Combined explanations of (g-2), (g-2)e and implications...

Combined explanations of (g − 2)µ, (g − 2)e

and implications for a large muon EDM

Martin Hoferichter

Institute for Nuclear Theory

University of Washington

Rencontres de Moriond: EW 2019

La Thuile, March 17

A. Crivellin, MH, P. Schmidt-Wellenburg PRD 98 (2018) 113002

M. Hoferichter (Institute for Nuclear Theory) Correlations between (g − 2)µ,e and muon EDM La Thuile, March 17, 2019 1

Lepton dipole moments

Dipole moments

H = −µℓ · B − dℓ · E ℓ = e, µ, τ

µℓ = −gℓ

e

2mℓ

S dℓ = −ηℓe

2mℓ

S aℓ =gℓ − 2

2

Usually quote aℓ for magnetic and dℓ =ηℓ2

e2mℓ

for electric dipole moment

Anomalous magnetic moment Schwinger 1948

aLOℓ =

α

2π= 0.00116 . . .

with fine structure constant α−1 = 137.035999 . . .

EDM violates CP symmetry, very small in SM Pospelov, Ritz 2014

|dSMe | = O(10−44e cm) |dequiv

e | = O(10−38e cm)

Any deviation from SM value could be a hint for BSM physics!


Lepton dipole moments: experimental status

Anomalous magnetic moments Hanneke et al. 2008, Bennett et al. 2006

aexpe = 1,159,652,180.73(28)× 10−12 aexp

µ = 116,592,089(63)× 10−11

Electric dipole moments Andreev et al. 2018, Bennett et al. 2009

|de| < 1.1 × 10−29e cm |dµ| < 1.5 × 10−19e cm 90%C.L.

Not much known about τ dipole moments, some limits from

e+

e−

τ+

τ−

e+

e−

e+

e−

τ+

τ−


A tension in (g − 2)e

For SM prediction need independent input for α, from atomic interferometry

Until recently:

Best value from Rb Bouchendira et al. 2010, leading to

∆ae = aexpe − aSM

e = −1.30(77)× 10−12 [1.7σ]

Uncertainty totally dominated by ∆aSMe = 0.72 × 10−12, i.e. ∆α


A tension in (g − 2)e

For SM prediction need independent input for α, from atomic interferometry

Until recently:

Best value from Rb Bouchendira et al. 2010, leading to

∆ae = aexpe − aSM

e = −1.30(77)× 10−12 [1.7σ]

Uncertainty totally dominated by ∆aSMe = 0.72 × 10−12, i.e. ∆α

Last year:

New measurement of α in Cs Parker et al. 2018

aexpe = 1,159,652,180.73(28)× 10−12

aSMe = 1,159,652,181.61(23)× 10−12

→ ∆ae = −0.88(36)× 10−12 [2.5σ]

Dominant uncertainty now in aexpe

-1.9 -1.4 -0.9 -0.4 0.1 0.6

(-1

/137.035999139 - 1) 109

Cs

g-2

Rb


On the Standard Model predictions

How robust is ∆ae?

Status ca. 2012: Giudice et al.

aSMe = 1,159,652,181.78(6)4-loop(4)5-loop(2)had(76)Rb × 10−12

Since then

Analytic cross check of heavy-fermion loops at 4 loops Kurz et al. 2014

Semi-analytic calculation of mass-independent terms at 4 loops Laporta 2017

Improved numerical calculation of 5-loop coefficient Kinoshita et al. 2018

aSMe = 1,159,652,181.61(1)5-loop(1)had(23)Cs × 10−12

aexpe = 1,159,652,180.73(28)× 10−12

→ from theory perspective: golden opportunity!

