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Low-functioning pinhead...
January 28, 2006
Ghost D-Branes and Renormalization
Catch a very interesting discussion over at Cosmic Variance of a paper by Evans, Morris and Rosten, relating Morris’s Exact Renormalization Group for large- Yang Mills to the “Ghost D-brane” proposal of Okuda and Takayanagi.
The claim is that, by embedding Yang-Mills in a larger (nonunitary) theory, whose gauge groups is the supergroup , spontaneously broken to , one can produce a gauge-invariant Pauli-Villars regulator, with which to implement the Exact RG. The latter theory, in turn, is what Okuda and Takyanagi argue is the world-volume theory of a stack of D-branes and ghost D-branes.
When the gauge symmetry is unbroken, the theory is equivalent to , as far as computing gauge-invariant observables. In particular, there is a perfect cancellation of diagrams for .
Turning on a nonzero Higgs VEV (separating the D-branes from the ghost D-branes) provides a cutoff for the original theory. Above the scale of the Higgs VEV, you get zero; far below it, the “original” degrees of freedom decouple from the ghost .
Evans et al propose and AdS/CFT geometry realization of this idea, with the hope of connecting, in a explicit way, the “holographic RG” (evolution in the radial coordinate of AdS) with the “exact RG” of Morris.
Anyway, Takuya Okuda is over there, fielding questions, so take advantage …
January 20, 2006
Gravity is Weak
I wasn’t going to post anything about the recent paper by Arkani-Hamed et al, figuring that Luboš is perfectly capable of explaining himself. But, after discussions with various people, it became clear that a few comments are in order.
They argue for two propositions which “must” be true in any theory of quantum gravity, but which are not obvious, at all, from the point of view of low-energy effective field theory. They both concern theories in which there is an unbroken gauge symmetry in the low-energy theory.
First, they argue that there must exist charged particle(s) in the theory, whose charge-to-mass ratio exceeds1 the extremal bound on the charge-to-mass ratio of blackholes. A-priori, you could imagine that one could twiddle the masses and charges of fundamental particles arbitrarily. However, as they make clear, in the absence of particles which exceed the bound, charged blackholes cannot radiate away all their charge, and one is left with a large (possibly infinite) number of charged remnants.
This is a perfectly solid result, and one which can be understood quite clearly, once one takes account of blackholes and their evaporation. From it, they abstract away the slogan, “Gravity is the weakest force.” Which leads them to their second conjecture.
The strength of the effective gravitational force grows like a power-law in the UV. The strength of gauge-interactions vary only logarithmically (growing in the UV for abelian gauge theories and falling in the UV for asymptotically-free nonabelian gauge theories). If we go to high enough energies, the gravitational force, therefore, comes to dominate, or would do so if the theory were not cut off.
If we ignore the slow logarithmic running of the gauge coupling, demanding that gauge-interactions dominate over gravitational ones puts a cutoff on effective field theory, not at , but at a lower scale, .
Now, it’s certainly true in all known string theories, 4D effective field theory breaks down below (often, well below) the 4D Planck scale. Indeed, as Arkani-Hamed knows well, the scale at which 4D effective field theory breaks down could be as low as several TeV. I firmly believe (along with the authors) that this is a general principle. But, to put a precise upper bound on the cutoff, at which 4D effective field theory must break down, does require taking account of the running of the gauge coupling.
To sharpen the conjecture, the authors assert that “” in the above formula is the low-energy value of the gauge coupling below the mass of the lightest charged particle. This is not directly related to the “high-energy” value of the coupling (close to the cutoff scale). The rate at which the coupling runs depends on massive charged species at intermediate scales. Not just the magnitude, but even the sign of the -function could change (if the abelian gauge theory is un-higgsed into a nonabelian one). So the scale at which gravity and gauge interactions become comparable in strength cannot be determined from low-energy data alone; it could be higher or lower than the “naïve” estimate of .
Indeed, in many string backgrounds, is unbroken, and the quarks and leptons are massless. (In the same approximation, supersymmetry is, frequently, also unbroken.) In that case, the gauge coupling is driven all the way to zero in the IR2. But that does not mean that effective field theory has zero range of validity.
One can imagine a self-consistent bound of the form . That’s the form of the bound that they actually check in examples. (It’s that is directly related to , not .) In that form, as they verify, the bound holds3. But it’s not a form that depends solely on low-energy data.
1 In the BPS case, saturates the bound.
2 The fact that a massless electron causes the gauge coupling to flow to zero in the IR does not contradict the previous argument about the charge-to-mass ratio. Even though both and vanish as , the latter vanishes more slowly, so we preserve the fact that we have particles whose charge-to-mass ratio exceeds the extremal bound.
3 After some back-and-forth over email, Nima Arkani-Hamed agrees that this, rather than , is the bound. My argument is that it directly expresses the idea that is the scale at which gravity and gauge interactions become comparable in strength. Nima had a more sophisticated argument, involving the evaporation of magnetically-charged blackholes.
January 15, 2006
More on Accesskeys
In my previous post, I argued that accesskey keyboard shortcuts, to be usable, need to be
- discoverable
- modifiable
and I suggested some techniques for achieving that. To make them discoverable, I suggested including on each page, in some artful way, a definition-list of the form
<dl id="AccessKeyList"> <dt>0</dt><dd><a href="/~distler/blog/accessibility.html" accesskey="0">Accessibility Statement</a></dd> <dt>1</dt><dd>Main Page</dd> <dt>2</dt><dd>Skip to Content</dd> <dt>3</dt><dd>List of Posts</dd> <dt>4</dt><dd>Search</dd> <dt>p</dt><dd>Previous (individual/monthly archive page)</dd> <dt>n</dt><dd>Next (individual/monthly archive page)</dd> </dl>
with the accesskeys for your site. To make them modifiable, I proposed a Javascript, which turns that definition-list into a template for editing the keybindings, and stores your modified keybindings for future visits.
But what abou