# Anomalous and linear holographic hard wall models
for light unflavored mesons

Rafael A. Costa-Silva<EMAIL_ADDRESS><EMAIL_ADDRESS>Henrique
Boschi-Filho<EMAIL_ADDRESS><EMAIL_ADDRESS>Instituto de Física,
Universidade Federal do Rio de Janeiro, 21.941-909 - Rio de Janeiro - RJ -
Brazil

###### Abstract

In this work we consider anomalous and linear holographic hard wall (HW)
models for light unflavored mesons inspired by the AdS/CFT correspondence. The
anomalous dimensions depend on the logarithm of the spin $S$ of the meson
state and come from a semiclassical analysis of gauge/string duality. The
anomalous HW model produces very good masses and good Regge trajectories for
mesons compared with PDG data. Inspired by this anomalous HW model we also
propose a phenomenological modification of the dimension of the boundary
operators such that the model produces asymptotic linear Regge trajectories.

##  I Introduction

The AdS/CFT correspondence Aharony:1999ti  establishes relations between Yang-
Mills (YM) and higher dimensional gravitational theories. One of the amazing
properties of the correspondence is that if YM theories are strongly coupled,
the gravitational one is weak coupled allowing for the solution of a myriad of
phenomena from the quark-gluon plasma to condensed matter (for reviews, see
e.g., Casalderrey-Solana:2011dxg ; Zaanen:2015oix ). In particular, hadronic
spectra were described by different proposals as the holographic hard wall
(HW) and the soft wall (SW) models Boschi-Filho:2002xih ; Boschi-Filho:2002wdj
; deTeramond:2005su ; Erlich:2005qh ; Boschi-Filho:2005xct ; Karch:2006pv ;
Colangelo:2008us . The HW model Boschi-Filho:2002xih ; Boschi-Filho:2002wdj ;
deTeramond:2005su ; Erlich:2005qh ; Boschi-Filho:2005xct  is well known for
producing reasonable hadronic masses but non-linear (parabolic) Regge
trajectories. On the other side, the SW model Karch:2006pv ; Colangelo:2008us
also give good masses with the advantage of generating linear Regge
trajectories.

In this work, we are going to consider the inclusion of anomalous dimensions
for the boundary operators describing light unflavored mesons with spins
$S\geq 1$. These anomalous dimensions depend on the logarithm of the spin $S$
of the mesonic state inspired by a semiclasical analysis of gauge/string
dualities Gubser:2002tv  and was used recently to discuss the case of even
spins glueballs Costa-Silva:2023vuu . We find very good masses and Regge
trajectories for the studied mesons when compared with PDG data PDG:2022pth ,
better than the ones from the HW and SW models. Furthermore, we discuss a
phenomenological modification of the dimension of the boundary operators
representing the mesonic states in order to obtain asymptotic linear Regge
trajectories, similar to the case of glueballs. Actually, the linearization of
the mesonic states require an additional parameter to obtain the best fit to
PDG data. For a recent account on non-holographic meson Regge trajectories,
see e.g. Chen:2021kfw .

This work is organized as follows. In Section II, we review the HW model for
light unflavored mesons and in Section III we discuss the introduction of
anomalous dimensions in the HW model and obtain the spectra for the mesonic
families starting with the states of the $\rho(770)$, $\omega(782)$,
$b_{1}(1235)$, and $h_{1}(1170)$. Then, in Section IV we modify the dimension
of the mesonic boundary operators such that we find asymptotic linear Regge
trajectories for the mesons already discussed in the previous section.
Finally, in Section V we present our conclusions.

##  II Original HW model for mesons

In this work we are interested in studying the spectra of light unflavored
mesons, and their Regge trajectories using anomalous and linear holographic HW
models. These mesons can be classified in four families according to their
behavior under $\mathcal{CP}$ transformations (even or odd), isospin $I=0$ and
$1$, and spins $S\geq 1$. These four families imply four distinct Regge
trajectories.

In order to describe such objects using the holographic HW model, we need to
study a massive vector field in $AdS_{5}$. We start with the action

$S=-\frac{1}{2}\int
d^{5}x\sqrt{-g}\bigg{[}\frac{1}{2}g^{pm}g^{qn}F_{mn}F_{pq}+M_{5}^{2}g^{pm}A_{p}A_{m}\bigg{]},$
(1)

where $F_{mn}=\partial_{m}A_{n}-\partial_{n}A_{m}$ is the stress tensor of the
Proca field with mass $M_{5}$ in the five-dimensional AdS space. Moreover, $g$
is the determinant of the metric of $AdS_{5}$,

$ds^{2}=g_{mn}dx^{m}dx^{n}=\frac{R^{2}}{z^{2}}(dz^{2}+\eta_{\mu\nu}dx^{\mu}dx^{\nu}),$
(2)

with $R$ being the $AdS$ radius, $z$ the holographic coordinate and
$\eta_{\mu\nu}$ is the Minkowski four-dimensional boundary metric. The
equation of motion that follows from the action, Eq. (1), is given by

$\partial_{z}\bigg{[}\bigg{(}\frac{1}{z}\bigg{)}F_{zn}\eta^{nq}\bigg{]}+\partial_{\mu}\bigg{[}\bigg{(}\frac{1}{z}\bigg{)}\eta^{m\mu}F_{mn}\eta^{nq}\bigg{]}-\frac{(M_{5}R)^{2}}{z^{3}}A_{n}\eta^{nq}=0.$
(3)

