Restoring and breaking SU(2)xU(1)

[arivero]

I think I am not the only one fascinated by the strange discussion what happened some years ago at Woit's
http://www.math.columbia.edu/~woit/wordpress/?p=3
between Distler and the owner of the blog, about the "turning off" of the different components of the standard model.

Actually nobody there took the point to discuss the parameters for such "turning off".

 

[arivero]

To start:

I think that the discussion was not really about SU(3)xU(1) but about U(1). It is true that it makes sense to speak globally of SU(3)xU(1) because it is the surviving, unbroken, part of the standard model, and it is non chiral. But this is the U(1) of electromagnetism, not the U(1) of the standard model. So this parragraph, to me, is confusing:

The left-handed fermions of the standard model form a real representation of SU(3)xU(1), and they form a pseudoreal representation of SU(2). It’s only when you look at the full SU(3)xSU(2)xU(1), that you find that they form a complex (ie, chiral) representation of the gauge group.

Is the U(1) in the complex representation the same U(1) that in the real representation?. I believe not. And then, is there some law about joining real plus pseudoreal to form complex representations? And in which sense a complex representation is chiral and a pseudoreal is not?

 

[arivero]

a first detour:

How unique is, at all, SU(2)xU(1)? How unique is SU(3)xU(1)? We know that they can be implemented on different representations. But if we think to get it from Kaluza Klein (we can do it for SU(3)xU(1) as it is non chiral, or we could use orbifolds for SU(2)xU(1)) then there is the point of how many different spaces do support an action of the group. For SU(3)xU(1) the corresponding space would be a quotient by a maximal subgroup, so M=SU(3)xU(1)/SU(2)xU(1), dim M= 8 + 1 - 3 - 1. Different embeddings of the subgroup will produce different manifolds, will them? Similarly, the action of a SU(2) group will be on SU(2)/U(1), this is unique. But for N=SU(2)xU(1)/U(1), with dim N=3+1-1, the embedding of U(1) in the big group again classifies the manifolds. In this case it can not be all the history because over compact spaces we can not get chiral theories, so we would need orbifolds or other gadgets.

But both questions, for M-manifolds and for N-manifolds, are interesting.

 

[daschaich]

...Is the U(1) in the complex representation the same U(1) that in the real representation?...

Looks to me (after an admittedly cursory inspection) like the latter [the one in SU(3)xU(1)] is electromagnetism, while the former [the one in SU(3)xSU(2)xU(1)] is weak hypercharge: not the same

 

[arivero]

Yep, one is EM and "nonchiral", the other is hypecharge and, if it is chiral, does it mean that it is being represented via a complex representation of U(1)?

Actually, I find that the clasification on complex, real and pseudoreal representations is not very useful, at the end what matters is if the chiral invariancy, the separation of Dirac Equation in two Weil spinor equations, is preserved or not.

 

[arivero]

Going back to the symmetry of the electroweak model. The point, anyway, is that we have three free parameters to play with, a dimensional "vacuum expectation value" $v$ and two adimensional coupling constants, $g, g'$ for the weak isospin SU(2) and the hypercharge U(1) of the product SU(2)xU(1). Of course, U(1) hypercharge acts differently in the right and left sectors; we define g' as the coupling constant for the left sector, $$Y_L=-1$$ and then the model imposes the coupling for the right sector to be 2 g', so we write $$Y_R=-2$$

But but ... this relationship comes from asking SU(2) to act only in the Left sector and then looking for a non-chiral massless photon, so its coupling to neutrino and electron is, respectively.

$$g' Y_L \cos \theta + g \sin \theta =0$$
$$g' Y_R \cos \theta = g' Y_L \cos \theta - g \sin \theta$$

When $g=0$ there is nothing coming out from the GWS model to demand U(1) to be different in the left and right sectors (In fact there is no photon: the hypercharge generator is completely mapped to the Z particle). It is during the limit process, while SU(2) has a coupling $g \neq 0$, that we have $$Y_R=2 Y_L$$.

Let's follow the convention $Y_L=-1$. The usual electric coupling of the photon is then
$$e= g \sin \theta =g' \cos \theta = { g g' \over \sqrt {g^2 + g'^2}}$$

So much for "switching off" a group. But symmetry restoration (resp full breaking) is a different bussiness, it is about moving the masses of the broken generators to zero (resp infinity). The masses are

$$m_W=g \ \frac v 2$$
$$m_Z= \sqrt {g^2 + g'^2} \ \frac v 2$$

If we keep v and g finite and take g' to infinity, the situation is very amusing: the Z boson again coincides with the hypercharge, and then the photon with the W3 boson of weak isospin, but now this W3 boson... is not chiral!

If we keep v and g', and we take g to infinity, then the Z boson is the W3, the photon coincides with the hypercharge... and again is not chiral.

 

[arivero]

The format of the above equations, where $g v$ and $g' v$ are akin to mass scales, invites to try to derive the GSW model from Kaluza Klein theory. We know that we are going to fail, because all the theories out of Kaluza Klein on homogeneus spaces are not chiral.

detour: I am not sure if we should really follow the gravity prescription. In usual KK, Newton constant provides the unit to cancel the compactification radius and produce adimensional coupling constants. Here instead of Newton constant we have Fermi constant already available. Its units are actually the same that Newton constant, but it act differently. A way to think on the four fundamental forces is to order according some dependence. Force is m.a, so enabling relativity and if we put c=1 it has dimensions of mass/length. Enabling quantum mechanics, with h=1, we can consider it to have units or mass square or inverse of area. So, the table of classical fundamental forces and the dimensions of their corresponding couplings is:

$F= G_N \ m^2 / r^2$ with the dimensions of "area" or "mass square"
$F= G_F \ q q' /r^4$ with dimensions of "area" too
$F= K_{em} \ q q' / r^2$ adimensional
$F= K_{s}$ with dimensions of "inverse of area"

 

[arivero]

gravity in 7 dimensions

To produce SU(2)xU(1) in KK, the minimal recipe is to build non-trivial homogeneus spaces from a subgroup of the acting group.

