Pushing dimensional reduction of QCD to lower temperatures Philippe - PowerPoint PPT Presentation

Intro DimRed Center symm. SU(2) Outlook Pushing dimensional reduction of QCD to lower temperatures Philippe de Forcrand ETH Zürich and CERN arXiv:0801.1566 with A. Kurkela and A. Vuorinen Really: hep-ph/0604100, A. Vuorinen and L. Yaffe university-logo GGI, Florence, June 2008 Ph. de Forcrand GGI, June 2008 DimRed

Intro DimRed Center symm. SU(2) Outlook Motivation QCD thermodynamics well understood at low T : hadron resonance gas 14.0 ε /T 4 12.0 10.0 8.0 6.0 4.0 2.0 T/T c 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 university-logo Karsch et al., hep-ph/0303108 Ph. de Forcrand GGI, June 2008 DimRed

Intro DimRed Center symm. SU(2) Outlook Motivation ... and at asymptotically high T : gas of free quarks and gluons What about T ∼ a few T c , ie. experimental range? 1.0 p/p SB 0.8 0.6 0.4 3 flavour 2 flavour 2+1 flavour 0.2 T/T c 0.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Karsch et al., hep-lat/9602007 university-logo Still far from non-interacting gas Ph. de Forcrand GGI, June 2008 DimRed

Intro DimRed Center symm. SU(2) Outlook Motivation ... and at asymptotically high T : gas of free quarks and gluons What about T ∼ a few T c , ie. experimental range? Pisarski, hep-ph/0612191 university-logo Far from leading order perturbation theory ( e − 3 p ) / T 4 ∼ log T Ph. de Forcrand GGI, June 2008 DimRed

Intro DimRed Center symm. SU(2) Outlook Perturbative expansion IR divergences → non-perturbative (Linde) 1.5 1.0 p/p SB 2 g 3 g 0.5 4 g 5 g 6 (ln(1/g)+0.7) g 4d lattice 0.0 1 10 _ 100 1000 T / Λ MS Kajantie et al., hep-ph/0211321 Spatial area law ↔ non-perturbative ∀ T university-logo Ph. de Forcrand GGI, June 2008 DimRed

Intro DimRed Center symm. SU(2) Outlook Brute force Solution: ( 3 + 1 ) d lattice simulations However: • N τ must be large ( O ( 10 ) ) to control a → 0 limit T c ? Fodor et al. ↔ Karsch et al. • Finite density ?? Alternative approach? university-logo Ph. de Forcrand GGI, June 2008 DimRed

Intro DimRed Center symm. SU(2) Outlook Pert. Results Dimensional reduction • dim ( d + 1 ) system with one compact dimension: looks like dim d at distances ≫ β = 1 T β • degrees of freedom are static modes φ 0 ( x ) � x , τ ) = T ∑ + ∞ φ ( n = − ∞ exp ( i ω n τ ) φ n ( x ) � � • Effective action: integrate out non-static modes � D φ 0 D φ n exp ( − S 0 ( φ 0 ) − S n ( φ 0 , φ n )) Z = � D φ 0 exp ( − S 0 ( φ 0 ) − S eff ( φ 0 )) = with � D φ n exp ( − S n ( φ 0 , φ n )) exp ( − S eff ( φ 0 )) ≡ • In practice? Goal is to reproduce Green’s fncts � φ 0 ( 0 ) φ 0 ( x | ≫ β � x ) � for | � � university-logo T is UV cutoff for S eff Ph. de Forcrand GGI, June 2008 DimRed

Intro DimRed Center symm. SU(2) Outlook Pert. Results Dimensional reduction for QCD Asymptotic freedom: g ( T ) ∼ 1 / log T causes separation of scales at high T : • hard modes, energy O ( T ) : non-static, esp. fermions (odd Matsubara) • soft modes, O ( gT ) : Debye mass � A 0 ( 0 ) A 0 ( x ) � • ultrasoft modes, O ( g 2 T ) : magnetic masses � A i ( 0 ) A i ( x ) � university-logo Ph. de Forcrand GGI, June 2008 DimRed

