Geneviève Harrouspaicheler

Male circumcision as HIV prevention in Africa : debates and resistance

Keywords: Male circumcision, HIV/AIDS, Africa, controversies, biopower, post-colonial context

While the practice of male circumcision (MC) is declining in Western countries, its promotion in order to prevent HIV/AIDS is currently implemented in Eastern and Southern Africa, where it is not frequently carried out and where the HIV epidemic is very widespread. However, in spite of the scientific demonstration of MC's efficacy, its advocacy as a method to prevent HIV/AIDS has generated controversy, with its most extreme opponents denouncing a conspiracy by an occult organisation of powerful "circumcisers". Drawing on an analysis of scientific papers and the responses they have elicited in the journals where they are published and on Internet sites, this article sheds light on this opposition and its connotations. It situates the distorted vision of Africans and of their sexuality -as expressed through these publications and reactions -in a post-colonial context where the ex-colonisers' domination over ex-colonised populations is perpetuated under different forms. Finally, it analyses the action designed in Western countries and exerted on African bodies as an instance of biopower involving medical experimentation transfers to Southern countries. When deprived of any symbolic dimension and practised with a prophylactic objective, MC appears emblematic of such biopower.

Introduction

Male circumcision (MC) is one of the oldest surgical operations in the world. Yet it remains highly controversial. Where some people identify an effective means for avoiding sexually transmitted diseases, others see useless surgery, unacceptable mutilation, both barbarous and reactionary, either performed for bad religious or cultural reasons or promoted by puritans opposed to sexual satisfaction. Whereas the results of randomised controlled trials (RCTs) carried out in 2005 and 2007 in Africa showed its efficacy in preventing HIV infection in men, its frequency has significantly decreased with newborns in the U.S. -a country where it was once widely practiced, notably, in higher social classes and on White men. While the neonatal MC rate was 56% in 2006 -already far lower than in the second half of the twentieth century: 84% of men born in the United States in the 1970s [START_REF] Xu | Prevalence of circumcision and herpes simplex virus type 2 infection in men in the United States: the National Health and Nutrition Examination Survey (NHANES), 1999-2004[END_REF] -it dropped by 32.5% in 2009 1 .

Moreover, under the impetus of a network of activists allied, notably, with natural health advocates, anti-circumcision leagues clinched enough signatures (over 7,200) to introduce a bill in the San Francisco and San Diego 2011 ballots prohibiting and penalising neonatal circumcision 2 . In the United Kingdom, where Queen Victoria had set the example by having her sons circumcised, the circumcision rate fell in the late twentieth century to 15.8% among men aged 16-44 years in 200016-44 years in (Dave et al., 2003) ) 3 . Western governments -notably, the United Kingdom, Finland, Sweden, the Netherlands, Australia, and New Zealand -tend to restrict the practice of neonatal circumcision, which was relatively widespread in the twentieth century. It must not be routine practice and can only be justified by medical or religious reasons, sometimes only in a hospital institution. Qualified persons and/or doctors should practise it under anaesthesia.

Beyond issues regarding integrity of the human body and free will, controversies on circumcision have proved very impassioned. MC is practised, on the one hand, for religious Notes 1 Review of circumcision data for 6,571,500 newborn U.S. boys in 2006-2009, drawn from a representative electronic health care reimbursement database, presented at the 18 th International AIDS Conference in Vienna, 2010, by Dr. El Bcheraoui, an epidemic intelligence service officer in the division of HIV/AIDS prevention at the Centers for Disease Control and Prevention. It is worth noting that the MC rate is difficult to assess with precision by merely referring to operations performed in a medical institution (in this case, on newborns), as an undetermined number of ritual circumcisions are not performed in such settings. 2 The MGM Bill (Male Genital Mutilation Bill) accessible on mgmbill.org (consulted June 2011). 3 A rate evaluated on the occasion of a broad national survey: the National Survey of Sexual Attitudes and Lifestyles (Natsal). and/or cultural reasons (which are deeply rooted but nonetheless denounced by its opponents); and, on the other hand, for medical reasons: malformations, preventing infections. It is supported by its proponents, and vilified by its adversaries who consider it as a sexual mutilation operated on a person who has not reached the age of consent.

In fact, current debates should be grasped and unravelled both in their local and global dimension. One cannot avoid asking why an increasing number of people contest MC, particularly in Western countries, at a time when international organisations -the WHO and UNAIDS -or North American -USAID and PEPFAR -are striving to promote this practice in order to curb the HIV epidemic in Eastern and Southern Africa, where its prevalence is very high and MC infrequent. Few countries, however, easily accept this promotion. In most concerned countries, the international entities cited above experience difficulties in finding political partners and relays: hence disseminating circumcision is, to say the least, a slow process. To these difficulties, one must add the fact that law in South Africa punishes circumcision before age 18. Thus, we face a paradox, which lies at the core of this article: why should different principles in terms of circumcision practice be applied in Africa and not in other areas, notably, in Western countries? Why should circumcision be considered as an African, exceptional, solution to prevent AIDS? To cast light on this paradox, we shall first consider developments in research and scientific discussions: it has taken a long time to demonstrate the efficacy of circumcision based on Random Controlled Trials (RCTs). As we shall see below, this demonstration has not succeeded in terminating scientific polemics on the issue. Then we shall examine the implications of the fact that circumcision is considered as an African solution to AIDS, i.e., an exception. Such implications have multiple dimensions. They unfold in different contexts: first, in a racial context involving a representation of superior or inferior races; and, secondly in a colonial legacy context. This has been developed in a huge body of so-called scholarly literature. Racial hierarchy have been used to explain why Africa was more severely affected by the AIDS epidemic, with a reference to uncontrolled sexuality. The argument was based on "evidences" that did not even need to be justified since they were such a massive consensus.

Furthermore, Africa has become a prime area for scientific globalisation: many clinical trials, notably with pharmaceutical products, are conducted -sometimes in highly questionable conditions. This fact may help explain why African populations tend to distrust scientific research of which they are the actual or potential subjects. Finally, promoting circumcision can be understood as exercising a bio-power, via government action, over human bodies whose sanitary, prophylactic dimension takes precedence over their symbolic dimension.

Male circumcision in the scientific arena

In a first section, I shall analyse the scientific research conducted in Africa on MC as HIV infection prophylaxis. For this purpose, the Pubmed, Medscape, Embase, Aidsline databases, as well as the databases of identified peer-reviewed journals, were searched for articles using statistical methods and reviews of such articles. The retrieved articles were all read and categorized manually to capture the range of comments, arguments and critics. My review seeks to follow the research process that had led to carrying out RCTs -a process based on arguments and counter-arguments relative to MC-and, subsequently, to examine the reactions to these RCTs. It does not address their scientific reliability nor the appropriateness of methods used4 .

Based on a sociological approach, it examines their discourse, their objectives and conclusions, as stated by the authors.

The randomized controlled trials

In order to understand the huge disparity of HIV-infection prevalence in Sub-Saharan Africa 5 , notably, the contrast between West Africa, relatively unaffected, and Southern and Eastern Africa6 , the need to carry out RCTs became obvious after a mass of heterogeneous data had accumulated7 , Three RCT with comparable designs showed that MC was effective against HIV acquisition. A first RCT was carried out between 2002 and 2005 in a periurban setting near Johannesburg, Orange Farm, in South Africa [START_REF] Auvert | Randomized, controlled intervention trial of male circumcision for reduction of HIV infection risk : the ANRS 1625 trial[END_REF]. 3,274 HIV-negative men, aged 18-24 years, were randomized, 1,617 to immediate and 1,656 to later circumcision. They received a compensation of 300 rands (about 45 US $) for their participation. The other benefit for their inclusion in the trial was a medical follow-up. After circumcision, men were advised to wait for complete healing (about one month) before resuming sexual activity. During the trial, they received repeated information on HIV prevention and other sexually transmitted diseases, as well as free condoms8 . They were followed up at regular intervals for sexually transmitted diseases and HIV infection, and their sexual behaviour was investigated by questionnaires to control the comparability between the two arms of the trial. The RCT was stopped early at 18 months after a planned interim analysis showed a significant 61% reduction in HIV risk. At endpoint, 20 circumcised participants were HIV infected versus 49 uncircumcised ones.

These results were confirmed in two trials with a similar design carried out in Kenya [START_REF] Bailey | Male circumcision for HIV prevention in young man in Kisumu, Kenya : a randomized control trial[END_REF] and in Uganda [START_REF] Gray | Male circumcision for HIV prevention in young men in Rakai, Uganda: a randomized controlled trail[END_REF], also stopped early as evidence emerged that MC provided significant benefits. The trial carried out at Kisumu in Kenya on 2,784 men aged 18-14 years, yielded results very close to those of [START_REF] Auvert | Randomized, controlled intervention trial of male circumcision for reduction of HIV infection risk : the ANRS 1625 trial[END_REF]: 60% reduction of HIV risk in the circumcised men. The trial conducted in a rural area at Rakai in Uganda [START_REF] Gray | Male circumcision for HIV prevention in young men in Rakai, Uganda: a randomized controlled trail[END_REF] on 4,996 men aged 15-49 years was similar to the preceding ones, except for its wider age range. It was also stopped early after an interim analysis showed significant efficacy of MC.

