Vacuum stability in the SM and the three-loop \beta-function for the Higgs self-interaction

In this article the stability of the Standard Model (SM) vacuum in the presence of radiative corrections and for a Higgs boson with a mass in the vicinity of 125 GeV is discussed. The central piece in this discussion will be the Higgs self-interaction $\lambda$ and its evolution with the energy scale of a given physical process. This is described by the $\beta$-function to which we recently computed analytically the dominant three-loop contributions. These are mainly the QCD and top-Yukawa corrections as well as the contributions from the Higgs self-interaction itself. We will see that for a Higgs boson with a mass of about 125 GeV the question whether the SM vacuum is stable and therefore whether the SM could be valid up to Planck scale cannot be answered with certainty due to large experimental uncertainties, mainly in the top quark mass.Comment: Extended version of a talk given at the ISSP 2012 in Erice, 23 June - 2 July 2012, part of the proceedings for this school; v2: references added; v3: references added; v4: references added, improved Fig. 1; v5: final version as submitted for publication, new Fig.

Top-Yukawa effects on the β\beta-function of the strong coupling in the SM at four-loop level

We present analytical results for the QCD $\beta$-function extended to the gaugeless limit of the unbroken phase of the Standard Model at four-loop level. Apart from the strong coupling itself we include the top-Yukawa contribution and the Higgs self-coupling. We observe a non-naive $\gamma_5$ contribution at order $y_t^4 g_s^4$, a feature not encountered in lower loop orders.Comment: v2: more sophisticated treatment and more detailed description of the non-naive \gamma_5 contribution; Ref. added. v3: this the version published in JHEP; references [49,50] fixed; v4: changed statement on p.8: a different gamma_5 treatment only leads to a factor 3, not a factor 6 in the non-naive part compared to the prescription used in this paper. Note added on recent developments (p. 12

Beta-function for the Higgs self-interaction in the Standard Model at three-loop level

The discovery of a Higgs particle has triggered numerous theoretical and experimental investigations concerning its production and decay rates and has led to interesting results concerning its interaction with fermions and gauge bosons. The self-interaction $\lambda$ of the Standard Model Higgs boson is particularly important due to its close connection with the stability of the SM vacuum. In this talk precision calculations for the evolution of this crucial coupling are presented and their impact on the question of vacuum stability is analysed. We also compare the theoretical precision resulting from the calculation of three-loop $\beta$-functions to the experimental uncertainties stemming from key parameters, such as the top mass, the Higgs mass and the strong coupling, and to the theoretical uncertainties introduced by the matching of experimental data to parameters in the theoretically favoured $\overline{\text{MS}}$ renormalization scheme.Comment: contribution to the proceedings of the European Physical Society Conference on High Energy Physics, 201

Four-loop QCD β\beta-function with different fermion representations of the gauge group

We present analytical results at four-loop level for the $\beta$-function of the coupling of a generic gauge group and any number of different quark representations. From this we can directly derive the gluino contribution to the strong coupling $\beta$-function of supersymmetric extensions of the Standard Model.Comment: v2: reference added, version accepted by JHEP, v3: typo fixed in (3.4

Leading QCD-induced four-loop contributions to the β\beta-function of the Higgs self-coupling in the SM and vacuum stability

We present analytical results for the leading top-Yukawa and QCD contribution to the $\beta$-function for the Higgs self-coupling $\lambda$ of the Standard Model at four-loop level, namely the part $\propto y_t^4 g_s^6$ independently confirming a result given in [1]. We also give the contribution $\propto y_t^2 g_s^6$ of the anomalous dimension of the Higgs field as well as the terms $\propto y_t g_s^8$ to the top-Yukawa $\beta$-function which can also be derived from the anomalous dimension of the top quark mass. We compare the results with the RG functions of the correlators of two and four scalar currents in pure QCD and find a new relation between the anomalous dimension $\gamma_0$ of the QCD vacuum energy and the anomalous dimension $\gamma_m^{SS}$ appearing in the RG equation of the correlator of two scalar currents. Together with the recently computed top-Yukawa and QCD contributions to $\beta_{g_s}$ [2,3] the $\beta$-functions presented here constitute the leading four-loop contributions to the evolution of the Higgs self-coupling. A numerical estimate of these terms at the scale of the top-quark mass is presented as well as an analysis of the impact on the evolution of $\lambda$ up to the Planck scale and the vacuum stability problem.Comment: v2: This is the version accepted by JHEP; extended discussion of the numerics and vacuum stability analysis; references added; plot adde

Quantum localization bounds Trotter errors in digital quantum simulation

A fundamental challenge in digital quantum simulation (DQS) is the control of an inherent error, which appears when discretizing the time evolution of a quantum many-body system as a sequence of quantum gates, called Trotterization. Here, we show that quantum localization-by constraining the time evolution through quantum interference-strongly bounds these errors for local observables, leading to an error independent of system size and simulation time. DQS is thus intrinsically much more robust than suggested by known error bounds on the global many-body wave function. This robustness is characterized by a sharp threshold as a function of the Trotter step size, which separates a localized region with controllable Trotter errors from a quantum chaotic regime. Our findings show that DQS with comparatively large Trotter steps can retain controlled errors for local observables. It is thus possible to reduce the number of gate operations required to represent the desired time evolution faithfully

Four-loop renormalization of QCD with a reducible fermion representation of the gauge group: anomalous dimensions and renormalization constants

We present analytical results at four-loop level for the renormalization constants and anomalous dimensions of an extended QCD model with one coupling constant and an arbitrary number of fermion representations. One example of such a model is the QCD plus gluinos sector of a supersymmetric theory where the gluinos are Majorana fermions in the adjoint representation of the gauge group. The renormalization constants of the gauge boson, ghost and fermion fields are analytically computed as well as those for the ghost-gluon vertex, the fermion-gluon vertex and the fermion mass. All other renormalization constants can be derived from these. Some of these results were already produced in Feynman gauge for the computation of the beta-function of this model, which was recently published. Here we present results for an arbitrary gauge parameter.Comment: v2: version accepted by JHEP, extended discussion of the treatment of Majorana spinor