A new approach to evaluate the leading hadronic corrections to the muon g-2
C. M. Carloni Calame, M. Passera, L. Trentadue, G. Venanzoni
AA new approach to evaluate the leadinghadronic corrections to the muon g -2 (cid:73) C. M. Carloni Calame a , M. Passera b , L. Trentadue c , G. Venanzoni d a Dipartimento di Fisica, Universit`a di Pavia, Pavia, Italy b INFN, Sezione di Padova, Padova, Italy c Dipartimento di Fisica e Scienze della Terra “M. Melloni”Universit`a di Parma, Parma, Italy andINFN, Sezione di Milano Bicocca, Milano, Italy d INFN, Laboratori Nazionali di Frascati, Frascati, Italy
Abstract
We propose a novel approach to determine the leading hadronic corrections to the muon g -2. It consistsin a measurement of the effective electromagnetic coupling in the space-like region extracted from Bhabhascattering data. We argue that this new method may become feasible at flavor factories, resulting in analternative determination potentially competitive with the accuracy of the present results obtained with thedispersive approach via time-like data.
1. Introduction
The long-standing discrepancy between experi-ment and the Standard Model (SM) prediction of a µ , the muon anomalous magnetic moment, haskept the hadronic corrections under close scrutinyfor several years [1–4]. In fact, the hadronic uncer-tainty dominates that of the SM value and is com-parable with the experimental one. When the newresults from the g -2 experiments at Fermilab andJ-PARC will reach the unprecedented precision of0.14 parts per million (or better) [5–7], the uncer-tainty of the hadronic corrections will become themain limitation of this formidable test of the SM.An intense research program is under way toimprove the evaluation of the leading order (LO)hadronic contribution to a µ , due to the hadronicvacuum polarization correction to the one-loop di-agram [8, 9], as well as the next-to-leading order(NLO) hadronic one. The latter is further dividedinto the O ( α ) contribution of diagrams contain-ing hadronic vacuum polarization insertions [10], (cid:73) This work is dedicated to the memory of our friend andcolleague Eduard A. Kuraev.
Email addresses: [email protected] (C. M. Carloni Calame), [email protected] (M.Passera), [email protected] (L. Trentadue), [email protected] (G. Venanzoni) and the leading hadronic light-by-light term, alsoof O ( α ) [2, 11, 12]. Very recently, even the next-to-next-to leading order (NNLO) hadronic contri-butions have been studied: insertions of hadronicvacuum polarizations were computed in [13], whilehadronic light-by-light corrections have been esti-mated in [14].The evaluation of the hadronic LO contribution a HLO µ involves long-distance QCD for which pertur-bation theory cannot be employed. However, us-ing analyticity and unitarity, it was shown longago that this term can be computed via a disper-sion integral using the cross section for low-energyhadronic e + e − annihilation [15]. At low energy thiscross-section is highly fluctuating due to resonancesand particle production threshold effects.As we will show in this paper, an alternativedetermination of a HLO µ can be obtained measuringthe effective electromagnetic coupling in the space-like region extracted from Bhabha ( e + e − → e + e − )scattering data. A method to determine the run-ning of the electromagnetic coupling in small-angleBhabha scattering was proposed in [16] and appliedto LEP data in [17]. As vacuum polarization in thespace-like region is a smooth function of the squaredmomentum transfer, the accuracy of its determi-nation is only limited by the statistics and by thecontrol of the systematics of the experiment. Also,1 a r X i v : . [ h e p - ph ] M a y s at flavor factories the Bhabha cross section isstrongly enhanced in the forward region, we willargue that a space-like determination of a HLO µ maynot be limited by statistics and, although challeng-ing, may become competitive with standard resultsobtained with the dispersive approach via time-likedata.
2. Theoretical framework
The leading-order hadronic contribution to themuon g -2 is given by the well-known formula [4, 15] a HLO µ = απ (cid:90) ∞ dss K ( s ) ImΠ had ( s + i(cid:15) ) , (1)where Π had ( s ) is the hadronic part of the photonvacuum polarization, (cid:15) > K ( s ) = (cid:90) dx x (1 − x ) x + (1 − x )( s/m µ ) (2)is a positive kernel function, and m µ is the muonmass. As the total cross section for hadron produc-tion in low-energy e + e − annihilations is related tothe imaginary part of Π had ( s ) via the optical theo-rem, the dispersion integral in eq. (1) is computedintegrating experimental time-like ( s >
0) data upto a certain value of s [2, 18, 19]. The high-energytail of the integral is calculated using perturbativeQCD [20].Alternatively, if we exchange the x and s integra-tions in eq. (1) we obtain [21] a HLO µ = απ (cid:90) dx ( x −
1) Π had [ t ( x )] , (3)where Π had ( t ) = Π had ( t ) − Π had (0) and t ( x ) = x m µ x − < x = (1 − β ) ( t/ m µ ) , with β = (1 − m µ /t ) / , and from eq. (3) we obtain a HLO µ = απ (cid:90) −∞ Π had ( t ) (cid:18) β − β + 1 (cid:19) dttβ . (5)Equation (5) has been used for lattice QCD calcu-lations of a HLO µ [22]; while the results are not yetcompetitive with those obtained with the disper-sive approach via time-like data, their errors areexpected to decrease significantly in the next fewyears [23]. The effective fine-structure constant at squaredmomentum transfer q can be defined by α ( q ) = α − ∆ α ( q ) , (6)where ∆ α ( q ) = − ReΠ( q ) . The purely leptonicpart, ∆ α lep ( q ), can be calculated order-by-orderin perturbation theory – it is known up to threeloops in QED [24] (and up to four loops in specific q limits [25]). As ImΠ( q ) = 0 for negative q ,eq. (3) can be rewritten in the form [26] a HLO µ = απ (cid:90) dx (1 − x ) ∆ α had [ t ( x )] . (7)Equation (7), involving the hadronic contributionto the running of the effective fine-structure con-stant at space-like momenta, can be further formu-lated in terms of the Adler function [27], defined asthe logarithmic derivative of the vacuum polariza-tion, which, in turn, can be calculated via a disper-sion relation with time-like hadroproduction dataand perturbative QCD [26, 28]. We will proceeddifferently, proposing to calculate eq. (7) by mea-surements of the effective electromagnetic couplingin the space-like region (see also [9]).
3. ∆ α had ( t ) from Bhabha scattering data The hadronic contribution to the running of α in the space-like region, ∆ α had ( t ), can be extractedcomparing Bhabha scattering data to Monte Carlo(MC) predictions. The LO Bhabha cross sectionreceives contributions from t - and s -channel pho-ton exchange amplitudes. At NLO in QED, it iscustomary to distinguish corrections with an ad-ditional virtual photon or the emission of a realphoton (photonic NLO) from those originated bythe insertion of the vacuum polarization correc-tions into the LO photon propagator (VP). Thelatter goes formally beyond NLO when the Dysonresummed photon propagator is employed, whichsimply amounts to rescaling the α coupling in theLO s - and t -diagrams by the factor 1 / (1 − ∆ α ( q ))(see eq. (6)). In MC codes, e.g. in BabaYaga [29],VP corrections are also applied to photonic NLOdiagrams, in order to account for a large part ofthe effect due to VP insertions in the NLO con-tributions. Beyond NLO accuracy, MC genera-tors consistently include also the exponentiation of(leading-log) QED corrections to provide a morerealistic simulation of the process and to improve2he theoretical accuracy. We refer the reader toref. [30] for an overview of the status of the mostrecent MC generators employed at flavor factories.We stress that, given the inclusive nature of themeasurements, any contribution to vacuum polar-ization which is not explicitly subtracted by theMC generator will be part of the extracted ∆ α ( q ).This could be the case, for example, of the contribu-tion of hadronic states including photons (which, al-though of higher order, are conventionally includedin a HLO µ ), and that of W bosons or top quark pairs.Before entering the details of the extraction of∆ α had ( t ) from Bhabha scattering data, let us con-sider a few simple points. In fig. 1 (left) we plot theintegrand (1 − x )∆ α had [ t ( x )] of eq. (7) using the out-put of the routine hadr5n12 [31] (which uses time-like hadroproduction data and perturbative QCD).The range x ∈ (0 ,
1) corresponds to t ∈ ( −∞ , x = 0 for t = 0. The peak of the integrandoccurs at x peak (cid:39) .
914 where t peak (cid:39) − . and ∆ α had ( t peak ) (cid:39) . × − (see fig. 1(right)). Such relatively low t values can be ex-plored at e + e − colliders with center-of-mass energy √ s around or below 10 GeV (the so called “flavorfactories”) where t = − s − cosθ ) (cid:18) − m e s (cid:19) , (8) θ is the electron scattering angle and m e is theelectron mass. Depending on s and θ , the inte-grand of eq. (7) can be measured in the range x ∈ [ x min , x max ], as shown in fig. 2 (left). Notethat to span low x intervals, larger θ ranges areneeded as the collider energy decreases. In thisrespect, √ s ∼ x interval [0 . , .
