Light-front holographic radiative transition form factors for light mesons

We predict the $\mathcal{V} \to \mathcal{P} \gamma$ decay widths and the $\mathcal{V} \to \mathcal{P} \gamma^{*}$ transition form factors, where $\mathcal{V}=(\rho, \omega, K^*, \phi)$ and $\mathcal{P}= (\pi,K, \eta,\eta^\prime)$, using spin-improved holographic light-front wavefunctions for the mesons. We find excellent agreement with the available data for both the decay widths and the timelike transition form factors extracted from the leptonic conversion decays $\mathcal{V} \to \mathcal{P} l^+ l^-$.


I. INTRODUCTION
The vector-to-pseudoscalar meson radiative transitions, V → Pγ ( * ) , are important probes of quark confinement dynamics, encoded in their transition form factors at zero or low momentum transfer. These non-perturbative form factors are universal and appear in other physical processes like the hadronic light-by-light contribution to the Standard Model prediction of the muon anomalous magnetic moment [1]. On the experimental side, there exists measurements of the V → Pγ decay widths [2] and of the V → Pγ * transition form factors for low-momentum timelike photons. The latter are extracted from the leptonic conversion decays V → Pl + l − : ω → π 0 µ + µ − in the Lepton-G and NA60 experiments [3][4][5]; ω → π 0 e + e − in the A2MM experiment [6] and φ → ηe + e − in the SND and KLOE experiments [7,8]. The discrepancy between the Lepton-G and NA60 data with the prediction of the Vector Meson Dominance (VMD) model has triggered considerable theoretical attention [9][10][11][12], and prompted the measurement of the φ → π 0 e + e − decay by the KLOE experiment [13]. The transition form factors have also been predicted using the Dyson-Schwinger Equations [14], the pQCD factorization approach [15] and, more relevant to this paper, in the light-front formalism [16][17][18], where they are expressed as overlap integrals of the meson light-front wavefunctions.
In our approach, the difference between pseudoscalar and vector mesons lies in the quark-antiquark helicity wavefunction that modifies their universal holographic wavefunction. The dynamical part of the latter satisfies the holographic Schrödinger Equation where ζ = x(1 − x)b ⊥ , with x = k + P + being the light-front momentum fraction carried by the quark and b ⊥ is the magnitude of the transverse separation, b = b ⊥ e iθ⊥ , between the quark and antiquark. Eq. (1) can be derived in light-front QCD in a semiclassical approximation where quark masses and quantum loops are neglected [28][29][30][31]. More interestingly, mapping ζ onto the fifth dimension, z, in anti-de Sitter spacetime, AdS 5 , Eq. (1) becomes the wave equation for the amplitude of spin-J string modes propagating in a modified AdS 5 spacetime, where (2 − J) 2 = L 2 − (µR) 2 with µ being the 5-d mass of the string modes and R the radius of curvature of AdS 5 [30]. The geometry of AdS 5 is distorted by a dilaton field ϕ(z) which drives the confining potential in physical spacetime: with ζ ↔ z. While Eq. (2) is true for an arbitrary dilaton field, only a quadratic confinement potential, U (ζ) = κ 4 ζ 2 , leaves the underlying action leading to Eq. (1) conformally invariant [32], and this, in turn, requires the dilaton field to be also quadratic, ϕ = κ 2 z 2 . Then, Eq.
(2) yields The mass scale κ which simultaneously sets the strength of the dilaton field in AdS 5 and the hadron mass scale in physical spacetime, is referred to as the AdS/QCD mass scale.
The supersymmetrization of Eq. (1) leads to the identification of mesons and baryons (considered as quark-diquark systems) as supersymmetric partners, provided that they differ by only one unit of orbital angular momentum [33][34][35]. In other words, the meson and baryon mass spectra are given by where S is the spin of the quark-antiquark in mesons and the lowest possible value of the diquark spin in baryons. The lightest hadron (with J = L = S = 0) is massless, with no supersymmetric partner, and is naturally identified with the pion. At this point, the only free parameter is the mass scale, κ, and it can be fixed by a simultaneous fit to the Regge slopes of light mesons and baryons. This fit yields κ = 523 ± 24 MeV [36], which we refer to as the universal AdS/QCD mass scale.
Solving Eq. (1) yields the dynamical part of the holographic meson wavefunction, and the complete meson wavefunction is given by [30] where X(x) is fixed by mapping the spacelike electromagnetic form factor of a generic spinless hadron in AdS 5 and in physical spacetime [37]. In AdS 5 , the form factor is given by an overlap integral of the ingoing and outgoing hadronic modes convoluted with the bulk-to-boundary propagator which maps onto the free electromagnetic current in physical spactime. In physical spacetime, the form factor is given by an integral overlap of the meson light-front wavefunctions, i.e. the Drell-Yan-West formula [38,39]. This procedure yields [30]. Matching of the AdS 5 and physical spacetime gravitational form factors gives an identical result [37].
The normalized holographic light-front wavefunction for mesons with n = L = 0 is given by or, in momentum space, where M 2 = k 2 ⊥ xx is the invariant mass of the quark-antiquark pair. Here k ⊥ is the magnitude of the two-dimensional transverse momentum k = k ⊥ e iθ k ⊥ which is the Fourier conjugate of the transverse distance b between the quark and the antiquark. For non-zero quark masses, this invariant mass should be M 2 where f andf ′ denote the flavours of the quark and antiquark respectively. This motivates a prescription [40] to account for light quark masses: replace M 2 by M 2 ff ′ in Eq. (8). Then, the holographic wavefunction becomes So far, the quark and antiquark helicity indices have been suppressed [30]. Making them explicit, we have where and With a universal AdS/QCD scale, this would lead to degenerate decay constants for the pseudoscalar and vector mesons, as well as degenerate decay constants for the longitudinally and transversely polarized vector mesons, in contradiction with experiment [2] and lattice QCD [41,42]. Indeed, in light-front holography, there is no distinction between the dynamical wavefunctions of light pseudoscalar and vector mesons: see Eq. (5).

II. DYNAMICAL SPIN EFFECTS
The above shortcomings can be addressed by taking into account dynamical spin effects.
The pseudoscalar and vector meson wavefunctions are then given by [19,20,22,23] where Ψ(x, k 2 ⊥ ) is the holographic wavefunction given by Eq. (9), and the Lorentz invariant spin structures are given by with where and where Eq. (15) is modelled upon the photon-quark-antiquark vertex and leads to a successful description of diffractive ρ and φ electroproduction [19,21]. On the other hand, Eq. (17) does not give a good description of the pseudoscalar meson data. However, since the individual terms of Eq. (17) are separately Lorentz invariant, we are able to use the more flexible structure, where A and B are dimensionless constants which quantify the importance of dynamical spin effects. Indeed, setting A = B = 0, we are left with the non-dynamical γ + γ 5 spin structure which yields Eq. (11). Refs. [22,23] choose A = 0, as required by the data, while the situation is less clear for B: the pion data favour B ≥ 1, the (charged) kaon data prefer B = 0. For the η η ′ system, the η η ′ → γγ * transition form factor data prefer B ≫ 1 while the η(η ′ ) → γγ decay widths data prefer B = 0 (B = 1). Consequently, we are compelled to treat B as a free parameter here.
Explicitly, the spin-improved holographic wavefunctions are given by [22,23] while [19,20] and The normalization constants N (L,T ) are fixed using which embodies the assumption that the meson consists only of a quark-antiquark pair.

III. RADIATIVE TRANSITION FORM FACTORS
The transition form factors, F VP (Q 2 ), are defined by [16] iF where P (P ′ ) is the 4-momentum of the vector (pseudoscalar) meson, q 2 = (P ′ − P ) 2 is the spacelike 4-momentum transfer, and J µ em (0) is the quark electromagnetic current. To leading order in α em , there are two contributions to the radiative transition matrix element, with the photon being either radiated by the quark or the antiquark, as shown in Fig. 1. Focusing on states with a specified flavour content, we can write: with and where, for notational simplicity, we have suppressed the helicity and colour indices. For the non-strange mesons, P = (π, η, η ′ ) and V = (ρ, ω, φ), Eq. (26) and Eq. (27) where G P,V and I P,V are the G-parity and isospin quantum numbers. For the non-strange mesons, the I G assignments are: π 0,± (1 − ), ρ 0,± (1 + ), η η ′ (0 + ), and φ ω(0 − ), implying that i.e. the two Feynman graphs of Fig. 1 differ only by a minus sign. This is not the case for transitions involving the strange mesons. To proceed, we choose the "good" current, J + em (0), in the Drell-Yan-West frame [38,39] where i.e. with q + = 0 and q 2 = −q 2 ⊥ < 0. This choice avoids the zero-mode contributions [45] but, at the same time, restricts the computation of the transition form factor to the spacelike region where Q 2 ≡ −q 2 > 0. However, it is possible to analytically continue the spacelike form factor to the timelike region using the prescription q ⊥ → iq ⊥ [18,46,47]. Note that the "good" current matrix element vanishes for λ = L, and therefore we must take λ = T (here we choose T = +) in order to extract F VP (Q 2 ). Fock expanding the meson states and using Eqs. (26) and (27), we find that where we have used the shorthand notation, ⨋ ≡ ∑ h,h ∫ . Using our spin-improved holographic wavefunctions, given by Eqs. (20) and (22), we obtain whereÑ ≡ N N T κ 2 (8π 2 ) and Inserting Eqs. (32) and (33) in Eq. (31), Eq. (24) leads to where For the non-strange mesons, m f = mf′, the two integrals in Eq. (35) are identical, i.e. where Eq. (37) is consistent with the model-independent expectation expressed by Eq. (29) and it implies that F ρ ± π ± (Q 2 ) = F ρ 0 π 0 (Q 2 ) and F ω 0 π 0 (Q 2 ) = 3F ρ 0 π 0 (Q 2 ). For the strange mesons, we must instead use Eq. (35), leading to the interesting possibility of destructive interference between the two Feynman diagrams of Fig. 1 for the K * ± → K ± γ * transition. We shall discuss this further in Section IV.
For the neutral mesons, (η, η ′ ) and (φ, ω), we need to account for mixing. Although the φ − ω mixing is small, it is essential to account for the φ → π 0 γ * transition. We use the SU(3) octet-singlet mixing scheme where with η 8 ω 8 ⟩ = 1 √ 6 (uū + dd − 2ss) and η 1 ω 1 ⟩ = 1 √ 3 (uū + dd + ss). It then follows that [48] ⎛ ⎜ ⎝ and cos θ V cos θ P − cos θ V sin θ P − sin θ V cos θ P sin θ V sin θ P cos θ V sin θ P cos θ V cos θ P − sin θ V sin θ P − sin θ V cos θ P sin θ V cos θ P − sin θ V sin θ P cos θ V cos θ P − cos θ V sin θ P sin θ V sin θ P sin θ V cos θ P cos θ V sin θ P cos θ V cos θ P where, using Eq. (37), with [23] ⎛ ⎜ ⎝ Evaluating the transition form factors at Q 2 = 0 allow us to predict the radiative decay widths: and, as mentioned before, to predict the timelike transition form factor, we use the prescription q ⊥ → iq ⊥ in Eq. (36) which then reads: As expected, Eq. (53) diverges for q 2 ⊥ ≥ 4m 2 u d , corresponding to the kinematic threshold for quark-antiquark production. Since we do not account for the latter here, we shall restrict our predictions in the timelike region below this threshold.
In order to reproduce the non-perturbative pole structure of the form factor in the timelike region, above the quark-antiquark production threshold, one must use the confined bulk-toboundary propagator, i.e. one which propagates in the dilation-modified AdS 5 spacetime and maps onto a "dressed" (i.e. incorporating higher Fock states) electromagnetic current in physical spacetime [30]. The resulting form factor also reproduces the VMD behaviour in the low momentum region, as well as the hard scattering power scaling behaviour at large Q 2 . This technique has been used to predict the pion electromagnetic form factor [30,49], the (π 0 , η, η ′ ) → γ * γ transition form factors [50,51] as well as the nucleon electromagnetic form factors in the spacelike region [52,53].
Our predictions for the (ρ, ω, φ) → πγ radiative decay widths are shown in Table I. As can be seen, B ≥ 1 is favoured by the data, corroborating the findings of Ref. [23] that B ≥ 1 is favoured for the pion. This is further supported by our predictions for the ω → π 0 γ * timelike transition form factor, as shown in Fig. 2. The empirical pole fit (dotted-green curve) is generated using Eq. (54) with Λ = 0.676 GeV, the average of the Lepton-G, A2MM and NA60 values, and it agrees very well with our B ≥ 1 predictions (solid-black and dot-dashed red curves). Our predictions for the φ → π 0 γ * timelike transition form factor are shown in  further adjustment of parameters.
Pole fit For the K * 0 → K 0 γ and K * ± → K ± γ decay widths, Table II shows that B = 1 accommodates the data for both the neutral and charged decay modes. Note that the theory uncertainty is amplified for the latter because of the destructive interference between the two Feynman graphs of Fig. 1. At first glance, the preference for B = 1 for the charged decay mode may seem in disagreement with the findings of Ref. [23], where B = 0 is reported to be preferred by decay constant, electromagnetic elastic form factor and radius data for charged kaons. However, we must emphasize that taking 0 < B ≪ 1, say B = 0.2, still fits the radiative width data in Table II, as well as all data in Ref. [23]. On the other hand, as can be seen in Table II, B < 1 is excluded for the neutral decay mode. As we mentioned before, destructive interference occurs only in the charged decay mode, leading to a zero (at leading order) in the transition form factor in the spacelike region. This is shown in Fig. 4. We note that the location of the zero is sensitive to the strength of SU(3) flavour symmetry breaking, shifting to lower Q 2 as the difference between m s and m q increases, as was pointed out previously in Refs. [17,56], although the precise location of the zero is very much model-dependent.  [2].
In Table III, we show our predictions for the radiative decays to η and η ′ where an additional theory uncertainty results from the η η ′ mixing angle. Clearly, B ≥ 1 is preferred by the data. This is consistent with the findings of Ref. [23] where it is reported that B ≥ 1 is also preferred by the η η ′ → γ * γ transition form factor data. In Fig. 5, we compare our predictions for the φ → ηγ * transition to KLOE and SND data. In this case, the data cannot discriminate between the B = 0 (dashed-blue curve) and B ≥ 1 (solid-black and dotdashed-red curves) predictions which start to differ only at large momentum transfer where the experimental error bars are much larger. Both the B = 0 and B ≥ 1 curves agree with the empirical pole fit (dotted-green curve) which is now generated with Λ PDG = 0.88 ± 0.04 GeV. Finally, we also predict the η ′ → (ρ, ω)γ decay widths given by Our results are shown in Table IV where we find that B ≥ 1 is again favoured by the data.

V. CONCLUSIONS
We have used the spin-improved holographic light-front wavefunctions for the light vector mesons (ρ, ω, K * , φ) and pseudoscalar mesons (π, K, η, η ′ ) to predict the radiative transition form factors and decay widths. We find excellent agreement with the available data for the decay widths as well as the timelike transition form factors in the low-momentum region.
Our findings support the idea that light pseudoscalar and vector mesons share a universal holographic light-front wavefunction which is modified differently by dynamical spin effects.