Soft-Collinear Effective Field Theory } Thomas Becher Bern University

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CUSOC Graduate2 Course, EPFL, Nov. 2015 Tools for QFT computations

Expansion Expansion in in the interaction scale ratios: strength: Effective Field perturbation Theories theory

Toy models: Numerical solvable models methods: SUSY theories lattice simulations AdS/CFT Jet physics at the LHC

Many scale hierarchies! ps pT M E m ⇤ Jet Jet out proton ⇠ QCD → Soft-Collinear Effective Theory (SCET) Outline of the course

• Invitation: perturbative QCD and effective ﬁeld theory

• The method of regions

• a simple example

• Scalar Sudakov form factor

• Introduction to perturbative QCD / introduction to EFT

• Scalar SCET

• factorization for the scalar Sudakov form factor in d=6

• Generalization to QCD Outline […]

• Sudakov form factor in QCD

• Resummation by RG evolution

• Drell-Yan process near partonic threshold

• Factorization for generic Drell-Yan

• Factorization constraints on infrared divergences in n-point amplitudes

• Factorization and resummation for cone-jet processes (1508.06645 with Neubert, Rothen and Shao). Literature A few selected original references are:

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] Sample collider physics applications:

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 1508.06645 Lecture Notes in Physics 896

Thomas Becher Alessandro Broggio Andrea Ferroglia Introduction to Soft-Collinear Ef ective Theory

arXiv:1410.1892 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-quark physics

• B-physics, top physics, unstable particle EFT

• 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. Alternative SCET introduction edX online course on effective ﬁeld theory by Iain Stewart, see https://www.edx.org/course/effective-ﬁeld- theory-mitx-8-eftx

• Video lectures on SCET and other EFTs

• includes a set of TASI lecture notes on SCET by Christian Bauer and Iain Stewart

• Uses and introduces the label formalism. R-ratio 6 46. Plots of cross sections and related quantities

+ σ and R in e e− Collisions

-2 10 ω φ / -3 J ψ 10 ψ(2S) ρ′ Υ -4 ρ 10 Z

-5 10 [mb]

σ -6 10 6 46. Plots of cross sections and related quantities -7 10 + − -8e e → hadrons:σ and R in e+e crossCollisions section 10 − 2 -2 1 10 10 10 ω φ J/ψ -3 Υ 10 3 / 10 J ψ ψψ(2S)) ρ′ Υ Z -4 ρ 10 Z 10 2 -5 φ 10 ω

R [mb]

σ 10 -6 10 ρ′

-7 101

-8 ρ 10 -1 10 2 1 10 10 2 1 10 10 √s [GeV] 3 Υ 10 J/ψ Figure 46.6: World data on the total cross section of e+e−ψ(2hadronsS) and the ratio R(s)=σ(e+e− hadrons, s)/σ(e+e− µ+µ−,s). σ(e+e− hadrons, s)istheexperimentalcrosssectioncorrectedforinitialst→ ate radiation and electron-positron→ Z vertex loops,→σ(e+e− µ+µ−,s)=4→ πα2(s)/3s.Dataerrorsaretotalbelow2GeVandstatisticalabove2GeV.Thecurvesareaneducativeguide:thebrokenone→ (green) is a naive2 quark-parton model prediction, and the solid one (red) is 3-loop pQCD prediction (see “Quantum Chromodynamics” section of this Review,Eq.(910 .7) or, for more details, K. G. Chetyrkin et al.,Nucl.Phys.B586,56(2000)(Erratumibid. B634,413(2002)).Breit-Wigner parameterizations of J/ψ, ψ(2S), and Υφ(nS),n =1, 2, 3, 4arealsoshown.Thefulllistofreferencestotheoriginaldata and the details of the R ratio extraction from themω can be found in [arXiv:hep-ph/0312114]. Corresponding computer-readable data ﬁles are available at http://pdg.lbl.gov/current/xsect/R .(CourtesyoftheCOMPAS(Protvino)andHEPDATA(Durham)Groups, May 2010.) 10 ρ′

1 ρ -1 10 2 1 10 10 √s [GeV]

