Abstract
Effects of the theta parameter are studied in Witten's model of holographic 4d Yang-Mills, where theta is the coefficient of the CP-breaking topological term. First, the gravity background, including the full backreaction of the RR form dual to the theta parameter, is revisited. Then, a number of observables are computed holographically: the ground-state energy density, the string tension, the 't Hooft loop, the light scalar glueball mass, the baryon mass scale, the critical temperature for deconfinement - and thus the whole (T, theta) phase diagram - and the entanglement entropy. A simple rule is provided to derive the theta corrections to (at least) all the CP-neutral observables of the model. Some of the observables we consider can and have been in fact studied in pure 4d Yang-Mills on the lattice. In that framework the results, obtained in the small theta regime, are given up to very few powers of theta^2. The corresponding holographic results agree qualitatively with available lattice data and signal an overall mass scale reduction by theta. Moreover, being exact in theta, they provide a benchmark for higher order corrections in Yang-Mills.
Highlights
Being lc,0 ∼ 1.288/MKK the critical length at Θ = 0. This expression is confirmed by the numeric comparison of the entanglement entropy of the disconnected solution with the connected one (l is not forced to be small in this case), see figure 3
The model allows to extract the exact θ dependence of a class of interesting observables, like the vacuum energy density, the string tension, the mass of the baryon vertex, the ’t Hooft loop, the confinement-deconfinement critical temperature, the mass of the 0++ glueball and the entanglement entropy
The model shows a common trend of all the mass scales of the theory: they get reduced by θ
Summary
The UV ’t Hooft coupling and the θ angle of the gauge theory, can be related to the gravity parameters by considering the low energy limit of the D4-brane action. Together with (2.5), it implies that the bare θ angle is related to the Θ parameter of the background by As it is suggested by eq (2.1), in the ’t Hooft limit, the corrections to the physics due to the θ parameter, w.r.t. the θ = 0 case, are weighed by the combination appearing in Θ. Since this parameter depends on k, what we get on the gravity side is an infinite family of solutions corresponding to possible field theory vacua As it was shown in [5] the curvature invariants of the background remain small if |Θ| λ14/4. Using the u → ∞ limit of the 10d background introduced above, going to the Euclidean frame, and performing the identification with the one-instanton action, one precisely gets the relations (2.9)
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