Effective Field Theoriesand Multiple Scales in

Quantum PhysicsThomas Mehen, Duke U.

CTMS WorkshopTheory and Applications of Multiscale Modelling

9/11/09

Units in Nuclear/Particle Physics

Quantum Mechanics L = n!Special Relativity E = mc2 c = 3.00 10!8m/s

! = 1.05 10!34J s

! = c = 1‘natural units’

all dimensionful quantities expressed in terms of energy

Length ! !c

Energy

Energy ! Momentum" c ! Mass" c2

Time ! !Energy

mass/energy: electron-Volt (eV)

electron massproton mass

me = 0.511 MeV/c2 = 9.11 10!31 kgmp = 938MeV/c2 = 1.67 10!27 kg

!c = 197.3 MeV 10!15m = 197.3 MeV fm

proton radius 0.88 fm

Large Mass/Energy Small Length/Time

The Many Mass Scales of Elementary Particle Physics

fermions (matter) gauge bosons(forces)

hadrons (quark composites)

fermions (matter)

Nuclear Physics proton mass :mp = 938.3 MeVneutron mass : mn = 939.6 MeVmp !mn = 1.3 MeV

deuteron (np) binding energy : 2.2 MeV

binding energy per nucleon (lead) : ! 9 MeV O

!m2

!

2mN

"pion mass : m! ! 137 MeV "# range of nuclear force $ 1.4 fm

Atomic Physicsme = 0.511 MeV fine structure constant:

size: binding energy:

(H2)

! =1

13712me!

2 = 13.6 eVa0 =1

me!= 0.5 10!10 m

Quantum Physics provides numerous problems with widely separated scales

Quantum Field Theory: Language of Particle Interactions

action : S[!] local field : !(x, t) classical e.o.m :"S

"!= 0

S[!] =!

d4x12("µ!"µ!!m2!2) + g!3 + #!4 + #!!5 + ...

kinetic (quadratic) terms interaction termsQuantization:

non-interacting particles w/ relativistic dispersion relationskinetic terms:

3 or more particles meeting at a point in space/time(local) interactions:

absorption/emission of quanta, collisions

E =!

p2c2 + m2c4

g!3 !"4

coupling constants : g, !,!!, ...

S[!] =!

d4xL[!] Lagrangian

Quantum Electrodynamics (QED)Feynman Diagrams: visual recipe for calculating transition amplitude

photons

electrons/positronsinteractions

Perturbation Theory for e!

e!

! e!

e!

O(!) O(!2)

free particle propagation !"

! ! =1

137

Draw all diagrams to up to O(!n) n

Basic Idea

, determined by accuracy required

QFT is a machine that computes physics given S[!]

How do you figure outS[!] =!

d4xL[!]?

Experiments and a lot of guesswork!

Symmetry

Dimensional Analysis - Renormalization Theory

Symmetry in QM

S-waves P-waves - multiplet of 3

Rotational Invariance of H2 H =!p 2

2 m!

e2

r

Degenerate (same energy) Multiplets

Constrains Dynamics -Wigner-EckartTheorem

!J, m|!r · !E|J !, m!" # CJ,J!;m,m!!J ||!r · !E||J !"

Symmetries in QFTGlobal symmetry, e.g. isospinHeisenberg noticed nuclear force (almost) invariant under n! p

N =!

np

"!" =

!

""+

"0

"!

#

$

N ! UN !" ! R!" U, R are SU(2) matrices

Gauge (local) symmetry

degenerate multiplets

(QED)

!(nn! nn) = !(np! np) = !(pp! pp)

electron! ! eie!(x)! Aµ ! Aµ + "µ!(x)

photon

(SM) gauge symmetry gauge bosons forces

Lorentz Invariance - Special Relativity form of kinetic terms for spin-0,1/2,1particles, S[!] is L.I. invariant

Symmetry is not enough!

LQED = !14Fµ!Fµ! + !̄(i/" !me)! ! e!̄/A!

Blue terms allowed by symmetry, but not required to due accurate atomic physics calculations, e.g.

