103

COSMOLOGY AND THE NEUTRON ELECTRIC DIPOLE MOMENT

John Ellis CERN - Geneva

ABSTRACT

There is a contribution to the neutron electric dipole moment d from the CP violating e vacuum parameter of QCD. Diagrams analogous to thosg responsible for the baryon number of the also contribute to e, providing an order of magnitude lower bound on dn in terms of the baryon-to-photon ratio n8!ny · GUTs sufficiently complicated to explain the observed nBlny predict that dn should be close to the present experimental upper limit. The comparison between dn and n8!ny gives us information about entropy generation after the epoch of baryon generation in the very early Universe . 104

1. - INTRODUCTION

l) 2) Much experimental effort , has been devoted to the search for an

electric dipole moment of the neutron, d which violates CP . Detection of a n ' non-zero d would add much to our knowledge of CP violation , confined hitherto n 2) to the K0-R0 system 3l and the present experimental limit 10-25 d 6 x e cm ( l) n < -

l) is already a strong constraint on different models of this phenomenon . Theo­

retical analysis of d has proceeded along two lines , the first being calcula­ n tion of perturbation theory diagrams in GP-violating weak interaction models .

More recently it has been realized 4 l that non-perturbative effects in QCD will violate CP if a hitherto unremarked parameter e is non-zero. While it is QCD not renormalized by the strong interactions alone , e is in general renorma­ QCD lized by the non-strong interactions , even if its bare value is zero .

Mary Gaillard , Demetre Nanopoulos , Serge Rudaz and I have recently em­

phasized that there are contributions to e from Higgs exchange diagrams 5l QCD in GUTs , which are closely analogous to those responsible for the baryon number H of the Universe produced in the decays of very heavy iggs particles &) These contributions to e give us an order of magnitude lower bound on d of QCD n

(2)

-lO If we take from a lower bound on n /n of 2xlO then we 7l 8 y find that

-28 (3) d > 5 l0 e-cm n "' x

to be compared with the upper bound (1) .

In this talk I will review our work S),B), starting with a reminder of the meaning of e its relation to Higgs couplings and its renormalization QCD ' g). Then we will see how n8/n is believed to be generated in GP-violating decays of Higgs bosons via dia rams similar to some contributing to e " In fact , 6l � QCD in many models e and hence d is mueh larger than the lower bound QCD n 5 l,S) 2) (2) or (3) . Finally we will see how the experimental upper limit on d n 105 already gives us useful information on the amount of entropy that could have been generated subsequent to the creation of a baryon-antibaryon asymmetry. The bound lO ) (1) already poses problems for the scenario of symmetry breaking by radiative corrections in the Weinberg-Salam model , and constrains S) deviations from the usual Robertson-Walker-Friedmann (RWF ) big bang cosmology with some implications ll) for models of galaxy formation

2. - THE QCD 6 PARAMETER

12 ) In perturbation theory , QCD automatically conserves CP However ,

there is a possible term 4l in the QCD Lagrangian

';[ QCD (4)

a where is the gluon field strength tensor and G µv is its dual . The e G v QCD term (4)� is a total derivative and hence does not show up in perturbation theory ,

but it is non-zero for non-perturbative field configurations such as instantons 4l. It conserves charge conjugation C because it is the product of two gauge field v strengths , but it violates parity because the dual tensor µ contains an p aa antisymmetric E vpo 'symbol. Therefore the 6 term (4) violates CP . µ QCD

However, it is not directly related to the CP violation seen 3 l in the

K0-K0 system since there it occurs in an effective interaction with l�sl = 2, whereas the interaction ( ) clearly conserves strangeness . It will , however , con­ 4 tribute to the neutron electric dipole moment d in an amount estimated 1 as n ' 3)

d (5) n e-cm

from both MIT baggery and chiral perturbation theory techniques . Using the most 2 recent ) upper limit on d we infer that (1) n

( 6)

- !This upper bound on e means that the 0(10 3 ) CP violation in the QCD K0-R0 system cannot be due to a combination of e and a CP conserving !�SI = 2 QCD interaction .J 106

l) For comparison , the best upper limit on d by an American group is n -24 d � xlo e-cm, and they are now starting an extensive series of experiments n 3 to improve this by as many as three orders of magnitude , A major problem of theo­ (6). retical is to determine why e is so small QCD

In pure QCD , e is an arbitrary parameter which is not renormalized QCD by the strong interactions but has no natural reason to be small or zero . When

one takes into account the non-strong interactions e is in general renorma­ ) QCD lized 9 . The most obvious contribution to renormalization of ' which seems eQCD to give the correct order of magnitude in general, arises from the renormalization

of the quark matrix which is necessary at each order of perturbation theory in the non-strong interactions . If we define an effective quark mass parameter m(q) from the inverse quark propagator :