The biggest possible subgroup is SU(2), the quotient is SU(2)xU(1) / SU(2) = U(1) = S^1, the one-dimensional circle, and the resulting theory is electromagnetism, the original Kaluza-Klein theory.

Next (and unique) idea is to use U(1), and here the surprise: the quotient SU(2)xU(1) /U(1) is ambiguous... because it depends on the way we inject the subgroup into the group. We get a one-parametric family of 3-dimensional spaces, interpolating from the sphere $S^3$ (=SU(2)) to the product $S^2 \times S^1$. Will this family of spaces produce something remotely similar to the GSW model?

 

[arivero]

We get a one-parametric family of 3-dimensional spaces, interpolating from the sphere $S^3$ (=SU(2)) to the product $S^2 \times S^1$. Will this family of spaces produce something remotely similar to the GSW model?

Hmm it seems I am wrong about one-parametric, in the sense of a continuous parameter across a continuous family of spaces. It is safer to follow Witten, who did a similar argument for D=7 spaces in his article "Towards a Realistic Kaluza-Klein Theory". Now the "essentially unique subgroup" of $G=SU(2) \otimes U(1)$ is U(1)xU(1), the two factors being generated by [itex]T_3[/tex] and $Y$. We want to discard, of this subgroup, a linear combination $qT_3+rY$ with the conditions that $q, r$, are two arbitrary integers which have no common divisor, and $r$ should be non-zero. The subgroup H is then going to be the U(1) generated with the combination orthogonal to that one

$$H=\frac 1r T_3+ \frac 1q Y$$

And we should define $M^{qr}=G/H$I guess that the condition that $r$ is non zero avoids the highly symmetric sphere $S^3$ (whose total symmetry should be $SU(2) \otimes SU(2) \subset SO(4)$). But Witten only remarks that "r should be non-zero to avoid obtaining a space on which U(1) is realized as the identity". Well of course in this case we lost our beloved U(1), but we gain a full extra SU(2).

 

[arivero]

I guess that the condition that $r$ is non zero ...

Here there is a discrepancy in notation between Witten and Castellani et al (the people who did the full review of the classification of 7 dim manifolds), as the later group allows for zero, and I tend to agree with them.

It seems that the spaces we get to interpolate between S^3 and S^2 \otimes S^1 are lens spaces. There is a non-commutative version of them, and we should verify that the parameter in the non-commutative quotient is akin to Weinberg angle. It is pretty exciting.

Of course to get the GWS model we need chiral fermions. I suspect this is going to be related to a decrease in the dimension of the space, from 3 to 2. Or I hope it.

Another intriguing touch: by quotient the extreme spaces by S^2, we get the circle S1 and the S1/Z2 orbifold, do we? I am not sure... S2/S1 is the equal to S1/Z2, but I have not checked if it is the same that S3/S2Edited: Hmm, actually it seems (more reading, or independent confirmation pending) that the lens spaces L(q,p) interpolate between S1xS2 and S2xS1, the S3 being the middle point p=q. No bad, specially if you think that in S3 the symmetry enhances to SU(2)xSU(2) (this is, SO(4) of course).

 

[arivero]

Slow advancement in the exersice. Both the notation for witten spaces Mpqr amd for lens spaces is muddled. So in the case somebody is following the thread I will give hints to bibliography:

After Witten's "realistic paper", Kreck and Stolz did in 1988 "A diffeomorphism classification of 7-dimensional homogeneous Einstein manifolds with SU(3)xSU(2)xU(1)-symmetry". They link to a review article of Duff Nilsson and Pope in 1986 Physics Reports, where they link to the work of Witten, of Castellani et al, and of them selves, particularly

"Stability Analysis of Compactifications of D=11 Supergravity with SU(3)xSU(2)xU(1) symmetry", by Don N. Page and C. N. Pope, Physics Letters B 145 p 337 (1984).

The reading of the review and associated papers brings interesting details. Everybody knows about lens spaces, but the fact that they interpolate between S3 and S2xS1 is not noticed. Of course we could ask if Witten has found a different family of spaces with the same extreme limits. More, nobody hints, ever as a footnote, of the relationship with Weinberg angle.

Other remarks are that one could expect solutions coming from non-homogeneous spaces, and such solutions are far from being classified. And that not all the range of p,q is stable, only certain values, around a solution which keeps supersymmetry, are stable. All the other, including the extreme cases, divide the extra dimensions in two submanifolds, one of them expands and the other contracts as time passes. If you remember that the size of the manifolds is related to the size of coupling constants in standard kaluza klein, then it is not a very good thing to happen.

 

[arivero]

It seems that the spaces we get to interpolate between S^3 and S^2 \otimes S^1 are lens spaces.

Steenrod explains that actually lens spaces classify all the fiber bundles over S2 with fiber S1. So the idea could be to look SU(2)xU(1) as a mixing of an action SU(2) in the basis and U(1) in the fiber. And for the Hopf fibration it gets extra generators becoming SU(2)xSU(2) (ie, SO(4)).