Neutrino Mass ( Inverse Seesaw Mechanism ) And
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Transcript of Neutrino Mass ( Inverse Seesaw Mechanism ) And
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Presented by
Mallika
Priyadarshini
Shivam
PHY14002
M.Sc 3rd sem
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THE STANDARD MODEL
) Fundamental forces are mediated by photon, gluons, W’s and Z’s (bosons)
Basic Ingredient are quarks
and the electron-like
objects (leptons
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THE STANDARD MODEL
) It provides a unified
framework for 3 of 4
(known) forces of
nature.
SU(3)× 𝑆𝑈(2) ×U(1)
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THE STANDARD MODEL
)
Strong (QCD)
SU(3)× 𝑆𝑈(2) ×U(1)
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THE STANDARD MODEL
)
Electroweak
(=weak +QED)
SU(3)× 𝑆𝑈(2) ×U(1)
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Neutrinos... Within Standard
Model Beyond Standard Model
Massless
Left handed
Three Flavours
𝜗𝑒 , 𝜗𝜇 , 𝜗𝜏
Neutrino Oscillations
𝑝(𝜗𝑒 → 𝜗𝜇) = sin2 2𝜃 sin2[∆𝑚2𝐿
4𝐸]
∆𝑚2 = 𝑚22 −𝑚1
2
must be non-zero if neutrino oscillation exists.
BSM phenomena (seesaw) explains its tiny mass.
e
.
.
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What’s meant by a gauge theory?
1.A theory described by a Lagrangian having local
symmetry properties (Invariant under local transformations)
2.Associated with each gauge symmetry is a conserved
quantity and a gauge field
[The symmetry is an internal symmetry in most gauge
theories]
Example: Electromagnetism
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The Lagrangian for a free electron field 𝜳(𝒙) is
𝑳 = Ѱ 𝒊𝜸𝝁𝝏𝝁 − 𝒎 𝝍(𝒙)
Considering local symmetry
𝜳(𝒙) → 𝜳/=𝒆−𝒊𝜽 𝒙 𝜳 𝒙
• 𝜳 𝒙 𝝏𝝁𝜳 𝒙 = 𝜳 𝒙 𝝏𝝁𝜳 𝒙 − 𝒊𝜳(𝒙)[𝝏𝝁𝜶 𝒙 ]𝜳(𝒙)
Not gauge invariant covariant derivative
Maxwell’s electromagnetic field appears due to the gauge invariance principle
𝑫𝝁𝜳 = (𝝏𝝁 + 𝒊𝒆𝑨𝝁)𝜳
𝑨𝝁 = 𝑨𝝁 +𝟏
𝒆𝝏𝝁𝜶(𝒙)
ABELIAN CASE
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Therefore the invariant lagrangian can be written as
𝑳/ = 𝜳𝒊𝜸𝝁 𝝏𝝁 + 𝒊𝒆𝑨𝝁 𝜳 − 𝒎𝜳𝜳
We add one kinetic energy term for the photon field
𝑳 = −𝟏
𝟒𝑭𝝁𝝑𝑭𝝁𝝑
Therefore the final lagrangian is
𝑳/ = 𝜳𝒊𝜸𝝁 𝝏𝝁 + 𝒊𝒆𝑨𝝁 𝜳 − 𝒎𝜳𝜳 −𝟏
𝟒𝑭𝝁𝝑𝑭𝝁𝝑
The following features of the equation are--
The photon is massless as the term 𝑨𝝁 𝑨𝝁is not Gauge
invariant.
The Lagrangian does not have a gauge field self
coupling.
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Non Abelian gauge field
Under SU(2)
𝑯𝒆𝒓𝒆 𝒘𝒆 𝒅𝒆𝒇𝒊𝒏𝒆 𝒗𝒆𝒄𝒕𝒐𝒓 𝒈𝒂𝒖𝒈𝒆 𝒇𝒊𝒆𝒍𝒅 𝒂𝒔
The gauge field here transforms as
Ѱ′(𝒙) = 𝒆𝒙𝒑[−𝒊𝝉.𝜽
𝟐]𝜳(𝒙)
𝑫𝝁 𝜳 𝒙 = 𝝏𝝁 − 𝒊𝒈𝝉. 𝑨𝝁
𝟐𝜳 𝒙
𝝏𝝁 − 𝒊𝒈𝝉.𝑨𝝁
′
𝟐𝐔 𝜽 𝜳 𝒙 = 𝐔(𝜽) 𝝏𝝁 − 𝒊𝒈
𝝉.𝑨𝝁
𝟐𝜳(𝒙)
𝑨𝝁𝒊′ = 𝑨𝝁
𝒊 + 𝜺𝒊𝒋𝒌𝜽𝒋𝑨𝝁𝒌 −
𝟏
𝒈(𝝏𝝁𝜽𝒊)
𝑭𝝁𝝑𝒊 = 𝝏𝝁𝑨𝝑
𝒊 − 𝝏𝝑𝑨𝝁𝒊 + 𝒈𝜺𝒊𝒋𝒌𝑨𝝁
𝒋𝑨𝝑
𝒌
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The complete gauge invariant lagrangian is
But we again got massless bosons because there is no mass term.
