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1. Search for exotic decays of the Higgs boson to a pair of pseudoscalars in the $$\upmu \upmu \text{ b } \text{ b } $$ and $$\uptau \uptau \text{ b } \text{ b } $$ final states

   https://doi.org/10.1140/epjc/s10052-024-12727-4  

   Hayrapetyan, A ; Tumasyan, A ; Adam, W ; Andrejkovic, J W ; Bergauer, T ; Chatterjee, S ; Damanakis, K ; Dragicevic, M ; Del_Valle, A Escalante ; Hussain, P S ; et al ( May 2024 , The European Physical Journal C)

   A search for exotic decays of the Higgs boson ($$\text {H}$$$\text{H}$) with a mass of 125$$\,\text {Ge}\hspace{-.08em}\text {V}$$$\text{Ge}\text{V}$to a pair of light pseudoscalars$$\text {a}_{1} $$${\text{a}}_1$is performed in final states where one pseudoscalar decays to two$${\textrm{b}}$$$\text{b}$quarks and the other to a pair of muons or$$\tau $$$\tau$leptons. A data sample of proton–proton collisions at$$\sqrt{s}=13\,\text {Te}\hspace{-.08em}\text {V} $$$\sqrt{s}=13\text{Te}\text{V}$corresponding to an integrated luminosity of 138$$\,\text {fb}^{-1}$$${\text{fb}}^{-1}$recorded with the CMS detector is analyzed. No statistically significant excess is observed over the standard model backgrounds. Upper limits are set at 95% confidence level ($$\text {CL}$$$\text{CL}$) on the Higgs boson branching fraction to$$\upmu \upmu \text{ b } \text{ b } $$$\mu \mu \text{b}\text{b}$and to$$\uptau \uptau \text{ b } \text{ b },$$$\tau \tau \text{b}\text{b},$via a pair of$$\text {a}_{1} $$${\text{a}}_1$s. The limits depend on the pseudoscalar mass$$m_{\text {a}_{1}}$$$m_{{\text{a}}_1}$and are observed to be in the range (0.17–3.3) $$\times 10^{-4}$$$\times {10}^{-4}$and (1.7–7.7) $$\times 10^{-2}$$$\times {10}^{-2}$in the$$\upmu \upmu \text{ b } \text{ b } $$$\mu \mu \text{b}\text{b}$and$$\uptau \uptau \text{ b } \text{ b } $$$\tau \tau \text{b}\text{b}$final states, respectively. In the framework of models with two Higgs doublets and a complex scalar singlet (2HDM+S), the results of the two final states are combined to determine upper limits on the branching fraction$${\mathcal {B}}(\text {H} \rightarrow \text {a}_{1} \text {a}_{1} \rightarrow \ell \ell \text{ b } \text{ b})$$$B(\text{H}\to {\text{a}}_1{\text{a}}_1\to \ell \ell \text{b}\text{b})$at 95%$$\text {CL}$$$\text{CL}$, with$$\ell $$$\ell$being a muon or a$$\uptau $$$\tau$lepton. For different types of 2HDM+S, upper bounds on the branching fraction$${\mathcal {B}}(\text {H} \rightarrow \text {a}_{1} \text {a}_{1} )$$$B(\text{H}\to {\text{a}}_1{\text{a}}_1)$are extracted from the combination of the two channels. In most of the Type II 2HDM+S parameter space,$${\mathcal {B}}(\text {H} \rightarrow \text {a}_{1} \text {a}_{1} )$$$B(\text{H}\to {\text{a}}_1{\text{a}}_1)$values above 0.23 are excluded at 95%$$\text {CL}$$$\text{CL}$for$$m_{\text {a}_{1}}$$$m_{{\text{a}}_1}$values between 15 and 60$$\,\text {Ge}\hspace{-.08em}\text {V}$$$\text{Ge}\text{V}$.

