Abstract

The measurements of the total cross section of the reaction from the VENUS, TOPAS, OPAL, DELPHI, ALEPH, and L3 collaborations, collected between 1989 and 2003, are used to perform a test to validate the current quantum electrodynamics (QED) theory and search for possible deviations with the direct contact term annihilation. By observing a deviation from the QED predictions on the total cross section of the reaction above  GeV, a non-QED direct contact term is introduced following the dimension 6 effective theory to explain the deviation. In the non-QED direct contact term, a threshold energy scale is included and explained to the finite interaction length in direct contact term and in consequence the size of the electron involved in the annihilation area. The experimental data of the total cross section is compared to the QED cross section by a test, which gives a best fit of the to be  GeV, corresponding to a finite interaction length of (cm). In the direct contact term annihilation, this interaction length is a measure of the size of an electron . By combining all the data results from the mentioned collaborations, we have at least 2 to 3 times more statistics than every single experiment at high region. This induces the best precision on compared to the previous measurements.

1. Introduction

The Standard Model (SM) of particle physics is the theory describing three of the four known fundamental forces (the electromagnetic (QED), weak, and strong interactions). After the discovery of the three families of fundamental particles, the gluons, photon, Z-boson, and W-boson, the discovery of Higgs in 2012 established the last cornerstone of the SM. Missing so far is the unification between the SM and the fourth interaction, the gravitation. It is for this reason essential to investigate deviations of the SM. All the three fundamental forces have a definite interaction length. In particular, the QED has an infinite interaction length. In this paper, we search for minimal interaction length. A minimal interaction length would be an indication of a deviation from the QED. The concept of a minimal interaction length suggested by path-integral quantisation [1], string theory [2, 3], black hole physics [4], and quantum gravity [57] has been introduced into quantum mechanics and quantum field theory through generalized uncertainty principle [8, 9] which restricts an accuracy of in measuring a particle position by a certain finite minimal length scale related to maximum resolution [1012] (for a review, see [13]). In gravity, the limiting quantum length is the Planck length  cm, the related energy scale  TeV. However, gravitational effects have only been tested up to 1 TeV scale [14] which corresponds to  cm [15]; therefore, minimal length could be in principle found within the range and [15]. In models with extra dimensions, the Planck length can be reduced to with  TeV, which results in modification of the cross sections of basic scattering processes ([16] and references therein).

In this paper, we summarize the results of investigating experimental data on the annihilation reaction motivated by the search for manifestation of the non-point-like behavior of fundamental particles.

In a first approach, the question of the intrinsic structure of a charged spinning particle like an electron has been discussed in the literature since its discovery by Thomson in 1897. In quantum field theory, a particle is assumed to be point-like, and classical models of point-like spinning particles describe them by various generalizations of the classical Lagrangian [1731]. Another type of point-like model [3242] goes back to Schrödinger’s suggestion that the electron spin can be related to its zitterbewegung motion [43]. The development of this approach ends today in the excellent experimental tested Standard Model of particle physics. The model is described by three families of point-like particles. The point-like structure of the fundamental particle with rest mass requests an unnatural density . The SM is a stand-alone model which does not include the fourth interaction, the gravity.