98] can be measuredvarying θ between ∼ ◦ and 28 ◦ . It is also worthremarking that data collected at flavor factories,such as DAΦNE (Frascati), VEPP-2000 (Novosi-birsk), BEPC-II (Beijing), PEP-II (SLAC) and Su-perKEKB (Tsukuba), and possibly at a future high-energy e + e − collider, like FCC- ee (TLEP) [32] orILC [33], can help to cover different and comple-mentary x regions.Furthermore, given the smoothness of the inte-grand, values outside the measured x interval maybe interpolated with some theoretical input. In par-ticular, the region below x min will provide a rela-tively small contribution to a HLO µ , while the regionabove x max may be obtained by extrapolating thecurve from x max to x = 1, where the integrand isnull, or using perturbative QCD. The analytic dependence of the MC Bhabha pre-dictions on α ( t ) (and, in turn, on ∆ α had ( t )) is nottrivial, and a numerical procedure has to be devisedto extract it from the data. In formulae, we haveto find a function α ( t ) such that dσdt (cid:12)(cid:12)(cid:12) data = dσdt (cid:16) α ( t ) , α ( s ) (cid:17)(cid:12)(cid:12)(cid:12) MC , (9)where we explicitly kept apart the dependence onthe time-like VP α ( s ) because we are only inter-ested in α ( t ). We emphasise that, in our analy-sis, α ( s ) is an input parameter. Being the Bhabhacross section in the forward region dominated bythe t -channel exchange diagram, we checked thatthe present α ( s ) uncertainty induces in this regiona relative error on the θ distribution of less than ∼ − (which is part of the systematic error).We propose to perform the numerical extractionof ∆ α had ( t ) from the Bhabha distribution of the t Mandelstam variable. The idea is to let α ( t ) varyin the MC sample around a reference value andchoose, bin by bin in the t distribution, the valuethat minimizes the difference with data. The pro-cedure can be sketched as follows:1. choose a reference function returning the valueof ∆ α had ( t ) (and hence α ( t )) to be used in theMC sample, we call it ¯ α ( t );2. for each generated event, calculate N MCweights by rescaling ¯ α ( t ) → ¯ α ( t ) + iN δ ( t ),where i ∈ [ − N, N ] and δ ( t ) is for example theerror induced on ¯ α ( t ) by the error on ∆ α had ( t ).Being done on an event by event basis, the fulldependence on α ( t ) of the MC differential crosssection can be kept;3. for each bin j of the t distribution, comparethe experimental differential cross section withthe MC predictions and choose the i j whichminimizes the difference;4. ¯ α ( t j ) + i j N δ ( t j ) will be the extracted value of α ( t j ) from data in the j th bin. ∆ α had ( t j ) canthen be obtained through the relation between α ( t ) and ∆ α had ( t ).We finally find, for each bin j of the t distribution, dσdt (cid:12)(cid:12)(cid:12) j, data = dσdt (cid:16) ¯ α ( t ) + i j N δ ( t ) , α ( s ) (cid:17)(cid:12)(cid:12)(cid:12) j, MC . (10) This was not the case for example in [16, 17]: there α ( t )was extracted from Bhabha data in the very forward regionat LEP, where the t channel diagrams are by far dominantand α ( t ) factorizes. x peak . · − .
98 10 . . t peak ∞ ( − x ) · ∆ α h a d (cid:16) x m µ x − (cid:17) × x | t | × (GeV ) x peak ’ . t peak ’ − . GeV ∆ α h a d (cid:16) x m µ x − (cid:17) × x Figure 1: Left: The integrand (1 − x )∆ α had [ t ( x )] × as a function of x and t . Right: ∆ α had [ t ( x )] × . x θ ( deg ) √ s = 1 GeV √ s = 3 GeV √ s = 10 GeV
10 20 30 40 50 60 70 80 90 d σ / d θ ( p b − d e g ) θ ( deg ) √ s = 1 GeV √ s = 3 GeV √ s = 10 GeV
Figure 2: Left: Ranges of x values as a function of the electron scattering angle θ for three different center-of-mass energies.The horizontal line corresponds to x = x peak (cid:39) . BabaYaga [29] asa function of θ for the same three values of √ s in the angular range 2 ◦ < θ < ◦ .
4e remark that the algorithm does not assumeany simple dependence of the cross section on α ( t ),which can in fact be general, mixing s , t channelsand higher order radiative corrections, relevant (ornot) in different t domains.In order to test our procedure, we perform apseudo-experiment: we generate pseudo-data usingthe parameterization ∆ α I had ( t ) of refs. [19, 34] andcheck if we can recover it by inserting in the MCthe (independent) parameterization ∆ α II had ( t ) (cor-responding to ¯ α ( t ) of eq. 10) of ref. [31] by meansof the method described above. For this exercise,we use the generator BabaYaga in its most com-plete setup, generating events at √ s = 1 .
