Pion electromagnetic form factor at high precision with implications to a ππ μ and the onset of perturbative QCD
PPion electromagnetic form factor at high precision with implications to a ππµ and theonset of perturbative QCD B.Ananthanarayan, Irinel Caprini, and Diganta Das Centre for High Energy Physics, Indian Institute of Science, Bangalore 560 012, India Horia Hulubei National Institute for Physics and Nuclear Engineering, P.O.B. MG-6, 077125 Magurele, Romania Department of Physics and Astrophysics, University of Delhi, Delhi 110007, India
We extend recently developed methods used for determining the electromagnetic charge radiusand a ππµ to obtain a determination of the electromagnetic form factor of the pion, F Vπ ( t ), in severalsignificant kinematical regions, using a parametrization-free formalism based on analyticity andunitarity, with the inclusion of precise inputs from both timelike and spacelike regions. On theunitarity cut, below the first inelastic threshold, we use the precisely known phase of the form factor,known from ππ elastic scattering via the Fermi-Watson theorem, and above the inelastic threshold aconservative integral condition on the modulus. We also use as input the experimental values of themodulus at several energies in the elastic region, where the data from e + e − → π + π − and τ hadronicdecays are mutually consistent, as well as the most recent measurements at spacelike momenta.The experimental uncertainties are implemented by Monte Carlo simulations. At spacelike values Q = − t > Q confirm the late onset ofperturbative QCD for exclusive quantities. From the predictions of | F Vπ ( t ) | on the timelike axisbelow 0.63 GeV, we obtain the hadronic vacuum polarization (HPV) contribution to the muonanomaly, a ππµ | ≤ .
63 GeV = (132 . ± . × − , using input from both e + e − annihilation and τ decay, and a ππµ | ≤ .
63 GeV = (132 . ± . × − using only e + e − input. Our determinations canbe readily extended to obtain such contributions in any interval of interest lying between 2 m π and0.63 GeV. PACS numbers:
I. INTRODUCTION
The electromagnetic form factor of the pion, F Vπ ( t ),defined by the matrix element (cid:104) π + ( p (cid:48) ) | J elm µ | π + ( p ) (cid:105) = ( p + p (cid:48) ) µ F Vπ ( t ) , (1)where t = q and q = p − p (cid:48) , is a fundamental observ-able of the strong interactions and a sensitive probe ofthe composite nature of the pion. An expansion near theorigin to linear order in t , F Vπ ( t ) = 1 + (cid:104) r π (cid:105) t/ r π = (cid:112) (cid:104) r π (cid:105) = (0 . ± . a ππµ to the anomalous magnetic moment of themuon was determined in a region where experimentaldata have significant lack of agreement. In this work, weadapt the methods introduced in these studies to the de-termination of the form factor itself in several kinematicregions of interest. In contrast to the prior investigations,where a single number was determined in each of them, inthe present work we obtain the values of the form factorat a large number of points.We recall that there is a large amount of information,both theoretical and experimental, on the pion vector form factor, making it one of the most investigated quan-tity in hadron physics. The form factor determination athigh precision is of utmost importance to several observ-ables including the low-energy dipion contribution to themuon g −
2, and poses a significant challenge to experi-ment as well as to theory. Theoretical studies are basedat low energies on nonperturbative approaches and effec-tive theories of the type first formulated by Weinberg [3],and at large energies on perturbative QCD. In the frame-work of chiral perturbation theory (ChPT), the effectiverealization of QCD at low energies first formulated at oneloop order with two [4] and three light quark flavours[5, 6], the pion vector form factor has been calculated upto two loops [7–11]. Lattice gauge theory has recentlybecome another useful nonperturbative tool for the cal-culation of the form factor at low energies [12, 13].The form factor is also a probe of energies at whichasymptotic QCD predictions are expected to set in. Per-turbative QCD predicts the behavior at large momentaalong the spacelike axis, where Q ≡ − t (cid:29) F V,LOπ ( − Q ) ∼ πF π α s ( Q ) Q , Q → ∞ , (2)where F π = 131 MeV is the pion decay constant and α s ( Q ) = 4 π/ [9 ln( Q / Λ )] is the running strong cou-pling to one loop with three active light quark flavors.NLO corrections to (2) have been calculated in [20, 21].The experimental information available on the pion a r X i v : . [ h e p - ph ] D ec form factor is very rich. This quantity was measuredat spacelike values Q > t ≥ m π , where the form factor is complex,its modulus has been measured from the cross section ofthe process e + e − → π + π − [34]-[46] and, using isospinsymmetry, from the τ → ππντ decay [47]-[51].Due to the extensive experimental and theoretical in-formation, the pion vector form factor is, compared withother hadronic quantities, a well-known function. How-ever, the precision does not reach the same level for alltimelike and spacelike momenta. A better precision is re-quired on the spacelike axis, for checking the consistencywith experimental data and for testing the calculationsprovided by lattice QCD at low momenta and perturba-tive QCD at larger momenta. On the timelike axis, atlow energies the phase of the form factor is well known,being equal by Fermi-Watson theorem [52, 53] to the ππ scattering P -wave phase shift, which has been calculatedwith high precision using ChPT and Roy equations [54–56]. However, the modulus is poorly known, due to thedifficulties of the experimental measurements in this re-gion: only two experiments, BABAR [38] and KLOE [40–42] reported data at low energies, and unfortunately theyare not consistent with each other.This situation drastically affects the calculation of thehadronic vacuum polarization (HVP) contribution to themuon anomaly, a µ = ( g − µ /
