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An Effective Holographic Approach To Nonlinear realization of chiral symmetry breaking in the holographic soft wall model To appear on arXiv Alfonso Ballon-Bayona In collaboration with Luis A. H. Mamani (UFABC - Brazil) WONPAQCD 2019 – UDELAR 1 Outline - Motivation - AdS/CFT and holographic QCD - Chiral symmetry breaking in holographic QCD. - Conclusions 2 Motivation ퟏ QCD: A challenging theory 푳 = 흍ഥ 풊 휸흁푫 − 풎 흍 − 퐓퐫 퐅 퐅흁흂 푸푪푫 풇 흁 풇 풇 ퟐ 흁흂 where 푫흁 = 흏흁 − 풊품푨흁 , 푭흁흂 = 흏흁푨흂 − 흏흂푨흁 − 풊품 푨흁, 푨흂 - Quarks are Dirac spinors 흍풇 with 풇 = ퟏ, . , 푵풇 and 푵풇 = ퟔ. - Gluons are non-Abelian gauge fields 푨흁 in the group 푺푼 푵풄 with 푵풄 = ퟑ. 품ퟑ ퟏퟏ ퟐ 휷 = − 푵 − 푵 < ퟎ - UV asymptotic freedom: ퟒ흅 ퟐ ퟑ 풄 ퟑ 풇 - IR confinement: 3 Nonperturbative approaches to QCD - Lattice QCD: Wilson 1974 -Dyson-Schwinger equations: Dyson 1949, 휹푺 흓 ( ቚ 휹 − 퐉 )퐙 퐉 = ퟎ Schwinger 1951 흓= 휹흓 휹푱 - Other approaches: Chiral Lagrangians, Nambu-Jona-Lasinio (NJL) model, quark-meson model, QCD sum rules, RG flows, Gribov-Zwanziger theories, etc. 4 AdS/CFT and holographic QCD - The large 푁푐 limit: ‘t Hooft 1974 - The AdS/CFT correspondence : Maldacena 1997 Gubser, Klebanov and Polyakov 1998, 풁푪푭푻 흓ퟎ = 풁푨풅푺[흓] Witten 1998 5 From AdS/CFT to HQCD - The top-down approach: ퟓ 4-d 푵 = ퟒ super Yang-Mills IIB string theory in 푨풅푺ퟓ × 푺 4-d QCD-like theories deformed string theories Klebanov-Witten (1998), Klebanov-Strassler (2000), Maldacena-Nunez (2000), Sakai-Sugimoto (2004),... - The bottom-up approach: 4-d CFT 푨풅푺ퟓ 4-d QCD as a deformed CFT 푨풅푺ퟓ deformed by fields Polchinski-Strassler (2000), Erlich et al (2005), Karch et al (2006), Csaki-Reece (2006), Gursoy et al (2007),… 6 Some progress in holographic QCD - Confinement: 푾 푪 = 풁풔풕풓풊풏품(흏횺 = 퐂) ퟐ ퟐ ퟐ ퟐ 5-d backgrounds of the form 풅풔 = 품풕풕(풛)풅풕 + 품풙풙(풛)풅풙풊 + 품풛풛(풛)풅풛 lead to 푽푸ഥ푸(푳) = 흈푳 at large L when 품풕풕품풙풙 has a nonzero minimum. Kinar, Schreiber and Sonnenschein 1998 - Chiral symmetry breaking: 4-d QCD operators 5-d fields 흁,풂 흁 풂 풎,풂 푱푳/푹 = 풒ഥ푳/푹 휸 푻 풒푳/푹 푨푳/푹 , where 풎 = 풛, 흁 . 