We use a plane wave solution as an ansatz for the five-dimensional vector
field $A_{m}(z,x^{\mu})$ with four-momentum $q_{\mu}$, and polarization four-
vector $\epsilon_{\rho}$,

$\displaystyle A_{\rho}(z,x^{\mu})=\epsilon_{\rho}v(z)e^{iq_{\mu}x^{\mu}},$
(4) $\displaystyle A_{z}=0,$ (5)

with $\partial_{\mu}A^{\mu}=0$. Defining $v(z)=zf(z)$, one obtains

$z^{2}\frac{d^{2}\psi(z)}{dz^{2}}+z\frac{d\psi(z)}{dz}-q^{2}z^{2}\psi(z)-[1+(M_{5}R)^{2}]f(z)=0.$
(6)

This is Bessel equation, whose solutions are linear combination of Bessel
($J_{\nu}$) and Neumann ($Y_{\nu})$ functions of order
$\nu=\sqrt{1+(M_{5}R)^{2}}$. Since we are interested only in regular solutions
in the bulk, we will consider just the Bessel solution

$f(z)\sim J_{\nu}(mz),$ (7)

where $m$ is an irrelevant constant in the conformal field theory defined on
the Minkowski boundary of the $AdS_{5}$ space, due to scale invariance, but
has a fundamental role for the HW model with the introduction of the cutoff in
the AdS space (see Eq. (15) below).

On the other hand, the mesons with spin $S$ with $S=1,2,3,...$, that we are
interested in are described by operators in the boundary Minkowski space as
Gubser:2002tv  111It is also possible to alternatively write the mesons with
spin $S$ as deTeramond:2005su  ${\cal
O}^{\mu\,\mu_{1}...\mu_{S}}_{2+S}=\;\bar{\psi}\gamma^{\mu}D_{\\{\mu_{1}\dots}D_{\mu_{S}\\}}\psi\;;\qquad(S=1,2,3,...)\,,$
(8) such that  $\Delta=2+S.$ (9) However, in this case, the results for the
AHW and LAHW are not so good as the ones discussed in Sections III and IV,
with $\Delta=3+S$.

${\cal
O}^{\mu_{1}...\mu_{S}}_{3+S}=\;\bar{\psi}D_{\\{\mu_{1}\dots}D_{\mu_{S}\\}}\psi\;;\qquad(S=1,2,3,...)\,,$
(10)

where $\psi$ and $\bar{\psi}$ represent quark and antiquark fields while
$D_{\mu_{i}}$ with $i=1,\dots,S$ are symmetrized covariant derivatives. Then,
the canonical dimensions of these operators are given by

$\Delta=\Delta_{\rm can.}=3+S.$ (11)

The link between the operators $\cal O$ and string excitations at $AdS_{5}$
can be constructed from its conformal dimension. From the AdS/CFT
correspondence, the vector field in $AdS_{5}$ that we are describing is a
$1$-form, so its holographic mass $M_{5}$ is related to the conformal
dimension $\Delta$ of boundary operators $\cal O$, is given by

$(M_{5}R)^{2}=(\Delta-1)(\Delta-3).$ (12)

Using this relation in the index $\nu=\sqrt{1+(M_{5}R)^{2}}$ of the Bessel
function, one sees that

$\nu=\Delta-2.$ (13)

Furthermore, combining Eqs. (13) and (11), one finds that the Bessel functions
describing the mesons represented by the operators $\cal O$, Eq. (10), have
order

$\nu=S+1.$ (14)

The HW model consists in the holographic relation between fields in AdS space
with an IR cutoff that breaks the conformal symmetry of the boundary theory.
That means a hard cutoff in the holographic $z$ coordinate, so we are working
in an AdS slice, given by $0\leq z\leq z_{\rm max}$, where

$z_{\rm max}=\frac{1}{\Lambda_{\rm QCD}},$ (15)

with $\Lambda_{\rm QCD}$ a convenient energy scale related to be fixed later.
Then the solution of the vector bulk field, Eq. (7), assumes the form

$f(z)\sim J_{\nu}(m_{\nu,k}z),$ (16)

where $m_{\nu,k}$ is identified with the mass of the boundary operator $\cal
O$ that lives in the four-dimensional Minkowski space, describing mesons. Note
that $k=1,2,3...$ denote radial excitations, where $k=1$ is the corresponding
ground state. In this work we are interested in angular momenta excitations,
so we take $k=1$ for all the objects from now on.

Imposing Dirichlet boundary conditions (for $k=1$) in Eq. (16) at $z=z_{\rm
max}$, one obtains

$J_{\nu}\bigg{(}\frac{m_{\nu,1}}{\Lambda_{\rm QCD}}\bigg{)}=0.$ (17)

Defining the first zero $\chi_{\nu,1}$ of the Bessel function of order $\nu$,
such that $J_{\nu}(\chi_{\nu,1})=0$, and making use of the relation Eq. (14),
we can write the mass of the mesonic state with spin $S$ as

$m_{\nu,1}=\chi_{\nu,1}\Lambda_{\rm QCD};\qquad\qquad(\nu=S+1).$ (18)

In particular, from the state with $S=1$ we fix

$\Lambda_{\rm QCD}=\frac{m_{2,1}}{\chi_{2,1}},$ (19)

taking $m_{2,1}$ as an input from PDG for the states $\rho(770)$,
$\omega(782)$, $b_{1}(1235)$, and $h_{1}(1170)$. The corresponding values of
$\Lambda_{\rm QCD}$ found here for each mesonic family are shown in Table 2.
The meson masses obtained from the HW model, using Eq. (18), are shown in
Table 1, together with the SW model predictions with the relative deviations
with respect to PDG data PDG:2022pth . In Table 2, we also show the equations
for the Regge trajectories found from PDG data for the same mesonic families.

