The many-body reciprocal theorem and swimmer hydrodynamics

We present a reinterpretation and extension of the reciprocal theorem for swimmers, extending its application from the motion of a single swimmer in an unbounded domain to the general setting, giving results for both swimmer interactions and general hydrodynamics. We illustrate the method for a squirmer near a planar surface, recovering standard literature results and extending them to a general squirming set, to motion in the presence of a ciliated surface, and expressions for the flow field throughout the domain. Finally, we present exact results for the hydrodynamics in two dimensions which shed light on the near-field behaviour.Comment: 6 pages, 6 figure

The Pinch Technique to All Orders

The generalization of the pinch technique to all orders in perturbation theory is presented. The effective Green's functions constructed with this procedure are singled out in a unique way through the full exploitation of the underlying Becchi-Rouet-Stora-Tyutin symmetry. A simple all-order correspondence between the pinch technique and the background field method in the Feynman gauge is established.Comment: 10 pages, 4 figures; one reference added, typos corrected; final version to match the pubblished on

Exact solutions for hydrodynamic interactions of two squirming spheres

We provide exact solutions of the Stokes equations for a squirming sphere close to a no-slip surface, both planar and spherical, and for the interactions between two squirmers, in three dimensions. These allow the hydrodynamic interactions of swimming microscopic organisms with confining boundaries, or each other, to be determined for arbitrary separation and, in particular, in the close proximity regime where approximate methods based on point singularity descriptions cease to be valid. We give a detailed description of the circular motion of an arbitrary squirmer moving parallel to a no-slip spherical boundary or flat free surface at close separation, finding that the circling generically has opposite sense at free surfaces and at solid boundaries. While the asymptotic interaction is symmetric under head-tail reversal of the swimmer, in the near field microscopic structure can result in significant asymmetry. We also find the translational velocity towards the surface for a simple model with only the lowest two squirming modes. By comparing these to asymptotic approximations of the interaction we find that the transition from near- to far-field behaviour occurs at a separation of about two swimmer diameters. These solutions are for the rotational velocity about the wall normal, or common diameter of two spheres, and the translational speed along that same direction, and are obtained using the Lorentz reciprocal theorem for Stokes flows in conjunction with known solutions for the conjugate Stokes drag problems, the derivations of which are demonstrated here for completeness. The analogous motions in the perpendicular directions, i.e. parallel to the wall, currently cannot be calculated exactly since the relevant Stokes drag solutions needed for the reciprocal theorem are not available.Comment: 27 pages, 7 figure

Electroweak pinch technique to all orders

The generalization of the pinch technique to all orders in the electroweak sector of the Standard Model within the class of the renormalizable 't Hooft gauges, is presented. In particular, both the all-order PT gauge-boson-- and scalar--fermions vertices, as well as the diagonal and mixed gauge-boson and scalar self-energies are explicitly constructed. This is achieved through the generalization to the Standard Model of the procedure recently applied to the QCD case, which consist of two steps: (i) the identification of special Green's functions, which serve as a common kernel to all self-energy and vertex diagrams, and (ii) the study of the (on-shell) Slavnov-Taylor identities they satisfy. It is then shown that the ghost, scalar and scalar--gauge-boson Green's functions appearing in these identities capture precisely the result of the pinching action at arbitrary order. It turns out that the aforementioned Green's functions play a crucial role, their net effect being the non-trivial modification of the ghost, scalar and scalar--gauge-boson diagrams of the gauge-boson-- or scalar--fermions vertex we have started from, in such a way as to dynamically generate the characteristic ghost and scalar sector of the background field method. The pinch technique gauge-boson and scalar self-energies are also explicitly constructed by resorting to the method of the background-quantum identities.Comment: 48 pages, 8 figures; v2: typos correcte

The pinch technique at two-loops: The case of mass-less Yang-Mills theories

The generalization of the pinch technique beyond one loop is presented. It is shown that the crucial physical principles of gauge-invariance, unitarity, and gauge-fixing-parameter independence single out at two loops exactly the same algorithm which has been used to define the pinch technique at one loop, without any additional assumptions. The two-loop construction of the pinch technique gluon self-energy, and quark-gluon vertex are carried out in detail for the case of mass-less Yang-Mills theories, such as perturbative QCD. We present two different but complementary derivations. First we carry out the construction by directly rearranging two-loop diagrams. The analysis reveals that, quite interestingly, the well-known one-loop correspondence between the pinch technique and the background field method in the Feynman gauge persists also at two-loops. The renormalization is discussed in detail, and is shown to respect the aforementioned correspondence. Second, we present an absorptive derivation, exploiting the unitarity of the $S$-matrix and the underlying BRS symmetry; at this stage we deal only with tree-level and one-loop physical amplitudes. The gauge-invariant sub-amplitudes defined by means of this absorptive construction correspond precisely to the imaginary parts of the $n$-point functions defined in the full two-loop derivation, thus furnishing a highly non-trivial self-consistency check for the entire method. Various future applications are briefly discussed.Comment: 29 pages, uses Revtex, 22 Figures in a separate ps fil

