Irrational Axions as a Solution of
Irrational Axions as a Solution of
arXiv:hep-th/9109040v1 23 Sep 1991RU-91-41Irrational Axions as a Solution ofThe Strong CP Problem in an Eternal UniverseTom Banks∗, Michael Dine∗∗and Nathan Seiberg∗∗Department of Physics and AstronomyRutgers University, Piscataway, NJ 08855-0849, USA∗∗Santa Cruz Institute for Particle PhysicsUniversity of California, Santa Cruz, CA 95064 USAWe exhibit a novel solution of the strong CP problem, which does not involve any masslessparticles. The low energy effective Lagrangian of our model involves a discrete spacetimeindependent axion field which can be thought of as a parameter labeling a dense set ofθ vacua.
In the full theory this parameter is seen to be dynamical, and the model seeksthe state of lowest energy, which has θeff = 0. The processes which mediate transitionsbetween θ vacua involve heavy degrees of freedom and are very slow.
Consequently, wedo not know whether our model can solve the strong CP problem in a universe which hasbeen cool for only a finite time. We present several speculations about the cosmologicalevolution of our model.Sep.
1991
1. Introduction and a 1+1 Dimensional ExampleThere have been a number of mechanisms proposed for resolving the strong CP prob-lem of Quantum Chromodynamics (QCD).
Most of them are either ruled out by experimentor only marginally consistent with the combination of experimental data and conventionalcosmology1. In the present note we would like to present a new solution of the strongCP problem.
We will exhibit a flat space-time quantum field theory in which the QCDθ parameter is “screened.” That is to say, although θ appears in the Lagrangian of themodel, the observable correlation functions in the true ground state do not depend on it.In particular, they are all CP invariant.We do not yet know whether our model provides a realistic solution of the strong CPproblem in the world, as opposed to a model of a flat eternal universe. It has a largenumber of nearly degenerate metastable vacuum states, and it is not clear that the systemwill ever find its ground state in a universe which has been large only for a finite time.The details of the discussion may also depend on the mechanism that ensures that thecosmological constant vanishes.
Since our understanding of these cosmological issues isnot complete, we can only offer a few speculations about them. These will be presented insection III.The idea for our solution of the strong CP problem is most easily demonstrated by firstexamining another theory with a θ parameter: 1+1 dimensional QED.
The Abelian gaugegroup of the theory can be either the compact group U(1) or its non-compact coveringgroup IR. The choice between IR and U(1) has significant consequences.The chargesof the matter fields in the IR theory are not restricted but they have to be integers inthe U(1) theory.
Another difference between these two theories is that unlike the U(1)gauge theory, the IR gauge theory on a compact two-dimensional parameter space has no θdependence. On a non-compact parameter space both theories exhibit θ dependence whichis interpreted as a background electric field originating from classical charges at infinity.This fact remains true even in the presence of dynamical matter fields whose charges arequantized.Now consider the case of dynamical matter fields whose charges are not quantized.
Inparticular, consider the Abelian Higgs model with Higgs field of charge one and a massive1 The single exception that we are aware of is the class of models [1] in which discrete sym-metries guarantee that the argument of the determinant of the quark mass matrix is zero at treelevel.1
fermion of irrational charge q. It is particularly interesting to examine the limit in which thefermion mass is much larger than the scale defined by the Higgs model.
Since the chargesare not quantized, the gauge group must be IR and there is no θ dependence on a compactparameter space. Perhaps more surprising is the fact that there is also no θ dependence ona non-compact parameter space.
The point is that any background electric field θ can bescreened by popping charges out of the vacuum. With the two charges, one and q, everyvalue of θ = 2π(n + mq) with n and m integers can be screened.
The screening processmay cost a lot of energy (and be very slow) because the irrationally charged fermions areso heavy, but this will always be compensated by gaining the constant field energy overthe infinite volume of space. Since the ground state energy is minimized when the externalfield is zero and since any value ofθ2π can be arbitrarily closely approximated2 by a numberof the form (n + mq), the ground state has vanishing background field regardless of thevalue of θ in the Lagrangian.
This is the essence of our solution of the strong CP problem.The bosonized version of this model can be immediately generalized to 3 + 1 dimensions.This will be done in the next section.2. Irrational Discrete AxionsConsider a model consisting of an axion, a, coupled to two nonabelian gauge fields inthe standard manner.
