Senior Research Fellow Kate M Fleming
email: kate.fleming@nottingham.ac.uk

Clinical Joe West

Consultant Guruprasad P Aithal

Astrid E Fletcher

Consultant Gastroenterologist

Hepatobiliary Physician

Abnormal liver tests in people aged 75 and over: prevalence and association with mortality

Keywords: liver tests, liver function tests, mortality, prevalence

Abnormal liver tests in people aged 75 and over:

prevalence and association with mortality.

Older people represent an increasingly large group of health-care users in the UK.

An estimated 40% of all 75-79 year olds have at least one prescription for statins per year [START_REF]NICE implementation uptake report: statins[END_REF] and liver tests are performed in this group often to determine their suitability for prescription of statins.[2] Many of the misconceptions regarding statins have since been challenged; idiosyncratic hepatototoxicity from statins is rare and prior liver test abnormalities are not now considered a contraindication to statin prescription. [START_REF] Bader | The myth of statin-induced hepatotoxicity[END_REF] A recent study suggested that majority of liver test abnormalities detected prior to statin prescription were due to non-alcoholic fatty liver disease and relative risk reduction in cardiovascular events from statins was greater in patients with prior liver test abnormalities compared with those with normal liver tests. [START_REF] Athyros | Safety and efficacy of long-term statin treatment for cardiovascular events in patients with coronary heart disease and abnormal liver tests in the Greek Atorvastatin and Coronary Heart Disease Evaluation (GREACE) Study: a post-hoc analysis[END_REF] However, liver tests are still routinely performed, and it is surprising that we do not have any robust information on the proportion of older people with elevated liver tests or what any abnormalities in these tests may mean for their future health. Currently primary care physicians and hepatologists use a pragmatic approach to the management of these abnormalities, in the absence of evidence to the contrary, whereby repeated measurements are made before further investigation is carried out. To date only four population based studies in The Netherlands [START_REF] Schindhelm | Alanine aminotransferase predicts coronary heart disease events: a 10-year follow-up of the Hoorn Study[END_REF], South Korea [START_REF] Kim | Normal serum aminotransferase concentration and risk of mortality from liver diseases: prospective cohort study[END_REF] the USA [START_REF] Lee | Serum aminotransferase activity and mortality risk in a United States community[END_REF] and Scotland [START_REF] Mclernon | Health outcomes following liver function testing in primary care: a retrospective cohort study[END_REF] have examined the relationship between liver tests and allcause mortality in adults with conflicting conclusions as to the magnitude of any association primarily due to the wide variation in populations studied and, with respect to the European studies, no truly population based approach. No studies have particularly concentrated on a population of older people.

We therefore aimed to estimate the prevalence of abnormal liver tests in a random sample of people aged 75 and over; to examine the association between abnormal liver tests and demographic, lifestyle and clinical characteristics in people aged 75 and over; and to determine the risk of all cause and cause-specific mortality in elderly people with abnormal liver tests.

Methods

Data source and participants

We used data collected as part of the Medical Research Council (MRC) Trial of Assessment and Management of Older People in the Community. Full details of the trial protocol and results have been reported [START_REF] Fletcher | The MRC trial of assessment and management of older people in the community: objectives, design and interventions [ISRCTN23494848[END_REF], [START_REF] Fletcher | Population-based multidimensional assessment of older people in UK general practice: a cluster-randomised factorial trial[END_REF]. Briefly, 106 practices were recruited through the MRC General Practice Research Framework, selected to represent the population of the UK based on mortality (Standardised Mortality Ratios) and deprivation [START_REF] Jarman | Underprivileged areas: validation and distribution of scores[END_REF]. All people aged 75 or over registered with these selected general practices, non-resident in nursing homes, and who were not known to be terminally ill, were eligible for this trial which comprised two arms comparing multidimensional assessment of the elderly. One arm (universal arm) invited all patients to an in-depth health assessment whilst the other arm (targeted arm) invited only selected patients. We used data only from participants in the universal arm (53 practices) as they were considered a representative sample of community dwelling older people. A total of 15 308 of 20 934 (73.1%) potential participants attended for baseline assessment in the universal arm. Nonresponders were more likely to be women and were older. [START_REF] Osborn | Factors associated with depression in a representative sample of 14 217 people aged 75 and over in the United Kingdom: results from the MRC trial of assessment and management of older people in the community[END_REF] Data collection took place between 1995 and 1999.

Local Research Ethics Committee approvals were obtained for each participating practice.

Data collection and definitions

Socio-demographic information, self-reported medical history and lifestyle data (including alcohol consumption and smoking history) were obtained at the in depth assessment by nurse interviews. Body Mass Index (BMI) was defined as weight (in kg) divided by height (in m) squared, each of which were measured once. Waist circumference was measured to the nearest 0.1cm midway between the iliac crest and the lower rib margin by using an insertion tape. Hip circumference was Hip and waist were both measured twice (having asked patients to roll down their undergarments) and an average of the two values used in the calculation of waist:hip ratio (WHR). Diabetes was classified as present following either a self-reported medical diagnosis, prescription of anti-diabetic medication or a single high random blood glucose measurement. Heart attack, hypertension, stroke, cancer and ulcer were defined as a positive self-reported medical diagnosis. We used the Rose Chest Pain questionnaire [START_REF] Rose | The diagnosis of ischaemic heart pain and intermittent claudication in field surveys[END_REF] and categorised definite angina as chest pain that limits exertion, is situated over the sternum or in the left chest and left arm, and is relieved within 10 minutes by rest. Severe dementia was defined as a score of <9 /30on the Mini-Mental State examination [START_REF] Folstein | Mini-mental state". A practical method for grading the cognitive state of patients for the clinician[END_REF]. In order to assess the impact of multiple illnesses a co-morbidity score was derived. A patient was given a score of zero (not present) or one (present ever before the date of interview) for each of the following self-reported medical conditions: asthma, stomach / digestive ulcer, hypertension, heart attack, stroke, fractured hip or spine (within last year only), Parkinson's disease, cancer, and for each of the following: angina, dementia and diabetes (as per definitions above). Scores were added together to give an overall score and then categorised into 0, 1, 2 and 3 or more comorbidities.

At the assessment participants provided a blood sample which was sent to the local laboratory for a standard blood examination including aspartate transaminase (AST), alkaline phosphatase (ALP) and bilirubin (which we will henceforth collectively refer to as liver tests). In order to determine whether a patient had an abnormal liver test we requested reference ranges from the laboratories that provided the testing services for the practices involved in this arm of the trial. Reference ranges were not collected at the time of the study and only half of the practices involved had any recorded reference ranges for the three liver tests of interest. For those practices (n=25) where we did not have a provided reference range a median reference range was imputed. Abnormal liver tests were determined as the result being above the upper limit of normal relative to the reported or imputed laboratory reference range.

Abnormal results were then categorised as being between 1 and 2x the upper limit of normal or above 2x the upper limit of normal.

All participants in the trial were registered with the NHS central register for mortality follow-up and coded cause of death, date of death or date of migration for all individuals were provided through the UK Office for National Statistics. Deaths were coded according to the International Classification for Diseases, 9 th revision (ICD9) until September 2002, and according to the 10 th revision (ICD10) afterwards.

Specifically examined causes of death were cardiovascular disease (ICD9 401-405, 410-414, 425-447; ICD10 I10-I15, I20-I25, I42-I52, I6-I77, I79), cancer (all cancers excluding liver cancer (ICD9 140-154, 156-239; ICD10 C0-C21, C23-D4)), respiratory disease (ICD9 46-51; ICD10 J) and liver disease (viral hepatitis, liver cancer and liver disease (ICD9 070, 155, 570-573; ICD10 B15-B19, C22, K7)).

Statistical analyses

Prevalences of abnormal liver tests were calculated as the number of people with an abnormal result divided by the number of subjects with a valid test or combination of tests. 95% CIs were calculated using the binomial distribution.

To examine the association between abnormal liver tests and demographic, lifestyle and medical characteristics we used logistic regression to calculate odds ratios and 95% CIs, with standard errors adjusted to account for potential clustering within practices. Alcohol consumption was categorised in groups of 7-unit increments. BMI was categorised using accepted definitions [START_REF] Organization | Physical Status: the use and interpretation of anthropometry[END_REF] and WHR was categorised in 0.1 unit bands. After univariate analyses, a priori confounders of age, sex and comorbidity plus additional statistically significant associations were built into a multivariate logistic regression model to account for potential confounding. In the mortality analysis subjects were considered as contributing time to the analysis from the date of interview until death, censoring (migration) or 5 November 2005 (the last date of available data from the ONS) whichever came earliest. All cause and cause-specific mortality were examined for each abnormal liver test in turn.

Associations between death and demographic, lifestyle and medical characteristics were examined in a control population consisting of all those elderly people without any liver test abnormality in order to identify appropriate confounding variables. Cox proportional hazards regression analysis was used to compare the survival of elderly people with normal liver tests to elderly people with abnormal liver tests. A priori confounders of age, sex and co-morbidity, as well as other identified confounders for individual liver tests, were included within multivariate cox proportional hazards models to provide adjusted hazard ratios. Proportional hazards assumptions were checked using Schoenfeld residuals and log-log plots.

Results

Prevalence of abnormal liver tests

Of 15 308 participants aged 75 and over in the universal arm of the trial, 13 276 (86.7%) had at least one valid measurement of a liver test. Participants who did not have a valid test result were more likely to be female, were slightly older and had a slightly greater co-morbidity index. Participants without valid measurements were also more likely to have missing data for other variables examined.

Overall, 16.1% of those with one or more valid measurement had at least one abnormal liver test. The most common abnormal liver test within this population was ALP with 9.2% (95%CI [8.7%, 9.7%]) of people having a result above the upper limit of normal (Table 1). Only 5.4% (95%CI [5.0%, 5.8%]) of people had an abnormal bilirubin test, whilst an even smaller proportion had an abnormal AST result, 3.3% (95%CI [3.0%, 3.7%]) of those with a valid measurement. The majority of subjects with an abnormal measurement had a test result less than or equal to 2x the upper limit of normal: 86.2% AST; 90.0% ALP, 93.8% bilirubin.

