Letters in High Energy Physics LHEP 125, 1, 2019 Group–Theoretical Origin of CP Violation Mu–Chun Chen and Michael Ratz University of California Irvine, Irvine CA 92697, USA Abstract This is a short review of the proposal that violation may be due to the fact that certain finite groups do CP not admit a physical transformation. This origin of violation is realized in explicit string compacti- CP CP fications exhibiting the Standard Model spectrum. Keywords: 1.2. and Clebsch–Gordan coefficients CP DOI: 10.31526/LHEP.1.2019.125 It turns out that some finite groups do not have such outer au- tomorphisms but still complex representations. These groups thus clash with ! Further, they have no basis in which all CP 1. INTRODUCTION Clebsch–Gordan coefficients (CGs) are real, and violation CP As is well known, the flavor sector of the Standard Model (SM) can thus be linked to the complexity of the CGs [3]. violates , the combination of the discrete symmetries and CP C . This suggests that flavor and violation have a common P CP origin. The question of flavor concerns the fact that the SM 2. CP VIOLATION FROM FINITE GROUPS fermions come in three families that are only distinguished by 2.1. The canonical transformation their masses. SU(2) interactions lead to transitions between CP L Let us start by collecting some basic facts. Consider a scalar these families, which are governed by the mixing parameters field operator in the CKM and PMNS matrices. These mixing parameters are completely unexplained in the SM. Furthermore, violation Z h i CP 3 1 i p x † i p x d f(x) = d p a(~p) e− · + b (~p) e · , (2.1) manifests in the SM through the non–zero phase q in the CKM 2E matrix [1]. In the lepton sector, the latest measurements from ~p T2K as well as the global fit for neutrino oscillation parameters where a annihilates a particle and b† creates an antiparticle. also hint at non–zero value for the Dirac phase d in the PMNS ` The operation exchanges particles and antiparticles, matrix [2], which will, if proved, establish violation of in CP CP the lepton sector. 1 (CP)− a(~p) CP = h b( ~p) , (2.2a) The observed repetition of families, i.e., the fact that the CP − 1 † † quarks and leptons appear in 3 generations, hints at a flavor (CP)− a (~p) CP = h∗ b ( ~p) (2.2b) CP − symmetry under which the generations transform nontrivially. 1 (CP)− b(~p) CP = h∗ a( ~p) , (2.2c) The main punchline of this review is the statement that certain CP − 1 † † flavor symmetries clash with [3, 4]. In other words, vi- (CP)− b (~p) CP = h a ( ~p) , (2.2d) CP CP CP − olation can be entirely group theoretical in origin. where h is a phase factor. On the scalar fields, transfor- CP CP 1.1. What is a physical transformation? mations act as CP Charge conjugation inverts, by definition, all currents. This CP C f(x) h f∗( x) . (2.3) implies that Standard Model representations R get mapped to 7−−! CP P their conjugates, R. Likewise, parity exchanges the (0, 1/2) At this level, h can be viewed as the freedom of rephasing the P and (1/2, 0) representations of the Lorentz group, which corre- field, i.e., a choiceCP of field basis. Later, when we replace h by sponds to complex conjugation at the level of SL(2, C). That some matrix U , this will still reflect the freedom to chooseCP CP is, at the level of GSM SL(2, C), is represented by the a basis. The important message here is that there is a well– × CP (unique) nontrivial outer automorphism. defined operation, the transformation, which exchanges CP This fact has led to the suspicion that any nontrivial outer particles with antiparticles. It is this very transformation which automorphism can be used to coin a valid transforma- is broken in the K0 K0 system, and whose violation is a pre- CP − tion [5]. However, this turns out not to be the case [4]. To see requisite for baryogenesis. this, let us review why we care about whether or not is vio- CP lated. One reason we care is that violation is a prerequisite CP 2.2. vs. outer automorphisms for baryogenesis [6], i.e., the creation of the matter–antimatter CP asymmetry of our universe. Therefore, a physical transfor- Next let us review what does in the context of most of the CP CP mation exchanges particles and antiparticles, a requirement an continuous (i.e., Lie) groups. If the representation