The estimated reduction of risk of HIV infection was 55%. The results of the three RCTs are remarkably close: MC more than halves the risk of infection by HIV but the protection is partial.

Therefore, the authors insist condoms remain the safest protection against HIV when they areproperly-used.

Prior research

The three RCTs produced their results after nearly two decades of intense debates on the protective role of MC against HIV, which I shall now examine to shed light on the long process of evidence search.

Biological evidence

In 1986, two clinicians, Fink and Alcena, assumed a link between HIV infection and absence of MC. 'The majority of men from Central Africa are not circumcised; there is frequent mini-ulceration of the foreskin of the penis. This represents an easy portal of entry for the virus.' (Alcena, 1986, 446). The biological basis for MC protection was later described later by [START_REF] Szabo | How does male circumcision protect against HIV infection?[END_REF]: the foreskin contains more Langerhans/dendritic cells and CD4/CD8 T lymphocytes, all target cells for HIV. By reducing the surface of the foreskin, MC reduces the number of these target cells. [START_REF] Weiss | Male circumcision for HIV prevention : Current research and programmatic issues[END_REF] cite recent researches that confirm this explanation.

Data on the protective function of MC before the RCTs Although indigenous African healers recommend MC for protection against sexually transmitted infections [START_REF] Green | Indigenous African healers promote circumcision for prevention of sexually transmitted diseases[END_REF], the first epidemiological studies on the prevalence of HIV infection in Africa found contrasting situations [START_REF] Moses | Geographical patterns of male circumcision practices in Africa: Association with HIV seroprevalence[END_REF]. The question was to find out if the observed differences were attributable to MC. More than forty observational studies were aimed at tackling this question. Among the first ones, [START_REF] Bongaarts | The relationship between male circumcision and HIV infection in African populations[END_REF] compared 409 African ethnic groups with an approximate method and concluded that MC provided an explanation for the disparity of HIV prevalence in Africa. Using a prospective methodology, [START_REF] Cameron | Female to male transmission of human immunodeficiency virus type 1: Risk factors for seroconversion in men[END_REF] confirmed the link between the absence of MC, genital ulcers -two interrelated factors -and HIV infection, studying the HIV-incidence of the clients of prostitutes with a very high HIV-prevalence (85%).

During the following decades, research intensified. Data on MC was synthesized in review articles. Analyzing a series of observational studies, [START_REF] Caldwell | The neglect of an epidemiological explanation for the distribution of HIV/AIDS in Sub-Saharan Africa : Exploring the male circumcision hypothesis[END_REF] already emphasized the controversial aspect of the protective effect of MC. 'The role of male non-circumcision has been presented in epidemiological studies which claimed a level of statistical association usually accepted as approaching proof in other investigations. These studies have been largely ignored for reasons that may not be entirely scientific' (Caldwell and Caldwell, 1994: 23). [START_REF] Moses | The association between lack of male circumcision and risk for HIV infection : a review of the epidemiological data[END_REF] found that 18 cross-sectional studies 9 , out of 26 included in their review, showed an association between MC and lower risk. They stressed the reliability of the data. 'Most of the Bradford-Hill criteria of causation are met: strength of association, consistency, temporality, biologic gradient and theoretical plausibility, coherence, analogy' (Moses et al., 1994: 207). [START_REF] Halperin | Male circumcision and HIV infection: 10 years and counting[END_REF] reviewed ten years of studies on the association between MC and reduction in HIV risk. The evidence seemed so clear to them that they recommended that 'the international health community [to] add MC services to the current limited armamentarium of AIDS prevention measures in countries with a high prevalence of heterosexually transmitted HIV and STDs… [Otherwise] medical professionals and public health authorities may inadvertently be harming the very individuals whom they are trying to help' (Halperin andBailey, 1999: 1814). But to confirm the link between MC and HIV protection, possible confounding factors such as religious, cultural practices and s e x u a l behaviours must be controlled. This was done in two articles comparing the prevalence of HIV infection in four African cities with contrasting rates of HIV infection: moderate in Cameroon and Benin, where MC is widespread or very high in Kenya and Zambia, where most men are not circumcised [START_REF] Auvert | Male circumcision and HIV infection in four cities in sub-Saharan Africa[END_REF]).

9 Cross-sectional studies provide a "snapshot" of the frequency of a disease in a population at a particular point in time.

Doubts and caution

From the onset of the epidemic in Africa, it was difficult to sort the wheat from the chaff in the accumulating data on MC. At a national level, the data was often not significant. African borders have been arbitrarily traced in the colonial era and ethnic groups with different circumcising tradition live in the same countries. Moreover, some ethnic groups practising MC showed higher rates of HIV infection. This could be explained later by the assumption that the later MC is practised the less protective it is, as men resume sexual activity too early. Besides, genital contact may be considered as a way to heal the MC wound.

Opinions against the benefits of MC were voiced. In an influential review article, Vincenzi and Mertens (1994) expressed several criticisms that were picked up and amplified in numerous subsequent publications. They contested the biological plausibility of MC: 'It is unclear what the portal of entry is for HIV' (Vincenzi and Mertens, 1994: 156). Data was judged heterogeneous and contradictory, lacking a coherent explanatory model. They cited numerous biases: hygiene practices, sexual behaviour, misclassification of exposure, unreliability of selfreports on MC. Such criticisms were acceptable but one of their arguments seems rather surprising, all the more as no effort was made to support it: scientific journals allegedly have a publication bias in articles supporting MC. In their concluding remarks, the authors wondered whether the interpretation of the MC data relied on faith10 or evidence. Rather -they concluded after throwing oil on the fire of subjectivity -the effects of MC should be subject to an objective scientific analysis and not be debated in an impassioned and emotional fashion.

Reservations were also voiced in several epidemiological studies and reviews confirming the beneficial effects of MC [START_REF] O'farrell | Circumcision in men and the prevention of HIV infection : A "meta-analysis" revisited[END_REF][START_REF] Weiss | Male circumcision and risk of HIV infection in sub-Saharan Africa: A systematic review and meta-analysis[END_REF]. Lowered risk of HIV infection could be due to religious proscriptions and strict rules of hygiene. In a cohort study carried out on 5,507 seronegative men and 410 serodifferent couples [START_REF] Gray | Male circumcision and HIV acquisition and transmission: Cohort studies in Rakai, Uganda, Rakai Project Team[END_REF], the data was considered confusing because some of the participants were Muslim. A letter reacting to this article [START_REF] Halperin | Response to Ronald Gray, Male circumcision and HIV acquisition and transmission: cohort studies in Rakai, Uganda[END_REF] backed the positive role of MC by evoking mounting biological evidence, and in their reply, [START_REF] Gray | Male circumcision and HIV acquisition and transmission: Rakai, Uganda (Correspondence)[END_REF] still recommended c a u t i o n a n d implementing RCTs. Reviews looked at confounding factors [START_REF] Bailey | Male circumcision : Current knowledge and future research[END_REF][START_REF] Hayes | Confounding, effect modulation and attributable fractions : Issues in interpreting circumcision effects[END_REF][START_REF] Quigley | Male circumcision as a measure to control HIV infection and other sexually transmitted diseases[END_REF] and concluded that the data in favour of MC was not sufficiently convincing.

A series of questions were raised, notably, the type of MC, its protective function against other sexually transmitted infections, the reasons for practicing MC, the impact of the age at which it is carried out, and its scientific foundations. Finally, a Cochrane meta-analysis11 [START_REF] Siegfried | Male circumcision for prevention of heterosexual acquisition of HIV in men[END_REF] emphasized the methodological heterogeneity of the studies, as well as their inconsistent quality, and concluded there was no strong evidence of the benefits of MC. The results of observational studies were limited by confounding factors and it was necessary to await results

of RCTs to get unquestionable data.

Scientific criticism: back to "real life"

As the gold standard of medical research12 , the RCTs on MC were expected to put an end to debate, at least as regards i t s s c i e n t i f i c a s p e c t s , s i n c e i s s u e s o f i m p l e m e n t a t i o n a n d acceptability may widen the gap between efficacy and efficiency. RCTs constitute a working model where researchers can exercise maximum control but in the real world, things may be very different. Moreover, 'there is an inevitable role of judgment in weighting evidence, even within the constraints of a formal statistical meta-analysis' (Marks, 2009, 98). And along with this judgment, there are 'conflicts over the quality of evidence and over the appropriate role of evidence in directing practice' (Ibid. 99).

The publication of the RCTs gave rise to numerous letters where critics repeatedly referred to the same arguments and questioned the methodology. The criticism was focused on the quality of evidence. The RCTs took place over a short period. What would be the impact of MC over the long term? Participants were provided medical follow-up and repeated counselling.

