Dipartimento di Fisica E. Fermi Corso di Laurea Magistrale in Fisica
Study of the calorimeter gain fluctuations of the
Muon g-2 experiment at Fermilab
Candidato:
Elia Bottalico
Matricola 508385
Relatore Interno:
Prof. Giovanni Batignani
Relatore Esterno:
Dott. Graziano Venanzoni
questo percorso, supportandomi sempre.
A Stella, compagna di vita, che `e riuscita a mostrarmi il bello delle cose
aiutandomi ad affrontare i momenti pi`
u difficili e a godere dei successi.
Al Dott. Graziano Venanzoni, fonte di ispirazione e conoscenza, che mi
ha introdotto nel mondo della ricerca passo dopo passo, rendendo
possibile questo traguardo.
1 The Anomalous Magnetic Moment 1
1.1 Magnetic moment of elementary particles . . . 1
1.1.1 Virtual loops . . . 3
1.2 Why measuring aµ? . . . 4
1.3 Measurement principle of aµ . . . 5
2 Theoretical Calculation of aµ 11 2.1 Summary of the main contributions . . . 11
2.2 The QED Contribution . . . 12
2.3 The EW Contribution . . . 13
2.4 The Hadronic Contribution . . . 15
2.5 Beyond Standard Model contribution . . . 17
3 The aµ measurement: early experiments 22 3.1 The CERN experiments . . . 24
3.1.1 CERN I experiment . . . 24
3.1.2 CERN II experiment . . . 26
3.1.3 CERN III experiment . . . 27
3.2 The BNL experiment . . . 29
4 The Muon g-2 experiment at Fermilab 33 4.1 Beam Structure . . . 35
4.2 The Storage Ring Magnet and the Inflector . . . 36
4.3 Kicker . . . 36
4.4 Vertical Focusing: Electrostatic Quadruples . . . 37
4.5 Measurement Principle . . . 38
4.6 ωa Measurement . . . 40
4.7 ωp Measurement . . . 44
4.8 The Calorimeter System . . . 45 iii
4.9 Tracker System . . . 52
4.10 Other Detectors . . . 53
5 The Laser Calibration System 55 5.1 Diffusing System . . . 57
5.2 Front Panel . . . 59
5.3 The double pulse setup . . . 60
5.4 Time Synchronization . . . 61
5.5 The laser monitoring system . . . 62
5.5.1 The Source Monitor . . . 62
5.5.2 Local Monitor . . . 65
5.5.3 Standard operation mode . . . 67
6 The Gain Corrections 69 6.1 Out-of-Fill Gain correction . . . 70
6.2 In-Fill Gain Corrections . . . 70
6.2.1 Run1 dataset . . . 72
6.2.2 Run1 ωa analysis Strategy . . . 72
6.2.3 Construction of the gain functions . . . 74
6.3 Short Term Double Pulse . . . 75
6.3.1 Reconstruction Software . . . 77
6.3.2 Energy Studies for Run1 correction . . . 78
6.3.3 STDP correction to the official 60h dataset . . . 83
6.4 IFG corrections for 60 hours dataset . . . 85
6.5 IFG systematic studies . . . 88
7 Long Term Double Pulse studies 90 7.1 LTDP Gain Function . . . 91
7.2 Numerical model . . . 93
7.3 Analytical model . . . 94
7.4 LTDP Energy Study . . . 96
7.5 LTDP analysis . . . 100
7.6 In-Fill Gain Comparison . . . 102
7.7 Energy Splash Measurement . . . 104
7.8 LTDP Corrections . . . 109
The Muon g-2 experiment at Fermilab (E989) aims to measure the anomalous magnetic moment of the muon with an accuracy of 0.14 ppm (parts per million). This precision, ob-tained by adding in quadrature a statistical and a systematic contribution of comparable value (0.1 ppm), will allow to reduce the experimental uncertainty (from the previous E821 experiment at BNL) of a factor of 4, and represents a high precision test of the Standard Model (SM) theory of elementary particles. The quantity which is measured is the muon magnetic anomaly aµ=gµ
−2
2 where gµ is the gyromagnetic factor of the muon. Dirac’s
equation predicts gµ = 2, while radiative corrections dominated by the QED contribution
due to an exchange of a virtual photon, cause a per-mille correction on this quantity. By including all the SM contributions, aµ is known at 0.5 ppm. E989 measures aµ injecting
positive muons with momentum of 3.1 GeV/c polarized longitudinally in bunches (called f ill) with an average rate of 12 Hz, in a storage ring of 14 meters diameter. Due to the parity violation in the weak muon decay, high energy positrons produced are emitted preferably in the muon’s spin direction. By counting the number of positrons with energy greater then 1.7 GeV in function of the time, the frequency precession of the muon spin is measured, that together with the measurement of the magnetic field, allow to extract aµ.
The positrons are detected with 24 electromagnetic calorimeters, that measure the energy and the arrival time of the positrons, made up of 1296 crystals of lead fluoride (PbF2)
read by a silicon photomultipliers (SiPM). Due to the large flux of particles that hits the calorimeters soon after the injection, the SiPMs suffer significant gain fluctuations that, if not correct, introduce as systematic error on the measure of the energy released in the calorimeters and eventually on ωa. These fluctuations are characterized by two different
time scales: a short one (ST, Short Time) with a maximum amplitude of the order of 5%, due to two pulses that hit the SiPM in an interval of tens of nanoseconds, where the first pulse inhibits a portion of the pixels of the SiPM before the arrival of the second pulse, and a long one (LT, Long Time) with maximum amplitude of 3%, in the scale of tens-hundreds microseconds, due to the recovery time of the SiPM Bias Voltage power supply. Since the total systematic error allowed for these corrections is 20 ppb (parts per billion),
these gain fluctuations must be known with the precision of 0.04% during the 700 µs of the fill. To monitor the gain fluctuation at the desidered value a laser system was built by INFN, composed of: a light source supplied by 6 laser heads, a distribution system (set of mirrors and fibers), capable of sending simultaneous light calibration pulses onto each of 1296 crystals of the electromagnetic calorimeter, and a monitoring system. To monitor the gain fluctuations two different operation modes are mainly used: the Standard Mode and the Double Pulse Mode. In the Standard Mode 3 laser pulses with a relative delay of 200 µs are sent in 10% of the fills (to not contaminate too much the data), each pulse delayed by 2.5 µs. The In-Fill Gain corrections (called IFG) are obtained. In the Double Pulse Mode by placing a movable mirror in front of each laser head it is possible to form 3 pairs of lasers, 1-2, 3-4 and 5-6, coupled together. By remotely controlling the position of the movable mirrors, it is possible to re-direct the light of each laser into the path of its paired one through a beam-splitter cube. The two laser beams are thus superimposed and injected, with comparable intensity, into the same fiber, so the lights of 2 laser heads hit the same crystal of the calorimeter.
The work of this Thesis consisted on the Study of Double Pulse mode to obtain the ST and LT corrections. Chapter 1 introduces the subject of the anomalous magnetic moment of the muon. Chapter 2 discusses the Standard Model prediction and possible new physics scenario. Chapter 3 presents previous measurements of the muon g-2 and Chapter 4 describes the E989 experiment, in which the experimental technique and the experimental apparatus are described focusing on the improvements necessary to reach the final goal on the measurement of aµ. The main subject of the Thesis is discussed
in the last 3 Chapters. Chapter 5 describes the Laser Calibration system built to keep under control the gain fluctuations at the 0.04%. Chapter 6 contains the Gain corrections procedures, with the original work on the study of In-Fill Gain and Short Term Double Pulse corrections. Chapter 7 is focused on the final part of my work on the Long Term Double Pulse (LTDP) procedure, used to perform the study of the IFG curves in function of energy and providing an independent determination of the IFG corrections. Particu-larly for the LTDP study an excellent agreement with the IFG corrections was obtained, with a 2 parameters model capable to describe the gain function of the SiPMs in function of the energy.
The Anomalous Magnetic Moment
1.1
Magnetic moment of elementary particles
A charged particle with spin has an intrinsic magnetic moment defined as: ~
µ= g e
2mS~ (1.1)
where e is the charge, m the particle mass, ~S the spin and g the gyromagnetic ratio1.
