IRC Safety vs. Calculability

Transcription

1 IRC Safety vs. Calculability Anrew Larkoski MIT AJL, J. Thaler ; AJL, I. Moult, D. Neill ; AJL, S. Marzani, G. Soyez, J. Thaler 1402.soon Rutgers, February 11, 2014

2 Reminer: How to compute ifferential cross sections in perturbation theory O = X n n = number of external particles O = observable Z Infrare an Collinear Safety The phase space constraints impose by the observable are smooth through real an virtual contributions Collinear safety Linear in energy Weighte by positive powers of angles Infrare safety Weighte by positive power of energy n M n O 2 Ô( n) M n = M 0 n + M 1 n + M 2 n + tree-level one-loop two-loop g n 2 g n 1 g n 2 Real 1 Q X i Virtual Poster Chil: Thrust E i sin i tan i 2

3 Reminer: How to compute ifferential cross sections in perturbation theory 3

4 Reminer: How to compute ifferential cross sections in perturbation theory Fixe-Orer Distribution breaks own Logarithms become large Dominant energy flow is along momentum axis Must resum logarithms for reliable preictions Soft-Collinear Effective Theory: Resummation by RG evolution Explicit summation of singular approximation to matrix element Fixe-Orer Distribution is Accurate No large logarithms Dominant emissions are away from soft/collinear region 4

5 Reminer: How to compute ifferential cross sections in perturbation theory s log 1 " X =( s + )exp n Leaing Logarithms Next-to-Leaing Logarithms n s log n+1 + n s log n + # = s + 2 s + 3 s + 1 5

6 Reminer: How to compute ifferential cross sections in perturbation theory s log 1 " X =( s + )exp Match istributions in ifferent phase space regions n Leaing Logarithms Next-to-Leaing Logarithms n s log n+1 + n s log n + = s + s 2 + s # 6

7 Reminer: How to compute ifferential cross sections in perturbation theory QCD Q Non-perturbative effects ominate s log 1 " X =( s + )exp Match istributions in ifferent phase space regions n = s + s 2 + s Leaing Logarithms Next-to-Leaing Logarithms n s log n+1 + n s log n + QCD Q Non-perturbative effects are power corrections (OPE) #

8 Ratio Observables in Perturbation Theory A r = A/B A, B: IRC safe observables Real contribution: ivergent for all r Virtual contribution: ivergent, proportional to δ(r) Singular region of phase space B 8

9 Ratio Observables in Perturbation Theory A r = A/B A, B: IRC safe observables Real contribution: ivergent for all r Virtual contribution: ivergent, proportional to δ(r) Singular region of phase space B IRC Unsafe!? Soyez, Salam, Kim, Dutta, Cacciari 2012 Stanar computation methos are useless for these observables A, B can be measure separately; why can t their ratio? This is a major practical issue 9

10 ( ) N Ratio Observables in Perturbation Theory Example: N-subjettiness = X p Ti min{r i1,r i2,...,r in } i2j 2,1 = ( ) 2 ( ) ( ) ( ) 1 3,2 = ( ) 3 ( ) 2 Powerful booste W tagger Selects for 2-subjet structures Powerful booste t tagger Selects for 3-subjet structures Other ratio observables use for jet substructure analysis: Energy correlation functions Angular correlation functions Planar flow Thaler, van Tilburg 2010 AJL, Salam, Thaler 2013 Jankowiak, AJL 2011 Almeia, Lee, Perez, et al Jets / 0.08 Jets / 0.08 MC / Data Data, L = 35pb Pythia Herwig++ anti-k t R=1.0 jets 300 < p < 400 GeV T = 1, y < 2 N PV ATLAS N-subjettiness τ Why oes Monte Carlo L = 35pb -1 moel ata so well? 2010 Data, Pythia Herwig++ ATLAS ATLAS

11 Ratio Observables in Perturbation Theory A r = A/B Singular region of phase space B Assume A B 0 r 1 Nee to regulate the singular region of phase space for calculability 11

12 Ratio Observables in Perturbation Theory A Bcut r = A/B Singular region of phase space 1) Explicit cut on enominator observable B May introuce unesire logarithmic sensitivity to Bcut B Assume A B 0 r 1 Nee to regulate the singular region of phase space for calculability 12

13 Ratio Observables in Perturbation Theory A Suakov suppression r = A/B Singular region of phase space 1) Explicit cut on enominator observable B May introuce unesire logarithmic sensitivity to Bcut B Assume A B 0 r 1 Nee to regulate the singular region of phase space for calculability 2) Inclue emissions to all-orers in perturbation theory Exponentially suppresses singular region organically 13

