Scale Invariant Optimized Perturbation at

������XXXXXXzero and �nite temperature

Jean-Loïc Kneur (Lab. Charles Coulomb, Montpellier)

RPP 2019, Clermont-ferrand, 25/01/2019

Context: QCD phase diagram/ Quark Gluon PlasmaComplete QCD phase diagram far from being con�rmed:

T = 0, µ = 0 well-established from lattice: no sharp phase transition,rather continuous crossover at Tc ≃ 154± 9 MeV (Aoki et al '06).

Goal: more analytical approximations, ultimately in regions not muchaccessible on the lattice: large density (chemical potential) due to thefamous �sign problem�

Tool: unconventional 'RG optimized' (RGOPT) resummation ofperturbative expansions

Outline

▶ Introduction/Motivation

▶ Optimized (or 'screened') perturbation (OPT ∼ SPT)

▶ RG-compatible OPT ≡ RGOPT

▶ Application: thermal nonlinear sigma model(many similarities with QCD but simpler)

▶ Application: thermal (pure gauge) QCD: hard thermal loops

▶ Conclusions

1. Motivation: problems of thermal perturbative expansion(QCD, ϕ4, ...)

poorly convergent and very scale-dependent (ordinary) perturbativeexpansions:

QCD (pure gauge) pressure at successive (standard) perturbation ordersshaded regions: scale-dependence for πT < µ < 4πT(illustration from Andersen, Strickland, Su '10)

Problems of thermal perturbation (QCD but generic)

Usual suspect: mix up of hard p ∼ T and soft p ∼ αST modes.

Thermal 'Debye' screening mass m2

D ∼ αST2 gives IR cuto�,

BUT ⇒ perturbative expansion in√αS in QCD

→ often advocated reason for slower convergence

Yet many interesting QGP physics features happen at not that largecoupling αS(∼ 2πTc) ∼ .5, (αS(∼ 2πTc) ∼ 0.3 for pure gauge)

Many e�orts to improve this (review e.g. Blaizot, Iancu, Rebhan '03):

Screened PT (SPT) (Karsch et al '97) ∼ Hard Thermal Loop (HTLpt)

resummation (Andersen, Braaten, Strickland '99); Functional RG, 2PI

formalism (Blaizot, Iancu, Rebhan '01; Berges, Borsanyi, Reinosa, Serreau '05)

Our RGOPT(T = 0) essentially treats thermal mass 'RG consistently':→ UV divergences induce mass anomalous dimension.

2. Optimized (or Screened) Perturbation (OPT/SPT)

Trick (T = 0): add and subtract a mass, consider m δ as interaction:

L(g ,m) → L(δ g ,m(1− δ)) (e.g. in QCD g ≡ 4παS)

0 < δ < 1 interpolates between Lfree and massless Lint ;→ m: arbitrary trial parameter

• Take any (renormalized) pert. series, (re)expand in δ after:

m → m (1− δ); g → δ gthen δ → 1 (to recover original massless theory):

BUT m-dependence remains at �nite δk -order:�xed by stationarity prescription: optimization (OPT):∂∂m (physical quantity) = 0 for m = mopt(g) = 0:

•T = 0: exhibits dimensional transmutation: mopt(g) ∼ µ e−const./g

•T = 0 similar idea: �screened perturbation� (SPT), or resummed �hardthermal loop (HTLpt)� (QCD) = expand around a quasi-particle mass.

Does this 'cheap trick' always work? and why?•Exponentially fast convergence of this procedure for D = 1 ϕ4 oscillator(cancels large pert. order factorial divergences!) Guida et al '95

•In Quantum Field Theories (QFT):

May be viewed as enforcing low order, best approximation of (all order)scale-invariance (massless limit!)

(NB genuine masses (e.g. mquarks = 0) neglected/treated as smallperturbations)

•OPT resums standard perturbation: in O(N),SU(N), · · · models,one-loop ≃ large-N approximation.

•But QFT multi-loop calculations (specially T = 0) (very) di�cult:→ empirical convergence? not clear

•Main problem (largely overlooked): OPT (=mass optimization) notmuch consistent with RG properties

•Other pb at higher order: OPT: ∂m(...) = 0 has multi-solutions (somecomplex!), how to choose right one, if no nonperturbative �insight�??