Demographic, lifestyle and medical characteristics associated with an abnormal liver test

Having an abnormal AST measurement was associated with a lower age and with alcohol consumption of 7 units or more in the previous week. (Table 2) Conversely, people with an elevated ALP measurement were more likely to be older and to have a lower alcohol consumption. A history of diabetes and an assessment of dementia were associated with both elevated AST and ALP, with a history of heart attack associated only with increased ALP. Categorisation of comorbidity score showed an association only with abnormal ALP. An abnormal bilirubin was strongly associated with male sex and being a current or ex-smoker appeared to confer a protective effect on having an abnormal bilirubin following adjustment for other variables. 3). This corresponded to an adjusted hazard ratio of 1.27, i.e. an abnormal AST was associated with a 27% (95% CI [9%, 47%]) increased hazard for all-cause mortality. There was an increased hazard of death with increasing levels of abnormality with patients with an abnormal AST over 2x the ULN having a 62% increased hazard of death. Abnormal AST was associated with increase in mortality from cancer (56% increased hazard) and mortality from liver disease (7-fold increased hazard) (Table 3). It should be noted that only 8 out of 429 subjects (1.8%) with an abnormal AST died from liver disease.

ALP

The median follow-up for mortality was 7.2 years (84,256 person-years of follow-up) with 7971 deaths. Mortality in subjects with an abnormal ALP was 138.8 per 1000 person-years of follow-up representing an increase of nearly 50 deaths per 1000 person years compared with subjects with a normal ALP measurement (Table 4). An abnormal ALP measurement was associated with a 47% increase in hazard of death, adjusted for age, sex, co-morbidity score and alcohol intake (hazard ratio 1.47; 95% CI [1.35, 1.61]). An abnormal ALP was associated with an increased hazard for death from CVD (34% increase), death from cancer (61% increase), death from respiratory disease (58% increase) and death from liver disease (nearly 6-fold increase) (Table 4). However, only 13 out of 1246 subjects (1.0%) with an abnormal ALP died from liver disease.

Bilirubin

The median follow-up for mortality was 7.2 years (79,285 person-years of follow-up) with 7479 deaths. Abnormal bilirubin was associated with a small increase in hazard of death (15%, 95%CI [2%, 30%]) following adjustment for the confounding factors of age, sex, comorbidity score, smoking status, alcohol intake and WHR group (Table 5). Abnormal bilirubin was not associated with a change in hazard of any causespecific mortality examined (Table 5).

Multiple elevations of liver tests

181 subjects had two or more elevated liver tests. Having two or more elevated liver tests was associated with an increased hazard of all-cause mortality (adjusted HR 1.54 (95%CI [1.30, 1.83])), a doubling in hazard of death from cancer and a 14-fold increase in hazard of death from liver disease compared to subjects with no abnormal liver tests (Table 6).

Discussion

We found that nearly 1 in 6 of the general population aged 75 and over had at least one elevated liver test of AST, ALP or bilirubin. Over half of these abnormalities can be accounted for by an elevated measurement of ALP (9.2% of all subjects) and the vast majority were within 1 x the ULN for the test measured. A single abnormal liver test in this elderly population was associated with a higher risk of death overall, and for AST and ALP in particular a higher risk of death from liver disease, compared to those elderly subjects with normal liver tests. The occurrence of two or more abnormal liver tests was associated with a 2-fold risk of death from cancer and a 14fold risk of death from liver disease. The highest absolute mortality rates were seen for cardiovascular disease both in subjects with and without abnormal liver tests.

Although frequently performed as part of standard laboratory blood tests and bracketed under the nomenclature of 'liver functions tests', individual elevations of AST, ALP or bilirubin arguably do not reflect impaired liver function with AST also being found in skeletal muscle and the heart and ALP being particularly concentrated also in bones and the kidneys. Elevations of these enzymes would be considered indicative of liver injury or impairment if found in combination with elevations in other liver tests which might be considered more specific to the liver such as alanine aminotransferase (ALT) or gamma-glutamyl transpeptidase (GGT). Unfortunately, neither ALT nor GGT measurements were collected as part of the panel of blood tests in the original study from which these data are taken. We do not therefore know whether there were concurrent abnormalities in either ALT or GGT in those patients with elevated AST, ALP or bilirubin.

In addition to an unsurprising increased risk of death from liver disease (albeit mediated through very small absolute numbers of deaths) both elevated AST and ALP were seen to be associated with an increased risk of death from cancer (excluding liver cancer). Elevated ALP was also associated with an increased risk of death from cardiovascular and respiratory disease with absolute excess mortality for both these causes being much greater than that observed for liver disease. Rather than being related to liver function per se this may reflect the fact that these enzymes are also present in the other body tissues referred to above and not just specific to the liver. Indeed the associations seen between these "liver function tests" and the presence of other non-liver diseases (diabetes, heart attack and dementia) support this. When we considered subjects with more than one elevated liver test a greater increased hazard of death from all causes and from cancer (adjusted HR 2.11 [1.59, 2.80]and liver disease [adjusted HR 17.0 [7.2, 40.2] respectively) was seen compared to that seen in patients with only one isolated elevated liver test. We have provided hazard ratios for death adjusted for self-reported comorbidity which may only have measured significant comorbidity as determined by the participant. It is possible that there may be other non-reported comorbidities associated with both abnormal liver tests and mortality leading to some residual confounding in our study.

Our study relied on a single measurement of each of AST, ALP and bilirubin.

Naturally in the clinical setting it would be desirable to have multiple measurements of these liver tests in order to assess whether a patient warranted further follow-up and investigation. Our findings imply though that if the test result is not markedly abnormal the absolute excess mortality following a single isolated abnormality for any given individual is small with absolute excess mortality rates of 7.2, 10.3 and 47.9 deaths per 1000 person years in patients with abnormal bilirubin, AST and ALP respectively.

As a result of the general practice design of this large study blood data were analysed across 48 laboratories. Unfortunately, not all laboratories were able to Information on the prevalence and consequences of abnormal liver function in the UK is surprisingly lacking. In this study we were able to assess, for the first time, some commonly used markers of liver biochemical activity (even if they are not truly specific for the liver) in a representative randomly selected community based sample of people aged 75 and over. The representative nature of our study enables us to provide a more accurate estimate of the prevalence of these abnormal liver tests in the general population than has been previously available. Our results show a slightly lower prevalence of abnormal transaminase (3.3% abnormal AST vs 7.0% abnormal AST or alanine transaminase), ALP (9.2% vs 10.4%) and bilirubin (5.4% vs 7.1%) than those reported in the recent study of adults of all ages from Scotland [START_REF] Mclernon | Health outcomes following liver function testing in primary care: a retrospective cohort study[END_REF]. This is expected as the population within this study was selected on the basis that subjects already had a measurement of liver test contained within their primary care records. The authors acknowledged that they did not know why liver tests were requested and that this was likely to include the investigation of suspected liver disease leading to a non-representative sample of the general population. In a study in America [START_REF] Lee | Serum aminotransferase activity and mortality risk in a United States community[END_REF], only a selected group of participants had blood tests, and it is likely that the true population prevalence of abnormal liver tests may have been over estimated. Indeed, 12.8% of those with a measurement of AST had an elevated test, nearly four times that seen in our study. Other studies from the USA also show a slightly higher prevalence of elevated AST to that which we have described in this study, with 3.7% and 4.9% of the population aged 20 or over having an elevated AST in 1988-1994 and 1999-2002 respectively [START_REF] Clark | The prevalence and etiology of elevated aminotransferase levels in the United States[END_REF], [START_REF] Ioannou | The prevalence and predictors of elevated serum aminotransferase activity in the United States in 1999-2002[END_REF]. However, given that our study looks only at people aged 75 and over, it may be that the prevalence of abnormal Our data show an increased risk of death with a single measurement of abnormal AST, consistent with those reported from America using the NHANES data [START_REF] Lee | Serum aminotransferase activity and mortality risk in a United States community[END_REF] and those in Scotland [START_REF] Mclernon | Health outcomes following liver function testing in primary care: a retrospective cohort study[END_REF]. A higher risk of death was seen with increasingly abnormal results in both these studies. Further studies from working populations in Germany and South Korea report greater increases in all-cause mortality than those shown in our data [START_REF] Arndt | Elevated liver enzyme activity in construction workers: prevalence and impact on early retirement and all-cause mortality[END_REF], [START_REF] Kim | Normal serum aminotransferase concentration and risk of mortality from liver diseases: prospective cohort study[END_REF].

We identified one study that specifically looked at the association between liver tests and mortality in the elderly, which came from a small population of 70 year-olds in Jerusalem [START_REF] Elinav | Low alanine aminotransferase activity in older people is associated with greater long-term mortality[END_REF]. This study did not show any significant associations between either AST or ALP and all-cause mortality. However, the categorisation of the liver test results as above or below the mean value in the study population makes the interpretation of these results difficult to compare with our study where we were specifically looking at abnormalities as defined by upper limits of normal reference ranges.

Irrespective of the evidence, or lack of, to support the role of statins in hepatotoxicity, current guidelines recommend liver tests should be carried out before and within 1-3 months of starting treatment on statins to monitor for signs of altered liver function [2].

With an estimated 40% of all 75-79 year olds having at least one prescription for statins annually [START_REF]NICE implementation uptake report: statins[END_REF], incidental findings of elevated liver tests in this section of the population in the absence of symptoms is likely to occur regularly. Many referrals to our hepatology clinics [GPA, JW] occur for investigation of abnormal "liver function tests" in the elderly and there is little existing work to guide practitioners as to the need for and the direction of further investigation. This is echoed by a report from Department of Health Quality Strategy Team on Liver Disease which called for guidance to be made available to general practitioners on the use and interpretation of "liver function tests". However, to do this we need to understand the natural history associated with the abnormalities found [START_REF] Dh | Liver disease: A scoping study into the nature and burden of the disease[END_REF]. Our study has shown that abnormalities in some liver enzymes are fairly common and they are associated with an increased risk in all-cause mortality in the elderly population when compared with a similarly elderly control population i.e. those subjects aged 75 and over with normal liver tests.

