already ≫ 1 (and for $r \sim 1/m_q$, the left hand side is sizeable whereas the tachyon is, by definition of $m_q$, $O(1)$).

For the sake of concreteness, let us consider the case $v_p = 2$ so that the tachyon has the $r$-dependence of (B.16) when the additional terms $\propto \lambda'$ are still small. Then (B.35) is saturated for

$$r \sim \frac{1}{m_q} \left[ \log \frac{m_q}{\Lambda_{\text{IR}}} \right]^{\frac{2}{\xi d_p}} \equiv r_c. \qquad (B.36)$$

That is, the tachyon behaves as $r^\xi$ only up to $r \sim r_c$, which is only enhanced with respect to $r \sim 1/m_q$ by a logarithmic term. For $r_c \ll r \ll 1/\Lambda_{\text{IR}}$ (i.e., for almost the whole interval $1/m_q \ll r \ll 1/\Lambda_{\text{IR}}$) we need to solve the tachyon EoM with different dominant terms. Assuming that the right hand side in (B.34) dominates over the left hand side, we obtain

$$e^{-2A} \kappa \lambda' \tau' \frac{\partial}{\partial \lambda} \log V_f \approx \frac{\partial}{\partial \tau} \log V_f \qquad (B.37)$$

which becomes for the current case

$$\tau' \tau^{d_p-1} \approx \frac{4 \xi b_0^{\text{YM}}}{r a_1} [\log(r \Lambda_{\text{IR}})]^2 . \qquad (B.38)$$

This equation is solved by

$$\tau \approx \left[ \frac{4 d_p \xi b_0^{\text{YM}}}{3 a_1} [\log(r \Lambda_{\text{IR}})]^3 + \tau_0 \right]^{\frac{1}{d_p}} \qquad (B.39)$$

so that the tachyon only increases logarithmically for $r_c \ll r \ll 1/\Lambda_{\text{IR}}$. Inserting this expression in (B.22), and recalling that $r_c$ is close to $1/m_q$, we see that the bottom of the Schrödinger potential is found at $r \sim 1/\Lambda_{\text{IR}}$, and the mass gap is again $O(\Lambda_{\text{IR}})$ up to logarithmic corrections.

In conclusion, the phenomenologically interesting large mass gap (with power law dependence on $m_q$) is obtained only when $v_p = 2$, $0 \le v_\ell \le 1$, and when $\frac{\partial}{\partial \lambda} \log V_f$ is suppressed (e.g., $d_p \le 0$). The last requirement means that for the critical exponential asymptotics $\log V_f \sim -a\tau^2$, the factor $a$ cannot$^{28}$ depend on $\lambda$.

Interestingly we notice that potentials I, the construction of which was motivated by using completely independent arguments in [30, 42], do produce a large mass gap. In this case we have

$$V_f(\lambda, \tau) = V_{f0}(\lambda) \exp(-a_0 \tau^2) \qquad (B.40)$$

where $a_0$ is indeed independent of $\lambda$. The value of $a_0$ was fixed in the UV (where the tachyon is small) to reproduce the UV dimension of the chiral condensate and quark mass so that

$$\xi = \frac{v_p a_0 \ell_0^2}{4 \kappa_0} = \frac{3 \ell_0^2}{4 \ell^2}. \qquad (B.41)$$

The value of $\xi$ is therefore slightly below the critical value $\xi = 1$ (because also $\ell_0^2/\ell^2 < 1$). This is not surprising since holographic models are not expected to work perfectly in the

<footnote>$^{28}$More precisely, only dependence which is highly suppressed in the UV is allowed, for example terms $\sim \exp(-\#/\lambda)$.</footnote>
UV region, where the coupling $\lambda$ is small, so fixing the parameters by UV arguments may lead to some tension in the model. Recall, for example, that the UV asymptotics of the pressure and the UV asymptotics of the correlators for the energy momentum tensor give slightly different numbers for the normalization of the glue action [147].

For potentials II, we used instead $a(\lambda)$ with nontrivial dependence on $\lambda$, and consequently the meson mass gap will be small.

Finally, let us point out that in all cases classified above, $H_A$ of (B.5) contributes at leading order to the Schrödinger potentials near their bottoms. This is seen from the estimates (B.22) and (B.23) when the bottom is at $r \sim 1/\Lambda_{IR}$, and by inserting the UV expansions in (B.5) if the bottom is at $r \sim 1/m_q$. Consequently, axial vectors and pseudoscalar mesons have larger mass gaps than vectors and scalars: the ratio of, say, the axial and the vector mass gap approaches a finite number which is larger than one in the limit of large quark mass.<fnref n="29" />

## B.2.2 Mass splittings

Let us then discuss the mass splittings between the lowest meson states for the various cases described above. When the Schrödinger potential $V_S(u)$ has a clear and regular minimum, the splittings can be estimated by computing the second derivative $V_S''(u_0)$ at the minimum $u = u_0$. Most of the potentials discussed above fall into two categories, where the Schrödinger potential has a clear minimum either at $r \sim 1/m_q$ or at $r \sim 1/\Lambda_{IR}$.

When the minimum is at $r \sim 1/m_q$, we found that the mass gap was $O(m_q)$. From the definition (B.2) of the Schrödinger coordinate we see that $u \sim r$ and the scale of the derivatives of all relevant functions is given by $d/dr \sim m_q$. Therefore $m_q$ is the only scale that enters the analysis, and the splitting is also $O(m_q)$, i.e., much larger than expected for nonrelativistic bound states. Notice that this also includes some of the asymptotics with critical $v_p = 2$, i.e., those with $0 < v_\ell \le 1$, and those with $v_\ell = 0$ and $\xi > 1$.

When the minimum is at $r \sim 1/\Lambda_{IR}$, the computation is more involved. We shall only discuss the case $v_p = 2$ and $v_\ell = 0$, for which $\tau \sim (rm_q)^\xi$ and the mass gap is large,

$$V_S(u_0) \sim \left. \frac{\tau^2}{r^2} \right|_{r \sim 1/\Lambda_{IR}} \sim \Lambda_{IR}^2 \left( \frac{m_q}{\Lambda_{IR}} \right)^{2\xi} . \qquad (B.42)$$

Recall that the minimum is at $r \sim 1/\Lambda_{IR}$ when $0 < \xi < 1$. At the minimum, all fields ($\tau$, $\lambda$, and $A$) have logarithmic or power-law dependence on the coordinate $r$ so the scale of the $r$-derivatives is $\Lambda_{IR}$. The relevant quantities are, however, the $u$-derivatives of the potential, which can be estimated by using the chain rule and $du/dr \sim \tau \sim (m_q/\Lambda_{IR})^\xi$. We find that

$$V_S''(u_0) \sim \Lambda_{IR}^4, \qquad V_S^{(n)}(u_0) \sim \Lambda_{IR}^{2+n} \left( \frac{m_q}{\Lambda_{IR}} \right)^{(2-n)\xi} \qquad (B.43)$$

where $n > 2$. The extent of the Schrödinger wave functions around the minimum is determined for the lowest fluctuation modes by $V_S''(u_0)$, and roughly given by $\Delta u = u$ –

<footnote n="29">²⁹When $1 < v_p < 2$, the term $H_A$ is actually leading by a logarithmic factor, so the mass gaps of the axials and pseudoscalars are logarithmically enhanced with respect to those of the vectors and the scalars.</footnote>
$u_0 \sim 1/\Lambda_{IR}$. Outside this region, the wave functions vanish very fast. The higher order derivatives in (B.43) vanish as $m_q \to \infty$, and therefore higher order terms in the Taylor expansion of the Schrödinger potential are suppressed (for $u-u_0 \sim 1/\Lambda_{IR}$) and the potential takes the Harmonic oscillator form. In particular, the masses are given by

$$m_n^2 = V_S(u_0) + \sqrt{2V_S''(u_0)} \left(n + \frac{1}{2}\right) + \mathcal{O}\left(\frac{\Lambda_{IR}^{2+\xi}}{m_q^{\xi}}\right), \quad (B.44)$$

where $n = 0, 1, 2, \dots$. From here we see that the mass splitting is suppressed as $\sim \Lambda_{IR} (\Lambda_{IR}/m_q)^\xi$ at large $m_q$.

In the remaining case ($v_p = 2$, $v_\ell = 0$, and $\xi = 1$) the bottom of the Schrödinger potential, as obtained from the leading tachyon solution, is flat for $1/m_q \ll r \ll 1/\Lambda_{IR}$, suggesting a relatively small level splitting. There are, however, subleading logarithmic corrections to the solution due to the YM RG flow, which we have not computed. They will be important, and are expected to cause a minimum either at $r \sim 1/m_q$ or at $r \sim 1/\Lambda_{IR}$. Then the mass splittings of the very lowest states will be as in one of the cases discussed above (with $\xi = 1$ if the minimum is at $r \sim 1/\Lambda_{IR}$), up to logarithmic corrections, unless the subleading corrections cancel miraculously.

### B.2.3 Decay constants

Finally let us discuss the mass dependence of the (vector/axial) decay constants, given in (B.8). The Schrödinger wave function vanishes very rapidly whenever $m_n^2 < V_S(u)$, which limits the integral to the “classically allowed” region.

