Axion monodromy. Albion Lawrence, Brandeis/NYU

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1 Axion monodromy Albion Lawrence, Brandeis/NYU arxiv: with Nemanja Kaloper (UC Davis), AL, and Lorenzo Sorbo (U Mass Amherst) Work in progress with Sergei Dubovsky (NYU), AL, and Matthew Roberts (NYU) and with Kaloper and AL 0/31

2 I. Introduction: high scale inflation in UVcomplete theories II. 4d models of axion monodromy III. Quantum corrections IV. Monodromy from strongly coupled QFT V. Conclusions 1

3 I. Introduction Scale of inflation Observational upper bound on GW: Close to unification scale V GeV M GUT α i = e 2 i /c α 3 α 2 See also: α 1 α grav p decay log 10 (E/GeV ) ν mass Couplings unify (assuming MSSM above 1 T ev ) at approximately GeV. Graph not to scale. If V near upper bound: detectable by PLANCK or ground-based CMB polarization experiments 3

4 Detectable primordial GWs require large inflaton range Single field slow-roll inflation with inflaton φ δρ ρ H2 φ 10 5 from observations Lyth, hep-ph/ N e = dth = dφ φ H = dφ H H 2 φ 60 upper bound on upper bound on V,H = V /m pl dφ dn, φ during inflation to match observed flatness ϕ m pl 4

5 Effective field theory and large φ Effective field theory: expansion in 1/M for some UV scale M V = n g n φn M n 4 generically g n 1 M m pl unless forbidden by symmetry Expansion breaks down for φ >M New degrees of freedom could become light Relevant d.o.f. very different 5

6 High scale inflation looks like a highly nongeneric theory Consider V m 2 φ 2 or V λφ 4 δρ/ρ 10 5,N e 60 m2 m 2 pl λ Corrections δv = n g n φn M n 4 all g n must be small: infinite fine tuning! else e.g. η = m 2 pl V V 1 Slow roll inflation requires approximate shift symmetry φ φ + a 6

7 Perturbative quantum corrections Small couplings m 2 m 2 pl, λ m pl -suppressed couplings to gravity loops of inflaton, graviton gives suppressed couplings V loop = V class F V m 4 pl, V m 2 pl,... Coleman and Weinberg; Smolin; Linde Slow roll inflation safe against inflaton, graviton loops perturbative corrections preserve symmetries 7

8 UV completions make slow roll difficult to maintain Continuous global symmetries like φ φ + a are always (we think) broken Gravity breaks continuous global symmetries (Hawking radiation/virtual black holes, wormholes,...) Holman et al; Kamionkowski and March-Russell; Barr and Seckel; Lusignoli and Roncadelli; Kallosh, Linde, and Susskind String theory: continuous global symmetries tend to be gauged, anomalous Anomalous symmetries broken by nonperturbative effects (e.g. Peccei-Quinn symmetry of axion) δv Λ 4 n c n cos(nφ/f φ ) 8

9 One attempt: pseudonatural inflation Use anomalous symmetry to generate potential V = Λ 4 cos φ f φ +... δv Λ 4 n>1 c n cos(nφ/f φ ) c n e ns, fφ M n Λ some dynamical scale; slow roll for f φ Λ large field if f φ m pl Problem: f φ >m pl with c n f φ M 1 small does not seem to be allowed Banks, Dine, Fox, and Gorbatov; Arkani-Hamed, Motl, Nicolis, and Vafa 9

10 Candidate solution: monodromy in field space Consider compact scalar field ϕ ϕ + f ; f m pl Silverstein and Westphal; McAllister, Silverstein, and Westphal Theory invariant under shift ϕ ϕ + f physical state need not be V (φ) n = 2 n =1 n = 1 n =0 f φ φ Let axion wind N times such that Nf φ m pl Compactness of field space seems to control quantum corrections 10

11 Cartoon Most models to date constructed within string theory But see Berg, Pajer, and Sors; Kaloper and Sorbo Illustrative example: type IIA with D4-brane wrapped on 2-torus τ has period =1 φ = m pl τ canonically normalized scalar τ C τ +1 = Shift τ τ + 1 is symmetry of torus, but stretches D-brane. V (φ) m4 s g s 1+(mpl φ) 2 n = # of D4 windings Shift τ n times; D-brane becomes n times as long. Doesn t quite work but illustrates point. Note potential flattens: V M 3 φ at large φ 11

12 Known string realizations seem to give flat potentials, with relatively small powers V M 4 p ϕ p<2 Seems to the result of coupling to moduli, KK modes Is a quadratic potential viable? Dong, Horn, Silverstein, and Westphal CMB data: p <=2 viable, smaller p more viable Quantum corrections studied model by model: these are complicated, and physical reason for flat potentials is not completely transparent. 14

13 Effective field theory approach Input basic fields, symmetries, topology of field space Expand action in powers of 1/M (M = UV scale), include all terms consistent with symmetries Pinpoints physics behind suppressing corrections to slow roll Isolates fine tuning required. Provides a framework for building new string models String theory has a complicated landscape Realistic models very hard to construct Quantum corrections difficult to compute 4d effective field theory analysis is always important 15

14 II. 4d models of axion monodromy Axion-four form model S class = d 4 x g Kaloper and Sorbo m 2 pl R 1 48 F ( ϕ)2 + µ 24 ϕ F F µνλρ = [µ A νλρ] U(1) gauge symmetry: δa µνλ = [µ Λ νλ] ϕ periodic: ϕ ϕ + f ϕ F does not propagate. U(1) quantized F µνλρ = ne 2 µνλρ ; n Z n can jump across domain walls/membranes 16

15 Dynamics Single massive scalar degree of freedom Dvali; Kaloper and Sorbo Hamiltonian: H tree = 1 2 p2 φ (p A + µφ) 2 + grav. Compact U(1): p A = ne 2 p A conserved by H tree Jumps by membrane nucleation Consistency condition: µf ϕ = e 2 V (φ) n = 2 n =1 Realizes monodromy inflation: theory invariant if ϕ ϕ + f ϕ ; n n 1 n = 1 n =0 f φ φ Good model for inflation: fits data well if + observable GW µ 10 6 m pl 17

16 Large-N gauge dynamics S class = d 4 x g m 2 pl R 1 4g 2 YM trg ( ϕ)2 + ϕ f ϕ trg G G: field strength for U(N) gauge theory with N large; strong coupling in IR Instanton expansion breaks down Witten; Giusti, Petrarca, and Taglienti nλ 2 + µϕ 2 H tree = H gauge p2 ϕ Λ strong coupling scale of U(N) theory µ = Λ 2 /f ϕ Can be related to 4-form version: F µνλρ tr G [µν G λρ] Dvali 18

17 III. Quantum corrections S class = d 4 x g m 2 pl R 1 48 F ( ϕ)2 + µ 24 ϕ F µ 10 6 m pl to match constraints on δρ/ρ, N e What are the possible corrections? Effective field theory: Allow all terms consistent with symmetries, topology of field space Dimenson-d operators suppressed by M d 4 uv Corrections controlled by: Compactness of scalar, U(1) Small coupling µ/m uv 1 Stability: Quantum jumps between branches mediated by membrane nucleation 19

18 Direct corrections to V (ϕ) Periodicity of ϕ quantum corrections to S must be Functions of n ϕ periodic functions of ϕ δv Λ 4 n>1 c n cos(nϕ/f ϕ ) V (ϕ) f φ m pl Monodromy potential modulated by periodic effects ϕ V corr 1 2 µ2 ϕ 2 Λ 4 M 4 gut η = m 2 pl V V 1 Λ4 f 2 ϕ V m 2 pl = H 2 Example: feasible if Λ.1 M gut,f>.01 m pl 20

19 Gauge dynamics: Λ =Λ QCD from couplings ϕ f ϕ tr G G instanton corrections take above form (if dilute gas approx good) strong coupling effects (when dilute gas aprox fails) δv Λ 4 min k F ϕ + k f ϕ multibranched function of ϕ When using this effect to generate monodromy potential: mixing between branches must be weak Witten; Giusti, Petrarca, and Taglienti! "#$#& "#$#!' When this generates corrections: mixing must be strong (else trapped in a fixed branch) "#$#!& "#$#% (! Gravitational dynamics: Λ 4 f n+4 ϕ m n pl gravitational instantons, wormholes, etc. 21

20 Caveat: moduli stabilization In any string theory: couplings in V will depend on moduli ψ V = V 0 (ψ)+ 1 2 µ2 ψ m pl ϕ 2 + Λ 4 n c n ψ m pl cos nϕ f ϕ Periodic corrections change sign many times since f φ m pl Moduli must be stabilized by different effects than instantons coupling to inflaton M 2 ψ V 0 (ψ) Λ4 m 2 pl Large ϕ m pl sources potential for ψ Stability requires M 2 ψ µ2 ϕ 2 /m 2 pl µ2 / H 2 22

21 Indirect corrections to V (ϕ) Additional corrections must respect periodicity of ϕ corrections to dynamics of four-form F S class = d 4 x g m 2 pl R 1 48 F ( ϕ)2 + µ 24 ϕ F Consider δl = n d n F 2n M 4n 4 Integrate out F: F µϕ +... δv eff = V class n=1 d n+1 V n class M 4n Safe if: M 4 V class Mgut 4 Corrections of the form δl = n=1 d n+1 F 2n M 4n ( ϕ) 2 Gives same effect after redefining ϕ to be canonically normalized 23

22 Small M not always fatal Many string theory scenarios: V (ϕ) =M ϕ2 M 2 2 M 2 m pl Silverstein and Westphal; McAllister, Silverstein, and Westphal For small ϕ V 1 2 µ2 ϕ 2 ; µ = M 4 1 M 2 2 For ϕ m pl V m 3 ϕ; m 3 = M 4 1 M 2 Out of range of 4d effective field theory; requires understanding of UV completion (eg 10d SUGRA) to compute 26

23 Example: backreaction on compactification Consider string modulus ψ determines KK scale: L 0 e ψ/m pl ; V D L D ; m 2 pl = md+2 V D L ψ = 1 2 ( ψ)2 1 2 M 2 ψ ψ2 + c ψ m pl F Integrate out ψ : ψ F m pl = c 2 Mψ 2 m2 pl c V m 2 pl M 2 ψ = c H2 M 2 ψ δm 2 pl m 2 pl H2 M 2 ψ Since η = m 2 pl V V ; = m2 pl (V ) 2 V 2 We must have δm 2 pl m 2 pl H2 M 2 ψ 1 Moduli coupling to inflaton must be fairly heavy If coupling to F is: (ψ ψ 0) 2 m 2 pl F 2 corrections proportional to ψ 0 m pl Dong, Horn, Silverstein, and Westphal ψ 0 m pl 1 also edge of validity of effective field theory 25

24 Example: Coleman-Weinberg corrections Consider scalar fields ψ n (e.g. moduli, KK states, etc.) δl 1 2 ( ψ n) M 2 nψ 2 n k d n,k F 2n M 4n 2 ψ 2 n Integrate out F: F 2 V class = 1 2 µ2 ϕ 2 Effective mass for ψ : M 2 eff = M 2 ψ + M 2 k d n,k V 2 Integrate out ψ n : δv CW (ϕ) M eff (φ) 4 ln M eff M Must include all such states with M 2 n <M 2 M 4n Corrections safe if n eff M 2 ψ M 2 ; V M 4 26

25 Kaluza-Klein corrections Roughly n eff = m2 pl m 2 ; m =(m s,m pl,10 ) M gut V CW = KK V D d 4 q ln q 2 + Mn,eff 2 d D+4 q ln q 2 + k m D+4 V D (ψ)+m 2 V D δv (ψ)+v tree F k Vtree M 4 d k d k Vtree k M 4k 4 m 2 pl V k tree M 4k 4 m 2 pl Corrections safe if V class M 4 NB if KK mode couples to F as (ψ n ψ 0,n ) 2 m 2 pl F 2 tree level corrections subleading if H 2 <M 2 KK ; ψ n,0 <m pl,10 27

26 Additional stringy light states W p Shift τ n times; D-brane becomes n times as long. Consider square torus with sides of length L; D4 wrapped n times m 2 W = m4 s L2 1+n 2 ; m 2 p = 1 L(1+n 2 ) ; n = ϕ f ϕ = F µf ϕ n >> 1: strings have spectrum of asymmetric torus with sides of length L W = n m 2 s L ; L p n L and volume V eff n2 m 2 s F 2 m 2 s e4 where e 2 = µf ϕ is unit of quantization of F flux 28

27 Leading quantum correction V CW = k,l d 4 q ln q 2 + m 2 W,k + m 2 p,k +... F 2 m 2 se 4 d 6 q lnq m4 s e 4 F Effect is to renormalize e 2 m 2 s M 2 gut 10 4 m 2 pl Dangerous: µ = 10 6 m pl to match observation f ϕ 10 2 m pl Must ensure renormalization of e is suppressed: f ϕ.1 m pl e 2 (.1M gut ) 2 29

28 NB model above is crude (and known not to work for other reasons) so this is a caveat and not a fatal flaw Even if µ 2 pushed above 10 6 m pl we may still get successful large field inflation of the form, e.g. V (ϕ) =M ϕ2 M 2 2 but this requires more than our 4d EFT can do at present 30

29 Quantum stability V (φ) n = 2 n =1 n = 1 n =0 f φ φ Success of monodromy inflation requires that transition between branches is slow compared to time scale of inflation (must complete 60 efolds before such transitions) 31

