# Running anomalous dimensions in holographic QCD:
from the proton to the sexaquark

Nick Evans  STAG Research Centre & Physics and Astronomy, University of
Southampton, Southampton, SO17 1BJ, UK  Matthew Ward  STAG Research Centre &
Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UK

###### Abstract

In holographic models of QCD, the running of the anomalous dimension of the
quark bilinear operator leads to chiral symmetry breaking when $\gamma=1$ and
the Breitenlohner-Freedman bound is violated. In that case, the running drives
the sigma meson mass tachyonic inducing the chiral symmetry breaking. Here we
include the running anomalous dimension in the computation of the spectrum of
bound states associated with other operators made of light quarks, such as the
nucleon and exotic sexaquark states. We show that including the one loop gauge
theory running can have substantial effects on the predictions. For example,
the nucleon mass to rho mass ratio is improved and lies much closer to the
observed value. A similar result is obtained for the $\Lambda$ and $\Xi$
baryon when strange quarks are included. A $uuddss$ sexaquark state with a low
enough mass to make it stable can be achieved, but this depends on the input
assumptions about the running dimension.

##  I Introduction

Holographic models Erlich:2005qh ; DaRold:2005mxj  of QCD are remarkably
simple, consisting of free scalars, vectors or fermions in a bulk Anti de-
Sitter (AdS) space. The fields’ bulk masses (in units of the AdS radius) are
set by the holographic dictionary Witten:1998qj , for example, the scalar mass
$m^{2}=\Delta(\Delta-4)$ where $\Delta$ is the scaling dimension of the dual
field theory operator. Simple AdS/QCD models, with the mass gap included via a
hard wall, are already successful in predicting the light mesonic spectrum at
the 15% level Erlich:2005qh ; DaRold:2005mxj . The pattern of baryon masses is
also well produced by treating them as spinors in AdS deTeramond:2005su  but
the nucleon to rho meson mass ratio typically comes out $\sim$50% too high.
These hadrons arise from the lowest energy normalisable modes in the bulk. The
justification for these models is provided by the Dirac-Born-Infeld (DBI)
action of probe branes in top-down models, such as the Sakai-Sugimoto model
Sakai:2004cn  or the D3/probe D7 system Karch:2002sh ; Kruczenski:2003be ;
Erdmenger:2007cm .

Top-down string constructions of holographic duals to chiral symmetry breaking
Babington:2003vm ; Kruczenski:2003uq  show that, upon deforming the AdS
background (to describe field theories beyond ${\cal N}=4$ super Yang-Mills),
the DBI action is altered in a way that provides a running anomalous dimension
to the quark operators it describes Alvares:2012kr . In bottom-up models of
QCD, the running of the $\bar{q}q$ state is usually imposed by hand, either
directly Alho:2013dka ; Erdmenger:2020flu  or via a constructed scalar
potential Jarvinen:2011qe . In the bulk, the dual field’s mass changes with
radial position, becoming tachyonic (i.e. violating the Breitenlohner-Freedman
(BF) bound Breitenlohner:1982jf ) in the infra-red. For a scalar, this
instability sets in when the anomalous dimension of the quark condensate
equals 1 or, equivalently, when the bulk mass falls to $-4$. As such, the
formation of a quark condensate offers a dynamical alternative to the hard
wall mechanism of the original models.

The vev of the $\bar{q}q$ operator in QCD implies that the $\sigma$ meson (the
scalar isospin singlet of the $SU(N)_{L}\times SU(N)_{R}$ chiral symmetry) has
become tachyonic as a result of strong interactions. Given the substantial
effect on the $\sigma$’s mass, one might imagine that including the running
dimension of operators is crucial when calculating the mass of further states
in the spectrum. However, this has scarcely been investigated beyond the
scalar sector in AdS/QCD. One reason is that vector and axial-vector states
are described by gauge fields with vanishing masses in the bulk, indicating
the absence of any such running.