The Euclidean Lagrangian has the formL = f 22 (∂a)2 +14g21F 21 +14g22F 22 + i(a + θ1)Q1 + i(qa + θ2)Q2(2.1)where the Qi are the topological charge densities of the two gauge fields, f is the axiondecay constant and q is a dimensionless parameter. The second gauge group is supposed torepresent QCD, while the first is another confining theory with a much larger confinementscale, Λ1 ≫Λ2 = ΛQCD.
We assume that the fermions coupled to these two gauge theorieshave no global symmetries that could be used to rotate away the topological terms in theLagrangian. Clearly, without loss of generality we can use the shift symmetry of the axionto set θ1 = 0.
The conventional wisdom is that this leaves us with no freedom to changethe value of θ2 and the model (2.1) has strong CP violation. However, a closer examination2 The fundamental result that is needed to prove this is that the error in a rational approx-imant to any number can be made to vanish like one over the square of the denominator in theapproximant [2].2
shows that after θ1 is set to zero we still have the freedom to shift a by 2πn for integer n.If the parameter q is irrational, we can use this freedom to set θ2 arbitrarily close to anydesired value. Therefore, the theory based on the Lagrangian (2.1) is independent of bothθ1 and θ2 and is equivalent toL = f 22 (∂a)2 +14g21F 21 +14g22F 22 + ia(Q1 + qQ2) .
(2.2)Unlike the conventional wisdom, one axion field a can remove more than one θ parameter.What we have just argued is that the theory based on (2.1) is equivalent to that basedon (2.2). The latter is obviously CP invariant, if the axion field a is defined to changesign under a CP transformation.What we would like to show now is that CP is notspontaneously broken and therefore all correlation functions are CP invariant.
To showthat we first integrate out all the non-zero modes of a. The result of this Gaussian integralisLeff =14g21F 21 (x) + 14g22F 22 (x) + qf 2ZdyQ1(x)G(x, y)Q2(y) + ia0(Q1(x) + qQ2(x)) (2.3)where G(x, y) is the propagator of a with the zero mode, a0, removed.
Now, integratingover the gauge fields we find the effective potential Veff(a0). Following the argument in[3], we see that the point a0 = 0 is a global minimum of Veff.
The integrand of the gaugefield functional integral is positive for a0 = 0 and carries a phase for any other value ofa0. The partition function is therefore maximized, and the energy minimized, for the CPconserving vacuum a0 = 0.Let us examine the theory in more detail in order to understand the physical mech-anism for resolving the strong CP problem.
We set θ1 = 0 and integrate out the heavygauge degrees of freedom. This fixes the VEV of the axion to 2πn for some integer n andthe axion acquires a mass of order Λ21f .
Typically, one ignores the integer n and sets itto zero. Then the low energy theory includes the light gauge fields (QCD), no axion anda θ parameter equal to θ2.
This is essentially the argument that one axion can removeonly one θ parameter. However, the general argument in the previous paragraphs showsthat when q is irrational the theory cannot depend on θ2.
The reason for that is that theinteger n cannot be ignored. After integrating out the massive gauge fields, the theory hasan infinite number of degenerate ground states labeled by n. The dynamics of the lightgauge fields breaks this degeneracy.
When f ≫Λi the term with the propagator in (2.3)3
is small and can be neglected. In this approximation the effective potential Veff(a0) hasthe formVeff(a) = Λ41E1(a) + Λ42E2(qa + θ2)(2.4)where we have dropped the subscript of a.The functions Ei are both periodic withperiod 2π and have their minimum when the argument vanishes [3].
Assuming that Ei arecontinuous at 0 mod 2π we can minimize the total effective potential by making both ofthe arguments of the Ei as close as possible to multiples of 2π. Thusa ≈2πn;qa + θ2 ≈2πm(2.5)These two equations are compatible iffθ ≈2π(m −qn) .
(2.6)These are of course just the equations that we discussed in the 1 + 1 dimensional Higgsmodel. If q is irrational, then we can satisfy this condition with arbitrary precision byappropriate choice of n, m.Our mechanism has an obvious generalization to the case of several gauge groups allcoupled to the axion a through some coefficients qi.
If all these coefficients qi are relativelyirrational, there is an infinite set of vacua where (qi⟨a⟩+ θi)mod2π < ǫi for any ǫi. Itmight however be important for cosmological reasons to note that the fraction of vacuasatisfying these inequalities goes like Qi ǫi when all the ǫi are small.
