An Effective field theory for non-relativistic Majorana ... · AnEffectivefield theory for...

An Effective field theory for

non-relativistic Majorana neutrinos

Simone Biondini

in collaboration with N. Brambilla, M. Escobedo, A. Vairo

BLV 2013 ,MPIK, HeidelbergGermany, 8th April

S. Biondini (TUM, T30f) EFT and Cosmology Heidelberg-MPIK, 8th April 1 / 24

Outline

1 Introduction and Motivation

2 Effective field theories and Majorana fermions

3 Finite temperature corrections

4 Conclusions


Introduction and Motivation

A pair of open problems in cosmology

Dark Matter...

84% of the matter in the Universe is believed to be Dark Matter

we need a suitable Dark Matter candidate in agreement with cosmologicalconstraints

QX = 0 non baryonic, stable MX 6= 0

Standard Model neutrinos ruled out..no galaxies clustering

Baryon Asymmetry...

the Standard Model and Standard Cosmology are not able to explain theBaryon Asymmetry in the Universe

Y thB ≪ Y exp

B

look for a dynamical process to generate such an asymmetry: Baryogenesis

⇒ physics beyond Standard Model is required



Dark Matter candidates

Why Dark Matter?

amount of ordinary matter is not able to explain the observed gravitationaleffects: rotational curves, galaxies formation ... ⇒ Dark Matter (?)

Some examples...

neutralino (SUSY)being the LSP → stable

gravitino (SUSY)being the LSP → stable

heavy neutrinos (See-Saw Type I)weakly coupled → stable

e−

e+

e

χ01

χ01

e+

e

d u

χ01

G

N

ℓ

φ

Common feature: Majorana fermions

ψM = (ψM)C where ψCM = Cγ0ψ∗ = iγ2ψ

∗



Baryon Asymmetry in the Universe

Experimental evidences

Cosmic rays: NP/NP is consistent with secondary process p + p → 3p + p

Matter-Antimatter in cluster of galaxies: detectable background ofγ-radiation, NOT DETECTED

The BAU is accurately determined by CMB and Anisotropy measurement

Y CMBB =

nb − nbs

= (8.75± 0.23)× 10−11

Starting with Y = 0 Standard Cosmology gives: YB ≃ 10−18 ≪ Y CMBB

Sakharov conditions, A. Sakharov (1967)

1 baryon number (B) violation

2 C and CP violation

3 processes out of thermal equilibrium



Baryogenesis via Leptogenesis

CP violation in quark sector is not enough, very high Trh

SPHALERONS : Baryons ⇆⇆⇆ Leptons

B and L well conserved at low temperature regime

T > TEW transition between vacua of non Abelian Gauge Theory (SU(2))

∆B = ∆L = nf∆Nv

100GeV ≤ T ≤ 1012 GeV: sphaleron transitions activated

due to sphalerons properties a Baryon Asymmetry can be generated

B = C · (B − L) , L = (C − 1) · (B − L)

Leptogenesis mechanism: L 6= 0 ⇒ (B − L) 6= 0 ⇒ B 6= 0

Look for lepton number violating processes...



Sterile Neutrinos Lagrangian

ν oscillation experiment ⇒ small mass to ν , but how?

Standard Model Extension

N singlet fermions NI (I = 1, ...,N ) MN1 ≤ MN2 ... ≤ MN

Q = 0 ; IW = 0 ; Y = 0 → sterile particles

renormalizable Lagrangian with Dirac-Majorana mass term

L = LSM + i NI∂µγµNI −

(

FαI LαNI Φ−MI

2Nc

I NI + h.c .

)

R. N. Mohapatra and G. Senjanovic (1981);M. Gell-Mann, P. Ramond and R. Slansky (1979)

Majorana mass → Lepton number violation

FαI Yukawa couplings: complex phases → CP violation source

out of equilibrium due to weak couplings



Leptogenesis M. Fukugita and T. Yanagida (1986)

Full fill Sakarov conditions (B → L), example: Sterile Neutrinos

(1) : N → ℓαφ† , (1) : N → ℓαφ ⇒ δℓ =

Γ(1) − Γ(1)

ΓTot

Different scales: Mi , T , ∆M , EW ... thermal environment

General: Ni decays are efficient in the regime T < MX

⇒ Possible hierarchy scale: M >> T >> EW → Effective field theory?


