text: 8 Di Cintio 8.5 0.40 ---- Schramm & Silverman 13 SIDM SIDM 0.35 8.0 CDM CDM 0.30 7.5 0.25 0.20 7.0 0.15 0.10 6.5 0.05 6.0 0.00 7 8 9 10 11 12 38 39 40 41 42 43 44 log(M*[M ]) log(L [erg/s]) ﬂ BH Figure 5. Global properties of the SMBHs population in SIDM and CDM galaxies. Only the most massive SMBH in each galaxy is shown. Left-hand panel: SMBH mass-host stellar mass relation, SMBHs in SIDM galaxies are indicated as circles and corresponding CDM ones as squares. Same colours refer to the same galaxy in the two runs. Right-hand panel: Normalized histograms and PDFs of SMBH luminosities, SIDM results are shown in blue, CDM ones in red. At a ﬁxed stellar mass, several SMBHs in the CDM run tend to be ies, while in the case of SIDM we found as many as 13 SMBHs more massive than the SIDM ones, reﬂecting the fact that SMBHs above such a distance. In the SIDM case SMBHs are found as far ∼ in a dense CDM halo can grow more than their SIDM counter- as 9 kpc from the host center. More speciﬁcally, 88% of all the part. The two most massive galaxies instead host SMBHs with a SMBHs identiﬁed in the CDM simulation have sunk to the center of similarly high mass of M >107.5M(cid:12). The overall distribu- their host galaxy and lie within 350 pc (corresponding to the reso- SMBH tion of SMBH luminosities in the right-hand panel of Fig. 5 shows lution limit of our simulation, (cid:15)) from it by z=0.5, while only about that the peak of luminosity of CDM-SMBHs happens at higher lu- 20% of the SMBHs in the SIDM run lie within the same distance minosities than the corresponding SIDM case: from the PDFs we from their host galaxy. All the SMBHs that lie within (cid:15) from the estimated a peak at L =1041.7erg/s in the CDM case, and center of SIDM galaxies are found in galaxies more massive than SMBH at L =1040.5erg/s in the SIDM case. While, of course, cau- M(cid:63)=1010M(cid:12), indicating that the mechanism responsible for keep- SMBH tion should be exercised due to the low number statistic, the reason ing SMBHs away from the centre is more dramatic in lower mass for this has to be searched in the SMBHs evolution and accretion: galaxies. Recall that our SMBH model is able to effectively take SMBHs forming within galaxies with a shallow central proﬁle, like into account the dynamical friction forces acting on the SMBHs, the ones of SIDM haloes, will accrete less mass and therefore have which are therefore not forced at the center of the galaxy as in pre- a lower luminosity than the corresponding CDM case, in which vious prescriptions (Di Matteo et al. 2005; Sijacki et al. 2007) but SMBHs accrete in a denser environment with higher average gas they are rather allowed to orbit and sink at the center of their hosts supply that can reach into the SMBH. over realistic timescales. 3.4 Black holes dynamics We can therefore understand the trends shown in Fig. 6 in Most interestingly, we found that the dynamics of SMBHs is terms of dynamical friction timescales (Chandrasekhar 1943; Bin- different in the two cosmologies and that, in the SIDM case, it has ney & Tremaine 2008; Read et al. 2006). Because the timescale is been affected by the modiﬁcation to the halo structure due to self- inversely proportional to the density of the surrounding medium, interactions between DM particles. In Fig. 6 we plot the distance of SMBHs sinking within lower density SIDM haloes have not yet SMBHs from host centre vs host stellar mass, with SIDM results reached the center of their host, and are ‘stalling’ within the core indicated as circles (left-hand panel) and CDM ones as squares of SIDM galaxies, in agreement with the analytic model presented (right-hand panel). As in the previous section, only the most mas- in Read et al. (2006) and Petts et al. (2016). The ‘stalling’ is due sive SMBH within each galaxy is shown, and same colours refer to to the failing of the Chandrasekar formula in the presence of a cen- the same galaxy in the two cosmologies. tral density core, and reﬂects the situation in which SMBHs effec- It appears that several SMBHs in SIDM galaxies have not yet tively do not feel any dynamical friction. In comparison, a much ∼ reached the galaxy centre, being found as far as 9 kpc from it, more centrally concentrated CDM halo will have a short enough while the vast majority of SMBHs in CDM lie within 1 kpc from dynamical friction timescale for the SMBHs to have sunk all the the centre of the host galaxy. We observe a striking difference in way to its center. Moreover, as already investigated in Section , the behaviour of SMBHs in SIDM and CDM, already anticipated the increased efﬁciency of gas outﬂows in the presence of an un- in Section : SMBHs in the SIDM run are on average further away derlying cored distribution further contributes to keep the SMBH from the galaxy center than their CDM counterparts. Only 4 outlier away from the galactic centre during vigorous SF episodes. Both SMBHs lie at or above 1 kpc from the host’s center in CDM galax- effects are clearly shown in Fig. 4, by comparing a SIDM and CDM MNRAS 000, 1–10 (2010) )] M[ M(gol ﬂ HB noitcarf ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: MNRAS 000, 1–7 (2016) Preprint 29 May 2020 Compiled using MNRAS LATEX style ﬁle v3.0 Using photo-ionisation models to derive carbon and oxygen gas-phase abundances in the rest UV Enrique Pérez-Montero,1 Ricardo Amorín,2 3 4 (cid:63) , , 1Instituto de Astrofísica de Andalucía. CSIC. Apartado de correos 3004. 18080, Granada, Spain. 2INAF-Osservatorio Astronomico di Roma, via di Frascati 33, I-00078, Monte Porzio Catone, Italy. 3Cavendish Laboratory, University of Cambridge, 19 JJ Thomson Avenue, Cambridge, CB3 0HE, UK. 4Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK Accepted XXX. Received YYY; in original form ZZZ ABSTRACT We present a new method to derive oxygen and carbon abundances using the ultraviolet (UV) lines emitted by the gas-phase ionised by massive stars. The method is based on the comparison of the nebular emission-line ratios with those predicted by a large grid of photo- ionisation models. Given the large dispersion in the O/H - C/O plane, our method ﬁrstly ﬁxes C/O using ratios of appropriate emission lines and, in a second step, calculates O/H and the ionisation parameter from carbon lines in the UV. We ﬁnd abundances totally consistent with those provided by the direct method when we apply this method to a sample of objects with an empirical determination of the electron temperature using optical emission lines. The proposed methodology appears as a powerful tool for systematic studies of nebular abundances in star-forming galaxies at high redshift. Key words: methods: data analysis – ISM: abundances – galaxies: abundances 1 INTRODUCTION lent width emission of Heii1640 Å and Ly . While these spectral α features could be quite common at very high redshifts (Stark et Establishing the gas-phase abundances of carbon, nitrogen and al. 2017), they provide tight constraints to the ionisation, age, and oxygen in galaxies through cosmic time is key to understand not metallicity properties of galaxies, as recently shown by emission- only their chemical enrichment, but also how galaxies assemble line diagnostics based on detailed photo-ionisation models (e.g. and evolve (Edmunds & Pagel 1978; Garnett et al. 1995; Henry Feltre, Charlot, & Gutkin 2016; Gutkin, Charlot, & Bruzual 2016; et al. 2000; Chiappini, Romano, & Matteucci 2003). Jaskot & Ravindranath 2016). In the last decade or so, large deep optical spectroscopic sur- veys using ground-based 8-10m-class telescopes (e.g. Steidel et al. The wealth of information contained in the UV spectra of 2003; Shapley et al. 2003; Lilly et al. 2007; Kurk et al. 2013; Le strong emission -line galaxies allow us to explore new techniques Fèvre et al. 2015) led to a signiﬁcant increase in the amount and for an accurate characterisation of the nebular chemical content of depth of rest-frame UV spectra of high redshift galaxies (z (cid:38)1-2). star forming galaxies based on the analysis of the carbon abun- These and other follow-up spectroscopy of galaxies in deep ﬁelds dance, the processes governing the C/O ratio, and its relation with (e.g. Erb et al. 2010; Karman et al. 2015; Contini et al. 2016), the total metallicity (e.g. Henry et al. 2000; Erb et al. 2010; Berg in particular gravitationally lensed systems (e.g. Christensen et al. et al. 2016). The use of photo-ionisation models assuming diﬀerent 2012; James et al. 2014; Stark et al. 2014; Vanzella et al. 2016a), combinations of C/O and O/H in an emitting ionised gas distribu- are pushing now UV studies towards the low-luminosity (mass) tion is hence an alternative to estimate the gas-phase metallicity in regime out to redshift z ∼3.5 (e.g. Patrício et al. 2016; Vanzella et those cases where very few UV emission lines can be measured. al. 2016b, 2017; Amorín et al. 2017) and beyond (Stark et al. 2017, These methodologies, entirely or partially based on the available and references therein). UV emission line ratios, are necessary to complement the widely Overall, the above studies consistently ﬁnd that the UV spec- used recipes to derive physical properties and chemical abundances trum of high redshift galaxies is systematically bluer and harder than from rest-frame optical spectra, which at ∼2-3 rely on the (usually z in most local galaxies. Thus, it includes a number of high ionisation, challenging) detection of faint emission lines in the near infra-red metal emission lines with relative large equivalent widths, such as (NIR, e.g. Steidel et al. 2014; Maseda et al. 2014; Troncoso et al. those of carbon (e.g. Ciii] 1908 Å, Civ1549 Å) and oxygen (Oiii] 2014; Amorín et al. 2014; Shapley et al. 2015; Onodera et al. 2016; 1664 Å), which are often related to the presence of high equiva- Sanders et al. 2016; Trainor et al. 2016). They would also be of great interest for metallicity studies of galaxies at the highest redshifts, where observing the optical rest-frame is extremely challenging. In E-mail: epm@iaa.es (EPM) particular, the future arrival of unprecedentedly deep observations (cid:63) © 2016 The Authors

text: MNRAS

label: paragraph

font: LMRTJF+TeXGyreTermes-Regular

font: NUBEUB+TeXGyreTermes-Bold

text: Preprint

text: May

text: Compiled

text: using

text: MNRAS

text: LATEX

text: style

text: ﬁle

text: Using

label: title

text: photo-ionisation

text: models

text: to

text: derive

text: carbon

text: and

text: oxygen

text: gas-phase

text: abundances

text: in

text: the

text: rest

text: UV

text: Enrique

label: author

text: Ricardo

font: CLQAFG+rtxmi

font: KXUBYU+TeXGyreTermes-Italic

text: de

text: Astrofísica

text: de

text: Andalucía.

