Matthew Schwartz

1
Preliminary results froma SCET-based event
generator

• Matthew Schwartz
• Johns Hopkins

Theory with Christian Bauer, LBNL
Implementation with Stephen Mrenna, Fermilab
and Johan Alwall, SLAC
hep-ph/0604065, PRL 97142001, 2006
work in progress
hep-ph/0607296, PRD to appear
2
The LHC turns on THIS YEAR!
are we ready?
The are two important differences between the LHC
and previous machines
1. High Energy 14 TeV

• Large logs are larger

2. New physics should show up in events with many
hard jets

• Interference effects are crucial

We must have a systematic way to improve
simulations of QCD backgrounds
SCET can do it!
3
Outline

• Introduction
• Monte Carlo event generators
• Matrix Elements vs. Parton Showers
• SCET approach
• Matching, running
• Practical implementation
• Preliminary Results for ee- ! jets
• Open questions
• Conclusion

4
Monte Carlo Uncertainty
ee- T 4 jets at ECM2 TeV
kT of fourth jet
5
Matrix Elements
Matrix elements means fixed order perturbative
QCD

• Calculate Feynman diagrams numerically or
  analytically

. . .
h QCD i

• Includes important interference effects

• spin correlations

• accurate distribution of many hard jets

• Limited number of final states

• Current limit is 2 T 6

• Amplitudes are divergent

• dependence on cuts

• no Sudakov resummation

• Implemented in MadEvent, Comphep, Alpgen, . . .

6
Parton Shower

• Branchings are approximated by a classical
  Markov process
• Emission probability given by splitting
  functions

• Probability of no-emission given by Sudakov
  Factors

Classical resummation -- all orders in as

• Implemented in Pythia, Herwig, . . .

7
Compare ME and PS
ee- T 4 jets at ECM2 TeV
pT of fourth jet
resummatation
interference effects
8
Combining ME and PS
CKKW
(Catani, Krauss, Kuhn, Webbber)

• Starts with fixed order ME calculation
• Clusters partons to deterimine dominant PS
  history
• Reweights event with Sudakov weight that PS
  would have given
• Showers event, and vetoes emissions to avoid
  double counting
• Understood theoretically

Issues

• Explicit jet definition and cutoff needed to
  regulate ME divergences
• Need to combine 2-, 3-, 4-jet, etc. samples.
• Uses classical Sudakov factors not extendable
  to next-to-leading log
• Cannot include finite parts of loop effects
  only leading large logs

9
CKKW
10
CKKW
11
Combining ME and PS
MLM
( Michaleangelo Mangano)

• Starts with fixed order ME calculation
• Showers distribution, and then clusters showered
  event.
• vetoes event if there is not a one-to-one
  correspondence between clustered jets and partons
  from the ME
• Sudakov factors come literally from no
  branching probability
• Simple to implement independent of method used
  for ME or PS

Issues

• All of the issues for CKKW (leading log / no
  loops/ cutoff dependence/ )
• Can be very inefficient most events are vetoed
• Difficult to analyze theoretically
• May not work for FSR

12
MLM
13
MLM
14
SCET Approach
1. Match to QCD at hard scale Q
Matrix elements
h QCD i S Cn(Q)h On i
2. Run coefficients Cn using renormalizion group
evolution
Q
Cn(m) Cn(Q) exp( gn dm )
!
Parton shower
m
3. Match across thresholds in efffective theory
4. Final matrix element given by
h SCETi S Cn(m)h On i
15
SCET
16
SCET
17
SCET Approach
Pt1
Pt2
C2(pt1)
C2(Q)
C3(pt2)
C3(pt1)
C4(pt2)
P2(Q, pt1) P3(pt1, pt2) . . .
C3(Q)
C3(pt2)
C4(pt2) P3(Q, pt2) . . .
h SCETi P2(Q, pt1) P3(pt1, pt2)(. ..) h O2 i
P3(Q, pt2)()h O3 i-h O2 i
18
Matching to QCD

• General condition

We can build up Cj systematically by choosing
states
n i qqi, qqg i ,
First, 2-jet matching
(normalization)
1-loop matching is finite
-
)
19
Running
In SCET


. . .
In QCD

uv finite

because
is a conserved current.
gQCD 0
20
Resummation
Wilson coefficient is
In parton shower, splitting function is
Sudakov factor is an integral over the splitting
function
Explicitly,
SCET and parton shower both sum leading logs.
21
3-jet matching
Three jet matching
Solved by
-
Effective SCET vertex
QCD vertex
22
How events are unweighted
In the Matrix Element approach (e.g. MadEvent)
Matrix Eements of feynman diagrams in QCD
1. Pick external momenta p1 . . . pn based on grid
2. Calculate
ds phase space x h QCD i
3. Refine grid so that chance of picking p is
proportional to ds
4. Once grid is acceptable, generate events
The SCET approach
1-4 as above, but with h SCET i instead of
h QCD i
6. Run parton shower on event
We need to calculate h SCET i as a function of
given external momenta
23
SCET matrix elements
QED vertex
SCET vertex
QCD vertex
h O2 i
h O3i
h QCDi
SCET
h
i
m min(pt) for all diagrams
P2(Q, pt1) P3(pt1, pt2)
P4(pt2,m) h O2 i
P3(Q, pt2) P4(pt2,m) (h O3 i -h O2 i)
explicit analytic functions
P4(Q,m) (h QCDi -h O3i)
SCET
SCET
h
i
h
i
h SCETi
. . .
24
Implementation
Start with MadGraph

• versitile matrix element calculator
• works for standard and beyond-the-standard model
  physics

