Chiral Perturbation Theory
This page is an attempt to allow access to many of the oneloop and
twoloop 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++.
 Basic Papers
 Introductory
 general EFT, Goldstone Boson
 Books
 Lectures in Les Houches 2017
 Some explanatory notes
 Explicit calculations, formulas, etc.
 More references or
directly as pdf
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
Introductory
 A Good starting point:
Note that in these good low level
material is not always well distinguished from more advanced (and more
difficult) comments. The below also have a large overlap between them.
 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 277538,
hepph/0210398
 S. Scherer and M. Schindler, Lectures at ECT*,
hepph/0505265
 S. Scherer and M. Schindler, book: A Primer for Chiral
Perturbation Theory,
Lect.Notes Phys. 830 (2012) 1249
 Somewhat complimentary to the above is Section 2 of my review article
J. Bijnens, Prog. Part. Nucl. Phys. 58 (2007) 521
hepph/0604043
 A. Pich, Les Houches Lectures,
hepph/9806303
 A. Pich, Lectures given at the
V Mexican School of Particles and Fields
Guanajuato, Mexico, December 1992,
hepph/9308351
 G.Ecker, Benasque lectures,
hepph/0011026
 H. Leutwyler, Hadrons 94 lectures,
hepph/9406283,
local ps.gz version (hepph version has
figures stored inside the tex file)
 H. Leutwyler, Boris Ioffe Festschrift
hepph/0008124
 G. Colangelo and G. Isidori, Frascati Spring school lectures,
hepph/0101264
 J. Gasser, Chiral Dynamics 1997 introduction,
hepph/9711503
 J. Gasser, Schladming Lectures,
hepph/0312367
 B. Kubis, Lectures at "Physics and Astrophysics of Hadrons and Hadronic
Matter"hepph/0703274
 S. Sharpe, Lectures given at Workshop on Perspectives in Lattice QCD,
heplat/0607016
 M. Golterman, lectures given at the 2009 Les Houches Summer School "Modern
perspectives in lattice QCD",
arXiv:0912.4042
General EFT, Goldstone Boson, etc.
 C. Burgess, Goldstone Boson primer,
hepph/9812468
 C. Burgess, Goldstone Boson review article,
hepth/9808176
 A. Manohar, Effective Field Theories (Schladming lectures),
hepph/9606222
 I. Rothstein, Lectures on Effective Field Theories (TASI lectures),
hepph/0308266
 G. Ecker, Effective field theories, Encyclopedia of Mathematical
Physics, hepph/0507056
 D.B. Kaplan, Five lectures on effective field theory,
nuclth/0510023
 D.B. Kaplan, Effective Field Theories,
nuclth/9506035
 G. Lepage, What is Renormalization?,
hepph/0506330
 G. Lepage, How to renormalize the Schrödinger equation,
nuclth/9706029
 T. Cohen, As Scales Become Separated: Lectures on Effective Field Theory,
arXiv:1903.03622
 A. Beekman, L. Rademaker, J. van Wezel, An Introduction to Spontaneous Symmetry Breaking
arXiv:1909.01820
 H. Watanabe, Counting Rules of NambuGoldstone Modes, arXiv:1904.00569
Books
There are a number of books dedicated to EFT and ChPT
 S. Scherer and M. Schindler: A Primer for Chiral
Perturbation Theory,
Lect.Notes Phys. 830 (2012) 1249
 A. Petrov and A.E. Blechmann: Effective Field Theories,
World Scientific 2016
 U.G Mei&ss;ner and A. Rusetksy: Effective Field Theories,
Cambridge University Press, 2022
 T. Brauner: Effective Field Theory for Spontaneously Broken Symmetry,
Springer 2024
Lect.Notes Phys. 1023 (2024), arxiv:2404.14518
Lectures at Les Houches 2017
 Aneesh Manohar, Introduction to Effective Field Theories,
1804.05863
 Antonio Pich, Effective Field Theory with NambuGoldstone Modes,
1804.05664
 Thomas Becher, Les Houches Lectures on SoftCollinear Effective Theory,
1803.04310
 C.P. Burgess, Intro to Effective Field Theories and Inflation,
1711.10592
Some explanatory notes
Explicit calculations
 Meson masses and decay constants
in the isospin limit
 Pi pi scattering in Three Flavour ChPT
 Pi K scattering in Three Flavour ChPT
 Isospin Breaking in K to 3pi Decays I:
Strong Isospin Breaking
