Marek Zrałek University of Silesia Katowice Workshop on Discrete Symmetries and Entanglement 10 06 2017 Kraków Outline Introduction Discrete symmetries in Space Time and charge c ID: 655425
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Slide1
Discrete Symmetries inFundamental Interaction
Marek ZrałekUniversity of Silesia, Katowice
Workshop on
Discrete Symmetries and Entanglement
10. 06. 2017,
KrakówSlide2
Outline Introduction Discrete symmetries in Space Time and charge conjugation symmetry Discrete symmetries in the Standard Model Beyond the Standard Model ConclusionsSlide3
Discrete symmetries play a key role in developing theories and models of basic interactions in nature. The current theory of elementary interactions - the Standard Model (SM) - does not answer a number of questions, so there is a widespread belief that this is only an effective theory and must be expanded. From the experimental point of view, to study future new interaction it is necessary in the precise way to understand, how discrete symmetries work in the present theory. In the lecture the C, P, T, CP, and CPT symmetries of the SM interactions are discussed by examining the symmetry transformations for the base fields of irreducible Lorentz group representations with spin 0, spin 1, as well as the left - and right - handed with spin ½. The quark and lepton sectors of the theory, with Dirac and Majorana neutrinos, are considered separately. Beyond the SM, only models with the CPT symmetry are studied. Ways to construct interactions with stronger CP (and T) symmetry violation, which help to understand the particle-antiparticle asymmetry in the Universe, are presented.AbstractSlide4
1. IntroductionSlide5
1954; Yang and Mills introduce local isospin transformations as an internal symmetry1964; Higgs and others find that for spontaneously broken gauge symmetries there are no Goldstone bosons but instead massive vector mesons (Higgs phenomenon)Milestones in the application of symmetry in physics1830; Group theory (Everiste Galois)1895-1910; Theory of group representation, Frobenius and Schur
1905; Einstein started to regard symmetry as the primary feature of nature1918; Emmy Noether theorem symmetries are connected with conservation laws
1927-28;
Fritz London and Weyl introduce gauge transformations into quantum
theory
1931;
Wigner theorem, discrete symmetries can give conservation law
1959-
61;
Heisenberg, Goldstone and Nambu spontaneous symmetry breaking
2012
;
ATLAS and CMS at LHC, Higgs boson discoveredSlide6
Global (conservation laws) Local Space – time{ Discrete or Continuous} Galileo Galilei and Poincare transformation
(Conservation laws exist or not) energy, momentum, angular momentum, centre of mass free movement,Parity (P), approximate conservation law,Time reverse (T) no conservation law ,……
Symmetries connected with
General Theory of Relativity
Space-time structure depends on a
mass distribution,…
Internal
{ Discrete
or
Continuous
}
Full:
Conservation
law of
charge, baryon number, lepton number,…
Spontaneously broken:
Goldstone particles appear for continuous symmetry, do not appear for discrete symmetry
Approximate:
Flavour, colour
, charge
parity(C)
,
isospin (I), strangeness (S),….
For full symmetry –
gauge particles
appear; W, Z, A,….
For spontaneously broken
symmetry:
Goldstone bosons disappear, some
of gauge particles become massive,
Unification of week and electromagnetic interaction,…..Slide7
1936; Heisenberg introduces charge conjugation (C) as a symmetry operation connecting particles and antiparticle states.The law of right-left symmetry was used in classical physics. But no conservation law for discrete symmetry. 1924; O. Laporte – energy levels of complex atoms can be classified into even and odd.1927; Wigner proved that empirical rule of Laporte is a consequence of the reflection symmetry. 1931;
Wigner introduces time reversal (T) symmetry into quantum theory and discover that this symmetry cannot give conservation law.Short history of discrete symmetriesP
T
CSlide8
1964; The CP breaking part of the weak interaction is found experimentally by J.W. Cronin and W.L.Fitch1957; CP-symmetry was proposed in 1957 by Lew Landau as a valid symmetry between matter and antimatter 1956-7; A parity breaking weak interaction is proposed by C.N. Yang and T.D Lee and verified experimentally by C.S. WuP
CPCP
1954-
5;
The PCT theorem is proved by Lüders and Pauli,
involving space inversion (P), charge conjugation
(C) and time reversal (T): in a local quantum field
theory the product PCT of these transformations is
always a symmetry.
