Codes and some additional notes related to the author papers on topic Minkowski space solution of
QCD and beyond"
(papers are available at ArXiV)
In the early days I was involved in the nonperturbative studies of quantum field theories,
which are based on the analytical assumptions known from perturbative studies of particle like systems.
Most of these studies relly on the validity of spectral representation for two point Green's functions and on the
Perturbation Theory Integral Representation in more general case. Alhough, the reflection positivity has never been required,
even so, these analytical assumptions turned to be too strong and overconstraining for Quantum Chromodynamics.
First stage -assuming standard analyticity-
Selected papers on solutions of Schwinger-Dyson and Bethe-Salpeter equations in Minkowski space:
V. Sauli, Implication of analyticity to solution of
Schwinger-Dyson equations in Minkowski space,
submitted to Few Body Systems,
V. Sauli, Minkowski solution of Dyson-Schwinger equations in
momentum subtraction scheme,
JHEP 0302 , 001 (2003); arXiv:[hep-ph/0209046]
V. Sauli, Running coupling and fermion mass in strong
coupling Quantum Electrodynamics,
J. Phys. G30, 739 (2004); arXiv:[hep-ph/0306081]
V. Sauli, J. Adam, Study of relativistic bound states for
scalar theories in the Bethe-Salpeter and Dyson-Schwinger formalism,
Phys. Rev. D67, 085007 (2003);arXiv: [hep-ph/0111433],
V. Sauli, Solving the Bethe-Salpeter equation for fermion-antifermion
pseudoscalar bound state in Minkowski space,
submitted to Phys. Rev. D;
V. Sauli, Some aspects of dynamical mass generation,
oral conference contribution, SMFT Bari 2004, Neutrino Prague 2004; arXiv:[hep-ph/0410167]
V. Sauli, Non-perturbative solution of metastable scalar
J. Phys. A36, 8703 (2003); arXiv:[hep-ph/0211221]
Confined gluon from Minkowski space continuation of PT-BFM SDE solution,
J. Phys. G.: Nucl. Part. Phys. 39 (2012); arXiv:[1102.5765]
Higgsonium in singlet extension of Standard Model
In all cases the applicability of methods were limited by some critical value of coupling. Above that value a disagreement with the assumptions, e.g.
some disagreements with the counter-partner solutions performed directly in the Euclidean space were found. Sometimes the convergence has stopped at this point.
Quite interestingly, in the models with fermions, the value of critical coupling agrees with the one for dynamical chiral symmetry breaking.
Within a huge effort I spent long time by trying to find QCD-like solution for quark gap equation within particle-like analycity and standard Feynmann prescription constraints. The best what I got together with my colaborators Pedro Bicudo and J. Adam in this respect was the solution obtained at the vicinity of critical coupling for a quite special and certainly simplified model. Characteristic amount of chiral symmetry breaking known for QCD has not been achieved by the method. See the paper:
Dynamical chiral symmetry breaking with Minkowski space integral representations
V. Sauli, J. Adam, P. Bicudo Phys. Rev. D75; 087701 (2007) ,arXiv:[hep-ph/0607196]
Second stage -Minkowski space QCD-
(papers are available at ArXiV)
From 2013 I started to calculate QCD Green's functions directly in Minkowski space.
In the begening of 90 (last century) there was a little believe that it is possible at all and that it is a badly defined numerical problem was taken
almost as granted. See the paper: S.J. Stainsby, R.T. Cahill, Phys. Lett. A146, 467 (1990) which has, aty elast for me
quite sceptical title "Is spacetime Ecludena inside the hadrons". The paper represent pionering work in SDE/BSE study
, noting the complex conjugated singularity structure of quark propagator when calculated in the Euclidean space.
The scepticism taht the numerical solution is impossible without the use of auxiliary Euclidean space can be trace
during the last decades, see the discussion in "C. D. Roberts, A. G. Williams PPNP 33, (1994) ( Dyson-Schwinger Equations and the Application to Hadronic Physics)", which represents one of the first overwiews written by londstanding gap equations leader and speaker american physicist Craig Roberts.
