Verena Varadi
email: v.varadi@dkfz-heidelberg.de

Melanie Bevier

Ewa Grzybowska

Robert Johansson

Kerstin Enquist

Roger Henriksson

Dorota Butkiewicz

Jolanta Pamula-Pilat

Karolina Tecza

Kari Hemminki

Asta Försti

Genetic variation in genes encoding for polymerase ζ subunits associates with breast cancer risk, tumour characteristics and survival

Keywords: Breast Cancer, SNPs, chromosomal instability, polymerases, prognostic marker

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

DNA polymerases have a central function in DNA metabolism as they are involved in DNA replication, repair, recombination and translesion synthesis and with that ensure faithful transmission of genetic information from one generation to the next [START_REF] Caldecott | Single-strand break repair and genetic disease[END_REF][START_REF] Friedberg | Suffering in silence: the tolerance of DNA damage[END_REF][START_REF] Loeb | DNA polymerases and human disease[END_REF][START_REF] Sweasy | DNA polymerases and human diseases[END_REF]. As a dysfunction within these genes might have severe influences, DNA polymerase genes represent potential candidates for cancer genes. A total number of 15 enzymes with a known DNA synthesis activity exist. Some of these polymerases, the so called classic DNA polymerases [START_REF] Loeb | DNA polymerases and human disease[END_REF], are involved in rapid, high-fidelity replication of most of the eukaryotic genome, but are not capable of working on sites of DNA damage [START_REF] Hubscher | Eukaryotic DNA polymerases[END_REF]. That often leads to a blockage of DNA synthesis with a stalled replication fork [START_REF] Wittschieben | Loss of DNA polymerase zeta causes chromosomal instability in mammalian cells[END_REF]. The arrest of DNA replication might even influence the ability of the cells to complete the cell cycle, and as a consequence it induces multiple mechanisms collectively referred to as DNA-damage-tolerance mechanism [START_REF] Friedberg | Suffering in silence: the tolerance of DNA damage[END_REF]. Several specialized DNA polymerases, the so called translesion synthesis (TLS) DNA polymerases [START_REF] Loeb | DNA polymerases and human disease[END_REF], are involved in this process, and they allow strand extension across template lesions [START_REF] Friedberg | Suffering in silence: the tolerance of DNA damage[END_REF]. Despite their special function across template lesions, when operating on undamaged DNA or non-cognate lesions, the TLS polymerases show a considerably reduced fidelity. Thus, both overand underexpression of these polymerases can lead to an increased mutation rate, suggesting that they could potentially act both as oncogenes and tumour suppressor genes, respectively [START_REF] Albertella | The overexpression of specialized DNA polymerases in cancer[END_REF][START_REF] Brondello | Novel evidences for a tumor suppressor role of Rev3, the catalytic subunit of Pol zeta[END_REF]. Loss of specific DNA polymerases has been reported to cause chromosomal instability in mammalian cells [START_REF] Wittschieben | Loss of DNA polymerase zeta causes chromosomal instability in mammalian cells[END_REF] and chromosomal unstable cancers are likely to have a poorer prognosis than diploid cancers [START_REF] Carter | A signature of chromosomal instability inferred from gene expression profiles predicts clinical outcome in multiple human cancers[END_REF][START_REF] Habermann | The gene expression signature of genomic instability in breast cancer is an independent predictor of clinical outcome[END_REF] with the degree of aneuploidy being in accordance with the severity of the disease [START_REF] Rajagopalan | Aneuploidy and cancer[END_REF].

We hypothesized, that common genetic variation in the form of single nucleotide polymorphisms (SNPs) in genes encoding classic or TLS DNA polymerases could affect DNA replication by influencing gene expression or protein configuration. This might increase the level of mutations and influence cellular transformation and in the end affect cancer risk and clinical outcome. We focused on four genes encoding the catalytic subunits of two classical polymerases, polymerase δ (POLD1) and polymerase β (POLB), and two TLS-polymerases polymerase ζ (REV3L) and polymerase θ (POLQ). These polymerases have already been reported to be involved in different types of cancer or causing chromosomal instability [START_REF] Albertella | The overexpression of specialized DNA polymerases in cancer[END_REF][START_REF] Brondello | Novel evidences for a tumor suppressor role of Rev3, the catalytic subunit of Pol zeta[END_REF][START_REF] Kawamura | DNA polymerase theta is preferentially expressed in lymphoid tissues and upregulated in human cancers[END_REF][START_REF] Srivastava | DNA polymerase beta expression differences in selected human tumors and cell lines[END_REF][START_REF] Sweasy | DNA polymerases and human diseases[END_REF][START_REF] Wittschieben | Loss of DNA polymerase zeta causes chromosomal instability in mammalian cells[END_REF]. In a second selection round, we selected SNPs in the genes MAD2L2 and REV1. The encoded proteins act as a subunit (MAD2L2) and a scaffold protein (REV1) for the polymerase ζ enzyme [START_REF] Srivastava | DNA polymerase beta expression differences in selected human tumors and cell lines[END_REF][START_REF] Szuts | REV1 restrains DNA polymerase zeta to ensure frame fidelity during translesion synthesis of UV photoproducts in vivo[END_REF].

We selected putatively functional SNPs and most informative tagging SNPs (tagSNPs) from these six genes and genotyped them in a population-based series of 783 Swedish BC cases with detailed clinical data and up to 15 years of follow-up together with 1562 age and gender matched controls. SNPs that showed a significant association in the Swedish population were also genotyped in a Polish population consisting of 506 familial/early onset BC cases with corresponding clinical data and 568 controls.

Material and Methods

Study populations

All selected SNPs were genotyped in a Swedish study population consisting of 783 BC cases and 1562 controls. Seven hundred forty nine cases with age and gender matched controls were drawn from the population-based Västerbotten intervention project and the mammary screening project, which contain blood samples collected between January 1990 and January 2001 from an ethnically homogenous population living in a geographically defined region in North Sweden [START_REF] Kaaks | Prospective study of IGF-I, IGF-binding proteins, and breast cancer risk, in northern and southern Sweden[END_REF]. Prospective cases were identified from the cohorts by record linkage to the regional cancer registry. The controls were selected from the same cohort as the corresponding case. They were matched with the case by age at baseline (+/-6 months) and the time of sampling (+/-2 months). The controls had to be alive at the time of diagnosis of the corresponding case and without any previous cancer diagnosis, except carcinoma in situ of cervix uteri. Thirty three samples were collected consecutively during the same time period from untreated patients referred to the Department of Oncology for newly diagnosed BC. Their controls, also matched for age and gender, were selected from the Västerbotten intervention cohort. The genotyping for this study was performed blinded by the casecontrol status of each sample using whole genome amplified DNA as described by Brendle et al 2009 [START_REF] Brendle | Single nucleotide polymorphisms in chromosomal instability genes and risk and clinical outcome of breast cancer: a Swedish prospective case-control study[END_REF]. Clinical data for the unselected BC cases was retrieved from the registry managed by the Northern Sweden Breast Cancer Group (Table 1). Information about the date of death was collected from the Swedish population register with a BCspecific follow-up until 31 st of December 2004. The median follow-up time was 4.7 years.

In order to investigate whether the SNPs that showed a significant association in the Swedish population also show an association in a population with familial/early onset BC, we genotyped a second population consisting of 506 familial/early onset BC cases and 568 regionally and ethnically matched controls [START_REF] Varadi | Polymorphisms in telomere-associated genes, breast cancer susceptibility and prognosis[END_REF]. The cases were collected through the Chemotherapy Clinics, the Genetic Counselling Service and the Surgery Clinics (Gliwice, Poland) between the years 1997-2009. Only cases without BRCA1/2 mutations were included in the study. Clinical data was available for 475 cases (Table 1). Follow-up was performed until 31 st December 2009. The median follow-up time after surgery was 4.1 years. We used event-free survival (EFS) as the clinical endpoint. EFS was defined as the time from surgery to the occurrence of a breast event (i.e. local or distant recurrence, contralateral BC or death due to any cause).

