PoS(LC2019)011 https://pos.sissa.it/ ∗† Speaker. I thank the organizers of Cone 2019 for their invitation and a very successful conference. I also thank the Light-front holography offers a successfultroscopy and first dynamics. We semiclassical review its approximation underlyingas to assumptions, attempts its hadronic to remarkable go predictions spec- beyond as the well semiclassicaldata approximation with in a order universal to AdS/QCD the mass describe scale. a wide range of † ∗ Copyright owned by the author(s) under the terms of the Creative Commons c Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). Natural Sciences and Engineering Research2017-00031) Council (NSERC) of Canada for funding (Discovery Grant No. SAPIN- Light Cone 2019 - QCD on16-20 the September light 2019 cone: from hadronsEcole to Polytechnique, heavy Palaiseau, France ions - LC2019 Ruben Sandapen Acadia University 15 University Avenue Wolfville NS Canada B4P 2R6 E-mail: [email protected] An overview of light-front holography PoS(LC2019)011 is ¯ h (1.4) (1.5) (1.6) (1.1) (1.2) (1.3) ) then be- 1.2 Ruben Sandapen its helicity ( h being the light-front ) . Eq. ( ¯ h 0 x , → h f , . m ⊥ | ) 2 k , ζ ( x M ( φ for the pion. Second that, in the 2 Ψ 2 ¯ h M − ¯ h ϕ , M h h i iL . S ) δ e ) = 0 ) h ) P 2 ) = ζ sets the scale for masses. ) ⊥ ( f x ( ¯ > h x ( ∝ k , Ψ m φ ]. In QCD, conformal invariance is explic-  , | ¯ X h h − x 1 2 h ] ,  ( QCD + 1 ) S 3 ) ⊥ ( M Λ πζ 2 ⊥ Ψ ζ ζ k x 2 ( k ( , = x φ 1 → eff i p ( = ) ) is its transverse momentum and U ¯ Ψ 0 . Since the Higgs-generated masses are much h P , ( ⊥ + → ! ) = h f k ) Ψ , 2 is the valence wavefunction, with ) and also beyond tree-level where renormalization gen- ϕ | m QCD f ⊥ | L ⊥ , i 2 2 Λ 4 k k ζ m ζ , Ψ , , | QCD 4 x LF − x x M ¯ ( h ( ( 1 H , eff Ψ h Ψ , − U ⊥ 2 2 + k ζ ]. To apply the dAFF procedure in QCD, we need to reduce the d , f 2 d ) 2 x x : m − ¯ − q  + is the 2-dimensional Fourier transform of q 1 h 2 ⊥ ( 2 ⊥ x k b )

) = x ¯ h , − h 1 , ( ⊥ x k , is momentum-independent. For example, = x ¯ h ( 2 h ζ S Ψ In the chiral limit and at tree-level, the QCD Lagrangian is invariant under conformal trans- Yet, there is another way to generate a scale for the hadron masses even within tree-level, , which describes the quark-antiquark interaction as well as the effects of higher Fock states eff Then, the wavefunction can be factorized as where comes strongly-coupled multi-parton problemproblem. in This QCD is to possible a in light-front two-parton QCD QM-like where bound [ state can be reduced to a Schrödinger-like equation for the lowest quark-antiquark Fock state: the helicity of the antiquark). At this point, there are noon approximations the and lowest the Fock effective potential state,necessary. is First essentially that unknown. the helicity To dependence make of further the progress, wavefunction is two non-dynamical, assumptions i.e. are formations. This underlying conformal symmetry ofaccording QCD to motivates the Maldacena’s search equation for (AdS=CFT)coupled its which gravitational theories dual refers in higher to dimensional anti-de an Sitterconformal equivalence (AdS) field space between theories and weakly- strongly-coupled (CFT) in physical [ An overview of LFH 1. Conformal symmetry in QCD itly broken by non-zero quark masseserates ( a scheme-dependent mass scale: momentum fraction carried by the quark, massless QCD. A massunderlying scale action can remains conformally appear invariant as in was(dAFF) first the shown in Hamiltonian by de and conformal Alfaro, Fubini equations QM and of Furlan [ motion while the smaller than hadron masses, it is usually stated that where where chiral limit, the confinement potential isi.e. a function of the invariant quark-antiquark mass squared, U PoS(LC2019)011 , z ↔ (1.7) (2.1) (2.2) (2.3) (2.4) (2.5) ζ which appears Ruben Sandapen ] κ 4 as the variable on is a 5-dimensional , ζ ) µ 2 ζ 2 κ . ( where L n ) z L 2 ) maps onto the classical wave ( ) 0  J ϕ 2 1.6 . 