Effective field theories and Majorana fermions

Setting up the tools

Dealing with problems involving more than one energy scale:

Effective Field Theories

1 a hierarchy of energy scales: separation of the scales, e.g. T << M2 identify the dynamical scale (T ) and integrate out high energy modes (M)3 organize an expansion of the operators in terms of

T

M→ power counting

4 dimensional analysis helps in building the effective Lagrangian

LFT → LEFT =∑

i

ciOn

i

Mn−4

EFT strategy

identify the symmetries of the low energy Lagrangian

identify the suitable degrees of freedom, ingredients of your system

write down the low energy Lagrangian exploiting the hierarchy of the scales



Defining the problem: thermal decay rate

Our physical system and degrees of freedom

hot plasma of SM particles at T ≫ EW GeV: mi ≪ T and ~pi ∼ T

Majorana neutrinos (N ,M) are almost not affected by T, being M >> T

⇒ N described by non-relativistic fields, Poincare symmetry

Different approaches:

1 Consider directly thermal field theory (ITF) without exploiting M >> T

M.Laine and Y. Schroder (2012), A.Salvio, P. Lodone and S. Strumia (2011)

complete two loops computation in the high energy theorymany term ∼ e

−M/T

2 EFT for heavy Majorana neutrinos

computation at T=0 via one loop diagrams M ≫ T hence T → 0thermal effects as correction via simple tadpole diagrams (RTF)



Non-relativistic Majorana fermions

A Majorana fermion full fills

ψM = (ψM)C ⇒ self-conjugate spinor

the relativistic propagator of a free Majorana particle are as follows

〈0|T{

ψa(x)ψb(y)}

|0〉 =

∫

d4p

(2π)4(/p +M)ab

p2 −M2 + iǫe−ip(x−y)

〈0|T {ψa(x)ψb(y)} |0〉 = −i

∫

d4p

(2π)4(/p +M)abC


〈0|T{

ψa(x)ψb(y)}

|0〉 = −i

∫

d4p

(2π)4C (/p +M)ab


In the low energy theory one needs

non-relativistic Majorana spinors + the non-relativistic propagator



Non-relativistic Majorana fermions

N =(

1+γ0

2

)

N +(

1−γ0

2

)

N = N< + N>

projector properties

N =(

1+γ0

2

)

N< +(

1−γ0

2

)

N>

hermitian conjugate expression and N = iγ2N†

N =(

1−γ0

2

)

iγ2N†< +

(

1+γ0

2

)

iγ2N†>

comparing one gets no disentanglement between particle and anti-particle

N< = iγ2N†> , N> = iγ2N

†<

N< contains only annihilation operator like other known EFTs and{

Na<(~x , t),N

b†< (~y , t)

}

= δ3(x − y) δab

{

Na<(~x , t),N

b<(~y , t)

}

={

Na†< (~x , t),Nb†

< (~y , t)}

= 0



EFT for heavy Majorana neutrinos

EFT strategy:

1 Poincare invariance, Gauge invariance

2 Non-relativistic spinors for Majorana neutrinos: N< ≡ N (|~p| ≪ M)

3 Low energy Lagrangian (Relevant operators for the Leptogenesis problem)

LEFT = N†∂0N +A

MN†Nφ†φ+

B

M3N†NψD0ψ +

C

M3N†NF 2 + ...

where φ is the Higgs doublet, ψ are fermions, F 2 ≃ (∂0Ai∂0Ai ) gauge bosons

Thermal correction of each term through dimensional analysis:

δΓ(N)φ ∝T 2

M, δΓ(N)ψ ∝

T 4

M3, δΓ(N)F ∝

T 4

M3

A, B, C called matching coefficients

the power counting + M ≫ T ⇒ expansion under control



EFT diagrams

the effective Lagrangian produces the following diagrams

LEFT = N†∂0N +A

MN†Nφ†φ+

B

M3N†Nψ∂0ψ +

C

M3N†NF 2 + ...

N

N

NN

N

N

φ ψ

Aµ

ψ

Aµ

φ†

A

C

B



The matching computation: an example

LEFT ,φ =a

MN†Nφ†φ+

b

M3N†ND0φD0φ

Lorentz gauge ⇒ determine a clear structure in powers of qµ

manifest contributions to each effective vertex: ∝ q0 or ∝ q2

+ + + +

=a b

+

High energy green function = EFT green function

Im(a) = −i3

8π|F |2λ , Im(b) = −i

5

32π(3g 2 + g ′2)|F |2


Finite temperature corrections

From T=0 to thermal corrections

We get the result for a as follows

a = −i3

8

|F |2λ

π⇒ LNNφφ = −i

3

8

|F |2λ

πMN†Nφ†φ

N N

φ

φ

N

N

φ

1 1 1

1 12

RTF + heavy particles ⇒ only type 1 (N. Brambilla et al (2008))

