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1. Search for the $$ {B}_s^0 $$ → μ+μ−γ decay

   https://doi.org/10.1007/JHEP07(2024)101  

   Aaij, R; Abdelmotteleb, A_S W; Abellan_Beteta, C; Abudinén, F; Ackernley, T; Adefisoye, A A; Adeva, B; Adinolfi, M; Adlarson, P; Agapopoulou, C; et al (July 2024, Journal of High Energy Physics)

   A<sc>bstract</sc> A search for the fully reconstructed$$ {B}_s^0 $$ $B_s^0$→ μ⁺μ⁻γdecay is performed at the LHCb experiment using proton-proton collisions at$$ \sqrt{s} $$ $\sqrt{s}$= 13 TeV corresponding to an integrated luminosity of 5.4 fb⁻¹. No significant signal is found and upper limits on the branching fraction in intervals of the dimuon mass are set$$ {\displaystyle \begin{array}{cc}\mathcal{B}\left({B}_s^0\to {\mu}^{+}{\mu}^{-}\gamma \right)<4.2\times {10}^{-8},& m\left({\mu}^{+}{\mu}^{-}\right)\in \left[2{m}_{\mu },1.70\right]\textrm{GeV}/{c}^2,\\ {}\mathcal{B}\left({B}_s^0\to {\mu}^{+}{\mu}^{-}\gamma \right)<7.7\times {10}^{-8},&\ m\left({\mu}^{+}{\mu}^{-}\right)\in \left[\textrm{1.70,2.88}\right]\textrm{GeV}/{c}^2,\\ {}\mathcal{B}\left({B}_s^0\to {\mu}^{+}{\mu}^{-}\gamma \right)<4.2\times {10}^{-8},& m\left({\mu}^{+}{\mu}^{-}\right)\in \left[3.92,{m}_{B_s^0}\right]\textrm{GeV}/{c}^2,\end{array}} $$ $\begin{array}{cc}B\left(B_s^0\to {\mu }^{+}{\mu }^{-}\gamma \right)<4.2\times {10}^{-8},& m\left({\mu }^{+}{\mu }^{-}\right)\in \left(2m_{\mu },1.70\right)\mathrm{GeV}/c^2,\\ B\left(B_s^0\to {\mu }^{+}{\mu }^{-}\gamma \right)<7.7\times {10}^{-8},& m\left({\mu }^{+}{\mu }^{-}\right)\in \left(\mathrm{1.70,\; 2.88}\right)\mathrm{GeV}/c^2,\\ B\left(B_s^0\to {\mu }^{+}{\mu }^{-}\gamma \right)<4.2\times {10}^{-8},& m\left({\mu }^{+}{\mu }^{-}\right)\in \left(3.92,m_{B_s^0}\right)\mathrm{GeV}/c^2,\end{array}$ at 95% confidence level. Additionally, upper limits are set on the branching fraction in the [2m_μ,1.70] GeV/c²dimuon mass region excluding the contribution from the intermediateϕ(1020) meson, and in the region combining all dimuon-mass intervals. 

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2. Strong and fragile topological Dirac semimetals with higher-order Fermi arcs

   https://doi.org/10.1038/s41467-020-14443-5  

   Wieder, Benjamin J.; Wang, Zhijun; Cano, Jennifer; Dai, Xi; Schoop, Leslie M.; Bradlyn, Barry; Bernevig, B. Andrei (January 2020, Nature Communications)

   Abstract Dirac and Weyl semimetals both exhibit arc-like surface states. However, whereas the surface Fermi arcs in Weyl semimetals are topological consequences of the Weyl points themselves, the surface Fermi arcs in Dirac semimetals are not directly related to the bulk Dirac points, raising the question of whether there exists a topological bulk-boundary correspondence for Dirac semimetals. In this work, we discover that strong and fragile topological Dirac semimetals exhibit one-dimensional (1D) higher-order hinge Fermi arcs (HOFAs) as universal, direct consequences of their bulk 3D Dirac points. To predict HOFAs coexisting with topological surface states in solid-state Dirac semimetals, we introduce and layer a spinful model of ans–d-hybridized quadrupole insulator (QI). We develop a rigorous nested Jackiw–Rebbi formulation of QIs and HOFA states. Employing ab initio calculations, we demonstrate HOFAs in both the room- (α) and intermediate-temperature (α″) phases of Cd₃As₂, KMgBi, and rutile-structure ($$ \beta ^{\prime} $$ ${\beta }^{\prime }$-) PtO₂. 

