

 24 Apr 1996

UM-P-96/26 Constraints on CP violating four-fermion interactions

\Lambda

Xiao-Gang He and Bruce McKellar School of Physics, University of Melbourne

Parkville, Vic. 3052, Australia

(April 1996)

Abstract It has been shown that CP violating electron-nucleon and nucleon-nucleon interactions can induce atomic electric dipole moments and are therefore constrained from experimental data. We show that using the experimental upper bounds on neutron and electron electric dipole moments, one can also obtain constraints, in some cases better ones, on these interactions. In addition stringent constraints can also be obtained for muon-quark and tauon-quark four-fermion CP violating interactions, which cannot be constrained from atomic electric dipole moment experiments.

Typeset using REVTEX \Lambda Work supported in part by Australian Research Council.

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A non-zero electric dipole moment (EDM) of an atom signals CP violation. There have been extensive studies of atomic EDM in the past several years both experimentally [1-3] and theoretically [4-8]. A non-zero atomic EDM can be induced by several CP violating interactions: electron and nucleon EDM's, CP violating nucleon-nucleon and electron-nucleon interactions, and etc. So far only upper bounds on atomic EDM have been obtained; these bounds constrain the underlying CP violating interactions. Here we concentrate on constraints on CP violating electron-nucleon and nucleon-nucleon interactions and their generalization to lepton-quark and quark-quark four-fermion interactions. The CP violating electron-nucleon and nucleon-nucleon interactions can be written as:

CS GFp2 i _N N _efl5e ; CP GFp2 i _N fl5N _ee ; CT GFp2 i _N oe_*N _eoe_*fl5e ;

jNN0 GFp2 i _N fl5N _N

0N 0 ; j0

NN0 G

Fp

2 i

_N oe_*N _N 0oe_*fl5N 0 ; (1)

where N and N

0 can be n or p. The best experimental constraints on these interactions

come from bound on the EDM of 199Hg [2,6] with jCSj ! 1 \Theta 10\Gamma 6 and jCT j ! 2 \Theta 10\Gamma 8. The constraint on CP is about one order of magnitude weaker than that on CS. The parameter jjj is constrained to be less than 0.1 [4,5]. No constraint has been obtained for j

0.

The CP violating interactions enumerated in eq.(1) are generated by some underlying CP violating interaction at the leptons, quarks and their interactions. In particular these interactions can be generated by four-fermion lepton-quark and quark-quark interactions. In this note we point out that CP violating interactions at this level can also be constrained by the experimental bounds on the electron EDM (de ! 10\Gamma 26 ecm [3]) and neutron EDM (dn ! 10\Gamma 25ecm [9]). In this paper we determine these bounds on the underlying four-fermion lepton-quark and quark-quark interactions, and the implied constraints on the parameters of the interactions of equation (1). It will be seen that this analysis leads to improved bounds on jpn.

At the elementary particle level, the interactions in eq.(1) are related to the following lepton-quark and quark-quark interactions

_efl5e_qq ; _ee_qfl5q ; _eoe_*e_qoe_*fl5q ;

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_qq _q

0fl

5q

0 ; _qoe_*q _q0oe_*fl

5q

0 ; (2)

by the relations: hN jmq _qqjN i = \Delta Nq _N N , hN jmq _qfl5qjN i = \Delta 0Nq _N fl5N , and the nonrelativistic valence quark model approximation, hnj_uoe_* ujni = (\Gamma 1=3)_noe_* n, hnj _doe_*djni = (4=3)_noe_* n. Here \Delta Nq can be determined from experimental data and calculations of the nucleon mass shift due to SU(3)-breaking quark masses, and \Delta 0Nq can be obtained from polarized proton and neutron experimental data [10]. Similarly, quark-quark interactions can be related to nucleon-nucleon interactions. We will confine ourselves to flavour conserving interactions, that is, we do not consider interactions which convert different lepton or quark generation. Flavour conserving CP violating interactions are very small (at higher than two loop level order) in the Standard Model (SM) [7,8]. If relatively large interactions are shown to exist experimentally, they are signals of physics beyond the SM.

