$\displaystyle S_{\rm UV}^{\rm connected}\sim\frac{M_{\rm UV}N_{f}^{\rm UV}\
{}^{\frac{4}{3}}{\cal R}_{\rm UV}^{4}\left(\log
N\right)^{\frac{4}{3}}\log{\cal R}_{\rm
UV}\alpha_{\theta_{2}}^{2}}{N^{\frac{1}{20}}\alpha_{\theta_{1}}^{6}}.$ (140)

One notes that in the large-$N$ limit, recalling ${\cal R}_{\rm UV}<\left(4\pi
g_{s}N\right)^{\frac{1}{4}}$ and setting
$\alpha_{\theta_{1}}^{3}=\alpha_{\theta_{2}}^{4}$, one sees that $S_{\rm
UV}^{\rm disconnected}=S_{\rm UV}^{\rm connected}$.

As $l$ increases, i.e., $r^{*}$ decreases and reaches ${\cal
R}_{D5/\overline{D5}}^{\rm th}$, $S_{\rm connected}$ changes from being
negative to vanishing and $S_{\rm disconnected}$ stays negative implying
disconnected region has lesser entropy. At $r^{*}=r_{\rm criticial}$, $S_{\rm
connected}=S_{\rm disconnected}$ and one sees that $r_{\rm critical}$ is the
solution of the equation: $\left(r^{*}\right)^{4}\log r^{*}=\gamma$, where
$\gamma\sim\frac{{g_{s}}^{4/3}M{N_{f}}^{8/3}\alpha_{\theta_{1}}^{3}\log^{2}(r_{0})(\log(N)-3\log(r_{0}))^{5/3}}{\pi^{4/3}{M_{\rm
UV}}{N_{f}^{\rm UV}}^{4/3}\alpha_{\theta_{2}}^{4}\log^{\frac{4}{3}}(N)}$. The
solution is
${r^{*}}=e^{\frac{1}{4}W(4\gamma)}\approx\sqrt{2}\sqrt[4]{\gamma}$. From Sec.
3, setting $M_{\rm UV},N_{f}\sim\frac{1}{\log N}$ and $\log
r_{0}=-\frac{f_{r_{0}}}{3}\log N$ (and setting $f_{r_{0}}\approx 1$ [1]), one
obtains:

$r_{\rm
critical}=\frac{\sqrt[3]{{g_{s}}}M{N_{f}}^{2/3}\alpha_{\theta_{1}}^{3/4}\log^{\frac{7}{6}}(N)}{\sqrt[4]{2}\sqrt[3]{\pi}\alpha_{\theta_{2}}}r_{0}.$
(141)

For $M=N_{f}=3,g_{s}=0.1-1$ (141) yields ${\cal
O}(1)\frac{\alpha_{\theta_{1}}^{\frac{3}{4}}}{\alpha_{\theta_{2}}}$. From
(95), one thus obtains by taking the 4D-limit effected via $M_{\rm
KK}\rightarrow 0$, or equivalently $r_{0}\rightarrow 0$:

$\displaystyle{\cal
O}(1)\frac{\alpha_{\theta_{1}}^{\frac{3}{4}}}{\alpha_{\theta_{2}}}\stackrel{{\scriptstyle\scriptsize
4D-{\rm limit}}}{{\longrightarrow}}\frac{\sqrt[4]{\frac{\kappa_{\rm GHY}^{{\rm
th},\ \beta^{0}}}{\kappa_{\rm GHY}^{{\rm bh},\ \beta^{0}}}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}\sqrt[4]{\frac{\log\left(\frac{{{\cal R}_{\rm
UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm th}}}\right)}{\log\left(\frac{{{\cal
R}_{\rm UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm bh}}}\right)}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm th}}},$ (142)

corresponding to the deconfinement temperature. As mentioned below, one would
require $\alpha_{\theta_{1}}^{3}\alpha_{\theta_{2}}^{2}=2$, which would for
$N=100,M=N_{f}=3,g_{s}=1$ would yield $l(r_{\rm critical})=3850$ (as obtained
numerically below) for $\alpha_{\theta_{2}}\sim{\cal O}(10)$.

Numerically, choosing $N=100,M_{\rm UV}=N_{f}^{\rm
UV}=0.01,M=N_{f}=3,\alpha_{\theta_{1,2}}:\alpha_{\theta_{1}}^{3}\alpha_{\theta_{2}}^{2}=2$,
and $r_{0}=N^{-\frac{f_{r_{0}}}{3}},f_{r_{0}}\approx 1$ [1], one obtains the
plots in Fig. 2.

Figure 2: Plot $S_{\rm connected}$ (blue) and $S_{\rm disconnected}$(orange)
versus $l(r^{*})$

Let us also discuss the possibility of $r^{*}\in[r_{0},{\cal
R}_{D5/\overline{D5}}]$, i.e., IR-valued $r^{*}$. and splitting
$\int_{r^{*}}^{{\cal R}_{\rm UV}}$ into $\left(\int_{r^{*}}^{{\cal
R}_{D5/\overline{D5}}}+\int_{{\cal R}_{D5/\overline{D5}}}^{{\cal R}_{\rm
UV}}\right)$, one obtains:

$\displaystyle\int_{r^{*}}^{{\cal
R}_{D5/\overline{D5}}}dr\frac{\sqrt{\sigma(r)}H(r)}{\sqrt{H(r)-H(r^{*})}}$
$\displaystyle\sim 2\sqrt{{g_{s}}}M^{2}{N_{f}}^{13/3}\int_{r^{*}}^{{\cal
R}_{D5/\overline{D5}}}r^{6}\sqrt{1-\frac{r_{0}^{4}}{r^{4}}}\log^{2}(r)(\log
N-3\log(r))^{11/3}\,dr$ $\displaystyle=\int_{r^{*}}^{{\cal
R}_{D5/\overline{D5}}}\left(2\sqrt{{g_{s}}}\log
N^{11/3}M^{2}{N_{f}}^{13/3}r^{6}\sqrt{1-\frac{r_{0}^{4}}{r^{4}}}\log^{2}(r)+O\left(\left(\log
N\right)^{8/3}\right)\right)dr$
$\displaystyle=\Biggl{[}\frac{2}{343}\sqrt{{g_{s}}}\log
N^{11/3}M^{2}{N_{f}}^{13/3}r^{7}\Biggl{(}7\log(r)\Biggl{(}7\log(r)\,_{2}F_{1}\left(-\frac{7}{4},-\frac{1}{2};-\frac{3}{4};\frac{r_{0}^{4}}{r^{4}}\right)$
$\displaystyle-2\,_{3}F_{2}\left(-\frac{7}{4},-\frac{7}{4},-\frac{1}{2};-\frac{3}{4},-\frac{3}{4};\frac{r_{0}^{4}}{r^{4}}\right)\Biggr{)}+2\,_{4}F_{3}\left(-\frac{7}{4},-\frac{7}{4},-\frac{7}{4},-\frac{1}{2};-\frac{3}{4},-\frac{3}{4},-\frac{3}{4};\frac{r_{0}^{4}}{r^{4}}\right)\Biggr{)}\Biggr{]}_{r^{*}}^{{\cal
R}_{D5/\overline{D5}}}$ $\displaystyle=\frac{4}{343}\sqrt{{g_{s}}}\log
N^{11/3}M^{2}{N_{f}}^{13/3}-\frac{2r_{0}^{4}\left(\sqrt{{g_{s}}}\log
N^{11/3}M^{2}{N_{f}}^{13/3}\right)}{27{{\cal
R}_{D5/\overline{D5}}}^{4}}+O\left(r_{0}^{5}\right)$
$\displaystyle-\frac{2}{7}\left(\sqrt{{g_{s}}}\log
N^{11/3}M^{2}{N_{f}}^{13/3}r^{*}\
{}^{7}\log^{2}(r^{*})\right)+\frac{1}{3}\sqrt{{g_{s}}}\log
N^{11/3}M^{2}{N_{f}}^{13/3}r_{0}^{4}r^{*}\
{}^{3}\log^{2}(r^{*})+O\left(r_{0}^{5}\right),$

and similarly,

$\displaystyle\int_{{\cal R}_{D5/\overline{D5}}}^{{\cal R}_{\rm
UV}}dr\frac{\sqrt{\sigma(r)}H(r)}{\sqrt{H(r)-H(r^{*})}}$
$\displaystyle=\int_{{\cal R}_{D5/\overline{D5}}}^{{\cal R}_{\rm
UV}}dr\frac{\sqrt[4]{\frac{1}{{g_{s}}}}\sqrt[6]{{g_{s}}}{M_{\rm
UV}}{N_{f}^{\rm
UV}}^{4/3}r^{3}\alpha_{\theta_{2}}^{2}\sqrt{1-\frac{r_{0}^{4}}{r^{4}}}\log(r)(\log
N-3\log(r))^{8/3}}{18\ 2^{2/3}3^{5/6}\pi^{25/12}\log
N^{4/3}\sqrt[20]{N}\alpha_{\theta_{1}}^{3}}+O\left(\left({N_{f}^{\rm
UV}}\right)^{2}\right)$ $\displaystyle=\int_{{\cal
R}_{D5/\overline{D5}}}^{{\cal R}_{\rm UV}}dr\frac{\log N^{4/3}{M_{\rm
UV}}{N_{f}^{\rm
UV}}^{4/3}r^{3}\alpha_{\theta_{2}}^{2}\sqrt{1-\frac{r_{0}^{4}}{r^{4}}}\log(r)}{18\
2^{2/3}3^{5/6}\pi^{25/12}\sqrt[12]{{g_{s}}}\sqrt[20]{N}\alpha_{\theta_{1}}^{3}}+{\cal
O}\left(\sqrt[3]{\log N}\right)$ $\displaystyle=\int_{{\cal
R}_{D5/\overline{D5}}}^{{\cal R}_{\rm UV}}dr\frac{\log N^{4/3}{M_{\rm
UV}}{N_{f}^{\rm UV}}^{4/3}r^{3}\alpha_{\theta_{2}}^{2}\log(r)}{18\
2^{2/3}3^{5/6}\pi^{25/12}\sqrt[12]{{g_{s}}}\sqrt[20]{N}\alpha_{\theta_{1}}^{3}}+O\left(\left(r_{0}^{4}\right)^{1}\right)$
$\displaystyle=\Biggl{[}\frac{\log N^{4/3}{M_{\rm UV}}{N_{f}^{\rm
UV}}^{4/3}r^{4}\alpha_{\theta_{2}}^{2}(4\log(r)-1)}{288\
2^{2/3}3^{5/6}\pi^{25/12}\sqrt[12]{{g_{s}}}\sqrt[20]{N}\alpha_{\theta_{1}}^{3}}\Biggr{]}_{{\cal
R}_{D5/\overline{D5}}}^{{\cal R}_{\rm UV}}$ $\displaystyle=\frac{\log
N^{4/3}{M_{\rm UV}}{N_{f}^{\rm UV}}^{4/3}{{\cal
R}_{D5/\overline{D5}}}^{4}\alpha_{\theta_{2}}^{2}}{288\
2^{2/3}3^{5/6}\pi^{25/12}\sqrt[12]{{g_{s}}}\sqrt[20]{N}\alpha_{\theta_{1}}^{3}}+\frac{\log
N^{4/3}{M_{\rm UV}}{N_{f}^{\rm UV}}^{4/3}{\cal R}_{\rm
UV}^{4}\alpha_{\theta_{2}}^{2}\log({\cal R}_{\rm UV})}{72\
2^{2/3}3^{5/6}\pi^{25/12}\sqrt[12]{{g_{s}}}\sqrt[20]{N}\alpha_{\theta_{1}}^{3}}.$
(144)

  * •

One hence notes that the “UV-divergent” terms in $S_{\rm
connected}^{r^{*}\in[{\cal R}_{D5/\overline{D5}},{\cal R}_{\rm UV}]},S_{\rm
disconnected}^{r^{*}=r_{0}},S_{\rm connected}^{r^{*}\in[r_{0},{\cal
R}_{D5/\overline{D5}}]}$ are the same, denoted by $S_{\rm UV}\sim\frac{\log
N^{4/3}{M_{\rm UV}}{N_{f}^{\rm UV}}^{4/3}{\cal R}_{\rm UV}^{4}\log({\cal
R}_{\rm UV})}{\sqrt[12]{{g_{s}}}\sqrt[20]{N}}$.

  * •

Further, $S_{\rm connected}^{r^{*}\in[r_{0},{\cal
R}_{D5/\overline{D5}}]}-S_{\rm UV}\sim\sqrt{{g_{s}}}\log
N^{11/3}M^{2}{N_{f}}^{13/3}$.