Lamb Shift (2S-2P splitting), ae, aµ, ...

Some dimensional analysis:

+b1 !̄"µ!Fµ!! + c1 (Fµ!Fµ!)2 + c2 (#µ!"#Fµ!F"#)2 + ...

(! = c = 1) [...] = mass dimension

[Aµ] = 1 [!µ] = 1 [Fµ! ] = 2 [!] = 3/2

[S[!]] = 0! [L] = 4

[me] = 1 [e] = 0 [b1] = !1 [c1] = [c2] = !4

Why?

Question: How do very heavy particles affect low energy physics at low energies?

! decay microphysics:

mp !mn = 1.3 MeV

mW = 80.4 GeV

EW !=!

p2 + M2W

virtual particle how far can it go?

!c

mW! 2.5 10!3 fm

Interaction looks local! Replace W boson with effective interaction

Fermi Theory:

All QFT’s are low energy Effective Field Theories:

string theory?

grand unified theory?

supersymmetry?

QED

QCD

electroweakStandard Model}

all terms in the Lagrangian allowed by symmetry are expected:“That which is not forbidden is compulsory’’, Gell-Mann

but not all terms are equally important (and thankfully? most are not)

energy scale of physics we’re interested inE :! : physical cutoff (e.g. lightest particle that’s being ignored)

Li =!

d4x giOi [Oi] = di [gi] = 4! di

relevant

marginal

irrelevant

but not all terms are equally important (and thankfully? most are not)

di ! 4 < 0di ! 4 = 0di ! 4 > 0

gi !1

!di!4

!d4xOi ! Edi!4 Li !

"E

!

#di!4

Path Integral formulation of QFT

Wilson: ‘integrating out’ high energy modes

Renormalization Group Equations

!d

d!gi = !ij({g})gj

quantum effects can modify scaling expectations from naive dimensional analysis

Running Coupling Constants

!

!

q2

q = p - p’ : momentum transfer

q

+ + +... !

!e!(q)

q2

p

p’

! ! log(|q2|/m2

e)

Vacuum Polarization screens bare charge

!(me) = 1/137

e!

e!

e+

!(90 GeV) = 1/128

|q| ! me

LEP : e+e!

! Z0 ! X

atomic physics

Heisenberg: q ! h̄/r

Key Difference: gluons colored and self-interact

Quantum Chromodynamics (QCD)

electrons/positrons - (+,-) charges

photon - neutral

quarks - three colors (r,g,b)

gluons - 8 colors (e.g., r g )

QED QCD

Screening

Anti-Screening

!s(Q)

Q(GeV )

Asymptotic Freedom(Gross, Politzer, Wilczek; Nobel 2004)

!(Q ! ") ! 0

IR - Strong Coupling

perturbation theory in UV!

"

#

Confinement ! Colorless HadronsChiral Symmetry Breaking!QCD " 500 MeV

ConfinementPhysical Strongly Interacting Particles (hadrons) are Colorless

Mesons - q q Baryons - q q q

Potential Energy between quark and antiquark

lattice QCD simulations

try to pull a quark-antiquark, create more mesons

force between quark and antiquark separated by ~1 fm equivalent to 14 tons!

Vqq̄(r) ! "

4

3

!s(1/r)

r+ " r

q q

Hydrogen atom

Meson (qq + ...)

Hydrogen a two-body system up to small corrections

many-body system: many Fock states contribute

Effective Field Theoriesisolate relevant degrees of freedom

small expansion parameter (ratio of scales):

Lagrangian is most general consistent w/ symmetries of underlying theory, i.e. QCD

Power Countingmust be able to count powers of expansion parameter in Lagrangian and in

Feynman diagrams so we can systematically compute to a given order

Physics at high scales (> 1 GeV) determines coefficients in Lagrangian

determine by matching (e.g. Fermi theory),or experimentally, or numerical simulations

RG Improved Perturbation Theory for Strong Interactions

b

c

dWeak decays of heavy quarks

b c u d

like Fermi theory for ! decay

large strong interaction corrections

u

~0.2

1 + c1 !s(mb) log!

m2W

m2b

"+ c2 !s(mb)2 log2

!m2

W

m2b

"+ ...

