2014 National Nuclear Physics Summer School Lectures on Effective Field Theory I. Removing heavy particles II. Removing large scales III. Describing Goldstone bosons IV. Interacting with Goldstone bosons Brian Tiburzi RIKEN BNL Research Center
On the foundations of chiral perturbation theory H. Leutwyler (Bern U.). Aug 1993. 52 pp. Published in Annals Phys. 235 (1994) 165-203 BUTP-93-24 DOI: 10.1006/aphy.1994.1094 e-print: hep-ph/9311274 PDF Effective Reviews Five lectures on effective field theory David B. Kaplan. Oct 2005. 79 pp. e-print: nucl-th/0510023 PDF TASI lectures on effective field theories Ira Z. Rothstein (Carnegie Mellon U.). Aug 2003. 90 pp. e-print: hep-ph/0308266 PDF Effective field theory for few nucleon systems Paulo F. Bedaque (LBL, Berkeley), Ubirajara van Kolck (Arizona U. & RIKEN BNL). Published in Ann.Rev.Nucl.Part.Sci. 52 (2002) 339-396 DOI: 10.1146/annurev.nucl.52.050102.090637 e-print: nucl-th/0203055 PDF
Effective Overview ( = c = 1) Effective (quantum) field theories exploit a separation of scales Ω ω Low-scale physics is largely insensitive to high-scale physics (long-distance) (short-distance) Ω 1/r ω 1/R Example: Electric field far from source R r R r physical distribution of charge
Effective Overview ( = c = 1) Effective (quantum) field theories exploit a separation of scales Ω ω Low-scale physics is largely insensitive to high-scale physics (long-distance) (short-distance) Ω 1/r r R Example: Electric field far from source r ω 1/R R multipolarity (power counting) r R ω Ω physical distribution of charge + + + L multipoles to describe E to p% effective multipole theory
Effective Overview ( = c = 1) Effective (quantum) field theories exploit a separation of scales Ω ω Low-scale physics is largely insensitive to high-scale physics (long-distance) (short-distance) Ω 1/r ω 1/R r R Goal: investigate power counting for various effective QFTs QFTs require a regulator and renormalization UV regulator alters high-scale physics to compute quantum effects Renormalization: absorb effects of regulator into theory parameters
Effective Overview ( = c = 1) Effective (quantum) field theories exploit a separation of scales Ω ω Low-scale physics is largely insensitive to high-scale physics (long-distance) (short-distance) Ω 1/r ω 1/R r R Goal: investigate power counting for various effective QFTs Top Down I. Removing heavy particles II. Removing large scales Start from perturbative high-scale theory, arrive at simpler low-scale EFT Bottom Up III. Describing Goldstone bosons IV. Interacting with Goldstone bosons Construct low-scale EFT using non-perturbative degrees of freedom
Effective Field Theory I. Removing heavy particles
Euler-Heisenberg EFT L =dim4 F ν =dim2 ψ =dim 3 2 EFT for low-frequency photons ω m e obtained by integrating out electron QED EFT m e ω L QED = 1 4 F νf ν + ψ(id/ m e )ψ Photon-photon scattering mediated by virtual electron in QED
Euler-Heisenberg EFT L =dim4 F ν =dim2 ψ =dim 3 2 EFT for low-frequency photons ω m e obtained by integrating out electron QED EFT m e ω L QED = 1 4 F νf ν + ψ(id/ m e )ψ Photon-photon scattering mediated by virtual electron in QED Photon is the only degree of freedom in the EFT L EFT = 1 4 F νf ν +... Low-frequency photon-photon scattering requires 4-photon operators Lorentz invariance, Gauge invariance, C, P, T F 4 =dim8
Euler-Heisenberg EFT L =dim4 F ν =dim2 ψ =dim 3 2 EFT for low-frequency photons ω m e obtained by integrating out electron QED EFT m e ω L QED = 1 4 F νf ν + ψ(id/ m e )ψ Photon-photon scattering mediated by virtual electron in QED Photon is the only degree of freedom in the EFT L EFT = 1 4 F νf ν + 1 c m 4 1 (F ν F ν ) 2 + c 2 (F ν F ν ) 2 e Low-frequency photon-photon scattering requires 4-photon operators Lorentz invariance, Gauge invariance, C, P, T F 4 =dim8 F ν = 1 2 ναβf αβ
