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The \(c\)-theorem probably proved in 4 dimensions

In 1986, Alexander Zamolodchikov of the Soviet Landau Institute – whom I know well from my Rutgers PhD student years more than a decade later – proved the so-called "Zamolodčikov \(c\)-theorem": Wikipedia. You may read the whole paper:

"Irreversibility" of the flux of the renormalization group in a 2D field theory (PDF, full text)
Saša is a hero of two-dimensional conformal (and, more generally, renormalizable) field theories.

What did he prove on those 3 pages? He proved that it's possible to define a generalized number counting "the number of degrees of freedom" \(c\) in such a way that the value of \(c\) always decreases if you extract the "effective theory for longer distances", relatively to the value at shorter distances. In other (experts') words, the function is non-increasing under the renormalization group flow.



The quantity \(c\) had to be defined as a clever function.




The function is
\[ c(g) = C(g) + 4\beta^i H_i - 6\beta^i \beta^j G_{ij} \] of \(C(g)\), the coefficient of the \(1/z^4\) divergence in the correlator of two copies of the stress-energy tensor; the beta-functions \(\beta^i\); the dimension-like quantities \(H_i\) extracted from the correlator of the stress-energy tensor and fields \(\Phi_i\) which are associated with the couplings \(g_i\) and their beta-functions \(\beta^i\).

Fine. I don't want to be too technical but if one does it in this clever way, one may prove – and Zamolodčikov did prove – that \(c\) is non-increasing as we run the "renormalization group flow" from short distances to long distances. Zamolodčikov needed to work hard to find the right proof but it's not "intuitively" hard to understand why something like that is true: as you go to lower energy scales (longer distance scales), you find out that previously relevant "particles" are suddenly too heavy and can't be produced. They no longer affect your long-distance physics so if the "number of particles" is counted in a proper way so that it works not only for free particles but also for particular interactions, a quantity expressing this number has to go down. At fixed points, i.e. when the theory is conformal (scale-invariant), the \(c\)-function coincides with the central charge but Zamolodčikov had to generalize it away from the fixed points, too.



May There Always Be Sunshine (Pusť vsěgdá búdět sólnce), a touching Soviet song that I still love despite its not-so-hidden "communism = peace" message, is appropriate here, isn't it? ;-)

That Russian result became rightfully celebrated. However, its detailed proof depended on some technicalities that seem to be special features of two-dimensional field theories, i.e. the class of theories that include the description of the world sheet dynamics of a string in string theory. Some of these features – e.g. that the group of conformal (angle-preserving) transformations of space is infinite-dimensional in \(d=2\) but not in higher dimensions – are important reasons why strings are more fundamental than higher-dimensional branes, at least perturbatively. Could a similar result "the degrees of freedom disappear under the RG flow" nevertheless be proved in other dimensions such as \(d=4\)?

In 1988, John Cardy asked this very question and conjectured that the answer was Yes. Well, you should't give him too much credit for this guess. He was probably not the only one who has asked the same question and made the same guess about the answer. You shouldn't even worship him for inventing the new name of the quantity that may be non-increasing in \(d=4\): he called it the \(a\)-function, in analogy with the \(c\)-function. Just a name, not a big deal. But he also showed that the trace of the stress-energy tensor could play the role of \(a\) in some approximation. (See also a 1997 paper by Anselmi, Freedman, Grisaru, Johansen for a similar "middle age" discussion.)

However, what happened in July 2011 is that a proof that still seems to stand was apparently found by Zohar Komargodski and Adam Schwimmer of the Weizmann Institute, Israel: the former author is also affiliated with the IAS Princeton. In the paper called
On Renormalization Group Flows in Four Dimensions,
they proved the counterpart of the \(c\)-theorem for \(d=4\), the \(a\)-theorem. Unlike the \(d=2\) case of the \(c\)-function, the \(a\)-function seems to be more "non-Abelian" which means that the action involves nonzero self-interactions of the dilaton. At any rate, the main result is unchanged: the function is non-increasing. It interpolates between the Euler anomalies at short and long distances and it leads to a positive-definite universal contribution of \(2\to 2\) dilaton scattering.

If a bug in the proof has already been found, I am not aware of it. I haven't verified the paper and I will probably only do so once I really need the result for something else. At this moment, the machinery seems straightforward enough and I see no good reason for thinking that those two competent physicists should have made a mistake.

See Nature News / Universe Today (pop-science presentation of the result published 2 weeks ago, with some reactions by several top and near-top theorists)

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