# The V-QCD baryon : numerical solution and baryon spectrum

M. Järvinen♯, E. Kiritsis♮♭, F. Nitti♮, E. Préau♮

♮ Université Paris Cité, CNRS, Astroparticule et Cosmologie, F-75006 Paris,
France

♭ Crete Center for Theoretical Physics, Institute for Theoretical and
Computational Physics, Department of Physics, P.O. Box 2208,
University of Crete, 70013, Heraklion, Greece

♯ Asia Pacific Center for Theoretical Physics, Pohang 37673, Republic of Korea
and Department of Physics, Pohang University of Science and Technology, Pohang
37673, Republic of Korea

###### Abstract:

The single baryon solution of V-QCD is numerically computed. The spectrum of
spin and isospin modes is also computed by quantizing the light fluctuations
around the baryon. It is shown that there is a partial restoration of chiral
symmetry at the baryon center.

††preprint: APCTP Pre2022 - 027
CCTP-2022-8
ITCP-2022/8

##  1 Introduction

The holographic correspondence, [1, 2, 3, 4] has offered from the start a new
tool to approach the strongly coupled low energy physics of QCD or QCD-like
theories in the large-$N$ limit [5, 6, 7, 8, 9, 10, 11, 12]. This applies in
particular to the description of baryon bound states. These have been
investigated both in top-down string-theoretical constructions [13, 14, 15,
16, 17, 18, 31, 32] and in simple bottom-up phenomenological models [33, 34,
35]. In all these models, the baryon can be understood as an instanton of the
bulk non-abelian gauge fields which are holographically dual to the
$U(N_{f})_{L}\times U(N_{f})_{R}$ flavor currents.

Within the bottom-up approach, one of the most complete phenomenological
frameworks is the V-QCD setup [36, 37, 38, 39, 40, 41, 42], a five-dimensional
holographic model which aims at describing the Veneziano limit (large $N_{c}$,
large $N_{f}$ with $N_{c}/N_{f}$ fixed) of QCD with $N_{c}$ colors and $N_{f}$
flavors.

The V-QCD model has 22 parameters in its CP-even part and one extra parameter
in the Chern-Simons term. Although upon a more complete fit it may turn out
that some parameters may not be as important, a substantial number of
parameters is necessary because the model has the ambition to describe a
rather complete set of observables that go well beyond other competing models.
The model can provide upon calculation, most $T=0$ mass spectra including
baryons, thermodynamic functions and phase diagram in a multidimensional space
of temperature, baryon and isospin chemical potentials, correlation functions
of several local operators at finite temperature and density, including
transport coefficients and quasinormal modes and eventually way out-of
equilibrium observables like quenches. It therefore can provide info that goes
well beyond the parameter input to define the theory.

There are several approaches and competing phenomenological models that we
summarize below.

  * •

Lattice Quantum Chromodynamics [19]. This is an ab-initio approach. Wherever
it is applicable, and the numerics are reliable, it gives the proper answer of
QCD. It has however, computational limitations: finite density is out of
direct reach as the probability density is not real and similar remarks apply
to real time dynamical processes. Clearly, phenomenological models like V-QCD
are useful if they can address issues that cannot be studied by Lattice
techniques. The problems mentioned above are in this class and as known,
holographic models are well-tuned to address them.

  * •

Chiral Perturbation Theory (CPTh), [20]-[22]. This is an Effective Theory
based on symmetry principles and as such reliable in the low-energy pion
sector. The addition of baryons is more tricky and although the formalism is
well-studied, it rests on shakier principles. That being said, this approach
is the workhorse for analysing low-energy phenomena and low-density phase
diagrams of the hadronic phase. It is however unsuitable for the study of
other phases where confinement and chiral symmetry are realized differently.
Therefore V-QCD is more appropriate to address static and dynamical issues in
the various plasma phases.

  * •

FRG improved CPTh, [23]. This is CPTh improved with the functional
Renormalization Group. It allows CPTh to extend a bit up in energy but has
similar limitations as CPTh.

  * •

PNJL models, [24, 25]. These are extensions of the NJL model,[26, 27], by
including the Polyakov loop as one extra order parameter. They are
phenomenological, but well-motivated and have made predictions on the phase
diagram of QCD. They are expected to work less well in phases where chiral
symmetry is unbroken, and quarks are not confined. V-QCD is well tuned for
dynamical questions in such phases.

  * •

Quark-Meson-Coupling models, [28, 29]. They are hybrids between MIT bags for
nucleons and meson exchanges and they were developed as extensions of nuclear
QFT Serot-Walecka models. They can describe well parts of nuclear
phenomenology in the hadron phase, and possibly in medium quark condensates,
but are not best suited for pure deconfined/quark-gluon plasma phases.

  * •

HTL quasiparticle models, [30]. This is a class of weakly-coupled models of
massive quasi-particles with improved HTL propagators, used to extrapolate
calculations of finite temperature static quantities to finite chemical
potential. Their use so far is restricted to static properties and their
success is localized.

Compared to all the approaches above, V-QCD is a model that can address a
wider variety of problems than any of them. Moreover, unlike other approaches,
it describes naturally both phases and dynamics with broken or unbroken chiral
symmetry, as well as confined and deconfined phases.

Up to very recently, a description of the baryon state in V-QCD was missing.
This was due in part to the incomplete understanding of Chern-Simons terms in
this theory. These terms are crucial for the construction of the baryon as an
instanton, since a) they contribute to stabilising the instanton size and
position in the bulk; b) they provide the correct identification between bulk
instanton number and boundary baryon number.

Recently, this gap was filled by the authors of the present work in [43],
where, in the limit of zero quark masses, a systematic analysis of the allowed
Chern-Simons terms in V-QCD was performed and the general construction of the
baryon solution was presented. In that work, it was shown that an axial
$SU(2)$ instanton ansatz satisfying appropriate boundary conditions
(normalizability close to the AdS boundary and regularity in the interior) has
indeed all the properties of a single localized baryon state: finite boundary
energy and unit boundary baryon charge. The latter is indeed a topological
quantity, which matches both the bulk instanton number and the boundary
Skyrmion number (written in terms of an appropriate unitary pion matrix).

###  1.1 Summary

The present work is the direct continuation of [43]. While that work presented
the general features of the solution, here we construct the baryon in a
specific model: we first obtain numerically the static instanton solution
(which corresponds to the baryon ground state), then we analyse the instanton
collective modes and their quantization (which correspond to baryon excited
states).

As important ingredient of this work, we present a specific model in the V-QCD
class which offers a good quantitative match to low-energy QCD parameters,
including some parameters in the flavor sector (like the pion decay constant)
which were not correctly reproduced in previous models.

We now give a summary of the results obtained in this work. We refer the
reader to the introduction of [43] for a more detailed discussion and an
extensive list of references on holographic baryons.

#### Fit to QCD data

We carry out an extensive comparison of the model predictions to experimental
and lattice QCD data in order to pin down the parameters of the V-QCD action
in section 2. The comparison consists of two main steps:

  1. 1.

Qualitative comparison to QCD physics. The action both in the limit of weak
and infinitely strong coupling, up to a few remaining parameters, can be
determined by requiring that the model respects known properties of QCD such
as confinement and asymptotic freedom. This work has been done in earlier
literature [36, 38, 40, 39, 44], and we simply review the results here.

  2. 2.

Quantitative comparison to QCD data. The details of the action at intermediate
coupling, as well as the few remaining weak and strong coupling parameters,
can be tuned so that the predictions of the model match with QCD data. In this
step, the dependence of the predictions on the exact values of the model
parameters is typically weak. Despite this, the model has been able to
describe various observables to a high precision [37, 45, 46].

As for the second step, the work in this article extends the earlier work
where the full V-QCD model was separately compared to data for thermodynamics
[45] and to meson spectra [46]:

  * •

We fit the model parameters to data for thermodynamics and spectra
simultaneously.

  * •

Unlike in [46] (where the model was fitted to a large number of excited meson
states), we stress the lowest lying meson states.

  * •

Importantly, we require a good match of the model with the experimental value
of the pion decay constant $f_{\pi}\approx 92~{}\mathrm{MeV}$, which was
poorly reproduced in both previous fits.

We now discuss the fit in more detail. The V-QCD model contains two sectors,
corresponding to gluons (improved holographic QCD [36]) and quarks (tachyon
Dirac-Born-Infeld actions for space filling branes [47, 12]). The former
sector can be separately compared to data from lattice analysis of pure Yang-
Mills theory [37]. In this article, we use the fit of [46] for the gluon
sector, and check explicitly that it reproduces both the lattice data for the
thermodynamics of Yang-Mills at $N_{c}=\infty$ (see Figure 1) and for the
glueball spectrum (see table 2) to a good precision.

Most of the freedom in the full V-QCD model is however in the quark sector. In
order to determine the model parameters in this sector, we compare the
predictions to

  * •

Lattice data for the thermodynamics of full QCD with $N_{f}=2+1$. We use both
the data for the equation of state at vanishing chemical potential (see Figure
2) and for the first nontrivial cumulant of the pressure at nonzero chemical
potential (see Figure 3).

  * •

Experimental data for lowest lying meson masses and the pion decay constant.
See table 4.

The final values of the model parameters are given in table 3. Apart from a
few exceptions, the model depends on these parameters through four different
functions of the coupling, which are shown in Figure 4 for the final fit.

The fit in Figures 2, 3 and table 4 has rather good quality. However, the
agreement when fitting the thermodynamics [45] and the spectra [46] separately
was significantly better. This is the case because the combined fit is
challenging: there is some clear tension between the fit to the properties of
the finite temperature state and the zero temperature vacuum state. It is
likely that this tension can be reduced by carrying out a simultaneous
numerical fit of all parameters to all data. We do not attempt to do this
technically demanding task here, but are planning to return to it in future
work. Notice also that, at least to our knowledge, an overall fit to QCD data
of the similar extent as presented in this article has not been attempted in
any other model in earlier literature.

#### Static baryon solution

Starting from the formalism introduced in [43], we compute in section 3 the
numerical bulk solution for a single static baryon, with the V-QCD potentials
presented in Section 2.111In addition, the solutions with a different choice
of potentials are discussed in Appendix D. In the Veneziano limit
$N_{c}\to\infty,\ N_{f}\to\infty$, the baryon contribution to the bulk action
is of order $N_{c}$, which is negligible compared to $N_{c}^{2}$ and
$N_{c}N_{f}$. This implies that the leading order baryon solution can be
treated as a “probe” on the background dual to the vacuum of the boundary
theory.

The numerical baryon solution is computed both at leading order and including
the first corrections to the background, which are of order
$\mathcal{O}(1/N_{c},1/N_{f})$. The leading order baryon solution is reliably
calculated, by ignoring both the tachyon and glue backreactions. The
backreaction of the baryon solution to the tachyon is computed by neglecting
the glue backreaction for simplicity. It is expected that the back-reaction on
the color sector, does not affect the qualitative results for the tachyon
backreaction.

For the leading order probe solution, the following results are obtained:

  * •

The instanton number and bulk Lagrangian densities (Figure 5) are confined to
a region of finite extent in the bulk, which confirms the solitonic nature of
the baryon solution. The integrals of these densities give respectively the
baryon number, which is confirmed to be equal to 1 numerically, and the
classical contribution to the nucleon mass. The latter is found to be
relatively close to the experimental nucleon mass for $N_{c}=3$ colors

$M_{0}\simeq\frac{N_{c}}{3}\times 1150\,\text{MeV}\,.$ (1.1)

We recall however, that the full result for the V-QCD baryon mass should also
include quantum corrections: these should be computed from the perturbations
around the baryon, together with an appropriate subtraction of the similar
fluctuations around the vacuum state. Computing these corrections goes beyond
the scope of this work. Although these corrections are subleading at large
$N_{c}$, starting at order $\mathcal{O}\big{(}N_{c}^{0}\big{)}$, they may give
a sizeable contribution when $N_{c}$ is set to 3. Note that this state of
affairs regarding quantum corrections is not particular to our model, and is
true also for both the Skyrme model and other holographic models.

  * •

In [43], it was found that, for the baryon solution, the pion matrix at the
boundary follows the Skyrmion hedgehog ansatz

$U_{P}(\xi)=\exp{\left(i\theta(\xi)\frac{x\cdot\sigma}{\xi}\right)}\,,$ (1.2)

with $\xi$ the 3-dimensional radius. Also, the baryon number was shown to be
equal to the skyrmion number for the pion matrix. This indicates that the
baryon solution in V-QCD is qualitatively similar to the Skyrme model skyrmion
solution222This should not come as a surprise, as we already know that the
dual boundary theory can be understood in the confined phase as a chiral
effective theory coupled to a tower of massive mesons.. To measure the
difference with the Skyrme model skyrmion, the pion phase $\theta(\xi)$ is
compared with the Skyrme result in Figure 6. This indicates that the two
solutions are quantitatively close.

At the next order in the large N expansion, the back-reacted solution provides
the following information:

  * •

The modulus of the chiral condensate
$\left|\left<\bar{\psi}\psi\right>\right|$ is observed to decrease towards the
baryon center, as shown in Figure 7. This signals the expected partial
restoration of the chiral symmetry inside the baryon.

  * •

The correction to the baryon Lagrangian density from the back-reaction is
calculated numerically and presented in Figure 8. The observed behavior is
well understood in terms of the chiral restoration, which has two effects.
First, the negative contribution in the UV is mainly understood as a direct
consequence of the decrease of the contribution from the chiral condensate to
the Lagrangian density. Second, a positive correction is observed in the IR,
which corresponds to a shift of the baryon towards the IR. This shift also
contributes to the negative region in Figure 8, and is understood as a
consequence of the weakening of the IR-repelling bulk force felt by the baryon
in the chirally broken background. The results also indicate that the
correction to the classical soliton mass is negative, and relatively small in
absolute value.

#### Rotating baryon solution and spin-isospin spectrum

The second part of this work is devoted to the quantization of the isospin
collective coordinates of the bulk soliton dual to a baryon. In the large
$N_{c}$ limit, the moment of inertia $\lambda$ of the baryon is of order
$\mathcal{O}(N_{c})$ and the quantization is that of a solid rotor. The result
of this procedure is the derivation of the spin-isospin baryon spectrum for
the V-QCD model considered in this work.

The starting point for the quantization of the collective coordinates is the
classical bulk solution obtained by a time-dependent isospin rotation of the
static soliton, parametrized by

$V(t)\equiv\exp{\left(it\,\omega^{a}\lambda^{a}\right)}\in SU(N_{f})_{L+R}$

with $\omega^{a}$ the rotation velocity. In this work we restrict to the
following regime

  * •

Only an SU(2) subgroup (the same where the static soliton sits) of the full
isospin subgroup is quantized. This means that we impose that $V(t)\in
SU(2)_{L+R}$. By doing so, we compute only a subset of the full spin-isospin
spectrum, corresponding to baryons composed of quarks with 2 flavors (or
equivalently, the states with strong hypercharge $Y=1$).

  * •

The rotation is assumed to be stationary and slow. In terms of the rotation
velocity $\omega^{a}$, this means that $\omega^{a}$ is assumed to be a
constant and obey $\omega^{2}\ll M_{0}/\lambda$. This regime describes well
the baryon states with spins

$s\ll N_{c}\ .$

The quantization therefore requires the calculation of the bulk solution
corresponding to a slowly rotating baryon. This calculation is done at linear
order in $\omega$. Already at this order, it turns out that a simple rotation
of the static soliton fields with $V(t)\in SU(2)_{L+R}$ is not a solution of
the bulk equations of motion. Instead, as soon as the soliton is made to
rotate, some new flavor fields are turned on in the bulk, at linear order in
$\omega$ [48]. These are the flavor equivalents of the magnetic field sourced
by a rotating charge.

The appropriate ansatz for the rotating fields is constructed in Section 5, by
imposing the same symmetries as for the static solution, apart from time-
reversal. These include 3-dimensional rotations and parity. Once this ansatz
is determined, the construction of the rotating soliton solution follows the
same steps as in the static case:

  * •

We derive the expression of the moment of inertia of the soliton in terms of
the ansatz fields.

  * •

We derive the full equations of motion for the fields of the rotating ansatz.

  * •

We identify the boundary conditions such that the moment of inertia is finite.
The boundaries here are 1) the near-AdS region UV boundary $r\to 0$, where the
solution should satisfy vev-like boundary conditions for all the fields; 2)
the boundary at spatial infinity $|\vec{x}|\to\infty$, where the fields have
to vanish fast enough for the moment of inertia to be finite.

  * •

We identify suitable regularity conditions in the IR region of the geometry.

The last part of this work presents the results of the numerical calculation
of the solution to the equations of motion for the rotating ansatz fields.
From this solution, the moment of inertia is calculated and the corresponding
spin-isospin baryon spectrum in Table 1.

Spin | V-QCD mass | Experimental mass
---|---|---
$s=\frac{1}{2}$ | $M_{N}\simeq 1170\,\text{MeV}$ | $M_{N}=940\,\text{MeV}$
$s=\frac{3}{2}$ | $M_{\Delta}\simeq 1260\,\text{MeV}$ | $M_{\Delta}=1234\,\text{MeV}$
Table 1: Baryon spin-isospin spectrum in the V-QCD model with the potentials
of Section 2, compared with experimental data.

We emphasize that these numbers where obtained by substituting $N_{c}=3$ in
the leading order large $N_{c}$ and $N_{f}$ result. In principle, at small
values of $N_{c}$ and $N_{f}$, one does not expect this result to be
quantitatively accurate.

###  1.2 Discussion and outlook

The V-QCD baryon solution we present here has several advantages with respect
to similar constructions in the literature, as well as some limitations. In
order to discuss them, we shall compare the results of this work with the two
main models of holographic QCD in which single-baryon solutions were analyzed.
These are the top-down Witten-Sakai-Sugimoto model (WSS) [11, 16] and the
bottom-up Hard-Wall model (HW) [49, 35].