All participants gave informed consent to the use of their samples for research purpose.

The study was approved by the ethical committees of the participating institutes.

Selection of SNPs

We focused on four genes encoding for the catalytic subunits of the polymerases β, δ, θ and ζ (POLB, POLD1, POLQ and REV3L, respectively), and two associated genes, MAD2L2 and REV1. SNPs with a minor allele frequency (MAF) of ≥0.1 located in promoters (up to 1kb upstream of transcription start), 3`UTRs, reported nonsynonymous (ns) SNPs and most informative tagSNPs, were selected in these genes using data for the CEU population (Utah residents with Northern and Western European ancestry from the CEPH -Centre d'Etude du Polymorphisme Human population) from the International HapMap Project (http://www.hapmap.org/) and the NCBI dbSNP database (http://www.ncbi.nlm.nih.gov/snp). TagSNPs were selected using the plug-in tagSNP picker incorporated in the HapMap function <Bulk data download> (http://hapmap.ncbi.nlm.nih.gov/downloads/index.html.en). They were sequenced for verification in a set of 32 BC samples as described earlier by Vaclavicek et al 2007 [START_REF] Vaclavicek | Genetic variation in the major mitotic checkpoint genes does not affect familial breast cancer risk[END_REF].

Promoter polymorphisms within or close to transcription factor binding sites may alter gene expression and contribute to tumorigenesis. To identify putative transcription factor binding sites we used the TESS-Transcription Element Search System tool (http://www.cbil.upenn.edu/tess/). SNPs located in the 3´UTR of the genes may have an effect on microRNA binding sites. MicroRNAs can down-regulate protein translation or even degrade mRNA. To identify putative microRNA binding sites we used the microRNA.org prediction tool (http://www.microrna.org/microrna/getGeneForm.do). nsSNPs were included in our analyses due to their potential effect on protein function.

The SIFT (Sorting Intolerant from Tolerant) prediction tool (http://sift.jcvi.org/) was used to predict a possible impact of an amino acid substitution on the structure and function of human proteins [START_REF] Ng | SIFT: Predicting amino acid changes that affect protein function[END_REF][START_REF] Ng | Predicting deleterious amino acid substitutions[END_REF]. Additionally, an estimation of the impact of the different SNPs according to their linkage disequilibrium (LD) with other SNPs was done using the Haploview software [START_REF] Teare | Genetic linkage studies[END_REF]. Verified SNPs and the SNPs selected for genotyping are documented in Table 2.

Genotyping

Genotyping was performed using an allele specific PCR based KASPar SNP Genotyping System purchased from KBiosciences (Hoddesdon, Great Britain).

Thermocycling was performed according to the KBiosciences` PCR conditions. Detection was performed using an ABI PRISM 7900 HT Sequence Detection System with SDS 2.2 software (Applied Biosystems, Weiterstadt, Germany). The Swedish sample set contained 104 duplicate samples as quality controls for the KASPar assays.

For less than 0.1% the genotypes of the duplicate samples did not agree with each other.

Statistical analysis

The observed genotype frequencies in the controls were tested for Hardy-Weinberg equilibrium (HWE) using the χ² test. Statistical significance for a different genotype distribution in cases and controls was determined by global χ² test. When the overall genotype effect of a SNP on the risk of BC or a clinical factor was statistically significant at the level of 0.05, the best model to represent the relationship between the genotypes and the risk of BC or a clinical factor, respectively, was selected based on likelihood ratio tests (LRTs). Odds ratios (ORs) and 95% confidence intervals (95% CIs) for associations between genotypes, BC risk and tumour characteristics were calculated by logistic regression (PROC LOGISTIC, SAS Version 9.2; SAS Institute, Cary, NC). Haplotypes and diplotypes were estimated and compared using the Haplotypes procedure of SAS/Genetics (SAS Version 9.1). The frequency of a specific haplotype or diplotype (haplotype pair) was estimated by summing the carrier probability for all women. The relationship between haplotypes/diplotypes and BC was assessed by logistic regression with the most common haplotype/diplotype as the reference. Gene-gene interaction was analysed by pairwise interaction using logistic regression.

The survival curves for the BC-specific survival were derived by Kaplan-Meier method (PROC LIFETEST, SAS Version 9.2, SAS Institute). The joint analysis was carried out using Mantel-Haenszel adjustment. The relative risk of death by BC was estimated as hazard ratio (HR) using Cox regression (PROC PHREG, SAS Version 9.2, SAS Institute). In the Swedish population, censoring events were death by another cause than BC, moving out of the study and 31 st of December 2004. In the Polish study, cases without a breast event on 31 st December 2009 were censored. The HRs were also calculated within subgroups of cases with a similar manifestation of a clinical factor. Furthermore, the hazard ratios were adjusted for the clinical factors (estrogen and progesterone receptor status, tumour size, lymph node metastasis, histologic grade) to determine the value of the genotypes as an independent prognostic marker.

Results

SNPs in REV3L and MAD2L2 associate with BC risk in the Swedish study population

The global χ 2 test indicated a significant association for two SNPs in REV3L and one SNP in MAD2L2 with BC susceptibility (Table 3). According to LRT, the best model to represent the relationship between the genotypes of the REV3L SNPs and the risk of BC was multiplicative for rs462779 (OR 0.79, 95%CI 0.68-0.92) and dominant for rs3204953 (OR 1.28, 95%CI 1.05-1.56). The haplotype and diplotype analysis of the three SNPs genotyped in REV3L confirmed the association by showing an inverse correlation of the haplotype CCG (rs11153292 T/C -rs462779 T/C -rs3204953 G/A: OR 0.81, 95% CI 0.70-0.95) and the diplotype CCG/CCG (OR 0.62, 95% CI 0.40-0.97) with BC (data not shown).

For the MAD2L2 SNP rs2233004, carriers of the minor allele had a significantly decreased BC risk (recessive model: OR 0.49, 95%CI 0.28-0.86). This result was confirmed by the haplotype and diplotype analysis in which the only haplotype containing the variant G allele of rs2233004, GAT, was associated with BC risk with an OR of 0.77 (95%CI 0.60-0.99) and the diplotype GAT/GAT with an OR of 0.50 (95%CI 0.28-0.90) (data not shown).

Gene-gene interaction studies between REV3L and MAD2L2 with risk associated alleles showed no evidence for epistasis (data not shown). SNPs in POLB, POLQ and POLD1

were not associated with BC susceptibility (Table 3).

None of the associations were replicated in the Polish population, which contained familial/early onset BC cases, at a statistically significant level. However, in the joint analysis the ORs remained statistically significant (Table 3).

Association of REV3L SNPs rs11153292 and rs462779 with survival

For the tagSNP rs11153292, located in intron 6 of REV3L, an association with BCspecific survival was observed. Homozygous carriers of the minor allele C had a significantly worse survival compared to the TT genotype carriers (HR 2.93, 95%CI 1.34-6.44) (Table 4, Fig 1A). After adjustment for tumour size, lymph node metastasis, histologic grade, estrogen and progesterone receptor status, the HR was 3.95 (95% CI 1.18-13.22) (Table 4), indicating the value of rs11153292 as an independent prognostic marker. The survival was poor especially for women with the CC genotype and No association with EFS in the Polish population was observed.

