Monday, December 10, 2007

Running Coupling Constants


[Figure: D.I. Kazakov, hep-ph/0012288 p 12 .]


Coupling constants in the quantum field theories of the Standard Model (SM) are not constant. The couplings, which set the strength for the interactions, change their value if one probes smaller distances with higher energies. This is due to contributions of virtual particles that cause a 'running' of the coupling with the energy scale. This energy scale is also referred to as the 'sliding scale'. If one evaluates the necessary Feynman diagrams to compute this effect, it turns out that the couplings run logarithmically with the sliding scale, and their slope depends on the particle content.

The left side of the above plot shows the inverse of the three SM couplings ╬▒i as a function of the sliding scale Q (in GeV) [see comment]. The thickness of the lines depicts the experimental error (LEP data '91). The scale on the x-axis is logarithmic, such that the curves become straight lines. This running of the coupling constants has been experimentally confirmed in the accessible energy range, but the more interesting thing here is that one can extrapolate the curves far beyond where we can test them experimentally. One sees then that these couplings form a triangle somewhere around 1016 GeV.

The plot to the right shows the running of the gauge couplings within the Minimal Supersymmetric extension of the Standard Model (MSSM). Since the particle content with Supersymmetry (SUSY) is different, the slope of the curves changes. Interestingly, the result is that the gauge couplings meet almost exactly (within the errorbars) in one point, somewhere around 1016 GeV, usually referred to as the GUT scale (which isn't too far off the Planck scale).

The above depicted fit of the curves depends on the scale where SUSY is (un)broken. Below this energy, the running is according to the SM, and only changes above the SUSY breaking scale. This is why in the plot to the right the curves have a kink and change slope around a TeV, which was assumed to be the SUSY breaking scale.

This calculation has first been done by Amaldi, de Boer and F├╝rstenau in 1991 (Phys. Lett. B 260 447-455, 1991), and this result is to be considered one of the most compelling arguments for SUSY.



This post is part of our 2007 advent calendar A Plottl A Day.

53 comments:

piscator said...

This plot should always have a health warning. The three lines shown unifying with supersymmetry are *not*, repeat are *not* those of the gauge couplings of the Standard Model. Specifically, the U(1) coupling is GUT-normalised. The ordinary U(1)_Y gauge coupling will miss SU(3) and SU(2) by a factor 5/3. In general, there is no unique U(1) gauge coupling.

What the plot is evidence for is not the MSSM in general, but specifically SUSY-GUTs.

Bee said...

yes. thanks for your comment.

alex said...

That's absolutely amazing! After running over 15 orders of magnitude in the energy scale! We are so used to seeing the second plot these days but I'd imagine the person who plotted it for the first time must have been completely ecstatic. Of course, there are other good reasons for TeV scale SUSY, i.e. REWSB caused by the large top Yukawa, a natural dark matter candidate, stablization of the hierarchy, etc but the precision gauge coupling unification is just so darn impressive. I can't believe some people still reject this and the other hints as evidence that the MSSM embedded in a SUSY GUT is the real deal.

Low Math, Meekly Interacting said...

I've been horrendously busy of late, but have taken the time to read the latest string of excellent (and I mean excellent) posts. Thanks so much!

Arun said...

What would it take to check experimentally whether 1/Alpha_2 runs with a negative rather than positive slope?

Bee said...

well, as piscator mentioned above the alpha_i are are not actually the couplings of the SM gauge groups. alpha_2 is related to the quantities we know and like as alpha_2 = alpha/sin^2(theta) where alpha is the fine structure constant (runs) and theta is the weak mixing angle. which is supposed to run, but the last time I heard about it, the data wasn't too convincing.

Bee said...

Hi Alex:

Well, you know, despite me writing this post I am not actually so impressed by the above. Probably exactly because as you say its an extrapolation over 15 orders of magnitude. I could imagine all kind of things happening till we reach that energy scale. Call it naive but if it was a unification I'd kind of expect the couplings to run together towards some attractor, and not just meet in one point. Best,

B.

Eric said...

"Call it naive but if it was a unification I'd kind of expect the couplings to run together towards some attractor, and not just meet in one point."

Yes, this is very naive. These coupling constants don't exist above the unification energy as they are replaced by the gauge coupling of the unified group. It doesn't make any sense to keep running the SM gauge couplings above the unification scale. One should run the gauge coupling of the unified group above this energy.

Bee said...

Yes, Eric, thanks. So, then these curves run together to one point and then they have a kink and proceed on one line all together? That's what I don't like about it. Best,

B.