In the absence of any evidence for an overwhelming increase in mortality following a single abnormal measurement of AST, ALP or bilirubin in isolation, it would seem sensible that current practise of referral and active investigation of patients only with multiple or persistent abnormalities should be continued.

Introduction

  In the UK liver biochemistry tests are frequently performed both in primary and secondary care and are often termed "liver function tests" frequently including measurements of serum aminotransferase, alkaline phosphatase and bilirubin. A finding of an elevation of one or more of these tests may be used as an indication of potential liver disease with subsequent follow-up and more invasive testing probable.Despite their common use what the occurrence and consequences of these abnormalities are remains unclear.

  widest circumference over the buttocks and below the iliac crest.

  All data management and analyses were conducted using Stata software [

  Mortality associated with abnormal liver tests.ASTThe median follow-up for mortality was 7.2 years (80,071 person-years of follow-up) with 7590 deaths. Mortality in subjects with an abnormal AST was 104.8 per 1000 person-years representing an increase of 10 deaths per 1000 person-years of followup compared to subjects with a normal AST (Table

  for the liver tests so we relied on imputation for 25/53 of our reference ranges. This means that we are unable to examine the association between absolute values of enzymatic activity and mortality as many different reference ranges were used.

  a population of older people is indeed lower than that found in the general adult population of all ages.

Table 1 . Prevalence of abnormal liver tests Abnormal test(s) (Number with valid test)

 1

		Number	% of valid tests
		abnormal	abnormal [95%
		(% of	CI]
		abnormal)
	All 3 within reference range	10025	83.6 [82.9, 84.2]
	(N=11994)
	Any abnormal (N=13546)	2175	16.1 [15.4, 16.7]
	AST (N=12826)	429 (100)	3.34 [3.04, 3.67]
	≤ 2x upper range	370 (86.2)	2.88 [2.60, 3.19]
	Above 2x upper range	59 (13.8)	0.46 [0.35, 0.59]
	ALP (N=13499)	1246 (100)	9.23 [8.75, 9.73]
	≤ 2x upper range	1122 (90.0)	8.31 [7.85, 8.79]
	Above 2x upper range	124 (10.0)	0.92 [0.76, 1.09]
	Bilirubin (N=12690)	690 (100)	5.44 [5.05, 5.85]
	≤ 2x upper range	647 (93.8)	5.10 [4.72, 5.50]
	Above 2x upper range	43 (6.2)	0.34 [0.25, 0.46]
	AST and ALP (N=12794)	90	0.70 [0.57, 0.86]
	AST and bilirubin (N=12021)	42	0.35 [0.25, 0.47]
	ALP and bilirubin (N=12648)	67	0.52 [0.41, 0.67]
	AST, ALP and bilirubin	9	0.07 [0.03, 0.14]
	(N=11994)

Table 2 Associations between demographic characteristics and co-morbidities and abnormal liver tests All confidence intervals are calculated following adjustment for potential clustering in practices.

 2 Multivariate model for AST includes sex, age, alcohol group, WHR group and comorbidity score **Multivariate model for ALP includes sex, age, alcohol group and comorbidity score ***Multivariate model for bilirubin includes sex, age, alcohol group, smoking group, WHR group and comorbidity score

		AST	ALP	Bilirubin
		OR [95% CI]	OR [95% CI]	OR [95% CI]	OR [95% CI]	OR [95% CI]	OR [95% CI]
		Univariate	Multivariate*	Univariate	Multivariate**	Univariate	Multivariate***
	Sex
	Female	0.96 [0.79, 1.18] 1.28 [0.94, 1.68] 1.05 [0.93, 1.19] 0.96 [0.83, 1.11] 0.37 [0.32, 0.44] 0.33 [0.26, 0.42]
	Male	1	1	1	1	1	1
	Age
	Yearly increase	0.97 [0.94, 0.99] 0.97 [0.94, 0.99] 1.05 [1.04, 1.07] 1.04 [1.03, 1.06] 0.99 [0.97, 1.01] 0.99 [0.97, 1.01]
	in age
	Alcohol
	7unit increase	1.22 [1.08, 1.37] 1.16 [1.02 1.32] 0.81 [0.74, 0.88] 0.83 [0.76, 0.91] 1.20 [1.09, 1.31] 1.06 [0.96, 1.17]
	in alcohol
	consumption in
	previous week
	Drink in last	0.84 [0.66, 1.08]		1.34 [1.17, 1.54]		0.85 [0.70, 1.04]
	year
	Smoking
	Never	1		1		1	1
	Ex	0.86 [0.70, 1.07]		0.82 [0.72, 0.94]		1.22 [1.03, 1.44] 0.79 [0.65, 0.97]
	Current	0.97 [0.69, 1.35]		1.06 [0.86, 1.29]		1.76 [0.56, 1.05] 0.47 [0.32, 0.68]
	BMI
	<18.5	0.73 [0.32, 1.69]		1.36 [0.93, 1.99]		0.95 [0.54, 1.66]
	18.5-	1		1		1
	25-	1.08 [0.86, 1.36]		0.85 [0.74, 0.98]		1.10 [0.92, 1.31]
	30-	1.26 [0.92, 1.72]		0.79 [0.64, 0.98]		0.91 [0.70, 1.19]
	35+	1.28 [0.72, 2.27]		1.52 [1.10, 2.10]		0.44 [0.22, 0.86]
	Missing
	Increase in BMI	1.12 [0.99, 1.26]		0.95 [0.88, 1.03]		0.94 [0.85, 1.04]
	group
	Waist:hip ratio

*

Table 3 Cox proportional hazards model for death (all-cause and cause-specific mortality) by AST test

 3 *Deaths from cancer in subjects with abnormal AST were digestive cancer[START_REF] Elinav | Low alanine aminotransferase activity in older people is associated with greater long-term mortality[END_REF], respiratory cancer[START_REF] Osborn | Factors associated with depression in a representative sample of 14 217 people aged 75 and over in the United Kingdom: results from the MRC trial of assessment and management of older people in the community[END_REF], breast cancer (6), genitourinary cancer[START_REF] Fletcher | The MRC trial of assessment and management of older people in the community: objectives, design and interventions [ISRCTN23494848[END_REF], and other cancers[START_REF] Ioannou | The prevalence and predictors of elevated serum aminotransferase activity in the United States in 1999-2002[END_REF].

		Events Person-	Mortality rate	Hazard ratio	Adjusted
			years	(per 1000	[95%CI]	Hazard ratio*
				person years)		[95%CI]
	All-cause mortality
	Normal AST	7319	77485	94.5	1	1
	Abnormal AST	271	2586	104.8	1.11	1.27
					[0.99, 1.26]	[1.09, 1.47]
	Abnormal AST	229	2267	101.0	1.07	1.17
	within 1x				[0.92, 1.24]	[1.02, 1.34]
	Abnormal AST	42	319	131.7	1.43	1.62
	over 2x				[1.04, 1.96]	[1.22, 2.16]
	Hazard ratio for increase in AST abnormality group	1.12	1.26
					[1.02, 1.24]	[1.12, 1.41]
	Cause-specific mortality
	CVD
	Normal AST	3097	77485	40.0	1	1
	Abnormal AST	109	2586	42.1	1.06	1.15
					[0.87, 1.28]	[0.92, 1.44]
	Cancer (excluding liver cancer)
	Normal AST	1359	77485	17.5	1	1
	Abnormal AST**	63	2586	24.4	1.40	1.56
					[1.09, 1.80]	[1.21, 2.01]
	Respiratory disease
	Normal AST	1235	77485	15.9	1	1
	Abnormal AST	33	2586	12.7	0.80	1.03
					[0.57, 1.13]	[0.70, 1.52]
	Liver disease
	Normal AST	38	77485	0.49	1	1
	Abnormal AST	8	2586	3.09	6.28	7.00
					[2.92, 13.5]	[2.88, 17.06]

*adjusted for age, sex, comorbidity group, alcohol intake and WHR group. *

Table 4 Cox proportional hazards model for death (all-cause and cause-specific mortality) by ALP test

 4 adjusted for age, sex, comorbidity group and alcohol intake. **Deaths from cancer in subjects with abnormal ALP were digestive cancer (51), respiratory cancer (26), skin cancer(5), breast cancer[START_REF] Organization | Physical Status: the use and interpretation of anthropometry[END_REF], genitourinary cancer (35), and other cancers (42).

