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

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 $\gamma$.

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)

$\Delta=3+\gamma$ 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)

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.

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.

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.

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

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.