PPrepared for submission to JHEP
Direct measurement of α QED ( m ) at the FCC-ee Patrick Janot
CERN, PH Department, Geneva, Switzerland
Abstract:
When the measurements from the FCC-ee become available, an improveddetermination of the standard-model "input" parameters will be needed to fully exploitthe new precision data towards either constraining or fitting the parameters of beyond-the-standard-model theories. Among these input parameters is the electromagnetic couplingconstant estimated at the Z mass scale, α QED ( m ) . The measurement of the muon forward-backward asymmetry at the FCC-ee, just below and just above the Z pole, can be usedto make a direct determination of α QED ( m ) with an accuracy deemed adequate for anoptimal use of the FCC-ee precision data. a r X i v : . [ h e p - ph ] N ov ontents The design study of the Future Circular Colliders (FCC) in a 100-km ring in the Genevaarea has started at CERN at the beginning of 2014, as an option for post-LHC particle ac-celerators. The study has an emphasis on proton-proton and electron-positron high-energyfrontier machines [1]. In the current plans, the first step of the FCC physics programmewould exploit a high-luminosity e + e − collider called FCC-ee, with centre-of-mass energiesranging from the Z pole to the t¯t threshold and beyond. A first look at the physics case ofthe FCC-ee can be found in Ref. [2].In this first look, an estimate of the achievable precision on a number of Z-pole observ-ables was inferred and used in a global electroweak fit to set constraints on weakly-coupled– 1 –ew physics up to a scale of 100 TeV [3]. These constraints were obtained under two assump-tions: (i) the precision of the pertaining theoretical calculations will match the expectedexperimental accuracy by the time of the FCC-ee startup; and (ii) the determination ofstandard-model input parameters – four masses: m Z , m W , m top , m Higgs ; and three couplingconstants: α s ( m ) , G F , α QED ( m ) – will improve in order not to be the limiting factorsto the constraining power of the fit. The determinations of the Higgs boson mass from theLHC data [4] and of the Fermi constant from the muon lifetime measurement [5] are al-ready sufficient for this purpose. It is argued in Refs. [2, 6] that the FCC-ee can adequatelyimprove the determination of the other three masses and of the strong coupling constantby one order of magnitude or more: the experimental precision targets for the FCC-ee are100 keV for the Z-boson mass, 500 keV for the W-boson mass, 10 MeV for the top-quarkmass, and 0.0001 for the strong coupling constant. (The FCC-ee also aims at reducing theHiggs boson mass uncertainty down to 8 MeV.)No mention was made, however, of a way to improve the determination of the elec-tromagnetic coupling constant evaluated at the Z mass, and it was simply assumed that afactor improvement with respect to today’s uncertainty – down to × − – could beachieved by the time of the FCC-ee startup. Today, α QED ( m ) is determined from α QED (0) (itself known with an accuracy of − ) with the running coupling constant formula: α QED ( m ) = α QED (0)1 − ∆ α ‘ ( m ) − ∆ α (5)had ( m ) . (1.1)Its uncertainty is dominated by the experimental determination of the hadronic vacuumpolarization, ∆ α (5)had ( m ) , obtained from the dispersion integral: ∆ α (5)had ( m ) = αm π Z ∞ m π R γ ( s ) s ( m − s ) ds, (1.2)where R γ ( s ) is the hadronic cross section σ (e + e − → γ ∗ → hadrons) at a given centre-of-mass energy √ s , normalized to the muon pair cross section at the same centre-of-massenergy. At small values of √ s , typically up to 5 GeV, and in the Υ resonance regionfrom 9.6 to 13 GeV, the evaluation of the dispersion integral relies on the measurementsmade with low-energy e + e − data accumulated by the KLOE, CMD-2/SND, BaBar, Belle,CLEO and BES experiments. The most recent re-evaluation [7, 8] gives ∆ α (5)had ( m ) =(275 . ± . × − , which leads to α − ( m ) = 128 . ± . , (1.3)corresponding to a relative uncertainty on the electromagnetic coupling constant, ∆ α/α ,of . × − . It is hoped that future low-energy e + e − data collected by the BES III andVEPP-2000 colliders will improve this figure to × − or better [9].In this study, it is shown that the FCC-ee can provide another way of determiningthe electromagnetic coupling constant with a similar or better accuracy, from the precisemeasurement of muon forward-backward asymmetry, A µµ FB , just above and just below theZ peak, as part of the resonance scan. This method does not rely on the experimental– 2 –etermination of the vacuum polarization ∆ α (5)had . Here, the point is not to extrapolate α QED ( m ) from α QED (0) , but to provide a direct evaluation of α QED at √ s ’ m Z , hencewith totally different theoretical and experimental uncertainties. This measurement wouldin turn be combined with other determinations for an even smaller uncertainty.This letter is organized as follows. In Section 2, the reasons for the choice of A µµ FB asan observable sensitive to α QED are given, and the sensitivity is determined as a functionof the centre-of-mass energy. The optimal centre-of-mass energies, as well as the integratedluminosities and running time needed to achieve a statistical uncertainty of a few − aredetermined in Section 3. Possible systematic uncertainties are discussed and evaluated inSection 4. At the FCC-ee, the muon pair production proceeds via the graph depicted in Fig. 1 througheither a Z or a γ exchange. - e + e , Z g - m + m Figure 1 . Tree-level Feynmann graph for µ + µ − production at the FCC-ee At tree level, the cross section σ µµ therefore contains three terms: (i) the γ -exchangeterm squared, proportional to α ( s ) ; (ii) the Z-exchange term squared, proportional to G (where G F is the Fermi constant); and (iii) the γ -Z interference term, proportionalto α QED ( s ) × G F . These three terms are denoted G , Z , and I in the following. Theirexpressions as a function of the centre-of-mass energy √ s can be found in Ref. [10] andreported below. G = c γ s , (2.1) Z = c ( v + a ) × s ( s − m ) + m Γ , (2.2) I = 2 c γ c Z v × ( s − m )( s − m ) + m Γ , (2.3)– 3 –ith the following definitions: c γ = r π α QED ( s ) , c Z = r π m π G F √ , a = − , v = a × (1 − θ W ) , (2.4)and where θ W is the effective Weinberg angle ( sin θ W ’ . ).An