Wegner-Houghton equation and derivative expansion

We study the derivative expansion for the effective action in the framework of the Exact Renormalization Group for a single component scalar theory. By truncating the expansion to the first two terms, the potential $U_k$ and the kinetic coefficient $Z_k$, our analysis suggests that a set of coupled differential equations for these two functions can be established under certain smoothness conditions for the background field and that sharp and smooth cut-off give the same result. In addition we find that, differently from the case of the potential, a further expansion is needed to obtain the differential equation for $Z_k$, according to the relative weight between the kinetic and the potential terms. As a result, two different approximations to the $Z_k$ equation are obtained. Finally a numerical analysis of the coupled equations for $U_k$ and $Z_k$ is performed at the non-gaussian fixed point in $D<4$ dimensions to determine the anomalous dimension of the field.Comment: 15 pages, 3 figure

Generalization of geometric phase to completely positive maps

We generalize the notion of relative phase to completely positive maps with known unitary representation, based on interferometry. Parallel transport conditions that define the geometric phase for such maps are introduced. The interference effect is embodied in a set of interference patterns defined by flipping the environment state in one of the two paths. We show for the qubit that this structure gives rise to interesting additional information about the geometry of the evolution defined by the CP map.Comment: Minor revision. 2 authors added. 4 pages, 2 figures, RevTex

Effective chiral lagrangian in the chiral limit from the instanton vacuum

We study the effective chiral Lagrangian in the chiral limit from the instanton vacuum. Starting from the nonlocal effective chiral action, we derive the effective chiral Lagrangian, using the derivative expansion to order $O(p^4)$ in the chiral limit. The low energy constants, $L_1$, $L_2$, and $L_3$ are determined and compared with various models and the corresponding empirical data. The results are in a good agreement with the data. We also discuss about the upper limit of the sigma meson, based on the present results.Comment: 14 pages, 5 figures, submitted to Phys.Rev.

A Pearson-Dirichlet random walk

A constrained diffusive random walk of n steps and a random flight in Rd, which can be expressed in the same terms, were investigated independently in recent papers. The n steps of the walk are identically and independently distributed random vectors of exponential length and uniform orientation. Conditioned on the sum of their lengths being equal to a given value l, closed-form expressions for the distribution of the endpoint of the walk were obtained altogether for any n for d=1, 2, 4 . Uniform distributions of the endpoint inside a ball of radius l were evidenced for a walk of three steps in 2D and of two steps in 4D. The previous walk is generalized by considering step lengths which are distributed over the unit (n-1) simplex according to a Dirichlet distribution whose parameters are all equal to q, a given positive value. The walk and the flight above correspond to q=1. For any d >= 3, there exist, for integer and half-integer values of q, two families of Pearson-Dirichlet walks which share a common property. For any n, the d components of the endpoint are jointly distributed as are the d components of a vector uniformly distributed over the surface of a hypersphere of radius l in a space Rk whose dimension k is an affine function of n for a given d. Five additional walks, with a uniform distribution of the endpoint in the inside of a ball, are found from known finite integrals of products of powers and Bessel functions of the first kind. They include four different walks in R3 and two walks in R4. Pearson-Liouville random walks, obtained by distributing the total lengths of the previous Pearson-Dirichlet walks, are finally discussed.Comment: 33 pages 1 figure, the paper includes the content of a recently submitted work together with additional results and an extended section on Pearson-Liouville random walk

Geochronology and isotope geochemistry of Eocene dykes intruding the Ladakh Batholith

In order to determine the extent and timing of dyke formation in the Ladakh Batholith we examined about 30 mostly andesitic dykes intruding the Ladakh batholith in a ca. 50 km wide area to the west of Leh (NW India). The dykes in the east of the area trend E-NE and those in the west trend N-NW. The difference in orientation is also evident in the petrography and isotopic signatures. The eastern dykes contain corroded quartz xenocrysts and show negative ε0(Nd) and positive ε0(Sr) values, where as the western dykes do not contain quartz xenocrysts and exhibit positive ε0(Nd) and near-zero ε0(Sr) values. The variability in Sr-Nd isotopes (ε0(Nd) = 3.6 to −9.6, ε0(Sr) = 0.4 to 143) and the quartz xenocrysts can best be explained by (differing degrees of) crustal assimilation of the parent magma of the dykes. Separated minerals from five dykes were dated by 40Ar-39Ar incremental heating: amphibole ages range between 50 and 54 Ma, and one biotite dated both by Rb-Sr and by 40Ar-39Ar gave an age of 45 Ma. One dated pseudotachylyte sample attests to brittle faulting at ca. 54 Ma. The combination of structural field evidence with petrographic, isotopic and geochronological analyses demonstrates that the dykes did not form from a single, progressively differentiating magma chamber, despite having formed in the same tectonic setting around the same time, and that processes such as crustal assimilation and magma mixing/mingling also played a significant role in magma petrogenesis

Geochronology and isotope geochemistry of Eocene dykes intruding the Ladakh Batholith

In order to determine the extent and timing of dyke formation related to possible E-W extension along the southern margin of Eurasia during Early Cenozoic time, we examined ca. 30 mostly andesitic dykes intruding the Ladakh batholith from 10 to 50km west of Leh (NW India). The dykes in the east of the area trend E-NE and those in the west trend N-NW. The difference in orientation is also evident in the petrography and isotopic signatures. The eastern dykes contain corroded quartz xenocrysts and show negative ε(Nd) and positive ε(Sr) values, whereas the western dykes do not contain quartz xenocrysts and exhibit positive ε(Nd) and near-zero ε(Sr) values. The variability in Sr-Nd isotopes (ε(Nd)=3.6 to -9.6, ε(Sr)=0.4 to 143) and the quartz xenocrysts can best be explained by (differing degrees of) crustal assimilation of the parent magma of the dykes. Separated minerals from five dykes were dated by Ar-Ar incremental heating: amphibole ages range between 50 and 54Ma, and one biotite separate dated both by Rb-Sr and Ar-Ar gave an age of 45Ma. One dated pseudotachylyte sample attests to brittle faulting at ca. 54Ma. The combination of structural field evidence with petrographic, isotopic and geochronological analyses demonstrates that the dykes did not form from a single, progressively differentiating magma chamber, despite having formed in the same tectonic setting around the same time, and that processes such as crustal assimilation and magma mixing/mingling, also played a significant role in magma petrogenesis