Lattice QCD and its interface to phenomenology · Lattice QCD and its interface to phenomenology...

Lattice QCDand its interface to phenomenology

Stephan Durr

University of WuppertalJulich Supercomputing Center

EINN 2015Paphos – 3.11.2015

,Lattice QCD (1): combined UV/IR regulator,

QCD Lagrangian contains quarks and gluons [Fritzsch, Gell-Mann, Leutwyler (1973)]

LQCD =1

2Tr(FµνFµν) +

Nf∑i=1

q(i)(D/+m(i))q(i) + iθ1

32π2εµνρσTr(FµνFρσ)

• QCD must be regulated both in the UV and in the IR to make it well-defined.

• The lattice does the job through a>0 and V =L4<∞, but other options arepossible. Each gauge/fermion action combination is a different regulator.

• For a→0 correlation lengths diverge, but ratios ξπ/ξΩ stay finite (renormalization).The extrapolations a→0 and V →∞ are performed in dimensionless observables.

• The result is independent of the action, thanks to universality [Wilson], but someEuclidean results must be rotated to Minkowskian (ill-posed problem).

• Does this Lagrangian-regulator-extrapolation package explain phenomena such asconfinement, spontaneous/anomalous symmetry breaking, hadron spectrum ?

S. Durr, BUW/JSC EINN 2015, Paphos, 3.11.2015 1

,Lattice QCD (2): effective masses and decay constants,A

eff(t

)=C

(t)

exp

(Meff(t

)t)

Meff(t

)=

1 2lo

g(C

(t−

1)

C(t

+1))

0 10 20 30 40 50 60

0.0515

0.052

0.0525

0.053

0.0535

0.054

meff_inft_ll

meff_cosh_ll

0 10 20 30 40 50 60

0.189

0.19

0.191

0.192

0.193

0.194

0.195

0.196

0.197

0.198

0.199

meff_inft_ls

meff_cosh_ls

0 10 20 30 40 50 60

0.05

0.0505

0.051

0.0515

0.052

0.0525

feff_inft_ll

feff_cosh_ll

0 10 20 30 40 50 60

0.0595

0.06

0.0605

0.061

0.0615

0.062

0.0625

0.063

feff_inft_ls

feff_cosh_ls


,Lattice QCD (3): scale hierarchies,

typical spacing: 0.04 fm≤a≤0.20 fm

1 GeV≤a−1≤5 GeV

typical length: 2 fm≤L≤6 fm

require (UV): amq 1

require ( IR ): MπL ≥ 4

u c (t)

d s b︷︸︸︷work near

︷︸︸︷interpolate to

︷︸︸︷extrapolate

12(mphys

u +mphysd ) mphys

s ,mphysc mb→mphys

b

In QCD with Nf flavors, 1+Nf observables utilized to set scale and quark masses.


Overview

• How to cure/assess systematic uncertainties

• T =0 topics: spectroscopy, quark masses, LECs in ChPT, hadron structure

• T =0 topics: FLAG compilations of flavor related quantities

• T >0 topics: pseudocritical Tc, EoS, transport coefficients, sign problem at µ 6=0

• Including electromagnetism and topics left out


,Systematic uncertainties (1): theory for a→ 0,

• Functional form of asymptotic effects via Symanzik analysis ∝ a, αQCD

a, a2.

• A priori no telling how close to the continuum Symanzik scaling sets in.

• A priori no knowledge about size of prefactor in Symanzik regime.

• Possibly competition with subleading order in accessible a-range, e.g.

O(αa) with O(a2) for fat clover, in such cases include both options.

