Thursday, June 24, 2010

Why string theory implies supersymmetry

 Human Spanish translation is available.

Future piece of evidence in favor of supersymmetry:
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Next: SUSY and hierarchy problem
Next: SUSY exists because the number 3/2 can't be missing
By far the most important argument in favor of supersymmetry is the fact that it seems to be implied by string theory, the only known - and, most likely, the only mathematically possible - consistent unifying theory of fundamental forces including gravity.

Let us look at this relationship a bit more closely.

In the 1960s, people would discover the "dual models" which would ultimately be known as "bosonic string theory". One-dimensional relativistic strings embedded in D spacetime dimensions carried spacetime coordinates on their world sheet. The vibrational energy eigenvalues corresponded to allowed masses of particles that a string is able to mimic.

Technicalities and vices of bosonic strings

In particular, the allowed physical states on the string must be invariant under the coordinate reparametrization symmetry: the world sheet theory is equipped with a small, two-dimensional theory of gravity itself. Even after some gauge-fixing, a part of this symmetry remains in the form of the so-called "Virasoro algebra".

Roughly speaking, it's the algebra that generates the group of reparametrizations of a circle. In fact, a closed string has two copies of such an algebra, one for left-moving waves and the other one for the right-moving waves.

In particular, the generators of the uniform repameterization symmetries are called L0 and L0~. Their difference, L0 - L0~, generates the uniform rotation of the circle. On the other hand, the sum is the total energy that generates the evolution in the world sheet time. The invariance of the string's physical states under L0 + L0~ tells you that
E2 - p2 = N / alpha'
where E is the total spacetime energy carried by the string, p is the total momentum, and N is an integer that counts the sum of (weighted) excitations of the Fourier modes that oscillate along the string. The constant alpha' is the "squared string scale" and determines the characteristic length and energy scale of the stringy vibrations.

This equation is nothing else than the dispersion relation for a particle - and N/alpha' is nothing else than the squared mass of the corresponding particle!

If you translate the wave function of the string (as a function of its zero-mode coordinates - the center-of-mass position and/or total energy-momentum of the string) from the momentum space to the position space, you will realize that the equation above is
Box Psi = m2 Psi,
nothing else than the Klein-Gordon equation. In fact, the string may also carry some angular momentum (an integer multiple of hbar) - the oscillations that propagate around the string may make it vibrate. So the equation above may also become Maxwell's equations for the electromagnetic potential or anything of the sort.

However, they're still second-order differential equations. And indeed, you will find out that bosonic strings only carry bosonic excitations. Fermions - such as leptons and quarks - can't be obtained from bosonic string theory. (Well, there are some exotic ways to achieve "fermion only" branes in bosonic string theory etc. but let's not discuss them here - it's unlikely they can become realistic models.)

That's bad for the realistic applications of the theory but there's another problem that is even more serious because it's about the very consistency of the theory: the ground state boson is a tachyon - a particle whose squared mass is negative. Classically, it moves faster than light. In terms of quantum field theory, it signals an instability of the vacuum. The vacuum can roll down the hill - i.e. produce as many tachyon-antitachyon pairs as it wants. It's just way too bad.

The musicians are wrong. They surely do need ten dimensions. Moreover, one can only "cut the strings" at the locus of a D-brane. ;-)

Going to superstrings

How can we get rid of the tachyon and add fermions to the theory? Well, one basic problem with the differential equation for Psi that we could derive was that it was a second-order equation. In fact, fermions such as electrons are described by a first-order differential equation, namely the Dirac equation:
( γμμ - m ) Psi = 0
How can we get the Dirac matrices and first-order equations? Well, the states of the string must clearly be allowed to transform as spacetime spinors and there must be allowed to be both bosonic and fermionic. To switch in between them, there must exist fermionic fields on the world sheet. They must non-trivially transform under the spacetime Lorentz group to allow us to produce different spins.