What would happen when these conditions were no longer present? [START_REF] Garenne | Male circumcision and HIV control in Africa[END_REF] doubted that the protective effects of MC could be maintained over the long term, using the analogy of insufficiently safe contraceptive methods that do not succeed in preventing u nwanted pregnancy. According to him, the 61% protection rate shown by [START_REF] Auvert | Randomized, controlled intervention trial of male circumcision for reduction of HIV infection risk : the ANRS 1625 trial[END_REF] only represents a decrease in the annual risk. MC will have less impact on HIV infection over several years [START_REF] Green | Male circumcision is not the vaccine we have been waiting for (Editorial)[END_REF]. Arguments against MC repeated criticisms about confounding factors from before RCTs [START_REF] Potterat | The protective effect of male circumcision as a faith lift for the troubled paradigm of HIV epidemiology in sub-Saharan Africa[END_REF] as though evidence provided by the latter was not valid.

'Male circumcision is not the vaccine we have been waiting for', stated [START_REF] Green | Male circumcision is not the vaccine we have been waiting for (Editorial)[END_REF] in the title of their article. 'Thousands of African men now line up to get circumcised in the mistaken belief that it will save them from HIV […] The public is misled; false hope is promoted from uncertain conclusions' (Green et al, 2008: 193). The world health community should scrutinize the data and examine 'several other factors that might have influenced and skewed the results' (ibid.), especially the supposed high proportion of non-sexual transmission. MC is dangerous because of contaminated medical instruments, exposure to blood, a false sense of security, and, last but not least, risk compensation behaviours, as African males would be too happy to do without condoms. [START_REF] Myers | Rolling out male circumcision and HIV prevention as a mass intervention seems neither justified or practicable[END_REF] again condemned a pro-circumcision lobby impatient to achieve their ends by the promotion of a license for unprotected sex. MC will foster out-of-control sexuality 13 .

Numerous experts -academicians, members of international organizations and major foundations -issued a rejoinder expressing their surprise 'at the degree of resistance to something which seems to them both self-evidently good and worthwhile, and also entirely consistent with the "scientific" biomedical paradigm within which they operate' (Wamai et al., 2008: 399). They restated the fact that scientific evidence was "overwhelming", that the HIV epidemic was devastating and the many other prevention strategies disappointing; early termination of the RCTs was due to the high level of efficacy of MC; the data was consistent and the observed effects were 'virtually identical to [those] seen in many previous observational studies' (ibid.: 400). As South African, Kenyan and Ugandan cohorts are being followed up for long-term risk reduction, 'there is no reason why the protective effect of foreskin removal would decline over time' (ibid.). Furthermore, there is no increase in reported risky behaviour of circumcised men, the so-called risk compensation. Addressing the recurrent statement that many infections were not due to heterosexual contacts -hence the protection by MC would be useless -, the authors replied that 'this theory has been repudiated by the WHO and virtually all reputable scientists' (ibid.). They repeated that no serious or permanent complications or aftereffects of MC were reported. With regard to confounding factors, RCTs, are designed to control them. The authors remind that they complied with ethical rules and that 'it is unethical to deny safe MC services in high prevalence settings' (ibid.: 401). In conclusion, 'as more and more people in sub-Saharan Africa become needlessly infected with HIV, the time has come for urgent and decisive leadership, not circular and unscientific arguments about an intervention whose efficacy has been proven beyond a reasonable doubt' (ibid.: 403). For [START_REF] Rennie | Male circumcision and HIV prevention : ethical, medical and public health tradeoffs in low-income countries[END_REF], it would be unethical not to take the preventive opportunities offered by MC seriously. It was apparently as effective as the vaccine everyone has been looking forward to [START_REF] Jeffrey | Is male circumcision as good as the HIV vaccine we are waiting for ? (Editorial)[END_REF]. Finally, a

Cochrane meta-analysis [START_REF] Siegfried | Male circumcision for prevention of heterosexual acquisition of HIV in men[END_REF] confirmed the protective effect of MC.

Outside the scientific arena: resistance and disputation

The detractors of MC would not cease their struggle. Their goal is henceforth to slow down or even prevent the scaling up that was carried out after 2007 mainly by WHO and UNAIDS with the involvement of USAID, PEPFAR, the Bill and Melinda Gates Foundation and the Global Fund to Fight AIDS, Tuberculosis and Malaria.

We now face two opposite coalitions of actors: in favour of MC promotion, a number of scientists working in the HIV-infection field and members of international organisations. Against circumcision promotion, the "intactivists" (as they call themselves) include, amongst persons with various political and religious leanings, clinicians, notably, paediatricians14 , who operate outside the HIV-infection field. They hold highly influential positions in anti-circumcision leagues and express their opposition on numerous Internet sites. The impact of these leagues is difficult to ascertain, but their presence on the Internet and intensive lobbying have resulted in legislative actions that may well spread.

The data I shall present below will draw on manual analysis of multiple articles and letters to scientific journals, and in order to put these into perspective, newspaper articles, websites and blogs where similar or close arguments were developed.

An arduous implementation: the long road from evidence to public policy

In 2007, a review meeting 15 was organized at Montreux by the WHO and UNAIDS to confront different points of view, notably those of social scientists. Many participants were critical about MC, and did their part in fuelling controversy. If the research evidence was clear, consensus was far from achieved [START_REF] Berer | Taking on the opposition, Male circumcision and other themes[END_REF].

Reacting to the f a c t t h a t M C i s ' j u s t a s n i p ' , a s d e s c r i b e d b y a p a r t i c i p a n t i n a f o r m e r WHO/UNAIDS meeting, [START_REF] Aggleton | Just a snip ?: A social history of male circumcision[END_REF] maintains that 'few if any investigations contain robust controls for confounding factors such as social background, sexual behaviour or penile hygiene' (Aggleton, 2007: 19). The strengthening advocacy in favour of MC he perceived was based on a collusion between public health and social control. Since prevention programs have failed, he argues, there is a backlash to biomedical interventions and their implicit moral constraint. The flawed results of the RCTs are promoted by a powerful coalition. 'Evidence from recent trials, which at the very least requires continued scientific scrutiny, is now trumpeted as "truth".

Opponents… have been silenced and marginalized under the onslaught. Curious alliances have arisen between clinicians, advocates, religious leaders and moral entrepreneurs' (ibid.: 20). [START_REF] Dowsett | Male circumcision and HIV prevention : Is there really enough of the same kind of evidence?[END_REF] echoed these statements. They evoked the "euphoria" of participants at the international conference in Toronto in 2006, while 'many were less sanguine' (Dowsett and Couch, 2007: 34) and pushed aside, notably the social scientists who were 'sceptical about the narrow form of "science" being touted as the only form of evidence needed […] The clamour for circumcision silenced many questions, overrode any misgivings and swept sceptics to the side lines' (ibid.). MC is a discriminatory procedure promoted as an African solution. It would surely lose its benefits in real world settings. The methodology of the RCTs was also disputed:

findings were taken out of context; the double-blind method was impossible, intensive counselling does not exist in real settings. The adverse effects of MC will develop over time.

These authors advise to remain 'wary of the indecent haste with which the discussion about MC has been swept up in a tide of enthusiasm based on only one, albeit significant, part of the evidence base needed for recommending such a radical public health initiative' (ibid.: 42). [START_REF] Buvé | Delivery of male circumcision services : "Festina lente[END_REF] warned against precipitation and advised to hurry up slowly (Festina Lente is the title of their article), considering that 'the biggest challenge is how to deliver a complex prevention package that combines a surgical procedure with a behaviour change intervention' (Buvé et al., 2007: 58). How could MC be practiced under proper conditions in insufficient health services? 'The biggest bottleneck when going to scale with MC will be the lack of human resources' (ibd.: 59). Actually, it is not the only bottleneck: political and popular resistance appears to be much more significant.

Despite the efforts of the international organizations involved in its promotion, one can only note that, on the field, the diffusion of MC is slow and political involvement sometimes reluctant16 ( K a g u m i r e , 2 0 0 8 ; K a t z a n d W r i g h t , 2 0 0 8 ) . In January 2010, Mary Shaba, the

The offensive against MC: uncovering a conspiracy

The issue of MC was caught between studies that attempted to untangle cause from effect on one hand, and strong resistance, on the other. Opponents of MC express their viewpoint mainly in letters reacting to published scientific articles in favour of MC and in a host of websites 19 . Most of their leaders are not researchers but rather clinicians, often paediatricians opposed to newborns' circumcision. Van Howe (1999) considered MC a s s c i e n t i f i c a l l y u n f o u n d e d a n d dangerous. He denied that studies carried out in Africa showed that circumcised men were less exposed to HIV risk: he argued that they have more partners and that condoms do not stay on their penises. In a letter to The Lancet, [START_REF] Darby | Male circumcision and risk of HIV-1 infection (Correspondence)[END_REF] be successful when the scientists involved in this endeavour can gather and analyze data objectively and rationally rather than use AIDS as yet another excuse to promote an old blood rite' (Van Howe et al., 2005: 264). But these authors' criticism is not confined to the scientific aspects. In an escalation of arguments, they accuse researchers, supposedly circumcised themselves and considering 'their incomplete penis… [as] superior to the intact penis' (ibid.) of proselytising in favour of a barbaric practice. 'Should healthy body parts be amputated to conform to cultural and religious practices of scientists from outside cultures?' (ibid.) In the same line, [START_REF] Boyle | Male circumcision and risk of HIV-1 infection (Correspondence)[END_REF] stated there was a positive bias for MC among circumcised researchers -if this was the case he prudently added.

