Presented by
Mallika
Priyadarshini
Shivam
PHY14002
M.Sc 3rd sem
THE STANDARD MODEL
) Fundamental forces are mediated by photon, gluons, Wโs and Zโs (bosons)
Basic Ingredient are quarks
and the electron-like
objects (leptons
THE STANDARD MODEL
) It provides a unified
framework for 3 of 4
(known) forces of
nature.
SU(3)ร ๐๐(2) รU(1)
THE STANDARD MODEL
)
Strong (QCD)
SU(3)ร ๐๐(2) รU(1)
THE STANDARD MODEL
)
Electroweak
(=weak +QED)
SU(3)ร ๐๐(2) รU(1)
Neutrinos... Within Standard
Model Beyond Standard Model
Massless
Left handed
Three Flavours
๐๐ , ๐๐ , ๐๐
Neutrino Oscillations
๐(๐๐ โ ๐๐) = sin2 2๐ sin2[โ๐2๐ฟ
4๐ธ]
โ๐2 = ๐22 โ๐1
2
must be non-zero if neutrino oscillation exists.
BSM phenomena (seesaw) explains its tiny mass.
e
.
.
Whatโs 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
The Lagrangian for a free electron field ๐ณ(๐) is
๐ณ = ัฐ ๐๐ธ๐๐๐ โ ๐ ๐(๐)
Considering local symmetry
๐ณ(๐) โ ๐ณ/=๐โ๐๐ฝ ๐ ๐ณ ๐
โข ๐ณ ๐ ๐๐๐ณ ๐ = ๐ณ ๐ ๐๐๐ณ ๐ โ ๐๐ณ(๐)[๐๐๐ถ ๐ ]๐ณ(๐)
Not gauge invariant covariant derivative
Maxwellโs electromagnetic field appears due to the gauge invariance principle
๐ซ๐๐ณ = (๐๐ + ๐๐๐จ๐)๐ณ
๐จ๐ = ๐จ๐ +๐
๐๐๐๐ถ(๐)
ABELIAN CASE
Therefore the invariant lagrangian can be written as
๐ณ/ = ๐ณ๐๐ธ๐ ๐๐ + ๐๐๐จ๐ ๐ณ โ ๐๐ณ๐ณ
We add one kinetic energy term for the photon field
๐ณ = โ๐
๐๐ญ๐๐๐ญ๐๐
Therefore the final lagrangian is
๐ณ/ = ๐ณ๐๐ธ๐ ๐๐ + ๐๐๐จ๐ ๐ณ โ ๐๐ณ๐ณ โ๐
๐๐ญ๐๐๐ญ๐๐
The following features of the equation are--
The photon is massless as the term ๐จ๐ ๐จ๐is not Gauge
invariant.
The Lagrangian does not have a gauge field self
coupling.
Non Abelian gauge field
Under SU(2)
๐ฏ๐๐๐ ๐๐ ๐ ๐๐๐๐๐ ๐๐๐๐๐๐ ๐๐๐๐๐ ๐๐๐๐๐ ๐๐
The gauge field here transforms as
ัฐโฒ(๐) = ๐๐๐[โ๐๐.๐ฝ
๐]๐ณ(๐)
๐ซ๐ ๐ณ ๐ = ๐๐ โ ๐๐๐. ๐จ๐
๐๐ณ ๐
๐๐ โ ๐๐๐.๐จ๐
โฒ
๐๐ ๐ฝ ๐ณ ๐ = ๐(๐ฝ) ๐๐ โ ๐๐
๐.๐จ๐
๐๐ณ(๐)
๐จ๐๐โฒ = ๐จ๐
๐ + ๐บ๐๐๐๐ฝ๐๐จ๐๐ โ
๐
๐(๐๐๐ฝ๐)
๐ญ๐๐๐ = ๐๐๐จ๐
๐ โ ๐๐๐จ๐๐ + ๐๐บ๐๐๐๐จ๐
๐๐จ๐
๐
The complete gauge invariant lagrangian is
But we again got massless bosons because there is no mass term.
THEN HOW DO PARTICLES GET MASS???
๐ณ = ๐ณ ๐๐ธ๐๐ซ๐๐ณ โ ๐๐ณ ๐ณ โ๐
๐๐ญ๐๐
๐ ๐ญ๐๐๐
Higgs Field and Symmetry Breaking
The presence of particle masses in the Standard model Lagrangian is prohibited by the SU(2)L ร 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.
