text:
\section{Introduction}

Since in 2021 the Muon $g-2$ Collaboration at Fermilab
\cite{Muong-2:2021ojo}
has succeeded in confirming and improving the result
of the E821/BNL measurement from 2006 \cite{Muong-2:2006rrc}
for the anomalous magnetic moment of the
muon \cite{Jegerlehner:2017gek}
and is under way on further increasing its accuracy,
the existing uncertainties in the disagreeing theoretical Standard Model
result \cite{Aoyama:2020ynm} need to be scrutinized and
also improved.

Whereas QED \cite{Aoyama:2012wk,*Aoyama:2017uqe,*Aoyama:2019ryr} and electroweak contributions \cite{Czarnecki:2002nt,Gnendiger:2013pva} are
sufficiently under control, the theoretical uncertainty
is dominated by hadronic effects
\cite{Melnikov:2003xd,Prades:2009tw,Kurz:2014wya,Colangelo:2014qya,Pauk:2014rta,Davier:2017zfy,Masjuan:2017tvw,Colangelo:2017fiz,Keshavarzi:2018mgv,Colangelo:2018mtw,Hoferichter:2019gzf,Davier:2019can,Keshavarzi:2019abf,Hoferichter:2018dmo,*Hoferichter:2018kwz,Gerardin:2019vio,Bijnens:2019ghy,Bijnens:2020xnl,Colangelo:2019lpu,Colangelo:2019uex,Danilkin:2019mhd,Blum:2019ugy,Chao:2021tvp,Hoferichter:2021wyj,Danilkin:2021icn}.
The largest contribution by far is the hadronic vacuum polarization (HVP),
where a recent lattice calculation \cite{Borsanyi:2020mff}
is at variance with the result of the 2020 White Paper (WP) of the Muon $g-2$ Theory Initiative \cite{Aoyama:2020ynm} beyond the
respective estimated errors, leading to a less strong deviation
from the experimental result if the lattice result is used
in place of the data-driven one obtained in the WP.
Once this discrepancy is resolved, it will be important to
also reduce the uncertainty in the contribution from hadronic light-by-light scattering (HLBL), which at present has errors at the level of 20\%,
which in absolute numbers are comparable to the small errors aimed for in the case
of HVP.

Besides the dominant pion pole contribution to HLBL, which by now
seems to be well understood, and where data-driven approaches and
lattice evaluations agree perfectly, and the similarly well determined contributions from $\eta$ and $\eta'$ mesons, other single-meson contributions
are much less under control. An important contribution is
expected in particular from axial vector mesons, which like
pseudoscalars have anomalous couplings to photons. However,
theoretical predictions from various hadronic models vary
a lot \cite{Bijnens:2001cq,Melnikov:2003xd,Pauk:2014rta,Jegerlehner:2017gek,Roig:2019reh,Masjuan:2020jsf}, which has led to a WP estimate
of the axial vector contribution with 100\% error.

Holographic QCD models motivated by the AdS/CFT correspondence \cite{Maldacena:1997re,Witten:1998zw,Aharony:1999ti}
have proved to be remarkably successful in qualitatively and also
quantitatively describing hadronic observables,
even those with a minimal set of
parameters and the simplest geometry of anti-de Sitter space with a hard-wall (HW) cutoff.
Such AdS/QCD models are not good enough to help with the
current discrepancy between different predictions for the HVP contribution, where sub-percent accuracy is required. However,
they are certainly of interest for the HLBL contributions.

In Ref.~\cite{Leutgeb:2019zpq}, we have revisited a previous
study \cite{Cappiello:2010uy} of the pion pole contribution to HLBL and its consequences
for the value of $a_\upmu=(g-2)_\upmu/2$ using simple bottom-up AdS/QCD models
in the chiral limit and we have found a satisfactory agreement
with the data-driven and lattice approaches. The transition
form factors obtained in AdS/QCD involve infinite towers of
vector mesons, realizing vector meson dominance (VMD) in a form
that is consistent with the asymptotic behavior derived
from perturbative QCD \cite{Hoferichter:2020lap}
for both, the singly and the doubly virtual case.

In \cite{Leutgeb:2019gbz,Cappiello:2019hwh}, also the
contribution from the infinite tower of axial vector mesons and
their anomalous coupling to photons has been calculated,
and it could be shown that this takes care of the
long-standing problem that simpler hadronic models had
with the Melnikov-Vainshtein (MV) constraint \cite{Melnikov:2003xd}
on the HLBL scattering
amplitude (see \cite{Colangelo:2021nkr} for
a review assessing its impact on $a_\upmu$).
In \cite{Leutgeb:2021mpu}, we have more recently
extended these calculations to include finite quark masses
in the flavor-symmetric case. Besides demonstrating that
the saturation of the MV constraint is entirely due to
axial vector mesons also away from the chiral limit, we have
confirmed the relatively large contribution obtained in
the chiral model.

In the present paper, we consider a minimal extension of the original
hard-wall AdS/QCD model \cite{Erlich:2005qh} due to Katz and Schwartz \cite{Katz:2007tf} for solving the U(1)$_A$ problem associated with
the relatively large $\eta'$ mass.
Going slightly beyond the setup of \cite{Katz:2007tf} by
including a nonvanishing gluon condensate, we find that a very accurate
match of the masses of $\eta$ and $\eta'$ mesons as well as their
coupling strength to photons can be achieved.
We then use this model to evaluate all contributions of
pseudoscalar and
axial vector meson excitations, and thereby also the effect of the MV short-distance constraint, to the HLBL contribution to $a_\upmu$.

\section{Katz-Schwartz model: Hard-wall AdS/QCD with solved $U(1)_A$ problem}

The model proposed by Katz and Schwartz \cite{Katz:2007tf,Schafer:2007qy} for solving the $U(1)_A$ problem builds upon the original HW AdS/QCD models of Ref.~\cite{Erlich:2005qh,DaRold:2005mxj} which have turned out to
provide a remarkably good approximation to the physics of light hadrons
while introducing a minimal set of parameters.

In these models, one keeps the background geometry of pure anti-de Sitter space
with metric
\begin{equation}
ds^2=z^{-2}(\eta_{\mu\nu}dx^\mu dx^\nu - dz^2),
\end{equation}
cut off by a hard wall at a finite value of the holographic radial coordinate at $z=z_0$ with suitable boundary conditions for the five-dimensional fields
that at the conformal boundary at $z=0$ represent sources for a set of
QCD operators of interest.
In addition to five-dimensional Yang-Mills fields $\mathcal{B}^{L,R}$ dual to left and right chiral
quark currents, a bifundamental scalar $X$ representing quark-antiquark bilinears is introduced for spontaneous symmetry breaking of $U(N_f)\times U(N_f)\to U(N_f)_V$.

Confinement is implemented by
a cutoff at some finite value of the radial coordinate $z_0$,
where boundary conditions for the five-dimensional fields are imposed.

The five-dimensional Yang-Mills action
\begin{eqnarray}
S_{\rm YM} &=& -\frac{1}{4g_5^2} \int d^4x \int_0^{z_0} dz\
\sqrt{g}\, g^{PR}g^{QS}
\nonumber\\
&&\qquad\text{tr}\left(\mathcal{F}^\mathrm{L}_{PQ}\mathcal{F}^\mathrm{L}_{RS}
+\mathcal{F}^\mathrm{R}_{PQ}\mathcal{F}^\mathrm{R}_{RS}\right),
\end{eqnarray}
where $P,Q,R,S=0,\dots,3,z$ and $\mathcal{F}_{MN}=\partial_M \mathcal{B}_N-\partial_N \mathcal{B}_M-i[\mathcal{B}_M,\mathcal{B}_N]$, is
augmented by
a Chern-Simons action $S_{\rm CS}=S_{\rm CS}^\mathrm{L}-S_{\rm CS}^\mathrm{R}$
to account for flavor anomalies, reading (in
differential form notation)
\begin{equation}\label{SCS}
S_{\rm CS}^\mathrm{L,R}=\frac{N_c}{24\pi^2}\int\text{tr}\left(\mathcal{B}\mathcal{F}^2-\frac{i}2 \mathcal{B}^3\mathcal{F}
-\frac1{10}\mathcal{B}^5\right)^\mathrm{L,R},
\end{equation}
(up to a boundary term at $z_0$ that needs to be subtracted \cite{Grigoryan:2007wn,Leutgeb:2021mpu}).

The bifundamental bulk scalar $X$ is
parametrized as
\cite{Abidin:2009aj}
\begin{equation}
X=e^{i\eta^a(x,z) t^a}X_0e^{i\eta^a(x,z)t^a},
\end{equation}
where $\eta^a$, $a=0,\ldots 8$ is a nonet of pseudoscalars excitations.
The five-dimensional mass of $X$ is fixed at $M_X=-3$ by the scaling
dimension of the dual operator $\bar q_L q_R$,\footnote{In \cite{Leutgeb:2021mpu}
we have also studied the generalization to other values of $M_X$ as
proposed in \cite{Domenech:2010aq}.}
leading to a vacuum solution
\begin{equation}
X_{0ij}=\frac12 m_{ij}z+\frac12 \sigma_{ij}z^3.
\end{equation}
Choosing $N_f=3$, we restrict ourselves to the isospin symmetric case
$m_u=m_d=m_q\not=m_s$ with
$X=\frac12\text{diag}(v_q,v_q,v_s)$.

For taking care of the $U(1)_A$ problem, a massless complex field $Y$ is
introduced, representing the gluon field strength squared $\alpha_s G_{\mu\nu}^2$ by its modulus and $\alpha_s G\tilde G$ by its phase, such that the Lagrangian of scalars reads
\begin{eqnarray}
&&\mathcal{L}_{X,Y}/\sqrt{g}=
\text{tr}\left[ |DX|^2+3|X|^2\right]\nonumber\\
&&+\frac1{2(\ln z\Lambda)^2}|DY|^2+\frac{\kappa_0}{4}\left[\bar{Y}^{N_f}
\det(X)+\text{h.c.}\right],\label{LXY}
\end{eqnarray}
where  the logarithm in front of the kinetic term for $Y$ accounts for
the fact that its dual operators approach scaling dimension 4 only
asymptotically.
The complex scalar field $Y$ is charged only under the singlet axial vector field and hence its coupling is given by
\begin{eqnarray}
 D_M Y= \partial_M Y+\frac{i}{\sqrt{2 N_f}}(\mathcal{B}^{L,0}_M-\mathcal{B}^{R,0}_M)Y.
\end{eqnarray}
Without the logarithm in \eqref{LXY}, the field equations for $Y$
would give a background $\langle Y\rangle = C + \Xi z^4$, where
$\Xi$ represents a gluon condensate.
After absorbing some numerical factors into $C$, the authors of \cite{Katz:2007tf} use the axial anomaly relation and the QCD operator product expansion (OPE fit) of the flavor-singlet axial vector correlation function to find
\begin{equation}
C=\frac{\alpha_s}{2\pi^2}\sqrt{2N_f}.
\end{equation}
In QCD $\alpha_s$ is a running coupling and since in holography energy is identified with $z^{-1}$, they argue that $\alpha_s$ should be made $z$-dependent.