First we notice that when the Schrödinger has its bottom at $r \sim 1/\Lambda_{IR}$, the decay constants of the lowest states are very small, because $\Xi$ in the above integral formula contains the factor $\sqrt{V_f}$ which is exponentially suppressed because the tachyon is also large near $r \sim 1/\Lambda_{IR}$. For the interesting case of $v_p = 2$, $v_\ell = 0$, and $0 < \xi < 1$ we find that

$$\frac{f_n^2}{N_f N_c} \sim \exp \left[ -\# \left( \frac{m_q}{\Lambda_{IR}} \right)^{2\xi} \right]. \quad (B.45)$$

One can also show that the pion decay constant has similar dependence on $m_q$. Recall that $H_V = 0$ for the vectors and $H_A$ is given in (B.5). The low-lying states are therefore asymptotically decoupled. Sizeable decay constants are only found for highly excited states. More precisely, the suppression factor disappears when the classically allowed region extends to $r \sim 1/m_q$ where the tachyon is no longer small. Since $V_S \sim 1/u^2$ in the UV, this requires $m_n \sim m_q$ (whereas the lowest states had $m_n \sim \Lambda_{IR}(m_q/\Lambda_{IR})^\xi$). Therefore the coupled states appear at the scale where mesons are expected in QCD. By using (B.8), it is possible to show that for such states the decay constants are $f_n^2/(N_c N_f) \sim \Lambda_{IR}^2$.

When the Schrödinger potential has its bottom at $r \sim 1/m_q$, it is easy to see from the above expressions that

$$\frac{f_n^2}{N_c N_f} \sim m_q^2 \quad (B.46)$$

for the lowest states, because $m_q$ is the only scale which enters the formulas.
# C Analysis of the free energy

## C.1 Chiral condensate as the derivative of free energy

The chiral condensate (including the sum over flavor) can be defined as<fnref n="30" />

$$ \langle \bar{q}q \rangle = \frac{\partial \mathcal{E}}{\partial m_q} = \frac{1}{V_4} \frac{\partial S_{\text{on-shell,E}}}{\partial m_q}, \qquad (C.1) $$

where $\mathcal{E}$ is the (zero temperature) energy density of QCD, and the subscript E denotes that Euclidean signature was used (so that the sign of the action is opposite with respect to the Minkowski signature, which was used in the main text).

By computing the renormalized on-shell action using the identity (C.1) one can find the relation between the chiral condensate and the coefficient $\sigma$ in the vev term of the tachyon. This is, however, slightly complicated in V-QCD. First, since there is full backreaction: changing the value of the quark mass will affect the geometry, possibly leading to nontrivial contributions in the $m_q$ derivative of (C.1). Second, the counterterms needed to regularize the on-shell action depend on the quark mass, and may also contribute in the derivative. As it turns out, these issues can be fully solved in the limit of zero quark mass.

Below we will first demonstrate how the free energy, and the backreaction in particular, can be analyzed by using the EoMs for the background. This will be compared to the direct computation of the regularized on-shell action done in appendix C.3. In the calculations below the fields shall be decomposed as

$$ A = A_0 + A_{\tau} + A_1 \qquad \lambda = \lambda_0 + \lambda_{\tau} + \lambda_1 \qquad \tau = \tau_0 + \tau_1 + \tau_q \qquad (C.2) $$

asymptotically in the UV (where $r \to 0$). Here the terms with subscript zero (one) are the source (vev) terms. The terms $A_{\tau}$ and $\lambda_{\tau}$ and are sourced by the leading quadratic terms in the tachyon in the UV ($\propto \tau_0^2$) and are therefore $\propto m_q^2$. The term $\tau_q$ is the leading nonlinear term of the tachyon. It is $\propto m_q^3$ and computed in appendix C.2. It is shown to be subleading with respect to the vev term $\tau_1$ and therefore irrelevant in the calculations below. It is also argued that the same is true for similar terms due to quartic and higher order tachyon perturbation in the expansions of $A$ and $\lambda$. That is, the terms sourced by $\tau_0\tau_q$ or $\tau_0^4$ are subleading with respect to the vev terms $A_1$ and $\lambda_1$. The UV expansions of the various terms will be given in appendix C.3.

### C.1.1 Chiral condensate at vanishing quark mass

Let us start by computing the chiral condensate at zero quark mass. This is the simplest case because the square of the source term of the tachyon does not contribute in the on-shell action, and as we shall see, there is no issue with extracting value of $\sigma$ (the proportionality coefficient of the vev term) from the UV asymptotics.

<footnote n="30">³⁰Recall that the quark mass is defined only up to a proportionality constant in the holographic model, which we have set to unity for simplicity. The inverse of this constant would appear in the definition of the chiral condensate.</footnote>
Consider a perturbation of a generic background (around $m_q = 0$) which keeps $\Lambda_{UV}$ (but not the quark mass) fixed. The variation of the background solution in the UV is found by using the UV expansions from appendix C.3:

$$ \delta A(r) \approx \delta A_1(r) = \delta G \frac{r^4}{\ell^3} \left[ 1 + \mathcal{O} \left( \frac{1}{\log r \Lambda} \right) \right] \qquad (C.3) $$

$$ b_0 \delta \lambda(r) \approx b_0 \delta \lambda_1(r) = \left[ -\frac{45}{8} \delta G - \frac{9}{32} B_{\sigma} \sigma \delta m_q \right] \frac{r^4}{\ell^3} \left[ 1 + \mathcal{O} \left( \frac{1}{\log r \Lambda} \right) \right] \qquad (C.4) $$

$$ \frac{\delta \tau(r)}{\ell} = \delta m_q r (-\log r \Lambda)^{-\rho} \left[ 1 + \mathcal{O} \left( \frac{1}{\log r \Lambda} \right) \right] + \delta \sigma r^3 (-\log r \Lambda)^{\rho} \left[ 1 + \mathcal{O} \left( \frac{1}{\log r \Lambda} \right) \right], \qquad (C.5) $$

where

$$ B_{\sigma} = xW_{0}\kappa_{0}\ell^{5}. \qquad (C.6) $$

The chiral condensate may then be computed by studying the variation of the on-shell action as seen from (C.1). The computation is analogous to the holographic derivation of the first law of thermodynamics at finite temperature and chemical potential: the differential of the free energy equals the variation of the action, and expressing this in terms of the various UV and IR boundary terms gives the terms in the desired expression. In order to be as precise as possible, we will formulate the computation in terms of a conserved infinitesimal current, which can also be easily used in the computations later. It can be found as follows.

The on-shell Lagrangian can be expressed as a total derivative [30]:

$$ L_{\text{on-shell}} \propto \frac{d}{dr} [-2A' e^{3A}] . \qquad (C.7) $$

But any leading variation of a generic Lagrangian around its on-shell value is a total derivative as well, given formally by<fnref n="31" />

$$ \frac{d}{dr} \left[ \sum_{i} \frac{\partial L}{\partial \varphi'_{i}} \delta \varphi_{i} \right], \qquad (C.8) $$

where the sum is over all fields in the Lagrangian. Requiring equality of the generic expression and the variation of (C.7), one identifies the following infinitesimal conserved “current”:

$$ J = 6e^{3A} A' \delta A - 6e^{3A} \delta A' - \frac{8e^{3A} \lambda' \delta \lambda}{3\lambda^2} - \frac{xe^{3A} V_{f\kappa\tau'} \delta \tau}{\sqrt{1 + e^{-2A} \kappa(\tau')^2}}. \qquad (C.9) $$

Indeed it is straightforward to check that $J' = 0$ by using the EoMs (and their variations).

<footnote n="31">³¹In the case of the gravitational action there is a complication because R contains second derivatives. It is, however, well known that the second derivatives can be isolated in another total derivative term, related to the Gibbons-Hawking boundary term.</footnote>
We will then require that the variation of the background is regular in the IR. By using the IR expansions of the background [30] in the expression (C.9) one sees that $J$ vanishes in the IR ($r \to \infty$) so it must vanish everywhere for regular variations:

$$J = 0. \qquad (C.10)$$

In the UV the above expansions are inserted, leading to

$$0 = \lim_{r \to 0} J = -15 \delta G - \frac{9}{4} B_{\sigma} \delta m_q. \qquad (C.11)$$

The expression for the free energy density in terms of $G$, $\sigma$, and $m_q$ can be extracted from the finite temperature computation in appendix C.3 (see the expression for the free energy (C.63)). At small quark mass and at $T = 0$ (so that also $C = 0$) one obtains the expression$^{32}$

$$\mathcal{E} = M^3 N_c^2 \left( 15G + \frac{1}{4} B_{\sigma} m_q \sigma \right). \qquad (C.12)$$

Inserting here (C.11),

$$\delta \mathcal{E} = -2M^3 N_c^2 B_{\sigma} \delta m_q. \qquad (C.13)$$

Therefore the chiral condensate is given by

$$\langle \bar{q}q \rangle = -2M^3 N_c^2 B_{\sigma} \sigma. \qquad (C.14)$$

The coefficient in this equation can be fixed by using the asymptotics of the scalar-scalar correlator (see appendix C in [42]), which leads to

$$\langle \bar{q}q \rangle = -\frac{N_f N_c}{2\pi^2} \sigma. \qquad (C.15)$$

### C.1.2 Chiral condensate at finite quark mass

Let us then try to generalize the above computation to finite $m_q$. Recall that there is an issue in the definition of the vevs $G$ and $\sigma$. In principle, they are well defined as the coefficients of the vev terms $A_1$ and $\tau_1$, respectively. However such definitions are useless in practice, because the source terms $A_0$ and $\tau_0$ cannot be solved perturbatively to high enough orders in order to separate them from the vev terms in the UV. In practice only differences of the vevs can be computed. Therefore we will define the values of $G$ and $\sigma$ with respect to some reference solutions. Let us first discuss how we define a reference solution for all values of $m_q$.