Thus if the dynamicsof the universe randomly chooses between all possible metastable states of the system, theprobability of not seeing any low energy CP violation goes rapidly to zero as the numberof gauge groups is increased.The failure of the decoupling theorem for this model stems from two separate sources.First of all, the high energy theory has an infinite set of degenerate vacua, and the degen-eracy is broken by QCD. Equally importantly, we are discussing the ground state of themodel, which means that we are willing to wait an arbitrarily long time for the system tosettle down.
The processes by which the system moves from one of the almost degeneratemetastable states with θQCD ̸= 0, to the true vacuum will be very slow. For the purposesof discussing local physics over finite time intervals the decoupling theorem is valid.An equivalent way of thinking about the model is the following.
We can include in ourlow energy effective Lagrangian a discrete field n which is independent of the coordinates,4
x, and takes values in the integers.3 This field represents the value of ⟨a⟩/2π and thus labelsthe almost degenerate ground states. It is crucial that by varying the value of n every valueof θQCD = (q⟨a⟩+θ2) mod 2π can be approximated arbitrarily well.
Therefore, the sum inthe functional integral of the low energy theory over n can be replaced by an integral over asingle continuous variable θQCD which is independent of x. Therefore, our theory looks likeordinary QCD with one more integration variable θQCD.
The standard problem with sucha theory is that ordinarily different values of θQCD correspond to different superselectionsectors. There are no physical processes or local operators which communicate betweenthese different sectors and therefore θQCD should not be integrated over.
The novelty inour theory is that θQCD does not label different superselection sectors. The high energytheory makes the barrier between these sectors finite and allows transitions between them.As with the standard axion, θQCD is a field which is integrated over and can relax tozero.
In our case, though, only the zero momentum mode of the field n and therefore alsoonly the zero momentum mode of θQCD exists. Hence, we do not have a massless or lightaxion.One might ask whether our model suffers from the U(1) problem when massless quarksare coupled to it.
Then it has an axial U(1) symmetry under which the coordinate θQCDis shifted by a constant.However, since θQCD is independent of x, it does not havea conjugate momentum and no charge generates this symmetry. From the high energypoint of view this follows from the fact that nearby values of θQCD are associated withfar separated points in field space.
Therefore, when the symmetry is broken there is noGoldstone boson and there is no U(1) problem. A simple toy model which exhibits such abehavior is based on the LagrangianLtoy = f 2η2 ∂µη(x)∂µη(x) + V (η(x) + θQCD)(2.7)where θQCD is an integration variable in the functional integral.
The field η(x) plays therole of the would be light boson of the U(1) problem. The theory (2.7) is invariant underthe broken U(1) symmetry η(x) →η(x) + α; θQCD →θQCD −α and the potential V canbe arbitrary.
We can use the U(1) symmetry to set θQCD = 0 in the Lagrangian and thenthe integral over θQCD factorizes. Clearly, the resulting theory does not have a massless ηparticle.3 It should be stressed that at the level of the low energy theory, this field has no dynamics.5
Technically, the violation of current algebra “theorems” relating the U(1) Goldstoneboson to θ dependence in QCD comes about in our model because the two point functionof topological charge density is discontinuous at zero momenta. The discontinuity comesfrom intermediate metastable vacuum states, which are exactly stable in the low energytheory.
We repeat that in the low energy theory we sum over superselection sectors. Thisis “forbidden” by the “axioms” of field theory, but in our model the high energy sectorprovides a dynamical rationale for summing over these sectors.
The full theory satisfies allrelevant axioms.3. Cosmological SpeculationsThe model that we have discussed so far would solve the strong CP problem in aneternal flat world in the absence of gravitation.
In the real world, an expanding universewhich has undoubtedly been at a temperature below the QCD scale for only a finite time,one must ask whether the system we have described will ever find its true ground state.The answer to this question may involve very complicated, perhaps chaotic or spin glass-like dynamics, and/or be connected to other deep cosmological puzzles. At the presenttime we have no clear picture of how the model behaves.
We will therefore simply suggestsome possible scenarios and leave a more serious discussion for future work.The first scenario is the simplest, and adheres most closely to the standard discussionof axion cosmology. It should be applicable for at least some range of parameters in ourmodel.In this standard scenario we assume that the axion field is sufficiently weaklycoupled that after inflation it simply falls into one of its classical vacua over regions muchlarger than the entire universe visible to us today.
It is easy to argue that when θeff =(qa + θ2) mod 2π is very small, the fraction of vacua with θ < θeff is linear in θeff, sothere is only one chance in 10−9 that any given region has θ small enough to be compatiblewith the current experimental bound on the neutron electric dipole moment. All is notlost however if we make the further assumption that the cosmological constant vanishes atthe true minimum of the axion potential.
