Measuring the muon magnetic anomaly 𝒂𝝁with the Muon g-2 experiment at Fermilab
Lomonosov Conference – 21 August 2021
Marco Incagli – INFN Pisa

What is “g-2” ?
𝑒𝜇⃗! = −𝑔!
𝑆⃗2𝑚!

• gP : proportionality constant between
spin and magnetic moment for particle P• aP : magnetic anomaly• aP = 0 at tree level (purely Dirac particle)

𝑔! − 2𝑎! =2• Using modern language, the term (g-2)/2 reflects themagnitude of the Feynmann diagrams beyond leading ordera=0+DiracµµB
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Schwinger+....

Standard Model determination of aµ+ ?
QEDKnown

WeakKnown

QED ( + `)HVP(lo) Davier17
QED ( + `)HVP(lo)KNT2017
HVP(lo) Davier17
HVP(ho)
KNT2017
HVP(lo)KNT2017
HLbL
Glasgow
HVP(ho) KNT2017
EW
HLbL SM
Glasgow
Total
Davier17
EW SM KNT17
Total
Total SM Davier17
Total SM KNT17

HLO
Data

HLbL

Models/Lattice

New Physics.
?

10Value(⇥10) units
QCD
11 658 471.8951 ± 0.0009 ± 0.0019 ± 0.0007 ± 0.0077↵
10Value (⇥ 10692.6) ±units
3.33
11 658 471.8951 ± 0.0009 ± 0.0019 ± 0.0007± 0.0077
693.9± 2.6↵
692.6
3.33
9.84 ±±0.07
693.9
2.6
10.5 ±±2.6
9.84
15.4 ±± 0.07
0.1
10.5±±4.2
2.6
11 659 181.7
15.4±±3.7
0.1
11 659 182.7
11 659 181.7 ± 4.2
11 659 182.7 ± 3.7

Theory Initiative White Paper (arXiv 2006:08443)
𝑎! = 116 591 810 ± 43 ×10"## → 370 ppb
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A rich history of g-2 Theory and Experiment

CERN

BNLTHEORY

EXPERIMENTTHEORY“Consolidation”

Tension between theory and experiment4

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The Fundamental Experimental Principle• Difference between spin precession and cyclotron revolution
for a muon (charged particle with spin) in a magnetic field*:𝑒𝑒
𝑔−2 𝑒𝑒𝜔" = 𝜔# − 𝜔$ = 𝑔
𝐵− 𝐵=
𝐵 = 𝑎% 𝐵
2𝑚𝑚2 𝑚𝑚*s and p are assumed to be in a plane perpendicular to B

• simple classical calculation• the relativistic approach provides the same result

g=25

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g>2How do we measure the spin direction?
• Use V-A structure of weak decays to build a polarized beam...

• ... and to measure the muon polarization looking for energeticpositrons

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Measuring the spin precession• The number of observed positrons above a threshold energy
oscillates with the wa/2p frequency due to spin precession

N (t) = N 0e −t/τ [1+ A cos(ω a t + φ )]time (µsec)
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• exponential decay
modulated by spinprecession• note that the x-axis"wraps up" every 100µsec for a total of~700 µs à ~10 muon
lifetimes

Extracting aµ(simplified)𝝎𝒂 = 𝒂𝝁 (𝒆⁄𝒎) 𝑩 à 𝒂𝝁 = 𝝎𝒂 ⁄𝑩 (𝒎⁄𝒆)by expressing B in terms of the (shielded) proton precession frequency:(𝐵 = ℏ𝜔!& ⁄2𝜇!& ):& 𝑚𝜇𝜔$ 𝜇%& 𝑚# 𝑔'% # 𝑔'&𝑎# = & = 𝑅# 𝜔,% 𝜇' 𝑚' 2
𝜇' 𝑚' 2𝜔#"
𝑅! =𝜔′$$

What wemeasure

External data

ratio of muon to proton precessions inthe same magnetic field𝜔′
1 ! = (shielded) Proton angular velocity weighted for the muon distribution
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Additional corrections• A beam of muons
1. requires focusing elements à electrostatic quadrupoles
2. undergoes betatron oscillations around the ideal trajectory• Additional terms have to be considered in the muon
precession formula:𝑒1𝛾⃗
𝜔" = −𝑎% 𝐵 − 𝑎% − '𝛽×𝐸 − 𝑎%𝑚𝑐
𝛾 −1𝛾+1Term due to ElectroStatic
Quadrupoles (ESQ)
Becomes ~0 at magic g~29.3
or 𝑝~3.1 GeV/c

921/08/21

Marco Incagli - INFN Pisa𝛽⃗ ⋅ 𝐵 𝛽⃗

Term due to beam verticaloscillations: pitch correction

Additional corrections• The ratio 𝑅%& requires additional corrections related to beam
dynamics and to magnetic transient fields:Corrections due to beam dynamics