First a (IR regular) solution with exactly zero tachyon (and therefore zero quark mass) is picked. This solution is chosen to have, by definition, $G = 0$ (and trivially $\sigma = 0$). Therefore $A_\tau$ and $A_1$ in (C.2) are zero, and the solution for the scale factor $A$ defines the source term $A_0$. Let us call this solution 1. Then solution 2 is chosen which has finite $m_q$, defines the nonnormalizable term $\tau_0$ of the tachyon in the UV, and has $\sigma = 0$ by definition.

<footnote>$^{32}$We omitted contributions from the reference background used to regulate this expression, but as we shall argue below, this does not affect the result at $m_q = 0$.</footnote>
Further one requires that $\mathcal{G} = 0$ also for solution 2. Since $A_0$ was already defined, this choice also defines $A_\tau$.

For $0 < x < x_c$, the solution 2 can be identified with the solution that was used to define the vanishing of $\sigma$ in the plots of figure 9. In principle any solution can be picked, but as we argued in section 5, a choice which avoids fine tuning is to pick the solution 2 is the crossover between regimes A and B when $0 < x < x_c$ and a solution with tiny quark mass in the conformal window.

We have argued that “nonlinear” terms of the tachyon are suppressed with respect to the vev terms. Similarly, the terms sourced by quartic tachyons in the UV expansions for A and $\lambda$ are suppressed with respect to the vev terms. Therefore fixing the values of $\mathcal{G}$ and $\sigma$ for solutions 1 and 2 is enough to define the vevs for all backgrounds. It is also possible construct a reference background which has vanishing $\mathcal{G}$ and $\sigma$ for arbitrary $m_q$, as a (generally not IR regular) combination of the two IR regular solutions 1 and 2 constructed above by appropriately scaling the constructed $A_\tau$ and $\tau_0$ to have the desired quark mass.

The above construction implies that the vevs can be written as

$$ \mathcal{G} = \hat{\mathcal{G}} - \mathcal{G}_1 - m_q^2 k_A, \quad \sigma = \hat{\sigma} - m_q k_\tau \qquad (C.16) $$

where the hatted quantities are the exact coefficients of the vev terms, and the quantities without hat are given by the above subtraction procedure. The coefficients $k_i$ and $\mathcal{G}_1$ are independent of $m_q$.

Let us then proceed with the calculation. Consider again a perturbation of a background with $\delta m_q \neq 0$, but now at generic value of $m_q$. The UV expansion of the perturbation has additional terms, in particular

$$ \begin{aligned} \delta A = & -\frac{1}{18} B_{\sigma} m_q \delta m_q \frac{r^2}{\ell^3} (-\log r \Lambda)^{-2\rho} \left[ 1 + \mathcal{O}\left( \frac{1}{\log r \Lambda} \right) \right] \\ & + \delta \hat{\mathcal{G}} \frac{r^4}{\ell^3} \left[ 1 + \mathcal{O}\left( \frac{1}{\log r \Lambda} \right) \right], \end{aligned} \qquad (C.17) $$

where we chose to use the hatted vevs. The variation of $b_0\lambda$ includes a term $\propto m_q dm_q$ which turns out to only contribute at subleading orders in the UV, and the term (compare to (C.4))

$$ b_0 \delta \lambda_1 = \left[ -\frac{45}{8} \delta \hat{\mathcal{G}} - \frac{9}{32} B_{\sigma} (\hat{\sigma} \delta m_q + m_q \delta \hat{\sigma}) \right] \frac{r^4}{\ell^3} \left[ 1 + \mathcal{O}\left( \frac{1}{\log r \Lambda} \right) \right]. \qquad (C.18) $$

It is again useful to study the current (C.9). As above, $J = 0$ because it vanishes in the IR limit. In the UV one obtains

$$ 0 = \lim_{r \to 0} J = -15 \delta \hat{\mathcal{G}} - \frac{1}{4} B_{\sigma} m_q \delta \hat{\sigma} - \frac{9}{4} B_{\sigma} \hat{\sigma} \delta m_q. \qquad (C.19) $$

In particular, we expect that no terms $\propto m_q \delta m_q$ arise in the UV because $\hat{\sigma}$ and $\hat{\mathcal{G}}$ are exactly the coefficients of the vev terms. This identity may be rewritten as

$$ \delta \left[ 15 \hat{\mathcal{G}} + \frac{1}{4} B_{\sigma} m_q \hat{\sigma} \right] = -2 B_{\sigma} \hat{\sigma} \delta m_q, \qquad (C.20) $$
where the expression in the square brackets has similar structure to the regularized vacuum energy given above in (C.12). The natural expectation is that indeed

$$ \mathcal{E} = M^3 N_c^2 \left( 15 \hat{G} + \frac{1}{4} B_\sigma m_q \hat{\sigma} \right), \qquad (C.21) $$

but in principle the expression could also contain additional terms $\propto m_q^2$ (or constants) which would cancel in the regularization.

We see that

$$ \delta\mathcal{E} = -2M^3 N_c^2 B_\sigma \hat{\sigma} \delta m_q = -2M^3 N_c^2 B_\sigma (\sigma + k_\tau m_q) \delta m_q, \qquad (C.22) $$

where extra terms $\propto m_q^2$ in the definition of $\mathcal{E}$ would effectively change the value of $k_\tau$ which remains unknown in any case.<fnref n="33" /> Therefore the chiral condensate is given by

$$ \langle \bar{q}q \rangle = -2M^3 N_c^2 B_\sigma \hat{\sigma} = -2M^3 N_c^2 B_\sigma (\sigma + k_\tau m_q) . \qquad (C.23) $$

### C.1.3 Free energy on the Efimov spiral

Let us then discuss how the free energy is computed for the configurations of figure 9, which are found when $0 < x < x_c$. Let us assume for simplicity that we are able to define the vevs such that the constant $k_\tau$ of (C.23) vanishes. In this case the chiral condensate is simply given by the derivative of the free energy density with respect to the quark mass:

$$ \langle \bar{q}q \rangle = \frac{\partial \mathcal{E}}{\partial m_q} = -2M^3 N_c^2 B_\sigma \sigma \equiv -N_f N_c c_\sigma \sigma . \qquad (C.24) $$

Inserting the asymptotic result of (5.14) in (C.24) and integrating, one can readily find the free energy for asymptotically small quark mass:

$$ \frac{1}{N_f N_c \Lambda_{\text{UV}}^4} (\mathcal{E} - \mathcal{E}_0) = - \frac{c_\sigma K_{\text{IR}}^2}{2 K_m K_\sigma \sin(\phi_m - \phi_\sigma)^2} e^{-4u} \qquad (C.25) $$

$$ \times \left[ -\sin(\phi_{\text{IR}} - \phi_{\sigma} - ku) \sin(\phi_{\text{IR}} - \phi_{m} - ku) + \frac{\nu}{4} \sin(\phi_{m} - \phi_{\sigma}) \right] $$

$$ = -\frac{c_\sigma m_q \sigma}{2 \Lambda_{\text{UV}} \Lambda_{\text{UV}}^3} - \frac{\nu c_\sigma K_{\text{IR}}^2}{8 K_m K_\sigma \sin(\phi_m - \phi_\sigma)} e^{-4u}, \qquad (C.26) $$

where $\mathcal{E}_0$ is the free energy of the solution having $m_q = 0 = \sigma$, and $u = \log \Lambda_{\text{UV}}/\Lambda_{\text{IR}}$. In general, free energy differences are given by the area between the spiral and the horizontal axis (see, for example, figure 12).

Interestingly, the result for both zero $m_q$ and zero $\sigma$ simplifies to

$$ \frac{1}{N_f N_c \Lambda_{\text{UV}}^4} (\mathcal{E} - \mathcal{E}_0) = - \frac{\nu c_\sigma K_{\text{IR}}^2}{8 K_m K_\sigma \sin(\phi_m - \phi_\sigma)} e^{-4u} < 0, \qquad (C.27) $$

where we used the handedness of the spiral in (5.16): $\sin(\phi_m - \phi_\sigma) > 0$. Therefore these solutions are dominant over the solution with zero tachyon.

