## August 2, 2008

### Noncommutative

From time to time, I’ve been asked what I think of the Connes-Chamseddine-et al “Noncommutative Standard Model”. I haven’t had anything terribly profound to say, but recently I got involved in a conversation with Urs Schreiber about it. So, perhaps it might be useful to summarize what was said.

We’ll work in Euclidean signature. Let $X$ be a Riemannian 4-manifold, and $V\to X$ a complex vector bundle with unitary connection. $D: \Gamma(S^+\otimes V) \to \Gamma(S^- \otimes V)$ is the usual chiral Dirac operator. Let’s also define $\tilde{D}: \Gamma(S^-\otimes V^*) \to \Gamma(S^+ \otimes V^*)$ and a nonderivative (scalar) operator, $\Phi\in\Gamma(Sym^2 V^*)$, or equivalently $\Phi: \Gamma(V)\to \Gamma(V^*)$ Putting these together, we can form $\mathcal{D} = \begin{pmatrix}\Phi&\tilde{D}\\ D&\Phi^\dagger\end{pmatrix}: \Gamma(S^+\otimes V\oplus S^-\otimes V^*)\to \Gamma(S^+\otimes V^*\oplus S^-\otimes V)$ and, finally, the self-adjoint operator $\hat{\mathcal{D}} = \begin{pmatrix}0&\mathcal{D}\\ \mathcal{D}^\dagger & 0\end{pmatrix}$

The Euclidean action for the fermions is

(1)$S_f = \int dVol(X) \tfrac{1}{2}(\Psi,\mathcal{D} \Psi)$

where $(\cdot,\cdot)$ denotes the natural skew-bilinear pairing, $S^\pm\otimes S^\pm \to \mathbb{C}$.

For the bosons one writes the nonlocal action

(2)$S_b = Tr\left(f(\hat{\mathcal{D}}/\Lambda)\right)$

for some positive, but otherwise unspecified function $f$.

The particular case of interest is to take $V$ to be associated to a $G=SU(3)\times SU(2)\times U(1)$ principal bundle over $X$, via the representation $R=R_0^{\oplus 3}$, where

(3)$R_0 = (3,2)_{1/6} \oplus (\overline{3},1)_{-2/3} \oplus (\overline{3},1)_{1/3} \oplus (1,2)_{-1/2} \oplus (1,1)_1 {\color{red}\oplus (1,1)_0 }$

For $\Phi$, one takes $\Phi = (\lambda'_d +\lambda'_e) \phi + (\lambda'_u {\color{red}+\lambda'_N}) \phi^* {\color{red}+M}$ for $\phi\in \Gamma(U)$, where $U$ is the bundle associated to the representation $R_h=(1,2)_{-1/2}$. The $\lambda'_{u,d,e,N}$ are $G$-invariant elements $\begin{gathered} \lambda'_d,\lambda'_e\in R_h^*\otimes Sym^2 R^* \\ \lambda'_u,{\color{red}\lambda'_N}\in R_h \otimes Sym^2 R^* \\ \end{gathered}$ That is, they are Clebsch-Gordon coefficients2 multiplied by coupling constants. ${\color{red}M}$ is a constant matrix, corresponding to the $G$-invariant subspace of $Sym^2 R^*$. It provides a Majorana mass term for the right-handed neutrinos, the ${\color{red} (1,1)_0 }\in R$.

Now, there’s a certain amount of mumbo-jumbo about interpreting $S^+\otimes V$ as the bundle of chiral spinors on a certain noncommutative spacetime and $\mathcal{D}$ as the associated chiral Dirac operator. But I don’t see that there’s much of anything to be gained thereby. Just take (1),(2) as the definition of the theory (at least at the classical level) and go from there.

Now, (2) is pretty uselessly nonlocal (not to mention under-specified). But you can do a heat-kernel expansion of it, and recover a local effective action, valid at low energies, $E\ll \Lambda$. The lowest-order terms in this expansion depend only on the first few moments of the function $f$. The complete expansion, up to dimension-4 operators, can be found in equation (3.41) of Chamseddine, Connes and Marcolli.

