Alexandra Vignolles
email: avignolles@groupeinseec.com

Paul-Emmanuel Pichon

A taste of nostalgia: links between nostalgia and food consumption

Keywords: nostalgia, food consumption, typology, qualitative study. Article classification: Research paper

Purpose:

The purpose of this article is to analyze the links between nostalgia and food consumption.

Design/methodology/approach: Based on an exploratory qualitative analysis of 100 respondents, the use of NVivo helps to substantiate and interpret the textual interview data.

Findings: This article suggests six themes of food nostalgia and a typology of food nostalgia consumptions.

Practical implications: Perspectives are given for the use of nostalgia in an advertising context and as a reinsurance factor.

Food consumption is the one area most likely to take people back into their past. It calls on different senses, so it can be a valuable vector for nostalgia. The link between nostalgia and food consumption is almost obvious and yet, very little research work has been conducted so far on their links [START_REF] Baker | My favourite recipes: recreating emotions and memories through cooking[END_REF]. That is why our paper aims at studying the links existing between nostalgia and food consumption. We particularly mean to draw possible parallels between the sociological mechanisms at work in food consumption and in nostalgia and to study how consumers' loss of confidence in modern food can be minimized if the brand, the product or the communication are granted a nostalgic touch.

As part of the research, a first exploratory study was conducted with 300 persons interviewed on the subject of nostalgic consumption. Out of the 300 answers, over a 100 allude to food consumption. Thus, a majority of consumers seem to establish a link between nostalgia and food. We shall see that it goes beyond the «Proust's little madeleine » effect, that is travelling back through a product into one's childhood memories with elation, but also sometimes with regret or with a bittersweet emotional reaction. What are the main features of the nostalgic food experience? Could experiencing nostalgia be a valuable vector to reassure consumers when purchasing? Can we talk about food nostalgia? Our research has many different objectives.

The first part of our paper will be devoted to elucidate the concepts related to food consumption and nostalgia. In the second part, we will bring into perspective the contribution of sociology to food consumption and nostalgia and we will highlight the common features of food behavior and nostalgia. In a third part, we will present the results of the exploratory qualitative study, especially those concerning the characterization and typology of nostalgic food consumption.

FOOD CONSUMPTION AND NOSTALGIA : ELUCIDATING THE CONCEPTS

This first part presents and elucidates the concepts used in this research work. Indeed, one must understand to what extent the current food context is troubled and how nostalgia can be a relevant concept to comfort consumers in their purchasing and consumption activities as well as in their everyday life.

Food consumption: a troubled context

In the last thirty years, food crises have often repeated themselves at a national and international level. Food and the ensuing risks are of great concern to consumers. According to a recent survey by the CREDOC -food behaviour and consumption (September 2005) 85% of the respondent households think that their food pattern has an impact on their health as against 79% in 2000 and 75% in 1997. So, the problems linked to food hazards have been of great interest to marketing researchers in the last few years (e.g. [START_REF] Kapferer | Les marques, base de la confiance ?[END_REF][START_REF] Sirieix | La consommation Alimentaire : Problématiques, Approches et Voies de Recherche[END_REF][START_REF] Miles | Investigating specific concerns about different food hazards[END_REF][START_REF] Holbrook | Nostalgic Bonding: Exploring the role of nostalgia in the consumption experience[END_REF].The latest research work consider food hazard as a vital consumption hazard, and as a current trend in our western society [START_REF] Bergadaà | Le risque alimentaire comme risque vital perçu de consommation : émergences, adaptation et gestion[END_REF].

The list of possible threats is long: BSE, the foot and mouth disease, the link between food patterns and heart diseases… Modern food is a source of deep anxiety; as a result consumers no longer trust what they eat. The fear of choosing the wrong product triggers an anxiety and a strong feeling of guilt [START_REF] Rozin | Attitudes to Food and the Role of Food in Life in the U.S.A, Japan, Flemish Belgium and France: Possible Implications for the Diet-Health Debate[END_REF]. This feeling of discomfort, anxiety and the loss of confidence in food products can be accounted for by four main factors [START_REF] Poulain | Les nouvelles pratiques alimentaires, entre commensalisme et vagabondage[END_REF]: lack of identification of products, food abundance, contradictory statements and the influence of the media. Some nutrients of the agri-food industry arouse suspicion and doubt in the consumers' minds whereas traditional and artisanal food is considered as safe. The consumers looking for authenticity in the origin, in the symbolic dimension and unicity of a product are presented with a dilemma: convenience and cheap food on the one hand and the fear resulting from the fact that very little is known on this type of food. As a concept of reassurance, nostalgia and its impact on consumer behavior deserve closer scrutiny. In the next paragraph, we will present its main features.

Nostalgia and food behaviour

The word "nostalgia" originates from the Greek words nostos (homecoming) and algos (ache); it first described the homesickness caused by geographical distance. The concept has long been part of the history of mankind, since it was already mentioned in Biblical books or in Greek mythology (Homer, the Odyssey).

The first research work about the concept of nostalgia was carried out in the field of medicine by [START_REF] Hofer | Medical dissertation on nostalgia[END_REF]. His thesis gives a clinical description of the physical and psychological symptoms of nostalgia. Gradually, the meaning of the word changed. Today, nostalgia is no longer considered as a medical disease, but commonly regarded as yearning for yesterday.

This concept has been of interest to marketing researchers for several years (e.g. [START_REF] Holbrook | Echoes of the Dear Departed Past: Some Work in Progress on Nostalgia[END_REF][START_REF] Holak | Nostalgia: An Exploratory Study of Themes and Emotions in the Nostalgia Experience[END_REF]. Hence, the definition proposed by several marketing and psychology researchers, as shown by the synthesis below (table 1: main definitions of nostalgia). Nostalgia is alternately considered as an emotion, a mood, a preference, in other words as an emotional reaction.

Table 1: Definitions of nostalgia

Nostalgia is necessary to the human being for different reasons, positive or negative functions.

According to [START_REF] Sedikides | Nostalgia: Conceptual Issues and Existential Functions, Handbook of Experimental Existential Psychology[END_REF], nostalgia fulfill existential functions and acts «as a stock of emotions and experiences which people resort to in order to cope with their existential fears ». Exploring the complexity and richness of the concept of nostalgia can provide additional tools to better understand consumers and purchasers' behavior.

Many agri-food businesses resort to products, design, packaging, outlets or communication based on the concept of nostalgia. As for packaging, the Bonne Maman jams with their Vichy patterned lids remind us of home-made jams. Other products recall an "à l'ancienne" design even if the recipe or the product is not at all old. The Paul bakeries gather all the necessary ingredients -furniture, employees' clothing, and smell of freshly baked bread -to evoke nostalgia of times past.

So, if we agree that nostalgia helps to reassure people, the study of its links with food consumption should be promising. Food consumption is largely characterized by risk-taking which is more often subjective than objective. Even if food hazards are objectively less nowadays, the general public has got the feeling that these hazards are more severe [START_REF] Kapferer | Les marques, base de la confiance ?[END_REF].Nostalgia can thus be a reassuring factor for the consumers.

After presenting the troubled context of food consumption and the relevance of studying nostalgia as a variable affecting consumers' behaviour, the second part of the paper will study the links between nostalgia and food consumption from a sociological point of view.

A SOCIOLOGICAL APPROACH OF THE LINKS BETWEEN FOOD

CONSUMPTION AND NOSTALGIA

Many researchers have attempted to understand the psychological and sociological mechanisms at work within consumers confronted all along their life and several times a day to feeding necessities. All this fundamental research work (e.g. [START_REF] Aron | Le mangeur du XIXème[END_REF][START_REF] Barthes | Pour une psychosociologie de l'alimentation contemporaine[END_REF][START_REF] Bourdieu | Distinction: A social critique of the judgement taste[END_REF][START_REF] Chiva | Le doux et l'amer[END_REF][START_REF] Fischler | L[END_REF] whether in sociology, general anthropology or anthroposociology, have emphasised the complexity of food activities. All these approaches are valuable contributions to marketing and consumers' behaviour research.

Nostalgia can be defined as "a bittersweet emotional reaction, that may be associated to reflection and experienced by an individual when external or internal stimuli take him back to an ideal past moment or event belonging or not to his living experience" [START_REF] Divard | La nostalgie: un thème récent dans la recherche en marketing[END_REF]. The emotions emerging from an idealised past (features inherent to nostalgia) are focused on inanimate objects, sounds, odours or tastes that have been discovered at the very moment the emotions were experienced [START_REF] Hirsch | Nostalgia: A Neuropsychiatric Understanding[END_REF]. The object that has been actually discovered or its substitute has the ability to revive emotions. This can be the case for a food product that has been really consumed or for its substitute (the Bonne Maman jams for some consumers whose parents or grandparents used to make jams) or for a product dating back to an idealised past.

One can draw parallels between the sociological functions of food consumption and those of nostalgia. They share in common a social and cultural function, they contribute to strengthening self-identity and they are both ambivalent. Even if these «common points » between nostalgia and food behaviour could also characterise other themes, it is relevant to describe the way they converge. [START_REF] Halbwachs | La classe ouvrière et les niveaux de vie, Recherche sur la hiérarchie des besoins dans les sociétés industrielles et contemporaines, Alacan, rèd[END_REF] was one of the first sociologists to analyze meals as "a social institution playing a fundamental part in socialising and transmitting the norms and values of social groups". Similarly, nostalgia regenerates and sustains a whole sense of cultural meanings: this function is fulfilled by identifying and supporting a common cultural worldview (Sedikides et al., 2005). Both nostalgia and food practices tend to increase people's feeling of belonging to a culture [START_REF] Baumeister | The need to belong: Desire for interpersonal attachments as a fundamental human motivation[END_REF]. Remembering people who were a meaningful part of our past strengthens our feeling of belonging to a social group and invigorates connectedness within the group.

A social and cultural act

Nostalgia establishes a symbolic connection with people or events from the past and consumption objects are significant evidence of the past [START_REF] Belk | The Role of Possessions in Constructing and Maintaining a Sense of Past[END_REF], even if the past is experienced indirectly [START_REF] Stern | Historical and personal nostalgia in advertising text: the Fin de siecle effect[END_REF].These consumption objects, such as family recipes or specific food products, can stir up nostalgic emotions and memories. [START_REF] Baker | My favourite recipes: recreating emotions and memories through cooking[END_REF] have described the nostalgic responses and associations related to various consumers' favourite recipes. The results show that these recipes are associated with rituals performed on feast days, significant events, birthdays, new seasons or recovering from illnesses. Recipes contribute to structuring families and support the intergenerational transmission of the knowhow.

After demonstrating that nostalgia and food practices are social and cultural acts, we are going to explain to what extent these two concepts contribute to the attainment of identity. Nostalgia strengthens and augments selfhood: it reduces sources of uncertainty, increases one's ability to deal with the present and restores self-worth by resorting, at least momentarily, to an idealised past [START_REF] Kleiner | On nostalgia[END_REF][START_REF] Kaplan | The psychopathology of nostalgia[END_REF].

Identity Attainment

Quite consistently with the part played by nostalgia in the attainment of identity, [START_REF] Baker | My favourite recipes: recreating emotions and memories through cooking[END_REF] show that family recipes are packed with many elements of selfhood. Favourite recipes are often related to happy or intense childhood memories, such as birthdays or time spent with people they loved. A food activity or a mere cooking smell stirs up memories and creates nostalgic associations. Just like these dear objects either owned or inherited [START_REF] Curasi | In Hope of an Enduring Gift: The Intergenerational Transfer of Cherished Possessions: A Special Case of Gift Giving[END_REF][START_REF] Price | Older Consumer's Disposition of Special Possessions[END_REF], recipes are left to the people we love, hoping they will also love them and will be reminded of special moments.

Nostalgic links and memories can be conveyed by food practices, whose symbolic function contributes to shaping families, increasing intergenerational links, strengthening selfhood or transmitting rituals. Food practices help bring back memories and past experiences through nostalgic bonds conveying the memory of people, places or events. [START_REF] Kessous | A semiotic analysis of nostalgia as a connection to the past[END_REF] point out that food products are particularly likely to stir nostalgic remembrances, especially from one's childhood.

A further feature shared by both nostalgia and food consumption can account for their similarity, that is the ambivalence existing between their positive and negative aspects.

Ambivalence

The consumers' complex behaviour concerning food may be characterized by their three ambivalent attitudes when eating, as described by [START_REF] Beardsworth | The management of food ambivalence: Erosion and Reconstruction?[END_REF]: ambivalent pleasuredispleasure approach, ambivalent health-disease approach, and life-death approach. The ambivalent pleasure-displeasure approach relies on taste-distaste. Indeed, eating can be a source of pleasure and joy, but may also cause distaste to the extent of vomiting. [START_REF] Poulain | Sociologies de l'alimentation[END_REF] considers that "this ambivalence accounts for the fact that food may be both a source of content and intense sensual pleasure but may also cause a range of unpleasant sensations, from mere unpleasantness to revolting disgust responsible for sickness, and even vomiting.