Association of REV1 SNPs rs6761390 and rs3792142 with tumour characteristics

When we examined the genotype distribution by tumour characteristics at the time of diagnosis, significant associations were observed for the REV1 SNPs rs6761390, rs3792136, rs3792142 and rs4341989 (supplementary table1+2). Minor allele carriers of the SNPs rs6761390 and rs3792142 had more often large tumours and tumours with high histologic grade and stage than the common homozygotes. LD analysis indicated that these two SNPs were in complete LD. This was previously not known as only genotype information from two different European populations, CEU for rs3792142 and CEPH for rs6761390, was available. For rs3792136, carriers of the minor allele were more likely to have small tumours and stage 0+I tumours than the CC carriers. Minor allele carriers of rs4341989 had more often hormone receptor positive tumours than the common homozygotes. These SNPs were not associated with tumour characteristics in the Polish population. No other associations with tumour characteristics were observed.

Discussion

Chromosomal instability is a characteristic of almost all human cancers [START_REF] Negrini | Halazonetis TD Genomic instability--an evolving hallmark of cancer[END_REF]. The degree of chromosomal instability has been shown to predict clinical outcome and discern prognostically relevant BC subtypes [START_REF] Habermann | The gene expression signature of genomic instability in breast cancer is an independent predictor of clinical outcome[END_REF]. The strength of our study lies in the selection of six polymerase subunit genes according to their previously reported association with chromosomal instability and/or tumorigenesis. We focused on SNPs with the highest probability of an influence on gene transcription or protein product, and on the most informative tagSNPs in order to cover a multitude of genetic variation within a gene.

With the present sample size of the Swedish study population we had a power of >90% to detect an OR of 1.5 for the risk of BC in SNPs with a MAF ≥ 0.1.

We showed that genetic variation in genes encoding for the polymerase ζ subunits REV3L, MAD2L2 and REV1 had a statistically significant influence on BC risk and the outcome of the disease. Two SNPs in REV3L associated with BC risk, one of these was additionally associated with survival for patients with aggressive tumours. A third SNP might even represent an independent prognostic marker. One SNP in MAD2L2 associated with BC risk. Several SNPs in REV1 associated with tumour characteristics.

REV3L encodes for the catalytic subunit of polymerase ζ. Wittschieben et. al. reported that loss of REV3L expression could contribute to genomic instability during neoplastic transformation and progression [START_REF] Wittschieben | Loss of DNA polymerase zeta causes chromosomal instability in mammalian cells[END_REF] and Brondello et. al. reported that REV3L depletion leads so genomic instability through double strand break accumulation [START_REF] Brondello | Novel evidences for a tumor suppressor role of Rev3, the catalytic subunit of Pol zeta[END_REF]. REV3L has also been reported to have a lower expression in tumour versus normal tissue from early to late stages of colon carcinoma [START_REF] Brondello | Novel evidences for a tumor suppressor role of Rev3, the catalytic subunit of Pol zeta[END_REF]. These findings provide evidence for a role of REV3L as a tumour suppressor gene.

The two SNPs in REV3L, which associated with BC risk, are nsSNPs: rs462279 leads to an amino acid change from threonine to isoleucine, rs3204953 from valine to isoleucine.

However, no functional effect on the coded protein was predicted for any SNP by the SIFT prediction tool, nor did the three 3`UTR SNPs that were captured by the two nsSNPs change the binding sites for any microRNA. The third SNP, rs11153292, in REV3L associated with survival with homozygous carriers of the minor allele C having a worse survival compared with the common homozygotes. The strongest association was observed in patients with aggressive tumours (lymph node metastasis and high stage tumours). rs11153292 is a highly informative intronic tagSNP capturing 41 other intronic SNPs, with unknown function.

In MAD2L2, an association of the SNP rs2233004 with BC risk was observed. This promoter SNP is not in LD with any other SNP. The G-allele of the SNP creates an additional binding site for the transcription factor CCAAT enhancer binding protein alpha (CEBPA) (TFSEARCH). However, the homozygous minor allele genotype that was associated with risk was very rare with a frequency of 2% and 4% in cases and controls, respectively, minimizing the power of this result. The two other promoter SNPs genotyped in MAD2L2 were not associated with BC risk. In our previous casecontrol study, the most common MAD2L2 promoter SNP rs2233006 was not associated with familial BC [START_REF] Vaclavicek | Genetic variation in the major mitotic checkpoint genes does not affect familial breast cancer risk[END_REF]. Thus, genetic variation in the promoter of MAD2L2 does not seem to affect strongly on BC risk.

In REV1, rs6761390 and rs3792142, which were in complete LD, associated with tumour size, histologic grade and stage. rs6761390 is located in the promoter of the REV1 gene, rs3792142 in intron 5. Additionally, they capture three intron 1 SNPs, rs4143760, rs1011633 and rs896249. Only for rs4143760 a change of a transcription factor binding site was predicted by TFSEARCH, with the A-allele of the SNP creating a sequence for binding of GATA transcription factors.

None of the associations between the genotyped SNPs and BC risk, tumour characteristics and survival in the Swedish study population were confirmed in the Polish study population at a statistically significant level. The main difference in the study populations lies in the selection criteria for the cases. The Swedish study is a population-based study, whereas the Polish study comprises of cases with a family history of BC or an early onset of the disease. The sample sets show a significant difference in the median age of the cases (Swedish 58 years; Polish 43 years).

Additionally, the survival probabilities for the Polish cases were calculated using any breast event as the clinical endpoint compared with death due to BC for the Swedish population. Thus, the lack of association in the Polish population may not necessarily indicate a lack of association with death due to BC, but a lack of association with BC progression. We selected the genes for the present study based on their known association with chromosomal instability. Chromosomal instability is common in both sporadic and hereditary cancer. In hereditary cancers, the presence of chromosomal instability has been linked to mutations in DNA repair genes [START_REF] Negrini | Halazonetis TD Genomic instability--an evolving hallmark of cancer[END_REF]. Germline mutations in BRCA1, BRCA2, PALB2, NBS1, WRN, BLM and RECQL4, all of which have been linked to the repair of DNA double-strand breaks or DNA interstrand cross links, have been reported to predispose to the development of various chromosomal instable hereditary cancers [START_REF] Bachrati | RecQ helicases: suppressors of tumorigenesis and premature aging[END_REF][START_REF] Kennedy | DNA repair pathways in clinical practice: lessons from pediatric cancer susceptibility syndromes[END_REF][START_REF] Ripperger | Breast cancer susceptibility: current knowledge and implications for genetic counselling[END_REF]. Unlike hereditary cancers, the molecular basis of chromosomal instability in sporadic cancers remains unclear [START_REF] Negrini | Halazonetis TD Genomic instability--an evolving hallmark of cancer[END_REF]. These data indicate that different mechanisms influence chromosomal instability in hereditary and sporadic cancers, giving a possible explanation for the difference in our results for the Polish and Swedish study population.

For the genes POLB, POLD1 and POLQ, there was no association of the selected SNPs with BC susceptibility, tumour characteristics or survival in the Swedish sample set.

Similar results were obtained by Han et al. [START_REF] Han | Genetic variation in DNA repair pathway genes and premenopausal breast cancer risk[END_REF]. Taking our results on mainly postmenopausal women and the results of Han et al in premenopausal women into account, we can exclude, with a high probability any effect of common genetic variation within these genes on BC susceptibility and prognosis.