풒ഥ풒 푿 Chiral symmetry breaking Gauge symmetry breaking Erlich, Katz, Son and Stephanov & Da Rold and Pomarol 2005 7 Finite temperature HQCD - The conformal plasma: 푻흁흂 = 휶 푻ퟒ 휼흁흂 + ퟒ풖흁풖흂 − ퟐ휼 흈흁흂 + ⋯ is dual to a 5-d black hole in AdS. Universal ratio : 휼 ퟏ 휼: shear viscosity, s: entropy density. = 풔 ퟒ흅 Policastro, Son and Starinets 2001, Kovtun, Son and Starinets 2004 -The non-conformal plasma: 푻흁흂 = 흆풖흁풖흂 + 푷 − 휻휽 휼흁흂 + 풖흁풖흂 − ퟐ휼 흈흁흂 + ⋯ 휻: bulk viscosity Gubser, Nellore, Pufu and Rocha 2008, Gursoy, Kiritsis, Mazzanti and Nitti 2008 QCD deconfinement transition 5-d Hawking-Page transition between thermal AdS and black hole AdS Witten 1998, Herzog 2006, B-B, Boschi-Filho, Braga and Pando Zayas 2007, Gursoy, Kiritsis, Mazzanti and Nitti 2008 8 Chiral symmetry breaking in holographic QCD Chiral symmetry breaking in the hard wall model Erlich, Katz, Son and Stephanov & Da Rold and Pomarol 2005 - 5d background: AdS ending in an IR hard wall ퟏ 풅풔ퟐ = [−풅풕ퟐ + 풅풙ퟐ + 풅풛ퟐ] , ퟎ < 풛 ≤ 풛 풛ퟐ 풊 ퟎ 푨풅푺ퟓ - This background satisfies the confinement criterion. - Adding flavour to the hard-wall model: ퟓ ퟐ ퟐ ퟐ ퟏ 푳 ퟐ ퟏ 푹 ퟐ 푺 = −∫ 풅 풙 −품 퐓퐫 [ 푫풎푿 + 풎푿 |푿| + ퟐ 푭풎풏 + ퟐ 푭풎풏 ] 품ퟓ 품ퟓ w/ (푳/푹) (푳/푹) (푳/푹) (푳/푹) (푳/푹) 푭풎풏 = 흏풎푨풏 − 흏풏푨풎 − 풊[푨풎 , 푨풏 ] , 푳 (푹) 푫풎푿 = 흏풎푿 − 풊푨풎 푿 + 풊 푿 푨풎 9 - Map between 5d fields and 4d operators (푳 /푹) (푳 /푹) 푨풎 ↔ 푱흁 푿 ↔ 풒ഥ푹풒푳 ퟐ 풎푿 = 횫 횫 − ퟒ = −ퟑ (tachyonic field) - Focus on light quarks: 푵풇 = ퟐ - Classical background ퟐ푿ퟎ 풛 = 풗 풛 ퟏퟐ×ퟐ (푳 /푹) 푨풎 = ퟎ - Tachyon field equation ퟐ ퟐ [풛 흏풛 − ퟑ 풛흏풛 + ퟑ ]풗(풛) = ퟎ ퟑ - Exact solution 풗 풛 = 풄ퟏ풛 + 풄ퟑ풛 - Coefficients 풄ퟏ and 풄ퟑ related to the quark mass and chiral condensate. - In the hard wall model 풄ퟑ is fixed by boundary conditions. This is different from QCD w/ we expect a chiral condensate generated dynamically. - The hard wall model is the simplest model t/ describes confinement and chiral symmetry breaking (explicit and spontaneous). 