One may wonder how the lepton-quark and quark-quark four-fermion interactions may be generated beyond the SM. The CS;P type of interactions can be generated, for example, by exchanging neutral Higgs scalars or leptoquark scalars at the tree level [7,8]. Interactions of the CT type can be generated by exchanging leptoquark scalars at tree level, and the j interaction can be generated by exchanging a di-quark scalar at the tree level [7,8]. In the following we will not restrict ourselves to particular models, but use an effective Lagrangian approach to analyze possible constraints on the strength of these interactions [11,12]. We assume that any new physics introduced beyond the SM, say at a scale \Lambda ? mZ, which may have a gauge symmetry different to that at the scale \Lambda , is such that symmetry breaking between \Lambda and mZ gives the SM gauge symmetry as a residual symmetry at the electroweak scale. We will first analyze all possible lepton-quark and quark-quark interactions which respect the SM gauge symmetry SU (3)C \Theta SU (2)L \Theta U (2)Y and identify the relevant CP violating interactions, and then constrain these interactions using experimental data.

The left-, and right-handed quarks qiL, uiR ; diR, and left-, and right-handed leptons liL, eiR transform under the SM gauge group SU (3)C \Theta SU (2)L \Theta U (2)Y as:

qiL(3; 2; 1=3); uiR(3; 1; 4=3); diR(3; 1; \Gamma 2=3)

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liL(1; 2; \Gamma 1); eiR(1; 1; \Gamma 2) ; (3) where i is the generation index.

There are four different chiral structures for four-fermion interactions, _LL _LL, _RR _RR, _LR _RL, and _LR _LR with appropriate Lorentz structures associated with them. Here L, and

R indicate left-, and right-handed fermions. We find that CP violation can only occur in operators with the _LR _RL and _LR _LR structure. Among these operators only the following ones contribute to fermion EDMs at one loop level,

Oijkl1 = _liLoe_* ejR _qkLoe_*ulR ; Oijkl2 = _qiLujR _qkLdlR ; Oijkl3 = _qiLoe_* ujR _qkLoe_* dlR ; Oijkl4 = _qiLT aujR _qkLT adlR ; Oijkl5 = _qiLoe_*T aujR _qkLoe_*T adlR ; (4)

where T a is the SU (3)C generator and is normalized as Tr(T aT b) = ffiab=2. The operators O1;3;5 are not listed in Ref. [11]. They are independent of O2 and O4 and should be considered in calculating the EDM. All the operators listed above have dimension six, and are expected to be generated at the high energy scale \Lambda by some unknown physics. Flavour conservation requires: i = j, k = l or i = l, j = k. For O1, only i = j, k = l is allowed. To reflect the scale at which these operators are generated and their dimensionality, we parameterize the effective Lagrangian as

L = *

ij 1

\Lambda 2 _l

i Loe_* e

i R _q

j Loe

_*uj

R + *

ij 2

\Lambda 2 _q

i Lu

j R _q

j Ld

i R + *

0ij 2

\Lambda 2 _q

i Lu

i R _q

j Ld

j R + *

ij 3

\Lambda 2 _q

i Loe_* u

j R _q

j Loe

_*di

R

+ *

0ij 3

\Lambda 2 _q

i Loe_* u

i R _q

j Loe

_*dj

R + *

ij 4

\Lambda 2 _q

i LT

auj

R _q

j LT

adi

R + *

0ij 4

\Lambda 2 _q

i LT

aui

R _q

j LT

adj

R

+ *

ij 5

\Lambda 2 _q

i Loe_* T

auj

R _q

j Loe

_*T adi

R + *

0ij 5

\Lambda 2 _q

i Loe_* T

aui

R _q

j Loe

_* T adj

R + h:c: ; (5)

where for *0ij, we impose the restriction i 6= j to avoid double counting, since the *ii and *0ii interactins are identical. To compare with the interaction strength of the standard Fermi interaction, *ij =\Lambda 2 is some times conveniently written as CijGF =p2 in the literature. The dimensionless parameter Cij indicates the relative strength of the new interaction. We prefer to use the parameterization in eq. (5) in our calculations to keep track of the energy scale. We will first use experimental data to obtain bounds on the parameters *ij , and then convert these bounds into the parameters CS;T and j.

