This research has been co-financed by the European Union (European Social Fund, ESF) and Greek national funds through the Operational Program

Transcription

1 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

2 1/10 Holographic models for QCD in the Veneziano limit Matti Järvinen University of Crete 30 July 2013 MJ, Kiritsis, arxiv: Arean, Iatrakis, MJ, Kiritsis, arxiv: , arxiv:1308.xxxx (Next talk: Alho, MJ, Kajantie, Kiritsis, Tuominen, arxiv: work in progress)

3 2/10 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

4 3/10 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 in the Veneziano limit with full backreaction V-QCD models [MJ, Kiritsis arxiv: ]

5 Defining V-QCD 4/10 Degrees of freedom (T = τi): The tachyon τ qq, and the dilaton λ TrF 2 λ = e φ is identified as the t Hooft coupling g 2 N c S V QCD = N 2 c M 3 N f N c M 3 d 5 x g [ R 4 ( λ) 2 ] 3 λ 2 + V g (λ) d 5 xv f (λ, τ) det(g ab + κ(λ) a τ b τ) V f (λ, τ) = V f 0 (λ) exp( a(λ)τ 2 ) ; ds 2 = e 2A(r) (dr 2 +η µν x µ x ν ) Need to choose V g, V f 0, a, and κ... Good IR singularity etc. The simplest and most reasonable choices do the job!

6 5/10 Phase diagram Fixing the potentials reasonably, at zero quark mass, after some analysis: 0 x ~4 x BZ=11/2 ChS c ChS QCD-like IR-Conformal Running Walking IRFP Banks- Zaks Meets standard expectations from QCD! Conformal transition at x 4 [Kaplan,Son,Stephanov;Kutasov,Lin,Parnachev]

7 Fluctuation analysis Study at qualitative level: [Arean, Iatrakis, MJ, Kiritsis, arxiv: , arxiv:1308.xxxx] 1. Meson spectra (at zero temperature and quark mass) Add gauge fields in SV QCD Four towers: scalars, pseudoscalars, vectors, and axial vectors Flavor singlet and nonsinglet (SU(Nf )) states 2. The S-parameter S d dq 2 q2 [ Π V (q 2 ) Π A (q 2 ) ] 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

8 Meson masses 7/10 Flavor nonsinglet masses m UV Masses of lowest modes Vectors 0.01 Axial vectors Scalars Pseudoscalars x ( ) κ m n exp xc x All masses show Miransky scaling as x x c

9 Existance of the dilaton 8/10 Mass ratios: Scalar singlet masses normalized to lowest one Lowest masses in each tower normalized to ρ mass m n m x S 0 NS 1 1 f Π x All ratios tend to constants as x x c : no dilaton

10 S-parameter 9/10 S For two choices of potentials d dq 2 q2 [ Π V (q 2 ) Π A (q 2 ) ] q 2 =0 SN c N f x c x SN c N f x c x The S-parameter increases with x: expected suppression absent Jumps discontinuously to zero at x = x c

11 10/10 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

12 Extra slides 11/10

13 QCD phases in the Veneziano limit 12/10 Expected structure at zero T, µ, and quark mass; finite x = N f /N c Phases: 0 < x < xc : 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 0 x c~4 x BZ=11/2 ChS ChS QCD-like IR-Conformal QED-like Running Walking IRFP Banks- Zaks

14 13/10 Matching to QCD In the UV ( λ 0): UV expansions of potentials matched with perturbative QCD beta functions λ(r) β 0 log r with r 1/µ 0 τ(r) m( log r) γ 0/β 0 r+σ( log r) γ 0/β 0 r 3 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

15 Other important features logσ 3 UV x ( ) κ qq σ exp xc x 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

16 Finite temperature definitions Lagrangian as before S V QCD = N 2 c M 3 N f N c M 3 d 5 x g [ R 4 ( λ) 2 ] 3 λ 2 + V g (λ) 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 ds 2 = e 2A(r) 2 f (r) f (r)dt2 + dx 2 Black hole thermodynamics: f (r) = 4πT (r h r) + O ( (r r h ) 2) ; s = 4πM 3 Nc 2 e 3A(r h) Also: Thermal gas solutions (f 1, zero T solutions) 15/10

17 Phase diagram: example 16/10 Phases on the (x, T )-plane (PotII) χs p 0 Black Hole χsb p = 0 Thermal Gas

18 Scalar singlet masses 17/10 Scalar singlet spectrum (PotII): In log scale Normalized to the lowest state m n UV x m n m x No light dilaton?