The replacement of the "L0" equation is a "G0" equation. The operator "G0" must square to "L0" in the very same sense as the Dirac operator squares into the box. But that makes a difference: by adding this "G0" which is a kind of a square root of "L0", we are extending the world sheet Virasoro algebra!

If you know bosonic strings but not superstrings, you should start to read Volume II of Polchinski's textbook or another source. At any rate, the inevitable consequence of this reasoning is that the world sheet includes fermions that transform either as spinors or as vectors under the spacetime Lorentz group; it turns out that both of these formalisms lead to completely equivalent theories (if the projections etc. are done correctly). And the Virasoro algebra must be extended into a superalgebra, the super-Virasoro algebra.

In other words, there must be supersymmetry on the world sheet if your theory is able to produce both spacetime bosons and spacetime fermions. "G0" was one such generator of this world sheet supersymmetry; yes, it had to be a fermionic generator.

There are many ways to choose the chirality, allowed boundary conditions, and allowed projections of the spectrum. Only some particular combinations lead to consistent theories (with mutually local operators and modular invariance) but the number of possibilities is still higher than one.

It turns out that the most consistent theories also lead to spacetime supersymmetry - not only world sheet supersymmetry that is really inevitable if spacetime bosons exist together with spacetime fermions. In the 1980s, people would find type I, type IIA, type IIB, heterotic SO(32), and heterotic E8 x E8 string theories in 10 dimensions. All of these five "theories" have spacetime supersymmetry.

The ten-dimensional vacua without spacetime supersymmetry may include fermions but they otherwise resemble the old bosonic theory in many respects. You may say that SUSY is broken "at the string scale" in all of them. Most of them, although not all of them, predict tachyons. Spacetime supersymmetry eliminates all spacetime tachyons (which is a good thing to do!) because the spacetime energy may be obtained as the square of a supersymmetry generator - so the energy is positively semidefinite.

In the 1990s, the five theories would be supplemented by a sixth limit, M-theory in 11 dimensions, and connected into a tight network by dualities which are equivalences. So it turned out that all the previous "theories" are just superselection sectors - labeled by points or limits in moduli spaces - of a single theory that we continue to call "string theory" although the descriptions strictly based on strings are just some of the limits of the "string theory" as we understand it today.

The duality revolution has shown us that the non-perturbative behavior of the previously called "string theories" is well-behaved. The reason is simple: people can actually describe all these "infinite coupling limits" by another theory that is weakly coupled. It's important to emphasize that virtually all these proofs rely on spacetime supersymmetry.

If spacetime supersymmetry were absent, the "intermediate coupling" regime would be full of curved spacetimes, potential instabilities, and - which is the most guaranteed thing - calculational nightmares that would prevent us from showing that one theory can be extrapolated to another one. Now, the only thing that is crystal clear is that the non-supersymmetric theories are "hard": it is not easy to prove that they're linked by well-defined dualities. It's not easy to calculate anything too accurately about them because many of the nice methods to calculate rely on SUSY. However, it's very likely that it's not just a calculational problem: the dualities probably don't exist for the non-supersymmetric vacua (at least most of them).

Now, nearly all the phenomenologically attractive vacua in string theory - those based on heterotic strings; heterotic M-theory; M-theory on G2 manifolds; F-theory with 7-branes on 4-cycles; most of the intersecting braneworlds - respect the laws of spacetime supersymmetry.

So the effective, point-like quantum field theories that you may derive as their limits are supersymmetric Grand Unified Theories (GUT) or other extensions of the Minimal Supersymmetric Standard Model (MSSM). It's assumed that supersymmetry is broken by methods that may be described in the field-theoretical language, too. This assumption is linked to the expected low energy of the supersymmetric breaking scale, a point that will be discussed later.

Some people may argue that string theory predicts many more non-supersymmetric vacua. They have to be there and they may be "generic".

Well, I don't know how the "genericity" may be counted in infinite sets and I don't know whether the fundamentally non-supersymmetric vacua in string theory exist at all and whether they're generic if they do. But I am pretty certain that whatever the answers are, they're not relevant for the search of the truth about the unification.