absolute measurement of the µ + µ − production cross section σ µµ = Z + I + G istherefore a priori sensitive to α QED through the interference term and the γ -exchange term.The cross section and the three contributing terms are displayed in Fig. 2 as a function of thecentre-of-mass energy √ s , with the inclusion of initial state radiation (ISR). In this figure,the effective collision energy after ISR, denoted √ s , is required to satisfy s > . s . Theimportance of such a requirement on s , together with the way to control it experimentally,is discussed in Section 4.3.2.At a given √ s , a small variation ∆ α of the electromagnetic coupling constant translatesto a variation ∆ σ µµ of the cross section : ∆ σ µµ = ∆ αα ( I + 2 G ) . (2.5)As is well visible in Fig. 2, the interference term can be neglected in the above equation.As a consequence, if the cross section can be measured with a precision ∆ σ µµ , the relativeprecision on the electromagnetic coupling constant amounts to ∆ αα ’ ∆ σ µµ G ’
12 ∆ σ µµ σ µµ (cid:18) ZG (cid:19) . (2.6)The target statistical precision of × − on α QED can therefore be achieved withmore than µ + µ − events and at centre-of-mass energy where the Z contribution to thecross section is much smaller than the photon contribution. These two conditions call fora centre-of-mass energy smaller than 70 GeV, where the cross section is both large anddominated by the photon contribution. Beside the fact that this centre-of-mass energy isnot in the current core programme of the FCC-ee and that the needed integrated luminosityof
50 ab − would require at least a year of running at this energy in the most favourableconditions, the measurement itself poses a number of intrinsic difficulties. Indeed, theabsolute measurement of a cross section with a precision of a few − requires the selectionefficiency, the detector acceptance, and the integrated luminosity to be known with thisprecision or better. Even if not impossible to meet, these requirements are exceedinglychallenging in the extraction of α QED from this method with the needed precision.The muon forward-backward asymmetry, A µµ FB , defined as A µµ FB = σ F µµ − σ B µµ σ F µµ + σ B µµ , (2.7)where σ F(B) µµ is the µ + µ − cross section for events with the µ − direction in the forward(backward) hemisphere with respect to the e − -beam direction, hence with σ F µµ + σ B µµ = σ µµ ,solves most of these obstacles. Indeed, it is a self-normalized quantity, which thus does not– 4 – (GeV) s50 60 70 80 90 100 110 120 130 140 150 C r o ss s e c t i on ( ab ) Total γ exchangeZ exchangeZ γ interf . Figure 2 . Cross section for the e + e − → µ + µ − process (red curve) and the three contributions,calculated from the analytical expressions of Ref.[10]: pure γ -exchange term (blue curve); pureZ-exchange term (green curve); and the absolute value of the γ -Z interference term (black curve).The initial-state radiation is included, and s /s is required to exceed 0.99. need the measurement of the integrated luminosity. Moreover, most uncertainties on theselection efficiency and the detector acceptance simply cancel in the ratio. This observableis therefore a good candidate for a measurement with an exquisite precision.At lowest order, and if the terms proportional to m µ /m ∼ − are neglected, theangular distribution of the µ − from the e + e − → µ + µ − production can be written in thefollowing way [11]: dσ µµ d cos θ ( s ) ∝ G ( s ) × (1 + cos θ ) + G ( s ) × θ, (2.8)where G ( s ) and G ( s ) can be expressed as a function of G , Z and I as follows: G ( s ) = G + I + Z and G ( s ) = a v (cid:26) I + 4 v /a (1 + v /a ) Z (cid:27) . (2.9)After integration over the muon polar angle θ , the forward-backward asymmetry thereforeamounts to: A µµ FB ( s ) = 34 G ( s ) G ( s ) . (2.10)– 5 –he variation of A µµ FB as a function of the centre-of-mass energy, as obtained fromEq. 2.10, is shown in Fig. 3. In the above expressions, the photon-exchange term is totallysymmetric, hence is absent from the numerator. Because v /a ’ × − , the Z-exchangeterm contribution to the asymmetry is minute, except at the Z pole where the interferenceterm vanishes and the asymmetry is small: A µµ FB , = (3 / × v a / ( a + v ) ’ . .The interference term, on the other hand, is almost 100% anti-symmetric and contributesmostly to the numerator. (The contribution of the interference term to the denominator, i.e. , to the total cross section, can be neglected as shown in Fig. 2.) (GeV) s50 60 70 80 90 100 110 120 130 140 150 µµ F B A -1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0 Figure 3 . The muon forward-backward asymmetry in e + e − → µ + µ − as a function of the centre-of-mass energy. The off-peak muon forward-backward asymmetry can therefore be expressed as follows: A µµ FB = A µµ FB , + 34 a v IG + Z . (2.11)At a given √ s , a small variation ∆ α of the electromagnetic coupling constant translates toa variation ∆ A µµF B of the muon forward-backward asymmetry: ∆ A µµ FB = ∆ αα × a v I ( Z − G )( G + Z ) = (cid:16) A µµ FB − A µµ FB , (cid:17) × Z − GZ + G × ∆ αα . (2.12)In first approximation, the asymmetry is therefore not sensitive to α QED when the Z-and photon-exchange terms are equal, i.e. , at √ s = 78 and GeV (Fig. 2), where the– 6 –symmetry is maximal (Fig. 3). Similarly, the sensitivity to the electromagnetic couplingconstant vanishes in the immediate vicinity of the Z pole. The red curve of Fig. 4 shows thevariation of A µµ FB for a relative change of α QED by +1 . × − , as a function of √ s . In otherwords, the red curves displays the absolute precision with which A µµ FB must be measured tostart improving the accuracy on α QED ( m ) with respect to today’s determination. (GeV) s50 60 70 80 90 100 110 120 130 140 150 µµ F B A (cid:54) -0.08-0.06-0.04-0.020.000.020.040.060.08 -3 × -4 × = +1.1 (cid:95) / (cid:95)(cid:54) With Statistical uncertainty
Figure 4 . The red curve shows the variation of the muon forward-backward asymmetry as afunction of √ s for a relative change of α QED ( s ) by +1 . × − . The asymmetry has no sensitivityto α QED when the red curve crosses the black horizontal line. The blue area represents the absolutestatisitical uncertainty with which the muon forward-backward asymmetry can be measured at theFCC-ee in one year of data taking at any given centre-of-mass energy.
