## Do quarks grant confinement?

21/07/2014

In 2010 I went to Ghent in Belgium for a very nice Conference on QCD. My contribution was accepted and I had the chance to describe my view about this matter. The result was this contribution to the proceedings. The content of this paper was really revolutionary at that time as my view about Yang-Mills theory, mass gap and the role of quarks was almost completely out of track with respect to the rest of the community. So, I am deeply grateful to the Organizers for this opportunity. The main ideas I put forward were

• Yang-Mills theory has an infrared trivial fixed point. The theory is trivial exactly as the scalar field theory is.
• Due to this, gluon propagator is well-represented by a sum of weighted Yukawa propagators.
• The theory acquires a mass gap that is just the ground state of a tower of states with the spectrum of a harmonic oscillator.
• The reason why Yang-Mills theory is trivial and QCD is not in the infrared limit is the presence of quarks. Their existence moves the theory from being trivial to asymptotic safety.

These results that I have got published on respectable journals become the reason for rejection of most of my successive papers from several referees notwithstanding there were no serious reasons motivating it. But this is routine in our activity. Indeed, what annoyed me a lot was a refeee’s report claiming that my work was incorrect because the last of my statement was incorrect: Quark existence is not a correct motivation to claim asymptotic safety, and so confinement, for QCD. Another offending point was the strong support my approach was giving to the idea of a decoupling solution as was emerging from lattice computations on extended volumes. There was a widespread idea that the gluon propagator should go to zero in a pure Yang-Mills theory to grant confinement and, if not so, an infrared non-trivial fixed point must exist.

Recently, my last point has been vindicated by a group that was instrumental in the modelling of the history of this corner of research in physics. I have seen a couple of papers on arxiv, this and this, strongly supporting my view. They are Markus Höpfer, Christian Fischer and Reinhard Alkofer. These authors work in the conformal window, this means that, for them, lightest quarks are massless and chiral symmetry is exact. Indeed, in their study quarks not even get mass dynamically. But the question they answer is somewhat different: Acquired the fact that the theory is infrared trivial (they do not state this explicitly as this is not yet recognized even if this is a “duck” indeed), how does the trivial infrared fixed point move increasing the number of quarks? The answer is in the following wonderful graph with $N_f$ the number of quarks (flavours):

From this picture it is evident that there exists a critical number of quarks for which the theory becomes asymptotically safe and confining. So, quarks are critical to grant confinement and Yang-Mills theory can happily be trivial. The authors took great care about all the involved approximations as they solved Dyson-Schwinger equations as usual, this is always been their main tool, with a proper truncation. From the picture it is seen that if the number of flavours is below a threshold the theory is generally trivial, so also for the number of quarks being zero. Otherwise, a non-trivial infrared fixed point is reached granting confinement. Then, the gluon propagator is seen to move from a Yukawa form to a scaling form.

This result is really exciting and moves us a significant step forward toward the understanding of confinement. By my side, I am happy that another one of my ideas gets such a substantial confirmation.

Marco Frasca (2010). Mapping theorem and Green functions in Yang-Mills theory PoS FacesQCD:039,2010 arXiv: 1011.3643v3

Markus Hopfer, Christian S. Fischer, & Reinhard Alkofer (2014). Running coupling in the conformal window of large-Nf QCD arXiv arXiv: 1405.7031v1

Markus Hopfer, Christian S. Fischer, & Reinhard Alkofer (2014). Infrared behaviour of propagators and running coupling in the conformal
window of QCD arXiv arXiv: 1405.7340v1

## f0(500) and f0(980) are not tetraquarks

27/06/2014

Last week I have been in Giovinazzo, a really beautiful town near Bari in Italy. I participated at the QCD@Work conference. This conference series is now at the 7th edition and, for me, it was my second attendance. The most striking news I heard was put forward in the first day and represents a striking result indeed. The talk was given by Maurizio Martinelli on behalf of LHCb Collaboration. You can find the result on page 19 and on an arxiv paper . The question of the nature of f0(500) is a vexata quaestio since the first possible observation of this resonance. It entered in the Particle Data Group catalog as f0(600) but was eliminated in the following years. Today its existence is no more questioned and this particle is widely accepted. Also its properties as the mass and the width are known with reasonable precision starting from a fundamental work by Irinel Caprini, Gilberto Colangelo and Heinrich Leutwyler (see here). The longstanding question around this particle and its parent f0(980) was about their nature. It is generally difficult to fix the structure of a resonance in QCD and there is no exception here.

