Dilaton and massive hadrons in a conformal phase
Luigi Del Debbio, Roman Zwicky
Abstract
A bstract As the number of fermion fields is increased, gauge theories are expected to undergo a transition from a QCD-like phase, characterised by confinement and chiral symmetry breaking, to a conformal phase, where the theory becomes scale-invariant at large distances. In this paper, we discuss some properties of a third phase, where spontaneously broken conformal symmetry is characterised by its Goldstone boson, the dilaton. In this phase, which we refer to as conformal dilaton phase, the massless pole corresponding to the Goldstone boson guarantees that the conformal Ward identities are satisfied in the infrared despite the other hadrons carrying mass. In particular, using renormalisation group arguments in Euclidean space, we show that for massless quarks the trace of the energy momentum tensor vanishes on all physical states as a result of the fixed point. This implies the vanishing of the gluon condensate and suggests that the scale breaking is driven by the quark condensate which has implications for the cosmological constant. In addition form factors obey an exact constraint for every hadron and are thus suitable probes to identify this phase in the context of lattice Monte Carlo studies. For this purpose we examine how the system behaves under explicit symmetry breaking, via quark-mass and finite-volume deformations. The dilaton mass shows hyperscaling under mass deformation, viz. $$ {m}_D=\mathcal{O}\left({m}_q^{1/\left(1+{\gamma}^{\ast}\right)}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>m</mml:mi><mml:mi>D</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>O</mml:mi><mml:mfenced><mml:msubsup><mml:mi>m</mml:mi><mml:mi>q</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mfenced><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:msup><mml:mi>γ</mml:mi><mml:mo>∗</mml:mo></mml:msup></mml:mrow></mml:mfenced></mml:mrow></mml:msubsup></mml:mfenced></mml:math> . This provides another clean search pattern.