So, anxiety has a sensory and hedonistic component".

Similarly, nostalgia is a bitter-sweet emotional reaction which can stir sadness or yearning as well as joy and solace. Hence depending on the occasion, this reaction can be positive or negative or both at the same time. Nostalgia is considered as a bittersweet emotion and as [START_REF] Dickinson | Nostalgia and Autobiography: The Past in the Present[END_REF] remind it: "Below the surface of the idealized memory of nostalgia there may be hidden conflicts, a point that may explain some of "the bitter" component of nostalgia".

The second part of our paper has emphasized the common points between food consumption and nostalgia from a theoretical point of view. The following exploratory qualitative study will definitely confirm the part played by nostalgia in food consumption.

EXPLORATORY QUALITATIVE STUDY OF NOSTALGIC FOOD CONSUMPTION ACTIVITIES

A qualitative study of the influence of nostalgia on consumer behavior has been performed.

Three hundred interviewees aged 16 to 84, half males, half females have been asked to complete a self-administered questionnaire. The respondents had to answer the following two questions with as many details as possible:

1. Describe a situation in which you have experienced nostalgia and describe all the emotions you have felt on this occasion 2. Describe a nostalgic situation associated with a consumption activity and all the related emotions you have experienced

We have tried to better understand individual perception, the situations and the emotions related to nostalgia as well as the recurring nostalgic consumption activities. Out of the 224 descriptions of nostalgic consumptions, approximately half of them (46%) deal with food consumption. In the next paragraph, we are going to describe the themes mentioned by the respondents and we are going to illustrate them with verbatim. In order to help analyse the qualitative data, we have used a computer-assisted analysis NVivo 2.0.

Themes characterizing nostalgic food consumption

The analysis of the results has pointed to six main themes:

-Food products related to childhood; -Yearning for products;

-Substitute products;

-Products related to homesickness;

-Products for special occasions; -Rediscovering products;

Nearly half of these food consumption activities involve food products related to childhood. Next comes the yearning for products which are no longer available or whose production has changed. This feeling is mostly found with people over 60 who think that things were better before: François (67 years old) points out «You no longer find our childhood sweets, which makes me feel nostalgic, it was better before », Maurice (75 years old) says: «When I buy tomatoes, I really yearn for those you could find 30 years ago. They were juicy fruit that you could eat just like that. I really feel sad, but also angry because today no product is genuine, everything is artificially redesigned». All these people are yearning for their past, which corresponds to the bitter dimension of nostalgia. This finding is in line with Davis' view (1979) who considers that elderly people find solace in the past to get comfort from their shrinking prospects.

Other people very well know that the products consumed are just copies of what they used to have; still, they remind them of happy times. These products act as substitutes filling a gap.

Sometimes, these substitutes replace products that people can no longer have because the person who used to prepare or cook them has gone away or are deceased. As Frederic reminds (28 years old) «I always feel nostalgic when I buy some Bonne Maman jam because it reminds me of my mother who used to give us bread with jam and we were always in such a hurry that we couldn't wait to sit down to eat them. Happy because the memory made me smile but also sad because my mother no longer prepares my breakfast ». Daniel (37 years old) declares: «Whenever I have some frozen moussaka, it always reminds me of the dish my mother and aunt used to cook, but it has nothing to do with theirs. I sadly miss the people who are no longer there ». We are dealing here with the nostalgia for family, family recipes and time spend with family rather than for brand products.

Although nostalgia is mainly caused by time elapsing [START_REF] Divard | La nostalgie: un thème récent dans la recherche en marketing[END_REF], it may also be due to homesickness. Some respondents who have left their homeland feel nostalgia because they cannot find in their host country the products of their country of Lastly, some respondents associate the nostalgic experience with a food product that they had not had for long and that they rediscover. Just like Laurent (29 years old): «Indeed! When, as an adolescent, I discovered that my childhood cookies were no longer sold, I felt so sad! And yet, now they're back into the shops, these Quaddro biscuits that my Dad used to buy me when I was in nursery school. It made me feel both nostalgic and happy as it reminded me of my childhood. I have bought some, I really felt like having some again ». Rediscovering a product long forgotten makes it easier for the consumers to buy it because they are willing to recreate the emotions experienced in the past and their associated memories. Juliette (27 years old) says «I was looking for sweets for my nephew when I came across some of the sweets I used to eat as a child and which I had not seen ever since. I thought those Frizzy Panzy lollipops that looked like tongues that you would dip into a bitter kind of powder were no longer for sale. I was overwhelmed with a feeling of well-being, from so far away, I could almost feel the taste. I bought two right away, one for me, one for my nephew ».These answers definitely support the empirical study by [START_REF] Sierra | Attitudes and emotions as determinants of nostalgia purchases: an application of social identity theory[END_REF] who investigate the determinants of nostalgic purchasing acts. Their results show that nostalgic consumption practices are influenced by an attraction for the past and a specific attitude in this respect.

Nostalgic food consumption activities described by the respondents involve branded products (38%) and unbranded products (62%) to a larger extent. People can travel back into the past thanks to unbranded products attached to some period of their childhood, for instance homemade cakes and pies but even pasta that are closely related to personal moments.

The design of the product more than its taste seems to evoke a feeling of nostalgia which once more triggers the will to buy or the purchasing act itself: «Banania re-marketed in limited numbers the original metal box of my childhood, which stirred up the memory of the chocolate my mother would give me and some yearning for time that flies so quickly, that time, long ago, when I used to live at my parents'» (Gregory, 30 years old). In some other cases, a long-lasting packaging will trigger the purchasing act much more readily than a revamping of the initial packaging: «One of my childhood products has kept almost the same packaging, symbols, colours and logos: the Nestlé milky sweets. I sometimes buy some, just to taste back childhood flavours. It's such an enjoyable feeling of being comfortable and safe » (Fabrice, 25 years old). These two elements, back to original packaging or long-lasting characteristics may well reassure consumers. After defining the themes inherent to food nostalgia, we are going to review the different types of food nostalgia referred to in our research as well as described in literature.

