Extending -front holographic QCD using the ’t Hooft Equation

Mohammad Ahmady,1, ∗ Harleen Dahiya,2, † Satvir Kaur,2, ‡ Chandan Mondal,3, 4, 5, § Ruben Sandapen,6, ¶ and Neetika Sharma7, ∗∗ 1Department of Physics, Mount Allison University, Sackville, New Brunswick, E4L 1E6, Canada. 2Department of Physics, Dr. B. R. Ambedkar National Institute of Technology, Jalandhar 144011, India 3Institue for Modern Physics, Chinese Academy of Sciences, Lanzhou-730000, China 4School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 100049, China 5CAS Key Laboratory of High Precision Nuclear Spectroscopy, Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China 6Department of Physics, Acadia University, Wolfville, Nova Scotia, B4P 2R6, Canada. 7Department of Physical Sciences, I K Gujral Punjab Technical University, Kapurthala-144603, Punjab, India. We show the ’t Hooft Equation and the light-front holographic Schr¨odingerEquation are comple- mentary to each other in governing the transverse and longitudinal dynamics of colour confinement in -antiquark . Together, they predict remarkably well the light, heavy-light and heavy- heavy spectroscopic data. The universal emerging hadronic scale of light-front holography, κ ≈ 0.5 GeV, controls the transverse dynamics of confinement in all these mesons. In heavy-heavy mesons, it also coincides numerically with the ’t Hooft coupling which governs longitudinal confine- ment, thus reflecting the restoration of manifest 3-dimensional rotational symmetry.

INTRODUCTION T´eramond[11, 14–16], exploits this conformal limit in the Hamiltonian formulation of (3 + 1)-dim QCD on the Although QCD is the accepted theory for the strong light front, with Nc = 3. The valence meson light-front interactions, it is not yet possible to predict the experi- wavefunction then factorizes as: mentally observed spectrum from first principles. φ(ζ) This is due to our incomplete understanding of the non- Ψ(x, ζ, ϕ) = √ eiLϕX(x) (2) perturbative aspects of QCD responsible for colour con- 2πζ

finement. While much progress is being made with nu- p iϕ merical simulations on the lattice [1], complementary in- where ζ = x(1 − x)b⊥ with b⊥ = b⊥e , being the sights into non-perturbative QCD can be obtained from transverse separation between the quark and the anti- quark. x = k+/P + is the fraction of the meson’s light- the AdS/CFT duality [2, 3] which refers to a correspon- + dence between in a higher dimensional anti de front momentum, P , carried by the quark and L is the Sitter (AdS) space and a conformal field theory (CFT) orbital angular momentum quantum number. The trans- in a lower-dimensional space. The prototypical example verse mode, φ(ζ), satisfies the holographic Schr¨odinger discovered by Maldacena [4] is the duality between su- Equation: persymmetric SU(Nc) Yang-Mills theory with gauge√ cou-  2 2  d 1 − 4L 2 pling, gs, in the ’t Hooft limit (Nc  1 with g = gs Nc − − + U (ζ) φ(ζ) = M φ(ζ) (3) dζ2 4ζ2 T T finite) and Type IIB on AdS5 × S5. The AdS/CFT correspondence may be extended to a more with general gauge-gravity duality which does not simultane- ously require supersymmetry, conformal symmetry and U (ζ) = κ4ζ2 + 2κ2(J − 1) (4) the ’t Hooft limit [5–13]. T arXiv:2105.01018v1 [hep-ph] 3 May 2021 QCD is not conformally invariant. The SUc(3) invari- where J is the meson’s spin. Eq. (4) is the holographic ant (3 + 1)-dim QCD Lagrangian is: potential at equal light-front time, x+ = 0. While the 1 derivation of Eq. (4) in QCD remains an open question, L = Ψ(¯ iγµD − m)Ψ − Ga Gaµν (1) QCD µ 4 µν its form is uniquely fixed by the underlying conformal symmetry and a holographic mapping to AdS5 [17]. In a a a a a where Dµ = ∂µ − igsAµT , and Gµν = ∂µAν − ∂ν Aµ + this mapping, the variable ζ is identified with the fifth di- abc b c a b abc c gsc AµAν with [T ,T ] = ic T , contains two mass mension of AdS5, and Eq. (3) becomes the scales: the Higgs-generated current quark mass, m, and for spin-J bosonic modes propagating in AdS5 the scale ΛQCD in the running coupling, generated af- distorted by a quadratic dilaton field [18]. The emerg- ter perturbative renormalization beyond tree-level. How- ing hadronic scale, κ, generates the meson masses in the ever, if we neglect quark masses (m → 0) and ignore absence of quark masses and ΛQCD. The longitudinal quantum loops (no ΛQCD), QCD possesses an underly- mode, X(x), is fixed by the holographic mapping of the ing conformal symmetry. electromagnetic (or gravitational) form factor in physical Light-front holography, pioneered by Brodsky and de spacetime, resulting in X(x) = px(1 − x) [19, 20]. 2

