Timelike and Spacelike evaluation of the leading order hadronic contribution to the muon g-2

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Dipartimento di Fisica Corso di Laurea in Fisica Teorica

Timelike and Spacelike evaluation of the leading

order hadronic contribution to the muon g−2

Relatori:

Dott. Graziano Venanzoni

Dott. Massimo Passera

Presentata da:

Marta Della Gatta

Sessione autunnale

Anno Accademico 2018/2019

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List of Figures

1. Chapter 1

1.1. 3-point function.

1.2. QED lower order contribution to aµ(taken from [4]).

1.3. QED diagrams contribuiting to the muon g-2 in order α2_{(taken from [4]).}

1.4. QED diagrams contribuiting to the muon g-2 in order α3_{(taken from [4]).}

1.5. Weak contributions to the muon g-2 (taken from [6]).

1.6. Hadronic contributions to the anomalous magnetic moment (taken from [6]). 1.7. Graphical representation of the calculation of aµ by dispersion relartions (taken

from [6]). 2. Chapter 2

2.1. Photon propagator (taken from [5]). 2.2. Optical theorem (taken from [5]).

2.3. Feyman diagram for the process e+_e−_{→ µ}+_µ−_.

2.4. Feyman diagram for the process e+_e−_{→ π}+_π−_.

2.5. Vacuum polarization diagram. 2.6. Leading hadronic contribution to aµ.

3. Chapter 3

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3.2. The behaviour of the final state radiation correction (taken from [12]). 3.3. Compilation of e+_e−_{→ π}+_π−_{data (taken from [2]).}

3.4. Distribution of contribution and square errors for aHLO

µ (taken from [2]).

3.5. Integrand of aHLO

µ superimposed with KLOE data.

µ superimposed with CMD2 data.

µ superimposed with BaBar data.

3.8. Fit of the Pion form factor with the GS parametrization for the KLOE data. 3.9. Fit of the Pion form factor with the GS parametrization for the CMD2 data. 3.10. Fit of the bare cross section with GS parametrization for the BaBar data. 3.11. (a),(b) Behaviour of ∆αhad_{for −10GeV}2 _{< t < 0}_{range for different datasets.}

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Acknowledgements

I would first like to thank my thesis advisors Dott. Venanzoni and Dott. Passera. They con-sistently allowed this paper to be my own work, but steered me in the right the direction whenever they thought I needed it. Finally, I must express my very profound gratitude to my parents for providing me with unfailing support and continuous encouragement through-out my years of study and through the process of researching and writing this thesis. This accomplishment would not have been possible without them. Thank you.

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Introduction

The measurement of the muon anomaly aµ = (gµ−2)

2 provides a powerful test of the

Stan-dard Model (SM) and gives important constraints on its extensions. Since more than 10 years there is an intriguing 3 ÷ 4 σ deviation between the most recent experimental measurement and the theoretical prediction which could point to possible contributions outside the SM. The knowledge of both quantities contributes to the significance of this comparison: a 0.54 ppm uncertainty on the experimental measurement and a 0.31 ppm theoretical uncertainty (accord-ing to the most recent evaluations) of the SM prediction. The uncertainty of aSMµ is dominated

by strong interaction effects, which cannot be computed perturbatively at low energies. These hadronic uncertainties are divided into those coming from the leading-order (LO) hadronic contribution to the muon g-2, aHLO

µ , and those from hadronic light-by-light (LbL)

contribu-tion, aHLbLµ . The dominant part aHLOµ is calculated with help of a dispersion integral over

experimental measurements of the total e+_e− _{→ hadrons cross-sections, where the}

contribu-tion to the central value and its error come mostly from the low energy region. Specifically the e+_e−_{→ π}+_π−_{channel below 2 GeV contributes with 73% to a}HLO

µ . To date the most precise

measurements of this cross section, which have achieved a systematic uncertainty up to 0.5%, come from the CMD2, KLOE and BaBar experiments. The calculation of aHLO

µ via dispersion

integral suffers from the complications due to the combination of different data sets with their systematics and correlations and the presence of resonances and threshold effects which make the cross sections highly fluctuating, especially at low energies ( <10 GeV). An alternative way to calculate aHLO

µ with experimental input from direct measurement of the hadronic part of

the photon vacuum polarization in the spacelike region has been recently proposed [1]. In par-ticular a proposal exists, the MUonE experiment [18], which plans to measure the hadronic part of the running of the electromagnetic coupling constant ∆αhadQED(t)as a function of the

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squared momentum transfer t = q2 _{< 0, by extracting it from the differential cross section of}

the elastic scattering µe → µe process. In this Thesis we will compare the two methodologies for the calculation of the leading order hadronic contribution to the muon g-2, aHLO

µ . In

Sec-tion 1 we will review the status of the theoretical calculaSec-tion of the muon g-2. In SecSec-tion 2 we will focus on the hadronic contribution, and will present the main formula for computing aHLO

µ in timelike and spacelike region. In Section 3 we will compute the 2π contribution of

aHLO_µ in the timelike and spacelike regions and we will compare the behaviours from different experiments. Finally in the Conclusions we will summarize the results and discuss the possible future developments of this work.

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Chapter 1 Muon magnetic moment

1.1 The muon g−2

All charged particles possess a magnetic moment that is connected with its intrinsic spin and it is given by the equation ~µ = g_2mce S. To understand this important link between the magnetic~ moment and the spin we begin this discussion with the emphasis on the spin of a particle . The spin and the magnetic moment of the electron became evident from the deflection of atoms in an inhomogeneous magnetic field and from the observation of the fine structure by optical spectroscopy. In quantum mechanics spin was considered as a new degree of freedom but in modern relativistic terms it is considered in a different way. In the Standard Model, particles and in particular leptons and quarks are considered to be massless originally, as required by chiral symmetry. All particles acquire their mass due to symmetry breaking via the Higgs mechanism. It gives a fermion mass term with mass mf =

Gf

√

2v, where Gf is the Yukawa

cou-pling and v is the non-zero vacuum expectation value of the scalar neutral Higgs field. In the massless state there are actually two independent electrons characterized by positive and neg-ative helicities (chiralities) corresponding to right−handed (R) and left−handed (L) electrons, respectively, which do not “talk” to each other. Helicity h is defined as the projection of the spin vector onto the direction of the momentum vector: h = S ·_|p_p_| and transform into each other by space-reflections P (parity). As we saw above, the magnetic moment is expressed in terms of the gyromagnetic ratio g and Dirac equation predicts g = 2 for fermions. The difference is the anomalous magnetic moment, denoted as a and defined as a = g−2₂ . Particles of definite

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mass, spin and charge quantized in units of the charge of electron e, have associated electro-magnetic form factors and in particular a definite electro-magnetic moment. However, for the proton, for example, the gyromagnetic ratio g = gP turns out to be gP ' 2.8 or aP = (gP

−2)

2 ' 0.4

(see [2]) showing that the proton is not a point particle but has internal structure. Only after a fermion has acquired a mass, helicity flip transitions as effectively mediated by an anoma-lous magnetic moment are possible. The anomaanoma-lous magnetic moment can only be a term induced by radiative corrections (in a renormalizable QFT theory it can’t appear as a term in the lagrangian) and in order for it not to vanish requires chiral symmetry to be broken by a corresponding mass term. This was first shown a long time ago by atomic beam magnetic deflection experiments, before the nature of the muon was clarified. Muon magnetic moment was the measurement at CERN in a sequence of experiments with higher precision and more recently at Brookhaven National Laboratory which yielded aµ= 0.00116592082(55)0.47ppm

[1] and revealed a∼ 3.5 standard deviation difference between measured value and Standard Model calculation. To measure the magnetic moment for a lepton we have to consider the mo-tion of a relativistic point particle with charge Qland mass mlin an external electromagnetic

field Aext

µ (x). The corresponding Dirac equations are :

(i~γµ∂µ+ Ql e cγ µ_(A µ+ Aextµ (x)) − mlc)ψl= 0 (1.1) (gµν− (1 − ξ−1)∂µ∂ν)Aν(x) = −Qleψl(x)γ µ_ψ l(x) (1.2)

In the non-relativistic limit1and in the weak field approximation we get the Pauli equation which reads: i~∂ ˆϕ ∂t = ˆH ˆϕ = 1 2m ~ p −e c ~ A 2 + eφ − e~ 2mc~σ · ~B ˆ ϕ (1.3)

This equation is the non−relativistic Schr¨odinger equation plus the spin term which has the form of a potential energy of a magnetic dipole in an external field. In leading order in 1

c the

lepton behaves as a particle which has a charge besides a magnetic moment: ~ µ = e~ 2mc~σ = e mc ~ S (1.4)

1_{In the non-relativistic limit the Dirac equation in an external electromagnetic field reduces to a Schrodinger}

equation, i∂ψ_∂ψ = Hψ, with a Hamiltonian H which contains an expansion in powers of the velocity of the particle, i.e. relativistic corrections (see [3]).

