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Transcript of 1 SCET for Colliders Matthias Neubert Cornell University LoopFest V, SLAC – June 21, 2006 Based on...
![Page 1: 1 SCET for Colliders Matthias Neubert Cornell University LoopFest V, SLAC – June 21, 2006 Based on work with Thomas Becher (FNAL) and Ben Pecjak (Siegen)](https://reader035.fdocuments.net/reader035/viewer/2022062806/56649dc85503460f94abe122/html5/thumbnails/1.jpg)
1
SCET for Colliders
Matthias NeubertCornell University
LoopFest V, SLAC – June 21, 2006
Based on work with Thomas Becher (FNAL) and Ben Pecjak (Siegen)
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SCET for Colliders …
Introduction Overview of SCET literature
(hard QCD processes outside B physics) Parton showers Factorization in DIS (x→1) Threshold resummation in momentum space
(DIS and Drell-Yan) Conclusions
[Bauer, Schwartz, hep-ph/0604065]
[Becher, MN, hep-ph/0605050]
[Becher, MN, Pecjak, to appear]
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Introduction
Generic problem in QCD: Resummation for processes with >1 scales Interplay of soft and collinear emissions
→ Sudakov double logarithms Jet physics: MX
2 « Q2
Soft:Soft: low momentum pμ→0 Collinear:Collinear: p || pX with p2→0
Examples: DIS, fragmentation, Drell-Yan, Higgs production, event shapes, inclusive B decays, …
MX
(see talk by T. Becher)
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Introduction
Problems of scale separation often best addressed using effective field theory
Natural framework for studying questions of factorization,factorization, resummation, resummation, and power corrections power corrections
Approach first developed for B physics, later applied to other hard QCD processes
Soft-Collinear Effective Theory[Bauer, Pirjol, Stewart (2000, 2001)]
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Introduction
SCET is not the invention of the wheel (given 20+ year history in this field)
Most of what can be done with SCET can be done with conventional techniques (in fact, we never use SCET Feynman rules!)
However, SCET may provide a novel perspective on factorization, scale separation, resummation, and power corrections in applications where interplay of soft and collinear radiation is relevant
Existing analyses just the beginning; much room for future work
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Overview of SCET literature
Factorization for π-γ form factor, light-meson form factors, DIS, Drell-Yan, and deeply virtual Compton scattering
Factorization (or “non-factorization”) and threshold resummation in DIS for x→1
pt resummation for Drell-Yan and Higgs production
Threshold resummation for Higgs production
[Bauer, Fleming, Pirjol, Rothstein, Stewart (2002)]
[Manohar (2003, 2005); Pecjak (2005); Chay, Kim (2005);Idilbi, Ji (2005); Becher, MN (2006); Becher, MN, Pecjak (in prep.)]
[Gao, Li, Liu (2005); Idilbi, Ji, Yuan (2005)]
[Idilbi, Ji, Ma, Yuan (2006); Idilbi, Ji, Yuan (2006)]
Ouch!
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Overview of SCET literature
Nonperturbative effects on jet distributions in e+e- annihilation
Universality of nonperturbative effects in event shapes
Parton showers
In this talk: Factorization and threshold resummation in DIS and Drell-
Yan production Parton showers (briefly…)
[Bauer, Manohar, Wise (2002); Bauer, Lee, Manohar, Wise (2003)]
[Lee, Sterman (2006)]
[Bauer, Schwartz (2006)]
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Parton Showers[Bauer, Schwartz, hep-ph/0604065]
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An interesting proposal
Process of parton showering as a sequence of hard matchings in SCET onto operators containing increasing number of hard-collinear fields
Sudakov logs resummed using RG equations
Straightforward to go beyond LL approximation
(Courtesy M. Schwartz)
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An interesting proposal
Leading effective operator (two collinear fields) same as in Drell-Yan 2-loop matching coefficient known (see below)
3-loop anomalous dimension known (see below)
Questions: Is this really an advance over existing approaches
(MC@NLO)? How to implement in a generator? Details of calculations (NLO and beyond)?
