Using Covariant Polarisation Sums in QCD

Covariant gauges lead to spurious, non-physical polarisation states of gauge bosons. In QED, the use of the Feynman gauge, $$\sum _{\lambda } \varepsilon _\mu ^{(\lambda )}\varepsilon _\nu ^{(\lambda )*} = -\eta _{\mu \nu }$$ ∑ λ ε μ ( λ ) ε ν ( λ ) ∗ = - η μ ν , is justified by the Ward identity which ensures that the contributions of non-physical polarisation states cancel in physical observables. In contrast, the same replacement can be applied only to a single external gauge boson in squared amplitudes of non-abelian gauge theories like QCD. In general, the use of this replacement requires to include external Faddeev–Popov ghosts. We present a pedagogical derivation of these ghost contributions applying the optical theorem and the Cutkosky cutting rules. We find that the resulting cross terms $$\mathcal {A}(c_1,\bar{c}_1;\ldots )\mathcal {A}(\bar{c}_1,c_1;\ldots )^*$$ A ( c 1 , c ¯ 1 ; … ) A ( c ¯ 1 , c 1 ; … ) ∗ between ghost amplitudes cannot be transformed into $$(-1)^{n/2}|\mathcal {A}(c_1,\bar{c}_1;\ldots )|^2$$ ( - 1 ) n / 2 | A ( c 1 , c ¯ 1 ; … ) | 2 in the case of more than two ghosts. Thus the Feynman rule stated in the literature holds only for two external ghosts, while it is in general incorrect.

Analysis of fabrics and metal threads from two Andalusian liturgical vestments from the seventeenth and eighteenth centuries: a multitechnical approach

This manuscript provided one of the first scientific studies performed on ancient Spanish liturgical vestments by using a multitechnical physicochemical approach. The interlining and lining fabrics were done by cotton and the core of threads by silk. Silver, gold and little amounts of copper were detected. The highest values of gold were found in the metal threads of the cope and the gallon of the chasuble. Two layers were identified: the internal composed by silver and the external by silver and gold. The fibers were proved to be degraded and also the surface of the metal, where scratches and silver-based compounds were detected due to environmental and anthropogenic actions. Indigo was employed as dye in the blue and green fabrics. Wax was detected in the yellow fibers, although other dyes could not be discarded. The materials used in the restoration processes were of minor quality.