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Landscape decay channels

One of the reasons why I think that a megalomanic amount of metastable de Sitter vacua in string theory should not exist is that they have a megalomanic number of ways how they can decay.

See also Resonance tunneling and landscape percolation (2007)
Of course, this particular argument can't eliminate the supersymmetric anti de Sitter vacua of the landscape because they are exactly stable. However, de Sitter vacua - which is what we eventually want to get to match reality - can generically decay to other vacua with smaller vacuum energy. Such a process is described by an instanton whose Minkowski interpretation is nothing else than membrane nucleation. A charged spherical domain wall is spontaneously created in space. The interior of this "bubble" carries a lower energy density (which is why it is allowed by energy conservation) and a different value of one type of the electric or magnetic field (a flux over the internal manifold).

If the action of this instanton is "S", the decay amplitude is suppressed roughly by "exp(-S)" and can become negligible if the action is comparable to hundreds or thousands.

But there are actually many types of domain walls that one can nucleate. Each of the basic "types" of the domain walls is able to change one type of the flux by a single unit. In principle, you may consider tunnelling to a much more distant vacuum elsewhere in the landscape. These more convoluted decay processes are suppressed by even greater actions; but there are many such decay channels. Who wins?

In the simplest models you can imagine - such as Shamit+Nima+Savas' model with many scalar fields with a quartic potential - the parameteric answer is actually a tie. If the instanton that induces the flip of a single scalar field from "+v" to "-v" or the other way around has the action "S", then the instanton that flips "k" fields simultaneously may be interpreted as a superposition of "k" copies of decoupled instantons of this type. Recall that the Hamiltonian for these scalar fields is a simple sum of contributions from the individual scalar fields.

And the action of the composite instantons is therefore "kS". If you take "k" to be comparable to "N", the total number of the scalar fields, there are "2^N" decay channels and each of them is suppressed by something like "exp(-NS)". If the elementary action "S" is of order one, the product may be close to one, too. However, it is natural to imagine that there are semi-isolated sectors in which the elementary action is much higher, e.g. 1000, and the instanton suppression wins. In such cases, we can indeed neglect all decay channels except for the fastest one. But note that both factors, the growing factor and the suppression factor, have the same, roughly exponential dependence on "N".

However, in reality, I assume that the action of the composite instanton is parameterically smaller than "kS". My guess is something like "sqrt(k)S". One needs to nucleate a composite domain wall but the tension of the composite domain wall is likely to be smaller than a simple sum of the contributions; a composite domain wall is a bound state of the "elementary" domain walls. If you take the guess "sqrt(k)S" for the action, justifiable by the Pythagorean theorem in the configuration space (landscape) or the analogy with the (p,q) strings, as your starting point, the total decay rate will go approximately like
  • 2^N exp(-sqrt(N)S) ~ (morally) ~ exp(N-sqrt(N)S)

You see that for really large values of "N", being many thousands, the positive term in the exponential dominates and the decay rate becomes fast. The minimum of the exponent appears at

  • N = S^2 / 4

If you assume that the elementary action "S" is, for example, 60, the resulting "N" that maximizes the lifetime will be around 1000. That's still large because it gives you 2^{1000}=10^{300} vacua, but still, you see an argument that the number of long-lived de Sitter vacua should not be allowed to grow indefinitely. You may either think that this argument is flawed or irrelevant, or you may think that in reality, the numbers are actually much more stringent - and closer to one - and your conclusion will be that the number of stable de Sitter vacua must be reasonable, too.

At any rate, the statement that there are googols of nearly stable de Sitter vacua is a rather strong statement - imagine how weird it would be to argue that there are 10^{500} stable states of the Hydrogen atom - and I would expect a rather extraordinary body of evidence, including a detailed refinement of the ideas above, before their existence is accepted as a consequence of string theory.

Once again, this argument does not affect the anti de Sitter vacua. It may be a bit puzzling to have zillions of anti de Sitter vacua - and their dual conformal field theories -too. But maybe this is how the (supersymmetric) life eventually works.

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