The MUonE experiment: a novel way to measure the hadronic contribution to the muon g-2

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Dipartimento di Fisica E. Fermi Corso di Laurea Magistrale in Fisica

The MUonE experiment: a novel way to measure

the hadronic contribution to the muon g-2

Candidato:

Riccardo Nunzio Pilato

Relatore:

Dott. Graziano Venanzoni

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Introduction

The muon magnetic anomaly is defined as aµ= (gµ−2)/2, where gµ is the muon gy-romagnetic ratio. It is a low energy observable which can be computed and measured with very high precision. The most recent value of aµ was measured with an accuracy of ∼ 0.54 ppm by the experiment E821 at BNL, in 2001. It exhibits a ∼ 3.7σ discrep-ancy from the theoretical prediction, and this makes aµ an important observable to search for physics beyond the Standard Model. For this reason, a new experiment is presently running at Fermilab, with the aim to improve the accuracy on aµ by a factor of four, corresponding to a precision goal of 0.14 ppm. Furthermore, a new experiment to measure the muon anomaly with a different approach is being developed at J-PARC, with a comparable precision goal.

Given this remarkable experimental effort, the theoretical prediction can become the main limitation for a precision test of the Standard Model. The accuracy on the Standard Model calculation is limited by the evaluation of the leading order hadronic contribution aHLO

µ , which cannot be computed perturbatively at low energies. Conse-quently, the hadronic contribution is traditionally determined by means of a dispersion integral on the annihilation cross section e+_e− _→_{hadrons. The hadronic cross section}

is densely populated by resonances and influenced by flavour threshold effects, which limit the final precision achievable by this method on the evaluation of aHLO

µ . Despite these difficulties, the calculation of aHLO

µ has reached a remarkable accuracy of ∼ 0.5%. To claim for possible new physics, it is important to crosscheck this calculation in an independent way.

The MUonE experiment has been recently proposed, with the aim to measure aHLO µ in a completely independent way. It is based on the measurement of the hadronic contribution to the running of the electromagnetic coupling constant (∆αhad) in the space-like region, by means of µ±_e− _{→ µ}±_e− _{elastic scattering. The measurement of}

the shape of the differential cross section provides direct sensitivity to ∆αhad, and it is carried out by scattering a high energy muon beam on a Beryllium target. A beam with the proper energy and intensity is available at CERN. It allows to achieve a

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sta-measurement of aµ competitive with the dispersive approach.

The challenge of the proposed measurement is to achieve a systematic accuracy at the same level of the statistical one. For this purpose, the differential cross section must be measured with a systematic uncertainty . 10 ppm. Systematic uncertainties arise both from experimental and theoretical aspects, such as: bad reconstruction of the elastic events, limited control of the experimental conditions, missing contributions in the computation of the theoretical cross section.

This Thesis focuses on the analysis of two different systematic effects: the multiple scattering and a constant, correlated systematic error on the signal events. This latter effect can be due to, for instance, a bad count of events in the normalization region required to perform the measurement.

The Thesis is structured as follows: Chapter 1 introduces the status of the muon anomalous magnetic moment, discussing both the experimental technique adopted to perform the measurement and the current Standard Model calculation, with a focus on the determination of aHLO

µ . Chapter 2 describes the MUonE experimental pro-posal, which has been recently submitted to the Super Proton Synchrotron Committee (SPSC) at CERN. The original contribution of the Thesis is presented in the following two chapters. A procedure to extract ∆αhad from the elastic scattering cross section at Leading Order is discussed in Chapter 3, together with a method to determine a correlated systematic error. Chapter 4 is instead dedicated to discuss the multiple scattering effects of the experimental apparatus on muons and electrons, and build an analytical parameterization of the scattered angle distribution as a function of electron energy. In this way, it will be possible to determine the level of accuracy needed on the multiple scattering distribution to accomplish the main goal of 10 ppm on the knowl-edge of the differential cross section. Finally, the conclusions summarize the obtained results.

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4.4.2 Non independency of the projected angles . . . 85 4.4.3 Parameterization of the joint distribution for the projected angles 86 4.4.4 Distribution of the unprojected angles for the incoming muons . 87 4.4.5 Distribution of the unprojected angles for the outgoing electrons 88 4.4.6 Effect of a miscalibration on the tails of the angular distribution 96

5 Conclusions 99

A Fit performed on the projected angular distribution for electrons 101

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Chapter 1 The anomalous magnetic moment

of the muon

1.1 Magnetic moments

The magnetic dipole moment ~µ of a particle determines how much torque it experiences when placed in an external magnetic field ~B. The corresponding potential energy U and the torque ~τ are

U = −~µ · ~B ~τ = ~µ × ~B (1.1)

The orbital magnetic moment of a particle with charge e and mass m orbiting in a region with a magnetic field is

µ= i · A = ev

2πrπr2 = evr

2 (1.2)

where i is the current caused by the motion of the charge, v is its velocity and A is the surface area defined by its circular trajectory of radius r. The angular momentum of this particle, with respect to the center of the circumference described by its motion, is L= |~r × ~p| = mvr. Therefore, this quantity can be connected to the orbital magnetic moment by the relation

~

µ= gl e

2m~L, gl = 1 (1.3)

Here the Landé factor gl has been introduced. It has a value equal to one, using classical considerations. The same relation is still valid in quantum mechanics, where the quantization of the angular momentum allows to describe the motion of a particle moving in a central field, for instance atomic electrons in the field of the nucleus, by means of three quantum numbers: n, eigenvalue which determines the energy level,

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l, eigenvalue associated to the operator L2_{, and m ∈ [−l, l], related to Lz. However,}

this description is not sufficient to remove the degeneration of two for each eigenstate, revealed since the first spectroscopy experiments. Therefore, a fourth quantic number is required to complete the picture. A solution to the problem was given by Goud-smith and Uhlenbeck in 1925 [1], who introduced the concept of intrinsic spin angular momentum ~S for the electron, where Sz has eigenvalues ±1

2. As a consequence of this,

the electron has an intrinsic magnetic moment ~µs,

~ µs = g

e

2mS~ (1.4)

with a gyromagnetic ratio g which is analogous to the Landé factor gl for the orbital magnetic moment.

This idea was also able to explain the famous experiment performed three years earlier by Stern and Gerlach [2], in which a beam of silver atoms was physically separated into two different bands, when passing through a region with a magnetic field. The origi-nal purpose of the experiment was to observe the quantization of the orbital angular momentum, but at the time they did not realize to be directly sensitive to the intrinsic magnetic moment of the electron, due to the properties of the silver atom. Quantita-tively, if we assume g = gl = 1 the results of the experiment are underestimated by a factor of 2. It might thus be empirically concluded that g = 2.