Altogether, all three genes encoding polymerase ζ subunits showed an association either with BC susceptibility, progression or survival. Further analysis to explain the functional consequences of the SNPs on gene expression and on BC progression are warranted.

  aggressive tumours (positive lymph node metastasis: HR 4.20, 95%CI 1.32-13.44, histologic grade 3: HR 4.70, 95%CI 1.52-14.57, stage II-IV: HR 3.20, 95%CI 1.23-8.32) (Fig.1 B, C, D).The REV3L nsSNP rs462779 was associated with survival in patients with aggressive tumours. The CC genotype carriers had worse survival than the common TT genotype carriers (positive lymph node metastasis: HR 4.15 95%CI 1.18-14.62; histologic grade 3: HR 3.37 95%CI 1.12-10.08; stage II-IV: HR 3.51 95%CI 1.20-10.23), although the association with disease-free survival was not statistically significant among all BC patients (HR 2.23 95%CI 0.78-6.38). None of the other SNPs associated with survival.

Figure 1 :

 1 Figure 1: Kaplan-Meier estimates of breast cancer-specific survival according to rs11153292 genotypes in the Swedish study population. A) all breast cancer patients, B) patients with lymph node metastasis, C) patients with histologic grade 3 tumours, D) patients with stage II -IV tumours.

Table 1 :

 1 Tumour characteristics of the Swedish and the Polish breast cancer samples at the time of diagnosis.

		Swedish samples	Polish samples
	Characteristics	N (%)	N (%)
	Age at diagnosis, mean (range)	58.0 (30.6-86.2)	43.1 (23.0-56.0)
	Estrogen receptor
	Positive	404 (51.50)	257 (50.79)
	Negative	133 (16.99)	189 (37.35)
	Missing	246 (31.42)	60 (11.86)
	Progesterone receptor
	Positive	283 (36.14)	262 (51.78)
	Negative	214 (27.33)	182 (35.97)
	Missing	286 (36.53)	62 (12.25)
	Estrogen/progesterone receptor
	ER+/PR+	229 (29.25)	224 (44.27)
	ER+/PR-	71 (9.07)	31 (6.13)
	ER-/PR+	12 (1.53)	38 (7.51)
	ER-/PR-	114 (14.56)	150 (29.64)
	Missing	357 (45.59)	63 (12.45)
	Tumour size in mm
	≤ 20 mm	501 (63.98)	190 (37.55)
	> 20 mm	228 (29.12)	289 (57.11)
	Missing	54 (6.90)	27 (5.34)
	Histologic grade
	Grade 1+2	516 (65.90)	NA
	Grade3	225 (28.74)	NA
	Missing	42 (5.36)	NA
	Regional lymph node metastasis
	Negative	469 (59.90)	265 (52.37)
	Positive	220 (28.10)	217 (42.89)
	Missing	94 (12.01)	24 (4.74)
	Stage at diagnosis
	0-I	407 (51.98)	NA
	II-IV	367 (46.87)	NA
	Missing	9 (1.15)	NA
	N= number of samples
	NA=not available

Table 2 :

 2 Summarized information with possible functional effect and linkage disequilibrium (LD) for SNPs in the selected polymerase and polymerase associated genes. All SNPs complying with our selection criteria that were confirmed by sequencing and/or database information with a minor allele frequency ≥0.1 are listed. The SNPs selected for genotyping are marked in bold.

	Gene	SNP	region	possible functional effect

Table 3 :

 3 Association of the SNPs selected in polymerase and polymerase associated genes with BC susceptibility in the Swedish study population. SNPs that showed a significant association in the Swedish population were also genotyped in the Polish population. For the SNPs with a significant association, the best model was calculated. For the SNPs with genotyping information for both populations, a joint analysis was performed.

				Swedish study population				Polish study population				Joint analysis
	Gene	SNP ID	Genotype	Cases	Controls	OR 95%CI	p-value global	Cases	Controls	OR 95%CI	p-value global	OR 95%CI	p-value
				N (%)	N (%)			p-value	N (%)	N(%)			p-value
	POLB	rs3136717	TT	589 (76.59) 1187 (77.18)		.
			TC	165 (21.46) 327 (21.26) 1.02 (0.83-1.27)	0.82
			CC	15 (1.95 )	24 (1.56 )	1.23 (0.65-2.36)	0.53	0.80
	POLD	rs1726801	GG	678 (89.56) 1335 (88.53)		.
			GA	79 (10.44)	173 (11.47) 0.89 (0.67-1.18)	0.41
			AA	0 (0)	0 (0)	.	.	0.41
	POLQ	rs487848	CC	671 (87.26) 1329 (86.81)		.
			CT	94 (12.22)	194 (12.67) 0.96 (0.74-1.26)	0.79
			TT	4 (0.52)	8 (0.52)	0.88 (0.26-2.95)	0.84	0.94
		rs3218651	AA	533 (69.13) 1069 (69.60)		.
			AG	217 (28.15) 424 (27.60) 1.03 (0.85-1.26)	0.77
			GG	21 (2.72)	43 (2.80)	1.00 (0.58-1.71)	0.99	0.96
		rs11713643	CC	320 (41.67) 618 (40.39)
			CT	338 (44.01) 704 (46.01) 0.94 (0.78-1.13)	0.49
			TT	110 (14.32) 208 (13.59) 1.02 (0.78-1.34)	0.88	0.72
	REV3L	rs11153292	TT	218 (28.42) 412 (26.82)
			TC	393 (51.24) 797 (51.89) 0.93 (0.75-1.14)	0.47
			CC	156 (20.34) 327 (21.29) 0.90 (0.70-1.16)	0.42	0.68
		rs462779	TT	492 (63.81) 886 (57.72)		.		307 (68.07)	315 (68.48)
			TC	250 (32.43) 562 (36.61) 0.80 (0.67-0.96)	0.02		129 (28.60)	129 (28.04) 1.03 (0.77-1.37)	0.86		0.86 (0.74-1.01)	0.06
			CC	29 (3.76)	87 (5.67)	0.59 (0.38-0.92)	0.02	0.01	15 (3.33)	16 (3.48)	0.96 (0.47-1.98)	0.92	0.98	0.68 (0.47-0.99)	0.04
		Best model multiplicative (number of C´s)	0.79 (0.68-0.92)	0.002				1.01 (0.79-1.28)	0.94		0.85 (0.75-0.96)	0.01
		rs3204953	GG	552 (71.60) 1170 (76.42)				326 (71.96)	317 (72.54)
			GA	199 (25.81) 330 (21.55) 1.27 (1.03-1.57)	0.02		115 (25.39)	110 (25.17) 1.02 (0.75-1.38)	0.92		1.19 (1.01-1.41)	0.04
			AA	20 (2.59)	31 (2.02)	1.34 (0.75-2.40)	0.33	0.05	12 (2.65)	10 (2.29)	1.17 (0.50-2.74)	0.72	0.94	1.30 (0.81-2.10)	0.28
		Best model dominant ((GA+AA) vs GG)	1.28 (1.05-1.56)	0.02				1.03 (0.78-1.38)	0.85		1.20 (1.02-1.41)	0.03
	MAD2L2	rs2233004	AA	511 (68.22) 983 (65.36)				274 (61.30)	287 (60.42)
			AG	222 (29.64) 455 (30.25) 0.92 (0.75-1.11)	0.37		146 (32.66)	153 (32.21) 1.00 (0.76-1.32)	1.00		0.96 (0.82-1.12)	0.59
			GG	16 (2.14)	66 (4.39)	0.48 (0.27-0.84)	0.01	0.03	27 (6.04)	35 (7.37)	0.81 (0.48-1.37)	0.43	0.72	0.61 (0.42-0.89)	0.01
		Best model recessive (GG vs (AA+AG))	0.49 (0.28-0.86)	0.01				0.81 (0.48-1.36)	0.42		0.62 (0.43-0.90)	0.01
		rs2233006	AA	322 (42.93) 644 (43.16)
			AT	339 (45.20) 684 (45.84) 1.00 (0.82-1.21)	0.98
			TT	89 (11.87)	164 (10.99) 1.13 (0.83-1.53)	0.43	0.69
		rs2233011	TT	453 (61.72) 943 (64.19)
			TG	246 (33.51) 469 (31.93) 1.08 (0.89-1.31)	0.43
			GG	35 (4.77)	57 (3.88)	1.26 (0.80-1.97)	0.32	0.5
	REV1	rs6761390	AA	181 (24.93) 388 (26.58)
			AC	362 (49.86) 703 (48.15) 1.12 (0.90-1.39)	0.32
			CC	183 (25.21) 369 (25.27) 1.07 (0.83-1.38)	0.61	0.61
		rs3792136	CC	502 (68.96) 985 (67.93)
			CT	205 (28.16) 420 (28.97) 0.98 (0.80-1.21)	0.85
			TT	21 (2.88)	45 (3.10)	0.96 (0.57-1.63)	0.88	0.97
		rs3792142	TT	182 (24.56) 391 (26.33)
			TG	371 (50.07) 724 (48.75) 1.11 (0.89-1.38)	0.36
			GG	188 (25.37) 370 (24.92) 1.10 (0.85-1.42)	0.46	0.63
		rs13409359	AA	555 (74.10) 1106 (74.23)
			AC	174 (23.23) 342 (22.95) 1.01 (0.82-1.24)	0.96
			CC	20 (2.67)	42 (2.82)	0.95 (0.53-1.69)	0.85	0.98
		rs4341989	CC	340 (46.45) 675 (45.89)
			CT	319 (43.58) 634 (43.10) 1.00 (0.82-1.21)	0.99
			TT	73 (9.97)	162 (11.01) 0.91 (0.66-1.24)	0.55	0.82
		rs3792152	CC	286 (38.65) 617 (41.47)
			CT	357 (48.24) 682 (45.83) 1.15 (0.95-1.39)	0.16
			TT	97 (13.11)	189 (12.70) 1.14 (0.85-1.52)	0.38	0.35

