QCD Simulations at Realistic Quark Masses: Probing the Chiral Limit - - PowerPoint PPT Presentation

QCD Simulations at Realistic Quark Masses: Probing the Chiral Limit

• G. Schierholz

Deutsches Elektronen-Synchrotron DESY – QCDSF Collaboration –

Special mention:

• M. G¨
• ckeler, T. Hemmert, R. Horsley, Y. Nakamura, D. Pleiter,

P.E.L. Rakow, W. Schroers, T. Streuer, H. St¨ uben and J. Zanotti

Objective

Solve QCD and probe the limits of the Standard Model · · ·

• Parameters of QCD

– ΛQCD resp. αs(Q2) – Quark masses – θ angle

• QCD in the wider world

– CKM matrix

• How does QCD work ?

– Hadron structure – Spectroscopy

• Fundamental properties

– χSB – Confinement · · · in concert with Exp & Phen

Problem: Chiral Extrapolation

Recently ChPT O(p4)

• Stat. error 5%

68.3% CL Need to reduce (scale) error to a few %

Outline Lattice Simulations Pion Sector Nucleon Sector Miscellaneous Conclusions & Outlook

Lattice Simulations

Action

Nf = 2 S = SG + SF SG = β X

x,µ<ν

“ 1 − 1 3Re Tr Uµν(x) ” SF = X

x

n ¯ ψ(x)ψ(x) − κ ¯ ψ(x)U†

µ(x − ˆ

µ)[1 + γµ]ψ(x − ˆ µ) − κ ¯ ψ(x)Uµ(x)[1 − γµ]ψ(x + ˆ µ) − 1 2κ cSW g ¯ ψ(x)σµνFµν(x)ψ(x)

• ∂µAimp

µ

= 2mqP

Clover Fermions

Advantages

• Local
• Transfer matrix
• O(a) improved
• Flavor symmetry

Prerequisite to making contact with SU(2) ChPT – Finite size corrections – Chiral extrapolation – Determination of low-energy constants

• Fast to simulate

Cost of Simulation

1000 Configurations ∝ L4.8 (mπ/mρ)−3.6 (r0/a)0.9 Hasenbusch, QCDSF, L¨ uscher, Urbach et al., · · ·

Compared to · · ·

Clark

Parameters

Nf = 2 ← a = 0.065 fm ← a = 0.077 fm 243 48 323 64 + 403 64 For gauge field sampling we use ‘ordinary’ HMC algorithm with Hasenbusch integration + 3 time scales

Obstructions ?

m mπ

q 2

O(a )

2

1st order transition

m mπ

q 2

O(a )

2

Aoki phase

← cold start ← hot start

Landscape

Minimal pion mass : mπ(L) = 3 2f 2

0L3

„ 1 + 2 4πf 2

0 L2 2.837

«−1 Leutwyler Hasenfratz & Niedermayer

Effect of Unquenching ?

Vector Ward Identity ? χtop ≡ Q2 V = Σ mq 2 “ 1 χtop ”2 = “ 2 Σ mq ”2 + “ 1 χ∞

top

”2 D¨ urr

Pion Sector

Pion Mass

Raw data ↑ NPRen No 1st order phase transition or Aoki phase ! NLO m2

PS = m2

» 1 + 1 2x ˆ l3 + O(x2) – mPS − mPS(L) mPS = − X

| n|=0

x 2λ h I(2)

mPS(λ)

+ xI(4)

mPS(λ)

i Colangelo, D¨ urr & Haefeli m2

0 =2Σmq , x=

m2 16π2f 2 , λ=mPS| n|L ˆ li =ln Λ2

i

m2

I(2)

mPS(x) = −B0(x)

I(4)

mPS(x) =

„ −55 18 + 4¯ l1 + 8 3 ¯ l2 − 5 2 ¯ l3 − 2¯ l4 « B0(x) + „112 9 − 8 3 ¯ l1 − 32 3 ¯ l2 « B2(x) + S(4)

mPS(x)