		Events Person-	Mortality rate	Hazard ratio	Adjusted
			years	(per 1000	[95%CI]	Hazard ratio*
				person years)		[95%CI]
	All-cause mortality
	Normal ALP	7068	77752	90.9	1	1
	Abnormal ALP	903	6503.9	138.8	1.58	1.47
					[1.47, 1.69]	[1.35, 1.61]
	Abnormal ALP	809	5895.7	137.2	1.56	1.42
	within 1x				[1.44, 1.68]	[1.32, 1.52]
	Abnormal ALP	94	608.3	154.5	1.76	1.44
	over 2x				[1.44, 1.68]	[1.13, 1.84]
	Hazard ratio for increase in ALP abnormality group	1.47	1.37
					[1.39, 1.56]	[1.26, 1.48]
	Cause-specific mortality
	Cardiovascular disease
	Normal ALP	3010	77752	38.7	1	1
	Abnormal ALP	358	6504	55.0	1.46	1.34
					[1.30, 1.62]	[1.17, 1.55]
	Cancer (excluding liver cancer)
	Normal ALP	1324	77752	17.0	1	1
	Abnormal ALP**	174	6504	26.8	1.61	1.61
					[1.37, 1.89]	[1.39, 1.86]
	Respiratory disease
	Normal ALP	1181	77752	15.2	1	1
	Abnormal ALP	161	6504	24.8	1.68	1.58
					[1.42, 1.98]	[1.32, 1.90]
	Liver disease
	Normal ALP	35	77752	0.45	1	1
	Abnormal ALP	13	6503.9	2.00	4.59	5.95
					[2.43, 8.69]	[2.83, 12.51]

*

Table 6 Cox proportional hazards model for death (all-cause and cause-specific mortality) by multiple elevated tests

 6

		Events Person-	Mortality	Hazard ratio	Adjusted Hazard
			years	rate	[95%CI]	ratio* [95%CI]
				(per 1000
				person
				years)
	All-cause mortality
	Normal	6952	74068	93.9
	2 or more abnormal 134	950	141.1	1.55 [1.30, 1.83]	1.54 [1.30, 1.83]
	Cause-specific mortality
	Cardiovascular disease
	Normal	2958	74068	39.9
	2 or more abnormal 46	950	48.4	1.24 [0.93, 1.66]	1.24 [0.88, 1.74]
	Cancer (excluding liver cancer)
	Normal	1284	74068	17.3
	2 or more abnormal 34	950	35.8	2.11 [1.50, 2.97]	2.11 [1.59, 2.80]
	Respiratory disease
	Normal	1169	74068	15.8
	2 or more abnormal 18	950	19.0	1.23 [0.77, 1.96]	1.23 [0.79, 1.90]
	Liver disease
	Normal	38	74068	0.51
	2 or more abnormal 7	950	7.4	14.6 [6.5, 32.6]	17.0 [7.2, 40.2]
	*adjusted for age, sex and comorbidity group


Fernand Grard
email: fernand.grard@umons.ac.

Jean Nuyts
email: jean.nuyts@umons.ac.

Kaluza-Klein towers for real vector fields in flat space

Keywords: vectorfinalJPhysG, tex PACS numbers: 04, 50, Cd

come

Introduction

Following, Norström [1], and Kaluza and Klein [2] , numerous articles have been devoted to the study of various fields propagating in higher dimensional spaces, especially in five dimensions with the fifth dimension compactified either on a strip on or a circle. With the discovery of the importance of higher dimensions in string theories, these study of the Kaluza-Klein models have been subject of renewed interest and scrutiny (see for example: [3], [START_REF] Appelquist | Modern Kaluza-Klein Theories[END_REF], [START_REF] Duff | Kaluza-Klein Theory in Perspective[END_REF], [START_REF] Overduin | [END_REF], [START_REF] Wesson | Space-Time-Matter, Modern Kaluza-Klein Theory[END_REF], [START_REF] Pope | Lectures on Kaluza-Klein Theory[END_REF], [START_REF] Nam | [END_REF], [START_REF] Rizzo | Introduction to extra dimensions[END_REF]). Following the seminal work of [START_REF] Arkhani-Ahmed | [END_REF], [12] in flat spaces and of [13], [14] in warped spaces, much attention has been focussed essentially in three directions. Firstly, the so called Kaluza-Klein mass towers have been considered extensively. It should be stressed here that the spectra of the mass squared towers are often accepted to be linearly spaced as a result of an implicit choice of periodic or antiperiodic boundary conditions (see many of the preceding citations [15] and [START_REF] Luk | Extra Dimensions and Kaluza-Klein Theories[END_REF]). Secondly, the impacts of the towers on the decay and on the scattering amplitudes of detectable particles have been evaluated theoretically [START_REF] Bhattacharyya | [END_REF], [18], [19], [20], [21]. They are also often obtained using a linear spectrum and degeneracy of the different spin states which occur at a given level (see instead one of our main conclusion in Sect.( 5)). Finally, experimentalists at high energy accelerators have already been looking and are planning to search for effects from the existence of the hypothesized Kaluza-Klein states, or directly for the occurrence of these states at the highest available energies [22], [23], [24], [25] (particularly at LHC). It has seemed to us, in view of the importance of the concept and of its consequences, should Kaluza-Klein effects be discovered, that a careful mathematical study of all the possibilities is worthwhile. In particular the investigation of all the allowed boundary conditions is of particular rele-vance as they lead directly to the prediction of all the possible forms of the Kaluza-Klein mass spectra. Needless to say, the knowledge of these spectra is essential to the experimentalists as it will help them in determining what they should look for to unravel and understand the structure of their data if higher dimensions turn out to be physical.

In previous publications [26], we have applied the Kaluza-Klein reduction procedure to various fields propagating in a five dimensional space with the fifth dimension compactified either on a strip or on a circle.

Our approach is based on an exhaustive study of the hermiticity properties of the operators defining the masses. This results in the determination of all the sets of allowed field boundary conditions which have to be applied in the extra dimension. We have successively considered scalar and spinor fields in flat and warped spaces and have established in each case all the corresponding Kaluza-Klein towers. Our main result is that, apart from the usual towers where the masses (or masses squared) are regularly spaced (essentially linearly), there appears new types of towers with masses which are solutions of specific transcendental functions.

We consider here the case of a free real vector field in the five dimensional flat space and discuss its most general free Lagrangian and related equations of motions (see Sect. (2.1)). Requiring the hermiticity of the relevant operators, the basic boundary relations for the field are established in Sect. (2.2). Adopting a K-K reduction which breaks SO(4,1) covariance while preserving SO (3,1) covariance, the boundary relations for the reduced fields in the fifth dimension are derived (see Sect.(2.3)), from which, taking into account the restrictions imposed by the 5-dimensional Lorenz gauge condition, all the sets of allowed boundary conditions are obtained (see (2.4)). Some physical considerations about the locality of the boundary conditions and about the possible extension to a closed circle in the extra-dimension are given in Sect.(3.1) while the physical relevance of the mass scales which have to be adopted in a flat space is discussed in Sect. (3.2). A complete analysis of the boundary conditions leads to the appearance of both 4-dimensional vector and scalar states in the resulting Kaluza-Klein mass towers as seen in the 4-dimensional branes(see Sect.( 4)). Some physical consequences are discussed and the main results are summarized in the conclusions (Sect.( 5))

2 Real vector field in five dimensional flat space.

Equations of motion. Kaluza-Klein reduction. Boundary relations and boundary conditions

Lagrangian. Equations of motion

Let B A be a free real vector field propagating in a five-dimensional flat space

x A = {x, x 5 } = {x µ , x 5 }, µ = 0, 1, 2, 3, s = x 5 (1)
with an infinite four dimensional Minkowski space x and the fifth dimension s compactified on a strip [0, 2πR] or on a circle of radius R. The metric is η µν = diag(1, -1, -1, -1), η µ5 = 0, η 55 = -1. The most general Lorentz invariant action, quadratic in the vector field B A , contains, apart from the bulk mass M , two (real) free parameters λ 1 and λ 2

A = 1 4 F AB F AB + λ 1 2 (∂ A B A ) 2 + λ 2 2 (∂ A B B )(∂ B B A ) - M 2 2 B A B A d 5 x (2)
with

F AB = ∂ A B B -∂ B B A . (3)
Since the two factors of λ 1 and λ 2 are equal up to a divergence, the equations of motion ( 14) depend only on λ with

λ = λ 1 + λ 2 . (4)
The so-called usual case corresponds to the choice λ = 0 and we will focus our attention on it. We will also discuss what we call the generalized case with λ = 0. Compared to the usual case, the generalized case involves, as we will explain later, essentially one more degree of freedom which is of scalar type. We will see that the five dimensional vector field leads, upon Kaluza-Klein reduction, to vector and scalar particles, mimicking, in some cases, 4-dimensional situations corresponding to a generalized case in the brane.

Equations of motion. Usual case (λ = 0)

The usual Lorentz invariant action (see however the discussion of the "Generalized case" below) corresponds to the particular choice λ = 0

A = 1 4 F AB F AB - M 2 2 B A B A d 5 x (5)
and leads to the equations of motion

( 5 + M 2 )B A -∂ A (∂ C B C ) = 0. ( 6
)
Usual Case (λ = 0) with M 2 = 0

For M 2 = 0, one deduces from the divergence of ( 6) the usual Lorenz condition

∂ A B A = 0 (7)
and the equations of motion

5 + M 2 B A = 0, (8)
which justify the interpretation of M 2 as the bulk mass squared of the transverse (Eq.( 7)) massive (Eq.( 8)) vector field with four degrees of freedom.

Usual Case (λ = 0) with M 2 = 0

For M 2 = 0, the Lagrangian is invariant under the gauge transformation

B A -→ B A + ∂ A Λ (9)
for any arbitrary Λ.

The Lorenz condition [START_REF] Wesson | Space-Time-Matter, Modern Kaluza-Klein Theory[END_REF] can be obtained by a suitable gauge transformation

(B ) A = B A + ∂ A Λ (10)
leading to

5 B A = 0 (11) ∂ A B A = 0 ( 12
)
provided Λ is chosen as

5 Λ = -∂ A B A (13)
which defines Λ up to an arbitrary function of null Dalembertian. In the five dimensional space, B A has three transverse degrees of freedom. Indeed, the longitudinal degree of freedom can be eliminated by a further suitable gauge transformation (10) of null Dalembertian.

In the main part of the text, we will use, for the usual case λ = 0, the equations ( 8) and ( 7) for all M 2 .

Equations of motion. Generalized case (λ = 0)

The equations of motion in the generalized case (2)-( 4), corresponding to λ = 0, are

( 5 + M 2 )B A + (λ -1) ∂ A (∂ B B B ) = 0. ( 14
)
Generalized Case (λ = 0) with M 2 = 0

For λ = 0 and M 2 = 0, defining the new vector field B A and the new scalar field B by

(B ) A = B A + λ M 2 ∂ A ∂ L B L B = ∂ A B A , (15)
one easily sees that they are the physical fields as they satisfy, from ( 14), the usual physical equations

5 + M 2 B A = 0 ∂ A B A = 0 5 + M 2 λ B = 0. ( 16
)
Let us stress that the bulk mass squared of the vector field is M 2 while the bulk mass squared of the scalar field is multiplied by 1/λ.