The holographic Schr¨odingerEquation admits analyt- Hooft Equation to go beyond the BdT prescription was ical solutions: first proposed in Ref. [35], with the goal of predicting the meson decay constants. Very recently, Refs. [36, 37] also −κ2ζ2  φ (ζ) ∝ ζ1/2+L exp LL (κ2ζ2) (5) go beyond the BdT prescription using a phenomenologi- nT L nT 2 cal longitudinal confinement potential, first proposed in and Ref. [38] in the context of basis light-front quantization. While both Refs. [36, 37] focus on the chiral limit and  J + L M 2 (n , J, L) = 4κ2 n + (6) the phenomenon of chiral symmetry breaking, Ref. [36] T T T 2 extends their analysis to heavy mesons in their ground state, and discusses the relation of their approach to the where J = L+S and S is the total quark-antiquark spin, 1 ’t Hooft Equation. i.e. S = 0 or 1. Importantly, Eq. (6) predicts that the In this letter, we show that the ’t Hooft Equation is pion is massless: complementary to, and consistent with, the holographic Schr¨odingerEquation. Together, they capture the main Mπ = MT (0, 0, 0) = 0 (7) features of 3-dimensional confinement dynamics in (non- just as expected in the chiral limit. Eq. (6) also correctly exotic) mesons and successfully predict their full spec- predicts the Regge-like linear dependence of the meson trum. mass squared on the radial and orbital quantum numbers. Light-front holography needs to be extended to accom- modate non-zero quark masses that generate the physical THE ’t HOOFT EQUATION pion mass. This was originally done using a prescription by Brodsky and de T´eramond(BdT) [21], resulting in a In an earlier approach [39], ’t Hooft derived a first-order shift to the mass spectrum given by Schr¨odinger-like equation for the longitudinal mode, starting from the QCD Lagrangian in (1 + 1)-dim in the 2 2 ! Z dx m m Nc  1 approximation. This Lagrangian now contains ∆M 2 = X2 (x) q + q¯ (8) BdT x(1 − x) BdT x 1 − x two mass scales: the quark mass and the gauge coupling. The resulting ’t Hooft Equation is: where 2 2 ! mq mq¯ 2 2 2 ! + X(x) + UL(x)X(x) = MLX(x) , (10) (1 − x)mq + xmq¯ x 1 − x X (x) = px(1 − x) exp − (9) BdT 2κ2x(1 − x) with so that M = ∆M . Similarly, M = ∆M when π BdT K BdT g2 Z |X(x) − X(y)| the strange quark is taken into account. Using the BdT UL(x)X(x) = P dy 2 (11) prescription, a global fit to the spectroscopic data of light π (x − y) 2 , using mu/d = 0.046 GeV and ms = 0.357 where P denotes the principal value prescription and GeV, yields κ = 0.523 ± 0.024 GeV [22]. The BdT pre- √ g = gs Nc is the (finite) ’t Hooft coupling with mass scription, together with a universal κ ≈ 0.5 GeV, have dimensions which plays the role of ΛQCD. Together with been widely used in a successful phenomenology of light the quark masses, it generates the meson masses. The mesons [25–29]. The same prescription has also been ’t Hooft potential, Eq. (11), is derived by summing an used to accommodate heavy , leading to the con- √ infinite number of planar ladder and rainbow diagrams clusion that κ ∝ mQ, where mQ is the heavy quark at x+ = 0, and in the light-front gauge, A+ = 0. mass, in order to be consistent with Heavy Quark Ef- Using the fact that, k+ = xP +, is conjugate to the fective Theory (HQET) [30] and spectroscopic data [31– light-front distance, x−, the Fourier transformation of 33]. In other words, when the BdT prescription is used Eq. (11) yields for heavy quarks, the universality of κ seems to be lost. Ref. [34] attempts to prevent this by using a new scale g2 U (x−) = P +|x−| , (12) λ 6= κ in Eq. (9), thus hinting at the possibility that L 2 the longitudinal mode is the solution of Schr¨odinger-like + Equation different from Eq. (3). The idea to use the ’t and, since x = 0, we can rewrite Eq. (12) as 2 + UL(bk) = g P |bk| , (13)

