Lisa Randall

Preskill 60th March 16 2013

IntroductionHonored to be hereGreat admiration for JohnBut finding suitable talk topic challengingJohns 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)

@preskill

@preskill

@preskill

@preskill

@preskill

Back to problem at handComplementarityJohns work and mine only have finite overlapAxionsNeutrino flavor symmetry: who knew?

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 tooSo Ill talk about duality of effective field theories: consistency check for more recent applications (w/Fitzpatrick, Kaplan, Katz)

Effective Field TheoryMuch of current modern physics understanding based on the notion of an effective field theoryUniversal predictions about long-distancesDecouple short-distance detailsAdS/CFT and warped 5d AdS (RS) provide a different context AdS/CMT, AdS/QCD leads to EFT in new regime

AdS/QCD Ehrlich, Katz, Son, StephanovAdS/CFT thought of as a way to study QCDHowever, formulation starts with strong coupling in UV and no asymptotic freedomNonetheless good laboratory for confinement and chiral symmetry breakingAnd can try to deform SQFT to give QCD Alternative: start with QCD and construct dual AdSChoose field content to holographically reproduce chiral symmetry breakingNeglect running of coupling, stringy physics, higher spinDont include infinitely many operators (4d), fields (5d)Just include ones corresponding to currents and order parameter

QCD Nonperturbative nature of QCD truly problematicAnd interestingExact solutions (2d QCD) sometimes give insightsSolitonic solutions (monopoles) give insightsSo does AdS/CFT in some casesBut necessary requirement is validity of EFT

More generallyMany physical contexts where one wants to include only finite number of bulk fieldsFinite number of CFT operators Duality in this case requires cutoff not just on energy, but on dimension of operators in CFT

Duality and CFTDuality connects mass to dimension in CFTDoes this different notion of effective field theory apply?Answer: yes but decoupling works differentlyand can be even more efficient

Goal of WorkShow how decoupling occurs in this different contextRequires conventional decouplingAnd decoupling of higher dimenision operatorsSometimes latter occur exponentiallyJustifying use of EFT even when only a small gap

Relevant geometryAdS space with boundary branes

AdS in bulkBroken CFT on braneHard wallGives boundary conditions And characterizes IR breaking of CFTRS: Warped extra-dimensional geometry

Holographic InterpretationFrom 5d vantage point, AdS with boundary branesFrom 4d vantage point, CFT with UV and IR cutoffsUV cutoff provides normalizable gravitonIR cutoff breaks CFT Weak scale mesons, baryons: CFT bound statesHere we are only interested in IR symmetry breakingno UV brane

Broken CFT: Two point function of ops with different dimensions doesnt generally vanish

Will demonstrate exponential decouping in some cases: Consider high and low dim opsWill find

Physics ApplicationsPhysics applications often based in low-energy effective theoriesWhen the 4d theory is dual to AdS on the Poincare patch, duality not entirely obvious

Hard Wall vs Soft WallOriginal RS model had a hard wallBoundary conditions on a braneFollow-up theories based on soft wallsSolutions to the equations of motion where fields get localized away from IRWe will explore what types of soft walls are possibleAnd show how decoupling occurs in both hard and soft wall cases

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

Set-upPoincare patch of AdSAdS slice ends at brane in IR zIRDual vantage point, brane is source of conformal symmetry breakingMost DOF are bulk fields in AdS that correspond to CFT ops filling out irreducible reps of conformal group in absence of CF symmetry breakingHave bulk and brane action

Hard WallWell first consider original hard wall scenarioAbrupt end at zIR corresponds to hard-wall model d+1-dim RS compatible with d-dim EFT of low-mass mesons and glueballs

Hard WallMany familiar featuresBound states (composites)Discrete spectrumChiral symmetry (or tuning) necessary for massless modesHere 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

Scalars with hard wall boundary m0~ : conventional decouplingCan see from eq of motionlooks like a potential: pushes wf against zIR with energy

Scalar Hard Wall DecouplingIn some sense trivialNo light states without tuningLow-lying states have masses proportional to dimension of bulk operator Decoupling conventional

Fermion Hard Wall DecoupingSimilar, but can be more interestingYou can have fewer light states in 4d theory than light states in bulk Due to chiral symmetriesBut 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

Decoupling of heavier bulk modesCan have zero modes even for heavy states in presence of chiral symmetryConsider when number of zero modes less than number of light modes see modes decouple from light modes as you increase their massEffectively overlap shrinks with mass

FermionsLift bulk masses of fermions that originally had large overlap with massless 4d modesConformal sym breaking mixes bulk fields, so generally massless mode has overlap with all light bulk fermions

Decoupling heavy bulk fermions from zero modesDo simple example with only two bulk modes