3. RG-compatible OPT (≡ RGOPT) (JLK, A. Neveu '2010)

Consider a physical quantity (perturbatively RG invariant)e.g. in thermal context the pressure P(m, g ,T ):

in addition to 'OPT' Eq.: ∂∂mP

(k)(m, g , δ = 1)|m≡m ≡ 0,

Require (δ-modi�ed!) P(m, g) at order δk to satisfy (perturbative)Renormalization Group (RG) equation:

RG

(P(k)(m, g , δ = 1)

)= 0

RG ≡ µd

d µ= µ

∂

∂µ+ β(g)

∂

∂g− γm(g)m

∂

∂m

β(g) ≡ −b0g2 − b1g

3 + · · · , γm(g) ≡ γ0g + γ1g2 + · · ·

→ combined with OPT, RG Eq. reduces to massless form:[µ

∂

∂µ+ β(g)

∂

∂g

]P(k)(m, g , δ = 1) = 0

•Using both OPT AND RG completely �x m ≡ m and g ≡ g .

→ Parameter-free determination!: intermediate optimal m(g ,T ) but�nal (physical) result from P(m, g ,T ) → P(ΛQCD

MS,T )

OPT + RG = RGOPT main features

•Basic interpolation: why not m → m (1− δ)a ?Most previous works (T = 0 OPT, T = 0 Screened PT, HTLpt) do a = 1but generally (we have shown) a = 1 spoils RG invariance!

•Standard OPT gives multiple m(g ,T ) solutions at increasing δk -orders

→ Our approach restores RG, +requiring matching to perturbation(i.e. Asymptotic Freedom (AF) for QCD):ln µ

m∼ 1

b0g+ · · · for g → 0, m ∼ ΛQCD

→ At successive orders, AF-compatible optimal m (often unique) onlyappears for a universal critical a:

m → m (1− δ)γ0b0 (in general γ0

b0= 1)

→ RG consistency goes beyond simple �add and subtract� trick

and removes any spurious solutions (incompatible with AF)

•But does not always avoid complex m solutions(artifacts of perturbative coe�cients, possibly cured by renormalizationscheme change)

NB: some previous results with RGOPT (T = 0)

Chiral symmetry breaking order parameter Fπ(mu,d,s = 0)/ΛQCDMS

:

Fπ exp input → Λnf =3

MS→ αMS

S (µ = mZ ).

N3LO: Fmq=0

π /Λnf =3

MS≃ 0.25± .01 → αS(mZ ) ≃ 0.1174± .001± .001

(JLK, A.Neveu, PRD88 (2013))

(compares well with αS lattice and world average values [PDG2016-17])

Also applied to ⟨qq⟩ at N3LO (using spectral density of Dirac operator):

⟨qq⟩1/3mq=0(2GeV) ≃ −(0.84± 0.01)ΛMS (JLK, A.Neveu, PRD 92 (2015))

compares well with latest lattice result

Digression: basic Thermal QFT calculations in a nutshellpartition function: Z = Tr e−βH , β ≡ 1/T ,derive from it free energy F ∼ −Pressure, entropy,... → equation of state

At equilibrium (no time dependence), connected with �eld theory(imaginary time formalism) by

•going to Euclidean time t → −iτ ;•restrict τ in a box of size β = 1/T⇒ p0 has discrete (Matsubara) modes (bosons):ωn = 2πn/β = 2πnT , n ∈ Z

→ e.g. scalar propagator:

1

p2 −m2→ − 1

ω2n + p2 +m2

Loop integration (dimensional regularization, MS-scheme):∫dp0 d

3p

(2π)4f (p0,p) → (µeγE )ϵ T

+∞∑n=−∞

∫d3−2ϵp

(2π)3−2ϵf (iωn,p) ≡

∑∫P

f (P)

Remark: beyond one-loop∑∫Pgenerally di�cult:

often done as systematic m/T expansions (when justi�ed).

