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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?

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SLIDE 1

Resummation

  • f Jet Rates

Andrew Hornig LANL Jan 1, 2016

In collaboration with Y. Chien, C. Lee (arXiv:1509.04287)

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SLIDE 2

What is a Jet?

❖ high-energy event:

= ??

❖ organizing principle (beyond fixed-order calculation)?

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SLIDE 3

What is a Jet?

❖ (soft & collinear) singularities ➝ organize through factorization

  • ❖ can be achieved via Effective Field Theory (in particular, Soft-

Collinear Effective Theory, or SCET)

=

Soft + power corrections jet (coll. splittings) hard process

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SLIDE 4

SCET & Factorization: Thrust

❖ thrust measures “jettiness” of e+e- events:

  • ❖ small thrust ⟹ all particles close to thrust axis (very jetty)

❖ fixed order calculation not possible in this region:

τ = τL + τR

= X

i∈L,R

Ei cos θL,R

i

ˆ t

L R

θR

i

θ

L i

1 σ0 dσ dτ = 1 + αs ⇣ a12 ln τ τ + a11 1 τ + a10 ⌘

+ α2

s

⇣ a23 ln3 τ τ + a22 ln2 τ τ + a21 ln τ τ + a20 ⌘ + · · ·

1 − τL,R

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SLIDE 5

SCET & Factorization: Thrust

Soft + power corrections jet function (coll. splittings) hard function

dσ dτ = H ∗ Jn ⊗ J¯

n ⊗ Sn¯ n

virtual

  • coll. real

soft real

µH = Q µJ = Q√τ µS = Qτ

❖ resummation via RGE: ❖ factorization:

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SLIDE 6

Factorization of Jet Rates

❖ “unmeasured jets” : tagged with algorithm but unprobed ❖ “measured jets” : probed with mass, angularity, etc

“jet shapes” (not the jet shape Ψ(r/R))

}

Ellis, Kunszt, Soper ’91, ‘92

µ µ µ µ

record rate (count events) dσ(R) dmJ

1 2

m2

J = (p1 + p2)2

R

}

E < Λ

}

E < Λ

σ(R, Λ) bin in (e.g.) mass

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SLIDE 7

Factorization of Jet Rates

❖ “unmeasured jets” : tagged with algorithm but unproved ❖ “measured jets” : probed with mass, angularity, etc

µ µ µ µ

dσ(R) dmJ

1 2

R

}

E < Λ

}

E < Λ

σ(R, Λ)

= H(Q) ∗ Junmeas(QR) ∗ Sunmeas(R, Λ/Q)

= H(Q) ∗ Jmeas(mJ, R) ∗ Smeas(R, Λ/Q, mJ)

Ellis, AH, Lee, Vermilion, Walsh 1001.0014

? ?

valid for R << 1

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SLIDE 8

Andrew Hornig, LANL SF Flavor WS Jan 11, 2016

Jet Rates from Integrating Shapes to αs

1

8

❖ can get rates directly from integrating shapes:

σcone

c

(τ =R2) = 1 + αsCF 2π ⇣ 8 ln R ln 2Λ Q 6 ln R ⌘ ⇣ + 6 ln 2 1 ⌘

σkT

c

⇣ τ = R2 4 ⌘ = 1 + αsCF 2π ⇣ 8 ln R ln 2Λ Q 6 ln R 2π2 ⌘ ⇣ + 5 2π2 3 ⌘

σ(R) = Z τ max(R) dτ dσ dτ =

}

= H(Q) ∗ Junmeas(QR) ∗ Sunmeas(R, Λ/Q)

= H ∗ Jmeas(τ, R) ∗ Smeas(R, Λ/Q, τ)

note: → part of Smeas(τ) is needed (more later!)

Z τ max(R) dτJmeas(τ, R) 6= Junmeas(QR)

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SLIDE 9

Andrew Hornig, LANL SF Flavor WS Jan 11, 2016

Jet Shapes to αs1

9

Jalg.

n

(tn, R, µ) = Jincl(tn, µ) + ∆Jalg.(tn, R),

∆Jcone(t, R) = αsCF 4π  θ(t)θ(Q2R2 t) 6 t + Q2R2

  • 2

2 ⇣

⌘  + θ(t Q2R2) t ⇣ 4 ln t Q2R2 + 3 ⌘ ,

→ power correction for τ << R, but needed in general!

