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String theory admits a vast number of vacuum solutions, which I gather come from the all the ways of compactifying the geometry of spacetime down to 3+1 dimensions using Calabi-Yau manifolds. I have always been unclear on the status of actually identifying/constructing solutions that exactly match experimental observations. This paper claims to construct $10^{15}$ solutions "that realize the exact chiral spectrum of the Standard Model (SM) of particle physics with gauge coupling unification in the context of F-theory." Previous papers cited in the introduction make similar claims, but it seems like they all actually construct the "minimally supersymmetric Standard Model," and I can't find any clear answer as to whether this is an important caveat (including in other stack exchange posts like this one), or if there are other important caveats.

I can imagine 3 possible scenarios:

  1. String theory vacua are known that should be able to correctly match all 19 free parameters of the Standard model, plus whatever free parameters are needed to accommodate neutrino masses. We still have the usual naturalness questions like the hierarchy problem, but no major new ones.

  2. We technically can get the full Standard Model, but not without introducing new severe issues like extreme fine tuning* or something else that should give us pause. Or we can get various aspects using different vacua, but don't yet know how to get everything at the same time with just one vacuum.

  3. There are certain parameters/features of the Standard Model that we fundamentally don't know how to construct from string theory yet.

I imagine working out all the details super-explicitly to actually construct the Standard Model vacuum solution may require infeasibly difficult calculations (and I don't see the actual SM parameters showing up in these papers). In this case, I suppose the right question is what the general consensus is regarding which of the 3 cases we are in.**

Let's ignore issues of the cosmological constant, inflation, the big bang..., as that's a separate can of worms.

*Perhaps the term 'fine tuning' is ambiguous in this context, since we're talking about picking $10^{15}$ vacua out of $10^{500}$ or whatever. But if something like pushing up the masses of supersymmetric partner particles or eliminating proton decay drastically reduces the number of vacuum solutions, then I suppose that would count.

**I suppose there is also a fourth possibility, which is that string theory cannot reproduce the Standard Model and will therefore be falsified.

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  • $\begingroup$ At least you could not say it's "not even wrong". $\endgroup$ Commented Apr 7, 2023 at 7:49

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Almost always in string theory papers, a "standard model vacuum" means a "supersymmetric standard model vacuum", meaning a scenario in which, at low energies, you have the superfield counterparts of the fields and interactions of the actual standard model.

Each such vacuum will then have its own values for the gauge and yukawa coupling constants; and supersymmetry may or may not be broken. For the real world to be one of these vacua, supersymmetry has to be broken, and the couplings have to turn out right.

But if you were to pick a particular one of these vacua, and then ask, so what are the couplings for this one - no one would be able to answer you. They depend on things like the expectation values of the metric for a specific Calabi-Yau space, that are quite beyond calculation at this time.

This 2021 thesis looks very state-of-the-art. It's considered a great achievement to obtain the "texture" of the yukawa matrices (the distribution of zeroes) for a given string vacuum - see equation 3.137. Textures are useful in theoretical particle physics because they can have implications like, the third-generation fermions are much heavier than the first two generations... It's considered an even greater achievement to get a handle on the actual magnitude of the yukawas - see chapter 5 of the thesis, at the bottom of page 118.

Quantitative calculations of this kind are presently only possible for quite specialized classes of vacua, that have some nice property that renders them amenable to computation. Most string phenomenology is still a struggle to obtain the right qualitative features, the right fields and interactions, without having too much other junk also showing up, without having the extra dimensions expand, and so on.

In this regard, there's no guarantee that people are working in the right direction. Most string phenomenology (e.g. that 2021 thesis) is pursued within the paradigm of supersymmetric grand unification, something which certainly fits string theory well, but which is, nonetheless, not an absolute logical necessity. It actually seems quite likely that the real world needs to be sought (and the computational problems solved) in realms of string theory that are relatively neglected, or even still unknown.

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  • $\begingroup$ Eq. (3.137) reads$$\lambda^{(u)}=\frac{i\pi^3}{6}\left(\begin{array}{ccc} 0 & 0 & 0\\ 0 & 0 & 1\\ 0 & 1 & 0 \end{array}\right).$$Are you sure that's the equation you meant? $\endgroup$
    – J.G.
    Commented May 2, 2023 at 8:54
  • $\begingroup$ @J.G. Yes, those are the "holomorphic Yukawa couplings" for up-type quarks in this model; still one step short of physical Yukawa couplings, see the discussion in the middle of page 10. $\endgroup$ Commented May 2, 2023 at 14:50
  • $\begingroup$ "Almost always in string theory papers, a "standard model vacuum" means a "supersymmetric standard model vacuum..." But isn't any supersymmetric standard model vacuum falsified by experiment? Or do you mean that there is enough supersymmetry breaking to push unobserved superpartners up to very high masses? $\endgroup$
    – user34722
    Commented May 27 at 20:02
  • $\begingroup$ If all of these models predict identical supersymmetric counterparts of all particles, then isn't that kind of a non-starter? Do people not know how to break enough supersymmetry to get the standard model at low energies? $\endgroup$
    – user34722
    Commented May 27 at 20:06
  • $\begingroup$ Supersymmetry here includes broken supersymmetry (there are also genuinely non-supersymmetric vacua in string theory, with no superpartners at all, but they have been studied much less). Many ways to break supersymmetry are known, but I don't know how hard it is to calculate the actual magnitude of Susy breaking for any given string vacuum. $\endgroup$ Commented May 29 at 2:13

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