J.R

olf,

S.S

int

[AL

PH

A],

JHE

P02

12(2

002)

007


,Systematic uncertainties (2): theory for L→∞,

ChPT predicts relative finite-volume effects of

MP (L), fP (L) [GasserLeutwyler 87]:

Mπ(L) = Mπ

1 +

1

2Nfξg1(MπL) +O(ξ2)

fπ(L) = fπ

1− Nf

2ξ g1(MπL) +O(ξ2)

with ξ = M2

π/(4πFπ)2 = M2π/(8π

2f2π) and

16 24 32

L/a

0.14

0.15

0.16

0.17

0.18

0.19

aM

π ignored in final analysis

MπL=4M

πL=3

Du

rret

al.[

BM

W],

1011

.271

1

g1(z) =24

zK1(z) +

48√2zK1(√

2z) +32√3zK1(√

3z) +24

2zK1(2z) + ...

K1(z) =

√π

2ze−z

1 +

3

8z− 3 · 5

2(8z)2+

3 · 5 · 21

6(8z)3− 3 · 5 · 21 · 45

24(8z)4+ ...

• Unlike in the Symanzik case, here coefficients [1/(2Nf), Nf/2] are known.

• For realistic masses/volumes this expansion converges slowly [CD 03, CDH 05].


,Systematic uncertainties (3): theory for mud→ mphysud ,

• Memo: correct values of mud or ms not known a priori, must be dialed by

adjusting M2π/X

2 and (2M2K−M2

π)/X2 to their physical values.

• Memo: X=fπ,MΩ, ... is the scale setting quantity.

• In Nf =2+... QCD, input values must be corrected for isospin-breaking effects.

0 0.01 0.02 0.03 am

0

0.02

0.04

0.06

0.08

(amπ )2

mπ∼676 MeV

484

381

294

0 0.02 0.04 0.06 0.08 (amπ)2

0

0.02

0.04

0.06

aFπ

676

484381

294

M.L

usc

her

,P

oS(L

at20

05)

002

• ChPT is standard tool to extrapolate; now simulations at Mπ'135 MeV feasible.


,Systematic uncertainties (4): histogramming method,

Include all reasonable ansaetze, and count spread towards systematic uncertainty !

1.188 1.1885 1.189 1.1895 1.19 1.1905 1.191 1.1915 1.1920

1000

2000

3000

4000

5000

6000

7000

distribution of symm in continuum

linear

quadratic

1.188 1.189 1.19 1.191 1.1920

200

400

600

800

1000

1200

1400

1600

1800

2000

distribution of symm in continuum

Mka_10

Mka_30

Mka_50

Mka_70

Mka_90

Multitude of “final” results (i.e. at a=0, L=∞,mphysud , ...) may be weighted:

• equal weight

• weighting with dof/χ2

• weighting with p-value

• weighting with AIC •S. Durr, BUW/JSC EINN 2015, Paphos, 3.11.2015 8

,QCD spectroscopy (1): ground states,

0

500

1000

1500

2000M

[MeV

]

p

K

r K* NLSX D

S*X*O

experiment

widthinput

QCD

After a→0, L→∞,Mπ=135 MeV agreement with experiment [S. Durr et al., Science 322, 1224 (2008)]


,QCD spectroscopy (2): anatomy of mq calculation,

1. Decide on observables to be utilized for setting scale and quark masses, e.g.Mπ,MK,MΩ in Nf = 2+1 QCD, and look up their “experimental” values in aworld without isospin splitting and without electromagnetism [Mπ=134.8(3) MeV,MK=494.2(5) MeV according to FLAG].

2. For a given bare coupling β (yields a) tune bare masses 1/κud, 1/κs such that theratios Mπ/MΩ,MK/MΩ assume their physical values.

3. Read off 1/κud,s or determine bare amud,s via AWI and convert them (perturbati-vely or non-perturbatively) to the scheme of your choice (e.g. MS at µ=3 GeV).

4. Repeat steps 2 and 3 for at least three lattice spacings and extrapolate the resultto the continuum by means of Symanzik scaling.

In practice, step 2 involves inter-/extrapolations to the physical quark masses.

In practice, step 3 may be technically involved [RI/MOM, SF renormalization].

In practice, finite-volume corrections will to be included.


