HVP contribution of the light quarks Davide Giusti to (g -2) - - PowerPoint PPT Presentation

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HVP contribution of the light quarks to (g𝝂-2) including QED corrections with twisted-mass fermions OUTLINE

Isospin breaking effects on the lattice (RM123 method) Results for the light quark contribution to a𝝂HVP In collaboration with:

• V. Lubicz, G. Martinelli, S. Romiti, F. Sanfilippo, S. Simula, C. Tarantino

XXXVI International Symposium on Lattice Field Theory East Lansing

22nd - 28th July

Davide Giusti

SLIDE 2

Muon magnetic anomaly

2

LO Had.

aµ

SM = 116 591 823 1

( ) 34 ( ) 26 ( )⋅10−11

QED+EW NLO/NNLO Had.

PDG 2018

µ q q

HVP

= (−i e) ¯ u(p0)  γµF1(q2) + iσµνqν 2m F2(q2)

• u(p)

aµ ≡ gµ − 2 2 = F

2 0

( )

muon anomalous magnetic moment: is generated by quantum loops; receives contribution from QED, EW and QCD effects in the SM; is a sensitive probe of new physics

(c) 3b

µ f γ Z Z

(c)

(e)

0.4ppm

ab-initio LQCD

error budget

dispersion relations e+e- hadrons

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The determination of some hadronic

• bservables in flavor physics has reached such

an accurate degree of experimental and theoretical precision that electromagnetic and strong isospin breaking effects cannot be neglected anymore

Phenomenological motivations

3

SLIDE 4

ISOSPIN BREAKING EFFECTS

Though small, IB effects can play a very important role (quark masses, Mn - Mp, leptonic decay constants, vector form factor)

Qu ≠ Qd : O(αe.m.) ≈ 1/100 mu ≠ md : O[(md-mu)/ΛQCD] ≈ 1/100 “Electromagnetic” “Strong” Isospin breaking effects are induced by:

Since electromagnetic interactions renormalise quark masses the two corrections are intrinsically related

Isospin symmetry is an almost exact property of the strong interactions

4

SLIDE 5

Hadronic Vacuum Polarisation

5

e+e- data

100%

lattice data

100% ates

550 600 650 700 750

a

µ HVP * 10 10

ETMC 18 RBC/UKQCD 18 BMW 17 CLS/Mainz 17 HPQCD 16 KNT18 no New Physics DHMZ 17 FJ 17 RBC/UKQCD 18

lattice + e+e-

~ 30% + 70% µ q q

≳ 2% ≳ 0.4%

h h )

Given the present exper. and theor. (LQCD) accuracy, an important source of uncertainty are long distance electromagnetic and SU(2) breaking corrections

19) 20) 21)

δaµ

HVP ~ 39 1

( )⋅10−11

estimate in sQED

• K. Melnikov, 2001

SLIDE 6

Isospin breaking effects on the lattice

RM123 method

SLIDE 7

A strategy for Lattice QCD: The isospin breaking part of the Lagrangian is treated as a perturbation Expand in:

arXiv:1110.6294

+

arXiv:1303.4896

RM123 Collaboration

αem md – mu

7

SLIDE 8

• Identify the isospin breaking term in the QCD action

S

m =

m

uuu + m ddd

⎡ ⎣ ⎤ ⎦

x

∑

= 1 2 m

u + m d

( ) uu + dd

( )− 1

2 m

d − m u

( ) uu − dd

( )

⎡ ⎣ ⎢ ⎤ ⎦ ⎥

x

∑

= = m

ud uu + dd

( )− Δm uu − dd ( )

⎡ ⎣ ⎤ ⎦

x

∑

= S

0 − Δm

ˆ

S

• Expand the functional integral in powers of Δm

O = Dφ O e

−S

0+Δm ˆ

S

∫

Dφ e

−S

0+Δm ˆ

S

∫

1st

!

