Axionic Band Structure of the Cosmological Constant
AAxionic Band Structure of the Cosmological Constant
Thomas C. Bachlechner
Physics Department and Institute for Strings, Cosmology and Astroparticle Physics,Columbia University, New York, NY 10027, USA
We argue that theories with multiple axions generically contain a large number of vacua that canaccount for the smallness of the cosmological constant. In a theory with N axions, the dominantinstantons with charges Q determine the discrete symmetry of vacua. Subleading instantons breakthe leading periodicity and lift the vacuum degeneracy. For generic integer charges the number ofdistinct vacua is given by (cid:112) det( Q (cid:62) Q ) ∝ e N . Our construction motivates the existence of a landscapewith a vast number of vacua in a large class of four-dimensional effective theories. INTRODUCTION
Einstein’s field equations famously couple the vacuumenergy density ρ Λ to the curvature of spacetime. Su-pernova data and cosmic microwave background observa-tions are well described by general relativity with a small,positive vacuum energy ρ Λ ≈ − M [1–3]. However,known contributions to the vacuum energy correct anybare cosmological constant by at least 10 − M [4, 5].This vast discrepancy in scales is known as the cosmo-logical constant problem. One of the best motivated ap-proaches to this apparent fine-tuning is to study a largenumber of populated vacua and consider selection effects:any experiment we perform is subject to a selection biasthat skews the distribution of observed outcomes [6–11].For example, Weinberg first pointed out that galaxieswill not form unless | ρ Λ | < ∼ − M . In an eternallyinflating universe with a diverse vacuum structure, theselection effect would be manifest in the form of a sur-prisingly small vacuum energy [12–15].Brown and Teitelboim (BT) suggested that a singlefour-form field strength might cancel a bare cosmologicalconstant to arbitrary accuracy [16–19]. Independently,Banks, Dine and Seiberg (BDS) noted that theories witha single axion, but irrational decay constants, also leadto a large landscape of vacua [20]. Unfortunately, bothproposals suffer from cosmological problems and resist anembedding in quantum gravity [16–24]. The BT mecha-nism was generalized to the multiple field strengths thatgenerically arise in string theory, which led to the dis-covery of a string landscape that famously allows for asmany as 10 vacua, accommodates a realistic cosmol-ogy and contains the desired selection effect of vacuumenergies [25]. In this work we extend the BDS approachto theories of multiple axions, such as string compact-ifications with fixed fluxes, and discover a field theorylandscape with an exponential number of vacua and con-sistent cosmological history. Crucially, our constructionmay be embedded in a theory of quantum gravity andprovides a new framework for the study of vacua in thestring landscape.The leading contributions to the axion potential are in-variant under some discrete translations Q ai θ i → Q ai θ i +1. A x i o n P o t e n t i a l ( a . u . ) θ / / / / FIG. 1. Left: Axion potential (dark blue) and a subleadingcontribution (light blue). Right: Vacuum energies. Selectioneffects may lead to preferred vacua in the green band.
