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Flavor Physics: Introduction to C, P and T symmetries David Delepine, Carlos Vaquera-Araujo. Conacyt DCI-Campus Le on Universidad de Guanajuato. October 25, 2018 (MSPF-Sonora) Flavor Physics October 25, 2018 1 / 31 Outline 1

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SLIDE 1

Flavor Physics: Introduction to C, P and T symmetries

David Delepine, Carlos Vaquera-Araujo.

Conacyt DCI-Campus Le´

  • n

Universidad de Guanajuato.

October 25, 2018

(MSPF-Sonora) Flavor Physics October 25, 2018 1 / 31

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Outline

1 Introduction 2 C, P and T in field theory 3 Standard Model 4 Cabibbo Kobayashi Maskawa Matrix 5 CP violation

(MSPF-Sonora) Flavor Physics October 25, 2018 2 / 31

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EW SM ingredients

R / G / B 2/3

1/2 2.3 MeV

up

u

R / G / B −1/3

1/2 4.8 MeV

down

d

−1

1/2 511 keV

electrn

e

1/2 < 2 eV

e neutrino

νe

R / G / B 2/3

1/2 1.28 GeV

charm

c

R / G / B −1/3

1/2 95 MeV

strange

s

−1

1/2 105.7 MeV

muon

µ

1/2 < 190 keV

µ neutrino

νµ

R / G / B 2/3

1/2 173.2 GeV

top

t

R / G / B −1/3

1/2 4.7 GeV

bottom

b

−1

1/2 1.777 GeV

tau

τ

1/2 < 18.2 MeV

τ neutrino

ντ

±1

1 80.4 GeV

W ±

1 91.2 GeV

Z

1

fotn

γ

c

  • l
  • r

1

glun

g

125.1 GeV

Higgs

H

Strong Interaction (color) Electromagnetic Interaction (charge) Weak Interaction (weak isospin)

Charge Color Mass Spin

quarks leptons fermions bosons everyday matter unstable matter Gauge Bosons Scalar Boson

1st 2nd 3rd Generation: (MSPF-Sonora) Flavor Physics October 25, 2018 3 / 31

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Flavor Physics

Flavor physics is the study of different types of quarks and leptons,

  • r flavors, their spectrum and the transmutations among them.

Flavor physics is very rich. Check out http://pdg.lbl.gov for the many different transition rates among hadrons with different quark content. We aim at understanding this wealth of information in terms of some simple basic principles.

(MSPF-Sonora) Flavor Physics October 25, 2018 4 / 31

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C, P and T

C, P and T are discrete spacetime transformations A priori, they have noting to do with flavor physics, as flavor has to do with internal symmetries. However, it turns out that in nature, all observations of CP violation happen to come along with flavor violation.

(MSPF-Sonora) Flavor Physics October 25, 2018 5 / 31

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C, P and T in Quantum Mechanics

  • Parity: P performs a spatial inversion through the origin x → −x

UP ψ(t, x) = ηP ψ(t, −x) Introduced by Wigner in 1927/28 Unitary transformation Applying parity twice restores the original state, U 2

P = 1 up to an

unobservable phase. From this the parity of the UP eigenfunctions has to be either even, ηP = +1, or odd, ηP = −1.

(MSPF-Sonora) Flavor Physics October 25, 2018 6 / 31

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  • Time reversal: T performs reversal of motion in time (sense of time

evolution). That is t → −t with exchange of initial and final states. AT ψ(t, x) = ηT ψ(−t, x) Introduced by Wigner in 1932 Antiunitary transformation AT = UT K (Necessary to preserve [xi, pj] = iδij). Antiunitary: unitary- for conserving probabilities, anti- for complex conjugation (antilinear).

