Theory of the neutron electric dipole moment

Nuclear Instruments and Methods in Physics Research A284 (1989) 33-39North-Holland, Amsterdam

THEORY OF THE NEUTRON ELECTRIC DIPOLE MOMENT

John ELLISCERN, Geneva, Switzerland

Predictions of various models for the neutron electric dipole moment are presented and compared with the experimental situationof CP violation.

1. What we know about CP violation

The first CP-violating observable measured [1] in thelaboratory was the mass mixing parameter in the K° -K° system [2] :

I E K I _ (2.26 ± 0 .02) X 10-3 ,

which can be accommodated in the Standard Model aswe discuss later. More recently, direct CP violation hasbeen measured [3] in K° -> 21T decay amplitudes :

'E'/'EK = (3 .3 ± 1.1) X 10-3 ,

(2)

which can also be accommodated in the Standard Model[4]. Another piece of circumstantial evidence for CPviolation is the baryon asymmetry of the Universe,corresponding to a present baryon-to-photon ratio

inferred [5] from the success of Big Bang nucleosynthe-sis calculations [6] . It seems very likely that explainingn B/n ., requires physics beyond the Standard Model.The limits on the neutron electric-dipole moment doquoted at this meeting were

which we will later compare with predictions fromvarious models of CP violation.

2. Approaches to calculating d� [9]

2.1 . Naive quark model

In this approach, one first calculates CP-violatinginteractions at the quark level, and then uses the non-

0168-9002/89/$03 .50 C Elsevier Science Publishers B.V.(North-Holland Physics Publishing Division)

33

relativistic quark model to calculate hadronic matrixelements of the quark operators. For example, onemight calculate the electric dipole moments du,, of theu and d quarks and then use a non-relativistic SU(6)wave function

d(") = 3 (4dd - d� ) .

(Sa)Or one might calculate the colour electric dipole mo-ments fu,d of the quarks and use the same SU(6) as-sumption to obtain

dn)

3e(3fd+ 3fu )'

(Sb)However, the successes of chiral symmetry in mesondynamics tell us that m� .d << AQCD, so that the u and dquarks are unlikely to be non-relativistic . Moreover,measurements of the iTN a-term [10] and of the spinstructure of the proton [111 tell us that the proton andneutron contain ss pairs . Therefore the naive quarkmodel with its non-relativistic SU(6) wave functionscontaining just valence quarks may not always be veryreliable .

2.2. Effective Lagrangian

In this approach one uses a phenomenologicalLagrangian for baryons and pseudoscalar mesons, in-cluding strong couplings

Y5= _~2 gBiYSB'P + (h.c .),

weak couplings

Y, = rf e- '0BB'P + (h.c .)

(6b)

and meson kinetic terms

(6a)

YI a = h e"'DMP'*D,,P + (h.c .) .

(6c)

The magnitudes of the couplings are fixed using phe-nomenology and/or some particular weak interactionmodel. Loop diagrams calculated with this effectiveLagrangian are the leading contributions as m2 -> 0(m,,,,, - 0) . There are two important classes of di-agrams, those with one weak vertex as in fig . 1(a), and

1 . NEW PHENOMENA

dn =( -1 .4±0.6)X10 -25 ecm [7]

= (-0.7 ± 0.4) X 10 -25 e cm [8], (4a)corresponding to

dn < 2.6 X 10 - ' e cm, (4b)

n,/n, = 10-11-10-s, (3a)

inferred [5] from direct observation, or

n B/n,,= (3-5) X 10 -1° (3b)

3 4

3. Model predictions for d �

3.1 . Standard model

where UKM may be parametrized by

UKM

k Sts2

a)

b)

Im MK KzEK -

I= XKS2S3SS .

MK, - MK,

J. Ellis / Theory of the neutron electric dipole moment

"---"M

Y

n 1- B

n

n

t

vM~,

~'M

+

M'~

__*~~

+

Fig. 1 . Classes of loop diagrams contributing to d� : (a) withone weak vertex, (b) with two weak vertices

those with two as in fig. 1(b) . The generic result ofcalculating the one-weak-vertex diagrams is [9]

d ;,'t= ~~ sin(-$)MB

fl't(mMlms), (7a)

The generic result of calculating the two-weak-vertexdiagrams in fig. 1(b) is [9]

d~2t_eghf

sin(B-$)1f(2) ( mMlmB)~4m2 ins

( C l

-SIC3-S 1S3

slc2

ctc2c3-

s2s3e's

clc2s3 + szcse's

cis2c3 + c2s3 e' s

cls2s3 - c2c3 e' s )

(7b)

Because of the logarithms in eq . (7) and the phenome-nological enhancement of some non-leptonic weak cou-plings due to the DI = i rule, the contributions (7) tod� can be larger than the naive quark model contribu-tions (6) in a given model.

