Download - Yu Maezawa (YITP, Kyoto University) - KEK
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…