,QCD spectroscopy (3): Nf =3 RI-running extrapolation for ZS,

Evolution ZRIS (µ)/ZRI

S (4 GeV) has no visible cut-off effects among three finest lattices:separate continuum limit with RRI

S (µ, 4 GeV) = limβ→∞ ZRIS,β(4 GeV)/ZRI

S,β(µ).

0 10 20 30 40

µ2[GeV

2]

0.5

0.6

0.7

0.8

0.9

Z^SR

I (µ2)

β=3.8

β=3.7

β=3.61

β=3.5

β=3.31

10 20 30 40

µ2[GeV

2]

0.8

0.9

1

1.1

1.2

ZR

I

S,n

onper

t(µ2)/

ZR

I

S,4

-loop(µ

2)

µ=

4G

eV

β=3.8

Plots taken from BMW-collaboration paper [arXiv:1011.2403,1011.2711].On the finest lattice contact within errors to 4-loop PT for µ ≥ 4 GeV.


,QCD spectroscopy (4): excited states,

Charmonium spectroscopy withvarious source/sink wave functions(including open charm states),with disconnected contributionsand GEP technology

G. Bali, S. Collins, C. Ehmann [arXiv:1110.2381]


,QCD spectroscopy (5): scattering states,

J. Dudek, R. Edwards, C. Thomas [1212.0830] I = 1 P-wave ππ at Mπ = 390 MeV •


,QCD meson structure (1): leptonic decays,

Leptonic decays of pseudoscalar mesons, with P = π,K,D,Ds, B,Bs, Bc, ..., via

Γ(P → `ν`) =G2F

8π|Vij|2f2

Pm2`MP (1−m2

`/M2P )2

where experiment determines product |Vij| fP , but lattice can determine fP .

JCCµ = (u, c, t)γµ

1

2[1−γ5]

( d′s′

b′

) d′

s′

b′

=

Vud Vus VubVcd Vcs VcbVtd Vts Vtb

︸︷︷︸

VCKM

dsb

〈0|(uγµγ5d)(x)|π−(p)〉 = ifπpµeipx

〈0|(uγµγ5s)(x)|K−(p)〉 = ifKpµeipx


,QCD meson structure (2): semi-leptonic decays,

q ≡ p−p′ momentum transfer

〈π(p′)|sγµu|K(p)〉 = f0(q2)M2K−M2

π

q2qµ + f+(q2)

[(p+p′)µ −

M2K−M2

π

q2qµ

]

• Experiment can determine |Vus|f+(0), lattice can determine f+(0) [RBC/UKQCD].

• CKM matrix element Vus determined from leptonic K decay and from semi-leptonic• K → π decay must agree, if SM is correct.

• Strong dynamics restricted to QCD matrix elements 〈0|Aµ|π〉, 〈0|Aµ|K〉 and form• factors 〈π|Vµ|K〉 with specified external currents.

• Natural generalization to BSM processes (matrix elements).

•S. Durr, BUW/JSC EINN 2015, Paphos, 3.11.2015 15

,FLAG effort (1): methodology and observables,

For each quantity FLAG provides [see http://itpwiki.unibe.ch/flag]:

• complete list of references

• summary of essential ingredients of each study (Nf , action, ... )

• averages for “mature” quantities (separately for each Nf)

• pressure on reader to cite original papers

1 st edition [arXiv:1011.4408, Eur.Phys.J. C71 (2011) 1695] covers:

1. light quark masses mud,ms

2. Vus and Vud via decay constants and form factors

3. chiral low-energy constants (LECs)

4. kaon bag parameter BK

2nd edition [arXiv:1310.8555, Eur.Phys.J. C74 (2014) 2890] in addition:

5. D-meson decay constants and form factors

6. B-meson decay constants, mixing parameters, and form factors

7. strong coupling constant


,FLAG effort (2): color coding in FLAG-2,

FLAG-2 [arXiv:1310.8555, Eur.Phys.J. C74 (2014) 2890] definitions as follows:

Chiral extrapolation:F Mπ,min < 200 MeV 200 MeV ≤Mπ,min ≤ 400 MeV Mπ,min > 400 MeV

Continuum extrapolation:F 3 or more lattice spacings and at least 2 points below 0.1 fm / in scaling window 2 or more lattice spacings and at least 1 point below 0.1 fm / in scaling window otherwise

Finite-volume effects:F (Mπ)minL > 4 or at least 3 volumes (Mπ)minL > 3 and at least 2 volumes otherwise

Renormalization (where applicable):F non-perturbatively or 2-loop well-convergent PT 1-loop (or higher) PT with reasonable error estimate otherwise


,FLAG effort (3): light quark masses,

2 3 4 5 6

=+

=pheno.

