Katiane De
email: mr.katiane@gmail.com

Moraes Rocha

Luc Trouche
email: luc.trouche@ens-lyon.fr

Documentational trajectory: a tool for analyzing the genesis of a teacher's resource system across her collective work

Keywords: Documentational approach to didactics, documentational trajectory, metaresource, reflective investigation, professional development

Our contribution to TWG 22 is dedicated to discussing teachers' interactions with resources for planning their classroom instruction, particularly in the context of collective work. Each teacher along her professional life uses and creates a lot of resources. For analyzing the history of teacher work with/for resources, we propose the notion of documentational trajectory. This idea is based on and aims to contribute the the development of the documentational approach to didactics. We present the data related to a teacher's work, which allows us building her documentational trajectory. We use it to analyze teacher's professional development. It evidences a strong particpation in collective work, in particular a collective named SESAMES, has an essential role. It contributes to the emergence of a particular resource, which is called as metaresource, structuring her documentation work.

Introduction

The new possibilities arising from communication and information technologies have had a significant impact on discussions on mathematics education, due to their impact on resources available to teachers and the way of designing them. It led to new conceptualization of teaching resources [START_REF] Adler | Conceptualising resources as a theme for teacher education[END_REF] and evidenced new teachers' relationships with these resources (Remilliard, 2005). In this thread, [START_REF] Gueudet | Teaching resources and teachers' professional development: towards a documentational approach of didactics[END_REF] analysed teacher professional development through the lens of resources: they (ibid, 2009) introduced the documentational approach to didactics aiming to analyze how teachers select, use, and produce their resources, along with a process named documentation work. Later, [START_REF] Gulay | Expert and novice teachers' classroom practices in a technological environment[END_REF] showed how digital resources structure classroom practices, evidencing five main features: working environment, resource system, activity format, curriculum script, and time economy. In this article, we rely on and articulate these approaches for analyzing teacher's history with resources. We propose a new concept for modeling this history: the teacher's documentational trajectory. We mainly discuss here which could be the effects of a teacher's collective work on her documentational trajectory? For doing this, we organize our contributions in four sections. In the first one, we introduce the concepts structuring our analysis. In the following one, we present our methodological choices. In the third one, we develop a case study based on a middle school teacher, Anna. And in the last section, we propose some considerations and perspectives.

Theoretical framework and research issues

We present and articulate in this section the theoretical approaches grounding our analysis: the documentational approach to didactics, the structuring features of teachers' practices, and the notion of thought collective. Then we present our own propositions.

As stated by Gueudet and Trouche (2012, p. 24) "teachers interact with resources, select them and work on them (adapting, revising, reorganizing, etc.) within processes where design and enacting are intertwined". This process is the central focus of the Documentational Approach to Didactics [START_REF] Gueudet | Teachers' Work with Resources: Documentational Geneses and Professional Geneses[END_REF] grounding our work. In this approach, the term resource is used in a broad sense, as everything that can nourish teacher's work. To prepare their teaching, teachers work on resources, and the result of this process is called a document, made of resources that have been modified and re-organized, and knowledge, both guiding, and produced by, teacher's work, grounding her professional development. The new resources generated take place in a very structured set of teacher's resources, called teacher's resource system. For developing this system, some resources play a critical role: metaresources, introduced by Prieur (2016) for designating resources supporting and guiding the creation of other ones and favoring a reflection on teacher's own documentation work.

Teacher's documentation work grounds teacher's classroom practices, structured, according to Gulay and Ruthven (2012) by five features: the working environment, where lessons class take place (infrastructure, social organization, etc); the activity format, which comprises the body of work in the classroom, such as routines and models of interactions between teacher and student along teaching and learning; the resource system, which gathers tools and materials for class. The curriculum script is to be understood in the cognitive sense of structured organization of activity guiding teacher's work in the classroom: goals, actions, activities, potential difficulties of students, among others. And time economy, based on the comparison between the teaching time and the learning time of students. Gulay and Ruthven use this framework for analyzing teacher's integration of new digital resources into classroom practices. We will extend this usage for analyzing teacher's design and usages of resources. Thus, we will retain the notion of resource system as proposed by [START_REF] Gueudet | Teachers' Work with Resources: Documentational Geneses and Professional Geneses[END_REF]:

"we consider here as resource system does not fully coincide with Ruthven's definition, because of the broader meaning of resources we retained. The resource system comprises material elements, but also other elements that are more difficult to collect, like conversations between teachers" (p. 27).

We consider that teachers' interactions with colleagues are likely to foster their documentation work and professional development [START_REF] Gueudet | Teachers' Work with Resources: Documentational Geneses and Professional Geneses[END_REF]. This is the reason why we give a primary importance to collectives. A collective is, for us, a place designed by a teacher where interactions develop with others actors of her teaching. We retain the broad definition of thought collective (Fleck, 1934, p. 44) existing when "two or more people are exchanging thoughts" and generating a thought style "characterized by standard features in the problems of interest to a thought collective, by the judgment which the thought collective considers evident, and by the methods which it applies as a means of cognition" (p. 99). We distinguish different types of collectives, according to their duration (stable vs. unstable), their organization (formal vs. informal) and type of participation (voluntary vs. required).

Finally, we present our proposition for modeling teacher' history with resources based on the notions of resource, collective, and event. An event is something happening in the professional life of a teacher, and remembered by her as important regarding her documentation work. We define the teacher's documentational trajectory [START_REF] Rocha | Uses of online resources and documentational trajectories: the cases of Sésamath[END_REF] as the interplay, over the time, between events and resources, this interplay being socially situated, because it happens in schools or collectives, or because the events or the resources themselves are social products. The design of a teacher's documentational trajectory is then a way to analyze when, where, why, how and which resources are created. We will focus in this article on the documentational trajectory as a tool for analyzing the genesis of a teacher' resource system across her collective work. We describe, in the following section, our methodological choices for such a design.

Methodological design

Our methodology is inspired by the four principles of reflective investigation presented by [START_REF] Gueudet | Teachers' Work with Resources: Documentational Geneses and Professional Geneses[END_REF]: "long-term follow-up", "in-and out-of-class follow-up", "reflective follow-up" and "broad collection of the material resources". This methodology also gives a major importance to specific drawings made by the teacher (for example the 'schematic representation of her resource system'). Retaining this way of reflective investigation, by the teacher, of her documentation work, we propose some changes: instead of the word "representation", we propose [START_REF] Rocha | Uses of online resources and documentational trajectories: the cases of Sésamath[END_REF] the word "mapping", integrating the metaphor of a progressive exploration of a new territory. And we propose two kinds or mapping: reflective (made by the teacher herself) vs. inferred (made by the researcher) mapping. We distinguish then Reflective, vs. Inferred, Mapping of teacher Resource System; Reflective, vs. Inferred, Mapping of teacher Documentational Trajectory. For the design and analysis of a documentational trajectory, we also use: interviews, follow-up of lesson preparation, and classroom observation, and a logbook filled by teachers.

Our current research is mainly based on cases study. For choosing teachers, we search teachers that had Sésamath textbooks as official textbooks in their class. Sésamath (http://www.sesamath.net/) is an association of mathematics teachers in France that collaboratively designs online resources (software, textbooks, etc.) at a very large scale, opening for us a window on advanced teachers from resources design and use. We present here the case of Anna, a middle school mathematics teacher, whose school had chosen a textbook designed by Sésamath. She has a strong partnership with a colleague from her school, Cindy, and both of them participate in various collectives and use a lot of digital resources beyond the Sésamath textbook.

Our work with Anna started in mars 2015, when we followed her 6 th grade class for three months. During this period, Anna created and shared with us a Dropbox folder, where she uploads resources that she used in or to prepare her lessons. In addition, we also had videorecorded four moments of interaction involving her. In the first one, she made a reflexive mapping of her documentational trajectory. In the second one, she reviewed her reflexive mapping (focusing on a particular resource and a specific year). In the third one, she prepared a lesson with Cindy about a new curricular subject. In the last one, she spoke about her usages of a particular digital tool, a padlet (https://padlet.com/) used to save and organize resources found online. Also, we used a logbook for complement our collect of data in the moments that we can not follow her documentation work.

We will explore in this paper different mappings of Anna's documentational trajectory: inferred and reflexive mapping. For obtaining the initial reflexive mapping, we have asked her to write down over an axis (representing time) the mains events having influenced her uses of resources for teaching, or the way of conceiving them, and to write down, under this axis, the resources associated with the event at stake. For helping Anna, we gave her a sample of possible events: the arrival of a new person in her school; the participation in a new collective, an unexpected interaction with a student or colleague; a change of program; a change of teaching level or of textbook; a training course or the discovery of a new resource related to mathematics teaching (book, movie, website, etc.). For obtaining inferred map we made a data crossing among all data.

Our method of analysis is in development. For data analyzing exposed in this article, we transcribed our first interview with Anna and we did a digital transposition of her reflexive mapping. After, we identified on the map, among the events she exposed, those related with collective work, following our hypothesis that knowledge is socially situated. Afterwards, we identified the effects of collective work on her documentation work through associated resources at the event. After we looked in Anna speech her role in which collective and collective's nature, this is based on our hypothesis that some features tell us as a collective effect teachers documentation work. For example, a collective where teachers are voluntary and have a long or permanent engagement nourish teachers work differently than a collective where they are required and they had a short time engagement. Then, we searched collectives that have an important status in her documentational work. For this purpose, we looked for the collectives that appeared more frequently and related to related to other events in the map. Afterwards, we analyzed how these collective nourished her documentational work exploring associated resources, and identifying in her words features relating resources and collective work. We will present, in the following section, the main results of this analysis, still in progress.

Analysis

This analyze is divided into two parts: Anna's participation in collectives along her documentational trajectory and the structuring role of a particular collective, SESAMES in it; and, the structuring role of a particular resource, "Mise en train", on Anna's individual and collective documentation work.

Anna's documentation work in collectives along her documentational trajectory and relationships with SESAMES."

