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Lecture 1: Basics of Lattice QCD

Lecture 1: Basics of Lattice QCD

Lecture 1: basics of lattice QCD

Peter Petreczky

• Lattice and gauge : Wilson gauge , doubling

• Different fermion formulations

• Meson correlation function and Wilson loops

• Scale setting, continuum limit and lines of constant physics (LCP)

• Numerical simulations : path integral,

• Improved actions and thermodynamics

• The integral method and from lattice QCD

1 Finite Temperature QCD and its Lattice Formulation

evolution operator in imaginary time

Integral over functions integral with very large (but finite) Lattice dimension ( > 1000000 )

Costs :

difficult to study real time properties: spectral functions, 2 transport coefficients and fields on a lattice

U (x) G†(x)U G(x + µ) µ ! µ

fermion doubling ! 16 d.o.f !

3 Wilson

Wilson (1975)

chiral symmetry is broken even in the massless case !

additive mass

Wilson Dirac operator is not bounded from below

difficulties in numerical simulations 4 Discretization errors ~ a g2 , used for study of properties, spectral functions Staggered fermions

Kogut, Susskid (1975)

different flavors, spin componets sit in different corners of the Brillouin zone or in hypercube 4-flavor theory

5 Chiral fermions on the lattice ?

We would like the following properties for the lattice Dirac operator:

Nielsen-Ninomiya no-go theorem : conditions one 1-4 cannot be satisfied simultaneously Nielsen, Ninomiya (1981)

Wilson fermion formulation gives up 4) Staggered fermion formulation gives up 3) 6 Ginsparg-Wilson fermions

Ginsparg, Wilson (1982)

• anti-commutation properties are recovered in the continuum limit (a->0) • the r.h.s. of the Ginsparg-Wilson relation is zero for the solutions

mildest way to break the chiral symmetry on the lattice : physical consequences of the chiral symmetry are mantained ( e.g. chiral perturbation theory )

7 Generalized chiral symmetry and topology

GW relation Luescher (1998)

flavor singlet transformation :

Luescher (1998)

Hasenfratz, Laliena, Niedermeyer (1998) for flavor non-singlet transformation no ! 8 Constructing chiral fermion action I

Overlap fermions :

Neuberger (1998)

using it can be shown that

GW relation with R=1/2

9 Constructing chiral fermion action II

Domain wall fermions : introduce the fictitious 5th dimension of extent :

Shamir (1993)

Extensively used in numerical simulations : (see P. Boyle, 2007 for review) 10 QCD at finite baryon density

Hasenfratz, Karsch, PLB 125 (83) 308

S ( µa (x) −µa + (x) ) (x) ( (x) + (x) ) am = ψ x e U 0 ψ x+0 −ψ x e U 0 ψ x−0 + ∑ηi ψ xU i ψ x+i −ψ xU i ψ x−i + ∑ψ xψ x x,i x

det M is complex => sign problem det M exp(-S) 11 cannot be a probability Meson correlators and Wilson loops Meson states are created by bilenear operators: J(x, y; ⌧)= ¯(⌧,x) (x, y) (⌧,y) U Fixes the quantum number of of mesons, Γ is one Of the Dirac matrices Most often one considers point operators x=y and their correlation function:

decay constant

Consider static quarks :

12 Static meson correlation function functions after integrating out the static quark fields:

τ x y

Static quark anti-quark potential

0 x y R String tension

n=2 and larger : hybrid potentials 13 Numerial results on the potentials

Static quark anti-quark potential Hybrid potentials

Sommer scale 14 Scale setting in lattice QCD and continuum limit

Hadron masses in lattice QCD are dimensionless: m=mphys a

Continuum limit:

Physics does not depend on the details on the regularization, e.g dimensionless ratios :

Should be independent of the lattice spacing

The gauge depends on the lattice spacing:

15 Lattice QCD calculations

Staggered fermions : we get 4nf flavors to get 1-flavor replace nf by ¼ (rooting trick) Monte-Carlo Methods, importance sampling:

sign problem

Costs : improved discretization schemes are needed: 16 p4, asqtad, stout, HISQ where W k,l denotes a symmetrized combination of k l Wilson loops in the (µ, ν)- µ,ν × plane of the lattice,