The magnetic moment is a measure of the torque and energy that a charged particle feels when is placed in a magnetic field: ~τ = ~µ × ~B and U = -~µ · ~B. In a classical model we can describe the magnetic moment as a rotation of the particle about own z-axis. To compute the magnetic moment we can consider the particle moving around a close path with radius r and speed v. The loop drawn by the charged particle has an area: A = πr2
and carries an electric current I = ev
2πr. From the equation µ = IA we obtain:
µ= πr2 ev 2πr = evr 2 (1.2) replacing L = mrv we obtain: ~ µ= e 2mL~ (1.3)
which shows g=1 for a classical description of a rotating particle. The mathematical description of the spin came in the 1928 from Dirac’s equation that attempted to create a relativistic equation for an electron that preserved linearity with respect to time:
i¯h∂ψ
∂t = Hψ (1.4)
1Also called g-factor or Land`e factor.
where:
H = βmc2+ ~α · ~pc →(i∂µγµ+ eAµγµ)ψ(x) = mψ(x) (1.5)
where ~α and β are the 4x4 Dirac’s matrix operator, ~p the operator −i¯h∇ and γµ the set
of matrices with the anticommutation relations defined as γ0 ≡ β and γi ≡ βαi The ψ
is a four-component column matrix and when solved for particle’s speed close to 0, we obtain Equation 1.5 for particle and anti-particle. When we introduce an electromagnetic field the Hamiltonian becomes:
H = βmc2+ ~α · ~pc+ eV (1.6)
where V is the scalar potential and ~A the vector potential included in the operator ~p = −i¯h∇ − e
cA. Using the algebraic rules for the Dirac’s matrices we can obtain in the~
non-relativistic limit the Hamiltonian H0:
H0 =
p2
2m+ eV − e¯h
2mc~σ · ~B (1.7)
the last term in the Hamiltonian represents the interaction of a particle with spin and external magnetic field. From the last term in e¯h
2mc~σ · ~B we can point out the spin operator
~
S = ¯h~σ/2, replacing ~S instead of ~Lin Equation 1.3 we obtain the magnetic moment:
~
µ= e¯h
2mc~σ (1.8)
comparing that equation with Equation 1.1 we obtain a value for g = 2 for point-like particle with spin equal to 1/2. Although the proton is a fermion with spin equal to 1/2, not being a point-like particle, it has gp ≈ 5.6. The Dirac’s theory breaks the degeneration
of the atomic levels due to the last term in Equation 1.7, going beyond the Bohr atomic model. This effect, called Zeeman’s effect, is observed by placing the atoms in a magnetic field giving rise to the fine and iperfine structure.
Two experiments done in 1947 showed 2 effects that the Dirac theory could not explain, the Lamb shift and the measurement of the electron g-factor. The first measurement observed a shift in the energy level 2S1
2 of the Hydrogen atom discovering that the 2P 1 2
state is slightly lower than the 2S1
2 state measuring a shift between the two states equal to
4.372 µeV, a correction of 0.003% of the orbital energy, which is know as Lamb Shift (from Willis Lamb who discovered this effect). The second experiment performed by Kusch and Foley[3] was done comparing the values of the gJ for the atomic states 2P3
2 and 2P 1
2 of
the Gallium atom by the measurement of the frequencies lines in the hyperfine spectra in a constant magnetic field. The results obtained for gL was 1 as expected in Equation
1.3 and for gS was 2×(1.00119 ± 0.00005), in disagreement with the value predicted by
Dirac (gS = 2).
The explanation to these results brought to a revolution in the framework of the Quan-tum Mechanics, combining the latter to the Special Relativity formulated by Einstein, giving birth to Quantum Electrodynamics (QED) or more in general to Quantum Field Theory (QFT). According to the new theory a charged particle that interacts with the magnetic field exchanges virtual photons that are continually emitted and re-absorbed. These photons cannot be revealed, but contribute in a significant way to the physical process, modifying the values of the mass and charge. This kind of corrections was called radiative. The first radiative correction was computed by Julian Schwinger in 1948, us-ing for the first time the renormalization concept. Afterwards, Tomonaga, Schwus-inger and Feynman elaborated, independently each from the others, some particular methods of renormalization, so that quantum electrodynamics became one of the most accurate the-ories that ever have been developed and tested, bringing the Nobel Prize in 1965. Defining the magnetic anomaly of the electron as: ae = ge2−2 the first value found by Schwinger was
incredibly elegant and simple because joined two natural constants: ae = 2πα = 0.00116
where α is the fine structure constant α−1 = 137.035999084(21).
1.1.1
Virtual loops
Figure 1.1 shows the basic Feynman diagrams which contribute to the magnetic anomaly of the muon (the same are for the electron). In particular the first radiative correction computed by Schwinger is represented in the third diagram in the Figure 1.1 (c), where a virtual photon is exchanged between the two lepton legs. Higher order corrections can be also due to a production by the virtual photon of a couple of particle anti-particle (loop). The principle that makes possible this effect is the Heisenberg’s Uncertainty principle describe by: ∆E · ∆t ≥ ¯h
2 together with the Einstein’s equation E = mc
2 which means
that the vacuum is not empty, but for a very short time pairs of particles can be produced and reabsorbed. In particular from, ∆t the life-time of this pairs of particles is inversely proportional to the particle masses. In light of this, aµ cannot be measured without
the influence from the effects of virtual particles exchanged in the vacuum state. This effect called ”Vacuum Polarisation” (VP) is the quantum analogous of the polarisation of molecules in a dielectric when an external electric field is applied. The distorted molecules produce an electric field that reduces the field in the medium. The same, at quantum level, the production of a e+e− pair causes a screening effect that reduces the strength of
the electromagnetic force carried by the exchanged photon. This effect leads to a small change (anomaly) in the measured magnetic moment from what would be expected for a
Figure 1.1: Feynman diagrams for (a) the magnetic moment, corresponding to g = 2, (b) the general form of diagrams that contribute to the anomalous magnetic moment aµ, and
(c) the Schwinger term [2]. bare charge. [4]
Moreover a difference from 2 of the lepton’s g-factor could be an hint of internal structure as happen for the proton and neutron.
1.2
Why measuring a
µ?
The charged leptons family is constituted by 3 particles (and their antiparticles): electron, muon and tau. We expect that the anomalous magnetic moment can be observed for all the leptons. The measurement of the ae is easier then aµ or aτ because the electron is
a stable particle, in fact the first measurement of the anomalous magnetic moment was performed measuring the frequency of the Gallium atom states as described above [3]. The short mean life of muons (τµ≈ 2.2 µs) and tau particles (ττ ≈ 0.29 ps) is a limitation
for the experimental approach especially for taus. Moreover the measurement of the ae
is 1000 times more accurate then aµ. Despite this ae is less sensitive to weak and strong
interaction and possible new physics, in fact it is a ideal test for the QED and useful for the determination of α (fine-structure constant), but to test the entire Standard Model we need to measure the anomalous moment of an heavier lepton. Indeed radiative corrections of the g-factor are sensitive to ”new physics” effects proportionally to the squared mass of the lepton δal ∝
m2l
Λ2 where ml is the mass of the lepton and Λ the scale of new physics.
With this in mind aµ is m2µ/m2e ≈ 43000 times more sensitive to new effects then ae [5].
For this reason to perform a complete and accurate test of the SM it is more suitable to use muons, as they offer the best compromise between ae (highest precision experiment,
less sensitive to heavy particle) and aτ (high sensitive to new physics, too short mean
life). The muon presents also a very clear decay channel µ− → e−ν¯
eνµ (µ+ → e+νeν¯µ)
with a branching ratio around 100% ([6]) instead of tau that decays in adronic states because of its mass that makes highly difficult at the present, with his short mean life, a measurement of aτ2.