<footnote n="33">³³A possible handle to control the value of this coefficient is to fix somehow the subleading linear corrections to the GOR relation discussed in appendix D.</footnote>
### C.1.4 Free energy with multi-trace deformations

Let us then study how the free energy is computed in the presence of multi-trace deformations. Recall that the UV boundary conditions are

$$ \alpha_m = m_q + \sum_{n=2}^{n_{\max}} g_n c_{\sigma}^{n-1} \beta_m^{n-1} \qquad (C.28) $$

$$ \beta_m = \sigma \qquad (C.29) $$

and that the identity (C.19) is interpreted as

$$ 0 = -15 \delta \mathcal{G} - \frac{1}{4} B_{\sigma} \alpha_{m} \delta \beta_{m} - \frac{9}{4} B_{\sigma} \beta_{m} \delta \alpha_{m}. \qquad (C.30) $$

By using the UV boundary conditions, this identity can be rearranged to read

$$ \delta \left[ \frac{15M^3}{x} \mathcal{G} + \frac{1}{8} c_{\sigma} \alpha_m \sigma + \sum_{n=2}^{n_{\max}} \left( \frac{n-1}{n} \right) g_n c_{\sigma}^n \sigma^n \right] = -c_{\sigma} \delta m_q - \sum_{n=2}^{n_{\max}} \frac{1}{n} c_{\sigma}^n \sigma^n \delta g_n . \qquad (C.31) $$

Therefore the quantity in the square brackets is identified as the new free energy (over $N_f N_c$). It matches with the free energy obtained by renormalizing the action (C.12) up to the last term involving the couplings $g_n$. This extra term is proportional to

$$ W - \beta_m \frac{\delta W}{\delta \beta_m(x)}, \qquad (C.32) $$

where $W$ is as in (6.2). Therefore the term agrees with that found in [112].

The condensates are given by

$$ \langle \mathcal{O} \rangle = \frac{1}{N_f N_c} \frac{\partial \mathcal{E}}{\partial m_q} = -c_{\sigma} \sigma \qquad (C.33) $$

$$ \langle \mathcal{O}^n \rangle = \frac{(-1)^n n}{N_f N_c} \frac{\partial \mathcal{E}}{\partial m_q} = (-c_{\sigma} \sigma)^n, \qquad (C.34) $$

where the normalization factors were read from (6.1). Therefore we find agreement with the large $N$ factorization of the expectation values.

The free energy on the Efimov spiral asymptotically close to the origin can be computed as above. That is, by inserting the spiral equations (6.8) to (C.33) and integrating, we obtain

$$ \frac{1}{N_f N_c \Lambda_{\text{UV}}^4} (\mathcal{E} - \mathcal{E}_0) = -\frac{c_{\sigma}}{2} \frac{m_q}{\Lambda_{\text{UV}}^3} \frac{\sigma}{\Lambda_{\text{UV}}^3} + \sum_{n=2}^{n_{\max}} \frac{n-2}{2n} g_n c_{\sigma}^n \sigma^n - \frac{\nu c_{\sigma} K_{\text{IR}}^2}{8 K_m K_{\sigma} \sin(\phi_m - \phi_{\sigma})} e^{-4u}. \qquad (C.35) $$

Notice that the term involving $g_n$ in (C.35) vanishes for $n = 2$, which is consistent with nonzero $g_2$ amounting to a redefinition of the parameters of the Efimov spiral. The higher order terms are suppressed as $u$ grows because $\sigma \sim \exp(-2u)$. Therefore we conclude as above that the solutions with $m_q = 0$ but $\sigma \neq 0$ dominate over the solution with $\sigma = 0$, and that the solution with the smallest $u$ (for $u \gg 1$ so that the parametrization of the spiral is accurate) has the lowest free energy.
## C.2 Contributions $\propto m_q^3$ in the on-shell action

The on-shell action may involve terms $\propto m_q^3$ which arise due to the nonlinear nature of the tachyon EoM. We will now check how these terms behave. We will need the UV asymptotics of the source and the vev terms of the tachyon which are given in equations (C.48) and in (C.54) in appendix C.3. Using results from [30], the tachyon EoM can be written in the UV as

$$ \tau'' - \frac{3}{r}\tau' + \frac{1}{r^2}\left(3 - \frac{2\rho}{\log(r\Lambda)}\right)\tau - \frac{4\kappa_0 r}{\ell^2}(\tau')^3 + \frac{3\kappa_0}{\ell^2}\tau(\tau')^2 \approx 0, \quad (C.36) $$

where $\kappa_0 = \kappa(0)$. All contributions suppressed by $1/\log(r\Lambda)^2$ or more are hidden in the coefficients of the terms linear in the tachyon, as well as all contributions suppressed by $1/\log(r\Lambda)$ in the nonlinear terms (including the complete terms $\propto \tau^2\tau'$ and $\propto \tau^2(\tau')^3$).

The “nonlinear” tachyon solution is then found by writing

$$ \tau(r) = \tau_0(r) + \tau_q(r), \quad (C.37) $$

where the UV expansion of $\tau_0$ is given in (C.48) and $\tau_q$ is the term which needs to be solved. Using the fact the $\tau_0$ solves the linear equation, one finds that

$$ \begin{aligned} \tau'' - \frac{3}{r}\tau' + \frac{1}{r^2}\left(3 - \frac{2\rho}{\log(r\Lambda)}\right)\tau_q &\approx -\frac{4\kappa_0 r}{\ell^2}(\tau_0')^3 + \frac{3\kappa_0}{\ell^2}\tau_0(\tau_0')^2 \\ &\approx \kappa_0 m_q^3 \ell (-\log(r\Lambda))^{-3\rho}. \end{aligned} \quad (C.38) $$

One finds that (dropping the terms corresponding to the source and vev terms, which arise as solutions to the homogeneous equation)

$$ \frac{\tau_q}{\ell} = \frac{\kappa_0 m_q^3}{2(4\rho - 1)} r^3 (-\log(r\Lambda))^{-3\rho+1} \left[ 1 + \mathcal{O}\left(\frac{1}{\log(r\Lambda)}\right) \right], \quad (C.39) $$

which agrees with the result of [89, 90] when $\rho = 0$.

Notice that the nonlinear term is subleading to the vev term in (C.54) provided that

$$ \rho > \frac{1}{4}, \quad (C.40) $$

which is satisfied for QCD in the Veneziano limit as

$$ \rho = \frac{\gamma_0}{b_0} = \frac{9}{22 - 4x} \ge \frac{9}{22} > \frac{1}{4}, \quad (C.41) $$

where $b_0$ and $\gamma_0$ are the leading coefficients of the beta function and the anomalous dimension of the quark mass, respectively. Therefore one can conclude that only linear terms of the tachyon EoM are relevant in the computation of the on-shell action.<fnref n="34" /> The conclusion is different from that of [89, 90] where the quark mass did not run. Notice, however, that these terms are only logarithmically suppressed and would be important for large values of $m_q$ if a finite value of the UV cutoff was used.

<footnote n="34">³⁴Notice that even though the nonlinearities in the tachyon asymptotics do not appear directly in the computation of the on-shell action, nonlinear terms of the EoM are important in general because they affect the values of the vevs such as the chiral condensate through IR boundary conditions.</footnote>
### C.3 Regularized on-shell action

The free energy is given by the on-shell value of the action. It is a UV divergent quantity and needs to be renormalized.<fnref n="35" /> We work here at finite temperature, and subtract the UV divergences by considering the difference with respect to a reference (zero temperature) background with the same values for the sources. We could also consider the difference between two different zero temperature backgrounds.

It is not difficult to show that the Lagrangian is a total derivative even in the presence of the tachyon (as already shown at zero temperature in [30]). After a straightforward computation, and taking into account the Gibbons-Hawking term, one finds that

$$S_{\text{on-shell}} = M^3 N_c^2 \beta V_3 (6A'f + f') e^{3A}|_{r=\epsilon} + \text{counterterms}. \qquad (\text{C.42})$$

When the quark mass is finite, the tachyon contributes at $O(r^2)$ (terms $\propto m_q^2$) and at $O(r^4)$ (terms $\propto m_q\sigma$) in the UV expansions. These contributions need to be taken into account in the holographic renormalization procedure. The calculation of [140, 141] at $N_f = 0$ used a reference (thermal gas) solution to subtract the divergences. We will use the same technique here. Denoting the fields and other variables of the reference solution by tildes, the following conditions need to be fulfilled at the UV cutoff:

$$\tilde{\beta}e^{\tilde{A}(\tilde{\epsilon})} = \beta e^{A(\epsilon)} \sqrt{f(\epsilon)}, \qquad \tilde{V}_3 e^{3\tilde{A}(\tilde{\epsilon})} = V_3 e^{3A(\epsilon)}, \qquad (\text{C.43})$$

$$\tilde{\lambda}(\tilde{\epsilon}) = \lambda(\epsilon), \qquad \tilde{\tau}(\tilde{\epsilon}) = \tau(\epsilon), \qquad (\text{C.44})$$

where the possibility that the cutoffs of the two solutions are different was included, $\tilde{\epsilon} \neq \epsilon$, as in [140, 141]. In the absence of the tachyon this was convenient since the UV scales $\Lambda = \Lambda_{UV}$ of the two solutions could be chosen to be the same, $\tilde{\Lambda} = \Lambda$. The fact that the present system has two scalars suggest that it is better to choose here $\tilde{\epsilon} = \epsilon$ and satisfy the last two conditions by varying the sources $\Lambda$ and $m_q$. However, it turns out to be convenient to still require that $\tilde{\Lambda} = \Lambda$ and vary the cutoff (and $m_q$) instead. This is so because the tachyon contributions are independent on whether one chooses to vary $\Lambda$ or $\epsilon$: the variation if suppressed by $O(\epsilon^4)$, i.e., $\tilde{\epsilon}/\epsilon = 1 + O(\epsilon^4)$, so that the effect on the tachyon contributions will be down by $O(\epsilon^6)$ and therefore negligible. Keeping $\Lambda$ fixed one can maintain very close contact to the computation of [140, 141].

