Yu Maezawa (YITP, Kyoto University)

with Peter Petreczky (Brookhaven National Lab.)

KEK:HadronandNuclearPhysicsin2017,Jan.7-10

PRD94 (2016) 034507 (1606.08798)

Heavymesoncorrela?onfunc?on

Longdistance+experimentsShortdistance+perturba?on andat�s(µ) mc(µ) µ = mc

welltuned&mc/ms , mb/mcmlatq

1.  Introduc?ontolaLceQCD

2.  Longdistance+experimentaldata

3.  Shortdistance+perturba?ontheory

4.  Summary

welltuned&mc/ms , mb/mcmlatq

andat�s(µ) mc(µ) µ = mc

FundamentalparametersofQCD:

Determina?on:mostelementalquestinQCD

�s, mu, md, ms, mc, mb, mt

36 9. Quantum chromodynamicsτ-d

ecayslattice

structu

refu

nctio

ns

e+e

- ann

ihilatio

n

hadron collider

electroweakprecision fits

Baikov

ABMBBGJR

MMHT

NNPDF

Davier

PichBoitoSM review

HPQCD (Wilson loops)

HPQCD (c-c correlators)

Maltmann (Wilson loops)

JLQCD (Adler functions)

Dissertori (3j)

JADE (3j)

DW (T)

Abbate (T)

Gehrm. (T)

CMS (tt cross section)

GFitter

Hoang (C)

JADE(j&s)

OPAL(j&s)

ALEPH (jets&shapes)

PACS-CS (vac. pol. fctns.)

ETM (ghost-gluon vertex)

BBGPSV (static energy)

Figure 9.2: Summary of determinations of αs(M2Z) from the six sub-fields

discussed in the text. The yellow (light shaded) bands and dashed lines indicate thepre-average values of each sub-field. The dotted line and grey (dark shaded) bandrepresent the final world average value of αs(M2

Z).

whereby the dominating contributions to the overall error are experimental (+0.0017−0.0018), from

parton density functions (+0.0013−0.0011) and the value of the top quark pole mass (±0.0013).

February 10, 2016 16:30

reviewinPDG(2015)

Precision:newphysicsfromStandardModel

e.g.)Higgsbranchingra?o1404:0319 h� bb̄, cc̄, �+��, gg, ��, WW �, ZZ�

�s, mb, mc, �, m� , mW , mZ

10-4--10-5 levelcomparablelevel:desired

Allparam.:definedinperturba?vescheme

pQCD+

experimentslaLceQCDsimula?ons…

R(Q) = �(e+e� � hadron)/�(e+e� � µ+µ�)

LaLceQCDsimula?ons�s(µ) and mq(µ)

Experimentslow~MeV(spectra)high~G-TeV(collision)

non-perturba5veperturba5ve

Theory

scale

G(t) = a6�

x

(amc0)2�j5(x, t)j5(0, 0)�Current-currentcorrela?onfunc?on

j5 = q̄�5q

�c�c

Shortdistance+perturba?ontheory

Longdistance+experimentaldata

distance

G(t) = a6�

x

(amc0)2�j5(x, t)j5(0, 0)�Current-currentcorrela?onfunc?on

j5 = q̄�5q

�c�c

Shortdistance+perturba?ontheory

Longdistance+experimentaldata

distance

LaLceQCDsimula?ons�s(µ) and mq(µ)

Experimentslow~MeV(spectra)high~G-TeV(collision)

non-perturba5veperturba5ve

Theory

scale

andat

�s(µ) mc(µ)µ = mc

welltuned&mc/ms , mb/mc

mlatq

:Strongnon-linearityandinfinite-dimensionalintegralQCD

regularizedonlaLceinEuclideanspace-?me

Monte-Carlosimula?onsbasedonimportancesampling

�O� =1Z

�Dq̄DqDAO(q̄, q, A) e�SQCD

=1

Nconf

Nconf�

{Ui}

O(Ui)±O(1�

Nconf)