The process described above is known as the holographic HW model and was used
in the past to obtain hadronic spectra among other properties. It is a well-
known fact that this model implies non-linear Regge trajectories.

In the next section we develop an improvement of the HW model, by considering
anomalous dimensions, in order to obtain better masses compared with the PDG
ones and better Regge trajectories for these meson families.

Table 1:  Masses in MeV for light unflavored mesons classified under even or odd $\mathcal{CP}$ and with isospin $I=0,1$, and spins $S\geq 1$ from PDG PDG:2022pth , from the HW Boschi-Filho:2002xih ; Boschi-Filho:2002wdj ; deTeramond:2005su ; Erlich:2005qh ; Boschi-Filho:2005xct  and SW Karch:2006pv ; Colangelo:2008us  models, and the percentual deviations $\delta_{\rm HW}$ and $\delta_{\rm SW}$ of the masses from these two models compared with PDG data. The HW masses are determined from Eq. (18), using the values of $\Lambda_{\rm QCD}$ for each family (same $\mathcal{CP}$ parity and Isospin) shown in Table 2, and the masses from the SW model come from the relation $m_{S}^{2}\sim S$ Karch:2006pv . The mass of the state with $J=S=1$ for each family is used as an input for the HW and SW models. $J^{PC}$ | ${CP}$ | $I$ | Meson | PDG | HW | $\delta_{\rm HW}$ | SW | $\delta_{\rm SW}$
---|---|---|---|---|---|---|---|---
$1^{--}$ | even | 1 | $\rho(770)$ | $775.26\pm 0.25$ | $775.26$ | $0\%$ | $775.26$ | $0\%$
$2^{++}$ | even | 1 | $a_{2}(1320)$ | $1316.9\pm 0.9$ | $963.13$ | $25.9\%$ | $1096.38$ | $16.7\%$
$3^{--}$ | even | 1 | $\rho_{3}(1690)$ | $1688.8\pm 2.1$ | $1145.52$ | $32.2\%$ | $1342.79$ | $20.5\%$
$4^{++}$ | even | 1 | $a_{4}(1970)$ | $1967\pm 16$ | $1324.12$ | $32.7\%$ | $1550.52$ | $21.2\%$
$5^{--}$ | even | 1 | $\rho_{5}(2350)$ | $2330\pm 35$ | $1499.93$ | $35.6\%$ | $1733.53$ | $25.6\%$
$6^{++}$ | even | 1 | $a_{6}(2450)$ | $2450\pm 130$ | $1673.57$ | $31.7\%$ | $1898.99$ | $22.5\%$
$1^{--}$ | even | 0 | $\omega(782)$ | $782.66\pm 0.13$ | $782.66$ | $0\%$ | $782.66$ | $0\%$
$2^{++}$ | even | 0 | $f_{2}(1270)$ | $1275.5\pm 0.8$ | $972.31$ | $23.8\%$ | $1106.85$ | $13.2\%$
$3^{--}$ | even | 0 | $\omega_{3}(1670)$ | $1667\pm 4$ | $1156.44$ | $30.6\%$ | $1355.61$ | $18.7\%$
$4^{++}$ | even | 0 | $f_{4}(2050)$ | $2018\pm 11$ | $1336.74$ | $33.8\%$ | $1565.32$ | $22.4\%$
$5^{--}$ | even | 0 | $\omega_{5}(2250)$ | $2250\pm 70$ | $1514.23$ | $32.7\%$ | $1750.08$ | $22.2\%$
$6^{++}$ | even | 0 | $f_{6}(2510)$ | $2465\pm 50$ | $1689.52$ | $31.5\%$ | $1917.12$ | $22.2\%$
$1^{+-}$ | odd | 1 | $b_{1}(1235)$ | $1229.5\pm 3.2$ | $1229.5$ | $0\%$ | $1229.5$ | $0\%$
$2^{-+}$ | odd | 1 | $\pi_{2}(1670)$ | $1670.6\pm 2.05$ | $1527.41$ | $8.57\%$ | $1738.78$ | $4.08\%$
$3^{+-}$ | odd | 1 | $b_{3}(2030)$ | $2032\pm 12$ | $1816.70$ | $10.6\%$ | $2129.56$ | $4.80\%$
$4^{-+}$ | odd | 1 | $\pi_{4}(2250)$ | $2250\pm 15$ | $2099.95$ | $6.67\%$ | $2459.00$ | $9.29\%$
$1^{+-}$ | odd | 0 | $h_{1}(1170)$ | $1166\pm 6$ | $1166$ | $0\%$ | $1166$ | $0\%$
$2^{-+}$ | odd | 0 | $\eta_{2}(1645)$ | $1617\pm 5$ | $1352$ | $16.4\%$ | $1648.97$ | $1.98\%$
$3^{+-}$ | odd | 0 | $h_{3}(2025)$ | $2025\pm 20$ | $1709$ | $15.6\%$ | $2019.57$ | $0.27\%$
$4^{-+}$ | odd | 0 | $\eta_{4}(2330)$ | $2328\pm 38$ | $1882$ | $19.2\%$ | $2332.00$ | $0.17\%$
Table 2:  The Regge trajectory in GeV units for each light unflavored meson family (same $\mathcal{CP}$ parity and same isospin $I$) from the best fit of PDG data shown in Table 1. We also show the mass of the state with $J=S=1$ for each family, and the corresponding $\Lambda_{\rm QCD}$ value, both in MeV, determined from Eq. (19). $J^{PC}$ | ${CP}$ | $I$ | Meson | PDG (MeV) | PDG Regge Trajectory (GeV) |  $\Lambda_{\rm QCD}$ (MeV)
---|---|---|---|---|---|---
$1^{--}$ | even | 1 | $\rho(770)$ | $775.26\pm 0.25$ | $J=(0.88\pm 0.04)M^{2}+0.47\pm 0.15.$ | $150.96$
$1^{--}$ | even | 0 | $\omega(782)$ | $782.66\pm 0.13$ | $J=(0.90\pm 0.02)M^{2}+0.47\pm 0.07$ | 152.24
$1^{+-}$ | odd | 1 | $b_{1}(1235)$ | $1229.5\pm 3.2$ | $J=(0.83\pm 0.04)M^{2}-0.30\pm 0.16$ | 239.41
$1^{+-}$ | odd | 0 | $h_{1}(1170)$ | $1166\pm 6$ | $J=(0.73\pm 0.02)M^{2}+0.03\pm 0.06$ | 227.04