text: CSIC.

text: Apartado

text: de

text: correos

text: Granada,

text: Spain.

text: Astronomico

text: di

text: Roma,

text: via

text: di

text: Frascati

text: Monte

text: Porzio

text: Catone,

text: Italy.

text: Laboratory,

text: University

text: of

text: Cambridge,

text: JJ

text: Thomson

text: Avenue,

text: Cambridge,

text: UK.

text: Institute

text: for

text: Cosmology,

text: University

text: of

text: Cambridge,

text: Madingley

text: Road,

text: Cambridge

text: UK

text: Accepted

text: XXX.

text: Received

text: YYY;

text: in

text: original

text: form

text: ZZZ

text: ABSTRACT

text: We

text: present

text: a

text: new

text: method

text: to

text: derive

text: oxygen

text: and

text: carbon

text: abundances

text: using

text: the

text: ultraviolet

text: (UV)

text: lines

text: emitted

text: by

text: the

text: gas-phase

text: ionised

text: by

text: massive

text: stars.

text: The

text: method

text: is

text: based

text: on

text: the

text: comparison

text: of

text: the

text: nebular

text: emission-line

text: ratios

text: with

text: those

text: predicted

text: by

text: a

text: large

text: grid

text: of

text: photo-

text: ionisation

text: models.

text: Given

text: the

text: large

text: dispersion

text: in

text: the

text: O/H

text: C/O

text: plane,

text: our

text: method

text: ﬁrstly

text: ﬁxes

text: C/O

text: using

text: ratios

text: of

text: appropriate

text: emission

text: lines

text: and,

text: in

text: a

text: second

text: step,

text: calculates

text: O/H

text: and

text: the

text: ionisation

text: parameter

text: from

text: carbon

text: lines

text: in

text: the

text: UV.

text: We

text: ﬁnd

text: abundances

text: totally

text: consistent

text: with

text: those

text: provided

text: by

text: the

text: direct

text: method

text: when

text: we

text: apply

text: this

text: method

text: to

text: a

text: sample

text: of

text: objects

text: with

text: an

text: empirical

text: determination

text: of

text: the

text: electron

text: temperature

text: using

text: optical

text: emission

text: lines.

text: The

text: proposed

text: methodology

text: appears

text: as

text: a

text: powerful

text: tool

text: for

text: systematic

text: studies

text: of

text: nebular

text: abundances

text: in

text: star-forming

text: galaxies

text: at

text: high

text: redshift.

text: Key

text: words:

text: methods:

text: data

text: analysis

text: ISM:

text: abundances

text: galaxies:

text: abundances

text: INTRODUCTION

text: lent

text: width

text: emission

text: of

text: Å

text: and

text: Ly

text: While

text: these

text: spectral

text: α

text: features

text: could

text: be

text: quite

text: common

text: at

text: very

text: high

text: redshifts

text: (Stark

text: et

text: Establishing

text: the

text: gas-phase

text: abundances

text: of

text: carbon,

text: nitrogen

text: and

text: al.

text: they

text: provide

text: tight

text: constraints

text: to

text: the

text: ionisation,

text: age,

text: and

text: oxygen

text: in

text: galaxies

text: through

text: cosmic

text: time

text: is

text: key

text: to

text: understand

text: not

text: metallicity

text: properties

text: of

text: galaxies,

text: as

text: recently

text: shown

text: by

text: emission-

text: only

text: their

text: chemical

text: enrichment,

text: but

text: also

text: how

text: galaxies

text: assemble

text: line

text: diagnostics

text: based

text: on

text: detailed

text: photo-ionisation

text: models

text: (e.g.

text: and

text: evolve

text: (Edmunds

text: Pagel

text: Garnett

text: et

text: al.

text: Henry

text: Feltre,

text: Charlot,

text: Gutkin

text: Gutkin,

text: Charlot,

text: Bruzual

text: et

text: al.

text: Chiappini,

text: Romano,

text: Matteucci

text: Jaskot

text: Ravindranath

text: In

text: the

text: last

text: decade

text: or

text: so,

text: large

text: deep

text: optical

text: spectroscopic

text: sur-

text: veys

text: using

text: ground-based

text: telescopes

text: (e.g.

text: Steidel

text: et

text: al.

text: The

text: wealth

text: of

text: information

text: contained

text: in

text: the

text: UV

text: spectra

text: of

text: Shapley

text: et

text: al.

text: Lilly

text: et

text: al.

text: Kurk

text: et

text: al.

text: Le

text: strong

text: emission

text: -line

text: galaxies

text: allow

text: us

text: to

text: explore

text: new

text: techniques

text: Fèvre

text: et

text: al.

text: led

text: to

text: a

text: signiﬁcant

text: increase

text: in

text: the

text: amount

text: and

text: for

text: an

text: accurate

text: characterisation

text: of

text: the

text: nebular

text: chemical

text: content

text: of

text: depth

text: of

text: rest-frame

text: UV

text: spectra

text: of

text: high

text: redshift

text: galaxies

text: (z

text: star

text: forming

text: galaxies

text: based

text: on

text: the

text: analysis

text: of

text: the

text: carbon

text: abun-

text: These

text: and

text: other

text: follow-up

text: spectroscopy

text: of

text: galaxies

text: in

text: deep

text: ﬁelds

text: dance,

text: the

text: processes

text: governing

text: the

text: C/O

text: ratio,

text: and

text: its

text: relation

text: with

text: (e.g.

text: Erb

text: et

text: al.

text: Karman

text: et

text: al.

text: Contini

text: et

text: al.

text: the

text: total

text: metallicity

text: (e.g.

text: Henry

text: et

text: al.

text: Erb

text: et

text: al.

text: Berg

text: in

text: particular

text: gravitationally

text: lensed

text: systems

text: (e.g.

text: Christensen

text: et

text: al.

text: et

text: al.

text: The

text: use

text: of

text: photo-ionisation

text: models

text: assuming

text: diﬀerent

text: James

text: et

text: al.

text: Stark

text: et

text: al.

text: Vanzella

text: et

text: al.

text: combinations

text: of

text: C/O

text: and

text: O/H

text: in

text: an

text: emitting

text: ionised

text: gas

text: distribu-

text: are

text: pushing

text: now

text: UV

text: studies

text: towards

text: the

text: low-luminosity

text: (mass)

text: tion

text: is

text: hence

text: an

text: alternative

text: to

text: estimate

text: the

text: gas-phase

text: metallicity

text: in

text: regime

text: out

text: to

text: redshift

text: z

text: (e.g.

text: Patrício

text: et

text: al.

text: Vanzella

text: et

text: those

text: cases

text: where

text: very

text: few

text: UV

text: emission

text: lines

text: can

text: be

text: measured.

text: al.

text: Amorín

text: et

text: al.

text: and

text: beyond

text: (Stark

text: et

text: al.

text: These

text: methodologies,

text: entirely

text: or

text: partially

text: based

text: on

text: the

text: available

text: and

text: references

text: therein).

text: UV

text: emission

text: line

text: ratios,

text: are

text: necessary

text: to

text: complement

text: the

text: widely

text: Overall,

text: the

text: above

text: studies

text: consistently

text: ﬁnd

text: that

text: the

text: UV

text: spec-

text: used

text: recipes

text: to

text: derive

text: physical

text: properties

text: and

text: chemical

text: abundances

text: trum

text: of

text: high

text: redshift

text: galaxies

text: is

text: systematically

text: bluer

text: and

text: harder

text: than

text: from

text: rest-frame

text: optical

text: spectra,

text: which

text: at

text: rely

text: on

text: the

text: (usually

text: z

text: in

text: most

text: local

text: galaxies.

text: Thus,

text: it

text: includes

text: a

text: number

text: of

text: high

text: ionisation,

text: challenging)

text: detection

text: of

text: faint

text: emission

text: lines

text: in

text: the

text: near

text: infra-red

text: metal

text: emission

text: lines

text: with

text: relative

text: large

text: equivalent

text: widths,

text: such

text: as

text: (NIR,

text: e.g.

text: Steidel

text: et

text: al.

text: Maseda

text: et

text: al.

text: Troncoso

text: et

text: al.

text: those

text: of

text: carbon

text: (e.g.

text: Ciii]

text: Å,

text: Å)

text: and

text: oxygen

text: (Oiii]

text: Amorín

text: et

text: al.

text: Shapley

text: et

text: al.

text: Onodera

text: et

text: al.

text: Å),

text: which

text: are

text: often

text: related

text: to

text: the

text: presence

text: of

text: high

text: equiva-

text: Sanders

text: et

text: al.

text: Trainor

text: et

text: al.

text: They

text: would

text: also

text: be

text: of

text: great

text: interest

text: for

text: metallicity

text: studies

text: of

text: galaxies

text: at

text: the

text: highest

text: redshifts,

text: where

text: observing

text: the

text: optical

text: rest-frame

text: is

text: extremely

text: challenging.