2, a quantity which playsan important role for testing the standard model andfinding possible signals of new physics. The great interestin the muon anomaly is motivated by the present discrep-ancy of about 3 to 4 σ between theory and experiment.New generation measurements of muon g − [57] and JPARC [58] are expected to produceresults with experimental errors at the level of 16 × − ,a factor of 4 smaller compared to the Brookhaven mea-surement [59]. This requires a precision at the same levelalso for the theoretical result: see for instance Ref. [58]for an updated review, Refs. [60, 61] for most recent phe-nomenological determinations, and Ref. [62] for a recentlattice calculation.Dispersion theory, which exploits analyticity and uni-tarity, is a powerful tool for performing the analytic con-tinuation of the form factor to energies where it is not pre-cisely known. The pion vector form factor is an analyticfunction in the complex t plane cut along the real axis for t ≥ t + , where t + = 4 m π is the first unitarity threshold.Moreover, it is normalized as F Vπ (0) = 1, and satisfiesthe Schwarz reflection property F Vπ ( t ∗ ) = ( F Vπ ( t )) ∗ . Itturns out that the standard dispersion relation, based onthe Cauchy integral, is not suitable for F Vπ ( t ), since it The E989 experiment at Fermilab has started its pilot runs andis gathering data at an accelerated pace. requires the knowledge of its imaginary part on the uni-tarity cut, which is not available in a straightforward way.On the other hand, as mentioned above, in the limit of ex-act isospin symmetry, the Fermi-Watson theorem [52, 53]states that below the first inelastic threshold, the phaseof F Vπ ( t ) is equal to the P -wave phase shift of ππ elas-tic scattering, which is better known. Many dispersionanalyses of the pion vector form factor have been basedon the so-called Omn`es representation, which amountsto reconstruct an analytic function from its phase on thecut. However, this approach involves some assumptionson the phase above the inelastic threshold, where it isnot known, and on the positions of the possible zerosin the complex plane. A related approach uses specificparametrizations which implement the analytic proper-ties of the form factor. Recent analyses based on thisapproach are [63, 64].In the present paper, we use a method based on an-alyticity and unitarity for calculating the form factor inkinematical regions where it is not precisely known, us-ing the more precise input available in other energy re-gions. We implement the phase of the form factor alonga part of the unitarity cut, where it is well known, andinformation on the modulus on the remaining part of thecut. Thus, our method is neither a standard dispersiverepresentation, nor a specific parametrization for the ana-lytic extrapolation in the complex momentum plane. Theadvantage is that we can implement only known input,avoiding to a large extent model-dependent assumptionsabout the behavior of the form factor in regions where itis less known. The price to be paid was the fact that wedo not obtain definite values for the extrapolated quan-tity, but only optimal allowed ranges for it, in terms ofthe phenomenological input. This shortcoming has beenovercome now as described below.This method, proposed in [65] and presented in detailin the review [66], has been applied already in severalpapers [67–70], where optimal bounds on the pion vectorform factor in various energy regions have been derived.An important improvement has been achieved by imple-menting the statistical distribution of the experimentalinput by Monte Carlo simulations, which converted theanalytic bounds into allowed intervals with definite con-fidence levels. This elaborate formalism was applied inRefs. [2] and [1] for the calculation with a remarkableaccuracy of the low-energy HVP contribution to muon g − II. EXTREMAL PROBLEM
Our aim is to make precision predictions for the pionvector form factor in several regions on both spacelikeand timelike axis. In particular, we will be interested inthe modulus | F Vπ ( t ) | in the low energy region t + ≤ t ≤ (0 .
63 GeV) where t + = 4 m π , which will allow a newdetermination of the pion-pion contribution to the muonanomaly a µ from this region. We will determine also theform factor F Vπ ( t ) in the unphysical region 0 ≤ t ≤ t + and at spacelike values t < t = 0, expressed by: F Vπ (0) = 1 . (3)An important ingredient is Fermi-Watson theorem [52,53] mentioned above. Since this theorem is valid in theexact isospin limit, we must first remove the main isospin-violating effect in the pion vector form factor, known toarise from ω − ρ interference. We shall follow standardapproach [71, 72] to do this, by defining a purely I = 1function F ( t ) as F ( t ) = F Vπ ( t ) /F ω ( t ) , (4)where F ω ( t ) includes the I = 0 contribution due to ω .Then Fermi-Watson theorem writes asArg[ F ( t + i(cid:15) )] = δ ( t ) , t + ≤ t ≤ t in , (5)where δ ( t ) is the phase-shift of the P -wave of ππ elasticscattering and t in is the first inelastic threshold.Above the inelastic threshold t in , where the phase isnot known, we shall use the phenomenological informa-tion available on the modulus at intermediate energies,and perturbative QCD at high energies. Since the pre-cision is not enough to impose the condition at each t above t in , we shall adopt a weaker condition, written as1 π (cid:90) ∞ t in dtρ ( t ) | F Vπ ( t ) | ≤ I, (6)where ρ ( t ) > I from the available information is possible.We shall use, in addition, the experimental value of theform factor at one spacelike energy: F Vπ ( t s ) = F s ± (cid:15) s , t s < , (7) and the modulus at one energy in the elastic region ofthe timelike axis, where it is known with precision fromexperiment: | F Vπ ( t t ) | = F t ± (cid:15) t , t + < t t < t in . (8)The aim of our work can be expressed as the follow-ing functional extremal problem: using as input the con-ditions (3)-(8), derive optimal upper and lower boundson | F Vπ ( t ) | on the unitarity cut below 0 .
