MD Vittorio Colletti
email: vittoriocolletti@yahoo.com

MD Marco Mandalà

MD Paolo Manganotti

Phd Stefano Ramat

MD Luca Sacchetto

Phd Liliana Colletti

INTRAOPERATIVE OBSERVATION OF CHANGES IN COCHLEAR NERVE ACTION POTENTIALS DURING EXPOSURE TO ELECTROMAGNETIC FIELDS GENERATED BY MOBILE PHONES

Keywords:

The rapid spread of devices generating electromagnetic fields has raised concerns as to the possible effects of this technology on humans. The auditory system is the neural organ most frequently and directly exposed to electromagnetic activity due to the daily use of mobile phones. In recent publications, a possible correlation between mobile phone usage and central nervous system tumors has been detected. Very recently a deterioration of otoacoustic emissions and of the auditory middle latency responses after intensive and long term magnetic field exposure in humans has been demonstrated. To determine with objective observations if exposure to cell phone electromagnetic fields affects acoustically evoked cochlear nerve compound action potentials (CNAPs), 7 patients suffering from Ménière's disease and undergoing retrosigmoid vestibular neurectomy were exposed to the effects of mobile phone placed over the craniotomy for 5 minutes. All patients showed a substantial decrease in amplitude and a significant increase in latency of CNAPs during the 5 minutes of exposure to electromagnetic fields. These changes lasted for a period of around 5 minutes after exposure. The possibility that electromagnetic fields can produce relatively long-lasting effects on cochlear nerve conduction is discussed and analyzed in light of contrasting previous literature obtained under non-surgical conditions. Limitations of this novel approach, including the effects of the anesthetics, craniotomy and surgical procedure, are presented in detail.

INTRODUCTION

Mobile phones are today an integral part of modern life and allow people to maintain continuous communication without restraining the freedom of movement. The exponential diffusion of cell phones bears witness to the enormous success these devices have enjoyed in recent years. This enthusiasm is offset by the widespread opinion that cell phones may constitute a human health hazard. Public opinion therefore urgently seeks reassurance regarding the safety of the electromagnetic fields (EMF) produced by these devices. To date the scientific community has come up with discordant data on this issue.

In 1996 the World Health Organization set up the International Electromagnetic Fields Project to assess the existing scientific evidence of the possible health effects of electromagnetic fields, including radiofrequency fields emitted by mobile phones and their base stations, and to promote further research [START_REF]Electromagnetic fields and public health. Mobile telephones and their base stations[END_REF]. There are a number of experimental investigations showing that EMF exposure induces a series of alterations at the cellular level [START_REF] Zhao | Studying gene expression profile of rat neuron exposed to 1800MHz radiofrequency electromagnetic fields with cDNA microassay[END_REF][START_REF] Ammari | GFAP expression in the rat brain following sub-chronic exposure to a 900 MHz electromagnetic field signal[END_REF][START_REF] Campisi | Reactive oxygen species levels and DNA fragmentation on astrocytes in primary culture after acute exposure to low intensity microwave electromagnetic field[END_REF][START_REF] Brillaud | Effect of an acute 900MHz GSM exposure on glia in the rat brain: a time-dependent study[END_REF]. Changes in blood-brain-barrier permeability, neurotransmitter function and metabolic changes in brain tissues after exposure to low-intensity EMF have been found both in experimental animals [START_REF] Nittby | Increased blood-brain barrier permeability in mammalian brain 7 days after exposure to the radiation from a GSM-900 mobile phone[END_REF][START_REF] Mausset-Bonnefont | Acute exposure to GSM 900-MHz electromagnetic fields induces glial reactivity and biochemical modifications in the rat brain[END_REF][START_REF] Meral | Effects of 900-MHz electromagnetic field emitted from cellular phone on brain oxidative stress and some vitamin levels of guinea pigs[END_REF] and humans [START_REF] Aalto | Mobile phone affects cerebral blood flow in humans[END_REF][START_REF] Söderqvist | Exposure to an 890-MHz mobile phone-like signal and serum levels of S100B and transthyretin in volunteers[END_REF]. Furthermore, it has recently been demonstrated that mobile phone bioactivity is related to its intensity or distance from the antenna: high for intensities down to less than 10 mW/cm and still evident as far as 1 mW/cm exhibiting 'window' effects [START_REF] Panagopoulos | Bioeffects of mobile telephony radiation in relation to its intensity or distance from the antenna[END_REF].

Changes in neuronal electrophysiology, evoked potentials and EEG traces have been reported [START_REF] Varró | Changes in synaptic efficacy and seizure susceptibility in rat brain slices following extremely low-frequency electromagnetic field exposure[END_REF]. Varrò et al. [START_REF] Varró | Changes in synaptic efficacy and seizure susceptibility in rat brain slices following extremely low-frequency electromagnetic field exposure[END_REF] described a decrease in basic synaptic activity as a reduction in amplitude of evoked potentials in rat brain slice preparation when exposed to an electromagnetic field.

In humans, auditory brainstem responses (ABR) do not seem to be affected by mobile short term phone exposure, suggesting that mobile phone EMF do not affect the transmission of sensory stimuli from the cochlea to the midbrain [START_REF] Arai | Thirty minutes mobile phone use has no short-term adverse effects on central auditory pathways[END_REF][START_REF] Oysu | Effects of the acute exposure to the electromagnetic field of mobile phones on human auditory brainstem responses[END_REF][START_REF] Stefanics | Short GSM mobile phone exposure does not alter human auditory brainstem response[END_REF][START_REF] Balbani | Mobile phones: influence on auditory and vestibular systems[END_REF][START_REF] Kwon | No effects of mobile phone electromagnetic field on auditory brainstem response[END_REF]. However, exposure to mobile phone EMF does affect neural function in humans (when evaluating EEG recordings) as a function of exposure duration and its effect vanishes after phone use ceases [START_REF] Croft | Acute mobile phone operation affects neural function in humans[END_REF]. Likewise, effects on human motor excitability have been reported using transcranial magnetic stimulation after EMF exposure [START_REF] Brignani | Modulation of cortical oscillatory activity during transcranial magnetic stimulation[END_REF].

From an epidemiological standpoint, Hardell et al. [START_REF] Hardell | Long-term use of cellular phones and brain tumours: increased risk associated with use for > or =10 years[END_REF][START_REF] Hardell | Mobile phones, cordless phones and the risk for brain tumours[END_REF] found that exposure to mobile phone EMF correlated with an increased incidence of astrocytoma [START_REF] Hardell | Long-term use of cellular phones and brain tumours: increased risk associated with use for > or =10 years[END_REF][START_REF] Hardell | Mobile phones, cordless phones and the risk for brain tumours[END_REF], acoustic neuroma and glioma [START_REF] Hardell | Meta-analysis of long-term mobile phone use and the association with brain tumours[END_REF] and in a recent review it was concluded that the "Current standard for exposure to microwaves during mobile phone use is not safe for long-term exposure and needs to be revised" [START_REF] Hardell | Epidemiological evidence for an association between use of wireless phones and tumor diseases[END_REF]. Very recently, long term and intensive mobile phones users demonstrated a significantly higher risk of having both cochlear (absent otoacustic emission or higher speech frequency thresholds) and auditory cortex alterations (lower middle latency response waves amplitude) as compared to controls [START_REF] Modi | Auditory Changes in Mobile Users: Is Evidence Forthcoming?[END_REF].

These conflicting data prompted us to investigate the effect of EMF exposure on cochlear nerve bioelectric activity using a near-field electrophysiological procedure.

It is known that the bioelectric activity of the cochlear nerve may be safely and reliably recorded during cerebellopontine angle surgery from the exposed human cochlear nerve [START_REF] Moller | Compound action potentials recorded intracranially from the auditory nerve in man[END_REF][START_REF] Colletti | Vulnerability of hearing function during acoustic neuroma surgery[END_REF][START_REF] Colletti | Iatrogenic damage of hearing during surgery for acoustic neuroma[END_REF].

The acoustically evoked cochlear compound nerve action potentials (CNAPs), directly recorded from the exposed nerve, are characterized by a stable morphology, amplitude and latency. The procedure is highly sensitive to damage to the cochlea and cochlear nerve and even minor stimulations, e.g. changes in temperature due to moderate cold or warm saline irrigation, cause significant transitory changes in latency of the potentials of the order of 0.2-0.3 ms that can be easily assessed [START_REF] Colletti | Vulnerability of hearing function during acoustic neuroma surgery[END_REF][START_REF] Colletti | Iatrogenic damage of hearing during surgery for acoustic neuroma[END_REF].

The possibility of obtaining real-time recordings directly from the auditory nerve, the neural structure which is most likely to be directly affected by EMF generated by cell phones, prompted us to conduct the present study to investigate the effect of electromagnetic fields due to cell phone exposure on bioelectric activity in response to acoustic stimulation.

MATERIALS AND METHODS

The cohort reported here is a patient-based clinical population seen from January 2009 to March 2010. The study was conducted at the Department of Otolaryngology (tertiary referral center) of the University of Verona (Verona, Italy). All participants were affected by definite Ménière's disease (MD), according to the criteria of the American Academy of Otolaryngology-Head and Neck Surgery [1995]; they had also received medical therapy (diuretics, betahistine, low salt diet) for at least 6 months. All participants performed a complete audiological evaluation with pure tone audiometry, speech audiometry, impedance audiometry, ABR, electrocochleography (ECoG) and underwent a complete neuro-otological evaluation including: eye-movement bedside examination, vestibular evoked myogenic potentials (VEMPs), caloric test before the procedure and at follow-up examinations up to 6 months. A 1.5 T MRI evaluation of the brain was obtained before the procedure in all patients.

Twenty-one patients met the entry criteria. Six patients were not enrolled in the study due to profound hearing loss in the ear to be operated on. Three patients refused surgery. Seven patients underwent retrosigmoid vestibular neurectomy (VN) for disabling definite unilateral MD (classes A and B) while being monitored for mobile phone EMF effects and 5 patients undergoing the same surgical procedure were enrolled in the study as a control group.

The 7 patients enrolled for EMF monitoring were first exposed for 5 minutes to the magnetic field generated by a cell phone during an active call placed over the craniotomy and then submitted to VN for Ménière's disease. Hearing was monitored intraoperatively using direct recording of CNAPs. The control group underwent 5 minutes' exposure in the same experimental conditions using the same mobile phone in stand-by mode.

All patients were duly informed regarding the aim and protocol of the experiment and gave their consent. The patients were operated on in the lateral position with the head rotated contralaterally to the operated ear. A retrosigmoid craniotomy with a diameter of 3-4 cm was carried out. The dura was opened in an "H" fashion. The cerebellum was gently depressed using a self-restraining retractor secured to the operating table. The entire area from the brainstem to the posterior wall of the petrous bone and the tentorium was exposed. Sharp dissection of the arachnoid allowed access to the eighth nerve.

Recording from the cochlear nerve was performed using a Teflon insulated silver electrode wire (Type Ag 7/10; Medwire Corporation, Mt. Vernon, NY, U.S.A.) with a small cotton wick sutured on its tip, which was uninsulated over a distance of 2-3 mm.

The electrode was located on the proximal portion of the eighth nerve (Figure 1).

A pledget of fibrin sponge covered the electrode in the CPA to stabilize and segregate it from the surgical field. The monopolar electrode (inverting electrode) was referenced to a subdermal platinum electrode (Type E-2; Grass Instrument Company, Quincy, MA, U.S.A.) placed in the ipsilateral tragus (non-inverting electrode). A subdermal ground electrode was positioned on the sternum.

To monitor auditory function (cochlea, cochlear nerve and cochlear nuclei) the ear undergoing the operation was stimulated by alternating click stimuli at 31 pulses/sec using a walkman-type earphone. Click intensity ranged from 100 to 120 dB sound pressure level, depending on the hearing level.

The recorded potentials were filtered through a 30 to 2,500-Hz bandpass filter, amplified (10,000x or 20,000x) and averaged with a 10 channel evoked potential system (Medelec Synergy N-EP, ViasysTM). The acquired large-amplitude potentials were easily visualized after averaging a few responses (100 repetitions), providing information on the function of the cochlea and cochlear nerve with a delay of only 2-3 seconds.

CNAPs latency of the first negative peak (N1) and normalized absolute amplitude of N1 were evaluated in all subjects.

Prior to vestibular nerve section, a Nokia 6310i mobile phone was positioned over the craniotomy area. It was activated (ongoing call but no sound was played from the speaker of the mobile phone) or kept in stand-by mode respectively for the subjects and controls for 5 minutes. The distance between the mobile phone antenna and the eighth nerve was measured in every trial.

The mobile phone used in the present study emits and receives radio signals in the region of 900 MHz, and the highest specific absorption rate (SAR) value for this model when tested for compliance against the standard was 0.82 W/kg [START_REF] Oysu | Effects of the acute exposure to the electromagnetic field of mobile phones on human auditory brainstem responses[END_REF].

To ensure the stability of the recording procedure and assess baseline behavior in all subjects, CNAPs were first recorded with the phone in stand-by mode over a 2 minute interval (T0), CNAPs were then continuously monitored throughout the entire 5 minutes of cochlear nerve exposure to the EMF (T1).

After the exposure to the cell phone, the potentials were recorded for 10 more minutes (T2) (time necessary to harvest an autologous abdominal fat graft) and surgery was then continued with the vestibular neurectomy.

ABR were also recorded in all subjects at the same time as the CNAPs recordings. ABR were obtained using differential recordings from the vertex (non-inverting electrode) and the ipsilateral earlobe (inverting electrode) referenced to the sternum (ground).

Needle electrodes were used to this end and the recorded activity was filtered (1000 to 2500 Hz), amplified (100,000x) and averaged (100 sweeps).

Three months after surgery an audiometric evaluation with pure tone audiometry was performed to evaluate the extent of hearing preservation.

Recordings at time T0 and at the end of the 5 minutes exposure were compared with the CNAPs recordings obtained from the control group (sham exposure) at the same times.

Differences between the EMF and sham groups were tested using Student's t-test and a non-parametric test (Wilcoxon rank-sum test), as appropriate based on the results of the Kolmogorov-Smirnov test for normal distribution. A nonparametric repeated-measures ANOVA procedure (Friedman test) in combination with a post hoc test (Dunn's multiple-comparison test) was used to assess the effects of 5 minutes EMF or sham exposure by comparing the data collected at each minute of the 15 minutes of CNAPs recording with that recorded at T0. Statistical significance threshold was set at p=0.05.

RESULTS

Demographic data of the population exposed to EMF and of the control group are reported in Table 1. No statistically significant differences in the distance measured from the mobile phone antenna to the eighth nerve were found between the study group and the control group (62±9 mm vs. 65±11 mm; p=0.6142; t-test).

No significant changes could be detected with V wave latency of the ABR recordings with EMF exposure or sham exposure (p>0.05; Wilcoxon test). Changes in wave I and III of the ABR could not be statistically analyzed due to their inconsistency in the recordings.

All CNAPs measurements showed stability and reproducibility at T0. A typical threephase response was obtained with a predominant negative peak (N1) generated by the depolarization wave propagation (Figure 2 and3). The basal CNAPs latency and amplitude measures for both the study and control groups showed no significant changes at T0 (p>0.05, Wilcoxon test).

During EMF exposure a significant decrease in CNAPs amplitude (Figure 3 and4) and an increase in latency (Figure 3), which started in T1 and lasted, albeit to a minor extent, over the T2 period, were observed in all patients. Figure 2 shows representative series of CNAPs recorded in two subjects submitted to EMF exposure and sham exposure, respectively. The effect of the 5' EMF exposure over the 15' of recording, was shown to be statistically significant both on CNAPs latency and amplitude (Friedman test, p< 0.0001; Figure 4) compared to the baseline values recorded at T0. A post hoc test (Dunn's multiple-comparison test of the data recorded at each minute with that at T0) indicated that significant changes both in CNAPs latency and amplitude (Figure 4) could be demonstrated from few minutes after the beginning EMF exposure to around the 10 th minute of recording (5' after EMF exposure ceased).

A substantial deterioration in amplitude and latency of N1 was observed after 30-60 seconds of EMF exposure in all subjects. Due to the progressive deterioration of amplitude and latency of the potentials during exposure, we decided to limit the exposure to 5 minutes. This decision was taken because it is known that a shift in latency above 0.5 msec may be critical for hearing preservation procedures during posterior fossa surgery [START_REF] Colletti | Vulnerability of hearing function during acoustic neuroma surgery[END_REF][START_REF] Colletti | Iatrogenic damage of hearing during surgery for acoustic neuroma[END_REF]. Latency shifts above these values increase the chance of postoperative hearing deterioration proportionally to the extent of the changes in latency.

In our experiment the maximum latency shift was thus observed at 5 minutes of exposure with a mean increase in latency of 0.55±0.18 msec (Table 1, Figure 4). The same trend was observed in terms of amplitude reduction, the maximal amplitude change being observed at 5 minutes of exposure with a mean decrease in amplitude of 34±16% (Table 1, Figure 4). When comparing CNAPs at the end of EMF or sham exposure (T1), results showed statistically significant differences in terms of latency (p<.0001; Wilcoxon test) and amplitude (p<.0273; Wilcoxon test) between the EMFexposed subjects and the control group (Table 1). Sham exposure to cell phones in stand-by mode on the 5 control subjects showed no significant differences in CNAPs latency and amplitude (p>0.05; Friedman test).

The three-month audiological follow-up showed no changes (p>0.05, Wilcoxon test) in hearing threshold when comparing the mean PTA of the whole population before (35±12 dB HL) and after surgery (42±9 dB HL). Furthermore, no statistically significant differences were found between the study group and the control group in terms of hearing changes (p>0.05, Wilconxon test) at the 3-month examination.

DISCUSSION

The new finding of this study is that short-term exposure (5 minutes) to the EMF produced by active cell phones causes significant CNAPs deterioration, occurring with a specific time course, both during and after exposure. Changes are observed in CNAPs latency and amplitude, which respectively increased and decreased substantially and continuously during the 5-minute exposure. It was decided to suspend the exposure after 5 minutes to avoid possible permanent damage to the auditory structures. Altered latency and amplitude values persisted, albeit in milder forms, after removal of cell phone exposure. The largest changes were observed after 5 minutes of exposure and consisted in a 0.55±0.18 msec latency increase and an 34±16% amplitude reduction. These changes were statistically significant (p<0.001). ABR analysis performed on the V wave showed no statistically significant differences related to the EMF exposure.

Changes in wave I and III of the ABR could not be statistically analyzed due to their inconsistency in the recordings and this may be due to the experimental setting which required a limited number of stimuli (100) not to prolong the surgical time and to the amount of sensorineural hearing loss that all MD patients suffered from. Postoperative audiological follow-up revealed no significant hearing threshold deterioration in any of our patients.

The findings resulting from this novel approach to CNAPs seem in contrast with our ABR findings and with most of the literature results, showing no effect related to mobile phone exposure on pure tone audiometry, ABR, transient and distortion products otoacoustic emissions. However the population of the present study differs among the population study of the quoted papers in several aspects. We have analyzed subjects with a significant degree of hearing loss and within a surgical setting whereas in all other studies normal hearing subjects and non-surgical setting conditions [START_REF] Oysu | Effects of the acute exposure to the electromagnetic field of mobile phones on human auditory brainstem responses[END_REF][START_REF] Stefanics | Short GSM mobile phone exposure does not alter human auditory brainstem response[END_REF][START_REF] Balbani | Mobile phones: influence on auditory and vestibular systems[END_REF][START_REF] Kwon | No effects of mobile phone electromagnetic field on auditory brainstem response[END_REF][START_REF] Mora | A study of the effects of cellular telephone microwave radiation on the auditory system in healthy men[END_REF][START_REF] Parazzini | Effects of GSM cellular phones on human hearing: the European project "GUARD[END_REF].

Furthermore the ABR procedure of our study limited the number of stimuli to 100 to limit the surgical time. Nevertheless recent studies highlight trends for permanent abnormalities in several parameter of hearing function within long-term mobile phone users [START_REF] Panda | Audiologic disturbances in long-term mobile phone users[END_REF] and even a significantly higher risk for absent otoacustic emission and higher speech frequency thresholds [START_REF] Modi | Auditory Changes in Mobile Users: Is Evidence Forthcoming?[END_REF].

The discrepancy between ABR and CNAPs results may be explained by different recording methodology and neurophysiological events. The direct recordings from the exposed cochlear nerve of CNAPs provide a real time measurement of extremely reliable, stable and relatively large amplitude signals. Therefore CNAPs are more sensitive to changes in the cochlea and/or cochlear nerve than ABR and offer functional information that heralds subclinical structural changes [START_REF] Colletti | Vulnerability of hearing function during acoustic neuroma surgery[END_REF][START_REF] Colletti | Iatrogenic damage of hearing during surgery for acoustic neuroma[END_REF][START_REF] Moller | Click evoked responses from the exposed intracranial portion of the eighth nerve during vestibular nerve section: bipolar and monopolar recordings[END_REF][START_REF] Yamakami | Hearing preservation and intraoperative auditory brainstem response and cochlear nerve compound action potential monitoring in the removal of small acoustic neurinoma via the retrosigmoid approach[END_REF]. Despite ABR offers the opportunity to investigate peripheral and central auditory pathways, some features as the average technique, the low number of trials, the possible synchronization of volley in the central nuclei and the supramaximal stimulation performed in intraoperative monitoring may make the methodology less feasible and useful in testing the effect of EMF exposure on auditory pathways [START_REF] Møller | Anatomy, Physiology, and Disorders of the Auditory System[END_REF][START_REF] Popper | The Mammalian Auditory Pathway: Neurophysiology[END_REF].

