Heavy and Strange Holographic Baryons

TL;DR

This paper extends holographic models to include heavy and strange baryons, predicting new exotic states and confirming some observed particles using a strong coupling approach.

Contribution

It introduces a holographic framework for heavy-light baryons with strangeness, providing predictions for new exotic states and spectra.

Findings

Confirmed existence of two low-lying charmed penta-quark states

Predicted many new states with charm and bottom

Suggested a quartet of neutral $\\Omega_c^0$ states heavier than observed ones

Abstract

We extend the $D4$-$D8$ holographic construction to include three chiral and one heavy flavor, to describe heavy-light baryons with strangeness and their exotics. At strong coupling, the heavy meson always binds to the bulk instanton in the form of a flavor zero mode in the fundamental representation. We quantize the ensuing bound states using the collective quantization method, to obtain the spectra of heavy and strange baryons with both explicit and hidden charm and bottom. Our results confirm the existence of two low-lying charmed penta-quark states with $\frac 12^-,\frac 32^-$ assignments, and predict many new ones with both charm and bottom. They also suggest a quartet of low-lying neutral $\Omega_c^0$ with assignments $\frac 12^\pm ,\frac 32^\pm$ that are heavier than the quintuplet of neutral $\Omega_c^0$ recently reported by LHCb.

Equations

$$S_{\mathrm{DBI}}\approx -\kappa \int d^{4}xdz\mathrm{Tr}\left(f(z){\mathbf{F}}_{\mu \nu }{\mathbf{F}}^{\mu \nu }+g(z){\mathbf{F}}_{\mu z}{\mathbf{F}}^{\nu z}\right)$$

$$f(z)=\frac{R^3}{4U_z},g(z)=\frac{9}{8}\frac{U_z^3}{U_{KK}}$$

$${\mathbf{F}}_{MN}=$$

$$\displaystyle\left(\begin{array}[]{cc}F_{MN}-\Phi_{[M}\Phi_{N]}^{\dagger}&\partial_{[M}\Phi_{N]}+A_{[M}\Phi_{N]}\\ -\partial_{[M}\Phi^{\dagger}_{N]}-\Phi^{\dagger}_{[M}A_{N]}&-\Phi^{\dagger}_{[M}\Phi_{N]}\end{array}\right)$$

$$\displaystyle{\bf A}=\left(\begin{array}[]{cc}A&\Phi\\ -\Phi^{\dagger}&0\end{array}\right)$$

$$\frac{1}{2}{m}_H^2\mathrm{Tr}\left({\Phi }_M^{†}{\Phi }_M\right)\sim \frac{1}{2}{v}^2\mathrm{Tr}\left({\Phi }_M^{†}{\Phi }_M\right)$$

$$S_{CS}=\frac{i{N}_c}{24{\pi }^2}{\int }_{M_5}\mathrm{Tr}\left(A{F}^2-\frac{1}{2}{A}^{3}F+\frac{1}{10}{A}^5\right)$$

$$J_8=\frac{N_c}{2\sqrt{3}}$$

$$S_{nCS}=S_{CS}$$

$$A{|}_{z\to \pm \infty }={\hat{A}}^{\pm h^{\pm }}=h^{\pm }(d+{\hat{A}}^{\pm })h^{\pm -1}$$

$$(h^{-},h^{+})\equiv \left(1,P{e}^{-{\int }_{-\infty }^{\infty }{A}_{z}dz}\right)$$

$$A_M=$$

$$A_0=$$

$$x_0\to x_0,x_M\to x_M/\sqrt{\lambda },\sqrt{\lambda }\rho \to \rho$$

$$(A_0,{\Phi }_0)\to (A_0,{\Phi }_0),$$

$$(A_M,{\Phi }_M)\to \sqrt{\lambda }(A_M,{\Phi }_M)$$

$$F_{MN}=\mathrm{diag}\left(2\frac{{\bar{\sigma }}_{MN}{\rho }^2}{(x^2+{\rho }^2{)}^2},0\right)$$

$$\mathcal{L}=a{N}_c\lambda {\mathcal{L}}_0+a{N}_c{\mathcal{L}}_1+{\mathcal{L}}_{CS}$$

$${\mathcal{L}}_0=$$

$${\mathcal{L}}_1=$$

$${\mathcal{L}}_{CS}=$$

$${\tilde{\mathcal{L}}}_1=$$

$$D_M{D}_M{\Phi }_N+2F_{NM}{\Phi }_M-D_N{D}_M{\Phi }_M=0$$

$$D_M(D_0{\Phi }_M-D_M{\Phi }_0)$$

$$-F^{0M}{\Phi }_M-\frac{{ϵ}_{MNPQ}}{64{\pi }^{2}a}{K}_{MNPQ}=0$$

$$K_{MNPQ}=$$

$$\frac{{\mathcal{L}}_{1,m}}{a{N}_c}=4i{m}_H{\varphi }_m^{†}{D}_0{\varphi }_m-2i{m}_H({\varphi }_0^{†}{D}_M{\varphi }_M-c.c.)$$

$$\frac{m_H{N}_c}{16{\pi }^2}{ϵ}_{MNPQ}{\varphi }_M^{†}{F}_{NP}{\varphi }_Q=\frac{m_H{N}_c}{8{\pi }^2}{\varphi }_M^{†}{F}_{MN}{\varphi }_N$$