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The Feynman diagram responsible for the fermion EDM at one loop level is shown in figure 1. ?From this we obtain, for the EDM of the leptons and the ui quark,

dli(*ij1 ) = N12ss2 eIm(*ij1 )muj 1\Lambda 2 ln \Lambda

2

m2uj ;

duj (*ij1 ) = \Gamma 18ss2 eIm(*ij1 )mli 1\Lambda 2 ln \Lambda

2

m2li ;

dui (*ji2 ) = 18 18ss2 eQdIm(*ji2 )mdj 1\Lambda 2 ln \Lambda

2

m2dj ;

dui (*ji3 ) = 2N ffii

1ffij1 \Gamma 1

2

1 8ss2 eQdIm(*

ji 3 )mdj 1\Lambda 2 ln \Lambda

2

m2dj ;

dui (*

0ij

3 ) = 2N \Gamma 12 18ss2 eQdIm(*

0ij 3 )mdj 1\Lambda 2 ln \Lambda

2

m2dj ;

dui (*ji4 ) = N

2 \Gamma 1

16N

1 8ss2 eQdIm(*

ji 4 )mdj 1\Lambda 2 ln \Lambda

2

m2dj ;

dui (*ji5 ) = \Gamma N

2 \Gamma 1

4N

1 8ss2 eQdIm(*

ji 5 )mdj 1\Lambda 2 ln \Lambda

2

m2dj ; (6)

where N = 3 is the number of colours. Here we have taken the cut-off in the loop integral to be the same as the scale \Lambda . The operators O2;3;4;5 also induce an EDM of the dj quark. The ddj are obtained by replacing dj , ui and Qd by ui, dj and Qu in the above equations. We use the valence quark model to relate the neutron EDM to the quark EDM with: dn = (4=3)dd \Gamma (1=3)du. We note that not all operators of eq.(4) contribute to the fermion EDM. Therefore the EDM bounds will constrain only some of the parameters of the effective Lagrangian.

The results are logrithmically divergent. This is a good indication that the results are reliable. In fact if leptoquark or di-quark exchange is responsible for the EDM, the scale \Lambda is just their masses [7,8]. The operators discussed here may indeed be generated by the exchange of some heavy particles at energy scale \Lambda . Several operators with _LR _RL and _LR _LR chiral structures can induce phases in the determinants of the quark mass matrices and therefore a non-zero strong QCD CP violating ` term. If one naively uses the constraint on `, one would obtain very stringent bounds on the strength of these operators. However, the phases calculated in this way are quadratically divergent. We regard this as unsatisfactory, and feel that until one has a better understanding of the physics at the scale \Lambda , these results

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should not be used to give bounds on the parameters of eq.(5).

In principle one can use the EDM bounds on the neutron, electron, muon and tauon to constrain the parameters. We find that only the neutron and electron EDM's can provide significant constraints. Assuming that there is no accidental cancellation among different contributions, we obtain constraints given in Table I.

Using experimental bounds on CS;T and j, we can also obtain constraints on the parameters *ij . These constraints, in most cases, are much less stringent than those obtained in Table I. To compare with constraints on the parameters from direct atomic EDM measurements, we write the constraints obtained in Table I in terms of the parameters CnT , jpn. We use the factorization approximation and the non-relativistic valence quark model to estimate the matrix elements. We have,

hnj _eoe_* e_uoe_* fl5ujni = \Gamma 13 _eoe_* e_noe_* fl5n ; hpnjO11112 jpnijCP = 14 2N + 12N [hnj_u1u1jnihpj _d1fl5d1jpi

+ hnj _d1d1jnihpj_u1fl5u1jpi] = 14 2N + 12N (\Delta nu1 \Delta

0

pd1 + \Delta nd1 \Delta

0 pu1)_nn_pfl5p ;

hpnjOijji2 jpnijCP = 18N [hnj_uiuijnihpj _dj fl5djjpi

+ hnj _dj djjnihpj_uifl5uijpi] = 18N (\Delta nui \Delta

0

pdj + \Delta ndi \Delta

0 pui)_nn_pfl5p ;

hnpjO11113 jnpijCP = 122N + 1 hnpjO11112 jnpijCP ; hnpjOijji3 jnpijCP = 12hnpjOijji2 jnpijCP ; hnpjO11114 jnpijCP = N

