A model for right-handed neutrino magnetic moments

arXiv:0911.4103v1 [hep-ph] 20 Nov 2009

Alberto Apari i a

, Ar adi Santamaria a

,José Wudka b

a

Departamentde Físi aTeòri a,Universitat de Valèn ia

andIFIC, Universitat de Valèn ia-CSIC

Dr. Moliner 50,E-46100 Burjassot(Valèn ia), Spain

b

Departmentof Physi s andAstronomy,Universityof California, RiversideCA

92521-0413, USA

Abstra t

A simple extension of the Standard Model providing Majorana magneti

moments to right-handed neutrinos is presented. The model ontains, in

additionto the Standard Modelparti lesand right-handed neutrinos, justa

singly hargeds alarand ave tor-like hargedfermion. The phenomenology

of the model is analysedand its impli ationsin osmology,astrophysi s and

lepton avour violatingpro esses are extra ted. Iflightenough, the harged

parti les responsible for the right-handed neutrino magneti moments ould

opiously be produ edat the LHC.

Key words: Neutrinos, magneti moments,ee tive Lagrangian, LHC

PACS: 14.60.St, 13.35.Hb,13.15.+g, 13.66.Hk

1. Introdu tion

In ref. [1℄ we studied the most general ee tive Lagrangian built with

the Standard Model(SM) eldsplus right-handed neutrinosup tooperators

of dimension ve. We found this Lagrangian ontains only three nonrenor-

malizableoperators,oneofthembeingthe wellknown Weinbergoperator[2℄

whi honlyinvolvestheSMleptondoubletsandtheHiggsdoublet. Theother

two ontain anintera tionofright-handedneutrinoswiththe SMHiggsdou-

blet and a Majorana ele troweak moment for the right-handed neutrinos.

This last operator is parti ularly interesting and an have a variety of phe-

nomenologi al onsequen es in osmology, astrophysi s and at olliders [1℄.

Of ourse, it is interesting to have expli it models in whi h these nonrenor-

malizable intera tions arise naturally be ause one an use them to he k

the general features of the ee tive Lagrangian approa h and extend them

portant if the parti lesresponsible forthe new intera tions are lightenough

as tobeprodu ed atthe next generation of olliders.

Here we present a very simple model whi h gives rise to right-handed

neutrino ele troweak moments; it in ludes, in additionto the SM elds and

the right-handed neutrinos, a harged s alar singlet and a harged singlet

ve tor-like fermion. We obtain the tree level and one-loop ontributions

tothe dimensionveee tiveLagrangian, andinparti ular we ompute the

ontributiontotheright-handedneutrinoele troweakmoments. Weperform

athorough phenomenologi alanalysisof themodel, payingspe ialattention

to the ase in whi h the new harged parti les are light enough to be pro-

du ed atthe Large HadronCollider (LHC). Thus, inse tion2 wedene the

modeland ompute the one-loop ontributiontothe ele troweakmomentof

right-handed neutrinos. The simplestversion of the model, in whi h several

ouplings areset tozeroby usingglobalsymmetries, ontainsstable harged

massive parti les(CHAMPs) whi h are stronglydisfavoured from osmolog-

i al and astrophysi al onsiderations. To avoid su h problems we extend

minimally the model by allowing a soft breaking of the symmetries, whi h

is enough toindu e CHAMP de ays; su h de ays are studiedin se tion 2.3.

The model also indu es some tree-level lepton avour violating (LFV) pro-

esses like

µ → 3e

^{whi h are studied in se tion 2.4. In se tion 3 we dis uss}

briey the one-loop ontributions of the model tothe ee tive Higgs-

ν R

^op-

erator. In se tion 4 we ompute the produ tion ross se tion of the harged

parti les at the LHC and dis uss their observability as a fun tion of their

masses. Finally,in se tion5 we present our on lusions.

2. The model

Asdis ussedinref.[1℄themostgeneraldimensionveintera tionsamong

SM elds and three right-handed neutrinos an be writtenas 1

L ⁵ = ν _R ^c ζσ ^µν ν R B µν +  ˜ℓφ χ  ˜φ ^† ℓ 

− φ ^† φ
ν _R ^c ξν R + h.c.

