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Standard model as string-inspired double field theory

Patrick of UC Davis told us not to overlook a new intriguing Korean hep-th paper

Standard Model Double Field Theory
by Choi and Park. They excite their readers by saying that it's possible to rewrite the Standard Model – or a tiny modification of it – as a special kind (or variation) of a field theory that emerged in string theory: the so-called double field theory.

The paper is rather short so you may try to quickly read it – I did – and I am a bit disappointed after I did. The abstract suggested that there is something special about the Standard Model (if it is not completely unique) that makes its rewriting as a double field theory more natural (or completely natural if not unavoidable). I couldn't find any fingerprint of this sort in the paper. It seems to me that what they did to the Standard Model could be done to any quantum field theory in the same class.

Double field theory is a quantum field theory but it has a special property that emerged while describing phenomena in string theory. But if you remember your humble correspondent's recent comments about "full-fledged string theory", "string-inspired research", and "non-stringy research", you should know that I would place this Korean paper to the middle category. I disagree with them that the features they are trying to cover or find are "purely stringy". It's a field theory based on finitely many point-like particle species – their composition is picked by hand, and so are various interactions and constrains taming these fields – so it is simply not string theory where all these technicalities (the field content, interactions, and constraints) are completely determined from a totally different starting point (and not "adjustable" at all). They're not solving the full equations of string theory etc. Again, I don't think that the theories they are describing should be counted as "string theory" although the importance of string theory for this research to have emerged is self-evident.

What is double field theory (DFT)? In string theory, there is this cute phenomenon called T-duality.

If a circular dimension is compactified on a circle of radius \(R\) i.e. circumference \(2\pi R\), the momentum becomes quantized in units of \(1/R\) i.e. \(p=n/R\), \(n\in\ZZ\). That's true even in ordinary theories of point-like particles and follows from the single-valuedness of the wave function on the circle. However, in string theory, a closed string may also be wrapped around the circle \(w\) times (the winding number). In this way, you deal with a string of the minimum length \(2\pi R w\) whose minimum mass is \(2\pi R w T\) where \(T=1/2\pi\alpha'\) is the string tension (mass/energy density per unit length).

So there are contributions to the mass of the particle-which-is-a-string that go like \(n/R\) and \(wR/ \alpha'\), respectively (note that the factors of \(2\pi\) cancel). Well, a more accurate comment is that there are contributions to \(m^2\) that go like \((n/R)^2\) and \((wR/ \alpha')^2\), respectively, but I am sure that you become able to fix all these technical "details" once you start to study the theory quantitatively.

There is a nice symmetry between \(n/R\) and \(wR/ \alpha'\). If you exchange \(n\leftrightarrow w\) and \(R\leftrightarrow \alpha'/R\), the two terms get interchanged. (The squaring changes nothing about it.) That's cool. It means that the spectrum (and degeneracies) of a closed string on a circle of radius \(R\) is the same as on the radius \(\alpha'/R\). This is no coincidence. The symmetry actually does exist in string theory and applies to all the interactions, too. In particular, something special occurs when \(R^2=\alpha'\). For this "self-dual" radius, the magnitude of momentum-like and winding-like contributions is the same and that's the point where string theory produces new enhanced symmetries. For example, in bosonic string theory on the self-dual circle, \(U(1)\times U(1)\) from the Kaluza-Klein \(g_{\mu 5}\) and B-field \(B_{\mu 5}\) potentials gets extended to \(SU(2)\times SU(2)\).

You may compactify several dimensions on circles, i.e. on the torus \(T^k\). The T-duality may be interpreted as \(n_i\leftrightarrow w_i\), the exchange of the momenta and winding numbers, which may be generalized "locally" on the world sheet to the "purely left-moving" parity transformation of \(X_i\). The reversal of the sign of the left-moving part of \(X_i\) only may also be interpreted as a Hodge-dualization of \(\partial_\alpha X_i\) on the world sheet.

In the full string theory, the theory has the \(O(k,k)\) symmetry rotating all the \(k\) compactified coordinates \(X_i\) and their \(k\) T-duals \(\tilde X_i\) if you ignore the periodicities as well as what the bosons are actually doing on the world sheet (left-movers vs right-movers). Normally, we only want to use either the tilded or the untilded fields \(X\). The double field theory is any formalism that tries to describe the string in such a way that both \(X_i\) and \(\tilde X_i\) exist at the same time. You have to "reduce" 1/2 of these spacetime coordinates at a "different place" not to get a completely wrong theory but it's possible to find such a "different place": some extra constraints on the string fields.

When the periodicities (circular compactification) are not ignored, the theory on \(T^k\) has the symmetry just \(O(k,k,\ZZ)\), a discrete subgroup, and the moduli space of the vacua is the coset\[

{\mathcal M}= (O(k,\RR)\times O(k,\RR)) \backslash O(k,k,\RR) / O(k,k,\ZZ)

\] because both compact simple \(O(k,\RR)\) transformations remain symmetries. This \(k^2\)-dimensional moduli space parameterizes all the radii and angles in the torus \(T^k\) as well as all the compact antisymmetric B-field components on it. It's not hard to see why there are \(k^2\) parameters associated with the torus: you may describe each torus using "standardized" periodic coordinates between zero and one, and all the information about the shape and the B-field may be stored in the general (non-symmetric, non-antisymmetric) tensor \(g_{mn}+B_{mn}\) which obviously has \(k^2\) components.