10 Chiral symmetry breaking in the soft wall model Karch, Katz, Son and Stephanov 2006 - 5d background: AdS and a scalar field 횽 (the dilaton) ퟏ 풅풔ퟐ = [−풅풕ퟐ + 풅풙ퟐ + 풅풛ퟐ] , 풛ퟐ 풊 푨풅푺ퟓ ퟐ 횽 풛 = 흓∞풛 횽(풛) - 횽(풛) responsible for conformal symmetry breaking in the gluon sector - Adding flavour to the soft wall model: ퟓ −횽 ퟐ ퟐ ퟐ ퟏ 푳 ퟐ ퟏ 푹 ퟐ 푺 = −∫ 풅 풙 −품 풆 퐓퐫 [ 푫풎푿 + 풎푿 |푿| + ퟐ 푭풎풏 + ퟐ 푭풎풏 ] 품ퟓ 품ퟓ - Tachyon field equation ퟐ ퟐ ퟐ [풛 흏풛 − (ퟑ + ퟐ흓∞풛 ) 풛흏풛 + ퟑ ]풗(풛) = ퟎ - Exact analytic solution consistent with UV asymptotics and regularity in the IR 흅 ퟏ 풗 풛 = 풄 풛 푼( , ퟎ; 흓 풛ퟐ) ퟐ ퟏ ퟐ ∞ 11 - Near the boundary, the solution behaves as ퟑ 풗푼푽 풛 = 풄ퟏ풛 + 풅ퟑ 풄ퟏ 푳풐품 풛 + 풄ퟑ 풄ퟏ 풛 + ⋯ w/ 풅ퟑ and 풄ퟑ are proportional to 풄ퟏ . AdS/CFT maps 풄ퟏ and 풄ퟑ to the quark mass 풎풒and chiral condensate 흈풒. The soft wall model leads to explicit chiral symmetry breaking ! - THIS WORK What happens when we include a nonlinear potential 푼 푿 for the tachyon? - Consider the action ퟓ −횽 ퟐ ퟐ ퟐ ퟒ ퟏ 푳 ퟐ ퟏ 푹 ퟐ 푺 = −∫ 풅 풙 −품 풆 퐓퐫 [ 푫풎푿 + 풎푿 푿 + 흀 푿 + ퟐ 푭풎풏 + ퟐ 푭풎풏 ] 품ퟓ 품ퟓ ퟐ w/ 풎푿 = −ퟑ and we consider the cases 흀 > ퟎ and 흀 < ퟎ - The tachyon field equation becomes nonlinear: 흀 [풛ퟐ흏ퟐ − (ퟑ + ퟐ흓 풛ퟐ) 풛흏 + ퟑ]풗(풛) − 풗ퟑ(풛) = ퟎ 풛 ∞ 풛 ퟐ 12 - Far from the boundary , the solution is regular −ퟐ 풗푰푹 풛 = 풄ퟎ + 풄ퟐ 풄ퟎ 풛 + ⋯ - Solutions for 풗(풛) found numerically. The UV coefficients 풄ퟏ and 풄ퟑ become functions of 풄ퟎ The quark mass 풎풒 and chiral condensate 흈풒 depend on a single parameter ! - We find saturation effects in the nonlinear cases. There is an upper bound in 풄ퟎ w/ 흀 > ퟎ and an upper bound in 풄ퟏ w/ 흀 < ퟎ 13 - For 흀 > ퟎ and 흀 < ퟎ the VEV coefficient 풄ퟑ becomes a nonlinear function of the source coefficient 풄ퟏ - In holographic QCD the coefficient 풄ퟑ is the main contribution to the chiral condensate 흈풒 and the coefficient 풄ퟏ is proportional to the quark mass 풎풒 흈풒 is a nonlinear function of 풎풒 . However, 흈풒 still vanishes w/ 풎풒 → ퟎ Chiral symmetry breaking remains explicit in the chiral limit. - The tachyon background has nonzero energy. The renormalized Hamiltonian varies as 푹풆풏 Consistent w/ the AdS/CFT dictionary 풅푯 = −흈풒풅풎풒 14 Meson spectrum 풗 풛 푿 = 풆흅 풙,풛 [ + 푺(풙, 풛)]풆흅 풙,풛 , - Consider the 5d field perturbations ퟐ 푳 푹 푨풎 = 푽풎(풙, 