Unstable Quantum Fields and Higgs Vacuum Bernd A. Kniehl II. Institut für Theoretische Physik, Universität Hamburg, Germany DOI: http://dx.doi.org/10.3204/PUBDB-2018-00782/B4 We summarize the progress of the research performed in project B4 of the SFB 676 and highlight the most important results obtained. Starting from the observation that mixing and instability of elementary particles concur in nature, one central goal was to construct a general renormalization scheme that rigorously incorporates both aspects and to explore its phenomenological consequences in the Standard Model (SM) of elementary particle physics and its most favorable extensions. Another central goal was to signicantly improve the analysis of the electroweak vacuum stability and the theoretical concepts necessary for the extraction of the top-quark mass from experimental data. 1 Overview The research of project B4 was divided into two areas. The rst research area, which was continuously worked on during all three funding periods, was dedicated to deepening our un- derstanding of the notions of mixing and instability of elementary particles in the context of quantum eld theory and is covered in Sec.2. The second research area, which was primarily worked on during the third funding period, was devoted to the stability of the electroweak (EW) vacuum and is covered in Sec.3. Project B4 has produced eighteen journal publications [118] and four proceedings contributions [1922] altogether. Three of these publications made it into Physical Review Letters [1,9,15]. One [15] of the latter three publications was distinguished as Editors' Suggestion and featured by Alexander Kusenko with the article Viewpoint: Are We on the Brink of the Higgs Abyss? in Physics.1 A related publication in Computer Physics Com- munications [17], in which the C++ library mr for the matching and running of the Standard Model parameters is described, was distinguished there as Highlighted Article and featured by Abigail Beall with the article A time machine to explore the universe.2 Key publications in preparation of the second research area include Refs. [2326]. Project B4 yielded one bachelor thesis [27], one master thesis [28], and one doctoral thesis [29]. 1https://physics.aps.org/articles/v8/108 2https://www.journals.elsevier.com/computer-physics-communications/highlighted-articles/a-time- machine-to-explore-the-universe SFB 676 Particles, Strings and the Early Universe 189 Bernd A. Kniehl 2 Mixing and instability in eld theory 2.1 Starting point The state of the art at the beginning of project B4 may be summarized as follows. An on-shell renormalization scheme had been formulated to all orders for the SM under the simplifying assumptions that all particles are stable, neutrinos are massless, and quark avors do not mix [30]. A complex-pole scheme of renormalization had been elaborated for the W ± [31], Z [32], and Higgs bosons [33, 34]. The complex poles of the propagators of the physical elds in the SM had been shown to be gauge independent via Nielsen identities [35]. Denitions of branching ratios and partial decay widths of unstable particles had been proposed that satisfy the basic principles of additivity and gauge independence to all orders [36, 37]. Various schemes for the renormalization of the CabibboKobayashiMaskawa (CKM) mixing matrix of stable quarks had been proposed at one loop, none of which was completely satisfactory and suitable for generalizations to higher orders and the case of instability. The renormalized CKM matrix was either gauge dependent [38], o-shell [39], not unitary [40], or complicated by vertex corrections [41]. Mixing renormalization had also been considered at one loop for mixed systems of stable fermions involving Majorana degrees of freedom [42]. An on-shell renormalization scheme had been formulated to all orders for the EW part of the minimal supersymmetric extension of the SM (MSSM) allowing for the physical particles to be unstable [43], which, however, left room for improvements and extensions. The central goal to establish a complex-pole scheme of mixing renormalization for unstable particles valid to all orders for the SM and its most attractive extensions requires generalized concepts for avor-changing propagators and vertices. In the SM with massless neutrinos, these are the propagator matrices of the up- and down-type quarks and their charged-current vertices. This pattern carries over to the lepton sector if the neutrinos are massive Dirac fermions, while things are more complicated in the presence of Majorana degrees of freedom, which typically give rise to avor-changing vertices involving the Z0 and Higgs bosons, too. In supersymmetric extensions of the SM, there is additional avor mixing for the superpartners of the fermions, 1 the spin-zero squarks and sleptons, and for the spin- =2 charginos and neutralinos. The results on avor-changing propagators and vertices achieved in project B4 are reported in Secs. 2.2 and 2.3, respectively. 