Add SCET

• add new numerical HELAS routines for SCET
  Feynman fules

• calculate matrix elements in SCET and QCD

• logs are resummed through renormalization group
  evolution

Unweight with MadEvemt

• efficiently samples phase space

Interface to PYTHIA

• start shower at event-dependent RG scale m
  min(pT)

• consistent, because SCET Parton Shower in
  collinear limit

25
Preliminary Results
Fourth jet pT
26
Preliminary Results
Energy of fourth jet
Fourth jet pT
27
Preliminary Results
Third jet pT
28
Preliminary Results
Shift in shape function due to soft effects
Thrust
29
Preliminary Results
Matching to PS
30
Preliminary Results
Matching to PS
31
Advantages of SCET

• SCET naturally combines Matrix Elements and
  Parton Showers
• Different philosophy from other approaches not
  merging, but calculating distribution correctly
• Systematically improvable
• Can resum subleading logs
• Can add finite parts of loop effects without
  negative weights
• Provides a forum to relate multiloop
  calculations to data
• Can improve matching to PDFs
• Reduces dependence on factorization scale
• Can improve matching to hadronization
• No explicit cutoff, no clustering
• RG kernels (sudakov factors) regulate
  divergences
• No jet cocktails
• If merging samples is desired, RG scale provides
  a nautral cutoff

32
SCET
33
Open Questions

1. NLL running
2. Threshold effects
3. SCET matrix elements
4. Implementing NLO effects
5. Soft effects
6. Factorization
7. Heavy mass threshholds

34
1. NLL running
The LL cusp anomalous dimension is
Some NLL effects are known

• at higher orders, there is mixing
• anomalous dimensions must be calculated/extracted
  from the literature
• explicit numerical implementation required

No other monte carlo can do this!
35
Preliminary Results
NLL uncertainty
Third jet pT
36
Partial NLL effects
37
2. Threshold effects
In matching to QCD
h SCETi m h QCDiQ

• match at m Q 2 TeV

• what is m in SCET? If m pT, we lose
  probability since pTlt Q/2
• should be canceled by NLO matching

In threshold matchings
m2
m1
these RG scales are in different SCETs

• should we choose m2 (1/2) m1?
• can we understand NLO threshold matching?
• is there a systematic way to address these
  issues?

38
Vary starting scale for running
39
3. SCET matrix elements
What does
mean?
Emission is neither strictly soft, nor strictly
collinear
h
i

p1s1, p2 s2, p3 em
numerical on-shell momenta and spins
What is pa

• In the collinear limit, h SCETi h QCDi Parton
  Shower

• Away from collinear limit (i.e. everywhere)

• must be smooth no explicit cuts like

• must avoid new poles for backwards emission

1 convention in our paper
2 h SCETi h QCDi
40
Convention dependence
Third jet pT
2
1
41
4. Implementing NLO effects

• The NLO wilson coefficients are finite. E.g.

• We can just put these into the Monte Carlo. E.g.

h SCETi C2 P2(Q, pt1) P3(pt1, pt2)(. ..) h O2 i
P3(Q, pt2)()h O3 i-h O2 i

• For other operators (more jets, heavy particles,
  etc.), NLO corrections affect the shape of
  distributes

• Extremely important for single top, Wjets,
  higgs,

What has been calculated?
42
Total cross section
Integral divergent in QCD
Integral convergent after resummation
43
Total cross section
First order in as
Finite virtual piece
1. 2-jets tree s C2(Q)2
loop s
Pure divergence
2. 3-jets tree s
First order in as (same as QCD)
Finite real piece
sT s0(1 )
44
Total cross section
Including LL resummation (i.e. in the monte carlo)
1. 2-jets tree s C2(mIR)2 a 1 (say, for
mIR 1 GeV)
2. 3-jets tree s
C2(Q)2 x Parton Shower C2(Q)2 x
dPno-branching(Q,m) C2(Q)2
s0(1 0.34 )
Thus,
interference
Just power corrections, convention dependent
what happened to the loops and the real infrared
divergence?
45
Toy model of NLO
phase space variable x pT
coupling constant
Parton Shower

• Start with a 2-parton event with probability 1
  (proportional to total s)

• First emission in shower given by

sudakov factor is classical no-branching
probability
integral is divergent
integral is 1
46
Toy model of NLO
Resummed result is convergent
But expansion is divergent
How do we do a systematically build an NnLO and
NnLL expansion at the same time?
47
5. Soft Effects

• Do soft gluons effect the distribution of hard
  jets?

• no

• maybe the hard parton-level matrix elements
  are not infrared safe

• For example, can the event shape results be
  reproduced in a monte carlo?

6. Factorization

• Is there a nice way to parametrize fragmentation
  in SCET?

• Can we interface a shape function to endpoint of
  a parton shower?

• Can we match to the parton-distribution-functions
  in a clean, improvable way

• how do we choose a factorization scale in pp?

7. Heavy mass thresholds

• Are there systematic improvements near the top
  mass?
• Can we say anything about W/Z/Higgs mass
  thresholds?

48
Conclusions
SCET is the key to systematically improving
simulations for the LHC

• SCET has elements of both matrix elements and
  parton showers

• cleanly solves the ME/PS merging problem of
  current monte carlos

• Straightforward implemention with MadEvent/Pythia

• Testing on ee- and pp collisions currently
  underway

• There remain a number of theoretical and
  practical issues to be resolved

1. NLL running
2. Threshold effects
3. SCET matrix elements
4. Implementing NLO effects
5. Soft effects
6. Factorization
7. Heavy mass threshholds
8. Name for monte carlo