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 δ φ(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 δ 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 = η 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 = η∗ b ( ~p) (2.2b) CP − symmetry under which the generations transform nontrivially. 1 (CP)− b(~p) CP = η∗ 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 = η a ( ~p) , (2.2d) CP CP CP − olation can be entirely group theoretical in origin. where η 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 φ(x) η φ∗( 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, η 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 η 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 φ φ as usual but CP CP 7−−→ ∗ SU(N) groups. x y x1 1∗ y1 1∗ CPg CPg x2 x3∗ , y2 y3∗ . (2.7) always involves outer automorphisms, 7−−→ 7−−→ 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 ρ u(g) = U ρ(g)∗ U g G . (2.8) CP CP ∀ ∈ 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 Φ(x) g U Φ ( 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 Φ 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 can now impose a so–called generalized transformation, ↑ -% ↑ φr Ur φr∗ CP i1 i1 i1 1 (CP)− a(~p) CP = UCP b( ~p) , (2.4a) ↓ .& ↓ − CPg 1 † † † , (CP)− a (~p) CP = b ( ~p) UCP (2.4b) ↑ -% ↑ φri 7−−→ Uri φr∗ − 2 2 i2 1 † (CP)− b(~p) CP = a( ~p) UCP , (2.4c) − ↓ .& ↓ 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 φria 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 Ψ = Ψ∗ for a The transformation law (2.10) disagrees with [5], where it was Dirac spinor Ψ. However, in the Weyl or Dirac basis, this condi- suggested that one can use any outer automorphism in order tion becomes Ψ = C Ψ∗ with some appropriate matrix C. That to define a viable transformation. CP is, the antiparticle of a particle described by Ψ is described by Therefore, the requirement that the candidate transforma- C Ψ∗, and not just Ψ∗. 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 φ will be inverting. In all known cases, u can be taken to be an automor- described by U φ∗, see (2.9) below. So, in a way, U is the phism of order two. Of course, this does not exclude the inter- CP CP analogy of the matrix C for Dirac fermions. esting possibility to make part of a higher–order transfor- CP As is evident from this argument and as pointed out in [5], mation [8]. generalizing may not be an option, but a necessity. To see CP this, consider a model in which G is A (or T ). Then a T – 4 0 0 2.5. vs. –like transformations invariant contraction/coupling is given by However,CP it isCP important to distinguish physical transfor- h i 2 CP φ1 (x3 y3) ∝ φ x y + ω x2 y2 + ω x3 y3 , (2.5) mations, and their proper generalizations, from –like trans- 2 11 1 1 CP ⊗ ⊗ 10 formations. Unfortunately, the latter have sometimes been 1 where ω = e2π i/3. Crucially, the canonical transformation called “generalized transformations” in the literature. CP CP maps this invariant contraction to something noninvariant, 1 CP CP CP More detailed comments on the literature can be found in Section VII.15 of x x & y y & φ φ . (2.6) [9]. 7−−→ ∗ 7−−→ ∗ 7−−→ ∗ 2 Letters in High Energy Physics LHEP 125, 1, 2019
However, some of the proposed “generalized transforma- Even though the steps in Fig. 2 may, at first sight, appear CP tions” do not warrant physical conservation. Thus, they do a bit cumbersome, one should remember that they allow us to CP not have a connection to the observed violation in the CKM uniquely determine, in an automatized way, whether or not a CP sector, nor to baryogenesis and so on. That is, the violation of symmetry has a basis in which all CG’s are real, or, if a symme- physical is a prerequisite of a nontrivial decay asymmetry, try clashes with . Of course, this analysis is independent of CP CP but the violation of a so–called “generalized transforma- bases, as it should be. CP tion” is not. That is to say some of the operations dubbed “gen- eralized transformation” in the literature are not physical CP 2.7. Three types of groups transformations, which is why we refer to them as “ – CP CP Given these tools, one can distinguish between three types of like”. groups [4]: Given all these considerations, it is a valid question whether or not one can impose a physical in any model. CP Case I: for all involutory automorphisms uα, i.e., automor- As mentioned above, this is not the case. Certain finite sym- phisms that square to unity, of the flavor group there is at metries