Aspects of Thermal QCD Phenomenology at Intermediate Gauge/'t Hooft Coupling: Appendix B

APPENDIX B

B.1 O(R4 ) Corrections to the M-theory metric of [15] in the MQGP limit near the ψ = 2nπ, n = 0, 1, 2- branches

where Σ1 has the following form:

B.2 Thermal fMN EOMs, their Solutions in the IR, 4DLimit and MχPT Compatibility

In the following appendix, we’re going to discuss the independent EOMs associated with the metric perturbations fMN of (3.2) near the IR cut-off r0 up to leading order in N, the solutions they provide and constraints as well as values for the same in the decompactification limit of a spatial direction (that has an important role for showing proof of an all-loop non-renormalization of Tc at O(R4 )). Through such a manner, we are capable to calculate the values of the metric perturbations (in the deep IR) across the three-cycle S 3 (θ1, x, z) - the delocalized version of S 3 (θ1, ϕ1, ψ) -strictly speaking across the fiber S 1 (z) (the S 3 (θ1, x, z) is a S 1 (z) fibration across the vanishing two-cycle S 2 (θ1, x)) as well as another two-cycle S 2 (θ1, z) (which has also a S 1 (z)-fibration). A unique linear combination of contributions from fMN |S3(θ1,x,z) , close to the Ouyang embedding within the parent type IIB dual, shows up frequently in Tc computations and the LECs of SU(3)χPT Lagrangian at O(p 4 ) from MχPT in chapter 2 (from [2]). In chapter 2, we found that the aforementioned combination of integration constant has a negative sign (2.30). Here we will derive this constraint. The EOMs of the metric perturbations fMN (r) are worked out as follows:

EOMs (B.3) have the following solutions:

Utilizing (B.5), around r = r0, we obtained:

• We solved (B.13) - (B.15) and obtained:

B.3 M-Theory Metric Associated with Rotating Cylindrical Black Hole and Thermal Backgrounds

• The following are O(β 0 ) components associated with the M-theory dual involving a rotating cylindrical black hole background in the MQGP limit differ from those derived in [15,66]. Apart from these components, rest of the components are same as given in (B.1) at O(β 0 ).

where

r 4 . (B.18) • The O(β) terms in the small ω-limit, remain the same as worked out in [1] and has been provide in (B.1).

B.4 O(β) Contribution to the On-Shell Action Densities for the Rotating QGP Tc Calculation

To derive the contribution that comes from the O(β) contributions in the on-shell action density for the black hole backdrop, write the metric in diagonal basis (3.118). The t − ϕ subspace’s unwarped metric could be expressed as:

where

and

We can now write the metric in (t, ϕ) subspace and O(R4 ) correction to the same, i.e., fMN (r), where (M, N = t, ϕ) (we are concentrating exclusively on the t, ϕ) subspace). The metric’s tt component could be written as:

implying

Similarly, the ϕt component of the metric is written as:

and therefore,

The same is also applicable for the ϕϕ component:

hence fϕϕ is obtained as:

In the small-ω limit, γ = 1, hence the equation (B.19) simplified as:

As a result, the structure of the M-theory metric for the black hole background in the small ω-limit in rotating quark-gluon plasma is the same as the structure of the M-theory metric without rotation. As a result, in the small ω-limit, the higher derivative corrections to the M-theory metric for the rotating cylindrical black hole and thermal backgrounds will be the same as obtained in [1]. As a result, the on-shell action densities associated with the rotating cylindrical thermal and black hole background will be identical to [3] (i.e. similar to 3.2) augmented by a factor of l, which is going to cancel out at the UV cut-off from both sides of the equation (3.131).

B.5 Holographic Renormalization of Rotating Gravitational Backgrounds

Here, we have discussed the holographic renormalization of rotating cylindrical black hole and thermal backgrounds used in 3.4.

• Rotating Cylindrical Black Hole Background: Divergent parts of the M-theory on-shell action of the rotating cylindrical black hole background at O(β 0 ) have the following structure:

UV-divergences appearing in the M-theory on-shell action of the rotating cylindrical black hole background at O(β 0 ) will be cancelled by the following term:

We found that UV-divergence appearing in equation (B.32) will be cancelled by the following counter term:

provided

• Rotating Cylindrical Thermal Background: Divergent part of the M-theory on-shell action at O (β0) in the UV have the following form:

where first and second terms in equation (B.36) are counter terms for the first and second terms in equation (B.35) and are given as below: