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CP Violation in Higher Dimensional Gauge Theories

CP Violation in Higher Dimensional Gauge Theories. @ A series of seminars at RIKEN (Dec. 5, ’09) . C.S. Lim （林　青司） (Kobe University) . Introduction In spite of the great success of Kobayashi-Maskawa model,

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CP Violation in Higher Dimensional Gauge Theories

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  1. CP Violation in Higher Dimensional Gauge Theories @ A series of seminars at RIKEN (Dec. 5, ’09) C.S. Lim （林　青司） (Kobe University)

  2. Introduction • In spite of the great success of Kobayashi-Maskawa model, • the origin of CP violation still seems to be not conclusive . In D = 4 space-time, the mechanism of CP violation: ・explicit (hard) breaking due to complex Yukawa coupling a la K-M ・spontaneous (soft) breaking due to complex VEV’s of Higgs (T.D. Lee, S. Weinberg) We may have some new mechanism to break CP, once space-time is extended, as in superstring theory or Kaluza-Klein type theories. In fact, CP violation due to compactification of extra space was discussed: C.S. Lim, Phys. Lett. B256(’91)233 (A.Strominger and E. Witten, Commun. Math Phys. 101(’85)231); T. Kobayashi and C.S.L., P.hys. Lett. B343(’95)122 (in orbifold string theory).

  3. However, e.g. in string theory or 10 dim. SUSY Y.-M. theory, as its low energy limit, to break CP is a non-trivial issue, as the theory does not have complex parameter, to start with. On the other hand, if it succeeds, it may provide a new type of mechanism of CP violation. The condition for CP violation may be utilized in order to select viable models and/or the manner of compactification. (N.B.) ・ The scenario of Gauge-Higgs Unification (GHU) , as an attractive candidate of New Physics, also has the same problem. 10 dim. SUSY Y.-M. theory is a sort of GHU, where Higgs originates form the extra space component of gauge field. ・In GHU, the VEV of the Higgs, Wilson line phase (Hosotani mechanism), can be a new source of the CP violation.

  4. (Higher dimensional C, P transf.s should be modified : C.S. L. ’91 ) We can easily find C matrix, for instance, in higher dim. space-time, so that it satisfies . ・Such defined higher dimensional C, P transf.s , however, do not correspond to the 4-dimensional ones, in general, and some modification is necessary. ・Interestingly, the modified CP transformation act on the extra space coordinates non-trivially: it acts as a complex conjugation of the complex homogeneous coordinates for the extra space. ・If the compactfied space has “complex structure”, the breaking of CP can be realized.

  5. Take D=6 case for the illustrative purpose. In the basis, where 6D spinor decomposes into two 4-D spinors, gamma matrices are given as The C matrix is found not to reduce to ordinary 4-D transf., , because of . Modifying C and P as

  6. Accordingly the transformation properties of a vector is uniquely determined and we find: (N.B.) The reason of the peculiar transf. under C was that extra dimensional gamma matrices are half symmetric and half anti-symmetric. Thus introducing, a complex coordinate as CP transf. is nothing but a complex conjugation:

  7. This peculiar property persists for higher (even) dimensions, as is easily seen by iterative construction: For instance, in 10D (N.B.) In this case, C should not be modified, as Majorana fermion exists in D = 2 ,4 (mod 8), and P causes the complex conjugation.

  8. Consider Type-I superstring theory with 6-dimensional Calabi-Yau manifold defined by a quintic polynomial for the coordinates of CP4 , “4 generation model” CP is broken only when the coefficient C is complex, since otherwise the above defining equation is invariant under ・In fact, resultant Yukawa couplings is known to have a CP violating phase for complex C (M. Matsuda, T. Matsuoka, H. Mino, D. Suematsu and Y. Yamada, Prog. Theor. Phys. 79(’88)174). ・If we impose phenomenological requirement, of no FCNC at the tree level, the CP phase disappears, unfortunately.