14 Ratio Observables in Perturbation Theory Definition: r = Z A B 2 A B r A B 2 A B is the funamental object Well-efine orer-by-orer in perturbation theory Follows from IRC safety of A an B To all-orers, singular region is exponentially suppresse by perturbative Suakov factor Marginalization is well-efine Ratio observable is Suakov safe 14

15 Outline Example: Ratio of Angularities IRC Unsafety at fixe orer Suakov Safety at all-orers Controlle non-perturbative sensitivity Looking Forwar Higher-orer effects Other examples of Suakov Safe observables Conclusions 15

16 Ex: Ratio of Angularities 16 AJL, J. Thaler AJL, I. Moult, D. Neill

17 Angularities Measure on Jets Want to measure: Z r = e e R0 Jet Axis Recoil-free jet axis (broaening, winner-take-all) AJL, Neill, Thaler e e r e e e = 1 E jet X i2jet i E i R 0 angle measure wrt jet axis Familiar angularities: α = 2, thrust α = 1, broaening/with/girth We take α > β so eα < eβ 0 r 1 Compute ouble ifferential cross section to ifferent accuracies 17

18 θ z << 1 broaening axis Phase space constraints: Energy conservation: Emission within jet: 1 2 e e Fixe-Orer Distribution ~1 z<1! e <e <R 0! e <e Singular matrix element: S(z,)z =2 s C F =2 s C F z z Double Differential Cross Section: e = z 18 R 0 z = e 1 e e e e (e e ) e a e = z e = e 1 R 0 Angularities Phase Space a = 2.0 b = 1.5 b = 1.0 b = 0.5 b = 0.2 R 0 e e b

19 1 2 e e Fixe-Orer Distribution =2 s C F 1 e e e e (e e ) r = e e e <e! e <r Measuring r oes not e <e! r<1 regulate eβ singularity! 1 r =2 s C F 1 r Z r 0 e e Logarithmic sensitivity to any lower boun Ratio observable cross section unefine at fixe orer: IRC unsafe 19

20 Double-Logarithmic Distribution Probability for an emission: Emissions are uniformly istribute in the plane 2 s C F z z =2 s C F log 1 log 1 z log 1 z soft soft-collinear collinear log R 0 20

21 Double-Logarithmic Distribution Probability for an emission: Emissions are uniformly istribute in the plane 2 s C F z z =2 s C F log 1 log 1 z log 1 = log 1 e z + log R 0 log 1 z log 1 e soft soft-collinear P (x <e )=e 2 s C F (e )=e s C F log 2 e 1 log 1 e collinear log R (e )

22 log 1 z log 1 e log 1 e soft Double-Logarithmic Distribution Probability for an emission: Emissions are uniformly istribute in the plane 2 s C F soft-collinear z z =2 s C F log 1 log 1 z log 1 = log 1 e z + log R 0 log 1 e = log 1 z + log R 0 P (x <e,y <e )=e 2 s C F (e,e )=e s C F log 2 e + log 2 e e! 1 log log e e collinear log R e e 2 (e,e )

23 (e,e )=e 1 r = LL r = Z e e! p s C F Double-Logarithmic Distribution s Z C F log 2 e + log 1 r 2 e 2 (e,e ) r e 1 2 s C F log 2 r erf e (phase space constraints suppresse) apple p s C p F log r +1 e s ( ) C F log 2 r 2 s C F log r r e s C F ( ) 2 log2 r Expaning in αs: 1 r = p p CF s 1 r + O ( p s ) 2 No Taylor expansion about αs = 0! 23

24 Double-Logarithmic Distribution 1 r = p p CF s 1 r + O ( p s ) 2 Consequences: IRC unsafe No Taylor series about αs = 0 Suakov safe : finite cross section with all-orers inclue Observations: Taylor series expansion about αs 0 Can this cross section be compute with CFT techniques? Connections: j 1: Anomalous imension of fragmentation function moments (j, s )= sc A 1 j 1 + O( 2 s) 24 j = 1: (j =1, s )= QCD Pink Book r s C A 2

25 Practical Consequences of Calculability: Monte Carlos an the ratio observable Monte Carlos approximate allorers exclusive cross sections Σ r Α,Β Monte Carlo Resummation Ratio r Α,Β, Α 1 Β 0.75: LL LL MC Β 0.5: LL LL MC Β 0.25: LL LL MC 1 Shoul accurately reprouce the cross section for the ratio near r = 0 Deviation for r ~ 1 where multiple emissions become important Σ r Α,Β r Α,Β e Α e Β Monte Carlo Resummation Ratio r Α,Β, Α 2 Β 1.5: LL LL MC Β 1.0: LL LL MC Β 0.5: LL LL MC r Α,Β e Α e Β

