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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/0505164 (Harvard University)
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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)
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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
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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
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D3/D7 in the probe approximation `t Hooft limit: Karch & Katz (2002)
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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 @ ½ f ` ( ½ ) + µ M 2 ( ½ 2 + L 2 ) 2 ¡ ` ( ` + 2 ) ½ 2 ¶ f ` ( ½ ) = 0 @ a µ ½ 3 " 3 ½ 2 + L 2 g a b @ b x 8 ; 9 ¶ = 0 Kruczenski et al. (2003)
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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 = 3 2 + ` ¡ 1 2 p 1 + M 2 R 4 = L 2 ! ¹ f ( ½ ) » ½ 3 + ` = ½ ¢
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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
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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
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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/0504041 where S f D 7 = ¿ D 7 2 Z d 8 » p ¡ g ^ ¹ ª P ¡ ¡ ^ A ( D ^ A + 1 8 i 2 ¢ 5 ! F ^ N ^ P ^ Q ^ R ^ S ¡ ^ N ^ P ^ Q ^ R ^ S ¡ ^ A ) ^ ª
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Equation of motion (part 1) Dirac equation on : decomposition: D = ^ ª + 1 8 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
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Equation of motion (part 2) spinorial harmonics on n-sphere: (for n=3) transform in the result: masses: ( S 3 :n = 3 ) m + ` = 5 2 + ` ; m ¡ ` = ¡ ( 1 2 + ` ) ¡ ( ` + 1 2 ; ` 2 ) an d ( ` 2 ; ` + 1 2 ) o f SO ( 4 )
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Dual operators? The dual operators must have the following properties: - spin ½ - mass-dimension relation: - Spin-½ operators: SU ( 2 ) R £ SU ( 2 ) © : ( ` + 1 2 ; ` 2 ) an d ( ` 2 ; ` + 1 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
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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 @ 2 z ¡ d z @ z ¡ m 2 R 2 + 6 + m R ° z ) ª ( x ¹ ; z ) = 0 A d S 5 :
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Spectrum of spin-½ fluctuations (part 2) solution: where spectrum: ¡ n + = j m ` j ¡ 1 2 p 1 + M 2 = L 2 ; ¡ n ¡ = ¡ n + + 1 ¡ m + ` = 5 2 + ` ; m ¡ ` = ¡ ( 1 2 + ` ) ¢ 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 )
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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
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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 + 3 2 N ¡ 3 2 )( n + 3 2 N ¡ 5 2 ) as in Teramond & Brodsky (2004/05)
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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
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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 )
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The warp factor h(r, , ) Poison equation: D. Vaman, I.K. (2005) Fourier expansion: Schrödinger-like equation with log-potential (for 0):, or,,
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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,...)
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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/0503156) gauge coupling and -angle: Results: nonconformal theories lead to (harmless) logarithmic tadpoles in the SUGRA background which reproduce the correct perturbative gauge theory parameters
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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
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