The effective field theory of axion monodromy

Albion Lawrence, Brandeis University

arxiv:1101.0026 with Nemanja Kaloper (UC Davis) and Lorenzo Sorbo (U Mass Amherst)

arxiv:1105.3740 with Sergei Dubovsky (NYU) and Matthew Roberts (NYU)

arxiv:1203.6656

arxiv:1404.2912 with NK

Work in progress with NK and Alexander Westphal

Thursday, August 28, 14

I. Large field inflation

��

mpl⇠

rr

0.01⇠

pV

(1.06⇥ 1016 GeV )2Turner; Lyth

“High scale”:

Why is this confusing?

V ⇠X

n

cn�n

Mn�4

M . mpl• UV scale• Slow roll inflation (canonical kinetic terms)

M = mpl : cn ⌧ 1 8n : c2 . 10�10; c4 . 10�12 . . .

V 1/4 ⇠ 1016 GeV ⇠ MGUT ) �� > mpl ⇠ 2⇥ 1018GeV

Thursday, August 28, 14

(1) loops of inflaton, graviton give suppressed couplings

Vloop = VclassF�

Vm4

pl, V �

m2pl

, . . .⇥

Coleman and Weinberg; Smolin; Linde

perturbative corrections preserve symmetries

Where is the danger?

� ! �+ a

(2) UV completion generically expected to break global symmetries

(absent some mechanism!)

Hawking radiation, wormholes, couplings to other sectors...

�L ⇠X

i

giAs.b.

i

m�i�4pl

; gi ⇠ O(1)

Thursday, August 28, 14

Natural/pseudonatural inflation

V = �4 cos�

�f�

⇥+ . . .

�V � �4�

n>1 cn cos(n⇥/f�)Problem:

Banks, Dine, Fox, and Gorbatov; Arkani-Hamed, Motl, Nicolis, and Vafa

9

Adams, Bond, Freese, Frieman, Olinto;Arkani-Hamed, Cheng, Randall

� ⌘ �+ 2⇡f� forbids direct corrections �V ⇠ �n

mn�4pl

Match to data: requires f� > mpl

cn ⇠ e�nS , S .✓mpl

f�

◆2

Dynamical, nonperturbative breaking

Thursday, August 28, 14

II. Axion monodromy inflation in 4 dimensionsKaloper and Sorbo; Kaloper, Lawrence, and Sorbo; Kaloper and Lawrence

Sclass =⇤

d4x⇥

g�m2

plR� 148F 2 � 1

2 (⇥�)2 + µ24��F

⇥

Compact U(1) gauge symmetry

F does not propagate.

Can jump across domain walls/membranes

Fµ⇤�⌅ = �[µ A⇤�⌅]

�Aµ⇤� = ⇥[µ � ⇤�]

' ⌘ '+ 2⇡f

Brown and Teitelboim

Combines chaotic, natural inflation

Effective field theory for original and recent string models, unwinding inflation

Marchesano, Shiu, and Uranga; Kaloper, Lawrence, and Westphal

But note UV completions still very important!MonodromyEva:

(Perhaps after some duality transformation)

Thursday, August 28, 14

Hamiltonian:

Compactness of gauge group: pA = ne2

Htree = 12p2

� + 12 (pA + µ⇥)2 + grav.

Consistency condition: µf� = e2

n jumps by membrane nucleation

Monodromy: theory invariant under

Absent transitions: equivalent to

+ observable GW

' ! '+ 2⇡f ;n ! n� 1

n=0

n=1 n=2

n=3

q

V(q

f 2f

V =1

2µ2'2

Fits data OK if: can still have f < mpl

Will assume this can be achieved by some natural mechanism: worry about corrections

e ⇠ MGUT )

µ ⇠ 10�5mpl

f ⇠ .1mp

Thursday, August 28, 14

Why does this work?

Basic points:

• still protects theory from direct corrections • Coupling of to F: governed by single small parameter' µ�F

Corrections:

(1) Instanton effects

�L = �f tr G⇤G

Couple to nonabelian gauge field ' G

Instanton corrections: �V ⇠ �4 cos 2�⇥f + . . .

Easy to make subleading: interesting signatures

V (�)

�

' ⌘ '+ 2⇡f

cf Eva’s talk

�V ⇠ �n

mn�4pl

Thursday, August 28, 14

(2) Corrections to 4-form energetics: �L = 1M4n�4F 2n

Effective potential: �V = (µ2�2)n

M4n�4

Corrections small if

stable if

(3) Coupling to moduli: V = V0(⇥) + 12µ2

��

mpl

⇥⇤2 + �4

⇤n cn

��

mpl

⇥cos

�n⇥f�

⇥

M� >p

Vinf

mp⇠ 1014 GeV

but see Dong, Horn, Silverstein, Westphal; McAllister, Silverstein, Westphal, and Wrase

M4 � Vinf ⇠⇠ (1016 GeV )4

and minimum of wrt within a Planck distance from minimum ofµ V0

Thursday, August 28, 14

Quantum stability

transition rate must be slow compared to time scale of inflation

(1) Jumps between branches

n ! n� 1

For “unwrapped” '

�� = f

For ruled out by observation

f > H ⇠ 1014 GeV

n=0

n=1 n=2

n=3

q

V(q

f 2f

Thursday, August 28, 14

Transitions occur by bubble nucleation. Let:

Constraints:

• T = tension of bubble wall• E = energy difference between branches

Decay probability:

Phenomenological bound on T: branch-changing bubbles

� = Nf� ; �� = f�

(thin wall) Coleman

�� 1⇤ T 1/3 ⇥�

227�2N3

⇥1/4V 1/4

f� � .1 mpl; N � 100;V �M4gutLet:

• Borderline; should check against explicit models• UV complete 2d models: T ~ M

T ⇥ (.2V 3)1/4 � (.9Mgut)3

N.B.: E larger for large V; transitions more likely early in inflation

AL

Extremely sensitive to energy scale

� ⇠ exp

✓�27⇡2

2

T 4

(�E)

3

◆

�E4 ⇠ V 0(')f' ⇠ V

N

Thursday, August 28, 14

L⇥�F = µ⇣

�mp

⌘�F

(2) Jumps of parameters

can have multiple local minima; transitions cause to jumpV (�)

Not restricted to monodromy inflation!

µ

Can occur during, after inflation

Interesting signatures?

14

�µ ⇠ 0.1µ ) Pjump < 1 if T 1/3 > .5MGUT

�E ⇠✓�µ

µV

◆1/4

) Jumps more likely at earlier times

Thursday, August 28, 14

Large-N gauge dynamics

Sclass =⇤

d4x⇤

g�m2

plR� 14g2

Y MtrG2 � 1

2 (⇥�)2 + �f�

trG ⇥G⇥

G: field strength for U(N) gauge theory with N large; strong coupling in IR

strong coupling scale of U(N) theory�

Witten; Giusti, Petrarca, and TaglientiInstanton expansion breaks down

H ⇠ Hgauge +12p

2� + 1

2

�n�2 + µ�

�2+ . . . for

monodromy potentialE(�) � N2E⇣

�Nf

⌘⇥

�/f ⌧ 1

µ = �2/f

Monodromy from strongly coupled gauge theory

trG ^G = F (4) Dvali

Thursday, August 28, 14

Example Witten; Dubovsky, Lawrence, and Roberts

Antiperiodic boundary conditions for fermions break SUSYN type IIA D4-branes wrapped on S1 with radius �

Massless sector: U(N) gauge theory

N ! 1,� = g24,Y MN fixed: theory has gravitational dual w/ monodromy Witten

0

0

�

EV3

Three Branches of Vacua

= 2�⇥f

EV3(x) = 2⇥N2

37⇤2�4

⇣1� 1

(1+x2)3

⌘x = �⇤

2⇥Nf

V ⇠ A� B�6 ) ns ⇠ 0.965 ; r ⇠ 8⇥ 10�4

Improved version of natural inflation

Match to CMBR

�� ⇠ 2mpl ; x > 1

Transitions between branches strongly suppressedsee also Dine, Draper, and Monteux

Thursday, August 28, 14

III. Corrections

“High scale” stringy models consistent with unification at

• motivates deeper study of “stringy” compactifications• Corrections due to UV physics at edge of being observable

Muv = (ms,m10,pl,m11,pl,mKK) =

✓1

few� few

◆⇥MGUT

MGUT

Heterotic models, M on G2, type II with branes,...

Mmoduli

. H =M2

GUT

mpl

without extra work

V ⇠ M4GUT

“Edge of respectability”

Kaloper and Lawrence

Thursday, August 28, 14

(1) Corrections to potential

V =1

2µ2'2 + c

�12µ

2'2�2

M4uv

+ . . .

=1

2µ2'2 ± �'4 + . . .

c ⇠ 1,Muv = 2.3 MGUT )⇢

r ⇠ .2�PSPS

⇠ ±.18Example

n, µ

Fluctuations in bubble walls

S = T

Zd3x(@X)2 ) �(

pTX) =

pH

T 1/3 ⇠ MGUT ) Fluctuations in bubble walls compete w/ inflaton fluctuations

(2) Nonperturbative jumps in

Fluctuations in bubble walls compete w/ inflaton fluctuations

Transition just before visible epoch hemispherical asymmetry)D’Amico, Gobetti, Kleban, and Schillo

Thursday, August 28, 14

IV. London equation for monodromy Kaloper, Lawrence,and Westphal

Dual of axion-4 form

• Integrate out H

L = � 1

48(F (4))2 � 1

2(@')2 � µ'⇤F

• Integrate out : ' H = dB(2)

A ! A+ d⇤

B ! B + ⇤

L = � 1

48(F (4))2 � 1

2µ2

✓A(3) � 1

µH(3)

◆2

+ '⇤dH

B can be gauge fixed to zero

L = � 1

48(F (4))2 � 1

2µ2A2

' dual to longitudinal mode of A

Should be renormalizable (cf massive QED)

Thursday, August 28, 14

Julia-Toulouse mechanism Julia and Toulouse; Quevedo and Trugenberger

Membranes electrically charged under A

L = � 1

48(F (4))2 � 1

2µ2A2

4-form coupled to membrane condensate?

• UV complete model (eg via string theory)?

D-brane condensates often dual to fundamental fieldsStrominger; Witten; ...

• Mechanism for small ?µ

Thursday, August 28, 14

2d analog: Schwinger model

Charged fermions = 2d domain walls

L = �1

4Fµ⌫F

µ⌫ + ̄�µ (@µ � iAµ)

Bosonization:

L = �1

4Fµ⌫F

µ⌫ � 1

2(@')2 � '✏µ⌫Fµ⌫

Lawrence; Seiberg

L = �1

4Fµ⌫F

µ⌫ � 1

2A2

Thursday, August 28, 14