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γ

µ(p1)

µ(p2)

Figure 1.1: It represents all contributions to the 3 point function.

where ~_{S = ~~s = ~}~σ

2. The orbital momentum is:

~ µorb= Q 2M ~ L = gl Q 2M ~ L (1.5)

where ~L = ~r × ~_{p = −i~~r × ~}_{∇ = ~~l. We can write now the total momentum :} ~ µT OT = Q 2M(gl ~ L + gsS) =~ me MµB(gl~l + gs~s) (1.6) where µB = _2me~_e_c is Bohr’s magneton. As a result for the electron: Q= −e, M= me, gl = 1

and gs = 2. The gyromagnetic ratio _mce is twice as large as the one from the orbital motion. In

the absence of electrical fields E, the quantum correction can be included in a single quantity the anomalous magnetic moment, which is the result of relativistic quantum fluctuations, usu-ally simply called radiative corrections (RC). In QED aµmay be calculated using perturbation

theory, by considering the matrix element: M(x; p) = hµ−(p2, r2)| jemµ (x) |µ −_(p

1, r1)iof the

electromagnetic current for the scattering of muons in the classical limit of zero momentum transfer: q2 _{= (p}

1 − p1)2 → 0 and q is the momentum of the photon in fig. 1.1. Using the

spacetime invariance and going to the momentum space we obtain the T-matrix given by: hµ−_(p

2)| jemµ (0) |µ −_(p

1)i. We can represent it as the diagram in the fig. 1.1 and write it in a

relativistic covariant decomposition of the form: hµ−(p2)| jemµ (0) |µ − (p1)i = (−ie)u(p2) γµFE(q2) + iσµν_q ν 2mµ FM(q2) u(p1) (1.7)

where, FE(q2) is the electric charge or Dirac form factor and FM(q2) is the magnetic or

Pauli form factor. The matrix σµν _{= i[γ}µ_{, γ}ν_]_{represents the spin} 1

2 angular momentum tensor.

In the static (classical) limit we have FE(0) = 1, FM(0) = aµ where the first relation is

the charge renormalization condition (in units of the physical positron charge e) while the second relation is the finite prediction for aµ, in terms of the form factor FM . At the leading

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order FM(0) = _2πα ' 0.0011614... (see [2]) as predicted by the theory. We will see now the

contributions to the magnetic moment that come from the diagram in the figure 1.1: we must distinguish the case of the electron magnetic moment from the one of the muon, because ae

is rather insensitive to strong and weak interactions. It provides a rigorous test of QED and leads to a precise determination to date of the fine-structure constant α, while aµ allows to

test the entire SM, as each of its sectors contribute in a significant way to the total prediction. Compared to ae, aµ is also much better suited to unveil or constrain ‘new physics’ effects.

Indeed for a lepton l(= {e, µ, τ }), al ' m2

l

Λ2 where ml is the mass of the lepton and Λ is the

scale of the new physics. Sincemµ

me

2

' 4 × 104 _{(see [4]), we have an enhancement of the}

sensitivity of the muon compared to ae. The τ particle would be the best choice to detect ”new

physics”, but its short lifetime makes such a measurement very difficult.

1.1.1 Electromagnetic contribution

Starting with the QED contribution to aµ, arises from a subset of SM diagrams containing only

leptons (= {e, µ, τ }) and photons. We can write : aQED_µ = A1 + A2 m_µ me + A2 m_µ mτ + A3 m_µ me ,mµ mτ (1.8) A1 comes from diagrams with only photons and muons, it is mass independent so it is same

for the QED contribution to the anomalous magnetic moment of all three charged leptons: ae = aµ = aτ. Since QED is renormalizable we can expand the functions Ai(i = 1, 2, 3) as

power series in α_π and they can be computed order by order : 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 + ... (1.9) At one loop we have the diagram in figure 1.2 and A(2)1 = 12, A

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2 = 0, A (2)

3 = 0(see [4]). At 2

loops we have the QED diagrams contributing to the muon g−2 in order α2 _{(see figure 1.3) .}

The mirror reflections (not shown) of the third and fourth diagrams must be included as well. Seven diagrams contribute to A(4)₁ , one to A(4)₂ mµ

me and one to A(4)₂ mµ mτ . The values for the muon–electron mass ratio mµ

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γ

µ µ

Figure 1.2: Lower order QED contribution to aµ.

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Figure 1.4: Examples of QED diagrams contributing to the muon g-2 in order α3_{: (a) a}

‘triple-cross’ diagram, (b) and (c) sixth-order muon vertex obtained by insertion of an electron or τ vacuum polarization (light-by-light) subdiagram and (d) graph with e and τ loops in the photon propagator.

105.6583692(94)MeV and mτ = 1776.99(29)MeV (see [4]) yields

A(4)₂ mµ me = 1.0942583111(84) (1.10) A(4)₂ mµ mτ = 0.000078064(25) (1.11)

(see [4]) where the uncertainties are due to the measurement uncertainties of the lepton mass ratios. The τ contribution in equation (1.11) provides a ' 42 × 10−11 _{(see [4]) contribution}

to aQED_{. As there are no two−loop diagrams containing both virtual electrons and taus,}

A(4)₃ mµ me, mµ mτ

= 0. For the 3 loops, putting m = mµ, M = me or M = mτ we have to

know that A(6)₂ _Mm can be split into two parts: the first one, A(6)₂ _Mm, vp, receives contri-butions from 36 diagrams containing electron or τ vacuum polarization loops (see (B) in the figure 1.4 ), whereas the second one, A(6)₂ _Mm, lbl, is due to 12 light−by−light scattering di-agrams with electron or τ loops (like the graph of fig. 1.4 (c)). This new kind of contribution consists of closed fermion loops with four photons attached.

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We can find that : A(6)₂ mµ me , vp = 1.920455130(33), (1.12) A(6)₂ mµ me , lbl = 20.94792489(16), (1.13) A(6)₂ mµ mτ , vp = 0.00178233(48), (1.14) A(6)₂ mµ mτ , lbl = 0.00214283(69) (1.15)

The sums of equations (1.12)−(1.13) and equations (1.14)−(1.15) are (see [4]): A(6)₂ mµ me = 22.86838002(20), (1.16) A(6)₂ mµ mτ = 0.00036051(21) (1.17)

to determine the uncertainties, the correlation of the terms to be added has been taken into account. Note the large contribution from the electron light-by-light diagrams, its leading term is 2₃ π2_lnmµ

me

. More generally, it was shown that the O(α2n+1₎_{contribution to a}QED µ , from

diagrams in which the electron light−by−light subgraph is connected with 2n + 1 photons to the muon, contains a large π2n_lnmµ

me

term with a coefficient of O(1). The analytic calculation of the three-loop diagrams with both electron and τ loop insertions in the photon propagator (see fig. 1.4 (d)) became available in 1999. This analytic result yields the numerical value ( [4]):

A(6)₃ mµ me ,mµ mτ = 0.00052766(17) (1.18)

providing a small 0.7×10−11(see [4]) contribution to aQED

µ so it induces a negligible O(10 −14₎

uncertainty in aQEDµ (see [4]). The error, 1.7×10−7, is caused by the uncertainty of the ratio mµ

mτ

[4]. For the fourth order term (4 loops) the total number of diagrams exceeds 1000. The univer-sal contribution in the fourth order is described by 891 diagrams. With perturbation theory, the leading contribution to C4 (which is the sum A41+ A42 + A43 = C4) comes from diagrams

with electron loops, A(8)2

_m

µ

me

. In total, there are 469 diagrams. Later, errors were found in the initial calculations related to the insufficient accuracy of the 8-byte real arithmetic and to insufficient statistics of Monte Carlo simulations. Some of the diagrams were calculated ana-lytically. The minor contribution of diagrams with τ -lepton loops was calculated numerically.

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The combination of all fourth-order contributions gives: A(8)₁ = −1.91298(84), (1.19) A(8)₂ mµ me = 132.6852(60), (1.20) A(8)₂ mµ mτ = 0.04234(12), (1.21) A(8)₃ mµ me ,mµ mτ = 0.06272(4) (1.22) C4 = 130.8734(60) (1.23)

The uncertainty of calculations is determined by the statistical accuracy of the numerical inte-gration by the Monte Carlo method [5]. The value for the QED contribution to the muon g−2 is: aQED

µ = 116584718.859(0.034) × 10−11[2]. The first error is due to the uncertainties of the

O(α2_{), O(α}4₎_{and O(α}5₎_{terms, and is strongly dominated by the last of them (the uncertainty}

of the O(α3_{)term is negligible).}

1.1.2 Electroweak contribution

The weak interaction contribution to g-2 attracted attention of theoreticians long time before it started to play a relevant role in the comparison with the experimental result. Weak inter-action effects are mediated by exchange of the heavy weak gauge bosons W±, which mediate charged current (CC) processes, and Z, which mediates the neutral current (NC) processes. Beyond the electroweak SU (2)L⊗ U (1)Y Yang-Mills gauge theory, a Higgs sector is required

which allows to generate the masses of the gauge bosons W and Z, as well as the masses of the fermions, without spoiling renormalizability. Thus the gauge symmetry is broken down SU (2)L⊗U (1)Y → U (1)emto the Abelian subgroup of QED, and an additional physical

parti-cle has to be taken into account, the recently discovered Higgs partiparti-cle. We should remark that before symmetry breaking the theory has the two gauge couplings g and g0 as free parameters, after the breaking we have in addition the vacuum expectation value (VEV) of the Higgs field v, thus three parameters in total, if we disregard the fermion masses and their mixing param-eters for the moment. The most precisely known paramparam-eters are the fine structure constant α(electromagnetic coupling strength), the Fermi constant Gµ(weak interaction strength) and

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the Z mass MZ. Some of these quantities are related by : α = e 2 4π, e = gsinΘw, tanΘw = g0 g and the mass generation by the Higgs mechanism yields:

MW = gv 2 , MZ = gv 2cosΘw

After the discovery of the Higgs, we also know the Higgs mass within a rather narrow error band: mH = 125.1 ± 0.3GeV (see [2]). Thus, for the first time, all relevant parameters of the

SM are known with impressive accuracy. The electroweak contribution (EW, shown in Fig. 1.5) is now calculated through two loops. The one loop result (see [6])

aEW (1)_µ = G√F 2 m2 µ 8π2 10 3 + 1 3(1 − 4sin 2_θ w)2− 5 3 +O _m2 µ M2 Z logM 2 Z m2 µ + m 2 µ M2 H Z 1 0 dx 2x 2_{(2 − x)} 1 − x + m2µ M2 H x2    = 194.8 × 10−11 (1.24)

was calculated by five separate groups shortly after the Glashow-Salam-Weinberg theory was shown by ’t Hooft to be renormalizable. Due to the small Yukawa coupling of the Higgs boson to the muon, only the W and Z bosons contribute at a measurable level in the lowest-order electroweak term. In the formula (1.24) the first term in the curly brackets is due to W, while the other two are due to Z. In the fig.1.5 there are : Weak contributions to the muon anomalous magnetic moment. Single-loop contributions from (a) virtual W and (b) virtual Z gauge bosons. These two contributions enter with opposite sign, and there is a partial cancellation. The 2 loop EW contribution can be split in the fermionic and bosonic part: the first one includes all the 2 loop electroweak corrections containing closed fermion loops while the second one groups all other contributions . If mf is the fermion mass scale (<< MW), aEW,H(2loop) is quite

important because it is enhanced by the quantity lnMZ,W

mf

. The hadronic uncertainties, estimated be ' 2 × 10−11 (see [4]), arise from two types of two−loop diagrams: hadronic

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Figure 1.5: Weak contributions to the muon anomalous magnetic moment.

photon−Z mixing, and quark triangle loops with the external photon, a virtual photon and a Z attached to them. The tiny hadronic γ−Z mixing terms can be evaluated either in the free quark approximation or via a dispersion relation using data from e+_e− _{annihilation into}

hadrons; the second type of diagrams (the quark triangle ones), calculated in the free quark approximation, is numerically more important. More in detail, the two-loop contributions fall into three categories: (c) fermionic loops which involve the coupling of the gauge bosons to quarks, (d) bosonic loops which appear as corrections to the one-loop diagrams, and (e) a new class of diagrams involving the Higgs where G is the longitudinal component of the gauge bosons. The × indicates the photon from the magnetic field. The two-loop electroweak contribution (see figs. 1.5(c-e)), which is negative, has been re-evaluated using the LHC value of the Higgs mass and consistently combining exact two-loop with leading three-loop results. The total electroweak contribution is aEW = (153.6 ± 1.0) × 10−11(see [6]) where the error

comes from hadronic effects in the second-order electroweak diagrams with quark triangle loops, along with unknown three-loop contributions. The leading logs for the next-order term have been shown to be small. The weak contribution is about 1.3 ppm (see [6]) of the anomaly, so the experimental uncertainty on aµof ±0.54 ppm (see [6]) now probes the weak scale of

the Standard Model.

1.1.3 Hadronic contribution

The evaluation of the hadronic LO contribution aHLOµ involves long-distance QCD for which

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Figure 1.6: Hadronic contributions to the anomalous magnetic moment.

shown long ago that this term can be computed via a dispersion integral using the cross sec-tion for low-energy hadronic e+_e− _{annihilation. At low energy this cross-section is highly}

fluctuating due to resonances and particle production threshold effects. As we will show in this thesis, an alternative determination of aHLO

µ can be obtained measuring the effective

elec-tromagnetic coupling in the spacelike region. The hadronic contribution to aµ is about 60

ppm of the total value (see [7]). The lowest-order diagram shown in fig. 1.6 (a) dominates this contribution and its error, but the hadronic light-by-light contribution (fig. 1.6 (e)) is also important. Physicists are doing a lot of efforts to improve the evaluation of the leading order (LO) hadronic contribution to aµ, due to the hadronic vacuum polarization correction to the

one-loop diagram (we will deal with its calculation in the next section). Concerning high order the O(α3₎_{hadronic contribution to the muon g-2 a}HHO

µ we find that :

aHHO

µ = aHHOµ (vp) + aHHOµ (lbl)

The first term is the contribution of diagrams containing hadronic vacuum polarization in-sertions. The second one is the light-by-light contribution. The hadronic LbL contributions, although small compared to the hadronic vacuum polarization sector, have, in the past, been determined through model-dependent approaches. New efforts into the prospects of deter-mining ahad,LbL

µ using dispersive approaches are also very promising ([19]). For aHLOµ the

main ingredients we need are those in fig. 1.7 because using the dispersion relations we can calculate the dominant hadronic contribution to the muon anomaly. In fig. 1.7 (a) is the “cut” hadronic vacuum polarization diagram , (b) is the e+_e−_{annihilation into hadrons and (c) is the}

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Figure 1.7: (a) The “cut” hadronic vacuum polarization diagram ; (b) is the e+e− annihilation into hadrons; (c) initial state radiation accompanied by the production of hadrons.

contribution to the muon g-2 is given by: ahad;HLO_µ = (αmµ 3π 2Z ∞ m2 π ds s2K(s)R(s) (1.25)

where R(s) ≡ σtot(e+e−→hadrons)

σ(e+_e−_→µ+_µ−₎ and K(s) is the a kinematic factor ranging from 0.4 at s = m2_π

to 1 at s = ∞. This dispersion relation relates the bare cross section for e+e− annihilation into hadrons to the hadronic vacuum polarization contribution to aµ. The contribution is

dom-inated by the two-pion final state. Two analyses using the e+e− → hadrons data (see in [7]) obtained: ahad;HLO

µ = (6923 ± 42) × 10

−11_{, a}had;HLO

µ = (6949 ± 43) × 10

−11_{respectively. The}

most recent evaluation of the next-to-leading order hadronic contribution shown in fig. 1.6 (b-d), which can also be determined from a dispersion relation, ahad;N LO

µ = (98.4±0.60.4)×10 −11

(see [7]). As we said above the hadronic light-by-light contribution (HLbL) can be calculated using hadronic models that correctly reproduce properties of QCD. A synthesis of the model contributions, which was agreed to by authors from each of the leading groups that have been working in this field, is aHLbL

µ = (105 ± 26) × 10

−11_{(see [7]).}

1.2 The running of the fine structure constant α(q

2

)

The new approach of the spacelike determination of aHLOµ relies on the determination of the

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of the squared momentum transfer t = q2 _{< 0. It may not be limited by statistics and,}

al-though challenging, may become competitive with standard results obtained with the disper-sive approach via timelike data. In the next section we will explain the very useful relation ImΠhad = _12π1 R(s)which allow us to express formula (1.25) in the following way:

aHLO_µ = α π2

Z ∞

0

ds

s K(s)ImΠhad(s + i) (1.26)

where Πhad(s)is the hadronic part of the photon vacuum polarization, > 0 and

K(s) = Z 1 0 dx x 2_{(1 − x)} x2_{+ (1 − x)} s mµ (1.27)

The dispersion integral in eq. (1.25) is computed integrating experimental timelike (s > 0) data up to a certain value of s (see [6]). The high-energy tail of the integral is calculated using perturbative QCD. We will see in the next section that if we exchange the x and s integrations in eq. (1.26) we obtain: aHLO_µ = α π Z 1 0 dx(x − 1)Πhad[t(x)] (1.28)

where Πhad(t) = Πhad(t) − Πhad(0) and t(x) = x2_m2

µ

x−1 < 0 is a spacelike squared

four-momentum. The effective fine-structure constant at squared momentum transfer s = q2 can be defined by

α(q2) = α

1 − ∆α(q2₎ (1.29)

where ∆α(q2) = −ReΠ(q2). The purely leptonic part, ∆αlep(q2), can be calculated

order-by-order in perturbation theory (it is known up to three loops in QED). As ImΠhad(q2) = 0for

negative q2, replacing Π with ∆α in (1.28), we can rewrite it in the form : aHLO_µ = α

π Z 1

0

dx(1 − x)∆αhad[t(x)] (1.30)

In the next section we will proceed calculating this equation by measurements of the effective electromagnetic coupling in the spacelike region.

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Chapter 2 Calculation of a

HLO

_µ

2.1 Hadronic leading order contribution

The strong interaction contribution to aµ, although being four orders of magnitude smaller

than the electromagnetic contribution, is very significant compared to the experimental errors. This contribution is about 10 times as high as the CERN III experimental error and 100 times as high as the measurement error of aµat BNL [5]. We will concern with the leading-order (LO)

term (from hadron vacuum polarization) aHLO

µ . Due to the smallness of the electromagnetic

and weak interaction coupling constants, the perturbation theory enables a high-accuracy calculation of the corresponding contribution. In the case of strong interactions, the pertur-bation theory is applicable only at high energies where the effective QCD coupling constant becomes small. In the calculation of aµ, the characteristic momentum transfer is of the order

of the muon mass, which is smaller than the characteristic energies for asymptotic freedom (about several GeV). For this reason we need an alternative way to calculate the hadronic con-tribution. A method of calculation of the leading contribution aHLO

µ due to hadronic vacuum

polarization based on the use of dispersion relations was proposed in the 1960s [20]. To date, it is the only method that allows achieving the required accuracy. Let see the main steps of this approach: the photon propagator can be represented as a sum of one-particle-irreducible blocks of loop diagrams, none of which can be decomposed into two independent blocks by cutting one photon line (see fig.2.1).