The main improvement with respect to both models mentioned above is that the
V-QCD background solution on which the baryon solution is constructed is a
more accurate description of the QCD vacuum. The V-QCD vacuum possesses a rich
structure, including the running of the Yang-Mills coupling and the
spontaneous breaking of the chiral symmetry in the chiral limit. Moreover, it
incorporates the back-reaction of the flavor sector onto the color sector, due
to the Veneziano limit. The model can have several parameters that can be
adjusted to experimental data if one wants to produce a precise
phenomenological model for strongly-coupled QCD [40, 41] although generically,
the dependence on these parameters is weak.

Let us now focus on the comparison with the HW model. In the HW model, a
baryon state was constructed as a bulk axial instanton for the chiral gauge
fields, using the same kind of ansatz that is considered in this work [34].
However, the main difference is that the bulk geometry was arbitrarily fixed
to AdS5, where a hard wall was placed in the IR for the boundary theory to be
confining. Because of the gravitational potential, the bulk soliton was found
to fall on the IR wall. This indicates that the hard wall model is too crude
to stabilize the baryon solution dynamically. Moreover, since its position is
at the very end of space, the properties of the soliton will strongly depend
on the IR boundary conditions. On the contrary, the baryon solution that is
constructed in the present work is well localized in the holographic
direction. It stands at a value of the holographic coordinate of the order of
the inverse of the soliton mass. This is due to the fact that, beyond the
metric, the V-QCD vacuum contains another field under which the baryon is
charged: the tachyon field333The baryon solution including a non-trivial
tachyon in the context of the HW model was considered in [50, 35]. In that
work it was also found, as in our model, that considering a non-trivial
tachyon resulted in a repulsive force on the baryon from the IR, although the
mechanism for this to happen is different (in our case the baryon is a probe
on the tachyon background). At large chiral condensate, this could eventually
make the baryon detach from the IR wall, but only a finite distance from it.
dual to the quark bilinear operator. The combined effect of the baryon
boundary conditions and the interaction of the gauge fields with the tachyon
field, result in a force that balances the gravitational attraction towards
the IR.

Let us now discuss the WSS model. There, the main drawback of the baryon
solution that was constructed in [16] is that the baryon size was found to be
parametrically small at large ’t Hooft coupling. Instead, the size of the
baryon solution that we derived in V-QCD is set by the mass scale of the
boundary theory, which roughly corresponds to $\Lambda_{\text{QCD}}$. Note
that this was not an obstacle to the calculation of meaningful baryon form
factors in the WSS model, as the latter were found to be related to the scale
set by the rho meson mass rather than the soliton mass [51]. Nevertheless, the
infinitesimal size of the soliton will be an issue for classical fields in the
bulk, such as the chiral condensate.

Another aspect where our construction is an improvement, compared with
previous settings, lies in the tachyon dependence of the bulk action. First,
the DBI form of the kinetic action for the flavor fields contains an infinite
sum of corrections compared with the quadratic action considered in the HW
model. In vacuum, such a square-root behavior was found to play an important
role to reproduce linear trajectories for the meson spectrum [40]. Although in
the WSS model the same kind of action was introduced in [52], the present work
is the first one in which a baryon solution is computed by keeping the full
DBI action for the tachyon444Note that the calculation is done here by
expanding the DBI action at quadratic order in the non-abelian fields, but
keeping the abelian part of the tachyon fully non-linear.. Second, and most
importantly, we consider for the fist time the tachyon dependence of the
topological Chern-Simons (CS) term. In our bottom-up approach, this term was
constructed in [43] as the most general topological action compatible with QCD
symmetries and chiral anomalies.

The approach we followed here presents also some limitations. Apart from the
usual drawbacks which are intrinsic in a bottom-up model (a certain amount of
indeterminacy in the action, no known embedding as a low energy approximation
of string theory), the most important limitation is that the solution
presented here is only valid in the exact chiral limit. This is related to the
CS term mentioned above: as was explained in [43], our construction only
applies in the limit of zero quark masses, as turning on non-zero quark masses
requires modifying both the CS term and the instanton ansatz. We refer the
reader to [43] for a more extended discussion of this point.

In this work, we focused on a small subsector of the baryon spectrum. This is
due to the specific ansatz considered for the quantization of the soliton
excitations:

  * •

We considered only the zero-modes, which resulted in the spin-isospin spectrum
of the baryons. However, the experimentally observed baryon spectrum contains
higher excited states for each isospin eigenvalue. These states are understood
in the holographic picture as non-zero modes of the soliton, that is modes
that are associated with a non-trivial potential, for which the soliton
solution sits at the minimum. These include for instance the dilation and
oscillating modes considered in the WSS model [16].

  * •

Within the isospin zero-modes, we focused on a subgroup containing $N_{f}=2$
flavors. For phenomenology, it is interesting to quantize higher subgroups, in
particular $N_{f}=3$. When introducing asymmetric masses for the quarks, this
will make it possible to discuss the properties of holographic hyperons.

  * •

We restricted to the case of a slowly rotating soliton, which is enough to
compute the spin-isospin spectrum at $s\ll N_{c}$. In particular, at linear
order in the rotation velocity $\omega$, there is no deformation of the static
fields, such that the cylindrical symmetry is preserved. The consequence on
the spectrum is that only states with equal spin and isospin $s=I$ are
reproduced. States that do not obey $s=I$ are observed experimentally, such as
N(1520) or $\Delta(1950)$. Reproducing such states will require a rotating
ansatz that deviates from cylindrical symmetry. This can be obtained, for
example, by computing the solution at next order in $\omega$.

  * •

The assumption of slow rotation also means that the linear Regge trajectories
that are observed experimentally cannot be reproduced. Namely, the spin-
isospin spectrum that we compute is that of the rigid rotor, for which the
masses go as $M\sim s^{2}$ at large spin, instead of $M\sim s^{1/2}$ for a
linear trajectory. It is expected that reaching the linear regime, if it can
be reached in this framework, will require to consider states with $s\gtrsim
N_{c}$. For such high spins, the relevant ansatz for the rotating soliton
should be fully non-cylindrical. In particular, it will reproduce the linear
Regge behavior if it turns out that the solution resembles a string at high
rotation velocity [53].

All the points above can be the subject of future improvements.

##  2 The V-QCD model: comparison to data

We start by briefly reviewing the V-QCD model [38]. First, we discuss the
definitions needed for the data comparison. We only discuss the main points,
see the companion article [43] and the review [54] for more details.

We then go on and determine the parameters of the model by comparing to QCD
data. This is done in two stages: In the first stage, we choose the
asymptotics of the functions so that the model has the potential to resemble
QCD. In the second stage, we fit the remaining parameters to QCD data. In
earlier work, this second step has been done for pure Yang-Mills [37] (see
also [55, 56]), as well as for full QCD by fitting the thermodynamics [45] and
meson spectrum [46] separately. Here we carry out a simultaneous fit to the
thermodynamic and spectrum data for full QCD, with a slightly higher weight on
the spectrum fit because it probes directly the vacuum phase which is relevant
for the current study. Apart from just combining the earlier thermodynamics
and spectra fits, we also choose a different set of observables for the latter
fit as compared to [46]. In this reference, the stress was on fitting a high
number of meson masses including relatively heavy states, in order to obtain a
good description of Regge trajectories. Here we focus on the mesons with
lowest masses and also fit the pion decay constant.

###  2.1 The action of V-QCD

V-QCD is a bottom-up holographic model for QCD with $N_{c}$ colors and $N_{f}$
flavors. The holographic description is obtained by working in the Veneziano
large-$N$ limit [57]:

$N_{c}\to\infty\quad\mathrm{and}\quad N_{f}\to\infty,\quad\mathrm{with}\quad
x\equiv{N_{f}\over N_{c}}\quad\mathrm{fixed}\ .$ (2.1)

The five-dimensional dynamical fields of the bulk theory are in one-to-one
correspondence with the lowest-dimensional gauge-invariant operators in QCD.
They are

  1. 1.

The metric $g_{MN}$, dual to the stress-tensor of QCD;

  2. 2.

The dilaton $\lambda$, dual to the operator
$\mathrm{Tr}\,G_{\mu\nu}G^{\mu\nu}$ (where $G_{\mu\nu}$ is the Yang-Mills
field strength);

  3. 3.

Non-Abelian gauge fields $\mathbf{L}_{M}$, $\mathbf{R}_{M}$, of the bulk gauge
group $U(N_{f})_{L}\times U(N_{f})_{R}$, dual to the currents

$(J^{(L)}_{\mu})^{i}_{j}=\bar{q}_{L}^{i}\gamma_{\mu}q_{L\,j},\qquad(J^{(R)}_{\mu})^{i}_{j}=\bar{q}_{R}^{i}\gamma_{\mu}q_{R\,j}\qquad
i,j=1\ldots N_{f}$ (2.2)

where $q_{L,R}$ are the left and right-handed quarks;

  4. 4.

The tachyon, i.e., a $N_{f}\times N_{f}$ complex scalar field ${T^{i}}_{j}$,
dual to the quark bilinear $\bar{q}_{R}^{i}q_{L\,j}$, which therefore
transforms in the bi-fundamental representation of $U(N_{f})_{L}\times
U(N_{f})_{R}$.

The five-dimensional action

$S_{V-QCD}=S_{g}+S_{DBI}+S_{CS}$ (2.3)

is the sum of three terms555The action generically contains an additional CP-
odd term that couples the flavor fields to the holographic axion, dual to the
$\text{Tr}(G\wedge G)$ operator. This term was derived in [58, 59] based on
the ideas of [12]. We do not write this term here because, as explained in
Section 5.1, the coupling to the axion is ignored in the following.. First is
the glue term $S_{g}$, equivalent to the action of IHQCD [36, 60]. The second
and third terms are the flavor terms inspired by a configuration of space-
filling D4-branes and $\overline{\rm D}_{4}$-branes. The flavor action
separates into a Dirac-Born-Infeld action (second term) and a Chern-Simons
action (third term).

We now discuss explicit expressions for the terms in the action. The first
term associated to pure Yang-Mills, is given by the five-dimensional Einstein-
dilaton gravity:

$S_{g}=M^{3}N_{c}^{2}\int d^{5}x\sqrt{-g}\left[R-{4\over
3\lambda^{2}}g^{MN}\partial_{M}\lambda\partial_{N}\lambda+V_{g}(\lambda)\right]\
.$ (2.4)

This action is that of IHQCD [36, 60, 61]: it can be obtained from noncritical
five-dimensional string theory, which also gives a prediction for the
potential $V_{g}$. However, in order to obtain a phenomenologically
interesting model, we need to switch to a bottom-up approach and choose a
potential which reproduces known features of Yang-Mills theory. We shall
discuss this in more detail below. For the vacuum background solution, the
dilaton field will be running from $\lambda=0$ at the UV boundary to
$\lambda=\infty$ in the IR (see Appendix B). Because the dilaton is dual to
the $\mathrm{Tr}\,G_{\mu\nu}G^{\mu\nu}$ operator, the corresponding source is
the ’t Hooft coupling $g_{\mathrm{YM}}^{2}N_{c}$. As the source term dominates
the background solution near the boundary, the dilaton can be in practice
identified with the ’t Hooft coupling of YM theory, in this region.

As for the flavor terms, the simplified DBI action introduced in [42] is
enough to discuss the properties of the background, assuming that it is
homogeneous and that all quark flavors have the same mass. For the full DBI
action see [43]. The simplified DBI action reads

$\displaystyle S_{DBI,simpl}=$ $\displaystyle-M^{3}N_{c}N_{f}\times$ (2.5)
$\displaystyle\times\int d^{5}x\
V_{f}(\lambda,\tau)\sqrt{-\mathrm{det}\left(g_{MN}+\kappa(\lambda)\partial_{M}\tau\partial_{N}\tau+w(\lambda)F_{MN}\right)}\,,$

where $\tau$ is the scalar tachyon field defined through
$T^{i}_{j}=\tau\delta^{i}_{j}$, with $i,j=1\ldots N_{f}$, and $F_{MN}$ is the
field strength of the Abelian vectorial gauge field $v_{M}$, defined through

$\left(\mathbf{L}_{M}\right)^{i}_{j}=\left(\mathbf{R}_{M}\right)^{i}_{j}=v_{M}\delta^{i}_{j}\
.$ (2.6)

This form of the DBI action follows the ideas of Sen [62] for the decay of a
pair of unstable D-branes. Accordingly, the tachyon potential should be chosen
to be exponentially suppressed at large $\tau$,
$V_{f}(\lambda,\tau)\sim\exp(-a\tau^{2})$, which models the annihilation of
the branes in the IR. As the tachyon is dual to the $\bar{q}q$ operator in
QCD, the bulk tachyon condensate driven by the exponential potential gives
rise to a chiral condensate and chiral symmetry breaking on the QCD side.

We do not present explicitly the CS term $S_{CS}$ here, as it is complicated
and not needed for the background analysis – please see [43] for details. It
is however essential for the baryon solutions that we construct below.

###  2.2 Comparing V-QCD to data: asymptotics

We now discuss in more detail how the potentials $V_{g}(\lambda)$,
$V_{f}(\lambda,\tau)$, $\kappa(\lambda)$ and $w(\lambda)$ are determined. We
start by considering the asymptotic constraints as $\lambda\to 0$ (UV) or
$\lambda\to\infty$ (IR), which arise from comparison to QCD.

In the UV, the field theory becomes weakly coupled and it is far from obvious
that gauge/gravity duality can provide useful predictions. In this region we
follow the usual practice for bottom-up models and adjust the model by hand so
that it mimics closely the behavior of QCD in the perturbative regime. Such
choices are expected to give good boundary conditions for the more
interesting, strongly coupled IR physics.

In order to write down the asymptotics as $\lambda\to 0$, we write down the
effective potential

$V_{eff}(\lambda,\tau)=V_{g}(\lambda)-{N_{f}\over N_{c}}V_{f}(\lambda,\tau)\
,$ (2.7)

as well as the expansion of the flavor potential at small tachyon:

$V_{f}(\lambda,\tau)=V_{f,0}(\lambda)\left[1-\hat{a}(\lambda)\tau^{2}+\mathcal{O}\left(\tau^{4}\right)\right]\
.$ (2.8)

The UV behavior is in general chosen such that the geometry is asymptotically
AdS5 and the asymptotics of the bulk fields match with the expected (free
field) UV dimensions of the dual operators. Moreover, one can require that the
logarithmic running of the dilaton agrees with the two-loop perturbative
running of the QCD coupling. This maps the subleading terms in the potential
$V_{g}$ and the effective potential $V_{eff}$ to the two-loop $\beta$
functions of the YM theory and full QCD, respectively. We implement this
constraint here for the gluon potential $V_{g}$ only, and choose to fit the
other higher order coefficients to low energy data instead. The explicit
definitions of the UV expansions take the form

$\displaystyle
V_{g}(\lambda)=V_{g,0}\left[1+V_{g,1}\lambda+O(\lambda^{2})\right]$
$\displaystyle
V_{eff}(\lambda,\tau=0)=V_{0}\left[1+V_{1}\lambda+O(\lambda^{2})\right]$
$\displaystyle
V_{f,0}(\lambda)=W_{0}\left[1+W_{1}\lambda+O(\lambda^{2})\right],$
$\displaystyle\kappa(\lambda)=\kappa_{0}\left[1+\kappa_{1}\lambda+O(\lambda^{2})\right],\qquad\qquad\lambda\to
0,$ (2.9) $\displaystyle
w(\lambda)=w_{0}\left[1+w_{1}\lambda+O(\lambda^{2})\right],$
$\displaystyle\hat{a}(\lambda)=1+a_{1}\lambda+\mathcal{O}(\lambda^{2})\,.$

Here the coefficients $V_{g,0}$ and $V_{0}$ are linked to the radii of the UV
AdS5 geometry in the absence of flavors, and for the full background,
respectively:

$V_{g,0}=\frac{12}{\ell_{g}^{2}}\ ,\qquad
V_{0}=V_{g,0}-xW_{0}=\frac{12}{\ell^{2}}\ .$ (2.10)

We fixed the normalization of the tachyon field such that
$\hat{a}(\lambda=0)=1$666This is without loss of generality, as we have left
the normalization of the tachyon kinetic term free for the moment..

The requirement that the asymptotic dimension of the $\bar{q}q$ operator
equals 3 sets

$\kappa_{0}=\frac{2\ell^{2}}{3}\ .$ (2.11)

The subleading coefficients $V_{g,1}$ and $V_{g,2}$ are determined by
comparing to the YM $\beta$ function, which sets

$V_{g,1}=\frac{11}{27\pi^{2}}\ ,\qquad V_{g,2}=\frac{4619}{46656\pi^{4}}\ .$
(2.12)

The rest of the parameters are left free at this point.

The IR ($\lambda\to\infty$) asymptotics of the various functions is chosen
such that the model agrees with various qualitative features. First, the
asymptotics of $V_{g}(\lambda)$ is adjusted such that the model has a discrete
glueball spectrum with Regge-like linear radial trajectories (i.e. that the
masses behave as $m_{n}^{2}\sim n$ as a function of the radial excitation
number $n$). This requirement automatically implies also color confinement and
magnetic screening, [36].

The geometry ends in a “good” IR singularity according to the classification
by Gubser [63]. Second, several requirements constrain the asymptotics of the
functions in the DBI action:

  * •

The IR singularity should remain fully repulsive, so that the boundary
conditions for the background and fluctuations are properly determined [38].

  * •

The meson mass spectrum needs to be discrete, admit linear Regge-like radial
trajectories, and all meson trajectories should have the same universal slope
[40]777There is a certain freedom in this direction. Tachyon condensation
induces a mass term for the axial gauge field in the bulk, and depending on
bulk IR asymptotics, it may change the slope of the radial trajectories for
the axial and vector mesons, [64]. It is not yet clear whether this is true in
QCD, [65].. The mass gap of, say, vector mesons should grow linearly with
quark mass at large values of the mass [39].

  * •

The flavor action should vanish in the IR for chirally broken backgrounds,
corresponding to the annihilation of the flavor branes in the IR, [62].
Moreover, the phase diagram as a function of $x$, $T$, $\mu$ and the
$\theta$-angle, should have the qualitatively correct structure.

These requirements can be met if we choose the flavor action which behaves as

$V_{f}(\lambda,\tau)\sim e^{-a(\lambda)\tau^{2}},\qquad\tau\to+\infty$ (2.13)

with $a(\lambda)>0$, and require that as $\lambda\to\infty$

$\displaystyle V_{g}\sim V_{IR}\lambda^{4/3}(\log\lambda)^{1/2},\quad
V_{f,0}\sim W_{IR}\lambda^{v_{p}},$ (2.14)
$\displaystyle\kappa\sim\kappa_{IR}\lambda^{-4/3}(\log\lambda)^{1/2},\quad
a\sim a_{IR},\quad w\sim w_{IR}\lambda^{-4/3}(\log\lambda)^{w_{l}},$

with $4/3<v_{p}<10/3$ and $w_{l}>1/2$ (see [40] for details).

###  2.3 Comparing V-QCD to data: fit of parameters

After the asymptotics of the potentials have been fixed, the remaining task is
to determine the leftover freedom by doing a more precise comparison to QCD
data. We shall be using lattice data for the thermodynamics of the YM theory
and QCD as well as glueball masses, and experimental data for meson masses and
decay constants.

The complete ansatz is given by

$\displaystyle
V_{g}(\lambda)=12\,\biggl{[}1+V_{g,1}\lambda+{V_{g,2}\lambda^{2}\over
1+c_{\lambda}\lambda/\lambda_{0}}+V_{\mathrm{IR}}e^{-\lambda_{0}/(c_{\lambda}\lambda)}\left(\frac{c_{\lambda}\lambda}{\lambda_{0}}\right)^{4/3}\\!\\!\sqrt{\log(1+c_{\lambda}\lambda/\lambda_{0})}\biggr{]}\
$ (2.15) $\displaystyle
V_{f}(\lambda,\tau)=V_{f0}(\lambda)\left(1+\tau^{4}\right)^{\tau_{p}}\exp\left(-a(\lambda)\tau^{2}\right)$
(2.16) $\displaystyle
V_{f0}(\lambda)=W_{0}+W_{1}\lambda+\frac{W_{2}\lambda^{2}}{1+c_{f}\lambda/\lambda_{0}}+W_{\mathrm{IR}}\left(1+\frac{\bar{W}_{1}\lambda_{0}}{c_{f}\lambda}\right)e^{-\lambda_{0}/(c_{f}\lambda)}(c_{f}\lambda/\lambda_{0})^{2}$
(2.17) $\displaystyle
a(\lambda)=1+\frac{1}{2}a_{\mathrm{st}}\left[1+\tanh\left(-a_{\mathrm{sh}}+\log(\lambda/\lambda_{0})\right)\right]$
(2.18)
$\displaystyle\kappa(\lambda)=\kappa_{0}\biggl{[}1+\bar{\kappa}_{0}\left(1+\frac{\bar{\kappa}_{1}\lambda_{0}}{c_{\kappa}\lambda}\right)e^{-\lambda_{0}/(c_{\kappa}\lambda)}\frac{(c_{\kappa}\lambda/\lambda_{0})^{4/3}}{\sqrt{\log(1+c_{\kappa}\lambda/\lambda_{0})}}\biggr{]}^{-1}$
(2.19) $\displaystyle
w(\lambda)=w_{0}\biggl{\\{}1+\frac{w_{1}c_{w}\lambda/\lambda_{0}}{1+c_{w}\lambda/\lambda_{0}}+$
(2.20) $\displaystyle\qquad\qquad\quad+\
\bar{w}_{0}e^{-\hat{\lambda}_{0}/(c_{w}\lambda)}\frac{(c_{w}\lambda/\lambda_{0})^{4/3}}{\log(1+c_{w}\lambda/\lambda_{0})}\left[1+w_{\mathrm{as}}\left(\log\left(1+c_{w}\lambda/\lambda_{0}\right)\right)^{4}\right]\biggr{\\}}^{-1}$

where $\lambda_{0}=8\pi^{2}$. We have also chosen $v_{p}=2$ in the IR, and set
$\ell_{g}=1$, so that the AdS radius is given by

$\ell=\frac{1}{\sqrt{1-xW_{0}/12}}$ (2.21)

and consequently

$\kappa_{0}=\frac{2}{3\left(1-xW_{0}/12\right)}\ .$ (2.22)

Requiring agreement with the two-loop $\beta$-function of Yang-Mills fixes the
subleading coefficients of $V_{g}$ in the UV to the values given in (2.12).

We now set $x=2/3$, roughly corresponding to QCD with $N_{f}=2$ light quarks
and $N_{c}=3$. The rest of the parameters are fitted to data. In addition to
the parameters of the potentials, we also fit the 5D Planck mass $M$ and the
characteristic scale $\Lambda$ of the background solutions, which is roughly
dual to the scale $\Lambda_{QCD}$ in field theory, see Appendix B.

The fit is carried out in steps. We fit the various functions sequentially to
appropriately chosen observables as we explain below. Although the best
strategy would be a global fit of all the parameters to all appropriate data,
this is numerically very demanding and we are not yet able to do it.

Figure 1: Our result for the thermodynamics of pure Yang-Mills theory (dashed curves) compared to lattice data interpolated to $N_{c}\to\infty$ [66] (solid curves and error bars). The red, blue, and green curves show the normalized energy density, pressure, and interaction measure, respectively.  Ratio | Model | Lattice ($N_{c}=3$) | Lattice ($N_{c}=\infty$)
---|---|---|---
$m_{0^{*++}}/m_{0^{++}}$ | 1.52 | $\begin{array}[]{c}1.603\pm 0.042~{}\cite[cite]{[\@@bibref{}{Lucini}{}{}]}\\\ 1.719\pm 0.016~{}\cite[cite]{[\@@bibref{}{Athenodorou}{}{}]}\end{array}$ | $\begin{array}[]{c}1.835\pm 0.032~{}\cite[cite]{[\@@bibref{}{Lucini}{}{}]}\\\ 1.903\pm 0.018~{}\cite[cite]{[\@@bibref{}{Athenodorou}{}{}]}\end{array}$
$m_{2^{++}}/m_{0^{++}}$ | 1.29 | $\begin{array}[]{c}1.346\pm 0.037~{}\cite[cite]{[\@@bibref{}{Lucini}{}{}]}\\\ 1.437\pm 0.011~{}\cite[cite]{[\@@bibref{}{Athenodorou}{}{}]}\end{array}$ | $\begin{array}[]{c}1.451\pm 0.048~{}\cite[cite]{[\@@bibref{}{Lucini}{}{}]}\\\ 1.497\pm 0.008~{}\cite[cite]{[\@@bibref{}{Athenodorou}{}{}]}\end{array}$
$T_{c}/m_{0^{++}}$ | 0.162 | $0.182\pm 0.004~{}\cite[cite]{[\@@bibref{}{Lucini,LuciniT}{}{}]}$ | $0.181\pm 0.003~{}\cite[cite]{[\@@bibref{}{Lucini,LuciniT}{}{}]}$
Table 2: Our results for glueball mass ratios and the ratio of the mass to the
critical temperature compared to lattice results.

The function $V_{g}$ is chosen to be the same as in the fit of [46] where it
was determined through a global fit to the QCD spectrum. Even if it was not
directly fitted to the data from Yang-Mills theory, it does produce a good
description of the lattice data for Yang-Mills thermodynamics [66] and the
glueball spectra [67, 68, 69], see Figure 1888In this figure, the Planck mass
$M$ and the scale parameter $\Lambda$ were fitted independently of the later
fits to full QCD data. This reflects the dependence on $x$ of these
parameters: the values for Yang-Mills are those with $x=0$, whereas for full
QCD we take $x=2/3$. and Table 2.

The next function to be fitted is $V_{f}(\lambda,\tau)$ at zero tachyon, i.e.,
$V_{f0}(\lambda)$. For this function, we mostly use the lattice data for QCD
thermodynamics at high temperatures and zero density in the chirally symmetric
phase. As we are working at zero quark mass in the holographic model, chiral
symmetry is fully restored in this phase, meaning that the tachyon is
identically zero. This means that the thermodynamics only depends on $V_{f0}$
through the effective potential

$V_{eff}(\lambda,\tau=0)=V_{g}(\lambda)-xV_{f0}(\lambda)\;.$

In order to present our results, we introduce a reference value
$M_{\text{UV}}$ for the Planck mass, defined by

$M_{\text{UV}}^{3}\equiv\frac{1}{45\pi^{2}\ell^{3}}\left(1+\frac{7x}{4}\right)\;\;\;,\;\;\;x\equiv{N_{f}\over
N_{c}}\,,$ (2.23)

where $x=2/3$. This expression is obtained by requiring agreement of the
pressure with the perturbative QCD result in the limit $T\to\infty$. We fitted
this effective potential to latice data for various fixed values of the Planck
mass $M$ while the other scale parameter $\Lambda$, which only affects the
temperature scale in the plots, was allowed to vary freely. See Figure 2 for
fits with $M^{3}/M_{\text{UV}}^{3}=1.5$ (dashed thin black curves) and
$M^{3}/M_{\text{UV}}^{3}=4$ (solid thin black curves).999In the right hand
plot disentangling the two curves is difficult because they almost overlap.
Note, that while the fit results are determined mostly by
$V_{eff}(\lambda,\tau=0)$, $M$, and $\Lambda$, they also depend on other
parameters of the theory via the critical temperature of the phase transition
between the chirally-unbroken and the chirally-broken vacuum.

When fitting, we therefore took the critical temperature as an additional fit
parameter. In the next step, the tachyon dependent functions were adjusted
such that the critical temperature is indeed close to the fitted value. The
two values will however be slightly different, and therefore the results for
the thermodynamics for the final fit parameters, which are also shown in
Figure 2 and will be discussed in detail below, differ from the direct fits of
the effective potential.

The best fits were obtained at low $M^{3}/M_{\text{UV}}^{3}\lesssim 2$.
However, in the analysis that follows we chose a high value
$M^{3}/M_{\text{UV}}^{3}\simeq 4$. The reason is that such high values lead to
a better description of the spectrum and in particular of the pion decay
constant, which, as we remarked above, are stressed in the fit. We also note
that there is a simple flat direction in this fit as the thermodynamics is
unchanged under uniform rescalings of the coupling $\lambda\mapsto c\lambda$
in the effective potential. This happens because the kinetic term in (2.4) is
invariant under such rescalings, so that a constant in $V_{eff}$ multiplying
$\lambda$ can be eliminated through a field redefinition. We use this freedom
to ensure that the constructed effective potential is consistent with the
choice of $V_{g}$, meaning, in particular, that $V_{f0}$ is set to be positive
and monotonic.

The most complicated step in the fit is the next step, where we choose the
form of $V_{f}(\lambda,\tau)$ at nonzero tachyon, including the function
$a(\lambda)$ in the exponential factor, and the tachyon kinetic term
$\kappa(\lambda)$. These functions are probed by the chirally broken vacuum,
which has nonzero bulk tachyon condensate. The main observables are the pion
decay constant, the mass of the $\rho$ meson, and the mass of the lightest
scalar flavor nonsinglet (i.e., isotriplet for $N_{f}=2$) state. When doing
the fit, it is important to keep an eye on the ratio of the meson masses to
the critical temperature.

As it turns out, fitting the thermodynamics and meson spectra simultaneously
leads to tensions in the choices of potentials. The basic issue is that it is
difficult to find a choice of function that would, at the same time, give high
enough pion decay constant, heavy enough scalar states, and the experimentally
observed meson mass to critical temperature ratio. In order to alleviate this
tension we introduced an ansatz for the tachyon dependence of $V_{f}$ in
(2.16) and (2.18), which is somewhat more detailed than those used in previous
studies. This ansatz has the following properties

  * •

It includes a new parameter $\tau_{p}$ in (2.16), controlling a term which
depends on the tachyon only. We find that increasing $\tau_{p}$ leads to
better fits, so we choose the value $\tau_{p}=1$ which is close to the maximal
possible value. This maximum arises because $V_{f}(\lambda,\tau)$ needs to be
monotonic in $\tau$ at small $\lambda$, otherwise no appropriate background
solutions exist.

  * •

The exponent in the tachyon exponential $a$ in (2.16), is taken to be a
function of the dilaton $a(\lambda)$. $a(\lambda)$ needs to be constant at
both large and small $\lambda$, but may have a step in the middle. The fit
result is that the IR value of $a(\lambda)$ should be significantly higher
than the UV value (that was normalized to one).

In addition, the function $\kappa(\lambda)$ has three free parameters (notice
that $\kappa_{0}$ in the ansatz (2.19) is given in (2.22)). One of these (in
practice $\bar{\kappa}_{1}$) is fitted such that the final critical
temperature is close to that obtained from the fit of $V_{eff}$ discussed
above. Due to tension with the fit to thermodynamics, we however choose a
value that is a bit higher than obtained in the fit. The other two parameters
$\bar{\kappa}_{0}$ and $c_{\kappa}$, are used to adjust the function such that
the pion decay constant and the scalar masses are optimal. This means, in
practice, taking $\bar{\kappa}_{0}$ to be close to the critical value beyond
which the model stops to be confining for mesons (see [40]), and adjusting
$c_{\kappa}$ according to the fit of the masses and the decay constant.

The remaining task is to fit the function $w(\lambda)$, parametrized in
(2.20). The spectra, in particular the vector and axial meson masses, do
depend on this parameter. But as it turns out, the dependence is rather weak.
Therefore, we fit this function to lattice data on the baryon number
susceptibility

$\chi_{B}(T)=\frac{1}{N_{c}^{2}}\frac{\partial^{2}p(T,\mu)}{\partial\mu^{2}}\Big{|}_{\mu=0}$
(2.24)

following [45]. Here $\mu$ is the quark chemical potential. For the parameter
$w_{\mathrm{as}}$ in (2.20), we choose a small value101010This value can also
be chosen to be zero without affecting the quality of the fits.

$w_{\mathrm{as}}=2\times 10^{-5}$ (2.25)

so that this fit is essentially independent of the IR modification (i.e., the
factor in the square brackets in (2.20)). Due to the weak dependence of the
spectrum on $w(\lambda)$ the last two steps of the fitting procedure need to
be done in part in parallel: in practice we determine first
$V_{f}(\lambda,\tau)$ and $\kappa(\lambda)$ using a choice of $w(\lambda)$
that produces a rough fit to the lattice data for the susceptibility. When
$V_{f}(\lambda,\tau)$ and $\kappa(\lambda)$ are known, we then tune
$w(\lambda)$ to obtain a good fit as the last step.

Parameter | Value
---|---
$M^{3}/M_{\mathrm{UV}}^{3}$ | 4.929 (3.8)
$\Lambda$ | 4107 MeV (3350 MeV)
$M^{3}/M_{\mathrm{UV}}^{3}|_{x=0}$ | 1.4
$V_{\mathrm{IR}}$ | 1.804
$c_{\lambda}$ | 2.833
$W_{0}$ | 2.376
$W_{1}$ | 0.04603
$W_{2}$ | 0.02546
$W_{\mathrm{IR}}$ | 1.783
$\bar{W}_{1}$ | 4.357
$c_{f}$ | 1.463

Parameter | Value
---|---
$\tau_{p}$ | 1
$a_{\mathrm{st}}$ | 2.5
$a_{\mathrm{sh}}$ | 1.5
$\bar{\kappa}_{0}$ | 2.429
$\bar{\kappa}_{1}$ | 0.32
$c_{\kappa}$ | 2.0
$w_{0}$ | 1.17
$w_{1}$ | 52.5
$\bar{w}_{0}$ | 200
$c_{w}$ | 0.18
$w_{\mathrm{as}}$ | $2\times 10^{-5}$

Table 3: Choices of model parameters, split in groups. For the first two
parameters, the Planck mass $M$ and the scale $\Lambda$, the first set of
values are determined by the $\rho$ mass and $f_{\pi}$, whereas the values in
parentheses are the values preferred by the fit to thermodynamics. We also set
$x=2/3$ and $b=10$ for the additional parameter appearing in the CS action
[43].

Figure 2: Fits to the thermodynamics lattice data [70] of QCD with 2+1 flavors. The thin black solid and dashed curves are direct fits of the effective potential $V_{eff}$. The dotted blue and solid magenta curves are final fits with scale parameters $M$ and $\Lambda$ optimized for thermodynamics and spectrum data, respectively, with fit parameters given in Table 3. These two fits have different transition temperature as the direct fits of the effective potential; see the text for details. Figure 3: Fits to the lattice data (with 2+1 flavors) [71] for the light quark susceptibility of QCD. The dotted blue and solid magenta curves are final fits with scale parameters optimized for thermodynamics and spectrum data, respectively. Quantity | Model | Experiment [72]
---|---|---
$f_{\pi}$ | 92 MeV∗ | 92 MeV
$m_{\rho}$ | 775 MeV∗ | 775 MeV
$m_{\rho^{*}}$ | 886 MeV |  $1465\pm 25$ MeV
$m_{a_{1}}$ | 1240 MeV |  $1230\pm 40$ MeV
$m_{\pi^{*}}$ | 1260 MeV |  $1300\pm 100$ MeV
$m_{a_{0}}$ | 639 MeV | –
Table 4: Fitted low lying (flavor nonsinglet) meson masses and the pion decay
constant. The values of the scale parameters $M$ and $\Lambda$ were chosen
such that the values of $f_{\pi}$ and $m_{\rho}$, marked with asterisks, match
exactly the experimental values.