1 W k,l (x)=1 Re TrL(k) L(l) L(k)+ L(l)+ +(k l) . (2.2) µ,ν − 2N ! x,µ x+kµ,ˆ ν x+lνˆ,µ x,ν ↔ "

(k) k−1 Here we have introduced the short hand notation for long links, Lx,µ = j=0 Ux+jµ,µˆ and x =(n ,n ,n ,n )denotesthesitesonanasymmetriclatticeofsize# N 3 N . 1 2 3 4 σ × τ

With a suitable choice of the coefficients ak,l it can be achieved that the gener- alized Wilson actions reproduce the continuum Euclidean Yang-Mills Lagrangian, = 1 F F ,uptosomeorderO(a2n)[1].Thestandardone-plaquetteWilsonac- L − 2 µ,ν µ,ν tion, S(1,1),witha =1anda =0forall(k, l) =(1, 1) receives O(a2)corrections 1,1 k,l ̸ in the naive continuum limit. Expanding the link variables, Ux,µ =exp(igaAµ(x)), in powers of a one findsImproved gauge action

1 S(1,1)(x)=1 Re TrU U U + U + µ,ν − N x,µ x+ˆµ,ν x+ˆν,µ x,ν 1 1 = g2a4 F F + a2F (∂2 + ∂2)F −2N ! µ,ν µ,ν 12 µ,ν µ ν µ,ν +O(a4) . (2.3) " can be eliminated by adding larger loops By adding an additional Wilson loop to the action one can achieve that corrections start only at O(a4). In particular we will consider here actions obtained by adding aplanar6-linkand8-linkloop,respectively.Thenon-vanishing coefficients in these cases are,

5 1 I (1, 2) : a = ,a= ≡ 1,1 3 1,2 −6 4 1 I (2, 2) : a = ,a= 17 (2.4) ≡ 1,1 3 2,2 −48

The action S(1,2) is a specific choice of the 6-link improved actions originallyproposed by Symanzik [1]. The action S(2,2) has recently also been discussed in the context of NRQCD calculations [15, 16]. Its generalization to largerquadraticloopsgivesa straightforward procedure to eliminate also higher order cut-off effects. Already in O(a4)thereexisttwoindependentoperatorsofdimensioneight,which contribute to the finite cut-off effects. In general, one thus needs three independent loops of length six and eight in order to eliminate all cut-off effects proportional to a2 and a4,respectively.Inanactionconstructedonlyfromquadraticloopsthedifferent

