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
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.
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)
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QCD: SU(Nc) gauge theory with Nf quark flavors (fundamental)
◮ Often useful: “quenched” or “probe” approximation, Nf ≪ Nc ◮ Here Veneziano limit: large Nf , Nc with x = Nf /Nc 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 = Nf /Nc
◮ The QCD thermodynamics as a function of x ◮ Phase diagram as a function of baryon density 2/10
The fusion:
[Gursoy, Kiritsis, Nitti; Gubser, Nellore]
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
Degrees of freedom (T = τI):
◮ The tachyon τ ↔ ¯
qq , and the dilaton λ ↔ TrF 2
◮ λ = eφ is identified as the ’t Hooft coupling g2Nc
SV−QCD = N2
c M3
3 (∂λ)2 λ2 + Vg(λ)
Vf (λ, τ) = Vf 0(λ) exp(−a(λ)τ 2) ; ds2 = e2A(r)(dr2+ηµνxµxν)
◮ Need to choose Vg, Vf 0, a, and κ . . .
◮ Good IR singularity etc.
◮ The simplest and most reasonable choices do the job! 4/10
Fixing the potentials reasonably, at zero quark mass, after some analysis:
Running Walking QCD-like IR-Conformal
c
BZ
ChS ChS IRFP
Banks- Zaks
x ~4
x =11/2
◮ Meets standard expectations from QCD! ◮ Conformal transition at x ≃ 4 [Kaplan,Son,Stephanov;Kutasov,Lin,Parnachev] 5/10
Study at qualitative level:
[Arean, Iatrakis, MJ, Kiritsis, arXiv:1211.6125, arXiv:1308.xxxx]
◮ Add gauge fields in SV−QCD ◮ Four towers: scalars, pseudoscalars, vectors, and axial vectors ◮ Flavor singlet and nonsinglet (SU(Nf )) states
S ∼ d dq2 q2 ΠV (q2) − ΠA(q2)
Open questions in the region relevant for “walking” technicolor (x → xc 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?
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Flavor nonsinglet masses
2 3 4 x 104 0.001 0.01 0.1 1 mUV
Masses of lowest modes
mn ∼ exp
κ √xc − x
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Mass ratios: Scalar singlet masses Lowest masses in each tower normalized to lowest one normalized to ρ mass
1.0 1.5 2.0 2.5 3.0 3.5 x 1 2 3 4 mnm1
1 1 0 NS 0 S 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 x 0.5 1.0 1.5 2.0 2.5 3.0 mmΡ
All ratios tend to constants as x → xc: no dilaton
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S ∼ d dq2 q2 ΠV (q2) − ΠA(q2)
For two choices of potentials
2 3 4 x 0.1 0.2 0.3 0.4 0.5 0.6 SNcN f xc
1.0 1.5 2.0 2.5 3.0 3.5 x 0.2 0.4 0.6 0.8 1.0 SNcN f xc
The S-parameter increases with x: expected suppression absent Jumps discontinuously to zero at x = xc
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◮ We explored bottom up models for QCD in the
◮ A class of models, V-QCD, was obtained by a
◮ V-QCD models meet expectations from QCD at
◮ Ongoing and future work:
10/10
11/10
Expected structure at zero T, µ, and quark mass; finite x = Nf /Nc
◮ Phases:
◮ 0 < x < xc: QCD-like IR, chiral symmetry broken ◮ xc ≤ x < 11/2: Conformal window, chirally symmetric
◮ Conformal transition at x = xc ◮ RG flow of the coupling: running, walking, or fixed point QED-like Running Walking QCD-like IR-Conformal
c
BZ
ChS ChS IRFP
Banks- Zaks
x ~4 x =11/2
12/10
In the UV ( λ → 0):
◮ UV expansions of potentials matched with perturbative QCD
beta functions ⇒ λ(r) ≃ − β0 log r τ(r) ≃ m(− log r)−γ0/β0 r+σ(− log r)γ0/β0 r3 with r ∼ 1/µ → 0 In the IR (λ → ∞):
◮ Vg(λ) chosen as for Yang-Mills, so that a “good” IR
singularity exists
◮ Vf 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
3.90 3.95 4.00 x 100 80 60 40 20 logΣUV
3
¯ qq ∼ σ ∼ exp
κ √xc − x
◮ E.g., The chiral condensate ¯
qq ∝ σ (From tachyon UV τ(r) ∼ mq(log r) r + σ(log r) r 3)
14/10
Lagrangian as before SV−QCD = N2
c M3
3 (∂λ)2 λ2 + Vg(λ)
A more general metric, A and f solved from EoMs ds2 = e2A(r) dr2 f (r) − f (r)dt2 + dx2
f (r) = 4πT(rh − r) + O
; s = 4πM3N2
c e3A(rh)
Also: Thermal gas solutions (f ≡ 1, ∼ zero T solutions)
15/10
Phases on the (x, T)-plane (PotII) χS p = 0 Black Hole χSB p = 0 Thermal Gas
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Scalar singlet spectrum (PotII): In log scale Normalized to the lowest state
0.5 1.0 1.5 2.0 2.5 3.0 3.5 x 0.01 0.05 0.10 0.50 1.00 5.00 mnUV
1.0 1.5 2.0 2.5 3.0 3.5 x 1 2 3 4 mnm1
No light dilaton?
17/10
Mass ratios (PotII): Lowest states normalized to ρ
0.5 1.0 1.5 2.0 2.5 3.0 3.5 x 0.5 1.0 1.5 2.0 2.5 3.0 mmΡ
All ratios tend to constants as x → xc: indeed no dilaton
18/10
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
◮ Vg(λ) with (two-loop) Yang-Mills beta function ◮ Vg(λ) − xVf 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: W0
Vf 0(λ) = W0 + W1λ + O(λ2)
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In the IR (λ → ∞), there must be a solution where the tachyon action ∝ e−a(λ)τ 2 → 0
◮ Vg(λ) chosen as for Yang-Mills, so that a “good” IR
singularity exists
◮ Vf 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 (!)
20/10
Analysis of the backgrounds (r-dependent solutions of EoMs) at zero temperature
◮ Expect two kinds of solutions, with
◮ Identify the dominant vacua ◮ Fully backreacted system ⇒ rich dynamics, complicated
numerical analysis . . .
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Sketch of behavior in the conformal window (x > xc):
◮ Tachyon vanishes
(no ChSB)
◮ Similar to IHQCD, different
potential ⇒ IR fixed point
◮ Dilaton flows between
UV/IR fixed points
Λ 1020 1015 1010 105 1 105 r 5 10 15 20 25 Λ
Here UV: r → 0, IR: r → ∞ As x goes below xc, this solution becomes unstable (tachyon BF bound)
22/10
Right below the conformal window (x < xc; |x − xc| ≪ 1)
◮ Dilaton flows very close
to the IR fixed point
◮ “Small” nonzero tachyon
induces an IR singularity
Λ log Τ 1018 1014 1010 106 0.01 r 40 30 20 10 10 20 30 Λ, log Τ
Result: “walking”