The final parameter values are collected in Table 3. The fits for the QCD
thermodynamics are shown in Figures 2 and 3. The results for the most
important meson masses and the $f_{\pi}$ are shown in Table 4.

Before discussing the details of the fit results, we give a summary of the
observables the various parameters were fitted to. Notice that the parameters
in Table 3 were grouped in six groups. In the first group (top left), we show
the final values fixed to $f_{\pi}$ and $m_{\rho}$ for the mass scales $M$ and
$\Lambda$, as well as the values preferred by the fit to the lattice results
for thermodynamics (values in parentheses). The value of $M$ at $x=0$ is the
one used in Figure 1 for the thermodynamics of pure Yang-Mills.

The parameters of $V_{g}$ (middle left group) were not fitted here but we used
the values from [46]. The thermodynamics and glueball masses from this choice
were however seen to agree well with lattice data (Figure 1 and Table 2).

The parameters of the $V_{f0}$ potential (the bottom left group) are fitted to
the thermodynamic data of Figure 2 and the parameters of the $w$ potential
(the bottom right group) are fitted to the susceptibility in Figure 3. The
remaining parameters of the tachyon potential $V_{f}$ (top right group) were
adjusted to obtain a spectrum that mimics that of QCD, as shown in Table 4.
The parameters of the $\kappa$ potential (middle right group) were fitted in
part to the thermodynamics and in part to the spectrum.

Apart from the parameters fitted to data here, there is a single parameter
arising from the CS sector, which was discussed in the companion article [43].
In [43] we derived the most general CS term which is compatible with known
constraints, and which contains four functions $f_{i}(\tau)$ which are only
known at $\tau=0$ and at $\tau=\infty$. We will use here the functions derived
in [12] arising from flat space string theory, up to the parameter $b$ which
corresponds to a rescaling of the tachyon field in the CS term. The precise
functions are given in equation (3.17) of [43]

$\displaystyle f_{1}(\tau)$ $\displaystyle=-\frac{1}{6}e^{-b\tau^{2}}\,,$
$\displaystyle f_{2}(\tau)$
$\displaystyle=\frac{i}{12}(1+b\tau^{2})e^{-b\tau^{2}}\,,$ $\displaystyle
f_{3}(\tau)$ $\displaystyle=-\frac{1}{12}e^{-b\tau^{2}}\,,$ $\displaystyle
f_{4}(\tau)$
$\displaystyle=\frac{1}{120}(2+2b\tau^{2}+b^{2}\tau^{4})e^{-b\tau^{2}}\,.$
(2.26)

Our results here turn out to have little dependence on different finite values
of this additional parameter; we set $b=10$ following [44].

We now discuss some additional details of the fit. Figures 2 and 3, show the
final fit to the thermodynamic data, with the values of $M$ and $\Lambda$
given in the parentheses in Table 3.

The dotted blue curves in Figure 2, show the results for the thermodynamic fit
to the equation of state. They differ from the direct fit of $V_{eff}$, the
solid black curve, because the transition temperature was adjusted differently
in the final fit, in order to obtain a better agreement with the experimental
meson spectrum. As we pointed out above, the final transition temperature is
determined, apart from the effective potential, by the values of
$\bar{\kappa}_{0}$, $\bar{\kappa}_{1}$ and $c_{\kappa}$ in Table 3.

Finally, the solid magenta curves show the fit for the values of $M$ and
$\Lambda$ that reproduce the values of $f_{\pi}$ and the $\rho$ mass, i.e.,
the values in Table 3 which are not in parentheses. Their difference to the
dashed blue curves therefore demonstrates the remaining tension between the
fits to thermodynamic and spectrum data. Because we are interested in baryons
in the zero temperature vacuum state in this article, we chose to use this
latter fit in the analysis of the properties of the baryon solution in the
rest of this paper.

Notice that we only fit the lattice data above the QCD crossover, and in the
deconfined phase of the holographic model, where the phases are separated by a
first order phase transition. The phase transition in the model, is at the
same time a deconfining transition as well as a chiral restoration transition.
Since we are in the massless quark case this is in agreement with universality
arguments, [41]. It is possible to obtain higher order phase transitions by
tuning the potentials [73], but such tuning would contradict the other
constraints we have set, in particular the requirement of linear radial
glueball trajectories. However, it is expected that stringy loop corrections,
which map to the pressure of pions and other light hadrons on the QCD side,
can make the transition continuous also in the current setup for the
holographic model [41]. Such corrections are neglected in the holographic
model, but may be added as in the second reference in [41]. After the
holographic model has been fitted to lattice data, even simple hadron
resonance gas models for the confined phase equation of state match almost
continuously with the model in the deconfined phase [74].

Regarding the meson spectrum, we compare in Table 4 the masses of the flavor
nonsinglet mesons (i.e. the fluctuation modes with vanishing trace in flavor
space) to the experimental values of isospin $I=1$ mesons from the particle
data group tables [72]. We also include the pion decay constant, and its value
as well as the $\rho$ meson mass are used to determine the final values of $M$
and $\Lambda$, as mentioned above.

Of the remaining mesons, the mass of the lowest axial vector and the mass of
the first pion excitation agree very well with the experimental values. The
mass of the excitation of the $\rho$ meson is however too low. As it turns
out, requiring the scalar mass to be high with respect to the $\rho$ mass
leads to a situation where excited states in all sectors are rather close to
the ground states. One should however also notice that the state in QCD that
we are comparing is suspected not to be a clean radial excitation of the
$\rho$ but to contain a significant hybrid component [72]. This may in part
explain the difference in the numbers.

As we remarked above, the mass of the lowest scalar $m_{a_{0}}$ is low: it is
slightly less than $m_{\rho}$. We do not attempt to compare this mass directly
to the experimental data as the scalar sector in QCD has a rather involved
structure with several states that resemble pion and kaon molecules.
Nevertheless our result is too low to be identified with any known state in
the spectrum. This is perhaps not surprising since similar issues often appear
in simple potential quark models. We do note, however, that the model of [64],
which is closely related to V-QCD, does produce a significantly heavier flavor
non-singlet scalar state. We also remark that we did not try to fit the pion
mass as we carried out the fit at zero quark mass. It would be simple to fit
the pion mass accurately by turning on a nonzero quark mass.

Figure 4: Potentials for the choice of parameters given in Table 3. Notice
that the left (right) plot uses linear (logarithmic) scale for the dilaton
$\lambda$.

Finally we show the potentials after the fit in Figure 4. We remark that all
the functions are simple, i.e. monotonic functions with no rapid changes in
behavior. Notice also that even though we list a high number of parameters in
Table 3, almost all of these parameters only appear through the functional
form of the functions shown in these plots. As the asymptotic form of the
functions is fixed by comparison of QCD properties independently of the values
of the parameters, they only affect the details of the functions in the
middle, i.e., for $\lambda/\lambda_{0}=\mathcal{O}(1)$. That is, despite the
high number of parameters, the details of the model and therefore predictions
for the observables are tightly constrained from the beginning, and the fit
basically amounts to small tuning of the final results.

##  3 The static soliton

We discuss in this section the bulk soliton dual to a static baryon state at
the boundary. A single static baryon is realised in the bulk as a Euclidean
instanton of the non-abelian bulk gauge fields extended in the three spatial
directions plus the holographic direction. We start by reviewing the main
ingredients of the formalism described in [43] that are necessary for the
present discussion. In particular, we describe the ansatz that is used to
compute the instanton solution in the bulk. We then present the numerical
results for the static soliton solution.

###  3.1 Ansatz for the instanton solution

The ansatz that is relevant for the instanton solution is obtained by
requiring that the solution is invariant (up to a global chiral
transformation) under a maximal set of symmetries of the V-QCD action (and
QCD) compatible with a finite baryon number. These symmetries are cylindrical
symmetry (rotations in the 3 spatial directions of the boundary), parity and
time-reversal. The parity and time-reversal symmetries act non-trivially on
the fields in addition to their usual action on space-time. The explicit
transformations can be found in [43].

### Ansatz for the glue sector

As explained in [43] and reviewed in the next subsection, at leading order in
$N_{c}$, the glue sector composed of the metric and dilaton is not affected by
the presence of the baryon, and is identical to the vacuum solution. The
latter depends only on the holographic coordinate $r$

$\mathrm{d}s^{2}=\mathrm{e}^{2A(r)}(-\mathrm{d}t^{2}+\mathrm{d}\textbf{x}^{2}+\mathrm{d}r^{2})\,,$
(3.1) $\lambda=\lambda(r)\,.$ (3.2)

### Gauge fields ansatz

The left and right handed gauge fields are denoted

$\mathbf{L}=L+\hat{L}\mathbb{I}_{N_{f}}\;\;\;,\;\;\;\mathbf{R}=R+\hat{R}\mathbb{I}_{N_{f}}\,,$
(3.3)

where $L$ and $R$ correspond to the $SU(N_{f})$ part of the gauge fields, and
$\hat{L}$ and $\hat{R}$ to the $U(1)$ part.

Because for any $N_{f}>1$, the homotopy groups of $U(N_{f})$ and $SU(2)$ are
equal

$\pi_{3}\left(U(N_{f})\right)=\pi_{3}\left(SU(2)\right)=\mathbb{Z}\,,$ (3.4)

a $U(N_{f})$ instanton can be constructed by embedding an $SU(2)$ instanton in
$U(N_{f})$. The $SU(2)$ subgroup couples to a $U(1)$ subgroup via the CS term,
in such a way that the baryon ansatz belongs to a $U(2)$ subgroup of
$U(N_{f})$

$\mathbf{L}=L^{a}\frac{\sigma^{a}_{L}}{2}+L^{\text{T}}\mathbb{I}_{2,L}\;\;\;,\;\;\;\mathbf{R}=R^{a}\frac{\sigma^{a}_{R}}{2}+R^{\text{T}}\mathbb{I}_{2,R}\,,$
(3.5)

where $\sigma^{a}_{L/R}$ are the Pauli matrices and $\mathbb{I}_{2,L/R}$ are
the unit matrices in the subgroups.

By imposing invariance under the cylindrical symmetry, time-reversal and
parity, the $U(2)$ instanton ansatz for the gauge fields is

$L_{i}^{a}=-\frac{1+\phi_{2}(\xi,r)}{\xi^{2}}\epsilon_{iak}x_{k}+\frac{\phi_{1}(\xi,r)}{\xi^{3}}(\xi^{2}\delta_{ia}-x_{i}x_{a})+\frac{A_{\xi}(\xi,r)}{\xi^{2}}x_{i}x_{a}\,,$
(3.6)
$R_{i}^{a}=-\frac{1+\phi_{2}(\xi,r)}{\xi^{2}}\epsilon_{iak}x_{k}-\frac{\phi_{1}(\xi,r)}{\xi^{3}}(\xi^{2}\delta_{ia}-x_{i}x_{a})-\frac{A_{\xi}(\xi,r)}{\xi^{2}}x_{i}x_{a}\,,$
(3.7)
$L_{r}^{a}=\frac{A_{r}(\xi,r)}{\xi}x_{a}\;\;\;,\;\;\;R_{r}^{a}=-\frac{A_{r}(\xi,r)}{\xi}x_{a}\,,$
(3.8)
$L^{\text{T}}_{0}=\Phi(r,\xi)\;\;\;,\;\;\;R^{\text{T}}_{0}=\Phi(r,\xi)\,,$
(3.9)

where $i,k=1,2,3$ refer to spatial indices,
$\xi\equiv\sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}$ is the 3-dimensional spatial
radius and $a=1,2,3$ is the index for the components in the $SU(2)$ basis.
Note that the gauge field ansatz is fully specified by 5 real functions

$A_{\bar{\mu}}\equiv(A_{\xi},A_{r})\;\;\;,\;\;\;\phi\equiv\phi_{1}+i\phi_{2}\;\;\;\text{and}\;\;\;\Phi\,,$
(3.10)

depending on the two variables $x^{\bar{\mu}}\equiv(\xi,r)$, that are used as
coordinates on a 2D space.

The choice of the ansatz partially fixes the gauge but there is still a
residual (axial) $U(1)$ invariance, corresponding to the $SU(2)$
transformation

$g_{L}=g_{R}^{\dagger}=\exp\left(i\alpha(\xi,r){x\cdot\sigma\over
2\xi}\right)\,,$ (3.11)

where

$x\cdot\sigma\equiv x^{a}\sigma^{a}\,.$ (3.12)

Under this gauge transformation, $A_{\bar{\mu}}$ is the gauge field, $\phi$
has charge +1 and $\Phi$ is neutral

$A_{\bar{\mu}}\to
A_{\bar{\mu}}+\partial_{\bar{\mu}}\alpha\;\;\;,\;\;\;\phi\to\mathrm{e}^{i\alpha}\phi\;\;\;,\;\;\;\Phi\to\Phi\,.$
(3.13)

### Tachyon ansatz

For the tachyon matrix restricted to the $SU(2)$ subgroup $T^{SU(2)}$, the
ansatz compatible with the symmetries of the V-QCD action is of the form

$T^{SU(2)}=\tau(r,\xi)\exp{\left(i\theta(r,\xi)\frac{x\cdot\sigma}{\xi}\right)}\,,$
(3.14)

which reproduces the Skyrmion ansatz for the unitary part of the tachyon. This
last point is more than a coincidence. Indeed, from the near-boundary behavior
at $r\to 0$ of the tachyon field at zero quark mass, the pion matrix in the
boundary theory is identified to be

$U_{P}(\xi)^{\dagger}\equiv\exp{\left(i\theta(0,\xi)\frac{x.\sigma}{\xi}\right)}\,.$
(3.15)

Also, the baryon number in the boundary theory can be shown to be equal to the
Skyrmion number for $U_{P}$ [43]. Note that under the residual $U(1)$ gauge
freedom (3.11), the tachyon phase in (3.14) transforms as

$\theta\to\theta-\alpha\,.$ (3.16)

To summarize the content of this subsection, the ansatz for the instanton
solution (3.6)-(3.9), and (3.14) contains 7 real dynamical fields

$\Phi(r,\xi)\;\;\;,\;\;\;\phi(r,\xi)\equiv\phi_{1}(r,\xi)+i\phi_{2}(r,\xi)\;\;\;,\;\;\;A_{\bar{\mu}}(r,\xi)\equiv\left(A_{\xi}(r,\xi),A_{r}(r,\xi)\right)\,,$
(3.17) $\tau(r,\xi)\;\;\;,\;\;\;\theta(r,\xi)\,,$

that depend on the two coordinates $x^{\bar{\mu}}=(\xi,r)$ and have a U(1)
gauge redundancy under which the fields transform as

$\phi\to\mathrm{e}^{i\alpha}\phi\;\;\;,\;\;\;A_{\bar{\mu}}\to
A_{\bar{\mu}}+\partial_{\bar{\mu}}\alpha\;\;\;,\;\;\;\theta\to\theta-\alpha\,.$
(3.18)

At this point, a useful observation is that there exists a redefinition of the
fields (3.17) such that, in the equations of motion for the flavor fields, the
phase $\theta$ in the tachyon ansatz (3.14) can be absorbed into the gauge
field. By doing so, the dynamical field content is reduced to a set of 6
fields invariant under the residual gauge freedom. In practice, if we define

$g(\theta)\equiv\exp{\left(i\theta\frac{x\cdot\sigma}{2\xi}\right)}\,,$ (3.19)

then we consider the following redefinition of the gauge fields

$\mathbf{L}_{M}\to\tilde{\mathbf{L}}_{M}\equiv
g(\theta)\mathbf{L}_{M}g(\theta)^{\dagger}+ig(\theta)\partial_{M}g(\theta)^{\dagger}\,,$
(3.20) $\mathbf{R}_{M}\to\tilde{\mathbf{R}}_{M}\equiv
g(\theta)^{\dagger}\mathbf{R}_{M}g(\theta)+ig(\theta)^{\dagger}\partial_{M}g(\theta)\,,$
(3.21)

which for the ansatz (3.6)-(3.9) is equivalent to

$A_{\bar{\mu}}\to\tilde{A}_{\bar{\mu}}\equiv
A_{\bar{\mu}}+\partial_{\bar{\mu}}\theta\;\;\;,\;\;\;\phi\to\tilde{\phi}\equiv\mathrm{e}^{i\theta}\phi\;\;\;,\;\;\;\Phi\to\Phi\,.$
(3.22)

From (3.18), we see that the gauge fields thus redefined are indeed invariant
under the residual gauge transformation (3.11).

In the following, we find it convenient to use the redefined fields in several
places. When they appear, we always write the tildes so that it is clear that
we are using the gauge-invariant fields.

#### Lorenz gauge

To compute the baryon solution, we need to solve the bulk equations of motion
with the ansatz (3.6)-(3.9). The equations of motion can be written in terms
of the redefined gauge fields (3.22). In particular, when solving the
equations of motion numerically, we find convenient to work with
$\tilde{\phi}$ instead of $\phi$ as a dynamical field.

However, as far as the 2-dimensional gauge field $A_{\bar{\mu}}$ is concerned,
the equations of motion written in terms of $\tilde{A}_{\bar{\mu}}$ are not
elliptic. While this is not problematic per se, the heat diffusion method that
we use to solve the equations numerically requires that they are in elliptic
form.

The equations of motion can be recast in elliptic form if we write them
instead in terms of the gauge variant fields (3.6)-(3.8), and then fix the
gauge with the Lorenz condition

$\partial_{r}A_{r}+\partial_{\xi}A_{\xi}=0\,.$ (3.23)

This is the gauge that we shall use in the following.