1)genera?ngconfigura?ons{Ui}withweight

2)numericalaveragewithsta?s?calerrorateachlaLcespacinga

3)extrapola?on:physicalvalueincon?nuumlimit

e�SQCD

�O�a

lima�0

�O�a

Ø  laLcear?facts:discre?za?on(quark∝O(a),doubler,…)

sophis?catedimprovementsandcarefulextrapola?ons:important

Prac?calsetup

ü  2+1flavorQCD:up/down&strangequarksasdynamical(sea)charmandbobomquarksonlyasvalenceü  fixingup/downquarkmasses:ml=ms/20

Genera?ngconfigura?ons(step1) HotQCDColl.PRD85(2012)054503PRD90(2014)094503

� Lattice TU a�1 [GeV]6.740 484 8K 1.816.880 484 8K 2.077.030 484 10K 2.397.150 483 � 64 8K 2.677.280 483 � 64 8K 3.017.373 483 � 64 9K 3.287.596 644 9.5K 4.007.825 644 9.5K 4.89

*moleculardynamic?meunits

*Ø  HighlyImprovedStaggeredQuarksØ  Tree-levelimprovedgluonsØ  Strangequarkms:fixedonphysicalLineofconstantphysics

Ø  Scale:viapiondecayconstant

r1 = 0.3106(18) fm

� = 10/g20

�c�c

Shortdistance+perturba?ontheory

distanceLongdistance+experimentaldata

Correla?onfunc?on

Groundstate:dominantatlargeτ1stexcitedstate(m1>m0)

G(�) =�

x

(amc0)2�j5(x, t)j5(0, 0)� = A0e�m0� + A1e

�m1� + · · ·

Groundstate:dominantatlongdistance()� ��welltuned&mc/ms , mb/mcmlat

q

unmixedetaspinaveragedmassBobomoniumss̄

M�ss̄ =�

2M2K �M2

� M�bM =14

�3MJ/� + M�c

�

Tuningquarkmasses

M2�ss̄

= B ms M = d + b mc M�h = dh + bh mh

(mc < mh <� mb)

StrangeCharmBobom

ExperimentalvaluesfromPDG

uncertain?es:absenceofEMeffectsanddisconnecteddiagram

mlatcmlat

s mlatb

M�b = 9.398(3) GeVM�ss̄ = 686.00(92) MeV M = 3.067(3) GeVPRD70,114501 PRD83,074504

determinedwithin0.4--0.7%errors

fixedonphysicalone,donebyHotQCD

es?matedbyextrapola?onfromtoamh < 1.0 mb

mc/ms

ü  a2anda2+a4 fitswelldoneandshowsimilarresults

Scheme&scaleindependent:mlat

c /mlats = mc/ms

combining&M2�ss̄

= B msM = d + b mc

M = 34MJ/� + 1

4M�c

LaLcevs.experiments

MPS =2.982(12) GeVMV =3.095(12) GeV

2.95

3

3.05

3.1

3.15

0 0.1 0.2 0.3 0.4 0.5

a2 [GeV

-2]

MV = 3.095(12) [GeV]

MPS = 2.982(12) [GeV]

[GeV]

2.95

3

3.05

3.1

3.15

0 0.1 0.2 0.3 0.4 0.5 2.95

3

3.05

3.1

3.15

0 0.1 0.2 0.3 0.4 0.5 2.95

3

3.05

3.1

3.15

0 0.1 0.2 0.3 0.4 0.5

MV

MPS

experiments

HyperfinespliLng

Experiments:

MV �MPS = 113.5(18)(7) MeV

MJ/� �M�c = 113.3(6) MeV

LaLce:~1.7%

~0.5%

mb/mc

ü  onlyfourfinestβpointsü  systema?cerrorduetoextrapola?onto:grayshadowM�b

Interpola?onorextrapola?onfrom0.7 � amh � 0.9

Difficultduetoheavymass:extrapola?onsfromamh < 1.0 < amb

M = d + b mcM�h = dh + bh mh &

M�h [GeV]

�c�c

Shortdistance+perturba5ontheory

distanceLongdistance+experimentaldata

G(t) = a6�

x

(amc0)2�j5(x, t)j5(0, 0)�

Gn =�

t

(t/a)n G(t) :n-thorderwith

ReducedmomentsusefultocancellaLcear?facts,withmomentsoffreecorrelators:Rn =

�Gn

G(0)n

�1/(n�4)

