Deconfinement Phase Transition in Thermal Qcd-Like Theories at Intermediate Coupling

Table of Links

Abstract

Acknowledgment

PART I

Chapter 1: Introduction

Chapter 2: SU(3) LECs from Type IIA String Theory

Chapter 3: Deconfinement Phase Transition in Thermal QCD-Like Theories at Intermediate Coupling in the Absence and Presence of Rotation

Chapter 4: Conclusion and Future Outlook

PART II

Chapter 5: Introduction

Chapter 6: Page Curves of Reissner-Nordström Black Hole in HD Gravity

Chapter 7: Entanglement Entropy and Page Curve from the M-Theory Dual of Thermal QCD Above Tc at Intermediate Coupling

Chapter 8: Black Hole Islands in Multi-Event Horizon Space-Times

Chapter 9: Multiverse in Karch-Randall Braneworld

Chapter 10: Conclusion and Future outlook

APPENDIX A

APPENDIX B

APPENDIX C

Bibliography

CHAPTER 3 - DECONFINEMENT PHASE TRANSITION IN THERMAL QCD-LIKE THEORIES AT INTERMEDIATE COUPLING IN THE ABSENCE AND PRESENCE OF ROTATION

3.1 Introduction

We can compute the corrections to the infinite-’t Hooft-coupling limit as done in [100] using gauge/gravity duality, but this time using an AdS background. They used terms quartic in the Weyl tensor to include higher-derivative corrections on the gravity side. From the gauge theory point of view, there is a SU(N) supersymmetric Yang-Mills plasma with N = 4 SUSY at an intermediate ’t Hooft coupling. They provided an explanation for the transport peak that appeared in the spectral function of the stress-energy tensor at zero spatial momentum in the small frequency zone. This is a characteristic of perturbative plasma generally.

In general theories of gravity, Wald provided a formula to determine black hole entropy [102]. He thought about the Lagrangian that results from the diffeomorphism invariant classical theory of gravity in n dimensions. Noether charge is a (n−2)-form in these theories of gravity, and the (n − 2)-form Noether charge’s integral across its bifurcate killing horizon will yield the black hole’s entropy. Therefore, for stationary black holes with bifurcate killing horizons, entropy equals Noether charge. A formula to calculate entropy for dynamical black holes was made in [103].

Every physical phenomenon has an energy/distance scale that defines it. RG group flow is used to move from UV to IR. However, short-range physics begins to interact with long-range physics in noncommutative field theories and string theory, a process known as UV/IR mixing. UV divergences in real ϕ 4 theory defined on commutative space are changed into infrared poles in the identical theory defined on noncommutative space as a result of UV/IR mixing, for example. There are further examples that are nicely explained in [104]. Theoretical models of gravity are nonlocal. As a result, UV/IR mixing might be present in such theories. In the gauge/gravity duality, the energy scale in the gauge theory side matches with the radial coordinate in gravitational theory. In the context of the AdS/CFT correspondence, the authors looked at the “I.R.-U.V.” connection and demonstrated how infrared phenomena in bulk theory are turned into ultravoilet phenomena in boundary theory [105]. In particular, the job of the ultraviolet regulator in the N = 4 super Yang Mills theory is played by the infrared regulator in the bulk theory. It’s fascinating that in our work, matching at the deconfinement temperature, M-theory actions dual to the thermal and black-hole backgrounds at the UV-cut-off, and obtaining a relationship between the O(R4 ) metric corrections in the IR, a specific type of UV-IR connection manifests itself.

In noncentral relativistic heavy ion collisions (RHIC), a fluid known as quark-gluon plasma (QGP) is created. According to experimental findings from the RHIC collaboration, the angular momentum and angular velocity of this rotating fluid are on the order of 103ℏ and ω ∼ 6 MeV, respectively. In [108], the authors applied hydrodynamic simulations of heavy ion collisions and found ω ∼ 20−40 MeV. Considering the results mentioned above, a top-down analysis employing gauge-gravity duality of the effect of rotation on the deconfinement temperature of thermal QCD-like theories will be very interesting, with this motivation we obtained deconfinement temperature of thermal QCD from a top-down approach in [5]. The authors in [109] examined the consequences of rotating plasma on the deconfinement temperature of QCD in Hard wall and Soft wall models, and they discovered that deconfinement temperature of QCD decreases as angular velocity of the plasma increases. Similar works using a bottom-up method are available in [110–112].