Eagerly await long paper …!
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SCET and DIS
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SCET analysis of DIS for x→1
Simplest example of a hard QCD process
SCET can be used to rederive elegantly all existing results
Provides much simplermuch simpler result than conventional approach for threshold resummation
Cross section: d2σ/dx·dQ2 ~ F2(x,Q2)
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SCET analysis of DIS for x→1
Will discuss: Factorization for x→1 Threshold resummation at NNLO (N3LL) Connection with conventional approach Numerical results
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Factorization
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Factorization for x→1
QCD factorization formula:
Most transparent to derive this in SCET: need hard-collinear, anti-collinear, and soft-collinear modes (called “soft” in the literature)
Resum threshold logarithms by solving RGEs of SCET in momentum space
[Sterman (1987); Catani, Trentadue (1989); Korchemsky, Marchesini (1992)]
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Factorization for x→1
Momentum modes in Breit frame (→ fields in SCET): Hard: ph ~ Q(1,1,1) Hard-collinear (final-state
jet): phc ~ Q(ε,1,√ε) Anti-collinear (initial-state
nucleon): pc ~ Q(1,λ2,λ) Soft-collinear (“soft”)
messengers: psc ~ Q(ε,λ2,λ√ε)
hc
scc
[Sterman (1987)]
c
(here ε=1-x and λ~Λ/Q)
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Step 1:Step 1: At hard scale μ~Q, match QCD vector current onto current operator in SCET
Step 2:Step 2: Hard-collinear and anti-collinear fields can interact via exchange of soft-collinear particles; at leading power, their couplings to hard-collinear fields can be removed by field redefinitions
Step 3:Step 3: After decoupling, vacuum matrix element of hard-collinear fields can be evaluated in perturbation theory (for μ~MX=Q√1-x)
Step 4:Step 4: Identify remaining nucleon matrix element over anti-collinear and soft-collinear fields with PDF in endpoint region (x→1)
[Becher, MN, Pecjak, to appear]
SCET factorization: Outline
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SCET factorization
Step 1:Step 1: current matching
Implication for hadronic tensor:Q2
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SCET factorization
Simplest to obtain hard matching coefficient from bare on-shell QCD form factor
Matching converts IR poles into UV poles (subtraction of scaleless SCET graphs):
[Kramer, Lampe (1987, E: 1989); Matsuura, van Neerven (1988);Gehrmann, Huber, Maitre (2005); Moch, Vermaseren, Vogt (2005)]
UV renormalization factor
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SCET factorization
2-loop result (with L=ln(Q2/μ2)):
with:
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SCET factorization
Step 2:Step 2: decoupling transformation
Vacuum matrix element over hard-collinear fields factorizes into a jet function:
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SCET factorization
Step 3:Step 3: compute jet function perturbatively (known at 2-loop order)
[Becher, MN, hep-ph/0603140]
quark propagator inlight-cone gauge
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SCET factorization
Step 4:Step 4: identify PDF in endpoint region
soft-collinear Wilson loop:
[Sterman (1987)][Korchemsky, Marchesini (1992)]
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Threshold Resummation
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Threshold resummation
Traditionally, resummation is performed in Mellin moment space Landau poles (in Sudakov exponent and Mellin
inversion) Mellin inversion only numerically Non-trivial matching with fixed-order calculations
in momentum space
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Threshold resummation
Define moments of structure function and PDF:
Short-distance coefficients CN can be written:
N-independent
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Threshold resummation
Resummed exponent:
Integrals run over Landau pole in running couplg. (ambiguity ~(Λ/MX)2 for DIS, ~Λ/MX for Drell-Yan)
Additional singularity encountered in Mellin inversion (physical scales in moment scales are Q2 and Q2/N)
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Threshold resummation
Solving RG equations in SCET, we obtain all-orders resummed expressions directly in momentum space momentum space (x space) Transparent physical interpretation, no Landau
poles, simple analytical expressions Reproduce moment-space expressions order by
order in perturbation theory Understand IR singularities of QCD in terms
of RG evolution (UV poles) in EFT
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Evolution of the hard function
RG functions: Sudakov exponent
Anomalous exponent
Functions of running couplings αs(μ), αs(ν)
RG equation:
Exact solution:
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3-loop anomalous dimension γV
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Evolution of the jet function
Integro-differential evolution equation:
Exact solution Exact solution (via Laplace transformation):
with:
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Evolution of the jet function
2-loop result:
with:[Becher, MN, hep-ph/0603140]
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3-loop anomalous dimension γJ
derived(see below)
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Evolution of the PDF
RG invariance of DIS cross section implies evolution equation for PDF for ξ→1:
with:
→ has been used to derive 3-loop coefficient of γJ
[Moch, Vermaseren, Vogt (2004)]
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Evolution of the PDF
Endpoint behavior can be parameterized as
where:
Will use this to perform final convolutions
running exponent!
