M esons and baryons in holographic soft-wall model Valery - PowerPoint PPT Presentation

M esons and baryons in holographic soft-wall model Valery Lyubovitskij Institut f¨ ur Theoretische Physik, Universit¨ at T¨ ubingen Kepler Center for Astro and Particle Physics, Germany in collaboration with Thomas Gutsche Ivan Schmidt Alfredo Vega based on PRD 80 (2009) 055014 PRD 82 (2010) 074022 PRD 83 (2011) 036001 Hadron 2011, 14 June 2011, M¨ unchen – p.1

Introduction • Holographic QCD (HQCD) – approximation to QCD: Hadron Physics in terms of fields/strings living in extra dimensions (AdS space) • Motivation: AdS/CFT correspondence 1998 (Maldacena, Polyakov, Witten et al) Dynamics of the superstring theory in AdS d +1 background is encoded in d conformal field theory living on the AdS boundary. ds 2 = R 2 “ dx µ dx µ − dz 2 ” • AdS metric Poincaré form z 2 z is extra dimension (holographic) coordinate; z = 0 is UV boundary AdS/CFT dictionary Gauge Gravity ˆ O Bulk field Φ( x, z ) Operator ˆ ∆ — scaling dimension of O m — mass of Φ( x, z ) ˆ Source of O Non-normalizable bulk profile near z = 0 � ˆ O� Normalizable bulk profile near z = 0 – p.2

Introduction • Towards to QCD: – Break conformal invariance and generate mass gap – Tower of normalized bulk fields (Kaluza-Klein modes) ↔ Hadron wave functions – Spectrum of Kaluza-Klein modes ↔ Hadrons spectrum • HQCD: Description of low-energy QCD • Bottom-up HQCD: hard-wall and soft-wall models • Hard-wall: AdS geometry is cutted by two branes UV ( z = ǫ → 0) and IR ( z = z IR ) Analogue of quark bag model, linear dependence on J ( L ) of hadron masses • Soft-wall: Soft cuttoff of AdS space by dilaton field exp( − ϕ ( z )) M 2 ∼ J ( L ) (Regge behavior) Analytical solution of EOM, – p.3

Introduction • Brodsky and de Téramond: Semiclassical 1st approximation to QCD based on combination of LF holography and correspondence of String Theory in AdS 5 and CFT in Min 4 . • LF holography EOM for propagation of spin-J modes in AdS are equivalent to Hamiltonian formulation of QCD on LF • Mapping of string mode in AdS 5th dimension z to the hadron LFWF depending on impact variable ζ — separation of quark and gluons inside hadron. • Objective: SW holographic approach for mesons and baryons with any n, J, L, S . – p.4

Approach: Fields propagating in AdS • Conformal group contains 15 generators: 10 Poincaré (translations P µ , Lorentz transformations M µν ), 5 conformal (conformal boosts K µ , dilatation D ): M µν = i ( x µ ∂ ν − x ν ∂ µ ) rotational symmetry D = i ( x ∂ ) energy P µ = i∂ µ raising energy K µ = 2 ix µ ( x ∂ ) − ix 2 ∂ µ lowering energy • Isomorphic to SO (4 , 2) – the isometry group of AdS 5 space • Fields in AdS 5 are classified by unitary, irreducible representations of SO (4 , 2) – p.5

Approach: Scalar Field • Action for scalar field Brodsky, Téramond Φ = 1 d d xdz √ g e ϕ ( z ) „ « Z ∂ N Φ + ∂ N Φ + − m 2 Φ 2 S + + 2 Our Φ = 1 „ « Z d d xdz √ g e − ϕ ( z ) ∂ N Φ − ∂ N Φ − − ( m 2 + ∆ U ( z )) Φ 2 S − − 2 ϕ ( z ) = κ 2 z 2 (Regge behavior of hadron masses), • dilaton • metric g MN ( z ) = ǫ a M ( z ) ǫ b N ( z ) η ab , g = | det g MN | • vielbein ǫ a M ( z ) = e A ( z ) δ a M , A ( z ) = log( R/z ) (conformal) ds 2 = g MN dx M dx N = e 2 A ( z ) ( g µν dx µ dx ν − dz 2 ) • interval • equivalence: bulk field redefinition Φ ± = e ∓ ϕ ( z ) Φ ∓ • potential ∆ U ( z ) = e − 2 A ( z ) [ ϕ ′′ ( z ) + 1 − d z ϕ ′ ( z )] – p.6

Approach: Scalar Field • Klebanov, Witten ˛ → z d − ∆ h i + z ∆ h i ˛ Φ 0 ( x ) + O ( z 2 ) A ( x ) + O ( z 2 ) Φ( x, z ) ˛ ˛ z → 0 Φ 0 ( x ) is source of the CFT operator ˆ O A ( x ) ∼ � ˆ O� is physical fluctuation • Towards to QCD Brodsky, Téramond ∆ ≡ τ = 2 + L scaling dimension of two-parton state with L = 0 , 1 . extended to any J and independent on J – p.7

