Hadronic contributions to
the muon anomalous
magnetic moment
Tom Blum (UConn/RBRC)
20th Lomonosov Conference on Elementary Particle Physics
Moscow State University/Zoom
August 21, 2021

Anomalous moment for the muon, Cern Courier May/June 2021

Credit: Mark Rayner/CERN

Muon g-2 experiments
CERN (1959-1979)

BNL E821(1997-2001)

FNAL E989 (2018-)
2

Conclusions

Introduction

Framework

Challenges

Window

Con

Overview• We have determined aµ to an unprecedented 460 ppb precision!
B. Toth, Lattice 2021

• The Run 1 result

BNL g-2
FNAL g-2– 6% of ultimate data sample

1.5 σ– 15% smaller error than BNL

BMWlattice QCD
Standard Model– 3.3s tension with SM

Experimental
Average

4.2 σWhite Paper
Standard Model

17.5

18

18.5

19
19.5
20
aµ × 109 – 1165900

20.5

21

21.5

• After 20 years, we confirm
experimental
2.1 σ thehigherBNL
than R-ratio
value [WP’20] results!
Consistent with experiment within 1.5 σ• Combining BNL/FNAL andcomparing to theory à 4.2s tension
30th Jul 20213

4/7/2021

Chris Polly | Muon g-2 Announcement

B. C. Tóth

Muon g-2: BMW calculation of HVPn (there are many off-shell form factors that enter which can not be measured). W
ations exist (see [9] for a summary), they are not systematically improvable. A dlatticeStandard
QCD where
all errors
areQED+EW+QCD
controlled is therefore desirable.
Model
Theory:n Sec. 2 we review the status of lattice calculationsof aµ (HVP). Section
3 is a
✓
◆
2)
F
(q
2
results for aµ (HLbL)
frameworkhµ(~p 0 )|J⌫computed(0)|µ(~p )i =in thee ū(~
p 0 ) F1 (q 2 ) ⌫ of
+ i lattice[ QCD+QED.
p ) Section⌫ , ⇢ ]q⇢ u(~4m
0usions and outlook for futureacalculations.
⌘
(g
2)/2
=
F
(0)
(q
=
p
p)
µ
2
…

Z

Z

…
...

W

…

1: Representative diagrams, up to order

3 , in the Standard Model that contribute4

:nmscmsr jhrsr SuShjSajc Ss DbhcmbcChpcbs

Ogxrhbr Qdonqsr

Muon g-2 Theory Initiative
intpmSj gnlcoSec/ vvv cjrcuhcp bnl.jnbSsc.ogwrpco(2006.04822[hep-ph])l_fmdshb lnldms ne sgd ltnm hm sgd Rs_mc_qc
Sgd _mnl_kntr
Lncdk

4
6+7⇥1011
Contribution
Value
S- @nx_l_ 0+1+2 + M- @rltrrdm 3 + L-Section
Adm_xntmEquation+ I- Ahimdmr 5 + S- Aktl+

116592061
L- Aqtmn+ H-E821+E989:
B_oqhmh + B-L- B_qknmh
B_k_ldEq.(41)+ (8.13)
L- BçExperiment(E821)
8

0/00

8+01+02

×

References

03 +×
0.35
+ 116
F- Bnk_mfdkn+ [1]592PPM089(63) Ref.