The problem arose from famous papers by Jaffe on 1977 (this one and this one) that using a quark-bag model introduced a low-energy nonet of states made of four quarks each. These papers set the stage for what has been the current understanding of the f0(500) and f0(980) resonances. The nonet is completely filled with all the QCD resonances below 1 GeV and so, it seems to fit the bill excellently.

Someone challenged this kind of paradigm and claimed that f0(500) could not be a tetraquark state (e.g. see here and here but also papers by Wolfgang Ochs and Peter Minkowski disagree with the tetraquark model for these resonances). The answer come out straightforwardly from LHCb collaboration: Both f0(500) and f0(980) are not tetraquark and the original view by Jaffe is no more supported. Indeed, people that know the Nambu-Jona-Lasinio model should know quite well where the f0(500) (or $\sigma$ ) comes from and I would also suggest that this model can also accommodate higher states like f0(980).

I should say that this is a further striking result coming from LHCb Collaboration. Hopefully, this should give important hints to a better understanding of low-energy QCD.

LHCb collaboration, R. Aaij, B. Adeva, M. Adinolfi, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S. Amato, S. Amerio, Y. Amhis, L. An, L. Anderlini, J. Anderson, R. Andreassen, M. Andreotti, J. E. Andrews, R. B. Appleby, O. Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma, M. Baalouch, S. Bachmann, J. J. Back, A. Badalov, V. Balagura, W. Baldini, R. J. Barlow, C. Barschel, S. Barsuk, W. Barter, V. Batozskaya, Th. Bauer, A. Bay, L. Beaucourt, J. Beddow, F. Bedeschi, I. Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M. -O. Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P. M. Bjørnstad, T. Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, M. Borsato, T. J. V. Bowcock, E. Bowen, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton, N. H. Brook, H. Brown, A. Bursche, G. Busetto, J. Buytaert, S. Cadeddu, R. Calabrese, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, D. Campora Perez, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, L. Cassina, L. Castillo Garcia, M. Cattaneo, Ch. Cauet, R. Cenci, M. Charles, Ph. Charpentier, S. -F. Cheung, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P. E. L. Clarke, M. Clemencic, H. V. Cliff, J. Closier, V. Coco, J. Cogan, E. Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, S. Coquereau, G. Corti, M. Corvo, I. Counts, B. Couturier, G. A. Cowan, D. C. Craik, M. Cruz Torres, S. Cunliffe, R. Currie, C. D’Ambrosio, J. Dalseno, P. David, P. N. Y. David, A. Davis, K. De Bruyn, S. De Capua, M. De Cian, J. M. De Miranda, L. De Paula, W. De Silva, P. De Simone, D. Decamp, M. Deckenhoff, L. Del Buono, N. Déléage, D. Derkach, O. Deschamps, F. Dettori, A. Di Canto, H. Dijkstra, S. Donleavy, F. Dordei, M. Dorigo, A. Dosil Suárez, D. Dossett, A. Dovbnya, F. Dupertuis, P. Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V. Egorychev, S. Eidelman, S. Eisenhardt, U. Eitschberger, R. Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, S. Ely, S. Esen, T. Evans, A. Falabella, C. Färber, C. Farinelli, N. Farley, S. Farry, D. Ferguson, V. Fernandez Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, M. Fiore, M. Fiorini, M. Firlej, C. Fitzpatrick, T. Fiutowski, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M. Frosini, J. Fu, E. Furfaro, A. Gallas Torreira, D. Galli, S. Gallorini, S. Gambetta, M. Gandelman, P. Gandini, Y. Gao, J. Garofoli, J. Garra Tico, L. Garrido, C. Gaspar, R. Gauld, L. Gavardi, E. Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, A. Gianelle, S. Giani’, V. Gibson, L. Giubega, V. V. Gligorov, C. Göbel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, C. Gotti, M. Grabalosa Gándara, R. Graciani Diaz, L. A. Granado Cardoso, E. Graugés, G. Graziani, A. Grecu, E. Greening, S. Gregson, P. Griffith, L. Grillo, O. Grünberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S. C. Haines, S. Hall, B. Hamilton, T. Hampson, X. Han, S. Hansmann-Menzemer, N. Harnew, S. T. Harnew, J. Harrison, T. Hartmann, J. He, T. Head, V. Heijne, K. Hennessy, P. Henrard, L. Henry, J. A. Hernando Morata, E. van Herwijnen, M. Heß, A. Hicheur, D. Hill, M. Hoballah, C. Hombach, W. Hulsbergen, P. Hunt, N. Hussain, D. Hutchcroft, D. Hynds, M. Idzik, P. Ilten, R. Jacobsson, A. Jaeger, J. Jalocha, E. Jans, P. Jaton, A. Jawahery, M. Jezabek, F. Jing, M. John, D. Johnson, C. R. Jones, C. Joram, B. Jost, N. Jurik, M. Kaballo, S. Kandybei, W. Kanso, M. Karacson, T. M. Karbach, M. Kelsey, I. R. Kenyon, T. Ketel, B. Khanji, C. Khurewathanakul, S. Klaver, O. Kochebina, M. Kolpin, I. Komarov, R. F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V. Kudryavtsev, K. Kurek, T. Kvaratskheliya, V. N. La Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert, R. W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch, B. Langhans, T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J. -P. Lees, R. Lefèvre, A. Leflat, J. Lefrançois, S. Leo, O. Leroy, T. Lesiak, B. Leverington, Y. Li, M. Liles, R. Lindner, C. Linn, F. Lionetto, B. Liu, G. Liu, S. Lohn, I. Longstaff, J. H. Lopes, N. Lopez-March, P. Lowdon, H. Lu, D. Lucchesi, H. Luo, A. Lupato, E. Luppi, O. Lupton, F. Machefert, I. V. Machikhiliyan, F. Maciuc, O. Maev, S. Malde, G. Manca, G. Mancinelli, M. Manzali, J. Maratas, J. F. Marchand, U. Marconi, C. Marin Benito, P. Marino, R. Märki, J. Marks, G. Martellotti, A. Martens, A. Martín Sánchez, M. Martinelli, D. Martinez Santos, F. Martinez Vidal, D. Martins Tostes, A. Massafferri, R. Matev, Z. Mathe, C. Matteuzzi, A. Mazurov, M. McCann, J. McCarthy, A. McNab, R. McNulty, B. McSkelly, B. Meadows, F. Meier, M. Meissner, M. Merk, D. A. Milanes, M. -N. Minard, N. Moggi, J. Molina Rodriguez, S. Monteil, D. Moran, M. Morandin, P. Morawski, A. Mordà, M. J. Morello, J. Moron, R. Mountain, F. Muheim, K. Müller, R. Muresan, M. Mussini, B. Muster, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N. Neri, S. Neubert, N. Neufeld, M. Neuner, A. D. Nguyen, T. D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, R. Niet, N. Nikitin, T. Nikodem, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, G. Onderwater, M. Orlandea, J. M. Otalora Goicochea, P. Owen, A. Oyanguren, B. K. Pal, A. Palano, F. Palombo, M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, C. Parkes, C. J. Parkinson, G. Passaleva, G. D. Patel, M. Patel, C. Patrignani, A. Pazos Alvarez, A. Pearce, A. Pellegrino, M. Pepe Altarelli, S. Perazzini, E. Perez Trigo, P. Perret, M. Perrin-Terrin, L. Pescatore, E. Pesen, K. Petridis, A. Petrolini, E. Picatoste Olloqui, B. Pietrzyk, T. Pilař, D. Pinci, A. Pistone, S. Playfer, M. Plo Casasus, F. Polci, A. Poluektov, E. Polycarpo, A. Popov, D. Popov, B. Popovici, C. Potterat, A. Powell, J. Prisciandaro, A. Pritchard, C. Prouve, V. Pugatch, A. Puig