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3. Multipliers on bi-parameter Haar system Hardy spaces

   https://doi.org/10.1007/s00208-024-02887-9  

   Lechner, R.; Motakis, P.; Müller, P_F_X; Schlumprecht, Th (May 2024, Mathematische Annalen)

   Abstract Let$$(h_I)$$ $\left(h_I\right)$denote the standard Haar system on [0, 1], indexed by$$I\in \mathcal {D}$$ $I\in D$, the set of dyadic intervals and$$h_I\otimes h_J$$ $h_I\otimes h_J$denote the tensor product$$(s,t)\mapsto h_I(s) h_J(t)$$ $(s,t)\mapsto h_I\left(s\right)h_J\left(t\right)$,$$I,J\in \mathcal {D}$$ $I,J\in D$. We consider a class of two-parameter function spaces which are completions of the linear span$$\mathcal {V}(\delta ^2)$$ $V\left({\delta }^2\right)$of$$h_I\otimes h_J$$ $h_I\otimes h_J$,$$I,J\in \mathcal {D}$$ $I,J\in D$. This class contains all the spaces of the formX(Y), whereXandYare either the Lebesgue spaces$$L^p[0,1]$$ $L^p[0,1]$or the Hardy spaces$$H^p[0,1]$$ $H^p[0,1]$,$$1\le p < \infty $$ $1\le p<\infty$. We say that$$D:X(Y)\rightarrow X(Y)$$ $D:X\left(Y\right)\to X\left(Y\right)$is a Haar multiplier if$$D(h_I\otimes h_J) = d_{I,J} h_I\otimes h_J$$ $D(h_I\otimes h_J)=d_{I,J}{h}_I\otimes h_J$, where$$d_{I,J}\in \mathbb {R}$$ $d_{I,J}\in R$, and ask which more elementary operators factor throughD. A decisive role is played by theCapon projection$$\mathcal {C}:\mathcal {V}(\delta ^2)\rightarrow \mathcal {V}(\delta ^2)$$ $C:V\left({\delta }^2\right)\to V\left({\delta }^2\right)$given by$$\mathcal {C} h_I\otimes h_J = h_I\otimes h_J$$ $C{h}_I\otimes h_J=h_I\otimes h_J$if$$|I|\le |J|$$ $\left|I\right|\le \left|J\right|$, and$$\mathcal {C} h_I\otimes h_J = 0$$ $C{h}_I\otimes h_J=0$if$$|I| > |J|$$ $\left|I\right|>\left|J\right|$, as our main result highlights: Given any bounded Haar multiplier$$D:X(Y)\rightarrow X(Y)$$ $D:X\left(Y\right)\to X\left(Y\right)$, there exist$$\lambda ,\mu \in \mathbb {R}$$ $\lambda ,\mu \in R$such that$$\begin{aligned} \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}-\mathcal {C})\text { approximately 1-projectionally factors through }D, \end{aligned}$$ $\begin{array}{c}\lambda C+\mu (\text{Id}-C)\text{approximately 1-projectionally factors through}D,\end{array}$i.e., for all$$\eta > 0$$ $\eta >0$, there exist bounded operatorsA, Bso thatABis the identity operator$${{\,\textrm{Id}\,}}$$ $\text{Id}$,$$\Vert A\Vert \cdot \Vert B\Vert = 1$$ $\Vert A\Vert ·\Vert B\Vert =1$and$$\Vert \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}-\mathcal {C}) - ADB\Vert < \eta $$ $\Vert \lambda C+\mu (\text{Id}-C)-ADB\Vert <\eta$. Additionally, if$$\mathcal {C}$$ $C$is unbounded onX(Y), then$$\lambda = \mu $$ $\lambda =\mu$and then$${{\,\textrm{Id}\,}}$$ $\text{Id}$either factors throughDor$${{\,\textrm{Id}\,}}-D$$ $\text{Id}-D$. 

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4. Restricted spaces of holomorphic sections vanishing along subvarieties

   https://doi.org/10.1007/s00209-025-03706-w  

   Coman, Dan; Marinescu, George; Nguyên, Viêt-Anh (May 2025, Mathematische Zeitschrift)

