Keywords: RF switch, Air microcavities, CMOS, SOI, Off-state capacitance, Vapor phase selective etching

This paper investigates the reduction of the off-state capacitance (COFF) of SOI-CMOS RF switches induced by the introduction of air microcavity in the back-end interconnection network. A detailed methodology combining electromagnetic and semiconductor transport simulations is used to separately evaluate the respective contributions of the interconnects and junction capacitances. A baseline switch and an optimized version of the same are studied to evaluate their ability to take advantage of air microcavities. Simulations show a reduction of 73 fF/mm regardless of the switch structure considered, resulting in a 24.6% and 30.6% improvement in COFF for baseline and optimized switches, respectively, bringing the optimized version to a 60fs record Ron × Coff. This concept was experimentally implemented using a partial etch process that resulted in a reduction of 21.7 fF/mm, i.e., 7.2%. Finally, the implementation of a more isotropic and selective etching process using HF in vapor phase is shown to approach the optimal configuration of air microcavities.

I. INTRODUCTION

HE constant demand for evolution in communications forces standards to push their limits in terms of data rate, latency or security. As such, the latest 5G standard is based on key technologies such as spectrum sharing, massive MIMO or carrier aggregation for optimized usage of available bandwidth. We also note the deployment of a millimeter wave frequency bands (mmW) referred to as FR2 band covering the 24.25-52. [START_REF] Im | Characterization and optimization of partially depleted SOI MOSFETs for high power RF switch applications[END_REF] GHz range to expand into high data rate marked at fixed locations [START_REF] Chen | The Standardization of 5G-Advanced in 3GPP[END_REF] [2]. As a consequence of the growing complexity of communication systems, an increase of switches becomes necessary to selectively propagate high frequency signals through the transmit/receive pathways. This constraint of complexity must of course be handled without impacting the systems in terms of insertion loss, linearity, power handling and isolation [START_REF] Balteanu | RF Front End Module Architectures for 5G[END_REF]. In order to limit the accumulation of losses along the propagation path, RF switches must offer low insertion losses reflected by a small on-state resistance with a sufficient isolation in the off-state. The two important parameters, RON and COFF, precisely reflect the performance of a switch in terms of insertion loss and isolation, respectively. Minimizing them also means reducing their product (RON × COFF), the main figure of merit (FoM) used to compare switches with various layouts implemented in different semiconductor technologies. Strategies for optimizing switch performance are manyfold, such as reducing the SOI film thickness to reduce COFF and downscaling the gate length to lower RON. Several reports have shown that these optimization levers can be implemented without compromising linearity and power handling [START_REF] Gianesello | Highly linear and sub 120 fs Ron × Coff 130 nm RF SOI technology targeting 5G carrier aggregation RF switches and FEM SOC[END_REF][5][6] [START_REF] Kim | High Power Ku-Band T/R and SP4T Switches in SOI CMOS[END_REF]. Nevertheless, these papers do not sidestep the point that the optimization of both insertion losses and isolation is a matter of compromise since any path leading to the reduction of RON inevitably degrades COFF and the RF breakdown voltage as well. Improving these two parameters together is all the more difficult given the level of optimization already achieved to date, e.g., by simultaneously reconsidering the junction design, introducing offset spacers, boosting the stress level in the Contact-Etch-Stop-Layer (CESL), reducing the gate length and thickness as well as optimizing the circuit layout and the routing strategy of metal interconnect tracks [START_REF] Gianesello | Advanced 200-mm RF SOI Technology exhibiting 78 fs[END_REF]. In that context, the use of air microcavities within the metal interconnect network, also known as back-end-of-the-line (BEOL), appears as a legitimate option to investigate in an attempt to lower COFF independently of RON. The contribution of BEOL to the overall COFF is significant and typically ranges from 30 to 40%. [START_REF] Gianesello | Highly linear and sub 120 fs Ron × Coff 130 nm RF SOI technology targeting 5G carrier aggregation RF switches and FEM SOC[END_REF] [START_REF] Jaffe | Improvements in SOI technology for RF switches[END_REF]. The replacement of the intermetal dielectric with air theoretically enables the reduction of the material permittivity from εr~4 to εr=1, thus holding the promise of a substantial reduction of COFF. In the present study, the impact of air microcavities in the BEOL is analyzed on two generations of switches, the first one serving as a baseline [START_REF] Gianesello | Highly linear and sub 120 fs Ron × Coff 130 nm RF SOI technology targeting 5G carrier aggregation RF switches and FEM SOC[END_REF] and the second one being a significant evolution integrating many optimization factors [START_REF] Gianesello | Advanced 200-mm RF SOI Technology exhibiting 78 fs[END_REF]. Through this comparison, the specific question that is addressed here is whether the same COFF reduction margin can be achieved on a baseline structure and on an optimized structure. In this perspective, this paper first presents a simulation strategy to finely evaluate the impact of air microcavities in BEOL of RF switches. It follows by detailing a selective etching method to create microcavities in a post-process sequence. Subsequently, the efficiency of the proposed approach is experimentally validated by extracting COFF from S-parameter measurements on switches incorporating microcavities. Finally, perspectives for improvement of the etching process using a selective vapor phase are discussed in order to perfect the etching of the intermetallic dielectric down to the CESL, i.e., at the bottom of the metal 1 (M1) interconnect layer.

II. RF SWITCH LAYOUT AND BEOL ROUTING

As outlined previously, the baseline switch, referred to as [A] is based on a previous 130nm technology [START_REF] Gianesello | Highly linear and sub 120 fs Ron × Coff 130 nm RF SOI technology targeting 5G carrier aggregation RF switches and FEM SOC[END_REF]. The improved switch structure, referred to as [B], is an upgrade of structure [A] in many points [START_REF] Gianesello | Advanced 200-mm RF SOI Technology exhibiting 78 fs[END_REF]. Here we focus on one of the Daniel Gheysens namely the organization of its layout and routing strategy of source/drain terminals and body contact. The respective layouts are available in Fig. 1. The structure [A] is composed of 25 MOS transistors in parallel, each with a gate width of 10 µm, i.e., a cumulative width of 250 µm. Several variations of this structure exist in terms of polysilicon gate pitch (0.86 µm 0.92µm) and gate length (0.14, 0.16, 0.18 and 0.22 µm). In the following of the study, we will restrict considerations on structure [A] with a grid length of 0.14 µm and a poly pitch of 0.92µm. The structure [B] is different in its layout and routing of the M1 layer. First, it consists of 50 transistors in parallel with a gate width of 5µm giving a cumulative width identical to structure [A]. The structure [B] also features variations on the polysilicon gate pitch and length. To obtain comparable and consistent results, a geometrical configuration similar to structure [A] is selected. Another main difference between the structures is the access to the active zone. On the one hand, structure [A], which has 4 metal layers, uses M1 and M2 levels to contact the source/drain terminals and for the body bias, resulting in a rather congested central region. Structure [B] has only 3 metal levels and uses only M1 over the active zone to reduce as much as possible parasitic capacitive coupling. This layout and routing simplification is achieved by using the active silicon layer to distribute the body bias along the chain of 50 transistors in parallel.