That being said, such effects may be important for spin-½ baryons composed of
$u$, $d$ and $s$ quarks, such as the proton, $\Lambda$ and $\Xi$ baryons.
Indeed, mass predictions for these states have typically been overestimated in
AdS/QCD, which we demonstrate by calculating the nucleon mass within a simple
hard wall model. Following this initial estimate, we construct a basic
holographic model based on the D3/probe D7 system that incorporates the
running anomalous dimension for both the $\bar{q}q$ operator and the baryons.
We use the one loop results from QCD which, as we will show, replaces these
runnings with a product of the anomalous dimensions for the individual quark
fields that constitute the operators. In this model, we find that our estimate
for the nucleon mass is brought down by over a third, aligning both it and the
$\Lambda$ and $\Xi$ baryon masses much more closely with their measured
values.

Following these results, we turn our attention to a proposed, but so far
undetected, particle in the QCD spectrum, the sexaquark Farrar:2017eqq ;
Farrar:2022mih . Modelling the tightly-bound $uuddss$ six-quark state as a
scalar in the AdS bulk, we consider a range of possible runnings for the
operator’s anomalous dimension. One can obtain a wide range of answers
depending on the choice of running but, interestingly, opting for the one loop
result (extended into the non-perturbative regime) leads to a mass that falls
within the proposed range for it to remain stable against decay into two
$\Lambda$s. We discuss some other aspects of the sexaquark’s holographic
phenomenology also.

Figure 1: Diagrams at one loop order contributing to the anomalous dimension
of a gauge invariant scalar operator with $n$ quark legs.

##  II Anomalous Dimensions at One Loop

The crucial new ingredient we wish to explore in AdS/QCD models is the
anomalous dimension of multi-quark operators. Let us therefore review the
theory of anomalous dimensions in QCD at one loop level (covered in more
detail in Grozin:2005yg ).

The dimension $d$ for a generic operator ${\cal O}=\tilde{\cal O}\mu^{d}$,
where $\tilde{\cal O}$ is the dimensionless component of ${\cal O}$ at the
scale $\mu$, can be expressed as

$d={1\over{\cal O}}~{}\mu{d{\cal O}\over d\mu}$ (1)

Upon renormalisation by $Z_{\cal O}$, the anomalous dimension $\gamma$ is then
given by

$\gamma_{\cal O}=-{1\over Z_{\cal O}}~{}\mu{dZ_{\cal O}\over d\mu}$ (2)

where we have included a minus sign such that $d=d_{0}+\gamma$. Computing
$Z_{\cal O}$ for a colour singlet gauge invariant operator comprised of $n$
quark fields requires one to consider two sets of Feynman diagrams (fig.1).
Firstly, there is a factor $Z_{\psi}$ originating from the wave function
renormalisation of the $n$ external legs, which diverges in the
$\epsilon\rightarrow 0$ limit as per the dimensional regularization procedure

$Z_{\psi}=1-C_{2}(R)~{}\xi~{}{\alpha\over 4\pi}~{}{1\over\epsilon}$ (3)

where $C_{2}(R)$ is the quadratic Casimir element. We also define the square
of the gauge coupling $\alpha=g^{2}/4\pi$ and the gauge fixing parameter
$\xi$, which specifies the form of the gluon propagator

$D^{0}_{\mu\nu}(p^{2})={-i\over
p^{2}}\left(g_{\mu\nu}-(1-\xi){p^{\mu}p^{\mu}\over p^{2}}\right)$ (4)

Secondly, one must consider diagrams describing the exchange of gluons between
any two external lines as permitted by the colour symmetry of the scalar
operator. We have

$Z_{V}=1+(3+\xi)~{}{\alpha\over 4\pi}~{}{1\over\epsilon}$ (5)

which, combined with (3), defines the renormalisation factor for the $n$ quark
operator

$Z_{{\cal O}n}=\left(1+C_{n}(3+\xi)~{}{\alpha\over
4\pi}~{}{1\over\epsilon}\right)Z_{\psi}^{n/2}$ (6)

where $C_{n}$ is a combinatoric colour factor discernible from the permitted
diagrams. To ensure that our operator remains gauge invariant, we impose that
$Z_{{\cal O}n}$ is independent of $\xi$, thereby fixing $C_{n}=nC_{2}(R)/2$.
This gives

$Z_{{\cal O}n}=1+{3n\over 2}{C_{2}(R)}~{}{\alpha\over
4\pi}~{}{1\over\epsilon}$ (7)

In the fundamental representation with $N_{c}=3$ colours, we find that
$C_{2}(F)=(N_{c}^{2}-1)/2N_{c}=4/3$. Combining this with the result of (2), we
can derive a general expression for the anomalous dimension of an $n$ quark
vertex

$\gamma_{{\cal O}n}(\mu)=-n~{}{\alpha(\mu)\over\pi}$ (8)

As (8) shows, the anomalous dimension is directly proportional to the number
of quark legs. Beyond one loop, this factorization is expected to break down
and operators will garner their own distinct running. In the following
sections, we will use this one loop result in AdS/QCD models which we have
extended into the non-perturbative regime. Although results at two loop and
beyond will begin to distinguish operators with different colour wave
functions, one would need the full non-perturbative running to correctly
understand the resultant splittings.