MeVMaltman 01Narison 06Dominguez 09PDG

CP-PACS 01JLQCD 02QCDSF/UKQCD 04SPQcdR 05QCDSF/UKQCD 06ETM 07RBC 07JLQCD/TWQCD 08AETM 10BDürr 11our estimate for =MILC 04, HPQCD/MILC/UKQCD 04HPQCD 05CP-PACS/JLQCD 07RBC/UKQCD 08PACS-CS 08MILC 09MILC 09AHPQCD 09APACS-CS 09Blum 10RBC/UKQCD 10AHPQCD 10MILC 10APACS-CS 10BMW 10ALaiho 11PACS-CS 12RBC/UKQCD 12our estimate for = +

70 80 90 100 110 120

=+

=pheno.

MeVVainshtein 78Narison 06Jamin 06Chetyrkin 06Dominguez 09PDG

CP-PACS 01JLQCD 02QCDSF/UKQCD 04ALPHA 05SPQcdR 05QCDSF/UKQCD 06ETM 07RBC 07ETM 10BDürr 11ALPHA 12our estimate for =MILC 04, HPQCD/MILC/UKQCD 04HPQCD 05CP-PACS/JLQCD 07RBC/UKQCD 08PACS-CS 08MILC 09MILC 09AHPQCD 09APACS-CS 09Blum 10RBC/UKQCD 10AHPQCD 10PACS-CS 10BMW 10ALaiho 11PACS-CS 12RBC/UKQCD 12our estimate for = +

• inconsistencies only among red results

• for each Nf all green points nicely consistent

• mphysud and mphys

s depend only mildly on Nf , ratio essentially Nf -independent

• reasonable consistency with non-lattice results


,FLAG effort (4): chiral low-energy constants,

200 250 300 350

=+

+=

+=

MeVJLQCD/TWQCD 07AJLQCD/TWQCD 08ACERN 08ETM 08ETM 09CJLQCD/TWQCD 10Bernardoni 10TWQCD 11ATWQCD 11ETM 12ETM 13Brandt 13

our estimate for =

RBC/UKQCD 08JLQCD/TWQCD 08BTWQCD 08MILC 09MILC 09A [SU(2) fit]MILC 09A [SU(3) fit]JLQCD 09RBC/UKQCD 10AJLQCD/TWQCD 10MILC 10ABorsanyi 12BMW 13

our estimate for = +

ETM 13

our estimate for = + +

/

3.5 4.0 4.5 5.0 5.5

=+

+=

+=

Gasser 84Colangelo 01

JLQCD/TWQCD 08AETM 08JLQCD/TWQCD 09ETM 09CTWQCD 11Bernardoni 11Brandt 13Gülpers 13

our estimate for =

RBC/UKQCD 08MILC 09AMILC 10ANPLQCD 11Borsanyi 12RBC/UKQCD 12BMW 13


ETM 10

our estimate for = + +

• inconsistency between RBC/UKQCD 10A and MILC 10A in case of Σ for Nf = 2+1

• discrepancy resolved by inflated FLAG error (−→ less precise than the better calc.)

• no warning through inconsistency in case of ¯4, but why non-monotonic in Nf ?