Figure 46.6: World data on the total cross section of e+e− hadrons and the ratio R(s)=σ(e+e− hadrons, s)/σ(e+e− µ+µ−,s). σ(e+e− hadrons, s)istheexperimentalcrosssectioncorrectedforinitialst→ ate radiation and electron-positron→ vertex loops,→σ(e+e− µ+µ−,s)=4→ πα2(s)/3s.Dataerrorsaretotalbelow2GeVandstatisticalabove2GeV.Thecurvesareaneducativeguide:thebrokenone→ (green) is a naive quark-parton model prediction, and the solid one (red) is 3-loop pQCD prediction (see “Quantum Chromodynamics” section of this Review,Eq.(9.7) or, for more details, K. G. Chetyrkin et al.,Nucl.Phys.B586,56(2000)(Erratumibid. B634,413(2002)).Breit-Wigner parameterizations of J/ψ, ψ(2S), and Υ(nS),n =1, 2, 3, 4arealsoshown.Thefulllistofreferencestotheoriginaldata and the details of the R ratio extraction from them can be found in [arXiv:hep-ph/0312114]. Corresponding computer-readable data ﬁles are available at http://pdg.lbl.gov/current/xsect/.(CourtesyoftheCOMPAS(Protvino)andHEPDATA(Durham)Groups, May 2010.) 6 46. Plots of cross sections and related quantities

+ σ and R in e e− Collisions

-2 10 ω φ / -3 J ψ 10 ψ(2S) ρ′ Υ -4 ρ 10 Z

-5 10 [mb]

σ -6 10

-7 10

-8 10 R-ratio 2 1 10 10

Υ 3 / 10 J ψ ψ(2S) Z

10 2 φ R ω 10 ρ′

1 ρ -1 10 2 1 10 10 √s [GeV]

Figure 46.6: World data on the total cross section of e+e− hadrons and the ratio R(s)=σ(e+e− hadrons, s)/σ(e+e− µ+µ−,s). σ(e+e− Thehadrons, green s)istheexperimentalcrosssectioncorrectedforinitialst dashed line is our LO→ predictionate radiation after and inclusion electron-positron→ of vertex quark loops,→σ (e+e− µ+µ−,s)=4→ πα2(s)/3s.Dataerrorsaretotalbelow2GeVandstatisticalabove2GeV.Thecurvesareaneducativeguide:thebrokenone→ (green) ismasses a naive quark-parton and Z model-exchange. prediction, and Red the solid line one (red) includes is 3-loop pQCD NNLO prediction (seeQCD “Quantum corrections. Chromodynamics” section of this Review,Eq.(9.7) or, for more details, K. G. Chetyrkin et al.,Nucl.Phys.B586,56(2000)(Erratumibid. B634,413(2002)).Breit-Wigner parameterizations of J/ψ, ψ(2S), and Υ(nS),n =1, 2, 3, 4arealsoshown.Thefulllistofreferencestotheoriginaldata and the details of the R ratio extraction from them can be found in [arXiv:hep-ph/0312114]. Corresponding computer-readable data ﬁles are available at http://pdg.lbl.gov/current/xsect/.(CourtesyoftheCOMPAS(Protvino)andHEPDATA(Durham)Groups, May 2010.) 46. Plots of cross sections and related quantities 7

R in Light-Flavor, Charm, and Beauty Threshold Regions

10 2 φ u, d, s ω 3looppQCD 10 Naive quark model ρ ρ′ 46. Plots of cross sections and related quantities 7 1

Sum of exclusive Inclusive R in Light-Flavor,measurements Charm, and Beauty Thresholdmeasurements Regions -1 10 0.5 1 1.5 2 2.5 3 1072 ψ(2S) J/ψ φ u, d,ψ s4160 c 6 ωMark-I Mark-I + LGW 3looppQCD ψ4415 Mark-II ψ4040 5 Naive quark model 10 PLUTO ψ3770 R ρ DASP ρ′ 4 Crystal Ball BES 13 b-quark threshold region

2 Sum of exclusive Inclusive measurements measurements -1 10 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 3 78 Υ(1S) ψ(2S) Υ(3S) b 7 J/ψ Υ(2S) ψ4160 c 6 Mark-I Υ(4S) 6 Mark-I + LGW ψ4415 Mark-II ψ4040 5 ψ3770 5 PLUTO R DASP 44 Crystal Ball BES 3 ARGUS CLEO CUSB DHHM 3 MD-1 2 Crystal Ball CLEO II DASP LENA

