Resummation Andrew Hornig LANL of Jet Rates Jan 1, 2016 In - PowerPoint PPT Presentation

Resummation Andrew Hornig � LANL � of Jet Rates Jan 1, 2016 � In collaboration with Y. Chien, C. Lee (arXiv:1509.04287)

What is a Jet? ❖ high-energy event: ?? = ❖ organizing principle (beyond fixed-order calculation)?

What is a Jet? ❖ (soft & collinear) singularities ➝ organize through factorization � jet (coll. splittings) � � = + power � Soft corrections � � hard process � � ❖ can be achieved via Effective Field Theory (in particular, Soft- Collinear Effective Theory, or SCET)

SCET & Factorization: Thrust ❖ thrust measures “jettiness” of e + e - events: � ˆ t R � L θ R i τ = τ L + τ R � � X E i cos θ L,R 1 − τ L,R L = i θ i i ∈ L,R � ❖ small thrust ⟹ all particles close to thrust axis (very jetty) � ❖ fixed order calculation not possible in this region: ln 3 τ ln 2 τ 1 d σ ln τ 1 ln τ ⇣ ⌘ ⇣ ⌘ + α 2 d τ = 1 + α s a 12 + a 11 τ + a 10 + a 22 + a 21 + a 20 + · · · a 23 s τ τ τ σ 0 τ

SCET & Factorization: Thrust ❖ factorization: jet function � (coll. splittings) d σ d τ = H ∗ J n ⊗ J ¯ n ⊗ S n ¯ n + power � Soft corrections � hard function virtual coll. real soft real ❖ resummation via RGE: µ H = Q µ J = Q √ τ µ S = Q τ

Factorization of Jet Rates ❖ “unmeasured jets” : tagged with algorithm but unprobed R E < Λ } record rate (count events) σ ( R, Λ ) ❖ “measured jets” : probed with mass, angularity, etc } “jet shapes” (not the jet shape Ψ (r/R)) 1 2 µ Ellis, Kunszt, Soper ’91, ‘92 E < Λ } d σ ( R ) µ bin in (e.g.) mass dm J m 2 J = ( p 1 + p 2 ) 2 µ µ

Factorization of Jet Rates Ellis, AH, Lee, Vermilion, Walsh 1001.0014 ❖ “unmeasured jets” : tagged with algorithm but unproved R E < Λ } ? σ ( R, Λ ) = H ( Q ) ∗ J unmeas ( QR ) ∗ S unmeas ( R, Λ /Q ) ❖ “measured jets” : probed with mass, angularity, etc 1 2 µ E < Λ } d σ ( R ) µ ? = H ( Q ) ∗ J meas ( m J , R ) ∗ S meas ( R, Λ /Q, m J ) dm J valid for R << 1 µ µ

1 Jet Rates from Integrating Shapes to α s ❖ can get rates directly from integrating shapes: ⇣ Z τ max ( R ) ( τ = R 2 ) = 1 + α s C F � 8 ln R ln 2 Λ ⌘ ⇣ d τ d σ } σ cone Q � 6 ln R + 6 ln 2 � 1 ⇣ c 2 π σ ( R ) = d τ = ⌘ τ = R 2 + 5 � 2 π 2 = 1 + α s C F � 8 ln R ln 2 Λ ⌘ ⇣ ⌘ ⇣ σ k T Q � 6 ln R 0 c 3 4 2 π 2 π 2 ⌘ = H ∗ J meas ( τ , R ) ∗ S meas ( R, Λ /Q, τ ) = H ( Q ) ∗ J unmeas ( QR ) ∗ S unmeas ( R, Λ /Q ) Z τ max ( R ) note: � d τ J meas ( τ , R ) 6 = J unmeas ( QR ) 0 → part of S meas ( τ ) is needed (more later!) 8 Andrew Hornig, LANL SF Flavor WS Jan 11, 2016

Jet Shapes to α s1 ❖ jet function with a jet algorithm (R dependence needed!): J alg. ( t n , R, µ ) = J incl ( t n , µ ) + ∆ J alg. ( t n , R ) , n  + θ ( t � Q 2 R 2 ) t ⌘� ∆ J cone ( t, R ) = α s C F  6 ⇣ θ ( t ) θ ( Q 2 R 2 � t ) 4 ln Q 2 R 2 + 3 , t 4 π t + Q 2 R 2 2 2 ⇣ ⌘� � → power correction for τ << R, � but needed in general! ❖ soft function: h ⇣ 4 π 4 Λ − π 2 µ 2  θ ( k i ) µR � 2 α s C F 1 ln k i 1 + α s C F ⌘i h ⇣ X , S ( k n , k ¯ n , Λ , R, µ ) = δ ( k ) 4 ln R ln − 3 µR k i µR 4 π 4 Λ 2 R π + i = n, ¯ n } 2 ⌘i  � X part associated with veto: � minimized for μ ~ 2 Λ R 1/2 � CLUE?? 9 Andrew Hornig, LANL SF Flavor WS Jan 11, 2016

