Wednesday, June 23, 2010

SUSY and gauge coupling unification

Future piece of evidence in favor of supersymmetry:
Prev: Why string theory implies supersymmetry
Next: SUSY and dark matter
Next: SUSY and hierarchy problem
Next: SUSY exists because the number 3/2 can't be missing
Because the LHC is hopefully going to study previously unseen realms of particle physics within months or a year, I plan to write a couple of articles explaining the basic points that make me believe that despite the negative opinions of the rest of the Internet, supersymmetry is likely to be seen at the LHC.

Let me begin with the gauge coupling unification.

Curiosity is one thing that makes us human. But let's admit that this chimp is more curious than most people. Via David Simmons-Duffin.

One of the most popular constants of Nature that armchair physicists (especially numerologists) want to explain is the fine structure constant,
alpha = e2 / (4 π ε0 hbar c) = 1 / 137.0359997...
When the squared electric charge is divided by some universal constants (in the SI units), it becomes a mysterious dimensionless ratio that all civilizations in this Universe would agree upon, regardless of their choice of units (because there aren't any units in the ratio).

It's nice and you may try to explain the number by a simple formula except that you should know that the number has no reason to be given by a simple formula. Why? Because it is no longer quite fundamental, primarily for these two reasons:
  1. the electric charge is no longer a natural quantity in our unified theories
  2. the values of similar constants depend on the energy scales - they run
Electroweak reparametrization of the couplings

Concerning the first point, the electroweak theory has unified the electromagnetic interaction with the weak nuclear force (responsible for the beta decay) into a U(1) x SU(2) gauge theory. However, the original electromagnetic U(1) group is not the same thing as the first factor in the U(1) x SU(2) product. Instead, its electromagnetic U(1) generator is a combination of the U(1) generator "Y" (hypercharge) and the third, z-rotation generator of the "isospin" SU(2):
Q = Y/2 + T3.
That's the combination that keeps the Higgs vacuum condensate (which carries both Y and T_3, but no electric Q) unchanged. This invariance is needed for the vacuum to stay electrically neutral - and for the photons to remain massless (and for the corresponding electromagnetic to remain a long-range force).

It means that you should rewrite the electric charge "e" in terms of more fundamental couplings g and g' associated with the U(1) and SU(2) groups, respectively:
e = g g' / sqrt(g2 + g'2)
You shouldn't expect a simple formula for the fine-structure constant or for "e" because it is now expressed in terms of more fundamental couplings g,g' of the U(1) x SU(2) gauge theory in this way. If there are simple formulae, they're formulae for g,g', not "e".

Renormalization group: running couplings

However, as the second point suggests, the observed values of g,g' shouldn't be expected to be "simple", either. It's because they "run". If you calculate the quantum effects - the loop Feynman diagrams induced by the creation and annihilation of temporary "virtual particle pairs" in the vacuum - and the one-loop diagrams are the most important ones in a weakly coupled approximation -, you will find out that the "fine-structure constants" are not constants, as expected classically. Instead, they are linear functions of the logarithm of the energy scale.

In some sense, you may imagine that the refined "Coulomb's laws" should get an additional, "quantum" factor that slowly - kind of logarithmically - depends on the distance between the sources. All the details can be calculated in a straightforward way. You may link the extra logarithmic dependence on the energy to the properly cured logarithmic divergences of the one-loop diagrams.

And it is the behavior of the coupling constants at very high energies rather than the observed values at very low energies that has a chance to be fundamental and to be expressed by a simple formula! So it's not too reasonable to expect any simple formula for the observed, low-energy gauge coupling constants.

Looking at the three running couplings

The Standard Model (SM) is based on a gauge group composed out of three pieces: U(1) and SU(2) from the electroweak force and SU(3) from the strong force (Quantum Chromodynamics describing the confining forces between the quarks and gluons). Their fine-structure constants run according to the following picture:

The y-axis is the fine structure constant itself, i.e. "1/g^2" for the three relevant constants "g"; the value of y would be 137 for the electromagnetic U(1) if it were included in the graph. However, we only include the fundamental coupling, including the hypercharge U(1) group. Its U(1) coupling is actually rescaled in the right factor, sqrt(3/5), that would incorporate the U(1) naturally into an SU(5) group.