19 Meson mass ratios Mass ratios (PotII): Lowest states normalized to ρ mm Ρ x All ratios tend to constants as x x c : indeed no dilaton 18/10

20 Matching to QCD 19/10 Similar strategy as in IHQCD Matching in the UV ( λ 0): Take analytic potentials at λ = 0 RG flow consistent with QCD (when A log µ) Require correct (naive) operator dimensions in the deep UV Match expansions of potentials with perturbative QCD beta functions V g (λ) with (two-loop) Yang-Mills beta function Vg (λ) xv f 0 (λ) with QCD beta function a(λ)/κ(λ) with the anomalous dimension of the quark mass/chiral condensate ( properly running quark mass!) After this, a single undetermined parameter in the UV: W 0 V f 0 (λ) = W 0 + W 1 λ + O(λ 2 )

21 20/10 In the IR (λ ), there must be a solution where the tachyon action e a(λ)τ 2 0 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 Working potentials often string-inspired power-laws, multiplied by logarithmic corrections (!)

22 21/10 Background analysis: zero temperature Analysis of the backgrounds (r-dependent solutions of EoMs) at zero temperature Expect two kinds of solutions, with 1. Nontrivial tachyon profile (chirally broken) 2. Identically vanishing tachyon (chirally symmetric) Identify the dominant vacua Fully backreacted system rich dynamics, complicated numerical analysis...

23 Backgrounds at zero quark mass 22/10 Sketch of behavior in the conformal window (x > x c ): Tachyon vanishes (no ChSB) Similar to IHQCD, different potential IR fixed point Dilaton flows between UV/IR fixed points Here UV: r 0, IR: r 25 Λ Λ r As x goes below x c, this solution becomes unstable (tachyon BF bound)

24 23/10 Right below the conformal window (x < x c ; x x c 1) 30 Dilaton flows very close to the IR fixed point Small nonzero tachyon induces an IR singularity Result: walking Λ log Τ r

25 Actual solutions 24/10 UV: r = 0 IR: r = A log µ log r x c Λ, A A 30 Λ r A log Τ 20 Λ Λ, A, log Τ r A log Τ Λ Λ, A, log Τ r

26 25/10 The BF bound and x c At an fixed point with τ(r) C 1 r + C 2 r 4 m 2 l 2 = (4 ) Requiring real gives the Breitenlohner-Freedman bound for the tachyon (Starinets lectures) m 2 l 2 = (4 ) 4 Saturated for = 2, then τ(r) C 1 r 2 + C 2 r 2 log r Violation of BF bound instability

27 26/10 Analysis of this instability of the tachyon x c Dependence on the UV parameter W 0 and IR choices for the potentials m 2 2 IR IR 4.5 Resulting variation of the edge of conformal window x c = Agrees with most of the other estimates x

28 Potentials I 27/10 V g (λ) = π 2 λ π 4 λ 2 (1 + λ/(8π 2 )) 2/3 1 + log(1 + λ/(8π 2 )) V f (λ, τ) = V f 0 (λ)e a(λ)τ 2 V f 0 (λ) = a(λ) = κ(λ) = 4(33 2x) x + 92x2 + 99π 2 λ π 4 λ 2 3 (11 x) 22 1 ( x 288π 2 ) 4/3 λ In this case the tachyon diverges exponentially: τ(r) τ 0 exp [ ] /6 (115 16x) 4/3 (11 x) r /6 R

29 Potentials II 28/10 V g (λ) = π 2 λ π 4 λ 2 (1 + λ/(8π 2 )) 2/3 1 + log(1 + λ/(8π 2 )) V f (λ, τ) = V f 0 (λ)e a(λ)τ 2 V f 0 (λ) = 12 4(33 2x) x + 92x π 2 λ π 4 λ 2 a(λ) = x π (11 x) λ + λ 2 /(8π 2 ) 2 22 (1 + λ/(8π 2 )) 4/3 κ(λ) = 1 (1 + λ/(8π 2 )) 4/3 In this case the tachyon diverges as τ(r) 27 23/4 3 1/4 r r R

