resummation of the d parameter - reed collegepeople.reed.edu/~larkoski/talks/ucla_2018.pdf ·...

Resummation of the D Parameter

Andrew Larkoski Reed College

work in progress with Andrea Banfi, Aja Procita

UCLA, April 5, 2018

2

Legacy of LEP Precision Measurements

Y

XZ

200 . cm.

Cen t r e o f sc r een i s ( 0 . 0000 , 0 . 0000 , 0 . 0000 )

50 GeV2010 5

Run : even t 4093 : 1000 Da t e 930527 T ime 20716 Ebeam 45 . 658 Ev i s 99 . 9 Emi ss - 8 . 6 V t x ( - 0 . 07 , 0 . 06 , - 0 . 80 ) Bz=4 . 350 Th r us t =0 . 9873 Ap l an=0 . 0017 Ob l a t =0 . 0248 Sphe r =0 . 0073

C t r k (N= 39 Sump= 73 . 3 ) Eca l (N= 25 SumE= 32 . 6 ) Hca l (N=22 SumE= 22 . 6 ) Muon (N= 0 ) Sec V t x (N= 3 ) Fde t (N= 0 SumE= 0 . 0 )

Event Display from OPAL

3


Y

XZ

200 . cm.


50 GeV2010 5



Event Display from OPAL

Event shape variables quantify energy flow distribution

Y

XZ

200 . cm.


50 GeV2010 5



4

τ

σ

dσ

dτ

τ

0.300.10 0.15 0.20 0.250.0

0.4

0.3

0.2

0.1

Fit at N LL3 ’

theory scan error

DELPHI

ALEPH

OPAL

L3

SLD

for & �


Abbate, Fickinger, Hoang, Mateu, Stewart 2010

Thrust at N3LL + NNLO

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Cpar �

d�

dCpar

ALEPH dataGGGH

LONLONNLO

⇠R 2 [0.5, 2]⇠R 2 [0.5, 2]⇠R 2 [0.5, 2]

pQ2 = 91.2GeV

↵s(Q2) = 0.118

0.951.01.05

GGGH

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Cpar

GGGH

0.951.01.05

SW

SW

C-parameter at NNLO

Del Duca, Duhr, Kardos, Somogyi, Szőr, Trócsányi, Tulipánt 2016

Precision data/theory comparisons enable 𝛼s extraction

5


Why thrust and C-parameter?NNLO:

2 → 3 at two-loopsNNLL/N3LL:

Small values restrict radiation

Lots of tools!

e+

e-

q

q

g

𝜏, C, B, first non-zero

EERAD3, CoLoRFulNNLO, etc.

𝜏, C, B ≪ 1

Cross section factorizesHard, Soft, Collinear functions

6


LEP measured more than just 𝜏 and C!See: http://hepdata.cedar.ac.uk/review/shapes/

D parameter at DELPHI, L3, OPAL

C → 0: linear event codimension 2

D → 0: planar event codimension 1

7


Challenges with D parameter: Fixed Order

NNLO: 2 → 4 at two-loops

Just starting to be calculated

State of the art NLO: 2 → 4 at one-loop

Badger, Brønnum-Hansen, Hartanto, Peraro 2017Abreu, Febres Cordero, Ita, Page, Zeng 2017

Nagy, Trócsányi 1997

D parameter

D

xµ = 1

xµ = 0.1

: L3 datalines : NLO QCD

0 0.2 0.4 0.6 0.8 110-4

10-3

10-2

10-1

100

8


Challenges with D parameter: Resummation

NLL: D → 0 has many regions

State of the art NLL: Restrict to near-planar region

Banfi, Dokshitzer, Marchesini, Zanderighi 2001

Can have the same value of D0.01

0.1

1

10

0 0.2 0.4 0.6 0.8 1

σ-1dσ/dD

D

PTPT+NP

y3 > 0.1Not inclusive over final state

9


Challenges with D parameter: Resummation

Banfi, Dokshitzer, Marchesini, Zanderighi 2001

0.01

0.1

1

10

0 0.2 0.4 0.6 0.8 1

σ-1dσ/dD

D

PTPT+NP

y3 > 0.1Inclusive over everything but D

≠

10

This Talk

Remove all vestigial restrictions on the event

Resum D parameter to NLL as proof-of-concept

Approach inspired by developments in jet substructureReview: AJL, Moult, Nachman 2017