Navarro, G. Punzi, W. Qian, B. Rachwal, J. H. Rademacker, B. Rakotomiaramanana, M. Rama, M. S. Rangel, I. Raniuk, N. Rauschmayr, G. Raven, S. Reichert, M. M. Reid, A. C. dos Reis, S. Ricciardi, A. Richards, M. Rihl, K. Rinnert, V. Rives Molina, D. A. Roa Romero, P. Robbe, A. B. Rodrigues, E. Rodrigues, P. Rodriguez Perez, S. Roiser, V. Romanovsky, A. Romero Vidal, M. Rotondo, J. Rouvinet, T. Ruf, F. Ruffini, H. Ruiz, P. Ruiz Valls, G. Sabatino, J. J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, V. Salustino Guimaraes, C. Sanchez Mayordomo, B. Sanmartin Sedes, R. Santacesaria, C. Santamarina Rios, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D. Savrina, M. Schiller, H. Schindler, M. Schlupp, M. Schmelling, B. Schmidt, O. Schneider, A. Schopper, M. -H. Schune, R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N. Serra, J. Serrano, L. Sestini, P. Seyfert, M. Shapkin, I. Shapoval, Y. Shcheglov, T. Shears, L. Shekhtman, V. Shevchenko, A. Shires, R. Silva Coutinho, G. Simi, M. Sirendi, N. Skidmore, T. Skwarnicki, N. A. Smith, E. Smith, E. Smith, J. Smith, M. Smith, H. Snoek, M. D. Sokoloff, F. J. P. Soler, F. Soomro, D. Souza, B. Souza De Paula, B. Spaan, A. Sparkes, F. Spinella, P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp, O. Stenyakin, S. Stevenson, S. Stoica, S. Stone, B. Storaci, S. Stracka, M. Straticiuc, U. Straumann, R. Stroili, V. K. Subbiah, L. Sun, W. Sutcliffe, K. Swientek, S. Swientek, V. Syropoulos, M. Szczekowski, P. Szczypka, D. Szilard, T. Szumlak, S. T’Jampens, M. Teklishyn, G. Tellarini, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S. Tolk, L. Tomassetti, D. Tonelli, S. Topp-Joergensen, N. Torr, E. Tournefier, S. Tourneur, M. T. Tran, M. Tresch, A. Tsaregorodtsev, P. Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, A. Ustyuzhanin, U. Uwer, V. Vagnoni, G. Valenti, A. Vallier, R. Vazquez Gomez, P. Vazquez Regueiro, C. Vázquez Sierra, S. Vecchi, J. J. Velthuis, M. Veltri, G. Veneziano, M. Vesterinen, B. Viaud, D. Vieira, M. Vieites Diaz, X. Vilasis-Cardona, A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Voß, H. Voss, J. A. de Vries, R. Waldi, C. Wallace, R. Wallace, J. Walsh, S. Wandernoth, J. Wang, D. R. Ward, N. K. Watson, D. Websdale, M. Whitehead, J. Wicht, D. Wiedner, G. Wilkinson, M. P. Williams, M. Williams, F. F. Wilson, J. Wimberley, J. Wishahi, W. Wislicki, M. Witek, G. Wormser, S. A. Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, Z. Xing, Z. Xu, Z. Yang, X. Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W. C. Zhang, Y. Zhang, A. Zhelezov, A. Zhokhov, L. Zhong, & A. Zvyagin (2014). Measurement of the resonant and CP components in
$\overline{B}^0\rightarrow J/ψπ^+π^-$ decays arXiv arXiv: 1404.5673v2
Irinel Caprini, Gilberto Colangelo, & Heinrich Leutwyler (2005). Mass and width of the lowest resonance in QCD Phys.Rev.Lett.96:132001,2006 arXiv: hep-ph/0512364v2
Jaffe, R. (1977). Multiquark hadrons. I. Phenomenology of Q^{2}Q[over ¯]^{2} mesons Physical Review D, 15 (1), 267-280 DOI: 10.1103/PhysRevD.15.267
Jaffe, R. (1977). Multiquark hadrons. II. Methods Physical Review D, 15 (1), 281-289 DOI: 10.1103/PhysRevD.15.281
G. Mennessier, S. Narison, & X. -G. Wang (2010). The sigma and f_0(980) from K_e4+pi-pi, gamma-gamma scatterings, J/psi,
phi to gamma sigma_B and D_s to l nu sigma_B Nucl.Phys.Proc.Suppl.207-208:177-180,2010 arXiv: 1009.3590v1