The rigorous theoretical description of the spin came in 1928 from Dirac’s theory [3], aimed to create a relativistic extension of Schrödinger’s equation which preserves linear-ity with respect to time. Together with the prediction of the existence of antiparticles, one of the most relevant results of Dirac’s equation is the prediction of the correct value for the gyromagnetic ratio. In fact, in the non relativistic limit, Dirac’s equation for a particle in an external magnetic field Aµ becomes [4]

i∂0ψ =   (~p − e ~A)2 2m − e 2m~σ · ~B+ eA0  ψ (1.5)

Here the second term on the right side represents the interaction with a magnetic field reported in Eq. 1.1. Therefore, comparing the two terms, and recalling that for an electron the spin can be written in terms of Pauli matrices ~σ as ~S = ~σ/2, we get

~ µ= e 2m~σ = e m ~ S = 2 e 2mS~ (1.6)

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Chapter 1. The anomalous magnetic moment of the muon

valid not only for the electron, but for all the point-like particles with spin 1/2. For almost 20 years Dirac’s theory managed to describe satisfactorily nature, and his prediction for the g-factor was confirmed by experiments. However, this was just the beginning of a long story. In 1948 Kush and Foley, while carrying out their mea-surements on the hyperfine structure of atomic spectra, showed the first evidence of a discrepancy from the value predicted by Dirac [5]. They measured a value

ae=

ge−2

2 = 0.00119 ± 0.00005 (1.7)

for the so called electron magnetic anomaly. Clearly, there could be no doubt about this result, given such a small error. Since Dirac’s theory describes the behaviour of point-like particles, in principle a deviation from g = 2 would be a strong indication of an internal structure for the electron, as already observed for the proton and for the neutron. Instead, the anomaly found by Kush and Foley can be explained in terms of electron interactions with virtual particles which arise from vacuum fluctuations, as illustrated by Schwinger in a paper presented in the same year [6]. The g-factor of the electron would have been identically 2, if not for these phenomena. Nevertheless, the magnetic moment cannot be measured without the influence of virtual particles, which surround all the real particles in a cloud acting a screening effect. This makes the anomaly unavoidable. Schwinger focused his attention on the lowest order electron self-interaction term, in which the lepton interacts with the electromagnetic field generated by itself (see Figure 1.1). His calculation, valid for any spin1

2 point-like particle, returns

a value which was in agreement with the experimental outcome: δµ µ = α0 2π '0.00116 (1.8) where α0 = e 2

4π ' 1/137 is the fine structure constant. It was one of the first

con-firmations of the validity of the renormalization procedure, developed by Feynman, Schwinger and Tomonaga to account for the divergencies found in Quantum Electro-dynamics (QED), which originate from higher order diagrams involving loops. Since then an intense, still vivid theoretical activity has begun, aimed to calculate contribu-tions to the magnetic anomaly from all known particles and interaccontribu-tions. Contextually, the experimental techniques designed to measure the anomalies with increasing accu-racy improved remarkably. Nowadays both electron and muon magnetic moments are two of the most precise measurements in physics, and can be used to test the validity of the Standard Model or to search effects of new physics.

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Figure 1.1: Lowest order Feynman diagram for the electron self interaction, calculated by

Schwinger. Figure from [7].

1.2 The electron anomaly in the Standard Model

The electron anomalous magnetic moment provides a stringent test of QED. In fact, according to the Standard Model prediction, it receives only small corrections from strong and weak interactions. The complete calculation can be developed as a sum over powers of the fine structure constant, of which Schwinger’s term represents the first order approximation. The most recent result comes from the computation up to the fifth order in α0 [8, 9, 10]:

ath_e = 0.5 _α 0 π −0.328478965579 _α 0 π 2 + 1.181241456α0 π 3 −1.912245764 _α 0 π 4 + 6.675(192)α0 π 5 + 1.6927(120) · 10−12 _{= 0.001 159 652 182 032(720) (1.9)}

where the last term takes into account the contributions from hadronic and electroweak sectors. On the experimental side, the measurement of ae using a Penning trap [11] has reached the astonishing precision of 0.24 ppb (part-per-billion):

aexp_e = 0.001 159 652 180 73(28) (1.10) Taking advantage of these results, it is possible to extract a measurement of the fine structure constant: forcing the equation ath

e = aexpe we get α

−1

0 = 137.035 999 149(33)

[10]. Here the error is mainly due to the experimental uncertainty in the determination of the electron anomaly, and corresponds to a 0.24 ppb precision on α0. This allows to

test the validity of QED at a very high level, if compared with other evaluations of the fine structure constant which are obtained independently from the measurement of the electron magnetic moment. One of the most recent results in this context is achieved using the recoil frequency of Cs atoms in a matter-wave interferometer, which leads

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to α−1

0 = 137.035 999 046(27) [12]. This value is in agreement within 2.4σ with the

measurement obtained from ae, and it currently gives a confirmation of the efficacy of QED in describing electromagnetic interactions.

1.3 The muon anomaly

The muon is usually considered as a heavier unstable version of the electron, with a mass about 207 times greater and a lifetime of 2.2 µs [13]. However, compared to its lighter brother, muon anomalous magnetic moment receives significant contributions also from weak and strong interactions, as already pointed out by Berestetskii et al in 1956 [14]. In fact, they considered that quantum fluctuations due to heavier particles modify the anomalous magnetic moment of a lepton, according to

δal al ∝ _ml M 2 (1.11) where ml is the lepton mass and M is the mass of a heavier Standard Model parti-cle. Therefore, the sensitivity to deviations from pure QED is enhanced by a factor _m

µ

me

2

'4 · 104 _{for the muon with respect to the electron. For the same reason, if}

we take M as the mass of a hypothetical beyond the Standard Model particle, or as an UV-cutoff where the Standard Model ceases to be valid, we can conclude that the muon is much more sensitive to effects due to new physics.

Following this argument, the τ lepton should represent in principle an even better probe. However, an experiment to measure aτ with high precision is still not achiev-able with current technology, because of its short lifetime of ' 290 fs and its many decay modes. Instead, muon’s longer lifetime permits an accurate measurement of all its characteristics, including the magnetic moment, and its single dominant decay mode

µ−→ νµ ¯νe e− µ+ → νe ¯νµ e+ (1.12) allows to exploit a relatively simple strategy aimed to measure the magnetic anomaly.

1.3.1 Experimental technique for the measurement of a

µ

Muon g − 2 experiments are based on the measurement of the muon spin precession in presence of a uniform magnetic field. When polarized muons are injected in a storage ring, the spin precesses around the direction of the magnetic field, as a consequence of the interaction between muons intrinsic magnetic moment and the magnetic field B.

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This rotation is referred to as Larmor precession, and has a frequency [15] ~ ωs = g e ~B 2m + (1 − γ) e ~B mγ (1.13)

Furthermore, muons will perform a circular motion with a cyclotron frequency

~

ωc= e ~B

mγ (1.14)

These relations clearly reveal that if g were exactly 2 the two frequencies would be equal, demonstrating the spin direction being locked at any time to the momentum direction, as shown in Figure 1.2a. Instead, since g is larger than 2 at the per mill level, the spin vector precession develops (1 + aµ) times faster than the rotation of the momentum vector, meaning that it rotates slightly more than 2π in a single cyclotron period (see Figure 1.2b). The difference between the frequencies of the two motions is known as the anomalous precession frequency:

~ ωa= ~ωs− ~ωc = e ~B m g 2 −1 = aµe ~B m (1.15)

This quantity shows two important features: first of all, it only depends on the muon anomaly, rather than on the full magnetic moment; moreover, it has a linear dependence on the applied magnetic field.