+d�

dD=

11

This Talk

Method: • Identify distinct configurations • Resum Separately • Sum together

+d�

dD=

d�di-jet

dD

d�tri-jet

dD

12

Spherocity Tensor

C and D are defined from the eigenvalues of the spherocity tensor

⇥↵� =1

Q

X

i

pi↵pi�Ei

C = 3(�1�2 + �1�3 + �2�3) =3

Q2

X

i<j

EiEj sin2 ✓ij

D = 27�1�2�3 =27

Q3

X

i<j<k

|pi · (pj ⇥ pk)|2

EiEjEk

�1 � �2 � �3 �1 + �2 + �3 = 1

13

Spherocity Tensor

C = 3(�1�2 + �1�3 + �2�3) =3

Q2

X

i<j

EiEj sin2 ✓ij

D = 27�1�2�3 =27

Q3

X

i<j<k

|pi · (pj ⇥ pk)|2

EiEjEk

C ≈ (area of triangle)2Ei

Ej

✓ij

D ≈ (volume of parallelpiped)2

pi

pjpk

14

Spherocity Tensor

Simultaneous measurement of C and D isolate configurations!

• Region 1: Large Areas, Small Volumes

• Region 2: Small Areas, Smaller Volumes

• Region 3: Small Areas, Small Volumes

Three parametric relations in D → 0 limit:

D ⌧ C2 ⇠ 1

D ⌧ C2 ⌧ 1

�3 ⌧ �2 ⇠ �1 ⇠ 1

�3 ⌧ �2 ⌧ �1 ⇠ 1

�3 ⇠ �2 ⌧ �1 ⇠ 1 D ⇠ C2 ⌧ 1

15

Spherocity Tensor

Simultaneous measurement of C and D isolate configurations!

d�

dD=

ZdC

d�I

dC dD+

d�II

dC dD+

d�III

dC dD

�End goal:

Each of these cross sections are easy to calculate!!!

16

Region 1 Resummation

C = 3(�1�2 + �1�3 + �2�3) ! 3�1�2

D = 27�1�2�3 = 27�21�

22

�3

�1�2⌧ C2

�3 ⌧ �2 ⇠ �1 ⇠ 1

D ⇠ ⇠ C ⇠ 1

Region considered by Banfi, et al., 2001

Only large logs are of D/C2 ~ D

Additional soft/collinear emissions do not affect event plane

17

Region 1 Resummation�3 ⌧ �2 ⇠ �1 ⇠ 1

D ⇠ ⇠ C ⇠ 1

Factorization:d�I

dx1 dx2 dD= H(x1, x2)J1(D)⌦ J2(D)⌦ J3(D)⌦ S123(D)

Ji(D)

H(x1, x2)

S123(D)

Resummation by RG evolution from hard scale to

soft and jet scales

C = 6(1� x1)(1� x2)(1� x3)

x1x2x3

18


D ⇠

Novel configuration

Large logs of both C and D/C2

Additional soft/collinear emissions do not affect event plane

�3 ⌧ �2 ⌧ �1 ⇠ 1

C = 3(�1�2 + �1�3 + �2�3) ! 3�2

D = 27�1�2�3 ! 27�22�3

�2⌧ C2

⇠ C ⌧ 1

19


D ⇠

Just a re-factorization of the hard function from region 1

Now, there is the 2 → 2 hard function H and the splitting

function H2→3

�3 ⌧ �2 ⌧ �1 ⇠ 1

⇠ C ⌧ 1

Factorization:d�II

dx1 dx2 dD= H(Q2)H2!3(x1, x2)J1(D)⌦ J2(D)⌦ J3(D)⌦ S123(D)

