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In this paper we show that the Cutkosky cutting rules are still valid term by term in the expansion in powers of $\kappa$ of the $\kappa$-deformed 1-loop correction to the propagator. We first present a general argument which relates each term in the expansion to a non-deformed amplitude containing additional propagators with mass $M>\kappa$. We then show the same thing more pragmatically, by reducing the singularity structure of the coefficients in the expansion of the $\kappa$-deformed amplitude, to the singularity structure of non-deformed loop amplitudes, by using algebraic and analytic identities. We will explicitly show this up to second order in $1/\kappa$, but the technique can be generalized to higher orders in $1/\kappa$. Both the abstract and the more direct approach easily generalize to different deformed theories. We will then compute the full imaginary part of the $\kappa$-deformed 1-loop correction to the propagator in a specific model, up to second order in the expansion in $1/\kappa$, highlighting the usefulness of the approach for the phenomenology of deformed models. This explicitly confirms previous qualitative arguments concerning the behaviour of the decay width of unstable particles in the considered model., Comment: 20 pages, 4 figures

In this paper we consider description of kaon -- anti-kaon interference in the context of a theory with deformed $\cal CPT$ symmetry. In the case of such theoretical models, deviations from the standard $\cal CPT$ invariance is related to the momentum carried by the particles; in particular the rest masses of particles and antiparticles are equal. We find that the decay intensity of kaon -- anti-kaon pair has three contributing terms: the correct-parity, the wrong-parity, and the interference between them, all of which are affected by deformation. Using the fact that the presence of such terms were not observed we estimate the magnitude of deformation parameter $\kappa \gtrsim 10^{18}$ GeV, very close to the expected Planck mass scale. This raises hopes that the effect, if exists, could be detected in a near future accelerator experiments. An uncertainty of the energy measurement is crucial for accuracy of these predictions., Comment: 17 pp

In the following work we will introduce and discuss in detail a particular model of complex $\kappa$-deformed scalar field, whose behaviour under C, P , T transformation is particularly transparent from both a formal and phenomenological point of view. We will begin by introducing the key mathematical structure at the basis of our investigation, namely the $\kappa$-Poincar\'e (Hopf) algebra and the $\kappa$-Minkowski spacetime. We will then investigate the behaviour of general two-particle states under deformed boost. After this we will introduce the action of our $\kappa$-deformed complex scalar field. From it, we will derive the equations of motion, as well as the Noether charges due to the continuous symmetries. The peculiar features of $\kappa$-deformation in general, and of our model in particular, allow for very non-trivial interaction between discrete and continuous symmetries, of which we will investigate the phenomenological consequences (particularly in terms of difference of lifetime of decaying particles). To conclude, we will obtain the $\kappa$-deformed propagator of the $\kappa$-deformed complex scalar field, and the imaginary part of the 1-loop contribution to it, ending with additional phenomenological consequences. The third chapter is new, unpublished work., Comment: 150 pages, submitted in fulfillment of the requirements for the degree of Doctor of Philosophy in the Department BP2, NCBJ, Warsaw

Quantum gravity phenomenology has been historically regarded as a difficult endeavour, due to the apparent scarcity of phenomena involving the required scales of length (Planck length $l_P$) and energy (Planck energy $E_P$). It was realized, however, that one can look for cumulative effects of a quantum theory of gravity at energies $E/E_P \ll 1$ if even tiny effects are amplified by the large time of flight or very high energies, common to astrophysical phenomena. In the this work, complementary to the COST action CA18108 White Book, we put forward a proposal that quantum gravity phenomenology can be fruitfully pursued also with the help of terrestrial particles accelerators. We first discuss the theoretical background, and then concentrate on deformed discrete symmetries C,P,T. We investigate possible experimental signatures of CPT deformation, particularly concerning the difference in decay time between particles and antiparticles, and fields interference., Comment: 13 pages, 2 figures

In this paper we consider the \k{appa}-deformed boost acting on a two-particles states. Using techniques developed in the case of infinitesimal boost we compute explicit expression for the components of the finite boost matrices acting on the first and second particle. We briefly discuss phenomenological consequences of our findings., Comment: 14 pages, 2 figures

It is often expected that one cannot treat spacetime as a continuous manifold as the Planck scale is approached, because of to possible effects due to a quantum theory of gravity. There have been several proposals to model such a deviation from the classical behaviour, one of which is noncommutativity of spacetime coordinates. In this context, the non-commutativity scale is seen as an observer-independent length scale. Of course, such a scale impose a modification of ordinary relativistic symmetries, which now need to be deformed to accommodate this fundamental scale. The $\kappa$-Poincar\'e algebra is an example of this deformation. In what follows I will briefly describe a construction of a $\kappa$-deformed complex scalar field theory, while at the same time shedding light on the behaviour of discrete and continuous symmetries in this formalism. This in turn will open the way to the study of the application of this formalism to actual physical processes. I will then conclude with some comments and prospects for the future., Comment: Presented at the 8th Conference of the Polish Society on Relativity

We will briefly describe how to build a field theory of a complex scalar field in the $\kappa$-Minkowski spacetime. After introducing the action, we will shortly describe its properties under both continuous and deformed symmetry transformations. We will then describe how to compute the charges and describe their non-trivial properties due to $\kappa$-deformation. We will conclude with the experimental significance of the model, particularly in the context of decay probability differences between particles and antiparticles., Comment: Presented at the Ninth Meeting on CPT and Lorentz Symmetry, Bloomington, Indiana, May 17-26, 2022