A series of additional factors may be responsible for the significant CNAPs deterioration with EMF exposure including anesthesia, increased cerebrospinal fluid shunting, volume intensity of the mobile phone, heat, metabolic changes and neurotransmitter alterations. The use of common anesthetics in surgery can produce a very low effect on the responses from auditory nerve fibers while it may impact the responses of the ascending auditory pathways form the cochlear nuclei [START_REF] Møller | Anatomy, Physiology, and Disorders of the Auditory System[END_REF]. CNAPs modifications may be also have been caused by an increased shunting of cerebrospinal fluid which could have accumulated around the recording electrode. However this possibility seems unlikely since the amplitudes were normalized after the end of EMF exposure, therefore increased shunting would not have affected the latency of the recorded CNAPs. Moreover, the volume intensity of the mobile phone could not have determined CNAPs changes since no sounds were played trough it and the external auditory canal was closed by the earphone and further covered by three sterile surgical drapes. Heat, on the other hand, might have been a contributing factor since the electrode placed on the auditory nerve could act as an antenna inducing local heating or local chemical reactions that could affect neural transmission. Yet, both these explanations appear unlikely since the temperature of cotton wick electrode measured before, during and after 5' of EMF exposure showed no changes and no Ag wires were in contact with the 8 th nerve. Recent studies based on a model of the human head during exposure to mobile phone EMF [START_REF] Chen | Temperature increase in human heads due to different models of cellular phones[END_REF][START_REF] Schmid | High-resolution numerical model of the middle and inner ear for a detailed analysis of radio frequency absorption[END_REF] have shown a very low increase in temperature in the ear and brain regions close to the phone. Furthermore, it has been shown that the presence of a cochlear implant array inside the cochlea produces negligible variations in the averaged SAR values and consequently in temperature, both in the head and in the cochlear tissues [START_REF] Sibella | Assessment of SAR in the tissues near a cochlear implant exposed to radiofrequency electromagnetic fields[END_REF].

The possibility that changes observed during EMF electromagnetic radiation could be due to interference with the stimulating and/or recording system was also considered. However, we found that the amplitude and latency shifts increased as a function of the duration of the exposure and slowly decreased after turning off the cell phone.

Interference with the stimulating or recording system would have caused instantaneous and substantial modifications throughout the exposure period and should have ceased immediately after exposure. Similarly, possible acoustic interference, i.e. a masking effect, can also be ruled out.

Metabolic alteration in the cochlear nerve or at the synaptic level may be responsible for the CNAPs changes. RF fields generated by cell phones can affect membrane proteins and can change the movement of ions, especially the efflux of calcium ions from brain tissue across membranes under normal conditions [START_REF] Maskey | Effect of 835 MHz radiofrequency radiation exposure on calcium binding proteins in the hippocampus of the mouse brain[END_REF]. This might cause subtle changes in cell function, but the significance of such effects for human health is uncertain. Thus, changes in calcium ion concentration might lead to alterations in neural functions. Other effects of mobile phone EMF on the nervous system are changes in blood-brain-barrier permeability due to hyperthermia, neurotransmitter functions, cellular metabolism and related electrophysiological modifications [START_REF] Valentini | Neurophysiological effects of mobile phone electromagnetic fields on humans: a comprehensive review[END_REF].

The available data suggest a complex reaction of the nervous system to EMF exposure.

Other parameters to be considered in EMF exposure are frequency, duration, waveform and amplitude modulation, which are important determinants of biological responses and affect the dose-response relationship [START_REF] Panagopoulos | Bioeffects of mobile telephony radiation in relation to its intensity or distance from the antenna[END_REF].

Despite evidence that mobile phone use can trigger brain evoked potentials [START_REF] Marino | The effects of mobile-phone electromagnetic fields on brain electrical activity: a critical analysis of the literature[END_REF] and induce CNAPs modifications, the crucial scientific question as to the pathophysiology of mobile-phone EMF effects on brain activity remains unanswered [START_REF] Carrubba | Mobile-phone pulse triggers evoked potentials[END_REF].

Two main limitations of the present investigation need to be discussed in detail. Due to the intraoperative recording methodology it is not, presently, possible to determine whether the EMF affect the cochlea, the cochlear nerve or both. CNAPs recordings (figure 2 and3) showed a decrease in amplitude and a latency shift of all components, which suggest a decreased excitation of the auditory nerve. This finding suggests an effect of our stimulation on the cochlea rather than on the auditory nerve. A conduction block of the VII nerve would have determined an increase in the positive peak that precedes the N1 and a decrease in N1 amplitude. An effect on hair cells would explain the observed changes in the CNAPs [START_REF] Møller | Anatomy, Physiology, and Disorders of the Auditory System[END_REF].

The presence of a mild transitory threshold shift due to temporary cochlear damage would have no significant and long lasting effect on the latency and amplitude of the potentials recorded from the cochlear nerve [START_REF] Colletti | Iatrogenic damage of hearing during surgery for acoustic neuroma[END_REF]. Severe cochlear damage could certainly cause such effects. In this event, however, the time course of the effects would have been longer and a systematic postoperative hearing loss might have been observed. This did not occur in the studied population and postoperative shifts in the hearing threshold were not statistically different between the EMF exposure group and patients who underwent sham exposure.

Another limitation is the surgical access performed to investigate the eighth nerve. The retrosigmoid craniotomy directly exposes the eighth nerve to mobile phone EMF without any biological structure in-between. This condition is clearly far from reproducing the real-life EMF absorption of neural structures, because it offers a reliable measurement of the direct effect of a known EMF on a sensitive nerve without the interposition of other structures (skin, skull, fat, muscle, blood, grey and white matter) that have different SAR [START_REF] Kwon | No effects of mobile phone electromagnetic field on auditory brainstem response[END_REF][START_REF] Gajsek | Parametric dependence of SAR on permittivity values in a man model[END_REF]. All these structures play an important role in determining the SAR of inner ear structures in responses to EMF exposure [START_REF] Parazzini | Effects of GSM cellular phones on human hearing: the European project "GUARD[END_REF].

The long-term effects of EMF exposure of the cochlear nerve could not be assessed in the present study. Nervous system conduction might adapt or severely deteriorate in response to the perturbation produced by repeated exposure to EMF.

Further investigations are needed to clarify these issues. Despite the fact that short-term exposure appears not to affect the human auditory system [START_REF] Parazzini | Effects of UMTS cellular phones on human hearing: results of the European project EMFnEAR[END_REF], audiologic abnormality trends have recently been described [START_REF] Modi | Auditory Changes in Mobile Users: Is Evidence Forthcoming?[END_REF][START_REF] Panda | Audiologic disturbances in long-term mobile phone users[END_REF].

To the best of our knowledge, this study, for the first time, adds important information on the transitory and short-term effects of electromagnetic fields on the auditory nervous structures through intraoperative monitoring, and reveals relevant clinical and general health implications. Subjects undergoing VN for Ménière's disease represent a valuable and unique model for investigating human auditory physiopathology.

Further studies on larger numbers of patients with different experimental conditions are necessary to validate the present findings and clarify the pathological and physiological substrate of the described effect. Table 1. Demographic data and latency and amplitude variations at the end of the 5' exposure for the EMF-exposed population and the control group.

Group

Figure 1 .

 1 Figure 1. Monopolar cotton-wick (arrow) electrode placed at the root entry zone of the eighth cranial nerve (asterisk) in a patient undergoing vestibular neurectomy via the retrosigmoid approach.

Figure 2 .

 2 Figure 2. Series of CNAPs recorded in one of the patients submitted to EMF exposure and one control subject (sham exposure). CNAPs recordings obtained before, during and after EMF exposure are shown. A significant reduction in amplitude and increase in latency may be seen in the EMF exposure group. T0, before EMF exposure; T1, during EMF exposure; T2, after EMF exposure.

Figure 3 .

 3 Figure 3. Series of CNAPs recorded in one of the patients submitted to EMF exposure.Recordings obtained before RF exposure (dotted lines) are superimposed to CNAPs obtained during and after exposure (continuous lines). Significant reduction in amplitude and increase in latency can be seen. T0, before EMF exposure; T1, during EMF exposure; T2, after EMF exposure.

Figure 4 .

 4 Figure 4. Mean (± SEM) values of latency shift and normalized amplitude variations from basal values in both groups of subjects. Patients submitted to EMF exposure show a significantly larger latency shift and amplitude reduction compared to controls. Results of the Friedman test (*) in combination with a post hoc Dunn's multiplecomparison test are showed for the EMF exposure group.

Acknowledgments, Competing interests, Funding: nothing to declare

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N Kaiser

Radiative corrections to lepton-lepton scattering revisited

Keywords: 12.20.-m, 12.20.Ds, 14.70.Bh

We calculate in closed analytical form the one-loop radiative corrections to electronelectron scattering e -e -→ e -e -. Explicit expressions (in terms of dimensionless Mandelstam variables) are given for the pertinent interference terms between the tree diagrams and the one-loop diagrams. Infrared finiteness of these virtual radiative corrections is achieved (in the standard way) by including soft photon radiation below an energy cut-off λ. We evaluate the remaining finite part of the soft photon correction factor in the centerof-mass frame. These results can be also utilized for Bhabha scattering e -e + → e -e + and for elastic muon scattering µ -µ ∓ → µ -µ ∓ . Furthermore, we calculate in the same way the one-loop radiative corrections to the scattering of unequal leptons e -µ -→ e -µ -and apply them likewise to the muon pair production process e -e + → µ + µ -. In contrast to most other calculations of these radiative corrections we keep the full dependence on the lepton masses. Effects from hard photon radiation which are also important for a direct comparison with (a specific) experiment are not treated explicitly.

Introduction and summary

For a correct interpretation of the experimental data obtained in elastic or inelastic lepton scattering processes it is essential to include in the analysis of these data the radiative corrections arising from virtual photon loops and (soft) photon bremsstrahlung. Radiative corrections to electromagnetic processes have been first calculated in their classical works by Schwinger [1] for electron scattering in an external static potential, and by Brown and Feynman [2] for the Compton effect. Tsai [3] has extended the calculation of radiative corrections to elastic electronproton scattering. The subject of radiative corrections has by now already a long history and we refer to the standard review papers of Maximon [4] and Mo and Tsai [5] which have been used in the interpretation of many electron scattering experiments. In recent years the issue of radiative corrections has received a renewed interest in connection with new precision experiments, performed e.g. at JLAB or MAMI. The two-photon exchange contribution [6,7,8] to the elastic electron-proton scattering plays in fact a crucial role in order to reconcile the apparent discrepancies for the ratio of the proton electric and magnetic form factors G p E /G p M as determined with the polarization transfer technique on the one hand side and via the (more traditional) Rosenbluth separation method on the other hand side. A comprehensive study of the analogous two-boson (γγ, γZ 0 ) exchange corrections for parity-violating electron-proton scattering has been presented very recently in refs. [9,10]. The complete treatment of the full excitation spectrum of the proton in the two-photon exchange remains still a challenge, and so far only the ground-state [7,8] and the low-lying ∆ + (1232) resonance [11] have been taken into account in the computation of the respective two-photon exchange diagrams. Likewise, the QED radiative corrections to virtual Compton scattering off the proton (e -p → e -pγ) have been calculated and studied in ref. [12]. It has been concluded that a very good understanding of these radiative corrections is indispensable if one wants to extract nucleon structure information (such as generalized polarizabilities or generalized parton distributions) from the e -p → e -pγ reaction. Also recently, the radiative corrections to (real and virtual) pion Compton scattering have been calculated in refs. [13,14]. These results are particularly relevant for analyzing the COMPASS experiment at CERN which aims at measuring the pion electric and magnetic polarizabilities (α π and β π ) with high statistics using the Primakoff effect (i.e. the scattering of high-energy pions in the Coulomb field of a heavy nucleus). The one-loop radiative corrections to the purely leptonic scattering process e -e -→ e -e - have been calculated some time ago by Furlan and Peressutti in ref. [15]. However, as noted in their follow-up paper [16] on numerical results that original paper [15] contains some (unspecified) mistakes and misprints. A corrected version of their analytical formulas for the radiative corrections to electron-electron scattering has however never been published. The related paper by Tsai [17] deals only with the case of high-energy electron-electron scattering, where the radiative corrections can be approximated by some dominant logarithmic terms. Likewise, the one-loop radiative corrections to the scattering of unequal leptons, e -µ -→ e -µ -, have been calculated shortly afterwards by Eriksson in ref. [18]. Also in that case the original paper [18] contains mistakes and misprints which have been corrected iteratively in subsequent papers [19,20] together with a presentation of some numerical results. Unfortunately, the comprehensibility of the analytical formulas written therein is not always straightforward since many auxiliary variables have been introduced although the process e -µ -→ e -µ -can be described by just two independent Mandelstam variables. Moreover, it is rather cumbersome to trace back which part of the analytical expressions corresponds to a particular loop diagram.