⁽¹⁾

1

The reader should note adieren e in notation respe t to [1℄, where we used

ν ^′

^to

denotetheneutrinoavoreigenelds. Asinthepresentworkwearenotgoingtodis uss

thediagonalisationoftheneutrinomassmatri eswewilljustuse

ν

^{torepresenttheavor}

eigenelds.

where

ℓ = ^ν _e ^L

L

denotesthe left-handedleptonisodoublet,

e R

^and

ν R

^{the or-}

respondingright-handed isosinglets, and

φ

^{the s alarisodoublet(familyand}

gauge indi es will be suppressed when no onfusion an arise). The harge-

onjugate elds are dened as

e ^c _R = C ¯ e ^T _R , ν _R ^c = C ¯ ν _R ^T

^and

ℓ = ǫC ¯ ˜ ℓ ^T , ˜ φ = ǫφ ^∗

where

ǫ = iσ 2

^{a ts on the}

SU(2)

^{indi es. The} hyper harges assignments are

φ : 1/2

^,

ℓ : −1/2

^,

e R : −1

^,

ν R : 0

^{. The}

SU(2)

^and

U(1)

^{gauge elds are}

denotedby

W

^and

B

respe tively(gluonandquarkseldswillnotbeneeded in the situations onsidered below). The ouplings

χ

^,

ξ

^,

ζ

^{have dimension}

of inverse mass, whi h is asso iated with the s ale of the heavy physi s re-

sponsible for the orresponding operator.

χ

^{, and}

ξ

^{are omplex symmetri}

3 × 3

^{matri es in avour spa e, while}

ζ

^{is a omplex} antisymmetri matrix proportionalto the right-handed neutrino ele troweak moments.

The dierent terms in eq. (1) and their phenomenologi al onsequen es

were dis ussed in [1℄. Here we are more interested in models that ould

give rise to

ζ

^{. This an only o ur at the one-loop level and the models}

should ne essarily involve either a s alar-fermion pair with opposite (non-

zero) hyper harges and havingYukawa ouplings with both

ν R

^and

ν _R ^c

^{, ora}

ve tor-fermionpair withthe same properties. Here wewill onsider onlythe

rst (simpler) possibility. Thus we enlarge the SM by adding a negatively

hargeds alar singlet

ω

^,

Y (ω) = −1

^{,and one negatively harged} ve tor-like fermion

E

^(two hiralities and nogeneration indi es) alsowith

Y (E) = −1

^.

We an then write the Lagrangian as

L = L ^SM + L ^NP ,

⁽²⁾

where

L ^SM

^{is the SM Lagrangian while the new physi s} Lagrangian,

L ^NP

^,

olle ts all the terms ontaining any of the new parti les, in luding among

them the right-handed neutrinos. Wewrite

L ^SM

^as

L ^SM = iℓ 6D ℓ + ie ^R 6D e ^R + (ℓY e e R φ + h.c.) + · · ·

⁽³⁾

with

Y e

^{theYukawa ouplingsof hargedleptonswhi hare ompletelygeneral}

3 × 3

^{matri es in avour spa e; the dots represent SM gauge boson, Higgs}

bosonandquarkkineti terms,quarkYukawaintera tionsand theSMHiggs

potential. Wedivide the new physi s ontribution,

L ^{N P}

^{,in dierent terms:}

L ^{N P} = L ^K + L ^Y − V ^{N P} + L ^Extra

⁽⁴⁾

L ^K

^{des ribes the kineti terms of the new parti les}

L ^K = D µ ω ^† D ^µ ω + iE 6D E − m ^E EE + i¯ ¯ ν R ∂ /ν R −  1

2 ν _R ^c M R ν R + h.c.



(5)

with

M R

^{theMajoranamasstermof}right-handedneutrinos,whi hisa om- plex symmetri matrix in avour spa e.

L ^Y

^{ontains the standard Yukawa}

intera tions of right-handed neutrinos and the Yukawa ouplings of right-

handed neutrinos with the parti lesneeded togenerate the ele troweak mo-

ments:

L ^Y = ℓY ν ν R φ + ν ˜ _R ^c h ^′ E ω ⁺ + ν R hE ω ⁺ + h.c.