For instance, the Yang-Mills kinetic terms just depend on $f_0\equiv f(0)$. One reads off the $SU(3)\times SU(2)\times U(1)$ gauge couplings

(4)$g_3^2 = g_2^2 = \frac{5}{3} g_1^2 = \frac{\pi^2}{2f_0}\equiv g^2$

The fact that the gauge couplings are “unified” is no surprise. $R$ consists of complete $SU(5)$ multiplets, so the “induced action”, (2) for the gauge fields is $SU(5)$-symmetric.

Slightly more interesting is that the kinetic term for the Higgs field is also proportional to $f_0$. The canonically-normalized Higgs fields $\varphi= \sqrt{\tfrac{a f_0}{2\pi^2}}\phi$ where $a = tr(\lambda'^\dagger\lambda')\equiv 3(\lambda'_u^\dagger \lambda'_u+\lambda'_d^\dagger \lambda'_d)+ \lambda'_e^\dagger \lambda'_e{\color{red}+\lambda'_N^\dagger \lambda'_N}$ So the physical Yukawa couplings $\lambda = \frac{\lambda'}{\sqrt{a f_0/2\pi^2}}$ satisfy

(5)$tr(\lambda^\dagger\lambda)\equiv 3(\lambda_u^\dagger \lambda_u+\lambda_d^\dagger \lambda_d)+ \lambda_e^\dagger \lambda_e{\color{red}+\lambda_N^\dagger \lambda_N} = 4 g^2$

The Higgs potential depends, as well, on $f_2$.

(6)$V(\varphi) = 4 g^2 tr(\lambda^\dagger\lambda)^2 {\left(|\varphi|^2 -v^2\right)}^2$

where

(7)$v^2 = \frac{\Lambda^2 f_2}{\pi^2 tr{(\lambda^\dagger\lambda)}^2}- \frac{tr(M^\dagger M \lambda_N^\dagger \lambda_N)}{4 g^2 tr{(\lambda^\dagger\lambda)}^2}$

The Einstein-Hilbert term also depends on $f_2$

(8)$\left(\frac{4 \Lambda^2 f_2 }{\pi^2}-\frac{tr(M^\dagger M)}{48g^2}\right)\int dVol(X) R$

There’s also a cosmological constant term, which depends on $f_4$, but let’s skip that for now.

What is interesting is that the heat-kernel expansion did not give us the most general possible gauge-invariant low energy effective action. Instead, a-priori independent couplings obeyed relations, given in (4),(5),(6)(7) and (8), for any choice of the function $f$.

While we have no clue how to quantize an action like (1)+(2), we do know how to treat an effective field theory. We should interpret above relations on the coupling constants as holding for the running coupling constant, evaluated at the cutoff scale, $\Lambda$. One might have hoped that renormalization would preserve the form of the action (1)+(2), and that divergences could be absorbed into a redefinition of the function $f$. Clearly, that does not happen in the effective theory. But perhaps that is just as well, because these relation clearly don’t hold in the real world at low energies.

Given the freedom that we do have, can we ensure that the restricted set of couplings, stemming from (1)+(2), give the correct physics, after being run down to low energies?

Somehow or other, by tuning $\Lambda f_2$ and $M$, we need to make (7) electroweak scale, while making (8) Planck-scale. That requires that $M$ be Planck scale (we shan’t worry about fine-tuning). But choosing $M$ be Planck scale would seem to preclude getting acceptable neutrino masses (see below) But let’s press on.

(4) suggests identifying $\Lambda$ as the GUT scale. Of course, that’s a little problematic, as the gauge couplings don’t actually unify (with just the SM degrees of freedom).

If the couplings did unify, then we could identify $g$ as the GUT coupling and (5) would give a sum rule on fermion masses-squared. Connes et al state this as an upper bound on the top mass, which is a rather odd way to state the result.