clash with . Here “clash” means that any physi- least one representation r for which FS (r ) = 0. Such CP i uα i cal transformation maps some G–invariant term(s) on non- CP discrete symmetries clash with . invariant terms, and thus does not comply with G, i.e., is not an CP automorphism thereof. We will discuss next how one may tell Case II: there exists an involutory automorphism u for which those symmetries that clash with apart from those which the FSu’s for all representations are non–zero. Then there CP do not. are two sub–cases: Case II A: all FS ’s are +1 for one of those u’s. In this 2.6. The Bickerstaff–Damhus automorphism (BDA) u case, there exists a basis with real Clebsch–Gordan In a more group–theoretical language the question whether or coefficients. The BDA is then the automorphism not one can impose can be rephrased as whether or not that corresponds to the physical transforma- CP CP a given finite group has a so–called Bickerstaff–Damhus auto- tion.. morphism (BDA) [10] u, Case II B: some of the FS ’s are 1 for all candidate u’s. u − † That means that there exists no BDA, and, as a con- ρr (u(g)) = Ur ρr (g)∗ U g G and i , (2.11) i i i ri ∀ ∈ ∀ sequence, one cannot find a basis in which all CG’s where U is unitary and symmetric. The existence of a BDA im- are real. Nevertheless, any of the u’s can be used to define a physical transformation. plies the existence of a basis in which all Clebsch–Gordan (CG) CP coefficients are real. In physics, such a basis is often referred to The distinction between the groups is illustrated in Fig. 3. as “ basis”. The connection between the BDA, the complex- CP ity of the CG’s, and has first been pointed out in [3].2 CP Of course, this raises the question whether or not one CP can tell if a given group has an BDA. There is a rather 3. VIOLATION WITH AN UNBROKEN simple criterion for this, based on the so–called extended CP TRANSFORMATION 3 twisted Frobenius–Schur indicator (see [10, 12] for the so– Having seen that there are finite groups that do not admit a called twisted Frobenius–Schur indicator), physical transformation, one may wonder about the follow- CP n 1 ing question: if one obtains this finite group from a continuous (n) (dim r ) FS (r ) := i − χ (g u(g ) g u(g )) , one by spontaneous breaking, at which stage does violation u i n ∑ ri 1 1 n n CP G g G ··· arise? That is, take an SU(N) gauge symmetry, impose , and | | i ∈ CP (2.12) break it down to a type–I subgroup. The obvious options how violation may come about include CP where χr denotes the character and i 1. gets broken by the VEV that breaks SU(N) to G, and ( CP ord(u)/2 if ord(u) is even, 2. the resulting setting always has additional symmetries n = (2.13) and does not violate . ord(u) if ord(u) is odd. CP Rather surprisingly, none of these are the true answer. As It has the crucial property demonstrated explicitly in an example in which an SU(3) sym- metry gets broken to T7 = Z3 o Z7, the outer automorphism of (n) ( ) T FSu (ri) = 1 i u is class–inverting . (2.14) SU 3 merges into the outer automorphism of 7, which how- ± ∀ ⇐⇒ ever does not entail conservation [13]. CP So one has to scan over all candidate automorphisms u to de- This leads to a novel way to address the strong prob- CP termine whether one of them is a BDA, a task that can be au- lem. Start with a theory based on SU(3) SU(3)F (and of C × tomatized. course the other gauge symmetries of the Standard Model). Now impose , which implies that the coefficient θ of the µν CP QCD GµνGe term vanishes. Next, break the continuous flavor 2However, the example used there, T , turns out not to be of the violating 0 CP symmetry down to a type I flavor symmetry. Then θ still van- type. ishes, but is violated in the flavor sector. This is required to 3Recall that the Frobenius–Schur indicator FS(r) allows one to distinguish be- CP solve the strong problem of the Standard Model. An explicit tween real, pseudo–real and complex representations, for which FS(r) takes the CP values 1, 1 and 0, respectively [11]. example will be discussed elsewhere. − 3 Letters in High Energy Physics LHEP 125, 1, 2019
G has G has only order G class–inverting | | no yes irreps of odd of G is odd involutory automorphisms dimension
no yes no yes
there is an there exists automorphism u there exists a basis no basis no yes with all FSu’s with real CG’s with real CG’s equal to +1
FIGURE 2: Sequence of steps to determine whether or not a group admits a basis in which all CG’s are real. From [4].
