[PPT] - The hadronic contribution to the running of the electroweak mixing PowerPoint Presentation

SLIDE 1

The hadronic contribution to the running

f the electroweak mixing angle

Marco Cè a,b Miguel Teseo San José Pérez a,b Antoine Gérardin c Harvey B. Meyer a,b Kohtaroh Miura a,d Konstantin Ottnad b Jonas Wilhelm b Hartmut Wittig a,b

a. Helmholtz-Institut Mainz, Johannes Gutenberg-Universität Mainz
b. PRISMA+ Cluster of Excellence and Institut für Kernphysik, Johannes Gutenberg-Universität Mainz
c. John von Neumann-Institut für Computing, DESY Zeuthen
d. Kobayashi-Maskawa Institute for the Origin of Particles and the Universe, Nagoya University

37th International Symposium on Lattice Field Theory 武汉, 17th June 2019

SLIDE 2

introduction – the electroweak mixing angle

the electroweak mixing (Weinberg) angle 𝜄W parametrizes the mixing between the SU(2)𝑀 and U(1)𝑍 sectors of the Standard Model. At tree level,

sin2 𝜄W = 𝑕′2 𝑕2 + 𝑕′2 ,

where 𝑕 and 𝑕′ are the SU(2)𝑀 and U(1)𝑍 coupling respectively

it is a free parameter of the Standard Model
sin2 𝜄W = 1 −

𝑁2

𝑋

𝑁2

𝑎

𝑎 vector coupling 𝑤𝑔 = 𝑈𝑔 − 2𝑅𝑔 sin2 𝜄efg

𝑔

weak charge of the proton 𝑅𝑋(𝑞) ∼ 1 − 4 sin2 𝜄W

the precise numerical value of sin2 𝜄W depends on the renormalization scheme and on the energy scale

Marco Cè (HIM, JGU Mainz) The hadronic contribution to the running of the electroweak mixing angle 17/06/2019 1 / 13

SLIDE 3

the running – a precision test of the SM

4 −

10

3 −

10

2 −

10

1 −

10 1 10

2

10

3

10

4

10 [GeV]

µ

0.225 0.23 0.235 0.24 0.245

) µ (

W

θ

2

sin

LHC LEP 1 SLC Tevatron NuTeV eDIS SLAC-E158 Qweak APV LHC LEP 1 SLC Tevatron NuTeV eDIS SLAC-E158 Qweak APV RGE Running Particle Threshold Measurements

[PDG 2018]

sin2 𝜄W(𝑅2) = sin2 𝜄0[1 + 𝛦 sin2 𝜄W(𝑅2)]

experiments: at high 𝑅2

measurements at colliders

at low 𝑅2, upcoming

MOLLER @ JLab
P2 @ MESA, Mainz

[Becker et al. 2018]

⇒ non-perturbative QCD efgects

[talk by J. Wilhelm, Had. Struct., Fri. 15:00]

theory: running, in the MS scheme

sin2 𝜄0 = 0.238 68(5) in the

Thomson limit [Erler, Ferro-Hernández 2017]

the running at scales ≲ 𝛭QCD is

afgected by non-perturbative QCD physics and global fjts to EW precision data

Marco Cè (HIM, JGU Mainz) The hadronic contribution to the running of the electroweak mixing angle 17/06/2019 2 / 13

SLIDE 4

the hadronic contribution

the leading hadronic contribution to the running of sin2 𝜄W is

[Jegerlehner 1986; 2011]