Approach: Scalar Field d d p (2 π ) d e − ipx Φ n ( p ) Φ n ( p, z ) • Kaluza-Klein expansion Φ( x, z ) = P R n • Substitution Φ n ( p, z ) = e − B ( z ) / 2 φ n ( p, z ) • Schrödinger-type EOM for φ n ( z ) = φ n ( p, z ) | p 2 = M 2 n : h dz 2 + 4 L 2 − 1 d 2 i φ n ( z ) = M 2 − + U ( z ) n φ n ( z ) 4 z 2 q Γ( n + L +1) κ L +1 z L +1 / 2 e − κ 2 z 2 / 2 L L 2Γ( n +1) • φ n ( z ) = n ( κ 2 z 2 ) n = 4 κ 2 “ ” • M 2 n + L , 2 • Massless pion M 2 π = 0 for n = L = 0 (Brodsky, Téramond) • Φ n ( z ) = z 3 / 2 φ n ( z ) ∼ z 2+ L (at small z) • Φ n ( z ) → 0 (at large z) – p.8

Approach: Higher J boson fields • Φ J = Φ M 1 ··· M J ( x, z ) – a symmetric, traceless tensor: Brodsky, Téramond „ « d d xdz √ g e ϕ ( z ) J ∂ N Φ J, + − µ 2 S + ∂ N Φ + J Φ + Φ = 1 J Φ J, + R 2 Our „ « d d xdz √ g e − ϕ ( z ) J ∂ N Φ J, − − ( µ 2 S − ∂ N Φ − J + ∆ U J ( z )) Φ − Φ = 1 J Φ J, − R 2 J R 2 = (∆ − J )(∆ + J − d ) • µ 2 J = L 2 − (2 − J ) 2 for d = 4 • ∆ = 2 + L and µ 2 • Effective potential ∆ U J ( z ) = e − 2 A ( z ) [ ϕ ′′ ( z ) + 1+2 J − d ϕ ′ ( z )] z – p.9

Approach: Higher J boson fields • Axial gauge Φ z... ( x, z ) = 0 d d P (2 π ) d e − iP x ǫ n • KK decomposition R Φ ν 1 ··· ν J ( x, z ) = P Φ nJ ( z ) ν 1 ··· ν J ( P ) n • Substitution ” 1 − d “ R 2 Φ nJ ( z ) = ϕ nJ ( z ) z • Schrödinger EOM for Φ nJ ( z ) : − d 2 h i ϕ nJ ( z ) = M 2 dz 2 + U J ( z ) nJ ϕ nJ ( z ) • Effective potential U J ( z ) U J ( z ) = κ 4 z 2 + 4 a 2 − 1 + 2 κ 2 “ ” b J − 1 . 4 z 2 • a = 1 d 2 + 4( µR ) 2 = ∆ − d b J = J + 4 − d q 2 , 2 2 – p.10

Approach: Higher J boson fields • Solutions at d = 4 : s 2 n ! ( n + L )! κ 1+ L z 1 / 2+ L e − κ 2 z 2 / 2 L L n ( κ 2 z 2 ) ϕ nJ ( z ) = n + L + J 4 κ 2 “ ” M 2 = nJ 2 M 2 nJ = 4 κ 2 ( n + J ) • At J ( L ) → ∞ Φ nJ = z 3 / 2 ϕ nJ ∼ z τ , • Scaling twist τ = 2 + L – p.11

Approach: Higher J boson fields (gauge-invariant) • Fradkin, Vasiliev, Metsaev, Buchbinder et al, Karch et al, · · · „ N Φ d d xdz √ g e − ϕ ( z ) M 1 ...MJ S Φ = 1 R ∇ N Φ M 1 ...MJ ∇ 2 « “ ” M 1 ...MJ µ 2 − J + U J ( ϕ ) Φ M 1 ...MJ Φ + · · · K K • ∇ N Φ M 1 ...MJ = ∂ N Φ M 1 ...MJ − Γ NM 1 Φ KM 2 ...MJ − Γ NM J Φ M 1 ...MJ − 1 K KL “ ∂g LM ∂g LN ∂g MN K ” • Affine connection Γ MN = 1 2 g + ∂x M − ∂x N ∂x K • Gauge constraints (transversity, traceless) M 1 Φ M 1 M 2 ...MJ = 0 M 1 M 2 Φ M 1 M 2 ...MJ = 0 ∇ and g J R 2 = ∆ J (∆ J − d ) − J = J 2 + J ( d − 5) + 4 − 2 d • Bulk mass µ 2 with ∆ J = J + d − 2 nJ = 4 κ 2 “ ” n + L + J M 2 Scaling Φ nJ ∼ z 2+ L • Mass spectrum 2 – p.12