Euler-Heisenberg EFT L =dim4 F ν =dim2 ψ =dim 3 2 EFT for low-frequency photons ω m e obtained by integrating out electron QED L QED = 1 4 F νf ν + ψ(id/ m e )ψ EFT m e ω Photon-photon scattering mediated by virtual electron in QED Photon is the only degree of freedom in the EFT L EFT = 1 4 F νf ν + α2 90m 4 e (F ν F ν ) 2 + 74 (F ν F ν ) 2 Low-frequency photon-photon scattering requires 4-photon operators Lorentz invariance, Gauge invariance, C, P, T F 4 =dim8 F ν = 1 2 ναβf αβ
Exercise Include low-energy virtual photons in the Euler-Heisenberg EFT. What new local operator of lowest dimension is required? Determine the coefficient of this operator from matching to QED at one-loop order. L EFT = 1 4 F νf ν +??? + (F α2 90m 4 ν F ν ) 2 + 74 (F F ν ν ) 2 e
Effective Lessons EFT as low-energy limit of a QFT Short-distance physics encoded in coefficients of local operators c 1 (F ν F ν ) 2 + c 2 (F ν F ν ) 2 Infinite tower of such higher-dimensional operators requires power counting F 2 ω 4, F 4 ω 8 γγ γγ ω4 m 4 e Can make predictions without employing full QFT, and can systematically improve Finitely many operators to a given order Operators built from effective d.o.f. Respect symmetries of underlying theory EFT is itself a QFT... compute radiative corrections (non-analytic) Coefficients of higher dimensional operators must be determined Top Down Perturbation Theory Bottom Up Experiment Non-perturbative (lattice QCD)
Effective Lessons EFT is low-energy limit of a QFT Short-distance physics encoded in coefficients of local operators c 1 (F ν F ν ) 2 + c 2 (F ν F ν ) 2 Operators built from effective d.o.f. Respect symmetries of underlying theory Infinite tower of such higher-dimensional operators requires power counting F 2 ω 4, F 4 ω 8 γγ γγ ω4 m 4 e Can make predictions without employing full QFT, and can systematically improve Coefficients of higher dimensional operators must be determined Top Down Finitely many operators to a given order EFT is itself a QFT... compute radiative corrections (non-analytic) Perturbation Theory Euler-Heisenberg EFT from finite one-loop diagram in QED perturbation theory
Standard Model as an EFT L =dim4 F ν =dim2 ψ =dim 3 2 Renormalizable interactions of SM are the low-energy limit of some high-energy theory Physics beyond SM encoded in a tower of higher-dimension ops. M TeV? L EFT = L SM + j=1 dim O (4+j) k k =4+j c (j) k M j O(4+j) k dim c (j) k =0 naturalness c (j) k 1 E EW
Standard Model as an EFT L =dim4 F ν =dim2 ψ =dim 3 2 Renormalizable interactions of SM are the low-energy limit of some high-energy theory Physics beyond SM encoded in a tower of higher-dimension ops. M TeV? L EFT = L SM + j=1 dim O (4+j) k k =4+j c (j) k M j O(4+j) k dim c (j) k =0 naturalness c (j) k 1 E EW Example: CP violation beyond SM O (4) = G ν Gν O (6) = ψ L σ ν Gν Φ ψ R O (6) = f abc ν Ga G b ν α G c α
L =dim4 Standard Model as an EFT F ν =dim2 ψ =dim 3 2 Renormalizable interactions of SM are the low-energy limit of some high-energy theory Physics beyond SM encoded in a tower of higher-dimension ops. M TeV? L EFT = L SM + j=1 dim O (4+j) k k =4+j c (j) k M j O(4+j) k dim c (j) k =0 naturalness c (j) k 1 E EW Example: CP violation beyond SM O (4) = G ν Gν c (4) θ 1 unnatural = strong CP problem Generate electric dipole moments O (6) = ψ L σ ν Gν Φ ψ R O (6) = f abc ν Ga G b ν α G c α EEW M 2 Neutron EDM d E S E QFT prejudice: associate energy scale with mass of new heavy particle
Exercise Enumerate all dimension-6 CP violating operators that respect Standard Model symmetries. How do these operators appear after electroweak symmetry breaking? O (6) = ψ L σ ν Gν Φ ψ R O (6) = f abc ν Ga G b ν α G c α
Φ, φ Integrating Out Heavy Particles M Φ m φ Perform path integral over heavy field Expand result in local operators built from light field e is eff[φ] = DΦ e i d 4 xl(φ,φ) DΦ e i d 4 xl(φ,0) Illustrative toy model (Gaussian path integral) M Φ L = 1 2 Φ Φ 1 2 M 2 ΦΦ 2 + Φ J <-- coupling to φ m φ Complete the square to deduce S eff [φ] = 1 d 4 xd 4 yj(x)g(x y)j(y) 2 Operator product expansion J(y) =J(x)+(y x) J(x)+... +...
