4. Some Implications
4.2 Electric dipole moment of neutron
The experimental upper bound for neutron EDM we mentioned in previous is 0.29 × 10−25e cm (CL = 90%), which is larger for comparing with the standard model prediction. In our model we will use the parameters like VEVs and Higgs mass in previous neutral meson mixing discussion as input to examine whether the neutral EDM we calculated can be close to the experimental upper bound.
At first we consider the one loop contribution to quark EDM. Note that in our model, the exchange of only one Higgs we obtained can not produce the quark EDM because all the couplings with H’s and a’s are real and pure imaginary, respectively.
So we discuss the contribution shown as Fig.4.2(a), in which the EDM contribution comes from the cross terms between H0i and ak. For example, with the flavor con-serving interaction, the u quark EDM is generated by exchange of a u quark in the loop. Using H10 and a1 as the example, the contribution is shown as
dHu10a1 = e(2/3) 32π2
mum2H0 1a1
m2H0
1 − m2a1(−2(v1mu v12v2
− s213v12mu
v1v2 )2)[f (m2H1, m2u) − f (m2a1, m2u)], (4.11) where f (x, y) = 2
Z 1
0
dz z2
x(1 − z) + z2y.
This formula shows that the one loop contribution is small because it is proportional to m3u which is small.
Fig. 4.2: The neutron EDM contribution from (a) quark EDM for q with one loop diagram.
The cross sign means the interaction between Hi0 and ak (b) quark EDM at two loop diagram, and (c) gluon color EDM
We therefore consider the three dominant two-loop contribution as shown in Fig.4.2(b), the electromagnetic operator Oγ [83, 84], the color EDM OC [83, 84], and the gluon color EDM operator Og in Fig.4.2(c) proposed by Weinberg [85, 86], which is often called Weinberg operator. These operators are written as
Oγ = −dq
2i¯qσµνγ5Fµνq, OC = −fq
2igsqσ¯ µνγ5Gµνq, Og = −1
6CfabcGaµνGbµαGecνα (4.12) The corresponding electric dipole moment contributions from Eq.(4.12) are written in the form [61, 62, 63]
dγn= ηd[4 3dd−1
3du]Λ ; dCn = eηf[4 9fd+2
9fu]Λ; (4.13) dgn ≈ eM
4π ξC, (4.14)
where dγnis the radiative contribution from Oγ; dCn is the gluon emitted contribution from OC. dq, fq are the contribution to neutron EDM from photon and gluon radiative contribution to quark q respectively, and the subscript Λ indicates that the hadronic energy scale. Eq.(4.14) is the approximation contribution for the color EDM of gluon operator OCg, and M = 1.190GeV indicates the scale related to the chiral symmetry breaking. The factor C will be defined later. The ηdand ηf [87, 88]
are
where g(Λ) = 4π/6 [85] is the strong coupling constant at hadronic scale.
The quark EDM qi, quark color EDM fi and the factor C in gluon color EDM formula Eq.(4.14) are written as follows
dq = eαemQq
24π3 mqGq; fq = αs
64π3mqGq; C = 1
8πHg, (4.17) where mq is the mass of quark and Qq is the charge of quark, and αem, αs are electromagnetic coupling constant and strong coupling constant respectively. The factor Gq and Hg are defined as
The functions f, g,and h are
f (z) = z
Summation of Eq.(4.13) and Eq.(4.14) is the totally contribution to the neutron EDM. That is
dn= dγn+ dCn + dgn. (4.20) and we only consider the flavor conserving interaction because the flavor violating contribution is suppressed by s12,s23,s13 for PDG parametrization, or s1,s2,s3 for KM parametrization.
From Eq.(3.40) to Eq.(3.54), we find that for mass mixing terms of scalar and pseudoscalar, there are m2H0
1a1, m2H0
1a2, m2H0
2a1, m2H0
3a1, and m2H0
4a1 which are nonzero.
We will not consider the H40 − a1 contribution because the the factor 1/vs in Yukawa couplings suppresses the H40 and a contribution. For model(a) with PDG parametrization, we use H30, a1 as an example. Writing down all Yij31 in following with i, j indicating quarks,
Using the input from previous D0−D0 discussion for this model with tan β = 40, v12 = 240GeV, and v3 = 10GeV. When the neutral Higgs mass is about order 100GeV, we substitute the functions f, g, h difference between input by m2t/m2Hl and m2t/m2ak by (∆f, ∆g, ∆h) = (1, 2, 0.1).
Substituting Eq.(4.21) into Eq.(4.17, 4.18, 4.13, 4.14), we obtain the relation dn(H30− a1) ≈ −3 × 10−25 m2H0
3a1
m2H0
3 − m2a1 e cm. (4.22) For the other three kinds of Higgs pair exchange
dn(H20− a1) ≈ −2 × 10−26 m2H0
So neutron electric dipole moment is dominated by the contribution of H30 − a1
exchange. At this moment λ31 = m2H0
3a1/(m2H0
3 − m2a1) . 0.1 is required.
In model(b) with PDG parametrization, we note that there is no up-type quarks coupling with H10 and a1. If we take the neutral Higgs mass to be with order TeV, we choose (∆f, ∆g, ∆h) = (0.2, 0.2, 0.03), and then treat the H20− a1 contribution as
dn≈ −1 × 10−26 m2H0 1a2
m2H0
1 − m2a2 e cm (4.24)
In KM parametrization of model(a), we consider the H10 − a2 process. If the choice for VEVs is v1 = v2 = v3, with the Higgs mass to be 100GeV, then the contribution to neutron EDM for H10 − a2 exchange is
dn≈ 7 × 10−26 m2H0 1a2
m2H0
1 − m2a2 e cm (4.25)
For small λ12 . 0.4 this contribution can saturated the upper bound of neutron EDM. Also note that from CP phenomenon in K0− K0 mixing H10− a1 contribution is small.
For model(b), H10 − a1 interaction are also not including the interaction with top quarks, so the this interaction will not give the dominate contribution. Taking Higgs mass 100GeV and v1 = v2 = v3. The H10− a2 gives
dn ≈ 1 × 10−25 m2H0 1a2
m2H0
1 − m2a2 e cm. (4.26)
If we choose Higgs mass about 300 GeV, which is the same condition as that for Bs0− Bs0 mixing discussion. The contribution to neutron EDM will be small
dn ≈ 4 × 10−26 m2H0 1a2
m2H0
1 − m2a2 e cm. (4.27)
Using λ12 . 0.7 the result can be close to the upper bound.
In above discussion, we treat the two loop contribution to neutron electric dipole moment. Using PDG parametrization in model(a), with effective neutral Higgs mass about 100GeV and λ31 . 0.1, the result can be close to the experimental bounds.
In model(b), the effective Higgs mass we choose is 1TeV, which is the same as that
in K0− K0 mixing. For KM parametrization, we take Higgs mass about 100GeV in model(a) with λ12. 0.4 and 300GeV in model(b) with λ12. 0.7 to get close results to experimental bound of neutron EDM.
The CKM matrix can not deal with problems from the baryogenesis, and also it can not deal with the question where CP violation come from. So another source for CP violation is required. With more than one Higgs doublets, these problems may be answered. CP violation can be a result of spontaneous symmetry breaking.
That is, spontaneous CP violation. With two Higgs doublets, these are three types model with two Higgs doublets, which is so-called Lee model. Type I and type II introduce the discrete symmetry, and it lead to the vanishing of spontaneous CP violating phase. Type III can make the spontaneous CP violation, and it has tree level FCNC with too many parameters arbitrary. The Weinberg model solves the problem for Lee model. It introduces three Higgs doublets which is the minimal model to have the spontaneous CP violating phase but without tree level FCNC process. However, the Weinberg model has been ruled out by the experimental data for sin 2β. This motivation makes us to study new models with the spontaneous CP violation. We summary our work in the following
• We introduce an idea that make the spontaneous CP violating phase be identi-cal to the CKM matrix phase. Two kinds of Yukawa interactions are discussed.
One is called model(a) where two Higgs doublet couple to the up-type quarks and one Higgs couples to the down-type quarks. Another one is model(b) with two Higgs doublets couple to the down-type quarks and one Higgs doublets couple to the up-type quarks.
• For model(a) using the PDG parametrization a phase is absorbed into the up-type quarks to make the CKM matrix with uniform phase δ13. We obtain
δ = −δ13,
and all coupling matrices are determined. Here δ13 is the phase causing spon-taneous CP violation.
The same process can be apply to KM parametrization. The phase relation is similar to that of PDG,
δ = −δKM.
The model(b) has the same phase relation as that of the model(a).
• We construct a model with three Higgs doublets and one Higgs singlet, with the Pessei-Quinn symmetry to make small enough neutron electric dipole mo-ment. The minimal condition of the Higgs potential makes the spontaneous CP violating phase δ be the only one phase in the Higgs potential, and the spontaneous CP violating phase is the source of CP violation.
• We extract the Goldstone boson eaten by W± and Z0, also the axion by appropriate rotation, and then we derive the corresponding Yukawa couplings.
From the couplings we find that the H40 and a interaction are neglected by the factor 1/vs. Tree level FCNC only occurs in the interaction by exchanging Higgs H10 and a1. The coupling matrices are related to the VCKM. When we choose an explicit parametrization for CKM matrix, all couplings can be written in terms of CKM parameters and quark masses.
• Using experimental data on meson and anti-meson mixing, the mass of effective neutral Higgs with the relation 1/m2eff = 1/m2H0
1 − 1/m2a1 are constrained.
• We use the result from the previous discussion of meson and anti-meson mix-ing to discuss the neutron electric dipole moment. It is well-known that the one loop contribution for quarks EDM with exchanging Higgs is small and negligible, so we calculate the two loop contribution from quark electric dipole moment, quark color electric dipole moment, and the gluon color electric dipole moment. The result is shown that the EDM could be close to the present upper bound for neutron electric dipole moment.
[1] S. L. Chen, N. G. Deshpande, X. G. He, J. Jiang and L. H. Tsai, Eur. Phys. J.
C 53, 607 (2008) [arXiv:0705.0399 [hep-ph]].
[2] M. Gell-Mann, Phys. Rev. 125, 1067 (1962).
[3] S. L. Glashow, Nucl. Phys. 22 (1961) 579.
[4] S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967).
[5] A. Salam, in: proc. 8th Nobel Symposium, ed. by N. Svartholm(Almquist and Wiskell, Stockholm, 1968).
[6] T. D. Lee and C. N. Yang, Phys. Rev. 104, 254 (1956).
[7] J. H. Christenson, J. W. Cronin, V. L. Fitch and R. Turlay, Phys. Rev. Lett.
13, 138 (1964).
[8] B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. 87, 091801 (2001) [arXiv:hep-ex/0107013].
[9] K. Abe et al. [Belle Collaboration], Phys. Rev. Lett. 87, 091802 (2001) [arXiv:hep-ex/0107061].
[10] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652 (1973).
[11] N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963).
[12] W. M. Yao et al. [Particle Data Group], J. Phys. G 33, 1 (2006).
[13] L. Wolfenstein, Phys. Rev. Lett. 51, 1945 (1983).
[14] C. Jarlskog, Phys. Rev. Lett. 55, 1039 (1985).
[15] J. C. Hardy, arXiv:hep-ph/0703165.
[16] F. Ambrosino et al. [KLOE Collaboration], JHEP 0804, 059 (2008) [arXiv:0802.3009 [hep-ex]].
[17] C. Aubin et al. [Fermilab Lattice Collaboration], Phys. Rev. Lett. 94, 011601 (2005) [arXiv:hep-ph/0408306].
[18] G. S. Huang et al. [CLEO Collaboration], Phys. Rev. Lett. 95, 181801 (2005) [arXiv:hep-ex/0506053].
[19] M. Artuso, Int. J. Mod. Phys. A 21, 1697 (2006) [AIP Conf. Proc. 842, 533 (2006)] [arXiv:hep-ex/0510052].
[20] F. J. Gilman et al. [Particle Data Group], Phys. Lett. B 592 793(2004).
[21] G. D. Lellis, P. Migliozzi and P. Santorelli, Phys. Rept. 399, 227 (2004) [Erratum-ibid. 411, 323 (2005)].
[22] A. Kayis-Topaksu et al. [CHORUS Collaboration], Phys. Lett. B 626, 24 (2005).
[23] V. B. Golubev, Y. I. Skovpen and V. G. Luth, Phys. Rev. D 76, 114003 (2007) [arXiv:hep-ph/0702072].
[24] B. Aubert et al. [BABAR Collaboration], Phys. Rev. D 73, 012006 (2006) [arXiv:hep-ex/0509040].