For a positive variation of ∆ α , the sign of ∆ A µµ FB , i.e. , the sign of (cid:16) A µµ FB − A µµ FB , (cid:17) × ( Z − G ) , changes at each of these centre-of mass energies: it is positive below 78 GeV, wherethe asymmetry is negative and the Z contribution is smaller than the photon contribution,becomes negative between 78 GeV and the Z pole, where the Z contribution dominates,then positive again from the Z pole all the way to 112 GeV because the asymmetry becomespositive, and negative for larger centre-of-mass energies where the photon contribution takesover. This interesting property, in particular the sign change around the Z pole, is fullyexploited in Section 4. Written the other way around and in a perhaps more useful mannerfor the following, the relative precision on the electromagnetic coupling constant amounts– 7 –o ∆ αα = ∆ A µµ FB A µµ FB − A µµ FB , × Z + GZ − G ’ ∆ A µµ FB A µµ FB × Z + GZ − G , (2.13)where the approximation in the last term of the equality is valid off the Z peak. The optimal centre-of-mass energies are those which minimize the statistical uncertainty on α QED ( s ) . For a given integrated luminosity L , the statistical uncertainty on the forward-backward asymmetry amounts to σ (cid:0) A µµ FB (cid:1) = s − A µµ FB2 L σ µµ . (3.1)The target luminosities for the FCC-ee in a configuration with four interaction points are × cm − s − per interaction point at the Z pole and × cm − s − per interactionpoint at the WW pair production threshold [12]. With effective seconds per year,the total integrated luminosity is therefore expected to be
86 ab − / year at the Z poleand . − / year at the WW threshold. Between these two points, the variation ofthe luminosity with the centre-of-mass energy is assumed to follow a simple power law: L ( √ s ) = L ( m Z ) × s a . The very large Z pole luminosity is achieved by colliding about60,000 bunches of electrons and positrons, which fill the entirety of the 400 MHz RF bucketsavailable over 100 km. It also corresponds to a time between two bunch crossings of 5 ns,which is close to the minimum value acceptable today for the experiments. With a constantnumber of bunches, the luminosity was therefore conservatively assumed to linearly decreasewith the centre-of-mass energy (and reach . for √ s = 0 . ), leading to the profile of Fig. 5.With the cross section of Fig. 2, the asymmetry of Fig. 3, and the integrated luminosityof Fig. 5, Eq. 3.1 leads to the statistical uncertainty on A µµ FB displayed as the blue area inFig. 4, for a one-year running at any given centre-of-mass energy. An improvement on thedetermination of α QED ( s ) is possible wherever the red curve lies outside the blue area, andis largest when the absolute value of the ratio between the red and blue curves is maximum.The corresponding relative accuracy for the α QED ( s ) determination is shown in Fig. 6.The best accuracy of ∼ × − is obtained for one year of running either just below orjust above the Z pole, specifically at √ s − ∼ . GeV and √ s + ∼ . GeV.The value of the electromagnetic coupling constant extracted from the muon forward-backward asymmetry measured at either energy, α − ≡ α QED ( s − ) and α + ≡ α QED ( s + ) , arethen extrapolated towards a determination of α ≡ α QED ( m ) with the running couplingconstant expression around the Z pole, valid at all orders in the leading-log approximation: α = 1 α ± + β log s ± m , (3.2)where β is proportional to the well-known QED β -function. In the standard model and atthe lowest QED/QCD order, it reads β = P f Q / π , where the sum runs over all activefermions at the Z pole (f = e, µ , τ , d, u, s, c b) and Q f is the fermion electric charge in– 8 – (GeV) s50 60 70 80 90 100 110 120 130 140 150 ) - I n t eg r a t ed Lu m i / Y ea r ( ab FCC-ee integrated luminosity / year (4 IP)
Figure 5 . Target integrated luminosities for the FCC-ee, in a scheme with four interaction points,for centre-of-mass energies between 50 and 150 GeV. unit of e . The standard model extrapolation correction from α ± to α therefore amounts to − . from the measurement below the Z pole, and +0 . from the measurement abovethe Z pole, corresponding to a relative correction of ± . × − in both cases, i.e. , an orderof magnitude larger than the targeted uncertainty on α . While this correction is knownwith an excellent precision in the standard model – the QED β -function is now known withQED corrections up to five loops and QCD corrections up to four loops [13, 14] –, it iscertainly preferable to remove this model dependence (and the residual theory uncertainty)from the determination of α .The dual measurements of α − and of α + solve this issue and yields the straightforwardcombination: α = 12 (cid:18) − ξα − + 1 + ξα + (cid:19) , where ξ = log s − s + /m log s − /s + ’ . , (3.3)without any model dependence related to the running of the electromagnetic constant. Thiscombination of a measurement below the Z peak and a measurement above the Z peak hasother advantages, the most important of which is the cancellation to a large extent of manysystematic uncertainties, as explained in the next section. With this weighted average, thetargeted precision of × − can be obtained from one year at . GeV and one year– 9 – (GeV) s50 60 70 80 90 100 110 120 130 140 150 (cid:95) ) / (cid:95) ( (cid:109) -5 -4 at FCC-ee µµ FB accuracy from A QED (cid:95)
Figure 6 . Relative statistical uncertainty for the α QED ( s ) determination from a measurementof the muon forward-backward asymmetry at the FCC-ee, with a one-year running at any givencentre-of-mass energy. The best accuracy is obtained for one year of running either just below orjust above the Z pole. at . GeV with the sole measurement of the muon forward-backward asymmetry. Therunning time at each energy can be reduced to six months – as is assumed in the following– if additional measurements are considered, e.g. , the tau forward-backward asymmetryand, possibly, the electron forward-backward asymmetry.