Marco Frasca (2010). Glueball spectrum and hadronic processes in low-energy QCD Nucl.Phys.Proc.Suppl.207-208:196-199,2010 arXiv: 1007.4479v2

## Looking for the Lucasian Professor

09/04/2014

Michael Green, the current Lucasian Professor, is now 67 and Cambridge University is looking for his substitute (see here). It is interesting that Wikipedia is now reporting rumours from researchers at DAMPT (see here)

As of 3rd April 2014, the University of Cambridge is recruiting the 19th Lucasian Professor. Rumour amongst DAMPT researchers is that the process is a required formality, but that unless there’s a last minute surprise the Lucasian Professorship will be held for the first time by a woman. It may also be the first time that the chair is not held by somebody born in the British Isles.

Wikipedia is generally not doing gossip so I report this here because I think this will not last too long. Anyhow, the procedure will be brief. The candidates should send their CV for 15 April and the chosen one should take up the appointment on 1 September this year.  I would like to remember that this is a really prestigious chair held by Newton, Dirac and Hawking to name just a few. Surely, Cambridge University will choose for the best once again.

## Evidence of the square root of Brownian motion

06/03/2014

A mathematical proof of existence of a stochastic process involving fractional exponents seemed out of question after some mathematicians claimed this cannot exist. This observation is strongly linked to the current definition and may undergo revision if nature does not agree with it. Stochastic processes are very easy to simulate on a computer. Very few lines of code can decide if something works or not. I and Alfonso Farina, together with Matteo Sedehi,  have introduced the idea that the square root of a Wiener process yields the Schroedinger equation (see here or download a preprint here). This implies that one has to attach a meaning to the equation

$dX=(dW)^\frac{1}{2}.$

In a paper appeared today on arxiv (see here) we finally have provided this proof: We were right. The idea is to solve such an equation by numerical methods. These methods are themselves a proof of existence. We used the Euler-Maruyama method, the simplest one and we compared the results as shown in the following figure

a) Original Brownian motion. b) Same but squaring the formula for the square root. c) Formula of the square root taken as a stochastic equation. d) Same from the stochastic equation in this post.

There is now way to distinguish each other and the original Brownian motion is completely recovered by taking the square of the square root process computed in three different ways. Each one of these completely supports the conclusions we have drawn in our published paper. You can find the code to recover this figure in our arxiv paper. It is obtained by a Monte Carlo simulation with 10000 independent paths. You can play with it changing the parameters as you like.

This paper has an important consequence: Our current mathematical understanding of stochastic processes should be properly extended to account for our results. As a by-product, we have shown how, using Pauli matrices, this idea can be generalized to include spin introducing a new class of stochastic processes in a Clifford algebra.

In conclusion, we would like to remember that, it does not matter what your mathematical definition could be, a stochastic process is always a well-defined entity on a numerical ground. Tests can be easily performed as we proved here.

