Two-loop neutrino masses with large R-parity violating...

Two-loop neutrino masses with large R-parityviolating interactions in supersymmetry

Paramita Dey

Regional Center for Accelerator-based Particle PhysicsHarish-Chandra Research Institute, Allahabad, India

Work done with Anirban Kundu, Biswarup Mukhopadhyaya and Soumitra Nandi

JHEP12(2008)100, [arXiv:0808.1523]

Paramita Dey HRI NuHoRIzons09 – p. 1

Basic ideas behind our work

It is an inescapable fact that neutrinos are massive andtheir physical states are mixtures of flavour eigenstates.




In standard model (SM) neutrinos are massless to allorders in perturbation theory (without νR and L-violation).





SM has to be extended; there are several possibilities(including introduction of νR).






One possibility (without νR) is a supersymmetric (SUSY)extension of SM with renormalizable R-parity violating( 6Rp) terms in the Lagrangian.






One possibility (without νR) is a supersymmetric (SUSY)extension of SM with renormalizable R-parity violating( 6Rp) terms in the Lagrangian.

Rp = (−1)3B+L+2S can be violated either by L-orB-violation, but not both simultaneously.


Basic ideas...The MSSM superpotential is:

WMSSM = µH1H2 + huijQ

iU jH2 + hdijQ

iDjH1 + hℓijL

iEjH2

The complete 6Rp-superpotential is:

W 6Rp= λijkLiLjE

ck + λ′

ijkLiQjDck + λ′′

ijkUci D

cjD

ck + ǫiLiH2




iU jH2 + hdijQ

iDjH1 + hℓijL

iEjH2


W 6Rp= λijkLiLjE

ck + λ′


ijkUci D

cjD

ck + ǫiLiH2

L violation results in small Majorana mass terms forneutrinos (∆L = 2).




iU jH2 + hdijQ

iDjH1 + hℓijL

iEjH2


W 6Rp= λijkLiLjE

ck + λ′


ijkUci D

cjD

ck + ǫiLiH2


We consider the trilinear λ′ijk-type couplings only.




iU jH2 + hdijQ

iDjH1 + hℓijL

iEjH2


W 6Rp= λijkLiLjE

ck + λ′


ijkUci D

cjD

ck + ǫiLiH2


We consider the trilinear λ′ijk-type couplings only.

λ′ijk

[

νiLdk

RdjL + dj

LdkRνi

L + (dkR)∗(νi

L)cdjL − ei

LdkRuj

L

−ujLdk

ReiL − (dk

R)∗(eiL)cuj

L

]

+ h.c.


Neutrino mass from λ′ijk-terms

Relevant terms : λ′ijk

[

djL dk

R νiL + (dk

R)∗ (νiL)c dj

L

]

+ h.c.

⊗dp d∗

p

νi dk νcj

(m1−loopν )ij ≃

3

8π2md

k mdp M

SUSY

1

m2q

λ′ipk λ′

jkp

Observed pattern of neutrino masses ⇒ λ′ ∼ 10−5 − 10−4 fora SUSY breaking mass scale ∼ 500 GeV.


Basic ideas...

Q: Does this mean

All trilinear 6Rp-couplings are destined to be so small?

Observation of any process, requiring large values(∼ 0.1 − 1.0) of some λ′-terms, necessarily indicate weneed additional mechanism to explain neutrino masspattern?


Basic ideas...

Q: Does this mean

All trilinear 6Rp-couplings are destined to be so small?

Observation of any process, requiring large values(∼ 0.1 − 1.0) of some λ′-terms, necessarily indicate weneed additional mechanism to explain neutrino masspattern?

A: No, so long as we can

Eliminate one-loop contributions, but

Allow two-loop ones,

through a limited number of large (∼ 0.1 − 1.0) λ′-terms.


Indications(?) of some large λ′’sLarge leptonic branching ratio of the Ds meson:

Expts : fDs= 273 ± 10 MeV. (Dobrescu and Kronfeld, arXiv:0803.0512

[hep-ph])

Lattice : fDs= 250 ± 15 MeV. (HPQCD Collab, arXiv:0706.1726

[hep-lat], Lubicz and Tarantino, arXiv:0807.4605 [hep-lat])

⇒ Large λ′223 for Ds → µν, λ′

323 for Ds → τν

(Kundu and Nandi, arXiv:0803.1898 [hep-ph])


Suppose such a hint comes insome other expt, even if Ds-data donot stay?