30 Bounds on membrane tension Transitions occur by bubble nucleation. Let: T = tension of bubble wall E = energy difference between branches Decay probability: Γ exp 27π2 2 T 4 E 3 (thin wall) Coleman Phenomenological bound on T ϕ = Nf ϕ ; ϕ = f ϕ E V V (ϕ)f ϕ V N N.B. E larger for large V; transitions more likely early in inflation Γ 1 T 1/3 2 27π 2 N 3 1/4 V 1/4 Let: f φ.1 m pl ; N 100; V M 4 gut T (.2V 3 ) 1/4 (.9M gut ) 3 Borderline; should check against explicit models 32

31 IV. Monodromy from strongly coupled QFT We wish to study monodromy in a setting where we have control over nonperturbative physics Understand flattening of potential. Understand stability of metastable branches. Look for strongly coupled gauge theory with gravitational dual

32 A nonsupersymmetric QFT N type IIA D4-branes wrapped on S 1 with radius β Antiperiodic boundary conditions for fermions break SUSY Bosons get mass from loops Massless sector: U(N) gauge theory g 2 5,Y M =4π2 α g s g 2 4,Y M = g2 5/2πβ θ angle from D-brane coupling to RR 1-form potential S WZ = S 1 R 4 C (1) TrF F For constant RR field polarized along S 1 θ = 2πC ββ α (Wilson line)

33 Decoupling limit and gravitational dual α 0,g s such that g 2 5,Y M,g2 4,Y M held fixed N, λ = g 2 4,Y M N fixed massless open strings decouple from closed strings, oscillator modes at low energies throat is locally u 0 u = radial direction R 4 S 1 R u S 4 dual to QFT energy scale Dual gravity solution for small found by Witten (1998) θ N/λ = g 2 4,Y M

34 Phases of theory (1) Throat is infinite -- no mass gap. Deconfined phase. Vacuum energy independent of θ (1I) Throat ends at u = u 0 Mass gap at Λ QCD u 0 /λ (u 0 λ/β for small θ) E(θ) λn 2 V x = λθ 4π 2 N Witten; DLR Less useful for studying 4d confinement (at small x) This always has lower energy Energy dependence implies monodromy potential for Think of θ as nondynamical axion θ = φ/f φ θ

35 Three Branches of Vacua 0 E V 3 0 Θ

36 Large-x behavior S 1 u= dχc (1) χ = dudχf uχ = θ +2πn For x λn 2πN 1 must take backreaction of 2-form flux into account Λ QCD u 0 λ 1 β(1+x 2 ) E V 3 x = Throat recedes into IR, glueballs become 4d objects λθ 4π 2 N = 2λN π 2 β 1 4 (1+x 2 ) 3 Potential flattens (response of E to θ depends on Λ QCD ) x 2λN π 2 β x 6

37 Stability at large x R u S 1 becomes long, thin cylinder u Winding modes about χ when x = λθ 4π 2 N λ1/3 Casimir forces dominate over RR 2-form flux when x 7 Nλ 1/2 Result in both cases is to pinch off cylinder for u>u 0 (x) But we already know a solution; branch with lower energy. Conjecture: a given branch with x = 0 at minimum ceases to exist at large x

38 Nonperturbative instabilities D6-brane is a source for RR 2-form charge. Two candidate domain wall solutions D6-brane wrapping S 4 sitting at u = u 0 Witten D6-brane wrapping S 3 (ϕ) S 4 S 3 (ϕ) filling R 4 ϕ appears as QFT mode ϕ S 4 θ =2π(n 1) + δ analogous to Kachru, Pearson, Verlinde Domain wall when ϕ varies in space θ =2πn + δ Nucleation of second domain wall has lower action at large x

39 E ϕ =0 ϕ = π Height of barrier E λ2 N β 4 x 11 at large x Scaling applied to DBI action of D6 S λ2 N x 11 metastable branch beginning at x = 0 should end when x 11 λ 2 N

40 V. Conclusions Check stability in explicit string models Interesting observational signals if a single branch-changing or mass-changing bubble nucleates early within our horizon? Kaloper and AL, in progress General issue: monodromy inflation does not seem parametrically safe. Should we worry? Perhaps this is interesting: Implies number of e-foldings could be close to lower bound Implications for measurements of curvature, pre-inflation transients Other interesting applications of axion monodromy Kerr black holes; axion condensation via Penrose process. Instability/ disappearance of branch can lead to observable axion decays Dubovsky and Gorbenko 33