G(0)n

Rn =�

r4 (n = 4)rn ·

�mlat

c /mc(µ)�

(n � 6) ,

LaLceincon?nuum Perturba?on

rn = 1 +3�

j=1

rnj(µ,mc)�js(µ)

Numericalnumbersatµ = mc (Nf = 4)

Effectsofcharmloops:0.7%forR4es?matedinperturba?ontheoryPRD78,054513correctlaLceR4inour2+1flavorsimula?ons

HPQCD(2008)(2010)(2015)

typicalscaleofcorrelators

tnG(t)

G(t) = a6�

x

(amc0)2�j5(x, t)j5(0, 0)�

Gn =�

t

(t/a)n G(t) :n-thorderwith

ReducedmomentsusefultocancellaLcear?facts,withmomentsoffreecorrelators:Rn =

�Gn

G(0)n

�1/(n�4)

G(0)n

Rn =�

r4 (n = 4)rn ·

�mlat

c /mc(µ)�

(n � 6) ,

LaLceincon?nuum Perturba?on

rn = 1 +3�

j=1

rnj(µ,mc)�js(µ)

Numericalnumbersatµ = mc (Nf = 4)

Effectsofcharmloops:0.7%forR4es?matedinperturba?ontheoryPRD78,054513correctlaLceR4inour2+1flavorsimula?ons

HPQCD(2008)(2010)(2015)

typicalscaleofcorrelators

tnG(t)

EPJC48(2006)107NPB824(2010)1Vacuumpolariza?onfunc?on

q2�PS(q2) = i

�dx eiqx�j5(x, t)j5(0, 0)� = q2 3

16�2

�

n>0

Cnzn z = q2/4m2

Perturba?oninMSschemeknownupto4-loop()�3s

Cn = C(0)n +

�s(µ)�

�C(10)

n + C(11)n lm

�

+�

�s(µ)�

�2 �C(20)

n + C(21)n lm + · · ·

�+

��s(µ)

�

�3 �C(30)

n + C(31)n lm + · · ·

�· · ·

lm = log(m2(µ)/µ2)

a0

a1a2

a3

a4

a5

a6

a7=

∑

bi,r0

c̃bi,0,r0

1 (d)r0b1

b2

b3

b4

b5+

∑

bi,r2

c̃bi,2,r2

1 (d)r2b1

b2

b3

b4

b5

+∑

bi,r0

c̃bi,0,r0

2 (d) r0

b1

b2

b3

b4

b5

+∑

bi,r2

c̃bi,2,r2

2 (d) r2

b1

b2

b3

b4

b5

Fig. 1. The figure above shows an example of eq. (17) for a four-loop tadpole. Notethat on the right hand side besides massless (dashed) lines and lines with mass m

(solid) also a line with mass 2m (double line) appears which is not present in theinitial integral. The next step would be to repeat the procedure for the two-loopself energy.

where Pm0(q2) was written as Pk(q2, m) with m0 = km. By convention we

choose rk ≥ 0 for k ̸= 0 so that the decomposition (15) is unique. Afterapplying these transformations the integral T takes the form

T =∫

ddq∑

p

Rp(q2, m)∑

z

∑

k

∑

rk

c̃p,k,rkz (d)P rk

k (q2, m)Sz(q2, m), (16)

which can be written more conveniently as

T =∑

k

∑

z

∑

p

∑

rk

c̃p,k,rkz (d)T k

z (ρp, rk, m) (17)

with

T kz (ρp, rk, m) =

∫

ddq Rp(q2, m)P rkk (q2, m)Sz(q

2, m). (18)

ρp denotes the set of propagator powers of the rest graph. That means theinitial integral T is expressed as a linear combination of integrals of the typeT k

z (ρp, rk, m) in which the self energy insertions appear only as master integralsand all cross talking momenta between the self energy and the rest graph areremoved. Figure 1 illustrates eq. (18) for a four-loop vacuum diagram.

