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] → { } Politzer-Wise

B D 1 mc Dπ d¯uc¯b B = NF → (0) dξ T (ξ, , Q, µ) φπ(ξ, µ) h | | 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. π 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 . ψ(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 † µ µ † 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

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

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 †, 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 † • 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 † 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

p p′

µ , A ν , B

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 Yn†

x where Yn = P exp ig ds n Aus(ns) , n DYn = 0, and Yn†Yn = 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 † • 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. J = ξn W Γ Yn†hv (0) (0) (0) (0) eg2. J = ξ¯nW Γ W †ξn = ξ¯n W Γ W †ξn Iain Stewart – p.20 Soft-Collinear Interactions soft gluons can not couple to collinear particles in a local way q = q + q Q(λ, 1, λ) q2 = Q2λ s c ∼ soft Wilson line Sn built up by integrating out offshell ﬂucutations C builds up ¯ S ξnSn† Γ W hv S C

¯ S S switches order ξnW Γ Sn† hv

C C

More general method (simpler too):

2 (0) (0) i) match QCD onto SCET with p QΛ: ξ¯n W Γ Y †h I c ∼ n v 2 2 ii) match SCET onto SCET with p Λ : ξ¯ W Γ S† h I II c ∼ n n v

Iain Stewart – p.21 Other People

Other work:

Rothstein, Chay, Kim, Manohar, Mehen, Fleming, Leibovich, Low, Beneke, Feldmann Chapovsky, Diehl position space formulation, ξn(x) Neubert, Hill ) Wyler, Lunghi Bosch, Lange Wise, Ligeti . . .

Iain Stewart – p.22 Why use two theories?

Useful for exclusive processes:

SCETI SCETII a local theory (except at Q) highly non-local (1/Λ) describes non-perturbative describes how particles in binding in soft and energetic SCETII communicate hadrons

tells us how operators factor all operators factor

sum logs Q2 > µ2 > QΛ sum logs QΛ > µ2

With SCETI can also analyze cases where αs(QΛ) is too large to trust perturbation theory

Iain Stewart – p.23 Matching Steps

+ N dz dx dr+ T (z, Q, µ0)J(z, x, r+, µ0, µ)φM (x, µ)φB(r+, µ) Z

i) At µ2 Q2 match QCD onto SCET , get C(µ, ¯) T (z, Q, µ) ∼ I P → ii) Factorize the usoft-collinear interactions with the ﬁeld redeﬁnitions,

(0) µ (0)µ ξn = Y ξn and An = Y An Y †.

iii) At µ2 QΛ match SCET onto SCET , get jet functions J 0 ∼ I II iv) Run to µ 1 GeV ( few Λ) if desired ∼ ∼ ×

SCETI ﬁelds contain SCETII collinear ﬁelds, two theories share • common states, equate S-matrix elements (as usual)

Iain Stewart – p.24 SCET Summary

collinear, soft, usoft degrees of freedom • SCET has hard coefficients C( ¯, µ), Wilson lines W, • I P factorization of usoft gluons by field redefinition with Y

SCETII has coefﬁcients J, collinear W and soft S needed • for gauge invariance Sometimes matching through SCET is trivial, so J = 1 • I

Iain Stewart – p.25 Factorization for B Dπ → History: B D Naive Factorization: O = NF → f h i π Bjorken (’89) Color Transparency Dugan & Grinstein (’91) LEET, = ξ¯ n¯/ in D ξ L 2 · B D Generalized Factorization: O = NF → f dx T (x)φ (x) h i π π Politzer & Wise (’91) Proposal R Beneke et al. (’00) Two loop proof, general arguments Bauer, Pirjol, I.S. (’01) All orders proof

π Q = m , m , E Λ b c π QCD B, D are soft, π collinear

B D

Iain Stewart – p.26 Reproduce 2-loop result

(1) (1) Non-trivial result from 1 hard & 1 collinear loop, dx T (x) φπ (x) 62 two-loop graphs calculated by Beneke, Buchalla,RNeubert, Sachrajda

