Spectroscopy of fermionic operatorsSpectroscopy of fermionic operators

in AdS/CFT with flavor 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)

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(Nc) super Yang-Mills (3-3 strings) coupled to Nf 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 limitNf const:;Nc ! 1 ) ¯ ¸

N =2 ! 0

D3/D7 in the probe approximation

`t Hooft limit:

Karch & Katz (2002)

4d N=4SU(N) type IIB SUGRA onSuper Yang-Mills theory AdS5 £ S5 +

coupled to D7-braneaction on4d N = 2hypermultiplets AdS5 £ S3

Spectroscopy of meson operators

Spin-0/spin-1 open string fluctuations on the D7-brane aredescribed by the bosonic part of the D7-brane action (DBI):

e.o.m.:

plane-wave ansatz:

eqn. for fluctuation:

SbD7 = ¡ T7

Zd8»

q¡ det(gP B

ab +Fab)

x8 = 0; x9 = L +f `(½)eik¢xY `(S3)

@2½f `(½) +

3½@½f `(½) +

µM 2

(½2 +L2)2¡

(`+2)½2

¶f `(½) = 0

@a

µ½3"3

½2 +L2gab@bx8;9

¶

= 0

Kruczenski et al. (2003)

Meson spectroscopy (part 2)

solution:

quantization condition:

mass spectrum:

dual scalar meson operator:

f `(½) =½

(½2 +L2)n+`+1F (¡ (n+`+1);¡ n;`+2;¡ ½2=L2)

M 2s =

4L2

R4(n+`+1)(n+`+2) (n;`> 0)

M A`s = ¹Ãi¾A

i j X`Ãj + ¹qmX A

V X `qm (i;m= 1;2)

¢ = 3+`

¡ n = 32 +`¡ 1

2

p1+M 2R4=L2 ! ¹f (½) » ½3+` = ½¢

U(1) chiral symmetry breaking

U(1)A chiral symmetry breaking:

- U(1) chiral sym. , , SO(2) isometry in x8, x9

- broken by quark condensate:

D3NONSUSY/D7:

screening effect: D7-branes repel from spont. U(1) breaking: m! 0 , c 0 singularity

X9

Babington, Erdmenger, Evans, Guralnik, I.K. (2003)

à ! e¡ i "à ~à ! e¡ i " ~Ã

c= hÃ~Ãi 6= 0

Meson spectrum and large Nc Goldstone boson (')

Consider fluctuations x8=f(r) sin(k¢x) , x9 =h(r) sin(k¢x) of the plane

wave type (M2=-k2) around the embedding solution x8=0, x9 = x9(r) ) meson spectrum M(m)

mexican hat for small m

(GMOR)

X9

X8

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»A world-volumecoordinates (A = 0;¢¢¢;7)

¡ A pullback of 10d gamma matrices;¡ A = ¡ M @AX M

ª 10d pos. chirality Majorana-Weyl spinor

FN P Q R S self-dual type IIB 5-form

DA covariant derivative

SfD 7 =

¿D 7

2

Zd8»

p¡ g¹ª P ¡ ¡ A (DA +

18

i2¢5!

FN P Q R S ¡N P Q R S ¡ A )ª

Equation of motion (part 1)

Dirac equation on :

decomposition:

D=ª +18

i2¢5!

¡ AFN P Q R S ¡N P QR S ¡ A ª = 0

AdS5 £ S3

ª =" ­ Â|{z}S5

­ ª|{z}AdS5

; Â = Âjj|{z}S3

­ Â?

{¡ M

5-form: FN P QRS =1R

"N P QRS ; Fnpqrs =1R

"npqrs

¡ M = ¾y ­ 14 ­ °M (M = 0;1;2;3;4)

¡ m = ¾x ­ °m ­ 14 (m= 5;6;7;8;9)

¡ M = ¾y ­ 14 ­ °M (M = 0;1;2;3;4)

¡ m = ¾x ­ °m ­ 14 (m= 5;6;7;8;9)

Equation of motion (part 2)

spinorial harmonics on n-sphere:

(for n=3) transform in the

result:

masses:

(S3 : n = 3)

m+` = 5

2 +`; m¡` = ¡ (1

2 + )

D=Sn §` = ¨ i¸`§

` = ¨ iR (`+ n

2 )§` (` > 0)D=Sn §

` = ¨ i¸`§` = ¨ i

R (`+ n2 )Â

§` (` > 0)

(D=AdS5 ¨ 1R (`+ 3

2) +1R|{z}

5¡ f orm

)ª §` =

((D=AdS5 ¡ 1

R (`+ 52))ª

+`

(D=AdS5 + 1R (`+ 1

2))ª¡`

)