III. SIMULATION STRATEGY AND RESULTS

Since the efficiency of air microcavities is very strongly related to the BEOL density of metal tracks in the central zone of the switch, structure [B] is expected to limit the potential COFF reduction as we start from an already optimized structure. To verify this assertion, a comprehensive simulation strategy was set up to appreciate with accuracy the impact of air microcavities. RF modeling is partitioned into two parts. The first consists in a 2D simulation of the transistors at the FEOL level using the IMPACT software to solve the small-signal transport equations based on the drift-diffusion (DD) approach. The second simulation task manages the complex interconnects in three dimensions (3D) due to the complexity of BEOL geometry. This is realized by solving Maxwell's equations with the Momentum module of the ADS software. In both simulation methods, COFF is extracted from the identification of the calculated Y-parameters to a lumped equivalent circuit [START_REF] Bhaskar | Substrate-Induced Dissipative and Non-Linear Effects in RF Switches: Probing Ultimate Performance Based on Laser-Machined Membrane Suspension[END_REF]. The effect of air microcavities is essentially captured by simulations of the BEOL part of the switch. The DD simulation of the transistor is therefore used to calibrate results against measurements as the overall COFF derives from the sum of the BEOL and FEOL contributions. Under these conditions, the modelling approach is therefore able to determine the relative contributions of FEOL and BEOL to COFF before the introduction of air microcavities. For structure [A], we obtain COFF-FEOL= 190.1 fF/mm (63%) and COFF-BEOL= 111.7 fF/mm (37%) for a total figure of 301.8 fF/mm. On the other hand, structure [B] features COFF-FEOL= 134.2 fF/mm (53%) and a COFF-BEOL= 116.5 fF/mm (47%) which yields a total COFF of 250.7 fF/mm. It is worth noting that the relative weight of the BEOL in the total capacitance is more important in structure [B] given the optimizations made to the intrinsic transistor. The relative contribution of air microcavities is therefore expected to be more consequent when compared to structure [A]. The electromagnetic simulation of the interconnect network (BEOL) is built from the circuit layout and the description of the metal/dielectric multilayer stack, each layer being described by its material and the relevant electrical parameters. The simulation methodology consists in replacing the inter-metal dielectric by air above a uniform etching front penetrating the interconnection network stack. The attenuation of the parasitic capacitive coupling induced by the BEOL can thus be appreciated as a function of the etching depth. The results of this simulation study are available in Fig. 2, from which two main conclusions can be drawn. First, it is clearly established that the attenuation of COFF is extremely sensitive as the etching front falls below the M2 level and reaches its maximum when the dielectric removal is complete down to the bottom of M1. This defines the optimum target area to reach by etching. The second point relates to the COFF reduction margin obtained for each structure. COFF of structure [A] is reduced from 301.8 fF/mm to 227.4 fF/mm representing a reduction of 24.6%. For the structure [B], a reduction of 30.5% is obtained from 250.7 fF/mm to 174.1fF/mm.

IV. AIR MICROCAVITY PROCESS

A post-process etching sequence has been set up for implementing the concept of air microcavities theoretically studied so far. The process flow is schematically illustrated in Fig. 4 and comprises 3 localized etching steps using a focused ion beam (FIB) and a final finishing step based on reactive ion etching (RIE). Among the FIB steps, the first two create rectangular cavities, 50 × 30 µm 2 and 30 × 10 µm 2 , respectively, as shown in Fig. 5(a). These two cavities enable to go down 9 µm and reach the top of the M2 metal level. The separation in two distinct steps is used to recalibrate the etching depth at each transition between two metal levels, here M4 and M3, successively. The third FIB step creates narrow trenches of 0.6 × 5 µm 2 between the fingers of the M2 level as shown in Fig. 5(b) and penetrate to a depth of 1.5 µm down the top of M1 level. This step is the most critical in the process as any misalignment of the trenches can lead to a degradation of the metal interconnections. Finally, the last step, carried out by RIE etching, removes as much oxide as possible from the bottom of the trenches. The RIE chemistry used is composed of CF4 and Ar in order to achieve etch selectivity with respect to the aluminum layers, the nitride CESL layer and the polysilicon gate fingers in case of CESL over etching. The result after RIE etching is given in Fig. 5 (c) and (d). It is worth noting that the obtained result is still imperfect but is sufficient to evaluate the impact of air microcavities. The COFF small signal characterization of the etched structure gives 279 fF/mm, i.e., an improvement of 22 fF/mm (7.3%) as shown by marker [A*] in Fig. 3. This figure does not match the simulated best result that reaches 227.4 fF/mm. To explain this discrepancy, the remaining volume of oxide due to incomplete etching at the M1 level must be carefully taken into account. It is possible to calculate an equivalent permittivity for each dielectric layer constituting the BEOL stack by considering the volume of oxide removed during the etching. The calculation is governed by the following formula:

. 1 .

After applying these modifications in the BEOL stack description, Momentum simulation gives a revised COFF of 274 fF/mm, in excellent agreement with the experimental result. This model can be used regardless of the shape of remaining oxide obtained during the process by introducing effective permittivities.

V. PROCESS OUTLOOK AND CONCLUSION

In order to completely remove the intermetallic oxide from the BEOL, a final vapor phase etching step is desirable to promote the isotropic penetration of the etchant. A second prerequisite is to ensure etch selectivity with respect to the metal interconnects. Hydrofluoric acid (HF) in vapor phase fulfills both of these requirements and has therefore been implemented in a simplified etching scheme given in Fig. 6 [11] [START_REF] Valle | Experimental Analysis of Vapor HF Etch Rate and Its Wafer Level Uniformity on a CMOS-MEMS Process[END_REF]. Fig. 6. Air microcavity etching process flow focused on HF etching Fig. 7 proposes pictures taken after vapor phase HF etching oxide into the cavity for 9 min. The implementation of this process highlights the efficiency deep into the cavity and the aggressiveness on top surface of the chip. This process matches expectations by reaching the polysilicon gates and by etching the oxide between the vias and contacts. Compared to the process using RIE etching, the efficiency of this process is closer to the complete oxide removal scenario on which simulations are based. No characterization could be performed on these chips due to the damage of vapor HF on the contact probing pads located at the top surface of the chip. To cope with this problem, the process has been revisited by adding a protective layer so that the surface of the chip is not exposed to vapor HF. A 60 nm thick Al2O3 layer meets all requirements and was implemented using atomic layer deposition (ALD) after the second FIB etching step. This additional protective layer requires a new FIB etching step to expose the cavity bottom to vapor HF. The final process given in Fig. 8 has been implemented as illustrated in Fig. 9 showing that the Al2O3 protective coating preserves the chip integrity. The degree of completion of the latest process scheme opens the way to a complete demonstration of the concept of air microcavities in the BEOL. The results obtained have allowed to appreciate how much the efficiency of this concept depends on a total elimination of the dielectric down to the base of the first metal level. Future work will focus on confirming the principle of air microcavities through a series of characterization on COFF, RON and the RF breakdown voltage. Particular attention will also be given to the impact of microcavities on thermal dissipation. Beyond its implementation in a post-process approach, this proof of concept must also be extended for an industrial deployment, which requires replacing the FIB steps, which is achievable through a modification of the BEOL process.

  structures [A] and [B],

Fig. 1 .

 1 Fig. 1. (a) Partial layout of the RF switch test structure [A] (b) Partial layout of the RF switch test structure [B]

Fig. 2 .

 2 Fig. 2. (a) Simulated COFF as a function of the air microcavity etching depth for (a) structure [A] and for (b) structure [B]These results show that air microcavities lead to a larger relative reduction of COFF on structure [B] despite an absolute improvement of about 73 fF/mm for the two styles of switch structures. In both cases, the simulations confirm the interest and efficiency of air microcavities for COFF reduction, even in the case of structure [B] whose layout had been optimized in

Fig. 3 .

 3 Fig. 3. RON x COFF figure of merit with the simulated impact of the cavities

Fig. 4 .

 4 Fig. 4. Air microcavity etching process flow focused on RIE etching

FIB

Fig. 7 .

 7 Fig. 7. SEM picture taken after vapor phase HF etching: (a) Tilted view of the complete structure. (b) Tilted view focused at the bottom of the structure

Fig. 8 .

 8 Fig. 8. Air microcavity etching process flow focused on HF etching with Al2O3 protection layer

  1,2 , Alain Fleury 2 , Stéphane Monfray 2 , Frédéric Gianesello 2 , Philippe Cathelin 2 , Jean-François Robillard 1 , David Troadec 1 and Emmanuel Dubois 1 1) Univ. Lille, CNRS, Centrale Lille, Univ. Polytechnique Hauts de France, JUNIA, UMR 8520 -IEMN, F-59000 Lille, France 2) STMicroelectronics, 850 rue Jean Monnet 38920 Crolles France Improving off-state capacitance of SOI-CMOS RF switches: how good are air microcavities?

ACKNOWLEDGEMENT

This work was supported by: i) the STMicroelectronics-IEMN joint laboratory ii) the French government through the National Research Agency (ANR) under program PIA EQUIPEX LEAF ANR-11-EQPX-0025 and iii) the French RENATECH network on micro and nanotechnologies.