We consider the simple case of abelian U(1) Gauge theory
๐ณ = ๐ซ๐๐โ๐ซ๐๐ โ ๐๐๐โ๐ โ ๐(๐โ๐)๐ โ๐
๐๐ญ๐๐๐ญ๐๐
There will be two cases ๐๐ > ๐ ๐๐๐ ๐๐ < ๐.
But since we want to generate the mass we are interested
in ๐๐ < ๐
Shifting the origin to ๐๐(๐) = ๐, ๐๐(๐) = ๐,
And expanding the lagrangian in terms of ๐ผ and ฮพ
๐ =๐
๐(๐ + ๐ผ ๐ + ๐๐ ๐ )
Then the Lagrangian will be ๐ณ =๐
๐(๐๐๐)๐ +
๐
๐(๐๐๐ผ)๐ โ
๐๐๐๐ผ๐ +๐
๐๐๐๐๐๐จ๐๐จ๐ โ ๐๐๐จ๐๐๐๐ โ
๐
๐๐ญ๐๐๐ญ๐๐ + โฏ ๐จ๐ญ๐ก๐๐ซ ๐ญ๐๐ซ๐ฆ๐ฌ
To remove this Goldstone boson we need to make the
following Gauge corrections.
๐ =๐
๐[๐ + ๐ผ]๐๐๐/๐
And, ๐จ๐ = ๐จ๐ +๐
๐๐๐๐๐
So the final Lagrangian after these transformations
becomes
๐ณ =๐
๐(๐๐๐ผ)๐ โ ๐๐๐๐ผ๐ +
๐
๐๐๐๐๐๐จ๐๐จ๐ โ ๐๐๐ผ๐ โ
๐
๐๐๐ผ๐ +
๐
๐๐๐๐จ๐
๐ + ๐๐๐๐จ๐๐๐ผ โ
๐
๐๐ญ๐๐๐ญ๐๐
Thus we see
Massless vector boson + Goldstone boson = Massive
Vector Boson
This is called the Higgs mechanism
โข The symmetry we use here is the
SU(2)รU(1) Gauge symmetry.
โข Spontaneous symmetry breaking
makes SU(2)รU(1)โ ๐ผ(๐)๐๐
โข From SU(2), we get 3 gauge bosons
and from U(1) we get one Gauge Boson,
โข Higgs mechanism gives mass to 3 of the
4 Gauge bosons.
HIGGS
MECHANISM
IN THE
STANDARD
MODEL
Under SU(2)รU(1) local Gauge transformation
๐ โ ๐๐๐ฝ๐๐ป๐+๐
๐๐ถ๐
๐
Now the Lagrangian of Higgs field can be written as
๐ณ๐ฏ๐ฐ๐ฎ๐ฎ๐บ =๐
๐๐ซ๐๐
ฯฏ(๐ซ๐๐) โ ๐๐(๐+๐)
Where, we define
๐ซ๐ = (๐๐ โ ๐๐๐พ๐๐๐ป๐ โ
๐
๐๐/๐ฉ๐๐)
A simple and useful form of the Higgs field is ฮฆ=๐๐
To generate masses we need to give a fluctuation to a
ฮฆ= ๐
๐ + ๐ผ
We do in steps, first we don't take the fluctuation and
generate the gauge boson masses as follows
๐ซ๐ ๐๐
= (-ig๐พ๐๐๐ป๐-
๐
๐๐โฒ๐ฉ๐Y)
๐๐
= ๐ซ๐๐๐
= -i๐
๐
๐๐พ๐+
โ๐๐พ๐๐ + ๐,๐ฉ๐
๐
๐๐ซ๐๐
๐=
๐๐
๐(๐๐๐พ๐
+๐พ๐โ + โ๐๐พ๐
๐ + ๐/๐ฉ๐๐) = ๐๐
๐ ๐พ๐+ +
๐
๐๐๐๐
๐
Where we define,
๐ง =โ๐๐๐
๐+๐/๐ฉ๐
๐๐+๐/๐ and ๐พ๐
+๐พ๐๐ = ๐พ๐
๐๐พ๐๐ + ๐พ๐
๐๐พ๐๐
We generated the masses of 3 bosons which are ๐+
. , Z.
๐๐พยฑ =๐๐๐๐
๐ ๐๐ =
(๐๐+๐โฒ๐)๐๐
๐
๐ด๐ field is orthogonal to Z
๐จ๐=๐/๐พ๐
๐+๐๐ฉ๐
๐๐+๐/๐
where , sin ๐๐ค =๐/
๐2+๐/2 andcos ๐๐ค =
๐
๐2+๐/2
Since there is no Mass term for the ๐ด๐ field So photon
remains massless in this theory also.