63 GeV, and on F Vπ ( t ) on the real axis for t < t + .The solution of the extremal problem and the algo-rithm for obtaining the bounds are presented for com-pleteness in the Appendix. It will be applied in Sec. IVfor making precise predictions on the form factor in theregions of interest. In Sec. III we shall describe the phe-nomenological information used as input. III. INPUT IN THE EXTREMAL PROBLEM
For the function F ω ( t ), which accounts for theisospin violation due to ω resonance, we shall use theparametrization proposed in [71, 72]: F ω ( t ) = 1 + (cid:15) t ( m ω − i Γ ω / − t , (9)with (cid:15) = 1 . × − . This function is normalized as F ω (0) = 1 and, due to the small value Γ ω = 8 .
49 MeV[73], is highly peaked around √ t = m ω = 782 .
65 MeV. Inour treatment, we first converted the experimental valuesof F Vπ ( t ) used as input in Eqs. (6), (7) and (8) to theisospin-conserving function F ( t ) defined in (4), solved theextremal problem for this function and finally reinsertedthe factor F ω ( t ) in the results. Actually, since we do notinclude the resonance region in our study, the correctionsdue to F ω ( t ) are very small in all the kinematical regionsconsidered, and are practically negligible for t ≤ t in for the pionform factor is due to the opening of the ωπ channel, i.e. , √ t in = m ω + m π = 0 .
917 GeV. Below this threshold, weuse in (5) the phase shift δ ( t ) obtained from dispersionrelations and Roy equations applied to ππ scattering inRefs. [54, 55] and [56], which we denote as Bern andMadrid phase, respectively. Actually, in the calculationof the Bern phase, for the P -wave phase shift some inputfrom previous data on the form factor was used at thematching point 0.8 GeV, which may raise doubts of acircular calculation (this problem was discussed recentlyalso in [64]). However, we note that the Bern value at0.8 GeV is practically identical to what has been called“constrained” fit to data (CFD) solution of the Madrid An alternative parametrization written as a dispersion relationin terms of the imaginary part of (9) leads practically to the sameresults. phase [56], which we adopt, and which is independentof form factor data. Actually, the error attached to thisinput to Bern phase is larger (more than double) than theuncertainty attached to the CFD solution, which reducesthe possible bias. Moreover, as we shall explain later,in our determination we take the simple average of theresults obtained with the two phase-shifts, which reducesfurther the potential bias produced by this input andpractically avoids the danger of circularity.We have calculated the integral (6) using the
BABAR data [38] from t in up to √ t = 3 GeV, smoothly contin-ued with a constant value for the modulus in the range3 GeV ≤ √ t ≤
20 GeV, and a 1 /t decreasing modulusat higher energies, as predicted by perturbative QCD[14, 15, 20, 21]. This choice is expected to overestimatethe true value of the integral (see Refs. [67, 68, 70] fora detailed discussion), which has the effect of leading toweaker bounds due to a monotonicity property discussedin the Appendix. As concerns the weight ρ ( t ), severalchoices have been investigated in [70], leading to stableresults in most of the investigated regions. In the presentwork, we have adopted the weight ρ ( t ) = 1 /t , for whichthe contribution of the range above 3 GeV to the inte-gral (6) is only of 1%. The value of I obtained with thisweight is [70] I = 0 . ± . , (10)where the uncertainty is due to the BABAR experimentalerrors. In the calculations we have used as input for I thecentral value quoted in Eq. (10) increased by the error,which leads to the most conservative bounds due to themonotonicity property mentioned above.On the spacelike axis at moderate and large Q theform factor is extracted indirectly, from experimentalmeasurements of the pion electro-production from a nu-cleon target, where a virtual photon couples to a pionin the cloud surrounding the nucleon. As a consequence,there are uncertainties associated with the off-shellnessof the struck pion and the consequent extrapolation tothe physical pion mass pole, which leads to uncertaintiesin the extraction of the form factor. The errors appearto be under control in the most recent determinations of F π Collaboration at JLab [31, 32], as shown in the sub-sequent analysis [33]. Therefore, as spacelike input (7)we have used the values [31, 32] F Vπ ( − .
60 GeV ) = 0 . ± . +0 . − . ,F Vπ ( − .
45 GeV ) = 0 . ± . +0 . − . . (11)We mention that we do not use as input the data on thespacelike axis near the origin, obtained from eπ scatteringby NA7 Collaboration [22]. We shall however compareour predictions for this region with the NA7 data andwith the lattice calculations [12].A major role in increasing the strength of the boundsis played by condition (8). We shall take 0 .
65 GeV ≤√ t t ≤ .
71 GeV, since in this region the modulus mea-sured by various experiments exhibits smaller variations
Experiment Number of pointsCMD2 [34] 2SND [37] 2
BABAR [38, 39] 26KLOE 2011 [41] 8KLOE 2013 [42] 8BESIII [46] 10CLEO [47] 3ALEPH [48, 49] 3OPAL [50] 3Belle [51] 2TABLE I: Number of points in the region 0 .