𝛦had sin2 𝜄W(𝑅2) = − 4π𝛽 sin2 𝜄W 𝛲𝑎𝛿

𝑆 (𝑅2),

𝛲𝑎𝛿

𝑆 (𝑅2) = 𝛲𝑎𝛿(𝑅2) − 𝛲𝑎𝛿(0),

proportional to the subtracted hadronic vacuum polarization

(𝑅𝜈𝑅𝜉 − 𝜀𝜈𝜉𝑅2)𝛲𝑎𝛿(𝑅2) = 𝛲𝑎𝛿

𝜈𝜉 (𝑅2) = ∫ d4𝑦 𝑓i𝑅𝑦⟨𝑘𝑎 𝜈 (𝑦)𝑘𝛿 𝜉(0)⟩

f the e.m. current and the vector part of the 𝑎 current

𝑘𝛿

𝜈 = 2

3 ̄ 𝑣𝛿𝜈𝑣 − 1 3 ̄ 𝑒𝛿𝜈𝑒 − 1 3 ̄ 𝑡𝛿𝜈𝑡 + 2 3 ̄ 𝑑𝛿𝜈𝑑, 𝑘𝑈3

𝜈 = 1

4 ̄ 𝑣𝛿𝜈𝑣 − 1 4 ̄ 𝑒𝛿𝜈𝑒 − 1 4 ̄ 𝑡𝛿𝜈𝑡 + 1 4 ̄ 𝑑𝛿𝜈𝑑, 𝑘𝑎

𝜈 = 𝑘𝑈3 𝜈 − sin2 𝜄W𝑘𝛿 𝜈,

can be extracted from phenomenology using dispersion relations
or can be computed ab initio on the lattice

[Burger et al. 2015; Gülpers et al. 2015]

similarly, the hadronic contribution to the running of 𝛽QED is given by 𝛲𝛿𝛿

𝑆 (𝑅2)

[next talk by M. T. San José Pérez]

Marco Cè (HIM, JGU Mainz) The hadronic contribution to the running of the electroweak mixing angle 17/06/2019 3 / 13

SLIDE 5

the time-momentum representation (TMR) method

introduced for the HVP contribution to (𝑕 − 2)𝜈

[Bernecker, Meyer 2011; Francis et al. 2013]

𝛲𝑎𝛿

𝑆 (𝑅2) = ∫ ∞

d𝑦0 𝐻𝑎𝛿(𝑦0)[𝑦2

0 − 4

𝑅2 sin2 ( 𝑅𝑦0 2 )], 𝐻𝑎𝛿(𝑦0) = −1 3 ∫ d3𝑦

3

∑

𝑙=1

⟨𝑘𝑎

𝑙 (𝑦)𝑘𝛿 𝑙(0)⟩,

⇒ using correlators from 𝑂f = 2 + 1 Mainz efgort in computing (𝑕 − 2)HVP

𝜈

[Gérardin et al. 2019; talk by A. Gérardin, Had. Struct., Tue. 14:40]

non-perturbatively 𝒫(𝑏)-improved vector currents

[Gérardin, Harris, Meyer 2018]

two discretizations: local-local and local-conserved
w.r.t. the (𝑕 − 2)HVP

𝜈

case, the kernel has a shorter range

expect 𝛦had sin2 𝜄W to be more sensitive at cut-ofg efgects, especially at high 𝑅2
but much simpler large-distance systematic

⇒ no loss of signal in the tail of the connected correlator

Marco Cè (HIM, JGU Mainz) The hadronic contribution to the running of the electroweak mixing angle 17/06/2019 4 / 13

SLIDE 6

lattice correlators

with SU(3)𝐺 notation, in the isospin-symmetric limit (light quark ℓ: either 𝑣 or 𝑒):

