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The Soft-Collinear Effective Theory

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The Soft-Collinear Effective Theory The Soft-Collinear Effective Theory Iain W. Stewart MIT Ringberg Workshop April 30, 2003 Iain Stewart – p.1 Outline Motiviation • Soft-Collinear Effective Theory • 1) Power counting 2) Gauge Symmetries 3) Lagrangian 4) Factorization = separation of scales/interactions Factorization for B Dπ, DIS • ! Reparameterization Invariance, Power Corrections • New: Results for B ? ? • ! Conclusions • Iain Stewart – p.2 Motivation Factorization: To describe hadronic processes we need to separate short (p Q) and long (p Λ ) distance contributions ∼ ∼ QCD Effective Field Theory: Useful for separating physics at different momentum scales Many processes have energetic hadrons Q Λ QCD B Decays: + B πeν, B ρeν, B K∗γ, B Ke e−, B γeν ! ( ) ! ! ! ! B D ∗ π, B ππ, B Kπ, ! ! ! B X eν, B X γ, B D∗X ! u ! s ! u Hard Scattering Processes: DIS, Drell Yan, jet production, γ∗p γX, 0 ! γ∗γ π , γ∗M M, ! ! ( ) γ∗p Xπ, γ∗p γ ∗ p0 ! ! Iain Stewart – p.3 What is Factorization? m m C (µ, m ) (µ, m ) 1 2 i i 1 Oi 2 P DIS structure functions [Q2 Λ2 , 1 x Λ =Q] QCD − QCD 2 1 1 F1(x; Q ) = x x dξ H(ξ=x; Q; µ) fi=p(ξ; µ) R 0 0 2 2 γ∗γ π form factor from e−γ e−π [Q Λ ] ! ! QCD Brodsky-Lepage 2 fπ 1 Fπγ (Q ) = Q2 0 dξ T (ξ; Q; µ) φπ(ξ; µ) R B Dπ decay rate [Q = mQ; Eπ ΛQCD] ! f g Politzer-Wise B D 1 mc Dπ d¯uc¯b B = NF ! (0) dξ T (ξ; ; Q; µ) φπ(ξ; µ) h j j i 0 mb R Iain Stewart – p.4 A Goal Traditionally factorization of collinear and soft gluons involves an analysis of diagrams Collins,Soper,Sterman Can this be made simpler? Iain Stewart – p.5 Kinematics Take: nµ = (1; 0; 0; 1), n¯µ = (1; 0; 0; 1) − eg1. p B D Pion has: pµ = (2:310 GeV; 0; 0; 2:306 GeV) = Q nµ π − collinear constituents: Q ΛQCD 2 + ΛQCD 2 (p ; p−; p?) ; Q; Λ Q(λ ; 1; λ) λ 1 ∼ Q QCD ∼ γ eg2. Xs B endpoint jet constituents: + 2 (p ; p−; p?) Λ ; Q; QΛ Q(λ ; 1; λ) λ 1 ∼ QCD QCD ∼ p Iain Stewart – p.6 Soft-Collinear Effective Theory Introduce fields for infrared degrees of freedom modes pµ = (+; ; ) p2 − ? collinear Q(λ2; 1; λ) Q2λ2 soft Q(λ, λ, λ) Q2λ2 usoft Q(λ2; λ2; λ2) Q2λ4 Offshell modes with p2 Q2λ2 are integrated out Developed in: C. Bauer, S. Fleming, M. Luke, hep-ph/0005275 (PRD) C. Bauer, S. Fleming, D. Pirjol, I. Stewart, hep-ph/0011336 (PRD) C. Bauer, I. Stewart, hep-ph/0107001 (PLB) C. Bauer, D. Pirjol, I.Stewart, hep-ph/0109045 (PRD) Builds on and extends earlier work: Hard Exclusive: Brodsky, Lepage, : : : Jet physics: Collins, Soper, Sterman, Korchemsky, : : : B-physics: Beneke, Buchalla, Neubert, Sachrajda, : : : Iain