QCD
description
Transcript of QCD
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QCDHsiang-nan Li
Academia Sinica, TaipeiPresented at AEPSHEP
Oct. 18-22, 2012
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Titles of lectures
• Lecture I: Factorization theorem• Lecture II: Evolution and resummation• Lecture III: PQCD for Jet physics• Lecture IV: Hadronic heavy-quark decays
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References
• Partons, Factorization and Resummation, TASI95, G. Sterman, hep-ph/9606312
• Jet Physics at the Tevatron , A. Bhatti and D. Lincoln, arXiv:1002.1708
• QCD aspects of exclusive B meson decays, H.-n. Li, Prog.Part.Nucl.Phys.51 (2003) 85, hep-ph/0303116
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Lecture IFactorization theorem
Hsiang-nan LiOct. 18, 2012
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Outlines
• QCD Lagrangian and Feynman rules• Infrared divergence and safety• DIS and collinear factorization • Application of factorization theorem• kT factorization
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QCD Lagrangian
See Luis Alvarez-Gaume’s lectures
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Lagrangian • SU(3) QCD Lagrangian
• Covariant derivative, gluon field tensor
• Color matrices and structure constants
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Gauge-fixing• Add gauge-fixing term to remove spurious
degrees of freedom
• Ghost field from Jacobian of variable change, as fixing gauge
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Feynman rules
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Feynman rules
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Asymptotic freedom• QCD confinement at low energy, hadronic
bound states: pion, proton,…• Manifested by infrared divergences in
perturbative calculation of bound-state properties
• Asymptotic freedom at high energy leads to small coupling constant
• Perturbative QCD for high-energy processes
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Infrared divergence and safty
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Vertex correction• Start from vertex correction as an
example
• Inclusion of counterterm is understood
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Light-cone coordinates• Analysis of infrared divergences simplified
• As particle moves along light cone, only one large component is involved
2
),,(30 lll
llll T
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Leading regions• Collinear region• Soft region• Infrared gluon• Hard region
• They all generate log divergences
),,(~~
),,(~),,(~),,(
22
2
EEEll
lEEllll T
log~~~ 4
4
4
4
4
4
EEdd
lld
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Contour integration
• In terms of light-cone coordinates, vertex correction is written as
• Study pole structures, since IR divergence comes from vanishing denominator
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Pinched singularity• Contour integration over l-
• collinear region
• Soft region
Non-pinch
1
3
1
3
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Double IR poles
• Contour integration over l- gives
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e+e- annihilation• calculate e+e- annihilation • cross section = |amplitude|2 • Born level
fermion charge
momentum transfer squared
final-state cut
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Real corrections• Radiative corrections reveal two types of
infrared divergences from on-shell gluons• Collinear divergence: l parallel P1, P2• Soft divergence: l approaches zero
overlap ofcollinear andsoft divergences
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Virtual corrections• Double infrared pole also appears in virtual
corrections with a minus sign
overlap of collinear andsoft divergences
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Infrared safety• Infrared divergences cancel between real and
virtual corrections• Imaginary part of off-shell photon self-energy
corrections• Total cross section (physical quantity) of
e+e- -> X is infrared safe
)(Im 22 pipi
propagatoron-shellfinal state
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KLN theorem• Kinoshita-Lee-Neuberger theorem:
IR cancellation occurs as integrating over all phase space of final states
• Naïve perturbation applies
• Used to determine the coupling constant
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DIS and collinear factorization
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Deep inelastic scattering• Electron-proton DIS l(k)+N(p) -> l(k’)+X• Large momentum transfer -q2=(k-k’)2=Q2 • Calculation of cross section suffers IR
divergence --- nonperturbative dynamics in the proton
• Factor out nonpert part from DIS, and leave it to other methods?
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Structure functions for DIS• Standard example for factorization theorem
LOamplitude
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NLO diagrams
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NLO total cross section
infrared divergenceplus function
LO term
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IR divergence is physical!
• It’s a long-distance phenomenon, related to confinement.
• All physical hadronic high-energy processes involve both soft and hard dynamics.