1.3
Measurement principle of a
µThe measurement of aµ is carried out producing polarized muons from pion decay and
injecting them to a storage ring. The spin precession of the muon when it’s immersed in a magnetic field is measured. In the rest frame the muon, spin rotates with a frequency proportional to the g-factor according to the Larmor precession formula:
~ ωS = g
e
2mB~ (1.9)
In the same time, the relativistic muons are collected in the storage ring orbiting with a frequency defined by the cyclotron frequency:
~ ωC =
e ~B
mγ (1.10)
The muon experiences a transverse centripetal acceleration that bring to the Thomas precession [9] of all observable in the lab frame. This effect can be describe as a Lorentz contraction of the rest frame axes causes the spin to turn at the frequency:
~
ωT = (γ − 1) ~ωC = (γ − 1)
e ~B
mγ (1.11)
instead the spin rotation frequency in the lab frame is: ~ ωS = g e 2mB~ + (γ − 1) e ~B mγ (1.12)
Jackson [10], citing Thomas [9] and Bargmann, Michel, and Telegdi [11], expands this expression to include electric as well as magnetic fields:
~ ωS = e m g 2− 1 + 1 γ ~ B − g 2− 1 γ γ+ 1(~β · ~B)~β − g 2 − γ γ+ 1 (~β × ~E) (1.13)
2A recent proposal to measure a
in this general treatment, the cyclotron frequency is written as: ~ ωC = 1 β2 ∂ ~β ∂t × ~β= e m 1 γ ~ B − 1 γβ2(~β · ~B)~β − γ γ2 − 1(~β × ~E) (1.14) from those equations we can define a new quantity as the difference between ~ωS and ~ωC
that represents the frequency of the rotation of the muon spin relative to its momen-tum, called ~ωa. This frequency (called ”anomalous precession frequency”) is the ”g-2
frequency” and represents with the measurement of the magnetic field, the most impor-tant observables of the g-2 experiment. Assuming that the motion of the muons is purely longitudinal, the momentum is perpendicular to the B field deleting the term ~β · ~B in Equation 1.14 and 1.13 ~ωa becomes:
~ ωa= ~ωS− ~ωC = e m aµB −~ aµ− 1 γ2− 1 (~β × ~E) (1.15) Motion of the spin vector relative to the momentum is represented in Figure 1.2 We can
Figure 1.2: In figure (a) is shown the spin procession for g=2, in figure (b) is shown the spin procession for g>2 [4].
simplify this expression cancelling the effect of the E-field, choosing a specific value for the Lorentz boost, computing γ from Equation 1.15 obtaining: γ = q1 + 1
aµ ≈ 29.3
that corresponds to a momentum pµ = 3.094 GeV/c called ”magic momentum”. The
experiments from CERN III (which will be discussed in the next section) to E989 use magic momentum to reduce at minimum the influence of the E-field on the muon beam in the storage ring. In this configuration the relationship between aµ and B is reduced to:
~ ωa = aµ
e ~B
from which to obtain aµ we need to know the ratio between ~ωa and the B-field.
Muon production
The charged pions produced from proton scattering on fixed target, decay in 2-body fi-nal state π+ → µ+ν
µ (π− → µ−ν¯µ) in the 99.9877 % of the time, and in π+ → e+νe
(π− → e−ν¯
e) in the 1.2 10−4 of the time [6]. After the discovery of the parity violation
in weak decay the pion’s decay channel became a easy way to produce polarized muons. Because of the spin-zero nature of the pion, the muon is emitted isotropically in the pion reference system. Due to the parity (P) violation the neutrino is emitted left-handed (due to the projection operator (1-γ5)/2 for V-A) and by the conservation of the angular
momentum, the muon is produced with helicity -1 (left-handed). In this way we obtain, from the pion decay, a polarized muon with spin opposite to the momentum. The muon thus produced, decays via the weak three-body decay µ−→ e−ν¯
eνµ (µ+→ e+νeν¯µ). The
positron (electron), in the regime of high momentum, has the the spin forward (backward) to its momentum. This is an effect of the P violation that prefers the positron (electron) in a right-handed (left-handed) helicity state. In particular on account of this positron are emitted with maximum probability when spin and momentum of the muon are parallel and minimum when are opposite.
Kinematics
In the pion’s rest frame the decay muon has energy equal to E∗
µ=
Mπ2+Mµ2
2Mπ and momentum
p∗µ= Mπ2−Mµ2
2Mπ . Computing a Lorentz transformation in laboratory frame the muon energy
becomes:
EµLAB = γπ(Eµ∗+ βπp∗µcosθ) (1.17)
where γπ = mEππ and βπ = mEππ (with Eπ, mπ and pµ energy, mass and momentum of the
pion) and θ is the angle between pion and muon momentum in the rest frame. Computing the distribution of the positron produced from the muon’s decay we obtain [8]:
d3Γ
dydcosθdφ ∝ n(y)[1 + a(y)cosθ] a(y) = 1 − 2y 3 − 2y n(y) = 2y2(3 − 2y)
(1.18)
where y = E
Emax is the normalized positron’s energy, a(y) is the asymmetry function,
de-pends on energy, n(y) is a factor and θ the angle between muon and positron’s direction. It’s important to know the distribution in Eq.1.18 is expressed in the muon reference
frame. To calculate Eq.1.18 in laboratory frame (the whole calculation is in [12]), we can write cosθ as ˆPe· ˆA, where ˆPe=cosθek+sinθˆ ecosφeˆi+sinφecosθeˆj is the positron’s
momen-tum unit vector (positron’s direction) and ˆA= cos(ωat)ˆk+sin(ωat)ˆi a generic
transforma-tion about y (ˆj) axis. So cosθ becomes:
cosθ = cosθecosωat+ sinθecosφesinωat (1.19)
we can replace cosθ in Eq.1.18 with Eq.1.19 and then integrate over φ from 0 to 2π because we are interested only on happens longitudinal plane (plane of the ring). Equation 1.18 becomes:
Γ(y, cos(ωat)) ∝ n(y)(1 + a(y)cos(ωat)ξ)dydξ (1.20)
where ξ=cosθe. We can approximate the energy of the electron in the laboratory frame
as ELab
e ∼yγE
∗
e(1+ξ) because the electron energy in the rest frame is much bigger than
its mass, so E∗
e ≈ p
∗
e, in the rest frame the maximum positron’s energy is E ∗
e=
M2
µ+m2e
2Mµ
(when positron is emitted forward muon’s direction and neutrinos’ backward), while the minimum energy is Emin
e =me (when the positron is emitted backward in the muon’s
direction and neutrino forward). We do then a simple transformation of Equation 1.20 from ξ to E:
Γ(y, E)dydE = P (y, ξ)dydξ Γ(y, E) = P (y, ξ)dξ dE Γ(y, E) = 2πn(y) 1 + a(y) E yγEmax e cos(ωat) 1 yγEmax e (1.21)
where we have now to integrate over y Eq.1.21 replacing a(y) and n(y) with function in Eq.1.18, because we want to know the number of positrons produced over a threshold y= E
Emax
e (generic electron energy). By integrating and rearranging all the terms we can
obtain the new values of A(y) and N(y) equal to: A(y) = −1 + 9y 2− 8y3 5 − 9y2+ 4y3 = 1 + y − 8y2 4y2− 5y − 5 N(y) = 2π 3γEmax e (y − 1)(4y2− 5y − 5) (1.22)
These functions represent the asymmetry and normalization in the Laboratory Frame and are compared with the ones in the Rest Frame in Fig.1.4 and Fig.1.3. Finally we can shown in Fig.1.5 the distribution of the positrons produced at a certain energy Eq.1.23, using the new expression found for N(y) and A(y), for different angles ωat showing the
different positron spectrum:
Γ(y) ∝ N (y)[1 + A(y)cos(ωat)] (1.23)
The distribution of positrons in function of time for different angles between muon’s
Figure 1.3: Function of asymmetry (a(y)) in Rest Frame and (A(y)) in Laboratory Frame.