We start analysing the first reflexive mapping drawn by Anna (Figure 1). The analysis of the events evidences Anna's strong involvement in collective work (she says: "I cannot work alone"). Eight (E6, E7, E8, E9, E10, E12, E13 and E14) over 14 events are related to collectives. For deepening the analysis, we study the properties of collectives, the roles of Anna in these collectives, and the functions of the resources that are designed.

The collectives have different natures:

-some of them are transient, as a short episode of coworking with Sésamath (E4), or Assist Me (E12, linked to a European project), or M@gistère (E13, linked to the design of a teacher training path); some of them are 'permanent' (meaning that, once Anna enters this collective, she stays in it), as APMEP (E10/E11, the French national mathematics teacher association), or LéA (E9, collective linking Anna' middle school and the French Institute of Education), or SESAMES1 (E6, a team associating researchers and teachers for renewing Algebra teaching ); -some of them are required (as E14, meeting with parents), some of them are voluntary (as E7 the close partnership with Cindy).

In theses collectives, Anna can have six different roles: member, reading and using their resources (E8 and E10); author, conceiving articles and resources in a group for outside this group (E6, E8, E10, among others); teacher trainer, training middle school teachers (E6, E8, and E10); teacher researcher, reflecting about mathematics teaching (E6, E12 and E8); partner, exchanging and co-producing resources with colleagues (E6 and E10). SESAMES2 had a big impact in Anna collective work. We can see (Figure 1) that she joined Assist me, M@gistère and Léa as a consequence of her engagement in SESAMES. It gave the opportunity for a new partnership with Camille. It opened a window on competencies at the heart of the French new curriculum, giving matter to the design of a teacher training path in IREM, and an occasion for joining this institute. These principles gave birth to resources emblematic of SESAMES thought style, guiding then the whole process of collaborative resource design in this group. It's exactly the characteristic of the metaresources we have already introduced in this paper. One of them is Mise en train, and we will analyze in the following section its impact on Anna's documentation work.

The structuring role of a metaresource on Anna's individual and collective documentation

The Mise en train (MET) corresponds to a specific activity format: it aims to organize teacher's work at the beginning (around 15 minutes) of each class. The expression Mise en train has three meanings: the direct one is warming up (like for an athlete at the beginning of its training); the second one derives from a literal translation, "put on a train", meaning 'cutting a mathematics subject in short successive parts (allowing to store them in the successive wagons of a train); the third meaning derives from an acronym (created by Anna): Travail de Recherche ou d'Approfondissement avec prise d'INitiative (Research and Deepening Work with Initiative Taken). Then the global meaning of Mise en Train has to be understood as the compilation of these three interpretations. In the following, we have chosen to keep this acronym MET, incorporating this global meaning. This global meaning evidences some features of MET resources design. MET is exactly a metaresource, as it gives a way to produce news resources and stimulates teacher's reflection on her documentation work and its effect on student's activity.

MET appears as emerging, in SESAMES, from the documentation work of Anna and her colleagues. Anna explains factors leading them to create this metaresource: the lost of time at the beginning of each class (teacher being mobilized by administrative tasks); the good experience with the short sections of reflecting calculation; and her exchange with English teachers dividing student's activity in short articulated moments for a more dynamics activity format.

Once created, MET deeply changed Anna's documentation work (cf. Figure 4). It affects all five structuring features of classroom practice. The working environment change, for example, students entering class lately do not disturb class activity. The activity format also is altered, because the class is divided in two moments: MET vs. main course. MET also affects Anna's work in other collectives: in her school, the new curricular script is shared by all teachers, as Cindy and Anna explain the principles of MET, and evidence their interest form their practice; outside of her school, Anna disseminates this metaresource in SESAMES training, IREM group, and training, APMEP group and training.

Finally, this metaresource, initially constructed in SESAMES group to teach Algebra, was enlarged to others mathematics subjects. For us, the metaresource MET is a point of convergence between Anna's need and SESAMES interest.

Final considerations and perspectives

Our original question was: which could be the effects of teacher's collective work on her documentational trajectory? Our initial analysis of Anna's documentation work gives us some clues.

Our exploration of Anna's documentation work in collectives allows understanding her resource system better. We saw a diversity of collectives that she participates or participated in, different roles and contribution to her work. Among them, SESAMES appears as an important collective, having a strong impact on Anna's documentational trajectory. It contributes to developing new collective work, resources and thought style. In this collective, she contributes to create a metaresource that structures afterwards her documentation work: this metaresource is exploited in various collectives and structures her way to create resources.

We proposed the concept of documentational trajectory for modelling the teacher's history with resources. In this modeling, the reflective and inferred mapping of documentational trajectory allows us to evidence some critical aspects of this history. It should be noted that these maps constitute a picture at a given moment, and in a given context. This temporal aspect is linked to the fact that her documentation work is still ongoing. The context aspect is also to the relationships the researcher can build with the teacher.

combination of the Structuring Features of Classroom Practice and the Documentary Approach to Didactics helps us to analyze teachers documentational trajectory, evidencing the structuring role of SESAMES and a metaresource associated to a though collective.

Figure 1 .

 1 Figure 1. Anna' reflective mapping (October, 22 nd 2016) of her documentational trajectory Each collective contributes in different ways to her documentation work. However, they are so entangled, that it is difficult to attribute a single function to each of them. The interviews of Anna help us to distinguish some structuring features of Anna's documentation work: elaborating her resources for teaching Algebra; elaborating activities for teaching mathematics notions and interpreting curriculum materials; elaborating resources for developing and evaluating students' competencies; creating lessons and curricular script for her class; reflecting about using digital resources; creating new resources according to pedagogical changes in the school, supporting her participation in other collectives outside school, writing papers, teachers training. These functions are not supported by all collectives, but all of them are exploited in Sésames, where Anna and Cindy use to work together. And SESAMES opens for Anna new possibilities for participating in new collectives and establishing new partnerships. Figure 2 evidences how SESAMES resources nourish the resources of other collectives. Develop a critical thinking on their practice. When she was invited to join SESAMES, Anna hesitated "that was a change, anyway. I accepted, finally, to join ... to join SESAMES. [...] It was a real challenge…".

Figure 2 .

 2 Figure 2. Inferred mapping about the collectives and resource system impact SESAMES has two sets of principles guiding Anna's documentation work (cf. the SESAMES website, names Pégame: http://pegame.ens-lyon.fr/), constituting the thought style of this collective. The first set is composed of three principles for teaching Algebra: justifying computation throughout algebraic rules; proposing proving activities and exploiting formula to introduce the notion of function. The second set is composed of four principles for teaching mathematics: providing students with sufficiently rich and open problems; giving them a chance to seek; giving them a chance to speculate; giving sense to concepts taught.

Figure 4 .

 4 Figure 4. The impact of MET on Anna's documentation workThe curriculum script is modified, including new goals and activities. Anna has then three possibilities for developing a lesson: MET then the main course; the main course, then MET; or beginning with MET… and going on with MET, for giving more responsibility to students for the advancement of the knowledge in the classroom. Regarding Anna's resource system, new resources are created (new curriculum script, new notebook for students, new lesson plans with MET activities, slides that contains MET activities linked to a given notion, new articles (APMEP, IREM, Pégame website) for disseminating SESAMES resources. Last, but not least, the time economy changed, for example, Anna removes the initial "call to students" at the beginning of each lesson.

In spite of the likeness of the acronyms, Sésamath, a mathematics teacher association designing resources at a large scale, is totally different of SESAMES, a small team gathering researchers and teachers for re-thinking Algebra teaching.

Science Education: Modeling Activities, Assessment, Simulation (SESAMES, Situations d'Enseignement Scientifique : Activités de Modélisation, d'Evaluation, de Simulation).


Monica Panero
email: monica.panero@ens-lyon.fr

Gilles Aldon
email: gilles.aldon@ens-lyon.fr

Jana Trgalová
email: jana.trgalova@univ-lyon1.fr

Luc Trouche
email: luc.trouche@ens-lyon.fr

Analysing MOOCs in terms of teacher collaboration potential and issues: the French experience

Keywords: MOOC, teachers' professional development, meta-didactical transposition, community, collaboration

The main aim of this paper is to analyse the experience of a MOOC for teacher training implemented in France, in parallel with a similar experience in Italy. The study focuses on teacher collaboration within such an online learning environment, in terms of co-working and co-learning. The Italian and the French teams outlined a common starting point of the research and some common research concerns (see the Introduction). Each team then reformulated the research questions and tried to answer them through specific theoretical lenses. In the case of the French MOOC eFAN Maths, we study the trainees' collaborative design of a pedagogical resource, by focusing on the efficiency of an evaluation grid designed by trainers to be used within a global process of peer evaluation. This analysis will be enriched by the comparison with the results of the Italian and other MOOC experiences shared within the TWG15.

Introduction

Internet communication tools have given the opportunity to develop new types of teaching methods that combine online courses, resources and discussions. MOOCs (Massive Open Online Courses) were born in 2008 from the initiative of prestigious Universities (MIT, Harvard, Stanford, ...) that find an interest in enlarging the courses they offer to a wide audience. Since then, the MOOC phenomenon has been regularly growing and the worldwide number of MOOCs doubled from 2014 to 2015 [START_REF] Shah | Less Experimentation, More Iteration: A Review of MOOC Stats and Trends in 2015[END_REF]. Although there is a wide choice of several different topics the range is limited when looking specifically for a MOOC aiming at teacher training, especially in mathematics. Nevertheless, there is a growing interest in MOOCs involving mathematics teachers as participants, as shown by TSG44 work during the 13th ICME1 . In particular, from our experiences, there is a need for designing and implementing a MOOC for teacher training in mathematics education with a focus on the development of communities of practice [START_REF] Wenger | Communities of practice: learning, meaning, and identity[END_REF] and collaborative work among teachers as the basis for their professional development. Indeed, when people co-work (work together collaboratively) they can also co-learn (learn together collaboratively), as highlighted in the ICME survey of [START_REF] Robutti | ICME international survey on teachers working and learning through collaboration: June 2016[END_REF]. The authors find that teachers can learn through discussion, conversation and reflection on their own teaching, on students' learning and on others' teaching. The methodology of the French MOOC eFAN Maths (as well as the Italian MOOC Geometria) aims at creating collaborative contexts for teachers' work, where they can learn from these kinds of practices. Taking into account the necessity for teachers to be supported in exploiting the technological affordances, the objectives of both MOOCs are: accompanying teachers in the production of teaching resources, by examples of activities and reflection on their ongoing resource design; fostering a reasoned use of technology, encouraging teachers to choose proper digital tools for the classroom. Such aims are related to the interest in the design and the implementation of teacher professional development programmes to include the role of teachers working and learning in communities [START_REF] Wenger | Communities of practice: learning, meaning, and identity[END_REF][START_REF] Jaworski | Inquiry community in an activity theory frame[END_REF].

