Vacuum state of string theory

[Spinnor]

So if string theory is to be correct it must be able to come up with the observed value for the energy density of empty space which is pretty close to zero?

A naive understanding of the compact spaces of string theory tells me that an "energy audit" of the highly curved compact manifolds of string theory must have both large negative and positive energy contributions associated with the compact manifolds such that their sum approximately equals zero?

Naively then I would expect that if there were zero point energy from all the fields of the standard model this "extra" energy density would be small compared with the energy density contributions from the curved compact manifolds at each point of our nearly flat space?

My naive thought is that the extra energy from possible zero point energy of the standard model fields while very large is small in comparison with the positive and negative contributions of the compact hidden dimensions of string theory at each point of our large space dimensions. --->

So even without supersymmetry the zero point energy of fields is "small change"?

Thanks for any clarification!

 

[Urs Schreiber]

That question is the source of a huge amount of activity, that changed the face of the string theory community starting around the turn of the millenium.

The importance of understanding how to choose perturbative string theory vacua to predict the cosmological constant was once highlighted in Witten 00.

But
a) there are many perturbative string theory vacua, each with its own cosmological constant;
b) the de Sitter vacua that seem to be required by observation are (as opposed to anti de Sitter vacua) typically metastable in string theory.

For some time it was unclear if realistc de Sitter vacua make sense at all in string theory. Then the result of Kachru-Kallosh-Linde-Trivedi 03, now widely known as the "KKLT construction", was regarded as solving this problem, while at the same time re-amplifying the presence of a "landscape of string vacua" (which was pointed out earlier, starting with Lerche-Lüst-Schellekens 86, but not finding much attention then).

For review of this see Kallosh 05, section 3.

The rest is history. In the popular domain, this is the source much of the informal chit-chat about the "multiverse" these days.

But as far as actual science goes, the situation remains very murky and inconclusive, because very little is known for sure about the moduli space of 2d superconformal field theories of central charge 15, aka the "landscape".

There is an interesting mathematical approach by Soibelman 11 to study the string landscape by serious tools from spectral geometry. There is a PF-Insights exposition of this at Spectral Standard Model and String Compactifications.

 

[mitchell porter]

@Spinnor, regarding your original question... Your "naive thought" is wrong. Let's talk first about the late-1980s models that were considered the first realistic thing to come out of string theory - heterotic string theory compactified on Calabi-Yaus. The Calabi-Yau is (Ricci) flat, there is no geometric contribution to the vacuum energy from curvature of the compact space. The vacuum energy comes entirely from the string modes, and this cancels out because of supersymmetry; the Calabi-Yaus were chosen in order to preserve supersymmetry.

Something else about those first models is that there was no explanation of why the Calabi-Yau would have a particular size and no proof that it would remain that size. The exact size of the extra dimensions was chosen on phenomenological grounds, and it was assumed or hoped that this was explained by cosmological initial conditions. There were probably handwaving arguments for why the extra dimensions would remain the same size but I don't know what they were, and in the following years, instability of the extra dimensions - e.g. the possibility that one of them would start expanding and we'd get an extra macroscopic dimension - was a major issue.

The KKLT paper that Urs cites was a landmark not just because it constructed string models with a net positive vacuum energy, but also because it stabilized the shape and size of the extra dimensions, by stuffing them with "flux". Also, the extra dimensions are no longer flat, they are "warped" and so their curvature does contribute to the vacuum energy, unlike the flat compactifications of the 1980s and 1990s.

But there's still no principle that the vacuum energy contribution from curvature is of greater magnitude than that coming from the vacuum energy contribution coming from strings. The curvature can be shallow and the string contribution can be dominant. It just depends on the details.

 

[mitchell porter]

As for how to obtain a small nonzero cosmological constant, there are two main ideas. One is that all the contributions to vacuum energy cancel for a simple reason like a symmetry; the other is anthropic selection - when they don't cancel out, you can't have atoms and galaxies and any kind of lasting physical structure.

