Analytic ranks of elliptic curves over number fields

Let $E$ be an elliptic curves over the rational numbers. Let $F$ be a cyclic extension of prime degree $l$. Then, we show that the average of analytic ranks of $E(F)$ over all cyclic extension of prime degree $l$ is at most $2+r_\mathbb{Q}(E)$, where $r_\mathbb{Q}(E)$ is the analytic rank of $E(\mathbb Q)$. This bound is independent of the degree of the cyclic extension. Also, we also obtain some average rank result over $S_d$-fields

Comment on Ds∗→Dsπ0D_s^* \to D_s \pi^0 Decay

We calculate the rate for $D_s^* \rightarrow D_s \pi^0$ decay using Chiral Perturbation Theory. This isospin violating process results from $\pi^0$ - $\eta$ mixing, and its amplitude is proportional to $(m_d - m_u)/\bigl(m_s-(m_u+m_d)/2 \bigr)$. Experimental information on the branching ratio for $D_s^* \rightarrow D_s \pi^0$ can provide insight into the pattern of $SU(3)$ violation in radiative $D^*$ decays.Comment: 7 pages with 2 figures not included but available upon request, CALT-68-191

Average analytic rank of elliptic curves with prescribed torsion

We show that average analytic rank of elliptic curves with prescribed torsion $G$ is bounded for every torsion group $G$ under GRH for elliptic curve $L$-functions.Comment: Major revision: now we consider all possible torsion subgroup

Strong and Electromagnetic Decays of Two New Lambdac∗Lambda_c^* Baryons

Two recently discovered excited charm baryons are studied within the framework of Heavy Hadron Chiral Perturbation Theory. We interpret these new baryons which lie 308 \MeV and 340 \MeV above the $\Lambda_c$ as $I=0$ members of a P-wave spin doublet. Differential and total decay rates for their double pion transitions down to the $\Lambda_c$ ground state are calculated. Estimates for their radiative decay rates are also discussed. We find that the experimentally determined characteristics of the $\Lambda_c^*$ baryons may be simply understood in the effective theory.Comment: 16 pages with 4 figures not included but available upon request, CALT-68-191

Strong Decays of Strange Charmed P-Wave Mesons

Goldstone boson decays of P-wave $D_s^{**}$ mesons are studied within the framework of Heavy Hadron Chiral Perturbation Theory. We first analyze the simplest single kaon decays of these strange charmed mesons. We derive a model independent prediction for the width of $D_{s2}$ and use experimental information on $D_{s1}$ to constrain the S-wave contribution to $D_1^0$ decay. Single and double pion decay modes are then discussed and shown to be significantly restricted by isospin conservation. We conclude that the pion channels may offer the best hope for detecting one strange member of an otherwise invisible P-wave flavor multiplet.Comment: 16 pages, 2 updated figures not included but available upon request, CALT-68-1902. (Revised estimates for error on $D_{s2}$ width and for isospin violating neutral pion decay of $D'_{s1}$.

On Nonperturbative Exactness of Konishi Anomaly and the Dijkgraaf-Vafa Conjecture

In this paper we study the nonperturbative corrections to the generalized Konishi anomaly that come from the strong coupling dynamics of the gauge theory. We consider U(N) gauge theory with adjoint and Sp(N) or SO(N) gauge theory with symmetric or antisymmetric tensor. We study the algebra of chiral rotations of the matter field and show that it does not receive nonperturbative corrections. The algebra implies Wess-Zumino consistency conditions for the generalized Konishi anomaly which are used to show that the anomaly does not receive nonperturbative corrections for superpotentials of degree less than 2l+1 where 2l=3c(Adj)-c(R) is the one-loop beta function coefficient. The superpotentials of higher degree can be nonperturbatively renormalized because of the ambiguities in the UV completion of the gauge theory. We discuss the implications for the Dijkgraaf-Vafa conjecture.Comment: 23 page

More on Chiral-Nonchiral Dual Pairs

Expanding upon earlier work of Pouliot and Strassler, we construct chiral magnetic duals to nonchiral supersymmetric electric theories based upon SO(7), SO(8) and SO(9) gauge groups with various numbers of vector and spinor matter superfields. Anomalies are matched and gauge invariant operators are mapped within each dual pair. Renormalization group flows along flat directions are also examined. We find that confining phase quantum constraints in the electric theories are recovered from semiclassical equations of motion in their magnetic counterparts when the dual gauge groups are completely Higgsed.Comment: 25 pages, harvmac and tables macros, 1 figur