Matching $z\partial_z\alpha_s(z)\simeq -\beta_0\alpha_s^2$ with the one-loop
QCD $\beta$ function where $\beta_0=9/2\pi$ for $N_c=N_f=3$
gives $\alpha_s^{-1}=\beta_0\ln(\Lambda z)$, which
is adopted for all $z<z_0=\Lambda^{-1}$.
Making $\alpha_s$ and therefore $C$ depend on $z$ is of course inconsistent with the equations of motion (without the logarithm), hence we use the modified version \eqref{LXY} including the logarithms.
The presence of the logarithm in the action modifies the field equations for $Y$, leading to the general solution%
\footnote{Here we also deviate from Ref.~\cite{Katz:2007tf},
where a gluon condensate, here parametrized by $D_1$ was neglected and only $C$ of the background solution had to be modified.}

\begin{equation}\label{yvev}
\langle Y\rangle=D_0 +D_1 z^4 \left[(\ln z\Lambda)^2 -\frac12\ln z\Lambda+\frac{1}{8} \right].
\end{equation}
For later convenience we define\footnote{This is the same redefinition that \cite{Katz:2007tf} use, except for the logarithm} $\tilde{Y}_0=\frac{2}{\sqrt{2 N_f}}(-\ln z \Lambda)^{-1}\langle Y\rangle
$, which is parameterized as
\begin{equation}
\tilde{Y}_0=\frac{C_0}{-\ln z \Lambda}-\Xi_0 z^4 \big( (\ln z\Lambda) -\frac{1  }{2}+\frac{1}{8\ln z \Lambda } \big)
\end{equation}
where $C_0=\sqrt{2N_f}/(2\pi^2\beta_0)=\sqrt{2/3}/(3\pi)$.
This background now naturally incorporates the running of $\alpha_s$ consistently and permits a nonvanishing gluon condensate through nonzero values of $\Xi_0$.

The coupling constant $g_5$ can be fixed by the OPE of the
vector current correlator as
\begin{equation}\label{g5LO}
g_5^2=12\pi^2/N_c=(2\pi)^2\quad \text{(OPE fit)},
\end{equation}
but we shall alternatively consider matching the
decay constant of the $\rho$ meson,
which in the hard-wall model leads to \cite{Leutgeb:2022cvg}
\begin{equation}\label{g5Frho}
g_5^2=0.894(2\pi^2)\quad \text{($F_\rho$-fit)}.
\end{equation}
The latter leads to a significant improvement of the holographic
result for the hadronic vacuum polarization:
With the leading-order OPE fit \eqref{g5LO}, there is a deviation
of 14\% from $N_f=2$ dispersive results, which is reduced to
about 5\% with \eqref{g5Frho} \cite{Leutgeb:2022cvg}.
A reduction of $g_5^2$ by about 10\% appears to be
warranted also by comparing with next-to-leading order QCD results
for the vector correlator at moderately large $Q^2$ values \cite{Shifman:1978bx,Melic:2002ij,Bijnens:2021jqo}.
It also brings our $N_f=2$ results for the pion pole contribution to $a_\upmu$
\cite{Leutgeb:2021mpu} in line with the WP result.

With $g_5$ and $C_0$ fixed by the UV behavior, the free parameters of the model
are i) the location of the hard wall, $z_0$, which can be identified with $\Lambda^{-1}$, and will be set by the $\rho$ meson mass,
ii) quark masses in $m_{ij}=\text{diag}(m_q,m_q,m_s)$,
iii) chiral condensates $\sigma_{ij}$, which we shall assume to be
given by a single parameter, $\sigma_{ij}=\sigma\delta_{ij}$,
and iv) $\Xi_0$, which corresponds to the gluon condensate $\alpha_s\langle G^2\rangle$.
The coupling constant $\kappa_0$, on the other hand, can be
set to some sufficiently large value, since it turns out that
for $\kappa_0\gg1$ all results depend only weakly on $\kappa_0$ \cite{Katz:2007tf}.

\section{Meson modes and transition form factors}

Vector meson dominance is naturally part of this model by
relating a nonzero boundary value $\mathcal{B}^V_\mu(0)=e\mathcal{Q}A_\mu^\mathrm{e.m.}$
to the background electromagnetic potential and setting
$\mathcal{Q}=\text{diag}(\frac23,-\frac13,-\frac13)$ according to the charges of up, down, and strange quarks.
Normalizable modes of $\mathcal{B}_V$ and $\mathcal{B}_A$
correspond to vector and axial vector mesons, the longitudinal polarizations
of the latter mixing with the pseudoscalars $\eta^a$ in $X$.
Ignoring scalar excitations, which in this model do not
couple to photons,\footnote{See Ref.~\cite{Cappiello:2021vzi} for extensions of the HW model where further interactions are switched on to have scalars couple to photons in order to study their potential contribution to $a_\upmu$ in AdS/QCD.} the additional $Y$ field also involves a
pseudoscalar $a$ through its phase
\begin{equation}
Y=\langle Y\rangle \exp\left[{i2a(x,z)/\sqrt{2N_f}}\right],
\end{equation}
which corresponds to a pseudoscalar glueball in the boundary theory ($G$) that
eventually couples to photons through mixing with the flavor-singlet pseudoscalar mesons.

To determine the pseudoscalar eigenmodes in the mixed $a=(0,8)$-sector, we consider the equations of motion\footnote{We assume a summation over $a=(0,8)$ for contracted flavor indices and work in the $A_z=0$ gauge.}
\begin{eqnarray}
\label{eq:phi}
     &&\partial_z\left(\frac{1}{z} \partial_z \varphi^a_n \right) + g_5^2 \frac{M_{ab}^2}{z^3} \left(\eta^b_n-\varphi^b_n\right) \nonumber\\
     &&\qquad+\delta^{a0}  g_5^2 \frac{\tilde{Y}_0^2}{z^3} \left(a_n-\varphi^0_n\right)=0,
\end{eqnarray}
\begin{eqnarray}
\label{eq:aeq}
    &&\partial_z\left(\frac{\tilde{Y}_0^2}{z^3} \partial_z a_n \right)  + m_n^2 \frac{\tilde{Y}_0^2}{z^3} \left(a_n-\varphi^0_n\right)   \nonumber\\
    &&\qquad+ { \kappa N_f}  \frac{v_q^2 v_s }{z^5 } \left(\frac{ \tilde{Y}_{00}}{4}\right)^{N_f}  \left(\eta^0_n-a_n\right)=0,
\end{eqnarray}
\begin{eqnarray}
    &&\frac{m_n^2}{g_5^2}  \frac{1}{z} \partial_z \varphi^a_n-  \delta^{a0}\frac{{\tilde{Y}_0^2}}{z^3} \partial_z a_n     -\frac{M^2_{ab}}{z^3}\partial_z\eta^b_n=0,
\end{eqnarray}
with the longitudinal component of the axial gauge field $\partial_\mu \varphi^a=A^{a\parallel}_\mu$ and an effective 5-dimensional mass term
\begin{equation}
    M^2_{ab} = \frac{1}{3}
    \begin{pmatrix}
        2 v_q^2+ v_s^2 & \sqrt{2}( v_q^2-v_s^2)\\
         \sqrt{2}(v_q^2-v_s^2) & v_q^2+ 2 v_s^2
    \end{pmatrix},
\end{equation}
with $v_{q,s}=m_{q,s} z + \sigma z^3$.
Here we have also absorbed numerical constants in $\kappa_0$ and renamed it to $\kappa$, and we defined $\tilde{Y}_{00}=-{\tilde{Y}_0 \ln z \Lambda}/{C_0}$.

The fields $\varphi^a$ are dual to QCD operators $-\partial_{\mu}J_A^{\mu, a}$ and the glueball field $a$ is dual to $-\sqrt{2N_f}K$, where K is the instanton density $K=\frac{\alpha_s}{8 \pi}G^a_{\mu \nu}\tilde{G}^{a\mu \nu}$. This new field $a$ allows then among other things to compute overlaps of the instanton density $K$ with pseudoscalar modes $\eta, \eta',...$ and the topological susceptibility.

All fields of the normalizable modes have Dirichlet boundary conditions in the UV at $z=0$, Neumann boundary conditions in the IR at $z=z_0$ and are canonically normalized by
\begin{equation}
    \int dz \left( \frac{M_{ab}}{z^3}(\eta^a_n(\eta^b_m-\varphi^b_m))+\frac{\tilde{Y_0}^2}{z^3}a_n(a_m-\varphi^0_m) \right) = \delta_{nm}.
\end{equation}

From the Chern-Simons term \eqref{SCS} we obtain the transition form factor (TFF)
\begin{eqnarray}
    &&F_n(Q_1^2, Q_2^2)=\text{tr}(t^a \mathcal{Q}^2) F_n^{a}(Q_1^2, Q_2^2),
\end{eqnarray}
with
\begin{eqnarray}
    &&F_n^a(Q_1^2, Q_2^2)= -\frac{N_c}{2 \pi^2} \bigg( \int dz \varphi'^a_n(z) \mathcal{J}(Q_1,z) \mathcal{J}(Q_2,z) \nonumber\\
    &&\qquad-\left.\left(\varphi^a_n(z)-\eta^a_n(z)\right) \mathcal{J}(Q_1,z) \mathcal{J}(Q_2,z)\right|_{z_0} \bigg),
\end{eqnarray}
where the vector bulk-to-boundary propagator
\begin{equation}\label{HWVF}
    \mathcal{J}(Q,z)=
    Qz \left[ K_1(Qz)+\frac{K_0(Q z_0)}{I_0(Q z_0)}I_1(Q z) \right]
\end{equation}
describes virtual photons
with spacelike momentum $q^2=-Q^2$.
We can also generalize the
sum relations obtained in \cite{Leutgeb:2021mpu} to the $a=0,3,8$-sector. Most importantly, we can derive the anomaly equation
\begin{equation}
    \sum_n f_n^a F_n^{a}(0,0) = \frac{N_c}{2 \pi^2} \text{tr}(t^a \mathcal{Q}^2), \quad a=(0,3,8)
\end{equation}
and the short-distance constraint (SDC) ($Q \gg 1$)
\begin{equation}
    F_n(Q^2 (1+w), Q^2 (1-w))=\frac{N_c}{2 \pi^2}\text{tr}(t^a \mathcal{Q}^2) g_5^2 f_n^a \frac{1}{Q^2} f(w),
\end{equation}
with the asymmetry function
\begin{equation}
    f(w)=\frac{1}{w^2}-\frac{1-w^2}{2w^3}\ln\frac{1+w}{1-w},
\end{equation}
and the pseudoscalar decay constants
\begin{equation}
f_n^a=\left.-g_5^{-2}\partial_z\varphi^{a}_n/z\right|_{z\to0}.
\end{equation}
Note that the decay rate associated with the glueball field,
\begin{equation}
\left. f_G^n=\tilde{Y}_0^2 \partial_z a_n / z^3\right|_{z\to0}
\end{equation}
does not contribute to the TFF. In QCD $ f_G^n$ computes $(-\sqrt{2 N_f})\langle \Omega|K|P_n\rangle$, where $P_n$ is the respective pseudoscalar particle.

