NuFact15 - Rio de Janeiro, Brazil - August 2015

The Muon g

2 Experiment at Fermilab⇤
Kevin R. Lynch1, 2, †(The Fermilab Muon g
1

2 Collaboration), ‡Department of Earth and Physical Sciences,

York College, City University of New York,
94-20 Guy R. Brewer Blvd, Jamaica, NY 11451 USA
2

The Graduate Center, City University of New York,365 Fifth Avenue New York, NY 10016 USA(Dated: April 18, 2016)438

NuFact15 - Rio de Janeiro, Brazil - August 2015

Abstract
There remains a tantalizing discrepancy between the Standard Model prediction for the muong2, and the value measured by the Brookhaven E821 Experiment. This discrepancy has driven

designs for multiple experiments. Here I present current progress on the construction of the Fermilab E989 experiment which will improve on the Brookhaven experiment by a factor of four, toa combined uncertainty of 0.14 ppm.

INTRODUCTION

Understanding the anomalous magnetic moment of the muon has consumed the energies
of generations of physicists, from the likes of Dirac and Schwinger, down to the presentday. Advances in the theory and measurement of g

2 have fed o↵ each other and haveproductively informed all other subfields of particle and nuclear physics.
The most recent chapter closed at the Brookhaven National Laboratory (BNL) in the
United States at the turn of the century, followed by publication of the final report of theMuon g2 Experiment E821 [1]. This experiment reached an ultimate combined uncer-

tainty of 0.54 ppm. This precision is sufficiently high to access the very small electroweak
contributions present at second order. Parallel developments at the same precision have

FIG. 1: This plot summarizes the current discrepancy between the consensus Standard Model
theory prediction for the muon anomaly and the BNL E821 experimental result.

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NuFact15 - Rio de Janeiro, Brazil - August 2015
been required in the theory. Consensus among contributors to the ongoing theory e↵orts
makes it appear likely that the results of E821 show a discrepancy from the Standard Model
prediction, at the level of 3.6 standard deviations; see Figure 1.
This unresolved state-of-a↵airs fairly begs for significant continued e↵ort on both thetheoretical and experimental fronts. The US High Energy Physics community has recognized
the importance of this measurement: Recommendation 22 in the recent report of the Particle
Physics Project Prioritization Panel (P5) [2] recommends completing the Fermilab E989
Muon g2 project under all funding scenarios. Here, I describe the current progresson bringing this next generation storage ring experiment to the data taking stage. Other
contributions in this workshop present progress on a radically di↵erent method for measuring
the muon magnetic anomaly at the Japan Proton Accelerator Research Center (JPARC).

THE METHOD

Modern electron g-factor experiments have dramatically improved uncertainties by goingto smaller experiments with very large observation times on individual particles in cyclotron
traps [3]. The brief muon lifetime prevents that approach, so muon experiments have bynecessity proceeded by a di↵erent path: the last four precision g

2 experiments all utilized

storage ring techniques, to measure time dependent ensemble spin precession. Understanding
this method is key to understanding the design of the experiment.
Muon beams are naturally highly polarized: muons are born in a two-body decay of spinzero pions via the V

A (left-handed) weak interaction. As neutrinos are only observedin left-handed chirality, angular momentum conservation demands that all muons have thesame spin orientation. In the case of muon g2, we maximize and utilize that beam

ensemble polarization to e↵ect our measurement.
To produce a muon beam, a proton beam impinges upon a pion production target. Thosepions are collected, momentum selected, and transported through a decay channel, along
which they decay to muons. Those muons are then injected into a storage ring with anextremely uniform magnetic field. For those muons which are stored in the ring, the momentum will precess at the cyclotron frequency
!C =

440

eB
,
mc

NuFact15 - Rio de Janeiro, Brazil - August 2015
while the spin undergoes both Larmor and Thomas precession
!S =

geB
+ (1
2mc

)

eB
.
mc

The di↵erence between these two frequencies,!a = !S

!C =

g

2 eB
eB
= aµ
,
2 mc
mc

is known as the anomaly frequency. Measuring all quantities in this expression (in particular,!a and B) gives the anomaly directly.
In practice, things are of course not so simple. Muons will not remain in a storage ringwith only a single uniform magnetic field. This field will retain particles by focusing in theplane perpendicular to the field, but will do nothing to prevent vertical losses (along the field
direction). Since we want an extremely uniform magnetic field across the entire volume ofstored muons, we can not use magnetic fields to perform vertical focusing. Instead, we utilize
an electrostatic quadrupole focusing system; the induced horizontal defocusing is overcomeby the magnetic field.
In the presence of vertical motions and the electric fields, our previous expression relating!a to aµ must be modified:
"
e
~
!
~a =
aµ B
mc