  * •

One therefore obtains (similar to some of the examples in [2]):

$\displaystyle S_{\rm connected}^{r^{*}\in[r_{0},{\cal
R}_{D5/\overline{D5}}]}-S_{\rm UV}>S_{\rm connected}^{r^{*}\in[{\cal
R}_{D5/\overline{D5}},{\cal R}_{\rm UV}]}>S_{\rm
disconnected}^{r^{*}=r_{0}}-S_{\rm UV},\ l(r^{*})>l(r_{\rm critical}),$
$\displaystyle S_{\rm connected}^{r^{*}\in[r_{0},{\cal
R}_{D5/\overline{D5}}]}-S_{\rm UV}>S_{\rm connected}^{r^{*}\in[{\cal
R}_{D5/\overline{D5}},{\cal R}_{\rm UV}]}-S_{\rm UV}<S_{\rm
disconnected}^{r^{*}=r_{0}}-S_{\rm UV},\ l(r^{*})<l(r_{\rm critical}).$

##  6 ${\mathscr{M}}\chi$PT Compatibility

In [1], we worked out one-loop renormalised LECs of $SU(3)$ chiral
perturbation theory Lagrangian up to ${\cal O}(p^{4})$ from type IIA string
dual of large-$N$ thermal QCD like-theory inclusive of ${\cal O}(R^{4})$
corrections. We constrained ourselves to the chiral limit due to which some
terms were absent in $SU(3)$ chiral perturbation theory Lagrangian of Gasser
and Leutwyler [5]. Therefore we were able to calculate
$L_{1,2,3,9,10}^{r},F_{\pi}$ and $g_{YM}$. We match our results with
phenomenological values of one-loop renormalised LECs as given in [32]. Let us
discuss this in some detail.

The meson sector in the type IIA dual background of top-down holographic type
IIB setup is given by the flavor $D6$-branes action. Restricting to the Ouyang
embedding (1) for a vanishingly small $|\mu_{\rm Ouyang}|$, one will assume
that the embedding of the flavor $D6$-branes will be given by
$\iota:\Sigma^{1,6}\Bigg{(}R^{1,3},r,\theta_{2}\sim\frac{\alpha_{\theta_{2}}}{N^{\frac{3}{10}}},y\Bigg{)}\hookrightarrow
M^{1,9}$, effected by: $z=z(r)$. As obtained in [39] one sees that
$z$=constant is a solution and by choosing $z=\pm{\cal C}\frac{\pi}{2}$, one
can choose the $D6/\overline{D6}$-branes to be at “antipodal" points along the
z coordinate. As in [39], we will be working with redefined $(r,z)$ in terms
of new variables $(Y,Z)$:

$r=r_{0}e^{\sqrt{Y^{2}+Z^{2}}}$ $z={\cal C}\arctan\frac{Z}{Y}.$

Vector mesons are obtained by considering gauge fluctuations of a background
gauge field along the world volume of the embedded flavor $D6$-branes (with
world volume
${\Sigma}_{7}(x^{0,1,2,3},Z,\theta_{2},\tilde{y})={\Sigma}_{2}(\theta_{2},\tilde{y})\times{\Sigma}_{5}(x^{0,1,2,3},Z)$).
Turning on a gauge field fluctuation $\tilde{F}$ about a small background
gauge field $F_{0}$ and the backround $i^{*}(g+B)[i:\Sigma_{7}\hookrightarrow
M_{10}$, $M_{10}$ being the ten-dimensional ambient space-time]. $Y=0$ is the
SYZ mirror of the Ouyang embedding [39]. Picking up terms quadratic in
$\tilde{F}$ in the DBI action:

${\rm
S}^{IIA}_{D6}=\frac{T_{D_{6}}(2\pi\alpha^{\prime})^{2}}{4}\left(\frac{\pi
L^{2}}{r_{0}}\right){\rm
Str}\int\prod_{i=0}^{3}dx^{i}dZd\theta_{2}dy\delta\Bigg{(}\theta_{2}-\frac{\alpha_{\theta_{2}}}{N^{\frac{3}{10}}}\Bigg{)}e^{-\Phi}\sqrt{-{\rm
det}_{\theta_{2}y}(\iota^{*}(g+B))}\sqrt{{\rm
det}_{{\mathbb{R}}^{1,3},Z}(\iota^{*}g)}g^{\tilde{\mu}\tilde{\nu}}\tilde{F}_{\tilde{\nu}\tilde{\rho}}g^{\tilde{\rho}\tilde{\sigma}}\tilde{F}_{\tilde{\sigma}\tilde{\mu}},$
(145)

where $\tilde{\mu}=i(=0,1,2,3),Z$. Expanding five dimensional gauge fields as:
$A_{\mu}(x^{\nu},Z)=\sum_{n=1}^{\infty}\rho^{(n)}_{\mu}(x^{\nu})\psi_{n}(Z)$
and $A_{Z}(x^{\nu},Z)=\sum_{n=0}^{\infty}\pi^{(n)}(x^{\nu})\phi_{n}(Z)$. After
a gauge transformation given below:

$\rho_{\mu}^{(n)}\rightarrow\rho_{\mu}^{(n)}+{\cal
M}_{(n)}^{-1}\partial_{\mu}\pi^{(n)}.$

Action simplified as given below:

$\displaystyle-\int d^{3}x\ \ {\rm
tr}\left[\frac{1}{2}\partial_{\mu}\pi^{(0)}\partial^{\mu}\pi^{(0)}+\sum_{n\geq
1}\left(\frac{1}{4}\tilde{F}^{(n)}_{\mu\nu}\tilde{F}^{(n)\mu\nu}+\frac{m_{n}^{2}}{2}\rho^{(n)}_{\mu}\rho^{(n)\mu}\right)\right].$
(146)

Working in the $A_{Z}(x^{\mu},Z)=0$-gauge, integrating out all higher order
vector and axial vector meson fields except keeping only the lowest vector
meson field [40]
$V_{\mu}^{(1)}(x^{\mu})=g\rho_{\mu}(x^{\mu})=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}}\left(\rho_{\mu}^{0}+\omega_{\mu}\right)&\rho_{\mu}^{+}&K_{\mu}^{*+}\\\
\rho_{\mu}^{-}&-\frac{1}{\sqrt{2}}\left(\rho_{\mu}^{0}-\omega_{\mu}\right)&K_{\mu}^{*0}\\\
K_{\mu}^{*-}&{\bar{K}}_{\mu}^{*0}&\phi_{\mu}\end{array}\right)$ and lightest
pseudo-scalar meson field i.e.
$\pi=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta_{8}+\frac{1}{\sqrt{3}}\eta_{0}&\pi^{+}&K^{+}\\\
\pi^{-}&-\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta_{8}+\frac{1}{\sqrt{3}}\eta_{0}&K^{0}\\\
K^{-}&{\bar{K}}^{0}&-\frac{2}{\sqrt{6}}\eta_{8}+\frac{1}{\sqrt{3}}\eta_{0}\end{array}\right)$
meson, the gauge field $A_{\mu}(x^{\nu},Z)$ up to ${\cal O}(\pi)$ is given by:

$\displaystyle
A_{\mu}(x^{\nu},Z)=\frac{\partial_{\mu}\pi}{F_{\pi}}\psi_{0}(Z)-V_{\mu}(x^{\nu})\psi_{1}(Z),$
(147)

where
$\psi_{0}(z)=\int^{Z}_{0}dZ^{\prime}\phi_{0}(Z^{\prime}),V_{\mu}^{(1)}(x^{\nu})=\rho^{(1)}_{\mu}-\frac{1}{{\cal
M}_{(1)}}\partial_{\mu}\pi^{(1)}$.

To introduce external vector ${\cal V}_{\mu}$ and axial vector fields ${\cal
A}_{\mu}$, one could use the Hidden Local Symmetry (HLS) formalism of [40] and
references therein, wherein
$\frac{1}{F_{\pi}}\partial_{\mu}\pi\rightarrow\hat{\alpha}_{\mu\perp}=\frac{1}{F_{\pi}}\partial_{\mu}\pi+{\cal
A}_{\mu}-\frac{i}{F_{\pi}}[{\cal V}_{\mu},\pi]+\cdots$ , and one also works
with $\hat{\alpha}_{\mu||}\equiv-V_{\mu}+{\cal
V}_{\mu}-\frac{i}{2F_{\pi}^{2}}[\partial_{\mu}\pi,\pi]+\cdots$. Mode expansion
of the gauge field contains infinite number of vector meson fields
$V_{\mu}^{(n)}(x^{\mu})$ and axial vector meson fields
$A_{\mu}^{(n)}(x^{\mu})$, therefore to obtain the low energy effective theory
of QCD, we need to truncate the KK spectrum in such a way that we are left
with lowest vector meson field $(V_{\mu}^{(1)}=g\rho_{\mu})$ and lightest
psuedo-scalar meson field ($\pi$ meson)[40],

$A_{\mu}(x^{\mu},z)=\hat{\alpha}_{\mu\perp}(x^{\mu})\psi_{0}(z)+(\hat{\alpha}_{\mu||}(x^{\mu})+V_{\mu}^{(1)}(x^{\mu}))+\hat{\alpha}_{\mu||}(x^{\mu})\psi_{1}(z),$
(148)

therefore,

$\displaystyle
F_{\mu\nu}=-V_{\mu\nu}\psi_{1}+v_{\mu\nu}(1+\psi_{1})+a_{\mu\nu}\psi_{0}-i[\hat{\alpha}_{\mu||},\hat{\alpha}_{\nu||}]\psi_{1}(1+\psi_{1})+i[\hat{\alpha}_{\mu\perp},\hat{\alpha}_{\nu\perp}](1+\psi_{1}-\psi_{0}^{2})$
$\displaystyle-i([\hat{\alpha}_{\mu\perp},\hat{\alpha}_{\nu||}]+[\hat{\alpha}_{\mu||},\hat{\alpha}_{\nu\perp}])\psi_{1}\psi_{0}.$
(149)

Based on [41], as regards a chiral power counting, one notes that
$M_{\rho}\equiv{\cal O}(p)$ implying $\hat{\alpha}_{\nu||}\equiv\frac{{\cal
O}(p^{3})}{M_{\rho}^{2}}\equiv{\cal O}(p),\hat{\alpha}_{\nu\perp}\equiv{\cal
O}(p)$. Further, $V_{\mu\nu},a_{\mu\nu}$ and $v_{\mu\nu}$ are of ${\cal
O}(p^{2})$. Hence, using (6), $\left(F_{\mu\nu}F^{\mu\nu}\right)^{m}$ is of
${\cal O}(p^{4m}),m\in\mathbb{Z}^{+}$. Defining parity as
$x^{i}\rightarrow-x^{i}$, $i$ indexing the conformally Minkowskian spatial
directions and $Z\rightarrow-Z$, given that $A_{\mu}(x,Z)$ will be odd,
$\alpha_{\mu\perp}$ will be even, $\alpha_{\mu||}$ will be odd and $V_{\mu}$
will be odd implies $\psi_{0}(Z)$ will be odd and $\psi_{1}(Z)$ will be even.
As coupling constants are assumed to be scalars and they are given by
$Z$-integrals, the $Z$-dependent terms in the action must be separately of
even-$Z$ parity. As $\psi_{0}$ has odd $Z$-parity and $\psi_{1}$ has even
$Z$-parity, therefore at ${\cal O}(p^{4})$, terms with
$(3\hat{\alpha}_{\mu||}s\ ,\ 1\hat{\alpha}_{\mu\perp}\ {\rm or}\
3\hat{\alpha}_{\mu\perp}s\ ,\ 1\hat{\alpha}_{\mu||})$, are dropped as they
involve coefficients of the type $\psi_{0}^{2m+1}\psi_{1}^{2n}(Z)$ for
appropriate postive integral values of $n,m$. Similarly, at ${\cal O}(p^{2})$,
$tr(\hat{\alpha}_{\mu\perp}\hat{\alpha}_{||}^{\mu})$ accompanied by
$\dot{\psi_{0}}\dot{\psi_{1}}(\ ^{.}\equiv\frac{d}{dZ})$ of odd-$Z$ parity, is
dropped. At ${\cal O}(p^{4})$, one hence obtains [40]:

$\displaystyle{\mathcal{L}}_{(4)}\ni y_{1}\,{\rm
tr}[{\hat{\alpha}}_{\mu\perp}{\hat{\alpha}}^{\mu}_{\perp}{\hat{\alpha}}_{\nu\perp}{\hat{\alpha}}^{\nu}_{\perp}]+y_{2}\,{\rm
tr}[{\hat{\alpha}}_{\mu\perp}{\hat{\alpha}}_{\nu\perp}{\hat{\alpha}}^{\mu}_{\perp}{\hat{\alpha}}^{\nu}_{\perp}]+y_{3}\,{\rm
tr}[{\hat{\alpha}}_{\mu||}{\hat{\alpha}}^{\mu}_{||}{\hat{\alpha}}_{\nu||}{\hat{\alpha}}^{\nu}_{||}]+y_{4}\,{\rm
tr}[{\hat{\alpha}}_{\mu||}{\hat{\alpha}}_{\nu||}{\hat{\alpha}}^{\mu}_{||}{\hat{\alpha}}^{\nu}_{||}]$
$\displaystyle+y_{5}\,{\rm
tr}[{\hat{\alpha}}_{\mu\perp}{\hat{\alpha}}^{\mu}_{\perp}{\hat{\alpha}}_{\nu||}{\hat{\alpha}}^{\nu}_{||}]+y_{6}\,{\rm
tr}[{\hat{\alpha}}_{\mu\perp}{\hat{\alpha}}_{\nu\perp}{\hat{\alpha}}^{\mu}_{||}{\hat{\alpha}}^{\nu}_{||}]+y_{7}\,{\rm
tr}[{\hat{\alpha}}_{\mu\perp}{\hat{\alpha}}_{\nu\perp}{\hat{\alpha}}^{\nu}_{||}{\hat{\alpha}}^{\mu}_{||}]$
$\displaystyle+y_{8}\,\left\\{{\rm
tr}[{\hat{\alpha}}_{\mu\perp}{\hat{\alpha}}^{\nu}_{||}{\hat{\alpha}}_{\nu\perp}{\hat{\alpha}}^{\mu}_{||}]+{\rm
tr}[{\hat{\alpha}}_{\mu\perp}{\hat{\alpha}}^{\mu}_{||}{\hat{\alpha}}_{\nu\perp}{\hat{\alpha}}^{\nu}_{||}\right\\}+y_{9}\,{\rm
tr}[{\hat{\alpha}}_{\mu\perp}{\hat{\alpha}}_{\nu||}{\hat{\alpha}}^{\mu}_{\perp}{\hat{\alpha}}^{\nu}_{||}]$
$\displaystyle+z_{1}\,{\rm tr}[v_{\mu\nu}v^{\mu\nu}]+z_{2}\,{\rm
tr}[a_{\mu\nu}a^{\mu\nu}]+z_{3}\,{\rm tr}[v_{\mu\nu}V^{\mu\nu}]+iz_{4}\,{\rm
tr}[V_{\mu\nu}{\hat{\alpha}}^{\mu}_{\perp}{\hat{\alpha}}^{\nu}_{\perp}]$
$\displaystyle+iz_{5}\,{\rm
tr}[V_{\mu\nu}{\hat{\alpha}}^{\mu}_{||}{\hat{\alpha}}^{\nu}_{||}]+iz_{6}\,{\rm
tr}[v_{\mu\nu}{\hat{\alpha}}^{\mu}_{\perp}{\hat{\alpha}}^{\nu}_{\perp}]+iz_{7}\,{\rm
tr}[v_{\mu\nu}{\hat{\alpha}}^{\mu}_{||}{\hat{\alpha}}^{\nu}_{||}]-iz_{8}\,{\rm
tr}\left[a_{\mu\nu}\left({\hat{\alpha}}^{\mu}_{\perp}{\hat{\alpha}}^{\nu}_{||}+{\hat{\alpha}}^{\mu}_{||}{\hat{\alpha}}^{\nu}_{\perp}\right)\right]$
(150)

where:

$v_{\mu\nu}=\frac{1}{2}\left(\xi_{R}{\cal
R}_{\mu\nu}\xi^{\dagger}_{R}+\xi_{L}{\cal
L}_{\mu\nu}\xi_{L}^{\dagger}\right)\hskip 14.22636pt{\rm and}\hskip
14.22636pta_{\mu\nu}=\frac{1}{2}\left(\xi_{R}{\cal
R}_{\mu\nu}\xi^{\dagger}_{R}-\xi_{L}{\cal
L}_{\mu\nu}\xi_{L}^{\dagger}\right),$ (151)

${\cal L}_{\mu\nu}=\partial_{[\mu}{\cal L}_{\nu]}-i[{\cal L}_{\mu},{\cal
L}_{\nu}]$ and ${\cal R}_{\mu\nu}=\partial_{[\mu}{\cal R}_{\nu]}-i[{\cal
R}_{\mu},{\cal R}_{\nu}]$ and ${\cal L}_{\mu}={\cal V}_{\mu}-{\cal A}_{\mu}$
where ${\cal R}_{\mu}={\cal V}_{\mu}+{\cal A}_{\mu}$, and
$\xi_{L}^{\dagger}(x^{\mu})=\xi_{R}(x^{\mu})=e^{\frac{i\pi(x^{\mu})}{F_{\pi}}}$;
also, $V_{\mu\nu}=\partial_{[\mu}V_{\nu]}-i[V_{\mu},V_{\nu}]$, and $y_{i}$s
and $z_{j}$s are given as radial integrals involving $\psi_{0}(Z)$ and
$\psi_{1}(Z)$ (as reviewed in [1]).

In the chiral limit, the ${\cal O}(p^{4})$ $SU(3)\ \chi$PT Lagrangian is given
by [5]:

$\displaystyle L_{1}\left({\rm
Tr}(\nabla_{\mu}U^{\dagger}\nabla^{\mu}U)\right)^{2}+L_{2}\left({\rm
Tr}(\nabla_{\mu}U^{\dagger}\nabla_{\nu}U)\right)^{2}+L_{3}{\rm
Tr}\left(\nabla_{\mu}U^{\dagger}\nabla^{\mu}U\right)^{2}$ $\displaystyle-
iL_{9}Tr\left({\cal L}_{\mu\nu}\nabla^{\mu}U\nabla^{\nu}U^{\dagger}+{\cal
R}_{\mu\nu}\nabla^{\mu}U\nabla^{\nu}U^{\dagger}\right)+L_{10}Tr\left(U^{\dagger}{\cal
L}_{\mu\nu}U{\cal R}^{\mu\nu}\right)+H_{1}Tr\left({\cal L}_{\mu\nu}^{2}+{\cal
R}_{\mu\nu}^{2}\right),$

where $\nabla_{\mu}U\equiv\partial_{\mu}U-i{\cal L}_{\mu}U+iU{\cal R}_{\mu},\
U=e^{\frac{2i\pi}{F_{\pi}}}$. $SU(3)$ chiral perturbation theory Lagrangian
(6) can be obtained from the HLS Lagrangian (6) by integrating out the $\rho$
mesons as done in [41]. One hence obtains, relationships between the LECs
$y_{i},z_{i}$ of (6) and the $L_{i}$s of (6). Using these relations we have
calculated LECs of $SU(3)$ $\chi$PT Lagrangian via gauge-gravity duality in
[1].

The parameters $L_{i}$ and $H_{i}$ are renormalized at one-loop level with all
vertices in one-loop diagrams arising from the ${\cal O}(p^{2})$ terms. Using
dimensional regularization and performing renormalizations of the parameters
via [5]:

$L_{i}=L_{i}^{r}(\mu)+\Gamma_{i}\lambda(\mu)\ ,\qquad
H_{i}=H_{i}^{r}(\mu)+\Delta_{i}\lambda(\mu)\ ,$ (153)

where $\mu$ is the renormalization point, and $\Gamma_{i}$ and $\Delta_{i}$
are certain numbers given later; $\lambda(\mu)$ is the divergent part given
by:

$\lambda(\mu)=-\frac{1}{2\left(4\pi\right)^{2}}\left[\frac{1}{\bar{\epsilon}}-\ln\mu^{2}+1\right]\
,$ (154)

where

$\frac{1}{\bar{\epsilon}}=\frac{2}{4-d}-\gamma_{E}+\ln 4\pi\ ,$ (155)

$d$ being the non-radial non-compact space-time dimensionality to be set to
four. The constants $\Gamma_{i}$ and $\Delta_{i}$ for $SU(3)$ $\chi$PT theory
were worked out in [5]:

$\begin{array}[]{ccccc}\Gamma_{1}=\frac{3}{32}\ ,&\Gamma_{2}=\frac{3}{16}\
,&\Gamma_{3}=0\ ,&\Gamma_{4}=\frac{1}{8}\ ,&\Gamma_{5}=\frac{3}{8}\ ,\\\
\Gamma_{6}=\frac{11}{144}\ ,&\Gamma_{7}=0\ ,&\Gamma_{8}=\frac{5}{48}\
,&\Gamma_{9}=\frac{1}{4}\ ,&\Gamma_{10}=-\frac{1}{4}\ ;\\\
\Delta_{1}=-\frac{1}{8}\ ,&\Delta_{2}=\frac{5}{24}\ .&&&\end{array}$ (156)

The analog of the 1-loop renormalization in $\chi$PT can be understood on the
gravity dual side by noting that the latter requires holographic
renormalization as discussed in Section 3.

In [1], we had shown how, in five steps, it is possible to match the
phenomenological values of the ${\cal O}(p^{4})$ $SU(3)$ $\chi$PT Lagrangian
[5] one-loop renormalized LECs $L_{1,9,10}^{r}$ as well as $F_{\pi}^{2},g_{\rm
YM}(\Lambda_{\rm QCD}=0.4{\rm GeV},\Lambda=1.1{\rm Gev},\mu=M_{\rho})$
exactly, where $\Lambda$ is the “HLS-QCD” matching scale [41] and $\mu$ is the
renormalization scale, as well as the order of magnitude and signs of
$L_{2,3}^{r}$.

On matching LECs computed holographically with their
phenomenological/experimental values, it turns out that a particular
combination of integration constants, $({\cal C}_{zz}^{\rm th}-{\cal
C}_{\theta_{1}z}^{\rm th}+2{\cal C}_{\theta_{1}x}^{\rm th})$, is appearing in
all the LECs of $SU(3)$ chiral perturbation theory Lagrangian up to ${\cal
O}(p^{4})$. Here, ${\cal C}_{MN}^{\rm th}$ are integration constants appearing
in solutions to ${\cal O}(R^{4})$ corrections as worked out in [1] to the MQGP
metric for the thermal gravitational dual correpsonding to (29). This
combination, $({\cal C}_{zz}^{\rm th}-{\cal C}_{\theta_{1}z}^{\rm th}+2{\cal
C}_{\theta_{1}x}^{\rm th})$, encodes information about the compact part of
non-compact four cycle around which flavor $D7$-branes are wrapping in type
IIB setup.

Now, as we are working up to ${\cal O}(\beta)$ and further due to the
smallness of $\epsilon$ assuming working up to ${\cal O}(\epsilon^{2})$, where
$\epsilon$ is defined via the relation
$a=\left(\frac{1}{\sqrt{3}}+\epsilon\right)r_{0}$, simplified form of
$L_{1,2,3}^{r}$, as worked out in [1], are given below:

$\displaystyle\hskip-28.90755ptL_{1}^{r}=\frac{L_{2}^{r}}{2}=-\frac{L_{3}^{r}}{6}=\frac{1}{g_{\rm
YM}^{2}}-z_{4}+y_{2}$
$\displaystyle\hskip-28.90755pt\sim\frac{1}{{f_{r_{0}}}{g_{s}}^{8}\log
NM^{4}N_{f}^{8}\alpha_{\theta_{1}}^{3}}\Biggl{\\{}3\pi({f_{r_{0}}}+1)N^{7/5}\Biggl{(}\frac{3{g_{s}}^{9}\log
r_{0}M^{4}N_{f}^{8}}{({f_{r_{0}}}+1)^{2}}$
$\displaystyle\hskip-28.90755pt-\frac{{f_{r_{0}}}{g_{s}}^{9}\log
NM^{4}N_{f}^{8}\alpha_{\theta_{1}}^{2}}{{f_{r_{0}}}+1}-\frac{576{f_{r_{0}}}{g_{s}}^{8}\log
NM^{4}N^{\frac{1}{3}}N_{f}^{8}\alpha_{\theta_{1}}^{5}}{\pi}\left(\frac{\epsilon^{2}}{\Omega}\right)\Biggr{)}\Biggr{\\}},$
(157)

where:

$\Omega\equiv 7\beta\left({\cal C}_{zz}^{\rm th}-2{\cal C}_{\theta_{1}z}^{\rm
th}+2{\cal C}_{\theta_{1}x}^{\rm
th}\right){f_{r_{0}}}^{2}\gamma^{2}{g_{s}}^{2}\left(\log
N\right)^{2}M^{4}+3456\epsilon^{2}({f_{r_{0}}}+1)N^{2}.$ (158)