Eric said...

Bee,
At the unification scale, there is a discontinuity where the SM gauge groups with three independent coupling constants get replaced by a single gauge group with a single gauge coupling. It's not that the SM gauge couplings run as a line after the unification scale. Above this energy, the SU(3), SU(2)_L, and U(1)_Y gauge coupling constants don't exist because the gauge group is no longer SU(3)x U(2)_L x U(1)_Y. The unified gauge symmetry, e.g. SU(5), which is manifest above the unification scale has it's own gauge coupling and there are no others.

Bee said...

Eric: there is a discontinuity where the SM gauge groups with three independent coupling constants get replaced by a single gauge group with a single gauge coupling

Yes, Eric, thanks again. This is exactly what I don't like about it. I didn't say it is wrong, or doesn't work, or can't be the way nature is organized, I just said it doesn't impress me much. Best,

B.

alex said...

"That's what I don't like about it"
Bee, could you please elaborate what exactly you don't like? The fact that they meet at a point and then run as a single unified coupling? As opposed to what?

I think one should really look at this backwards from high to low energy scale, i.e before the GUT is broken you have a single coupling. Then at around 10^16 GeV the GUT symmetry is spontaneously broken and the couplings start running separately. What's your alternative?

Eric said...

Bee,
The discontinuity is no different than what happens in condensed matter when a material undergoes a phase change. In fact, this is exactly what's happening at the unification scale.

Bee said...

Hi Alex:

Well, if I had an alternative, I'd go post a paper with my GUT model :-) What I am trying to say is that I would like to see the different couplings run together towards one coupling and not just the above extrapolation until the curves meet, and then, bang, the symmetry is unbroken and we have the GUT.

Hi Eric:

Yes. The necessity for 3 different discontinuities doesn't make me like it better though.

Best,

B.

alex said...

"...until the curves meet, and then, bang, the symmetry is unbroken and we have the GUT."
There is no "bang", it's just that at around 10^16 GeV the additional degrees of freedom, like the Higgs triplets, for instance, start to contribute into the running modifying the beta-function coeffitients and force the couplings to run together as a single GUT coupling.

Mike said...

I have a question about this graph I've been meaning to ask for some time.

As I understand it, the logarithmic form of the running couplings follows from only considering 1-loop effects. Why should I believe that over 15 orders of magnitude in the scale factor, higher order terms will be of no importance?

Thanks.

Bee said...

Hi Alex:

it's just that at around 10^16 GeV the additional degrees of freedom, like the Higgs triplets, for instance, start to contribute into the running modifying the beta-function coeffitients and force the couplings to run together

It's just that I don't find that scenario particularly compelling. You don't have to share my opinion, but maybe you could just accept it. It seems to be rather useless to repeat myself. Best,

B.

Bee said...

Hi Mike:

Because the coupling itself doesn't change significantly. What you actually do to compute the running is to sum up an infinite series of loops that contributes to the propagator. Best,

B.

Bee said...

One can do the higher order contributions though. It's more important in some scenarios than in others. I am sure one finds that somewhere on the arxiv.

Mike said...

Thanks Bee. When I'm busy not being busy, I'll have to go digging.

Bee said...

:-) I can really recommend Weinberg's QFT book, volume 2, chapter 18.

alex said...

"As I understand it, the logarithmic form of the running couplings follows from only considering 1-loop effects. Why should I believe that over 15 orders of magnitude in the scale factor, higher order terms will be of no importance?"
Two loop effects and threshold corrections have certainly been accounted for and there are lots of papers on this.

Bee, I still fail to see what your objection is. There is a kink at around TeV when the superpartner contributions kick in. So, at the GUT scale you also get extra light states which complete the multiplets and make the couplings run together.

What's your objection precisely?

Bee said...

Hi Alex:

I have no 'objection'. I didn't say there is anything wrong with it, and if you think that's the way nature works, fine. I will believe in susy if it is found at the LHC. Best,

B.

Plato said...

Bee,

Hopefully this comes forward from the layman properly.

Alex said:"I think one should really look at this backwards from high to low energy scale, i.e before the GUT is broken you have a single coupling. Then at around 10^16 GeV the GUT symmetry is spontaneously broken and the couplings start running separately. What's your alternative?

It is like asking a cosmologist whether there is ever a before to the current universe? Qui! Non? :)

Pierre Ramond has a nice plot I showed earlier.

That Eric mentioned the condensed matter theorist situation, is one Robert Laughlin refers too often as to the "building blocks of the universe" being what ever you want them to be.

