1 Invariant Principles and Conservation laws Kihyeon Cho April 26, 2011 HEP

2 Contents Symmetry Parity (P) Gauge invariance Charge (C) CP Violation

3 Line Symmetry Shape has line symmetry when one half of it is the mirror image of the other half. Symmetry exists all around us and many people see it as being a thing of beauty.

4 Is a butterfly symmetrical?

5 Line Symmetry exists in nature but you may not have noticed.

6 At the beach there are a variety of shells with line symmetry.

7 Under the sea there are also many symmetrical objects such as these crabs and this starfish.

8 Animals that have Line Symmetry Here are a few more great examples of mirror image in the animal kingdom.

9 THESE MASKS HAVE SYMMETRY These masks have a line of symmetry from the forehead to the chin. The human face also has a line of symmetry in the same place.

10 Human Symmetry The 'Proportions of Man' is a famous work of art by Leonardo da Vinci that shows the symmetry of the human form.

11 REFLECTION IN WATER

12 The Taj Mahal Symmetry exists in architecture all around the world. The best known example of this is the Taj Mahal.

13 This photograph shows 2 lines of symmetry. One vertical, the other along the waterline. (Notice how the prayer towers, called minarets, are reflected in the water and side to side).

14 2D Shapes and Symmetry After investigating the following shapes by cutting and folding, we found:

15 an equilateral triangle has 3 internal angles and 3 lines of symmetry.

16 a square has 4 internal angles and 4 lines of symmetry.

17 a regular pentagon has 5 internal angles and 5 lines of symmetry.

18 a regular hexagon has 6 internal angles and 6 lines of symmetry.

19 a regular octagon has 8 internal angles and 8 lines of symmetry.

20

21 Conservation Rules Conserved Quantity WeakElectromagneticStrong I(Isospin)No (  I=1 or ½) NoYes (No in 1996) S(Strangeness)No (  S=1,0) Yes C(charm)No (  C=1,0) Yes P(parity)NoYes C(charge)NoYes CPNoYes

22 Conserved Quantities and Symmetries Every conservation law corresponds to an invariance of the Hamiltonian (or Lagrangian) of the system under some transformation. We call these invariances symmetries. There are 2 types of transformations: continuous and discontinuous Continuous  give additive conservation laws x  x+dx or   +d  examples of conserved quantities: electric charge momentum baryon # Discontinuous  give multiplicative conservation laws parity transformation: x, y, z  (-x), (-y), (-z) charge conjugation (particle  antiparticle): e -  e + examples of conserved quantities: parity (in strong and EM) charge conjugation (in strong and EM) parity and charge conjugation (strong, EM, almost always in weak)

23 Conserved Quantities and Symmetries Example of classical mechanics and momentum conservation. In general a system can be described by the following Hamiltonian: H=H(p i,q i,t) with p i =momentum coordinate, q i =space coordinate, t=time Consider the variation of H due to a translation q i only. For our example dp i =dt=0 so we have: Using Hamilton’s canonical equations: We can rewrite dH as: If H is invariant under a translation (dq) then by definition we must have: This can only be true if: Thus each p component is constant in time and momentum is conserved. Read Perkins: Chapters 3.1

24 Conserved Quantities and Quantum Mechanics In quantum mechanics quantities whose operators commute with the Hamiltonian are conserved. Recall: the expectation value of an operator Q is: How does change with time? Recall Schrodinger’s equation: Substituting the Schrodinger equation into the time derivative of Q gives: H + = H *T = hermitian conjugate of H Since H is hermitian ( H + = H ) we can rewrite the above as: So then is conserved. Read Perkins: Chapters 3.1

25 Three Important Discrete Symmetries Parity, P –Parity reflects a system through the origin. Converts right-handed coordinate systems to left-handed ones. –Vectors change sign but axial vectors remain unchanged x   xL  L Charge Conjugation, C –Charge conjugation turns a particle into its anti-particle e   e  K   K   Time Reversal, T –Changes the direction of motion of particles in time t  t CPT theorem –One of the most important and generally valid theorems in quantum field theory. –All interactions are invariant under combined C, P and T transformations. –Implies particle and anti-particle have equal masses and lifetimes  