##  III Hard Wall Model

The simplest AdS/QCD model DaRold:2005mxj ; Erlich:2005qh  involves the study
of bulk fields in $AdS_{5}$ space with metric

$ds^{2}=r^{2}dx_{\mu}dx^{\mu}+\frac{dr^{2}}{r^{2}}$ (9)

where $x^{\mu}\in\mathbb{R}^{1,3}$ and $r$ is the radial ordinate of the
curved AdS space. In the dual field theory, this radial direction corresponds
to the renormalisation group (RG) scale, allowing us to insert a mass gap into
QCD by placing a hard wall boundary at $r=1$.

A simple example of extracting dynamics from the hard wall model is to
consider a $SU(N_{f})$ vector field in $AdS_{5}$ space with the action

$S=\int d^{5}x\textrm{Tr}\sqrt{-g}F^{2}$ (10)

The equation of motion in the $A^{r}=0$ gauge where $A_{\perp}=A(r)e^{ikx}$
and $k^{2}=-M^{2}$ is

$\partial_{r}r^{3}\partial_{r}A+{M^{2}\over r}A=0$ (11)

For large $r$, the solutions take the form
$A=A_{0}+\bar{q}\gamma^{\mu}q/r^{2}$, with the two constants playing the role
of background flavour gauge fields and the vector current respectively.
Solutions that asymptote to $A=0$ pick out the rho meson states. In units
where the wall is at $r=1$, the $\rho$ meson ground state has bulk mass
$M^{2}=5.9$. Throughout this paper, we will use the computation of this mass
in a given model to set the QCD scale. For these calculations, we use
$M_{\rho}=770$ MeV.

###  III.1 Prediction of the Nucleon Mass

A spin-half baryon can be described by a bulk spinor field $\psi$ with mass
$m_{\psi}$ deTeramond:2005su ; Abt:2019tas . One can then square the linear
Dirac equation to find the Klein-Gordon equation for the two components
projected by $\gamma^{0}$; the linear equation linking the form of the
solutions for each. In $AdS_{5}$, the suitable Klein-Gordon equation is

$\left(\partial_{r}^{2}+{6\over r}\partial_{r}+{6-m_{\psi}^{2}+m_{\psi}\over
r^{2}}+{M_{B}^{2}\over r^{4}}\right)\psi=0$ (12)

where $m_{\textrm{eff}}^{2}=6-m_{\psi}^{2}+m_{\psi}$ defines an effective
mass. The holographic dictionary then allows one to express this mass as a
function of the scaling dimension associated with the boundary operator which,
for a fermion field in $AdS_{5}$, takes the form
$m_{\textrm{eff}}^{2}=\Delta(5-\Delta)$. Solving this equation for a three
quark state (i.e. $\Delta=9/2$), the bulk fermion mass is given by
$m_{\psi}=5/2$ and the solution to (12) takes the following form (as given in
Abt:2019tas )

$\psi={JM_{B}\over 4}{1\over r^{5-\Delta}}+{O\over r^{\Delta}}$ (13)

Assuming the solution approaches the hard wall smoothly and is normalised to
unity, i.e. the boundary conditions $\psi(1)=1$ and $\psi^{\prime}(1)=0$ are
imposed, we can tune the $M_{B}$ mass such that $\psi$ vanishes in the UV
(i.e. the source $J$ is set to zero). Relative to the rho mass, we find that
$M_{B}=2.2M_{\rho}\approx 1.7$ GeV.