So you push back perspective to a time in the universe's expression(this a has been progressive) and your response on Zajc(Microseconds) is a case in point.

So Alex asks again, still not convinced?

Plato said...

Bee said,I have no 'objection'. I didn't say there is anything wrong with it, and if you think that's the way nature works, fine. I will believe in susy if it is found at the LHC.

Robert Lauglin:Likewise, if the very fabric of the Universe is in a quantum-critical state, then the stuff that underlies reality is totally irrelevant-it could be anything, says Laughlin. Even if the string theorists show that strings can give rise to the matter and natural laws we know, they won't have proved that strings are the answer-merely one of the infinite number of possible answers. It could as well be pool balls or Lego bricks or drunk sergeant majors.


Just seen your response.

We know that the experimental process is limited while in a cosmological view the energy values are still much higher.Constant referrals to GZK just doesn't cut it :)

The very act of pushing back these view on the cosmos do not relegate them to insignificant because of what you see in the experimental process and are waiting for.

I was looking for a animation about "the sand and pebble on the beach for you" so you get the picture.

Bee said...

Hi Plato:

I am not actually sure what you want to convince me of. You are probably not going to convince me to start working on SUSY models. If I look at the curves backwards I see that one introduces two symmetry breaking scales which results in three curves getting kinks and reaching three values they have today within the required experimental precision.

I hope 'Lego' theory doesn't become fashionable ;-) Best,

B.

alex said...

I forgot to mention another impressive prediction from SUSY GUT's, namely the value of the Weinberg angle. It is completely fixed by group theory at the GUT scale and when one runs it down to the electroweak scale it agrees with the experiment to 1% accuracy. The threshold effects at the GUT scale (when included) fix even this tiny discrepancy.

Moreover, by the seesaw mechanism, the electroweak scale comes out to be the geometric mean of the GUT scale (predicted by the running) and the neutrino mass scale, which puts the neutrino mass in O(1eV) range - consistent with experimental bounds.

So, it's not just that second graph that's appealing about SUSY GUT models.

Bee said...

the value of the Weinberg angle

Is this specific to susy GUTs?

alex said...

"Is this specific to susy GUTs?"
The GUT scale value is fixed by group theory. I think the value for SU(5) is sin^2(\theta)=3/8, if I recall correctly.

The presence of superpartners in the spectrum affects the running of sin^2(\theta) and hence, its value at the electroweak scale. The experimental value measured at the EW scale is sin^2(\theta)~0.2312.

So, for an SU(5) non-SUSY GUT the predicted value for sin^2(\theta) at the electroweak scale is ~0.206 whereas for the susy GUT one gets ~0.23 - an impressive agreement with the experimental value.

Bee said...

Hi Alex:

I forgot to mention another impressive prediction from SUSY GUT's, namely the value of the Weinberg angle.

In how far is this actually another prediction? Isn't 3/5 tan^2 theta just alpha_1/alpha_2?

Best,

B.

PS: Completely off topic, I just heard an advertisement on the radio for the 'four course Italian fest', announced with appropriate Italian accent. What I thought I head was however 'something for causality fest', I think I'm working too much
:-)

alex said...

"Isn't 3/5 tan^2 theta just alpha_1/alpha_2?"

Yes, it is.

"In how far is this actually another prediction?"

So, in order to make a "real prediction" from a susy GUT one would have to find a parameter that could be directly computed in the GUT theory. Since we cannot compute the value of alpha_GUT from first principles yet, we can instead reliably compute the ratio of the SU(2) and U(1) gauge couplings. Then we run them DOWN to the EW scale and get a very robust prediction which agrees with the experiment. Note that in this high scale to low scale running the only "input" was the MSSM spectrum.

Now, this is quite different from the way the second plot in your post is obtained. In that case, we take experimentally measured values of alpha_1, alpha_2 and alpha_3 and run them UP to high scale and observe that they unify to a high degree of accuracy.
So this is nice but suppose they did not unify but you still had the match for the Weinberg angle.
That would mean that you don't have just the MSSM all the way to the GUT scale and you would need to unclude some extra multiplets to fix the unification.

So do you see how one feature, i.e. the unification of couplings is not necessarily correlated with the other (the correct ratio of alpha_1/alpha_2)?

Now, bear in mind that the MSSM was not invented to deal with the gauge coupling unification and fixing the prediction for the Weinberg angle from GUTs. These came as very nice surprises.

alex said...

"Then we run them DOWN..."
Oops, I meant run it (sin^2\theta) down...

constantly on the run said...

Thankyou for explaining the "Running Coupling Constants" to the masses.