In the Standard Model, CP violation is due to asingle complex phase S in the Cabibbo-Kobayashi-Maskawa matrix UKM [12] :

where

c, ( s, ) = cos B,

(sin 0,)

(t = 1, 2, 3).

Note

thatthere is a five-fold ambiguity in the phase convention :qr - e`°qr, and that all physical quantities are propor-tional to the combination s2s3 sin S, which can inprinciple be determined from EK [4,9] :

(10)

The hadronic matrix element factor is only poorlyknown, although it is in principle calculable once thetop quark mass (m,) is known. Fig. 2(a) shows thevalue of s2 and the allowed range of s3 for a plausibleset of values of hadronic matrix elements, and fig. 2(b)shows the corresponding range of sin S, as functions ofm, [4]. One finds that

2 x 10 -4 < s2s3 stn S < 2 X 10-3 ,

(11)

where the lower limit comes from mI < 200 GeV asindicated by precision neutral current experiments [13],and the upper limit comes from mt > 41 GeV as indi-cated by the published UAl search [14] at the CERNPp collider. Fig. 3 shows that the value of 'O f K predic-ted [4] in the Standard Model on the basis of eqs. (10)and (11) agrees well with the experimental value (2) [3].The precision of the agreement is not high, but could beimproved with time .

It was soon realized [15,16] that Standard Modeldiagrams containing only two W± vertices could notgive any CP violation, as they were proportional to

(UKM),(UKM);, which is real . It is possible to obtain CPviolation with four W± vertices, since (UKM)~(UkM)k(Ui+cM)i(UKM); is complex in general. However, all theinternal quark masses must be different, and one mustbeware of symmetrizatton that can cancel out the CPviolation. For example, the sum of diagrams with dustone internal quark line and no gluons is real [17] . Oneneeds at least two quark lines and/or several stronginteraction corrections . The resulting contribution to d�is small

when calculated in the naive quark model.The fact that one needs four W f vertices means that

in the effective Lagrangian approach one needs di-agrams with at least two weak vertices such as f, h ineqs. (7b), (7c). Experimental results on A and Y_ decaystell us that [9]

whilst a theoreti~al calculation indicates that

(A= O.OSs2s3 sin S .

(13b)

The rate of K° - 2,Tr decays tells us that

whilst another theoretical calculation indicates that

a = 0.3s2s3 sin S.

(14b)

It should be noted that both If I and

I h I are muchlarger than would be estimated in the naive quarkmodel, because of the 01 = ' rule . Using (13) and (14)in the generic two-weak-vertex formula (7b) one finds

d __ eghfsin(B- (p

)f(2)lmM/mB)

4 Ir 2ms

(12)

i

-(10 -z9-10-2x )s2s3 sin 3 e cm .

(15)

0 08m

N0 06

014

012

0 10

0 04

002

020 40 60 80 100 120 140 160 180 200

rit (GeV)

J. Elhs / Theory of the neutron electric dipole moment 3 5

i

i

i

i

i

i

i

- 1

100 120 140 160 180 200mt (GeV)

Fig. 2 . (a) The value of sz and the range of s3, and (b) the value of sin S extracted from a Standard Model analysis [4] of B-B mixingfor a range of m ,

I . NEW PHENOMENA

36

0XYW

W

After substituting the range (11) for s,s3 sin 8 into eq .(15), one finds [9]

d. _ (10 -33-2 x 10-31) e cm,

(16)

which is somewhat larger than the nâive quark modelestimate (12), but still far below the present experimen-tal sensitivity .

3.2. QCD 0 parameter

2'm loo"

mumdms(m �md + mdms + msm � )


2_ -0

G GI`° ; Gw°-EU v al6G

Qo° 32ô

IT 2

~p

I

~

I

I20 40 60 80

In the presence of non-perturbative effects in QCD,one must allow [18] for a term

8K = 0

L_-

- --___-__-_____ .._____I

I

I

I

I

I120 120 140 160 180 200

ni t (GeV)Fig . 3 . Standard Model prediction [4] for E'/EK compared with experiment [3] .

