Holography and the conformal window in the Veneziano limit Matti J - PowerPoint PPT Presentation

Holography and the conformal window in the Veneziano limit Matti J¨ arvinen ENS, Paris SCGT15 – Nagoya – 5 March 2015 1/23

Outline 1. Brief introduction and motivation 2. Basic properties of V-QCD ◮ Definition of the model ◮ Conformal transition in V-QCD 3. Results and applications ◮ Miransky scaling ◮ Hyperscaling ◮ Light scalars ◮ The S-parameter ◮ Four fermion deformations 2/23

1 . Introduction 3/23

QCD phases in the Veneziano limit Veneziano limit: large N f , N c with x = N f / N c fixed . Conformal x c 0 window x QCD−like In the Veneziano limit (discrete) N f replaced by (continuous) x = N f / N c ◮ Transition expected at some x = x c Computations near the transition difficult ◮ Schwinger-Dyson approach, . . . ◮ Lattice QCD ◮ Holography (?) → This talk 4/23

Our approach: general idea A holographic bottom-up model for QCD in the Veneziano limit ◮ Bottom-up, but trying to follow principles from string theory as closely as possible More precisely: ◮ Derive the model from five dimensional noncritical string theory with certain brane configuration ⇒ some things do not work (at small coupling) ◮ Fix model by hand and generalize → arbitrary potentials ◮ Tune model to match QCD physics and data ◮ Effective description of QCD Last steps so far incomplete: model not yet tuned to match any QCD data! 5/23

2 . V-QCD 6/23

Holographic V-QCD: the fusion The fusion: 1. IHQCD: model for glue inspired by string theory (dilaton gravity) [Gursoy, Kiritsis, Nitti; Gubser, Nellore] 2. Adding flavor and chiral symmetry breaking via tachyon brane actions [Klebanov,Maldacena; Bigazzi,Casero,Cotrone,Iatrakis,Kiritsis,Paredes] Consider 1 . + 2 . in the Veneziano limit with full backreaction ⇒ V-QCD models [MJ, Kiritsis arXiv:1112.1261] 7/23

Defining V-QCD Degrees of freedom ◮ The tachyon τ , and the dilaton λ ◮ λ = e φ is identified as the ’t Hooft coupling g 2 N c ◮ τ is dual to the ¯ qq operator ( ∂λ ) 2 � � � d 5 x √ g R − 4 S V − QCD = N 2 c M 3 + V g ( λ ) λ 2 3 � − N f N c M 3 d 5 xV f ( λ, τ ) � − det( g ab + κ ( λ ) ∂ a τ∂ b τ ) ds 2 = e 2 A ( r ) ( dr 2 + η µν x µ x ν ) V f ( λ, τ ) = V f 0 ( λ ) exp( − a ( λ ) τ 2 ) ; Need to choose V f 0 , a , and κ . . . A simple strategy works (!): ◮ Match to perturbative QCD in the UV (asymptotic AdS 5 ) ◮ Logarithmically modified string theory predictions in the IR 8/23

Phase diagram of V-QCD ◮ Choose reasonable potentials ◮ Ansatz τ ( r ), λ ( r ), A ( r ) in equations of motion ◮ Construct numerically all vacua (various IR geometries) Desired phase diagram obtained: x ~4 x =11/2 0 ChS ChS c BZ QED-like QCD-like IR-Conformal IRFP Running Banks- Walking Zaks ◮ Matching to QCD perturbation theory → Banks-Zaks ◮ Conformal transition (BKT) at x = x c ≃ 4 (With tuned potentials, the phase diagram may change) 9/23

How does the phase structure arise? Turning on a tiny tachyon in the conformal window τ ( r ) ∼ m q r γ ∗ +1 + σ r 3 − γ ∗ ( IR , r → ∞ ) Breitenlohner-Freedman (BF) bound for γ ∗ at the IRFP ( γ ∗ + 1)(3 − γ ∗ ) = ∆ ∗ (4 − ∆ ∗ ) = − m 2 τ ℓ 2 ∗ ≤ 4 Violation of BF bound ⇒ instability ⇒ tachyon/chiral condensate ◮ ⇒ bound saturated at the conformal phase transition ( x = x c ) ◮ γ ∗ = 1 at the transition ◮ BF bound violation leads to a BKT transition quite in general ◮ Predictions near the transition to large extent independent of model details 10/23

3 . Results 11/23

Energy scales (at zero quark mass) V-QCD reproduces the picture with Miransky scaling: 1. QCD regime: single energy scale Λ 2. Walking regime ( x c − x ≪ 1): two scales related by Miransky/BKT scaling law 3 � log � Σ � � UV x 3.85 3.90 3.95 4.00 � � � � 20 � � Λ UV � κ � � 40 � √ x c − x ∼ exp � � 60 Λ IR � � 80 � � 100 � 3. Conformal window ( x c ≤ x < 11 / 2): again one scale Λ, but slow RG flow 12/23