⁽⁶⁾

Y ν

^isageneral

3 × 3

^{omplexmatrix and, ifthere isjustone}

E

^,

h

^and

h ^′

^are

ve tors ingeneration spa e. The

ω

ontributions to the s alar potentialare

V N P = m ^′2 _ω |ω| ² + λ ω |ω| ⁴ + 2λ ωφ |ω| ² φ ^† φ , m ² _ω = m ^′2 _ω + λ ωφ v ²

⁽⁷⁾

Where

v

^{istheva uum}expe tationvalueoftheHiggsdoublet,

hφ ^† φi = v ² /2

^,

andthe

λ

^{'sarequarti s alar ouplings. Weassume}

λ, λ ω > 0

^and

λλ ω > λ ² _ωφ

to insure global (tree-level)stability, as well as

m ² _ω > 0

^{inorder to preserve}

U(1) em

^{. It is important to remark that with only one Higgs doublet there}

annot be trilinear ouplings between the doubletand the singlet,

ω

^{. Then,}

the potentialhas two independent

U(1)

symmetries, one for the singlet and one forthe doublet.

Inaddition,theSMsymmetriesallowthefollowingYukawa ouplingsand

mass terms

L ^Extra = ¯ E L κe R + ℓY E E R φ + ˜ ℓf ℓω ⁺ + ¯ e R f ^′ ν _R ^c ω + h.c.

⁽⁸⁾

whi h an beset tozeroby imposinga dis retesymmetrywhi hae tsonly

the new parti les

E → −E , ω → −ω

⁽⁹⁾

In this ase all low-energy physi s ee ts will be loop generated[3℄. Noti e

that the resultingLagrangian has a larger ontinuous symmetry

E → e ^iα E , ω → e ^iα ω

⁽¹⁰⁾

whi h is not anomalous, therefore there is a harge, arried only by

E

^and

ω

^{whi h is exa tly onserved. In that ase, the lightest of the}

E

^or

ω

^will

ν R ν _R ^c ω ⁻

B

E ⁻ E ⁻ ν R ν _R ^c

E ⁻

B ω ⁻ ω ⁻

(a) (b)

Figure1: Contributingdiagramsto theright-handedneutrinoele troweakmoment.

be ompletely stable be oming a CHAMP, whi h ould reate serious prob-

lems in standard osmology s enarios. However, su h problems an easily

be evaded by allowing some of the terms in eq. (8). We will return to this

issue afterverifyingthatthe modelindeedgeneratesaright-handedneutrino

magneti moment.

2.1. The

ν R

^{magneti moment}

Inthe model onsideredwe have two diagrams,depi ted ingure 1, on-

tributingtothe

ν R

^Majoranaele troweakmoment: a)loopwiththe

B

^gauge

boson atta hed to the

E

^{and b) loop with the}

B

^{gauge boson atta hed to}

the s alar

ω

^.

For

M R ≪ m ^E , m ω

^{we an negle t allexternal momentaand masses and}

the al ulation of the diagrams simplies onsiderably. The nal result an

be astasa ontributiontotheee tivemagneti momentoperatorineq.(1).

We nd

ζ ij = g ^′ f (r) (4π) ² 4m E

h ^′ _i h ^∗ _j − h ^′ j h ^∗ _i

(11)

with

r = (m ω /m E ) ²

^,

g ^′

^the

B µ

^{gauge ouplingand}

f (r) = 1

1 − r + r

(1 − r) ² log(r) →







1 , r ≪ 1 1/2 , r = 1 (log(r) − 1)/r , r ≫ 1

(12)

Foranestimatewe antake,forinstan e,

m ω = m E

^,and

h ^′ _i h ^∗ _j − h ^′ j h ^∗ _i
= 0.5

^while

g ^′ = √

α4π/c W ≈ 0.35

^{, then}

ζ ≈ 10 ⁻⁴ /m E

^(for

m E ≫ m ^ω

^there

willbeafa tor

2

enhan ementand for

m E ≪ m ^ω

^therewillbeasuppression byroughly a fa tor

(m E /m ω ) ²

^{);these values are in agreement with the esti-}

mates obtained using ee tive eld theory. In terms of

Λ N P ≡ 1/ζ

^wehave

Λ N P = 10 ⁴ m E

^{. PresentboundsfromLEPandTevatrongive}

m E & 100 GeV

^,

whi himply

Λ N P & 10 ⁶ GeV

^{. This anbe omparedwithdire tboundsthat}

an beset ontheright-handed neutrinoele troweakmomentsderived in[1℄.