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Figure 2.1: Photon propagator representation as a sum of one-particle-irreducible blocks, each of which has the corresponding polarization operator.

sum of the series of the fig. 2.1 yelds: iD(q2) = − i

q2_{(1 − Π(q}2₎₎ = −

i

q2(1 + Π(q

2₎₎ _(2.1)

We could write the last equality because we kept only one irreducible block, because our goal is to study the diagram in fig. 1.6 (a). From the causality principle we can use the analytic properties of Π(s) and we can write the dispersion relation as:

−Π(q 2₎ q2 = Z ∞ 0 ds s 1 πImΠ(s) 1 q2_{− s} (2.2)

This formula can be interpreted in the following way: the part −Π(q_q22) of photon propagator

(2.1) corresponding to vacuum polarization is equivalent to the sum of contributions due to massive photons m2

γ = swith the propagator q21_−s integrated with the weight

ImΠ(s) πs . The

contribution to aµfrom the diagram with a massive photon exchange, m2γ = s, yelds :

am

2 γ=s

µ =

α

πK(s) where K(s) is explicitated in eq. (1.27) (2.3) Using the integral representation (2.2) for the photon propagator, it is possible to calculate the contribution of vacuum polarization to the muon anomalous magnetic moment as a sum of contributions due to massive photon exchange (2.3) integrated with the weight ImΠ(s)_πs :

aV P_µ = α π Z ∞ 0 ds s 1 πImΠ(s)Kµ(s) (2.4)

The optical theorem, which results from the scattering matrix unitarity, allows relating the imaginary part of the polarization operator to the total cross section of particle production in electron-positron annihilation (see fig. 2.2).

ImΠ(s) = αs

4π|α(s)|2σ(e +

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Figure 2.2: Optical theorem.

The polarization operator Π(s) contains the contributions of both lepton and hadron vacuum polarizations. The calculation of the contribution due to strong interactions in the leading order yields : aHLO_µ = α 2 3π2 Z ∞ 4m2 π ds s R(s)Kµ(s) (2.6)

where R(s) is the normalized total hadron production cross section in electron-positron anni-hilation: R(s) = σ(e +_e−_{→ γ∗ → hadrons)(s)} 4π|α(s)|2 (3s) (2.7) and K(s) = 3s m2 µ Kµ(s) (2.8)

which is a slowly varying monotonic function asymptotically tending to unity. This relation shows that eq. (2.6) it is nothing more than eq. (1.25). The denominator of eq. (2.7) is σ(e+_e− _→

µ+_µ−_{) =} 4π|α(s)|2

(3s) or equivalently σ

0_(e+_e−_{→ µ}+_µ−_{) =} 4π|α(s)|2

(3s) because we consider the mass

of the muons equal to zero ( for the real muon the term compensating a change in the phase space must be added in the cross section). Formula (1.25) allows calculating aHLO

µ using a

known R(s). R(s) corresponds to all diagrams in which the virtual photon decays into one or more hadrons and an arbitrary number of other particles. Therefore R(s) corresponds to the hadron production via one virtual photon. Processes with several photon exchanges or photon emission by initial particles should not be taken into account in R(s), but the observed hadron production cross sections should be supplemented with radiative corrections, which

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are known to an accuracy of 0.1−1% [5]. Processes with photon emission by final hadrons should be taken into account in R(s). R(s) corresponds to one-particle-irreducible insertions in the photon propagator, and therefore hadron production should proceed via one virtual photon without loop insertions. In the observed process e+_e−_{→ γ∗ → hadrons, the virtual photon is}

described by the full propagator with the vacuum polarization taken into account. Therefore, the vacuum polarization contribution should be excluded from the observed cross section. One way to do it is normalizing the hadron cross sections to the cross section of e+_e− _{→ µ}+_µ−_.

Therefore it is necessary to add corrections for the emission of real photons, but the vacuum polarization contribution is automatically canceled. If we first integrate over x we find the well known standard representation (2.6) as an integral along the cut of the vacuum polarization amplitude in the timelike region, while an interchange of the order of integrations yields an integral over the hadronic shift of the fine structure constant ∆α(s) = −Re (Π0(s) − Π0(0)) in the spacelike domain :

aHLO_µ = α π Z 1 0 dx(1 − x)∆α(5)_had(t(x)) where − t(x) ≡ x 2 1 − xm 2 µ (2.9)

2.1.1 Cross section calculation

We will start calculating the cross sections σ(e−e+ _{→ µ}−_µ+₎ _{and σ(e}−_e+ _{→ π}−_π+_{). The}

corresponding Feynman diagram for the first process is: e− e+ µ+ µ− p p0 k0 k γ q −→

Figure 2.3: Feynman diagram for the process e−e+ _{→ µ}−_µ+_.

We have to write the amplitude: Mf i = vs 0 (p0)(−ieγµ)us(p) −igµν q2 ur(k)(−ieγν)vr0(k0) = ie 2 q2(v s0 (p0)γµus(p))(ur(k)γµvr 0 (k0)) ≡ iM

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and squaring M:

| M |2₌ e4

q4(v(p 0

)γµu(p)u(p)γνv(p0))(u(k)γµv(k0)v(k0)γνu(k))

Summing over all the spins and remembering the following relations: X s us(p)us(p) = /p + m X s vs(p)vs(p) = /p − m we obtain: 1 4 X spins | M |2= e 4 4q4tr[( /p 0_{− m} e)γµ( /p0+ me)γν]tr[(/k + mµ)γµ( /k0− mµ)γν]

Using the Dirac matrices algebra and neglecting the electron-positron masses, we arrive at: 1 4 X spins | M |2₌ 8e4 q4 [(p · k)(p 0_{· k}0_{) + (p · k}0_)(p0_{· k) + m}2 µ(p · p 0_)]

In the center of Mass frame we have:

q2 _{= (p + p}0₎2 _{= 4E}2_{, p · p}0 _{= 2E}2_{, p · k = p}0_{· k}0 _{= E}2_{− E | ~k | cosθ , p · k}0 _{= p}0_{· k = E}2_{+ E |}

~k | cosθ with these relations and remembering that | ~k |= pE2_{− m}2

µand ~k · ˆz =| ~k | cosθ ,

the squared amplitude becomes: 1 4 X spins | M |2= 8e 4 4q4[(E 2_{− E | ~k | cosθ)}2 + (E2+ E | ~k | cosθ)2+ m2_µ2E2] = 8e4 16E4[2E 4_{+ 2E}2 _{| ~k}2 _|2 _(cosθ)2_{+ m}2 µ2E2] = 8e4 16E4[2E 4 _{+ 2E}2_(E2_{− m}2 µ) 2_(cosθ)2_{+ m}2 µ2E 2_{] =} 8e4 16E4 2E4 1 + m 2 µ E2 + 2E4 1 −m 2 µ E2 (cosθ)2 = e4 1 + m 2 µ E2 + 1 − m 2 µ E2 (cosθ)2

We are in the CM frame, remember that s = E2

cm = (2E)2 = q2, so the differential cross

section is: dσ dΩ = 1 2E2 cm | ~k | 16π2_E cm 1 4 X spins | M |2₌ 1 2E2 cm pE2 _{− m}2 µ 16π2_E cm e4 1 + m 2 µ E2 + 1 −m 2 µ E2 (cosθ)2 =

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α2 4E2 cm r 1 −m 2 µ E2 1 + m 2 µ E2 + 1 − m 2 µ E2 (cosθ)2 (2.10) Now we can calculate the cross section and for semplicity we define: a = _4Eα22

cm q 1 −m2µ E2, b = 1 + m2µ E2 , c =1 − m2µ E2 σ = Z dσ dΩdΩ = Z 2π 0 Z π 0

a[b + c(cosθ)2]dφsinθdθ] = 2π Z π 0 absinθdθ + Z π 0 ac(cosθ)2sinθdθ = 2π 2ab + ac2 3 = 4πahb + c 3 i we arrive at: σ = 4π α 2 4E2 cm r 1 − m 2 µ E2 1 + m 2 µ E2 +1 3 1 − m 2 µ E2 = 4πα 2 3E2 cm r 1 − m 2 µ E2 1 + m 2 µ 2E2

For the process e−e+ → π−π+we have the following Feynman diagram: e− e+ π+ π− p p0 k0 k γ q −→

Figure 2.4: Feynman diagram for the process e−e+ _{→ π}−_π+_.

The associated lagrangian is: L = −1 4FµνF µν _{+ (D} µφ)∗(Dµφ) − m2πφ ∗ φ + ψ(i /D − me)ψ = −1 4FµνF µν _{+ (∂} µ− ieAµ)φ∗(∂µ+ ieAµ)φ − m2πφ ∗ φ + ψ(i /∂ − e /A_µ− me)ψ

We find the interaction vertex of pions:

ie[(∂µφ∗)Aµφ − Aµφ∗(∂µφ)] + e2AµAµφ∗φ

going into the momentum space we change: ∂µ→ ipµand we obtain the vertex : e(k0− k)µ.