Table 4 :

 4 Association of REV3L SNPs rs11153292 and rs462779 with BC-specific survival in the Swedish study population and with event-free survival (EFS) in the Polish study population. For the Swedish population event = death due to breast cancer; for the Polish study population event = local or distant recurrence, contralateral breast cancer or death due to any cause

	SNP	Genotype	No	No event (%)	HR (95%CI)	p value	HR (95%CI)	p value
							adjusted for size, grade,
			Swedish study population			regional lymph node

*

Acknowledgements

We thank Asa Agren (Department of Public Health and Clinical Medicine/Nutritional Research, Umeå University, Sweden) for her efficiency and skill in keeping track of samples and data. The Northern Sweden Breast Cancer Group is appreciated for providing the clinical data.

LD POLB rs2307158 promoter putative TFBS* rs3136717 (D`=1, r 2 =0.55) rs3136717 intron 1 tag SNP tags 4 out of 5 listed HapMap SNPs POLD1 rs3219281 promoter putative TFBS* rs1726801 (D`=1, r 2 =1) rs3219282 promoter putative TFBS* rs1726801 (D`=1, r 2 =1) rs1726801 exon 4 R119H tolerated POLQ rs587553 promoter putative TFBS* rs11713643 (D`=1, r 2 =0.52) rs3806614 promoter putative TFBS* rs11713643 (D`=1,r 2 =1) rs13065220 promoter putative TFBS* rs11713643 (D`=1,r 2 =0.79) -798 ATG promoter putative TFBS* rs11713643 (D`=1, r 2 = 0.50) rs11713643 intron 1 tag SNP tags 23 out of 53 listed HapMap SNPs rs487848 exon 11 A581V tolerated rs3218649 exon 16 T982R tolerated rs11713643 (D`=1,r 2 =1) rs3218651 exon 16 H1201R tolerated rs532411 exon 24 A2304V tolerated rs487848 (D`=1,r 2 =1) rs1381057 exon 28 Q2513R tolerated rs11713643 (D`=1,r 2 =0.79) rs3218634 exon 29 L2538V possibly damaging rs487848 (D`=1,r 2 =0.81) REV3L rs11153292 intron 6 tag SNP tags 41 out of 108 listed HapMap SNPs rs462779 exon 15 T1224I tolerated rs3204953 exon 33 V3064I tolerated rs465646 3`UTR putative microRNA binding site rs462779 (D`=1,r 2 =0.70) rs1061388 3`UTR putative microRNA binding site rs3204953 (D`=1,r 2 =1) rs181294 3`UTR putative microRNA binding site rs462779 (D`=1,r 2 =0.76) MAD2L2 rs2233004 promoter putative TFBS* rs2233006 promoter putative TFBS* rs2233011 promoter putative TFBS* rs2272978 promoter putative TFBS* rs2233011 (D`=1,r 2 =1) REV1 rs28369925 promoter putative TFBS* rs3792136 (D`=1,r 2 =1) rs6761390 promoter putative TFBS* rs28369929 intron 1** putative TFBS* rs3792136 (D`=1,r 2 =1) rs6714244 intron 1** putative TFBS* rs3792152 (D`=1,r 2 =1) rs1017697 intron 1** putative TFBS* rs13409359 (D`=1,r 2 =1) rs4143760 intron 1** putative TFBS* rs3792142 (D`=1,r 2 =1) rs10173883 intron 1** putative TFBS* rs13409359 (D`=1,r 2 =1) rs6542882 intron 1** putative TFBS* rs3792152 (D`=1,r 2 =1) rs13430962 intron 1** putative TFBS* rs13409359 (D`=1,r 2 =1) rs1011633 intron 1** putative TFBS*


Amand Faessler

Rastislav Hodák

Sergey Kovalenko

Fedor Šimkovic

Tritium and Rhenium as a Probe of Cosmic Neutrino Background

Keywords: numbers: 98.80.Es, 23.40.Bw; 23.40.Hc Keywords: relic neutrinos, neutrino capture, beta decay

come

Tritium and rhenium as a probe of cosmic neutrino background

I. INTRODUCTION

The cosmological relic neutrinos are one of the most important and abundant constituents of the Universe predicted by Big Bang cosmology. There are about 10 87 neutrinos per flavor in the visible Universe [1], which corresponds to the number density per flavor of relic (anti-)neutrinos in average over the Universe about η ∼ 56 cm -3 [START_REF] Giunti | Fundamentals of Neutrino Physics and Astrophysics[END_REF]. In number, neutrinos exceed the constituents of ordinary matter (electrons, protons, neutrons) by a factor of ten billion. On the other hand their existence has not yet been confirmed by direct observations. This represents a challenging problem of modern cosmology and experimental astroparticle physics. Many proposals in the literature aim to observation of indirect astrophysical manifestations of relic neutrino sea [START_REF] Weiler | [END_REF][4][5][6][7][8]. However laboratory experiments would be the most robust probe of this component of the Universe. There are various strategies discussed for laboratory searches of relic neutrinos [9], [10][11][12][13][14][15][16][START_REF] Cocco | [END_REF][18][19][20][21][START_REF] Hodák | AIP Conf. Proc[END_REF][START_REF] Hodák | [END_REF]. The neutrino capture reaction, [START_REF] Hodák | AIP Conf. Proc[END_REF] ν e + (A, Z) → (A, Z + 1) + e -,