As expe ted, olliderlimitson

E

^{produ tionare mu hmore} restri tivethan olliderlimitsderivedfromtheindu ed ele troweakmomentintera tion. Af-

ter all, the ele troweak moment intera tion is generated at one loop. How-

ever,iftheright-handedneutrinosarerelativelylight(below

10 MeV

^)bounds

fromtransitionmagneti moments omingfromsupernova ooling(whi hare

Λ N P & 4 × 10 ⁶ GeV

^{) orred giant ooling(whi hare}

Λ N P & 4 × 10 ⁹ GeV

^for

m N . 10 keV

^{) an be mu h stronger.}

2.2. E or

ω

^{as CHAMPs}

The model as des ribed so far ontains only the ouplings ne essary to

generate the right-handed neutrino Majorana ele troweak moments. But it

is lear that the trilinear verti es

¯ ν R Eω ^†

^and

ν ¯ _R ^c Eω ^†

^{alone annot indu e}

de ays for both the

E

^{and the}

ω

^{. The lightest of the two willremain stable}

and ould then a umulate in the galaxy lusters, appearing as ele tri ally

hargeddarkmatter. Theideathatdarkmatter ouldbe omposedmostlyof

hargedmassiveparti leswasproposedin[4,5℄anditisstrongly onstrained

from very dierent arguments[6, 7, 8, 9, 10℄. One might still onsider the

possibility of having massive stable

E

^or

ω

^{parti les within the rea h of}

the LHC, but with a osmi abundan e lower than the one required for

dark matter. Unfortunately, su h s enario seems also to be ex luded: if

one assumes, as in [4℄, that the

E

^{'s and}

ω

^{'s were produ ed in the early}

universe through the standard freeze-out me hanism [11℄, the bounds from

interstellar alorimetry [10℄ and terrestrial sear hes for super-heavy nu lei

[7, 8℄ ompletely lose the window of under-TeV CHAMP abundan es.

There is, however, a way to es ape all these bounds. A re ent paper

[12℄ notes that CHAMPs, if very massive or arrying very small harges,

are expelled from the gala ti disk by the magneti elds. That situation

prevents any terrestrial or gala ti dete tion and leaves room for CHAMPs

to exist. The bound spe i ally states that parti leswith

100(Q/e) ² TeV . m . 10 ⁸ (Q/e) TeV

^{are depleted fromthe disk, and in fa tour model(if we}

forbidthe termsineq. (8))doesnotx thehyper harge of

E

^and

ω

^{,so they}

an be milli harged. Unfortunately, this situation is not interesting for our

magneti moments and wouldn't show up in the future a elerators, either

due to their heavy masses orto their small ouplings.

In on lusion, we need anadditionalme hanism for

E

^or

ω

^{de ays. The}

easiest way toa omplish this isby allowingone ormore of the ouplingsin

eq. (8), whi h an be taken small, if needed, by arguing that (10) is an al-

most exa t symmetry. Wedis ussone of the possibilitiesinse tion2.3. The

s enario of de aying CHAMPs has, onits own, a number of advantages and

drawba ks. Some re ent papers [13, 14℄ have pointed out that the presen e

ofamassive, hargedand olourlessparti leduringthepro essofprimordial

nu leosynthesis might lead to an explanation for the osmi lithium prob-

lem. Also, the de ay of massiveparti les during nu leosynthesis ould have

a dramati inuen e in the nal abundan es of primordial elements, whi h

provides us with bounds on the lifetime and abundan e of CHAMPs that

ould be useful.

2.3. Allowing for CHAMP de ays

Iftheparti leshavetode aythe globalsymmetry (10)has tobebroken,

and forthat it is enoughto allowsome of the termsin eq. (8). For the sake

of simpli ity, we will onsider only the ase where the symmetry is softly

broken by

E L

^

e R

^mixing

2

L ^κ = ¯ E L κe R + h.c.

⁽¹³⁾

This termwill indu ede ays of

E

^{intoSMparti les mu hlike the heavy}

neutrinode aysinseesawmodels,sin eonlythismixinglinksthe

E

^totheSM

degrees offreedom. Afterdiagonalisationof the harged leptonmass matrix

one obtains intera tions that onne t the

E

^to

W + ν

^,

Z + ℓ ^±

^and

H + ℓ ^±

^.