The originality of our research based on the data collected from two similar MOOCs, one in Italy and one in France, is twofold. First, it focuses on the specific dynamics of online interactionsamong trainees and between trainees and trainers -to study the trainees' use and appropriation of a tool (an evaluation grid) designed by trainers for supporting peer evaluation of collaborative projects. It is topical and urgent to analyse the efficiency of such tools and interactions in the context of distance learning, because of the increased interest in this approach in recent years. Second, it analyses such dynamics according to the cultural constraints that shape MOOCs design and development. French and Italian school environments have some remarkable differences and one of the most palpable is a wider freedom that institutional regulations traditionally give to the Italian teachers, compared to the major institutional constraints met by the French teachers. The Italian Indicazioni2 (guidelines) highlight for each discipline the fundamental learning goals that students have to achieve at the end of each cycle of instruction (two or three scholastic grades). They have the character of general didactic guidelines and deliver to teachers the responsibility of choosing and linking the specific mathematical contents to be developed in the classroom in order to reach the established learning goals. The French Programmes 3 (syllabus) are also based on competences for a given cycle of instruction, but they appear to be more detailed and normative: for each mathematical content, they provide some examples of activities. Moreover, they are accompanied by several additional resources intended to support the curriculum implementation in the classroom.

In this paper, in parallel with the Italian one, drawing on the common theoretical element of the Meta-Didactical Transposition [START_REF] Arzarello | Meta-didactical transposition: A theoretical model for teacher education programmes[END_REF], we intend to highlight how the concepts of community and of collaborative work evolve, taking new and different forms, and how this impacts on teachers' professional learning. During the conference the two teams, who are working together for comparing the data from the two MOOCs, will discuss the relevance of cultural and institutional aspects in the specific dynamics of the two experiences.

Description of the MOOC eFAN Maths

The MOOC eFAN Maths4 was delivered on a French national platform (FUN, France Universités Numérique) and its target was the French-speaking community all over the world, namely mathematics teachers and teacher educators willing to improve their practices in using technology in their classrooms. The second season of the MOOC, which is reported in this paper, lasted from the 8th of March to mid-April 2016 and was organised in five weeks. The MOOC eFAN Maths is part of the Stratégie Mathematiques of the French Ministry of Education, which stresses the relationship of mathematics with other sciences and with the world, and aims at training teachers in this perspective in order to give students a refreshed image of mathematics. More specifically, this season of the MOOC was created with a double institutional aim: to support teachers and teacher educators in understanding and implementing the new French curriculum (since September 2016 in all French primary and secondary schools) and to promote collaboration within the French-speaking community. Courses, activities and discussion were especially focused on the new themes involved in the French curriculum, namely algorithmics and interdisciplinary work. The MOOC was grounded on a project-based pedagogy, aimed at the design and the analysis of a classroom activity involving the use of digital tools. Every week, trainees took courses on specific topics of mathematics education from three video-based lessons, answered the related quizzes and worked on specific activities. The courses were constructed to provide trainees with elements to develop their reflections and projects about teaching and learning mathematics with technology. They showed brief episodes of classroom observation and their analysis, or were based on existing resources, showing and commenting animations or programs created with technology. Regarding the activities, collective work was strongly encouraged among trainees. For this purpose, they were invited to join the "MOOC eFAN Maths 2016" group, created on Viaéduc5 , a professional social network specifically designed for teachers' exchanges. Viaéduc essentially allows members to post comments, to create subgroups, to create and publish documents and to comment/recommend/share them. Group members can work collaboratively either asynchronously, being authors of the same online document, or synchronously, writing on the same online collaborative board (padlet).

During week 0, on Viaéduc, trainees were invited to propose a theme, an idea or a project to work on and to start establishing relationships with others. Week 1 was devoted to the characterisation of digital or non-digital resources that can support teachers' and teacher educators' work. On Viaéduc, trainees gathered together around a project constituting public subgroups of the main "MOOC eFAN Maths 2016" group, so that any trainee could read the work of any subgroup and follow any discussion. Week 1 activity consisted in selecting resources deemed as relevant for the project of the group in order to constitute its "toolkit". Week 2 was devoted to the analysis of students' activity using technology in mathematical situations. On Viaéduc, each group had to design a mathematical situation and to analyse it from the student's point of view using an analysis grid proposed by the trainers. Week 3 was devoted to the analysis of the teacher's role in the designed mathematical situation. An analysis grid focusing on instrumental orchestration was presented through the courses and trainees were invited to apply it to their situation. During week 4 trainers organised a process of peer evaluation of the different projects (submitted as versions 0) and proposed an evaluation grid grounded on the theoretical frames presented in the courses underpinning the analysis of digital resource quality [START_REF] Trgalová | Analyse de ressources comme moyen de développement professionnel des enseignants[END_REF]. Finally, every group was supposed to refine the project, using the received feedback, and to submit the version 1.

Theoretical framework

In order to analyse the MOOC eFAN Maths we combine three main theoretical frameworks: the Meta-Didactical Transposition [START_REF] Arzarello | Meta-didactical transposition: A theoretical model for teacher education programmes[END_REF], the documentational approach to didactics [START_REF] Gueudet | Towards new documentation systems for mathematics teachers?[END_REF] and the concept of communities of practice [START_REF] Wenger | Communities of practice: learning, meaning, and identity[END_REF].

The Meta-Didactical Transposition (MDT) model, as well as the documentational approach to didactics, giving major importance to the collective aspects of teachers' work, needed a theory for analysing the development of teachers (as well as researchers) collectives. Both approaches chose the theory of communities of practice (CoP), mainly because these communities are structures where learning occurs. CoP are formed by people who engage in a process of collective learning in a shared domain of human endeavour. For Wenger (1998), a condition for the development of such communities is to equilibrate participation and reification, where reification means producing resources, symbols, stories etc., recognised by the whole community as common goods. Such communities may develop by themselves, or be "cultivated" [START_REF] Wenger | Cultivating communities of practice: a guide to managing knowledge[END_REF], i.e. encouraged, supported by an organisation. Indeed, the knowledge of an organisation lives in a constellation of CoP each taking care of a specific aspect of the competence that the organisation needs.

The MDT model captures dynamic interactions between teachers' and researchers' practices when these two communities interplay, typically in training contexts, and in particular in the case of training programmes in mathematics education. Using the MDT lens, we can address research questions that imply the influence of the practices of one community on the other. Such practices are described through the notion of praxeology [START_REF] Chevallard | L'analyse des pratiques enseignantes en théorie anthropologique du didactique[END_REF]: a praxeology for a given task consists of a practical block comprising techniques to accomplish the task, and a theoretical block justifying and supporting these techniques. In a teacher education programme, trainees and trainers bring into play the components of their respective praxeologies. The objective of the programme is to transpose, in the sense of [START_REF] Chevallard | La transposition didactique : du savoir savant au savoir enseigné[END_REF], some components of the trainers' research practices into the teachers' practices, taking into account the classroom reality and teachers' expertise for effectively enacting such components. Thus, the two communities together contribute to creating a shared praxeology, which both communities can adapt in their future practices. This occurs through the phenomenon of internalisation: a community internalises a component of the praxeology of the other community, that was previously external to it, entailing an evolution of practices.

The documentational approach to didactics [START_REF] Gueudet | Towards new documentation systems for mathematics teachers?[END_REF] analyses teacher professional development through the interplay of practices and resources. This interplay is modelled as a documentational genesis, extending the notion of instrumental genesis introduced by [START_REF] Verillon | Cognition and artifacts: A contribution to the study of thought in relation to instrument activity[END_REF] between artefact and instrument. A documentational genesis entails different steps, such as looking for resources, selecting/designing mathematical tasks, planning their sequence, managing available artefacts, etc., for reaching a given teaching goal. This process gives birth to a document, which is a mixed entity composed of the revised and recombined resources and the associated usage scheme. Each documentational genesis is then a means of triggering teacher professional development. The genesis of a document combines two processes: instrumentation, when the affordances and constraints of the resources influence the subject's activity, and instrumentalization, when the subject shapes the resources that he/she appropriates. The documentational approach to didactics, from its beginning, and even more in its recent developments [START_REF] Pepin | Re-sourcing teachers' work and interactions: A collective perspective on resources, their use and transformation[END_REF], gives a major importance to the collective aspects of teachers' work with resources, evidencing the importance of interactions within teacher collectives for spurring documentational genesis and teachers' professional development.

The combination of these three frameworks allows us to analyse what happened in the MOOC, seen as a constellation of cultivated CoP. These communities are not created at once, they emerge in the dynamics of a shared project. First, the community of trainers emerges to design and implement a new teacher education programme. Then the communities of trainees emerge, each one developing all along the advancement of its own project. The elaboration of a project consists in designing a pedagogical activity from selected existing resources that are adapted, modified and combined by the group members. This process can thus be seen as a collective documentational genesis. In such a process, we are interested in analysing the efficiency of the tools designed by the trainers, as elements of their meta-didactical praxeologies, to foster both collaboration and project development. We study the phenomenon of internalisation in interaction with the process of documentational genesis as a reification process, addressing the following research questions: How does each trainees' CoP emerge through the process of documentational genesis according to CoP criteria? How do the CoP of trainers and each trainees' CoP interact through the MDT lens?

