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Neutron electric dipole moment and spontaneous CP breaking
24 September 1998
Physics Letters B 436 Ž1998. 369–371
Neutron electric dipole moment and spontaneous CP breaking Paul H. Frampton, Masayasu Harada Department of Physics and Astronomy, UniÕersity of North Carolina, Chapel Hill, NC 27599-3255, USA Received 2 June 1998 Editor: H. Georgi
Abstract In a model where CP is spontaneously broken, the aspon model, it is shown by extending previous arguments about B decay and the value of ReŽ e Xre . in the kaon system, that the neutron electric dipole moment Dn is bounded from below by Dn G 10y29 e.cm. q 1998 Elsevier Science B.V. All rights reserved.
The electric dipole moment D of a neutron ŽNEDM. must be proportional to its spin S and is hence odd under T or CP and under P. The energy D P E is odd under CP and even under P because E is odd under P and even under CP. Thus a nonvanishing NEDM would be an explicit example of CP violation, outside of the kaon system. The discovery of a NEDM will be very important in formulating the theory of CP violation where the problem is the fact that only one Žcomplex. parameter, e K , has been accurately measured. The second parameter in the kaon system, ReŽ e Xre ., remains uncertain – it is still consistent with zero – and the CP asymmetries in B meson decays have yet to be determined. Any further experimental measurement of CP violation will constrain the viable theories. The experimental limit on Dn is presently Dn F 10y2 5 e.cm. w1,2x but it is possible w3x that this limit will be pushed back by several orders of magnitude in the forseeable future, even a limit of 10y3 0 e.cm. being conceivable. On the other hand, the prediction of the KM mechanism within the framework of the Standard Model is Dn , 2 = 10y3 2 e.cm. w4–7x which is beyond the reach of any planned experiments.
In a model of spontaneously broken CP symmetry w8,9x one may simultaneously solve the strong CP problem Žwithout an axion. and accommodate the observed CP violation in the kaon system. In an economical model, the Standard Model is augmented by an additional UŽ1. gauge group under which all the usual three generations of quarks and leptons are neutral, as are the established twelve gauge bosons Žg ,Z,W ", g a . and the single Higgs doublet. One adds a non-chiral quark doublet ŽU, D . which carries the new charge, and two complex Higgs scalar singlets x a. These x a Ž a s 1,2. carry an equal new charge. With this arrangement, the generalized Ž4 = 4. quark mass matrix has a real determinant and hence at tree-level the value of Q is zero if the underlying theory respects CP symmetry. Denoting the Žcomplex. VEVs of the x a by ² x a : and the Yukawa coupling constants to the ith generation by h ai useful parameters are as2
xi s
Ý
h ai ² x a :
Ž 1.
as1
These x i are complex, but for convenience we shall
0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 8 6 2 - 4
P.H. Frampton, M. Haradar Physics Letters B 436 (1998) 369–371
370
write x to denote x s < x i <, the modulus, and taken to be generation independent since the limits on x are not sensitive to the generation considered. In w10x, it was shown that the predicted value of Q is estimated from a one-loop diagram as:
Qs
lx2
Ž 2.
16p 2
This follows from the diagram shown in Fig. 1. In the notation of w9x the imaginary part gives a contribution:
Q Ž up . s
1
1
'2 Ž 4p .
Fig. 2. One loop < f < 2 < x < 2 counterterm for the coupling l.
as2,ls3 2
Ý as1,ls1
h al Im x l) la
k M
Ž 3.
which gives the estimate of Eq. Ž2., from which it follows that l x 2 is less than ; 10y8 . Here la is the coefficient of the quartic interaction between the two types of Higgs < f < 2 < xa < 2 , and l with no subscript is an average value. ŽActually there are three independent l corresponding to indices 11, 12 q 21, 22 but our estimates will not distinguish these.. The neutron electric dipole Dn has been calculated in terms of Q long ago w11,12x with the result that Dn , 10y1 5Q e.cm.
Ž 4.
and so we know from Dn F 10y2 5 e.cm. empirically that Q F 10y1 0 . Here we seek to establish a lower limit on the prediction for Q in the present model.
Firstly, we look at the kaon system for which the parameter < e K < is given by w9,13x: < eK < s
1
'2 D m K
mK
f K2 2 3 k2
x4
Ž 5.