Comparing the axial vector sector to \cite{Leutgeb:2021mpu}, the $a=(0,8)$ equations of motion are changed to
\begin{eqnarray}\label{psiAnHW}
    \partial_z\left(\frac1z \partial_z \psi^a_{A,n}(z)\right)+\frac1z m_{A,n}^2 \psi^a_{A,n}(z) \nonumber
    \\
-\frac{g_5^2( M_{ab}^2+\delta_{0a} \delta_{0b}\tilde{Y}_0^2)}{z^3} \psi^b_{A,n}(z)=0,
\end{eqnarray}
and we obtain the corresponding asymptotic behavior of the axial vector TFF%
\footnote{See Ref.~\cite{Leutgeb:2019gbz} for the precise relation to the amplitude $\gamma^*\gamma^*\to A$.}
\begin{eqnarray}
    &&A_n(Q_1^2, Q_2^2)=\text{tr}(t^a \mathcal{Q}^2) A_n^{a}(Q_1^2, Q_2^2),
\end{eqnarray}
with
\begin{equation}\label{TFFAn}
    A_n^a(Q_1^2,Q_2^2) = \frac{2g_5}{Q_1^2} \int_0^{z_0} \!\!\! dz \left[ \frac{d}{dz} \mathcal{J}(Q_1,z) \right]
    \mathcal{J}(Q_2,z) \psi^{A,a}_n(z)
\end{equation}
and the SDC
\begin{equation}
    A_n^a(Q^2 (1+w), Q^2 (1-w)) \to \frac{g_5^2 F^a_{A,n}}{Q^4} f_A(w) ,
\end{equation}
with the decay constants
\begin{equation}
F_n^a=\left.-g_5^{-2}\partial_z\psi^{a}_{A,n}/z\right|_{z\to0}
\end{equation}
and the asymmetry function
\begin{equation}
    f_A(w) = \frac1{w^4} \left[ w(3-2w  + \frac12 (w+3)(1-w)\ln\frac{1-w}{1+w}\right],
\end{equation}
in agreement with the asymptotic form derived from QCD in \cite{Hoferichter:2020lap}.

The most general expression for axial vector amplitudes has actually one further asymmetric structure function \cite{Pascalutsa:2012pr,Roig:2019reh,Zanke:2021wiq}, which is set to zero in the holographic model and whose phenomenological importance has not yet been established; see Ref.~\cite{Zanke:2021wiq} for
a compilation of the available phenomenological information.

\section{Results}

\subsection{Parameter settings}

As one of the input data which we fit, we take the $\rho$ meson mass\footnote{A shortcoming of the minimal HW models considered here is that the strange quark mass modifies the vector meson masses too little compared to reality: $\rho$, $\omega$ and $\phi$ mesons are degenerate, the mass of $K^*$ is raised to only 0.79 GeV.}
$m_\rho=\gamma_{0,1}z_0^{-1}=2.40483\ldots z_0^{-1}$, where $\gamma_{0,1}$
is the first zero of the Bessel function $J_0$. Following Ref.~\cite{Abidin:2009aj}, we have chosen $z_0^{-1}=0.3225$ GeV corresponding to $m_\rho=775.556$ MeV. This fixes the location
of the hard wall, $z_0$, and $\Lambda$ in
the expression for $\alpha_s$.
The coupling $g_5$ is either set by the leading-order OPE result \eqref{g5LO} or the slightly reduced value \eqref{g5Frho} obtained by fitting the $\rho$ meson decay constant, where the TFFs reach only 89.4\% of the OPE and Brodsky-Lepage limits,
thereby coming closer to next-to-leading order results at moderately large, experimentally relevant energy scales.

The isospin-symmetric quark mass parameter $m_q$ and the chiral condensate parameter $\sigma$ are chosen such that $m_\pi=134.97$ MeV and $f_\pi=92.21$ MeV
\cite{FlavourLatticeAveragingGroupFLAG:2021npn}; the strange quark mass
parameter $m_s$ is chosen such that \cite{Brunner:2015oga}
\begin{eqnarray}
m_K^2&=&\frac12(m_{K_\pm}^2+m_{K_0}^2)-
\frac12(m_{\pi_\pm}^2-m_{\pi_0}^2)\nonumber\\
&&= (495.007\mathrm{MeV})^2
\end{eqnarray}
in order to minimize isospin-breaking contributions.

For the two choices of $g_5$, we consider the model with and without a gluon condensate parameter $\Xi_0$. When $\Xi_0=0$ (referred to as model version v0 in the following), we obtain predictions for $m_\eta$
and $m_\eta'$ that are around 10\% lower than the real-world values, in accordance
with Ref.~\cite{Katz:2007tf} who had omitted to turn on a nonzero $\Xi_0$.
Fitting $\Xi_0$ such that $(1-m^{exp}_{\eta}/m^{th}_{\eta})^2+(1-m^{exp}_{\eta'}/m^{th}_{\eta'})^2$ is minimized (model v1), $m_\eta$
and $m_\eta'$ can be matched at the percent level, as shown in
Table \ref{tab:etas}.
In the four versions of our model, we have chosen a large value of  $\kappa=700$, in order to be in the regime where the dependence on $\kappa$
is rather weak.

\subsection{Decay constants and photon coupling}

Up to the slightly different choice of $f_\pi$, the results for the mesons in the isotriplet sector, where $\Xi_0$ does not play a role, are identical to the
HW1m model presented in \cite{Leutgeb:2021mpu} for $g_5=2\pi$. Table \ref{tab:pia1}
generalizes this to the case where $g_5$ is fitted to match $F_\rho$.

In Tables \ref{tab:etas} and \ref{tab:AV}, detailed results for the two versions v0 and v1 are given for the first few pseudoscalar and axial vector modes in the isosinglet sector, showing their mixing behavior in the decay constants $f^8$, $f^0$, $f_G$ for the $\eta$'s, and $F_A^8$, $F_A^0$ for the $f_1$'s, as well
as in the coupling to real photons given by $F(0,0)$ and $A(0,0)$, respectively.
All results are given in units of GeV raised to the appropriate power [note that
the mass dimension of $f^8$ and $f^0$ is 1, but that of $f_G$ is 3; $F(0,0)$ and
$A(0,0)$ have mass dimensions $-1$ and $-2$, respectively].

The pseudoscalars $\eta$, $\eta'$ and a third ground-state $\eta''$ meson arise from mixing of flavor octet and flavor singlet degrees of freedom with the pseudoscalar glueball $G$, each followed by an infinite tower of excited states.
The ground state modes are dominantly flavor octet, singlet, and
glueball judging from the corresponding decay constants evaluated at $z\to0$,
while the first excited triplet $\eta^{(3)}$ to $\eta^{(5)}$ shows a more
involved mixing behavior.
The decay constants for $\eta$ and $\eta'$ agree reasonably well with the
recent lattice results of Ref.~\cite{Bali:2021qem}, where also pseudoscalar matrix
elements have been evaluated. Our results for $f_G$ correspond to $\sqrt{N_f/2}a$ in \cite{Bali:2021qem} and also agree reasonably well. The ratio $a_{\eta'}/a_\eta$ is between 2 and 2.5 depending on the renormalization scale. This is better in line with our model v1 that includes a nonzero gluon condensate, where $f_{G,\eta'}/f_{G,\eta}=2.60$ and 2.66 for the two choices of $g_5$, while
model v0 has 1.46 and 1.30.

Without gluon condensate (v0), the results for $F(0,0)$ show rather poor agreement
with experimental results for the $\eta$ meson with deviations of around 30\%, while
those for $F_{\eta'\to\gamma\gamma}(0,0)$ are much better. With gluon condensate (model v1), where the masses of $\eta$ and $\eta'$ agree with experimental data
at the percent level,
both couplings turn out to agree remarkably well with the experimental values.

For isosinglet axial vector mesons (Table \ref{tab:AV}), both model versions predict
generally too high values of $f_1$ and $f_1'$ masses (+8\% to +28\% compared to PDG data \cite{ParticleDataGroup:2022pth}). The $f_1$ and $f_1'$ mesons are obtained as
dominantly flavor octet and flavor singlet, respectively.
In the holographic model, the mixing angle is an energy or $z$ dependent
quantity. In the case of the $f_1$ mesons, it is usually
extracted from equivalent photon decay rates at zero virtuality, where the experimental results from
the L3 experiment read \cite{Achard:2001uu,Achard:2007hm}
\begin{equation}
\tilde\Gamma_{\gamma\gamma}=\left\{
\begin{array}{cc}
    3.5(8) \;\text{keV} &  \text{for}\;f_1=f_1(1285)\\
    3.2(9) \;\text{keV} & \text{for}\;f_1'=f_1(1420)
\end{array}
\right.
.
\end{equation}
With the definition
\begin{equation}
f_1=\cos\theta_A f^0+\sin\theta_A f^8
\end{equation}
and the assumption that
$\tilde\Gamma_{\gamma\gamma}\propto m_A$, one has
\begin{equation}
\tan^2(\theta_A-\arcsin\frac13)=\frac{m_{f_1}\tilde\Gamma_{\gamma\gamma}^{f_1'}}{m_{f_1'}\tilde\Gamma_{\gamma\gamma}^{f_1}},
\end{equation}
leading to  \cite{Hoferichter:2020lap}
$\theta_A=62(5)^\circ$, superficially agreeing with model version v0.
However, in the holographic model we have $\tilde\Gamma_{\gamma\gamma}\sim m_A(m_A/\Lambda)^4$, resulting in $\theta_A=56(5)^\circ$
for the experimental value, which
does not fit to the results for either v0 or v1, the latter
disagreeing even more than the former.\footnote{It would be interesting to revisit this issue in other holographic
QCD models, in particular ones that are closer to a string-theoretic top-down construction such as the models of Ref.~\cite{Casero:2007ae,Arean:2013tja}.}
While the mixing angle depends rather strongly on $\Xi_0$,
the combination $\sqrt{[A^8(0,0)]^2+[A^0(0,0)]^2}$ changes only slightly
between models v0 and v1, and it
is also close to the value of $A(0,0)$ in the isotriplet sector,
as well as to the same quantity in the chiral hard-wall model
\cite{Leutgeb:2019gbz}, $(21.04\text{GeV})^{-2}$.
Matching $A(0,0)$ with $\tilde\Gamma_{\gamma\gamma}\propto m_A(m_A/\Lambda)^4$
to the L3 results
leads to a value of 15.2(2.0) GeV$^{-2}$ so that the holographic results, which
read 20-21 GeV$^{-2}$ when $g_5=2\pi$ and 19-20 GeV$^{-2}$ for the reduced $g_5$, are somewhat too high for $f_1$ and $f_1'$, but
not excluded for $a_1$, for which Ref.~\cite{Roig:2019reh} has a concordant
estimate of 19.3(5.0) GeV$^{-2}$.