  9. Our purpose: To realize CP violation in the framework of higher dimensional field (gauge) theories, not string theories, with much simpler compact spaces, such as a circle or orbifold. More precisely, what we have in our mind as the higher dimensional gauge theory is GHU scenario. Note that as long as higher dimensional gauge theory itself is CP invariant, without phase, CP violation should be a sort of “spontaneous” breaking. We have discussed two possibilities: 1. CP violation due to compactification (Z4orbifold) (w./ N. Maru and K. Nishiwaki, arXiv:0910.2314 [hep-ph] ) 2. CP violation due to the VEV of the Higgs (w./ Y. Adachi and N. Maru, arXiv:0905.1022 [hep-ph] , Phys. Rev. D 80(‘09)055025)

  10. I. Gauge-Higgs unification (GHU) unification of gravity (s=2) & elemag (s=1) (A. Einstein) Kaluza-Klein theory unified theory of gauge (s=1) & Higgs (s=0) interactions “Gauge-Higgs unification” : realized in higher dimensional gauge theory extra dimension 5D gauge field Higgs 4D gauge-field 4D space-time

  11. the idea of gauge-Higgs unification itself is not new: ・N.S. Manton, Nucl. Phys. 58(’79)141. ・Y.Hosotani, Phys. Lett. B126 (‘83) 309 : ``Hosotani mechanism” The scenario was revived: ・H. Hatanaka , T. Inamiand C.S.L., Mod. Phys. Lett. A13(’98)2601 ( the main points) ・ The quantum correction to mH is finite because of the higher dimensional gauge symmetry →Anew avenue to solve the hierarchy problemwithout invoking SUSY ・The sum over all K-K modes is essential to get the finite（for arbitrary dimensions）Higgs mass (N.B.) The scenario may also shed some light on the arbitrariness problem in the interactions of Higgs.

  12. II. Issues related to GHU ・dimensional deconstruction (N. Arkani-Hamed, A.-G. Cohen, H. Georgi, Phys.Lett. B513(’01)232) : latticized 5D gauge theory , @ N → ∞ limit, the effective potential for H coincides with what we obtained. ・Little Higgs model : 4D theory, where G/H of global symmetry provides Higgs as a N-G, may be “dual” to 5D GHU, where Ay associated with G/H of higher dimensional local gauge symmetry provides Higgs (holographic principle). ・GHU may be related even with the supersymmetry (!) (T. Nagasawa, M. Sakamoto, H. Sonoda and C.S.L., Phys. Rev. D72(‘05) 064006) We have demonstrated that N=2 quantum mechanical (QM) SUSY is hiddeninthe gauge-Higgs unification scenario, which is due to Higgs-like mechanism for massive K-K modes. (On R-S background, the ``warp-factor” just corresponds to a superpotential .)

  13. ・(ultra) natural inflation (N. Arkani-Hamed, H.-C. Cheng, P. Creminelli and L. Randall, Phys.Rev.Lett. 90(’03)221302; T. Inami, Y. Koyama, S. Minakami &C.S.L., Progr. Theor. Phys. (09), to appear) : Ay(0) may be a natural candidate for the inflaton, as the local gauge symmetry stabilizes the potential under the quantum correction

  14. ・ “Minimal GHU standard model”: SU(3) onM4 x (S1/Z2) (Kubo, C.S.L. andH. Yamashita, Mod. Phys. Lett. A17(’02)2249) (N.B.) In the GHU, gauge group should be enlarged, as the Higgs belongs to adjoint repr., while SM Higgs is SU(2) doublet. Recall that in the heterotic string theory, Higgs belonging to the fundamental repr. of E6 comes from adjoint repr. of E8 . SU(3) → SU(2) x U(1) breaking due to non-trivial Z2-parity assignment(Kawamura): Zero-modes of Gauge-Higgs sector : Exactly what we need for the SU(2) x U(1) SM !

