## Friday, January 17, 2014

### Coupling constants II

One form of the LC&P sum rule is

2 λ + g2/4 + (g2 + g'2)/4 + yt2/2 ~ 1

... based on their equation 2, and neglecting yukawa couplings for fermions other than the top quark.

As they remark (but I didn't notice until Andrew pointed it out), the contributions from bosons and fermions are almost equal. So we can also say that

2 λ + g2/2 + g'2/4  ~  yt2/2  ~  1/2

The "fermionic part" of this makes sense, if we recall that yt ~ 1. But the bosonic part

2 λ + g2/2 + g'2/4  ~  1/2

... just considered by itself, seems to be very notable new numerology, connecting electromagnetic and weak couplings with the Higgs self-coupling λ.

edit: Actually, if I think about it for a moment, I remember that g is small and g' (the weak coupling) is even smaller. So the bosonic part reduces to

2 λ  ~  1/2

i.e. λ  ~  1/4. I noted almost a year ago that this is implied by the fact that the Higgs VEV / electroweak scale is approximately twice the Higgs boson mass.

edit #2: Study of the literature (e.g. PDG 2013 Higgs review) makes it clear that

λ  ~  1/8

is closer to the truth. Apparently there are some factors of √2 that I missed. But now I don't understand why LC&P works.

(Or are we just dealing with different conventions? Remedial study of Higgs-sector basics is in order...)

1. The cosW = mW/mZ and the extradimensions, hyperbolic geometry
One hyperboloid is a cut torus at middle of it.

L7 ( lenght larger dimensionless seven dimensions ) = [ { 2( 2pi )⁷ }/(16(Pi)³ /15]¹/9 = 3.05790095610237

R7 ( small length dimensionless seven dimensions ) = [ { 4( 2pi )⁷ }/8(16(Pi)³ /15]¹/8 = 2.95694905822489

a) 1/sinh( L7/Pi ) =cosW =0.8814410171

b) [ cosh( L7/Pi)-1] =0.512317278 mH/v ; mH = 126.17 +- 0.03 Gev v= 246.221202 Gev

La = sqr( alpha⁻1/4Pi) ; sqr( [La - R7]/La )= sin(2theta13) = 0.3233734916

theta13 neutrino 13 mixing angle = 9.433531267º

[ (1- sin(2theta13))/3 ]= 0.2255421695 = sin( Cabibbo angle = 13.04º )

e/[La - R7] = cosEffW = 0.8769328952

Bests regards

2. The simpler Higgs relationship is as follows:

The sum of the square of the mass of each of the Standard Model particles is exactly equal to the square of the Higgs vev (I was going over these numbers last weekend to do some sensitivity analysis). Similarly, if you divide each of the squared masses by the Higgs vev squared, you will get a set of dimensionless coupling constants that sum to one.

By convention, for fermions this number if converted to its Yukawa coupling by multiplying by two and taking the square root. For the Higgs self-coupling you divided by two and don't take the square root. And, for the electroweak boson couplings you make other conversions. But, it is easy to get muddled about g and g'. I have a post or comment to a post where I worked it out and will have to look it up.

The fermonic and bosonic contributions to the Higgs vev are nearly equal, but not exactly equal. The fermionic contribution is dominated by the mass of the top quark and very sensitive to it. A 0.02 GeV shift in the mass of the top quark is roughly equivalent to the impact of an entire extra 3 GeV particle. In GeV^2 units, an uncertainty of 0.1 GeV in the top quark mass contributes 34 GeV of uncertainty to the sum of the squared masses, with the runner up being a 4.3 GeV^2 uncertainty in the Higgs vev mass due to uncertainty in the W boson mass.

There are three masses - the W, the Z and the Higgs boson mass that contribute materially to the boson side. Also, uncertainty in the W boson mass contributes to uncertainty in the value of the Higgs vev. Indeed, uncertainty in the W boson mass is the second biggest source of uncertainty overall since this uncertainty is the biggest source of experimental uncertainty in the measured value of the Higgs vev, because it directly impacts the W boson mass, and because I used the formula 2H=2W+Z to determine the Higgs boson mass.

The non-equality of the fermion side to the boson side exceeds present experimental uncertainty in the relevant mass measurements, although they are close. Indeed, one way that the Higgs boson mass could have been derived pre-discovery would have been to use Higgs vev squared minus sum of each of the fermion masses squared minus W+ boson mass squared, minus W- boson mass squared, minus Z boson mass squared. One can think of the Higgs boson mass as a fudge factor the reconcile all of the other masses to Higgs vev squared.

Of course, given the accuracy of H=2W+Z+photon mass/sqrt(4) as a formula, it might make more sense to think of all of the boson masses as a function of an overall electroweak mass scale and a weak mixing angle.

I remain curious about the energy scale at which the sum of squared boson masses equals the sum of squared fermion masses, which would seem significant but is beyond my capacity to calculate. But, I'm not sure that this ever happens because the Higgs mass goes to zero at about the GUT scale and I don't think that the fermion masses go to zero at that point, which would imply that there is no mass scale from 2 GeV to the GUT scale at which the fermion masses collectively are less than the boson masses collectively. Perhaps to get the discrepancy between the fermion and boson masses to disappear out you would instead have to work in the other direction and bring 2 GeV or pole masses down to a hypothetical (and unphysical) 0 GeV scale or even some hypothetical negative energy scale.