Figure 1.4: Function of normalization (n(y)) in Rest Frame and (N (y)) in Laboratory Frame.
spin and momentum is shown in the following equation: N(t) = N (y)e−t/γτµ[1 + A(y)cos(ω
at+ φa)] (1.24)
where the γτµ ≈ 64.4 µs is the muon mean life in the laboratory frame, while A(y) is the
asymmetry factor defined as the probability to produce a positron in function of energy for a given angle between spin and momentum of the muon, N(y) is the normalization constant and φa is the initial phase of the muon spin relative to its momentum, at the
Figure 1.5: Energy spectrum of positron in muon decay, for different ωat angles.
moment of the muon injection.
Figure 1.6 shows the number of positron in function of time for different values of energy. We expect that for E such that A(y)=0 (y∼0.42), Eq.1.24 is a simple exponential due to the muon’s decay distribution. For y>0.42 we have exponential modulated proportionally to cos(ωat). With this in mind, in the experimental data we have a mean amplitude
integrated from 1.7 to 3.1 GeV, or from 0.58 to 1 in y.
Figure 1.6: Number of positrons produced from muon’s decay, in function of time, for 3 different values of positron’s energy.
Theoretical Calculation of a
µ
From the first calculation of the radiative correction of the electron g-factor in 1948 computed by Julian Schwinger continuous progress have been done and a large community of physicists working with the aim to reach a new level of precision on the theoretical prediction on the aµ.
2.1
Summary of the main contributions
As discussed in the previous chapter the muon magnetic anomaly is due to the exchange of virtual particles. To compare the experimental results with the theoretical prediction we need to compute the contribution to aµ from different sectors of the SM, i.e.
Quan-tum Electrodynamics (QED), Electroweak theory (EW) and QuanQuan-tum Chromodynamics (QCD). aSM
µ can be therefore written as:
aSMµ = aQED µ + a EW µ + a HV P µ + a HLBL µ + a HOHV P µ (2.1)
where the QCD contribution, that brings the highest uncertainty, is divided in 3 different parts. The lowest contribution arises from the hadronic vacuum polarization (HVP) where a loop of quarks is inserted in the virtual photon line, the higher order to this correction (HOHVP) that contains the HVP with an additional loop; the ”hadronic light-by-light” (HbLb) scattering and it’s cause of large uncertainty, because depends on the phenomenological model used to compute it. The final precision reached by the recent muon g-2 experiment in BNL (E821) is:
δaexpµ = 6 × 10−10(0.54ppm) (2.2)
SM Term aT hµ × 1011 Ref. aQED µ 116 584 718.951 ± 0.009 [16] aEWµ 154± 1 [16] aHlBl µ 105 ± 26 [16] aHV Pµ 6923 ± 42 [16] aHOHV P µ -98.4 ± 0.7 [16]
aT OTµ 116 591 828 ± 43HLO ± 26HHO ± 2other (± 50tot) [16]
Table 2.1: Values of the SM term that define the aµ prediction compared with the best
experimental error in E821 experiment.
that is useful to compare with the accuracy reached in the theoretical prediction, shown in Table 2.1 From Table 2.1 appears that the QED contribution is the greater, but also the most precise, while the error comes almost entirely from the hadronic contribution, for this reason most of the efforts are directed to increase the precision in this sector.
2.2
The QED Contribution
The QED contribution arises from the subset of the SM diagrams containing only leptons (e, µ, τ ) and photons. This contribution can be summarize in this form:
aQEDµ = A1+ A2(mµ/me) + A2(mµ/mτ) + A3(mµ/me, mµ/mτ) (2.3)
where me, mµ and mτ are the masses of the leptons. The first term A1 in Equation
2.3 arises from the diagrams that contain only muon and photon, it is mass independent and it is the same for the anomalous magnetic moment of the three charged leptons. While, the terms A2 and A3 are mass dependent (mass ratios, to be exact), by the graphs
that contain also electrons and tauons.The renormalizability of QED guarantees that the functions Ai (i = 1, 2, 3) can be expanded as power series n α/π and computed order by
order [5]: Ai = A (2) i α π + A(4)i α π 2 + A(6)i α π 3 + A(8)i α π 4 + A(10)i α π 5 + .... (2.4) The diagram shown in Figure 2.1 is the only one involved in the calculation of the lowest-order contribution, it is the famous diagram that Julian Schwinger computed for the first correction of the electron g-factor. from Equation 2.4 at the first order is A(2)1 = 1 2
while A(2)2 = 0 and A(2)3 = 0. The next order includes a two-loop correction (fourth-order) in which contribute 9 different diagrams, seven to A(4)1 , one to A
(4)
Figure 2.1: Diagram of the lowest QED correction of aµ [5].
one to A(4)2 (mµ/mτ). The three-loop correction (sixth-order) is the last one computed
analytically that required approximately 30 years and was completed in the second part of 1990s, by Remiddi and his collaborators [18]. The diagrams that contribute to A(6)1 are 72, while the term A(6)2 (mµ/me/τ) is divide in two different contributions, one heavy due
to the e or τ vacuum polarization loops, described by 36 diagrams and one light due to light-by-light scattering with e or τ loops. The four-loop correction to aµ was calculated
by Kinoshita and his collaborators in the early 1980s ([19][20]). Only few terms of this eight-order correction was computed analytically. Finally there is the five-loop correction, tenth-order term, that was provided again by Kinoshita and collaborators. We can sum up all the QED correction known until the tenth order in the final result of the theoretical prediction based on the SM, knowing the fine-structure constant α with an uncertainty of 3.3 ppb [21]:
α−1 = 137.03599911(46) (2.5)
the value of the correction on aµ by QED is:
αQEDµ = 116584718.8(0.3)(0.4) × 10−11. (2.6) in particular the first error is due to the uncertainty of O(α2), O(α4) and O(α5) terms and
is strongly dominated by the last one (the error on O(α3) is negligible). The second error
is caused by the 3.3 ppb on the α uncertainty. Combining the uncertainty in quadrature we obtain: δaQED
µ = 0.5 × 10 −11
2.3
The EW Contribution
This contribution is due to the coupling with the gauge bosons as Z, W± and Higgs.
In fact this term is suppressed as the ratio between the lepton mass and the boson mass (mµ/MW)2 respect to the QED contribution. For this reason, as explained in the previous
Chapter, the sensitivity to a different contribution from the QED is bigger for heavier lepton, like muon or tau instead of electron. The one-loop contribution was computed in 1972 from many authors [22], at those time the experimental uncertainty was 3 times larger than this contribution, now is less than one-third of those result.
The analytical expression of the one-loop EW correction due to the diagrams in Figure 2.2, containing only the coupling to vector bosons, is defined by the following equation:
aEWµ = 5Gµm 2 µ 24√2π2 1 + 1 5(1 − 4sin 2 θW)2+ O m2 µ M2 Z,W,H (2.7) where Gµ = 1.16637(1) × 10−5 GeV−1 is the Fermi constant MZ, MW and MH are the
Figure 2.2: Diagram of the one-loop EW correction of aµ, due to Z and W bosons [5].
masses of Z, W and Higgs bosons and θW the mixed angle in weak interaction. Using the
SM definition for sin2θ
W1: sin2θW ≡ 1 −
M2
W
M2
Z ' 0.231 and the particles masses kept from
PDG [6] mµ= 105.685 MeV/c2, MZ=91.188 GeV/c2, MW=80.385 GeV/c2and MH=125.7
GeV/c2, using the approximation where the O
m2 µ M2 Z,W,H
are neglected the contributions of the 3 gauge bosons are [23]:
a(2)EWµ (W ) = √ 2Gµm2µ 16π2 10 3 ' (388.70 ± 0.10) × 10 −11 a(2)EWµ (Z) = √ 2Gµm2µ 16π2 (−1 + 4sin2θ W)2− 5 3 ' (−193.89 ± 2) × 10 −11 a(2)EWµ (H) = √ 2Gµm2µ 4π2 Z 1 0 dy (2 − y)y 2 y2+ (1 − y)(M H/mµ)2 √ 2Gµm2µ 4π2 m2 µ M2 H lnM 2 H m2 µ ∼ 5 × 10−14 (2.8)
using the parameters defined before the one-loop EW contribution predicted is:
a(2)EWµ = (194.82 ± 0.02) × 10−11, where the error is due to the uncertainty on sin2θW.