RGOPT(T = 0) generic features (JLK, M.B Pinto, PRL116 '16)

1. Start from pert. 1-, 2-loop free energy/pressure F(m = 0,T = 0)

2. One crucial point [before doing δ-expansion +RG-optimization]:

F(m = 0,T = 0) involves leading order scale dependence: ∼ m4 lnµ

compensated by E0 �nite T-independent 'vacuum energy' subtraction:

E0(g ,m) = −m4

(s0g+ s1 + s2g + · · ·

)determined order by order such that µ d

d µE0 cancels the lnµ dependence:

s0 =1

2(b0−4γ0)= 8π2; s1 =

(b1−4γ1)8γ0 (b0−4γ0)

= −1, ...:

Well-known at T = 0, BUT MISSED by T = 0 PT, SPT, HTLpt(QCD):largely explains the (huge) scale dependence seen at higher order in theseapproaches! (more on this below)

3. Next expand in δ after m2 → m2(1− δ)a ; g → δg ; δ → 1:RG only consistent for a = 2γ0/b0 (= 1/3 e.g.for ϕ4 while a = 1 in SPT)

•All together, RGOPT gives much better residual scale dependence

4. Not quite QCD: O(N) D = 2 nonlinear σ model (NLSM)•Shares nice properties with QCD (asymptotic freedom, mass gap).•T = 0: pressure, trace anomaly, etc have QCD-like shapes

•Nonperturbative T = 0 results available for comparison:(lattice (N = 3)[Giacosa et al '12], 1/N-expansion [Andersen et al '04])

L0 =1

2(∂πi )

2 +g(πi∂πi )

2

2(1− gπ2i )− m2

g

(1− gπ2i

)1/2Two-loop pressure from:

•Advantage w.r.t. QCD: exact T -dependence at 2-loops:

Ppert.2loop = − (N − 1)

2

[I0(m,T ) +

(N − 3)

4m2gI1(m,T )2

]+E0,

I0(m,T ) =1

2π

(m2(1− ln

m

µ) + 4T 2K0(

m

T)

)K0(x) =

∫∞0

dz ln(1− e−

√z2+x2

), I1(m,T ) = ∂I0(m,T )/∂m2

One-loop RGOPT pressure (NLSM, but generic features)

�Exact� (T -dependence) mass gap m(g ,T ) from ∂mP(m) = 0:

lnm

µ= − 1

b0 g(µ)− 2

∂

∂m2K0(

m

T), (bnlsm

0=

N − 2

2π)

•Exhibits �exact� (one-loop) scale invariance (generic RGOPT feature):since one-loop running g−1(µ) = g−1(M0) + b0 ln

µM0

)

T = 0: m(T = 0) = µe− 1

b0 g(µ) = Λ1−loopMS

T ≫ m: m(T )

T=

π b0 g

1− b0 g LT, (LT ≡ ln

µ eγE

4πT)

•Other remarkable property: running g(µ ∼ T ) emerges automatically:

PRGOPT1L

Pideal(T ≫ m) ≃ 1− 3

2b0g(

4πT

eγE)

while in standard perturbation/SPT/HTLpt/... µ ∼ 2πT is imposed

e.g. to get correct ideal gas limit for large T

NLSM pressure [G. Ferrari, JLK, M.B. Pinto, R.0 Ramos, 1709.03457,PRD]

P/Pideal (N = 4, g(M0) = 1): RGOPT 1- and 2-loop versus large N (LN)and SPT (≡ ignoring RG-induced subtraction + m2 → m2(1− δ)):

0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

T�M0

P�P

SB

LN

RGOPT 1L

RGOPT 2L

SPT

PSB (= Pideal ) =π

6

(N − 1)T2

(shaded range: scale-dependence πT < µ ≡ M < 4πT )

→ At two-loops a moderate scale-dependence reappears, although lesspronounced than 2-loop standard PT, SPT.

NLSM interaction measure (trace anomaly)

(normalized) ∆NLSM ≡ (E − P)/T 2 ≡ T∂T (PT2 )

0.0 0.1 0.2 0.3 0.4 0.50.02

0.05

0.10

0.20

0.50

1.00

T�M0

D

LN

RGOPT 1L

RGOPT 2L

SPT

N = 4, g(M0) = 1 (shaded regions: scale-dependence πT < µ = M < 4πT )

•∆SPT : perturbative monotonic behaviour + sizeable scale dependence.

