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Solutions to the Strong-CP Problem in a World with Gravity

arXiv:hep-ph/9203206v1 9 Mar 1992NSF–ITP–92–06CMU–HEP92–05FNAL–PUB–92/34–AHUTP–92A011VAND–TH–92–2January, 1992Solutions to the Strong-CP Problem in a World with GravityRichard Holman,(a,b) Stephen D. H. Hsu,(a,c) Thomas W. Kephart,(d)Edward W. Kolb,(a,e) Richard Watkins,(e) Lawrence M. Widrow(a,f)(a)Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106(b)Physics Department, Carnegie Mellon University, Pittsburgh, PA15213(c)Lyman Laboratory of Physics, Harvard University, Cambridge, MA02138(d)Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 37235(e)NASA/Fermilab Astrophysics CenterFermi National Accelerator Laboratory, Batavia, IL 60510and Department of Astronomy and Astrophysics, Enrico Fermi InstituteThe University of Chicago, Chicago, IL60637(f)Canadian Institute for Theoretical AstrophysicsUniversity of Toronto, Toronto, Ontario, M5S 1A1, CANADAAbstractWe examine various solutions of the strong-CP problem to determine theirsensitivity to possible violations of global symmetries by Planck scale physics.While some solutions remain viable even in the face of such effects, violationsof the Peccei–Quinn (PQ) symmetry by non-renormalizable operators of di-mension less than 10 will generally shift the value of θ to values inconsistentwith the experimental bound θ <∼10−9. We show that it is possible to con-struct axion models where gauge symmetries protect PQ symmetry to therequisite level.

It is well known that there are two contributions to CP violation in the standardmodel. First, QCD instantons induce a term LQCD = θ tr G eG in the effective Lagrangian,which violates both P and CP [1].

Here, θ is a dimensionless coupling constant, which onemight naively expect to be of order unity. Second, the quark mass matrix can be complex,leading to a CP-violating phase in the Kobayashi-Maskawa mixing matrix.

The phase ofthe quark mass matrix gives rise to an additional contribution θQFD = arg det Mq to thecoeffiecient of tr G eG. The degree of strong-CP violation is controlled by the parameterθ = θ + arg det Mq, which is constrained by measurements of the electric dipole momentof the neutron to be less than 10−9 [2].

The strong-CP problem is that there is no reasonfor these two contributions, which arise from entirely different sectors of the standardmodel, to sum to zero to such high accuracy.The solutions that have been proposed for the strong-CP problem fall into threegeneral classes. First, there are those that rely on the existence of an extra global U(1)Asymmetry.

This symmetry arises naturally if one or more of the quark masses are zero[3]. In this case, it can be shown that the QCD θ parameter becomes unobservable.

Thissolution is considered unattractive, since experimental evidence implies that it is unlikelythat any of the quarks are massless. Peccei and Quinn [4] (PQ) proposed a solution tothe strong-CP problem in which they introduced an auxiliary, chiral U(1)PQ symmetrythat is spontaneously broken at a scale fa, giving rise to a Nambu–Goldstone boson aknown as the axion [5].

This symmetry is explicitly broken by instanton effects. Thisexplicit breaking generates a mass for the axion of order ma ∼Λ2/fa, where Λ is theQCD scale.

The important point is that the effective potential for the axion has itsminimum at ⟨a/fa⟩= −θ. It follows that when the axion field relaxes to its minimum,the coefficient of tr G eG is driven to zero.

This solution has received the most attentionand has been explored by many authors.A second class of solutions involve models where an otherwise exact CP is either softly1

or spontaneously broken. Specific models have been proposed where θ is calculably smalland within the experimental limits [6].A third class of solutions involve the action of wormholes [7].

As we will argue below,wormholes can break global symmetries explicitly, thus giving rise to potentially largecontributions to θ. However, under certain assumptions, it can be shown that wormholesactually have the effect of setting θ = 0 [7].In this letter, we address the question of whether these solutions to the strong-CPproblem can remain viable if Planck scale effects break global symmetries explicitly.There are many arguments suggesting that all global symmetries are violated at somelevel by gravity.First, no-hair theorems tell us that black holes are able to swallowglobal charge.

This allows for a gedanken experiment in which a quanta with globalcharge “scatters” with a black hole, leaving only a slightly more massive black hole, butone with indeterminate global charge as dictated by the no-hair theorem. Heuristically,if one considers “virtual” black hole states of mass M arising from quantum gravity, onecan integrate them out to yield global charge violating operators suppressed by powersof M, where M might be as small as MP l, the Planck mass.Another indication that gravity might not respect global symmetries comes fromwormhole physics [8].

Wormholes are classical solutions to Euclidean gravity that describechanges in topology. Integrating over all wormholes (with a cutoffon their size) yields alow-energy effective action that contains operators of all dimensions that violate globalsymmetries [9].