Didier Bessis
email: didierbessis@gmail.com

Sophie Trouillet-Assant

Léo-Paul Secco

Nathalie Bardin

Brigitte Blanc

Véronique Blatière

Christine Chable-Bessia

Christophe Delfour

Céline Girard

Jean-Christophe Richard

V Eronique Blati Ere

C Eline Girard

Nathalie Gros

Vincent Le Moing

Nicolas Molinari

Val Erie Pallure

Amandine Pisoni

Nadia Raison-Peyron

Elisa Reynaud

Emilie Schwob

R Emi Pescarmona

Quentin Samaran

Marjolaine Willems

Thierry Vincent

Mircea T Sofonea

Alexandre Belot

E ´douard Tuaillon

COVID-19 pandemic-associated chilblains: more links for SARS-CoV-2 and less evidence

 [START_REF] Garc Ia | Are the chilblain-like lesions observed during the COVID-19 pandemic due to severe acute respiratory syndrome coronavirus 2? Systematic review and meta-analysis[END_REF]Likewise, the suspected pivotal pathogenic role of upregulation of interferon type I (IFN-I) is mainly indirectly supported by assessment of in situ immune response. [START_REF] Arkin | From your nose to your toes: a review of severe acute respiratory syndrome coronavirus 2 pandemic-associated pernio[END_REF] From April 2020 to January 2022, we prospectively assessed children and adults with new-onset CAC seen in the dermatology, infectious diseases, and adult and paediatric emergency departments, as well as the intensive care unit of the University Hospital of Montpellier. The study was approved by the local institutional review board (ID: 202000442).

Overall, 50 consecutive outpatients were seen, mostly white European and with a female predominance (sex ratio 1Á5), a median age of 21 years (range 6-74) and a median body mass index of 19Á8 kg m À2 (range 14Á8-26Á6). CAC involved the feet and the hands, respectively, in 86% and 66% of patients. Raynaud phenomenon, BASCULE syndrome (Bier anaemic spots, cyanosis, urticaria-like eruption) and subungual splinter haemorrhages (Figure 1a) were present in 18%, 13% and 8%, respectively. Complete recovery, persistence and recurrence of CAC were noted, respectively, in 93% (median duration 12 weeks, range 1-30), 7% (12 months of followup) and 42Á9%.

To assess the causal role of SARS-CoV-2 in CAC, we analysed SARS-CoV-2 real-time PCR in nasopharyngeal swabs and SARS-CoV-2 IgG, IgM and/or IgA serologies that were positive, in 10% and 23% of cases, respectively. Neutralizing antibody response using microneutralization assay was present in one case of isolated IgA antibodies. Serum SARS-CoV-2 nucleocapsid antigen detection was positive in 13%. Given all the virological tests, formal proof of SARS-CoV-2 infection could be established in 22% of patients, including 32% of children and 18% of adults. Using multivariate analysis, the comparison of the pertinent clinical, histologic and biologic characteristics of the patients with and without virologically proven SARS-CoV-2 infection was not significantly discriminating.

In total, 30 patients with CAC were assessed for wholeblood interferon-stimulated gene (ISG) expression, showing an increase of ISG compared with controls. To determine whether this ISG upregulation was different between CAC and other COVID-19 phenotypes, we analysed available values of the ISG scores within the first 50 days after onset of infectious symptoms in the cohorts of patients with CAC (n = 19), patients with COVID in the intensive care unit (n = 72) and nonhospitalized patients with COVID-19 and without CAC (n = 8). The latter group had COVID-19 confirmed by realtime PCR nasopharyngeal swab, and had a benign condition not requiring oxygenation, antibiotics, steroids or antiviral agents. The results are represented in Figure 1(b,c). In the CAC cohort, the first ISG measures at day 10 were similar to those of acute infection, and the overall kinetics revealed an identical shape compared with mild or severe COVID-19, turning to normal values around day 20. Two of 47 patients had circulating anti-IFN-a2 antibodies above the cutoff (> 34 ng mL À1 ).

In our series, the characteristics of CAC, including an overall balanced sex ratio, predominance of patients with low BMI, and clinical presentation and outcome, were broadly similar to those previously described. [START_REF] Garc Ia | Are the chilblain-like lesions observed during the COVID-19 pandemic due to severe acute respiratory syndrome coronavirus 2? Systematic review and meta-analysis[END_REF][START_REF] Gehlhausen | Lack of association between pandemic chilblains and SARS-CoV-2 infection[END_REF] We found significant high frequencies of Raynaud phenomenon, BASCULE syndrome and subungual splinter haemorrhages, possibly explained by the prospective study design and an absence of clinical details in some of the large series.

The role of SARS-CoV-2 infection as a trigger of CAC has been widely debated due to the low levels of positive realtime PCR nasopharyngeal and IgG serologies, which are within the ranges, respectively, of 5Á5-13Á3% and 7Á2-11Á8%. [START_REF] Garc Ia | Are the chilblain-like lesions observed during the COVID-19 pandemic due to severe acute respiratory syndrome coronavirus 2? Systematic review and meta-analysis[END_REF][START_REF] Rocha | COVID toes': a meta-analysis of case and observational studies on clinical, histopathological, and laboratory findings[END_REF] In our series, by using a microneutralization assay for isolated IgA antibodies and serum SARS-CoV-2 nucleocapsid antigen detection, we found an overall virological confirmation of SARS-CoV-2 infection in 22% of patients, including 32% in children. This latter result is consistent with the frequency of 30-37% of seroconversion observed in previous series of children with asymptomatic and/or mild SARS-CoV-2 infection during the acute and convalescent periods, [START_REF] Toh | Comparison of seroconversion in children and adults with mild COVID-19[END_REF][START_REF] Tagarro | Dynamics of reverse transcription-polymerase chain reaction and serologic test results in children with SARS-CoV-2 infection[END_REF] adding new arguments for a causative role of SARS-CoV-2 in CAC.

We confirmed that IFN-I signalling was increased in whole blood of patients with CAC but the kinetics of ISG over time were like those of SARS-CoV-2 infection without CAC. These results underscore the dynamics of systemic immune response to SARS-CoV-2 infection. Yet, previous histopathological findings revealed that CAC comprise a highly lymphocytic inflammatory process with an IFN-I microenvironment, as evidenced by extensive epidermal and dermal expression of MxA (myxovirus resistance protein) and pJAK1 (phosphorylated Janus kinase 1) in the endothelial cells and surrounding tissue, and rich presence of plasmacytoid dendritic cells. [START_REF] Magro | The differing pathophysiologies that underlie COVID-19-associated perniosis and thrombotic retiform purpura: a case series[END_REF][START_REF] Aschoff | Type I interferon signature in chilblain-like lesions associated with the COVID-19 pandemic[END_REF] Thus, if a pivotal role of IFN-I should be envisaged in the pathogenesis of CAC, it may be rather through local tissue-specific susceptibility to a physiological antiviral IFN-I response than to systemic overexpression of ISGs after COVID-19.

Submitted on 3

 3 Oct 2023     COVID-19 pandemic-associated chilblains: more links for SARS-CoV-2 and less evidence for high interferon type I systemic response COVID-19 pandemic-associated chilblains: more links for SARS-CoV-2 and less evidence for high interferon type I systemic response DEAR EDITOR, Since the onset of the SARS-CoV-2 pandemic, the direct causative role of the virus in COVID-19-associated chilblains (CAC) has remained under question due to the low rate of positivity to SARS-CoV-2 nasopharyngeal polymerase chain reaction (PCR) and blood serology.

Figure 1

 1 Figure 1 (a) Examples of clinical manifestations of COVID-19-associated chilblains. Left to right: (i) typical chilblain pattern with circumscribed red macules and papules overlying the dorsal and distal phalanges of the toes; (ii) acral erythema pattern with well-delineated symmetrical erythema and slight oedema located on the distal part of the toes; (iii) close-up view of typical chilblains and Raynaud phenomenon of the distal phalanges of the toes; (iv) chilblains of the toes associated with BASCULE syndrome, characterized by Bier anaemic macules, cyanosis, and multiple irregular red-orange macules (white arrows); (v) dilated capillaries and subungual splinter haemorrhages on the distal part of the nail plate. (b) Whole-blood interferon (IFN) score and relative expression of the six interferon-stimulated genes (ISGs) used for the IFN signature in healthy volunteers (HV; n = 24), a cohort of patients with COVID 19-associated chilblains (CAC; n = 19), a cohort of critically ill patients with COVID-19 in the intensive care unit (ICU; n = 72) and a cohort of nonhospitalized patients with nonsevere COVID-19 (benign; n = 8). Relative quantification of ISGs was performed using the nCounter analysis system (NanoString Technologies, Seattle, WA, USA). (c) Plasma ISG score (IFN signature). The cutoff value for positive IFN signature was > 3Á2 (lower positive = 2Á3-3Á2, negative < 2Á3). From left to right: patients with COVID 19-associated chilblains (n = 19); nonhospitalized patients with nonsevere COVID-19 without CAC (n = 8); and critically ill patients with COVID-19 in the intensive care unit (n = 72). ISGs were analysed using restricted cubic splines and are displayed as continuous variables.