As the urrentbound onheavy hargedleptonsrequire that

m E > 100 GeV

^,

the

W

^and

Z

^{will be produ ed on-shell; the Higgs hannel may or may not}

beopen depending onthe a tualvalue of the Higgsand

E

^masses3.

The

ω

^{, on the other hand, has to de ay through the Yukawa}

Eν ¯ R ω

^ver-

ti es; either dire tly to

E + ν R

^if

m ω > m E

^{or to}

e + ν R

^{suppressed by the}

2

Sin e this hoi e breaks(10)softly, noneofthe othertermsin eq.(8)needbeintro-

du ed forthemodelto remainrenormalizable.

3

Notethat,as

U(1) em

^isnotbroken,avour- hangingverti esinvolvingaphoton annot appearattreelevel;

Γ(E → eγ)

^{mustbeatleastaone-loopee t,andthereby}suppressed.

mixing

κ

^{. The simplest situation then arises if}

m ω > m E

^{, for in that ase}

the

ω

^{'s will de ay into on-shell}

E

^{'s, whi h in turn will de ay in the afore-}

mentionedway. Inwhatremains,forsimpli ity,weshallrestri tourselvesto

this spe i ase.

Ingure2wepresentthebran hingratiosforthede aysofthe

E

^{. Asthe}

de ays are ontrolled by the would-beGoldstone part of the

W

^and

Z

^(and

theHiggsbosonifallowedkinemati ally)theyarealwaysproportionaltothe

Yukawa ouplings of the harged leptons; therefore, if all the

κ

^{'s are of the}

sameorder,the

E

^{willde aymainlytotheleptonsofthethirdfamily. We an}

seethatforrelativelylowmassesthedominant hannelis

E → W ν ^τ

^whilefor

verylargemassestheratiostendtothe equivalent-Goldstoneapproximation:

0.5

^forthe

W

^hanneland

0.25

^forthe

Z

^and

H

^hannels.

W Ν Τ

Z Τ

H Τ

100 200 300 400 500

0.0 0.2 0.4 0.6 0.8

m E HGeVL

BR HE ™ X L

Figure 2: Dominant de ay bran hing ratios of the ve tor-like fermion

E

^{. The de ays}

aresuppressed bythemassofthe harged leptons,thuswehaveonlyrepresentedde ays

into the third family. The Higgs boson mass has been taken to the present best t,

m ^H = 129 GeV

^.

Thede ayratesofthe

E

^{fermionarepresented ingure3for}

κ τ = 1 GeV

^.

Noti e that the rates de rease for large

m E

^{. This is be ause the de ays}

pro eed through the mixing

E

^

τ

^{and this is suppressed by fa tors}

m τ /m E

^;

thusthein reaseinphasespa eforlarge

m E

^is ompensatedbythesefa tors.

For the hosen value of

κ τ

^{the de ay widths are of the order of the eV. For}

widths of this order of magnitude the

E

^{'s will not be present at the time}

de ay rates depend on

κ ² _τ

^{, and}

κ τ

^{is relatively free, thus the de ay rates}

an vary in several orders of magnitude depending on the value of

κ τ

^{. For}

κ τ < 10 ⁻⁷ GeV

^{the CHAMPs will ae t} nu leosynthesis and, as ommented above, might help to solve the osmi lithium problem [13, 14℄. We also

require

κ τ > 10 ⁻¹⁶ GeV

^{toavoidCHAMPs at the present epo h.}

W Ν Τ

Z Τ

H Τ

100 200 300 400 500

0 1 2 3 4 5

m E HGeVL

G HE ™ X L HeV L

Figure 3: Dominantde ayratesof theve tor-likefermion

E

^withthesameassumptions madein gure2. Fortheseestimateswehavetaken

κ ^τ = 1 GeV

^.

2.4. Lepton Flavour Violating pro esses

For general

κ

^{'s and Yukawa ouplings}

Y e

^{, family lepton avour is not}

onserved; one mightthen worryabout possiblebounds set bypro esses like

µ → 3e

^,

µ → eγ

^or

τ → 3µ

^{. Wenowdeterminewhether theboundsonthose}

rare pro esses an impose restri tionson the parameters of our model.