Φ, φ Integrating Out Heavy Particles M Φ m φ Perform path integral over heavy field Expand result in local operators built from light field e is eff[φ] = DΦ e i d 4 xl(φ,φ) DΦ e i d 4 xl(φ,0) Illustrative toy model (Gaussian path integral) M Φ L = 1 2 Φ Φ 1 2 M 2 ΦΦ 2 + Φ J <-- coupling to φ m φ Complete the square to deduce S eff [φ] = 1 d 4 xd 4 yj(x)g(x y)j(x)+... 2 Operator product expansion J(y) =J(x)+(y x) J(x)+... +...
Φ, φ Integrating Out Heavy Particles M Φ m φ Perform path integral over heavy field Expand result in local operators built from light field e is eff[φ] = DΦ e i d 4 xl(φ,φ) DΦ e i d 4 xl(φ,0) Illustrative toy model (Gaussian path integral) M Φ L = 1 2 Φ Φ 1 2 M 2 ΦΦ 2 + Φ J <-- coupling to φ Complete the square to deduce S eff [φ] = 1 d 4 xj(x)j(x) 2 d 4 yg(y)+... m φ Operator product expansion J(y) =J(x)+(y x) J(x)+... +...
Φ, φ Integrating Out Heavy Particles M Φ m φ Perform path integral over heavy field Expand result in local operators built from light field e is eff[φ] = DΦ e i d 4 xl(φ,φ) DΦ e i d 4 xl(φ,0) Illustrative toy model (Gaussian path integral) M Φ L = 1 2 Φ Φ 1 2 M 2 ΦΦ 2 + Φ J <-- coupling to φ Complete the square to deduce S eff [φ] = 1 d 4 xj(x)j(x) 2 d 4 yg(y)+... m φ d 4 yg(y) = G(k = 0) = 1 M 2 Φ Operator product expansion J(y) =J(x)+(y x) J(x)+... +...
Φ, φ Integrating Out Heavy Particles M Φ m φ Perform path integral over heavy field Expand result in local operators built from light field e is eff[φ] = DΦ e i d 4 xl(φ,φ) DΦ e i d 4 xl(φ,0) Illustrative toy model (Gaussian path integral) M Φ L = 1 2 Φ Φ 1 2 M 2 ΦΦ 2 + Φ J <-- coupling to φ m φ Complete the square to deduce S eff [φ] = 1 2M 2 Φ d 4 xj(x)j(x)+... d 4 yg(y) = G(k = 0) = 1 M 2 Φ Operator product expansion J(y) =J(x)+(y x) J(x)+... +...