𝑅#& =𝜔$ 1 + 𝐶' + 𝐶% + 𝐶() + 𝐶%$- 𝑓,)-,*
𝜔′,%
1 + 𝐵* + 𝐵+Corrections due to transient magnetic fields• 𝑓$()$* = blinding frequency ; described later in the talk10

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Four articles on PRx and on arXiv for details
PRAB

Run1 ResultPRAPRD

PRLBeam Dynamics
Proton Precession
Muon Precession
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The key ingredients

wa=muon spin precession respect tomomentum (in B field)

𝝎𝒂N / 149.2 ns• Neglecting the corrections, the three key ingredients tomeasure the muon magnetic anomaly are:
108

Data
Fit

107
106
105
104
103

𝜔
#"
𝑅! = ′ ~𝜔$$102

Fermilab Muon g-2 Experiment
Combined Run-1 Data

101
020

40
60
80
100
Time after injection modulo 102.5 [μs]·𝝎𝒑 (𝒙, 𝒚, 𝜑)𝝎𝒑 (𝒙, 𝒚, 𝜑)𝑴(𝒙, 𝒚, 𝜑)

1 𝒑′ = 𝝎𝒑′ 𝒙, 𝒚, 𝜑 D 𝑴(𝒙, 𝒚, 𝜑)𝝎1221/08/21

w’p=proton precession frequency

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M=muon spatial distribution

mMuon g-2RINGuonsInflector

FIELD

QUADS

DETECTORS

24 Calorimeters + 2 trackers located all around the ring

NMR probes and electronics located all around the ring

Kicker13

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Measuring 𝜔$ : 5 parameters fit function• Fit with simple positron oscillation:
𝑁4 𝑡 = 𝑁5 exp − 𝑡⁄𝜏 % [1 + 𝐴𝑐𝑜𝑠 𝜔" 𝑡 + 𝜑 ]• This simple fit is clearly not sufficient and well definedresonances are observed in the residuals
muon lifetime: tµ=gt0µ = 64.33 µsec
CBO = Coherent Betatron Ocillations
VW = Vertical Waist (oscillations)RESIDUALS (in frequency space)1421/08/21

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The complete 22 parameters fit function
𝜔" , 𝜔#$ vertical oscillations
𝜔%&' , 𝜔(%&' radial oscillations

Red = free parameters
Blue= fixed parameters

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Lost muons (µ hittingcollimators)

Final fit to get 𝜔$16

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Marco Incagli - INFN PisaMagnetic field 𝑩 determination
The “trolley”

The "trolley" inside the beam "pipe”

The map

Fixed Probes

17 TrolleyProbes• A Trolley (Cylinder with 17 NMR probes) runs inside the ring
every 2-3 days to map the field experienced by muons• A set of 378 fixed probes, located in 72 azimuthal positions,continuosly measures the field• Absolute probes for calibration tested at Argonne (ANL) magnet
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𝜔′% → 𝜔′, % : muon distribution inside the Ring
• Two tracker stations, made ofstraw tube modules, placed at
f~180o and f~270o, are used to
trace back positrons and get themuon distribution• Use Beam Dynamics models toextrapolate the distribution all
around the ring
• Systematic uncertainties mostly
due to Beam Dynamics models
used for extrapolation and to
tracker alignment

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Final uncertainties from Run 1, ~8×109 positrons•

462 ppb overall error
434 ppb statistical
157 ppb systematic
25 ppb external inputs•

Results for Run 1 are vastly
dominated by statistical error•

At 157 ppb systematic error– Nearly half of BNL– Not quite to 100 ppb goal

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Final uncertainties from Run 1• Correction larger thantotal error• Dominated by ElectricField correction 𝐶4• Related to non-centeredradial distribution due tolow kicker field (sparks!)• Fixed in Run-2

𝑒1⃗𝜔) = −𝑎* 𝐵 − 𝑎* − (𝛽×𝐸𝑚𝑐
𝛾 −1(..) = 0 for 𝛾 = 𝛾+),-.
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Low Momentum

g < gmHigh Momentum

g > gmFinal uncertainties from Run 1• Two largest systematics:– phase acceptance– quadrupole field

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ElectroStatic Quadrupoles (ESQ) transient field Bq• Mechanical vibration in the plates
due to the ESQ charge/discharge
which generates magnetic
perturbations• Special NMR probes measure theperturbation Bq at several positions• Uncertainty dominated by limited
number of measurements in Run-1• Reduced in Run-2 by moremeasurementst [µs]

Quad Plates inside Vacuum Chamber

𝑩𝒒 ~𝟐𝟎 𝒑𝒑𝒃, 𝜹𝑩𝒒 ~𝟗𝟎 𝒑𝒑𝒃
Muon fills
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The blinding
𝑅#& = 𝑓,)-,* •𝜔$ 1 + 𝐶' + 𝐶% + 𝐶() + 𝐶%$𝜔′,%
1 + 𝐵* + 𝐵+

Clock frequency 𝑓!"#!$ uncalibrated by Joe Lykken andGreg Bock (FNAL Directorate) Feb 22 2018
– stop in each week to check clock and sealing•

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Secret envelopes kept until physics analysis complete
and ready to be revealed Feb 25 2021

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aµ: Unblindingt [µs]On February 25 2021 the Collaboration
met for the unblinding:1) The box (envelope) was opened
2) The number was plugged into twoindependent programs3) And the result was….