\subsection{Transition form factors}

For the HLBL contribution of single mesons to $a_\upmu$, their singly and doubly virtual TFFs are of
critical importance.

As in the chiral HW model \cite{Leutgeb:2019zpq},  we find excellent agreement
of the singly virtual result for the pion TFF with available experimental data, see Fig.~\ref{fig53comp}. At virtualities relevant for $a_\upmu$, the
results with $g_5$ fitted to $F_\rho$, where the asymptotic limit
is 89.4\% of the Brodsky-Lepage value, seem to give the best match.

\begin{figure}
\bigskip
\centerline{$Q^2 F_{P\gamma^*\gamma}(Q^2,0)$ [GeV]\hfill}
\includegraphics[width=0.38\textwidth]{pi0TFF+.pdf}
\centerline{$Q^2$ [GeV$^2$]}

\caption{Holographic results for the single virtual TFF $Q^2 F(Q^2,0)$ for $\pi^0$, plotted on top of experimental data as compiled in Fig.~53 of Ref.~\cite{Aoyama:2020ynm}
for $g_5=2\pi$ (OPE fit, blue) and the reduced value (red)
corresponding to a fit of $F_\rho$.
 (For $\pi^0$ results for the model with and without gluon condensate coincide.)}
\label{fig53comp}
\end{figure}

For the symmetric doubly virtual TFF the comparison is made with
the dispersive result of Ref.~\cite{Hoferichter:2018kwz} and the lattice result of Ref.~\cite{Gerardin:2019vio}
in Fig.~\ref{figpi0TFFd}. Both choices of $g_5$ are within the error band
of the dispersive result, while the result for the reduced $g_5$ is
also within the error band of the lattice result and moreover
happens to coincide with
the central values of the dispersive approach within line thickness of the plot
throughout the entire range of $Q^2$.

\begin{figure}
\bigskip
\includegraphics[width=0.45\textwidth]{pi0TFFd+l.pdf}
\caption{Holographic results for the doubly virtual $F_{\pi^0\gamma^*\gamma^*}$ compared to dispersive result of Ref.~\cite{Hoferichter:2018kwz} (green band) and the lattice result of Ref.~\cite{Gerardin:2019vio} (yellow band); the OPE limit given by dashed horizontal line. Full lines are with gluon condensate (version v1), dashed lines without (v0); blue color corresponds to $g_5=2\pi$ (OPE fit) and red to the reduced value $g_5$ ($F_\rho$-fit).
}
\label{figpi0TFFd}
\end{figure}

With $\eta$ and $\eta'$ mesons, there is a rather strong
dependence on the parameter $\Xi_0$ representing a gluon condensate.
With this parameter turned on, the masses of $\eta$ and $\eta'$ can be
matched to percent level accuracy, and the resulting prediction for
$F_{P\gamma\gamma}(0,0)$ is then in complete agreement with experiment for $g_5=2\pi$ (see Table \ref{tab:etas}), while with reduced $g_5$ this value is
slightly underestimated in the case of $\eta'$.
For the singly virtual TFF of $\eta$, only the results with nonzero $\Xi_0$ are
close to the experimental data, see Fig.~\ref{fig54comp}. They match those at low $Q^2$ quite well,
but are generally larger at higher virtualities.
In the case of $\eta'$, all model versions agree with the low-$Q^2$ date
due to L3, while at higher $Q^2$ the results without gluon condensate
agree with more of the data points, but only with unreduced $g_5=2\pi$.

\begin{figure}
\bigskip
\centerline{$Q^2 F_{P\gamma^*\gamma}(Q^2,0)$ [GeV]\hfill}
\includegraphics[width=0.38\textwidth]{etaTFF+.pdf}
\includegraphics[width=0.38\textwidth]{etaprimeTFF+.pdf}
\centerline{$Q^2$ [GeV$^2$]}

\caption{Holographic results for the single virtual TFF $Q^2 F(Q^2,0)$ for $\eta$ and $\eta'$ plotted on top of experimental data as compiled in Fig.~54 of Ref.~\cite{Aoyama:2020ynm}
for $g_5=2\pi$ (OPE fit, blue) and the reduced value (red)
corresponding to a fit of $F_\rho$. Full lines are with gluon condensate (version v1), dashed lines without (v0).
}
\label{fig54comp}
\end{figure}

\begin{figure}
\bigskip
\includegraphics[width=0.45\textwidth]{BBetapggV.pdf}
\caption{Holographic results for the doubly virtual $F_{\eta'\gamma^*\gamma^*}$ compared to BABAR data points (black) and a simple VMD model fitted with singly virtual data (cyan circles) \cite{BaBar:2018zpn}. Full lines are with gluon condensate (version v1), dashed lines without (v0); blue color corresponds to $g_5=2\pi$ (OPE fit) and red to the reduced value $g_5$ ($F_\rho$-fit).
}
\label{figBBetacomp}
\end{figure}

In contrast to the case of $\pi^0$, there are also several experimental data points for the doubly virtual TFF of $\eta'$. As opposed to the simple VMD model
considered in \cite{BaBar:2018zpn} and represented by the cyan circles in Fig.~\ref{figBBetacomp}, the holographic results are within 1-2 standard deviations.
For the lowest virtualities $Q^2_1=Q_2^2=6.48 \text{GeV}^2$, which are the most significant for $a_\upmu$, all versions of the model come close to the experimental result. With gluon condensate, the agreement is better with the reduced $g_5$, whereas without gluon condensate, a reduction of $g_5$ to fit $F_\rho$ moves the prediction slightly outside the error bar.

All in all, the model with gluon condensate and reduced $g_5$ seems to
be the optimal choice regarding pseudoscalar TFFs.

\begin{table*}[]
    \centering
\begin{tabular}{lccccccp{12pt}cccccc}
\toprule
& \multicolumn{6}{c}{$g_5^2=(2\pi)^2$}                                    & & \multicolumn{6}{c}{$g_5^2=0.894(2\pi)^2$}    \\
              & $\pi^0$ &  $\pi^*$ & $a_1$ & $a_1^*$ & {$a_1^{**}$} & $a_1^{***}$  & &  $\pi^0$ &  $\pi^*$ & $a_1$ & $a_1^*$ & {$a_1^{**}$} & $a_1^{***}$  \\
              \hline
$m$ & 0.135$^*$ & 1.891 & 1.363 & 2.137 & {2.987} & 3.935 && 0.135$^*$ & 1.841  & 1.278 & 2.047 & {2.936} & 3.902 \\
$f\;\vee\;F_A/m_A$ & 0.09221$^*$ & 0.00157 & 0.175 & 0.204 & {0.263} &0.311 && 0.09221$^*$ &0.00173&0.173 &0.217&0.280 & 0.329\\
$F(0,0)\vee A(0,0)$  &0.277 &-0.203 & 20.96& 3.31& {-0.336}& 2.16&& 0.276&-0.199&19.46&4.87&-0.413&2.05 \\ \hline
$a_\upmu\times 10^{11}$ & 66.1 & 0.7 & 7.8 & 1.2 & {0.4} & 0.3 &&
63.4 & 0.7 & 7.1 & 1.5 & 0.4 & 0.3 \\
\botrule
\end{tabular}
    \caption{Results for pseudoscalar and axial vector mesons in the isotriplet sector (the gluon condensate parameter $\Xi_0$ does not play a role here). All quantities in units of (powers of) GeV.}\label{tab:pia1}
\end{table*}

\begin{table*}
\centering\bigskip
\begin{tabular}{lccccccp{12pt}cccccc}
\toprule
\textbf{(v0)}         & \multicolumn{6}{c}{$\Xi_0=0$}                                 && \multicolumn{6}{c}{$\Xi_0=0$}                                 \\
& \multicolumn{6}{c}{$g_5^2=(2\pi)^2$}                                    & & \multicolumn{6}{c}{$g_5^2=0.894(2\pi)^2$}    \\
              & $\eta$  & $\eta'$ & $G/\eta''$     & $\eta^{(3)}$ & {$\eta^{(4)}$}  & $\eta^{(5)}$  && $\eta$  & $\eta'$ & $G/\eta''$      &  $\eta^{(3)}$ & $\eta^{(4)}$ & $\eta^{(5)}$ \\ \colrule
$m$  & 0.513     & 0.840     & 1.862 & 1.999    & {2.257}  & 2.705  && 0.503     & 0.819   & 1.764  & 1.948     & {2.207}  & 2.638  \\
${m}-{m^\mathrm{exp}}$ & -6.4\% & -12.3\% & & & & && -8.2\% & -14.5\% & & \\
$f^8$         & 0.0917 & -0.0565  & 0.00197 & 0.0266 & {0.0121} &0.0080 && 0.0902 & -0.0624 & 0.00405 & 0.0293  & {0.0132}&0.00837 \\
$f^0$         & 0.0394 & 0.0945 & -0.0212 & -0.00823 & {-0.0390} &0.0362 && 0.0446 & 0.0952 & -0.0224 & -0.00802  & {-0.0416} &0.0337 \\
$f_G$         & -0.0264  & -0.0385  & 0.0674 & -0.0400  & {0.154} &-0.310 && -0.0265 & -0.0344  & 0.0600 & -0.0454   & {0.156} &-0.280 \\
$F^8(0,0)$         & 1.46   & -0.674  & 0.177 & -1.18    & {0.00233} &0.236 && 1.41   & -0.737 & 0.0640  & -1.16     & {0.0239}&0.241 \\
$F^0(0,0)$         & 0.776  & 1.42  & 0.169 & 0.0383 & {1.08} &0.229 && 0.828  & 1.34   & 0.00310  & 0.00492 & {1.10}&0.253   \\
$F(0,0)$           & 0.351  & 0.322 & 0.0629 & -0.103   & {0.293} &0.0851 && 0.361  & 0.295  & 0.00700 & -0.110    & {0.302}&0.0922  \\
${F}-{F^\mathrm{exp}}$ & +28(2)\% & -6(2)\% & & & & && +32(2)\% & -14(2)\% & & \\
\hline
$a_\upmu\times 10^{11}$ & 32.8 & 15.7 & 0.055 & 0.14 & {0.79} & 0.16 && 34.0 & 13.3 & 0.003 & 0.16 & 0.85 & 0.16 \\
\botrule
\end{tabular}