Data availability statement: The data that supports the findings of this study are available in the supplementary material of this article.

Conflicts of interest: the authors declare they have no conflicts of interest.

Supporting Information

Additional Supporting Information may be found in the online version of this article at the publisher's website:


Laurent Nottale
email: laurent.nottale@obspm.fr

Scale-relativistic correction to the muon g -2 and its hadronic contribution

The anomalous magnetic moment (AMM) of the muon a µ = (g -2)/2 is one of the most precisely measured quantities in physics. Its experimental value shows, in 2023, a 5.2 σ discrepancy δa µ = (249 ± 48) × 10 -11 with its theoretical value calculated in the standard model framework, using a data-driven (R ratio) dispersive method to calculate the Hadron Vacuum Polarization (HVP) contribution. Meanwhile, lattice QCD numerical calculations of this contribution (L) have also yielded a significant discrepancy with respect to the data-driven value (R), reaching ∼ 5 σ for the ratio of their best determinations (in reduced windows), a HVP µL /a HVP µR = 1.0257 ± 0.0052. We suggest here a common solution to these two problems.

In standard quantum mechanics, mass ratios and inverse Compton length ratios are identical. This is no longer the case in the special scale-relativity (SSR) framework, in which the Planck length-scale is identified with a lower limit scale, invariant under dilations and replacing the zero point. Consequently a generalized form of Compton relation holds in this theory.

Regarding the HVP contribution to the muon g -2, the lattice QCD calculation is performed in position space-time while the data-driven result is calculated in terms of energy-momentum. The lattice QCD result should therefore be corrected for SSR effects. We estimate this correction to amount to a factor ρ SSR = 1.022 ± 0.002, which is fully compatible with the observed excess. Once corrected, the lattice and R ratio HVP contributions agree within uncertainties.

As regards the muon g -2 theoretical calculation, it involves a mass-dependent contribution which comes from two-loop vacuum polarization insertions due to electron-positron pairs and depends on the electron to muon mass ratio x = m e /m µ . Using the renormalization group approach, we show that, in this relation, ln x logarithmic terms depend on mass, while linear x terms are expected to actually depend on inverse Compton lengths. By defining the constant C 0 = ln(m P /m 0 ) in terms of the Planck mass m P and of a reference mass m 0 , the resulting scale-relativistic correction writes δa µ = -α 2 (x ln 3 x)/(8 C 2 0 ), where α is the fine structure constant. For m 0 = m µ , the numerical values of this correction, δa µ = (230 ± 15) × 10 -11 , would fully account for the observed experiment-theory difference.

Introduction

The Dirac equation predicts a muon magnetic moment, M µ = g µ (e/2m µ )S, with gyromagnetic ratio g µ = 2 and spin operator S. Quantum loop effects lead to a small calculable deviation, parameterized by the anomalous magnetic moment a µ = (g µ -2)/2 [1].

That quantity can be accurately measured and precisely predicted within the framework of the Standard Model (SM). Hence, comparison of experiment and theory tests the SM at its quantum loop level.

While the theoretical and experimental values of the electron anomalous magnetic moment agree within uncertainties [2,3], a e (th) = 0.00115965218178(77), a e (exp) = 0.00115965218091 (26), (1) on the contrary, the muon anomalous magnetic moment exhibits since the E821 Brookhaven experiment [START_REF] Bennett | Muon g -2 Collaboration), Final report of the muon E821 anomalous magnetic moment measurement at BNL[END_REF] about 20 years ago, a difference of (268 ± 72) × 10 -11 [1,5,6], statistically significant at the 3.7 σ level:

a µ (th) = 116591823(36) × 10 -11 , a µ (exp) = 116592091(63) × 10 -11 .

(

) 2
This effect constitutes one of the main discrepancies between the SM theoretical predictions and experiments. It is all the more puzzling that it is not found in the high energy realm of today's particle physics frontier (≈ 10 TeV), but instead at the atomic-nuclear scale (≈ 100 MeV) which was up to now thought to be fully understood.

In 2021, the first results (run 1) of the Fermilab National Accelerator Laboratory (FNAL) Muon g-2 E989 experiment [7] have been found to be in excellent agreement with the previous BNL E821 measurement. They obtained a µ (FNAL) = 116592040(54)×10 -11 , leading to a combined BNL+FNAL average result:

a µ (exp) = 116592061(41) × 10 -11 , (3)
increasing to 4.2 standard deviations the tension between experiment [7] and theory [START_REF] Aoyama | Muon g-2 Theory Initiative "White Paper[END_REF] (White Paper from the Muon g -2 Theory Initiative [WP]), δa µ = (251 ± 59) × 10 -11 . This difference has been obtained by deriving the Hadron Vacuum Polarization (HVP) contribution, which dominates the uncertainty of the Standard Model (SM) prediction, from a data-driven approach using the experimental e + e -→ hadrons R-ratio through a dispersion relation. In 2023 the new result from the FNAL E989 runs 2&3 has yielded a two times more precise world average value

a µ (exp) = 116592059(22) × 10 -11 . (4)
The resulting difference with the theoretical SM prediction, δa R µ = (249 ± 48) × 10 -11 , now reaches the 5.2 σ level using the R ratio, seemingly pointing to possible new physics beyond the standard model.

However, in the course of these studies, a second statistically highly significant discrepancy has emerged. A numerical calculation of the HVP contribution to the muon g -2 from lattice QCD by Borsanyi et al [BMW collaboration] [START_REF] Borsanyi | Leading hadronic contribution to the muon magnetic moment from lattice QCD[END_REF] has reached for the first time a level of precision similar to the data-driven method. They have found a HVP-L µ = (7075 ± 55) × 10 -11 , which differed by 2.1 σ from the R-ratio WP recommended value a HVP-R µ = (6931 ± 40) × 10 -11 .

Although the cause of this difference between the data driven and lattice QCD SM prediction remained misunderstood, a recent new method has allowed to point out more precisely its origin. The window observable method [10] amounts to restrict the numerical integration to a subinterval which essentially removes the regions of strong statistical fluctuations, large finite-volume effects and large lattice artefacts [START_REF] Wittig | Progress on (g -2) µ from Lattice QCD[END_REF]. In addition, it is possible to evaluate a win µ also using the R-ratio, so that a more precise view of the origin of the discrepancy can be acquired by comparing the two determinations.

The window results have confirmed and reinforced the existence of the HVP discrepancy. Davies et al [START_REF] Davies | Windows on the hadronic vacuum polarization contribution to the muon anomalous magnetic moment[END_REF] have used one-sided windows of various widths contributing to increasing percentages of the whole result and have found (a w µL -a w µR )/a w µR = 0.016 ± 0.007, 0.020 ± 0.005 and 0.021 ± 0.0075 in windows covering respectively 14%, 43% and 70% of the whole difference, thus reaching an almost 4σ effect in the second window.