text: In

text: E-mail:

text: epm@iaa.es

text: (EPM)

text: particular,

text: the

text: future

text: arrival

text: of

text: unprecedentedly

text: deep

text: observations

text: The

text: Authors


text: 2 Enrique Pérez-Montero & Ricardo Amorín using the James Webb Space Telescope (JWST) and the new gener- Table 1. References and number of Hii regions, local starbursts (LSB), and ation of 30m-class telescopes, will increase by a huge factor current Lyman-break galaxies (LBG) used in this work. galaxy samples at (cid:38)3, extending detailed UV spectroscopic stud- z ies towards the cosmic dawn. Reference Number of objects Type For this aim, in this paper we present an adapted version of de Barros et al. (2016) 1 LBG Bayliss et al. (2014) 1 LBG the code Hii-Chi-mistry (Pérez-Montero 2014, hereinafter HCm), Berg et al. (2016) 7 LSB originally designed to derive O/H and N/O from optical emission Christensen et al. (2012) 3 LBG lines, that deals with UV emission lines to derive O/H and C/O Erb et al. (2010) 1 LBG ratios. This new semi-empirical method is fully consistent with Garnett et al. (1995) 6 LSB direct estimations based on the electron temperature of star form- Garnett et al. (1997) 2 LSB ing regions and provide a potentially powerful tool to constrain the Garnett et al. (1999) 6 Hii metallicity of galaxies out to very high redshifts. Even considering Izotov et al. (1999) 3 LSB the inherent limitations aﬀecting all methods for estimating chem- James et al. (2014) 1 LBG ical abundances in galaxies from their integrated spectra, such as, Kurt et al. (1995) 2 Hii e.g. the fact that these are based on spatially unresolved data (e.g. Kobulnicky et al. (1997) 3 LSB Kobulnicky & Skillman (1998) 3 LSB Iglesias-Páramo et al. 2016) and rely on the simplistic assumption of Steidel et al. (2016) 1 (stack) LBG a single Hii region for the analysis of the observed emission lines, Thuan et al. (1999) 1 LSB it is still possible to ﬁnd characteristic abundance ratios (i.e. O/H, Vanzella et al. (2016a) 1 LBG N/O, Pérez-Montero et al. 2016) that correlate with the integrated Villar-Martín et al. (2004) 1 LBG properties of galaxies, such as stellar mass, thus providing useful constraints for their assembly histories and chemical evolution. Our paper is organized as follows; In Section 2 we describe the cases where the ﬂux of the [Oii] 3727 Å line were not available to sample of objects with available emission lines both in the optical calculate the O+ abundances, we resorted to the empirical relation and in the UV whose O and C abundances could be derived using between O/H and ([Oiii]) given by Amorín et al. (2015), which is te the direct method. In Section 3 we describe the models and the valid for a wide range of metallicity and ionisation values. method used to derive the abundances and we compare our results Using the pyneb code, we also deﬁned expressions for the tem- with the direct abundances. Finally, in Section 4 we present our perature and carbon abundances using UV emission lines. Comput- results and in Section 5 we summaries our conclusions. ing the emission line ratio: Throughout this paper the following convention is adopted: O/H is 12+log(O/H), C/O is log(C/O), Oiii] 1664Å represents the λ I(5007 Å) total ﬂux of 1661 and 1666Å, Ciii] λ 1908Å represents the total ﬂux R 3 = (1) of lines at 1907 and 1909Å, and Civ 1549Å represents the total O I(1664 Å) λ ﬂux of lines at 1548 and 1551Å. we derive the [Oiii] electron temperature using the same set 2 CALIBRATION SAMPLE AND ABUNDANCES of atomic parameters as in Pérez-Montero (2014): We compiled from the literature a sample of Hii regions and starburst galaxies at diﬀerent with optical and UV lines emitted by the gas ionised by episodes ofzmassive star formation. Owing to the te([Oiii]) = 1.9263 + 6.433 · 10−5 · R 3 − 0.4221 · log(R 3) (2) O O scarce number of objects with simultaneous measurement of optical and UV lines, our sample consists on a combination of integrated valid in the range ([Oiii]) = 0.8 to 2.5 in units of 104 K and that spectra observed using a variety of instruments and techniques. Our te is nearly independent on electron density (i.e. a change from 10 to sample was selected in such a way that the O/H and C/O abundance 1000 cm−3 implies a change in of less than 0.1%). Nevertheless, ratios could be derived using the direct method from the estimation te due to the very long wavelength baseline between the involved of the [Oiii] electron temperature, ([Oiii]). This was obtained te emission lines and the large extinction correction in the UV, the from the ratio between [Oiii]5007 and 4363Å and, in some cases, used ratio has a large uncertainty if the extinction correction is big. from the ratio between [Oiii]5007 and 1664Å. Our calibration For instance assuming an extinction of 1 mag at 1664 Å implies sample includes objects without observed ﬂuxes in the [Oii] 3727 Å an electron temperature of about 10% larger using a Calzetti et al. line. For these objects the the total oxygen abundance was derived (2000) than a Cardelli et al. (1989) extinction law. by applying an empirical calibration between O/H and ([Oiii]) te The carbon ionic ratios based on UV lines were derived using (see below). For C/O determinations using the direct method, we the following expressi(cid:18)ons and assum(cid:19)ing the derived [Oiii]: compiled objects with measurements of at least Oiii] 1664 Å and te Ciii] 1909 Å, and also Civ1549 Å in very high-excitation objects. Aetlallt.h(e19c8o9m)pexilteidncﬂtiuoxnelsawwebraesreeddodnenthinegexctoinrrcetciotendcouesiﬃngcieanCtsagrdiveellni log(C2+/O2+) = log OCiiiiii]] 11960684 ÅÅ − 0.8361+ in these references. The list of references and the number of objects are listed in Table 1. 0 4801 We recalculated the physical properties and the chemical abun- − . t − 0.0358 · te + 0.2535 · log(te) (3) dances of the selected objects using expressions based on the pyneb e (cid:18) (cid:19) software (Luridiana, Morisset, & Shaw 2015). The O/H abundances were calculated using the derived electron temperatures and the bright optical emission lines [Oii] 3727 Å and [Oiii] 4959,5007 Civ1549 Å Å and the expressions given in Pérez-Montero (2014). In those log(C3+/O2+) = log Oiii] 1664 Å − 1.5625+ MNRAS 000, 1–7 (2016) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Enrique

text: Pérez-Montero

text: Ricardo

text: Amorín

text: using

text: the

text: James

text: Webb

text: Space

text: Telescope

text: (JWST)

text: and

text: the

text: new

text: gener-

text: Table

text: References

text: and

text: number

text: of

text: Hii

text: regions,

text: local

text: starbursts

text: (LSB),

text: and

text: ation

text: of

text: telescopes,

text: will

text: increase

text: by

text: a

text: huge

text: factor

text: current

text: Lyman-break

text: galaxies

text: (LBG)

text: used

text: in

text: this

text: work.

text: galaxy

text: samples

text: at

text: extending

text: detailed

text: UV

text: spectroscopic

text: stud-

text: z

text: ies

text: towards

text: the

text: cosmic

text: dawn.

text: Reference

text: Number

text: of

text: objects

text: Type

text: For

text: this

text: aim,

text: in

text: this

text: paper

text: we

text: present

text: an

text: adapted

text: version

text: of

text: de

text: Barros

text: et

text: al.

text: LBG

text: Bayliss

text: et

text: al.

text: LBG

text: the

text: code

text: Hii-Chi-mistry

text: (Pérez-Montero

text: hereinafter

text: HCm),

text: Berg

text: et

text: al.

text: LSB

text: originally

text: designed

text: to

text: derive

text: O/H

text: and

text: N/O

text: from

text: optical

text: emission

text: Christensen

text: et

text: al.

text: LBG

text: lines,

text: that

text: deals

text: with

text: UV

text: emission

text: lines

text: to

text: derive

text: O/H

text: and

text: C/O

text: Erb

text: et

text: al.

text: LBG

text: ratios.

text: This

text: new

text: semi-empirical

text: method

text: is

text: fully

text: consistent

text: with

text: Garnett

text: et

text: al.

text: LSB

text: direct

text: estimations

text: based

text: on

text: the

text: electron

text: temperature

text: of

text: star

text: form-

text: Garnett

text: et

text: al.

text: LSB

text: ing

text: regions

text: and

text: provide

text: a

text: potentially

text: powerful

text: tool

text: to

text: constrain

text: the

text: Garnett

text: et

text: al.

text: Hii

text: metallicity

text: of

text: galaxies

text: out

text: to

text: very

text: high

text: redshifts.

text: Even

text: considering

text: Izotov

text: et

text: al.

text: LSB

text: the

text: inherent

text: limitations

text: aﬀecting

text: all

text: methods

text: for

text: estimating

text: chem-

text: James

text: et

text: al.

text: LBG

text: ical

text: abundances

text: in

text: galaxies

text: from

text: their

text: integrated

text: spectra,

text: such

text: as,

text: Kurt

text: et

text: al.

text: Hii

text: e.g.

text: the

text: fact

text: that

text: these

text: are

text: based

text: on

text: spatially

text: unresolved

text: data

text: (e.g.

text: Kobulnicky

text: et

text: al.

text: LSB

text: Kobulnicky

text: Skillman

text: LSB

text: Iglesias-Páramo

text: et

text: al.

text: and

text: rely

text: on

text: the

text: simplistic

text: assumption

text: of

text: Steidel

text: et

text: al.

text: (stack)

text: LBG

text: a

text: single

text: Hii

text: region

text: for

text: the

text: analysis

text: of

text: the

text: observed

text: emission

text: lines,

text: Thuan

text: et

text: al.

text: LSB

text: it

text: is

text: still

text: possible

text: to

text: ﬁnd

text: characteristic

text: abundance

text: ratios

text: (i.e.

text: O/H,

text: Vanzella

text: et

text: al.

text: LBG

text: N/O,

text: Pérez-Montero

text: et

text: al.

text: that

text: correlate

text: with

text: the

text: integrated

text: Villar-Martín

text: et

text: al.

text: LBG

text: properties

text: of

text: galaxies,

text: such

text: as

text: stellar

text: mass,

text: thus

text: providing

text: useful

text: constraints

text: for

text: their

text: assembly

text: histories

text: and

text: chemical

text: evolution.

text: Our

text: paper

text: is

text: organized

text: as

text: follows;

text: In

text: Section

text: we

text: describe

text: the

text: cases

text: where

text: the

text: ﬂux

text: of

text: the

text: [Oii]

text: Å

text: line

text: were

text: not

text: available

text: to

text: sample

text: of

text: objects

text: with

text: available

text: emission

text: lines

text: both

text: in

text: the

text: optical

text: calculate

text: the

text: O+

text: abundances,

text: we

text: resorted

text: to

text: the

text: empirical

text: relation

text: and

text: in

text: the

text: UV

text: whose

text: O

text: and

text: C

text: abundances

text: could

text: be

text: derived

text: using

text: between

text: O/H

text: and

text: ([Oiii])

text: given

text: by

text: Amorín

text: et

text: al.

text: which

text: is

text: te

text: the

text: direct

text: method.

text: In

text: Section

text: we

text: describe

text: the

text: models

text: and

text: the

text: valid

text: for

text: a

text: wide

text: range

text: of

text: metallicity

text: and

text: ionisation

text: values.

