A bstract We report the measurement of the twophoton decay width of χ c 2 (1 P ) in twophoton processes at the Belle experiment. We analyze the process γγ → χ c 2 (1 P ) → J/ψγ , J/ψ → ℓ + ℓ − ( ℓ = e or μ ) using a data sample of 971 fb − 1 collected with the Belle detector at the KEKB e + e − collider. In this analysis, the product of the twophoton decay width of χ c 2 (1 P ) and the branching fraction is determined to be $$ {\Gamma}_{\gamma \gamma}\left({\chi}_{c2}(1P)\right)\mathcal{B}\left({\chi}_{c2}(1P)\to J/\psi \gamma \right)\mathcal{B}\left(J/\psi \to {\ell}^{+}{\ell}^{}\right)=14.8\pm 0.3\left(\textrm{stat}.\right)\pm 0.7\left(\textrm{syst}.\right) $$ Γ γγ χ c 2 1 P B χ c 2 1 P → J / ψγ B J / ψ → ℓ + ℓ − = 14.8 ± 0.3 stat . ± 0.7 syst . eV, which corresponds to Γ γγ ( χ c 2 (1 P )) = 653 ± 13(stat.) ± 31(syst.) ± 17(B.R.) eV, where the third uncertainty is from $$ \mathcal{B} $$ B ( χ c 2 (1 P ) → J/ψγ ) and $$ \mathcal{B} $$ B ( J/ψ → ℓ + ℓ − ).
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Averages Along the Primes: Improving and Sparse Bounds
Abstract Consider averages along the prime integers ℙ given by {\mathcal{A}_N}f(x) = {N^{  1}}\sum\limits_{p \in \mathbb{P}:p \le N} {(\log p)f(x  p).} These averages satisfy a uniform scalefree ℓ p improving estimate. For all 1 < p < 2, there is a constant C p so that for all integer N and functions f supported on [0, N ], there holds {N^{  1/p'}}{\left\ {{\mathcal{A}_N}f} \right\_{\ell p'}} \le {C_p}{N^{  1/p}}{\left\ f \right\_{\ell p}}. The maximal function 𝒜 * f = sup N 𝒜 N f  satisfies ( p , p ) sparse bounds for all 1 < p < 2. The latter are the natural variants of the scalefree bounds. As a corollary, 𝒜 * is bounded on ℓ p ( w ), for all weights w in the Muckenhoupt 𝒜 p class. No prior weighted inequalities for 𝒜 * were known.
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 Award ID(s):
 1800689
 NSFPAR ID:
 10176650
 Date Published:
 Journal Name:
 Concrete Operators
 Volume:
 7
 Issue:
 1
 ISSN:
 22993282
 Page Range / eLocation ID:
 45 to 54
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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