##  III Anomalous HW model

In an interacting theory, the dimension of an operator is no longer canonical,
thanks to renormalization that implies in anomalous contributions
($\Delta_{\rm Anom}$). So, the total dimension of an operator is given by its
canonical plus the anomalous dimensions:

$\Delta_{\rm Total}=\Delta_{\rm can.}+\Delta_{\rm Anom}\,.$ (20)

From a well known semi-classical gauge/string duality analysis Gubser:2002tv ,
it is found that the anomalous dimension increases logarithmically with the
spín $S$ of the boundary operators as

$\Delta_{\rm
Anom}=\frac{\sqrt{\lambda}}{\pi}\ln{\bigg{(}\frac{S}{\sqrt{\lambda}}\bigg{)}}+\mathcal{O}(S^{0}),$
(21)

where $\lambda$ is ’t Hooft coupling. Inspired by this result, we consider an
anomalous dimension given by

$\Delta_{\rm Log}=a\ln(S),$ (22)

where $a$ is a constant that will be determined by best fit for each meson
family.

Then, the anomalous HW (AHW) model consists in considering bulk fields in AdS
slice, Eq. (15), with total dimension given by Eq. (20) such that the mass
associated with boundary operators, now takes the form

$m_{\nu,1}=\chi_{\nu,1}\Lambda_{\rm QCD};\qquad\qquad(\nu=S+1+\Delta_{\rm
Log}),$ (23)

where we used the fact that $\Delta_{\rm can.}=3+S$ for these mesonic states
and that the order of the corresponding Bessel function is $\nu=\Delta_{\rm
Total}-2$. The value of $\Lambda_{\rm QCD}$ is determined for each meson
familiy using as inputs the masses of the states with $S=1$ from PDG, namely
$\rho(770)$, $\omega(782)$, $b_{1}(1235)$, and $h_{1}(1170)$. These values of
$\Lambda_{\rm QCD}$ coincide with the ones used in the usual HW model, listed
in Table 2, once Eq. (22) implies a vanishing anomalous dimension for $S=1$.

In order to obtain the best numerical fit, let us define the percentual
deviation:

$q_{i}=\frac{|m_{i}-M_{i}|}{M_{i}},$ (24)

where $m_{i}$ is the mass of a determined particle obtained by our model, from
Eq. (23), and $M_{i}$ its mass obtained from PDG, listed in Table 1. Further,
we define the function $Q$ as

$Q=\sqrt{\sum_{i}q_{i}^{2}}.$ (25)

So, the best fit method adopted here corresponds to minimize the function $Q$
for each meson family in the AHW model.

We will also consider an uncertainty for the parameter $a$, defined in Eq.
(22), which implies in uncertainties for the masses ${m_{i}}$, obtained by
error propagation, accordingly to the formula:

$\displaystyle\delta{m_{i}}=\sqrt{\left(\frac{\chi_{\nu_{i},1}}{\chi_{2,1}}\right)^{2}\left(\delta
m_{0}\right)^{2}+\left(\frac{m_{0}}{\chi_{2,1}}\right)^{2}\left(\delta\chi_{\nu_{i},1}\right)^{2}},$
(26)

where $\delta m_{0}$ and $\delta\chi_{\nu_{i},1}$ are the uncertainties in the
mass of the state with $S=1$ and in the zero of Bessel function that is
related to the mass $m_{i}$. The uncertainty $\delta m_{0}$ of the state with
$S=1$ for each meson family in our model will be the arithmetic average of the
percentual uncertainties of PDG masses $M_{i}$ for that given family. Defined
our criterion for best fit and how we compute the uncertainties
$\delta{m_{i}}$ for each mass $m_{i}$, we are able to tackle with the AHW
model for light unflavored mesons.