Secret offset
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aµ: Unblinding and result

3.7 s
3.3 s

4.2 s

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A new prediction based on lattice• a recent (published April 7!) result has reduced theuncertainty of the lattice calculation from 2-3% to 0.8%!
• breakthrough which requires further scrutiny• uncertainty similar to the standard approach based on e+eàhadrons data via dispersion relation (0.6%)
Lattice calculation
closer to experimentalvalue than standard
approach26

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The Theory Initiative• ~130 physicists collaborated for 3 years [June 2017 – June2020] in seven workshops to produce a reference number for
am to be used by FNAL g-2 experiment as a benchmark

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The future of E989 Muon g-2 (Fnal)• RUN-1 is only 6% of thefinal dataset• Analysis of RUN-2/3 inprogress (factor ~2 inprecision)• RUN-4 (November2020-July 2021), which
was recently
completed, and Run5(2021-2022) will allow
to reduce by a factor~4 current error

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TDR
RUN5

RUN4RUN3
RUN1RUN2

Conclusions: experiment
1. FNAL Muon g-2 :–––aµ measured at 0.46 ppm à 4.2s discrepancy!
data already available to reduce error to ~0.25 ppm
Run4 (just finished) and Run5, starting later this year 2021, willallow to reach ~0.15 ppm

2. A new type of experiment projected at J-Parc using low
energy muons (p~300 MeV/c)–––

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new techniqueunder constructionfinal goal ~0.4 ppm

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Conclusions: theory
1. close scrutiny of lattice calculations to establish its solidity–how to reconcile it with dispersion approach? and withStandard Model global EW fit? à next talks

2. Use the dispersive approach with t-channel data (muonelectron scattering), instead of the standard s-channel–

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Letter Of Intents submitted at CERN: Muone (mu-on-escattering)

21/08/21µµee

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BACKUP SLIDES

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aµHLO : dispersion integral
HLOµa••∞
1=σ(s)K(s)ds3 ∫ 4m 2 e+e− →hadrπ4πKernel function: 𝐾(𝑠) ∝ F⁄#
Due to the 1/s term, the low
energies most important
K (s) ~33

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1s

Marco Incagli - INFN Pisaσ had
R= 0σ µµ

aµHAD from μ + e à μ + e scattering• NEW IDEA:• measure aµHAD using µeàµe (t-channel) instead of eeàpp !
µµα (t) =t=q2<0ee1HLOµa34α (0)1− Δα LEP (t) − Δα HAD (t)21/08/21α=(1− x)Δα had (x)dx∫π 0Marco Incagli - INFN Pisa

x 2 mµt=< 0;x −1(0 ≤ x < 1);t=0t=-¥

wa principle of measurementnnµ+ Center of Massµ+e+ Max E: e+ parallel to µ spinn Min E: e+ antiparallel to µ spinne+(N (t) = N 0e −t τ 1+ A cos(ω a t + ϕ )• positron emission correlatedwith muon spin direction• correlation depends on positronenergy : the Asymmetry A(E)can be positive, null or negative35

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Optimizing the statistical sensitivity– by decreasing the threshold theasymmetry decreases but thenumber of events increases àmax sensitivity for Ethr~1.7 GeV• A-method = each positron isweighted by the value of its
asymmetry A(E)– optimize statistical sensitivity

normalize sensitivity• T-method = counting positrons
above Ethr vs time

(N (t) = N 0e −t τ 1+ A cos(ω a t + ϕ )A-methodT-method𝑦 = 𝐸 ⁄𝐸/01(𝐸!"# = 3.1𝐺𝑒𝑉)• In theory the A-method can use all decay positrons, in practice,
due to calorimeter acceptance and to low A(E) value at lowenergies, only positrons with Ethr >1.1 GeV (y>0.3) are used36

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CPA – Phase acceptance error• But if the phase of the muon population changes in time :cos 𝜔" 𝑡 + 𝜙 𝑡≈ cos 𝜔" 𝑡 + 𝜙 & 𝑡 + 𝜙5 = cos((𝜔" +𝜙 & )𝑡 + 𝜙5 )• The extracted wa is shifted by f’ !
• The phase depends on the muon
decay position (x,y)• Not a big issue if the muon
distribution remains stable with time37

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CPA – Phase acceptance error• 2 quadrupoles HV resistors failed àchanging E-field à beam verticalmean and width changed• 𝐶!" = −158 𝑝𝑝𝑏, 𝛿!" = 75 𝑝𝑝𝑏• Faulty resistors fixed before Run-2

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GOODBAD