Transcript 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.

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(M2/Q2,1,M/Q) ~ Q(l2,1,l)
• l << 1, and p2 ~ l2
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
SCET describes the light and energetic particles
SCET is QCD in a limit
Kinematics
B
p
Pion momentum:
p
p m = (2.640 GeV,0,0,-2.636 GeV)
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  p0 ,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. B255: 583, 1991
Politzer & Wise: Factorization (proposed) Phys. Lett. B257: 399, 1991
Beneke, Buchalla, Neubert, Sachrajda: QCD factorizaton (proved to 2 loops) Nucl. Phys. B591: 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/x2 singularity
kT factorization, Sudakov suppression
Light-cone sum rules
Symmetry relations: z(E), 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
}
q2
HQET
SCET
SCET factorization: all orders in as, leading order in l:
F(M2) = 12 fB fM
Bauer, Pirjol. Stewart:
hep-ph/0211069
∫ dz ∫ dx ∫ dr+ T(z,M,m0) J(z,x,r+,M, m0,m) jM(x,m) jB+(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: Q2 range where SCET is valid
mr = 770 Mev
Remember for SCET to be valid we need Q >> LQCD , mr
Q2 (GeV2)
0
0.25
1
2.25
4
6.25
E (GeV)
2.70
2.67
2.60
2.48
2.32
2.10
P (GeV)
2.58
2.56
2.48
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
0.155
0.330
0.165
0.360
0.180
Too
Big?!?!
Heavy-to-light factorization in SCET:
Details
F(Q2) =
∫ dz ∫ dx ∫ dr+ T(z,Q,m0) J(z,x,r+,Q m0,m) jM(x,m) jB+(r+,m)


1 f f
2 B M
Decay constants
Calculable
Light-cone
wave-functions
+ Ck(Q,m) xk(Q,m)
Calculable
Soft form factor
• T(z,Q,m0) & Ck(Q,m): Expansion in as(Q)
2/(2m )}
Q
~
{m
,E=m
-q
b
b
b
• J(z,x,r+,Q m0,m): Expansion in as(Q L )
}
Factorization in B  p p (K)
QCD Factorization Proposed:
F(M) = fBp(0)
Beneke, Buchalla, Neubert, Sachrajda: Phys. Rev. Lett. 83: 1914, 1999
Nucl. Phys. B591: 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
Phenomenology in heavy-to-light semileptonic decays
• Forward backward asymmetry
• Extraction of form factors
• Decay rates