Typology of food nostalgia

Through the respondents' answers, three different types of food nostalgia, in-keeping with the definition of nostalgia, emerge. All researchers agree on a definition of nostalgia as being an ambivalent emotional reaction [START_REF] Belleli | Une émotion ambiguë : la nostalgie[END_REF][START_REF] Holbrook | Echoes of the Dear Departed Past: Some Work in Progress on Nostalgia[END_REF][START_REF] Batcho | Nostalgia: A Psychological Perspective[END_REF][START_REF] Divard | La nostalgie: un thème récent dans la recherche en marketing[END_REF]. Indeed, nostalgia may be defined as a sweet, bitter or bittersweet emotional reaction. Through nostalgia, people may travel back to some specific past time and their feeling of nostalgia may trigger emotions or feelings experienced then. The respondents' answers have been encoded into three categories and defined as:

-Positive food nostalgia (sweet): a food product is linked to positive memories and to emotional reactions such as joy, happiness, comfort or peacefulness -Negative food nostalgia (bitter); a food product is linked to negative memories and to emotional reactions such as regret, sadness or even anger.

-Ambivalent food nostalgia (bittersweet): a food product is linked to both positive and negative memories, emotional reactions are ambivalent and imply contradictory emotions such as joy and sadness.

Most answers belong to the positive food nostalgia (51%) class. The products mentioned are products that people have not consumed for long, sometimes since childhood and which evoke happy moments. Most of the time, people have bought these products again to travel back into their memories. Sweets, biscuits, cakes, jam are the most often quoted products whatever the age of the respondents. Sweet food is associated to the sweet pleasures of childhood. To a lesser extent, some salted products are quoted, mainly by elderly respondents or in connection with homesickness.

In decreasing order of importance, food consumption linked to ambivalent food nostalgia comes next (33%

CONCLUSION

This research highlights the significant part played by the concept of nostalgia in food consumption. This close bond between food and nostalgia can be accounted for by two factors. First of all, food consumption addresses our five senses, and more particularly the senses of smell and taste which imprint the consumers' mind with strong memories, hence their bond with nostalgia. According to [START_REF] Hirsch | Nostalgia: A Neuropsychiatric Understanding[END_REF], the sense of smell has the deepest imprint. On the other hand, eating is a complex social act and consumers who, today, lack traditional landmarks can be troubled by the great number of food crises. Many researchers stress that nostalgia contributes to people's feeling of safety, as a food product which has already been consumed and is associated with positive memories reduces consumers' risktaking experience [START_REF] Hirsch | Nostalgia: A Neuropsychiatric Understanding[END_REF][START_REF] Stern | Historical and personal nostalgia in advertising text: the Fin de siecle effect[END_REF][START_REF] Baker | Death by Nostalgia: A Diagnostic of Context-Specific-Cases[END_REF][START_REF] Holbrook | Nostalgic Bonding: Exploring the role of nostalgia in the consumption experience[END_REF].

Concerning theoretical contributions, this first exploratory work characterises nostalgic food consumption through 6 themes: childhood, yearning, substitution, homesickness, special occasions and rediscovery. Besides, three types of food nostalgia consumption emerge:

positive, negative and ambivalent nostalgia. Respondents mostly express positive nostalgia.

Hence, the food processing industry should definitely rely on this type of nostalgia to promote their products. The typology of food nostalgia supports the theory of emotional transfer according to which the advertising content may influence consumers' behavior concerning the brand, the product and purchasing intention [START_REF] Bettman | Memory Factors in Consumer Choice: A Review[END_REF]Stayman and Aaker, 1998;[START_REF] Muehling | The Power of Reflection: An Empirical Examination of Nostalgia Advertising Effects[END_REF]. A product which has remained the same throughout its life cycle can be an indication of consumers' long-lasting preferences. Childhood memories will most certainly be used for a better understanding of what brands mean to the consumers and the way they relate to them [START_REF] Braun-Latour | Using Childhood Memories to Gain Insight into Brand Meaning[END_REF].

Managerial applications concern a variety of fields: communication strategies of food brands, product design and packaging. In the field of communication strategies, one or several themes characterising food nostalgia can be used to promote products. The use of childhood, happy family times and of intergenerational transmission is particularly suitable for the communication strategies of food brands. Communication can support the notion of intergenerational transmission of «good products » by emphasizing the nostalgic touch related to childhood. When it comes to the design of products and packaging, consumers pay a particular attention to products and packaging which last over time and thus act as reassuring landmarks. Therefore, brands should be very careful when it comes to logo or packaging changes.

Yet, this preliminary research work has some limitations which may due to the fact that the qualitative exploratory study may have failed to identify all the features characterizing food nostalgia despite the diversity and size of our sample. Conducting a larger scale study and checking whether its results substantiate our preliminary comments and conclusions opens new avenues for future qualitative and quantitative research.

For

  instance, Patrice (32 years old) said «In a supermarket, I came across the Balisto chocolate bars that used to be my snack when in primary school. They are sweet memories of innocence and relying on my parents. », or Céline (28 years old) «Upon seeing marshmallows in a supermarket, I felt a mixture of joy and sadness ». These memories may also convey time spent in the family, more particularly with the mother or the grandmother. Martine (35 Years old) remembers: «While I was purchasing a pack of biscuits, I stopped in front of the Lu strawberry cookies, because these were what my mother used to give me for my tea. I felt both joy and yearning». Similarly, Claire (22 years old) is aware of her nostalgic purchasing behaviour: «Whenever I buy some Benco, it reminds me of my childhood, when my mother would give us, my cousin and I, bread with butter and Benco, that we would eat under the veranda while playing cards. What I feel is a mixture of comfort, happiness, but at the same time of yearning for this time gone past ». The respondents' accounts clearly show the importance of rituals in nostalgic food consumption. These rituals imply a common place, a precise time and the presence of people around (family or friends).

  origin. Soraya (20 years old) explains: «I am from Morocco, I go back to my parents' once a year, but in the meantime in France, I keep buying Garbit couscous, even if it is not very trendy. It comforts me while waiting to go back ». Yves (49 years old) wrote: «When I was living in the USA, I would sometimes go to a bakery which sold croissants, which was not very common over there. It reminded me of the good French bakeries and I would always feel both happy and proud. ». These nostalgic recollections are quite similar to nostalgia as defined by[START_REF] Hofer | Medical dissertation on nostalgia[END_REF] who considered it as equivalent to homesickness. Some products are consumed on special occasions. As an example, the respondents mention the products they have on feast days or on special occasions, just like what René (78 years old): «The "pain au chocolat" was only bought on one special occasion, which was the annual visit to the eye specialist in Paris with my parents; it was exceptional and that brought intense pleasure ». These scarce moments are engraved in people's minds and can be idealized in their autobiographic memory but they are always associated with intense and highly emotional occasions, triggering precise recollections such as the date, the name of the persons attending the event or the emotions experienced.