S123(D)Ji(D)

H(Q2)H2!3(x1, x2)

20

Combining Regions 1 and 2

Option 1: Just additively match

d�

I&II

dx1 dx2 dD=

d�

I

dx1 dx2 dD+

d�

II

dx1 dx2 dD� d�

I!II

dx1 dx2 dD

Exact to any logarithmic accuracy

21


Option 2: Set scales to smoothly interpolate

d�I&II


When C ~ 1, H(Q2) → 1

When C ≪1, H2→3 → region 2 value

22


Option 2: Set scales to smoothly interpolate

d�I&II


µH2!3 =

8><

>:

✓A1

(2�x1�x2)C6

1+⇣A1� 6

(2�x1�x2)C1

⌘(2�x1�x2)C

6

◆1/2

Q , C < C1 ,

Q , C > C1 .

Exact to NLL accuracy

May not hold at higher orders

23


D ⇠

Novel configuration

Additional soft/collinear emissions can affect event plane

�3 ⇠ �2 ⌧ �1 ⇠ 1

C = 3(�1�2 + �1�3 + �2�3) ! 3(�2 + �3)

D = 27�1�2�3 ! 27�2�3 ⇠ C2

Only large logs are of C2 ~ D

⇠ C2

24


Issues:

D is not additive; convolution doesn’t work

Want for Factorization:�3 ⇠ �2 ⌧ �1 ⇠ 1

d�III

dC dD= H(Q2)J1(C,D)⌦ J2(C,D)⌦ S12(C,D)

Because emissions that set D contribute to C, need to keep track of all emissions

May be related to recent work on non-global logarithmsCaron-Huot 2015; AJL, Moult, Nachman 2015; Becher, Neubert, Shao 2016; Ángeles Martínez, De Angelis, Forshaw, Plätzer, Seymour 2018

25

Region 3 Resummation Redux

Throw out factorization!�3 ⇠ �2 ⌧ �1 ⇠ 1

p(C,D) = p(C)p(D|C)

d�

dC dD=

d�

dC

d�(C)

dD

Exploit conditional probabilities to make progress

d�

dCCalculable to high fixed

and resummed order

d�(C)

dDHas all of the subtleties

of this region

26


To make progress, we calculate what we can�3 ⇠ �2 ⌧ �1 ⇠ 1

Exact to NLL:

Misses some logs at NLL

d�NLL

dC dD=

d�NLL

dC

d�(C)NLL

dD

Approximate NLL: d�NLL

dC dD⇡ d�NLL

dC

d�(C)↵2s

dD

Think it gets all logs to 𝛼sn logn+1

27


Need to restrict to D ~ C2�3 ⇠ �2 ⌧ �1 ⇠ 1

Then, can additively match with regions 1 and 2

Advantages

d�III

dC dD⇡ d�NLL

dC

"d�(C)↵

2s

dD� d�(C)↵

2s,D⌧C2

dD

#

DownsidesGets a result

Systematically improvableNo formal logarithmic accuracy

Requires new definition of formal accuracy?

28

Complete Result

Now just combine and integrate over C!

d�

dD=

Z 3/4

p43D

dC

d�I&II

dC dD+

d�III

dC dD

�

Maximum value of C when D → 0

From constraint in region 3 C ≈ 3(λ2+λ3) D ≈ 27 λ2λ3

All overlaps have been subtracted

Accomplishes fully inclusive resummation of the D parameter!

29

Conclusions

The D parameter is one of the legacy precision measurements from LEP

Expect comparisons to LEP data soon!

Due to its complicated event selection, only near-planar region had been resummed

Inclusive resummation accomplished secondary measurement that simplifies event configuration

resummation of the d parameter - reed collegepeople.reed.edu/~larkoski/talks/ucla_2018.pdf ·...

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