Discrete Symmetries An example of a discrete transformation is the operation of inverting all angles:   -  In contrast a rotation by an amount  is a continuous transformation. Reminder: Discrete symmetries give multiplicative quantum numbers. Continuous symmetries give additive quantum numbers. The three most important discrete symmetries are: Parity (P)(x,y,z)  (-x,-y,-z) Charge Conjugation (C)particles  anti-particles Time Reversal (T)time  -time Other not so common discrete symmetries include G parity: G=CR=C exp (i π I 2 ) = (-1) l+S+I G parity is important for pions under the strong interaction. Note: discrete transformations do not have to be unitary transformations ! P and C are unitary transformations T is not a unitary transformation, T is an antiunitary operator! Read Perkins: Chapters 3.3, 3.4 Read Perkins: Chapters 4.5.1

27 Read Perkins: Chapters 3.3, 3.4

28 Parity Quantum Number

Parity Let us examine the parity operator (P) and its eigenvalues. The parity operator acting on a wavefunction is defined by: P  (x, y, z) =  (-x, -y, -z) P 2  (x, y, z) = P  (-x, -y, -z) =  (x, y, z) Therefore P 2 = I and the parity operator is unitary. If the interaction Hamiltonian (H) conserves parity then [H,P]=0, and: P  (x, y, z) =  (-x, -y, -z) = n  (x, y, z) with n = eigenvalue of P P 2  (x, y, z) = PP  (x, y, z) = nP  (x, y, z) = n 2  (x, y, z)  (x, y, z) = n 2  (x, y, z)  n 2 =  1 so, n=1 or n=-1. The quantum number n is called the intrinsic parity of a particle. If n= 1 the particle has even parity. If n= -1 the particle has odd parity. In addition, if the overall wavefunction of a particle (or system of particles) contains spherical harmonics (Y L m ) then we must take this into account to get the total parity of the particle (or system of particles). The parity of Y L m is: PY L m = (-1) L Y L m. For a wavefunction  (r, ,  )=R(r)Y L m ( ,  ) the eigenvalues of the parity operator are: P  (r, ,  )=PR(r)Y L m ( ,  ) = (-1) L R(r)Y L m ( ,  ) The parity of the particle would then be: n(-1) L Note: Parity is a multiplicative quantum number M&S pages Read Perkins: Chapters 3.3, 3.4

Parity The parity of a state consisting of particles a and b is: (-1) L n a n b where L is their relative orbital momentum and n a and n b are the intrinsic parity of each of the two particle. Note: strictly speaking parity is only defined in the system where the total momentum (p) =0 since the parity operator (P) and momentum operator anticommute, (Pp=-p). How do we know the parity of a particle ? By convention we assign positive intrinsic parity (+) to spin 1/2 fermions: +parity: proton, neutron, electron, muon (  - ) Anti-fermions have opposite intrinsic parity -parity: anti-proton, anti-neutron, positron, anti-muon (  + ) Bosons and their anti-particles have the same intrinsic parity. What about the photon? Strictly speaking, we can not assign a parity to the photon since it is never at rest. By convention the parity of the photon is given by the radiation field involved: electric dipole transitions have + parity magnetic dipole transitions have - parity We determine the parity of other particles ( , K..) using the above conventions and assuming parity is conserved in the strong and electromagnetic interaction. Usually we need to resort to experiment to determine the parity of a particle. Read Perkins: Chapters 3.3, 3.4