04+05
E- Btqbh_qdkkn+ G- Byxś 06 + H- C_mhkjhm 01 + L- C_uhdq 07 + + B-S-G- C_uhdr 08 ++HVP LO (e e )Sec.× 2.3.7 Eq. (2.33)
6931(40) Refs. [2–7]×
L- Cdkk_+ Lnqsd 1/ + R-H- Dhcdkl_m 10+11 + + @-W- Dk,Jg_cq_ 12+13 + + @- Fèq_qchm 14 +HVP NLO (e e15)+16Sec. 2.3.8 Eq. (2.34)
98.3(7) Ref.
17
18
2/
03 [7]
CFhtrsh+
LFnksdql_m+
RsdudmFnsskhda+
UFúkodqr+
EG_fdkrsdhm+
HVP NNLO (e e ) 20+1
Eq. 22(2.35)
12.4(1) Ref. +[8]21 Sec. 2.3.8
23
Gdqcnëy_ + Sec.
C-V-3.5.1
Gdqsynf+ @- Gndbjdq + 7116(184) Refs. [9–17]
HVPLLOG_x_j_v_(lattice, udsc)03++24FEq. (3.49)
+×
25
01+02
L-(phenomenology)
Gnedqhbgsdq+ A-,K- Gnhc Sec.
+ Q-I-4.9.4
Gtcrohsg+ E- Hfm_snu 10 + 92(19) Refs. [18–30]
HLbL
Eq.
(4.92)
26+7
6+7
28
3/+30
S- NLO
Hytatbgh+ E- Idfdqkdgmdq 27 + KIhm4.8+ @- Jdrg_u_qyh+ S- Jhmnrghs_2(1)+ Ref. [31]HLbL(phenomenology)Sec.
Eq. (4.91)
25
10
31+32
03
15+26 +×
14
A- (lattice,
Jtahr uds)
+ @- Jtohbg + @- Jtoőý Sec. 5.7+ K- K_ta+ B- Kdgmdq+ K- Kdkkntbg+ [32]HLbL
Eq. (5.49)
79(35) Ref.
10
4
33+34
35+36
H- Knf_rgdmjn
L_k_drbt + J-Sec.
L_ksl_m+ L-J- L_qhmjnuhý 90(17)+HLbL(phenomenology++A-lattice)8
Eq. (8.10)
Refs. [18–30, 32]
37+38
26
01+02
0 +×
01+02+2
O- L_rit_m+ @-R- Ldxdq + G-A- Ldxdq+ S- Lhad + J- Lhtq_+
QED
Sec.41
6.5
Eq. (6.30) 01 +×116 584 718.931(104)
Refs. [33, 34]
4/
1+40
+42
01
R-D- Lúkkdq + L- Mhn+ C- Mnltq_+ @- Mxeedkdq+ U- O_rb_ktsr_ +Electroweak
Sec. 7.4
Eq. (7.16)
153.6(1.0) Refs. [35, 36]
L- +O_rrdq_ 43 + D- Odqdy cdk Qhn 44 + R- Odqhr 37+38 + @- Onqsdkkh 2/ + L- Oqnbtq_ 45 +HVP (e e , LO + NLO+ NNLO) 46 +× Sec. 8
Eq. (8.5)38
6845(40)
Refs. [2–8]01
10B-EQdcldq+
A-KQnadqsr+
ORàmbgdy,Otdqs_r+
RRdqdcmx_jnu+
HLbL (phenomenology+ lattice 16+ NLO) Sec. 8 47 Eq. (8.11)
92(18) 48Refs. [18–32]
1047
ARgv_qsy+
RRhltk_+CRsóbjhmfdq+
GRsóbjhmfdq,Jhl+
ORsneedq Refs.
+ [2–8, 18–24, 31–36]
Total SM Value 5/ +×
Sec. 8Eq. (8.12) 01+02 116 591 810(43)1350
S- Sdtamdq+ Q- U_m
cd V_sdq Sec.
+ L-8U_mcdqg_dfgdm+ F- Udm_mynmh+Di↵erence:aµ := aexp
aSM
Eq. (8.14)279(76)
µ01
µ
01+02
07
10
51
F- unm Ghoodk + G- Vhsshf+ Y-E821+E989:
Yg_mf + L-M- @bg_rnu+ @A_rghq +0.37 PPM
251
(59)
36
52
01
14
53+54
M- B_qcnrn contributions+ A- Bg_jq_anqsx+ D-,GBg_n + I-number
Bg_qkdr+ @- Bqhudkkhm+
SM . After
Table 1: Summary of thetothe experimental
from 56
E821, the first block gives
µ
01
01+a
02
55
57 the main results for the hadronic
N- Cdhmdj_
Cdmhf+ B- CdS_q+ B-@Cnlhmftdy+ @-D- Cnqnjgnu+ QCD constructed in Sec. 8. The
contributions
from Secs. 2 +to@5 as
well as the combined
result for
HLbL
scattering from phenomenology
and lattice
10
58+36
6/
60
61
U-O- summarizes
Cqtyghmhm+ F- Dhbgl_mm+ L- E_dk SM+ B-RD- Fàlhy+
second block
the quantities
entering our recommended
value,Ehrbgdq
in particular,+ the
total HVP contribution,
evaluated from e+ e data,12862
and the total
HLbL number.
constructionof the total HVP and HLbLY- Fdkydq+ I-Q-The
Fqddm+ R- Ftdkk_sh,Jgdkhe_+ C-contributions
G_ssnm 08takes+ into account correlations among the terms at di↵erent
03 digits at intermediate
63 The HVP evaluation
0 on the experimental Refs. [37–orders, andfinal rounding includes subleading
stages.
is mainly based
M- theGdql_mrrnm,Sqtdcrrnm+ R- Gnky 25 + A- Góqy
+ L- Jmdbgs 14 + I- Jnonmdm+
89]. In addition, the HLbL 13
evaluation uses64experimental input31from Refs. [90–109]. The
13lattice QCD calculation
26of the HLbL contribution builds on@-R- Jqnmedkc+ I- K_hgn+[110–116].
R- Kdtonkc+ O-AL_bjdmyhd
V-I- L_qbh_mn+
crucial methodologicaladvancesfrom Refs.
Finally,the
QEDvalue uses
the+fine-structure
constant
obtainedfrom atom-interferometry
65
01+02
03
66
57
BLbMdhkd+
CLngkdq+
ILnmm_qc+
D-SMdhk+@-UMdrsdqdmjn+
measurements of the Cs atom [117].
J- Nssm_c 01 + U- O_tj 01 + @-D- Q_cyg_anu 67 + D- cd Q_e_dk 14 + J- Q_x_ 68 + @- Qhrbg 01 +@- Qncqëftdy,Ràmbgdy 5 + O- Qnhf 7/ + S- R_m Inrè 01+02 + D-O- Rnkncnu 10 + Q- Rtf_q 70 +0. Executive
Summary10 + @- U_hmrgsdhm 71 + @- U_ptdqn @uhkèr,B_rbn 55 + D- Vdhk 60 +J- Xt- Sncxrgdu
I- Vhkgdkl 01 + Q- Vhkkh_lr 60 + @-R- Ygduk_jnu 67 5
The current tension between the experimental and the theoretical values of the muon magnetic anomaly, a ⌘Hadronic vacuum polarization (HVP) I: data driven (e+e-)
Need magnetic part:

had,VPaµ: data analysis

HVP crosssectioninput
Hadronic
cross
section
input
2

↵
LO VP
ahad,
=
µ
3⇡ 2

Credit: T. Teubner

Z 1

0
ds
had, (s)R(s)K(s), where R(s) =4⇡↵2 /3s
sth s

10000

Υ(1s−6s)

Non-perturbative(Experimental data,isopsin, ChPT...)





J/ψ

Perturbative(pQCD)

ψ(2s)FIG. 5. The upper diagrams from left to right show the leadingquark-disconnected, LO QED corrections, and an example of nNondiagrams. The lower diagrams
show the leading quark-connect-perturbative/
perturbative(right) contributions to the
HLbL contribution. Subleading diag
(Experimental
data, shown.
in light-by-light scattering
are not
Credit: T. Teubner
1000

R(s)

100

φ

ρ/ω

10

1

6

pQCD,
Breit-Wigner...)