where we have chosen the notation x3 ≡ b . Therefore, 1 In light-front holography, the quark spin wavefunction is assumed k + to decouple from the confinement dynamics. in the meson’s rest frame, where P = M, the ’t Hooft 2 Supersymmetric light-front holography [22–24] provides a unified potential corresponds to the Coulomb potential which is framework for and mesons/tetraquarks. linear in one space dimension. 3

We start by investigating the consistency of the ’t PREDICTING THE MESON SPECTRUM Hooft Equation with light-front holography by taking the conformal limit: (m, g) → 0. The end-point anal- We compute the meson mass spectrum using [35, 38] ysis of the ’t Hooft Equation with mq = mq¯ = m, using β β 2 2 2 the ansatz X(x) = x (1 − x) yields the transcendental M (nL, nT , J, L) = MT (nT , J, L) + ML(nL) (19) equation: 2 2 where MT (nT , J, L) and ML(nL) are the eigenvalues of m2π Eq. (3) and Eq. (10) respectively. Using the light- − 1 + πβ cot πβ = 0 . (14) g2 front parity and charge conjugation operators given in Ref. [46], we predict the parity and charge conjugation L+1 In the conformal limit, Eq. (14) predicts that β = 1/2 if quantum numbers to the meson to be P = (−1) and C = (−1)L+S+nL respectively. Before showing our nu-  g  √ merical predictions, we make two comments that are im- lim = π . (15) g→0 m portant to interpret our results. m→0 First, we use the universal holographic mass scale: Therefore, the ’t Hooft Equation reproduces the light- κ = 0.523 GeV for all mesons. Besides the successful me- front holographic wavefunction, X(x) = px(1 − x), pro- son phenomenology using similar values [25–29] and the MS vided the constraint, Eq. (15), is satisfied. Note that, fact that it also correctly predicts ΛQCD [47], consistency in a carefully constrained conformal limit, the ’t Hooft with HQET also hints at a universal κ (without fixing Equation possesses a gravity dual in AdS3 [40]. its numerical value). According to HQET, the masses Away from the conformal limit, Eq. (15) need not be of heavy-light pseudoscalar and vector mesons in their satisfied. For instance, in the chiral limit, when m → 0 ground state, scale as with g fixed, it is known [36, 41] that the ’t Hooft Equa- P/V 2 MqQ ∼ mQ (20) tion predicts that Mπ ∝ mu/d, which is consistent with the Gell-Mann-Oakes-Renner relation [42]. It is also with known [43] that, in the heavy quark limit, the ’t Hooft −1/2 1 Equation predicts that fM ∝ m , as expected from V P Q MqQ − MqQ ∼ . (21) Heavy-Quark-Effective-Theory (HQET) [44]. In our ap- mQ proach, these results carry over to (3 + 1)-dim since the Using Eq. (19), we predict that holographic Schr¨odingerEquation gives no contribution to the pion mass (see Eq. (7)) and the meson decay con- κ2 M V − M P ∼ (22) stant is only sensitive to the meson wavefunction, Eq. qQ qQ m (2), evaluated at ζ = 0 [19]. Q Unlike the holographic Schr¨odingerEquation, the ’t which is consistent with Eq. (21) if κ is universal and Hooft Equation must be solved numerically. Following does not scale with mQ. Ref. [35], we expand the longitudinal mode onto a Jacobi Second, we expect that g ≈ κ for heavy-heavy mesons polynomial basis: and g to deviate from κ for light and heavy-light mesons. This can be understood starting from the general relation X X(x) = cnfn(x) (16) between a light-front and an instant-form potential in n the confinement region (where kinetic energy is minimal): [48] with 2 ULF = VIF + 4mVIF . (23) β1 β2 (2β2,2β1) fn(x) = Nnx (1 − x) Pn (2x − 1), (17) Eq. (23) is independently true for the light-front holo-