A second approach works with extended particle models. Early electron models based on the concept of an extended electron, proposed by Abraham and Lorentz more than a hundred years ago [44, 45], encountered the problem of preventing an electron from flying apart under the Coulomb repulsion. Theories based on geometrical assumption about the “shape” or distribution of a charge density were compelling to introduce cohesive forces of nonelectromagnetic origin (the Poincaré stress) [46]. A new review of models is discussed in [47]. In this paper, we apply this model [47]. To find evidence for an extended particle picture, we used available data from experiments performed to search for a non-point-like behavior, which focuses on characteristic energy scales related to characteristic length of interaction region in reference [4854]. Experimental limits on the size of a lepton in reference [4852] appear to be much less than its classical radius which suggests the existence of a relatively small characteristic length scale related to gravity in reference [4852]. A non-point-like behavior of fundamental particles would open a window to the fundamental problem in physics beyond the SM for unification between the SM and the gravity in terms of general relativity. The different concepts of extended particle models discussed in reference [4447], in particular, the paper “Image of the Electron Suggested by Nonlinear Electrodynamics coupled to Gravity” [7] and “Gravity vs. quantum theory: is electron really point like?” [47] in connection with the generalized uncertainty principle [8, 9], show an example of a path to include the gravity in physics beyond the SM. Essential is that the inclusion of gravity in the different models allows a transition between the SM and the general relativity. The unnatural density of the fundamental particle would disappear. To confirm the theoretical approach of a non-point-like behavior of fundamental particles, it is essential in an experiment to detect a minimal interaction length.

To investigate the pure electromagnetic interaction using differential cross data from VENUS, TOPAS, ALEPH, DELPHI, L3, and OPAL sets the limit on a maximal resolution at scale  TeV by the character length  cm with a 5 significance [53, 54]. An earlier report set a 2.6 on an axial-vector contact interaction in the data on at center-of-mass energies 192-208 GeV [55].

The available data from the accelerators favor two experiments to test the finite interaction length the and reactions. Both reactions are shown in Figure 1.

The QED reaction is testing the behavior of the electrodynamics long-range force of the reaction. The two ’s in the final state of the reaction are indistinguishable. The reaction performs for this reason via the t- and u-channels. The s-channel is forbidden, by the law of angular momentum conservation. The two ’s in the final state are left-handed and right-handed polarized. They couple to total spin zero. Under these circumstances, the s-channel with spin one for and is highly suppressed.

The Bhabha reaction is not only sensitive to the long-range force of the electromagnetic reaction, in addition via the also to the short-range force of the electroweak interaction. The reaction proceeds, via scattering in the s-channel and t-channel. The and in the initial state and final state are identical. The gammas in the final state of the reaction disappear. The high charge sensitivity of the involved detectors allows to suppress the background reaction, even under the circumstances that at the pole, the total cross section of the reaction is a factor two bigger as the total cross section of the reaction.

After the commissioning of the high energy accelerators 1986 TRISTAN at KEK, the VENUS collaboration (1989) initiated the first experiments at  GeV to investigate the total and differential cross section of the reaction. The experiments continued until LEP was closed 2000 at  GeV.

In detail, the reaction was investigated by the VENUS collaboration [56] from energies  GeV to 57 GeV, OPAL collaboration [57] at the pole at  GeV, TOPAS collaboration [58] at  GeV, ALEPH collaboration [59] at the pole at  GeV, DELPHI collaboration [6062] from 1994 to 2000 at energies  GeV to 202 GeV, L3 collaboration [63] from 1991 to 1993 at the pole range from  GeV to 93.7 GeV, L3 collaboration [64] from  GeV to 207 GeV, and OPAL collaboration [65] from  GeV to 209 GeV.

The experimental data of the differential cross section of these six collaborations from  GeV to 207 GeV are compared to the theoretical predicted QED differential cross section. Possible deviations from QED were studied in terms of contact interaction and excited electron exchange shown in Figure 2.

The deviation to the QED reaction is visible in the angular distribution of the gammas from the experiment to the QED theory and in the experimental total cross section data to the QED. It is necessary to perform, for example, a test or similar statistical tests, to search in the test for a minimum or limit for a scale parameter (GeV). This parameter finally allows to define a finite radius of the electron or the mass of an excited heavy electron. It is important to notice that a signal of deviation from QED is not visible in new final state particles like for the HIGGS search; it is hidden only in the angular distribution of the differential cross section and in the energy dependence of the total cross section to the QED values.