โข Fermion masses
โข For Fermion masses we consider the interaction Lagrangian
๐ณ๐๐๐ = -๐ฎ๐(๐ณ ๐ฑ๐น + ๐น ๐ฑ+๐ณ)
โข ๐ณ๐ณฮฆ= ๐๐ ๐ ๐ณ
๐
๐ฑ๐ +๐(๐)
๐
๐ณ ๐ณฮฆ๐ณ๐น =๐ ๐ณ ๐ฑ๐ +๐(๐)
๐๐๐น
Similarly ๐ณ ๐น๐ฑ+๐ณ๐ณ= ๐ ๐น ๐ฑ๐ +๐(๐)
๐๐๐ณ
โข ๐ณ๐๐๐= -๐ฎ๐๐ฑ๐(๐ ๐ณ๐๐น + ๐ ๐น๐๐ณ)- ๐ฎ๐๐(๐)
๐(๐ ๐ณ๐๐น + ๐ ๐น๐๐ณ)
โข Thus electron acquires a mass m = ๐ฎ๐๐ฑ๐
โข 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 !!!!
Beyond Standard Model
But Why??
RIGHT HANDED NEUTRINOS
ARE INSERTED BY HAND..
We get three neutrino mass
termsโ
1. ๐ณ๐๐๐๐๐ซ =
๐
๐ (๐๐ซ๐ ๐น๐๐ณ +
๐๐ซ๐ ๐ณ๐๐๐น
๐ ) +h.c
2. ๐ณ๐๐๐๐๐ณ =
๐
๐๐๐ณ๐ ๐ณ
๐๐๐ณ + ๐. ๐
3. ๐ณ๐๐๐๐๐น =
๐
๐๐๐น๐ ๐น
๐ ๐๐น + ๐. ๐
. ๐ณ๐๐๐๐ = ๐ณ๐๐๐๐๐ซ + ๐ณ๐๐๐๐
๐ณ + ๐ณ๐๐๐๐๐น
= ๐ ๐ณ๐ ๐ ๐น
๐๐ณ ๐๐ซ
๐๐ซ๐ป ๐๐น
๐๐ณ
๐๐น๐
The above mass matrix is ๐ ๐๐ซ
๐๐ซ๐ป ๐๐น
๐๐ ๐๐ณ=0 .
After diagonalizing the matrix the following mass eigen states are obtained---
๐๐ โ ๐๐น โ ๐๐๐๐ ๐ฎ๐๐ฝ
๐๐ โ๐๐ซ
๐
๐๐น
๐๐ซ๐๐นโ๐๐๐ซ
๐ป =๐๐๐ร๐๐๐
๐๐๐๐ โ ๐. ๐ ๐๐ฝ
.INVERSE SEESAW MODEL
โข Here small neutrino masses arise as a result of new Physics at TeV scale .
โข May be probed at LHC , unlike TYPE I.
โข 3 right handed neutrinos ๐๐ + the three extra SM gauge singlet neutral fermions S + the three active neutrinos ๐๐ฟ
โข =1
2๐๐ฟ ๐๐
๐ ๐๐
0 ๐๐ท 0
๐๐ท๐ 0 ๐๐ ๐
0 ๐๐ ๐๐ ๐
๐๐ฟ๐
๐๐
๐
. A diagonalisation of the 9ร ๐ matrix leads to the
effective light neutrino mass matrix ie.
๐๐= ๐๐ซ๐ป ๐ด๐น๐บ
๐ป โ๐๐ ๐ด๐น๐บ
โ๐๐๐ซ๐ป
Or, ๐๐
๐.๐ ๐๐ฝ =
๐๐ซ
๐๐๐ ๐ฎ๐๐
๐ ๐
๐ ๐ฒ๐๐ฝ
๐ด๐น๐
๐๐ ๐ป๐๐ฝ
โ๐
Thus we see that Standard neutrinos with mass at sub ev scale are obtained for ๐๐ซ at electroweak scale and ๐ด๐ at Tev scale .
ISS is also called DOUBLE SEESAW .
24
Dark matter-connection
[1]R. N. Mohapatra and G. Senjanovic, Phys. Rev. Lett., 44, 912,
1980.
[2] Halzen, Francis, and Alan D Martin, โQuarks and
Leptonsโ,John Wiley & Sons(1984
[3] Moriyasu,K., โAn Elementary Primer for Gauge Theories,โ
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 1562โ1567
References