Parity Example: determination of the parity of the  using  - d  nn. For this reaction we know many things: a) s  =0, s n =1/2, s d =1, orbital angular momentum L d =0, => J d =L d +s d =0+1=1 b) We know (from experiment) that the  is captured by the d in an s-wave state. Thus the total angular momentum of the initial state is just that of the d (J=1). c) The isospin of the nn system is 1 since d is an isosinglet and the  - has I=|1,-1> note: a |1,-1> is symmetric under the interchange of particles. (see below) d) The final state contains two identical fermions and therefore by the Pauli Principle the wavefunction must be anti-symmetric under the exchange of the two neutrons. Let’s use these facts to pin down the intrinsic parity of the . i) Assume the total spin of the nn system =0. Then the spin part of the wavefunction is anti-symmetric: |0,0> = (2) -1/2 [|1/2,1/2>|1/2-1/2>-|1/2,-1/2>|1/2,1/2>] To get a totally anti-symmetric wavefunction L must be even (0,2,4…) Cannot conserve momentum (J=1) with these conditions! ( since J=L+s => 1  0+0,2,) ii) Assume the total spin of the nn system =1. Then the spin part of the wavefunction is symmetric: |1,1> = |1/2,1/2>|1/2,1/2> |1,0> = (2) -1/2 [|1/2,1/2>|1/2-1/2>+|1/2,-1/2>|1/2,1/2>] |1,-1> = |1/2,-1/2>|1/2,-1/2> ( since J=L+s => 1=1+1) To get a totally anti-symmetric wavefunction L must be odd (1, 3, 5…) L=1 consistent with angular momentum conservation: nn has s=1, L=1, J=1  3 P 1 The parity of the final state is: n n n n (-1) L = (+)(+)(-1) 1 = - The parity of the initial state is: n  n d (-1) L = n  (+)(-1) 0 = n  Parity conservation gives: n n n n (-1) L = n  n d (-1) L  n   = - Read Perkins: Chapters 3.3.1

Parity There is other experimental evidence that the parity of the  is -: the reaction  - d  nn  0 is not observed the polarization of  ’s from  0  Some use “spin-parity” buzz words: buzzwordspinparityparticle pseudoscalar 0 - , k scalar 0 +higgs (none observed) vector 1 -  pseudovector 1 +A1 (axial vector) How well is parity conserved? Very well in strong and electromagnetic interactions ( ) not at all in the weak interaction! The  puzzle and the downfall of parity in the weak interaction In the mid-1950’s it was noticed that there were 2 charged particles that had (experimentally) consistent masses, lifetimes and spin = 0, but very different weak decay modes:  +  +  0  +  +  -  + The parity of  +  = + while the parity of  +  = - Some physicists said the  +  and  + were different particles, and parity was conserved. Lee and Yang said they were the same particle but parity was not conserved in weak interaction! Lee and Yang win Nobel Prize when parity violation was discovered. Note:  +  + is now known as the K +. M&S pages Read Perkins: Chapters 3.3, 3.4

Discrete Symmetries, Parity Parity and nature: The strong and electromagnetic interactions conserve parity. The weak interaction does not. Thus if we consider a Hamiltonian to be made up of several pieces: H = H s + H EM + H W Then the parity operator (P) commutes with H s and H EM but not with H W. The fact that [P, H W ]  0 constrains the functional form of the Hamiltonian. What does parity do to some common operations ? vector or polar vector x  - x or p  - p. axial or pseudo vectors J = x  p  J. time (t) t  t. nameformparity scalarrr+ pseudoscalarx(y  z)- vectorr- axial vectorr x p+ TensorF uv indefinite According to special relativity, the Hamiltonian or Lagrangian of any interaction must transform like a Lorentz scalar. Read Perkins: Chapters 3.5

Parity Violation in  -decay Co J=5 60 Ni* J=4 60 Ni* J=4 B-field J z =1 pvpv pvpv p e- YES NO Classic experiment of Wu et. al. (Phys. Rev. V105, Jan. 15, 1957) looked at  spectrum from: followed by: Note: 3 other papers reporting parity violation published within a month of Wu et. al.!!!!!   detector   detector   detector  counting rate depends on  p e which is – under a parity transformation Read Perkins: Chapters 3.5

게이지 대칭성 게이지대칭성을 가진다는 것은 측정 기준, 척도 혹은 측정을 하는 시공간의 위치에 변환이 생겨도 변하지 않는 것을 말한다. 게이지 대칭성은 우리 눈에 보이는 성질에 관한 대칭성이 아니라, 내부에 숨어 있는 성 질에 관한 대칭성이다. P175, LHC 현대물리학의 최전선 P Spring 2003 L6 Richard Kass