Turning on the quark mass will modify the UV expansions of $A$ and $\lambda$. at this point it is useful to write down the complete UV expansion including all relevant terms. One can decompose, as $r \to 0$,

$$A = A_0 + A_{\tau} + A_1, \qquad \lambda = \lambda_0 + \lambda_{\tau} + \lambda_1, \qquad f = 1 + f_1, \qquad \tau = \tau_0 + \tau_1. \qquad (\text{C.45})$$

The source terms have the expansions [30]

$$A_0 = -\log r + \log \ell + \frac{4}{9 \log r \Lambda} + \mathcal{O}\left(\frac{1}{(\log r \Lambda)^2}\right) \qquad (\text{C.46})$$

$$b_0 \lambda_0 = -\frac{1}{\log r \Lambda} + \mathcal{O}\left(\frac{1}{(\log r \Lambda)^2}\right) \qquad (\text{C.47})$$

$$\frac{\tau_0}{\ell} = m_q r (-\log r \Lambda)^{-\rho} \left[ 1 + \mathcal{O}\left(\frac{1}{\log r \Lambda}\right) \right]. \qquad (\text{C.48})$$

<footnote n="35">³⁵For detailed analysis of the holographic renormalization of dilaton gravity, see [110, 111, 148, 149].</footnote>
and $\rho = \gamma_0/b_0$ with $\gamma_0$ being the leading coefficient of the anomalous dimension of the quark mass. The $O(r^2)$ terms $A_\tau$ and $\lambda_\tau$ were also included here which are sourced by the $O(r^2)$ tachyon perturbation.<fnref n="36" /> As it turns out, $\lambda_\tau$ is suppressed by logarithms of $\log r\Lambda$ with respect to $A_\tau$ so that it will not enter the calculation and its expansion will not be needed. $A_\tau$ is given by

$$A_{\tau} = -\frac{1}{36} B_{\sigma} m_{q}^{2} \frac{r^{2}}{\ell^{3}} (-\log r \Lambda)^{-2\rho} \left[ 1 + \mathcal{O}\left(\frac{1}{\log r \Lambda}\right) \right], \qquad (C.49)$$

where $W_0 = V_{f0}(0)$, $\kappa_0 = \kappa(0)$, and as above

$$B_{\sigma} = xW_{0}\kappa_{0}\ell^{5}. \qquad (C.50)$$

The vev terms have the expansions (compare to [140, 141])

$$A_1(r) = \mathcal{G} \frac{r^4}{\ell^3} \left[ 1 + \mathcal{O} \left( \frac{1}{\log r \Lambda} \right) \right] \qquad (C.51)$$

$$b_0 \lambda_1(r) = \left( -\frac{45}{8} \mathcal{G} - \frac{9}{32} B_{\sigma} m_q \sigma \right) \frac{r^4}{\ell^3} \left[ 1 + \mathcal{O} \left( \frac{1}{\log r \Lambda} \right) \right] \qquad (C.52)$$

$$f_1(r) = -\frac{C}{4} \frac{r^4}{\ell^3} \left[ 1 + \mathcal{O} \left( \frac{1}{\log r \Lambda} \right) \right] \qquad (C.53)$$

$$\frac{\tau_1(r)}{\ell} = \sigma r^3 (-\log r\Lambda)^\rho \left[ 1 + \mathcal{O}\left(\frac{1}{\log r\Lambda}\right) \right]. \qquad (C.54)$$

Let us then go on with the renormalization procedure. In the expressions below only the differences of the vevs $\mathcal{G}$ and $\sigma$ with respect to the values of the reference solutions will appear. Therefore, without loss of generality, one can set the vevs of the reference solution to zero. As motivated above, one can choose $\tilde{\Lambda} = \Lambda$. Then the relation between $\tilde{\epsilon}$ and $\epsilon$ is fixed<fnref n="37" /> by the first condition in (C.44):

$$\frac{\tilde{\epsilon}}{\epsilon} = 1 + \left( -\frac{45}{8} \mathcal{G} - \frac{9}{32} B_{\sigma} m_q \sigma \right) \frac{\epsilon^4}{\ell^3} (-\log \epsilon \Lambda)^2 \left[ 1 + \mathcal{O} \left( \frac{1}{\log \epsilon \Lambda} \right) \right], \qquad (C.55)$$

whereas the second condition sets

$$\tilde{m}_q = m_q + \sigma \epsilon^2 (-\log \epsilon \Lambda)^{2\rho} \left[ 1 + \mathcal{O} \left( \frac{1}{\log \epsilon \Lambda} \right) \right]. \qquad (C.56)$$

It is then straightforward to calculate the renormalized pressure. Inserting the expression of the (unrenormalized) on-shell action from eq. (C.42) one finds that

$$-\beta V_3 p = \lim_{\epsilon \to 0} [S(\epsilon) - \tilde{S}(\tilde{\epsilon})] \qquad (C.57)$$

$$= M^3 N_c^2 \lim_{\epsilon \to 0} \left[ \beta V_3 (6A'(\epsilon)f(\epsilon) + f'(\epsilon))e^{3A(\epsilon)} - 6\tilde{\beta}\tilde{V}_3\tilde{A}'(\tilde{\epsilon})e^{3\tilde{A}(\tilde{\epsilon})} \right]. \qquad (C.58)$$

<footnote n="36">³⁶There are also $O(r^4)$ terms which arise due to the perturbation from the quartic terms in the tachyon. As we argued above, these contributions are subleading with respect to those arising from the vev terms.</footnote>

<footnote n="37">³⁷It can be checked that the variation of $m_q$, which would enter through the term $\lambda_\tau$ is subleading.</footnote>
After eliminating $\tilde{\beta}$ and $\tilde{V}_3$ by using the conditions (C.43), the expression for the pressure reads

$$p = M^3 N_c^2 \lim_{\epsilon \to 0} e^{3A(\epsilon)} \left[ 6\tilde{A}'(\tilde{\epsilon})e^{A(\epsilon)-\tilde{A}(\tilde{\epsilon})} \sqrt{f(\epsilon)} - 6A'(\epsilon)f(\epsilon) - f'(\epsilon) \right]. \quad (C.59)$$

The relation (C.55) implies that

$$\tilde{A}'(\tilde{\epsilon})e^{-\tilde{A}(\tilde{\epsilon})} = \tilde{A}'(\epsilon)e^{-\tilde{A}(\epsilon)} + \left(\frac{5}{2}\mathcal{G} + \frac{1}{8}B_{\sigma}m_q\sigma\right) \frac{\epsilon^4}{\ell^4} \left[1 + \mathcal{O}\left(\frac{1}{\log \epsilon \Lambda}\right)\right] \quad (C.60)$$

where the expansion (C.46) was used. Further, notice that the variation of $m_q$ enters through

$$A_{\tau}(\epsilon) = \tilde{A}_{\tau}(\epsilon) + \frac{1}{18} B_{\sigma} m_q \sigma \frac{\epsilon^4}{\ell^3} \left[ 1 + \mathcal{O}\left( \frac{1}{\log \epsilon \Lambda} \right) \right] \quad (C.61)$$

so that

$$A(\epsilon) = \tilde{A}(\epsilon) + \left[ \mathcal{G} + \frac{1}{18} B_{\sigma} m_q \sigma \right] \frac{\epsilon^4}{\ell^3} \left[ 1 + \mathcal{O}\left( \frac{1}{\log \epsilon \Lambda} \right) \right]. \quad (C.62)$$

A similar result can be found for the derivative of $A$. Inserting these relations and the expansion of $f$ in (C.59) one obtains the final result<fnref n="38" />

$$p = M^3 N_c^2 \left( \frac{1}{4} C - 15 \mathcal{G} - \frac{1}{4} B_{\sigma} m_q \sigma \right). \quad (C.63)$$