4 2.1 Improvement of at tree level

4

3.5 py=pz=0 E(p) 3 momentum2.5 space we find then

2 (0) ¯ SF = i ψ(p) γµhµ(p)+m ψ(p) , p " µ $ 1.5 ! # N 2 1 with S˜(0) = i c − ψ¯(p) γ h(2)(p) ψ(p) , F 1 µ µ 2Nc p µ AAppendix ! # (1) ¯ STANDARDˆ(1) 2 2 SF0.5 = i ψ(p) γµSµ (NAIKp, k) ψ(p k) , sF (p)= hµ(p)+m . p k µ − We give here explicit expressionsµ as well as some technical details of the perturbative ! ! # P4 p ! calculations. It is organized as follows: In part A.1 we definethefundamentalterms with(2)0 i ¯ with ˆ(2) SF 0 = 0.5 1 ψ(p) 1.5 γµSµ (2p, k1,k22.5) ψ(p k13 k2) , and functions we refer to in the following parts. Part A.2 and part A.3 contain some −2 p k1 k2 µ − − ! ! ! Improved# staggered fermion actionsdetails of the determination of the tree-level and 1-loop coefficients respectively. Figure 1: Dispersion relation E2= E(p⃗)with2 ⃗p =(s (p,p)=0, 0) for theh standard2 (p)+m2 staggered.A.2 Tree level coefficients for improved rotational symme- withsF (p)= h (p)+m . F µ 2 In part A.4 we give the explicit results of the 1-loop calculation of the fermionic Standard staggeredµ action has discretization errorsµ ~ a Heller, Karsch, Sturm, fermionwhere action (blue), the Naik-actionµ (green) and the p4-action! (red) comparedcontribution to to the free energy density. ! PRD60try (1999) 114502 the continuumEliminate dispersion those relation usingE higher= p.Solidanddashedlinesdenoterealand order difference scheme 2 2 s(1)F (p)= h (p)+m . complex polesˆ of the propagatorµ respectively. 4 Sµ (p, k)=µ Kµ;ρ(p,A.2 k) Aρ( Treek) , level coefficientsAn for expansion improved of the rotational free inverse symme- fermion up to order p yields A.2 Tree level! coeρ fficients for improved rotationalA.1 symme- General 1-loop results and definitions # try The inverseˆ(2) free fermion propagator of the action 2.1 in momentum space takes Sµ (p,try k1,k2)= Lµ;ρ,σ(p, k1,k2) Aρ(k1)Aσ(k2) . the form ρ,σ ForD the(0)( trigonometricp)= h ( functionsp)h (p)= we useAp the2 shortA A hand+2 notatiB p2 on:+2B p2 + (p6) , A.2 Tree level coe# fficientsAn expansion for improved of the free rotationalinverse fermion symme- propagator up toµ orderµp4 yields µ ⎛ 1 µ 2 ν⎞ µ µ ν̸ µ O An expansion of the free inverse fermion propagator up to order p4 yields ! ! != (0)−1 tryi µ γµhµ(p)+m i µ γµhµ(p)+m ⎝ ⎠ Here∆ we(p use)= the− following definitions= − also referred to in appe, ndices A.2 -(2.6) A.4:sµ(p) sin(pµ) , F Free quark 2propagator:2 (0) (0) 2 2 2≡ 2 6 !ρ hρ(p)+m !D D(p)+(p)=m hµ(p)hµ(p)= ApwithA coeA +2fficientsB p +2B p + (p ) , µ ⎛ cµ(p) 1 µ cos(2 pµ) ν.⎞ (0) 2 µ 2 2 µ 4 6 ≡ ν≠ µ O AnD expansion(p)=! ofh theµ(p free)hµ(p inverse)= fermionAp(0)µA propagatorA(0)+2B!1pµ +2 upB to2 orderp(0)ν!p +yields(p ) , ! hµ(p)=2sµ(p) c , +2⎛ c , cν (2p) +2c , ⎞sµ(3p) ,⎝ (A.2) ⎠ µ µ 1 0 1 2 ν≠ µ 3 0 O where the functions hµ!