under con- sideration is real, the canonical does the job. For complex arbitrary outer automorphism may or may not fulfill. As dis- CP cussed in detail in [4], transformations are linked to class– representations, involves a nontrivial outer automorphism CP CP inverting outer automorphisms. (cf. Figure 1). In particular, in the context of the Standard Model gauge group and the usual theories of grand unification (GUTs), G = SU(3) SU(2) U(1) SU(5) SO(10) E SM C × L × Y ⊂ ⊂ ⊂ 6 1 Letters in High Energy Physics LHEP 125, 1, 2019 1 2 n 2 n 1 Hence, the canonical transformation is not an (outer) au- − − CP tomorphism of T0(in this basis). Therefore, in order to warrant conservation, one needs to impose a so–called generalized CP CPg FIGURE 1: acts as the unique nontrivial outer automorphism on the transformation CPg under which f f as usual but CP CP 7−−! ∗ SU(N) groups. 0 1 0 1 0 1 x 0 1 y x1 1∗ y1 1∗ CPg B C CPg B C @ x2 A B x3∗ C , B y2 C B y3∗ C . (2.7) always involves outer automorphisms, 7−−! @ A @ A 7−−! @ A CP x y 3 x2∗ 3 y2∗ × ⊂ 2.4. Constraints on generalized transformations In order for a transformationCP not to clash with the group, ⊂ CP i.e., in order to avoid mapping something that is invariant un- der the symmetry transformations to something that isn’t (cf. (2.6)), it has to be an automorphism u : G G of the group. ! ⊂ An automorphism u corresponding to a physical transfor- CP mation has to fulfill the consistency condition [5] (see also [7]) One may thus expect that this is also true for discrete (fam- † ily) symmetries. However, this is an accident, and is already not r u(g) = U r(g)∗ U g G . (2.8) CP CP 8 2 the case for SO(8), the only Lie group with a non–Abelian outer Here, U is a unitary matrix that enters the generalized automorphism group, namely S . Which of those outer auto- CP 3 transformation, morphisms, if any, corresponds to the physical transforma- CP CP tion? As we shall discuss next, in particular for finite groups, it CP F(x) g U F ( x) , (2.9) is not true that there is a unique outer automorphism. In addi- 7−−! CP ∗ P tion, not all non-trivial outer automorphism qualify as a physi- where F denotes collectively the fields of the theory/model, cal transformation [3, 4]. and (t,~x) = (t, ~x) as usual. In particular, each represen- CP P − tation gets mapped on its own conjugate, i.e., UCP is block– 2.3. Generalized transformations diagonal in Equation (2.9), To see this, considerCP a setting with discrete symmetry G. One 0 1 0 1 0 1 can now impose a so–called generalized transformation, " -% " Bfr C B Ur C Bfr∗ C CP B i1 C B i1 C B i1 C 1 B C B C B C (CP)− a(~p) CP = UCP b( ~p) , (2.4a) B # C B.& C B # C − B C CPg B C B C 1 † † † B C B C B C , (CP)− a (~p) CP = b ( ~p) UCP (2.4b) B " C B -% C B " C fri 7−−! Uri fr∗ − B 2 C B 2 C B i2 C 1 † B C B C B C (CP)− b(~p) CP = a( ~p) UCP , (2.4c) B C B C B C − @ # A @ .& A @ # A 1 † † . .. (CP)− b (~p) CP = U a ( ~p) , (2.4d) . CP − . where a is a vector of annihilation operators and a† is a vector (2.10) of creation operators. UCP is a unitary matrix. where the Uria are unitary matrices that depend on the rep- The reader may wonder whether or not the need to “gener- resentation ri only. The a subscripts in fria label the particles alize” is specific to the transformation. This is not the case. CP whereas the ia subscripts indicate the representations, i.e., dif- A very close analogy is the Majorana condition. In the Majo- ferent particles can furnish the same representations under G. rana basis, it boils down to the requirement that Y = Y∗ for a The transformation law (2.10) disagrees with [5], where it was Dirac spinor Y. However, in the Weyl or Dirac basis, this condi- suggested that one can use any outer automorphism in order tion becomes Y = C Y∗ with some appropriate matrix C. That to define a viable transformation. CP is, the antiparticle of a particle described by Y is described by Therefore, the requirement that the candidate transforma- C Y∗, and not just Y∗. Likewise, in the above discussion around tion is a physical transformation, which exchanges particles (2.4), the conjugate (i.e., antiparticle up to a transformation CP CP and their antiparticles, amounts to demanding that u be class– of the spatial coordinates) of a scalar described by f will be inverting.