'HIV transmission is heavily dependent on certain sexual behaviours, not anatomy' [START_REF] Van Howe | Male circumcision and HIV prevention: Some science would not have gone amiss (Letter)[END_REF]: 1467). [START_REF] Szabo | How does male circumcision protect against HIV infection?[END_REF] biological arguments are attributed to a retrograde and puritanical Victorian period, to backward tribal customs, to decades o f p r e s s u r e b y circumcisers playing on fears: the foreskin cannot be an 'error of nature' [START_REF] Cruz | Nature has not made a design error (letter)[END_REF](Cruz, : 1468)). The criticized authors reacted by condemning their opponents' dogmatic position. 'It would be unfortunate if the zealous opponents of neonatal male circumcision in developed countries… distracted attention from the glaring fact that in Central and Southern Africa, where 24.5 million people are infected with HIV, circumcision could offer some immediate protection against spread of the disease' [START_REF] Szabo | How does male circumcision protect against HIV infection?[END_REF]Short, 2000: 1594).

Increasing the escalation, [START_REF] Boyle | Issues associated with the introduction of circumcision into a noncircumcising society[END_REF] describes numerous negative effects of MC. It provokes an 'emotional defence against (one's) own painful feelings of grief for a lost body part and reduced sexual function' (Boyle, 2003: 427). The catalogue of these undesirable effects was eclectic: unhappiness, anger, sadness, feeling incomplete, cheated, hurt, concerned, frustrated, abnormal, and violated. Circumcised men would be prone to alcohol dependence and the use of drugs, as well as solitude, marital problems, anti-social behaviours, domestic violence, rape, sexual abuse of children, theft and suicide. In front of this bleak picture, the practice of MC is described as 'wishful fantasy' generating 'a calamitous worsening of the HIV/AIDS epidemic' (ibid.). For [START_REF] Hill | HIV and circumcision : New factors to consider (Letter)[END_REF], members of Doctors Opposing Circumcision, 90% of the sub-Saharan African epidemic was not due to heterosexual transmission but to iatrogenic transmission through blood that could only be increased by MC practiced under poor hygienic conditions. They state that only 30% of infections are attributable to sexual activity. Besides, MC could change sexual behaviours by increasing risky practices. The authors raised the threat of what 'African males' may think or do, 'abused and exploited by scientists who recommend the circumcision policy, sensitive of previous colonial exploitation and suspicious of the biological warfare origin of the virus' (Hill and Denniston, 2003: 496) 20 . They warned against the political consequences of the probable failure of MC: 'African males would have sacrificed their erogenous tissue for a false hope of preventing HIV infection' (ibid.).

As more evidence accumulated in favour of MC, the criticisms became sharper. For Hellsten ( 2004), the promotion of MC, a 'genital mutilation', absurd and irrational, will allow all sorts of other mutilations: 'We would have no justification for stopping (parents) cutting off their children's ears, fingers or noses if their religion or cultural beliefs demand it' (Hellsten, 2004: 249). It will also foster all sort of evils: 'a sexual disinhibition, and for women, unsafe, maybe also forced, sex' (ibid.:251). He too points toward a conspiracy: the reasons to practice MC are a 'mere smokescreen to cover up the actual social, political or economic reasons that are behind the preservation of genital mutilation in any given cultural context' (ibid.: 250). But the worst is yet to come. 'In a modern, American, market oriented society male circumcision became a form of commercial exploitation of children when physicians, in cooperation with transnational biotechnology corporations, looked for the sales of marketable and economically profitable products made from harvested human foreskins that could further be used in the pharmaceutical industry' (ibid.:251). Medicine is in league with the powerful and dangerous global pharmaceutical industry in an illegal trade. And this in the name of a science without ethical bounds that 'can be a double edge sword that readily lends itself as an alibi for strongly held preferences and cultural biases' (ibid.). 20 Here, Hill and Denniston repeat word for word a sentence from an article by Ntozi (1997, 99) that puts forward a measured position on MC and recommends proceeding to RCTs.

Backing the opposition to MC, a conspiracy theory is clearly voiced and its xenophobic dimension is hardly concealed. Its anti-Semitic stance becomes quite clear in two titles of the "Foreskin Man" MGMBill comic book series, two Kindle eBooks21 , condemned by the Antidefamation League because of its 'age-old anti-Semitic canards such as …the accusation that Jews ritually murder Christian children… [or] when a character complains that the "procircumcision lobby" has 'all the well-connected doctors and lawyers'22 .

Understanding the resistances: a contextual analysis

As seen above, the discourse against male circumcision is articulated around three main arguments: an arbitrary power over Black African bodies exerted by big international organizations and its colonial reminiscences; an inadequate scientific research and the distrust it fosters; the exceptionality of the African situation. Situating these arguments in their context helps to understand the resistances to the implementation of the practice.

Blaming the victims

Drawing on a long tradition of racist and eugenic discourses, a common explanation of AIDS dissemination throughout Africa considers that the disease is the consequence of an overabundant, uncontrolled, primitive sexuality, compounded by disintegrating social structures and ensuing deculturation [START_REF] Vaughan | Colonial power and African illness[END_REF]. 'The disease was deflected onto Africa as primal other, Africa as an icon of dangerous desire, Africa as the projection of a self never fully tameable' (Comaroff, 2007, 197). Such assertions are so widely shared they do not even need to be proven.

Furthermore, a demonstration of AIDS growth in Africa as anti-scientific as [START_REF] Rushton | Population differences in susceptibility to AIDS: An evolutionary analysis[END_REF], based on a pseudo-evolutionist theory arguing that Blacks are unable to control their impulses, was published in a review as prestigious as Social Science and Medicine.

Subsequently, the distorted and stigmatising nature of the scholarly discourse on AIDS in Africa was highlighted [START_REF] Bibeau | L'Afrique, terre imaginaire du sida. La subversion du discours scientifique par le jeu des fantasmes[END_REF][START_REF] Vaughan | Colonial power and African illness[END_REF]. Over a decade later, such criticism is still relevant. 'Century-old stereotypes that emphasize exotic and exceptional sexuality encumber the attempts to understand the intensity of the HIV/AIDS epidemic' (Stillwagon, 2003, 811).

Assumptions without investigation have been made about the higher number of sexual partners but surveys have showed that people in Africa do not have more partners then Westerners23 .

Challenged by this rather unflattering description, Africans have attempted to reverse the stigma: thus, South African president Thabo Mbeki rejected the association between AIDS and sexually transmitted disease on the grounds that it 'perpetuates Western racist stereotypesand the Euro-American propensity to use African bodies for experimentation and profit' (Comaroff, 2007, 214). Africans' resistance to Western attempts at controlling their sexuality is quite understandable indeed.

Using and controlling African bodies

African countries are increasingly used as operational fields for medical and pharmaceutical research, public health actions, and mass therapies. This trend is not new but its intensification is obvious. Via such interventions, many international organisations as well as powerful pharmaceutical firms control, or even replace existing low-resource power structures and political authorities. Hence, they are able to escape any form of control and violate current ethical rules enforced in Western countries. From the Western world's viewpoint, bio-medical actions are justified forms of sanitary interference motivated by humanitarian emergency, poor resources, and political shortcomings [START_REF] Vinh | Government-by-exception: Enrolment and experimentality in mass HIV treatment programmes in Africa[END_REF]. But viewed by those who are the targets of these actions, they generate distrust, suspicion and multiple rumours that attempt to explain such dubious condescendence. 'Two logics are at work: an economy of resentment, whereby the past constitutes an inexhaustible reservoir of painful memory, and an economy of suspicion, whereby the present is interpreted through the lens of an intense mistrust of anyone making any claim to authority' (Fassin, 2007: XIX). The infamous Tuskegee syphilis experiment , a « clinical trial » that followed over four decades the natural (i.e., with no treatment) evolution of syphilis in a Black Population segment from Alabama fuelled a conspiracy theory, according to which 'AIDS is a form of racial genocide employed by Whites to exterminate Blacks' (Jones, 1993, 2). Likewise, in Africa, many rumours circulate on condoms24 , treatments, drugs tested in clinical trials, which are all considered as vectors for conspiracies schemed by Whites to harm Africans, undermine their fecundity [START_REF] Kaler | Health interventions and the persistence of rumour: The circulation of sterility stories in African public health campaigns[END_REF] and spread diseases25 .