text: method

text: used

text: to

text: derive

text: the

text: abundances

text: and

text: we

text: compare

text: our

text: results

text: Using

text: the

text: pyneb

text: code,

text: we

text: also

text: deﬁned

text: expressions

text: for

text: the

text: tem-

text: with

text: the

text: direct

text: abundances.

text: Finally,

text: in

text: Section

text: we

text: present

text: our

text: perature

text: and

text: carbon

text: abundances

text: using

text: UV

text: emission

text: lines.

text: Comput-

text: results

text: and

text: in

text: Section

text: we

text: summaries

text: our

text: conclusions.

text: ing

text: the

text: emission

text: line

text: ratio:

text: Throughout

text: this

text: paper

text: the

text: following

text: convention

text: is

text: adopted:

text: O/H

text: is

text: C/O

text: is

text: log(C/O),

text: Oiii]

text: represents

text: the

text: λ

text: Å)

text: total

text: ﬂux

text: of

text: and

text: Ciii]

text: λ

text: represents

text: the

text: total

text: ﬂux

text: R

text: of

text: lines

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text: • Are the neutrino mass squareds normal ordered (NO)? • What is the lightest neutrino mass value? • Are the neutrino masses of the Dirac or Majorana type? Before entering into such details about the neutrino mass and mixing, and the emerging hints arising from the latest data for the what this pattern looks like, it is important the emphasise that we are dealing with BSM physics. To understand why this is evidence for BSM physics, we recall that, in the SM, neutrinos are massless for three reasons: • There are no RH (sterile) neutrinos ν in the SM; R • In the SM there are no Higgs in SU(2) triplet representations; L • The SM Lagrangian is renormalisable. In the SM, there are three neutrinos ν , ν , ν which all massless and are distinguished by separate e µ τ lepton numbers L , L , L . The neutrinos and antineutrinos are distinguished by total lepton number e µ τ L = ±1. Clearly we must go beyond the SM to understand the origin of the tiny neutrino masses, so that at least one of the above should not apply. For instance, if RH (sterile) neutrinos ν are included, then the usual Higgs mechanism of the SM R yields Dirac neutrino mass in the same way as for the electron mass m . This would break L , L , e e µ L , but preserve L. According to this simplest possibility, the Yukawa term would be Y LHν , in the τ ν R standard notation, and Y ∼ 10−12. By comparison, the electron mass has a Yukawa coupling eigenvalue ν Y of about 10−6. e Alternatively, neutrinos may have a Majorana mass which would break L, leading to neutrinoless double beta decay. Indeed, having introduced RH neutrinos (also called sterile neutrinos, since they are SM singlets), something must prevent (large) Majorana mass terms M ν ν where M could take R R R R any value up to the Planck scale. A conserved symmetry such as U(1) would forbid RH neutrino B−L masses, but if gauged (in order to be a robust symmetry) it would have to be broken at the TeV scale or higher, allowing Majorana masses at the U(1) breaking scale. B−L According to the above argument, SM singlet RH neutrino Majorana masses seem diﬃcult to avoid. However, it is also possible to generate left-handed (LH) Majorana neutrino masses, which may arise even without RH neutrinos. Such masses are allowed below the electroweak (EW) scale since neutrinos do not carry electric charge. For instance, introducing a Higgs triplet ∆ (written as a 2 × 2 matrix), LH Majorana neutrino masses arise from the term y LT (∆)L, where y is a dimensionless coupling. M M Majorana masses occur once the lepton doublets L are contracted with the neutral component of the Higgs triplet which develops a vacuum expectation value (VEV). Alternatively, Majorana mass can arise from dimension ﬁve operators ﬁrst proposed by Weinberg [4], (cid:18) (cid:19) 1 λ − LT (HH)L, (1) 2 Λ where λ is a dimensionless coupling constant, Λ is a mass scale and (HH) is an SU(2) triplet L combination of two Higgs doublets (written as a 2 × 2 matrix). This is a non-renormalisable operator, 4 ##LTLine## ##LTLine## ##LTLine##

label: list

text: Are

text: the

text: neutrino

text: mass

text: squareds

text: normal

text: ordered

text: (NO)?

text: What

text: is

text: the

text: lightest

text: neutrino

text: mass

text: value?

text: Are

text: the

text: neutrino

text: masses

text: of

text: the

text: Dirac

text: or

text: Majorana

text: type?

text: Before

text: entering

text: into

text: such

text: details

text: about

text: the

text: neutrino

text: mass

text: and

text: mixing,

text: and

text: the

text: emerging

text: hints

text: arising

text: from

text: the

text: latest

text: data

text: for

text: the

text: what

text: this

text: pattern

text: looks

text: like,

text: it

text: is

text: important

text: the

text: emphasise

text: that

text: we

text: are

text: dealing

text: with

text: BSM

text: physics.

text: To

text: understand

text: why

text: this

text: is

text: evidence

text: for

text: BSM

text: physics,

text: we

text: recall

text: that,

text: in

text: the

text: SM,

text: neutrinos

text: are

text: massless

text: for

text: three

text: reasons:

text: There

text: are

text: no

text: RH

text: (sterile)

text: neutrinos

text: ν

text: in

text: the

text: SM;

text: R

text: In

text: the

text: SM

text: there

text: are

text: no

text: Higgs

text: in

text: triplet

text: representations;

text: L

text: The

text: SM

text: Lagrangian

text: is

text: renormalisable.

text: In

text: the

text: SM,

text: there

text: are

text: three

text: neutrinos

text: ν

text: ν

text: ν

text: which

text: all

text: massless

text: and

text: are

text: distinguished

text: by

text: separate

text: e

text: µ

text: τ

text: lepton

text: numbers

text: L

text: L

text: L

text: The

text: neutrinos

text: and

text: antineutrinos

text: are

text: distinguished

text: by

text: total

text: lepton

text: number

text: e

text: µ

text: τ

text: L

text: Clearly

text: we

text: must

text: go

text: beyond

text: the

text: SM

text: to

text: understand

text: the

text: origin

text: of

text: the

text: tiny

text: neutrino

text: masses,

text: so

text: that

text: at

text: least

text: one

text: of

text: the

text: above

text: should

text: not

text: apply.

text: For

text: instance,

text: if

text: RH

text: (sterile)

text: neutrinos

text: ν

text: are

text: included,

text: then

text: the

text: usual

text: Higgs

text: mechanism

text: of

text: the

text: SM

text: R

text: yields

text: Dirac

text: neutrino

text: mass

text: in

text: the

text: same

text: way

text: as

text: for

text: the

text: electron

text: mass

text: m

text: This

text: would

text: break

text: L

text: L

text: e

text: e

text: µ

text: L

text: but

text: preserve

text: L.

text: According

text: to

text: this

text: simplest

text: possibility,

text: the

text: Yukawa

text: term

text: would

text: be

text: Y

text: LHν

text: in

text: the

text: τ

text: ν

text: R

text: standard

text: notation,

text: and

text: Y

text: By

text: comparison,

text: the

text: electron

text: mass

text: has

text: a

text: Yukawa

text: coupling

text: eigenvalue

text: ν

text: Y

text: of

text: about

text: e

text: Alternatively,

text: neutrinos

text: may

text: have

text: a

text: Majorana

text: mass

text: which

text: would

text: break

text: L,

text: leading

text: to

text: neutrinoless

text: double

text: beta

text: decay.

text: Indeed,

text: having

text: introduced

text: RH

text: neutrinos

text: (also

text: called

text: sterile

text: neutrinos,

text: since

text: they

text: are

text: SM

text: singlets),

text: something

text: must

text: prevent

text: (large)

text: Majorana

text: mass

text: terms

text: M

text: ν

text: ν

text: where

text: M

text: could

text: take

text: R

text: R

text: R

text: R

text: any

text: value

text: up

text: to

text: the

text: Planck

text: scale.

text: A

text: conserved

text: symmetry

text: such

text: as

text: would

text: forbid

text: RH

text: neutrino

text: B−L

text: masses,

text: but

text: if

text: gauged

text: (in

text: order

text: to

text: be

text: a

text: robust

text: symmetry)

text: it

text: would

text: have

text: to

text: be

text: broken

text: at

text: the

text: TeV

text: scale

text: or

text: higher,

text: allowing

text: Majorana

text: masses

text: at

text: the

text: breaking

text: scale.

text: B−L

text: According

text: to

text: the

text: above

text: argument,

text: SM

text: singlet

text: RH

text: neutrino

text: Majorana

text: masses

text: seem

text: diﬃcult

text: to

text: avoid.

text: However,

text: it

text: is

text: also

text: possible

text: to

text: generate

text: left-handed

text: (LH)

text: Majorana

text: neutrino

text: masses,

text: which

text: may

text: arise

text: even

text: without

text: RH

text: neutrinos.

text: Such

text: masses

text: are

text: allowed

text: below

text: the

text: electroweak

text: (EW)

text: scale

text: since

text: neutrinos

text: do

text: not

text: carry

text: electric

text: charge.

text: For

text: instance,

text: introducing

text: a

text: Higgs

text: triplet

text: (written

text: as

text: a

text: matrix),

text: LH

text: Majorana

text: neutrino

text: masses

text: arise

text: from

text: the

text: term

text: y

text: LT

text: (∆)L,

text: where

text: y

text: is

text: a

text: dimensionless

text: coupling.

text: M

text: M

text: Majorana

text: masses

text: occur

text: once

text: the

text: lepton

text: doublets

text: L

text: are

text: contracted

text: with

text: the

text: neutral

text: component

text: of

text: the

text: Higgs

text: triplet

text: which

text: develops

text: a

text: vacuum

text: expectation

text: value

text: (VEV).

text: Alternatively,

text: Majorana

text: mass

text: can

text: arise

text: from

text: dimension

text: ﬁve

text: operators

text: ﬁrst

text: proposed

text: by

text: Weinberg

text: λ

text: LT

text: (HH)L,

text: Λ

text: where

text: λ

text: is

text: a

text: dimensionless

text: coupling

text: constant,

text: Λ

text: is

text: a

text: mass

text: scale

text: and

text: (HH)

text: is

text: an

text: triplet

text: L

text: combination

text: of

text: two

text: Higgs

text: doublets

text: (written

text: as

text: a

text: matrix).

