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As explained in the answer to this question, in string theory there are massless and massive states. The massive states have their masses at the Planck scale so we do not observe them (at least with current colliders), so it is the massless states that correspond to particles in the standard model.

The particles in the standard model are indeed fundamentally massless, and get their masses from interaction with the Higgs field after Electroweak symmetry breaking. So that fits nicely, except for the Higgs particle itself - how does he gets his mass? If it was a string massive state then the mass should have been at the Planck scale, so I assume that's not the answer.

A comment in the linked question pointed to this paper titled "Calculating the Higgs Mass in String Theory" which I tried to read, but it is well outside the scope of my knowledge. Can anyone explain this in terms accessible to non string theory experts?

Qmechanic
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Steerpike
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2 Answers2

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In accessible terms:

Branes are surfaces in the projective space of extra (small) dimensions that we can not observe. In a theoretical model where the Higgs field is generally localised to specific intersecting branes, the recombination of these branes after breaking would yield the properties of the Higgs boson.

Plus:

a simple deformation of N = 4 SYM with large N can reduce at low energy to an effective gauge theory with few string-like Higgs modes coupled to chiral fermions, interpreted in terms of a gauge theory on intersecting branes in 6 extra dimensions. Moreover, the possible quantum numbers include those of the Standard Model

Ref: https://arxiv.org/pdf/1803.07323 Sperling, Steinacker 2018 (page 56)

Wookie
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To a first approximation, the Higgs boson gets its mass in string theory, the same way it gets its mass in field theory.

In the standard model, if you just focus on the Higgs field, it is a scalar field with two complex (or four real) components, with particular transformation properties under $SU(2)_L$ and $U(1)_Y$. The rules of QFT construction are that once you know the fields and their transformation properties, the Lagrangian consists of every renormalizable combination of the field operators, and then the non-renormalizable combinations define the additional terms of an "effective field theory".

If you apply this algorithm, starting just with the Higgs field, in the scalar potential you get the quadratic and quartic self-interaction terms, with coefficients that are free parameters; if the quadratic coefficient is negative, you get a nonzero "vacuum expectation value" (the famous sombrero potential), and the field resolves itself into three massless Goldstone degrees of freedom, and a residual scalar field obeying the Klein-Gordon equation. In the context of the rest of the standard model, the Goldstones give mass to the massive gauge bosons, and the Higgs boson is the quantum of that residual scalar field.

Bringing this back to string theory, the point is that the Higgs field can be realized in a variety of ways in string theory, but in the field-theory limit (where we treat the string as having zero length), it must reduce to that same set of four real scalar components. (If it's a string model which contains grand unification, which is most but not all of them, this will be a subset of a larger multiplet, e.g. the 5-dimensional scalar representation of SU(5).) And they will interact as described above. So the abstract mechanism by which the Higgs boson gets its mass is the same in all stringy implementations.

The paper that you link to, by Abel and Dienes, actually does unearth distinctive features of such calculations, when you do them in the full string theory context. They are especially interested in how the EFT perspective on the non-renormalizable terms looks different, when instead you do a stringy calculation involving all the massive states of the string, which are connected to the lighter states via modular symmetry. In particular, they say it can prevent the divergent heavy contributions to the Higgs boson mass, which led people to assume that the LHC would see, at the Fermi scale, some new symmetry like supersymmetry.

Mitchell Porter
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