S(kn, k¯

n, Λ, R, µ) = δ(k)

h 1 + αsCF 4π ⇣ 4 ln R ln µ2 4Λ2R

2 ⌘i

X 

  • h

4π ⇣ 4Λ − π2 3 ⌘i − X

i=n,¯ n

2αsCF π 1 µR θ(ki)µR ki ln ki µR

  • +

,

part associated with veto: minimized for μ ~ 2 Λ R1/2 CLUE??

❖ jet function with a jet algorithm (R dependence needed!): ❖ soft function:

}

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SLIDE 10

Andrew Hornig, LANL SF Flavor WS Jan 11, 2016

The αs2 Result

10 ¯ K(2)

TC(τω, ω, r → 0, µ) = CACF

  • −176

9 ln3 µ Qτω

  • +
  • −88 ln(r)

3 + 8π2 3 − 536 9

  • × ln2

µ Qτω

  • +
  • −44

3 ln2(r) + 8 3π2 ln(r) − 536 ln(r) 9 + 56ζ3 + 44π2 9 − 1616 27

  • × ln

µ Qτω

  • +
  • −44

3 ln2(r) − 8 3π2 ln(r) + 536 ln(r) 9 − 44π2 9

  • ln

µ 2ω

  • + 88

3 ln(r) × ln2 µ 2ω

  • − 8

3π2 ln2 Qτω 2rω

  • +
  • −16ζ3 − 8

3 + 88π2 9

  • ln

Qτω 2rω

  • + 4

3π2 ln2(r) −268 ln2(r) 9 − 682ζ3 9 + 109π4 45 − 1139π2 54 − 1636 81

  • + CFnfTF

64 9 ln3 µ Qτω

  • +

32 ln(r) 3 + 160 9

  • ln2

µ Qτω

  • +

16 ln2(r) 3 + 160 ln(r) 9 − 16π2 9 + 448 27

  • ln

µ Qτω

  • −32

3 ln(r) ln2 µ 2ω

  • +

16 ln2(r) 3 − 160 ln(r) 9 + 16π2 9

  • ln

µ 2ω

  • +

16 3 − 32π2 9

  • × ln

Qτω 2rω

  • + 80 ln2(r)

9 + 248ζ3 9 + 218π2 27 − 928 81

  • .

(5.14)

Manteufell, Schabinger, Zhu 1309.3560

❖ large logs at μ ~ 2 Λ R1/2

❖ “refactorization??” (but not clear any set of scales will work)

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SLIDE 11

Andrew Hornig, LANL SF Flavor WS Jan 11, 2016

SCET+

11

q3 q1 q2

⇒

m2 /Q m Q SCET QCD soft (a) All jets equally separated. q3 q1 q2

⇒

m2 /Q m2 / √ t √ t m Q SCET QCD SCET+ soft+ soft (b) Two jets close to each other.

❖ originally used for when jets get close: ❖ requires a new “csoft” mode

pcs ∼ Q(λ2, η2, ηλ) ,

λ = m Q .

η = λ λt = m √ t .

Bauer, Tackmann, Walsh, Zuberi 1106.6047

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SLIDE 12

Andrew Hornig, LANL SF Flavor WS Jan 11, 2016

SCET+ for Jet Rates

12

❖ we also fix small component and decrease ⊥

p⊥ ∝ p−/R

fixed by τ meas small R

}

p+ = Qτ

inside R (the jet) virtuality increased due to R!

p = Qτ(1, 1/R2, 1/R)

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SLIDE 13

Andrew Hornig, LANL SF Flavor WS Jan 11, 2016

O .

p+

g = Rp− g

p+

g = p− g /R

2Λ 2Λ p−

g

p+

g

ss

scn sc¯

n

=

−

The Soft-Collinear Mode (new!)

13

(Λ, Λ, Λ)

soft anywhere

Λ(1, R2, R)

soft but confined to jet “soft-collinear”

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SLIDE 14

Andrew Hornig, LANL SF Flavor WS Jan 11, 2016

Re-Factorization

14

µS = Qτ/R Hard scale Jet scale Global soft (veto) scale µH = Q µΛ = 2Λ

µsc = 2ΛR µJ = Q√τ Csoft scale Soft-collinear
 scale

Q(1, τ, √τ)

∼ (Q, Q, Q)

Qτ ⇣ 1 R2 , 1, 1 R ⌘

(Λ, Λ, Λ)

Λ(1, R2, R)