65 GeV ≤ √ t ≤ .
71 GeV where the modulus is measured by the e + e − anni-hilation and τ -decay experiments considered in the analysis. than in other energy regions and a higher degree of mu-tual consistency. Moreover, this region is close to theregion of interest and therefore has a stronger effect onimproving the bounds than the input from higher ener-gies. The e + e − data are taken below 0.705 GeV andthe τ -decay data below 0.710 GeV, with the exception ofone datum from CLEO that corresponds to an energy of0.712 GeV. Since this last datum is at an energy that isonly marginally higher than the upper limit of the afore-mentioned energy range, it is included in the analysis.The numbers of experimental points from various ex-periments, used as input in our analysis, are summarizedin Table I. We emphasize that in this region the e + e − -annihilation and τ -decay experiments are fully consis-tent, so it is reasonable to use all the experiments on anequal footing.The extraction of the values of timelike modulus | F Vπ ( t ) | from the cross-section of the process e + e − → π + π − and the spectral function measured in τ -decay ex-periments requires the application of several corrections,described in detail in Appendix B of [2]. In particular,for the e + e − experiments the isospin correction due to ω has been applied as discussed above, and the vacuumpolarization has been removed from the data. IV. DETERMINATION OF THE FORMFACTOR AND ITS UNCERTAINTY
Using the algorithm presented in the Appendix, we ob-tain an allowed range for the value of F Vπ ( t ) (or | F Vπ ( t ) | )at an arbitrary point t < t in for every set of specificvalues of the input quantities. However, with the excep-tion of the exact condition F Vπ (0) = 1, the input quan-tities are known only with some uncertainties. One ofthe key aspects of our calculation is the proper statisti-cal treatment of the errors. This is achieved by randomlysampling each of the input quantities with specific dis-tributions: the phase of F Vπ ( t ), which is the result of atheoretical calculation, is assumed to be uniformly dis-tributed, while for the spacelike and the timelike data,which are known from experimental measurements, weadopt Gaussian distribution with the measured centralvalue as mean and the quoted error (the biggest errorfor spacelike data where the errors are asymmetric) asstandard deviation.For each point from the input statistical sample, if theinput values are compatible, we calculate from Eq. (A16)upper and lower bounds on F Vπ ( t ) (or | F Vπ ( t ) | ). Sinceall the values between the extreme points are equally al-lowed, we uniformly generate values of F Vπ ( t ) (or | F Vπ ( t ) | )in between the bounds. For convenience, the minimalseparation between the generated points was set at 10 − and for allowed intervals smaller than this limit no in-termediate points were created. In this way, for eachinput from one spacelike t s < t t inthe region (0 . − .
71) GeV, we obtain a large sampleof values of F Vπ ( t ) (or | F Vπ ( t ) | ). The results proved tobe stable against the variation of the size of the randomsample and the minimal separation mentioned above.In Fig. IV, we present for illustration the distributionsof the output values of the form factor at several pointsof interest (two spacelike points in the upper panel, andtwo timelike points, one below and the other above theunitarity threshold, in the lower panel). The histogramshave been obtained using as input the Bern phase, thevalue at the spacelike point t s = − . , and themodulus at one timelike point measured by BABAR [38].Similar results have been obtained using as input theMadrid phase and other experimental data. One can seethat the distributions are very close to a Gaussian andallow the extraction of the mean value and the standarddeviation (defined as the 68.3% confidence limit (CL)intervals) for the values of interest F Vπ ( t ) or its modulus.The next step is to take the average of the results ob-tained with input from various measurements. Since thedegrees of correlations between the measurements at dif-ferent energies are expected to vary from one experimentto another, we perform first the average of the values ob-tained with input from each experiment. As argued in[74], the most robust average of a set of n measurements a i is the weighted average¯ a = n (cid:88) i =1 w i a i , w i = 1 /δa i (cid:80) nj =1 /δa j , (12)where δa i is the error of a i .For the best estimation of the error in the case of un-known correlations, the prescription proposed in [74] isto define a function χ ( f ) χ ( f ) = n (cid:88) i,j =1 ( a i − ¯ a )( C ( f ) − ) ij ( a j − ¯ a ) (13)in terms of the covariance matrix C ( f ) with elements C ij = (cid:40) δa i δa i if i = j,f δa i δa j if i (cid:54) = j. (14) The parameter f denotes the fraction of the maximumpossible correlation: for f = 0 the measurements aretreated as uncorrelated, for f = 1 as fully (100%) corre-lated.If χ (0) < n −
1, the data might indicate the existenceof a positive correlation. The prescription proposed in[74] is to increase f until χ ( f ) = n −
1. With the solution f of this equation, the standard deviation σ (¯ a ) of ¯ a isdetermined from the variance [74] σ (¯ a ) = n (cid:88) i,j =1 ( C ( f ) − ) ij − . (15)On the other hand, a value χ (0) > n − χ (0) / ( n −
1) is not very far from 1, the proce-dure suggested in [73, 74] is to rescale the variance σ (¯ a )calculated with (15) by the factor χ (0) / ( n − f derived from data.Then the results obtained with the two phases, Bern andMadrid, were combined in a conservative way by tak-ing the simple average of the central values and of theuncertainties. The same conservative average was usedfor combining the results obtained with the two spacelikedata (11).The last step was to combine the individual values ob-tained with measurements by the different experimentslisted in Table I. Again, the error correlation for thesevalues is difficult to assess a priori . Therefore, we haveapplied the same data-driven procedure described abovefor finding the correlations. Since, as discussed in [2], thedata from e + e − -annihilation and τ -decay experimentsare consistent in the region 0 . − .
71 GeV, the resultsfrom all the 10 experiments in Table I can be combinedinto a single central value and standard deviation whichwe quote as the error.
V. RESULTS
We have applied the procedure described above for de-riving central values and standard deviations for F Vπ ( t )in three energy regions: small spacelike momenta Q = − t ≤ .
