The effective field theory of axion monodromy

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1 The effective field theory of axion monodromy Albion Lawrence, Brandeis University arxiv: with Nemanja Kaloper (UC Davis) and Lorenzo Sorbo (U Mass Amherst) arxiv: with Sergei Dubovsky (NYU) and Matthew Roberts (NYU) arxiv: arxiv: with NK Work in progress with NK and Alexander Westphal

2 I. Large field inflation m pl r r 0.01 p V ( GeV ) 2 Turner; Lyth High scale : V 1/ GeV M GUT ) >m pl GeV Why is this confusing? V X n n c n M n 4 UV scale M. m pl Slow roll inflation (canonical kinetic terms) M = m pl : c n 1 8n : c ; c

3 Where is the danger? (1) loops of inflaton, graviton give suppressed couplings V V loop = V class F, V,... m 4 pl m 2 pl perturbative corrections preserve symmetries! + a Coleman and Weinberg; Smolin; Linde (2) UV completion generically expected to break global symmetries Hawking radiation, wormholes, couplings to other sectors... (absent some mechanism!) L X i g i A s.b. i m i 4 pl ; g i O(1)

4 Natural/pseudonatural inflation +2 f forbids direct corrections V V = 4 cos f +... Match to data: requires f >m pl Adams, Bond, Freese, Frieman, Olinto; Arkani-Hamed, Cheng, Randall n m n 4 pl Dynamical, nonperturbative breaking Problem: V 4 n>1 c n cos(n /f ) c n e ns, S. 2 mpl f Banks, Dine, Fox, and Gorbatov; Arkani-Hamed, Motl, Nicolis, and Vafa 9

5 II. Axion monodromy inflation in 4 dimensions Kaloper and Sorbo; Kaloper, Lawrence, and Sorbo; Kaloper and Lawrence Combines chaotic, natural inflation S class = d 4 x g m 2 pl R 1 48 F ( )2 + µ 24 F F µ = [µ A ] A µ = [µ ] Compact U(1) gauge symmetry ' ' +2 f F does not propagate. Can jump across domain walls/membranes Brown and Teitelboim Effective field theory for original and recent string models, unwinding inflation (Perhaps after some duality transformation) Marchesano, Shiu, and Uranga; Kaloper, Lawrence, and Westphal Eva: Monodromy But note UV completions still very important!

6 Hamiltonian: H tree = 1 2 p (p A + µ ) 2 + grav. Compactness of gauge group: p A = ne 2 n jumps by membrane nucleation Consistency condition: µf = e 2 Monodromy: theory invariant under '! ' +2 f ; n! n 1 V( n=1 n=2 Absent transitions: equivalent to n=0 n=3 V = 1 2 µ2 ' 2 f 2f Fits data OK if: µ 10 5 m pl can still have f <m pl e M GUT ) f.1m p + observable GW Will assume this can be achieved by some natural mechanism: worry about corrections

7 Why does this work? Basic points: ' ' +2 f still protects theory from direct corrections Coupling of ' to F: µ F governed by single small parameter Corrections: (1) Instanton effects Couple ' to nonabelian gauge field L = f tr G G Instanton corrections: V 4 cos 2 f +... Easy to make subleading: interesting signatures G V ( ) cf Eva s talk V n m n 4 pl

8 (2) Corrections to 4-form energetics: L = 1 M 4n 4 F 2n Effective potential: V = (µ2 2 ) n M 4n 4 Corrections small if M 4 V inf (10 16 GeV ) 4 (3) Coupling to moduli: V = V 0 ( )+ 1 2 µ2 m pl n c n mpl cos n f stable if M > p Vinf m p GeV and minimum of µ wrt within a Planck distance from minimum of V 0 but see Dong, Horn, Silverstein, Westphal; McAllister, Silverstein, Westphal, and Wrase

9 Quantum stability V( n=1 n=2 (1) Jumps between branches n! n 1 n=0 f 2f n=3 For unwrapped ' = f For f>h GeV ruled out by observation transition rate must be slow compared to time scale of inflation

