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When we define C symmetry, the CHC−1. The operation of CPT on a particle already interactions are considered as a part of the equations, implies C symmetry(H = H). If the particle remains

µ itself under P , the Hamiltonian should be also be the (i∂/ − m)ψ = ±eAµγ ψ for defining C same but the practical calculation of lifetime is based on the H = P HP −1 free and interactions in separate, which neglects their relative signs and since the mass of a particle is the same regardless of z component of angular , m can be ignored (i∂/ − m)ψ = 0 and ± eA γµψ for lifetime µ and thus and the opposite signs are known from other independent −1 physical observations indicating two opposite charges. Hmass = THmassT for the mass term Now we have

III. PROOF OF MASS EQUALITY −1 −1 −1 −1 ΘHmass Θ = CPTHmassT P C −1 −1 = CPHmassP C Let us review how the mass and lifetime equality be- −1 tween particle and antiparticle are proved and investigate = CHmassC these proofs[3, 4]. = Hmass Theorem. Mass equality between particles and antipar- by definition H ≡ CHC−1 and thus H = H. ticles However, this proof would fail with C violation even if CPT symmetry is conserved. For example, if CPT Proof. Let |pim be the state of an at theorem holds(ΘHΘ −1 = H ) but C is violated in a way rest with its z component angular momentum m, Since −1 the mass of the particle is given by the expectation value, H ≡ CHC = −H, then the mass of particle and an- tiparticle is different. massp = hp|H|pim massp = hp|H|pim where H is the total Hamiltonian, real and independent −1 −1 = −hp|C CHC C|pi of m. Hence it equals its complex conjugation. We have m = −hp|H|pim ∗ −1 −1 m massp = hp|H|pim = hp|Θ ΘHΘ Θ|pi . ≡ −massp where the operator Θ is defined to be And in order to investigate whether the masses of particle Θ ≡ CPT . and antiparticle, the equation of particle and antiparticle is of interest, not that of CPT transformed particle is −1 −1 Apart from a multiplicative phase factor, under C the unless it is identified as antiparticle(ΘHΘ = CHC ). state becomes |pim, under P it remains itself, but under T , m is changed into −m, as given by (i∂/ − m)ψ = 0 for particle CP T ∗ (i∂/ − m)ψ = 0 for CPT transformed particle T |j, mi = U |j, mi = U |j, mi = eiπJy |j, mi T T (i∂/ − m)ψc = 0 for antiparticle = (−1)j+m|j, −mi Therefore, the mass of particle and antiparticle should Therefore, be proved by the conservation of C(H = H ≡ CHC−1), iθ not by the CPT theorem, as in Θ|pim = e |pi−m

Since ΘHΘ −1 = H by the CPT theorem, the above ex- massp = hp|H|pim −1 −1 pression can also be written as = hp|C CHC C|pim

= hp|H|pim massp = hp|H|pi−m ≡ massp massp ≡ hp|H|pi−m Therefore, To be more accurate, the equality of mass is pre- requisite for C symmetry since the transformation of −1 massp = massp equation(CHC ) cannot be equated to H unless its mass and coupling constants consisting of the equation when H = H. are the same. The free Dirac equation(or the Dirac 3

Hamiltonian), for example, cannot be transform from one Theorem. The lifetimes of A and A are the same, even to another(L0 → L0) unless m = m. if Hweak is not invariant under CPT. µ L0 = ψ(iγ ∂µ − m)ψ Proof. The Hamiltonian H commutes with Θ ≡ CPT by µ L0 = ψ(iγ ∂µ − m)ψ the CPT theorem 1 Therefore, the mass of particle and antiparticle is re- ΘHΘ − =+H quired to be the same for the conservation of C sym- metry (H = H ≡ CHC−1) and the proper definition of and if CPT is violated, then we have mass representation should show that C symmetry, not −1 the CPT theorem, implies the mass equality. ΘHΘ = −H

Following the same steps as before, one obtains IV. PROOF OF LIFETIME EQUALITY ∗ −1 −1 hB|Hweak|Ai = −hTB|C P HweakP C|T Ai,

