Measurement of the running of the fine structure constant below 1 GeV with the KLOE detector

The precision measurement of the $d\sigma(e^+e^- \rightarrow \mu^+\mu^- \gamma)/d\sqrt{s}$ cross section with the photon emitted in the initial state with the KLOE detector has been used to measure the running of the QED coupling constant $\alpha(s)$ in the energy range 0.6 $<\sqrt{s}<$ 0.975 GeV in the time-like region. We were able to achieve a significance of the hadronic contribution to the running of $\alpha(s)$ of more than 5$\sigma$ with a clear contribution of the $\rho - \omega$ resonances to the photon propagator. The real and imaginary part of the shift $\Delta\alpha$ to the running has been estracted and a fit of the real part allowed us to measure the branching fraction $BR(\omega \rightarrow \mu^+\mu^-)$ = (6.6$\pm$1.4$_{stat}$$\pm$1.7$_{syst}$ )$\cdot$10$^{-5}$.


it of the rea
part allowed us to measure the branching fraction BR(ω → µ + µ − ) = (6.6±1.4stat±1.7syst)•10 −5 .

Introduction

Tests of the Standard Model (SM) as well as establishing possible new physics deviations from it require the very precise knowledge of a set of input parameters like the fine structure constant α, the Fermi constant G µ and the Z boson mass M Z .In QED the electromagnetic coupling constant α(s) depends logarithmically on the energy scale due to the vacuum polarization that causes a partial screening of the charge in the low energy limit (Thomson limit) while at higher energy the strenght of the electromagnetic interaction grows.Thus, the classical charge has to be replaced by a "running charge":
e 2 → e 2 (q 2 ) = e 2 1 + (Π ′ γ (q 2 ) − Π ′ γ (0))(1)
where, in the perturbation theory, the lowest order diagram which contributes to Π ′ γ (q 2 ) is the vacuum polarization diagram which describes the virtual creation and reabsorption of fermion pairs: γ * → e + e − , µ

Introduction
Tests of the Standard Model (SM) as well as establishing possible new physics deviations from it require the very precise knowledge of a set of input parameters like the fine structure constant α, the Fermi constant G µ and the Z boson mass M Z .In QED the electromagnetic coupling constant α(s) depends logarithmically on the energy scale due to the vacuum polarization that causes a partial screening of the charge in the low energy limit (Thomson limit) while at higher energy the strenght of the electromagnetic interaction grows.Thus, the classical charge has to be replaced by a "running charge": where, in the perturbation theory, the lowest order diagram which contributes to Π ′ γ (q 2 ) is the vacuum polarization diagram which describes the virtual creation and reabsorption of fermion pairs: γ * → e + e − , µ + µ − , τ + τ − , uū, d d, ... → γ * at the leading order 1 .In terms of the QED coupling constant α = e 2 /4π: n terms of the QED coupling constant α = e 2 /4π:
α(q 2 ) = α (1 − ∆α) ; ∆α = −Re(Π ′ γ (q 2 ) − Π ′ γ (0)). (2)
The various contributions to the shift in the fine structure constant come from the leptons (lep=e,µ and τ The various contributions to the shift in the fine structure constant come from the leptons (lep=e,µ and τ ), the 5 light quarks (u,b,s,c and the corresponding hadrons =had) and from the top quark: ∆α = ∆α lep + ∆ (5) α had + ∆α top + ... .The experimental difficulties in the measurement of the running of the coupling constant are related to the evaluation of the hadronic contribution ∆α had because the low energy contributions of the five light quarks u,d,s,c, and b cannot be reliably calculated using perturbative quantum chromodynamics (p-QCD) due to the non-perturbative behaviour of the strong interaction at low energies; perturbative QCD only allows us to calculate the high energy tail of the hadronic (quark) contributions.In the lower energy region the hadronic contribution can be evaluated through a dispersion relation over the measured e + e − → hadrons cross section.Therefore, it is clear that the dominant uncertainty in the evaluation of ∆α is given by the experimental data accuracy.In the following the measurement of the running of the QED coupling constant in the range 0.6 < √ s < 0.975 GeV in the time-like region will be reported together with the extraction, for the first time, of the real and imaginary part of ∆α.
(u,b,s,c and the corresponding hadrons =had) and from the top quark: ∆α = ∆α lep + ∆ (5) α had + ∆α top + ... .The experimental difficulties in the measurement of the running of the coupling constant are related to the evaluation of the hadronic contribution ∆α had because the low energy contributions of the five light quarks u,d,s,c, and b cannot be reliably calculated using perturbative quantum chromodynamics (p-QCD) due to the non-perturbative behaviour of the strong interaction at low energies; perturbative QCD only allows us to calculate the high energy tail of the hadronic (quark) contributions.In the lower energy region the hadronic contribution can be evaluated through a dispersion relation over the measured e + e − → hadrons cross section.Therefore, it is clear that the dominant uncertainty in the evaluation of ∆α is given by the experimental data accuracy.In the following the measurement of the running of the QED coupling constant in the range 0.6 < √ s < 0.975 GeV in the time-like regi

will be reported
ogether with the extraction, for the first time, of the real and imaginary part of ∆α.


