. The obtained nonzero amplitude implies a nonvanishing probability for detecting a recoil electron outside the light cone of the initial muon. Transposing this Minkowski-spacetime scattering result to a near-horizon spacetime region of a massive Schwarzschild black hole and referring to a previously proposed Gedankenexperiment, we conclude that, in principle, it is possible to have information transfer from inside the black-hole horizon to outside.

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Recently it was shown that the scaling dimension of the operator \(\phi ^n\) in \(\lambda (\bar \phi \phi )^2\) theory may be computed semiclassically at the Wilson–Fisher fixed point in \(d=4-\epsilon \), for generic values of \(\lambda n\), and this was verified to two-loop order in perturbation theory at leading and subleading \(n\). In subsequent work, this result was generalised to operators of fixed charge \(\bar Q\) in O\((N)\) theory and verified up to three loops in perturbation theory at leading and subleading \(\bar Q\). Here, we extend this verification to four loops in O\((N)\) theory, once again at leading and subleading \(\bar Q\). We also investigate the strong-coupling regime.

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Whenever our basic understanding of the fundamental laws of physics improves, when more unified formalisms are uncovered, these advances are branded by subtle reformulations of the so-called Big Questions. More understanding comes with new questions, asked in a better way than before. When the renormalisation procedure for quantum field theories was finally unravelled, theoreticians realised that these gave new views on how the basic forces among elementary particles all could have a common, unified, origin. One elementary quantum field model stood out, which was dubbed the ‘Standard Model’, and the question was asked to what extent this model could describe all we know. Are there physical phenomena that suggest further improvement? Such questions could be asked to experimenters, but also from a purely theoretical point of view, one could ask what shortcomings the model has and what strategy should be followed to find better pathways. This paper briefly reviews some Big Questions of the past and asks how to use our deepest insights to rephrase the questions of the present.