3.2 Deconfinement Phase Transition in Thermal QCDLike Theories in the Absence of Rotation

In this section, we will discuss the deconfinement phase transition of thermal QCD-like theories at intermediate coupling without including the effect of rotation.

3.2.1 Tc from Semi-classical Method

Here, we’ll detail how we implemented Witten’s proposal [4] to obtain the deconfinement temperature thermal QCD by applying the gauge/gravity duality and holographic renormalization of the eleven-dimensional supergravity action via 3.2.1.1, 3.2.1.2, and 3.2.1.3.

The M-theory dual for high temperatures, that is, temperatures that are higher than the critical temperature Tc, will entail a black hole with the metric presented by:

that results in an EOM:

3.2.1.1 Black Hole Background Uplift (Relevant to T > Tc) and Holographic Renormalization of On-Shell D = 11 Action

In this chapter, we worked near the Ouyang embedding implemented by the following coordinate patches [66]:

The following expression is what we computed the IR Einstein-Hilbert term:

Performing now the angular integration for the previously stated equation,

which simplifies near (3.10) as given below:

We derived the following expression for the action density (for Einstein-Hilbert term) in the IR by utilizing equation (3.13) and carrying out radial integration of (3.11):

The version of the Einstein-Hilbert term pertaining to the ultraviolet (UV) region is:

The following is the part that make up the EH action’s contribution to UV divergent:

Up to O(β), the UV-finite boundary Gibbons-Hawking-York contribution is found to be:

The GHY term’s UV-divergent contribution is as follows:

We showed that the contribution from the higher derivative term is as follows, up to O(β0):

The higher derivative term that gives rise to the UV finite contribution was found to be:

In addition, we observe that the surface counter term that neutralizes UV divergence is emerging via higher derivative term is provided by the expression that follows when r is held constant.

Holographic Renormalization When T > Tc: It has been found that the form of the UV-divergent part of the on-shell action (3.9) for the black hole backdrop is as follows:

where the second term of (3.33) is worked out in (3.28), and

3.2.1.2 Thermal Background Uplift (Relevant to T < Tc) and Holographic Renormalization of On-Shell D = 11 Action

In the limit of large N, as well as in the infrared, the fMN EOMs associated with the thermal background become algebraic. Writing below just those components that receive a non-trivial O(β) contributions, below is the O(β)-corrected MQGP metric for the thermal background in the IR in the ψ = 2nπ, n = 0, 1, 2-coordinate patches.

The Einstein-Hilbert term pertaining to the thermal background, which works in the infrared close to (3.10), is:

In the ultraviolet (UV), the equivalent Einstein-Hilbert term is:

implying:

Hence, the UV finite Einstein-Hilbert term, after carrying out the radial and angular integrals of the preceding equation is obtained as:

The Einstein-Hilbert divergent term for the thermal background in the UV region is:

For a thermal background up to O(β), the UV-finite portion of the boundary GibbonsHawking-York term is calculated to be:

The Gibbons-Hawking-York term in thermal background has an additional UV divergence component, which is:

implying:

In addition, the contribution arising from the higher derivative term in the UV near (3.10) is:

The UV-differential portion of the equation (3.46) was further deduced as:

We found that:

Let us assume the following in relation to (3.48) and the fourth equation in (3.49):

The following can be deduced from (3.49):

3.2.1.3 Tc Inclusive of O(R4) Corrections

Here, we’ll use the on-shell action comparison between black hole and thermal backgrounds to determine the deconfinement temperature using information gathered from 3.2.1.1 and 3.2.1.2. We will also describe a variant of the UV-IR connection that emerges from the O(R4) terms.

Following is the on-shell action that corresponds to the thermal background uplift:

this results in,

) In the context of this discussion, the term “UV-IR connection” refers to the equations (3.62) and (3.59). Because this is providing a connection that connects the integration constant appearing in the O(R4 ) corrections to the thermal background along the M-theory circle and a specific combination of integration constants appearing in the O(R4 ) corrections to the black hole background. The combination figuring in the black hole background is along the compact part S3 of the non-compact four cycle (locally) Σ(4) = R+ × S 3 wrapping the flavor D7-branes of the type IIB dual [14], hence this has the information of flavor branes in M-theory uplift which has no branes and this is known as “Flavor Memory” effect in this context.