(p)aregiveninappendixA.1(Eq.A.2).!% with coefficientsν≠ µ & ! The inverse gluon(0) propagator(0) of the(0) 1 2actionwascalculatedin[2].Forthe ⎝ # ⎠ A =2c1,0 +12c1,2 +6c3,0× , In order to obtain(0) a(2) rotational invariant free(2) 2 propagator(2) we2 should(2) achieve2 standard that6 plaquette action as for the 1 2actionthepropagatorcanbewrittenas with coeD ffi(cientshp)=µ (p)=2hµ(p)hµ(ps)=µ(p) c1Ap,0 +2µA cA1,2+2Bc1νp(2µ p+2) +2B2 c3,0psνµ(3+p) ,(p ) , 1 × D(0)(p)= h (p)h (p)isafunctionofp2 only.⎛ For the standard one-link⎞ prop-O (0) (0) (0) µ µ µ µ %µ ν≠ µ (0) & (0)ν≠ µ (0) B1 = c1,0 2c1,2 9c3,0 , ! ! A =2# c1,0 +12c1!,2 +6c3,0 , −3 − − agator of the staggered fermion action(0) fat this requirement⎝ (0) for(3,0)arotationalinvariant(0)⎠ (1,2) ! K (p, k)=(0) c(0) (0)(ω; p, k)+c 1 (0)(p, k)+(0)c (0) (p,∆ k)Gµ,, ν (k)=(0)DGµ(k)δµ,ν EGµ,ν(k)+ξ gµ(k)gν(k) , µ;ρ 4 1,0 µ;ρ (0) i.e.3,0 µ;ρ 1,2 µ;ρ B = 8c . − propagatorwith is violated coeA fficients=2 at c1(,0p+12). Expandingc1,2 +6A c3,0 D, B(p1), = theA trigonometricc1,0 2c1,2 9 functionscA3,0 , 2 1,2 2 2 2 2 O (0) fat −(0)3 (3−,0) − (0) D(1G,µ2)(k)=4− a1,1 sν (k/2) 16 a1,2 sν(k/2) 2 sν (k/2) sµ(k/2) , appearingL inµ;hρ,σ((pp,)(seeappendixA.1),inordersof k1,k2)=1 (0) c1,(0)0 µ ρ,σ(0)(ω; p, k1,kp afurtherconstrainttotheco-2)+c3,0 (0) µ ρ,σ(p, k1,k2)+c1,2 µ ρ,σ(p, k1,k2) , ν − ν − − Theµ different staggered which; flavorsB sit= in different8c ;. corners of the Brillouin; zone " # (0) B1 = c1,0 2c1,2B 9c3,0 , 2 1B,2 B ! ! −3 − − − EGµ,ν (k)=4a1,1 sµ(k/2)sν(k/2) efficients ci,j canAreA be completely determined=2c(0) equivalent+12 suchc(0) that+6 inc the(0) the free, free propagator theory => is flavor rotationa symmetrylinvariant 4 1(0),0 1,2 3,0 Obviously the condition B1 = B2 has to be satisfied2 to achieve2 rotational symmetry up towhere order p NotB,fordetailsseeappendixA.1.Theconstraintis2 the= case8 cin1, 2the. interacting theory: 4 16 a1,2 sµ(k/2)sν(k/2) 2 sν(k/2) sµ(k/2) , − 1 (0) (0) Obviously(0) the condition B1 = B2 hasup to to be order satisfiedp to,whichleadstotheconstraint achieve− rotational symmetry − − exchangeB1 = with cgluons, 2c with, momenta9c , , ~ π/a can change the quark flavor (taste) as " # −3 1 0 − 1 2 − up3 0 to order p4,whichleadstotheconstraint (0) (0) (0) (0) 18 it bringsfat it to another(0) corner of the Brillouin zone c1,0Obviously+27c3,0 the+6(ω condition;cp,1,2 k=24)=2Bc1,2=.Bcµ(hasp tok/ be2) satisfied to achieve rotationalwith(2.7) symmetrya1,1 1, a1,2 0forthestandardWilsonone-plaquetteactionanda1,1 5/3, B2µ;ρ = 8c1,2 .1 2 (0) ≡ (0)≡ (0) (0) ≡ A 4 − − a , c +271/12c for the+6 1 c2action.Inbothcaseswechoosethegaugefixingterm=24c . up to order p ,whichleadstotheconstraint4ω (0) (0) (0) (0)1 2 ≡−1,0 3,0 ×1,2 1,2 c1,0 +27c23,0 +6c1,2 =24c1g,2 (k. )=2s (k/2) and Feynman gauge ξ =1. δµ,ρ δµ,ρ sν(k/2) (1 δµ,ρ) sµ(k/µ 2)sρ(k/µ 2) , Two simple choices which eliminate·' − the1+6 bendedω ( or straight three− link− terms in 2.1, )* Obviously the condition B1 = B2 has to beν satisfied≠ µ to achieve rotational symmetry (0) (0) (0) (0) # The basis of our 1-loop calculations is an expansion of the fermion action in 4 (0) (0) upfat toc1 order,0 +27pc,whichleadstotheconstraint3,0 +6c1,2 =24c1,2 . The expansioni.e. shows that setting c 0, i.e. the Naik-action, leads to vanishing µ;ρ,σ(ω; p, k1,k2)=2sµ(5pThek1 expansion/2 k2/2) shows that setting c1,2 powers0, ofthe the Naik-action, bare coupling leadsg: to vanishing1,2 ≡ B − 4 − (≡p4)coefficients B = B 2 0, which corresponds even to an (a2)improvement. (4pω)coefficients B1 = B2 0, which corresponds even to1 an (a2 )improvement. δ δ O δ δ s2(k ≡/2+k /O2) O ≡ O (0) (0) (0)µ,ρ µ,σ (0) (0) i.e.µ,ρ µ,σ ν 1 2 (0) (1) 2 (2) 2 ˜(0) 3 The expansionc1,0 +27 showsc3,0 +6 that·'c1,2 setting=24−c1+61,2 c1,2ω0,(. the Naik-action,ν≠ µ leads to vanishingSF = SF + gSF + g SF + g SF + (g ) . (A.1) 4 ≡ # 2 O (p )coefficients B1 = B2 0, which corresponds even to an (a )improvement. 2 +2A.3 (1 δ Improvedµ,ρ) δρ,σ sµ(k rotational1/2)sµ(k2/2) symmetrycρ(k1/2+k2/ at2) (g ) O ≡ − O A.3 Improved rotational symmetry at (g2) (0) To achieve that,O we expand the exponential representation ofthelinkvariables δµ,ρ (1 δi.e.µ,σ) sµ(k2/2) icµ(k2/2) sσ(k1 + k2/2) 2 O The expansion shows that setting c1,2 0, the Naik-action, leads to vanishing Nc −1 b b − ≡ − { − 2 U}µ(x)=exp(igaAµ(x)), with the gauge fields Aµ A λ and normalization (p4)coefficients B = B 0, whichThe explicite corresponds expressions even to for an the(a2()improvement.g )contributionstothefermionpropagatorare: ≡ b=1 µ 1 2 2 O 2Tr λaλb = δ for the group generators λ2a.Thelatticespacingissettoa =1.In A.3O Improved rotational≡ δ symmetryµ,σ (1 δµ,ρ) atsµ(k1/(g2)) + icOµ(k1The/2) explicitesσ(k2 + kab expressions1/2) , for the (g )contributionstothefermionpropagatorare:$ − − { O } )* O (0) D (p)= hµ(p) , 17 2 µ − The explicite expressions for the (g )contributionstothefermionpropagatorare:2 2 (0) A.3 Improved rotationalO symmetry18 (2) at N(c g )1 (2) Dµ (p)= hµ(p) , Dµ (p)= O − hµ (p) , − − 2Nc N 2 1 (0) Σ (p)= (p)+ (p) , (2) c (2) TheD expliciteµ (p)= expressionshµ(p for) , the (g2)contributionstothefermionpropagatorare:µ µ µ Dµ (p)= − hµ (p) , − K L − 2Nc 2 O (2) Nc 1 (2) Σ (p)= (p)+ (p) , Dµ (p)= − hµ (p) , 20 µ µ µ (0) − 2Nc K L Dµ (p)= hµ(p) , Σµ(p)= −µ(p)+ µ(p) , K 2 L 20 (2) Nc 1 (2) Dµ (p)= − hµ (p) , − 2Nc 20 Σ (p)= (p)+ (p) , µ Kµ Lµ 20 2.1 Improvement of rotational symmetry at tree level