글로벌 대칭성 위치에 관계없이 모든 사람들이 한꺼번에 똑같이 움직이는 광역 변환에 대해 변하지 않는 것을 광역 대칭성  예 ) 바둑판 위의 정열 => 좌향좌 또는 우향우  예 ) 내재된 대칭성 - 주머니속의 시계 36

Ex) 게이지 장 (Guage Field) 게이지 대칭성을 유지하려면, 한 사람이 시계 바늘을 돌려서 기준을 바꾸면, 다른 사람에게 알려야 한다. 모든 사람의 시계바늘을 가는 줄로 연결 ⇒ 한 사람이 시계바늘을 돌리면, 다른 모든 이들도 알게 됨  시계바늘 돌림 ( 게이지변환 ) 은 상쇄되어 대칭성 유지  시계줄과 같은 매체 필요 ( 게이지 장 )  맥스웰의 전자기학 ( 전기장, 자기장 포텐셜 -> 게이지 장 ) 37

국소적 게이지변환과 글로벌 게이지 변환 위상이 공간의 위치에 따라 다르면 국소적 게이지라 부르고 위상이 공간의 어느 위치에서나 똑같으면 글로벌 게이지라 부른다. 38

39 Assessment Global gauge invariance implies the existence of a conserved current, according to Noether’s theorem. Local gauge invariance requires the introduction of massless vector bosons, restricts the form of the interactions of gauge bosons with sources, and generates interactions among the gauge bosons if the symmetry is non-Abelian. - Chris Quigg P63

40 Conservation of electric charge and gauge invariance Conservation of electric charge:  Q i =  Q f Evidence for conservation of electric charge: Consider reaction e -  v e which violates charge conservation but not lepton number or any other quantum number. If the above transition occurs in nature then we should see x-rays from atomic transitions. The absence of such x-rays leads to the limit:  e > 2x10 22 years There is a connection between charge conservation, gauge invariance, and quantum field theory. Recall Maxwell’s Equations are invariant under a gauge transformation: A Lagrangian that is invariant under a transformation U=e i  is said to be gauge invariant. There are two types of gauge transformations: local:  =  (x,t) global:  =constant, independent of (x,t) Conservation of electric charge is the result of global gauge invariance Photon is massless due to local gauge invariance Maxwell’s EQs are locally gauge invariant Read Perkins: Chapters 3.6

41 Gauge invariance, Group Theory, and Stuff Consider a transformation (U) that acts on a wavefunction (  ):  U  Let U be a continuous transformation then U is of the form: U=e i    is an operator. If  is a hermitian operator (  =  *T ) then U is a unitary transformation: U=e i  U + =(e i  ) *T = e -i  *T = e -i   UU + = e i  e -i  =1 Note: U is not a hermitian operator since U  U + In the language of group theory  is said to be the generator of U There are 4 properties that define a group: 1) closure: if A and B are members of the group then so is AB 2) identity: for all members of the set I exists such that IA=A 3) Inverse: the set must contain an inverse for every element in the set AA -1 =I 4) Associativity: if A,B,C are members of the group then A(BC)=(AB)C If  = (  1,  2,  3,..) then the transformation is “Abelian” if: U(  1 )U(  2 ) = U(  2 )U(  1 ) i.e. the operators commute If the operators do not commute then the group is non-Abelian. The transformation with only one  forms the unitary abelian group U(1) The Pauli (spin) matrices generate the non-Abelian group SU(2) S= “special”= unit determinant U=unitary n=dimension (e.g.2) Read Perkins: Chapters 3.6

42 Global Gauge Invariance and Charge Conservation The relativistic Lagrangian for a free electron is:  is the electron field (a 4 component spinor) m is the electrons mass  u = “gamma” matrices, four (u=0,1,2,3) 4x4 matrices that satisfy  u  v +  v  u =2g uv  u          t  x  y  z  Let’s apply a global gauge transformation to L By Noether’s Theorem there must be a conserved quantity associated with this symmetry! This Lagrangian gives the Dirac equation: Read Perkins: Chapters 3.6

