Lisa Randall Preskill 60 th March 16 2013

1. Dual EFTs Decoupling of High Dim Ops from Low Energy Sector Lisa Randall Preskill 60th March 16 2013

2. Introduction • Honored to be here • Great admiration for John • But finding suitable talk topic challenging • John’s expertise: • QCD, Solitons, Black Holes, Quantum Computing,… • My topics: Flavor, Precision EW, Baryogenesis, Dark Matter, GUTs, LHC tools, Supersymmetry, Warped Extra Dimension,… • Clearly there is a problem • (for me)

3. @preskill John Preskill ‏@preskill Higgs boson would enter 48-year ultra-marathon. Starts in hinterland in '64, ends in the stadium in '12 before adoring throngs. #higgsgold

4. @preskill John Preskill ‏@preskill Ran out of shaving cream last weekend, but used the dribble I could shake out of the can for eight days. #hanukkahmiracle#gillettenightmare

5. @preskill Lisa Randall ‏@lirarandall "Higgs Discovery" in itunes store too. Should be for all ereaders. If you like it tweet! Im curious if this new format works. 25 JulJohn Preskill ‏@preskill @lirarandalliBooks version looks fine; Kindle edition produced download error on my iPad. Bought both, so send me a check.

6. @preskill 25 JulLisa Randall ‏@lirarandall @preskill Thanks for clarifying. Hope you enjoy but u won't learn much for either price--except about my vacation! Expand 25 JulJohn Preskill ‏@preskill @lirarandall Thanks, I was looking for good vacation ideas. I'm off to Greece ...

7. @preskill John Preskill ‏@preskill Heilemann's wrong --- we introverts really do like people. Honest, we do. Is that why we tweet? http://andrewsullivan.thedailybeast.com/2012/09/electing-an-introvert.html …

8. Back to problem at hand • Complementarity • John’s work and mine only have finite overlap • Axions • Neutrino flavor symmetry: who knew?

9. Duality to Rescue Warped extra dimension (Poincare patch of d+1-dim AdS) related: To conformal theory on boundary (related to d-dim QCD) Field theory point of common interest too So I’ll talk about duality of effective field theories: consistency check for more recent applications (w/Fitzpatrick, Kaplan, Katz)

10. Effective Field Theory • Much of current modern physics understanding based on the notion of an effective field theory • Universal predictions about long-distances • Decouple short-distance details • AdS/CFT and warped 5d AdS (RS) provide a different context • AdS/CMT, AdS/QCD leads to EFT in new regime

11. AdS/QCD Ehrlich, Katz, Son, Stephanov • AdS/CFT thought of as a way to study QCD • However, formulation starts with strong coupling in UV and no asymptotic freedom • Nonetheless good laboratory for confinement and chiral symmetry breaking • And can try to deform SQFT to give QCD • Alternative: start with QCD and construct dual AdS • Choose field content to holographically reproduce chiral symmetry breaking • Neglect running of coupling, stringy physics, higher spin • Don’t include infinitely many operators (4d), fields (5d) • Just include ones corresponding to currents and order parameter

12. QCD Nonperturbative nature of QCD truly problematic And interesting Exact solutions (2d QCD) sometimes give insights Solitonic solutions (monopoles) give insights So does AdS/CFT in some cases But necessary requirement is validity of EFT

13. More generally Many physical contexts where one wants to include only finite number of bulk fields Finite number of CFT operators Duality in this case requires cutoff not just on energy, but on dimension of operators in CFT

14. Duality and CFT • Duality connects mass to dimension in CFT • Does this different notion of effective field theory apply? • Answer: yes but decoupling works differently • and can be even more efficient

15. Goal of Work Show how decoupling occurs in this different context Requires conventional decoupling And decoupling of higher dimenision operators Sometimes latter occur exponentially Justifying use of EFT even when only a small gap

16. Relevant geometry AdS space with boundary branes

17. RS: Warped extra-dimensional geometry • AdS in bulk • Broken CFT on brane • “Hard wall” • Gives boundary conditions • And characterizes IR breaking of CFT

18. Holographic Interpretation • From 5d vantage point, AdS with boundary branes • From 4d vantage point, CFT with UV and IR cutoffs • UV cutoff provides normalizable graviton • IR cutoff breaks CFT • Weak scale “mesons”, “baryons”: CFT bound states • Here we are only interested in IR symmetry breaking—no UV brane

19. Will demonstrate exponential decouping in some cases: Consider high and low dim ops Will find Broken CFT: Two point function of ops with different dimensions doesn’t generally vanish

20. Physics Applications Physics applications often based in low-energy effective theories When the 4d theory is dual to AdS on the Poincare patch, duality not entirely obvious

21. Hard Wall vs Soft Wall • Original RS model had a hard wall • Boundary conditions on a brane • Follow-up theories based on soft walls • Solutions to the equations of motion where fields get localized away from IR • We will explore what types of soft walls are possible • And show how decoupling occurs in both hard and soft wall cases

22. Essential for many of these theories is a valid EFT Generally, EFT follows from integrating out heavy/high momentum states In this case duality with the higher-dimensional theory means that higher-dimensional operators should decouple from the light states We will find first notion applies with hard walls, second with soft walls

23. Set-up Poincare patch of AdS AdS slice ends at brane in IR zIR Dual vantage point, brane is source of conformal symmetry breaking Most DOF are bulk fields in AdS that correspond to CFT ops filling out irreducible reps of conformal group in absence of CF symmetry breaking Have bulk and brane action