$${\mathcal{L}}_0=-\frac{1}{2}{\left|f_{MN}-\star f_{MN}\right|}^2$$

$$f_{MN}-\star f_{MN}=0and{D}_M{\varphi }_M=0$$

$${\sigma }_M{D}_M\psi =D\psi =0with\psi ={\bar{\sigma }}_M{\varphi }_M$$

$${\psi }_{\alpha \beta }^a={ϵ}_{\alpha a}{\chi }_{\beta }\frac{\rho }{(x^2+{\rho }^2{)}^{\frac{3}{2}}}witha=1,2$$

$$\Phi \to V(a_I(t))\Phi (X_0(t),Z(t),\rho (t),\chi (t))$$

$$h^{-}=\mathrm{diag}\left({a_I(t)}^{-1},1\right)$$

Full text

I Introduction

Recently the Belle collaboration BELLE and the BESIII collaboration BESIII have reported many multiquark exotics uncommensurate with quarkonia, e.g. the neutral $X(3872)$ and the charged $Z_{c}(3900)^{\pm}$ and $Z_{b}(10610)^{\pm}$. These exotics have been also confirmed by the DO collaboration at Fermilab DO , and the LHCb collaboration at CERN LHCb . Also recently, the same LHCb collaboration has reported new pentaquark states $P_{c}^{+}(4380)$ and $P_{c}^{+}(4450)$ through the decays $\Lambda_{b}^{0}\rightarrow J\Psi pK^{-},J\Psi p\pi^{-}$ LHCbx , and five narrow and neutral excited $\Omega^{0}_{c}$ baryon states that decay primarily to $\Xi_{c}^{+}K^{-}$ LHCbxx . These flurry of experimental results support new physics involving heavy-light multiquark states, a priori outside the canonical classification of the quark model.

Some of the tetra-states exotics maybe understood as molecular bound states mediated by one-pion exchange much like deuterons or deusons MOLECULES ; THORSSON ; KARLINER ; OTHERS ; OTHERSX ; OTHERSZ ; OTHERSXX ; LIUMOLECULE . Non-molecular heavy exotics were also discussed using constituent quark models MANOHAR , heavy solitonic baryons RISKA ; MACIEK2 , instantons MACIEK3 and QCD sum rules SUHONG . A flurry of quark-based descriptions of the reported neutrals $\Omega_{c}^{0}$ states have also been proposed OMEGAC following earlier descriptions EARLIER , including a recent lattice simulation LATTICEC .

The penta-states exotics reported in LHCbx have been foreseen in MAREK and since addressed by many using both molecular and diquark constructions MANY , as well as a bound anti-charm to a Skyrmion PENTARHO . String based pictures using string junctions VENEZIANO have also been suggested for the description of exotics, including a recent proposal in the context of the holographic inspired string hadron model COBI .

In QCD the light quark sector (u, d, s) is dominated by the spontaneous breaking of chiral symmetry, while the heavy quark sector (c, b, t) exhibits heavy-quark symmetry ISGUR . Both symmetries are at the origin of the chiral doubling in heavy-light mesons MACIEK ; BARDEEN , as measured by both the BaBar collaboration BABAR and the CLEOII collaboration CLEOII . As most of the heavy hadrons and their exotics exhibit radiative decays through light or heavy-light mesons it is important to formulate a non-perturbative model of QCD that honors both chiral and heavy quark symmetry.

The initial holographic construction offers a framework for addressing chiral symmetry and confinement in the double limit of large $N_{c}$ and large t′Hooft coupling $\lambda=g^{2}N_{c}$. A concrete model was proposed by Sakai and Sugimoto SSX using a $D4$-$D8$ brane construction. The induced gravity on the probe $N_{f}$ $D8$ branes due to the large stack of $N_{c}$ $D4$ branes, causes the probe branes to fuse in the holographic direction, providing a geometrical mechanism for the spontaneous breaking of chiral symmetry. The Dirac-Born-Infeld ( DBI) action on the probe branes yields a low-energy effective action for the light pseudoscalars with full global chiral symmetry, where the vectors and axial-vector light mesons are dynamical gauge particles of a hidden chiral symmetry HIDDEN . This construction was recently extended to accomodate heavy mesons with explicit heavy quark symmetry LIUHEAVY . The construction makes use use of an additional heavy probe $D8$ brane in bulk LIUHEAVY .

In the $D4$-$D8$ brane construction, baryons are identified with small size instantons by wrapping $D4$ around $S^{4}$, and are dual to Skyrmions on the boundary SSXB ; SSXBB . Remarkably, this identification provides a geometrical description of the baryonic core that is so elusive in most Skyrme models SKYRME . A first principle description of the baryonic core is paramount to the understanding of heavy hadrons and their exotics since the heavy quarks bind over their small Compton wavelength. In a recent analysis we have shown how heavy baryons and their exotics can be derived from the zero modes of bulk instantons using two light flavors LIUBARYON . This paper extends this analysis to the case of three light and one heavy flavors with both chiral and heavy quark symmetry. A key new feature of the three flavor case is the subtle form of the Chern-Simons term CSLIGHT ; CSTHREE and its importance in fixing the baryonic hypercharge in the presence of a heavy flavor. The effect of the light strange quark mass is introduced using a bulk instanton holonomy and treated perturbatively WORLD . The model allows for the description of heavy baryons as a bound instanton zero mode in the double limit of strong coupling followed by a large heavy quark mass. This approach will extend the bound state approach developed in the context of the Skyrme model with heavy mesons PENTARHO ; SKYRMEHEAVY to holography. We note that alternative holographic models for the description of heavy hadrons have been developed in FEWX ; BRODSKY without the dual strictures of chiral and heavy quark symmetrty.