✓

aµ

1
2

1

◆

~⇥E
~

aµ

+1

⇣

⌘

#

~·B
~ ~ ,

The first term in the square brackets is our original result. The second encodes the e↵ect of
the electrostatic quadrupoles. The final term comes from the vertical momentum component.
Since the deviations from the horizontal plane will be small, let me ignore the final term fornow; in the experiment, we must correct for the vertical motions, but this pitch correction
is small and well controlled. The key insight here is that the electric field e↵ects vanish fora particular “magic” value of the momentum, given bys
11+.
magic =aµ
For muons this corresponds to a momentum pmagic = 3.091 GeV/c. Producing large numbersof muons at this momentum demands only quite modest proton beam energy; at Fermilab,
we will use an 8 GeV beam. Of course since there is a distribution of momenta in the stored
beam, corrections must be applied to the final result; again, this E-field correction turns
out to be small and well controlled.

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NuFact15 - Rio de Janeiro, Brazil - August 2015
Key to the experiment then is measurement of !a . The chiral structure of the weakinteractions gives us direct access to the spin direction, at least on a statistical basis. In thethree body decay of the muon, the electron is not emitted isotropically. In the rest frame of
the muon, with the spin aligned along a coordinate axis, the di↵erential decay distribution
for the electron momentum direction is given (in units of the dimensionless electron energy
y = Ee /Emax ) byd 2 µ±= n(y) (1 ⌥ a(y) cos ✓) ,dyd⌦with n(y) = y 2 (2y) and a(y) = (2y1)/(32y). This results in the highest energy

positrons (electrons) being emitted preferentially parallel (anti-parallel) to the muon spinorientation. In the lab frame the combination of spin precession and energy dependent asymmetry is observed as a sinusoidal variation in the observed number of positrons (electrons)
above a threshold energy
f (t) = N0 (Ethresh )e t/⌧ [1 + A(Ethresh ) cos(!a t + )](1)

This threshold is chosen to maximize the statistical power of the collected data, which is
proportional to the asymmetry A(E) (rising with energy) and the square of the observed
number of electrons N (E) (falling with energy); A(E) and N (E) are the integrals of a(y)and n(y) from the chosen threshold to Emax . During analysis, we will provide a multilevel blinding process to the analysis so that neither the Collaboration as a whole, nor the
analyzers individually, will be able to steer (whether intentionally or not) the analysis toward
a “preferred” value.
To extract the anomaly, we must know the absolute value of the magnetic field distribution, integrated over the storage volume, in addition to !a . A number of pulsed NMR
probes, including fixed, plunging, and traveling, are used to measured the proton Larmor
precession frequency in the field
!P =

eB
gp
2mP

Like the !a analysis, this frequency is also hidden by a blinding procedure, further protectingfrom unintentional bias.
The combination of !a and !P allows calculation of aµ .
aµ =!a /!P,µµ /µP !a /!P

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(2)

NuFact15 - Rio de Janeiro, Brazil - August 2015

TABLE I: A summary of the !a systematics in both the BNL E821 and expected improvements
for the FNAL E989 experiment [4].
Category

E821 E989 Improvement Plans

Goal

[ppb]

[ppb]

Detector Gain changes

120 Better laser calibration
low-energy threshold

Pileup20

80 Low-energy samples recorded
calorimeter segmentation

40

Lost muons

90 Better collimation in ring

20

Coherent Betatron Oscillations

70 Higher n value (frequency)
Better match of beamline to ring < 30

Field and pitch corrections

50 Improved tracker
Precise storage ring simulations

Total

180 Quadrature sum

30
70

where the ratio of magnetic moments was measured at the Los Alamos National Laboratory(LANL) in the United States utilizing the same spherical calibration probe used in the E821
NMR calibration procedure.

IMPROVING THE SYSTEMATICS

To improve the overall error relative to E821 by a factor of four will require improvements not only in statistics (a factor of at least twenty), but also systematics. In the BNL
experiment, systematic errors contributed 0.28 ppm, and statistics 0.46 ppm, to the final,
combined 0.54 ppm result.
When discussing improvements to the systematics, it helps to recall that we are performing two parallel measurements - !a and !P - and we divide our systematic studies along
these lines. We intend to improve substantially on both of these, to better than 0.1 ppm ineach category. The major !a systematics are illustrated in Table I, and !P systematics inTable II.
What do we need to do to reach these goals? Since it is impossible to discuss the443

NuFact15 - Rio de Janeiro, Brazil - August 2015

TABLE II: A summary of the !P systematics in both the BNL E821 and expected improvements
for the FNAL E989 experiment [4].
Category

E821 Main E989 Improvement Plans

Goal

[ppb]