Inversely, if B A and B are given such that they satisfy [START_REF] Luk | Extra Dimensions and Kaluza-Klein Theories[END_REF], then B A defined by

B A = B A - λ M 2 ∂ A B ( 17
)
satisfies the initial equation (14). This shows that the physical fields B A , B and the initial fields B A are equivalent ways of describing the theory.

Generalized Case (λ = 0) with M 2 = 0

For M 2 = 0, the equations of motion in the generalized case (λ = 0) reduce to

5 B A + (λ -1) ∂ A (∂ B B B ) = 0. ( 18
)
Let us note that the full gauge invariance (Eq.( 9) with Λ arbitrary) of the usual Lagrangian (λ = 0) is lost, but there is a remaining "partial gauge" invariance provided that

5 Λ = 0. ( 19
)
It is convenient to define the new fields B A and B by

(B ) A = B A + ∂ A Σ (20) B = ∂ A B A (21)
with Σ chosen to satisfy

5 Σ = (λ -1) ∂ A B A . ( 22
)
Remark that Σ is determined up to a function of null Dalembertian i.e. (B ) A can still be changed by a partial gauge transformation.

The new fields then obey the equations

5 B A = 0 ( 23
) 5 B = 0 (24) ∂ A B A = λB . ( 25
)
The interpretation of B as a massless physical scalar field poses no problem. The interpretation of B A as a massless vector field is more delicate as the Lorenz gauge condition is missing. In momentum space (with p 2 = 0 by ( 23)), the five degrees of freedom of B A decompose as follows. The divergence part ∂ A B A is equivalent to the scalar B by (25). Any longitudinal part (in the direction of p A ) is physically irrelevant as it can be removed by a suitable partial gauge transformation (19). The truly transverse part with three degrees of freedom describes a physical massless vector field. The transverse part is defined up to an arbitrary longitudinal part i.e. belongs to the equivalence class, in the mathematical sense, of fields differing by partial gauge transformations.

In the main part of the paper, a complete discussion of the usual case will be given. The changes which follow from the generalized case (λ = 0) are outlined. In order to guarantee the reality of the bulk mass squared M 2 (see [START_REF] Pope | Lectures on Kaluza-Klein Theory[END_REF] and ( 11)), following the approach discussed and justified at great length in our preceding articles, [26], we impose that the operator 5 is an observable and hence hermitian (more precisely symmetric)

Basic boundary relations

(C, OB) = (OC, B) (26) O = 5 (27)
for the invariant scalar product of real vector fields

(C, B) = +∞ -∞ d 4 x 2πR 0 ds C A (x, s)B A (x, s). (28)
In this paper, we restrict ourselves to this scalar product, which is currently used by mathematicians. We do not consider generalized scalar products which for finite intervals may include boundary terms as, for example,

d 4 x s 00 C A (x, 0)B A (x, 0) + s 0R C A (x, 0)B A (x, 2πR) + ... .
Such scalar products would conceivably lead to boundary conditions equivalent to those obtained when adding localized kinetic terms to the Lagrangian, as was considered for instance in reference [27].

The scalar product (28) leads to

d 5 x ∂ A C B (∂ A B B ) -(∂ A C B )B B = 0 (29)
which, taking into account the decrease of the fields at infinity in the x space (essentially faster than 1/ | x |), becomes the basic boundary relation

d 4 x (∂ s C A ) B A -C A (∂ s B A ) s=2πR s=0 = 0. ( 30
)
This boundary condition is consistent with the principle of least action, if we impose the following linear boundary conditions on the fields B and C, δC 2πR = (αC + β∂ The physical condition of real bulk mass is empty when the mass is zero. In that case, requiring the differential operator 5 to be symmetric is not the most general ansatz, and it would be interesting to study the zero bulk mass case more thoroughly in future. Here, we restrict the discussion to symmetric operators.

In the next section, upon the Kaluza-Klein reduction, the boundary conditions for the reduced fields will be obtained from reduced boundary relations compatible with (30).

Basic boundary relations. Generalized case (λ = 0)

In the generalized case, the equations of motion ( 14) are written

Ω A B B B = -M 2 B A (31)
with the operator

Ω A B = 5 δ A B + (λ -1) ∂ A ∂ B . (32)
In order for the eigenvalue M 2 to be real, the hermiticity of this operator is imposed

(ΩC, B) = (C, ΩB) (33)
for the scalar product (28). This leads to the boundary relation

d 4 x C A ∂ s B A -(∂ s C A ) B A + (λ -1) C S ∂ A B A -∂ A C A B S 2πR 0 = 0 (34)
which generalizes (30), since for λ = 0, ∂ A B A = 0 by ( 7)).

2.3 Kaluza-Klein reduction. Mass towers. Boundary conditions

2.3.1 Kaluza-Klein reduction. Usual case (λ = 0)
We perform the following Kaluza-Klein reduction

B µ (x, s) = r B [V ] r (s) λ µ r (x) (35) B 5 (x, s) = p B [S] p (s) φ p (x) (36)
which breaks the SO(4,1) covariance and respects the SO(3,1) covariance in any brane (at fixed s).

The functions B

[V ] r (s) and B

[S] r (s) (35),(36) are supposed to satisfy

∂ 2 s B [V ] r (s) = -v [V ] r 2 B [V ] r (s) (37) ∂ 2 s B [S] p (s) = -v [S] p 2 B [S] p (s) (38)
in such a way that the basic equations ( 8), applied for each r (or p), reduce to

4 + m [V ] r 2 λ µ r (x) = 0 (39) 4 + m [S] p 2 φ p (x) = 0. ( 40
)
The 4-dimensional masses for the 4-vector λ µ r (x) and for the 4-scalar φ p (x) are then given by

m [V ] r 2 = v [V ] r 2 + M 2 m [S] p 2 = v [S] p 2 + M 2 . ( 41
)
The indices r and p in ( 35) and (36) refer respectively to the 4-dimensional masses squared m Imposing the hermiticity (26) of the operator ∂ 2 s appearing in (37), (38) for the scalar product between functions of s

(G, F ) = 2πR s=0 G(s)F (s) ds (42)
ensures the reality of its eigenvalues and so of the corresponding 4-dimensional masses squared (which could be negative). This leads to boundary relations for the reduced functions

B [V ] r , B [S] p (defined from B A in (35),( 36
)), C [V ]
m and C

[S]

q (defined similarly from

C A ) (∂ s C [V ] m ) B [V ] r -C [V ] m (∂ s B [V ] r ) 2πR 0 = 0 (43) (∂ s C [S] q ) B [S] p -C [S] q (∂ s B [S] p ) 2πR 0 = 0. ( 44
)
These boundary relations are compatible with the basic boundary relations (30).

Boundary conditions result from these boundary relations under the hypothesis that, on one side, all the reduced fields related to B µ and C µ live in some common domain in the Hilbert space and that, on the other side, all the reduced fields in B 5 and C 5 live in some, eventually different from the first one, common domain. Hence, the reduced fields are submitted to boundary conditions which must be independent of the index r in (35), independent of the index p in (36), but may depend on the [V ] or [S] index. Thus, for any field B the boundary conditions have to be expressed by two linear homogeneous relations, independent of r, among the following quantities

B [V ] r (0) , B [V ] r (2πR) , ∂ s B [V ] r (0) , ∂ s B [V ] r (2πR) (45)
and similarly by two such relations, independent of p, among

B [S] p (0) , B [S] p (2πR) , ∂ s B [S] p (0) , ∂ s B [S] p (2πR). ( 46
)
For B

[S]

n , all the sets of allowed boundary conditions compatible with the boundary relations (44) are listed in Table (1). The boundary conditions for

B [V ] n are identical to those for B [S]
n with the parameters {α

[S] 1 , . . . } replaced by {α [V ] 1 , . . . }.
Remark that all these boundary conditions are essentially the same as those allowed for a free real scalar field (see the first reference in [26]).

The eigenvalues (v

[V ]
r ) 2 of Eq.(37) corresponding to a chosen set of boundary conditions for B {n} φ {n} = 0).

Depending on {n}, any of these three situations may occur.

2.3.2 Kaluza-Klein reduction. Generalized case (λ = 0)

In the generalized case λ = 0, starting from the equations of motion ( 16) or ( 23)-( 25) for the physical fields B A and B , we perform the following Kaluza-Klein reduction

B µ (x, s) = r B [V ] r (s) λ µ r (x) (47) B 5 (x, s) = p B [S] p (s) φ p (x) (48) B (x, s) = v B v (s) ψ v (x). (49)
Imposing the hermiticity of ∂ 2 s and following the usual procedure, one obviously finds that the reduced fields B

(∂ s C [V ] m ) B [V ] r -C [V ] m (∂ s B [V ] r ) 2πR 0 = 0 (50) (∂ s C [S] q ) B [S] p -C [S] q (∂ s B [S] p ) 2πR 0 = 0 (51) (∂ s C v ) B w -C v (∂ s B w ) 2πR 0 = 0. ( 52
)
The derived boundary conditions must be the same for all the reduced components of each of the fields. In other words, the boundary conditions in terms of the fields

B [V ]
r (s), B

[S] p and B v (s) evaluated at s = 0 and s = 2πR (see the discussion of (45)) must be independent of the indices r, p and v respectively.

Refering to [START_REF] Bhattacharyya | [END_REF], it is natural, in order to obtain definite boundary conditions for B A , to impose the same boundary conditions for B A and ∂ A B B [V ] r (s) same boundary conditions as

B v (s) (53) B [S]
r (s) same boundary conditions as

∂ s B v (s) (54)
independently of r and v.

