Hadron spectroscopy using the light-front holographic Schr\"odinger equation and the 't Hooft equation

Light-front holographic QCD provides a successful first approximation to hadron spectroscopy in the chiral limit of $(3+1)$-dim light-front QCD, where a holographic Schr\"odinger-like equation, with an emerging confining scale, $\kappa$, governs confinement in the transverse direction. In its supersymmetric formulation, light-front holography predicts that each baryon has two superpartners: a meson and a tetraquark, with their degenerate masses being generated by the same scale, $\kappa$. In nature, this mass degeneracy is lifted by chiral symmetry breaking and longitudinal confinement. In this paper, we show that the latter can be successfully captured by the 't Hooft equation of $(1+1)$-dim, large $N_c$, QCD. Together, the holographic Schr\"odinger equation and the 't Hooft equation, provide a good global description of the data across the full hadron spectrum with a universal $\kappa$.


I. INTRODUCTION
In light-front (3 + 1)-dim QCD, the mass of a quark-antiquark meson is given by [1] where Ψ(x, b ⊥ ) is the meson light-front wave function, x the light-front momentum fraction carried by the quark, and b ⊥ (or b ⊥ e iϕ in polar representation) the transverse distance between the quark and the antiquark. In Equation (1), m q and mq are the effective quark and antiquark masses respectively [1]. The "interactions" contain all the complicated QCD dynamics of a bound state, including the effect of higher Fock sectors on the valence sector.
Introducing the light-front variable, allows a factorization of the wave function into a transverse mode, φ(ζ), and a longitudinal mode, X(x): with L = L max z being the light-front orbital angular momentum, and X(x) = x(1 − x)χ(x).
Here the QCD bound-state dynamics are encoded in the potentials U ⊥ (ζ) and U ∥ (x). Their exact derivation from first principles remains an open question.
Light-front holography [1-4] considers the chiral limit, i.e., massless quarks, and neglects longitudinal confinement, thus implying that M ∥ = 0. Remarkably, the form of the transverse confinement potential, U ⊥ (ζ), is uniquely fixed by the underlying conformal symmetry and a holographic mapping to anti-deSitter AdS 5 , resulting in [1][2][3][4] U LFH ⊥ (ζ) = κ 4 ζ 2 + 2κ 2 (J − 1) (8) where J = L + S. In this holographic mapping, the variable ζ maps onto the fifth dimension of AdS 5 . Equation (5), which can be rewritten as maps onto the wave equation for the amplitude of spin-J modes propagating in AdS 5 modified by a quadratic dilaton field. In light-front holography, the longitudinal mode, X(x), is hence not dynamical, i.e., undetermined by Equation (6). It is instead fixed by the holographic mapping of the electromagnetic (or gravitational) pion form factor in physical spacetime and AdS 5 [5, 6], resulting in χ(x) = 1, i.e., The supersymmetric formulation of light-front holography [7-11] provides a unified description of baryons and mesons/tetraquarks where each baryon (viewed as a quarkdiquark system) possesses two superpartners: a (quark-antiquark) meson and a (diquarkantidiquark) tetraquark. This supersymmetric connection stems from the fact that a diquark can be in the same SU c (3) representation as an antiquark, and an antidiquark can be in the same SU c (3) representation as a quark. The supersymmetric holographic Schrödinger equation reads [9] H φ⟩ = M 2 ⊥ φ⟩ (11) where with U M (ζ) = κ 4 ζ 2 + 2κ 2 (L M + S M − 1) , U B (ζ) = κ 4 ζ 2 + 2κ 2 (L B + S D ) (14) where S M is the total quark-antiquark spin and S D is the diquark spin. The 4-plet, φ⟩, is given by where ψ + and ψ − are the two components of the baryon wave function and φ M T is the meson/tetraquark wave function. Equation (11) admits analytical solutions, with its eigenvalues given by and where n ⊥ is the principal quantum number that emerges when solving the holographic Schrödinger equation, and S T is total diquark-antidiquark spin.

II. CHIRAL SYMMETRY BREAKING AND LONGITUDINAL CONFINEMENT
The effects of nonzero quark masses were originally taken into account using a prescription by Brodsky and de Téramond (BdT) [12], which relies on the fact that the holographic ground state wave function depends on the invariant mass of the qq pair. Indeed, in mo- where is the invariant mass of the qq pair, with k being the transverse momentum of the quark.