All the collaborations search for bounds on effective interactions from the reaction , for example, [66]. Cutoff parameter is used to set mass scales of different dimensional interactions. Data of differential cross section are used to set limits on compositeness scales and in the direct contact interaction of the diagram shown in Figure 2(b) and search for excited electrons in the t- and u-channels of the diagram shown in Figure 2(c). For example, the L3 collaboration published four papers (1992, 1996, 1997, and 2000) and set (1992) [6770] limits on  GeV and  GeV and a limit on the mass of an excited electron to  GeV.

1.1. Outline of This Paper

The VENUS, TOPAS, OPAL, DELPHI, L3, and ALEPH collaborations used the differential cross section of the reaction to search for a deviation from QED. This was performed for certain energy ranges and luminosities.

Even all the mentioned experiments above are closed and there is no more new data come in, it is possible to combine all these data results from each single experiment and perform a global test. This combination can give us 2-3 times more data in high region, e.g.,  GeV, thus yielding much better results. It is also important to mention that the scale parameter allows to define a finite radius of electrons through the generalized uncertainty principle as discussed in Section 2.5. The limits on are not presented in any of the previous single experiment result. In this study, we motived to analyze the combined data results from all the 6 experiments and give the first limits on through the process.

We have performed a global FIT, using the differential data from these six research projects to investigate , , and for energies from  GeV to 207 GeV including the associated luminosities [51, 54, 71]. That analysis allowed to set an approximately limit on the finite size of the electron (cm) and on the mass of an exited electron of (GeV). The deviation in the differential experimental cross section from the QED values of approximately 4% was only visible in the fit results but not direct in the comparison of the experimental and theoretical QED cross section.

The aim of this instigation is to prove in a fit that the use of only the total cross section of all these data implies a similar result. First, it is necessary to discuss the theoretical framework of the calculation of the total QED cross section, in particular, the fact that an analytic precise QED cross section must be calculated via a Monte Carlo program. Second, as no total cross section for the QED reaction from  GeV to 207 GeV exists, we introduced a model for a total cross section using the data from the 6 collaborations. Including this information, it is possible to perform a total fit with all data. Finally, it is possible to discuss the results.

2. Theoretical Framework

To test the point structure of the electron, request a high precision of the theoretical QED calculation of the differential and total cross section of the reaction.

The interactions of particles are dictated by symmetry principles of local gauge invariance and conserved physical quantities. Mathematically, the Lagrangian formalism is used to connect symmetries and conservation laws.

The Euler Lagrange equation is used to describe a free particle with spin 1/2. The minimum of the action path integral is the Lagrangian density of the Dirac equation. where is the fermion field, is its adjoint spinor, is the gamma matrices, is the covariant derivative, and is the mass of the particle.

A particle with interaction is described by the local gauge invariance QED Lagrangian function in where is the gauge field, the mass , is the charge of the electron, is the interaction term, and .

2.1. The Lowest Order Cross Section of

The interaction term of the QED Lagrangian allows to calculate the lower case Born cross section of the reaction. The mathematical formalism uses the matrix of where stands for the final state and stands for the initial state. The derivation of the differential cross section uses the square of the matrix, including the t- and u-channels of the Feynman graphs of Figure 2(a) and neglecting the electron mass for high energies.

is a statistical factor, is the center-of-mass energy, the momentum , for high energies , and . The angle is the photon-scattering angle with respect to the beam axis.

The total Born cross section is the integral over the angle and the azimuth angle .

The precision of the Born cross sections is absolutely not sufficient to search for non-point-like behavior of the electron in a fit of the reaction.

2.2. Radiative Corrections of the QED Cross Section

All 6 collaborations used radiative corrections of the QED cross section for virtual, soft, and hard photons [54, 72, 73]. In total, 22 corrections are implemented [73, 74].

The first set of eight virtual photon corrections is shown in the Feynman graphs of Figure 3 [75].

The second set of four soft real photon initial state corrections, including the six hard photon corrections, is shown in the Feynman graphs of Figure 4 [75].