All two-loop graphs reproduced by 3 one-loop SCET graphs ⇒ p p' p p' p p' q

p+q p+q q p'-q q Results agree p'+q

v v v v' v v' a) b) c)

Iain Stewart – p.27 Proof for B Dπ → Steps

2 2 Match at µ Q onto SCET ( ¯ = ¯† + ¯) • ∼ I P+ P P (c) (b) (d) (u) c¯b d¯u h¯ 0 hv ξ¯ 0 W C0( ¯ )W †ξn,p = v n,p P+ A ¯ A ¯(c) A (b) ¯(d) ¯ A (u) c¯T b d T u ) ⇒ ( h 0 T hv ξ 0 W C8( +)T W †ξn,p v n,p P

Iain Stewart – p.28 Proof for B Dπ → Steps

Match at µ2 Q2 onto SCET [Decouple ξ Y ξ(0)] • ∼ I → (c) (b) (0) (0) (0) (0) c¯b d¯u h¯ 0 hv ξ¯ 0 W Y †Y C0( ¯ )W †ξn,p = v n,p P+ A ¯ A ¯(c) A (b) ¯(0) (0) ¯ A (0) (0) c¯T b d T u ) ⇒ ( h 0 T hv ξ 0 W C8( +)Y †T Y W †ξn,p v n,p P

Iain Stewart – p.28 Proof for B Dπ → Steps

Match at µ2 Q2 onto SCET [Decouple ξ Y ξ(0)] • ∼ I → (c) (b) (0) (0) (0) (0) c¯b d¯u h¯ 0 hv ξ¯ 0 W C0( ¯ )W †ξn,p = v n,p P+ A ¯ A ¯(c) A (b) ¯(0) (0) ¯ A (0) (0) c¯T b d T u ) ⇒ ( h 0 Y T Y † hv ξ 0 W C8( +)T W †ξn,p v n,p P Match at µ2 QΛ onto SCET • ∼ II (c) (b) h¯ 0 h ξ¯ 0 W C0( ¯ )W †ξ v v n,p P+ n,p (c) A (b) A h¯ 0 ST S† h ξ¯ 0 W C8( ¯ )T W †ξ v v n,p P+ n,p Take matrix elements • (0) (0) (0)† (0) i πn ξ¯ 0 W C0( ¯+)W ξ 0 = fπEπ dx C[2Eπ(2x 1)]φπ(x) n,p P n,p 2 − Z B→D 0 ¯ 0 Dv hv Γhhv B v = F (0) h | | i 1 B D Dπ c¯bud¯ B = N F → dx T (x, µ) φ (x, µ) h | | i π Z0 Iain Stewart – p.28 ( ) B D ∗ X phenomenology → ¯0 + 0 Proof Applies to: B D π−, B− D π−, (and π ρ), → → → (and D D∗), →

0 ( )+ B¯ D ∗ P − decays agree within errors • → 0 ( )0 0 B¯ D ∗ π small as expected • → B D∗ρ helicity amplitudes agree with Factorization (CLEO’03) • → 0 35% power corrections for B− D π− matrix elements • Nonz∼ ero strong phase δ 27 deg→rees ∼ Large N ? • c

Iain Stewart – p.29 Inclusive Processes – DIS

4 iq y B.F.P.R.S. e−p e−X T = i d y e · p T J †(y)J (0) p → µν h | { µ ν }| i R q q P 2 = Q2(1 x)/x X − standard OPE Q2 ∼ P endpoint region QΛ X ∼ QCD p resonance region Λ2 ∼ QCD qµ = Q (n¯µ nµ), n¯ p = Q/x, X hard, p collinear 2 − · µν 1 µν µν ¯ 0 1 ¯ Match O1 = Q g ξn,p W C ( +)W † ξn,p, O2 = . . . ⊥ P 1 p.d.f. p ξ¯ 0 W δ(ω ¯ ) n¯/ W †ξ p = n¯ p dξ δ(ω 2ξ n¯ p)f (ξ) n,p −P+ n,p · − · i/p Z0