[ppb]

Absolute field calibra- 50 Special 1.45 T calibration magnet with 35
tion

thermal enclosure; additional probes;
better electronics

Trolley probe calibra- 90 Plunging probes that can cross cali- 30
tions

brate o↵-central probes; better position
accuracy by physical stops and/or optical survey; more frequent calibrations

Trolley measurements of 50 Reduced position uncertainty by fac- 30
B0

tor of 2; improved rail irregularities;
stabilized magnet field during measurements*

Fixed probe interpola- 70 Better temperature stability of the 30
tionmagnet; more frequent trolley runs

Muon distribution

30 Additional probes at larger radii; im- 10
proved field uniformity; improved muontracking

Time-dependent

exter-–

nal magnetic fields
Others †Direct measurement of external fields;

5

simulations of impact; active feedback
100 Improved trolley power supply; trolley 30
probes extended to larger radii; reduced
temperature e↵ects on trolley; measure
kicker field transients

Total systematic error 170

70

on !p

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NuFact15 - Rio de Janeiro, Brazil - August 2015
experimental requirements along with all of our plans in the short space available here, I
choose to concentrate on just a few issues. It should be understood, however, that much
more work has been done to date than I have room to discuss.

Implementation details

The FNAL implementation of the experiment is an evolutionary upgrade of the E821
model. The majority of the major devices, in fact, have been transplanted from BNL to
FNAL, and form the backbone of the experiment. Despite this, nearly all of the subsystems
are undergoing major upgrades in functionality and/or precision to meet the stringent new
requirements.
At FNAL, the experiment is part of a larger muon physics program - which includes themuon-to-electron conversion (Mu2e) search experiment - in a new Muon Campus facility
close to Wilson Hall. The g

2 experiment occupies the new MC1 experimental hall and isserved by a new beamline. Proton delivery to the muon campus requires modest upgrades tothe accelerator complex which are compatible with the ongoing neutrino program. Protons
will continue to be accelerated by the Booster ring to 8 GeV. Bunches will be extracted
from the Booster and transported to the Recycler ring. After extraction from the Recycler,
the protons will impinge on an upgraded production target in the AP0 hall - occupying theformer Tevatron antiproton production facility. Pions will be focused by a lithium lens andmomentum selected by a pulsed dipole; these devices served the antiproton program andrequired significant upgrades to operate reliably with the more demanding g

2 pulse rates.

Pions will be sent to the Delivery ring - which will serve as part of an 800 m long decayline, e↵ectively eliminating pions from the beam - and will be extracted after one turn fordelivery to the g

2 storage ring. Upgrades to the Delivery ring (the former antiproton

Debuncher) are designed to accommodate both single-turn muon extraction for g

2 andresonant proton extraction for Mu2e. Extracted beam will traverse a new line to the g2

Storage Ring, which was moved - with much fanfare - from BNL to FNAL in the summerof 2013 [5].
Once the muons have reached the storage ring, the work of the accelerator physicists
gives way to the particle physicists. Of course, the beam must be transported from theoutside to the inside of the ring; the storage volume is a torus with radius 7.11 m, with an

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NuFact15 - Rio de Janeiro, Brazil - August 2015
unobstructed circular cross sectional of diameter 90 mm. Over this region, the storage field,
averaged over azimuth and the weighted by the muon storage distribution, must be uniform
to better than 1 ppm. To reach the storage region, the fringe fields must be cancelled without
destroying the field uniformity in the neighboring storage region. This key task is the jobof a superconducting magnetic inflector. The baseline design for the FNAL experiment isto refurbish and reuse the E821 inflector. This device is a truncated double cosine magnet
with an external superconducting shield that both maintained a field free channel within
the fringe field of the storge ring and reduced the impact of the injection channel field on
the storage field at the required level. Despite the success of this device, some design choices- in particular, that windings cover the ends of the beam channel - significantly impact theefficiency of beam injection. Signficant R&D is ongoing to design a new inflector with open
ends and a wider bore as a possible option should scope contingency be available at a later
date.
The exit of the E821 inflector is centered 77 mm from the center of the storage region.
Without additional work, the muon ensemble would not store. The task of moving thebeam on-orbit is handled by three fast kicker modules, each consisting of a pair of plates
and a high voltage, high-speed pulse forming network. The current sheets created during
the kick locally suppress the magnetic field by an amount sufficient to deflect magic momentum muons by 10.8 mrad, sufficient to move the distribution onto the nominal orbit. New
fast kicker modules are being designed for the FNAL e↵ort, based on a Blumlein triaxial
transmission line design. This should enable a signifiantly shorter, higher amplitude current
pulse, improving storage efficiency. Simultaneously, the new design will retain significant
margin to allow a higher kick for compatibility with the proposed increase in the inflectorbore, which will shift the centroid of the muon bunch further outward radially, requiring astronger kick deflection.
As mentioned in the overview above, maintaining a stored beam requires electrostatic
quadrupole fields for vertical focusing. This is accomplished by a four quadrant system,
which will be largely reused. A number of upgrades are in progress to improve various
aspects of the system. Of perhaps greatest importance, the system will be operated at ahigher field index to move the frequency of coherent betatron oscillations far away fromthe second harmonic of the g