In this situation it is still meaningful and helpful to reconsider the radiative corrections to lepton-lepton scattering. The purpose of the present paper is to document the results of such a careful rederivation. Our paper is organized as follows. In section 2, we calculate in closed analytical form the one-loop radiative corrections to electron-electron scattering e -e -→ e -e -. Explicit expressions (in terms of suitable dimensionless Mandelstam variables) are given for the pertinent interference terms between the tree diagrams and the one-loop diagrams summed over the electron spin states. Infrared finiteness of these virtual radiative corrections is achieved (in the standard way) by including soft photon radiation (off the in-or out-going electrons) below an energy cut-off λ. We evaluate the remaining finite part of the soft photon correction factor in the center-of-mass frame, assuming an isotropic photon emission therein. Figures with numerical results are presented in order to demonstrate the size as well as the energy and angular dependence of these radiative corrections to electron-electron scattering. By making use of crossing symmetry the radiative corrections to electron-positron (Bhabha) scattering e -e + → e -e + can be obtained from the same analytical formulas. We consider also the elastic muon scattering processes µ -µ ∓ → µ -µ ∓ where the electronic vacuum polarization enters as a new and special feature. In section 3, we calculate then in the same way the one-loop radiative corrections to the scattering of unequal leptons e -µ -→ e -µ -. In that case the squared mass ratio r = (m e /m µ ) 2 comes into play as an additional parameter thus making the calculation a bit more challenging. After presenting numerical results for the radiative corrections to e -µ -→ e -µ -scattering we turn via crossing symmetry to those for the muon pair production process e -e + → µ + µ -. For the purpose of comparison, we treat in section 4 the analogous radiative corrections for (point-like) charged scalar bosons. In the appendix, we collect the analytical results for the pertinent one-loop integrals with two, three and four propagators. We use dimensional regularization to treat both ultraviolet and infrared divergences (where the latter are caused by the vanishing photon mass). The imaginary parts of the loop functions are also given there. Together with a Kramers-Kronig dispersion relation these imaginary parts are most useful in order to control the correct analytical continuation of the real parts of the loop functions above their respective branch points.

In contrast to many other (modern) calculations of these QED radiative corrections we keep the full dependence on the lepton masses in all results. We do, however, not treat effects from hard photon radiation or hadronic vacuum polarization, which can also be important for a (direct) comparison with experiments.

(A) (B)

Radiative corrections to electron-electron scattering

We start out with calculating the radiative corrections to electron-electron scattering. The inand out-going four-momenta of the reaction e -(p 1 ) + e -(p 2 ) → e -(p 3 ) + e -(p 4 ) give rise to the Lorentz-invariant (dimensionless) Mandelstam variables (s, t, u) defined as follows:

s m 2 e = (p 1 + p 2 ) 2 = (p 3 + p 4 ) 2 , t m 2 e = (p 1 -p 3 ) 2 = (p 2 -p 4 ) 2 , u m 2 e = (p 1 -p 4 ) 2 = (p 2 -p 3 ) 2 . (1)
Since m 2 e has been factored out they obey the numerical constraint s + t + u = 4, and in the physical region the inequalities s > 4, t < 0, u < 0 hold. In the case of unpolarized scattering the squared T-matrix for e -e -→ e -e -has to be summed over all the electron spin-states. This fourfold sum is most efficiently performed via Dirac-traces: . In order to keep the technical complications of the loop calculation as low as possible the integration over the loop-momenta in a photon-loop diagram is carried out after the summation over the electron spin-states. With that approach in mind the unpolarized differential cross section for electron-electron scattering in the center-of-mass frame, including radiative corrections of relative order α, can be represented in the following compact form:

dσ dΩ cm = α 2 2m 2 e s A t + B u ⊗ A t + B u +2 Re I + II + III + IV ⊗ A t + B u + (t ↔ u) , (2)
with α = 1/137.036 the fine-structure constant. Here A and B denote the direct and crossed tree diagram (shown in Fig. 1) and I, II, III, IV stand for the four (classes of) contributing one-loop diagrams (shown in Figs. 2,3). The product symbol ⊗ designates the interference term between the T-matrices from two diagrams with the sums over the electron spins already carried out via Dirac-traces. The symmetrization prescription +(t ↔ u) in eq.( 2) generates the additional contributions from the crossed one-loop diagrams (i.e. the diagrams in Figs. 2,3 with crossed out-going electron lines). The advantage of working with the dimensionless variables (t, u) shows up already when evaluating the Born terms. The two tree diagrams in Fig. 1 lead to the following simple polynomial expressions:

A ⊗ A = t 2 + 2tu + 2(u -2) 2 , (I) (II)
A ⊗ B = B ⊗ A = (t + u) 2 -4 , B ⊗ B = 2(t -2) 2 + 2tu + u 2 .
(3)

Note that the (relative) minus-sign belonging to the crossed tree diagram B, which originates from the exchange of two identical fermions, is already included in the interference term A ⊗ B = B ⊗ A. Specifying to the center-of-mass kinematics, t = (4 -s)(1cos θ cm )/2 and u = (4 -s)(1 + cos θ cm )/2 with θ cm the scattering angle,1 the terms in the first line of eq.( 2) produce the well-known Møller cross section for electron-electron scattering:

dσ (1γ) dΩ cm = α 2 m 2 e s 4(s -2) 2 (s -4) 2 (4η 2 -3η) + 4η + 1 , η= 1 sin 2 θ cm . (4)

Evaluation of one-loop diagrams

In this section, we present analytical expressions (of order α) for the interference terms between the one-loop diagrams and the tree diagrams for electron-electron scattering. We use dimensional regularization to treat both ultraviolet and infrared divergences (where the latter are caused by the masslessness of the photon). The method consists in calculating loop integrals in d spacetime dimensions and expanding the results around d = 4. Divergent pieces of one-loop integrals generically show up in form of the composite constant:

ξ = 1 d -4 + 1 2 (γ E -ln 4π) + ln m e µ , (5)
containing a simple pole at d = 4. In addition, γ E = 0.5772 . . . is the Euler-Mascheroni number and µ an arbitrary mass scale introduced in dimensional regularization in order to keep the mass dimension of the loop integrals independent of d. Ultraviolet (UV) and infrared (IR) divergences are distinguished by the feature of whether the condition for convergence of the d-dimensional integral is d < 4 or d > 4. We discriminate them in the notation by putting appropriate subscripts, i.e. ξ UV and ξ IR . In order to simplify all calculations, we employ the Feynman gauge where the photon propagator is directly proportional to the Minkowski metric tensor g µν . Let us now enumerate the analytical results as they emerge from the four (classes of) one-loop diagrams shown in Figs. 2,3. The (on-shell) vertex corrections in both diagrams of class I are comprised by the onephoton loop form factors F 1,2 (t) γ-loop of the electron. These Dirac and Pauli form factors are normalized at zero momentum transfer t = 0 to: F 1 (0) γ-loop = 0 and F 2 (0) γ-loop = α/2π (anomalous magnetic moment). Putting all pieces together the pertinent interference terms with the (direct and crossed) tree diagrams read:

I ⊗ A = α πt t 2 + 2tu + 2(u -2) 2 t -2 √ -t √ 4 -t 4ξ IR √ -t L(-t) + Φ(-t) +2ξ IR -2 + L(-t) √ 4 -t (4 -t)(3t 2 + 16) + 2u(8 -3t)(t + u -4) , (6)
I ⊗ B = α πt (t + u) 2 -4 t -2 √ -t √ 4 -t 4ξ IR √ -t L(-t) + Φ(-t) +2ξ IR -2 + L(-t) √ 4 -t 3(4 -t)(t + u) 2 + 2t 2 -4u 2 -32 . (7)
Here, we have introduced the frequently occurring logarithmic loop function:

L(x) = 1 √ x ln √ 4 + x + √ x 2 , (8)
and the auxiliary function:

Φ(x) = Li 2 (v(x)) -Li 2 (1 -v(x)) + 1 2 ln 2 v(x) - 1 2 ln 2 (1 -v(x)) , (9)
composed of dilogarithms and squared logarithms of the argument:

v(x) = 1 2 1 - x 4 + x , (10)
where Li

2 (v) = ∞ n=1 n -2 v n = v ∞ 1 dy[y(y -v)
] -1 ln y denotes the conventional dilogarithmic function. Note that in addition to the photon-loop also the appropriate counterterm [12] of quantum electrodynamics which eliminates the ultraviolet divergence ξ UV in the Dirac form factor F 1 (t) γ-loop has been included in eqs. (6,7).

Z 1 -1 = Z 2 -1 = α(2ξ IR + ξ UV -2)/2π
Diagram II involves a vacuum polarization at the exchanged photon. Together with the counterterm Z 3 -1 = 2α ξ UV /3π [12] the contribution from the electronic vacuum polarization takes the form:

II e ⊗ A = α 3πt 2 t 2 + 2tu + 2(u -2) 2 (2t + 4) √ 4 -t L(-t) - 5t 3 -4 , (11)
II e ⊗ B = α 3πt 2 (t + u) 2 -4 (2t + 4) √ 4 -t L(-t) - 5t 3 -4 . (12)
The additional (small) effect from the muonic vacuum polarization at the exchanged photon is readily obtained by substituting t → rt, with r = (m e /m µ ) 2 the squared mass ratio, in the polarization function:

II µ ⊗ A = α 3πt 2 t 2 + 2tu + 2(u -2) 2 2t + 4 r √ 4 -rt L(-rt) - 5t 3 - 4 r , (13)
II µ ⊗ B = α 3πt 2 (t + u) 2 -4 2t + 4 r √ 4 -rt L(-rt) - 5t 3 - 4 r . (14)
Next, we come to the (planar and crossed) two-photon exchange box diagrams shown in Fig. 3. For these diagrams it is most advantageous to perform the (multiple) spin sums via Dirac-traces before the (d-dimensional) loop-integration. In this way of proceeding the subsequent integrand emerges as a Lorentz-scalar constructed solely from the external four-momenta p 1 , p 2 , p 3 , p 4 and the loop momentum l. It can be profitably decomposed into partial fractions, thus reducing greatly the number of independent loop integrals to be solved. All of these loop integrals are ultraviolet convergent but some of them generate additional infrared divergences (III) (IV) (∼ ξ IR ). Employing the basic loop integrals H(s), K(t) and Ψ(s, t) with three and four propagators, defined and solved in the appendix, one finds for the interference terms of the planar box diagram III with the tree diagrams the following analytical expressions:

III ⊗ A = α π 3t 2 + 6tu + 4u 2 -14t -24u + 24 + 16u 4 -t K(t) + 4u 4 -t -1 - u 2 ln(-t) + L(-s) √ 4 -s (s 2 + st -6s + 8) + 2 -s √ 4 -s L(-s) 3t + 6u -8 + 4 t (u -2) 2 ln(-t) +4ξ IR t + 2u + 2 t (u -2) 2 + (2 -s)(2s + t) 2 √ -s √ 4 -s Φ(-s) , (15)
III ⊗ B = α π 2 (t + u) 2 -t -2u -6 + 4u 4 -t K(t) + 2 L(-s) √ 4 -s 2u + 6 -s t (s -2) 2 -2s -u ln(-t) +2ξ IR 6 -s t (s -2) 2 + 2u ln(-t) 4 -t + 2s + u √ -s √ 4 -s Φ(-s) , (16)
with the t-dependent auxiliary function:

K(t) = 1 √ -t √ 4 -t π 2 3 -t L 2 (-t) + Li 2 2 -t - √ -t √ 4 -t 2 . ( 17
)
In the same way one obtains for the interference terms of the crossed two-photon exchange box diagram IV with the tree diagrams the similar results:

IV ⊗ A = α π 8u -t 2 -2tu -4u 2 -2t -8 + 16u 4 -t K(t) + 4u 4 -t -3 + s 2 ln(-t) + L(-u) √ 4 -u (6u -tu -u 2 -8) + u -2 √ 4 -u L(-u) t + 2u + 4 t (u -2) 2 ln(-t) +4ξ IR t + 2u + 2 t (u -2) 2 + (u -2)(t + 2u) 2 √ -u √ 4 -u Φ(-u) , (18)
IV ⊗ B = α π 4 -t 2 -2tu -2u 2 -4u + 8u 4 -t K(t) + L(-u) √ 4 -u 4ξ IR (u -2) t + 2u + u 2 -4 t + (2 -t)(4 + u) -u 2 + tu + 2u 2 -6u -8 + 2 t (u + 2)(u -2) 2 ln(-t) + 2u 4 -t -1 + s 2 ln(-t) + 2u 2 + (4 -u)(2 -t) 2 √ -u √ 4 -u Φ(-u) . (19)
We note that it is instrumental to keep the factorized square roots in eqs. (6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) as they stand. If one would combine them into a single square root than a wrong sign could be obtained for some terms. We have checked via the Kramers-Kronig dispersion relation and the imaginary parts of the loop functions (given in the appendix) that the factorized square roots lead in all cases to the correct analytical continuation of the (only relevant) real parts of the loop functions along their branch cuts. When written with factorized square roots the correct sign often emerges as i • i = -1 from two negative radicands. It is important to have a representation with the correct analytical continuation built in, if one wants to carry over the results of the loop calculation from electron-electron scattering to electron-positron scattering (via the crossing transformation s ↔ u).

Infrared finiteness and numerical results

In the next step we have to consider the infrared divergent terms proportional to ξ IR . Inspection of eqs. (6,7,15,16,18,19) reveals that these scale with the Born terms A ⊗ A/t and A ⊗ B/t given in eq.( 3). As a consequence of that feature, the infrared divergent loop corrections multiply the differential cross section dσ (1γ) /dΩ cm at leading order by a (t ↔ u crossing-symmetric) factor:

δ (IR) virt = 8α π ξ IR 1 2 + t -2 √ 4 -t L(-t) + u -2 √ 4 -u L(-u) + Re 2 -s √ 4 -s L(-s) . (20)
The unphysical infrared divergence ξ IR gets canceled at the level of the (measurable) cross section by the contributions from soft photon bremsstrahlung. In its final effect, the (single) soft photon radiation off the in-or out-going electrons multiplies the Møller cross section dσ (1γ) /dΩ cm by a factor:

δ soft = α µ 4-d | l |<λ d d-1 l (2π) d-2 l 0 2p 1 • p 3 p 1 • l p 3 • l + 2p 2 • p 4 p 2 • l p 4 • l + 2p 1 • p 4 p 1 • l p 4 • l + 2p 2 • p 3 p 2 • l p 3 • l - 2p 1 • p 2 p 1 • l p 2 • l - 2p 3 • p 4 p 3 • l p 4 • l - m 2 e (p 1 • l) 2 - m 2 e (p 2 • l) 2 - m 2 e (p 3 • l) 2 - m 2 e (p 4 • l) 2 , (21)
which depends on a small photon energy cut-off λ. Working out this momentum space integral by the method of dimensional regularization (with d > 4) one finds that the infrared divergent correction factor δ

(IR)
virt ∼ ξ IR in eq.( 20) gets eliminated and the following finite radiative correction factor remains: with the abbreviations W t = s -4 + 4tx(1 -x) and W u = s -4 + 4ux(1 -x). We remark that the exact cancellation of infrared divergences ξ IR works out by reason of the identity:

δ (cm) real = 2α π 4 1 2 + t -2 √ 4 -t L(-t) + u -2 √ 4 -u L(-u) + s -2 √ s L(s -4) ln m e 2λ +2 √ s L(s -4) + 1/2 0 dx √ s(t -2) [1 -tx(1 -x)]W t ln √ s + W t √ s -W t + √ s(u -2) [1 -ux(1 -x)]W u ln √ s + W u √ s -W u + 2s -4 √ s 2 -4s (4 -s) L 2 (s -4) + √ s -4 ln(s -4) L(s -4) + π 2 6 -Li 2 s -2 - √ s 2 -4s 2 , (22)
Re [(2 -s)L(-s)/ √ 4 -s ] = (s -2)L(s -4)/ √ s,
where the sign change emerges from two negative radicands. Furthermore, we note that the terms beyond those proportional to ln(m e /2λ) are specific for the evaluation of the soft photon correction factor δ soft in the center-of-mass frame with λ an infrared cut-off therein. As it is written in eq.( 22), δ (cm) real refers to an (idealized) experimental situation where all undetected soft photon radiation fills a small sphere of radius λ in the center-of-mass frame. In a real experiment the momentum space region of undetected photons can be of different (e.g. non-isotropic) shape with no sharp boundaries due to detector efficiencies etc. Such additional experiment specific radiative corrections can be accounted for and calculated by integrating the fivefold differential cross section for e -e -→ e -e -γ over the appropriate region in phase space. By construction this region excludes the infrared singular domain | l | < λ and thus leads to a finite result. At this point one should note that for comparison with a real experiment the treatment of hard photon bremsstrahlung is also important.

We are now in the position to present some numerical results for the radiative corrections to electron-electron scattering e -e -→ e -e -. The radiative correction factor is δ

(cm)
real written in eq.( 22) plus the sum of all finite interference terms (second line in eq.( 2) with ξ IR → 0) divided by the Born terms (first line in eq.( 2)). Fig. 4 shows in percent the radiative corrections arising from loops alone (discarding the infrared divergent ξ IR terms) at seven selected center-of-mass energies √ sm e = (3, 6, 12, 24, 48, 96)m e . One notices that these (mostly negative) photonloop corrections become maximal for perpendicular scattering (θ cm = 90 • ) and that they grow approximately linearly with the total center-off-mass energy √ sm e . The effect from the muonic vacuum polarization (see eqs. (13,14)) is negligibly small in the energy range considered. Let us also mention that we confirm with good accuracy the numerical results of Furlan and Peressutti presented in ref. [16]. The full lines in Fig. 5 show the radiative corrections to electron-electron scattering with inclusion of the soft photon bremsstrahlung. For the sake of having a concrete case we have set the infrared cutoff to the value λ = √ sm e /200, thus modelling an (idealized) experimental situation where the electron energies can be resolved within 1% accuracy. One observes that the soft photon bremsstrahlung significantly increases the (negative) radiative corrections to electron-electron scattering. An approximate logarithmic scaling with the total center-off-mass energy √ sm e can now be recognized. Similar features apply also to the radiative corrections to (electron and pion) Compton scattering [2,13]. Clearly, as soon as the order α radiative corrections exceed about 20% in magnitude one has to worry about the two-loop and double bremsstrahlung effects of order α 2 . At higher energies the hadronic vacuum polarization will even start to play a role. Pursuing such higher order and heavy mass effects goes beyond the scope of the present work.

Radiative corrections to Bhabha scattering

Next, we want to discuss the radiative corrections to electron-positron (Bhabha) scattering. The reaction e -e + → e -e + can be obtained from electron-electron e -(p 1 )+e -(p 2 ) → e -(p 3 )+e -(p 4 ) via the replacement p 2 ↔ -p 4 of in-and out-going four-momenta. In terms of the invariant Mandelstam variables this involves the crossing transformation s ↔ u. The crossed one-photon exchange diagram B in Fig. 1 gets this way converted into the annihilation diagram. At leading order the Bhabha cross section takes the form:

dσ (e -e + ) dΩ cm = α 2 m 2 e s (s -2) 2 (s -4) 2 ζ 2 + 4 -2s 2 s(s -4) ζ + 2 + 4 s + 1 s 2 s -2 + (4 -s) ζ 2 , (23)
where ζ = sin 2 (θ cm /2) with θ cm the scattering angle in the center-of-mass frame. Note that there is missing a factor 1/4 in the second term in eq.( 6.47) of the textbook by Itzykson and Zuber [START_REF] Itzykson | Quantum Field Theory[END_REF]. While the results for the one-loop corrections to electron-positron scattering can be directly taken over from subsection 2.1 via the substitution s ↔ u this does not hold in the same way for the contributions from the soft photon bremsstrahlung. In the reaction e -e + → e -e + the soft photon is emitted either from a negatively or positively charged particle and this leads to a sign change in some (quantum mechanical) interference terms. In comparison to eq.( 22) the s and u dependent interference terms change sign and (after cancelling the infrared divergence ξ IR ) the finite part of the soft photon correction factor for electron-positron scattering reads:

δ (cm) real = 2α π 4 1 2 + t -2 √ 4 -t L(-t) + 2 -u √ 4 -u L(-u) + 2 -s √ s L(s -4) ln m 2λ +2 √ s L(s -4) + 1/2 0 dx √ s(t -2) [1 -tx(1 -x)]W t ln √ s + W t √ s -W t + √ s(2 -u) [1 -ux(1 -x)]W u ln √ s + W u √ s -W u + 2s -4 √ s 2 -4s (s -4) L 2 (s -4) - √ s -4 ln(s -4) L(s -4) - π 2 6 + Li 2 s -2 - √ s 2 -4s 2 . ( 24
)
Let us clarify instantaneously that no further s ↔ u crossing must be applied to the soft photon correction factor written in eq.( 23). Returning to eq.( 20) with permuted variables s ↔ u one finds that now the exact cancellation of infrared divergences ξ IR works out by reason of the identity: Re

[(s -2)L(-s)/ √ 4 -s ] = (2 -s)L(s -4)/ √ s.
The dashed lines in Fig. 6 show in percent the radiative corrections to electron-positron scattering arising from (photon and electron-positron) loops only. In comparison to electronelectron scattering (see Fig. 4) the size of these radiative corrections has significantly increased, in particular for scattering under backward angles, 90 • < θ cm < 180 • . Let us also mention that we confirm the numerical results of Furlan and Peressutti presented in ref. [16]. The full lines in Fig. 7 show the radiative corrections to electron-positron scattering with inclusion of the soft photon bremsstrahlung. The infrared cut-off has been set to the value λ = √ sm e /200, with the intension to model an (idealized) experimental situation where the electron and positron energies can be resolved within 1% accuracy. It is evident from Fig. 7 that the soft photon bremsstrahlung effects, which have not been considered by Furlan and Peressutti in ref. [16], add sizeably to the radiative corrections to electron-positron scattering. It should again be stressed that for a comparison with real experiments the hard photon contributions also have to be included. Generally, the hard bremsstrahlung is not cut completely, but treated as part of the QED radiative corrections. We do not pursue these (experimentspecific) contributions here further. As a consequence of that omission the numerical results presented in all figures should be considered only as indicative.