###  III.1 AHW model for mesons with even ${\cal CP}$, $I=1$, and $S\geq 1$

We start applying the AHW model for mesons with even ${\cal CP}$, isospin
$I=1$, and spins $S\geq 1$, taking the mass of the $\rho(770)$ state from PDG
as an input, which fixes $\Lambda_{\rm QCD}=150.96$ MeV. We minimize the
parameter $Q$, given by Eq. (25), with respect to PDG masses from Table 1,
finding the logarithm anomalous dimensions, Eq. (22), to be

$\Delta_{\rm Log}(S)=(2.77\pm 0.12)\ln(S).$ (27)

The masses with its uncertainties obtained within this model, from Eq. (23),
with these anomalous dimensions, are presented in Table 3, together with
percentual deviations in comparison with PDG data. The value of $\Delta_{\rm
Log}(S)$ of the above equation for each spin $S$ is also shown in the last
column of this Table. The Regge trajectory obtained by doing linear regression
with the obtained masses from this AHW Log model is given by

$J=(0.88\pm 0.01)M^{2}+0.50\pm 0.03,$ (28)

with $\chi^{2}/ndf=4.32/4=1.08$. This trajectory is presented in the upper
left panel of Fig. 1 as the blue line, and can be compared with the Regge
trajectory of this family from PDG data, shown in Table 2. The red dots are
the values in the $J\times M^{2}$ plane of the mesons $\rho(770)$,
$a_{2}(1320)$, $\rho_{3}(1690)$, $a_{4}(1970)$, $\rho_{5}(2350)$, and
$a_{6}(2450)$ in this AHW Log model, with their uncertainties.

###  III.2 AHW model for mesons with even ${\cal CP}$, $I=0$, and $S\geq 1$

Here, we discuss the case of mesons with even ${\cal CP}$, isospin $I=0$, and
spins $S\geq 1$. Minimizing $Q$, given by Eq. (25), we obtain $a=2.61\pm 0.09$
for the anomalous dimension, Eq. (22), which reads

$\Delta_{\rm log}(S)=(2.61\pm 0.09)\ln(S).$ (29)

Using the mass of the state $\omega(782)$ from PDG as an input, which implies
$\Lambda_{\rm QCD}=152.24$ MeV for this case, we calculate the tower of masses
with the uncertainties in this AHW Log model and show these results in Table
3. The percentual deviations of these masses in comparison with PDG are also
presented, as well as the value of the anomalous dimension $\Delta_{\rm Log}$,
for each meson state. The Regge trajectory obtained by doing linear regression
with the masses obtained by the AHW Log model is given by

$J=(0.90\pm 0.01)M^{2}+0.48\pm 0.03,$ (30)

with $\chi^{2}/ndf=8.88/4=2.22$. This trajectory is presented in the upper
right panel of Fig. 1 as the blue line. The red dots are the values in the
$J\times M^{2}$ plane of the mesons with even ${\cal CP}$, isospin $I=0$, and
spins $S\geq 1$, in this AHW Log model, with their uncertainties. This
trajectory can be compared with the one obtained from PDG for the
$\omega(782)$ family, presented in Table 2.

###  III.3 AHW model for mesons with odd ${\cal CP}$, $I=1$, and $S\geq 1$

Now we present our results for mesons with odd ${\cal CP}$, $I=1$, and $S\geq
1$. The minimization of $Q$, Eq. (25), gives $a=0.56\pm 0.03$ for the the
anomalous dimension, Eq. (22), which becomes

$\Delta_{\rm Log}(S)=(0.56\pm 0.03)\ln(S).$ (31)

The masses obtained in this AHW Log model are shown in Table 3, with their
uncertainties, where we used the mass of the state $b_{1}(1235)$ as an input,
so that $\Lambda_{\rm QCD}=239.41$ MeV. The Regge trajectory obtained by
linear regression with these masses is given by

$J=(0.78\pm 0.02)M^{2}-0.13\pm 0.08,$ (32)

with $\chi^{2}$/ndf$=27/2=13.5$. This trajectory is represented in the lower
left panel of Fig. 1 as the blue line and can be compared with the one
obtained from PDG data, shown in Table 2. The red dots are the values obtained
from this AHW Log model, with their corresponding uncertainties.

Table 3:  Masses in MeV for mesons classified under even or odd $\mathcal{CP}$ with isospin $I=0,1$ and spins $S\geq 1$ from the AHW Log model, determined from Eq. (23), and the percentual deviations $\delta_{\rm Log}$ of the masses from this model compared with PDG data. For each family the mass of the state with $J=S=1$ is taken as an input from PDG. In the last column we show the anomalous dimension $\Delta_{\rm Log}$ for each state determined by Eq. (22). $J^{PC}$ | $\mathcal{CP}$ | $I$ | Meson | AHW Log | $\delta_{\rm Log}$ | $\Delta_{\rm Log}$
---|---|---|---|---|---|---
$1^{--}$ | even | 1 | $\rho(770)$ | $775.26\pm 10.08$ | $0\%$ | $0$
$2^{++}$ | even | 1 | $a_{2}(1320)$ | $1309.95\pm 22$ | $0.53\%$ | $1.92$
$3^{--}$ | even | 1 | $\rho_{3}(1690)$ | $1681.02\pm 31$ | $0.46\%$ | $3.04$
$4^{++}$ | even | 1 | $a_{4}(1970)$ | $1988.75\pm 37$ | $1.11\%$ | $3.84$
$5^{--}$ | even | 1 | $\rho_{5}(2350)$ | $2262.37\pm 43$ | $2.90\%$ | $4.46$
$6^{++}$ | even | 1 | $a_{6}(2450)$ | $2514.58\pm 47$ | $2.64\%$ | $4.96$
$1^{--}$ | even | 0 | $\omega(782)$ | $782.66\pm 7.83$ | $0\%$ | $0$
$2^{++}$ | even | 0 | $f_{2}(1270)$ | $1302.58\pm 17$ | $2.12\%$ | $1.81$
$3^{--}$ | even | 0 | $\omega_{3}(1670)$ | $1666.4\pm 23$ | $0.04\%$ | $2.87$
$4^{++}$ | even | 0 | $f_{4}(2050)$ | $1969.63\pm 28$ | $2.40\%$ | $3.62$
$5^{--}$ | even | 0 | $\omega_{5}(2250)$ | $2240.19\pm 32$ | $0.44\%$ | $4.20$
$6^{++}$ | even | 0 | $f_{6}(2510)$ | $2490.23\pm 36$ | $1.02\%$ | $4.68$
$1^{+-}$ | odd | 1 | $b_{1}(1235)$ | $1229.5\pm 5.0$ | $0\%$ | $0$
$2^{-+}$ | odd | 1 | $\pi_{2}(1670)$ | $1640.57\pm 8.35$ | $1.80\%$ | $0.39$
$3^{+-}$ | odd | 1 | $b_{3}(2030)$ | $1991.55\pm 11$ | $1.99\%$ | $0.62$
$4^{-+}$ | odd | 1 | $\pi_{4}(2250)$ | $2316.73\pm 13$ | $2.97\%$ | $0.78$
$1^{+-}$ | odd | 0 | $h_{1}(1170)$ | $1166\pm 10$ | $0\%$ | $0$
$2^{-+}$ | odd | 0 | $\eta_{2}(1645)$ | $1629.93\pm 18$ | $0.80\%$ | $0.66$
$3^{+-}$ | odd | 0 | $h_{3}(2025)$ | $2003.12\pm 25$ | $1.08\%$ | $1.04$
$4^{-+}$ | odd | 0 | $\eta_{4}(2330)$ | $2339.01\pm 30$ | $0.47\%$ | $1.32$