The α s2 Result Manteufell, Schabinger, Zhu 1309.3560 � µ + 8 π 2 � − 176 � � − 88 ln( r ) − 536 � K (2) ¯ 9 ln 3 TC ( τ ω , ω , r → 0 , µ ) = C A C F + 3 3 9 Q τ ω � µ + 56 ζ 3 + 44 π 2 � � � − 44 3 ln 2 ( r ) + 8 3 π 2 ln( r ) − 536 ln( r ) − 1616 × ln 2 + Q τ ω 9 9 27 � µ � µ − 44 π 2 � � − 44 3 ln 2 ( r ) − 8 3 π 2 ln( r ) + 536 ln( r ) � + 88 � × ln + ln 3 ln( r ) 9 9 2 ω Q τ ω × ln 2 � µ 3 + 88 π 2 � Q τ ω � � � � Q τ ω � − 8 − 16 ζ 3 − 8 + 4 � 3 π 2 ln 2 3 π 2 ln 2 ( r ) + ln 2 ω 2 r ω 9 2 r ω � µ − 268 ln 2 ( r ) + 109 π 4 − 1139 π 2 − 682 ζ 3 − 1636 � � 64 � 9 ln 3 + C F n f T F 9 9 45 54 81 Q τ ω � µ � µ � 16 ln 2 ( r ) − 16 π 2 � 32 ln( r ) � � � � + 160 + 160 ln( r ) + 448 ln 2 + + ln 3 9 Q τ ω 3 9 9 27 Q τ ω 3 ln( r ) ln 2 � µ � µ � 16 ln 2 ( r ) + 16 π 2 3 − 32 π 2 − 32 − 160 ln( r ) � � 16 � � � + ln + 2 ω 3 9 9 2 ω 9 + 80 ln 2 ( r ) + 218 π 2 � Q τ ω � � + 248 ζ 3 − 928 × ln (5.14) . 2 r ω 9 9 27 81 ❖ large logs at μ ~ 2 Λ R 1/2 ❖ “refactorization??” (but not clear any set of scales will work) 10 Andrew Hornig, LANL SF Flavor WS Jan 11, 2016

SCET + Bauer, Tackmann, Walsh, Zuberi 1106.6047 ❖ originally used for when jets get close: QCD QCD q 1 Q Q SCET q 1 √ SCET t SCET + q 3 q 3 ⇒ ⇒ m m soft + q 2 m 2 √ soft / t soft m 2 m 2 /Q /Q q 2 (a) All jets equally separated. (b) Two jets close to each other. ❖ requires a new “csoft” mode p cs ∼ Q ( λ 2 , η 2 , ηλ ) , = m η = λ λ = m t . Q . √ λ t 11 Andrew Hornig, LANL SF Flavor WS Jan 11, 2016

SCET + for Jet Rates ❖ we also fix small component and decrease ⊥ small R } p + = Q τ fixed by τ meas p = Q τ (1 , 1 /R 2 , 1 /R ) virtuality increased due to R! p ⊥ ∝ p − /R inside R (the jet) 12 Andrew Hornig, LANL SF Flavor WS Jan 11, 2016

The Soft-Collinear Mode (new!) O . p + p + g = p − g /R g = − 2 Λ sc ¯ n p + g = Rp − g s s p − sc n g 2 Λ soft but confined to jet � soft anywhere “soft-collinear” ( Λ , Λ , Λ ) Λ (1 , R 2 , R ) 13 Andrew Hornig, LANL SF Flavor WS Jan 11, 2016

⬅ Re -Factorization } } µ H = Q Hard scale ∼ ( Q, Q, Q ) SCET+ Jet scale µ J = Q √ τ Q (1 , τ , √ τ ) Bauer, Tackmann, Walsh, Zuberi SCET ++ 1106.6047 ⇣ 1 µ S = Q τ /R Csoft scale R 2 , 1 , 1 ⌘ Q τ Chien, AH, Lee R 1509.04287 Global soft µ Λ = 2 Λ ( Λ , Λ , Λ ) (veto) scale depends on choice Soft-collinear of shape Λ (1 , R 2 , R ) Soft-collinear 
 µ sc = 2 Λ R mode (new!) scale radiation everywhere d σ d τ = H ( Q ) ∗ J meas ( τ , R ) ∗ S meas ( R, Λ /Q, τ ) w/ E ~ Λ radiation in jet w/ E ~ Λ S meas ( R, Λ /Q, τ ) → S in ( Q τ R ) S s ( Λ ) S c ( Λ R ) 14 Andrew Hornig, LANL SF Flavor WS Jan 11, 2016