(2β2,2β1) graphic potential and the ’t Hooft potential. This means where P are the Jacobi polynomials and [45] n that s ˜ ˜ 2 ˜ ˜ n!Γ(n + β1 + β2) UT = V⊥ (24) Nn = (2n + β1 + β2) (18) Γ(n + β˜ + 1)Γ(n + β˜ ) 1 2 and ˜ ˜ with β1 ≡ 2β1 and β2 ≡ 2β2 + 1. The resulting ma- 2 UL = Vk + 4mVk , (25) trix representation of Eq. (10) can then be diagonalized numerically. Note that we require our predictions to be where V⊥ and Vk are the instant-form potentials cor- independent of the choice of basis, i.e. to remain stable responding to the holographic and ’t Hooft potentials with respect to variations in β1,2. respectively. For heavy-heavy mesons, the linear term 4

Mesons g mu/d ms mc mb n=2 Light 0.128 0.046 0.357 - - 6 n=2 n=1 Heavy-light 0.680 0.046 0.357 1.370 4.640 ]

2 n=1 4 π2 (1880) ▲n=0 Heavy-heavy 0.523 - - 1.370 4.640 ▲ n=0 ρ(1700) ▲ a4 (1970) GeV ▲ ▲ [ π(1800) ▲ ▲ a2 (1700) ρ (1690) 2 2 π (1670) ▲ 3 ▲ ▲ 2 ▲ TABLE I: The quark masses and ’t Hooft couplings in M ρ(1450) a (1320) π(1300) 2 b1 (1235) ▲ GeV. Note that we use κ = 0.523 GeV for all mesons. 0 ▲ ρ(770) π(140)

n=2 6 n=2 n=1 dominates the right-hand-side of Eq. (25): UL ≈ 4mQVk. ]

2 n=1 + 4 ▲n=0 Using Eq. (13) with, P = M ≈ 2mQ, we deduce that n=0 * ▲ K4 (2045) 2 4 2 GeV ▲ ▲ ▲ V ≈ (g /2)b . On the other hand, UT ≈ (κ /4)b (since [ K*(1680) K* (1780) k k ⊥ 2 3 2 ▲ K2 (1820) ▲ ▲ ▲ * M K (1410) * x ≈ 1/2 in heavy-heavy mesons), and thus Eq. (24) im- K2 (1770) K2 (1430) K1 (1400) ▲ 2 ▲ K (1270) K*(892) plies that V⊥ ≈ (κ /2)b⊥. The 3-dim rotational symme- 0 K(494) 1 try is restored if V⊥ = Vk, i.e. when g ≈ κ. The same n=3 arguments do not apply to heavy-light and light mesons. 8 n=2 n=3

Our numerical values for quark masses and ’t Hooft ] n=2 2 6 n=1 ϕ(2170) n=1 couplings are shown in Table I. Notice that we purpose- n=0 ▲ ▲▲ f2 (2300) GeV

[ ▲n=0 fully choose the same effective light quark masses as in 4 ▲ 2 ω(1650) ▲ f2 (1950) f4 (2050) ▲ ϕ (1850) M 3 ▲ ▲ light-front holography. The corresponding predictions for 2 ϕ(1680) ▲ ω3 (1670) ω(1420) ▲ the Regge trajectories of light mesons is shown in Fig. ▲ f2 (1270) ϕ(1020) ▲ 1. The agreement with data is very good. The qual- 0 ω(782) ity of agreement with data is similar to that achieved 0 1 2 3 0 1 2 3 L L using the BdT prescription for light mesons [11]. The BdT prescription can then be thought as resulting from FIG. 1: Our predictions for the Regge trajectories of the ’t Hooft Equation with a weak longitudinal coupling, light mesons. Data from the Particle Data Group [49]. g  κ: see Table I. Our predictions for the heavy-heavy mesons are shown in Fig. 2. The agreement with data is also very good, with g = κ, as we anticipated from the 25 restoration of rotational symmetry. Finally, our predic- n=2 n=2 ] 2 tions for heavy-light mesons are shown in Fig. 3. As can 20 n=1 n=1 GeV be seen, the agreement with data is good, although less [ n=0 ▲▲ n=0 2 15 ψ(3S) ▲

M χ (2P) impressive (even though the maximum discrepancy never ▲▲ ▲ c2 ▲ ψ(2S) ▲ ηc(2S) χ (1P) exceeds 10%) than for the light and heavy-heavy mesons. hc(1P) c2 10 ▲ ▲ J/ψ(1S) For heavy-light mesons, the data prefer g > κ: see Table ηc(1S) I. Interestingly, g deviates from κ in opposite directions 120 for light and heavy-light mesons. The underlying reason n=2 χb2(3P) n=2 ▲ ] 110 2 n=1 n=1 for this remains to be explored. hb (2P) ▲ χb2(2P) ▲▲ Y(3S) ▲ n=0 n=0 GeV ▲ We emphasize that, except for pseudoscalar mesons in [ 100 ▲ ηb (2S) ▲ Y(2S) ▲ 2 hb (1P) χb2(1P) their ground states (for which MT = 0), the precise lo- M 90 ▲ cations and slopes of the Regge trajectories of all other ▲ Y(1S) ηb (1S) mesons are sensitive to both g and κ. Therefore the uni- versality of κ across the full spectrum is non-trivial. 0 1 2 0 1 2 L L

FIG. 2: Our predictions for the Regge trajectories of CONCLUSIONS heavy-heavy mesons. Data from the Particle Data Group [49]. We have shown that the meson spectrum can be very well described by using the holographic Schr¨odinger Equation in conjunction with the ’t Hooft Equation. We find that the emerging hadronic scale of light-front holog- ACKNOWLEDGEMENTS raphy remains universal across the full spectrum. For heavy-heavy mesons, it coincides with the ’t Hooft cou- RS and MA are supported by individual Discov- pling, as expected from the restoration of manifest 3- ery Grants (SAPIN-2020-00051 and SAPIN-2021-00038) dimensional rotational symmetry in the non-relativistic from the Natural Sciences and Engineering Research limit. Council of Canada (NSERC). CM thanks the Chinese 5

∗∗ 10 n=0 [email protected] n=0 [1] B. Jo´o,C. Jung, N. H. Christ, W. Detmold, R. Edwards,

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