Like vaccination resisters in the Western countries [START_REF] Hobson-West | Trusting blindly can be the biggest risk of all': organized resistance to childhood vaccination in the UK[END_REF], people in Southern and Eastern African countries resist health technologies, such as MC, which is not supposed to cure but to prevent partially the unknown risk of a disease: surgery on a very sensitive -and symbolic-part of a sane male body. Of course, such resistance is difficult to understand and may appear illogical to researchers who have dedicated their energy studying MC, as well as to actors in charge of promoting it. While they acknowledge that health public action requires time, they have no doubts about the fact that the practice of circumcision should extend for the well being of all, as the emergency is to curb the devastating epidemic. The exceptionality of AIDS legitimises a form of therapeutic domination, reminiscent of a colonial past, which, decades after its official end, still serves as a matrix to explain the present [START_REF] Nguyen | Social Context and determinants of HIV transmission[END_REF][START_REF] Vinh | The republic of therapy. Triage and sovereignty in West Africa's time of AIDS[END_REF]. 'This framing of AIDS as a humanitarian crisis is empirically justified; however doing so mobilizes a range of political and therapeutic technologies that highlight the biopolitical dimension of HIV interventions and point to how they effectively enact under the rubric of AIDS a government-by-exception' (Nguyen, 2009, 202).

The deepening discrepancy between a policy of promoting MC in Africa and a tendency to curb its frequency in the Western world threatens to severely jeopardise its acceptance by populations for which its signification is prophylactic rather than symbolic; for these populations, it is far from obvious that health is a value superior to others [START_REF] Fassin | When bodies remember: Experiences and politics of AIDS in South Africa[END_REF], and over which each person should exert individual control, in the name of a "biological citizenship" -a corollary of Western biomedicine. 'In the West novel practices of biological choice are taking place within a 'regime of the self' as a prudent yet 26 enterprising individual, actively shaping his or her life course through acts of choice' (Rose and Novas, 2005, 460). In lowerresource countries where all sorts of uncontrollable dangers threaten people's daily lives, this 'regime of the self' is hardly in tune with the way individuals cope with their immediate problems. Seeking to compel them to conform to this approach, and blaming them when they resist, amounts to exerting a form of symbolic violence against them.

Conclusion: A turning point, MC versus the promotion of anti HIV treatments

As is the case with other medical actions, such as vaccinations, the implementation of MC in Southern and Eastern Africa faces a number of obstacles, which we have endeavoured to highlight. Indeed, a behavioural modification as loaded with significance as MC is not easily accepted by populations for which it is alien. Those who strive to promote circumcision are animated by the best will in the world; they trust that with patience, a good strategy and an appropriate, efficient, painless technology, they will succeed in overcoming these obstacles, especially if local political authorities cooperate. However, one can understand why such acceptance by populations is not guaranteed. In countries where HIV/AIDS infection prevalence is highest, MC promoters are caught between the emergency that dictates their action and slow progress in terms of disseminating the practice. Nonetheless, epidemiological data is clear, whether in terms of prevalence in countries where circumcision is widespread or not, or in RCT results. But local political motivation is deficient, and men are recalcitrant. This situation would be difficult to grasp without using the conceptual elements we have developed, and which vividly highlight the root causes of such reluctance: reminiscences of colonial domination, stigmatisation, and strategies aimed at controlling the human body, interventions parachuted from Western countries. Moreover, how can one accept a recommendation that is simultaneously and increasingly challenged precisely in the very countries attempting to export it? Either MCs are seldom practised in Western countries or its frequency is decreasing. There are calls to subject MC to legal restrictions or even to prohibit the practice with minors. The growing anti-circumcision movement in the United States needs to globalise its fight by opposing MC in African countries too. Equated by its advocates with deleterious female genital mutilations or to other harmful actions perpetrated on human bodies, MC is seen as reflecting the perpetuation of a primitive state of mind.

Moreover, MC is an inexpensive operation that is performed once in a lifetime. There are no profits to be made, in particular for the powerful pharmaceutical industry, whether it manufactures patented or generic drugs. On the other hand, by disseminating the therapeutic as well as preventive use (by the so-called pre-exposure prophylaxis -PrEP) of antiretroviral drugs, big pharma groups would extend considerably their market shares and profits (Greene, 2007) -even if they cut prices. They could, furthermore, count on international solidarity to finance the dissemination of anti-retroviral treatments in poor countries. Praiseworthy efforts by the Global Fund to Fight AIDS, Tuberculosis and Malaria enable increasing numbers of people in Africa to access the treatment, hence curbing the epidemic, since a properly treated person has practically no risk of transmitting HIV infection. A great number of clinical trials are currently dedicated to PrEP, which is promoted by major organisations or foundations. Many scientists believe PrEP would enable actors to drop behavioural prevention, to rehabilitate biomedicine's primacy in the realm of AIDS, without having to wait for a hypothetical vaccine. It would indeed supplant MC. No further need to carry out arduous negotiations with human beings to lead them to control their health, just a 'magic bullet' that is supposed to be easily acceptable. A 'magic bullet', a vector of permanent power over human bodies that would, moreover, generate substantial profits. This is the hope -and probably the illusion -looming at the onset of the fourth decade of the AIDS pandemic.

  Malawian secretary to the Office of the President, responsible for HIV/AIDS and Nutrition, decided not to promote MC because of a lack of evidence supporting the practice 17 . Conversely, Kenya performed 91,300 MC between 2009-June 2010 18 , but except in the Nyanza Province where the MCs were performed, the majority of Kenyan men are traditionally circumcised. On the reverse, only 350 MC were reported in Namibia at the same period and 542 in Rwanda. The MM., there is indeed a long way to go before achieving a significant impact on the epidemic. Even if a public policy needs time to gain acceptance, one may wonder if the slow development of MC in eastern and southern African countries where it is promoted does not reflect a mounting opposition in Western countries.

	countries with the highest HIV rate m a d e s o m e e f f o r t s : 6,180 MC in Botswana (24,8%
	prevalence) and 10,000 in Swaziland (25,9% prevalence). With a population of respectively 2
	MM. and 1,2

  repeated the same methodological arguments: selection bias, religion as a confounding factor, inadequate statistical analysis.

	Likewise, for Van Howe et al. (2005), epidemiological research on MC wrongfully depicted itself
	as scientific. 'Several opinion pieces published in the medical literature have been portrayed as
	"studies"…Scientific efforts to understand, contain and prevent HIV infection are more likely to
	17 http://www.nyasatimes.com/health/shaba-says-malawi-cannot-follow-rwanda-on-circumcision-to-
	fight-aids.html , accessed 10 June 2010.

18 UNAIDS. Global Report 2010. www.unaids.org/globalreport/global_report.htm, accessed 9 June 2011.

19

Some websites opposed to

MC: cirp.org; circumcisionandhiv.com; nocirc.org; doctorsopposingcircumcision.org; stopinfantcircumcision.org; intactamerica.org; mgmbill.org.

Analyses of the internal validity of the research projects are found in the literature and integrated into our approach.

The UNAIDS global report on the AIDS epidemic (2010) shows how contrasted is the prevalence in different countries: 25.9% in Swaziland, 24,8% in Botswana, 23,6% in Lesotho versus 0,7% in Mauritania, 0,8% in Niger, 0,9% in Senegal. http://www.unaids.org/globalreport/global_report.htm (consulted June 2010).

Even between countries of Southern and Eastern Africa, sharp contrasts can be observed: for[START_REF] Swidler | Responding to AIDS in Sub-Saharan Africa: Culture, Institution and Health[END_REF] who compares the contrasted HIV prevalence rates in Botswana (23,6%) and Uganda (7%), "what is missing [in Botswana] is precisely the activation of social solidarities, the sense of community and the mobilization of collective identities" (144).

The UNAIDS global report on the AIDS epidemic (2010) shows how contrasted is the prevalence in different countries: 25.9% in Swaziland, 24,8% in Botswana, 23,6% in Lesotho versus 0,7% in Mauritania, 0,8% in Niger, 0,9% in Senegal. http://www.unaids.org/globalreport/global_report.htm (consulted June 2010).

As in the following two RCTs, the ethical aspects were taken into consideration(Cleaton-Jones, 2005).

The use of this word doubtless relates to the religious dimension of MC, not mentioned directly here.

The Cochrane collaboration aims to be a global resource for systematic review and a critical appraisal of research articles. Set up by experts according to a highly codified procedure, Cochrane reviews are a basic reference.

'Because the randomized trial and especially the systematic review of several randomized trials is so much more likely to inform us and so much less likely to mislead us, it has become the 'gold standard' for judging whether a treatment does more harm than good'(Sackett et al., 1996: 72).

This assertion is infirmed by the data of the RCT carried out in Kisumu, Kenya[START_REF] Mattson | Risk compensation is not associated with male circumcision in Kisumu, Kenya : A multifaceted assessment of men enrolled in a randomized controlled trial[END_REF].

The American Academy of Paediatrics issued a 1999 statement opposed to newborn circumcision that had a dramatic impact on parents' decisions and induced several states to withdraw Medicaid coverage. See http://www.aap.org

www.who.int/hiv/pub/malecircumcision/meetingreport_nov07/en/index.html/

See the website Plusnews Global HIV/AIDS news and analysis (plusnews.org, accessed 10 June 2011): 3 December 2009: South Africa: Male circumcision, why the delay ? 23 July 2009: Africa: Male circumcision slowly taking off; 2 March 2010: Tracking the male circumcision roll-out.

www.foreskinman.com (accessed 15 June 2011).

www.adl.org/PresRele/ASUS_12/6064_12.htm (posted 3 June 2011, accessed 15 June 2011).