On the observability of the neutrino charge radius

It is shown that the probe-independent charge radius of the neutrino is a physical observable; as such, it may be extracted from experiment, at least in principle. This is accomplished by expressing a set of experimental neutrino-electron cross-sections in terms of the finite charge radius and two additional gauge- and renormalization-group-invariant quantities, corresponding to the electroweak effective charge and mixing angle.Comment: 10 pages, 1 figure; a typo in Eq.1 corrected, some comments adde

The Two-Loop Pinch Technique in the Electroweak Sector

The generalization of the two-loop Pinch Technique to the Electroweak Sector of the Standard Model is presented. We restrict ourselves to the case of conserved external currents, and provide a detailed analysis of both the charged and neutral sectors. The crucial ingredient for this construction is the identification of the parts discarded during the pinching procedure with well-defined contributions to the Slavnov-Taylor identity satisfied by the off-shell one-loop gauge-boson vertices; the latter are nested inside the conventional two-loop self-energies. It is shown by resorting to a set of powerful identities that the two-loop effective Pinch Technique self-energies coincide with the corresponding ones computed in the Background Feynman gauge. The aforementioned identities are derived in the context of the Batalin-Vilkovisky formalism, a fact which enables the individual treatment of the self-energies of the photon and the $Z$-boson. Some possible phenomenological applications are briefly discussed.Comment: 50 pages, uses axodra

Low-Energy Constraints on New Physics Revisited

It is possible to place constraints on non-Standard-Model gauge-boson self-couplings and other new physics by studying their one-loop contributions to precisely measured observables. We extend previous analyses which constrain such nonstandard couplings, and we present the results in a compact and transparent form. Particular attention is given to comparing results for the light-Higgs scenario, where nonstandard effects are parameterized by an effective Lagrangian with a linear realization of the electroweak symmetry breaking sector, and the heavy-Higgs/strongly interacting scenario, described by the electroweak chiral Lagrangian. The constraints on nonstandard gauge-boson self-couplings which are obtained from a global analysis of low-energy data and LEP/SLC measurements on the Z pole are updated and improved from previous studies. Replaced version: tables and figures of Section VIb recalculated. There were roundoff problems, especially in Fig. 8. Text unchanged.Comment: \documentstyle[preprint,aps,floats,psfig]{revtex}, 10 figures, postscript version available from ftp://ftp.kek.jp/kek/preprints/TH/TH-51

Gauge-Independent Off-Shell Fermion Self-Energies at Two Loops: The Cases of QED and QCD

We use the pinch technique formalism to construct the gauge-independent off-shell two-loop fermion self-energy, both for Abelian (QED) and non-Abelian (QCD) gauge theories. The new key observation is that all contributions originating from the longitudinal parts of gauge boson propagators, by virtue of the elementary tree-level Ward identities they trigger, give rise to effective vertices, which do not exist in the original Lagrangian; all such vertices cancel diagrammatically inside physical quantities, such as current correlation functions or S-matrix elements. We present two different, but complementary derivations: First, we explicitly track down the aforementioned cancellations inside two-loop diagrams, resorting to nothing more than basic algebraic manipulations. Second, we present an absorptive derivation, exploiting the unitarity of the S-matrix, and the Ward identities imposed on tree-level and one-loop physical amplitudes by gauge invariance, in the case of QED, or by the underlying Becchi-Rouet-Stora symmetry, in the case of QCD. The propagator-like sub-amplitude defined by means of this latter construction corresponds precisely to the imaginary parts of the effective self-energy obtained in the former case; the real part may be obtained from a (twice subtracted) dispersion relation. As in the one-loop case, the final two-loop fermion self-energy constructed using either method coincides with the conventional fermion self-energy computed in the Feynman gauge.Comment: 30 pages; uses axodraw (axodraw.sty included in the src); final version to appear in Phys. Rev.