Methodology of the study

In this paper, we focus on the last week of the MOOC and in particular on the use of the evaluation grid by the trainees. Indeed, this tool was constructed by the trainers to encompass all the phases of the pedagogical design, developed in the MOOC week after week. Analysing the way trainees use it can give us insights into the way they have understood and interiorised the principles of each phase of design. Moreover, this moment represented an interesting dynamic between an individual activity (the evaluation of another project through the grid) and a collective work (of each group on the delivered version 0 for improving it).

The grid was structured around the following four criteria:

1. Accuracy of the definition and description of the project.

2. Relevance of the mobilised digital resources with respect to the educational goal of the designed mathematical task.

3. Relevance of the analysis of the students' activity.

4. Relevance of the analysis of the teacher's role.

For each criterion, aimed at evaluating the work done by a group during a specific week, some guiding questions are proposed with a double objective: to foster the production of justified feedback and to deepen the reflection carried out in the previous weeks of the MOOC. The grid finally asked for a brief global feedback on the project and some suggestions to improve the work.

Trainers provided this tool to support trainees in the process of peer evaluation, with the implicit aim of facilitating the internalisation of the evaluation criteria. Each trainee was invited to use the grid individually to evaluate the project of another group, by answering each guiding question with an appreciation: very good, satisfactory, fragile or insufficient, accompanied by a justification. Trainers gradually collected feedback and comments in a table and shared it in a specific space on Viaéduc, called "Project evaluation", so that all the trainees could access them.

Data analysis

We collected a great amount of data coming from different sources: answers to the final questionnaire sent to all the trainees; interventions on the forum; discussions on Viaéduc within the "MOOC eFAN Maths 2016" group and within the different subgroups. Among the 2500 people enrolled in the MOOC, more than 700 registered on Viaéduc and about 11% of them contributed to the work of a group. For analysing how trainees used the evaluation grid, we focused on those who participated to the work of a group. Drawing specifically on Viaéduc discussions, we illustrate in this section the main finding of this study: the trainees' internalisation of the evaluation criteria, through the tools provided and the process established by the trainers.

The evaluation grid became a resource for trainees. They used it both for providing feedback to other projects (instrumentation) and for reflecting on and refining their own project (instrumentalization). Let us illustrate this double action -that of the grid on the trainees and that of the trainees on the grid -through some excerpts of Viaéduc discussions. The comments, written by some trainees (EE, CC, CM) on the wall of the "MOOC eFAN Maths 2016" group, show a formative value of the peer evaluation and specifically refer to an introspective use of the grid.

EE

Yes, actually, "evaluating" the project of your peers is ultra-formative. It inevitably sends us back to ourselves and stimulates a lot of reflections. I encourage all those who can to participate in it.

CC

It is clear that the peer evaluation of the projects is also an exercise [...] I think that the aim is not marking "very good" everywhere, so I try to use the grid with its criteria that I start to understand [...]. A difficulty that I encounter is that, when I perceive a small flaw in one of the aspects of the whole structure of the project, this flaw seems to impact on several items of the grid...? [...]" CM [answering to EE] Indeed, by evaluating another project, you discover much better how to improve yours. The evaluation grid is a great support but I join CC on the domino effect of some points.

In particular, CC's and CM's comments show a well-thought-out use of the grid, especially the awareness that all the criteria are interrelated to analyse the work. The double process of instrumentation and instrumentalization of the grid shows that trainees internalise the evaluation criteria using the grid also for reflecting on the quality of their own document.

Moreover, the remote collaboration on Viaéduc allowed trainees to make the version 0 of the document evolve into the version 1, taking into account both peers' external feedback and each member's introspective feedback. An example of this step in the collective documentational genesis is represented by the use of the padlet "TO DO List" within one of the groups. In this padlet, the group members organise the different tasks to be done in order to make the common document (version 0), seen as a resource, evolve into a new document (version 1). When a comment is ticked off and an author and a date are specified, this is the sign that the task has been done. This to-do list consists of feedback coming from peers but also of some personal comments, such as "For the resources I think that we must orient the reading of the first ones according to the soma cube activity". We can reconstruct the story of this proposal of reorganisation of the project resources, thanks to the parallel discussions occurred in the group. Such discussions show that trainees (JP in this case) benefited from both peers' feedback and introspective feedback, coming from the action of evaluating other projects.

JP

[on the wall of the group] Hello, after having read several projects I actually expected that someone "criticises" a little bit our profusion of resources. This being said, as noticed by our (unique, sniff!) evaluator, AM's work is very rich and it would be a pity to modify our toolkit that will maybe carry on after the MOOC. Perhaps, we could prune it in the v1 of the project by keeping only those that are actually usable in the SOMA cube activity. What do you think?

In terms of remote collaboration, it is worth noting that some groups used such collaborative tools for organising their remote work. This organisation guided the transformation of the version 0 into the version 1 of the project, as a reification of the collaborative participation within the CoP.

Conclusions

Each community of practice benefited from the others' feedback and from the introspective reflections that the members, who were engaged in the design process, made during the evaluation process. Crossing the external and the introspective feedback allows the trainees to work collaboratively on the refinement of their version 0 and to produce the version 1 of their project. This new version of the document is both a stage in the documentational genesis and the result of the internalisation of the evaluation criteria which occurred through the participation in the evaluation process.

On the one hand, the evaluation grid is a technical tool of trainers' meta-didactical praxeologies, based on the theoretical concepts tackled in the courses and justified within the global process of peer evaluation. On the other hand, the trainers' choice of collaborative tools, seen as a trainers' technique, is grounded on the trainers' objectives to foster the emergence of communities of practice among trainees. Our analysis shows how these praxeological choices influence the trainees' work when they improve the version 0 of their project into the version 1. In return, this analysis influences the trainers' meta-didactical praxeologies in the perspective of a re-design of the new season of the MOOC. In particular, the organisation of the MOOC schedule will be modified taking into account the emergent question of time. Communities of practice need time for establishing an effective remote collaboration, so that participation and reification equilibrate as much as possible.

At the same time, the evaluation process and the documentational genesis need time for being effectively carried out. In that sense, we can observe the double phenomenon of internalisation: from trainers to trainees as well as from trainees to trainers.

For more information, see http://www.icme13.org/files/tsg/TSG_44.pdf

http://www.indire.it/lucabas/lkmw_file/licei2010/indicazioni_nuovo_impaginato/_decreto_indicazioni_nazionali.pdf

For further information, see https://www.fun-mooc.fr/courses/ENSDeLyon/14003S02/session02/about

See www.viaeduc.fr


Q Duret
email: duret@lpthe.jussieu.fr

& B Machet
email: machet@lpthe.jussieu.fr

MIXING ANGLES OF QUARKS AND LEPTONS AS AN OUTCOME OF SU (2) HORIZONTAL SYMMETRIES

Keywords: 11.30.Hv, 11.40.-q, 12.15.Ff, 12.15.Hh, 14.60.Pq

We show that all mixing angles are determined, within experimental uncertainty, by a product of SU (2) horizontal symmetries intimately linked to the algebra of weak neutral currents. This concerns: on one hand, the three quark mixing angles; on the other hand, a neutrino-like pattern in which θ 23 is maximal and tan(2θ 12 ) = 2. The latter turns out to exactly satisfy the "quark-lepton complementarity condition" θ c + θ 12 = π/4. Moreover, among all solutions, two values for the third neutrino mixing angle arise, which satisfy the bound sin 2 (θ 13 ) ≤ 0.1: θ 13 = ±5.7 10 -3 and θ 13 = ±0.2717.

Introduction

Following the study of neutral kaons done in [START_REF] Machet | Binary systems of Neutral Mesons in Quantum Field Theory[END_REF], we have shown in [START_REF] Duret | Mixing angles and non-degenerate coupled systems of particles[END_REF] and [START_REF] Duret | The emergence of the Cabibbo angle in non-degenerate coupled systems of fermions[END_REF] that:

* in Quantum Field Theory (QFT), mixing matrices linking flavour to mass eigenstates for non-degenerate coupled systems should never be parametrized as unitary. Indeed, assuming that the effective renormalized quadratic Lagrangian is hermitian at any q2 and that flavour eigenstates form an orthonormal basis, different mass eigenstates, which correspond to different values of q 2 (poles of the renormalized propagator) belong to different orthonormal bases 1 ; * when it is so, the properties of universality for diagonal neutral currents and absence of flavor changing neutral currents (FCNC) which are systematically implemented, for the Standard Model (SM), in the space of flavour eigenstates, do not automatically translate anymore into equivalent properties in the space of mass eigenstates. In the case of two generations of fermions, imposing them for mass eigenstates yields two types of solutions for the mixing angles 2 of each doublet with identical electric charge: Cabibbo-like solutions 3 which reduce to a single unconstrained mixing angle, and a set of discrete solutions, unnoticed in the customary approach, including in particular the so-called maximal mixing π/4 ± kπ/2; * for any of these solutions one recovers a unitary mixing matrix; but, as said above, very small deviations are expected due to mass splittings, which manifest themselves as a tiny departure from the exact two previous conditions. In particular, in the neighborhood of a Cabibbo-like solution, these deviations become of equal strength for a value of the mixing angle extremely close to the measured Cabibbo angle

tan(2θ c ) = 1/2. (1)
This success was a encouragement to go further in this direction. We present below the outcome of our investigation of neutral current patterns in the case of three generations of fermions. In particular, we show that the requested scheme of unitarity violation of the mixing matrices trivially interprets in terms of a product of SU (2) horizontal symmetries, implemented at the level of neutral weak currents.

Hence, the values of all mixing angles, in the quark as well as in the leptonic sector, are controlled by this symmetry.