Similarly, (6) would yield a prediction for the Higgs mass. Again, eliding the small problem that the gauge couplings don’t actually unify, the claimed value for the Higgs mass is $m_H =171.6\pm 5\text{GeV}$

More fundamental, though, than the difficulty of getting the electroweak scale to come out right, or the lack of coupling constant unification, is that we don’t have a clue how the treat (1)+(2) as a quantum theory. Indeed, without a principle for choosing $f$, we don’t even know what theory we wish to quantize.

#### Update (8/4/2008): Uniqueness

I forgot to mention one really elementary point about their construction that confuses me. They start with the involutive algebra $\mathcal{A}_{LR} = \mathbb{C}\oplus \mathbb{H}\oplus \mathbb{H}\oplus M_3(\mathbb{C})$ The direct sum of all irreducible odd $\mathcal{A}_{LR}$ bimodules, $\mathcal{M}_F$, can be written as $\mathcal{M}_F = \mathcal{E} \oplus \overline{\mathcal{E}}$ where $\mathcal{E}$ is a bimodule3 of complex dimension 16, and $\mathcal{M}_F$ has a real structure, given by the antilinear map $J(\xi,\overline{\eta}) = (\eta,\overline{\xi})$ which satisfies $J^2 =1,\quad u b = J b^* J u,\quad \forall\, u\in \mathcal{M}_F,\, b\in \mathcal{A}_{LR}$

The way Connes et al read off the gauge group (and fermion representation) is to pick a certain subalgebra $\mathcal{A}_{F}\subset \mathcal{A}_{LR}$. The gauge group is then the group of unitaries, $G = SU(\mathcal{A}_{F})$ $U(\mathcal{A}) = \{u\in \mathcal{A} | u u^* = u^* u =1\}$ and $SU(\mathcal{A}_{F})\subset U(\mathcal{A}_{F})$ is the subgroup of determinant 1, when acting on $\mathcal{E}$.

But which subalgebra to choose? They take

(9)$\mathcal{A}_{F} = \mathbb{C}\oplus \mathbb{H}\oplus M_3(\mathbb{C})$

embedded as $(\alpha, q, m)\mapsto (\alpha, q,\alpha,m)$, which yields $G= SU(3)\times SU(2)\times U(1)$, with $\mathcal{E}$ transforming as the representation $R_0$, defined above. But a different choice, which also seems to satisfy all of their criteria, is to take

(10)$\mathcal{A}_{F'} = \mathbb{C}\oplus \mathbb{C}\oplus \mathbb{H}\oplus M_3(\mathbb{C})$

embedded as

(11)$(\alpha, \beta, q, m) \mapsto (\alpha, q,\beta, m)$

This leads to the gauge group $G' = SU(3)\times SU(2)\times U(1) \times U(1)_{B-L}$ So I don’t really understand the uniqueness theorem (proposition 2.11 of Chamseddine, Connes and Marcolli) that leads to (9).

Evidently, I’m missing something …

#### Update (8/5/2008): The Electroweak Scale

Perhaps it’s worth spelling out more explicitly the problem, alluded to above, with getting the electroweak scale and neutrino masses to work out. Start with (8), which says $\frac{1}{8\pi G_N} = \frac{4 f_2 \Lambda^2}{\pi^2} - \frac{tr(M^\dagger M)}{48 g^2},$ solve for $f_2 \Lambda^2$, and plug into (7):

(12)$v^2 = \frac{1}{tr(\lambda^\dagger\lambda)^2}\left[\frac{1}{8\pi G_N} + \frac{tr(M^\dagger M) -48 tr(M^\dagger M \lambda_N^\dagger \lambda_N)}{48g^2}\right]$

We want $v\sim 250$GeV, while $\frac{1}{8\pi G_N}\sim (2.4\times 10^{18} \text{GeV})^2$. Now, the neutrino masses are $m_\nu \sim \frac{\lambda_N^2 v^2}{M}$ To get $m_\nu \sim 5\times 10^{-11}$ GeV, with $|\lambda_N|^2 \lesssim 1$ (for a perturbative analysis to be valid), we need $M \lesssim 10^{15}$ GeV. But that’s incompatible with (12), which requires $M\sim 10^{17}$ GeV.