q
qg
t=-infty t=0, when hard scattering occurs
Soft dynamics
Hard dynamics
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Collinear divergence• Integrated over final state kinematics, but
not over initial state kinematics. KLN theorem does not apply
• Collinear divergence for initial state quark exists. Confinement of initial bound state
• Soft divergences cancel between virtual and real diagrams (proton is color singlet)
• Subtracted by PDF, evaluated in perturbation hard kernel or Wilson coefficient
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Assignment of IR divergences
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Parton distribution function• Assignment at one loop
• PDF in terms of hadronic matrix element reproduces IR divergence at each order
splitting kernel
Wilson links
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Factorization at diagram levelEikonal approximation
lnn
lklkP
PPPlklk
lPPP
P
lklk
lPP
P
lllklk
lPlP
P
PPkklklk
lPlP
P
q
qqqq
qqq
q
q
qqq
2
22
2
22
22
)(
0,)(2
2
)(2
,)()(
,,)()(
k
Pq
l
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Effective diagrams• Factorization of collinear gluons at leading
power leads to Wilson line W(y-,0) necessary for gauge invariance
• Collinear gluons also change parton momentum
~
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Wilson links
0
y-
loop momentum does not flow through the hard kernel
loop momentum flows through the hard kernely-
0
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Factorization in fermion flow• To separate fermion flows for H and for
PDF, insert Fierz transformation
• goes into definition of PDF. Others contribute at higher powers
ji k
l
2)(2)( ljlj
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Factorization in color flow• To separate color flows for H and for PDF,
insert Fierz transformation
• goes into definition of PDF
ji k
l
Clj NI
for color-octet state, namelyfor three-parton PDF
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Parton model• The proton travels huge space-time,
before hit by the virtual photon• As Q2 >>1, hard scattering occurs at point
space-time• The quark hit by the virtual photon
behaves like a free particle• It decouples from the rest of the proton• Cross section is the incoherent sum of the
scattered quark of different momentum
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Incoherent sum
i2
i2
holds after collinearfactorization
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Factorization formula• DIS factorized into hard kernel (infrared finite,
perturbative) and PDF (nonperturbative)
• Universal PDF describes
probability of parton f carrying momentum fraction in nucleon N
• PDF computed by nonpert methods, or extracted from data
)0,0,( TPk
)()()()( 1 Nffxf xHdxF
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Expansion on light cone• Operator product expansion (OPE): expansion
in small distance • Infrared safe
• Factorization theorem: expansion in • Example: Deeply inelastic scattering (DIS)• Collinear divergence in longitudinal direction
exists (particle travels) finite
)0()( iii OyCXee
0 y
2y
y
y
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Factorization scheme• Definition of an IR regulator is arbitrary,
like an UV regulator: (1) ~1/IR+finite part
• Different inite parts shift between and H correspond to different factorization schemes
• Extraction of a PDF depends not only on powers and orders, but on schemes.
• Must stick to the same scheme. The dependence of predictions on factorization schemes would be minimized.
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2
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Extraction of PDF• Fit the factorization formula F=HDIS f/N to
data. Extract f/N for f=u, d, g(luon), sea
CTEQ-TEA PDFNNLO: solid colorNLO: dashedNLO, NNLO meansAccuracy of H
Nadolsky et al.1206.3321
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PDF with RG
see Lecture 2
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Application of factorization theorem
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Hard kernel• PDF is infrared divergent, if evaluated in
perturbation confinement• Quark diagram is also IR divergent.• Difference between the quark diagram and
PDF gives the hard kernel HDIS
_HDIS=
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Drell-Yan process• Derive factorization theorem for Drell-Yan
process N(p1)+N(p2)->+-(q)+X
1 p1
2 p2
p1
p2
+
-
f/N
f/N
X
X
Same PDF
*
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Hard kernel for DY
• Compute the hard kernel HDY
• IR divergences in quark diagram and in PDF must cancel. Otherwise, factorization theorem fails
HDY = _
Same as in DIS
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Prediction for DY
• Use DY=f1/N HDY f2/N to make predictions for DY process
f1/N
HDY
f2/N
DY=
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Predictive power• Before adopting PDFs, make sure at
which power and order, and in what scheme they are defined Nadolsky et
al.1206.3321
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kT factorization
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Collinear factorization• Factorization of many processes
investigated up to higher twists• Hard kernels calculated to higher orders• Parton distribution function (PDF)
evolution from low to high scale derived (DGLAP equation)
• PDF database constructed (CTEQ)• Logs from extreme kinematics resummed• Soft, jet, fragmentation functions all
studied
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Why kT factorization• kT factorization has been developed for
small x physics for some time• As Bjorken variable xB=-q2/(2p.q) is small,
parton momentum fraction x > xB can reach xp ~ kT . kT is not negligible.
• xp ~ kT also possible in low qT spectra, like direct photon and jet production
• In exclusive processes, x runs from 0 to 1. The end-point region is unavoidable
• But many aspects of kT factorization not yet investigated in detail
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Condition for kT factorization • Collinear and kT factorizations are both
fundamental tools in PQCD• x 0 (large fractional momentum exists)
is assumed in collinear factorization.• If small x not important, collinear
factorization is self-consistent • If small x region is important
• , expansion in fails• kT factorization is then more appropriate
yx 0 2y
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Parton transverse momentum• Keep parton transverse momentum in H • dependence introduced by gluon
emission• Need to describe distribution in
TT
TNfTfTxf
lkl
kkxHkddxF
,
),(),()()( 21
Tk
Tk
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Eikonal approximation
lnn
lklkP
PPlklk
lPPP
P
lklk
lPP
P
lllklk
lPlP
P
PPkklklk
lPlP
P
q
qqq
qqq
q
q
qqq
2
2
2
22
22
)(
0,)(2
2
)(2
,)()(
,,)()(
k
Pq
l
drop lT in numeratorto get Wilson line
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Effective diagrams• Parton momentum• Only minus component is neglected• appears only in denominator • Collinear divergences regularized by• Factorization of collinear gluons at leading
power leads to Wilson links W(y-,0)
~
),0,( TkPk
kT
Tk2Tk
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Factorization in kT spaceUniversal transverse-momentum-dependent (TMD) PDF describes probability of parton carrying momentum fraction and transverse momentumIf neglecting in H, integration over can be worked out, giving
),( TNf k
)(),( /2 NfTNfT kkd
TkTk
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Summary• Despite of nonperturbative nature of QCD,
theoetical framework with predictive power can be developed
• It is based on factorization theorem, in which nonperturbative PDF is universal and can be extracted from data, and hard kernel can be calculated pertuebatvely
• kT factorization is more complicated than collinear factorization, and has many difficulties