2.3. The Total Cross Section for Hard Core and Soft Core Radiative Corrections

The differential cross section for the soft- and hard-Bremsstrahlung process is not analytically known. The corrections of the Feynman diagrams from Figures 3 and 4 are calculated by numerical simulations. The details of the differential cross section for the numerical approach are shown in Appendix A and B.

An analytic equation of total integrated cross section, adding the plus reactions can be calculated with (6) and (7).

The parameter is as a function of the mass of the electron and the .

2.4. The Numerical Calculation of the Cross Sections

For practical reasons, the differential and total cross sections were used from the numerical calculation for the fit.

2.4.1. The Calculation of the Differential Cross Sections

The 6 collaborations under discussion used an event generator [73, 74] for the reaction .

The lowest order differential cross section is corrected numerically by adding the higher order correction of Figures 3 and 4 to in where is the virtual correction and and are the soft- and hard-Bremsstrahlung corrections.

If the energy of the photons from initial state radiation (soft Bremsstrahlung) are too small to be detected, the reaction can be treated as 2-photon final state. This is valid if the parameter of the reaction is fulfilled (11).

For the third-order differential cross section, the program generates three events sorted after the energies , with the correct mixture for soft and hard QED corrections shown in Figures 3 and 4. The angle between the and event is connected to the scattering angle by .

The differential cross section at an angle , an energy , and an angle bin width can be expressed with

The scattering angle is , where is the scattering angle of and is the scattering angle of , is the number of events in an angle bin width , and is the total amount of events used in the generator. The Monte Carlo generator together with (12) is used to calculate distributions of differential cross sections as a function of including the five discussed parameters.

In the past, the Monte Carlo generator [73, 74] was used from all 6 collaborations, to generate the QED cross section of the reaction. Our interest is focused on a new Monte Carlo generator BabaYaga@nlo [76]. To use the BabaYaga QED generator, request a comparison of QED cross section data from both generators. The new generator implements the same radiative corrections of Figures 3 and 4. To generate BabaYaga@nlo events, the following seven parameters are used:

To perform a fit for a finite size of the electron, an analytical expression for the differential cross section (12) is needed. The angular distribution of the differential cross section (12) is fitted by a fit using a polynomial with 6 parameters to shown in (14).

To calculate cross sections for different detectors, it is necessary for the calculation of the QED cross sections to regard the various parameters how every detector is able to measure events. We normalize all cross section measurements and calculation of the QED cross section to the L3 parameters [64].

Considering the experimental data of the ratio of the total measured cross section divided by the total QED cross section of the L3 collaboration [64] shown in Figure 2, a small deviation from the QED ratio from 90 GeV to 207 GeV is visible. To test the QED deviation as a function of , we performed two sets of tests: one set from 55 GeV to 207 GeV with 17 energy points and the other set from 91.2 GeV to 200 GeV with 7 energy points.

Including the parameters of (15), it is possible to calculate and fit all 17 angular distributions of the energies under request from  GeV to 207 GeV. The 6 fit parameters to (14) are summarized in Table 8 (upper part) in Appendix C. The same parameters for 7 energies from  GeV to 200 GeV of the generator [73, 74] are summarized in Table 8 (lower part) in Appendix C.

For an example at  GeV, the differential QED cross section fitted to (14) is shown in Figure 5.

Including the fit parameters from Table 8 together with (16), it is possible to calculate the total QED BabaYaga cross section from  GeV to 207 GeV in Figure 6. The points are the cross section.

The red line is a fit using (17) and (18) to form an analytic expression to the total QED cross section. The two fitting parameters are and .

To compare all the experimental data to the QED data, the six collaboration used the Monte Carlo QED generator [73, 74]; in this analysis, we used the generator BabaYaga@nlo [76]. For this reason, it was necessary to search for deviation between both QED generators.