  ). Products recall nice memories but are kind of blurred with a feeling of sadness or regret because those times are over.«I always feel nostalgic when I buy some Bonne Maman jam because it reminds me of mymother who used to give us bread with jam and we were always in such a hurry that we am nostalgic of the taste of the old «flan » When I eat one today, I feel disgusted, fed up because it no longer tastes the same » (Denis, 65 years old). The food products, even if they've been enjoyed in the past, can no longer be seen in a positive light because they are associated to people who are no longer present or a too distant past.

couldn't wait to sit down to eat them. Happy because the memory made me smile, but also sad because my mother no longer prepares my breakfast » (Nawel, 27 years old). This ambivalent nostalgia is similar to what

[START_REF] Batcho | Nostalgia: A Psychological Perspective[END_REF]

has reported in her work, that is a lack, something missing vis-à-vis people, feelings or objects. Lack has thus been described as being part of ambivalent nostalgia, for instance «When I went into a sweetshop, I found those chewing-gum tubes that I hadn't seen for so long. It reminded me of nursery and primary school where you had to share them with all our friends. I was happy and at the same time I missed the simple pleasures of my generation» (Carole, 30 years old). These answers clearly illustrate the bittersweet feature of ambivalent food nostalgia.

Lastly, negative food nostalgia (16%) when people evoke the yearning for a product or a brand no longer available or produced in a different manner. This feeling is linked to yearning for savors of yore and also to distaste for current products. Reactions can be very strong. «I


P Lugan

A Aspect

L Sanchez-Palencia

D Delande

B Grémaud

C A Müller

C Miniatura

One-dimensional Anderson localization in certain correlated random potentials

Keywords: numbers: 03.75.-b, 42.25.Dd, 72.15.Rn

We study Anderson localization of ultracold atoms in weak, one-dimensional speckle potentials, using perturbation theory beyond Born approximation. We show the existence of a series of sharp crossovers (effective mobility edges) between energy regions where localization lengths differ by orders of magnitude. We also point out that the correction to the Born term explicitly depends on the sign of the potential. Our results are in agreement with numerical calculations in a regime relevant for experiments. Finally, we analyze our findings in the light of a diagrammatic approach.

I. INTRODUCTION

Anderson localization (AL) of single electron wave functions [1], first proposed to understand certain metalinsulator transitions, is now considered an ubiquitous phenomenon, which can happen for any kind of waves propagating in a medium with random impurities [START_REF] Akkermans | Mesoscopic Physics of Electrons and Photons[END_REF][START_REF] Van Tiggelen | Wave Diffusion in Complex Media[END_REF]. It can be understood as a coherent interference effect of waves multiply scattered from random defects, yielding localized waves with exponential profile, and resulting in complete suppression of the usual diffusive transport associated with incoherent wave scattering [START_REF] Lee | [END_REF]. So far, AL has been reported for light waves in diffusive media [5,6] and photonic crystals [7,8], sound waves [9], or microwaves [10]. Ultracold atoms have allowed studies of AL in momentum space [11,12] and recently direct observation of localized atomic matter waves [13,14].

In one-dimensional (1D) systems, all states are localized, and the localization length is simply proportional to the transport mean-free path [15]. However, this strong property should not hide that long-range correlations can induce subtle effects in 1D models of disorder, in particular those whose power spectrum has a finite support [16,17]. Examples are random potentials resulting from laser speckle and used in experiments with ultracold atoms [13,18,19]. Indeed, by construction [START_REF] Goodman | Speckle Phenomena in Optics: Theory and Applications[END_REF], speckles have no Fourier component beyond a certain value 2k c , and the Born approximation predicts no back-scattering and no localization for atoms with momentum k > k c . This defines an effective mobility edge at k = k c [17], clear evidence of which has been reported [13].

Beyond this analysis -relevant for systems of moderate size [13,17]-study of AL in correlated potentials beyond the effective mobility edge requires more elaborated approaches. In Ref. [START_REF] Tessieri | [END_REF], disorder with symmetric probability distribution was studied, and examples were exhibited, for which exponential localization occurs even for k > k c although with a much longer localization length than for k < k c . It was also concluded that for Gaussian disorder, there is a second effective mobility edge at 2k c , while for non-Gaussian disorder, it is generally not so. These results do not apply to speckle potentials whose probability distribution is asymmetric. Moreover, although speckle potentials are not Gaussian, they derive from the squared modulus of a Gaussian field, and, as we will show, the conclusions of Ref. [START_REF] Tessieri | [END_REF] must be reexamined. Hence, considering speckle potentials presents a twofold interest. First, they form an original class of non-Gaussian disorder which can inherit properties of an underlying Gaussian process. Second, they are easily implemented in experiments with ultracold atoms where the localization length can be directly measured [13]. In this work, we study AL in speckle potentials beyond the Born approximation, using perturbation theory [22], numerical calculations, and diagrammatic methods. We find that there exist several effective mobility edges at k (p) c = pk c with integer p, such that AL in the successive intervals k (p-1) c < k < k (p) c results from scattering processes of increasing order. Effective mobility edges are thus characterized by sharp crossovers in the k dependence of the Lyapunov exponent (see Fig. 1). We prove this for the first two effective mobility edges by calculating the three lowest-order terms, and give general arguments for any p. In addition, we discuss the effect of odd terms that appear in the Born series due to the asymmetric probability distribution of speckle potentials.

II. SPECKLE POTENTIALS

Let us first recall the main properties of speckle potentials. Optical speckle is obtained by transmission of a laser beam through a medium with a random phase profile, such as a ground glass plate [START_REF] Goodman | Speckle Phenomena in Optics: Theory and Applications[END_REF]. The resulting complex electric field E is a sum of independent random variables and forms a Gaussian process. In such a light field, atoms experience a random potential proportional to the intensity |E| 2 . Defining the zero of energies so that V = 0, the random potential is thus

V (z) = V R × |a(z/σ R )| 2 -|a(z/σ R )| 2 (1)
where the quantities a(u) are complex Gaussian variables proportional to the electric field E, and σ R and V R feature characteristic length and strength scales of the random potential (The precise definition of V R and σ R may depend on the model of disorder; see below). In contrast, V (z) is not a Gaussian variable and its probability distribution is a decaying exponential, i.e. asymmetric. The sign of V R is thus relevant and can be either positive or negative for "blue"-and "red"-detuned laser light respectively. However, the random potential V (z) inherits properties of the underlying Gaussian field a(u). For instance, all potential correlators c n are completely determined by the field correlator c a (u) = a(0) * a(u) via

a * 1 ...a * p × a 1 ...a p = Π a * 1 a Π(1) ... a * p a Π(p) , (2)
where

a p ′ = a(z p ′ /σ R ) and Π describes the p! permuta- tions of {1, ..., p}. Hence, c 2 (u) = |c a (u)| 2 and defining a(u) so that |a(u)| 2 = 1, we have V (z) 2 = |V R |.
Also, since speckle results from interference between light waves of wavelength λ L coming from a finite-size aperture of angular width 2α, the Fourier transform of the field correlator has no component beyond k c = 2π sin α/λ L , and c a has always a finite support:

ĉa (q) = 0 for |q| > k c σ R ≡ 1. (3)
As a consequence, the Fourier transform of the potential correlator also has a finite support: ĉ2 (q) = 0 for |q| > 2.