Theeasiestwayto al ulatetheamplitudesforthesepro essesisbyusing

an ee tiveLagrangian obtained by integration ofthe

E

^{eld. This integra-}

tion is performed by using the equations of motion for

E

^{and expanding in}

powers of

1/m E

^{(foradetailedexampleofthe}integrationofasingly harged s alar see [15℄). One then obtains

L ^LFV = − 1

m ⁴ _E e R κκ ^† i 6D ³ e R + · · ·

⁽¹⁴⁾

breaking leadstoalepton avour violatingintera tionof the

Z

^{gauge boson}

with left-handed hargedleptons,

L ^LFV = e 2s W c W

Z µ e L C LFV γ ^µ e L , C LFV ≈ v ²

2m ⁴ _E Y e κκ ^† Y _e ^† .

⁽¹⁵⁾

C LFV

^{isamatrix inavorspa e whi h isnot,in general,diagonal;therefore,}

eq. (15) will indu e pro esses su h as

µ → 3e

^and

τ → 3µ

^{. Without loss of}

generalitywe antake

Y e

^{diagonalwithelements}proportionaltothe harged lepton masses; then we an estimate the bran hing ratio for the

µ → 3e

pro ess as

BR(µ → 3e) = Γ(µ → 3e) Γ(µ → eν ¯ν) ≈

 

m ^e κκ ^†

eµ m µ

  

2

m ⁸ _E

⁽¹⁶⁾

Our ee tive Lagrangian is anexpansion inpowers of

1/m E

^{whi h ould be}

ompensated, inpart,by

κκ ^†

^{fa torsinthe numerator; thus, for} onsisten y, we should require

κ < m E

^{whi h allows us to establish an upper bound}

for the bran hing ratio. Re alling also that the present limit on the mass

of harged heavy leptons is around

100 GeV

^{, and therefore we should have}

m E > 100 GeV

^{, we obtain}

BR(µ → 3e) <

 m µ m e

(100 GeV) ²

 2

< 10 ⁻¹⁶

⁽¹⁷⁾

to be ompared with present bounds 4

whi h are of the order of

10 ⁻¹²

^{. If}

we apply the same reasoning to

τ → 3µ

^{we see that the bran hing ratio is}

enhan ed by a

(m τ /m e ) ²

^{fa tor}

R(τ → 3µ) ≡ Γ(τ → 3µ)

Γ(τ → µν ¯ν) <

 m τ m µ

(100 GeV) ²

 ²

< 10 ⁻¹⁰

⁽¹⁸⁾

whi hisstillunderthe presentsensitivityforthis ratio,whi hisabout

10 ⁻⁷

^.

Anotherveryrestri tivepro essis

µ → eγ

^{,whi hisbounded atthe}

10 ⁻¹¹

level,

BR(µ → eγ) < 1.2 × 10 ⁻¹¹

^{. This limit will be improved in a lose}

future by the MEG experiment by two orders of magnitude [17℄. However,

4

All experimentallimitsare takenfrom[16℄.

therefore, we do not expe t stringent bounds from it. The ontributions to

the obliqueparametersare suppressed by powers of thefermionsmasses and

are toosmallto be observed atthe urrently available pre ision.

Finally,

µ

^

e

^{onversioninnu leialsoprovidesstronglimitsingeneral;for}

instan e,

µ

^

e

^{onversion on}

Ti

^gives

σ(µ ⁻ Ti → e ⁻ Ti)/σ(µ ⁻ Ti → capture) <

4.3 ×10 ⁻¹²

^{. Inourmodel,the pro essisindu edbyexa tlythesameintera -}

tion (15)thatgives

µ → 3e

^{,andweagaindonotexpe t,atpresent,astrong}

boundfrom

µ

^

e

onversion. However, given the future plans toimprove the limitsby several ordersof magnitude, then perhaps

µ

^

e

^{onversion willpro-}

vide the best bound for LFV pro esses in this model. In any ase, urrent

data on LFVpro esses annot onstrain this me hanismfor

E

^{de ays.}

3. The

ν ^R

^{mass and the ee tive Higgs boson} intera tion with

ν ^R

The model we have dis ussed ontains several sour es of lepton number

non- onservation: the right-handed neutrino Majorana mass and the

h

^and

h ^′

^{ouplings(ifbothofthemaredierentfromzero). Thenitis}interestingto ask what is the naturalsize of the right-handed neutrino Majorana masses,

sin e, even if they are set to zero by hand, radiative orre tions involving

ouplingsthat donot onserveleptonnumberwillgenerate them. Infa t,by

removing the photon line in the diagrams that give rise to the ele troweak

moments, gure 1, one obtains a renormalization of the right-handed neu-

trino Majorana mass. The diagrams are logarithmi ally divergent and give

orre tions of the type

δM R ∼ h ^′ h

(4π) ² m E

⁽¹⁹⁾

(ifthe s alar

ω

^{ismu hheavierthanthe}

E

^,this ontributionwillhaveanex- trasuppression

(m E /m ω ) ²

^{). Itisthennaturaltorequire}

M R & h ^′ hm E /(4π) ²

^.