Using Ž D m K rm K . s 7.0 = 10y1 5, f K s 0.16 GeV gives the relationship between x 2 and the UŽ1. new breaking scale:
krx 2 s 2.9 = 10 7 GeV.
Ž 6.
Thus, if we insist that the UŽ1. new is broken above the electroweak breaking scale Ž; 250 GeV. then: x 2 R 10y5 .
Ž 7.
From Eq. Ž2., this means that l - 10y3 . In w10x, it was further argued on the basis of naturalness that l ) 10y5 which Žif taken at face value. would imply that Q ) 10y1 2 and hence Dn ) 10y2 7 e.cm. But here we use what is a more solid, and more conservative, bound to achieve our lower limit on Dn . Our bound follows from the fact that the < f < 2 < xa < 2 interaction receives a one-loop correction from the quark loop box where three sides are the top quark and the fourth is the heavy U quark Žsee Fig. 2.. The full l is given by:
l s ltreeŽ bare. q l1 - loop Ž including counterterm. q higher loops. Fig. 1. One loop diagram of which the imaginary part contributes to Q .
Ž 8.
and the 1-loop finite contribution, for the dominant diagram ŽFig. 2., neglecting the quark masses and
P.H. Frampton, M. Haradar Physics Letters B 436 (1998) 369–371
taking h 3a , g t as the respective Yukawa couplings to xa and f of the third generation
l1 - loop ,
d4 k
H Ž 2p .
4
< h 3a < 2 < g t < 2
1 k4
lR
16p 2
that at least one of these different CP breakdown mechanisms must be experimentally refuted with the advent of B Factories.
q counterterm. Ž 9 .
which implies that the lowest value for l Žwithout accidental cancellations. is: x2
371
Ž 10 .
Acknowledgements This work was supported in part by the US Department of energy under Grant No. DE-FG0585ER-40219.
Combining Eqs. Ž2. and Ž10. then gives the estimate for Q of
QG
x4 16p 4
References
Ž 11 .
which implies that x 2 F 10y3 Žincidentally in full agreement with w10x. and that 10y1 0 G Q G 10y14 . From Eq. Ž4., this then give our required lower limit on the neutron electric dipole moment of Dn G 10y2 9 e.cm.
Ž 12 .
This is more than two orders of magnitude greater than the prediction of the KM mechanism and thus provides yet another distinguishing feature of spontaneous CP violation. Recall that this spontaneous CP violation model predicts, as reflects its superweak nature, that ReŽ e Xre . , 10y5 w13x Žcompared to O Ž10y3 . in the KM mechanism. and that the angles b ,g of the unitarity triangle measurable in B Factories are a few milliradians w10x Žrather than O Ž1. as in the KM mechanism.. It is very exciting to realize
w1x N.F. Ramsey, Ann. Rev. Nucl. Sci. 32 Ž1982. 211. w2x K.F. Smith et al., Phys. Lett. B 234 Ž1990. 191. w3x M.S. Freedman, G.R. Ringo, T.W. Dombeck, A Proposed Method for Measuring the Electric Dipole Moment of the Neutron Using Acceleration in an Electric-Field Gradient and Ultracold Neutron Interferometry. nucl-exr9804003. w4x E.P. Shabalin, Sov. J. Nucl. Phys. 31 Ž1980. 864. w5x N. Deshpande, Phys. Lett. B 108 Ž1982. 42. w6x I.B. Khriplovich, A.R. Zhitnitsky, Phys. Lett. B 109 Ž1982. 490. w7x B. Gavela et al., Phys. Lett. B 109 Ž1982. 215. w8x P.H. Frampton, T.W. Kephart, Phys. Rev. Lett. 66 Ž1991. 1666. w9x P.H. Frampton, D. Ng, Phys. Rev. D 43 Ž1991. 3034. w10x P.H. Frampton, S.L. Glashow, Phys. Rev. D 55 Ž1997. 1691. w11x V. Baluni, Phys. Rev. D 19 Ž1979. 2227. w12x R.J. Crewther, P. Di Vecchia, G. Veneziano, E. Witten, Phys. Lett. B 88 Ž1979. 123. w13x P.H. Frampton, M. Harada, UNC-Chapel Hill Report IFP757-UNC Ž1998. hep-phr9803416.