## D Gell-Mann-Oakes-Renner relation

In this appendix we check the GOR relation explicitly. The starting point is the normalization integral (E.13) for the pion wave function. When $m_q$ is small, it is dominated<fnref n="39" /> at small $u \simeq r$. Also the wave function $\hat{\psi}_P$ is constant in the UV up to corrections $\mathcal{O}(r^2)$ (see appendix G in [42]). Therefore integral one needs to compute is (the UV contribution to)

$$I = \int_0^1 dr \frac{1}{V_f \kappa e^{3A \tau^2}}. \quad (D.1)$$

Let us first try to compute the integral by using the UV expansions of the various fields, given in appendix C.3. We obtain

$$I = \frac{1}{W_0 \kappa_0 \ell^3} \int_0^\infty dr \frac{r^3}{\tau(r)^2} \left[ 1 + \mathcal{O}\left(\frac{1}{\log r \Lambda}\right) \right] \\ = \frac{1}{W_0 \kappa_0 \ell^5} \int_0^\infty dr \frac{r^3}{\left\{ m_q r (-\log r \Lambda)^{-\rho} \left[ 1 + \mathcal{O}\left(\frac{1}{\log r \Lambda}\right) \right] + \sigma r^3 (-\log r \Lambda)^{\rho} \left[ 1 + \mathcal{O}\left(\frac{1}{\log r \Lambda}\right) \right] \right\}^2} \quad (D.2)$$

where $\Lambda = \Lambda_{UV}$ and $\kappa_0 = \kappa(0)$. As $m_q \to 0$ this integral is dominated by the regime with $r \sim \sqrt{m_q/\sigma}$. Substituting here $v \simeq r(-\log r \Lambda)^\rho$ one obtains

$$I = \frac{1}{W_0 \kappa_0 \ell^5} \int_0^\infty dv \frac{v^3 \left[ 1 + \mathcal{O}\left( \frac{1}{\log v \Lambda} \right) \right]}{(m_q v + \sigma v^3)^2} = \frac{1}{2W_0 \kappa_0 \ell^5 m_q \sigma} \left[ 1 + \mathcal{O}\left( \frac{1}{\log(m_q \Lambda^2 / \sigma)} \right) \right]. \quad (D.3)$$

<footnote n="38">³⁸Recall that the vevs here should be interpreted as their differences with respect to the values of the reference solutions, because we set the vevs of the reference solution to zero.</footnote>

<footnote n="39">³⁹As can be verified numerically, this holds for the pion state but not for higher pseudoscalar states.</footnote>
Notice that in the walking regime corrections are suppressed only when $m_q$ is a small perturbation, i.e., (A.10) holds. Also, the result (D.3) is not valid in the conformal window for any $m_q$.

From (D.3) we see that the UV RG flow leads to correction terms which are suppressed only by logarithms of $m_q$. It is however possible to show that such corrections vanish to all orders. First we notice that the tachyon EoM (B.12) can be written as

$$ \tau'' + \frac{d}{dr} \log (e^{3A} V_{f0\kappa}) \tau' + \frac{2e^{2A}a}{\kappa} \tau = 0 \qquad (D.4) $$

up to corrections suppressed by $\tau^2$ in the UV. Let us denote by $\tau_\sigma$ the solution having $m_q = 0$ and $\sigma = 1$, and let $\tau$ be a generic solution with $m_q \neq 0$. The Wronskian

$$ W \equiv \tau \tau'_{\sigma} - \tau_{\sigma} \tau' \qquad (D.5) $$

satisfies

$$ W' + \frac{d}{dr} \log (e^{3A} V_{f0\kappa}) W = 0 \qquad (D.6) $$

with the solution

$$ W = \tau \tau'_{\sigma} - \tau_{\sigma} \tau' = \frac{C_W}{e^{3A} V_{f0\kappa}}. \qquad (D.7) $$

By inserting the UV expansions for all fields we see that the constant is given by

$$ C_W = 2W_0 \kappa_0 \ell^5 m_q. \qquad (D.8) $$

Notice that (D.7) may be written as

$$ \frac{d}{dr} \frac{\tau_{\sigma}}{\tau} = \frac{C_W}{e^{3A} V_{f0\kappa} \tau^2}. \qquad (D.9) $$

By integrating this identity, the UV contribution to the integral (D.1) becomes

$$ \int_0^{r_{\text{cut}}} \frac{1}{e^{3A} V_{f0\kappa} \tau^2} = \frac{1}{C_W} \left. \frac{\tau_{\sigma}}{\tau} \right|_0^{r_{\text{cut}}} = \frac{1}{C_W} \frac{\tau_{\sigma}(r_{\text{cut}})}{\tau(r_{\text{cut}})}. \qquad (D.10) $$

Choosing the cutoff such that $r_{\text{cut}} \gg \sqrt{m_q/\sigma}$ (but so that it is not too large so that our approximation are still valid) we obtain

$$ I = \int_0^{r_{\text{cut}}} \frac{1}{e^{3A} V_{f0\kappa} \tau^2} = \frac{1}{C_W \sigma} \left[ 1 + \mathcal{O} \left( \frac{m_q}{\sigma r_{\text{cut}}^2} \right) \right]. \qquad (D.11) $$

Our approximations will break down at $r_{\text{cut}} \sim 1/\Lambda_{\text{UV}}$, where the r-dependence of the tachyon changes qualitatively. In order to complete the calculation, one should estimate the contributions to the normalization integral (E.13) for $r \gtrsim 1/\Lambda_{\text{UV}}$. But in this regime the dependence on the tachyon is regular and one can just do a Taylor expansion at $m_q = 0$. The cutoff dependence should cancel against the UV contribution given in (D.11). As the dust clears, we can effectively set $r_{\text{cut}} \sim 1/\Lambda_{\text{UV}}$, in (D.11), obtaining

$$ I = \frac{1}{C_W \sigma} \left[ 1 + \mathcal{O} \left( \frac{m_q \Lambda_{\text{UV}}^2}{\sigma} \right) \right]. \qquad (D.12) $$
The normalization condition (E.13) becomes for small quark mass

$$1 = \hat{\psi}_{P,\pi}^2(0)I \left[ 1 + \mathcal{O}\left( \frac{m_q \Lambda_{UV}^2}{\sigma} \right) \right] = \frac{\hat{\psi}_{P,\pi}^2(0)}{2\ell^5 W_0 \kappa_0 m_q \sigma} \left[ 1 + \mathcal{O}\left( \frac{m_q \Lambda_{UV}^2}{\sigma} \right) \right], \quad (D.13)$$

where we restricted to the lowest pseudoscalar mode, the pion. Solving for $\hat{\psi}_{P,\pi}(0)$ and inserting in (E.21) one obtains the GOR relation

$$f_{\pi}^{2} m_{\pi}^{2} \approx 2 M^{3} N_{f} N_{c} \ell^{5} W_{0} \kappa_{0} m_{q} \sigma = 2 M^{3} N_{c}^{2} B_{\sigma} m_{q} \sigma \approx -m_{q} \langle \bar{q} q \rangle \quad (D.14)$$

with corrections suppressed by $m_q \Lambda_{UV}^2/\sigma$. It can be checked that the proportionality factor is correct for our definitions of $f_\pi$ and $\langle \bar{q}q \rangle$.

## E Fluctuation equations, $f_\pi$, and the S-parameter

The radial wave functions for the flavor nonsinglet fluctuations satisfy the following equations [42]:

$$\partial_u(C_V \partial_u \psi_V) = C_V q^2 \psi_V \quad (E.1)$$

$$\partial_u(C_V \partial_u \psi_A) = C_V(H_A + q^2)\psi_A \quad (E.2)$$

$$\partial_u(C_V \partial_u \psi_L) = C_V H_A (\psi_L - \psi_P) \quad (E.3)$$

$$H_A \partial_u \psi_P = -q^2 \partial_u \psi_L \quad (E.4)$$

where

$$\frac{du}{dr} = \sqrt{1 + e^{-2A} \kappa(\tau')^2} \equiv G, \qquad H_A = \frac{4\tau^2 \kappa e^{2A}}{w^2}, \quad (E.5)$$

$$C_V = V_f w^2 e^A . \quad (E.6)$$

In addition it is convenient to define the pseudoscalar wave function

$$\hat{\psi}_P = -C_V \partial_u \psi_L \quad (E.7)$$

which satisfies the single equation

$$\partial_u(C_P \partial_u \hat{\psi}_P) = C_P(H_A + q^2) \hat{\psi}_P \quad (E.8)$$

where

$$C_P = \frac{4}{C_V H_A} = \frac{1}{V_f \tau^2 \kappa e^{3A}}. \quad (E.9)$$

At nonzero quark mass the pion decay constant may be defined in terms of the pole of $\Pi_L$ in (7.16) at $q^2 = -m_\pi^2$. As the computation is similar also for higher modes, a generic pseudoscalar fluctuation can be considered. To compute the decay constants one needs to study the fluctuations for small values of

$$\delta q^2 = q^2 + m_n^2 \quad (E.10)$$
where $m_n^2$ is the mass of the fluctuation mode. The wave functions are written as

$$ \psi_L = k_L \left[ \psi_{L,n} + \delta q^2 \tilde{\psi}_L + \mathcal{O}((\delta q^2)^2) \right] \quad (\text{E.11}) $$

$$ \psi_P = k_L \left[ \psi_{P,n} + \delta q^2 \tilde{\psi}_P + \mathcal{O}((\delta q^2)^2) \right], \quad (\text{E.12}) $$

where $k_L$ is a normalization constant which will be fixed below.