This prediction is likely too large because we have neglected to include the
running dimension of the bound state operator. The dimension of each quark in
$\bar{q}q$ has fallen to one at the hard wall in order to trigger chiral
symmetry breaking. By (8), the dimension of a three quark operator should
therefore have fallen to $\Delta=3$ in the IR, i.e. $m_{\textrm{eff}}^{2}=6$.
If one just uses this lower bulk mass over the full range of $r$, then the
predicted baryon mass falls to $M_{B}=1.5$ GeV.

This naive result certainly suggests that the anomalous dimension of a quark
operator can play a role in reducing the predicted mass of its associated
bound state. In the following section, we seek a more accurate estimate by
generating the IR scale dynamically via a running $\gamma$.

##  IV Dynamic AdS/QCD Model

A more advanced AdS/QCD model would need to incorporate the running anomalous
dimension $\gamma$ of the quark anti-quark bilinear, which triggers a BF bound
violation and dynamically generates the mass gap, as well as the running
dimensions of all the operators that source our bound states. We will work in
the spirit of the Dynamic AdS/QCD model outlined in Alho:2013dka . The model,
analogous to some of the earliest hard wall models, attempts to make the
minimum number of changes to a basic AdS/QCD model that will exemplify the
physics under discussion without it being a first principle holographic
description of QCD. Mostly, we import insights from the D3/probe D7 system
Karch:2002sh ; Kruczenski:2003be ; Erdmenger:2007cm .

###  IV.1 Anomalous Dimension for the Quark Condensate

To describe the quark condensate dynamics, we allow ourselves to be guided by
the D3/probe D7 system, where the $\bar{q}q$ operator is dual to a scalar
field $L$ of dimension one (i.e. $L=r\chi$) in AdS space - the field $\chi$
has bulk mass $m^{2}=-3$ ($\Delta=3)$. By considering the action

$S=\int d^{4}xdr~{}\left(r^{3}(\partial_{r}L)^{2}+r\Delta m^{2}L^{2}\right)$
(14)

one can see that, upon rewriting the first term in terms of $\chi$ and
integrating by parts, we arrive at the canonical Lagrangian for the
dimensionless AdS field $\chi$ with $m^{2}=-3+\Delta m^{2}$. Since the BF
bound is $m^{2}\geq-4$ in the $AdS_{5}$ spacetime, it is clear that a scalar
mass deviation of $\Delta m^{2}<-1$ will trigger the violation we are looking
for.

The equation of motion associated with (14) is given by

$\partial_{r}\left[r^{3}\partial_{r}L\right]+r\Delta m^{2}L=0$ (15)

Substituting (at the level of the equation of motion)

Figure 2: Plot of the bulk scalars $L_{ud}$ and $L_{s}$ as functions of the RG
scale $r$, with the black line indicating the on-mass-shell condition. In the
UV, these fields take on the values of the up/down quark mass (set to zero)
and the strange mass (100 MeV) respectively. At lower energies, a quark
condensate forms, contributing to the mass of each scalar in the IR.

$\Delta=3+\gamma$ and expanding $m^{2}=\Delta(\Delta-4)$ for small $\gamma$
then gives

$\Delta m^{2}=2\gamma=-\frac{4}{\pi}\alpha$ (16)

i.e. a scalar bulk mass that depends on $r$ is dual to the running anomalous
dimension of its associated boundary operator. Using (8), we then express
$\gamma$ in terms of $\alpha$, which is specified by the following RG flow at
one loop

$\mu{d\over d\mu}\alpha=-b_{0}\alpha^{2},\hskip 28.45274ptb_{0}={1\over
6\pi}\left(11C_{2}(G)-4N_{f}T(F)\right)$ (17)

For an $SU(N_{c})$ gauge theory like QCD, the quadratic Casimir for the
adjoint representation specifies the number of colours, i.e. $C_{2}(G)=N_{c}$.
In the fundamental representation, the trace over group generators is
normalised by $T(F)=1/2$.

Naively, one would assume $\alpha$ is a function of $r$ only, but this would
remove any stable solutions below where the BF bound is violated. The natural
resolution (found in the D3/probe D7 system) is to set $r^{2}\rightarrow
r^{2}+L^{2}$, which is both dimensionally sensible and means that, if $L$
grows sufficiently, the BF bound violation is removed and a stable solution is
allowed to form.