For the forward-backward asymmetry measurement to be relevant in the determination ofthe electromagnetic coupling constant, all possible systematic uncertainties must be keptwell below the statistical uncertainty aimed at in Section 3. Systematic uncertainties maybe of experimental, parametric, and theoretical origin, and are studied in turn below.
Circular e + e − colliders have the unique feature of providing the possibility to measure thebeam energy with the resonant depolarization method [15] with an outstanding accuracy.– 10 –t the FCC-ee [16], this accuracy has been estimated [2, 17] to be of the order of 50 keVaround the Z pole, of which 45 (23) keV are (un)correlated between all energy points,corresponding to a total relative uncertainty of − . The derivative of the muon forward-backward asymmetry with respect to the centre-of-mass energy, however, is largest aroundthe Z pole, as can be seen from Fig. 3. It is therefore important to check that this expectedprecision is indeed sufficient.At √ s − and √ s + , the photon contribution is only 5% of the total cross section andvaries slowly with the centre-of-mass energy around the Z pole (Fig. 2): this contributioncan be considered as a second order effect in the uncertainty evaluation. Equations 2.11and 2.13 therefore simplify to A µµ FB ( s ± ) ’ a v IZ and ∆ α ± α ± ’ ∆ A µµ FB A µµ FB . (4.1)The dependence of I and Z on s and m Z is given at the beginning of Section 2. Theforward-backward asymmetry dependence on s and m Z in the vicinity of the Z pole issimply A µµ FB ( s, m Z ) ∝ ( s − m ) / ( sm ) . (4.2)The uncertainties on √ s and m Z both amount to 95 keV, are dominated by the un-certainty of the beam energy measurement, and are largely correlated as indicated above.The uncorrelated variables are therefore the difference D = √ s − m Z and the average Σ = ( √ s + m Z ) / . , with uncertainties of σ D = 46 keV and σ Σ = 94 keV, respectively. Astraightforward error propagation yields σ ( A µµ FB ) A µµ FB ’ √ sm Z r(cid:0) s + m − √ sm Z (cid:1) σ D D + (cid:0) s + m + √ sm Z (cid:1) σ Σ , (4.3)which in turn simplifies to, at √ s ± , σ ( α ± ) α ± ’ σ D ± D ± , with D ± = √ s ± − m Z , (4.4)after neglecting the much smaller term proportional to ( σ Σ / Σ) . Numerically, the relativeuncertainties on α ± , or equivalently on /α ± , arising from the beam energy measurementboth amount to . × − and are uncorrelated. The uncertainty on the coefficient ξ ( ± . ) was found to have a totally negligible contribution ( ± × − ) to the relativeuncertainty on α . Only the (uncorrelated) errors on α − and α + contribute. As a conse-quence, the relative uncertainty on α QED ( m ) arising from the beam energy measurementamounts to σ ( α ) α ’ s (1 − ξ ) σ ( α − ) α − + (1 + ξ ) σ ( α + ) α ’ × − . (4.5) At the FCC-ee, the relative beam energy spread δ for centre-of-mass energies around theZ pole is expected [12] to be of the order of 0.12%, i.e. , two orders of magnitude larger– 11 –han the accuracy of the (average) beam energy measurement. The relative centre-of-massenergy spread δ is √ times smaller, i.e. , of the order of 0.08%. The shift ∆ A µµ FB between thepredicted asymmetry and its measured value at a centre-of-mass energy √ s ± is therefore ∆ A µµ FB ( s ± ) = 1 √ πs ± δ Z A µµ FB ( s ) exp − (cid:0) √ s − √ s ± (cid:1) s ± δ d √ s − A µµ FB ( s ± ) , (4.6)which yields, with the functional form of A µµ FB ( s ) given in Eq. 4.2 expanded around s ± : ∆ A µµ FB A µµ FB ( s ± ) ’ m s ± − m δ , (4.7) i.e. , numerically ∆ A FB A FB ( s − ) = − . × − and ∆ A FB A FB ( s + ) = +3 . × − , (4.8)under the reasonable assumption that the beam energy spread values are similar at √ s ± and m Z .The relative changes of A FB ( s ± ) are of the order of the statistical uncertainty, andlarger than the uncertainty originating from the beam energy measurement. These changesare, however, of opposite sign, and lead to a remarkable cancellation by more than one orderof magnitude in the determination of α . Indeed, the combination of Eqs. 2.13 and 3.3 leadsto the following estimate of the bias on α : ∆ α α ’ .
528 ∆ A FB A FB ( s − ) + 0 .