43 E-L equation in 1D Global Gauge Invariance and Charge Conservation We need to find the quantity that is conserved by our symmetry. In general if a Lagrangian density, L=L( ,  x u ) with  a field, is invariant under a transformation we have: For our global gauge transformation we have: Plugging this result into the equation above we get (after some algebra…) The first term is zero by the Euler-Lagrange equation. The second term gives us a continuity equation. Result from field theory Read Perkins: Chapters 3.7

44 Global Gauge Invariance and Charge Conservation The continuity equation is: Recall that in classical E&M the (charge/current) continuity equation has the form: Also, recall that the Schrodinger equation give a conserved (probability) current: If we use the Dirac Lagrangian in the above equation for L we find: This is just the relativistic electromagnetic current density for an electron. The electric charge is just the zeroth component of the 4-vector: Therefore, if there are no current sources or sinks (  J=0) charge is conserved as: (J 0, J 1, J 2, J 3 ) =( , J x, J y, J z ) Result from quantum field theory Conserved quantity

45 The Dirac Equation on One Page A solution (one of 4, two with +E, two with -E) to the Dirac equation is : The function U is a (two-component) SPINOR and satisfies the following equation: The Dirac equation: Spinors are most commonly used in physics to describe spin 1/2 objects. For example: Spinors also have the property that they change sign under a rotation!

Helicity 46 Read Perkins: Chapters 3.6 Dirac Eq. with mass=0

카이랄 대칭성 좌우 대칭이 아닌 개체, 즉, 거울상과 실제 모습이 다른 개체에 대하여 우리는 그 개체 가 “Chirality 를 가지고 있다 ” 라고 표현한다.  한 쌍으로 나타남 => 각각 오른손잡이, 왼손잡이  예 ) 양 손 => 각각 Chirality 갖고 있음  예 ) 구두 한 켤레, 아미노산 … 좌우 대칭적 개체는 그에 반하여 “achiral (not chiral)” 이라고 표현한다 47

카이럴 대칭성 48 질량이 없는 광자만 카이럴 대칭성을 가진다. => “ 카이럴 대칭성 ” 파괴는 질량을 가진다. Nanbu 노벨상 2008 카이럴 대칭성 => 입자의 진행 방향에 대하여 스핀이 오른쪽 회전 (+1) 스핀이 왼쪽 회전 (-1)

49 Charge-conjugation Quantum Number

Charge Conjugation Charge Conjugation (C) turns particles into anti-particles and visa versa. C(proton)  anti-proton C(anti-proton)  proton C(electron)  positron C(positron)  electron The operation of Charge Conjugation changes the sign of all intrinsic additive quantum numbers: electric charge, baryon #, lepton #, strangeness, etc.. Variables such as spin and momentum do not change sign under C. The eigenvalues of C are  1 and, like parity, C is a multiplicative quantum number. It the interaction conserves C then C commutes with the Hamiltonian, [H,C]A=0. note: c is sometimes called the “charge parity” of the particle. Most particles are NOT eigenstates of C. Consider a proton with electric charge = q. Let Q= charge operator, then: Q|q>=q|q> and C|q>=|-q> CQ|q>=qC|q> = q|-q> and QC|q>=Q|-q> = -q|-q> [C,Q]|q>=2q|q> Thus C and Q do not commute unless q=0. We get the same result for all additive quantum numbers! Only particles that have all additive quantum numbers = 0 are eigenstates of C. e.g.  These particle are said to be “self conjugate”. strong and EM conserve C weak violates C M&S pages Read Perkins: Chapters 3.7

Charge Conjugation How do we assign c to particles that are eigenstates of C? a) photon. Consider the interaction of the photon with the electric field. As we previously saw the interaction Lagrangian of a photon is: L EM =J u A u with J u the electromagnetic current density and A u the vector potential. By definition, C changes the sign of the EM field and thus J transforms as: CJC -1 =-J (this is how operators transform in QM) Since C is conserved by the EM interaction we have: CL EM C -1 = L EM CJ u A u C -1 = J u A u CJ u C -1 C A u C -1 = -J u C A u C -1 = J u A u C A u C -1 = -A u Thus the photon (as described by A) has c = -1. A state that is a collection of n photons has c = (-1) n. b) The  o : Experimentally we find that the  o decays to 2  s and not 3  s. This is an electromagnetic decay so C is conserved. Therefore we have: C  o = (-1) 2 = +1 particles with the same quantum numbers as the photon (  ) have c = -1. particles with the same quantum numbers as the  o (  ) have c = +1. Read Perkins: Chapters 3.8