1 The “$\tfrac{1}{2}$” indicates that the path integral over the fermions is a section of the Pfaffian line bundle, $PFAFF(\mathcal{D})$. When the anomalies cancel, $PFAFF(\mathcal{D})$ can be trivialized, and we can interpret the fermion path integral as a function on the parameter space. If we take $\Phi\equiv 0$, then we reduce to the more familiar case, $PFAFF(\mathcal{D})=DET(D)$.

The slight awkwardness of the above description is associated to the usual doubling of the fermion degrees of freedom, when we pass to Euclidean signature. In a Minkowski signature spacetime, the description would have been a bit more straightforward.

2 There are two independent $G$-singlets in $R_h^*\otimes R_0^*\otimes R_0^*$, corresponding to $\lambda'_d$ and $\lambda'_e$. Without the ${\color{red}(1,1)_0}\in R_0$, there would be just a single $G$-singlet in $R_h\otimes R_0^*\otimes R_0^*$ (corresponding to $\lambda'_u$) With it, there is a second $G$-singlet, corresponding to ${\color{red}\lambda'_N}$.

3 If $\mathcal{M}$ is a bimodule for $\mathcal{A}$, then $\overline{M}$ is the bimodule $(a,b): \overline{\xi} \mapsto \overline{b^* \xi a^*},\quad \forall \xi\in \mathcal{M}$

Posted by distler at August 2, 2008 3:14 AM

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### Re: Noncommutative

Thanks for the useful review.

Now, there’s a certain amount of mumbo-jumbo about interpreting [this] as the bundle of chiral spinors on a certain noncommutative spacetime […]. But I don’t see that there’s much of anything to be gained thereby.

I see where you are coming from, but am wondering then to which degree you would apply similar reasoning to Kaluza-Klein theory in general.

Consider a statement such as

“Kaluza-Klein theories interpret gravity coupled to gauge theory as pure gravity on a Riemannian bundle over spacetime. But nothing is gained by this (except that it yields lots of complications) since one can just as well consider gravity coupled to gauge theory directly.”

For certain practical purposes this statement may very accurately reflect the situation. But of course it would miss somehow the point of Kaluza-Klein theory.

Of course plain vanilla usual Kaluza-Klein theory by itself is always unrealistic.

If one has any reasonable concept of generalized geometry one can try to see what Kaluza-Klein theory would mean in that generalized context. That’s what Connes’ model does. He shows that in a context of generalized geometry there are KK-models which come pretty close to being realistic.

Certainly after the fact the interpretation in terms of generalized geometry can be ignored for practical purposes. In a similar way that one can ignore the geometric meaning of Einstein gravity, if one insists, or the cohomological meaning of Maxwell theory or the 6-dimensional geometric interpretation of S-duality in 4-dimensional super Yang-Mills. In all these cases the formulas are in the end not affected by the way in which we interpret them. But geometric interpretations tend to be regarded as promsing for theoreticl purposes. Connes does provide a geometric interpretation for some of the structure seen in the standard model. This may or may not turn out to be on the right track, but clearly the idea is that what could be gained by it is a deeper understanding of the fundamental theory.

And, as I said before #, I keep being struck by how close the picture emerging from Connes’ work actually is to that in String theory. Clearly, his hope is very similar in principle to String phenomenology: embed the existing gauge and gravity theory in a unified whole and hope to get answers from that. Connes is hoping that the unified picture is NCG. The idea that one sees in his articles and those of his collaborators is that his work on the standard model indicates that there is fundamental theory of spectral triples where not only the internal KK-factor is noncommutative, but also the external spacetime, and that introducing this noncommutativity will solve the problems of quantum gravity.