2 \Gamma 1

2N (2N + 1) hnpjO

1111 2 jnpijCP ;

hnpjOijji4 jnpijCP = N

2 \Gamma 1

2N hnpjO

ijji 2 jnpijCP ;

hnpjO11115 jnpijCP = 6(N

2 \Gamma 1)

N (2N + 1) hnpjO

1111 2 jnpijCP ;

hnpjOijji5 jnpijCP = 6(N

2 \Gamma 1)

N hnpjO

ijji 2 jnpijCP : (7)

In our numerical evaluations, we will use the values mu = 4:2 MeV, md = 7:5 MeV, and ms = 150 MeV:

\Delta nu = 18MeVm

u ; \Delta

nd = 18MeVm

d ; \Delta

ns = 247MeVm

s ;

6

\Delta nh = 48MeVm

h ; \Delta

0 pu = 432MeVmu ; \Delta

0 d = \Gamma 419MeVmd ;

\Delta

0

ns = \Gamma 165MeVms ; \Delta

0 ph = \Gamma 63MeVmh ;

where h indicates a heavy quark [10]. The operators with coefficients *03 do not contribute to jpn. The constraints on CnT and jpn are shown in Table II.

The constraints on CnT are much weaker than those obtained from the upper bound on the atomic EDM. However, the constraints on jpn in most cases are much better than those obtained from atomic EDM. In addition, we obtained constraints on muon-quark and tauon-quarks interactions which cannot be obtained from atomic electric dipole moment measurements.

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REFERENCES [1] T.G. Vold et al., Phys. Rev. Lett. 52, 2229(1984); S.A. Murthy et al., Phys. Rev. Lett.

63, 965(1989); P. Cho, K. Sangster and E.A. Hinds, Phys. Rev. Lett. 63, 2559(1989).

[2] J.P. Jacobs et al., Phys. Rev. Lett. 71, 3782(1993). [3] K. Abdullal et al., Phys. Rev. Lett. 65, 2347(1990). [4] P. Sanders and R. Sternheimer, Phys. Rev. A 11, 473(1975); V. Flambaum, Sov. J. Nucl.

Phys. 24, 199(1976); W. Johnson et al., Phys. Rev. A 34, 1043(1986); I. Khriplovich, Sov. Phys. JETP 44, 25(1976); P. Convey and P. Sanders, J. Phys. B16, 3727(1983); V. Dzuba, V. Flambaum and P. Silvestron, Phys. Lett. B154, 93(1985); V. Flambaum and I. Khriplovich, Sov. Phys. JETP 62, 872(1985); P. Sanders, Phys. Lett. 14, 194(1965); 22, 290(1966); V. Ignatouich, Sov. Phys. JETP 29, 1084(1969); M. Player and P. Sanders, J. Phys. B2, 1620(1970).

[5] O. Sushkov, V. Flambaum and I. Khriplovich, Sov. Phys. JETP 60, 873(1984); V.

Flambaum, I. Khriplovich and O. Sushkov, Phys. Lett. B162, 213(1985); Nucl. Phys. A49, 750(1986).

[6] C. Bouchiat Phys. Lett. B57, 284(1975); E. Hinds, C. Loving and P. Sanders, Phys.

Lett. B62, 97(1976); Ann-Marie Martensson-Pendrill, Phys. Rev. Lett. 54, 1153(1985).

[7] X.-G. He and B. McKellar, Phys. Rev. D46, 2131(1992); Int. J. Mod. Phys. A8,

209(1993).

[8] X.-G. He, B. McKellar and S. Pakvasa, Phys. Lett. B283, 348(1992). [9] K. Smith et al., Phys. Lett. B234, 234(1990). [10] A. Anselm et al., Phys. Lett. B152, 116(1985); T. P. Cheng and L.F. Li, Phys. Lett.

B214, 165(1990); H.-Y. Cheng, Int. J. Mod. Phys. A7, 1059(1992)

[11] W. Buchmuller and D. Wyler, Nucl. Phys. B268, 621(1986).