As shown in longitudinal studies in South Africa, Uganda and Zimbabwe, three countries with high prevalence:Todd, Cremin, McGrath and al., 2008.

Notably, White doctors who infect the condoms they manufacture with the HIV virus (in the form of 'tiny worms')(Stadler, 2003).

Narratives revolving around blood abound in the rumours circulating during clinical trials, whether it is taken from young children in Ghana[START_REF] Newton | Drawing blood from young children: lessons learned from a trial in Ghana[END_REF] or at a trial on the PRO 2000 microbicide[START_REF] Stadler | Rumours about blood and reimbursements in a microbicide gel trial[END_REF] -where a women said: 'We are always giving blood, we will end up having no blood at all' (p. 349) -or, even worse, stolen during a pneumococcal vaccine trial in the Gambia[START_REF] Fairhead | Where techno-science meets poverty: Medical research and the economy of blood in The Gambia, West Africa[END_REF].

Acknowledgments

This work was supported by the French National Agency for AIDS and Viral Hepatitis research (ANRS). I thank James Bryant for his help for the translation and editing work.


Carl Ganter

Walter Schirmacher
email: walter.schirmacher@ph.tum.de

Euclidean random matrix theory: low-frequency non-analyticities and Rayleigh scattering

Keywords: theoretical, disorder, phonons, hopping transport, diffusion Rayleigh scattering topolocigal disorder, euclidean random matrix, Rayleigh scattering, diffusion, long-time tails, velocity autocorrelation function, low-frequency nonanalyticity

come

Introduction

Rayleigh scattering [START_REF] Strutt | Third Baron Rayleigh[END_REF], i.e. the fact that the mean-free path of weakly scattered waves varies as ω -(d+1) in a d-dimensional disordered medium as ω → 0, is widely believed to be a general property of quenched disordered matter [2][3][4][5][6]. However, R.1 recently it has been claimed [7] that a harmonic system with displacements u i (t) obeying

d 2 dt 2 u i (t) = - j t ij u i (t) -u j (t) , (1)
where i and j denote random sites r i,j in d-dimensional space, would have wavelike excitations, which have a line-width (inverse mean-free path), varying with ω 2 instead of ω 4 in d = 3. t ij are force constants, divided the mass at the node i, which are assumed to depend on the distance, i.e. t ij = t(r ij ). The claim of absence of Rayleigh scattering had been substantiated by a high-density expansion and a diagrammatical analysis [7]. This claim is not only astonishing with respect to the mentioned general view on waves in disordered media, but it is also in contradiction with the known analytic properties of the analogous diffusion system. If one replaces the double time derivative in [START_REF] Strutt | Third Baron Rayleigh[END_REF] equation of a random walk among the sites i, j:

d dt n i (t) = - j t ij n i (t) -n j (t) , (2)
where n i (t) give the odds for the walker to be at i at time t and t ij = t(r ij ) is the hopping probability per unit time. Eq. ( 2) describes e.g. the motion of electrons hopping among shallow impurities in a semiconductor [8,9,[START_REF] Shklovskii | Electroic Properties of Doped Semiconductors[END_REF][START_REF] Böttger | Hopping Conduction in Solids[END_REF]. Such a random walk is known [10] to exhibit a long-time tail of the velocity-autocorrelation function (VAF) varying as Z(t) ∝ t (d+2)/2 for t → ∞ [10,[START_REF] Ernst | [END_REF], a feature shared with Lorentz models [START_REF] Ernst | [END_REF][14][15]. In fact, the Laplace transform of the VAF is the frequency-dependent diffusivity D(z = iω + ), which has, according to the Tauberian theorems [START_REF] Feller | An Introduction to Probability Theory and Its Applications[END_REF] a low-frequency singularity D(z) → z d/2 , |z| → 0. Now, in the analogous vibrational problem this quantity corresponds to the square of a frequency-dependent sound velocity D(z = -ω 2 + i ) = v 2 (z). The imaginary part v (ω) of the latter is related to the mean-free path via

1 (ω) = 2ωv (ω) |v(z)| 2 (3)
This gives ∝ ω -(d+1) , i.e. Rayleigh scattering. We conclude that the long-time tail of the VAF in the diffusion problem is mathematically equivalent to the Rayleighscattering property.

In the following we calculate all irreducible diagrams (self-energy diagrams) up to second order in the inverse density ρ -1 = V /N , where N is the number of sites and V the volume. We show that to this order the self energy is proportional to k 2 z d/2 , z = iω + (diffusion) or z = -ω 2 + i (sound). and not as claimed in Refs. [7] ∝ k 2 z (d-2)/2 . We also show, why the so-called cactus approximation for a self-consistent theory erroneously leads to a non-analyticity z (d-2)/2 instead of z d/2 .

Formalism

As in refs. [7] we start from a high-frequency (z) and high-density (ρ = N/V ) expansion of the averaged propagator

G(k, z) = 1 N mn e ikrmn [z1 -M] -1 mn = 1 z + ∞ p=1 1 z p+1 1 N i0...ip e ikri 0 i 1 M i0i1 . . . e ikri p-1 ip M ip-1ip
Here M is a matrix with off-diagonal elements M ij = t ij and diagonal elements

M ii = - =i t i . t(k) = t(k) is the d-dimensional
Fourier transform of t(r). The configurationally averaged Green's function can now be expressed in terms of the irreducible self energy Σ (k, z) as follows The frequency-dependent diffusivity/sound velocity is given by

G (k, z) = 1 z -ρ [ t (k) -t (0) ] -Σ (k, z) k→0 = 1 z + D(z)k 2 (4)
D(z) = v(z) 2 = - 1 2 ∂ 2 ∂k 2 t(k) + Σ(k, z) k→0 (5)
For simplicity, we assume complete site disorder (i.e. the radial pair correlation function g (r) ≡ 1). Therefore t (k) is simply the Fourier transform of the transition rate t (r) [17].

It is possible, to consider a more general case, e.g. by replacing t n (r ij ) → g (r ij ) t n (r ij ) (n > 0) in every diagram (Kirkwood factorization). Without proof, it can be demonstrated that the analytical properties of the first and second order diagrams would remain unaffected by such an extension.

R.4

We denote the unrenormalized part of the Green's function by G 0 :

G 0 (k, z) := 1 z -ρ [ t (k) -t (0) ] (6)
In analogy to the approach in Ref. [7], it is helpful to consider a high-density expansion of the propagator, which is in turn determined by an analogous expansion the self energy Σ (k, z)

Σ (k, z) =: ∞ n=1 ρ -n Σ (n) (k, z) , (7)
As outlined in [7], the index n counts repetitions of sites in the high frequency / high density expansion.

In the following, we will derive exact results for n = 1 and n = 2.

To this end, we will use diagrammatic representations (as explained in Figure 1), to distinguish topologically different contributions to the self energy.

In the following, we calculated the diagrams for the special case d = 3, but the results remain valid for arbitrary dimensions.

R.3

3. First-order diagrams: Σ (1) (k, z)

This case is comparably trivial and requires the addition of four diagrams (cf. Figure 1), since the first and last connection can refer to an off-diagonal (O) or a diagonal (D) transition rate.

OO = ρ dp (2π) 3 t 2 (p) G 0 (p, z) OD = -ρ dp (2π) 3 t (k -p) t (p) G 0 (p, z) DO = -ρ dp (2π) 3 t (k -p) t (p) G 0 (p, z)
= [ z + ρ t (0) ] -1 , which is formally obtained as G 0 (z) = lim k→∞ G 0 (k, z).
In most cases, this requires no special attention. Exceptions, when these terms need to be explicitly excluded to preserve irreducibility will be mentioned below. Open circles will always indicate start and end points of a diagram.

DD = ρ dp (2π) 3 t 2 (k -p) G 0 (p, z)
Added together:

ρ dp (2π) 3 [ t (k -p) -t (p) ] 2 G 0 (p, z) (8)
Since we are mainly interested in the imaginary part of

lim z→0 lim k→0 Σ (k, z) (9)
we will have to examine the bracket in Eq. ( 8) in the limit k → 0

lim k→0 t (k -p) -t (p) = t (p) p • kp + 1 2 t (p) • p -t (p) p 3 • [ kp ] 2 + 1 2 t (p) p • k 2 + O k 3 (10) We therefore obtain lim p→0 lim k→0 [ t (k -p) -t (p) ] 2 = c • [ kp ] 2 ∝ p 2 +O k 3 , p 3 (11)
with some usually nonzero constant c. With the additional p 2 factor from the threedimensional integral, we obtain from the diffusion pole of G 0 (k, z) 4. Second-order diagrams:

lim z→0 lim k→0 Im Σ (1) (k, z) ∝ z 3/2 k 2 (12)
Σ (2) (k, z)
This case is considerably more complex. It turns out to be advantegeous to consider topologically different groups of irreducible diagrams separately:

• Σ (2) α (k, z): (1 * 2 • • • 2 • • • 1) • Σ (2) β (k, z): (1 2 • • • 2 • • • 1) • Σ (2) γ (k, z): (1 * 2 • • • 1 • • • 2) • Σ (2) δ (k, z): (1 2 • • • 1 • • • 2) • Σ (2) ε (k, z): (1 • • • 1 • • • 1)
Unlike in • • • at least one additional site index (differnt from 1 and 2) needs to be contained in * .