In order to focus the muon beam in the vertical direction, g − 2 experiments use a quadrupolar electric field, which modifies the value of ~ωa:

~ ωa= e m " aµB −~ aµ− 1 γ2₋1 ! ~ β × ~E # (1.16)

The contribution of the electric field can be cancelled tuning the muon beam momentum at the so called "magic value" pmagic = m/√aµ ' 3.09 GeV/c, which corresponds to γ ' 29.3. Because of beam spread, a small contribution due to the electric field remains, and will require a second order correction in data analysis stage [15].

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Figure 1.2: Muon spin and momentum vectors rotating in a magnetic field, in case of

g = 2 (a), or g > 2 (b).

A fundamental requirement to perform the measurement is to find a mechanism which produces a polarized muon beam. The most common way to produce muons is as a result of the charged pion decay. Due to parity violation, it seems to be the ideal process for this. Let us start by considering the decay π− _{→ µ}− _¯νµ _{in the rest frame}

of the pion: the antineutrino is always a right-handed eigenstate of helicity1, thus its

spin points in the same direction of its momentum. Since the pion is a scalar meson, the conservation of angular momentum imposes the muon to be also right-handed, as depicted in Figure 1.3. This condition endures in the boosted laboratory frame, for those muons emitted in the forward direction. So a spin polarized muon beam is achieved taking advantage of the decay itself.

At this stage a way to measure muons spin precession is essential, and parity violation proves to be again very helpful. Muons decay with ∼ 100% probability following the reaction µ− _{→ e}− _ν

µ ¯νe. As shown in Figure 1.4, in the rest frame of the muon the highest energy electrons come from the scenario in which the two neutrinos are emitted

1_{In the approximation of massless neutrino.}

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in the same direction, opposite to electron’s one. In this case, half of the total energy is carried out by the electron (Ee,max '53 MeV), while the remaining is shared between the two neutrinos. Since these two particles have to be in helicity eigenstates, the antineutrino is right-handed, while the neutrino is left-handed. Following the angular momentum conservation, electrons emitted in this configuration are forced to carry the spin of the parent muon. Furthermore, the V − A nature of the weak decay prefers to couple to left-handed electrons, so in the discussed situation the high energy electrons are emitted in a direction opposite to their spin. The scheme is, of course, analogous with regard to positive muons: µ+ _{obtained from π}+ _{decay are left-handed, and the}

emitted maximum energy positrons which carry muons spin have their momentum pointing in the same direction.

Figure 1.4: Pictorial representation of muon decay.

Moving to the laboratory frame, this means that it is possible to determine muons level of polarization and monitor their spin direction by measuring the instantaneous number of high energy electrons emitted at a fixed direction. The anomaly shows up as a modulation of the number of detected electrons as a function of time, which overlaps the usual exponential decay law. This is evident in the so called "wiggle plot", represented in Figure 1.5. To extract the anomalous precession frequency, the number of decay electrons N(t) is modeled with the following function:

N(t) = N0e−t/τ [1 + A cos(ωat+ φ)] (1.17)

where N0 is the total population at the time t = 0, τ is the muon lifetime, A represents

the asymmetry in the direction of the decaying particle and the phase φ depends on the initial polarization of the muon beam. The latter two parameters depend on the energy threshold adopted to select the high energy component of the decay electrons

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Figure 1.5: Effect of the muon anomaly on the number of decay electrons as a function of

time. Figure from [16].

spectrum.

A precise knowledge of the magnetic field, averaged on its spatial distribution and on the time of data taking, is required to perform the measurement of aµ, as recalled in Eq. 1.15. The field is determined using the technique of the nuclear magnetic resonance, employing protons at rest. Consequently, the second observable needed to measure the anomaly becomes the proton precession frequency ωp. It is thus possible to determine aµ taking advantage of the relationship [15]

aµ=

ωa/ωp λ+− ωa/ωp

(1.18)

where λ+ = µµ+/µ_p = 3.183345107(84) is the muon-to-proton magnetic moment ratio,

measured from muonium (µ+_e−_{) hyperfine structure [13].}

This strategy is the result of several decades of experimental efforts and improve-ments, starting with the three experiments at CERN (1965, 1968, 1979) up to the experiment at BNL (2001), aimed to perform the measurement of aµ with increasing accuracy over the years.

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To date, the reference value is the one obtained by the BNL E821 experiment [17]:

aBN L_µ = 11 659 208.0(5.4)(3.3) × 10−10 (1.19) where statistical and systematic uncertainties are given respectively, to get a combined uncertainty of 0.54 ppm (part-per-million). This result shows a discrepancy of ∼ 3.7σ from the current Standard Model value [18], whose calculation will be discussed more in detail in the following section.

This remarkable difference between experimental and theoretical values makes the measurement of aµ of central importance in the present physics landscape. For this reason, two new g − 2 experiments have been designed to improve the experimental accuracy. The E989 experiment at Fermilab [15] aims to get a 0.14 ppm precision on the measurement of the muon anomaly, improving by a factor of four the BNL achievement. The estimated error includes a 100 ppb statistical component in the measurement of ωa, which is expected to be achieved taking 21 times the total BNL statistics, and two equal systematic uncertainties of 70 ppb coming from ωa and ωp [19]. The E989 experiment has recently completed the second run of data taking, while a first result, obtained from an amount of statistics comparable to the BNL total, is expected to be released soon. In addition, a completely new low energy approach is being developed by the E34 collaboration at J-PARC, which aims to reach a precision comparable to the Fermilab experiment with completely different systematics [20]. The present situation on the muon anomaly measurement is illustrated in Figure 1.6, with the 3.7σ difference between Standard Model prediction and measured value. If the new g − 2 experiment will confirm the BNL value with a four-fold improved precision, then it would correspond to a 7σ discrepancy. This would represent a compelling signal of physics beyond Standard Model.

1.3.2 Present Standard Model prediction of a

µ

As anticipated in the previous section, the muon anomaly receives significant contri-butions from electroweak and strong interactions, in addition to the pure QED part. Therefore we can write

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Figure 1.6: Comparison between theoretical and experimental values of aµ. The Standard Model predictions are quoted in chronological order from the top to bottom. The most recent result is KNT18 [18], which defines the uncertainty band other evaluations are compared to. The BNL measurement defines the blue error band, while the light grey band corresponds to a situation in which the Fermilab experiment would get the same mean value as BNL for aµ, but with the expected four-fold improved precision. Figure from [18].

to obtain the total value of the anomaly. The most recent calculation for the Standard Model value of aµ is [18]:

aSM_µ = (11 659 182.04 ± 3.56) × 10−10 . (1.21) QED and electroweak contributions can be computed with a very high precision us-ing perturbative calculations, whilst the major source of uncertainty comes from the hadronic contribution. It mainly originates from two terms: the leading order hadronic vacuum polarization, which relies on an experimental data based evaluation, and the hadronic light-by-light contribution, which represents the second largest error in the theoretical calculation.

