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November 10, 2004

A Fearful Symmetry

Longtime readers of this blog will know that I think that discrete symmetries (particularly, those that suppress proton decay and flavour-changing neutral currents) are a serious challenge for ideas about the Landscape and, more generally, applications of the Anthropic Principle to String Theory Vacua (see, for instance, these three and posts).

If supersymmetry is broken in the visible sector below the GUT scale, then there are, generically, dimension-4 operators in the supersymmetric theory which violate baryon number. Below the supersymmetry-breaking scale, these induce the usual dimension-6 baryon number operators, suppressed, not by M GUT 2M_{\text{GUT}}^{-2}, but by M q˜ 2M_{\tilde q}^{-2}, where M q˜M_{\tilde q} is the squark mass. To suppress these bad operators (note that the observed proton lifetime is at least 20 orders of magnitude longer than the anthropic bound), one generally invokes a discrete symmetry. For instance, R-parity arises as a 2\mathbb{Z}_2 global symmetry, under which the quark and lepton chiral multiplets are odd, while the Higgs chiral multiplets are even.

Such discrete symmetries typically only occur on subspaces of the moduli space of very high codimension. In flux compactifications, not only must the vacuum lie on this invariant subspace, but the fluxes must be chosen to respect the symmetry as well (they appear as coupling constant in the low-energy theory). This drastically reduces the number of flux vacua.

For concreteness, let us work in the orientifold limit, with a Calabi-Yau 3-fold, XX. The standard estimate of the number of flux vacua is roughly L b 3(X)L^{b_3(X)}, where LL is a measure of the D3-brane charge that must be cancelled. If the desired discrete symmetry requires restricting to a subspace of the moduli space of complex dimension nn, then the naïve estimate of the number of flux vacua respecting the symmetry is L 2(n+1)L^{2(n+1)} (the extra 1 comes from H (3,0)H (0,3)H^{(3,0)}\oplus H^{(0,3)}). This is vastly smaller than L b 3(X)L^{b_3(X)}.

There’s a very interesting recent paper which does a more careful estimate. What they find is that — in the class of examples they look at — the number of vacua respecting the discrete symmetry can be enhanced, for number-theoretic reasons, over the above naïve estimate by powers of log(L)\log(L).

Where does number theory come in? Pick a point, zz, in the complex structure moduli space and a value, ϕ\phi, for the axio-dilaton. View the equations D αW=0D_\alpha W=0 as a set of (linear) equations for the fluxes. If the fluxes were real-valued, we could always solve those equations. But they’re integer-valued. For a generic point, there will be no integer-valued solutions for the allowed fluxes. For special points, the periods take values in some finite extension of \mathbb{Q}, and it is possible to adjust the (integer) fluxes, so as to find a solution. Consider the case where the field extension is of degree1 dd and ϕ\phi is of height2 H(ϕ)H(\phi) in this extension. If the degree of the extension and the height, H(ϕ)H(\phi), are small enough, then one gets N vacua(L (1d/4)H(ϕ) d/2) b 3 N_{\text{vacua}}\sim \left(\frac{L^{(1-d/4)}}{H(\phi)^{d/2}}\right)^{b_3} as an estimate of the number of flux vacua at such a point.

At least in the simple examples they consider, summing over such points respecting the discrete symmetry produces a result enhanced by powers of log(L)\log(L) over the naïve one.

They also address a related question: how many such vacua also satisfy W tree=0W_{\text{tree}}=0 (i.e. are supersymmetric in 4D flat space, in the no-scale approximation)? Such vacua are non-generic3 as the system of equations in overdetermined. But their abundance is closely related to the statistical likelihood (if you believe in such statistical games) of low-scale supersymmetry breaking. Again, DeWolfe et al give an estimate N W=0vacua(L 1/2H(ϕ)) d(L (1d/4)H(ϕ) d/2) b 3 N_{W=0\,\text{vacua}}\sim \left(\frac{L^{1/2}}{H(\phi)}\right)^{-d} \left(\frac{L^{(1-d/4)}}{H(\phi)^{d/2}}\right)^{b_3} of the number of W tree=0W_{\text{tree}}=0 vacua at a point where the periods lie in a degree-dd extension of the rationals.


1 A field KK is an extension of a field FF if FF is a subfield of KK. The degree of the extension is the dimension of KK, considered as a vector space over FF, d(K/F)=dim F(K) d(K/F)= dim_{F}(K)

2 The height of a rational number, p/qp/q, for p,qp,q coprime is max(|p|,|q|)\max(|p|,|q|). If ϕ\phi is valued in a finite extension of the rationals, view that extension as a vector space over \mathbb{Q}, as in the previous footnote, and hence ϕ\phi as a dd-tuple, ϕ=(p 1/q 1,,p d/q d)=(m 1/gcd(q i),,m d/gcd(q i))\phi= (p_1/q_1, \dots, p_d/q_d)= (m_1/\gcd(q_i),\dots,m_d/\gcd(q_i)). Then H(ϕ)=max(gcd(q i),|m 1|,,|m d|)H(\phi)=\max(\gcd(q_i),|m_1|,\dots,|m_d|).

3 The more general supersymmetric vacuum has W=W 00W= W_0\neq 0. These typically lead to supersymmetric vacua in anti-de Sitter space after the no-scale structure is broken by the generation of a superpotential for the Kähler moduli.

Posted by distler at November 10, 2004 12:08 PM

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Re: A Fearful Symmetry

Jacques,

I am not sure what are the rules of the game, I am wondering what is your opinion:

Did you expect huge suppression of the number of (candidates for) vacua if you demand the existence of any discrete symmetry? it seems likely that demanding that the discrete symmetries be the right ones (do the job) will provide an additional supression, perhpas a more significant one. More generally, the more you demand of your vacuum, the less likely you are to get a huge number of vacua. Is the game in your mind detecting if there is a large number of vacua left that resemble some semi-realistic phenomenology?

What do we do if the answer is yes? what do we do if the answer is no?

Posted by: Moshe Rozali on November 11, 2004 11:24 AM | Permalink | Reply to this

Les Règles du Jeu

Did you expect huge suppression of the number of (candidates for) vacua if you demand the existence of any discrete symmetry?

Yes. And, up to these logarithmic enhancements (a surprise to me), that is confirmed.

More generally, the more you demand of your vacuum, the less likely you are to get a huge number of vacua.

A believer in anthropic reasoning would demand only those things required on anthropic grounds.

As far as I can tell, our universe is quite atypical among (what one expects would be) anthropically-allowed ones.

Is the game in your mind detecting if there is a large number of vacua left that resemble some semi-realistic phenomenology?

I believe that is a crucial question. It determines whether we should

  1. go about search for “the” vacuum (or a small number of vacua which might) describe our world.
  2. develop some statistical tools to extract statistical predictions about some large number of candidate vacua which might describe our world.

I find the latter extraordinarily unlikely, but it seems to a rather popular view right now.

Posted by: Jacques Distler on November 11, 2004 11:52 AM | Permalink | PGP Sig | Reply to this

Re: Les Règles du Jeu

Thanks Jacques, let me ask you one more thing,

So, option 1 does make complete sense to me, the statistical method is then just an efficient way to look for a needle in a haystack. It would be even more efficient if we talked about a subset of significant measure in the space of all string vacua, but one has to start somewhere.

The first possibility seems to me inconsistent: if we are already discovering that to prevent proton decay we are at an unlikely corner (even allowing for anthropic reasoning), how can we have confidence in statistical predictions regarding other quantities?
(once one starts with conditional probabilities, those can be forever tweaked with).

best,

Moshe

Posted by: Moshe Rozali on November 11, 2004 12:11 PM | Permalink | Reply to this

Re: Les Règles du Jeu

[I]f we are already discovering that to prevent proton decay we are at an unlikely corner (even allowing for anthropic reasoning), how can we have confidence in statistical predictions regarding other quantities?

For a while, Nima had me half-convinced that there might be a more indirect anthropic explanation for the observed proton lifetime. I now think I had been bamboozled.

Posted by: Jacques Distler on November 11, 2004 12:22 PM | Permalink | PGP Sig | Reply to this

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