The x-axis is the base-ten logarithm of the exchanged momentum expressed in GeV/c. The left side of the picture corresponds to energies comparable to 100 GeV - the state-of-the-art collider energies. The right parts of both graphs describe much higher energies that are directly relevant for the unifying theories of Nature - i.e. for physics at very short, more fundamental distance scales.

You see that there are two graphs above. The left graph describes the Standard Model - the model including all particles that have been clearly seen plus one God particle. The right graph is the Minimal Supersymmetric Standard Model (MSSM) - which has the same well-known particles, five God particles, and the superpartners of all the previous ones.

The colors of the U(1), SU(2), SU(3) couplings are cyan, purple, yellow: note that the colors are sorted alphabetically. ;-)

Location of the three lines in the Standard Model

You may see that the cyan, U(1) curve is decreasing in both graphs. Because it corresponds to the inverse coupling, the corresponding interaction is actually getting stronger at higher energies. The effect, previously linked to one-loop Feynman diagrams, is called "screening". You may imagine that at shorter distances (higher energies), the electrons are "sharper", i.e. more charged, and their charges are diluted by gluing virtual positrons on top of them, an effect that becomes more important at longer distances and that weakens the low-energy strength of the interaction.

At extremely high energies, well behind the right end of the graphs above, the hypercharge U(1) fine-structure constant would go to zero and the coupling would therefore diverge. This pathological point is called the Landau pole. Unless new physical phenomena modify the behavior of the coupling (and the associated particles), the theory would become inconsistent. To say the least, the perturbative expansion would break down above the Landau pole energy scale. But it's likely that this inconsistency is also a problem nonperturbatively.

On the other hand, the purple SU(2) and yellow SU(3) lines in the Standard Model graph are increasing. That means that the interactions are getting weaker with increasing energy; this effect is known as antiscreening and may be explained by a heuristic argument in which you create ever bigger magnets which strengthen the interaction at longer distances in this case.

Antiscreening is particularly important for the colorful SU(3) group because it leads to the asymptotic freedom - the main feature of Quantum Chromodynamics (QCD) that brought the well-deserved Nobel prize to Gross, Wilczek, and Politzer. Quarks at very short distances - inside the protons and neutrons - move almost like free particles, an effect that had been known from the experiments that saw the "deep inelastic scattering" a few years before QCD was born.

The Standard Model shows three straight lines and three random lines have no reason to intersect at one point. Instead, you see that just like you would expect, there are three pairwise intersections, corresponding to the fine-structure constants between 40 and 45 and energy scales between 10^{13} and 10^{17} GeV. They kind of want to meet but as you may expect from generic three lines, they won't meet.

Fixing the botched intersection by the MSSM

The right picture, one of the Minimal Supersymmetric Standard Model, looks nicer.

First, you may notice that all the lines become "more decreasing". Because they're the fine-structure constants (i.e. inverted squared couplings), the actual couplings are actually "more quickly increasing with energy" - because the "screening" effect is larger due to the larger number of charged particle species (the superpartners and new Higgses and higgsinos).

The antiscreening can never be amplified by new particles because it only comes from the gauge bosons themselves and their number can't change; there is always one gauge boson for one generator of the gauge group. While the SU(3) coupling still remains decreasing with energy (asymptotic freedom), although the slope is reduced by 50% or so (and there's not much space for new particles unless you want to sacrifice asymptotic freedom), the SU(2) coupling has actually switched the sign relatively to the Standard Model. The SU(2) interaction is now getting stronger with the increasing energy, much like the U(1) coupling.

More importantly, you see that the three lines intersect nicely, within the error that can be identified with the experimental and theoretical errors. Things seem to work. The probability of their intersection that is correspondingly accurate (and is a priori distributed in a dozen of orders of magnitude) is a few percent. And it works. This match is not an indisputable proof of SUSY but it is surely a powerful hint - one that is surely enough to "annihilate" one negative argument against SUSY based on a 1% fine-tuning.

You may try to play the devil's advocate and discuss the possibilities.

The couplings at low energies are what they are - the left beginning of the three lines in the graph. You may extrapolate them to the right. At least the U(1) coupling will be getting stronger - the U(1) fine-structure constant will decrease with energy.