26 Practical Consequences of Calculability: Non-perturbative Corrections Stanar lore: IRC safe observables have controlle sensitivity to non-perturbative physics Ex: Thrust 1 Q X i E i sin i tan i 2 OPE region: = pert + O QCD Q QCD Q Dominate by non-perturbative physics QCD Q 26

27 Practical Consequences of Calculability: Non-perturbative Corrections Assume: r = Z /Q 2 e e e e = apple 2 2 pert e e pert e e + Z /Q e e + O 2 e e + O QCD Q QCD Q r e e r e e QCD e,e Q QCD Q Dominate by NP effects NP effects are power corrections Direct non-perturbative contribution is small for: Q e p 2 p p s CF QSu Requires finite αs NP effects are power-suppresse at large Q for r ~ 1 For reasonable values of the parameters, QSu ~ 250 GeV 27

28 Practical Consequences of Calculability: Non-perturbative Corrections Σ Pythia 8 Distributions Ratio r Α,Β, Α 2, Q 450,550 GeV Β 1.5: Parton Haron Β 1.0: Parton Haron Β 0.5: Parton Haron Σ Pythia 8 Distributions Ratio r Α,Β, Α 1, Q 450,550 GeV Β 0.75: Parton Haron Β 0.5: Parton Haron Β 0.25: Parton Haron r Α,Β 3 r Α,Β Σ r Α,Β r Α,Β e Α e Β (a) Pythia 8 Distributions Ratio r Α,Β, Α 2, Q 4500,5500 GeV Β 1.5: Parton Haron Β 1.0: Parton Haron Β 0.5: Parton Haron 500 GeV Σ r Α,Β r Α,Β e Α e Β (b) Pythia 8 Distributions Ratio r Α,Β, Α 1, Q 4500,5500 GeV Β 0.75: Parton Haron Β 0.5: Parton Haron Β 0.25: Parton Haron Power-suppresse non-perturbative corrections! r Α,Β e Α e Β 5000 GeV r Α,Β e Α e Β 28

29 Review Ratio observables are IRC unsafe but calculable: Suakov safe Cross section series in αs ½ Monte Carlo shoul escribe these observables accurately Inclues all-orers approximation to matrix element Non-perturbative corrections are power-suppresse Similar behavior as with IRC safe observables 29

30 Going Further Calculating ouble ifferential cross section to higher accuracy Fixe-Orer Corrections Schematic form: 2 match e e = 2 resum e e + 2 FO e e 2 sing e e Bulk contribution to ratio cross section Small (power-suppresse) corrections Σ Log R Matching Ratio r Α,Β, Α 2 Β 1.5: LL LL LO Β 1.0: LL LL LO Β 0.5: LL LL LO Σ Log R Matching Ratio r Α,Β, Α 1 Β 0.75: LL LL LO Β 0.5: LL LL LO Β 0.25: LL LL LO r Α,Β 1.5 r Α,Β r Α,Β e Α e Β r Α,Β e Α e Β

31 Going Further Calculating ouble ifferential cross section to higher accuracy Higher-orer Resummation AL, Moult, Neill 2014 Har Q Jet Q τ ½ μ Soft Q τ ΛQCD Factorization theorem: ' H(µ) J(,µ) S(,µ) 31

32 Going Further Calculating ouble ifferential cross section to higher accuracy Higher-orer Resummation AL, Moult, Neill 2014 Har Jet Q Q τ ½ Constraints: (e,e = e / )= (e ) (e = e,e )= (e ) Soft μ Q τ ΛQCD Factorization theorem: ' H(µ) (e,e ) 32 e =e / e =e (e,e ) = e = e Define by interpolating function that satisfies bounary conitions! Unique up to O(αs 4 ) Holographic Factorization Theorem

33 Going Further Other Examples of Suakov Safety: Groome Energy Loss Soft Drop! apple AJL, Marzani, Soyez, Thaler 1402.soon! apple Grooming E E 0 E g E 0 Original jet: E0 energy-rop ( E) = log z cut B i log E B i αs expansion: + energy-rop ( E) =1! Groome jet: Eg 2C i s (log E B i ) 2 s C i log 2 z cut 1 e 2 s E + O C i log z cut E log s 2! 1 E +B i β = 0: energy-rop ( E) =0 = log z cut B i log E B i inepenent of αs!? 33

34 Conclusions What other examples of Suakov safe observables are there? Can these techniques be applie to observables like N-subjettiness? Do we nee a new efinition of IRC safety/calculability in perturbation theory? Can techniques from CFTs help in unerstaning these observables? Is QCD better approximate by a free theory, a weakly couple CFT or something else? 34