## Igor Suslov and the beta function of the scalar field

21/02/2011

I think that blogs are a very good vehicle for a scientist to let his/her work widely known and can be really helpful also for colleagues doing research in the same field. This is the case of Igor Suslov at Kapitza Institute in Moscow. Igor is doing groundbreaking research in quantum field theory and, particularly, his main aim is to obtain the beta function of the scalar field in the limit of a very large coupling. This means that the field of research of Igor largely overlaps mine. Indeed, I have had some e-mail exchange with him and we cited our works each other. Our conclusions agree perfectly and he was able to obtain the general result that, for very large bare coupling $\lambda$ one has

$\beta(\lambda)=d\lambda$

where d is the number of dimensions. This means that for d=4 Igor recovers my result. More important is the fact that from this result one can draw the conclusion that the scalar theory is indeed trivial in four dimensions, a long sought result. This should give an idea of the great quality of the work of this author.

On the same track, today  on arxiv Igor posted another important paper (see here). The aim of this paper is to get higher order corrections to the aforementioned result. So, he gives a sound initial explanation on why one could meaningfully take the bare coupling running from 0 to infinity and then, using a lattice formulation of the n components scalar field theory, he performs a high temperature expansion.  He is able to reach the thirteenth order correction! This is an expansion of $\beta(\lambda)/\lambda$ in powers of $\lambda^{-\frac{2}{d}}$ and so, for d=4, one gets an expansion in $1/\sqrt{\lambda}$. Again, this Igor’s result is in agreement with mine in a very beautiful manner. As my readers could know, I have been able to go to higher orders with my expansion technique in the large coupling limit (see here and here). This means that my findings and this result of Igor must agree. This is exactly what happens! I was able to get the next to leading order correction for the two-point function and, from this, with the Callan-Symanzik equation, I can derive the next to leading order correction for $\beta(\lambda)/\lambda$ that goes like $1/\sqrt{\lambda}$ with an opposite sign with respect to the previous one. This is Igor’s table with the coefficients of the expansion:

So, from my point of view, Igor’s computations are fundamental for all the understanding of infrared physics that I have developed so far. It would be interesting if he could verify the mapping with Yang-Mills theory obtaining the beta function also for this case. He did some previous attempt on this direction but now, with such important conclusions reached, it would be absolutely interesting to see some deepening. Thank you for this wonderful work, Igor!

I. M. Suslov (2011). Renormalization Group Functions of \phi^4 Theory from High-Temperature
Expansions J.Exp.Theor.Phys., v.112, p.274 (2011); Zh.Eksp.Teor.Fiz., v.139, p.319 (2011) arXiv: 1102.3906v1

Marco Frasca (2008). Infrared behavior of the running coupling in scalar field theory arxiv arXiv: 0802.1183v4

Marco Frasca (2010). Mapping theorem and Green functions in Yang-Mills theory arxiv arXiv: 1011.3643v2

## Ashtekar and the BKL conjecture

18/02/2011

Abhay Ashtekar is a well-known Indian physicist working at Pennsylvania State University. He has produced a fundamental paper in general relativity that has been the cornerstone of all the field of research of loop quantum gravity. Beyond the possible value that loop quantum gravity may have, we will see in the future, this result of Ashtekar will stand as a fundamental contribution to general relativity. Today on arxiv he, Adam Henderson and David Sloan posted a beautiful paper where the Ashtekar’s approach is used to reformulate the Belinski-Khalatnikov-Lifshitz (BKL) conjecture.

Let me explain why this conjecture is important in general relativity. The question to be answered is the behavior of gravitational fields near singularities. About this, there exist some fundamental theorems due to Roger Penrose and Stephen Hawking. These theorems just prove that singularities are an unavoidable consequence of the Einstein equations but are not able to state the exact form of the solutions near such singularities. Vladimir Belinski, Isaak Markovich Khalatnikov and Evgeny Lifshitz put forward a conjecture that gave them the possibility to get the exact analytical behavior of the solutions of the Einstein equations near a singularity: When a gravitational field is strong enough, as near a singularity, the spatial derivatives in the Einstein equations can be safely neglected and only derivatives with respect to time should be retained. With this hypothesis, these authors were able to reduce the Einstein equations to a set of ordinary differential equations, that are generally more treatable, and to draw important conclusions about the gravitational field in these situations. As you may note, they postulated a gradient expansion in a regime of a strong perturbation!