there is a u group G with Type II: defines u for which yes a physical automorphisms u (n) CP no FSu is 0 transformation
no there is an involutory u yes for which all (1) FSu are +1
no Type I: generic settings based on Type II A: there G do not allow is a basis Type II B: there is CP for a physical in which all no basis in which CP transformation CG’s are real all CG’s are real
(n) FIGURE 3: The regular and extended twisted Frobenius–Schur indicators FSu and FSu allow us to distinguish between the three types of groups. Here, n is n = ord(u)/2 for even and n = ord(u) for odd ord(u). From [4].
4. CP VIOLATION FROM STRINGS sive states. In particular, the winding strings (see Figure 4) give rise to ∆(54) doublets. Of course, there are alternatives to embedding the discrete fla- The presence of these doublets leads to violation [22]. vor symmetry G into a continuous gauge symmetry (in four CP This can be made explicit by finding a basis–invariant contrac- spacetime dimensions). In fact, anomaly considerations seem tion (see [23]) that has a nontrivial phase. Of course, at this level to disfavor this possibility: an SU(3) symmetry with the fam- F the flavor symmetry is unbroken, and there is no direct con- ilies transforming as 3–plets has un-cancelled anomalies (see nection between the phase of the contraction presented in [22] e.g. [14]). On the other hand, non–Abelian discrete flavor sym- and the violation in the CKM matrix or baryogenesis. One metries may originate from extra dimensions [15]. In particular, CP would have to study, how, in explicit models (e.g. [24]) in which orbifold compactifications of the heterotic string lead to various the flavor symmetry gets broken and potentially realistic mass flavor groups [16, 17]. These symmetries originate from gauge matrices arise, this violation from strings manifests itself in symmetries in higher dimensions [18, 19], as they should [20]. CP the low–energy effective theory. This has not yet been carried As it turns out, already the very first 3–generation orbifold out. Nevertheless, it is clear that if is broken at the orbifold model [21] has a ∆(54) flavor symmetry [16], which is accord- CP point, it won’t un-break by moving away from it by e.g., giving ing to the classification [4] type I and thus –violating. There- CP the flavons VEVs. fore, is violated in such models [22]. CP More recently, an additional amusing observation has been When establishing explicitly that is violated, it was no- CP made [25]. In orbifold compactifications (without the so–called ticed that at the massless level only 1- and 3–dimensional repre- Wilson lines [26]), the flavor symmetry G is simply the outer sentations of ∆(54) occur. There exist outer automorphisms of automorphism group of the space group S. In a bit more detail, ∆(54) which map all these representations on their conjugates. the states of an orbifold correspond to conjugacy classes of the However, this is no longer the case when one includes the mas-
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Z Z Z Z
X Y X Y X Y X Y (a) (b) (c) (d)
FIGURE 4: Winding strings. Linear combinations of 4a–4c give rise to three ∆(54) doublet representations while 4d leads to the fourth and last one. space group, and can be represented by space group elements beautiful city. The work of M.C.C. was supported by, in part, k k (θ , nαeα), where θ stands for a discrete rotation and nαeα an by the National Science Foundation under Grant No. PHY- element of the underlying torus lattice. These conjugacy classes 1620638. The work of M.R. is supported by NSF Grant No. form multiplets under the outer automorphism group of the PHY-1719438. space group, thus G = out(S). This leads to the picture of “out of out”, References out(G) = out out(S) . (4.1) CP ∈ [1] M. Kobayashi and T. Maskawa, Prog.Theor.Phys. 49 Let us also mention that other orbifold geometries come (1973), 652. with different flavor symmetries. The probably simplest op- [2] Z. Maki, M. Nakagawa, and S. Sakata, Prog. Theor. Phys. 28 (1962), 870. tion is a Z2 orbifold plane, which leads to a D4 family symme- D [3] M.