𝐻33

𝜈𝜉(𝑦) = 1

2𝐷ℓ,ℓ

𝜈𝜉 (𝑦),

𝐻88

𝜈𝜉(𝑦) = 1

6[𝐷ℓ,ℓ

𝜈𝜉 (𝑦) + 2𝐷𝑡,𝑡 𝜈𝜉 (𝑦) + 2𝐸ℓ−𝑡,ℓ−𝑡 𝜈𝜉

(𝑦)], 𝐻08

𝜈𝜉(𝑦) =

1 2√3[𝐷ℓ,ℓ

𝜈𝜉 (𝑦) − 𝐷𝑡,𝑡 𝜈𝜉 (𝑦) + 𝐸2ℓ+𝑡,ℓ−𝑡 𝜈𝜉

(𝑦)],

where the connected and disconnected Wick’s contractions are

𝐷𝑔1,𝑔2

𝜈𝜉

(𝑦) = −⟨Tr{𝐸−1

𝑔1 (𝑦, 0)𝛿𝜈𝐸−1 𝑔2 (0, 𝑦)𝛿𝜉}⟩,

𝐸𝑔1,𝑔2

𝜈𝜉

(𝑦) = ⟨Tr{𝐸−1

𝑔1 (𝑦, 𝑦)𝛿𝜈} Tr{𝐸−1 𝑔2 (0, 0)𝛿𝜉}⟩,

the 𝑎𝛿 correlator is given by

𝐻𝑎𝛿 = ( 1 2 − sin2 𝜄W)(𝐻𝛿𝛿) − 1 6√3 𝐻08, 𝐻𝛿𝛿 = 𝐻33 + 1 3𝐻88,

where 𝐻𝛿𝛿 is the e.m. current correlator, relevant for e.g. 𝛦had𝛽QED(𝑅2), 𝑏HVP

𝜈

, …

Marco Cè (HIM, JGU Mainz) The hadronic contribution to the running of the electroweak mixing angle 17/06/2019 5 / 13

SLIDE 7

ensembles

from the CLS initiative

[Bruno et al. 2015, Bruno, Korzec, Schaefer 2017]

tree-level Lüscher-Weisz gauge action, non-perturbatively 𝒫(𝑏)-improved Wilson fermions, open BCs in time

𝑈 /𝑏 𝑀/𝑏 𝑏 [fm] 𝑀 [fm] 𝑛𝜌 [MeV] 𝑛𝐿 [MeV] 𝑛𝜌𝑀

H101

96 32 0.086 2.8 415 415 5.8

H102

96 32 2.8 355 440 5.0

H105*

96 32 2.8 280 460 3.9

N101

128 48 4.1 280 460 5.8

C101*

96 48 4.1 220 470 4.6

S400

128 32 0.076 2.4 350 440 4.3

N401*

128 48 3.7 285 460 5.3

H200

96 32 0.064 2.1 420 420 4.4

N202

128 48 3.1 410 410 6.4

N203*

128 48 3.1 345 440 5.4

N200*

128 48 3.1 285 465 4.4

D200*

128 64 4.1 200 480 4.2

E250§

192 96 6.2 130 490 4.1

N300

128 48 0.050 2.4 420 420 5.1

N302*

128 48 2.4 345 460 4.2

J303

192 64 3.2 260 475 4.2

* disconnected contribution available, § periodic BCs in time

Marco Cè (HIM, JGU Mainz) The hadronic contribution to the running of the electroweak mixing angle 17/06/2019 6 / 13

SLIDE 8

preliminary results

E250: physical meson masses, 𝑏 = 0.064 26(74) fm

1 2 3 4 5 𝑢0𝑅2 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 𝛲𝑆(𝑅2) 33 88, conn. 08, conn. 2 4 6 8 10 𝑅2 [GeV2]

at 𝑅2 = 1 GeV2 l.c. l.l.

33 0.035 70(36) 0.035 29(36) 88 0.026 00(12) 0.025 59(12) 08 0.008 26(21)

𝛦had sin2 𝜄W(𝑅2) = ⎧ ⎪ ⎨ ⎪ ⎩ −0.002 484(39) 𝑅2 = 0.24 GeV2 −0.005 888(40) 𝑅2 = 1 GeV2 −0.010 329(41) 𝑅2 = 4.22 GeV2

Marco Cè (HIM, JGU Mainz) The hadronic contribution to the running of the electroweak mixing angle 17/06/2019 7 / 13

SLIDE 9

preliminary results – including disconnected

N200: 𝑁𝜌 ≈ 285 MeV, 𝑏 = 0.064 26(74) fm

0.00 0.02 0.04 0.06 0.08 𝛲𝑆(𝑅2) 33 88, conn. 08, conn. 1 2 3 4 5 𝑢0𝑅2 −0.002 −0.001 0.000 𝛲𝑆(𝑅2) 88, disc. 08, disc. 2 4 6 8 10 𝑅2 [GeV2]

at 𝑅2 = 1 GeV2 l.c. l.l.