III. PHASE FORMALISM

Consider now a particle of energy E in a 1D random potential V (z) with zero statistical average [V (z) need not be a speckle potential here]. The particle wave function φ can be written in phase-amplitude representation

φ(z) = r(z) sin [θ(z)] ; ∂ z φ = kr(z) cos [θ(z)] , (4)
which proves convenient to capture the asymptotic decay of the wave function (here k = 2mE/ 2 is the particle wave vector in the absence of disorder). It is easily checked that the Schrödinger equation is then equivalent to the coupled equations

∂ z θ(z) = k 1 -(V (z)/E) sin 2 (θ(z)) (5) ln[r(z)/r(0)] = k z 0 dz ′ (V (z ′ )/2E) sin (2θ(z ′ )) .( 6
)
Since Eq. ( 5) is a closed equation for the phase θ, it is straightforward to develop the perturbation series of θ in increasing powers of V . Reintroducing the solutions at different orders into Eq. ( 6) yields the corresponding series for the amplitude r(z) and the Lyapunov exponent:

γ(k) = lim |z|→∞ ln[r(z)] |z| = n≥2 γ (n) (k). ( 7
)
The nth-order term γ (n) is thus expressed as a function of the n-point correlator

C n (z 1 , ..., z n-1 ) = V (0)V (z 1 )...V (z n-1 ) of the random potential, which we write C n (z 1 , ..., z n-1 ) = V n R c n (z 1 /σ R , ..., z n-1 /σ R ). Up order n = 4, we find γ (n) = σ -1 R ǫ R kσ R n f n (kσ R ) (8)
where

ǫ R = 2mσ 2 R V R / 2 and f 2 (κ) = + 1 4 0 -∞ du c 2 (u) cos(2κu) (9)
f 3 (κ) = - 1 4 0 -∞ du u -∞ dv c 3 (u,v) sin(2κv) (10) f 4 (κ) = - 1 8 0 -∞ du u -∞ dv v -∞ dw c 4 (u, v, w) (11) × 2 cos(2κw)+cos[2κ(v+w-u)] .
Note that the compact form ( 11) is valid provided that oscillating terms, which may appear from terms in c 4 that can be factorized as c 2 correlators, are appropriately regularized at infinity. Note also that in Eq. ( 8), the coefficients (ǫ R /kσ R ) n diverge for k → 0, while the exact γ(k) remains finite for any ǫ R [23]. This signals a well-known breakdown of the perturbative approach. Conversely, the perturbative expansion is valid when γ(k) ≪ k (for k → 0), i.e. when the localization length exceeds the particle wavelength, a physically satisfactory criterion. 12) and ( 13) and the Appendix] and comparison with numerical calculations (points with error bars). The inset is a magnification of function f4 around κ = 2.

IV. ONE-DIMENSIONAL ANDERSON LOCALIZATION IN SPECKLE POTENTIALS A. Analytic results

Let us now examine the consequences of the peculiar properties of speckle potentials in the light of the above perturbative approach. For clarity, we restrict ourselves to 1D speckle potentials created by square diffusive plates as in Refs. [13,18] for which c a (u) = sin(u)/u and ĉa (q) ∝ Θ(1 -|q|) where Θ is the Heaviside step function [24]. Using Eqs. ( 9) and (10), we find

f 2 (κ) = π 8 Θ(1 -κ)(1 -κ) (12)
f 3 (κ) = - π 4 Θ(1 -κ) [(1-κ) ln (1-κ) + κ ln (κ)] ( 13
)
The functions f 2 and f 3 are simple and vanish for κ ≥ 1 (see Fig. 2). This property is responsible for the existence of the first effective mobility edge at k = k c [17], such that γ

(k)σ R ∼ (ǫ R /kσ R ) 2 for k σ -1 R while γ(k)σ R = O(ǫ R /kσ R ) 4 for k σ -1 R .
The fact that f 3 vanishes in the same interval (κ ≥ 1) as f 2 exemplifies the general property that odd-n terms cannot be leading terms in any range of k because γ(k) must be positive whatever the sign of V R . For κ < 1 however, f 3 (κ) is not identically zero owing to the asymmetric probability distribution in speckle potentials. The term γ (3) can thus be either positive or negative depending on the sign of V R [22].

The function f 4 is found similarly from Eq. (11). While its expression is quite complicated (see the Appendix), its behavior is clear when plotted (see Fig. 2). Let us emphasize some of its important features. First, there is a discontinuity of the derivative of f 4 at κ = 1/2. Second, we find a very narrow logarithmic divergence, f 4 (κ) ∼ -(π/32) ln |1 -κ| at κ = 1, which signals a singularity of the perturbative approach (note that it does not appear in Fig. 1 due to finite resolution of the plot). Finally, the value κ = 2 corresponds to the boundary of the support of f 4 , showing explicitly the existence of a second effective mobility edge at

Γ Σ R Ε R 0.10 Ε R 0.02 0.5 1 0 1 2 Γ 3 Γ 2 Γ Γ 2Γ 2 Ε R 0.10
k = 2σ -1 R . Hence, while γ(k)σ R ∼ (ǫ R /kσ R ) 4 for k 2σ -1 R , we have γ(k)σ R = O(ǫ R /kσ R ) 6 for k 2σ -1 R , since f 4 (κ)
as well as f 5 (κ) vanish for κ ≥ 2.

B. Numerics

In order to test the validity of the perturbative approach for experimentally relevant parameters, we have performed numerical calculations using a transfer matrix approach. The results are plotted in Fig. 3: ǫ R = 0.02 corresponds to V R / = 2π × 16Hz in Fig. 3 of Ref. [13] and ǫ R = 0.1 to V R / = 2π × 80Hz in Fig. 3 and to Fig. 4 of Ref. [13]. For ǫ R = 0.02, the agreement between analytical and numerical results is excellent. The effective mobility edge at k = σ -1 R is very clear: we find a sharp step for γ(k) of about 2 orders of magnitude. For ǫ R = 0.1, we find the same trend but with a smoother and smaller step (about one order of magnitude). In this case, although the Born term for k σ -1 R and the fourth-order term for k σ -1 R provide reasonable estimates (within a factor of 2), higher-order terms -which may depend on the sign of V R -contribute significantly.