This shift symmetry is broken by subleading contribu-tions to a periodicity θ i → θ i + 1. The fundamen-tal region of the leading potential has a volume of size1 / (cid:112) det( Q (cid:62) Q ), such that after including the remainingsubleading terms, the vacuum energies are split intoabout N r ∝ (cid:112) det( Q (cid:62) Q ) ∝ e cN distinct vacua, for someconstant c . If the charges of the dominant instantonsdo not form a primitive basis of the integer lattice, wefind c >
0, leading to an exponentially large number ofnon-degenerate vacua. For example, consider the caseof diagonal charges, Q ai = nδ ai . The leading potential isinvariant under discrete shifts θ i → θ i + 1 /n , but thissymmetry is broken by subleading terms to the originalunit periodicity, giving rise to n N distinct vacua. Thisbasic mechanism is illustrated in Figure 1 for the one-axion case. THE VACUA OF N AXIONS
Consider a theory of N axions θ i , whose continuousshift symmetries are broken only by non-perturbative ef-fects to the discrete shift symmetry θ i → θ i + 1. Theinstanton effects generate an axion potential of the form a r X i v : . [ h e p - t h ] O c t L = 12 K ij ∂θ i ∂θ j − (cid:88) a Λ a (cid:2) − cos (cid:0) π Q ai θ i + δ a (cid:1)(cid:3) − V , (1)where the sum runs over all contributions to the axionpotential, V is some constant energy density, K ij is thefield space metric and Q a denotes the integer charge vec-tor of the a th instanton contribution.We are interested in the vacuum distribution in thistheory. In general, the location of minima is difficult toobtain: the critical point equation consists of N coupled,non-linear equations. To simplify the problem, let usdenote the number of terms that give rise to the leadingcontributions by P . We can now decompose the chargematrix according to that choice: Q a (cid:12)(cid:12) a =1 ,... = (cid:18) Q Q r (cid:19) a (cid:12)(cid:12)(cid:12)(cid:12) a =1 ,... , (2)where Q is a full rank, rectangular P × N matrix and Q r isa rectangular matrix of rank R , containing all remainingcharges. The leading potential is given by V Q = P (cid:88) a =1 Λ a [1 − cos (2 π Q a θ + δ a )] , (3)and we call the typical scale of this potential Λ Q . Thecharges Q specify the periodicity of vacua in terms of thelattice L ( B Q ) with basis B Q , that is L ( B Q ) ≡ (cid:8) B Q n | n ∈ Z N (cid:9) = P (cid:92) a =1 { θ | Q a θ ∈ Z } . (4)In the special case of a square matrix Q , the periodicityis generated by the basis Q − . To make the periodicityof the potential manifest we now perform the GL ( N )transformation, φ = B − Q θ , (5)such that the leading potential is invariant under the dis-crete shifts φ i → φ i +1. Let us call the number of distinctvacua of the leading potential N Q ≥
1. These vacua arelocated at field values φ ∗ α, n = φ ∗ α + n , n ∈ Z N , (6)where the index α labels distinct vacua. The scale of thevacuum energies is set by the leading dynamical scales,i.e. V ( φ ∗ α ) ∼ Λ Q . THE AXIONIC BAND STRUCTURE
While the leading terms in the potential exhibit degen-erate vacua that are invariant under discrete shifts on the lattice L ( B Q ), this symmetry is broken by the sublead-ing potential. As in (4), let us refer to the basis thatgenerates the lattice of rank R , under which the sublead-ing potential is invariant, as B r . In terms of the field φ ,the subleading potential is symmetric under the discreteshifts [26] B − B Q φ → B − B Q φ + n , n ∈ Z N . (7)Therefore, when including the subleading terms, thevacua of V Q are periodic on the lattice generated by thebasis B − B Q , modulo Z R , L ( B ) = (cid:8) ( B − B Q ) n | n ∈ Z N (cid:9) / Z R , (8)where we call the corresponding basis B . Each of the N r sites of the lattice L ( B ), that are located within the unit R -hypercube, corresponds to a distinct, non-degeneratevacuum. The fundamental parallelepiped of the latticewas defined through a quotient by Z R , so it is a tiling ofthe unit R -hypercube. Therefore, the number of distinctvacua is given by the inverse volume of the fundamentalparallelepiped, N r = (cid:113) det ( B (cid:62) B ) − . (9)This is the main result of our work. Each of the N Q vacuaof the leading potential is split into an energy band ofwidth Λ , containing N r vacua. We refer to this vacuumdistribution as the axionic band structure . If any of thevacua of the leading potential vacua are within about Λ of zero, there exist vacua with energies as low as Λ / N r .Returning to the simple example of P = N leadingterms and we take Q r to generate the integer lattice, suchthat B r = , we immediately find the number of distinctvacua to be N r = (cid:113) det( Q (cid:62) Q ) . (10) RANDOM AXION THEORIES