\medskip
\begin{tabular}{lccccccp{12pt}cccccc}
\toprule
\textbf{(v1)}         & \multicolumn{6}{c}{$\Xi_0=0.01051$}                                 && \multicolumn{6}{c}{$\Xi_0=0.01416$}                                 \\
& \multicolumn{6}{c}{$g_5^2=(2\pi)^2$}                                    & & \multicolumn{6}{c}{$g_5^2=0.894(2\pi)^2$}    \\
              & $\eta$  & $\eta'$ & $G/\eta''$     & $\eta^{(3)}$ & {$\eta^{(4)}$} & $\eta^{(5)}$  && $\eta$  & $\eta'$ & $G/\eta''$      &  $\eta^{(3)}$ & $\eta^{(4)}$ & $\eta^{(5)}$ \\ \colrule
$m$ & 0.557     & 0.950     & 1.992    & {2.390}    & 2.954 & 3.214 && 0.561     & 0.947     & 1.943    & 2.428  & 2.914 & 3.317 \\
${m}-{m^\mathrm{exp}}$  & +1.7\% & -0.8\% & & {} & & && +2.4\% & -1.1\% & & \\
$f^8$         & 0.101  & -0.0385  & -0.0267  & {0.0116} & -0.0228 &-0.0049 && 0.103  & -0.0393  & -0.0299 & 0.0112 & -0.0253&-0.00767 \\
$f^0$         & 0.0272 & 0.113  & 0.0049  & {-0.0492}  &-0.00115 &-0.0214 && 0.0298 & 0.121  & 0.00761  & -0.0522 & 0.00320&-0.0128 \\
$f_G$         & -0.0298   & -0.0774  & 0.053  & {0.233}  & 0.1483 &0.269 && -0.0313  & -0.0821  & 0.048 & 0.260 & 0.1236& 0.214 \\
$F^8(0,0)$         & 1.55   & -0.431   & 1.19   & {-0.0478}  &-0.887 & 0.167&& 1.53   & -0.442   & 1.149  & -0.0312 & -0.877&0.129 \\
$F^0(0,0)$         & 0.468  & 1.40   & 0.0051 & {0.904}  & 0.0300 &0.0867 && 0.444  & 1.31   & -0.000026  & 0.837 & 0.0307&0.130 \\
$F(0,0)$           & 0.276  & 0.340  & 0.116  & {0.241}  & -0.0772 &0.0397 && 0.268  & 0.313  & 0.111  & 0.225 & -0.0760& 0.0477 \\
${F}-{F^\mathrm{exp}}$ & +1(2)\% & -0(2)\% & & {} & & && +2(2)\% & -8(2)\% & & \\
\hline
$a_\upmu\times 10^{11}$ & 19.3 & 16.9 & 0.19 & {0.53} & 0.043 & 0.008 && 17.6 & 14.9 & 0.18 & 0.45 & 0.039 & 0.007 \\
\botrule
\end{tabular}
\caption{Results for the isoscalar pseudoscalar sector, for the model with (v1) and without (v0) gluon condensate, and for two choices of $g_5$: $g_5=2\pi$ corresponding to matching the vector correlator to the LO UV-behavior in QCD, and the reduced value corresponding to a fit of $F_\rho$. All dimensionful quantities in units of (powers of) GeV.}
\label{tab:etas}
\end{table*}

\begin{table}[]
    \centering
\begin{tabular}{lccp{12pt}cc}
\toprule
\textbf{(v0)}         & \multicolumn{2}{c}{$\Xi_0=0$}                                 && \multicolumn{2}{c}{$\Xi_0=0$}                                 \\
& \multicolumn{2}{c}{$g_5^2=(2\pi)^2$}                                    && \multicolumn{2}{c}{$g_5^2=0.894(2\pi)^2$}    \\
               & $f_1$ &  {$f_1'$}  && $f_1$ &  {$f_1'$}      \\
              \colrule
$m$ & 1.460 & {1.651} && 1.388 & {1.598} \\
${m}-{m^\mathrm{exp}}$ & +14\% & +16\% && +8\% & +12\% \\
$F_A^8/m$  & 0.163 & -0.0732 && 0.165 & -0.0627 \\
$F_A^0/m$  & 0.0743 & 0.169 && 0.0690 & 0.180 \\
$A^8(0,0)$  & 19.27 & -8.649 && 18.38 & -7.194 \\
$A^0(0,0)$  & 8.676 & 19.21 && 7.310 & 18.62 \\
$\theta_A$  & $65.8^\circ$ & -24.2$^\circ$ && 68.3$^\circ$ & -21.1$^\circ$ \\
$A(0,0)$  & 4.22 & 4.40 && 3.76 & 4.37 \\
$m^*$ & 2.241 & 2.614 && 2.147 & 2.561 \\
$m^{**}$ & 3.056 & 3.580 && 2.999 & 3.535 \\ \hline
$a_\upmu\times 10^{11}$ & 11.0 & {10.8} && 9.1 & 11.0 \\
$a^*_\upmu\times 10^{11}$ & 0.6 & 1.5 && 0.6 & 1.5 \\
$a^{**}_\upmu\times 10^{11}$ & 0.2 & 1.1 && 0.2 & 1.0 \\
\botrule
\end{tabular}

\medskip
\begin{tabular}{lccp{12pt}cc}
\toprule
\textbf{(v1)}         & \multicolumn{2}{c}{$\Xi_0=0.01051$}                                 && \multicolumn{2}{c}{$\Xi_0=0.01416$}                                 \\
& \multicolumn{2}{c}{$g_5^2=(2\pi)^2$}                                    && \multicolumn{2}{c}{$g_5^2=0.894(2\pi)^2$}    \\
               & $f_1$ &  {$f_1'$}  && $f_1$ &  {$f_1'$}      \\
\colrule
$m$ & 1.481 & 1.810  &&  1.410 & 1.820 \\
$m-m^\mathrm{exp}$ & +15\% & +27\% && +10\% & +28\% \\
$F_A^8/m_A$  &0.176 &-0.0299 &&0.176 & -0.0167\\
$F_A^0/m_A$  & 0.0365&0.201 && 0.0292&0.219 \\
$A^8(0,0)$  &20.77 &-3.842 && 19.58&-2.556 \\
$A^0(0,0)$  &3.857 &20.07 && 2.690& 19.00\\
$\theta_A$  &79.5$^\circ$ &-10.8$^\circ$ &&82.2$^\circ$ &-7.7$^\circ$ \\
$A(0,0)$  &3.05 & 5.09&& 2.62&4.93 \\
$m^*$ & 2.246 & 2.862 && 2.153 & 2.891 \\
$m^{**}$ & 3.058 & 3.869 && 3.004 & 3.907 \\ \hline
$a_\upmu\times 10^{11}$ & 5.7 & 14.3 && 4.3 & 13.6 \\
$a^*_\upmu\times 10^{11}$ & 0.3 & 0.9 && 0.3 & 0.9 \\
$a^{**}_\upmu\times 10^{11}$ & 0.1 & 1.1 && 0.05 & 1.0 \\
\botrule
\end{tabular}
    \caption{Results for the isoscalar axial vector sector, for the model with (v1) and without (v0) gluon condensate, and the two choices $g_5$(OPE fit) and
    $g_5$($F_\rho$-fit). Here $\theta_A\equiv\arctan(A^8(0,0)/A^0(0,0))$
    for both $f_1$ and $f_1'$. All dimensionful quantities in units of (powers of) GeV. In the $a_\upmu$ contributions, about 58\% are due to the longitudinal part of the axial vector meson propagator which contributes to the MV constraint.}
    \label{tab:AV}
\end{table}

\subsection{HLBL contribution to $a_\upmu$}

Tables \ref{tab:pia1} and \ref{tab:etas} include also the individual contributions
of the listed pseudoscalar and axial vector meson modes to $a_\upmu$, which are
collected in Table \ref{tab:total} for the model with nonzero gluon condensate (v1) with $g_5=2\pi$ (OPE-fit)
and the reduced value \eqref{g5Frho} from fitting the $\rho$ meson decay. Only with
the extra parameter $\Xi_0$ for the gluon condensate, the predictions for $F_{P\gamma\gamma}(0,0)$ and masses of $\eta$ and $\eta'$ match experimental data with good accuracy. With reduced $g_5$ ($F_\rho$-fit), the predictions for
$a_\upmu^{\pi^0}$ and $a_\upmu^{\eta'}$ are
extremely close to the central values adopted by the White Paper
\cite{Aoyama:2020ynm}, and those for $\eta$ agree within $1\sigma$.

The holographic model also includes a third ground-state $\eta$ meson, which we called $\eta''$, the result of mixing with the pseudoscalar glueball $G$.
It contributes only $0.2\times 10^{-11}$, but
there is also a whole tower of excited $\eta$ modes, which together with excited pion modes contribute around $1.5\times 10^{-11}$ so that the total pseudoscalar poles prediction
for model v1($F_\rho$-fit) is close to the upper end of the WP prediction, whereas
the result for model v1(OPE fit) is 2.5$\sigma$ higher.

The main aim of this study is of course the experimentally less well constrained axial vector meson contribution, which in holographic QCD has been shown to
take into account the Melnikov-Vainshtein short-distance constraint \cite{Leutgeb:2019gbz,Cappiello:2019hwh}, also away from the chiral limit \cite{Leutgeb:2021mpu}. The holographic result thus presents an alternative
estimate of the combined contribution of axial vector mesons, for which
the WP estimate is $6(6)\times 10^{-11}$, and of short-distance contributions\footnote{In the symmetric high-energy limit, the holographic results
for the HLBL scattering amplitude have the correct dependence on $Q^2$, but reproduce the OPE value only at the level of 81\% when $g_5=2\pi$, where the asymmetric
MV limit is saturated fully \cite{Cappiello:2019hwh,Leutgeb:2021mpu}.},
estimated in the WP as $15(10)\times 10^{-11}$. With errors added linearly,
the WP value is at $21(16)\times 10^{-11}$.