When $\delta q^2 = 0$ the wave functions are normalizable. One can therefore choose the fields $\psi_{I,n}$ to satisfy the usual normalization condition

$$ 1 = \int_0^\infty du C_P \hat{\psi}_{P,n}^2 = \int_0^\infty du \frac{4C_V}{H_A} (\partial_u \psi_{L,n})^2, \quad (\text{E.13}) $$

with $\hat{\psi}_{P,n}$ defined as in (E.7) for the normalizable mode. At finite but small $\delta q^2$ we impose the standard normalization condition in the UV:

$$ 1 = \psi_L(0) \approx k_L \delta q^2 \tilde{\psi}_L(0); \quad 0 = \psi_P(0) \propto \tilde{\psi}_P(0), \quad (\text{E.14}) $$

where we used the fact that the normalizable wave functions vanish in the UV.

Expanding the fluctuation equations at small $\delta q^2$ gives

$$ \partial_u(C_V \partial_u \tilde{\psi}_L) = C_V H_A(\tilde{\psi}_L - \tilde{\psi}_P) \quad (\text{E.15}) $$

$$ H_A \partial_u \tilde{\psi}_P = m_n^2 \partial_u \tilde{\psi}_L - \partial_u \psi_{L,n}. \quad (\text{E.16}) $$

By using these equations and the fluctuation equations for $\psi_{I,n}$ one finds the identity

$$ \begin{aligned} & \partial_u \left[ m_n^2 C_V (\tilde{\psi}_L \partial_u \psi_{L,n} - \psi_{L,n} \partial_u \tilde{\psi}_L) + H_A C_V (\tilde{\psi}_P \partial_u \psi_{P,n} - \psi_{P,n} \partial_u \tilde{\psi}_P) \right] \\ & = H_A C_V \psi_{P,n} (\psi_{P,n} - \psi_{L,n}) \end{aligned} \quad (\text{E.17}) $$

Integrating this over $u$, using the boundary conditions from above, and further using the fluctuation equations to simplify the result one obtains

$$ C_V \tilde{\psi}_L \partial_u \psi_{L,n} \Big|_{u=0} = - \frac{1}{m_n^2} \int_0^\infty du \, \psi_{P,n} \partial_u \hat{\psi}_{P,n}. \quad (\text{E.18}) $$

Inserting (E.14), integrating partially, and using again fluctuation equations this relation simplifies to

$$ C_V \partial_u \psi_{L,n} \Big|_{u=0} = - \frac{k_L \delta q^2}{4} \int_0^\infty du C_P \hat{\psi}_{P,n}^2 = - \frac{k_L \delta q^2}{4}. \quad (\text{E.19}) $$

By using this result and the definitions from (E.11), $\Pi_L$ from (7.16) can be written as

$$ \begin{aligned} \Pi_L(q^2) &= \frac{M^3 N_f N_c}{4q^2} C_V \partial_u \psi_L \Big|_{u=0} \approx \frac{M^3 N_f N_c}{4q^2} C_V k_L \partial_u \psi_{L,n} \Big|_{u=0} \\ &= -\frac{M^3 N_f N_c}{q^2 \delta q^2} (C_V \partial_u \psi_{L,n})^2 \Big|_{u=0} \end{aligned} \quad (\text{E.20}) $$

for small $\delta q^2$. The decay constants are given in terms of the residue at $\delta q^2 = 0$:

$$ f_n^2 m_n^2 = M^3 N_f N_c (C_V \partial_u \psi_{L,n})^2 \Big|_{u=0} = M^3 N_f N_c (\hat{\psi}_{P,n})^2 \Big|_{u=0}. \quad (\text{E.21}) $$
From (E.20) we see that for $q^2 \sim m_\pi^2$, as $m_q \to 0$,

$$ \Pi_L(q^2) \approx \frac{f_\pi^2 m_\pi^2}{q^2(q^2 + m_\pi^2)} = \frac{f_\pi^2}{q^2} - \frac{f_\pi^2}{q^2 + m_\pi^2} \qquad (E.22) $$

so that $S_0$ (defined as the zero momentum value of $q^2\Pi_L(q^2)$) approaches $f_\pi^2$. The same factor for the axial-axial correlator does not have a pion node, so that simply

$$ \Pi_A(q^2) \approx \frac{f_\pi^2}{q^2} \qquad (E.23) $$

in the same scaling limit. Consequently, the axial-axial correlator has the correct structure at small momentum:

$$ (q^2\eta^{\mu\nu} - q^\mu q^\nu) \Pi_A(q^2) + q^\mu q^\nu \Pi_L(q^2) \approx f_\pi^2 \left( \eta^{\mu\nu} - \frac{q^\mu q^\nu}{q^2 + m_\pi^2} \right) . \qquad (E.24) $$

That is, the longitudinal term arises from the coupling of the pion to the axial current.

Let us then derive a few integral representations which are useful when computing observables numerically. First we analyze

$$ D(q^2) \equiv q^2 \Pi_A(q^2) - q^2 \Pi_A(q^2) = \frac{1}{4} M^3 N_f N_c C_V \left[ \partial_u \psi_V(u, q^2) - \partial_u \psi_V(u, q^2) \right]_{u=\epsilon} , \qquad (E.25) $$

where we inserted (7.16), and used the fact that $r \simeq u$ near the boundary. Recalling the normalization $\psi_V(u = \epsilon) = 1 = \psi_V(u = \epsilon)$ we obtain

$$ \begin{aligned} C_V [\partial_u \psi_V - \partial_r \psi_V]_{u=\epsilon} &= \int_{\epsilon}^{\infty} du \, \partial_u [\psi_V C_V \partial_u \psi_A - \psi_A C_V \partial_u \psi_V] \\ &= \int_{\epsilon}^{\infty} du \, \psi_V H_A C_V \psi_A, \end{aligned} \qquad (E.26) $$

where fluctuation equations (E.1) were used at the last step. Taking $\epsilon \to 0$, it follows that

$$ D(q^2) = \frac{1}{4} M^3 N_f N_c \int_0^\infty du \, \psi_V H_A C_V \psi_A . \qquad (E.27) $$

A rather similar formula can be derived for the S-parameter. Let us denote

$$ \left. \frac{\partial}{\partial q^2} \psi_{V/A} \right|_{q^2=0} = \dot{\psi}_{V/A} . \qquad (E.28) $$

Then the S-parameter (7.18) can be written as

$$ S = \pi M^3 N_f N_c C_V \partial_u \left( \dot{\psi}_A - \dot{\psi}_V \right)_{u=\epsilon} . \qquad (E.29) $$

Derivating the fluctuation equations with respect to $q^2$ results in

$$ \begin{aligned} \partial_u(C_V \partial_u \dot{\psi}_V) &= C_V \psi_V \\ \partial_u(C_V \partial_u \dot{\psi}_A) &= C_V \psi_A + C_V H_A \dot{\psi}_A \end{aligned} \qquad (E.30) $$
where it is understood that $\psi_{V,A}$ are evaluated at $q^2 = 0$. The combination appearing in the S-parameter can then be written as

$$ C_V \partial_u [\dot{\psi}_A - \dot{\psi}_V]_{u=\epsilon} \approx \int_{\epsilon}^{\infty} du \, \partial_u [\psi_V C_V \partial_u \dot{\psi}_V - \psi_V C_V \partial_u \psi_V - \psi_A C_V \partial_u \dot{\psi}_A + \dot{\psi}_A C_V \partial_u \psi_A] \quad (E.31) $$

$$ = \int_{\epsilon}^{\infty} du \, C_V (\psi_A^2 - \psi_V^2), \quad (E.32) $$

where we dropped higher order terms in $\epsilon$ in the first step (noticing that $\dot{\psi}_{V,A}$ vanish fast in the UV) and used the fluctuation equations (E.1) as well as equations (E.30) in the second step. Taking $\epsilon \to 0$, we obtain the final result

$$ S = \pi M^3 N_f N_c \int_0^\infty du \, C_V [\psi_A^2 - \psi_V^2]_{q^2=0}. \quad (E.33) $$

Here one could also insert that $\psi_V = 1$ when $q^2 = 0$.

## F Free field (one-loop) computation of the S-parameter

The free field result for the vector-vector and axial-axial correlators is given by

$$ \int d^4 x \, e^{-iqx} \langle 0 | T \{ J_{\mu}^{a}(V/A)(x) J_{\nu}^{b}(V/A)(0) \} | 0 \rangle = \frac{N_c \delta^{ab}}{2} \int \frac{d^4 k}{(2\pi)^4} \frac{\text{Tr} [\gamma^{\mu}(\gamma_5)(\not{\!k} + m)\gamma^{\nu}(\gamma_5)(\not{\!k} - \not{\!q} + m)]}{(k^2 + m^2 - i\epsilon) [(k-q)^2 + m^2 - i\epsilon]}, \quad (F.1) $$

where the $\gamma_5$'s are only present in the axial-axial correlator. The loop is divergent but the contribution to the S-parameter will be finite.