Stewart – p.7 Soft-Collinear Effective Theory Introduce fields for infrared degrees of freedom modes pµ = (+; ; ) p2 − ? collinear Q(λ2; 1; λ) Q2λ2 soft Q(λ, λ, λ) Q2λ2 usoft Q(λ2; λ2; λ2) Q2λ4 Offshell modes with p2 Q2λ2 are integrated out Goals of EFT: simplify power counting in λ • make symmetries explicit • sum IR logarithms, Sudakov double logarithms • understand factorization in a universal way • model independent description of power corrections • Iain Stewart – p.7 Soft-Collinear Effective Theory Introduce fields for infrared degrees of freedom modes pµ = (+; ; ) p2 − ? collinear Q(λ2; 1; λ) Q2λ2 soft Q(λ, λ, λ) Q2λ2 usoft Q(λ2; λ2; λ2) Q2λ4 Offshell modes with p2 Q2λ2 are integrated out Typically either: usoft pµ Λ SCETI λ = Λ=Q ∼ collinear p2 QΛ c ∼ p SCET λ = Λ=Q soft pµ Λ II ∼ collinear p2 Λ2 c ∼ Iain Stewart – p.7 1) Power counting 2) Gauge Symmetries 3) Lagrangian 4) Factorization Iain Stewart – p.8 Separate Momenta label residual µ µ µ HQET P = mbv + k hv(x) (Georgi) µ µ µ SCET P = p + k ξn;p(x) (1; λ) Q Collinear Quarks k ip x . (x) e− · ξn;p(x) Q λ ! p p . n= ξn;p = P0 µ 2 . @ ξn;p (Qλ ) ξn;p ∼ Q λ2 But labels are changed q • by SCET interactions p p′ Iain Stewart – p.9 Separate Momenta Introduce Label Operator µ y µ µ y φ φp = (p +: : : q : : :) φ φp P q1 · · · 1 · · · 1 − 1 − q1 · · · 1 · · · Can pull phases to front of operators • µ ip x ip x µ µ i@ e− · φ (x) = e− · ( + i@ )φ (x) p P p Iain Stewart – p.10 Power Counting + Type (p ; p−; p?) Fields Field Scaling 2 collinear (λ ; 1; λ) ξn;p λ + 2 (An;p, An;p− , An;p? ) (λ , 1, λ) 3=2 soft (λ, λ, λ) qs;p λ µ As;p λ 2 2 2 3 usoft (λ ; λ ; λ ) qus λ µ 2 Aus λ n¯= Make kinetic terms order λ0 d4X ξ¯ 0 in @ + : : : ξ n;p 2 · n;p 0 4 2 λ = R λ− λ λ λ At leading power only λ0 interactions are required • n¯ A n¯ q λ0 operators are f(n¯ A ; n¯ q ) • · n;q ∼ · i ∼ · n;q · i Iain Stewart – p.11 More Power Counting In an arbitrary graph, Euler identities allow all powers of λ to be associated just with vertices Bauer, Pirjol, I.S. π SCET graphs λδ I ∼ B D δ = 4u + 4 + (k 4)V c + (k 8)V us − k − k k i X k where Vk = # of order λ operators Power counting is gauge invariant • Iain Stewart – p.12 More Power Counting In an arbitrary graph, Euler identities allow all powers of λ to be associated just with vertices Bauer, Pirjol, I.S. π SCET graphs λδ II ∼ B D δ = 4 + (k 4)(V c + V s + V scE) + (k 2)V scL − k k k − k k i X k where Vk = # of order λ operators Power counting is gauge invariant • Iain Stewart – p.12 1) Power counting 2) Gauge Symmetries 3) Lagrangian 4) Factorization Iain Stewart – p.13 Wilson Lines and Coefficients Consider matching for heavy-to-light current, u¯ Γ b ξ¯n;p Γ hv u integrate out offshell quarks b n¯ A one-gluon: ξ¯ ( g) · n;q Γ h n;p − n¯ q v · ( g)k n¯ A n¯ A all orders: ξ¯ − · n;q1 · · · · n;qk Γ h n;p k! k v perms [n¯ q1] [n¯ i=1 qi] Xk X · · · · · P W y W is like W (y; ) = P exp ig ds n¯ A (sn¯µ) −∞ · n Z−∞ Iain Stewart – p.14 Wilson Lines and Coefficients Consider matching for heavy-to-light current, u¯ Γ b ξ¯n;p Γ hv u integrate out offshell quarks b n¯ A one-gluon: ξ¯ ( g) · n;q Γ h n;p − n¯ q v · ( g)k n¯ A n¯ A all orders: ξ¯ − · n;q1 · · · · n;qk Γ h n;p k! k v perms [n¯ q1] [n¯ i=1 qi] Xk X · · · · · P W 1 W = exp g n¯ An;q perms − ¯ · P P h i Iain Stewart – p.14 Gauge Invariance J = ξ¯n;pW Γhv is invariant Consider collinear gauge transformation U, @µ U(y) Q(λ2; 1; λ) ∼ . ξn Uξn, W UW , h ! h (since!Uh is far offshell) v ! v v Consider ultrasoft gauge transformation U, @µ U(y) Qλ2 ∼ . ξ Uξ , W UW U y, h Uh n ! n ! v ! v U, U transformations are homogeneous in λ • matching argument not spoiled by loop corrections • Hard-Collinear Factorization: f( ¯ + gn¯ A ) = W f( ¯)W y • P · n;q P Wilson coefficients are only functions of ¯ = n¯ λ0, and their • P ·P ∼ location is restricted by gauge invariance e.g. C(µ, ¯) ξ¯ W Γh P n;p v Iain Stewart – p.15 1) Power counting 2) Gauge Symmetries 3) Lagrangian 4) Factorization Iain Stewart – p.16 Collinear Lagrangian (0) 1 n¯= ¯ 0 c c ξξ = ξn;p n i@+gn Aus +gn An;q + iD= W W y iD= ξn;p L · · · ? ¯ ? 2 P most general order λ0 gauge invariant action • µ µ µ n Aus n An;q but others suppressed, iDc? = + gAn;q? • · ∼ · P? k has multipole expansion for usoft kµ • p in= n¯ p final propagator = · 2 [ (n k + n p) n¯ p + p2 + i] · · · ? (0) µ cg (A ; n A ) L n;q · us order λ0, gauge invariant, also multipole expanded • A enters like a background field for A • us n;q Usoft and Soft actions are just like QCD at LO Iain Stewart – p.17 Feynman Rules µ , A µ , A p p′ µ , A ν , B q p p′ ξ describe fluctuations of quark about direction n • n (0): has particles & antiparticles, pair creation Lξξ • includes spin-dependent effects, k2 effects, binding • ? Iain Stewart – p.18 1) Power counting 2) Gauge Symmetries 3) Lagrangian 4) Factorization Iain Stewart – p.19 Factorization of Usoft Gluons Consider the following field redefinitions in SCET (0) (0) ξn;p = Yn ξn;p ; An;q = Yn An;q Yny x where Yn = P exp ig ds n Aus(ns) , n DYn = 0, and YnyYn = 1 −∞ · · Find R (0) (0) = ξ¯ 0 in D + : : : ξ = ξ¯ 0 in @ + : : : ξn;p • Lq n;p · n;p ) n;p · (0) h i W = Y W Y y • n n µ (0) (0)µ (ξ ; A ; n A ) = (ξn;p; An;q ; 0) • L n;p n;q · us L Moves all usoft gluons to operators, simplifies cancellations ¯(0) (0) eg1.
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