2 9.5 10 10.5 11 3 3.5√s [GeV] 4 4.5 5

Figure 46.7: 8R in the light-ﬂavor, charm, and beauty threshold regions. Dataerrorsaretotalbelow2GeVandstatisticalabove2GeV. The curves are the same as in Fig. 46.6.Υ(1Note:S) CLEO data above Υ(4S)werenotfullycorrectedforradiativeeΥ(3S) ﬀects,b and we retain them on the plot7 only for illustrative purposes with a normalΥ(2izationS) factor of 0.8. The full list of references to the original data and the details of the R ratio extraction from them can be found in [arXiv:hep-ph/0312114].Thecomputer-readabledataareavailableatΥ(4S) http://pdg.lbl.gov/current/xsect/.(CourtesyoftheCOMPAS(Protvino)andHEPDATA(Durham)Groups, May 2010.) 6

3 ARGUS CLEO CUSB DHHM MD-1 2 Crystal Ball CLEO II DASP LENA

9.5 10 10.5 11 √s [GeV]

Figure 46.7: R in the light-ﬂavor, charm, and beauty threshold regions. Dataerrorsaretotalbelow2GeVandstatisticalabove2GeV. The curves are the same as in Fig. 46.6. Note: CLEO data above Υ(4S)werenotfullycorrectedforradiativeeﬀects, and we retain them on the plot only for illustrative purposes with a normalization factor of 0.8. The full list of references to the original data and the details of the R ratio extraction from them can be found in [arXiv:hep-ph/0312114].Thecomputer-readabledataareavailableat http://pdg.lbl.gov/current/xsect/.(CourtesyoftheCOMPAS(Protvino)andHEPDATA(Durham)Groups, May 2010.) 46. Plots of cross sections and related quantities 7

R in Light-Flavor, Charm, and Beauty Threshold Regions

10 2 φ u, d, s ω 3looppQCD 10 Naive quark model 46. Plots of cross sections and related quantities 7 ρ ρ′

1 R in Light-Flavor, Charm, and Beauty Threshold Regions Sum of exclusive Inclusive 2 measurements measurements 10-1 10 0.5 1φ 1.5u, d, 2 s 2.5 3 7 ω ψ(2S) 3looppQCD J/ψ ψ4160 c 106 Mark-I Naive quark model Mark-I + LGW ρ ′ ψ4415 Mark-II ψ4040 5 ρ PLUTO ψ3770 R DASP 41 Crystal Ball BEScharm region 3 Sum of exclusive Inclusive measurements measurements -1 102 0.5 1 1.5 2 2.5 3 7 3 3.5 4 4.5 5 ψ(2S) 8 J/ψ ψ4160 c 6 Mark-I Υ(1S) Υ(3S) b 7 Mark-I + LGW Υ(2S) ψ4415 Mark-II ψ4040 Υ(4S) 5 ψ3770 6 PLUTO R DASP 54 Crystal Ball BES 4 3 3 ARGUS CLEO CUSB DHHM MD-1 22 Crystal Ball CLEO II DASP LENA

39.5 3.5 10 4 10.5 4.5 11 5

8 √s [GeV] Υ(1S) Υ(3S) b Figure 46.7: R7in the light-ﬂavor, charm, and beauty thresholdΥ(2 regions.S) Dataerrorsaretotalbelow2GeVandstatisticalabove2GeV. The curves are the same as in Fig. 46.6. Note: CLEO data above Υ(4S)werenotfullycorrectedforradiativeeΥ(4S) ﬀects, and we retain them on the plot only for illustrative purposes with a normalization factor of 0.8. The full list of references to the original data and the details of the 6R ratio extraction from them can be found in [arXiv:hep-ph/0312114].Thecomputer-readabledataareavailableat http://pdg.lbl.gov/current/xsect/.(CourtesyoftheCOMPAS(Protvino)andHEPDATA(Durham)Groups, May 2010.) 5

3 ARGUS CLEO CUSB DHHM MD-1 2 Crystal Ball CLEO II DASP LENA

9.5 10 10.5 11 √s [GeV]

R in Light-Flavor, Charm, and Beauty Threshold Regions

10 2 φ u, d, s ω 3looppQCD 10 46. PlotsNaive of cross quark sections model and related quantities 7 ρ ρ′