Dφ O e

−S

0 1+ Δm ˆ

S

( )

∫

Dφ e

−S

0 1+ Δm ˆ

S

( )

∫

! O

0 + Δm O ˆ

S 1+ Δm ˆ S = O

0 + Δm O ˆ

S

• At leading order in Δm the corrections only appear in the

valence quark propagators:

(disconnected contractions of ūu and ƌd vanish due to isospin symmetry)

The (md-mu) expansion

Advantage: factorised out

8

Ŝ = Σx(ūu-ƌd)

for isospin symmetry

SLIDE 9

The QED expansion for the quark propagator

In the electro-quenched (qQED) approximation:

9

SLIDE 10

Results for the light quark contribution to a

aµ

HVP

SLIDE 11

HVP from LQCD

11

aµ

HVP = 4α em 2

dQ2

∞

∫

1 mµ

2 f

Q2 mµ

2

⎛ ⎝ ⎜ ⎞ ⎠ ⎟ Π Q2

( )− Π 0

( )

⎡ ⎣ ⎤ ⎦

Πµν Q

( ) =

d 4x eiQ⋅x

∫

Jµ x

( )Jν 0 ( ) = δ µνQ2 −QµQν

⎡ ⎣ ⎤ ⎦ Π Q2

( )

V t

( ) ≡ 1

3 d! x

∫

i=1, 2, 3

∑

Ji ! x,t

( )Ji 0 ( )

µ q q

5000 10000 15000 20000 0.0001 0.001 0.01 0.1 1 Q2 [GeV2] pheno.

• F. Jegerlehner, “alphaQEDc17”

f Q2 mµ

2

⎛ ⎝ ⎜ ⎞ ⎠ ⎟ Π Q2

( )− Π 0

( )

⎡ ⎣ ⎤ ⎦⋅1010

Q2 ! mµ

2 4

• B. E. Lautrup and E. de Rafael, 1969; T. Blum, 2002

aµ

HVP = 4α em 2

dt f

! t

( )

∞

∫

V t

( )

Time-Momentum Representation

• D. Bernecker and H. B. Meyer, 2011

quark-connected terms only

t ≤ Tdata < T/2 (avoid bw signals) t > Tdata > tmin (ground-state dom.)

aµ

HVP =

4α em

2

f

! t

( )V f t ( )+

f

! t

( ) GV

f

2MV

f e−MV

ft

t=Tdata+a ∞

∑

t=0 Tdata

∑

⎧ ⎨ ⎪ ⎩ ⎪ ⎫ ⎬ ⎪ ⎭ ⎪

f =u,d,s,c

∑

lattice data analytic representation

up to 10% for light quarks local vector currents

SLIDE 12

Details of the lattice simulation

We have used the gauge field configurations generated by ETMC,

European Twisted Mass Collaboration, in the pure isosymmetric QCD theory with Nf=2+1+1 dynamical quarks

• Gluon action: Iwasaki
• Quark action: twisted mass at maximal twist

(automatically O(a) improved) OS for s and c valence quarks

12

ensemble

• V/a4

aµud aµ aµ Ncf aµs M⇡ MK (MeV) (MeV) A40.40 1.90 403 · 80 0.0040 0.15 0.19 100 0.02363 317(12) 576(22) A30.32 323 · 64 0.0030 150 275(10) 568(22) A40.32 0.0040 100 316(12) 578(22) A50.32 0.0050 150 350(13) 586(22) A40.24 243 · 48 0.0040 150 322(13) 582(23) A60.24 0.0060 150 386(15) 599(23) A80.24 0.0080 150 442(17) 618(14) A100.24 0.0100 150 495(19) 639(24) A40.20 203 · 48 0.0040 150 330(13) 586(23) B25.32 1.95 323 · 64 0.0025 0.135 0.170 150 0.02094 259 (9) 546(19) B35.32 0.0035 150 302(10) 555(19) B55.32 0.0055 150 375(13) 578(20) B75.32 0.0075 80 436(15) 599(21) B85.24 243 · 48 0.0085 150 468(16) 613(21) D15.48 2.10 483 · 96 0.0015 0.1200 0.1385 100 0.01612 223 (6) 529(14) D20.48 0.0020 100 256 (7) 535(14) D30.48 0.0030 100 312 (8) 550(14)

Pion masses in the range 220 - 490 MeV 4 volumes @ and Mπ ! 320 MeV

a ! 0.09 fm

MπL ! 3.0 ÷ 5.8

SLIDE 13

Light quark contribution

13

250 300 350 400 450 500 550 600 650 0.1 0.2 0.3 0.4 0.5 0.6

β = 1.90, L/a = 20 β = 1.90, L/a = 24 β = 1.90, L/a = 32 β = 1.90, L/a = 40 β = 1.95, L/a = 24 β = 1.95, L/a = 32 β = 2.10, L/a = 48

a

µ HVP(ud) * 10 10

M

π (GeV) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 5 10 15 20 25 30 35 A80.24 B55.32 D30.48

