Volume 209, number 2,3 PHYSICS LETTERS B 4 August 1988

UPPER BOUND ON COMPOSITENESS SCALE FROM STRONG CP VIOLATION

Ann NELSON ’ Institute qf Theoretical Physics and Department ofPh_vsics, Stanford University, Stanford, CA 94305.4060, USA

Received 17 May 1988

In this letter I estimate the size of corrections to the neutron electric dipole moment in composite-technicolor models. The

natural size of the electric dipole moment is about lo-” e cm u/3.5 TeV)z4, where fis the compositeness scale. The current experimental bound of 1.2X lO-z5 e cm on the electric dipole moment of the neutron provides an upper bound on the composite-

ness scale of 3.5 TeV.

1. Introduction

The stringent experimental limit on the size of the electric dipole moment of the neutron (EDMN) is one of the strongest constraints on new physics, as well as a puzzle for the standard model [ 11. The small size of the EDMN in the standard model is attributed to the conservation of CP by the strong interactions, which requires the CP-violating strong interaction parameter 0 to be less than 2 x 1 O- lo. e is renormal- ized by weak interaction radiative corrections, how- ever, the size of the finite corrections is only of order < 1 O- I6 [ 2 1. @also receives infinite radiative correc- tions at the seven-loop level, and so understanding its small size requires new physics at high scales. In any theory with a spontaneously broken Peccei-Quinn symmetry, the vacuum dynamically adjusts itself in such a way as to make 8 approximately zero [ 3,4]. Another possibility is that CP is a good symmetry which becomes spontaneously broken at some high scale, and so radiative corrections in the full theory to all CP-violating quantities, such as @, are finite. B then becomes a calculable quantity, which is small enough in some models to agree with experiment [ 5 1.

In this paper I estimate the radiative corrections to r?, and so the EDMN, in a class of composite-techni- color models recently proposed by Chivukula and Georgi [ 61. Although Chivukula and Georgi have not completely constructed any specific examples of their

’ Supported by NSF Grant #Phy 88-01426.

models, they have succeeded in using the require- ment that the models should have a GIM [ 71 mech- anism for suppressing flavor-changing neutral currents to constrain the low energy phenomenology, and to place upper and lower limits on the compos- iteness scale. In this paper I show that one can get an upper bound on the compositeness scale simply from the requirement that compositeness corrections to 8 not exceed the experimental bound. Furthermore, this upper bound is within the experimental reach of compositeness searches at e+e- machines.

2. Review of Chivukula-Georgi composite models

Recently Chivukula and Georgi have come up with a useful set of rules for constructing the low energy effective lagrangian in composite-technicolor models [ 6 1. Their scheme is based on the assumption that the SU (2) X U ( 1) gauge group is broken by techni- color [ 8 ] , and that both techniquarks and ordinary fermions are composites made of the same preons. Quarks and leptons are kept light, and flavor chang- ing neutral currents are suppressed, if the preon dy- namics approximately preserves an [ SU (3 ) X U ( 1) ] 5 global symmetry. This is just the symmetry the stan- dard model would have if the Yukawa couplings to the Higgs were turned ofc there would then be inde- pendent SU( 3) x U( 1) transformations on each of the five types of fermion fields (the left-handed quark doublets, right-handed up quarks, right-handed down

0370-2693/88/$03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

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Volume 209, number 2,3 PHYSICS LETTERS B 4 August 1988

quarks, lepton doublets, and right-handed charged leptons). It is this symmetry which is responsible for the absence of flavor-changing neutral currents in the standard model at tree level. The [ SU (3) × U ( 1 ) ] 5 symmetry is broken, however, by three preon mass matrices, just as in the standard model the flavor symmetry is broken by three Yukawa coupling matri- ces. Hence the Chivukula-Georgi model provides no new insight into the origins of the ratios of fermion masses. It does have the great advantage over the standard model of being natural, as no elementary scalars are required. A technidoublet of fermions condenses at the weak scale, giving mass to the W and Z and, through four-fermion operators present in the low energy effective theory, to the ordinary quarks and leptons. The effective lagrangian is constructed using the rules of naive dimensional analysis (NDA) [ 9 ], with an unknown cons tan t fas the relevant scale for the preon dynamics.

The fields present in the low energy theory b e l o w f are given in table 1, with their quantum numbers un- der the gauge and global symmetries. It is also neces- sary to include the three preon mass matrices dgu, d~D, and die in the effective lagrangian (just as the light quark masses need to be included in low energy chiral theories of pions and kaons). These mass matrices and their transformations are also given in table 1.

The philosophy of naive dimensional analysis is to include all possible terms in the effective lagrangian consistent with the structure of symmetries and sym-

merry breaking terms, with coefficients of order

( 4 ~ r ) 2 f 4 ( 1 / 4 r c f 3 ) " ( 1 / 4 r r f ) b + C ( g f f 4 z t j ) d' , (1)

where a is the number of pairs of fermion fields, b is the number o f derivatives, c the number o f symme- try-breaking preon mass matrices required, di is the number of gauge fields of type i, and the g, are gauge couplings.

To lowest order in preon mass matrices, the terms responsible for fermion masses are

41f3QLJ/{u UR (~U ( P L T R ) + ~ g ( T R M L ) ) +h.c.

(2)

1 4rcf3 QL'////DDR (~D (I~'[LTR) "1- ~b ("[RPL) ) + h . c . ,

(3)

1 LLddzER(~r (PLT R ) q " ~ ('i~RML) ) +h.c. 47~f3

(4)

where the ~'s are parameters of order 1, determined by the preon dynamics. The quark and lepton masses are approximately proportional to the preon mass matrices, with a constant of proportionality of order

( P T ) ( ' I ' M ) 4~v 3 V 3 47cf3 -- 4rcf3 -- 47~f3 -- f 3 , (5)

Table 1

Left-handed Gauge quantum numbers Global fermion field SU(3) × SU(2) ×U( 1 ) ×SU (N) ~ [SU(3) ]2

Q (3,2,1/6,1) (3,1,1,1,1) u c (],1,-2/3,1) (1,3,1,1,1) D c (3,1,1/3,1) (1,1,3,1,1) L (1,2,-1/2,1) (1,1.1,3,1) E c (1,1,1,1) (1,1,1,1,3) T c (1,2,0,~7) singlet P ( 1,1,1/2,N) singlet M ( 1,1, - 1/2,N) singlet symmetry global [ SU ( 3 ) ] 5 breaking term ,¢/u (3,3,1,1,1) ,~/D (3,1,3,1,1) ,'#E (1,1,1,3,3)

,,7 For definiteness, I have taken technicolor to be SU(N); other choices are possible.

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Volume 209. number 2,3 PHYSICS LETTERS B 4 August 1988

where v is the scale of weak symmetry breaking, ap- proximately GF”*.

To higher order in symmetry breaking, the masses are more complicated. For instance the up quark mass matrix has the form

M”=(ao+a,h”+a*h,+a,h:+a,h2,

+a~h”hD+...)(v3/f3)A”)

where

(6)

and the a’s are parameters of order 1. The quark mass matrices are approximately proportional to preon mass matrices, with corrections proportional to pow- ers of the hermitian matrices hu and hp.

3. Estimating the neutron electric dipole moment

Potentially the largest contribution to the EDMN comes from the strong interaction parameter @ which is defined 8~ 6’ - arg det Mo, where Mo is the quark mass matrix and 0 is the coefficient of (g2/ I 6n2)Tr GG in the QCD lagrangian. This combina- tion is invariant under phase redefinitions of the quark fields, and is a physical parameter. The EDMN has been estimated to be about 5 x 1 O- ’ 6 8e cm [ 10 1, and so @must be quite small. Other contributions to the neutron dipole moment could come from quark electric dipole moments or from U-violating inter- actions between the quarks.