Farina, A., Frasca, M., & Sedehi, M. (2013). Solving Schrödinger equation via Tartaglia/Pascal triangle: a possible link between stochastic processing and quantum mechanics Signal, Image and Video Processing, 8 (1), 27-37 DOI: 10.1007/s11760-013-0473-y

Marco Frasca, & Alfonso Farina (2014). Numerical proof of existence of fractional Wiener processes arXiv arXiv: 1403.1075v1

## Nailing down the Yang-Mills problem

22/02/2014

Millennium problems represent a major challenge for physicists and mathematicians. So far, the only one that has been solved was the Poincaré conjecture (now a theorem) by Grisha Perelman. For people working in strong interactions and quantum chromodynamics, the most interesting of such problems is the Yang-Mills mass gap and existence problem. The solutions of this problem would imply a lot of consequences in physics and one of the most important of these is a deep understanding of confinement of quarks inside hadrons. So far, there seems to be no solution to it but things do not stay exactly in this way. A significant number of researchers has performed lattice computations to obtain the propagators of the theory in the full range of energy from infrared to ultraviolet providing us a deep understanding of what is going on here (see Yang-Mills article on Wikipedia). The propagators to be considered are those for  the gluon and the ghost. There has been a significant effort from theoretical physicists in the last twenty years to answer this question. It is not so widely known in the community but it should because the work of this people could be the starting point for a great innovation in physics. In these days, on arxiv a paper by Axel Maas gives a great recount of the situation of these lattice computations (see here). Axel has been an important contributor to this research area and the current understanding of the behavior of the Yang-Mills theory in two dimensions owes a lot to him. In this paper, Axel presents his computations on large volumes for Yang-Mills theory on the lattice in 2, 3 and 4 dimensions in the SU(2) case. These computations are generally performed in the Landau gauge (propagators are gauge dependent quantities) being the most favorable for them. In four dimensions the lattice is $(6\ fm)^4$, not the largest but surely enough for the aims of the paper. Of course, no surprise comes out with respect what people found starting from 2007. The scenario is well settled and is this:

1. The gluon propagator in 3 and 4 dimensions dos not go to zero with momenta but is just finite. In 3 dimensions has a maximum in the infrared reaching its finite value at 0  from below. No such maximum is seen in 4 dimensions. In 2 dimensions the gluon propagator goes to zero with momenta.
2. The ghost propagator behaves like the one of a free massless particle as the momenta are lowered. This is the dominant behavior in 3 and 4 dimensions. In 2 dimensions the ghost propagator is enhanced and goes to infinity faster than in 3 and 4 dimensions.
3. The running coupling in 3 and 4 dimensions is seen to reach zero as the momenta go to zero, reach a maximum at intermediate energies and goes asymptotically to 0 as momenta go to infinity (asymptotic freedom).

Here follows the figure for the gluon propagator

and for the running coupling

There is some concern for people about the running coupling. There is a recurring prejudice in Yang-Mills theory, without any support both theoretical or experimental, that the theory should be not trivial in the infrared. So, the running coupling should not go to zero lowering momenta but reach a finite non-zero value. Of course, a pure Yang-Mills theory in nature does not exist and it is very difficult to get an understanding here. But, in 2 and 3 dimensions, the point is that the gluon propagator is very similar to a free one, the ghost propagator is certainly a free one and then, using the duck test: If it looks like a duck, swims like a duck, and quacks like a duck, then it probably is a duck, the theory is really trivial also in the infrared limit. Currently, there are two people in the World that have recognized a duck here:  Axel Weber (see here and here) using renormalization group and me (see here, here and here). Now, claiming to see a duck where all others are pretending to tell a dinosaur does not make you the most popular guy  in the district. But so it goes.

These lattice computations are an important cornerstone in the search for the behavior of a Yang-Mills theory. Whoever aims to present to the World his petty theory for the solution of the Millennium prize must comply with these results showing that his theory is able to reproduce them. Otherwise what he has is just rubbish.

