Arithmetic properties of the Ramanujan function

We study some arithmetic properties of the Ramanujan function $\tau(n)$, such as the largest prime divisor $P(\tau(n))$ and the number of distinct prime divisors $\omega(\tau(n))$ of $\tau(n)$ for various sequences of $n$. In particular, we show that \hbox{$P(\tau(n)) \geq (\log n)^{33/31 + o(1)}$} for infinitely many $n$, and \begin{equation*} P(\tau(p)\tau(p^2)\tau(p^3)) > (1+o(1))\frac{\log\log p\log\log\log p} {\log\log\log\log p} \end{equation*} for every prime $p$ with \hbox{$\tau(p)\neq 0$}.Comment: 8 page

Perturbativity Constraints in BSM Models

Phenomenological studies performed for non-supersymmetric extensions of the Standard Model usually use tree-level parameters as input to define the scalar sector of the model. This implicitly assumes that a full on-shell calculation of the scalar sector is possible - and meaningful. However, this doesn't have to be the case as we show explicitly at the example of the Georgi-Machacek model. This model comes with an appealing custodial symmetry to explain the smallness of the $\rho$ parameter. However, the model cannot be renormalised on-shell without breaking the custodial symmetry. Moreover, we find that it can often happen that the radiative corrections are so large that any consideration based on a perturbative expansion appears to be meaningless: counter-terms to quartic couplings can become much larger than $4\pi$ and/or two-loop mass corrections can become larger than the one-loop ones. Therefore, conditions are necessary to single out parameter regions which cannot be treated perturbatively. We propose and discuss different sets of such perturbativity conditions and show their impact on the parameter space of the Georgi-Machacek model. Moreover, the proposed conditions are general enough that they can be applied to other models as well. We also point out that the vacuum stability constraints in the Georgi-Machacek model, which have so far only been applied at the tree level, receive crucial radiative corrections. We show that large regions of the parameter space which feature a stable electroweak vacuum at the loop level would have been - wrongly - ruled out by the tree-level conditions.Comment: 64 pages, 20 figure

On Kaon production in e+e- and Semi-inclusive DIS reactions

We consider semi-inclusive unpolarized DIS for the production of charged kaons and the different possibilities to test the conventionally used assumptions s-\bar=0 and D_d^{K^+-K^-}=0. The considered tests have the advantage that they do not require any knowledge of the fragmentation functions. We also show that measurements of both charged and neutral kaons would allow the determination of the kaon FFs D_q^{K^++K^-} solely from SIDIS measurements, and discuss the comparison of (D_u-D_d)^{K^+-K^-} obtained independently in SIDIS and e+e- reactions. All analysis are performed in LO and NLO in QCD. The feasibility of the tests to HERMES SIDIS data is considered.Comment: 7 pages, NLO analysis for all presented tests and feasibility to HERMES data adde

Spontaneous Charge Breaking in the NMSSM - Dangerous or not?

We investigate the impact of charge-breaking minima on the vacuum stability of the NMSSM. We find that, in contrast to Two-Higgs-Doublet Models like the MSSM, at both tree- and loop-level there exists global charge-breaking minima. Consequently, many regions of parameter space are rendered metastable, which otherwise would have been considered stable if these charge-breaking minima were neglected. However, the inclusion of these new scalar field directions has little impact on otherwise metastable vacuum configurations.Comment: 7 pages, 4 figure

On two functions arising in the study of the Euler and Carmichael quotients

We investigate two arithmetic functions naturally occurring in the study of the Euler and Carmichael quotients. The functions are related to the frequency of vanishing of the Euler and Carmichael quotients. We obtain several results concerning the relations between these functions as well as their typical and extreme values

The Ultraviolet Landscape of Two-Higgs Doublet Models

We study the predictions of generic ultraviolet completions of two-Higgs doublet models. We assume that at the matching scale between the two-Higgs doublet model and a ultraviolet complete theory -- which can be anywhere between the TeV and the Planck scale -- arbitrary but perturbative values for the quartic couplings are present. We evaluate the couplings down from the matching scale to the weak scale and study the predictions for the scalar mass spectrum. In particular, we show the importance of radiative corrections which are essential for both an accurate Higgs mass calculation as well as determining the stability of the electroweak vacuum. We study the relation between the mass splitting of the heavy Higgs states and the size of the quartic couplings at the matching scale, finding that only a small class of models exhibit a sizeable mass splitting between the heavy scalars at the weak scale. Moreover, we find a clear correlation between the maximal size of the couplings and the considered matching scale.Comment: 16 pages, 10 figure