Naively, it might appear unlikely that the axionicbands contain a large number of non-degenerate vacua.In the one-axion case, a large number can only beachieved by an equally large tuning of two axion de-cay constants to an almost irrational ratio [20]. We nowconsider an ensemble of random multi-axion theories de-fined through the measure on the space of charge matrices Q . In particular, we will consider charge matrices withentries consisting of independent, identically distributed(i.i.d.) random integers. This choice is motivated fromexplicit flux compactifications on Calabi-Yau manifolds[27, 28] or from gravitational instantons [29, 30]. Eventhough the charge matrix may be sparse, it rapidly ap-proaches its universal limit when a small fraction > ∼ /N of entries are non-vanishing [27, 31]. In the universallimit, the matrix Q (cid:62) Q is well described by the Gaus-sian orthogonal Wishart ensemble, i.e. the ensemble ofmatrices W W = A (cid:62) · A , (11)where the entries of A are real, i.i.d. random numberswith variance σ and vanishing mean. For a sparse,square matrix A with δN non-vanishing integer entriesthe variance in the universal regime is given by σ ≈ δNN > ∼ N . (12)The determinant of Wishart matrices follows a productchi-squared distribution, so that the expected value ofthe determinant is given by [32, 33] (cid:10) det( Q (cid:62) Q ) (cid:11) = σ N Q Γ( N + 1) . (13)Let us now obtain a simple estimate for the number ofnon-degenerate vacua. The dynamical scales Λ a are gen-erated non-perturbatively, so we expect them to be dis-tributed uniformly on a logarithmic scale. This largehierarchy implies a small number of leading terms, so wecan take P = N . In the context of string compactifica-tions, the scales are set by Λ a ∼ exp( −Q ai τ i ) for somevolumes τ i . Naively, one might worry that the leadingcharges are small and form a primitive basis for the in-teger lattice. However, depending on the volumes andother possible constraints on the charges, the dominantcontributions in general are non-trivial. On the otherhand, there is a vast number of subleading instantonswith unconstrained charges. These charges do form aprimitive basis for the integer lattice, so that we can take B r = . With (10) and (13) we immediately obtain theexpected number of non-degenerate vacua (cid:10) N (cid:11) = σ N Q Γ( N + 1) > ∼ √ πN (cid:18) e (cid:19) N . (14)For generic, square charge matrices Q with more thanabout 3 N non-vanishing integer entries, there exists anexponentially large number of non-degenerate vacua. Togive a concrete example, in the presence of 500 axions,a charge matrix with 2% non-vanishing integer entriessuffices to generate more than 10 distinct vacua. COSMOLOGICAL CONSIDERATIONS
In the previous sections we argued for the existence ofan exponentially large number of vacua in theories con-taining multiple axions. However, a successful theory forthe observed value of the cosmological constant not onlyrealizes a classically stable vacuum in the right energyrange, but also connects to a realistic cosmology. First,the theory must admit a consistent quantum gravity com-pletion and accommodate (metastable) vacua at energies vastly smaller than the natural scale of the theory. Sec-ond, vacua with small cosmological constant are stableon time-scales of the age of the universe. Finally, thetheory allows for sufficient energy to drive inflation andreheating. We now briefly discuss these considerations inturn.We considered generic theories of multiple axions anddemonstrated the existence of a vast number of vacua,distributed uniformly over an energy range given by Λ in a potential with typical scale Λ Q > ∼ Λ . Therefore,classically stable vacua with small vacuum energy are notatypical. Due to the discrete shift symmetry in the axionsector, the vacuum distribution is decoupled from highenergy physics. However, a domain wall that interpo-lates between two vacua may have access to high-energydegrees of freedom that change the effective axion po-tential. The extra degrees of freedom do not change theaxion symmetry and therefore the number of vacua ineach axionic band is unchanged. Finally, a large canoni-cal field displacement during a vacuum transition may bein conflict with quantum gravity or resist an embeddingin string theory [34–36]. The potential is approximatelyperiodic under discrete shifts φ i → φ i + 1. Therefore,the canonical separation between two vacua cannot ex-ceed √ N ξ N , where ξ N is the largest eigenvalue of the ki-netic matrix for the axions φ . While the precise form ofquantum gravitational constraints on field ranges are cur-rently under debate, the displacement √ N ξ N may wellbe sub-Planckian and therefore decoupled from quantumgravity [33, 37]. We conclude that theories containingmultiple axions can accommodate classically stable solu-tions with exponentially small vacuum energies.Let us now ask whether the vacua are sufficiently sta-ble against decay. Given the age of our universe, asufficient condition for stability is a small decay rate,Γ (cid:28) e − M . In general, it is difficult to estimatethe tunneling rate between distinct vacua of the leadingpotential V Q , but we can obtain an upper limit for thetunneling rate between the vacua split by the subleadingpotential contributions. Using the thin-wall approxima-tion