A Short Introduction to the SoftCollinear Effective Theory

A Short Introduction to the Soft-Collinear Effective Theory Sean Fleming Carnegie Mellon University |Vxb| and |Vtx| A workshop on semileptonic and radiative B decays SLAC, December 2002

SCET Bauer, Fleming, Luke, Phys. Rev. D 63: 014006, 2001 Bauer, Fleming, Pirjol, Stewart, Phys. Rev. D 63: 114020, 2001 Bauer & Stewart, Phys. Lett. B 516: 134, 2001 Bauer, Pirjol, Stewart, Phys. Rev. D 65: 054022, 2002 Effective field Theory of highly energetic particles that have a small invariant mass E >> M: Near the lightcone • p = (p+, p-, p ) ~ Q(M 2/Q 2, 1, M/Q) ~ Q(l 2, 1, l) • l << 1, and p 2 ~ l 2 SCET has the right degrees of freedom for describing energetic particles interacting with soft “stuff” Analogous to HQET: Effective Field Theory of Heavy and soft degrees of freedom--describes heavy particles interacting with soft “stuff”

If you only remember one thing… Remember this picture: Heavy: HQET p B p Light and energetic: SCET describes the light and energetic particles SCET is QCD in a limit

Kinematics B p Pion momentum: p p m = (2. 640 Ge. V, 0, 0, -2. 636 Ge. V) p m m ≈Qn n = (1, 0, 0, -1) = (0, 2, 0 ) LC coordinates Corrections are small ~ LQCD , mp relative to Q • Expansion in LQCD Q or LQCD √ Q

Motivation Systematic: power counting in small parameter l Understand Factorization in a universal way § Key to separate hard contributions from soft & collinear § Systematic corrections to factorization (power counting) Symmetries § Reduce the number form-factors § In HQET where there is only the Isgur-Wise function Sum infrared logarithms § Sudakov logarithms

So…what’s it good for? SCET couple to HQET can be used for any decays involving stationary heavy, and fast light particles: B Dp, B pen, B ren, B K* g, B K e+ e- , B pp, B Kp, B Xu e n, B Xsg , U X g … DIS, Drell Yan, g*g p 0 , g p p, …

B D p factorization J. D. Bjorken: Color-transparency, Nucl. Phys. B (Proc. Suppl. ) 11, 1989, 325 Dugan & Grinstein: Factorization in LEET Phys. Lett. B 255: 583, 1991 Politzer & Wise: Factorization (proposed) Phys. Lett. B 257: 399, 1991 Beneke, Buchalla, Neubert, Sachrajda: QCD factorizaton (proved to 2 loops) Nucl. Phys. B 591: 313, 2000 Bauer, Pirjol, Stewart: SCET (proved to all orders in as) Phys. Rev. Lett. 87: 201806, 2001 Heavy: HQET p B D Light & Fast: SCET i 2 mb Ep fp FB D(0) ∫ dx T(x, m) fp(x, m) Soft B D form factor Hard coefficient calculate in PT: as(Mb) Light-cone pion wavefunction: nonperturbative

Semi-leptonic heavy-to-light Selected history: Brodsky et. al. (1990) Li & Yu (1996) Bagan, Ball, Braun (1997) Charles et. al. (1998) Beneke & Feldman (2000) Bauer et. al. (2000) Descotes, Sachradja (2001) Bauer, Pirjol, Stewart (2002) Pirjol & Stewart (2002) Hard part, 1/x 2 singularity k. T factorization, Sudakov suppression Light-cone sum rules Symmetry relations: z(E), z||(E) O(as) corrections, factorization proposal Collinear gluons, Ci(P), soft factorization More on Sudakov suppression Factorization in SCET Details of factorization in SCET

Semi-leptonic heavy-to-light e. g. B r l n at large recoil r B n } q 2 HQET SCET factorization: all orders in as, leading order in l: F(M 2) = 12 f. B f. M Bauer, Pirjol. Stewart: hep-ph/0211069 ∫ dz ∫ dx ∫ dr+ T(z, M, m 0) J(z, x, r+, M, m 0, m) j. M(x, m) j. B+(r+, m) + Ck(M, m) xk(q, m) Non-factorizable piece Non-perturbative form factors (restricted by symmetries in SCET) Factorizable piece Non-perturbative parameters: decay constants, LC wave functions Note both the pieces are same order in power counting!

B r l n: Q 2 range where SCET is valid mr = 770 Mev Remember for SCET to be valid we need Q >> LQCD , mr Q 2 (Ge. V 2) 0 0. 25 1 E (Ge. V) 2. 70 2. 67 2. 60 P (Ge. V) 2. 58 2. 56 2. 48 2. 25 4 6. 25 2. 48 2. 32 2. 10 2. 36 2. 19 1. 96 mr /E or mr /2 E 0. 286 0. 143 0. 288 0. 144 0. 300 0. 150 0. 310 0. 330 0. 360 0. 155 0. 165 0. 180 Too Big? !? !

Heavy-to-light factorization in SCET: Details F(Q 2) = ∫ dz ∫ dx ∫ dr+ T(z, Q, m 0) J(z, x, r+, Q m 0, m) j. M(x, m) j. B+(r+, m) f. B f. M 1 2 Decay constants Calculable Light-cone wave-functions + Ck(Q, m) xk(Q, m) Calculable Soft form factor • T(z, Q, m 0) & Ck(Q, m): Expansion in as(Q) 2/(2 m )} Q ~ {m , E=m -q b b b • J(z, x, r+, Q m 0, m): Expansion in as( Q L ) }

Factorization in B p p (K) QCD Factorization Proposed: F(M) = f. B p(0) Beneke, Buchalla, Neubert, Sachrajda: Phys. Rev. Lett. 83: 1914, 1999 Nucl. Phys. B 591: 313, 2000 ∫ dx T (x) F (x) + ∫ dx dx dy T (x, x, y) F (x) F (y) I II p p • Was shown to hold to order as Perturbative QCD: F(M) = 0 + Keum, Li, Sanda: hep-ph/0201103 ∫ dx dx dy T (x, x, y) F (x) F (y) II p p • Sum Sudakov logarithms No proof in SCET yet • It is not a given that this will give the above formula • Wait and see… p

What’s to come ? Proof of factorization in B p p • Phenomenology in heavy-to-light semileptonic decays • Forward backward asymmetry • Extraction of form factors • Decay rates