Physics Letters B 284 (1992) 77-80 Norlh-Holland PHYSICS LETTERS 13

Composite axion models and Planck scale physics

Lisa Randall Center fi~r Theorettcal Phvst~, Massa~hu~ett~ Instttute oJ Technology, CambrMge. ~14 02139. US,I

Received 20 February 1992, revised manuscript received 20 March 1992

Recently tt has been argued that global symmetry vtolatmg effects at the Planck scale could upset ax~on based solutions to the strong CP problem ffthere ~s no gauge symmetry to prevent the generation of global s:rmmetry vtolatmg operators We show that tt ~s straightforward to construct composite axion models for which global symmetr~ violating effects are very suppressed The PQ symmetry of these models can be legitimately regarded as an approximate acodental symmetr~ Simdar considerations apply to the texture scenario, although models are more constrained

Axton models are well known as a potentml reso- lution of the strong C P problem [ 1,2 ]. These models reqmre the extstence of a global U ( 1 ) symmetry which ts spontaneously broken at the PQ scale, f. In general, global symmetries may be constdered some- what artlfictal, unless they are acctdental conse-

quences of some gauge symmetry. Thts pomt has been cmphastzed in two recent works [3,4] whtch point

out a specific mechamsm for global symmetry break- mg effects, namely h~gher dtmenslon operators in- duced by gravtty.

However, these papers constder only scalar axlon models, and not those constructed sold) wtth fer- mions. It ts well known that constructing modcls wtth

scalars is problematic. The hierarchy problem means that mvtslble ax~on models constructed wtth scalars

arc suspect from the start, since one requtres the ex- tstence of two well separated symmetry breakmg scales, namely the weak scale and the PQ scale, sep- arated by many orders of magmtude. This in ttself poses a stgntficant fine tuning problem for an axton model. Thts problem can however be resolved m su- persymmetnc axion models.

The authors of refs. [3,4] point out that the degree to which the global symmetry must bc exact poses sttll

Th~s work ~s supported m pall by funds provided by the tJS Department of Energy (DOE) under contract # DE-AC02- 76ER03069 On leave from the Harvard Sooety of Fellows

more severe constramts on the parameters of the model. Reconstdermg a point made m ref. [ 5 ], they show that htgher dtmenslon operators, even tf sup- pressed by the Planck scale, would need to be about dlmenston l 0 or greater if one ts to generate an axton potential which ts minimized at sufficiently small d (wtthout fine tunmg the coupling), or equtvalently have a coeffictent suppressed by about l0 -44. They

cite spectfic mechanisms whtch m~ght generate global

symmetry vtolatmg effects, namely black holes ore wormholes. But they point out that m light of our ig- norance of Planck scale physics, we mtght view all theortes below this scale as effecttve theortes, and m- elude all htgher dimension opcrators, even those whtch violate the approxtmate symmetries of the lower energ~ approxtmate theory So the fine tunmg problem whtch was already very severe ts even worse.

Rcf. [3] points out that gauge symmetrtes might guarantee sufficiently suppressed global symmetry violation They mention supersymmctrtc models which meet their constramt. However, the scalar based models they constdered are not very elegant. Wc constder instead compostte axton models. Thts allows for many new models which preserve an ap- proxtmate PQ symmetry broken only by operators of htgh dtmenston. We present a rather stmple compos- tte axton model for which gauge symmetrtes prevent PQ symmetry breaking operators of low order. More- over, ttts obvtous that these models do not suffer from the hierarchy problem alluded to earher, so they arc

0370-2693/92/$ 05 00 © 1992 Elsevier Science Pubhshers B V All rights reserved 77

~¢olume 284. number 1.2 PHYSICS LETTERS B 18 June 1992

indeed a natural solution to the strong CP problem in any case.