10 Constraints: Transitions occur by bubble nucleation. Let: T = tension of bubble wall E = energy difference between branches 27 2 T Decay probability: 4 exp (thin wall) 2 ( E) 3 Phenomenological bound on T: branch-changing bubbles = Nf ; = f E 4 V 0 (')f ' V N Extremely sensitive to energy scale Coleman N.B.: E larger for large V; transitions more likely early in inflation 1 T 1/ N 3 1/4 V 1/4 Let: f.1 m pl ; N 100; V M 4 gut T (.2V 3 ) 1/4 (.9M gut ) 3 Borderline; should check against explicit models UV complete 2d models: T ~ M AL

11 (2) Jumps of parameters L F = µ mp F V ( ) can have multiple local minima; transitions cause µ to jump Not restricted to monodromy inflation! Can occur during, after inflation µ 0.1µ ) P jump < 1ifT 1/3 >.5M GUT E µ µ V 1/4 ) Jumps more likely at earlier times Interesting signatures? 14

12 Monodromy from strongly coupled gauge theory Large-N gauge dynamics S class = d 4 x g m 2 pl R 1 4g 2 YM trg ( )2 + f 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 H H gauge p n 2 + µ E( ) N 2 E strong coupling scale of U(N) theory µ = 2 /f Nf monodromy potential for /f 1 trg ^ G = F (4) Dvali

13 Example Witten; Dubovsky, Lawrence, and Roberts N type IIA D4-branes wrapped on S 1 with radius Antiperiodic boundary conditions for fermions break SUSY Massless sector: U(N) gauge theory N!1, = g 2 4,Y M N fixed: theory has gravitational dual w/ monodromy Witten E V 3 0 Three Branches of Vacua Improved version of natural inflation x = 2 Nf E V 3 (x) = 2 N (1+x 2 ) 3 Match to CMBR V A B 6 ) n s ; r = 2 f Transitions between branches strongly suppressed 2m pl ; x>1 see also Dine, Draper, and Monteux

14 III. Corrections Kaloper and Lawrence High scale stringy models consistent with unification at M GUT Heterotic models, M on G2, type II with branes,... M uv =(m s,m 10,pl,m 11,pl,m KK )= M moduli. H = M 2 GUT m pl 1 few without extra work few M GUT V MGUT 4 motivates deeper study of stringy compactifications Corrections due to UV physics at edge of being observable Edge of respectability

15 (1) Corrections to potential V = 1 2 µ2 ' 2 + c 1 2 µ2 ' 2 2 Muv = 1 2 µ2 ' 2 ± ' Example c 1,M uv =2.3 M GUT ) r.2 P S P S ±.18 (2) Nonperturbative jumps in n, µ Fluctuations in bubble walls Z S = T d 3 x(@x) 2 ) ( p TX)= p H T 1/3 M GUT ) Fluctuations in bubble walls compete w/ inflaton fluctuations Fluctuations in bubble walls compete w/ inflaton fluctuations Transition just before visible epoch ) hemispherical asymmetry D Amico, Gobetti, Kleban, and Schillo

16 IV. London equation for monodromy Kaloper, Lawrence, and Westphal Dual of axion-4 form L = 1 48 (F (4) ) µ2 2 A (3) 1 µ H(3) + ' dh Integrate out H L = 1 48 (F (4) ) (@')2 µ' F Integrate out ' : H = db (2) A! A + d B! B + B can be gauge fixed to zero L = 1 48 (F (4) ) µ2 A 2 ' dual to longitudinal mode of A Should be renormalizable (cf massive QED)

17 Julia-Toulouse mechanism Julia and Toulouse; Quevedo and Trugenberger Membranes electrically charged under A L = 1 48 (F (4) ) µ2 A 2 4-form coupled to membrane condensate? UV complete model (eg via string theory)? D-brane condensates often dual to fundamental fields Mechanism for small µ? Strominger; Witten;...

18 2d analog: Schwinger model Lawrence; Seiberg Charged fermions = 2d domain walls L = 1 4 F µ F µ + µ (@ µ ia µ ) Bosonization: L = 1 4 F µ F µ 1 2 (@')2 ' µ F µ L = 1 4 F µ F µ 1 2 A2