OK, what do you do with the Standard Model?

I said that the spacetime coordinates are effectively "doubled" when we add all the T-dual coordinates at the same moment. In this "Standard Model" case, it's done with all the spacetime coordinates, including – and especially – the 3+1 large dimensions. So instead of 3+1, we get 3+1+1+3 = 4+4 dimensions (note that the added dimensions have the opposite signature so the "sum" always has the same number of time-like and space-like coordinates).

The parent spacetime is 8-dimensional and the parent Lorentz group is \(O(4,4)\). This is broken to \(O(3,1)\times O(1,3)\). We obviously don't want an eight-dimensional spacetime. The authors describe some (to my taste, ad hoc) additional constraints that make all the fields in the 8-dimensional spacetime independent of 1+3 coordinates. So they only depend on the 3+1 coordinates we know and love.

They work hard to rewrite all the normal Standard Model to fields in this 8-dimensional parent spacetime with some extra restrictions and claim that it can be done. They just make some "very small" comments that their formalism bans the \(F\wedge F\) term in QCD – which would solve the strong CP-problem – as well as some quark-lepton couplings (an experimental prediction about the absence of some dimension-six operators). I don't quite get these claims. And their indication that the quarks transform under the first \(O(3,1)\) while the leptons transform under the other \(O(1,3)\), but they may also transform under the "same" factor of the group, sound scary to me. Depending on this choice, one must obtain very different theories, right?

Aside from the very minor (I would say) issue concerning the \(\theta\)-angle, I think it's fair to say that they present no evidence that the Standard Model is "particularly willing" to undergo this doubling exercise.

Even though I "independently discovered" the basic paradigm of the double field theory before others published it, I do share some worries with my then adviser who was discouraging me. The \(O(k,k,\RR)\) symmetry is really "totally broken" at the string scale, by the stringy effects. Some of the bosonic components are left-moving, others (one-half) are right-moving. This is no detail. The left-movers and right-movers are two totally separate worlds. So there isn't any continuous symmetry that totally mixes them.

In some sense, the \(O(k,k,\RR)\) symmetry is an illusion that only "seems" to be relevant in the field-theoretical limit but it's totally broken at the string scale. This fate of a symmetry is strange because we're used to symmetries that are restored at high energies and broken at low energies. Here, it seems to be the other way around.

After all, the symmetry is brutally broken in the double field theory Standard Model, too. The eight spacetime dimensions aren't really equal. Things can depend on four of them but not the other four. Maybe this separation is natural and may be done covariantly – they make it look like it is the case. But I still don't understand any sense or regime in which the \(O(k,k,\RR)\) symmetry could be truly unbroken which is why it seems vacuous to consider this symmetry physical.

Maybe such a symmetry may be useful and important even if it can never be fully restored. I just don't follow the logic. I don't understand why this symmetry would be "necessary for the consistency" or otherwise qualitatively preferred. That's why I don't quite see why we should trust things like \(\theta=0\) which follow from the condition that the Standard Model may be rewritten as a double field theory even though this rewriting doesn't seem to be "essential" for anything.

But at the end, I feel that they have some chance to be right that there's something special and important about theories that may be rewritten in this way. The broader picture of \(O(4,4)\)-symmetric theories reminds me of many ideas especially by Itzhak Bars who has been excited about "theories with two time coordinates" for many years. Here, we have "four times".

Quite generally, more than "one time coordinate" makes the theory inconsistent if you define it too ordinarily. The plane spanned by two of the time coordinates contains circles – and, as you may easily verify, they are closed time-like curves, the source of logical paradoxes (involving your premature castration of your grandfather). So the new time coordinates cannot be quite treated on par with the time coordinate we know. There have to be some gauge-fixing conditions or constraints that only preserve the reality of one time coordinate.

The idea is that there may be some master theory with a noncompact symmetry, \(O(4,4)\) or \(O(\infty,\infty)\) or something worse, which has some huge new "gauge" symmetry that may be fixed in many ways and the gauge fixing produces the "much less noncompact" theories we know – theories with at most one time and with compact Yang-Mills gauge groups. Is this picture really possible? And if it is, are the "heavily noncompact" parent theories more than some awkward formalism that teaches us nothing true? Can these "heavily noncompact" parent theories unify theories that look very different in the normal description? And if this unification may be described mathematically, should we believe that it's physically relevant, or is it just a bookkeeping device that reshuffles many degrees of freedom in an unphysical way?

I am not sure about the answers to any of these questions. Many questions in physics are open and many proposals remain intriguing yet unsettled for a very long time. But I also want to emphasize that it is perfectly conceivable that these questions may be settled and will soon be settled. And they may be settled in both ways. It may be shown that this double field theory formalism is natural, important, and teaches us something. But it may also be shown that it is misguided. Before robust enough evidence exists in either direction, I would find it very dangerous and unscientific to prematurely discard one of the possibilities. The usefulness, relevance, or mathematical depth of the double field theory formalism is just a working hypothesis, I think, and the amount of evidence backing this hypothesis (e.g. nontrivial consistency checks) is in no way comparable to the evidence backing the importance and validity of many established concepts in string theory (or physics).

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