Physical implication (λ = 0)

Since B

[V ] r and B v are submitted to the same boundary conditions (53), their eigenvalue spectrum v 2 n are identical. Hence, the scalar and vector mass towers of the λ = 0 case are the scalar and vector mass towers of the λ = 0 case accompanied by a new scalar mass tower (with fields ψ p (x)) with all their masses, according to [START_REF] Luk | Extra Dimensions and Kaluza-Klein Theories[END_REF], shifted by

Square mass shift = (m [ψ] n ) 2 -(m [V ] n ) 2 = 1 -λ λ M 2 . ( 55
)
2.4 Gauge condition 2.4.1 Gauge condition. Usual case (λ = 0)

The Lorenz condition [START_REF] Wesson | Space-Time-Matter, Modern Kaluza-Klein Theory[END_REF], restricted to one {n}, reduces to

B [V ] {n} ∂ µ λ µ {n} + ∂ s B [S] {n} φ {n} = 0. ( 56
)
In order to have a non-zero vector and/or a non-zero scalar part {n} in the tower (not both zero), this equation implies in general (taking into account an arbitrary normalisation) one of the following four distinct physical cases Generic Case 1 :

B [V ] {n} = 0 , ∂ s B [S] {n} = B [V ] {n} , ∂ µ λ µ {n} + φ {n} = 0 φ {n} = 0 (57) Generic Case 2 : B [V ] {n} = 0 , no scalar , ∂ µ λ µ {n} = 0 (58) Special Case 1 : B [V ] {n} = 0 , B [S] {n} = 1 , ∂ µ λ µ {n} = 0 φ {n} = 0 (59) Special Case 2 : no vector , B [S] {n} = 1 , φ {n} = 0. ( 60
)
From these equations, the physical content of the tower at level {n} can be easily deduced for every case (57)-(60). In general, the Generic Cases are related to tower states while the Special Cases correspond (within a tower) to specific states characterized by v 2 {n} = 0.

• Generic Case 1 when m 2 {n} = M 2 + v 2 {n} = 0 (see (57)). For any such {n}, there is a 4-dimensional vector field A µ {n} and a 4-dimensional scalar field φ {n} given by Generic Case 1 :

   vector field A µ {n} scalar field φ {n} m 2 {n} = 0          A µ {n} = λ µ {n} - ∂ µ φ {n} m 2 {n} ( 4 + m 2 {n} )φ {n} = 0 ( 4 + m 2 {n} )A µ {n} = 0 ∂ µ A µ {n} = 0 . ( 61
)
Note that the vector A µ {n} is a true physical vector field, which satisfies as expected the 4-dimensional Lorenz condition.

• Generic Subcase 1 when, for a given {n}, m 2 {n} = M 2 +v 2 {n} = 0. There is then a 4-dimensional massless scalar field φ {n} , and a 4-dimensional massless vector field A µ {n} which satisfy Generic Subcase 1 :

   vector field A µ {n} scalar field φ {n} (m A {n} ) 2 = (m φ {n} ) 2 = 0        A µ {n} = λ µ {n} 4 A µ {n} = 0 4 φ {n} = 0 ∂ µ A µ {n} = -φ {n} . ( 62
)
The equations for A µ {n} and φ {n} in four dimensions are of the same form as the equations ( 23)-(25) for respectively B A and B in five dimensions for λ = -1. They belong to a generalized 4-dimensional case with λ = 0. Refering to the discussion following (25), we conclude that φ {n} describes a scalar massless particle in the brane and that A µ {n} describes a particle whose 4-dimensional divergence is proportional to φ {n} , whose longitudinal part can be removed by a suitable partial gauge transformation and whose two truly transverse degrees of freedom describe a physical massless vector field in the brane.

• Generic Case 2 with m 2 {n} = M 2 + v 2 {n} (see ( 58)). For any {n} with m 2 {n} = v 2 {n} + M 2 , there is a true 4-dimensional vector field A µ {n} and no scalar field.

Generic Case 2 :

   vector field A µ {n} no scalar field m 2 {n} = v 2 {n} + M 2        A µ {n} = λ µ {n} ( 4 + m 2 {n} )A µ {n} = 0 ∂ µ A µ {n} = 0 no φ {n} . (63)
• Special Case 1 (59). Both a 4-dimensional vector A µ {n} and a 4-dimensional scalar φ {n} field are present in the brane and, from (38), v 2 {n} = 0. Their mass squared is equal to the bulk mass squared. Special Case 1 :

   vector field A µ {n} scalar field φ {n} m 2 {n} = M 2        A µ {n} = λ µ {n} ( 4 + M 2 )φ {n} = 0 ( 4 + M 2 )A µ {n} = 0 ∂ µ A µ {n} = 0 . ( 64
)
• Special Case 2 (60). There is a no 4-dimensional vector state but a lonely 4dimensional scalar state φ {n} is present and, from (38), v 2 {n} = 0. This means that the scalar has a brane mass squared equal to the bulk mass squared. Special Case 2 :

   no vector field scalar field φ {n} m 2 {n} = M 2 ( 4 + M 2 )φ {n} = 0 no λ µ {n} . (65)
2.4.2 Gauge condition. Generalized case (λ = 0)

Remembering the discussion concerning Generic Case 1 (57) and Generic Case 2 (58) and taking into account Eq.( 17), ( 53), (54), we have the general possibilities Generalized Generic Case 1 : same BC for (67)

B [V ] r , ∂ s B [S]
The discussion of the Lorenz condition (56) (for B [V ] and B [S] ) and of the the Extended Special Cases 1 and 2 follow the same line as in the λ = 0 case with obvious changes.

From here on, we concentrate on the usual case (λ = 0).

Final sets of allowed boundary conditions

The sets of boundary conditions (abbreviated BC in this subsection) for B

[S] {n} and B

[V ] {n} (see Table ( 1)) have to be made compatible with each other, taking into account the eigenvalue equations (37),(38) and in particular the Lorenz condition (56). The results, as obtained in App.(A), are summarized hereafter. The generic cases are related to BC which lead to towers composed of an infinite number of states. The special cases are related to BC which lead to the presence of specific states in the towers.

• Generic Case 1

In the Generic Case 1 (57), the allowed sets of BC for B

[V ] n and B

[S]

n are given in the following Table Sets

G1                              Set G1a B [V ] n (2πR) ∂sB [V ] n (2πR) B [S] n (2πR) ∂sB [S] n (2πR) = = = = αB [V ] n (0) 1 α ∂sB [V ] n (0) 1 α B [S] n (0) α∂sB [S] n (0) G1b B [V ] n (0) = 0 B [V ] n (2πR) = 0 ∂sB [S] n (0) = 0 ∂sB [S] n (2πR) = 0 G1c ∂sB [V ] n (0) = 0 ∂sB [V ] n (2πR) = 0 B [S] n (0) = 0 B [S] n (2πR) = 0 G1d B [V ] n (0) = 0 ∂sB [V ] n (2πR) = 0 ∂sB [S] n (0) = 0 B [S] n (2πR) = 0 G1e ∂sB [V ] n (0) = 0 B [V ] n (2πR) = 0 B [S] n (0) = 0 ∂sB [S] n (2πR) = 0 . ( 68
)
• Generic Case 2

In the Generic Case 2 (58), the allowed sets of BC are

Sets G2      B [S]
{n} = 0 special solution compatible with any set of BC of Table (1) for B

[V ] {n} = 0 any set of BC of Table (1) .

(69)

• Special Case 1

In the Special Case 1 (59), the allowed sets of BC are Set S1a

               B [S]
{n} = 1 special solution of a subset of the set A1 of Table ( 1)

B [S] {n} (2πR) = B [S] {n} (0) + α [S] 2 ∂ s B [S] {n} (0) ∂ s B [S] {n} (2πR) = ∂ s B [S] {n} (0) for B [V ] {n} = 0 any set of BC of Table (1) (70) Set S1b                B [S] {n} = 1 special solution of a subset of the set A2 of Table (1) ∂ s B [S] {n} (0) = 0 ∂ s B [S] {n} (2πR) = 0 for B [V ] {n} = 0 any set of BC of Table (1)
.

(71)

• Special Case 2

In the Special Case 2 (60), the allowed sets of BC are Set S2a

                   B [S]
{n} = 1 special solution of the subset of the set A1 of Table ( 1)

B [S] {n} (2πR) = B [S] {n} (0) + α [S] 2 ∂ s B [S] {n} (0) ∂ s B [S] {n} (2πR) = ∂ s B [S] {n} (0) B [V ] {n} = 0 special solution of any set of BC of Table (1) (72) Set S2b                    B [S] {n} = 1 special solution of the subset of the set A2 of Table (1) ∂ s B [S] {n} (0) = 0 ∂ s B [S] {n} (2πR) = 0 B [V ] {n} = 0 special solution of any set of BC of Table (1) . ( 73
)
3 Physical considerations

Physical discussion of the boundary conditions. Closure to a circle

If one looks at the boundary conditions of Table (1), one sees that there are two very different situations. For the Sets A2, A3, A4, A5, the boundary conditions are local.

The values of the fields are related at the same point, either at s = 0 or at s = 2πR. The boundary conditions A1 are of a rather different kind as they connect values of the fields evaluated at these two different points of the s domain. In this latter case, the field explores in fact its full domain at once. This is tantamount to action at a distance or to non locality. In the third article of [26], we noted that this was not in contradiction with quantum mechanics.

Under the A1 boundary conditions with α 2 = α 3 = 0, the strip can be closed into a circle by identifying the points s = 0 and s = 2πR. With α 1 = α 4 = 1 this leads to the periodic boundary conditions, and with α 1 = α 4 = -1 to the antiperiodic boundary conditions. They are both of local type.

Mass considerations

For completeness, we will give, in the next subsection, a description of all the allowed mass equations from which the mass towers are derived. We remind that we are here restricted to a five dimensional flat space with a compactified fifth coordinate. In general, an evaluation of the low lying mass eigenvalues give values of the form 1/R multiplied by a factor of order 1. For mass eigenvalues of the order of 1 TeV, this requires R to be of the order 10 -16 cm. This value is far away from the R ≈ 10 10 cm originating from the relation

M 2 P l ≈ V {n} M n+2 * (74)
between the Planck mass M P l , the fundamental scale M * and the volume V 5 = (2πR) 5 under the hypothesis that Gauss law is valid in the five dimensional space. This idea has been discussed at length in the litterature [START_REF] Arkhani-Ahmed | [END_REF], but the results are still under investigation.