Initially, this conjecture met with skepticism. People simply have no reason to believe to it and, apparently, there was no reason why spatial variations in a solution of a non-linear equation with a strong non-linearity should have to be neglected. I had the luck to meet Vladimir Belinski at the University of Rome “La Sapienza”. I was there to follow some courses after my Laurea and Vladimir was teaching a general relativity course that I took. The course showed the BKL approach and gravitational solitons (another great contribution of Vladimir to general relativity). Vladimir is also known to have written some parts of the second volume of the books of Landau and Lifshitz on theoretical physics. After the lesson on the BKL approach I talked to him about the fact that I was able to get their results as their approach was just the leading order of a strong coupling expansion. It was on 1992 and I had just obtained the gradient expansion for the Schroedinger equation, also known in literature as the Wigner-Kirkwood expansion, through my approach to strong coupling expansion. The publication of my proof happened just on 2006 (see here), 14 years after our colloquium.

Back to Ashtekar, Henderson and Sloan’s paper, this contribution is relevant for a couple of reasons that go beyond application to quantum gravity. Firstly, they give a short but insightful excursus on the current situation about this conjecture and how computer simulations are showing that it is right (a gradient expansion is a strong coupling expansion!). Secondly, they provide a sound formulation using Ashtekar variables of the Einstein equations that is better suited for its study. In my proof too I use a Hamiltonian formulation but through ADM formalism. These authors have in mind quantum gravity instead and so ADM formalism could not be the best for this aim. In any case, such a different approach could also reveal useful for numerical simulations.

Finally, all this matter is a strong support to my view started with my paper on 1992 on Physical Review A. Since then, I have produced a lot of work with a multitude of applications in almost all areas of physics. I hope that the current trend of confirmations of the goodness of my ideas about perturbation theory will keep on. As a researcher, it is a privilege to be part of this adventure of humankind.

Ashtekar, A. (1986). New Variables for Classical and Quantum Gravity Physical Review Letters, 57 (18), 2244-2247 DOI: 10.1103/PhysRevLett.57.2244

Abhay Ashtekar, Adam Henderson, & David Sloan (2011). A Hamiltonian Formulation of the BKL Conjecture arxiv arXiv: 1102.3474v1

Marco Frasca (2005). Strong coupling expansion for general relativity Int.J.Mod.Phys. D15 (2006) 1373-1386 arXiv: hep-th/0508246v3

Frasca, M. (1992). Strong-field approximation for the Schrödinger equation Physical Review A, 45 (1), 43-46 DOI: 10.1103/PhysRevA.45.43

## Updated paper

18/03/2009

After a very interesting analysis about classical solutions of Yang-Mills equations, in this blog and elsewhere in the web, and having recognized that a paper of mine was in great need for corrections (see here) I have finally done it.

I have replaced the paper on arxiv a few moments ago (see here). I do not know if it is immediately available or you have to wait for tomorrow morning. In any case, the only new result added, with respect to material already discussed in this blog, is the first order correction to the propagator of the massless scalar theory. This goes like $1/\lambda$ making all the argument consistent. This asymptotic series should be modified as the limit $\lambda\rightarrow\infty$ becomes more and more difficult to be applied and this should be in a kind of intermediate region that, presently, I have no technique to manage. This is matter for future work. The perspective is the ability to recover the solution of a scalar field theory for all energy range.