eff

t / a

aM eff

t / a

a3V u,d

t / a 160 stoch. sources / gauge conf. StN: ∝ e

− Mρ −Mπ

( )t

• G. Parisi, 1984;
• G. P

. Lepage, 1989

µ q q

aµ

HVP s

( ) = 53.1 2.5 ( )⋅10−10

aµ

HVP c

( ) = 14.75 56 ( )⋅10−10

DG et al., 2017

aµ

HVP ud

( ) = 619.4 12.7 ( )stat+ fit 6.8 ( )chir 6.2 ( )FVE 5.4 ( )disc ⋅10−10

= 619.4 16.6

( )⋅10−10

talk by S. Simula @ (g𝝂-2) plenary workshop, Mainz 2018

preliminary

quark-connected terms only

450 500 550 600 650 700

a

µ HVP(ud) * 10 10

ETMC 18 RBC/UKQCD 18 BMW 17 CLS/Mainz 17 HPQCD 16 ETMC 14

SLIDE 14

LIB corrections

14

h h )

tad

δaµ

HVP QCD

( ) = 4α em

2

f

! t

( )δV QCD t ( )

t=0 ∞

∑

δaµ

HVP QED

( ) = 4α em

2

f

! t

( )

δVf

QED t

( )

f =u,d,s,c

∑

t=0 ∞

∑

md − mu

( ) ZP

3 0 T Ji

† x

! ,t

( ) qd

2ψ dψ d − qu 2ψ uψ u

⎡ ⎣ ⎤ ⎦ Ji 0

( )

{ }

x ! ,y

∑

i=1,2,3

∑

δaµ

HVP = δaµ HVP QCD

( )+δaµ

HVP QED

( )

QEDL photon zero-mode:

• M. Hayakawa and S. Uno, 2008

md − mu

[ ] MS,2 GeV

( ) = 2.38 18

( ) MeV

DG et al., 2017

QCD/QED separation is scheme and scale dependent

m f MS,2 GeV

( ) = m f

0 MS,2 GeV

( )

• J. Gasser et al., 2003

RM123 method

10 20 30 40 50

t / a

• 200

200 400

a

3 δV QCD*10 10

Mπ ! 260 MeV a ! 0.06 fm

D20.48

δaµ

HVP QCD

( ) = 4α em

2

f

! t

( )δV QCD t ( )

t=0 ∞

∑

quark-connected terms only

• G. M. de Divitiis et al.,

2012; 2013 qQED approximation

SLIDE 15

LIB corrections

15

h h ) pol

δV exch t

( )+δV self t ( )+δV tad t ( )+δV PS t ( )+δV S t ( )+δV ZA t ( )

δaµ

HVP QED

( ) = 4α em

2

f

! t

( )

δVf

QED t

( )

f =u,d,s,c

∑

t=0 ∞

∑

UV and IR finite

e.m. shift of the critical mass

• G. M. de Divitiis et al., 2013

δV ZA t

( ) = −2.51406 α emq f

2 ZA fact V t

( )

ZA

fact = 0.95 5

( )

preliminary

O α emα s

n

( )

RI-MOM

• G. Martinelli and

Y.-C. Zhang, 1982

10 20 30 40

t / a

200 400 600

a

3 δV QED*10 10

Mπ ! 260 MeV a ! 0.06 fm

D20.48

u-,d-quark contributions

ZA = ZA

( ) 1− 2.51406 α emq f

2 ZA fact

( )+O α em

2

( )

Mtm-LQCD setup + local Ji(x) with opposite Wilson r-parameters

perturbative estimate at LO

SLIDE 16

LIB corrections

16

h h )

δaµ

HVP = 4α em 2

f

! t

( )δV t ( )+

f

! t

( )δ

GV

f

2MV

f e−MV

ft

⎡ ⎣ ⎢ ⎤ ⎦ ⎥

t=Tdata+a ∞

∑

t=0 Tdata

∑

⎧ ⎨ ⎪ ⎩ ⎪ ⎫ ⎬ ⎪ ⎭ ⎪

δV t

( )

V t

( )

t≫a

⎯ → ⎯ δGV GV − δ MV MV 1+ MVt

( )

RM123 method

lattice data analytic repr.

t / a

δV QCD V

Mπ ! 260 MeV a ! 0.06 fm D20.48

• G. M. de Divitiis et al., 2012; 2013

t / a

δV QED V

Mπ ! 260 MeV a ! 0.06 fm D20.48

u-,d-quark contributions

SLIDE 17

17

LIB corr.: results

h h )

FVEs expected to start at (neutral mesons with vanishing charge radius)

O 1 L3

( ) δaµ

HVP ud

( )

aµ

HVP ud

( ) = δ 0 1+δ1mud +δ 2χ + Da2 + FVE

⎡ ⎣ ⎤ ⎦ phenomenological fitting function

FVE = Fe−MπL

FVE = F ! L3

χ = mud

2

χ = mud log mud

( )

FVE = F

!