The KLOE detector

Data corresponding to an integrated luminosity of 1.7 fb −1 were collected by the KLOE detector at DAΦNE, the Frascati e + e − collider, which operates at a center of mass energy W = m φ ∼ 1020 MeV .The KLOE detector consists of a large cylindrical drift chamber (DC), surrounded by a fine sampling lead-scintillating fibers electromagnetic calorimeter (EMC) inserted in a 0.52 T magnetic field.The DC 2 , 4 m diameter and 3.3 m long, has full stereo geometry and operates with a gas mixture of 90% helium and 10% isobutane.Momentum resolution is σ(p ⊥ )/p ⊥ ≤ 0.4%.Position resolution in r -φ is 150 µm and σ z ∼ 2mm.Charged tracks vertices are reconstructed with an accuracy of ∼ 3 mm.The EMC 3 is divided into a barrel and two endcaps, for a total of 88 modules and covers 98% of the solid angle.Cells close in time and space are grouped into a calorimeter cluster.The cluster energy E is the sum of the cell energies, while the cluster time t and its position r are energy weighted

The KLOE detector
Data corresponding to an integrated luminosity of 1.7 fb −1 were collected by the KLOE detector at DAΦNE, the Frascati e + e − collider, which operates at a center of mass energy W = m φ ∼ 1020 MeV .The KLOE detector consists of a large cylindrical drift chamber (DC), surrounded by a fine sampling lead-scintillating fibers electromagnetic calorimeter (EMC) inserted in a 0.52 T magnetic field.The DC 2 , 4 m diameter and 3.3 m long, has full stereo geometry and operates with a gas mixture of 90% helium and 10% isobutane.Momentum resolution is σ(p ⊥ )/p ⊥ ≤ 0.4%.Position resolution in r -φ is 150 µm and σ z ∼ 2mm.Charged tracks vertices are reconstructed with an accuracy of ∼ 3 mm.The EMC 3 is divided into a barrel and two endcaps, for a total of 88 modules and covers 98% of the solid angle.Cells close in time and space are grouped into a calorimeter cluster.The cluster energy E is the sum of the cell energies, while the cluster time t and its position r are energy weighted averages.The respective resolutions are σ E /E = 5.7%/ E(GeV ) and σ t = 57ps/ E (GeV ) ⊕ 100 ps.

verages.The respective resolutio
s are σ E /E = 5.7%/ E(GeV ) and σ t = 57ps/ E (GeV ) ⊕ 100 ps.


Measurement of the running of α

The running of α(s) has been obtained from the ratio between the precise measurement of the Initial State Radiation (ISR) process e + e − → µ + µ − γ and the Monte Carlo (MC)

