94. Quantum scale invariance on the lattice

Posted by Dmitry

November 13, 2008

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Arguably, the most interesting paper in archives today is the one by M. Shaposhnikov and I. Tkachev. As the authors state, they have found a scheme leading to a non-perturbative definition of lattice field theories scale invariant on the quantum level. I have so many problems understanding this paper, that I don’t even have a slightest idea where to begin… Since I know Igor personally (and he is an extraordinary guy), this probably just shows that I am pretty bad, but I shall still pose my questions – one of the most attracting blogging Powers is that you can ask dumb questions, isn’t it?

1. Introduction: QFTs and continuum limit

High energy physics is the world described in terms of QFTs. Different QFTs are written on different pages in the book of high energy physics – some of those QFTs are renormalizable (like QED and QCD), some – non-renormalizable but effective QFTs (like 4-fermion Fermi field theory describing weak interactions at energy scales much lower than the mass of the M_W boson). But the only thing we are certainly sure of is that particles colliding at LHC (well, after they finally get it running) and particles produced in those collisions are elementary excitations of quantum fields – strongly fluctuating continuous variables.

Now, is it necessary for the world to work like this? As we know from condensed matter theory, no – take a generic sample of condensed matter, say, a magnet. Correct degrees of freedom at the microscopic level are discrete (they are spins), and their interaction is described in the first approximation by Heisenberg or Ising model, not a field theory. Only if the temperature of the system is getting close to the temperature of the phase transition (second order!), the correlation length in the system grows and it starts to be effectively describable in terms of a field theory (for the case of the Ising model the latter is just scalar field theory with $\lambda \phi^4$ self-interaction and spontaneous symmetry breaking, i.e., negative mass term). This is in fact the only regime described by a QFT in a condensed matter system!

What if $T\gg T_c$ or $T\ll T_c$ ? No way, your correct degrees of freedom are spins, not fields. What if the first order phase transition is of the first kind, not the second order? Again, no, there is no way for a field theory to adequitely describe the physics of your system.

If the physics you are interested in is not described by a field theory, be prepared that observables do depend on various interesting scales (like correlation length, which is of the order of the distance between spin degrees of freedom). On the other hand, near $T\sim T_c$ correlation length grows (it actually may become of the same order as the macroscopic size of the system) and all the information about physics at small scales is lost. Finally, I present you with the last piece of lore that comes from condensed matter – if it turned out that your field theory is nonrenormalizable, then there is a 100% chance that the partition function for discrete degrees of freedom you have started with does not have a continuum limit.

2. Scale invariance and conformal field theories

One of the most interesting facts we learn studying perturbation theory in QFT is that physical, measurable, quantities are getting renormalized, i.e., they do depend on the scale. However, is it always the case? Does the fact that our physics is described by QFT automatically imply that interaction between relevant degrees of freedom is scale-dependent?

Runig couplings for QED, QCD and weak interaction in the absence of SUSY

The answer is again negative, and again condensed matter lore helps. If one reaches T=T_c scale in a system with second order phase transition, the field theory symmetry is actually promoted to conformal (the mass scale of the field theoretic collective excitations is proportional to |T-T_c| ). On the field theoretic language this means that the beta function of the coupling has reached the fixed point, and the running of the coupling constant has stopped. There are many examples of conformal field theories in Life, apart from N=4 SYM the authors mention – since there are many different second order phase transitions in Nature (they differ by critical exponents).

3. What do they want to do?

It seems that Igor and Mikhail are scared by quadratic renormalizations of mass scales in many field theories (EW theory, for one) as well as the renormalization of the cosmological constant scale. To fight quadratic divergences, they want to propose a theory with exact but spontaneously broken scale invariance.

As I explained above, the very statement of the problem may not even be meaningful – there are systems in Nature that are not described by discrete degrees of freedom at small scales and field theoretic ones (block spins) at large scales. Although the field theory at large scales is not conformal, it does not make the behaviour of the system pathological at small scales. It could very well be that correct degrees of freedom of the very high energy theory as well as gravity are discrete ones. But let us stick with Shaposhnikov-Tkachev conjecture and follow their logic to find a theory that has a spontaneously broken scale invariance.

The main obstruction for a quantum field theory to be scale invariant is of course identified to be the cutoff scale $\Lambda$ itself (inverse lattice spacing). To remove the dependence of all interesting quantities on this scale, why not to make it field dependent itself? The cutoff is then another field that is allowed to fluctuate (they call it dilaton). The particular model for achieving their goal is the following:

1) in continuum limit one has a Lagrangian

$L=\frac{1}{2}(\partial_\mu \chi)^2+\frac{1}{2}(\partial_\mu h)^2 – V(h,\chi)$ , (1)

where

$V(h,\chi)=\lambda (h^2-\zeta^2\chi^2)^2+\beta \chi^4$ .

The action is scale invariant at the classical level. If $\beta=0$ , then there is a flat direction in the potential, and scale invarnace is spontaneously broken by a particular choice for $\chi$ . Next, they say – if I appropriately modify the renormalization scale:

$\mu^{2\epsilon}\to (c_1 \chi^2 + c_2 h^2 )^{\frac{\epsilon}{1-\epsilon}}=\omega^\frac{2\epsilon}{1-\epsilon}$ ,

then the flat direction is not uplifted by quantum corrections (at one loop level or for all loops???). Honestly, I am currently completely mad about this idea. How would you write down a renormalization group equations for such a setup? How would you write down equations of motion following from the 1-loop effective action that does depend on $\mu$ ? Many, many other questions, too many to put them here.

But let us blindly accept this.

2) At the next step the authors want to write the discretized version of (1). How would it look like? Well, you may guess:

$S=\Sigma_i\left( \frac{\Delta \chi_i^2+\Delta h_i^2}{2\omega_i^2}-\frac{V(h_i,\chi_i)}{\omega_i^4} \right)$ (2)

They now say: as follows from the scaling analysis of the action (2), the corresponding partition function gives you a scale invariant theory! Well, they of course forgot to present us with the measure in the functional integral with field dependent cutoff, but who cares about such things nowadays.

Well, you know, guys – you should care. Depending on the form of the measure there may or may not exist the very continuum limit (1) you started with. The continuum limit of the partition function with the action (2) does not exist if we use the usual $D\chi Dh$ measure, and the theory (2) is not renormalizable.

Oooo, at this point I am so mad that I should stop, otherwise I have a good chance to get a stroke in the very young age, and this Blog is supposed to make me happy and support my well-being.

Mad. Not Agata :-)

In conclusion: guys, if you called some scalar field a dilaton, it does not become a dilaton because of your linguistic exercises, you have to provide a corresponding measure for the functional integral.

Cosmology, Journal club, Master Mason, Quantum field theory, String theory

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94. Quantum scale invariance on the lattice

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