) <( - rn(q) (7

then it gets renormalized in each order of perturbation theory by diagrams shown Ren generically in Fig. la . The renormalized mass matrix m should then be restored

to real and diagonal form at each order in perturbation theory. This entails phase rotations on the left- and right-handed quark fields which feed through via the

QCD U(l) anomaly into a renormalization of e by an amount QCD en arg det rnR (8)

The leading contribution to comes from one particle irreducible diagrams o8QCD F1 making a transition between left- and right-handed quarks :

Ren rn "' = arg det - 1 F QCD "' Im Tr(m ) (9) lie o 1

where m is the zeroth order quark mass matrix . Since the effective mass para­ �en (7) meter m is a function of the momentum scale q , so also will be o8 QCD ' The observation (or non-observation) of d constrains the effective value of n renormalized at a momentum scale of order 1 GeV or so. Presumably there is

an eventual theory of everything (TOE ) which fixes e at some enormous scale 1 QCD q � O (lo 9) GeV . Between this enormous scale and the scale of the neutron e QCD is renormalized by all the non-strong interactions , and we can distinguish 5 ) the following contributions : 107

(10)

1 We denote by o the renormalization due to GUTs at energy scales �10 5 GeV , 8GUT and by o the renormalization due to conventional six-quark Kobayashi-Maskawa 8KM 1 (KM) weak interactions at momentum scales <10 5 GeV. These latter effects have been estimated 9 ) as

(11)

-31 -3 resulting in a contribution to d of order l0 to lo 2 e-cm. This contri­ n bution to d is in fact dwarfed by the direct contribution to d from pertur- n n . 9) 14) bation theory in the strong and weak interactions which has been estimated ' as

m2 2 �d c s S s (12) n sin 1 2 3

in the standard KM model . We see that the standard KM weak interaction contribu­ tions to d are very small. n

As yet we do not know how to estimate e but we can calculate TOE ' o This is derived from the quark mass matrix, which is in turn given by a 8GUT' Higgs vacuum expectation value multiplied by a Higgs coupling matrix H as il­ 1 lustrated in Fig. 1 : F = v H . For convenience we will work directly with the 1 1 coupling matrix of this so-called m -Higgs . We will see that in general a GUT sufficiently complicated to explain the observed n /n will yield a oe B y GUT >> oe In the absence of a TOE we will assume that there is no conspiratorial >> KM' cancellation between 8 and o8 so that in order of magnitude TOE GUT

(13) and we will now go on to consider the relation of to in some detail . 3 . - THE COSMIC CONNECTION

) In his talk at this meeting, Demetre Nanopoulos 6 has already described

to you the mechanisms offered by GUTs for generating a baryon-antibaryon asymmetry

in the very early Universe . It seems most likely that the net baryon number ori­

ginates from the C- and CP- violating decays of some very heavy particles , typically 14 15 gauge bosons , Higgs bosons or fermions with masses 0(10 to 10 ) GeV. Figure 2 ) - shows how 15 a net baryon asymmetry of order 10 9 can be built up at high tempe-

ratures if one postulates heavy particles X with a suitable B- , C- and CP-viola- ting asymmetry in their decays . Superheavy fermions are not present in the simplest ) GUTs , though they are commonplace in larger ones 6 Most quantitative analyses of baryon number generation have focused on the decays of gauge and/or Higgs bosons ,

and we shall concentrate on them here. Their contributions to in simple 1 ) models have been calculated 6 numerically to be

(1 01 ev ) 160  for gauge bosons 10-1' I £G [ -1·3 ( nB/ny) :\: J (14) for Higgs bosons f"0.5 X 10-l•I EH The factor in square parentheses for gauge bosons in (14 ) represents the suppres­ sion effects of 2-2 interactions which tend to dilute the asymmetry generated

by gauge boson decays and inverse decays . There are no such factors for Higgs 14 -4 bosons in (14 ) if m > 2xlo GeV and the g /4TI a 10 as seems plau­ H � = H � 6 sible. The factors E and E in ( 14) are the CP-violating asymmetries in G H ) decays of gauge and Higgs particles and antiparticles . In simple models 6 E G ' is one order higher in a than is � Furthermore , the parenthesized sup­ GUT H 14 1 ) pression factor in ( 14) is quite fierce if m x10 GeV as expected 7 in G � 6 ) minimal SU ( 5 ). We are therefore inclined 6 to expect that Higgs boson decays would dominate the production of a net baryon asymmetry, and estimate

( 15)