Charge Conjugation and Parity Experimentally we find that all neutrinos are left handed and anti-neutrinos are right handed (assuming massless neutrinos). By left or right handed we mean: left handed: spin and z component of momentum are anti-parallel right handed: spin and z component of momentum are parallel If neutrinos were not left handed, the ratio would be > 1! M&S pages This left/right handedness is illustrated in  +  l + l decay: p S p e,  S e,   Angular momentum conservation forces the charged lepton (e,  ) to be in the “wrong” handed state, e.g. a left handed positron (e + ). The probability to be in the wrong handed state ~m l 2 handedness~2x10 -5 phase space ~5 W+W+ e+e+ v p  =0 S  =0 Read Perkins: Chapters 3.8, 7.8 H=-1 Axial Vector Current

‘CP’ 란 무엇인가 ? 53

54

Charge Conjugation and Parity parity C C CPCP  p  p right handed  p left handed  p In the strong and EM interaction C and P are conserved separately. In the weak interaction we know that C and P are not conserved separately. BUT the combination of CP should be conserved! Consider how a neutrino (and anti-neutrino) transforms under C, P, and CP. Experimentally we find that all neutrinos are left handed and anti-neutrinos are right handed. So, CP should be a good symmetry Read Perkins: Chapters 3.8

56 C, P, T violation? Since early universe… “Alice effect” Intuitively… Boltzmann and S=kln  C is violated P is violated T is violated

57 Charge conjugate and Parity CP is the product of two symmetries: C for charge conjugation, which transforms a particle into its antiparticle, and P for parity, which creates the mirror image of a physical system.symmetriescharge conjugationantiparticleparity C

58 C P CP

59 Mirror symmetry  Parity P All events should occur in exactly the same way whether they are seen directly or in mirror. There should not be any difference between left and right and nobody should be able to decide whether they are in their own world or in a looking glass world Charge symmetry  Charge C Particles should behave exactly like their alter egos, antiparticles, which have exactly the same properties but the opposite charge Time symmetry  Time T Physical events at the micro level should be equally independent whether they occur forwards or backwards in time. There are 3 different principles of symmetry in the basic theory for elementary particles Three principles of symmetry Violation in

60 Mirror symmetry  Parity P All events should occur in exactly the same way whether they are seen directly or in mirror. There should not be any difference between left and right and nobody should be able to decide whether they are in their own world or in a looking glass world Charge symmetry  Charge C Particles should behave exactly like their alter egos, antiparticles, which have exactly the same properties but the opposite charge Time symmetry  Time T Physical events at the micro level should be equally independent whether they occur forwards or backwards in time. There are 3 different principles of symmetry in the basic theory for elementary particles Three principles of symmetry Violation in Violation in

61 C and P violation! Experiments show that only circled ones exist in Nature C and P are both maximally violated! But, CP and T seems to be conserved, or is it? CP We can test this in 1 st generation meson system: Pions

62 CP and T violation! For 37 years, CP violation involve Kaons only! Is CP violation a general property of the SM or is it simply an accident to the Kaons only? CP violation T violation K 0   +   - We can test this in 2 nd generation meson system: Kaons Need 3 rd generation system: B-mesons and B-factories

