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Soft-Collinear Effective Field Theory and Collider Physics
Thomas Becher Bern University
Higgs Centre School of Theoretical Physics, U. of Edinburgh, May 2016
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Tools for QFT computations
Numerical methods:
lattice simulations
Expansion in the interaction
strength: perturbation
theory
Toy models: solvable models SUSY theories
AdS/CFT
Expansion in scale ratios:
Effective Field Theories
Benefits of EFTs• Simplification: expansion in scale ratios simplifies computations
• Factorization of physics at different energy scales
• Dimensional analysis
• Separate perturbative from non-perturbative physics
• Perturbation theory works.
• It cannot be applied in multi-scale problems due to large logarithms.
• RG improved perturbation theory resums large logarithms.
• Symmetries
• emergent: heavy-quark symmetry
• approximate: chiral symmetry
• General framework also for cases where the full theory is not known, or cannot be used for computations
Jet physics at the LHC
Many scale hierarchies!
→ Soft-Collinear Effective Theory (SCET)
ps � pT
Jet
� MJet
� Eout
� mproton
⇠ ⇤QCD
Example: Drell-Yan process
The total DY cross section involves physics associated with two scales
• hard scales q2=M2, …
• low-energy scale ΛQCD~1 GeV (hadronic bound state effects)
lepton pair
X (arbitrary hadrons)
protonγ*, Z0
q2 = M2
`1
`2
q = p`1 + p`2
proton
Cross section factorizes into parton distribution functions (PDFs) convoluted with partonic amplitudes
Can be obtained from an analysis within SCET.
• PDFs are operator matrix elements in SCET
• Partonic cross sections are Wilson coefficients of the operators
After this introduction to the basic properties of QCD, we now turn to review the application of perturbative QCD in high-energy collisions
The QCD processes that we will study are the following:
Juan Rojo University of Oxford, 06/05/2014
Electron-positron annihilationNo hadrons in initial state
Deep-inelastic scatteringOne hadrons in initial state
Hadron collisionsTwo hadrons in initial state
Hadron collisionsTwo hadrons in initial state
Parton showersRealistic hadronic final state
partonic cross section i + j → l+ l- + X
high-energy part
perturbatively calculable
PDFs
low-energy part
non-perturbative universal
d4σ
dq4=
∫ 1
0
dx1
∫ 1
0
dx2 fi(x1, µ) fi(x2, µ)d4σ̂ij(x1, x2, µ)
d4q+O
(
Λ2QCDq2
)
(1)
1
qT spectrum of Z-bosons
For qT ≪ M the partonic cross section contains two widely separated scales.
qT resummationPhenomenology
Appendix
IntroductionFactorizationResummation
ObservableConsider:N1 + N2 ! F (q) + X
F = l+l�, Z , W , H, Z 0, . . .
Test SM to high precision.
d�/dqT in region q2T ⌧ M2.
) Need to resum largelogarithms.
) Transverse PDFs (TPDFs)(Beam functions).
) F recoils against initial stateradiation
pp→Z+X @ 7TeV|pT
l |>20GeV,|ηl |
The integrated cross section
has for low qT an expansion of the form ( )
leading logarithms, next-to-leading logarithms, …
Sudakov logarithms1 Formluae
⌃(qT ) =Z qT
0dq0T
1
�
d�
dq0T(1)
1
1 Formlulae
⌃(qT ) =
Z qT
0dq0T
1
�
d�
dq0T(1)
⌃(qT ) = 1+↵s
✓c2 ln
2 q2T
M2+ c1 ln
2 q2T
M2+ c0
◆+↵2s
✓c4 ln
4 q2T
M2+ c3 ln
3 q2T
M2+ . . .
◆+O(↵3s,
q2TM2
)
⌃(qT ) = 1+↵s
✓c2 ln
2 q2T
M2+ c1 ln
2 q2T
M2+ c0
◆+↵2s
✓c4 ln
4 q2T
M2+ c3 ln
3 q2T
M2+ . . .
◆+. . .