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Results
Exact all-orders momentum-space formula:
No integrals over Landau poles! Physical scales μh ~ Q and μi ~ Q√1-x cleanly
separated from factorization scale μf
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Results
Performing final convolution integral yields the K-factor:
Explicit dependence on physical scales Q and MX
Factor (1-x)η is source of huge K-factor if μf>μi (i.e., η<0)
MX2
(MX/Q)2η
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Results
Analogous result obtained for Drell-Yan:
Straightforward to expand these results order by order in RG-resummed perturbation theory (known to NNLO = N3LL)
MX2
(MX/√s)2η
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Connection with conventional approach
Recall conventional formula (moment space):
Work out how g0, Aq, and Bq are related to objects in SCET (anomalous dimensions and Wilson coefficients)
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Connection with conventional approach
Find (with = d/dlnμ2):
Bq (as well as g0) not related to simple field-theoretic objects in EFT, but to complicated combinations of anomalous dimensions and matching coefficients
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Connection with conventional approach
It has been claimed that resummation in x-space is plagued by strong factorial growth of expansion coefficients not related to IR renormalons
Leads to “unphysical” power corrections ~ (Λ/Q)γ with γ = 1.44 / 0.72 for Drell-Yan in MS / DIS scheme, and γ = 0.16 for heavy-quark production in gluon-gluon fusion
[Catani, Mangano, Nason, Trentadue (1996)]
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Connection with conventional approach
In our approach this problem has been overcome!
Indeed, perturbative convergence is better in x-space than in N-space (see below)
Physical IR renormalon poles (unavoidable) arise in matching conditions only and are commensurate with power corrections from higher-dimensional operators in SCET: CV(Q,μ) → (Λ/Q)2 at hard scale j(L,μ) → (Λ/MX)2 at jet scale
~
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Connection with conventional approach
Absence of unphysical power corrections follows from very existence of effective theory Difference with Catani et al. is that we fix the
intermediate scale μi~MX at the end, after all integrals are performed
Also, their LL approximation does not correspond to any consistent truncation in EFT approach
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Numerical Results
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Resummed vs. fixed-order PT
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Perturbative uncertainties
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Resummation in x- vs. N-space
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Conclusions
Methods from effective field theory provide powerful, efficient tools to study factorization, resummation, and power corrections in many hard QCD processes
Have resummed Sudakov logarithms directly in momentum space by solving RGEs
Results agree with traditional approach at every fixed order in perturbation theory, but are free of spurious Landau-pole singularities
Easier to match with FOPT results for differential cross sections away from threshold region
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Conclusions
What else can SCET do for you? Will try to get more mileage out of resummation Possible to study power corrections
systematically (often messy) SCET approach to parton showers appears
promising! Understand miracles of N=4 SUSY Yang-Mills? … ?
More at LoopFest VI