As is well-known the problem has been cured elegantly by Randall and Sundrum [13] by the use of warped spaces (see also [28]) as will be discussed in a subsequent publication.

Mass considerations. Experimental aspects

The presence or absence of scalars states in the Kalauza-Klein towers can roughly be summarized as follows

(1) λ = 0, Generic Case 1 (61)

Accompanying the 4-dimensional vector tower there are two 4-dimensional scalar towers. The masses in one of the scalar tower are identical to the masses in the vector tower while the masses in the second scalar tower are shifted from the masses in the vector tower by the fixed amount (55).

(2) λ = 0, Generic Case 2 (63)

Accompanying the 4-dimensional vector tower there is one 4-dimensional scalar tower. The masses in the scalar tower are shifted from the masses in the vector tower by the fixed amount (55).

(3) λ = 0, Generic Case 1 (61)

There is a 4-dimensional vector tower and one 4-dimensional scalar tower. The masses in the scalar tower are identical to the masses in the vector tower.

(4) λ = 0, Generic Case 2 (63)

There is one 4-dimensional vector tower and no 4-dimensional scalar tower.

An examination of the masses of the known bosons as given by the Particle Data Group [22], reveals at first sight that, to any set of observed vector (or pseudo-vector) particles, there is no sign of a related set of scalar (or pseudo-scalar) particles either with identical masses or with masses squared shifted by a fixed amount.

Hence, if a Kaluza-Klein tower of vector mesons exists and some of its low states already appear in the Particle Data Group tables, the theory must belong to the case (4) above i.e. must correspond to a Lagrangian with λ = 0 while the boundary conditions must belong to the Generic Case 2.

Towers

In this section, we give the main properties of the vector and related scalar towers which follow from specific choices of allowed boundary conditions (( 68) and ( 69)). The detailed towers are obtained by following the procedure used in the first article of [26]

Towers in the Generic Case G1a

The general solution of B

[S] {n} (s) for v 2 n > 0 is a superposition of sine and cosine functions (see (75)). For v 2 n < 0 it is a superposition of hyperbolic sine and cosine functions while for v n = 0 it is a linear function in s. It is easy to see that the hyperbolic solution is incompatible with the G1a boundary conditions in Table (68) for any α, while the linear solution can only be a constant and is compatible with α = 1 only. We thus focuss of the sine and cosine solution.

Writing the general solution B

[S] {n} (s) of (38) (taking v {n} > 0) as

B [S] {n} (s) = σ [S] {n} sin(v {n} s) + τ [S] {n} cos(v {n} s), (75)
with

B [V ] {n} (s) = ∂ s B [S]
{n} (s), the set of boundary conditions G1a (68) leads to the eigenvalue equation for

v {n} cos(2πv {n} R) = 2 α + 1 α (76) with sin(2πv {n} R) = α 2 -1 α 2 + 1 , 2 = 1. ( 77
) If α 2 = 1, the field parameters σ [S]
{n} and τ

[S]
{n} are related by

τ [S] {n} = -ασ [S] {n} (78)
while if α 2 = 1 they are arbitrary.

Calling

w = 1 π arccos 2 α + 1 α , 0 ≤ w ≤ 1 (79)
the mass tower m 2 {n} is composed of the = +1 and = -1 intertwined branches

n = 2p + 1 , m 2 {2p+1} = M 2 + (2p + w) 2 4R 2 , p ≥ 0 n = 2p , m 2 {2p} = M 2 + (2p -w) 2 4R 2 , p ≥ 1. (80)
Note that the index {n} is defined in such a way that the first state in the tower corresponds to n = 1 and that the masses are ordered in such a way that they increase with n. Remark also that for α = ±1, every state in the tower is twice degenerate except for α = 1 where the first state with m 2 {1} is single. Indeed, for α = 1 (w = 0) one has m 2 {2q} = m 2 {2q+1} , q ≥ 0 and for α = -1 (w = 1) one has m 2 {2q+1} = m 2 {2(q+1)} , q > 0.

The only possibility for having v {1} = 0 is a constant

B [S]
{n} (s). It implies α = 1 and corresponds to the Special Case 1 (59) or Special Case 2 (60) (see (64) or ( 65)).

The lowest mass squared in the tower (41) is always larger than the bulk mass squared M 2 if α = 1 and equal to it if α = 1. Note that the bulk mass squared can a priori be negative which means that the original five-dimensional field is tachyonic. This could lead, at the bottom of the tower, to the appearance of one or more four-dimensional tachyons. A zero mass state (eventually twice degenerate for α 2 = 1) can occur provided the boundary parameter α is such that M 2 equals -v 2 {n} i.e. the negative of one of the eigenvalues.

Towers in the Generic Case G1b

The boundary conditions in the Case G1b of (68), when imposed to the sine and cosine solution (75) for B

[S] {n} (s) (v {n} > 0), lead to the well-known regular tower

v {n} = n 2R for n = 1, 2, . . . (81)
as the solutions of the equation

sin(2πRv {n} ) = 0. ( 82
)
There is no hyperbolic sine or cosine solution (v 2 {n} < 0). The case v {n} = 0, corresponding to a linear B {n} (s) = 1 by normalisation. This corresponds to the lowest state (n = 0) in the tower which is of the (59) form. The resulting values of the masses squared of all the states in the tower are finally

m 2 {n} = M 2 + n 2R 2 for n = 0, 1, . . . . (83)

Towers in the Generic Case G1c

Analogously to the preceding G1b case, the boundary conditions in the Case G1c of (68) lead for B

[S] {n} (s) to the sine and cosine solution when v 2 {n} > 0 but neither to a constant solution for v {n} = 0 nor to a hyperbolic solution for v 2 {n} < 0. The same regular tower (83) results except that the n = 0 state is absent.

Towers in the Generic Cases G1d, G1e

Analogously to the preceeding G1b, G1c cases, the boundary conditions in the Case G1d and G1e of (68) lead to the sine and cosine solution for v 2 {n} > 0 but neither to a constant solution for v 2 {n} = 0 nor to a hyperbolic solution for v 2 {n} < 0 . The v {n} > 0 must satisfy the equation cos(2πRv {n} ) = 0 (84)

and the masses squared of the states in the tower are

m 2 {n} = M 2 + 1 4R 2 n + 1 2 2 for n = 0, 1, . . . . (85)

Towers in the Generic Case G2

Since the

B [V ]
{n} are allowed to satisfy any set of boundary conditions (69) as summarized in Table (1

) (with [P ] replaced by [V ]) and since B

[S] {n} = 0, there are only vector towers and no corresponding scalar towers. The allowed towers are summarized in Table (2). They are in complete correspondance with the towers obtained for scalar states in a flat space (see the first article in [26]). In general, one finds that the masses in the towers are derived from equations involving trigonometric functions and polynomials. These equations depend in a specific way on the parameters defining the boundary conditions and lead to less regular towers than the usual linear forms (83), (85) which were originally associated to Kaluza-Klein towers.

Conclusions

The occurence of Kaluza-Klein towers for vector fields propagating in a five dimensional compactified flat space might be interesting to look at, at high energies. In this paper, we have analyzed and presented the results of a careful study of all the sets of allowed boundary conditions which follow from the physical requirement of hermiticity (symmetry or self-adjointness) of the operators whose eigenvalues are related to the Kaluza-Klein mass tower states, as seen in a 4-brane.

Our approach starts from the most general free Lagrangian quadratic in the bulk vector field and its derivatives. This Lagrangian depends on two free parameters, the bulk mass M and one extra parameter λ (see Eq.( 2) and Eq.( 4)). The sets of allowed boundary conditions are restricted by the all important Lorenz 5-dimensional gauge condition.

In the case of the generalized Lagrangian (λ = 0), there appears, for any set of boundary conditions and apart from the main tower constituted by vector states (with possibly an associated tower of scalar states with the same mass spectrum), an extra tower of scalar particles with masses squared which differ from the main tower masses squared by a universal additive term (1 -λ)M 2 /λ which does not depend on the parameters chosen for the boundary conditions.

For the main tower, there are essentially two distinct situations. In the first one, the four-dimensional vector tower is associated to a four-dimensional scalar tower with coinciding masses. Their spectrum is simple and regular (see (80), ( 83) and ( 85)). In the second situation, there is no associated scalar tower. The allowed vector mass spectra are much richer (see Tables (2), ( 3), ( 4)) and similar to the spectra found for the scalar and spinor towers (see the first and fourth articles of [26]).

It may be too naive to try to associate the towers for vector states that we have obtained from a flat five-dimensional compactified space to experimentally existing particle masses (3.3). Indeed, as is well-known, the mass scale given by the parameter 1/R, which should be of the order of Tev for a flat space, suffers from severe defects (3.2). Randall and Sundrum [13] elegantly cured this problem by the consideration of warped spaces with an exponential behavior of the metric in the extra dimension. We intend to address this issue in a forthcoming article.

Regarding the photon, it is believed that it has no associated massless scalar partner, as an axion like particle [29] has up to now escaped observation. If it is then a low lying state of a vector tower, boundary conditions of the Generic Case 2 type must apply. Moreover, the zero mass of the photon would require very precise relations (fine tuning!) between the free parameters, including the bulk mass.

A Discussion of the joined boundary conditions for B

[V ] {n} and B

[S]

{n} resulting from the gauge conditions

The a priori independent allowed sets of boundary conditions for B

[S] {n} and B

[V ]

{n} are given in Table (1). The gauge conditions (56) imply that these functions and hence their respective boundary conditions are related (see (57)-( 60)).