To solve (15) for the vacuum configuration of the theory, we must impose
sensible boundary conditions on $L$ in the IR region. We use the regularity
condition $\partial_{\rho}L=0$ proposed by the D3/probe D7 system, thus
allowing $L$ to be interpreted as the RG flow of the constituent mass.

We should also integrate any quarks from the theory which fail to satisfy the
on-mass-shell condition $L\leq r$. As such, we begin the solution on the line
$L=r$ with $\partial_{\rho}L=0$ and only explore scales above that bound. We
show vacuum solutions that asymptote to either zero quark mass or the strange
mass in the UV in fig.2.

Goldstone bosons, associated with the broken axial symmetry, correspond to the
phase of the field $L$. As usual in the D3/probe D7 system Kruczenski:2003uq ,
the pion wave functions $\pi(r)$ are described precisely by the vacuum $L$
solutions above (for zero UV quark mass). The additional interpretation of
these solutions does, however, require that they be properly normalised in
order to have a canonically kinetic term in the 4D Lagrangian. As such, we
impose the following normalisation condition on the pion wave function

$\int dr{r^{3}\over(r^{2}+L^{2})^{2}}\pi(r)^{2}=\frac{1}{2}$ (18)

###  IV.2 Nucleon Mass Revisited

In order to calculate the spin-half baryon mass in the dynamical model, we
must first recalculate the scalar mass of the $\rho$ meson ground state. The
equation of motion that determines the $\rho$ meson mass is now given by

$\partial_{r}r^{3}\partial_{r}A+{r^{3}M^{2}\over(r^{2}+L^{2})^{2}}A=0$ (19)

We solve (19) with $A(r_{\rm min})=1$ and $A^{\prime}(r_{\rm min})=0$ with
$r_{\rm min}$ specifying where the quark mass goes on-shell. We find a new
value for the bulk mass of the $\rho$ meson that will set the QCD scale.

As for the nucleon, the equation of motion for a bulk fermion in the Dynamic
AdS/QCD model was computed in Abt:2019tas . The nucleon wave function $\psi$
is given by the solution of

$\begin{array}[]{c}\left(\partial_{r}^{2}+\mathcal{P}_{1}\partial_{r}+\frac{M^{2}_{B}}{(r^{2}+L^{2})^{2}}+\mathcal{P}_{2}\frac{1}{(r^{2}+L^{2})^{2}}\right.\\\
\\\ \left.\hskip
28.45274pt-\frac{m_{\psi}^{2}}{(r^{2}+L^{2})}+\mathcal{P}_{3}\frac{m_{\psi}}{(r^{2}+L^{2})^{3/2}}\right)\psi=0\end{array}$
(20)

where $M_{B}$ is the baryon mass, $m_{\psi}=5/2$ as before and the pre-factors
are given by

$\displaystyle\mathcal{P}_{1}$
$\displaystyle=\frac{6}{r^{2}+L^{2}}\left(r+L~{}\partial_{r}L\right),$
$\displaystyle\mathcal{P}_{2}$
$\displaystyle=2\left((r^{2}+L^{2})L\partial^{2}_{r}L+(r^{2}+3L^{2})(\partial_{r}L)^{2}+4rL\partial_{r}L\right.$
$\displaystyle\left.\hskip 56.9055pt+3r^{2}+L^{2}\right),$
$\displaystyle\mathcal{P}_{3}$
$\displaystyle=\left(r+L~{}\partial_{r}L\right).$

We can add a contribution to the running anomalous dimension of the three
quark operator by allowing the fermion mass to be $r$ dependent. Using the
relation between $m_{\psi}$ and the dimension of the operator $\Delta$, as
well as the one loop running result for a three quark operator, we find that

$\Delta m_{\psi}=\gamma=-{3\over\pi}\alpha$ (21)

The normalisation of the nucleon wave function is

$\int dr{r^{3}\over(r^{2}+L^{2})^{1/2}}\psi(r)^{2}=\frac{1}{2}$ (22)

Amending $m_{\psi}$ with the running anomalous dimension (21), we can solve
(20) subject to the usual IR boundary conditions. Then, tuning $M_{B}$ such
that $\psi$ vanishes in the UV (the normalised wave function is plotted in
fig.3), we find that $M_{B}=1.40M_{\rho}=1.08$ GeV, which is within 15% of the
measured proton mass ($938$ MeV).