text: This

text: is

text: a

text: non-renormalisable

text: operator,

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: consequently the following consistency equation (as in Eq.80) has to be satisﬁed for ZSU, 2 X ρ∗(SU)X−1 = ρ (SU) . (83) ν 3 ν 3 There are four consistent possible solutions for X , ν X = ρ (1), ρ (S), ρ (U), ρ (SU) . (84) ν 3 3 3 3 The light neutrino mass matrix m is constrained by the residual family symmetry ZSU and residual ν 2 CP symmetry X as [113]: ν ρT (SU)m ρ (SU) = m , (85a) 3 ν 3 ν XT m X = m∗ , (85b) ν ν ν ν where the second of these equations follows from Eq.78. For X = ρ (S), ρ (U), the lepton mix- ν 3 3 ing angles and CP phases are determined to be a special case of TM mixing, with maximal atmo- 1 spheric mixing angle and maximal Dirac CP violation δ = ±π. The Majorana phases are trivial with CP 2 α , α = 0, π. The other two cases in Eq.84 predict zero CP violation. 21 31 Finally we note that the Littlest Seesaw neutrino mass matrix in Eqs.54, 73 satisﬁes Eq.85a (after multiplying L by a minus sign) but can only satisfy Eq.85b for η = 0, which is not acceptable, therefore 2 that model does not possess any remnant CP symmetry in the neutrino sector. Instead the LS prediction η = ±2π/3 arises from an extra Z symmetry of the ﬂavon potential, as explained below Eq.73. 3 6 Uniﬁcation: Grand Uniﬁed Theories of Flavour We have argued that neutrino masses and mixing angles are a part of the ﬂavour puzzle, which includes charged leptons and quarks. However lepton mixing angles are quite large, which seems to suggest discrete family symmetry. When the type I seesaw mechanism is also included, as a mechanism for small neutrino masses, then large scales may become involved, possibly as large as the GUT scale. In such a framework the origin of all quark and lepton masses and mixing could be related to some GUT symmetry group G , which uniﬁes the fermions within each family and therefore relates neutrino GUT masses to charged quark and lepton masses. Indeed, the inclusion of GUTs requires the problem of neutrino masses and the problem of quark and lepton masses to be tackled simultaneously. The choice of GUT group is quite large, but some possible candidate gauge groups are shown in Fig. 15. In this section we shall focus mainly on SU(5) [132], the Pati-Salam gauge group SU(4) × SU(2) × SU(2) [133] C L R and SO(10) [134] (shown in pale blue in Fig. 15). 6.1 SU(5) We ﬁrst consider the gauge group SU(5) [132], which is rank 4 and has 24 gauge bosons which trans- form as the 24 adjoint representation. A LH lepton and quark fermion family is neatly accommodated into the SU(5) representations F = 5 and T = 10, where 38 ##LTLine## ##LTLine##

text: consequently

text: the

text: following

text: consistency

text: equation

text: (as

text: in

text: has

text: to

text: be

text: satisﬁed

text: for

text: ZSU,

text: X

label: equation

text: ρ

text: (SU)

text: ν

text: ν

text: There

text: are

text: four

text: consistent

text: possible

text: solutions

text: for

text: X

text: ν

text: X

text: ρ

text: ρ

text: (S),

text: ρ

text: (U),

text: ρ

text: (SU)

text: ν

text: The

text: light

text: neutrino

text: mass

text: matrix

text: m

text: is

text: constrained

text: by

text: the

text: residual

text: family

text: symmetry

text: ZSU

text: and

text: residual

text: ν

text: CP

text: symmetry

text: X

text: as

text: ν

text: ρT

text: (SU)m

text: ρ

text: (SU)

text: m

text: ν

text: ν

text: XT

text: m

text: X

text: m∗

text: ν

text: ν

text: ν

text: ν

text: where

text: the

text: second

text: of

text: these

text: equations

text: follows

text: from

text: For

text: X

text: ρ

text: (S),

text: ρ

text: (U),

text: the

text: lepton

text: mix-

text: ν

text: ing

text: angles

text: and

text: CP

text: phases

text: are

text: determined

text: to

text: be

text: a

text: special

text: case

text: of

text: TM

text: mixing,

text: with

text: maximal

text: atmo-

text: spheric

text: mixing

text: angle

text: and

text: maximal

text: Dirac

text: CP

text: violation

text: δ

text: ±π.

text: The

text: Majorana

text: phases

text: are

text: trivial

text: with

text: CP

text: α

text: α

text: π.

text: The

text: other

text: two

text: cases

text: in

text: predict

text: zero

text: CP

text: violation.

text: Finally

text: we

text: note

text: that

text: the

text: Littlest

text: Seesaw

text: neutrino

text: mass

text: matrix

text: in

text: satisﬁes

text: (after

text: multiplying

text: L

text: by

text: a

text: minus

text: sign)

text: but

text: can

text: only

text: satisfy

text: for

text: η

text: which

text: is

text: not

text: acceptable,

text: therefore

text: that

text: model

text: does

text: not

text: possess

text: any

text: remnant

text: CP

text: symmetry

text: in

text: the

text: neutrino

text: sector.

text: Instead

text: the

text: LS

text: prediction

text: η

text: arises

text: from

text: an

text: extra

text: Z

text: symmetry

text: of

text: the

text: ﬂavon

text: potential,

text: as

text: explained

text: below

text: Uniﬁcation:

text: Grand

text: Uniﬁed

text: Theories

text: of

text: Flavour

text: We

text: have

text: argued

text: that

text: neutrino

text: masses

text: and

text: mixing

text: angles

text: are

text: a

text: part

text: of

text: the

text: ﬂavour

text: puzzle,

text: which

text: includes

text: charged

text: leptons

text: and

text: quarks.

text: However

text: lepton

text: mixing

text: angles

text: are

text: quite

text: large,

text: which

text: seems

text: to

text: suggest

text: discrete

text: family

text: symmetry.

text: When

text: the

text: type

text: I

text: seesaw

text: mechanism

text: is

text: also

text: included,

text: as

text: a

text: mechanism

text: for

text: small

text: neutrino

text: masses,

text: then

text: large

text: scales

text: may

text: become

text: involved,

text: possibly

text: as

text: large

text: as

text: the

text: GUT

text: scale.

text: In

text: such

text: a

text: framework

text: the

text: origin

text: of

text: all

text: quark

text: and

text: lepton

text: masses

text: and

text: mixing

text: could

text: be

text: related

text: to

text: some

text: GUT

text: symmetry

text: group

text: G

text: which

text: uniﬁes

text: the

text: fermions

text: within

text: each

text: family

text: and

text: therefore

text: relates

text: neutrino

text: GUT

text: masses

text: to

text: charged

text: quark

text: and

text: lepton

text: masses.

text: Indeed,

text: the

text: inclusion

text: of

text: GUTs

text: requires

text: the

text: problem

text: of

text: neutrino

text: masses

text: and

text: the

text: problem

text: of

text: quark

text: and

text: lepton

text: masses

text: to

text: be

text: tackled

text: simultaneously.

text: The

text: choice

text: of

text: GUT

text: group

text: is

text: quite

text: large,

text: but

text: some

text: possible

text: candidate

text: gauge

text: groups

text: are

text: shown

text: in

text: Fig.

text: In

text: this

text: section

text: we

text: shall

text: focus

text: mainly

text: on

text: the

text: Pati-Salam

text: gauge

text: group

text: C

text: L

text: R

text: and

text: (shown

text: in

text: pale

text: blue

text: in

text: Fig.

text: We

text: ﬁrst

text: consider

text: the

text: gauge

text: group

text: which

text: is

text: rank

text: and

text: has

text: gauge

text: bosons

text: which

text: trans-

text: form

text: as

text: the

text: adjoint

text: representation.

text: A

text: LH

text: lepton

text: and

text: quark

text: fermion

text: family

text: is

text: neatly

text: accommodated

text: into

text: the

text: representations

text: F

text: and

text: T

text: where

text: ##LTLine##

text: ##LTLine##


text: The A Clebsch-Gordan coeﬃcients can be obtained from these expressions by simply dropping all 4 S primes and identifying the components of the S doublet 2 as the 1(cid:48)(cid:48) and 1(cid:48) representations of A . 4 4 4 We thus ﬁnd the non-trivial A products, explicitly, 4 1(cid:48) ⊗ 1(cid:48)(cid:48) → 1 αβ ,   β 3 1(cid:48) ⊗ 3 → 3 α β  , 1 β 2   β 2 1(cid:48)(cid:48) ⊗ 3 → 3 α β  , 3 β 1 3 ⊗ 3 → 1 α β + α β + α β , 1 1 2 3 3 2 3 ⊗ 3 → 1(cid:48) α β + α β + α β , 3 3 1 2 2 1 3 ⊗ 3 → 1(cid:48)(cid:48) α β + α β + α β , 2 2 3 1 1 3     2α β − α β − α β α β − α β 1 1 2 3 3 2 2 3 3 2 3 ⊗ 3 → 3 + 3 2α β − α β − α β  + α β − α β  . 3 3 1 2 2 1 1 2 2 1 2α β − α β − α β α β − α β 2 2 3 1 1 3 3 1 1 3 Although the table in Eq.59 shows the diagonal T basis of A , it is sometimes convenient to work 4 in diagonal S basis in which all matrices are real in the triplet representation [27],     1 0 0 0 1 0 S =  0 −1 0  , T =  0 0 1  . (125) 0 0 −1 1 0 0 From these generators one may obtain all 12 real 3 × 3 matrix group elements after multiplying these two matrices together in all possible ways [27]. Note that although the basis in Eq.125 diﬀers from Eq.59, in both bases T is traceless since 1 + ω + ω2 = 0 and is said to have zero character in all bases, while S has a character (or trace) of −1 in all bases. In the basis of Eq.125 one has the following Clebsch rules for the multiplication of two triplets, 3 ⊗ 3 = 1 ⊕ 1(cid:48) ⊕ 1(cid:48)(cid:48) ⊕ 3 ⊕ 3 , with 1 2 (ab) = a b + a b + a b ; 1 1 1 2 2 3 3 (ab) = a b + ω2a b + ωa b ; 1(cid:48) 1 1 2 2 3 3 (ab) = a b + ωa b + ω2a b ; (126) 1(cid:48)(cid:48) 1 1 2 2 3 3 (ab) = (a b , a b , a b ) ; 31 2 3 3 1 1 2 (ab) = (a b , a b , a b ) , 32 3 2 1 3 2 1 where a = (a , a , a ) and b = (b , b , b ) are the two triplets and ω3 = 1. These diﬀer from the 1 2 3 1 2 3 Clebsch rules in the diagonal (but complex) T basis given earlier, showing that, although the Kronecker product decomposition is valid in all bases, the Clebsch rules are basis dependent. References 55