2 1 1 2Qξ Result T1(x, Q ) = x 0 dξ C1 x , Q, µ fi/p(ξ, µ) R

Iain Stewart – p.30 Power Corrections

Iain Stewart – p.31 Reparameterization Invariance

Lorentz invariance restored by invariance Chay and Kim under changes in n, n¯ in SCET Manohar, Mehen, Pirjol, I.S. (Luke & Manohar, v in HQET) any choice with n2 = n¯2 = 0, n¯ n = 2 equally good • · connects pµ + kµ • Three classes act within and between orders in λ

n n + ∆⊥ n n n (1 + α) n (I) µ → µ µ (II) µ → µ (III) µ → µ  n¯ n¯  n¯ n¯ + ε⊥  n¯ (1 α) n¯  µ → µ  µ → µ µ  µ → − µ (0)  constrains collinear operators, eg. in c rules out • L 1 n¯/ ¯ µ µ µ µ ξn iDc⊥ W W † iDcµ⊥ ξn (with D⊥ = + gA⊥ ) ¯ 2 c P n,q n P o ⊥ relates coefﬁcients of operators at different orders in λ • (1) 1 n¯/ = ξ¯ iD/⊥ W W † iD/⊥ ξ + h.c. Lξξ n us ¯ c 2 n P n o Iain Stewart – p.32 Plus Gauge Invariance

Gauge complete pµ + kµ

µ µ 1) Dc + Dus, mixes orders in λ in gauge transformations Chay, Kim 2) Other possibilities are related by ﬁeld redeﬁnitions, and choices exist that do not mix orders in λ Beneke, Feldmann

3) A nice one is: Bauer, Pirjol, I.S.

in¯ D + W in¯ D W † · c · us µ µ iDc, + W iDus, W † ⊥ ⊥

Iain Stewart – p.33 Plus Gauge Invariance

Gauge complete pµ + kµ

µ µ 1) Dc + Dus, mixes orders in λ in gauge transformations Chay, Kim 2) Other possibilities are related by ﬁeld redeﬁnitions, and choices exist that do not mix orders in λ Beneke, Feldmann

3) A nice one is: Bauer, Pirjol, I.S.

in¯ D + W in¯ D W † · c · us µ µ iDc, + W iDus, W † ⊥ ⊥ Example: (1) 1 n¯/ = ξ¯ iD/⊥ W W † iD/⊥ ξ + h.c. Lξξ n us ¯ c 2 n n P o

Iain Stewart – p.33 Plus Gauge Invariance

Gauge complete pµ + kµ

µ µ 1) Dc + Dus, mixes orders in λ in gauge transformations Chay, Kim 2) Other possibilities are related by ﬁeld redeﬁnitions, and choices exist that do not mix orders in λ Beneke, Feldmann

3) A nice one is: Bauer, Pirjol, I.S.

in¯ D + W in¯ D W † · c · us µ µ iDc, + W iDus, W † ⊥ ⊥ Example: (1) 1 n¯/ = (ξ¯ W )iD/⊥ W † iD/⊥ ξ + h.c. Lξξ n us ¯ c 2 n nP o

Iain Stewart – p.33 Power Corrections

(λ) heavy-to-light currents (v⊥ = 0) O Chay, Kim Beneke,Chapovsky Diehl,Feldmann (1a) 1 (d)α J (ω) = ξ¯ i←D−⊥ W Υ h , Pirjol, I.S. i n c α ω ¯ i v P† (1b) 1 (d)α 1 J (ω1, ω2) = ξ¯nW Θ W †igB⊥ W hv , i ω1 i ¯ c α mb ω2 P Basis is valid for matching at any order in αs (λ2) heavy-to-light currents, usoft-collinear Lagrangian O µ (1) ¯ 1 , a ξq = ig ξn B/c⊥W qus + h.c. L in¯ Dc · q 1 (2a) = ig ξ¯ M/ W q + h.c. Lξq n in¯ D us (p, k) · c (2b) = . . . Lξq igB/⊥ = [in¯ D , iD/⊥], igM/ = [in¯ D , iD/ + n¯/n A /2] c · c c · c us · n Iain Stewart – p.34 B ? ? →

