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.