Figure 1: Regge trajectories (blue lines) for mesons, with masses $m$
expressed in GeV, from the AHW Log model. Red dots are the values obtained
from Eq. (23) in the AHW Log model with uncertainties shown in Table 3. The
blue lines are obtained by linear regression of these points. Upper left
panel: mesons with even ${\cal CP}$, $I=1$ ($\rho(770)$ family); Upper right
panel: mesons with even ${\cal CP}$, $I=0$ ($\omega(782)$ family); Lower left
panel: mesons with odd ${\cal CP}$, $I=1$ ($b_{1}(1235)$ family); Lower right
panel: mesons with odd ${\cal CP}$, $I=0$ ($h_{1}(1170)$ family).

###  III.4 AHW model for mesons with odd ${\cal CP}$, $I=0$, and $S\geq 1$

For mesons with odd ${\cal CP}$, $I=0$, and $S\geq 1$, we minimize the
function $Q$, Eq. (25), finding $a=0.95\pm 0.06$ for the anomalous dimension,
Eq. (22), obtaining

$\Delta_{\rm Log}(S)=(0.95\pm 0.06)\ln(S).$ (33)

The masses obtained in this AHW Log model are presented, with uncertainties,
in Table 3, where we used the mass value of the state $h_{1}(1170)$ from PDG
as an input, such that $\Lambda_{\rm QCD}=227.04$ MeV. The Regge trajectory
obtained by doing linear regression with these masses is given by

$J=(0.73\pm 0.01)M^{2}+0.04\pm 0.05,$ (34)

with $\chi^{2}$/ndf$=3.52/2=1.76.$ This trajectory can be compared with the
one obtained from PDG data shown in Table 2 and is presented in the lower
right panel of Fig. 1 as the blue line. The red dots are the values of the
masses obtained within this AHW Log model with their uncertainties.

##  IV Linear AHW for mesons

Recently, we have shown Costa-Silva:2023vuu  that one can obtain
asymptotically linear Regge trajectories for glueballs with even spins $S$
from an AHW model, when considering a total dimension $\Delta_{\rm Total}$
proportional to $\sqrt{S}$. In order to find asymptotically linear Regge
trajectories for light unflavored mesons with spins $S\geq 1$, we apply a
similar procedure here with a small modification in the power of the spins $S$
as

$\Delta_{\rm Lin}=a_{\rm Lin}(S^{\frac{1}{2}+\epsilon}-1)+4,$ (35)

where in the ideal case, $\epsilon\rightarrow 0$. Note that when $S=1$, which
are the lowest spin state for each mesonic family considered here, we recover
the canonical dimension $\Delta=4$, compatible with Eq. (11).

Here, the parameter $a_{\rm Lin}$ is obtained exactly as in the AHW Log model,
by minimizing $Q$ in Eq. (25). The parameter $\epsilon$ that represents a
little deviation of the ideal case $\epsilon=0$ is chosen to obtain the best
asymptotic linear Regge trajectory for each meson family. The value of
$\epsilon$ is the one that minimizes the total deviation of the masses from
the straight line representing the Regge trajectory.

The masses that follows from this linear AHW model are determined by

$m_{\nu,1}=\chi_{\nu,1}\Lambda_{\rm QCD};\qquad\qquad(\nu=\Delta_{\rm
Lin}-2),$ (36)

where $\Delta_{\rm Lin}$ is defined in Eq. (35). Since $\Delta_{\rm Lin}=4$
for $S=1$, the values of $\Lambda_{\rm QCD}$ here in this model, for each
meson family, are the same as those in the AHW Log model, as well as in the
original HW presented in Table 2.

As in the case of logarithmic anomalous dimension, we will analyze the
families of light unflavored mesons with even or odd $\cal CP$, isospins
$I=0,1$, and spins $S\geq 1$. The Regge trajectories that we obtain within
this model are asymptotically linear for each meson family with the
appropriate choice of parameters.