Iain Stewart – p.35 Factorization for Color-Suppressed Decays

0 ( )0 0 Mantry, Pirjol, I.S. (unpublished) B¯ D ∗ M →

new SCET factorization mechanism for color suppressed channels • quite predictive

Iain Stewart – p.35 Factorization for Color-Suppressed Decays

Mantry, Pirjol, I.S. (unpublished) First consider large Nc: "Tree" "Color suppressed" "Exchange"

π b c c D u d B D u b u, d b c d , u d B u,d B D d π π u u , d 0 + − 0 0 + B¯ D π B− D π− B¯ D π− → 0 0 → 0 0 0 → 0 0 B− D π− B¯ D π B¯ D π → → → 0 (Nc) 1/Nc 1/Nc

1 ( )0 0 ( ) ¯ ¯ ∗ 1/2 D ∗ π (db)(cu¯ ) Bd = F0 + Nc h | | i · · ·

1 ( )0 0 a a 1 ( ) ¯ ¯ ∗ 1/2 D ∗ π (dT b)(cT¯ u) Bd = N1 + Nc h | | i Nc · · ·

Iain Stewart – p.35 Factorization for Color-Suppressed Decays

Mantry, Pirjol, I.S. (unpublished) First consider large Nc: "Tree" "Color suppressed" "Exchange"

π b c c D u d B D u b u, d b c d , u d B u,d B D d π π u u , d 0 + − 0 0 + B¯ D π B− D π− B¯ D π− → 0 0 → 0 0 0 → 0 0 B− D π− B¯ D π B¯ D π → → → 0 (Nc) 1/Nc 1/Nc

¯0 0 0 A(B D∗ π ) (C2 + C1/Nc)F0∗ + (2C1/Nc)N1∗ Rπ | 0 → 0 0 | = = ? ≡ A(B¯ D π ) (C2 + C1/Nc)F0 + (2C1/Nc)N1 | → |

Iain Stewart – p.35 Factorization for Color-Suppressed Decays

Mantry, Pirjol, I.S. (unpublished) First consider large Nc: "Tree" "Color suppressed" "Exchange"

π b c c D u d B D u b u, d b c d , u d B u,d B D d π π u u , d 0 + − 0 0 + B¯ D π B− D π− B¯ D π− → 0 0 → 0 0 0 → 0 0 B− D π− B¯ D π B¯ D π → → →

0 + − 1 2 A(B¯ D π ) = A3/2 + A1/2 = T + E → √3 r 3 A(B− D0π−) = √3A = T + C → 3/2 0 0 0 2 1 1 A(B¯ D π ) = A3/2 A1/2 = (C E) A0 → r 3 − √3 √2 − ≡

Iain Stewart – p.35 Take Eπ Λ QCD Mediated by a single class of SCET operators T , (1), (1) I {O Lξq Lξq } (a) (b) b c b c u d , u d d , u d d , u u d , u

When matched onto SCETII we ﬁnd a factorization formula:

∗ D( ) ( ) + + (i)( ) (i) + + (i) + + A0 = N0∗ dx dz dk1 dk2 TL,R∗ (z) J (z, x, k1 , k2 ) SL,R(k1 , k2 ) φM (x) Z new non-perturbative soft B D functions (i = 0, 8) → (0,8) ¯(c) h a (b) ¯ a O = (hv0 S)Γ 1, T (S†hv ) (d S)k+ Γs 1, T (S†u)k+ { } 1 { } 2 ( )0 (0h ,8) 0 (0,8) + + i D ∗ O B¯ S (k , k ) same for D and D∗ h | | i → 1 2

