Analytical study of Yang-Mills theory from first principles by a massive expansion

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1 Analytical study of Yang-Mills theory from first principles by a massive expansion Department of Physics and Astronomy University of Catania, Italy Infrared QCD APC, Paris Diderot University, 8-10 November 2017

2 PREAMBLE What is "perturbative" and what is not? It depends on the Expansion Point

3 PREAMBLE What is "perturbative" and what is not? It depends on the Expansion Point 0 (p) = 1 p 2 = The limit p 2 0 is NP

4 PREAMBLE What is "perturbative" and what is not? It depends on the Expansion Point While if we take 0 (p) = 1 p 2 0 (p) = 1 p 2 + m 2 = The limit p 2 0 is NP = PT works well in the IR (Tissier and Wschebor, 2010, 2011)

5 PREAMBLE What is "perturbative" and what is not? It depends on the Expansion Point While if we take 0 (p) = 1 p 2 0 (p) = 1 p 2 + m 2 = The limit p 2 0 is NP = PT works well in the IR (Tissier and Wschebor, 2010, 2011) Lattice m 0 in the IR: = The exact YM theory must predict a mass

6 PREAMBLE What is "perturbative" and what is not? It depends on the Expansion Point While if we take 0 (p) = 1 p 2 0 (p) = 1 p 2 + m 2 = The limit p 2 0 is NP = PT works well in the IR (Tissier and Wschebor, 2010, 2011) Lattice m 0 in the IR: = The exact YM theory must predict a mass Any variational ansatz for 0 (p) works well provided it is massive (FS, 2014,2015): two-step approach

7 Outline Gaussian Effective Potential and mass generation Massive expansion around the best vacuum

8 Outline Gaussian Effective Potential and mass generation Massive expansion around the best vacuum Thermal effects as a check of physical consistency: Analytic properties at finite T Gaussian Free Energy and deconfinement at finite T

9 Gaussian Effective Potential (GEP) A toy model for mass generation L = [ 1 2 ϕ ( 2 m 2) ] [ ] λ ϕ 4! ϕ4 m 2 ϕ 2 1 st Order Effect Potential = Vac Energy by a Gaussian Funct [J M Cornwall, R Jackiw and E Tomboulis (1974); PM Stevenson (1985); JM Cornwall (1982)] + + SCALAR SU(N)

10 Gaussian Effective Potential (GEP) Gap equation and renormalization δv δm 2 = 0 = { m 2 = 8π 2 α J(m) Σ (1) = 0 Self consist pole where α = λ/(16π 2 ) and d 4 p J(m) = (2π) 4 0(p) = d 4 p 1? (2π) 4 p 2 + m 2 =

11 Gaussian Effective Potential (GEP) Gap equation and renormalization δv δm 2 = 0 = { m 2 = 8π 2 α J(m) where α = λ/(16π 2 ) and d 4 p J(m) = (2π) 4 0(p) = J(m) = m 2 16π 2 log m2 m2 16π 2 [ 2 ϵ Λ 2 + C Σ (1) = 0 Self consist pole d 4 p 1? (2π) 4 p 2 + m 2 = (derivate and integrate back) + log µ2 m 2 ] = m2 16π 2 log m2 Λ 2 ϵ m 2 log m2 16π 2 Λ 2 IR ( UV finite by subtraction of DR zeros)

12 Gaussian Effective Potential (GEP) Gap equation and renormalization δv δm 2 = 0 = { m 2 = 8π 2 α J(m) where α = λ/(16π 2 ) and d 4 p J(m) = (2π) 4 0(p) = J(m) = m 2 16π 2 log m2 m2 16π 2 [ 2 ϵ Λ 2 + C Σ (1) = 0 Self consist pole d 4 p 1? (2π) 4 p 2 + m 2 = (derivate and integrate back) + log µ2 m 2 ] = m2 16π 2 log m2 Λ 2 ϵ m 2 log m2 16π 2 Λ 2 IR m2 p 2 +m 2 p 2 +m 2 p 2 p 4 ( UV finite by subtraction of DR zeros) (leading and sub-leading)

13 Gaussian Effective Potential (GEP) Renormalized Effective Potential in units of the best mass m 0 [ ) 2 V(m) = m4 128π 2 α (log m2 + 2 log m2 m 2 0 m ] α s = 04 α s = 08 α s = m 2 2 /m 0 From the gap eq: Λ IR = m 0 exp( 1/α) The vacuum energy does not depend on Λ IR and α: V(m 0 ) = m π 2 < 0 Λ IR 0 Triviality?

14 Gaussian Effective Potential (GEP) Renormalized Effective Potential in units of the best mass m 0 [ ) 2 V(m) = m4 128π 2 α (log m2 + 2 log m2 m 2 0 m ] From the gap eq: Λ IR = m 0 exp( 1/α) The vacuum energy does not depend on Λ IR and α: -1 α s = 04 α s = 08 α s = m 2 2 /m 0 V(m 0 ) = m π 2 < 0 Λ IR 0 Triviality? Gluon mass generation: the same identical result for SU(N) Yang-Mills Theory in any covariant ξ-gauge if α = 9Nα s /(8π)

15 SU(N) Yang-Mills Expanding around the best vacuum of the GEP Add and subtract a transverse mass term in the exact Faddev-Popov Lagrangian in ξ-gauge: µν 0 (p) = 1 p 2 + m 2 tµν (p) + ξ p 2 lµν (p) (free propagator) Exact since Π L = 0 δγ µν = m 2 t µν (p) (2-point vertex) = Gauge invariant GEP and mass generation

16 SU(N) Yang-Mills Expanding around the best vacuum of the GEP Add and subtract a transverse mass term in the exact Faddev-Popov Lagrangian in ξ-gauge: µν 0 (p) = 1 p 2 + m 2 tµν (p) + ξ p 2 lµν (p) (free propagator) Exact since Π L = 0 δγ µν = m 2 t µν (p) (2-point vertex) = Gauge invariant GEP and mass generation Σ = + The pole shift cancels at tree level Π = (1a) (1b) (1c) (1d) (2a) + (2b) + (2c) + All spurious diverging mass terms cancel without counterterms and/or parameters Standard UV behavior

17 UNIVERSAL SCALING RS Optimized Perturbation Theory Ignoring RG effects, setting α Nα s Σ(p) = ασ (1) (p) + α 2 Σ (2) (p, N) + Σ (1) = p 2 F(p 2 /m 2 ); (p) = Σ(p) αp 2 Z p 2 Σ(p) = J(p) p 2 Setting Z = z (1 + αδz) (one-loop): = F(p2 /m 2 ) + O(α) z J(p) 1 = 1 + α [ F(p 2 /m 2 ) δz ] + O(α 2 ) z J(p) 1 = 1 + α [ F(p 2 /m 2 ) F(µ 2 /m 2 ) ] + O(α 2 ) Must exist x, y, z: z J(p/x) 1 + y = F(p 2 /m 2 ) + F 0 + O(α)

18 UNIVERSAL SCALING GLUON INVERSE DRESSING FUNCTION (Landau gauge ξ = 0) Duarte et al, SU(3) Cucchieri and Mendes, SU(2) Bogolubsky et al, SU(3) F(p 2 /m 2 )+F 0 J(p) p (GeV)

19 UNIVERSAL SCALING GLUON INVERSE DRESSING FUNCTION (Landau gauge ξ = 0) Duarte et al, SU(3) Cucchieri and Mendes, SU(2) Bogolubsky et al, SU(3) F(p 2 /m 2 )+F 0 J(p) p (GeV)

20 UNIVERSAL SCALING GHOST INVERSE DRESSING FUNCTION (Landau gauge ξ = 0) Denoting by G(s) the ghost universal function (F(s) G(s)) χ(p) Bogolubsky et al 07 Duarte et al Cucchieri-Mendes SU(2) 06 Ayala et al N f = 0 Ayala et al N f = 2 05 Ayala et al N f = G(p 2 /m 2 )+G p (GeV)