The next step is to construct Integration-by-Parts identities for the integralsT k

z in which the self energy insertions are treated as objects depending onlyon their external momenta. The identities have the form

0 =∫

ddq∂

∂kµ

ℓµRp(q2, m)P rk

k (q2, m)Sz(q2, m) (19)

= δkℓ d T kz (ρp, rk, m) +

∫

ddq[

ℓµ∂

∂kµ

]

IRp(q2, m)P rk

k (q2, m)Sz(q2, m)

= δkℓ d T kz (ρp, rk, m)

+∫

ddq

(

[

ℓµ∂

∂kµ

]

IRp(q2, m)

)

P rkk (q2, m)Sz(q

2, m)

6

Gn =gn(�s(µ), µ/mc)

(amc(µ))n�4

Momentswith g2k+2 = (2m(µ))2k 12�2

k!

��

�q2

�k

�PS(q2)

�����q2=0

G(t) = a6�

x

(amc0)2�j5(x, t)j5(0, 0)�

Gn =�

t

(t/a)n G(t) :n-thorderwith

ReducedmomentsusefultocancellaLcear?facts,withmomentsoffreecorrelators:Rn =

�Gn

G(0)n

�1/(n�4)

G(0)n

Rn =�

r4 (n = 4)rn ·

�mlat

c /mc(µ)�

(n � 6) ,

LaLceincon?nuum Perturba?on

rn = 1 +3�

j=1

rnj(µ,mc)�js(µ)

Numericalnumbersatµ = mc (Nf = 4)

Effectsofcharmloops:0.7%forR4es?matedinperturba?ontheoryPRD78,054513correctlaLceR4inour2+1flavorsimula?ons

HPQCD(2008)(2010)(2015)

typicalscaleofcorrelators

tnG(t)

EPJC48(2006)107NPB824(2010)1Vacuumpolariza?onfunc?on

q2�PS(q2) = i

�dx eiqx�j5(x, t)j5(0, 0)� = q2 3

16�2

�

n>0

Cnzn z = q2/4m2

Perturba?oninMSschemeknownupto4-loop()�3s

Cn = C(0)n +

�s(µ)�

�C(10)

n + C(11)n lm

�

+�

�s(µ)�

�2 �C(20)

n + C(21)n lm + · · ·

�+

��s(µ)

�

�3 �C(30)

n + C(31)n lm + · · ·

�· · ·

lm = log(m2(µ)/µ2)

a0

a1a2

a3

a4

a5

a6

a7=

∑

bi,r0

c̃bi,0,r0

1 (d)r0b1

b2

b3

b4

b5+

∑

bi,r2

c̃bi,2,r2

1 (d)r2b1

b2

b3

b4

b5

+∑

bi,r0

c̃bi,0,r0

2 (d) r0

b1

b2

b3

b4

b5

+∑

bi,r2

c̃bi,2,r2

2 (d) r2

b1

b2

b3

b4

b5

Fig. 1. The figure above shows an example of eq. (17) for a four-loop tadpole. Notethat on the right hand side besides massless (dashed) lines and lines with mass m

(solid) also a line with mass 2m (double line) appears which is not present in theinitial integral. The next step would be to repeat the procedure for the two-loopself energy.

where Pm0(q2) was written as Pk(q2, m) with m0 = km. By convention we

choose rk ≥ 0 for k ̸= 0 so that the decomposition (15) is unique. Afterapplying these transformations the integral T takes the form

T =∫

ddq∑

p

Rp(q2, m)∑

z

∑

k

∑

rk

c̃p,k,rkz (d)P rk

k (q2, m)Sz(q2, m), (16)

which can be written more conveniently as

T =∑

k

∑

z

∑

p

∑

rk

c̃p,k,rkz (d)T k

z (ρp, rk, m) (17)

with

T kz (ρp, rk, m) =

∫

ddq Rp(q2, m)P rkk (q2, m)Sz(q

2, m). (18)

ρp denotes the set of propagator powers of the rest graph. That means theinitial integral T is expressed as a linear combination of integrals of the typeT k

z (ρp, rk, m) in which the self energy insertions appear only as master integralsand all cross talking momenta between the self energy and the rest graph areremoved. Figure 1 illustrates eq. (18) for a four-loop vacuum diagram.