is one of the most discussed possibilities in the literature [9,[START_REF] Cocco | [END_REF][18][19][20][21][START_REF] Hodák | AIP Conf. Proc[END_REF][START_REF] Hodák | [END_REF]. Since relic neutrinos have very low energy this reaction should have no threshold implying that the initial nucleus is β-radioactive. This idea was first proposed many years ago by Weinberg in Ref. [9], but in the context of massless neutrinos with large chemical potential. The latter proved to be inconsistent with the Big Bang cosmology. The discovery of neutrino mass m ν revived the interest to the above reaction, since at the endpoint of the electron spectrum the energy difference between β-decay and neutrino capture electrons should be around ∼ 2m ν . In principle, this effect can be observed with a detector having an energy resolution less than this difference. However, the existing experimental techniques are unable to detect relic neutrinos via neutrino capture if their number density in our vicinity does not significantly differ from the global average number density η ∼ 56 cm -3 [18][19][20][21][START_REF] Hodák | AIP Conf. Proc[END_REF][START_REF] Hodák | [END_REF]. As it was pointed out recently [18,24,25], the gravitational clustering of neutrinos in our galaxy or galaxy cluster may drastically enhance the relic neutrino density in comparison to the overall average η making its detection more realistic. In this paper, unlike in previous articles [9,[START_REF] Cocco | [END_REF][18][19][20][21], where the capture rate is derived on the basis of neutrinonucleus scattering cross-sections, we calculate the capture rate directly [START_REF] Hodák | AIP Conf. Proc[END_REF][START_REF] Hodák | [END_REF]. Neutrino velocity does not enter to our formulas and the information about the distribution in velocities of the relic neutrinos is not needed. For the relativistic wave function of an electron we present a detailed derivation of both β-decay and neutrino capture rates for 3 H and 187 Re nuclei. In the case of 187 Re this is a complex task as β-decay of rhenium is first unique forbidden β-decay process. An explicit form of the corresponding nuclear matrix elements for both nuclei is given. We focus on the calculation of the event rate of the reaction (1) for tritium and rhenium targets and estimate typical detection rates of the relic neutrino capture in the Karslruhe tritium neutrino mass (KATRIN) experiment [26,27] and the experiment, called the "Microcalorimeter Arrays for a Rhenium Experiment" (MARE) [28], by taking into account Confidential: not for distribution. Submitted to IOP Publishing for peer review 5 April 2011 possible effect of gravitational clustering [18]. To our knowledge, prospects of the MARE experiment for the detection of relic neutrinos has been not studied in the literature yet. We propose to use future data from the KATRIN (under construction) and the MARE (under preparation) experiments to set upper limits on the density of relic neutrinos.

In Sec. II we present our approach to nuclear β-decay with an emphasis on tritium 3 H and rhenium 187 Re radioactive isotopes. In Sec. III we apply these approach to analysis of relic neutrino capture on these isotopes and discuss KATRIN 3 H [26,27] and MARE 187 Re experiments [28] dedicated to measurement of electron neutrino mass at sub-eV scale. We examine prospects of these experiments for searching for the relic neutrinos via neutrino capture on the corresponding nuclei.

II. NUCLEAR β-DECAY. BASIC INGREDIENTS

Here we discuss theoretical aspects of single β-decay of nuclei with special emphasis on tritium and rhenium isotopes [29]. We introduce nuclear structure ingredients used in the subsequent sections for analysis of relic neutrino capture by these nuclei.

Let us consider general case of nuclear β-decay

(A, Z) → (A, Z + 1) + e -+ ν e , (2)
driven by the standard weak β-decay Hamiltonian

H β (x) = G β √ 2 ē(x)γ µ (1 -γ 5 )ν e (x)j µ (x) + h.c.. (3)
Here, G β = G F cos θ C , where cos θ C is the Cabbibo angle. e(x) and ν e (x) are the electron and neutrino fields, respectively. The strangeness conserving free nucleon charged current is

j µ (x) = p(x)γ µ (g V -g A γ 5 )n(x), (4)
where p(x) and n(x) are the proton and neutron fields, respectively. g V = 1.0 and g A = 1.25. Single β-decay occurs in the first order in the weak interaction. The corresponding S-matrix element is given by:

f |S (1) |i = 2πδ(E f + E e + E ν -E i ) f |T (1) |i = = 2πδ(E f + E e + E ν -E i )(-i) G β √ 2 × ψe (x, E e )γ µ (1 -γ 5 )ψ c ν (x, E ν ) A, Z + 1|J µ (0, x)|A, Z dx. (5)
Here, J µ is the nuclear weak charged current in the Heisenberg representation. In our notation E i , E f , E e and E ν are energies of initial and final nuclei, electron and antineutrino, respectively. The wave functions of outgoing electron and antineutrino are denoted as ψ e , ψ c ν . We use non-relativistic impulse approximation for the nuclear current:

J µ (0, x) = A n=1 τ + n [g V g µ0 + g A (σ k ) n g µk ]δ(x -x n ). (6)
¿From now on it is convenient to consider β-decay of tritium and rhenium separately.

A. β-decay of tritium

In the tritium β-decay the spin and parity of initial and final nuclei are equal:

3 H((1/2) + ) → 3 He((1/2) + ) + e -+ ν e , (7)
In this case the dominant contribution to the decay rate is determined by s 1/2 wave-states of outgoing electron and antineutrino

ψ e (x, E e ) ≈ F 0 (Z + 1, E e ) u(P e ) ( 8 ) ψ c ν (x, E ν ) ≈ u(-P ν ) ( 9
)
where x is the coordinate of the lepton. F 0 is the relativistic Fermi function (for more details see Appendix) [30]. Normalization of the spinor is u † (P )u(P ) = 1. P e ≡ (E e , p e ) and P ν ≡ (E ν , p ν ) are 4-momenta of electron and antineutrino, respectively. Substituting Eqs. ( 8), ( 9) and ( 6) into Eq. ( 5) we get for T-matrix element an expression

f |T (1) |i = (-i) G β √ 2 F 0 (Z + 1, E e ) × ū(P e )γ µ (1 -γ 5 )u(-P ν ) g µ0 M F + g A g µk (M GT ) k , (10)
where Fermi and Gamow-Teller nuclear matrix elements are defined as

M F (m, m ) = 3 He (1/2) + m | n τ + n |(1/2) + m 3 H , M GT (m, m ) = 3 He 1/2 + m | n τ + n σ n |(1/2) + m 3 H . (11)
Here, m, m = ±1/2. Now we sum up over the lepton polarizations, over the projection of angular momenta, m , of final nucleus and average over projection of angular momenta, m, of initial nucleus. Evaluating the corresponding traces for the squared T-matrix we have

| f |T (1) |i | 2 = 2 G β √ 2 2 F 0 (Z + 1, E e ) B F ( 3 H) + B GT ( 3 H) (12)
with

B F ( 3 H) = |M F | 2 = 1 2 m,m |M F (m, m )| 2 , B GT ( 3 H) = g 2 A |M GT | 2 = g 2 A 1 2 m,m M GT (m, m ) • M * GT (m, m ). (13)
We neglected terms proportional to the space components of the lepton momenta as they vanish after integration over angles in calculation of the total β-decay rate.