Let us comment on some recent related works. In the work by Fleischer et al. [START_REF] Fleischer | [END_REF] the oneloop photonic corrections to Bhabha scattering have been calculated in d spacetime dimensions. We have checked our results against theirs and found in the case d = 4 satisfactory agreement for all calculated terms. Actually, the two-loop radiative corrections to high-energy Bhabha scattering (approximated by the dominant logarithms in the electron mass m e ) are also known by now from the work of Penin [23]. We find that in the low-energy region considered here the subleading effects in the lepton mass still play a role.

Radiative corrections to elastic muon scattering

Having available the analytical results for the one-loop radiative corrections to electron-electron (and electron-positron) scattering one can furthermore apply them to the elastic muon scattering processes µ -µ ∓ → µ -µ ∓ . Only the role of the electron and the muon needs to be interchanged (via the substitution of masses m e ↔ m µ ) in the analytical expressions presented so far. The electronic vacuum polarization to the photon-exchange given accordingly by eqs. (13,14) is now written in terms of the large variable t/r = t(m µ /m e ) 2 = 42753 t. Despite that huge enhancement factor the effect of the electronic vacuum polarization is rather modest because of its weak (asymptotic) logarithmic t/r dependence. The estimate -(2α/3π) ln r 1.65% suggests a positive contribution of the order of a few percent from the electronic vacuum polarization in elastic muon scattering. This expectation is confirmed by the curves in Figs. 8,9 which show the calculated radiative corrections to µ -µ -→ µ -µ -and µ -µ + → µ -µ + scattering. In order to be comparable with the electron case the infrared cut-off for soft photon bremsstrahlung has been set now to the value λ = √ sm µ /200. Indeed one finds by comparison with Figs.

µ -µ --> µ -µ - λ = s 1/2 m µ /200

Radiative corrections to electron-muon scattering

In this section, we calculate the one-loop radiative corrections to the scattering of unequal leptons (e.g. electrons and muons). The in-and out-going four-momenta of the reaction e -(p 1 ) + µ -(p 2 ) → e -(p 3 ) + µ -(p 4 ) give rise to the (dimensionless) Mandelstam variables:

s = (p 1 + p 2 ) 2 /m 2 µ , t = (p 1 -p 3 ) 2 /m 2 µ , u = (p 1 -p 4 ) 2 /m 2 µ
, where we have divided by the square of the (large) muon mass m 2 µ . These convenient dimensionless variables satisfy the constraint s + t + u = 2 + 2r with r = (m e /m µ ) 2 the squared mass ratio. In analogy to eq.( 2) the unpolarized differential cross section for electron-muon scattering (e -µ -→ e -µ -) in the center-of-mass frame, including radiative corrections of relative order α, takes the following form:

dσ (e -µ -) dΩ cm = α 2 2m 2 µ s A ⊗ A t 2 + 2 t Re (I + II + III + IV ) ⊗ A . (25)
The momentum transfer variable t is given as t = [s 2 -2s(1 + r) + (1 -r) 2 ](cos θ cm -1)/2s in terms of the center-of-mass scattering angle θ cm and the total center-of-mass energy √ sm µ . In the case of unequal lepton scattering only the direct photon exchange diagram A (see Fig. 1) is possible and the corresponding spin-summed Born term reads:

A ⊗ A = 2(s -1 -r) 2 + 2st + t 2 = 2(u -1 -r) 2 + 2ut + t 2 . ( 26
)
Note the invariance of A ⊗ A under the crossing transformation s ↔ u. It expresses the obvious fact that at leading order the cross sections for e -µ -→ e -µ -and e -µ + → e -µ + are equal. In eq.( 25), I , II , III , IV stand for the four (classes of) loop diagrams shown in Fig. 2,3 interpreted now as diagrams for electron-muon scattering. By the prime we distinguish them notationally from those for electron-electron scattering. The product symbol ⊗ designates again the interference term between the T-matrices from two diagrams with the sums over electron and muon spins already carried out via Dirac-traces:

1 8 tr[O 1 (/ p 1 + m e )O 1 (/ p 3 + m e )]• tr[O 2 (/ p 2 + m µ )O 2 (/ p 4 + m µ )].

Evaluation of one-loop diagrams

In this subsection we present analytical expressions (of order α) for the interference terms between the one-loop diagrams and the tree diagram A for electron-muon scattering.

Considering the diagrams of class I shown in Fig. 2 one can have a vertex correction either at the (left) electron line or at the (right) muon line. In both cases it is given by the respective one-photon loop form factors F γ-loop 1,2 (t) for electrons or muons. The pertinent interference term of the electronic vertex correction with the tree diagram A takes the form:

I e ⊗ A = α 2πt A ⊗ A L(-tr -1 ) √ 4r 2 -rt (4ξ IR + 2 ln r)(t -2r) + 8r -3t + (t -2r)Φ(-tr -1 ) √ -t √ 4r -t + 2ξ IR -2 + ln r + 4t √ r(2 + t) L(-tr -1 ) √ 4r -t , (27)
while that of muonic vertex correction reads:

I µ ⊗ A = α 2πt A ⊗ A L(-t) √ 4 -t 4ξ IR (t -2) + 8 -3t + (t -2)Φ(-t) √ -t √ 4 -t + 2ξ IR -2 + 4t(2r + t) L(-t) √ 4 -t . ( 28
)
Note that the infrared divergence ξ IR (see eq.( 5)) is now redefined with the muon mass m µ under the logarithm ln(m µ /µ). This leads to the additional ln r terms for I e ⊗ A written in eq.( 27). The functions L(-t) and Φ(-t) have been defined in eqs. (8,9,10).

The vacuum polarization diagram II shown in Fig. 2 comes also in two versions, either as electronic vacuum polarization:

II e ⊗ A = α 3πt 2 2 √ r (t + 2r) √ 4r -t L(-tr -1 ) - 5t 3 -4r A ⊗ A , (29)
or as muonic vacuum polarization:

II µ ⊗ A = α 3πt 2 (2t + 4) √ 4 -t L(-t) - 5t 3 -4 A ⊗ A . ( 30
)
More tedious to evaluate are the (planar and crossed) two-photon exchange box diagrams shown in Fig. 3. We take advantage of performing first the spin-sums (via Dirac-traces) and of decomposing in the next step the resulting Lorentz-scalar loop integrand into partial fractions.

In this form the majority of terms has either only one electron propagator or only one muon propagator and few terms involve the product of both. The latter loop integrals depend in a non-trivial way on the squared mass ratio r = (m e /m µ )2 . Putting all pieces together one finds for the interference term of the planar two-photon exchange diagram III with the tree diagram A the following analytical expression:

III ⊗ A = α π 8 4 -t (r -1 -s) + t 2 2 + t + st + 2(s -r) 2 + 2 K(t) + 8r 4r -t (1 -r -s) + t 2 2r + t r (s + r) + 2 r (s -1) 2 + 2r K(tr -1 ) + 2 4 -t (r -1 -s) + 2r 4r -t (1 -r -s) + 1 2 (s + t + 1 + r) ln(-t) + r -1 s (1 + r + t) + 8r 4r -t (s + r -1) + 2 -4r -s -t ln r 4 + 1 + r -s √ ρ + -s √ ρ --s 4ξ IR 2s + t + 2 t (s -1 -r) 2 + 2s + t + 4 t (s -1 -r) 2 ln(-t) ln √ ρ + -s + √ ρ --s 2r 1/4 + s 2 + st -3s(1 + r) + 3r 2 + 2r + 3 - (1 -r) 2 s (1 + r + t) × 1 √ ρ + -s √ ρ --s ln √ ρ + -s + √ ρ --s 2r 1/4 + 1 2 (2s + t) ×(1 + r -s) ∞ ρ + dx ln[x -2 -2r + (1 -r) 2 x -1 ] x 2 -2x(1 + r) + (1 -r) 2 (x -s) , (31)
with the abbreviations ρ ± = 1 + r ± 2 √ r. For the numerical evaluation of the (only relevant) real part of III ⊗ A the spectral integral ∞ ρ + dx/(x -s) . . . in the last line of eq.( 31) has to be treated as a principal value integral (if s > ρ + ). It can be conveniently decomposed into a sum of two non-singular integrals by the following formula:

- ∞ ρ + dx f (x) x -s = 2s-ρ + ρ + dx f (x) -f (s) x -s + ∞ 2s-ρ + dx f (x) x -s . ( 32
)
Finally, we are left with the contribution of the crossed two-photon exchange box diagram IV shown in Fig. 3. Its interference term with the tree diagram A can be obtained via crossing (s → u = 2 + 2r -s -t) as follows:

2 IV ⊗ A = -III ⊗ A | s→u , (33)
with III ⊗ A given in eq.( 31). We have actually verified this relation by an explicit calculation of IV ⊗ A as it arises from the crossed two-photon exchange diagram IV . The physical reason behind eq.( 33) can be easily explained and understood. When turning the right muon line in the crossed two-photon exchange diagram IV upside-down one gets the planar two-photon exchange diagram III for electron-antimuon scattering (e -µ + → e -µ + ). The same manipulation applied to the tree diagram A introduces an additional minus sign due to the (single) opposite electric charge, and to close the argument, e -µ + → e -µ + and e -µ -→ e -µ -are connected with each other by s ↔ u crossing. For the same reason the relation IV ⊗ A = -III ⊗ A | s→u holds also for the expressions written in eqs. (15,18).

Infrared finiteness and numerical results

The infrared divergent terms proportional to ξ IR in eqs.(27,28,31,33) follow again the pattern of the Born term A ⊗ A /t. The leading order differential cross section gets therefore multiplied by the (infrared divergent) factor:

δ (IR) virt = 4α π ξ IR 1 + t -2 √ 4 -t L(-t) + t -2r √ 4r 2 -rt L(-tr -1 ) +Re 2(1 + r -s) √ ρ + -s √ ρ --s ln √ ρ + -s + √ ρ --s 2r 1/4 + 2(u -1 -r) √ ρ + -u √ ρ --u ln √ ρ + -u + √ ρ --u 2r 1/4 . ( 34
)
On the other hand, the (single) soft photon radiation from an electron or a muon yields the multiplicative factor:

δ soft = α µ 4-d | l |<λ d d-1 l (2π) d-2 l 0 2p 1 • p 3 p 1 • l p 3 • l + 2p 2 • p 4 p 2 • l p 4 • l + 2p 1 • p 4 p 1 • l p 4 • l + 2p 2 • p 3 p 2 • l p 3 • l - 2p 1 • p 2 p 1 • l p 2 • l - 2p 3 • p 4 p 3 • l p 4 • l - m 2 e (p 1 • l) 2 - m 2 µ (p 2 • l) 2 - m 2 e (p 3 • l) 2 - m 2 µ (p 4 • l) 2 , ( 35
)
which includes exactly the same infrared divergence (∼ ξ IR ) but with the opposite sign. The remaining finite radiative correction factor for electron-muon scattering (e -µ -→ e -µ -) reads:

δ (cm) real = α π 4 1 + t -2 √ 4 -t L(-t) + t -2r √ 4r 2 -rt L - t r + 2 √ P (s -1 -r) ln √ s -ρ + + √ s -ρ - 2r 1/4 + 2(u -1 -r) √ ρ + -u √ ρ --u ln √ ρ + -u + √ ρ --u 2r 1/4 ln m µ 2λ + 2 √ P × (s -1 + r) ln s -1 + r + √ P 2 √ sr + (s + 1 -r) ln s + 1 -r + √ P 2 √ s + 1/2 0 dx (t -2)(s + 1 -r) [1 -tx(1 -x)] √ R t ln s + 1 -r + √ R t s + 1 -r - √ R t + (t -2r)(s -1 + r) [r -tx(1 -x)] √ R t ln s -1 + r + √ R t s -1 + r - √ R t + 1 0 dx (s -1 -r)[s + (1 -r)(1 -2x)] (1 -2x)[sx(1 -x) + (1 -2x)(1 -x -rx)] √ P
× ln s + (1 -2x)(1 -r + √ P ) s + (1 -2x)(1 -r - √ P ) + (u -1 -r)[s + (1 -r)(1 -2x)] [1 + (r -1)x -ux(1 -x)] √ R u ln s + (1 -r)(1 -2x) + √ R u s + (1 -r)(1 -2x) - √ R u , (36)
where we have introduced the polynomials:

P = s 2 -2s(1 + r) + (1 -r) 2 , R t = P + 4stx(1 -x), R u = P + 4x(1 -x)[su -(1 -r) 2 ]
. We note that the terms beyond those proportional to ln(m µ /2λ) are specific for the evaluation of the soft photon correction factor δ soft in the centerof-mass frame with λ an infrared cut-off therein. Setting r = 1 (i.e. ρ + = 4, ρ -= 0) one recovers from eq.(36) the soft photon correction factor δ (cm) real for electron-electron scattering written in eq. [START_REF] Fleischer | [END_REF]. Fig. 10 shows in percent the radiative corrections to electron-muon scattering arising from loops only (discarding the infrared divergence ξ IR ). A striking difference to the electron-electron case (see Fig. 4) is that these radiative corrections are now positive and larger in magnitude. Moreover, with increasing total center-of-mass energy √ sm µ the angular dependence of these radiative corrections changes. The maximum around θ cm 90 • turns gradually into a local minimum. At this point it should be stressed that these finite loop corrections are by themselves not physical since they have been obtained by dropping the (scheme-dependent) infrared divergence ξ IR , which is defined here with the muon mass m µ under the logarithm ln(m µ /µ). When choosing a different mass scale, the loop corrections will change accordingly. The curves in Fig. 11 show the (full) radiative corrections to electron-muon scattering (e -µ -→ e -µ -) with inclusion of the soft photon bremsstrahlung. For the sake of having a concrete case we have set the infrared cutoff to the value λ = √ sm µ /200. With inclusion of the soft photon bremsstrahlung the radiative corrections turn back to (sizeable) negative values and their energy and angular dependences follow more closely the pattern observed for electron-electron scattering (see Fig. 5). We do not discuss here in further detail the radiative corrections to electron-antimuon scattering e -µ + → e -µ + . These can be obtained through the crossing transformation (s ↔ u) in the loop amplitudes and a sign change in some interference terms of the soft photon correction factor δ soft (see eqs.(35,36)). For guidance one can compare eq.( 22) with eq.( 24) which exhibit this feature in the case of equal leptons.

Radiative corrections to lepton pair production

As a final application of our calculation we consider the radiative corrections to lepton pair production. The reaction e -(p 1 ) + e + (p 2 ) → µ + (p 3 ) + µ -(p 4 ) is obtained from electron-muon scattering via the crossing transformation s ↔ t. We define the center-of-mass scattering angle θ cm through the (uncrossed) variable u = (p 1 -p 4 ) 2 /m 2 µ = 1+r + ( √ s -4 √ s -4r cos θ cm -s)/2 as the angle between the momentum vectors of negatively charged particles. The differential and total cross section for e -e + → µ + µ -at leading order have the following well-known form:

dσ (pair) dΩ cm = α 2 4m 2 µ s s -4 s -4r 1 + cos 2 θ cm + 4 s (1 + r) sin 2 θ cm + 16r s 2 cos 2 θ cm , (37) σ (pair) tot
= 4πα 2 3m 2 µ s s -4 s -4r 1 + 2 s 1 + 2r s , (38)
with √ sm µ the total center-of-mass energy and r = (m e /m µ ) 2 . The analytical expressions for the tree and loop contributions to lepton pair production e -e + → µ + µ -can be directly taken over from eqs.(25-32) via the crossing transformation s ↔ t. However, the corresponding soft photon correction factor δ soft , obtained formally from eq.( 35) by the permutation p 2 ↔ p 3 , needs to be reevaluated for the center-of-mass kinematics under consideration. After cancelling the infrared divergences ξ IR against those from the loops the finite remainder of the soft photon correction factor for lepton pair production e -e + → µ + µ - reads: Figure 12: Radiative corrections to lepton-pair production e + e -→ µ + µ -arising from photon loops only. The numbers (3,6,12,24,48) on the curves correspond to the total center-of-mass energy √ sm µ in units of m µ .

δ (cm) real = α π 4 1 + 2 -s √ s L(s -4) + 2r -s √ sr L s r - 4
+ 2(1 + r -t) √ ρ + -t √ ρ --t ln √ ρ + -t + √ ρ --t 2r 1/4 + 2(u -1 -r) √ ρ + -u √ ρ --u ln √ ρ + -u + √ ρ --u 2r 1/4 ln m µ 2λ +2 s r L s r -4 + 2 √ s L(s -4) + 1/2 0 dx √ s(2 -s)(s -4) -1/2 (1 -2x)[1 + (s -4)x(1 -x)] ln √ s + (1 -2x) √ s -4 √ s -(1 -2x) √ s -4 + √ s(2r -s)(s -4r) -1/2 (1 -2x)[r + (s -4r)x(1 -x)] ln √ s + (1 -2x) √ s -4r √ s -(1 -2x) √ s -4r + 1 0 dx √ s(1 + r -t) [1 + (r -1)x -tx(1 -x)] √ C t ln √ s + √ C t √ s - √ C t + √ s(u -1 -r) [1 + (r -1)x -ux(1 -x)] √ C u ln √ s + √ C u √ s - √ C u , (39)
with the polynomials

C t = s -4 + 4(1 -r)x + 4tx(1 -x) and C u = s -4 + 4(1 -r)x + 4ux(1 -x).
The dashed curves in Fig. 12 show the radiative corrections to muon pair production e -e + → µ + µ -arising from loops only (discarding the infrared divergent terms ξ IR ). One notices that the forward-backward symmetry (θ cm → 180 • -θ cm or t ↔ u) of the tree level cross section (see eq.( 37)) is broken by the loop corrections. This feature originates from the negative sign between the planar and crossed two-photon exchange contributions as exhibited in eq.( 33) and explained afterwards. Also by interchanging the role of particle and antiparticle for µ -and µ + one gets a relative minus sign between the one-photon and two-photon exchange contributions. Their (quantum mechanical) interference can therefore distinguish the two alternatives of fixing the scattering angle θ cm between the electron and µ -direction or between the electron and µ + direction. This way a forward-backward asymmetry is generated by the two-photon exchange corrections. The full curves in Fig. 13 show the radiative corrections to muon pair production e -e + → µ + µ -with inclusion of the soft photon bremsstrahlung. The infrared cutoff has been set to the value λ = √ sm µ /200. One observes that the previous (scheme-dependent) positive radiative corrections from loops turn back into sizeable negative values by the soft photon bremsstrahlung. The forward-backward asymmetry becomes even more pronounced. The reason for that additional effect is that some interference terms of the soft photon emission are different for emission from positively and negatively charged muons. The expressions in the second and third as well as in the last two lines of eq.( 39) demonstrate explicitly the asymmetry under t ↔ u or equivalently cos θ cm →cos θ cm .

Finally, we show in Fig. 14 the radiative corrections to the total cross section for muon pair production e -e + → µ + µ -as a function of the total center-of-mass energy √ sm µ . In this case all t and u dependent contributions from two-photon exchange and soft photon bremsstrahlung (see eq.( 39)) drop out, because they odd under t ↔ u and thus integrate to zero. The analytical expression for the radiative corrections to the total cross section σ tot (s) (see eq.( 38)) can therefore be directly read off from eqs.(27-30) substituting t → s, and from eq.(39) dropping the t and u dependent terms. The pertinent angular average of the Born term A ⊗ A for muon pair production e + e -→ µ + µ -is of the form: 2(s + 2)(s + 2r)/3. This concludes our presentation and (schematic) discussion of the radiative corrections of order α to lepton-lepton scattering processes. It should be stressed again that for a comparison with real experiments the contributions from hard bremsstrahlung also have to be considered. The one-loop radiative corrections to muon-pair (and quark-pair) production in e -e + collisions in the region of the Z 0 -resonance have been calculated some time ago by Berends et al. in ref. [24]. The additional and new elements there are photonic vertex corrections to the Z 0exchange and the γZ 0 box diagrams.

Radiative corrections to scalar boson scattering

For the sake of comparison, we treat in this section the radiative corrections to the scattering of (identical) charged scalar bosons. The new element in scalar quantum electrodynamics is

dσ (b -b -) dΩ cm = α 2 4m 2 b s |A(t, u) + A(u, t)| 2 , (40)
where A(t, u) is the scattering amplitude arising from diagrams with uncrossed (external) boson lines only. The corresponding tree-level Born term (as given by one-photon exchange) reads:

A(t, u) (tree) = s -u t . (41)
Loop diagrams of scalar quantum electrodynamics contribute in the form of vertex corrections:

A(t, u) (vert) = α πt (s -u) 2ξ IR -2 + t -2 √ -t √ 4 -t 4(ξ IR -1) √ -t L(-t) + Φ(-t) , (42)
and via a (scalar boson) vacuum polarization:

A(t, u) (vap) = α 6πt 2 (s -u) 4 - 4t 3 -(4 -t) 3/2 L(-t) . (43)
Specific for scalar quantum electrodynamics are the two-photon exchange bubble and triangle diagrams shown in Fig. 15. Their individual contributions to the scattering amplitude A(t, u) read:

A(t, u) (bub) = α π -4ξ UV + 3 -2 ln(-t) , (44)
A(t, u) (tria) = α π 2ξ UV -2 + 2 ln(-t) + (8 -t)K(t) . (45)
with the function K(t) defined in eq.( 17) and ξ UV denotes the ultraviolet divergence in dimensional regularization. Likewise, the planar two-photon exchange box diagram (see Fig. 3) for scalar bosons leads to the contribution:

A(t, u) (box) = α π 1 2 (1 -ξ UV ) + 3s -8 + 4s 4 -t K(t) + 4L(-s) t √ 4 -s ξ IR (s -2)(u -s) -(s -2) 2 ln(-t) - 1 2 √ 4 -s L(-s) - u ln(-t) 4 -t + (2 -s) Φ(-s) √ -s √ 4 -s , (46)
while that of the crossed two-photon exchange box diagram reads:

A(t, u) (crb) = α π 1 2 (1 -ξ UV ) + 3u -8 + 4u 4 -t K(t) + 4L(-u) t √ 4 -u ξ IR (u -2)(s -u) -(u -2) 2 ln(-t) - 1 2 √ 4 -u L(-u) - s ln(-t) 4 -t + (2 -u) Φ(-u) √ -u √ 4 -u . (47)
As suggested by turning the charged boson line upside down, these last two contributions are related by the crossing transformation s ↔ u, i.e. A(t, u) (crb) = A(t, s) (box) with s = 4 -t -u.