The natural scale of violation in this case is the wormhole scale, usuallythought to be very near (within an order of magnitude or so) MP l.Without explicit calculations of these effects, we are left with the following prescrip-tion: Due to our lack of understanding of physics at the Planck scale, we have no choicebut to interpret theories that do not include gravity in a quantum mechanically con-sistent way as effective field theories with a cutoffat MP l. If we adhere rigorously to2

this principle, we are then required to add all higher dimension operators (suppressed bypowers of MP l) consistent with the symmetries of the full theory at MP l. As discussedabove, it seems very unlikely that the full theory respects global symmetries. We notethat it would be particularly surprising if the entire theory respects U(1)PQ, since thissymmetry is already explicitly broken by instanton effects.

We should note that similarideas were noted briefly in the prescient paper of Georgi, Hall, and Wise [10]; however,we are now in a position to be somewhat more specific about the nature of the Planckscale effects in question and to explore their consequences.We consider first the implications for the axion model.To be specific, we con-sider a generic invisible axion model [11] in which an electroweak singlet φ, chargedunder U(1)PQ, is responsible for spontaneous breaking of the PQ symmetry. We mayparametrize φ on the vacuum manifold as φ = (fa/√2) exp(ia/fa), where a is the axionfield.

The effects of the QCD anomaly are to generate a mass for the axion of orderma ∼Λ2/fa, where Λ is the QCD scale. A variety of astrophysical and cosmologicalconstraints on the axion force fa into a narrow range of 109GeV <∼fa <∼1012GeV forstandard axions, or in a still narrower range around 107GeV for hadronic axions [12].The instanton induced potential for a takes the form [4]:V (a) = Λ4 cos(a/fa + θ).

(1)where θ is the QCD theta angle in a basis where the quark mass matrix is real. Whiledominating the path integral with instantons is probably a bad approximation in an un-broken gauge theory like QCD, there are rigorous results [13] showing that the minimumof V (a) occurs at strong-CP conserving values.One possibility is that gravity does not respect U(1)PQ at all, as is the case if wormholeeffects are large.

In this case, one should include renormalizable operators such as∆V (φ) ∼M2Wφ2 + h.c.(2)3

Here MW is the wormhole scale, which is expected to be of the order of the Planck mass.With the addition of these operators, the PQ symmetry is strongly broken and axionsnever arise at all.A second possibility is the U(1)PQ is only broken through non-renormalizable oper-ators of higher dimension. This can occur if either wormhole effects are suppressed orif the PQ symmetry is automatic, i.e., it is present “automatically” when one includesall renormalizable terms consistent with a given gauge group.

As we shall see below,higher dimension operators will also spoil the axion solution to the strong-CP problemexcept possibly in the case of an automatic PQ symmetry, where gauge symmetries caneliminate operators up to some required high dimension.We now explore the effect upon the axion potential of dimension D operators such asOD =αDMP lD−4 φ∗aφb + h.c.(a ̸= b;a + b = D),(3)which explicitly break U(1)P Q. Operators of dimension D will modify the axion potentialof Eq.

(1):V (a) = Λ4 cos(a/fa +θ)+X∆n cos(na/fa + δn)(n = D, D−2, D−4, . .

. ),(4)where ∆n ∼αDf Da /MP lD−4, and δn is a phase angle.

Let us simply analyze the n = 1contribution. The extra contribution will shift the minimum of the axion potential awayfrom the strong-CP conserving minimum of ⟨a/fa⟩= −θ.

Unless ǫ = ⟨a/fa⟩+ θ is lessthan 10−9 the amount of CP violation obtained will be in conflict with experiment. Theminimum of the axion potential is now determined by faV ′(a) ≃Λ4ǫ+∆1 sin(ǫ−θ+δ1) =0.

The magnitude of sin(ǫ −θ + δ1) will not, in general, be small, and ǫ ∼∆1/Λ4.Since we know ǫ < 10−9, ∆1 < 10−9Λ4.For dimension D operators, we expect∆1 ∼αDf Da /MP lD−4. Using Λ = 10−1GeV, the limit on ǫ translates into the following4

limit on the dimension D of the operator as a function of fa and αD:D <∼89 + log αD9 −log (fa/1010GeV). (5)If Eq.

(5) is satisfied, it is very simple to show that the higher-dimension operators willhave an insignificant effect on the axion mass. In fact, the zero temperature axion massis just ma ∼Λ2(1 + ǫ)/f.

However, we should note that the temperature dependenceof the axion mass is quite different in the presence of higher dimensional operators. Inparticular, the mass induced by the higher dimension operators is always “turned on.”This may affect axion cosmology in interesting ways.

We are currently investigating thistopic, as well as such effects on other theories (such as Majoron models) relying uponNambu–Goldstone boson physics [14].These results at first seem puzzling, since low-energy physics is not in general sensitiveto physics at the Planck scale. However, Nambu–Goldstone bosons have the peculiarproperty that although they are massless (or very light in the case of pseudo-Nambu–Goldstone bosons such as the axion), they are not, properly speaking, part of the low-energy theory as evidenced by the fact that self-couplings, and couplings to light fieldsare suppressed by a power of a large mass scale.