⌃(qT ) = 1+↵s
✓c2 ln
2 q2T
M2+ c1 ln
2 q2T
M2+ c0
◆+↵2s
✓c4 ln
4 q2T
M2+ c3 ln
3 q2T
M2+ . . .
◆+. . .
⌃(qT ) = 1+↵s�c2L
2+ c1L+ c0
�+↵2s
�c4L
4+ c3L
3+ . . .
�+↵3s
�c6L
6+ c5L
6+ . . .
�+. . .
L = lnq2TM2
(2)
⌃(pT ) = exp�Lg1(↵sL) + g2(↵sL) + ↵sg3(↵sL) + ↵
2sg4(↵sL) + . . .
�
1
1 Formlulae
⌃(qT ) =
Z qT
0dq0T
1
�
d�
dq0T(1)
⌃(qT ) = 1+↵s
✓c2 ln
2 q2T
M2+ c1 ln
2 q2T
M2+ c0
◆+↵2s
✓c4 ln
4 q2T
M2+ c3 ln
3 q2T
M2+ . . .
◆+O(↵3s,
q2TM2
)
⌃(qT ) = 1+↵s
✓c2 ln
2 q2T
M2+ c1 ln
2 q2T
M2+ c0
◆+↵2s
✓c4 ln
4 q2T
M2+ c3 ln
3 q2T
M2+ . . .
◆+. . .
⌃(qT ) = 1+↵s
✓c2 ln
2 q2T
M2+ c1 ln
2 q2T
M2+ c0
◆+↵2s
✓c4 ln
4 q2T
M2+ c3 ln
3 q2T
M2+ . . .
◆+. . .
⌃(qT ) = 1+↵s�c2L
2+ c1L+ c0
�+↵2s
�c4L
4+ c3L
3+ . . .
�+↵3s
�c6L
6+ . . .
�+. . .
L = lnq2TM2
(2)
⌃(pT ) = exp�Lg1(↵sL) + g2(↵sL) + ↵sg3(↵sL) + ↵
2sg4(↵sL) + . . .
�
1
For large L ~ 1/αs the standard perturbative expansion breaks down: need to (re)sum the large logs to all orders!
Soft-collinear factorization
Basis for factorization and higher-log resummation.
H
J J
J J
S
Collins, Soper, Sterman, ...
collinear emissions
soft emissions
virtual corrections
Soft-Collinear Effective Theory
Implements interplay between soft and collinear into effective field theory Hard
Collinear
Soft
Correspondingly, EFT for such processes has two low-energy modes: collinear fields describing the energetic partons propagating in each direction of large energy, and
soft fields which mediate long range interactions among them.
} high-energy
} low-energy part
Bauer, Pirjol, Stewart et al. 2001, 2002; Beneke, Diehl et al. 2002; ...
Diagrammatic FactorizationThe simple structure of soft and collinear emissions forms the basis of the classic factorization proofs, which were obtained by analyzing Feynman diagrams.
Advantages of the the SCET approach:
Simpler to exploit gauge invariance on the Lagrangian level
Operator definitions for the soft and collinear contributions
Resummation with renormalization group
Can include power corrections
J.C. Collins, D.E. Soper / Back-to-back lets
/ . I
421
I _ J
Fig. 7.2. Dominant integration region for e+e annihilation for small wr. In both fig. 7.1 and this figure, the soft gluon subgraphs may be disconnected.
We begin by considering the slightly simpler process a + b ~ A + B + X, where a and b are quarks with momenta k~, and k~ respectively. Let k~, be collinear (as defined in subsect. 4.2) in the v~ direction and let k~ be collinear in the v~ direction. Then the dominant integration regions are as shown in fig. 7.3.
Consider a graph G for this process. A subgraph T of G will be called a tulip if G can be decomposed into subgraphs as indicated in fig. 7.3 with T being the central (possibly disconnected) S subgraph connecting the "jet" subgraphs J a and Jn. The jet subgraphs must be connected and be one particle irreducible in their gluon legs.