Fermi Theory as an EFT L =dim4 F ν =dim2 ψ =dim 3 2 Weak interactions are so because of large masses of W and Z bosons g g +... M Z,M W 1 q 2 M 2 W g 2 M 2 W 1+O q 2 M 2 W Four-Fermion interaction L eff = G F 2 ψψ V A ψψ V A m f Non-renormalizability of Fermi theory implies energy scale G F =dim( 2) matching to standard model reveals G F = 2g 2 8M 2 W Efficacy of Fermi theory controlled by power counting, e.g. ß-decay (u L γ d L )(e L γ ν L ) n p + e + ν e (δm N ) 2 power corrections M 2 W 10 10
Beyond Tree Level Top-down uses perturbation theory to integrate out heavy particles In addition to power corrections, there are perturbative corrections M Z,M W + power corrections Fortunately QED and QCD are perturbative at the weak scale m f α(m Z ) 1 128 + α s (M Z ) 0.1 QED QCD Introduces renormalization scale and scheme dependence in EFT c O(x) c()o(x, )
Exercise Are there pqcd corrections to the ß-decay operator in Fermi EFT? If so, characterize them. If not, explain why. (u L γ d L )(e L γ ν L ) n p + e + ν e
q W ±,Z 0 q The Dirtiest Corner of Standard Model q q Hadronic weak interaction observable through processes that violate QCD symmetries M Z,M W Λ QCD Flavor changing s =1, K ππ [Particle Physics] Quark weak interactions known at weak scale c() ππ O s=1 (x, ) K Flavor conserving but parity violating s =0, p+ p PV p + p p + 4 He PV p + 4 He + PV nuclear reactions [Nuclear Physics] Must determine EFT at low scales using renormalization group... c() pp O PV (x, ) pp will be Non-perturbative matrix elements computed at QCD scales with lattice
q W ±,Z 0 q The Dirtiest Corner of Standard Model q q Hadronic weak interaction observable through processes that violate QCD symmetries M Z,M W Flavor changing s =1, K ππ [Particle Physics] Quark weak interactions known at weak scale Flavor conserving but parity violating s =0, p+ p PV p + p p + 4 He PV p + 4 He + PV nuclear reactions [Nuclear Physics] m b m c Must determine EFT at low scales using renormalization group... c() ππ O s=1 (x, ) K c() pp O PV (x, ) pp will be Non-perturbative matrix elements computed at QCD scales with lattice New features: Multiple scales (treat one at a time) Λ QCD Operator mixing (coeffients --> matrix) Matching to lattice (scheme dependence)
q W ±,Z 0 q The Dirtiest Corner of Standard Model q q Hadronic weak interaction observable through processes that violate QCD symmetries M Z,M W Flavor changing s =1, K ππ [Particle Physics] Quark weak interactions known at weak scale Flavor conserving but parity violating s =0, p+ p PV p + p p + 4 He PV p + 4 He + PV nuclear reactions [Nuclear Physics] m b m c Must determine EFT at low scales using renormalization group... c() ππ O s=1 (x, ) K c() pp O PV (x, ) pp will be Non-perturbative matrix elements computed at QCD scales with lattice New features: Multiple scales (treat one at a time) Λ QCD Operator mixing (coeffients --> matrix) Matching to lattice (scheme dependence)
Hadronic Parity Violation Beyond Trees Standard Model has isotensor PV interactions Fermi PV EFT described by one operator L I=2 PV = G F 2 C()O() O =(qτ 3 γ q) L (qτ 3 γ q) L 1 3 (qτ γ q) L (qτ γ q) L {L R} QCD Renormalization at one loop C tree sin 2 θ W M Z,M W Full theory Effective theory m b m c Λ QCD log M W 2 UV finite p 2 UV divergent log 2 p 2 log M W 2 2 2 = log log p2 p2 M 2 W EFT will reproduce full theory by altering the low-energy coefficient C() = γ O α s () 4π log 2 M 2 W EFT @ M W has no log, EFT @ m b has large log --> sum using RG
Hadronic Parity Violation Beyond Trees Standard Model has isotensor PV interactions Fermi PV EFT described by one operator L I=2 PV = G F 2 C()O() O =(qτ 3 γ q) L (qτ 3 γ q) L 1 3 (qτ γ q) L (qτ γ q) L {L R} QCD Renormalization at one loop C tree sin 2 θ W N f =5 N f =4 M Z,M W m b m c Effective theory Solution to RG evolution β 0 = 11 2 3 N f 4π d d C() =γ O α s () C() C( )=U(,)C() U( αs ( ),)= α s () N f -dependent γo /2β 0 Multiple scales: once below heavy quark threshold integrate out, then match N f 1 and EFTs at the scale N f m Q N f =3 Λ QCD C(Λ QCD )= αs (Λ QCD ) α s (m c ) 6 27 6 α s (m c ) 25 6 23 α s (m b ) C(MW ) α s (m b ) α s (M W )
Effective Summary I. Removing heavy particles Heavy particles can be systematically integrated out resulting in EFTs power corrections mφ M Φ n αs (m φ ) perturbative corrections α s (M Φ ) γo /2β 0 EFT coefficients determined from matching top-down N f =5 N f =4 M Z,M W m b m c log M W 2 2 2 = log log p2 p2 Theories have different UV behavior M 2 W Only IR behavior is shared and thus cancels in matching Computations in EFT are simpler (one scale at a time) EFT involves only d.o.f. relevant to energy regime N f =3 Λ QCD Standard Model is an EFT... arises from integrating out heavy new particles?