## Quantum field theory and gradient expansion

21/02/2009

In a preceding post (see here) I showed as a covariant gradient expansion can be accomplished maintaining Lorentz invariance during computation. Now I discuss here how to manage the corresponding generating functional

$Z[j]=\int[d\phi]e^{i\int d^4x\frac{1}{2}[(\partial\phi)^2-m^2\phi^2]+i\int d^4xj\phi}.$

This integral can be computed exactly, the theory being free and the integral is a Gaussian one, to give

$Z[j]=e^{\frac{i}{2}\int d^4xd^4yj(x)\Delta(x-y)j(y)}$

where we have introduced the Feynman propagator $\Delta(x-y)$. This is well-knwon matter. But now we rewrite down the above integral introducing another spatial coordinate and write down

$Z[j]=\int[d\phi]e^{i\int d\tau d^4x\frac{1}{2}[(\partial_\tau\phi)^2-(\partial\phi)^2-m^2\phi^2]+i\int d\tau d^4xj\phi}.$

Feynman propagator solving this integral is given by

$\Delta(p)=\frac{1}{p_\tau^2-p^2-m^2+i\epsilon}$

and a gradient expansion just means a series into $p^2$ of this propagator. From this we learn immeadiately two things:

• When one takes $p=0$ we get the right spectrum of the theory: a pole at $p_\tau^2=m^2.$
• When one takes $p_\tau=0$ and Wick-rotates one of the four spatial coordinates we recover the right Feynman propagator.

All works fine and we have kept Lorentz invariance everywhere hidden into the Euclidean part of a five-dimensional theory. Neglecting the Euclidean part gives us back the spectrum of the theory. This is the leading order of a gradient expansion.

So, the next step is to see what happens with an interaction term. I have already solved this problem here and was published by Physical Review D (see here). In this paper I did not care about Lorentz invariance as I expected it would be recovered in the end of computations as indeed happens. But here we can recover the main result of the paper keeping Lorentz invariance. One has

$Z[j]=\int[d\phi]e^{i\int d\tau d^4x\frac{1}{2}[(\partial_\tau\phi)^2-(\partial\phi)^2-m^2\phi^2-\frac{\lambda}{2}\phi^4]+i\int d\tau d^4xj\phi}$

and if we want something not trivial we have to keep the interaction term into the leading order of our gradient expansion. So we will break the exponent as

$Z[j]=\int[d\phi]e^{i\int d\tau d^4x\frac{1}{2}[(\partial_\tau\phi)^2-\frac{\lambda}{2}\phi^4]-i\int d\tau d^4x\frac{1}{2}[(\partial\phi)^2+m^2\phi^2]+i\int d\tau d^4xj\phi}$

and our leading order functional is now

$Z_0[j]=\int[d\phi]e^{i\int d\tau d^4x\frac{1}{2}[(\partial_\tau\phi)^2-\frac{\lambda}{2}\phi^4]+i\int d\tau d^4xj\phi}.$

This can be cast into a Gaussian form as, in the infrared limit, the one of our interest, one can use the following small time approximation

$\phi(x,\tau)\approx\int d\tau' d^4y \delta^4(x-y)\Delta(\tau-\tau')j(y,\tau')$

being now

$\partial_\tau^2\Delta(\tau)+\lambda\Delta(\tau)^3=\delta(\tau)$

that can be exactly solved giving back all the results of my paper. When the Gaussian form of the theory is obtained one can easily show that, in the infrared limit, the quartic scalar field theory is trivial as we obtain again a generating functional in the form

$Z[j]=e^{\frac{i}{2}\int d^4xd^4yj(x)\Delta(x-y)j(y)}$

being now

$\Delta(p)=\sum_n\frac{A_n}{p^2-m^2_n+i\epsilon}$

after Wick-rotated a spatial variable and having set $p_\tau=0$. The spectrum is proper to a trivial theory being that of an harmonic oscillator.

I think that all this machinery does work very well and is quite robust opening up a lot of possibilities to have a look at the other side of the world.