(MSPF-Sonora) Flavor Physics October 25, 2018 7 / 31

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  • Charge Conjugation: C reverses the sign of the electric charge, colour

charge and magnetic moment of a particle. Introduced by Kramers in 1937. Requires quantum field theory, as it is better understood as particle-antiparticle interchange

(MSPF-Sonora) Flavor Physics October 25, 2018 8 / 31

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Maxwell Equations, C, P and T

L(Aµ) = −1 4FµνF µν + jµAµ; Fµν = ∂µAν − ∂νAµ Equations of Motion: ∂µF µν = jν, ∇ · E = ρ, ∇ · B = 0, ∇ × E = −∂B ∂t , ∇ × B = j + ∂E ∂t , E = −∇φ − ∂A ∂t , B = ∇ × A are invariant under: PARITY: x → −x TIME REVERSAL t → −t CHARGE CONJUGATION ρ → −ρ

(MSPF-Sonora) Flavor Physics October 25, 2018 9 / 31

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P T C P T C t +

  • +

x

  • +

+ xµ xµ −xµ xµ ρ + +

  • j
  • jµ

jµ jµ −jµ φ + +

  • A
  • Aµ

Aµ Aµ −Aµ E

  • +
  • B

+

  • F µν

Fµν −Fµν −F µν

(MSPF-Sonora) Flavor Physics October 25, 2018 10 / 31

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Fermion Fields

The fermion fields transformation rules under C, P and T symmetry follow from LQED = −1 4FµνF µν + i ¯ ψγµ(∂µ − iQAµ)ψ − ¯ ψmψ PLQED(t, x)P −1 = LQED(t, −x) CLQED(t, x)C−1 = LQED(t, x) TLQED(t, x)T −1 = LQED(−t, x) with ¯ ψ = ψ†γ0 and using γµγν + γνγµ = 2gµν γ0㵆γ0 = γµ

(MSPF-Sonora) Flavor Physics October 25, 2018 11 / 31

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ψP (t, x) = Pψ(t, x)P −1 = Pψ(t, −x) ψC(t, x) = Cψ(t, x)C−1 = C ¯ ψT (t, x) ψT (t, x) = Tψ(t, x)T −1 = T ψ(−t, x) where P and C are unitary operators and T is a anti-unitary operator. We obtain (in the chiral representation): PγµP−1 = (γµ)† = γµ → P = γ0 C−1γµC = −(γµ)T → C = −iγ0γ2 = −CT T γµT −1 = (γµ)T → T = iγ1γ3 = −T ∗

(MSPF-Sonora) Flavor Physics October 25, 2018 12 / 31

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Using the 16 Dirac matrices which form a complete basis for the Clifford Algebra, one can build the corresponding bilinear forms: s12(x) = : ¯ ψ1(x)ψ2 : p12(x) = : ¯ ψ1(x)iγ5ψ2 : vµ

12(x)

= : ¯ ψ1(x)γµψ2 : aµ

12(x)

= : ¯ ψ1(x)γµγ5ψ2 : tµν

12 (x)

= : ¯ ψ1(x)σµνψ2 : where σµν = i

2[γµ, γν] and γ5 = iγ0γ1γ2γ3

(MSPF-Sonora) Flavor Physics October 25, 2018 13 / 31

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s12(xρ) p12(xρ) vµ

12(xρ)

aµ

12(xρ)

tµν

12 (xρ)

P s12(xρ) −p12(xρ) v12

µ (xρ)

−a12

µ (xρ)

t12

µν(xρ)

T s12(−xρ) −p12(−xρ) v12

µ (−xρ)

a12

µ (−xρ)

−t12

µν(−xρ)

C s21(xρ) p21(xρ) −vµ

21(xρ)

aµ

21(xρ)

−tµν

21 (xρ)

(MSPF-Sonora) Flavor Physics October 25, 2018 14 / 31

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CPT Theorem

Under CPT, any hermitian local Poincar´ e invariant theory described by L satisfies L(x) → (CPT)L(x)(CPT)−1 = L†(−x) = L(−x) Thus, the action is invariant under CPT. ⇒ Particles and antiparticles have equal masses, equal total lifetimes and opposite charges |CPTα >≡ CPT|α >≡ |¯ α > mα = m¯

α,

τ(α) = τ(¯ α), Q(α) + Q(¯ α).

(MSPF-Sonora) Flavor Physics October 25, 2018 15 / 31

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Status of discrete symmetries

No evidence for CPT violation

  • mK0 − mK0

mK0

  • < 10−18,
  • Γ(K0) − Γ(K

0)

mK0

  • < 10−17,
  • Q(p) + Q(¯

p) e

  • < 10−21.