A.1 Generic Case 1

In the Generic Case 1 (57), the functions

B [V ] {n} and B [S]
{n} are related by

∂ s B [S] {n} = B [V ] {n} . (86)
This implies the following compatibility conditions

• Set A1 for B [V ] {n} (see Table (1)). If the boundary conditions for B [V ] {n} are of the form A1 B [V ] {n} (2πR) = α [V ] 1 B [V ] {n} (0) + α [V ] 2 ∂ s B [V ] {n} (0) ∂ s B [V ] {n} (2πR) = α [V ] 3 B [V ] {n} (0) + α [V ] 4 ∂ s B [V ] {n} (0) α [V ] 1 α [V ] 4 -α [V ] 2 α [V ] 3 = 1, (87)
those deduced naively from (86) for B

[S]

{n} , taking into account (38), give also boundary conditions of the form A1 for B

[S] {n} with α [S] 1 = α [V ] 4 α [S] 2 = - α [V ] 3 v 2 {n} α [S] 3 = -v 2 {n} α [V ] 2 α [S] 4 = α [V ] 1 . (88)
Since the boundary parameters must be independent of {n}, and taking into account that α

[S] 1 α [S] 4 -α [S] 2 α [S]
3 must be equal to 1, the values of all the parameters are given in terms of one parameter only, say α α

[S] 2 = α [S] 3 = α [V ] 2 = α [V ] 3 = 0 α [V ] 1 = α , α [V ] 4 = 1 α , α [S] 1 = 1 α , α [S] 4 = α (89)
leading to the set G1a in (68).

• Set A2 for B

[V ] {n} (see Table (1)). Applying the same reasoning as for the set A1, one finds that the only allowed case A2 for B

[V ] {n} is with ρ [V ] 1 = 0 and ρ [V ] 2 = 0. It implies the case A5 for B [S]
{n} . The set G1c in (68) follows.

• Set A3 for B [V ]
{n} (see Table (1)). Applying the same reasoning as for the set A1, one finds that the only allowed case A3 for B {n} (see Table (1)). Applying the same reasoning as for the set A1, one finds that the only allowed case A4 for B

[V ] {n} is with ζ [V ] = 0. It implies the case A3 for B [S]
{n} with κ [S] = 0. The set G1e in (68) follows.

• Case A5 for B [V ]
{n} (see Table (1)). Applying the same reasoning as for the set A1, one finds that the only allowed case for B • The results for the Generic Case 1 are summarized in (68) in Sect.(2.5).

A.2 Generic Case 2

The Generic Case 2, corresponding to B

A.3 Special Cases 1 and 2

The Special Cases 1 and 2, corresponding to the function B

{n} (2πR) = B [S] {n} (0) + α [S] 2 ∂ s B [S] {n} (0) ∂ s B [S] {n} (2πR) = ∂ s B [S] {n} (0) (90) Boundary Conditions Sb ∂ s B [S] {n} (0) = 0 ∂ s B [S] {n} (2πR) = 0 . ( 91
(2πR) = α [P ] 1 B [P ] {n} (0) + α [P ] 2 ∂ s B [P ] {n} (0) ∂ s B [P ] {n} (2πR) = α [P ] 3 B [P ] {n} (0) + α [P ] 4 ∂ s B [P ] {n} (0) α [P ] 1 α [P ] 4 -α [P ] 2 α [P ] 3 = 1 A2 ∂ s B [P ] {n} (0) = ρ [P ] 1 B [P ] {n} (0) ∂ s B [P ] {n} (2πR) = ρ [P ] 2 B [P ] {n} (2πR) A3 B [P ] {n} (0) = 0 ∂ s B [P ] {n} (2πR) = κ [P ] B [P ] {n} (2πR) A4 B [P ] {n} (2πR) = 0 ∂ s B [P ] {n} (0) = ζ [P ] B [P ] {n} (0) A5 B [P ] {n} (0) = 0 B [P ] {n} (2πR) = 0

2. 2 . 1

 21 Basic boundary relations. Usual case (λ = 0)

= 0 and ∂ x 5

 5 x 5 C) 2πR δB 2πR = (-αB + β∂ x 5 B) 2πR δC 0 = (αC + β∂ x 5 C) 0 δB 0 = (-αB + β∂ x 5 B) 0 , and provided ∂ x 5 C| 2πR 0

r 2 increasing with r and m [S] p 2 increasing

 22 with p. Together they build up the B A tower.

  define the corresponding Kaluza-Klein mass tower for B µ , and analogously the (v [S] p ) 2 define the mass tower for B S . The full B A tower is composed of all the vector (m [V ] r ) 2 and scalar (m [S] p ) 2 squared masses (41) organized in such a way that m 2 {n} increases with some index which we call {n}. There are two possibilities. Either the same set of boundary conditions is imposed to B [V ] r and to B [S] p , in which case the corresponding spectra (37),(38) of eigenvalues coincide. Or different sets of boundary conditions are imposed to B [V ] r and to B [S] p , in which case the Kaluza-Klein tower is composed of the two corresponding spectra. This conforms to the idea that SO(4,1) is broken to SO(3,1) both by the compactification of the s coordinate and by the Kaluza-Klein reduction (35)-(36). In general, for a given m 2 {n} , there are three possible situations (1) a vector (B [V ] {n} λ µ {n} = 0) and no scalar particle ( B [S] {n} φ {n} = 0) (2) no vector (B [V ] {n} λ µ {n} = 0) and a scalar particle (B [S] {n} φ {n} = 0) (3) a vector (B [V ] {n} λ µ {n} = 0) and a scalar particle (B [S]

  p and B v (s) must satisfy boundary relations of the form (43)-(44)

  same BC for B[V ] r and B v .

  {n} (s), leads to a constant field, say B[S]

  {n} is with κ [V ] = 0. It implies the case A4 for B [S] {n} with ζ [S] = 0. The set G1d in (68) follows. • Set A4 for B [V ]

  {n} arising from the case A5 forB [V ]{n} is the case A2 with ρ [S] 1 = ρ [S] 2 = 0. The set G1b in (68) follows.

  {n} = 0 (58), is compatible with the boundary conditions of all the sets of Table (1) for [P ] = [V ].

  {n} = 0, are compatible with any allowed boundary conditions forB [V ]{n} and with two subsets only of boundary conditions for B[S] {n} (originating from the Sets A1 and A2 of Table(1

1 gc2

 1 {n} sin(v {n} s) + τ [V ] {n} cos(v {n} s), B [S] {n} (s) = 0). The upper index [V ] has been omitted for the parameters σ {n} = 0, see (58)) Boundary Conditions and Towers for Real Vector Fields withv {n} ≥ v {n} (α 1 +α 4 ) cos 2πv {n} R + α 2 v 2 {n} -α 3 sin 2πv {n} R -2v {n} = 0 ∂sB [V ] {n} (2πR) = α 3 B [V ] {n} (0) + α 4 ∂sB [V ] {n} (0) sin 2πv {n} R -α 2 v {n} σ {n} = α 1 -cos 2πv {n} R τ {n} α 1 α 4 -α 3 α 2 = v {n} (ρ2 -ρ 1 ) cos(2πv {n} R) + v 2 {n} + ρ 1 ρ 2 sin(2πv {n} R) = 0 ∂sB [V ] {n} (2πR) = ρ 2 B [V ] {n} (2πR) v {n} σ {n} = ρ 1 τ {n} gc2-A3-tw B {n} cos(2πv {n} R) -κ sin(2πv {n} R) v {n} cos(2πv {n} R) + ζ sin(2πv {n} R) v {n} σ {n} = ζ τ {n} gc2-A5-tw B [V ] {n} (0) = 0 sin(2πv {n} R) = 0 B [V ]{n} (2πR) = 0 τ {n} = 0

1 + α 4 - 2 ) -2πR α 3 α 4 )

 14234 {n} (s) = 0). The upper index [V ] has been omitted for the parameters A {n} = 0, see (58)) Boundary Conditions for a Real Vector State withv {n} = A = α 3 B α 1 α 4 -α 3 α 2 = 1 gc2-A2-0 ∂sB [V ] {n} (0) = ρ 1 B [V ] {n} (0) (ρ1 -ρ 2 ) -2πR ρ 1 ρ 2 = 0 ∂sB [V ] {n} (2πR) = ρ 2 B

1 gc2

 1 {n} sinh(w {n} s) + τ[V ]{n} cosh(w {n} s), B [S] {n} (s) = 0). The upper index [V ] has been omitted for the parameters σ {n} = 0, see (58)) Boundary Conditions and Towers for a Real Vector State with w2 {n} = w {n} (α 1 +α 4 ) cosh 2πw {n} R -α 2 w 2 {n} +α 3 sinh 2πw {n} R -2w {n} = 0 ∂sB [V ] {n} (2πR) = α 3 B [V ] {n} (0) + α 4 ∂sB [V ] {n} (0) sinh 2πw {n} R -α 2 w {n} σ {n} = α 1 -cosh 2πw {n} R τ {n} α 1 α 4 -α 3 α 2 = w {n} (ρ 1 -ρ 2 ) cosh(2πw {n} R) + w 2 {n} -ρ 1 ρ 2 sinh(2πw {n} R) = 0 ∂sB [V ] {n} (2πR) = ρ 2 B [V ] {n} (2πR) w {n} σ {n} = ρ 1 τ {n} gc2-A3-tac B {n} cosh(2πw {n} R) -κ sinh(2πw {n} R) w {n} cosh(2πw {n} R) + ζ sinh(2πw {n} R) w {n} σ {n} = ζ τ {n} gc2-A5-tac B [V ] {n} (0) = 0 sinh(2πw {n} R) = 0 → impossible B [V ]{n} (2πR) = 0 τ {n} = 0

Table 1 :

 1 ) Table of allowed boundary conditions for B

	[P ] {n} with [P ] = [S] or [P ] = [V ], for

Table 2 :

 2 Table of Kaluza-Klein vector towers v 2 {n} > 0 for the Generic Case 2 (B

Table 3 :

 3 Table of Kaluza-Klein vector states v 2 {n} = 0 for the Generic Case 2 (B

	[V ] {n} (s) =

Table 4 :

 4 Table of Kaluza-Klein vector state v 2 {n} < 0 (w 2 {n} = -v 2 {n} > 0) for the Generic Case 2 (B

Acknowledgment: The authors would like to thank Professors David Fairlie and Anne Taormina for discussions on general solutions of the equations of motion, and Professor Nicolas Boulanger for discussions in relation with particles of higher spins.