Of ourse these type of ontributions an be renormalized into

M R

^{whi h,}

after all,is afree parameter of the theory.

In addition, similar diagrams with a vertex

(φ ^† φ)|ω| ²

^{atta hed to the}

ω

eld (seegure4)giveanite ontributiontothe

φ ^† φ
ν _R ^c ξν R

^operatorthat

annot be avoided. A simple al ulation gives

ξ ij = λ ωφ f φ (r) (4π) ² 4m E

h ^′ _i h ^∗ _j + h ^′ _j h ^∗ _i

(20)

ν _R ν ^c _R E ⁻

ω ⁻ ω ⁻

φ φ

Figure4: Diagram ontributingtothe

φ ^† φ
ν R ^c ξν R

^operator.

where

f φ (r)

^{an be written in terms of}

f (r)

^{, dened in eq. (12):}

f φ (r) = 4f (1/r)/r

^{. After} spontaneous symmetry breaking this operator gives addi- tional ontributions tothe right-handed Majorana neutrino mass

δM R ∼ λ ωφ h ^′ hv ² (4π) ² 4m E

(21)

Therefore, atleast, one should require

M R > λ ωφ h ^′ hv ²

(4π) ² 4m E ∼ λ ωφ h ^′ h

(4π) ² 100 GeV ∼ 1 MeV

⁽²²⁾

where we took

h ^′ = h = λ ωφ = 0.1

^{. By taking smaller ouplings, smaller}

right-handed neutrino masses would be natural (for instan e for

h ^′ = h = λ ωφ = 0.01

^{one obtains}

M R > 1 keV

^).

4. The Model at olliders

In spite of the fa t that the new parti les are

SU(2)

^{singlets and only}

have Yukawa ouplings to right-handed neutrinos,they are harged and an

be opiously produ ed at the LHC, if light enough (

< 1 TeV

^{), through the}

Drell-Yanpro ess.

The ross se tionsfor proton-proton ollisions an be omputed interms

of the partoni ross se tions using the parton distribution fun tions of the

proton (for a very lear review see for instan e [18℄); in gure 5 we present

theresults 5

fortheprodu tiontotal rossse tionsattheLHC(

√ s = 14 TeV)

5

WehaveusedtheCTEQ6Mpartondistributionsets[19℄. One ouldalsoin ludenext-

Σ H pp™E ⁺ E ^- + XL

Σ Hpp™Ω ⁺ Ω ^- + XL

100 150 200 300 500 700 1000

0.01 0.1 1 10 100 1000

m HGeVL

Σ Hfb L

Figure5: Produ tion rossse tionsofthe hargedparti lesattheLHC(

√ s = 14 TeV)

^as

afun tion oftheirmasses.

m

^{representseither}

m ^E

^or

m ^ω

^{dependingon thepro essand}

X

^{represents that other hadroni orleptoni produ ts are expe ted in a} proton-proton ollision.

as afun tionof the

E

^and

ω

^masses,

m E

^and

m ω

^(both represented by

m

ⁱⁿ

the gure). Sin e the parti lesare produ edby

γ

^and

Z

^{ex hange,there are}

nounknown free parametersex ept the masses of the parti les. We see that

rossse tionsfrom

1 fb

^to

1 pb

^{are easilyobtainedfortheprodu tionof}

E

^for

masses between

700 GeV

^and

100 GeV

^{. Forthe same masses the produ tion}

ross se tion for

ω

^{is roughly one order of magnitude smaller.}

On e produ ed in pairs, the parti les have to be dete ted and identi-

ed. The hara teristi signatures for this identi ation are very dierent

depending on the lifetimes of the parti les, mostly be ause if the

E

^and

ω

are long-lived they an be tra ked dire tly in the dete tors or, at least, be

identied through adispla ed de ay vertex. The parameter relevant for this

behavior is

κ

^,the

E − e

^mixing.