• scarcity of non-lattice results


,FLAG effort (5): fD and fDs,

180 200 220 240

=+

+=

+=

ETM 09

ETM 11A

ETM 13B

our average for =

FNAL/MILC 05

HPQCD/UKQCD 07

HPQCD 10A

PACS-CS 11

FNAL/MILC 11

HPQCD 12A

our average for = +

FNAL/MILC 12B

FNAL/MILC 13

ETM 13F

230 250 270

=+

+=

+=

MeV

ETM 09

ETM 11A

ETM 13B

our average for =

FNAL/MILC 05

HPQCD/UKQCD 07

HPQCD 10A

PACS-CS 11

FNAL/MILC 11

HPQCD 12A

our average for = +

FNAL/MILC 12B

FNAL/MILC 13

ETM 13F

1.10 1.15 1.20 1.25

=+

+=

+=

ETM 09ETM 11AETM 13B

our average for =

FNAL/MILC 05HPQCD/UKQCD 07PACS-CS 11FNAL/MILC 11HPQCD 12A

our average for = +

FNAL/MILC 12BFNAL/MILC 13ETM 13F

/

• several good-quality calculations available or underway

• roughly fDs/fD ' fK/fπ ' 1.19(1)

• so far consistent with CKM unitarity (cf. next slide)


,FLAG effort (6): D → K and D → π form factors,

0.55 0.65 0.75

=+

+

=+

=

ETM 11B

FNAL/MILC 04

HPQCD 11 / 10B

our average for = +

+ ( )

0.65 0.70 0.75 0.80

=+

+=

+=

ETM 11

FNAL/MILC 04

HPQCD 10B

our average for = +

+ ( )

0.20 0.22 0.24

=+

=

CKM unitarity

neutrino scattering

ETM 13B

our average for =

HPQCD 11/10B

FNAL/MILC 11

HPQCD 12A/10A

our average for = +

| |

0.95 1.05

=+

+=

+=

CKM unitarity

neutrino scattering

ETM 13B

our estimate for =

HPQCD 10B

FNAL/MILC 11

HPQCD 10A


| |

• semi-leptonic / leptonic comparison tests CKM paradigm

• good precision challenging, better proposal for q2 available [FNAL/MILC, HPQCD]

• so far no threat for CKM unitarity


,QCD baryon structure (1): lattice technology,

Lots of interesting topics [1406.4310]:

• nucleon sigma terms

• rms charge radius of proton

• transition form factors, GPDs, ...

• signal for strange/charmed baryons

→ talk by Giannis Koutsou


,QCD baryon structure (2): from quarks to nuclei,

HalQCD collaboration [1408.4892]:nuclei from quarks via Bethe-Salpeter

• NN -potential for S,D-waves• at Mπ ' 470 MeV

• Single particle levels in 16O and 40Ca• at Mπ ' 470 MeV

• E versus A of doubly magic nuclei• follows Bethe-Weizsacker (straight line)

→ talk by Takashi Inoue •S. Durr, BUW/JSC EINN 2015, Paphos, 3.11.2015 23

,QCD thermodynamics (1): transition temperature,

• At physical values of mud,ms, ... QCD shows crossover.

• Transition reasonably sharp, hence pseudo-critical Tc meaningful concept.

• Tc depends on observable, e.g. P , 〈ψψ〉, ..., but each one is well-defined.

! !!!!!!!!

!!!!!! ! !!!

""

""""

"

""

"""""" "

!!!!!!!

!!!!! !

! ! ! !

##

##

#

#

##!!""!!

ContinuumNt"16Nt"12Nt"10Nt"8

100 120 140 160 180 200 220

0.2

0.4

0.6

0.8

1.0

T !MeV"

#l,s

! !!!!!!!!

!!!!!!

! !!!

""

"""""

"""""""

" "

!!!!!!!

!!!! !

! !! ! !

####

##

##!!""!!

ContinuumNt"16Nt"12Nt"10Nt"8

100 120 140 160 180 200 2200.0

0.1

0.2

0.3

0.4

T!MeV"#$$%R

S.B

orsa

nyi

etal

,ar

Xiv

:100

5.35

08


,QCD thermodynamics (2): equation of state,

• Apparent discrepancy for trace anomaly• ε− 3p between Wuppertal-Budapest• and HotQCD in the past.