The complete and exact second-order self energy is then just the sum of these partial contributions:

Σ (2) (k, z) = Σ (2) α (k, z) + Σ (2) β (k, z) + Σ (2) γ (k, z) + Σ (2) δ (k, z) + Σ (2) ε (k, z) (13) 4.1. Σ (2) α (k, z): Irreducible Diagrams (1 * 2 • • • 2 • • • 1)
Here, the 16 diagrams in Figure 2 need to be distinguished [18]:

OOOO = ρ 2 dp dq t 3 (p) t (q) G 2 0 (p) G 0 (q) OOOD = -ρ 2 dp dq t (k -p) t 2 (p) t (q) G 2 0 (p) G 0 (q) OODO = -ρ 2 dp dq t 3 (p) t (p -q) G 2 0 (p) G 0 (q) OODD = ρ 2 dp dq t (k -p) t 2 (p) t (p -q) G 2 0 (p) G 0 (q) ODOO = -ρ 2 dp dq t 2 (p) t (p -q) t (q) G 2 0 (p) G 0 (q) ODOD = ρ 2 dp dq t (k -p) t (p) t (p -q) t (q) G 2 0 (p) G 0 (q) ODDO = ρ 2 dp dq t 2 (p) t 2 (p -q) G 2 0 (p) G 0 (q)
α (k, z) ODDD = -ρ 2 dp dq t (k -p) t (p) t 2 (p -q) G 2 0 (p) G 0 (q) DOOO = -ρ 2 dp dq t (k -p) t 2 (p) t (q) G 2 0 (p) G 0 (q) DOOD = ρ 2 dp dq t 2 (k -p) t (p) t (q) G 2 0 (p) G 0 (q) DODO = ρ 2 dp dq t (k -p) t 2 (p) t (p -q) G 2 0 (p) G 0 (q) F o r P e e r
DODD = -ρ 2 dp dq t 2 (k -p) t (p) t (p -q) G 2 0 (p) G 0 (q) DDOO = ρ 2 dp dq t (k -p) t (p) t (p -q) t (q) G 2 0 (p) G 0 (q) DDOD = -ρ 2 dp dq t 2 (k -p) t (p -q) t (q) G 2 0 (p) G 0 (q) DDDO = -ρ 2 dp dq t (k -p) t (p) t 2 (p -q) G 2 0 (p) G 0 (q) DDDD = ρ 2 dp dq t 2 (k -p) t 2 (p -q) G 2 0 (p) G 0 (q)
Added together:

Σ (2) α (k, z) = (14)
ρ 2 dp dq [ t (k -p) -t (p) ] 2 [ t (p -q) -t (p) ] [ t (p -q) -t (q) ] G 2 0 (p) G 0 (q)
Two cases need to be distinguished:

p is small: Because of Eq. ( 11), the first squared bracket delivers a factor p 2 . Additional p 2 factors result from the third bracket and the 3D integration, respectively, so that we finally arrive at a p 6 factor. From the identity G 2 0 (p, z) ∝ ∂ ∂z G 0 (p, z), we obtain a nonanalyticity ∝ z 3/2 . Note that uneven occurrences of p and/or q, such as an isolated product pq, are not rotation invariant and therefore do not contribute to the integral. q is small: Now the second bracket delivers an additional nonanalyticity [19] q 2 , which again produces a z 3/2 nonanalyticity.

We therefore conclude for this group of diagrams

lim z→0 lim k→0 Im Σ (2) α (k, z) ∝ z 3/2 k 2 (15) 4.2. Σ (2)
β (k, z): Irreducible Diagrams (1 2 • • • 2 • • • 1)
Since sites 1 and 2 are connected directly via t o 12 or t d 12 , the diagrams contain only three t's and two G 0 's. Therefore, only eight diagrams are possible (cf. Figure 3): Added together:

OOO = ρ dp dq t 2 (p) t (q) G 0 (p) G 0 (q)
β (k, z) OOD = -ρ dp dq t (k -p) t (p) t (q) G 0 (p) G 0 (q) ODO = -ρ dp dq t 2 (p) t (p -q) G 0 (p) G 0 (q) ODD = ρ dp dq t (k -p) t (p) t (p -q) G 0 (p) G 0 (q) DOO = -ρ dp dq t (p) t (p -q) t (q) G 0 (p) G 0 (q) DOD = ρ dp dq t (k -p) t (p -q) t (q) G 0 (p) G 0 (q) DDO = ρ dp dq t (p) t 2 (p -q) G 0 (p) G 0 (q) DDD = -ρ dp dq t (k -p) t 2 (p -q) G 0 (p) G 0 (q) F o r P e e r
Σ (2) β (k, z) = (16) -ρ dp dq [ t (k -p) -t (p) ] [ t (p -q) -t (p) ] [ t (p -q) -t (q) ] G 0 (p) G 0 (q)
Because of (10), the first bracket gives us only a k 2 . Depending on whether p or q are small, the third, respectively second, bracket delivers the required additional p 2 , respectively q 2 , to obtain:

lim z→0 lim k→0 Im Σ (2) β (k, z) ∝ z 3/2 k 2 (17)
Note a particular property of diagrams DDO and DDD: If the propagator in the middle of the diagram collapses to the diagonal G 0 , as explained above, these diagrams are factorizable at site 1 and therefore not irreducible anymore. To avoid double counting, the diagonal term G 0 must therefore be subtracted from this propagator.

It can be easily verified, however, that the z 3/2 -nonanalyticity is not affected by this.

4.3. Σ (2) γ (k, z): Irreducible Diagrams (1 * 2 • • • 1 • • • 2)
The crossover topology of these 16 diagrams (Figure 4) leads to more intricate convolution integrals.

Based on the following abbreviations

α := k -p -q β := p γ := q a := p + q b := k -p c := k -q (18) we obtain OOOO = ρ 2 dβ dγ G 0 (α) G 0 (β) G 0 (γ) • t (α) t (β) t 2 (γ) OOOD = -ρ 2 dβ dγ G 0 (α) G 0 (β) G 0 (γ) • t (a) t (β) t 2 (γ) OODO = -ρ 2 dβ dγ G 0 (α) G 0 (β) G 0 (γ) • t 2 (α) t (a) t (γ)
γ (k, z) OODD = ρ 2 dβ dγ G 0 (α) G 0 (β) G 0 (γ) • t (b) t (c) t 2 (β) ODOO = -ρ 2 dβ dγ G 0 (α) G 0 (β) G 0 (γ) • t (α) t (a) t (β) t (γ) ODOD = ρ 2 dβ dγ G 0 (α) G 0 (β) G 0 (γ) • t 2 (a) t (β) t (γ) ODDO = ρ 2 dβ dγ G 0 (α) G 0 (β) G 0 (γ) • t (b) t (c) t (β) t (γ) F o r P e e r R e v i e w O n l y ODDD = -ρ 2 dβ dγ G 0 (α) G 0 (β) G 0 (γ) t (b) t 2 (c) t (β) DOOO = -ρ 2 dβ dγ G 0 (α) G 0 (β) G 0 (γ) • t (α) t (a) t (β) t (γ) DOOD = ρ 2 dβ dγ G 0 (α) G 0 (β) G 0 (γ) • t (α) t (b) t (c) t (β) DODO = ρ 2 dβ dγ G 0 (α) G 0 (β) G 0 (γ) • t (α) t 2 (a) t (γ) DODD = -ρ 2 dβ dγ G 0 (α) G 0 (β) G 0 (γ) • t 2 (b) t (c) t (β) DDOO = ρ 2 dβ dγ G 0 (α) G 0 (β) G 0 (γ) • t (α) t (b) t (c) t (γ) DDOD = -ρ 2 dβ dγ G 0 (α) G 0 (β) G 0 (γ) • t (α) t (b) t 2 (c) DDDO = -ρ 2 dβ dγ G 0 (α) G 0 (β) G 0 (γ) • t 2 (b) t (c) t (γ) DDDD = ρ 2 dβ dγ G 0 (α) G 0 (β) G 0 (γ) • t 2 (b) t 2 (c)

Collecting Terms

To derive a usable expression for the sum of these 16 diagrams, we have to exploit the symmetries of the problem.

Transforming variables p, q → p, q allows to arbitrarily permute α, β, γ under the boundary condition that a, b, c perform the same permutation, as indicated in the following table: For example, the transformation

p := k -p -q und q := p (20) leads to k -p -q → p (21) p → q (22) q → k -p -q (23) p + q → k -p (24) k -p → k -q (25) k -q → p + q (26)
and thus the permutation α β γ a b c → β γ α b c a.