The 2-loop correction was performed for the first time in 1992 [24], after the measurement
of the Higgs mass at LHC the prediction was re-evaluated in 2013 [25]:
a(4)EWµ = (−19.97 ± 0.03) × 10−11 (2.9)
where the error is due to the experimental precision on the Higgs mass (mH = 125.7 ± 0.4
GeV/c2 [6]). The higher order contributions were found to be negligible.
2.4
The Hadronic Contribution
Considering that hadronic effects in the two-loop EW contribution are included in aEW
µ ,
discussed in the previous section, the leading correction is due to hadronic vacuum po-larization to the internal photon propagator of the one-loop diagram. The evaluation of this term involves long-distance QCD for which perturbation theory cannot be applied. In 1961 Bouchiat and Michel [26] understood that this term can be calculated using analyticity and unitarity (the optical theorem of the scattering matrix) using the e+e−
annihilation in hadrons data computing the dispersion integral ([26], [27]): aHLOµ = 1 4π3 Z ∞ 4m2 π dsK(s)σ(0)(s) = α2 3π2 Z ∞ 4m2 π dsK(s)R(s) (2.10)
where σ(0)(s) is the experimental total cross section for e+e−annihilation into any hadronic
state, R(s) is the ratio between σ(0)(s) and the high-energy limit of the Born cross section
of the µ-pair production: R(s) = σ(0)(s)/(4πα2/3s), while K(s) is the kinematic factor
ranging from 0.4 at s=mπ to 0 for s=∞ which can be expressed:
K(s) = Z 1 0 dx x 2(x − 1) x2+ (1 − x)s/m2 µ (2.11) From one of the last analyses [28] using e+e−→ hadrons the results to hadronic leading
order correction is: aHLO
µ = (6923 ± 42) × 10
−11.
The higher order correction of the hadronic contribution aHHLO
µ is divided in 2 different parts: aHHLOµ = aHHO µ (vp) + a HHO µ (lbl) (2.12)
where the first term (order O(α3)) contribution contains all the diagrams with hadronic
vacuum polarization, including also other couplings as shown in Figure 2.3. The most recent evaluation of the next-to-leading order hadronic contribution, determined by the
Figure 2.3: Different contribution of the next to leading order for the hadronic vacuum polarization contributing to aµ [5].
dispersion relation is:
aHV Pµ = (−98.4 ± 0.6exp± 0.4rad) × 10−11 (2.13)
Very recently, also the next-to-next-to-leading order hadronic contribution has been eval-uated in [29] with a result of the order of the expected future experimental uncertainty. The second term in Equation 2.12 is the hadronic light by light scattering that it cannot be determined from data, but it could be computed using hadronic models that correctly reproduce properties of QCD. This contribution is shown in Figure 2.4 (a) it is dominated
Figure 2.4: (a)The Hadronic Light-by-Light (LbL) contribution. (b) The pseudoscalar meson contribution to aµ prediction. [30].
by the long distance contribution shown in Figure 2.4 (b).
The final result which was agreed by all the authors of the leading groups that worked on that [33] is:
aHlblµ = (105 ± 26) × 10−11 (2.14)
Comparing the theoretical prediction:
aSMµ = 116591802 ± 42H−LO± 26H−HO± 2other(±49tot) (2.15)
with the last experimental result in E821 shows a discrepancy: ∆aµ= aE821µ − aSMµ = (26.86 ± 7.24) × 10
−10
(2.16) which corresponds to 3.6 standard deviations. If the Muon g-2 experiment at Fermilab will confirm the result obtained in E821 with the design experimental uncertainty of 0.14 ppm [30], the discrepancy could reach 7 σ that it would be a very strong statistical evidence for possible contribution beyond Standard Model.
2.5
Beyond Standard Model contribution
The discrepancy observed with the experimental measurement of the aµ could be due to
statistical fluctuation, systematic bias (in theory or experiment) or possible contribution beyond SM. The muon g-2 is sensitive to a large NP (New physics) energy scale within different models. In fact the are many possibilities to explain, if confirmed in the new g-2 experiment, the discrepancy on aµ value, for example compositeness of the muon, rare
decays that break the lepton flavour conservation (i.e. µ− → e−γ), coupling to
axion-like particles (which may constitute the Dark Matter), interaction with dark photon (a propagator similar to the photon for the electromagnetism but potentially connected to the dark matter), contribution to the loop from supersimmetric particles and finally the existence of the electric dipole moment (EDM). It’s interesting to review some possible contributions cited above.
Muon Compositeness
The hadrons as proton, neutrons and mesons are particle composited by quarks. The evidence of this, for example, is in the values of g-factor of proton and neutron extremely different from 2 (gp=5.6 and gn=-3.8). The aµ is a tiny correction (∼ 10−3) from 2, then
this could be an evidence that muons are composed of smaller constituents than quarks. This constituents could be new fundamental particles known as ”preons” belonging to another generation of leptons. In Figure 2.5 are shown the tree-level Feynman diagrams that represent the model of the muon compositeness. The contribution to the aµ
correc-Figure 2.5: Feynman diagrams [34] for (a) the leading-order effect of compositeness, which must be canceled out in a workable model; (b) a form factor at each µγ interaction vertex; (c) excited lepton states; (d) four-fermion contact interactions.
tion is linear with the ratio between muon mass mµand Λ (characteristic scale constant).
This model is not untenable because already from the measure in CERN III the limit on Λ was > 2000 TeV. The self term interaction brings to unphysical muon mass, so any reasonable model must delete this term. One natural way to achieve this cancellation is to build a chirally symmetric wavefunction in which the left-handed and right-handed interaction terms exactly balance [35]. Once the linear contribution as been removed, substructure affects aµ in three way. First any vertex of interaction of the muon with
any particle should be multiplied by a from factor (1 −Λq22) to take in account the charge
spatial distribution. Second, the muon should be in excited state and the component may acquired an orbital angular momentum. Third, there may be contact interactions among the constituents that do not correspond to the usual exchange of gauge bosons. Feynman diagrams representative of each of these categories of effects are shown in Figures 2.5(b)-(d). Clearly the numerical result depends on the model used, in each case any results is proportional to m2µ
Λ2 [34].
Electric dipole moment (EDM)
The electric moment dipole (EDM) for a point-like particle has never been measured, if it were present would case a precession of the radial component of the particle’s spin. In the muon g-2 experiment we measure the ωa spin precession frequency, that is affected
by EDM in this way:
∆ ~ωa= −2dµβ × ~~ B −2dµE~ (2.17)
in the experimental design we measure the magnitude of ωa=| ~ωa|, which is sensitive to
the combined effect of the magnetic and electric dipole moments. Usually we neglect the E-field term because is very small compare to B-field one. Considering the contribution of the EDM we can write ωa as:
ωa = B s e mµ aµ 2 + (2dµ)2 (2.18)
neglecting dµ we obtain the typical expression in Equation (1.16) of ωa. The existence
of the EDM for a elementary particle violates both parity (P) and time reversal (T) symmetries. Defining the EDM operator as:
~ D=
Z ~
rρ(~r) ~dr (2.19)
that is a vectorial quantity, in fact an elementary particle in the rest frame, the only vector property is the spin, so D is proportional to the spin. Spin is antisymmetric respect to T and symmetric under P , while ~D is opposite: symmetric under T and antisymmetric under P. In the SM the CPT symmetry should be unbreakable, so the T violation is similar to the CP violation. The only mechanism known in the standard model for the CP violation is the complex phase in CKM mixing matrix. Several models that include also supersymmetric particles, include additional CP violation diagrams that produce an EDM at one-loop order. If we consider the same model for electron and muon the EDM scales only proportionally with the mass. The Standard Model value of the EDM contribution to ωa is dµ = −3 × 10−36e cm, calculated in [36]. Assuming muon-electron universality
we can define from the current limit on electron EDM a bound, model dependent on the muon EDM:
dµ<9.1 × 10−25 e cm (90%) C.L. [37] (2.20)
The g-2 experiment is not a perfect probe to measure the EDM, because the effect on ωais
dominated by the B-field that reduces heavily the sensibility on the EDM. An experiment to measure the EDM has been proposed [38] in which we want to remove all the effect that introduce spin precession (E-field and B-field) of the bunch and observe the spin precession due only to EDM effect. The designed sensitivity of this experiment is around 10−24 e cm, which is comparable with the value in Equation 2.20.