0.1
1

10
√s [GeV]100

tksr sdmcr sn ektbst_sd q_mcnlkx-

Data-driven a -HVP

ldqhbZk qdrtksr- Sgd mtladqr qdpthqdc sn hlokdldms sgµ
shbtk_q+ S_akd 4- Vd nas_hm
GUO+ KN

Zλ+ 582)0&1)7(dwo &1)7(rxr &/)6(CU*PBC · 0/

0/

0/

+
582
)
0&3
)
/(
·
0/
)
More precise than lattice determination. Total error

larger than DHMZ and KNT separately.
r u_ktd hr a_rdc nm Qder- Z1ż6[+ vghbg rgntkc ad bhsdc hm _ts eqnl Qder- Z26ż78[- Sgd ehqrs dqqnq hm Dp- &1-22( qdedqr
Data from BABAR, BESIII,
0/ CMD-2, KLOE, SND
gd 1ν bg_mmdk &0)8 · 0/ (+ enkknvdc ax sg_s _rrnbh_s
odqhldms_k
rxrsdl_shb
bnqqdk_sdc“Merged”
value fromtmbdqs_hmshdr
DHMZ, KNT, andCHHKSadsvddm
sdl_shbdqqnq
hr md_qkx
bnlokdsdkx
r_stq_sdc
sgd 1ν b(simpleaveragein eachchannel for
central ax
value,
g_keconservative
sgd cheedqdmbd
adsvddm of
du_kt_shnmr
vhsgnts
A@A@Q
combination
errors). Errors
statisticalg_keandsgdsystematic
cheedqdmbd adsvddm sgd CGLY _mc JMS du_kt_s
7

HVP contribution to muon g-2 from lattice QCD
Hadronic vacuum polarization II: Lattice QCD Blum 2002

× 10

-10

Use lattice QCD [Blum, 2003, Lautrup et al., 1971]

500
400
300
200
100
0

Light+Strange (64I)
R-ratioZ

0+

Blobs: Non-Perturbative quark loop

RBC/UKQCD 20182ˆ⇧µ⌫ (q) List
= of ensembles
d 4 x e iqx hj(all
(x)j
(0)i
=
⇧(q
qµ q⌫
µ at ⌫physical pion )mass)⇣ ↵ ⌘2 Z 1
ˆ 2)
aµHVP =
dq 2 f (q 2 ) ⇧(q
⇡
0

0.5

1

q 2 µ⌫

1.5

2
2.5
3
3.5
Time Momentum Representation (TMR) [Bernecker and Meyer, 2011]t / fm

4

12/20

4.5

X left to right show
G. 5. The upper diagramsfrom
1 X the leading-order
HVPaµ=w (t)C (t),C (t) =hji (~x , t)ji (0)i3t96, 5.5fm box
48I: 48 ×corrections,64I: 64 and× 128, 5.5fmbox
rk-disconnected,
LO
QED
an
of data
next-toi,~
xexample
FIG. 4. Comparison of wt C(t) obtained using
R-ratiograms.
lower
diagrams
the leading quark-connected (lef[1] and The
latticedataon our show
64I ensemble.
ht) contributions to the HLbL contribution. Subleading diagrams w
Calculate C(t) on 4d
ight-by-lightscattering
Euclidean space-timelattice are not shown.
3

3

lation presented here, we only include diagram M. For
24D: 24 × 64, 4.8fm box 32Dfine: 32 × 64, 4.8fm box 32D: 32 × 64, 6.4fm box
8
thecontribution
meson masses
this corresponds
to neglecting
the experimseaDcalculated
in Ref. [147].
Although the333

aµ-HVP Summary
LM 2020BMW 2020
ETM 2018/19
Mainz/CLS 2019
FNAL-HPQCD-MILC 2019
PACS 2019
RBC/UKQCD 2018
BMW 2017(0.75%)HVP (BMW-20): aµ = 707.5 (5.5) x 10-10(2.6%)HVP (Lattice): aµ = 711.6 (18.4) x 10-10

KNT 2019
DHMZ 2019(0.58%)BDJ 2019
Jegerlehner 2018

HVP (pheno): aµ = 693.1 (4.0) x 10-10

RBC/UKQCD 2018
Muon g-2 Theory Initiative 2020
LQCD
Pheno.
Pheno+LQCD

600

No New Physics(E821+E989)