•RGOPT qualitatively more similar to 4D lattice QCD, showing a peak:originates from nonperturbative RGOPT mass gap from T = 0 to T = 0.

(But no phase transition in 2D NLSM (Mermin-Wagner-Colemantheorem): just re�ects broken conformal invariance (mass gap)).

5. Thermal (pure gauge) QCD: hard thermal loop (HTL)QCD(glue) adaption of OPT → HTLpt [Andersen, Braaten, Strickland '99]:

same trick now operates on a gluon mass term [Braaten-Pisarski '90]:

LQCD(gauge)−m2

2Tr

[Gµα⟨

yαyβ

(y .D)2⟩yGµ

β

], Dµ = ∂µ−ig Aµ, yµ = (1, y)

(e�ective, explicitly gauge-invariant but nonlocal Lagrangian):

originally describes screening mass m2 ∼ αST2 + other HTL

contributions [dressing gluon vertices and propagators]

But here m arbitrary: to be determined by RGOPT optimization.

PHTL

1-loop = − (N2

c − 1)

2

∑∫P

{(d − 1) ln[P2 +ΠT (P)] + ln[p2 +ΠL(P)]

}with HTL-dressed T,L propagators:

ΠL(P) = m2(1−TP); ΠT (P) =

m2

(d − 1)n2P

(TP−1+n2P); TP ≡∫

1

0

dcP2

0

P2

0+ c2p2

+HTL dispersion relation e.g.: k2 +m2

[1− ωL

2kln(ωL+k

ωL−k)]= 0; · · ·

•Exact 2-loop? daunting task...

NB possibly simpler e�ective gluon mass models/prescriptions exist...[e.g. Reinosa, Serreau, Tissier, Weschbor '15, see J. Maelger's previous talk]

...But HTLpt advantage: calculated up to 3-loops α2

S (NNLO)

e.g. at two-loops:

BUT only as m/T expansions [Andersen et al '99-'15]

Drawback: HTLpt ≡ high-T approximation by de�nition.

PHTLpt

1-loop,MS

= Pideal ×[1− 15

2m2 + 30m3 +

45

4m4(ln

µ

4πT+ γE − 7

2+

π2

3)

]m ≡ m

2πT Pideal = (N2

c − 1)π2 T4

45

Standard HTLpt results

Pure gauge at NNLO (3-loops) [Andersen, Strickland, Su '10]:

Reaching closer to lattice results (down to T ∼ 2− 3Tc) only emerges atNNLO (3-loop) for lower end of scale µ ∼ πT − 2πT .

Main HTLpt issue: drastically increasing scale dependence at NNLO orderMoreover HTLpt perturbative mass prescription: m → m

pertD (αS)

[rather than ∂mP(m) = 0, to avoid complex solutions]

[NB when including quarks, 3-loop HTLpt in agreement with lattice onlyfor central scale, but even larger scale dependence (Andersen et al '13-16)]

RGOPT adaptation of HTLpt =RGOHTL

Main points/changes:• Crucial RG invariance-restoring subtractions in Free energy (pressure):PHTLpt → PHTLpt −m4( s0

αS+ s1 + · · · ): re�ects its anomalous dimension.

• Interpolate with RG-preserving m2(1− δ)γ0b0 , where gluon mass

anomalous dimension de�ned from (available) counterterm.

→ Scale dependence improves at higher orders since RG invariancemaintained at all stages

•SPT,HTLpt,... do not ful�ll (had missed) this:RG-inconsistent, yet moderate scale dependence up to 2-loops: screenedbecause the (leading order) RG-unmatched term O(m4 lnµ) isperturbatively '3-loop' O(α2

S) from m2 ∼ αS T2.

→ But explains why HTLpt scale dependence dramatically resurfaces at3-loops!

