Discrete Symmetries in Fundamental Interaction - PowerPoint Presentation

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