We can write now the amplitude and the squared amplitude summed over all the spins: iMf i= vs 0 (p0)(−ieγµ)us(p) −igµν q2 e(k0 − k)ν

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1 4 X spins | Mf i|2= 1 4 (−i)2e 2 q2(v(p 0_)γµ_u(p)(k0 _{− k)} µ (−i)2e 2 q2u(p)γ ν_v(p0_)(k0_{− k)} ν

Now we use the approximation that the electron mass in negligible: = e 4 4q4 tr( /p 0 γµ_/pγν)(k0− k)µ(k0 − k)ν = e4 q4p 0 αpβ(k0− k)µ(k0− k)ν[gαµgβν − gαβgµν+ gανgβµ] = e4 q4[p 0µ pν(k0− k)µ(k0− k)ν− (p0· p)(k0− k) · (k0 − k) + p0ν(k0− k)νpµ(k0− k)µ] = 2e4 q4 [(p 0_{· k}0 )(p · k0) − (p0· k0)(p · k) − (p0 · k)(p · k0) + (p0· k)(p · k) − (p · p0)(m2_π − k · k0)] As we did before, in the Centre of mass frame we have: q2 _{= (p + p}0₎2 _{= 4E}2 _{, p · p}0 _{= 2E}2_,

p · k = p0· k0 _{= E}2_{− E | ~k | cosθ , p · k}0 _{= p}0_{· k = E}2_{+ E | ~k | cosθ}_{with these relations and}

remembering that | ~k |=pE2_{− m}2

πand ~k · ˆz =| ~k | cosθ, the squared amplitude becomes:

2e4

q4 [2(E

4_{− E}2 _{| ~k |}2

(cosθ)2) − 2(E4+ E2 | ~k |2 (cosθ)2) − 4E2(m2_π− E2)] = 4e4 q4 [(E 4_{− E}2 _{| ~k |}2 (cosθ)2) − (E4+ E2 | ~k |2 (cosθ)2) − 2E2(m2_π − E2)] = 4e4 q4 [2E 4_{− 2E}2 m2_π − 2E2 | ~k |2 (cosθ)2)] = 8e4 q4 [E 4_{− E}2_m2 π− E2(E2− m2π)(cosθ)2] = 8e4 q4 E 2_(E2_{− m}2 π)[1 − (cosθ)2] = e4 2 1 − m 2 π E2 (sinθ)2 Putting this formula inside the differential cross section formula we obtain:

dσ dΩ = 1 2E2 cm | ~k | 16π2_E cm 1 4 X spins | M |2= 1 2E2 cm pE2_{− m}2 π 16π2_E cm e4 2 1 − m 2 π E2 (sinθ)2 = α2 8E2 cm r 1 −m 2 π E2 1 − m 2 π E2 (sinθ)2

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For the total cross-section: σ = Z dσ dΩdΩ = Z 2π 0 Z π 0 A(sinθ)2dφsinθdθ = 2π Z π 0 A(sinθ)3dθ = A2π4 3 = α2_2π 6E2 cm 1 − m 2 π E2 3₂ we arrive at: σ = α 2_π 3E2 cm 1 −m 2 π E2 3₂

The quark structure of the charged pion is taken care of by introducing a pion form factor e → eFπ(q2), e2 → e2 | Fπ(q2) |2. σ = α 2 _{| F} π(q2) |2 π 3E2 cm 1 −m 2 π E2 3₂ (2.11)

2.1.2 1 loop correction to the photon propagator calculation

The first term of the expansion of the photonic 2-point function < Aµ(x)Aν(y) >is g_kµν2 (here

we use the Feynman gauge ε = 1), but if we want to consider the higher orders, say the 1 loop order, there are other diagrams. Consider the 1PI diagram, so at 1 loop we have the vacuum polarization diagram: It can be viewed as a modification to the photon structure by a virtual

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electron-positron pair. The integral associated with the vacuum polarization diagram is: iΠµν₂ = −(−ie)2 Z dk4 (2π)4tr γµ i(/k + m) (k2_{− m}2₎γ ν i(/k + /q + m) (k + q)2_{− m}2 = −e2 Z dk4 (2π)4 tr[γµγαγνγβ]kα(k + q)β + tr[γµγν]m2 [k2_{− m}2_{][(k + q)}2_{− m}2_] = −4e2 Z _dk4 (2π)4 [gµα_gνβ_{− g}αβ_gµν_{+ g}να_gµβ_]k α(k + q)β + gµνm2 [k2_{− m}2_{][(k + q)}2_{− m}2_] = −4e2 Z dk4 (2π)4 kµ(k + q)ν+ kν(k + q)µ− gµν_{[k · (k + q) − m}2_] [k2_{− m}2_{][(k + q)}2_{− m}2_]

Now we proceed manipulating the denominator using the Feynman parameters: 1 [k2_{− m}2_{][(k + q)}2 _{− m}2_] = Z 1 0 dx 1 {[(k + q)2 _{− m}2_{]x + [k}2_{− m}2_{](1 − x)}}2 = Z 1 0 dx 1 [k2_{+ 2kqx + q}2_{− m}2_]2 = Z 1 0 dx 1 [(k + qx)2_{+ q}2_{x(1 − x) − m}2_]2 Call l = k + qx, ⇒ Z 1 0 dx 1 [l2_{+ q}2_{x(1 − x) − m}2_]2

Put all together: −4e2 Z _dk4 (2π)4 kµ_{(k + q)}ν _{+ k}ν_{(k + q)}µ_{− g}µν_{[k · (k + q) − m}2_] [k2_{− m}2_{][(k + q)}2_{− m}2_] = −4e2 Z 1 0 dx Z _dk4 (2π)4 kµ_{(k + q)}ν _{+ k}ν_{(k + q)}µ_{− g}µν_{[k · (k + q) − m}2_] [(k + qx)2_{+ q}2_{x(1 − x) − m}2_]2

Changing variable also in the numerator, we obtain: 2lµ_lν_−gµν_l2_−2qµ_qν_x(1−x)+q2_gµν_x(1−

x) + m2_gµν_{+(linear terms in l). Going in the Euclidean space l}0 _{→ il}0

E and calling ∆ = m2_{− x(1 − x)q}2_{we obtain:} iΠµν₂ (qE) = −4ie2 Z 1 0 dx Z _dl4 E (2π)4 −gµν₂ l2 E + gµνl2E − 2x(1 − x)qµqν + gµν[x(1 − x)q2+ m2] [l2 E + ∆]2

This integral is divergent so we need the dimensional regularization, it means that we have to switch from the 4-dimensional space to a d dimension space and after that take the Euclidean

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limit: iΠµν₂ (qE) = −4ie2 Z 1 0 dx Z _dld E (2π)d −2g_dµνl2 E+ gµνl2E − 2x(1 − x)qµqν + gµν[x(1 − x)q2+ m2] [l2_E + ∆]2 = −4ie2 Z 1 0 dx Z dld_E (2π)d gµνl_E2 −2 d+ 1 − 2x(1 − x)q µ_qν _{+ g}µν_{[x(1 − x)q}2_{+ m}2_] [l2 E + ∆]2

for the first term: Z _dld E (2π)d gµν_l2 E − 2 d+ 1 [l2 E + ∆]2 = gµν −2 d + 1 Γ 1 +d₂ Γ 1 − d 2 (4π)d2Γ(2)Γ d 2 ∆ d 2−1 = 1 (4π)d2 Γ 2 −d 2 1 ∆ 2−d₂ (−∆gµν)

the other terms don’t depend on lE so we can considerate only the integral of the denominator:

Z _dld E (2π)d 1 [l2 E + ∆]2 = Γ d 2 Γ 2 − d 2 (4π)d2Γ(2)Γ d 2 ∆ d 2−2

writing the numerator : −4ie2 Z 1 0 dx Z dld_E (2π)d −2x(1 − x)qµ_qν _{+ g}µν_{[x(1 − x)q}2_{+ m}2_] [l2 E + ∆]2 = −4ie2 Z 1 0 dx Γ 2 − d 2 (4π)d2∆2− d 2 {−2x(1 − x)qµ_qν _{+ g}µν_{[x(1 − x)q}2_{+ m}2_]}

putting all the terms together: −4ie2 Z 1 0 dx Γ 2 − d 2 (4π)d2∆2− d 2 gµν_{[x(1 − x)q}2_{− m}2_{] − 2x(1 − x)q}µ_qν _{+ g}µν_{[x(1 − x)q}2_{+ m}2₌ −4ie2 Z 1 0 dx Γ 2 − d 2 (4π)d2∆2− d 2 2(gµνq2− qµ_q2_{)x(1 − x) ≡ (q}2_gµν_{− q}µ_qν_)iΠ 2(q2) we find that: Π2(q2) = −8e2 Z 1 0 dx Γ 2 − d 2 (4π)d2∆2− d 2 x(1 − x) Since d = 4 − , performing the limit d → 4 we obtain:

Γ 2 − d₂ (4π)d2∆2− d 2 = Γ 2 (4π)2_(4π)−2 ∆ 2 = 1 (4π)2Γ 2 4π ∆ ₂ ' 1 (4π)2 2 − γE + O() 1 + 2 (ln(4π) − ln∆) = 1 (4π)2 2 − γE+ ln(4π) − ln∆

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With these approximations our integral becomes: Π2(q2) = −8e2 Z 1 0 dxx(1 − x) (4π)2 2 − γE + ln(4π) − ln∆ = −2α π Z 1 0 dxx(1 − x) 2 − γE+ ln(4π) − ln∆ = −2α π Z 1 0 dxx(1 − x) 2 − γE + ln(4π) − ln(m 2_{− x(1 − x)q}2 ) Π2(0) = − 2α π Z 1 0 dxx(1 − x) 2 − γE + ln(4π) − ln(m 2 )

We need to subtract the quantity Π2(0)to remove the −dependence.