Iain Stewart – p.36 Results

Find both C and E suppressed by Λ/Q relative to T • S is complex, gives non-perturbative strong phase φ which is • independent of M and choice of D vs. D∗

predict ∗ equal strong phases δD = δD D D equal amplitudes A0 = A0 ∗

corrections to this are αs(mb), Λ/Q

At tree level T (i)= constant • α (µ ) φ (x) S(i)(k+, k+) J (i) s 0 so get π 1 2 ∼ x k+ k+ x k+ k+ 1 2 Z Z 1 2

Iain Stewart – p.37 Test and Predictions

#1), #2) Non-perturbative functions drop out:

∗ 0 0 0 D π A(B¯ D∗ π ) δ R | → | = 1 , = 1 π ≡ A(B¯0 D0π0) δDπ | → | Expt (pdg average of Belle and CLEO):

◦ ¯0 0 0 3 Dπ +8 Br(B D π ) = (0.29 0.05) 10− , δ = 30◦ 14◦ → × − 0 0 0 3 D π Br(B¯ D∗ π ) = (0.25 0.07) 10− , δ ∗ = 30◦ 6◦ 1.0 → × = D = D* 0.5 Fit moments seff = seff eiφ | | Im seff , agree! 0.0 D D∗ (GeV) expect seff Λ , | | ∼ QCD -0.5 φ 1 ∼ A0 -1.0 tan(δ) = 3 | T | sin(φ) -1 -0.5 0. 0.5 1. − Re seff (GeV) Iain Stewart – p.38 Test and Predictions

#3), #4) Predictions for ρ (not post-dictions):

∗ 0 0 0 D ρ A(B¯ D∗ ρ ) δ R | → | = 1 , = 1 ρ ≡ A(B¯0 D0ρ0) δDρ | → | Experimental limits:

0 0 0 3 Br(B¯ D ρ ) < 0.39 10− 0 → 0 0 × 3 Br(B¯ D∗ ρ ) < 0.56 10− → ×

Iain Stewart – p.38 Test and Predictions

#5) Predict ρ to π ratios (number preliminary):

A(B¯0 D0ρ0) f x 1 ρ − ρ uncertainty Rρ | 0 → 0 0 | = # h 1i = 1.6 ( 20% ) ≡ A(B¯ D π ) fπ x π ∼ | → | h − i R2 = 2.56 1.0 ρ This prediction has corrections from jet scale. Experiment:

Br(D0ρ0) < 1.34 Br(D0π0)

Agreement with limit is not so good.

Iain Stewart – p.38 Test and Predictions

#6) 0 A(B− D π−) rDπ = | → | = 1.31 0.09 ¯0 + | | A(B D π−) ∗ | → | rD π = 1.24 | | rDρ = 1.25 | ∗ | rD ρ = 1.39 ∗| | rD ππππ = 1.02 0.27 | | From our factorization formula we can predict r’s at subleading order since C E √2A r = 1 + − + . . . = 1 + 0 + . . . T T Magnitude for rDπ comes out right with ﬁt for seﬀ Λ • ∼ QCD Predict them to be somewhat universal since f /f 1.6 cancels • ρ π ∼ 1 (C E) fM x− M 1 − = # h i = # x− M can extract moments T fM h i → Iain Stewart – p.38 Test and Predictions

This mechanism can also be used for other color suppressed decays

Iain Stewart – p.38 Test and Predictions

a ? 2 →

Iain Stewart – p.38 Other Processes

Inclusive Processes (operator formulation, arbitrary gauge):