21 UNIVERSAL SCALING GHOST INVERSE DRESSING FUNCTION (Landau gauge ξ = 0) The ghost universal function is just G(s) = [ s 2s log s + 1 s 2 (1 + s) 2 (2s 1) log (1 + s) ] χ(p) Bogolubsky et al 07 Duarte et al Cucchieri-Mendes SU(2) 06 Ayala et al N f = 0 Ayala et al N f = 2 05 Ayala et al N f = G(p 2 /m 2 )+G p (GeV)

22 UNIVERSAL SCALING GHOST INVERSE DRESSING FUNCTION (Landau gauge ξ = 0) The ghost universal function is just G(s) = [ s 2s log s + 1 s 2 (1 + s) 2 (2s 1) log (1 + s) ] χ(p) Bogolubsky et al Duarte et al Cucchieri and Mendes SU(2) Ayala et al N f = 0 Ayala et al N f = 2 Ayala et al N f = G(p 2 /m 2 )+G p (GeV)

23 TABLE of OPTIMIZED RENORMALIZATION CONSTANTS: z J(p/x) 1 + y = F(p 2 /m 2 ) + F 0 arxiv: Data set N N f x y z y z Bogolubsky et al Duarte et al Cucchieri-Mendes Ayala et al Ayala et al Ayala et al Table: Scaling constants x, y, z (gluon) and y, z (ghost) The constant shifts F 0 = 105, G 0 = 024 and the mass m = 073 GeV are optimized by requiring that x = 1 and y = y = 0 for the lattice data of Bogolubsky et al (2009)

24 ANALYTIC CONTINUATION arxiv: GLUON PROPAGATOR - SU(3) Lattice Real Part Imaginary Part? p 2 (GeV 2 ) Lattice data are from Bogolubsky et al (2009)

25 ANALYTIC CONTINUATION arxiv: GLUON PROPAGATOR - SU(3) Lattice Real Part Imaginary Part m 2 = (016 ± i 060) GeV p 2 (GeV 2 ) Lattice data are from Bogolubsky et al (2009)

26 ANALYTIC CONTINUATION Imaginary part of the dressed gluon propagator In the long wave-length limit p 2 = ω 2 k 2 ω 2 the poles are at ω = ±(m ± iγ) where m = 063 GeV and γ = 048 GeV Ιm Re ω (GeV) Ιm ω (GeV) (Using m 0 = 073 GeV and F 0 = 105 in the Landau gauge)

27 ANALYTIC CONTINUATION Imaginary part of the dressed gluon propagator 0

28 CONFINEMENT Intrinsic gluon lifetime No violation of unitarity and casuality (Stingl, 1996) (ω) [ ] 1 R ± ω (m ± iγ) 1 ω + (m ± iγ) ± (t) = where R ± = R e ±iϕ dω 2π (ω)eiωt 2e γ t R sin (m t + ϕ) Short-lived quasigluons with lifetime τ = 1/γ (canceled from asymptotic states) During its short life the quasigluon eigenstate with energy m Mass and damping rate are physical or artefacts of the expansion?

29 Finite T Gluon propagator in the Landau gauge The extension to finite T is straightforward (but tedious!) Lucky enough: many explicit details by Reinosa, Serreau, Tissier and Wschebor (2014) New crossed graphs by a simple derivative Set m 0 = 073 GeV as for T = 0

30 Finite T Gluon propagator in the Landau gauge L (p) The extension to finite T is straightforward (but tedious!) Lucky enough: many explicit details by Reinosa, Serreau, Tissier and Wschebor (2014) New crossed graphs by a simple derivative Set m 0 = 073 GeV as for T = 0 T=198 T=298 T=595 T=893 T=198 1loop T=298 1loop T=595 1loop T=893 1loop L (p) T=198 T=298 T=595 T=893 T=198 1loop T=298 1loop T=595 1loop T=893 1loop p (GeV) 001 Lattice data are from RAouane et al(2012) p (GeV)

31 Finite T Trajectory of poles in the complex plane In the limit k 0 the pole ω = ±(m ± iγ) is the same for L, T Using m 0 = 073 GeV and F 0 = 105 (optimal at T = 0): m (GeV) 065 γ (GeV) T (GeV) T (GeV) The line is the fit γ = γ 0 + bt with γ 0 = 0295 GeV and b = 112 (Hard thermal loops: γ/t = 33α s )