This result is smaller than we achieved in the hard wall model in which the
dimension of the three quark operator was fixed at $\Delta=3$. This is because
the value of the anomalous dimension $\gamma$ is greater in this model to the
extent that it violates the BF bound very close to the chiral symmetry
breaking scale. Such a violation within a short region of $r$ need not be
sufficient to trigger an instability. Instead, the contribution provided by
the ‘kinetic’ derivative terms in the $r$ direction may counter mass terms in
the field potential, leading to an overall solution that remains stable.

With the stability of the solution assumed, the phenomenological effect of the
running dimension is to drive the mass of the bound state down. The fact that
the one loop result for $\gamma$ reproduces a nucleon mass that is closer to
the physical value suggests that a large running of the anomalous dimension
may indeed be present. Of course, one could tune $\gamma$ in the non-
perturbative regime in a way that reproduces the observed mass exactly, though
without an analytical understanding of this phenomena it would be difficult to
motivate any kind of precise ansatz.

###  IV.3 Strange Baryon Masses

We next turn our attention to QCD bound states containing strange quarks, such
as the $\Lambda$ baryon. For states containing yet heavier quarks, we expect
that the running anomalous dimension has a negligible impact on the bound
state masses. This is because the dimension of the quark operator will only
run as far as its mass scale which, for those heavier than the strange quark,
is large enough to suppress the effects the light quarks see from the sudden
pole in the QCD coupling.

In top-down models with flavour branes, the strange and light quark branes
will separate in the bulk space and mixed heavy-light states would appear as
complicated stringy states tied between them Erdmenger:2007vj . We do not try
to reproduce this structure here. The key point is that the bound state
operators see the $L(r)$ functions associated with the quarks they contain,
thus making them aware of the constituent’s masses. A simple phenomenological
approach is to write

$L^{2}\rightarrow f_{ud}L_{ud}^{2}+f_{s}L_{s}^{2}$ (23)

where $f_{i}$ is the fraction of the quarks of the type $i$ in the hadron.

For example, we consider a uds $\Lambda$ bound state whose equation of motion
follows the same general form as that of the nucleon (20) but with an amended
field structure $L^{2}\rightarrow\frac{2}{3}L_{ud}^{2}+\frac{1}{3}L_{s}^{2}$
reflecting the composition of the $\Lambda$ baryon in terms of first and
second generational quarks. The distinction made is that, for the scalar dual
to the strange quark $L_{s}$, the solution asymptotes to a non-zero bulk mass
in the UV limit (fig.2). This field redefinition changes both the pre-factors
of the equations of motion and the deformed AdS radius, which is now given by
$(r^{2}+\frac{2}{3}L_{ud}^{2}+\frac{1}{3}L_{s}^{2})^{1/2}$.

Solving the equation of motion for the $\Lambda$ returns a wave function
solution $\psi_{\Lambda}$ which vanishes in the UV limit for a tuned baryon
mass of $M_{\Lambda}=1.49M_{\rho}=1.15$ GeV (fig.3). This prediction is within
3% of the measured $\Lambda$ mass ($1115$ MeV).

Moving to heavier bound states, we can once again amend the field structure
and deformed AdS radius to describe particles comprised of two strange quarks,
such as $uss$. In this case, we take (23) with $f_{ud}=1/3$ and $f_{s}=2/3$.
Solving a similar equation of motion as with the $\Lambda$, we find a $\Xi$
mass of 1.22 GeV, which is within 8% of its measured value (1315 MeV). Again,
the running anomalous dimension of the quark fields is key to producing these
closer fits with experimental data.

##  V Sexaquarks in AdS/QCD

Given the apparent importance of running anomalous dimensions in the baryonic
sector, it is interesting to look for other light quark states in which the
same mechanism might be present. Looking towards novel forms of matter, it has
been suggested that the spectrum of QCD may include a deeply bound six-quark
state (see for example Farrar:2017eqq ; Farrar:2022mih ; Azizi:2019xla  to
motivate this and review the phenomenology). The $uuddss$ state forms a colour
singlet that is antisymmetric in colour, flavour and spin, endowing it with a
spatially symmetric wave function. Where the state is considered as a loosely
bound state of two $\Lambda$ baryons, the $uuddss$ state has historically been
referred to as the H dibaryon Jaffe:1976yi .