###  IV.1 Linear AHW for mesons with even ${\cal CP}$, $I=1$, and $S\geq 1$

For mesons with even ${\cal CP}$, isospin $I=1$, and spin $S\geq 1$, the
parameters in Eq. (35) that minimizes $Q$ are $a_{\rm Lin}=6.87\pm 0.16$ and
$\epsilon=0$, so the total dimension reads

$\Delta_{\rm Lin}=(6.87\pm 0.16)\sqrt{S}-2.87\pm 0.16.$ (37)

The masses obtained here, using Eq. (36), are presented in Table 4, with
respectives uncertainties, using the mass of the state $\rho(770)$ as an input
from PDG. The Regge trajectory for these mesons is given by

$J=(0.87\pm 0.01)M^{2}+0.51\pm 0.02,$ (38)

with $\chi^{2}$/ndf$=8.68/4=2.17$ and is presented in the upper left panel of
Fig. 2 as the blue line. The red dots represent these masses with their
uncertainties.

In order to show that the linear AHW model indeed present asymptotic linear
Regge trajectories we also plot $J\times M^{2}$ for these mesons with even
$\mathcal{CP}$, isospin $I=1$, and $S\geq 1$, for the first 50 states. The
trajectory in this case becomes

$J=(0.8479\pm 0.0002)M^{2}+0.60\pm 0.01,$ (39)

with $\chi^{2}$/ndf$=72.96/48=1.52.$ This trajectory is presented in the upper
left panel of Fig. 3. Both trajectories can be compared with the one obtained
from PDG data shown in Table 2 and are in good agreement with it.

Table 4:  Masses in MeV for mesons classified under even or odd $\mathcal{CP}$ with isospin $I=0,1$ and spins $S\geq 1$ from the Linear AHW model, determined from Eq. (36), and the percentual deviations $\delta_{\rm Lin}$ of the masses from this model compared with PDG data. For each family the mass of the state with $J=S=1$ is taken as an input. In the last column we show the total dimension $\Delta_{\rm Lin}$ for each state determined by Eq. (35). $J^{PC}$ | $\mathcal{CP}$ | I | Meson | Linear AHW | $\delta_{\rm LAHW}$ | $\Delta_{\rm Lin}$
---|---|---|---|---|---|---
$1^{--}$ | even | 1 | $\rho(770)$ | $775.26\pm 10.08$ | $0\%$ | $0$
$2^{++}$ | even | 1 | $a_{2}(1320)$ | $1296.75\pm 21$ | $1.53\%$ | $1.85$
$3^{--}$ | even | 1 | $\rho_{3}(1690)$ | $1678.61\pm 30$ | $0.60\%$ | $3.03$
$4^{++}$ | even | 1 | $a_{4}(1970)$ | $1993.84\pm 38$ | $1.36\%$ | $3.87$
$5^{--}$ | even | 1 | $\rho_{5}(2350)$ | $2268.03\pm 44$ | $2.66\%$ | $4.49$
$6^{++}$ | even | 1 | $a_{6}(2450)$ | $2513.71\pm 51$ | $2.60\%$ | $4.96$
$1^{--}$ | even | 0 | $\omega(782)$ | $782.66\pm 7.83$ | $0\%$ | $0$
$2^{++}$ | even | 0 | $f_{2}(1270)$ | $1294.28\pm 16$ | $1.47\%$ | $1.76$
$3^{--}$ | even | 0 | $\omega_{3}(1670)$ | $1669.09\pm 22$ | $0.13\%$ | $2.88$
$4^{++}$ | even | 0 | $f_{4}(2050)$ | $1978.53\pm 28$ | $1.96\%$ | $3.67$
$5^{--}$ | even | 0 | $\omega_{5}(2250)$ | $2247.67\pm 33$ | $0.10\%$ | $4.24$
$6^{++}$ | even | 0 | $f_{6}(2510)$ | $2488.81\pm 38$ | $0.97\%$ | $4.67$
$1^{+-}$ | odd | 1 | $b_{1}(1235)$ | $1229.5\pm 5.0$ | $0\%$ | $0$
$2^{-+}$ | odd | 1 | $\pi_{2}(1670)$ | $1667.30\pm 7.89$ | $0.20\%$ | $0.48$
$3^{+-}$ | odd | 1 | $b_{3}(2030)$ | $2000.60\pm 10.75$ | $1.55\%$ | $0.65$
$4^{-+}$ | odd | 1 | $\pi_{4}(2250)$ | $2281.03\pm 13.31$ | $1.38\%$ | $0.65$
$1^{+-}$ | odd | 0 | $h_{1}(1170)$ | $1166\pm 10$ | $0\%$ | $0$
$2^{-+}$ | odd | 0 | $\eta_{2}(1645)$ | $1646.87\pm 17$ | $1.85\%$ | $0.72$
$3^{+-}$ | odd | 0 | $h_{3}(2025)$ | $2009.96\pm 24$ | $0.74\%$ | $1.07$
$4^{-+}$ | odd | 0 | $\eta_{4}(2330)$ | $2314.19\pm 31$ | $0.59\%$ | $1.22$

Figure 2: Regge trajectories (blue lines) for mesons, with masses $m$
expressed in GeV, from the Linear AHW model, with total dimension given by Eq.
(35). Red dots are the values obtained from Eq. (36) with uncertainties and
the blue lines are obtained by linear regression of these points, for each
meson family. Upper left panel: mesons with even ${\cal CP}$, $I=1$
($\rho(770)$ family); Upper right panel: mesons with even ${\cal CP}$, $I=0$
($\omega(782)$ family); Lower left panel: mesons with odd ${\cal CP}$, $I=1$
($b_{1}(1235)$ family); Lower right panel: mesons with odd ${\cal CP}$, $I=0$
($h_{1}(1170)$ family).