25 GeV , where measurements are available fromNA7 experiment [22], larger spacelike momenta, up to Q ≤ . , and the unphysical timelike region 0 63 GeV, and have used these results for (cid:45) t (cid:61) F Π (cid:72) t (cid:76) D i s t r i bu t i on V (cid:45) t (cid:61) F Π (cid:72) t (cid:76) D i s t r i bu t i on V t (cid:61) F Π (cid:72) t (cid:76) D i s t r i bu t i on V t (cid:61) (cid:200) F Π (cid:72) t (cid:76) (cid:200) D i s t r i bu t i on V FIG. 1: Statistical distributions of the output values of the form factor at two spacelike points (upper panel) and two timelikepoints, one below and the other above the unitarity threshold t + (lower panel). In the calculation, we used the Bern phase, theinput from the spacelike point t s = − . , and the modulus at the timelike point √ t t = 0 . 699 GeV measured by BABAR [38]. The vertical lines indicate the 68.3% confidence limit (CL) intervals). a new determination of the HVP contribution from en-ergies below 0 . 63 GeV to the muon g − 2. In the fol-lowing subsections we present the results for each kine-matical region, namely small spacelike momenta, largespacelike momenta, unphysical timelike momenta, andtimelike momenta on the unitarity cut below 0.63 GeV.The implications of these determinations are also studiedin each of these subsections. A. Small spacelike momenta At small spacelike momenta squared, Q ≤ . 25 GeV ,the pion form factor has been measured from ep elasticscattering by the NA7 experiment [22], considered fora long time a landmark experiment. Recently, the ETMcollaboration [12] reported the most precise lattice calcu-lations of F Vπ ( − Q ) for small Q . The comparison withthe lattice results has been actually the main motivationfor choosing this kinematical region in our study. It turnsout that our predictions for the form factor in this regionare very precise: the errors, obtained by the procedure described in the previous section, vary from 0.0005 nearthe origin to 0.003 at the end of the region.In Fig. 2, we present the values of the form factorcalculated in this work at a number of spacelike pointsbelow 0 . 25 GeV . Also shown are the experimental datafrom Ref. [22] and the results of the lattice calculationreported in Ref. [12], shown as a band which includesall the uncertainties. One can see that our results areconsistent with the lattice values, and are much moreprecise. It is a challenge for the future lattice calcula-tions to increase the precision to the level reached bythe phenomenological determination based on analytic-ity and unitarity.It may be noted that our procedure can be extendedfurther as there is no real constraint on the range of valuesto be probed in this sector, but for practical purposes, ourdetermination has been limited to the same range as inthe lattice study and in the NA7 experiment. [GeV ] F π V (- Q ) ETM 2018NA7 1986this work FIG. 2: The predictions for the pion form factor in the space-like region near the origin derived in this work, comparedwith the experimental results of the NA7 Collaboration [22]and the lattice calculations of the ETM Collaboration [12]. B. Large spacelike momenta and the onset ofperturbative QCD It has long been known that in the case of the pionform factor the asymptotic regime described by the dom-inant term (2) of perturbative QCD sets in quite slowly,due to the complexity of soft, nonperturbative processesin QCD in the intermediate Q region. Several nonper-turbative approaches have been proposed for the studyof the pion form factor, including QCD sum rules [75],quark-hadron local duality [76–79], extended vector me-son dominance [80], light-cone sum rules [81–83], sumrules with nonlocal condensates [84–86], AdS/QCD mod-els [87, 88], k T factorization method [89], dispersion rela-tions treatment [90], covariant spectator theory [91], andDyson-Schwinger equation framework [92]. In particu-lar, the onset of the asymptotic regime in the presence ofSudakov corrections [93] and large N c Regge approaches[94] is expected to be quite slow. Constructing a fullyvalid model to describe the form factor at intermediateenergies in fundamental QCD still remains a major the-oretical challenge.Measurements of the spacelike form factor for space-like momenta are reported in Refs. [23–32], the mostprecise being the recent results of the JLab collaboration[31, 32] quoted in Eq. (11). The lack of precise experi-mental data in the higher Q region is a major obstacle toconfirm or discard the theoretical models available. Thecalculation presented in this work provides an alternativeway for testing the onset of the asymptotic QCD regimeand the validity of various theoretical models proposedfor intermediate energies.In the left panel of Fig. 3, the predictions of this workfor Q < , represented as a cyan band which in-cludes the full error, are compared with some of the ex-perimental data. We recall that in our calculation the only input from the spacelike axis consists of the pointsgiven in Eq. (11), denoted as Horn in Fig. 3. Theincreased precision of our determinations is due to thetimelike information. One can see that, except for afew points, the experimental measurements are in gen-eral agreement with our determinations.At higher spacelike momenta, the precision of our pre-dictions starts to diminish, since the extrapolation ismore sensitive to the values of the form factor at interme-diate timelike energies, for which no precise informationis available. To account for this, we have adopted the con-servative, weaker condition (6). Up to Q around 8 GeV ,the precision nevertheless is acceptable, allowing us toprobe the onset