For the muon, long-standing 3–4σ tension ∆aµ = aexpµ − aSM

µ ∼ 270(85)× 10−11

→ SM theory: Muon g − 2 Theory Initiative upcoming white paper prior to E989


EFT analysis

Effective dipole operators Heff = cℓf ℓiR ℓfσµνPRℓiF

µν + h.c.

aℓ = −4mℓ

eRe cℓℓ

R dℓ = −2 Im cℓℓR Br[µ → eγ] =

m3µ

4π Γµ

(

|ceµR

|2 + |cµeR

|2)

→ in general only one power in mℓ for aℓ

Consequences

Phase of ceeR

much better constrained than phase of cµµ

R

∣

∣

∣

∣

Im ceeR

Re ceeR

∣

∣

∣

∣

. 6 × 10−7

∣

∣

∣

∣

Im cµµR

Re cµµR

∣

∣

∣

∣

. 600

For models that fulfill ceµR

=√

ceeR

cµµR

Br[µ → eγ] =αm2

µ

16meΓµ|∆aµ∆ae| ∼ 8 × 10−5

µR, eR µL, eL

γL

φ, V

→ violates MEG bound Br[µ → eγ] < 4.2 × 10−13 by 8 orders of magnitude!


Possible BSM explanations

BSM contribution large

aBSMµ ∼ 270(85)× 10−11 aEW

µ = 153.6(1.0)× 10−11

→ need some form of enhancement

Light new particle: dark photon, light scalar, . . .

Dark (axial) photon yields positive (negative) sign

Light scalar Davoudiasl, Marciano 2018: interplay of one- and two-loop diagrams

cℓℓR

by construction real, i.e. no EDM

Chiral enhancement

Higgs coupling larger than mℓ/v : tan β in MSSM, mt/mℓ for LQs, . . .

Chirality flip does not come from mℓ, but from new heavy fermion in the loop

cℓℓR

can be complex, with a priori arbitrary phase


Model with new vector-like fermions

ℓR ℓRℓL ℓL

γLj

W,Z

γ

h

Lj

Vector-like fermions + Higgs

LM = −MLLLLR − ME ELER + h.c.

LH = −κLLLHER − κE LRHEL − λLLLℓRH − λE ERHℓL + h.c.

Chirally enhanced by κL,Rv

mµ


Model with new vector-like fermions

Works for ae but tensions with aµ

Modifications

New heavy scalar to explain aµ

Impose Abelian flavor symmetry, e.g. Lµ − Lτ , to avoid µ → eγ, correlations with

b → sℓℓ anomalies? Altmannshofer et al. 2014, . . .

Minimal model if ae explained with SM Higgs and aµ with new heavy scalar

In all cases: no correlations between ceeR and cµµ

R

→ phase of cµµ

R not constrained by |de| and thus |dµ| could be sizable


Indirect limit on muon EDM

Minimal flavor violation

|dMFVµ | =

mµ

me|de| < 2.3 × 10−27e cm

Direct limit E821

|dµ| < 1.5 × 10−19e cm 90%C.L.

Indirect limit from electron EDM ACME 2018

|dµ| ≤

[

(

15

4ζ(3)−

31

12

)

me

mµ

(

α

π

)3]

−1

|de|

≤ 0.9 × 10−19e cm 90%C.L.

µ

e


Future experimental sensitivity

Current limit E821: |dµ| < 1.5 × 10−19e cm

Fermilab/J-PARC (g − 2)µ experiments will be sensitive to |dµ| ∼ 10−21e cm

Proposal for a dedicated muon EDM experiment at PSI, could reach

|dµ| ∼ 5 × 10−23e cm


Conclusions

∆aµ > 0

small ∆ae > 0

Minimal flavor

violationsmall |dµ|

∆aµ > 0

sizeable ∆ae < 0

Generic chiral

enhancement|dµ| unconstrained

∆aµ > 0

sizeable ∆ae > 0

Light

particles|dµ| zero


On the Standard Model prediction: muon

aµ

[

10−11]

∆aµ

[

10−11]

experiment 116 592 089. 63.

QED O(α) 116 140 973.32 0.08

QED O(α2) 413 217.63 0.01

QED O(α3) 30 141.90 0.00

QED O(α4) 381.01 0.02

QED O(α5) 5.09 0.01

QED total 116 584 718.95 0.08

electroweak, total 153.6 1.0

HVP (LO) ∼ 6 930. ∼ 40.

HVP (NLO) −98. 1.

HLbL (LO) ∼ 100. ∼ 40.

HVP (NNLO) 12.4 0.1

HLbL (NLO) 3. 2.

theory ∼ 116 591 820. ∼ 57.

Schwinger 1948



aµ

[

10−11]

∆aµ

[

10−11]

experiment 116 592 089. 63.