The intricate system of trigonometric equations has been solved by successive approximations, starting from configurations in which θ 13 is vanishing. We will see that this approximation, obviously inspired by the patterns of mixing angles determined from experimental measurements, turns out to be a very good one. Indeed, we precisely show, without exhibiting all the solutions of our equations, that the presently observed patterns of quarks as well as of neutrinos, do fulfill our criterion. While the three angles of the Cabibbo-Kobayashi-Maskawa (CKM) solution are "Cabibbo-like", the neutrino-like solution tan(2θ 12 ) = 2 ⇔ θ 12 ≈ 31.7 o , θ 23 = π/4, θ 13 = ±5.7 10 -3 or θ 13 = ±0.2717

is of a mixed type, where θ 23 is maximal while θ 12 and θ 13 are Cabibbo-like. Two significant features in these results must be stressed. First, the values for the third neutrino mixing angle θ 13 given in (2) are predictions which take into account present (loose) experimental constraints. Only two possibilities survive: an extremely small value θ 13 ∼ V ub ∼ a few 10 -3 , and a rather "large" one, at the opposite side of the allowed range. Secondly, our procedure yields in an exact, though quite simple way, the well-known "quark-lepton complementarity relation" [START_REF] Raidal | Prediction theta c + T heta sol = pi/4 from flavor physics: a new evidence for grand unification?[END_REF] for 1-2 mixing:

θ 12 + θ c = π/4, (3)
where θ 12 is the leptonic angle, and θ c the Cabibbo angle for quarks. for leptons; they include in particular the charged leptons detected experimentally, since their identification proceeds through the measurement of their charge/mass ratio in a magnetic field; * the neutrinos that couple to mass eigenstates of charged leptons in charged weak currents. These are the usual "electronic", "muonic" and "τ " neutrinos ν e , ν µ , ν τ considered in SM textbooks [5]: they are indeed identified by the outgoing charged leptons that they produce through charged weak currents, and the latter are precisely mass eigenstates (see above). These states read (see Appendix D)

     ν e ν µ ν τ      = K † ℓ      ν ef ν µf ν τ f      = (K † ℓ K ν )      ν em ν µm ν τ m      , (4)
where K ℓ and K ν are the mixing matrices respectively of charged leptons and of neutrinos (i.e. the matrices that connect their flavour to their mass eigenstates). Note that these neutrinos coincide with flavour eigenstates when the mixing matrix of charged leptons is taken equal to unity K ℓ = 1, i.e. when the mass and flavour eigenstates of charged leptons are aligned, which is often assumed in the literature.

Neutral currents and SU (2) symmetry

The principle of the method is best explained in the case of two generations. This in particular makes the link with our previous work [START_REF] Duret | The emergence of the Cabibbo angle in non-degenerate coupled systems of fermions[END_REF].

Let us consider for example the (d, s) channel 4 . The corresponding neutral currents in the basis of mass eigenstates are controlled by the product C † C of the mixing matrix C between d and s with its hermitian conjugate (see [START_REF] Duret | Mixing angles and non-degenerate coupled systems of particles[END_REF]). Requesting the absence of non-diagonal currents and universality for diagonal currents (that we call hereafter the "unitarization" conditions) selects two classes of mixing angles [START_REF] Duret | Mixing angles and non-degenerate coupled systems of particles[END_REF]: a first class that we call "Cabibbo-like" which is unconstrained at this level, and a second class made of a series of discrete values, maximal or vanishing. As soon as C departs from unitarity, the Lagrangian for neutral currents reads, in the basis of mass eigenstates

L ∝ W 3 µ α dm γ µ L d m + β sm γ µ L s m + δ dm γ µ L s m + ζ sm γ µ L d m . (5)
The condition that was shown in [START_REF] Duret | The emergence of the Cabibbo angle in non-degenerate coupled systems of fermions[END_REF] to fix the value of the "Cabibbo" angle, i.e. that the universality of dm γ µ L d m and sm γ µ L s m currents is violated with the same strength as the absence of dm γ µ L s m and sm γ µ L d m currents, now simply reads δ = α -β = ζ, which states that L in ( 5) is invariant by the SU (2) symmetry which rotates d and s. Eq. ( 5) indeed trivially rewrites, then,

L ∝ W 3 µ (α + β) dm γ µ L d m + sm γ µ L s m 2 + (α -β) dm γ µ L d m -sm γ µ L s m 2 + dm γ µ L s m + sm γ µ L d m , (6)
in which all components of the triplet of SU (2) currents 1 2 dm γ µ L d m -sm γ µ L s m , dm γ µ L s m , sm γ µ L d m , the corresponding (vector) charges of which make up the relevant SU (2) algebra, have the same coefficient (α -β). The work [START_REF] Duret | The emergence of the Cabibbo angle in non-degenerate coupled systems of fermions[END_REF] states accordingly that the "Cabibbo angle" is controlled by this SU (2) symmetry.

The generalization to three generations is now straightforward. Neutral currents are controlled by the product K † K of the 3 × 3 mixing matrix K with its hermitian conjugate; for example, the (left-handed) neutral currents for quarks with electric charge (-1/3) read

     d f s f b f      γ µ L      d f s f b f      =      d m s m b m      γ µ L K † d K d      d m s m b m      . ( 7
)
Requesting SU (2) symmetry in each (i, j) channel is trivially equivalent to the condition that, in this channel, universality for the diagonal currents is violated with the same strength as the absence of nondiagonal currents. We will show that all presently known mixing angles, in the quark as well as in the leptonic sectors, satisfy this criterion.

Mixing matrices. Notations

We write each mixing matrix K as a product of three matrices, which reduce, in the unitarity limit, to the basic rotations by θ 12 , θ 23 and θ 13 (we are not concerned with CP violation)

K =      1 0 0 0 c 23 s 23 0 -s 23 c23      ×      c 13 0 s 13 0 1 0 -s 13 0 c13      ×      c 12 s 12 0 -s 12 c12 0 0 0 1      . (8)
We parametrize each basic matrix, which is a priori non-unitary, with two angles, respectively (θ 12 , θ12 ), (θ 23 , θ23 ) and (θ 13 , θ13 ). We deal accordingly with six mixing angles, instead of three in the unitary case (where θij = θ ij ). We will use throughout the paper the notations s ij = sin(θ ij ), sij = sin( θij ), and likewise, for the cosines, c ij = cos(θ ij ), cij = cos( θij ).

To lighten the text, the elements of K † K will be abbreviated by

[ij], i, j = 1 . . . 3 instead of (K † K) [ij] ,
and the corresponding neutral current will be noted {ij}. So, in the quark case, {12} stands for ūm γ µ L c m or dm γ µ L s m , and, in the neutrino case, for νem γ µ L ν µm or ēm γ µ L µ m .

The unitarization conditions

They are five: three arise from the absence of non-diagonal neutral currents for mass eigenstates, and two from the universality of diagonal currents. Accordingly, one degree of freedom is expected to be unconstrained.

Absence of non-diagonal neutral currents of mass eigenstates

The three conditions read: * for the absence of {13} and {31} currents:

[13] = 0 = [31] ⇔ c 12 c 13 s 13 -c13 s13 (c 2 23 + s 2 23 ) -c13 s12 (c 23 s 23 -c23 s23 ) = 0; (9)
* for the absence of {23} and {32} currents:

[23] = 0 = [32] ⇔ s 12 c 13 s 13 -c13 s13 (c 2 23 + s 2 23 ) + c13 c12 (c 23 s 23 -c23 s23 ) = 0;
(10) * for the absence of {12} and {21} currents:

[12] = 0 = [21] ⇔ s 12 c 12 c 2 13 -s12 c12 (c 2 23 + s2 23 ) + s 12 c 12 s2 13 (s 2 23 + c2 23 ) + s13 (s 12 s12 -c 12 c12 )(c 23 s 23 -c23 s23 ) = 0. (11)

Universality of diagonal neutral currents of mass eigenstates

The two independent conditions read: * equality of {11} and {22} currents:

[11] -[22] = 0 ⇔ (c 2 12 -s 2 12 ) c 2 13 + s2 13 (s 2 23 + c2 23 ) -(c 2 12 -s2 12 )(c 2 23 + s2 23 ) +2s 13 (c 23 s 23 -c23 s23 )(c 12 s12 + s 12 c12 ) = 0; (12)
* equality of {22} and {33} currents:

[22] -[33] = 0 ⇔ s 2 12 + c2 12 (c 2 23 + s2 23 ) -(s 2 23 + c2 23 ) + (1 + s 2 12 ) s2 13 (s 2 23 + c2 23 ) -s 2 13 +2s 12 s13 c12 (c 23 s23 -c 23 s 23 ) = 0. ( 13
)
The equality of {11} and {33} currents is of course not an independent condition.

Solutions for θ 13 = 0 = θ13

In a first step, to ease solving the system of trigonometric equations, we shall study the configuration in which one of the two angles parametrizing the 1-3 mixing vanishes 5 , which is very close to what is observed experimentally in the quark sector, and likely in the neutrino sector. It turns out, as demonstrated in Appendix A, that the second mixing angle vanishes simultaneously. We accordingly work in the approximation (the sensitivity of the solutions to a small variation of θ 13 , θ13 will be studied afterwards)

θ 13 = 0 = θ13 . (14)
Eqs. ( 9), ( 10), ( 11), ( 12) and ( 13), reduce in this limit to Accordingly, the two following sections will respectively start from: * θ 12 and θ 23 Cabibbo-like (and, in a first step, vanishing θ 13 ), which finally leads to a mixing pattern similar to what is observed for quarks; * θ 23 maximal and θ 12 Cabibbo like (and, in a first step, vanishing θ 13 ), which finally leads to a mixing pattern similar to the one observed for neutrinos.

-s 12 (c 23 s 23 -c23 s23 ) = 0, (15a) c12 (c 23 s 23 -c23 s23 ) = 0, (15b)
s 12 c 12 -s12 c12 (c 2 23 + s2 23 ) = 0, (15c)
(c 2 12 -s 2 12 ) -(c 2 12 -s2 12 )(c 2 23 + s2 23 ) = 0, (15d)
s 2 12 + c2 12 (c 2 23 + s2 23 ) -(s 2 23 + c2 23 ) = 0. (15e)

The quark sector; constraining the three CKM angles

Mass splittings entail that the previous general conditions, which, when exactly satisfied, correspond de facto to unitary mixing matrices, cannot be exactly fulfilled. We investigate the vicinity of their solutions, and show that the same violation pattern that led to an accurate determination of the Cabibbo angle in the case of two generations, is also satisfied by the CKM angles in the case of three generations.

