Beyond Standard Model from Neutrinoskimcs.yonsei.ac.kr/.../seminar/2017a_schedule/10-2.pptx · 2017. 5. 31. · Beyond Standard Model from Neutrinos Sunhaeng Hur Yonsei University

Beyond Standard Modelfrom Neutrinos

Sunhaeng Hur

Yonsei University

May 30, 2017

Contents Main Body Conclusion

Contents

I The importance of neutrinoI Dirac vs Majorana massI Mixing matrix about quarksI Standard modelI Problem of Standard ModelI Various mechanism and models

Sunhaeng Hur BSM from neutrino


The importance of neutrino

1. Are neutrinos with definite masses Majorana or Dirac particles?

2. What is the value of the CP phase?

3. What is the character of the neutrino mass spectrum?Is it normalwithsmaller mass-squared difference between lighter neutrinos or inverted withsmaller mass squared difference between heavier neutrinos?

4. What are the absolute values of the neutrino masses?

5. Is the number of massive neutrinos equalto the number of the flavorneutrinos (three) or larger than three?In other words do so called sterileneutrinos exist?



Dirac vs Majorana masss

The Dirac equation(i γµ∂µ − m)ψ = 0

for a fermion fieldψ = ψL + ψR

is equivalent to the equations

{ i γµ∂µψL = mψRi γµ∂µψR = mψL

for the chiralfields ψL and ψR , whose space-time evolutions are coupled by themass m.If a fermion is massless, the two equations are decoupled:

{ i γµ ∂µψL = 0i γµ∂µ ψR = 0

Hence, a massless fermion can be described by a single chiralfield (left-handedor right-handed), which has only two independent components.The equationsare called the Weylequations and the spinors ψL and ψR are called Weylspinors.



This means that the two equations must be two ways of writing the sameequation for one independent field, say ψL. In order to obtain i γµ ∂µψL = mψRfrom i γµ∂µ ψR = mψL, we take the Hermitian conjugate of i γµ∂µ ψR = mψLand we multiply it on the right with γ0. By using the property inγ0γµ† γ0 = γµ of Dirac matrices, we obtain

−i∂µψRγµ = mψL

Now, in order to obtain the same structure asi γµ∂µψL = mψR , we take thetranspose of −i ∂µψRγµ = mψL and multiply on the left with the chargeconjugation matrix C.Using the defining property in CγT

µ C−1 = −γµ of C, wefinally obtain

i γµ∂µCψRT = mCψL

T

This equation has the same structure as i γµ ∂µψL = mψR and we can considerthem as identicalif we set

ψR = ξCψLT

where ξ is an arbitrary phase factor (|ξ|2 = 1). This is the Majorana relationbetween ψL and ψR , which makes sense, because CψL

T is right-handed:usingCγ5T C−1 = −γ5 we have PLC = CPT

L , which leads to

PL(CψLT ) = C(ψLPL)T = C[(PRψL)†γ0]T = 0



From iγµ ∂µψL = mψR and ψR = ξCψLT we obtain the Majorana equation for

the chiralfield ψL:i γµ ∂µψL = mξCψL

T

We can eliminate the phase factor ξbyrephasingthefield ψL as

ψL → ξ12 ψL

Then the Majorana equation for the rephased chiralfield ψL reads

i γµ∂µ ψL = mCψLT

The Majorana condition for the field ψ in ψ = ψL + ψR,

ψ = ψL + ψR = ψL + CψLT

isψ = CψL

T



Summarize

I Lorentz Invariance allows two kinds of mass terms for fermions:ψLψR orψT

L C−1ψL(or L↔R)I Note under a symmetry transformation ψ → eiα ψ, the first mass is

invariant whereas the second term is notI Fermions only with the first kind of mass are called Dirac fermions and

those with both kinds are called Majorana fermionsI Dirac fermion unlike Majorana requires an extra symmetry:e.g.for

e,µ,q..,extra symmetry is U(1)em; since Q(ν) = 0, no such symmetry isthere for ν

I Hence for neutrinos, Majorana-ness is more natural; also smallmass iseasier for Majorana neutrino.

I For Majorana mass, ν = ν



Mixing matrix about quarksTo begin with, we have to know Uckm which is matrix to explain quarks mixing.The fermion charged current jρ

W is the sum of the leptonic and quark chargedcurrents,

jρW = jρW ,L + jρW ,Q

given byjρW ,L = 2(ν0

eLγρe0

L + ν0µL γρµ0

L + ν0τLγρτ0

L)jρW ,Q = 2(u0

Lγρd0

L + c0Lγ

ρs0L + t0Lγ

ρb0L)

jρW ,Q also can be expressed by another way

q0UL ≡

u0

Lc0

Lt0L

, q0UR ≡

u0

Lc0

Rt0R

, q0DL ≡

d0

Ls0L

b0L

, q0DR ≡

d0

Rs0R

b0R

Also, where VDL ,VDR ,VU

L , and VUR are four appropriate 3×3 unitary matrices.

