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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