A couple conferences going on this week have already put some of the talks online.

SLAC has a summer school each year, aimed more at experimentalists than theorists. This year’s topic is “Nature’s Greatest Puzzles” and there are quite a few interesting talks already online there.

The Michigan Center for Theoretical Physics is hosting this year’s String Phenomenology 2004 conference. The “Landscape” seems to be a big topic; two online talks are Michael Douglas’s, which is more or less the same as his one at Strings 2004 a few weeks ago, and Michael Dine’s. Dine seems optimistic that the Landscape will lead to predictions, saying

“If we adopt the anthropic viewpoint, we may be lead to predictions – perhaps the first predictive framework for string theory”

Before taking this too seriously, one should note that Dine has been giving review talks about “superstring phenomenology” and claiming that predictions are right around the corner since before most of our incoming students at Columbia were born (see his 1986 Erice lectures, no, they’re not online, this was way before the arXiv).

And I’m headed off soon for a short vacation out of the range of the internet, back late next week.

Just to be silly about the “137 loop” QED argument in my previous post, let’s change one of the assumptions.

In the case of an exploding number of Feynman diagrams at n-loop order, if we have instead have (n!)^2 diagrams at the nth order of perturbation theory (instead of n! diagrams in my last post), then the A_n’s should be proportional to (n!)^2 after all the Feynman diagrams are summed up at n-th order.

In this case the R_n ratios will become

R_n ~= alpha * n^2

(with all other assumptions the same).

For divergence the R_n’s are greater than 1.

So we now have n^2 greater than 1/alpha, which now says the “QED series” will start to diverge after n = 11 loops.

Peter,

I thought about the “137 loop” assertion thing for QED. Here’s sort of a naive argument that I can think of offhand.

We start with a QED perturbative series

“QED series” = sum_{n=0 to infinity} C_n

where

C_n = A_n * alpha^n,

alpha = fine structure constant

From a naive look at the explosion of the number of Feynman diagrams at each order of n-loops, there appears to be n! number of diagrams for the nth order in perturbation theory. Just summing up all the Feynman diagrams at n-loop order, the coefficient A_n should possibly be proportional to n!. So we factor out n! from A_n as

A_n = B_n * n!

So now C_n = B_n * n! * alpha^n

Now we look at the ratio R_n between succesive orders in perturbation theory as

R_n = C_n/C_{n-1} , where n = 1 to infinity

R_n = [B_n/B_{n-1}] * [n!/(n-1)!] * alpha^{n-(n-1)}

R_n = [B_n/B_{n-1}] * n * alpha

For convergence, we want R_n to be always less than 1. If we naively say that all the B_n’s (left over after factoring out the alpha^n and n!, from the C_n’s) have roughly the same order of magnitude each, then the ratios [B_n/B_{n-1}] will be approximately an order of magnitude 1.

(If the B_n’s all have the same order of magnitude that is not of “order of magnitude” 1, then the overall “order of magnitude” can be factored out from the overall “QED series” expansion).

With these naive assumptions, now the ratios R_n become

R_n ~= n*alpha

For divergence, R_n will be greater than 1.

n will be greater than 1/alpha = 137.

Hence above n=137 loops the “QED series” will start to diverge.

The hierarchy problem is really just dimensional analysis. It doesn’t depend on any details of a theory of quantum gravity. It’s just that, in natural units, the natural scale of things is the Planck scale and one might like an explanation of why the ew scale is so much less than the natural scale.

The technical naturalness problem also doesn’t depend on the theory of qg. The divergence in the Higgs mass is there in the low energy theory.

Lastly, some theories of SUSY breaking do have an intermediate scale.

Hi Aaron,

“There may be no hierarchy with the GUT scale, but there certainly still is one with the Planck scale.”

Umm, maybe that’s why my initial comment that you objected to saying that the hierarchy problem went away if you didn’t believe in GUTs had a parenthetical caveat “ignoring gravity…”

The hierarchy problem for GUTs is a well-defined one since a GUT is a well-defined theory that includes the standard model with electroweak symmetry breaking. To have a hierarchy problem involving the electroweak scale and the Planck scale, first you need a well-defined unified theory that includes quantum gravity and the standard model, including an explanation of the electroweak scale. Some of us don’t believe this exists. All supposed candidates for such a theory have a far more serious problem, the vacuum energy problem. You have to first figure out why the absolute value of the effective potential is completely wrong before worrying about its second derivative.

Then again, the latest trend seems to be to announce that the anthropic principle solves all such problems, so we can stop worrying about them.

Aaron,

Will the hierarchy problem with the Planck scale disappear if gravity is just strictly a classical force with no quantum counterpart?