CPT
C
TSlide9
Our system is symmetric if, probabilitiesandaverage values of any physical quantity, do not change after symmetry transformation
Definition of Symmetry in quantum physicsSlide10
If the action I[φA] is invariant under a continuous group of transformations depending smoothly on independent parameters εi , ( i = 1, 2, ...,p ), then there exist p conservation lawsEmmy Noether theorem If there exist unequivocal mapping between states from our state space:
such, that for any and probability is conserved
then for the states it is possible to choose the phases in such a way, that the mapping exist
:
Wigner theorem
where the operator is
linear and unitary or antilinear and unitary (antiunitary)Slide11
But what can be conserved in the case of discrete groups??In general Tg are unitary operators, they are not hermitian and cannot be observables.But there are some symmetry groups for which Tg are unitary and hermitian.Consider a symmetry group with two elements:But T is unitary:
And from
it follows:
For such groups we obtain multiplicative conservation law – conserved quantum numbers are multiplicative.
There is additional requirement – symmetry operators must be linear not antilinear.
There
is
one
symmetry which is represented by antilinear
operator
- time
r
eversal symmetry. Slide12
2. Discrete symmetries in Space Time and charge conjugation symmetrySlide13
P and T transformations are part of Full Lorentz Groupproperinproperortochronous
nonortochronous
T
T
P
PSlide14
Lorentz group --- 6 parameter, non compact , Lie group Pure Lorentz transformations;
Rotations;Six generators +2:
Irreducible representations;Slide15
scalar
right-handed spinor
left-
handed spinor
vector
Important irreducible representations
,
In Quantum Field Theory – the fields transformation:
Linear
:
Antilinear:Slide16
Discrete transformation for spinor fieldsPrecise look for the P transformation:Slide17
Then first order equations for spinors consistent with Lorentz invariance are the next :
w
here:Slide18
For charge conjugation complex conjugationSlide19
In the same way for all transformations (without complex conjugation):P
C
T
CP
CPTSlide20
Usually theories are formulated in the language of four component spinors (bispinors), we defineDirac spinors:andtwo type of Majorana bispinors:
We need Dirac gamma
matrices (in
Weyl
representation):Slide21
P
C
T
CP
CPT
C,P,T transformation for
bispinors
(
with complex conjugation for antilinear operations)Slide22
Discrete symmetries for various
terms in the SM LagrangianCP Violation, Gustavo C. Branco, Luís Lavoura
, João
Paulo
Silva, Oxford Science Publications, 1999Slide23
Discrete symmetries for scalar and vector fieldsVector fieldsScalar fieldsPCT
CPCPTSlide24
3. Discrete symmetries in the Standard ModelSlide25
Any theory has discrete symmetry if Lagrangian of this theory satisfies the conditions:
P
C
T
CP
CPTSlide26
We have to construct the SM Integral Action:For any symmetry group G we have a group representation U(G)
If it is possible to define a new fields: in such a way that:
t
hen we say that the SM possess a symmetry G. Slide27
In order to find where the discrete symmetries in the SM are violated we have to look for full Lagrangian (without kinetic energy):
Notice the differencesSlide28
Seesaw I typeSlide29
1) Parity2
) Charge Conjugation
QED and QCD conserve parity.
Weak interaction are not invariant due to the spatial inversion.
QED and QCD conserve parity (comment about QCD).
Weak
interaction are not invariant due to the
charge conjugation transformation.Slide30
But
In order to have C invariance we have to assume that:Slide31
For such gluon fields transformation the gluon field strength tensors
For such transformation for the field strength tensors, the kinetic energy term is also C invariant:
and thus full QCD Lagrangian (and the integral action) is C invariant:
have proper C
transformation and
is possible to check that
:Slide32
3) Time ReversalTime reversal operator is anti-unitary and usually is parameterized in the way: where is unitary and complex conjugates any c- complex number.
If there is no any phase in a
Lagrangian
,
theory is T symmetric, so QED and QCD are time reversal invariant.
The phase(s) appears in the charge current of the week interaction, so the GWS theory has not T symmetry. Slide33
As we know:
After the T transformation:And we are able to define the T transformation for bosons in such a way that (for the Action):
So the GWS theory has not T-invariance in the quark sector (CKM matrix), as well as in the lepton sector (PMNS matrix).