of the asymptotic regime predicted byfactorization and perturbative QCD. In the right panelof Fig. 3, we compare our predictions shown in cyanband with perturbative QCD at LO and NLO, and withsome theoretical models. The gray band corresponds tothe NLO result obtained by varying the renormalizationscale in suitable range following [21]At first sight we note that perturbative QCD at LOcan not reliably describe the form factor at Q ≤ .Though the description improves at NLO it is still un-reliable for Q ≤ . . We limit ourselves only tothese conservative statements, since precisely at the ener-gies where the NLO and our predictions start to becomecompatible, our procedure meets its natural limitations.This can be seen in the fact that our band hits the x -axisin right panel of Fig. 3, while there are strong argu-ments (cf. for instance Ref. [71]) that this form factorcannot have zeros on the spacelike axis. Therefore, werefrain from making definite statements for higher Q , inview of the fact that this is the region where our methodlacks the precision that it has in the other three regimesconsidered in this work.As we discussed above, there are many theoreticalmodels in the literature for addressing the properties ofthe form factors in this region. For illustration, we haveconsidered the predictions from four of these as typicalexamples. For instance, the theoretical models proposedin [78, 82] appear to be consistent with the phenomeno-logical band, while the predictions of [88, 94] appear tobe too high.We note finally that the results derived in this work areconsistent with those derived in our previous work [67],being more precise, since we now included informationon the modulus of the form factor on the timelike axisand used extensive Monte Carlo simulations for the erroranalysis. C. Unphysical timelike region No experimental information or QCD lattice calcula-tions are available for the pion form factor in the unphys-ical timelike region between the origin and the unitaritythreshold t + . For this region our method allows to makevery precise predictions. In Table II, we list the cen- Q [GeV ] Q F π V (- Q ) [ G e V ] this workAckermannAmendoliaBebekBrauelBrownHornVolmer Q [GeV ] Q F π V (- Q ) [ G e V ] this workpQCD @ LOpQCD@NLORuiz Arriola, Broniowski (2008)Braun, Khodjamirian, Maul (2000)Radyushkin (2001)Brodsky, de Teramond (2008) FIG. 3: The pion form factor calculated in this work on the spacelike axis, represented as a band which includes the total error.Left panel: comparison with experimental data. Right panel: comparison with perturbative QCD at LO and NLO, and withseveral nonperturbative models. tral values and the errors on F Vπ ( t ) at several unphysicaltimelike points. This region cannot be accessed by ex-periment, but it can be by the lattice, in principle, soour results can be viewed as a benchmark for lattice pre-dictions in the future. √ t GeV F Vπ ( t )0.140 1 . ± . . ± . . 242 1 . ± . . 279 1 . ± . F Vπ ( t ) in the timelikeregion below the unitarity threshold √ t + = 2 m π . In this region the predictions of chiral perturbationtheory are expected to be most accurate. The precisedeterminations in table II can be used to determine thecurvature c and higher shape parameter d of the Taylorexpansion F Vπ ( t ) = 1 + (cid:104) r π (cid:105) t/ ct + dt + O ( t ). D. Low energies above the unitarity threshold andthe contribution to muon g − As mentioned in the Introduction, above the unitaritythreshold, where the form factor is a complex function,its modulus is extracted from the cross section of the e + e − → π + π − process, or, using isospin symmetry, fromthe hadronic decay of the τ lepton. The τ decay has beenfor a long time the most precise source of information,in spite of the nontrivial corrections that are requiredto convert the measured spectral functions into genuinevalues of | F Vπ ( t ) | . However, the accuracy of the e + e − experiments improved gradually, the extraction of the modulus being based at present almost exclusively onthem.At low energies, the modulus of the form factor ispoorly known, due to the difficulties of the experimen-tal measurements in this region: only two experiments, BABAR [38] and KLOE [40–42] reported data at low en-ergies, and unfortunately they are not consistent amongthem. Our method allows a precise determination of | F Vπ ( t ) | at low energies. In Fig. 4 we present our re-sults, together with the experimental values of BABAR [38] and KLOE [41, 42]. For convenience, we show thevalues of the modulus squared, which enter directly intothe calculation of the two-pion contribution to the muonmagnetic anomaly. One can see that our predictions aremuch more precise than the available experimental re-sults, especially at energies below 0.5 GeV. [GeV]246810 | F π V ( t ) | this workBABAR 2009KLOE 2010KLOE 2013 FIG. 4: Our predictions for the modulus squared of thepion form factor on the cut below 0.63 GeV, compared with BABAR and KLOE experimental data. For completeness, we list in Table III the central valuesand the uncertainties of the modulus squared of the formfactor at several energies below 0 . 63 GeV. √ t GeV | F Vπ ( t ) | √ t GeV | F Vπ ( t ) | . ± . 004 0 . 437 2 . ± . . ± . 004 0 . 455 2 . ± . . ± . 004 0 . 472 2 . ± . . ± . 004 0 . 490 3 . ± . . ± . 004 0 . 507 3 . ± . . ± . 005 0 . 525 4 . ± . . ± . 006 0 . 542 4 . ± . . ± . 007 0 . 560 5 . ± . . ± . 008 0 . 577 6 . ± . . ± . 009 0 . 595 7 . ± . . ± . 011 0 . 612 8 . ± . . ± . 012 0 . 