33 0.030 11(11) 0.029 72(11) 88 0.025 40(5) 0.025 00(5) 08 0.003 93(6) 88 −0.000 32(7) −0.000 32(7) 08 −0.001 09(29)

𝛦had sin2 𝜄W(𝑅2) = ⎧ ⎪ ⎨ ⎪ ⎩ −0.002 115(10) 𝑅2 = 0.24 GeV2 −0.005 427(14) 𝑅2 = 1 GeV2 −0.009 874(15) 𝑅2 = 4.22 GeV2

connected only!

Marco Cè (HIM, JGU Mainz) The hadronic contribution to the running of the electroweak mixing angle 17/06/2019 8 / 13

SLIDE 10

preliminary results – including disconnected

N200: 𝑁𝜌 ≈ 285 MeV, 𝑏 = 0.064 26(74) fm

0.00 0.02 0.04 0.06 0.08 𝛲𝑆(𝑅2) 33 88, conn. 08, conn. 1 2 3 4 5 𝑢0𝑅2 −0.002 −0.001 0.000 𝛲𝑆(𝑅2) 88, disc. 08, disc. 2 4 6 8 10 𝑅2 [GeV2]

at 𝑅2 = 1 GeV2 l.c. l.l.

33 0.030 11(11) 0.029 72(11) 88 0.025 40(5) 0.025 00(5) 08 0.003 93(6) 88 −0.000 32(7) −0.000 32(7) 08 −0.001 09(29)

𝛦had sin2 𝜄W(𝑅2) = ⎧ ⎪ ⎨ ⎪ ⎩ −0.002 138(13) 𝑅2 = 0.24 GeV2 −0.005 457(16) 𝑅2 = 1 GeV2 −0.009 905(17) 𝑅2 = 4.22 GeV2

Marco Cè (HIM, JGU Mainz) The hadronic contribution to the running of the electroweak mixing angle 17/06/2019 8 / 13

SLIDE 11

preliminary results – chiral and continuum extrapolation

at 𝑅2 = 1 GeV2

, ∶𝛲33

𝑆 ⇒

▾, ▴ ∶𝛲88

𝑆 ⇒

∶𝛲08

𝑆 ⇒ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 𝜚2 = 8𝑢0𝑁2

𝜌

0.00 0.01 0.02 0.03 0.04 0.05 𝛲𝑆(𝑅2) 140 200 240 280 315 340 370 395 420 𝑁𝜌 [MeV] 0.086 fm 0.076 fm 0.064 fm 0.050 fm

Marco Cè (HIM, JGU Mainz) The hadronic contribution to the running of the electroweak mixing angle 17/06/2019 9 / 13

SLIDE 12

preliminary results – chiral and continuum extrapolation

at 𝑅2 = 1 GeV2, local-conserved only

∶𝛲33

𝑆 ⇒

▴ ∶𝛲88

𝑆 ⇒ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 𝜚2 = 8𝑢0𝑁2

𝜌

0.024 0.026 0.028 0.030 0.032 0.034 0.036 𝛲𝑆(𝑅2) 140 200 240 280 315 340 370 395 420 𝑁𝜌 [MeV] 0.086 fm 0.076 fm 0.064 fm 0.050 fm

Marco Cè (HIM, JGU Mainz) The hadronic contribution to the running of the electroweak mixing angle 17/06/2019 9 / 13

SLIDE 13

fjnite-size correction

added to the 𝐽 = 1 correlator 𝐻33(𝑢), with 𝑢𝑗 = (𝑛𝜌𝑀/4)2/𝑛𝜌

[Gérardin et al. 2019; talk by A. Gérardin, Had. Struct., Tue. 14:40]

𝑢 < 𝑢𝑗: correction from scalar QED (a.k.a. NLO χPT)

[Francis et al. 2013; Della Morte et al. 2017]