3.1

The simplified case θ 13 = 0 = θ13

In the neighborhood of the solution with both θ 12 and θ 23 Cabibbo-like, we write

θ12 = θ 12 + ǫ, θ23 = θ 23 + η. (16)
The pattern (θ 13 = 0 = θ13 ) can be reasonably considered to be close to the experimental situation, at least close enough for trusting not only the relations involving the first and second generation, but also the third one.

Like in [START_REF] Duret | The emergence of the Cabibbo angle in non-degenerate coupled systems of fermions[END_REF], we impose that the absence of {12}, {21} neutral currents is violated with the same strength as the universality of {11} and {22} currents. It reads

|2ηs 12 c 12 s 23 c 23 + ǫ(c 2 12 -s 2 12 )| = | -2ηs 23 c 23 (c 2 12 -s 2 12 ) + 4ǫs 12 c 12 |. ( 17
)
We choose the "+" sign for both sides, such that, for two generations only, the Cabibbo angle satisfies tan(2θ 1 as a function of θ 12 , together with the experimental intervals for θ 23 and θ 12 . There are two [START_REF] Particle | Review of Particle Physics[END_REF] for θ 12 ; the first comes from the measures of V ud (in black on Fig. 1)

V ud ∈ [0.9735, 0.9740] ⇒ θ 12 ∈ [0.2285, 0.2307], (20)
and the second from the measures of V us (in purple on Fig. 1) The measured value for θ 23 is seen on Fig. 1 to correspond to θ 12 ≈ 0.221, that is cos(θ 12 ) ≈ 0.9757. Our prediction for cos(θ 12 ) is accordingly 1.7 10 -3 away from the upper limit of the present upper bound for V ud ≡ c 12 c 13 [START_REF] Hardy | The status of V ud[END_REF] [START_REF] Particle | Review of Particle Physics[END_REF]; it corresponds to twice the experimental uncertainty. It also corresponds to sin(θ 12 ) = 0.2192, while V us ≡ s 12 c 13 is measured to be 0.2247(19) [START_REF] Spadaro | KLOE results on kaon decays and summary status of V us[END_REF] [START_REF] Particle | Review of Particle Physics[END_REF]; there, the discrepancy is 2/100, only slightly above the 1.8/100 relative width of the experimental interval.

V us ∈ [0.
The approximation which sets θ 13 = 0 = θ13 is accordingly reasonable, though it yields results slightly away from experimental bounds. We show in the next subsection that relaxing this approximation gives results in excellent agreement with present experiments.

Going to

(θ 13 = 0, θ13 = 0)
Considering all angles to be Cabibbo-like with, in addition to [START_REF] Kniehl | Simple On-Shell Renormalization Framework for the Cabibbo-Kobayashi-Maskawa Matrix[END_REF] {22} currents, one gets ǫ/η as a function of θ 12 , θ 23 , θ 13 7 . This expression is plugged in the relation (24b) = (-)(24e) 8 , which expresses the same condition for the (2, 3) channel; from this, one extracts ρ/η as a function of θ 12 , θ 23 , θ 13 9 . The expressions that have been obtained for ǫ/η and ρ/η are then inserted into the third relation, (24a) = (24f) , which now corresponds to the (1, 3) channel. This last step yields a relation F 0 (θ 12 , θ 23 , θ 13 ) = 1 between the three angles θ 12 , θ 23 , θ 13 .

It turns out that ∂F 0 (θ 12 ,θ 23 ,θ 13 ) ∂θ 13 = 0, such that, in this case, a condition between θ 12 and θ 23 alone eventually fulfills the three relations under concern θ 23 is plotted on Fig. 2 as a function of θ 12 , together with the experimental intervals for θ 23 and θ 12 (the intervals for θ 12 come respectively from V ud (eq. ( 20)) and V us (eq. ( 21))).

The precision obtained is much better than in Fig. 1 since, in particular, for θ 23 within its experimental range, the discrepancy between the predicted θ 12 and its lower experimental limit coming from V us is smaller than the two experimental intervals, and even smaller than their intersection.

The general solution

The principle for solving the general equations ( 23) is the same as above. One first uses the relation (23c) = (-) (23d) to determine ρ/ǫ in terms of η/ǫ. The result is plugged in the relation (23b) = (-) (23e), which fixes η/ǫ, and thus ρ/ǫ as functions of (θ 12 , θ 23 , θ 13 ). These expressions for η/ǫ and ρ/ǫ are finally plugged in the relation (23a) = (23f) , which provides a condition F (θ 12 , θ 23 , θ 13 ) = 1. When it is

7 ǫ η = s13(c 2 23 -s 2 23 ) + 2s23c23 s12c12 + c 2 12 -s 2 12 4c12s12 -(c 2 12 -s 2 12 ) ; (25)
ǫ/η has a pole at tan(2θ12) = 1/2, the predicted value of the Cabibbo angle for two generations. 8 There, again, the (-) sign has to be chosen so as to recover approximately (19).

9 ρ η = 2c23s23 s13 -c12 2 (c12s12 + c 2 12 -s 2 12 ) 4s12c12 -(c 2 12 -s 2 12 ) - 1 + c 2 12 c12s12 + 1 s12 c 2 23 -s 2 23 2s23c23 . ( 26
)
ρ/η has a pole at tan(2θ12) = 1/2 and, for θ13 = 0, it vanishes, as expected, when θ12 and θ23 satisfy the relation (19), which has been deduced for θ13(≡ θ13 + ρ) = 0 = θ13.

fulfilled, the universality of each pair of diagonal neutral currents of mass eigenstates and the absence of the corresponding non-diagonal currents are violated with the same strength, in the three channels (1, 2), (2, 3) and (1, 3).

The results are displayed in Fig. 3; θ 23 is plotted as a function of θ 12 for θ 13 = 0.004 and 0.01. The present experimental interval is [START_REF] Particle | Review of Particle Physics[END_REF] V ub = sin(θ 13 ) ≈ θ 13 ∈ [4 10 -3 , 4.6 10 We conclude that: * The discrepancy between our predictions and experiments is smaller than the experimental uncertainty; * a slightly larger value of θ 13 and/or slightly smaller values of θ 23 and/or θ 12 still increase the agreement between our predictions and experimental measurements; * the determination of θ 12 from V us seems preferred to that from V ud .

Another confirmation of the relevance of our criterion is given in the next section concerning neutrino mixing angles.

A neutrino-like pattern; quark-lepton complementarity

In the "quark case", we dealt with three "Cabibbo-like" angles. The configuration that we investigate here is the one in which θ 23 is, as observed experimentally [START_REF] Particle | Review of Particle Physics[END_REF], (close to) maximal, and θ 12 and θ 13 are Cabibbo-like (see subsection 2.5).

The case θ 13 = 0 = θ13

We explore the vicinity of this solution, slightly departing from the corresponding unitary mixing matrix, by considering that θ12 now slightly differs from θ 12 , and θ23 from its maximal value

θ12 = θ 12 + ǫ, θ 23 = π/4 , θ23 = θ 23 + η. (29)
The l.h.s.'s of eqs. (9) (10) (11) ( 12) and ( 13) no longer vanish, and become respectively

-
η(1 + c 2 12 ) -2ǫs 12 c 12 (1 + η), (30d)
showing by which amount the five conditions under scrutiny are now violated. Some care has to be taken concerning the accurateness of equations (30). Indeed, we imposed a value of θ 13 which is probably not the physical one (even if close to). It is then reasonable to consider that channel (1, 2) is the less sensitive to this approximation and that, accordingly, of the five equations above, (30c) and (30d), marked with an " * ", are the most accurate 10 .

The question: is there a special value of θ 12 = θ12 Cabibbo-like for which small deviations (ǫ, η) from unitarity entail equal strength violations of * the absence of {12}, {21} non-diagonal neutral currents; * the universality of {11} and {22} neutral currents ?

gets then a simple answer

s 12 c 12 = c 2 12 -s 2 12 ⇒ tan(2θ 12 ) = 2. (31)
We did not take into account the terms proportional to ǫ because we assumed that the mass splittings between the first and second generations (from which the lack of unitarity originates) are much smaller that the ones between the second and the third generation 11 .

In the case of two generations, only ǫ appears, and one immediately recovers from (30c) and (30d) the condition fixing tan(2θ c ) = 1/2 for the Cabibbo angle.

Accordingly, the same type of requirement that led to a value of the Cabibbo angle for two generations very close to the observed value leads, for three generations, to a value of the first mixing angle satisfying the quark-lepton complementarity relation (3) [START_REF] Raidal | Prediction theta c + T heta sol = pi/4 from flavor physics: a new evidence for grand unification?[END_REF].

The values of θ 12 and θ 23 determined through this procedure are very close to the observed neutrino mixing angles [START_REF] Particle | Review of Particle Physics[END_REF].