3 ]. The confining QCD potential is − . ]. The mass scale ζ 5 ) 2 6 z 2 1 2 − 1 2  κ J = 2 S − − 2 + ( , Eq. ( 2 J 2 | 5 +  ( 2 + ) ) 2 L 2 ζ R ζ ) κ 4 , µ z exp ) then yields + 2 x κ ( ( ( L 0 n ¯ h + + , 2.1 ϕ  2 h 2 2 ↔ / ) = 4 1 2 ζ 1 Ψ 2 ζ κ 4 | ζ ( L x 4 κ ! ) + d ) eff . Eq. ( z = b 2 ! L ( U 2 z 00 S n ) = , d 2 string modes in AdS [ + ) maps onto the conformal QM Hamiltonian considered by 2 L , and ϕ ζ , κ 5 n ) is referred to as the holographic Schrödinger Equation. Its 2 ( ]. Here, we consider the latter to also include the assumption n J 1 2 Z ( 4 1.6 ¯ h eff M , 1.6 h ∑ ) = s U z ) = L ( + ζ 1 ( ϕ κ eff in the Hamiltonian without spoiling the conformal invariance of the ), Eq. ( U κ ) = 2.2 ζ is the radius of curvature of AdS ( R nL φ dimension of AdS while its eigenfunctions, th S + L 0, the Hamiltonian in Eq. ( ): no quantum loops, zero quark masses and the identification of is the 5 ) embodies the assumption that the consists only of the leading quark-antiquark ]. Then, following dAFF, taking = 2 z J = 1.7 2.5 In the semiclassical approximation, light-front QCD possesses a gravity dual. With eff U introduces a mass scale which the confinement potentialcalled depends. semiclassical approximation [ These three assumptions are encapsulated in the so- mass parameter and To recover the quadratic effective potential dictated bybe the chosen dAFF to mechanism, the be dilation quadratic, field i.e. must As can be seen,constant while spin the term quadratic emerges form fromin the of both mapping the the to potential dilaton AdS field is space andpotential [ fixed the given by confinement by potential the Eq. is dAFF ( called mechanism, the AdS/QCD the mass scale. With its where are normalized so that Fock state. underlying action. It is remarkable thatfixed the by quadratic the form dAFF of mechanism the of confinementto conformal potential symmetry AdS is breaking uniquely space). (at this point,Eq. For there is this no ( reason, reference it is important to recap the underlying assumptions leading to eigenvalues are: where then determined by the dilaton field which breaks conformal invariance in AdS: [ Eq. ( of a non-dynamical spin wavefunction. 2. Light-front holography An overview of LFH If dAFF [ equation for freely propagating spin- PoS(LC2019)011 ) ]. = D 7 S n (3.1) (2.6) (2.8) (2.7) = T S , B ] L 4 = Ruben Sandapen ) lie on identical T D L S ( = ]. M ) then generalizes to 4 S . These are interpreted as [ ) ) 2.3 D ) x S 1 , − 1 and B − 1 L ( + D ( x S B ) is completely fixed by supersym- 2 L ], the partner potential to a harmonic p κ 8 F 2 2 ) + 2.7  = 1 1 2 ln ) = M ∂κ ) + + x ∂ L + ) is the potential for . Explicitly ( 2 4 B ) is broken by non-zero quark masses. Light f κ X κ L , 2.6 + (( ζ 2.8 ]. Therefore, in addition to , there are 2 ( n . This implies that mesons and baryons differing 4 ] = κ ) n M 3  2 M U m 2 ), is a first approximation to the meson light-front S + κ 4 2 ,... − 1.5 ) = 1 ζ 1 2 = 4 ( m . On the other hand, light-front holography tells us that , resulting in κ [ f 5 κ 2 5 , , 2 . The supersymmetrization of the holographic Schrödinger 2 / n ) ζ 1 M 2 ( M ) = B ∆ κ − 2 , 1 respectively [ U κ M ζ , ( + L ζ M 0. This state is naturally identified with the pion since the pion mass B ( L B U ) = = U D 2 S ) = and M ζ + ( B 1 L ( eff ] and references therein. 