Other authors have used a two-sided window covering (0.4 -1.0) fm contributing for 33% of the result. An excellent self-consistency has been found between these various lattice calculations [START_REF] Bruno | Hadronic contributions to the muon g -2 from lattice QCD[END_REF][START_REF] Wittig | Progress on (g -2) µ from Lattice QCD[END_REF][START_REF] Blum | An update of Euclidean windows of the hadronic vacuum polarization[END_REF], therefore validating the initial BMW claim [START_REF] Borsanyi | Leading hadronic contribution to the muon magnetic moment from lattice QCD[END_REF]. Their average results for R [START_REF] Colangelo | [END_REF][START_REF] Borsanyi | Leading hadronic contribution to the muon magnetic moment from lattice QCD[END_REF][START_REF] Aubin | Light quark vacuum polarization at the physical point and contribution to the muon g -2[END_REF] and L [START_REF] Borsanyi | Leading hadronic contribution to the muon magnetic moment from lattice QCD[END_REF][START_REF] Blum | An update of Euclidean windows of the hadronic vacuum polarization[END_REF][START_REF] Alexandrou | Lattice calculation of the short and intermediate time-distance hadronic vacuum polarization contributions to the muon magnetic moment using twisted-mass fermions[END_REF][START_REF] Cè | Window observable for the hadronic vacuum polarization contribution to the muon g -2 from lattice QCD[END_REF] determinations also lead to a similar statistical significance, with a relative difference 0.0285 ± 0.0072. The effect reaches ∼ 5 σ using their best determinations yielding (a w µL -a w µR )/a w µR = 0.0257 ± 0.0052 [Aubin et al /CL/KNT19 vs. RBC/UKQCD23 in Fig. 11 of ref. [START_REF] Blum | An update of Euclidean windows of the hadronic vacuum polarization[END_REF]). Alexandrou et al (ETMC22, [START_REF] Alexandrou | Lattice calculation of the short and intermediate time-distance hadronic vacuum polarization contributions to the muon magnetic moment using twisted-mass fermions[END_REF]) even quote a 6.1σ effect and conclude that this striking low energy anomaly definitely needs to be understood.

On the experimental side, although some difficulties have also appeared (see e.g. [START_REF] Bruno | Hadronic contributions to the muon g -2 from lattice QCD[END_REF]) which may point to an underestimation of the experimental errors in the determination of the R ratio, some authors have studied the deviation which should be imposed to the data in order to cancel the observed differences in a µ values [START_REF] Keshavarzi | [END_REF]. They have found that shifts in the e + e -to hadrons cross section needed to bridge it are excluded above s > 0.7 GeV at the 95% C.L., while prospects for ∆a µ originating below that energy are deemed improbable given the required increases in the hadronic cross section. This conclusion would also apply to the L -R difference, which is of the same order of size.

The overall conclusion seems now to be that the (R) and (L) HVP contributions to the muon g -2 cannot be reconciled, neither from the data-driven R-ratio side, nor from the numerical calculation lattice side, so that one is led to conclude that a new genuine effect has appeared, linked to the muon g -2 discrepancy but different from it and now reaching the same highly significant statistical level.

Special Scale-Relativity

We suggest here that both the muon g -2 discrepancy and the new HVP problem can be explained in the special scale-relativity (SSR) framework. This is achieved by accounting for the correction to the relation between mass ratios and Compton length ratios which occurs in this theory, where one introduces a generalized Compton relation based on the true nature of the Planck length as being a limit-length invariant under scale transformations.

We have long ago argued [20,21,22,[START_REF] Nottale | Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity[END_REF] that the geometry of space-time has no reason to remain Minkowskian (i.e. classical) in the quantum domain, at scales smaller than the de Broglie scale, owing to the fundamentally quantum nature of all existing objects at these scales. We have shown that the quantum space-time [21], on the contrary, is expected to be scale-dependent, i.e. fractal in a general meaning [START_REF] Nottale | Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity[END_REF][START_REF] Nottale | Scale Relativity and Fractal Space-Time: a New Approach to Unifying Relativity and Quantum Mechanics[END_REF] in the quantum realm.

Moreover, the principle of relativity, which was up to now applied, in current theories of relativity, to origin, orientation and motion transformations of the coordinate system [START_REF] Einstein | [END_REF], has been extended to apply also to scale transformations of the measurement resolutions, which are added to the variables that characterize its (relative) state [21,22,[START_REF] Nottale | Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity[END_REF][START_REF] Nottale | Scale Relativity and Fractal Space-Time: a New Approach to Unifying Relativity and Quantum Mechanics[END_REF].

The scale-relativity theory is the general framework built from this first principle [START_REF] Nottale | Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity[END_REF][START_REF] Nottale | Scale Relativity and Fractal Space-Time: a New Approach to Unifying Relativity and Quantum Mechanics[END_REF] including the construction of new scale laws of log-Lorentzian form [22]; a geometric foundation of quantum mechanics [26] and of gauge fields [27] in terms of a nondifferentiable and therefore fractal space-time continuum whose geodesics define wave functions; and the suggestion of the possibility of a new macroscopic quantum-type mechanics (based on a constant different from ) relevant to chaotic systems beyond their horizon of predictability [START_REF] Nottale | Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity[END_REF]28,29,30,31,32], in particular to turbulence in velocityspace [33].

Here we are concerned with only the log-Lorentzian scale laws aspect of the theory, specifically, with the fact that inverse length-scales and mass-scales, which are identical in standard quantum mechanics (QM), become different in the SSR framework.

We have mathematically proved [22], [START_REF] Nottale | Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity[END_REF]Chapt. 6], [START_REF] Nottale | Scale Relativity and Fractal Space-Time: a New Approach to Unifying Relativity and Quantum Mechanics[END_REF]Chapt. 4.4] that the general solution to the special relativity problem (i.e., find the linear laws of transformations which come under the principle of relativity) is, as well for motion as for scales, the Lorentz transformation. This proof is based on only two axioms, internality of the composition law and reflection invariance, which are both expressions of the only principle of relativity.

We know since Poincaré and Einstein that the special relativity law of composition of two velocities u and v writes w = (u + v)/(1 + u v/c 2 ). In the same way, the general law of composition of length-scale ratios, r → r = ρ × r, writes in special scale-relativity theory [22] W = (U + V)/(1 + UV/C 2 ), where U = ln(r/λ), V = ln ρ and W = ln(r /λ) and C is a constant whose meaning will be specified in the following.

Another relevant result of special (motion)-relativity is that the Galilean relation of proportionality between velocity and momentum, p/mc = v/c, becomes

p mc = v/c 1 -(v/c) 2 . ( 5
)
In the same way as velocity characterizes the state of motion of the reference system, we consider in SSR that the length and time measurement resolutions characterize its state of scale [21,22,[START_REF] Nottale | Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity[END_REF][START_REF] Nottale | Scale Relativity and Fractal Space-Time: a New Approach to Unifying Relativity and Quantum Mechanics[END_REF]. The difference is that motion transformations constitute an additive group, while scale transformations are a multiplicative group. However, using the Gell-Mann-Levy method, one can show [START_REF] Nottale | Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity[END_REF][START_REF] Nottale | Scale Relativity and Fractal Space-Time: a New Approach to Unifying Relativity and Quantum Mechanics[END_REF] that the natural variables for describing length-scales and their transformations are the logarithm of a scale ratio, V = ln(r/λ), so that one recovers an additive group in terms of these variables.

In analogy with the various levels of theories of motion-relativity, one can first define a "Galilean scale-relativity" (GSR) framework, which just corresponds to the usual laws of dilation and contraction, expressed by the direct product r /λ = ρ×r/λ. The standard de Broglie law which relates momentum and length-scales in quantum mechanics, p = /λ, can be established from Noether's theorem in this framework. This is similar to the obtention, in motion-relativity theory, of the standard relation p = mv from uniformity of space. It may be also written as a direct equality ln(p/p 0 ) = ln(λ 0 /λ), with p 0 λ 0 = . The generalization of this relation in SSR involves a log-Lorentz factor [22,[START_REF] Nottale | Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity[END_REF][START_REF] Nottale | Scale Relativity and Fractal Space-Time: a New Approach to Unifying Relativity and Quantum Mechanics[END_REF]:

ln p p 0 = ln(λ 0 /λ) 1 -ln(λ 0 /λ) 2 /C 2 . ( 6
)
The usual GSR law is clearly recovered in the limit C → ∞. The meaning of this constant can be clarified by expressing it also in terms of the reference scale λ 0 :

C 0 = ln λ 0 λ P . (7)
This introduces a minimal scale λ P which is invariant under dilations and contractions, unreachable and unpassable, whatever the scale λ 0 which has been taken as reference [22,[START_REF] Nottale | Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity[END_REF]. Moreover, momentum-energy now tends to infinity when the length-time scale tends to this limit, which therefore plays the role of the zero scale interval of the standard theory. This remarkable property has naturally led us [22] to identify it with the Planck length-scale,

λ P = G c 3 . (8)
Let us now analyse the current situation of the muon g -2 in the light of the SSR framework. The fact that there is no strong effect on the electron but only on the muon points toward a manifestation of the mass-scale increase by a factor ≈ 200 between the electron and the muon, which is the only difference between these two particles (apart from their consequently different lifetimes).

