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1. Search for the lepton-flavour violating decays B0 → K*0τ±μ∓

   https://doi.org/10.1007/JHEP06(2023)143  

   Aaij, R.; Abdelmotteleb, A. S.; Abellan Beteta, C.; Abudinén, F.; Ackernley, T.; Adeva, B.; Adinolfi, M.; Afsharnia, H.; Agapopoulou, C.; Aidala, C. A.; et al (June 2023, Journal of High Energy Physics)

   A bstract A first search for the lepton-flavour violating decays B 0 → K *0 τ ± μ ∓ is presented. The analysis is performed using a sample of proton-proton collision data, collected with the LHCb detector at centre-of-mass energies of 7, 8 and 13 TeV between 2011 and 2018, corresponding to an integrated luminosity of 9 fb − 1 . No significant signal is observed, and upper limits on the branching fractions are determined to be $$ \mathcal{B}\left({B}^0\to {K}^{\ast 0}{\tau}^{+}{\mu}^{-}\right)<1.0(1.2)\times {10}^{-5} $$ B B 0 → K ∗ 0 τ + μ − < 1.0 1.2 × 10 − 5 and $$ \mathcal{B}\left({B}^0\to {K}^{\ast 0}{\tau}^{-}{\mu}^{+}\right)<8.2(9.8)\times {10}^{-6} $$ B B 0 → K ∗ 0 τ − μ + < 8.2 9.8 × 10 − 6 at the 90% (95%) confidence level. 

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2. Search for the dark photon in B0 → A′A′, A′ → e+e−, μ+μ−, and π+π− decays at Belle

   https://doi.org/10.1007/JHEP04(2021)191  

   Park, S.-H.; Kwon, Y.-J.; Adachi, I.; Aihara, H.; Al Said, S.; Asner, D. M.; Atmacan, H.; Aushev, T.; Ayad, R.; Babu, V.; et al (April 2021, Journal of High Energy Physics)

   null (Ed.)

   A bstract We present a search for the dark photon A ′ in the B 0 → A ′ A ′ decays, where A ′ subsequently decays to e + e − , μ + μ − , and π + π − . The search is performed by analyzing 772 × 10 6 $$ B\overline{B} $$ B B ¯ events collected by the Belle detector at the KEKB e + e − energy-asymmetric collider at the ϒ(4 S ) resonance. No signal is found in the dark photon mass range 0 . 01 GeV /c 2 ≤ m A ′ ≤ 2 . 62 GeV /c 2 , and we set upper limits of the branching fraction of B 0 → A ′ A ′ at the 90% confidence level. The products of branching fractions, $$ \mathrm{\mathcal{B}}\left({B}^0\to A^{\prime }A^{\prime}\right)\times \mathrm{\mathcal{B}}{\left(A\prime \to {e}^{+}{e}^{-}\right)}^2 $$ ℬ B 0 → A ′ A ′ × ℬ A ′ → e + e − 2 and $$ \mathrm{\mathcal{B}}\left({B}^0\to A^{\prime }A^{\prime}\right)\times \mathrm{\mathcal{B}}{\left(A\prime \to {\mu}^{+}{\mu}^{-}\right)}^2 $$ ℬ B 0 → A ′ A ′ × ℬ A ′ → μ + μ − 2 , have limits of the order of 10 − 8 depending on the A ′ mass. Furthermore, considering A ′ decay rate to each pair of charged particles, the upper limits of $$ \mathrm{\mathcal{B}}\left({B}^0\to A^{\prime }A^{\prime}\right) $$ ℬ B 0 → A ′ A ′ are of the order of 10 − 8 –10 − 5 . From the upper limits of $$ \mathrm{\mathcal{B}}\left({B}^0\to A^{\prime }A^{\prime}\right) $$ ℬ B 0 → A ′ A ′ , we obtain the Higgs portal coupling for each assumed dark photon and dark Higgs mass. The Higgs portal couplings are of the order of 10 − 2 –10 − 1 at $$ {m}_{h\prime}\simeq {m}_{B^0} $$ m h ′ ≃ m B 0 ± 40 MeV /c 2 and 10 − 1 –1 at $$ {m}_{h\prime}\simeq {m}_{B^0} $$ m h ′ ≃ m B 0 ± 3 GeV /c 2 . 