Naively, one might estimate the H dibaryon mass to be three times that of the
rho meson plus an additional contribution of 200 MeV arising from the larger
mass of the two strange quarks. This leads to a mass of around 2.5 GeV. A
previous bag model analysis placed this estimate a little lower at 2.15 GeV
Jaffe:1976yi . Both these predictions lie above the $\sim$2 GeV stability
bound which causes the dibaryon to decay with a typical weak interaction
lifetime of $10^{-10}\textrm{s}$. Lattice QCD simulations NPLQCD:2012mex ;
PadmanathMadanagopalan:2021exb  are currently unable to work at the physical
quark masses. One could consider models with larger quark masses, but this may
exclude the key regime around the scale of chiral symmetry breaking thus
impacting the mass predictions.

At present, experimental searches for the dibaryon remain unsuccessful. If the
state were more tightly bound, its mass could sit below 2 GeV, causing it to
potentially remain undetected against the large neutron background in collider
experiments Farrar:2017eqq . If confirmed, it is proposed that this tightly
bound sexaquark would not only be stable, but it might also be a good
candidate for dark matter Farrar:2018hac  or a stable cosmological relic
Kolb:2018bxv .

We take this discussion as motivation to investigate the sexaquark state in
holography. A previous estimate of the sexaquark mass within holography was
conducted in the Sakai-Sugimoto model Suganuma:2016lmp , where the sexaquark
was treated as an instanton-like solution arising as a bound state of baryons
- that study predicted a mass of 1.7 GeV. Instead, we will model the sexaquark
as a scalar in the bulk of the Dynamic AdS/QCD model. If there is a deeply
bound state, this may be the more appropriate description, distinct from
treating it as a bound state of two $\Lambda$s. The scalar’s bulk mass will
run in our description, dual to the anomalous dimension of the $uuddss$
operator.

###  V.1 Sexaquarks in the Hard Wall Model

We can represent the sexaquark operator by inserting a scalar field $\Phi$
into $AdS_{5}$ bulk space with Klein-Gordon action

$S=\int
d^{5}x\sqrt{-\textrm{det}g}\left((\partial_{\mu}\Phi)(\partial_{\nu}\Phi^{*})g^{\mu\nu}+m^{2}\Phi^{2}\right)$
(24)

Writing the field as a plane wave $\Phi(r,x)=\phi(r)e^{ikx}$ with
$k^{2}=-M^{2}$, the resulting equation of motion

$\partial_{r}\left[r^{5}\partial_{r}\phi\right]-r^{3}m^{2}\phi+M^{2}r\phi=0$
(25)

admits the following large $r$ solution for $m^{2}=\Delta(\Delta-4)$

$\phi(r)=\frac{C_{1}}{r^{\Delta}}+\frac{C_{2}}{r^{4-\Delta}}$ (26)

Using the UV scaling dimension of the six quark state $\Delta=9$, we conclude
that the bulk mass $m^{2}=45$.

Taking the IR boundary condition $\phi^{\prime}({\rm wall})=0$, $M^{2}$ is
again tuned such that $\phi$ vanishes in the UV, leading to a sexaquark mass
of $M=3.77M_{\rho}\approx 2.90$ GeV. This large estimate assumes that the
dimension of the quark is fixed at its maximum value throughout the strong
coupling regime. Alternatively, one could assume that the IR dimension
$\Delta=6$ (as suggested by the one loop calculation) holds for all energies
instead. In this case, the sexaquark mass is substantially lower at 1.90 GeV.

With two very distinct estimates for the sexaquark mass residing on either
side of the 2 GeV stability condition, this simple hard wall model
demonstrates the importance that the specific form of the running dimension
likely has on the calculation of bound state masses composed of a greater
number of quarks. As such, a prediction of the sexaquark mass in Dynamic
AdS/QCD may be particularly worthwhile.

