This research has been co-financed by the European Union (European - PowerPoint PPT Presentation

This research has been co-financed by the European Union (European Social Fund, ESF) and Greek national funds through the Operational Program ”Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF), under the grants schemes ”Funding of proposals that have received a positive evaluation in the 3rd and 4th Call of ERC Grant Schemes” and the EU program “Thales” ESF/NSRF 2007-2013.

Holographic models for QCD in the Veneziano limit Matti J¨ arvinen University of Crete 30 July 2013 MJ, Kiritsis, arXiv:1112.1261 Arean, Iatrakis, MJ, Kiritsis, arXiv:1211.6125, arXiv:1308.xxxx (Next talk: Alho, MJ, Kajantie, Kiritsis, Tuominen, arXiv:1210.4516 + work in progress) 1/10

Motivation QCD: SU ( N c ) gauge theory with N f quark flavors (fundamental) ◮ Often useful: “quenched” or “probe” approximation, N f ≪ N c ◮ Here Veneziano limit: large N f , N c with x = N f / N c fixed ⇒ backreaction Important new features can be captured in the Veneziano limit: ◮ Phase diagram of QCD (at zero temperature, baryon density, and quark mass), varying x = N f / N c ◮ The QCD thermodynamics as a function of x ◮ Phase diagram as a function of baryon density 2/10

Holographic V-QCD: the fusion The fusion: 1. IHQCD: model for glue by using 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] 3/10

Defining V-QCD Degrees of freedom ( T = τ I ): ◮ The tachyon τ ↔ ¯ and the dilaton λ ↔ Tr F 2 qq , ◮ λ = e φ is identified as the ’t Hooft coupling g 2 N c ( ∂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 g , V f 0 , a , and κ . . . ◮ Good IR singularity etc. ◮ The simplest and most reasonable choices do the job! 4/10

Phase diagram Fixing the potentials reasonably, at zero quark mass, after some analysis: x ~4 x =11/2 0 ChS ChS c BZ QCD-like IR-Conformal IRFP Running Banks- Walking Zaks ◮ Meets standard expectations from QCD! ◮ Conformal transition at x ≃ 4 [Kaplan,Son,Stephanov;Kutasov,Lin,Parnachev] 5/10

Fluctuation analysis Study at qualitative level: [Arean, Iatrakis, MJ, Kiritsis, arXiv:1211.6125, arXiv:1308.xxxx] 1. Meson spectra (at zero temperature and quark mass) ◮ Add gauge fields in S V − QCD ◮ Four towers: scalars, pseudoscalars, vectors, and axial vectors ◮ Flavor singlet and nonsinglet ( SU ( N f )) states 2. The S-parameter d dq 2 q 2 � Π V ( q 2 ) − Π A ( q 2 ) � S ∼ q 2 =0 Open questions in the region relevant for “walking” technicolor ( x → x c from below): ◮ The S-parameter might be reduced ◮ Possibly a light “dilaton” (flavor singlet scalar): Goldstone mode due to almost unbroken conformal symmetry. The 125 GeV state seen at the LHC? 6/10

Meson masses Flavor nonsinglet masses m � � UV � � � � � � � � � � � � � � � � � � � � � � � � 1 � � � � � � � � � � � � 0.1 � � � Masses of lowest modes � Vectors 0.01 � Axial vectors � Scalars � 0.001 Pseudoscalars � � � � � 10 � 4 x � � � � 1 2 3 4 � κ � m n ∼ exp − √ x c − x ◮ All masses show Miransky scaling as x → x c 7/10

Existance of the dilaton Mass ratios: Scalar singlet masses Lowest masses in each tower normalized to lowest one normalized to ρ mass m � m Ρ m n � m 1 3.0 � ����������������� � � 4 � 2.5 � � � ����������������� � � � � 0 �� S � � � 3 2.0 � � 0 �� NS � � � ����������������� � � � � � � � � � � 1.5 � � 1 �� 2 � � � � �� � � � 1.0 � � � � � � � � � � � � � 1 �� � � � �� � �� � 1 ����������������� � � � � � � � � 0.5 f Π �� � � � � 3.5 x � x � � � 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 All ratios tend to constants as x → x c : no dilaton 8/10

