Hadron Spectroscopy and Form Factors in AdS/QCD for Experimentalists Guy F. de T´ eramond University of Costa Rica and SLAC High Energy Physics Group Imperial College London October 4, 2010 HEP , Imperial College, October 4, 2010 Page 1
I. Introduction Lattice QCD Gravity Holographic Correspondence II. Gauge/Gravity Correspondence and Light-Front QCD Higher Spin Modes in AdS Space III. Light Front Dynamics Light-Front Fock Representation Semiclassical Approximation to QCD in the Light Front Light-Front Holographic Mapping Light Meson and Baryon Spectrum IV. Light-Front Holographic Mapping of Current Matrix Elements Electromagnetic Form Factors V. Higher Fock Components Detailed Structure of Space-and Time Like Pion Form Factor HEP , Imperial College, October 4, 2010 Page 2
I. Introduction • QCD fundamental theory of quarks and gluons • QCD Lagrangian follows from the gauge invariance of the theory � T a , T b � ψ ( x ) → e iα a ( x ) T a ψ ( x ) , = if abc T c • Find QCD Lagrangian L QCD = − 1 4 g 2 Tr ( G µν G µν ) + iψD µ γ µ ψ + mψψ where D µ = ∂ µ − igT a A a µ , G a µν = ∂ µ A a ν − ∂ ν A a µ + f abc A b µ A c ν • Quarks and gluons interactions from color charge, but ... gluons also interact with each other: strongly coupled non-abelian gauge theory → color confinement • Most challenging problem of strong interaction dynamics: determine the composition of hadrons in terms of their fundamental QCD quark and gluon degrees of freedom HEP , Imperial College, October 4, 2010 Page 3
Lattice QCD • Lattice numerical simulations at the (resolution ∼ L/a ) teraflop/sec scale • Sums over quark paths with billions of dimensions • LQCD (2009) > 1 petaflop/sec – a – ← L → • Dynamical properties in Minkowski space-time not amenable to Euclidean lattice computations HEP , Imperial College, October 4, 2010 Page 4
Gravity • Space curvature determined by the mass-energy present following Einstein’s equations R µν − 1 2 R g µν = κ T µν ���� � �� � mater geometry R µν Ricci tensor , R space curvature ( ds 2 = g µν dx µ dx ν ) g µν metric tensor T µν energy-momentum tensor κ = 8 πG/c 4 , • Matter curves space and space determines how matter moves ! Annalen der Physik 49 (1916) p. 30 HEP , Imperial College, October 4, 2010 Page 5
Holographic Correspondence HEP , Imperial College, October 4, 2010 Page 6
II. Gauge Gravity Correspondence and Light-Front QCD • The AdS/CFT correspondence [Maldacena (1998)] between gravity on AdS space and conformal field theories in physical spacetime has led to a semiclassical approximation for strongly-coupled QCD, which provides analytical insights into the confining dynamics of QCD • Light-front (LF) quantization is the ideal framework to describe hadronic structure in terms of quarks and gluons: simple vacuum structure allows unambiguous definition of the partonic content of a hadron, exact formulae for form factors, physics of angular momentum of constituents ... • Light-front holography provides a remarkable connection between the equations of motion in AdS and the bound-state LF Hamiltonian equation in QCD [GdT and S. J. Brodsky, PRL 102 , 081601 (2009)] • Isomorphism of SO (4 , 2) group of conformal transformations with generators P µ , M µν , K µ , D, with the group of isometries of AdS 5 , a space of maximal symmetry, negative curvature and a four-dim boundary: Minkowski space Isometry group: most general group of transformations which leave invariant the distance between two points ( d +1)( d +2) Dim isometry group of AdS d +1 is 2 HEP , Imperial College, October 4, 2010 Page 7
• AdS 5 metric: = R 2 � − dz 2 � ds 2 η µν dx µ dx ν z 2 ���� � �� � L AdS L Minkowski • A distance L AdS shrinks by a warp factor z/R as observed in Minkowski space ( dz = 0) : L Minkowski ∼ z R L AdS • Since the AdS metric is invariant under a dilatation of all coordinates x µ → λx µ , z → λz , the variable z acts like a scaling variable in Minkowski space • Short distances x µ x µ → 0 maps to UV conformal AdS 5 boundary z → 0 • Large confinement dimensions x µ x µ ∼ 1 / Λ 2 QCD maps to large IR region of AdS 5 , z ∼ 1 / Λ QCD , thus there is a maximum separation of quarks and a maximum value of z • Use the isometries of AdS to map the local interpolating operators at the UV boundary of AdS into the modes propagating inside AdS HEP , Imperial College, October 4, 2010 Page 8