Bauer, Fleming, Luke ('00) B Xsγ and B Xueν → → Bauer, Pirjol, I.S.('00) factorization and reproduce summation of logarithms • + ( ) e−p e−X , pp¯ X` `− , e−p γ ∗ p0 → → → Bauer, Fleming, Pirjol, Rothstein, I.S.('01) reproduces traditional factorization theorems • Υ Xγ → Bauer,Chiang,Fleming,Leibovich,Low('01) Fleming, Leibovich('02) resum logs for color singlet and octet mechanisms • + e e− jets → Bauer, Manohar, Wise ('02) smeared distributions are given by two non-perturbative • parameters

Iain Stewart – p.39 Other Processes

Exclusive Processes:

Korchemsky, Pirjol, Yan ('99) B γ`ν → Descotes-Genon, Sachrajda ('02) Lunghi, Pirjol, Wyler ('02) Bosch, Hill, Lange, Neubert ('03) no k convolutions at one-loop • ⊥

γ∗γ π, γ∗π π and power corrections Rothstein ('03) → → 2 enhanced power corrections in the f.f. αs(√QΛ) Λ • ∼ α(Q) Q

Iain Stewart – p.40 Conclusions

Factorization emerges as a property of effective theory • SCET simpli®es proofs of factorization formulae • Ward identities, invariant Lagrangian, pinched surfaces power counting Universal, applies to many inclusive/exclusive • processes

Allows power corrections to be addressed in a model • independent way

Results presented for • . B¯0 D(∗)0π0 →

Iain Stewart – p.41 B Xsγ near endpoint → 4 iq x µ Optical Thm: Γ Im d x e− · B T J †(x)J (0) B ∼ h | { µ }| i R q q P 2 = m (m 2E ) X B B − γ s standard OPE m2 ∼ B b PX b endpoint region mBΛQCD p ∼ 2 B resonance region Λ ∼ QCD

For EndPoint: E & 2.2 GeV, X collinear, B usoft, λ = ΛQCD γ s mB q Decay rate is given by factorized form Korchemsky, Sterman ('94)

Λ¯ 1 dΓ + + + = H(mb, µ) dk S(k , µ) J(k +mb 2Eγ , µ) Γ dE − − 0 γ Z2Eγ mb

Iain Stewart – p.42 Steps 1, 2

Bauer, Pirjol, I.S. i(mbv ) x Match: s¯Γ b e −P · C( ¯)ξ¯ W γ⊥P h µ → P n,p µ L v

n¯ − · † µ 4 i(mb 2 q) x label conservation Tµ = d x e B T Jeﬀ (x)Jeﬀ (0) B ¯ mb Z D E P →

¯ ¯(0) (0) Factor usoft: ξnW Γµhv = ξn W ΓµYn†hv

n¯ µ 2 4 i(mb −q)·x † T = C(mb) d x e 2 B T h¯vY (x) Y hv (0) B µ | | Z D E (0)† (0) (0) (0) µ 0 T W ξ (x) ξ¯ W (0) 0 Γµ Γ × n n × ⊗ D E

Iain Stewart – p.43 Step 3

Convolution: 0 0 JP (k+) h | · · · | i Deﬁne Fourier Transforms B B S(l) S(l+) h | · · · | i

2 Im T µ = C(m ) dl+S(l+) Im J (m 2E + l+) µ b P b − γ Z

where S(l+) = B h¯ δ(in D l+)h B h | v · − v| i Reproduces K.S. result CLEO '01 1850801-007 Data Spectator Model 2 Vub 40 Useful for extracting | ∗ | 2 V Vtb | ts | using B Xsγ and B Xueν spectrums→ →

Weights / 100 MeV 0 Neubert, Bigi et al., Leibovich, Low, Rothstein

1.5 2.5 3.5 4.5 E (GeV)

Iain Stewart – p.44 Colors

This is blue This is red This is brown This is magenta This is Dark Green This is Dark Blue This is Green This is Cyan Test how this color looks Test how this color looks Test how this color looks Test how this color looks Test how this color looks

Iain Stewart – p.45