Those 1980s-vintage models have zero vacuum energy so long as they remain supersymmetric. Once supersymmetry breaks, they will develop a nonzero vacuum energy, which is why Witten liked the idea that the world might somehow be secretly supersymmetric.

Then we have the KKLT-like models of the 2000s, like the G2 compactifications of M-theory that Gordon Kane has promoted, in which there's some enormously flexible detail of the model, and the argument is that this flexibility allows a "coincidental" cancellation of all the contributions to vacuum energy, without spoiling the part of the model that is to account for the physics that we actually see.

 

[Urs Schreiber]

@Spinnorthere is no geometric contribution to the vacuum energy from curvature of the compact space. The vacuum energy comes entirely from the string modes, and this cancels out because of supersymmetry

It seems to me that the common idea of computing the vacuum energy by adding up QFT zero-mode contributions is not viable, since the vacuum energy is subject to renormalization. This means that by field theory arguments alone it may take any value, since after adding up all these zero mode-contributions there is still a renormalization constant freely to be chosen.

The beauty of perturbative string theory is that the string scattering S-matrix for a given 2d SCFT is (supposedly) order-wise UV-finite, hence has already chosen its own renormalization for us. Given a 2d SCFT, we read off the renormalized vacuum energy from it, be it what it may.

For practical purposes people usually imagine their 2d SCFT is a sigma-model defined on a classical gravity background spacetime, which then by self-consistency comes out as the effective field theory background for the corresponding string perturbation series, so that the vacuum energy is read off from the classical contributions to the cosmological "constant" in that classical background. That's why there are so many arguments squarely in classical (super-)gravity in this business.

 

[Urs Schreiber]

It seems to me that the common idea of computing the vacuum energy by adding up QFT zero-mode contributions is not viable, since the vacuum energy is subject to renormalization. This means that by field theory arguments alone it may take any value, since after adding up all these zero mode-contributions there is still a renormalization constant freely to be chosen.

I have collected some citations for this statement: here.

The most pronounced statement of the renormalization freedom in the cosmological constant is in Hack 15, section 3.2.1.

 

[Spinnor]

no geometric contribution to the vacuum energy from curvature of the compact space

Is that zero as in zero or zero as in something plus something (plus maybe more terms) = 0

An interesting fact that I have wondered about. It seems funny how something, these compact spaces, that are so severely twisted and curved have no gravitational energy, or is it

something plus something (plus maybe more terms) = 0 gravitational energy?

Thanks for your help!

 

[Spinnor]

It seems to me that the common idea of computing the vacuum energy by adding up QFT zero-mode contributions is not viable, since the vacuum energy is subject to renormalization. This means that by field theory arguments alone it may take any value, since after adding up all these zero mode-contributions there is still a renormalization constant freely to be chosen.

So is there a consensus among string theorists that zero point fluctuations of the standard model fields do not exist? Thank you for the links.

Is there in theory such a thing as zero point fluctuations of the gravitational field?

Thanks!

 

[Urs Schreiber]

So is there a consensus among string theorists that zero point fluctuations of the standard model fields do not exist?

Since perturbative string theory by definition is an S-matrix theory it has no fundamental concept of spacetime field fluctuations (or even of spacetime!). Perturbative string theory is a formula that reads in a 2d SCFT worldsheet QFT and spits out probability amplitudes that are to be interpreted as scattering amplitudes of something scattering somewhere.

In order to make sense of this, one looks for effective quantum field theories (see chapter 16. here in the PF QFT notes) that have the same scattering amplitudes at a given UV-cutoff scale. This then describes what the given string vacuum "looks like" in that approximation.

The effective QFT in turn is encoded by its field content, the field interaction and a choice of renormalization. In particular the vacuum expectation value of its stress-energy tensor, hence the "vacuum energy" is determined by this. But in matching the effective QFT scattering amplitudes to the presribed string scaterring matrix at the given scale, these parameters get prescribed by the string vacuum. In this way one, indirectly, reads off what the vacuum energy of a given perturbative string vacuum is.