For massive quarks, the invariant mass becomes Thus, after Fourier transforming back to position space, the BdT prescription amounts to a modification of the longitudinal mode: where Treating the kinetic energy of massive quarks as a first-order perturbation to the holographic potential, Equation (8), the resulting first-order shift to M = M ⊥ due to quark masses is then given by so that the pion mass, M π = ∆M BdT . Similarly, the kaon mass, M K = ∆M BdT when the strange quark is taken into account. Equation (24) can be generalized for baryons and tetraquarks [11]: where with λ = κ 2 . Note that Equation (25) coincides with Equation (24) for n = 2.
Strictly speaking, Equation (24) is only accurate for light hadrons in their ground states.
However, as a first approximation, it is not unreasonable to assume the same correction for excited states. At the same time, this guarantees that the predicted Regge trajectories remain linear. Indeed, Equation (24), together with Eqs. (16), (17), and (18), have been used to fit the light hadron spectrum in Ref. [11], resulting in a universal κ = 0.523 ± 0.024 GeV, with m u d = 0.046 GeV and m s = 0.357 GeV. The use of Equation (24) has also been extrapolated to heavy quarks in order to predict the heavy-light and heavy-heavy hadron spectra in Refs. [7,8,10], with the requirement that κ is no longer universal in the heavy hadron sector. In an attempt to protect the universality of κ in the heavy sector, Ref. [13] introduces another mass scale (instead of κ) in Equation (23).
A theoretical shortcoming of Equation (24) is that, in the chiral limit, it predicts [14] M 2 π ∝ 2m 2 u d (ln κ 2 m 2 u d −γ E ) where γ E ≈ 0.577216 is Euler's constant, which is not consistent with the Gell-Mann-Oakes-Renner (GMOR) relation [15] which states that M 2 π ∝ m u d in the chiral limit. An improved ansatz, consistent with the GMOR relation, has been proposed in Ref. [16]. Recently, this shortcoming has been more rigorously addressed in Refs. [14,17] by solving Equation (6) with a longitudinal confining potential [18,19] where σ is a mass scale. Reference [14] studies light mesons including their excited states while Ref. [17] focuses on the ground states of light and heavy mesons. While using Equation Ref. [20], with the unique goal of predicting the meson decay constants while leaving the predicted mass spectrum unchanged. The latter constraint is imposed by subtracting M 2 ∥ from the 't Hooft potential.
In this paper, we use the 't Hooft equation (without shifting the 't Hooft potential as in Ref. [20]), together with the holographic Schrödinger equation, to compute the full hadron spectrum, i.e., the masses of the ground and excited states of mesons, baryons, and tetraquarks, including those with one or two heavy quarks. In Ref. [21], only the meson spectrum was studied. We shall also compare our results with the mass spectrum obtained using the alternative longitudinal potential given by Equation (27).

III. THE 't HOOFT EQUATION
The 't Hooft equation is derived from the QCD Lagrangian in (1 + 1)-dim and in the large N c approximation where only planar diagrams contribute. The result is [22]: where g = g s √ N c is the (finite) 't Hooft coupling and P denotes the principal value prescription. Note that Equation (28) can also be derived in the continuum limit of discretized (1 + 1)-dim light-front QCD [23]. We use the 't Hooft equation for baryons and tetraquarks by making the transformationq → [qq] for baryons, followed by q → [qq] for tetraquarks.
Since the color interaction is invariant under these transformations, the longitudinal confinement scale, g, remains the same within a family of superpartners. The 't Hooft equation has been extensively studied in the literature [24][25][26][27][28][29][30][31][32] and it is worth noting that, in the conformal limit, it possesses a gravity dual on AdS 3 [33]. Here, we only highlight how it is both complementary and consistent with the holographic Schrödinger equation First, assuming a 't Hooft wave function of the form [17] then, near the endpoints, x → 0, 1, Equation (30) implies that, in the chiral limit, m i → 0, and Since, via Equation (16), the holographic Schrödinger equation predicts a massless pion, it follows that the only contribution to the pion mass is generated by the 't Hooft equation.