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UV: r = 0 IR: r = ∞ A ∼ log µ ∼ − log r xc ≃ 3.9959
Λ A 30 20 10 10log r 10 10 20 30 Λ, A
x 4 IR fixed point
Λ A log Τ 30 25 20 15 10 5 5 log r 30 20 10 10 20 30 Λ, A, log Τ
x 3.9 walking
Λ A log Τ 15 10 5 5 log r 20 10 10 20 30 Λ, A, log Τ
x 2 running
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At an fixed point τ(r) ∼ C1r∆ + C2r4−∆ with −m2ℓ2 = ∆(4 − ∆) Requiring real ∆ gives the Breitenlohner-Freedman bound for the tachyon (Starinets’ lectures) −m2ℓ2 = ∆(4 − ∆) ≤ 4
◮ Saturated for ∆ = 2, then τ(r) ∼ C1r2 + C2r2 log r ◮ Violation of BF bound ⇒ instability 25/10
Analysis of this instability of the tachyon ⇒ xc Dependence on the UV parameter W0 and IR choices for the potentials Resulting variation of the edge of conformal window xc = 3.7 . . . 4.2
4.0 4.5 5.0 5.5 x 3.5 4.0 4.5 mIR
2 IR 2
Agrees with most of the other estimates
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Vg(λ) = 12 + 44 9π2 λ + 4619 3888π4 λ2 (1 + λ/(8π2))2/3
Vf (λ, τ) = Vf 0(λ)e−a(λ)τ2 Vf 0(λ) = 12 11 + 4(33 − 2x) 99π2 λ + 23473 − 2726x + 92x2 42768π4 λ2 a(λ) = 3 22 (11 − x) κ(λ) = 1
288π2 λ
4/3 In this case the tachyon diverges exponentially: τ(r) ∼ τ0 exp
812944 21/6 r R
Vg(λ) = 12 + 44 9π2 λ + 4619 3888π4 λ2 (1 + λ/(8π2))2/3
Vf (λ, τ) = Vf 0(λ)e−a(λ)τ2 Vf 0(λ) = 12 11 + 4(33 − 2x) 99π2 λ + 23473 − 2726x + 92x2 42768π4 λ2 a(λ) = 3 22 (11 − x) 1 + 115−16x
216π2 λ + λ2/(8π2)2
(1 + λ/(8π2))4/3 κ(λ) = 1 (1 + λ/(8π2))4/3 In this case the tachyon diverges as τ(r) ∼ 27 23/431/4 √ 4619
R 28/10
For solutions with τ = τ∗ = const S = M3N2
c
3 (∂λ)2 λ2 + Vg(λ) − xVf (λ, τ∗)
Veff(λ) = Vg(λ) − xVf (λ, τ∗) = Vg(λ) − xVf 0(λ) exp(−a(λ)τ 2
∗ )
Minimizing for τ∗ we obtain τ∗ = 0 and τ∗ = ∞
◮ τ∗ = 0: Veff(λ) = Vg(λ) − xVf 0(λ);
fixed point with V ′
eff(λ∗) = 0 ◮ τ∗ → ∞: Veff(λ) = Vg(λ) (like YM, no fixed points) 29/10
Ongoing work: the dependence σ(m) of the chiral condensate on the quark mass
◮ For x < xc spiral structure
1.5 1.0 0.5 0.5 1.0 1.5m 0.5 1.0 1.5 Σ
15 10 5 5 10 logm 15 10 5 5 10 15 20 logΣ
◮ Dots: numerical data ◮ Continuous line: (semi-)analytic prediction
Allows to study the effect of double-trace deformations
30/10
Example: PotII at x = 3, W0 = 12/11 TΛh, Τ 0 TΛh, Τh0Λh, mq 0 Λh Λ Λend 1 10 100 1000 104 105 0.0 0.5 1.0 1.5 2.0 Simple phase structure: 1st order transition at T = Th from thermal gas to (chirally symmetric) BH
31/10
More complicated cases: PotII at x = 3, W0 SB PotI at x = 3.5, W0 = 12/11
Ts(Λh Tb(Λh Th Tend
0.01 0.1 1 10 100 1000
Λh
1 1000 106 109 1012 1015
T
10 100 1000 0.03 0.05 0.1
Ts(Λh Tb(Λh T12 Tend Th
0.01 1 100 104 106 108
Λh
1 100 104 106
T
100 10000 0.06 0.07 0.08 0.09 0.1 0.11 0.12
◮ Left: chiral symmetry restored at 2nd order transition with
T = Tend > Th
◮ Right: Additional first order transition between BH phases
with broken chiral symmetry Also other cases . . .
