Supplementary Slides II

Transcription

1 s } C 1 C b } Introduction 2 to SCET: 1 } C a Supplementary Slides II } C 2 Thomas Becher Bern University Lectures on Soft-Collinear Effective Field Theory Les Houches Summer School, July 2017

2 Thrust

3 Thrust and thrust axis T = 1 Q max n n p i =0 i =1 T = 1 3 =1 1 3 =0.42 =0.48 = M M 2 2 Q 2 3

4 Logarithmically enhanced contributions 1 d 0 d The LO thrust distribution has the form = 2 s (1 ) = 2 s 4 ln 3 + d regular ( ) 3 singular terms ln 1 2 Integral over the end-point is R( ) = 0 d 1 0 d d = 2 s 3 2 ln 2 3 ln +... Sudakov double logarithm 4

5 Thrust measurement by ALEPH ALEPH 15 ds dt Q = 91.2 ds dt ds dt 10 t t t 5

6 Resummed vs fixed order 20 fixed fixedorder 20 resummed + matched resummed and matched dσ dt 10 dσ dt T 1 T τ=1-t 6

7 2.0 fixed fixedorder 2.0 resummed + matched resummed and matched dσ dt 1.0 dσ dt T 1 T For αs(mz)=0.118 This is the region relevant for αs determination 7

8 Precision determination of αs τ σ 0.4 dσ dτ Fit at N3 LL for &! 0.3 theory scan error DELPHI ALEPH OPAL L3 SLD Ω τ Λ QC α s (m Z )= ± (0.0002) expt ± (0.0005) Ω1 ± (0.0009) pert (hadronisation) Abbate, Fickinger, Hoang, Mateu and Stewart

9 20 Q 91.2 GeV 1 d Σ Σ d T st order 4 th order PYTHIA hadrons PYTHIA partons T 9

10 50 Q 1 TeV 1 d Σ Σ d T st order 4 th order PYTHIA hadrons PYTHIA partons T 10

11 Another example: qt resummation at LHC CuTe 2.0 TB, Lübbert, Neubert, Wilhelm Cannot use fixed-order Good agreement computation within data: peak region. ATLAS hep-ex/ Z/ 20.3 f Cuts for d fiducial /dq T :66< M ll /GeV 11

12 Cone jets & NGLs 12

13 al observables all have similar structure, key feature are m Factorization theorem tracking hard partons. scuss resummation. TB, Neubert, Rothen, Shao 15 16, see also Caron-Huot 15 Hard function. m hard partons along fixed directions {n1,, nm} l trouble with event generators? Soft function with m Wilson lines ( )= 1X m=2 Hm ({n},q,µ) S m ({n},q,µ), color trace 16 integration over the m directions First all-order factorization theorem for non-global observable. Achieves scale separation!

14 present case additional UV divergences in the real-emission diagramsarisebecause ion is achieved by evolving the Wilson coefficients of these operators from soft radiation is not 1X constrained inside the jet. It is precisely those types of diverge Q down to the scale where the low-energy physics takes place. Le which lead to NGLs. Furthermore, the upper triangular form of Z he wide-angle ( )= cross section Hm ({n},q,µ) for which the factorization S m ({n},q lm H implies that hig theorem,µ) has, multiplicity soft functions are needed to absorb divergences of matrix elements been In fewer ourwilson effective lines. theory, m=2 The symbol the hard ˆ indicates functions that Hin m (2.37) are one thehas Wilson tointegrate coefficien over (m l) additionaldirectionsoftheunresolvedpartonsonwhichthe bare function depends. High-E physics Low-E physics ctive field theory matrix elements contain UV divergences since the short The scale dependence Wilson coefficients of the renormalized hard andeft softoperator functions is governed by e of the full theory is not resolved. The corresponding 1/ϵ poles can be RG equations ine matrix elements S m and we regularize both quantities in d =4 2ϵ dim malizing the hard Wilson coefficients according to Renormalization of hard Wilson coefficients d m 16 H m ({n},q,δ, ϵ) = m d ln µ H m({n},q,δ,µ)= d d ln µ S l({n},qβ, δ,µ)= l=2 H l ({n},q,δ,µ) Γ H lm ({n},q,δ,µ), (2 Hl=2 l ({n},q,δ,µ) Zlm H ({n},q,δ, ϵ,µ). m=l Γ H lm ({n},q,δ,µ) ˆ S m ({n},qβ, δ,µ), (2 ice, it is easiest Same to Z-factor obtain the must bare render WilsonSm coefficients finite! from on-shell ons, whichwhere ensure that the poles the cross arise section from(2.15) IR isscaleindependent.theanomalous-dimen divergences. However, Associated anomalous dimension Γ H these IR pol matrix is obtained from the standard relation ne correspondence to UV divergences since the effective-theory loop-int tching computations d are scaleless, see e.g. [13] foradetailedexplanatio ithin SCET. d ln µ ZH km We ({n},q,δ, have discussed ϵ,µ)= m Zkl H this ({n},q,δ, correspondence ϵ,µ) ˆ Γ H lm ({n},q,δ,µ), (2 after (2.15). It imp l=k 14 understand the UV divergences of H m from the structure of the IR div and it has linear dependence on ln(q/µ) asisfamiliarfromsudakov-typeproblems.h

15 Resummation by RG evolution X Wilson coefficients fulfill renormalization group (RG) equations d d ln µ H m(q, µ) = 1. Compute Hm at a characteristic high scale µh ~ Q 2. Evolve Hm to the scale of low energy physics µl ~ Qβ mx l=2 H l (Q, µ) Q H lm (Q, µ) Qβ RG evolution Avoids large logarithms αs n ln n (β) of scale ratios which can spoil convergence of perturbation theory.

16 RG = Parton Shower X X Ingredients for LL H 2 (µ = Q) = 0 H m (µ = Q) = 0 for m>2 S m (µ = Q) = 1 RG S Z X X Γ (1) = d dt H m(t) =H m (t)v m + H m 1 (t)r m 1. Z X Equivalent to parton shower equation V 2 R V 3 R V 4 R V t = Z (Q) (µ) d ( ). 4 H m (t) =H m (t 1 )e (t t 1)V n + Z t t 1 dt 0 H m 1 (t 0 )R m 1 e (t t0 )V n 16

17 1-loop anomalous dimension V m = Γ (1) m,m =2 (ij) R m = Γ (1) m,m+1 = 4 (ij) dω(nk ) (T i,l T j,l + T i,r T j,r ) 4π T i,l T j,r W m+1 ij Θ n n in (n m+1 ). W k ij, Contain dipoles dipole shower W k ij = n i n j n i n k n j n k Trivial color structure at large Nc : T i T j N c 2 δ j,i±1 17

18 ( ) Δ = e + e 2 jets rapidity gap Δy=1 Preliminary parton shower Equivalent to the dipole shower used by Dasgupta and Salam

19 Work in progress Finite Nc nontrivial color structure, interference MC over colors? Expand in 1/Nc? Plätzer, ) Sjödahl 12, Plätzer 13 Subleading logarithms Γ (2) = v 2 r 2 d v 3 r 3 d v 4 r v Hadronic collisions and super-leading logs 19 vm: two-loop virtual rm: real-virtual dm: double real see Caron-Huot 15