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3. Modular invariance in superstring theory from $$ \mathcal{N} $$ = 4 super-Yang-Mills

   https://doi.org/10.1007/JHEP11(2020)016  

   Chester, Shai M.; Green, Michael B.; Pufu, Silviu S.; Wang, Yifan; Wen, Congkao (November 2020, Journal of High Energy Physics)

   A bstract We study the four-point function of the lowest-lying half-BPS operators in the $$ \mathcal{N} $$ N = 4 SU( N ) super-Yang-Mills theory and its relation to the flat-space four-graviton amplitude in type IIB superstring theory. We work in a large- N expansion in which the complexified Yang-Mills coupling τ is fixed. In this expansion, non-perturbative instanton contributions are present, and the SL(2 , ℤ) duality invariance of correlation functions is manifest. Our results are based on a detailed analysis of the sphere partition function of the mass-deformed SYM theory, which was previously computed using supersymmetric localization. This partition function determines a certain integrated correlator in the undeformed $$ \mathcal{N} $$ N = 4 SYM theory, which in turn constrains the four-point correlator at separated points. In a normalization where the two-point functions are proportional to N 2 − 1 and are independent of τ and $$ \overline{\tau} $$ τ ¯ , we find that the terms of order $$ \sqrt{N} $$ N and $$ 1/\sqrt{N} $$ 1 / N in the large N expansion of the four-point correlator are proportional to the non-holomorphic Eisenstein series $$ E\left(\frac{3}{2},\tau, \overline{\tau}\right) $$ E 3 2 τ τ ¯ and $$ E\left(\frac{5}{2},\tau, \overline{\tau}\right) $$ E 5 2 τ τ ¯ , respectively. In the flat space limit, these terms match the corresponding terms in the type IIB S-matrix arising from R 4 and D 4 R 4 contact inter-actions, which, for the R 4 case, represents a check of AdS/CFT at finite string coupling. Furthermore, we present striking evidence that these results generalize so that, at order $$ {N}^{\frac{1}{2}-m} $$ N 1 2 − m with integer m ≥ 0, the expansion of the integrated correlator we study is a linear sum of non-holomorphic Eisenstein series with half-integer index, which are manifestly SL(2 , ℤ) invariant. 

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4. Uniqueness and Symmetry for the Mean Field Equation on Arbitrary Flat Tori

   https://doi.org/10.1093/imrn/rnaa109  

   Gu, Guangze; Gui, Changfeng; Hu, Yeyao; Li, Qinfeng (May 2020, International Mathematics Research Notices)

   Abstract We study the following mean field equation on a flat torus $$T:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau )$$: $$\begin{equation*} \varDelta u + \rho \left(\frac{e^{u}}{\int_{T}e^u}-\frac{1}{|T|}\right)=0, \end{equation*}$$where $$ \tau \in \mathbb{C}, \mbox{Im}\ \tau>0$$, and $|T|$ denotes the total area of the torus. We first prove that the solutions are evenly symmetric about any critical point of $$u$$ provided that $$\rho \leq 8\pi $$. Based on this crucial symmetry result, we are able to establish further the uniqueness of the solution if $$\rho \leq \min{\{8\pi ,\lambda _1(T)|T|\}}$$. Furthermore, we also classify all one-dimensional solutions by showing that the level sets must be closed geodesics. 

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5. Maps from Feigin and Odesskii's elliptic algebras to twisted homogeneous coordinate rings

   https://doi.org/10.1017/fms.2020.60  

   Chirvasitu, Alex; Kanda, Ryo; Smith, S. Paul (January 2021, Forum of Mathematics, Sigma)

   null (Ed.)

   Abstract The elliptic algebras in the title are connected graded $$\mathbb {C}$$ -algebras, denoted $$Q_{n,k}(E,\tau )$$ , depending on a pair of relatively prime integers $$n>k\ge 1$$ , an elliptic curve E and a point $$\tau \in E$$ . This paper examines a canonical homomorphism from $$Q_{n,k}(E,\tau )$$ to the twisted homogeneous coordinate ring $$B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$$ on the characteristic variety $$X_{n/k}$$ for $$Q_{n,k}(E,\tau )$$ . When $$X_{n/k}$$ is isomorphic to $E^g$ or the symmetric power $S^gE$ , we show that the homomorphism $$Q_{n,k}(E,\tau ) \to B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$$ is surjective, the relations for $$B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$$ are generated in degrees $$\le 3$$ and the noncommutative scheme $$\mathrm {Proj}_{nc}(Q_{n,k}(E,\tau ))$$ has a closed subvariety that is isomorphic to $E^g$ or $S^gE$ , respectively. When $$X_{n/k}=E^g$$ and $$\tau =0$$ , the results about $$B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$$ show that the morphism $$\Phi _{|\mathcal {L}_{n/k}|}:E^g \to \mathbb {P}^{n-1}$$ embeds $E^g$ as a projectively normal subvariety that is a scheme-theoretic intersection of quadric and cubic hypersurfaces. 

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