Now please, explain the masses.

Thomas Larsson said...

Moreover, by the seesaw mechanism, the electroweak scale comes out to be the geometric mean of the GUT scale (predicted by the running) and the neutrino mass scale, which puts the neutrino mass in O(1eV) range - consistent with experimental bounds.

Aren't neutrino masses rather in the 10^-3 - 10^-2 eV range, so the EW scale is the geometric mean between the Planck and neutrino scales?

Bee said...

Hi Thomas:

I think you are talking about the differences of the neutrino masses, not the absolute masses.

Hi Alex:

So do you see how one feature, i.e. the unification of couplings is not necessarily correlated with the other (the correct ratio of alpha_1/alpha_2)?

Actually, no. You are saying one could have theta being correct without having alpha_1/alpha_2 correct. what I am saying is, if you have alpha_1/alpha_2 correct, then theta follows, so in how far is this actually 'another impressive prediction'.

Then we run them DOWN to the EW scale and get a very robust prediction which agrees with the experiment.

Well, if the angle runs when you go DOWN it seems to me its value depends on how far you let it run. Since the GUT scale is an input parameter, can't you just shift the start point of the running around such that at the EW scale the value comes out right?

Best,

B.

alex said...

"You are saying one could have theta being correct without having alpha_1/alpha_2 correct."

No, I was saying that sin^2(\theta) is fixed by the embedding of the U(1) into the SU(5) and does not care if ALL THREE couplings unify. You can get alpha_1 and alpha_2 to cross at a point and get the right value of sin^2(\theta) WITHOUT alpha_3 crossing at the same point.
Just add some extra multiplets that are only charged under SU(3) and you can screw up the unification of ALL THREE couplings without changing sin^2(\theta).

alex said...

"Well, if the angle runs when you go DOWN it seems to me its value depends on how far you let it run. Since the GUT scale is an input parameter, can't you just shift the start point of the running around such that at the EW scale the value comes out right?"

Sure, so you choose the GUT scale to get the right value for sin^2(\theta) and get agreement. But then you also take the experimental values of ALL THREE couplings at the EW scale and run them UP to see if they ALL actually unify at the SAME input GUT scale you had chosen to get the agreement for the Weinberg angle.

Do you see that these two tests are different?

alex said...

Oh, and you can imagine another scenario. Suppose you add extra matter in complete GUT multiplets to the MSSM somewhere in between the EW and the GUT scale. In that case you still have the unification of all three couplings but the Weinberg angle at the GUT scale would not match the value 3/8. So, having the MSSM only spectrum all the way to the GUT scale is crucial in getting the agreement.

Bee said...

Hi Alex:

Thanks for agreeing that it's fixing the GUT scale that gives the right Weinberg angle. Regarding the question of whether it is 'another prediction'. It seems I have made myself somewhat unclear so let me try it again.

Alex: So do you see how one feature, i.e. the unification of couplings is not necessarily correlated with the other (the correct ratio of alpha_1/alpha_2)?

You can get alpha_1 and alpha_2 to cross at a point and get the right value of sin^2(\theta) WITHOUT alpha_3 crossing at the same point.
Just add some extra multiplets [...]


That was not my question. What I said was if alpha_1 and alpha_2 have the correct, measured values at EW scale, and tan^2 theta is a ratio of both, then the value of theta at the EW scale is not another prediction, but a consequence of the previous ones. If you are saying you can have tan^2 theta correct without having the correct ratio of alpha_2 to alpha_2, or alpha_3 not crossing at the same point or other constructions, that doesn't make it better. I.e. what I am saying is B follows from A and you are saying no, A does not follow from B. Best,

B.

Bee said...

Oh, and you can imagine another scenario. Suppose you add extra matter in complete GUT multiplets to the MSSM somewhere

I was not talking about other scenarios. I am pretty sure if one adds further parameters, on can fit further parameters independently.

Bee said...

thinking about it again, it seems you are actually saying C does not follow from B. Translate A -> ratio of a_1/a_2, B -> sin^2 theta, C -> unification of couplings.

alex said...

"I.e. what I am saying is B follows from A"

No is does not necessarly follow. You can have unification and not get the agreement for the Weinberg angle. It is only when you have the MSSM spectrum ONLY all the way to the GUT scale do you get the agreement with sin^2(\theta)=3/8.

Bee said...

No is does not necessarly follow. You can have unification and not get the agreement for the Weinberg angle.