Figure 3: Regge trajectories for mesons, with masses $m$ expressed in GeV,
from the linear AHW model, with total dimension given by Eq. (35). Red dots
are the values obtained from Eq. (36) with uncertainties and the blue lines
are the Regge trajectories, obtained by linear regression of these 50 points,
for each meson family. Upper left panel: mesons with even ${\cal CP}$, $I=1$
($\rho(770)$ family); Upper right panel: mesons with even ${\cal CP}$, $I=0$
($\omega(782)$ family); Lower left panel: mesons with odd ${\cal CP}$, $I=1$
($b_{1}(1235)$ family); Lower right panel: mesons with odd ${\cal CP}$, $I=0$
($h_{1}(1170)$ family).

###  IV.2 Linear AHW for mesons with even ${\cal CP}$, $I=0$, and $S\geq 1$

For mesons in the linear AHW model with even ${\cal CP}$, $I=1$, and $S\geq
1$, the parameters in Eq. (35) that minimizes $Q$ are $a_{\rm Lin}=6.67\pm
0.13$ and $\epsilon=0$. So, the total dimension reads

$\Delta_{\rm Lin}=(6.67\pm 0.13)\sqrt{S}-2.67\pm 0.13.$ (40)

The masses obtained by this model, using Eq. (36), are presented with
respectives uncertainties in Table 4, taking the mass of the state
$\omega(782)$ as an input from PDG. Furthermore, we also present in this table
the total dimension and the percentual deviations from PDG data for each
mesonic state. The Regge trajectory obtained by linear regression of these
masses (first six states) is presented in the upper right panel of Fig. 2 and
is given by

$J=(0.89\pm 0.01)M^{2}+0.49\pm 0.02,$ (41)

with $\chi^{2}$/ndf$=10.12/4=2.53.$ The upper right panel of Fig. 3 presents
the Regge trajectory for the first 50 sates with even ${\cal CP}$, $I=0$, and
$S\geq 1$, and it is given by

$J=(0.8766\pm 0.0003)M^{2}+0.52\pm 0.01,$ (42)

with $\chi^{2}$/ndf$=32.16/48=0.67.$ Both equations can be compared with the
one obtained from PDG data, shown in Table 2, and are in good agreement with
it.

###  IV.3 Linear AHW for mesons with odd ${\cal CP}$, $I=1$, and $S\geq 1$

For mesons with odd ${\cal CP}$, $I=1$, and $S\geq 1$ ($b_{1}(1235)$ family),
the parameters in Eq. (35) that minimizes $Q$ are $a_{\rm Lin}=3.19\pm 0.03$
and $\epsilon=0.05$, so the total dimension becomes

$\Delta_{\rm Lin}=(3.19\pm 0.03)S^{0.55}+0.81\pm 0.03.$ (43)

The masses obtained in this model, using Eq. (36), are presented in Table 4.
The Regge trajectory obtained for the first four points is presented in Fig. 2
and is given by

$J=(0.81\pm 0.01)M^{2}-0.24\pm 0.03,$ (44)

with $\chi^{2}$/ndf$=2.86/2=1.43.$

For 50 points the trajectory is in Fig. 3 and is given by

$J=(0.8549\pm 0.0003)M^{2}-0.42\pm 0.01,$ (45)

with $\chi^{2}$/ndf$=163/48=3.39.$ Both equations, considering the
uncertainties, are in agreement with the Regge trajectory obtained from PDG
data, shown in Table 2.

###  IV.4 Linear AHW for mesons with odd ${\cal CP}$, $I=0$, and $S\geq 1$

The parameters in Eq. (35) that minimizes $Q$ for meson with mesons with odd
${\cal CP}$, $I=0$, and $S\geq 1$ ($h_{1}(1170)$ family) are $a_{\rm
Lin}=3.79\pm 0.08$ and $\epsilon=0.04$, so the total dimension is

$\Delta_{\rm Lin}=(3.79\pm 0.08)S^{0.54}+0.21\pm 0.08.$ (46)

The masses obtained in this model, using Eq. (36), are presented in Table 4.
The Regge trajectory for the first four points is presented in Fig. 2 and is

$J=(0.7509\pm 0.033)M^{2}-0.03\pm 0.01,$ (47)

with $\chi^{2}$/ndf$=0.22/2=0.11.$ The Fig. 3 presents the Regge trajectory
for the first 50 points, given by

$J=(0.7667\pm 0.0003)M^{2}-0.06\pm 0.01,$ (48)

with $\chi^{2}$/ndf$=2.20/48=0.046.$ Both trajectories are in good agreement
the one from PDG shown in Table 2.

##  V Conclusions

In this work, we proposed anomalous and linear HW models for light unflavored
mesons. We have shown that the masses and Regge trajectories produced by these
models are better than that obtained from the original HW and SW models.

In particular, the linear AHW model produce asymptotically linear Regge
trajectories, circumventing the well-kown problem of non-linearity of the
original HW model.

Finally, we expect to succesfully extend the analysis presented here to other
particles as heavy and heavy-light mesons, as well as baryons. This is
presently under investigation.

## Acknowledgments

R.A.C.-S. is supported by Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq) and Coordenação de Aperfeiçoamento de Pessoal de Nível
Superior (CAPES) under finance code 001. H.B.-F. is partially supported by
Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) under
grant $\\#$ 310346/2023-1.

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