The total cross section from the Monte Carlo QED generator [73, 74] and the [76] is in the range of the statistical error bar the same. At energies between  GeV and 207.0 GeV, the middle value of [73, 74] is (pb) and [76] is (pb), a total deviation of (pb). This is approximately the statistical error of the QED generator [73, 74] of (pb). The total cross section of [76] above  GeV is approximately 0.9% smaller as in [73, 74].

2.5. Deviations from QED

If the QED is a fundamental theory, the experimental parameters of the reaction should be correctly calculated by the Monde Carlo generator [73, 74] or [76] up to the grand unification scale. All collaborations [56] to [6770] investigated deviations from the QED reaction from  GeV up to  GeV. New non-QED phenomenon could become visible at high energy scales and may be below the grand unification scale. An energy scale characterized by a cutoff parameter (GeV) can be used as a threshold point for a QED breakdown and for the underlying new physics. Different models are discussed in [7781]. Deviations from QED with the reaction [8286] are investigated in a program initiated between 1991 and 2020 by the Swiss Federal Institute of Technology in Zurich (ETHZ) and USTC Hefei (University of Science and Technology of China). In this paper, we focus on a model of finite size of an interaction length in the reaction.

2.5.1. In the Direct Contact Term Annihilation, a Finite Interaction Length Is a Measure for the Size of the Electron

The QED differential and total cross sections of the reaction are calculated under the condition that the electron is point-like, without limited interaction length and coupled to the vacuum as shown in Figures 3 and 4. So far, no experiments exist to test, in particular, the non-limited interaction length of the electron up to the Planck scale. If at a certain energy scale between in the reaction, a finite interaction length appears, this defines a size of an object where the annihilation occurs. It is possible to calculate the size of the object via the generalized uncertainty principle [1012] or via the electromagnetic energy and wavelength [87] of the light that the object submits.

It is well known that the wavelength of the gammas must be smaller or equal to the size of the interaction area. If the energy of the size of the interaction area is known, the frequency of the gammas is known via . This frequency is connected to the size of the object via the wavelength to the equation . The energy scale defines under these circumstances the size of the interaction area and in consequence the size of the electron involved in the annihilation area.

It is possible to construct several effective Lagrangians containing nonstandard or couplings which are gauge invariant and only differ in their dimensions [7781]. In the lowest order effective Lagrangian, this reaction contains operators of dimension 6, 7, and 8 [66]. In the further discussion, we concentrate on the simplest operator of dimension 6, with the effective Lagrangian of

The coupling constant is related to the mass scale , is the QED covariant derivative, and is the dual of the electromagnetic tensor . The differential cross section corresponds to is shown in

We use , and higher order terms like or of are omitted.

To search for a deviation of QED, it is common to use tests. This test compares the QED cross section to the experimental measured cross section. In the test, the QED cross section is modified by a non-QED direct contact term threshold energy scale after equation (20). If the test indicates a minimum of a finite threshold energy scale , the energy of the cutoff parameter, it defines via the two equations and a finite size of the area where the annihilation must occur. For , this is a measure for size of the electron shown in (21), where is the Planck constant and is the speed of light.

Equation (21) is generic, the calculation using the generalized uncertainty principle generates the same equation. It is interesting to notice that in equation (21) for , the size of the object will be . In consequence, the point-like QED would be correct to infinite energies.

3. The Measurement of the Total Cross Section

The reaction initiates in a storage of ring accelerators; it is a very clean signal in the detector. For example, Figure 7 shows an event from LEP in the L3 detector. Shown are the position and energy storage of an event perpendicular to the beam axis in the electromagnetic BGO calorimeter. The charge sensitive detectors of the inner trackers, the outer hadron calorimeter, and the muon chambers are free of any signal.

The total cross section can be described as a function of the number of events in an angular range of the scattering angle , the luminosity , and the efficiency of the detector for events.