In the following we present the current state of art of for each term which contributes to the anomaly, focusing particularly on the hadronic contribution.

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Figure 1.7: Second order QED diagrams which contribute to the muon anomaly. Figure

from [7].

QED contribution

The QED contribution to aµ can be expressed in the perturbation series of the fine structure constant, analogously to what was done for the electron anomaly. The two contributions are different because of the different mass of the two leptons, which leads to separate computations starting from the two loop diagrams. From the most recent evaluation of aQED µ we get [13] aQED_µ = 0.5 _α 0 π + 0.765857425(17)α0 π 2 + 24.05050996(32)α0 π 3 + 130.8796(63)α0 π 4 + 753.3(1.0)α0 π 5 aQED_µ = 11658471.895(0.008) × 10−10 (1.22) Here the error results mainly from the uncertainty in the evaluation of α0, which in

this computation is taken from the measurement of Rb atoms oscillations in an optical lattice [21].

Electroweak contribution

According to Eq. 1.11, the electroweak contribution is suppressed by a factor ∼_m

µ

mW

2 with respect to the pure QED term, and represents the smallest contribution to aµ.

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In fact, the sensitivity on this term was reached only by the g − 2 experiment at BNL [7]. One loop term was calculated in 1972 [22], and includes contributions from Z, W and Higgs bosons (see Figures 1.8a, b, c). Two loop corrections are not negligible, and they are negative. They include also a hadronic part, which comes from quark loops (see an example in Figure 1.8d) and represents the main source of uncertainty on the total electroweak contribution, which is [13]

aEW_µ = (15.36 ± 0.10) × 10−10 (1.23)

Figure 1.8: Leading electroweak contributions to aµ in (a), (b), (c). Example of two loops diagram involving quarks in (d). Figure from [7].

1.4 Hadronic contribution to the muon g − 2

The evaluation of the hadronic terms cannot be performed applying the perturbation theory, differently from what is done for QED and EW contributions. The reason for that lies in the calculation involving strong interactions at low energies, for which the perturbative approach cannot be employed. The hadronic contribution ahad

µ can be divided in three parts, regarding the leading and next to leading order vacuum polarization effects, respectively aHLO

µ and aHN LOµ , and the light-by-light contribution aHLbL

µ :

ahad_µ = aHLO_µ + aHN LO_µ + aHLbL_µ (1.24) The largest hadronic effect comes from a vacuum polarization insertion in the virtual photon line of the Schwinger term diagram, as shown in Figure 1.9. It represents also the main source of uncertainty on the Standard Model prediction on aµ. The addition of another vacuum polarization loop, being both leptonic or hadronic, contributes to the next to leading order effect. The difficulties associated to the calculation of this latter term are similar to aHLO

µ . Nevertheless, it does not affect significantly the overall error, due to the suppression by one power in α0 with respect to the leading order [7].

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Figure 1.9: Hadronic leading order contribution to the muon anomaly.

Figure 1.10: Schematic representation of the optical theorem.

Given the difficulties associated to the QCD direct calculation, aHLO

µ is traditionally obtained via a dispersion integral, whose derivation is based on the general principles of causality and unitarity, which imply respectively analyticity of quantum field theory and validity of the optical theorem. The obtained dispersion integral is [7]:

aHLO_µ = α0 π ∞ Z 0 ds πImΠhad(s) K(s) s (1.25)

In this relation Πhad(s) is the hadronic part of the photon vacuum polarization, as a function of the time-like squared momentum transfer s = q2 _> _{0, while K(s) is the}

kernel function K(s) = 1 Z 0 dx x 2(1 − x) x2+ (1 − x)s/m2 µ (1.26) which behaves approximately as 1/s. Taking advantage of the optical theorem, dia-grammatically represented in Figure 1.10, the imaginary part of the hadronic vacuum polarization function is related to the annihilation cross section e+_e− _→ _{hadrons, or}

rather to the hadronic ratio R(s):

ImΠhad(s) = σ(e+e−_4πα/s→hadrons) = α₃R(s), R(s) = σ(e

+_e− _→_hadrons)

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Figure 1.11: The hadronic R(s) ratio as a function of√s.

The leading hadronic contribution to the muon anomaly is hence given by

aHLO_µ = α 2 0 3π2 ∞ Z 4m2 π dsK(s) s R(s) (1.28)

This method is known as the time-like dispersive approach, and relies on the knowledge of the experimental data for the determination of R(s). In fact this quantity cannot be calculated from first principles in the low energy limit, because of the lack of appropriate methods for QCD non perturbative calculations2. Consequently, the common strategy

is to use experimental data only up to a certain energy cut, above which, due to the asymptotic freedom of the QCD, it is safe to perform the perturbative calculation. For instance, in his 2017 evaluation Fred Jegerlehner applies perturbative QCD after the J/ψ resonances region, from 5.2 GeV to 9.46 GeV, and above the Υ series, namely for √

s >11.5 GeV [23].

2_{Significant progress has been done in the recent years on lattice calculation of a}HLO µ .

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Figure 1.12: Contributions to the total hadronic ratio from the different final states. In

yellow the π+π− channel, which dominates the region below 1 GeV. Figure from [18].

In the low energy region R(s) is highly fluctuating, as dramatically illustrated in Figure 1.11, because of hadronic resonances and flavour threshold effects. Moreover, this is the region which mainly contributes to the final value of aHLO

µ , and this is the reason why the hadronic term is affected to such a large uncertainty, if compared to the others. Indeed, in Eq. 1.28 the hadronic ratio is weighted by the factor K(s)/s ∼ 1/s2. In

particular, as can be seen in Figures 1.12 and 1.13, about 75% of the total value comes from the region 2mπ <√s <1 GeV, dominated by the π+π− channel. It is the largest contribution to the error on aHLO

µ , because of the presence of ρ and ω resonances. Merging measurements exploited by different experiments in different energy in-tervals is necessary to get a complete estimation of aHLO

µ . This represents one of the central issues related to the time-like method. Therefore, the proper handling of the uncertainties associated to each cross section measurement is fundamental. Statistical errors are usually assumed to be Gaussian, hence they are added in quadrature. On the other hand, the treatment of systematic errors is much less easy, since this kind of uncertainties strongly depends on the single machine and detector adopted. Exper-imentally, the measurement of the hadronic ratio performed at e+_e− _{colliders is the}

result of a scan in the center of mass energy. As a consequence, even experimental points coming from the same detector might be laborious to manage, since detector conditions can evolve between different data taking periods. Therefore, the proper evaluation of the correlations between these measurements, which are collected at dif-ferent√svalues, is still a problem. A new technique to resolve this issue was pioneered

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Figure 1.13: Pie chart showing the fractional contribution to the total aHLO_µ value, and to its error squared. Note the green contribution in the zone of ρ and ω resonances (0.6 GeV < √

s < 0.9 GeV). Figure from [18].

by the KLOE experiment [24]. It consists of considering only events with a hard pho-ton emitted from the initial state, so as to cover a wide range of √s values without changing the energy of the beams. In this way, the so called ISR method allowed to take data always in the same detector conditions, reducing the systematics related to the behaviour of the machine [25].