THEN HOW DO PARTICLES GET MASS???
𝑳 = 𝜳 𝒊𝜸𝝁𝑫𝝁𝜳 − 𝒎𝜳 𝜳 −𝟏
𝟒𝑭𝝁𝝑
𝒊 𝑭𝝁𝝑𝒊
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Higgs Field and Symmetry Breaking
The presence of particle masses in the Standard model Lagrangian is prohibited by the SU(2)L × U(1)Y gauge symmetry of the electroweak interaction.
The Higgs mechanism has been suggested which leads to spontaneous breakdown of the electroweak symmetry by condensation of a scalar Higgs field.
Particles acquire momentum (mass) by interacting with this field.
Particles that interact strongly with the Higgs field are heavy, while those that interact weakly are light.
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We consider the simple case of abelian U(1) Gauge theory
𝑳 = 𝑫𝝁𝝋∗𝑫𝝁𝝋 − 𝝁𝟐𝝋∗𝝋 − 𝝀(𝝋∗𝝋)𝟐 −𝟏
𝟒𝑭𝝁𝝑𝑭𝝁𝝑
There will be two cases 𝝁𝟐 > 𝟎 𝒂𝒏𝒅 𝝁𝟐 < 𝟎.
But since we want to generate the mass we are interested
in 𝝁𝟐 < 𝟎
Shifting the origin to 𝝋𝟏(𝒙) = 𝒗, 𝝋𝟐(𝒙) = 𝟎,
And expanding the lagrangian in terms of 𝜼 and ξ
𝝋 =𝟏
𝟐(𝒗 + 𝜼 𝒙 + 𝒊𝝃 𝒙 )
Then the Lagrangian will be 𝑳 =𝟏
𝟐(𝝏𝝁𝝃)𝟐 +
𝟏
𝟐(𝝏𝝁𝜼)𝟐 −
𝒗𝟐𝝀𝜼𝟐 +𝟏
𝟐𝒆𝟐𝒗𝟐𝑨𝝁𝑨𝝁 − 𝒆𝒗𝑨𝝁𝝏𝝁𝝃 −
𝟏
𝟒𝑭𝝁𝝑𝑭𝝁𝝑 + ⋯ 𝐨𝐭𝐡𝐞𝐫 𝐭𝐞𝐫𝐦𝐬
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To remove this Goldstone boson we need to make the
following Gauge corrections.
𝝋 =𝟏
𝟐[𝝑 + 𝜼]𝒆𝒊𝝃/𝝑
And, 𝑨𝝁 = 𝑨𝝁 +𝟏
𝒆𝝑𝝏𝝁𝝃
So the final Lagrangian after these transformations
becomes
𝑳 =𝟏
𝟐(𝝏𝝁𝜼)𝟐 − 𝒗𝟐𝝀𝜼𝟐 +
𝟏
𝟐𝒆𝟐𝒗𝟐𝑨𝝁𝑨𝝁 − 𝝀𝝑𝜼𝟑 −
𝟏
𝟒𝝀𝜼𝟒 +
𝟏
𝟐𝒆𝟐𝑨𝝁
𝟐 + 𝝑𝒆𝟐𝑨𝝁𝟐𝜼 −
𝟏
𝟒𝑭𝝁𝝑𝑭𝝁𝝑
Thus we see
Massless vector boson + Goldstone boson = Massive
Vector Boson
This is called the Higgs mechanism
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• The symmetry we use here is the
SU(2)×U(1) Gauge symmetry.