A very, simple composite axlon model was pro- posed by Klm [6] and later considered in detail by Kaplan [7]. The model we consider below adds a single addit ional nonabehan gauge group, which IS spontaneously broken along with the PQ symmetry

at the PQ scale .-a

Recall that all that is reqmred for the successful implementat ion of the axlon scenario is the existence ofa U ( 1 ) s~mmetry broken only by the color anom-

aly [8] In the composite axlon model discussed be- low. only exotic fermlons carry this charge. In addi- tion to the gauge symmetries of the standard model, there is an SU(N) which becomes strong at high en- erg~ and a gauged S U ( m ) which is spontaneousl~ broken at the PQ scale. A U ( 1 ) global s~mmetr~ which is anomal~ free under the SU (N) and SU (m) gauge groups but is howe~,er broken b,~ the anomaly of SU (3)~ exists as an accidental consequence of the structure of the low dimension operators. This U ( 1 ) s,~ m m e t ~ is spontaneousl~ broken at the scale where SU(N) gets strong. Therefore this scale is associated with the PQ scale,./.

The fermlon content of the model is the standard fermlons plus left-handed exotic fermlons transform- l n g u n d e r S U ( V ) × S U ( m ) × S U ( 3 ) c as

(A', m, 3) + 3(3,', m, 1 ) + m ( N , 1, 3)

+3re(N, 1, 1) , ( I )

where the prefactor refers to the number of copies of the relevant representation. It is assumed that SU (N) becomes strong, inducing fermaon condensat ion in such a way that SU (3)c is preserved, SU (m) is bro- ken however. That this particular model is indeed rather simple can be observed by comparing it to the symmet~, breaking of the electroweak theor), b~ chlral condensat ion due to a strong SU(N) group but ~,llh SU ( 2 ) replaced by SU (m) .

Consider now the U ( 1 ) symmetD under which the colored fermlons have charge + 1 and the neutral fer- talons have charge - I . This symmetD has an SU(3)c anomaly but not an SU(N) or S U ( m ) anomal~ and therefore is the PQ symmetry. In addl-

"~ I am gratetul 1o Da'.]d Kaplan and Ann Nelson for discussing th~s and other models

lion, there exist exact global symmetries which are

irrelevant to our anal,~sls When SU(N) becomes strong, these symmetries break leading to exact Goldstone bosons which arc not coupled to the stan- dard fermlons. Because they are not coupled to the standard fermions, their temperature can be differ- ent from standard model particles at the time of nu-

cleosynthesls, so their energy density is not necessar- ll~ important.

,kfter spontaneous symmetry breaking by chlral condensation, the PQ symmetry is broken and the

axlon survives as the pseudo-Goldstone boson field which couples to Gao, which is all that is needed for

an axlon model [8]. Now it is clear that there is no hierarchy problem in this model, since there are no fundamental scalars Moreover. it is readily seen that there are no low dimension operators which preserve the gauge symmetries but v.olate the U ( I ) s~mme- try. The gauge s~mmel~ is such that the U ( 1 ) sym- metry is an accidental consequence of the gauge sym- me t~ of the theory.

The lowest dimension operator consistent with the gauge s}mmelry and Lorcntz lnvarlance which does not respect the U ( 1 ) symmetry IS the operator w~th 2m color neutral ferm~ons, half of them N ' s and half

,~?'s. (Similar operators could also break the anomal~ free symmetries by a small amount . )

Let us see how big a contr ibution to the axlon po- tential wc get from the above operator. We approxi- mate the condensate b,', 4r~/ ~. Of course, this is non- perlurbatlve and moreover the normalizat ion of flS ambiguous, so that the scale / can differ somewhat from the Peccei -Qulnn scale as constrained by astro- physics, experiment, and cosmology. Moreover we neglect the charge of the operator which would yield a logarithmic correction to the required dimension of the operator. These approximations should be ade- quate to convince the reader that there is no serious problem for this model for m greater than or equal to 3 or 4, depending on what one believes is the axlon scale. Our estimate gl', es an operator of size

(4n) '" (2)

Recall that the standard axlon potential, which IS minimized at tq=0 has size approximately in~ f~. Thc mi n i mum of the total potential, including the global

78

Volume 284, number 1,2 PHYSICS LETTERS B 18 June 1992

symmetry breaking operator above, is not minimized at 67=0. However, if the inequality

f ~m (4n) '" < 10 -" 2 - - m . f ~ (3)

M 3p] ,,-n

is satisfied, the neutron electric dipole moment should be sufficiently small. Notice the composite axlon model is of the hadronlc type, with no coupling to physical leptons, so we can take, /~ 107 GeV consis-

tent with existing axlon constraints [9]. We then find that for m>~ 3, the higher dimension operator is suf- ficiently suppressed. This is hardly a constraint on the

composite axlon model. It is seen that the global sym- m e t ~ breaking effects are automatically suppressed. Even for hadronlc axlon models, the window around

f ~ 107 GeV is quite small [9] If one believes the

composite axlon scale should be more like 10 ~" GeV, one concludes that m >/4. So although it is reasonable to conclude that scalar axlon models are proble- matic, it would certainly be a leap to conclude that all axlon models with sufficlentl~ exact U( 1 ) s~mmetry are complicated The model given above is very sim- ple, and elegantly leads to an axlon scenario.

We briefly comment also that the same reasoning we applied above to axlon models can be applied also

to composite models of "'texture" [10] However, here the potential is still more constrained [11], leading to a more stringent constraint on the neces- sary size of the gauge symmetry group. Recall that the texture scenario requires breaking a global symmetry G to a subgroup H such that n~ (G/H ) is nontrlvlal, allowing for the possibility of an unstable topological defect as the unl ,erse evolves past the s~mmetry breaking phase transition. In ref [ 1 1 ], the authors demonstrate that global symmetry breaking effects destro~ the texture scenario, unless the eft'eels are ~e~' strongly suppressed. If one assumes a coefficient of l0 -2, one discovers the operator must be dimension 35 [ l l ] or greater if the symmetry breaking scale is of the order of the G U T scale, which seems to give the approximately correctly-sized denslt~ perturba-

tions. Can one make a reasonablc texture scenario con-

structed only with fermlons? Notice that because of the three copies of lcft- and right-handed leptons (really there are 3m on the right), the exact symme- try group has a factor SU ( 3 ) × SU (3m) which breaks to an SU ( 3 ), guaranteeing that x~(G/H ) is nontrlv-

ial. making this a viable model for texture. Using ref. [ 1 1 ] and constructing a global symmetry operator as we did above forces us to conclude that in this model with a simple gauge group, that m >/14. This is per-

haps uncomfortably large but of course one can gen- erate arguably aesthetically superior scenarios where

the gauge structure is the product of simple groups, each with small m.

We conclude that even with global symmetry breaking effects generated at the Planck scale taken

into account, composite models both of axlons and texture are superior to their fundamental scalar counterparts. Fine tuning problems are much less se- xere. In fact the simple composite axlon model we

considered automatlcalb gives the necessa~ scale suppression for the axlon scenario to go through

unscathed Moreover. it was pointed out b~ Kaplan [7] that

composite axlon models might be favored on phe- nomenologlcal grounds as well m that they have no electron couplings and potentially small photon cou- plings, making astrophysical axlon bounds less sig- nificant. Because the constraints were less severe, the axlon window is a little wider for these models, which is what permits the low fvalue of 10" GeV. It would also be worthwhile to investigate the effects of Planck

scale physics on chlral symmetries. In s u m m a ~ , composite axlon models might be the most elegant and natural class of axlon models.

I am especially grateful to David Kaplan and Ann Nelson for discussing this and other models I also thank Mike Dugan, Tom Imbo, and Bob Jaffe for useful discussions I thank Steve Hsu for describing his work and pro~ ldlng refs. [ 3,4 ] and Mitch Golden

for comments on the manuscript.

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Volume 284, number 1,2 PHYSICS LETTERS B 18 June 1992

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