Soft Collinear Effective Theory: An Overview

Transcription

1 Soft Collinear Effective Theory: An Overview Sean Fleming, University of Arizona EFT09, February 1-6, 2009, Valencia Spain

2 Background

3 Before SCET there was QCD Factorization Factorization: separation of long- and short-distance dynamics for interactions involving massless particles Classify sources of long-distance behavior in perturbation theory To identify and calculate short-distance quantities Scalar Vertex as Example (Dim. Reg.) 1 ( ɛ) 2 3 2( ɛ) Double and Single (InfraRed) Poles!

4 Origin of IR Poles in the Scalar Vertex All components of the loop momentum vanish together k µ 0 Soft region One light-cone component remains finite, all other components vanish k fixed k i 0 k + 0 Collinear region Scaling: Soft Collinear I II k µ λ 2 q 2 k µ λ q 2 k q 2 k + λ 2 q 2 k i λ q 2

5 All Order Analysis: Landau Equations All order behavior of diagrams is analyzed using the Landau Equations One loop results hold at all orders: IR singularities arise from Soft and Collinear regions Diagrams of IR regions: Reduced Diagrams E.g. Electromagnetic Form Factor J H: k µ q 2 H S J: Collinear S: Soft J

6 Soft Collinear Effective Theory Effective Field Theory of long-distance modes Recasts perturbative factorization as an EFT SCET Degrees of Freedom modes p µ = (+,, ) p 2 fields collinear Q(λ 2, 1, λ) Q 2 λ 2 ξ n, A µ n soft Q(λ, λ, λ) Q 2 λ 2 q s, A µ s usoft Q(λ 2, λ 2, λ 2 ) Q 2 λ 4 q us, A µ us

7 Soft Collinear Effective Theory SCET I Energetic jets usoft p µ Λ collinear p 2 c QΛ, λ = Λ/Q SCET II soft p µ Λ collinear p 2 c Λ 2, λ = Λ/Q Energetic hadrons

8 Soft Collinear Effective Theory Split QCD into two sectors: QCD Collinear Soft Describes interactions among the energetic particles L s = ψ s i/d s ψ s L c = ξ n {in D c + i/d c Interactions between soft and collinear particles 1 i n D c i/d c + gn A s Describes interactions among the soft particles } / n 2 ξ n

9 Before: SCET Soft Collinear Factorization L c = ξ n {in D c + i/d c 1 i n D c i/d c Field redefinition by eikonal phase: + gn A s } / n 2 ξ n After: ξ n,p Y n ξ n,p, A µ n,p Y n A µ n,py n Y n (x) = P exp ( ig 0 L (0) c = ξ n {in D c + id/ c ) ds n A s (ns+x) 1 id/ c in D c } n/ 2 ξ n

10 SCET Factorization Factorization of Hard from long-distance modes comes from Matching W = dω H(ω,µ)W SCET (ω,µ) Factorization Collinear from Soft by decoupling W SCET (ω,µ)= dl + J(ω, l +,µ)s(l +,µ)

11 Recent Progress:

12 Dynamical Threshold Enhancement in Drell-Yan T. Becher, M. Neubert, G. Xu τ = M 2 z = M 2 s ŝ τ 1 ln 2 (1 τ) Soft scale not M(1 τ) but Mλ 1 (1 τ) with λ 10! Large Logarithms at Hadronic Threshold : z 1 τ 1 Large Logarithms at Partonic Threshold but!

13 Dynamical Threshold Enhancement in Drell-Yan Both Effects Due to same source: Dynamical enhancement from the parton distributions τ 1 Then x 1 Simple parameterization of PDF: f q/n (x, µ) x 1 N q (µ)(1 x) b q(µ) Analytic K-factor at NLO: K(M 2,τ) ln M 2 λ 1 (1 τ) 2 µ 2 s λ 2b q 10

14 Dynamical Threshold Enhancement in Drell-Yan Dynamical enhancement important even for small τ 1 τ Fixed Order Resummed Fixed Order Resummed τ 0.04 τ 0.16

15 Resummation for Higgs Production at Hadronic Colliders V. Ahrens, T. Becher, M. Neubert, L.L. Yang For Higgs that are not too heavy use effective local interaction t H H L eff = C t (m 2 t,µ 2 ) H v G µν,ag µν a SCET G µν,a G µν a C S (Q 2,µ 2 )Q 2 g µν A µ,a n, Aν,a n, Q 2 = q 2

16 Resummation for Higgs Production at Hadronic Colliders H(m 2 H,µ 2 )= C S ( m 2 H iɛ, µ 2 ) 2 Sum logarithms by running this down to µ = m H There is a problem! space-like C S (Q 2,Q 2 ) = α s (Q 2 ) 0.152α 2 s(q 2 )+... time-like ln 2 Q 2 µ 2 ln2 Q 2 Q 2 =0 C S ( q 2,q 2 ) = α s (Q 2 ) ( i)α 2 s(q 2 )+ ln 2 q 2 iɛ µ 2 ln 2 q 2 iɛ q 2 = π 2

17 Resummation for Higgs Production at Hadronic Colliders Run to µ 2 = q 2 C S ( q 2, q 2 ) = α s ( q 2 ) 0.152α 2 s( q 2 )+... Analytically continue to large µ 2 α s (µ 2 ) α s ( µ 2 ) = 1 i β 0 4 α s(µ 2 ) + β 1 α s ( µ 2 ) β 0 4π ln [ 1 i β ] 0 4 α s(µ 2 ) + O(α 2 s)

18 Resummation for Higgs Production at Hadronic Colliders

19 Electroweak Corrections at High Energy J. Chiu, F. Golf, R. Kelley, A. Manohar Electroweak Sudakov Form Factor Di-jet Production Lepton Pair Production t t Production Squark Pair Production

20 Electroweak Sudakov Logarithms Electromagnetic form factor ( Q 2 m e 0) F E (Q) = q p 2 p 1 Sizable Electroweak radiative corrections ( s 4 TeV ) α 4π sin 2 θ W ln 2 ( s M 2 W,Z ) 0.15

21 Electroweak Sudakov Logarithms Sum Logarithms using an Infrared Evolution Equation J. Collins Series for ln F E takes on a simple form ln F E = Lf 0 (αl)+f 1 (αl)+αf 2 (αl)+... ln F E (Q 2 ) = ln F 0 (a(m)) + Q 2 dµ 2 M µ 2 2 [ ζ(a(µ)) + ξ(a(m)) + dµ 2 ] M µ 2 Γ(a(µ )) 2 µ 2

22 Electroweak Sudakov Logarithms Previous computation: all gauge bosons have common mass Conceptual problems with symmetry breaking and SU(2) U(1) mixing which leads to E.g. what do you do about ln Q2 Q2 ln M 2 MW 2 M γ =0 Use EFT methods + ln Q2 M 2 Z + ln Q2 M 2 γ M W M Z

23 EFT Approach Electroweak Sudakov Logarithms C(Q, µ) Full Theory SCET (M = 0) Running µ = Q D(M, µ) SCET (M) SCET (without massive gauge bosons) µ = M

24 Electroweak Sudakov Logarithms Q All double logarithms in ln F E of the form L ln are summed by the evolution M W,Z Q Single L ln appears in D(M, µ) at M W,Z Does this invalidate perturbation theory?!?!?! µ M No! Proof that there is at most a single logarithm in D, and the matching condition D = D 0 (M, µ)+d 1 (M, µ) ln Q2 µ 2 holds to all orders in perturbation theory

25 Event Shapes in e + e annihilation High energy collisions usually produce jets of hadrons Use event shapes to pick-out two-jet events Event shapes eare observables where a two-jet event has e 0

26 Event Shapes in e + e annihilation Precision determination of strong coupling Newest Fit: T. Becher, M. Schwartz

27 Thrust T = 1 Q max ˆt τ =1 T Event Shapes in Jet Invariant Mass ( MA,B 2 = ˆt p i i A,B Jet Broadening B = 1 ˆt p i Q i p i ) 2 e + e annihilation Two-jet limit τ 0 MA,B 2 0 B 0

28 Event Shapes in e + e annihilation C. Bauer, S. Fleming, C. Lee, G. Sterman Derivation of factored cross section in SCET First derivation that does not rely on quark-hadron duality

29 Event Shapes in e + e annihilation T. Becher, M. Schwartz Extraction of strong coupling using SCET

30 Event Shapes in e + e annihilation A. Hornig, C. Lee, G. Ovanesyan Factorization of Generalized Event Shape: Angularities

31 Event Shapes in e + e annihilation A. Hornig, C. Lee, G. Ovanesyan

32 More to come... Look for Updates at the 2009 SCET Workshop