650

700

750

Lattice – pheno ≈ 18.5 18.8
BMW-20 – pheno ≈ 14.4 (6.8)
9'-.**28,2?*
Towards precise comparisons: the window methodx(cf. Section 3.1.3) in that the Taylor expansion of f˜ (x2000 ) in the integrand of Eq. (3.22) yields the sum over time mome
Bnmshmttl thatkhlhsSgd
bnmshmttl
khlhs
ne
‘
rdkdb,
HVP,
LO
gives aµin that method.
For a discussion of150other related methods see Ref. [384].
V[RBC/UKQCD-18]
m ne khfgs,pt‘qj vhmcnv bnmsqhatshnmr Zα hr rgnvm hm
100f­ 7­ Vd mnsd sg‘s
sgd qdrtksr nm sgd bn‘qrd k‘sshbd chΩdq
3.1.5. Windows in euclidean time
50
l sgd bnmshmttl khlhs nmkx ‘s sgd kdudk ne ‘ edv odq,
In the aµ integral in Eq. (3.22), it is useful to consider
different time regions in order to separate the short0
Rtookdldms‘qx
ms­ Vd ‘ssqhatsd sghr lhkc bnmshmttl
khlhs Hmenql‘shnm
sn sgd e‘, z R00
0.5 volume,1
1.5 etc.).
2
2.5
3
3.5
4the4.5
long-distance systematic lattice effects (discretization, finite
To thisend,
RBC/UKQCD collaborationq‘akd oqnodqshdr ne sgd cnl‘hm,v‘kk chrbqdshy‘shnm trdc
t / fmproposed the window method [11], which breaks the time integral into three parts:
sghr vnqj­ Sghr450
hr hm bnmsq‘rs sn ‘ q‘sgdq rsddo
C(t) wbnmshm,
Σt C(t) wt
tC(t) wt θ(t,1.5fm,0.15fm)
Σt C(t) wt θ(t,1.5fm,0.15fm)
1E+05
HVP, LO
SD
W
LD
400
l dwsq‘onk‘shnm sg‘s
trhmf
rs‘ffdqdc‘r!!! "!#" ! " ! ! !$%"#$!"!$#& " '(Σt C(t) wt [1-θ(t,0.4fm,0.15fm)][1-θ(t,0.4fm,0.15fm)]
aµnbbtqr=aC(t)aµ + aµpt‘qjr
,
µ wt+
1E+04
m+
Z31“­
⇣ ↵ ⌘2 Z 1d, d­f­+ hm Qde­ 350
1E+03
e
300
(%#)
aSD
=
dx0*A'C (x⇥ (xt0 , )] ,Sgdkhlhs
enq khfgs
pt‘qj
bnmsqhat,
ms lhkc bnmshmttl
G-H32
@E
!%
0 )f (x0 )[1
0 ,2+1+1
µ &'&
⇡
1E+02
0
mr hr bnmrhrsdms250
vhsg ‘ m‘hud onvdq,bntmshmf
drshl‘sd
Z⇣⌘
1@E *A' !% I J32@4E !4 1E+01
2↵ G-H32’ZΘ(1 ¡ /σ/4 vhsg
Θ ¡W
3//& LdU‘mc rtffdrsr sg‘s200CD9 )" .>?@A5B%1 6E
(%$$
aµ =
dx0 C (x0 )ef (x0 )[⇥ (x0 , t0 , ) ⇥ (x0 , t1 , )] ,1E+00b,l‘hmhmf
chrbqdshy‘shnm
dqqnqr l‘x
ad rl‘kk­
Rhmbd vd
⇡ 78
150
0
&#
Z rhmfkd⇣⌘
1
chmrtbg ‘ lhkc adg‘uhnq
mns
itrs
enq
‘
pt‘mshsx
1E-01
2
↵
100
LDV
!'$
(%!'
dx0 eqnl
C (x0 )ef/σ2(x0 )⇥ (x0 , t1 1E-02
, ),
(3
&#
sdqenq ‘kk rstchdc 50
u‘ktdr neaµZα=vhsg98+s, q‘mfhmf⇡
dq,
sn /σ4 el ‘mc s0 q‘mfhmf eqnl /σ2 el0 sn 1σ5 el+ vd
1E-03
0
GH<;IJK<- )" .>‘,ffdrs
sg‘s hs hr q‘sgdq
sgd&#
lhkc
0
0.5tmkhjdkx
1 !'( 1.5 sg‘s
2CD8
2.5
3 ,>?@A3B'/
3.5 adg‘u,4
4.54E*E 12AF5
(%)*
56
0.1
1
10
100 Reports xxx (xxxx) xxx
dc
T. Aoyama,
N. Asmussen,
M. b‘mbdkk‘shnm
Benayount /etPhysics
fm al.ne ghfgdq,nqcdq
hr qdrtksne ‘m‘bbhcdms‘ksqrt(s) / GeV
K9<;LMN<O!$
m, hm ‘m dwo‘mrhnm hm Z1 ­ Sghr kdmcr
Σt C(t) wt sn ntq
lrrtoonqs2+1!')
Σt C(t) wt θ(t,1.5fm,0.15fm)
1E+05
(%)
‘r
3
Σ
C(t)
wt [1-θ(t,0.4fm,0.15fm)]tnsdc chrbqdshy‘shnm
dqqnq
a‘rdc
nm
‘m
!’Z
(
drshl‘sd­
PQ8< &# ,>?@A3B'5
1E+04
etstqd vnqj+1E+03
sghr vhkk ad rtaidbs sn etqsgdq rbqtshmx ax
LM9< )" .>?@A5B%7
EHF­ 8­ Vhmcnv ne Q,q‘shn c‘s‘ hm Dtbkhcd‘m onrhshnm ro‘bd!'&
Aubin, et al.-19t,(%()
chmf ‘ c‘s‘,onhms‘s ‘m ‘cchshnm‘k8*32:;<7=k‘sshbd ro‘bhmf­’sno(‘mc sgd dΩdbs ne sgd vhmcnv hm sdqlr ne qd,vdhfgshmf&#
J
I?@40',>?@A3B'5
1E+02
sd
!
dmdqfx qdfhnmr ’anssnl(­
1E+01‘s
8*32:;<7=,>?@A3B'5
Dmdqfw qdfivdhfgshmf- Sgd!sno o‘mdk
ne Ehf­ 8 &#
rgnvr
(%"$
1E+00
vd
d vdhfgsdc bnqqdk‘snq ts /’s( enq sgd etkk Zα ‘r vdkk ‘rsx
1E-01
9+54:;<=, )" .>?@A
RBC/UKQCD-18ZQC ‘mc ZHC#'$ oqnidbshnmr
nqs,chrs‘mbd ‘mc knmf,chrs‘mbd
α
α
σ2
1E-02
"%&'
l‘x
‘ssdlossnbnmsq‘rssgd
Q,q‘shnc‘s‘
chqdbskx
vhsg
s
¡
/σ3
el
‘mc
s
¡
0σ4
el­
Sgd
anssnlo‘mdk
ne,0
KI?*E31 R A*EE30@
vd
1E-03‘m dwbktrhud rstcx ne sgd knv,kxhmf µµ rs‘sdr hm sgd k‘sshbd
f­u,8 rgnvr sgd bnqqdronmchmf
bnmsqhatshnmr
KNT-19– lattice s, c, … sn Zα rdo,#'("b‘kbtk‘shnm­
Sghr hr &##kdes sn etstqd vnqj­‘sdcq­ Vd 1!"#
mnshbd sg‘s+10‘r dwodbsdc+
0.1
100
!$#
!%#
&!#
!"
!#
dq ax dmdqfx rb‘kd
Csqrt(s) / GeV
tqg‘r qdctbdc bnmsqhatshnmr eqnl knv,dmdqfx rb‘kdr ‘mc
,+
!#
C
+
* ,-./
0122/
3415
6
!#
sd­g‘r qdctbdc bnmsqhatshnmr eqnl ghfg,dmdqfx rb‘kdr­
RsSshrshbr ne khfgsfiptSqi bnmsqhatshnm- Vd trd ‘m!
!
sgd khlhs
neoqnidbshnm
sn
rtflbhdmskx
knmf
chrs‘mbdr+
vdax
hloqnudc rs‘shrshb‘k drshl‘snq hmbktchmf ‘ etkk knv,lncd
EHF­ 8­ Vhmcnv ne Q,q‘shn c‘s‘ hm Dtbkhcd‘m onrhshnm ro‘bd
(ud
0.4
1.0
fm
window)W
Fig.’sno(
47. (Left)Continuumof the intermediateaµ‘udq‘fd(ud) ( =.15 fm,t0 = connected,
0.4 fm,t1 = 1 fm)
for –DW(circles)and HISQ (squares)‘mc sgddΩdbs nelimit
sgd vhmcnv
hm sdqlr ne window
qd,vdhfgshmf
enq0sgd
khfgs,pt‘qj
bnmmdbsdc
bnmsqhatshnm
hm sgddmdqfxLines
qdfhnmr’anssnl(­fermions.
denotefits linear in a2 for the a ! 0 limit (bursts).
R-ratio result(cross,Ref. [445]) hm
is also
It corresponds to thehrnrohmTherxlldsqhb
khlhs‘r chrbtrrdc
sgd shown.
l‘hm sdws­
300
t
=
0.4,
t
=
0.9,
Light
result based on Ref. [3] and the sumof all but the isospin-symmetric light-quark lattice results of Ref. [11].
vr difference of the total R-ratio0 window
1
Enq sghr 10
drshl‘snq+
vd ffmc sg‘s vd ‘qd ‘akd sn r‘stq‘sd
t0 = 0.4, t1 = 1.0, Light280