1 R in Light-Flavor, Charm, and Beauty Threshold Regions

Sum of exclusive Inclusive 10 2 measurements measurements -1 10 φ 0.5 1 1.5u, d, 2 s 2.5 3 7 ω ψ(2S) 3looppQCD J/ψ ψ4160 c 106 Mark-I Naive quark model ρ Mark-I + LGW ′ ψ4415 Mark-II ψ4040 ρ 5 7 PLUTO ψ3770 46. Plots of cross sections and related quantities R DASP 41 Crystal Ball BES BelowR in Light-Flavor,Sum of exclusivecharm Charm, and Beauty threshold ThresholdInclusive Regions 3 measurements measurements -1 1022 10 0.5 1 1.5 2 2.5 3 7 3 3.5φ 4 4.5 5 ψ(2S) u, d, s J/ψ ω ψ4160 c 8 6 Mark-I 3looppQCD Υ(1S) Υ(3S) b 10 Mark-I + LGW Naive quark model 7 Υ(2S) ψ4415 ρ Mark-II ψ4040 Υ(4S) 5 ψ3770 ′ 6 PLUTO ρ R DASP 54 Crystal Ball 1 BES 4 3 Sum of exclusive Inclusive 3 measurementsARGUS CLEO CUSBmeasurements DHHM -1 MD-1 10 22 Crystal Ball CLEO II DASP LENA 0.5 1 1.5 2 2.5 3 7 39.5 3.5 10 4 10.5 4.5 11 5 ψ(2S) 8 J/ψ √s [GeV] ψ4160 c 6 Mark-I Υ(1S) Υ(3S) b Figure 46.7: R7in the light-ﬂavor, charm,Mark-I and beauty + LGW threshold regions. Dataerrorsaretotalbelow2GeVandstatisticalabove2GeV. The curves are the same as in Fig. 46.6. Note: CLEO data aboveΥ(2ΥS(4)S)werenotfullycorrectedforradiativeeψ4415 ﬀects, and we retain Mark-II ψ4040 them on the plot5 only for illustrative purposes with a normalization factor of 0.8. The full list of referencesΥ(4 toS the) original data and PLUTO ψ3770 the details of the 6R ratio extraction from them can be found in [arXiv:hep-ph/0312114].Thecomputer-readabledataareavailableat http://pdg.lbl.gov/current/xsect/R .(CourtesyoftheCOMPAS(Protvino)andHEPDATA(Durham)GrDASP oups, May 2010.) 4 5 Crystal Ball BES

3 4

3 ARGUS CLEO CUSB DHHM 2 MD-1 2 Crystal Ball CLEO II DASP LENA 3 3.5 4 4.5 5 9.5 10 10.5 11 8 Υ(1S) √sΥ[GeV](3S) b 7 Υ(2S) Figure 46.7: R in the light-ﬂavor, charm, and beauty threshold regions. Dataerrorsaretotalbelow2GeVandstatisticalabove2GeV.Υ(4S) The curves are6 the same as in Fig. 46.6. Note: CLEO data above Υ(4S)werenotfullycorrectedforradiativeeﬀects, and we retain them on the plot only for illustrative purposes with a normalization factor of 0.8. The full list of references to the original data and the details of the R ratio extraction from them can be found in [arXiv:hep-ph/0312114].Thecomputer-readabledataareavailableat http://pdg.lbl.gov/current/xsect/5 .(CourtesyoftheCOMPAS(Protvino)andHEPDATA(Durham)Groups, May 2010.)

3 ARGUS CLEO CUSB DHHM MD-1 2 Crystal Ball CLEO II DASP LENA

9.5 10 10.5 11 √s [GeV]

Figure 46.7: R in the light-ﬂavor, charm, and beauty threshold regions. Dataerrorsaretotalbelow2GeVandstatisticalabove2GeV. The curves are the same as in Fig. 46.6. Note: CLEO data above Υ(4S)werenotfullycorrectedforradiativeeﬀects, and we retain them on the plot only for illustrative purposes with a normalization factor of 0.8. The full list of references to the original data and the details of the R ratio extraction from them can be found in [arXiv:hep-ph/0312114].Thecomputer-readabledataareavailableat http://pdg.lbl.gov/current/xsect/.(CourtesyoftheCOMPAS(Protvino)andHEPDATA(Durham)Groups, May 2010.) 34th International Conference on High Energy Physics, Philadelphia, 2008