For leptonsSlide34
4) CP symmetry
So in the charged current:
And once more we can define the CP transformation for gauge bosons, in such a way that (of course we should think about the Action):Slide35
5) CPT symmetry
+
L
ccSlide36
CPT theorem (Pauli, 1955)If nature is described by a theory, for which a Lagrangian is:---- local,---- Lorentz invariant,---- with the useful connection between spin and statistics,---- hermitianthenthe Integral Action of such theory is always invariant under the combined application of C, P, and T transformation. Slide37
Complex current interaction breaks: P, C, CP and T; CPT is not breaking
In neutral current interaction C and P is not conserved;T, CP and CPT symmetries are satisfied
In electromagnetic interaction all symmetries are satisfied Slide38
5. Beyond the Standard ModelSlide39
Up to now Standard Model is consistent with all dataBUTThe Gauge symmetry problem--Three groups—three different couplings,-- Charge quantization, why charge ,
The Fermion problem-- Only first family of fermion ( e
-
, ν
e
, u , d) has visible role in
nature, why tree family exist?
-- No explanation of fermion masses , ,Slide40
-- neutrino - Majorana or Dirac?-- completely different mixing matrices for quarks and leptonsThe Higgs - hierarchy problem-- MH ≈ MW, MZ; but if we calculate the Higgs mass we getand Λ is large Λ ≈ 1014 GeV, Λ≈ 1019 GeV. So natural value for MH
is O(Λ) and we must fine-tune.
The strong CP problem
-- To the QCD Lagrangian we can add term which break CP
symmetry, Why this term, if exist, is so small?Slide41
The Gravity problem-- No quantum theory of gravitySM requires a number of new ingredients-- mechanism for small neutrino mass-- explain the baryon asymmetry in the Universe-- explain the dark matter-- explain the dark energy (acceleration of the Universe),-- FCNC, proton decay, particle dipole electric moment.Slide42
PDG 2016
In the quark sectorMechanism of CP T symmetry breaking Slide43
In the lepton sector
0.810 – 0.8290.539 – 0.5620.147 – 0.169
(- 0.485) – (- 0.479)
0.467 – 0.563
0.669 – 0.743
0.278 – 0.339
(-0.683) – (0.626)
0.647 – 0.728
PDG 2016
For Majorana neutrinos ---
two additional CP violating phasesSlide44
Left –handed neutrino states
Right –handed neutrino statesSlide45
There are several possibilities to extend the neutrino sector in the SM1) Only left handed neutrinos
2) Left handed and Majorana right-handed neutrinosSlide46
3)Majorana left handed and right-handed neutrinosUp to now no experimental information about heavy neutrinosBUT IF THEY EXIST See-saw mechanism –we understand why masses of observed neutrino are so small
Heavy neutrino exist they can explain part (maybe all) of dark matter phenomenaMass matrix is larger (e.g. 6x6) and more CP violating parameters are workingSlide47
6. ConclusionsSlide48
P and T symmetry are the part of full Lorentz symmetry group and properties of quantum field transformation follows from the group structureIn Quantum Field Theory where antiparticles exist naturally it is possible to define charge conjugation transformationIn the Standard Model the P and C symmetry are maximally violated but only in weak interactionAs CPT symmetry is naturally satisfied, CP and T symmetries are equivalent and both are violated in weak interactionIn quark sector CP is violated (but weakly), in the lepton sector CP violation is stronger but interaction are weak.Slide49
Thank youSlide50
We introduce the quantum numbers which characterize components of the L - doublet and R – singlet:Week isospin operator with eigenvalues T3i:
Week hipercharge operator with eigenvalues Y:
Particles
Charge
Q
i
Week isospin
T
3i
Hpercharge
Y
ν
L
0
1/2
-1/2
e
L
-1
-1/2
-1/2
ν
R
0
0
0
e
R
-1
0
-1
u
L
2/3
1/2
1/6
d
L
-1/3
-1/2
1/6
u
R
2/3
0
2/3
d
R
-1/3
0
-1/3
Particles Quantum numbersSlide51
Let us assume that our subsystem states are eigenstates of the symmetry operator alone:
So,
our
system, which
consists of two
subsystems,
is also the eigenstate
of the symmetry
operation and:
And we obtained multiplicative conservation law