2.2 Flavor-changing propagators We made signicant progress in the SM. We established the all-order renormalization of the propagator matrix of a mixed system of unstable Dirac fermions in three stages: (i) unstable fermion without mixing [3]; (ii) stable fermions with mixing [8]; (iii) general case [9, 11]. In Ref. [3], the concepts of pole mass and width, previously elaborated for scalar [33, 34] and vector bosons [31, 32], were extended to unstable fermions in the general framework of parity-nonconserving gauge theories, such as the SM. In contrast with the conventional on-shell denitions, these concepts are gauge independent and avoid severe unphysical singularities, such as threshold [44,45] or power-like infrared singularities [46], properties of great importance since most fundamental fermions in nature are unstable particles. General expressions for the unrenormalized and renormalized dressed propagators of unstable fermions and their wave- function renormalization (WFR) constants were presented. It was shown that the masses and widths of the left- and right-handed fermion elds coincide to all orders and that their pole 190 SFB 676 Particles, Strings and the Early Universe Unstable Quantum Fields and Higgs Vacuum residues cannot be arranged to be both unity, with a relative phase remaining. In Ref. [8], we derived general and explicit expressions for the unrenormalized and renormal- ized dressed propagators of stable Dirac fermions in parity-nonconserving theories with inter- generation mixing. The mass eigenvalues, the corresponding mass counterterms, and the eect of inter-generation mixing on their determination were discussed. Invoking the AokiHioki KawabeKonumaMuta [47] renormalization conditions and employing a number of very useful relations from Matrix Algebra, we showed explicitly that the renormalized dressed propagators satisfy important physical properties. In Refs. [9, 11], we showed how to renormalize the propagator matrix of a mixed system of unstable Dirac fermions to all orders in the pole scheme. We presented closed expressions for the pole masses, comprising both the physical masses and total decay widths, and the WFR matrices in terms of the scalar, pseudo-scalar, vector, and axial-vector form factors of the one- particle-irreducible self-energies, and also listed their expansions through two loops, which had not previously been available. We explicitly demonstrated that the poles of the left- and right- handed parts of the dressed propagator matrices coincide as they should, and that the resonating diagonal elements of the latter have unit residues. We identied residual degrees of freedom in the WFR matrices and proposed an additional renormalization condition to exhaust them. As a consequence of instability, the WFR matrices of the in and out states bifurcate in the sense that they are no longer related by Hermitian conjugation, in contrast to text book wisdom. This feature had already been anticipated in a one-loop analysis [48] triggered by the applicant. By contrast, if the WFR of the out state is enforced to be the Hermitian conjugate of that of the in state, then the residues of the left- and right-handed parts of the resonating propagator cannot both be unity, but are related by a phase factor [3]. In contrast to the conventional on-shell scheme, the complex-pole scheme leads to gauge-invariant denitions of the masses and total decay widths of physical particles and avoids severe unphysical singularities, of threshold [33] or power-like infrared [31] nature. In the case of stable Dirac fermions, the well-known one-loop results were recovered [42]. In Ref. [12], generalized the analysis of Refs. [9,11] to unstable Majorana fermions in a gen- eral parity-nonconserving theory. In contrast to the case of unstable Dirac fermions, the WFR matrices of the in and out states are uniquely xed, while they again bifurcate in the sense that they are no longer related by pseudo-Hermitian conjugation. We again presented closed analytic expressions, as well as their expansions through two loops, for the renormalization constants in terms of the scalar, pseudoscalar, vector, and pseudovector parts of the unrenor- malized self-energy matrix. The well-known one-loop results for stable Majorana fermions [42] were reproduced. To formulate the WFR conditions in the pole scheme underlying Refs. [9, 11, 12], it was necessary to generalize the Dirac equations and spinors in momentum space to free unstable 1 spin- =2 fermions taking into account the fundamental requirement of Lorentz covariance. In Ref. [13], we thus derived the generalized adjoint Dirac equations and spinors, and explained the very simple relation that exists, in our formulation, between the unstable and stable cases. As an application of the generalized spinors, we evaluated the probability density. We also discussed the behavior of the generalized Dirac equations under time reversal. 2.3 Flavor-changing vertices In Refs. [1,2], we developed an explicit on-shell framework to renormalize the CKM quark mixing matrix Vij at the one-loop level.