Spectroscopy of fermionic operators in AdS/CFT with flavor Ingo Kirsch Workshop „QCD and String Theory“ Ringberg Castle, Tegernsee, July 2-8, 2006 I. K., hep-th/0607xxx D. Vaman, I.K., hep-th/ (Harvard University)

Outline 1. Holographic meson spectroscopy - review on AdS/CFT with flavor (fundamentals) in the probe approximation (neglect backreaction of probe brane) - D3/D7 intersection, meson spectroscopy 2. Spectroscopy of spin-1/2 fluctuations in the D3/D7 system - fermionic action for the D7-brane - Dirac-like equations for spin-1/2 fluctuations 3. Beyond the probe approximation: - construction of the fully localized D3/D7 supergravity solution (including the backreaction of the D7-brane)

The D3/D7 brane intersection Set-up: preserves: 8 supersymmetries SO(4) x SO(2) isometry Field theory: N=4 SU(N c ) super Yang-Mills (3-3 strings) coupled to N f N=2 hypermultiplets (3-7 strings) SU(2) R x U(1) R R-symmetry + SU(2)  global sym. quark mass: separate branes in 89 by a distance L ~ m

More on the N=2 field theory perturbative beta function: running gauge coupling: UV Landau pole: probe approximation: conformal limit N f cons t :; N c ! 1 ) ¯ ¸ N = 2 ! 0

D3/D7 in the probe approximation `t Hooft limit: Karch & Katz (2002)

Spectroscopy of meson operators Spin-0/spin-1 open string fluctuations on the D7-brane are described by the bosonic part of the D7-brane action (DBI): e.o.m.: plane-wave ansatz: eqn. for fluctuation: S b D 7 = ¡ T 7 Z d 8 » q ¡ d e t ( g PB a b + F a b ) x 8 = 0 ; x 9 = L + f ` ( ½ ) e i k ¢ x Y ` ( S 3 2 ½ f ` ( ½ ) + 3 ½ f ` ( ½ ) + µ M 2 ( ½ 2 + L 2 ) 2 ¡ ` ( ` + 2 ) ½ 2 ¶ f ` ( ½ ) = a µ ½ 3 " 3 ½ 2 + L 2 g a b x 8 ; 9 ¶ = 0 Kruczenski et al. (2003)

Meson spectroscopy (part 2) solution: quantization condition: mass spectrum: dual scalar meson operator: f ` ( ½ ) = ½ ` ( ½ 2 + L 2 ) n + ` + 1 F ( ¡ ( n + ` + 1 ) ; ¡ n; ` + 2 ; ¡ ½ 2 = L 2 ) M 2 s = 4 L 2 R 4 ( n + ` + 1 )( n + ` + 2 )( n ; ` > 0 ) M A ` s = ¹ Ã i ¾ A ij X ` Ã j + ¹ q m X A V X ` q m ( i ; m = 1 ; 2 ) ¢ = 3 + ` ¡ n = ` ¡ 1 2 p 1 + M 2 R 4 = L 2 ! ¹ f ( ½ ) » ½ 3 + ` = ½ ¢

U(1) chiral symmetry breaking U(1) A chiral symmetry breaking: - U(1) chiral sym.,, SO(2) isometry in x 8, x 9 - broken by quark condensate: D3 NONSUSY /D7: screening effect: D7-branes repel from spont. U(1) breaking: m ! 0, c  0 singularity X9X9 Babington, Erdmenger, Evans, Guralnik, I.K. (2003) à ! e ¡ i " à ~ à ! e ¡ i " ~ à c = h à ~ à i 6 = 0

Meson spectrum and large N c Goldstone boson (  ') Consider fluctuations  x 8 =f(r) sin(k ¢ x),  x 9 =h(r) sin(k ¢ x) of the plane wave type (M 2 =-k 2 ) around the embedding solution x 8 =0, x 9 = x 9 (r) ) meson spectrum M(m) mexican hat for small m (GMOR) X9X9 X8X8

Spectroscopy of fermionic operators Spin-1/2 open string fluctuations on the D7-brane are described by the fermionic part of the D7-brane action: Martucci et al., hep-th/ where S f D 7 = ¿ D 7 2 Z d 8 » p ¡ g ^ ¹ ª P ¡ ¡ ^ A ( D ^ A i 2 ¢ 5 ! F ^ N ^ P ^ Q ^ R ^ S ¡ ^ N ^ P ^ Q ^ R ^ S ¡ ^ A ) ^ ª

Equation of motion (part 1) Dirac equation on : decomposition: D = ^ ª i 2 ¢ 5 ! ¡ ^ A F ^ N ^ P ^ Q ^ R ^ S ¡ ^ N ^ P ^ Q ^ R ^ S ¡ ^ A ^ ª = 0 A d S 5 £ S 3 ^ ª = " ­ Â |{z} S 5 ­ ª |{z} A d S 5 ; Â = Â jj |{z} S 3 ­ Â ? { ¡ ^ M 5 - f orm: F NPQRS = 1 R " NPQRS ; F npqrs = 1 R " npqrs