(17)

that is C-even (because it has two gluon field strengthsG,�,) and P-odd (because of the E,_,q) and hence violatesCP . By making an anomalous global U(1) transforma-tion on the quark fields, one can replace the 0-term (17)by a phase in the quark mass matrix :

X (tlysu+dys d+sygs) .

(18)

The contribution this violation of CP makes to do hasbeen calculated m the bag model [19] :

dn = 2.7 x 10-I'BQCD e cm

(19a)

F_T7

and using pion-nucleon loops [20] :

dn = - 3.8 x 10 -I6BQCD e cm .

(19b)

Including all pseudoscalar meson-baryon loops andboth yw and omnq° couplings of the proton, it has beenestimated that [9]

2x10-I6<d

IO

cm<5x10- '6 .

(19c)QCD e

The experimental upper limit (4b) therefore tells us that

I OQCD I -< 10-9 (20)

and the big question is : why?In the Standard Model there is a complex renormal-

ization of the quark mass matrix at the two-loop levelwhich has been estimated [21] to yield

180QCD I =10-'6

(21)

and hence according to (19c)

Idol = few x 10 -32 e cm,

(22)

which is comparable to the previous Standard Modelcontributions (12) and (16) . However, other models caneasily give S0Qc, = 0 in one-loop order. In a GrandUnified Theory (GUT), the dominant source of thebaryon asymmetry (3) is often believed to be GUTHiggs decay [22], via the generic diagrams in fig . 4(a). Inthe minimal SU(5) GUT the baryon asymmetry only

f

H

f

arises in three-loop order, and is far too small [22] .However, it may well be that any model giving a largeenough baryon asymmetry will also give a large contri-bution to BQCD [23] . This is because there is a renormal-ization of the quark mass matrix which is proportionalto a Higgs coupling (see fig. 4(b)) analogous to thatappearing in the baryon asymmetry diagram of fig . 4(a) .This possible connection suggests that [23]

180QCD I ° 10(nB/n1,)(gH9q)z.

(23)

Putting in (23) the estimate (3b) of n B/n y and eitherthe Higgs coupling to the b quark ()nb -- 5 GeV) or tothe t quark (m i - 100 GeV) gives the range

I Sd � 1 - (6 X 10-z8-2 X 10-25) e cm,

(24)

which approaches the experimentally accessible range.One reason why BQCD may be very small is that it is

actually zero, thanks to the Peccei-Quinn mechanism[24] . According to this, there is an approximate U(1)global symmetry realized with an almost massless pseu-doscalar boson, the axion. Its mass and couplings

3 1/2( m gA QCD) mf

Ma-

f

baff =fa '

(25)a

are inversely scaled by a large axion decay constant fa

that is often related to a large Higgs v.e .v. Astrophysics,including most stringently the supernova 1987a, andcosmology [26] tell us that

10 11 GeV <_fa < 10 12 GeV

(26)

leaving a narrow window in which the axion couldprovide an interesting and observable amount of darkmatter. Detection of this dark matter would presumablyrule out BQCD as a source of a large d� .

3.3 . Two Higgs doublet model

The requirement of natural flavour conservation im-poses that one Higgs doublet give masses to all the


a)

b)Fig. 4 . The possible connection [23] between BQCD and the baryon number of the Universe : (a) GUT Higgs decay diagram, and (b)

contribution to BQCD .

charge +

quarks, and one to all the charge - ''3

3quarks [27] . This means that there is no spontaneous CPviolation, and that the only source of CP violation is thefamiliar phase 8 in the Kobayashi-Maskawa matrix .The only difference from the Standard Model calcula-tion of d � is in the extra H ± exchange diagrams . Sincewe to any case took the values of the weak vertices fromA, 2 and K decays, their only possible effect could beto alter the theoretical predictions for the phases 0, 0(13b) and (14b). However, since Higgs couplings to lightquarks are small, this effect is so small, and one esti-mates [9,28]

d� = (10 -33-2 X 10-31) e cm

(27)

as in the Standard Model.

3.4. Sipersymmetry

H

37

There are two new sources of CP violation in aminimal supersymmetric extension of the StandardModel, namely phases in the gluino mass [29]

m,=

mg j e-2, o8

(28)

and in the squark mass matrix [30], which may beparametrized as

XMd Mâ +c~M�M�

(A*M)d

(AM+ )d

mâ+cdMd Md)

for the charge - ; squarks, and similarly for the charge+ squarks. We can diagonalize the conventional quarkmass matrices Md and M. by the Kobayashi-Maskawamatrix UKM, but the squark mass matrix (29) will ingeneral require an extra diagonalization by a matrix Uthat can introduce additional flavour and CP violationinto the qgq couplings.