Mπ

2

16π 2 fπ

2

e−MπL MπL

( )

n

0.01 0.02 0.03 0.04 0.05

mud (GeV)

0.005 0.01 0.015 0.02

δaµ

HVP(ud) / aµ HVP(ud)

β=1.90, L=20 β=1.90, L=24 β=1.90, L=32 β=1.90, L=40 β=1.95, L=24 β=1.95, L=32 β=2.10, L=48 physical point

δaµ

HVP ud

( )

aµ

HVP ud

( )

systematics

DG et al., 2017

in the ratio various systematics cancel out δaµ

HVP ud

( )

aµ

HVP ud

( ) = 0.011 3 ( )stat+ fit 2 ( )chir 2 ( )FVE 1 ( )disc 1 ( )ZA ... ( )qQED

= 0.011 4

( )

preliminary

quark-connected terms only

aµ

HVP ud

( ) = 619.4 16.6 ( )⋅10−10

δaµ

HVP ud

( ) = 7 2 ( )⋅10−10

SLIDE 18

18

IB corr.: comparison

h h )

δaµ

HVP ud

( ) = 7 2 ( )⋅10−10

δaµ

QCD MS,2 GeV

( ) = 5.6 2.0

( )⋅10−10

δaµ

QED MS,2 GeV

( ) = 1.3 0.9

( )⋅10−10

u-,d-quark contributions

δaµ

HVP s

( ) = −0.018 11 ( )⋅10−10

δaµ

HVP c

( ) = −0.030 13 ( )⋅10−10

DG et al., 2017

∼80% due to strong IB

δaµ

HVP ud

( ) = 7.8 5.1 ( )⋅10−10

δaµ

HVP ud

( ) = 9.5 10.2 ( )⋅10−10

δaµ

HVP ud;QCD

( ) = 9.0 4.5 ( )⋅10−10

negligible s-,c-quark contributions quark-connected terms only

• Sz. Borsanyi et al., 2018
• T. Blum et al., 2018
• B. Chakraborty et al., 2018

estimate from π 0γ,ηγ, ρ −ω mixing,Mπ ±

strong IB only

x

p r e l i m i n a r y

SLIDE 19

Conclusions

19

550 600 650 700 750 a µ HVP * 10 10 ETMC 18 RBC/UKQCD 18 BMW 17 CLS/Mainz 17 HPQCD 16 KNT18 no New Physics DHMZ 17 FJ 17 RBC/UKQCD 18

The HVP contribution is currently one of the most important sources

• f the theoretical uncertainty to the muon (g-2)

We have completed our lattice calculation of , by determining the light quark contribution at and (s- and c-quark contributions

already published on JHEP). The results will appear soon on arXiv

aµ

HVP

O α em

2

( )

O α em

3

( )

evaluation of the quark-disconnected terms and relaxation of the qQED approximation non-perturbative determinations of the e.m. corrections to the RCs of bilinear operators use of the new ETMC lattice setup @ the physical pion point systematic study of FVEs in the strong and QED IB corrections

RM123 method

LQCD

preliminary

In progress…

Future perspectives

aµ

HVP = 683 18

( )⋅10−10

δaµ

HVP = 7 2

( )⋅10−10

SLIDE 20

Backup slides

SLIDE 21

IB corr.: s-,c-quark contributions

• 0.012
• 0.008
• 0.004

0.000 0.004 0.008 10 20 30 40 50 total self tad + PS exch scalar Z

A

δa

µ s (t) * 10 10

t / a

D20.48 strange contribution

• 0.04
• 0.03
• 0.02
• 0.01

0.00 0.01 0.02 0.03 5 10 15 20 total self tad + PS exch scalar Z

A

δa

µ c (t) * 10 10

t / a

D20.48 charm contribution

• r

ated

SLIDE 22

Tuning the critical mass

The Dashen theorem: in the massless theory, the neutral pion and kaon are Goldstone bosons even in the presence of electromagnetic interactions: 1

lim

mf →0 Mπ 0 = lim mf →0 M K0 = 0

With twisted mass fermions, one can extend the method used also in the isosymmetric QCD case, based on a specific Ward-Takahashi identity: 2

∇µ Vµ

1(x)P 5 2(0) = 0

More precise: it does not require a chiral extrapolation