The Form Factors of the Gauge-Invariant Three-Gluon Vertex

The gauge-invariant three-gluon vertex obtained from the pinch technique is characterized by thirteen nonzero form factors, which are given in complete generality for unbroken gauge theory at one loop. The results are given in $d$ dimensions using both dimensional regularization and dimensional reduction, including the effects of massless gluons and arbitrary representations of massive gauge bosons, fermions, and scalars. We find interesting relations between the functional forms of the contributions from gauge bosons, fermions, and scalars. These relations hold only for the gauge-invariant pinch technique vertex and are d-dimensional incarnations of supersymmetric nonrenormalization theorems which include finite terms. The form factors are shown to simplify for $N=1,2$, and 4 supersymmetry in various dimensions. In four-dimensional non-supersymmetric theories, eight of the form factors have the same functional form for massless gluons, quarks, and scalars, when written in a physically motivated tensor basis. For QCD, these include the tree-level tensor structure which has prefactor $\beta_0=(11N_c-2N_f)/3$, another tensor with prefactor $4N_c-N_f$, and six tensors with $N_c-N_f$. In perturbative calculations our results lead naturally to an effective coupling for the three-gluon vertex which depends on three momenta and gives rise to an effective scale which governs the behavior of the vertex. The effects of nonzero internal masses are important and have a complicated threshold and pseudo-threshold structure. The results of this paper are an important part of a gauge-invariant dressed skeleton expansion and a related multi-scale analytic renormalization scheme. In this approach the scale ambiguity problem is resolved since physical kinematic invariants determine the arguments of the couplings.Comment: 53 pages, 10 figures. v2: added reference

Two loop mass effects in the static position space QCD-potential

The perturbatively calculable short distance QCD potential is known to two loops including the effect of massive quarks. Recently, a simple approximate solution in momentum space was utilized to obtain the potential in coordinate space. The latter is important in several respects. A comparison with non-perturbative lattice results is feasible in the overlap regime using light $\bar{MS}$ masses. This might be even more promising employing the concept of the force between the heavy color singlet sources, which can be easily derived from the potential. In addition, the better than two percent accuracy bottom mass determination from $\Upsilon$-mesons is sensitive to massive charm loops at the two loop order. We summarize recent results using exact one loop functions and explicit decoupling parametrizations.Comment: Version to appear in Proceedings of QCD0

Moduli of Tropical Plane Curves

We study the moduli space of metric graphs that arise from tropical plane curves. There are far fewer such graphs than tropicalizations of classical plane curves. For fixed genus $g$, our moduli space is a stacky fan whose cones are indexed by regular unimodular triangulations of Newton polygons with $g$ interior lattice points. It has dimension $2g+1$ unless $g \leq 3$ or $g = 7$. We compute these spaces explicitly for $g \leq 5$.Comment: 31 pages, 25 figure

Fourth Cranial Nerve Palsy and Brown Syndrome: Two Interrelated Congenital Cranial Dysinnervation Disorders?

Based on neuroimaging data showing absence of the trochlear nerve, congenital superior oblique palsy is now classified as a congenital cranial dysinnervation disorder. A similar absence of the abducens nerve is accompanied by misinnervation to the lateral rectus muscle from a branch of oculomotor nerve in the Duane retraction syndrome. This similarity raises the question of whether some cases of Brown syndrome could arise from a similar synkinesis between the inferior and superior oblique muscles in the setting of congenital superior oblique palsy. This hypothesis has gained support from the confluence of evidence from a number of independent studies. Using Duane syndrome as a model, we critically review the accumulating evidence that some cases of Brown syndrome are ultimately attributable to dysgenesis of the trochlear nerv

Commodity Indexed Securitization and Infrastructural Change: Turkey\u27s Role in Emerging Economies

In view of these issues, this paper proposes an optimal approach to design and regulation of commodity contingent instruments for private enterprises. The design of these instruments is likely to significantly alleviate the capital constraints in emerging markets, particularly in Eurasia. A commodity contingent security usually consists of a combination of a traditional debt security (a bond) and several units of a financial instrument, the payoff of which is in some well-defined way linked to the price of a traded commodity. Although commodity contingent securitization can, in theory, be applied at both the national and the private level, the proposals in this paper are directed mainly toward private sector funding. Government plays an important and complementary role in determining the success of these instruments. Government should provide tax subsidies to external sources of private sector capital, design a regulatory framework that preserves the integrity and professional reputation of these infant markets, and, more generally, foster a social consensus for entrepreneurship and growth

Commodity Indexed Securitization and Infrastructural Change: Turkey\u27s Role in Emerging Economies

In view of these issues, this paper proposes an optimal approach to design and regulation of commodity contingent instruments for private enterprises. The design of these instruments is likely to significantly alleviate the capital constraints in emerging markets, particularly in Eurasia. A commodity contingent security usually consists of a combination of a traditional debt security (a bond) and several units of a financial instrument, the payoff of which is in some well-defined way linked to the price of a traded commodity. Although commodity contingent securitization can, in theory, be applied at both the national and the private level, the proposals in this paper are directed mainly toward private sector funding. Government plays an important and complementary role in determining the success of these instruments. Government should provide tax subsidies to external sources of private sector capital, design a regulatory framework that preserves the integrity and professional reputation of these infant markets, and, more generally, foster a social consensus for entrepreneurship and growth