Are there any simple ways to “preserve” naturalness when SUSY is broken?

I haven’t checked the literature lately, but has anyone showed that supergravity can easily deal with the hierarchy/naturalness problems? Or is supergravity coupled to Yang-Mills theory just another disaster with the same hierarchy problems between the electroweak, GUT, and Planck scales?

As a side note, are there any speculative particle models which have an “intermediate” scale that’s higher than the electroweak scale but lower than the GUT scale? If there are any from the past, have they all vanished into the dustbins of physics history?

There may be no hierarchy with the GUT scale, but there certainly still is one with the Planck scale.

There are really two kinds of naturalness problems. One is to ask what stabilizes the Higgs mass against loop corrections which tend to drive it up to the Planck scale. The other is what explains the hierarchy between the weak scale and the GUT scale.

SUSY has some nice features regarding the naturalness problems (and nice features not involving the naturalness problem) which I think I’ve discussed here before. As I think I’ve also said, some of these nice features have proved to be less nice than they first appeared. But such is life. When you break SUSY, for example, I think the naturalness problem can easily reappear, but I don’t want to make any big pronouncements about SUSY breaking because I’m not really up on it.

Peter,

What’s the exact argument for the QED perturbative series diverging beyond 137 loop order? I remember seeing this “137 loop” assertion over the years as if it was some folklore wisdom. I never figured out an easy semi-rigorous argument to justify it.

It almost certainly is just an asymptotic expansion. What determines the running of the coupling constants is the beta-function and presumably the perturbation expansion for this is just an asymptotic expansion. But one expects the situation to be like in QED, where the expansion is expected to be good up to an order that goes like the inverse of the coupling constant, i.e. 137 in QED. So, roughly in this case one expects the perturbation calculation to be good up to an order given by the inverse of the coupling constants in the range that you want to follow how they run. This should be less than 137, but still probably much larger than the order to which one can realistically do the calculation.

Peter,

If the 3-loop calculation (or for that matter higher loops) changes the 2-loop result significantly, will some folks try to argue that peturbative expansion is really just an “asymptotic series” that starts to break down early and diverge? Besides being rhetorical, are there any good estimates as to what loop order the perturbative SUSY calculations will start to break down due to asymptotic nature of the series expansion?

Hi Aaron,

I’m not sure what you mean by the “hierarchy problem”. I was referring to the problem of keeping the electroweak and GUT scales separate, i.e., the electroweak scale being so small in GUT scale units is technically “unnatural” and requires fine tuning. By definition this problem disappears if there is no GUT scale. Supersymmetry solves it (kind of, there’s something called the “mu-problem” of it reappearing) by pairing the electroweak Higgs with a fermion whose mass is small because of chiral symmetries. Technicolor solves it because running of the coupling constant makes the coupling strong at exponentially smaller scales than the unification scale (also this is why the small size of the strong scale in GUT units is not a problem). But if there is no GUT with symmetry breaking occuring at a high GUT scale, there is no problem.

The coupling constant unification you mention is far and away the best evidence for a GUT (specifically a supersymmetric GUT), but, using the two-loop results, it is far from convincing. I don’t know if anyone has done 3-loops but one wouldn’t expect 3-loops to change the 2-loop results much. If it did one would have to worry about whether the whole perturbative set-up was making any sense.

Aaron,

Has anyone calculated the SUSY 3-loop result, or for that matter any other higher loop result? If so, does it improve or worsen the coupling constant convergence scenario?

If you ignore GUTs, the hierarchy problem does

notdisappear.The reason a lot of people believe in GUTs is that the couplings of the standard model, when you run them to high energy,

almostmeet. Now, there’s no particular reason for three lines to almost meet, and no one likes coincidences. Thus, it’s not implausible to believe that the couplings really do unify, that the reason for this is a GUT, and that some higher energy stuff fixes the running of the couplings.For a long time, supersymmetry seemed to be exactly that higher energy stuff. It causes the agreement of the couplings to improve significantly. The one loop calculation works almost perfectly. Unfortunately, as I understand it, the two loop calculation ruins the agreement which is one reason why people are somewhat less sanguine about SUSY these days.

Supergravity was certainly the most popular idea about unifying gravity and the standard model during the late seventies, early eighties, but it never dominated things the way string theory has. By 1984 it was kind of killed off by

1. Witten showing that you couldn’t get the kind of chiral spectrum you needed if you compactified 7 of the 11 dimensions of 11d supergravity on a smooth manifold.

2. Calculations showed that potential counterterms appeared in higher orders, so presumably supergravity was not a renormalizable theory.