563 ∆ A FB A FB ( s + ) ’ +1 . × − . (4.9)The uncertainty on this small bias (which is to be corrected for) depends on the accuracywith which the beam energy spread in known. For example, the measurement of bunchlength from the distribution of the µ + µ − event primary vertices determined directly bythe FCC-ee experiments would allow a precise determination of the beam energy spread.A precision of 2.5% could be reached with this method at LEP [18], yielding a negligibleuncertainty on the α QED ( m ) determination. In Eq. 2.10, the asymmetry is determined under the assumption of a 100% muon identi-fication efficiency and a π detector acceptance. This equation is still valid for a smallerefficiency, with the condition that it is independent of the muon polar angle. If instead theidentification efficiency times the detector acceptance is a non-trivial function of the polarangle, ε (cos θ ) , the measured muon angular distribution gets modified accordingly, and sodoes the measured forward-backward asymmetry.This issue can be solved experimentally with the observation [19] that a e + e − → µ + µ − event contains not only a negative muon but also a positive muon, the measured angulardistributions of which are given by Eqs. 2.8 and 2.10 modified with ε (cos θ ) : d N ± d cos θ ∝ (cid:26) θ ± A µµ FB cos θ (cid:27) × ε (cos θ ) , (4.10)– 12 –here ε (cos θ ) is assumed to be independent on the muon electric charge. This very rea-sonable assumption can be verified with an adequate accuracy from the × Z → µ + µ − events collected at √ s = 91 . GeV, by tagging one of the two muons in each event, andprobing the other to determine ε (cos θ ) separately for positive and negative muons. Theratio of the difference to the sum of the numbers of negative and positive muons detecteda given cos θ bin, N − (cos θ ) and N + (cos θ ) , therefore amounts to N − (cos θ ) − N + (cos θ ) N − (cos θ ) + N + (cos θ ) = 43 2 cos θ θ A µµ FB , (4.11)which allows the muon forward-backward asymmetry to be determined in each bin as fol-lows: A µµ FB = 34 × N − (cos θ ) − N + (cos θ ) N − (cos θ ) + N + (cos θ ) × θ θ , (4.12)an expression from which ε (cos θ ) has simplified away in the ratio, hence without anyimpact on the measurement uncertainty. The muon forward-backward asymmetry for thecomplete event sample is then obtained by the statistically-weighted average of the bin-by-bin determination over the detector acceptance. Any systematic effect related to the choiceof the bin size – related for example to the muon angular resolution – can be eliminated bythe use of an unbinned likelihood instead. The asymmetry of the sample of events where both muon charges are wrongly measuredequals − A µµ FB . The relative change of the asymmetry arising from double charge inversiontherefore amounts to ∆ A µµ FB A µµ FB = − f ± , (4.13)where f ± is the probability for a muon to be measured with the wrong charge sign. Forthis effect to be relevant ( i.e. , larger than × − ), f ± would need to exceed 0.5% – atypical value for LEP detectors. If FCC-ee detectors were similar to LEP detectors, f ± would be measured with an outstanding precision from the several million µ ± µ ± eventscollected at √ s ± , thus allowing the effect to be corrected with no additional uncertainty onthe asymmetry. On the other hand, the next generation of detectors for FCC-ee is likely toprovide a track momentum resolution better than that of the LEP detectors by up to oneorder of magnitude, reducing f ± to ridiculously small values, with no sizable effect on theasymmetry. The background from e + e − → τ + τ − , where the two taus decay into µν e ν µ has a crosssection of the order of 3% of the e + e − → µ + µ − cross section. It can be greatly reducedby cuts on the muon impact parameters, on the angle between the two muons, and onthe muon momenta. The very small residual contribution from this process is however notan issue, as the tau forward-backward asymmetry is expected to be identical to the muonforward-backward asymmetry. No additional uncertainty is therefore expected from thissource. – 13 – .2 Parametric uncertainties The cross section given in Section 2 depends solely on four parameters – beside α QED ( s ) –namley the Fermi constant G F , the Z boson mass and width, m Z and Γ Z , and the Weinbergangle, sin θ W . The precision with which these parameters are known is the source ofadditional uncertainties for the muon forward-backward asymmetry, and in turn, on theelectromagnetic coupling constant. These uncertainties are examined in turn below. The uncertainty on the Z mass is fully correlated to the uncertainty on the beam energy. Itseffect on the forward backward asymmetry is already taken into account in Section 4.1.1.
The Z width simplifies away from the ratio given in Eq. 4.1, which contains only the domi-nant contributions from I and Z to the asymmetry. To exhibit the sub-leading dependenceon Γ Z , it is necessary to go back to the more complete expression given in Eq. 2.11, whichcontains also the G term. The uncertainty arising from the knowledge of Γ Z is thereforenot expected to be dominant. Straightforward error propagation yields σ ( A µµ FB ) A µµ FB = 2 GZ × m Γ ( s − m ) + m Γ × σ Γ Z Γ Z . (4.14)The uncertainty on Γ Z is dominated by the energy calibration error and amounts to σ D =46 keV, i.e., to about × − Γ Z . At √ s ± , the photon contribution G is about 5% ofthe Z contribution Z , itself about 50% of its pole value. As a consequence, the relativeuncertainties on α ± are equal and amount to − with a 100% correlation. The relativeuncertainty on α QED ( m ) arising from the Z width is therefore at the same level of − . Only the terms proportional to I and Z in the complete asymmetry expression (Eqs. 2.9and 2.10) vary with sin θ W , through the vectorial coupling v . In the vicinity of the Z pole,the small photon contribution can anyway be neglected, and the asymmetry expressionsimplifies to A µµ FB ( s ) = 3 v a ( v + a ) + c γ c Z s − m s a ( v + a ) . (4.15)The derivative of A µµ FB ( s ) with respect to sin θ W can be obtained analytically, yielding ina straightforward manner ∆ A µµ FB A µµ FB ( s ) = av v + a × a − v − c γ c Z s − m s v + c γ c Z s − m s × ∆ sin θ W , (4.16) i.e. , numerically for s = s ± : ∆ A µµ FB A µµ FB ( s − ) ’ − .
92∆ sin θ W and ∆ A µµ FB A µµ FB ( s + ) ’ +4 .
87∆ sin θ W . (4.17)– 14 –he propagation to α from Eq. 4.9 leads to a partial cancellation by almost one order ofmagnitude: ∆ α α ’ .
528 ∆ A FB A FB ( s − ) + 0 .
563 ∆ A FB A FB ( s + ) ’ − .