and neglecting gravity, the decay rate for tunnelingbetween two (local) minima located at Φ i and Φ f is givenby Γ ∼ e − B where, [38, 39] B = 27 π σ (cid:15) . (15)Here, the surface tension of the Coleman-De Luccia in-stanton is given by σ = (cid:90) Φ f Φ i d Φ (cid:112) V (Φ) − V (Φ i )] , (16)the integration is performed along the path of extremalaction and (cid:15) is the difference in energy density betweenthe two vacua. Adjacent vacua are separated by a canon-ical distance of at least ξ , where ξ is the smallest eigen-value of the corresponding kinetic matrix. The differencein energy density between the two vacua is constrainedby the scale of the subleading potential, while a typicalfield trajectory interpolates through a potential of scaleΛ Q . By demanding B > ∼ we then find a rough boundfor the scale of the smallest axion decay constant, ξ > ∼ Λ Λ Q . (17)Gravitational contributions to the decay rate are neg-ligible in the relevant regime. Therefore, theories thatsatisfy the constraint in (17) are expected to have vacuathat are stable on time scales long compared to the ageof the universe.Following Coleman-De Luccia decay to a vacuum withsufficiently small cosmological constant, the universe ex-hibits negative spatial curvature and may undergo slow-roll inflation. Since the vacua within one axionic bandare spread over a range of Λ , the tunneling process toour current vacuum will leave the axions at an energy ofabout Λ , evading the empty universe problem for suffi-ciently large energy bands. To understand this importantpoint, consider tunneling from a penultimate vacuum at φ ∗ α, n (cid:48) to our current vacuum with very small cosmologicalconstant V ( φ ∗ α, n ) (cid:28) Λ r . For simplicity, let us consideronly one subleading term in the axion potential with dy-namical scale Λ r . The degeneracy between the two vacuais lifted by the subleading potential, δV α ; n , n (cid:48) ≈ Λ r cos (cid:16) πQ r B Q n (cid:48) + ˜ δ α (cid:17) , (18)where we neglected the small vacuum energy in the finalvacuum and ˜ δ α is a phase. The sum inside the cosineconsists of N order one terms. Therefore, neighboringvacua, will differ by about Λ r in energy density. Findinga vacuum with energy density in a given range is a dif-ficult problem: we need to test an exponential numberof vacua before finding one with a suitably small energy.Since tunneling between widely separated vacua is ex-ponentially suppressed, the energy released during thefinal transition is of order Λ r . This feature is due to themulti-dimensionality and absent in theories with a sin-gle axion. This energy density can lead to a subsequentphase of slow-roll inflation, while the tunneling event maygive rise to observable features [40, 41].It is curious to point out that when considering theLagrangian (1) with N = 500 axions and P = N leadingcontributions of scale Λ a ∼ . M pl , some subleading con-tributions at the GUT scale and 2% non-vanishing orderone entries in the charge matrix, we expect a sufficientnumber of vacua to account for the observed smallnessof the cosmological constant and, at the same time, largefield axion inflation via kinetic alignment, which wouldsolve the flatness problem [27, 42]. However, the infla-tionary dynamics are highly sensitive to heavy fields, so itis not clear if this model can be embedded in a consistenttheory of quantum gravity. THE STRONG CP PROBLEM
The action of QCD famously contains a CP-violatingterm that is proportional to the Yang-Mills instantonnumber, δS = θ QCD π (cid:90) d x Tr( F µν ˜ F µν ) . (19)By measuring the electric dipole moment of the neutronone obtains an upper bound on the coupling parameter | θ QCD | < ∼ − . The smallness of this dimensionlessparameter constitutes the strong CP problem. One ofthe most compelling approaches to this problem is topromote the coupling constant to a dynamical field witha continuous shift symmetry that is broken to a discreteshift symmetry via its coupling to the QCD anomaly [43].This generates a potential of the form V QCD = Λ QCD cos(2 πθ QCD ) , (20)where Λ QCD ≈ f π m π . In the absence of additional cou-plings the axion is dynamically stabilized at θ QCD = 0.However, upon embedding QCD in a theory of quantumgravity, such as string theory, there are additional contri-butions to the QCD axion potential. These high energycontributions take the form δV ∼ Λ UV cos(2 πθ QCD + δ ),where δ is some order one phase, set at a high energy.Since the expectation value for the QCD axion is nowdominated by the high-energy effects this leads to a largeCP violating phase δ [44, 45]. This is precisely the situa-tion we have when considering a general coupling betweenthe axions θ i in Lagrangian (1) to QCD, with charges q QCD . However, as argued above, in a multi-axion the-ory, we expect a large number of possible values for theCP violating phase. While this theory does not dynam-ically favor CP conservation, the band structure is suffi-cient to accommodate a small CP violating phase.
