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