32 Gaussian Free Energy At finite temperature the GEP becomes predicitive! V G (m) = 3N Am 4 128π 2 [ ) 2 α (log m2 + 2 log m2 m 2 0 m ] = F G (T, m) 128 π 2 V G (m) / (3N A m 0 4 ) α s = 04 α s = 08 α s = m 2 2 /m 0 m 0 = m(t) such that m(0) = m 0 Just add the thermal part to the same vacuum graphs: + + SCALAR SU(N)

33 Gaussian Free Energy The deconfinement transition (Landau gauge) GComitini and FS, arxiv: m(t) in units of m 0 0 T = 015 m T = 025 m 0 1 F G / m T = 030 m 0 T = 035 m 0 T = 040 m 0 m(t) / m T / m m / m 0 here α s = 09 but not too much sensitive to α s in the range 04 < α s < 12

34 Gaussian Free Energy Scale/coupling invariance of T c GComitini and FS, arxiv: Λ IR = m 0 exp( 1/α) T c / m T c (MeV) T c stationary at α s 09 (m 0 = 073 GeV) α s Compares well with the lattice value T c = 270 MeV found by PJ Silva et al (2014)

35 Gaussian Free Energy Equation of State (GComitini and FS, arxiv: ) p = [F G (T, m(t)) F G (0, m 0 )] 20 Lattice α s = α s = s = T F G(T, m(t)) Lattice α s = 06 α s = p / T c 4 10 s / T c T / T c Not a fit! (no free parameters) Lattice data are from L Giusti and M Pepe (2017) T / T c

36 Outlook How far from a fully consistent perturbative approach to YM theory? An incomplete wish list: A consistent criterion for optimization? Are the poles gauge invariant? What about higher loops? (A general criterion for truncation?) RG matching with UV limit Gribov copies (expected to be relevant deep in the IR) Bound states (BS equation)

37 Summary What we have done so far A simple variational argument for mass generation by the GEP

38 Summary What we have done so far A simple variational argument for mass generation by the GEP A perturbative expansion around the optimal vacuum yields analytical results at one-loop

39 Summary What we have done so far A simple variational argument for mass generation by the GEP A perturbative expansion around the optimal vacuum yields analytical results at one-loop Analytic continuation in the complex plane: prediction of dynamical properties

40 Summary What we have done so far A simple variational argument for mass generation by the GEP A perturbative expansion around the optimal vacuum yields analytical results at one-loop Analytic continuation in the complex plane: prediction of dynamical properties Thermal effects as a check of physical consistency for the poles

41 Summary What we have done so far A simple variational argument for mass generation by the GEP A perturbative expansion around the optimal vacuum yields analytical results at one-loop Analytic continuation in the complex plane: prediction of dynamical properties Thermal effects as a check of physical consistency for the poles At finite T the GEP becomes predictive: first order transition and EoS

42 Summary What we have done so far A simple variational argument for mass generation by the GEP A perturbative expansion around the optimal vacuum yields analytical results at one-loop Analytic continuation in the complex plane: prediction of dynamical properties Thermal effects as a check of physical consistency for the poles At finite T the GEP becomes predictive: first order transition and EoS Overall: good physical picture without free parameters (from first principles)

43 Summary What we have done so far A simple variational argument for mass generation by the GEP A perturbative expansion around the optimal vacuum yields analytical results at one-loop Analytic continuation in the complex plane: prediction of dynamical properties Thermal effects as a check of physical consistency for the poles At finite T the GEP becomes predictive: first order transition and EoS Overall: good physical picture without free parameters (from first principles) THANK YOU

44 BACKUP SLIDES

45 Jensen inequality with ghost fields Averaging over free-boson fields and using Jensen inequality F exact = F0 A T log e S int A DetM FP (A) F1 A + F gh where F gh = T log DetM FP (A) 0 F gh 1Loop In any linear covariant gauge M FP (A) = G δm(a) F gh = T [Tr log G 0 ]+ T 2 Tr [G 0δM(A)G 0 δm(a)] 0 + = F gh gh 1Loop +F2Loop + where F gh 2Loop α G 0 m G 0, etc, so that F G = F A 1 + F gh 1Loop F exact δf where δf = F gh F gh 1Loop 0 and by Jensen inequality again: F gh T [Tr log M FP (A) 0 ] = T [Tr log G 0 ] = F gh 1Loop so that δf 0