Non-Renormalization of the Deconfinement Temperature (Tc) from Semiclassical Method: We now turn to Green and Gutperle’s argument that a SL(2, Z) completion of an effective R4 interaction results into a fascinating non-renormalization theorem that prohibits perturbative corrections to this term beyond one loop in the zero-instanton sector [73]. term that occurs in (1.47) bilinear in the tensor tˆ bears the same structure just like terms that occur in the zero instanton sector via both the one-loop four-graviton amplitude and a (α ′ ) 3 effect at tree level [115]. Thus, in the Einstein frame, one gets the expression that follows for the total effective R4 action, which can be formally represented by:

wherein the summation denotes the sum of all positive and negative p, n values except p = n = 0. Now:

The sum over p, utilizing the Poisson resummation formula

3.3 Wald Entropy, Tc from Entanglement Entropy and MχPT Compatibility

3.3.1 Wald Entropy

As a result, we are able to notice that the contribution (iii) is the one that predominates in the MQGP limit, and the black hole entropy from the higher derivative terms is obtained as:

As a result, we found that linear constraints must be placed on the constants of integration that appear in the solutions to the EOMs of the metric corrections at O(R4 ) in the directions that contain the “memory” of the compact part of the non-compact four-cycle “wrapped” by the flavor D7-branes in the type IIB dual [14] of the large-N thermal QCD-like theories.

3.3.2 Deconfinement from Entanglement Entropy

where ϕ denotes the profile of the dilaton. Due to the absence of a dilaton in a M-theory dual, the equation (3.75) will be considered to be replaced with the following expression:

Consider now the gravity dual’s metric in string frame, which may be stated as:

Hence,

where H(r) is defined as:

Entanglement entropy will be reduced to the following using equations (3.81) and (3.83).

Let us discuss the implications of (3.87).

Equations (3.2) and (3.77) allow us to deduce the following:

wherein λ1 and λ2 have been defined as follows:

After that, rewriting σ(r) as follows:

then

Entanglement Entropy: The formula for the connected region entanglement entropy that takes into account both tiny l and large r is as follows:

wherein:

and,

3.3.3 MχPT Compatibility

3.4 Deconfinement Temperature of Rotating QGP at Intermediate Coupling from M-Theory

This part is based on [5]. In this paper, we have obtained the deconfinement temperature of thermal QCD-like theories in the presence of rotation. We will discuss the holographic construction of rotating QGP from a top-down approach in 3.4.1. Using this holographic dual, we will discuss the calculation of Tc in the presence of rotation in 3.4.2 and UV-IR mixing, Flavor Memory effect, and non-renormalization of Tc in 3.4.3

3.4.1 Top-Down Holographic Dual of Rotating QGP

When T > Tc on the gravity dual side and T < Tc on the gauge theory side, we could investigate the affect of rotation in thermal QCD-like theories by introducing a rotating cylindrical black hole and thermal backgrounds on the gravity dual side. In order to obtain the background of a rotating cylinder black hole, we need to make x 3 periodic by replacing it with lϕ, where l represents the length of the cylinder and 0 ≤ ϕ ≤ 2π. This will allow us to obtain the rotating cylinder black hole background. After completing the following Lorentz transformation around the cylinder of length l, one can obtain the holographic dual of rotating quark-gluon plasma. This can be done by following [116, 117]:

where ω represents the angular velocity of the rotating cylindrical black hole in the gravity dual, which is connected to the rotation of quark-gluon plasma by means of the gauge-gravity duality. Only when ωl < 1, the Lorentz transformation, also known as the Lorentz-boost, considered to be trustworthy. Following the application of the Lorentz transformation, which is denoted by the equation (3.117), the equation (3.1) is rewritten as follows:

In order to produce the rotating cylindrical thermal background, we substitute x3 by lϕ and carry out the Lorentz transformation (3.117) of (3.2) in the same manner as we did to the black hole background. As a result of these steps, the metric of the rotating cylindrical thermal background simplified as:

3.4.2 Deconfinement Temperature of Rotating QGP from M-Theory Dual

Using a semi-classical method [4], we will compute the deconfinement temperature of a rotating QGP at intermediate coupling in this section similar to 3.2.1. On-shell action, in this case, will be the same as (3.9), and hence we computed the various terms appearing in (3.9) for the rotating cylindrical black hole (3.118) and thermal (3.122) backgrounds, and integrated over the angular coordinates similar to 3.2.1. In the upcoming pages, we will not discuss every step, and we will write the final results because the procedure is the same as 3.2.1. Only metric is different for the rotating cylindrical black hole and thermal backgrounds compared to 3.2.1.