Physics Letters B 436 Ž1998. 369–371
Neutron electric dipole moment and spontaneous CP breaking Paul H. Frampton, Masayasu Harada Department of Physics and Astronomy, UniÕersity of North Carolina, Chapel Hill, NC 27599-3255, USA Received 2 June 1998 Editor: H. Georgi
Abstract In a model where CP is spontaneously broken, the aspon model, it is shown by extending previous arguments about B decay and the value of ReŽ e Xre . in the kaon system, that the neutron electric dipole moment Dn is bounded from below by Dn G 10y29 e.cm. q 1998 Elsevier Science B.V. All rights reserved.
The electric dipole moment D of a neutron ŽNEDM. must be proportional to its spin S and is hence odd under T or CP and under P. The energy D P E is odd under CP and even under P because E is odd under P and even under CP. Thus a nonvanishing NEDM would be an explicit example of CP violation, outside of the kaon system. The discovery of a NEDM will be very important in formulating the theory of CP violation where the problem is the fact that only one Žcomplex. parameter, e K , has been accurately measured. The second parameter in the kaon system, ReŽ e Xre ., remains uncertain – it is still consistent with zero – and the CP asymmetries in B meson decays have yet to be determined. Any further experimental measurement of CP violation will constrain the viable theories. The experimental limit on Dn is presently Dn F 10y2 5 e.cm. w1,2x but it is possible w3x that this limit will be pushed back by several orders of magnitude in the forseeable future, even a limit of 10y3 0 e.cm. being conceivable. On the other hand, the prediction of the KM mechanism within the framework of the Standard Model is Dn , 2 = 10y3 2 e.cm. w4–7x which is beyond the reach of any planned experiments.
In a model of spontaneously broken CP symmetry w8,9x one may simultaneously solve the strong CP problem Žwithout an axion. and accommodate the observed CP violation in the kaon system. In an economical model, the Standard Model is augmented by an additional UŽ1. gauge group under which all the usual three generations of quarks and leptons are neutral, as are the established twelve gauge bosons Žg ,Z,W ", g a . and the single Higgs doublet. One adds a non-chiral quark doublet ŽU, D . which carries the new charge, and two complex Higgs scalar singlets x a. These x a Ž a s 1,2. carry an equal new charge. With this arrangement, the generalized Ž4 = 4. quark mass matrix has a real determinant and hence at tree-level the value of Q is zero if the underlying theory respects CP symmetry. Denoting the Žcomplex. VEVs of the x a by ² x a : and the Yukawa coupling constants to the ith generation by h ai useful parameters are as2
xi s
Ý
h ai ² x a :
Ž 1.
as1
These x i are complex, but for convenience we shall
0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 8 6 2 - 4
P.H. Frampton, M. Haradar Physics Letters B 436 (1998) 369–371
370
write x to denote x s < x i <, the modulus, and taken to be generation independent since the limits on x are not sensitive to the generation considered. In w10x, it was shown that the predicted value of Q is estimated from a one-loop diagram as:
Qs
lx2
Ž 2.
16p 2
This follows from the diagram shown in Fig. 1. In the notation of w9x the imaginary part gives a contribution:
Q Ž up . s
1
1
'2 Ž 4p .
Fig. 2. One loop < f < 2 < x < 2 counterterm for the coupling l.
as2,ls3 2
Ý as1,ls1
h al Im x l) la
k M
Ž 3.
which gives the estimate of Eq. Ž2., from which it follows that l x 2 is less than ; 10y8 . Here la is the coefficient of the quartic interaction between the two types of Higgs < f < 2 < xa < 2 , and l with no subscript is an average value. ŽActually there are three independent l corresponding to indices 11, 12 q 21, 22 but our estimates will not distinguish these.. The neutron electric dipole Dn has been calculated in terms of Q long ago w11,12x with the result that Dn , 10y1 5Q e.cm.
Ž 4.
and so we know from Dn F 10y2 5 e.cm. empirically that Q F 10y1 0 . Here we seek to establish a lower limit on the prediction for Q in the present model.
Firstly, we look at the kaon system for which the parameter < e K < is given by w9,13x: < eK < s
1
'2 D m K
mK
f K2 2 3 k2
x4
Ž 5.