Call Π2(q2) = Π2(q2) − Π2(0)and now we calculate this quantity:

Π2(q2) = − 2α π Z 1 0 dx x(1 − x)ln m2 m2_{− x(1 − x)q}2 = 2α π Z 1 0 dx x(1 − x)ln 1 − x(1 − x) q 2 m2

In the limit −q2 _m2_{our expression becomes:}

Π2(q2) ' 2α π Z 1 0 dx x(1 − x)ln x(1 − x) − q 2 m2 + O m 2 q2 = 2α π Z 1 0 dxx(1 − x) ln(x(1 − x)) + ln −q 2 m2 + O m 2 q2

The integral non-depending on x is: Z 1 0 dxx(1 − x)ln −q 2 m2 = ln −q 2 m2 1 6 while for the other term we have to integrate by parts:

Z 1 0 dxx(1 − x)ln(x(1 − x)) = − Z 1 0 dx(x − 1)xln(x(1 − x)) = −  (...) |1₀ − Z 1 0 (1 − 2x)x₃3 − x2 2 (1 − x)x dx  

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other term: u = x − 1 1 6 Z 1 0 (2x − 1)(2x3− 3x2₎ (x − 1)x dx = 1 6 Z 1 0 dxx(2x − 1)(2x − 3) (x − 1) = 1 6 Z 1 0 du(u + 1)(2u − 1)(2u + 1) u = 1 6 Z 1 0 du(u + 1)(4u 2_{− 1)} u = 1 6 Z 1 0 du(4u 3_{− u + 4u}2_{− 1)} u = 1 6 Z 1 0 du 4u2 − 1 + 4u − 1 u = 1 6(4 u3 3 − u + 2u 2_{− ln(u)) |}1 0= −x 6 + 2(x − 1)3 9 + (x − 1)2 3 − 1 6ln | x − 1 | + 1 6 |1 0= − 5 18 Putting our result together:

Π2(q2) ' 2α π Z 1 0 dxx(1 − x) ln(x(1 − x)) + ln −q 2 m2 + O m 2 q2 = 2α π ln − q 2 m2 1 6 − 5 9+ O m2 q2 ⇒ ˆΠ2(q2) = α 3π ln −q 2 m2 − 5 3+ O m2 q2 (2.12) We know that: α(q2) = α 1 − ∆α

where by convention: ∆α = −ReΠ2(q2) = Re[Π2(q2) − Π2(0)]

⇒ α(q2₎ ef f = α 1 −_3πα h ln −_mq22 − 5 3 i = (2.13) α 1 − _3παln −_Amq22 (2.14) where A = e53 (see [9]).

2.1.3 1 loop K(s) calculation

The magnetic moment calculation with strong contributions is made using the dispersion re-lations and the optical theorem. In this section we perform the calculation step by step. As in the QED case we need the vertex correction diagram but now we have a blob in the photon line. The blob is the hadronic contribution at every order but here we will perform the calcu-lation with the 1 loop correction to the photon propagator we obtained in the last section. In

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Figure 2.6: Leading hadronic contribution to aµ.

fig. 2.6 is shown the diagram we’re going to consider. For the vacuum polarization diagram in fig. (2.5), the associated integral reads:

Πµν₂ (k) = −[k2gµν − kµ_kν_]Π(k2₎ _(2.15)

We know from the previous section that iΠµν(k) is the sum of all 1-particle irreducible in-sertion into the photon propagator and the resummation gives: iΠµν_{(k) =} −igµν

k2_[1+Π(k2_)] ' −igµν ₁ k2(1 − Π) = −igµν _k12 − Π

k2 The integral corresponding to the correction vertex

is:

iΓ = −e3 Z

d4xd4yd4zAµ(y)∆1(x − z)ψ(x)γτS(x − y)γµS(y − z)γτψ(z)

= −e3 Z d4xd4yd4zAµ(y) dp4₁ (2π)4 dp4₂ (2π)4e

−ip1(x−y)_e−ip2(y−z)_ψ(x)Λµ_(p

1, p2)ψ(z) = −e3 Z d4y dp 4 1 (2π)4 dp4 2 (2π)4e i(p1−p2)y_A µ(y)ψ(p1)Λµ(p1, p2)ψ(p2) where Λµ(p1, p2) = ∆1(x − z)γτS(x − y)γµS(y − z)γτ = (2.16) Z dk4 (2π)4 1 k2_{+ i} − Π k2_{+ i} γτ ( /p1− /k) + m (p1− k)2− m2 + i γµ ( /p2− /k) + m (p2− k)2− m2+ i γτ (2.17)

We already know the first term in_k12, it corresponds to the QED correction to the magnetic

mo-ment calculation with the photon propagator without corrections and it yields α

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the calculation for the new part we must use the dispersion relation : −Π(k 2₎ k2 = 1 π Z ds s ImΠ(s) 1 k2_{− s}

Putting this expression in the equation (2.16) and exchanging the integrals, we obtain: Λµ_(Π)(p1, p2) = ...+ 1 π Z ds s ImΠ(s) Z dk4 (2π)4 1 k2_{− s + i}γ τ ( /p1− /k) + m (p1− k)2− m2+ i γµ ( /p2− /k) + m (p2− k)2− m2+ i γτ

It looks like we are dealing now with a ”massive” photon. We can write this equation (for the moment consider only the inner integral) in a different way:

Λµ(p1, p2) = 4 Z _dk4 (2π)4 [(p1· p2) − (p1 + p2) · k + k 2 2 ]γ µ_{− mk}µ_{+ (p} 1+ p2− k)µ/k (k2_{− s + i)[k}2_{− 2p} 1k + i][k2− 2p2k + i] (2.18) we can put the numerator in this form by doing a little bit of algebra; from equation (2.17) we can see that Λµ_(p

1, p2)is a symmetric function of p1, p2 that takes the form of :

(2.19) Λµ(p1, p2) = G(q2)γµ+ (p1 + p2)µH(q2)

where qµ_{= (p}

1− p2)µis the momentum transfer. We know that

(2.20) ψ(p1)Λµ(p1, p2)ψ(p2) = ψ(p1)γµψ(p2)F1(q2) − iψ(p1)σµνψ(p2)qνF2(q2)

from the fact that

hp0| Jµ em|pi = F1(q2)u2γµu1+ F2(q2) i 2mqνu2σ µν_u 1

At the tree level the values of the form factors are F1(q2) = 1, F2(q2) = 0. F1(q2)contains UV

and IR divergences. Vacuum polarization, the wave function renormalization for the fermion and the term proportional to F1(q2)define the renormalization coupling constant in the q2 → 0

limit. Hence F1(q2)doesn’t modify the gyromagnetic moment of the particle; instead F2(q2)

is finite and doesn’t contain IR divergences, it determines the correction to the gyromagnetic moment of fermions if it’s calculated for q2 _{= 0}_{(see below). Using equations of the motion}

we can write in momentum space:

(2.21) ψ(p1)γµψ(p2) = 1 2m(p1+ p2) µ_ψ(p 1)ψ(p2) + i 2mψ(p1)σ µν_ψ(p 2)qν

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Going back to equation (2.20): ψ(p1)Λµ(p1, p2)ψ(p2) = 1 2m[(p1+ p2) µ_ψ(p 1)ψ(p2)+ iψ(p1)σµνψ(p2)qν]F1(q2) − iψ(p1)σµνψ(p2)qνF2(q2) = 1 2m(p1+ p2) µ_ψ(p 1)ψ(p2)F1(q2) + i 2mψ(p1)σ µν_ψ(p 2)qν[F1(q2) − 2mF2(q2)] (2.22)

which has to be equal to

ψ(p1)Λµ(p1, p2)ψ(p2) = ψ(p1)[G(q2)γµ+ (p1+ p2)µH(q2)]ψ(p2) = G(q2)ψ(p1)γµψ(p2) + ψ(p1)(p1+ p2)µH(q2)ψ(p2) = G(q2) 1 2m(p1+ p2) µ ψ(p1)ψ(p2) + i 2mψ(p1)σ µν ψ(p2)qν + ψ(p1)(p1+ p2)µH(q2)ψ(p2) = 1 2mG(q 2_{) + H(q}2₎ (p1+ p2)µψ(p1)ψ(p2) + G(q2) i 2mψ(p1)σ µν_ψ(p 2)qν (2.23)

Comparing equation (2.22) with the equation (2.23), we obtain: F1(q2) 2m = 1 2mG(q 2_{) + H(q}2₎ F1(q2) − 2mF2(q2) = G(q2) ⇒ F1(q2) = G(q2) + 2mH(q2) (2.24) F2(q2) = H(q2) (2.25)

To calculate F2(q2 = 0)we have to consider the part of Λµthat doesn’t depend on γµ(see eq

