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3 I. Soft-Collinear Effective Theory The soft-collinear effective theory is a low energy effective theory for collinear and soft particles. (Bauer, Stewart, et al.; Beneke, Neubert….) (1) It simplifies the proof of factorization theorem at the Lagrangian and operator level. (2) The summation of large-logs can be performed in a new way , KITPC, Beijing
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11
Sudakov and heavy-to-light Sudakov and heavy-to-light form factors in SCETform factors in SCET
Zheng-Tao Wei Zheng-Tao Wei
Nankai UniversityNankai University
2009.9.9, KITPC, Beijing2009.9.9, KITPC, Beijing
2009.9.9, KITPC, Beijing2009.9.9, KITPC, Beijing 22
Wei, PLB586 (2004) 282,Wei, hep-ph/0403069,Lu, et al., PRD (2007)
Introduction to SCET
Sudakov form factor
Heavy-to-light transition form factors
Summary
33
I. Soft-Collinear Effective TheoryI. Soft-Collinear Effective Theory
The soft-collinear effective theory is a low energy effective theory for collinear and soft particles. (Bauer, Stewart , et al.; Beneke, Neubert….)
(1) It simplifies the proof of factorization theorem at the Lagrangian and operator level.
(2) The summation of large-logs can be performed in a new way.
2009.9.9, KITPC, Beijing2009.9.9, KITPC, Beijing
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. regluon whesoft a with interactsquark energetican :processelmentary an Consider
QCDQ
• Diagrammatic analysis and effective Lagrangian
)0,,0( Qpc
eikonal approximation
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Transforming the diagrammatic analysis into an effective Lagrangian
0.WD where :onredefiniti Field ss0 inWs
}.)( exp{)( line Wilson The x
tnAigndtigPxW
LEET
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• Power counting
),,(~ :ultrasoft
),,(~ :soft),1,(~ :collinear
222
2
us
s
c
p
pp
MomentumField
23
2/3
2
~ ,~
~ ,~
),1,(~ ,~
usus
ss
c
Aq
Aq
A
• Degrees of freedom
SCET(I)for /
SCET(II)for /
QCD
QCD
Q
Q
Reproduce the full IR physics
2009.9.9, KITPC, Beijing2009.9.9, KITPC, Beijing
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),,Q(~
) ,1 ,(~
,~ :SCET(I)
222
2
QCD2
us
c
c
p
Qp
Qp
The effective Lagrangian:
• The effective interaction is non-local in position space.
• Two different formulae: hybrid momentum-position space and position space representation.
/QCD Q
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Gauge invariance
• The ultrasoft field acts as backgroud field compared to collinear field.
The collinear and ultrasoft gauge transformation are constrained in corresponding regions,
2009.9.9, KITPC, Beijing2009.9.9, KITPC, Beijing
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Wilson lines
ccc
us
WinWDin
W
11
on redefiniti Field0
Gauge invariant operators: (basic building blocks)
No interaction with usoft gluons
sv , , qWhWW ususcc
. , :ionfactorizatusoft -collinear 00 uscuscsu WAWAW
2009.9.9, KITPC, Beijing2009.9.9, KITPC, Beijing
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),,Q(~ ),1,(~
,~ :SCET(II)2
22
s
c
c
pQp
p /QCD Q
Matching: mismatch? New mode, such as soft-collinear mode proposed by Neubert et al.?
Endpoint singularity?
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1111
Q
Q
SCET(I)
SCET(II)
Two step matching:
1. Integrate out the high momentum fluctuations of order Q,
2. Integrate out the intermediate scale (hard-collinear field)
SCET(II)SCET(I)QCD 21
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2009.9.9, KITPC, Beijing2009.9.9, KITPC, Beijing 1212
Form factor
The matrix elements of current operator between initial and final states are represented by different form factors.
Form factors are important dynamical quantity for describing the inner properties of a fundamental or composite particle.
III. Sudakov form factorIII. Sudakov form factor
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The form factor (only the first term F1(Q2)) in the asymptotic limit q2→∞ is called Sudakov form factor. (in 1956)
).()](2
)()['( 22
21 puqF
mqi
qFpu
The interaction of a fermion with EM current is represented by
At q2=0 , the g-factor is given by and the anomalous magnetic moment is
sgme
qFg
2
)0(2
2 22
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The large double-logarithm spoils the convergence of pertubative expansion. The summation to all orders is an exponential function. The form factor is strongly suppressed when Q is large. In phenomenology, it relates to most high energy process in certain momentum regions, DIS, Drell-Yan, pion form factor, etc.