2 ) is fixed by light-front holography, Eq. ( U / 10 1 2.2 + B L . This is done by matching the electromagnetic or gravitational form factor of composite is the lowest possible spin of the diquark in the . Eq. ( ) 0, is massless: = x D ( f S = X The superconformal symmetry leading to Eq. ( The holographic wavefunction, Eq. ( Up to now, we have considered only quark-antiquark states, i.e. mesons. To make this ex- The first striking prediction is that the lowest lying bound state, with quantum numbers S ]. Note that the massless pion does not have a baryon superpartner. = 9 Equation unifies mesons and baryons (considered as diquark-quark systems) as superpartners [ where our notation has anticipated the fact that Eq. ( metry without further reference toa AdS baryon has equal probabilityangular of momentum being in the positive and negative states, with orbital 3. Beyond the semiclassical approximation by only one unit of orbital angular momentum (i.e. with second bosonic superpartners to baryons with quantum numbers quark masses can be treated as a small perturbation shifting the hadronic masses by [ plicit, let us write Just like in the supersymmetric formulation ofoscillator ordinary potential QM is [ a shifted harmonic oscillator potential: Regge trajectories. This degeneracy, although notthat exact, is once indeed Eq. seen in ( spectroscopic data. Note tetraquarks (considered as diquark-antidiquark systems) with quantum numbers wavefunction which is a crucialstants, input diffractive to cross-sections predict as various well as observableston non-perturbative like quantities Distribution charge like radii, Functions Form decay Factors (PDFs), (FF), con- Distributiontions Par- Amplitudes (GPDs) (DAs), and Generalized Transverse Parton Momentumon Distribu- Distributions predictions (TMDs). using In the whatholography, holographic follows, see meson [ we shall wavefunction. focus For other applications of light-front An overview of LFH L is expected to vanish into chiral fix limit. To completely specifystates the in light-front physical wavefunction, spacetime we and need in AdS where where [ PoS(LC2019)011 ]. φ = 20 357 540 Γ (3.4) (3.2) (3.3) and = = ]. The can be s ρ κ 24 MeV 13 κ ] and m ± 22 46 MeV. The ), the HERA , 523 Ruben Sandapen 21 = 3.4 d = / is a dimensionless u quark masses using κ B m )) were used only for d / ! 2.5 u 2 i i 1 components of the pion x m = ) i n 494 MeV, yielding L ∑ k ] and, indeed in previous work , 2 1 + x κ 17 ) = ) √ s for vector mesons [ xP − k 0 and ] even though the more recent work ( m , γ h

, · x = u 15 d ( ε / ¯ , is the polarization 4-vector of the vector h L Γ , u exp h ) µ = electroproduction can also be successfully m 14 S 140 MeV. This gives k ε ( ) ! Γ φ 1 − k , this leads to degenerate decay constants for M ], despite the fact that conformal symmetry is , , κ ) = ∆ x − x + d ( i 15 / P x , and indeed the latter can be predicted from the = − u ) Ψ 4 ]. Here i x n 1 m K 14 ∑ ( → 24 − √ M meson have the same holographic wavefunction since

, QCD ) M 1 ¯ h δ ρ Λ ∆ , 23 (( n ¯ h x h ¯ = v , d and the best simultaneous description of diffractive k π ] that, with a spin structure given by Eq. ( ... κ , 1 x M ) = x ( 21 k d 3 for baryons. This allows us to fix the Ψ , 1 0. With a universal x 0 = ( Z ¯ h = n , h L S ] = and/or normalization condition (instead of Eq. ( 2 κ κ [ is universal for light , it depends on the hadron mass (as expected 1 and F meson electroproduction prefer the holographic wavefunction with κ = ρ ]. These shortcomings can be overcome by taking into account dynamical spin n is the 4-momentum (mass) of the pseudoscalar meson and for pseudoscalar