e!0,-"LY R(*,@@2%

x 10

-10

November 2020 workshop on HVP, Harmut Wittig[$+-7:b8o.-Y7-; -,/,-,32, `)7355,*

Finite-volume (DWF and HISQ) and taste-breaking (HISQ) corrections are included to NLO in ChPT. Lattice spacing uncertainties, added in quadrature!

non-charm’ background from experimental results for
12
1.4725+R(eTowardsehadrons)
to comparisons:leave Rc for Eq. the(35). window
Rc is
precisemethod[RBC/UKQCD-18]
13
1.4805defined to be the result from diagrams with a charm
14
1.4610latticespacingquark loopContinuum
connected extrapolation
to a photon- Whatat bothends[64] is fine14† enough?
1.4702.e. the quark-line connected vector current-current cor15
1.457220 - lightquarklattice.
window The subtracted
relator thatBMW
we studyon theArticlebackgroundincludes QED effects for the non-charm and214
no reason to believe are collectiv
singlet (quark-line
disconnected) contributions. The reneed further investigation, alt
mainder, Rc thenincludestheQEDeffects
associated212that the tensions observed her
with the cc loop. The dominant source of uncertainty
is usually considered experime210
n Rc comes from the charmonium resonance (J/ and(3σ) and much smaller than wh
SRHO improvement
discovery (5σ).
) region and208
is set by the uncertainty
in ee for these
As a first step in that directio
states. The fractional
uncertainty
isapproximately
the206
ified observable, where the cosame for all moments [64, 70]. When the (n 2)th root isinterval by a smooth window fu
204taken the fractional uncertainty then falls with increasing
denote as aµ,win, is obtained muc
202
Its shorter-distance nature makn.
No improvement
noise and to finite-volume effec
Good agreementis
seenbetweenthephenomenological200
fermions, it has reduced discr
results and our198
new lattice results for n = 6, 8 and 10, alFig. 4, where the light, connec
0.005
0.010 at the0.015upper
0.020
a function of a2. Because the dthough the lattice results 0.000
are systematicallya2 (fm2)
not require overcoming manyend of the phenomenological range. The largest discrepother lattice groups have obt
Fig.
4|
Continuum
extrapolationoftheisospin-symmetric,
light,
3.7
tensionbetweenBMW20
and
R-ratiofor
Window!
ancy is a 2.8 tension for the 4th moment between us and19,20
ours28.
. Thisallows a sharper, (aconnected component of the window observable a) iso. The data FIG.
Extrapolationtothe resultspointsof [70]for
theirminimalselectionofdatasets.
this challenging, light-quark
are extrapolated to the infinite-volume limit. Central values areconnectedcharm
HVP co
light
Red
line
for
comparison
with
next
slidesSlide
credit:Christoph
LehnerOurdiffersby
0.2σ and 2
a µ,winmedians;
error forbars ares.e.m.
Two different
ways to performthe continuum
The tension
is 2.4thestandard
selection,
and
below
netic
momentof the Moreomuon
19
11 and another withextrapolations are shown: one without improvement,
and
ref.
,
respectively.
2 for the maximal selection and for the results of [64].
This work

R-ratio/latticeBlum et al.19(alight
P,win )iso

Aubin et al.20

BMW 20 light quark window

µ,win

light
µ,win

HPQCD 20 c
arXiv:2005.01

Towards precise comparisons: the window method [RBC/UKQCD-18]• Is lattice continuum limit incompatible with R-ratio result?
• Or do we need even finer lattices?
• Fits to ud connected, 0.4 – 1.0 fm window:
Log corrected
212
210

aµ +𝒃 𝒂𝟐

208

preliminary

206
204
202
200
198
0.000

Aubin et al.
12aµ+ 𝒃 𝒂𝟐

𝟏𝒄 − log 𝑎"

0.005

0.010

0.015

0.020

0.025

Strange and charm contributions
Seem to be in good shape
Physics Reports xxx (xxxx) xxx

BMW-20 53.393(89)(68)

BMW-20 14.6(0)(1)

Fig. 45. Comparisons of lattice results for flavor-specific contributions to aLO
(Upper-Right) Strange-quark connected contribution aHVP,(s). (Lower-Lefµ
HVP, LO

13

disconnected contribution aµ,disc . The lattice results in each panel are gro
the underlying calculations, where ‘‘Nf = 2 + 1 + 1’’ (circles) labels ensembl(squares) charm quarks are not included in the sea, while for ‘‘Nf = 2’’, (indicate results included in the lattice averages of Section 3.5.1, which are
been updated or superseded, see Table 9 for further details.
Source: Adapted from Ref. [443].