32/10
PotI∗ W0 SB PotII∗ W0 SB
Conformal window Tcrossover Th Tend No chiral symmetry breaking phase here
1 2 3 4
xf
1.00 0.50 2.00 0.30 1.50 0.70
T
Th Tend Tcrossover
1 2 3 4
xf
0.5 1.0 2.0 5.0 10.0
T
Backgrounds with zero quark mass, x < xc ≃ 3.9959 (λ, A, τ)
15 10 5 5 logr 20 20 40 Λ, A, logT x 3 20 15 10 5 5 logr 20 20 40 Λ, A, logT x 3.5 30 25 20 15 10 5 5 logr 20 20 40 Λ, A, logT x 3.9 40 30 20 10 logr 20 20 40 Λ, A, logT x 3.97
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Beta functions along the RG flow (evaluated on the background), zero tachyon, YM xc ≃ 3.9959
20 40 60 80 100 120 Λ 120 100 80 60 40 20 ΒΛ x 2 10 20 30 40 50 Λ 50 40 30 20 10 ΒΛ x 3 5 10 15 20 25 30 35 Λ 35 30 25 20 15 10 5 ΒΛ x 3.5 5 10 15 20 25 Λ 25 20 15 10 5 ΒΛ x 3.9
35/10
Generalization of the holographic RG flow of IHQCD β(λ, τ) ≡ dλ dA ; γ(λ, τ) ≡ dτ dA linked to dgQCD d log µ ; dm d log µ The full equations of motion boil down to two first order partial non-linear differential equations for β and γ
36/10
“Good” solutions numerically (unique)
37/10
As x → xc from below: walking, dominant solution
◮ BF-bound for the
tachyon violated at the IRFP
◮ xc fixed by the BF
bound: ∆ = 2 & γ∗ = 1 at the edge of the conformal window
UV Walking IR Half−period 1017 1014 1011 108 105 0.01 r 30 20 10 10 20 30 40 Λ, logT 1UV 1IR ◮ τ(r) ∼ r2 sin(κ√xc − x log r + φ) in the walking region ◮ “0.5 oscillations” ⇒ Miransky/BKT scaling,
amount of walking ΛUV/ΛIR ∼ exp(π/(κ√xc − x))
38/10
As x → xc ¯ qq∼σ∼exp(−2π/ (κ√xc − x)) with known κ ΛUV/ΛIR ∼exp(π/ (κ√xc − x))
3.90 3.95 4.00 x 100 80 60 40 20 logΣUV
3
0.050 0.100 0.200x 5 10 20 50 100 logΣUV
3
3.85 3.90 3.95 4.00 x 10 20 30 40 50 60 logUVIR
105 1010 1015 1020 1025UVIR 1041 1032 1023 1014 105 ΣUV
3
39/10
Comparison to other guesses V-QCD (dashed: variation due to W0) Dyson-Schwinger 2-loop PQCD All-orders β
[Pica, Sannino arXiv:1011.3832]
4.0 4.5 5.0 5.5 x 0.2 0.4 0.6 0.8 1.0 Γ
40/10
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/ΛQCD
◮ In this model, after rescalings, this parameter can be mapped
to a parameter (τ0 or r1) that controls the diverging tachyon in the IR
◮ x has become continuous in the Veneziano limit 41/10
All “good” solutions (τ = 0) obtained varying x and τ0 or r1 Contouring: quark mass (zero mass is the red contour) “Potentials I” ↔ T0 “Potentials II” ↔ r1
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T0 m T0 m
Conformal window (x > xc)
◮ For m = 0, unique
solution with τ ≡ 0
◮ For m > 0, unique
“standard” solution with τ = 0 Low 0 < x < xc: 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|)
43/10
◮ Tachyon oscillates over
the walking regime
◮ ΛUV/ΛIR increased wrt.
“standard” solution
1017 1014 1011 108 105 0.01 r 30 20 10 10 20 30 40 Λ, logT 1UV 1IR 44/10
Start from Banks-Zaks region, τ∗ = 0, chiral symmetry conserved (τ ↔ ¯ qq), Veff(λ) = Vg(λ) − xVf 0(λ)
◮ Veff 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(<xc) or as x →0
Banks-Zaks Conformal Window x → 11/2 x > xc x < xc ??