The slope of all the lines may be modified by the addition of new particles - and you may speculate about completely new kinds of physical phenomena (which are not known these days). But it is kind of natural to use Occam's razor and assume the minimum spectrum that is compatible with everything you know. In the Standard Model, the couplings simply won't meet. That's strange because it would make the short-distance physics disordered. Why should the short-distance physics look even more messed up than the observed long-distance physics?

In the supersymmetric theory, the couplings do meet. Moreover, the very high-energy scale around 10^{16} GeV, where they meet at a fine-structure constant close to alpha=1/24 or so (a number that may ultimately be explained by string theory, e.g. F-theory), is a natural scale for many other reasons. First, it is just slightly below the Planck scale where string theory mostly likely adds gravity to the unifying picture of all forces: so at slightly sub-Planckian scales, string theory produces its first viable field-theoretical approximation, a supersymmetric Grand Unified Theory.

(The lines shouldn't be trusted on the right side from the intersection: new particles exist and the running of all the constants is modified.)

Also, the grand unified scale around 10^{16} GeV is natural for the neutrino masses. Unprotected right-handed neutrinos are likely to acquire masses comparable to 10^{16} GeV. By the seesaw mechanism, with the electroweak scale being the geometric average (the center of the seesaw), the theory produces another, very low-energy scale slightly below 1 eV which is indeed (the right order of magnitude for) the typical mass scale of the observed neutrino masses (known from the oscillation experiments: at least the mass differences seem to be that big).

Note that 10^{16} GeV = 10^{25} eV and 300 GeV = 10^{11.5} eV which is exactly in between 10^{25} eV and 10^{-2} eV on the log scale.

Those things don't have to be right, after all. For example, there can be many more charged particles than those that are assumed in the SM or the MSSM. However, some collections of particles may naturally come in "full multiplets" of the grand unified group - such as SU(5), SO(10), or E_6. If that's so, the relative running of the three lines is not affected. If you add full GUT multiplets, the previous common intersection won't disappear (or won't appear) if it was there (or wasn't there) to start with. The intersection may just move to a different value of the fine-structure constant (i.e. down).

In the recent decades, many models (e.g. those with large or curved extra dimensions and/or with intersecting branes) have been studied where the unification of couplings - the intersection of the three lines - doesn't exist. It's good that they have been studied because they're completely new possibilities that had been previously overlooked or underestimated. And they may bring interesting, qualitatively new effects and explanations of the observed patterns of Nature. And to a limited extent, they have brought them to us.

However, when I look at all these models without any hype and outside the wind of the fashionable trends, I still think it is far more sensible and natural to believe that the gauge coupling unification at the grand unified energies is a genuine feature of Nature and it is real because the spectrum of charged particles up to these very high energies is related to the spectrum of the MSSM in a pretty simple way.


  1. Dear Lubos,
    You wrote:
    "It's nice and you may try to explain the number by a simple formula except that you should know that the number has no reason to be given by a simple formula. Why? Because it is no longer quite fundamental, primarily for these two reasons:
    1. the electric charge is no longer a natural quantity in our unified theories
    2. the values of similar constants depend on the energy scales - they run"

    So far it seems that no free stable particle exists with charge
    smaller than "e".
    There must be something fundamental in it.
    Does Standard model account for this or it predicts the possibility of existence of a free particle with fractional charge?

  2. Dear Stefan,

    the elementary electric charge as a notion is "fundamental" because it is the elementary electric charge. A kind of circular reasoning, right?

    But what I wrote - and demonstrated in detail - is that there is nothing fundamental about the value of "e" which means "sqrt(4.pi.alpha)" where "alpha" is the fine-structure constant, so there is no reason why it should be expressed by a simple formula.

    And I think that I wrote clearly why it's not fundamental - more precisely, why it is less fundamental than the values of "g" and "g'" of the U(1) x SU(2) couplings. The electromagnetic U(1) is just a diagonal group "mixed" from the two gauge groups - the latter are clearly more fundamental at the "deepest level" than their electromagnetic mixture.

    Moreover, these two couplings g,g' measured at low energies are less fundamental than their values at high-energies - because high-energy physics is fundamental and the low-energy physics is derived. And because they run, this difference is important.

    In this sense, the value of the fine-structure constant is analogous to the density of liquid hydrogen or any other complex quantity of this kind. This liquid looks fundamental but the calculation to find its density is complex - it's an "environmental" problem that depends on many detailed features of the arrangement. It's the most elementary particles whose parameters are fundamental and may be constrained by concise formulae - everything else is derived and messy.