It is difficult to estimate errors for any holographic result, but we expect
our results for $a_\upmu$ to be in good shape despite some deviations in its
ingredients.
The holographic results for axial vector mesons have turned out to overestimate the masses of $f_1$ and $f_1'$ by 8-28\%, where the models with gluon condensate have the higher deviations. On the other hand, all our models have an equivalent real photon coupling $A(0,0)$
that is 20-28\% too large compared to the value derived from L3 data for $f_1$ and $f_1'$, albeit
in good agreement with the estimate of Ref.~\cite{Roig:2019reh} for $a_1(1260)$.
The mixing angles for $f_1$ and $f_1'$ are poorly predicted, and even worse
when the gluon condensate is turned on. However, the prediction for the amplitude $\sqrt{(A^8)^2+(A^0)^2}$
appears to be fairly robust and only weakly dependent on $\Xi_0$.
A different modeling of the gluon condensate could perhaps lead to better
predictions for the mixing with similar overall amplitude.
Our summary in Table \ref{tab:total} therefore lists the more reliable
combined contribution of $f_1$ and $f_1'$.
Since the contribution to $a_\upmu$ decreases with increasing axial vector meson mass by approximately two inverse powers while the amplitude $A$ enters quadratically, we expect that the errors in the predictions of both will largely cancel out, so that the holographic results can still be a reasonably good
prediction for the axial vector meson contributions to $a_\upmu$.
For our favored model v1($F_\rho$-fit), the contribution from the
ground-state axial vector mesons is $a_\upmu^{a_1+f_1+f_1'}=25.0\times 10^{-11}$,
about 4 times the WP estimate. The contribution from $f_1+f_1'$ is 2.5 times that
of $a_1$, somewhat reduced from the flavor-U(3)-symmetric value of 3 that was
assumed in our previous estimates in Ref.~\cite{Leutgeb:2021mpu}. For this contribution, Pauk and Vanderhaeghen \cite{Pauk:2014rta}
have estimated a value of only $6.4(2.0)\times 10^{-11}$, much smaller than
our holographic prediction of $17.9\times 10^{-11}$. A crucial difference of the TFF assumed in \cite{Pauk:2014rta} is that it is obtained from a factorized ansatz
that unlike the holographic result does not have the correct asymptotic
behavior \cite{Hoferichter:2020lap} in the doubly virtual case, where it
falls off as $1/Q^{4}$ instead of $1/Q^2$.

In the holographic models, the excited axial vector mesons ensure agreement
with the longitudinal (Melnikov-Vainshtein) short-distance constraint. This
constraint derived from the axial anomaly is satisfied to 100\% in the model v1(OPE fit), and to 89.4\% in the case of v1($F_\rho$-fit). The latter should
provide a better approximation at large but still physically relevant energy
scales, where typically $\sim 10\%$ of next-to-leading order pQCD corrections apply \cite{Melic:2002ij,Bijnens:2021jqo}.

In the chiral HW1 model and in the U(3)-symmetric massive HW1m model that we have investigated in Refs.~\cite{Leutgeb:2019gbz,Leutgeb:2021mpu}, we have obtained 9.2 and $9.4\times10^{-11}$
from excited axial vectors, where 25\% are due to $a_1$ by U(3) symmetry.
The contribution of excited $a_1$'s in our present models are essentially
the same as in the HW1m model (up to a slightly different fit value of $f_\pi)$,
but the excited isoscalars remain below the extra factor of 3 expected from U(3) symmetry.\footnote{In order to approximate the sum of contributions from the infinite tower of axial vector mesons we have used the observation that in the chiral HW models as well as in the HW1m model the infinite series of contributions can be roughly approximated by a geometric one with $a_{n+1}/a_n\approx 0.6$ for $n>2$. The full sum can thus be approximated by multiplying the last contribution of a truncated sum by a factor of 1/(1-0.6)=2.5. In the case of excited pseudoscalars, which do not contribute to the longitudinal short-distance constraint \cite{Leutgeb:2021mpu}, the contributions drop much more quickly. Our results for those are obtained simply from the sum of the first few modes.} Instead, the latter provide only 1.4  and 1.2 times the contributions from excited $a_1$'s in the case of v1(OPE fit) and v1($F_\rho$-fit), respectively.

The total contribution from axial vector
mesons is thus significantly smaller than the estimates we have come up with
in the flavor-symmetric case of Ref.~\cite{Leutgeb:2021mpu}: 34 and 31$\times 10^{-11}$ for the two choices of $g_5$ (instead of 40.8 and 38.8$\times10^{-11}$ for HW1m and HW1m with reduced $g_5$, respectively). Comparing this to the combined estimate of axial vector mesons and short-distance contributions in the WP, $21(16)\times10^{-11}$, we find values that are about 50\% higher, but well within the estimated error.

\begin{table}[t]
\bigskip
\centering
\begin{tabular}{lccc}
\toprule
$a_\upmu^{...}\times 10^{11}$ & v1(OPE fit) &  v1($F_\rho$-fit) & WP \\
 \hline
 $\pi^0$ & 66.1 &  63.4 & 62.6$^{+3.0}_{-2.5}$ \\
 $\eta$ & 19.3 &  17.6 & 16.3(1.4) \\
 $\eta'$ & 16.9 &  14.9 & 14.5(1.9) \\
 $G/\eta''$ & 0.2 &  0.2 \\
 $\sum_{PS^*}$ & 1.6 & 1.4 \\[4pt]
 \hline
 PS poles total & 104 &  97.5 & 93.8(4.0) \\
 \hline
 $a_1$ & 7.8 &  7.1 \\
 $f_1+f_1'$  & 20.0 &  17.9 \\
 $\sum_{a_1^*}$ & 2.4 &  2.5 \\
 $\sum_{f_1^{(')*}} $ & 3.4 &  2.9 \\[4pt]
 \hline
 AV+LSDC total & 33.6 &  30.5 & 21(16) \\
 \hline
 total & 138 & 128 & 115(16.5) \\
 \botrule
\end{tabular}
    \caption{Summary of the results for the different contributions to $a_\upmu$ in comparison with the White Paper \cite{Aoyama:2020ynm} values.}
    \label{tab:total}
\end{table}

\section{Conclusion}

In this paper, we have upgraded our previous studies of the HLBL contribution
in HW AdS/QCD models to 2+1 flavors with strange quark mass $m_s>m_u=m_d$
plus a Witten-Veneziano mass for the flavor singlet degree of freedom
generated by interaction terms involving a pseudoscalar glueball with
the latter that implement the anomalous Ward identities of the $U(1)_A$ symmetry
in the line of Ref.~\cite{Katz:2007tf,Schafer:2007qy}.

In holographic QCD, the Melnikov-Vainshtein constraint on the HLBL scattering
amplitude is naturally satisfied, to the same degree that TFFs satisfy the
Brodsky-Lepage and OPE limits. All these are saturated at the level of 100\%
for the standard value of $g_5=2\pi$ in HW1 models.\footnote{The simpler Hirn-Sanz (HW2) model, which omits the bifundamental scalar $X$, reaches 62\% when $f_\pi$ and $m_\rho$ are fitted.} However, because these
models do not involve a running coupling in the UV, the UV-limits of TFFs
are approached too quickly, likely
leading to overestimated HLBL contributions to $a_\upmu$.
Next-to-leading-order gluonic corrections in pQCD suggest a reduction
by about 10\% at large but still experimentally relevant virtualities.
Precisely such a correction is obtained by fitting $g_5$ such that the
decay constant of the $\rho$ meson is matched instead of the OPE result
for the vector correlator. In Ref.~\cite{Leutgeb:2022cvg}, we have found
that this also brings the comparably crude result of HW AdS/QCD models for the HVP contribution
better in line with dispersive results.

In Ref.~\cite{Leutgeb:2019gbz,Leutgeb:2021mpu} we have shown that the MV short-distance constraint
is realized by the infinite tower of axial vector mesons, with the excited
axial vector mesons adding about a third of the contribution from the ground-state
axial vectors in the flavor-symmetric case. A much smaller contribution
comes from excited pseudoscalars, which do not contribute to the
longitudinal short-distance behavior at leading order.

In our present study with $U(1)_A$ anomaly included, where we have
obtained a remarkably accurate fit of the masses of $\eta$ and $\eta'$ mesons
as well as of their $F_{P\gamma\gamma}(0,0)$ values when including a
nonzero gluon condensate that was omitted in \cite{Katz:2007tf},
we have found a reduction of the ratio 3:1 for the isoscalar:isotriplet
contributions of axial vector mesons to about 2.5:1. For excited
mesons (axial vector as well as pseudoscalar), we have obtained an even
more pronounced reduction, which reduces our prediction for the $a_\upmu$
contribution of axial vector mesons in the U(3)-symmetric case
from around 41 and 39$\times 10^{-11}$ to 34 and 31$\times 10^{-11}$
for $g_5$(OPE) and $g_5$($F_\rho$-fit), respectively.
These values are above the estimate of the White Paper \cite{Aoyama:2020ynm}
for the contribution of (ground-state) axial vector mesons plus
short-distance constraints, but well within the error given there.

The pseudoscalar contributions obtained in our model v1($F_\rho$-fit)
agree completely with the WP results for $\pi^0$, $\eta$, and $\eta'$,
however this model also has a contribution of $1.6\times 10^{-11}$ from excited pseudoscalars, where the tower of $\eta$'s mixes with a pseudoscalar glueball.