02 GeV,requiring 10 ◦ < θ ± < ◦ , E ± > . ◦ . We choose δ ( t ) to be theerror induced on α ( t ) by the 1- σ error on ∆ α had ( t ),which is returned by the routine of ref. [31], weset N = 150, and we produce distributions with200 bins. We note that in the present exercise α ( s )and all the radiative corrections both in the pseudo-data and in the MC samples are exactly the same,because we are interested in testing the algorithmrather than assessing the achievable accuracy, atleast at this stage.In fig. 3, ∆ α extrhad is the result extracted with ouralgorithm, corresponding to the minimizing set of i j : the figure shows that our method is capableof recovering the underlying function ∆ α had ( t ) in-serted into the “data”. As the difference between∆ α I had and ∆ α extrhad is hardly visible on an absolutescale, in fig. 3 all the functions have been dividedby ∆ α II had to display better the comparison between∆ α I had and ∆ α extrhad .In order to assess the achievable accuracy on∆ α had ( t ) with the proposed method, we remarkthat the LO contribution to the cross section isquadratic in α ( t ), thus we have (see eq. (6))12 δσσ (cid:39) δαα (cid:39) δ ∆ α had (11)Equation (11) relates the absolute error on ∆ α had with the relative error on the Bhabha cross section.From the theoretical point of view, the present ac-curacy of the MC predictions [30] is at the levelof about 0 . (cid:104) , which implies that the precisionthat our method can, at best, set on ∆ α had ( t ) is δ ∆ α had ( t ) (cid:39) · − . Any further improvementrequires the inclusion of the NNLO QED correc-tions into the MC codes, which are at present notavailable (although not out of reach) [30]. From the experimental point of view, we remarkthat a measurement of a HLO µ from space-like datacompetitive with the current time-like evaluationswould require an O (1%) accuracy. Statistical con-siderations show that a 3% fractional accuracy onthe a HLO µ integral can be obtained by sampling theintegrand (1 − x )∆ α had [ t ( x )] in ∼
10 points aroundthe x peak with a fractional accuracy of 10%. Giventhe value of O (10 − ) for ∆ α had at x = x peak , thisimplies that the cross section must be known withrelative accuracy of ∼ × − . Such a statisticalaccuracy, although challenging, can be obtained atflavor factories, as shown in fig. 2 (right). With anintegrated luminosity of O (1), O (10), O (100) f b − at √ s = 1, 3 and 10 GeV, respectively, the angu-lar region of interest can be covered with a 0.01%accuracy per degree. The experimental systematicerror must match the same level of accuracy.A fraction of the experimental systematic errorcomes from the knowledge of the machine luminos-ity, which is normalized by calculating a theoreticalcross section in principle not depending on ∆ α had .We devise two possible options for the normaliza-tion process:1. using the e + e − → γγ process, which has nodependence on ∆ α had , at least up to NNLOorder;2. using the Bhabha process at t ∼ − GeV ( x ∼ . α had isof O (10 − ) and can be safely neglected.Both processes have advantages and disadvantages;a dedicated study of the optimal choice goes beyondthe scope of this paper and will be considered in afuture detailed analysis.
4. Conclusions
We presented a novel approach to determine theleading hadronic correction to the muon g -2 usingmeasurements of the running of α ( t ) in the space-like region from Bhabha scattering data. Althoughchallenging, we argued that this alternative deter-mination may become feasible using data collectedat present flavor factories and possibly also at a fu-ture high-energy e + e − collider. The proposed de-termination can become competitive with the ac-curacy of the present results obtained with the dis-persive approach via time-like data.5 .9850.990.99511.0051.011.015 -1 -0.8 -0.6 -0.4 -0.2 ∆ α i h a d ( t ) / ∆ α II h a d ( t ) t (GeV ) ∆ α II had − δ ∆ α II had + δ ∆ α I had ∆ α extr had Figure 3: The extracted function ∆ α extrhad ( t ) compared to the function ∆ α I had ( t ) used in the pseudo-data (see text). Thefunctions ∆ α II had ( t ) ± δ ( t ) are shown to display the range spanned by the MC samples. All functions have been divided by∆ α II had ( t ). The tiny difference between ∆ α I had and ∆ α extrhad is due to the binning discretization. Acknowledgements
We would like to thank G. Degrassi, G.V. Fedo-tovich, F. Jegerlehner and M. Knecht for useful cor-respondence and discussions. We would like also tothank G. Montagna, F. Piccinini and O. Nicrosinifor constant interest in our work and useful dis-cussions. We acknowledge the hospitality of theGalileo Galilei Institute in Florence, where part ofthis work has been carried out during the workshop“Prospects and Precision at the LHC at 14 TeV”.C.M.C.C. is fully supported by the MIUR-PRINproject 2010YJ2NYW. L.T. also acknowledges par-tial support from the same MIUR-PRIN project.M.P. also thanks the Department of Physics andAstronomy of the University of Padova for its sup-port. His work was supported in part by the MIUR-PRIN project 2010YJ2NYW and by the EuropeanProgram INVISIBLES (PITN-GA-2011-289442).
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