Renormalization of the Inverse Square Potential

The quantum-mechanical D-dimensional inverse square potential is analyzed using field-theoretic renormalization techniques. A solution is presented for both the bound-state and scattering sectors of the theory using cutoff and dimensional regularization. In the renormalized version of the theory, there is a strong-coupling regime where quantum-mechanical breaking of scale symmetry takes place through dimensional transmutation, with the creation of a single bound state and of an energy-dependent s-wave scattering matrix element.Comment: 5 page

The unfolding clinical spectrum of holoprosencephaly due to mutations in SHH, ZIC2, SIX3 and TGIF genes

Holoprosencephaly is a severe malformation of the brain characterized by abnormal formation and separation of the developing central nervous system. The prevalence is 1:250 during early embryogenesis, the live-born prevalence is 1:16 000. The etiology of HPE is extremely heterogeneous and can be teratogenic or genetic. We screened four known HPE genes in a Dutch cohort of 86 non-syndromic HPE index cases, including 53 family members. We detected 21 mutations (24.4%), 3 in SHH, 9 in ZIC2 and 9 in SIX3. Eight mutations involved amino-acid substitutions, 7 ins/del mutations, 1 frame-shift, 3 identical poly-alanine tract expansions and 2 gene deletions. Pathogenicity of mutations was presumed based on de novo character, predicted non-functionality of mutated proteins, segregation of mutations with affected family-members or combinations of these features. Two mutations were reported previously. SNP array confirmed detected deletions; one spanning the ZIC2/ZIC5 genes (approx. 100 kb) the other a 1.45 Mb deletion including SIX2/SIX3 genes. The mutation percentage (24%) is comparable with previous reports, but we detected significantly less mutations in SHH: 3.5 vs 10.7% (P=0.043) and significantly more in SIX3: 10.5 vs 4.3% (P=0.018). For TGIF1 and ZIC2 mutation the rate was in conformity with earlier reports. About half of the mutations were de novo, one was a germ line mosaic. The familial mutations displayed extensive heterogeneity in clinical manifestation. Of seven familial index patients only two parental carriers showed minor HPE signs, five were completely asymptomatic. Therefore, each novel mutation should be considered as a risk factor for clinically manifest HPE, with the caveat of reduced clinical penetrance