# Monte Carlo
Renormalization Group for Minimal Walking Technicolor

This project aims to extract (1) the running of the bare
lattice coupling and (2) the anomalous mass dimension in Minimal Walking Technicolor
using Monte Carlo Renormalization Group techniques. This research
continues detailed studies of SU(2) lattice gauge
theory with Nf=2 Dirac fermion flavors in the adjoint representation, motivated by its apparent proximity
to the ``conformal window'' and therefore potential use as a technicolor gauge
theory with walking behavior. Because this ``minimal walking technicolor'' has a small number of flavors, it is believed
that the S-parameter will likewise be small, as is required by electroweak precision
constraints. It is part of a class of walking technicolor
models that is currently under intense investigation within the lattice
community, a follow-up to older works that were inconclusive.

### Background: Electroweak symmetry is one of the
cornerstones of the Standard Model of particle physics. However, we know that
it is spontaneously broken. All of the
elementary particle masses that we have observed in nature arise from this
breaking of electroweak symmetry.
Restricting to four-dimensional theories, there are two competing ideas
for how electroweak symmetry is broken.
One is that there is a fundamental scalar field, the Higgs field, that has a nonzero value in the ground state of the
universe. The alternative is that new
particles, technifermions, form a condensate in the
ground state. This phenomenon already is
known to occur in quantum chromodynamics (leading to
spontaneous chiral symmetry breaking) and
superconductivity. This breaking of
electroweak symmetry due to dynamics of new fermions (and hence a new force in
nature) is called technicolor.

### The challenge: We are involved in the study of technicolor theories from first principles. This is challenging for a number of
reasons. First, the technicolor
theory is strongly interacting, and so a nonperturbative
approach such as lattice gauge theory must be used in order to obtain
meaningful results. Second, the technicolor theories that we are interested have dynamics
that scan a wide range of scales (walking technicolor). One cannot directly simulate a system that
incorporates these scales all at once, and a renormalization group approach is
essential in order to circumvent this problem.
Third, the fermions in the technicolor theory
are massless, which makes them very expensive to study on a computer, because
the matrix problem becomes ill-conditioned.

### Goals: (1) Use Monte Carlo renormalization group to
identify the infrared fixed point in Minimal Walking Technicolor. (2) Compute the anomalous mass dimension in
this theory. (3) Show that we are in the
basin of attraction of the Gaussian fixed point. What do these things mean? For this, we turn to the…

### Method: We start with the action for the theory: , which is a sum of lattice operators built out of
lattice fields, with “couplings” , which are constant coefficients. A simulation is then done on a “fine” lattice
of size , where 2N is the number of lattice sites in each of
the four dimensions. We measure observables , blocked n times, from the results of these
simulations. Here “blocking” refers to
an intelligent sort of averaging of the lattice fields. Next simulations are done on “coarse”
lattices of size , so that the number of sites in each direction is
half of what it was on the “fine” lattice.
These simulations are done with couplings , which will in general be different from . We measure observables , blocked n-1 times.
We repeat this until we find the couplings such that the
observables measured on the fine lattice agree with those measured on the
coarse lattice. Once this is achieved,
we say that the couplings and are
“matched.” This determines the renormalization group flow of the bare couplings, and it is
done using Monte Carlo simulations (Monte Carlo integration to estimate the
path integral that describes the quantum field theory); hence the name Monte
Carlo renormalization group.

### Fixed Point: In Minimal Walking Technicolor, it is now
believed that an infrared fixed point exists.
This means that under renormalization group flow, the coupling approach
a point in parameter space where the cease to change. This would be indicated by a zero of the bare
step scaling function (discrete beta function) . So far all
that we have found is that is small and
could be zero once systematic uncertainties are taken into account. We are currently working to reduce these
uncertainties by: (1) going to larger
lattices where more blocking steps can be taken, (2) using O(a)
improved actions (adding the clover term).
Both of these would reduce “scaling violations,” which are the source of
disagreement between different observables as to the matching of and .

### Anomalous Mass Dimension: This quantity, , characterizes how the running mass behaves with
respect to the renormalization group. It
also dictates the quantum mass dimension of the technifermion bilinear . Other
methods, including our own finite size scaling analysis, show that is about
0.4. However, Monte Carlo
renormalization group has been giving us confusing results: about 0 if we assume that we are near the fixed
point where , and if we take
into account our uncertainties in . We are
currently working to include fermionic observables in the matching, in addition
to reducing the scaling violations as mentioned above. We are hopeful that these improvements will
lead to more definitive answers.

### Simulations and
Measurements: These are demanding
computations, and we are currently using the CCNI and Department of Energy
computers at Fermi National Laboratory (as part of the USQCD collaboration) in order to do our work.

### Publications:

### · Simon Catterall (Syracuse U.), Luigi Del Debbio (Edinburgh U.), Joel Giedt and Liam Keegan
(Edinburgh U.), “MCRG Minimal Walking Technicolor,” PoS LATTICE2010 (2010) 057 [arXiv:1010.5909].

### · Simon Catterall (Syracuse U.), Luigi Del Debbio (Edinburgh U.), Joel Giedt and Liam Keegan (Edinburgh
U.), “MCRG Minimal Walking Technicolor,” submitted to Phys. Rev. D [arXiv:1108.3794].

### · Simon Catterall (Syracuse U.), Luigi Del Debbio (Edinburgh U.), Joel Giedt and Liam Keegan
(Edinburgh U.), “Systematic Errors of the MCRG Method,” PoS LATTICE2011 (2011) 068 [arXiv:1110.1660]