    Apologies but I don't see how this can be a subject for polemics. What I wrote was not just a statement that can be controversial: I also wrote a detailed explanation - a sketched proof - of the assertion that allows me to assume that everyone who doesn't get the point after having read the text is unfortunately incapable to understand basic physics concepts. There is no "democracy" between my statements and their negations. My statements are important insights about the electroweak unification and running couplings; their denial is just a symptom of a misunderstanding of physics.

    Concerning your question, in the Standard Model, the minimum nonzero electric charge of particles is +-e/3 which is the charge of the down-type quarks or their anti-quarks. They can't be isolated - so that the isolated objects which are electrically neutral inevitably have charges that are integer multiples of "e".

    However, at very short distances, you could say that the quarks are "free" (due to asymptotic freedom) and their charges are multiples of e/3.

    In some stringy models beyond the Standard Model, there can also exist other fractional charges such as e/5 or e/7 etc. They correspond to topologically nontrivial wrappings of strings (and, maybe, branes in some more modern vacua). They're extremely heavy, their discovery would be a pretty much clear proof of string theory, but I think that this detection won't occur, and the stringy vacua with these exotic particles are unlikely to be relevant for the Universe, anyway.

    When you ask whether the Standard Model accounts for "this", I didn't understand what was exactly this. If it were the statement that there should be a simple formula for alpha because it is fundamental, the Standard Model, on the contrary, accounts for your statement's being untrue, and I have shown the proof why it is so in detail.

    Best wishes

  3. The problem is that from the fact that the value of alpha is not fundamental you seem to imply that it can not be a quantity calculable only from considerations at low energy. But nothing in the running down from GUT scale tell us where should the electroweak symmetry break. And it is the breaking, the thing that determines the value of alpha.

    The peculiar value of the yukawa top coupling (and, btw, the smallness of all the others) seem to indicate that the values of the yukawas are related to the electroweak symmetry breaking scale, not to the GUT scale.

  4. Come on, Alejandro, you seem to misunderstand everything here.

    When I say alpha is not fundamental, I mean that it is a derived quantity that can be calculated, using already known things, from more fundamental things, namely g,g', and the latter should be calculated by running their more fundamental high-scale values down to low energies.

    You: "But nothing in the running down from GUT scale tell us where should the electroweak symmetry break."

    Of course that it does. Everything about physics is encoded in the high-scale theory. The most relevant things are the coefficients of the Higgs potential that also exist at the GUT scale. They must be carefully run to low energy to find out what the Higgs vev etc. will be.

    You: "The peculiar value of the yukawa top coupling (and, btw, the smallness of all the others) seem to indicate that the values of the yukawas are related to the electroweak symmetry breaking scale, not to the GUT scale."

    Nope. Every parameter that runs - and the Yukawa coupling runs - has a more natural value at the high-energy (GUT) scales than at low energies.

    You seem to "disagree" with the very fundamental point that the high-energy physics is fundamental why the low-energy physics is derived from it.

  5. Perhaps we are confusing naturality and calculability. I agree that the high-energy physics is fundamental and the low-energy physics is derived. What does not follow is the implication that every calculable low-energy expression needs to involve the values of high-energy physics. Normally the effects of phase transitions, fixed points, symmetry breakings etc help to hide the high energy and make the low energy relevant.

    Actually, your argument should run in reverse: because the electromagnetic coupling is not fundamental, we could hope it to be calculable in the low energy regime, without involving the high energy values.

  6. Dear Alejandro,

    what you write sounds very bizarre. The reason why these two things are being "confused" is that they are the very same thing, a fact that you seem to be confused about.

    Some low-energy quantities may be independent of the details of high-energy physics - and of the parameters relevant in high-energy physics. There can exist some universal behavior whose properties (some properties) may be calculable without the knowledge of the details of high-energy physics.

    However, it is completely manifest that it is not the case of the fine-structure constant because this constant clearly *does* depend on the U(1) and SU(2) couplings at the high energy scales. These couplings have to be run down to low energies and reparameterized from g,g' to e along the formula that I wrote and that everyone with the knowledge of the electroweak theory knows.

