![Page 1: 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.,](https://reader038.vdocuments.net/reader038/viewer/2022110319/56649c775503460f9492c131/html5/thumbnails/1.jpg)
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)
![Page 2: 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.,](https://reader038.vdocuments.net/reader038/viewer/2022110319/56649c775503460f9492c131/html5/thumbnails/2.jpg)
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)
![Page 3: 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.,](https://reader038.vdocuments.net/reader038/viewer/2022110319/56649c775503460f9492c131/html5/thumbnails/3.jpg)
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
![Page 4: 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.,](https://reader038.vdocuments.net/reader038/viewer/2022110319/56649c775503460f9492c131/html5/thumbnails/4.jpg)
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
![Page 5: 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.,](https://reader038.vdocuments.net/reader038/viewer/2022110319/56649c775503460f9492c131/html5/thumbnails/5.jpg)
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
![Page 6: 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.,](https://reader038.vdocuments.net/reader038/viewer/2022110319/56649c775503460f9492c131/html5/thumbnails/6.jpg)
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)
![Page 7: 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.,](https://reader038.vdocuments.net/reader038/viewer/2022110319/56649c775503460f9492c131/html5/thumbnails/7.jpg)
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¢
![Page 8: 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.,](https://reader038.vdocuments.net/reader038/viewer/2022110319/56649c775503460f9492c131/html5/thumbnails/8.jpg)
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
![Page 9: 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.,](https://reader038.vdocuments.net/reader038/viewer/2022110319/56649c775503460f9492c131/html5/thumbnails/9.jpg)
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
![Page 10: 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.,](https://reader038.vdocuments.net/reader038/viewer/2022110319/56649c775503460f9492c131/html5/thumbnails/10.jpg)
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
![Page 11: 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.,](https://reader038.vdocuments.net/reader038/viewer/2022110319/56649c775503460f9492c131/html5/thumbnails/11.jpg)
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)
![Page 12: 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.,](https://reader038.vdocuments.net/reader038/viewer/2022110319/56649c775503460f9492c131/html5/thumbnails/12.jpg)
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)
![Page 13: 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.,](https://reader038.vdocuments.net/reader038/viewer/2022110319/56649c775503460f9492c131/html5/thumbnails/13.jpg)
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 +`
![Page 14: 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.,](https://reader038.vdocuments.net/reader038/viewer/2022110319/56649c775503460f9492c131/html5/thumbnails/14.jpg)
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
![Page 15: 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.,](https://reader038.vdocuments.net/reader038/viewer/2022110319/56649c775503460f9492c131/html5/thumbnails/15.jpg)
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¡
![Page 16: 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.,](https://reader038.vdocuments.net/reader038/viewer/2022110319/56649c775503460f9492c131/html5/thumbnails/16.jpg)
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
![Page 17: 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.,](https://reader038.vdocuments.net/reader038/viewer/2022110319/56649c775503460f9492c131/html5/thumbnails/17.jpg)
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)
![Page 18: 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.,](https://reader038.vdocuments.net/reader038/viewer/2022110319/56649c775503460f9492c131/html5/thumbnails/18.jpg)
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
![Page 19: 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.,](https://reader038.vdocuments.net/reader038/viewer/2022110319/56649c775503460f9492c131/html5/thumbnails/19.jpg)
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)
![Page 20: 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.,](https://reader038.vdocuments.net/reader038/viewer/2022110319/56649c775503460f9492c131/html5/thumbnails/20.jpg)
The warp factor h(r,,)
Poison equation: D. Vaman, I.K. (2005)
Fourier expansion:
Schrödinger-like equation with log-potential (for 0):
,
or, ,
![Page 21: 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.,](https://reader038.vdocuments.net/reader038/viewer/2022110319/56649c775503460f9492c131/html5/thumbnails/21.jpg)
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,...)
![Page 22: 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.,](https://reader038.vdocuments.net/reader038/viewer/2022110319/56649c775503460f9492c131/html5/thumbnails/22.jpg)
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
![Page 23: 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.,](https://reader038.vdocuments.net/reader038/viewer/2022110319/56649c775503460f9492c131/html5/thumbnails/23.jpg)
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