###  3.2 Probe and back-reacting solutions

The instanton dual to a baryon state is a configuration of the ansatz of
equations (3.6)-(3.9) and (3.14) that obeys the bulk equations of motion.
These are written in Appendix F of [43].

As explained in the previous subsection, we shall consider a baryon whose
flavor quantum numbers are a $U(2)$ subgroup of the $U(N_{f})$ flavor group.
This implies that the flavor action (composed of the DBI and CS actions in
(2.3)) for the baryon ansatz does not depend on $N_{f}$ and is of order
$N_{c}$. On the other hand, the glue action is of order $N_{c}^{2}$. Likewise,
the tachyon modulus background contributes a factor $N_{f}$ more than the
baryon fields to the bulk action, which can be seen explicitly from (3.25)
below. So, at leading order in the Veneziano limit, both the glue sector
(metric and dilaton) and the tachyon modulus $\tau$ are not affected by the
presence of the baryon, and remain identical to the vacuum solution.

We start by computing the numerical baryon solution in this leading order
probe regime. In this case, the dynamical fields are those listed in (3.17)
with the exception of the tachyon modulus $\tau$, which is fixed to its
background value. The equations of motion obeyed by those fields are written
in Appendix F.1 of [43].

In the Veneziano limit, the back-reaction on the background starts at order
$\mathcal{O}(1/N_{c})$. At this order, the correction to the glue sector
(metric and dilaton) and tachyon modulus $\tau$ can be computed by solving the
linearized Einstein-dilaton equations sourced by the probe baryon solution,
together with the linearized equations for $\tau$. Qualitatively, we do not
expect a dramatic effect on the glue sector from the presence of the baryon.
Correspondingly, the back-reaction on the glue sector is not expected to
affect much the flavor structure of the baryon, which is its most important
dynamical property. This motivates the approximation that we consider in the
following, where the baryon is assumed to back-react only on the tachyon
background. The equations of motion in this case are written in Appendix F.2
of [43].

In summary, we consider two different regimes for the baryon solution, with
different treatments of the tachyon modulus:

  * •

The _probe baryon_ solution, where the tachyon modulus $\tau$ is fixed to its
vacuum value. This corresponds to the solution at leading order in the
Veneziano expansion. At this order, the chiral condensate profile around the
baryon is trivial, but the other flavor properties of the baryon are expected
to be qualitatively correct.

  * •

The _back-reacted tachyon_ regime, where the equations of motion for $\tau$
are solved, with the gauge fields and tachyon phase $\theta$ fixed to the
probe baryon solution. In the Veneziano limit, this will reproduce the leading
order $\mathcal{O}(1/N_{f})$ correction to the tachyon background, assuming no
back-reaction on the glue sector. In this case, the solution obtained for the
tachyon modulus should give a good idea of the qualitative behavior of the
chiral condensate in presence of the baryon.

###  3.3 Boundary conditions

$\xi\to 0$ | $\xi\to\infty$ | $r\to 0$ | $r\to r_{\text{IR}}\to\infty$
---|---|---|---
$\partial_{\xi}\tilde{\phi}_{1}-(A_{\xi}+\partial_{\xi}\theta)\to 0$ | $\xi^{1/2}\tilde{\phi}_{1}\to 0$ | $\tilde{\phi}_{1}\to\sin{\theta}$ | $\tilde{\phi}_{1}\to 0$
$\frac{1+\tilde{\phi}_{2}}{\xi}\to 0$ | $\xi^{1/2}\left(\tilde{\phi}_{2}-1\right)\to 0$ | $\tilde{\phi}_{2}\to-\cos{\theta}$ | $\partial_{r}\tilde{\phi}_{2}\to 0$
$\partial_{\xi}A_{\xi}\to 0$ | $\partial_{\xi}A_{\xi}\to 0$ | $A_{\xi}\to 0$ | $A_{\xi}\to 0$
$A_{r}\to 0$ | $\xi^{3/2}\left(A_{r}-\frac{\pi}{r_{\text{IR}}}\right)\to 0$ | $\partial_{r}A_{r}\to 0$ | $\partial_{r}A_{r}\to 0$
$\partial_{\xi}\Phi\to 0$ | $\Phi\to 0$ | $\Phi\to 0$ | $\partial_{r}\Phi\to 0$
$\theta\to 0$ | $\theta\to\pi\left(1-\frac{r}{r_{\text{IR}}}\right)$ | $\partial_{r}\theta+A_{r}\to 0$ | $\theta\to 0$
$\partial_{\xi}\tau\to 0$ | $\tau\to\tau_{b}(r)$ | $\tau\to 0$ | $\tau\to\tau_{b}(r_{\text{IR}})$
Table 5: Boundary conditions in Lorenz gauge. Here, $\tau_{b}(r)$ is the
vacuum profile for the tachyon modulus. $L$ and $r_{\text{IR}}$ are the
spatial and IR cut-offs, respectively. These are introduced to solve the
equations of motion numerically.

The equations of motion for the fields of the instanton ansatz can be derived
in the form presented in Appendix F of [43]. These equations must be subject
to appropriate boundary conditions both at spatial infinity $\xi\to+\infty$
and at the UV boundary $r\to 0$. The appropriate conditions were derived in
[43] from the requirement that the baryon mass be finite and the baryon number
equal to 1. Moreover, it was observed that certain (generalized) regularity
conditions must be imposed at the center of the instanton $\xi=0$ and in the
bulk interior. We list in Table 5 the conditions that are imposed on the
fields of the ansatz (3.17) in Lorenz gauge and refer to [43] for more details
about the derivation. The conditions are the same for the probe and back-
reacted cases.

###  3.4 Baryon mass

Once the soliton solution is found, several static properties of baryons can
be computed [34, 51]. The most elementary of these properties is the nucleon
mass. This mass is the sum of a classical contribution and quantum corrections

$M_{\text{nucleon}}=M_{0}+\delta M_{Q}\,.$ (3.24)

The classical contribution is computed from the bulk on-shell action evaluated
on the soliton solution [43]. In terms of the ansatz fields (3.17), its
expression in the approximation where the back-reaction on the glue sector is
neglected is given by

$M_{0}=N_{c}\Big{(}N_{f}S_{\tau}+S_{B}\Big{)}-E_{DBI,vac}\,,$ (3.25)

where $E_{\text{DBI,vac}}$ is the DBI contribution to the vacuum energy

$E_{DBI,vac}=\int\mathrm{d}r\mathrm{d}\xi\,4\pi\xi^{2}\rho_{DBI,vac}\;\;\;,\;\;\;\rho_{DBI,vac}\equiv
M^{3}\sqrt{1+\mathrm{e}^{-2A}\kappa(\partial_{r}\tau_{0})^{2}}\,V_{f}(\lambda,\tau_{0})\mathrm{e}^{5A},$
(3.26)

with $\tau_{0}(r)$ the vacuum profile of the tachyon field. The baryon
contribution to the bulk action is split into two pieces

$S_{\tau}=\int\mathrm{d}r\mathrm{d}\xi\,4\pi\xi^{2}\rho_{\tau}\;\;\;,\;\;\;S_{B}=\int\mathrm{d}r\mathrm{d}\xi\,4\pi\xi^{2}\rho_{B}\,,$
(3.27) $\rho_{\tau}\equiv
M^{3}\sqrt{1+\mathrm{e}^{-2A}\kappa\left((\partial_{r}\tau)^{2}+(\partial_{\xi}\tau)^{2}\right)}\,V_{f}(\lambda,\tau)\mathrm{e}^{5A}\,,$
(3.28) $\displaystyle\rho_{B}$ $\displaystyle\equiv
M^{3}\sqrt{1+\mathrm{e}^{-2A}\kappa\left((\partial_{r}\tau)^{2}+(\partial_{\xi}\tau)^{2}\right)}\,V_{f}(\lambda,\tau)\,\mathrm{e}^{A}\times$
$\displaystyle\hphantom{=+4\pi
M^{3}N_{c}}\times\left(\mathrm{e}^{2A}\kappa(\lambda)\tau^{2}\left(\mathrm{e}^{2A}\Delta_{rr}\tilde{A}_{r}^{2}+\left(1-\mathrm{e}^{2A}\Delta_{\xi\xi}\right)\tilde{A}_{\xi}^{2}+\frac{(\tilde{\phi}+\tilde{\phi}^{*})^{2}}{2\xi^{2}}-\right.\right.$
$\displaystyle\hphantom{=+4\pi
M^{3}N_{c}\times\left(\mathrm{e}^{2A}\xi^{2}\kappa(\lambda)\tau^{2}\left(\mathrm{e}^{2A}\Delta_{rr}\tilde{A}_{r}^{2}+\left(1-\mathrm{e}^{2A}\Delta_{\xi\xi}\right)\tilde{A}_{\xi}^{2}\right.\right.}-2\mathrm{e}^{2A}\Delta_{\xi
r}\tilde{A}_{r}\tilde{A}_{\xi}\Bigg{)}+$ $\displaystyle\hphantom{=+4\pi
M^{3}N_{c}\times\Bigg{(}}+w(\lambda)^{2}\left(\frac{1}{8}\mathrm{e}^{2A}\left[\Delta_{rr}\left(1-\mathrm{e}^{2A}\Delta_{\xi\xi}\right)-\mathrm{e}^{2A}\Delta_{\xi
r}^{2}\right](F_{\bar{\mu}\bar{\nu}})^{2}+\right.$
$\displaystyle\hphantom{=+4\pi
M^{3}N_{c}\times\Bigg{(}+w(\lambda)^{2}\bigg{(}}+\frac{1}{2\xi^{2}}\left(\left(1-\mathrm{e}^{2A}\Delta_{\xi\xi}\right)\left|\mathrm{D}_{\xi}\phi\right|^{2}+\mathrm{e}^{2A}\Delta_{rr}\left|\mathrm{D}_{r}\phi\right|^{2}\right)+$
$\displaystyle\hphantom{=+4\pi
M^{3}N_{c}\times\Bigg{(}+w(\lambda)^{2}\bigg{(}}+\frac{\left(1-|\phi|^{2}\right)^{2}}{4\xi^{4}}-\frac{1}{2\xi^{2}}\mathrm{e}^{2A}\Delta_{\xi
r}(D_{r}\phi^{*}D_{\xi}\phi+h.c.)-$ $\displaystyle\hphantom{=+4\pi
M^{3}N_{c}\times\Bigg{(}+w(\lambda)^{2}\bigg{(}}-\Big{(}\mathrm{e}^{2A}\Delta_{rr}(\partial_{r}\Phi)^{2}+\left(1-\mathrm{e}^{2A}\Delta_{\xi\xi}\right)(\partial_{\xi}\Phi)^{2}-$
$\displaystyle\hphantom{=+4\pi
M^{3}N_{c}\times\Bigg{(}+w(\lambda)^{2}\bigg{(}-\frac{N_{f}}{2}\xi^{2}\Big{(}\mathrm{e}^{2A}\Delta_{rr}(\partial_{r}\Phi)^{2}}-2\mathrm{e}^{2A}\Delta_{\xi
r}\partial_{\xi}\Phi\partial_{r}\Phi\Big{)}\bigg{)}\Bigg{)}+$
$\displaystyle\hphantom{=}+\frac{1}{\pi^{2}\xi^{2}}\epsilon^{\bar{\mu}\bar{\nu}}\partial_{\bar{\mu}}\Phi\times$
$\displaystyle\hphantom{=+}\times\left[(f_{1}(\tau)+f_{3}(\tau))\left(\tilde{A}_{\bar{\nu}}+\frac{1}{2}(-i\phi^{*}D_{\bar{\nu}}\phi+h.c.)+\frac{1}{4i}\partial_{\bar{\nu}}(\tilde{\phi}^{2}-(\tilde{\phi}^{*})^{2})\right)+\right.$
$\displaystyle\hphantom{=+\times\bigg{[}}\left.+\frac{1}{2}(3if_{2}(\tau)-f_{1}(\tau)-f_{3}(\tau))(\tilde{\phi}+\tilde{\phi}^{*})^{2}\tilde{A}_{\bar{\nu}}\right]\,.$
(3.29)

The covariant quantities $F_{\bar{\mu}\bar{\nu}}$ and $D_{\bar{\mu}}\phi$ are
defined in Appendix A.5 and the symbol $\Delta_{\bar{\mu}\bar{\nu}}$ in
(E.14)-(E.16). The $f_{i}(\tau)$ are the Chern-Simons potentials, whose
expressions are given in (2.26). Note that, as far as flavor fields are
concerned, $S_{\tau}$ depends only on the tachyon modulus $\tau$, whereas
$S_{B}$ contains the dependence on the baryon fields. In particular, in the
leading order probe baryon regime, only $S_{B}$ contributes to $M_{0}$. The
total bulk Lagrangian density will be denoted by $\rho_{M}$

$\rho_{M}\equiv N_{c}(N_{f}\rho_{\tau}+\rho_{B})-\rho_{DBI,vac}\,.$ (3.30)

Computing the quantum corrections $\delta M_{Q}$ requires to take the sum of
the ground state energies for the infinite set of bulk excitations on the
instanton background, and subtract the vacuum energy. It is not known how to
do this calculation, so we can only assume that the classical mass gives the
dominant contribution. Note that, in terms of the expansion in $N_{c}$, the
classical mass is of order $\mathcal{O}(N_{c})$, whereas the quantum
corrections start at order $\mathcal{O}(1)$. So the statement that the
classical contribution dominates is correct at least in the large $N_{c}$
limit.

The experimental spectrum of baryons contains the nucleons, but also many
excited states, such as the isobar $\Delta$. The calculation of the spin
dependence of the baryon mass spectrum is the subject of Section 4.

###  3.5 Numerical results

We present in this subsection the numerical solution for the static baryon
configuration. The equations of motion written in Appendix F of [43] are
solved with the gradient descent method111111The name heat diffusion method
also appears in the literature., imposing the boundary conditions of Table 5.
The same kind of method was used in [32, 35] to compute baryon solutions in
other holographic models. We focus here on the results and give more details
about the numerical method in Appendix C. We start by presenting the leading
order probe baryon solution and then discuss the back-reaction. We recall that
the back-reacting solution is computed assuming no back-reaction on the color
sector (metric and dilaton).

####  3.5.1 Probe baryon solution

We start with the numerical results obtained for the probe baryon solution. In
this case the modulus of the tachyon field $\tau$ is fixed to its background
value, and the equations of motion take the form presented in Appendix F.1.3
of [43].

The instanton number and bulk Lagrangian density in the $(\xi,r)$-plane are
presented in Figure 5, where all dimensionful quantities are expressed in
units of the classical soliton mass (3.25). The bulk Lagrangian density is
given by (3.30) , whereas the expression for the instanton number density can
be obtained by dividing equation (6.7) of [43] by $4\pi\xi^{2}$

$\rho_{N_{i}}\equiv\frac{1}{8\pi^{2}\xi^{2}}\epsilon^{\bar{\mu}\bar{\nu}}\Big{[}F_{\bar{\mu}\bar{\nu}}+\partial_{\bar{\mu}}\big{(}-i\phi^{*}D_{\bar{\nu}}\phi+h.c.\big{)}\Big{]}\,.$
(3.31)

Figure 5 shows the expected behavior for a solitonic configuration, that is
the densities are confined to a region of finite extent in the bulk. The size
of this lump in the $\xi$ direction gives an estimate of the baryon size,
which is of the order of $M_{0}^{-1}$. The numerical value for the classical
soliton mass $M_{0}$ is obtained by integrating the Lagrangian density in
Figure 5

$M_{0}\simeq\frac{N_{c}}{3}\times 1150\,\text{MeV}\,.$ (3.32)

This number is expected to give the leading contribution to the nucleon mass
in the V-QCD model with the parameters of Table 3. As discussed above, the
full result for the nucleon mass also receives quantum corrections (3.24)
whose evaluation is an unsolved problem.

In Figure 6, we also plot the profile at the boundary ($r=0$) for the non-
abelian phase $\theta$ of the tachyon field (3.14). As stated above, the pion
matrix in the boundary theory (3.15) reproduces121212This comparison is well
defined even though the phase $\theta$ transforms under the residual $U(1)$
gauge freedom: the boundary gauge transformations of $U_{P}$ match exactly
those of the pion matrix in the Skyrme model, and the gauge is fixed in both
cases by requiring the absence of sources for the gauge fields. the skyrmion
hedgehog ansatz, and the associated skyrmion number is equal to the baryon
number. Figure 6 should therefore be compared with the corresponding plot of
the pion field in the Skyrme model skyrmion solution in the chiral limit [75].
This plot is reproduced131313Notice that $\theta$ goes from 0 to $\pi$,
instead of $\pi$ to 0 for the skyrmion, because the boundary value of the
tachyon field is the conjugate of the pion matrix (3.15). So it is actually
$\pi-F_{\text{skyrmion}}$ that is plotted in Figure 6. in Figure 6. It makes
it clear that the shape of the boundary skyrmion is close to that of the
Skyrme model. Note, in particular, that the asymptotic behavior is the same:

$\theta(0,\xi)\underset{\xi\to
0}{\sim}\xi\;\;\;,\;\;\;\theta(0,\xi)-\pi\underset{\xi\to\infty}{\sim}\frac{1}{\xi}\,.$
(3.33)

This can be seen from the asymptotic analysis in Appendix G of [43].

\begin{overpic}[scale={0.49}]{figures/rhoNi_probe_v8_3D_M0}
\end{overpic}\begin{overpic}[scale={0.49}]{figures/rhoM_probe_v8_3D_M0}
\end{overpic}

Figure 5: Instanton number (3.31) (left) and bulk Lagrangian (3.30) (right)
density for the static soliton solution in the probe baryon regime. All
quantities are expressed in units of the classical mass of the soliton (3.25).
The center of the soliton is located at $\xi=0$ where the density diverges as
$\xi^{-1}$. The UV boundary is located at $r=0$. The green line in the right
plot indicates the boundary of the region over which the mean value is
computed to define the relative difference in Figure 8.
\begin{overpic}[scale={0.75}]{figures/thx_UV_probe_v8_M0} \end{overpic} Figure
6: Radial profile of the non-abelian phase of the tachyon field (3.14) at the
UV boundary (blue line). The dashed gray line indicates the asymptotic value
$\pi$. For comparison, we also plotted in black the profile for the pion
fields in the skyrmion solution of the Skyrme model. The parameters of the
Skyrme model were chosen such that $f_{\pi}$ and the soliton mass are equal to
those of the V-QCD model, with the parameters of Table 3.