Amongst the several aspects to take into account in combining data, some of the most relevant are how to interpolate and model the behaviour of R(s) between two different data sets, how to perform the integration3_{, or how to determine in which}

energy intervals use the theory. All these considerations lead to define an analysis procedure which is not univocal, but rather can change depending on the different methods used. To give an example, Keshavarzi et al (KNT group) in their new anal-ysis [18] made a comparison with other works, exploited on similar data inputs. The study made by Davier et al in 2017 (DHMZ group) gives a total estimation of aHLO

µ which is in good agreement with the value obtained by KNT [26], as can be seen in Figure 1.14. Nevertheless, there are some larger differences in the contributions of the individual channels to the total value: this is the case of the π+_π− _channel,

which shows a deviation & 1σ between the two works. Indeed, the values achieved by the two groups are aHLO

µ (π+π

−_{) = (503.74 ± 1.96) × 10}−10 _{for the KNT group, and}

aHLO µ (π+π

−_{) = (507.14 ± 2.58) × 10}−10 _{for DHMZ. As stated by KNT, this difference}

«is unexpected when considering the data input for both analyses are likely to be simi-lar and, therefore, points to marked differences in the way the data are combined» [18]. As a result of this situation, a certain freedom of choice is allowed in the estimation of

3_{Two possible procedures could be fitting data with a smooth function prior to integration, or}

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Figure 1.14: Comparison of recent evaluations of aHLO

µ applying the time-like method [18].

systematic uncertainties. This leads to a final error evaluation which can be more or less conservative, depending on the different analyses.

The three most recent values of aHLO µ are

aHLO_µ = (688.1 ± 4.1) × 10−10 [FJ17], [27] (1.29) aHLO_µ = (693.1 ± 3.4) × 10−10 [DHMZ17], [26] (1.30) aHLO_µ = (693.3 ± 2.5) × 10−10 [KNT18], [18] (1.31) The latter value is the one which leads to the theoretical prediction of aµ quoted in Eq. 1.21. If compared to the uncertainties related to the QED term (0.008 · 10−10₎

and the EW term (0.1 · 10−10_{), it is clear that the leading hadronic contribution is}

by far the main source of uncertainty on the Standard Model prediction. Therefore, given the present experimental effort, an improvement in the determination of aHLO

µ is highly desirable, in order to allow a robust comparison between the theoretical and measured value of the muon anomaly. Particularly, a refinement on the current pre-cision of ∼ 0.4 ÷ 0.5% achievable with the time-like approach is considered unlikely, because of the difficulties connected to handling the various measurements of R(s).

In the last years, alternative evaluations of aHLO

µ have been obtained using lattice QCD [28], which allows a numerical computation of the hadronic contribution starting

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from first principles. However, in spite of their continuous progress, these lattice com-putations have reached an accuracy of ∼ 2÷3%, thus they are not yet competitive with the dispersive ones obtained via time-like data. The innovative approach proposed by the MUonE experiment allows to get a new evaluation of aHLO

µ , competitive with the time-like method. In this way, it will be possible to perform a crosscheck between the two procedures, which have completely different systematics.

Hadronic light-by-light contribution

Figure 1.15: Feynman diagram of the hadronic light-by-light contribution.

The hadronic light-by-light process is the second largest source of uncertainty on aµ, even though it contributes to the anomaly at the same level of the electroweak term. Otherwise from the vacuum polarization contribution, this higher order term cannot be related to experimental data, so it has to be calculated from hadronic models which correctly reproduce properties of QCD in its non perturbative region. A synthesis of different models, known as "Glasgow consensus", gives the following value for the light-by-light contribution [29]:

aHLbL_µ = (10.5 ± 2.6) × 10−10 (1.32) Therefore, the absolute error on the evaluation of this term is similar to the uncertainty on aHLO

µ , but its contribution on aµ is more than one order of magnitude lower if compared to the vacuum polarization term.

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Figure 1.16: Effect of the vacuum polarization on the photon propagator. The result of all

the loop insertions can be absorbed in a redefinition of the coupling constant. The second diagram represents the first order correction to the photon propagator.

1.5 Vacuum polarization in QED and running

cou-pling constant

As we have seen in the previous sections, vacuum polarization phenomena consist of the emission and consequent reabsorption of a pair particle-antiparticle by a virtual photon. The first experimental evidence of these effects dates back to 1947, with the precise measurement of the energy difference between the 2_S

1/2 and 2P1/2 levels of the

hydrogen atom, the so called Lamb shift [30]. This difference should vanish according to Dirac’s theory, but the measurement of a non zero value reveals a further influence of virtual effects on real particles, which adds to the observations of Kush and Foley. The non degeneration of the two energy levels is mainly due to electron self energy, but it also includes a contribution from vacuum polarization effects, which is decisive to explain the experimental result.

The simplest photon vacuum polarization diagram is drawn in Figure 1.16, and represents the lowest order photon self-energy term. The integration over the loop momentum leads to divergent results. They can be handled by means of the renor-malization procedure in a redefinition of the electromagnetic coupling constant4_{, which}

acquires a non trivial dependence on the squared momentum transfer q2. The result of

the adopted technique is schematically depicted in Figure 1.16. Here the infinite sum over all the self-energy terms of the photon propagator at fixed coupling, indicated by the blob, is replaced by a simple propagator with an effective coupling constant

α(q2) = α0

1 − ∆α(q2) (1.33)

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where ∆α(q2_{) represents the contribution of all the possible virtual pairs to the running}

of α. The corresponding physical effect is that the strength of the electromagnetic force varies depending on the momentum transfer in a single interaction.

This is the quantum analogue to what happens in a dielectric, in which the presence of an external magnetic field polarizes the molecules of the material. As a consequence, the interaction between two charges in a dielectric is reduced. Similarly, at quantum level the real particles are surrounded by the virtual vacuum polarization pairs, which cause a screening effect and reduce the strength of the electromagnetic force. On the contrary, the strong interaction coupling constant αS decreases with rising of q2_{. This}

behaviour is due to the gluon-gluon virtual pairs, which can occur since the gluon itself has a colour charge. The gluon loops produce an anti-screening effect, thus the net result is a decrease of αS when the momentum transfer increases. This allows to use perturbation theory for QCD calculations at high energies.

Figure 1.17: Charge screening due to polarization effects.