[25] B. Aubert et al. [BABAR Collaboration], Phys. Rev. D 77, 032002 (2008) [arXiv:0705.4008 [hep-ex]].
[26] S. Hashimoto, A. S. Kronfeld, P. B. Mackenzie, S. M. Ryan and J. N. Simone, Phys. Rev. D 66, 014503 (2002) [arXiv:hep-ph/0110253].
[27] M. Okamoto, PoS LAT2005, 013 (2006) [arXiv:hep-lat/0510113].
[28] A. Gray et al. [HPQCD Collaboration], Phys. Rev. Lett. 95, 212001 (2005) [arXiv:hep-lat/0507015].
[29] S. Aoki et al. [JLQCD Collaboration], Phys. Rev. Lett. 91, 212001 (2003) [arXiv:hep-ph/0307039].
[30] A. J. Buras, A. Czarnecki, M. Misiak and J. Urban, Nucl. Phys. B 631, 219 (2002) [arXiv:hep-ph/0203135].
[31] M. Neubert, Eur. Phys. J. C 40, 165 (2005) [arXiv:hep-ph/0408179].
[32] V. M. Abazov et al. [D0 Collaboration], Phys. Lett. B 639, 616 (2006) [arXiv:hep-ex/0603002].
[33] V. M. Abazov et al. [D0 Collaboration], Phys. Rev. Lett. 98, 181802 (2007) [arXiv:hep-ex/0612052].
[34] E. A. Andriyash, G. G. Ovanesyan and M. I. Vysotsky, Phys. Lett. B 599, 253 (2004) [arXiv:hep-ph/0310314].
[35] H. Burkhardt et al. [NA31 Collaboration], Phys. Lett. B 206, 169 (1988).
[36] A. B. Carter and A. I. Sanda, Phys. Rev. Lett. 45, 952 (1980).
[37] A. B. Carter and A. I. Sanda, Phys. Rev. D 23, 1567 (1981).
[38] B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. 99, 171803 (2007) [arXiv:hep-ex/0703021].
[39] B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. 99, 021603 (2007) [arXiv:hep-ex/0703016].
[40] B. Aubert et al. [BABAR Collaboration], Phys. Rev. D 76, 091102 (2007) [arXiv:0707.2798 [hep-ex]].
[41] A. Kusaka et al. [Belle Collaboration], Phys. Rev. Lett. 98, 221602 (2007) [arXiv:hep-ex/0701015].
[42] A. Poluektov et al. [Belle Collaboration], Phys. Rev. D 73, 112009 (2006) [arXiv:hep-ex/0604054].
[43] The Heavy Flavor Averaging Group (HFAG), 2008. http://www.slac.stanford .edu/xorg/hfag/rare/winter08/acp/OUTPUT/HTML/acp table2.html.
[44] J. Chay, H. n. Li and S. Mishima, arXiv:0711.2953 [hep-ph].
[45] M. Neubert, Phys. Lett. B 424, 152 (1998) [arXiv:hep-ph/9712224].
[46] M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, arXiv:hep-ph/0007256.
[47] CKMfitter Group (J. Charles et al.), Eur. Phys. J. C41, 1-131 (2005) [hep-ph/0406184], updated results and plots available at: http://ckmfitter.in2p3.fr [48] A. G. Cohen, D. B. Kaplan and A. E. Nelson, Ann. Rev. Nucl. Part. Sci. 43,
27 (1993) [arXiv:hep-ph/9302210].
[49] S. M. Barr, G. Segre and H. A. Weldon, Phys. Rev. D 20, 2494 (1979).
[50] T. D. Lee, Phys. Rev. D 8, 1226 (1973).
[51] T. D. Lee, Phys. Rept. 9, 143 (1974)
[52] S. Weinberg, Phys. Rev. Lett. 37, 657 (1976).
[53] G. C. Branco, Phys. Rev. Lett. 44, 504 (1980).
[54] N. G. Deshpande and E. Ma, Phys. Rev. D 16, 1583 (1977).
[55] D. Chang, X. G. He and B. H. J. McKellar, Phys. Rev. D 63, 096005 (2001) [arXiv:hep-ph/9909357].
[56] P. Krawczyk and S. Pokorski, Nucl. Phys. B 364, 10 (1991).
[57] Y. Grossman and Y. Nir, Phys. Lett. B 313, 126 (1993) [arXiv:hep-ph/9306292].