(2017)020

hat the disconnected contribution is small where it can be resolved does
mply that it is negligible. Using our data we can derive an upper boundhich arises if one were to neglect the disconnected contribution altogether.
is useful to recall the isospin decompositionof the electromagnetic currentff’2.13),
which gives rise to the iso-vector (I = 1) correlator Gρρ and its(D.9)
I=0 (see eq. (2.15)). The iso-vector correlator Gρρ (x ) contains
terpart GSeem0
to be in good shapenected diagrams; it is related to the connected light quark contribution

Disconnected contributions

6ons

H. Wittig @ Lattice HVP workshop
elow the9 udρρ
According
G (x0 ) = G (x0 ).
FHM10
20

(preliminary)(D.7)

Mainz/CLS20
e iso-scalar correlator
GI=0 contains
both connected (preliminary)and disconnected conBMW 20

G(x0 )(D.10)

I=0

1 ud= G (x0 ) + Gs (x0 ) − Gdisc (x0 ). Mainz/CLS 19
10

RBC/UKQCD
ostic
theonewindow
1.0] fmof eqs. quantity:(D.3) and (D.7)
derives18the[0.4,expression(D.11)
BMW 17!
"

Gdisc (x0 )G(x0 ) − Gρρ (x0 ) 1
Gs (x0 )− -30− -151 +9 ρρ-10 -7.5.
-27.5= -25 -22.5-20 -17.5
-12.5
ρρ
ρρ
G (x0 )
G (x0 )9
G (x0 )

-5

(D.8)

-2.5 (D.9)
010ehaviourQuark-disconnected
(ahvp
)
·
10
disc
contribu6onsµ

ortant to realize that the iso-scalar spectral
function vanishes below the
14
I=0

−3m x

Isospin-breaking contributions

Lattice HVP: Isospin corrections

Needs further study, improvement

V. Gülpers @ Lattice HVP workshop
Introduction

Overview of published results - contributions to aµ ◊ 1010
BMW
RBC/UKQCD≠0.55(15)(11)≠1.27(40)(33)
5.9(5.7)(1.7)≠6.9(2.1)(2.0)

1.1(1.0)

ETM• Some tensio
individual co• Large cancecontributionBMW
RBC/UKQCD

δaμIB ≲ 1 %• Ongoing ef
Fermilab-HP≠0.0095(86)(99)

0.42(20)(19)

6.59(63)(53)BMW
RBC/UKQCD

10.6(4.3)(6.8)
6.0(2.3)
7.7(3.7)

9.0(2.3)

9.0(0.8)(1.2)

0.011(24)(14)BMW≠4.63(54)(69)≠0.047(33)(23)

BMW

BMW

ETM

BMW [arXiv:2002.12347]

FHM

RBC/UKQCD [Phys.Rev.Lett. 121 (2018) 2, 022003]
ETM [Phys. Rev. D 99, 114502 (2019)]

LM

FHM [Phys.Rev.Lett. 120 (2018) 15, 152001]
LM [Phys.Rev.D 101 (2020) 074515]

Vera Gülpers (University of Edinburgh)

HVP from LQCD - workshop

15

18 Nov 2020

5/6

HVP: to reach desired precision (2-5 per-mil)• Strange, charm, (bottom) contributions in good shape(will not resolve issues)• FV corrections (𝐿 > 6 fm) reliable (NNLO 𝜒PT, LLGS, HP)
Important to have a big box (BMW, PACS use L = 10 fm)• Statistical precision top priority for DW, TM, Wilson (in progress)
Improved bounding method, low-lying states for long distance tail• Physical masses(most groups already)• More, more precise disconnected and IB calculations needed• Continuum limit and scale setting (per-mil) are crucial
16

HVP updates soon
• RBC/UKQCD• 3rd lattice spacing, a-1=2.7 GeV, 6 fm box• Window continuum limit w/3 lattice spacings by Fall
• 2+1 2+1+1 correction (expected to be small)• Full result w/3 lattice spacings by end of year
• Error on total below 1%• Aubin et al.
• NNLO FV and taste breaking (discretization) corrections• Improved window and full connected results by end of year17

aµ-HLbL from data and models
S- :nxZlZ+ M- :rltrrdm+ L- AdmZxntm ds Zk-

Ogxrhbr Qdonqsr 776 1/1/( 0ż055

Ehf- 5.- Bnlo_qhrnm ne sgd ν / SEE eqnl chrodqrhnm sgdnqx Z10+386[ &qdc(+ B@ Z08[ &a&kdes( _mc sgd cntakx,uhqst_k &qhfgs( enql e_bsnqr-

Good agreement between
data/dispersive and lattice approaches

Ehf- 46- Sgd ordtcnrb_k_q,onkd bnmsqhatshnm9 sgd c_rgdc khmdr rs_mc enq sgd ordtcnrb_k_q ldrnm+ vghkd sgd aknar b_m ad tm_lahftntrkx qdk_sdc sn
sgd SEErRntqbd9 Qdoqhmsdc eqnl Qde- Z08[-