Let us call the Higgs boson with the CP-violating decay asymmetry E H the d Higgs . The lowest order in which E can become non-zero is fourth­ H order, via diagrams like those in Fig. 3a, which appear in models with J2 Higgs multiplets coupling to fermions. An individual contribution to the decay asymmetry E can be written as the product of a Higgs emission vertex a and an absorption H d vertex b� , at least one of which contains a final state interaction . These are represented in Fig. 3b alongside the typical lowest order example of Fig. 3a. The 109

fermion lines in Fig. 3 must be summed over all fermion generation indices yielding us the trace of the product I, of coupling matrices indicated in Fig. 4a. This gives us a decay asymmetry

) r + Im Tr(I 1 Im T ( ad b d) ( 16) T (h h T h h+ ) r d �) r( d d where h is the lowest order coupling matrix of the d Higgs . In general the d d Higgs is a subset of a GUT multiplet ¢ of Higgs fields , e.g. , a colour tri­ d plet � in a five-dimensional representation of SU ( 5 ) Higgses. The m Higgs is also in general a subset of a GUT multiplet ¢ ' e.g. , an SU (2) doublet m � in a of SU ( ). the simplest minimal SU ( ) with just one of Higgs 2 5 In 5 2 fields ¢, the m and d Higgses must be partners in the same GUT multiplet ¢. However , this is not in general true but there will usually be a non-trivial over­ lap between the SU ( 5 ) multiplet ¢ containing the m Higgs and that (¢ ) m d containing the d Higgs . In a basis where the d Higgs is pure :

(17)

where we will suppose that is non-zero and of order one . We recall from udm Eq . that Im Tr (m 1F ), which we can rewrite in terms of the Higgs ( 9) 68GUT ::: � 1 coupling matrix H as l

(18)

where h is the lowest order coupling matrix of the m Higgs . Figure 4b m demonstrates that there is a contribution to of the form

(19)

which we can insert into Eq . (18 ) finding that

(20) llO

A specific exajllple of this lower bound to the renormalization of e by GUTs is QCD shown in Fig . 5 : part (a) shows the fourth order contribution to ( n /n ) while 8 y part (b) shows a corresponding contribution to the renormalization of e OCD "

Combining the expectations (13 ) and (20) we have

(21)

and the expression (16) for therefore means that EH

(22)

It seems reasonable to suppose that an order of magnitude estimate of

is given by the corresponding quantity Tr(h h ) m �

(23)

Using the results (15 ) of numerical calculations we therefore deduce from (22) and (23 ) that

(24)

8 QCll( lGeV)

5 ) Putting in the connection ( 5 ) between e and d we finally reach the OCD n promised bound (2) on d ' which leads in turn to the cosmological lower bound n ( 3 ) of

d > 5 10-28 n "' x e-cm

if we accept the astrophysical estimate 7) that

, ) It is worth mentioning 5 l S that this bound is not saturated in many of

the models we have investigated . For example , in minimal SU ( 5 ) with a single 2 2 of Higgs the renormalization Of takes place in order ( a/11 ) , whereas 8ocD Im Tr I only in eighth order. As for SU ( ) with two or more of Higgs , l i 0 5 2 2 is of order while Im Tr I is of order ( /11 ) In SU ( ) 68GUT (C

with a and a of Higgs , the lowest order contribution to 8 is 5 45 6 GUT -I la/n) ) larger than our bound. It may well be , therefore , that d is 0( n -l 0( ( a/TT) ) larger than the lower bound (3), and hence within an order of magni- tude of the present experimental upper limit , in any GUT which is sufficiently complicated to explain the observed baryon-to-photon ratio .

4. - A COSMIC SEISMOMETER

The neutron electric dipole moment is unique among observable low-energy

GP-violating parameters in being sensitive to aspects of physics at short dis­ tance scales and high energy scales up to 1015 GeV and beyond . From the point of view of a cosmologist , high energies translate into high temperatures and hence very early times . The neutron electric dipole moment is therefore unique in its potential for probing CF-violating processes at these very early times , and what­ ever may have happened subsequently during the expansion of the Universe . In par­ ticular, since the bound on 8 relates it to the primordial generation of QCD baryon number , any subsequent generation of an extra factor E of entropy would dilute the observed n8!n1 by l/E relative to the GUT calculation . The bound on d would therefore be a factor E larger : n

n d > 2.5 10-!8 E(_]_) (25) n "' x n y ) The experimental upper bound 2 on d already constrains E rather severely n

n 2 .4 10-7/E (26) (_]_) "'< x n y

-l (n !n > O we deduce and if we accept that 8 1 ) 2xlO S)

1.2 103 (27) E '\,< x

Thus there is not much scope after the creation of the baryon asymmetry for entropy generation, e.g. , during one of the subsequent phase transitions in the Universe . lO ) For example , it has been pointed out that if the SU ( 2 )xU(l) Weinberg-Salam symmetry is spontaneously broken by radiative corrections, then one should expect a strongly first order phase transition with much supercooling. It has been 112

lO) estimated that during the reheating of the Universe subsequent to the phase 4 transition a factor E > 10 of entropy would be generated . Our bound ( 27 ) may

therefore mean trouble for the scenario of Weinberg-Salam symmetry breaking by

radiative corrections. This problem would be exacerbated if the present experi- mental upper bound on d were improved , or if the favoured GUT turns out to be n one in which 6 is renormalized by a graph of lower order than those related ocD to the baryon number in Section 3.