QED O(α) 116 140 973.32 0.08

QED O(α2) 413 217.63 0.01

QED O(α3) 30 141.90 0.00

QED O(α4) 381.01 0.02

QED O(α5) 5.09 0.01

QED total 116 584 718.95 0.08


HVP (LO) ∼ 6 930. ∼ 40.

HVP (NLO) −98. 1.

HLbL (LO) ∼ 100. ∼ 40.

HVP (NNLO) 12.4 0.1

HLbL (NLO) 3. 2.

theory ∼ 116 591 820. ∼ 57.

e, µ, τ

Sommerfield, Petermann 1957



aµ

[

10−11]

∆aµ

[

10−11]

experiment 116 592 089. 63.

QED O(α) 116 140 973.32 0.08

QED O(α2) 413 217.63 0.01

QED O(α3) 30 141.90 0.00

QED O(α4) 381.01 0.02

QED O(α5) 5.09 0.01

QED total 116 584 718.95 0.08


HVP (LO) ∼ 6 930. ∼ 40.

HVP (NLO) −98. 1.

HLbL (LO) ∼ 100. ∼ 40.

HVP (NNLO) 12.4 0.1

HLbL (NLO) 3. 2.

theory ∼ 116 591 820. ∼ 57.

I(a) I(b) I(c) I(d) I(e)

I(f) I(g) I(h) I(i) I(j)

II(a) II(b) II(c) II(d) II(e)

II(f) III(a) III(b) III(c) IV

V VI(a) VI(b) VI(c) VI(d) VI(e)

VI(f) VI(g) VI(h) VI(i) VI(j) VI(k)

Kinoshita et al. 2012



aµ

[

10−11]

∆aµ

[

10−11]

experiment 116 592 089. 63.

QED O(α) 116 140 973.32 0.08

QED O(α2) 413 217.63 0.01

QED O(α3) 30 141.90 0.00

QED O(α4) 381.01 0.02

QED O(α5) 5.09 0.01

QED total 116 584 718.95 0.08


HVP (LO) ∼ 6 930. ∼ 40.

HVP (NLO) −98. 1.

HLbL (LO) ∼ 100. ∼ 40.

HVP (NNLO) 12.4 0.1

HLbL (NLO) 3. 2.

theory ∼ 116 591 820. ∼ 57.

Z ν

WW

h γ, Zt

1-loop: Jackiw, Weinberg and others 1972

2-loop: Kukhto et al. 1992, Czarnecki, Krause, Marciano

1995, Degrassi, Giudice 1998, Knecht, Peris, Perrottet, de

Rafael 2002, Vainshtein 2003, Heinemeyer, Stockinger,

Weiglein 2004, Gribouk, Czarnecki 2005

Update after Higgs discovery: Gnendiger et al. 2013



aµ

[

10−11]

∆aµ

[

10−11]

experiment 116 592 089. 63.

QED O(α) 116 140 973.32 0.08

QED O(α2) 413 217.63 0.01

QED O(α3) 30 141.90 0.00

QED O(α4) 381.01 0.02

QED O(α5) 5.09 0.01

QED total 116 584 718.95 0.08


HVP (LO) ∼ 6 930. ∼ 40.

HVP (NLO) −98. 1.

HLbL (LO) ∼ 100. ∼ 40.

HVP (NNLO) 12.4 0.1

HLbL (NLO) 3. 2.

theory ∼ 116 591 820. ∼ 57.

Bouchiat, Michel 1961, . . .

Davier et al. 2017, Keshavarzi et al. 2018

Jegerlehner 2018



aµ

[

10−11]

∆aµ

[

10−11]

experiment 116 592 089. 63.

QED O(α) 116 140 973.32 0.08

QED O(α2) 413 217.63 0.01

QED O(α3) 30 141.90 0.00

QED O(α4) 381.01 0.02

QED O(α5) 5.09 0.01

QED total 116 584 718.95 0.08


HVP (LO) ∼ 6 930. ∼ 40.

HVP (NLO) −98. 1.

HLbL (LO) ∼ 100. ∼ 40.

HVP (NNLO) 12.4 0.1

HLbL (NLO) 3. 2.

theory ∼ 116 591 820. ∼ 57.

Calmet et al. 1976, . . .

Keshavarzi et al. 2018



aµ

[

10−11]

∆aµ

[

10−11]

experiment 116 592 089. 63.