In order to apply the SSR framework to the muon g -2 problem, we need to specify the nature of two fundamental scales. The first is the scale at which occurs the transition between Galilean scale relativity (GSR) and Lorentzian special scale relativity (SSR). A natural identification of this transition is with the Compton scale of the electron λ e [22,[START_REF] Nottale | Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity[END_REF][START_REF] Nottale | Scale Relativity and Fractal Space-Time: a New Approach to Unifying Relativity and Quantum Mechanics[END_REF]. Indeed, physics changes drastically at scales smaller than λ e , in a way that is directly related to our purpose. (1) As remarked by Landau [START_REF] Landau | Relativistic Quantum Theory[END_REF], the very nature of positions and lengths is fundamentally changed at these scales, because of the inevitable production of electron-positron pairs in the process of measuring the coordinates of an electron. This formation of new particles in a way which cannot be detected by the process itself renders meaningless the measurement of the electron coordinates. (2) The various physical quantities, in particular masses and charges, become explicitly dependent on scale below λ e (running), a behaviour that is currently accounted for in terms of vacuum polarization and radiative corrections and well described by the renormalization group equations.

The second fundamental scale that should be known is the reference scale defining the constant C 0 . It is naturally given by the Compton length of the particle studied [22]. When the particle considered is the electron, both scales coincide. But here we are concerned with the muon, so that the SSR constant is given by the Compton length of the muon, i.e. C µ = ln(λ µ /λ P ) ≈ ln(m P /m µ ) = 46.196. This is a new configuration, in which the electron Compton length is larger than the reference length. This is justified by the fact that the e + e -pairs considered here are virtual, i.e. they are interpreted as being part of the set of fractals geodesics which constitutes the muon as a whole.

A possible intervention of SSR corrections can therefore be expected, since the ratio of the muon and electron Compton lengths is slightly different from their inverse mass ratio in this framework. Taking m = m 0 as reference scale, the SSR generalized Compton relation [22,[START_REF] Nottale | Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity[END_REF][START_REF] Nottale | Scale Relativity and Fractal Space-Time: a New Approach to Unifying Relativity and Quantum Mechanics[END_REF] writes:

ln λ µ λ e = ln(m e /m µ ) 1 + ln 2 (m e /m µ )/C 2 0 . (9)
Therefore, the length-scale ratio becomes y = λ µ /λ e = x 1/ √ 1+(ln x/C 0 ) 2 ≈ 1/200 when C 0 = C µ instead of the mass ratio x = m e /m µ ≈ 1/207. We recover as expected the standard Compton relation λ µ = /m µ c in the limit C 0 → ∞, i.e. λ P ≈ 0.

SSR correction to the HVP contribution to muon g -2

Let us first apply the SSR theory to the discepancy of the muon g -2 HVP contribution between the data-driven (R) and lattice QCD (L) approaches. Its applicability to this problem is straightforward owing to the fact that the lattice QCD calculation is performed in space-time while the R-ratio measurements and the contribution to the muon g -2 derived from it are performed in terms of energy-momentum. In the SSR view, the two calculations are not identical and there must therefore exist a transformation between them, so that the lattice result should be corrected.

As regard the light-by-light contribution, the data-driven and lattice QCD results are in excellent agreement [START_REF] Borsanyi | Leading hadronic contribution to the muon magnetic moment from lattice QCD[END_REF]: this comes in support of our interpretation, since there is no mass scale directly involved in that case, so that we do not expect any SSR effect.

Actually, the lattice QCD numerical calculation is made in position space while there is no possible direct measurement of length and time intervals at the muon scale and below. The smallest scale where direct length measurements have been performed is the Bohr atomic scale, while, from the Landau argument recalled above, they become impossible below the Compton electron scale. The lattice calculation therefore relies on the assumption of the universality of the standard Compton relation λ × m = , although it is a mere postulate of quantum mechanics which cannot be directly established at high energies. In the scale-relativity theory, on the contrary, one derives it (and more generally the Einstein-de Broglie relations) from first principles and space-time symmetries [26,[START_REF] Nottale | Scale Relativity and Fractal Space-Time: a New Approach to Unifying Relativity and Quantum Mechanics[END_REF] as recalled hereabove. The passage to Lorentzian-form laws is then a natural generalization in both scale and motion cases and can be directly deduced from the principle of relativity itself in its special version [22,[START_REF] Nottale | Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity[END_REF][START_REF] Nottale | Scale Relativity and Fractal Space-Time: a New Approach to Unifying Relativity and Quantum Mechanics[END_REF].

We have estimated the SSR correction to the lattice QCD results from the fact, emphasized by BMW, that a relative error of the lattice spacing a propagates into about twice as large a relative error on a µ [START_REF] Borsanyi | Leading hadronic contribution to the muon magnetic moment from lattice QCD[END_REF]. As long as the SSR correction remains small enough it can be treated as such an error, so that one expects δa µ = 2a µ δa/a. The left figure shows the original BMW result (Fig. 26 of [START_REF] Borsanyi | Leading hadronic contribution to the muon magnetic moment from lattice QCD[END_REF], where we have omitted the error bars on the individual points). The right figure shows the same points corrected for the SSR effect following Eq. 10. The blue and red curves are linear and quadratic fits of these points allowing to extrapolate the continuum limit, using the same method as BMW. The resulting corrected continuum limit is a µ ≈ 6270 (down dashed horizontal line), smaller than the BMW value 6403 (up dashed line) by a factor ≈ 1/1.022 and in good agreement with the corresponding data-driven R-ratio estimate.

We therefore predict an effect given, in terms of the variable a 2 , by:

δa µ a µ = 2     -1 + / √ a 2 m µ c 1-1/ 1+ln 2 / √ a 2 mµc /C 2 µ     . (10)
(Recall that, when the mass-energy is expressed in MeV and the length in fm, the conversion constant is c = 197.327 MeV fm). In order to estimate the SSR correction, we have considered the individual calculation points of BMW Fig. 26 [START_REF] Borsanyi | Leading hadronic contribution to the muon magnetic moment from lattice QCD[END_REF], which exemplifies the fit method of continuum extrapolation in the case of the light quark contribution a HVP,light µ . We have first verified that we recover the BMW result just using linear (a 2 ) and quadratic (a 2 , a 4 ) fits on the two series of points. We obtain a µ = 6400 ± 39 (omitting the 10 -11 factor here and in what follows) from a weighted average of these four fits, which yields a good estimation of the BMW result (itself obtained from 258048 fits [START_REF] Borsanyi | Leading hadronic contribution to the muon magnetic moment from lattice QCD[END_REF]), a µ = 6403 ± 44.

Then we have applied the SSR correction to each of the individual points and used the same fitting procedure on the corrected points. The result is shown in Fig. 1. We find a SSR µ = 6267 ± 44 (where we have kept the BMW error), i.e. a difference of ≈ 130 with the BMW result. This means that the SSR effect on the lattice QCD determination of the HVP contribution to the muon g -2 amounts to a factor ρ SSR = 1.0217.

Actually this value of the correction should be considered up to now as just an estimate, since it relies on an approximative relation between the deviations on a and on a µ . We intend to improve the situation in future works. The precise value of the correction depends also on the slopes of the fits and on the smallest lattice spacing and may therefore vary from a lattice calculation to another. For example, by applying the same method to the window ETMC22 results [START_REF] Alexandrou | Lattice calculation of the short and intermediate time-distance hadronic vacuum polarization contributions to the muon magnetic moment using twisted-mass fermions[END_REF], for which the smallest lattice spacing a 2 = 0.0035 is smaller than the BMW one (a 2 = 0.004 fm 2 ), we find a slightly larger effect ρ SSR = 2065/2017 = 1.024. Using RBC/UKQCD23 window data [START_REF] Blum | An update of Euclidean windows of the hadronic vacuum polarization[END_REF] we obtain ρ SSR = 2065/2025 = 1.020. This leads to an estimated correction of ρ SSR = 1.022 ± 0.002 for the presently available lattice QCD calculations.

