Volume 136B, number 5,6 PHYSICS LETTERS 15 March 1984

NATURALLY WEAK CP VIOLATION

Ann NELSON Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA

Received 14 December 1983

An SU(5) GUT model is proposed in which the observed CP violation is due to spontaneous symmetry breakdown. The effective field theory below the GUT scale is simply the standard model. Matching conditions at the GUT scale give a non- zero contribution to 0- at the one-loop level. This contr~ution is proportional to ratios of superheavy fermion masses over the GUT scale, which can be naturally small, in the sense of 't Hooft.

1. Introduction. How can we naturally explain the absence of strong CP violation? In the standard model, the strong interactions contain a physical CP violating parameter 0 = 0QC D + arg det Mq where Mq is the quark mass matrix and 0QC D is the coefficient of g2/327r2 F~'. Experimental measurement of the elec- tric dipole moment of the neutron determines 0 to be less than 10 -9 [1]. ' t Hooft has convincingly argued [2] that a physical parameter may only be very small if replacing it by zero increases the symmetry of the theory. Then, any renormalization of the parameter will be proportional to the parameter itself. Other- wise, the theory is said to be unnatural. In the stan- dard model, however, CP is explicitly broken in the weak interactions by dimension-four operators. probably receives an infinite renormalization even if it is zero at the tree level [3]. Thus it seems 0 cannot be naturally small.

One possible explanation for the observed absence of a neutron electric dipole moment is that the up quark is massless. A massless quark in the theory means that 0 is undefined and so can have no physical effect. However, current algebra estimates of the up quark mass make this explanation unlikely [4].

Another possible solution is due to Peccei and Quinn [5]. If the lagrangian possesses a spontaneously broken global U(1) symmetry with a color anomaly,

is dynamically determined to be zero. Such a theory predicts the existence of a pseudo-Goldstone boson, called an axion, whose couplings are inversely propor-

0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

tional to the Peccei-Quinn symmetry breaking scale. Experimental searches for the axion and astrophysical requirements force this scale to be larger than 109 GeV [6]. Recent cosmological arguments [7] place an up- per limit on this scale of 1012 GeV. An unnatural fine tuning seems to be required to give the axion just the right amount of invisibility.

A third and promising approach is that CP is violated softly or spontaneously. 0 then receives only finite, calculable contributions. It is difficult but not impos- sible to find models where these contributions are suf- ficiently small. Several rather complicated examples exist in the literature [8]. Some of these contain a low energy left-right symmetry, and most rely on addi- tional charged scalars in the low energy theory to pro- duce the observed CP violation in the weak interac- tions. In this paper we present an example of a model with spontaneously broken CP where the low energy effective theory is the standard model with no extra undesirable symmetries or scalar fields.

Our model is essentially the same as one proposed by the author in a recent paper [9] on flavor symme- try and proton decay. We extend the Georgi-Glashow SU(5) model [10] to include an SO(3) flavor symme- try and some additional heavy fermions. We impose a U(1) global chiral symmetry and CP invariance. At the GUT scale SU(5) spontaneously breaks to SU(3) ® SU(2) ® U(1) and all the global symmetries get broken. All fermions and scalars except for the three light fami- lies and the Weinberg-Salam Higgs doublet get masses.

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This model contains the usual fine tuning to keep the Higgs doublet light. At the tree level, the heavy fer- mion masses are all real. The light quark-Higgs Yuka- wa coupling matrices are complex but have real deter- minants. 0- is induced in the effective theory by one- loop and higher order matching conditions at the GUT scale. However, these corrections turn out to be pro- portional to ratios of the heavy fermion masses over the GUT scale. These masses can naturally be small because when they are zero some chiral symmetries are restored. In practice the heavy fermions need to have masses g l 0 -(3-4) times the GUT scale for 0 to be within experimental limits.

2. The model The symmetry group of our model is SU(5)Gauge ® SO(3) ® U(1)Global # CP. Under this group the fermion transform as

[10,3,0] • [5,3,0] • [10,1,11 • [10,1,-11

e [5 ,1 ,1] • [ 5 , 1 , - 1 ] .

The usual Higgs doublet is in a (5,1,0) rep, along with a superheavy SU(3)C triplet. The symmetry breaking at the GUT scale is done by scalars in reps (ri,3,1), where the ri are SU(5) reps containing SU(3) ® SU(2) # U(1) singlets, such as 1,24, and 75.

The scalar-fermion Yukawa couplings and fermion masses are;

Xu [10,3,0] [10,3,01(5,1,0)

+ Xd [10,3,01 [5,3,01(5,1,0)*

+ hi[lO,3,0] [1~ 1,-1](ri ,3,1)

+ j~ [5,3,0] [5 ,1 , -1 ] (ri,3,1)

+ ml [10,1,1] [1-0, I , - 1 ] + m2 [5,1,1] [5 ,1 , -1]

+ h.c. (1)

Note that we have used square brackets to designate fermions and round ones for scalars. Since we assume CP to be a symmetry of the theory, all masses and coupling constants are real. The VEVs of the complex fields (ri,3,1) will break SU(5), SO(3), U(1), and CP invariance, and cause mixing between SO(3) triplet and singlet fermions. This mixing will give the struc- ture of the light quark and lepton mass matrices, which are examined in detail in ref. [9]. Each SU(3) ® SU(2) ® U(1) multiplet in the [10,1 , -1] and [5,1,

-1 ] will pair up with a linear combination of fields in the [I0,3,0], [10,1,1], [5,3,0], and [5,1,1] reps to get a mass.

We can make a chiral rotation of the quark fields to give the heavy quarks real diagonal masses. We can choose this rotation to have determinant 1 so as not to contribute to FF. In the appendix we show that the Yukawa coupling matrix of the quarks to the light Higgs is complex but has a real determinant. 0 receives no contribution at tree level.

3. Effective fieM theory calculation o f O. At ener- gies sufficiently below the GUT scale, the effective theory is the standard SU(3) ® SU(2) ® U(1) gauge theory, containing only renormalizable interactions of light fields. Above M w, 0 is defined to be 0QC D + arg(det X u det X d) ,1, which is invariant under chiral transformations of the quarks. Below Mw the quarks are massive, and d is defined in the usual way. Ellis and Gaillard have shown [3] that finite corrections to 0 in the standard model are very small, of order 10 -16 . Contributions to the 0/3-function do not occur until at least the seven-loop order and so any running of d is incredibly slow and may be neglected. Thus the which we calculate from matching conditions at the GUT scale is, to a very good approximation, the 0 we measure from the neutron electric dipole moment.

We have shown in the previous section that there is no tree level contribution to the matching conditions for 0. In this section we will estimate the higher order corrections. In general there are two sources of CP violation in the full theory which contribute to the matching conditions for O. One source of CP violation is complex gauge and scalar couplings to the mass eigenstates. The other source is complex contributions to the heavy SO(3) triplet scalar mass matrix from the VEVs of the scalar fields. When this mass matrix is made real by redefinitions of the scalar fields, more complex phases are introduced into the couplings of the SO(3) triplet scalars to the fermions.

In calculating the corrections to 0QC D we will use the method of Georgi et al. [ 11 ]. At the one-loop level there is no contribution as all our heavy quarks have real tree level masses.

There will, however, be one-loop contributions to

,1 Here and below h u and X d are the Yukawa coupling ma- trices defined in eq. (3) of the appendix.

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Volume 136B, number 5,6 PHYSICS LETTERS 15 March 1984

(a) (b )

5 " . . . . . 5 " i i

I 1

I c )

. * * * * * * ~ S U ( 5 ) heavy gauge boson • * * o * * * heavy S 0 ( 3 ) tr iplet boson

. . . . light Higgs doublet

x × ~ × ~ - - Fermion I i

Fig. 1. One-loop contribution to h~ and hd. The crosses on the fermion lines represent the fact that these graphs only contribute to arg det h ff there are at least two mass insertions.

GUT scale by a factor of 1 0 - 3 - 1 0 -4. This is not un- natural! From (1) we can see that these fermions are light i f m l and m2 are ~I'/GU T and hi, fi ~g. In the limit h i andj~ -+ 0 we gain a new anomaly free chiral symmetry under which the [10,1,1] has charge 1 and the [10 ,1 , -1 ] has charge - 1 . When ml and m2 --> 0 we gain two more. Under one chiral symmetry the [10, 1,1] has charge 1 and the [5,1,1] has charge - 3 . Under the other, the [10 ,1 , -1 ] and [5 ,1 , -1 ] also have charges o f 1 and - 3 . Thus at the one-loop level can naturally be small.

The same arguments can be extended to the two- loop level and beyond. We will argue that all contribu- tions to 0 are suppressed by fermion masses or Yuka- wa couplings. For example, let us look at the two-loop

det ~u and det ~d from the graphs in fig. 1. These con- tributions could be infinite if there were no fermion mass insertions and yet we know all contributions to

are finite. However, it is only the chiral rotations performed to diagonalize the fermion masses which introduce CP violating phases into the couplings of the heavy gauge bosons and the light Higgs doublet. Therefore if there are no mass insertions in figs. 1 a and lb, these graphs conserve CP, and simply give a real renormalization of ~u and xd. The graph in fig. 1 c vanishes with no mass insertions, because the SO(3) triplet scalars only couple the light particles to the SO(3) singlet fermions, which have no tree level cou- plings to the light Higgs.

In the limit that the heavy fermions are lighter than the GUT scale the sum of all contributions to from fig. 1 type graphs with arbitrary numbers of mass insertions gives:

80 = ~ Ai~2--~ln---~ i=1,2,3,4

f2 M1M2" M1M2.~ fh M3M4, M3M4 + B 3 - ~ 2 gt2 327r 2 I n ~ - ~ . m - - ~ -ft . /.t2

Ai, B and C are factors of O(1) coming from phases in the couplings of the gauge and Higgs bosons to mass eigenstates. M1, M2, )143 and M 4 are the masses of the heavy [3,2,1/6], [3, I , - 2 / 3 ] , [ 1 ,2 , -1 /2 ] and [1,1,1] fermions./a is the GUT scale.