####  3.5.2 Back-reacted tachyon

We now discuss the numerical results obtained when taking into account the
back-reaction on the tachyon field. In this case, the gauge fields and tachyon
phase $\theta$ are fixed to the probe baryon solution, and the equation of
motion for the tachyon modulus $\tau$ is solved on this background. The
corresponding equation of motion for $\tau$ is written in Appendix F.2 of
[43].

For a back-reacted tachyon, the chiral condensate profile around the baryon
can be computed from the near-boundary behavior of the tachyon modulus

$\tau(r,\xi)\underset{r\to 0}{\sim}\Sigma(\xi)\,r^{3}\,,$ (3.34)

where $\Sigma$ is proportional to the modulus of the chiral condensate
$\left|\left<\bar{\psi}\psi\right>\right|$. The relative difference of
$\Sigma(\xi)$ with the vacuum value $\Sigma(\infty)$ is plotted in Figure 7.

\begin{overpic}[scale={0.75}]{figures/Xcond_NewPot_v8_x} \end{overpic} Figure
7: Chiral condensate profile around the static soliton solution in the back-
reacted tachyon regime, where the baryon center is at $\xi=0$. The plotted
quantity is the relative difference between the modulus of the chiral
condensate and the chiral condensate in vacuum. In the Veneziano limit
$N_{c},N_{f}\to\infty$, the difference is of order $\mathcal{O}(1/N_{f})$, so
that it has to be multiplied by $N_{f}$ in order to obtain a finite result. We
also divide by 3, which means that what the figure shows is the large $N_{f}$
result, where the value $N_{f}=3$ is substituted for the number of flavors. As
discussed in the text, there is no reason for this result to be quantitatively
accurate for $N_{f}=3$, but it gives an indication of the qualitative
behavior.

This shows the expected behavior, where the chiral symmetry tends to be
restored inside the baryon. Note that the result that is shown is valid in the
limit of large $N_{c}$ and large $N_{f}$. There is a priori no guarantee for
it to be a quantitatively accurate approximation when a small number of
flavors (for example $N_{f}=2$ or 3) is substituted in the leading large
$N_{f}$ result. So, at small $N_{f}$ and $N_{c}$, Figure 7 should not be
considered as more than an indication of the qualitative behavior of the
chiral condensate in presence of the baryon.

Another interesting information that can be extracted from this back-reacted
tachyon solution, is the effect of the back-reaction on the soliton mass
(3.25). The leading order correction to the on-shell Lagrangian density due to
the back-reaction on $\tau$ can be expressed as

$N_{c}^{-1}\delta\rho_{M}=\frac{1}{2}N_{f}\delta\tau^{2}\frac{\delta^{2}S_{\tau}}{\delta\tau^{2}}\bigg{|}_{\text{probe}}+\delta\tau\frac{\delta
S_{B}}{\delta\tau}\bigg{|}_{\text{probe}}+\mathcal{O}\big{(}N_{f}^{-2}\big{)}\,,$
(3.35)

where $\delta\tau$ refers to the order $\mathcal{O}\big{(}N_{f}^{-1}\big{)}$
correction to $\tau$, and $S_{\tau}$ and $S_{B}$ are defined in (3.27). Note
that we dropped the terms that vanish on-shell

$\frac{\delta S_{\tau}}{\delta\tau}\bigg{|}_{\text{probe}}=\frac{\delta
S_{B}}{\delta\varphi_{B}}\bigg{|}_{\text{probe}}=0\;\;\;,\;\;\;\varphi_{B}\in\\{\Phi,\phi,A_{\bar{\mu}},\theta\\}\,.$
(3.36)

Equation (3.35) can be simplified by using the back-reacted equations of
motion for the tachyon modulus

$\frac{\delta(N_{f}S_{\tau}+S_{B})}{\delta\tau}\bigg{|}_{\text{back-
react}}=0\,,$ (3.37)

which, at leading order in $N_{f}$, implies that

$\frac{\delta
S_{B}}{\delta\tau}\bigg{|}_{\text{probe}}+N_{f}\delta\tau\frac{\delta^{2}S_{\tau}}{\delta\tau^{2}}\bigg{|}_{\text{probe}}=\mathcal{O}\big{(}N_{f}^{-1}\big{)}\,,$
(3.38)

and finally

$N_{c}^{-1}\delta\rho_{M}=-\frac{1}{2}N_{f}\delta\tau^{2}\frac{\delta^{2}S_{\tau}}{\delta\tau^{2}}\bigg{|}_{\text{probe}}+\mathcal{O}\big{(}N_{f}^{-2}\big{)}\,.$
(3.39)

Although the correction to the mass (3.39) is suppressed by a factor
$\mathcal{O}(1/N_{f})$ in the Veneziano limit, it can be sizeable when a
realistic value is substituted for $N_{f}$. Figure 8 shows the relative
difference between the bulk Lagrangian density for the back-reacted solution
and the probe baryon solution, when setting $N_{f}=3$ in the leading large N
result. As for the chiral condensate, there is no reason for the result to be
quantitatively accurate at $N_{f}=3$, but it gives an indication of the
qualitative behavior.

We should also emphasize that the definition of the relative difference which
is shown in Figure 8 is not the standard one, where the difference of the two
quantities that are compared is divided by the quantity of reference (as in
(3.43) for instance). The usual definition of the relative difference is not
appropriate to compare the densities over the $(\xi,r)$ plane, since the place
where the densities go to zero is not exactly the same for the probe and back-
reacted solutions. Instead, we define the relative difference by dividing the
difference of the two densities by a reference value $\bar{\rho}_{M}$

$\Delta_{\text{rel}}\rho_{M}\equiv\frac{\rho_{M,\text{back-
reacted}}-\rho_{M,\text{probe}}}{\bar{\rho}_{M}}\,.$ (3.40)

$\bar{\rho}_{M}$ is defined as the mean value of the probe density over the
region of the bulk where most of the density is contained. To be more precise,
the criterion that we used to define the relevant region is given by

$\frac{3}{N_{c}}\rho_{M}\geq 0.7M_{0}^{4}\,,$ (3.41)

whose boundary is shown by the green line in the right of Figure 5. In
practice, the mean value is then computed numerically by averaging over the
cells contained in the given region, denoted $\mathcal{A}$ here

$\bar{\rho}_{M}=\frac{1}{N_{\mathrm{cells}}}\sum_{i\in\mathcal{A}}\rho_{M}(i)\,,$
(3.42)

where $N_{\text{cells}}$ is the number of grid cells contained in
$\mathcal{A}$.

Figure 8 indicates that there is a region near the UV boundary where the back-
reacted Lagrangian density decreases with respect to the probe solution. This
is understood easily as coming from the decrease of the tachyon modulus in
presence of the baryon, which is dual to the decrease of the chiral condensate
observed in Figure 7. Another noticeable feature of Figure 8 is the shift of
the baryon Lagrangian density towards the IR. This can be understood as
another consequence of the partial chiral restoration at the baryon center.
Indeed, the interaction of the baryon with the tachyon modulus results in a
repulsive force from the IR. So a decrease of the tachyon modulus weakens this
force, and implies the observed shift towards the IR.

\begin{overpic}[scale={0.75}]{figures/DrhoM_mean_NewPot_v8_3D_M0}
\end{overpic} Figure 8: Relative difference of bulk Lagrangian density for the
static soliton solution in the probe baryon and the back-reacted tachyon
regime. The relative difference is defined as the difference of the two
densities divided by the mean value of the probe density (3.42). The mean
value is taken over the area delimited by the green line in the right of
Figure 5, which is the region where the density is substantially different
from zero. The ratio is multiplied by $N_{f}$ in order to obtain something
finite in the Veneziano limit. The UV boundary is located at $r=0$ and the
baryon center at $\xi=0$.

Even at small values of $N_{f}$, the relative difference between the probe and
back-reacted solutions is observed to be relatively small numerically, of the
order of a few $10\%$. This is also the case at the level of the soliton
masses

$\frac{N_{f}}{3}\frac{M_{0,\text{back-
reacted}}-M_{0,\text{probe}}}{M_{0,\text{probe}}}\simeq-10\%\,.$ (3.43)

Here again, the number in (3.43) is the leading large $N_{f}$ result, that
cannot precisely be trusted for small $N_{f}$ and should be considered as
indicative.

##  4 Quantization of the isospin collective modes

The baryon states of equal half-integer spin and isospin are found by
quantizing the soliton collective coordinates (or zero modes) around the
static solution [75]. These modes are

  * •

The spatial position of the soliton $\vec{X}=(X^{1},X^{2},X^{3})$ .

  * •

The isospin orientation of the soliton, encoded in an $SU(N_{f})_{V}$
matrix141414The relevant collective coordinates are actually only a subgroup
of the isospin group $SU(N_{f})_{V}$. For instance, for $N_{f}=2$ they are the
elements of $SU(2)_{V}/\mathbb{Z}_{2}$ and for $N_{f}=3$, the elements of
$SU(3)_{V}/U(1)_{Y}$, where $U(1)_{Y}$ refers to the strong hypercharge
subgroup.  $V$.

The baryon solution can be deformed in many ways in addition to these modes,
such as changing the position of the soliton in the holographic direction, or
the size of the soliton [16]. Quantizing such modes leads to a tower of
excited baryon states in each spin sector. In the following, we focus on the
lowest states of these towers and consider only the quantization of the zero
modes.

There is no guarantee in principle that the rotation modes can be studied as
those of a rigid rotor, independently from the dilation mode of the soliton.
In [16], it was actually found that the geometry of the collective modes
manifold for dilation and isospin rotation had to be such that the modes are
rather quantized as a 4D harmonic oscillator, with energy levels given by
equation (5.24) in this reference. The rigid rotor is a good approximation to
the harmonic oscillator only when its fundamental frequency is very large
compared with its inverse moment of inertia. In QCD, this condition is
fulfilled in the large $N_{c}$ limit. This property is reproduced in the
holographic QCD model of [16], as is manifest from the large $N_{c}$ limit of
the energy levels, equation (5.31) in [16]. Their equation (5.24) also
indicates that the rigid rotor (large $N_{c}$) approximation is better for
lower spins. In the following, we assume that the rotation modes can be
treated as those of a rigid rotor. According to the previous discussion, once
we set the number of colors to its physical value $N_{c}=3$, this should not
induce too large errors for the low spin modes that exist in real QCD ($s=1/2$
and $3/2$). Note that the same rigid rotor approximation was considered in the
context of the hard wall model [48] .

The spatial position of the baryon is actually irrelevant to the study of
baryon states, so it will be kept fixed at $\vec{X}=0$. We are therefore left
with the problem of quantizing the isospin rotation mode of the soliton. To do
so, we consider a configuration where the soliton isospin orientation $V(t)$
evolves with time, but sufficiently slowly to be approximated by the ansatz

$\mathbf{L}(t)=V(t)\mathbf{L}^{\text{(sol)}}\left(r,\xi\right)V(t)^{\dagger}-i\mathrm{d}V(t)V(t)^{\dagger}\,,$
(4.1)
$\mathbf{R}(t)=V(t)\mathbf{R}^{\text{(sol)}}\left(r,\xi\right)V(t)^{\dagger}-i\mathrm{d}V(t)V(t)^{\dagger}\,,$
(4.2) $T(t)=V(t)T^{\text{(sol)}}\left(r,\xi\right)V(t)^{\dagger}\,,$ (4.3)

where the superscript (sol) refers to the field evaluated in the static
soliton solution. $V(t)$ is parametrized as

$V(t)\equiv\exp{\left(it\,\omega^{a}\lambda^{a}\right)}\,,$ (4.4)

the $\lambda^{a}$’s being the generators of $SU(N_{f})$, and the angular
velocity is identified to be

$\omega^{a}\lambda^{a}=-iV(t)^{\dagger}\frac{\mathrm{d}V(t)}{\mathrm{d}t}\,.$
(4.5)

We assume that the rotation is stationary

$\frac{\mathrm{d}\omega^{a}}{\mathrm{d}t}=0\,,$ (4.6)

and slow, so that $\omega$ can be treated as a perturbation on top of the
static soliton background. Also, _from now on we restrict to the case of
$N_{f}=2$_. This means that we shall quantize only a subset of the full
isospin rotations. Specifically, we restrict to $V(t)\in SU(2)_{V}$, in the
same subgroup as the baryon solution.

The starting point for the quantization of the isospin rotation modes is the
classical Lagrangian that controls their dynamics. The latter is obtained in
the next section by substituting the slowly rotating ansatz (4.1)-(4.3) into
the bulk action (2.3) and evaluating it on-shell for the slowly rotating
soliton solution

$L_{\text{rot}}=-M_{0}+\frac{1}{2}\lambda\vec{\omega}^{2}\,,$ (4.7)

where $M_{0}$ is the mass of the static soliton and $\lambda$ its moment of
inertia. The classical Hamiltonian is then computed, and quantized in the
canonical way described in Appendix F. As a result, the eigenstates of the
Hamiltonian are shown to have same spin and isospin and its eigenvalues are
given by

$E_{s}=M_{0}+\frac{1}{2\lambda}s(s+1)\,,$ (4.8)

where $s$ refers to the spin. In particular, the nucleon states correspond to
$s=1/2$ and the isobar $\Delta$ to $s=3/2$.

##  5 The rotating soliton

This section is dedicated to the calculation of the rotating soliton solution,
from which can be computed the moment of inertia $\lambda$ that controls the
splitting of the baryon energy levels (4.8). We start by determining the
ansatz relevant to the solution, before deriving the equations of motion for
the fields of the ansatz as well as the boundary conditions they should obey.
We finally describe the numerical solution for the rotating solution. We work
with a slowly rotating soliton, at first order in the rotation velocity
$\omega$. Also, we recall that we only consider the quantization of the
subsector of the chiral group that contains the 2 flavors of the soliton
solution.

###  5.1 Ansatz for the rotating instanton

Substituting the naive ansatz (4.1)-(4.3) with $V(t)\in SU(2)$ into the
equations of motion, reveals that this ansatz in itself cannot solve the time-
dependent equations of motion. The reason is that the components
$(L/R)^{a}_{0}$ and $(L^{\text{T}}/R^{\text{T}})_{r,i}$ in (3.5) are turned on
at linear order in $\omega$ [48], as is the abelian phase of the tachyon
$(\det T-1)$. Accordingly, the ansatz (4.1) for the gauge fields should be
supplemented by

$L_{0}(t)=V(t)L_{0}^{\text{(rot)}}\left(r,\xi;\vec{\omega}\right)V(t)^{\dagger}-i\partial_{0}V(t)V(t)^{\dagger}\,,$
(5.1) $L^{\text{T}}_{r,i}(t)=L^{\text{T,(rot)}}_{r,i}(r,\xi;\vec{\omega})\,,$
(5.2)

and likewise for the right-handed fields. Also, the ansatz for the tachyon
field (4.3) should be modified to

$T(t)=V(t)T^{\text{(rot)}}(r,\xi;\vec{\omega})V(t)^{\dagger}\,,$ (5.3)

where $\mathbf{L}^{\text{(rot)}}$ and $(\det T^{\text{(rot)}}-1)$ start at
linear order in $\omega$.