A garden is a nested set of tulips. In analogy with subsect. 5.5, we define a regularized version GR of G by
G R = G + ~. ( - 1 ) N S ( T 1 ) S ( T 2 ) • • • S ( T n ) G . (7.2) inequivalent
gardens
Here the operator S ( T ) makes the soft approximation on the attachments to the jets J A and JB of the gluons leaving tulip T. The soft approximation for attachments
Collins, Soper, Sterman 80’s ...
Collins and Soper ‘81
Outline of the course• Invitation: EFTs for soft photons
• γγ scattering at low energy (“Euler-Heisenberg Theory”)
• Soft photons in electron scattering (YFS ’61, “HQET”)
• The method of regions
• a simple example
• Scalar Sudakov form factor
• Scalar SCET
• factorization for the scalar Sudakov form factor in d=6
Outline […]• Generalization to QCD
• Sudakov form factor in QCD
• Resummation by RG evolution
• An application, e.g.
• Drell-Yan process near partonic threshold
• Factorization for DIS or DY
• Factorization constraints on infrared divergences in n-point amplitudes
• Factorization and resummation for cone-jet processes (1508.06645,1605.02737 with Neubert, Rothen and Shao).
Literature
Original SCET papers (using the label formalism): • C.W. Bauer, S. Fleming, D. Pirjol and I.W. Stewart, An effective
field theory for collinear and soft gluons: Heavy to light decays, PRD 63, 114020 (2001) [hep-ph/0011336]
• C.W. Bauer, S. Fleming, D. Pirjol and I.W. Stewart, Soft-Collinear Factorization in Effective Field Theory, PRD 65, 054022 (2002) [hep-ph/0109045]
SCET in position space: • M. Beneke, A.P. Chapovsky, M. Diehl and T. Feldmann,``Soft-
collinear effective theory and heavy-to-light currents beyond leading power,'' Nucl. Phys. B643, 431 (2002) [arXiv:hep-ph/0206152]
A few selected original references are:
Factorization analysis for DIS, Drell-Yan and other processes • C.W. Bauer, S. Fleming, D. Pirjol, I.Z. Rothstein and I.W. Stewart, Hard scattering
factorization from effective field theory, PRD 66, 014017 (2002) [hep-ph/0202088].
Threshold resummation in momentum space using RG evolution, threshold resummation for Drell-Yan
• TB and M. Neubert, Threshold resummation in momentum space from effective field theory, PRL 97, 082001 (2006), [hep-ph/0605050]
• TB, M. Neubert and G. Xu, Dynamical Threshold Enhancement and Resummation in Drell-Yan Production,’' JHEP 0807, 030 (2008) [0710.0680]
EFT analysis of the IR structure of gauge theory amplitudes • TB and M. Neubert, Infrared singularities of scattering amplitudes in perturbative QCD,
PRL 102, 162001 (2009) [0901.0722]; On the Structure of Infrared Singularities of Gauge-Theory Amplitudes, JHEP 0906, 081 (2009) [0903.1126]; Infrared singularities of QCD amplitudes with massive partons, PRD 79, 125004 (2009) [0904.1021]
Factorization for cone-jet processes • TB, M. Neubert, L. Rothen and D.Y. Shao, An effective field theory for jet processes,
PRL 116 (2016) [1508.06645], Factorization and Resummation for Jet Processes 1605.02737
Literature for sample collider physics applications:
arXiv:1410.1892
Lecture Notes in Physics 896
Thomas BecherAlessandro BroggioAndrea Ferroglia
Introduction to Soft-Collinear Eff ective Theory
Will follow this book for parts of the lecture, in particular for the construction of the EFT.
The book also contains a chapter with a review of the many application of SCET
• Heavy-particle decays
• B-physics, top physics, unstable particle EFT, dark matter
• Collider physics
• Event shapes, jets, threshold resummation, transverse momentum resummation, electroweak Sudakov resummation
• Others: Heavy-ion collisions, soft-collinear gravity, …
and a guide to the associated literature.