text: The

text: A

text: Clebsch-Gordan

text: coeﬃcients

text: can

text: be

text: obtained

text: from

text: these

text: expressions

text: by

text: simply

text: dropping

text: all

text: S

text: primes

text: and

text: identifying

text: the

text: components

text: of

text: the

text: S

text: doublet

text: as

text: the

text: and

text: representations

text: of

text: A

text: We

text: thus

text: ﬁnd

text: the

text: non-trivial

text: A

text: products,

text: explicitly,

text: αβ

text: β

text: α

text: β

text: β

text: β

text: α

text: β

text: β

text: α

text: β

text: α

text: β

text: α

text: β

text: α

text: β

text: α

text: β

text: α

text: β

text: α

text: β

text: α

text: β

text: α

text: β

text: 2α

text: β

text: α

text: β

text: α

text: β

text: α

text: β

text: α

text: β

text: 2α

text: β

text: α

text: β

text: α

text: β

text: α

text: β

text: α

text: β

text: 2α

text: β

text: α

text: β

text: α

text: β

text: α

text: β

text: α

text: β

text: Although

text: the

text: table

text: in

text: shows

text: the

text: diagonal

text: T

text: basis

text: of

text: A

text: it

text: is

text: sometimes

text: convenient

text: to

text: work

text: in

text: diagonal

text: S

text: basis

text: in

text: which

text: all

text: matrices

text: are

text: real

text: in

text: the

text: triplet

text: representation

text: S

text: −1

text: T

text: −1

text: From

text: these

text: generators

text: one

text: may

text: obtain

text: all

text: real

text: matrix

text: group

text: elements

text: after

text: multiplying

text: these

text: two

text: matrices

text: together

text: in

text: all

text: possible

text: ways

text: Note

text: that

text: although

text: the

text: basis

text: in

text: diﬀers

text: from

text: in

text: both

text: bases

text: T

text: is

text: traceless

text: since

text: ω

text: ω2

text: and

text: is

text: said

text: to

text: have

text: zero

text: character

text: in

text: all

text: bases,

text: while

text: S

text: has

text: a

text: character

text: (or

text: trace)

text: of

text: −1

text: in

text: all

text: bases.

text: In

text: the

text: basis

text: of

text: one

text: has

text: the

text: following

text: Clebsch

text: rules

text: for

text: the

text: multiplication

text: of

text: two

text: triplets,

text: with

text: (ab)

text: a

text: b

text: a

text: b

text: a

text: b

text: (ab)

text: a

text: b

text: b

text: ωa

text: b

text: (ab)

text: a

text: b

text: ωa

text: b

text: b

text: (ab)

text: (a

text: b

text: a

text: b

text: a

text: b

text: (ab)

text: (a

text: b

text: a

text: b

text: a

text: b

text: where

text: a

text: (a

text: a

text: a

text: and

text: b

text: (b

text: b

text: b

text: are

text: the

text: two

text: triplets

text: and

text: ω3

text: These

text: diﬀer

text: from

text: the

text: Clebsch

text: rules

text: in

text: the

text: diagonal

text: (but

text: complex)

text: T

text: basis

text: given

text: earlier,

text: showing

text: that,

text: although

text: the

text: Kronecker

text: product

text: decomposition

text: is

text: valid

text: in

text: all

text: bases,

text: the

text: Clebsch

text: rules

text: are

text: basis

text: dependent.

text: References


text: [132] H. Georgi and S. L. Glashow, Phys. Rev. Lett. 32 (1974) 438. [133] J. C. Pati and A. Salam, Phys. Rev. D 10 (1974) 275 Erratum: [Phys. Rev. D 11 (1975) 703]. [134] H. Fritzsch and P. Minkowski, Annals Phys. 93 (1975) 193; H. Georgi, in Proceedings of the American Institute of Physics, edited by C.E. Carlson, New York (1975) 575. [135] H. Georgi and C. Jarlskog, Phys. Lett. B 86 (1979) 297. [136] G. Ross and M. Serna, Phys. Lett. B 664 (2008) 97 [arXiv:0704.1248]. [137] S. Antusch and M. Spinrath, Phys. Rev. D 79 (2009) 095004 [arXiv:0902.4644]; S. Antusch, S. F. King and M. Spinrath, Phys. Rev. D 89 (2014) 5, 055027 [arXiv:1311.0877 [hep-ph]]. [138] X. Zhang and B. -Q. Ma, Phys. Rev. D 86 (2012) 093002 [arXiv:1206.0519]. [139] S. Antusch and V. Maurer, Phys. Rev. D 84 (2011) 117301 [arXiv:1107.3728]. [140] D. Marzocca, S. T. Petcov, A. Romanino and M. Spinrath, JHEP 1111, 009 (2011) [arXiv:1108.0614]. [141] R. Gatto, G. Sartori and M. Tonin, Phys. Lett. B 28 (1968) 128. [142] A. Anandakrishnan, S. Raby and A. Wingerter, Phys. Rev. D 87 (2013) no.5, 055005 [arXiv:1212.0542 [hep-ph]]. [143] B. Brahmachari, S. Choubey and M. Mitra, Phys. Rev. D 77 (2008) 073008 Erratum: [Phys. Rev. D 77 (2008) 119901] [arXiv:0801.3554 [hep-ph]]. [144] E. Ma, A. Natale and A. Rashed, Int. J. Mod. Phys. A 27 (2012) 1250134 [arXiv:1206.1570]; [145] G. Altarelli, F. Feruglio, L. Merlo and E. Stamou, JHEP 1208 (2012) 021 [arXiv:1205.4670 [hep- ph]]. [146] B. D. Callen and R. R. Volkas, Large lepton mixing angles from a 4+1-dimensional SU(5) × A 4 domain-wall braneworld model, Phys. Rev. D 86 (2012) 056007 [arXiv:1205.3617]. [147] S. Antusch, S. F. King and M. Spinrath, Phys. Rev. D 87 (2013) no.9, 096018 [arXiv:1301.6764 [hep-ph]]. [148] I. K. Cooper, S. F. King and C. Luhn, JHEP 1206 (2012) 130 [arXiv:1203.1324 [hep-ph]]; I. K. Cooper, S. F. King and C. Luhn, Phys. Lett. B 690 (2010) 396 [arXiv:1004.3243 [hep-ph]]. [149] F. Bjorkeroth, F. J. de Anda, I. de Medeiros Varzielas and S. F. King, JHEP 1506 (2015) 141 [arXiv:1503.03306 [hep-ph]]. [150] S. F. King and M. Malinsky, Phys. Lett. B 645 (2007) 351 [hep-ph/0610250]. 66

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text: Pati

text: and

text: A.

text: Salam,

text: Phys.

text: Rev.

text: D

text: Erratum:

text: [Phys.

text: Rev.

text: D

text: H.

text: Fritzsch

text: and

text: P.

text: Minkowski,

text: Annals

text: Phys.

text: H.

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text: in

text: Proceedings

text: of

text: the

text: American

text: Institute

text: of

text: Physics,

text: edited

text: by

text: C.E.

text: Carlson,

text: New

text: York

text: H.

text: Georgi

text: and

text: C.

text: Jarlskog,

text: Phys.

text: Lett.

text: B

text: G.

text: Ross

text: and

text: M.

text: Serna,

text: Phys.

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text: B

text: S.

text: Antusch

text: and

text: M.

text: Spinrath,

text: Phys.

text: Rev.

text: D

text: S.

text: Antusch,

text: S.

text: F.

text: King

text: and

text: M.

text: Spinrath,

text: Phys.

text: Rev.

text: D

text: [hep-ph]].

text: X.

text: Zhang

text: and

text: B.

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text: and

text: V.

text: Maurer,

text: Phys.

text: Rev.

text: D

text: D.

text: Marzocca,

text: S.

text: T.

text: Petcov,

text: A.

text: Romanino

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text: M.

text: Spinrath,

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text: Gatto,

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text: Sartori

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text: Tonin,

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text: Wingerter,

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text: B.

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text: S.

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text: Erratum:

text: [Phys.

text: Rev.

text: D

text: [hep-ph]].

text: E.

text: Ma,

text: A.

text: Natale

text: and

text: A.

text: Rashed,

text: Int.

text: J.

text: Mod.

text: Phys.

text: A

text: G.

text: Altarelli,

text: F.

text: Feruglio,

text: L.

text: Merlo

text: and

text: E.

text: Stamou,

text: JHEP

text: [hep-

text: ph]].

text: B.

text: D.

text: Callen

text: and

text: R.

text: R.

text: Volkas,

text: Large

text: lepton

text: mixing

text: angles

text: from

text: a

text: A

text: domain-wall

text: braneworld

text: model,

text: Phys.

text: Rev.

text: D

text: S.

text: Antusch,

text: S.

text: F.

text: King

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text: M.

text: Spinrath,

text: Phys.

text: Rev.

text: D

text: [hep-ph]].

text: I.

text: K.

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text: F.

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text: F.