HPQCD + Karlsruhe 08 Kuehn, Steinhauser, Sturm 07 lattice + pQCD low-moment sum rules, NNNLO Kuehn, Steinhauser, Sturm 07 Pineda, Signer 06 low-moment sum rules, NNNLO Ψ sum rules, NNLL (not complete) Della Morte et al. 06 Buchmueller, Flaecher 05 lattice (ALPHA) quenched B decays α 2β s 0 Buchmueller, Flaecher 05 Hoang, Manohar 05 B decays α 2β B decays α 2β s 0 s 0 Mc Neile, Michael, Thompson 04 Hoang, Jamin 04 lattice (UKCD) NNLO moments deDivitiis et al. 03 deDivitiis et al. 03 lattice quenched lattice quenched Penin, Steinhauser 02 Rolf, Sint 02 Ψ(1S), NNNLO lattice (ALPHA) quenched Pineda 01 Becirevic, Lubicz, Martinelli 02 Ψ(1S), NNLO lattice quenched Kuehn, Steinhauser 01 Kuehn, Steinhauser 01 low-moment sum rules, NNLO low-moment sum rules, NNLO Hoang 00 Ψ sum rules, NNLO QWG 2004 QWG 2004 PDG 2006 PDG 2006

0.8 0.9 1 1.1 1.2 1.3 1.4 4.1 4.2 4.3 4.4 4.5 4.6 4.7

mc(3 GeV) mb(mb)

Figure 1: Comparison of recent determinations of mc(3 GeV) and mb(mb). sensitive to non-perturbative contributions from condensates, to the Coulombic higher order eﬀects, the variation of

µ and the parametric αs dependence. For n =1:

mc(3 GeV) = 0.986(13) GeV . (3)

The moment with n =2islesssensitivetodataforR(s)fromthecontinuumregionabove5GeV,whereexperimental results are scarce and the aforementioned theory uncertainties are still relatively small. The agreement between n =1andn =2(mc(3 GeV) = 0.976(16) GeV), together with the nice convergence with increasing order in αs can be considered as additional confirmation of this approach. exp + − Instead of measuring the moments Mn in e e annihilation they can also be determined in lattice simulations. This approach has recently been pioneered in [10] using the Highly Improved Staggered Quarks (HISQ) discretization of the quark action in combination with four-loop perturbative results [3, 4, 5, 9, 11]. The final result, mc(3 GeV) = 0.986(10) GeV corresponds to a scale-invariant mass mc(mc)=1.268(9) GeV and is in excellent agreement with the determinations based on e+e− data. + − The approach based on e e data is also applicable to the determination of mb.Thethreeresultsbasedonn =1, 2 and 3 are of comparable precision. The relative size of the contributions from the threshold and the continuum region decreases for the moments n =2and3.Ontheotherhand,thetheoryuncertaintyisstillsmall. Therefore the result from n =2wastakenasthefinalanswer[5],despitethefactthatC¯2 was not yet known. The result, mb(10 GeV) = 3.609(25) GeV, corresponds to mb(mb)=4.164(25) GeV. The recent evaluation [9] of C¯2 has lead to adecreaseofthecentralvalueby2MeVandareductionoftheerror from 25 MeV to 19 MeV. A comparison of a few selected mb-andmc-determinations is shown in Fig. 1.

2. The strong coupling constant

One of the most precise and theoretically safe determinationofαs is based on measurements of the cross section for electron-positron annihilation into hadrons [12]. These have been performed in the low-energy region between 2GeVand10GeVand,inparticular,atandaroundtheZ resonance at 91.2 GeV. Conceptually closely related is the measurement of the semileptonic decay rate of the τ-lepton, leading to a determination of αs at a scale below 2GeV[13].TheperturbativeexpansionfortheratioR(s) ≡ σ(e+e− → hadrons)/σ(e+e− → µ+µ−)innumerical form is given by

2 2 3 R =1+as +(1.9857 − 0.1152 nf ) as +(−6.63694 − 1.20013nf − 0.00518nf ) as 2 3 4 +(−156.61 + 18.77 nf − 0.7974 nf +0.0215 nf ) as . (4)

2 4 Here as ≡ αs/π and the normalization scale µ = s.Theas corrections are conveniently classiﬁed according to their 4 3 power of nf ,withnf denoting the number of light quarks. The asnf term is part of the “renormalon chain”, the

2 R = KN e2 and K¯ =1 K c q q X