GAUGE COUPLINGS CALCULATED FROM MULTIPLE POINT CRITICALITY YIELD α-1 = 137 ± 9: AT LAST, THE ELUSIVE CASE OF U(1)
Authors: BENNETT D. L. , NIELSEN H. B. 
Published: April 2003

We calculate the U(1) continuum gauge coupling using the values of action parameters coinciding with the multiple point. This is a point in the phase diagram of a lattice gauge theory where a maximum number of phases convene. We obtain for the running inverse fine structure constant the values [Formula: see text] and [Formula: see text] at the Planck scale and the MZ scale, respectively. The gauge group underlying the phase diagram in which we seek multiple point parameters is what we call the Anti-grand-unified theory (AGUT) gauge group SMG 3, which is the Cartesian product of three Standard Model Groups (SMG's). There is one SMG factor for each of the N gen =3 generations of quarks and leptons. In our model, this gauge group SMG 3 is the predecessor of the usual SMG. The latter arises as the diagonal subgroup surviving the Planck scale breakdown of SMG 3. This breakdown leads to a weakening of the U(1) coupling by a N gen -related factor. For N gen =3, this factor would be N gen (N gen +1)/2=6 if phase transitions between all the phases convening at the multiple point were purely second order. The factor N gen (N gen +1)/2=6 corresponds to the six gauge-invariant combinations of the N gen =3 different U(1)'s that give action contributions that are second order in Fμν. The factor analogous to this N gen (N gen +1)/2=6 in the case of the earlier considered non-Abelian couplings reduced to the factor N gen =3 because action terms quadratic in Fμν that arise as contributions from two different of the N gen =3 SMG factors of SMG 3 are forbidden by the requirement of gauge symmetry.

Actually we seek the multiple point in the phase diagram of the gauge group U(1) 3 as a simplifying approximation to the desired gauge group SMG 3. The most important correction obtained from using multiple point parameter values (in a multiparameter phase diagram instead of the single critical parameter value obtained, say, in the one-dimensional phase diagram of a Wilson action) comes from the effect of including the influence of also having at this point phases confined solely w.r.t. discrete subgroups. In particular, what matters is that the degree of first-orderness is taken into account in making the transition from these latter phases at the multiple point to the totally Coulomb-like phase. This gives rise to a discontinuity Δγ eff in an effective parameter γ eff . Using our calculated value of the quantity Δγ eff , we calculate the above-mentioned weakening factor to be more like 6.5 instead of the N gen (N gen +1)/2=6, as would be the case if all multiple point transitions were purely second order. Using this same Δγ eff , we also calculate the continuum U(1) coupling corresponding to the multiple point of a single U(1). The product of this latter and the weakening factor of about 6.5 yields our Planck scale prediction for the continuum U(1) gauge coupling, i.e. the multiple point critical coupling of the diagonal subgroup of U(1) 3∈ SMG 3. Combining this with the results of earlier work on the non-Abelian gauge couplings leads to our prediction of α-1-137 ± 9 as the value for the fine structure constant at low energies.