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text: [hep-ph]].

text: S.

text: F.

text: King

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text: Malinsky,

text: Phys.

text: Lett.

text: B


text: m m 2 2 (cid:105)e (cid:105) µ (cid:105) (cid:111) m 2 m 2 3 2 solar~7 10 5eV2 7.5×. ⇥ 1 (cid:60)0  5 . eV2 m 2 atmospheric 1 ~2 10 3eV2 2..5 × ⇥ 1(cid:60) 0  3 e V. 2 atmospheric m22 2..~ . 25 × ⇥ 1 0 1(cid:60) 30 e  V 3 2 e ..V2 solar~7 10 5eV2 m 2 7.5×. ⇥ (cid:60) 1 0  .5 eV2 m 2 1 3 ? ? 0 0 Figure 1: On the left is the normal ordering (NO), while on the right is the inverted ordering (IO). The probability that a neutrino state of mass (squared) m2 contains each of (ν , ν , ν ) is proportional to the length i e µ τ of the respective coloured band. Oscillation experiments only determine m2 − m2. i j global ﬁts of [38, 39] 2 The non-zero reactor angle excludes tri-bimaximal mixing. The alternative tri- bimaximal-reactor mixing is evidently excluded by about two sigma. In addition, there is weak evidence for a non-zero CP violating phase. Present data (slightly) prefers a normal ordered (NO) neutrino mass pattern, with a CP phase δ = −100◦ ± 50◦, and (more signiﬁcantly) non-maximal atmospheric mixing. The meaning of the CP phase δ is discussed below. 2.2 Comparing the CKM and PMNS mixing matrices The PDG [41] standardizes the parameterisation of the CKM and the PMNS mixing matrices in terms of unitary matrices consisting of a product of matrices R U R : 23 13 12       1 0 0 c 0 s e−iδ c s 0 13 13 12 12  0 c s   0 1 0   −s c 0  = 23 23 12 12 0 −s c −s eiδ 0 c 0 0 1 23 23 13 13   c c s c s e−iδ 12 13 12 13 13  −s c − c s s eiδ c c − s s s eiδ c s  (4) 12 23 12 13 23 12 23 12 13 23 13 23 s s − c s c eiδ −c s − s s c eiδ c c 12 23 12 13 23 12 23 12 13 23 13 23 where δ ≡ δ is the CP violating phase, relevant to the particular sector (either quark or lepton). CP We follow the short-hand notation s = sin θ , etc, with small quark mixing, 13 13 2At the time of writing [40] has not yet been updated. 9 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: m

font: default

text: m

text: µ

text: m

text: m

text: m

text: atmospheric

text: e

text: V.

text: atmospheric

text: e

text: V

text: e

text: m

text: m

text: Figure

text: On

text: the

text: left

text: is

text: the

text: normal

text: ordering

text: (NO),

text: while

text: on

text: the

text: right

text: is

text: the

text: inverted

text: ordering

text: (IO).

text: The

text: probability

text: that

text: a

text: neutrino

text: state

text: of

text: mass

text: (squared)

text: contains

text: each

text: of

text: (ν

text: ν

text: ν

text: is

text: proportional

text: to

text: the

text: length

text: i

text: e

text: µ

text: τ

text: of

text: the

text: respective

text: coloured

text: band.

text: Oscillation

text: experiments

text: only

text: determine

text: i

text: j

text: global

text: ﬁts

text: of

text: The

text: non-zero

text: reactor

text: angle

text: excludes

text: tri-bimaximal

text: mixing.

text: The

text: alternative

text: tri-

text: bimaximal-reactor

text: mixing

text: is

text: evidently

text: excluded

text: by

text: about

text: two

text: sigma.

text: In

text: addition,

text: there

text: is

text: weak

text: evidence

text: for

text: a

text: non-zero

text: CP

text: violating

text: phase.

text: Present

text: data

text: (slightly)

text: prefers

text: a

text: normal

text: ordered

text: (NO)

text: neutrino

text: mass

text: pattern,

text: with

text: a

text: CP

text: phase

text: δ

text: and

text: (more

text: signiﬁcantly)

text: non-maximal

text: atmospheric

text: mixing.

text: The

text: meaning

text: of

text: the

text: CP

text: phase

text: δ

text: is

text: discussed

text: below.

text: Comparing

text: the

text: CKM

text: and

text: PMNS

text: mixing

text: matrices

text: The

text: PDG

text: standardizes

text: the

text: parameterisation

text: of

text: the

text: CKM

text: and

text: the

text: PMNS

text: mixing

text: matrices

text: in

text: terms

text: of

text: unitary

text: matrices

text: consisting

text: of

text: a

text: product

text: of

text: matrices

text: R

text: U

text: R

text: c

text: s

text: e−iδ

text: c

text: s

text: c

text: s

text: −s

text: c

text: −s

text: c

text: −s

text: eiδ

text: c

text: c

text: c

text: s

text: c

text: s

text: e−iδ

text: −s

text: c

text: c

text: s

text: s

text: eiδ

text: c

text: c

text: s

text: s

text: s

text: eiδ

text: c

text: s

text: s

text: s

text: c

text: s

text: c

text: eiδ

text: −c

text: s

text: s

text: s

text: c

text: eiδ

text: c

text: c

text: where

text: δ

text: δ

text: is

text: the

text: CP

text: violating

text: phase,

text: relevant

text: to

text: the

text: particular

text: sector

text: (either

text: quark

text: or

text: lepton).

text: CP

text: We

text: follow

text: the

text: short-hand

text: notation

text: s

text: sin

text: θ

text: etc,

text: with

text: small

text: quark

text: mixing,

text: the

text: time

text: of

text: writing

text: has

text: not

text: yet

text: been

text: updated.

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: v2 v1 v6 v3 v5 v4 Figure 3: A digraph of order 6 having no cut-vertex 1. Let G be a digraph with no cut-vertex. Also, assume b(G) denotes the number of blocks in G. Then \ ≥ \ ﬁnd out a vertex v, such that, b(G v) b(G u) for all u in G. Degree of a vertex v is number of vertices adjacent to it. In most of the digraphs having no cut-vertex, removing a vertex having maximum degree introduce maximum number of blocks in the resultant digraph. This observation is true for all digraphs of order less than 6,[29] but there are counterexamples in higher order. For example in digraph in ﬁgure 3: vertex v has degree 3, and vertex v has degree 4. But, removing v results in 2 6 2 2 blocks while removing v results in 1 block. 6 2. Matrices are classiﬁed into a number of families possessing diﬀerent properties. A rigorous study can be done on their characteristic polynomials with this combinatorial methods. It will provide us a combinatorial description of those particular properties. 3. Finding singular graphs is an important problem due to their applications in quantum chemistry, and Huckel’s molucular orbital theory. This article provides a new direction of research to ﬁnd out singular graphs. 4. The Schur’s complement method is not applicable for ﬁnding out permanent of a matrix. It makes this problem complicated. Here we have calculated permanent for graphs with a number of cut vertiecs. It can be extended for further investigations for those diagraphs having no cut vertex. 5. The Laplacian matrices of graphs provide a pictorial description of quantum states in the quantum mechanics and information theory. We may do the similar study where the Laplacian matrix is involved instead of graph adjacency matrix. It will be helpful to represent quantum states with weighted digraphs. References [1] Alireza Abdollahi. Determinants of adjacency matrices of graphs. Transactions on Combinatorics, 1(4):9–16, 2012. [2] Ravindra B Bapat. Graphs and matrices. Springer, 2010. [3] RB Bapat. A note on singular line graphs. Bull. Kerala Math. Assoc, 8(2), 2011. [4] RB Bapat and MI Beg. Order statistics for nonidentically distributed variables and permanents. Sankhy¯a: The Indian Journal of Statistics, Series A, pages 79–93, 1989. 13 ##LTFigure##

text: Figure

text: A

text: digraph

text: of

text: order

text: having

text: no

text: cut-vertex

text: Let

text: G

text: be

text: a

text: digraph

text: with

text: no

text: cut-vertex.

text: Also,

text: assume

text: b(G)

text: denotes

text: the

text: number

text: of

text: blocks

text: in

text: G.

text: Then

text: ﬁnd

text: out

text: a

text: vertex

text: v,

text: such

text: that,

text: b(G

text: v)

text: b(G

text: u)

text: for

text: all

text: u

text: in

text: G.

text: Degree

text: of

text: a

text: vertex

text: v

text: is

text: number

text: of

text: vertices

text: adjacent

text: to

text: it.

text: In

text: most

text: of

text: the

text: digraphs

text: having

text: no

text: cut-vertex,

text: removing

text: a

text: vertex

text: having

text: maximum

text: degree

text: introduce

text: maximum

text: number

text: of

text: blocks

text: in

text: the

text: resultant

text: digraph.

text: This

text: observation

text: is

text: true

text: for

text: all

text: digraphs

text: of

text: order

text: less

text: than

text: but

text: there

text: are

text: counterexamples

text: in

text: higher

text: order.

text: For

text: example

text: in

text: digraph

text: in

text: ﬁgure

text: vertex

text: v

text: has

text: degree

text: and

text: vertex

text: v

text: has

text: degree

text: But,

text: removing

text: v

text: results

text: in

text: blocks

text: while

text: removing

text: v

text: results

text: in

text: block.

text: Matrices

text: are

text: classiﬁed

text: into

text: a

text: number

text: of

text: families

text: possessing

text: diﬀerent

text: properties.

text: A

text: rigorous

text: study

text: can

text: be

text: done

text: on

text: their

text: characteristic

text: polynomials

text: with

text: this

text: combinatorial

text: methods.

text: It

text: will

text: provide

text: us

text: a

text: combinatorial

text: description

text: of

text: those

text: particular

text: properties.

text: Finding

text: singular

text: graphs

text: is

text: an

text: important

text: problem

text: due

text: to

text: their

text: applications

text: in

text: quantum

text: chemistry,

text: and

text: Huckel’s

text: molucular

text: orbital

text: theory.

text: This

text: article

text: provides

text: a

text: new

text: direction

text: of

text: research

text: to

text: ﬁnd

text: out

text: singular

text: graphs.

text: The

text: Schur’s

text: complement

text: method

text: is

text: not

text: applicable

text: for

text: ﬁnding

text: out

text: permanent

text: of

text: a

text: matrix.

text: It

text: makes

text: this

text: problem

text: complicated.

text: Here

text: we

text: have

text: calculated

text: permanent

text: for

text: graphs

text: with

text: a

text: number

text: of

text: cut

text: vertiecs.

text: It

text: can

text: be

text: extended

text: for

text: further

text: investigations

text: for

text: those

text: diagraphs

text: having

text: no

text: cut

text: vertex.

text: The

text: Laplacian

text: matrices

text: of

text: graphs

text: provide

text: a

text: pictorial

text: description

text: of

text: quantum

text: states

text: in

text: the

text: quantum

text: mechanics

text: and

text: information

text: theory.

text: We

text: may

text: do

text: the

text: similar

text: study

text: where

text: the

text: Laplacian

text: matrix

text: is

text: involved

text: instead

text: of

text: graph

text: adjacency

text: matrix.

text: It

text: will

text: be

text: helpful

text: to

text: represent

text: quantum

text: states

text: with

text: weighted

text: digraphs.

