A Is the weak interaction asymptotically free?

[HomogenousCow]

Is the $SU(2)_L$ part of the SM asymptotically free like a typical non-Abelian gauge theory? I've been made to understand that confinement does not occur for $SU(2)_L$ because the spontaneous symmetry breaking scale is above the confinement scale, however I can't find any information on whether this affects asymptotic freedom.

 

[malawi_glenn]

The sign of the beta function for SU(N) (unbroken) with n_f fermions is determined of the expression $\dfrac{-11N}{6} + \dfrac{n_f}{3}$, note that this is for fermions in the fundamental representation of SU(N). With N = 2 we get that $n_f < 11$ then SU(2) gauge theory is asymptotical free

But this is would be a strange theory, the gauge-bosons would be massless, it would not be the weak-interaction we know of.

Anyway, back to the standard model.

The weak force indeed becomes weaker at shorter distances (higher energy scales) [note - the vertical axis is 1/strenght]

 

[ohwilleke]

The sign of the beta function for SU(N) (unbroken) with n_f fermions is determined of the expression $\dfrac{-11N}{6} + \dfrac{n_f}{3}$, note that this is for fermions in the fundamental representation of SU(N). With N = 2 we get that $n_f < 11$ then SU(2) gauge theory is asymptotical free

But this is would be a strange theory, the gauge-bosons would be massless, it would not be the weak-interaction we know of.

Anyway, back to the standard model.
View attachment 303989
The weak force indeed becomes weaker at shorter distances (higher energy scales) [note - the vertical axis is 1/strenght]

It is worth noting that the strength of the strong force isn't a straight line as the chart depicts in the very low infrared. Instead, the red line should surge up on the far left from close to 1 or maybe a bit more than 1 to the top left of the chart.

Instead gets stronger down to a peak energy scale, below which it gets weaker to zero or near zero in the limit of zero energy, which is why we say that the strong force becomes asymptotically free at some point within hadrons as shown here:

From:

I.L. Bogolubsky, et al., "Lattice gluodynamics computation of Landau-gauge Green's functions in the deep infrared" 676 Phys.Lett.B 69-73 (2009). https://doi.org/10.1016/j.physletb.2009.04.076 (conformed free preprint at https://arxiv.org/abs/0901.0736).

There is some debate in QCD phenomenology over whether the strong force coupling constant has a low energy limit of zero or of a small finite value more than zero, which has significant theoretical implications.

Thus, the strong force coupling constant is asymptotically zero both at zero and at infinite energy scales with a peak value at q² = ca. 300 MeV.

Of course, we have no good reason to think that the weak force behaves in this manner, and the electromagnetic force certainly does not.

 

[malawi_glenn]

It is worth noting that the strength of the strong force isn't a straight line as the chart depicts in the very low infrared. Instead, the red line should surge up on the far left from close to 1 or maybe a bit more than 1 to the top left of the chart.

Logarithmic scale?

There is some debate in QCD phenomenology over whether the strong force coupling constant has a low energy limit of zero or of a small finite value more than zero, which has significant theoretical implications.

indeed, this is beyond perturbation theory calculation

 

[ohwilleke]

indeed, this is beyond perturbation theory calculation

If the question is "Is the weak interaction asymptotically free?" it is necessary to consider all cases and not just those that are subject to perturbative theory calculation, although this is essentially a null set in the case of the weak force.

[Not trying to shout. Can't figure out how to unbold the text.]
[Fixed it for you (berkeman)]

 

[malawi_glenn]

If the question is "Is the weak interaction asymptotically free?" it is necessary to consider all cases and not just those that are subject to perturbative theory calculation, although this is essentially a null set in the case of the weak force.

Well, you started the QCD-spinn off...

If you want to discuss the validation of the running of the SU(2)_L Gauge-coupling calculations for low energy regimes, that is another story.

 

[Vanadium 50]

If you don't have confinement, what can asymptotic freedom possibly mean except a beta function such that the coupling constant goes down with increasing energy?

Further, at high energy, "the weak force" is a poor description - it is better to discuss hypercharge and weak isospin, and one increases with energy and the other decreases.