Approach: Higher J fermion fields (gauge-invariant) M 1 ...MJ − 1 / 2 „ « d d xdz √ g e − ϕ ( z ) Ψ a Γ a D M − µ J − ϕ ( z ) • S Ψ = ǫ M R Ψ M 1 ...MJ − 1 / 2 R + · · · D M = ∇ M − 1 8 ω ab ω ab M = A ′ ( z ) ( δ a z δ b M − δ b z δ a • M [Γ a , Γ b ] , M ) • Relation of spin and affine connection “ ∂ M ǫ Kb + ǫ Nb Γ K ” ω ab M = ǫ a K MN • Gauge constraints (transversity, traceless) M 1 Ψ M 1 M 2 ...MJ − 1 / 2 = 0 , M 1 Ψ M 1 M 2 ...MJ − 1 / 2 = 0 , ∇ Γ M 1 M 2 Ψ M 1 M 2 ...MJ − 1 / 2 = 0 g • Bulk mass µ J R = ∆ J − d/ 2 with ∆ J = J + d − 2 Metsaev • Toward QCD: ∆ J ≡ τ + 1 / 2 = 7 / 2 + L independent on J and gives correct scaling of nucleon FF – p.13

Approach: Higher J fermion fields (gauge-invariant) » – 2 γ 5 − µR − ϕ ( z ) • EOM iz � ∂ + γ 5 z∂ z − d Ψ a 1 ··· a J − 1 / 2 ( x, z ) = 0 Ψ L/R = 1 ∓ γ 5 • Ψ( x, z ) = Ψ L ( x, z ) + Ψ R ( x, z ) , Ψ 2 d d p (2 π ) d e − ipx Ψ L/R ( p ) F n • Ψ L/R ( x, z ) = P R L/R ( p, z ) n L/R ( p, z ) = e − A ( z ) · d/ 2 f n • F n L/R ( p, z ) » – z + κ 4 z 2 + 2 κ 2 “ ” + µR ( µR ± 1) µR ∓ 1 − ∂ 2 f n L/R ( z ) = M 2 n f n • L/R ( z ) z 2 2 • For d = 4 and µR = L + 3 / 2 q Γ( n + L +3) κ L +3 z L +5 / 2 e − κ 2 z 2 / 2 L L +2 2Γ( n +1) f n ( κ 2 z 2 ) L ( z ) = n q Γ( n + L +2) κ L +2 z L +3 / 2 e − κ 2 z 2 / 2 L L +1 2Γ( n +1) f n ( κ 2 z 2 ) R ( z ) = n n = 4 κ 2 “ ” L ( z ) ∼ z 9 / 2+ L , M 2 F n F n R ( z ) ∼ z 7 / 2+ L n + L + 2 , – p.14

Approach: Hadronic Wave Function • Correspondence of holographic coordinate z to the impact variable ζ in LF ζ 2 = b 2 • Two parton case: q 1 ¯ q 2 mesons z → ζ, ⊥ x (1 − x ) ζ - impact variable; b ⊥ - impact separation (conjugate to k ⊥ ) • Mapping Φ nJ ( z ) to the transverse mode of LFWF • ψ nJ ( x, ζ, m 1 , m 2 ) = ψ T ( ζ ) · ψ L ( x ) · ψ A ( ϕ ) ψ T = Φ nJ ( ζ ) — transverse (from AdS/QCD) ψ L = f ( x, m 1 , m 2 ) = e − m 2 1 / (2 xλ 2 ) − m 2 2 / (2(1 − x ) λ 2 ) — longitudinal ψ A = e imϕ — angular mode λ - additional scale parameter 1 „ m 2 m 2 n + L + J « Z nJ = 4 κ 2 “ ” M 2 1 2 f 2 ( x, m 1 , m 2 ) + dx + 2 x 1 − x 0 – p.15

Approach: Choice of parameters • Constituent quark massses: m = 420 MeV , m s = 570 MeV , m c = 1 . 6 GeV , m b = 4 . 8 GeV • dilaton parameter κ = 550 MeV • Dimensional parameters λ in the longitudinal WF are fitted as: λ qq = 0 . 63 GeV , λ qs = 1 . 20 GeV , λ ss = 1 . 68 GeV , λ qc = 2 . 50 GeV , λ sc = 3 . 00 GeV λ qb = 3 . 89 GeV , λ sb = 4 . 18 GeV , λ cc = 4 . 04 GeV , λ cb = 4 . 82 GeV , λ bb = 6 . 77 GeV – p.16