Zero modesNo mixing: massless zero mode sits in state with mass M2However when nonzero mixing, zero mode will sit in state with mass M1 as M2 taken to infinity

Soft wallsThis feature of wavefunction overlap being relevant persists in soft wall scenariosWorks differently though States localized at different locations in the bulkBut before proceeding, interesting to study possible nature of soft walls

Now consider soft wallsAsympotically UV AdSIR breaking at single scaleBut soft wallspace doesnt suddenly endAssume 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 IRIn any case often present in top-down or bottom-up modelsBulk fields X dual to CFT ops OStandard EFT Lagrangian, expansion in cutoffFind C(z) and (Z) satisfying null energy condition

Scalar WF in Soft-Wall Models and Null Energy Condition

Scalar PotentialSolve Schrodinger eigensystem to get masses and bulk wavefunctionsDynamics controlled by balance between AdS term M2/(kz)2 in small z region and remaining soft wall potential

Constraints on metricPotential function of metric, dilaton But constraints on metric from NECEinstein tensor and constraint:Non-increasing

Either a new CFT or a Wall That Ends SpaceTo avoid large (definite) curvature, F cant decrease below some definite valueSince F non-increasing, must asymptote to greatest lower boundThe only such solution where F=-C2/C isC(z)->Finfty/zAsymptotically AdS in IR as well as UVOtherwise hard wallOr effective soft wall

Effective Soft WallsCurvature singularities imply bulk spacetime endsRegions beyond boundary encapsulated in boundary conditionsExtra compact dimensions, brane,But can have effective soft-wall modelsIf potential rises sharply enough at singularity, lowl-ying modes would have to tunnel through barrier so act as if in soft wallEffective soft wall requires small UV curvatureGzz has to grow before reaching singularitySchrodinger potential then can become large at singularity and push light KK modes away

ExamplePotential blows up power law smaller than curvatureSo small k compared to cutoff allows for light modes to be pushed away from curvature singularity

Example: Solvable Linear SpectrumSquared masses are linear in n and dimension Follows from particular metric (AdS) and dilaton profileThis illustrates competing terms in potentialLeading to localization at larger z for larger M

Soft Wall DecouplingArgument for exponential decouplingFocus on bulk zero modesV(z) with 0(1) coeffcients; mesons with ~1 are localized near z*~1Approximate as 2-dependence from AdS part of metricWhen V(z) does not depend strongly on find -dependent z If V(z) had strong -dependence, would have traditional decoupling

Suppressed WF OverlapsZero mode wavefunctions centered at

Approximate wavefunctions as harmonicsThen wf overlap between state corresponding to heavy mode and one corresponding to light mode is

Max OverlapHard wall: Soft wall: Exponent grows with Mass growing generic consequence of: if

Linear PotentialsExact result: Find Gives wf overlapCorrect exponential suppression

Decoupling much more interestingSoft wall decoupling can be exponentialDue to small wavefunction overlapJustifies keeping only low-lying states and operators

Aside: Comment on Meson SizesNote that overlap not small due to conventional association of position and sizeLook at form factor: eg scalar probeTransverse scalar charge distribution

SizesAlready known with this measure, mesons in hard-wall models have same sizeSoft-wall models with linear spectrumWe know grows with dimension, nNot true however for size

SizesMesons grow only logarithmically w/Seems robust resultSo mesonseven separated in bulkhave similar sizesNot responsible for decouplingAlso, though soft-wall bulk EFTs can produce Regge spectrumDoesnt give confining string picture for growth of states

DiscussionCFTs can have two distinct EFT descriptionsD-dim EFT standard WilsonianIntegrate out momentum shells along RG flowSecond notion from AdS/CFT dualityIntegrate out states based on scaling dimension or conformal CasimirScaling dims dual to bulk masses, std EFT on d+1-dim side

Discussion A priori, only loosely relatedWhen CFT preserved, spectral decomposition of arbitarily massive field has a continuum of light states down to zero massAfter CFT breaking d-dim modes have a discrete spectrumCFT ops of different dimensions mixCan then ask if and how quickly high dim ops decouple from lowest-mass sates

ConclusionStudied CFT breaking and consequences for KK modes of heavy bulk fieldsAside from tuned scenarios, two constructions closely relatedHeavy bulk fields decouple from physics of lightest d-dimensional resonancesHigh-dim ops in broken CFT decouple from physics of low-mass particlesApplies to many systems based on UV CFT broken in IREFTs with light resonances, low-d ops (small conformal Casimir)

Here weve seen a bit about the strong decoupling of higher-dimension operatorsWould be nice to also have a direct 4d derivation in a broken CFT

@lirarandall@preskill Impressive show from family and friends. Continue to enjoy your birthday. And continue the wisdom.

*