Further, as the $\log r_{0}$ in (6) is in fact $\log\left(\frac{r_{0}}{{\cal
R}_{D5/\overline{D5}}}\right)$ \- ${\cal R}_{D5/\overline{D5}}>r_{0}$ being
the $D5-\overline{D5}$ separation - one sees from (6) that in order to obtain
a positive value (as required from phenomenological value of $L_{1}^{r}$),
$\Omega<0$. Note, as shown in [1], matching with the experimental value of the
pion decay constant $F_{\pi}$ requires an $N$-suppression in
$\alpha_{\theta_{1}}$, implying the $N$ enhancement in the last term in (6) is
artificial. So, to ensure one does not pick up an ${\cal
O}\left(\frac{1}{\beta}\right)$ contribution in $L_{1}$ from
$\frac{\epsilon^{2}}{\Omega}$ in (6) and also to ensure that the third term in
(6) required to partly compensate the first two negative terms in the same (as
explained above) to produce a positive term, is not vanishingly small, from
(158), one needs to set:

$\epsilon=\lambda_{\epsilon}\frac{\sqrt{\beta}}{N}.$ (159)

Finally, combining the above observations with the requirement to match the
experimental value $L_{1}^{\rm exp}=0.64\times 10^{-3}$, as shown in [1], one
therefore requires to implement the following constraint (which follows from
the discussion above):

$\displaystyle\left({\cal C}_{zz}^{\rm th}-2{\cal C}_{\theta_{1}z}^{\rm
th}+2{\cal C}_{\theta_{1}x}^{\rm
th}\right)=-\frac{493.7(\delta+1)({f_{r_{0}}}+1)\lambda_{\epsilon}^{2}}{{f_{r_{0}}}^{2}\gamma^{2}{g_{s}}^{2}\left(\log
N\right)^{2}M^{4}},$
$\displaystyle\delta=\frac{0.053\alpha_{\theta_{1}}^{3}N^{-\frac{2{f_{r_{0}}}}{3}-1}}{{g_{s}}}.$
(160)

We hence see from (159) that $\epsilon$ provides an expansion parameter
connecting the $\frac{1}{N}$ and $\beta$ expansions. Also, together with (6),
as was noted in [1], this was the first connection between large-$N$ and
higher derivative corrections in the context of ${\cal M}$-theory dual of
large-$N$ thermal QCD-like theories.

Similarly, it was showin in [1] that:

$\displaystyle L_{9}^{r}=\frac{1}{8}\left(\frac{2}{g_{\rm
YM}^{2}}-2z_{3}-z_{4}-z_{6}\right).$
$\displaystyle=-\frac{0.0031({f_{r_{0}}}-1){g_{s}}N^{\frac{2{f_{r_{0}}}}{3}+\frac{4}{5}}}{({f_{r_{0}}}+1)\alpha_{\theta_{1}}^{3}},$
(161)

where,

${f_{r_{0}}}=1-\omega\alpha_{\theta_{1}}^{3},$ (162)

and

$\omega=\frac{4.6N^{-\frac{2{f_{r_{0}}}}{3}-\frac{4}{5}}}{{g_{s}}}.$ (163)

One sees one gets a match with the phenomenological/experimental value
$L_{9}^{\rm exp}=6.9\times 10^{-3}$. Substituting (162) - (163) into (6), one
obtains:

$\left({\cal C}_{zz}^{\rm th}-2{\cal C}_{\theta_{1}z}^{\rm th}+2{\cal
C}_{\theta_{1}x}^{\rm
th}\right)\approx-\frac{987.4\lambda_{\epsilon}^{2}}{\gamma^{2}{g_{s}}^{2}\left(\log
N\right)^{2}M^{4}}<0.$ (164)

From equation (164), it is clear that combination of constants of integration
required to have definite sign which is negative. In appendix A, we have
obtained values of these integration constants explicitly by taking the
decompactification limit (i.e. $M_{\rm KK}\rightarrow 0$ limit) of the spatial
circle $S^{1}(x^{3})$ appearing as part of the ${\mathscr{M}}$-theory metric
used in [1], i.e., in $S^{1}(x^{0})\times R^{2}\times
S^{1}\left(\frac{1}{M_{KK}}\right)$ to recover 4D QCD-like theory. In fact,
using those values we derived that the abovementioned combination of
integration constants which appears in all the LECs of $SU(3)$ $\chi$PT
Lagrangian up to ${\cal O}(p^{4})$, indeed can be made to have a negative sign
as required by matching with the phenomenological/experimental values of the
one-loop renormalised LECs of $SU(3)$ $\chi$PT Lagrangian as discussed above.
In particular from (• ‣ B), we see that close to the Ouyang embedding of the
flavor $D7$-branes in the parent type IIB dual, there occurs a delicate
cancelation between the contributions arising from the metric corrections at
${\cal O}(R^{4})$ in the ${\mathscr{M}}$ theory uplift along the
$S^{1}(\psi/z)$-fiber (considering the $S^{3}(\theta_{1},\phi_{1}/x,\psi/z)$
as an $S^{1}$-fibration over the vanishing two-cycle
$S^{2}(\theta_{1},\phi_{1}/x)$) and $S^{2}(\theta_{1},\psi/z)$ resulting in a
non-zero contribution only along $S^{2}(\theta,\phi_{1}/x)$ surviving : ${\cal
C}_{zz}^{\rm th}-2{\cal C}_{\theta_{1}z}^{\rm th}+2{\cal C}_{\theta_{1}x}^{\rm
th}=2{\cal C}_{\theta_{1}x}^{\rm
th}\sim\frac{\left(\frac{1}{N}\right)^{7/6}\Sigma_{1}}{\epsilon^{11}{g_{s}}^{9/4}\log
N^{4}{N_{f}}^{3}{r_{0}}^{5}\alpha_{\theta_{1}}^{7}\alpha_{\theta_{2}}^{6}}$
which is negative for $\Sigma_{1}<0$. Once again, the above discussion too is
a manifestation of the "Flavor Memory" effect discussed in Section 4.

##  7 Summary

Let us summarise our main results. In this paper, in the context of a top-down
holographic evaluation of the deconfinement temperature $T_{c}$ in QCD at
intermediate coupling, we obtain $T_{c}$ from the ${\mathscr{M}}$-theory dual
of large-$N$ thermal QCD-like theories (belonging to the class of theories
that display IR confinement, UV conformality with quarks in the fundamental
representation of the flavor and color symmetry groups) inclusive of the
${\cal O}(R^{4})$ corrections. In this process, there are the following
conclusions that are arrived upon.

  * •

UV-IR Mixing and Flavor Memory: Performing a semiclassical computation as
advocated in [16], by matching the actions at the deconfinement temperature of
the ${\mathscr{M}}$-theory uplifts of the thermal and black-hole backgrounds
at the UV cut-off, one sees that one obtains a relationship in the IR between
the ${\cal O}(R^{4})$ corrections to the ${\mathscr{M}}$-theory metric along
the ${\mathscr{M}}$-theory circle in the thermal background and the ${\cal
O}(R^{4})$ correction to a specific combination of the ${\mathscr{M}}$-theory
metric 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 ${\cal M}$-theory
uplift.

  * •

Non-Renormalization of $T_{c}$:

    * –

Semiclassical computation: We further show that the LO result for $T_{c}$ also
holds even after inclusion of the ${\cal O}(R^{4})$ corrections. The dominant
contribution from the ${\cal O}(R^{4})$ terms in the large-$N$ limit arises
from the $t_{8}t_{8}R^{4}$ terms (along with the sub-dominant
$\epsilon_{11}\epsilon_{11}R^{4}$ term), which from a type IIB perspective in
the zero-instanton sector, correspond to the tree-level contribution at ${\cal
O}\left((\alpha^{\prime})^{3}\right)$ as well as one-loop contribution to
four-graviton scattering amplitude and obtained from integration of the
fermionic zero modes. As from the type IIB perspective, the $SL(2,\mathbb{Z})$
completion of these $R^{4}$ terms [18] suggests that they are not renormalized
perturbatively beyond one loop in the zero-instanton sector, this therefore
suggests the non-renormalization of $T_{c}$ at all loops in
${\mathscr{M}}$-theory at ${\cal O}(R^{4})$.

    * –

$T_{c}$ from Entanglement entropy: With an obvious generalization of [2] to
${\mathscr{M}}$-theory, we calculated the entanglement entropy between two
regions by dividing one of the spatial coordinates of the thermal
${\mathscr{M}}$-theory background into a segment of finite length ${\it l}$
and its complement. Like [2], there are two RT surfaces - connected and
disconnected. There is a critical value of ${\it l}$ which is denoted by ${\it
l_{crit}}$ such that if one is below the critical value ${\it l_{crit}}$ then
it is the connected surface that dominates the entanglement entropy, and if
one is above the critical value ${\it l_{crit}}$ then it is the disconnected
surface that dominates the entanglement entropy; ${\it l}<{\it l_{crit}}$
corresponds to confining phase of large $N_{c}$ gauge theories whereas ${\it
l}>{\it l_{crit}}$ corresponds to deconfining phase of the same. This is
interpreted as confinement-deconfinement phase transition in large $N_{c}$
gauge theories.

Remarkably, when evaluating the deconfinement temperature from an entanglement
entropy computation in the thermal gravity dual, due to an exact and delicate
cancelation between the ${\cal O}(R^{4})$ corrections from a subset of the
abovementioned metric components, one sees that there are consequently no
corrections to $T_{c}$ at quartic order in the curvature supporting the
conjecture made in Section ${\bf 4}$ \- and summarized above - on the basis of
a semiclassical computation.

  * •

Deriving ${\mathscr{M}}\chi$PT-Phenomenology compatibility: As shown in [1],
matching the phenomenological value of the 1-loop renormalized coupling
constant corresponding to the ${\cal O}(p^{4})$ $SU(3)\ \chi$PT Lagrangian
term “$\left(\nabla_{\mu}U^{\dagger}\nabla^{\mu}U\right)^{2}$" with the value
obtained from the type IIA dual of thermal QCD-like theories inclusive of the
aforementioned ${\cal O}(R^{4})$ corrections, required the ${\cal O}(R^{4})$
corrections arising from contributions due to the very same abovementioned
combination of ${\mathscr{M}}$-theory metric components evaluated at the IR
cut-off, to have a definite sign (negative). The thermal supergravity
background dual to type IIB (solitonic) $D3$-branes at low temperatures,
includes $\mathbb{R}^{2}\times S^{1}(\frac{1}{M_{\rm KK}})$. By taking the
$M_{\rm KK}\rightarrow 0$ limit (to recover a boundary four-dimensional QCD-
like theory after compactifying on the base of a $G_{2}$-structure cone),
remarkably, we obtain the values of the aforementioned ${\cal O}(R^{4})$
corrections to the ${\mathscr{M}}$-theory uplift’s metric and therefore derive
the M$\chi$PT requirement of the sign of their specific relevant combination
evaluated at the IR cut-off. Close to the Ouyang embedding of the flavor
$D7$-branes in the parent type IIB dual, there occurs a delicate cancellation
between the contributions arising from the metric corrections at ${\cal
O}(R^{4})$ in the ${\mathscr{M}}$ theory uplift along the
$S^{1}(\psi/z)$-fiber (considering the $S^{3}(\theta_{1},\phi_{1}/x,\psi/z)$,
the compact part of the four-cycle $S^{3}\times\mathbb{R}_{>0}$ "wrapped" by
the flavor $D7$-branes in the parent type IIB dual of [6], as an
$S^{1}(\psi/z)$-fibration over the vanishing two-cycle
$S^{2}(\theta_{1},\phi_{1}/x)$) and $S^{2}(\theta_{1},\psi/z)$ resulting in a
non-zero contribution only along $S^{2}(\theta,\phi_{1}/x)$ surviving. This
further reinforces the "Flavor Memory" discussed earlier.

  * •

Wald Entropy at ${\cal O}(R^{4})$: Equivalence with Wald entropy for the black
hole in the high-temperature ${\mathscr{M}}$-theory dual at ${\cal O}(R^{4})$
imposes a linear constraint on the same linear combination of the
abovementioned metric corrections.

## Acknowledgement

GY is supported by a Senior Research Fellowship (SRF) from the Council of
Scientific and Industrial Research, Govt. of India. AM was partially supported
by a grant from the Council of Scientific and Industrial Research, Government
of India, grant number CSR-1477-PHY. We thank A. Ravichandran for verifying
some portion of the calculations of Sec. 3 as part of his UG research project.