This is a write-up of two lectures delivered at COST CA18108 First Training School. They cover the motivations and some basic technical results in the field of Doubly Special Relativity and Relative Locality. The energy-dependent speed of light is a recurring theme here. The soccer ball problem is also briefly described. The video recording of these and other lectures can be found at https://www.youtube.com/channel/UChRfzgjTkO9sOkwq358kHZw/videos., Comment: 24 pages

In this paper we revisit the model of $\kappa$-deformed complex scalar field. We find that this model possesses ten conserved Noether charges that form, under commutators, a representation of (undeformed) Poincar\'e algebra. It follows that the theory is relativistic and does not break Lorentz invariance. However the spacetime representation of boosts is not standard, and contains a non-local translation, different for positive and negative energy modes. It then follows that although the masses of particles and anti-particles are equal, the theory violates CPT symmetry in a subtle way. We explain why the Jost-Wightman-Greenberg theorem of equivalence of the Poincar\'e symmetry and CPT fails in our case. Finally, we discuss the phenomenological consequences of the theory and its possible observational signatures., Comment: 41 pages. arXiv admin note: text overlap with arXiv:2009.03135 The published version

We present a construction of $\kappa$-deformed complex scalar field theory with the objective of shedding light on the way discrete symmetries and CPT invariance are affected by the deformation. Our starting point is the observation that, in order to have an appropriate action of Lorentz symmetries on antiparticle states, these should be described by four-momenta living on the complement of the portion of de Sitter group manifold to which $\kappa$-deformed particle four-momenta belong. Once the equations of motions are properly worked out from the deformed action we obtain that particle and antiparticle are characterized by different mass-shell constraints leading to a subtle form of departure from CPT invariance. The remaining part of our work is dedicated to a detailed description of the action of deformed Poincar\'e and discrete symmetries on the complex field., Comment: Coincides with the published version

Using the failure forecast method we describe a first assessment of failure time on present-day unrest signals at Campi Flegrei caldera (Italy) based on the horizontal deformation data collected in [2011, 2020] at eleven GPS stations.

We summarize the physical equations and analytic solutions of three versions of the box model equations, suitable for the integral formulation of axisymmetric gravity-driven particle currents with constant volume. The first model is based on a simple constant resisting stress, while the second and third models assume flow dilution by particle deposition. The third model is characterized by assuming an interstitial fluid lighter than the ambient fluid. All the calculations are performed on a flat topography. Ambient fluid entrainment and cooling effects are not considered. All particles are assumed to deposit at the same velocity., Comment: 5 pages, 0 figures

We present a new statistically driven method for analyzing the modeling of geophysical flows. Many models have been advocated by different modelers for such flows incorporating different modeling assumptions. Limited and sparse observational data on the modeled phenomena usually does not permit a clean discrimination among models for fitness of purpose, and, heuristic choices are usually made, especially for critical predictions of behavior that has not been experienced. We advocate here a methodology for characterizing models and the modeling assumptions they represent, using a statistical approach over the full range of applicability of the models. Such a characterization may then be used to decide the appropriateness of a model and modeling assumption for use. We present our method by comparing three different models arising from different rheology assumptions, and the data show unambiguously the performance of the models across a wide range of possible flow regimes. This comparison is facilitated by the recent development of the new release of our TITAN2D mass flow code that allows choice of multiple rheologies The quantitative and probabilistic analysis of contributions from different modeling assumptions in the models is particularly illustrative of the impact of the assumptions. Knowledge of which assumptions dominate, and, by how much, is illustrated in two different case studies: a small scale inclined plane with a flat runway, and the large scale topography on the SW slope of Volc\'{a}n de Colima (MX). A simple model performance evaluation completes the presentation., Comment: 38 pages, 21 figures, 1 ancillary file

We introduce a doubly stochastic method for performing material failure theory based forecasts of volcanic eruptions. The method enhances the well known Failure Forecast Method equation, introducing a new formulation similar to the Hull-White model in financial mathematics. In particular, we incorporate a stochastic noise term in the original equation, and systematically characterize the uncertainty. The model is a stochastic differential equation with mean reverting paths, where the traditional ordinary differential equation defines the mean solution. Our implementation allows the model to make excursions from the classical solutions, by including uncertainty in the estimation. The doubly stochastic formulation is particularly powerful, in that it provides a complete posterior probability distribution, allowing users to determine a worst case scenario with a specified level of confidence. We apply the new method on historical datasets of precursory signals, across a wide range of possible values of convexity in the solutions and amounts of scattering in the observations. The results show the increased forecasting skill of the doubly stochastic formulation of the equations if compared to statistical regression., Comment: 27 pages, 13 figures

Inspired by coarea formula in geometric measure theory, an occupation time formula for continuous semimartingales in $\mathbb{R}^{N}$ is proven. The occupation measure of a semimartingale, for $N\geq2$, is singular with respect to Lebesgue measure but it has a bounded density "transversal" to a foliation, under proper assumptions. In the particular case of the foliation given locally by the distance function from a manifold, the transversal density is related to a geometric local time of the semimartingale at the manifolds of the foliation.