When summing the terms in eqs.(44-47) an ultraviolet divergence of -3α ξ UV /π remains which in the end gets canceled by an appropriate counterterm of scalar quantum electrodynamics. Fig. 16 shows the radiative corrections to equally charged scalar boson scattering arising from loops only. In comparison to the spin-1/2 case shown in Fig. 4, one observes that the transition from positive to negative loop corrections with increasing center-of-mass energy √ sm b proceeds somewhat differently. The full lines in Fig. 17 show the radiative corrections with inclusion of the effects due to soft photon bremsstrahlung (given in identical form by eq.( 22)). Altogether, one notices by comparison with Fig. 5 that the radiative corrections to the scattering of (equally charged) scalar bosons are somewhat smaller in magnitude than those for spin-1/2 fermions (at the same kinematical conditions).

Finally, we show in Figs. 18,19 the radiative corrections to the scattering of oppositely charged scalar bosons. As in the case of Bhabha scattering (see section 2.3) these are obtained through the crossing transformation s ↔ u applied to the amplitude A(t, u) + A(u, t) in eq.( 40) and the corresponding soft photon correction factor δ (cm) real is given by eq.( 24). Making the comparison with Figs. 6,7 one is lead to the conclusion that for oppositely charged particles the radiative corrections are larger in magnitude for scalar bosons than for spin-1/2 fermions. Although not of direct practical relevance (since elementary charged scalar bosons do not exist in nature) this is an interesting observation.

In passing we note that the results obtained here in scalar QED cannot be applied e.g. to the pion. The electromagnetic form factor of the pion is dominated by the ρ 0 (770)-resonance leading to an enhancement factor of up about 45 in the π + π -→ γ * → π + π -transition (in the ρ 0 region). In addition, the hard part of the photon spectrum is radiated by quarks, not by the mesons themselves. Actually, the two-photon exchange contribution to the elastic electron-pion scattering has been calculated recently in ref. [25] taking into account the structure of the pion through a monopole-type (charge) form factor as suggested by vector-meson dominance model. where t m 2 = q 2 = 2p 1 • q and p 2 1 = m 2 . Somewhat surprisingly, this loop integral is infrared convergent and the result has the analytical form:

K(t) = 1 √ -t √ 4 -t π 2 3 + Li 2 2 -t - √ -t √ 4 -t 2 + ln 2 √ 4 -t + √ -t 2 . ( 56
)
The corresponding imaginary part along the cut t > 0 consists of two pieces:

Im K(t) = π √ 4t -t 2 arccos √ t 2 , for 0 < t < 4 , (57)
Im

K(t) = π √ t 2 -4t ln √ t + √ t -4 2 , for t > 4 , (58)
which join smoothly at t = 4. Eventually, the two-photon exchange diagram leads to the following loop integral with four propagators (two massive, two massless):

d d l (2π) d i (4πm 2 ) 2 µ 4-d [m 2 -(p 1 -l) 2 ][m 2 -(p 2 + l) 2 ][-(q -l) 2 ][-l 2 ] = Ψ(s, t) + O(d -4) , (59)
where the dimensionless variables (s, t) are defined by s m 2 = (p 1 + p 2 ) 2 , t m 2 = q 2 = 2p 1 • q = -2p 2 • q and p 2 1 = p 2 2 = m 2 . In analogy to eq.( 44) the vanishing photon mass generates again an infrared divergence ξ IR and the complete result reads:

Ψ(
with ρ ± = (1 ± m e /m µ ) 2 . The importance of the imaginary parts of the loop functions lies in the fact that they determine via a Kramers-Kronig dispersion relation the corresponding real parts. On that basis we checked in detail (numerically) that the expressions with factorized square roots given in this appendix provide the correct analytical continuation of the real parts of the loop functions above their respective branch points. The master integral occurring in the calculation of the soft photon correction factor δ soft is:

1 -1 dx (1 -x 2 ) d-4 2 (E + xp) 2 = 2 E 2 -p 2 1 + (d -4) ln 2 - E 2p ln E + p E -p , (64)
carrying the expansion in d -4 up to linear order.

Figure 1 :

 1 Figure 1: Tree diagrams, (A) and (B), for electron-electron scattering.

Figure 2 :

 2 Figure 2: One-loop diagrams, (I) and (II), for electron-electron scattering.

Figure 3 :

 3 Figure 3: One-photon loop diagrams, (III) and (IV), for electron-electron scattering.

Figure 4 :

 4 Figure 4: Radiative corrections to electron-electron scattering arising from loops only. The numbers (3, 6, 12, 24, 48, 96) on the curves correspond to the total center-of-mass energy √ sm e in units of m e .

Figure 5 :

 5 Figure 5: Radiative corrections to electron-electron scattering including soft photon bremsstrahlung. The numbers (3, 6, 12, 24, 48, 96) on the curves correspond to the total centerof-mass energy √ sm e in units of m e . The infrared cutoff has been set to λ = √ sm e /200.

Figure 6 :

 6 Figure 6: Radiative corrections to electron-positron scattering arising from loops only. The numbers (3, 6, 12, 24, 48, 96) on the curves correspond to the total center-of-mass energy √ s in units of m e .

Figure 7 :

 7 Figure 7: Radiative corrections to electron-positron scattering including soft photon bremsstrahlung. The numbers (3, 6, 12, 24, 48, 96) on the curves correspond to the total centerof-mass energy √ s in units of m e . The infrared cutoff has been set to λ = √ sm e /200.

Figure 8 :

 8 Figure 8: Radiative corrections to elastic muon scattering µ -µ -→ µ -µ -. The numbers (3, 6, 12, 24, 48, 96) on the curves correspond to the total center-of-mass energy √ sm µ in units of m µ . The infrared cutoff has been set to λ = √ sm µ /200.

Figure 9 :

 9 Figure 9: Radiative corrections to muonic Bhabha scattering µ -µ + → µ -µ + . The numbers (3, 6, 12, 24, 48, 96) on the curves correspond to the total center-of-mass energy √ sm µ in units of m µ . The infrared cutoff has been set to λ = √ sm µ /200.

Figure 10 :

 10 Figure 10: Radiative corrections to electron-muon scattering arising from loops only. The numbers (1.5, 3, 6, 12, 24, 48) on the curves correspond to the total center-of-mass energy √ sm µ in units of m µ .

Figure 11 :

 11 Figure 11: Radiative corrections to electron-muon scattering including soft photon bremsstrahlung. The numbers (1.5, 3, 6, 12, 24, 48) on the curves correspond to the total centerof-mass energy √ sm µ in units of m µ . The infrared cutoff has been set to λ = √ sm µ /200.

Figure 13 :

 13 Figure 13: Radiative corrections to lepton-pair production e + e -→ µ + µ -including soft photon bremsstrahlung. The numbers (3, 6, 12, 24, 48) on the curves correspond to the total center-ofmass energy √ sm µ in units of m µ . The infrared cutoff has been set to λ = √ sm µ /200.

Figure 14 :

 14 Figure 14: Radiative corrections to the total cross section for lepton pair production e + e -→ µ + µ -. The numbers (3, 6, 12, 24, 48) on the curves correspond to the total center-of-mass energy √ sm µ in units of m µ . The infrared cutoff has been set to λ = √ sm µ /200.

Figure 15 :

 15 Figure 15: Additional (types of) photon-loop diagrams generated by the two-photon contact vertex of scalar quantum electrodynamics.

Figure 16 :

 16 Figure 16: Radiative corrections to equally charged scalar boson scattering arising from loops only. The numbers (3, 6, 12, 24, 48, 96) on the curves correspond to the total center-of-mass energy √ sm b in units of m b .

Figure 17 :

 17 Figure 17: Radiative corrections to equally charged scalar boson scattering including soft photon bremsstrahlung. The numbers (3, 6, 12, 24, 48, 96) on the curves correspond to the total center-of-mass energy √ sm b in units of m b . The infrared cutoff has been set to λ = √ sm b /200.

Figure 18 :

 18 Figure 18: Radiative corrections to oppositely charged scalar boson scattering arising from loops only. The numbers (3, 6, 12, 24, 48, 96) on the curves correspond to the total center-ofmass energy √ sm b in units of m b .

Figure 19 :

 19 Figure 19: Radiative corrections to oppositely charged scalar boson scattering including soft photon bremsstrahlung. The numbers (3, 6, 12, 24, 48, 96) on the curves correspond to the total center-of-mass energy √ sm b in units of m b . The infrared cutoff has been set to λ = √ sm b /200.

  In the case of electron-muon scattering one encounters generalizations of the loop integrals H(t) and Ψ(s, t) (see eqs.(52,59)) with different masses p 2 1 = m 2 e and p 2 2 = m 2 µ in the propagators and m 2 µ in the numerator. The corresponding generalizations of the imaginary parts read: (s -ρ + )(s -ρ -) 2ξ IR + ln(-t) , for s > ρ + , t < 0 ,

	s, t) =	-t	√	4 -s √	4 -s	2ξ IR + ln(-t) ln	√	4 -s + 2	√	-s	,	(60)
	together with the imaginary part:
	Im Ψ(s, t) =	-t	2π √ s 2 -4s	2ξ IR + ln(-t) ,	for s > 4 , t < 0 .	(61)
	Im H eµ (t) =	π (t -ρ + )(t -ρ -)	2ξ IR + ln	(t -ρ + )(t -ρ -) t	,	for t > ρ + ,	(62)
	Im Ψ eµ (s, t) =	-t		2π

Since there are two identical particles in the final state the physical region is restricted to 0 < θ cm ≤ 90 • .

Because of this property all the s-independent terms in the first four lines of eq.(31) could be dropped without affecting the sum (III + IV ) ⊗ A .

Appendix: Loop integrals

In this appendix we collect the results for the (basic) one-loop integrals one encounters in the calculation of the radiative corrections to lepton-lepton scattering. We are using dimensional regularization to treat both ultraviolet and infrared divergences. Divergent pieces of one-loop integrals show up in form of the constant:

containing a simple pole at d = 4 and the logarithm of an arbitrary mass scale µ. Ultraviolet (UV) and infrared (IR) divergences are distinguished by the feature of whether d < 4 or d > 4 is the condition for convergence of the d-dimensional integral. In the following all propagators are understood to have an infinitesimal negative imaginary part: -i0 + .

The vacuum polarization diagram leads to the following loop integral with two (massive) propagators:

where the dimensionless variable t is defined by t m 2 = q 2 . In addition to the ultraviolet divergence -2ξ UV one obtains the logarithmic function:

whose imaginary part along the branch cut t > 4 is:

The vertex correction diagram introduces the following loop integral with three propagators (two massive, one massless):

where t m 2 = (p 1 -p 2 ) 2 and p 2 1 = p 2 2 = m 2 . The mass zero of the photon generates an infrared divergence ξ IR and the complete result can be written as:

in terms of logarithms and dilogarithms where 1 -2v = -t/(4 -t). By convention, Li

] -1 ln y denotes the dilogarithmic function. We note as an aside that when choosing a fictitious photon mass m γ as an infrared regulator the infrared divergence appears always in the form ξ IR = ln(m/m γ ). The imaginary part of H(t) along the branch cut t > 4 is:

Im

The two-photon exchange diagram introduces another loop integral with three propagators (one massive, two massless):