4

3.5 py=pz=0 E(p) 3

2.5

2

1.5

1

STANDARD 0.5 NAIK P4 p 0 0 0.5 1 1.5 2 2.5 3

Figure 1: Dispersion relation E = E(p⃗)with⃗p =(p, 0, 0) for the standard staggered fermion action (blue), the Naik-action (green) and the p4-action (red) compared to the continuum dispersion relation E = p.Solidanddashedlinesdenoterealand complex poles of the propagator4 respectively.

3.5 p =p ,p =0 The inverse free fermion propagator4 of the actionx y 2.1z in momentum space takes E(p) the form 3 3.5 px=py,pz=0 fat 1 † Uµ (x)= Uµ(x)+ω Uν(x)Uµ(x2.5+ˆν)UνE(p)(x +ˆµ) −1 i γ h (p)+m i γ h (p)+m 1+6ω ! (0) ν≠ µ µ µ µ 3 µ µ µ ∆F "(p)=# − 2 2 = − (0) 2 , (2.6) !ρ hρ(p2)+m !D (p)+m † 2.5 +Uν (x νˆ)Uµ!(x νˆ)Uν(x +ˆµ νˆ) . − − 1.5 − % where the functions hµ(p)aregiveninappendixA.1(Eq.A.2).2 $ In order to obtain a rotational1 invariant free propagator we should achieve that (0) 2 The fat-link improved one-link term, the linear 3-link term 1.5and the angular 3-linkSTANDARD D (p)= µ hµ(p)hµ(p)isafunctionof0.5 p only. For the standard one-link prop- term appear with coefficients whose dependence on the gauge coupling we haveNAIK agator of the! staggered fermion action this requirementP4 for arotationalinvariantp 1 parameterized as follows a propagator is violated at (0p4). Expanding D(0)(p), i.e. the trigonometric functions O 0 0.5 1 1.5 2 2.5 3 3.5 4 appearing in hµ(p)(seeappendixA.1),inordersofp afurtherconstrainttotheco-STANDARD 2 0.5 NAIK (0) 2 (0)Nc Figure1 (2) 2: Dispersion relation E = E(p⃗)withp⃗ =(p/√2,p/√2, 0) for the standard efficients ci,j can be determined such that the free propagator isP4 rotationalinvariant ci,j = ci,j + g −staggeredci,j . fermion action (green), the Naik-action (blue) and the p4-actionp (red) up to order p24N,fordetailsseeappendixA.1.Theconstraintisc 0 compared to the continuum0 0.5 dispersion1 1.5 relation2 E =2.5p.Solidanddashedlines3 3.5 4 Together with the fat-link-weight ω there is thusdenote a total real number and complex of poles 7 parameters of the propagator in respectively. Rotational symmetry at order p4 : B = B (0) (0) (0) (0) √ √ the action. In order to reproduce1 2 the correctc1,0 +27 naivec3,0 +6Figure continuuc1,2 2:=24 Dispersionmlimitthecoec1,2 . relationffiEcients= E(p⃗)withp⃗ =(p/ 2,p/ 2(2.7), 0) for the standard staggered fermion action(0) (green), the Naik-action (blue)Naik-action and the p4-action (red) have to satisfy two constraints respectively, are to set either c1,2 0whichyieldsthefamiliar C1.0 + C comparedNaik to action: the continuum ≡ dispersion relation E = p.Solidanddashedlines 3,0 Two simple choices which eliminate the bended or straight three link terms in 2.1, denote(0) real9 and(0) complex1 poles of the propagator respectively. (0) (0) (0) c1,0 = ,c3,0 = , (2.8) Normalization: c1,0 +3c3,0 +6c1,2 =1/2 , 16 −48 (2.3) 5 (0) (2) (2) (2) respectively, are to set either c1,2 0whichyieldsthefamiliarNaik-action c1,0 +3c3,0 +6c1,2 =0. or to set c(0) 0whichleadsto (2.4)≡ 3,0 ≡ (0) 9 (0) 1 (0) c1,03= (0) ,c13,0 = , (2.8) Our aim is to fix these coefficients such that the rotationalc1,0 = sym,c161metry,2 = of, the−48 fermion (2.9) propagator is improved up to one loop order. 8 48 In a first step we consider the free fermion propagator.whichor to we set will Herec(0) call w theederivefurthercon-0whichleadstop4-action. (0) p43,0 action: straints for the tree level coefficients c .InparticularwewillconsiderthetwoSince for the Naik-action≡ the (p4)termsarecompletelyeliminatedonegetseven i,j O an (a2)improvement,whichseemsquiteaccidentalinthisapproach. cases where only one or the other of the three linkO terms contr3 ibutes.1 Then the c(0) = ,c(0) = , (2.9) constraints fix the tree level coefficients to certainAfirstimpressionofhowtheimprovementworksisobtainedby values.1,0 8 1,2 48 inspecting the dispersion relation E = E(px,py,pz)resultingfromthepolesofthefreepropagator In the second step we calculate the self energy contributionstothefermionpropaga-−1 (2) for massless fermions ∆(0) ,whicharesolutionsofD(0)(iE, ⃗p)=0.Figures1and tor. Here the one-loop coefficients ci,j come into playwhich and we canwill call be tunedtheF p4-action to improve. 2showthedispersionrelationsforon-axismomentaSince for the Naik-action the (p4)termsarecompletelyeliminatedonegetsevenp⃗ =(p, 0, 0) and momenta also the rotational symmetry in one-loop order. √ O√ onan the planar(a2)improvement,whichseemsquiteaccidentalinthisapproac diagonal p⃗ =(p/ 2,p/ 2, 0) respectively for the standard fermionh. Taste symmetryAs improvement: gluon part of theOrginos action et weal, PRD60 choose (1999) the tree054503 levelaction, improved the Naik-action 1 2-action: and the p4-action in comparison to the continuum relation O × Afirstimpressionofhowtheimprovementworksisobtainedbyinspecting the no taste breaking at 6 2Nc 5 1 dispersion relation E = E(px,py,pz)resultingfromthepolesofthefreepropagator S = 1 Re Tr (x) (0)−1 G 2 forµν massless fermions ∆ ,whicharesolutionsofD(0)(iE, ⃗p)=0.Figures1and Fat (smeared) link: g & x,=ν >µ 3 ' − N ( F " 2showthedispersionrelationsforon-axismomenta19 p⃗ =(p, 0, 0) and momenta Projection to U(3) => HISQ action 1 1 on the planar diagonal p⃗ =(p/√2,p/√2, 0) respectively for the standard fermion + 1 Re Tr (x)+ (x) . (2.5) ⎛ ⎛ action,µν the Naik-action⎞⎞ and the p4-action in comparison to the continuum relation 6 − 2N µν - ⎝ ⎝ ⎠⎠ 6 We note that also here the tree-level coefficients 5/3and1/6couldbefurtherim- proved at (g2). These corrections, however, are higher order correctionsforthe O analysis of the fermion contributions to thermodynamic observables which we will present in the following.