The situation of the QED is represented in Figure 1.17. In the quantum case, a smaller distance between two interacting particles means a larger momentum transfer, hence the intensity of the electromagnetic interaction grows as q2 _{increases. This}

corresponds to a scenario in which the real particles have been able to go through the virtual cloud, at least partially. It is confirmed also by the fact that ∆α(q2_{) > 0 at any}

momentum transfer. For this reason, we shall refer to the classical value of the electron charge as a dressed value, and to α0 as the electromagnetic coupling constant only in

the Thomson low energy limit. Since the electron is the lightest charged particle, the e+e−pairs are those which mainly contribute to vacuum polarization effects, hence the Thomson limit corresponds to q2 _m2

e.

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of the leptonic and hadronic contributions to the vacuum polarization:

∆α(q2_{) = ∆αlep(q}2_{) + ∆αhad(q}2₎ _(1.34)

The leptonic contribution can be calculated in perturbation theory. For high values of momentum transfer it has a logarithmic behaviour [7]:

∆αlep(q2_{) =} α0 3π X l=e,µ,τ " log |q2| m2 l ! −5 3 # , |q2| m2_l (1.35)

∆αlep(q2_{) is known so far with very high precision. The next-to-leading order}

contribu-tion has been calculated including the full mass dependence, while the next-to-next-to leading order (three loops) is analytically available as an expansion in terms of m2

l/q2 [31]. It is also known up to four loops in specific q2 _{limits. For instance, it is ∼ 2 · 10}−8

at the Z boson mass scale [32].

On the contrary, the hadronic contribution is affected by the same problems of aHLO µ , related to the non perturbative nature of QCD at low energies. The contribution of the top quark is usually separated from the others. This is because vacuum polar-ization effects interesting for particle physics are far below the t¯t threshold, thus the contribution of the heaviest quark is small. Besides, given its heavy mass, the top quark contribution can be calculated by means of perturbative QCD. Its analytical expression is hence the same for the leptonic contribution, up to multiplicative factors taking into account the top quark charge and the corresponding SU(3) colour factors. Instead, vacuum polarization effects due to the remaining 5 quarks, ∆α(5)

had(q2_{), can be}

determined taking advantage of a dispersion relation, similarly to what was employed for aHLO µ [33]: ∆α(5) had(q 2_{) = −}α0 q2 3π P ∞ Z m2 π ds0 R(s 0₎ s0(s0 _{− q}2) (1.36)

From this relation, it follows that in this case the hadronic ratio R(s) is weighted by a function which behaves like 1/s, instead of 1/s2 _{of Eq. 1.28. This means that the high}

energy region, computed using perturbative QCD, has a larger impact on the deter-mination of the hadronic contribution to the running, if compared to aHLO

µ . However, the error on ∆αhadis still dominated by the statistical and systematic uncertainties of the experimental data.

As well as for the evaluation of aHLO

µ , also for ∆αhad several parameterizations are currently available, which differ from each other for the way in which the results from

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Figure 1.18: Running of the electromagnetic coupling constant as a function of momentum

transfer in the time-like (q2 > 0) and the space-like (q2 < 0) regions (red line). The blue line

represents the leptonic contribution.

various experiments are combined. In this Thesis the compilation implemented by Fedor Ignatov will be used [34, 35]. In Figure 1.18 the behaviour of ∆α(q2_{) is}

rep-resented, both for time-like and space-like transferred momenta, obtained from this parameterization. The influence of the hadronic resonances on the vacuum polariza-tion effects is as well evident, whereas in the space-like region the running of α is a smooth function. This behaviour will be exploited by the MUonE experiment to get an alternative evaluation of aHLO

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Chapter 2 The MUonE experiment

2.1 A novel approach to measure a

HLO_µ

The MUonE experiment aims to determine aHLO

µ using a new method, based on the direct measurement of ∆αhad in the space-like region. In the time-like approach, aHLO

µ can be written as aHLO_µ = α0 π ∞ Z 0 ds π ImΠhad(s) K(s) s (2.1)

where Πhad(s) is the hadronic part of the photon vacuum polarization and K(s) given by Eq. 1.26. If the integration order between x and s is exchanged, we get

aHLO_µ = α0 π2 1 Z 0 dx m2_µx2 ∞ Z 0 ds s ImΠhad(s) s+x2m2µ 1−x (2.2)

The integral over the s variable can be simplified, taking advantage of the dispersion relation [7]

¯Πhad(t) ≡ Πhad(t) − Πhad(0) = t π ∞ Z 0 ds s ImΠhad(s) s − t (2.3) to obtain aHLO_µ = α0 π 1 Z 0 dx(x − 1)¯Πhad[t(x)] (2.4)

where the space-like squared four momentum transfer −∞ < t < 0 is defined as

t(x) = x

2_m2

µ

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Figure 2.1: Integrand (1 − x)∆αhad[t(x)] × 105 as a function of x and t (upper scale) [37]. The running of the electromagnetic coupling constant is linked to the photon vacuum polarization function by the relation ∆α(q2_{) = −Re¯Π(q}2_{). Since Im¯Π(q}2_{) = 0 for}

q2 = t < 0, we finally get [36] aHLO_µ = α0 π 1 Z 0 dx(1 − x)∆αhad[t(x)] (2.6)

This is the master formula of the MUonE experiment, and allows to calculate the hadronic contribution to the muon anomaly starting from the measurement of ∆αhad(t). The peak of the integrand occurs at xpeak '0.914, which corresponds to a momentum transfer value tpeak' −0.108 GeV2_{. Here ∆αhad(tpeak) ' 7.86 · 10}−4_.

The main feature of this integral is that it is a smooth function free of resonances, as can be seen in Figure 2.1, in contrast with the dispersive relation in Eq. 1.28 which uses time-like data. Accordingly, the computation of the integral is remarkably simplified. A further advantage is that the running of α in the region of interest for the evaluation of aHLO

µ can be measured by a single scattering experiment. For this reason, the space-like approach is not affected by the systematic uncertainties due to handling data from different experiments, which instead turn out to be the main limitation of the time-like method. Therefore, it allows a completely independent estimation of aHLO

µ , which can be compared with time-like approach and lattice QCD results towards a firmer prediction of aµ.

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Chapter 2. The MUonE experiment

Figure 2.2: Schematic view of the MUonE experimental apparatus (not to scale) [39].

2.2 The experimental approach

The MUonE experiment aims to extract ∆αhad from the measurement of the differ-ential cross section of the µ± _e− _{→ µ}± _e− _{elastic scattering [37]. It is performed by}

scattering a muon beam of Eµ= 150 GeV, available at the CERN M2 beamline [38], on the atomic electrons of a fixed Beryllium target. This process has the proper charac-teristics to determine aHLO

µ . In fact, it has the great advantage to be a pure t-channel process, allowing for an unambiguous identification of the momentum transfer of the reaction and making the differential cross section at Leading Order (LO) proportional to |α(t)/α0|2: dσ dt = dσ0 dt α(t) α0 2 (2.7) where dσ0/dt is the effective Born (LO) cross section1. Moreover, the simple

kine-matics of a two-body elastic process makes the scattering angles of electron and muon correlated, as will be shown in Section 2.3. This constraint is important to select signal events using only the angles of the outgoing particles. It is also possible in this way to minimize systematic effects in the determination of t, as well as to reject background events coming from radiative or inelastic processes.