24. Hard Wall • We’ll first consider original hard wall scenario • Abrupt end at zIR corresponds to hard-wall model d+1-dim RS compatible with d-dim EFT of low-mass “mesons” and “glueballs”

25. Hard Wall • Many familiar features • Bound states (composites) • Discrete spectrum • Chiral symmetry (or tuning) necessary for massless modes • Here standard decoupling will apply • Large mass Md+1 states dual to high-d CFT ops • In this case reate states with d-dim mass • md~Md+1

26. Scalars with hard wall boundary m0~ Δ: conventional decoupling Can see from eq of motion—looks like a potential: pushes wf against zIR with “energy”

27. Scalar Hard Wall Decoupling In some sense trivial No light states without tuning Low-lying states have masses proportional to dimension of bulk operator Decoupling conventional

28. Fermion Hard Wall Decouping • Similar, but can be more interesting • You can have fewer light states in 4d theory than light states in bulk • Due to chiral symmetries • But you can still see decoupling of heavy states when number of 4d zero modes less than number of light modes, as you take light modes heavy

29. Decoupling of heavier bulk modes • Can have zero modes even for heavy states in presence of chiral symmetry • Consider when number of zero modes less than number of light modes • see modes decouple from light modes as you increase their mass • Effectively overlap shrinks with mass

30. Fermions Lift bulk masses of fermions that originally had large overlap with massless 4d modes Conformal sym breaking mixes bulk fields, so generally massless mode has overlap with all light bulk fermions

31. Decoupling heavy bulk fermions from zero modes Do simple example with only two bulk modes

32. Zero modes No mixing: massless zero mode sits in state with mass M2 However when nonzero mixing, zero mode will sit in state with mass M1 as M2 taken to infinity

33. Soft walls • This feature of wavefunction overlap being relevant persists in soft wall scenarios • Works differently though • States localized at different locations in the bulk • But before proceeding, interesting to study possible nature of soft walls

34. Now consider soft walls • Assume a dilaton field w/ profile For example, running coupling corresonds to bulk field with space-dependent profile that breaks AdS isometries to Poincare group in IR In any case often present in top-down or bottom-up models Bulk fields X dual to CFT ops O Standard EFT Lagrangian, expansion in cutoff Find C(z) and Φ(Z) satisfying null energy condition Asympotically UV AdS IR breaking at single scale But soft wall—space doesn’t suddenly end

35. Scalar WF in Soft-Wall Models and Null Energy Condition

36. Scalar “Potential” Solve Schrodinger eigensystem to get masses and bulk wavefunctions Dynamics controlled by balance between AdS term M2/(kz)2 in small z region and remaining soft wall potential

37. Constraints on metric Non-increasing Potential function of metric, dilaton But constraints on metric from NEC Einstein tensor and constraint:

38. Either a new CFT or a Wall That Ends Space To avoid large (definite) curvature, F can’t decrease below some definite value Since F non-increasing, must asymptote to greatest lower bound The only such solution where F=-C2/C’ is C(z)->Finfty/z Asymptotically AdS in IR as well as UV Otherwise “hard wall” Or “effective soft wall”

39. Effective Soft Walls • Curvature singularities imply bulk spacetime ends • Regions beyond boundary encapsulated in boundary conditions • Extra compact dimensions, brane,… • But can have “effective soft-wall” models • If potential rises sharply enough at singularity, lowl-ying modes would have to tunnel through barrier so act as if in soft wall • Effective soft wall requires small UV curvature • Gzz has to grow before reaching singularity • Schrodinger potential then can become large at singularity and push light KK modes away

40. Example Potential blows up power law smaller than curvature So small k compared to cutoff allows for light modes to be pushed away from curvature singularity

41. Example: Solvable Linear Spectrum Squared masses are linear in n and dimension Δ Follows from particular metric (AdS) and dilaton profile This illustrates competing terms in potential Leading to localization at larger z for larger M

42. Soft Wall Decoupling Δ-dependence from AdS part of metric When V(z) does not depend strongly on Δ find Δ-dependent zΔ If V(z) had strong Δ-dependence, would have traditional decoupling Argument for exponential decoupling Focus on bulk zero modes V(z) with 0(1) coeffcients; mesons with Δ~1 are localized near z*~1 Approximate as Δ2

43. Suppressed WF Overlaps Approximate wavefunctions as harmonics Then wf overlap between state corresponding to heavy mode and one corresponding to light mode is Zero mode wavefunctions centered at

44. Max Overlap Soft wall: Exponent grows with Hard wall: Mass growing generic consequence of: if

45. Linear Potentials Exact result: Find Gives wf overlap Correct exponential suppression

46. Decoupling much more interesting Soft wall decoupling can be exponential Due to small wavefunction overlap Justifies keeping only low-lying states and operators

47. Aside: Comment on Meson Sizes Transverse scalar charge distribution Note that overlap not small due to conventional association of position and size Look at form factor: eg scalar probe

48. Sizes • Already known with this measure, mesons in hard-wall models have same size • Soft-wall models with linear spectrum • We know <z2> grows with dimension, n • Not true however for size

49. Sizes • Mesons grow only logarithmically w/ • Seems robust result • So mesons—even separated in bulk—have similar sizes • Not responsible for decoupling • Also, though soft-wall bulk EFTs can produce Regge spectrum • Doesn’t give confining string picture for growth of states

50. Discussion • CFTs can have two distinct EFT descriptions • “D-dim” EFT standard Wilsonian • Integrate out momentum shells along RG flow • Second notion from AdS/CFT duality • Integrate out states based on scaling dimension or conformal Casimir • Scaling dims dual to bulk masses, std EFT on d+1-dim side