63 Ref. 원은일교수 (2008)

64 Ref. 원은일교수 (2008)

Why CP violation? 66

67 노벨 물리학상 2008 Citation: 5483

68 Ref. 원은일교수 (2008)

69 Ref. 원은일교수 (2008)

70 Ref. 원은일교수 (2008)

Neutral Kaons and CP violation In 1964 it was discovered that the decay of neutral kaons sometimes (10 -3 ) violated CP! Thus the weak interaction does not always conserve CP! In 2001 CP violation was observed in the decay of B-mesons. CP violation is one of the most interesting topics in physics: The laws of physics are different for particles and anti-particles! What causes CP violation ? (it is put into Standard Model) Is the CP violation observed with B’s and K’s the same as the cosmological CP violation? To understand how CP violation is observed with k’s and B’s need to discuss MIXING. Mixing is a (QM) process where a particle can turn into its anti-particle! As an example, lets examine neutral kaon mixing. We will discuss B-meson mixing later... The neutral kaon is a bound state of a quark and an anti-quark: In terms of quark content these are particle and anti-particle. The k 0 has the following additive quantum numbers: strangeness = +1 charge = baryon # = lepton # = charm = top = 0 I 3 = -1/2 note: the k 0 ’s isospin partner is the k +. Also, I 3 changes sign for anti-particles. The k 0 and k 0 are produced by the strong interaction and have definite strangeness. Thus they cannot decay via the strong or electromagnetic interaction. M&S pages Read Perkins: Chapters 3.13

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Neutral Kaons, Mixing, and CP violation The neutral kaon decays via the weak interaction, which does not conserve strangeness. Let’s assume that the weak interaction conserves CP. Then the k 0 and k 0 are NOT the particles that decay weakly since they are not CP eigenstates. However, we can make CP eigenstates out of a linear combination of k 0 and k 0. If CP is conserved in the decay of k 1 and k 2 then we expect the following decay modes: k 1  two pions (  +  - or  0  0 ) (CP = +1 states) k 2  three pions (  +  -  0 or  0  0  0 ) (CP = -1 states) In 1964 it was found that every once in a while (  1/500) k 2  two pions ! This is just a QM system in two different basis. M&S Read Perkins: Chapters 3.13, 7.14

Neutral Kaons, Mixing, and CP violation The k 0 and k 0 are eigenstates of the strong interaction. These states have definite strangeness, are NOT CP eigenstates, and are particle/anti-particle. They are produced in strong interactions (collisions) e.g. The k 1 and k 2 are eigenstates of the weak interaction, assuming CP is conserved. These states have definite CP but are not strangeness eigenstates. Each is its own anti-particle. These states decay via the weak interaction and have different masses and lifetimes. In 1955 (!) Gell-Mann and Pais pointed out that the k 1 and k 2 lifetimes should be very different since there was more decay energy (phase space) available for k 1 than k 2. m k -2m   219 MeV/c 2 m k -3m   80 MeV/c 2 Thus, expect k 1 to have the shorter lifetime. The lifetimes were measured to be:  1  9x sec ( )  2  5x10 -8 sec (Lande et. al. Phys. Rev. V103, 1901 (1956)) We can use the lifetime difference to produce a beam of k 2 ’s from a beam of k 0 ’s. 1) produce k 0 ’s using  - p  k 0 . 2) let the beam of k 0 ’s propagate in vacuum until the k 1 component dies out. 1 GeV/c k 1 travels on average  5.4 cm 1 GeV/c k 2 travels on average  3100 cm Need to put detector m away from target. m 2 - m 1 =3.5x10 -6 eV M&S pages Read Perkins: Chapters 3.13, 7.14

Neutral Kaons, Mixing, and CP violation How do we look for CP violation with a k 2 beam? Look for decays that have CP = +1 ! k 2  (  +  - or  0  0 ) Experimentally we find: k 2   +  - about 0.2% of the time (Christenson et. al. PRL V13, 138 (1964)) k 2   0  0 about 0.1% of the time Can also look for differences in decays that involve matter and anti-matter: Nature differentiates between matter and antimatter ! CP violation has recently (2001) been unambiguously measured in the decay of B-mesons. This is one of the most interesting areas in HEP and we will discuss this topic later in the course. Observing CP violation with B-mesons is much more difficult than with kaons! Unfortunately (for us) the B S and B L have essentially the same lifetime and as a result there is no way to get a beam of B L and look for “forbidden” decay modes. In addition, it is much harder to produce large quantities of B-mesons than kaons. Read Perkins: Chapters 3.13, 7.14

78 Conservation Rules Conserved Quantity WeakElectromagneticStrong I(Isospin)No (  I=1 or ½) NoYes (No in 1996) S(Strangeness)No (  S=1,0) Yes C(charm)No (  C=1,0) Yes P(parity)NoYes C(charge)NoYes CPNoYes