   Abstract LetXbe a compact normal complex space of dimensionnandLbe a holomorphic line bundle onX. Suppose that$$\Sigma =(\Sigma _1,\ldots ,\Sigma _\ell )$$ $\Sigma =({\Sigma }_1,\dots ,{\Sigma }_{\ell })$is an$$\ell $$ $\ell$-tuple of distinct irreducible proper analytic subsets ofX,$$\tau =(\tau _1,\ldots ,\tau _\ell )$$ $\tau =({\tau }_1,\dots ,{\tau }_{\ell })$is an$$\ell $$ $\ell$-tuple of positive real numbers, and let$$H^0_0(X,L^p)$$ $H_0^0(X,L^p)$be the space of holomorphic sections of$$L^p:=L^{\otimes p}$$ $L^p:=L^{\otimes p}$that vanish to order at least$$\tau _jp$$ ${\tau }_{j}p$along$$\Sigma _j$$ ${\Sigma }_j$,$$1\le j\le \ell $$ $1\le j\le \ell$. If$$Y\subset X$$ $Y\subset X$is an irreducible analytic subset of dimensionm, we consider the space$$H^0_0 (X|Y, L^p)$$ $H_0^0\left(X\right|Y,L^p)$of holomorphic sections of$$L^p|_Y$$ $L^p{|}_Y$that extend to global holomorphic sections in$$H^0_0(X,L^p)$$ $H_0^0(X,L^p)$. Assuming that the triplet$$(L,\Sigma ,\tau )$$ $(L,\Sigma ,\tau )$is big in the sense that$$\dim H^0_0(X,L^p)\sim p^n$$ $dim{H}_0^0(X,L^p)\sim p^n$, we give a general condition onYto ensure that$$\dim H^0_0(X|Y,L^p)\sim p^m$$ $dim{H}_0^0\left(X\right|Y,L^p)\sim p^m$. WhenLis endowed with a continuous Hermitian metric, we show that the Fubini-Study currents of the spaces$$H^0_0(X|Y,L^p)$$ $H_0^0\left(X\right|Y,L^p)$converge to a certain equilibrium current onY. We apply this to the study of the equidistribution of zeros inYof random holomorphic sections in$$H^0_0(X|Y,L^p)$$ $H_0^0\left(X\right|Y,L^p)$as$$p\rightarrow \infty $$ $p\to \infty$. 

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5. Lower Bounds Against Sparse Symmetric Functions of ACC Circuits: Expanding the Reach of #SAT Algorithms

   https://doi.org/10.1007/s00224-022-10106-8  

   Vyas, Nikhil; Williams, R_Ryan (November 2022, Theory of Computing Systems)

   Abstract We continue the program of proving circuit lower bounds via circuit satisfiability algorithms. So far, this program has yielded several concrete results, proving that functions in$$\mathsf {Quasi}\text {-}\mathsf {NP} = \mathsf {NTIME}[n^{(\log n)^{O(1)}}]$$ $\mathrm{Quasi}-\mathrm{NP}=\mathrm{NTIME}\left[n^{{\left(\log n\right)}^{O\left(1\right)}}\right]$and other complexity classes do not have small circuits (in the worst case and/or on average) from various circuit classes$$\mathcal { C}$$ $C$, by showing that$$\mathcal { C}$$ $C$admits non-trivial satisfiability and/or#SAT algorithms which beat exhaustive search by a minor amount. In this paper, we present a new strong lower bound consequence of having a non-trivial#SAT algorithm for a circuit class$${\mathcal C}$$ $C$. Say that a symmetric Boolean functionf(x₁,…,x_n) issparseif it outputs 1 onO(1) values of$${\sum }_{i} x_{i}$$ ${\sum }_i{x}_i$. We show that for every sparsef, and for all “typical”$$\mathcal { C}$$ $C$, faster#SAT algorithms for$$\mathcal { C}$$ $C$circuits imply lower bounds against the circuit class$$f \circ \mathcal { C}$$ $f\circ C$, which may bestrongerthan$$\mathcal { C}$$ $C$itself. In particular:#SAT algorithms forn^k-size$$\mathcal { C}$$ $C$-circuits running in 2ⁿ/n^ktime (for allk) implyNEXPdoes not have$$(f \circ \mathcal { C})$$ $(f\circ C)$-circuits of polynomial size.#SAT algorithms for$$2^{n^{{\varepsilon }}}$$ $2^{n^{\epsilon }}$-size$$\mathcal { C}$$ $C$-circuits running in$$2^{n-n^{{\varepsilon }}}$$ $2^{n-n^{\epsilon }}$time (for someε> 0) implyQuasi-NPdoes not have$$(f \circ \mathcal { C})$$ $(f\circ C)$-circuits of polynomial size. Applying#SAT algorithms from the literature, one immediate corollary of our results is thatQuasi-NPdoes not haveEMAJ∘ACC⁰∘THRcircuits of polynomial size, whereEMAJis the “exact majority” function, improving previous lower bounds againstACC⁰[Williams JACM’14] andACC⁰∘THR[Williams STOC’14], [Murray-Williams STOC’18]. This is the first nontrivial lower bound against such a circuit class. 

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