No evidence for C, P, or T violation in purely electromagnetic or strong interactions.

(MSPF-Sonora) Flavor Physics October 25, 2018 16 / 31

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P and C maximally broken in weak interactions Violation of CP and T has been observed in weak interactions. The amount of CP and T observed is small

(MSPF-Sonora) Flavor Physics October 25, 2018 17 / 31

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Standard Model

Gauge symmetry is GSM = SU(3)C × SU(2)L × U(1)Y There are three fermion generations: QLi ∼ (3, 2, 1/6), uRi ∼ (3, 1, 2/3), dRi ∼ (3, 1, −1/3), LLi ∼ (1, 2, −1/2), eRi ∼ (1, 1, −1) The scalar representation is given by φ ∼ (1, 2, 1/2)

(MSPF-Sonora) Flavor Physics October 25, 2018 18 / 31

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The pattern of symmetry breaking is given by: GSM → SU(3)C × U(1)EM The SM lagrangian is the most general renormalizable lagrangian consistent with the gauge symmetry and the given particle content: LSM = Lkin + LHiggs + LYukawa −LLeptons

Yukawa

= λij

E ¯

LLiφeRj + h.c., −LQuarks

Yukawa

= λij

D ¯

QLiφdRj + λij

U ¯

QLi ˜ φuRj + h.c., where ˜ φ = iτ2φ∗.

(MSPF-Sonora) Flavor Physics October 25, 2018 19 / 31

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Global accidental symmetry: SM is invariant under the accidental symmetry U(1)B × U(1)e × U(1)µ × U(1)τ Including neutrino masses, the accidental symmetry is reduced to U(1)B × U(1)L. Taking into account the chiral anomaly and the topological gauge structure of the SM which implies that (B + L) is significantly violated through instantons and sphalerons at early times of the Universe, the accidental symmetry is reduced to U(1)B−L. P (C) explicitly and maximally broken. CP violation not obvious, since both P and C transformations take left- and right-handed fields into one another.

(MSPF-Sonora) Flavor Physics October 25, 2018 20 / 31

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Cabibbo Kobayashi Maskawa

After SSB φ = (0, v/ √ 2)T (suppressing flavor indices): − Lm = v √ 2uLλUuR + v √ 2dLλDdR + v √ 2eLλEeR + h.c. Diagonalization (quark sector): Field redefinition (flavor eigenstates → mass eigenstates) uR → VuRuR, uL → VuLuL, dR → VdRdR, dL → VdLdL. V †

uLλUVuR = λ′ U,

V †

dLλDVdR = λ′ D .

Here the matrices λ′

U and λ′ D, are diagonal, real and positive, and the

transformation matrices Vu,dL,R are unitary.

(MSPF-Sonora) Flavor Physics October 25, 2018 21 / 31

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Then from −Lm = v √ 2

  • uLλ′

UuR + dLλ′ DdR + eLλEeR + h.c.

  • = v

√ 2

  • uλ′

Uu + dλ′ Dd + eλEe

  • we read off the diagonal mass matrices, mU = vλ′

U/

√ 2, mD = vλ′

D/

√ 2 and mE = vλE/ √ 2.

  • However, in general the field redefinitions in are not symmetries of

the Lagrangian. We must check the induced Lagrangian dependency

  • n Vu,dL,R.

(MSPF-Sonora) Flavor Physics October 25, 2018 22 / 31

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Kinetic Terms are invariant uLi/ ∂uL → (uLV †

uL)i/

∂(VuLuL) = uL(V †

uLVuL)i/

∂uL = uLi/ ∂uL Electromagnetic and weak neutral currents are invariant (GIM mechanism) uL / ZuL → (uLV †

uL)/

Z(VuLuL) = uL(V †

uLVuL)/

ZuL = uL / ZuL Charged currents are not invariant uL / W +dL + dL / W −uL → uL(V †

uLVdL) /

W +dL + dL(V †

dLVuL) /

W −uL

(MSPF-Sonora) Flavor Physics October 25, 2018 23 / 31

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A relic of our field redefinitions has remained in the form of the unitary matrix V = V †

uLVdL.