The purpose of physics is not to search for generic vacua of string theory - much like before string theory, the purpose of physics was not to search for generic theories. The purpose of physics is to search for theories and vacua that explain the real observed world around us.

So they should be consistent with all the observations. And if we have several candidate vacua that are (possibly/likely) consistent with all the observations, we should still prefer the explanations that actually explain some features of the Universe instead of assuming them (by being selected from a wider class where the vacua that satisfy and those that don't satisfy the conditions are treated democratically because they don't differ qualitatively).

So whether or not we use string theory, the unification of couplings is still an advantage of a vacuum - much like it was an advantage of a Quantum Field Theory before people realized that it was just a low-energy limit of string theory.

SUSY breaking scale in string theory

At any rate, supersymmetry has to be spontaneously broken to agree with the real world where the particles and their superpartners obviously have different masses. A question that is crucial for all the experimental predictions - but one that is not necessarily paramount for the conceptual issues - is the actual energy scale at which the supersymmetry is broken. You may imagine that this scale is the maximum mass difference between a particle and its superpartner (taken over all particle species).

Supersymmetry is capable to solve the hierarchy problem - to explain why the Higgs boson (and therefore W bosons, Z boson, and others) remain much lighter than the Planck scale (or another very high energy scale). In non-supersymmetric theories, it's natural for the Higgs to acquire huge mass corrections from the quantum loops (Feynman diagrams with loops in them). These corrections typically push the total Higgs mass towards the Planck scale. Unless you fine-tune the initial Higgs mass with a relative accuracy of 10^{-15} or so, you will end up with insanely huge Higgs masses and, consequently, huge W and Z boson masses, too.

Supersymmetry is a method to cure this harmful impact of the loops.

The Higgs is paired with a higgsino and the higgsino, being a chiral fermion, naturally stays massless because it has no right-handed partner to team up with and to create a Dirac (non-chiral) fermion. The Higgs and higgsino masses are therefore close to the supersymmetry breaking scale. Another way to explain why SUSY protects Higgs is to note that the divergent loop contributions to the Higgs mass from particles cancel between these particles and their superpartners. Once again, this cancellation is accurate up to the supersymmetry breaking scale.

This explanation why SUSY keeps the Higgs light is only OK if the supersymmetry breaking scale is close to the Higgs mass. If the SUSY breaking scale is K times higher than the Higgs mass, the explanation becomes K times less attractive and you should say that the odds that the real world follows is just 1-in-K. ;-) So K may be 10 or so and I don't see anything terribly wrong if K equals ten - and if the superpartners have masses of a few TeV. But K shouldn't be thousands or millions - otherwise the explanation in the form above breaks down.

Note that these field-theoretical arguments hold in string theory, too. For the class of vacua with SUSY breaking scale E, most of them will have the Higgs mass comparable to E. To be more precise, this is not quite the right counting of the probabilities we need. We should assume what we know - namely that the Higgs is light - and deduce what the superpartner masses may be.

But if you assume e.g. that the SUSY breaking scales are uniformly distributed on the log(energy) axis and for each scale, the Higgs mass is uniformly distributed between 0 and the SUSY breaking scale, it's still true that most of the vacua will predict both mass scales being close to each other. That's why it's legitimate to revert the relationship and predict that the superpartner masses should be comparable to the Higgs mass, too.

Cosmological constant

The cosmological constant problem has often been presented as the greatest mystery of the contemporary high-energy physics. However, the role of supersymmetry has often been obscured.

In the Planck units, the observed cosmological constant is very tiny, something like 10^{-123}. An easy way to cancel the cosmological constant would be to have an exact supersymmetry. In that case, arguments showing that the C.C. is still equal to zero can work. The vacuum graphs (loops) with bosons would cancel against their superpartners.

However, SUSY is broken and the SUSY breaking scale is at least 300 GeV or so. Is that enough to make the cosmological constant tiny?