The complete contribution from summing pseudoscalar and axial vector contributions
thus turns out to be close to but below the upper end of the WP contribution for this model, which we consider our currently best estimate obtained from AdS/QCD.
\phantom{XXXXXXX}
\phantom{XXXXXXX}
\phantom{XXXXXXX}
\phantom{XXXXXXX}

\begin{acknowledgments}
We would like to thank
Gilberto Colangelo, Martin Hoferichter, and Elias Kiritsis for helpful discussions.
J.~L.\ and J.~M.\ have been supported by the Austrian Science Fund FWF, project no. P 33655,
and by the FWF doctoral program
Particles \& Interactions, project no. W1252-N27.
\end{acknowledgments}

\raggedright
\bibliographystyle{JHEP}

redpajama_set_name: RedPajamaArXiv


The two Latin mottos read Speramus Meliora and Resurget Cineribus, meaning "We hope for better things" and "It will rise from the ashes", which was written by Gabriel Richard after the fire of 1805. The seal is a representation of the Detroit fire which occurred on June 11, 1805. The fire caused the entire city to burn with only one building saved from the flames. The figure on the left weeps over the destruction while the figure on the right gestures to the new city that will rise in its place.
Don't be paranoid, don't read this.


text:
Jamarion Miller Christian McCaffrey Bryce Young Najee Harris DJ Dale Trey Sanders Roydell Williams Emmanuel Henderson Jaylen Moody Aaron Jones Kendall Randolph Derrick Henry Brian Robinson Jahmyr Gibbs Jase McClellan Athlete health Athlete injuries College sports College football Sports Football
SEC Alabama
'Electric' tailback Jahmyr Gibbs set to make Alabama debut
By JOHN ZENOR - Sep. 02, 2022 01:51 AM EDT
FILE - Alabama running back Jahmyr Gibbs (13) runs the ball during the first half of Alabama's A-Day NCAA college football scrimmage on April 16, 2022, in Tuscaloosa, Ala. Gibbs gained 5.2 yards per carry each of his two seasons at Georgia Tech and had a combined 1,206 yards rushing and 773 yards receiving during that stretch. He now figures to have a featured role for Alabama, which had two running backs selected in the first round and two more taken in the third round over the last four drafts. (AP Photo/Vasha Hunt, File)
The superlatives have been flying from Jahmyr Gibbs' new Alabama teammates.
They've called the Georgia Tech transfer "definitely electric," "a special talent" and "very dynamic." It's a lot to live up to for the top-ranked Crimson Tide's new tailback, who is set to make his 'Bama debut Saturday night against Utah State.
"Jahmyr Gibbs is definitely electric, fast," Tide offensive lineman Kendall Randolph said. "He's got some quick feet so he can make the cuts. To block around him is a great honor first and foremost, because he's a great athlete. We do what we can to make him successful."
Gibbs is the headliner of a deep backfield that has produced star after star with Derrick Henry, Najee Harris and Brian Robinson Jr. among the most recent crop.
While they each had to wait their turns to some degree or other, Gibbs has seemingly been penciled in as the lead back since his arrival in Tuscaloosa. A 75-yard touchdown run in the spring A-Day scrimmage in front of fans certainly didn't hurt his case.
Gibbs brings versatility to a backfield that also has Jase McClellan and Roydell Williams back from injuries, along with Trey Sanders and two highly rated freshmen. The 5-foot-11, 200-pounder is also returning kicks and he had 36 catches last season for Georgia Tech.
In fact, Gibbs earned All-Atlantic Conference honors at three positions: all-purpose player (first team), specialist (second team) and running back (third team.)
He figures his receiving ability is his biggest strength.
"That's what the NFL likes so I try to do my best to model my game after Christian McCaffrey or Aaron Jones," Gibbs said. "Players like that who catch out of the backfield."
The precise pecking order and roles in the backfield are to be determined. Robinson was the bruising 225-pound workhorse at a position that was decimated by injuries, forcing two players to switch position for depth.
Like Gibbs, McClellan proved to be a receiving threat out of the backfield before a season-ending knee injury. Williams averaged 5.9 yards per carry before his season ended with his own knee injury.
McClellan said the backs are happy to have Gibbs, even if it could mean diminished roles.
"We just welcomed him in like a brother," McClellan said. "He wanted to come join us, so we welcomed him. We're thankful for him, happy he wanted to come here. He adds something special to the group."
Freshmen Emmanuel Henderson and Jamarion Miller also join the mix. But Gibbs is the X-factor with the explosive speed and big-play ability in three different facets of the game.
"He's different. Very dynamic, can do anything," Tide defensive lineman DJ Dale said. "Run the ball, catch the ball, he's shifty. He got me (recently) at practice in team run. He made me stop my feet and he stopped and kept going and he was gone. He's a very dynamic piece to the offense and I'm glad we have him."
The backfield depth and talent could mean at least some more balance on offense. With Heisman Trophy winner Bryce Young racking up huge numbers, Alabama passed for nearly 3,000 yards more than it rushed.
Not exactly the traditional 'Bama balance, though the Tide's top three receivers are gone.
Enter the do-it-all Gibbs.
"He's a special talent man," Alabama linebacker Jaylen Moody said. "One of a kind. Really fast. Really slippery. He's special."

redpajama_set_name: RedPajamaCommonCrawl


text:
Biografia
Suo padre fu ucciso dal sultano Selim III, così il giovane Pasvandoglu dichiarò guerra ai Turchi e fece di Vidin la sua roccaforte. Le truppe del sultano non riuscirono ad espugnare la città e furono perciò costrette a capitolare (1798). Pasvandoglu diventò dunque pascià di Vidin.

Altri progetti

Collegamenti esterni

Storia della Valacchia
Morti a Vidin
Nati a Vidin

redpajama_set_name: RedPajamaWikipedia


•Your inquiry related to our product & price will be replied within 24hours.
•1 year warranty of products and good after-sale service offered, please get back if you got question.
Our website lists just some of the products we have successfully supplied,but please note this is just representative and an example of our products&service,so,if you cannot see the product you are looking for, please contact us today!!!


Thats right - ONLY SIX SWIMS!!!!! I knew the weekend would be a bit tricky to fit in a swim on both days with the tour on. On Saturday after racing we headed down to the Cambridge pool at about 6.15pm thinking it closed at 7pm but it had closed at 6. So the intention was there and being the only pool in Cambridge I was a bit stuck. But I swam in between stages the next day while all my competitors were resting up so there is some commitment for you. Sea swim was in my pointzero3 muppet suit, I thought I should be brave and face the cold sea and it was a better workout than with the buoyancy of the Helix. Will have to try and do a double up swim next week to make up for my lost one.
Had my little tour this week so will put a little race report up for that soon with some photos. The rest of the week was pretty standard, did a couple of 5min efforts but sucked big time and threw a few 15sec sprints in to prepare me for club racing.
Only 2 runs this week because I didn't run in the weekend but leggie seems to be getting better with each run. I don't feel like I'm running very nicely, my other leg feels like my knee keeps rolling in so will need to start some VMO strengthening to try and fix it.


text:
Most Americans huddle indoors as coronavirus deaths keep spiking
President Donald Trump said he saw no need for the federal government to issue a nationwide decree, with 39 states and the District of Columbia now requiring residents to stay home except for essential outings.
The governors of Florida, Georgia, Mississippi and Nevada each instituted the strict policies on a day when the death toll from COVID-19 shot up by 925 to more than 4,800 nationwide. Reuters
Published at : April 2, 2020
Updated at : April 2, 2020 11:26
Four new states imposed sweeping stay-at-home directives on Wednesday in response to the coronavirus pandemic, putting over 80% of Americans under lockdown as the number of deaths in the United States nearly doubled in three days.
The governors of Florida, Georgia, Mississippi and Nevada each instituted the strict policies on a day when the death toll from COVID-19 shot up by 925 to more than 4,800 nationwide, with 214,000 confirmed cases, according to a Reuters tally.
President Donald Trump said he saw no need for the federal government to issue a nationwide decree, with 39 states and the District of Columbia now requiring residents to stay home except for essential outings to the doctor or grocery store.
He also told a White House briefing on Wednesday he was considering a plan to halt flights to coronavirus hot spots.
"We're certainly looking at it, but once you do that you really are clamping down on an industry that is desperately needed," Trump told a White House news briefing.
Such a plan might conceivably shut down traffic at airports in hard-hit New York, New Orleans and Detroit.
"We're looking at the whole thing," Trump said of curtailing domestic flights already greatly reduced as demand has fallen.
White House medical experts have forecast that even if Americans hunker down in their homes to slow the spread of COVID-19, some 100,000 to 240,000 people could die from the respiratory disease caused by the coronavirus.
A Pentagon official who spoke to Reuters on condition of anonymity said the U.S. Department of Defense was working to provide up to 100,000 body bags for use by civilian authorities in the coming weeks.
Since 2010, the flu has killed between 12,000 and 61,000 Americans a year, according to the U.S. Centers for Disease Control and Prevention. The 1918-1919 flu pandemic killed 675,000 in the United States, according to the CDC.
New York state remained the epicenter of the outbreak, accounting for more than a third of the U.S. deaths. Governor Andrew Cuomo told police on Wednesday to enforce rules more aggressively for social distancing.
"Young people must get this message, and they still have not gotten the message. You still see too many situations with too much density by young people," Cuomo, a Democrat, said in imposing rules to close playgrounds, swing sets, basketball courts and similar spaces.
"How reckless and irresponsible and selfish for people not to do it on their own," Cuomo said.
California cases surge
New York City Mayor Bill de Blasio told a news conference the city was contracting with hotels as part of a massive effort to add 65,000 additional hospital beds by the end of the month.
De Blasio, also a Democrat, said the city had arranged to add 10,000 beds at 20 hotels, which have lost most of their guests as travel has stopped.
"This is going to be an epic process during the month of April to build out all that capacity," de Blasio said. "But this goal can be reached."
California saw the number of coronavirus cases surge by roughly 1,300 over the day before to nearly 10,000 as Governor Gavin Newsom warned that even as stay-at-home policies appeared to be having some effect, the state would run out of intensive- care hospital beds equipped with ventilators within six weeks.
Newsom said California could still manage to "bend" the state's infection curve more, saving the need for additional beds, if residents were rigorous in staying home and avoiding contact with others.
"We are in a completely different place than the state of New York and I hope we will continue to be, but we won't unless people continue to practice physical distancing and do their part," the Democratic governor told a news conference in the state capital, Sacramento.
But Americans under lockdown and largely unable to work struggled with making ends meet as rent came due on Wednesday, the first day of the month.
In Oakland, California, Alfa Cristina Morales said she had been surviving on money saved for a U.S. citizenship application since losing her job at a coffee shop. Morales had sought unemployment benefits to support her two-year-old son.
"We're worried that it won't be enough," she said.
Connecticut Governor Ned Lamont said a six-week-old baby had died from COVID-19, in what he called "a reminder that nobody is safe from this virus."
Florida Governor Ron DeSantis told Fox News that Broward County would likely allow two cruise ships with coronavirus outbreaks carrying a total of 2,500 people to dock in Fort Lauderdale, despite his misgivings about potentially contagious foreign nationals.
"We were concerned about a deluge into the hospitals, but I think it turns out that there will probably be some who need to go, but it's very manageable and the local hospital system thinks that they can handle it," DeSantis, a Republican, told Fox.
At Fort Lauderdale, Floridians aboard the ships would be taken home and flights arranged for foreigners, he said.
Vaccine nationalism puts world on brink of 'catastrophic moral failure': WHO chief
Music producer Phil Spector, convicted of murder, dead at 81: media
Gunmen kill two female Supreme Court judges in Afghanistan: Police
India kicks off 'world's largest' vaccination campaign
International Covid-19 vaccine poll shows higher mistrust of Russia, China shots
Most Read from World
Elon Musk leaves behind Amazon's Bezos to become world's richest person - Bloomberg News
Explained: Where are we in the COVID-19 vaccine race?
India gears up for 'world's biggest vaccination drive'