mesons [ ) 19 is the holographic wavefunction in momentum space and 5 P γ ) ] with the same value of ) tells us that the pion and the ) M k 2 for mesons and P ]. While ( ], a different , 22 x 2.4 µ ( = 12 18 P BM 1 , n Ψ ] seem to provide a more economical way (i.e. with less free parameters) to account for heavy + Only the unpolarized pion TMD can be predicted with the holographic wavefunction with no dynamical spin [ Eq. ( The assumption is that bound state effects are fully captured by the holographic wavefunction 15 9 ), with no short-distance corrections as in [ 1 γ , , ] which we refer to as the universal AdS/QCD scale. Being scheme-independent, · P 17 16 3.1 11 strange quark mass can thenMeV. be A fixed global using fit[ to all light meson and baryon spectroscopic data yields for both mesons, the two mesons, a predictionpion that is is a in special contradiction case since with[ it experiment. does It not fall has on been a argued Regge that trajectory the [ is the (point-like) meson-quark-antiquark vertexDirac in structure light-front between perturbation the light-front theory. spinors In is( general, given the meson, so that the spinHence, wavefunction is that of a point-like meson coupling to a quark-antiquark pair. where the fact that the physical pion mass viewed as being more fundamental than latter observation was already noted in earlier research [ . In particular, the only( correction to the heavystrongly hadron broken mass by spectrum heavy is quarks. that given by Eq. the pion. Nevertheless, these approachesMulders cannot TMD accommodate which a results non-zero from holographic the overlap pion between Boer- the effects. free parameter. It wasdata shown on in diffractive [ MeV (consistent with the universal value). Diffractive from Heavy Quark Effective Theory) for hadrons containing at least one heavy quark [ former [ predicted [ An overview of LFH where [ wavefunction [ where PoS(LC2019)011 φ , , ρ ] and Nuovo , 25 Int. J. , ]. Light-Front . ]. Ruben Sandapen ]. 1, the available kaon ]. Alternative ansatzes ≥ 1501.00959 [ B 27 (1981) 513 ] cautions against using a 1407.8131 23 ]. [ Threefold Complementary Superconformal Baryon-Meson ]. and B188 system (even though the meson- 0 hep-ph/9705477 η [ MeV. The holographic DAs for / (2015) 1 ] η (2015) 085016 ] reports an excellent description of the ]. MeV. Ref. [ 1302.4105 584 ] 140 [ Nucl. Phys. 24 28 , D91 1802.09652 , ] and lattice data [ ]. [ ]. 46 500 26 , (1998) 299 ]. Conformal Invariance in Quantum Mechanics ] = [ 5 (2014) 3 330 s 301 Phys. Rept. Light-Front Holography: A First Approximation to m Phys. Rev. . , , , d ] = [ / s B729 u 0809.4899 Hadronic superpartners from a superconformal and [ m 1). Further support for the spin-improved wavefunction m , (2018) 114001 1606.04638 [ d [ ≥ / u Phys. Rept. B D97 , m [ Phys. Lett. 1912.12578 , and hep-th/9711200 [ κ (2009) 081601 The Large N limit of superconformal field theories and supergravity Phys. Rev. , 2019, , (2016) 1630029 Color Confinement and Supersymmetric Properties of Hadron Physics from 102 . 0 while the situation is less clear for the A31 Dynamical Breaking of Supersymmetry = (1999) 1113 B leading to a successful description of a wide set of light meson data with a universal 38 (1976) 569 b for the pseudoscalar meson nonet: while the pion data prefer B Phys. Rev. Lett. A34 , have also been predicted and used to compute various observables in B decays: see [ ∗ Superconformal light-front holography successfully predicts the main features of hadronic K à posteriori QCD Approach to Holographic QCD Int. J. Mod. Phys. Theor. Phys. 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