We call this the Cabibbo-Kobayashi-Maskawa (CKM) matrix. This is the place where CP violation and flavor meet. CP can be broken by the terms uLV / W +dL + dLV † / W −uL. To see this, recall that under CP uLγµdL

CP

− − → −dLγµuL, W +µ CP − − → −W −

µ .

Hence CP invariance requires V † = V T , or V ∗ = V . This condition can be read as “physical non-zero phase” = “CP violation”.

(MSPF-Sonora) Flavor Physics October 25, 2018 24 / 31

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How does flavor enter the picture? Number of generations: NG. A general NG × NG unitary matrix V is characterized by N2

G real

parameters: NG(NG − 1)/2 moduli and NG(NG + 1)/2 phases. the case of V , many of these parameters are irrelevant because we can always choose arbitrary quark phases. Under the phase redefinitions ui → eiφi ui and dj → eiθj dj, the mixing matrix changes as Vij → Vij ei(θj−φi); thus, 2NG − 1 phases are unobservable. The number of physical free parameters in the quark-mixing matrix then gets reduced to (NG − 1)2: NG(NG − 1)/2 moduli and (NG − 1)(NG − 2)/2 phases.

(MSPF-Sonora) Flavor Physics October 25, 2018 25 / 31

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  • Cabibbo: NG = 2.

In this simple case, V is determined by a single parameter. One then recovers the Cabibbo rotation matrix V =

  • cos θC

sin θC − sin θC cos θC

  • .

This matrix satisfies V = V ∗, ⇒ NO CP violation induced by the field redefinition.

(MSPF-Sonora) Flavor Physics October 25, 2018 26 / 31

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  • Kobayashi-Maskawa: NG = 3.

The CKM matrix is described by three angles and one phase. It is useful to label the matrix elements by the quarks they connect: V =   Vud Vus Vub Vcd Vcs Vcb Vtd Vts Vtb   . Standard CKM parameterization:

V =    c12 c13 s12 c13 s13 e−iδ −s12 c23 − c12 s23 s13 eiδ c12 c23 − s12 s23 s13 eiδ s23 c13 s12 s23 − c12 c23 s13 eiδ −c12 s23 − s12 c23 s13 eiδ c23 c13    .

Here cij ≡ cos θij and sij ≡ sin θij , with cij ≥ 0 , sij ≥ 0 and 0 ≤ δ ≤ 2π .

(MSPF-Sonora) Flavor Physics October 25, 2018 27 / 31

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Notice that δ is the only complex phase in the SM Lagrangian. Therefore, it is the only possible source of CP-violation phenomena. In fact, it was for this reason that the third generation was assumed to exist! With two generations, the SM could not explain the observed CP violation in the K system.

(MSPF-Sonora) Flavor Physics October 25, 2018 28 / 31

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Manifestly basis-independent form of the CP violating phase: Jarlskog invariant J = Im (VudV ∗

cdVcbV ∗ ub) .

In the standard parameterization: J = c12c23c2

13s12s23s13 sin δ.

(MSPF-Sonora) Flavor Physics October 25, 2018 29 / 31

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Conditions for CP violation

Let’s assume the following form for a physical amplitude: A = A1eiδ1 + A2eiδ2 where A1,2 are two complex partial weak amplitudes with CP-conserving dynamical phases δ1,2. A CP − − → ¯ A = A∗

1eiδ1 + A∗ 2eiδ2 = A∗

(MSPF-Sonora) Flavor Physics October 25, 2018 30 / 31

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The CP-asymmetry in decay widths is: ACP ≡ Γ − ¯ Γ Γ + ¯ Γ = |A|2 − | ¯ A|2 |A|2 + | ¯ A|2 = −2Im(A1A∗

2) sin(δ1 − δ2)

|A1|2 + |A2|2 + 2Re(A1A∗

2) cos(δ1 − δ2)

A non-zero CP-asymmetry requires at least two partial amplitudes with

1 a relative CP-violating phase (weak phase) 2 a relative dynamical CP-conserving phase (strong phase) (MSPF-Sonora) Flavor Physics October 25, 2018 31 / 31