91∆ sin θ W . (4.18)For the current precision of the effective Weinberg angle determination, of the order . × − , this uncertainty on α QED is large and amounts to . × − . At the FCC-ee,however, the measurement of the asymmetry at the Z pole (insensitive to the electromag-netic coupling constant) can be used to improve the precision of the effective Weinbergangle [20] by a factor 30 to × − – an uncertainty dominated by the absolute calibrationof the beam energy – thus reducing the uncertainty on α QED to × − . The Fermi constant is known to a relative accuracy of × − [21], turning into a relativeuncertainty on α QED ( m ) of × − . Theoretical uncertainties on the muon forward-backward asymmetry arise from the lack ofhigher orders in the calculation of the muon angular distribution. The dominant higher-order effects on the muon angles originate from QED corrections, namely (i) initial-stateradiation (ISR), i.e. , the emission of one or several photons by the incoming beams; (ii) final state radiation (FSR), i.e. , the emission of photons from the outgoing muons; and (iii) the interference between ISR and FSR (IFI), which becomes significant when the muonsare produced at small angles with respect to the beam axis. The effect of these threeQED corrections on the muon angular distributions and on the muon forward-backwardasymmetry have been studied analytically in Ref. [22] with a complete O ( α ) calculationand soft-photon exponentiation, and in a more pragmatic way by the OPAL experiment inRef. [23]. Their conclusions are summarized and the relevant effects of the O ( α ) correctionsare examined in this section. Other electroweak corrections are discussed at the end of thesection. Final-state radiation is mostly collinear and is symmetric around the muon directions, atall orders in α . The effect on the forward-backward asymmetry is therefore expected to beunmeasurably small [23]. It was checked in Ref. [22] that the effect is rigorously 0 at order α (with soft-photon exponentiation) if no cut is applied on the final-state photon energy, andvanishingly small if a tight upper cut is applied to the final-state photon energy, typicallyof the order of απ E cut √ s , i.e. , ∼ × − for E cut ∼ MeV. The theoretical uncertainty onthis effect due to the O ( α ) corrections is typically one-to-two orders of magnitude smallerthan that. It is therefore ignored in the following.– 15 – .3.2 Initial-state radiation (ISR) Initial-state radiation corrections are known up to order O ( α ) with soft-photon exponenti-ation [24]. Unlike FSR, ISR has a macroscopic influence on the forward-backward asymme-try. Photons from ISR are emitted mostly along the beam axis, with a twofold consequence: (i) the centre-of-mass frame of the muon pair therefore acquires a longitudinal boost, whichmodifies the angular distribution of both muons in a non trivial way; and (ii) the effectivecentre-of-mass energy of the collision is reduced to √ s where s = (1 − x − )(1 − x + ) s and x ± = E γ ± / √ s are the fractional radiated energies by the e ± beams, which modifies theasymmetric term of the cross section through A µµ FB ( s ) . As A µµ FB ( s ) varies quite fast with √ s , as displayed in Fig. 3 and expressed in Eq. 4.15, a large, negative, variation of themeasured asymmetry is indeed to be expected.When only one ISR photon is radiated by one of the two beams, the effects can belargely mitigated. In the vast majority of the cases, the photon is radiated exactly alongthe beam axis. The polar angles of the outgoing muons, denoted θ ± , suffice in that case todetermine the effective centre-of-mass energy √ s : s s = sin θ + + sin θ − − | sin( θ + + θ − ) | sin θ + + sin θ − + | sin( θ + + θ − ) | , (4.19)as well as the µ + direction in the centre-of-mass frame of the muon pair cos θ ∗ = sin( θ + − θ − )sin θ + + sin θ − . (4.20)In this simplest configuration, the use of cos θ ∗ entirely corrects for the effect of the longitu-dinal boost on the angular distribution, and the forward-backward asymmetry dependenceon s /s can be studied explicitly. Furthermore, the events relevant for the determinationof α ± can be selected by requiring s /s to be close to unity. If the initial-state photon isradiated with a finite angle with respect to the beam axis, however, Eqs. 4.19 and 4.20 nolonger hold, but the corresponding events can be rejected by requiring the two muons tobe back-to-back in the plane transverse to the beam axis.In rare cases, both beams can radiate photons, which render these two equations onlyapproximate, and may still create a bias in the forward-backward asymmetry. To deter-mine the effect of this approximation, large samples of µ + µ − events were generated at √ s ± . The simulation of ISR was performed with the REMT package [25] modified to include O ( α ) correcctions with soft-photon exponentiation, and the possibility to radiate up totwo photons. For the reasons just explained, only events with s /s in excess of . and anacoplanarity angle between the two muons smaller than 0.35 mrad were considered. Thesetwo cuts typically select about 80% of the cross section in Fig. 2, and tremendously increasethe purity towards events without ISR. The blue histograms in Figs. 7 show, for √ s = √ s − and √ s + , the relative biasses on A µµ FB ( s ) with respect to the standard model prediction,as a function of − s /s and for a perfect muon angular resolution, σ θ = σ φ = 0 .Events with only one ISR photon would lead to a blue straight line at ∆ A FB /A FB ≡ . ,as s /s can be exactly determined in that case from Eq. 4.19. The possibility to radiatephotons from the two beams, however, induces a visible systematic effect on the measured– 16 – -s'/s0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 -3 · F B / A F B A D -4-3-2-101234 -3 · ) = 0.0 mrad f , q ( s ) = 0.1 mrad f , q ( s = 87.9 GeV - s -3 · F B / A F B A D -4-3-2-101234 -3 · ) = 0.0 mrad f , q ( s ) = 0.1 mrad f , q ( s = 94.3 GeV + s Figure 7 . Relative bias and statistical uncertainty on the muon forward-backward asymmetrywith respect to the standard-model expectation as a function − s /s (after the cut s /s > . isapplied) for √ s = √ s − (left) and √ s + (right). The blue histogram is obtained with a perfect muonangular resolution, while σ θ and σ φ are assumed to amount to 0.1 mrad in the red histogram. Inboth cases, s /s is obtained from the measured angles with Eq. 4.19. asymmetry, up to 0.1%, as soon as the energy of these photons is non-zero (i.e., in all butthe first bin of the blue histogram). The sign of this effect can be understood as follows: theemission of two photons in opposite directions reduces the effective boost of the µ + µ − pair,causing the value of s , as determined from Eq. 4.19, to be larger than the true effectivecentre-of-mass energy. The corresponding event migration towards the left of the bluehistogram therefore tends to reduce the value of the forward-backward asymmetry (hence arelative increase below the Z peak and a relative decrease above). The effect in the first binis much smaller than in the other bins because it contains the vast majority of the events,hence is little affected by the migration from bins with a much smaller number of events.A non-perfect muon angular resolution also affects the determination of s /s fromEq. 4.19 for all events (with or without ISR), hence has a non-trivial effect on the measuredasymmetry, as shown in the red histograms of Figs. 7 for a uniform angular resolution σ θ = σ φ = 0 . mrad. Because a worse angular resolution would lead to a proportionallylarger systematic bias, the measurement of α QED can be used as a benchmark to definethe muon reconstruction performance of the FCC-ee detectors. (As an example, the trackangular resolution of the pixel detector of SiD [26], with a hit resolution of µ m , a thicknessof . of a radiation length, a radius of 6 cm, and a length of 34 cm is better than 0.1 mradover the whole acceptance.) For most of the events, without significant ISR photons, themeasured angle between the two muons can only decrease from its maximal value of π rad,yielding a smaller value of s from Eq. 4.19. Part of these events therefore migrate to theother bins of the distribution, and therefore increase significantly the value of the measuredforward-backward asymmetry in these bins, by up to 0.4% (with a relative variation negativebelow the Z peak, and positive above). The residual migration from events with initial stateradiation, but with two wrongly measured back-to-back muons, has the opposite effect inthe first bin of the distribution, albeit with a much smaller amplitude, because of the muchlarger number of events originally in this bin.