What appears in the sight is also the proof of existence of the theory. Having two trivial fixed points, the theory is Gaussian in these limits exactly as the scalar field theory. A Gaussian theory is the simplest example we know of a quantum field theory that is proven to exist. Could one recover the missing part between the two trivial fixed points as also happens for the scalar theory? In the end, it is possible that a Yang-Mills theory is just the vectorial counterpart of the well-known scalar field, the workhorse of all the scholars in quantum field theory.

Axel Maas (2014). Some more details of minimal-Landau-gauge Yang-Mills propagators arXiv arXiv: 1402.5050v1

Axel Weber (2012). Epsilon expansion for infrared Yang-Mills theory in Landau gauge Phys. Rev. D 85, 125005 arXiv: 1112.1157v2

Axel Weber (2012). The infrared fixed point of Landau gauge Yang-Mills theory arXiv arXiv: 1211.1473v1

Marco Frasca (2007). Infrared Gluon and Ghost Propagators Phys.Lett.B670:73-77,2008 arXiv: 0709.2042v6

Marco Frasca (2009). Mapping a Massless Scalar Field Theory on a Yang-Mills Theory: Classical
Case Mod. Phys. Lett. A 24, 2425-2432 (2009) arXiv: 0903.2357v4

Marco Frasca (2010). Mapping theorem and Green functions in Yang-Mills theory PoS FacesQCD:039,2010 arXiv: 1011.3643v3

## Nature already patched it

09/02/2014

Dennis Overbye is one of the best science writer around. Recently, he wrote a beautiful piece on the odd behavior of non-converging series like $1+2+3+4+\ldots$ and so on to infinity (see here). This article contains a wonderful video, this one

where it shown why $1+2+3+4+\ldots=-1/12$ and this happens only when this series is taken going to infinity. You can also see a 21 minutes video on the same argument from these authors

This is really odd as we are summing up all positive terms and in the end one gets a negative result. This was a question that already bothered Euler and is generally fixed with the Riemann zeta function. Now, if you talk with a mathematician, you will be warned that such a series is not converging and indeed intermediate results become even more larger as the sum is performed. So, this series should be generally discarded when you meet it in your computations in physics or engineering. We know that things do not stay this way as nature already patched it. The reason is exactly this: Infinity does not exist in nature and whenever one is met nature already fixed it, whatever a mathematician could say. Of course, smarter mathematicians are well aware of this as you can read from Terry Tao’s blog. Indeed, Terry Tao is one of the smartest living mathematicians. One of his latest successes is to have found a problem in the presumed Otelbaev’s proof of the existence of solutions to Navier-Stokes equations, a well-known millennium problem (see the accepted answer and comments here).

This idea is well-known to physicists and when an infinity is met we have invented a series of techniques to remove it in the way nature has chosen. This can be seen from the striking agreement between computed and measured quantities in some quantum field theories, not last the Standard Model. E.g. the gyromagnetic ratio of the electron agrees to one part on a trillion with the measured quantity (see here). This perfection in the computations was never seen before in physics and belongs to the great revolution that was completed by Feynman, Schwinger, Tomonaga and Dyson that we have inherited in the Standard Model, the latest and greatest revolution seen so far in particle physics. We just hope that LHC will uncover the next one at the restart of operations. It is possible again that nature will have found further ways to patch infinities and one of these could be $1+2+3+4+\ldots=-1/12$.

So, we recall one of the greatest principles of physics: Nature patches infinities and use techniques to do it that are generally disgusting mathematicians. I think that diverging series should be taught at undergraduate level courses. Maybe, using the standard textbook by Hardy (see here). These are not just pathologies in an otherwise wonderful world but rather these are the ways nature has chosen to behave!

