Chiral Perturbation Theory

This page is an attempt to allow access to many of the one-loop and two-loop calculations done in chiral perturbation theory and the programs related to those calculations. They are available for general use but the relevant papers as mentioned with each output/program should be cited. Note that all these programs are provided as is and no guarantees of any kind are given.

Since this page has been created for the work of myself and collaborators there has been no attempt made to have a complete set of references. For these, please consult the papers mentioned.

1. Basic Papers
2. Introductory
3. general EFT, Goldstone Boson
4. Explicit calculations, formulas, etc.
5. More references

Basic Papers

• Departures from chiral symmetry: a review, Heinz Pagels, Phys. Rept. 16 (1975) 219
• Phenomenological lagrangians, Steven Weinberg, Physica A96 (1979) 327
• Approaching the chiral limit in qcd, J. Gasser and A. Zepeda, Nucl. Phys. B174 (1980) 445
• Chiral perturbation theory to one loop, J. Gasser and H. Leutwyler, Annals Phys. 158 (1984) 142
• Chiral perturbation theory: expansions in the mass of the strange quark, J. Gasser and H. Leutwyler, Nucl. Phys. B250 (1985) 465

• A. Pich, Les Houches Lectures, hep-ph/9806303
• G.Ecker, Benasque lectures, hep-ph/0011026
• H. Leutwyler, Hadrons 94 lectures, hep-ph/9406283, local ps.gz version (hep-ph version has figures stored inside the tex file)
• H. Leutwyler, Boris Ioffe Festschrift hep-ph/0008124
• G. Colangelo and G. Isidori, Frascati Spring school lectures, hep-ph/0101264
• S. Scherer, Introduction to chiral perturbation theory, in Advances in Nuclear Physics, (Editors: J.W. Negele and E. Vogt, Kluwer Academic / Plenum Publishers New York, 2003), Vol. 27 pages 277-538, hep-ph/0210398
• J. Gasser, Chiral Dynamics 1997 introduction, hep-ph/9711503
• J. Gasser, Schladming Lectures, hep-ph/0312367
• S. Scherer and M. Schindler, Lectures at ECT*, hep-ph/0505265
• B. Kubis, Lectures at "Physics and Astrophysics of Hadrons and Hadronic Matter"hep-ph/0703274
• S. Sharpe, Lectures given at Workshop on Perspectives in Lattice QCD, hep-lat/0607016
• M. Golterman, lectures given at the 2009 Les Houches Summer School "Modern perspectives in lattice QCD", arXiv:0912.4042

1. C. Burgess, Goldstone Boson primer, hep-ph/9812468
2. C. Burgess, Goldstone Boson review article, hep-th/9808176
3. A. Manohar, Effective Field Theories (Schladming lectures), hep-ph/9606222
4. I. Rothstein, Lectures on Effective Field Theories (TASI lectures), hep-ph/0308266
5. G. Ecker, Effective field theories, Encyclopedia of Mathematical Physics, hep-ph/0507056
6. D.B. Kaplan, Five lectures on effective field theory, nucl-th/0510023

1. Meson masses and decay constants in the isospin limit
2. Pi pi scattering in Three Flavour ChPT
3. Pi K scattering in Three Flavour ChPT
4. Isospin Breaking in K to 3pi Decays I: Strong Isospin Breaking
5. Isospin Breaking in K to 3pi Decays II: Radiative Corrections
6. Isospin Breaking in K to 3pi Decays III: Bremsstrahlung and Fit to Experiment
7. Kell3 decays in ChPT
8. Two Loop results for masses and decay constants in PQChPT
9. Two Loop results for the doublepole residue in PQChPT
10. Electromagnetism in PQChPT
11. Meson-meson scattering in QCD like theories

1. Meson masses and decay constants in the isospin limit

Note that due to a bug in a FORM program the expressions for the decay constants are with nonrenormalized masses and decay constants in the NLO parts, contrary to what is stated in the text.

The masses were quoted correctly with the decay constant and the masses at NLO renormalized correctly to the physical Fpi and physical masses.

The mistake was noted by Moussallam and described in Kl4 Form-factors and pi-pi Scattering, G.Amorós, J. Bijnens and P. Talavera LU TP 00-11, 51 p., hep-ph/0003258, Nucl. Phys. B585 (2000) 293-352, Erratum-ibid. B598 (2001) 665-666

FORM output with the expressions:

• Masses, unrenormalized
• Decay Constants, unrenormalized
• Masses and Decay Constants, renormalized

2. Pi pi scattering in Three Flavour ChPT

Form Input and the Perl script used to convert it into LaTeX

3. Pi K scattering in Three Flavour ChPT

Form Input

4. Isospin Breaking in K to 3pi Decays I: Strong Isospin Breaking

Note that factors C and F_0 and e^2 are often pulled into various other constants

Form Input for K to 3pi and K to 2pi and Perl script used to convert it to LaTeX.

5. Isospin Breaking in K to 3pi Decays II: Radiative Corrections

Note that factors C and F_0 and e^2 are often pulled into various other constants. The I amplitudes are the photon loop ones and the E amplitudes are the photon reducible ones.

C++ Program.

6. Isospin Breaking in K to 3pi Decays III: Bremsstrahlung and Fit to Experiment

Note that factors C and F_0 and e^2 are often pulled into various other constants.

C++ Program.

7. Kell3 decays in Chiral Perturbation Theory

An often requested number which is not included in the paper is the parametric dependence of f_+(0) on the constants L^r_i. This dependence on the subtraction constants can be parametrized at µ = 770 MeV as
f_+(0) = f_0(0)(L^r_i=0)+d_i L^r_i+e_ij L^r_i L^r_j
The nonzero coefficients for the two relevant cases are listed in the table below.

	K^0 decay	K^+ decay
d_1	-3.49657152	-3.5412938
d_2	-1.74828576	-1.7706469
d_3	-0.475756828	-0.482456193
d_4	4.45782399	4.7338465
d_5	-0.413559992	-0.388474391
e_55	5715.10788	5580.70217

8. Masses and Decay Constants at Two Loops in Partially Quenched Chiral Perturbation Theory (PQχPT)

Two Sea Quark Flavours

The results are for two-sea quark flavours but with different masses.

Number of different valence+sea quark masses
1+1	decay	mass
1+2	decay	mass
2+1	decay	mass
2+2	decay	mass

The links in the table lead to FORM outputs, numerical programs can be obtained from the authors.

Three Sea Quark Flavours

The results are for three-sea quark flavours but with different masses.

The links in the table lead to FORM outputs, numerical programs can be obtained from the authors.

9. Doublepole Residue at Two Loops in Partially Quenched Chiral Perturbation Theory (PQχPT)

10. Electromagnetic Corrections in Partially Quenched Chiral Perturbation

11. Meson-meson scattering in QCDlike theories