In a hot plasma particles are thermally excited ⇒ propagators affected by

i∆11(x − y) =

∫

d4K

2π4

[

i

K 2 −m2 + iǫ+ 2πnB(|k0|)δ(K

2 −m2)

]

e−iK(x−y)



Thermal production rate at O(T2

M2)

Thermal correction:

the first term: mφ ⇒ no scale ⇒ 0 in D-regularization

the second term: finite contribution → thermal correction

N N

φ

= − |F |2λπM

38

∫

d4K2π4 2πnB(|k0|)δ(K 2 −m2)

T is entering in Bose-Einstein distribution (mφ = 0)

nB = 1ek/T−1

⇒∫∞

0dk k

ek/T−1, k = xT

Hence one gets

ΓN(T ) =M |F |2

8π

[

1− λT 2

M2+O

(

T

M

)4]

, δΓ(N)φ ∝T 2

M



Thermal correction at O(T4

M4 )

Thermal corrections:

Add the contribution of gauge boson and fermions

Aµ , Bµ , ℓα , να , t , b

Consider the Fermi-Dirac thermal distribution for fermions

N N

Aµ

N N

ψ

Finally we get the decay rate with thermal corrections

M. Laine, Y. Schroder (2012)

Γ =|F |2M

8π

{

1− λ

(

T

M

)2

−π2

80

(

T

M

)4

(3g 2 + g ′2)−7π2

60

(

T

M

)4

|λtb|2

}



What is next? Lepton asymmetry

thermal effects in the N decays may be important for leptogenesis

get in touch with an observable

YB ≃135ζ(3)

4π2× ǫ× η × C

G.F. Giudice, A.Notari, M. Raidal, A. Riotto, A. Strumia (2004)

EFT formalism → Thermal corrections to lepton asymmetry

ǫ→ ǫ(T )

N

ℓα

φ

→



Thermal correction to lepton asymmetry,resonant case

Leading thermal correction

L =a′

MN†Nφ†φ+

∑

i

ciMn

Oi , n > 2

the definition of lepton asymmetry from N decays is

ǫ =∑

i ,α

Γ(Ni → ℓα)− Γ(Ni → ℓα)

Γ(Ni → ℓα) + Γ(Ni → ℓα)= 2

Im(B)Im[

(F1F∗2 )

2]

|F1|2

+

1) 2)

thermal corrections in Im(B) due to the thermal Higgs (tadpole as before)

⇒ ǫ(T ) ≃ ǫ[

1 + λ(

TM

)2]


Conclusions

Conclusions

Dark matter and Leptogenesis as open problems in Cosmology

Majorana particles involved

an EFT for non-relativistic Majorana fermions is built

focus on heavy neutrinos considered so far

the thermal decay rate for one kind of heavy neutrino is reproduced

the leading thermal correction to the lepton asymmetry via EFTs (work inprogress)


Conclusions

Backup slides


Conclusions

Thermal production of Dark Matter (X )

Thermal approach to the Dark Matter problem

X produced in a hot dense plasma at high temperature

Cold Dark Matter ≃ O(100) GeV or warm Dark Matter ≃ O(10)KeV

decays of X ′ play an important role in the regime T < M ′

⇒ finite temperature treatment,T 6= 0


Conclusions

Low energy Lagrangian and Wilsoncoefficient

the low energy Lagrangian in more details reads

Lint =aM3N

†Nφ†φ+ bM3N

†ND0φ†D0φ+ c1

(

NaR Lα) (

iD0LβaLN†)

+

c2M3

(

NaRγµγν Lα) (

iD0LβγνγµaLN

†)

+ c3M3N

†N(

taLγ0 iD0t

)

+

c4M3N

†N(

qaLγ0 iD0q

)

+ d1M3 tr

{

T aT b}

N†NF a0iF

b0i +

d2M3N

†NW0iW0i

the Wilson coefficients are

decay rate ⇒ imaginary part

a = −i 38π |F |

2λ , b = −i 532π (3g

2 + g ′2)|F |2

c1 = i 38π |λtb |

2|F |2 − i 316π (3g

2 + g ′2)|F |2 , c2 = i 1384π (3g

2 + g ′2)|F |2

c3 = −i 124π |λtb|

2|F |2 , c4 = −i 148π |λtb|

2|F |2

d1 = −i 148πg

2|F |2 , d2 = −i 196πg

′2|F |2 |F |2 = F1,αF∗1,α


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