(2.19)): qµ _{= 0}_{implies p}

1 = p2and Λµ(p1, p2) = 2pµ1H(0). With the condition p1 = p2we can

rearrange the denominator of (2.17) in the same form of the equation (2.18) and using the fact that p2

1 = m2 we write the numerator of eq. (2.18) in this form:

Λµ(p1, p2) = 4 Z dk4 (2π)4 [m2− 2p1· k + k 2 2]γ µ_{− mk}µ_{+ (2p} ]1 − k)µ/k (k2_{− s + i)[k}2_{− 2p} 1k + i]2 (2.26)

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The first term is proportional to γµ_{so we can neglect it. The remaining terms are relevant for}

the H(0) calculation. First we have to adjust the denominator using the Feynman parameters: 1 (k2_{− s + i)[k}2 _{− 2p} 1k + i]2 = 2 Z 1 0 dx (1 − x) [(k2_{− s + i)x + (k}2 _{− 2p} 1k + i)(1 − x)]3 = 2 Z 1 0 dx (1 − x) [k2_{− sx − 2p} 1k(1 − x) + i]3 (2.27) We can manipulate the denominator in this way:

[k2− sx − 2p1k(1 − x) + i]3 = [k2− sx − 2p1k(1 − x) + i + p21(1 − x)2−

p2₁(1 − x)2]3 = [(k − p1(1 − x))2− p21(1 − x)2− sx + i]3 =

[k2− m2_{(1 − x)}2_{− sx]}3

in the last step we made the following change of variables: k → k + p1(1 − x). It means a

change also in the numerator: −mkµ_{+ (2p} 1− k)µ/k → −m(kµ+ (1 − x)pµ1) + (2p1− k − (1 − x)p1)µ(/k + (1 − x)/p₁) = −m(1 − x)pµ₁ − mkµ+ 2pµ₁k + 2p/ µ₁(1 − x)/p₁− kµ/k − kµ (1 − x)/p₁− (1 − x)pµ₁k − (1 − x)/ 2pµ₁/p 1 = −m(1 − x)pµ₁ + pµ₁(1 − x2)/p₁

we neglected the linear terms in kµ_{because they don’t contribute and the term k}µ_/_{k =} k2

4 γ µ

because there is the γµ_{−dependence. From ψ(p}

1)/p₁ = mψ(p1) + O(e) we can substitute /p₁

with m. Going back to the Λµ_{equation we have:}

Λµ= 8mpµ₁ Z 1 0 dx(1 − x)2x Z dk4 (2π)4 1 [k2_{− m}2_{(1 − x)}2_{− sx + i]}3

Going in the Euclidean space we can calculate the inner integral: −i Z dk_E4 (2π)4 1 [k2 E + m2(1 − x)2+ sx]3 = −i Γ(2)Γ(1) 2Γ(2)(4π)2_Γ(2)((1 − x) 2 m2+ sx)−1 = −i 32π2_m2 1 (1 − x)2_{+ (} s m2)x

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Λµ_{equation becomes:} Λµ= −i8mp µ 1 32π2_m2 Z 1 0 dx(1 − x)2x 1 (1 − x)2_{+ (} s m2)x Doing the the following change of variables: x → 1 − x we obtain:

(2.28) Λµ= −ip µ 1 4π2_m Z 1 0 dx x 2_{(1 − x)} x2_{+ (} s m2)(1 − x) Remembering that: F2(0) = H(0) = Λµ₍noγ) 2pµ₁ and the 1 π R _ds

sImΠ(s)factor (see equations (2.19)

and (2.25)), we had at the start: ⇒ F2(0) = −i 8π2_m 1 π Z ds s ImΠ(s) Z 1 0 dx x 2_{(1 − x)} x2 _{+ (} s m2)(1 − x) = −i 8π2_m 1 π Z _ds s ImΠ(s)K(s) (2.29)

We call _π1 R ds_sImΠ(s) = Afor the moment. Going back to the integral equation correspond-ing to our Feynman diagram:

(2.30) iΓ = −e3 Z d4y dp 4 1 (2π)4 dp4 2 (2π)4e i(p1−p2)y_A µ(y)ψ(p1)Λµ(p1, p2)ψ(p2)

In the limit where the external field is constant in space and time, it holds Aµ= 1₂yσFσµwith

Fσµ independent from y (see [14] ).

Fµν = ∂µAν − ∂νAµ= ∂µ( 1 2y σ_F σν) − ∂ν( 1 2y σ_F σµ) = 1 2δ σ µFσν − 1 2δ σ νFσµ = 1 2Fµν− 1 2Fνµ= Fµν

Putting equation (2.20) in equation (2.30) and looking only to the spin part: iΓspin = ie3 Z d4y dp 4 1 (2π)4 dp4₂ (2π)4e i(p1−p2)y_A µ(y)ψ(p1)σµνψ(p2)qνF2(q2) = e 3 2 Z d4y dp 4 1 (2π)4 dp4 2 (2π)4 ∂ ∂p1,σ ei(p1−p2)y Fσµψ(p1)σµνψ(p2)qνF2(q2) = e 3 2 Z dp4₁ (2π)4 dp4₂ (2π)4 ∂ ∂p1,σ Z d4yei(p1−p2)y Fσµψ(p1)σµνψ(p2)qνF2(q2)

Remembering that R d4_yei(p1−p2)y = δ4_(p

1− p2), we have a derivative in p1 of the delta, so

we can integrate by parts: iΓspin = − e3 2 Z _dp4 1 (2π)4 dp4 2 (2π)4δ 4_(p 1− p2) ∂ ∂p1,σ Fσµψ(p1)σµνψ(p2)qνF2(q2)

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where qν = (p1− p2)ν. iΓspin = − e3 2 Z dp4₁ (2π)4 dp4₂ (2π)4δ 4_(p 1− p2) Fσµ ∂ψ(p1) ∂p1,σ σµνψ(p2)qνF2(q2) + Fσµψ(p1)σµνψ(p2)F2(q2) = −e 3 2 Z _dp4 1 (2π)4Fσµψ(p1)σ µν_ψ(p 1)δνσF2(0) = e3 2 Z _dp4 1 (2π)4Fµνψ(p1)σ µν ψ(p1)F2(0)

Writing explicitly F2(0)we find:

Γspin = −ie3 2 Z dp4₁ (2π)4Fµνψ(p1)σ µν_ψ(p 1)F2(0) = −e3_A 16π2_m Z dp4₁ (2π)4Fµνψ(p1)σ µν_ψ(p 1)K(s) = − e 2m e2 4π_2πA Z dp4 1 (2π)4Fµνψ(p1)σ µν_ψ(p 1)K(s) = −µ0 2 α πAK(s) Z dp4₁ (2π)4Fµνψ(p1)σ µν_ψ(p 1) where µ0 = _2me and α = e 2

4π. Putting all the contributions together we obtain:

Γspin = − µ0 2 1 + α πAK(s) Z dx4Fµνψ(x)σµνψ(x) (2.31)

Notice that if we put s = 0 we obtain the known result because K(s = 0) = 1₂, thus: Γspin = − µ0 2 1 + α 2π Z dx4Fµνψ(x)σµνψ(x) (2.32)

Using equation (2.31), finally we arrive at: aHLO_µ = Aα πK(s) = (2.33) 1 π Z _ds s ImΠ(s) α π Z 1 0 dx x 2_{(1 − x)} x2_{+ (} s m2)(1 − x) (2.34)

2.2 Spacelike region and ∆α

We summarize the main results of this section:

• The running coupling constant (see eq. (1.29)) and ∆α = −Re(Π(q2_{) − Π(0))}

• The dispersion relation:

Π(q2) − Π(0) = q 2 π Z ∞ 0 ds ImΠ(s) s(s − q2_{− i)} (2.35)

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• From the optical theorem we obtain: ImΠ(s) = e2Im ˆΠ(s) = α 3R(s) = αs 4π|α(s)|2σtot(e +_e− → γ∗ → anything) (2.36) where R(s) is in eq. (2.7). Using these relations we can write for the Rhad(s):

Πhad_ren(q2) = αq 2 3π Z ∞ 4m2 π ds Rhad(s) s(s − q2_{− i)} (2.37)

and ∆α(5)_had= −Πhad_ren(q2) = −αq

2 3π Z ∞ 4m2 π ds Rhad(s) s(s − q2 _{− i)} (2.38)

because for the hadronic case we need to use the experimental cross-section, we can’t use perturbation theory. Πhadren(q2)means the subtraction Πhad(q2) − Πhad(0)and the (5) in ∆α

(5) had

stands for the quark loops but not for the quark top. Dealing with the diagram in fig. 2.6 and from the contribution from the blob ( it has the full propagator ) to g-2, in the last sections we arrived at the formula:

aHLO_µ = Aα πK(s) = 1 π Z ds s ImΠ(s) α π Z 1 0 dx x 2_{(1 − x)} x2_{+ (} s m2)(1 − x) = (2.39) exchanging the integrations

= α π Z 1 0 dx(1 − x) Z ∞ 4m2 π dsImΠ(s) x 2 x2₊ s m2 µ(1 − x) (2.40)

calling −t(x) ≡ _1−xx2 m2_µ the new variable, noticing that _x2₊ sx2 m2µ(1−x)

= −t_s−t1 and using the optical theorem, we can write eq (2.40) in this form:

aHLO_µ = α π Z 1 0 dx(1 − x) Z ∞ 4m2 π dsImΠ(s) x 2 x2₊ s m2 µ(1 − x) = (2.41) α π Z 1 0 dx(1 − x) Z ∞ 4m2 π ds s x2 x2₊ s m2 µ(1 − x) α 3πR(s) = (2.42) −α π Z 1 0 dx(1 − x) Z ∞ 4m2 π ds s t s − t α 3πR(s) = (2.43) −αt 3π Z 1 0 dx(1 − x) Z ∞ 4m2 π ds s R(s) s − t (2.44)

remembering eq. (2.28) we can write eq. (2.44) in terms of ∆α(5)_hadand we arrive at the spacelike expression for the anomalous magnetic moment (see eq. (2.9), (1.28) and (1.30)):

aHLO_µ = α π

Z 1

0

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Chapter 3 Spacelike and timelike determination of

the 2π contribution at a

HLO

_µ

3.1 Timelike and Spacelike a

HLO_µ

calculation

The purpose of this chapter is compare the two methods described earlier for the evaluation of the hadronic leading order contribution to aµ. In particular we will focus on the dominant

π+_π−_{channel whose cross section has been provided by KLOE [15], BaBar[16] and CMD2[17]}

experiments. For each dataset we can summarize our procedure in the following steps: 1. Evaluation of aHLO

µ in the timelike region by an integration of the data points using the

trapezoidal rule.

