LECTURE 4 LECTURE 4 Introduction to the Parton Model and - PowerPoint PPT Presentation

CTEQ-MCnet school on QCD Analysis and Phenomenology and the Physics and Techniques of Event Generators LECTURE 4 LECTURE 4 Introduction to the Parton Model and Perturbative QCD Fred Olness (SMU) Lauterbad (Black Forest), Germany 26 July - 4 August 2010

We already studies DIS Now we consider Drell-Yan Process e + e - Important for Tevatron and LHC

What is the Explanation hadron n o t p e Drell-Yan l e + e - → 2 jets lepton hadron DIS Drell-Yan and e + e - have an interesting historical relation

A Drell-Yan Example: Discovery of J/Psi very narrow width The Process: p + Be → e + e - X ⇒ long lifetime at BNL AGS

The November Revolution: 1973 related by crossing ... q q J / ψ e e J / ψ + + e - e - q q e + e - Production Drell-Yan SLAC SPEAR Brookhaven AGS Frascati ADONE  e −  hadrons  R =  e = 3 ∑ 2 Q i  e −    −   e i

We'll look at Drell-Yan Specifically W/Z production

Side Note: From pp →γ / Z /W , we can obtain pp →γ /Z/W → l + l - Schematically:  l −  = d   q q   ∗  ∗  l  l −  d  qq  l × d  For example: d  −  = d    l ∗  ×  qq  l t  qq   2 d  d  2 3  Q dQ t

Kinematics in the hadronic CMS

Kinematics for Drell-Yan P 1 =  s 2 = 0  1,0,0,  1  P 1 2 P 1 = x ) P 2 =  s 1 k + k 2 2 = 0  1,0,0, − 1  P 2 k 1 ( = 1 P 1 q 2 P 2 2 = 0 k 1 = x 1 P 1 k 1 k 2 = x 2 P 2 2 = 0 k 2 = x 2 P 2 k 2 d  = ∑ { q  x 1  q  x 2  q  x 2  q  x 1  }   dx 1 dx 2 q ,q Partonic Parton Hadronic cross distribution cross section functions section

Kinematics for Drell-Yan 2  = x 1 x 2 =  s s ≡ Q  =  s s 2 = Therefore s =  P 1  P 2  s  x 1 x 2 Fractional energy 2 between partonic and hadronic system d x 1 d x 2 = d  dy Using: d  d  dy = ∑ { q  x 1  q  x 2  q  x 2  q  x 1  }   q ,q

Rapidity & Longitudinal Momentum Distributions 2 ln { E 12 − p L } E 12  p L y = 1 The rapidity is defined as: Partonic CMS has longitudinal momentum w.r.t. the hadron frame p 12 = p 1  p 2 = E 12 , 0,0, p L  E 12 =  s 2  x 1  x 2  p 1 = x 1 P 1 p 2 = x 2 P 2 p L =  s 2  x 1 − x 2  ≡  s 2 x F p 12 x F is a measure of the longitudinal momentum 2 ln { E 12 − p L } = 1 2 ln { x 2 } E 12  p L x 1 y = 1

Kinematics for W / Z / Higgs Production 12 T e v a t r o n L H C x 2 Z W H Z W x 1

Kinematics for W production at Tevatron & LHC LO W + Luminosities LO W + Luminosities tot tot Tevatron LHC ud ud cs cs us us cd cd y y 2  − M LO luminosities d  = ∫ dx 1 ∫ dx 2 ∫ d  { q  x 1  q  x 2  q  x 2  q  x 1  }  S  d  d  = dL d   

Kinematics in a Hadron-Hadron Interaction: The CMS of the parton-parton system is moving longitudinally relative to the hadron-hadron system How do we measure the W-boson mass?   e   u  d  W Can't measure W directly e + u Can't measure ν directly θ Can't measure longitudinal momentum d ν We can measure the P T of the lepton

The Jacobian Peak e + u Suppose lepton distribution is uniform in θ d The dependence is actually (1+cos θ ) 2 , but we'll worry about that later ν What is the distribution in P T ? Number of Events Max P T transverse direction Min P T beam direction max ≈ M W /2 We find a peak at P T

