Holographic heavy-light mesons
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(2) Outline 1. Holographic meson spectroscopy (single flavor) - review on AdS/CFT with flavor (fundamentals) in the probe (quenched) approximation - D3/D7 intersection, meson spectroscopy 2. Heavy-light mesons (two flavors) - Method: non-Abelian Dirac-Born-Infeld action HL mesons in - non-confining N=2 theory (D3/D7) : compare w/ HH spectrum new term of O(λ0 ) - confining N=1 theory (D(-1)/D3/D7): spin-dependence: scalar vs. vector mesons - “QCD-like“ N=0 theory (D3def/D7) 3. Conclusions. 2.
(3) References Paredes, Talavera (2004) Erdmenger, Evans, Grosse (2006) Erdmenger, Ghoroku, I.K. (2007). this talk. Herzog, Stricker, Vuorinen (2008). 3.
(4) Part 1: Single flavor mesons. `t Hooft limit. N → ∞, gs → 0, λ ∼ gs N = const.. (R4 = λα′2 ). AdS/CFT duality. 4d N=4 SU(N) Super Yang-Mills theory. type IIB SUGRA on AdS5 × S 5. 4.
(5) Part 1: Single flavor mesons Karch, Katz (2002). `t Hooft & probe brane limit Nf /N. 0: ignore D7 backreaction. AdS/CFT duality. 4d N=4 SU(N) Super Yang-Mills theory coupled to N=2 fund. hypermultiplets. type IIB SUGRA on AdS5 × S 5 + DBI action on AdS5 × S 3 5.
(6) Spectroscopy of single flavor operators Spin-0/spin-1 open string fluctuations on a single D7-brane are described by the abelian Dirac-Born-Infeld action:. SDBI = −T7 scalar e.o.m.:. ∂a. . 3. Kruczenski, Mateos, Myers, Winters (2003). PB + F ) d8 ξ − det(gab ab. ρ ε3 ab 8,9 g ∂ x b ρ2 + L2. . =0. plane-wave ansatz:. x8 = 0 , eqn. for fluctuation:. x9 = L + fℓ (ρ)eik·x Y ℓ (S 3 ) ,. M 2 = −k2 meson mass M(L, λ),. quark mass L 2 4 3 ℓ(ℓ + 2) M R ∂ρ2 + ∂ρ − + 2 fℓ (ρ) = 0 2 2 2 ρ ρ (ρ + L ). 6.
(7) Heavy-heavy meson spectroscopy (part 2) solution:. ρℓ 2 2 fℓ (ρ) = 2 F (−(n + ℓ + 1), −n; ℓ + 2; −ρ /L ) (ρ + L2 )n+ℓ+1. quantization condition: −n =. 3 2. mass spectrum:. +ℓ− 1 + M 2 R4 /L2 1 2. →. 4L2 M = 4 (n + ℓ + 1)(n + ℓ + 2) R dual ∆ = 3 + ℓ scalar meson operator: 2. ¯i σ A X ℓ ψj + q¯m X A X ℓ q m = ψ MAℓ s ij V. f¯(ρ) ∼ ρ3+ℓ = ρ∆. (n, ℓ 0). L ∼ mq R4 ∼ λ. (i, m = 1, 2). 7.
(8) Part 2: Heavy-light mesons Erdmenger, Ghoroku, I.K. (2007). Heavy-light mesons require two D7-branes. Their (asympt.) position determines the quark masses: mH = w1 |ρ→∞ , mL = w2 |ρ→∞ Diagonal embedding:. “heavy” brane. ¯8 = 0 , X. ¯9 = X. . w1 0. 0 w2. “light” brane. Fluctuations: X 8 = φ8 ,. ¯ 9 + φ9 X9 = X. w 1 x9. HH mesons. φa =. . HL mesons. φa+ φa21. φa12 φa−. LL mesons. w2. 8.
(9) Non-Abelian Dirac-Born-Infeld action 4. Myers (1999), Tseytlin (1997). nA-DBI expanded to O(X ): SNf = τp dp+1 ξ STr − det(Grs + Gab Dr X a Ds X b + T −1 Frs ) 1 2 T Gac [X c , X b ] × 1− 4 covariant derivative & symmetrized trace: Dr X a = ∂r Xa + i[Ar , X a ] 1 STr (A1 ...An ) = n! Tr(A1 ...An + all permutations). 9.
