Three-loop field renormalization for scalar field theory with Lorentz violation

Applying the counterterm method in minimal subtraction scheme we calculate the three-loop quantum correction to field anomalous dimension in a Lorentz-violating O($N$) self-interacting scalar field theory. We compute the Feynman diagrams using dimensional regularization and $\epsilon$-expansion techniques. As this approximation corresponds to a three-loop term, to our knowledge this is the first time in literature in which such a loop level is attained for a LV theory.Comment: 12 page

The locomotion pattern of Baurusuchus salgadoensis Carvalho, Nobre & Campos, 2005 and the distribution of Baurusuchidae in Gondwanaland

Baurusuchus salgadoensis Carvalho, Nobre & Campos, 2005 is a ziphodont mesoeucrocodylian closely related to the Sebecosuchia. Cranial features indicate that this species had terrestrial predatory habits. The postcranial skeleton of B. salgadoensis show distinctive characteristics when compared to extant Crocodylomorpha. The dorsal vertebrae are close articulated with short and craniocaudally expanded neural spines, specially the posterior ones. The sacral vertebrae are robust and have the lateral processes highly fused to the ilium. The anterior caudal vertebrae are robust as the sacral ones but their neural spines are more stout, anteriorlly expanded and higher. The close articulation of the dorsal vertebrae may indicate a less flexible dorsal spine, ideal to limb-driven methods of locomotion. The high and expanded neural spines of the posterior dorsal and sacral vertebrae are associated to muscle attachment from the osteoderms and the pelvic musculature, all used in the limb-driven locomotion. The appendicular bones of B. salgadoensis are long and stout. They show a straight aspect of their diaphysis and very well- developed epiphysis. The femur is long when compared to extant crocodilians, showing a straight aspect in lateral view and a slight sigmoid aspect in frontal view. There is a small axial torsion at its proximal end. The fourth trochanter is pronouced and posteriorlly oriented. The proximal end possess a mesial projection that articulates itself to the illium, similar to those of Protosuchia and thecodont archosaurs as the Rauisuchia. The illium of B. salgadoensis present a lateral and posteriorlly expanded postaccetabular crest, similar to those observed in Rauisuchia and Protosuchia, referred as overhanging ilium. This pattern of articulation is observed in the rauisuchian thecodonts and interpreted as a characteristic trait of those able erect-posture and limb-driven predators of Triassic environments. B. salgadoensis exoskeleton has only two dorsal osteoderm rows that run from the neck to the tip of the tail. Most of the osteoderms are wider than longer, with a round lateral portion that does not articulate to any flank osteoderm. The only morphological difference appears at the caudalmost portion of the tail, where the osteoderms are craniocaudally elongated. The medial portion has little variation along the scutes rows and may bear medial lamellar dorsoventral structures of articulation to the adjacent bony plate. The anterior articular facet is discreet and the osteoderms are not as imbricated as occur in other crocodyliforms and even may not be imbricated at all. The pelvic region osteoderms have the tallest keel of the row that runs from the anterior articular facet to posteriosmost portion of the osteoderm. B. salgadoensis has a light exoskeleton and then is less encumbranced by it. Therefore, becaming more agile. The imbrication of dorsal armor assists the limb-driven locomotion of many crocodyliform by reducing the flexibility of the dorsal spine during the high-walk (erect stance). Therefore the long and stout limb bones, overhanging crest of the illium and the light weighted armor, also allowed to B. salgadoensis to have the limb-driven locomotion without this pattern of osteoderms. These anatomical data may indicate how the Baurusuchidae had a wide distribution in Gondwanaland as they were able to move across large distances