    

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2. Characterizing Schwarz maps by tracial inequalities

   https://doi.org/10.1007/s11005-023-01636-4  

   Carlen, Eric ; Müller-Hermes, Alexander ( February 2023 , Letters in Mathematical Physics)

   Let$$\phi $$$\varphi$be a positive map from the$$n\times n$$$n\times n$matrices$$\mathcal {M}_n$$$M_n$to the$$m\times m$$$m\times m$matrices$$\mathcal {M}_m$$$M_m$. It is known that$$\phi $$$\varphi$is 2-positive if and only if for all$$K\in \mathcal {M}_n$$$K\in M_n$and all strictly positive$$X\in \mathcal {M}_n$$$X\in M_n$,$$\phi (K^*X^{-1}K) \geqslant \phi (K)^*\phi (X)^{-1}\phi (K)$$$\varphi \left(K^{\ast }{X}^{-1}K\right)⩾\varphi {\left(K\right)}^{\ast }\varphi {\left(X\right)}^{-1}\varphi \left(K\right)$. This inequality is not generally true if$$\phi $$$\varphi$is merely a Schwarz map. We show that the corresponding tracial inequality$${{\,\textrm{Tr}\,}}[\phi (K^*X^{-1}K)] \geqslant {{\,\textrm{Tr}\,}}[\phi (K)^*\phi (X)^{-1}\phi (K)]$$$\text{Tr}\left[\varphi \left(K^{\ast }{X}^{-1}K\right)\right]⩾\text{Tr}\left[\varphi {\left(K\right)}^{\ast }\varphi {\left(X\right)}^{-1}\varphi \left(K\right)\right]$holds for a wider class of positive maps that is specified here. We also comment on the connections of this inequality with various monotonicity statements that have found wide use in mathematical physics, and apply it, and a close relative, to obtain some new, definitive results.

    

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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)

   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. The geometric distribution of Selmer groups of elliptic curves over function fields

   https://doi.org/10.1007/s00208-022-02429-1  

   Feng, Tony ; Landesman, Aaron ; Rains, Eric M. ( September 2022 , Mathematische Annalen)

   Fix a positive integernand a finite field$${\mathbb {F}}_q$$$F_q$. We study the joint distribution of the rank$${{\,\mathrm{rk}\,}}(E)$$$\mathrm{rk}\left(E\right)$, then-Selmer group$$\text {Sel}_n(E)$$${\text{Sel}}_n\left(E\right)$, and then-torsion in the Tate–Shafarevich group Equation missing<#comment/>asEvaries over elliptic curves of fixed height$$d \ge 2$$$d\ge 2$over$${\mathbb {F}}_q(t)$$$F_q\left(t\right)$. We compute this joint distribution in the largeqlimit. We also show that the “largeq, then large height” limit of this distribution agrees with the one predicted by Bhargava–Kane–Lenstra–Poonen–Rains.

    

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5. A cluster of results on amplituhedron tiles

   https://doi.org/10.1007/s11005-024-01854-4  

   Even-Zohar, Chaim ; Lakrec, Tsviqa ; Parisi, Matteo ; Sherman-Bennett, Melissa ; Tessler, Ran ; Williams, Lauren ( September 2024 , Letters in Mathematical Physics)

   The amplituhedron is a mathematical object which was introduced to provide a geometric origin of scattering amplitudes in$$\mathcal {N}=4$$$N=4$super Yang–Mills theory. It generalizescyclic polytopesand thepositive Grassmannianand has a very rich combinatorics with connections to cluster algebras. In this article, we provide a series of results about tiles and tilings of the$$m=4$$$m=4$amplituhedron. Firstly, we provide a full characterization of facets of BCFW tiles in terms of cluster variables for$$\text{ Gr}_{4,n}$$${\text{Gr}}_{4,n}$. Secondly, we exhibit a tiling of the$$m=4$$$m=4$amplituhedron which involves a tile which does not come from the BCFW recurrence—thespuriontile, which also satisfies all cluster properties. Finally, strengthening the connection with cluster algebras, we show that each standard BCFW tile is the positive part of a cluster variety, which allows us to compute the canonical form of each such tile explicitly in terms of cluster variables for$$\text{ Gr}_{4,n}$$${\text{Gr}}_{4,n}$. This paper is a companion to our previous paper “Cluster algebras and tilings for the$$m=4$$$m=4$amplituhedron.”

    

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