(10) N=2: Scalar heavy-light mesons (e.o.m.) E.o.m. for off-diagonal fluctuations: 3 l(l + 2) 8 ∂ρ2 + ∂ρ − + M + M φ 1 2 12 = 0 ρ ρ2 where M 2 − v2 4 Mi = R 2(ρ2 + wi2 )2. (i = 1, 2) ,. v = w1 − w2. quark masses. w1,2 = (2πls2 )mH,L R4 = λls4 HL meson mass M = M (mH , mL ; λ) (can be extended to include non-trival dilaton ( N=0,1 theories)) 10.
(11) N=2: Scalar heavy-light mesons (plots) Scalar mass spectrum in N=2 theory (D3/D7): 2 MHL. m2L 2 = 8π n + (mH − mL )2 (for large mH , n; l = 0).
(12). λ O(λ0 ). Numerical mass spectrum:. analytic. (mL = 1, T = 1, n = l = 0) 11.
(13) Wilson loop for heavy-light meson quark-antiquark potential from Wilson loop:. W =. N1 TrP ei. . A0 dt. ∼ e−S S = NambuNambu-Goto action. 4d flux tube. q. Q. 5th dimension. q. recover O(λ0 ) term: E(L = 0) = mH − mL. (A) HL meson (B) LL meson (C) HH meson. 12.
(14) Comparison with experimental data at large `t Hooft coupling: MHL − MHL′ = |mL − mL′ | = MH ′ L − MH ′ L′ experimental values (PDG ‘07): MBs0 − MB = 87 MeV ms − mu,d ≈ 90 MeV MDs± − MD± = 99 MeV. ≈ 90 − 100 MeV. similar findings in Herzog et al. (2008). 13.
(15) N=1: Vector vs. scalar mesons Liu-Tseytlin D(-1)/D3 background (1999). Mesons in confining N=1 theory (D(-1)/D3/D7): dashed: vector mesons solid : scalar mesons R=2, q=10 (q=gauge condensate). vector masses are larger than the scalar masses mH ≫ 1 : vector spectrum degenerates with scalar spectrum again (N=2 susy restoration). 14.
(16) N=0: Mesons in non-susy theory Deformed D3/D7 (e.g. Constable-Myers, dilaton flow, etc.):. Babington, Erdmenger, Evans, Guralnik, I.K. (2003). U(1)A chiral symmetry breaking: - U(1) chiral sym. ψ → e−iε ψ, - broken by quark condensate:. ψ˜ → e−iε ψ˜ ⇔ SO(2) isometry in x8, x9 ˜ = 0 c = ψ ψ. Embedding profiles + condensate: X9. screening effect: D7D7-branes repel from singularity. spont. U(1) breaking: m→ m→ 0 , c ≠ 0. 15.
(17) Meson spectrum and large Nc Goldstone boson (η') Consider plane-wave fluctuations δx9 = f(ρ) sin(k x) (M2 = -k2) around the embedding solution x8 = 0, x9 = x9 (ρ) ⇒ single-flavor meson spectrum M(m). X9 w5. X8. η’ meson GMOR at small m:. Goldstone boson due to spont. U(1) chiral SB (anomaly suppressed at large N) 16.
(18) N=0: HH & HL meson spectrum Consider off-diagonal plane-wave fluctuations: ⇒ heavy-light meson spectrum M(mH ) (mL = 0). λ. numerical HL mass:. λ≫1. M 2 (λ) →. 1 2πα′ (w1. − w2 )|ρ=0 < mH − mL. ⇒ O(λ0 ) term also present in N=0 theory, but smaller than in supersymmetric theories 17.
(19) Conclusions We studied heavy-light meson spectra in N=2, N=1, N=0 QCD-related theories using AdS/CFT. We used the non-Abelian Dirac-Born-Infeld action to describe two separate D7-branes corresponding to a heavy and a light quark. We found an order O(1) contribution to the HL meson mass which is dominant at large `t Hooft coupling: 2 MHL. m2L 2 n + (mH − mL )2 = 8π.
(20). λ O(λ0 ). Verfied this term by computing the corresponding Wilson loop and comparing to experimental values. The O(1) term is less dominant in non-susy theories (but still present!) Also discussed the spin-dependence of mesons in N=1.. 18.
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