LSZ-reduction, resonances and non-diagonal propagators: fermions and scalars

We analyze in details the effects associated with mixing of fermionic fields. In a system with an arbitrary number of Majorana or Dirac particles, a simple proof of factorizability of residues of non-diagonal propagators at the complex poles is given, together with a prescription for finding the "square-rooted" residues to all orders of perturbation theory, in an arbitrary renormalization scheme. Corresponding prescription for the scalar case is provided as well.Comment: 37 pages, 1 figur

Vector-scalar mixing to all orders, for an arbitrary gauge model in the generic linear gauge

I give explicit fromulae for full propagators of vector and scalar fields in a generic spin-1 gauge model quantized in an arbitrary linear covariant gauge. The propagators, expressed in terms of all-order one-particle-irreducible correlation functions, have a remarkably simple form because of constraints originating from Slavnov-Taylor identities of Becchi-Rouet-Stora symmetry. I also determine the behavior of the propagators in the neighborhood of the poles, and give a simple prescription for the coefficients that generalize (to the case with an arbitrary vector-scalar mixing) the standard $\sqrt{\mathcal{Z}}$ factors of Lehmann, Symanzik and Zimmermann. So obtained generalized $\sqrt{\mathcal{Z}}$ factors, are indispensable to the correct extraction of physical amplitudes from the amputated correlation functions in the presence of mixing. The standard $R_\xi$ guauges form a particularly important subclass of gauges considered in this paper. While the tree-level vector-scalar mixing is, by construction, absent in $R_\xi$ gauges, it unavoidably reappears at higher orders. Therefore the prescription for the generalized $\sqrt{\mathcal{Z}}$ factors given in this paper is directly relevant for the extraction of amplitudes in $R_\xi$ gauges.Comment: 17 pages, 0 figures [v2: misprint in Eq.35 corrected

LSZ-reduction, resonances and non-diagonal propagators: gauge fields

We analyze in the Landau gauge mixing of bosonic fields in gauge theories with exact and spontaneously broken symmetries, extending to this case the Lehmann-Symanzik-Zimmermann (LSZ) formalism of the asymptotic fields. Factorization of residues of poles (at real and complex values of the variable $p^2$) is demonstrated and a simple practical prescription for finding the "square-rooted" residues, necessary for calculating $S$-matrix elements, is given. The pseudo-Fock space of asymptotic (in the LSZ sense) states is explicitly constructed and its BRST-cohomological structure is elucidated. Usefulness of these general results, obtained by investigating the relevant set of Slavnov-Taylor identities, is illustrated on the one-loop examples of the $Z^0$-photon mixing in the Standard Model and the $G_Z$-Majoron mixing in the singlet Majoron model.Comment: 54 pages, 7 figures, (published version

Two-loop RGE of a general renormalizable Yang-Mills theory in a renormalization scheme with an explicit UV cutoff

We perform a systematic one-loop renormalization of a general renormalizable Yang-Mills theory coupled to scalars and fermions using a regularization scheme with a smooth momentum cutoff $\Lambda$ (implemented through an exponential damping factor). We construct the necessary finite counterterms restoring the BRST invariance of the effective action by analyzing the relevant Slavnov-Taylor identities. We find the relation between the renormalized parameters in our scheme and in the conventional $\overline{\rm MS}$ scheme which allow us to obtain the explicit two-loop renormalization group equations in our scheme from the known two-loop ones in the $\overline{\rm MS}$ scheme. We calculate in our scheme the divergences of two-loop vacuum graphs in the presence of a constant scalar background field which allow us to rederive the two-loop beta functions for parameters of the scalar potential. We also prove that consistent application of the proposed regularization leads to counterterms which, together with the original action, combine to a bare action expressed in terms of bare parameters. This, together with treating $\Lambda$ as an intrinsic scale of a hypothetical underlying finite theory of all interactions, offers a possibility of an unconventional solution to the hierarchy problem if no intermediate scales between the electroweak scale and the Planck scale exist.Comment: updated references, 90 pages, many figure