For

κ . 1 MeV

^{, the}

E

^{'s willhave de ay lengths roughly over}

1

^entime-

to-leading-order orre tions by multiplying by a

K

^{-fa tor whi h typi ally would hange}

rossse tionsby

10 −20%

^{. Resultshavebeen he kedagainsttheCompHEPprogram[20,}

21℄.

ter 6

, infa t, for

κ < 0.2 MeV

^{, they will gothrough the dete tor and behave}

as a heavy ionizing parti le. A lot of work has been arried to analyse the

signatures of CHAMPs inside the dete tor (see, for example, [22℄, and [23℄

for a re ent improvement), and also displa ed verti es have been dis ussed

(see,forexample,[24,25℄). If

κ > 1 MeV

^the

E

^{'swillde aynearthe ollision}

pointand behave asa fourth generation harged lepton.

Dis overingthe

ω

^{'s anbemu hharder,be ausetheywillbeprodu edat}

a signi antly lower rate and the signatures of their de ays depend strongly

onthedetailsofthemodel. Inthe

m ω > m E

^{s enario,theywillde ayqui kly}

intoan

E

^{and aheavy neutrino(atleast ifwewant}

h

^and

h ^′

^{large enoughto}

havesigni antele troweakmoments)and thenone hastorelyagain onthe

dete tion of

E

^{'s unless the heavy neutrino provides a leaner signal, whi h}

is unlikely. In any ase, we think that the

E

^{'s, produ ed in a mu h greater}

number, should be onsidered the signature of this model, and perhaps the

doorway tounderstand the

ω

^{and heavy neutrino de ays.}

5. Con lusions

We have presented asimple modelthat generates right-handed neutrino

magneti moments and studiedits phenomenology. The simplest version of

the model ontainsCHAMPs ( hargedmassivestable parti les)whi h ould

presentsomeproblemswithstandard osmologi als enarios. Theseproblems

an easily be evaded by allowing additional ouplings in the Lagrangian.

The model an then give rise to various LFV pro esses at tree level su h

as

µ → 3e

^{; however, we have veried that the rates of these pro esses are}

stronglysuppressed andarewellbelowpresentand near-futureexperimental

onstraints.

The same intera tions that generate the right-handed neutrino magneti

moments will also generate, at one loop, the last operator in eq. (1) whi h

provides alepton numbernon- onserving intera tion between neutrinos and

the SMHiggsboson. Thisintera tiongivesanadditional ontributiontothe

right-handed neutrino Majorana mass; it is also interesting be ause ould

lead to an invisible Higgs de ay [1℄. We have omputed it and dis ussed

some of its onsequen es.

6

Note that there's room in the parameter spa e for this kind of ee ts even if one

requires thatCHAMPsdonotae ttheprimordialnu leosynthesis,forif

κ > 100 eV

^all

the

E

^{'swillhavede ayedbefore}nu leosynthesis.

neti moment are harged, if light enough they an opiously be produ ed

atthe LHCthrough theDrell-Yanpro ess. Wefound thatthe rossse tions

for Drell-Yan produ tion of

E

^{'s range from}

1 fb

^to

1 pb

^{for masses between}

700 GeV

^and

100 GeV

^{. For the same range of masses the produ tion ross}

se tion for

ω

^{is roughly one order of magnitude smaller.}

Inshort, weshowed thata very simplemodelgivingrise toright-handed

neutrino magneti models ompatiblewith allexisting onstraints an easily

be onstru ted. Iftheright-handedneutrinosarerelativelyheavy(

& 10

^MeV)

bounds on

ν R

^{magneti moments from red giants or supernovae do not ap-}

ply[1℄andthe hargedparti lesresponsibleforthemagneti moments ould

belight enoughas tobe produ ed and dete ted atthe LHC.

A knowledgments

This work has been supported in part by the Ministry of S ien e and

Innovation (MICINN) Spain, under the grant number FPA2008-03373, by

theGeneralitatValen ianagrantPROMETEO/2009/128,bytheEuropean

Union within the Marie Curie Resear h & Training Networks, MRTN-CT-

2006-035482 (FLAVIAnet), and by the U.S. Department of Energy grant

No. DE-FG03-94ER40837. A.A. is supported by the MICINN under the

FPU program.

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