Furthermore, due to the high energy muon beam employed, the kinematics in the lab-oratory frame is highly boosted in the forward direction. This makes possible to use a single detector to cover the whole acceptance, for the reason that the elastic events which are interesting for the experiment correspond to scattering angles contained within ∼ 50 mrad for the electron and ∼ 5 mrad for the muon.

The experimental apparatus consists of a sequence of 40 identical stations, as depicted in Figure 2.2. Each station is made up of a 15 mm thick Beryllium target, followed by a tracking system with a lever arm of ∼ 1 m, used to measure the scattering angles with high precision. The apparatus is also equipped with an electromagnetic calorimeter, placed downstream all the stations. Its main role is to provide particle identification (PID) in the region around ∼ 2 ÷ 3 mrad, in order to solve the ambiguity

1_{This is not true at higher orders, since the momentum transfer on the electron leg can be different}

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between muons and electrons. The contamination of pions in the M2 muon beam at CERN is . 10−6 _{[40], and will be controlled by a muon filter, instrumented with muon}

chambers and placed downstream the calorimeter.

2.2.1 Workflow and requested precision

The strategy used to extract aHLO

µ can be summarized by the following steps:

• Measurement of the scattering angles (θe, θµ), respectively associated with the outgoing electron and muon. They are measured with respect to the incoming muon direction.

• Count of the number of signal events as a function of the momentum transfer t. • Extraction of the hadronic contribution to the α running. Starting from Eq. 2.7

we get dσdata/dt dσno VP M C /dt = dNdata/dt L · dσno VP M C /dt = 1 |1 − ∆αlep(t) − ∆αhad(t)|2 (2.8)

where dNdata/dt and dσdata/dtare the number of events and the differential cross section respectively, while dσno VP

M C /dt is the differential cross section determined by Monte Carlo, imposing α(t) = α0. L is the luminosity of the machine, which

can be obtained using the same elastic process, as will be shown in the following. If at leading order this ratio can be expressed analytically, at higher orders it becomes, instead, a complex expression and is evaluated numerically by Monte Carlo methods. For this reason, the extraction of ∆αhad is carried out by a template fit method [39]. It is a powerful fitting tool which can be adopted in situations where an analytical expression of the cross section is not available. • Computation of the integral in Eq. 2.6 to get aHLO

µ . In Chapter 3 the procedure to determine aHLO

µ will be described in detail, considering the leading order cross section.

The aim of the MUonE collaboration is to measure aHLO

µ with a statistical un-certainty of ∼ 0.3% and a comparable systematics [37]. The space-like measure-ment will thus be competitive with the latest time-like results. The achievemeasure-ment on the statistical precision can be reached in two years of data taking with a run-ning time of ∼ 2 × 107 _{s/yr, which allows to collect an integrated luminosity of about}

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On the other hand, the big challenge is represented by the ultimate goal on the sys-tematic accuracy. This is equivalent to count the number of events with a syssys-tematic uncertainty of ∼ 10 ppm at the peak of the integrand function [37]. Such an accu-racy can be achieved only with a twofold effort, both on theoretical and experimental sides. From the experimental point of view, there are several crucial requirements to be satisfied. One of the most important issues is represented by the multiple scattering effects on the detected particles. Given the purely angular nature of the measurement, multiple scattering effects need to be described to high accuracy, since they break the muon-electron angular correlation. The experimental apparatus has been designed to minimize this kind of phenomena: that is the reason why the total target thickness of 60 cm was divided into 40 thin slices, each instrumented with its own tracking system. In so doing, multiple scattering effects will be kept under control without any loss on the collected statistics. The impact of the multiple scattering effects will be discussed in Chapter 4. Further relevant aspects are the control on the longitudinal alignment of the tracking system, which has to be performed with a 10 µm precision, and the knowledge of the average beam energy, which needs to be determined with extreme precision, of the order of few MeV [39].

On the theoretical side, the development of high precision Monte Carlo tools is required, including all the relevant radiative corrections to the differential cross section. They must be calculated taking into account all the contribution up to the next-to-next-to leading order (NNLO) [39]. The full set of NLO QED and electroweak corrections is presently completed, together with the corresponding Monte Carlo generator [41]. Furthermore, first results have been recently obtained for the NNLO QED and NNLO hadronic corrections [42, 43, 44, 45].

2.3 The µ

±

e

−

→ µ

±

e

−

scattering process

A deep knowledge of the elastic process and its kinematic properties is required, in order to have a full understanding of the experiment. At Born level, the µ±_e− _elastic

scattering is schematically represented by the interaction

µ±(p1) e−(p2) → µ±(p3) e−(p4) (2.9)

p1 and p3 are the four momenta of the initial and final muon respectively, while p2

and p4 are the ones of the initial and final electron. The scattering initiated by both

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(a) (b)

Figure 2.3: Feynman diagram for the LO µ±e− scattering (a). First order vacuum polar-ization correction (b).

Figure 2.4: µ±e− elastic scattering in the laboratory reference frame.

North Area [38]. The Feynman diagram which describes the tree level interaction is shown in Figure 2.3a, while the higher order diagram with the vacuum polarization inserted is represented in Figure 2.3b. As seen in Section 1.5, its effect is accounted to the running of α, which leads to the relation in Eq. 2.7. Figure 2.4 gives a schematical representation of the process µ± _e− _{→ µ}± _e− _{in the laboratory reference frame. Here}

the four momenta are given by:

p1 = (Eµ, pµ, 0, 0) p2 = (me, 0, 0, 0) p3 = (Eµ0, p 0 µcos θµ, p 0 µsin θµ, 0) (2.10)

p3 = (Ee, p0ecos θe, p

0

esin θe, 0) where pµ =q

E2

µ− m2µ is the incoming muon momentum, while (Eµ0, p0µ) and (Ee, p0e) are energies and momenta of the outgoing muon and electron respectively.

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Figure 2.5: Differential cross section of the µ±e− elastic scattering at LO, as a function of the momentum transfer.