RGOPT vs HTLpt: one-loop pressure

1 2 3 4 50.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

T�Tc

P�P

ideal

RGOPT 1 loopHm_optL

RGOPT 1loopHm_DebyeL

HTLpt 1loop

bands: Π T < Μ < 4Π T

NB: the bending of PRGOPT for small T/Tc is essentially due to−PRGOPT (T = 0) = 0, absent from HTLpt.

2-loop RGOHTL: needs new complicated calculations...Crucial RG-consistent 2-loop subtractions determined by αSm

4 lnµ term(non-logarithmic terms also very relevant).

These are O(αST4m

4

T4 ): not calculated in HTLpt as perturbatively O(α3

S)Involves ∼ 30 independent integrals, ∼ 20 being (very) complicated, e.g.

∑∫P,Q

TPTQ(p + q)2

p2P2q2Q2(P + Q)2; TP ≡

∫1

0

dcP2

0

P2

0+ c2p2

(P2 = P2

0+ p2), c ≡HTL angle (averaging).

→ formally �2-loop� but e�ectively 4-loop (HTL-dressed propagators)

(Techniques: contour integration + D-dim int. by parts+ other tricks)

Present status: our calculations in good progress but need more checks,specially the di�cult non-logarithmic parts (i.e. �nite parts in dim.reg.)

NB high-T ↔ T = 0 correspondance:

C20 ln2 µT+ C21 ln

µT+ C

(T≫m)22

↔ C20 ln2 µm+ C21 ln

µm+ C

(T=0)22

•Leading Log (LL) C20 determined simply from RG from one-loop.

•Similarly: 2-loop (perturbative) scale invariance guaranteed by RGgiving s1 simply in terms of C21 independently of precise C21 value!

(Preliminary) RGO(HTL) results (2-loop, pure gauge)illustrated here for simplest LL approx.: C21 = C22 = 0but RG-consistent subtraction s1(C21 = 0) = 0

Moderate scale-dependence reappears at 2-loops...but sensible improvement wrt HTLpt

1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0

1.2

T�Tc

P�P

ideal

RGOPT 2loop

HTLpt 2loop

HTLpt 3loop

Lattice HBoyd et al ’96L

Π T < Μ < 4Π T

[JLK, M.B Pinto, to appear soon]

NB scale dependence should further improve at 3-loops, generically:

RGOPT at O(αkS) → m(µ) appears at O(αk+1

S ) for any m, but

m2 ∼ αST2 → P ≃ m4

G/αS + · · · : leading µ-dependence at O(αk+2

S ).

• Warning: low T ∼ Tc genuine P(T ) shape sensitive to true C21,C22

(crucially needed before possibly comparing RGOPT with lattice...)

Preliminary RGO(HTL) approximate 3-loop results3-loops: exact m4α2

S terms need extra complicated calculations, butP3l

RGOHTL ∼ P2l

RGOHTL +m4α2

S(C30 ln3 µ2πT

+ C31 ln2 µ2πT

+ C32 lnµ

2πT+ C33):

3-lopp LL and NLL coe�cients C30,C31 fully determined from lowerorders from RG invariance

Within NLL approximation and in T/Tc >∼ 2 range where it is moretrustable:

5 10 15 20

0.01

0.02

0.05

0.10

0.20

0.50

1.00

T�Tc

DP�P

RGOPT 2loop

HTLpt 2loop

HTLpt 3loop

RGOPT 3loop HLL approxL

Π T < Μ < 4Π T

Hpreliminary!L

We assume that the true coe�cients will not spoil this improved scaledependence.

Summary and Outlook

•RGOPT includes 2 major di�erences w.r.t. previousOPT/SPT/HTLpt... approaches:

1) OPT +/or RG optimizations �x optimal m and possibly g = 4παS :→ parameter free determination in terms of only ΛQCD

2) Maintaining RG invariance uniquely �xes the basic interpolationm → m(1− δ)γ0/b0 : discards spurious solutions and acceleratesconvergence.

• At T = 0, exhibits improved stability + drastically improved scaledependence (with respect to standard PT, but also w.r.t. HTLpt)

•Paves the way to extend such RG-compatible methods to full QCDthermodynamics specially for exploring also �nite density