630 10 . ± . | F Vπ ( t ) | in the rangefrom two-pion threshold to 0.63 GeV. We shall use now these results for making a new de-termination of the low-energy pion-pion contribution tothe muon anomaly. The leading order (LO) two-pioncontribution to a µ from energies below √ t up , which doesnot contain the vacuum polarization but includes one-photon final-state radiation (FSR), is expressed in termsof F Vπ ( t ) as a ππµ | ≤ √ t up = α m µ π (cid:90) t up t + dtt K ( t ) β π ( t ) F FSR ( t ) | F Vπ ( t ) | . (16)In this relation, β π ( t ) = (1 − m π /t ) / is the two-pionphase space relevant for e + e − → π + π − annihilation, K ( t ) = (cid:90) du (1 − u ) u ( t − u + m µ u ) − (17)is the QED kernel function [95], which exhibits a drasticincrease at low t , and F FSR ( t ) = (cid:16) απ η π ( t ) (cid:17) (18)is the FSR correction, calculated in scalar QED [96, 97].Using the central values of | F Vπ ( t ) | given in Table III,the integral (16) gives (132 . ± . × − , where wequoted an uncertainty due to the method of integration.In order to estimate the statistical error σ a µ of this result,we shall apply the standard error propagation, expressedin our case as σ a µ = (cid:34)(cid:90) t up t + (cid:90) t up t + dtdt (cid:48) Cov( t, t (cid:48) ) W ( t ) W ( t (cid:48) ) (cid:35) / , (19)where W ( t ) = α m µ π K ( t ) t β π ( t ) F FSR ( t ) , (20) and Cov( t, t (cid:48) ) is the covariance matrix describing the cor-relation of the errors on | F Vπ | at two points t and t (cid:48) . Fora most conservative estimate, we assume fully correlatederrors, which means that we takeCov( t, t (cid:48) ) = σ ( t ) σ ( t (cid:48) ) , (21)where σ ( t ) is the error on | F Vπ ( t ) | , determined by theprocedure described in Sec. IV. Then the integral (19)gives an error of 0 . × − . Adding to this the inte-gration error quoted above, we finally obtain a ππµ | ≤ . 63 GeV = (132 . ± . × − . (22)For further comparison, we quote also the result ob-tained using the timelike input on the modulus in therange (0.65-0.71) GeV only from the e + e − experiments: a ππµ | ≤ . 63 GeV = (132 . ± . × − . (23)The values (22) and (23) are fully consistent with ourprevious results (133 . ± . × − and (133 . ± . × − , respectively, obtained in [2] for the samequantities with a slightly different method. The differ-ence stems from the fact that in Ref. [2] we generated thestatistical distribution of the integral (16) directly fromMonte Carlo simulations, without deriving the modulussquared of the form factor at each energy below 0.63 GeV.We quote also the result a ππµ | ≤ . 63 GeV = 132 . . × − of the recent analysis [64], which exploits analyt-icity and unitarity by using an extended Omn`es repre-sentation of the form factor in a global fit of the phe-nomenological data on e + e − → π + π − from energiesbelow 1 GeV and the NA7 experiment. We note alsothat the direct integration of the interpolation of the e + e − data below 0.63 GeV, proposed in [61], gives a ππµ | ≤ . 63 GeV = (131 . ± . × − .It may be noted that the explicit values listed in TableIII for this region allow an evaluation of the dipion con-tribution to the muon anomaly in any interval of interest. VI. DISCUSSION AND CONCLUSIONS In the present work we have obtained high-precisionpredictions for the pion electromagnetic form factor inseveral kinematical regions of interest. We have useda method based on analyticity and unitarity, whichdoes not involve standard dispersion relations or specificparametrizations. The input, summarized in Sec. II,consists of the phase of the form factor on a part of theunitarity cut and a conservative integral condition on themodulus squared on the remaining part. The experimen-tal values at some discrete points on the timelike and thespacelike axes are also included. We thank T. Teubner for this calculation. g − Q our results are much more precisethan the recent lattice calculations [12], and at larger Q we confirm our previous conclusion [67] that the asymp-totic regime of perturbative QCD is away from the region Q ≤ .On the timelike axis, we derived high-precision valuesof the modulus squared of the form factor on the unitaritycut below 0 . 63 GeV, shown in Fig. 4 and Table III. Ourpredictions are much more precise than the experimentalvalues available in this region from BABAR and KLOEexperiments, especially below 0.5 GeV. Also, in Sec. V,the determinations we provide in the unphysical timelikeregion could serve as a benchmark for theoretical probesin this region.From the precise values given in Table III, we haveperformed a new determination of the two-pion contri-bution from low energies to the muon g − 2. Our re-sults for a ππµ | ≤ . 63 GeV are given in Eqs. (22) and (23),where the first uses the input from both e + e − and τ ex-periments, and the second only from e + e − experiments.These results are consistent with the values derived inour previous work [2], where the technique of rigorousanalytic bounds and Monte Carlo simulations has beenapplied in a slightly different way, by deriving a statisticaldistribution directly for the quantity a ππµ | ≤ . 63 GeV .As seen from the values quoted at the end of the pre-vious section, our results are consistent with the predic-tion of the recent analysis [64] based on analyticity andunitarity, while the result obtained from the direct inte-gration of the data [61] is slightly lower. We emphasizethat we do not use as input experimental data from ener-gies below 0.63 GeV or from NA7 experiment. Thus, ourprediction for a ππµ | ≤ . 