In a procedure analogous to that of 3.2.1, the UV-finite and holographic IR regularized on-shell action density for the M-theory rotating cylindrical black hole background (3.118) was calculated as follows:

At the UV cut-off [3]:

The following is a solution to the equation presented above:

In the context of gauge-gravity duality, the Hawking Page phase transition, which occurs at T = TH(γ) on the gravity side between a thermal and black hole background is dual to the deconfinement phase transition, which occurs at T = Tc(γ) in gauge theories , i.e.,

Comparison with Other Models: The authors found identical behavior from NambuJona-Lansinio model in [120] because of chiral condensate suppression, which was investigated as well from lattice simulations in [121, 122]. Lattice simulations with different kinds of boundary conditions (open, periodic, and Dirichlet) have been carried out in rotating reference frames. In both gluodynamics and the theory of dynamical fermions, the authors investigated the impact of rotation on the critical temperature. The result found that the decrease in deconfinement temperature of thermal QCD from a rise of rotation of quark gluon plasma was previously studied using gauge-gravity duality from a bottom-up approach in [109–112]. In contrast to our finding and other holographic findings mentioned earlier, the critical temperature for gludynamics exhibits the opposite behavior. However, the authors found that the critical temperature for dynamical fermions decreases as rotation increases. The hard-wall and soft-wall models were used for the bottom-up holographic study in [109], and the Einstein-Maxwell-Dilaton system is involved in the gravity dual of [110]. The authors of [111, 112] considered the gravity dual of rotating QGP to be the Kerr-AdS black hole background in five dimensions. The deconfinement temperature of rotating QGP from a top-down model is not covered in any published article. Because top-down models compactify the initial type IIB/IIA string theory or M-theory on an internal manifold to produce four-dimensional QCD at finite temperature, they are more fundamental than bottom-up models. We successfully studied rotating QGP from a top-down holographic dual for the first time and were able to confirm the non-zero angular velocity of rotating QGP found in hydrodynamic simulation [108] and STAR collaboration [123].

3.4.3 UV-IR Mixing, Flavor Memory Effect and Non-Renormalization of Tc in Rotating QGP

Focusing on the outcomes presented in the appendix B.4. At O(β), the UV finite on-shell action densities for the rotating cylindrical black hole and thermal backgrounds are as follows:

We got the following results using the equations (3.130) and (3.131):

3.5 Summary

We obtained the following results in this chapter.

• UV-IR Mixing and Flavor Memory: During the computation of Tc from the semiclassical method [4], we obtained “UV-IR” mixing described as the connection between the O(β) corrections to the M-theory metric along the M-theory circle in thermal background and the O(β) correction to a specific combination of the M-theormetric components along the compact part of the four-cycle “wrapped” by the flavor D7-branes of the parent type IIB (warped resolved deformed) conifold geometry - the latter referred to as “Flavor Memory” in the M-theory uplift on equating the O(β) portion of the on-shell actions of the black hole and thermal backgrounds in the absence and presence of rotation both.

• Non-Renormalization of Tc:

• Wald Entropy at O(R4): In the high-temperature M-theory dual, matching of O(R4 ) computation of Wald entropy with the black hole entropy puts a linear constraint on the same linear combination of the aforementioned integration constants.

• Holographic Dual of Rotating QGP: By applying Lorentz transformations (3.117) along (t, ϕ) coordinates on the gravity dual side, we are able to derive the rotating QGP on the gauge theory side via gauge-gravity duality. Therefore, for T > Tc and T < Tc on the gauge theory side (where Tc serves as the deconfinement temperature of thermal QCD-like theories), the gravity dual includes a rotating cylinder black hole and thermal backgrounds.

• Tc in Rotating QGP: We found that the deconfinement temperature of rotating QGP (3.129) is inversely proportional to the Lorentz factor, γ, by equating the O(β 0 ) terms for the rotating cylindrical black hole and the cylindrical thermal backgrounds at the UV cut-off (3.125). This means that when the angular velocity of rotating QGP increases, the deconfinement temperature of thermal QCD-like theories decreases and vice-versa. Experimental results from the noncentral Relativistic Heavy Ion Collider (RHIC) have shown that the Quark Gluon Plasma created in these collisions has an angular velocity, and that the deconfinement temperature decreases with increasing rotation [123]. To ensure that the deconfinement temperature of rotating QGP computed using a top-down technique is in agreement with actual results, our study serves as a good check for the top-down duals of [14] and [1].