The contribution of the odd terms can be extracted by taking γ + -γ -, where γ ± are the Lyapunov exponents obtained for positive and negative disorder amplitude of same modulus |V R |, respectively. As shown in the inset of Fig. 3, the odd terms range from 30% to 70% of the Born term for 0.6 kσ R 0.9 and ǫ R = 0.1, and are of the order of γ (3) in weak disorder and away from the divergence at k = σ -1 R . This shows that the first correction γ (3) to the Born term can be relevant in experiments.

For completeness, we have calculated the f n (κ) as the a) coefficients of fits in powers of ǫ R /kσ R using series of calculations of γ(k) at fixed k and various ǫ R . As shown in Fig. 2, the agreement with the analytic formulas is excellent. In particular, the numerics reproduce the predicted kink at κ = 1/2. The logarithmic singularity around κ = 1 being very narrow, we did not attempt to study it.

⊗ ⊗ ⊗ ⊗ b) ⊗ ⊗ ⊗ ⊗ c) ⊗ ⊗ ⊗ ⊗ d) • • • •

V. DIAGRAMMATIC ANALYSIS

Let us finally complete our analysis using diagrammatic methods, which allow us to exhibit momentum exchange in scattering processes as compact graphics, and thus to identify effective mobility edges in a quite general way. In 1D, the localization length can be calculated from the backscattering probability of |ψ| 2 using quantum transport theory. The irreducible diagrams of elementary scattering processes in speckle potentials have been identified in Ref. [25].

To lowest order in ǫ R (Born approximation), the average intensity of a plane wave with wave vector k backscattered by the random potential is described by

U 2 (k) = • • q + k q -k k k k k =: ⊗ ⊗ 2k k k k k . ( 14
)
The upper part of the diagram represents ψ (particle) and the lower part its conjugate ψ * (hole). The dotted line

• q

• = ǫ R ĉa (q) represents the field correlator; simple closed loops over field correlations can be written as a potential correlation ⊗ ⊗ . Backscattering requires diagram (14) to channel a momentum 2k, entering at the particle, down along the potential correlations to the hole. Therefore, the diagram vanishes for kσ R > 1.

At order ǫ 3 R , the only possible contribution is

U 3 (k) = • • • p q + k q -k k k k k + c.c. (15)
The straight black line stands for the particle propagator [E k -E p + i0] -1 at intermediate momentum p. Diagram (15) features two vertical field correlation lines, just as diagram (14), and thus vanishes at the same threshold k = σ -1 R . Evaluating two-loop diagram (15), we recover precisely contribution (13).

Many diagrams contribute to order ǫ 4 R . First there are the usual backscattering contributions with pure intensity correlations [Fig. 4 R . Performing the integration, we find that this diagram reproduces those contributions to f 4 (κ) for κ ∈ [1, 2] that contain factorized correlators (see the Appendix). Second, there are nine more diagrams, all with non-factorizable field correlations [25]. A single one has not two, but four vertical field correlation lines, shown in Fig. 4(d), and contributes for kσ R ∈ [1,[START_REF] Akkermans | Mesoscopic Physics of Electrons and Photons[END_REF]. Carrying out the three-loop integration, we recover exactly the non-factorizable contributions to f 4 (κ) for κ ∈ [1,[START_REF] Akkermans | Mesoscopic Physics of Electrons and Photons[END_REF].

VI. CONCLUSION

We have developed perturbative and diagrammatic approaches beyond the Born approximation, suitable to study 1D AL in correlated disorder with possibly asymmetric probability distribution. In speckle potentials, the k dependence of the Lyapunov exponent exhibits sharp crossovers (effective mobility edges) separating regions where AL is due to scattering processes of increasing order. We have shown it explicitly for k = σ -1 R and k = 2σ -1 R , and we infer that there is a series of effective mobility edges at k = pσ -1 R with integer p since, generically, diagrams with 2p field correlations or p intensity correlations can contribute up to k = pσ -1 R . This is because, although speckles are not Gaussian, they derive from a Gaussian field. Finally, exact numerics support our analysis for experimentally relevant parameters, and indicate the necessity to use higher-order terms in the Born series, even for k < σ -1 R . Hence, important features that we have pointed out, such as odd terms in the Born series for k < σ -1 R and exponential localization for k > σ -1 R , should be observable experimentally.

Figure 1 :

 1 Figure 1: (Color online) Lyapunov exponent γ calculated two orders beyond the Born approximation for particles in 1D speckle potentials created with a square diffusive plate, versus the particle momentum k and the strength of disorder ǫR = 2mσ 2 R VR/ 2 (VR and σR are the amplitude and correlation length of the disorder). The solid blue lines correspond to ǫR = 0.1 and ǫR = 0.02.

Figure 2 :

 2 Figure 2: (Color online) Functions fn for n = 2, 3 and 4 for a speckle potential created with a square diffusive plate [solid lines; see Eqs. (12) and (13) and the Appendix] and comparison with numerical calculations (points with error bars). The inset is a magnification of function f4 around κ = 2.

Figure 3 :

 3 Figure 3: (Color online) Lyapunov exponent γ(k) versus the particle momentum k as determined numerically (solid red lines) and by perturbation theory up to order 4 (dashed blue lines), for a speckle potential created with a square plate. The dotted green lines are the Born term. Inset: comparison of odd and even contributions in the Born series for ǫR = 0.1.

Figure 4 :

 4 Figure 4: Relevant fourth-order backscattering contributions. Contrary to the case of uncorrelated potentials [26, 27], the sum of diagrams (a)-(c) does not give zero for speckle potentials; only diagrams (b) and (d) contribute for kσR ∈ [1, 2].