For these contributions to be less than 10 -9 re- quires that the heavy fermion masses be less than the

(o)

LGc~xTc

(b)

i Gc X YX Tc

Fig. 2. Two-loop contr~ution to F/~. The circles with crosses in them represent space dependence of the gluon potentials and of the CP violating phases in the couplings. The details of this calculational method are given in ref. [11]. This space dependence must be included because otherwise the graph vanishes.

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'r'In ° odr"

o o o

°o o

o o o ° ° ° ° ° O o o o

o o o

o ° o

2 o

/ /

t /

/ /

Fig. 3. This is an example of a complex two-loop contribution to hi] which is suppressed by small Yukawa couplings instead of fermion masses, fondly known as the dead duck graph.

out to also be able to explain the absence of a neutron electric dipole moment. Two experimental signatures of this model, which may or may not rule it out in the near future, were discussed in the previous paper. One is the prediction for the top quark mass o f m t Oc(mb/m~ mc ~ 150 GeV, and another is the predic- tion that the dominant decay modes of the proton are p ~ Ke and p ~ 7r/1.

With slight modifications, such as the addition of more heavy fermions, this model loses its predictive power about fermion masses and proton decay modes, but can continue to explain the smallness of 0. I be- lieve it would still be simpler than other models of spontaneously broken CP.

contribution to F F in fig. 2. The graph in fig. 2a with no mass insertions does not contribute to d for the same reason the graphs in figs. la and lb do not con- tribute - the gauge boson couplings do not violate CP unless the effects of the fermion masses are included. Fig. 2b at first seems a little more dangerous since there may be complex contributions to the heavy scalar-fermion couplings from diagonalizing the com- plex scalar mass matrix. With no mass insertions, how- ever, any complex coupling on one side of the graph gets multiplied by its complex conjugate on the other side. Thus both of these graphs give contributions to which are suppressed by powers of fermion masses over boson masses.

The graph in fig. 3 gives a complex renormalization of hi~ even if no fermion masses are inserted because of the complex scalar masses. It is, however, suppressed by a couple of powers of the SO(3) triplet scalar Yukawa couplings, which are small if the fermion masses are small.

Clearly any other contributions to the 0 matching conditions from the complex SO(3) triplet scalar mass matrix will also be proportional to their small Yukawa couplings. Similarly, any contributions which are due to the complex couplings of the heavy gauge boson fields to the mass eigenstates will be suppressed by powers of fermion masses over gauge boson masses. In this model, 0 can naturally be as small as desired.

4. Conclusion. The model presented above and in ref. [9] has some remarkable characteristics. It was originally proposed to explain the experimental ab- sence of the proton decay mode p ~ zre and it turned

I thank Howard Georgi, Andrew Cohen, and David Kaplan for useful conversations and David Kaplan for naming the dead duck graph.

This work was supported in part by the National Science Foundation under Grant No. PHY-82-15249.

Appendix. The heavy linear combination of fields in the [10,3,0], [10,1,1], [5,3,0] and [-5,1,1] reps may be represented by a four-vector with the first three components corresponding to an SO(3) three- vector and the fourth to an SO(3) singlet. For exam- ple, we will label the linear combination of the [3,2, 1/6]'s in the [10,3,0] and [10,1,1] which combine with the [3,2,-1/6] in the [i-0,1,-1] by a four-vec- tor qSH

q~H_ ~ 1" 2* ^* - ~ . ( f i o i /Ml, fioi /Ml, fio? /MI, ml/M1),(1) l

where

M a = ( E 1 ~ 3 ~ o a [ 2 + m 2)1/2 a=l,2, 3 i

= the tree level mass of the heavy [3,2,1/6] quark.

The o a are complex VEVs of the SO(3) triplet scalar fields contributing to the heavy [3,2,1/6] quark mass. Since ml is real, q~H may be written without loss of generality as:

(s3s2sl, s3s2cl, s3c2, c3),

where ci = c~ and Isil 2 +Ici l 2 = 1. I f m 1 were complex, we could not require both c 3 and the heavy quark mass to be real.

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We can find three directions orthogonal to 4 H, which correspond to light quarks

(c3s2sl, c3S2Cl, c3c2, -s;~ 4 = ~ S l C 2 , elc2, -S~, ; f t . (2)

\ e l , - s ~ , 0 ,

Note that this chiral transformation has determinant 1 and so does not contribute to F/~. The coupling con- stant matrices of the light quarks to the Higgs doublet are:

)t~]" = )t u ~ q~ugi=J , xd.=Xd ~ 4 d £ . (3) #=1,2,3 #=1,2,3

B~ inspection o f (2) and (3) we can see that X}} and )t# are the products of three-by-three matrices having real determinants, and so they also have real determi- nants. I f m l were not real and so c3 were complex, X~. and X d" would no longer have real determinants and would get a tree level contribution.

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[3] J. Ellis and M.K. Galliard, Nucl. Phys. B150 (1979) 141. [4] S. Weinberg, unpublished. [5] R. Peccei and H. Quinn, Phys. Rev. Lett. 38 (1977)

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