The reason for me to write about this matter is linked to a beautiful work I did with my colleagues Alfonso Farina and Matteo Sedehi on the way the Tartaglia-Pascal triangle generalizes in quantum mechanics. We arrived at the conclusion that quantum mechanics arises as the square root of a Brownian motion. We have got a paper published on this matter (see here or you can see the Latest Draft). Of course, the idea to extract the square root of a Wiener process is something that was disgusting mathematicians, mostly Didier Piau, that was claiming that an infinity goes around. Of course, if I have a sequence of random numbers, these are finite, I can arbitrarily take their square root. Indeed, this is what one sees working with Matlab that easily recovers our formula for this process. So, what does it happen to the infinity found by Piau? Nothing, but nature already patched it.

So, we learned a beautiful lesson from nature: The only way to know her choices is to ask her.

A. Farina,, M. Frasca,, & M. Sedehi (2014). Solving Schrödinger equation via Tartaglia/Pascal triangle: a possible link between stochastic processing and quantum mechanics Signal, Image and Video Processing, 8 (1), 27-37 DOI: 10.1007/s11760-013-0473-y

## Back to work

02/02/2014

I would like to have a lot more time to write on my blog. Indeed, time is something I have no often and also the connection is not so good as I would like in the places I spend most of it. So, I take this moment to give an update of what I have seen around in these days.

LHC has found no evidence of dark matter so far (see here). Dark matter appears even more difficult to see and theory is not able to help the search. This is also one of our major venues to go beyond the Standard Model. On the other side, ASACUSA experiment at CERN produced the first beam of antihydpogen atoms (see here, this article is free to read). We expect no relevant news about the very nature of Higgs until, on 2015, LHC will restart. It must be said that the data collected so far are saying to us that this particle is behaving very nearly as that postulated by Weinberg on 1967.

In these days there has been some fuss about the realization in laboratory of a Dirac magnetic monopole (see here).  Notwithstanding this is a really beautiful experiment, nobody has seen a magnetic monopole so far. It is a simulation performed with another physical system: A BEC. This is a successful technology that will permit us an even better understanding of physical systems that are difficult to observe. Studies are ongoing to realize a simulation of  Hawking radiation in such a system.  Even if this is the state of affairs, I have read in social networks and in the news that a magnetic monopole was seen in laboratory. Of course, this is not true.

The question of black holes is always at the top of the list of the main problems in physics. Mostly when a master of physics comes out with a new point of view. So, a lot of  fuss arose from this article in Nature involving a new idea from Stephen Hawking that the author published in a paper on arxiv (see here). Beyond the resounding title, Hawking is just proposing a way to avoid the concept of firewalls that was at the center of a hot debate in the last months. Again we recognize that a journalist is not making a good job but is generating a lot of noise around and noise can hide a signal very well.

Finally, we hope in a better year in science communication. The start was somewhat disappointing.

Kuroda N, Ulmer S, Murtagh DJ, Van Gorp S, Nagata Y, Diermaier M, Federmann S, Leali M, Malbrunot C, Mascagna V, Massiczek O, Michishio K, Mizutani T, Mohri A, Nagahama H, Ohtsuka M, Radics B, Sakurai S, Sauerzopf C, Suzuki K, Tajima M, Torii HA, Venturelli L, Wu Nschek B, Zmeskal J, Zurlo N, Higaki H, Kanai Y, Lodi Rizzini E, Nagashima Y, Matsuda Y, Widmann E, & Yamazaki Y (2014). A source of antihydrogen for in-flight hyperfine spectroscopy. Nature communications, 5 PMID: 24448273

M. W. Ray,, E. Ruokokoski,, S. Kandel,, M. Möttönen,, & D. S. Hall (2014). Observation of Dirac monopoles in a synthetic magnetic field Nature, 505, 657-660 DOI: 10.1038/nature12954

Zeeya Merali (2014). Stephen Hawking: ‘There are no black holes’ Nature DOI: 10.1038/nature.2014.14583

S. W. Hawking (2014). Information Preservation and Weather Forecasting for Black Holes arXiv arXiv: 1401.5761v1