The organization of the paper is as follows: In section 2 and 3 we briefly recall the geometrical set up for the derivation of the heavy-light effective action for three flavors in terms of the bulk DBI and CS actions. We detail the heavy-meson interactions to the flavor instanton, and the ensuing heavy meson bound state to the instanton in bulk in the double limit of large coupling and heavy meson mass. In section 4 and 5, we use the collective quantization approach to derive the pertinent spectra for holographic heavy baryons and their exotics with strangeness. Our conclusions are in section 6. In the Appendix we briefly review the collective quantization of the light baryons for $N_{f}=2,3$.

II Holographic effective action

II.1 D-brane set up

The $D4$-$D8$ construction proposed by Sakai and Sugimoto SSX for the description of the light hadrons is standard and will not be repeated here. Instead, we follow LIUHEAVY and consider the variant with $N_{f}-1$ light $D8$-$\bar{D}8$ (L) and one heavy (H) probe branes in the cigar-shaped geometry that spontaneously breaks chiral symmetry. A schematic description of the set up for $N_{f}=3$ is shown in Fig. 1. We assume that the L-brane world volume consists of $R^{4}\times S^{1}\times S^{4}$ with $[0-9]$-dimensions. The light 8-branes are embedded in the $[0-3+5-9]$-dimensions and set at the antipodes of $S^{1}$ which lies in the 4-dimension. The warped $[5-9]$-space is characterized by a finite size $R$ and a horizon at $U_{KK}$.

II.2 DBI action

The effective action on the probe L-branes consists of the non-Abelian DBI and CS action. After integrating over the $S^{4}$, the leading contribution in $1/\lambda$ to the DBI action is

[TABLE]

Our conventions are $(-1,1,1,1)$ with $A_{M}^{\dagger}=-A_{M}$. The warping factors are

[TABLE]

with $U_{z}^{3}=U_{KK}^{3}+U_{KK}z^{2}$, and $\kappa\equiv a\lambda N_{c}$ and $a=1/(216\pi^{3})$ SSX . The effective fields in the field strengths are ($M,N$ run over $(\mu,z)$)

[TABLE]

The matrix valued 1-form gauge field is

[TABLE]

For $N_{f}$ coincidental branes, the $\Phi$ multiplet is massless. However, their brane world-volume supports an adjoint and traceless scalar $\Psi$ in addition to the adjoint gauge field $A_{M}$, which we have omitted from the DBI action for notational simplicity. The scalar admits a quartic potential with finite extrema and a vev $v$ for the diagonal of $\Psi$ MEYERS , leading to a Higgs-like mass for the $\Phi$ multiplet

[TABLE]

The vev is related to the separation between the light and heavy branes MEYERS , which is about the length of the HL string. Below, $m_{H}$ will be taken as the heavy meson mass for the heavy-light $(0^{-},1^{-})$, i.e. $(D,D^{*})$ for charm and $(B,B^{*})$ for bottom. The introduction of a finite non-zero strange quark mass will be discussed also below.

II.3 Chern-Simons action

For $N_{f}>2$, the naive Chern-Simons 5-form

[TABLE]

fails to reproduce the correct transformation law under the combined gauge and chiral transformations CSLIGHT . In particular, when addressing the $N_{f}=3$ baryon spectra, (10) fails to reproduce the important hypercharge constraint CSLIGHT

[TABLE]

This issue was recently revisited in CSTHREE where boundary contributions were added to (10) to address these shortcomings. Specifically, the new Chern-Simons (nCS) contribution is CSTHREE

[TABLE]

Here $N_{5}$ is a 5-dimensional manidold whose boundaries are $\partial N_{5}=\partial M_{5}=M_{4+\infty}-M_{4-\infty}$, with the asymptotic flavor gauge field

[TABLE]

The gauged 4-form $\alpha_{4}$ is given in CSTHREE . $\hat{A}^{\pm}$ refer to the external gauge fields, and $h|_{\partial M_{5}}=(h^{+},h^{-})$. A is assumed to be well defined throughout $M_{5}$ and produces no-boundary contributions. In other words, in this gauge all topological information is moved to the holographic boundaries at $z=\pm\infty$. We can actually work in the $A_{z}=0$ gauge, and for the instanton profile (as discussed below) we have

[TABLE]

Note that in our case $A\rightarrow{\bf A}$ as defined in (8). As a result, the contributions from (10) are similar to those in the $N_{f}=2$ case discussed in LIUHEAVY . The contributions from the new terms in (12) will be detailed in the quantization approach below.