Though we only considered the two equations that are a priori the least sensitive to our choice of a vanishing third mixing angle (which is not yet confirmed experimentally), it is instructive to investigate the sensitivity of our solution to a small non-vanishing value of θ 13 . This is done in Appendix C in which, for this purpose, we made the simplification θ13 ≈ θ 13 . It turns out that the terms proportional to s 13 in the two equations [12] = 0 = [21] and [START_REF] Gonzalez-Garcia | Phenomenology with massive neutrinos[END_REF] = [22] are also proportional to (c 2 23 -s 2 23 ), such that our solution with θ 23 maximal is very stable with respect to a variation of θ 13 around zero. This may of course not be the case for the other three equations, which are expected to be more sensitive to the value of θ 13 . 10 The limitation of this approximation also appears in the fact that (30b), of second order in η, is not compatible with (30e), which is of first order. 11 Since the three angles play a priori symmetric roles, the simultaneous vanishing of θ and θ, which we demonstrated for θ13 and θ13 (see Appendix A), should also occur for the other angles. Two competing effects accordingly contribute to the magnitude of the parameters ǫ, η . . . : on one hand, they should be proportional to (some power of) the corresponding θ, and, on the other hand, one reasonably expects them to increase with the mass splitting between the fermions mixed by this θ. So, in the quark sector, that the violation of unitarity should be maximal for θ13 is not guaranteed since the corresponding mixing angle is also very small (as expected from hierarchical mixing matrices [START_REF] Machet | Hierarchies of quark masses and the mixing matrix in the standard theory[END_REF]). A detailed investigation of this phenomenon is postponed to a further work. In the neutrino sector, however, since θ23 is maximal (large), the assumption that the mass splitting between the second and third generation is larger than between the first and second is enough to guarantee ǫ ≪ η.

Prediction for θ 13

We now consider, like we did for quarks, the general case θ 13 = 0 = θ13 (ρ = 0), θ12 = θ 12 (ǫ = 0), θ23 = θ 23 (η = 0), while assigning to θ 12 and θ 23 their values predicted in subsection 4.1.

We investigate the eight different relations between θ 12 , θ 23 and θ 13 which originate from the 2×2×2 possible sign combinations in the conditions (27) (the r.h.s. is now replaced by a condition F (θ 12 , θ 23 , θ 13 ) = 1 involving the three mixing angles), where each modulus can be alternatively replaced by "+" or "-". Among the solutions found for θ 13 , only two (up to a sign) satisfy the very loose experimental bound sin 2 (θ 13 ) ≤ 0.1.

(32)

They correspond respectively to the sign combinations (+/-/-), (+/+/+), (-/+/+) and (-/-/-) θ 13 = ±0.2717 , sin 2 (θ 13 ) = 0.072,

θ 13 = ±5.7 10 -3 , sin 2 (θ 13 ) = 3.3 10 -5 . ( 33
)
The most recent experimental bounds can be found in [START_REF] Gonzalez-Garcia | Phenomenology with massive neutrinos[END_REF]. They read

sin 2 (θ 13 ) ≤ 0.05, (34)
which only leaves the smallest solution in (33) 12 .

Future experiments will confirm, or infirm, for neutrinos, the properties that we have shown to be satisfied with an impressive accuracy by quark mixing angles.

Comments and open issues

How close are mixing matrices to unitarity? Mixing angles and mass spectrum

An important characteristic of the conditions that fix the mixing angles is that they do not depend on the strength of the violation of the two properties under scrutiny, namely, the absence of non-diagonal neutral currents and the universality of the diagonal ones in the space of mass eigenstates. Since only their ratio is concerned, each violation can be infinitesimally small. This is, on one side, fortunate since we have not yet been able to calculate the magnitude of the violation of the unitarity of the mixing matrices from, for example, mass ratios. The issue, for fundamental particles, turns indeed to be much more difficult conceptually than it was for composite particles like neutral kaons [START_REF] Machet | Binary systems of Neutral Mesons in Quantum Field Theory[END_REF].

But, on the other side, this blurs the relation between the mixing pattern and the fermionic mass spectrum 13 . This was already blatant with the emergence of maximal mixing as a special set of solutions of the unitarization equations in [START_REF] Duret | Mixing angles and non-degenerate coupled systems of particles[END_REF], which did not depend of any special type of mass hierarchy. The question now arises of finding, if any, properties of the mass spectrum, which are, through the products K † K of mixing matrices, compatible with an SU (2) symmetric pattern of weak neutral currents.

Which mixing angles are measured

The results that have been exposed are valid for fermions of both electric charges. They concern the mixing angles which parametrize * for quarks, the mixing matrix K u of u-type quarks as well as K d of d-type quarks; * for leptons, the mixing matrix K ν of neutrinos as well as that of charged leptons K ℓ , and we have shown that our approach allows to obtain on purely theoretical grounds the values of the mixing angles which are experimentally determined.

However, a problem arises : the measured values of the mixing angles are commonly attached, not to a single mixing matrix, e.g. K u or K d , but to the product K = K † u K d which occurs in charged currents when both quark types are mass eigenstates. Thus, in the standard approach, they are a priori related to an entanglement of the mixing angles of quarks (or leptons) of different charges. This problem gets easily solved by the following argumentation. Considering, for example, semi-leptonic decays of pseudoscalar mesons in the approach where one of the constituent quarks is "spectator", we show that only one of the two mixing matrices is involved and measured. Indeed, while the two-fold nature (flavor and mass) of the neutral kaons has always been acknowledged, this step has never been taken yet for other mesons. This is what we do now, in a rather naive, but efficient way, that consists of distinguishing a [q i,f q j,f ] "flavor" meson from the mass eigenstate [q i,m q j,m ] (q i,j being the constituent quarks). Consider for example, the decay K 0 → π -e + ν e . The K 0 that decay semi-leptonically being produced by strong interactions cannot be but a flavor meson [s f d f ], while its decay product π -, which is identified by its mass and charge, is a mass eigenstate [ū m d m ]. At the quark level, the weak transition occurs accordingly between a flavour eigenstate (s f ) to a mass eigenstate (ū m ), which only involves one mixing matrix, K u , and not the product K † u K d . As for the spectator quark, the transition from its flavor state d f to its mass state d m involves the cosine of the corresponding mixing angle, which is always close to 1. It thus appears that the mixing angles that are measured in such processes are the ones of K u or K d (up to a cosine factor very close to 1), which fits with our symmetric prediction.

The same problem is expected in the leptonic sector. Its solution depends on the nature of the neutrino eigenstates that are produced and detected. Let us consider for example the case of solar neutrinos. If the flux predicted in solar models concerns flavour neutrinos, and if the detection process also counts flavour neutrinos, the sole mixing matrix which controls their evolution and oscillations is K ν , because it is the only matrix involved in the projection of flavour states onto mass states. This is the most likely situation. Indeed, the production mechanism inside the sun occurs through nuclear beta decay, in which the protons and neutrons, being bound by strong forces, are presumably, together with their constituent quarks, flavour eigenstates. The detection (for example the transition from chlorine to argon) also occurs through nuclear (inverse) beta decay, which accordingly also counts the number of ν ef reaching the detector. The situation would be different if the comparison was made between the fluxes of the eigenstates ν e , ν µ , ν τ defined in subsection 2.1 (see also appendix D); since their projections on the mass eigenstates involve the product K † ℓ K ν , their oscillations are now controlled by an entanglement of the mixing angles of neutrinos and charged leptons.

A multiscale problem

Recovery of the present results by perturbative techniques (Feynman diagrams) stays an open issue. All the subtlety of the problem lies in the inadequacy of using a single constant mass matrix; because nondegenerate coupled systems are multiscale systems, as many mass matrix should be introduced as there are poles in the (matricial) propagator [START_REF] Novikov | Binary systems in QM and in QFT: CPT[END_REF] 14 .

The existence of different scales makes the use of an "on-shell" renormalized Lagrangian [START_REF] Kniehl | Simple On-Shell Renormalization Framework for the Cabibbo-Kobayashi-Maskawa Matrix[END_REF] hazardous, because each possible renormalization scale optimizes the calculation of parameters at this scale, while, for other scales, one has to rely on renormalization group equations.

Unfortunately, these equations have only been approximately solved with the simplifying assumption that the renormalized mass matrices are hermitian 15 and that the renormalized mixing matrices are unitary [START_REF] Kniehl | Simple On-Shell Renormalization Framework for the Cabibbo-Kobayashi-Maskawa Matrix[END_REF]. Performing the same job dropping these hypotheses looks rather formidable and beyond the scope of the present work. It also unfortunately turns out that, as far as the Yukawa couplings are concerned, the expressions that have been obtained at two loops for their β functions (which start the evolution only up from the top quark mass) [START_REF] Balzereit | The Renormalization Group Evolution of the CKM Matrix[END_REF] have poles in (m i -m j ), which makes them inadequate for the study of subsystems with masses below the top quark mass.

Using a q 2 -dependent renormalized mass matrix

Departure from the inappropriate Wigner-Weisskopf approximation can also be done by working with an effective renormalized q 2 -dependent mass matrix M (q 2 ). It however leads to similar conclusions as the present approach.

Its eigenvalues are now q 2 -dependent, and are determined by the equation det[M (q 2 ) -λ(q 2 )] = 0 16 . Let them be λ 1 (q 2 ) . . . λ n (q 2 ). The physical masses satisfy the n self-consistent equations q 2 = λ 1...n (q 2 ), such that m

2 1 = λ 1 (m 2 1 ) . . . m 2 n = λ n (m 2 n ). At each m 2 i , M (m 2
i ) has n eigenvectors, but only one corresponds to the physical mass eigenstate; the others are "spurious" states [START_REF] Machet | Binary systems of Neutral Mesons in Quantum Field Theory[END_REF]. Even if the renormalized mass matrix is hermitian at any given q 2 , the physical mass eigenstates corresponding to different q 2 belong to as many different orthonormal sets of eigenstates and thus, in general, do not form an orthonormal set. The discussion proceeds like in the core of the paper.

Determining the exact form of the renormalized mass matrix could accordingly be a suitable way to recover our predictions via perturbative techniques (like was done in [START_REF] Machet | Binary systems of Neutral Mesons in Quantum Field Theory[END_REF] for the quantitative prediction of the ratio ǫ S /ǫ L ). As already mentioned, the difficulty is that hermiticity assumptions should be dropped, which open the possibility of departing from the unitarity of the mixing matrix. This is currently under investigation.