  (a)-4(c)]. Both Figs. 4(a) and 4(c) have a single vertical intensity correlation and vanish for k > σ -1 R . In contrast, the crossed diagram [4(b)] has two vertical intensity correlation lines and can thus accommodate momenta up to k = 2σ -1

three terms with different supports, f 4 ( 1 - 1 2 +

 4112 lives on the interval κ ∈ [α, β], andf 2κ) -(4 -2κ) ln(κ) -5 κ -3κ ln(1 -κ) + 1 κ + κ ln(1 + κ) -(4 -8κ) ln 2 (κ) + 22(1 -κ) ln 2 (1 -κ) + (18 + 14κ) ln 2 (1 + κ) -16(1 -κ) ln(1 -κ) ln(κ) -4(1 -κ) ln(1 -κ) ln(1 + κ) -32(1 + κ) ln(κ) ln(1 + κ) -24(1 + κ)Li 2 (κ) + 32(1 + κ)Li 2 κ κ ln(κ -1) -2(κ -1) ln 2 (κ -1) + 4κ ln(κ -1) ln(κ)where Li2 (z) = 0 z dt ln(1 -t)/t = ∞ k=1 z k /k 2 is the dilogarithm function.

ACKNOWLEDGMENTS

Stimulating discussions with P. Bouyer, V. Josse, T. Giamarchi and B. van Tiggelen are acknowledged. This research was supported by the French CNRS, ANR, MENRT, Triangle de la and IFRAF.

APPENDIX

Here, we give the explicit formula of the function f 4 (κ) for a speckle potential created by a square diffusive plate, such that the fourth-order term in the Born expansion of the Lyapunov exponent γ reads γ


Arnaud Hubert

Marion Taconne

Bogdan A Popescu ²

Erwan Donal
email: erwan.donal@chu-rennes.fr

Diastolic function and its non-invasive assessment. The quest for the Holy Grail continues

The assessment of left ventricular diastolic function is key for the diagnosis of heart failure with preserved ejection fraction (HFpEF) yet it is still limited by practical difficulties and inaccuracies. Therefore, the search of new methods to improve this assessment is ongoing.

Kertsens et al (1) sought to explore a very promising tool. They have compared left ventricular strain-volume loop (SVL) characteristics and diastolic dysfunction (DD) in patients with suspected HFpEF. Patients with good quality images and strain and volume analysis, were classified according to the current diastolic function recommendations (2) in two groups: DD (n=91) and control (n=65). SVL analysis showed a different longitudinal strain contribution to volume change in DD compared to controls, i.e. 'uncoupling of the SVL', which was independently associated with DD. This approach has already been looked at and discussed(3-5). There are other new innovative approaches with other limitations that are under investigation. In a pig model, it has been demonstrated that shear wave speed after mitral valve closure was influenced by both acute changes in loading but also myocardial mechanical properties. It might be a window for reflecting changes in operating myocardial stiffness(6).

Firstly, the authors have to be congratulated. They are pursuing on testing the practical use of strain-volume loops [START_REF] Hulshof | Exploratory assessment of left ventricular strain-volume loops in severe aortic valve diseases[END_REF]. Hubert et al have opened the field for an automatization of this analysis [START_REF] Hubert | New expectations for diastolic function assessment in transthoracic echocardiography based on a semi-automated computing of strain-volume loops[END_REF]. It has been nicely challenged in an editorial,(4) but of course, it is important in such an endeavour to hold the helm even if the results are not perfect. 383 patients were screened but only 156 were finally included in the statistical analysis. The feasibility has been an issue. Patients who were not in sinus rhythm were excluded, and so were patients whose diastolic function was classified as indeterminate by the current recommendations. These two categories include a significant number of patients for whom we still need a better solution. Left atrial reservoir strain seems a promising tool to solve part of the issues we still have in "estimating" diastolic function and filling pressures [START_REF] Popescu | How to assess left ventricular filling pressures by echocardiography in clinical practice[END_REF][START_REF] Smiseth | Multimodality imaging in patients with heart failure and preserved ejection fraction: an expert consensus document of the European Association of Cardiovascular Imaging[END_REF]. Some may argue that instead of looking for newer parameters, cardiologists should look more carefully at conventional ones, such as pulmonary vein flow signals, or response of mitral flow to a standardized Valsalva maneuver. This may be true in selected cases with feasible signals but will not work in others. The method used in the current study [START_REF] Kerstens | Left Ventricular Strain-Volume Loops and Diastolic Dysfunction in Suspected Heart Failure with Preserved Ejection Fraction[END_REF] has improved since the publication of the first reports, is more automated and could become more robust, averaged over several cardiac cycles and perhaps also including 3D-echocardiography for volume measurements [START_REF] Donal | The future of the diastolic function assessment will take advantage of the past and of the automatization[END_REF]. Already now, only the strain traces have to be checked by physicians (the human factor) and then all data are extracted from the industrial software and treated by an automated algorithm.

We previously found a difference between patients with DD and controls but this could be attributed to a difference in stroke volume between the two groups(3). Interestingly, Kertsens et al (1) demonstrated, on top of our previous work, that a real difference between SVL of patients with DD and controls exists even after normalizing for stroke volume. The underlying potential of the method exists and it has to be further tuned to further improve.

Journal Pre-proof

Conversely to the "classical" parameters advocated by the ASE/EACVI recommendations(2), SVL is an automated tool which could be easily integrated in machine learning algorithms to improve the capability of echocardiography to diagnose DD computing automated quantification of multiple parameters. The SVL will not solve every issue on their own but could be part of the solution. Our team has already proposed methods that have demonstrated value in predicting the result of a cardiac resynchronization system implantation just by looking automatically at the amount of information that strain curves provide about cardiac mechanics [START_REF] Gallard | Characterization of Responder Profiles for Cardiac Resynchronization Therapy through Unsupervised Clustering of Clinical and Strain Data[END_REF][START_REF] Gallard | Echocardiographic view and feature selection for the estimation of the response to CRT[END_REF].

The present work is a step forward but unresolved issues remain [START_REF] Kerstens | Left Ventricular Strain-Volume Loops and Diastolic Dysfunction in Suspected Heart Failure with Preserved Ejection Fraction[END_REF]. Explaining the difference in uncoupling between DD patients and controls will require further studies. Validation in animal experiments and/or clinical studies with invasive measurements of diastolic function properties will be needed. Also, the incremental value of this method over the existing ones will have to be tested prospectively in large cohorts looking at the impact of treatments and at the outcome.

It is an interesting study but it is still "work in progress" and it is too early to consider this approach for a change in the paradigm of the recommendations on left ventricular diastolic function assessment [START_REF] Popescu | How to assess left ventricular filling pressures by echocardiography in clinical practice[END_REF][START_REF] Smiseth | Multimodality imaging in patients with heart failure and preserved ejection fraction: an expert consensus document of the European Association of Cardiovascular Imaging[END_REF].

Figure 1

 1 Figure 1

Conflict of interest

There are no commercial products involved in this editorial.

None of the authors have any conflict of interest related to this work