  15. IV. CP violation due to compactification (w./ N. Maru and K. Nishiwaki, arXiv:0910.2314 [hep-ph] ) How to break CP symmetry is a challenging issue in the scenario of GHU, where the Higgs interactions are governed by gauge principle. One of a few possibilities to break CP symmetry is to invoke to the manner of compactificaion, which determines the vacuum state of the theory: “spontaneous breaking”.

  16. In Type I superstring theory, CP is broken only when the coefficient C to define Calabi-Yau manifold, complex, since otherwise the above defining equation is invariant under In our paper we consider much simpler compactification; we discuss the CP violation in the 6-dimensional U(1) GHU model due to the compactification on the orbifold We easily know that CP transfomration is not compatible with the condition of orbifolding.

  17. (orbifolding) (CP trensf.) : “orientation-changing operator” (Strominger and Witten) In terms of a complex coordinate the orbifold condition is written as After the CP transf., the condition reads as Thus, CP tranf. is not compatible with orbifolding condition , and CP symmetry is broken.

  18. The model: 6D QED Z4 orbifold conditions:

  19. (double ) K-K mode expansions: The zero-mode sector recovers ordinary QED, with The presence of factor i signals the CP violation.

  20. The mass eigenstates for fermions: where , By use of a freedom of unitary transformation due to the mass degenaracy between , the roles of L and R can be changes. ←　parity symmetry

  21. The interaction vertices of non-zero K-K photons:

  22. Even the interaction vertices for non-zero K-K photons generally have CP violating phases: ・Such obtained CP violating phases are confirmed to survive even after the re-phasing of the fields. ・we also have identified “Jarlskog parameter” in our model. The EDM of electron, as a typical CP violating observable, however, is found to vanish at 1-loop level. Unfortunately, we anticipate that we cannot get a non-vanishing contributions even at higher loops.

  23. This comes from the L-R symmetry due to the freedom of unitary transformation. Or, if we recall that P symmetry is not violated by the compactification, as is naively expected in QED. Since, EDM necessitates both of P and CP violations, we anticipate EDM vanishes in our model, although we expect EDM will get contributions in a realistic theory including the SM, since P should be violated anyway in such a realistic theory.

  24. 2. CP violation due to the VEV of the Higgs (w./ Y. Adachi and N. Maru, arXiv:0905.1022 [hep-ph] , Phys. Rev. D 80(‘09)055025) Another possibility to break CP is due to the VEV of some field which has odd CP eigenvalue. We argue that the VEV of the Higgs , or the VEV of Wilson-loop plays the role ( : “timeon” ?). We show that neutron EDM gets contribution already at 1-loop level in the model, though we assume the presence of only 1 generation. (The model) 5-D SU(3) GHU model compactified on an orbifold with a massive bulk fermion in a fundamental representation. In this case, the orbifold is too simple to break CP, thus only possibility seems to be due to

  25. (N.B.) To get EDM, both P and CP have to be broken. P symmetry, however, is broken anyway by the orbifolding. In 5D CP transf. can be defined just as in the 4D case: The CP transf. is known to be consistent with the orbifolding condition as and commute with each another: Correspondingly, the transformations of space-time and fields are fixed as, Thus we realize that has odd CP eigenvalue and the VEV may lead to CP violation.

  26. Actually, when the Z2 – odd bulk mass term is switched off , we can perform a chiral rotation for , so that the coupling of becomes scalar type and therefore has even CP eigenvalue. Hence, to get physical CP violating effects, the interplay between the VEV and the bulk mass is crucial. (The neutron EDM) In this mechanism of CP violation, EDM appears already at 1-loop level, though we have only 1 generation.

  27. ・We have confirmed that EDM appears only when both of M and the VEV of Ay are non-zero. ・Let us note that in K-M model, the EDM arises only at the 3-loop order.

  28. Comparing the contribution of the non-zero KK modes with the experimental upper bound on the EDM, we get the lower bound on the compactification scale,

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