S-parameter d dq 2 q 2 � Π V ( q 2 ) − Π A ( q 2 ) � S ∼ q 2 =0 For two choices of potentials S �� N c N f � S �� N c N f � x c x c 0.6 1.0 0.5 �� � � � � � 0.8 � 0.4 � � 0.6 � 0.3 � � 0.4 � � 0.2 ������������� � � � 0.2 0.1 � � � � ���� x x 1 2 3 4 0.5 1.0 1.5 2.0 2.5 3.0 3.5 The S-parameter increases with x : expected suppression absent Jumps discontinuously to zero at x = x c 9/10

Summary ◮ We explored bottom up models for QCD in the Veneziano limit ◮ A class of models, V-QCD, was obtained by a fusion of IHQCD with tachyonic brane action ◮ V-QCD models meet expectations from QCD at qualitative level ◮ Ongoing and future work: finite µ (next talk) and quantitative fits to QCD 10/10

Extra slides 11/10

QCD phases in the Veneziano limit Expected structure at zero T , µ , and quark mass; finite x = N f / N c ◮ Phases: ◮ 0 < x < x c : QCD-like IR, chiral symmetry broken ◮ x c ≤ x < 11 / 2: Conformal window, chirally symmetric ◮ Conformal transition at x = x c ◮ RG flow of the coupling: running, walking, or fixed point x ~4 x =11/2 0 ChS ChS c BZ IR-Conformal QED-like QCD-like Running IRFP Banks- Walking Zaks 12/10

Matching to QCD In the UV ( λ → 0): ◮ UV expansions of potentials matched with perturbative QCD beta functions ⇒ λ ( r ) ≃ − β 0 τ ( r ) ≃ m ( − log r ) − γ 0 /β 0 r + σ ( − log r ) γ 0 /β 0 r 3 log r with r ∼ 1 /µ → 0 In the IR ( λ → ∞ ): ◮ V g ( λ ) chosen as for Yang-Mills, so that a “good” IR singularity exists ◮ V f 0 ( λ ), a ( λ ), and κ ( λ ) chosen to produce tachyon divergence: several possibilities ( → Potentials I and II) ◮ Extra constraints from the asymptotics of the meson spectra 13/10

Other important features 3 � log � Σ � � UV x 3.85 3.90 3.95 4.00 � � � � 20 � � � 40 � � � 60 � � 80 � � 100 � � κ � √ x c − x � ¯ qq � ∼ σ ∼ exp − 1. Miransky/BKT scaling as x → x c from below ◮ E.g., The chiral condensate � ¯ qq � ∝ σ (From tachyon UV τ ( r ) ∼ m q (log r ) r + σ (log r ) r 3 ) 2. Unstable Efimov vacua observed for x < x c 3. Turning on the quark mass possible 14/10

Finite temperature – definitions Lagrangian as before ( ∂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 ) A more general metric, A and f solved from EoMs � dr 2 � ds 2 = e 2 A ( r ) f ( r ) − f ( r ) dt 2 + d x 2 Black hole thermodynamics: ( r − r h ) 2 � s = 4 π M 3 N 2 c e 3 A ( r h ) � f ( r ) = 4 π T ( r h − r ) + O ; Also: Thermal gas solutions ( f ≡ 1, ∼ zero T solutions) 15/10

Phase diagram: example Phases on the ( x , T )-plane (PotII) χ S p � = 0 Black Hole χ SB p = 0 Thermal Gas 16/10

Scalar singlet masses Scalar singlet spectrum (PotII): In log scale Normalized to the lowest state m n � m 1 m n � � UV 5.00 ����������������� � ����������������� � 4 ����������������� � ����������������� � � � � ����������������� � � � ����������������� � � 1.00 � � � � � � � 3 � � � � 0.50 ����������������� � � � � � � � � � � � � � � � � � � � 2 � � � 0.10 � � � � 0.05 � � 1 � ����������������� � � � � � � � � 0.01 3.5 x 3.5 x � � � � 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0 No light dilaton? 17/10

Meson mass ratios Mass ratios (PotII): Lowest states normalized to ρ m � m Ρ 3.0 � � 2.5 � � � 2.0 � � � � � � 1.5 � �� � � � 1.0 � � � � � � � � � � � � � � � �� � �� � 0.5 �� � � � � � x � � � 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 All ratios tend to constants as x → x c : indeed no dilaton 18/10