III Heavy-Light Baryons

III.1 Bulk instanton

In the original Sakai and Sugimoto model SSX light baryons are identified with small size flavor instantons in bulk SSXB . This construction carries to our current set up as we have recently shown for the $N_{f}=2$ case in LIUHEAVY . For the present $N_{f}=3$ case shown in Fig. 1, a small size instanton translates to a flat space 4-dimensional instanton in the $[1-4]$ directions. Specifically, the SU(3) flavor instanton $A_{M}$ and its time components are  CSLIGHT

[TABLE]

where the rescaling

[TABLE]

was used. From here on $M,N$ runs only over $1,2,3,z$ unless specified otherwise. The instanton solution $A_{M}$ in (15) carries a field strength

[TABLE]

III.2 Heavy-light effective action

To order $\lambda^{0}$ the rescaled contributions describing the interactions between the light gauge fields $A_{M}$ and the heavy fields $\Phi_{M}$ to quadratic order split to several contributions

[TABLE]

with each contribution given by

[TABLE]

and

[TABLE]

The additional boundary contributions in (12) do not generate any new heavy meson contribution besides those generated by the standard Chern-Simons contributions quoted in (III.2).

III.3 $\Phi$ equation of motion

We now consider the bound state solution of the heavy meson field $\Phi_{M}$ in the (rescaled) instanton background 15). We note that the field equation for $\Phi_{M}$ is independent of $\Phi_{0}$ and reads

[TABLE]

while the (contraint) field equation for $\Phi_{0}$ depends on $\Phi_{M}$ through the Chern-Simons term

[TABLE]

with $K_{MNPQ}$ defined as

[TABLE]

III.4 Heavy meson limit

In the heavy meson mass limit it is best to redefine $\Phi_{M}=\phi_{M}e^{-im_{H}x_{0}}$ for particles. The anti-particle case follows through $m_{H}\rightarrow-m_{H}$ with pertinent sign changes. In the double limit of $\lambda\rightarrow\infty$ followed by $m_{H}\rightarrow\infty$, the leading contributions are of order $\lambda m_{H}^{0}$ from the ligh effective action, and of order $\lambda^{0}m_{H}$ from the heavy-light interaction term ${\cal L}_{1}$ in (III.2)

[TABLE]

and the standard Chern-Simons term in (III.2)

[TABLE]

The constraint equation (III.3) simplifies considerably to order $m_{H}$, that is $D_{M}\phi_{M}=0$ and implying that $\phi_{M}$ is transverse in leading order in the double limit.

III.5 Zero-Mode

We now observe that the heavy field equation (21) in combination with the constraint equation (III.3) are equivalent to the vector zero-mode equation in the fundamental representation. For that, we recall that the field strength (17) is self-dual, and ${\cal L}_{0}$ in (III.2) can be written in the compact form

[TABLE]

using the Hodge $\star$ product, with $f_{MN}=\partial_{[M}\phi_{N]}+A_{[M}\phi_{N]}$. Therefore, the second order field equation (21) can be replaced by the anti-self-dual condition (first order) and the transversality condition (first order),

[TABLE]

which are equivalent to

[TABLE]

The spinor zero-mode $\psi$ is unique, and its explicit matrix form reads

[TABLE]

with explicitly $\phi_{M}=(\bar{\sigma}_{M}f(x)\chi,0)$. Here $\chi_{\alpha}$ is a constant two-component spinor. We have to understand that only the first two component of the spin-zero modes are non-zero. It can be checked explicitly that $\phi_{M}$ is a solution to the first order equations (27). In the presence of the instanton, the spin-1 vector field binds and transmutes to a spin $\frac{1}{2}$ spinor.

IV Quantization

The classical bound instanton-zero-mode breaks iso-rotational, rotational and translational symmetries. To quantize it, we promote the solution to a slowly moving and rotating solution. The leading contribution for large $\lambda$ is purely instantonic and its quantization is standard and can be found in SSXBB , so we will assume it here. The quantization of the subleading $\lambda^{0}m_{H}$ contribution involves the zero-mode and for $N_{f}=2$ was recently addressed in LIUBARYON . Here, we will address the new elements of the quantization for $N_{f}=3$.

The collective quantization method proceeds by first slowly rotating and translating the instanton configuration in bulk using

[TABLE]

with $\Phi_{0}=0$. Here $X_{0}$ is the center in the 123 directions and Z is the center in the $z$ directon. $a_{I}$ is the SU(3) gauge rotation moduli. The moduli is composed of the collective coordinates $X_{\alpha}\equiv(X,Z,\rho)$ and by the collective SU(3) rotation $a_{I}$. The time-dependent configuration is then introduced in the heavy-light effective action described earlier and expanded in leading order in the time-derivatives as we now detail.

IV.1 The new Chern-Simons contributions

The additional Chern-Simons contributions in (12) picks up from the collectively quantized instanton by defining

[TABLE]

We now note that the field ${\bf A}$ composed of the instanton solution $A$ plus the zero-mode solution ${\bf\Phi}$, carries the same topological number as the field with the instanton solution $A$ but ${\bf\Phi}=0$. Therefore, $h_{0}$ in (IV.1) can be represented by only the latter. With this in mind, we insert (IV.1) in the new contributions in (12) to obtain

[TABLE]

The heavy-light contributions from $S_{CS}$ are those in (III.2), while the new second contribution is identical to the one obtained in the light sector CSTHREE

[TABLE]

When combined to terms emerging from the heavy sector it will give rise to the correct hypercharge constraint as we will show next.

