2. Discrete Symmetries in Fundamental Interaction
Workshop on
Discrete Symmetries and Entanglement
, Kraków
True can be only such things which is beauty and simple
Marek Zrałek
University of Silesia, Katowice

3. Outline
Introduction
Discrete symmetries in Space Time and charge conjugation symmetry
Discrete symmetries in the Standard Model
Beyond the Standard Model
Conclusions

4. Abstract
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.

5. 1. Introduction

6. Milestones in the application of symmetry in physics
1830; Group theory (Everiste Galois)
; Theory of group representation, Frobenius and Schur
1905; Einstein started to regard symmetry as the primary feature of nature
1918; Emmy Noether theorem symmetries are connected with conservation laws
; Fritz London and Weyl introduce gauge transformations into quantum theory
1931; Wigner theorem, discrete symmetries can give conservation law
1954; Yang and Mills introduce local isospin transformations as an internal symmetry
; Heisenberg, Goldstone and Nambu spontaneous symmetry breaking
1964; Higgs and others find that for spontaneously broken gauge symmetries there are no Goldstone bosons but instead massive vector mesons (Higgs phenomenon)
2012; ATLAS and CMS at LHC, Higgs boson discovered

7. Local (conservation laws) { Discrete or Continuous} Internal
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
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,…..

8. Short history of discrete symmetries
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. P
1931; Wigner introduces time reversal (T) symmetry into quantum theory and discover that this symmetry cannot give conservation law. T
1936; Heisenberg introduces charge conjugation (C) as a symmetry operation connecting particles and antiparticle states. C

9. 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 P
1956-7; A parity breaking weak interaction is proposed by C.N. Yang and T.D Lee and verified experimentally by C.S. Wu C
1957; CP-symmetry was proposed in 1957 by Lew Landau as a valid symmetry between matter and antimatter CP CP
1964; The CP breaking part of the weak interaction is found experimentally by J.W. Cronin and W.L.Fitch T

10. Definition of Symmetry in quantum physics
Our system is symmetric if,
probabilities
and
average values of any physical quantity,
do not change after symmetry transformation

11. Emmy Noether theorem Wigner theorem
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 laws
Wigner 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:
where the operator is linear and unitary or antilinear and unitary (antiunitary)

12. 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 reversal symmetry.

13. 2. Discrete symmetries in Space Time and charge conjugation symmetry

14. P T T P P and T transformations are part of Full Lorentz Group
ortochronous T T
nonortochronous P
proper
inproper

15. Lorentz group --- 6 parameter, non compact , Lie group
Pure Lorentz transformations;
Rotations;
Six generators +2:
Ineger
half integer
Irreducible representations;

16. Important irreducible representations
scalar
right-handed spinor
left-handed spinor
vector ,
In Quantum Field Theory – the fields transformation:
Linear:
Antilinear:

17. Discrete transformation for spinor fields
Precise look for the P transformation:

18. Then first order equations for spinors consistent with Lorentz invariance are the next :
where:

19. For charge conjugation  complex conjugation

20. In the same way for all transformations (without complex conjugation):CP
CPT

21. Usually theories are formulated in the language of four component spinors (bispinors), we define
Dirac spinors:
and
two type of Majorana bispinors:
We need Dirac gamma matrices (in Weyl representation):

22. C,P,T transformation for bispinors
(with complex conjugation for antilinear operations) P C T CP
CPT

23. Discrete symmetries for various terms in the SM Lagrangian
CP Violation, Gustavo C. Branco, Luís Lavoura, João Paulo Silva, Oxford Science Publications, 1999

24. P C T CP CPT Discrete symmetries for scalar and vector fields
Scalar fields P C T CP
CPT

25. 3. Discrete
symmetries in the
Standard Model

26. Any theory has discrete symmetry if Lagrangian of this theory satisfies the conditions:P C T CP
CPT

27. 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:
then we say that the SM possess a symmetry G.

28. 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 differences

29. Seesaw I type

30. 1) Parity 2) Charge Conjugation QED and QCD conserve parity.
Weak interaction are not invariant due to the spatial inversion.
2) Charge Conjugation
QED and QCD conserve parity (comment about QCD).
Weak interaction are not invariant due to the charge conjugation transformation.

31. But
In order to have C invariance we have to assume that:

32. For such gluon fields transformation the gluon field strength tensors
have proper C transformation and is possible to check that :
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:

33. 3) Time Reversal
Time 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.

34. After the T transformation:
As we know:
For leptons
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).

35. 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):

36. 5) CPT symmetry
Lcc +

37. 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,
---- hermitian
then
the Integral Action of such theory is always invariant under the combined application of C, P, and T transformation.

38. In electromagnetic interaction all symmetries are satisfied
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

39. 5. Beyond the
Standard Model

40. Up to now Standard Model is consistent with all data
BUT
The Gauge symmetry problem
--Three groups—three different couplings,
-- Charge quantization, why charge ,
The Fermion problem
-- Only first family of fermion ( e-, νe , u , d) has visible role in nature, why tree family exist?
-- No explanation of fermion masses , ,

41. The Higgs - hierarchy problem
-- neutrino - Majorana or Dirac?
-- completely different mixing matrices for quarks and leptons
The Higgs - hierarchy problem
-- MH ≈ MW, MZ; but if we calculate the Higgs mass we get
and 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?

42. SM requires a number of new ingredients
The Gravity problem
-- No quantum theory of gravity
SM 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.

43. Mechanism of CP T symmetry breaking
In the quark sector
PDG 2016

44. For Majorana neutrinos --- two additional CP violating phases
In the lepton sector
0.810 – 0.829
0.539 – 0.562
0.147 – 0.169
( ) – ( )
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 phases

45. Left –handed neutrino states
Right –handed neutrino states

46. There are several possibilities to extend the neutrino sector in the SM
1) Only left handed neutrinos
2) Left handed and Majorana right-handed neutrinos

47. Up to now no experimental information about heavy neutrinos
3)Majorana left handed and right-handed neutrinos
Up to now no experimental information about heavy neutrinos
BUT 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 phenomena
Mass matrix is larger (e.g. 6x6) and more CP violating parameters are working

48. 6. Conclusions

49. P and T symmetry are the part of full Lorentz symmetry group and properties of quantum field transformation follows from the group structure
In Quantum Field Theory where antiparticles exist naturally it is possible to define charge conjugation transformation
In the Standard Model the P and C symmetry are maximally violated but only in weak interaction
As CPT symmetry is naturally satisfied, CP and T symmetries are equivalent and both are violated in weak interaction
In quark sector CP is violated (but weakly), in the lepton sector CP violation is stronger but interaction are weak.

50. Thank you

51. Particles Quantum numbers Week isospin operator with eigenvalues T3i:
We introduce the quantum numbers which characterize components of the L - doublet and R – singlet:
Particles Quantum numbers
Week isospin operator with
eigenvalues T3i:
Particles
Charge Qi
Week isospin
T3i
Hpercharge Y νL
1/2
-1/2 eL -1 νR eR uL
2/3
1/6 dL
-1/3 uR dR
Week hipercharge operator with eigenvalues Y:

52. And we obtained multiplicative conservation law
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