• Spontaneous symmetry breaking
makes SU(2)×U(1)→ 𝑼(𝟏)𝒆𝒎
• From SU(2), we get 3 gauge bosons
and from U(1) we get one Gauge Boson,
• Higgs mechanism gives mass to 3 of the
4 Gauge bosons.
HIGGS
MECHANISM
IN THE
STANDARD
MODEL
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Under SU(2)×U(1) local Gauge transformation
𝝋 → 𝒆𝒊𝜽𝒂𝑻𝒂+𝒊
𝟐𝜶𝒀
𝝋
Now the Lagrangian of Higgs field can be written as
𝑳𝑯𝑰𝑮𝑮𝑺 =𝟏
𝟐𝑫𝝁𝝋
ϯ(𝑫𝝁𝝋) − 𝝁𝟐(𝝋+𝝋)
Where, we define
𝑫𝝁 = (𝝏𝝁 − 𝒊𝒈𝑾𝝁𝒂𝑻𝒂 −
𝒊
𝟐𝒈/𝑩𝝁𝒀)
A simple and useful form of the Higgs field is Φ=𝟎𝒂
To generate masses we need to give a fluctuation to a
Φ= 𝟎
𝒂 + 𝜼
We do in steps, first we don't take the fluctuation and
generate the gauge boson masses as follows
𝑫𝝁 𝟎𝒂
= (-ig𝑾𝝁𝒂𝑻𝒂-
𝒊
𝟐𝒈′𝑩𝝁Y)
𝟎𝒂
= 𝑫𝝁𝟎𝒂
= -i𝒂
𝟐
𝒈𝑾𝝁+
−𝒈𝑾𝝁𝟑 + 𝒈,𝑩𝝁
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𝟏
𝟐𝑫𝝁𝝋
𝟐=
𝒂𝟐
𝟖(𝒈𝟐𝑾𝝁
+𝑾𝝁− + −𝒈𝑾𝝁
𝟑 + 𝒈/𝑩𝝁𝟐) = 𝒎𝒘
𝟐 𝑾𝝁+ +
𝟏
𝟐𝒎𝒛𝒛
𝟐
Where we define,
𝑧 =−𝒈𝒘𝝁
𝟑+𝒈/𝑩𝝁
𝒈𝟐+𝒈/𝟐 and 𝑾𝝁
+𝑾𝝁𝟏 = 𝑾𝝁
𝟏𝑾𝝁𝟏 + 𝑾𝝁
𝟐𝑾𝝁𝟐
We generated the masses of 3 bosons which are 𝑊+
. , Z.
𝒎𝑾± =𝒈𝟐𝒂𝟐
𝟖 𝒎𝒛 =
(𝒈𝟐+𝒈′𝟐)𝒂𝟐
𝟒
𝐴𝜇 field is orthogonal to Z
𝑨𝝁=𝒈/𝑾𝝁
𝟑+𝒈𝑩𝝁
𝒈𝟐+𝒈/𝟐
where , sin 𝜃𝑤 =𝑔/
𝑔2+𝑔/2 andcos 𝜃𝑤 =
𝑔
𝑔2+𝑔/2
Since there is no Mass term for the 𝐴𝜇 field So photon
remains massless in this theory also.
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• Fermion masses
• For Fermion masses we consider the interaction Lagrangian
𝑳𝒊𝒏𝒕 = -𝑮𝒆(𝑳 𝜱𝑹 + 𝑹 𝜱+𝑳)
• 𝜳𝑳Φ= 𝝑𝒆 𝒆 𝑳
𝟎
𝜱𝟎 +𝒉(𝒙)
𝟐
𝜳 𝑳Φ𝜳𝑹 =𝒆 𝑳 𝜱𝟎 +𝒉(𝒙)
𝟐𝒆𝑹
Similarly 𝜳 𝑹𝜱+𝜳𝑳= 𝒆 𝑹 𝜱𝟎 +𝒉(𝒙)
𝟐𝒆𝑳
• 𝑳𝒊𝒏𝒕= -𝑮𝒆𝜱𝟎(𝒆 𝑳𝒆𝑹 + 𝒆 𝑹𝒆𝑳)- 𝑮𝒆𝒉(𝒙)
𝟐(𝒆 𝑳𝒆𝑹 + 𝒆 𝑹𝒆𝑳)
• Thus electron acquires a mass m = 𝑮𝒆𝜱𝟎
• Thus STANDARD MODEL is a powerful synthesis that successfully explains all the masses of gauge bosons and
fermions, but failed in the problem of neutrino mass !!!!