IV.2 Heavy contributions in leading order

There are four contributions to order $\lambda^{0}m_{H}$ from the heavy meson sector, namely

[TABLE]

The second contribution is from the $A_{0}$ coupling, and the third contribution simplifies for the zero-mode

[TABLE]

The last contribution originates from the heavy terms in naive CS term, and also simplifies using the instanton field strength and the zero-mode

[TABLE]

In addition to the terms retained in (IV.2) the $\chi^{\dagger}\chi$ coupling to the U(1) flavor gauge field $A_{0}$ induces a Coulomb-like correction of the form $(\chi^{\dagger}\chi)^{2}$ as we have shown in LIUBARYON . With this in mind and after using the rescaling $\chi\rightarrow\chi/\sqrt{4aN_{c}m_{H}}$ in (IV.2) we obtain

[TABLE]

where the parameters $\eta,\mu$ are given by

[TABLE]

Here ${\cal L}_{0}[a_{I},X_{\alpha}]$ refers to the effective action density on the moduli stemming from the contribution of the light degrees of freedom in the instanton background without the $a^{8}$ term SSXB .

The term linear in $a^{8}$ in (37) couples to the hypercharge $J_{8}=\frac{N_{c}}{2\sqrt{3}}(1-\frac{\chi^{\dagger}\chi}{N_{c}})$. So (37) can be seen as an action density of light and heavy degrees of freedom supplemented by a hypercharge constraint, namely

[TABLE]

From (38) we note that $\eta\approx 3$ and $\mu\approx 1$ which are remarkably close to the same parameters derived in LIUBARYON for the $N_{f}=2$ case. These terms are inertial and not sensitive to the value of $N_{f}$.

IV.3 Heavy-light spectra

The quantization of (IV.2) follows the same arguments as those presented in SSXB ; CSLIGHT for ${\cal L}_{0}[a_{I},X_{\alpha}]$ as we briefly recall in the Appendix. Let $H_{0}$ be the Hamiltonian associated to ${\cal L}_{0}[a_{I},X_{\alpha}]$, then the full heavy-light Hamiltonian for (IV.2) is

[TABLE]

with the new quantization rule for the spinor and the hypercharge constraint

[TABLE]

We recall that the statistics and parity of $\chi$ were fixed in LIUBARYON . Specifically, we note the symmetry transformation

[TABLE]

since $U^{-1}\bar{\sigma}_{M}U=\Lambda_{MN}\bar{\sigma}_{N}$. So a rotation of the spinor $\chi$ is equivalent to a spatial rotation of the heavy vector meson field $\phi_{M}$. Since $\chi$ is in the spin $\frac{1}{2}$ representation it should be quantized as a fermion. Its parity is opposite to that of $\phi_{M}$, hence positive. With this in mind, the total spin ${\bf J}$ is given by

[TABLE]

Here for a general SU(3) representation, $\vec{\bf I}_{SU(2)}$ means the induced representation for the first three generators, $J_{1,2,3}$ as noted in the Appendix.

The spectrum of (IV.3) follows from the one discussed in SSXB ; CSLIGHT and recalled in the Appendix, with two key modifications

[TABLE]

and the change of the hypercharge as obtained in (IV.3). The quantum states with a single bound state $N_{Q}=\chi^{\dagger}\chi=1$ and the general $(p,q)$ representation for SU(3) and spin $j$ are labeled by

[TABLE]

with $n_{z}=0,1,2,..$ counting the number of quanta associated to the collective motion in the holographic direction, and $n_{\rho}=0,1,2,..$ counting the number of quanta associated to the radial breathing of the instanton core, a sort of Roper-like excitations. Following SSXB , we identify the parity of the heavy baryon bound state as $(-1)^{n_{z}}$. Using (44), the mass spectrum for the bound heavy-light states is

[TABLE]

with

[TABLE]

with $M_{KK}$ the Kaluza-Klein mass and $M_{0}/M_{KK}=8\pi^{2}\kappa$ the bulk instanton mass. The Kaluza-Klein scale is usually set by the light meson spectrum and is fit to reproduce the rho mass with $M_{KK}\sim m_{\rho}/\sqrt{0.61}\sim 1$ GeV SSX .

(IV.3) is to be contrasted with the mass spectrum for baryons with no heavy quarks or $N_{Q}=0$, where the nucleon state is idendified as $N_{Q}=0,l=1,n_{z}=n_{\rho}=0$ and the Delta state as $N_{Q}=0,l=3,n_{z}=n_{\rho}=0$ SSXB . The radial excitation with $n_{\rho}=1$ can be identified with the radial Roper excitation of the nucleon and Delta, while the holographic excitation with $n_{z}=1$ can be interpreted as the odd parity excitation of the nucleon and Delta.

IV.4 Single-heavy baryons

Since the bound zero-mode transmutes to a spin $\frac{1}{2}$, the lowest heavy baryons with one heavy quark are characterized by $n_{z},n_{\rho}=0,1,N_{Q}=1$, and $(p,q,j)=(0,1,0)$ for $\bar{\bf 3}$ and $(p,q,j)=(2,0,1)$ for $\bf 6$ . The $\bar{\bf 3}$-plet states have spin and parity $\frac{1}{2}^{+}$. We identify them with $\Lambda_{Q},\Xi_{Q}(\bar{\bf 3})$. The ${\bf 6}$-plet states have $J=\frac{1}{2},\frac{3}{2}$. We identify them with $\Sigma_{Q},\Xi_{Q}({\bf 6}),\Omega_{Q}$ and $\Sigma_{Q}^{\star},\Xi_{Q}({\bf 6})^{\star},\Omega_{Q}^{\star}$, respectively. In the absence of symmetry breaking, the mass spectra are degenerate

[TABLE]

or equivalently

[TABLE]

with the mass splitting $M_{{\bf 6}}-M_{\bar{\bf 3}}=0.334\,M_{KK}$.

