Last week we (Chamseddine-Connes-van Suijlekom) posted a
preprint on grand unification in the spectral Pati–Salam model which I summarize here (and
here).
The paper builds on two recent discoveries in the noncommutative geometry approach to particle physics: we
showed
how to obtain inner fluctuations of the metric without having to
assume the order one condition on the Dirac operator. Moreover the
original argument by
classification of finite geometries F that can provide the fine structure of Euclidean space-time as a product
M×F (where
M is a usual 4-dimensional Riemannian space) has now been replaced by a much stronger
uniqueness statement. This new result shows that the algebra
M2(ℍ)⊕M4(ℂ)
where
ℍ
are the quaternions, appears uniquely when writing the higher analogue
of the Heisenberg commutation relations. This analogue is written in
terms of the basic ingredients of noncommutative geometry where one
takes a spectral point of view, encoding geometry in terms of operators
on a Hilbert space
. In this way, the inverse line element is an unbounded self-adjoint operator
D. The operator
D is the product of the usual Dirac operator on
M and a `finite Dirac operator’ on
F, which is simply a hermitian matrix
DF.
The usual Dirac operator involves gamma matrices which allow one to
combine the momenta into a single operator. The higher analogue of the
Heisenberg relations puts the spatial variables on similar footing by
combining them into a single operator
Y
using another set of gamma matrices and it is in this process that the
above algebra appears canonically and uniquely in dimension 4.
This leads without arbitrariness to the Pati–Salam gauge group
SU(2)R×SU(2)L×SU(4), together with the corresponding gauge fields and a scalar sector, all derived as inner perturbations of
D.
Note that the scalar sector can not be chosen freely, in contrast to
early work on Pati–Salam unification. In fact, there are only a few
possibilities for the precise scalar content, depending on the
assumptions made on the finite Dirac operator.
tr(f(D/Λ))
where
Λ is a cutoff scale and
f an even and positive function. In the present case, it can be expanded using heat kernel methods,
tr(f(D/Λ))∼F4Λ4a0+F2Λ2a2+F0a4+⋯
where
F4,F2,F0 are coefficients related to the function
f and
ak are Seeley deWitt coefficients, expressed in terms of the curvature of
M
and (derivatives of) the gauge and scalar fields. This action is
interpreted as an effective field theory for energies lower than
Λ.
One important feature of the spectral action is that it
gives the usual Pati–Salam action with unification of the gauge couplings. Indeed, the scale-invariant term
F0a4 in the spectral action for the spectral Pati–Salam model contains the terms
F02π2∫(g2L(WαμνL)2+g2R(WαμνR)2+g2(Vmμν)2).
Normalizing this to give the Yang–Mills Lagrangian demands
F02π2g2L=F02π2g2R=F02π2g2=14,
which requires gauge coupling unification. This is very similar to the case of the
spectral Standard Model
where there is unification of gauge couplings. Since it is well known
that the SM gauge couplings do not meet exactly, it is crucial to
investigate the running of the Pati–Salam gauge couplings beyond the
Standard Model and to find a scale
Λ where there is grand
unification:
gR(Λ)=gL(Λ)=g(Λ).
This would then be the scale at which the spectral action is valid as
an effective theory. There is a hierarchy of three energy scales: SM,
an intermediate mass scale
mR where symmetry breaking occurs and which is related to the neutrino Majorana masses (
1011−1013GeV), and the GUT scale
Λ.
In
the paper,
we analyze the running of the gauge couplings according to the usual
(one-loop) RG equation. As mentioned before, depending on the
assumptions on
DF,
one may vary to a limited extent the scalar particle content,
consisting of either composite or fundamental scalar fields. We will not
limit ourselves to a specific model but consider all cases separately.
This leads to the following three figures:
Running of coupling constants for the spectral Pati–Salam model with composite Higgs fields
Running of coupling constants for the spectral Pati–Salam model with fundamental Higgs fields
Running of coupling constants for the left-right symmetric spectral Pati–Salam model
In other words, we establish grand unification for all of the scenarios with unification scale of the order of
1016 GeV, thus confirming validity of the spectral action at the corresponding scale, independent of the specific form of
DF.