30 Effective potential 29/10 For solutions with τ = τ = const S = M 3 Nc 2 d 5 x [ g R 4 ( λ) 2 ] 3 λ 2 + V g (λ) xv f (λ, τ ) IHQCD with an effective potential V eff (λ) = V g (λ) xv f (λ, τ ) = V g (λ) xv f 0 (λ) exp( a(λ)τ 2 ) Minimizing for τ we obtain τ = 0 and τ = τ = 0: V eff (λ) = V g (λ) xv f 0 (λ); fixed point with V eff (λ ) = 0 τ : V eff (λ) = V g (λ) (like YM, no fixed points)

31 Efimov spiral 30/10 Ongoing work: the dependence σ(m) of the chiral condensate on the quark mass For x < x c spiral structure 1.5 Σ logσ m logm Dots: numerical data Continuous line: (semi-)analytic prediction Allows to study the effect of double-trace deformations

32 Black hole branches Example: PotII at x = 3, W 0 = 12/ TΛ h,τ 0 TΛ h,τ h0 Λ h, m q Λ end Λ Λ h Simple phase structure: 1st order transition at T = T h from thermal gas to (chirally symmetric) BH 31/10

33 32/10 More complicated cases: T PotII at x = 3, W 0 SB PotI at x = 3.5, W 0 = 12/11 T s(λ h T end T h T b(λ h T T s(λ h T end T 12 T h T b(λ h Λ h Λ h Left: chiral symmetry restored at 2nd order transition with T = T end > T h Right: Additional first order transition between BH phases with broken chiral symmetry Also other cases...

34 Phase diagrams on the (x, T )-plane 33/10 PotI W 0 SB PotII W 0 SB T T No chiral symmetry breaking phase here T crossover T h T end Conformal window T crossover T h T end Conformal window x f x f

35 Backgrounds in the walking region 34/10 Backgrounds with zero quark mass, x < x c (λ, A, τ) x 3 Λ, A, logt x Λ, A, logt logr logr x 3.9 Λ, A, logt x 3.97 Λ, A, logt logr logr

36 35/10 Beta functions along the RG flow (evaluated on the background), zero tachyon, YM x c ΒΛ 0 ΒΛ x Λ x Λ ΒΛ ΒΛ x x Λ Λ

37 Holographic beta functions 36/10 Generalization of the holographic RG flow of IHQCD linked to β(λ, τ) dλ da ; dg QCD d log µ ; dτ γ(λ, τ) da dm d log µ The full equations of motion boil down to two first order partial non-linear differential equations for β and γ

38 Good solutions numerically (unique) 37/10

39 Miransky/BKT scaling 38/10 As x x c from below: walking, dominant solution BF-bound for the tachyon violated at the IRFP x c fixed by the BF bound: = 2 & γ = 1 at the edge of the conformal window Λ, logt UV UV Walking IR Half period IR r τ(r) r 2 sin(κ x c x log r + φ) in the walking region 0.5 oscillations Miransky/BKT scaling, amount of walking Λ UV /Λ IR exp(π/(κ x c x))

40 39/10 As x x c with known κ logσ 3 UV log UV IR x x qq σ exp( 2π/(κ x c x)) Λ UV /Λ IR exp(π/(κ x c x)) logσ 3 UV Σ UV x UV IR

41 γ in the conformal window 40/10 Comparison to other guesses V-QCD (dashed: variation due to W 0 ) Dyson-Schwinger 2-loop PQCD All-orders β [Pica, Sannino arxiv: ] Γ x

42 Parameters Understanding the solutions for generic quark masses requires discussing parameters YM or QCD with massless quarks: no parameters QCD with flavor-independent mass m: a single (dimensionless) parameter m/λ QCD In this model, after rescalings, this parameter can be mapped to a parameter (τ 0 or r 1 ) that controls the diverging tachyon in the IR x has become continuous in the Veneziano limit 41/10

43 Map of all solutions 42/10 All good solutions (τ 0) obtained varying x and τ 0 or r 1 Contouring: quark mass (zero mass is the red contour) Potentials I T 0 Potentials II r 1

44 43/10 Mass dependence and Efimov vacua m m T 0 T 0 Conformal window (x > x c ) For m = 0, unique solution with τ 0 For m > 0, unique standard solution with τ 0 Low 0 < x < x c : Efimov vacua Unstable solution with τ 0 and m = 0 Standard stable solution, with τ 0, for all m 0 Tower of unstable Efimov vacua (small m )

45 44/10 Efimov solutions Tachyon oscillates over the walking regime Λ UV /Λ IR increased wrt. standard solution Λ, logt UV IR r