CONCLUSIONS
We studied the vacuum distribution of a theory con-taining multiple axions. In general, it is difficult to pre-cisely determine the number and location of vacua ina multi-dimensional potential. In order to simplify theproblem, we separated the leading and the sub-leadingcontributions to the axion potential: the leading termsdetermine the vacuum periodicity, while sub-dominantcontributions break this discrete shift symmetry and liftthe vacuum degeneracy. In the universal regime, thenumber of discrete vacua scales exponentially with thenumber of axions. The splitting of the degenerate energylevels is small compared to the typical scale of the poten-tial energy, giving rise to energy bands containing stablevacua. If one of the energy bands spans zero energy,there exist vacua with very small cosmological constant.In an eternally inflating universe, this vacuum is popu-lated which leads to a small observed vacuum energy.The simple observations made in this work may haveprofound implications. We demonstrated that a suffi-ciently complex landscape to accommodate the cosmo-logical constant and a small CP violating phase can berealized in a large class of four dimensional effective theo-ries. The energy density of domain walls that arise duringvacuum transitions can be low enough to decouple fromunknown, heavy degrees of freedom. The only require-ment for a large number of vacua is that the leading in-stanton charges do not form a primitive basis for the inte-ger lattice. An intriguing observation is that the simplestmulti-axion model that accommodates the smallness ofthe cosmological constant also gives rise to an extendedperiod of large-field inflation via kinetic alignment. Fi-nally, our work motivates a more detailed study of theaxion sector in flux compactifications and provides a newframework to pursue de Sitter vacua in string theory.
ACKNOWLEDGEMENTS
I would like to thank Raphael Bousso, Frederik Denef,Michael Douglas, Cody Long, Liam McAllister, Eve Vav-agiakis, Erick Weinberg, and Claire Zukowski for usefuldiscussions. This work was supported by DOE undergrant no. de-sc0011941. [1] S. Perlmutter et al. (Supernova Cosmology Project),Astrophys. J. , 565 (1999), arXiv:astro-ph/9812133[astro-ph].[2] A. G. Riess et al. (Supernova Search Team), Astron. J. , 1009 (1998), arXiv:astro-ph/9805201 [astro-ph].[3] P. A. R. Ade et al. (Planck), (2015), arXiv:1502.01589[astro-ph.CO].[4] S. Weinberg, Rev. Mod. Phys. , 1 (1989).[5] S. M. Carroll, Living Rev. Rel. , 1 (2001), arXiv:astro-ph/0004075 [astro-ph].[6] T. Banks, Phys. Rev. Lett. , 1461 (1984).[7] A. D. Linde, Rept. Prog. Phys. , 925 (1984).[8] J. D. Barrow and F. J. Tipler, The Anthropic Cosmolog-ical Principle (Oxford U. Pr., Oxford, 1988).[9] S. Weinberg, Phys. Rev. Lett. , 2607 (1987).[10] J. Polchinski (2006) arXiv:hep-th/0603249 [hep-th].[11] R. Bousso (2006) arXiv:hep-th/0610211 [hep-th].[12] R. Bousso, R. Harnik, G. D. Kribs, and G. Perez, Phys.Rev. D76 , 043513 (2007), arXiv:hep-th/0702115 [hep-th].[13] A. De Simone, A. H. Guth, M. P. Salem, and A. Vilenkin,Phys. Rev.