QED O(α) 116 140 973.32 0.08

QED O(α2) 413 217.63 0.01

QED O(α3) 30 141.90 0.00

QED O(α4) 381.01 0.02

QED O(α5) 5.09 0.01

QED total 116 584 718.95 0.08


HVP (LO) ∼ 6 930. ∼ 40.

HVP (NLO) −98. 1.

HLbL (LO) ∼ 100. ∼ 40.

HVP (NNLO) 12.4 0.1

HLbL (NLO) 3. 2.

theory ∼ 116 591 820. ∼ 57.

Hayakawa, Kinoshita, Sanda 1995

Bijnens, Pallante, Prades 1995

Knecht, Nyffeler 2001

Jegerlehner, Nyffeler 2009

. . .



aµ

[

10−11]

∆aµ

[

10−11]

experiment 116 592 089. 63.

QED O(α) 116 140 973.32 0.08

QED O(α2) 413 217.63 0.01

QED O(α3) 30 141.90 0.00

QED O(α4) 381.01 0.02

QED O(α5) 5.09 0.01

QED total 116 584 718.95 0.08


HVP (LO) ∼ 6 930. ∼ 40.

HVP (NLO) −98. 1.

HLbL (LO) ∼ 100. ∼ 40.

HVP (NNLO) 12.4 0.1

HLbL (NLO) 3. 2.

theory ∼ 116 591 820. ∼ 57.

e

Kurz, Liu, Marquard, Steinhauser 2014



aµ

[

10−11]

∆aµ

[

10−11]

experiment 116 592 089. 63.

QED O(α) 116 140 973.32 0.08

QED O(α2) 413 217.63 0.01

QED O(α3) 30 141.90 0.00

QED O(α4) 381.01 0.02

QED O(α5) 5.09 0.01

QED total 116 584 718.95 0.08


HVP (LO) ∼ 6 930. ∼ 40.

HVP (NLO) −98. 1.

HLbL (LO) ∼ 100. ∼ 40.

HVP (NNLO) 12.4 0.1

HLbL (NLO) 3. 2.

theory ∼ 116 591 820. ∼ 57.

e

Colangelo, MH, Nyffeler, Passera, Stoffer 2014



aµ

[

10−11]

∆aµ

[

10−11]

experiment 116 592 089. 63.

QED O(α) 116 140 973.32 0.08

QED O(α2) 413 217.63 0.01

QED O(α3) 30 141.90 0.00

QED O(α4) 381.01 0.02

QED O(α5) 5.09 0.01

QED total 116 584 718.95 0.08


HVP (LO) ∼ 6 930. ∼ 40.

HVP (NLO) −98. 1.

HLbL (LO) ∼ 100. ∼ 40.

HVP (NNLO) 12.4 0.1

HLbL (NLO) 3. 2.

theory ∼ 116 591 820. ∼ 57.

aexpµ − a

SMµ ∼ 270(85)× 10

−11[3.2σ]

Hadronic uncertainties dominant

∆aexpµ and ∆aSM

µ same size

Fermilab: new number at E821

level next year, final goal factor 4

J-PARC: new approach based on

ultracold muons

→ huge challenge for theory!


Hadronic vacuum polarization

General principles yield direct connection with experiment

Gauge invariance

= −i(

k2gµν − kµkν)

Π(

k2)

Analyticity

Πren = Π(

k2)

− Π(0) =k2

π

∞∫

4M2π

dsImΠ(s)

s(

s − k2)

Unitarity

ImΠ(s) =s

4πασtot

(

e+e− → hadrons)

=α

3R(s)

1 Lorentz structure, 1 kinematic variable, no free parameters

Dedicated e+e− program under way, hopefully new results from CMD3 and

BaBar soon


Hadronic vacuum polarization: two-pion channel

HVP accuracy goal: 0.6% (present) → 0.2% (experiment)