That may or may not work out this way, or maybe at least not without a less naive refinement of this idea, but this is clearly what people think can be gained from the NCG standard model. There is a large industry looking into QFT on noncommutative spacetime as eventually a possible solution to quantum gravity. Connes’s NCG standard model may seem to provide a more first-principle rationale for such efforts.

And all of these efforts may very well be vain without further refinements, which in turn may show that spectral triple are themselves just a degeneration limit of something richer. But still then, I’d see a gain in Connes’ model in that it demonstrates that adding a little noncommutativity when compactifying down from higher dimensions may be very helpful for semi-realistic model building.

Posted by: Urs Schreiber on August 2, 2008 7:22 AM | Permalink | Reply to this

### Re: Noncommutative

Now, there’s a certain amount of mumbo-jumbo about interpreting [this] as the bundle of chiral spinors on a certain noncommutative spacetime […]. But I don’t see that there’s much of anything to be gained thereby.

I see where you are coming from, but am wondering then to which degree you would apply similar reasoning to Kaluza-Klein theory in general.

In Kaluza-Klein theory, one has some definite action for the higher dimensional theory (traditionally, the Einstein-Hilbert action; more generally, perhaps, some supergravity action) which one dimensionally-reduces to 4 dimensions.

There is, in that case, some physical content in saying that the resulting theory “comes from” the Kaluza-Klein reduction of the higher-dimensional theory.

Here, we have a perfectly arbitrary, unknown, bosonic action. So I don’t see how there’s any physical content in saying that it is the “Kaluza-Klein reduction” of some other, equally-unknown higher-dimensional theory.

In particular, it’s not like doing Kaluza-Klein theory. There, one really does have a local (albeit higher dimensional) Lagrangian description above the compactification scale.

Here, of course, we can’t expect a strictly local Lagrangian. Instead (if the noncommutative picture is to be useful, in the same way that the Kaluza-Klein picture is useful), one should expect some sort of noncommutative deformation of a local theory.

People have studied noncommutative field theories and, while they are not strictly local, they are – in many ways – nearly as good as local theories. In particular, there’s a notion of renormalizability.

The spectral action (in addition to being under-specified) seems to be much further from being local, and much further from being renormalizable.

I think that one has to adopt Connes’ attitude (expressed in the email that you quote) that this should just be seen as a cutoff theory, good for energy scales $E\ll\Lambda$.

That’s the attitude I have expressed here, and it’s very different from what one does in Kaluza-Klein theory.

Posted by: Jacques Distler on August 2, 2008 10:59 AM | Permalink | PGP Sig | Reply to this

### Re: Noncommutative

Maybe a different analogy would be helpful. There is a class of noncommutative field theories which has been much studied by physicists. It arises when one deforms the ordinary commutative product on the algebra of functions to the noncommutative Moyal product.

It is possible to take the resulting noncommutative field theory and expand it in a derivative expansion about an ordinary commutative field theory. This derivative expansion (the analogue of the heat kernel expansion that Connes et al apply to (2)) is perfectly good for studying processes at low momenta. But it breaks down, when one tries to study physics at short distances, precisely where the effects of noncommutativity become important.

In that case (unlike the present one), however, it is possible to treat the noncommutative theory directly (not in a derivative expansion about a commutative one). One can compute scattering amplitudes, one can carry out renormalization (which preserves the form of the noncommutative action), etc. And one can really study physics in the noncommutative regime.

Here, by contrast, we have a perfectly conventional effective field theory at scales $E\ll \Lambda$. And at scales $E\gtrsim \Lambda$, the whole formalism just breaks down.

Posted by: Jacques Distler on August 3, 2008 10:47 AM | Permalink | PGP Sig | Reply to this

### Re: Noncommutative

Yea this is something that confused me when I looked at the original papers some time ago.