Dark Photon
The dark photon is an hypothetical particle of an extra U(1) gauge group, which can ex-plain the discrepancy observed on aµ. Gauge bosons of an extra U(1) group are searched
for from the lowest up to the highest energies at the LHC. The mass range for a vector particle in the MeV to GeV scale has been in the focus of interest due to the obser-vation that particles of such a mass scale might explain a surprisingly large number of astrophysical anomalies besides the magnetic anomaly. Figure 2.6 shows two possible Feynman diagrams with Dark Photon interaction. The upper diagram shows the kinetic
Figure 2.6: Two different models of interaction via a hypothetical dark photon γ0. U p:
kinetic mixing model, Down: interaction between SM and Dark Sector. [39]
mixing between the SM photon and the hypothetical γ0 Dark Photon through a loop of
charged leptons (L). The mixing is established by a parameter . The NA48/2 Collab-oration’s 2015 publication has shown the limit on parameter, by defining an exclusion region, leaving many possibilities of light dark photon models, see Figure 2.7. For ex-ample a scalar or axion-like pseudoscalar particles (m0
γ < 1 GeV/c2) could explain the
aµ discrepancy [40]. The lower diagram represents the annihilation of dark matter in
standard model particle pair (as e+e− couple), it might give rise to the positron excess
presently seen in the spectrum of cosmic rays. A recently anomaly in Beryllium decay has been observed, showing the first hint of 17 MeV Dark Photon signal. Many experiments are working to independent confirmation of the anomaly. This could be an indication, together with the (g-2)µ anomaly, that the dark photon may be a good model to explain
Figure 2.7: Parameter space of the dark photon model relevant to muon g-2, reproduced from [41]. On the x-axis is the dark photon mass and on the y-axis is the parameter governing the mixing between the dark photon and the Standard Model photon. The red band is the region of parameter space that resolves the muon g-2 discrepancy
The a
µ
measurement: early
experiments
The first experiment thought to measure the muon g-factor was carried out after the discovery of parity (P) violation in weak decay, by Lee and Yang in the 1956 [44], who suggested several experiments able to observe this new phenomenon, that was measured by Madame Wu in the same year studying the β decay of the Cobalt atom [43]. The P violation allows to obtain polarized muon in the pion decay. The Lee and Young prediction didn’t stop there, they pointed out the problem that also the charge conjugation (C) and time reversal (T), the fundamental symmetries, could be violated by the weak decay. In the 1957 R. L. Garwin, L. M. Lederman and M. Weinrich made the first experiment to measure the muon g-factor and demonstrate the failure of parity and charge conjugation [42]. To do that they took pions from the Nevis cyclotron (Columbia University) with momentum of 85 MeV/c stopped in a carbon target and measuring the number of electrons (positrons) emitted from the muon’s decay. The absorber is immersed in a magnetic field, that induce a muon spin rotation, the Larmor precession frequency, measurable through the variation of the magnetic field strength. In fact changing the amplitude of the magnetic field the count of the number of electrons (positrons) changes, measuring the ratio of the counts with and without the field in function of the magnetic field strength they observe the precession. The result is shown in Figure 3.1 (from the original paper). The precision reached in this experiment on the muon g-factor was around the 10% not enough to show the anomaly, but sufficient to verify the P violation. In the 1963 another experiment was performed by Hutchinson and collaborators [45] with the intent to measure the muon g-factor measuring the number of electron emitted, from stopped muon, in function of the field magnitude, in a similar way of Garwin experiment. Hutchinson and collaborators, to compare the value with the theory prediction, had the foresight to
Figure 3.1: Historical plots showing Larmor precession data from Garwin experiment [42]
measure also the Larmor precession frequency of the proton, using nuclear measurement, with a NMR (Nuclear Magnetic Resonance) probe of polarized water. They reached a great precision, expressing the result as:
λ= ωµ ωp
= µµ µp
= 3.18338 (4) (3.1)
The uncertainty was 10 ppm (parts per million), but the comparison with the theoretical value was limited by the uncertainty on the muon mass (about 100 ppm). From Equation 1.9 we can extrapolate the muon g-factor value from Larmor precession:
gµ= 2mµ
| ~ωS|
e| ~B| (3.2)
The precision was enough to observe the first order correction to the measurement of the muon g-2: gµ− 2 = 2( 1 2 α π + ...) (3.3)
as explained in Chapter 2 the factorizability on the aµ prediction allows us to write each
contribute, of higher order, as power of α
π times a constant that define the amplitude of
the correction.
It’s important to know the experimental evolution on the measure of the g-2 of the muon, to understand the experimental and theoretical steps that led to the modern experiments as the new one at Fermilab that could lead to a revolution in particle physics breaking down the Standard Model.
3.1
The CERN experiments
From 1961 a series of experiments of increasing precision was performed at CERN mea-suring aµusing polarized muon produced by pion decay. They understood the powerful of
this kind of high precision measurement, able to breakdown the QED calculation pointing out some differences between theoretical prediction and experimental results. The idea was the same as the previous experiment, to measure the distribution of the muon decay products (electron/positron) in a magnetic dipole of the muon with an external magnetic field.
3.1.1
CERN I experiment
CERN I was the first experiment done in the Swiss laboratory of CERN, producing muons from the synchro-cyclotron that accelerates protons to 600 MeV, from proton scattering on beryllium target. The main goal of the first CERN experiment designed and made from Charpak et al. [47] was to reach an experimental precision of 10−5 on g
µ or 1% on
aµ. The theoretical prediction on aµ achieved in that moment was till the second order:
aµ= (gµ− 2)/2 = 1 2 α π + 0.75 α π 2 + ... = 0.001165 [47] (3.4) Theoretically to measure aµ to that precision they should know the muon mass with a
precision < 10−5. This precision on muon mass was impossible to reach in that time, but
fortunately they exploited a principle employed also for the electron g-factor measurement, which has been discussed in previous chapters. If muon circulate in a magnetic field (B) the spin turns (1+γa) times as fast as the muon momentum. For this reason after an interval of time, t, the spin has reached an angle respect to momentum θ = aµBωCtwhere
ωC is the cyclotron frequency of the muon beam. Figure 3.2 shows the muon path that
comes in the 6 metres magnet. The stored time is from 2.0 µs to 6.5 µs. The signal of muon is defined by 4-5-660¯7 coincidence and gated by 1-2-3 and forward or backward electron signal, in fact a muon that decays in electron, stopped in the methylene-iodide target (T in Figure 3.2), can emit electrons forward or backward dependently to the angle between its spin and momentum. In fact 660 and 770 are the signals of backward and forward emission respectively. By counting, during the acquisition time, the number of electrons emitted along the muons direction cn+ and those emitted in the opposite direction cn−
Figure 3.2: T op: The M bending magnet, Q a set of focusing quadrupoles, 1-Be-2-3 are a set of beryllium moderator and counter, to the other side T is a methylene-iodide target and 660 and 770 are the electron forward and backward telescope. The signal of stored and ejected muon is 4-5-660¯7 and gated by 1-2-3 signal and forward or backward electron signal. Bottom: Curve A (left-hand scale for ordinate) represent the muon distribution during the storage time, Curve B (right-hand scale for ordinate) represents the asymmetry values in function of storage time, where the black solid line is the best-fit in Equation 3.5 [47]
asymmetry for single interval of storage time (tn) parameter can be obtained as:
An=
cn+− cn−
cn++ cn−
= Asin(aBωCtn) (3.5)
The best fit of the data, reported in Figure 3.2, showed the following result:
aexpµ (CERN I) = 0.001162(5) → ± 5264 ppm (3.6) in agreement with the theoretical prediction:
3.1.2
CERN II experiment
The second experiment brought many improvements. A brand new accelerator was im-plemented at CERN the PS (proton synchrotron) machine capable to produce protons with 28 GeV of energy with an higher luminosity then the synchro-cyclotron. For the first time they had the idea to use a storage ring (idea exploited until today) of 5 metres of diameter with a C-shape cross section to absorb the proton beam that interacts with the target as shown in Figure 3.3. The new accelerator allowed to produce relativistic
Figure 3.3: Lef t: Plan of 5 m diameter ring magnet. The proton beam enters in the yoke and hits a target inside the magnetic field. Right: The distribution of decay electrons as a function of time. A) from 20 to 45 µs, B) from 65 to 90 µs, C) from 105 to 130 µs(upper scale time). The lower curve show the rotation frequency of the muon (lower scale time)
[48].