##  Appendix A ${\cal O}(R^{4})$ Corrections to the ${\mathscr{M}}$-theory
metric of [3] in the MQGP limit near the $\psi=2n\pi,n=0,1,2$-branches

The ${\cal O}(\beta)$-corrected ${\mathscr{M}}$-theory metric of [3] in the
MQGP limit near the $\psi=2n\pi,n=0,1,2$-branches up to ${\cal
O}((r-r_{h})^{2})$ [and up to ${\cal O}((r-r_{h})^{3})$ for some of the off-
diagonal components along the delocalized $T^{3}(x,y,z)$] - the components
which do not receive an ${\cal O}(\beta)$ corrections, are not listed in (A) -
was worked out in [11] and is given below:

$\displaystyle G_{tt}=G^{\rm
MQGP}_{tt}\Biggl{[}1+\frac{1}{4}\frac{4b^{8}\left(9b^{2}+1\right)^{3}\left(4374b^{6}+1035b^{4}+9b^{2}-4\right)\beta
M\left(\frac{1}{N}\right)^{9/4}\Sigma_{1}\left(6a^{2}+{r_{h}}^{2}\right)\log({r_{h}})}{27\pi\left(18b^{4}-3b^{2}-1\right)^{5}\log
N^{2}{N_{f}}{r_{h}}^{2}\alpha_{\theta_{2}}^{3}\left(9a^{2}+{r_{h}}^{2}\right)}(r-{r_{h}})^{2}\Biggr{]}$
$\displaystyle G_{x^{1,2,3}x^{1,2,3}}=G^{\rm
MQGP}_{x^{1,2,3}x^{1,2,3}}\Biggl{[}1-\frac{1}{4}\frac{4b^{8}\left(9b^{2}+1\right)^{4}\left(39b^{2}-4\right)M\left(\frac{1}{N}\right)^{9/4}\beta\left(6a^{2}+{r_{h}}^{2}\right)\log({r_{h}})\Sigma_{1}}{9\pi\left(3b^{2}-1\right)^{5}\left(6b^{2}+1\right)^{4}\log
N^{2}{N_{f}}{r_{h}}^{2}\left(9a^{2}+{r_{h}}^{2}\right)\alpha_{\theta_{2}}^{3}}(r-{r_{h}})^{2}\Biggr{]}$
$\displaystyle G_{rr}=G^{\rm
MQGP}_{rr}\Biggl{[}1+\Biggl{(}-\frac{2\left(9b^{2}+1\right)^{4}b^{10}M\left(6a^{2}+{r_{h}}^{2}\right)\left((r-{r_{h}})^{2}+{r_{h}}^{2}\right)\Sigma_{1}}{3\pi\left(-18b^{4}+3b^{2}+1\right)^{4}\log
NN^{8/15}{N_{f}}\left(-27a^{4}+6a^{2}{r_{h}}^{2}+{r_{h}}^{4}\right)\alpha_{\theta_{2}}^{3}}$
$\displaystyle+{\cal C}_{zz}^{\rm bh}-2{\cal C}_{\theta_{1}z}^{\rm bh}+2{\cal
C}_{\theta_{1}x}^{\rm bh}\Biggr{)}\beta\Biggr{]}$ $\displaystyle
G_{\theta_{1}x}=G^{\rm
MQGP}_{\theta_{1}x}\Biggl{[}1+\Biggl{(}-\frac{\left(9b^{2}+1\right)^{4}b^{10}M\left(6a^{2}+{r_{h}}^{2}\right)\left((r-{r_{h}})^{2}+{r_{h}}^{2}\right)\Sigma_{1}}{3\pi\left(-18b^{4}+3b^{2}+1\right)^{4}\log
NN^{8/15}{N_{f}}\left(-27a^{4}+6a^{2}{r_{h}}^{2}+{r_{h}}^{4}\right)\alpha_{\theta_{2}}^{3}}+{\cal
C}_{\theta_{1}x}^{\rm bh}\Biggr{)}\beta\Biggr{]}$ $\displaystyle
G_{\theta_{1}z}=G^{\rm
MQGP}_{\theta_{1}z}\Biggl{[}1+\Biggl{(}\frac{16\left(9b^{2}+1\right)^{4}b^{12}\beta\left(\frac{(r-{r_{h}})^{3}}{{r_{h}}^{3}}+1\right)\left(19683\sqrt{3}\alpha_{\theta_{1}}^{6}+3321\sqrt{2}\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{3}-40\sqrt{3}\alpha_{\theta_{2}}^{4}\right)}{243\pi^{3}\left(1-3b^{2}\right)^{10}\left(6b^{2}+1\right)^{8}{g_{s}}^{9/4}\log
N^{4}N^{7/6}{N_{f}}^{3}\left(-27a^{4}{r_{h}}+6a^{2}{r_{h}}^{3}+{r_{h}}^{5}\right)\alpha_{\theta_{1}}^{7}\alpha_{\theta_{2}}^{6}}+{\cal
C}_{\theta_{1}z}^{\rm bh}\Biggr{)}\Biggr{]}$ $\displaystyle
G_{\theta_{2}x}=G^{\rm
MQGP}_{\theta_{2}x}\Biggl{[}1+\Biggl{(}\frac{16\left(9b^{2}+1\right)^{4}b^{12}\left(\frac{(r-{r_{h}})^{3}}{{r_{h}}^{3}}+1\right)\left(19683\sqrt{3}\alpha_{\theta_{1}}^{6}+3321\sqrt{2}\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{3}-40\sqrt{3}\alpha_{\theta_{2}}^{4}\right)}{243\pi^{3}\left(1-3b^{2}\right)^{10}\left(6b^{2}+1\right)^{8}{g_{s}}^{9/4}\log
N^{4}N^{7/6}{N_{f}}^{3}\left(-27a^{4}{r_{h}}+6a^{2}{r_{h}}^{3}+{r_{h}}^{5}\right)\alpha_{\theta_{1}}^{7}\alpha_{\theta_{2}}^{6}}+{\cal
C}_{\theta_{2}x}^{\rm bh}\Biggr{)}\beta\Biggr{]}$ $\displaystyle
G_{\theta_{2}y}=G^{\rm
MQGP}_{\theta_{2}y}\Biggl{[}1+\frac{3b^{10}\left(9b^{2}+1\right)^{4}M\beta\left(6a^{2}+{r_{h}}^{2}\right)\left(1-\frac{(r-{r_{h}})^{2}}{{r_{h}}^{2}}\right)\log({r_{h}})\Sigma_{1}}{\pi\left(3b^{2}-1\right)^{5}\left(6b^{2}+1\right)^{4}\log
N^{2}N^{7/5}{N_{f}}\left(9a^{2}+{r_{h}}^{2}\right)\alpha_{\theta_{2}}^{3}}\Biggr{]}$
$\displaystyle G_{\theta_{2}z}=G^{\rm
MQGP}_{\theta_{2}z}\Biggl{[}1+\Biggl{(}\frac{3\left(9b^{2}+1\right)^{4}b^{10}M\left(6a^{2}+{r_{h}}^{2}\right)\left(1-\frac{(r-{r_{h}})^{2}}{{r_{h}}^{2}}\right)\log({r_{h}})\left(19683\sqrt{6}\alpha_{\theta_{1}}^{6}+6642\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{3}-40\sqrt{6}\alpha_{\theta_{2}}^{4}\right)}{\pi\left(3b^{2}-1\right)^{5}\left(6b^{2}+1\right)^{4}{\log
N}^{2}N^{7/6}{N_{f}}\left(9a^{2}+{r_{h}}^{2}\right)\alpha_{\theta_{2}}^{3}}$
$\displaystyle+{\cal C}_{\theta_{2}z}^{\rm bh}\Biggr{)}\beta\Biggr{]}$
$\displaystyle G_{xy}=G^{\rm
MQGP}_{xy}\Biggl{[}1+\Biggl{(}\frac{3\left(9b^{2}+1\right)^{4}b^{10}M\left(6a^{2}+{r_{h}}^{2}\right)\left(\frac{(r-{r_{h}})^{2}}{{r_{h}}^{2}}+1\right)\log({r_{h}})\alpha_{\theta_{2}}^{3}\Sigma_{1}}{\pi\left(3b^{2}-1\right)^{5}\left(6b^{2}+1\right)^{4}\log
N^{2}N^{21/20}{N_{f}}\left(9a^{2}+{r_{h}}^{2}\right)\alpha_{\theta_{2l}}^{6}}+{\cal
C}_{xy}^{\rm bh}\Biggr{)}\beta\Biggr{]}$ $\displaystyle G_{xz}=G^{\rm
MQGP}_{xz}\Biggl{[}1+\frac{18b^{10}\left(9b^{2}+1\right)^{4}M\beta\left(6a^{2}+{r_{h}}^{2}\right)\left(\frac{(r-{r_{h}})^{2}}{{r_{h}}^{2}}+1\right)\log^{3}({r_{h}})\Sigma_{1}}{\pi\left(3b^{2}-1\right)^{5}\left(6b^{2}+1\right)^{4}\log
N^{4}N^{5/4}{N_{f}}\left(9a^{2}+{r_{h}}^{2}\right)\alpha_{\theta_{2}}^{3}}\Biggr{]}$
$\displaystyle G_{yy}=G^{\rm
MQGP}_{yy}\Biggl{[}1-\frac{3b^{10}\left(9b^{2}+1\right)^{4}M\left(\frac{1}{N}\right)^{7/4}\beta\left(6a^{2}+{r_{h}}^{2}\right)\log\left(\frac{r_{h}}{{\cal
R}_{D5/\overline{D5}}^{\rm
bh}}\right)\Sigma_{1}\left(\frac{(r-{r_{h}})^{2}}{r_{h}^{2}}+1\right)}{\pi\left(3b^{2}-1\right)^{5}\left(6b^{2}+1\right)^{4}\log
N^{2}{N_{f}}{r_{h}}^{2}\left(9a^{2}+{r_{h}}^{2}\right)\alpha_{\theta_{2}}^{3}}\Biggr{]}$
$\displaystyle G_{yz}=G^{\rm
MQGP}_{yz}\Biggl{[}1+\Biggl{(}\frac{64\left(9b^{2}+1\right)^{8}b^{22}M\left(\frac{1}{N}\right)^{29/12}\left(6a^{2}+{r_{h}}^{2}\right)\left(\frac{(r-{r_{h}})^{3}}{{r_{h}}^{3}}+1\right)\log\left(\frac{r_{h}}{{\cal
R}_{D5/\overline{D5}}^{\rm
bh}}\right)}{27\pi^{4}\left(3b^{2}-1\right)^{15}\left(6b^{2}+1\right)^{12}{g_{s}}^{9/4}\log
N^{6}{N_{f}}^{4}{r_{h}}^{3}\left({r_{h}}^{2}-3a^{2}\right)\left(9a^{2}+{r_{h}}^{2}\right)^{2}\alpha_{\theta_{1}}^{7}\alpha_{\theta_{2}}^{9}}$
$\displaystyle\times\left(387420489\sqrt{2}\alpha_{\theta_{1}}^{12}+87156324\sqrt{3}\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{9}+5778054\sqrt{2}\alpha_{\theta_{2}}^{4}\alpha_{\theta_{1}}^{6}-177120\sqrt{3}\alpha_{\theta_{2}}^{6}\alpha_{\theta_{1}}^{3}+1600\sqrt{2}\alpha_{\theta_{2}}^{8}\right)+{\cal
C}_{yz}^{\rm bh}\Biggr{)}\beta\Biggr{]}$ $\displaystyle G_{zz}=G^{\rm
MQGP}_{zz}\Biggl{[}1+\Biggl{(}{\cal C}_{zz}^{\rm
bh}-\frac{b^{10}\left(9b^{2}+1\right)^{4}M\left({r_{h}}^{2}-\frac{(r-{r_{h}})^{3}}{{r_{h}}}\right)\log\left(\frac{r_{h}}{{\cal
R}_{D5/\overline{D5}}^{\rm
bh}}\right)\Sigma_{1}}{27\pi^{3/2}\left(3b^{2}-1\right)^{5}\left(6b^{2}+1\right)^{4}\sqrt{{g_{s}}}\log
N^{2}N^{23/20}{N_{f}}\alpha_{\theta_{2}}^{5}}\Biggr{)}\beta\Biggr{]}$
$\displaystyle G_{x^{10}x^{10}}=G^{\rm
MQGP}_{x^{10}x^{10}}\Biggl{[}1-\frac{27b^{10}\left(9b^{2}+1\right)^{4}M\left(\frac{1}{N}\right)^{5/4}\beta\left(6a^{2}+{r_{h}}^{2}\right)\left(1-\frac{(r-{r_{h}})^{2}}{{r_{h}}^{2}}\right)\log^{3}({r_{h}})\Sigma_{1}}{\pi\left(3b^{2}-1\right)^{5}\left(6b^{2}+1\right)^{4}\log
N^{4}{N_{f}}{r_{h}}^{2}\left(9a^{2}+{r_{h}}^{2}\right)\alpha_{\theta_{2}}^{3}}\Biggr{]},$
(A1)

where $\Sigma_{1}$ is defined as:

$\displaystyle\hskip-57.81621pt\Sigma_{1}\equiv
19683\sqrt{6}\alpha_{\theta_{1}}^{6}+6642\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{3}-40\sqrt{6}\alpha_{\theta_{2}}^{4},$
(A2)

and $G^{\rm MQGP}_{MN}$ are the ${\mathscr{M}}$-theory metric components in
the MQGP limit at ${\cal O}(\beta^{0})$ [33]. The explicit dependence on
$\theta_{10,20}$ of the ${\mathscr{M}}$-theory metric components up to ${\cal
O}(\beta)$, using (32), is effected by the replacemements:
$\alpha_{\theta_{1}}\rightarrow N^{\frac{1}{5}}\sin\theta_{10},\
\alpha_{\theta_{2}}\rightarrow N^{\frac{3}{10}}\sin\theta_{20}$ in (A).