46 Effective potential: zero tachyon 45/10 Start from Banks-Zaks region, τ = 0, chiral symmetry conserved (τ qq), V eff (λ) = V g (λ) xv f 0 (λ) V eff defines a β-function as in IHQCD Fixed point guaranteed in the BZ region, moves to higher λ with decreasing x Fixed point λ runs to either at finite x(<x c ) or as x 0 Banks-Zaks Conformal Window x 11/2 x > x c x < x c?? Β Λ Λ Β Λ Β Λ Λ Λ

47 Effective potential: what actually happens 46/10 Banks-Zaks Conformal Window x 11/2 x > x c x < x c Β Λ Λ Β Β 200 Λ Λ 100 Λ Λ τ 0 τ 0 τ 0 For x < x c vacuum has nonzero tachyon (checked by calculating free energies) The tachyon screens the fixed point In the deep IR τ diverges, V eff V g dynamics is YM-like

48 Where is x c? 47/10 How is the edge of the conformal window stabilized? Tachyon IR mass at λ = λ quark mass dimension m 2 IR l2 IR = IR(4 IR ) = 24a(λ ) κ(λ )(V g (λ ) xv 0 (λ )) γ = IR 1 Breitenlohner-Freedman (BF) bound (horizontal line) m 2 IR l2 IR = 4 γ = 1 defines x c m 2 2 IR IR x c x

49 48/10 Why γ = 1 at x = x c? No time to go into details... the question boils down to the linearized tachyon solution at the fixed point For IR (4 IR ) < 4 (x > x c ): τ(r) m q r IR + σr 4 IR For IR (4 IR ) > 4 (x < x c ): τ(r) Cr 2 sin [(Im IR ) log r + φ] Rough analogy: Tachyon EoM Gap equation in Dyson-Schwinger approach Similar observations have been made in other holographic frameworks [Kutasov, Lin, Parnachev arxiv: , ]

50 Mass dependence 49/10 For m > 0 the conformal transition disappears The ratio of typical UV/IR scales Λ UV /Λ IR varies in a natural way m/λ UV = 10 6, 10 5,..., 10 x = 2, 3.5, 3.9, 4.25, 4.5 UV IR 10 8 UV IR x m UV

51 sqcd phases 50/10 The case of N = 1 SU(N c) superqcd with N f quark multiplets is known and provides an interesting (and more complex) example for the nonsupersymmetric case. From Seiberg we have learned that: x = 0 the theory has confinement, a mass gap and N c distinct vacua associated with a spontaneous breaking of the leftover R symmetry Z Nc. At 0 < x < 1, the theory has a runaway ground state. At x = 1, the theory has a quantum moduli space with no singularity. This reflects confinement with ChSB. At x = 1 + 1/N c, the moduli space is classical (and singular). The theory confines, but there is no ChSB. At 1 + 2/N c < x < 3/2 the theory is in the non-abelian magnetic IR-free phase, with the magnetic gauge group SU(N f N c) IR free. At 3/2 < x < 3, the theory flows to a CFT in the IR. Near x = 3 this is the Banks-Zaks region where the original theory has an IR fixed point at weak coupling. Moving to lower values, the coupling of the IR SU(N c) gauge theory grows. However near x = 3/2 the dual magnetic SU(N f N c) is in its Banks-Zaks region, and provides a weakly coupled description of the IR fixed point theory. At x > 3, the theory is IR free.

52 Saturating the BF bound (sketch) 51/10 Why is the BF bound saturated at the phase transition (massless quarks)?? 24a(λ ) IR (4 IR ) = κ(λ )(V g (λ ) xv 0 (λ )) For IR (4 IR ) < 4: τ(r) m q r 4 IR + σr IR For IR (4 IR ) > 4: τ(r) Cr 2 sin [(Im IR ) log r + φ] Saturating the BF bound, the tachyon solutions will engtangle required to satisfy boundary conditions Nodes in the solution appear trough UV massless solution

53 52/10 Saturating the BF bound (sketch) Does the nontrivial (ChSB) massless tachyon solution exist? Two possibilities: x > x c : BF bound satisfied at the fixed point only trivial massless solution (τ 0, ChS intact, fixed point hit) x < x c : BF bound violated at the fixed point a nontrivial massless solution exist, which drives the system away from the fixed point Conclusion: phase transition at x = x c As x x c from below, need to approach the fixed point to satisfy the boundary conditions nearly conformal, walking dynamics

54 Gamma functions 53/10 Massless backgrounds: gamma functions ΓT x = 2, 3, 3.5, Λ γ τ = d log τ da ΓT log r