Doing the integral with dimensional regularization one obtains

$$ \Pi_V(q^2) = \frac{2N_f N_c}{(4\pi)^2} \int_0^1 dx \, x(1-x) \times \left[ \frac{2}{\epsilon} - \log(m^2 + x(1-x)q^2) - \gamma + \log 4\pi + \mathcal{O}(\epsilon) \right] \quad (F.2) $$

$$ \Pi_A(q^2) = \frac{2N_f N_c}{(4\pi)^2} \int_0^1 dx \left[ x(1-x) + \frac{m^2}{q^2} \right] \times \left[ \frac{2}{\epsilon} - \log(m^2 + x(1-x)q^2) - \gamma + \log 4\pi + \mathcal{O}(\epsilon) \right] \quad (F.3) $$

$$ \Pi_L(q^2) = \frac{2N_f N_c m^2}{(4\pi)^2 q^2} \times \int_0^1 dx \left[ \frac{2}{\epsilon} - \log(m^2 + x(1-x)q^2) - \gamma + \log 4\pi + \mathcal{O}(\epsilon) \right] . \quad (F.4) $$

Consequently

$$ D(q^2) - D(0) = - \frac{2N_f N_c m^2}{(4\pi)^2} \int_0^1 dx \log \left[ 1 + x(1-x) \frac{q^2}{m^2} \right], \quad (F.5) $$

where the value at $q^2 = 0$ was subtracted in order to remove the logarithmic divergence.
For the S-parameter one obtains the well-known result

$$S = -4\pi D'(0) = \frac{N_f N_c}{2\pi} \int_0^1 dx \, x(1-x) = \frac{N_f N_c}{12\pi}, \qquad (F.6)$$

whereas the corresponding finite difference reads

$$-4\pi \frac{D(q^2) - D(0)}{q^2} = \frac{N_f N_c}{2\pi} \frac{m^2}{q^2} \int_0^1 dx \log \left[ 1 + x(1-x) \frac{q^2}{m^2} \right] \qquad (F.7)$$

$$= \frac{N_f N_c}{\pi} \frac{m^2}{q^2} \left[ \sqrt{1 + 4 \frac{m^2}{q^2}} \operatorname{arctanh} \frac{1}{\sqrt{1 + 4 \frac{m^2}{q^2}}} - 1 \right] \qquad (F.8)$$

in agreement with [121–123]. The series as $q^2 \to 0$ and as $m^2 \to 0$ are given by

$$-4\pi \frac{D(q^2) - D(0)}{q^2} = \frac{N_f N_c}{12\pi} \left[ 1 - \frac{1}{10} \frac{q^2}{m^2} + \frac{1}{70} \frac{q^4}{m^4} + \dots \right] \qquad (F.9)$$

$$= \frac{N_f N_c}{\pi} \left[ -\frac{m^2}{q^2} - \frac{m^2}{2q^2} \log \frac{m^2}{q^2} + \frac{m^4}{q^4} - \frac{m^4}{q^4} \log \frac{m^2}{q^2} + \dots \right]$$

## G Details on numerics

The numerical results in this paper were computed for sets of potentials termed “potentials I” and “potentials II”. They are exactly the sets given in [42], but we repeat their definitions here for completeness:

* Both Potentials I & II.

$$V_g(\lambda) = V_0 \left[ 1 + V_1\lambda + V_2\lambda^2 \frac{\sqrt{1 + \log(1 + \frac{\lambda}{\lambda_0})}}{\left(1 + \frac{\lambda}{\lambda_0}\right)^{2/3}} \right],$$

$$V_{f0}(\lambda) = W_0 \left[ 1 + W_1\lambda + W_2\lambda^2 \right]. \qquad (G.1)$$

* Potentials I.

$$a(\lambda) = a_0, \qquad \kappa(\lambda) = \frac{1}{\left(1 + \frac{3a_1}{4}\lambda\right)^{4/3}}. \qquad (G.2)$$

* Potentials II.

$$a(\lambda) = a_0 \frac{1 + a_1 \lambda + \frac{\lambda^2}{\lambda_0^2}}{\left(1 + \frac{\lambda}{\lambda_0}\right)^{4/3}}, \qquad \kappa(\lambda) = \frac{1}{\left(1 + \frac{\lambda}{\lambda_0}\right)^{4/3}}. \qquad (G.3)$$

Here the most of the coefficients are fixed by matching to perturbative QCD. Exceptions include the normalizations factors $V_0$, which fixes the UV AdS radius, and $W_0$, which remains as a free parameter. We also set $\ell(x = 0) = 1$, and choose the parameter $\lambda_0$, which only affects the higher order coefficients of the UV expansions, such that the higher order
coefficients have approximately the same relative size as with standard scheme choices in perturbative QCD. Explicitly, the coefficients satisfy

$$V_0 = 12,$$

$$W_1 = \frac{24 + (11 - 2x)W_0}{27\pi^2 W_0},$$

$$a_0 = \frac{12 - xW_0}{8},$$

$$V_1 = \frac{11}{27\pi^2},$$

$$W_2 = \frac{24(857 - 46x) + (4619 - 1714x + 92x^2) W_0}{46656\pi^4 W_0};$$

$$a_1 = \frac{115 - 16x}{216\pi^2},$$

$$V_2 = \frac{4619}{46656\pi^4};$$

$$\lambda_0 = 8\pi^2. \qquad (G.4)$$

Two qualitatively different choices for $W_0$ are possible: either constant $W_0$, which satisfies

$$0 < W_0 < 24/11, \qquad (G.5)$$

or $W_0$ fixed such that the pressure agrees with the Stefan-Boltzmann (SB) result at high temperatures [37] (without the need to include $x$ dependence in the normalization of the action). The latter option is given explicitly (when $\ell(x = 0) = 1$) by

$$W_0 = \frac{12}{x} \left[ 1 - \frac{1}{\left(1 + \frac{7}{4}x\right)^{2/3}} \right] \quad (\text{Stefan-Boltzmann}), \qquad (G.6)$$

so that the AdS radius is

$$\ell(x) = \sqrt[3]{1 + \frac{7}{4}x}. \qquad (G.7)$$

In this article we have always chosen $W_0 = 3/11$ for potentials I, and the SB normalized $W_0$ for potentials II. For the coupling of the gauge fields $w(\lambda)$ which is required for the computation of the vector correlators, we used $w(\lambda) = \kappa(\lambda)$ for potentials I and $w(\lambda) = 1$ for potentials II.

The numerical result were mostly computed as detailed in [30, 37, 42]: coupled ordinary differential equations were solved by shooting from the IR, either starting from near the IR singularity (at zero temperature) or near the horizon (at finite temperature). In order to obtain accurate and reliable results, some tricks had to be used in various cases, in particular near the BZ point $x = x_{BZ}$ and at large quark masses.

In general, the difficulties of the numerics in V-QCD arise from two sources: the IR singularity and the logarithmic corrections to the UV asymptotics. The latter easily lead to sizeable numerical errors when one tries to extract the values of the sources or vevs near the boundary. To improve the accuracy of the numerical analysis, we did the following:

* The scale factor $A$ was used as the coordinate instead of $r$. This makes it easier to analyze the UV asymptotics, because the mapping from $r$ to $A$ is logarithmic and expands the details near the boundary.

* Near the IR singularity (at zero temperature), where possible, EoMs were implemented such that they do not contain the extremely small factors $\propto \exp(-a\tau^2)$. In many cases, this could be done by writing all expression in terms of $V_{\log}(\lambda, \tau) = \log V_f(\lambda, \tau) = -a\tau^2 + \log V_{f0}(\lambda)$ rather than in terms of $V_f$.
* Near the BZ region and at very large $m_q$, the zero temperature backgrounds were constructed by shooting from the IR toward the UV in four steps. Very close to the singularity, where the tachyon is large and completely decouples the flavors, YM solution was used for the geometry and the dilaton, whereas the tachyon was solved by using a simplified EoM with only the dominant terms on top of the YM background. In the next step, the complete tachyon EoM together with the decoupled YM flow for the geometry was used. In the third step, all fields were solved from the full coupled EoMs. In the final step, the tachyon was again solved separately since it is decoupled from the other fields near the boundary. Actually $\hat{\tau} = e^A\tau$ was used as the field, because it decreases much slower than $\tau$, and the flow can then be tracked closer to the boundary.

The first two steps were necessary because the tachyon EoM becomes stiff in the IR, in particular for large $m_q$, so that the tachyon takes larger values in the IR than otherwise. The last step was useful in particular at large $x$, where the RG flow of the fields is slower, and one needs to solve them closer to the boundary in order to control the leading logarithmic behavior of the various terms.

* For the finite temperature backgrounds in the BZ region and at large $m_q$, the background was solved in two steps, which were analogous to the two last steps of the zero temperature construction. That is, the full system was used in the IR, and the tachyon was treated as a decoupled field near the UV.

* The S-parameter was computed by using the integral formula (7.21) rather than UV asymptotics of the fluctuation wave functions.

* The nonsinglet meson wave functions were computed by rewriting the fluctuation equations into a system of two coupled first order equations (rather than a second order equation). Again the equations were written in a form which does not contain explicitly the tiny factors $\exp(-a\tau^2)$.

**Open Access.** This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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