IV.5 Double-heavy baryons: $QQ$

While the binding of a pair of heavy mesons with $QQ$ or $Q\bar{Q}$ content is always BPS-like to leading order in $1/\lambda$, the Chern-Simons contribution is twice more attractive with the $QQ$ content than with the $Q\bar{Q}$ content (see below), although the Coulomb induced contribution penalizes the former and not the latter. With this in mind, heavy baryons with two heavy quarks follow the same construct with $N_{Q}=2$ or $\chi^{\dagger}\chi\rightarrow 2$ in (IV.3-IV.3) and $J^{8}=1/2{\sqrt{3}}$. As a result, the lowest heavy baryons with two bound heavy mesons are now characterized by $n_{z},n_{\rho}=0,1$ and $(p,q,j)=(1,0,0)$ for the flavor ${\bf 3}$-plet with assignment $\frac{1}{2}^{+}$, which we identify as $\Xi_{QQ}$ with $u,d$ light content, and $\Omega_{QQ}$ with $s$ content. To this order, their degenerate masses are given by

[TABLE]

IV.6 Double-heavy baryons: $Q\bar{Q}$

For heavy baryons containing also anti-heavy quarks we note that a rerun of the preceding arguments using instead the reduction $\Phi_{M}=\phi_{M}e^{+im_{H}x_{0}}$, amounts to binding an anti-heavy-light meson to the bulk instanton also in the form of a zero-mode in the fundamental representation of spin, much like the heavy-light meson binding. Most of the results are unchanged except for pertinent minus signs. For instance, when binding one heavy-light and one anti-heavy-light (IV.2) now reads

[TABLE]

with the hypercharge constraint

[TABLE]

The mass spectrum for baryons with $N_{Q}$ heavy-quarks and $N_{\bar{Q}}$ anti-heavy quarks is the same as in (IV.3) with the substitution $N_{Q}\rightarrow N_{Q}-N_{\bar{Q}}$ to the present order of the analysis or $\lambda^{0}m_{H}$. For $N_{Q}=N_{\bar{Q}}=1$ the hypercharge constraint is simply $J_{8}=\sqrt{3}/2$. Therefore the lowest states carry $(p,q,j)=(1,1,1/2)$ and are identified with the baryonic states in the ${\bf 8}$-plet representation with the $J^{\pi}$ assignments ${\frac{1}{2}}^{-}$ and ${\frac{3}{2}}^{-}$, and $(p,q,j)=(3,0,3/2)$ in the ${\bf 10}$-plet representation with $J^{\pi}$ assignments (one) ${\frac{5}{2}}^{-}$, (two) ${\frac{3}{2}}^{-}$ and (one) ${\frac{1}{2}}^{-}$. Their masses are given by

[TABLE]

with the mass splitting $M_{\bar{Q}Q}^{{\bf 10}}-M_{\bar{Q}Q}^{{\bf 8}}=0.386\,M_{KK}$.

V Strange quark mass correction

To compare the previous results for single-heavy and double-heavy baryons to some of the reported physical spectra, we need to address the role of a finite strange quark mass. In so far, the light flavor branes $D\bar{8}$-$D8$ only connect at $U_{KK}$ because of the bulk gravity induced by $D4$, thereby spontaneously breaking chiral symmetry. To break explicitly chiral symmetry, say by introducing a finite strange quark mass, an additional bulk $D6$ brane can be introduced to connect $D\bar{8}$ to $D8$ WORLD ; KOJI . For the $N_{f}=3$ case with $m_{u}=m_{d}=0$ and finite $m_{s}$, the worldsheet instanton in $D6$ interpolating $D\bar{8}$ to $D8$, induces an explicit light mass breaking term for the light baryons, which takes the following form on the moduli KOJI

[TABLE]

with $\tau\approx|m_{s}\left<\bar{s}s\right>|$. Aside from the dependence on the moduli parameter through $\rho^{3}$, the explicit symmetry breaking term (55) is standard. An estimate of $\tau$ follows from holography, but here we will use $\tau$ as a free parameter to be adjusted below through the baryonic spectrum. (55) will be treated in perturbation theory by averaging $\rho^{3}$ using the radial baryonic wavefunctions $\phi_{{n_{\rho}},{\bf K}}$ discussed in the Appendix. For $n_{\rho}=n_{z}=0$, the averaged result is

[TABLE]

The emergence of the pion decay constant $f_{\pi}=93$ MeV follows from the holographic $\rho$-wavefunction as discussed in the Appendix. For the $\bar{\bf 3}$-plet and ${\bf 6}$-plet representations, we have specifically

[TABLE]

The corresponding mass shifts induced by the explicit symmetry breaking term (55) on the heavy-light baryonic spectra is then

[TABLE]

with the representation dependent parameters

[TABLE]

For the specific representations of relevance to our analysis we have

[TABLE]

V.1 Single-heavy baryon spectrum

Combining all the previous results for the heavy-light masses, including the correction induced by the strange quark mass symmetry breaking term (55) yield the following mass spectrum for the single-heavy baryons

[TABLE]

In the original Sakai and Sugimoto analysis, the Kaluza-Klein parameter is fixed by the light rho mass as indicated earlier with $M_{KK}\approx 1$ GeV. Although we will use this value for all the heavy-light baryon masses to follow, we note that this value of $M_{KK}$ was noted to be large in SSXB ; CSLIGHT . The nucleon mass $m_{N}=938$ MeV is set to its empirical value. The symmetry breaking parameter $m_{0}$ will be fitted to reproduce the mass splitting between the nucleon in the octet and the $\Omega^{-}=sss$ in the decuplet as it is the baryon with the largest strangeness. Specifically, we set

[TABLE]

which fixes $m_{0}=22.6$ MeV.