Well, it's not. The most natural value of the cosmological constant is something like 10^{-60} times the natural value that you would predict from the SUSY breaking at 300 GeV. And SUSY breaking can't be much lower than 300 GeV because the superpartners would otherwise be easily seen without big colliders - but they're not seen.

So with SUSY, the numerical problem survives. In fact, SUSY makes things more controllable so the statement that the natural value of the C.C. is different than the observed one becomes even more justifiable by mathematics.

Nevertheless, there is one point that pretty much everyone overlooks: 10^{-60} is less unnatural than 10^{-123}. By requiring low-energy SUSY, the fine-tuning problem for the cosmological constant has been reduced by 63 orders of magnitude or so. So much (10^{63} times) higher a fraction of the vacua with low-energy SUSY have a chance to predict a tiny enough vacuum energy so that it agrees with the observations.

The only counter-arguments I am aware of follow Susskind 2004 and assume an ad hoc factor in the prior probability that unjustly punishes the low-energy SUSY-breaking vacua for breaking SUSY at low energies. ;-)

So even if you use some kind of a statistical treatment and if you accept that some anthropic selection will be needed at the end, it seems a legitimate prediction of string theory to say that the SUSY breaking scale should be as small as allowed by the existing observations, i.e. it should be close to the electroweak scale. In all this reasoning, I urge the reader to assume everything we have already observed, to maximize the chances that we find the truth.

In some sense, you may view it as a "super anthropic principle" because we assume everything about us and our world that we know. In another sense, it's nothing else than the old-fashioned science because theories used to be constructed to match all the observations - and scientists often "cheated" by looking into Nature :-) - and I think that the approach during the search for the right string vacuum shouldn't be any different.

When it comes to similar counting, the only genuine difference between the anthropic people and the non-anthropic people is their answer to the question:
Are we allowed to discriminate against numerous vacua that don't explain anything because they only have their viable and realistic features by chance?
My answer is a resounding Yes: we must always prefer theories - and vacua - that actually explain patterns of Nature without assuming these patterns. The answer of the anthropic believers is No: in their opinion, we must throw all the garbage into trash can, together with all the valuable and beautiful theories we have found, treat all of the things in the trash can democratically, select a random piece of trash, and claim that it must be us. ;-)

That's the ultimate "inelegant universe" approach to the reality. Well, I think that the opinion that the Universe is elegant (and has crisp and unique reasons for its patterns) is more than just a belief: it is a consequence of a properly executed Bayesian inference. It is a part of a scientist's good taste.

Supersymmetry is a unique mathematical structure that can be defined by a few words. The laws of physics that respect the N=1 d=4 supersymmetry could be so constrained that no supersymmetry breaking could make the predictions compatible with the current observations. It happens that they're actually consistent and this fact is extremely nontrivial. If it is possible to add new symmetry-based constraints and given these additional constraints, you will still find viable models, you should take these models very seriously. They may contain more particle species but they're actually more constrained rather than less constrained.

Supersymmetry follows from string theory - the only way how to reconcile gravity and other forces described by renormalizable QFTs. Statistically speaking, low-energy supersymmetry seems to be a likely consequence of string-theoretical phenomenology, too. These are damn good reasons why it is sensible to expect it at a collider that should extend the reach of experimental particle physics by an order of magnitude.

And that's the memo.


  1. Hi,
    if you believe in supersymmetric particles, why don't you bet on their
    discovery at

    Cheers, Alexander

  2. Dear Ernst, I have written the reason in the article. The volume is just too low to justify the creation of the account and other "paperwork".

    Also, the odds are less profitable than what I can get elsewhere. A foe of SUSY, a professional particle physicist, made a bet with me and he will (hopefully) pay me ten thousand dollars one year after the discovery of SUSY. I will only pay $100 if it is not found. I would be willing to extend such a bet by a factor of 10 - to get a million. But 4:1 is not so attractive although it's a fairer ratio. Well, obviously, I am not interested in hets that are "completely fair" - that would be like buying lottery tickets.