[58] G. Beall and N. G. Deshpande, Phys. Lett. B 132, 427 (1983).
[59] C. A. Baker et al., Phys. Rev. Lett. 97, 131801 (2006) [arXiv:hep-ex/0602020].
[60] I. I. Y. Bigi and A. I. Sanda, Phys. Rev. Lett. 58, 1604 (1987).
[61] N. G. Deshpande, G. Eilam and W. L. Spence, Phys. Lett. B 108, 42 (1982).
[62] X. G. He, B. H. J. McKellar and S. Pakvasa, Int. J. Mod. Phys. A 4, 5011 (1989) [Erratum-ibid. A 6, 1063 (1991)].
[63] B. H. J. McKellar, S. R. Choudhury, X. G. He and S. Pakvasa, Phys. Lett. B 197, 556 (1987).
[64] R. D. Peccei and H. R. Quinn, Phys. Rev. D 16, 1791 (1977).
[65] R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. 38, 1440 (1977).
[66] A. R. Zhitnitsky, Sov. J. Nucl. Phys. 31, 260 (1980) [Yad. Fiz. 31, 497 (1980)].
[67] M. Dine, W. Fischler and M. Srednicki, Phys. Lett. B 104, 199 (1981).
[68] X. G. He and R. R. Volkas, Phys. Lett. B 208, 261 (1988) [Erratum-ibid. B 218, 508 (1989)].
[69] C. Q. Geng, X. D. Jiang and J. N. Ng, Phys. Rev. D 38, 1628 (1988).
[70] J. E. Kim, Phys. Rev. Lett. 43, 103 (1979).
[71] M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B 166, 493 (1980).
[72] B. Pontecorvo, Sov. Phys. JETP 6, 429 (1957) [Zh. Eksp. Teor. Fiz. 33, 549 (1957)].
[73] Z. Maki, M. Nakagawa and S. Sakata, Prog. Theor. Phys. 28, 870 (1962).
[74] H. Fusaoka and Y. Koide, Phys. Rev. D 57, 3986 (1998) [arXiv:hep-ph/9712201].
[75] B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. 98, 211802 (2007) [arXiv:hep-ex/0703020].
[76] K. Abe et al. [BELLE Collaboration], Phys. Rev. Lett. 99, 131803 (2007) [arXiv:0704.1000 [hep-ex]].
[77] M. Staric et al. [Belle Collaboration], Phys. Rev. Lett. 98, 211803 (2007) [arXiv:hep-ex/0703036].
[78] M. Ciuchini, E. Franco, D. Guadagnoli, V. Lubicz, M. Pierini, V. Porretti and L. Silvestrini, Phys. Lett. B 655, 162 (2007) [arXiv:hep-ph/0703204].
[79] E. Barberio et al. [Heavy Flavor Averaging Group (HFAG) Collaboration], arXiv:0704.3575 [hep-ex].
[80] A. Lenz and U. Nierste, JHEP 0706, 072 (2007) [arXiv:hep-ph/0612167].
[81] X. G. He and G. Valencia, Phys. Rev. D 74, 013011 (2006) [arXiv:hep-ph/0605202].
[82] K. Cheung, C. W. Chiang, N. G. Deshpande and J. Jiang, Phys. Lett. B 652, 285 (2007) [arXiv:hep-ph/0604223].
[83] S. M. Barr and A. Zee, Phys. Rev. Lett. 65, 21 (1990) [Erratum-ibid. 65, 2920 (1990)].
[84] J. F. Gunion and D. Wyler, Phys. Lett. B 248, 170 (1990).
[85] S. Weinberg, Phys. Rev. Lett. 63, 2333 (1989).
[86] S. Weinberg, Phys. Rev. D 42, 860 (1990).
[87] D. Chang, W. Y. Keung and T. C. Yuan, Phys. Lett. B 251, 608 (1990).
[88] D. Chang, X. G. He, W. Y. Keung, B. H. J. McKellar and D. Wyler, Phys.
Rev. D 46, 3876 (1992) [arXiv:hep-ph/9209284].
[89] E. Braaten, C. S. Li and T. C. Yuan, Phys. Rev. Lett. 64, 1709 (1990).
[90] E. Braaten, C. S. Li and T. C. Yuan, Phys. Rev. D 42, 276 (1990).