2. Fit of the bare cross section by a Gounaris-Sakurai parametrization of the Pion form factor. In this way we can extrapolate the cross section outside the range of experimental data.

3. Comparison of the data integration evaluation of the 2π contribution to aHLO

µ (a2πµ )with

the one obtained by the integral of the analytical parametrization.

4. Calculation of ∆αhad_(t)_{in the spacelike region using the analytical parametrization of}

the bare cross section. 5. Calculation of a2π

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3.1.1 The bare σ

0

(e

+

e

−

→ π

+

_π

−

₎

_{cross section}

The total e+_e− _{→ hadrons cross section gives an input through dispersion integral to the}

calculation of the anomalous magnetic moment of the muon. Its main theoretical uncer-tainty comes from the hadronic part of the anomaly, where the value and error are domi-nated by low energy R(s) below √s < 2 GeV and the e+_e− _{→ π}+_π− _{channel gives the}

biggest contribution (73% of aµ, see [11]). The hadronic cross section will be written as

σ0

had,γ(s) ≡ σ0(e+e

− _{→ γ∗ → hadrons + γ) where the subscript 0 denotes the bare cross}

section (undressed of all vacuum polarization effects) and the subscript γ indicates the inclu-sion of effects from final state photon radiation. This undressed and radiation-corrected cross section will be used in the ratio R(s), see eq.(2.7), calculated to the lowest order. The value of R(s) has been measured by many experiments at different energies (see fig. 3.1). At low ener-gies, R(s) is calculated by use of the experimental data and at high energies it can be calculated by perturbative QCD. Matching between the two regions is performed at energies of a few GeV, where both approaches for the determination of R(s) are in fair agreement. For√s ≤ 1.4 GeV, the total hadronic cross section is a sum of about 25 exclusive final states (see [13]). At the present level of precision, a careful treatment of the radiative corrections is required. The radiative corrections include the effects of both initial (ISR) and final state radiation (FSR) and vacuum polarization terms (both leptonic and hadronic). Togheter with Initial state radiation, which is part of next to leading order (NLO) correction, vacuum polarization (VP) effects must be removed from the observed cross sections. Final state radiation of hadrons should be kept because it cannot be separated in an unambiguous way in the measured hadronic cross sec-tions (see [8]). Therefore, while formally of higher order in α, FSR photons have to be taken into account in the definition of the one particle irreducible hadronic blobs and will already be included as part of the leading order hadronic VP contributions (see fig. 3.2). The bare cross section σ(e+_e−_{→ π}+_π−_{(γ)), written in eq. (2,11) can be therefore expressed as:}

σ_ππ(γ)0 = α 2_π 3s β 3_|F π(s)|2 · |1 − Π|2· 1 + α πη(s) (3.1) The |1 − Π|2 _{factor with the polarization operator Π(s) excludes the effect of leptonic and}

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over the whole allowed kinematical region, can be expressed as (see [20]) : η(s) = 1 + β 2 β 4Li2 1 − β 1 + β + 2Li2 −1 − β 1 + β − 3log 2 1 + β × (3.2) × log 1 + β 1 − β − 2 log(β)log 1 + β 1 − β − 3log 4 1 − β2 − 4log(β) + 1 β3× × 5(1 + β 2₎2 4 − 2 log 1 + β 1 + β + 3 2 (1 + β2₎ β2 where Li2(z) = − Rz 0 dx

xln(1 − x). The behaviour of η(s) is shown in fig. (3.2). The real part

of the vacuum polarization operator can be related to the running coupling constant.

σhad(s) = σ_had0 (s) |1 − Π|2 = σ 0 had α(s) α 2 (3.3)

3.2 Timelike evaluation of a

HLO_µ

As we said before, aHLO

µ can be calculated by a combination of experimental cross-section

data, involving e+e-annihilation to hadrons up to certain energy threshold (usually few GeV), and perturbative QCD. aHLO_µ =αmµ 3π 2 Z E 2 cut m2 π0 dsR data γ (s) ˆK(s) s2 + Z ∞ E2 cut dsR pQCD γ (s) ˆK(s) s2 ! (3.4) where K(s) =ˆ 3s m2 µ K(s)

This formula allows to evaluate energy-squared dispersion integral from the π0_γ _{threshold to}

infinity. The kernel occurring in this integral causes the hadronic vacuum polarization contri-butions to be dominated by the low energy domain, as the function K(s) behaves as ' 1_s, and there is an additional1

s suppression. This 1

s2-enhancement of contributions from low energies

together with the existence of the pronounced ρ0 resonance in the π+π− cross–section are

responsible for the fact that pion pair production e+_e− _{→ π}+_π−_{gives the by far largest}

con-tribution to aHLO

µ . About 75% of the lowest order hadronic contribution and 60% of the total

uncertainty-squared are given by the π+_π−_(γ)_{final state (see [21]). The compilation of the}

e+_{e → π}+_π _{data is shown in Fig. 3.3. The relative importance of various regions is illustrated}

in Fig. 3.4. In our work we calculated the 2π contribution to aHLO

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Figure 3.1: R-ratio shown in the range√s ≤ 11.2GeV with resonances showed.

Figure 3.2: The behaviour of the inclusive FSR correction, η(s), for the process e+_e− _{→ π}+_π−_.

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Figure 3.3: Compilation of e+_{e → π}+_π_{data which produce the ρ-resonance. The ρ − ω mixing}

caused by isospin breaking (mumd6= 0) is distorting the ideal Breit-Wigner resonance shape

of the ρ.

Figure 3.4: The distribution of contributions (left) and the square of errors (right) in % for aHLO µ

for different energy regions.

KLOE, BaBar and CMD2. To perform the numerical integration of each data set we used the following formula for K(s):

K(s) = x 2 2 (2 − x 2_{) +} (1 + x2)(1 + x)2 x2 ln(1 + x) − x +x 2 2 + (1 + x) (1 − x)x 2_ln(x) _(3.5)

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EXPERIMENT #of data point RANGE (GeV2₎ _a2π µ

KLOE 60 0.35 ÷ 0.95 (385.1 ± 1.1stat± 2.7sys) × 10−10

CMD2 118 0.14 ÷ 1.90 (493.6 ± 3.4stat± 3.8sys) × 10−10

BaBar 337 0.93 ÷ 8.70 (513.3 ± 2.2stat± 3.1sys) × 10−10

Table 3.1: π+π−contribution to aHLOµ for KLOE, CMD2 and BaBar experiments obtained by

trapezoidal integral data. Systematic errors are taken by ref. [15], [16] and [17].

which can be written in terms of the variables: x = 1 − βµ 1 + βµ , βµ = r 1 − 4m 2 µ s (3.6)

This representation of K(s) is valid for the muon (or electron) where we have s> 4m2 µin the

domain of integration s > 4m2

π, and x is real, and 0 ≤ x ≤ 1. The bare cross sections provided

by the experiments have been then multiplied for K(s) and numerically integrated using the trapezoidal rules. The results are shown in the table (3.1) for the three different experiments and the behaviour of eq. 3.4 (product of the kernel function for the π+π− cross section) as function of the center of mass energy squared is given in figs. 3.5-3.7. The superimposed analytical curve in fig. 3.5-3.7 are obtained by the fit of the cross sections as described in the next section.

3.3 Fit of the pion form factor

As can be seen from equations (1.30)-(2.45) the spacelike evaluation of aHLO

µ for negative

mo-mentum transfer can be obtained as an integral on the hadronic shift to the running of alpha (∆αhad_{(t)). As the dispersion relation (eq. 2.38) relates a single negative t values of ∆α}had_(t)

to an integral of the timelike cross section data, to estimate the 2π contribution to ∆αhad_(t)

in the spacelike domain, the cross section in the whole energy range is necessary. As exper-iments provide cross sections only in limited energy regions, an extrapolation outside these ranges is needed and can be obtained by a parametrization of the data with a well defined analytical function. This is what we will do in the following section.

Timelike and Spacelike evaluation of the leading order hadronic contribution to the muon g-2