text:
Disney+ Debuts Trailer For 'Save Our Squad With David Beckham'
Skyler Shuler October 13, 2022 2 min read
Disney+ today debuted the official trailer for the UK Original series, Save Our Squad with David Beckham, which premieres exclusively on Disney+ on Wednesday, November 9th.
Co-produced by BAFTA and RTS Award-winning production company Twenty Twenty, and Studio 99, the global content and production studio co-founded by Beckham, Save Our Squad with David Beckham is a heart-warming series that sees David Beckham return to his East London roots to mentor Westward Boys, a young, grassroots team who are at the bottom of their league, facing relegation. However, this isn't just any league… this is the same league that David played in as a young boy. He'll try to turn the team's fortunes around by working with the head coaches.
David Beckham mentors the young squad, sharing stories from his own playing career which spanned over 20 years and saw him play for some of the most successful clubs in the world, to help them learn valuable lessons about practice, ambition, and teamwork. Off the pitch, Beckham gets the chance to spend time with players and their families, learning about their lives, hopes, and concerns and talking to them about the importance of collaboration, work ethic, and community.
In the exclusive trailer, we see the first time the squad meet David Beckham, with one member of the team quizzing him on missed penalties. For David, this role is about giving back as we see him mentor the boys, give them inspiring team talks and even get his boots back on to try to score a trademark Beckham free kick against our young team.
Save Our Squad with David Beckham builds on the company's commitment to source, develop and produce original productions. Across EMEA alone, Disney's International Content and Operations team plan to create 60 local productions by 2024 as it continues to work with outstanding creators and premium producers.
Tags: Disney Disney plus Disney+ save our squad with david beckham
Previous After 'Werewolf By Night', Here Are Some Other MCU 'Special Presentations' We'd Like To See
Next 'She-Hulk: Attorney At Law' Episode Nine Review: "Whose Show Is This?"
'Star Wars: The Bad Batch – Season 2' Episode Five Review: "Entombed"
January 27, 2023 Maxance Vincent
Disney's 'Percy Jackson' Series Finally Finds Its Zeus and Poseidon
SEE IT: Leaked Photos Of Kathryn Hahn & Cast On Set For 'Agatha: Coven of Chaos'
'The Bear' Star Ayo Edebiri Joins Cast of 'Thunderbolts'
January 26, 2023 JeremyWrites
Hulu, 20th Television Animation Cut Ties With Justin Roiland


text:
byVictor Tangermann
/ Earth & Energy
Tree Wrap
Firefighters Wrap World's Largest Tree in Foil to Protect From Raging Fire
It's estimated to be 2,500 years old.
National Parks Service
Sequoia Safety
Massive wildfires in California are forcing firefighters to wrap massive sequoia trees in fire-resistant blankets, including General Sherman, a 275-foot specimen that is believed to be the world's largest tree by volume.
It's a climate change-driven disaster endangering some of the oldest and biggest living trees on our planet. General Sherman is estimated to be about 2,500 years old, the BBC reports. Tragic images shared online by the National Park Service show the base of the gigantic trunk wrapped in tinfoil-like blankets.
The world's largest tree was wrapped in foil to protect it from California wildfires. "General Sherman" is 2,500 years old and 275 feet tall.
A wildfire last year killed about 10,000 sequoia trees, some thousands of years old. Now the Colony fire is threatening 2,000 more. pic.twitter.com/vYkCGZBqQo
— AJ+ (@ajplus) September 17, 2021
Tinfoil Wrap
More than 350 firefighters were battling two fires stretching over parts of Sequoia National park.
"It's a very significant area for many, many people, so a lot of special effort is going into protecting this grove," Rebecca Paterson, a spokeswoman for Sequoia and Kings Canyon National Parks, told the LA Times.
Some of the massive trees "are being prepped the same way that we would prep structures," Patterson added.
Beyond wrapping the trees in aluminum blankets, nearby vegetation was also cleared to make sure the fire didn't spread.
The blankets are made out of the same material that is used to wrap homes in wildfire-endangered areas, and used by firefighters as a last resort in case they ever find themselves surrounded by fire, according to the LA Times.
It's a sad state of affairs. It would be a national travesty of our own making if we lost one of the largest trees in the world to record-breaking wildfires.
READ MORE: California fires: General Sherman and other sequoias given blankets [The BBC]
More on wildfires: 2021 Wildfires Spewed Record Breaking Amount of C02
The Skies in California Basically Look Like the Apocalypse Right Now
Going Solar
California Now Requires Solar Panels on All New Homes
How California's Incredible New Solar Panel Law Is Historic for Green Energy


Mountain Goat have made heaps of IPAs over the years in pretty much every way imaginable. But we're not sure they've ever brewed one as big as this. Trip Trap is a beer that layers it on thick: 100 IBU [in other words, heaps] of American hops and 10 percent of booze.
Described as "delightfully bitter" by the people responsible for it, we're told it pours a "deep golden with a slight red tint" and possesses an "intense hop aroma of citrus, tropical fruit and pine". It's keg only, so if you like your beers on the large and hoppy side, hunt one down quick.


Brilliantly boho! Kate Hudson arrives at LAX to jet out of LA in grey long-sleeved top, ripped jeans and a cream colored sweater over her shoulders. Kate puled some of her blonde hairs back into a high messy bun.


Read the first three chapters of Head Above Water HERE!
Listen to my latest interview HERE.
If you are a member of a book club or are interested in reviewing Head Above Water, shoot me an email by clicking HERE. I'm looking to help get the word out.
Welcome to the online home of Benjamin Jones, author of the Humanity's Ultimate Battle (H.U.B.) series of novels as well as the just released novel Head Above Water and the upcoming The Girl In The Red Balloon as well as an upcoming non-fiction book about Hip-Hop music that he's co-authoring.


text:
Q: How do you taper values off to zero in a numpy array? I have a very large array of 180000 columns, but for simplicity it looks something like this:
delta_v = [[3.4123 3.4213 3.4303 0.49377 0.49878 0.50381]]

What I want to do is to create a new 500x6 array, which will take each value and evenly increment down to a value of zero in the last row. I know how I'd do this if all values in the row were the same, but seeing as they're all different and would require different step sizes, I'm finding it a bit hard to figure out. So something that would look like this:
delt_var = [[3.4123 3.4213 3.4303 0.49377 0.49878 0.50381]
              ...     ...    ...    ...     ...     ...
            [  0       0      0      0       0       0   ]]

Cheers!

A: You can simply multiply all values to an array that decrement them evenly:
arr = np.vstack((delta_v, np.arange(0,1,1/499)[::-1][:,None]*delta_v))

stacking is due to the fact that in python, last index is excluded, so you might need to stack original array (or a row of 1s to np.arange) depending on number of rows you want.

redpajama_set_name: RedPajamaStackExchange


I have a small title loan through a title loan company here in South Carolina. I have made a couple payments over the phone to them in the past few months with my bank debit card. I just found out a couple weeks ago that they kept this debit card info on file without my permission, which makes me very upset. Secondly, I made a payment to them yesterday of $100, which I authorized. Then I find out that they charged my debit card another $100 today without my permission. I have had it with them and their sneaky practices.
What a nightmare! I hope you were able to clear up the situation with your bank quickly.
For the benefit of readers who are not familiar, a title loan is a short-term, high-interest loan that uses the title to your vehicle as collateral. Title loans, along with other short-term, high-interest loans such as payday loans, have been an area of great concern for the Consumer Financial Protection Bureau. Consumer groups have pressed the agency to place limitations on practices like the ones you experienced.
The CFPB has been considering a proposal that would require lenders to notify borrowers before accessing a consumer's bank or prepaid account. Industry trade groups have been fighting against it. Nonetheless, the agency is preparing to unveil a new rule covering small-dollar loans with high interest rates and has been trying to build support for this measure, according to a recent article in the Wall Street Journal.
If you want to make a complaint against the title loan company, your first step should be to re-read the contract you signed when you took out the loan. Look for a section that mentions an automatic repayment plan. This is when someone authorizes a lender to take regular payments directly from a bank account or debit card the day a payment is due. It is possible that, buried in the fine print, there is a clause that lets the lender take automatic deductions from your account.
What if you can't understand the contract because it is written in legal jargon? Hiring a lawyer would probably not be worth it, given that we're talking about a $100 charge. However, depending on your income, getting free legal help may be an option. LawHelp.org refers people with low and moderate incomes to free or low-cost legal help.
The FTC says the lender must give you a copy of your authorization of the recurring automatic debits. The commission also notes that it is illegal for a lender to require that you repay the loan through pre-authorized automatic transfers.
If it looks like your lender did not follow the rules, then consider making a complaint with the CFPB. To do this, use the submission box online. Go to the category that says "Other consumer loan," which covers title loans. The agency will forward your complaint to the company and try to get a response from them. The CFPB will also share the information you submit with state and federal agencies that oversee financial products and services.
If the lender did follow all the rules, there's not much you can do except pay back the loan as quickly as possible. Going forward, I'd recommend avoiding title loans if you can. Some people lose their cars by taking out these loans.
If you're not making enough money to stay current on your bills from your job, consider starting a small side business for yourself that you can fit into your life around your other responsibilities. There are many possibilities in the freelance economy, ranging from driving for Uber or Lyft to finding other types of gigs on Craigslist, TaskRabbit or Thumbtack. In today's world, many people need multiple income streams to survive. By earning even a little bit more, you can free yourself of lenders who prey on people who are in a financial crunch.


text:
Q: many Datpickers with same class getDate I've got a question, got couple of input datepickers like
<input name="datepicker_one" class="datepicker" readonly type="text" value="12-12-2013" />
<input name="datepicker_two" class="datepicker" readonly type="text" value="13-12-2013" />
<input name="datepicker_three" class="datepicker" readonly type="text" value="14-12-2013" />

and I'm trying to get values of clicked datepicker, while trying to avoid writing different methods for each of those inputs, so I wonder if there is a way to get datepicker value just by clicking on input with class .datepicker like:
$(".datepicker").click(function(e){ $(this).val(); });

but this gives me undefined value also:
$(".datepicker").click(function(e){ $(this).datepicker("getDate"); });

Does not seems to work.

A: You aren't using your value. The statment:
$(this).val();

Is meaningless as it doesn't do anything. You get the value, but didn't use it. If you did
console.log($(this).val());

you would actually see the result.

A: Try this:
    $(".datepicker").click(function(){
        alert($(this).val());
        return false;
    });


The aim of this training is to introduce delegates on the issues related to explosion and fire risks on an onshore mining installation and to equip them with the basic emergency response knowledge and skill.
Persons performing activities on an onshore mining installation.
To familiarize and train delegates with various types of small extinguishers.
To decide which extinguishing agent should be used in the given situation.
The effects of extinguishing and combustible materials present.
Extinguishing a small fire with various types of portable extinguishers.
Half (½) a day for both basic and refresher training.
A valid NOGEPA 0.5 or a 0.6 certificate.