What we have to fit

sin2(2θ12) = 0.86+0.03−0.04, sin2(2θ23) > 0.92, sin2(2θ13) < 0.19

Normal hierarchy: m3 >> m2 & m1

m22 = (7.60 ± 0.35) × 10−5 eV2,

∣

∣m23 − m2

2

∣

∣ = (2.50 ± 0.27) × 10−3 eV2

Inverted hierarchy: m2 & m1 >> m3

m22 − m2

1 = (7.60 ± 0.35) × 10−5 eV2,∣

∣m22 − m2

3

∣

∣ ≃∣

∣m21 − m2

3

∣

∣ = (2.50 ± 0.27) × 10−3 eV2

Degenerate neutrinos: m1 ≃ m2 ≃ m3 ≃ O(10−1) eV


What we need?

A ‘minimal’ set of ‘large’ λ′’s so thatthere are no one-loop neutrinomasses.


Minimal set of λ′’s: PropertiesNo less than three λ′ type couplings, each with a differentleptonic index, for the three neutrinos.



Couplings like λ′ijj =⇒ generate diagonal entries of

neutrino mass matrix at one-loop.


1-loop masses from λ′ijj

Relevant terms : λ′ijj

[

djL dj

R νiL

]

and λ′ijj

[

(djR)∗ (νi

L)c djL

]

⊗dj d∗

j

νi dj νci

(m1−loopν )ii ≃

3

8π2md

j mdj M

SUSY

1

m2q

λ′ijj λ′

ijj




neutrino mass matrix at one-loop =⇒ forbidden.





Combinations like λ′ilkλ

′jkl =⇒ generate diagonal (i = j)

and off-diagonal (i 6= j) entries of neutrino mass matrix atone-loop.


1-loop masses from λ′ilkλ

′jkl comb.

Relevant terms : λ′ilk

[

dlL dk

R νiL

]

and λ′jkl

[

(dlR)∗ (νj

L)c dkL

]

⊗dl d∗

l

νi dk νcj


3

8π2md

k mdl M

SUSY

1

m2q

λ′ilk λ′

jkl







and off-diagonal (i 6= j) entries of neutrino mass matrix atone-loop =⇒ forbidden.









′jkm =⇒ generate one-loop masses

with the mass insertion δLRlm .


1-loop masses from λ′ilkλ

′jkm comb.

Relevant terms : λ′ilk

[

dlL dk

R νiL

]

and λ′jkm

[

(dmR )∗ (νj

L)c dkL

]

⊗dl d∗

m

νi dk νcj


3

8π2md

k mdl M

SUSY

1

m2q

λ′ilk λ′

jkm ∆LRlm

where ∆LRlm = δLR

lm /mlMSUSY.










with the mass insertion δLRlm =⇒ Cannot avoid having at

least one element of neutrino mass matrix with l = m =⇒

forbidden.










with the mass insertion δLRlm =⇒ Cannot avoid having at

least one element of neutrino mass matrix with l = m =⇒

forbidden.

The chosen non-zero λ′’s can lie in the range 0.1 − 1.


What we need?

We need at least 3 λ′’s, each with adifferent leptonic index, and 2 ∆LR’sin a minimal set.


The minimal set chosen

Choose λ′223 and λ′

323 in the minimal set =⇒ partially motivated byexplanation of results on Ds decays.





Possible choices for the third λ′ are: λ′112, λ′

121, λ′113, λ′

131, λ′123.






121, λ′113, λ′

131, λ′123.

Cannot choose λ′112

One-loop contributions to Mν(1, 2) and Mν(1, 3) driven by ∆LR13 .

Phenomenological constraints don’t allow it to be large.






121, λ′113, λ′

131, λ′123.




Similarly, cannot choose λ′131.






121, λ′113, λ′

131, λ′123.




Similarly, cannot choose λ′131.

If λ′123 is chosen, we have

No one-loop contributions.