muons from 10.5 GeV proton beam onto a target inside the storage ring. The muons thus produced from the pions decay, had momentum equal to pµ = 1.27 GeV/c, captured by
a magnetic field of 1.7 T produced by a series of 40 independent bending magnets. The boosted experienced by the muons is γ = 12.06. The storage ring and the relativistic γ allowed Bailey et al. [48] to extend the mean life of the muons accumulating more statistics, with a new kind and very efficient injection in which pions (produced by the protons) and protons, that didn’t interact with the target, entered in the storage ring. This amount of lost particles was a great source of background (called hadronic f lash) that compromised the detectors, and limited the sensitivity of this experiment. This was one of the limitation of the second CERN experiment together with the small numbers of
muons stored.
The data acquisition was made by 6 counters placed in a quarter of the ring perimeter (reducing the acceptance), the detector measured the energy and time arrival of positron produced by the muon’s decay. The positron’s arrival time spectrum is described by:
N(t) = N0e−t/γτµ
1 + A(cosωat+ φ)
(3.8) where the parameters are already discussed in Section 1.3. The data fitted with Equation 3.8, between 20.5 µs and 189 µs, are shown in Figure 3.3 [48]. The positron spectrum allows to measure the ωavalue while the proton spin measurement (ωp) is performed with
NMR probe. The final result of CERN II experiment was:
aexpµ (CERN II) = 0.00116616 (31) → ±270 ppm (3.9) The theoretical prediction that reached the third order correction (α3
π3) was:
athµ(1969) = 0.00116587 (3) → ±25 ppm. (3.10) This experiment was the heart of the future experiments because introduced the modern experimental setup that with subsequent improvements brought to one of the most precise experiments in particle physics.
3.1.3
CERN III experiment
The third and last experiment of the CERN series was completed in 1979 [49]. Many improvements were introduced both on the engineering side and physical side. The ex-perimental apparatus is shown in Figure 3.4. From the engineering side was built a new storage ring with 14 metres diameter immersed in a dipole magnet field of 1.5 T was built, the beam target was moved out of the storage ring to reduce the hadronic f lash, an inflector magnet was introduced to push the muons on the right path inside the ring and, by transporting the pions inside the ring through a beamline a narrow range of pion momenta could be selected obtaining a better polarization of the muon beam. Finally quadrupole magnets were introduced to exert a vertical strong focusing on the muon beam, using the weak focusing provided by the dipole magnet along the horizontal axis and 24 electromagnetic calorimeters (a sandwich of lead and scintillator) all around the ring to improve the detector acceptance. From the physical side a particular momentum for muons, the so-called magic momentum equal to pµ = 3.049 GeV/c was chosen to
Figure 3.4: T op: Overview on CERN III apparatus showing the 14 m diameter ring magnet. The pion beam produced outside the magnet come in the storage ring via a beam line. Bottom: The distribution of decay electrons as a function of time, from 16 to 534 µs [49]
reduce the influence of the E-field, as explained in Section 1.3. Furthermore, the magic momentum implies a relativistic γ = 29.3 which leads to a dilated mean life equal to 64.4 µs, increasing the statistical precision respect to CERN II extending the storage time from 130 µs to 500 µs as shown in Figure 3.4. The data fit was performed with the same function used in CERN II experiment and written in Equation 3.8. These improvements brought an incredible experimental precision on the aµ. They were able to do the same
measure for µ− and µ+. Combining the results for positive and negative muons (with 10
ppm accuracy) the final value of aµ was:
aexpµ (CERN III) = 0.001165924 (8.5) → ±7.3ppm (3.11) where the error is completely dominated by the statistical error.
The theoretical predictions in those years reached a new level of precision introducing also the hadronic contribution-at one loop- substituting in the QED one-loop correction a couple of quark or gluon instead of a couple of lepton/anti-lepton. In fact only the experimental precision reached in CERN III could confirm the importance of hadronic polarization. The value on this prediction was:
athµ(1977) = 0.001165921(13) → ±11ppm (3.12) In agreement with the experimental value.
3.2
The BNL experiment
The next experiment after CERN III that measured the muon g-2 was E821 at Brookhaven National Laboratory (BNL). The necessity to realize a new experiment came from the incredible improvement in the theoretical prediction. In fact in 1984 Kinoshita et al [19] performed an accurate calculation of the fourth order on α/π for the QED prediction that until then was dominated by the hadronic contribution. Kinoshita and collaborators found a new way to increase the precision by computing the R(s) function defined in Chapter 2, using the experimental results in e+e− collider. Thanks to them a group of
physicist that included also many of the original CERN collaborators met at BNL to dis-cuss the possibility to build a new experiment at AGS (Alternating Gradient Synchrotron) accelerator. After many discussions they fixed the design precision at 0.35 ppm. A sketch of the storage ring is reported in Figure 3.5.
From CERN III they knew that the 7 ppm uncertainty was mostly due to the statistics, so the first goal was produce an higher rate of muons, reaching a factor of 20 times the
Figure 3.5: Diagram of the storage ring used in the BNL experiment.[4]
CERN III statistic, increasing the muon flux by approximately a factor of 400. It was possible thanks to the high luminosity of AGS, 20 times higher than the PS at CERN used for CERN III experiment. Another improvement was performed on the muon beam, producing it outside the storage ring, building a channel decay for pions, in such way they could injected only muon in the ring, increasing also the number of muon stored per AGS cycle. The new storage ring was equipped also with a system of 3 pulsed electromagnetic kickers, to avoid the interference between muon and pion beams and to inject the muons in the ideal orbit after the inflector. The magnetic field was produced by a superconduct-ing rsuperconduct-ing, composed by 3 continuously superconductor yokes with a very high homogeneity and uniformity. Finally a system of NMR probes was built to monitor and measure the magnetic field in situ. The positron signal reported in Figure 3.6 in function of the time was recorded by waveform digitizers and stored for later analyses.
Again the fit of the data was performed with the five-parameters function 3.8, reaching this time 700 µs of storage time instead of 500 µs of CERN III thanks to the improvements applied in the new experiment.
The result reached in the last analysis in 2001 on aµ measurement is:
Figure 3.6: The positron decay spectrum in function of time.[4]
compared with the best estimation of aµ considering the fifth order on (α/π)5, EW and
hadronic corrections [50] we obtain: aexpµ (BN L) − ath
µ = (26.86 ± 7.24) × 10 −10
(3.14) pointing out a not negligible discrepancy of 3.6 σ, as discussed in Chapter 2, this evidence could be explained using BSM model that could change the point of view on the known Universe. Table 3.1 summarizes the history of aµmeasurements. It’s interesting to notice
how by measuring the aµanomaly of µ+and µ−it is possible to verify the CPT symmetry.