= 0(D=AdS5 ¨ 1R (`+ 3

2) +1R|{z}

5¡ f orm

)ª §` =

((D=AdS5 ¡ 1

R (`+ 52))ª

+`

(D=AdS5 + 1R (`+ 1

2))ª¡`

)

= 0¡

( `+12 ; `

2) and ( `2;

`+12 ) of 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)©: ( `+12 ; `

2) and ( `2;

`+12 )

F `® » ¹qX ` ~Ãy

®+ ~îX `q;

G® » ¹Ãi¾Bi j ¸®C X `Ãj + ¹qmX B

V ¸®C X `qm (B;C = 1;2)

Ãi = (Ã; ~Ãy) [(0;0)];qm = (q; ¹~q) [(0; 12)] fundamentals

¸®A [(12;0)];X

` = X f i1 ¢¢¢X i l g [( `2;

`2)] adjoint ¯elds

¢ = jm§` j +2=

(92 +`52 +`

¢ = jm§` j +2=

(92 +`52 +`

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) = eiP ¹ x¹f `(r) ; M 2 = ¡ P ¹ P¹

(z = R2=r)

(z2@2z ¡ dz@z ¡ m2R2 +6+mR°z)ª (x¹ ;z) = 0

AdS5:

(D=AdS5 ¡ m§` )ª §

` = 0(D=AdS5 ¡ m§` )ª §

` = 0

³@2r + 6

r @r + 1r 2 (¡ jm`j2R2 +6+ jm`jR°r ) + M 2R 4

(r2+L 2)2

´f `(r) = 0

³@2r + 6

r @r + 1r 2 (¡ jm`j2R2 +6+ jm`jR°r ) + M 2R 4

(r2+L 2)2

´f `(r) = 0

Spectrum of spin-½ fluctuations (part 2)

solution:

where

spectrum:

¡ n+ = jm`j ¡ 12

p1+M 2=L2 ; ¡ n¡ = ¡ n+ +1

¡m+

` = 52 +`; m¡

` = ¡ (12 + )

¢

M 2G =

4L2

R4(n+ +`+2)(n+ +`+3) (n+ > 0;`> 0)

M 2F =

4L2

R4(n¡ +`+1)(n¡ +`+2) (n¡ > 0;`> 0)

f `(r) = r jm` j¡ 3(L2 +r2)12 ¡ jm` j¡ n+

2F 1

³12 ¡ jm`j ¡ n+;¡ n+; jm`j + 1

2;¡r 2

L 2

´a+

+r jm` j¡ 2(L2 +r2)¡12 ¡ jm` j¡ n ¡

2F 1

³¡ 1

2 ¡ jm`j ¡ n¡ ;¡ n¡ ; jm`j + 32;¡

r 2

L 2

´a¡

f `(r) = r jm` j¡ 3(L2 +r2)12 ¡ jm` j¡ n+

2F 1

³12 ¡ jm`j ¡ n+;¡ n+; jm`j + 1

2;¡r 2

L 2

´a+

+r jm` j¡ 2(L2 +r2)¡12 ¡ jm` j¡ n ¡

2F 1

³¡ 1

2 ¡ jm`j ¡ n¡ ;¡ n¡ ; jm`j + 32;¡

r 2

L 2

´a¡

Supermultiplets in the D3/D7 theory

Masses of supermultiplets: Kruczenski et al. (2003)

Field content:

M 2 =4L2

R4(n+`+1)(n+`+2) (n;`> 0)

8(`+1) bosons+ fermions

°uctuation (j 1; j 2)q spectrum op. ¢bos. 1 scalar ( `

2,`2 +1)0 M I ;¡ (n;`+1) (` ¸ 0) CI ` 2

2 scalars ( `2,

`2)2 Ms(n; ) (` ¸ 0) M A`

s 31 scalar ( `

2,`2)0 M I I I (n; ) (` ¸ 1) J ¹ `

B 31 vector ( `

2,`2)0 M I I (n; ) (` ¸ 0)

1 scalar ( `2,

`2 ¡ 1)0 M I ;+(n;` ¡ 1) (` ¸ 2) { 4

ferm. 1Dirac ( `2,

`+12 )1 MF (n; ) (` ¸ 0) F `

®52

1Dirac ( `2,

`¡ 12 )1 MG(n;` ¡ 1) (` ¸ 1) G`

®92

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)

B0 =1

pN!

"i1 i2:::iN Ãi1 :::ÃiN (¢ = 32N)

AdS5:

(D=AdS5¡ m)ª = 0; m= ¢ ¡ 2= 3

2N ¡ 2

) MB » N

M 2B = 4L 2

R 4 (n+ 32N ¡ 3

2)(n+ 32N ¡ 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 Nf 0 series expansion ansatz: Gesztesy and Pittner (1978)

solution:

recursion relation for pn(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/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

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