Theres a lot of fancy jargon going around, but it also seemed to me if we treat things as an effective theory we do not preserve the form of 1+2 (b/c additional terms will be generated that conform to the full symmetry), and instead just get something that is for all intents and purposes the standard model, except now you have all sorts of additional hierarchies.

Nor is it clear to me why some of the additional relations(eg 6) are supposed to hold in the effective theory to actually give the predictions Connes et al want. Why wouldn’t they be violated by quantum effects?

Posted by: Haelfix on August 2, 2008 4:08 PM | Permalink | Reply to this

### Re: Noncommutative

Nor is it clear to me why some of the additional relations(eg 5) are supposed to hold in the effective theory to actually give the predictions Connes et al want. Why wouldn’t they be violated by quantum effects?

I believe the statement they wish to make is that (the heat kernel expansion of) the bosonic action (2) is just the initial condition for the RG flow. In particular, the relations (4),(5),(6)(7),(8) hold for the running couplings at the scale $\Lambda$, and the running couplings at lower scales are just given by the standard RG evolution, starting with those initial conditions.

Apparently, we are not supposed to inquire, even as a gedanken experiment, what physics looks like at energy scales above the GUT scale.

Posted by: Jacques Distler on August 2, 2008 4:35 PM | Permalink | PGP Sig | Reply to this

### Re: Noncommutative

Barring the rather unfortuitous Higgs bound that seems to rule out his theory (as seen elsewhere around the blogosphere), its still not clear to me why at least some of those relations can’t simply be rescaled… He’d probably have to drop some version of minimality in his model, but I don’t understand why the whole thing is fixed like it is.

On another note, theres been claims in the past for the absense of proton decay in his model. Well, obviously since he restricts to dimension 4 operators they won’t appear in the effective theory, but whats stopping those higher dim terms from being generated at the GUT scale.

Posted by: haelfix on August 5, 2008 3:52 PM | Permalink | Reply to this

### Proton Decay

On another note, theres been claims in the past for the absense of proton decay in his model. Well, obviously since he restricts to dimension 4 operators they won’t appear in the effective theory, but whats stopping those higher dim terms from being generated at the GUT scale.

The bosonic action (2) preserves baryon and lepton number (since neither the gauge bosons, nor the Higgs, carry B or L charge). The fermionic actions is defined to be purely quadratic in the fermions. The Majorana mass for the right-handed neutrino violates L, but it doesn’t violate B.

So the whole action, at the GUT scale, (by construction) preserves B, and the proton is stable. Running it down to lower energies won’t change that.

He’d probably have to drop some version of minimality in his model, but I don’t understand why the whole thing is fixed like it is.

It’s certainly true that there are fewer independent parameters in the heat kernel expansion of (2) than there are renormalizable coupling constants. Hence, whatever the detailed matter content of the model, there are “predictions” here. Whether any of those can be made compatible with observations is another story …

Posted by: Jacques Distler on August 5, 2008 7:09 PM | Permalink | PGP Sig | Reply to this

### Re: Proton Decay

Does this mean B is conserved to all orders perturbatively, or really exactly conserved even after the instantons that pop up in the SM?
In other words, did U(1)_B become non-anomalous somewhere?

Baryogenesis is a touchy subject if B is exactly conserved…

Posted by: Thomas D on August 6, 2008 7:15 AM | Permalink | Reply to this

### Re: Proton Decay

Of course baryon number is still anomalous, and violated by electroweak instantons. (That much is a robust property of the infrared physics!)

I was discussing the perturbative behaviour of the theory, which — by fiat — conserves B at the classical level, and hence also perturbatively.

Posted by: Jacques Distler on August 6, 2008 8:08 AM | Permalink | PGP Sig | Reply to this

### SU(5)

The action is SU(5) symmetric and there are no extra bosons? Is this an usual situation in GUT model building, or is it a merit to be considered?

Posted by: Alejandro Rivero on August 3, 2008 10:58 AM | Permalink | Reply to this

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