The Inverse Amplitude Method in ππ\pi\pi Scattering in Chiral Perturbation Theory to Two Loops

The inverse amplitude method is used to unitarize the two loop $\pi\pi$ scattering amplitudes of SU(2) Chiral Perturbation Theory in the $I=0,J=0$, $I=1,J=1$ and $I=2,J=0$ channels. An error analysis in terms of the low energy one-loop parameters $\bar l_{1,2,3,4,}$ and existing experimental data is undertaken. A comparison to standard resonance saturation values for the two loop coefficients $\bar b_{1,2,3,4,5,6}$ is also carried out. Crossing violations are quantified and the convergence of the expansion is discussed.Comment: (Latex, epsfig) 30 pages, 13 figures, 8 table

An example of resonance saturation at one loop

We argue that the large-Nc expansion of QCD can be used to treat a Lagrangian of resonances in a perturbative way. As an illustration of this we compute the L_10 coupling of the Chiral Lagrangian by integrating out resonance fields at one loop. Given a Lagrangian and a renormalization scheme, this is how in principle one can answer in a concrete and unambiguous manner questions such as at what scale resonance saturation takes place.Comment: 9 pages, 5 figures. Enlarged discussion, results unchanged. To be published in Phys. Rev.

Splitting Strong and Electromagnetic Interactions in K(L4) Decays

We recently considered $K_{\ell 4}$ decays in the framework of chiral perturbation theory based on the effective Lagrangian including mesons, photons, and leptons. There, we published analytic one-loop-level expressions for form factors $f$ and $g$ corresponding to the mixed process, $K^0\to\pi^0\pi^-\ell^+\nu_{\ell}$. We propose here a possible splitting between strong and electromagnetic parts allowing analytic (and numerical) evaluation of Isospin breaking corrections. The latter are sensitive to the infrared divergence subtraction scheme and are sizeable near the $\pi\pi$ production threshold. Our results should be used for the extraction of the $P$-wave iso-vector $\pi\pi$ phase shift from the outgoing data of the currently running KTeV experiment at FNAL.Comment: 47 pages, LaTeX, 6 postscript figure

Strong Interactions at Low Energy

The lectures review some of the basic concepts relevant for an understanding of the low energy properties of the strong interactions: chiral symmetry, spontaneous symmetry breakdown, Goldstone bosons, quark condensate. The effective field theory used to analyze the low energy structure is briefly sketched. As an illustration, I discuss the implications of the recent data on the decay $K\to \pi\pi e\nu$ for the magnitude of the quark condensate.Comment: Lectures given at the school of physics "Understanding the structure of hadrons", Prague, July 2001, 20 p

Non-Perturbative Study of the Light Pseudoscalar Masses in Chiral Dynamics

We perform a non-perturbative chiral study of the masses of the lightest pseudoscalar mesons. In the calculation of the self-energies we employ the S-wave meson-meson amplitudes taken from Unitary Chiral Perturbation Theory (UCHPT) that include the lightest nonet of scalar resonances. Values for the bare masses of pions and kaons are obtained, as well as an estimate of the mass of the \eta_8. The former are found to dominate the physical pseudoscalar masses. We then match to the self-energies from Chiral Perturbation Theory (CHPT) to O(p^4), and a robust relation between several O(p^4) CHPT counterterms is obtained. We also resum higher orders from our calculated self-energies. By taking into account values determined from previous chiral phenomenological studies of m_s/\hat{m} and 3L_7+L^r_8, we determine a tighter region of favoured values for the O(p^4) CHPT counterterms 2L^r_6-L^r_4 and 2L^r_8-L^r_5. This determination perfectly overlaps with the recent determinations to O(p^6) in CHPT. We warn about a likely reduction in the value of m_s/\hat{m} by higher loop diagrams and that this is not systematically accounted for by present lattice extrapolations. We also provide a favoured interval of values for m_s/\hat{m} and 3L_7+L^r_8.Comment: 26 pages, 9 figures. Original new material is included. Major rewriting when comparing with lattice QC

The Pure Virtual Braid Group Is Quadratic

If an augmented algebra K over Q is filtered by powers of its augmentation ideal I, the associated graded algebra grK need not in general be quadratic: although it is generated in degree 1, its relations may not be generated by homogeneous relations of degree 2. In this paper we give a sufficient criterion (called the PVH Criterion) for grK to be quadratic. When K is the group algebra of a group G, quadraticity is known to be equivalent to the existence of a (not necessarily homomorphic) universal finite type invariant for G. Thus the PVH Criterion also implies the existence of such a universal finite type invariant for the group G. We apply the PVH Criterion to the group algebra of the pure virtual braid group (also known as the quasi-triangular group), and show that the corresponding associated graded algebra is quadratic, and hence that these groups have a (not necessarily homomorphic) universal finite type invariant.Comment: 53 pages, 15 figures. Some clarifications added and inaccuracies corrected, reflecting suggestions made by the referee of the published version of the pape

The electromagnetic effects in Ke4K_{e4} decay

The final state interaction of pions in $K_{e4}$ decay allows one to obtain the value of the isospin and angular momentum zero $\pi\pi$ scattering length $a_0^0$.We take into account the electromagnetic interaction of pions and isospin symmetry breaking effects caused by different masses of neutral and charged pions and estimate the impact of these effects on the procedure of scattering length extraction from $K_{e4}$ decay.Comment: 8 pages,1 figur

Meson-meson Scattering in QCD-like Theories

We discuss meson-meson scattering at next-to-next-to-leading order in the chiral expansion for QCD-like theories with general $n$ degenerate flavours for the cases with a complex, real and pseudo-real representation. I.e. with global symmetry and breaking pattern $SU(n)_L\times SU(n)_R\to SU(n)_V$, $SU(2n)\to SO(2n)$ and $SU(2n)\to Sp(2n)$. We obtain fully analytical expressions for all these cases. We discuss the general structure of the amplitude and the structure of the possible intermediate channels for all three cases. We derive the expressions for the lowest partial wave scattering length in each channel and present some representative numerical results. We also show various relations between the different cases in the limit of large $n$.Comment: 61 page