To determine relevant ansätze for $\mathbf{L}^{\text{(rot)}}$ and
$T^{\text{(rot)}}$, we proceed as in the case of the static soliton [43] and
impose the maximal number of symmetries of the bulk action. In the rotating
case, this includes the cylindrical symmetry and parity. For $N_{f}=2$
flavors, the cylindrical symmetry of the static soliton solution (3.6)-(3.9)
implies that a constant isospin rotation of the soliton is equivalent to a
constant spatial rotation. So the soliton rotating in isospin space can be
seen as rotating instead in physical space, with angular momentum
$\vec{\omega}$. In particular, $\vec{\omega}$ transforms as a pseudo-vector in
3-dimensional space151515Strictly speaking, $\vec{\omega}$ is a definite
3-dimensional vector, and the rotation breaks the cylindrical symmetry.
However, as is standard for broken symmetries, the appropriate ansatz can be
derived by assuming that $\vec{\omega}$ transforms as a pseudo-vector (in that
case, $\vec{\omega}$ is regarded as a field, called a spurion).. At linear
order in $\vec{\omega}$, the cylindrically symmetric ansatz for the gauge
fields of the rotating soliton is then [18, 48]

$(L/R)_{i}=V(t)(L/R)_{i}^{\text{(sol)}}V(t)^{\dagger}\;\;\;,\;\;\;(L/R)_{r}=V(t)(L/R)_{r}^{\text{(sol)}}V(t)^{\dagger}\,,$
(5.4) $(L/R)^{\text{T}}_{0}=(L/R)_{0}^{\text{T,(sol)}}\,,$
$\displaystyle(L/R)_{0}$
$\displaystyle=V(t)\bigg{(}\omega_{b}\left[\chi_{1}^{(L/R)}(r,\xi)\epsilon^{abc}\frac{x^{c}}{\xi}+\chi_{2}^{(L/R)}(r,\xi)\left(\frac{x^{a}x^{b}}{\xi^{2}}-\delta^{ab}\right)\right]+$
(5.5)
$\displaystyle\hphantom{=V(t)\bigg{(}}+\frac{v^{(L/R)}(r,\xi)}{\xi^{2}}(\vec{\omega}.\vec{x})x^{a}\,+\omega^{a}\bigg{)}\frac{\sigma^{a}}{2}V(t)^{\dagger}\,,$
$(L/R)^{\text{T}}_{i}=\frac{\rho^{(L/R)}(r,\xi)}{\xi}\left(\omega_{i}-(\vec{\omega}\cdot\vec{x})\frac{x_{i}}{\xi^{2}}\right)\\!+\\!B_{\xi}^{(L/R)}(r,\xi)(\vec{\omega}\cdot\vec{x})\frac{x_{i}}{\xi^{2}}+Q^{(L/R)}(r,\xi)\epsilon_{ibc}\omega^{b}\frac{x^{c}}{\xi},$
(5.6)
$(L/R)^{\text{T}}_{r}=B_{r}^{(L/R)}(r,\xi)\frac{\vec{\omega}\cdot\vec{x}}{\xi}\,,$
(5.7)

where the superscript (sol) refers to the field in the static soliton
configuration161616At order $\mathcal{O}(\omega^{2})$, the static fields will
receive corrections from the rotation. These could in principle contribute to
the moment of inertia in (5.21). It is not the case because the static fields
sit at a saddle point of the static action. The leading contribution to the
Lagrangian (5.21) from the $\mathcal{O}(\omega^{2})$ correction to the static
fields therefore starts at order $\mathcal{O}(\omega^{4})$, corresponding to
an $\mathcal{O}(\omega^{2})$ correction to the moment of inertia., and we
introduced the new 2-dimensional fields

$\chi_{1,2}^{(L/R)}\;\;\;,\;\;\;v^{(L/R)}\;\;\;,\;\;\;\rho^{(L/R)}\;\;\;,\;\;\;B_{\bar{\mu}}^{(L/R)}\;\;\;,\;\;\;Q^{(L/R)}\,.$
(5.8)

Imposing symmetry under the parity transformation (the full transformation
$P=P_{1}\cdot P_{2}$ in terms of the definitions in Appendix A)

$P\,:\quad\vec{x}\to-\vec{x}\;\;\;,\;\;\;L\leftrightarrow
R\;\;\;,\;\;\;\vec{\omega}\to\vec{\omega}\,,$ (5.9)

relates the right-handed and left-handed fields as

$\chi_{1}\equiv\chi_{1}^{L}=-\chi_{1}^{R}\;\;\;,\;\;\;\chi_{2}\equiv\chi_{2}^{L}=\chi_{2}^{R}\,,$
$v\equiv v^{L}=v^{R}\,,$ $\rho\equiv\rho^{L}=-\rho^{R}\,,$ (5.10)
$B_{\bar{\mu}}\equiv B_{\bar{\mu}}^{L}=-B_{\bar{\mu}}^{R}\,,$ $Q\equiv
Q^{L}=Q^{R}\,.$

For the tachyon field, the ansatz that has the right transformation properties
under 3-dimensional rotations and parity takes the form

$T=V(t)\exp{\left(i\zeta(r,\xi)\frac{\vec{\omega}\cdot\vec{x}}{\xi}\right)}T^{(sol)}V(t)^{\dagger}\;\;\;,\;\;\;\zeta(r,\xi)\in\mathbb{R}\,.$
(5.11)

The ansatz thus defined is invariant under a $U(1)_{s}\times U(1)_{r}$
residual gauge freedom, where the first factor was already present in the
static case (3.11), as denoted by the subscript “$s$”, and the second factor
appears in the rotating solution, as denoted by the subscript “$r$”. This new
factor is an axial $U(1)$ gauge freedom that is a subgroup of the chiral
$U(1)_{A}$

$\hat{g}_{L}=\hat{g}_{R}^{\dagger}=\exp{\left(i\beta(r,\xi)\frac{\vec{\omega}\cdot\vec{x}}{\xi}\right)}\,,$
(5.12)

under which only $B_{\bar{\mu}}$, $\rho$ and $\zeta$ transform, with
transformation rules

$B_{\bar{\mu}}\to
B_{\bar{\mu}}+\partial_{\bar{\mu}}\beta\;\;\;,\;\;\;\rho\to\rho+\beta\;\;\;,\;\;\;\zeta\to\zeta-2\beta\,.$
(5.13)

Also, the complex scalar field

$\chi\equiv\chi_{1}+i\chi_{2}\,,$ (5.14)

transforms as a charge 1 complex scalar field under $U(1)_{s}$ in (3.11).

In the case of a rotating soliton, one should in principle consider the
coupling to the holographic axion $\mathfrak{A}$, dual to the boundary Yang-
Mills instanton density operator $\text{Tr}\left(G\wedge G\right)$ [36, 58,
59]. This coupling appears because of the additional residual $U(1)_{A}$ gauge
freedom (5.12), which turns on the abelian phase of the tachyon $\zeta$ and
the axial part of the abelian gauge field $B_{\bar{\mu}}$ and $\rho$. However,
as for the other color fields, the action of the baryon on the axion is
suppressed by a factor $\mathcal{O}\big{(}N_{c}^{-1}\big{)}$ in the large
$N_{c}$ limit. The rotating baryon solution will therefore decouple from the
axion field at leading order in $N_{c}$. At next-to-leading order, the axion
will contribute to the order $\mathcal{O}\big{(}N_{c}^{-1}\big{)}$ correction
to the moment of inertia171717The $g_{0i}$ component of the metric will also
be turned on by rotation, and contribute at order
$\mathcal{O}\big{(}N_{c}^{-1}\big{)}$. In the following, we consider the same
approximation as in the static case and ignore the action on the color sector.
This implies in particular that we set the axion to 0.

As in the case of the static soliton, there exists a redefinition of the
ansatz fields such that the tachyon phases $\theta$ and $\zeta$ are absorbed
into the gauge fields

$\tilde{\chi}\equiv\mathrm{e}^{i\theta}\chi\;\;\;,\;\;\;\tilde{B}_{\bar{\mu}}\equiv
B_{\bar{\mu}}+\frac{1}{2}\partial_{\bar{\mu}}\zeta\;\;\;,\;\;\;\tilde{\rho}\equiv\rho+\frac{1}{2}\zeta\,.$
(5.15)

The resulting fields are invariant under the residual gauge freedom (3.11) and
(5.12). In addition to this field redefinition, for later use it is also
convenient to define the field strength for the $B_{\bar{\mu}}$ gauge field

$B_{\bar{\mu}\bar{\nu}}\equiv\partial_{\bar{\mu}}B_{\bar{\nu}}-\partial_{\bar{\nu}}B_{\bar{\mu}}\,,$
(5.16)

and the covariant derivative for the $\rho$ and $\chi$ field

$D_{\bar{\mu}}\rho=\partial_{\bar{\mu}}\rho-
B_{\bar{\mu}}\;\;\;,\;\;\;D_{\bar{\mu}}\chi=(\partial_{\bar{\mu}}-iA_{\bar{\mu}})\chi\,.$
(5.17)

#### Lorenz gauge

Because we obtain the static soliton solution by fixing the residual gauge
freedom (3.11) to the Lorenz gauge

$\partial_{r}A_{r}+\partial_{\xi}A_{\xi}=0\,,$ (5.18)

we work in the same gauge for the rotating soliton. Also, we fix the
additional gauge freedom (5.12) by imposing a similar Lorenz condition for
$B_{\bar{\mu}}$

$\partial_{r}B_{r}+\partial_{\xi}B_{\xi}=0\,.$ (5.19)

For this choice, the equations of motion for $B_{\bar{\mu}}$ (E.8) and (E.9)
are elliptic and can be solved numerically by a heat diffusion method upon
setting the right boundary conditions.

Note that the Lorenz condition (5.19) leaves a residual gauge freedom of the
form

$B_{\bar{\mu}}\to
B_{\bar{\mu}}+\partial_{\bar{\mu}}\mathfrak{f}\;\;\;,\;\;\;\partial_{r}^{2}\mathfrak{f}+\partial_{\xi}^{2}\mathfrak{f}=0\;\;\;,\;\;\;\mathfrak{f}(0,\xi)=0\,.$
(5.20)

Part of the choice of boundary conditions will correspond to the choice of
residual gauge freedom (5.20). The explicit choice that we make is discussed
below, in Section 5.3.

###  5.2 Moment of inertia and equations of motion

The Lagrangian for the rotational collective modes of the soliton is obtained
by substituting the ansatz (5.4)-(5.7) and (5.11) into the bulk action (2.3).
This yields the Lagrangian of a rigid rotor

$L_{rot}=-M_{0}+\frac{1}{2}\lambda\vec{\omega}^{2}\,,$ (5.21)

where $M_{0}$ is the mass of the static soliton (3.25), and we defined the
moment of inertia

$\lambda\equiv\lambda_{DBI}+\lambda_{CS}\,,$ (5.22)
$\lambda_{DBI}=\int\mathrm{d}r\mathrm{d}\xi\,4\pi\xi^{2}\rho_{\lambda,DBI}\;\;\;,\;\;\;\lambda_{CS}=\int\mathrm{d}r\mathrm{d}\xi\,4\pi\xi^{2}\rho_{\lambda,CS}\,,$
(5.23) $\displaystyle\rho_{\lambda,DBI}$
$\displaystyle\equiv-\frac{2}{3}M^{3}N_{c}V_{f}(\lambda,\tau)\mathrm{e}^{A}\sqrt{1+\mathrm{e}^{-2A}\kappa\left((\partial_{r}\tau)^{2}+(\partial_{\xi}\tau)^{2}\right)}\times$
$\displaystyle\hphantom{=}\times\bigg{(}\mathrm{e}^{2A}\kappa(\lambda)\tau^{2}\bigg{(}4\mathrm{e}^{2A}\Delta_{rr}\tilde{B}_{r}^{2}+4(1-\mathrm{e}^{2A}\Delta_{\xi\xi})\left[\tilde{B}_{\xi}^{2}+\frac{2}{\xi^{2}}\tilde{\rho}^{2}\right]-$
$\displaystyle\hphantom{=\times\bigg{(}\mathrm{e}^{2A}\xi^{2}\kappa(\lambda)\tau^{2}\bigg{(}}-8\mathrm{e}^{2A}\Delta_{\xi
r}\tilde{B}_{\xi}\tilde{B}_{r}-2\tilde{\chi}_{1}^{2}\bigg{)}-$
$\displaystyle\hphantom{=\times\bigg{(}}-w(\lambda)^{2}\bigg{(}\frac{1}{2}(1-\mathrm{e}^{2A}\Delta_{\xi\xi})|D_{\xi}\chi|^{2}+\frac{1}{2}\mathrm{e}^{2A}\Delta_{rr}|D_{r}\chi|^{2}-$
$\displaystyle\hphantom{=\times\bigg{(}-w(\lambda)^{2}}-\frac{1}{2}\mathrm{e}^{2A}\Delta_{\xi
r}\left(D_{\xi}\chi^{*}D_{r}\chi+h.c.\right)+\frac{1}{4}(1-\mathrm{e}^{2A}\Delta_{\xi\xi})(\partial_{\xi}v)^{2}+$
$\displaystyle\hphantom{=\times\bigg{(}-w(\lambda)^{2}}+\frac{1}{4}\mathrm{e}^{2A}\Delta_{rr}(\partial_{r}v)^{2}-\frac{1}{2}\mathrm{e}^{2A}\Delta_{\xi
r}\partial_{r}v\partial_{\xi}v+$
$\displaystyle\hphantom{=\times\bigg{(}-w(\lambda)^{2}}+\frac{1}{2\xi^{2}}(v^{2}+|\chi|^{2})(1+|\phi|^{2})-v(\chi\phi^{*}+h.c.)-$
$\displaystyle\hphantom{=\times\bigg{(}-w(\lambda)^{2}}-2\mathrm{e}^{2A}\Delta_{rr}\xi^{-2}(D_{r}\rho)^{2}-2(1-\mathrm{e}^{2A}\Delta_{\xi\xi})\xi^{-2}(D_{\xi}\rho)^{2}+$
$\displaystyle\hphantom{=\times\bigg{(}-w(\lambda)^{2}}+4\mathrm{e}^{2A}\Delta_{\xi
r}\xi^{-2}D_{r}\rho
D_{\xi}\rho-2\mathrm{e}^{2A}\Delta_{rr}(\partial_{r}Q)^{2}-$
$\displaystyle\hphantom{=\times\bigg{(}-w(\lambda)^{2}}-2(1-\mathrm{e}^{2A}\Delta_{\xi\xi})(\partial_{\xi}Q)^{2}+4\mathrm{e}^{2A}\Delta_{\xi
r}\partial_{r}Q\partial_{\xi}Q-$
$\displaystyle\hphantom{=\times\bigg{(}-w(\lambda)^{2}}-2\xi^{-2}Q^{2}\Big{[}2-\mathrm{e}^{2A}\xi\left(\partial_{\xi}\Delta_{\xi\xi}+\mathrm{e}^{2A}\partial_{r}(\mathrm{e}^{-2A}\Delta_{\xi
r})\right)\Big{]}-$
$\displaystyle\hphantom{=\times\bigg{(}-w(\lambda)^{2}}-\frac{1}{2}\Big{[}\mathrm{e}^{2A}\Delta_{rr}(1-\mathrm{e}^{2A}\Delta_{\xi\xi})-\mathrm{e}^{4A}\Delta_{\xi
r}^{2}\Big{]}(B_{\bar{\mu}\bar{\nu}})^{2}\bigg{)}\bigg{)}\,,$ (5.24)

where the $\Delta$ symbol is defined in Appendix E,

$\displaystyle\rho_{\lambda,CS}$
$\displaystyle\equiv\frac{2N_{c}}{3\pi^{2}\xi^{2}}\epsilon^{\bar{\mu}\bar{\nu}}\bigg{(}D_{\bar{\mu}}\rho\Big{(}(f_{1}(\tau)+f_{3}(\tau))(D_{\bar{\nu}}\phi\,\chi^{*}+h.c.)-$
$\displaystyle\hphantom{=\times\bigg{(}D_{\bar{\mu}}\rho\Big{(}}-2(2f_{3}(\tau)-f_{1}(\tau))\left(D_{\bar{\nu}}\tilde{\phi}+h.c.\right)\tilde{\chi}_{1}-$
$\displaystyle\hphantom{=\times\bigg{(}D_{\bar{\mu}}\rho\Big{(}}-4(f_{1}(\tau)-f_{3}(\tau)-if_{2}(\tau))\tilde{A}_{\bar{\nu}}\tilde{\phi}_{1}\tilde{\chi}_{2}\Big{)}+$
$\displaystyle\hphantom{=\times\bigg{(}}+\tilde{\rho}\Big{(}2(f_{3}(\tau)-if_{2}(\tau))\big{(}-F_{\bar{\mu}\bar{\nu}}\,\tilde{\phi}_{1}\tilde{\chi}_{2}+\tilde{A}_{\bar{\mu}}(i\partial_{\bar{\nu}}\phi\chi^{*}+h.c.)\,\big{)}-$
$\displaystyle\hphantom{=\times\bigg{(}+\rho}-\partial_{\bar{\nu}}\tau
f_{1}^{\prime}(\tau)(D_{\bar{\mu}}\phi\chi^{*}+h.c.)+\partial_{\bar{\nu}}\tau
f_{3}^{\prime}(\tau)(D_{\bar{\mu}}\tilde{\phi}\tilde{\chi}+h.c.)-$
$\displaystyle\hphantom{=\times\bigg{(}+\rho}-2i\partial_{\bar{\nu}}\tau
f_{2}^{\prime}(\tau)\tilde{A}_{\bar{\mu}}\tilde{\phi}_{1}\tilde{\chi}_{2}\,\Big{)}+$
$\displaystyle\hphantom{=\times\bigg{(}}+2\partial_{\bar{\nu}}\tau(f_{1}^{\prime}(\tau)-f_{3}^{\prime}(\tau))D_{\bar{\mu}}\rho\,\tilde{\chi}_{1}\tilde{\phi}_{1}+$
$\displaystyle\hphantom{=\times\bigg{(}}+\partial_{\bar{\mu}}(\xi
Q)\Big{(}(f_{1}(\tau)+f_{3}(\tau))\left(i\chi^{*}D_{\bar{\nu}}\phi+h.c.\right)+$
$\displaystyle\hphantom{=\times\bigg{(}+\partial_{\bar{\mu}}(\xi
Q)\Big{(}}+4(f_{1}(\tau)+f_{3}(\tau)-3if_{2}(\tau))\tilde{A}_{\bar{\nu}}\,\tilde{\phi}_{1}\tilde{\chi}_{1}\Big{)}+$
$\displaystyle\hphantom{=\times\bigg{(}}+2\partial_{\bar{\nu}}\tau(f_{1}^{\prime}(\tau)+f_{3}^{\prime}(\tau))\partial_{\bar{\mu}}(\xi
Q)\tilde{\phi}_{1}\tilde{\chi}_{2}-$
$\displaystyle\hphantom{=\times\bigg{(}}-8(f_{1}(\tau)+f_{3}(\tau))\xi
Q\,D_{\bar{\mu}}\rho\partial_{\bar{\nu}}\Phi+8\partial_{\bar{\nu}}\tau(f_{1}^{\prime}(\tau)+f_{3}^{\prime}(\tau))\xi
Q\tilde{\rho}\partial_{\bar{\mu}}\Phi+$
$\displaystyle\hphantom{=\times\bigg{(}}+v\Big{(}B_{\bar{\mu}\bar{\nu}}\Big{[}\frac{1}{2}(f_{1}(\tau)+f_{3}(\tau))\,(|\phi|^{2}-1)+(f_{1}(\tau)-f_{3}(\tau)-if_{2}(\tau))\tilde{\phi}_{1}^{2}\Big{]}+$
$\displaystyle\hphantom{=\times\bigg{(}+v}+(f_{1}(\tau)+f_{3}(\tau))\xi
QF_{\bar{\mu}\bar{\nu}}-2(f_{3}(\tau)-if_{2}(\tau))\tilde{B}_{\bar{\mu}}(D_{\bar{\nu}}\tilde{\phi}+h.c.)\tilde{\phi}_{1}\Big{)}+$
$\displaystyle\hphantom{=\times\bigg{(}}+v\partial_{\bar{\nu}}\tau\Big{(}(f_{1}^{\prime}(\tau)+f_{3}^{\prime}(\tau))\big{(}\tilde{B}_{\bar{\mu}}(1-|\phi|^{2})-2\xi
Q\tilde{A}_{\bar{\mu}}\big{)}+2if_{2}^{\prime}(\tau)\tilde{B}_{\bar{\mu}}\tilde{\phi}_{1}^{2}\Big{)}\,\bigg{)}.$
(5.25)

We recall that the tildes on the static fields refer to the redefined fields
that contain the non-abelian tachyon phase $\theta$ (3.22). The total bulk
Lagrangian density for the rotating fields is denoted by $\rho_{\lambda}$

$\rho_{\lambda}\equiv\rho_{\lambda,DBI}+\rho_{\lambda,CS}\,.$ (5.26)

The equations of motion for the fields of the rotating soliton ansatz are
obtained by extremizing the moment of inertia (5.22) with respect to small
deformations of the fields. They are presented in Appendix E.