Actually, what people are almost always doing when they say they are working on “M-theory” is really just 11d supergravity. Now it’s back with a vengeance, people just stopped worrying that the compactification manifold can’t be smooth.

Peter,

On a similar note, was there any silly hoopla surrounding supergravity in the mid-late 1970’s and early 80’s, before string theory got popular? Only thing I can recall offhand was reading some 1980 speech Stephen Hawking made which advocated supergravity as some sort of “grand unified theory”, when he first got the Lucasian chair.

Oops, correction, SO(10) is rank 5.

The choice of those GUT groups predates their use in string theory. To fit SU(3)xSU(2)xU(1) into a simple group in a way where they commute with each other, you need a group of at least rank 4 (the sum of the ranks of those 3 groups). SU(5) is probably the simplest example of such a group. SO(10) is also of rank 4, and has the added advantage that you can fit all the particles of a single generation in one irreducible representation (the 16 dim spinor rep).

The simplest examples like SU(5) just with the Higgs sector needed to get the standard model can be ruled out because they predict something (proton decay) that doesn’t happen. The problem with most possible GUTs is that they don’t really predict anything. Depending on how you break the symmetry you can get pretty much anything. It’s kind of bogus to claim that these are really “unified” models, because you have to assume some complicated Higgs mechanism to break the symmetry. You end up adding lots of new parameters to the standard model, not explaining any of the ones you’ve got.

Yes, if you don’t believe in GUTs (count me in..), the hierarchy problem goes away (ignoring gravity…), and thus one of the main motivations for supersymmetry.

Pink Floyd and Led Zeppelin certainly bring back memories from high school (“Stairway to Heaven” was the theme song of my High School prom, shudder….). I think Kiss appealed mainly to those a bit younger than me, or with different interests. Never paid much attention to Blue Oyster Cult.

Peter,

Is the only reason why anybody bothers paying attention to GUTs like SU(5), SO(10), E_6, etc … along with their SUSY versions, is mainly due to the gauge groups which show up in string theory like SO(32), E_8 x E_8, etc …? Have any of these GUTs beyond SU(5) been experimentally ruled out yet? Or are there so many free parameters in these GUTs that anybody can “fudge” their way through and get an answer that isn’t ruled out by experiment? I always found adding in tons of free parameters to be somewhat on the distasteful side.

If GUTs were wrong from the very beginning with no basis in experiment or even reality, wouldn’t this remove one of the major theoretical arguments in favor of low energy supersymmetry (ie. naturalness)?

On a different note, were rock bands like Pink Floyd, Led Zeppelin, Kiss, and Blue Oyster Cult very popular when you were in high school?

Actually SU(5) was even before my time, 1974, when I was still in high school. Throughout the late seventies there was a lot of work on SU(5) and other GUTs, but they were nowhere near as dominant as string theory is today. They were just one of several popular ideas to work on (instantons, supergravity, lattice gauge theory, etc.). Once the experimental results started coming in (around 1980?) showing that protons didn’t decay at the rate SU(5) predicted, interest went down, although there continues to be lots of research in GUTs, partly since they are supposed to be the effective field theory for whatever string theory is supposed to explain the world.

Peter,

When the SU(5) GUT was first around, did it have the same hoopla and propaganda surrounding it, like string theory in the 80’s and 90’s? Was it taken seriously at all at the time? Or did Sheldon Glashow have such a huge influence that people were willing to read and/or believe anything he said and wrote about?

The problem, as with Douglas, is how to weight the probability of all these vacuum states. In KKLT, the hope is that all continuous parameters are fixed, so your only choices are discrete, and you can assign every choice the same weight (although without a more fundamental theory that determines how the universe settles into a single vacuum, there’s no real reason to believe this).

If you start with a GUT with a large gauge group and try and break its symmetry with a Higgs sector, problem is there are lots of parameters associated with the Higgs. How you choose these determines what the low energy physics looks like. How do you weight these possible choices of parameters? Best guess is they should all be around the one scale in the problem, the GUT symmetry breaking scale, but then you can’t explain why the electroweak scale is so low. This is the problem supersymmetry is supposed to address, but that solution doesn’t really work.

Re the Douglas talk : suppose we take a QFT with a very large gauge group, with many possible ways of symmetry-breaking. We then try to catalog vacuums by the same criteria that is suggested – a symmetry-breaking resulting in particular low energy parameters is more “natural” if many vacua produce these parameters. From a QFT perspective, will it be true then that we live in a unnatural vacuum?

It sure sounds like Dine is taking the same approach that many economists take when doing economic forcasts. Many economists frequently just keep on making the exact same forcast year after year according to their own particular biases, hoping that it will become “true” one day. This type of mentality is just like an old broken clock that is always right two times every day.