Defining

qUL = VU†

L q0UL ≡

uLcLtL

, qUR = VU†R q0U

R ≡

uLcRtR

qDL = VD†

L q0DL ≡

dLsLbL

, qDR = VD†R q0D

R ≡

dRsRbR



Then, the quark weak charged current is written in the matrix formjρW ,Q = 2q0U

L γρq0DL

Now using VDL ,VDR ,VU

L , and VUR :

jρW ,Q = 2qUL V U†

L γρV DL qD

L = 2qUL γρV U†L V D

L qDL

Hence, the quark weak charged current does not depend separately on thematrices VUL and VD

L ,but only on their productV = V U†

L V DL

The matrix V is the quark mixing matrix, also called theCabibbo-Kobayashi-Maskawa(CKM) matrix, which embodies the physicaleffects of quark mixing.

The understanding of quark mixing is usefulin neutrino physics because thetreatment of neutrino mixing is analogous to that of quark mixing in the case ofDirac neutrinos and follows similar methods in the case of Majorana neutrinos.



The CKM mixing matrix of quarks is a unitary 3×3 matrix given by

V = V U†L V D

L =

V11 V12 V13V21 V22 V23V31 V32 V33

All properties of the quark mixing matrix V can be extended to the leptonmixing matrix U in the case of mixing of three Dirac neutrinos,with thereplacements

V → U = V l †L V ν

L =

U11 U12 U13U21 U22 U23U31 U32 U33

≡

Ue1 Ue2 Ue3Uµ1 Uµ2 Uµ3Uτ1 Uτ2 Uτ3



Physicalparameters in the mixing matrix

In general, a unitary N×N matrix depends on N2 independent realparameters.These parameters can be divided into

N(N − 1)2 mixing angles,

andN(N + 1)

2 phases.

Hence, the quark mixing matrix with N=3 can be written in terms of threemixing angles and six phases.However, not allthe phases are physicalobservables, because the only physicaleffect of the quark mixing matrix occursthrough its presence in the weak charged current of quarks

jµW ,Q = 2qU

L γµVqDL



Apart from the weak charged current, the Lagrangian is invariant under globalphase transformations of the quark fields of the type

qUα → eiψ U

α qUα , qD

k → e iψ Dk qD

k

with α = u, c, t and k = d, s, b.Performing this transformation, the quarkcharged current becomes

jµW ,Q = 2

Xα=u,c,t

Xk=d,s,b

qUαL γµe−iψ U

α Vαk eiψ Dk qD

kL

which can be written asjµW ,Q = 2 e−i(ψ U

c −ψ Ds )

| {z }1

Xα=u,c,t

Xk=d,s,b

qUαL γµ e−i(ψ U

α −ψ Uc )

| {z }N−1=2

Vαk ei(ψ Dk −ψ D

s )qDkL| {z }

N−1=2

where we have factorized an arbitrary phase e−i (ψUc − ψD

s ) and we haveindicated the number of independent phases in each term.



From this expression, it is clear that there are1 + (N − 1) + (N − 1) = 2N − 1 = 5

arbitrary phases of the quark fields that can be chosen to eliminate five of thesix phases of the quark mixing matrix.The reason why 2N-1 and not 2Nphases of the mixing matrix can be eliminated is that a common rephasing ofall the quark fields leaves the quark charged current invariant.Such aninvariance is related, through Noether’s theorem, to the conservation of baryonnumber.Thus, the quark mixing matrix contains

N(N + 1)2 − (2N − 1) =(N − 1)(N − 2)

2 = 1 physicalphase,



Parameterization of the mixing matrix-Two generationsA 2 × 2 unitary matrix can be written as

V = cosθeiω1 sinθe(i ω2+η)

−sinθe(i ω1−η) cosθeiω2

in terms of one mixing angle θ and three phases ω1, ω2, η.The equivalent form

V = ω1 00 ω2

ei η 00 1

cosθ sinθ−sinθ cosθ

e−iη 0

0 1shows explicitly that allthree phases can be eliminated by rephasing the quarkfields.Indeed, in the quark charged weak current

jρW ,Q = 2qU

L γρVqDL

for two generations we have

qUL = uL

cL, qD

L = dLsL

,

The three phases ω1, ω2, η in V can be eliminated by the rephasinguL → e i (ω1+η)uL, cL → e iω2cL, dL → e i ηdL