– 17 –hese two systematic biasses are much larger, by two orders of magnitude, than thetarget precision with which the forward-backward asymmetry needs to be measured. Thesebiasses can be corrected for if (i) the energy and angular distributions of initial-state radia-tion can be predicted with an accuracy better than 1%, which is probably the case alreadytoday; and if (ii) the muon angular resolution can be mapped with a precision of a few permil over the whole detector acceptance, which is probably feasible with the large samplesof K ’s, Λ ’s and even J/ψ ’s expected at the FCC-ee.The predicted relative biasses, however, appear to be quasi-"universal", in the sensethat they are similar in amplitude below and above the Z peak, albeit in opposite directions.The combination of the two measurements towards a determination of α QED ( m ) withEq. 4.9 exhibits an almost perfect cancellation in all bins, as displayed in Fig. 8 as afunction of − s /s , with the same vertical scale as in Figs. 7. When integrated overall bins, the total relative bias on α QED ( m ) amounts to − × − , i.e. , well below thetarget statistical precision of a few × − . The aforementioned theoretical knowledge ofinitial-state radiation and the in-situ determination of the angular resolution would allowthis residual bias to be predicted and corrected for, with a precision at least an order ofmagnitude better. -3 · a / aD -4-3-2-101234 -3 · ) = 0.0 mrad f , q ( s ) = 0.1 mrad f , q ( s Figure 8 . Relative bias and statistical uncertainty on the electromagnetic coupling constantestimated at the Z mass scale, as a function − s /s , from measurements of the muon forward-backward asymmetry at √ s ± , with a perfect muon angular resolution (blue histogram) and with σ θ = σ φ = 0 . mrad (red histogram). – 18 – .3.3 Interference between initial- and final-state radiation (IFI) While initial-state radiation does not change the functional form of the muon angulardistribution, the interference between initial-state and final-state occurs preferably when thefinal state muons are close to the initial state electrons, hence does affect their distributionin the forward and backward directions beyond the usual (1 + cos θ ∗ ) + 8 / A FB cos θ ∗ formula.It is shown in Ref. [23] that the muon angular distribution is be modified by a multiplica-tive factor with a characteristic logarithmic dependence on cos θ ∗ , and can be parameterizedas d σ µµ d cos θ ∗ ( s ) ∝ (cid:26) θ ∗ + 83 A FB ( s ) cos θ ∗ (cid:27) × (cid:26) f (cid:18) s s (cid:19) log 1 + cos θ ∗ − cos θ ∗ (cid:27) , (4.21)in presence of a tight muon acoplanarity cut as suggested in the previous section. Themultiplicative factor contains an additional asymmetric term, which enhances the integratedmuon forward-backward asymmetry. The tight cut on s /s aimed at rejecting ISR alsoreduces IFI in similar proportions. To mitigate the very small residual effect of IFI on theangular distribution, the specific shape of the additional contribution can be fitted away,as was done at LEP and with the benefit of the much larger data samples expected at theFCC-ee. On the other hand, this additional contribution appears to be "universal", ( i.e. ,with an amplitude that depends only on s /s , similarly to what is observed for ISR), hencecancels out in the determination of α QED ( m Z ) from a combination of the measurements atthe two centre-of-mass energies, with no loss of statistical power. Other electroweak corrections have so far "only" been computed off-peak with completeone-loop calculation [27]. One-loop box corrections lead to relative changes of − × − at √ s − and − × − at √ s + from the improved Born approximation of A µµ FB − A µµ FB , , henceto a shift of α QED ( m Z ) at the per-mil level. A shift of similar size arises from one-loopvertex corrections. The theoretical uncertainty arising from the missing higher orders inthe asymmetry calculation, estimated to be at the level of a few − [28], was adequate atthe time of LEP but is insufficient today to match the precision offered by the FCC-ee.An order of magnitude improvement would be achievable today, with proven techniques,by including the dominant two-loop and leading three-loop corrections, and would representa major breakthrough towards the FCC-ee targets. Meeting these targets might require acomplete three-loop calculation, including three-loop box corrections, perhaps a seriouschallenge with the current techniques, and definitely beyond the scope of the present work.It is not unlikely, however, that a large part of these missing corrections affect in the sameway the asymmetry at 87.9 GeV and the asymmetry at 94.3 GeV. If it were the case, the α QED ( m Z ) determination would enjoy a cancellation similar to the that observed for QEDcorrections, which could suffice even without a complete three-loop calculation.– 19 – Conclusions and outlook
In this paper, it has been shown that the measurement of the muon forward-backwardasymmetry at the FCC-ee, with six months of data taking just below ( √ s = 87 . GeV) andjust above ( √ s = 94 . GeV) the Z peak, as part of the Z resonance scan, would open theopportunity of a direct measurement of the electromagnetic constant α QED ( m ) , with arelative statistical uncertainty of the order of × − .A comprehensive list of sources for experimental, parametric, theoretical systematicuncertainties has been examined. Most of these uncertainties have been shown to be undercontrol at the level of − or below, as summarized in Table 1. A significant fractionof those benefits from a delicate cancellation between the two asymmetry measurements.The knowledge of the beam energy, both on- and off-peak, turns out to be the dominantcontribution, albeit still well below the targeted statistical power of the method.Type Source Uncertainty E beam calibration × − E beam spread < − Experimental Acceptance and efficiency negl.Charge inversion negl.Backgrounds negl. m Z and Γ Z × − Parametric sin θ W × − G F × − QED (ISR, FSR, IFI) < − Theoretical Missing EW higher orders few − New physics in the running . Total Systematics . × − (except missing EW higher orders) Statistics × − Table 1 . Summary of relative statistical, experimental, parametric and theoretical uncertainties tothe direct determination of the electromagnetic coupling constant at the FCC-ee, with a one-yearrunning period equally shared between centre-of-mass energies of 87.9 and 94.3 GeV, correspondingto an integrated luminosity of
85 ab − . The fantastic integrated luminosity and the unique beam-energy determination are the key breakthrough advantages of the FCC-ee in the perspective of a precise determination ofthe electromagnetic coupling constant. Today, the only obstacle towards this measurement– beside the construction of the collider and the delivery of the target luminosities – stemsfrom the lack of higher orders in the determination of the electroweak corrections to theforward-backward asymmetry prediction in the standard model. With the full one-loopcalculation presently available for these corrections, a relative uncertainty on A µµ FB of theorder of a few − is estimated. An improvement deemed adequate to match the FCC-eeexperimental precision might require a calculation beyond two loops, which may be beyond– 20 –he current state of the art, but is possibly within reach on the time scale required by theFCC-ee.A consistent international programme for present and future young theorists musttherefore be set up towards significant precision improvements in the prediction of all elec-troweak precision observables, in order to reap the rewards potentially offered by the FCC-ee. Acknowledgements
I thank Alain Blondel for enlightening discussions throughout the development of this anal-ysis. I am indebted to Roberto Tenchini for his expert suggestions and to Gigi Rolandi forhis subtle comments and his consistent checking of all equations in the paper. I am gratefulto Ayres Freitas for his careful reading, and for providing me with the predictions for A µµ FB from ZFITTER with the effects of the full one-loop calculation included.