While the singularity structure of Greens functions in unconfinig quantum field theory prevents -at least naive- numerical search
in Minkowski space the situation in QCD is drastically different. The numerical solution can be and has been gained directly in the physical Minkowski space since the analytical structure of quark propgators allows it. Even the continuation of Euclidean space solutions shows that
the poles are located very far from the real axis, which means that the integral kernel of SDEs remain nicely regular and tractable by
a numerical integrations. From 2013 I wrote several papers devoted to numerical solution of
Schwinger-Dyson and Bethe-Salpeter equations. However there is one unpleasant consequnce, which is not presented in quantum field theories defined with the Euclidean metrics: As the poles turn to be complex,the QCD vertices, propagators and
the all mesonic BS wave functions turns to be complex. The only poles which remain real are associtaed with the propagators of colorless physical composite object.i.e. with mesons and baryons, they do not spoil the numerics as they do not appear at low orders (skeleton graphs) of Schwinger-Dyson equations.
The numerics is sometimes cumbersome and it took me long time to get a stable results and figure out what are the main weak points of numerics. Minkowski metric enforced me to use a relatively large number of integrations point, which prevents me from exploatatiuon of popular determinant method and matrix inversion method. Roughly speaking, the BSE and the DSE do not show convergence for small number of mesh points and the numerics is usually time consuming. Insted of matrix method I allways use the iterations. As it is well known and as it has been experinced painfully by many,
the convergence of iteration is not automatically guaranted for nonlinear systems and the success depends on the details which is even hard to specify or understand. Happily, if convergence is achieved once one has already starting point. Mainly due to this not appealing numerical beahviour, I leave almost all main numerical codes public bellow. In some cases also a raw numerical datas are enclosed. Anybody can use the codes for its own purpose. The codes are not written in user-friendly manner, so do not hesitate to write an email if interested in. Of course, the second purpose is a safe storage of my codes here.
V. S., Pions and excited scalars in Minkowski space DSBSE formalism
The code that calculated the scalars: ( superscalar12.for)
the code that goes in reverse direction .i.e. form higher mass to... whatever (including unphysical tahyonic bound state):
Raw sample output for the second case (for C=1/80):
Raw sample output for the superscalar12.for case (for C=1/80):
The data are ploted in the Figure at refered paper.
For C=1/90 (unpublished case): ( iterscalar.dat)
The results are quite similar to previous, but the error is about two or three orders worse (it exhibits how the numerics is cumbersome, note the coupling is now about 10% smaller then in previous case, note also the quark propagator is the same, expecting only a small variation due to the change of the coupling).
The code, which calculated pions spectra in the paper above is here: ( sample data +code)
The ascii data file includes raw data -three columns output- mass, error, and lambda, then the body of the code follows (superpion2.for in my database).
The stability of numerics have been checked many times, e.g. by changin the number of iterations, steps etc. examples of working codes:
The code for quark gap equation, the first part is just the data: ( quargraph138.dat)
There are 1200 lines with p2 ,Re M and Im M as a columns, the the code named quark138.for in my database follows, this is the same code which provides
S^+ result for the data in the paper "Lattice Inspired..."
The convergence of the iterations was pretty fast: ( iter138.dat)
Code for the numerical fit of the quark propagator as described in the text: ( fit "embecko.for")
V. S., Intriguing solutions of the Bethe-Salpeter equation for radially excited pseudoscalar charmonia
where, appart more standard since Euclidean, the first Minkowski space solution hass been published for charmonium.
The codes: will be added tomorrow :)
The iteration method has been genuinelly checked for the Euclidean BSE as well,
It provides excited states without any problem. The code for an Euclidean BSE:
Codes related with the paper
The code which calculates the first(?) excited state of the pion:
with the out-put ( iterpion201n.dat)
Raw data are in the internal units where energy kernel period in cosine function is $sinsqrtp^2/bm^2 bm=1$, which corresponds with M=9.29bm and M=9.6bm
, errors are about 10^-3 (there are two numerical solutions)... uv part is not presented in the kernel.
PhD thesis (defended in 19-th September 2005)
Schwinger-Dyson approach to field models with strong coupling
(Supervisor: J. Adam)
Abstract of PhD thesis: