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Spin effects in the pion holographic light-front wavefunction

We account for dynamical spin effects in the holographic light-front wavefunction of the pion in order to predict its mean charge radius, decay constant, spacelike electromagnetic form factor, twist-2 Distribution Amplitude and the photon-to-pion transition form factor. Using a universal fundamental AdS/QCD scale of 523 MeV and a constituent quark mass of 330 MeV, we find a remarkable improvement in describing all observables.


I. INTRODUCTION
Hadronic light-front wavefunctions (LFWFs) provide the underlying link between the fundamental degrees of freedom of QCD, i.e. quarks and gluons, and their asymptotic hadronic states. LFWFs thus encode both the physics of confinement and chiral symmetry breaking, which are fundamental, intimately related [1] and yet not fully understood, emergent properties of QCD. In phenomenology, LFWFs are extremely important since all hadronic properties can, in principle, be derived from them. For example, in the exclusive decays of the B meson to light mesons, which are under intense investigation at the LHCb experiment, the theoretical non-perturbative inputs, i.e. the meson decay constants, Distribution Amplitudes and transition form factors can all be computed if the LFWFs of the mesons are known. These non-perturbative inputs are in fact the major source of theoretical uncertainties in current Standard Model predictions [2].
In principle, LFWFs are obtained by solving the LF Heisenberg equation for QCD: [3] H LF QCD |Ψ(P ) = M 2 |Ψ(P ) (1) where H LF QCD = P + P − − P 2 ⊥ is the LF QCD Hamiltonian and M is the hadron mass. At equal light-front time (x + = 0) and in the light-front gauge A + = 0, the hadron state |Ψ(P ) admits a Fock expansion, i.e. |Ψ(P + , P ⊥ , S z ) = n,h i where Ψ n (x i , k ⊥i , h i ) is the LFWF of the Fock state with n constituents and the integration measures are given by LFWFs are quantum mechanical probability amplitudes that depends on the momenta fraction x i = k + i /P + , the transverse momenta k ⊥i , and the helicities h i of the constituents. In practice, it is difficult, if not impossible, to solve Eq. (1) since it contains an infinite number of strongly coupled integral equations. Various approximation schemes involve truncating the Fock expansion, using discretized light-front quantization or solving the equations in a lower number of spatial dimensions. For a review of light-front quantum field theories, we refer to [4].
A remarkable breakthrough during the last decade is the discovery by Brodsky and de Téramond [5][6][7][8] of a higher dimensional gravity dual to a semiclassical approximation of light-front QCD. The result is a relativistic Schrödinger-like wave equation for mesons that can be solved analytically to predict meson spectroscopy and LFWFs in terms of a single mass scale κ. The approach can also include baryons [7] and more recently has been extended to a unified framework for baryons and mesons considered as conformal superpartners [9].
This gauge/gravity duality is referred to as light-front holography (LFH) and is reviewed in Ref. [3].
In the semiclassical approximation, quark masses and quantum loops are neglected, the LFWFs depend on the invariant mass M 2 = ( n i k i ) 2 of the constituents rather than on their individual momenta k i . For the valence (n = 2 for mesons) Fock state, the invariant mass of the qq pair is M 2 qq = k 2 ⊥ /x(1 − x) and the latter is the Fourier conjugate to the impact variable ζ 2 = x(1−x)b 2 where b is the transverse separation the quark and antiquark.
The valence meson LFWF can then be written in a factorized form: where the helicity indices have been suppressed [3]. We note that this suppression of the helicity indices is legitimate if either the constituents are assumed to be spinless or if the helicity dependence decouples from the dynamics. It can then be shown that Eq. (1) reduces to a 1-dimensional Schrödinger-like wave equation for the transverse mode of LFWF of the valence (n = 2 for mesons) state, namely: where all the interaction terms and the effects of higher Fock states on the valence state are hidden in the confinement potential U (ζ). The latter remains to be specified and, at present, this cannot be done from first principles in QCD.
However, Brodsky and de Téramond found that Eq. (5) maps onto the wave equation for the propagation of spin-J string modes in the higher dimensional anti-de Sitter space, AdS 5 , if the impact light-front variable ζ is identified with z 5 , the fifth dimension of AdS space and the light-front orbital angular momentum L 2 is mapped onto (mR) 2 − (2 − J) 2 where R and m are the AdS radius and mass respectively. For this reason, we refer to Eq. (5) as the holographic LF Schrödinger equation. In this AdS/QCD duality, the confining potential in physical spacetime is driven by the deformation of the pure AdS 5 geometry. Specifically, the potential is given by where ϕ(z 5 ) is the dilaton field which breaks conformal invariance in AdS space. A quadratic dilaton, ϕ(z 5 ) = κ 2 z 2 5 , profile results in a light-front harmonic oscillator potential in physical spacetime: since z 5 maps onto the LF impact variable ζ. Remarkably, Brodsky, Dosch and de Téramond have shown that the quadratic form of the AdS/QCD potential is unique [10]. In fact, starting with a more general dilaton profile ϕ ∝ z s 5 and requiring the pion to be massless, uniquely fixes s = 2 [11]. More formally, applying the mechanism of de Alfaro, Furbini and Furlan [12] (which allows the emergence of a mass scale in the Hamiltonian of a conformal 1-dimensional QFT while retaining the conformal invariance of the underlying action) to semiclassical LF QCD uniquely fixes the quadratic form of the AdS/QCD potential [10].
With the confining potential specified, one can solve the holographic Schrödinger equation to obtain the meson mass spectrum, which, as expected, predicts a massless pion. The corresponding normalized eigenfunctions are given by To completely specify the holographic meson wavefunction, we need the analytic form of the longitudinal mode X (x). This is obtained by matching the expressions for the pion EM or gravitational form factor in physical spacetime and in AdS space. Either matching consistently results in X (x) = x(1 − x) [5,13]. The meson holographic LFWFs for massless quarks can thus be written in closed form: with the corresponding meson masses lying on linear Regge trajectories as given by Eq. (8).
The reasons why a solution to a quantum field theory could reduce to a solution of a simple, one-dimensional differential equation are explored in Ref. [14].
For phenomenological applications, it is necessary to restore both the quark mass and helicity dependence of the holographic LFWF. In fact, it has recently been shown in Ref.
[15] that non-zero light quark masses drastically improves the description of data on the photon-to-pion, photon-to-η and photon-to-η transition form factors. On the other hand, for the pion and kaon EM form factors, the description of data actually worsens unless 3 data points at large Q 2 are excluded for the kaon form factor [15]. Accounting for non-zero light quark masses means going beyond the semiclassical approximation, and this is usually done following the prescription of Brodsky and de Téramond [16]. For the ground state pion, this leads to where N is a normalization constant fixed by requiring that where P qq is the probability that the meson consists of the leading quark-antiquark Fock state.
Note Eq. (8) tells us that the AdS/QCD scale κ can be chosen to fit the experimentally measured Regge slopes. Ref. [3] reports κ = 590 MeV for pseudoscalar mesons and κ = 540 MeV for vector mesons. A recent fit to the Regge slopes of mesons and baryons, treated as conformal superpartners, yields κ = 523 MeV [9]. On the other hand, the AdS/QCD scale κ can be connected to the scheme-dependent pQCD renormalization scale Λ QCD by matching the running strong coupling in the non-perturbative (described by lightfront holography) and the perturbative regimes [17]. With κ = 523 MeV and the β-function of the QCD running coupling at 5-loops, Brodsky, Deur and de Téramond recently predicted the QCD renormalization scale, Λ M S QCD , in excellent agreement with the world average value [18]. Furthermore, light-front holographic wavefunctions have also been used to predict diffractive vector meson production [19,20]. A fit to the HERA data on diffractive ρ electroproduction, with m u/d = 140 MeV, gives κ = 560 MeV [19] and using κ = 550 MeV 46[140] MeV) leads to a good simultaneous description of the HERA data on diffractive ρ and φ electroproduction [20]. These findings hint towards the emergence of a universal fundamental AdS/QCD scale κ ∼ 550 MeV. In the most recent application of LFH to predict nucleon EM form factors [21], it is pointed out that this universality holds up to 10% accuracy. In this paper, we shall use the value of κ = 523 MeV which fits the meson/baryon Regge slopes and accurately predicts Λ M S QCD [9,18]. In earlier applications of LFH with massless quarks, much lower values of κ were required to fit the pion data: κ = 375 MeV in Ref. [8] in order to fit the pion EM form factor data and κ = 432 MeV (with P qq = 0.5) to fit the photon-to-pion transition form factor data simultaneously at large Q 2 and Q 2 = 0 (the latter is fixed by the π 0 → γγ decay width) [22]. Note that in Ref. [8], the EM form factor is computed, both in the spacelike and timelike regions, as a convolution of normalizable hadronic modes with a non-normalizable EM current which propagates in the modified infrared region of AdS space and generates the non-perturbative pole structure of the EM form factor in the timelike region. Alternatively, the spacelike EM form factor can be computed using the Drell-Yan-West formula [23,24] in physical spacetime with the holographic pion LFWF. The latter approach is taken in Refs. [15,[25][26][27]. In Ref. [15], with m u/d = 330 MeV, the authors use a universal κ = 550 MeV for all mesons but fix the wavefunction normalization for the pion so as to fit the decay constant. Consequently, this implies that P qq = 0.61 only for the pion. In Ref. [26], a higher value of κ = 787 MeV is used with m u/d = 330 MeV and the authors predict P qq = 0.279, implying an important contribution of higher Fock states in the pion. In Ref.
[27], a universal AdS/QCD scale κ = 550 MeV is used for all mesons, together with a constituent quark mass m u/d = 420 MeV, but P qq = 0.6 is fixed for the pion only: for the kaon, P qq = 0.8 and for all other mesons, P qq = 1.
All these previous studies seem to indicate that a special treatment is required at least for the pion either by using a distinct AdS/QCD scale κ or/and relaxing the normalization condition on the holographic wavefunction, i.e. invoking higher Fock states contributions.
This may well be reasonable since the pion is indeed unnaturally light and does not lie on a Regge trajectory, as pointed out in Ref. [25]. However, we note that in the previous studies [15,26,27] where the pion observables are predicted using the holographic wavefunction, given by Eq. (11), the helicity dependence of the latter is always assumed to decouple from the dynamics, i.e. the helicity wavefunction is taken to be momentum-independent. This is actually consistent with the semi-classical approximation within which the AdS/QCD correspondence is exact. Consequently, Ref. [27] derives a single formula to predict simultaneously the vector and pseudoscalar meson decay constants, so that using a universal scale κ and P qq = 1 for all mesons inevitably leads to degenerate decay constants in conflict with experiment.
In this paper, we show that it is possible to achieve a better description of the pion observables by using a universal AdS/QCD scale κ and without the need to invoke higher Fock state contributions. We do so by taking into account dynamical spin effects in the holographic pion wavefunction, i.e. we use a momentum-dependent helicity wavefunction.
This approach goes beyond the semiclassical approximation, just like the inclusion of light quark masses in the holographic wavefunction. However, it does support the idea of the emergence of a universal, fundamental AdS/QCD scale. A similar approach was taken previously for the ρ meson, leading to impressive agreement to the HERA data on diffractive ρ electroproduction [19].

II. DYNAMICAL SPIN EFFECTS
To restore the helicity dependence of the holographic wavefunction, we assume that where the helicity wavefunction S hh (x, k) corresponds to the helicity wavefunction for a point-like meson-qq coupling. For vector mesons, the helicity wavefunction is therefore similar to that of the point-like photon-qq coupling, i.e.
S π hh (x, k) =vh This minimal γ 5 structure is used in Refs. [41,42] but other more general choices can be found in the literature [43,44]. The factor i ensures the positive-definiteness of the LFWF and the factor M π implies that our wavefunction vanishes in the chiral limit which we do not consider in this paper. Evaluating the right-hand-side of Eq. (15) using the light-front spinors given in Ref. [28], we obtain where k = ke iθ k . Note that this helicity wavefunction allows configurations in which the quark and antiquark have same helicities. These configurations are expected to be present in the pion LFWF [45,46]. In the non relativistic limit, k m f and x ≈ 1/2, only the configurations in which the quark and antiquark have opposite helicities survive, i.e.
where we have normalized the helicity wavefunction such that hh |S

π[nr]
h,h | 2 = 1. Eq. (17) is the helicity structure assumed for the meson (both pseudoscalar and vector) holographic wavefunction in Refs. [15,26] and we shall refer to it as the non-relativistic helicity wavefunction.
A two-dimensional Fourier transform of Eq. (18) and Eq. (19) to impact space gives and respectively.
We now fix the normalization constant N appearing in Eq. (4) by requiring that where Note that Eq. (22) reduces to the normalization condition given by Eq. (12) (with P qq = 1) if we substitute the original holographic wavefunction, Eq. (21), in Eq. (22). Imposing our normalization condition, Eq. (22), implies that we assume that the pion consists only of the leading quark-antiquark Fock state.
In Figures 1 and 2, we show the dynamical spin effects in the holographic wavefunction squared, after summing over all helicities. As can be clearly seen, the effect is a broadening of the wavefunction, i.e. a smooth enhancement of the end-point contributions.

III. EM FORM FACTOR AND RADIUS
Having specified the pion holographic LFWF, we are now in a position to compute several pion observables. We start with the EM form factor defined as where P = P + q, Q 2 = −q 2 and the EM current J µ em (z) = f e fΨ (z)γ µ Ψ(z) with f =d, u and ed ,u = 1/3, 2/3. The EM form factor can be expressed in terms of the pion LFWF using the Drell-Yan-West formula [23,24]: where |Ψ π (x, b)| 2 is given by Eq. (23). Note that Eq. (25) implies that F π (0) = 1 if the pion LFWF is normalized according to Eq. (22).
Our predictions for the EM form factor using the original (dashed-blue curve) and our spin-improved (continuous-red curve) are compared with the data from CERN [47], CEA [48], Cornell [49][50][51], Jlab [52,53] and CLEO [54,55]. As can be seen, the agreement with data is very much improved with our spin-improved holographic wavefunction. In fact, we achieve excellent agreement with data from the lowest Q 2 datum to Q 2 ≈ 7 GeV 2 . For Q 2 > 7 GeV 2 , our predictions with the original and spin-improved holographic wavefunctions coincide and they both undershoot the precise CLEO data [54]. This is the short-distance regime where perturbative corrections, not taken into account in the purely non-perturbative holographic wavefunction, become important. It is worth highlighting that agreement with the precise data in the non-perturbative region, Q 2 ≤ 1 GeV 2 , is excellent with our spinimproved holographic wavefunction.
The slope of the EM form factor at Q 2 = 0 is related to the mean radius of the pion [8]: so that, using Eq. (25), we have Our predictions for the pion radius are compared to the measured value in Table I. As can be seen, we achieve a much better agreement with our spin-improved holographic wavefunction, reducing the percentage difference between the theory prediction and the central measured value from 19% to 2%.
Note that if we compute the pion radius using the original holographic wavefunction but with κ = 540 MeV and m f = 330 MeV as in Ref. [15], we obtain r 2 π = 0.530 fm which is to be compared with the prediction of Ref. [15]: r 2 π = 0.529 fm. In Ref. [15], the authors obtain the pion radius from the slope of the EM pion form factor F π (Q 2 ) at Q 2 = 0 with the constraint that F (0) = 1. We note that the latter constraint on the form factor is automatically satisfied if the pion wavefunction is normalized with P qq = 1. Thus, although the authors of Ref. [15] imply P qq = 0.62 in order to fit the decay constant, they implicitly assume P qq = 1 when computing the EM form factor. This is why we are able to reproduce their prediction for the pion radius even though we assume P qq = 1.
For completeness, we also compute the transverse size, R ⊥ , of the pion defined as [43] R ⊥ = 1 where is the mean value of the quark (or antiquark) transverse momentum squared. Our results are shown in Table II. It can be seen that for both holographic wavefunctions, we predict k 2 ∼ m f > M π which, as noted in Ref. [43], confirms the highly relativistic nature of the pion. As expected, our spin-improved holographic wavefunction gives a more reasonable prediction (as expected from the measured radius) for the transverse size of the pion.

IV. DECAY CONSTANT AND DISTRIBUTION AMPLITUDE
The pion decay constant, f π , is defined by [28] Taking µ = + and expanding the left-hand-side of Eq. (30), we obtain The light-front matrix element in curly brackets can readily be evaluated: Original 114 Spin-improved 135 Experiment [56] 130.4 ± 0.04 ± 0.2 TABLE III. Our predictions for the pion decay constant with κ = 0.523 GeV and m f = 330 MeV.
which implies that only the opposite-helicities term in the holographic wavefunction contributes to the decay constant. We note, however, that the same-helicities term affects the normalization of our wavefunction and thus our prediction for the decay constant. Using our spin-improved wavefunction, Eq. (20), we deduce that On the other hand, using the original holographic wavefunction, Eq. (21), we obtain where we have written the momentum-space expression to point out that, up to a factor of √ 4π, it coincides with the formula for the decay constant given in Ref. [57] and widely used in the literature, as for example, in Refs. [15,27,58]. The √ 4π factor mismatch is consistent with the fact that our normalization in momentum-space (Eq. (12)) differs from the conventional light-front normalization [28] by a factor of 4π.
Our predictions for the pion decay constant are shown in Table III. As can be seen, we also achieve a much better agreement for this observable with our spin-improved holographic wavefunction, reducing the percentage difference between the theory prediction and the central measured value [56] from 13% to 4%.
We also predict the twist-2 holographic pion DA, ϕ π (x, µ), defined as [28,59] where z 2 = 0. The DA is conventionally normalized as dxϕ(x, µ) = 1 (36) such that taking the limit of local operators (z → 0) in Eq. (35), we recover the definition of the pion decay constant given by Eq. (30) (with µ = +). Proceeding in the same manner as for the decay constant, we are able to show that when using our spin-improved holographic wavefunction, and when using the original holographic wavefunction.
In Figure 4, we compare our spin-improved holographic DA given by Eq. (37) to the original holographic DA given by Eq. (38) and to the asymptotic DA as predicted in pQCD: . It can be seen that our spin-improved holographic DA (continuousred curve) is broader than both the original holographic DA (dashed-blue curve) and the asymptotic DA (dotted-black curve). The holographic DAs are both generated with µ = 1 GeV and we note they hardly evolve for µ > 1 GeV. In other words, our holographic DAs lack the hard, perturbative evolution given by the Efremov-Radyushkin-Brodsky-Lepage (ERBL) equations [60][61][62]. In Figure 5, we show the soft evolution of our spin-improved holographic DA between µ = 0.3 GeV and µ = 1 GeV. Implementing the ERLB evolution, as is done in Ref. [63], will allow our spin-improved holographic DA to evolve beyond µ = 1 GeV onto the asymptotic DA.
In order to compare our holographic DAs with the predictions of standard non-perturbative methods such as lattice QCD and QCD Sum Rules, we compute the moments defined as and its inverse moment is given by Our predictions for the first two non-vanishing moments ξ 2 and ξ 4 as well as the inverse moment are shown in Table IV. As can be seen, with the spin-improved holographic DA, we achieve better agreement with the predictions of lattice QCD and QCD Sum Rules. However, our predicted moments turn out to be smaller than the predictions of all non-perturbative

V. PHOTON-TO-PION TRANSITION FORM FACTOR
Using our holographic DAs, we are able to predict the photon-to-pion transition form factor (TFF) which, to leading order in pQCD, is given as [28] F γπ (Q 2 ) = We note that, even when computing the TFF in the perturbative region Q 2 ≥ 1 GeV 2 , the DA itself is probed at a scale µ = xQ, which can be low if x is close to its end-points. Figure  µ [GeV] 6 shows that our spin-improved holographic DA (solid-red curve) does a much better job than the original holographic DA (dashed-blue curve) to describe the available data over the whole Q 2 range. We note that the agreement with data extends to large Q 2 > 10 GeV 2 although neither holographic DA can fit the BaBar (2009) data [73]. The BaBar (2009) data indicate a strong scaling violation in disagreement with the Brodsky-Lepage limit: 2f π obtained by substituting the asymptotic DA in Eq. (41) and shown as the dotted-black curves in Figure 6. The more recent Belle (2012)   data are from CELLO [79], CLEO [80], BaBar [74] and Belle [74].

VI. CONCLUSIONS
We have accounted for dynamical quark spin effects in the holographic pion light-front wavefunction and found a remarkable improvement in the description of pion radius, decay constant, EM form factor and photon-to-pion TFF. We have used a universal AdS/QCD scale κ = 523 MeV and a constituent quark mass of 330 MeV together with the assumption that the pion consists only of the leading quark-antiquark Fock state. Our results suggest that it is possible to have a unified treatment of all light mesons, including the pion, with a universal fundamental AdS/QCD scale which fits the baryon and meson Regge slopes and also accurately predicts the non-perturbative QCD scale Λ M S QCD . We also found that the predicted moments for the spin-improved holographic pion twist-2 DA are in better agreement with the predictions of standard non-perturbative methods such as lattice QCD and QCD Sum Rules, although they remain smaller than the latter. This suggests that the simple ansatz we use for modelling the dynamical spin effects in the pion, might not be capturing all such effects. It does nevertheless bring a significant improvement in the description of all available experimental data without necessarily having to use a much smaller AdS/QCD scale and/or invoke higher Fock states contributions for the pion. Our findings support the idea of the emergence of a universal AdS/QCD confinement scale κ.

VII. ACKNOWLEDGEMENTS
The work of M.A and R.S is supported by a team grant from National Science and Engineering Research Council of Canada (NSERC). F.C and R.S thank Mount Allison University for hospitality where parts of this work were carried out.