The differential unpolarized cross section at LO in QED, whose behaviour as a function of the momentum transfer is reported in Figure 2.5, is [41]

dσ0 dt = 4πα2 0 t2_λ(s, m2 µ, m2e) " (s − m2 µ− m 2 e) 2_{+ st +} t2 2 # (2.11)

where s and t are the usual Mandelstam variables and λ is the Källén function:

s= (p1+ p2)2 = (p3 + p4)2 = m2µ+ m 2 e+ 2meEµ'(405.5 MeV)2 t= (p1− p3)2 = (p2− p4)2 = 2me(me− Ee) (2.12) λ(s, m2_µ, m2e) = (s − m2_µ− m2_e)2₋_4m2 µm 2 e = 4m 2 e(E2 µ− m 2 µ)

The momentum transfer t ranges in the interval [tmin,0], with tmin = −λ(s, m2

µ, m2e)/s ' −0.142 89 GeV2_{. It corresponds to xmax} _'_{0.93212. This means that a muon beam of}

Eµ = 150 GeV allows to cover the 87% of the integral which leads to aHLOµ , as graph-ically shown in Figure 2.6. The full value can be obtained by extrapolating ∆αhad in the region kinematically not accessible to the experiment (x ∈ [0.93212, 1]), using an appropriate parameterization for the hadronic contribution to the vacuum polarization. This aspect will be discussed in more detail in Chapter 3.

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Figure 2.6: Coverage of aHLO_µ for E_µ = 150 GeV (left). Fraction of aHLO_µ covered as a function of muon beam energy (right).

transverse momentum:

sin θµ= p0esin θe p0 µ = sin θe v u u t E2 e(θe) − m2e [Eµ+ me− Ee(θe)]2 _{− m}2 µ (2.13)

where the expression which relates θe to the electron energy is shown in Eq. 2.14. The so called "elasticity curve" is drawn in Figure 2.7. It represents the main experimental signature for the elastic process, together with the requirement of the planarity of the scattering events. Figure 2.7 shows that θµ is kinematically limited within ∼ 5 mrad. Moreover, the aforementioned ambiguity between the two scattering angles can be observed in the region of ∼ 2 ÷ 3 mrad, where also the particles momenta are similar. Such an ambiguity will be solved by means of PID, using calorimetric information.

The scattered electron energy depends only on its angular deflection θe, once the beam energy is known. It can be obtained from the four momentum conservation, to get Ee= me 1 + r2_cos2_θ e 1 − r2_cos2_θ e , r= q E2 µ− m2µ Eµ+ me (2.14) The function is represented in Figure 2.8a. The maximum energy value occurs at θe = 0, or rather t = tmin, and it corresponds to Eemax '139.8 GeV for the selected beam energy. Given this relation, it is possible to derive the connection between the momentum transfer and the electron scattering angle:

t(θe) = 2me[me− Ee(θe)] = 4m

2

er2cos2θe r2cos2_θ

e−1

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Figure 2.7: Correlation curve between muon and electron scattering angles, for a 150 GeV/c

muon beam momentum. Blue triangles indicate some values of the two observables together with the corresponding values of electron energy and x variable [37].

(a) (b)

Figure 2.8: Outgoing electron energy as a function of its scattering angle (a). Momentum

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which is shown in Figure 2.8b. Eq. 2.15 allows the differential cross section to be plotted as a function of the experimental observable θe. It is represented in Figure 2.9, and is given by dσ0 dθe = dt dθe dσ0 dt = 4m2 er2sin 2θe (r2cos2_θ e−1)2 dσ0 dt (2.16)

Figure 2.9: Differential cross section of the µ − e− elastic scattering at LO, as a function of the electron scattering angle.

The region θe . 30 mrad is used to extract ∆αhad, namely it is the signal region. The large angles region θe & 30 mrad, instead, will be employed to obtain the luminosity of the machine. This aspect will be analyzed in more details in the following.

2.3.1 Higher order corrections

As stated in the previous sections, the radiative corrections to the differential cross section must be computed up to NNLO, in order to match the 10 ppm experimental accuracy. Accordingly, the final theoretical cross section needed for the experiment can be schematically expressed as the sum of the following terms

σ(µ± e− → µ± e−) ≈ LO(QED) + LO(EW)

+ NLO(QED) + NLO(HAD) + NLO(EW) (2.17)

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Figure 2.10: Ratio between (NLO+LO) and LO QED cross sections as a function of tee (left) and t_µµ (right) [41].

Here LO(QED) is the Born cross section in Eq. 2.11, while NLO and NNLO(QED) are the higher order QED contributions; LO(EW) is the electroweak contribution domi-nated by a Z exchange, which needs to be considered since its effect is O(10−5_{) [41].}

The next-to leading order NLO(EW) is, instead, highly suppressed with respect to NLO(QED), and can be safely neglected to increase the NLO Monte Carlo speed gen-eration [39]. NLO(HAD) is the term we intend to extract, given by the hadronic vacuum polarization insertion in the photon propagator. Finally, the NNLO(HAD) correction exhibits a complex evaluation, due to the presence of non factorizable hadronic loops [45]. This latter contribution is of the order of 10−4_÷₁₀−5_{, playing thus an important}

role in the MUonE data analysis.

Calculations lead to different results for µ+_e− _{and µ}−_e− _{elastic processes beyond the}

leading order, since the radiative corrections are different in the two cases. Further-more, the momentum transfer evaluated on the electron leg (tee) differs from the muon leg one (tµµ) as well, starting from the NLO. Such a discrepancy is due to the radiative events µ±_{e → µ}±_{e γ}_{. Results of the NLO QED calculation [41] are reported in Figure}

2.10. The mentioned differences involving positive and negative muons are clearly visi-ble. The ratios between the (NLO+LO) with respect to the LO cross section are shown in the plots, without any elasticity cuts. Figure 2.10 shows that the NLO contribution is > 10% for high momentum transfer values. Although this ratio is expected to be much smaller once the elasticity cuts will be applied [41, 39], the knowledge of the NLO contribution is fundamental for the experiment.

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Figure 2.11: Running of α as a function of x and t (left) [37], and as a function of θe(right). ∆αhad in red, ∆αlep in blue.

2.3.2 Sensitivity to

∆α

had

High values of momentum transfer correspond to high values of the electron energy, and therefore to low values of the electron scattering angle. Since ∆αhad increases with the momentum transfer, as shown in Figure 2.11, it follows that the sensitivity to the hadronic contribution is enhanced in the small angle region, corresponding to θe _{. 10 mrad.}

More generally, in the whole MUonE kinematic range ∆αlep . 10−2and ∆αhad. 10−3. As shown in Figure 2.11, the evolution of the leptonic contribution is moderate, whilst the hadronic part variation is much more pronounced. It decreases up to ∆αhad ∼10−5

in the angular region θe & 30 mrad, which corresponds to electrons scattered at ener-gies Ee . 1 GeV.

The sensitivity to the hadronic contribution can be studied in detail exploiting the ratio

Ri =

dσi(∆αhad(t) 6= 0)

dσi(∆αhad(t) = 0) (2.18) which is displayed in Figure 2.12. According to the definition, it stands for the ratio between the cross section including the hadronic contribution and the same cross sec-tion without it [41]. The index i represents the order up to which the calculasec-tion is performed. The analysis of this quantity confirms that at LO the sensitivity to the hadronic part of the running of α is maximum in the small θeregion, whilst it decreases at large electron scattering angles. Furthermore, at NLO the sensitivity is reduced due

The MUonE experiment: a novel way to measure the hadronic contribution to the muon g-2