So for $n_{z}=n_{\rho}=0$, the lowest heavy-light mass spectra corrected in first order by the light strange quark symmetry breaking, with their $J^{\pi}$ assignments are

[TABLE]

The lowest excited states of these heavy-light baryons carry finite $n_{\rho},n_{z}$. For instance, for $n_{\rho}=1$, $n_{z}=0$ we have the even-parity or Roper-like excitation corresponding to $\Omega_{EQ}(\frac{1}{2})^{+}$, and for $n_{\rho}=0$ and $n_{z}=1$ we have the odd-parity excitation corresponding to $\Omega_{Q}(\frac{1}{2})^{-}$. Their masses are

[TABLE]

The masses of the single-heavy light baryons with charm follow by setting the charm heavy meson mass $m_{H}$ to its empirical value $m_{H}=m_{D}=1870$ MeV, and similarly for the bottom heavy meson mass $m_{H}=m_{B}=5279$ MeV. The specifics mass values are quoted below in [MeV] with the measured masses from PDG indicated in bold numbers.

V.1.1 Charm baryon masses [MeV]

[TABLE]

V.1.2 Bottom baryon masses [MeV]

[TABLE]

V.2 Double-heavy baryon spectrum

The double-heavy baryons with hidden charm or bottom are currently referred to as pentaquarks. Their masses in the ${\bf 8}$-plet of the flavor representation (IV.6) corrected by the strange quark mass are

[TABLE]

The penta-quark masses in the ${\bf 10}$-plet representation corrected by the strange quark mass are

[TABLE]

The double heavy baryons consisting of two heavy bound mesons with explicit charm or bottom will be referred to by $\Xi_{QQ}$ and $\Omega_{QQ}$ in the flavor ${\bf 3}$-plet representation as we noted earlier. Their strangeness corrected masses are

[TABLE]

It is clear, that the holographic construct also describes their excited Roper-like with even parity as well as their odd parity partners, which can be retrieved from our formula.

V.2.1 Charm penta-quark masses ${\rm[MeV]}$

[TABLE]

V.2.2 Mixed penta-quark masses ${\rm[MeV]}$

[TABLE]

V.2.3 Bottom penta-quark masses ${\rm[MeV]}$

[TABLE]

V.2.4 Charm and bottom ${\bf 3}$-plet masses ${\rm[MeV]}$

[TABLE]

VI Conclusions

We have presented a top-down holographic approach to the single- and double-heavy baryons in the variant of $D4$-$D8$ we proposed recently LIUHEAVY (first reference). To order $\lambda m_{H}^{0}$, the heavy baryons emerge from the zero mode after binding a heavy meson in the multiplet $(0^{-},1^{-})$ to the instanton. Remarkably, in the bulk instanton field the spin 1 and odd parity heavy meson transmutes equally to a spin $\frac{1}{2}$ and even parity massless fermion and anti-fermion. At subleading order, the Chern-Simons term is attractive for the bound meson with a heavy quark content and repulsive for the bound meson with heavy anti-quark content.

One of the key differences between the $N_{f}=2$ and $N_{f}=3$ case is the role played by the amended form of the Chern-Simons term which results in a good hypercharge quantization rule CSLIGHT ; CSTHREE . We have shown that the rule gets modified by the presence of the bound zero mode states, leading to a rich heavy-light spectra for single-heavy and double-heavy baryons with hidden charm and bottom. In particular, the formers follow from the $\bar{\bf 3}$ and ${\bf 6}$ flavor representations, while the latters from the ${\bf 8}$ and ${\bf 10}$ representations for the lowest states. The holographic set up allows for a simple description of the low-lying odd-parity and Roper-like excitations of the heavy baryons. Our results for $N_{f}=3$ with massive strangeness confirm and extend our previous findings for massless $N_{f}=2$.

To compare our results with currently known heavy-light charm and meson spectra, it is necessary to account for the light strange quark mass. In holography this is induced by a worldsheet instanton that connects $D8$ and $D\bar{8}$ WORLD . By accounting for this correction in leading order perturbation theory, we have found reasonable agreement for the lowest single-heavy baryons with a single adjustable parameter, namely the overall strength of the symmetry breaking term. The holographic model describes 2 neutral $\Omega_{c}^{0},\Omega_{c}^{*0}$ states with $\frac{1}{2}^{+},\frac{3}{2}^{+}$ assignments as the odd parity partners of the lowest $\Omega_{c}^{0},\Omega_{c}^{*0}$ states, and 2 Roper-like neutral states with $\frac{1}{2}^{+},\frac{3}{2}^{+}$ assignments as the even parity partners also of the lowest $\Omega_{c}^{0},\Omega_{c}^{*0}$ states. The $\frac{1}{2}^{-}\frac{3}{2}^{-}$ are predicted to be lighter than the excited $\frac{1}{2}^{+}\frac{3}{2}^{+}$ states, however both pairs are found to be heavier than the 5 neutral $\Omega_{c}^{0}$ states reported recently by the LHCb collaboration.