Consider a Hamiltonian −1 −1 −1 if CPT is violated( C P T HweakT P C = −Hweak H = Hstrong + Hweak ). If Hweak commutes (or anticommutes) with P , then where both terms are invariant under a proper Lorentz ∗ −1 hB|Hweak|Ai = ∓hTB|C HweakC|T Ai. transformation and Hstrong is assumed to be invariant under C,P (), and T ( reversal), for example, For a spinless system, −1 CHstrongC = Hstrong ∗ hB|Hweak|Ai = ∓hB|Hweak|Ai. Theorem. If a particle A decays through the interaction Hweak, and if the particle and its antiparticle A do not This shows that the lifetimes of A and A are the same. decay into the same final products (as e.g. when A is charged), then to the lowest order of Hweak the lifetimes of A and A are the same, even if Hweak is not invariant These theorems are limited as they exclude C viola- under charge conjugation. tions of different mass and coupling constants(m 6= m or e 6= e) where we cannot find a proper transformation Proof. Consider particle A with spin zero and the final of equations(H 6= CHC−1 and thus |Ai 6= C|Ai). If C states B and B in the decays would also have spin zero. symmetry is violated with different coupling constants, A → B, A → B, then the lifetimes of particle and antiparticle would be different. However, if particle and antiparticle have the Using the identity same mass and coupling constants, the lifetimes of par- ∗ ticle and antiparticle are the same regardless of C and hψ1|ψ2i = hT ψ1|T ψ2i, CPT symmetry if no interference is assumed one obtains 2 2 Γ ∝ |Mw| + |Mg| ∗ −1 hB|Hweak|Ai = hTB|THweakT |T Ai = hTB|C−1P −1H P C|T Ai, where the signs of S matrix Mw, Mg are irrelevant to weak the physical observation of lifetime. by the CPT theorem. If Hweak commutes (or anticom- mutes) with P , then ∗ −1 V. CONCLUSION hB|Hweak|Ai = ±hTB|C HweakC|T Ai. For a spinless system, The definitions and differences of C symmetry and par- ticle and antiparticle symmetry are discussed and the |T Ai = |Ai, |TBi = |Bi. proofs of mass and lifetime equality are reviewed and crit- Hence icized for their ambiguity and exclusiveness. The conser- vation of C, not CPT , symmetry requires the mass and ∗ −1 hB|Hweak|Ai = ±hB|C HweakC|Ai coupling constants to be the same between particle and antiparticle and the lifetime could be the same regardless = ±hCB|Hweak|CAi = ±hB|Hweak|Ai. of C and CPT violation. Therefore, we can conclude that This shows that the lifetimes of A and A are the same. the CPT theorem does not imply the equality of mass and lifetime and particle and antiparticle symmetry, not Following this proof, one can also show the lifetime CPT , is appropriate when the equality of mass and life- equality of particle and antiparticle under CPT violation. time is implied [1, 5, 6, 7, 8, 9, 10]. 4

[1] H. Murayama, Phys. Lett. B 597 (2004) 73. [7] V. Barger, D. Marfatia and K. Whisnant, Int. J. Mod. [2] C. Itzykson and J.B. Zuber, Quantum Theory, Phys. E 12 (2003) 569. McGraw-Hill, Inc., (1980) [8] KamLAND Collaboration, K. Eguchi et al., Phys. Rev. [3] T.D. Lee, and Introduction to Field The- Lett. 90 (2003) 021802. ory, Harwood Academic Publishers, (1981). [9] KTeV Collaboration, A. Alavi-Harati et al., Phys. Rev. [4] T.D. Lee and C.N. Yang, Phys. Rev. 106, (1957) 340. D 67 (2003) 012005. [5] R.D. McKeown and P. Vogel, Phys. Rept.394 (2004) 315. [10] G. Barenboim, L. Borissov, J. Lykken and A. Yu. [6] John N. Bahcall, M.C. Gonzalez-Garcia and Carlos Pena- Smirnov, JHEP 0210 (2002) 001. Garay, JHEP 0408 (2004) 016.