Measurement of the running of α
The running of α(s) has been obtained from the ratio between the precise measurement of the Initial State Radiation (ISR) process e + e − → µ + µ − γ and the Monte Carlo (MC) simulation without the vacuum polarization (VP) contribution, in other words, setting α(s) = α(0): = dσ data (e + e − → µ + µ − γ(γ))| ISR /d √ s dσ 0 M C (e + e − → µ + µ − γ(γ))| ISR /d √ s .(3)
The sample of µµγ events is selected requiring a photon and two tracks of opposite curvature; the photon is emitted at small angle (SA),i.e.within a cone of θ γ < 15 • around the beamline and the two charged muons are emitted at large polar angle, 50 • < θ µ < 130 • 4 .The experimental ISR µ + µ − γ cross section is ob The sample of µµγ events is selected requiring a photon and two tracks of opposite curvature; the photon is emitted at small angle (SA),i.e.within a cone of θ γ < 15 • around the beamline and the two charged muons are emitted at large polar angle, 50 • < θ µ < 130 • 4 .The experimental ISR µ + µ − γ cross section is obtained from the observed number of µµγ events (N obs ) and the background estimate (N bckg ) as: as:
dσ(e + e − → µ + µ − γ(γ)) d √ s ISR = N obs − N bkg ∆ √ s • (1 − δ F SR ) ǫ( √ s) • L ,(4)
where (1 − δ F SR ) is the correction applied to remove the Final State Radiaton (FSR) contribution, ǫ is global the efficiency, and L is the integrated luminosity.To separate the electrons from the pions or muons we used a particle identification estimator (L±), based on a pseudo-likelihood function using time-of-flight and calorimeter information (size and shape of the energy deposit).The muons were distiguished from the pions essentially by means of two selection cuts: the first one on the M T RK (M T RK <115 MeV) that is a variable computed requiring the energy and momentum conservation and the second on the σ M T RK that is constructed event by event with the error matrix of the fitted tracks at the point of closest approach (PCA).Cutting the high values of σ M T RK the bad reconstructed tracks are rejected allowing a reduction of the ππγ events contamination.The residual background is estimated by fitting the observed M T RK spectrum with a superposition of MC simulation distributions describing signal and π + π − γ, π + π − π 0 and e + e − γ events.Additional background from the e + e − → e + e − µ + µ − process has been evaluated using the NEXTCALIBUR MC generator.The maximum contribution is 0.7% at √ s=0.6 GeV.The contribution from e + e − → e + e − π + π − has been evaluated with the EKHARA generator and found to be negligible 4 .The measured µ + µ − γ cross-section with only ISR is then compared with the corresponding NLO QED calculation from PHOKHARA generator including the VP effects.The agreement between the two cross sections is excellent; the average ratio, using only the statistical errors, is 1.0006±0.0008.By setting in the MC α(s) = α(0), the hadronic contribution to the photon propagator, with its characteristic ρ − ω i where (1 − δ F SR ) is the correction applied to remove the Final State Radiaton (FSR) contribution, ǫ is global the efficiency, and L is the integrated luminosity.To separate the electrons from the pions or muons we used a particle identification estimator (L±), based on a pseudo-likelihood function using time-of-flight and calorimeter information (size and shape of the energy deposit).The muons were distiguished from the pions essentially by means of two selection cuts: the first one on the M T RK (M T RK <115 MeV) that is a variable computed requiring the energy and momentum conservation and the second on the σ M T RK that is constructed event by event with the error matrix of the fitted tracks at the point of closest approach (PCA).Cutting the high values of σ M T RK the bad reconstructed tracks are rejected allowing a reduction of the ππγ events contamination.The residual background is estimated by fitting the observed M T RK spectrum with a superposition of MC simulation distributions describing signal and π + π − γ, π + π − π 0 and e + e − γ events.Additional background from the e + e − → e + e − µ + µ − process has been evaluated using the NEXTCALIBUR MC generator.The maximum contribution is 0.7% at √ s=0.6 GeV.The contribution from e + e − → e + e − π + π − has been evaluated with the EKHARA generator and found to be negligible 4 .The measured µ + µ − γ cross-section with only ISR is then compared with the corresponding NLO QED calculation from PHOKHARA generator including the VP effects.The agreement between the two cross sections is excellent; the average ratio, using only the statistical errors, is 1.0006±0.0008.By setting in the MC α(s) = α(0), the hadronic contribution to the photon propagator, with its characteristic ρ − ω interference structure, is clearly visible in the data to MC ratio, as shown in Fig. 1.

erference structure, is clearly visible in the data to MC ra
io, as shown in Fig. 1.
Energy

Extraction of Real and Imaginary part of ∆α and fit of Re∆α

Since the VP function Π(q 2 ) is complex, both ∆α and α(q 2 ) are complex quantities.Although usually the real part of Π(q 2 ) is considered, which makes the effective coupling α(q 2 ) real, this approximation is not sufficient in presence of resonances, like the ρ.In this case the imaginary part become non-negligible and s