###  5.3 Boundary conditions

$\xi\to 0$ | $\xi\to L\to\infty$ | $r\to 0$ | $r\to r_{\text{IR}}\to\infty$
---|---|---|---
$v+\tilde{\chi}_{2}\to 0$ | $v\to-1$ | $v\to-1$ | $\partial_{r}v\to 0$
$\tilde{\chi}_{1}\to 0$ | $\tilde{\chi}_{1}\to 0$ | $\tilde{\chi}_{1}\to-\sin{\theta}$ | $\tilde{\chi}_{1}\to 0$
$\partial_{\xi}\tilde{\chi}_{2}\to 0$ | $\tilde{\chi}_{2}\to-1$ | $\tilde{\chi}_{2}\to\cos{\theta}$ | $\partial_{r}\tilde{\chi}_{2}\to 0$
$\partial_{\xi}B_{\xi}\to 0$ | $\partial_{\xi}B_{\xi}\to 0$ | $B_{\xi}\to 0$ | $B_{\xi}\to 0$
$B_{r}\to 0$ | $B_{r}\to 0$ | $\partial_{r}B_{r}\to 0$ | $\partial_{r}B_{r}\to 0$
$\partial_{\xi}\rho-B_{\xi}\to 0$ | $\rho\to 0$ | $\rho\to 0$ | $\rho\to 0$
$Q\to 0$ | $Q\to 0$ | $Q\to 0$ | $\partial_{r}Q\to 0$
$\zeta\to 0$ | $\zeta\to 0$ | $r\zeta\to 0$ | $\zeta\to 0$
Table 6: Boundary conditions for the rotating soliton solution in Lorenz
gauge.

We present in Table 6 the boundary conditions that are imposed on the fields
of the rotating soliton ansatz (5.4)-(5.7) and (5.11), which obey the
equations of motion (E.5)-(E.12). We discuss separately the 4 boundaries of
the $(\xi,r)$ space

  * •

UV : In the UV limit $r\to 0$, the condition that
$v,\tilde{\chi},B_{\xi},\rho$ and $Q$ should vanish comes from requiring that
there is no source for the gauge fields at the boundary. Moreover, the
condition for $B_{r}$ comes from imposing the Lorenz gauge.

The condition for $\zeta$ is somewhat more subtle than the other fields. The
reason is that, because the quark mass is set to 0, there is no source term
for the tachyon field. The abelian phase $\zeta$ is therefore not associated
with any source. At the level of the near-boundary behavior, it translates
into the fact that the boundary value of $\zeta$ appears in front of the vev
term for the tachyon field

$T(r,\xi)\underset{r\to
0}{=}\ell\Sigma(\xi)\exp{\left(i\zeta(0,\xi)\frac{\vec{\omega}\cdot\vec{x}}{\xi}+i\theta(0,\xi)\frac{x\cdot\sigma}{\xi}\right)}r^{3}(-\log{(r\Lambda)})^{-c}\left(1+\cdots\right),$
(5.27)

where the dots refer to terms that go to 0 near the boundary; $\Sigma(\xi)$ is
proportional to the modulus of the chiral condensate in the boundary theory
$\left|\left<\bar{\psi}\psi\right>\right|$. The UV condition for $\zeta$ will
therefore not come from a choice of source at the boundary, but rather from
the requirement that the solution be regular. The equation of motion for
$\zeta$ (E.12) has two linearly independent solutions, one of which behaves as
$r^{-2}$ near the boundary and the other as $r^{0}$. Requiring that

$r\zeta\underset{r\to 0}{\to}0\,,$ (5.28)

will therefore select the regular behavior.

  * •

$\xi\to L\,(L\to\infty)\,:$ Requiring that the moment of inertia (5.22) should
be finite imposes that, as $\xi\to\infty$

$\xi^{3/2}\tilde{\chi}_{1}\to
0\;\;\;,\;\;\;\xi^{3/2}\partial_{\bar{\mu}}\tilde{\chi}_{2}\to
0\;\;\;,\;\;\;\xi^{3/2}\partial_{\bar{\mu}}v\to 0\,,$ (5.29)
$\xi^{3/2}\left(B_{\bar{\mu}}+\frac{1}{2}\partial_{\bar{\mu}}\zeta\right)\to
0\;\;\;,\;\;\;\xi^{1/2}\left(\rho+\frac{1}{2}\zeta\right)\to
0\;\;\;,\;\;\;\xi^{3/2}\partial_{\bar{\mu}}Q\to 0\,.$ (5.30)

For regularity $\partial_{\xi}\zeta$ should go to 0 as $\xi\to\infty$ so
$B_{\xi}$ should also tend to 0 from (5.30). Then the Lorenz gauge condition
in the limit where $\xi\to\infty$ reads

$\partial_{r}B_{r}=0=-\frac{1}{2}\partial_{r}^{2}\zeta\,,$ (5.31)

so at $\xi\to\infty$, $\zeta$ should take the form

$\zeta\underset{\xi\to\infty}{\to}\zeta^{(0)}+r\zeta^{(1)}\,.$ (5.32)

In the UV, from (5.30)

$\underset{\xi\to\infty}{\text{lim}}\zeta(\xi,0)=-2\underset{\xi\to\infty}{\text{lim}}\rho(\xi,0)=0\,,$
(5.33)

because the source for $\rho$ is set to 0. We shall further fix the residual
gauge freedom (5.20) such that $\zeta=0$ in the IR. With this choice, the
linear part also vanishes in (5.32) and

$\zeta\underset{\xi\to\infty}{\to}0\,.$ (5.34)

The final condition for $B_{\xi}$, $\partial_{\xi}B_{\xi}\to 0$ rather than
$B_{\xi}\to 0$, is chosen to impose Lorenz gauge even at finite $L$ in the
numerical solution.

  * •

IR : As for the static soliton [43], the conditions in the IR limit $r\to
r_{\text{IR}}$ are regularity conditions. In [43] it was found that the
resulting conditions for the gauge fields were equivalent to the conditions
imposed in the hard-wall model for chiral symmetry to be broken on the IR wall

$\left.\left(\mathbf{L}-\mathbf{R}\right)\right|_{r_{\text{IR}}}=0\;\;\;,\;\;\;\left.\left(\mathbf{F}^{(L)}_{\mu
r}+\mathbf{F}^{(R)}_{\mu r}\right)\right|_{r_{\text{IR}}}=0\,.$ (5.35)

In the following, our strategy is to assume that the IR regularity conditions
for the gauge field are still equivalent to (5.35) in the case of the rotating
soliton. This assumption will be confirmed numerically if a solution can be
found with this behavior in the IR. It can also be checked analytically by
studying the IR asymptotics of the equations of motion. With the ansatz
(5.4)-(5.7), (5.35) translates to

$\left.\chi_{1}\right|_{\text{IR}}=0\;\;\;,\;\;\;\left.B_{\xi}\right|_{\text{IR}}=0\;\;\;,\;\;\;\left.\rho\right|_{\text{IR}}=0\,,$
(5.36)
$\left.\partial_{r}\chi_{2}\right|_{\text{IR}}=0\;\;\;,\;\;\;\left.\partial_{r}Q\right|_{\text{IR}}=0\,.$
(5.37)

Finally, the condition for $B_{r}$ comes from the Lorenz gauge (5.19) and, as
stated in the previous point, the residual gauge freedom (5.20) is chosen such
that $\zeta=0$ in the IR.

  * •

$\xi=0\,:$ The boundary conditions in the limit where $\xi$ goes to 0 come
from requiring that $\tilde{\mathbf{L}}$ and $\tilde{\mathbf{R}}$ are well
defined vectors at $\xi=0$ and that the field strength (E.1)-(E.4) is a well
defined 2-tensor. The additional constraint on $B_{\xi}$ is imposed by the
choice of Lorenz gauge. The final condition $\zeta\underset{\xi\to 0}{\to}0$
fixes what remained of the residual gauge freedom (5.20) after setting $\zeta$
to 0 in the IR. Note that for this choice the tachyon matrix (5.11) is well
defined at $\xi=0$.

###  5.4 Numerical results : the spin-isospin spectrum

We present in this subsection the numerical solution for the slowly rotating
soliton configuration. The equations of motion written in Appendix E are
solved with the gradient descent method, imposing the boundary conditions of
Table 6.

At linear order in $\omega$, the rotating solution is a probe on the static
background. At leading order in $N_{f}$, the static background will correspond
to the probe baryon solution presented in Section 3.5.1. Order
$\mathcal{O}\big{(}N_{f}^{-1}\big{)}$ corrections to the rotating solution
will come from including the back-reaction on the tachyon in the static
background, as discussed in Section 3.5.2. We start by presenting the leading
order probe baryon solution and then discuss the back-reaction. We recall that
the back-reacting solution is computed assuming no back-reaction on the color
sector (metric and dilaton).

####  5.4.1 Probe baryon background

We start with the numerical results obtained for the probe baryon background.
In this case the modulus of the tachyon field $\tau$ is fixed to its vacuum
value, and the equations of motion take the form presented in Appendix E.3.

\begin{overpic}[scale={0.5}]{figures/rhol_probe_v8_3D_M0}
\end{overpic}\begin{overpic}[scale={0.5}]{figures/rholr_probe_v8_3D_M0}
\end{overpic}

Figure 9: Left: bulk Lagrangian density for the rotating fields (5.26) in the
probe baryon approximation. All quantities are expressed in units of the
classical mass of the static soliton $M_{0}$ (3.25). The center of the soliton
is located at $\xi=0$ where the density diverges as $\xi^{-1}$. The UV
boundary is at $r=0$, and the green line indicates the boundary of the region
over which the mean value is computed to define the relative difference in
Figure 10. Right: Same as the left figure, but at some given value of the
3-dimensional radius $\xi$. This figure makes it clear that the Lagrangian
density eventually reaches 0 as $r\to 0$, as it should in absence of sources.

The bulk Lagrangian density for the rotating fields (5.26) in the
$(\xi,r)$-plane is presented in Figure 9, where all dimensionful quantities
are expressed in units of the classical soliton mass $M_{0}$ (3.25). As for
the static fields, Figure 9 shows the expected behavior for a solitonic
configuration, that is the densities are confined to a region of finite extent
in the bulk. The size of the lump is again of the order of $M_{0}^{-1}$.

Note that the density is observed to have a maximum very close to the UV
boundary. This is associated with the flavor gauge fields of the rotating
solution having the same kind of sharp behavior near the boundary. However, it
is possible to check that the smooth vev-like behavior is recovered
asymptotically as one goes closer to $r=0$

$\mathbf{A}_{\mu}\sim r^{2}\times\text{vev}\,.$

To check this, one needs to go very close to the boundary $r=0$, so log
coordinates $u=\log(r)$ are more appropriate. We have explicitly checked that
the vev behavior is recovered very near the boundary. We believe that this
feature is a peculiarity of the UV behavior of our choice of V-QCD potentials,
and that it can be avoided with a better choice. In particular, this behavior
is not observed for the potentials of [45], for which the solution is analyzed
in Appendix D.

The numerical value for the classical moment of inertia density $\lambda$ in
(5.22) is obtained by integrating the Lagrangian density in Figure 9

$\frac{1}{\lambda}\simeq\frac{3}{N_{c}}\times 60\,\text{MeV}\,.$ (5.38)

From this result, the spin-isospin spectrum of the baryons (4.8) can be
computed and compared with experimental QCD data, as shown in Table 7. We set
$N_{c}=3$ in the large N result to make this comparison. We recall that the
estimation (3.32) for the soliton mass gives only the classical contribution,
which can receive sizeable $\mathcal{O}\big{(}N_{c}^{0}\big{)}$ quantum
corrections at finite $N_{c}$. Likewise, (5.38) is the leading order
contribution to the moment of inertia in the Veneziano limit. Also, the V-QCD
potentials presented in Section 2 were not fitted to baryonic properties, but
rather to QCD thermodynamics and mesonic properties. In light of these
remarks, the precise numerical value for the baryon spectrum presented in
Table 7 should not be taken too seriously, but rather as an indicative result.
In particular, we shall not mention the numerical accuracy of the result as it
is much better than the theoretical uncertainty.

Spin | V-QCD mass | Experimental mass
---|---|---
$s=\frac{1}{2}$ | $M_{N}\simeq 1170\,\text{MeV}$ | $M_{N}=940\,\text{MeV}$
$s=\frac{3}{2}$ | $M_{\Delta}\simeq 1260\,\text{MeV}$ | $M_{\Delta}=1234\,\text{MeV}$
Table 7: Baryon spin-isospin spectrum in the V-QCD model with the potentials
of Section 2, compared with experimental data.

####  5.4.2 Back-reacted tachyon background

We now discuss the slowly rotating soliton solution computed on the static
background that takes into account the back-reaction on the tachyon field. The
corresponding equations of motion for the rotating fields are written in
Appendix E.2.

We are interested in the effect of the back-reaction on the soliton moment of
inertia (5.22). In the large $N_{f}$ limit, the correction is of order
$\mathcal{O}\big{(}N_{f}^{-1}\big{)}$

$\delta\lambda=\delta\tau\frac{\delta\lambda}{\delta\tau}\bigg{|}_{\text{probe}}+\mathcal{O}\big{(}N_{f}^{-2}\big{)}\,,$
(5.39)

where $\delta\tau$ refers to the order $\mathcal{O}\big{(}N_{f}^{-1}\big{)}$
correction to $\tau$. Figure 10 shows the relative difference between the bulk
Lagrangian density for the back-reacted and probe backgrounds, when setting
$N_{f}=3$ in the large N result. The definition of the relative difference is
analogous to the static case (5.40)

$\Delta_{\text{rel}}\rho_{\lambda}\equiv\frac{\rho_{\lambda,\text{back-
reacted}}-\rho_{\lambda,\text{probe}}}{\bar{\rho}_{\lambda}}\,,$ (5.40)

where the criterion defining the region over which the mean value
$\bar{\rho}_{\lambda}$ is computed is now given by

$\frac{3}{N_{c}}\rho_{\lambda}\geq M_{0}^{2}\,.$ (5.41)

This region is delimited by the green line in Figure 9.

The dominant effect of the tachyon back-reaction observed in Figure 10 is the
same as in the static case shown in Figure 8: the density decreases in the UV
and is shifted towards the IR.

\begin{overpic}[scale={0.75}]{figures/Drhol_mean_NewPot_v8_3D_M0}
\end{overpic} Figure 10: Relative difference of bulk Lagrangian density for
the slowly rotating soliton computed on the probe baryon and the back-reacted
tachyon backgrounds. The relative difference is defined as the difference of
the two densities divided by the mean value of the probe density. The mean
value is taken over the area delimited by the green line in the left of Figure
9, which is the region where the density is substantially different from zero.
The ratio is multiplied by $N_{f}$ in order to obtain something finite in the
Veneziano limit. The UV boundary is located at $r=0$ and the baryon center at
$\xi=0$.

Even more strikingly than for the soliton mass in the static case, even at
small values of $N_{f}$, the relative difference between the probe and back-
reacted solutions is observed to be small numerically, of the order of a few
percent. At the level of the soliton moment of inertia we obtain

$\frac{N_{f}}{3}\frac{\frac{1}{\lambda_{\text{back-
reacted}}}-\frac{1}{\lambda_{\text{probe}}}}{\frac{1}{\lambda_{\text{probe}}}}\simeq
0.37\%\,.$ (5.42)

As for the static mass, we would like to emphasize here again that precise
quantitative results such as (5.42) cannot be trusted when substituting a
small number of flavors in the large N result. The result in (5.42) should be
considered as an indication, that the moment of inertia does not seem to be
affected much by the back-reaction of the baryon on the background.

## Acknowledgements

We thank D. Arean, P. Figueras, A. Krikun, J. L. Mañes, A. Pich, C. Rosen, P.
Sutcliffe and P. Yi for discussions and/or correspondence. This work was
supported in part by CNRS contract IEA 199430. The work of MJ was supported by
an appointment to the JRG Program at the APCTP through the Science and
Technology Promotion Fund and Lottery Fund of the Korean Government. MJ was
also supported by the Korean Local Governments – Gyeongsangbuk-do Province and
Pohang City – and by the National Research Foundation of Korea (NRF) funded by
the Korean government (MSIT) (grant number 2021R1A2C1010834). FN would like to
thank the Asian Pacific Center for Theoretical Physics in Pohang, Korea, where