a Note that the Nc dependence of the 1-loop coefficients has been factored out here explicitely.

4 Why improved actions ?

Pressure of the ideal gluon Pressure of the ideal quark gas

1.4 (0) (0) f (N)/fcont 1.2

1

0.8 (1x1) 0.6 (1x2) RG 0.4 N 0.2 4 6 8 10 12 14 16

2.0 p/T4

1.5 20

1.0 (1x1) (1x2) (1x1), cont (1x2)tad 0.5 RG

T/Tc 0.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Mass splitting of pseudo-scalar mesons

Only one out of 16 PS mesons has zero mass in the chiral limit, the quadratic mass splitting is the measure of flavor symmetry breaking

asqtad HISQ

PS meson splittings in HISQ calculations are reduced by factor ~ 2.5 compared to asqtad at the same lattice spacing and are even smaller than for stout action => discretizations effects

for Nτ=8 HISQ calculations are similar to those in Nτ=12 asqtad calculations

21 Glossary of improved staggered actions p4 = std. staggered Dslash with 3-step (fat3) link +p4 term asqtad = std. staggered Dslash with 7-step (fat7) link + Naik term HISQ = std. staggered Dslash with re-unitarized doubly smeared 7-step (fat7) link stout = std. staggered Dslash with re-unitarized doubly smeared 3-step (fat7) link

22 p4, asqtad, HISQ, stout Integral method: bulk thermodynamics in SU(3)

In Monte-Carlo simulations ln Z(T) cannot be determined but only its derivatives

Boyd et al., Nucl. Phys. B496 (1996) 167

computational cost go as because of the vacuum subtraction 23 large cutoff effects !

Boyd et al., Nucl. Phys. B496 (1996) 167

the free gas limit overestimates cutoff effects Wilson gauge action discretization errors => corrections to the pressure 24 Boyd et al., Nucl. Phys. B496 (1996) 167 Karsch et al, EPJ C 6 (99) 133 Wilson gauge action Luescher-Weisz gauge action: continuum extrapolation large reduction of cutoff effects

25