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Corrigendum to "Direct measurement of α QED ( m ) at the FCC-ee" [JHEP 02 (2016) 053] Patrick Janot
CERN,EP Department, Geneva, Switzerland
E-mail: [email protected] a r X i v : . [ h e p - ph ] N ov ontents Equation 4.6 in Section 4.1.2 of Ref [1], which gives the muon forward-backward asymmetry( A µµ FB ) bias in presence of a finite beam energy spread δ , is valid only under the assumptionthat the number of e + e − → µ + µ − events is uniformly distributed over the centre-of-mass-energy interval covered by the beam energy spread. This assumption is not upheld byreality, as the dimuon production cross section σ µµ follows the Z resonant lineshape, andtherefore changes steeply with √ s around the two optimal centre-of-mass energies, √ s ± .The equation should therefore read: ∆ A µµ FB ( s ± ) = R A µµ FB ( s ) σ µµ ( s ) exp − ( √ s −√ s ± ) s ± δ d √ s R σ µµ ( s ) exp − ( √ s −√ s ± ) s ± δ d √ s − A µµ FB ( s ± ) . (1.1)The effect of the forward-backward asymmetry reweighing by the production crosssection amounts to increase (decrease) the average centre-of-mass energy above √ s − (below √ s + ), hence significantly change the average asymmetry towards smaller absolute values.Numerically, for a standard-model Z lineshape, and a relative beam energy spread of 0.12%,the corresponding relative biases on the measured forward-backward asymmetries amountto ∆ A FB A FB ( s − ) = − . × − and ∆ A FB A FB ( s + ) = − . × − , (1.2)which differ from Eq. 4.8 of Ref. [1] both quantitatively: the effect is almost two orders ofmagnitude larger; and qualitatively: the previously claimed cancellation no longer occurs inthe sum of Eq. 4.9. As a consequence, the bias on α QED , which changes from +1 . × − [1]to − . × − , needs to be corrected for, and therefore determined with a relative precisionbetter than 1%, to induce an uncertainty on α QED of the same order as or smaller thanthat arising from the beam energy measurement.A method to monitor the centre-of-mass energy spread with this precision every coupleminutes at the FCC-ee, from the angular analysis of the same e + e − → µ + µ − events as thoseused for the forward-backward asymmetry measurement, is written up in the forthcomingFCC Conceptual Design Report [2]. In Table 1 of Ref. [1], the line " E beam spread" isconsequently modified from < − to < − .The conclusions of Ref. [1] are unchanged after these corrections.– 1 – Interference between Initial and Final state radiation (IFI)
In Section 4.3.3 of Ref. [1], it is stated that the tight cut on s /s aimed at rejecting ISR alsoreduces IFI in similar proportions. This statement is incomplete. While rejecting indeedevents with energetic photons in similar proportions with or without IFI, a tight cut on s /s has no effect on soft photons, which in turn have a macroscopic effect (several per-centincrease) on the numerical value of the muon forward-backward asymmetry when IFI isincluded. As pointed out in Ref. [3], an important cancellation, already exploited in Ref. [1]for ISR, still occurs for IFI between the two centre-of-mass energies √ s − and √ s + .The precision with which this cancellation can be predicted is being evaluated and isthe topic of two forthcoming publications [4, 5]. Acknowledgments
I would like to thank very much Mogens Dam for kindly pointing out that Eq. 4.6 ofRef. [1] was missing an important contribution, and for motivating me to ascertain therobustness of the centre-of-mass energy spread determination with a sub-per-cent precisionat the FCC-ee.
References [1] P. Janot,
Direct measurement of α QED ( m Z ) at the FCC-ee , JHEP (2016) 053,[ arXiv:1512.05544 ].[2] The FCC-ee design study group, The FCC CDR Volume 5, Lepton Collider (Comprehensive) , CERN Yellow report in preparation (2018).[3] S. Jadach, “QED effects in muon charge asymmetry near Z peak.” Available at https://indico.cern.ch/event/438866/contributions/1085158/attachments/1256136/1854356/2016-rome-fccweek.pdf . Tak given at the FCC Week, Rome, 11-15 April 2016.[4] S. Jadach and S. A. Yost,
QED interference in charge asymmetry near the Z resonance infuture electron-positron colliders , in preparation (2018).[5] S. Jadach and P. Janot, Evaluation of the bias from the interference between initial- andfinal-state radiation on the measurement of α QED ( m Z ) at the FCC-ee , in preparation (2018).(2018).