##  Appendix B Thermal $f_{MN}$ EOMs, their Solutions in the IR, $4D$-Limit
and ${\mathscr{M}}\chi$PT Compatibility

In this appendix, we discuss the independent EOMs for the metric perturbations
$f_{MN}$ of (29) close to the IR cut-off $r_{0}$ up to LO in $N$, their
solutions and constraints and values of the same in the decompactification-
limit of a spatial direction (that plays a crucial role in providing evidence
of an all-loop non-renormalization of $T_{c}$ at ${\cal O}(R^{4})$). In this
way, we are able to obtain the values of the metric perturbations (in the deep
IR) along the three-cycle $S^{3}(\theta_{1},x,z)$ \- the delocalized version
of $S^{3}(\theta_{1},\phi_{1},\psi)$ -strictly speaking along the fiber
$S^{1}(z)$ (the $S^{3}(\theta_{1},x,z)$ is an $S^{1}(z)$ fibration over the
vanishing two-cycle $S^{2}(\theta_{1},x)$) and along another two-cycle
$S^{2}(\theta_{1},z)$ (which is also an $S^{1}(z)$-fibration). A specific
linear combination of the contributions from
$\left.f_{MN}\right|_{S^{3}(\theta_{1},x,z)}$, near the Ouyang embedding in
the parent type IIB dual, appears ubiquitously in computations of $T_{c}$ in
this work and the LECs of $SU(3)\ \chi$PT Lagrangian at ${\cal O}(p^{4})$ from
${\mathscr{M}}\ \chi$PT in [1]. We will prove that the negative sign of this
combination, as required by matching with the phenomenological values of the
LECs in the $\chi$PT Lagrangian at ${\cal O}(p^{4})$ in [1], can hence be
derived; the bonus is the actual values of the contributions appearing in the
aforementioned linear combination which further demonstrates a delicate
cancellation with the contribution along the vanishing two-cycle
$S^{2}(\theta_{1},x)$ being the only one surviving. These are manifestations
of the "Flavor Memory" as discussed in Sections 4, 6.

The EOMs are given below:

$\displaystyle{\rm EOM}_{tt}:$
$\displaystyle-\frac{\beta\left(\frac{1}{N}\right)^{9/4}\left(19683\sqrt{6}\alpha_{\theta_{1}}^{6}+6642\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{3}-40\sqrt{6}\alpha_{\theta_{2}}^{4}\right)\log({r_{0}})}{156728328192\pi^{3}{g_{s}}^{5}\log
N^{4}M{N_{f}}^{3}\epsilon^{10}\alpha_{\theta_{2}}^{3}}$
$\displaystyle\times\left(-\frac{98\pi^{3}(2{f_{zz}}({r_{0}})-3{f_{x^{10}x^{10}}}({r_{0}})-4{f_{\theta_{1}z}}({r_{0}})-5{f_{\theta_{2}y}}({r_{0}}))}{\log^{2}({r_{0}})}-\frac{1728{g_{s}}^{3}M^{2}\left(\frac{1}{N}\right)^{2/5}{N_{f}}^{2}({f_{yy}}({r_{0}})-2{f_{yz}}({r_{0}}))}{\alpha_{\theta_{2}}^{2}}\right)=0$
$\displaystyle{\rm EOM}_{\theta_{1}\theta_{1}}:$
$\displaystyle\frac{81\alpha_{\theta_{1}}^{2}\left(\frac{1152{g_{s}}^{3}M^{2}\left(\frac{1}{N}\right)^{2/5}{N_{f}}^{2}\log^{2}({r_{0}})\left(\alpha_{\theta_{2}}{f_{zz}}({r_{0}})-2\alpha_{\theta_{2}}{f_{yz}}({r_{0}})+\alpha_{\theta_{2}}{f_{yy}}({r_{0}})\right)}{\pi^{3}}-98\alpha_{\theta_{2}}^{3}(2{f_{x^{10}x^{10}}}({r_{0}})+{f_{\theta_{2}y}}({r_{0}}))\right)}{1024\alpha_{\theta_{2}}^{5}}$
$\displaystyle+\frac{\beta
M\left(\frac{1}{N}\right)^{19/10}\left(-19683\sqrt{6}\alpha_{\theta_{1}}^{6}-6642\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{3}+40\sqrt{6}\alpha_{\theta_{2}}^{4}\right)}{4782969\pi^{5/4}\sqrt[4]{{g_{s}}}\log
N^{2}{N_{f}}{r_{0}}^{2}\epsilon^{7}\alpha_{\theta_{2}}^{5}}=0$
$\displaystyle{\rm EOM}_{\theta_{1}\theta_{2}}:$
$\displaystyle-\frac{441\sqrt{3}{r_{0}}\alpha_{\theta_{1}}^{6}(2{f_{x^{10}x^{10}}}({r_{0}})+{f_{\theta_{2}y}}({r_{0}}))}{\alpha_{\theta_{2}}^{3}}-\frac{64\sqrt{2}{g_{s}}^{3/2}M{N_{f}}{fr}({r_{0}})}{\pi^{3/2}\sqrt[5]{N}}=0$
$\displaystyle{\rm EOM}_{\theta_{2}\theta_{2}}:$
$\displaystyle\frac{{f_{zz}}({r_{0}})-{f_{x^{10}x^{10}}}({r_{0}})-2{f_{\theta_{1}z}}({r_{0}})-{fr}({r_{0}})}{9\log
N^{2}\alpha_{\theta_{1}}^{4}}+\frac{9\sqrt{6}{g_{s}}^{3/2}M{N_{f}}{r_{0}}\log^{2}({r_{0}})({f_{\theta_{1}y}}({r_{0}})-{f_{yz}}({r_{0}}))}{\pi^{3/2}\log
N^{2}N^{2/5}\alpha_{\theta_{2}}^{3}}=0$ $\displaystyle{\rm
EOM}_{\theta_{2}y}:$
$\displaystyle\frac{-\frac{49\sqrt{2}\pi^{3}\alpha_{\theta_{2}}^{3}(7{f_{zz}}({r_{0}})-15{f_{x^{10}x^{10}}}({r_{0}})-14{f_{\theta_{1}z}}({r_{0}})+5{f_{\theta_{2}y}}({r_{0}}))}{\log^{2}({r_{0}})}-864\sqrt{2}{g_{s}}^{3}M^{2}\left(\frac{1}{N}\right)^{2/5}{N_{f}}^{2}\left(2\alpha_{\theta_{2}}{f_{yz}}({r_{0}})-\alpha_{\theta_{2}}{f_{yy}}({r_{0}})\right)}{10368\sqrt[4]{\pi}{g_{s}}^{13/4}\log
N^{2}M^{2}{N_{f}}^{2}\epsilon^{2}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}^{2}}$
$\displaystyle+\frac{32\beta\left(\frac{1}{N}\right)^{3/2}\left(19683\sqrt{6}\alpha_{\theta_{1}}^{6}+6642\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{3}-40\sqrt{6}\alpha_{\theta_{2}}^{4}\right)}{3486784401{g_{s}}^{2}\log
N^{2}{N_{f}}^{2}\epsilon^{8}\alpha_{\theta_{1}}^{6}\alpha_{\theta_{2}}}=0$
$\displaystyle{\rm EOM}_{xx}:$
$\displaystyle\frac{-\frac{1024\pi^{3/2}{g_{s}}^{3}\alpha_{\theta_{2}}^{3}({f_{zz}}({r_{0}})-2{f_{\theta_{1}z}}({r_{0}})+2{f68}({r_{0}})-{fr}({r_{0}}))}{\alpha_{\theta_{1}}^{4}}-\frac{11907\pi^{9/2}\left(\frac{1}{N}\right)^{2/5}\alpha_{\theta_{2}}^{5}(3{f_{x^{10}x^{10}}}({r_{0}})+5{f_{\theta_{2}y}}({r_{0}}))}{\log
N^{2}M^{2}{N_{f}}^{2}\log^{2}({r_{0}})}}{279936\pi^{2}{g_{s}}^{7/2}\epsilon^{2}\alpha_{\theta_{2}}^{5}}$
$\displaystyle+\frac{64\beta\left(\frac{1}{N}\right)^{23/20}}{43046721\sqrt[4]{\pi}{g_{s}}^{9/4}\log
N^{3}{N_{f}}^{2}\epsilon^{7}\alpha_{\theta_{1}}^{4}\alpha_{\theta_{2}}^{4}}=0$
$\displaystyle{\rm EOM}_{yy}:$
$\displaystyle\frac{49\pi^{5/2}\left(16{f_{zz}}({r_{0}})+3{f_{x^{10}x^{10}}}({r_{0}})+4{f_{\theta_{1}z}}({r_{0}})+\frac{18\alpha_{\theta_{2}}^{2}{f_{\theta_{1}y}}({r_{0}})}{\sqrt[5]{N}\alpha_{\theta_{1}}^{2}}+5{f_{\theta_{2}y}}({r_{0}})-36{f_{yz}}({r_{0}})+18{f_{yy}}({r_{0}})\right)}{1152{g_{s}}^{7/2}\log
N^{2}M^{2}{N_{f}}^{2}\epsilon^{2}\log^{2}({r_{0}})}$
$\displaystyle+\frac{2\beta\left(19683\alpha_{\theta_{1}}^{6}+1107\sqrt{6}\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{3}-40\alpha_{\theta_{2}}^{4}\right)}{387420489\sqrt[4]{\pi}{g_{s}}^{9/4}\log
N^{3}N^{3/2}{N_{f}}^{2}\epsilon^{9}\alpha_{\theta_{1}}^{4}\alpha_{\theta_{2}}^{2}}=0.$

The solutions to (B) are given as under:

$\displaystyle f_{t}(r)=f_{t}(r_{0}),$ $\displaystyle f(r)=f(r_{0}),$
$\displaystyle
f_{r}(r)=-\frac{99\sqrt{\frac{3}{2}}\beta{g_{s}}^{3/2}M\sqrt[5]{\frac{1}{N}}{N_{f}}{r_{0}}\alpha_{\theta_{1}}^{6}{f_{x^{10}x^{10}}}({r_{0}})\log^{2}({r_{0}})}{2\pi^{3/2}\alpha_{\theta_{2}}^{5}},$
$\displaystyle f_{\theta_{1}\theta_{1}}(r)=f_{\theta_{1}\theta_{1}}(r_{0}),$
$\displaystyle f_{\theta_{1}\theta_{2}}(r)=f_{\theta_{1}\theta_{2}}(r_{0}),$
$\displaystyle
f_{\theta_{1}x}(r)=-\frac{99\sqrt{\frac{3}{2}}{g_{s}}^{3/2}M\sqrt[5]{\frac{1}{N}}{N_{f}}{r_{0}}\alpha_{\theta_{1}}^{6}{f_{x^{10}x^{10}}}({r_{0}})\log^{2}({r_{0}})}{4\pi^{3/2}\alpha_{\theta_{2}}^{5}}-{f_{x^{10}x^{10}}}({r_{0}}),$
$\displaystyle f_{\theta_{1}y}(r)=f_{\theta_{1}y}(r_{0}),$ $\displaystyle
f_{\theta_{1}z}=\frac{539\pi^{3}N^{2/5}\alpha_{\theta_{2}}^{2}{f_{x^{10}x^{10}}}({r_{0}})}{1728{g_{s}}^{3}M^{2}{N_{f}}^{2}\log^{2}({r_{0}})}-\frac{185{f_{x^{10}x^{10}}}({r_{0}})}{108},$
$\displaystyle f_{\theta_{2}\theta_{2}}(r)=f_{\theta_{2}\theta_{2}}(r_{0}),$
$\displaystyle f_{\theta_{2}x}(r)=f_{\theta_{2}x}(r_{0}),$ $\displaystyle
f_{\theta_{2}y}(r)=\frac{352{g_{s}}^{3}M^{2}\left(\frac{1}{N}\right)^{2/5}{N_{f}}^{2}{f_{x^{10}x^{10}}}({r_{0}})\log^{2}({r_{0}})}{49\pi^{3}\alpha_{\theta_{2}}^{2}}-2{f_{x^{10}x^{10}}}({r_{0}}),$
$\displaystyle f_{\theta_{2}z}(r)=f_{\theta_{2}z}(r_{0}),$ $\displaystyle
f_{xx}(r)=f_{xx}(r_{0}),$ $\displaystyle f_{xy}(r)=f_{xy}(r_{0}),$
$\displaystyle f_{xz}(r)=f_{xz}(r_{0}),$ $\displaystyle
f_{yy}(r)=\frac{N^{2/5}{f_{x^{10}x^{10}}}({r_{0}})\left(32\sqrt{6}\pi^{3/2}{g_{s}}^{3/2}M{N_{f}}\alpha_{\theta_{2}}^{3}-4851\pi^{3}{r_{0}}\alpha_{\theta_{1}}^{4}\alpha_{\theta_{2}}^{2}\right)}{7776{g_{s}}^{3}M^{2}{N_{f}}^{2}{r_{0}}\alpha_{\theta_{1}}^{4}\log^{2}({r_{0}})}+\frac{55{f_{x^{10}x^{10}}}({r_{0}})}{27}$
$\displaystyle\hskip 43.36243pt+{f_{\theta_{1}y}}({r_{0}}),$ $\displaystyle
f_{yz}(r)=\frac{\pi^{3/2}N^{2/5}\alpha_{\theta_{2}}^{3}{f_{x^{10}x^{10}}}({r_{0}})}{81\sqrt{6}{g_{s}}^{3/2}M{N_{f}}{r_{0}}\alpha_{\theta_{1}}^{4}\log^{2}({r_{0}})}+\frac{{f_{\theta_{1}y}}({r_{0}})}{2},$
$\displaystyle
f_{zz}(r)=\frac{539\pi^{3}N^{2/5}\alpha_{\theta_{2}}^{2}{f_{x^{10}x^{10}}}({r_{0}})}{864{g_{s}}^{3}M^{2}{N_{f}}^{2}\log^{2}({r_{0}})}-\frac{77{f_{x^{10}x^{10}}}({r_{0}})}{54}(r_{0}),$
$\displaystyle f_{x^{10}x^{10}}(r)=f_{x^{10}x^{10}}(r_{0}).$ (B2)