D78 , 063520 (2008), arXiv:0805.2173 [hep-th].[14] R. Bousso and I.-S. Yang, Phys. Rev.
D75 , 123520(2007), arXiv:hep-th/0703206 [hep-th]. [15] T. Clifton, S. Shenker, and N. Sivanandam, JHEP ,034 (2007), arXiv:0706.3201 [hep-th].[16] L. F. Abbott, Phys. Lett. B150 , 427 (1985).[17] J. D. Brown and C. Teitelboim, Phys. Lett.
B195 , 177(1987).[18] J. D. Brown and C. Teitelboim, Nucl. Phys.
B297 , 787(1988).[19] J. L. Feng, J. March-Russell, S. Sethi, and F. Wilczek,Nucl. Phys.
B602 , 307 (2001), arXiv:hep-th/0005276[hep-th].[20] T. Banks, M. Dine, and N. Seiberg, Phys. Lett.
B273 ,105 (1991), arXiv:hep-th/9109040 [hep-th].[21] T. Banks, M. Dine, and L. Motl, JHEP , 031 (2001),arXiv:hep-th/0007206 [hep-th].[22] M. R. Douglas and S. Kachru, Rev. Mod. Phys. , 733(2007), arXiv:hep-th/0610102 [hep-th].[23] R. Bousso, Gen. Rel. Grav. , 607 (2008),arXiv:0708.4231 [hep-th].[24] R. Kallosh, A. Linde, and B. Vercnocke, Phys. Rev. D90 ,041303 (2014), arXiv:1404.6244 [hep-th].[25] R. Bousso and J. Polchinski, JHEP , 006 (2000),arXiv:hep-th/0004134 [hep-th].[26] Here, the inverse only acts on the R -dimensional sub-space on which the lattice L ( B r ) is defined.[27] T. C. Bachlechner, C. Long, and L. McAllister, (2014),arXiv:1412.1093 [hep-th].[28] C. Long, L. McAllister, and P. McGuirk, JHEP , 187(2014), arXiv:1407.0709 [hep-th].[29] S. B. Giddings and A. Strominger, Nucl. Phys. B306 ,890 (1988).[30] N. Arkani-Hamed, J. Orgera, and J. Polchinski, JHEP , 018 (2007), arXiv:0705.2768 [hep-th].[31] P. M. Wood, Ann. Appl. Probab. , 1266 (2012).[32] N. R. Goodman, Ann. Math. Statist. , 178 (1963).[33] T. C. Bachlechner, C. Long, and L. McAllister, (2015),arXiv:1503.07853 [hep-th].[34] N. Arkani-Hamed, L. Motl, A. Nicolis, and C. Vafa,JHEP , 060 (2007), arXiv:hep-th/0601001 [hep-th].[35] C. Cheung and G. N. Remmen, Phys. Rev. Lett. ,051601 (2014), arXiv:1402.2287 [hep-ph].[36] T. Banks, M. Dine, P. J. Fox, and E. Gorbatov, JCAP , 001 (2003), arXiv:hep-th/0303252 [hep-th].[37] B. Heidenreich, M. Reece, and T. Rudelius, (2015),arXiv:1506.03447 [hep-th].[38] S. R. Coleman, Phys. Rev. D15 , 2929 (1977), [Erratum:Phys. Rev.D16,1248(1977)].[39] S. R. Coleman and F. De Luccia, Phys. Rev.
D21 , 3305(1980).[40] M. Kleban and M. Schillo, JCAP , 029 (2012),arXiv:1202.5037 [astro-ph.CO].[41] R. Bousso, D. Harlow, and L. Senatore, JCAP ,019 (2014), arXiv:1404.2278 [astro-ph.CO].[42] T. C. Bachlechner, M. Dias, J. Frazer, and L. McAllister,Phys. Rev.
D91 , 023520 (2015), arXiv:1404.7496 [hep-th].[43] R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. , 1440(1977).[44] T. Banks and M. Dine, Nucl. Phys. B505 , 445 (1997),arXiv:hep-th/9608197 [hep-th].[45] P. Svrcek and E. Witten, JHEP06