𝐻33(𝑢, 𝑀) − 𝐻33(𝑢, ∞) = 1 3 ( 1 𝑀3 ∑

⃗ 𝑙

− ∫ d3⃗ 𝑙 (2π)3 ) ⃗ 𝑙2 + 𝑛2

𝜌

⃗ 𝑙2 e−2𝑢√⃗

𝑙2+𝑛2

𝜌

𝑢 > 𝑢𝑗: correction from GS model of 𝐺𝜌(𝜕)

[Gounaris, Sakurai 1968]

𝐻33(𝑢, ∞) = ∫

∞

d𝜕 𝜕2𝜍(𝜕2)e−𝜕𝑢 𝜍(𝜕2) =

1 48π2 (1 − 4𝑛2

𝜌

𝜕2 )

3 2 |𝐺𝜌(𝜕)|

2

and the corresponding fjnite-volume correlator

[Lüscher 1991; Lellouch, Lüscher 2000; Meyer 2011]

𝐻33(𝑢, 𝑀) = ∑

𝑜

|𝐵𝑜|

2e−𝜕𝑜𝑢

with Lüscher’s 𝜕𝑜 and LL’s 𝐵𝑜

Marco Cè (HIM, JGU Mainz) The hadronic contribution to the running of the electroweak mixing angle 17/06/2019 10 / 13

SLIDE 14

at 𝑅2 = 1 GeV2, local-conserved and local-local, with fjnite-size correction

, ∶𝛲33

𝑆 ⇒

▾, ▴ ∶𝛲88

𝑆 ⇒

∶𝛲08

𝑆 ⇒ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 𝜚2 = 8𝑢0𝑁2

𝜌

0.00 0.01 0.02 0.03 0.04 0.05 𝛲𝑆(𝑅2) 140 200 240 280 315 340 370 395 420 𝑁𝜌 [MeV] 0.086 fm 0.076 fm 0.064 fm 0.050 fm fjt ansatz and preliminary results in the appendix

Marco Cè (HIM, JGU Mainz) The hadronic contribution to the running of the electroweak mixing angle 17/06/2019 12 / 13

SLIDE 20

conclusions & outlook

the leading hadronic contribution to the running sin2 𝜄W can be computed on the lattice

with ≈ 1 % errors, competitive with phenomenology
including the disconnected contribution, with sub-percent determination
lattice provides fmavour separation ⇒ input for the dispersive approach
correction for fjnite-size efgects is essential
include the valence charm contribution (correlators already available)
investigate the systematics of the chiral continuum extrapolation
isospin breaking efgects

[talk by A. Risch, Had. Struct., Tue. 14:20]

Marco Cè (HIM, JGU Mainz) The hadronic contribution to the running of the electroweak mixing angle 17/06/2019 13 / 13

SLIDE 21

thanks for your attention! questions?

SLIDE 22

backup slides

SLIDE 23

preliminatry fjts

input: 𝑏2/𝑢0, 𝜚2 = 8𝑢0𝑁2

𝜌, 𝜚4 = 8𝑢0(𝑁2 𝐿 + 𝑁2 𝜌/2)

𝑢0 from CLS scale-setting paper

[Bruno, Korzec, Schaefer 2017]

𝑏𝑁𝜌 from Mainz 𝑂f = 2 + 1 (𝑕 − 2)𝜈 paper

[Gérardin et al. 2019]

𝑏𝑁𝐿 form CLS scale-setting paper and dedicated (preliminary!) measurements

fjt ansatz, at fjxed 𝑅2:

𝑔(𝑏2/𝑢0, 𝜚2, 𝜚4; 𝑞0, 𝜀1, 𝜀2, 𝛿1, 𝛿2, 𝛿4, 𝛿9) = 𝑞0 + 𝜀1𝑏2/𝑢0 + 𝜀2(𝑏2/𝑢0)3/2 + 𝛿1(𝜚2 − 𝜚0

2) + 𝛿2(log 𝜚2 − log 𝜚0 2) + 𝛿4(𝜚2 − 𝜚0 2)2 + 𝛿9(𝜚4 − 𝜚0 4)

combined fjt of the local-local and local-conserved discretizations
no error on 𝑏2/𝑢0, 𝜚2 = 8𝑢0𝑁2