S(4)

mPS(x) = 13

3 g0B0(x) − 1 3 (40g0 + 32g1 + 26g2) B2 + · · · B0(x) = 2K1(x) , B2(x) = 2K2(x)/x , ¯ li = ln Λ2

i

m2

PS

Λi , gi from hep-lat/05030142

FS corrected Corrections r0f0 = 0.179(2) , r0Λ3 = 1.82(7)

Pion Decay Constant

FS corrected Corrections fPS = f0 h 1 + x ˆ l4 + O(x2) i fPS − fPS(L) fPS = X

| n|=0

x λ h I(2)

fPS(λ) + xI(4) fPS(λ)

i r0f0 = 0.179(2) r0Λ4 = 3.32(6) fPS ← NPRen

I(2)

fPS(x) = −2B0(x)

I(4)

fPS(x) =

„ −7 9 + 2¯ l1 + 4 3 ¯ l2 − 3¯ l4 « B0(x) + „112 9 − 8 3 ¯ l1 − 32 3 ¯ l2 « B2(x) + S(4)

fPS(x)

S(4)

fPS(x) = 1

6 (8g0 − 13g1) B0(x) − 1 3 (40g0 − 12g1 − 8g2 − 13g3) B2 + · · · Colangelo, D¨ urr & Haefeli

Partially Quenched

mPS ≡ mSS

PS → mAB PS ,

fPS ≡ f SS

PS → f AB PS ,

A, B ∈ {V, S|V = S}

0.0 1.0 2.0 3.0 ξ 0.000 0.005 0.010 0.015 κ=0.1362 0.000 0.005 0.010 0.015 κ=0.1355 0.0 1.0 2.0 3.0 ξ κ=0.13632 κ=0.1359

ˆ R ≡ R mSS 2

PS

= f V S

PS

mSS 2

PS

q f V V

PS f SS PS

= − 1 8(4πr0f0)2 ln mV V 2

PS

mSS 2

PS

− mV V 2

PS

mSS 2

PS

+ 1 ! , ξ = mV V 2

PS

mSS 2

PS

Sharpe

Nucleon Sector

Nucleon Mass

FS corrected Corrections r0f0 = 0.179(2) g0

A = 1.15

r0 = 0.45(3) fm r0m0 = 2.00 c1/r0 = −0.43

mN = m0 − 4c1m2

PS − 3g0 2 A

32πf 2 m3

PS +

" e1(µ) − 3 64π2f 2 “g0 2

A

m0 − c2 2 ” − 3g0 2

A

32π2f 2 “g0 2

A

m0 − 8c1 + c2 + 4c3 ” ln mPS µ # m4

PS +

3g0 2

A

256πf 2

0 m2

m5

PS + O(m6 PS)

mN − mN(L) = −3g0 2

A m0m2 PS

16π2f 2 X

| n|=0

Z ∞ dzK0 „q m2

0z2 + m2 PS(1 − z)|

n|L « − 3m4

PS

4π2f 2 X

| n|=0

" (2c1 − c3)K1(mPS| n|L) mPS| n|L + c2 K2(mPS| n|L) (mPS| n|L)2 # + O(m5

PS)

Axial Coupling

Preliminary ↓

0.0 0.16 0.32 0.48 0.64 0.8

m 2 [GeV2]

0.8 1.0 1.2 1.4

gA

=5.20 =5.25 =5.29 =5.40

0.0 0.16 0.32 0.48 0.64 0.8

m 2 [GeV2]

0.8 1.0 1.2 1.4

gA

=5.20 =5.25 =5.29 =5.40

χPT O(p3) 68.3% CL

Miscellaneous

Rho Mass

Not FS corrected

Delta Mass

Not FS corrected

Conclusions & Outlook

• Simulations at pion masses of O(300) MeV with

Wilson-type fermions feasible now

• Improvement of algorithms
• Increase ofcomputing power
• Extrapolation to chiral limit and infinite volume

greatly improved FS corrections surprisingly well described by ChPT

• First meaningful lattice determination of low energy constants :

Preliminary ! r0 f0 Λ3 Λ4 0.45(3) fm 79(5) MeV 0.80(5) GeV 1.46(10) GeV

• Major investment in FS corrections (including partially quenched data) and δ expansion

needed