A set of four independent parameters: λ′223, λ′

323, λ′123 and ∆LR

23 .

Difficult to fit Mν with experimental data.


The minimal set...



A set of five independent parameters: λ′223, λ′

323, λ′113, ∆LR

23 and∆LR

13 .

=⇒ We could fit Mν with experimental data.


The minimal set...




323, λ′113, ∆LR

23 and∆LR

13 .

=⇒ We could fit Mν with experimental data.

λ′121 could also be chosen



323, λ′121, ∆LR

23 and∆LR

21 .

=⇒ But this choice is severely restricted by absence of leptonic flavorviolating decays π0 → eµ, φ → eµ, B → e(µ)τ etc.


Constraints: [ λ′223 λ′

323 λ′113 ∆LR

13 ∆LR23 ]

λ′223 and λ′

323 . 1.

Recent CLEO constraint on Υ → µτ consistent with this upper

limit (arXiv:0807.2695 [hep-ex]).

|λ′113| ≤ 0.15 at 99% confidence level.

charged current universality, processes like π+ → e+νe

(V.D. Barger, G.F. Giudice and T. Han, Phys. Rev. D 40, 2987 (1989)).

∆LR13 < 5.2

From B0-B0 mixing.

∆LR23 < 1

From inclusive b → sγ branching ratio.


Two-loop diagrams: 1⊗

dp d∗p′

νi dk un dk′ νcj

W+, φ+, H+

Overall vertex factors: λ′ipk λ′

jk′p′ ∆LRpp′ Vundk

V ∗undk′

[Mν ]11 ∼mbMSUSY

(m2d − m2

b)∆LR

13 V ∗tsVtb md

[Mν ]ij ∼mbMSUSY

(m2s − m2

b)∆LR

23 V ∗tdVtb ms i, j = 2, 3


Two-loop diagrams: 2⊗dp d∗

p′

νi dk uk′ ℓcj νc

j

φ+, H+


jk′p′ ∆LRpp′ Vuk′dk

[Mν ]11 ∼ Vub

mbMSUSY

m2W

∆LR13

m2u

m2W

m2e

m2W

md

[Mν ]ii ∼ Vcb

mbMSUSY

m2W

∆LR23

m2c

m2W

m2ℓi

m2W

ms i = 2, 3


Two-loop diagrams: 3

dk d∗p′

νi ℓci

up dk′ νcj

W+, φ+, H+


jk′p′ ∆LRkp′ Vupdk′

We chose [λ′223 λ′

323 λ′113] =⇒ k = p′ = 3 =⇒ ∆kp′ = 1

[Mν ]11 ∼ V ∗ud

m2u

m2W

me [Mν ]ii ∼ V ∗cs

m2c

m2W

mℓii = 2, 3


Diagrams contributing...

Based on lepton/quark mass dependence

of loop integrals

For a given element of Mν, contributions

of diagrams of type 1 > type 3 > type 2.

For a given diagram, contribution to the

(i, j)-elements (i, j = 2, 3) of Mν are

more than the rest of the elements.


Diagrams not contributing: 1

νi χ0k

νcj

νi νj

×

×

×

bL

bR

bR

bL

bL,R

bL,R

×

×

bL

bR

bR

bL



These diagrams contribute only when there is amass-splitting between CP-even and CP-odd sneutrinos(F. Borzumati and J.S. Lee, Phys. Rev. D 66, 115012 (2002) [arXiv:hep-ph/0207184]).

We choose a mass ≃ 500 GeV for all sleptons.



×

×

×

νi νjc

ν

χ0j

bL

bRbR

bL

bL

νi νjc

dp

dk dn

ν

d∗p′

These diagrams involve λ′ilkλ

′jkl combination, one which

generates one-loop masses.

Some are induced by trilinear L-violating soft SUSYbreaking terms.