63 GeV is to a certain extent comple-mentary to the determination of the analysis performed in [64], which exploits analyticity and unitarity in a dif-ferent way and uses as input low-energy data.This work represents the state of the art in an im-portant low-energy sector of the Standard Model, whichis going to be tested at the upcoming Fermilab experi-ment E969. In contrast to our prior publications [1, 2],which were focused on the determination of a single num-ber, here we have obtained an extensive tabulation of thevalues of the electromagnetic form factor in several sig-nificant kinematical regions. Using these determinations,the value of the dipion contribution to the muon anomalyremains consistent with the value reported earlier, prov-ing the robustness of the approach.The present work combines strong theoretical inputswith modern Monte Carlo methods along with high pre-cision experimental information and phase shift informa-tion in regions where experiments are in agreement toshed light on regions where either experiments do nothave sufficient precision or where there are significantdisagreements, or regions which are not directly acces-sible by experiment. It also offers a test to theoreticalpredictions based on very different approaches to stronginteraction phenomenology. Acknowledgments We would like to thank Urs Wenger for useful cor-respondence. B.A. acknowledges support from theMSIL Chair of the Division of Physical and Math-ematical Sciences, Indian Institute of Science. I.C.acknowledges support from the Ministry of Researchand Innovation of Romania, Contract No. PN18090101/2018. D.D. is supported by the DST, Gov-ernment of India, under INSPIRE Fellowship (No.DST/INSPIRE/04/2016/002620). Appendix A: Solution of the extremal problem Using the approach proposed in [65], the extremalproblem formulated at the end of Sec. II can be re-duced to a standard analytic interpolation problem [98](also known as a Meiman problem [99]). We review inwhat follows the main steps of the proof. As discussedin Sec. III, we first remove from the form factor theisospin-violating correction F ω ( t ), so in what follows weshall consider the function F ( t ) defined in (4). The nextstep is to introduce a function h ( t ) by writing F ( t ) = O ( t ) h ( t ) , (A1)where O ( t ) is the Omn`es function defined as O ( t ) = exp (cid:32) tπ (cid:90) ∞ t + dt (cid:48) δ ( t (cid:48) ) t (cid:48) ( t (cid:48) − t ) (cid:33) . (A2)In this relation, δ ( t ) is equal to δ ( t ) at t ≤ t in and is anarbitrary smooth (Lipschitz continuous) function above t in , which approaches asymptotically π .1From the Fermi-Watson theorem (5), it follows that h ( t ) is real on the real axis below t in , since the phase of F ( t ) is exactly compensated by the phase of O ( t ). Tak-ing into account the fact that h ( t ) satisfies the Schwarzreflection property, this implies that it is holomorphicon the real axis below t in , having a branch cut only for t ≥ t in .In terms of h ( t ), the equality (6) can be written as1 π (cid:90) ∞ t in dt ρ ( t ) |O ( t ) | | h ( t ) | ≤ I. (A3)This relation can be written in a canonical form if weperform the conformal transformation,˜ z ( t ) = √ t in − √ t in − t √ t in + √ t in − t , (A4)and express the factors multiplying | h ( t ) | in terms of anouter function, i.e. a function analytic and without zerosin the unit disk | z | < 1. In practice, it is convenient toconstruct it as a product of two outer functions [65, 66]:the first one, denoted as w ( z ), has the modulus equal to (cid:112) ρ ( t ) | d t/ d˜ z ( t ) | . For the choice ρ ( t ) = 1 /t , it is given bythe simple expression w ( z ) = (cid:114) − z z . (A5)The second outer function, denoted as ω ( z ), has the mod-ulus equal to |O ( t ) | , and can be calculated by the integralrepresentation ω ( z ) = exp (cid:32) (cid:112) t in − ˜ t ( z ) π (cid:90) ∞ t in ln |O ( t (cid:48) ) | d t (cid:48) √ t (cid:48) − t in ( t (cid:48) − ˜ t ( z )) (cid:33) . (A6)If we define the function g ( z ) by g ( z ) = w ( z ) ω ( z ) h (˜ t ( z )) , (A7)where ˜ t ( z ) is the inverse of z = ˜ z ( t ) defined in Eq.(A4),the condition (A3) can be written with no loss of infor-mation as 12 π (cid:90) π d θ | g ( ζ ) | ≤ I, ζ = e iθ . (A8)This condition leads to rigorous correlations among thevalues of the analytic function g ( z ) and its derivatives atpoints inside the holomorphy domain, | z | < D ≥ D defined as D = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I − g (0) ξ ξ ξ ξ z − z z z − z z z z − z z ξ z z − z z z − z z z − z z ξ z z − z z z z − z z z − z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (A10) where the real values z n ∈ ( − , 1) are defined as z n = ˜ z ( t n ) , (A11)in terms of the two points t = t s and t = t t used asinput and the value t where we want to calculate boundson the form factor, and ξ n = g ( z n ) − g (0) , ≤ n ≤ . (A12)The inequality (A9) defines an allowed domain for thereal values g ( z n ). For n = 1 and n = 3 with t < t + , wehave from Eqs. (A1) and (A7) g ( z n ) = w ( z n ) ω ( z n ) F ( t n ) / O ( t n ) , (A13)while for n = 2 and n = 3 with t > t + g ( z n ) = w ( z n ) ω ( z n ) | F ( t n ) | / |O ( t n ) | , (A14)where the modulus |O ( t ) | of the Omn`es function is ob-tained from (A2) by the principal value (PV) Cauchyintegral |O ( t ) | = exp (cid:32) tπ PV (cid:90) ∞ t + dt (cid:48) δ ( t (cid:48) ) t (cid:48) ( t (cid:48) − t ) (cid:33) . (A15)One can show that for each specific input, the deter-minant (A10) is a concave quadratic function of the un-known value F ( t ) for t < t + , or the modulus | F ( t ) | for t > t + , so the inequality (A9) can be written as Ax + 2 Bx + C ≥ , A ≤ , (A16)where x = F ( t ) or x = | F ( t ) | . This inequality leads toa definite allowed range for x if B − AC ≥ B − AC < 0. 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