Physics Letters B at last!

Yesterday, the Editor of Physics Letters B communicated to me that my paper (see here) was accepted for publication. This was great for at least one reason: Physics Letters B is one of the most important journals in our area of activity and the paper that was accepted gives a sensible mathematical proof of the form of the gluon and ghost propagators in the infrared and relative mass spectrum that implies the very existence of a mass gap for the Yang-Mills theory. The key theorem is what I called the “mapping theorem” where a SU(N) Yang-Mills theory is mapped on a \lambda\phi^4 theory whose solution in the low energy limit I presented here and Physical Review D (see here). This analysis is in perfect agreement with the scenario emerging from lattice computations but we have the nice situation of explicit formulas for the gluon propagator and the spectrum permitting explicit computations wherever needed.

Also in this case the peer-review system worked at best. Both Editor and referees permitted to correct what appeared a serious difficulty in the proof of the mapping theorem. Once I solved this the paper was straightforwardly approved for publication. I take this chance to thank them all publicly.

I give here the formulas for the gluon propagator in the Landau gauge (the ghost propagator is that of a free particle) and the spectrum:

D_{\mu\nu}^{ab}(p^2)=\delta_{ab}\left(\eta_{\mu\nu}-\frac{p_\mu p_\nu}{p^2}\right)\sum_{n=0}^\infty\frac{B_n}{p^2-m_n^2+i\epsilon}




m_n = (2n+1)\frac{\pi}{2K(i)}\left(\frac{Ng^2}{2}\right)^{1 \over 4}\Lambda

being \Lambda an integration constant of Yang-Mills theory, arising from conformal invariance, to be fixed experimentally and K(i) an elliptic integral that is about 1.3111028777. From the mass spectrum is clearly seen the mass gap when n=0 is taken. Nature decides what \Lambda is but an higher order theory should be able to derive it. We see that the spectrum of the theory is made of massive excitations that should not be called gluons. I think that glueballs is more appropriate.

So, this is a key moment for Yang-Mills theory. It implies a great understanding of the behavior of the theory in a regime not accessible before. Knowing the gluon propagator means that a Nambu-Jona-Lasinio model describes correctly the phenomenology at low energies. This we proved quite recently (see this post).


17 Responses to Physics Letters B at last!

  1. Ervin Goldfain says:

    Congratulations on your acceptance letter and your nice work!

    Here are two questions:

    1) apart from deriving a mass gap, is your work capable of making direct connection to experimental data (not lattice computations, but actual physical observations)? For example, can you consistently derive the hadron mass spectrum from the gluon propagator in the Landau gauge? How about explaining the emergence of chiral symmetry breaking or the spectrum of magnetic moments?

    2) how do you reconcile the perturbative behavior of Yang-Mills in the infrared with the existence of quantum chaos in QCD? (as proven numerically by Markum, Plessas, Pascalutsa and others).

    Best regards,


  2. mfrasca says:

    In all of your communications there is a substantial confusion about full QCD and pure Yang-Mills theory. The understanding we derived from studying pure Yang-Mills theory tell us the very existence of quarks makes the ground state of the theory sensibly lower. What I expect from my computations on pure Yang-Mills theory is to get a glueball spectrum. But this spectrum is biased by mixing with quarks. This is currently an hot line of research aimed to identify glueball states from the rich spectrum, yet to be understood, of light scalar mesons. My work does a very precise prediction and this is that the sigma resonance is a glueball. This is in agreement with recent works by Narison, Ochs, Mennessier and Minkowski. But it disagrees with other researchers thinking sigma being a tetraquark state. Of course, if it will be proved that I am right this will be a big hit as I would have foreseen a state that not even lattice computations are able to see and will be a serious starting point for the understanding of light scalar spectrum. About breaking of chiral symmetry you should consider full QCD. Nambu-Jona-Lasinio model has this and this fact is known since the inception of this model.

    The question of quantum chaos in Yang-Mills theory is known since eighties due to the beautiful works of Savviddy et al. What I have done is to identify a class of classical solutions that are not chaotic producing the right solution. This is done through the mapping theorem. Hints of this solution were given by Smilga (see or his beautiful book Chaotic solutions do not produce a sensible quantum field theory and should be thrown away as unphysical.



  3. ervin says:


    Thanks for your reply.

    I understand that your aim is not full QCD but pure Yang-Mills theory with a limited objective in mind to get the glueball spectrum. But on what basis can one make the argument that pure Yang-Mills theory can be completely decoupled from the quark sector in the infrared limit? If it is true that your model has validity and it points in the right direction (as you claim), one should be able to derive predictions that are in agreement with experiment and not with lattice computations only. So, I am asking again: is there agreement between your model and experimental data? Is your model testable?

    Regarding quantum chaos in QCD, I respectfully disagree with you. I am not referring to deterministic chaos in Yang-Mills theory as studied by Savvidy, Matynian et al. in the eigthies but to quantum chaos as manifested in the distribution of level spacings (see the works of Pascalutsa, Markum, Plessas et al on the Wigner surmise, Gaussian Orthogonal Ensembles in the Hadron mass spectrum). Their findings are in perfect harmony with experimental data, so your argument that “chaotic solutions do not produce a sensible quantum field theory and should be thrown away as unphysical” appears to be invalid.