Renormalization of an arbitrary renormalizable model in a gauge symmetry violating regularization

Streszczenie Praca dotyczy badania niestandardowej regularyzacji, polegającej na zastąpieniu każdej pochodnej w działaniu wg przepisu $\partial_\mu\mapsto\exp\left\{{\partial^2/(2\Lambda^2)}\right\}\partial_\mu$. W rozdziale 1 wyjaśniamy, co było motywacją dla podjętych badań. W rozdziale 2 podajemy praktyczne metody przeprowadzania rachunków w przyjętej regularyzacji oraz wprowadzamy oparty na niej schemat minimalnego odjęcia ($\Lambda$-\anti{MS}) dla ogólnego modelu renormalizowalnego. Ze względu na naruszenia symetrii cechowania, definicja schematu wymaga sprecyzowania, które wierzchołki są ``minimalnie renormalizowane" -- przyjmujemy, że są to wszystkie wierzchołki bez pól wektorowych. Jest to naturalnym uogólnieniem ``ścisłego" minimalnego odjęcia w modelach bez symetrii lokalnych. Zgodnie z analizą przedstawioną w podrozdziale 2.4, taki wybór jednoznacznie określa działanie 1PI, jeśli narzucimy dodatkowo warunek minimalnej renormalizacji ``wierzchołka" $A_\mu\partial^2\!A^\mu$ oraz wszystkich wierzchołków z polami o niezerowej liczbie duchowej. W praktyce, obliczenie zrenormalizowanych funkcji 1PI w rzędzie $\hbar$ przebiega w czterech etapach: (1) wyznaczenie naruszenia tożsamości Slavnova-Taylora przez zregularyzowane diagramy jednopętlowe, (2) minimalna renormalizacja \emph{wszystkich} wierzchołków, (3) wyznaczenie naruszenie tożsamości ST przez funkcje minimalnie zrenormalizowane, (4) wyznaczenie dodatkowych przeciwczłonów dla wierzchołków z polami wektorowymi z warunków przywrócenia tożsamości ST. Procedura ta przeprowadzona jest w rozdziale 3, gdzie podajemy pełen zestaw dodatkowych przeciwczłonów w przybliżeniu jednopętlowym. W rozdziale 4 wykazujemy, że $\Lambda$-\anti{MS} jest równoważny ze schematem \anti{MS} regularyzacji wymiarowej z ``naiwną" $\gamma^5$ (DimReg-\anti{MS}). W tym celu sprawdzamy, że w jednej pętli funkcje jednocząstkowo-nieprzywiedlne w obu schematach powiązane są przez ``skończoną renormalizację" pól i sprzężeń. Uzyskane związki między zrenormalizowanymi parametrami posłużą nam do wyznaczenia (w rozdziale 5) dwupętlowych funkcji $\beta$ w $\Lambda$-\anti{MS} przy użyciu dostępnych w literaturze wzorów na ich odpowiedniki w DimReg-\anti{MS}. W ramach dodatkowego testu spójności schematu $\Lambda$-\anti{MS}, wyznaczamy w rozdziale 6 rozbieżne części dwupętlowych diagramów próżniowych, pokazując, że $\Lambda$-\anti{MS} poprawnie usuwa przekrywające się rozbieżności. Pozwala to na \emph{bezpośrednie} obliczenie funkcji $\beta^{(2)}$ dla skalarnych stałych sprzężenia w schemacie $\Lambda$-\anti{MS}, w pełnej zgodności z wynikami rozdziału 5 opartymi na związku z regularyzacją wymiarową. Zastosowania tych wyników w kontekście problemu hierarchii omawiamy w rozdziale 7. Dodatki A, I, H oraz C.2 zawierają rachunki stanowiące integralną część pracy. Pozostałe dodatki mają charakter uzupełniający.Summary The thesis concerns loop calculations in a nonstandard regularization defined by the replacement $$\partial_\mu\mapsto\exp\left\{{\partial^2/(2\Lambda^2)}\right\}\partial_\mu$$ for all derivatives in the Lagrangian. In Chapter 1 we explain our main motivations. In Chapter 1 we develop practical methods for calculations in this regularization. We also introduce an appropriate minimal subtraction scheme ($\Lambda$-\anti{MS}) for an arbitrary renormalizable model. Because of violations of the gauge symmetry induced by regularization, we have to decide which vertices are minimally renormalized -- by our choice these are all vertices without vector fields. In Section 2.4 we show that this choice determines 1PI effective action unambiguously provided that the ``vertex" $A_\mu\partial^2\!A^\mu$ as well as all vertices containing fields carrying nonzero ghost number are also minimally renormalized. There are four stages of calculation of 1PI Green functions at the $\hbar$ order: (1) determination of violation of a Slavnov-Taylor identity for regularized one-loop diagrams, (2) minimal renormalization of \emph{all} vertices, (3) determination of violation of the ST identity for minimally renormalized 1PI functions, (4) determination of additional counterterms for vertices with vector fields from the condition of restoration of the ST identity. This procedure is performed in Chapter 3, and the complete set of additional counterterms is determined at one-loop. In Chapter 4 we show that $\Lambda$-\anti{MS} is equivalent with \anti{MS} scheme of dimensional regularization with ``naive" $\gamma^5$ (DimReg-\anti{MS}). To this end we check that the one-loop 1PI functions in both schemes are related through ``finite renormalization" of fields and couplings. So obtained relations between renormalized parameters allow for the conversion of well-known two-loop $\beta$ functions in DimReg-\anti{MS} into their counterparts in $\Lambda$-\anti{MS} -- this is done in Chapter 5. As an additional consistency check we calculate -- in Chapter 6 -- divergent parts of two-loop vacuum diagrams in $\Lambda$-\anti{MS}. We show that overlapping divergences are removed correctly in $\Lambda$-\anti{MS}. Moreover, this calculation allows for a \emph{direct} calculation of two-loop $\beta$ functions in $\Lambda$-\anti{MS} -- the result is consistent with the one obtained in Chapter 5. In chapter 7 we discuss possible applications of our results in the context of the hierarchy problem. Appendices A, I, H and C.2 contain calculations that are integral parts of the thesis. Remaining appendices contain a supplementary material

Conformal Standard Model with an extended scalar sector

We present an extended version of the Conformal Standard Model (characterized by the absence of any new intermediate scales between the electroweak scale and the Planck scale) with an enlarged scalar sector coupling to right-chiral neutrinos. The scalar potential and the Yukawa couplings involving only right-chiral neutrinos are invariant under a new global symmetry SU(3)$_N$ that complements the standard U(1)$_{B-L}$ symmetry, and is broken explicitly only by the Yukawa interaction, of order $10^{-6}$, coupling right-chiral neutrinos and the electroweak lepton doublets. We point out four main advantages of this enlargement, namely: (1) the economy of the (non-supersymmetric) Standard Model, and thus its observational success, is preserved; (2) thanks to the enlarged scalar sector the RG improved one-loop effective potential is everywhere positive with a stable global minimum, thereby avoiding the notorious instability of the Standard Model vacuum; (3) the pseudo-Goldstone bosons resulting from spontaneous breaking of the SU(3)$_N$ symmetry are natural Dark Matter candidates with calculable small masses and couplings; and (4) the Majorana Yukawa coupling matrix acquires a form naturally adapted to leptogenesis. The model is made perturbatively consistent up to the Planck scale by imposing the vanishing of quadratic divergences at the Planck scale (`softly broken conformal symmetry'). Observable consequences of the model occur mainly via the mixing of the new scalars and the standard model Higgs boson.Comment: version accepted for publication in the JHEP, 41 pages, 1 figur

Error estimates in approximate deconvolution models

International audienceWe consider general Approximate Deconvolution Models (ADM). We estimate the error modeling as a function of the residual stress $\tau_N$ and we compute the rate of convergence to the mean Navier-Stokes Equations in terms of the deconvolution order $N$