Extraction of Real and Imaginary part of ∆α and fit of Re∆α
Since the VP function Π(q 2 ) is complex, both ∆α and α(q 2 ) are complex quantities.Although usually the real part of Π(q 2 ) is considered, which makes the effective coupling α(q 2 ) real, this approximation is not sufficient in presence of resonances, like the ρ.In this case the imaginary part become non-negligible and should be taken into account.To evaluate the real part of ∆α we used this simple relation: he real part of ∆α we used this simple relation:
Re ∆α = 1 − |α(0)/α(s)| 2 − (Im ∆α) 2(5)
defined in terms of the measured quantity |α(s)/α(0)| 2 and of the imaginary part that has been evaluated considering that for the optical theorem it can be related to the total cross section σ(e + e − → γ defined in terms of the measured quantity |α(s)/α(0)| 2 and of the imaginary part that has been evaluated considering that for the optical theorem it can be related to the total cross section σ(e + e − → γ * → anything) ("anything" means any possible state), where the precise relation reads 1 : ), where the precise relation reads 1 :
Im∆α = − α 3 R(s) with R(s) = σ tot / 4πα(s) 2
3s .R(s) takes into account the leptonic an 3s .R(s) takes into account the leptonic and hadronic contributions R(s) = R lep (s) + R had (s), where the leptonic part is given by: R lep (s) = 1 − onic part is given by: R lep (s) = 1 −
4m 2 l s 1 + 2m 2 l s
, (l = e, µ, τ ) while for the evaluation of the hadronic part we use only the 2π hadronic contribution measured by KLOE 5 which dominates in this region:
R had (s) = 1 4 1 − 4m 2 π s 3 2 |F 0 π (s)| 2(6)
where the Pion Form Factor must be deconvoluted by the VP effects:  (the exp data are the red points) compared with the , (l = e, µ, τ ) while for the evaluation of the hadronic part we use only the 2π hadronic contribution measured by KLOE 5 which dominates in this region: where the Pion Form Factor must be deconvoluted by the VP effects:  (the exp data are the red points) compared with the values given by the R had (s) compilaton of Ref. 6 (blue solid line).The real part is shown on the right.
values given by the R had (s) compilaton of Ref. 6 (blue solid line).The real part is shown on the right.
|F 0 π (s)| 2 = |F π (s)| 2 | α(0) α(s) | 2 .
The Re∆α has been fitted by a sum of the leptonic and hadronic contributions, where the hadronic The Re∆α has been fitted by a sum of the leptonic and hadronic contributions, where the hadronic contribution is parametrized as a sum of ρ(770), ω(782) and φ(1020) resonances components and a non resonant term (param.by a first-order polynomial).
ontribution is parametrized as a sum of ρ(770), ω(782) and φ(1020 For the ω and φ resonances a Breit-Wigner description was used 4 resonances a Breit-Wigner description was used 4
Re ∆α V =ω,φ = 3 BR(V → e + e − ) • BR(V → µ + µ − ) αM V s(s − M 2 V )Γ V (s − M 2 V ) 2 + sΓ 2 V(7)
where M V and Γ V are the mass and the total width of the mesons V = ω and φ while for the ρ we use a Gounaris-Sakurai parametrizatio where M V and Γ V are the mass and the total width of the mesons V = ω and φ while for the ρ we use a Gounaris-Sakurai parametrization BW GS

Conclusions
usions

We present the first precision measurement of the running of α(s) in the energy region 0.6 < √ s < 0.975 and the strongest direct evidence of the hadronic contribution to α(s) achieved in both time-and space-like regions by a single experiment.For the first time also the real and imaginary part of ∆α(s) have been extracted showing clearly the importance of the role of the imaginary part.By fitting the real part of ∆α(s) and assumming the lepton universality, the branching fraction BR(ω → µ + µ − ) = (6.6±1.4 stat ±1.7 syst )•10 −5 has also been obta We present the first precision measurement of the running of α(s) in the energy region 0.6 < √ s < 0.975 and the strongest direct evidence of the hadronic contribution to α(s) achieved in both time-and space-like regions by a single experiment.For the first time also the real and imaginary part of ∆α(s) have been extracted showing clearly the importance of the role of the imaginary part.By fitting the real part of ∆α(s) and assumming the lepton universality, the branching fraction BR(ω → µ + µ − ) = (6.6±1.4 stat ±1.7 syst )•10 −5 has also been obtained.

Figure 1 -
Figure1-The square of the modulus of the running α(s) in units of α(0) compared with the prediction (provided by the alphaQED package) as a function of the dimuon invariant mass.The red points are the KLOE data with statistical errors; the violet points are the theoretical prediction for a fixed coupling (α(s) = α(0)); the yellow points are the prediction with only virtual lepton pairs contributing to the shift ∆α(s) = ∆α(s) lep , and finally the points with the solid line are the full QED prediction with both lepton and quark pairs contributing to the shift ∆α(s) = ∆α(s) lep+had .

Figure 2 -
Figure 2 -Left: Im ∆α extracted from the KLOE data compared with the values provided by alphaQED routine (without the KLOE data) for Im ∆α = Im ∆α lep (yellow points) and Im ∆α = Im ∆α lep+had only for ππ channels (blue solid line).Right: Re ∆α extracted from the experimental data with only the statistical error included compared with the alphaQED prediction (without the KLOE data) when Re ∆α = Re ∆α lep (yellow points) and Re ∆α = Re ∆α lep+had (blue solid line).

ming
The results obtained for the imaginary part of ∆α(s) (Im ∆α) are shown in left panel of the Fig.2