  4. mfrasca says:

    Let me repeat: Yang-Mills theory is part of QCD and as such its spectrum should be seen in experiments. Besides this, understanding Yang-Mills theory permits to solve a long-standing puzzle named confinement and generally considered one of the most important problems in this field. You could have heard of the infamous “mass gap” problem. Mixing of glueballs with quarks is for sure but if I know the gluon propagator I can evaluate this mixing. This happens through a Nambu-Jona-Lasinio model that is known to produce the right hadronic spectrum (check literature, please). These facts are well-known otherwise a lot of effort done by people both in phenomenology and lattice since eighties would be meaningless and this is not the case. Of course, as you pretend to be so skilled in QCD, if you are able to tell to those people where glueballs lie in the light meson spectrum where a lot of people is squeezing his brain to understand you are welcome. Then send your paper to a respectable journal and try to get it published. Easy, isn’t it?

    About the question of the spectrum of QCD we are not talking about the same thing. I am saying that I cannot use classical chaotic solutions to build a quantum field theory. This is not quite the same to say that QCD has a chaotic spectrum.



  5. carlbrannen says:

    Thanks for the link to the link to the Smilga lectures. I haven’t reached the parts you’re talking about, but found the lovely statement:

    “Gauge symmetry just does not act on the Hilbert space of the physical states, it exhibits itself only in the lagrangian formulation involving some extra unphysical variables which can be disposed of, in principle. In other words: gauge symmetry is not a symmetry, but rather a convenient way to describe constrained systems.”

    For the U(1) symmetry, another way of writing the constraint is that it must be possible to define a quantum state using density operators.

  6. ervin says:


    I apologize if my questions have irritated you. Please understand that this is far from being my intent. I don’t want to be confrontational. All I was looking for was an honest appraisal on whether or not 1) your model is testable and 2) is backed up by objective evidence.

    Whether or not I am skilled in QCD is completely irrelevant for the questions I posed to you. Regarding your comment on the NJL model, as far as I know, there is currently no QCD approach in the infrared that reproduces the entire spectrum of hadron properties from first principles, without fixing additional parameters in the theory (please correct me if I am wrong).



  7. mfrasca says:

    No problem.

    The aim of my work is to fix that parameters in the Nambu-Jona-Lasinio model through those of QCD Lagrangian. This I did (see and my post This is possible only when the gluon propagator is known. This is just what I am trying to explain and I regret if I am not able to let my thought understood.



  8. Daniel says:

    I thought the mass gap is related to what is called dimensional transmutation !
    Also I dn’t understand the statement that gauge symmetry is not a symmetry, if not why is it called so then ?

    I also saw in a book for mathematicians something like gauge symmetry is a redundancy not a symmetry am not sure what that means either….I always wonder if renormalization group equations had anything to do with group theory, unfortunately I hvn’t that explained in anybook I know of

    Any enlightments will be greately appreciated


  9. Daniel says:

    sorry last line should have been

    I havn’t “seen” that explained in any book I know of


  10. mfrasca says:


    In order to have a clear understanding of what a gauge is you should read the beautiful post of Terry Tao where you can find the link at

    My view about a mass gap is similar to that of Schwinger for D=1+1 QED. It is a dynamical effect due to the self-interacting term of the theory. Its existence is granted for a coupling large enough.

    Renormalization group is rather a semigroup than a group due to the fact that is has not an inverse. But group properties are not the reason why is so important. This remains rather a matter of nomenclature.


  11. […] and a full spectrum of glueballs does exist that can contribute to this computation (see my post and my […]

  12. […] choice and the mapping theorem After the acceptance of my paper (see here) I wondered what should have been Smilga’s choice for the SU(3) given its existence. Let me […]

  13. Mr. Peavey says:

    Neils Bohr and his friends developed a theory of physics that has left us blind to reality and now we are left feeling around the universe with equations that have no real physical meaning. One small visual insight into the reality of the universe is worth more than all the equation mathematicians can produce. Physics is a physical science not mathematical art. Mathematics is a language.

  14. mfrasca says:

    Normally I am tolerant enough to let also a crank like you post freely here. This is a must in view of the fact that people should be aware about what science is and what is not. Of course, we would like to understand by your side how we should interpret the numbers our measurements give back without mathematics. But, mostly important, why one should care about someone making a claim like yours if not just for entertainment.


  15. mfrasca says:

    Normally I am tolerant enough to let also a crank like you post freely here. This is a must in view of the fact that people should be aware about what science is and what is not. Of course, we would like to understand by your side how we should interpret the numbers our measurements give back without mathematics. But, mostly important, why one should care about someone making a claim like yours if not just for entertainment.


  16. carlbrannen says:

    Many many years ago, when my aunt found out that I was in physics grad school, her comment was “everything is made from waves, isn’t it.” I swore to myself that someday I would write a paper and put a reference to her comment as a “personal communication.” I should probably add that she was an artist.

  17. mfrasca says:

    Your aunt perception was really stunning.


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