In estimating the EDMN, I will begin by assuming that all CP violation in the low energy theory origi- nates from phases in the preon mass matrices. The observed CP violation in the kaon system is due to the Kobayashi-Maskawa mechanism. I will assume that 0 is zero and that the preon mass matrices have real determinant (several ways of producing com- plex mass matrices with real determinant have been found [ 51). The quark mass matrices will not have exactly real determinant, however, and 0 will not be exactly zero. (Note that the CP violation in the com- posite theory is soft, and so eis a calculable, finite quantity. )

The size of @can be estimated from (6) to be

& arg det MU + arg det Md

= arg det -A,, + arg det A$

+argdet (-l+-AU+-hp+-h:+-Ah+...)

=O+Im Tr f-h, +-A,,+ -hL + -hh

+-h”hp+-hnh”+...)) (7)

where the - indicates that terms of order 1 have been dropped. In order to get any non-zero contribution to eit is necessary to expand to very high order in h, and hD, because of the identities

ImTrh;hf;hGh”,=O, (8)

and

ImTrh;;h;,=O. (9)

(Similar cancellations occur in the renormalization of gin the standard model, which is why the infinite renormalization requires seven loops. ) The largest possible contribution to @is of order

8-imTrh~h~h,h,. (10)

Note that h,,D are dimensionless matrices which grow

with the compositeness scale.

(11)

Now we have 12

XImTr[ (M,Mt)*(MDM~)*MuM=MDM~]

(12)

mfmzmfm: f (47Lv)‘* v 0

24

Im(LKX,X) (13)

_(~)24(_&_)4(~)(250~)‘2

x8x 10-38. (14)

While there is a lot of uncertainty in this estimate, the limit on f will turn out to be quite firm, due to the enormous sensitivity of @to the value off: For v= 250 GeV, the experimental limit of 2x 1O-1o on @gives

an upper bound on f of 3.5 TeV. What of other contributions to the EDMN? The low

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Volume 209, number 2,3 PHYSICS LETTERS B 4 August 1988

energy theory below the compositeness scale is just the standard model with the Higgs replaced by techni- quarks, and we know the radiative corrections to the EDMN are quite small unless the effects of compos- iteness are included. Compositeness effects could contribute directly to other CP-violating operators, such as electric dipole moments for the light quarks, but these turn out to be tiny, suppressed by both the small size of CP violation in the K - M matrix and the light masses. If the preon mass matrices are the only source of CP violation in the theory then the only possible observable contribution to the EDMN comes from O.

What if the high energy theory contains a Peccei- Quinn symmetry and so 0 is a dynamically deter- mined parameter? It is still possible to get a bound on the compositeness scale from the EDMN. There are new CP-violating terms in the effective potential for O which may shift the m i n i m u m away from 0=0. Even in the standard model, the presence of weak CP violation causes O to settle down dynamically to a value of ~ 10 17 [ 1 1 ]. Unfortunately the new cor-

rections to the Opotential, which come mainly from flavor-conserving but CP-violating quark interac- tions, are very small in these models, suppressed by many powers of light quark masses/compositeness scale, and so the bound o n f i s uninteresting.

4. Conclusion

In this letter I have examined CP-violating correc- tions to the neutron electric dipole moment in a class of composite models proposed by Chivukula and Georgi. I find that if the high energy theory contains a Peccei -Quinn symmetry (and so the low energy theory contains an axion [4] ) these corrections are not observable. However, it is possible to solve the strong CP problem without an axion, in models where CP is spontaneously broken at high energies. In ax- ion-less theories I find that the value of the electric dipole moment naturally grows with the composite-

ness s c a l e f a s f 24, allowing one to place a sharp cutoff of f of 3.5 TeV. Because of the high power of f in- volved, this l imit is not as sensitive to unknown pa- rameters as the other upper bounds on f found by Chivukula and Georgi. For instance, they find an ap- proximate upper bound o n f o f about 2.8 TeV from the lower bound on the top mass, but rn~ is only sup- pressed by two powers o f f They also find a contri- but ion to Ap proportional t o f 6 which gives an upper bound on f of 3.3 TeV, however, the experimental bound on Ap is subject to theoretical uncertainty.

In conclusion, if compositeness is the source of the quark and lepton masses, the bound on strong CP vi- olation requires that the compositeness scale should be lower than 3.5 TeV unless there is an axion. Such a low scale could be accessible in compositeness searches at e +e machines.

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