-C. Chen and K. Mahanthappa, Phys. Lett. B681 (2009), try [27, 16]. 4 is a type II group, meaning that here one cannot arXiv:0904.1721 immediately conclude that is violated. On the other hand, 444, [hep-ph]. CP it entails a ZR symmetry, which solves several shortcomings of [4] M.-C. Chen, M. Fallbacher, K. Mahanthappa, M. Ratz, 4 and A. Trautner, Nucl. Phys. B883 (2014), 267, the supersymmetric Standard Model at once [28, 29, 30, 31]. In arXiv:1402.0507 particular, it solves the µ problem and explains the longevity [hep-ph]. [5] M. Holthausen, M. Lindner, and M. A. Schmidt, JHEP of the proton and the stability of the LSP. All these examples arXiv:1211.6953 illustrate the impact of properties of compact dimensions on 1304 (2013), 122, [hep-ph]. particle phenomenology. [6] A. Sakharov, Pisma Zh.Eksp.Teor.Fiz. 5 (1967), 32. Arguably, it is rather amusing that violation can be tied [7] F. Feruglio, C. Hagedorn, and R. Ziegler, JHEP 1307 (2013), CP arXiv:1211.5560 to the presence of states that are required anyway for complet- 027, [hep-ph]. [8] I. P. Ivanov and J. P. Silva, Phys. Rev. D93 (2016), no. 9, ing the models in the ultraviolet. One may thus say that, at least arXiv:1512.09276 in these models, consistency in the ultraviolet requires to be 095014, [hep-ph]. CP violated. [9] M. Fallbacher, Discrete Groups in Model Building and the Def- inition of CP, Ph.D. thesis, Munich, Tech. U., 2015-08-05. [10] R. Bickerstaff and T. Damhus, International Journal of 5. SUMMARY Quantum Chemistry XXVII (1985), 381. [11] J. F. Cornwell, GROUP THEORY IN PHYSICS. VOL. 1, violation may originate from group theory. We have re- CP 1985. viewed the observation that there are certain finite groups that [12] N. Kawanaka and H. Matsuyama, Hokkaido Math.J. 19 clash with in the sense that, if these groups are realized as CP (1990), 495. (flavor) symmetries, is violated. To the best of our knowl- CP [13] M. Ratz and A. Trautner, JHEP 02 (2017), 103, edge, this is a situation that is not too ubiquitous in theory arXiv:1612.08984 [hep-ph]. space. What usually happens is that an extra symmetry results [14] R. Zwicky and T. Fischbacher, Phys. Rev. D80 (2009), from imposing a symmetry. Here, the opposite happens: arXiv:0908.4182 CP 076009, [hep-ph]. can get broken because another (flavor) symmetry is imposed [15] G. Altarelli, F. Feruglio, and Y. Lin, Nucl. Phys. B775 or emerges. (2007), 31, arXiv:hep-ph/0610165 [hep-ph]. These –breaking symmetries emerge from explicit string CP [16] T. Kobayashi, H. P. Nilles, F. Plöger, S. Raby, and M. Ratz, models. Even the earliest 3–generation string models in the lit- Nucl. Phys. B768 (2007), 135, hep-ph/0611020. erature have a violating discrete symmetry. In the string CP [17] Y. Olguin-Trejo, R. Pérez-Martínez, and S. Ramos-Sánchez, models, all symmetries have a clear geometric interpretation, arXiv:1808.06622 [hep-th]. which is why it is fair to say that the origin of violation CP [18] F. Beye, T. Kobayashi, and S. Kuwakino, Phys. Lett. B736 described in this review deserves to be called “geometric”. (2014), 433, arXiv:1406.4660 [hep-th]. [19] F. Beye, T. Kobayashi, and S. Kuwakino, JHEP 03 (2015), Acknowledgements 153, arXiv:1502.00789 [hep-ph]. arXiv:1710.01791 We are indebted to Dillon Berger for valuable comments on this [20] E. Witten, Nature Phys. 14 (2018), 116, review. We would like to thank Ernest Ma for pushing us to [hep-th]. write this up, and the Valencia group for inviting us to their
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