Using Ž D m K rm K . s 7.0 = 10y1 5, f K s 0.16 GeV gives the relationship between x 2 and the UŽ1. new breaking scale:
krx 2 s 2.9 = 10 7 GeV.
Ž 6.
Thus, if we insist that the UŽ1. new is broken above the electroweak breaking scale Ž; 250 GeV. then: x 2 R 10y5 .
Ž 7.
From Eq. Ž2., this means that l - 10y3 . In w10x, it was further argued on the basis of naturalness that l ) 10y5 which Žif taken at face value. would imply that Q ) 10y1 2 and hence Dn ) 10y2 7 e.cm. But here we use what is a more solid, and more conservative, bound to achieve our lower limit on Dn . Our bound follows from the fact that the < f < 2 < xa < 2 interaction receives a one-loop correction from the quark loop box where three sides are the top quark and the fourth is the heavy U quark Žsee Fig. 2.. The full l is given by:
l s ltreeŽ bare. q l1 - loop Ž including counterterm. q higher loops. Fig. 1. One loop diagram of which the imaginary part contributes to Q .
Ž 8.
and the 1-loop finite contribution, for the dominant diagram ŽFig. 2., neglecting the quark masses and
P.H. Frampton, M. Haradar Physics Letters B 436 (1998) 369–371
taking h 3a , g t as the respective Yukawa couplings to xa and f of the third generation
l1 - loop ,
d4 k
H Ž 2p .
4
< h 3a < 2 < g t < 2
1 k4
lR
16p 2
that at least one of these different CP breakdown mechanisms must be experimentally refuted with the advent of B Factories.
q counterterm. Ž 9 .
which implies that the lowest value for l Žwithout accidental cancellations. is: x2
371
Ž 10 .
Acknowledgements This work was supported in part by the US Department of energy under Grant No. DE-FG0585ER-40219.
Combining Eqs. Ž2. and Ž10. then gives the estimate for Q of
QG
x4 16p 4
References
Ž 11 .
which implies that x 2 F 10y3 Žincidentally in full agreement with w10x. and that 10y1 0 G Q G 10y14 . From Eq. Ž4., this then give our required lower limit on the neutron electric dipole moment of Dn G 10y2 9 e.cm.
Ž 12 .
This is more than two orders of magnitude greater than the prediction of the KM mechanism and thus provides yet another distinguishing feature of spontaneous CP violation. Recall that this spontaneous CP violation model predicts, as reflects its superweak nature, that ReŽ e Xre . , 10y5 w13x Žcompared to O Ž10y3 . in the KM mechanism. and that the angles b ,g of the unitarity triangle measurable in B Factories are a few milliradians w10x Žrather than O Ž1. as in the KM mechanism.. It is very exciting to realize
w1x N.F. Ramsey, Ann. Rev. Nucl. Sci. 32 Ž1982. 211. w2x K.F. Smith et al., Phys. Lett. B 234 Ž1990. 191. w3x M.S. Freedman, G.R. Ringo, T.W. Dombeck, A Proposed Method for Measuring the Electric Dipole Moment of the Neutron Using Acceleration in an Electric-Field Gradient and Ultracold Neutron Interferometry. nucl-exr9804003. w4x E.P. Shabalin, Sov. J. Nucl. Phys. 31 Ž1980. 864. w5x N. Deshpande, Phys. Lett. B 108 Ž1982. 42. w6x I.B. Khriplovich, A.R. Zhitnitsky, Phys. Lett. B 109 Ž1982. 490. w7x B. Gavela et al., Phys. Lett. B 109 Ž1982. 215. w8x P.H. Frampton, T.W. Kephart, Phys. Rev. Lett. 66 Ž1991. 1666. w9x P.H. Frampton, D. Ng, Phys. Rev. D 43 Ž1991. 3034. w10x P.H. Frampton, S.L. Glashow, Phys. Rev. D 55 Ž1997. 1691. w11x V. Baluni, Phys. Rev. D 19 Ž1979. 2227. w12x R.J. Crewther, P. Di Vecchia, G. Veneziano, E. Witten, Phys. Lett. B 88 Ž1979. 123. w13x P.H. Frampton, M. Harada, UNC-Chapel Hill Report IFP757-UNC Ž1998. hep-phr9803416.