The naïve power counting is strongly modified (at tree level F=1).
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• Methods of momentum regions (by Beneke, Smirnov, etc)
The basic idea is to expand the Feynman diagram integrand in the momentum regions which give contributions in dimensional regularization. Each region is involved by one scale.
). , ,(~ :regionsoft the)4(
); ,1 ,(~ :region B-collinear the)3(
); , ,1(~ :regionA -collinear the(2)
);1 ,1 ,1(~gluon virtual the:region hard the(1) :from come onscontributi thefactor, formquark In the
222
2
2
Qk
Qk
Qk
Qk
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Regularization method
Introduce a cutoff scale δ in both k+ and k-.
DOF
),,Q(~
),1,(~
,~ :SCET(I)
222
2
2
us
c
c
p
Qp
Qp
Bauer (2003)
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Factorization
Step 1: the separation of hard from collinear contributions.
Step 2: the separation of soft from collinear functions.
. , :onredefiniti Field 00 scscs WAWAW
Q,at SCET ofoperator theonto QCDin current theMatching AB
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Evolution
The anomalous dimension depends on the renormalization scale. The exponentiation is due to the RGE. The suppression is caused by the positive anomalous dimension.
Two-step running:
]lnexp[]lnexp[)()( 2
Q S
Q
Q C ddQCQF
2009.9.9, KITPC, Beijing2009.9.9, KITPC, Beijing
2020
Exponentiation and scaling
.dg')(g') ,(g'Exp)C()C(
0,(g) and ) (g, :III Case
ion.approximat LLin ,)()()()C(
0,(g) and (g) :II Case
law.-power )()C(
0,(g) and zero-nonbut constant, :I Case
.ln
)(ln :RGE
)(
)g(0
)2/(
00
00
0
00
g
s
sC
C
dCd
Exponential of logs can be considered as a generalized scaling.
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Comparisons with other works
1. The leading logarithmic approximation method sums the leading contributions from ladder graphs to all orders. The ladder graphs constitutes a cascade chain: qq->qq->…->qq. There are orderings for Sudakov parameters.
2. Korchemsky et al. used the RGE for a soft function whose evolution is determined by cusp dimension. The cusp dimension contains a geometrical meaning.
).( spacen Minkowskiain anglewith
)( ),(
gg cuspcusp
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3. CSS use a diagrammatic analysis to prove the factorization. The RGEs are derived from gauge-dependence of the jet and hard function. The choice
of gauge is analogous to the renormalization scheme.
).(2)(lnln
);,/(),/(lnln
ggddK
ddG
gQGgmKQF
cuspK
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IV. Heavy-to-light transition form factorIV. Heavy-to-light transition form factor
The importance of heavy-to-light form factors:
CKM parameter Vub
QCD, perturbative, non-perturbative basic parameters for exclusive decays in QCDF or SCET new physics…
At large recoil region q2<<mb2, the light meson moves
close to the light cone.
QCD2
QCD ~ ,~ ,/~ PmPmP bb
Light cone dominance
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Hard gluon exchange: soft spectator quark → collinear quark Perturbative QCD is applicable.
2QCDQCD
2 ~)(
momentumgluon exchanged The
bcs mpp
Hard scattering
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Endpoint singularity
1
0
1
0 2 ...)()()1()()()12(
xxx
xxddxF BBB
0,0
x
endpoint singularity
Factorization of pertubative contributions from the non-perturbative part is invalid. There are soft contributions coming from the endpoint region.
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Hard mechanism -- PQCD approach
The transverse momentum are retained, so no endpoint singularity. Sudakov double logarithm corrections are included.
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Momentum of one parton in the light meson is small (x->0). Soft interactions between spectator quark in B and soft quark in light meson.
Methods: light cone sum rules, light cone quark model… (lattice QCD is not applicable.)
Soft mechanism
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Spin symmetry for soft form factor
In the large energy limit (in leading order of 1/mb),
The total 10 form factors are reduced to 3 independent factors. 3→1 impossible!