Drell-Yan Cross Section and the Scaling Form 2  0 = 4   x 2 −  s = 1 2 2 −   Q i  Q  and Using: 9  s s x 1 x 1 we can write the cross section in the scaling form: 1 dx 1 2 4 d  2 = 4  ∑ 2 ∫  { q  x 1  q / x 1  q  x 1  q / x 1  } Q Q i 9 x 1 dQ  q ,q Notice the RHS is a function of only τ , not Q. This quantity should lie on a universal scaling curve. Cf., DIS case, & scattering of point-like constituents

e + e - R ratio  e −  hadrons  R = e  e −    −   e

e + e - Ratio of hadrons to muons q 3 quark colors e + e - q  e −  hadrons  2 [ 1  s  ] R = e = 3 ∑ Q i  e −    −    e e + µ + i e - µ − NLO correction

e + e - NLO corrections

e + e - to 3 particles final state p 1 q e + p 3 e - p 2 Define the energy fractions E i : Energy Conservation: Range of x: Exercise: show 3-body phase space is flat in dx 1 dx 2

3-Particle Phase Space 1 x 3 =0 x 1 +x 2 +x 3 =2 x 2 0 p 1 x 1 x 3 =1 0 1 q e + p 3 d Γ ~ dx 1 dx 2 e - p 2

3-Particle Configurations 1 3-Jet x 2 Collinear 0 x 1 0 1 Soft After symmetrization

Singularities cancel between 2-particle and 3-particle graphs Same result with gluon mass regularization

e + e - Differential Cross Sections

Differential Cross Section p 1 What do we do about soft and q e + collinear singularities???? p 3 e - p 2 Introduce the concept of “Infrared Safe Observable” The soft and collinear singularities will cancel ONLY if the physical observables are appropriately defined.

Infrared Safe Observables Observables must satisfy the following requirements: Soft Collinear

Infrared Safe Observables Soft Collinear

Examples: Infrared Safe Observables Infrared Safe Observables: ● Event shape distributions ● Jet Cross sections Un-Safe Infrared Observables: ● Momentum of the hardest particle ● (affected by collinear splitting) ● 100% isolated particles ● (affected by soft emissions) ● Particle multiplicity ● (affected by both soft & collinear emissions)

Infrared Safe Observables: Define Jets Soft Collinear

Infrared Safe Observables: Define Jets Jet Cone Let's examine this definition a bit more closely

Jet Cone

Pseudo-Rapidity vs. Angle Pseudo-Rapidity θ η 0 90 ° 1 40.4 ° 2 15.4 ° 5.7 ° 3 2.1 ° 4 0.8 ° 5

D0 Detector Schematic

ATLAS Detector Schematic 1.0 1.5 2.0 2.5 3.0

homework

HOMEWORK: Jet Cone Definition η 1 GeV y 100 GeV

HOMEWORK: Light-Cone Coordinates & Boosts

Rapidity vs. Pseudo-Rapidity η y

HOMEWORK: Rapidity vs. Pseudo-Rapidity η y

Infrared Safe Observables: Define Jets Jet Cone Problem: The cone definition is simple, BUT it is too simple See talk by Such configurations can be mis- Ken Hatakeyama identified as a 3-jet event (Jets)

End of lecture 4: Recap ● Drell-Yan: Tremendous discovery potential ● Need to compute 2 initial hadrons ● e + e - processes: ● Total Cross Section: ● Differential Cross Section: singularities ● Infrared Safe Observables ● Stable under soft and collinear emissions ● Jet definition ● Cone definition is simple: ● ... it is TOO simple

Final Thoughts Scaling, Dimensional Analysis, Factorization, Regularization & Renormalization, Infrared Saftey ...

Can you find the Nobel Prize??? Hi ET Jet Excess p + N → µ + µ − + X conflusions.com CDF Collaboration, PRL 77, 438 (1996) Hi Q Excess cross section M µµ GeV H1 Collaboration, ZPC74, 191 (1997) ZEUS Collaboration, ZPC74, 207 (1997)

Thanks to ... Thanks to: Dave Soper, George Sterman, Steve Ellis for ideas borrowed from previous CTEQ introductory lecturers Thanks to Randy Scalise for the help on the Dimensional Regularization. Thanks to my friends at Grenoble who helped with suggestions and corrections. Thanks to Jeff Owens for help on Drell-Yan and Resummation. To the CTEQ and MCnet folks for making all this possible. and the many web pages where I borrowed my figures ...

END OF LECTURE 4