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]