I . NEW PHENOMENA

38

The latter we can factorize out as a diagonal phasematrix e'~: U= Ue'~ (det U real) in the couplings sothat

dgd = i>2gsd+S1V+ e' 0 ( 1-YS

)11

2

+ V e - '~ ( 1 2 Ys ) ] d,(30a)

where

is the observable relative phase, which is present even ifthere are only two generations .

There are then one-loop diagrams [31] like those infig . 5 which contribute to the quark electric dipolemoments

dd

2 e mg(2q))f(mass ratios),

etc.

(31)9 IT

and similarly to the colour dipole moments. These canbe used directly m the naive quark model formulae (5)to obtain an estimate of d � . Alternatively, one canestimate the CP-violating part of the BB'P vertex anduse it in the effective Lagrangian formula (7a) . One canestimate the order of magnitude by assuming

It should be emphasized that the estimate (33) is sensi-tive to the assumption (32) on the particle masses, buttaken at face value it suggests when combined with theupper limit (4b) that

< O(10-3 ) = O(a/ir) .

(34)

This is not necessarily a problem for realistic supersym-metric models, where q) often appears at the one-looplevel .

3 .5 . Weinberg multi-Higgs model

If there are three or more Higgs doublets, theirv.e .v .'s may be relatively complex: Im o,*v, = 0, which isa possible source of spontaneous CP violation even ifUKM is real [33] . The observed value of EK (1) can beused to constrain the model parameters, which can then

metric model.

J Ellis / Theory of the neutron electric dipole moment

W HR

-,i-, Htt

v

Fig. 6. One-loop diagrams contributing to d � m a left-rightsymmetric model.

be used to calculate quark electric dipole moments,colour dipole moments and inputs to the hadron-loopcalculations . In the latter case, the largest contributioninvolves the KIn coupling, and is [34]

d � = 10 -24 e cm .

(35)

This looks too large, but the calculation could be wrongby some factor, so it might be premature to reject theWeinberg model completely. There is also the possibil-ity of a cancellation between different diagrams, so weconclude that Weinberg is almost dead ; but . . .

3.6. Left-right supersymmetric model

In a model with gauge group SU(2) L X SU(2) R XU(1) there are two generalized Kobayashi-Maskawamatrices UK, and UKM. The number of independentparameters, and in particular the number of phases,depends on the degree of left-right symmetry assumed,but to general there are more phases than the StandardModel [35] . For example, if the model has "pseudo-manifest" symmetry, so that gL = gR and BL = BR (t =

1, 2, 3), there are still (n - 1)(n - 2)/2 + (2n - 1) = n(n+ 1)/2 phases . In this case, one does need not threegenerations to obtain CP violation . The dominant con-tributions to d � are from one-loop diagrams involvingleft-right mixing as in fig. 6. Since the model hasseveral phases, they are not completely constrained byEK(1) and e'(2), and there is a relatively large possiblerange for d � [36] :

I do I - (10-25-10-26) e cm .

(36)

However, it should be said that getting I d� I as large as10 -25 requires quite a special choice of parameters .

Table 1Summary of model predictions

Model Prediction for d � What if -

9 d (e cm)

d i d Standardmodel (10-33 -2X10-31 ) X

\v / + QCD 0 parameter 3X10- 16 6QCD (0-10-9 )d d 9 d Two-Higgs model (10-33-2 X10-31) X

1,9 119 Supersymmetry 10-22 p (,p-10 -3 )Weinberg model 10-24 (X)

Fig . 5 . One-loop diagrams contributing to d � in a supersym- Left-right symmetric 10 -27-10 -25

2tnL-m2R=m g = (32)

to which case [32]

d � = 10-22¢ e cm . (33)

4. What if . . . ?

What would be the implications for the various

models discussed in section 3 if d� were to turn out to

be _ 10 -25 cm? Table 1 summarizes the predictions of

the various models, and gives a verdict on them in this

hypothetical case . The only clear conclusion is that a

value of d� in this range would be unambiguous evi-

dence for some physics beyond the Standard Model.

Which physics is a different matter, and largely a matter

of taste at this point. My own favourite explanations

would be the QCD 0 parameter or supersymmetry.

However, we should not fantasize too freely : d� might

well be many orders of magnitude below the present

experimental sensitivity.

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I. NEW PHENOMENA

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Theory of the neutron electric dipole moment