We emphasize that we have no idea why this isso, but that this is the standard assumption about the cosmological constant.Given these assumptions, the cosmology of our model is fairly standard until temper-atures of order Tc ∼θ12 ΛQCD. In particular, since the axion potential is of order Λ4H, the6
energy density is not dominated by nonrelativistic axions which overclose the universe.4 In-stead, cosmology follows the standard Robertson Walker scenario until the universe reachestemperatures of order Tc. At this point, the universe becomes cosmological constant dom-inated, with a cosmological constant ∼θ2Λ4QCD.
Weinberg [4] has shown that a positivecosmological constant greater than about 103 times the present observational limit willprevent the formation of galaxies. Thus, the only regions in which galaxy formation cantake place are those in whichθ2Λ4QCD ≤10−9eV 4(3.1)Since ΛQCD ∼108eV , regions containing galaxies have θ ≤10−20.
In other words, con-ventional inflationary axion dynamics, coupled with the standard assumption that thecosmological constant vanishes at the absolute minimum of the potential, implies that theonly regions in our model universe which contain galaxies are those with θ much smallerthan the bound from the neutron electric dipole moment. This seems to us to be a rea-sonably attractive resolution of the strong CP problem.
Perhaps its greatest defect is thatonly about 1/30 of the metastable domains containing galaxies will have a cosmologicalconstant consistent with the present limit (despite the fact that we have fine tuned thetrue ground state energy to zero).We are not at all sure that our model behaves as we have described in the previousparagraph. Our alternative scenarios are harder to analyze and none seem to work verywell.They involve the assumption that the presently observable piece of the universeconsists of multiple domains in which the effective value of θ is different.
One must thenanalyze the dynamics of these domains as the temperature falls below the QCD scale.Those with very small values of θ are energetically favored, and begin to expand relativeto the others.5 On the other hand, those with higher values of θ become dominated bytheir cosmological constants and expand exponentially. There are possible contributionsto the energy density from domain walls, and nonrelativistic axion gases.
The situationis made more complicated by the bizarre nature of the potential. States that are close inenergy are far away in field space.
Even a single classical variable with such a potential has4 At least, it is possible to choose a wide range of parameters for which the axion lifetime isshort enough that this problem does not arise.5 Here we assume that the bubbles are larger than their critical size when the QCD temperatureis reached.The critical size is quite large because the surface tension in the domain walls isdetermined by the heavy scale ΛH.7
chaotic behavior and we are dealing with a field theory full of such degrees of freedom. Atthe present time we believe that the expansion of large θ domains due to their cosmologicalconstants is the dominant effect.
It is hard to see how a universe built in such a mannercould resemble our own. Nonetheless, we feel that these complicated scenarios should beunderstood more fully.
The wild speculation that the axion domains in such a chaoticsystem might have something to do with the foamlike large scale structure that has beenrecently observed [5] is immensely attractive. Indeed, because of the scarcity of states withsmall θ one might imagine that the most probable multiple domain configurations withgalaxies would have some domains just above and some just below the Weinberg bound.It is amusing to speculate that the famous voids in Bootes and other parts of the skyare regions in which the effective cosmological constant was too large to allow for galaxyformation.
For lack of talent and insight, we will have to leave such cosmic fantasies for afuture publication.Another speculative application of the irrational axion idea is to an anthropic solutionof the cosmological constant problem6. Imagine that in some version of supergravity it isnatural for the cosmological constant to be at the SUSY breaking scale M. Now considerthe SUSY version of the model of this paper with both gauge groups also at the SUSYbreaking scale.
The total effective potential is M 4(K + F(a)). F(a) has a set of minimawith energies that fill an interval of order 1 densely.
Thus there are many vacua in whichK is cancelled to an accuracy sufficient to allow galaxy formation, and in a fraction 10−3of those, the cosmological constant is as small as that observed in our universe. A typicalstate with small cosmological constant will not be sufficiently metastable to serve as amodel for our universe, however some fraction of these vacua will be.4.
ConclusionsIn this paper we provide an example of a flat space field theory which solves thestrong CP problem without massless particles of any kind. The model violates the usualdecoupling theorems in an interesting way.