The holographic set up for the heavy baryons is remarkable by the limited number of parameters it carries. Once the initial parameter $\kappa$ is traded for the pion decay constant $f_{\pi}$, only the symmetry breaking parameter $m_{0}$ was left to be fixed in either the light or heavy sector. We choose the latter to fix it. Clearly, the model can and should be made more realitic through the use of improved holographic QCD KIRITSIS .

The shortcomings of the heavy-light holographic approach stem from the triple limits of large $N_{c}$, strong ′t Hooft coupling $\lambda=g^{2}N_{c}$, and heavy meson mass. The corrections in $1/m_{H}$ are straighforward but laborious and should be studied as they shed important light on the hyperfine type splittings. Also, it should be useful to explore the sensitivity of our results by relaxing the value of $M_{KK}$ as fixed in the light meson sector and addressing the strangeness mass correction beyond leading order perturbation theory. The one-meson radiative decays of the heavy baryons and their exotics can be addressed in this model for further comparison with the experimentally reported partial widths.

VII Acknowledgements

This work was supported by the U.S. Department of Energy under Contract No. DE-FG-88ER40388.

VIII Appendix

In this Appendix we briefly recall the key steps in the collective quantization of the holographic light baryons for both $N_{f}=2,3$ SSXB ; CSLIGHT ; KOJI . For $N_{f}=2$ and no heavy-meson (1) describes the light meson sector. In the large $\lambda$ limit using the same rescaling (III.1) to re-write the contributions of the light gauge fields, we have

[TABLE]

Here $A$ refers to the SU(2) part of the light gauge field, and $\hat{A}$ to its U(1) part. The equation of motion for $A_{M},\hat{A}_{M}$ are at leading order of $\lambda$

[TABLE]

They are solved using the flat instanton $A_{M}$ and 0 for $\hat{A}_{M}$. The equation of motion for the time components are subleading

[TABLE]

To obtain the spectrum we promote the moduli of the solution to be time dependent, i.e.

[TABLE]

Here ${a}_{I}$ refers to the moduli of the global SU(2) gauge transformation. In order to satisfy the constraint equation (52) (Gauss’s law) we need to impose a further gauge transformation on the field configuration

[TABLE]

Inserting the transformed field configuration in the constraint equation, we find that $V$ is solved by

[TABLE]

with $\chi^{a}[{a}_{I}]=-i{\rm Tr}(\tau^{a}a_{I}^{-1}\partial_{t}a_{I})$. Putting the resulting slowly moving field configuration back in the action, allows for the light collective Hamiltonian SSXB

[TABLE]

So for $N_{f}=2$, the eigenstates of $H_{\rho}$ are given by $T^{l}(a)R_{l,n_{\rho}}(\rho)$, where $T^{l}$ are the spherical harmonics on $S^{3}$. Under $SO(4)=SU(2)\times SU(2)/Z_{2}$ they are in the $(\frac{l}{2},\frac{l}{2})$ representations, where the two SU(2) factors are defined by the isometry ${a}_{I}\rightarrow V_{L}{a}_{I}V_{R}$. The left factor is the isospin rotation, and the right factor is the space rotation. This quantization describes ${I}=J=\frac{l}{2}$ states. The nucleon is realized as the lowest state with $l=1$ and $n_{\rho}=n_{z}=0$.

For the SU(3) case most of the analysis remains the same except for two differences: 1/ the Chern-Simons term needs amendment as explained in main text; 2/ both $A_{0}$ and $\hat{A}_{0}$ need to be solved to a non-zero value at the static level as also explained in the main text. With this in mind, a general time-dependent SU(3) rotation $a_{I}$ generates the new collective Hamiltonian $H_{\rho}$ as CSLIGHT

[TABLE]

We note that in holography, the inertia in the $1,2,3$ directions is twice larger than the inertia in the $4,5,6,7$ directions reflecting on the inherent SU(2) character of the flavor instanton in bulk. The $J_{a}$ are the generators of the right representation on the group manifold associated to $a_{I}$. Given a representation $(p,q)$ and right-spin $j$, we have

[TABLE]

The radial wavefunctions and energies associated to the full Hamiltonian

[TABLE]

are found in the form

[TABLE]

The combination $m_{y}\omega_{\rho}\equiv 16\pi^{2}\kappa/\sqrt{6}$ if we remember to unwind the rescaling $\sqrt{\lambda}\rho\rightarrow\rho$ from (III.1). The value of $\kappa$ is fixed by the pion decay constant $f_{\pi}^{2}/M_{KK}^{2}=\kappa/(54\pi^{4})$ SSX . The explicit wavefunctions for the SU(3) representation with assignment $\mu=(p,q)$ are given by

[TABLE]

and the total state with one spinor attached (for a single-heavy baryon) follows by re-coupling

[TABLE]

A similar re-coupling holds for the double-heavy baryons. When evaluating the symmetry breaking contribution through $\left<D_{88}\right>$, we note that the Clebsch-Gordon coefficients play no role since they depend only on $\mu,YI,Y_{R}J_{s}$ and not on $M_{s}$.