All seven collaborations measured the total cross section at different energies . To perform a total fit of all experimental data, a model is needed to sum over all the information together in one total cross section at one . According to Table 11 in Appendix C, seventeen data sets exist from  GeV to 207 GeV. Table 11 shows the luminosities and the references that the total cross section is published. At LEP, nine data sets exist measuring from more than one detector. For the eight data sets, only one detector has measured . A total test would have in total 17 degrees of freedom, and if more than one detector is involved in one energy , the statistical error would decrease.

All the detectors of the six collaborations use different scattering angles , luminosities , and efficiencies for the events. In common, all the collaborations used for tests the CERNLIB program MINUIT [89, 90] or Monte Carlo programs using the same radiative correction as discussed in Figures 3 and 4. The Monte Carlo program generates event number including different scattering angle ranges and efficiency . This fact allows to normalize the experimental data of the six collaborations to an imaginary L3 detector using the QED cross section of the different collaborations.

3.1. Calculation of , , Ratio , and for More than One Detector

At LEP, nine data sets exist measuring from more than one detector. To calculate the total experimental cross section (26), the statistical error (27), the ratio (28), and the statistical error of this ratio (30), the following input parameters are needed: the total experimental cross section (pb) from VENUS, TOPAS, ALEPH, DELPHI, OPAL, and total L3- event rate from Table 9 in Appendix C; the total QED cross section (pb) from VENUS, TOPAS, ALEPH, DELPHI, OPAL, and total L3- event rate from Table 10 in Appendix C; and the luminosity used from the VENUS, TOPAS, ALEPH, DELPHI, L3, and OPAL experiment from Table 11 in Appendix C.

Under this condition in a first step, the model normalizes the experimental total cross section of a at one energy to the L3 normalized experimental total cross section (23) via the total QED cross section of L3 to the total QED cross section of the detector at under investigation. For the detailed calculation, numerical values of summarized in Table 9 (including the statistical error ) are needed.

The values of are summarized in Table 10. The value of is calculated via (17) and (18), and it is also added in Table 10 for simplicity.

Second, the model introduces a to-L3 QED normalized efficiency , including the L3 QED event numbers of the total cross section at the energy under investigation, the total QED cross section , and the L3 luminosity at (24). The numerical values of are taken from [64]. Inserted from this reference are the numbers from page 33 Table 1: “the expected 2 events,” which agrees with QED-Monte Carlo generators. The 3 setup has been implicitly included in the generator with the parameter and (deg). The values for L3 luminosity at the energy under investigation are taken from Table 11.

Third, the to-L3 normalized total QED cross section of the detector at under investigation (23), the L3 QED efficiency (24), and the luminosity of the different detectors open the possibility to calculate the experimental counting rate of every detector normalized to L3 of this detector (25). The numerical values for are summarized in Table 11. In this table, the important references for Tables 9 and 10 are included.

Including the detailed numerical numbers from (23)–(25), it is possible to sum over and to calculate for every under investigation the total summed cross section (26) and the statistical error (27).

The total summed cross section (26) divided by the total QED cross section of L3 allows to calculate the very important ratio at the under investigation (28).

To calculate the statistical error (30), the most conservative approach via the maximal possible error is used (29).

Not all different collaborations mark for a systematic error; for this reason, (27) and (30) include only the statistical error, which is dominating in this analysis. All collaborations generate with a Monte Carlo generator many millions of events to calculate the differential or total QED cross sections. This allows us to keep the systematic error originated from MC negligible, compared to the data statistical error. This analysis investigated one more possible important systematic error. We investigated two different Monte Carlo generators and used different energy ranges for the test to study the systematic error. In the following sections, we will see that the interaction radius from all the four different tests are in the range of the statistical error the same (Table 2). It is interesting that even the completely independent analyses of different cross section agree with each other on this interaction radius (Table 2). It proves that the systematic error does not change the result of this analysis.

3.2. Calculation of , , Ratio , and for One Detector

The input data in the case only one detector contribute to the calculation of (31)–(35), and the detailed information is again included in Tables 9–11. In this case, eight data sets exist of .