Is there in theory such a thing as zero point fluctuations of the gravitational field?

The only context where it is known how to answer this is perturbative quantum gravity. There the perturbative field of gravity contributes like any other field. I recommend Hack 15, section 3.2.1.

 

[Haelfix]

I have collected some citations for this statement: here.

The most pronounced statement of the renormalization freedom in the cosmological constant is in Hack 15, section 3.2.1.

His statement is a bit too strong, when you calculate the cosmological constant within say SUGRA, you get an absolute value that is not subject to an additional renormalization parameter. In any event, its not clear to me whether the situation is improved by having a value that is predicted to be 60 orders of magnitude away from experiment, is better or worse than a theory where two (actually many) free parameters that aren't related by any obvious physical process or symmetry must conspire to cancel to 120 orders of magnitude but that nevertheless we can do that operation by hand...

Incidentally, as far as I'm aware most reviews of the CC problem do include statements about the freedom to shift the renormalization parameter, I'm not seeing why rigorous paqft is mentioned as being different.

 

[Urs Schreiber]

when you calculate the cosmological constant within say SUGRA, you get an absolute value that is not subject to an additional renormalization parameter

You mean after susy breaking? Do you have a source for that?

 

[Urs Schreiber]

free parameters that aren't related by any obvious physical process or symmetry must conspire to cancel to 120 orders of magnitude

Beware that the space of renormalization choices at each order is an affine space (theorem 16.14). This means that it has no canonical origin, and hence there is no intrisnic sense in which a renormalization constant is small or large.

It it may seem otherwise after a choice of renormalization scheme has been made, thus picking an origin and identifying the affine space of renormalization choices with vector spaces. But this is an arbitrary human choice, a choice of coordinates, that has nothing to do with the intrinsic physics.

i'm not seeing why rigorous paqft is mentioned as being different.

It is not impossible to arrive at correct conclusions via careless arguments, but it gets increasingly difficult as the topic becomes more complex. I am mentioning rigorous pAQFT because there the statement in question is unambiguous and evident, laid out for everyone to mechanically check, and we don't have to rely on opinion, hearsay, folklore, authority.

 

[Spinnor]

it has no fundamental concept of spacetime field fluctuations (or even of spacetime!)

So spacetime is not there even when String Theorists look?!

Thank you for your help, much to read and learn.

 

[Urs Schreiber]

So spacetime is not there even when String Theorists look?!

Think about it: The only way that you know about spacetime is from particle scattering: mostly photons scattering into your eyes. In an S-matrix theory such as perturbative string theory, this scattering is the only fundamental concept. Everything else is derived.

The idea that also quantum field theory should have a fundamental description just by scattering processes encoded in an abstract S-matrix, not assuming spacetime and action functionals as a fundamental concept, is an old one, with an interesting history: it is known as the S-matrix bootstrap.

In its first life the S-matrix bootstrap program had this face:

• Geoffrey Chew,
  "Quark or Bootstrap: Triumph or Frustration for Hadron Physics",
  Physics Today, May 1970
  (pdf)

Then the program died.

Then it re-incarnated in the guise of perturbative string theory.

A funny account is at the beginning of Shankar 99.

 

[Haelfix]

You mean after susy breaking? Do you have a source for that?

I mean for unbroken SUGRA. There the value of the CC is identified as the minimum of the effective scalar potential, and is typically negative (with the canonical kinetic term). The potential is then unaltered by various nonrenormalization theorems in SUSY. This is pretty standard textbook stuff, so it shouldn't be that difficult to find. Bailin and Love has a pretty good discussion for instance.

Of course once you break SuSY, this is no longer strictly speaking true. Depending on the details of how you (softly) break SuSY, phenomenologically acceptable models typically require a hidden sector to mediate the breaking, and this will reintroduce the renormalization parameters somewhere.

Anyway, my point was just that there exists a counterexample to the claims that all QFTs coupled to gravity have arbitrary values of the CC. Various theoretical structures can and do pin down values.