}

SCET+

Soft-collinear mode (new!) ⬅

}

SCET++

Chien, AH, Lee 1509.04287

depends on choice

  • f shape

radiation everywhere w/ E ~ Λ

radiation in jet w/ E ~ Λ

Smeas(R, Λ/Q, τ) → Sin(Qτ R ) Ss(Λ)Sc(ΛR) dσ dτ = H(Q)∗Jmeas(τ, R)∗Smeas(R, Λ/Q, τ)

Bauer, Tackmann, Walsh, Zuberi 1106.6047

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SLIDE 15

Andrew Hornig, LANL SF Flavor WS Jan 11, 2016

Predicting the αs2 Result

15

Sc(k, Λ, R, µ) = SCF (k, Λ, R, µ) + ⇣αs 4π ⌘2 S(2)

nA(k, Λ, R, µ) ,

SCF (k, Λ, R, µ) = 1 + αs 4π h 2Γ0 ⇣ − ln2 µR k + ln R ln µ2 4Λ2R ⌘ − π2 3 CF i + ⇣αs 4π ⌘2⇢ 2(Γ0)2⇣ − ln2 µR k + ln R ln µ2 4Λ2R ⌘2 + 2Γ0 π2 3 CF ⇣ ln2 µR k − ln R ln µ2 4Λ2R ⌘ − 4π2 3 (Γ0)2⇣ ln2 µR k + ln2 R ⌘ − 16ζ3Γ2

0 ln µR

k + c(2)

CF

  • , (65)

S(2)

nA(k, Λ, R, µ) = 4

3Γ0β0 ⇣ − ln3 µR k + ln3 µ 2Λ − ln3 µ 2ΛR ⌘ + 2Γ1 ⇣ − ln2 µR k + ln R ln µ2 4Λ2R ⌘ + Sc(2)

ng (k, Λ, R, µ)

+ 2(γ1

in + 2β0c1 in) ln µR

k + (γ1

ss + 2β0c1 ss) ln µ

2Λ + 2(γ1

sc + 2β0c1 sc) ln

µ 2ΛR + c(2)

nA .

❖ comparison to α2 result ⇒ all logs of 2Λ, 2ΛR, and Qτ/R!

❖ this also gives the anom. dimensions to α2 for free!!

γ1

ss = −2γ1 in = −2γ1 sc

= CF h⇣1616 27 − 56ζ3 ⌘ CA − 448 27 TF nf − 2π2 3 β0 i . (70)

γhemi = γin = γsc = −γss 2 .

❖ can argue to all orders (ingredients known to α3)!!!

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SLIDE 16

Andrew Hornig, LANL SF Flavor WS Jan 11, 2016

How the Modes Integrate

16

µS = Qτ/R Csoft scale Qτ ⇣ 1 R2 , 1, 1 R ⌘

❖ complete EFT over all physical values of τ

Jet scale µJ = Q√τ

Q(1, τ, √τ)

τ → τ max ∼ R2

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SLIDE 17

Andrew Hornig, LANL SF Flavor WS Jan 11, 2016

Jet Rate Factorization (Proof)

17

these modes coincide @ τmax ~ R2

σ(R, Λ) → H(Q)Junmeas(QR)Ss(Λ)Sc(ΛR)

❖ now we have:

Junmeas(QR) = Z τ max(R) dτ Jmeas(τ, R)Sin(Qτ R ) dσ dτ = H(Q)∗Jmeas(τ, R)∗Smeas(R, Λ/Q, τ) Sin(Qτ R ) Ss(Λ)Sc(ΛR)

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SLIDE 18

Andrew Hornig, LANL SF Flavor WS Jan 11, 2016

Plots

18

0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.0 0.2 0.4 0.6 0.8 1.0 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.0 0.2 0.4 0.6 0.8 1.0

No s-c refactorization

τ

NLL NNLL

R = 0.2, Λ = 10 GeV, Q = 100 GeV

With s-c refactorization

τ

σc(τ, Λ, R) NLL NNLL

R = 0.2, Λ = 10 GeV, Q = 100 GeV

❖ reduced normalization and scale uncertainty:

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SLIDE 19

Andrew Hornig, LANL SF Flavor WS Jan 11, 2016

Conclusions

19

❖ can resum logs or R with 2 additional modes:

  • 1. “csoft” mode of SCET+
  • 2. soft-collinear mode (new) } SCET++

❖ all anomalous dimensions known to α3 ❖ can integrate jet shapes to get jet rates

  • 1. jet rate fact. thms now proven (with Junmeas)
  • 2. understand relation of unmeas. and meas. funcs