• Nonconformal metric dual to a confining gauge theory ds 2 = R 2 z 2 e ϕ ( z ) � η µν dx µ dx ν − dz 2 � where ϕ ( z ) → 0 at small z for geometries which are asymptotically AdS 5 • Gravitational potential energy for object of mass m V = mc 2 √ g 00 = mc 2 R e ϕ ( z ) / 2 z • Consider warp factor exp( ± κ 2 z 2 ) • Plus solution: V ( z ) increases exponentially confining any object in modified AdS metrics to distances � z � ∼ 1 /κ HEP , Imperial College, October 4, 2010 Page 9
Higher Spin Modes in AdS Space (Frondsal, Fradkin and Vasiliev) � x M = ( x µ , z ) � • Lagrangian for scalar field in AdS d +1 in presence of dilaton background ϕ ( z ) � d d x dz √ g e ϕ ( z ) � � g MN ∂ M Φ ∗ ∂ N Φ − µ 2 Φ ∗ Φ S = Φ P ( x µ , z ) = e − iP · x Φ( z ) • Factor out plane waves along 3+1: � � 2 � � e ϕ ( z ) � � µR − z d − 1 Φ( z ) = M 2 Φ( z ) e ϕ ( z ) ∂ z z d − 1 ∂ z + z where P µ P µ = M 2 invariant mass of physical hadron with four-momentum P µ • Define spin- J mode Φ µ 1 ··· µ J with all indices along 3+1 and shifted dimensions Φ J ( z ) ∼ z − J Φ( z ) • Find AdS wave equation � e ϕ ( z ) � � 2 � � � µR − z d − 1 − 2 J Φ J ( z ) = M 2 Φ J ( z ) ∂ z z d − 1 − 2 J ∂ z + e ϕ ( z ) z HEP , Imperial College, October 4, 2010 Page 10
III. Light Front Dynamics • Different possibilities to parametrize space-time [Dirac (1949)] • Parametrizations differ by the hypersurface on which the initial conditions are specified. Each evolve with different “times” and has its own Hamiltonian, but should give the same physical results • Instant form : hypersurface defined by t = 0 , the familiar one • Front form : hypersurface is tangent to the light cone at τ = t + z/c = 0 x + = x 0 + x 3 light-front time x − = x 0 − x 3 longitudinal space variable k + = k 0 + k 3 ( k + > 0 ) longitudinal momentum k − = k 0 − k 3 light-front energy 2 ( k + x − + k − x + ) − k ⊥ · x ⊥ k · x = 1 On shell relation k 2 = m 2 leads to dispersion relation k − = k 2 ⊥ + m 2 k + HEP , Imperial College, October 4, 2010 Page 11
Light-Front Fock Representation • LF Lorentz invariant Hamiltonian equation for the relativistic bound state system � � P µ P µ | ψ ( P ) � = P − P + − P 2 | ψ ( P ) � = M 2 | ψ ( P ) � ⊥ • State | ψ ( P ) � is expanded in multi-particle Fock states | n � of the free LF Hamiltonian � | ψ � = ψ n | n � , | n � = { | uud � , | uudg � , | uudqq � , · · · } n i , k i = ( k + i , k − with k 2 i = m 2 i , k ⊥ i ) , for each constituent i in state n i ) independent of P + and P • Fock components ψ n ( x i , k ⊥ i , λ z ⊥ and depend only on relative partonic coordinates: momentum fraction x i = k + i /P + , transverse momentum k ⊥ i and spin λ z i n n � � x i = 1 , k ⊥ i = 0 . i =1 i =1 HEP , Imperial College, October 4, 2010 Page 12
Semiclassical Approximation to QCD in the Light Front [ GdT and S. J. Brodsky, PRL 102 , 081601 (2009)] • Compute M 2 from hadronic matrix element � ψ ( P ′ ) | P µ P µ | ψ ( P ) � = M 2 � ψ ( P ′ ) | ψ ( P ) � • Find � k 2 � � � ⊥ ℓ + m 2 � � � | ψ n ( x i , k ⊥ i ) | 2 + interactions M 2 = �� d 2 k ⊥ i ℓ dx i x q n ℓ • Semiclassical approximation to QCD: � � ( k 1 + k 2 + · · · + k n ) 2 ψ n ( k 1 , k 2 , . . . , k n ) → φ n � �� � M 2 with k 2 i = m 2 n i for each constituent • Functional dependence of Fock state | n � given by invariant mass n k 2 ⊥ a + m 2 � � 2 � � a M 2 k µ n = = a x a a a =1 Key variable controlling bound state: off-energy shell E = M 2 −M 2 n HEP , Imperial College, October 4, 2010 Page 13
• In terms of n − 1 independent transverse impact coordinates b ⊥ j , j = 1 , 2 , . . . , n − 1 , � � n − 1 −∇ 2 b ⊥ ℓ + m 2 � � � � M 2 = ℓ dx j d 2 b ⊥ j ψ ∗ n ( x i , b ⊥ i ) ψ n ( x i , b ⊥ i ) + interactions x q n j =1 ℓ • Relevant variable conjugate to invariant mass n − 1 � � � x � � � ζ = x j b ⊥ j � � 1 − x j =1 the x -weighted transverse impact coordinate of the spectator system ( x active quark) • For a two-parton system ζ 2 = x (1 − x ) b 2 ⊥ • To first approximation LF dynamics depend only on the invariant variable ζ, and hadronic properties are encoded in the hadronic mode φ ( ζ ) from ψ ( x, ζ, ϕ ) = e iMϕ X ( x ) φ ( ζ ) √ 2 πζ factoring angular ϕ , longitudinal X ( x ) and transverse mode φ ( ζ ) HEP , Imperial College, October 4, 2010 Page 14
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