sgd enql e_bsnq mnql_khy_shnm enq sgd dwodqhldms_k OqhlDw qdrtks Z417

S- :nxZlZ+ M- :rltrrdm+ L- AdmZxntm ds Zk-

Ogxrhbr Qdonqsr 776 1/1/( 0ż055

3-2-3- QZchZshud bnqqdbshnmr Zmc Lnmsd BZqkn dudms fdmdqZsnqr
Vhsg sgd dqqnq nm &f1(/λ,onkd
bnlhmf eqnl sgd GUO _mc sgd GKaK1bnmsqhatshnmr_klnrs
600 _s sgd r_ld kdudk Z1+2+5+6[+
sgd qdkdu_mbd ne sgd dqqnq qdctbshnm hm sgd GKaK hr _r hlonqs_ms _r sgd dqqnq qdctbshnm hm GUO- @r _ bnmrdptdmbd+ _kk
1 0hlonqs_ms hmenql_shnm nm sgdrd bnmsqhatshnmr
onrrhakd rntqbdr ne sgd dqqnq g_ud sn ad rbqtshmhydc nmbd lnqd- Sgd lnrs
&×( &×(bnldr eqnl dwodqhldms_k c_s_ nm
β β " g_cqnmr _lokhstcdr+ vgdqd nmd ne sgd rntqbdr ne sgd rxrsdl_shb dqqnq hr/ ,onkd
00 ne sgd &f 1(λ _qd sgd
sgd _bbtq_bx ne sgd Lnmsd B_qkn dudms fdmdq_snqr- Sgd lnrs hlonqs_ms nmd enq sgd du_kt_shnm
β &×( β &×( " ordtcnrb_k_q&r( _lokhstcdr- Hm sgd ro_bdkhjd qdfhnm sgdx vdqd qdonqsdc hm Qder- Z8/+80+85+86+88[- @ rsqnmf
sdmrhnm v_r entmc adsvddm sgd A@A@Q c_s_ Z85[ _mc nsgdq ld_rtqdldmsr Z8/+80+88[ ne sgd ohnm SEE/ ,onkd
Enq sgd qdbdms dwodqhldms_k
ld_rtqdldmsr svn dudms fdmdq_snqr vdqd trdc9 SQDORORS Z88+480[
00 hm sgd ADKKD _m_kxrhr_mc FFQDRQB Z481[ hm sgd A@A@Q _m_kxrhr- Ansg ne sgdl hmbktcd q_ch_shud bnqqdbshnmr ax ld_mr ne _ rsqtbstqd etmbshnm
ldsgnc- Qdbdmskx sgd dudms fdmdq_snq DJG@Q@ Z482+483[ v_r tofq_cdc Z484[ _mc sgd PDC bnqqdbshnmr vdqd hmbktcdc
dw_bskx hmsn sgd bncd- Sgd DJG@Q@ bncd oqdchbsr ltbg ahffdq+ to sn 1/"+ q_ch_shud bnqqdbshnmr enq sgd rhmfkd s_f dudms
rdkdbshnm ne A@A@Q dwodqhldms _r bnlo_qdc sn sgd FFQDRQB dudms fdmdq_snq/ oqdchbshnm+ vghbg hr ne nqcdq ne 0"He sgd oqdchbshnmr ax sgd DJG@Q@ fdmdq_snq _qd bnqqdbs+ qd_m_kxrdr ne sgd dwodqhldms_k c_s_ _qd mdbdrr_qx- Hs hr mns
rsq_hfgsenqv_qc sn fds hmenql_shnm nm gnv sghr k_qfd bg_mfd hm sgd q_ch_shud/ bnqqdbshnmr hmektdmbdr sgd ld_rtqdldmsr+ _r
hs _eedbsr mns nmkx sgd q_ch_shud bnqqdbshnm e_bsnq+ ats _krn sgd dwodqhldms_k deehbhdmbhdr: _mc sn nas_hm sgd dwodqhldms_k
deehbhdmbhdr sgd cdsdbsnq rhltk_shnmr _qd mdbdrr_qx- Hfmnqhmf sgd deedbs nm sgd deehbhdmbhdr _mc s_jhmf hmsn _bbntms nmkx
sgd q_ch_shud bnqqdbshnm e_bsnq+ sgd ohnm SEE dwsq_bsdc eqnl sgd A@A@Q c_s_ hr _ants 1/" ghfgdq- Gnvdudq+ _r b_m ad rddm/
Dp- &3-08( _mc
Ehfv0.1 c_lodm
sgd
hmektdmbd
ne k_sshbd
sgd A@A@Q
c_s_
sn Zνλ ,onkd_r sgdx
Ehf-eqnl
5.- Bnlo_qhrnm
ne sgdν / 47+
SEE sgd
eqnlvdhfgshmf
chrodqrhnm e_bsnqr
sgdnqx Z10+386[&qdc(+ B@
Z08[&aktd(+ _mc
PBC Z11[&xdkknv(Vd +rgnv
ansg_qd
sgd rhmfkx,
11
nmkx_s P /&qhfgs(
3/
FdUenql- Mdudqsgdkdrr+
sgdrd bnmrhcdq_shnmr rsqnmfkx rtffdrs sg_s qd_m_kxrdr ne sgd dwodqhldms_k&kdes(_mc_u_hk_akd
sgd cntakx,uhqst_ke_bsnqr∝3 _r/ sgd q_ch_shud bnqqdbshnmr00c_s_ _mc _ bnmehql_shnm ne sgd qdrtksr,onkd
ne Qde- Z484[ _qd mdbdrr_qx+
lhfgs _eedbs sgd ld_rtqdc
enql e_bsnqr _s _ rb_kd qdkdu_ms enq sgd &f 1(λ oqdchbshnmr25&chro( + 52)/! )) · 0/

Zνλ

Zνλ

Zνλ

.

&B@( + 52)5&1)6( · 0/

&k_sshbd( + 51)2&1)2( · 0/.
)

Lnqdnudq+ _kk sgqdd tmcdqkxhmf ν SEEr _qd bnmrhrsdms vhsg d_bg nsgd
chrodqrhud du_kt_shnm ne sgd ν vhsg sgd B@ enq ζ _mc ζ∝ + vghbg _qd hm
cdsdqlhm_shnmr+ vd _qqhud _s sgd enkknvhmf btqqdms drshl_sd enq sgd orZνλ !ζ!ζ

+ 82)7! )) · 0/28
18
sgd3-3enql
e_bsnq mnql_khy_shnm
enqZmcsgdnsgdq
dwodqhldms_k
OqhlDw
Bnmsqhatshnm
ne sgd ohnm onkd
ordtcnrbZkZq
onkdr qdrtks Z417[9.

aµ-HLbL from data and models
S- :nxZlZ+ M- :rltrrdm+ L- AdmZxntm ds Zk-

Ogxrhbr Qdonqsr 776 1/1/( 0ż055

RVakd 04
Bnlo_qhrnm ne svn eqdptdmskx trdc bnlohk_shnmr enq GKaK hm tmhsr ne 0/ 00 eqnl 1//8 _mc _ qdbdms toc_sd vhsg ntq drshl_sd- Kdfdmc9 OcQU +Oq_cdr+ cd Q_e_dk+ U_hmrgsdhm &——Fk_rfnv bnmrdmrtr‘‘(: M.IM + Mxeedkdq . Idfdqkdgmdq+ Mxeedkdq: I + IdfdqkdgmdqBnmsqhatshnm

OcQU&/8( Z364[

M.IM&/8( Z365+485[

I&06( Z16[

Ntq drshl_sd

ν / . ζ. ζ∝ ,onkdr
ν. J ,knnor.anwdr
R,v_ud νν qdrb_ssdqhmf

003&02(08&08(
6&6(

88&05(
08&02(
6&1(

84)34&01)3/(
1/&4(
4)87&0)1/(

82)7&3)/(05)3&1(
7&0(• Huge improvement in pole contributions
rtasns_k
77&13(
62&10(
58)4&02)3(• All contributions
computedor estimatedrb_k_qr• Errors added in quad for dispersive results
sdmrnqr0-0&0(_wh_k udbsnqr
04&0/(
11&4(
6)44&1)60(•
Errorsaddedlinearlyformodel-dependent
results
t. c. r,knnor . rgnqs,chrs_mbd
10&2(
1/&3(
b,knno1-2

sns_k

0/4&15(005&28(58)3&3)0(

!