The product G 0 (α) G 0 (β) G 0 (γ) is invariant with respect to these permutations and because of

dα dβ = dα dγ = dβ dγ (27)
the integration variables can be chosen freely. After suitably regrouping the tfactors, we obtain for the sum of all diagrams the expression

K := OOOO + • • • + DDDD = ρ 2 dβ dγ G 0 (α) G 0 (β) G 0 (γ) × × [ t (β) -t (b) ] • [ t (γ) -t (c) ] • [ t (γ) -t (β) ] • [ t (α) + t (β) ]
Since we integrate over two variables only, we have to eliminate one G 0 -factor. To this end, we apply a partial fraction decomposition

ρ [ t (γ) -t (β) ] • G 0 (β) G 0 (γ) = [ G 0 (γ) -G 0 (β) ] (28)
Some further permutations and regrouping lead to

K = ρ 2 dβ dγ G 0 (β) G 0 (γ) × × [ t (α) -t (a) ] • [ t (α) -t (γ) ] • [ t (γ) -t (c) ]
After reinserting the above definitions, we can expand for small k:

t (α) -t (a) = t (k -p -q) -t (p + q) = t (p + q) |p + q| • k (p + q) + O k 2 t (c) -t (γ) = t (k -q) -t (q) = t (q) q • kq + O k 2
We thus obtain lim k→0 K ∝ k 2 , as required. Since we only consider the lowestorder term, we may use

t (α) -t (γ) = t (a) -t (γ) + O (k) (29)
and set

t (α) -t (γ) ≈ t (p + q) -t (q) (30)
We thus finally arrive at

K = ρ 2 dp dq G 0 (p) G 0 (q) × (31)
× t (p + q) |p + q| • k (p + q) A • [ t (p + q) -t (q) ] B • t (q) q • kq C +O k 3
For the analytical properties, we again have to consider two cases:

q is small:

Bracket C gives us a factor q, but brackets A and B approach a finite value for q → 0.

But: After setting q = 0 in A and B, both the integral over p (because of the kp term in A) and over q (due to the kq in C) vanish due to lack of rotational invariance. Consequently, we have to expand the fraction in A to first order in q, which provides us with an additional factor pq ∝ q. Now rotational invariance is preserved and a z 3/2 nonanalyticity is obtained.

p is small: Bracket C remains finite and B leads to a factor pq ∝ p. Setting p = 0 in A leads to the following structure

dp dq f (q) • [ pq ] • [ kq ] 2 (32)
In order to restore rotational invariance, we again have to expand in A to first order in p, which yields an additional pq ∝ p and leads to a z 3/2 nonanalyticity. We thus conclude that lim

z→0 lim k→0 Im Σ (2) γ (k, z) ∝ z 3/2 k 2 (33)
holds.

Σ

(2)

δ (k, z): Irreducible Diagrams (1 2 • • • 1 • • • 2)
Similar to Σ β above, 8 diagrams need to be considered (Figure 5):

OOO = ρ dp dq t (k -p + q) t (p) t (q) G 0 (p) G 0 (q) OOD = -ρ dp dq t (k -p + q) t (k -p) t (q) G 0 (p) G 0 (q) ODO = -ρ dp dq t (k -p + q) t (p) t (p -q) G 0 (p) G 0 (q) ODD = ρ dp dq t (k -p + q) t (k -p) t (p -q) G 0 (p) G 0 (q) DOO = -ρ dp dq t (k -p) t (p) t (q) G 0 (p) G 0 (q)
ε (k, z) DOD = ρ dp dq t 2 (k -p) t (q) G 0 (p) G 0 (q) DDO = ρ dp dq t (k -p) t (p) t (p -q) G 0 (p) G 0 (q) DDD = -ρ dp dq t 2 (k -p) t (p -q) G 0 (p) G 0 (q)
Added together:

Σ (2) δ (k, z) ∝ (34) ρ dp dq [ t (k -p) -t (p) ] [ t (k -p + q) -t (k -p) ] [ t (p -q) -t (q) ] G 0 (p) G 0 (q)
The first bracket is proportional to k 2 . Depending on whether p or q is small, the third or second bracket yields the required additional p 2 or q 2 , respectively.

Thus we have here too:

lim z→0 lim k→0 Im Σ (2) δ (k, z) ∝ z 3/2 k 2 (35) 4.5. Σ (2) ε (k, z): Irreducible Diagrams (1 • • • 1 • • • 1)
Here only 4 cases are possible (Figure 6).

OO = -ρ dp dq t (p) t (p -q) t (q) G 0 (p) G 0 (q) OD = ρ dp dq t (k -p) t (p -q) t (q) G 0 (p) G 0 (q) F o r P e e r R e v i e w O n l y DO = ρ dp dq t (k -q) t (p) t (p -q) G 0 (p) G 0 (q) DD = -ρ dp dq t (k -p) t (k -q) t (p -q) G 0 (p) G 0 (q)
Added together:

Σ (2) ε (k, z) ∝ (36) ρ dp dq [ t (k -p) -t (p) ] [ t (k -q) -t (q) ] t (p -q) G 0 (p) G 0 (q)
For order O k 2 , because of (10), the first two brackets muts be proportional to ∝ kp and kq, respectively. For rotational invariance, the last bracket needs to be expanded and thus provides the additional required pq term, to obtain:

lim z→0 lim k→0 Im Σ (2) ε (k, z) ∝ z 3/2 k 2 (37)
With ( 13) it follows immediately

lim z→0 lim k→0 Im Σ (2) (k, z) ∝ z 3/2 k 2 (38)

Conclusion

Working out term by term in the second-order self energy we have convinced ourselves that to this order the nonalytic behavior (38), which both leads to Rayleightype sound attenuation and to the correct long-time tail in the analogous diffusion problem is recovered. This is in contrast to the claims in the publications [7]. In these publications a self-consistent equation for the self-energy is advocated, which consists in making the first-order result (8) self-consistent, i.e. replacing the 0-thorder Green's function by the full one. Now, in performing a high-density expansion of this equation one easily convinces oneself that the corresponding diagrams are

• the entire sum Σ

α (k, z); • the diagrams OOO, OOD, ODO, ODD of Σ As the partial sums do not give the correct analytic properties, this is the reason, why the self-consistent scheme advocated by [7] does not lead to Rayleigh scattering. We shall publish shortly a self-consistent scheme, which includes Rayleigh scattering. Irreducible diagrams in Σ (1) (k,z).

F

Off-diagonal matrix elements, t o ij ≡ t(rij), associated with a site change, are represented by solid lines. Dashed lines do not effect a site change. The associated transition rates t d ij ≡ -t(rij) belong to the diagonal matrix elements and are a consequence of the sum rule.

The unrenormalized propagator [G0]ij ≡ G0(rij) is shown as a double-line. Note that the propagator contains a diagonal part (for i = j the diagram has length zero) G0(z) := [z + ρt(0)] -1 , which is formally obtained as G0(z) = limk→∞ G0(k,z). In most cases, this requires no special attention. Exceptions, when these terms need to be explicitly excluded to preserve irreducibility will be mentioned below.

1 Figure 1 .

 11 Figure 1. Irreducible diagrams in Σ (1) (k, z). Off-diagonal matrix elements, t o ij ≡ t (r ij ), associated with a site change, are represented by solid lines. Dashed lines do not effect a site change. The associated transition rates t d ij ≡ -t (r ij ) belong to the diagonal matrix elements and are a consequence of the sum rule. The unrenormalized propagator [ G 0 ] ij ≡ G 0 (r ij ) is shown as a double-line. Note that the propagator contains a diagonal part (for i = j the diagram has length zero) G 0 (z) := [ z + ρ t (0) ] -1, which is formally obtained as G 0 (z) = lim k→∞ G 0 (k, z). In most cases, this requires no special attention. Exceptions, when these terms need to be explicitly excluded to preserve irreducibility will be mentioned below. Open circles will always indicate start and end points of a diagram.

Figure 2 .

 2 Figure 2. Irreducible Diagrams in Σ (2)

Figure 3 .

 3 Figure 3. Irreducible Diagrams in Σ (2)

Figure 4 .

 4 Figure 4. Irreducible Diagrams in Σ (2)

  α β γ a b c α γ β a c b β α γ b a c β γ α b c a γ α β c a b γ β α c b a (19)

Figure 5 .

 5 Figure 5. Irreducible Diagrams in Σ (2) δ (k, z)

1 Figure 6 .

 16 Figure 6. Irreducible Diagrams in Σ (2)

( 2 )

 2 β (k, z), but not the remaining four diagrams;• the diagrams DOO, DOD, DDO, DDD of Σ

( 2 )

 2 δ (k, z), but not the remaining four diagrams.

  Open circles will always indicate start and end points of a diagram.156x35mm (600 x 600 DPI)Irreducible Diagrams in Σβ(2) (k,z) 161x74mm (600 x 600 DPI)

  by a single one, one obtains the

	September 12, 2010	17:12	Philosophical Magazine	ganter-schirmacher-rev-1
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