###  V.2 Dynamic AdS/QCD and the Sexaquark

We will treat the sexaquark operator as a scalar in the spirit of (14) but
with a non-zero spatial derivative term

$S=\int
d^{4}xdr~{}r^{3}\left[(\partial_{r}L)^{2}+\frac{(\partial_{x}L)^{2}}{(r^{2}+L^{2})^{2}}+\frac{\Delta
m^{2}L^{2}}{r^{2}}\right]$ (27)

Assuming a plane wave solution and adjusting the factor of $L^{2}$ to average
over the quark constituents, we arrive at the following equation of motion,

$\partial_{r}\left[r^{3}\partial_{r}L\right]-r\Delta
m^{2}L+{r^{3}M^{2}\over(r^{2}+{2\over 3}L_{ud}^{2}+{1\over
3}L_{s}^{2})^{2}}L=0$ (28)

where $M^{2}$ is the sexaquark mass. To enforce a UV bulk mass of $m^{2}=45$
and the IR running from the one loop analysis, we set

$\Delta m^{2}=(3+m^{2})+14\gamma=48-{84\over\pi}\alpha$ (29)

where we have used (8) with $n=6$.

As in the previous cases, we now seek a solution to (28) for which the UV
field vanishes. We find that this is satisfied for $M_{S}=2.27M_{\rho}\approx
1.75$ GeV. This result would support the stable deeply bound hypothesis,
although it is clearly very dependent on the assumptions made about the
running dimension of the six-quark operator in the

Figure 3: Plot of the normalised wave functions for the proton, $\Lambda$
baryon, sexaquark and $\pi$ meson as a function of RG scale $r$. For each wave
function, the value of the bound state mass has been tuned such that the
solution vanishes in the far UV.

IR. To stress that point, one could perform a similar analysis for a di-
neutron state and get an equally low mass, which appears to be at odds with
experimental data. Whether the unique wave function of the sexaquark would
enter beyond one loop to justify the assumptions here is unclear.
Nevertheless, holography can entertain the possibility that the sexaquark is
stable.

Similarly to the pion wave function (18), the normalisation of the sexaquark
wave function is fixed by

$\int dr{r^{3}\over(r^{2}+{2\over 3}L_{ud}^{2}+{1\over
3}L_{s}^{2})^{2}}L(r)^{2}=\frac{1}{2}$ (30)

We plot the normalised wave functions for the nucleon, $\Lambda$, sexaquark
and pion in fig.3. We note that any interactions between these states will be
set by overlap integrals of these wave functions; the ratios of which all
appear to be of order 1. One should also not expect any large factors from the
calculation of these integrals which act to raise or lower particular
interactions. A priori, the sexaquark does not seem particularly decoupled
from the rest of the hadronic spectrum. However, it is important to stress
that these calculations, whose origin is in the large $N_{c}$ limit, do not
include secondary interactions such as the size of pion clouds surrounding
each state.

##  VI Discussion

In this paper we have employed some simple holographic models to explore the
importance of the running anomalous dimension of operators on the predicted
spectrum of bound state masses in QCD. Similar considerations have previously
been made for the sigma meson in models that dynamically generate chiral
symmetry breaking. In those models, the running dimension drives the sigma to
become tachyonic, triggering the chiral symmetry breaking. This suggests that
the running for other light quark operators might be important.

Our first investigation was of the nucleon mass. In hard wall models using the
UV dimension of the three-quark operator, the nucleon mass is typically high
relative to the mass of the rho meson. We instead enacted a model that
includes the running of the anomalous dimension for the quark condensate and
dynamically describes chiral symmetry breaking in place of the hard wall.
Using an ansatz for the bulk fermion, we found that the nucleon mass was
brought down to 1.1 GeV. Our ansatz matches the bulk mass to the one loop QCD
running of the operator dimension, but then extrapolates that beyond the UV
regime in an uncontrolled fashion. Even so, we conclude that the running can
play a significant role in the predicted nucleon mass and should not be
neglected. A similar analysis for the $\Lambda$ ($usd$) and $\Xi$ ($uss$)
baryons also brings their masses into line with experimental values at the 3
and 8 percent level respectively.

We also considered a more exotic singlet sexaquark state ($uuddss$). Including
the one loop running of $\gamma$ in this case generated a mass prediction of
1.75 GeV, a value similar to other holographic models and low enough to make
the state stable against decay into two $\Lambda$ baryons. Whilst the precise
value depends on the assumed running, such a stable state does not seem
incompatible with holography.

Overall, we conclude that the running anomalous dimensions of multi-quark
operators is important when determining the bound state spectrum and
holographic models should seek to include these in the future.

Acknowledgements: NE’s work was supported by the STFC consolidated grant
ST/T000775/1.

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