 

[Haelfix]

Beware that the space of renormalization choices at each order is an affine space (theorem 16.14). This means that it has no canonical origin, and hence there is no intrisnic sense in which a renormalization constant is small or large.
It it may seem otherwise after a choice of renormalization scheme has been made, thus picking an origin and identifying the affine space of renormalization choices with vector spaces. But this is an arbitrary human choice, a choice of coordinates, that has nothing to do with the intrinsic physics.

Lets discuss the hierarchy problem instead, and there again you can make the same arguments about the sensitive cancellations within the renormalization parameters and you can make the above argument that there is no canonical choice about how to view something as large or small. Ok.
The point is that if you UV complete your theory to something that explains the mass of the Higgs, what was strange coincidences at the level of the effective field theory, are now promoted to actual algebraic expressions involving *real physical* quantities that are associated with the new scale. There everything is fixed by the physics of the problem and we *can* say that something is large or small. Now depending on what those algebraic structures are, you can often make the large cancellations in the effective field theory, look quite natural, and that is really what we mean when we try to 'solve' the hierarchy problem.

The correct way to view the quadratic divergences of the hierarchy problem is thus as a harbinger of something funny happening in the UV completed theory, nothing more.

 

[Urs Schreiber]

The point is that if you UV complete

Exactly, that was just my point above: Right before you entered the discussion I had been objecting to the argument that to determine the cc of a string background we should start counting zero-mode contributions of the effective fields it has. The objection I raised is that the effective field theory does not help with computing the cc, since from the point of view of the effective field theory the cc is a free renormalization parameter. Instead, to determine the cc of a perturbative string background, we have have to unpack the latter as what it is.

After all, "to UV complete" means (maybe in particular) to make renormalization choices .

 

[Urs Schreiber]

I mean for unbroken SUGRA.

That's what I thought. I am not sure if this is what people were trying to discuss here, but maybe I am wrong.

By the way, this is not inconsistent with the results by Hack and others that I had given pointers to: If we pick a symmetry of the background and then demand that under renormalization this symmetry be preserved, this is a renormalization condition. Imposing extra renormalization conditions in general shrinks the space of available choices, of course.

 

[mitchell porter]

I am not sure if this is what people were trying to discuss here.

At the start of #3, I did mean the order-by-order calculation of the cosmological constant in perturbative string theory - but I assumed that, for regimes with a geometric interpretation, the contributions could be separated into a part coming from curvature of the background geometry (which would be zero, when the compactification manifold is flat), and a part which has more to do with string excitations per se (which would also be zero, when supersymmetry is unbroken).

I still think that is a reasonable assumption, but even after a literature search, I would have a hard time defending it! In search of a string calculation of the cc in curved space, the closest thing I could find is a very obscure paper from 1989. There are very interesting papers by Dienes, and more recently Abel, on the cc for non-supersymmetric string theory, which emphasize that modular invariance, not supersymmetry, is the important symmetry at string level. Perhaps it will all become clearer if I keep digging.

 

[bluecap]

Question. According to https://en.wikipedia.org/wiki/String_theory_landscape

"In string theory the number of false vacua is thought to be somewhere between 10^10 to 10^500.[1] The large number of possibilities arises from choices[clarification needed] of Calabi–Yau manifolds and choices[clarification needed] generalized magnetic fluxes over various[clarification needed] homology cycles."

Is it like saying that if superstrings theory were not true. Our universe is in true vacuum while if superstrings theory was true, our universe is in false vacuum? The latter seems to be artificial.. like by manipulating different Calabi-Yau manifolds.. you can produce different vacua. What would it then be like to live in the true vacua or true vacuum.. or what is the true vacuum of supersting theory composed of versus the true vacuum without superstring theory?

 

[Spinnor]

or what is the true vacuum of supersting theory composed of versus the true vacuum without superstring theory?

This lecture might interest you, "Where in the World are SUSY & WIMPS? - Nima Arkani-Hamed"