Hence, it is convenient to write the mixing matrix V without the unphysicalphases:

V = cosθc sinθc−sinθc cosθc

where the mixing angle θc is the Cabibbo angle.Note, however, that the choice of parameterization of the two-generation quarkmixing matrix is not unique:one could choose, for example,

V = cosθ0c sinθ0c−sinθ0c cosθ0c

or V = sinθ00c cosθ00

c−cosθ00

c sinθ00c

All the different parameterizations are related(for example,θ0

c = −θc, θ00c = φ

2 − θc) and give the same physicalresults.In the case of three generations, as one can imagine, the situation is morecomplicated because there are more possible parameterizations.A large degreeof arbitrariness comes from the many possible choices of the phases of theelements of the mixing matrix, which correspond to only one physicalphase.



Standard Model

Quarks and leptons are elementary particles to constitute matter.They areexpressed as

Lepton doubletL i =

νi

ei

!∼ (1c, 2)−1; DDDµ L i = (∂µ + iWµ − i

2Bµ )L i

Lepton singletei ∼ (1c, 2)2; DDDµ ei = (∂µ + iBµ )ei

Quark doubletQQQi =

ui

di

!∼ (3c, 2)1

3; DDDµQQQi = (∂µ+iGµ+iWµ + i

3Bµ )Qi

Quark singletui ∼ (3c, 1)− 43; DDDµui = (∂µ − iG∗

µ − 2i3 Bµ )ui

Quark singletdi ∼ (3c, 1)23; DDDµdi = (∂µ − iG∗

µ + i3Bµ )di

νi do not have a right handed neutrinos!



Problem of SM

In the Standard Model, massive spin12 particles allhave corresponding Dirac

mass terms in the totalLagrangian.These terms always couple the left handedand right handed fields of the massive particle.It is assumed in the StandardModelthat only left handed neutrinos exist, and therefore there can be noDirac mass terms for neutrinos.If we were, however to introduce three righthanded, singlet fields similar to those of the charged leptons, we can stillgenerate mass by putting higgs doublet

φ = φaφb

→ φ =r

12

0γ for the minimum φ

Then, the neutrinos masses is given by

L Y = − νe e Lφaφb

νR − νR φa φbνee L

and upon breaking the symmetry we get:

− γ√ 2(νLνR + νRνL)



We can see the appearance of a Dirac Mass term just as we saw for thecharged leptons.However, upon inserting the right handed singlets, we end upunintentionally breaking the globalSU(2) symmetry of select other terms in thetotalLagrangian.We therefore lose globalB-L number.

Thus, to introduce right-handed neutrino, Standard Modelmust be extended!It is not the matter of the order of mass of right-handed neutrino.



Various mechanism and models - Seesaw mechanism

This modelproduces a light neutrino, for each of the three known neutrinoflavors, and a corresponding very heavy neutrino for each flavor, which has yetto be observed.The simple mathematicalprinciple behind the seesaw mechanism is thefollowing property of any 2 × 2 matrix

A = 0 MM B

where B is taken to be much larger than M.It has two very disproportionate eigenvalues:

Λ± = B ±√

B2 + 4M2

2The larger eigenvalue, λ+, is approximately equalto B, while the smallereigenvalue is approximately equalto

λ− ≈ −M2

B



Thus, | M | is the geometric mean of λ+ and -λ− , since the determinant equalsλ+λ− = −M2

If one of the eigenvalues goes up, the other goes down, and vice versa.This isthe point of the name "seesaw" of the mechanism.This mechanism serves to explain why the neutrino masses are so small.Thematrix A is essentially the mass matrix for the neutrinos.The Majorana masscomponent B is comparable to the GUT scale and violates lepton number;while the components Dirac mass M, are of order of the much smallerelectroweak scale, the VEV below.



Various mechanism and models - loop model

Loop models populate weak scale with many new particles e.g.singly anddoubly charged bosons- not motivated by any other physics; then one cangenerate two loop Majorana masses that are small.



Reference

I Carlo Giunti, Chung W. Kim Fundamentals of Neutrino Physics andAstrophysics

I Pierre Ramond, TASI: Standard ModelNeutrino MassesI Thomas Campbell, Massive Neutrinos and the See Saw MechanismI R. N. Mohapatra, Physics of Neutrino Mass



THANK YOU!


Documents

Beyond Standard Model from Neutrinoskimcs.yonsei.ac.kr/.../seminar/2017a_schedule/10-2.pptx · 2017. 5. 31. · Beyond Standard Model from Neutrinos Sunhaeng Hur Yonsei University