~5

RGE for Fermi theory can be used to resum series to all-orders easier to do with EFT than full Theory

mW = 80.4 GeV mb ! 4" 5 GeV

Chiral Theories of Low Energy QCD

Chiral Symmetrychirality = helicity (for a massless particle)

!s !s

!p !p!s · p̂ = !1/2 !s · p̂ = +1/2

q = qL + qR

qL ! ULqL qR ! URqR

SUL(3) ! SUR(3)

Symmetry of LQCD , but not the vacuum!0|q̄LqR + q̄RqL|0" #= 0

qLqR

Spontaneous Chiral Symmetry Breaking

Vacuum: !q̄LqR" #

Excited State: !q̄LU†L(x)UR(x)qR" #

! = 2"/k

recover vacuum as ! ! " (k ! 0) E(k) ! k

relativity: E(k) =!

k2 + m2

Spontaneous Symmetry Breaking Massless Particle!

Goldstone Boson

real world mu, md, ms != 0 pseudo-Goldstone Bosons

Light hadrons of QCD: pions, kaons, and eta’s

m2

PGB ! mq"0|q̄q|0#

m! ! 140 MeV mK ! 496 MeV m" = 547 MeV

Chiral Symmetry Breaking determines the low lying

degrees of freedom of QCD, constrains their self-

interactions and interactions with other hadrons

Interactions vanish as k ! 0 derivatively coupled

weakly interacting at low energies:

perturbative expansion in Q2

!2!

!! = 4!f" ! 1 GeV

Q4

!4!

Q2

!2!

!

Q ! (p, m!)

!

!

!

Chiral Perturbation Theory (ChiPT)

L =f2

!

8Tr !µ!!µ!† +

f2!B0

4Tr(mq! + mq!

†) + ...

Coupling to nucleons

! !

N N

(within 15%-25%)

Long range nuclear spin-tensor force

aI!N = CI

m!

f2!

g2

A

2f2!

!q · !" !q · !"

!q 2 + m2!

S-wave scattering length

Large Scattering Lengths

Quantum Mechanics of shallow bound statesV (r)

!EB

!(r) ! e!r/a

rEB =

12µa2

R

a! R

universal properties

! =4"

p2 + 1/a2

since physics observables can all be computed in terms of a single parameter, simple EFT with single relevant operator can reproduce this physics

2- and 3-body scattering easily calculated within this EFT

originally developed for few-body nuclear physics

deuteron - weakly bound neutron and proton

higher order corrections: expansion in R/a

R - range of force - scattering lengtha

EB = 2.2 MeV

neutron-deuteron scattering at low energies

NLO EFT: 5 parameters, 1-D coupled integral equations

dashed line:NLO EFT

red dots:potential model plus

Faddev equations

Potential model: 20> parameters in 2+3 body potentials

Faddeev equations (9-dimensional Schroedinger Eq.)

By focusing on relevant degrees of freedomEFT reduces parameters, computational complexity

(H. Hammer, T.M. PLB516:353(2001))

Current Topical Applications

Cold Trapped Atomsexperimental atomic physicist can tune scattering length at will using Feshbach Resonances (Prof. John Thomas, Duke U.)

Non-Relativistic Conformal Symmetry as a!"T.M., I. Stewart, M. Wise, PLB474:145 (2000)

D.T. Son, Y. Nishida, PRD76:086004 (2007)

T.M., PRA 78:013614 (2008)

X(3872) (discovered in 2003)

shallow bound state of D(cu) mesons EB < 0.7 MeV

S.Fleming, M. Kusunoki, T.M., U. van Kolck, PRD 76:034006 (2007)

S.Fleming, T.M., PRD 78:094019 (2008)XEFT:

Conclusions

Focusing on relevant degrees of freedom, interactions at a given scale useful for

systematic analysis of physics problems

Effective interactions - ‘coarse graining’ - can simplify computational problems

Examples: resumming strong interaction logarithms neutron-deuteron scattering at low energies