0&2(

5&5(
04&0/(

1-2&1(

2&0(0//)3&17)1(

81&08(

3-8-1- TmbdqsZhmsx drshlZsd
Fhudm sgd btqqdms q_sgdq tmr_shre_bsnqx sgdnqdshb_k rs_str ne sgd b_kbtk_shnm ne rnld bnmsqhatshnmr _mc vg_s hr _s rs_jd&_ onrrhakd chrbnudqx ne ogxrhbr adxnmc sgd RL(+ hs hr vnqsgvghkd chrbtrrhmf sgd tmbdqs_hmsx drshl_sd hm rnld cds_hk _mc
vgdsgdq hs rgntkc ad l_cd lnqd bnmrdqu_shud- Bkd_qkx+ sgd sgdnqdshb_kkx lnqd cheehbtks drshl_sdr bnmbdqm sgd enkknvhmf
entq bnmsqhatshnmr9 rb_k_qr+ sdmrnqr+ _wh_k udbsnqr+ _mc rgnqs19chrs_mbd- @r dwok_hmdc _anud+ rhmfkd,qdrnm_mbd,dwbg_mfd

Lattice HLbL

Blum, et al. (RBC) PRL 124 (2020)
Editor’s Suggestion
Point Source Photon M
30
aµ × 1010

25

+ ···

• Included in Muon g-2 Theory Initiative average• 92(19)x 10-11 (phenomenology)
detail in Ref. [31], and the diagrams to be computed are•90(17)x 10-11 (phenomenology+lattice)
shown in Figs. 2 and 3. It is still not possible to do all• Unlikely
explaindiscrepancy withexperimentof thesumstoover
coordinate
space
vertices exactly withcurrently available compute resources. Therefore we re20x,

32D
24D-32D48I
inf &

15
z,

10

xsrc
y,
0.05

0
0

z,

0.1

0.15

xsnk xx,
0.2

1/(mµ L)2

aµ × 1010'()
FIG. 1. Leading
hadronic
light-by-light𝑎#$%&% contributions= 7.87 ± 3.06 ±from
1.77×10
scattering to the muon anomaly. The shaded circles represent •quark
loops
containing
QCD
interactions
RBC: firstlatticecalculation withall errors
controlled. to all orders.
Horizontal lines represent muons. Quark- connected (left) and• 1 G core-hours
on ALCF’s Mira (BG/Q).
disconnected(right)diagrams are shown. Ellipsis denote di• 1stobtained
HLbL calculation
was done on
USQCD
resources
agramsby permuting
thephotoncontractions with
(Blum, et al., PRL 114 (2015))
the muonsand diagrams with three and four quark loops withphotoncouplings(See Model
Fig. 3).
• Crucialfor Standard
Comparison

y,

5

0
−2
−4
−6
−8
−10
−12
−14
−16
−18
−20

• Point source photons at x an

FIG. 2. Connected dputed stochastically.32D
24DMajor contribution comes48I
ternal vertex xop areinf &In fact,
can doneevaluateexamuon
linewearedistance less than a certa
to all orders are not s• Importance sampling is used

• Moment method for xop. Eva
0

0.05

10
8

0.1

0.15

1/(mµ L)xop , ν12aµ × 1010

+

xop ,

20

0.2

2

Method published in Phys.Rev. D
previous approach[Phys.Rev.Lett.
z, κ
y, σ

x, ρ

6
4

xsrc

z ! , κ!

y! , σ!

x! , ρ!

32D
24D48Iinf &xsnk x2
xop , ν

0
0

0.05

0.1
1/(mµ L)

0.15

0.2

2

z, κ

y, σ

HLbL: Comparison

HLbL lattice-pheno comparision
HLbL
aµCredit: A. El-Khadra, Lattice 2021

models of QCD +EFT,
large Nc

Glasgow consensus (09)N/JN09J17
Mainz21 (+ charm-loop)

Lattice QCD + QED

not used in WP20

RBC/UKQCD19(+ charm-loop)

data + dispersive
approach

WP20 data-driven
dispersive

WP20
0

20

40

60

80

100

120

140

160

HLbL
11
aµ
× 10

21
Now well-determined in two independent
approaches, systematically improvab

aµ-HLbL outlook• More data for data-driven approach• RBC: QED∞ calculation, 10-20% accuracy (5 years)• New result from Mainz group (arXive:2104.02632)• Other lattice groups starting (BMW, FHM, …)• Combined 10% result (or better) within 5 years possible
22Outlook• Standard Model remarkable success• Combining BNL and Fermilab experiments
and using muon g-2 theory initiative SM value,
discrepancy grew from 3.7 to 4.2 s• Using BMW20 lattice value for HVP,
discrepancy is 1.5 s• Consistency of data-driven and lattice HVP
calculations crucial for discovery of new physics• HLbL in good shape- can’t rescue SM• Improved lattice calculations in progress23

Acknowledgements• Work partially supported by US Departmentof Energy• Computations done under ALCC at Argonne
Large Computing Facility and XSEDE (NSF) at
Texas Advanced Computing Center24

Further viewing/reading• Trevor Noah: Physics may be a lie (2:09 minute mark)• The Muon g–2 Anomaly Explained• Fermilab E989 announces first results• An anomalous moment for the muon
• Muon g-2 Theory Initiative white paper

25