The fact that a light particle such asthe axion is part of the high-energy sector accounts for its interesting properties, but alsorenders it susceptible to high-energy corrections.In a generic invisible-axion model, there is no reason why a term such as φ5/MP lcould not be generated (here φ is a gauge-singlet field). This term would give rise tounacceptable shifts in θ unless αD <∼10−44−log(fa/1010GeV), which is remarkably small.

Isthere any to avoid this problem?There are, in fact, ways to construct axion models which suppress higher dimensionaloperators as needed. This construction is based on the notion of automatic PQ symme-tries [10], as described above.

We first consider a supersymmetric automatic model based5

on the gauge group E6 ×U(1)X [15]. The superfield content of the model is some numberof 27’s with X charges ±1 and a 351 with X charge 0.

The most general renormalizable,gauge-invariant superpotential will only contain terms of the form 271·27−1·3510, wherethe subscripts denote the U(1)X charges. This automatically gives rise to a PQ symme-try in which the 27’s have PQ charge +1 and the 351 has PQ charge −2.

The lowestdimension operators consistent with gauge invariance in the superpotential that breakthe PQ symmetry are terms like 27 6, 3516, and (27 · 27 · 351)4. These will then giverise to dimension 10 operators in the effective Lagrangian.

Furthermore, it is relativelyeasy to see that we can break the gauge symmetries and the PQ symmetry spontaneouslyin such a way so that the final PQ symmetry (a linear combination of the original PQsymmetry and some broken gauge symmetries) is broken around 1010 GeV.It is also possible to construct automatic PQ models based on supersymmetric SU(N)GUT’s that suppress higher dimension operators to any desired level for sufficiently largeN. Models of this type without exotic fermions must all have at least four differentchiral matter irreducible representations whose Young tableaux consist of a single column.Needless to say, these are exceedingly unattractive models.

They will tend to have manyextra families, which in addition to a host of phenomenological problems, will possiblydestroy the asymptotic freedom of QCD.Planck scale physics may also significantly affect the other solutions for the strong-CPproblem [16]. As described above, the second class of solutions are based upon modelswhere CP is softly or spontaneously broken.

How they fare under Planck scale physicsdepends on whether dimension four operators are generated, or whether only higherdimension operators appear.If renormalizable operators can be generated, then theviolation of CP by Planck scale effects will give rise to a tr G eG term, thus regenerating thestrong-CP problem (we should note, however, that the coefficient of such a term could beexponentially suppressed if it appeared in some controlled semiclassical expansion about6

some classical configuration [9]).Let us next consider the case in which only non-renormalizable operators are generatedby Planckian physics. In this case, all models with fields that acquire vacuum expectationvalues well below the Planck scale (typically the weak scale), will generate correctionsto θ that are highly suppressed by powers of MP l. In essence, this is nothing more thana restatement of the effective field theory philosophy: as long as we consider physics atenergies below the cutoffof our theory, the dominant effects come from the renormalizableoperators in the theory.

This way of thinking about effective field theories explains whythe PQ solution is so susceptible to possible effects of gravity. The problem is that thePQ scale is too close to MP l while the constraints on θ are too tight.Although we have seen that wormholes are troublesome for models that claim tosolve the strong-CP problem, there is some indication that wormhole effects themselvesmight drive the QCD θ parameter to a CP conserving value [7].

Within the frameworkof Coleman’s wormhole calculus [17] (which has since been shown to be naive in somerespects [18]), θ became a function of the wormhole parameters. The implementation ofColeman’s prescription for determining the value of these parameters was then shown toset θ to a CP conserving value.

It is not impossible that a more sophisticated approachto the wormhole calculus would still lead to a similar situation. However, until a betterunderstanding of wormholes and quantum gravity in general is reached, this will remaina conjecture.In conclusion, we see that Planck-scale physics can have dramatic effects on axionphysics.

If one wants to pursue the axion solution to the strong-CP problem, automaticmodels such as those presented here are probably the only consistent approach that canbe taken. We have also argued that the other known solutions are essentially unaffectedby gravity.

The essential difference between the PQ and the non-axionic solutions is dueto the sensitivity of the Nambu–Goldstone boson to physics at energies near the scale7

of spontaneous symmetry breaking. It remains to be seen whether other facets of theaxion scenario, such as the axion energy density crisis [19] will be modified by the effectsconsidered here.In the course of this work we learned that the effect of gravity on the Peccei–Quinnmechanism is also being considered by Kamionkowski and March-Russell [20], and byBarr and Seckel [21].

We would like to thank them for calling their work to our attention.It is a pleasure to thank S. Giddings, S. J. Rey and A. Strominger for useful dis-cussions. This research was supported in part by the National Science Foundation un-der grant No.

PHY89–04035. SDH acknowledges support from the National ScienceFoundation under grant NSF–PHY–87–14654, the state of Texas under grant TNRLC–RGFY106, and from the Harvard Society of Fellows.

EWK and RW were supported bythe NASA (through grant NAGW–2381 at Fermilab) and by the DOE (at Chicago andFermilab), RH was supported in part by DOE grant DE–AC02–76ER3066. TWK wassupported by the DOE (grant DE–FG05–85ER40226).References1.

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