46 Gaussian Free Energy Equation of State (GComitini and FS, arxiv: ) Lattice α s = 01 α s = 06 α s = 09 p / T c T / T c Lattice data are from L Giusti and M Pepe (2017)

47 Running Coupling Pure Yang-Mills SU(3) RG invariant product (Landau Gauge MOM-Taylor scheme): α s (µ) = α s (µ 0 ) J(µ)χ(µ)2 J(µ 0 )χ(µ 0 ) 2 What if δf 0 = δg 0 = ±25%? α s µ (GeV) µ 0 = 2 GeV, α s = 037, data of Bogolubsky et al(2009)

48 ANALYTIC CONTINUATION Ghost dressing function: G(p 2 ) = χ(p2 ) p Z G Reχ(p), -Z G Ιmχ(p) p 2 (GeV 2 ) Lattice data are from Bogolubsky et al (2009)

49 CHIRAL QCD Quark sector Σ q = but The counterterm δγ = M cancels the mass at tree-level A massive propagator from loops S(p) = A new parameter x = M/m Z(p) p M(p)

50 CHIRAL QCD Quark sector Σ q = but The counterterm δγ = M cancels the mass at tree-level A massive propagator from loops S(p) = A new parameter x = M/m Agreement not as good as for pure YM theory (Z(p) is decreasing) Z(p) p M(p) M(p) depends on α s Optimization is not easy without RG corrections!

51 One-Loop third-order double expansion (Landau Gauge) Yang-Mills FS, Nucl Phys B (2016) QCD FS, arxiv: Σ gh = + Π = Σ q = + + +

52 CHIRAL QCD Gluon sector Optimized by the Lattice N f = 2, m = 08 GeV M =? Lattice, N f = 2 M (GeV) = REAL PART p 2 (GeV 2 ) IMAGINARY PART p 2 (GeV 2 ) M (GeV) = Lattice data are for two light quarks, from Ayala et al (2012) What about poles? 2 pairs of compex conjugated poles

53 CHIRAL QCD Gluon sector Optimized by the Lattice: m = 08 GeV, M = 065 GeV m 2 1 = (054 ± 052i) GeV2, m 2 2 = (169 ± 01i) GeV2 02 Re G(p) 01 Ιm G(p), Re G(p) p = m Ιm G(p) p = 2M p 2 (GeV 2 )

54 CHIRAL QCD Quark sector: ANALYTIC CONTINUATION TO MINKOWSKY SPACE Quark propagator: S(p) = S p (p 2 ) p + S M (p 2 ) NO COMPLEX POLES = Standard Dispersion Relations S(p) = ρ M (p 2 ) = 1 π Im S M(p 2 ) ρ p (p 2 ) = 1 π Im S p(p 2 ) 0 dq 2 ρ p(q 2 ) p + ρ M (q 2 ) p 2 q 2 + iε Positivity Conditions: ρ p (p 2 ) 0, p ρ p (p 2 ) ρ M (p 2 ) 0

55 CHIRAL QCD Quark sector: N f = 2, M = 065 GeV, m = 07 GeV ρ p, Re[S p ] (GeV -2 ); ρ M, Re[S M ] (GeV -1 ) (032 GeV) ρ p ρ M Re[S p ] Re[S M ] p = M p = m+m m = 07 GeV M = 065 GeV α s = p 2 (GeV 2 ) Positivity Conditions: ρ p (p 2 ) 0, p ρ p (p 2 ) ρ M (p 2 ) 0

56 CHIRAL QCD Quark sector: N f = 2, M = 065 GeV, m = 07 GeV p ρ p (p) - ρ M (p) (GeV -1 ) m = 07 GeV M = 065 GeV α s = p 2 (GeV 2 ) Positivity Conditions: ρ p (p 2 ) 0, p ρ p (p 2 ) ρ M (p 2 ) 0