The next step is to construct Integration-by-Parts identities for the integralsT k

z in which the self energy insertions are treated as objects depending onlyon their external momenta. The identities have the form

0 =∫

ddq∂

∂kµ

ℓµRp(q2, m)P rk

k (q2, m)Sz(q2, m) (19)

= δkℓ d T kz (ρp, rk, m) +

∫

ddq[

ℓµ∂

∂kµ

]

IRp(q2, m)P rk

k (q2, m)Sz(q2, m)

= δkℓ d T kz (ρp, rk, m)

+∫

ddq

(

[

ℓµ∂

∂kµ

]

IRp(q2, m)

)

P rkk (q2, m)Sz(q

2, m)

6

Gn =gn(�s(µ), µ/mc)

(amc(µ))n�4

Momentswith g2k+2 = (2m(µ))2k 12�2

k!

��

�q2

�k

�PS(q2)

�����q2=0

n rn1 rn2 rn3

4 0.7427 0.0088 �0.02966 0.6160 0.4976 �0.09298 0.3164 0.3485 0.0233

10 0.1861 0.2681 0.0817

Atµ = mc ,

ü  datawellextrapolatedby a2anda2+a4 fits

ü  discre?za?oninHISQ:�sa2

Severalfitswithboostedcouplinguncertaintyduetocon?nuumextrap.

c.f.)R4=1.281(5) HPQCD’08 R4=1.282(4) HPQCD’10

�bs(1/a) =

14�

g20

u40

R4 = 1.2743(40)(47)Fromweobtaincouplingconstant:inthelowestenergyscalesofar

“trun.”:trunca?onerrorscomingfromterms�4s

R4 = r4(�s;µ = mc)

ü  Linear-likebehaviorü  uncertaintyduetocon?nuumextrap.

nosignificantdependence

From

weobtaincharmquarkmass:

mc(mc) =r6(�s;µ = mc)

R6/mlatc

R6

mlatc

= 1.0191(27)

PRD94 (2016) 034507 (1606.08798)

G(t) = a6�

x

(amc0)2�j5(x, t)j5(0, 0)�Current-currentcorrela?onfunc?on

j5 = q̄�5q

�c�c

Shortdistance+perturba?ontheory

Longdistance+experimentaldata

distance

PRD94 (2016) 034507 (1606.08798)

G(t) = a6�

x

(amc0)2�j5(x, t)j5(0, 0)�Current-currentcorrela?onfunc?on

j5 = q̄�5q

�c�c

Shortdistance+perturba?ontheory

Longdistance+experimentaldata

distance

�s = 0.3945(85)mc = 1.267(12) GeVat µ = mc, nf = 3

mc/ms = 11.877(91)mb/mc = 4.528(57)

welltunedmlatq

mesonspectra moments

evolvingwith4-loopsPTinMSscheme:RunDeCpackage

~0.72%~1.8%~0.93%~2.1%

ü  Determina?onwith10-2-10-3precisions:achievedü  charmquark:treatedcompletelyonlaLceü  bobomquark:applicableinfuture

�s(MZ , nf = 5) = 0.11622(84)ms(µ = 2GeV, nf = 3) = 92.0(1.7) MeVmc(µ = mc, nf = 3) = 1.267(12) GeVmb(µ = mb, nf = 5) = 4.184(89) GeV

mc/msmb/mc

PRD94 (2016) 034507 (1606.08798)

1.2 1.25 1.3 1.35

mc(mc)

Nf = 2

Nf = 2 + 1

Nf = 2 + 1 + 1

HPQCD’15

ETMC’14

our stury

JLQCD’16

χQCD’15

HPQCD’10

ETMC’10

mc(mc)�s(MZ)

**

*

ü  basiclaLceapproach:same(tunemqinnon-p->pQCD)ü  setup:different(ac?on,dynamicalquarks,parameters,pQCD,…)ü  *sameconfs.,butdifferentpQCD:QQ-barpoten?alandmomentsü  **samepQCD,butdifferentconfs.laLce~1.7%(10-2)uncertain?es…