07/02/2009

Due to the relevance of the argument, after a nice discussion with a contribution of Carl Brannen, I decided to pursue this matter further. Indeed, the only way to have a covariant formulation of a gradient expansion is adding a time variable and taking the true time variable Wick rotated. In this way, for d=1+1 wave equation you will use d=2+1 wave equation and so on. In d=3+1 you will use d=4+1 wave equation. Let me explain with some equations what I mean. I consider again d=1+1 case as

$\partial^2_{tt}u-\partial^2_{xx}u=0$

but, instead to apply a gradient expansion to it, I apply this to the equation

$\partial^2_{tt}u-\Delta_2u=0$

being $\Delta_2 = \partial_{xx}+\partial_{yy}$. As usual, I rescale time variable as $t\rightarrow\sqrt{\lambda}t$ and I take a solution series

$u=u_0+\frac{1}{\lambda}u_1+\frac{1}{\lambda^2}u_2+\ldots.$

Now I will get the set of equations

$\partial^2_{tt}u_0=0$

$\partial^2_{tt}u_1=\Delta_2u_0$

$\partial^2_{tt}u_2=\Delta_2u_1$

and so on. Let us note that, in this case, we can introduce two new spatial variables as $z=x+iy$ and $\bar z=x-iy$. These are conjugate variables as you know. So, already at the leading order I have solved my equation. Indeed, I note that

$\Delta_2=\partial_z\partial_{\bar z}$

and so the Laplacian has the solution $f(z)+g(\bar z)$ being f and g arbitrary functions. In this case the gradient expansion gives immediately the exact result making its application trivial as should be. Indeed, I take $t=0$ in the perturbation series and put $iy=t$ and I get

$u=f(x+t)+g(x-t)$

that is the exact solution. Nice, it works! This means that a quantum field theory using gradient expansion exists and it is a strong coupling expansion. This result is surely less trivial than the one obtained above.

## QCD with two colors

30/10/2008

Having an understanding of Yang-Mills theory grants the possibility to make QCD truly manageable and amenable to a perturbation treatment also in the infrared limit. A very easy example of this can be obtained working out the equations of QCD with two colors. In this case the gauge group is SU(2) and algebra is not too much involved. The relevant simplification is obtained via the “mapping theorem” (see my paper here). This theorem grants the existence of a leading order solution of Yang-Mills theory to do perturbation computations in the infrared limit by mapping it on a quartic massless scalar field through the so-called Smilga’s choice (see here). In turn this implies that in all QCD computations we have to manage just a scalar field making things simpler. In QCD with two colors we are reduced to the following action

$S=\int d^4x[\sum_q \bar q(i\gamma\cdot\partial+\frac{g}{2}\phi\Sigma-m_q)q+\frac{3}{2}(\partial\phi)^2-3\frac{2g^2}{4}\phi^4]$

being $\Sigma=\sigma_1\gamma^1+\sigma_2\gamma^2+\sigma_3\gamma^3$ with $\sigma_i$ Pauli matrices and $\gamma^i$ Dirac matrices. So, classical equations of motion are

$(i\gamma\cdot\partial+\frac{g}{2}\phi\Sigma-m_q)q=0$

$\partial^2\phi+2g^2\phi^3=\frac{g}{6}\bar q\Sigma q$

and we can do a strong coupling expansion by rescaling time as $\tau=\sqrt{2}gt$ leaving us with the non-trivial leading order equations

$i\gamma^0\partial_\tau q_0+\frac{1}{2\sqrt{2}}\phi_0\Sigma q_0=0$

$\partial_\tau^2\phi_0+\phi_0^3=0$

and this set of equations is easily solved. We observe that at the leading order quarks can be considered massless (chiral simmetry) and the spectrum of the Yang-Mills theory is part of observational QCD. Finally, a discrete spectrum for quarks is also obtained whose ground state is zero, an expression of chiral symmetry. So, at this order, we expect pion mass to be zero and glueballs having no decay.

These are quite interesting results but higher order corrections should be exploited to have a clear understanding of all physics. Essentially, we would like to compute the pion mass, i.e. to see the breaking of the chiral symmetry seen at the leading order, and the decay width of the lowest glueball state. I will exploit computations to higher orders to reach such aims.