This points to a potential problem: the SSR correction is expected to increase in a divergent way when a → 0. However, when looking at the relative contributions from the various energies in the R-ratio version of the HVP calculation, one finds that the contributions become negligible beyond the b quark, in agreement with the lattice QCD calculations taking account only of the u, d, s and c quarks. Then there is no reason to consider lattice spacings smaller that a 2 ≈ 0.0035, corresponding to energies larger than ≈ 3 GeV. Moreover our present estimation relies on the hypothesis of a small relative deviation on lattice spacing, and it is therefore expected to become irrelevant at high energies. It will be possible to clarify this point by integrating the SSR correction directly in the numerical lattice QCD calculation. µ to the muon g -2, from the lattice (L) and from data-driven dispersion relation (R), with windows of various widths. It is compared to its theoretical estimation in the SSR theory (red dashed horizontal line), ρ SSR -1 ≈ 0.022. The abscissa yields the fraction of the contribution for each window. The fractions 14%, 43% and 70% correspond to one-sided windows [START_REF] Davies | Windows on the hadronic vacuum polarization contribution to the muon anomalous magnetic moment[END_REF]. The 100% full result is the BMW estimate [START_REF] Borsanyi | Leading hadronic contribution to the muon magnetic moment from lattice QCD[END_REF]. The fraction 33% gives the best result for the two-sided window 0.4 -1 fm, using current most precise R value (quoted as Aubin et al 2019/CN/KNT in [START_REF] Blum | An update of Euclidean windows of the hadronic vacuum polarization[END_REF] and L value [START_REF] Blum | An update of Euclidean windows of the hadronic vacuum polarization[END_REF].

The observed factor from the BMW calculation compared to the WP recommended value is ρ obs = (7075 ± 55)/(6931 ± 40) = 1.021 ± 0.010 and is therefore in full agreement with the SSR theoretical prediction. The one-sided window values of Davies et al [START_REF] Davies | Windows on the hadronic vacuum polarization contribution to the muon anomalous magnetic moment[END_REF] are of the same order (see Fig. 2), while the two-sided window yields, using mean R and L values, ρ obs = (2362 ± 11)/(2295 ± 13) = 1.0285 ± 0.0072, within one σ of our theoretical prediction. The best R and L values (cf. [START_REF] Blum | An update of Euclidean windows of the hadronic vacuum polarization[END_REF] Fig. 11) yield a compatible and even more precise result reaching ∼ 5σ significance, ρ obs = 1.0257 ± 0.0052. All these estimates are in full agreement with the theoretical SSR prediction.

The one-sided window results are compatible with a constant ratio between the L and R values of the HVP effect, independant of the window size [START_REF] Davies | Windows on the hadronic vacuum polarization contribution to the muon anomalous magnetic moment[END_REF]. Such a constancy is also predicted in the SSR interpretation of the effect. This allows us to estimate the expected full HVP contribution from the best determined ratio. One finds a HVP µ = 1.0257 × 6931 × 10 -11 = 7109 × 10 -11 , slightly larger than the BMW result. This leads after SSR correction to a HVP,L µ SSR = (6924 ± 57) × 10 -11 , which agrees with the R result a HVP,R µ = (6931 ± 40) × 10 -11 [START_REF] Aoyama | Muon g-2 Theory Initiative "White Paper[END_REF] (see Fig. 4). We conclude that, once corrected for the SSR effects, the lattice QCD and data-driven estimates of the HVP contribution to the muon g -2 are expected to agree with each other within uncertainties.

SSR correction to muon g -2

Let us now remind how one can apply the SSR theory to the full muon g -2 itself [START_REF] Nottale | Scale-relativistic corrections to the muon anomalous magnetic moment[END_REF]. Since it implies corrections to the Compton relation between mass-scales and lengthscales, one is led to look for a possible SSR effect only among the mass-dependent terms of the muon g -2 calculation [5]. There is no such contribution to 1-loop level, i.e. in Schwinger's original α/2π correction to the magnetic moment [5]. To 2-loop, the main mass-dependent and electron-dependent term comes from the Feynman diagram shown in Fig. 3, which corresponds to the insertion of a closed lepton loop (e + e -pair). Let us first recall how the standard calculation of this contribution is performed without SSR correction. It has been calculated in 1949 by Karplus and Kroll (KK) [START_REF] Karplus | [END_REF]. It reads in position space (we denote by X the position coordinate so as not to confuse it with x = m e /m µ in what follows):

M IIe = - eα 2 4 c d 4 X 0 d 4 X 1 d 4 X 2 A e µ (X 0 ) D(2) F (X 2 -X 1 ) ψ(X 1 )γ ν S F (X 0 -X 1 )γ µ S F (X 2 -X 0 )γ ν ψ(X 2 ). (11)
The bar on M IIe and DF indicates that the renormalization terms have been removed. The Feynman function D F describes the properties of a virtual photon as modified by its interaction with the electron-positron field. The leading term is obviously

D (1) F (X) = - 2i (2π) 4 e -ipX d 4 p p 2 . (12)
The radiative corrections to this function arise from the ability of the virtual photon to create pairs. The first term D

F is simply due to the creation and annihilation of one pair, as can be seen in Fig. ??. After removal of the renormalization term and performing its Fourier transform, it reads D( 2)

F (p) = α 2π 1 0 dv 2v 2 (1 -v 2 /3) p 2 (1 -v 2 ) + 4κ 2 e , (13)
where κ e is the electron mass term. Feynman's S F (X) function describes the propagation of the particle for which the magnetic moment is computed (by looking at its elastic scattering by an electromagnetic field). This particle is an electron in the original KK calculation and a muon in the case considered here:

S F (X, κ µ ) = - 2i (2π) 4 d 4 p e -i p X i γ p -κ µ p 2 + κ 2 µ , (14)
where the mass term κ µ is now the muon one.

The subsequent calculations are continued by KK in momentum space. The momentum p 1 is used to denote the momentum of the final state and p 2 the momentum of the initial state:

ψ(X) = e ip 2 X ψ(p 2 )d 4 p 2 , (iγp 2 + κ µ )ψ(p 2 ) = 0, (15) ψ
(X) = e ip 1 X ψ(p 1 )d 4 p 1 , ψ(p 1 )(iγp 1 + κ µ ) = 0. (16)
Then KK find, for the contribution considered, the final expression:

M IIe = - 2ieα 2 c d 4 p 1 d 4 p 2 A e µ (p 1 -p 2 ) d 4 k 1 0 dv 2v 2 (1 -v 2 /3) k 2 (1 -v 2 ) + 4κ 2 e × ψ(p 1 )γ ν i γ (p 1 -k) -κ µ (p 1 -k) 2 + κ 2 µ γ µ i γ (p 2 -k) -κ µ (p 2 -k) 2 -κ 2 µ γ ν ψ(p 2 ), (17)
in which we have now clearly identified the muon mass term κ µ and the electronpositron pair mass term 2κ e .

The resulting muon vacuum polarization insertion was explicitly computed in the late 1950's [37,38]. It is given by the double integral:

a (4) µ = α π 2 1 0 du u 2 (1 -u) 1 0 dv v 2 (1 -v 2 /3) u 2 (1 -v 2 ) + 4x 2 (1 -u) , (18)
where x = κ e /κ µ = m e /m µ is the electron to muon mass ratio in the SM framework. An exact integration has been performed by Elend in 1966 [39] and the final expression has been written in compact form by Passera [40] as:

a (4) µ = α π 2 - 25 36 - 1 3 ln x + x 2 (4 + 3 ln x) + x 4 π 2 3 -2 ln x ln 1 x -x -Li 2 x 2 + x 2 1 -5x 2 π 2 2 -ln x ln 1 -x 1 + x -Li 2 (x) + Li 2 (-x) . (19)
A complete expansion of this expression has been given by Li et al. [41], for both cases x > 1 and x < 1. Keeping only the leading terms, it yields in terms of ln x and x:

a (4) µ = α π 2 - 25 36 - 1 3 ln x + π 2 4 x + O(x 2 ln x) . (20)
In the SSR framework, this formula can no longer be correct. Indeed, the muon Compton length-scale and mass scale, when they are referenced to the electron scale, are no longer strictly inverse quantities. One should therefore make the difference between x = m e /m µ and y = λ µ /λ e , whose relationship is given in Eq. ( 9).

An analysis of the way the above mass dependent contribution to the muon g -2 is obtained shows that it depends on both mass and Compton length. The original KK work starts from position space then ends the calculation in momentum space through the standard assumption of Fourier transform between position and momentum representations. This calculation relies on the previous work of Dyson [42], who explicitly specifies that the mass term κ in Feynman's S F (X) function (Eq. 14) is given by the reciprocal Compton length, not the mass itself. On the other hand, we also know that masses enter as such in the g -2 calculation, in particular through the threshold 2m e for pair creation.