text: References

text: Alireza

text: Abdollahi.

text: Determinants

text: of

text: adjacency

text: matrices

text: of

text: graphs.

text: Transactions

text: on

text: Combinatorics,

text: Ravindra

text: B

text: Bapat.

text: Graphs

text: and

text: matrices.

text: Springer,

text: RB

text: Bapat.

text: A

text: note

text: on

text: singular

text: line

text: graphs.

text: Bull.

text: Kerala

text: Math.

text: Assoc,

text: RB

text: Bapat

text: and

text: MI

text: Beg.

text: Order

text: statistics

text: for

text: nonidentically

text: distributed

text: variables

text: and

text: permanents.

text: Sankhy¯a:

text: The

text: Indian

text: Journal

text: of

text: Statistics,

text: Series

text: A,

text: pages

text: ##LTFigure##


text: similarity of A. Determinant of permutation matrices is equal to 1. Thus, relabelling on vertex set keep the determinant and the permanent unchanged. We frequently use this idea in this section. Lemma 2.1. Let G be a digraph, having B as a (cid:16)pendant block(cid:17)and v be the cut-vertex of G in B . Let the 1 1 weight of loop at vertex v be α. The following recurrence relation holds for the characteristic polynomial, × \ \ × \ \ − × \ × \ φ(G) = φ(B ) φ(G B ) + φ(B v) φ G (B v) + (λ α) φ(B v) φ(G B ). 1 1 1 1 1 1 Proof. Let A(G) be the matrix corresponding to the digraph G having n vertices. Let the number of vertices in block B is n . With suitable reordering of vertices in G, let block B has vertices with the labels { }1 1 1 1,2,. . .,n . Here, n -th vertex be the cut-vertex v of G in B . Let x,and z are column vectors of order − 1 − 1 1 (n 1), and (n n ), respectively, such that (x, α, z) formulates n -th column vector of A(G). Similarly, 1 1 − − 1 w, and y are row vectors of order (n1  1), and (n n1), respectively, such that(w, α, y) formulates n1-th row vector of A(G). Now,   \ − A(B v) λI x O − 1 − A(G) λI = w α λ y . (1) \ − O z A(G B ) λI 1 Here, O is the zero matrix of appropriate size. Let x , w , z , and y denote the i-th entry of the vectors i i i {i } x, w, z, and y, respectively. Using theorem 1.1, let us ﬁx set S = 1, 2, . . . , n , then T is n -subset of − − 1 1 n columns in A(G) λI. Note that, T must have numbers 1, 2, 3 . . . , (n 1) otherwise it will give zero ≤ ≤ − 1 contribution to φ(G). This is because if any number 0 i (n 1) is missing in T, then column − (cid:48) 1 − (cid:48) corresponding to i will be in submatrix [A(G) λI] . As this column in submatrix [A(G) λI] has S,T S,T all zero entries, hence its determinant is zero. Thus, only following sets of S, T have possible non-zero contribution in φ(G). { } { } 1. S = 1, 2, . . . , n , T = 1, 2, . . . , n . It contributes the following to φ(G) 1 1 × \ φ(B ) φ(G B ) (2) 1 1 { } { − } { } 2. S = 1, 2, . . . , n , T = 1, 2, . . . , n 1, r , where r is a value from set n + 1, n + 2, . . . , n . So −1 1 { − }1 1 − there will be n n such possible set of T. Let T = 1, 2, . . . , n 1, n + i for i = 1, 2, . . . , n n . 1 i 1 1 1 \ − Let c be the i-th column of A(G B ) λI, then contribution of these sets to φ(G) is i (cid:88) 1 − n n1 − − × − (cid:48) (cid:88) ( 1)w(S,Ti) det[(cid:20)A(G) λI]S,T det(cid:21)[A(G) (cid:104) λI](cid:16)S,T (cid:17) (cid:105) i=1 − \ − n n1 − (cid:32)A(B v) λI O × \ − \ (cid:33) = ( 1)w(S,Ti) det (cid:88)1 w y det (cid:104)z (cid:16)A(G B1) λI (cid:17) ci (cid:105) i i=1 − \ − × (cid:32) n n1 − × × \ − \ (cid:33) = det(B v λI) (cid:16)( 1)w(S,Ti) y (cid:17)det z A(G B(cid:16) ) λI c (cid:17) (3) 1 i 1 i i=1 \ −(cid:32) × \ \ − (cid:33)− × \ − = det(B v λI(cid:16)) det G(cid:17) (B v) λI + (λ α) det A(G B ) λI 1 1 1 \ × \ \ − \ =φ(B v) φ G (B v) + (λ α)φ(G B ) . 1 1 1 Hence, (cid:16) (cid:17) × \ \ × \ \ − × \ × \ φ(G) = φ(B ) φ(G B ) + φ(B v) φ G (B v) + (λ α) φ(B v) φ(G B ) (4) 1 1 1 1 1 1 4

text: similarity

text: of

text: A.

text: Determinant

text: of

text: permutation

text: matrices

text: is

text: equal

text: to

text: Thus,

text: relabelling

text: on

text: vertex

text: set

text: keep

text: the

text: determinant

text: and

text: the

text: permanent

text: unchanged.

text: We

text: frequently

text: use

text: this

text: idea

text: in

text: this

text: section.

text: Lemma

text: Let

text: G

text: be

text: a

text: digraph,

text: having

text: B

text: as

text: a

text: v

text: be

text: the

text: cut-vertex

text: of

text: G

text: in

text: B

text: Let

text: the

text: weight

text: of

text: loop

text: at

text: vertex

text: v

text: be

text: α.

text: The

text: following

text: recurrence

text: relation

text: holds

text: for

text: the

text: characteristic

text: polynomial,

text: φ(G)

text: φ(B

text: φ(G

text: B

text: φ(B

text: v)

text: φ

text: G

text: (B

text: v)

text: (λ

text: α)

text: φ(B

text: v)

text: φ(G

text: B

text: Proof.

text: Let

text: A(G)

text: be

text: the

text: matrix

text: corresponding

text: to

text: the

text: digraph

text: G

text: having

text: n

text: vertices.

text: Let

text: the

text: number

text: of

text: vertices

text: in

text: block

text: B

text: is

text: n

text: With

text: suitable

text: reordering

text: of

text: vertices

text: in

text: G,

text: let

text: block

text: B

text: has

text: vertices

text: with

text: the

text: labels

text: .,n

text: Here,

text: n

text: -th

text: vertex

text: be

text: the

text: cut-vertex

text: v

text: of

text: G

text: in

text: B

text: Let

text: x,and

text: z

text: are

text: column

text: vectors

text: of

text: order

text: (n

text: and

text: (n

text: n

text: respectively,

text: such

text: that

text: (x,

text: α,

text: z)

text: formulates

text: n

text: -th

text: column

text: vector

text: of

text: A(G).

text: Similarly,

text: w,

text: and

text: y

text: are

text: row

text: vectors

text: of

text: order

text: and

text: (n

text: respectively,

text: such

text: that(w,

text: α,

text: y)

text: formulates

text: row

text: vector

text: of

text: A(G).

text: Now,

text: A(B

text: v)

text: λI

text: x

text: O

text: A(G)

text: λI

text: w

text: α

text: λ

text: y

text: O

text: z

text: A(G

text: B

text: λI

text: Here,

text: O

text: is

text: the

text: zero

text: matrix

text: of

text: appropriate

text: size.

text: Let

text: x

text: w

text: z

text: and

text: y

text: denote

text: the

text: i-th

text: entry

text: of

text: the

text: vectors

text: i

text: i

text: i

text: {i

text: x,

text: w,

text: z,

text: and

text: y,

text: respectively.

text: Using

text: theorem

text: let

text: us

text: ﬁx

text: set

text: S

text: n

text: then

text: T

text: is

text: n

text: -subset

text: of

text: n

text: columns

text: in

text: A(G)

text: λI.

text: Note

text: that,

text: T

text: must

text: have

text: numbers

text: (n

text: otherwise

text: it

text: will

text: give

text: zero

text: contribution

text: to

text: φ(G).

text: This

text: is

text: because

text: if

text: any

text: number

text: i

text: (n

text: is

text: missing

text: in

text: T,

text: then

text: column

text: corresponding

text: to

text: i

text: will

text: be

text: in

text: submatrix

text: [A(G)

text: λI]

text: As

text: this

text: column

text: in

text: submatrix

text: [A(G)

text: λI]

text: has

text: S,T

text: S,T

text: all

text: zero

text: entries,

text: hence

text: its

text: determinant

text: is

text: zero.

text: Thus,

text: only

text: following

text: sets

text: of

text: S,

text: T

text: have

text: possible

text: non-zero

text: contribution

text: in

text: φ(G).

text: S

text: n

text: T

text: n

text: It

text: contributes

text: the

text: following

text: to

text: φ(G)

text: φ(B

text: φ(G

text: B

text: S

text: n

text: T

text: n

text: r

text: where

text: r

text: is

text: a

text: value

text: from

text: set

text: n

text: n

text: n

text: So

text: −1

text: there

text: will

text: be

text: n

text: n

text: such

text: possible

text: set

text: of

text: T.

text: Let

text: T

text: n

text: n

text: i

text: for

text: i

text: n

text: n

text: i

text: Let

text: c

text: be

text: the

text: i-th

text: column

text: of

text: A(G

text: B

text: λI,

text: then

text: contribution

text: of

text: these

text: sets

text: to

text: φ(G)

text: is

text: i

text: n

text: λI]S,T

text: n

text: v)

text: λI

text: O

text: det

text: w

text: y

text: det

text: λI

text: ci

text: i

text: n

text: det(B

text: v

text: λI)

text: y

text: z

text: A(G

text: λI

text: c

text: i

text: i

text: det(B

text: v

text: det

text: (B

text: v)

text: λI

text: (λ

text: α)

text: det

text: A(G

text: B

text: λI

text: =φ(B

text: v)

text: φ

text: G

text: (B

text: v)

text: (λ

text: α)φ(G

text: B

text: Hence,

text: φ(G)

text: φ(B

text: φ(G

text: B

text: φ(B

text: v)

text: φ

text: G

text: (B

text: v)

text: (λ

text: α)

text: φ(B

text: v)

text: φ(G

text: B