Phase diagram: example at finite T Phases on the ( x , T )-plane χ S Black Hole χ SB Thermal Gas Loop effects may affect the order of the transition [Alho,MJ,Kajantie,Kiritsis,Tuominen, arXiv:1210.4516, 1501.06379] 13/23

“Hyperscaling” relations In the conformal window all low lying masses obey the “hyperscaling” relations 1 1+ γ ∗ m ∼ m ( m q → 0) q 3 − γ ∗ 1+ γ ∗ � ¯ qq � ∼ m ( m q → 0) q [Kiritsis, MJ arXiv:1112.1261; MJ arXiv:1501.07272 ] ◮ Appear independently of the details of the Lagrangian ◮ Also demonstrated in the “dynamic AdS/QCD” models [Evans, Scott arXiv:1405.5373] 14/23

“Phase diagram” on the ( x , m q )-plane: m q � � UV 10.00 x c x BZ 5.00 C 1.00 0.50 B A 0.10 0.05 x 0 1 2 3 4 5 Re � Γ � � 1.0 0.8 Pot I Hyperscaling seen in “regime B”: 0.6 extends to x < x c Pot II 0.4 0.2 5.5 x 4.0 4.5 5.0 15/23

Example: masses for the walking case x c − x ≪ 1, Masses in units of IR (glueball) scale m � � IR � 100 � m Π � m Ρ � 10 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��� � � � m ss 1 � � 0.1 m q � 10 � 14 10 � 8 10 10 10 16 0.01 10 4 � UV ◮ All masses have the same behavior at intermediate m q (regime B) ◮ Meson masses enhanced wrt glueballs at large m q 16/23

Meson mass ratios as a function of x Lowest states of various sectors, normalized to m ρ m � m Ρ 1.5 � �� � � � � � � � � � 1.0 � � � � � � � � � �� � � � � � � � � � � � � �� 0.5 � � � � � � � � � � � �� � � � � � � � � x 0 1 2 3 4 All ratios tend to constants as x → x c : no technidilaton mode [Arean,Iatrakis,MJ,Kiritsis arXiv:1211.6125, 1309.2286] 17/23

Interpreting the absence of the dilaton What have we shown? ◮ Violation of BF bound does not automatically yield a light dilaton .. ◮ .. while Miransky scaling and hyperscaling relations are reproduced (GMOR and Witten-Veneziano relations also ok) However . . . ◮ Analytic analysis: scalar fluctuations “critical” in the walking region, suggesting a light state ◮ But criticality not enough: presence of such a light state is sensitive to IR Could this be a computational error or numerical issue? ◮ Scalar singlet fluctuations are a real mess .. ◮ .. but we did nontrivial checks and all results look reasonable Notice: easy to obtain light (but not parametrically light) scalars 18/23

S-parameter S �� N c N f � 0.30 x c x BZ 0.25 0.20 m q = 0 0.15 m q = 10 − 6 0.10 0.05 0.00 x 1 2 3 4 5 ◮ Discontinuity at m q = 0 in the conformal window ◮ Qualitative agreement with field theory expectations [Sannino] 19/23

Scaling of the S-parameter As m q → 0 in the conformal window, � m q � ∆ FF − 4 γ ∗ +1 S ( m q ) ≃ S (0+) + c Λ UV ◮ Limiting value S (0+) = lim m q → 0+ S ( m q ) is finite and positive (while S (0) = 0) ◮ ∆ FF is the dimension of tr F 2 at the fixed point � S � m q � � S � 0 � ��� N c N f 0.0100 � � 0.0070 � � 0.0050 � � � 0.0030 � � 0.0020 � � 0.0015 m q � � � 0.001 1 10 � 9 10 � 6 � UV 20/23

Chiral condensate The dependence of σ ∝ � ¯ qq � on the quark mass ◮ For x < x c spiral structure [MJ arXiv:1501.07272] 3 � Σ �� � UV 10 9 3 Σ � � UV � 0.15 � 10 6 � � � � � 1000 �� � 0.10 � � m q � � � � � ������ 10 � 10 10 � 7 10 � 4 0.1 100 � UV � � � � � 0.001 0.05 � � � � � � � � � ����� � � � � � 10 � 6 � � � � � �� � � � � � � m q � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 10 � 9 � 0.002 � � � � � � � � 0.002 0.004 � � UV ◮ Dots: numerical data ◮ Continuous line: (semi-)analytic prediction Allows to study the effect of double-trace deformations 21/23

Four-fermion operators Witten’s recipe: modified UV boundary conditions for the tachyon For interaction term in field theory ( O = ¯ qq ) � d 4 x O ( x ) + g 2 � d 4 x O ( x ) 2 W = − m q 2 At zero m q : g 2 Chirally Chirally broken symmetric x x c x BZ 0 Chirally broken 22/23

Conclusions ◮ V-QCD agrees with field theory results for QCD at qualitative level ◮ Most results close to the conformal transition independent of details ◮ Next step: tuning the model to match quantitatively with experimental/lattice QCD data 23/23

Extra slides 24/23