Values of parameters used

tan β = 10

Charged Higgs mass = 500 GeV

Squark mass (all) = 500 GeV

mt = 172.5 GeV

|Vtd| = (8.12 ± 0.88) × 10−3

|Vts| = (40.67 ± 1.3) × 10−3

|Vcb| = (40.8 ± 0.6) × 10−3

sin 2β = 0.755 ± 0.040 where β = arg(V ∗td)


Mixing matrix, θ23 = π/4, θ13 = 0

m1c2 + m2s

2 cs√2(−m1 + m2)

cs√2(m1 − m2)

cs√2(−m1 + m2)

1

2(m1s

2 + m2c2 + m3)

1

2(−m1s

2 − m2c2 + m3)

cs√2(m1 − m2)

1

2(−m1s

2 − m2c2 + m3)

1

2(m1s

2 + m2c2 + m3)



m1c2 + m2s

2 cs√2(−m1 + m2)

cs√2(m1 − m2)

cs√2(−m1 + m2)

1

2(m1s

2 + m2c2 + m3)

1

2(−m1s

2 − m2c2 + m3)

cs√2(m1 − m2)

1

2(−m1s

2 − m2c2 + m3)

1

2(m1s

2 + m2c2 + m3)

Look at (2, 2) and (3, 3) elements

Recall the dominant contributions;

[Mν ]22

∼ λ′223 λ′

223

mbMSUSY

(m2s − m2

b)

∆LR

23 V ∗tdVtb ms

[Mν ]33

∼ λ′323 λ′

323

mbMSUSY

(m2s − m2

b)

∆LR

23 V ∗tdVtb ms



m1c2 + m2s

2 cs√2(−m1 + m2)

cs√2(m1 − m2)

cs√2(−m1 + m2)

1

2(m1s

2 + m2c2 + m3)

1

2(−m1s

2 − m2c2 + m3)

cs√2(m1 − m2)

1

2(−m1s

2 − m2c2 + m3)

1

2(m1s

2 + m2c2 + m3)

Look at (2, 2) and (3, 3) elements

Recall the dominant contributions;

[Mν ]22

∼ λ′223 λ′

223

mbMSUSY

(m2s − m2

b)

∆LR

23 V ∗tdVtb ms

[Mν ]33

∼ λ′323 λ′

323

mbMSUSY

(m2s − m2

b)

∆LR

23 V ∗tdVtb ms

=⇒ λ′223 ≈ λ′

323.


Correlation plots for NH, θ13 = 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5

∆ 23

λ’223

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.1 0.2 0.3 0.4 0.5

∆ 13

λ’113

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5

λ’11

3

λ’223

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

∆ 13

∆23


Correlation plots for IH, θ13 = 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

∆ 23

λ’223

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

∆ 13

λ’113

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

λ’11

3

λ’223

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

∆ 13

∆23


Correlation plots for DN, θ13 = 0

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6

∆ 23

λ’223

0.5

1

1.5

2

2.5

3

3.5

4

0.4 0.5 0.6 0.7 0.8 0.9 1

∆ 13

λ’113

0.4

0.5

0.6

0.7

0.8

0.9

1

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6

λ’11

3

λ’223

0.5

1

1.5

2

2.5

3

3.5

4

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

∆ 13

∆23


Correlation plots for NH, θ13 = 10◦

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5

∆ 23

λ’223

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.1 0.2 0.3 0.4 0.5

∆ 13

λ’113

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5

λ’11

3

λ’223

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

∆ 13

∆23


ConclusionsLarge trilinear R-parity violating interactions are notnecessarily an impediment to the explanation of neutrinomasses and mixing.



One can reconcile large R-parity violating couplings andneutrino masses, if only a subset of all possible couplingsof such nature exist.




If there is indication of large couplings, the subset mustfurther be determined by the impossibility of generatingone-loop neutrino masses.




If there is indication of large couplings, the subset mustfurther be determined by the impossibility of generatingone-loop neutrino masses.

Two-loop contributions are tenable in such situations, andthey can fit the entire neutrino mass matrix answering tothe experimental constraints.


ConclusionsFor our choice of the minimal set with three λ′-couplingsand two squark flavour violating parameters, we couldreproduce the NH scenario.


ConclusionsFor our choice of the minimal set with three λ′-couplingsand two squark flavour violating parameters, we couldreproduce the NH scenario.

IH and DN cases could be reproduced either withadditional R-parity violating terms in the superpotential,or including the bilinear R-parity violating terms.


Thank You


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