SU-ITP 02-11

hep-th/0204027

Supersymmetry Breaking in the Anthropic Landscape

L. Susskind

Department of Physics

Stanford University

Stanford, CA 94305-4060

In this paper I attempt to address a serious criticism of the “Anthropic Landscape” and “Discretuum” approach to cosmology, leveled by Banks, Dine and Gorbatov. I argue that in this new and unfamiliar setting, the gauge Hierarchy may not favor low energy supersymmetry. In a added note some considerations of Douglas which substantially strengthen the argument are explained.

## 1 The Banks Dine Gorbatov Argument

Let’s begin by reviewing a successful use of the anthropic principle. In 1987 Steven Weinberg predicted that if the anthropic principle was correct, the cosmological constant should not be very much smaller than the bound provided by galaxy formation [2]. The argument is straightforward: a much smaller value would require fine tuning of the type which the anthropic principle was supposed to eliminate. Since the anthropic upper bound was only a couple of orders of magnitude above the empirical upper bound, Weinberg argued that the anthropic principle predicted that a cosmological constant would soon be discovered. And it was.

It is worth noting that there are also anthropic bounds on the weak scale that might be strong enough to require a gauge hierarchy like the one we actually observe. Increasing the Higgs expectation value keeping everything else fixed would among other things, increase the strength of gravity, making stars, galaxies and the universe evolve much faster. Alternatively we could try to keep particle masses fixed by decreasing Yukawa couplings. This would decrease the strength of weak interactions, thereby having many effects on the creation of heavy elements in stars as well as the primary mechanism for dispersing the elements, namely, supernovae.

Banks, Dine and Gorbatov (BDG) [1] have recently argued that similar logic can be applied to proton stability and that the exceptionally long life of the proton falsifies the Anthropic Principle. The context of the Banks Dine Gorbatov argument is the “Landscape” or “Discretuum” hypothesis [3, 4, 5, 6]. They argue (correctly I think) that the anthropic bound on the proton lifetime is about years. Therefore the vast majority of anthropically acceptable landscape sites have proton lifetimes many orders of magnitude shorter than the experimental lower limit years.

One way out of the Banks Dine Gorbatov argument is to note that in the non-supersymmetric standard model, there is no mechanism for proton decay. If the scale signaling the breakdown of the standard model is high enough, there is no problem. Thus BDG begin by first arguing that the vast majority of anthropically acceptable vacua have a very low supersymmetry breaking scale. Their argument goes as follows:

Without supersymmetry both the cosmological constant and the Higgs mass scale, must be fine tuned. The combined fine tuning is about one part in . If the supersymmetry breaking scale is called then the natural scale for radiative corrections to the cosmological constant is of order . If the actual cosmological constant is then the likelihood of radiative corrections cancelling and leaving the small value is of order

(1.1) |

Thus making as small as possible will yield the largest number of anthropically acceptable vacua. Similarly they argue that since the Higgs mass is quadratically sensitive to the actual measure of fine tuning is

(1.2) |

For of order the Planck mass we get but for of order the weak scale, the fine tuning is only of order . There is therefore a very strong statistical bias toward low energy supersymmetry breaking.

But low energy supersymmetry breaking is not without its problems. Many of the things which the standard model solved so neatly–proton stability, neutral strangeness changing currents and the like– are neither automatic nor especially natural in supersymmetric extensions. In particular, dimension 5 baryon violating operators are dangerous. Without some special non-generic symmetry, the proton lifetime in supersymmetric theories is several orders of magnitude too small. I might point out that dimension 4 operators can also occur in generic supersymmetry models. These lead to extremely short lifetimes unless R-parity forbids them. But unlike the dimension 5 operators, the dimension 4 operators are so bad that they can be ruled out anthropically.

The Banks Dine Gorbatov criticism is quite serious in my opinion, and needs to be addressed. In this note I will argue that Banks Dine Gorbatov overlooked one important statistical factor in their analysis which, when included, may change the conclusion. Here is the way I think the argument should go:

What we want to compute is the conditional probability that the supersymmetry breaking scale is , given that the cosmological constant and weak scale are in the anthropic range. This is not the function in 1.2. Rather the function is the conditional probability that the cosmological constant and weak scale are in the allowed range, given that the supersymmetry breaking scale is . The two distributions are related by Bayes’ theorem. Define the probability that the cosmological constant and weak scale are in the allowed range, given that the supersymmetry breaking scale is to be

and the probability that the supersymmetry breaking scale is , given that the cosmological constant and weak scale are in the anthropic range to be

Also define the unconditional probabilities for given supersymmetry breaking scale and for the values to be

Then Bayes’ theorem relates these probabilities

(1.3) |

Thus in comparing the likelihood of different values of the factor should be multiplied by . The factor in the denominator of 1.3 can be ignored since it is independent of . The question is, what is the value of the unconditional probability ?

I believe it is likely that goes to zero as tends to zero. The reason is that the conditions for supersymmetry generally pick out a subspace of the landscape whose dimensionality may be a good deal lower than the number of available dimensions including those that parameterize supersymmetry breaking. A reasonable guess is that should go to zero as a power of in which case it could overwhelm the factor . To illustrate the point I will concoct an illustrative example within a framework similar to that of [4]. The supersymmetry breaking mechanism and the discrete fine tunings of the cosmological constant and Higgs scale take place in different sectors of the theory. In particular the supersymmetry breaking sector is located at the infrared end of a warped compactification while the tunings of the cosmological constant and are done through the choice of fluxes in the ultraviolet part of the compactification manifold. The only modification of [4] is to suppose that there are throats instead of just one.