J. Charles, et al., PRD60 (1999) 014001.
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Definition
)*)((2| |
)v*)((2| |
)(2| |
v
||v5
Pv
EEBhV
EmiBhV
EEBhP
V
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QCDF and SCET
In the heavy quark limit, to all orders of αs and leading order in 1/mb,
)()(),( )()( 2 yxyxTdxdECqF BMiiii
Sudakovcorrections
Soft form factors,with singularity andspin symmetry Perturbative, no singularity
The factorization proof is rigorous. The hard contribution ~ (Λ/mb)3/2, soft form factor ~ (Λ/m b)2/3 (?) About the soft form factors, study continues, such as zero-bin method…
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Zero-bin method by Stewart and Manohar
A collinear quark have non-zero energy. The zero-bin contributions should be subtracted out. After subtracting the zero-bin contributions, the remained is finite and can be factorizable.
For example,
)/(ln)0(')0(')0()()( 1
0 2
1
0 2
pdx
xxxdx
xx
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Soft overlap mechanism
The soft part form factor is represented by the convolution of initial and final hadron wave functions.
Fock
jjLiiBjjiijikxkxdkdxdkdxq ),(),()(
,
2
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Dirac’s three forms of Hamiltonian dynamics( S. Brodsky et al., Phys.Rep.301(1998) 299 )
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Advantage of LC framework
LC Fock space expansion provides a convenient description of a hadron in terms of the fundamental quark and gluon degrees of freedom. The LC wave functions is Lorentz invariant. ψ(xi, k┴i ) is independent of the bound state momentum. The vacuum state is simple, and trivial if no zero-modes. Only dynamical degrees of freedom are remained. for quark: two-component ξ, for gluon: only transverse components A┴.
Disadvantage In perturbation theory, LCQCD provides the equivalent results as the covariant form but in a complicated way.
It’s difficult to solve the LC wave function from the first principle.
2009.9.9, KITPC, Beijing2009.9.9, KITPC, Beijing
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LC Hamiltonian Kinetic Vertex
Instantaneous interaction
LCQCD is the full theory compared to SCET.
Physical gauge is used A+=0.
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LC time-ordered perturbation theory
Diagrams are LC time x+-ordered. (old-fashioned) Particles are on-shell. The three-momentum rather than four- is conserved in each vertex. For each internal particle, there are dynamic and instantaneous lines.
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Perturbative contributions:
Only instantaneous interaction in the quark propagator.
The exchanged gluons are transverse polarized.
Instantaneous, no singularitybreak spin symmetry
have singularity,conserve spin symmetry
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Basic assumptions of LC quark model
Valence quark contribution dominates.
The quark mass is constitute mass which absorbs some dynamic effects.
LC wave functions are Gaussian.
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LC wave functions
2MPPH LC
In principle, wave functions can be solved if we know the Hamiltonian (T+V).
Choose Gaussian-type
Power law: )/exp(-~)( QCD2
bmq
The scaling of soft form factor depends on the light meson wave function at the endpoint.
2009.9.9, KITPC, Beijing2009.9.9, KITPC, Beijing
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Melosh rotation 22
0
'0
'
)(
)(2
),()',()(
pMxm
npiMxmm
spuspuR
ii
ssiii
iiDssM
meson. for vector
),,()(),(~21
meson;ar pseudoscalfor
),,(),(~21
:) ,( seigenstatehelicity LC ofout ),(spin of of state a constructs
2221
110
1
225110
00
21
21
21
21
kvwkkku
MR
kvkuM
R
SSR
V
S
zSS
z
z
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Numerical results
The values of the three form factors are very close, but they are quite different in formulations.
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Comparisons with other approaches
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SummarySummary
SCET provides a model-independent analysis of processes with energetic hadrons: proof of factorization theorem, Sudakov resummation, power corrections.
SCET analysis of Sudakov form factor emphasizes the scale point of view.
LC quark model is an appropriate non-perturbative method to study the soft part heavy-to-light form factors at large recoil.
How to treat the endpoint singularity is still a challenge.
2009.9.9, KITPC, Beijing2009.9.9, KITPC, Beijing
ThanksThanks
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