We will now show consistency between the solutions to the EOMs for ${\cal
O}(R^{4})$ metric perturbations as given in (B) and the ones obtained by
taking $\tilde{g}(r)\rightarrow 1$-limit of the ${\cal O}(R^{4})$ corrections
to (29) as obtained in [1]. In the $M_{\rm KK}\rightarrow 0$ limit yielding 4D
gauge theory after dimensional reduction to $M_{5}(r,t,x^{1,2,3})$,
consistency of the aforementioned pair of solutions $f_{MN}$-wise requires:

$\displaystyle
f_{t}(r)=f_{t}(r_{0})=\left.-\frac{1}{4}\frac{4b^{8}\left(9b^{2}+1\right)^{4}\left(39b^{2}-4\right)M\left(\frac{1}{N}\right)^{9/4}\beta\left(6a^{2}+{r_{0}}^{2}\right)\log({r_{0}})\Sigma_{1}}{9\pi\left(3b^{2}-1\right)^{5}\left(6b^{2}+1\right)^{4}\log
N^{2}{N_{f}}{r_{0}}^{2}\left(9a^{2}+{r_{0}}^{2}\right)\alpha_{\theta_{2}}^{3}}(r-{r_{0}})^{2}\right|_{r=r_{0}}$
$\displaystyle=f(r)=f(r_{0})=f_{x^{3}x^{3}}$
$\displaystyle=\left.\frac{1}{4}\frac{4b^{8}\left(9b^{2}+1\right)^{3}\left(4374b^{6}+1035b^{4}+9b^{2}-4\right)\beta
M\left(\frac{1}{N}\right)^{9/4}\Sigma_{1}\left(6a^{2}+{r_{0}}^{2}\right)\log({r_{0}})}{27\pi\left(18b^{4}-3b^{2}-1\right)^{5}\log
N^{2}{N_{f}}{r_{0}}^{2}\alpha_{\theta_{2}}^{3}\left(9a^{2}+{r_{0}}^{2}\right)}(r-{r_{0}})^{2}\right|_{r=r_{0}},$
$\displaystyle
f_{r}(r)=-\frac{99\sqrt{\frac{3}{2}}\beta{g_{s}}^{3/2}M\sqrt[5]{\frac{1}{N}}{N_{f}}{r_{0}}\alpha_{\theta_{1}}^{6}{f_{x^{10}x^{10}}}({r_{0}})\log^{2}({r_{0}})}{2\pi^{3/2}\alpha_{\theta_{2}}^{5}}$
$\displaystyle=-\frac{2\left(9b^{2}+1\right)^{4}b^{10}M\left(6a^{2}+{r_{0}}^{2}\right){r_{0}}^{2}\Sigma_{1}}{3\pi\left(-18b^{4}+3b^{2}+1\right)^{4}\log
NN^{8/15}{N_{f}}\left(-27a^{4}+6a^{2}{r_{0}}^{2}+{r_{0}}^{4}\right)\alpha_{\theta_{2}}^{3}}$
$\displaystyle+{\cal C}_{zz}^{\rm th}-2{\cal C}_{\theta_{1}z}^{\rm th}+2{\cal
C}_{\theta_{1}x}^{\rm th},$ $\displaystyle
f_{\theta_{1}\theta_{1}}(r)=f_{\theta_{1}\theta_{1}}(r_{0})=0,$ $\displaystyle
f_{\theta_{1}\theta_{2}}(r)=f_{\theta_{1}\theta_{2}}(r_{0})=0,$ $\displaystyle
f_{\theta_{1}x}(r)=-\frac{99\sqrt{\frac{3}{2}}{g_{s}}^{3/2}M\sqrt[5]{\frac{1}{N}}{N_{f}}{r_{0}}\alpha_{\theta_{1}}^{6}{f_{x^{10}x^{10}}}({r_{0}})\log^{2}({r_{0}})}{4\pi^{3/2}\alpha_{\theta_{2}}^{5}}-{f_{x^{10}x^{10}}}({r_{0}})$
$\displaystyle=-\frac{\left(9b^{2}+1\right)^{4}b^{10}M\left(6a^{2}+{r_{0}}^{2}\right){r_{0}}^{2}\Sigma_{1}}{3\pi\left(-18b^{4}+3b^{2}+1\right)^{4}\log
NN^{8/15}{N_{f}}\left(-27a^{4}+6a^{2}{r_{0}}^{2}+{r_{0}}^{4}\right)\alpha_{\theta_{2}}^{3}}+{\cal
C}_{\theta_{1}x}^{\rm th},$ $\displaystyle
f_{\theta_{1}y}(r)=f_{\theta_{1}y}(r_{0})=0,$ $\displaystyle
f_{\theta_{1}z}=\frac{539\pi^{3}N^{2/5}\alpha_{\theta_{2}}^{2}{f_{x^{10}x^{10}}}({r_{0}})}{1728{g_{s}}^{3}M^{2}{N_{f}}^{2}\log^{2}({r_{0}})}-\frac{185{f_{x^{10}x^{10}}}({r_{0}})}{108}$
$\displaystyle=\frac{16\left(9b^{2}+1\right)^{4}b^{12}\beta\left(19683\sqrt{3}\alpha_{\theta_{1}}^{6}+3321\sqrt{2}\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{3}-40\sqrt{3}\alpha_{\theta_{2}}^{4}\right)}{243\pi^{3}\left(1-3b^{2}\right)^{10}\left(6b^{2}+1\right)^{8}{g_{s}}^{9/4}\log
N^{4}N^{7/6}{N_{f}}^{3}\left(-27a^{4}{r_{0}}+6a^{2}{r_{0}}^{3}+{r_{0}}^{5}\right)\alpha_{\theta_{1}}^{7}\alpha_{\theta_{2}}^{6}}+{\cal
C}_{\theta_{1}z}^{\rm th},$ $\displaystyle
f_{\theta_{2}\theta_{2}}(r)=f_{\theta_{2}\theta_{2}}(r_{0})=0,$ $\displaystyle
f_{\theta_{2}x}(r)=f_{\theta_{2}x}(r_{0})$
$\displaystyle=\frac{16\left(9b^{2}+1\right)^{4}b^{12}\left(19683\sqrt{3}\alpha_{\theta_{1}}^{6}+3321\sqrt{2}\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{3}-40\sqrt{3}\alpha_{\theta_{2}}^{4}\right)}{243\pi^{3}\left(1-3b^{2}\right)^{10}\left(6b^{2}+1\right)^{8}{g_{s}}^{9/4}\log
N^{4}N^{7/6}{N_{f}}^{3}\left(-27a^{4}{r_{0}}+6a^{2}{r_{0}}^{3}+{r_{0}}^{5}\right)\alpha_{\theta_{1}}^{7}\alpha_{\theta_{2}}^{6}}+{\cal
C}_{\theta_{2}x}^{\rm th},$ $\displaystyle
f_{\theta_{2}y}(r)=\frac{352{g_{s}}^{3}M^{2}\left(\frac{1}{N}\right)^{2/5}{N_{f}}^{2}{f_{x^{10}x^{10}}}({r_{0}})\log^{2}({r_{0}})}{49\pi^{3}\alpha_{\theta_{2}}^{2}}-2{f_{x^{10}x^{10}}}({r_{0}})$
$\displaystyle=\frac{3b^{10}\left(9b^{2}+1\right)^{4}M\beta\left(6a^{2}+{r_{0}}^{2}\right)\log({r_{0}})\Sigma_{1}}{\pi\left(3b^{2}-1\right)^{5}\left(6b^{2}+1\right)^{4}\log
N^{2}N^{7/5}{N_{f}}\left(9a^{2}+{r_{0}}^{2}\right)\alpha_{\theta_{2}}^{3}},$
$\displaystyle
f_{\theta_{2}z}(r)=f_{\theta_{2}z}(r_{0})=\frac{3\left(9b^{2}+1\right)^{4}b^{10}M\left(6a^{2}+{r_{0}}^{2}\right)\log({r_{0}})}{\pi\left(3b^{2}-1\right)^{5}\left(6b^{2}+1\right)^{4}{\log
N}^{2}N^{7/6}{N_{f}}\left(9a^{2}+{r_{0}}^{2}\right)\alpha_{\theta_{2}}^{3}}$
$\displaystyle\hskip
108.405pt\times\left(19683\sqrt{6}\alpha_{\theta_{1}}^{6}+6642\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{3}-40\sqrt{6}\alpha_{\theta_{2}}^{4}\right)+{\cal
C}_{\theta_{2}z}^{\rm th},$ $\displaystyle f_{xx}(r)=f_{xx}(r_{0})=0,$
$\displaystyle
f_{xy}(r)=f_{xy}(r_{0})=\frac{3\left(9b^{2}+1\right)^{4}b^{10}M\left(6a^{2}+{r_{0}}^{2}\right)\log({r_{0}})\alpha_{\theta_{2}}^{3}\Sigma_{1}}{\pi\left(3b^{2}-1\right)^{5}\left(6b^{2}+1\right)^{4}\log
N^{2}N^{21/20}{N_{f}}\left(9a^{2}+{r_{0}}^{2}\right)\alpha_{\theta_{2l}}^{6}}+{\cal
C}_{xy}^{\rm th},$ $\displaystyle
f_{xz}(r)=f_{xz}(r_{0})=\frac{18b^{10}\left(9b^{2}+1\right)^{4}M\beta\left(6a^{2}+{r_{0}}^{2}\right)\log^{3}({r_{0}})\Sigma_{1}}{\pi\left(3b^{2}-1\right)^{5}\left(6b^{2}+1\right)^{4}\log
N^{4}N^{5/4}{N_{f}}\left(9a^{2}+{r_{0}}^{2}\right)\alpha_{\theta_{2}}^{3}},$
$\displaystyle
f_{yy}(r)=\frac{N^{2/5}{f_{x^{10}x^{10}}}({r_{0}})\left(32\sqrt{6}\pi^{3/2}{g_{s}}^{3/2}M{N_{f}}\alpha_{\theta_{2}}^{3}-4851\pi^{3}{r_{0}}\alpha_{\theta_{1}}^{4}\alpha_{\theta_{2}}^{2}\right)}{7776{g_{s}}^{3}M^{2}{N_{f}}^{2}{r_{0}}\alpha_{\theta_{1}}^{4}\log^{2}({r_{0}})}+\frac{55{f_{x^{10}x^{10}}}({r_{0}})}{27}+{f_{\theta_{1}y}}({r_{0}})$
$\displaystyle=-\frac{3b^{10}\left(9b^{2}+1\right)^{4}M\left(\frac{1}{N}\right)^{7/4}\beta\left(6a^{2}+{r_{0}}^{2}\right)\log({r_{0}})\Sigma_{1}}{\pi\left(3b^{2}-1\right)^{5}\left(6b^{2}+1\right)^{4}\log
N^{2}{N_{f}}{r_{0}}^{2}\left(9a^{2}+{r_{0}}^{2}\right)\alpha_{\theta_{2}}^{3}},$
$\displaystyle
f_{yz}(r)=\frac{\pi^{3/2}N^{2/5}\alpha_{\theta_{2}}^{3}{f_{x^{10}x^{10}}}({r_{0}})}{81\sqrt{6}{g_{s}}^{3/2}M{N_{f}}{r_{0}}\alpha_{\theta_{1}}^{4}\log^{2}({r_{0}})}+\frac{{f_{\theta_{1}y}}({r_{0}})}{2}$
$\displaystyle=\frac{64\left(9b^{2}+1\right)^{8}b^{22}M\left(\frac{1}{N}\right)^{29/12}\left(6a^{2}+{r_{0}}^{2}\right)\log({r_{0}})}{27\pi^{4}\left(3b^{2}-1\right)^{15}\left(6b^{2}+1\right)^{12}{g_{s}}^{9/4}\log
N^{6}{N_{f}}^{4}{r_{0}}^{3}\left({r_{0}}^{2}-3a^{2}\right)\left(9a^{2}+{r_{0}}^{2}\right)^{2}\alpha_{\theta_{1}}^{7}\alpha_{\theta_{2}}^{9}}$
$\displaystyle\hskip-21.68121pt\times\left(387420489\sqrt{2}\alpha_{\theta_{1}}^{12}+87156324\sqrt{3}\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{9}+5778054\sqrt{2}\alpha_{\theta_{2}}^{4}\alpha_{\theta_{1}}^{6}-177120\sqrt{3}\alpha_{\theta_{2}}^{6}\alpha_{\theta_{1}}^{3}+1600\sqrt{2}\alpha_{\theta_{2}}^{8}\right)$
$\displaystyle+{\cal C}_{yz}^{\rm th},$ $\displaystyle
f_{zz}(r)=\frac{539\pi^{3}N^{2/5}\alpha_{\theta_{2}}^{2}{f_{x^{10}x^{10}}}({r_{0}})}{864{g_{s}}^{3}M^{2}{N_{f}}^{2}\log^{2}({r_{0}})}=-\frac{b^{10}\left(9b^{2}+1\right)^{4}M{r_{0}}^{2}\log({r_{0}})\Sigma_{1}}{27\pi^{3/2}\left(3b^{2}-1\right)^{5}\left(6b^{2}+1\right)^{4}\sqrt{{g_{s}}}\log
N^{2}N^{23/20}{N_{f}}\alpha_{\theta_{2}}^{5}}$ $\displaystyle\hskip
180.67499pt+{\cal C}_{zz}^{\rm th},$ $\displaystyle
f_{x^{10}x^{10}}(r)=f_{x^{10}x^{10}}(r_{0})$
$\displaystyle=-\frac{27b^{10}\left(9b^{2}+1\right)^{4}M\left(\frac{1}{N}\right)^{5/4}\beta\left(6a^{2}+{r_{0}}^{2}\right)\log^{3}({r_{0}})\Sigma_{1}}{\pi\left(3b^{2}-1\right)^{5}\left(6b^{2}+1\right)^{4}\log
N^{4}{N_{f}}{r_{0}}^{2}\left(9a^{2}+{r_{0}}^{2}\right)\alpha_{\theta_{2}}^{3}}.$
(B3)