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[12] C.P. Burgess and J.A. Robinson, in BNL Summer Study 1990:205-248

(QCD161:S791:1990)

9 TABLES TABLE I. The upper bounds on jIm*ijj from the electric dipole moments of electron and neutron for \Lambda = 1TeV.

de

jIm(*eu1 )j jIm(*ec1 )j jIm(*et1 )j 2:0 \Theta 10\Gamma 4 1:2 \Theta 10\Gamma 6 3:3 \Theta 10\Gamma 8 dn

jIm(*eu1 )j jIm(*_u1 )j jIm(*_u1 )j 8:3 \Theta 10\Gamma 2 6:5 \Theta 10\Gamma 4 5:3 \Theta 10\Gamma 5

jIm(*du2 )j jIm(*dc2 )j jIm(*dt2 )j jIm(*su2 )j jIm(*bu2 )j 3:7 \Theta 10\Gamma 2 2:1 \Theta 10\Gamma 4 5:9 \Theta 10\Gamma 6 1:0 \Theta 10\Gamma 2 5:4 \Theta 10\Gamma 4

jIm(*du3 )j jIm(*dc3 )j jIm(*dt3 )j jIm(*su3 )j jIm(*bu3 )j 1:6 \Theta 10\Gamma 2 5:3 \Theta 10\Gamma 5 1:5 \Theta 10\Gamma 6 2:5 \Theta 10\Gamma 3 1:4 \Theta 10\Gamma 4

jIm(*0cd3 )j jIm(*0td3 )j jIm(*0us3 )j jIm(*0ub3 )j 8:8 \Theta 10\Gamma 6 2:5 \Theta 10\Gamma 7 4:1 \Theta 10\Gamma 4 2:2 \Theta 10\Gamma 5 jIm(*du4 )j jIm(*dc4 )j jIm(*dt4 )j jIm(*su4 )j jIm(*bu4 )j 2:8 \Theta 10\Gamma 2 1:6 \Theta 10\Gamma 4 4:4 \Theta 10\Gamma 6 0:75 \Theta 10\Gamma 2 4:1 \Theta 10\Gamma 4

jIm(*du5 )j jIm(*dc5 )j jIm(*dt5 )j jIm(*su5 )j jIm(*bu5 )j 0:69 \Theta 10\Gamma 2 3:9 \Theta 10\Gamma 4 1:1 \Theta 10\Gamma 6 1:9 \Theta 10\Gamma 3 1:0 \Theta 10\Gamma 4

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TABLE II. The upper bounds on CnT and jpn with \Lambda 5 = 1 TeV. de

jCnT (*eu1 )j 0:8 \Theta 10\Gamma 8 dn

CnT (*eu1 )j 3:4 \Theta 10\Gamma 3

jjpn(*du2 )j jjpn(*dc2 )j jjpn(*dt2 )j jjpn(*su2 )j jjpn(*bu3 )j

0:28 8:3 \Theta 10\Gamma 6 1:8 \Theta 10\Gamma 9 1:7 \Theta 10\Gamma 2 5:2 \Theta 10\Gamma 6 jjpn(*du3 )j jjpn(*dc3 )j jjpn(*dt3 )j jjpn(*su3 )j jjpn(*bu3 )j

0:21 2:5 \Theta 10\Gamma 5 5:5 \Theta 10\Gamma 9 5:1 \Theta 10\Gamma 2 1:6 \Theta 10\Gamma 5 jjpn(*du4 )j jjpn(*dc4 )j jjpn(*dt4 )j jjpn(*su4 )j jjpn(*bu4 )j 4:0 \Theta 10\Gamma 2 8:5 \Theta 10\Gamma 6 1:8 \Theta 10\Gamma 9 1:7 \Theta 10\Gamma 2 5:3 \Theta 10\Gamma 6

jjpn(*du5 )j jjpn(*dc5 )j jjpn(*dt5 )j jjpn(*su5 )j jjpn(*bu5 )j

0:12 2:5 \Theta 10\Gamma 5 5:4 \Theta 10\Gamma 9 5:1 \Theta 10\Gamma 2 1:5 \Theta 10\Gamma 5

11

FIGURES FIG. 1. One loop Feynman diagram for fermion EDM.

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