• Apparent discrepancy disappeared with• improved HotQCD action and analysis.

A. Bazavov et al [HotQCD], 1407.6387


,QCD thermodynamics (3): troubles with µbary 6= 0,

At non-zero baryon density (equivalent: chemicalpotential µ 6=0) the fermion determinant becomescomplex, which creates a major difficulty to theconcept of importance sampling.

A clear establishment/measurement of a second-order endpoint would be a major leap forward.

0 200 400 600 800 1000 1200 1400 1600 1800 2000

µ (MeV)

0

20

40

60

80

100

120

140

160

180

200

T (

MeV

)

© 2012 Andreas Kronfeld/Fermi Natl Accelerator Lab.

hadrons

quark-gluon plasma

n

other phases plo

tby

A.K

ron

feld

In QCD many approaches to solve the sign problem have been tried:

• absorb phase in observable [ancient]• two-parameter reweighting from µ=0 [Fodor Katz]• work at imaginary µ and continue [Philipsen deForcrand]• compute Taylor coefficients at µ=0

In QCD-inspired models many tricks/reformulations become possible.

•


Summary

• LQCD is the full definition of QCD, requires 1+Nf input observables

• Systematics non-trivial, in particular when matching to EFT-framework

• Main topics in T = 0 studies:

spectroscopy, quark masses, CKM elements, LECs in ChPT, hadron structure, ...

FLAG compilations [e.g. arXiv:1310.8555] serve as interface to phenomenology

• Main topics in T > 0 studies:

pseudocritical Tc, EoS, transport coefficients, sign problem at µbary 6= 0, ...

• New frontier of including electromagnetic interactions


,Outlook (1): simulations with electromagnetism,

• 2002-20??:

Nf = 2+1 QCD requires 3 polished input values [e.g. Mπ,MK,MΩ in theory withmu,md → 1

2(mu+md) and e→ 0]

−→ isospin analysis suggests Mπ=134.8(3)MeV,MK=494.2(5)MeV [FLAG]

• 2010-2???:

Nf = 2+1+1 QCD requires 4 polished input values [ditto and MDs in theory withmu,md → 1

2(mu+md) and e→ 0]

−→ charm unquenched, but no conceptual change on isospin issue

• 2014-????:

Nf = 1+1+1+1 QCD requires 5 input variables [e.g. Mπ±,MK±,MK0,MDs,MΩ]

−→ requires disconnected contribution to flavor-singlet quantities−→ analysis of π0-η-η′ mixing mandatory to extract physical masses−→ QED and QCD renormalization intertwined (ms/md is RGI, mu/md is not)

−→ final word on mu?=0 [in QCD+QED] will be possible


,Outlook (2): disentangling mass splittings,

Simulate Nf =1+1+1+1 QCD with non-compact QED. Use quark charges 2e/3 and−1e/3 with e varying between 0 and 1.41.

After algorithmic challenges are solved and finite-volume effects are under control,determine physical isospin splittings on each ensemble, and extrapolate results a→ 0,L→∞, mq → mphys

q for q ∈ d, u, s, c, αem → 1/137.036.

Compare to exp, test Coleman-Glashow relation ∆CG ≡ ∆MN −∆MΣ + ∆MΞ = 0.

∆qmMN

=+

2.52

(17)

(24)

MeV

∆emMN

=−

1.0

0(07

)(14

)M

eV

∆la

tMN

=+

1.51

(16)

(23)

MeV

∆expMN

=+

1.2

9333

21(4

)M

eV

0

2

4

6

8

10

ΔM

[MeV

]

ΔN

ΔΣ

ΔΞ

ΔD

ΔCG

ΔΞcc

experimentQCD+QEDprediction

BMW 2014 HCH BM

Wco

llab

orat

ion

,S

cien

ce34

7(2

015)

1452


Lattice QCD and its interface to phenomenology · Lattice QCD and its interface to phenomenology...

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