# Effective Field Theories - Chiral Perturbation Theory and Non-relativistic QFT

The central part of our research concerns *effective field theories*. These allow one to investigate underlying theories through effective Lagrangians that are relevant for a specific range of energies. Examples are Chiral Perturbation Theory, which is the effective field theory of QCD at low energy (below 1 GeV), and several of its variants.

This approach is complementary to the lattice implementation of QCD, and fruitful exchanges between the two fields exist since several years and continue to flourish.

Our institute coordinates the efforts of the FLAG Working Group,which reviews lattice results concerning low-energy particle physics.

In the past our group has been involved in a detailed study of *π π scattering*, and via a matching of the Roy equations solutions to the chiral representation, a very accurate description of the this reaction below 1 GeV has been achieved. In particular the scattering lengths have been predicted by our group with remarkably small uncertainties [hep-ph/0007112]:

a00 = 0.22 ± 0.005 a20 = -0.0444 ± 0.001

This high level of accuracy is also the basis for a precise determination of the sigma resonance parameters [hep-ph/0512364], which has put a decades-long debate on the existence of this scalar resonance to rest. The fact that the precision in our knowledge of the scattering lengths has a direct impact on the resonance parameters relies on the dispersive representation and unitarity.

In recent years different experiments have measured these quantities with high accuracy: BNL-E865, DIRAC, and NA48/2. Given the very high precision, it is important to have radiative corrections and all other sources of isospin breaking under control. These effects are specific for each experiment, and their understanding and removal requires a collaboration among experimentalists and theorists. Some recent papers of our group which address these issues are:

- Extraction of the scattering lengths from the pionium lifetime (DIRAC) [hep-ph/9910438]
- Extraction of the scattering lengths from a cusp effect in K → 3 π (NA48/2) [hep-ph/0604084]
- Extraction of the scattering lengths from K
_{e4}decays (E865, NA48/2) [hep-ph/0710.3048]

A similar approach based on a matching between the chiral and a dispersive representation has been later applied to several other processes:

- η → 3 π, [hep-ph/1301.7282, hep-ph/1305.6839]
- K
_{l4}, PoS CD12 (2013) 058 (and in progress) - as well as K-Kbar (in progress)
- and hadronic vacuum polarization [hep-ph/0312017].

More recently, our group has proposed a dispersive treatment of the *hadronic light-by-light contribution* to the anomalous magnetic moment (g-2) of the muon. For more than a decade, a discrepancy of about 3σ between the Standard-Model prediction and the measurement of the muon g-2 has persisted. It is expected that within a few years the hadronic light-by-light contribution will dominate the theory uncertainty. The proposed framework [hep-ph/1402.708, hep-ph/1408.2517] will allow a data-driven and hence less model-dependent evaluation of this contribution to the g-2 of the muon.

Another activity of our group concerns the analysis of finite volume effects in lattice calculations with the help of chiral perturbation theory [3]. If the volume available to a physical system is not so small that only the short-distance physics can take place, it is mostly the light degrees of freedom which "feel" the presence of boundaries. In QCD this means the pions. Chiral perturbation theory, which describes the large-distance interactions of pions is the appropriate tool to evaluate these effects quantitatively.

In a series of Diploma, PhD theses and papers, we have studied these phenomena and made predictions for the size of the finite volume effects for different masses and decay constants [hep-lat/0311023, hep-lat/0503014, hep‑lat/0602017].