In [4] the mechanism for supersymmetry breaking is one or more antibranes placed at the end of the throat. The supersymmetry breaking scale is the combined mass of the branes. Thus

(1.4) |

where is the antibrane mass and N is the number of antibranes.

If there are throats a given total number of antibranes can be distributed among the throats in many ways. For example antibranes can be placed in the first throat, in the second throat and so on. The number of ways of partitioning the mass among throats is of order . Thus if the number of throats exceeds six, the distribution favors high energy supersymmetry breaking. [8].

Which kind of throat structure–many or one– is favored by statistics of the landscape? This may be a delicate question. With no condition of supersymmetry breaking, one or even no throats may dominate. But you can’t live in a supersymmetric world. Given a scale of supersymmetry breaking, the statistics may be tilted toward many throats just because there are more ways to break supersymmetry if there are many throats.

If supersymmetry breaking is at a high scale then it would seem that the apparent unification of coupling constants is accidental. Dimopoulos and Arkani-Hamed have discussed a possible framework in which supersymmetry can be broken at a high scale but the coupling constant unification not be destroyed [7].

Finally, I should point out that the problem of the QCD theta parameter, pointed out by Banks Dine Gorbatov is not obviously eliminated by the considerations of this paper but the problem may be less serious if supersymmetry is broken at a very high scale. String theory contains many axion-like objects but they generally belong to super-multiplets that contain scalar moduli. If supersymmetry is an approximate symmetry then fixing the moduli will result in a large axion mass. But if supersymmetry is broken at a high scale and the moduli are fixed at a lower scale then there is no reason why the axions must get a large mass.

If nothing else, I hope this note shows that the measure on the landscape is a very subtle issue that demands better quantitative methods of the kind being developed by Douglas and collaborators [6]. The whole idea of fine-tuning has to be re-evaluated and redefined in this new unfamiliar framework.

After this paper was written I became aware of the fact that Denef, Douglas and Florea have discussed the problem of supersymmetry breaking in the Landscape. These authors find that high scale supersymmetry breaking may be favored [8].

## 2 Note Added

After this paper was written M Douglas put out a paper addressing the same problem [9]. Douglas’ conclusion is a good deal stronger than mine. The difference lies in formulas 1.1 and 1.2. Douglas would replace 1.1 by

(2.1) |

where is the Planck or string scale. The argument assumes that the cosmological constant satisfies a formula like

(2.2) |

where the first term is classical supergravity contribution and is the exact radiative correction. Assume that the discretuum gives a spectrum of values of that is more or less uniform over the range from to (My mistake was to assume the spectrum is bounded by instead of ). Also assume that the radiative corrections are of order . If no more than that is known then we can write

(2.3) |

where and have a dense spectrum lying between and Douglas gives strong arguments why the spectrum of is featureless with no singularity at Under these circumstances 2.1 is correct. Thus Douglas concludes that 1.2 should be replaced by

(2.4) |

In other words the only the gauge hierarchy fine-tuning favors low energy supersymmetry breaking. In this case we would need fewer powers of in to overwhelm the bias toward low scale breaking.

However in conversations with Douglas and Dimopoulos we realized that the same logic may apply to the hierarchy as to the cosmological constant in which case the bias would be nonexistent altogether. In this case would favor high scale breaking even with no enhancement from .

## 3 Acknowledgements

I would like to thank Tom Banks, Michael Dine, Ben Freivogel, Savas Dimopoulos Steve Shenker, Scott Thomas Shamit Kachru and Michael Douglas for very useful discussions.

## References

- [1] Tom Banks, Michael Dine, Elie Gorbatov, Is There A String Theory Landscape, hep-th/0309170
- [2] Steven Weinberg, ANTHROPIC BOUND ON THE COSMOLOGICAL CONSTANT, Phys.Rev.Lett.59:2607,1987
- [3] Raphael Bousso, Joseph Polchinski, Quantization of Four-form Fluxes and Dynamical Neutralization of the Cosmological Constant, hep-th/0004134, JHEP 0006 (2000) 006
- [4] Shamit Kachru, Renata Kallosh, Andrei Linde, Sandip P. Trivedi, de Sitter Vacua in String Theory, hep-th/0301240
- [5] Leonard Susskind, The Anthropic Landscape of String Theory, hep-th/0302219
- [6] Michael R. Douglas, The statistics of string/M vacua, hep-th/0303194, JHEP 0305 (2003) 046 Sujay Ashok, Michael R. Douglas, Counting Flux Vacua, hep-th/0307049 Michael R. Douglas, Bernard Shiffman, Steve Zelditch, Critical points and supersymmetric vacua math.CV/0402326
- [7] Nima Arkani-Hamed, Savas Dimopoulos, Supersymmetric Unification Without Low Energy Supersymmetry And Signatures for Fine-Tuning at the LHC, hep-th/0405159
- [8] F. Denef, M. R. Douglas and B. Florea, Building a better racetrack, arXiv:hep-th/0404257
- [9] Michael R. Douglas, Statistical analysis of the supersymmetry breaking scale, hep-th/0405279