The Fermi and Gamow-Teller beta strengths in Eq. ( 13) can be represented in terms of reduced nuclear matrix elements:

B F ( 3 H) = 1 2 3 He (1/2) + n τ + n (1/2) + 3 H 2 , (14)
B GT ( 3 H) = g 2 A 1 2 3 He (1/2) + n τ + n σ n (1/2) + 3 H 2 . ( 15
)
With our normalization of the lepton wave functions (8), ( 9) the differential decay rate is

dΓ β = | f |T (1) |i | 2 2πδ(E ν + E e + E f -E i ) dp e (2π) 3 dp ν (2π) 3 . ( 16
)
Inserting Eq. ( 12) for squared T-matrix element and integrating over the phase space we find the total β-decay rate for the reaction (7)

Γ β ( 3 H) = 1 2π 3 m e G β m 2 e 2 B F ( 3 H) + B GT ( 3 H) I β ( 3 H) ( 1 7 )
with the phase space integral

I β ( 3 H) = 1 m 5 e Ei-E f me F 0 (Z + 1, E e )p e E e (E i -E f -E e ) 2 dE e . (18)
The dependence of I β ( 3 H) on small neutrino mass is ignored. For tritium numerical integration in (18) gives

I β ( 3 H) = 2.88 × 10 -6 . (19)
The nuclear matrix elements for tritium are known in the literature. We take for these matrix elements the values [31]. Then we find

|M F | 2 = 1, |M GT | 2 = 3 derived in Ref.
T β 1/2 ( 3 H) = ln 2 Γ β ( 3 H) = 12.32 y, (20)
which is very close to the measured value of tritium β-decay half-life. Vice versa, using Eqs. ( 17)- (19), one can extract a value of

|M F | 2 + g 2 A |M GT |
2 in a model independent way from the experimental value T β 1/2 ( 3 H) ≈ 12.33 y. In this way we obtain

|M F | 2 + g 2 A |M GT | 2 ≈ 5.645. ( 21
)
This model independent estimate we will use in discussion of relic neutrino capture by tritium.

B. β-decay of rhenium

Single β-decay of rhenium

187 Re((5/2) + ) → 187 Os((1/2) -) + e -+ ν e . ( 22
)
is more complicated than the tritium β-decay mainly because of the fact that the initial and final nuclear spins and parities are different. In [START_REF] Dvornický | AIP Conf.Proc[END_REF] it was shown that the dominant contribution to this decay is given by the emission of electron and antineutrino in p 3/2 and s 1/2 wave states, respectively. The p 3/2 wave function of electron has the form

Ψ p 3/2 (x, E e ) i F 1 (Z + 1, E e ) x • p e + 1 3 γ • x γ • p e u(P e ). ( 23
)
F 1 is the relativistic Fermi function (for more details see Appendix) [30]. Now, we are to analyze the squared matrix element in Eq. ( 5) with the electron and neutrino wave functions given in Eqs. ( 9), [START_REF] Hodák | [END_REF]. Towards this end we consider the quantity

M = ψe (x, E e )γ µ (1 -γ 5 )ψ c ν (x, E ν ) × (1/2) -m | n τ + n (g V g µ0 + g A g µk (σ n ) k )|(5/2) + m δ(x -x n ), (24)
which is the integrand of the space integral in Eq. ( 5). Carrying out summation over polarizations of leptons we keep only the terms surviving after integration over phase space. Then, for the squared T-matrix element we obtain [29] | f |T (1) |i

| 2 = G 2 β g 2 A M (2 -) GT 2 (p e R) 2 3 F 1 (Z + 1, E e ). ( 25
)
where the squared nuclear matrix element includes summation over all the spin orientations of the final nucleus m and averaging over the spin orientations of the initial nucleus m. The β-strength for 187 Re takes the form

B GT ( 187 Re) = g 2 A M (2 -) GT 2 = 1 6 R 2 (1/2) - n τ + n {σ n ⊗ r n } 2 (5/2) + 2 . ( 26
)
Inserting the above expression to Eq. ( 16) and integrating over the phase space of final electron and neutrino we find the total β-decay rate of rhenium

Γ β ( 187 Re) = m e (G β m 2 e ) 2 2π 3 (Rm e ) 2 3 B GT ( 187 Re) I β ( 187 Re), (27)
where the phase space integral is

I β ( 187 Re) = 1 m 7 e Ei-E f me p e E e (E i -E f -E e ) 2 F 1 (Z + 1, E e )p 2 e dE e . (28)
where E e = E i + E ν -E f . With our neutrino wave function normalization both expressions correspond to capture rate per neutrino per 3 H atom. For the number density of cosmic neutrinos η ν we replace 1/V → η ν in (36) and get the capture rate per atom

Γ ν ( 3 H) = 1 π G 2 β F 0 (Z + 1, E e ) p e E e B F ( 3 H) + B GT ( 3 H) η ν η ν η ν . ( 37
)
Here η ν is the local cosmic neutrino number density which could be significantly larger than the average over the universe η ν ∼ 56 cm -3 due to gravitational clustering [18,24] η ν η ν ∼ 10 3 -10 4 (38)

assuming m ν = 1eV and baryon density η b = 10 -3 -10 -4 cm -3 in a cluster of galaxies.

A combination of the nuclear matrix elements in Eq. ( 37), shown in curl brackets, we previously encountered in β-decay rate and found its numerical value from the half-life T 1/2 of 3 H. The corresponding value is given in Eq. ( 21) which we substitute in (37) and obtain

Γ ν ( 3 H) = 4.2 × 10 -25 η ν η ν y -1 . ( 39
)
This value is compatible with the result of Ref. [START_REF] Cocco | [END_REF]18] for the case η ν = η ν .

The KATRIN experiment, dedicated to measurement of electron neutrino mass from endpoint electron energy spectrum of β-decay of tritium, can, in principle, also search for cosmic neutrinos via neutrino capture reaction (34). The KATRIN experiment (in construction phase) aims to measure m ν with sensitivity of 0.2 eV [27] using about 50 µg of tritium corresponding to 5 × 10 18 T 2 molecules [25]. For this amount of target nuclei we find from Eq. (39) the number of neutrino capture events

N ν capt (KAT RIN ) ≈ 4.2 × 10 -6 η ν η ν y -1 . (40)
Considering this estimate we conclude that observation of relic neutrino capture in KATRIN experiment looks rather unrealistic. Further comments on this subject will be given in sec. IV.

B. Neutrino capture on rhenium

The cosmic neutrino capture by rhenium

ν e + 187 Re((5/2) + ) → 187 Os((1/2) -) + e - (41)
we analyze in the way similar to the previous case of tritium. The T-matrix determining this process coincide with the rhenium β-decay matrix element shown in Eq. (25). After neglecting the part of the electron wave function associated with emission of the s 1/2 -electron, which is very small, the squared and summed over polarizations T-matrix takes the form

| f |T (1) |i | 2 = G 2 β B GT ( 187 Re) (p e R) 2 3 F 1 (Z + 1, E e ). (42)
Inserting this expression in Eq. ( 33), for the total capture rate per cosmic neutrino we obtain

Γ ν ( 187 Re) = 1 V 1 π G 2 β B GT ( 187 Re) F 1 (Z + 1, E e ) (p e R) 2 3 p e E e (43)
with

E e = E i + E ν -E f .
As in the case of tritium we replace 1/V with the neutrino number density η ν and get the capture rate per atom of rhenium

Γ ν ( 187 Re) = 1 π G 2 β F 1 (Z + 1, E e ) (p e R) 2 3 B GT ( 187 Re) p e E e η ν η ν η ν . (44)
Substituting to this equation numerical values of nuclear matrix element from (31) and other constants we obtain for the capture rate per atom of 187 Re

Γ ν ( 187 Re) = 2.75 × 10 -32 η ν η ν y -1 . (45)
For a detector with 100 g of 187 Re, i.e. with 3.2 × 10 23 rhenium atoms, we find the number of cosmic neutrino capture events

N ν capt 8.9 × 10 -9 η ν η ν y -1 . (46)
The MARE project will investigate the β-decay of 187 Re with absorbers of metallic rhenium or AgReO 4 . It foresees a 760 grams bolometer. For this amount of rhenium the number of neutrino capture events is

N ν capt (MARE) 6.7 × 10 -8 η ν η ν y -1 . (47)
The MARE detector technology can, in principle, be scaled up. With about 4 orders more rhenium the capture rate would become about two orders of magnitude larger than in case of the KATRIN experiment. Note that the KATRIN experiment can hardly be significantly scaled up due to safety limitations on the amount of highly radioactive tritium.

