LECTURE 3 Large LECTURE 3 Medium Small How does f change - PowerPoint PPT Presentation

Recap: Parton Model, Factorization, Evolution CTEQ-MCnet school on QCD Analysis and Phenomenology µ dependence must balance and the Physics and Techniques of Event Generators LECTURE 3 Large µ LECTURE 3 Medium µ Small µ How does f change Introduction to the Parton Model and Perturbative QCD with scale µ ??? Fred Olness (SMU) Lauterbad (Black Forest), Germany 26 July - 4 August 2010 DGLAP Evolution Equation DIS at NLO γ DIS Electron a c Proton f P → a AT NLO Sample NLO contributions to DIS

DIS NLO Kinematics Mandelstam Variables {s,t,u} 1 1 3 3 k 3 1 3 s u p=k 2 t q=k 1 θ 2 4 2 2 k 4 4 4 Exercise {s,t,u} are partonic Homework Homework Part 2 1) Let's work out the general 2 → 2 kinematics for general masses. a) Start with the incoming particles. p 3 Show that these can be written in the general form: p 1 p 1 2 = m 1 2 p 1 =  E 1 , 0 , 0 ,  p  p 1 θ p 2 p 2 2 = m 2 2 p 2 =  E 2 , 0 , 0 , − p  p 2 p 4 ... with the following definitions: 2 ∓ m 2 2 2, m 2 2  E 1,2 =  s ± m 1 p =   s ,m 1 p 3 2   2   s s Hint: by using invariants you can keep it simple. p 1 I.e., don't do it the way Goldstein does.  a ,b ,c  =  a 2  b 2  c 2 − 2  ab  bc  ca  θ p 2 The power of invariants Note that ∆ (a,b,c) is symmetric with respect to its arguments, p 4 and involves the only invariants of the initial state: s, m 1 2 , m 2 2 . b) Next, compute the general form for the final state particles, p 3 and p 4 . Do this by first aligning p 3 and p 4 along the z-axis (as p 1 and p 2 are), and then rotate about the y-axis by angle θ .

Matrix element: NLO DIS The Plan For the real 2 → 2 graphs Singular at z=1 Singular at x=1 Soft Singularity Collinear Singularity Separate infinity, absorb in PDF Separate infinity, cancel with virtual graphs Collinear Divergences Soft Singularities Plan Plan 1) Separate ∞ at z=1 1) Separate ∞ at x=1 Looks like a PDF splitting function 2) “Absorb” into PDF 2) Cancel between Real and Virtual graphs Method Method Need to regulate ∞ Need to regulate ∞ Choices Choices 1) Dimensional Regularization 1) Dimensional Regularization 2) Quark Mass 2) Gluon Mass 3) ... 3) θ Cut

We'll use a simple example to illustrate the key points: Infinite Line of Charge r x r =  x 2  y 2 Dimensional Regularization y meets Freshman E&M 1 dQ = Q / y dV = 4  0 r  1 ∞ 4  0 ∫ V = 2 = ∞ dy  x −∞ 2  y M. Hans, Am.J.Phys. 51 (8) August (1983). p.694 C. Kaufman, Am.J.Phys. 37 (5), May (1969) p.560 B. Delamotte, Am.J.Phys. 72 (2) February (2004) p.170 Note: ∞ can Regularization, Renormalization, and Dimensional Analysis: be very useful Dimensional Regularization meets Freshman E&M. Olness & Scalise, arXiv:0812.3578 [hep-ph] Scale Invariance Cutoff Method  1  L 4  0 ∫ V = dy r  x x − L 2  y 2 V(x) depends on artificial regulator L log [ 2 ] We cannot remove the regulator L  L   L 2  x 2  V = y − L   L 4  0 2  x V  kx = =  1 ∞ 4  0 ∫ dy   kx  −∞ 2  y 2 All physical quantities are independent of the regulator: d  k  =  y 1 ∞ 4  0 ∫  x E  x =− dV   −∞ 2  y / k  2 L dx = 2  Electric Field Note: ∞ + c = ∞  L 2  0 x 2  0 x =  1 2  x ∞ 4  0 ∫ dz ∴ ∞ - ∞ = ∞ = c  x ∴ ∞ −∞ 2  z 2 log [ 2 ] 2 x 2 = V  x    How do we distinguish  V = V  x 1 − V  x 2  Energy this from 4  0 L ∞ x 1 Naively Implies: ∞ - ∞ = = c+17 V  kx = V  x  ∞ Problem solved at the expense of an extra scale L V(kx) – V(x) = 0 AND we have a broken symmetry: translation invariance