Main contender: ππ channel

Current status Colangelo, MH, Stoffer 2018

485 490 495 500 505

1010 × aππ

µ|≤1GeV

all e+e−, NA7

all e+e−

energy scan

KLOE′′

BaBar

CMD-2

SND

Keshavarzi et al. 2018


Hadronic light-by-light scattering

So far: hadronic models, inspired by various QCD limits, but

error estimates difficult

Our suggestion: use again analyticity, unitarity, crossing,

and gauge invariance for data-driven approach Colangelo, MH,

Procura, Stoffer 2014, 2015

For simplest intermediate states: relation to π0→ γ

∗

γ∗

transition form factor and γ∗

γ∗

→ ππ partial waves


Towards a data-driven analysis of HLbL: our plan from 2013

e+e− → e+e−π0 γπ → ππγπ → ππ

e+e− → π0γe+e− → π0γ ω, φ → ππγ e+e− → ππγ

ππ → ππ

Pion transition form factor

Fπ0γ∗γ∗

(

q21, q2

2

)

Partial waves for

γ∗γ∗ → ππe+e− → e+e−ππ

Pion vectorform factor F V

π

Pion vectorform factor F V

π

e+e− → 3π pion polarizabilitiespion polarizabilities γπ → γπ

ω, φ → 3π ω, φ → π0γ∗ω, φ → π0γ∗

Colangelo, MH, Kubis, Procura, Stoffer 2014

Reconstruction of γ∗

γ∗

→ ππ,π0: combine experiment and theory constraints

Implementation

π0 pole done MH et al. 2018

First results for ππ Colangelo, MH, Procura, Stoffer 2017


The π0 pole: a data-driven calculation

aπ0-poleµ = 62.6(1.7)Fπγγ

(1.1)disp(2.21.4)BL(0.5)asym × 10−11

= 62.6+3.0−2.5

× 10−11

Fπγγ : π0 → γγ decay width

disp: systematics of the formalism

BL: Brodsky–Lepage limit and BaBar/Belle

tension

asym: transition point to pQCD0 5 10 15 20 25 30 35 40

0

0.05

0.1

0.15

0.2

0.25

0.3

Q2[

GeV2]

Q2Fπ0γ∗γ∗(−

Q2,0)[G

eV]

CLEO

CELLO

BaBar

Belle

MH, Hoid, Kubis, Leupold, Schneider 2018


Hadronic contributions to (g − 2)µ: where to go from here

Example: π0 pole, compare to preliminary lattice number Gerardin et al.

aπ0-poleµ

∣

∣

disp= 62.6+3.0

−2.5× 10−11 aπ0-pole

µ

∣

∣

lattice= 60.4(3.4)× 10−11

→ agree within uncertainties well below Fermilab goal

Suppose there is a > 5σ effect, how can we make the case most convincing?

Independent phenomenological and lattice numbers for both HVP and HLbL

→ mutual (global) cross check

More detailed cross checks, e.g. π0 pole in HLbL or particular energy regions in HVP

(“window method” RBC/UKQCD 2018)

Combinations to improve precision

Fermilab now has collected twice the BNL statistics, exciting times ahead!


Hadronic vacuum polarization: two-pion channel

Direct integration: local error inflation wherever tensions between data sets arise

In QCD: analyticity and unitarity imply strong relation between pion form factor

and ππ scattering

→ defines global fit function, very few parameters Colangelo, MH, Stoffer 2018

Data can be described with an acceptable χ2 in this way

Results close to direct integration

Should consolidate the ππ uncertainty

0

5

10

15

20

25

30

35

40

45

50

−0.2 0 0.2 0.4 0.6 0.8 1

|FV π(s)|2

s [GeV2]

| |

total errorfit error

NA7SND

CMD-2BaBar

KLOE08KLOE10KLOE12

−0.1

−0.05

0

0.05

0.1

0.15

0.6 0.65 0.7 0.75 0.8 0.85 0.9

|FV π(s)|

2 data

|FV π(s)|

2 fit

−1

√

s [GeV]

total errorfit error

SNDCMD-2

BaBarKLOE08KLOE10KLOE12


Impact on HLbL contribution

Numbers

aπ0-poleµ = 62.6+3.0

−2.5× 10−11

aπ-boxµ + a

ππ,π-pole LHCµ,J=0

= −24(1) × 10−11

Well-defined contributions with controlled error estimates

Plan towards a full evaluation of HLbL

η, η′

KK and ππ beyond π-pole LHC and S-waves

Resonance estimates for higher intermediate states

Asymptotics of HLbL tensor, matching to pQCD


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Combined explanations of (g-2), (g-2)e and implications...

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