Another example of an irregularity in the early Universe that is probed

by the neutron electric dipole moment is the inhomogeneous shear proposed by some ll ) authors as a way of generating isothermal density fluctuations which could give

rise to galaxies . If the Universe were shear-dominated at the time of baryon gene­ ) ration , grand unified viscosity 18 would help to damp it down shortly afterwards ,

and some entropy would be generated during this dissipation. People have pointed out previously that the value of (n /n ) now should enable one to bound the 8 y amount of primordial shear. Our connection with d enables us to establish a n quantitative bound , stating that the energy density at the epoch of baryon genera­ 1 tion cannot have been shear-dominated by a factor larger than 0110 8 ), about E twenty orders of magnitude better than the previous best limit from cosmological lg) nucleosynthesis . This does not , however, rule out the scenario of galaxy formation from isothermal fluctuations due to shear inhomogeneities , which can ll) - 3 . work if oE (and hence El is as small as 10

These examples may serve to demonstrate the utility of our cosmic con­

nection 5l,3l between baryon generation and the neutron electric dipole moment. This low enei•gy observable is unique in being related to physics at ultra-high

energies. As we gradually refine our knowledge of d (n /n ) and narrow down n ' 8 y the field of possible GUTs this connection will become ever more quantitative and restrictive .

FLOREAT THE INTERPLAY BETWEEN AND ASTROPHYSICS

ACKNOWLEDGEMENTS

This talk is based on work done in collaboration with Mary Gaillard , K. Demetre Nanopoulos and Serge Rudaz . It is a true pleasure to thank them for the enjoyable times we have had working together. Also I thank Jean Audouze , Phil Crane ,

Tom Gaisser and Dan Hegyi for arranging such a stimulating opportunity to discuss with astrophysicists, and J. Silk for interesting conversations. Finally thanks are due to Tran Thanh Van for making such meetings possible. 113

REFERENCES

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2) I.S. Altarev et al . - Pis 'ma Zh .Eksp. Teor .Fiz . 29 (1979 ) 794 and Leningrad Nuclear Physics Institute Preprint 636 (19Bl) .

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6) See the talk of D.V. Nanopoulos at this meeting for a review, and also : J. Ellis , M.K. Gaillard and D.V. Nanopoulos - "Unification of the Fundamental Particle Interactions", ed ., S. Ferrara , J. Ellis and P. van Nieuwenhuizen (Plenum Press , N.Y. , 19BO) , p. 461 .

7) D.N. Schramm - Enrico Fermi Institute Preprint EFI-Bl-03 11981) , and talk at this meeting based on work to be published with J. Yang, K.A. Olive and G. Steigman .

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17 ) See my talk at the Particle Physics Meeting held in parallel with this one .

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19 ) J.D. Barrow - Mon.Nat.Roy .Ast .Soc. 175 (1976) 359 D.W. Olson and J. Silk - Ap .J. 226 (1978) 50 .

FIGURE CAPTIONS

(a) Diagrams contributing to the renormalization of the quark mass matrix and thence to e ' which are related OCD (b) to the renormalization of the coupling of the m Higgs .

lS ) Illustration how the baryon-antibaryon asymmetry can be built up in the decays of superheavy particles X(Y = (n +n ) /n l' + x x y

Y (n -n ) /n ' Y_ = (n -n ) /n ). B q q y x x y

(a) Typical fourth order contributions to € ' and H (bl the generic vertices a and bt . d d

(a) The C- and GP-violating imaginary part of the trace of I 1 which gives € ' and H (b) an analogous contribution to the m Higgs coupling matrix .

(a) Fourth order contribution to Im Tr (I 1 in an SU (5) model 1 with two .2_'s of Higgs (the solid lines are lO 's of fermions , the zigzags are S's of fermions, the dashed lines are d Higgs and the dot-dashed lines are other Higgses ), with (b) an analogous contribution to the m Higgs coupling nt-trix . 115

I mHig�/

F, � = x v R L R A L ..__,._... H,

(al (bl

Fig. 1

Development of number densities

Y IO o 1--- "'-+ ------

-2 1 0

4 1 0-

15 10-

8 10-

40 1 0

12 10- +17 10 10•15

Temperature ( GeV l 116

---- d Higgs c· --...... l d Higgs :

(a) ( b)

mHi<;B; I R: __ - ...... _ " ,,,,. I/ � mH199s I ' � ' -� d �L : (a ) ( b)

(a) (b)