2 4 6 8 10 Λ 0.2 0.2 0.4 0.6 Β Λ 5 10 15 20 Λ 0.2 0.2 0.4 0.6 Β Λ 50 100 150 200Λ 50 50 100 150 200 250 Β Λ
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Banks-Zaks Conformal Window x → 11/2 x > xc x < xc
2 4 6 8 10 Λ 0.2 0.2 0.4 0.6 Β Λ 5 10 15 20 Λ 0.2 0.2 0.4 0.6 Β Λ 50 100 150 200Λ 200 100 100 200 Β Λ
τ ≡ 0 τ ≡ 0 τ = 0
◮ For x < xc vacuum has nonzero tachyon (checked by
calculating free energies)
◮ The tachyon screens the fixed point ◮ In the deep IR τ diverges, Veff → Vg ⇒ dynamics is YM-like 46/10
How is the edge of the conformal window stabilized? Tachyon IR mass at λ = λ∗ ↔ quark mass dimension −m2
IRℓ2 IR = ∆IR(4 − ∆IR) =
24a(λ∗) κ(λ∗)(Vg(λ∗) − xV0(λ∗)) γ∗ = ∆IR − 1 Breitenlohner-Freedman (BF) bound (horizontal line) −m2
IRℓ2 IR = 4 ⇔ γ∗ = 1
defines xc
xc 4.0 4.5 5.0 5.5 x 3.5 4.0 4.5 mIR
2 IR 2
47/10
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 > xc): τ(r) ∼ mqr∆IR + σr4−∆IR
◮ For ∆IR(4 − ∆IR) > 4
(x < xc): τ(r) ∼ Cr2 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:1107.2324, 1201.4123] 48/10
For m > 0 the conformal transition disappears The ratio of typical UV/IR scales ΛUV/ΛIR varies in a natural way m/ΛUV = 10−6, 10−5, . . . , 10 x = 2, 3.5, 3.9, 4.25, 4.5
2.0 2.5 3.0 3.5 4.0 4.5 x 1 100 104 106 108 UVIR 106 104 0.01 1 100mUV 1 100 104 106 UVIR 49/10
The case of N = 1 SU(Nc) superQCD with Nf 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 Nc distinct vacua associated with a spontaneous breaking of the leftover R symmetry ZNc . ◮ 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/Nc, the moduli space is classical (and singular). The theory confines, but there is no ChSB. ◮ At 1 + 2/Nc < x < 3/2 the theory is in the non-abelian magnetic IR-free phase, with the magnetic gauge group SU(Nf − Nc) 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
Banks-Zaks region, and provides a weakly coupled description of the IR fixed point theory. ◮ At x > 3, the theory is IR free. 50/10
Why is the BF bound saturated at the phase transition (massless quarks)?? ∆IR(4 − ∆IR) = 24a(λ∗) κ(λ∗)(Vg(λ∗) − xV0(λ∗))
◮ For ∆IR(4 − ∆IR) < 4:
τ(r) ∼ mqr4−∆IR + σr∆IR
◮ For ∆IR(4 − ∆IR) > 4:
τ(r) ∼ Cr2 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 51/10
Does the nontrivial (ChSB) massless tachyon solution exist? Two possibilities:
◮ x > xc: BF bound satisfied at the fixed point ⇒ only trivial
massless solution (τ ≡ 0, ChS intact, fixed point hit)
◮ x < xc: 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 = xc As x → xc from below, need to approach the fixed point to satisfy the boundary conditions ⇒ nearly conformal, “walking” dynamics
52/10
Massless backgrounds: gamma functions
γ τ = d log τ dA
20 40 60 80 100 Λ 3.0 2.5 2.0 1.5 1.0 0.5 0.0 ΓT 30 25 20 15 10 5 5 log r 3.0 2.5 2.0 1.5 1.0 0.5 ΓT
x = 2, 3, 3.5, 3.9
53/10