Under these conditions, the total experimental cross section normalized to L3 (31) is like (23), a function of the total experimental cross section of one detector and the ratio of the total QED L3 cross section to the total QED detector cross section . It is similarly possible to calculate from the experimental error of detector via the ratio to the to-L3 normalized error (32). The ratio (33) is a function of (31) and the total QED L3 cross section . The cancels in (33), which replaces by and by . To calculate the statistical error (35) for simplicity, the maximum value of (34) is used to form the difference between and . The sum of (31) and (32) divided by is shown in (34).

Not all different collaborations mark for a systematic error; for this reason, (32) and (34) include only the statistical error.

3.3. Numerical Calculation of , , Ratio , and

Inserting the numerical values from Tables 9–11 in (23)–(35) allows to calculate , , ratio , and shown in Table 3.

The values from  GeV to 207 GeV together with of Table 3 compared to the total QED cross section (17) is displayed in Figure 8.

Figure 8 shows a good agreement between the experimental measured values including the statistical error and the total QED cross section in the range of sensitivity.

To search for deviation between measured values and the total QED cross section , the graphic of Figure 8 is not sensitive enough because the deviation is on the % level. A more sensitive graphic (Figure 9) is plotted to display the ratio of the reaction of all detectors as a function of center-of-mass energy . In this plot, it is clearly seen that from  GeV, the ratio points are systematically under value 1.0, which means the data that measured total cross sections are systematically smaller than the QED predictions. The deviation is in general at % level and will be studied by the test in the following sections.

In addition, it is also visible that the statistical errors for the 8 data points are much bigger than those for the 9 data points. The decrease of the statistical errors originated from the fact that at LEP, the nine data sets exist measuring from more than one detector (27) and the eight data sets exist with that only one detector measuring . For example, in the used model at  GeV, the statistical error is (pb), compared to the statistical error of L3 (pb) at the same  GeV [64]. This is a decrease of approximately 42%.

4. Search for Finite Size of Electron Using a Test of the Ratio

The signal for the finite size of the electron is weak, not visible in the graphic of the total experimental cross section compared with the QED total cross section in Figure 8. A deviation of approximately some % is visible in the graphic of the ratio of the reaction in Figure 9. In accordance with the sensitivity between experiment and QED, a test on this ratio (36) is performed to search for a minimum in .

The ratio of the experimental data is in accordance with Table 3, the ratio of at a energy . The statistical error is the error at energy , in accordance with Table 3. The term to search for a deviation of a finite size of the electron is (37). The fit parameter is included, working as a function of the interaction size of the electron (21).

It is essential for the whole program of the test to use the fit parameter which is sensitive to the theoretical calculation for a deviation from the QED differential cross section for positive and negative interference. A parameter would test only the positive interference and cut out the negative part. This problem is visible in (37) and, in particular, (38). The parameter would keep the sign in front of always positive independent of the sign from . As a consequence, it always happens that and . The test would not be able to find values . For this reason, a negative interference could not be detected.

The integrals ((37) and (38)) included the differential QED cross section .

To test the angular contribution part of function, it is possible to integrate (37) over the and use only the contribution of the angular distribution of the direct contact term (20) like

The energy contribution to is in the range of % level, which opens the possibility to investigate the test under the assumption that the constant equals approximately .

4.1. Numerical Calculation of the Tests

To investigate the sensitivity of the test to the Monte Carlo generator BabaYaga@nlo [76] and the generator [73, 74], a separate numerical calculation for both tests was performed. In addition, an approximation was used to test only the impact of the direct contact term (20) in the test.

The mathematical details of different possible calculation of significance and error of the interaction radius are studied in Appendix D.

4.1.1. Calculation of the Test with QED BabaYaga

The generators under discussion generate numbers of events respecting to L3 parameters (15). The events per angular range are used to fit a differential cross section as a function of and . An example of such a differential cross section is shown at  GeV in Figure 5. The numerical parameters of this fit to defined in (14) for the 17 energies from 55 GeV to 207 GeV are summarized in Table 8 (upper part). The test on the ratio (36) as a function of for the reaction is displayed in Figure 10. The relevant center-of-mass energy ranges from  GeV to 207 GeV, and the used QED predictions are from Monte Carlo generator BabaYaga@nlo.