Predicting the α s2 Result ⇣ α s ⌘ 2 S (2) S c ( k, Λ , R, µ ) = S C F ( k, Λ , R, µ ) + nA ( k, Λ , R, µ ) , 4 π ⌘ 2 ⇢ h ⇣ ⌘ i ⇣ α s 2( Γ 0 ) 2 ⇣ ⌘ 2 µ 2 − π 2 µ 2 S C F ( k, Λ , R, µ ) = 1 + α s − ln 2 µR − ln 2 µR 2 Γ 0 k + ln R ln + k + ln R ln 3 C F 4 Λ 2 R 4 Λ 2 R 4 π 4 π � ⇣ ⌘ 3 ( Γ 0 ) 2 ⇣ ⌘ π 2 k − ln R ln µ 2 − 4 π 2 ln 2 µR ln 2 µR 0 ln µR k + ln 2 R k + c (2) − 16 ζ 3 Γ 2 + 2 Γ 0 , (65) 3 C F C F 4 Λ 2 R ⇣ ⌘ ⇣ ⌘ µ 2 nA ( k, Λ , R, µ ) = 4 − ln 3 µR k + ln 3 µ µ − ln 2 µR S (2) 2 Λ − ln 3 + S c (2) + 2 Γ 1 k + ln R ln ng ( k, Λ , R, µ ) 3 Γ 0 β 0 4 Λ 2 R 2 Λ R in ) ln µR ss ) ln µ µ 2 Λ R + c (2) + 2( γ 1 in + 2 β 0 c 1 k + ( γ 1 ss + 2 β 0 c 1 2 Λ + 2( γ 1 sc + 2 β 0 c 1 sc ) ln nA . ❖ comparison to α 2 result ⇒ all logs of 2 Λ , 2 Λ R, and Q τ /R! ❖ this also gives the anom. dimensions to α 2 for free!! γ 1 ss = − 2 γ 1 in = − 2 γ 1 sc 27 T F n f − 2 π 2 h⇣ 1616 C A − 448 ⌘ i = C F − 56 ζ 3 . 3 β 0 27 (70) ❖ can argue to all orders (ingredients known to α 3 )!!! γ hemi = γ in = γ sc = − γ ss . 2 15 Andrew Hornig, LANL SF Flavor WS Jan 11, 2016

How the Modes Integrate ❖ complete EFT over all physical values of τ ⇣ 1 Csoft scale µ S = Q τ /R R 2 , 1 , 1 ⌘ Q τ R τ → τ max ∼ R 2 Jet scale Q (1 , τ , √ τ ) µ J = Q √ τ 16 Andrew Hornig, LANL SF Flavor WS Jan 11, 2016

Jet Rate Factorization (Proof) these modes coincide @ τ max ~ R 2 d σ d τ = H ( Q ) ∗ J meas ( τ , R ) ∗ S meas ( R, Λ /Q, τ ) S in ( Q τ R ) S s ( Λ ) S c ( Λ R ) ❖ now we have: σ ( R, Λ ) → H ( Q ) J unmeas ( QR ) S s ( Λ ) S c ( Λ R ) Z τ max ( R ) d τ J meas ( τ , R ) S in ( Q τ J unmeas ( QR ) = R ) 0 17 Andrew Hornig, LANL SF Flavor WS Jan 11, 2016

Plots ❖ reduced normalization and scale uncertainty: With s-c refactorization No s-c refactorization 1.0 1.0 R = 0 . 2 , Λ = 10 GeV , Q = 100 GeV 0.8 0.8 σ c ( τ , Λ , R ) 0.6 0.6 0.4 0.4 NLL NLL 0.2 0.2 NNLL NNLL R = 0 . 2 , Λ = 10 GeV , Q = 100 GeV 0.0 0.0 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 τ τ 18 Andrew Hornig, LANL SF Flavor WS Jan 11, 2016

Conclusions ❖ can resum logs or R with 2 additional modes: 2. soft-collinear mode (new) } SCET ++ 1. “csoft” mode of SCET + � ❖ all anomalous dimensions known to α 3 ❖ can integrate jet shapes to get jet rates 1. jet rate fact. thms now proven (with J unmeas ) � 2. understand relation of unmeas. and meas. funcs 19 Andrew Hornig, LANL SF Flavor WS Jan 11, 2016