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Beyond Standard Model
But Why??
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RIGHT HANDED NEUTRINOS
ARE INSERTED BY HAND..
We get three neutrino mass
terms—
1. 𝑳𝒎𝒂𝒔𝒔𝑫 =
𝟏
𝟐 (𝒎𝑫𝝑 𝑹𝝑𝑳 +
𝒎𝑫𝝑 𝑳𝒄𝝑𝑹
𝒄 ) +h.c
2. 𝑳𝒎𝒂𝒔𝒔𝑳 =
𝟏
𝟐𝒎𝑳𝝑 𝑳
𝒄𝝑𝑳 + 𝒉. 𝒄
3. 𝑳𝒎𝒂𝒔𝒔𝑹 =
𝟏
𝟐𝒎𝑹𝝑 𝑹
𝒄 𝝑𝑹 + 𝒉. 𝒄
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. 𝑳𝒎𝒂𝒔𝒔 = 𝑳𝒎𝒂𝒔𝒔𝑫 + 𝑳𝒎𝒂𝒔𝒔
𝑳 + 𝑳𝒎𝒂𝒔𝒔𝑹
= 𝝑 𝑳𝒄 𝝑 𝑹
𝒎𝑳 𝒎𝑫
𝒎𝑫𝑻 𝒎𝑹
𝝑𝑳
𝝑𝑹𝒄
The above mass matrix is 𝟎 𝒎𝑫
𝒎𝑫𝑻 𝒎𝑹
𝒂𝒔 𝒎𝑳=0 .
After diagonalizing the matrix the following mass eigen states are obtained---
𝒎𝟐 ≈ 𝒎𝑹 ≈ 𝟏𝟎𝟏𝟒 𝑮𝒆𝑽
𝒎𝟏 ≈𝒎𝑫
𝟐
𝒎𝑹
𝒎𝑫𝒎𝑹−𝟏𝒎𝑫
𝑻 =𝟏𝟎𝟐×𝟏𝟎𝟐
𝟏𝟎𝟏𝟒 ≈ 𝟎. 𝟏 𝒆𝑽
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.INVERSE SEESAW MODEL
• Here small neutrino masses arise as a result of new Physics at TeV scale .
• May be probed at LHC , unlike TYPE I.
• 3 right handed neutrinos 𝑁𝑅 + the three extra SM gauge singlet neutral fermions S + the three active neutrinos 𝜗𝐿
• =1
2𝜗𝐿 𝑁𝑅
𝑐 𝑆𝑐
0 𝑚𝐷 0
𝑚𝐷𝑇 0 𝑀𝑅𝑆
0 𝑀𝑅𝑆𝑇 𝜇
𝜗𝐿𝑐
𝑁𝑅
𝑆
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. A diagonalisation of the 9× 𝟗 matrix leads to the
effective light neutrino mass matrix ie.
𝒎𝝑= 𝒎𝑫𝑻 𝑴𝑹𝑺
𝑻 −𝟏𝝁 𝑴𝑹𝑺
−𝟏𝒎𝑫𝑻
Or, 𝒎𝝑
𝟎.𝟏 𝒆𝑽 =
𝒎𝑫
𝟏𝟎𝟎 𝑮𝒆𝒗
𝟐 𝝁
𝟏 𝑲𝒆𝑽
𝑴𝑹𝒔
𝟏𝟎 𝑻𝒆𝑽
−𝟐
Thus we see that Standard neutrinos with mass at sub ev scale are obtained for 𝒎𝑫 at electroweak scale and 𝑴𝒔 at Tev scale .
ISS is also called DOUBLE SEESAW .
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24
Dark matter-connection
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[1]R. N. Mohapatra and G. Senjanovic, Phys. Rev. Lett., 44, 912,
1980.
[2] Halzen, Francis, and Alan D Martin, “Quarks and
Leptons”,John Wiley & Sons(1984
[3] Moriyasu,K., “An Elementary Primer for Gauge Theories,”
World Scientific, (1983)
[4] S. F. King, arXiv:hep-ph/0208266.
[5] Carlo Giunti, arXiv:hep-ph/020572
[6] G. Altarelli and F. Feruglio, arXiv:hep-ph/0206077
[7]Y Fukuda et al. 1998 Evidence for oscillation of atmospheric
neutrinos Phys. Rev. Lett. 81 1562–1567
References