We expect that the ratio between ωa and ωp (R) for µ+ and µ− measurements should be
the same. From the last measures at BNL the CPT test gave the result:
Rµ− − Rµ+ = 0.003 707 208 3(26) − 0.003 707 204 8(25) = (3.5 ± 3.4) × 10−9 (3.15)
Experiment Years Particle aµ× 1010 Precision [ppm] Ref. CERN I 1961 µ+ 11 450 000(220 000) 4300 [47] CERN II 1962-1968 µ+ 11 661 600(3100) 270 [48] CERN III 1974-1976 µ+ 11 659 100(110) 10 [49] CERN III 1975-1976 µ− 11 659 360(120) 10 [49] BNL 1997 µ+ 11 659 251(150) 13 [51] BNL 2000 µ+ 11 659 204(9) 0.7 [51] BNL 2001 µ− 11 659 214(8) 0.7 [51] Average 11 659 203(8) 0.7
The Muon g-2 experiment at
Fermilab
The new experiment E989 at Fermilab National Laboratory (FNAL) has the purpose to measure aµwith a precision of 0.14 ppm, a fourthfold improvement respect to the previous
E821 experiment. If the new result will confirm the previous measurement, a more then 5 σ discrepancy with SM can be achieved, which could be strong statistical evidence that the Standard Model is not sufficient at this level of precision and new physics is required. To reach this precision a statistics 20 times greater than BNL experiment is needed. In fact the design error of 140 ppb (parts per billion) is obtained by a statistical contribution of 100 ppb and a systematic one of equal value. The latter is made up of two contributions: 70 ppb on ωa and 70 ppb on ωp. Considering that the BNL budget on systematic error
was 180 ppb, the real challenge in E989 is to reduce this value by a factor of about 3. One of them is the use of a state-of -art Laser Calibration System to take under control the gain fluctuations of electromagnetic calorimeters that, as shown in Table 4.2, was the major source of systematic uncertainty in E821. This new and high precision system will be discussed in the following Chapter 5. The experimental setup is inherited in large part from the previous one, in fact the superconductor storage ring is the same of BNL, but new and sophisticated elements to replace or upgrade the old ones are installed in E989. Some of them are for example three new powerful electromagnetic kickers, 24 segmented electromagnetic calorimeters constitutes of 54 PbF2 crystals matched Silicon
Photo Multiplier (SiPM) and the Laser calibration system. Improvements were made also in the accelerator complex:
• higher proton rate with less protons per bunch, to reach the annually delivery 2.3·10208 GeV proton on an Inconel target1 target. This statistics should be enough
1Metal alloy composed by chrome and nickel, specially created to resist at beam interaction stress.
to reach the desired number of detected positron 1.8·1011 with energy major than
1.8 GeV achievable in less than 2 years of data taking [58];
• increasing of the number of muon per proton, obtaining a rate of 11.44 Hz, which is the average rate of muon f ill that consist of sequences of successive 700 µs f ill separated by 11 ms of spill-separation, compared to the BNL rate of 4.4 Hz. To reduce the systematic uncertainty on ωa the following improvements were made:
Figure 4.1: Muon Campus beamline, in red, pions are produced from 8 GeV protons in the AP0 target hall.[30]
• to obtain a clean muon beam the pion decay line is improved respect to BNL exper-iment. In fact a limiting factor in the previous experiment was a 120 m beamline between the pions producing target and the storage ring, since pions with momen-tum of 3.11 GeV/c have a decay length ∼ 173 metres. It means that in BNL a not negligible number of undecayed pions was injected together with the muons. In Fig-ure 4.1 is shown the beamline scheme used in E989. Following the FigFig-ure 4.1 pions are produced in the AP0 target hall and guided along M2 line, in this part produced muons of high energy are directed along M2 and M3 lines and after transferred in a facility called Delivery Ring (DR). The positive particles that enter in this long ring (505 metres) have momentum distributed around the magic momentum 3.094
GeV/c with uncertainty of ±10%. The beam now is composed of muons, protons and 20 % pions that didn’t decay in the M2 and M3 lines. Protons of 3.094 GeV/c momentum have γp ∼ 3.3, while muons have γp ∼ 29.3, so muons speed is 99.9 %
of light speed respect to protons that fly at 95% of c. Thanks to that after a turn inside the DR protons fall 25 metres behind the muons. After 4 turns the distance is 100 metres, with a time delay of 200 ns, given that the beam temporal width is around 120 ns, this time separation is enough to kick out the protons in the abort channel. Additionally, 4 orbits inside the DR correspond to 12 pions decay lengths, that are sufficient to leave to decay virtually all the pions produced. When the muon beam exits from the DR reaching the final lines before the storage ring (M4-M5) the muon beam is contaminated by 30-40 % of positrons. The final step before to enter in the storage ring is the inflector (placed out of the ring) that allows to clean the muon beam from the positrons contamination and to comes in the ring cancelling the fringe field of the ring itself, that otherwise would deflect the beam into the magnet iron, reducing dramatically the injection efficiency;
• detectors and electronics are improved, to reach a precision on ωa of 70 ppb, a great
improvement over the E821 experiment where the total systematic error on ωa was
180 ppb [4]. To make most of this improvement a new kind of electromagnetic calorimeters sensible to Cherenkov light and able to distinguish events that fall in a range of few nanoseconds, to correct for pileup effect, has been constructed. In addition a new tracking system that allows to monitor the muon distribution inside the storage ring, useful to correct the ωa value has been installed in the vacuum
ring. Finally a new laser calibration system has been realized to monitor the SiPM gain fluctuations at the required level (see Table 4.2);
• better shimming to reduce B-field variations to reach the target precision on ωp (70
ppb) that is approximately 2.5 times smaller by placing critical Nuclear Magnetic Resonance (NMR) probes at strategic locations around the ring and shimming the magnetic field to achieve a high uniformity, in addition to other incremental adjust-ments [30];
4.1
Beam Structure
The muon f ill in the ring of E989 experiment occurs with a particular time sequence, where the main clock cycle has a period of 1.4 seconds. At each cycle 2 groups of 8 bunches each are sent to the storage ring with a frequency of 100 Hz. Each bunch is
observed in the ring for approximately 700 µs f ill time. The bunch time structure is shown in Figure 4.2.
Figure 4.2: Bunch structure.[30]
4.2
The Storage Ring Magnet and the Inflector
The muon survived extracted by the beamline are injected in the storage ring built for E821 and used also in the new experiment. The 14 meters of ring’s diameter is composed by 3 superconducting coils. The continuous ”C” magnet yoke is built from twelve 30◦
segment of iron, which were designed to eliminate the end effects present in lumped magnets. This construction eliminates the large gradients that would make a precision determination of the average magnetic field < B > very difficult. Furthermore, a small perturbation in the yoke can effect the field at the ppm level at the opposite side of the ring. Thus every effort is made to minimize holes in the yoke, and other perturbations. The only penetrations to the yoke are to permit the muon beam to enter the magnet as shown in Figure 4.3-a, and to connect cryogenic services and power to the in inflector magnet and the outer radius coil, Figure 4.3-b. The beam enters through a hole in the ”back-leg” of the magnet and then crosses into the inflector magnet, which provides an almost field free region, delivering the beam to the edge of the storage region.
4.3
Kicker
Once that the beam is injected it requires to be kicked otherwise it impacts against the inflector after one turn. The kick required to put magic momentum muons onto a stable orbit centered at the magic radius is on the order of 10 mrad. There are strictly requirements on the muon kicker:
• Since the magnet is continuous, any kicker device has to be inside the precision magnetic field region;
Figure 4.3: (a) Plan view of the beam entering the storage ring. (b) Elevation view of the storage-ring magnet cross section.[30]
• The kicker hardware cannot contain magnetic elements such as ferrite, because they affects the precision uniform magnetic field;
• Any eddy currents produced in the vacuum chamber, or in the kicker electrodes by the kicker pulse must be negligible by 10 to 20 µs after injection, or must be well known and corrected for in the measurement;
• Any kicker hardware has to fit within the place that was occupied by the E821 kicker. The available space consists of three consecutive 1.7 m long spaces;
• The kicker pulse should be shorter than the cyclotron period of 149 ns.
The kicker plates are connected to the High Voltage transformer by the Blumlein gen-erator that is realized as a tri-axial line of concentric conducting tubes. The thyratron CX1724X is chosen as the commutator. It allows up to 70 kV peak forward anode voltage with maximal current up to 15 kA with the rate of rise 300kA/µs, and a repetition rate limited at 2kHz. These requirements are necessary to inject muons along the trajectory crossing the 7.11 m radius design orbit at an angle of 10.8 mrad, 1/4 betatron wavelength downstream from the inflector exit. [55]
4.4
Vertical Focusing: Electrostatic Quadruples
The storage ring acts as a weak-focusing betatron, with the vertical focusing provided by electrostatic quadrupoles. The beam in the ring is injected at magic momentum