Intro DimRed Center symm. SU(2) Outlook Pert. Results Perturbative approach Asymptotic freedom allows/enforces evaluation of S eff by perturbation theory • Degrees of freedom are static A i , A 0 , ie. 3 d YM with adjoint Higgs • Adjust couplings of S eff to match Green’s fncts in perturbation theory → after integrating out hard modes: EQCD � 1 0 + λ E A 4 S EQCD = d 3 x 2 F 2 ij + Tr D i A 0 D i A 0 + m 2 E A 2 � � 0 with F ij = ∂ i A j − ∂ j A i − ig 3 [ A i , A j ] , and g 2 3 = g 2 T m E ( T ) , λ E ( T ) fixed by perturbative matching → after integrating out soft modes: MQCD S MQCD = d 3 x 1 2 F 2 � ij university-logo Ph. de Forcrand GGI, June 2008 DimRed

Intro DimRed Center symm. SU(2) Outlook Pert. Results Successes and limitations of EQCD Screening masses down to ∼ 2 T c 8 SU(2) SU(3) 7 6 M/T 5 4 3 2 ++ 0 - ++ 0 - +- J R -+ -+ PC 0 + 0 + 0 - Laermann & Philipsen, hep-ph/0303042 university-logo Ph. de Forcrand GGI, June 2008 DimRed

Intro DimRed Center symm. SU(2) Outlook Pert. Results Successes and limitations of EQCD Screening masses at finite density ( T = 2 T c ) µ real, J P =0 + µ real, J P =0 − 10 10 M/T M/T 5 5 + + − − 0 0 + + + − − + 0 0 − − 0 0 0 1 2 3 4 0 1 2 3 4 | µ |/T | µ |/T Hart et al., hep-ph/0004060 university-logo Ph. de Forcrand GGI, June 2008 DimRed

Intro DimRed Center symm. SU(2) Outlook Pert. Results Successes and limitations of EQCD Spatial string tension down to ∼ T c ? r 0 T 1 0.5 2 1/2 (T) T/ σ s 0.8 0.7 0.6 0.5 N τ =4 N τ =6 N τ =8 0.4 1 1.5 2 2.5 3 3.5 4 4.5 5 T/T 0 Karsch et al., 0806.3264 university-logo Ph. de Forcrand GGI, June 2008 DimRed

Intro DimRed Center symm. SU(2) Outlook Pert. Results Successes and limitations of EQCD Wrong phase diagram ⇒ must fail near T c phase diagram β G = 12 symmetric phase 0.06 1st order 0.04 pert. theory 4d matching line xy pert. theory 0.02 2nd order tricritical point 0.00 broken symmetry phase −0.02 0.00 0.10 0.20 0.30 0.40 x � 1 0 + λ E A 4 S EQCD = d 3 x 2 F 2 ij + Tr D i A 0 D i A 0 + m 2 E A 2 � � 0 Symmetry is A 0 ↔ − A 0 , ie. Z 2 university-logo Matching line is in the wrong phase Ph. de Forcrand GGI, June 2008 DimRed

Intro DimRed Center symm. SU(2) Outlook Root of the problem • Perturbation theory ⇒ small fluctuations around one vacuum A 0 = 0 • YM vacuum is N c -degenerate: center symmetry (spontaneously broken for T > T c ) e τ ) = exp ( i 2 π A µ ( x ) → s ( x )( A µ ( x )+ i ∂ µ ) s ( x ) † , with s ( x + β ˆ N c k ) s ( x ) � β P ( x ) ≡ : exp ( i 0 d τ A 0 ( x , τ )) : 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 0 0 0 0 −0.05 −0.05 −0.05 −0.05 −0.1 −0.1 −0.1 −0.1 −0.1 0 0.1 −0.1 0 0.1 −0.1 0 0.1 −0.1 0 0.1 university-logo Effective action should respect symmetries of original action Ph. de Forcrand GGI, June 2008 DimRed

Intro DimRed Center symm. SU(2) Outlook Polyakov loop vs coarse-grained Polyakov loop • Use Polyakov loop P ( x ) instead of A 0 in S eff Pisarski But: P ( x ) ∈ SU ( N ) → non-renormalizable (non-linear σ -model) cf. PNJL • Here: degree of freedom is coarse-grained Polyakov loop Yaffe T Z ( x ) ≡ d 3 y U ( x , y ) P ( y ) U ( y , x ) � V block university-logo renormalizability preserved: easy T → ∞ matching with perturbation theory Ph. de Forcrand GGI, June 2008 DimRed

