Nearly conformal gauge theories in finite volume

doi:10.1016/j.physletb.2009.10.040

Physics Letters B

Volume 681, Issue 4, 9 November 2009, Pages 353-361

https://doi.org/10.1016/j.physletb.2009.10.040 Get rights and content

Abstract

We report new results on nearly conformal gauge theories with fermions in the fundamental representation of the $SU (3)$ color gauge group as the number of fermion flavors is varied in the $N_{f} = 4 – 16$ range. To unambiguously identify the chirally broken phase below the conformal window we apply a comprehensive lattice tool set in finite volumes which includes the test of Goldstone pion dynamics, the spectrum of the fermion Dirac operator, and eigenvalue distributions of random matrix theory. We also discuss the theory inside the conformal window and present our first results on the running of the renormalized gauge coupling and the renormalization group beta function. The importance of understanding finite volume zero momentum gauge field dynamics inside the conformal window is illustrated. Staggered lattice fermions are used throughout the calculations.

Introduction

The Large Hadron Collider will probe the mechanism of electroweak symmetry breaking. It is an intriguing possibility that new physics beyond the Standard Model might take the form of some new strongly-interacting gauge theory. In one scenario, the Higgs sector of the electroweak theory is replaced by a so-called technicolor theory, whose dynamics provides the required spontaneous symmetry breaking [1], [2], [3]. These models avoid the fine-tuning problem and may lead to a heavy composite Higgs particle on the TeV scale. Although attractive, the challenge is to extend a technicolor theory to include fermion mass generation, while satisfying the various constraints of electroweak phenomenology. This idea has lately been revived by new explorations of the multi-dimensional theory space of nearly conformal gauge theories [4], [5], [6], [7], [8], [9]. The terminology of technicolor in this report will refer in a generic sense to these investigations. Exploring the new technicolor ideas has to be based on non-perturbative studies which are only becoming feasible now with the advent of new lattice technologies.

Model building of a strongly interacting electroweak sector requires the knowledge of the phase diagram of nearly conformal gauge theories as the number of colors $N_{c}$ , number of fermion flavors $N_{f}$ , and the fermion representation R of the technicolor group are varied in theory space. For fixed $N_{c}$ and R the theory is in the chirally broken phase for low $N_{f}$ and asymptotic freedom is maintained with a negative β function. On the other hand, if $N_{f}$ is large enough, the β function is positive for all couplings, and the theory is trivial. If the regulator cut-off is removed, we are left with a free non-interacting continuum theory. There is some range of $N_{f}$ for which the β function might have a non-trivial zero, an infrared fixed point, where the theory is in fact conformal [10], [11]. This method has been refined by estimating the critical value of $N_{f}$ , above which spontaneous chiral symmetry breaking no longer occurs [12], [13], [14].

Interesting models require the theory to be very close to, but below, the conformal window, with a running coupling which is almost constant over a large energy range [15], [16], [17], [18], [19], [20]. The non-perturbative knowledge of the critical $N_{f}^{crit}$ separating the two phases is essential and this has generated much interest and many new lattice studies [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47].

Our goal to unambiguously identify the chirally broken phase below the conformal window requires the application and testing of a comprehensive lattice tool set in finite volumes which includes the test of Goldstone pion dynamics, the spectrum of the fermion Dirac operator, and eigenvalue distributions of Random Matrix Theory (RMT). Inside the conformal window we investigate the running coupling and the β function. We report new results at $N_{f} = 4, 8, 9, 12, 16$ for fermions in the fundamental representation of the $SU (3)$ technicolor gauge group. We find $N_{f} = 4, 8, 9$ to be in the chirally broken phase and $N_{f} = 16$ is consistent with the expected location inside the conformal window. To resolve the $N_{f} = 12$ phase from our simulations will require further analysis.

Section snippets

Chiral symmetry breaking below the conformal window

We will identify in lattice simulations the chirally broken phases with $N_{f} = 4, 8, 9$ flavors of staggered fermions in the fundamental $SU (3)$ color representation using finite volume analysis. The staggered fermions are deployed with a special 6-step exponential (stout) smearing procedure [48] in the lattice action to reduce well-known cutoff effects with taste breaking in the Goldstone spectrum. The presence of taste breaking requires a brief explanation of how staggered chiral perturbation theory

Simulations results in the p-regime

The tree level improved Symanzik gauge action was used in our simulations. The link variables in the staggered fermion matrix were exponentially smeared with six stout steps at $N_{f} = 4, 8$ and four stout steps at $N_{f} = 9$ . The RHMC algorithm was deployed in all runs but rooting of the fermion determinant only affected the $N_{f} = 9$ simulations. The results shown in Fig. 3 are from the p-regime of the chirally broken phase with the conditions $M_{π} \cdot L_{s} ≫ 1$ and $F_{π} \cdot L \sim 1$ when the chiral condensate begins to follow the

Epsilon-regime, Dirac spectrum and RMT

If the bare parameters of a gauge theory are tuned to the ϵ-regime in the chirally broken phase, the low-lying Dirac spectrum follows the predictions of random matrix theory. The corresponding random matrix model is only sensitive to the pattern of chiral symmetry breaking, the topological charge and the rescaled fermion mass once the eigenvalues are also rescaled by the same factor $Σ_{cond} V$ . This idea has been confirmed in various settings both in quenched and fully dynamical simulations. The

Inside the conformal window

We start our investigation and simulations of the conformal window at $N_{f} = 16$ which is the most accessible for analytic methods. We are particularly interested in the qualitative behavior of the finite volume spectrum of the model and the running coupling with its associated beta function which is expected to have a weak coupling fixed point around $g^{* 2} \approx 0.5$ , as estimated from the scheme independent two-loop beta function [63].

Acknowledgements

We are thankful to Claude Bernard and Steve Sharpe for help with staggered perturbation theory and to Ferenc Niedermayer for discussions on rotator dynamics. We also wish to thank Urs Heller for the use of his code to calculate Wilson loops in lattice perturbation theory, and Paul Mackenzie for related discussions. We are grateful to Sandor Katz and Kalman Szabo for helping us in using the Wuppertal RHMC code. In some calculations we use the publicly available MILC code, and the simulations

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Nearly conformal gauge theories in finite volume

Abstract

Introduction

Section snippets

Chiral symmetry breaking below the conformal window

Simulations results in the p-regime

Epsilon-regime, Dirac spectrum and RMT

Inside the conformal window

Acknowledgements

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