풛) ± 푨풎(풙, 풛) - The original action can be expanded as 푺 = 푺ퟎ + 푺ퟐ + ⋯ ퟓ −횽 ퟏ 풂 풎풏 ퟏ 풂 풎풏 ퟏ ퟐ 풂 퐚 ퟐ 푺ퟐ = ∫ 풅 풙 −품 풆 { − ퟐ 풗풎풏풗풂 − ퟐ 풂풎풏풂풂 − 풗 퐳 푨풎 − 흏풎흅 품ퟓ 품ퟓ ퟐ ퟏ ퟑ − 흏 푺 ퟐ − [풎ퟐ + 흀 풗ퟐ(풛)]푺ퟐ} ퟐ 풎 푿 ퟐ w/ 풗풎풏 = 흏풎푽풏 − 흏풏푽풎 and 풂풎풏 = 흏풎푨풏 − 흏풏푨풎 Euler-Lagrange equations −횽 풎풏 흏풎 −품풆 풗풂 = ퟎ , −횽 풎풏 ퟐ −횽 ퟐ 풂 퐚 흏풎 −품풆 풂풂 − 품ퟓ −품풆 풗 퐳 (푨풎 − 흏풎흅 ) = ퟎ −횽 ퟐ 풎 풎 흏풎 −품풆 풗 퐳 (푨풂 − 흏 흅풂) = ퟎ ퟑ 흏 −품풆−횽흏풎푺 + −품풆−횽 풎ퟐ + 흀 풗ퟐ 풛 푺 = ퟎ 풎 푿 ퟐ 15 - (풛, 흁) decomposition and Lorentz decomposition ⊥ 푽흁ෝ = 푽흁ෝ , 푽풛 = ퟎ , ⊥ 풂 푨흁ෝ = 푨흁ෝ , 푨풛 = −흏풛흓 Equations for the normal modes −ퟏ ퟐ ⊥,풏 Vector and axial-vector mesons [−풛흏풛 풛 흏풛 − 풎푽풏]푽흁ෝ = ퟎ , −ퟏ ퟐ ⊥,풏 [−풛흏풛 풛 흏풛 − 풎푨풏 + 휷(풛)]푨흁ෝ = ퟎ ퟑ 횽 −ퟑ −횽 ퟐ −ퟐ ퟐ ퟑ ퟐ 풏 Scalars and pseudoscalar mesons 풛 풆 흏 풛 풆 흏 + 풎 풏 − 풛 풎 − 흀 풗 풛 푺 = ퟎ 풛 풛 푺 푿 ퟐ −ퟏ 풏 풏 풏 풛흏풛 풛 흏풛흓 + 휷 풛 흅 − 흓 = ퟎ , 풏 ퟐ 풏 휷 풛 흏풛흅 − 풎흅풏흏풛흓 = ퟎ ퟐ −ퟐ −횽 ퟐ w/ 휷 풛 = 품ퟓ 풛 풆 풗 (풛) ퟐ - Soft wall background Linear Regge trajectories 풎풏 ∼ 풏 16 Results: - The masses of vector mesons 풎푽풏 are not sensitive to chiral symmetry breaking. ퟐ The mass of the 휌 meson 풎흆 = ퟕퟕퟔ 퐌퐞퐕 is used to fixed the soft wall parameter 흓∞ = ퟑퟖퟖ 퐌퐞퐕 - Masses of axial-vector mesons 풎푨풏 , pseudoscalar mesons 풎흅풏 and scalar mesons 풎푺풏 vary with the quark mass 풎풒 (Results for 흀 = ퟐ) - In the chiral limit 푐1 → 0 we have 푣 푧 = 0 chiral symmetry is restored - We obtain a degeneracy between the vector (scalar) mesons and axial-vector (pseudoscalar) mesons - The first pseudoscalar meson never becomes a Goldstone boson 17 Decay constants 흁 흁 - Holographic currents < 푱푽(풙) > = 품푽풏푽풏(풙) , 흁 흁 흁 풏 < 푱푨 풙 > = 품푨풏푨풏 풙 − 풇흅풏흏 흅 (풙) - Leading to a dictionary for the 4d decay constants −ퟏ −ퟏ −횽 품푽풏 = 품ퟓ 풛 풆 흏풛풗퐧 퐳 , −ퟏ −ퟏ −횽 품푨풏 = 품ퟓ 풛 풆 흏풛풂퐧 퐳 , −ퟏ −ퟏ −횽 Preliminary results 풇흅풏 = −품ퟓ 풛 풆 흏풛흓퐧 퐳 - For the vector mesons the decay constants 품푽풏 are independent of the quark mass. For the 휌 meson we obtain 푔휌 = 260.1 MeV , far from the experimental result 푔휌 = 346.2 ± 1.4 MeV - The axial-vector meson decay constants 품푨풏 increase with the quark mass.
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