Broken Translational Symmetry Dimensional Regularization n y = d  n n − 1 dy Compute in n-dimensions dy  d y Shift: y → y' = y – c 2 r x y=[+L+c, -L+c] y n / 2  n = ∫ d  n = 2  2 }  1,2,3,4 ={ 2,2  , 4  , 2   n / 2  -L +L Each term is individually n − 1   y dy 1 ∞  L  c 4  0 ∫ 4  0 ∫ 0 dimensionaless V = dy V = d  n  x − L  c 2  y 2  x n − 1  2  y 2 log [ 2 ]  L  c    L  c  2  x 2  V = − L − c    L − c  4  0 2  x New scale µ   4  0  In QFT, 2  V(r) depends on “y” coordinate!!!   [] V = gauge symmetries 2  x  are important. E.g., Ward identies Dimensional Regularization Why do we need an extra scale µ µ ??? r All physical quantities are independent of the regulators: x r =  x 1  2  ] 2  y 2 4  0 [ 2  [] E  x =− dV  2   1  Electric Field dx =  x 2  0 x  0  y log [ 2 ] 2 x 2   V = V  x 1 − V  x 2   Energy 4  0 1 dQ  0 x 1 = Q / y dV = 4  0 r Problem solved at the expense of an extra scale µ AND regulator ε  V = f  x  4  0 Translation invariance is preserved!!! Dimensional Regularization respects symmetries

Renormalization Connection to QFT 4  0 [ 2 ] ]  ln [  ]  ln [ − E 2   1 e The was the potential from our “Toy” calculation: Original V  x 4  0 [ 2 ] ]  ln [ 1  ]  ln [ − E 2  1 e  V  4  0 [ 2 ] ]   ln [  ]  ln [ x − E 2  1 e  MS V  x This is a partial result from 4  0 [ 2 ] ]  ln [  ]  ln [ a real NLO Drell-Yan Calculation: − E 2 MS-Bar  1 e   1 − 2   [ =  Cf., B. Potter V  2  2 ] ]  ln [ 4  ]  ln [ 2 − E 2 x D   1 − 4  1 e   Q Q Physical quantities are independent of renormalization scheme! V MS  x 1 − V MS  x 2 = V = V MS  x 1 − V MS  x 2  But only if performed consistently: V MS  x 1 − V MS  x 2 ≠ V ≠ V MS  x 1 − V MS  x 2  Recap Regulator provides unique definition of V, f, ω Apply Cutoff regulator L: simple, but does NOT respect symmetries Dimensional regulator ε : Dimensional respects symmetries: translation, Lorentz, Gauge invariance introduces new scale µ All physical quantities (E, dV, σ ) are independent of the regulator Regularization AND the new scale µ Renormalization group equation: d σ /d µ =0 We can define renormalized quantities (V,f, ω ) to QFT Renormalized (V,f, ω ) are scheme dependent and arbitrary Physical quantities (E,dV , σ ) are unique and scheme independent if we apply the scheme consistently

Homework: Part 1 D-Dimensional Phase Space #1) Show: 1 3 3 p 4 p d d   p 2 − m 2  = 4  2   3 2 E  2   2  1-particle 2 4 This relation is often useful as the RHS is manifestly Lorentz invariant Final state Final state #2) Show that the 2-body phase space can be expressed as: 3 p 3 3 p 4 d d 4  p 1  p 2 − p 3 − p 4  = d cos  4  d  =  2  Enter, µ scale 16  3 2 E 3 3 2 E 4  2   2  All the pieces Note, we are working with massless partons, and θ is in the partonic CMS frame Soft Singularities Soft Singularities REAL VIRTUAL From Soft Finite To be canceled phase Singularity remainder by virtual space diagram This only makes sense under the integral

KLN (Kinoshita, Lee, Nauenberg) Theorem Collinear Singularities virtual real virtual real Collinear Singularity How do we know what to “absorb” into PDFs ??? This should be This is finite “absorbed” for z=[-1,1] in the PDF ... looks like a splitting kernel Key 1) “Absorb 1/ ε into PDF Compute NLO Subtractions Points 2) This defines how to regularize PDF for a partonic target 3) Need to match schemes of ω and PDF ... MS, MS-Bar, DIS, ... 4) Note we have regulator ε and extra scale µ

Application of Factorization Formula at Leading Order (LO) Application of Factorization Formula at NLO Basic Factorization Formula Basic Factorization Formula At First Order: At Zeroth Order: Higher Twist 1 = f 1 ⊗ 0  f 0 ⊗ 1  0  O  0 = f 0 ⊗ 2 / Q 2   1 = 1 ⊗ 0  1  f f 0 f 0 f 1 f 1 Use: f 0 = δ for a parton target. We used: f 0 = δ for a parton target. for parton target 1 = 1 − f 1 ⊗ 0 Therefore: Therefore:  0 = f 0 ⊗ 0 =⊗ 0 = 0  ω 1 = 0 = 0  σ 1 f 1 ⊗ σ 0 P (1) defined by Warning: This trivial result leads to many misconceptions at higher orders scheme choice Application of Factorization Formula at NLO HOMEWORK PROBLEM: NNLO WILSON COEFFICIENTS Combined Result: Use the Basic Factorization Formula Complete NLO Term: ω 1 At Second Order (NNLO): 0   1 =  0   1 − f 1 ⊗ 0  NLO LO Therefore: SUB TOT Subtraction 2 = ???  Include Fragmentation Compute ω 2 at second order. Functions d TOT = LO + NLO − SUB Make a diagrammatic representation of each term.