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.

Some of the results included below are available via the program package CHIRON which has its own webpage. This is written in C++.

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

Basic Papers

Introductory

General EFT, Goldstone Boson, etc.

  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
  7. D.B. Kaplan, Effective Field Theories, nucl-th/9506035
  8. G. Lepage, What is Renormalization?, hep-ph/0506330
  9. G. Lepage, How to renormalize the Schrödinger equation, nucl-th/9706029

Explicit calculations

  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
  12. Meson masses and decay constants at finite volume
  13. Meson masses and decay constants at finite volume in PQChPT
  14. Meson masses, decay constants and vacuum expectation value at finite volume and partially quenched in QCD-like theories

Beware, in some of the formulas below the NNLO order constants have dimension -2 in mass units.

1. Meson masses and decay constants in the isospin limit

These were calculated in:
Two-point Functions at Two Loops in Three Flavour Chiral Perturbation Theory, G.Amorós, J. Bijnens and P. Talavera hep-ph/9907264, Nucl. Phys. B568 (2000) 319-363

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:

Numerical programs in C++ included in CHIRON

2. Pi pi scattering in Three Flavour ChPT

These were calculated in:
pi pi Scattering in Three Flavour ChpT
Johan Bijnens, Pierre Dhonte and Pere Talavera, LU TP 04-02, hep-ph/0401039, JHEP 0401 (2004) 050

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

3. Pi K scattering in Three Flavour ChPT

These were calculated in:
pi K Scattering in Three Flavour ChpT
Johan Bijnens, Pierre Dhonte and Pere Talavera, LU TP 04-19, hep-ph/0404150, JHEP 0405 (2004) 036

Form Input

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

These were calculated in:
Isospin Breaking in K to 3pi Decays I: Strong Isospin Breaking
Johan Bijnens and Fredrik Borg, LU TP 04-20, hep-ph/0405025, Nucl.Phys. B697 (2004) 319-342

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

These were calculated in:
Isospin Breaking in K to 3pi Decays II: Radiative Corrections
Johan Bijnens and Fredrik Borg, LU TP 04-37, hep-ph/0410333, Eur.Phys.J. C39 (2005) 347-357

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

These were calculated in:
Isospin Breaking in K to 3pi Decays III: Bremsstrahlung and Fit to Experiment
Johan Bijnens and Fredrik Borg, LU TP 04-40, hep-ph/0501163, Eur.Phys.J. C40 (2005) 383-394

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

These results can be found in:
Kell3 decays in Chiral Perturbation Theory
J. Bijnens and P. Talavera, LU TP 03-10, hep-ph/0303103, Nucl. Phys. B669 (2003) 341-362.

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 decayK^+ 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_555715.10788 5580.70217
The formulas for the form factor f+(t) and f-(t) are in KL3all_2.log.

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

Two Sea Quark Flavours

These results were published in Masses and Decay Constants of Pseudoscalar Mesons to Two Loops in Two-Flavor Partially Quenched Chiral Perturbation Theory, J. Bijnens and T. Lähde, hep-lat/0506004, Phys. Rev. D72 (2005) 074502

The results are for two-sea quark flavours but with different masses.
Number of different
valence+sea quark masses
1+1decay mass
1+2decay mass
2+1decay mass
2+2decay mass
The links in the table lead to FORM outputs, numerical programs in FORTRAN can be obtained from the authors.

Three Sea Quark Flavours

These results were published in:
The Pseudoscalar Meson Mass to Two Loops in Three-Flavor Partially Quenched χPT, J. Bijnens, N. Danielsson and T. Lähde, hep-lat/0406017, Phys. Rev. D70 (2004) 111503;
Decay Constants of Pseudoscalar Mesons to Two Loops in Three-Flavor Partially Quenched χPT, J. Bijnens and T. Lähde, hep-lat/0501014, Phys. Rev. D71 (2005) 094502;
Three-flavor partially quenched chiral perturbation theory at NNLO for meson masses and decay constants, J. Bijnens, N. Danielsson and T. Lähde, hep-lat/0602003, Phys. Rev. D73 (2006) 074509.

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

Number of different
valence+sea quark masses
1+1decay mass
1+2decay mass
1+3decay mass
2+1decay mass
2+2decay mass
2+3decay mass

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

Numerical programs in C++ included in CHIRON

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

The eta mass and NNLO Three-Flavor Partially Quenched Chiral Perturbation Theory, J. Bijnens and N. Danielsson, LU TP 06-23, hep-lat/0606017, Phys. Rev. D 74 (2006) 054503
Form output.

10. Electromagnetic Corrections in Partially Quenched Chiral Perturbation

Electromagnetic Corrections in Partially Quenched Chiral Perturbation, J. Bijnens and N. Danielsson, LU TP 06-38, hep-lat/0610127, Phys.Rev. D75 (2007) 014505

Masses and Decay Constants.

11. Meson-meson scattering in QCDlike theories

J. Bijnens and J. Lu, LU TP 11-07, arXiv:1102.0172, JHEP 1103 (2011) 028

All amplitudes and scattering lengths

12. Meson masses and decay constants at finite volume

These were calculated in Finite Volume at Two-loops in Chiral Perturbation Theory, J. Bijnens, T. Rössler, LU TP 14-38, arXiv:1411.6384, JHEP 1501 (2015) 034

The algebraic expressions are in massdecayfinitevolume.log

Numerical programs in C++ included in CHIRON

13. Meson masses and decay constants at finite volume in PQChPT

These were calculated in PQChPT at finitevolume at two loops, J. Bijnens, T. Rössler, Finite Volume for Three-Flavour Partially Quenched Chiral Perturbation Theory through NNLO in the Meson Sector, LU TP 15-29, arXiv:1508.07238

The algebraic expressions are in finalresultmassdecayPQV.log

Numerical programs in C++ included in CHIRON v0.53 and up

14. Meson masses, decay constants and vacuum expectation value at finite volume and partially quenched in QCD-like theories

These were calculated in PQChPT at finitevolume at two loops, J. Bijnens, T. Rössler, Finite Volume and Partially Quenched QCD-like Effective Field Theories, LU TP 15-34, arXiv:1509.nnnnn

The algebraic expressions are in analyticalresultsQCDlikePQV.txt

Numerical programs in C++ included in CHIRON v0.54 and up

More references

I keep a list of references for my own use with some comments here and there. It has no claim for completeness, correctness and it should definitely not be taken as a review article.