    So I can't understand how you can argue that the number is independent of those high-energy parameters if we can explicitly write what the dependence is.

    You could conjecture that there are two ways to calculate it - one that starts from high energies, makes RG and EW symmetry breaking. And one that avoids everything and leads to the same result. However, this is a kind of conspiracy theory. The EW symmetry breaking and low-energy manifestations of the U(1) x SU(2) group *is* messy, so there can't be any dual picture that is simple.

    Best wishes

  7. Dear Lubos,

    Going to the particular case of the fine structure constant, allow me to recall a triviality: that it also depends of the point where the electroweak symmetry breaks.

    So from the low energy point of view, alpha depends of M_W, M_Z and G_F. We do not know what is the exact mechanism producing the breaking of symmetry. If the breaking of symmetry implies any extra relationship between M_W, M_Z and G_F, it implies also a relationship to predict alpha.

  8. Dear Alejandro,

    why are you inventing this verbal and meaningless fog? Clearly, whatever you say is completely irrelevant, but let me analyze your sentences, anyway.

    First, whether alpha depends on "something" depends on what are the other things that it can depend upon. For example, one can tautologically say that alpha depends on alpha and nothing else. :-)

    If you want to express alpha in terms of high-scale quantities, of course that the calculation will need to know all important transitions in the RG flow, including the point where the EW symmetry breaks. That was my second point. Low-energy alpha is not fundamental because it runs, and it is its high-scale counterparts that are fundamental.

    So your claim in the first paragraph is either ill-defined or it just confirms that the running is important.

    Now, you say that alpha depends on M_W, M_Z, and G_F. This is an extremely bizarre parameterization, but let's check it by rewriting these things in terms of more fundamental parameters, to see the actual relations.

    G_F is

    sqrt(2)/8 g^2 / M_W^2

    so if you know M_W and G_F, it's the same as knowing M_W and g.

    Knowing M_W and M_Z is equivalent to knowing M_W and the Weinberg angle thetaW because cos(thetaW) = M_W/M_Z.

    So your three numbers are equivalent to M_W, theta_W, and g. Indeed, with these three things, you may also calculate g' (of the U(1) coupling) and therefore e, alpha. In fact, you clearly don't need M_W because it is dimensionful and it must therefore drop out.

    So your convoluted second paragraph essentially says that the same old thing that e=g.g'/sqrt(g^2+g'^2).

    Why did you deliberately obscure this thing? If you want to claim that G_F is more fundamental than the gauge couplings, then you are totally wrong. G_F and masses of some particular particles are effect of more fundamental physics. The more fundamental physics is about gauge theory which is governed by the dimensionless couplings - and the scale is determined by the Higgs vev. Everything else - especially the other masses - are derived from that.

    Best wishes

  9. The point is that formula e=g.g'/sqrt(g^2+g'^2) also hides a fact: that the running of g and g' (and e) is very different when the energy becomes a lot greater or a lot smaller than G_F. A calculation of the infrared value of alpha needs to give g, g' and G_F.
    We do not know the actual mechanism that produces G_F. Imagine, for instance, the unrealistic mechanism "let G_F be the scale such that the low energy alpha becomes 1/193.3455". Perhaps there is a mechanism "let G_F be the scale such that the yukawa top is 1 at this scale". Or perhaps there is even a mechanism "let G_F be the scale such that alpha runs down to 1/137 from it".

  10. Dear Alejandro, the dependence of the beta-functions (running rate) on the energy scale is not being "hidden": it's the very whole point why the running is important.

    If the dependence were trivial and independent, the running could just produce a simple modification of the couplings.

    However, your parameterization by G_F is totally silly. The reason why g of SU(2) changes its running at mass scale M is that there is some particle of mass M that contributes to running above M, but not below M.

    For example, the running of alpha almost totally stops below the mass of the electron, i.e. the lightest electrically charged particle. In the same way, the beta functions "jump" whenever a particle with nonzero charges under the same coupling disappears from the spectrum.

    This has nothing whatsoever to do with G_F, the weak interaction decay constant, except that G_F is close to a power of the W mass or Z mass. But W and Z are just two particles that influence the beta functions. There are many other particles that do.

    Your pretending that it is "G_F" that matters, or even "only G_F" that matters, shows that you have never actually calculated anything about the running coupling constants. You don't know how it works.