𝜌, 𝜚4 = 8𝑢0(𝑁2 𝐿 + 𝑁2 𝜌/2) contributing to the 𝜓2 (yet)

preliminary results:

fjt of 𝛲33

𝑆 , with 𝛿1 = 0: 𝜓2/#dof = 26.99/24 = 1.12, 𝑞-value = 0.30

fjt of 𝛲88

𝑆 , with 𝛿2 = 0: 𝜓2/#dof = 73.40/24 = 3.06, 𝑞-value < 0.01

fjt of 𝛲𝛿𝛿

𝑆 , with 𝛿1 = 0: 𝜓2/#dof = 32.28/24 = 1.35, 𝑞-value = 0.12

Marco Cè (HIM, JGU Mainz) The hadronic contribution to the running of the electroweak mixing angle 17/06/2019 1 / 3

SLIDE 24

renormalization and 𝒫(𝑏) improvement

for the local current

[Bhattacharya et al. 2006, […], Gérardin, Harris, Meyer 2018]

𝑊 3

𝜈,𝑆 = 𝑎𝑊(1 + 3 ̄

𝑐𝑊𝑏𝑛av

𝑟 + 𝑐𝑊𝑏𝑛𝑟,ℓ)𝑊 3,𝐽 𝜈 ,

𝑊 8

𝜈,𝑆 = 𝑎𝑊[(1 + 3 ̄

𝑐𝑊𝑏𝑛av

𝑟 + 𝑐𝑊 𝑏(𝑛𝑟,ℓ+2𝑛𝑟,𝑡) 3

)𝑊 8,𝐽

𝜈

+ (

𝑐𝑊 3 + 𝑔𝑊) 2𝑏(𝑛𝑟,ℓ−𝑛𝑟,𝑡) √3

𝑊 0,𝐽

𝜈 ],

𝑊 0

𝜈,𝑆 = 𝑎𝑊𝑠𝑊[(1 + (3 ̄

𝑒𝑊 + 𝑒𝑊)𝑏𝑛av

𝑟 )𝑊 0,𝐽 𝜈

+ 𝑒𝑊

𝑏(𝑛𝑟,ℓ−𝑛𝑟,𝑡) √3

𝑊 8,𝐽

𝜈 ]

where

𝑊 𝑏,𝐽

𝜈

= 𝑊 𝑏

𝜈 + 𝑏𝑑𝑊𝜖0𝑈 𝑏 0𝜈,

𝑊 0,𝐽

𝜈

= 𝑊 0

𝜈 + 𝑏 ̄

𝑑𝑊𝜖0𝑈 0

0𝜈.

while for the conserved current

𝑊 𝑏

𝜈,𝑆 = 𝑊 𝑏 𝜈 + 𝑏𝑑𝑑𝑡 𝑊 𝜖0𝑈 𝑏 0𝜈,

𝑊 0

𝜈,𝑆 = 𝑊 0 𝜈 + 𝑏 ̄

𝑑𝑑𝑡

𝑊 𝜖0𝑈 0 0𝜈.

⇒ we use only the conserved vector current for the fmavour-singlet component, and we set 𝑔𝑊 = 0, ̄ 𝑑𝑑𝑡

𝑊 = 𝑑𝑑𝑡 𝑊 .

Marco Cè (HIM, JGU Mainz) The hadronic contribution to the running of the electroweak mixing angle 17/06/2019 2 / 3

SLIDE 25

fjnite-size correction – an example

using the TMR kernel to compute (𝑕 − 2)𝜈

10 20 30 40 𝑢/𝑏 −1 1 2 3 𝑏𝜈 ×10−10

H200, 𝑁𝜌𝑀 = 4.4

10 20 30 40 𝑢/𝑏

N202, 𝑁𝜌𝑀 = 6.4

lattice correlator error scalar QED correction GS model correction

Marco Cè (HIM, JGU Mainz) The hadronic contribution to the running of the electroweak mixing angle 17/06/2019 3 / 3