The low energy theory has a discrete globalvariable, n, labeling a set of quasi degenerate vacua.The dynamics that allows thesevacua to transform into one another and settle down into the true ground state cannot be6 Some time ago, L. Abbott [6] suggested a scheme for cancelling the cosmological constantalso involving axions. Unlike the present proposal, it was necessary to introduce an extremelysmall energy scale, and the low energy theory contained a light particle.8
understood without appealing to the high energy theory. The decoupling theorems are stillvalid in the weak sense that local dynamics in each metastable ground state is describedcompletely in terms of the low energy Lagrangian.
The discrete variable does allow us toevade the usual argument connecting the strong CP problem to the U(1) problem.On the negative side, the basic Lagrangian of our model is nonrenormalizable, and itis easy to show that it cannot be the effective theory of any renormalizable field theory.In addition we have not been able to find a string compactification which leads to anaxion with such irrational couplings. Thus, the fundamental basis for our model remainsobscure.
A possible origin for the irrational couplings central to our model may be found inthe novel nonperturbative behavior of string theory that has been pointed out by Shenker[7]. He argued that intrinsically stringy nonperturbative effects will behave as e−1g insteadof the e−1g2 characteristic of field theory, and has speculated that this behavior could beunderstood in terms of “instantons of continuous topological charge” [8].
If such instantonsindeed exist in string theory, they might provide the irrational axion couplings that werequire.The general features of our mechanism may be applicable to other fine tuning problemsin particle physics. Its fundamental characteristic is that it allows us to turn couplingsinto dynamical variables without invoking massless particles7.
One need only have a highenergy field with a discrete set of degenerate ground states whose integer label is irrationallyrelated to a parameter in the low energy Lagrangian (so that the parameter can be given adense set of values by appropriate choice of the integer). Low energy dynamics resolves thedegeneracy and high energy processes mediate the transitions between states with differentvalues of the effective coupling.
We have already described a crude version of how sucha mechanism might help us to understand the cosmological constant problem. It is tobe hoped that one can do better than this.
In string theory, our mechanism might helpto resolve the problem of determining the string coupling. Conventionally it is said thatany potential which allows the fine structure constant to be weak, and does not force itto vary significantly over geological time scales, implies the existence of a scalar particleof very small mass [9].
Since string theory also determines the couplings of this particleto be about gravitational strength, it is ruled out by astrophysical considerations. Wenow envisage the possibility of generating an “irrational” potential for the string coupling,which could determine it (or allow it to be set as an initial condition for our part of the7 Or wormholes!9
universe) without requiring any massless particles. This exciting idea is also left for futurework.We do not know whether we have found a solution to the strong CP problem in the realworld.
Conventional assumptions about axion dynamics in an inflationary cosmology, andabout the value of the cosmological constant, lead to a correlation between the existence ofgalaxies in the observable part of the universe and the fact that θ is so small. Alternativeassumptions about the spatial configuration of the axion field in our universe might leadto an explanation of foam like large scale structure, and great voids.
At present, thesetwo pictures do not seem compatible with each other, and the second probably leads to ahighly inhomogeneous universe. However, our present understanding of the cosmology ofthis model is such that we can still hope for the best of all possible worlds.AcknowledgementsIt is a pleasure to thank S. Coleman, M. Douglas, D. Friedan, E. Martinec, G. Moore,R.
Leigh, S. Shenker, L. Susskind and A.B. Zamolodchikov for several useful discussions.This work was supported in part by DOE grants DE-FG05-90ER40559 and DE-AM03-76SF00010.10
References[1]A. Nelson, Phys.Lett. 136B (1984) 387; S. Barr, Phys.
Rev. Lett.
53 (1984) 329, Phys.Rev. D30 (1984) 1805; R. Mohapatra and K. Babu, Phys.
Rev. D41 (1990) 1286[2]I. Niven, Numbers: Rational and Irrational, Random House, New York, p.94[3]C. Vafa and E. Witten, Phys.
Rev. Lett.
53 (1984) 535.[4]S. Weinberg, Phys.
Rev. Lett.
59 (1987) 2607.[5]H. Rood, Ann.
Rev. Astron.
Astrophys. 26 (1988) 245, and references cited therein.[6]L.
Abbott, Phys. Lett.
150B (1985) 427.[7]S. Shenker, in the Proc.
of the Cargese meeting, Random Surfaces, Quantum Gravityand Strings, (1990).[8]S. Shenker, Private Communication.[9]M.
Dine and N. Seiberg, Phys. Lett.
162B (1985) 299.11
출처: arXiv:9109.040 • 원문 보기