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Register a new userThe scalar glueball operator, the a-theorem, and the onset of conformality
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We show that the anomalous dimension $gamma_G$ of the scalar glueball operator contains information on the mechanism that leads to the onset of conformality at the lower edge of the conformal window in a non-Abelian gauge theory. In particular, it distinguishes whether the merging of an UV and an IR fixed point -- the simplest mechanism associated to a conformal phase transition and preconformal scaling -- does or does not occur. At the same time, we shed light on new analogies between QCD and its supersymmetric version. In SQCD, we derive an exact relation between $gamma_G$ and the mass anomalous dimension $gamma_m$, and we prove that the SQCD exact beta function is incompatible with merging as a consequence of the $a$-theorem; we also derive the general conditions that the latter imposes on the existence of fixed points, and prove the absence of an UV fixed point at nonzero coupling above the conformal window of SQCD. Perhaps not surprisingly, we then show that an exact relation between $gamma_G$ and $gamma_m$, fully analogous to SQCD, holds for the massless Veneziano limit of large-N QCD. We argue, based on the latter relation, the $a$-theorem, perturbation theory and physical arguments, that the incompatibility with merging may extend to QCD.
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We discuss properties of non-Abelian gauge theories that change significantly across the lower edge of the conformal window. Their probes are the topological observables, the meson spectrum and the scalar glueball operator. The way these quantities change tells about the way conformal symmetry is lost.
We compute the Green function of the massless scalar field theory in the infrared till the next-to-leading order, providing a fully covariant strong coupling expansion. Applying Callan-Symanzik equation we obtain the exact running coupling for this case by computing the beta function. This result is applied using a recently proved mapping theorem between a massless scalar field theory and Yang-Mills theory. This beta function gives a running coupling going to zero as $p^4$ in agreement with lattice results presented in Boucaud et al. [JHEP 0304 (2003) 005] and showing that the right definition of the running coupling for a Yang-Mills theory in the infrared is given in a MOM scheme. The emerging scenario is supporting a quantum field theory based on instantons.
The center-of-gravity rule is tested for heavy and light-quark mesons. In the heavy-meson sector, the rule is excellently satisfied. In the light-quark sector, the rule suggests that the $a_0(980)$ could be the spin-partner of $a_2(1320)$, $a_1(1260)$, and $b_1(1235)$; $f_0(500)$ the spin-partner of $f_2(1270)$, $f_1(1285)$, and $h_1(1170)$; and $f_0(980)$ the spin-partner of $f_2(1525)$, $f_1(1420)$, and $h_1(1415)$. From the decay and the production of light scalar mesons we find a consistent mixing angle $theta^{rm s}=(14pm4)^circ$. We conclude that $f_0(980)$ is likely octet-like in SU(3) with a slightly larger $sbar s$ content and $f_0(500)$ is SU(3) singlet-like with a larger $nbar n$ component. The $a_0(1450)$, $K^*_0(1430)$, $f_0(1500)$ and $f_0(1370)$ are suggested as nonet of radial excitations. The scalar glueball is discussed as part of the wave function of scalar isoscalar mesons and not as additional intruder. It seems not to cause supernumerosity.
The content of two additional Ward identities exhibited by the $U(1)$ Higgs model is exploited. These novel Ward identities can be derived only when a pair of local composite operators providing a gauge invariant setup for the Higgs particle and the massive vector boson is introduced in the theory from the beginning. Among the results obtained from the above mentioned Ward identities, we underline a new exact relationship between the stationary condition for the vacuum energy, the vanishing of the tadpoles and the vacuum expectation value of the gauge invariant scalar operator. We also present a characterization of the two-point correlation function of the composite operator corresponding to the vector boson in terms of the two-point function of the elementary gauge fields. Finally, a discussion on the connection between the cartesian and the polar parametrization of the complex scalar field is presented in the light of the Equivalence Theorem. The latter can in the current case be understood in the language of a constrained cohomology, which also allows to rewrite the action in terms of the aforementioned gauge invariant operators. We also comment on the diminished role of the global $U(1)$ symmetry and its breaking.
We prove a theorem in QCD stating that in the limit of strong coupling, $gtoinfty$, the observed spectrum of glueballs in QCD is the same of a pure Yang-Mills theory, being mixing effects due to the next-to-leading order. A full effective theory for QCD is obtained and the width of the $sigma$ resonance decay is straightforwardly computed. This appears as the lowest glueball state. Vacuum gluon condensate is computed that consistently support studies on the identification of this meson as a glueball.
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