IV. DISCUSSIONS AND CONCLUSIONS

As was shown in the previous sections both with 3 H and 187 Re target nuclei the relic neutrino capture rate Eqs. ( 40)-(47) event numbers are extremely small and unobservable in the present and near future experiments. The gravitational clustering of relic neutrinos at the level of η ν / η ν 10 3 -10 4 [18,24] can hardly change this conclusion. On the other hand this effect could be significantly stronger leading to a clustering about η ν / η ν 10 13 discussed in Ref. [33]. Then the capture rate becomes large reaching to about N ν capt ∼ 10 7 y -1 which looks very promising. However, a key point is distinguishing the relic neutrino capture signal from the β-decay background. In this respect the main quantity to be analyzed is the ratio λ ν /λ β of the the partial rates of β-decayλ β and relic neutrino capture λ ν with the electrons in the energy interval around the endpoint Q β of the β-decay electron energy spectrum. Despite the β-decay spectrum ends at Q β -m ν and the electrons from the neutrino capture locates at Q β + m ν in a realistic experiment they overlap due to finite energy resolution ∆ of a detector. As it was shown in Refs. [START_REF] Cocco | [END_REF]18] the neutrino capture events can be discriminated from the β-decay background if m ν /∆ ∼ 2 or smaller. This condition is very challenging for experiments and is nearly independent of target nucleus and very weakly depend on the clustering η ν / η ν . Only a strong clustering η ν / η ν ∼ 10 13 , considered in Ref. [33], may relax this condition and simultaneously significantly increase the total event rate of the relic neutrino capture. We note also that the considered searches for relic neutrinos via neutrino capture on β-decaying nuclei even been unable to detect this process may set useful limits on the actual relic neutrino density η ν / η ν in the Earth vicinity. These possibility requires additional study. However it should be pointed out that there is a big uncertainty in this sort of studies due to unknown absolute mass scale of neutrino. If the forthcoming β-decay experiments [26,27], [28] set the value of m νe within their sensitivities then the energy resolution necessary for successful searches of relic neutrinos will be within reasonable values like ∆ ∼ 0.1eV [START_REF] Cocco | [END_REF], which could be achieved in future experiments.

In conclusion, we carried out a detailed analysis of nuclear physics and kinematical aspects of single nuclear βdecay and relic neutrino capture on β-radioactive nuclei. We focussed on 3 H and 187 Re isotopes to be used in the forthcoming KATRIN and in the planned MARE experiments, respectively.

As to the prospects of relic neutrino direct detection through neutrino capture on β-decaying nuclei we concluded that they are rather pessimistic with the present experimental methods and techniques. Even taking into account the gravitational clustering, which greatly enhances the relic neutrino number density in galaxy clusters, does not improve this situation: the detection rate remains very small and hardly observable in the near future. However we noted that scaling MARE experiment up to several hundreds of kilograms of rhenium would offer an event rate significantly larger than in KATRIN experiment. This possibility, however, remains technically very questionable.

Despite our rather pessimistic conclusions we point out that searches for cosmic neutrino capture via beta decaying nuclei deserves to be carried out. After all this is the only known direct way to probe this component of the universe and from non-observation of the capture process to extract limits on the local neutrino number density η ν , which could be complementary to the known cosmological and astrophysical limits. On the other hand one can never exclude that we are passing through a neutrino clump with a density much higher than predicted by the gravitational clustering [18,24,33]. This would eventually make observation of cosmic neutrino capture realistic in even the forthcoming experiments.

Acknowledgments

Authors are grateful to V. Egorov, S. Bilenky and Yu. Kamyshkov for stimulating discussions. F. Š. and R.H. acknowledge support of the Deutsche Forschungsgemeinschaft within the project 436 SLK 17/298 and of the VEGA Grant agency of the Slovak Republic under the contract No. 1/0639/09. S.K. acknowledges the support of FONDECYT grant 1100582, and Centro Científico-Tecnológico de Valparaíso PBCT ACT-028.

We neglected the effect of small neutrino mass. For rhenium numerical integration in (28) gives

With this value of the phase space integral and using the experimental half-life T β 1/2 ( 187 Re) = 4.35 × 10 10 years (30) we obtain the value of rhenium nuclear matrix element

We will use this value in our analysis of neutrino capture on rhenium where the same quantity will appear.

III. NEUTRINO CAPTURE WITH BETA DECAYING NUCLEI

Now let us consider processes, similar to the previously considered β-decays, but with neutrino in the initial state of reaction:

This neutrino can be of any origin. An interesting possibility is a neutrino from the cosmic neutrino background. Thus the above reaction may represent one of the very few possible ways to detect this important component of the Universe. Since the energy of cosmic neutrinos is negligibly small, the above neutrino capture reaction takes place only if there is no threshold. This implies that the target nucleus should be β-radioactive. In the previous sections we considered in details two candidates of this type, tritium and rhenium. Below we study capture of cosmic neutrinos with these radioactive nuclei.

The differential rate of the reaction ( 32) is given by

for the normalization of incoming neutrino plane wave to be one particle per volume V . The kinetic energy of electron produced in β-decay is continuously distributed within the interval 0 ≤ E e -m e ≤ Q β -m ν while in the reaction [START_REF] Dvornický | AIP Conf.Proc[END_REF] the final electron is monoenergetic with E e -m e = Q β + E rel ν , where E rel ν is the relic neutrino energy. Thus the signatures of β-decay and cosmic neutrino capture are quite different. However the energy difference between β-decay and neutrino capture emitted electrons is very small ∼ 2m ν which is challenging for energy resolution of detectors used in experiments searching for this process. ¿From the viewpoint of matrix element calculations neutrino capture reaction [START_REF] Dvornický | AIP Conf.Proc[END_REF] with very low energy neutrinos, such as the cosmic neutrinos, is very similar to β-decay of the target nucleus.

Below we consider two target nuclei tritium and rhenium which are planned to be used in the experimental setups KATRIN with 3 H and MARE with 187 Re aimed to measure neutrino mass in sub-eV range.

A. Neutrino capture on tritium

The cosmic neutrino capture reaction by tritium,

is characterized by the T-matrix element of the same form as in the case of single β-decay. Within the same approximations and summing over the polarizations we have

Inserting (35) to (33), for the total capture rate we find

APPENDIX: PARTIAL ELECTRON WAVE FUNCTIONS

Here, for convenience, we show several terms of the partial wave decomposition of the relativistic electron wave function in the Coulomb field of a uniform charge distribution of a nucleus:

In our notation we follow Ref. [30]. Keeping the leading terms in r we have for the partial wave functions: s 1/2 wave: The relativistic Fermi function F 0 (Z, E e ) (F 1 (Z, E e )) takes into account the Coulomb interaction of emitted s 1/2 and p 1/2 (p 3/2 ) electrons with the nucleus and is given, for instance, in Ref. [30]. The Dirac spinors are normalized as u † (P )u(P ) = 1.