 Isospin Breaking in K to 3pi Decays II:
Radiative Corrections
 Isospin Breaking in K to 3pi Decays III:
Bremsstrahlung and Fit to Experiment
 Kell3 decays in ChPT
 Two Loop results for masses and decay constants in
PQChPT
 Two Loop results for the doublepole residue in
PQChPT
 Electromagnetism in PQChPT
 Mesonmeson scattering in QCD like theories
 Meson masses and decay constants at finite volume
 Meson masses and decay constants at finite volume in PQChPT

Meson masses, decay constants and vacuum expectation value
at finite volume and partially quenched in QCDlike theories
 The p^{8} mesonic chiral Lagrangian
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:
Twopoint Functions at Two Loops in Three Flavour Chiral Perturbation
Theory,
G.Amorós, J. Bijnens and P. Talavera
hepph/9907264, Nucl. Phys. B568 (2000) 319363
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 Formfactors and pipi Scattering,
G.Amorós, J. Bijnens and P. Talavera
LU TP 0011, 51 p., hepph/0003258, Nucl. Phys. B585 (2000) 293352,
Erratumibid. B598 (2001) 665666
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 0402, hepph/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 0419, hepph/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 0420, hepph/0405025,
Nucl.Phys. B697 (2004) 319342
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 0437, hepph/0410333,
Eur.Phys.J. C39 (2005) 347357
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 0440, hepph/0501163,
Eur.Phys.J. C40 (2005) 383394
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 0310, hepph/0303103,
Nucl. Phys. B669 (2003) 341362.
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 
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 TwoFlavor Partially Quenched Chiral Perturbation Theory,
J. Bijnens and T. Lähde,
heplat/0506004, Phys. Rev. D72 (2005) 074502
The results are for twosea quark flavours but with different masses.
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 ThreeFlavor
Partially Quenched χPT,
J. Bijnens, N. Danielsson and T. Lähde, heplat/0406017,
Phys. Rev. D70 (2004) 111503;
Decay Constants of Pseudoscalar Mesons to Two Loops in ThreeFlavor
Partially Quenched χPT,
J. Bijnens and T. Lähde, heplat/0501014, Phys. Rev. D71 (2005) 094502;
Threeflavor partially quenched chiral perturbation theory at NNLO
for meson masses and decay constants,
J. Bijnens, N. Danielsson and T. Lähde,
heplat/0602003, Phys. Rev. D73 (2006) 074509.
The results are for threesea quark flavours but with different masses.
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 ThreeFlavor Partially Quenched Chiral
Perturbation Theory,
J. Bijnens and N. Danielsson, LU TP 0623, heplat/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 0638, heplat/0610127,
Phys.Rev. D75 (2007) 014505
Masses and
Decay Constants.
11. Mesonmeson scattering in QCDlike theories
J. Bijnens and J. Lu, LU TP 1107, 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 Twoloops in Chiral Perturbation Theory,
J. Bijnens, T. Rössler, LU TP 1438, 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 ThreeFlavour
Partially Quenched Chiral Perturbation Theory
through NNLO in the Meson Sector, LU TP 1529, 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 QCDlike theories
These were calculated in PQChPT at finitevolume at two loops,
J. Bijnens, T. Rössler, Finite Volume and
Partially Quenched QCDlike Effective Field Theories, LU TP 1534, arXiv:1509.04082
The algebraic expressions are in
analyticalresultsQCDlikePQV.txt
Numerical programs in C++ included in CHIRON v0.54 and up
15. The p^{8} mesonic chiral Lagrangian
This was determined in
J. Bijnens, N. HermanssonTruedsson, S. Wang,
The p^{8} mesonic chiral Lagrangian, LU TP 1834,
arXiv:1810.06834
The pdf, LaTeX or FORM version.
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.