Intro DimRed Center symm. SU(2) Outlook Setup Matching Lattice Results Simplest: SU ( 2 ) Yang-Mills • sum of SU ( 2 ) matrices is multiple of SU ( 2 ) matrix Z = λ Ω , Ω ∈ SU ( 2 ) , λ > 0 2 (Σ 1 + i Π a σ a ) • Parametrization: Z = 1 � 1 L eff = g − 2 2 Tr F 2 ij + Tr ( D i Z † D i Z )+ V ( Z ) � 3 • Include all Z 2 -symmetric super-renormalizable terms: V ( Z ) = b 1 Σ 2 + b 2 Π 2 a + c 1 Σ 4 + c 2 (Π 2 a ) 2 + c 3 Σ 2 Π 2 a Local gauge invariance Z ( x ) → Ω( x ) Z ( x )Ω − 1 ( x ) Global Z 2 symmetry: Z → − Z (actually Σ → − Σ , Π → − Π indep.) university-logo Ph. de Forcrand GGI, June 2008 DimRed

Intro DimRed Center symm. SU(2) Outlook Setup Matching Lattice Results Perturbative matching • Determine [almost] parameters { b 1 , b 2 , c 1 , c 2 , c 3 } using perturbation theory • Split potential into hard and soft pieces: V ( Z ) = V h + g 2 3 V s scales ∼ T (magnitude of coarse-grained Pol.) • Hard potential → V h = h 1 Tr Z † Z + h 2 ( Tr Z † Z ) 2 O ( 4 ) symmetric EQCD at high T • Soft potential → V s = s 1 Tr Π 2 + s 2 ( Tr Π 2 ) 2 + s 3 Σ 4 university-logo Ph. de Forcrand GGI, June 2008 DimRed

Intro DimRed Center symm. SU(2) Outlook Setup Matching Lattice Results Leading order • Classical solution: Σ( x ) = ¯ Σ , Π( x ) = ¯ Π δ a , 3 which minimize g − 2 Σ 2 + ¯ Σ 2 + ¯ V class = 4 (¯ Π 2 )( 2 h 1 + h 2 (¯ Π 2 )) 3 Two possible cases ( h 2 > 0 for stability): h 1 < 0 → deconfined h 1 > 0 → confined Π = − h 1 Σ = ¯ ¯ Σ = ¯ ¯ h 2 , Tr Z ≡ v Π = Tr Z = 0 university-logo Ph. de Forcrand GGI, June 2008 DimRed

Intro DimRed Center symm. SU(2) Outlook Setup Matching Lattice Results 1-loop effective potential • At 1-loop, U ( 1 ) symmetry of potential is broken: Σ = v cos ( πα ) , ¯ ¯ Π = v sin ( πα ) V eff = s 1 v 2 sin 2 ( πα )+ s 2 v 4 sin 4 ( πα )+ s 3 v 4 cos 4 ( πα ) − v 3 3 π | sin ( πα ) | 3 + O ( g 2 3 ) 2 4 university-logo Ph. de Forcrand GGI, June 2008 DimRed

Intro DimRed Center symm. SU(2) Outlook Setup Matching Lattice Results Matching to EQCD Decompose fluctuations into radial + angular: � 1 2 v 1 + g 3 ( 1 2 φ 1 + i χ ) � Z = ± � ( ∂ i φ ) 2 + m 2 φ φ 2 � � ( D i χ ) 2 + m 2 χ χ 2 � + V s ( φ , χ ) 2 Tr F 2 L = 1 ij + 1 + Tr 2 m 2 φ = 8 v 2 c 1 = − 2 h 1 , heavy m 2 χ = 2 ( b 2 + v 2 c 3 ) = g 2 3 ( s 1 − 4 v 2 s 3 ) , light 0 + λ E A 4 2 F 2 ij + Tr D i A 0 D i A 0 + m 2 E A 2 L EQCD = 1 0 λχ 4 = λ E A 4 χ is A 0 → ˜ 2 equations m 2 χ = m 2 university-logo E , 0 Ph. de Forcrand GGI, June 2008 DimRed