# Can we calculate the coupling constants?

1. Jan 31, 2016

### friend

I know that QFT puts in the coupling constants by hand, based on experiment. And many of the coupling constants are suppose to become equal at the grand unification energy. But I wonder if there is any principle that would allow us to calculate the present day coupling constants based the GUT constants (of 1?)? I've seen the graph that shows the coupling constants meeting at high energies. Is that graph based on calculations? Or is it based on data? Or is it just a schematic?

2. Feb 1, 2016

### Orodruin

Staff Emeritus
The behavior of the coupling constants running with energy is one of the fundamental predictions of QFT. Nobody has ever made an experiment at the GUT scale.

3. Feb 1, 2016

### friend

It sounds like it's a calculation. If so, can we calculate backwards to low energy values if we were to assume that the constants do meet at the GUT scale? Or have we already done this and found we need something like supersymmetry to the running of the constants to line up.

4. Feb 1, 2016

### Orodruin

Staff Emeritus
This has already been done. You need something extra for the couplings to unify. Whether this extra is SUSY or something else remains to be seen.

5. Feb 1, 2016

### friend

Thank you so much. A little more detail would be nice. What techniques were used? Was this a renormalization technique up to various energies? Was it done by taking into account some screening effect? Was this starting as low energy values and working towards higher energy levels? Can this technique be used to start from some assumed GUT level and work to the present low energy values? Thanks.

6. Feb 1, 2016

### Orodruin

Staff Emeritus
It is called renormalisation group running. You should be able to find it in any QFT textbook.

7. Feb 1, 2016

### ohwilleke

First, to clear up a misconception, when the coupling constants, which are dimensionless numbers, unify, for example, in the Minimal Supersymmetric Model at the GUT scale, the value of those constants is not generally equal to 1. Instead, in the MSSM, for example, the value of the gauge coupling constants at the GUT scale where they unify is on the order of 0.04. See, e.g., http://backreaction.blogspot.com/2007/12/running-coupling-constants.html

Also, while the running of the coupling constants with energy scale looks log-linear in typical chart of their running with energy scale, the strong force coupling constant peaks at a value near unity at a very low energy scale (corresponding to a typical confined hadron) below which and above which it gets weaker. There is no scale at which the other two coupling constants every attain numerical values close to 1 in any mainstream BSM theory, or in the SM, and there is no high energy scale at which the strong force coupling constant approaches 1.

The ordinary approach has been to use the measured value of the coupling constants at particular energy levels (easily available, for example, from the Particle Data Group) and then apply both the Standard Model beta functions of each of these three constants, and to compare those results to the beyond the Standard Model beta functions. The Standard Model beta functions have been confirmed up to certain scales, but one of the things that the LHC is looking at is whether or not the experimental data will match or deviated from the Standard Model beta functions at higher energy scales. (Incidentally, the physical constants that appear in the beta functions can be computed exactly and generally consist of integer or rational number values with the odd pi thrown in, from the Standard Model or beyond the Standard Model equations that are involved and the mathematics of the renormalization scheme adopted - they are not experimentally determined constants).

BSM theories with gauge coupling unification usually assume that there is a kink in the beta functions at a certain energy scale where the BSM effects kick in, below which the SM beta functions apply to the level of experimental measurement accuracy (usually somewhere in the low single digit TeVs).

The Wikipedia article on beta functions explains the basics of how the running of coupling constants is calculated and gives the one loop approximations of those functions for QED and QCD. https://en.wikipedia.org/wiki/Beta_function_(physics)

An important thing to note about the relevant experiments is that coupling constant values which are measurably distinct at low energy scales tend to converge at higher energy scales. For example, the QCD coupling constant measured at the Z boson mass energy has a PDG value of about 0.1184 world average (a dimensionless number). At the Z boson mass energy scale it is very feasible to distinguish between a value of .1170 and a value of .1190. But, if you assume that .1170 is the correct value and use the beta function to determine the value of that coupling constant at say, 900 GeV, and then repeat the procedure assuming that .1190 is the correct value at 900 GeV, it is very hard to distinguish experimentally the two respective high energy values of the coupling constants because the values are numerically much closer together and the precision of the high energy measurement is lower.

More generally, one has to know the identical numerical value shared by the three coupling constants to a very high precision at the GUT scale, in order to make an accurate prediction using the beta functions of the three respective coupling constants, to calculate their values to any useful level of precision at the scales that we can measure directly in experiments.

It is trivial to get two coupling constants to converge at high enough energies, but it is not trivial to get three to do so at the same energy scale. One the other hand, a low single digit percentage tweak to one or two beta functions would be sufficient to make them converge. For example, there is good reason to believe that considering quantum gravity effects at very high energies could tweak the relevant beta functions, possibly even making the Standard Model beta functions converge.

It is also possible to start from a given energy scale at which a convergence is assumed and work the other direction back to low energies, if you know the strength of the coupling constants at the GUT scale. Generally, what would make more sense to do would be to set the shared coupling constant value at a particular GUT scale energy level using the low energy measurement of the electromagnetic coupling constant (since it is known most precisely) using the electromagnetic force beta function in the theory you are examining, fix the other two coupling constants to that value, and then use their beta functions to calculate back down to the measurable prediction.

Measuring beta functions at low energies empirically is a very useful way to explore very high energy physics because beta functions are sensitive to a wide array of global components of the theory in much the same way, for example, as the magnetic moment of the muon is sensitive to a wide array of global components of the theory even at high energies. You can't dramatically tweak the rest of the model without impacting, for example, the way that the electomagnetic force coupling constant runs with energy scales that are within experimental reach. For example, the running of the QCD constant is sensitive to the number of quark flavors that are present in the theory.

Last edited: Feb 1, 2016
8. Feb 2, 2016

### friend

Thank you for that. This ought to give me a hobby for a while.

9. Feb 7, 2016