The minimum of the test including the error in Figure 10 is , which corresponds to  GeV. The value at the minimum is . The fit uses 17 degrees of freedom according to Table 3.

The important result of the fit is that the fit parameter has a negative sign. According to Figure 9, the fit is sensitive to the fact that the total QED cross section is bigger than the experimental total cross section above approximately 180 GeV. The discussed direct contact interaction term (20) has a negative sign. This indicates a negative interference of the direct contact interaction in the reaction.

From the fit results, the significance of the fit is essential. After international rules, the physic community accepts a as a discovery of new physics and a as a hint of new physics.

A detailed mathematical calculation of significance is discussed in Appendix D.1. A first approximation of the significance of the fit can be directly estimated from the error bars of the fit . In a second way to calculate the significance , a statistical probability function (D.1) is used. The value of the discussed test for 17 degrees of freedom (with minimum ) is equal to . According to Figure 16, the significance is approximately .

Similar to the significance, a detailed mathematical calculation of the error of the interaction size of the electron in the test is discussed in the Appendix D.2. According to (21), the size of the interaction term is (cm) and the error (cm). The summary of all these results is given in Table 1.

4.1.2. Calculation of the Test with QED Generator [73, 74]

The VENUS, TOPAS, OPAL, DELPHI, ALEPH, and L3 collaborations used for the QED cross section of the reaction and the generators [73, 74]. As discussed in Section 2.4.1, the deviation between BabaYaga and [73, 74] is approximately under the condition that both generators used the same L3 parameters (15). Similar to the QED BabaYaga version, the events per angular range are used to fit a differential cross section as a function of and . The numerical parameters of this fit to are defined in (14) for the 7 energies from 91.2 GeV to 200 GeV and summarized in the lower section of Table 8. Using QED predictions from Monte Carlo generator [73, 74], the test on the ratio (36) as a function of for the reaction is displayed in Figure 11. The relevant center-of-mass energy ranges from  GeV to 200 GeV.

The minimum of the test including the error in Figure 11 is . The value at the minimum is . The fit uses 7 degrees of freedom.

Similar to the QED test with BabaYaga, in the QED test [73, 74] is negative. A first direct approximation of the significance is . In a second approximation to calculate the significance , a statistical probability function formula (D.1) is used. The value of the discussed test for 7 degrees of freedom (with minimum ) is equal to . Using Figure 16, this corresponds to a significance of approximately .

According to (21), the size of the interaction term is (cm). The error is (cm) from equation (D.10). The summary of all these results is given in Table 4.

4.1.3. Numerical Calculation of the Test Using Only Direct Contact Term

The tests performed with the QED BabaYaga and [73, 74] generator request the calculation of the differential QED cross section and fit this data with (14).

Equation (39) opens the possibility to test straight the direct contact term (20). The deviation between the measured and is on the % level. This allows us to assume that the constant factors and are approximately the same, .

According to Table 8, two data sets exist with 7 degrees of freedom from  GeV to 200 GeV and 17 degrees of freedom from  GeV to 207 GeV. Both data sets are tested.

4.1.4. Numerical Calculation of the Test with 7 Degrees of Freedom in Approximation

The test on the ratio (36) as a function of for the reaction from center-of-mass energy  GeV to 200 GeV is displayed in Figure 12.

Inserted are the 7 energies from Table 8 (lower part) and the relevant parameters in Table 3, including also equations (36), (37), and (39) under the assumption .

The minimum of the test including the error in Figure 12 is . The value at the minimum is . The fit uses 7 degrees of freedom.

Similar to the QED test BabaYaga and [73, 74], in this approximation is also negative. A first direct approximation of the significance is