Equation of motion (part 2) spinorial harmonics on n-sphere: (for n=3) transform in the result: masses: ( S 3 :n = 3 ) m + ` = ` ; m ¡ ` = ¡ ( ` ) ¡ ( ` ; ` 2 ) an d ( ` 2 ; ` ) o f SO ( 4 )

Dual operators? The dual operators must have the following properties: - spin ½ - mass-dimension relation: - Spin-½ operators: SU ( 2 ) R £ SU ( 2 ) © : ( ` ; ` 2 ) an d ( ` 2 ; ` ) F ` ® » ¹ q X ` ~ Ã y ® + ~ Ã ® X ` q ; G ` ® » ¹ Ã i ¾ B ij ¸ ® C X ` Ã j + ¹ q m X B V ¸ ® C X ` q m ( B ; C = 1 ; 2 ) Ã i = ( Ã ; ~ Ã y )[( 0 ; 0 )] ; q m = ( q ; ¹ ~ q )[( 0 ; 1 2 )] f un d amen t a l s ¸ ® A [( 1 2 ; 0 )] ; X ` = X f i 1 ¢¢¢ X i l g [( ` 2 ; ` 2 )] a d j o i n t ¯ e ld s

Spectrum of spin-½ fluctuations (part 1) Dirac equation on Mück &Viswanathan (1998) second order equation: plane-wave ansatz: e.o.m. for fluctuations: ª ` ( x ; r ) = e i P ¹ x ¹ f ` ( r ) ; M 2 = ¡ P ¹ P ¹ ( z = R 2 = r ) ( z 2 z ¡ d z ¡ m 2 R m R ° z ) ª ( x ¹ ; z ) = 0 A d S 5 :

Spectrum of spin-½ fluctuations (part 2) solution: where spectrum: ¡ n + = j m ` j ¡ 1 2 p 1 + M 2 = L 2 ; ¡ n ¡ = ¡ n ¡ m + ` = ` ; m ¡ ` = ¡ ( ` ) ¢ M 2 G = 4 L 2 R 4 ( n + + ` + 2 )( n + + ` + 3 )( n + > 0 ; ` > 0 ) M 2 F = 4 L 2 R 4 ( n ¡ + ` + 1 )( n ¡ + ` + 2 )( n ¡ > 0 ; ` > 0 )

Supermultiplets in the D3/D7 theory Masses of supermultiplets: Kruczenski et al. (2003) Field content: M 2 = 4 L 2 R 4 ( n + ` + 1 )( n + ` + 2 )( n ; ` > 0 ) 8 ( ` + 1 ) b osons + f erm i ons fluctuation

Baryons in a phenomenological model Consider a large N baryon: Dirac equation on baryon spectrum: at large N as expected from FT, Witten (1979) B 0 = 1 p N ! " i 1 i 2 ::: i N Ã i 1 ::: Ã i N ( ¢ = 3 2 N ) A d S 5 : ( D = A d S 5 ¡ m ) ª = 0 ; m = ¢ ¡ 2 = 3 2 N ¡ 2 ) M B » N M 2 B = 4 L 2 R 4 ( n N ¡ 3 2 )( n N ¡ 5 2 ) as in Teramond & Brodsky (2004/05)

Leaving the quenched approximation… Quenched approximation: lattice QCD: fermion determinant:, 10-20% error ) quark-loops in QCD correlation functions are ignored AdS/CFT: quenched = probe approximation: no backreaction of the “flavor'' (D7-)brane on the geometry Beyond the quenched approximation: lattice QCD: logarithm of the fermion determinant is nonlocal ) dramatic slow-down of the Monte Carlo algorithms Grassmann variables difficult to handle on computers ) difficult to go beyond the quenched approximation! AdS/CFT: Easier! Take into account the backreaction of the “flavor“ brane, ie. construct fully localized brane intersections

The D3/D7 sugra background Susy-preserving ansatz by Polchinski and Grana (2001): metric: axion-dilaton: singularities: - curvature singularity at  =0 - dilaton divergence at  =  L ( ! Landau pole  L )

The warp factor h(r, ,  ) Poison equation: D. Vaman, I.K. (2005) Fourier expansion: Schrödinger-like equation with log-potential (for   0):, or,,

The warp factor h(r, ,  ) -- solution Solution for For N f  0 series expansion ansatz: Gesztesy and Pittner (1978) solution: recursion relation for p n (x): (n=0,1,2,...)

Logarithmic tadpoles and one-loop vacuum amplitudes Open string one-loop amplitude (to quadr. order in F): Di Vecchia et al. (e.g. hep-th/ ) gauge coupling and  -angle: Results: nonconformal theories lead to (harmless) logarithmic tadpoles in the SUGRA background which reproduce the correct perturbative gauge theory parameters

Summary and Outlook Two extensions of holography with flavor 1) Spectra of fermionic operators: - computed the mass spectrum of spin-½ operators in the D3/D7 theory from the fermionic part of the D7-brane action 2) Beyond the probe approximation - fully localized D3/D7 solution - completed the solution by providing an analytic expression for the warp factor h(r,  ) in terms of a convergent series - related the pathology of the D3/D7 background (dilaton divergence) to the Landau pole in the gauge theory Outlook: - The techniques discussed in this talk should be useful for the holographic computation of baryon spectra including Witten‘s string theory realization of a baryon vertex