But I never said that. What I am saying all the time is if a_1/a_2 has the correct value at the EW scale then doesn't the Weinberg angle follow from that, so in how far is this 'another impressive prediction' (besides not being a prediction that is). I am not interested in whether you can have a unification model that does not reproduce measured data. Best,

B.

alex said...

"What I am saying all the time is if a_1/a_2 has the correct value at the EW scale then doesn't the Weinberg angle follow from that, so in how far is this 'another impressive prediction' (besides not being a prediction that is)."

Computing the Weinberg angle at the EW scale tells you nothin illiminating about the underlying GUT theory. In order to compare it with the value predicted by the GUT you have to run it up to the GUT scale. The unification of the couplings alone does not guarantee that if you run the ratio up to the GUT scale you'll get a match with the prediction sin^2(\theta)=3/8. This works ONLY if the MSSM spectrum enters the running and nothing else.

alex said...

I said: "The unification of the couplings alone does not guarantee that if you run the ratio up to the GUT scale you'll get a match with the prediction sin^2(\theta)=3/8."
Ok, now that I've thought about it carefully, I have to admit that the above statement is wrong. You are right that as long as the couplings unify you always get the match for the Weinberg angle, no matter at what scale and at what value of the coupling unification takes place.

Sorry about the long exchange. I do not work on model building myself but I'm used to seeing talks where the Weinberg angle prediction is displayed as a separate statement.

Bee said...

Hi Alex:

Thanks for the interesting discussion, I certainly learned something! Best,

B.

constantly on the run said...

constantly on the run said...

`Thankyou for explaining the "Running Coupling Constants" to the masses.

Now please, explain the masses.'

Any ideas why the particles have the masses they do?

Bee said...

plenty, but so far none of them worked.

a quantum diaries survivor said...

Bee, please explain to me one thing... Concealed down this thread, I can express my ignorance: why is it a value to have a single scale being where the curves meet, rather than two different unification scales, whereby two curves meet, then follow at another slope to meet with the third, as E increases ?

PS I do not buy the argument as evidence for anything other than a lot of work by theorists trying to pay their bills.

Cheers,
T.

Bee said...

Hi Survivor :-) You're asking the wrong person. Concealed in the above comments you might find my lack of enthusiasm about this result. Though I have to admit the plot kind of looks nice. However, here as in other cases I wonder whether our built in sense for beauty and elegance is a good guide. Best,

B.

Anonymous said...

That could have been a possibility Survivor, but there would have been absolutely no way you would know what the new slope of the united two curves would be so as to hit the third absent knowledge about the intermediate scale physics. Not to mention its far less elegant.

No the real miracle imo isn't that the slopes hit a priori, its that the fit improves quite noticeably from the SM case. Of all the drastic things that supersymmetry naively could do, its completely nontrivial that it leads to a better fit after so many orders of magnitude and so many extra degrees of freedom.

Warren D Smith said...

"this result is to be considered one of the most
compelling arguments for SUSY."

--If this is really one of the most compelling SUSY arguments, that is utterly pathetic!!
(1) As piscator points out, the 3-way meet is bullshit because it is based on a redefinition intentionally designed to make it happen.
(2) As the post points out it depends on a complete guess about the value of the SUSY symbreak energy, and if that guess were adjusted, we'd lose or regain the 3-way meet. When Maxwell (or whoever) noticed the speed of light agreed numerically with prediction from magnetic & electric permeability constants, THAT was an impressive agreement. If however
that result had been "they might agree, or they might not, it depends on an adjustable and wholy-guessed parameter called 'the phase of the moon'"
then this result would have been 100% unimpressive!
(3) SO WHAT? Let's suppose SUSY really
predicts the 3 curves all exactly meet, and without caveats 1 & 2.
Why should this impress me? If there were some microscopic reason such as superstrings, that they ought to meet, then that's one thing. If however this is just a complete coincidence
with no underlying reason it should be so (it just happens to be so), that, it seems to me, is more of an indictment of SUSY than support for it. It indicates SUSY is clearly an incomplete
explanation of Nature.

So... what!?!?! Is this really "one of the most convincing arguments for SUSY? This really is the best you physicists have got? Because it seems pathetic to me.

--Warren D. Smith
warren.wds AT gmail.com


Warren D Smith said...

[continuing previous comment/question]

When you originally posted in 2007, it may have been reasonable to postulate as you did that the SUSY symbreak scale was 1 TeV.
But it is now 2013. We have the LHC now which is designed to run at 14 TeV. In view of this is it still tenable to contend the SUSY symbreak scale is 1 TeV? Or must we now contend it is >14 TeV? And if so, does that destroy the 3-way meet?