2 oscillation frequency. Second, during injection, the BNL

muon beam passed through the first outer quadrupole plate and insulating supports; ongoing446

NuFact15 - Rio de Janeiro, Brazil - August 2015
redesign of this part of the system should significantly reduce losses resulting from scattering.
The decay electrons have lower energy than the muons, with correspondingly smaller
orbital radii in the magnetic storage field. Therefore, decay positron paths will curve inward,
towards the center of the ring. The magnet yoke is C-shaped, with the open side toward
the ring center, and the beam vacuum vessel is scalloped to minimize scattering during
the inward spiral. Twenty-four segmented PbF2 calorimeters will be located symmetricallyaround the ring. The output signal waveforms from each segment will be digitized andstored. O✏ine processing produces the energy and time histograms needed to fit the physics
and extract !a .

Analysis improvements

In E821, oscillation spectra were constructed from individual event timings, the so called“T-method” of analysis. The output signals from the calorimeters were waveform digitized,
zero suppressed, and the over-threshold samples were stored in onboard RAM until read out
by the data acquisition system. Individual pulses were fit to a template, below threshold
pulses were rejected, and histograms were built. Construction cost and memory limitations
were severe constraints when the digitizer modules were designed necessitating this approach.
In the intervening years, the impact of Moore’s law on transient and persistent data
storage, network data transmission, and computation resources, have been very large, while
the physics have changed little. These gains permit us to reimagine the entire data collection and processing scheme. We building a new set of highly segmented calorimeters, andcustom waveform digitizers which will be able to store and readout every sample during
the measurement period, for every segment of every calorimeter. Advances in parallel anddistributed computation and in high speed networks promise significantly higher throughput
data acquisition and o✏ine analysis systems.
Access to the full data record for each channel enables fabulous possibilities undreamedof at the E821 design stage. We intend to again pursue the T-method, but access to thefull data stream enables many improvements in the areas of threshold and pedestal stability
tracking, as well as vast improvements in pileup reconstruction. In addition, the full data
stream permits a new integrated charge collection, or “Q method” approach. Here, we
integrate data from all data samples from early to late, and fit the resulting oscillatory447

NuFact15 - Rio de Janeiro, Brazil - August 2015
signal. Because this method sums over all data samples, it is impacted di↵erently by various
systematic errors. The complementarity of these two methods combined with the blindedanalysis will provide significant strength to our final result.

IN CONCLUSION

As of this workshop, the Fermilab Muon g 2 Project had reached a number of importantmilestones: the ring had been shipped from BNL to FNAL and installed in the new MC1
experiment hall at the Fermilab Muon Campus. As the workshop got underway, the storage
ring superconducting magnets had just been energized for the first time in fourteen years,
validating the condition of the full system; a few small issues had been discovered and fixes
were being planned and scheduled, and the field shimming plans for 2015 were being finalized.
A host of other construction and R&D activities were underway at many Laboratory andUniversity sites both inside and outside the USA. Under the current budget profile, theexperiment will begin full commissioning and data taking at the beginning of calendar year
2017, followed by three years of data taking.
The speaker’s work on the Muon g2 e↵ort is supported by the U.S. Department ofEnergy, Office of Science, Office of High Energy Physics under Award Number DE-FG0213ER41931.

⇤

Presented at NuFact15, 10-15 Aug 2015, Rio de Janeiro, Brazil [C15-08-10.2]†

klynch@york.cuny.edu; Speaker‡

URL: http://muon-g-2.fnal.gov/[1] G. W. Bennett et al. (Muon G-2), Phys. Rev. D73, 072003 (2006), hep-ex/0602035.
[2] Particle Physics Project Prioritization Panel (P5) (2014), URL http://science.energy.gov/~/media/hep/hepap/pdf/May-2014/FINAL_P5_Report_053014.pdf.
[3] D. Hanneke, S. Fogwell, and G. Gabrielse, Phys. Rev. Lett. 100, 120801 (2008), 0801.1134.
[4] J. Grange et al. (Muon g-2) (2015), 1501.06858.
[5] Fermi National Accelerator Laboratory, The Big Move (2013), URL http://muon-g-2.fnal.
gov/bigmove/.

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