Essential dimension of simple algebras with involutions

Let $1\leq m \leq n$ be integers with $m|n$ and \cat{Alg}_{n,m} the class of central simple algebras of degree $n$ and exponent dividing $m$. In this paper, we find new, improved upper bounds for the essential dimension and 2-dimension of \cat{Alg}_{n,2}. In particular, we show that \ed_{2}(\cat{Alg}_{16,2})=24 over a field $F$ of characteristic different from 2.Comment: Sections 1 and 3 are rewritte

Essential dimension of simple algebras in positive characteristic

Let $p$ be a prime integer, $1\leq s\leq r$ integers, $F$ a field of characteristic $p$. Let \cat{Dec}_{p^r} denote the class of the tensor product of $r$ $p$-symbols and \cat{Alg}_{p^r,p^s} denote the class of central simple algebras of degree $p^r$ and exponent dividing $p^s$. For any integers $s<r$, we find a lower bound for the essential $p$-dimension of \cat{Alg}_{p^r,p^s}. Furthermore, we compute upper bounds for \cat{Dec}_{p^r} and \cat{Alg}_{8,2} over $\ch(F)=p$ and $\ch(F)=2$, respectively. As a result, we show \ed_{2}(\cat{Alg}_{4,2})=\ed(\cat{Alg}_{4,2})=\ed_{2}(\gGL_{4}/\gmu_{2})=\ed(\gGL_{4}/\gmu_{2})=3 and 3\leq \ed(\cat{Alg}_{8,2})=\ed(\gGL_{8}/\gmu_{2})\leq 10 over a field of characteristic 2.Comment: Any comments are welcom

Dark matter contribution to b→sμ+μ−b\to s \mu^+ \mu^- anomaly in local U(1)Lμ−LτU(1)_{L_\mu-L_\tau} model

We propose a local $U(1)_{L_\mu-L_\tau}$ model to explain $b \to s \mu^+ \mu^-$ anomaly observed at the LHCb and Belle experiments. The model also has a natural dark matter candidate $N$. We introduce $SU(2)_L$-doublet colored scalar $\widetilde{q}$ to mediate $b \to s$ transition at one-loop level. The $U(1)_{L_\mu-L_\tau}$ gauge symmetry is broken spontaneously by the scalar $S$. All the new particles are charged under $U(1)_{L_\mu-L_\tau}$. We can obtain $C_9^{\mu,{\rm NP}} \sim -1$ to solve the $b \to s\mu^+\mu^-$ anomaly and can explain the correct dark matter relic density of the universe, $\Omega_{\rm DM} h^2 \approx 0.12$, simultaneously, while evading constraints from electroweak precision tests, neutrino trident experiments and other quark flavor-changing loop processes such as $b \to s \gamma$ and $B_s-\overline{B}_s$ mixing. Our model can be tested by searching for $Z'$ and new colored scalar at the LHC and $B \to K^* \nu \overline{\nu}$ process at Belle-II.Comment: 15 pages, 6 figure