However, the two contributions are not separated in the KK calculation of the electron g -2, nor in the muon calculation of Suura-Wichmann [37] and Peterman [38] derived from it. We therefore need to use another approach where the two contributions are separated. The renormalization group approach provides us with such a separation. It has been shown by Lautrup and de Raphael [43] that the equation for the contribution to the muon g -2 from electron vacuum polarization insertions takes the form of a Callan-Symanzik equation

m e ∂ ∂m e + β(α) ∂ ∂α a µ m e m µ , α = R m e m µ , (21)
where the right hand side of this equation is found to be vanishing as x = m e /m µ instead of the naive expectation x 2 from Eq. [START_REF] Cè | Window observable for the hadronic vacuum polarization contribution to the muon g -2 from lattice QCD[END_REF]. In this expression, no difference is yet made between the mass ratio and inverse Compton length ratio. However, the leading logarithmic term of Eq. ( 20), -1 3 ln x, is now provided by the left hand side of this equation [5], while the linear term π 2 4 x comes from its right hand side. In the l.h.s. of Eq. ( 21), β(α) is the standard QED β-function, β(α) = 2α 2 /3π to one-loop. The meaning of this equation is that the origin of the leading logarithm term - 1 3 ln x is just charge screening of the electromagnetic charge [5] and that it comes from the mere running QED coupling at muon scale,

α µ = α 1 + 2α 3π ln m µ m e . ( 22
)
This contribution is therefore generated by the electron-muon mass ratio x = m e /m µ . On the contrary, if one now considers the r.h.s. of Eq. ( 21), a detailed analysis by Lautrup and de Raphael [43] shows that it finds its origin in the S F (X) functions in which the mass terms are actually defined as inverse Compton lengths [42]. We are therefore led to differentiate between ln x logarithmic contributions which depend on mass, and x linear contributions which we assume to actually depend on inverse Compton lengths.

Therefore the Callan-Symanzik equation for the muon g -2 can now be written (keeping only the leading terms) as

x ∂ ∂x + β(α) ∂ ∂α a µ (x, α) = α 2 2 y, (23)
where y = λ µ /λ e while x = m e /m µ . The solution to order α 2 of this equation finally yields a correction to the muon g -2, δa SSR µ = (y -x)α 2 /4, which is well approximated by

δa SSR µ = -α 2 x ln 3 x 8 C 2 µ = 230 × 10 -11 . (24)
Assuming that higher order power contributions x k should also be corrected and replaced by y k , one finds that the 2-loop x 2 power terms of Eq. ( 19), coming from e + e -pairs, yield a small correction δa µ = 4 × 10 -11 , while x 2 ln x terms yield -20 × 10 -11 . Hadron loops contribute also by ∼ (m µ /m) 2 terms, yielding a SSR correction ≈ +17 × 10 -11 , so that these possible higher order effects cancel each other and are anyway smaller than current theoretical uncertainties (43 × 10 -11 ). Three-loop effects as well as tau lepton loops yield corrections smaller than 1 × 10 -11 . We therefore estimate the special scale-relativistic correction to δa µ = (230 ± 16) × 10 -11 , yielding a theoretically estimated (SM+SSR) muon g -2

a µ (th) = 116592040(43) × 10 -11 , (25)
in excellent agreement with the present experimental value a µ (exp) = 116592061(41) × 10 -11 [7]).

We finally find an SSR correction:

δa SSR µ = (230 ± 15) × 10 -11 , (26)
where the error bar is an estimation of the effects of the possible inclusion of higher order mass terms. This special scale-relativity correction agrees within uncertainties with the observed difference between the experimental and theoretical muon g -2, δa µ = (249 ± 48) × 10 -11 (using the data-driven HVP contribution). It is also now in agreement with the SM prediction using lattice QCD for the HVP contribution once the SSR correction of this contribution is also taken into account. The future decrease by a new factor 2 of the experimental measurement of a µ at FNAL when all runs will be taken into account is not expected to change this conclusion (see Fig. 4).

Conclusion

We have shown in the present paper that the two highly significant discrepancies affecting the muon anomalous magnetic moment (a 5.2σ difference between its theoretical and experimental values and another 5σ difference between the lattice QCD calculation of the HVP contribution to g -2 and its data-driven estimation derived from the R ratio) can be both explained in the framework of the special theory of scale-relativity (Fig. 4). It will be possible to test these proposals with improved precision in the next years, since the Muon g -2 Fermilab experiment E989 aims to finally reduce the experimental the SSR framework. The green band to the left stands for the SM prediction and its error bar in the case when the HVP contribution is derived from the R ratio experimental measurement. The yellow band to the right is today's (2023) state of experimental measurements from BNL and FNAL runs 1, 2 and 3. The difference between them reaches a 5.2 σ statistical significance (which may be slightly overevaluated due to possible under estimations of experimental uncertainties on the R ratio). The black arrow is the theoretically predicted SSR correction δa SSR µ = (230 ± 15) × 10 -11 . The open point is the SM-R prediction corrected for special scale-relativity effects. The green point is the SM-L prediction where the HVP contribution comes from windowed lattice QCD numerical integration, doubly corrected for SSR effects (correction to the lattice length-scale and correction to closed electron loop insertion).

uncertainty on the experimental value of the muon g -2 by a factor of four [START_REF] Grange | Muon g-2 Collaboration[END_REF]. If this goal is attained (experimental error ≈ 13×10 -11 ), the main uncertainty in the comparison between theory and experiment will come from the hadronic theoretical contribution, for which improvements are also expected [START_REF] Keshavarzi | [END_REF].

Provided the experimental result and SM theoretical prediction are confirmed in the future, one can reversely consider the two effects of muon g -2 (theory-experiment) difference and HVP contribution (L lattice QCD -R ratio) difference as tests of the special scale-relativity theory. These tests are both positively satisfied at the 5 σ level in terms of the most precise data and numerical calculations. From the observed effect, one can derive the value of the constant C 0 , C 0 = -α 2 x ln x 3 8 δa µ = 44.3 +5.0 -3.7 .

The value of C µ lies within 0.4 σ of this theoretical prediction and C e at 1.4 σ. Expressed in terms of scale ratio, one finds ρ P0 = e C 0 = 2 × 10 19±2 , which clearly points toward the Planck mass-scale m P = 1.22 × 10 19 GeV. This brings direct support to our deduction according to which the new length-scale, naturally introduced in SSR as being invariant under scale transformations of the reference system, must be identified with the Planck length-scale [22]. Such a length-scale, which is unreachable, uncrossable toward smaller scales, invariant under dilatations and contractions (including the Lorentz length contraction and time dilatation), plays the same role for resolution-scales as played by the velocity of light for velocities, since c is also unreachable by massive particles, unsurpassable and invariant under motion transformations of the reference system. Moreover, the theory could be newly put to the test in the future by performing in position space SM numerical calculations of other quantities of particle physics expected to be perturbed by SSR effects and comparing them to their data-driven counterpart derived in momentum space.

Figure 1 :

 1 Figure 1: SSR correction to lattice QCD calculation of the HVP contribution of light quarks to muon g -2.The left figure shows the original BMW result (Fig.26of[START_REF] Borsanyi | Leading hadronic contribution to the muon magnetic moment from lattice QCD[END_REF], where we have omitted the error bars on the individual points). The right figure shows the same points corrected for the SSR effect following Eq. 10. The blue and red curves are linear and quadratic fits of these points allowing to extrapolate the continuum limit, using the same method as BMW. The resulting corrected continuum limit is a µ ≈ 6270 (down dashed horizontal line), smaller than the BMW value 6403 (up dashed line) by a factor ≈ 1/1.022 and in good agreement with the corresponding data-driven R-ratio estimate.

Figure 2 :

 2 Figure 2: Fractional differences between determination of the HVP contribution a w

Figure 3 :

 3 Figure 3: Feynman diagram for the leading mass-dependent term, involving e + e -pairs, of the fourth-order radiative correction to the scattering of a muon by an electromagnetic field, which plays a central role in the SSR correction to the theoretical calculation of the muon g -2.

Figure 4 :

 4 Figure 4: Illustration of the present state of the muon g -2 discrepancies and of their solution in