From (B), near $r=r_{0}$, one sees one can set:

$f_{t}=f=f_{\theta_{i}\theta_{j}}=f_{\theta_{1}y}=f_{xx}=0,$ (B4)

and for $r_{0}$-dependent values of ${\cal C}_{\theta_{2}x}^{\rm th},{\cal
C}_{\theta_{2}z}^{\rm th},{\cal C}_{xy}^{\rm th}$ determined by (B),

$f_{\theta_{2}x}=f_{\theta_{2}z}=f_{xy}=0.$ (B5)

which is what has been used in our calculations. By matching near $r=r_{0}$ of
$f_{r}(r),f_{\theta_{1}x},f_{\theta_{1}z},f_{zz}$, one can solve for ${\cal
C}_{\theta_{1}x}^{\rm th},{\cal C}_{\theta_{1}z}^{\rm th},{\cal C}_{zz}^{\rm
th},f_{x^{10}x^{10}}(r_{0})$.

The function $f_{x^{10}x^{10}}$ can independently be determined from:

  * •

matching $f_{\theta_{2}y}$: Substituting $b=\frac{1}{\sqrt{3}}+\epsilon$ [11],
[1] this yields

$f_{x^{10}x^{10}}(r_{0})\sim-\frac{M\Sigma_{1}(\alpha_{\theta_{1}},\alpha_{\theta_{2}})\log
r_{0}}{N^{\frac{7}{5}}\log^{2}NN_{f}\epsilon^{5}\alpha_{\theta_{2}}^{3}}.$
(B6)

  * •

matching $f_{yy}$: This yields

$f_{x^{10}x^{10}}(r_{0})\sim-\frac{g_{s}^{\frac{3}{2}}M^{2}\log^{3}r_{0}\alpha_{\theta_{1}}^{4}\Sigma_{1}(\alpha_{\theta_{1}},\alpha_{\theta_{2}})}{\epsilon^{5}\log^{N}r_{0}\alpha_{\theta_{2}}^{6}N^{\frac{43}{20}}}.$
(B7)

One can show that (B6) and (B7) are mutually consistent provided the IR cut-
off $r_{0}$:

$r_{0}=W\left(\frac{1}{2\sqrt{a}}\right)\approx
ae^{-\frac{4\log^{2}(-\log(a))}{\log^{2}(a)}}\log^{2}(a),$ (B8)

$W$ being the Lambert’s product log function and
$a\equiv\frac{81\sqrt{6}{g_{s}}^{3/2}M\left(\frac{1}{N}\right)^{3/4}{N_{f}}\alpha_{\theta_{1}}^{4}}{\pi^{3/2}\alpha_{\theta_{2}}^{3}}$.

  * •

matching $f_{x^{10}x^{10}}$: This yields

$f_{x^{10}x^{10}}(r_{0})\sim-\frac{M\log^{3}r_{0}\Sigma_{1}(\alpha_{\theta_{1}},\alpha_{\theta_{2}})}{\epsilon^{5}r_{0}^{2}\log^{N}N_{f}\alpha_{\theta_{2}}^{3}N^{\frac{5}{4}}}.$
(B9)

Numerically, e.g., for $M=N_{f}=3,N=10^{2},g_{s}=0.1$ and ${\cal O}(1)$
$\alpha_{\theta_{1,2}}$, one can show that (B9) is consistent with (B6) and
(B7).

  * •

matching $f_{t}(r)$

$\displaystyle-\frac{2\left(9b^{2}+1\right)^{4}b^{10}M{r_{0}}^{2}\Sigma_{1}\left(6a^{2}+{r_{0}}^{2}\right)}{3\pi\left(-18b^{4}+3b^{2}+1\right)^{4}\log
NN^{8/15}{N_{f}}\alpha_{\theta_{2}}^{3}\left(6a^{2}{r_{0}}^{2}-27a^{4}+{r_{0}}^{4}\right)}+{\cal
C}_{zz}^{\rm th}-2{\cal C}_{\theta_{1}z}^{\rm th}+2{\cal C}_{\theta_{1}x}^{\rm
th}$
$\displaystyle=\frac{99\sqrt{\frac{3}{2}}{g_{s}}^{3/2}M\sqrt[5]{\frac{1}{N}}{N_{f}}{r_{0}}\alpha_{\theta_{1}}^{6}{f_{x^{10}x^{10}}}({r_{0}})\log^{2}({r_{0}})}{2\pi^{3/2}\alpha_{\theta_{2}}^{5}}.$
(B10)

  * •

matching $f_{\theta_{1}x}$

$\displaystyle{{\cal C}^{\rm
th}_{\theta_{1}x}}-\frac{b^{10}\left(9b^{2}+1\right)^{4}M{r_{0}}^{2}\Sigma_{1}\left(6a^{2}+{r_{0}}^{2}\right)}{3\pi\left(-18b^{4}+3b^{2}+1\right)^{4}\log
NN^{8/15}{N_{f}}\alpha_{\theta_{2}}^{3}\left(6a^{2}{r_{0}}^{2}-27a^{4}+{r_{0}}^{4}\right)}$
$\displaystyle=-\frac{99\sqrt{\frac{3}{2}}{g_{s}}^{3/2}M\sqrt[5]{\frac{1}{N}}{N_{f}}{r_{0}}\alpha_{\theta_{1}}^{6}{f_{x^{10}x^{10}}}({r_{0}})\log^{2}({r_{0}})}{4\pi^{3/2}\alpha_{\theta_{2}}^{5}}-{f_{x^{10}x^{10}}}({r_{0}}).$
(B11)

  * •

matching $f_{\theta_{1}z}$

$\displaystyle\frac{8\sqrt{2}\left(9b^{2}+1\right)^{4}b^{12}\Sigma_{1}}{243\pi^{3}\left(1-3b^{2}\right)^{10}\left(6b^{2}+1\right)^{8}{g_{s}}^{9/4}\log
N^{4}N^{7/6}{N_{f}}^{3}\alpha_{\theta_{1}}^{7}\alpha_{\theta_{2}}^{6}\left(6a^{2}{r_{0}}^{3}-27a^{4}{r_{0}}+{r_{0}}^{5}\right)}+{{\cal
C}^{\rm th}_{\theta_{1}z}}$
$\displaystyle=\frac{539\pi^{3}N^{2/5}\alpha_{\theta_{2}}^{2}{f_{x^{10}x^{10}}}({r_{0}})}{1728{g_{s}}^{3}M^{2}{N_{f}}^{2}\log^{2}(r)}.$
(B12)

  * •

matching $f_{zz}$

$\displaystyle\hskip-50.58878pt{\cal C}_{zz}^{\rm
th}-\frac{b^{10}\left(9b^{2}+1\right)^{4}M{r_{0}}^{2}\Sigma_{1}\log({r_{0}})}{27\pi^{3/2}\left(3b^{2}-1\right)^{5}\left(6b^{2}+1\right)^{4}\sqrt{{g_{s}}}\log
N^{2}N^{23/20}{N_{f}}\alpha_{\theta_{2}}^{5}}=\frac{539\pi^{3}N^{2/5}\alpha_{\theta_{2}}^{2}{f_{x^{10}x^{10}}}({r_{0}})}{864{g_{s}}^{3}M^{2}{N_{f}}^{2}\log^{2}({r_{0}})}.$
(B13)

  * •

One can solve (• ‣ B) - (B13) to obtain:

$\displaystyle{\cal C}_{zz}^{\rm
th}\sim\frac{\left(\frac{1}{N}\right)^{3/4}{r_{0}}^{2}\Sigma_{1}}{\epsilon^{5}{g_{s}}^{7/2}\log
N^{2}M{N_{f}}^{3}\alpha_{\theta_{2}}^{3}\log({r_{0}})}$ $\displaystyle{\cal
C}_{\theta_{1}z}^{\rm
th}\sim\frac{\left(\frac{1}{N}\right)^{3/4}{r_{0}}^{2}\Sigma_{1}}{2\epsilon^{5}{g_{s}}^{7/2}\log
N^{2}M{N_{f}}^{3}\alpha_{\theta_{2}}^{3}\log({r_{0}})}$ $\displaystyle{\cal
C}_{\theta_{1}x}^{\rm
th}\sim\frac{\left(\frac{1}{N}\right)^{7/6}\Sigma_{1}}{\epsilon^{11}{g_{s}}^{9/4}\log
N^{4}{N_{f}}^{3}{r_{0}}^{5}\alpha_{\theta_{1}}^{7}\alpha_{\theta_{2}}^{6}}.$
(B14)

This thus confirms that ${\cal C}_{zz}^{\rm th}-2{\cal C}_{\theta_{1}z}^{\rm
th}+2{\cal C}_{\theta_{1}x}^{\rm th}=2{\cal C}_{\theta_{1}x}^{\rm th}<0$ (as
$\Sigma_{1}<0$), which in [1] was argued by requiring compatibility with
phenomenological value of the 1-loop renormalized LEC appearing in the ${\cal
O}(p^{4})$ $SU(3)\ \chi$PT
$\left(\nabla_{\mu}U^{\dagger}\nabla_{\mu}U\right)^{2}$.

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