Conclusion and perspective

This work does not, obviously, belong to what is nowadays referred to as "Beyond the Standard Model", since it does not incorporate any "new physics" such as supersymmetry, "grand unified theories (GUT)" or extra-dimensions. However it does not strictly lie within the SM either, even if it is very close to. Of course, it shares with the latter its general framework (mathematical background and physical content), and also borrows from it the two physical conditions of universality for diagonal neutral currents and absence of FCNC's, which play a crucial role in the process. But, on the basis of the most general arguments of QFT, we make a decisive use of the essential non-unitarity of the mixing matrices, whereas only unitary matrices are present in the SM. This property may be considered, in the SM, as an "accidental" characteristic of objects which are intrinsically non-unitary.

The mixing angles experimentally observed get constrained in the vicinity of this "standard" situation, a slight departure from which being due to mass splittings. Hence our approach can be considered to explore the "Neighborhood of the Standard Model", which is likely to exhibit low-energy manifestations of physics "Beyond the Standard Model".

While common approaches limit themselves to guessing symmetries for the mass matrices (see for example [START_REF] Ma | Lepton family symmetry and the neutrino mixing matrix[END_REF] and references therein), we showed that special patterns are instead likely to reveal themselves in the violation of some (wrongly) intuitive properties 17 . In each given (i, j) channel of mass eigenstates, the characteristic pattern that emerges is that two a priori different violations follow from a precise horizontal continuous symmetry, which is the most intuitive SU (2) group attached to this pair of fermions. One simply falls back on an, up to now unraveled, manifestation of "old Current Algebra" [START_REF] Gell-Mann | The Eightfold Way[END_REF]. It is remarkable that the same symmetry underlies both the quark and leptonic sectors, which was never suspected before; they only differ through the 0th order solution to the unitarization equations, the two-foldness of which was recently uncovered in [START_REF] Duret | Mixing angles and non-degenerate coupled systems of particles[END_REF]. We have in particular learned that symmetries relevant for flavour physics should not be looked for, or implemented, at the level of the mass matrices and Yukawa couplings, but at the level of the weak currents.

We have also argued that, unlike what is generally assumed, the mixing angles that are measured are (up to a cosine) the ones of a single mixing matrix, and not of the product K † u K d or K † ℓ K ν . Our scheme then appears entirely coherent, and agrees with experimental data.

To conclude, the present work demonstrates that flavor physics satisfies very simple criteria which had been, up to now, unnoticed. Strong arguments have been presented in both the quark and leptonic sectors, which will be further tested when the third mixing angle of neutrinos is accurately determined.

D Charged weak currents

Charged weak currents can be written in six different forms that are all strictly equivalent, but nonetheless refer to different physical pictures. As an example, for two generations of leptons :

  ν ef ν µf   W + µ γ µ L   e - f µ - f   =   ν em ν µm   W + µ γ µ L K † ν K ℓ   e - m µ - m   =   ν ef ν µf   W + µ γ µ L   K ℓ   e - m µ - m     =   K ν   ν em ν µm     W + µ γ µ L   e - f µ - f   =   ν em ν µm   W + µ γ µ L   K † ν   e - f µ - f     =   K † ℓ   ν ef ν µf     W + µ γ µ L   e - m µ - m   . (37)
In the case where one of the SU (2) partners, for example the charged lepton, is undoubtedly a mass eigenstate20 , the last expression of (37) shows that it is coupled to the so-called electronic and muonic neutrinos

ν e = (K † ℓ K ν ) 11 ν em + (K † ℓ K ν ) 12 ν µm = K † ℓ,11 ν ef + K † ℓ,12 ν µf , ν µ = (K † ℓ K ν ) 21 ν em + (K † ℓ K ν ) 22 ν µm = K † ℓ,21 ν ef + K † ℓ,22 ν µf . (38)
The latter are neither flavour eigenstates, nor mass eigenstates, but a third kind of neutrinos, precisely defined as the ones which couple to electron and muon mass eigenstates in the weak charged currents

  It is shown in Appendix B that the only solutions are θ 12 and θ 23 Cabibbo-like ( θ12,23 = θ 12,23 + kπ) or maximal (θ 12,23 = π/4 + nπ/2, θ12,23 = π/4 + mπ/2).

Fig. 1 :

 1 Fig. 1: θ 23 for quarks as a function of θ 12 ; simplified case θ 13 = 0 = θ13

1 =Fig. 2 :

 12 Fig. 2: θ 23 for quarks as a function of θ 12 ; neglecting terms quadratic in θ 13

Fig. 3 :

 3 Fig. 3: θ 23 for quarks as a function of θ 12 , general case. θ 13 = 0 (red), 0.004 (blue) and 0.01 (green)

B

  (θ 12 , θ 23 ) solutions of eqs. (9) (10) (11) (12) (13) for θ 13 = 0 = θ13 Excluding θ12 = 0, (15a) and (15b) require sin(2θ 23 ) = sin(2 θ23 ) ⇒ θ23 = θ 23 + kπ or θ23 = π/2 -θ 23 + kπ. • for θ23 = θ 23 + kπ Cabibbo-like, (15c) requires sin(2θ 12 ) = sin(2 θ12 ) ⇒ θ12 = θ 12 + nπ or θ12 = π/2 -θ 12 + nπ; (15d) requires cos(2θ 12 ) = cos(2 θ12 ) ⇒ θ12 = ±θ 12 + pπ; (15e) requires s 2

symmetries 2.1 The different basis of fermions

  Three bases will appear throughout the paper: * flavour eigenstates, that we note (u f , c f , t f ) and (d f , s f , b f ) for quarks, (e f , µ f , τ f ) and (ν ef , ν µf , ν τ f ) for leptons; * mass eigenstates that we note (u m , c m , t m ) and (d m , s m , b m ) for quarks, (e m , µ m , τ m ) and (ν em , ν µm , ν τ m )

2 Neutral currents of mass eigenstates and SU

[START_REF] Duret | Mixing angles and non-degenerate coupled systems of particles[END_REF]

  In the r.h.s. of (19), we have assumed that θ 12 is close to its Cabibbo value tan(2θ 12 ) ≈ 1/2. θ 23 is seen to vanish with [tan(2θ 23 ) -1/2]. The predicted value for θ 23 is plotted in Fig.

	12 ) = +1/2. (17) yields the ratio η/ǫ, that we then plug into the condition equivalent to (17) for
	the (2, 3) channel.
		|ηc 12 (c 2 23 -s 2 23 )| = |2ηs 23 c 23 (1 + c 2 12 ) -2ǫs 12 c 12 |.			(18)
	(17) and (18) yield
	tan(2θ 23 ) =	1 + c 2 12 -2s 12 c 12	c 12 (s 12 c 12 + c 2 12 -s 2 12 ) 4s 12 c 12 -(c 2 12 -s 2 12 )	≈	2 -	4 5	c 12 tan(2θ 12 ) -s 12 c 12	2 1	.	(19)

Since at any given q

, the set of eigenstates of the renormalized quadratic Lagrangian form an orthonormal basis, the mixing matrix with all its elements evaluated at this q 2 is unitary and the unitarity of the theory is never jeopardized.[START_REF] Duret | Mixing angles and non-degenerate coupled systems of particles[END_REF] For two generations, one is led to introduce two mixing angles to parametrize each 2 × 2 non-unitary mixing matrix.

Cabibbo-like angles can only be fixed by imposing conditions on the violation pattern of the unitarity of the mixing matrix in its vicinity.

"Channel (i, j)" corresponds to two fermions i and j with identical electric charge; for example, "channel (2, 3)" corresponds to (d, b), (c, t), (µ -, τ -) or (νµ, ντ ).

By doing so, we exploit the possibility to fix one degree of freedom left a priori unconstrained by the five equations; see subsection 2.2.

The (-) signs ensures that tan(2θ12) ≈ (+)1/2.

Our predictions substantially differs from the ones in[12], which mainly focuses on special textures for the product of the quark and neutrino mixing matrices[START_REF] Xing | Nontrivial correlation between the CKM and MNS matrices[END_REF].

A rigorous investigation of this connection was done in[START_REF] Machet | Hierarchies of quark masses and the mixing matrix in the standard theory[END_REF]. It however rests on the assumption (incorrect in QFT) that a system of coupled fermions can be described by a unique constant mass matrix, which is diagonalized by a bi-unitary transformation. Then the so-defined "fermion masses" are not the eigenvalues of the mass matrix, which makes all the more tricky the connection with the poles of the full propagator in QFT.

In QFT, as opposed to a Quantum Mechanical treatment (in which a single constant mass matrix is introduced -this is the Wigner-Weisskopf approximation-), a constant mass matrix can only be introduced in a linear approximation to the inverse propagator in the vicinity of each of its poles[START_REF] Machet | Binary systems of Neutral Mesons in Quantum Field Theory[END_REF]. When several coupled states are concerned, the (matricial) propagator having several poles, as many (constant) mass matrices should be introduced[START_REF] Novikov | Binary systems in QM and in QFT: CPT[END_REF]; only one of the eigenstates of each of these mass matrices corresponds to a physical (mass) eigenstate.

One can go to hermitian mass matrices by rotating right-handed fermions as far as they are not coupled; however, at two loops, the charged weak currents also involve right-handed fermions, which cannot be anymore freely rotated.

This is the simple case of a normal mass matrix, which can be diagonalized by a single (q 2 -dependent) unitary matrix. When it is non-normal, the standard procedure uses a bi-unitary diagonalization (see footnote 13).

For a (constant unique) mass matrix, unitarity of the mixing matrix has commonly been linked with the unitarity of the theory. See also footnote 1.

= 0

This is the case inside the sun[5] where, because of the limited available energy, only massive electrons can be produced, and also in the detection process of neutrinos on earth, which always proceeds via charged currents and the detection of produced physical (mass eigenstates) charged leptons.

Acknowledgments: Discussions with A. Djouadi, J. Orloff and M.I. Vysotsky are gratefully acknowledged.

Appendix

A θ13 = 0 ⇒ θ 13 = 0

Using the notations of section 2, we start with the following system of equations:

From equation (35a), we have c 2 23 + s2 23 = 0, which entails c 2 13 = 0 18 . Let us study the consequence on the two equations (35b) and (35c).

• the two sides of (35b) vanish for cos(2θ 12 ) = 0 = cos(2 θ12 ), i.e.