79 References Class P by Richard Kass (2003) B.G Cheon’s Summer School (2002) S.H Yang’s Colloquium (2001) Class by Jungil Lee (2004) PDG home page (

80 Back-up

81 Conservation Laws When something doesn’t happen there is usually a reason! Read Perkins: Chapters 1.4, 1.10 That something is a conservation law ! A conserved quantity is related to a symmetry in the Lagrangian that describes the interaction. (“Noether’s Theorem”) A symmetry is associated with a transformation that leaves the Lagrangian invariant. time invariance leads to energy conservation translation invariance leads to linear momentum conservation rotational invariance leads to angular momentum conservation Familiar Conserved Quantities Quantity Strong EM Weak Comments energyYYYsacred linear momentumYYYsacred ang. momentumYYYsacred electric chargeYYYsacred

Discrete Symmetries, Parity Thus if H conserves parity then it should transform as like a scalar. If H does not conserve parity then it must contain some pseudoscalar terms. Fermi’s original theory of weak interactions (  -decay) considered the Hamiltonian to be made up of bilinear combination of vector operators (V,V). The observation of Parity violation showed that this was wrong ! A more general form of a weak Hamiltonian that does not conserve parity is of the form: H W = (S,S) + (S,PS) + (V,V) + (V,AV) + () It is an experimental fact that the weak interactions where a charged lepton turns into a neutrino (“charged current”) can be described by a Hamiltonian of the form (sometimes called a “V-A” interaction): H W = (V,V) + (V,AV) This is parity violating since (V,V) has + parity but (V,AV) has - parity. Examples: In QED the current is of the form: which transforms like a vector. In weak interactions the charged current (involves a W boson) is of the form: which contains both vector and axial vector terms, i.e. does not conserve parity.

83 Local Gauge Invariance and Physics Some consequences of local gauge invariance: a) For QED local gauge invariance implies that the photon is massless. b) In theories with local gauge invariance a conserved quantum number implies a long range field. e.g. electric and magnetic field However, there are other quantum numbers that are similar to electric charge (e.g. lepton number, baryon number) that don’t seem to have a long range force associated with them! Perhaps these are not exact symmetries!  evidence for neutrino oscillation implies lepton number violation. c) Theories with local gauge invariance can be renormalizable, i.e. can use perturbation theory to calculate decay rates, cross sections, etc. Strong, Weak and EM theories are described by local gauge theories. U(1) local gauge invariance first discussed by Weyl in 1919 SU(2) local gauge invariance discussed by Yang&Mills in 1954 (electro-weak)  e i  (x,t)   is represented by the 2x2 Pauli matrices (non-Abelian) SU(3) local gauge invariance used to describe strong interaction (QCD) in 1970’s  e i  (x,t)   is represented by the 3x3 matrices of SU(3) (non-Abelian)

84 Local Gauge Invariance and QED Consider the case of local gauge invariance, = (x,t) with transformation: The relativistic Lagrangian for a free electron is NOT invariant under this transformation. The derivative in the Lagrangian introduces an extra term: We can MAKE a Lagrangian that is locally gauge invariant by adding an extra piece to the free electron Lagrangian that will cancel the derivative term. We need to add a vector field A u which transforms under a gauge transformation as: A u  A u +  u  (x,t) with (x,t)=-q  (x,t) (for electron q=-|e|) The new, locally gauge invariant Lagrangian is:

85 The Locally Gauge Invariance QED Lagrangian Several important things to note about the above Lagrangian: 1) A u is the field associated with the photon. 2) The mass of the photon must be zero or else there would be a term in the Lagrangian of the form: m  A u A u However, A u A u is not gauge invariant! 3) F uv =  u A v -  v A u and represents the kinetic energy term of the photon. 4) The photon and electron interact via the last term in the Lagrangian. This is sometimes called a current interaction since: In order to do QED calculations we apply perturbation theory (via Feynman diagrams) to J u A u term. 5) The symmetry group involved here is unitary and has one parameter  U(1) e-e- e-e-  AuAu JuJu