Thus, together, the holographic Schrödinger equation and the 't Hooft equation correctly predicts the GMOR relation, M 2 π ∼ m u d . Notice also that in the chiral limit, β i → 0 and χ(x) = 1 as in light-front holography.
Second, as already noted in Ref. [21], the 't Hooft potential is consistent with the holographic potential in that they both correspond to a linear confining potential in the non-relativistic limit, with g = κ [21]. To show this, first note that the 't Hooft potential in position space reads where x − = x 0 −x 3 is the light-front longitudinal distance, and P + is the light-front longitudinal momentum of the meson. The second equality follows from the fact that Using the general relation between a frame-invariant light-front (LF) potential and a center of mass (CM) frame instant-form (IF) potential [35]: it follows from Equation (34) that the (chiral-limit) quadratic light-front holographic potential, Equation (8), corresponds to a linear instant form potential. The light-front 't Hooft potential, Equation (33), also corresponds to a linear instant form potential in the heavy quark (non-relativistic) limit where m Q ≫ g and P + → 2m Q , where m Q is the heavy quark mass. In this non-relativistic limit, it is also appropriate to take x → 1 2 in Equation (8).
Therefore, we see that, in the non-relativistic limit, the holographic potential and the 't Hooft potential are both equivalent to CM-frame linear instant-form potentials: and It then follows that V ⊥ = V ∥ , i.e., rotational symmetry is restored in the CM frame of a heavy-heavy meson if g = κ.
Third, when solving the 't Hooft equation, an additional quantum number, which we shall label as n ∥ , emerges. Thus, each hadronic state is identified by four quantum numbers, n ⊥ , L, S, n ∥ , with their squared masses given by and their parity given by In addition, where appropriate, the charge conjugation of mesons/tetraquarks is given by A posteriori, we find that i.e., in any hadron, an orbital and/or radial excitation in the transverse dynamics is always accompanied by an excitation in the longitudinal dynamics. This is a signature of the [20], and already used in Ref. [21].  Table I, together with κ = 0.523 GeV for all hadrons. In comparison to Ref. [21], we allow the light quark masses to vary between light and heavy-light hadrons, with the goal of achieving a better global description of the full hadron spectrum. We adopt the simplest assumption that the (anti)diquark mass is the sum of the (anti)quark masses, i.e., the (anti)diquark is essentially a cluster of two (anti)quarks.
Note that we identify the superpartners as in Ref. [9]. Our results are shown in Table III for the light hadrons, in Table IV for hadrons with one heavy quark, and in Table V for hadrons with two heavy quarks.
In general, our meson and baryon masses are in good agreement with the data, with a discrepancy not exceeding 13%, except for η ′ (958), where it is 21%. For the tetraquark candidates, the discrepancy does not exceed 18%, with notable exceptions for f 0 (500) and f 0 (980)(a 0 (980)), where the discrepancies are very large. The agreement for mesons and baryons is thus more impressive than for tetraquarks. In fact, the large discrepancies for the tetraquark candidates, f 0 (500) and f 0 (980)(a 0 (980)), are already present before any supersymmetry breaking by longitudinal dynamics. This can be seen by comparing the values of M ⊥ to the physical masses of f 0 (500) and f 0 (980)(a 0 (980)) in Table III. Thus these discrepancies are mostly inherited from the original formulation of supersymmetric light-front holography, and are not alleviated by longitudinal dynamics.
It is instructive to compare our results with those obtained using the alternative longitudinal potential given by Equation (27). In this case, an analytical formula can be derived for the meson mass spectrum [14]: Reference [17] shows that the restoration of rotational symmetry in heavy-heavy mesons implies the constraint: which is to be contrasted with the analogous constraint, g = κ, for the 't Hooft potential: see Equation (35)   We now proceed to show the Regge trajectories obtained using the 't Hooft potential: We can see that the Regge slopes are significantly changed with this variation in κ. These observations are also typical for the other Regge trajectories. Therefore, the precise slopes and locations of Regge trajectories, for a given set of quark masses, are sensitive to both g and κ. It is worth highlighting that, in our approach, κ remains universal across the full hadron spectrum.
In Figs. 9 and 10, we compare our results for all baryons, including those with no identified superpartners from the PDG. Using κ = 0.500 ± 0.024 GeV, we achieve very good agreement with the data. 2021PM0023. [