1. Event Shapes, Jet Substructure and S at HERA
Adam Everett
University of Wisconsin
On Behalf of the ZEUS and H1 Collaborations

2. Unique opportunity to study hadron-lepton collisions
HERA Description
920 GeV p+
(820 GeV before 1999)
27.5 GeV e- or e+
318 GeV cms
Equivalent to a 50 TeV Fixed Target
Instantaneous luminosity max: 1.8 x 1031 cm-2s-1
220 bunches
96 ns crossing time
IP~90mA p
Ie~40mA e+
HERA Circumference: 6336 m
Synchrotron radiation is lin. pol. in plane of acceleration.
Mean energy loss per revolution: for Electrons: ~8MeV
Fixed target: ECM=S=2E1m2
Colliding Beam: ECM = S=4E1E2
Luminosity: L =f n1 n2/4 xy
N1,n2 number of particles in bunch
f=frequency
xy characterize beam profiles in horizontal and vertical direction
~1011 Protons/Bunch
~3.5x1010 e/Bunch
Current: 1.6x10-19 q/C
Vacuum Pressure: 3x10-11 Torr
e+ beam: T B field
P beam: 4.65 T B field
Spaces left in beam so kicker magnets can dump beam, study beamgas
Sokholov-Ternov Effect: e- polarized with spin antiparallel to dir. of bending field (transverse)
DESY Hamburg, Germany
Unique opportunity to study hadron-lepton collisions

3. HERA Kinematic Variables
Center of Mass Energy of ep system squared:
s = (p+k)2 ~ 4EpEe
Photon Virtuality (4-momentum transfer squared at electron vertex):
q2 = -Q2 = (k-k’)2
Fraction of Proton’s Momentum carried by struck quark:
x = Q2/(2p·q)
Fraction of e’s energy transferred to Proton in Proton’s rest frame:
y = (p·q)/(p·k)
Variables are related:
Q2 = sxy
W: invariant mass of the system recoiling against the scattered lepton
y: measures inelasticity of interaction. y=1 – E’_e/E_e in proton rest frame
Related to scattering angle **************
y=0: totally elastic
x=0 when y=1. y=0 when x=1.

4. Methods to Study QCD QCD Effects – Gluons Jets Event Shapes
Direct – gluons observed as jets
Complexity of jet reconstruction and identification
Jet cross sections and substructure
Event Shapes
Energy and particle flow
Direct – gluon radiation changes event shapes
Reduce dependence on hadronization
We have already briefly discussed 1 and 2. Now we will focus on 3.

5. Jets
Colored partons evolve to a roughly collinear “spray” of colorless hadrons
 JETS
Partons => Hadrons => Detector: schematically:
As produced
As observed

6. Jet Finding: Longitudinally Invariant KT Algorithm
In ep: kT is transverse momentum with respect to beamline
For every object i and every pair of objects i, j compute
d2i = E2T,i (distance to beamline in momentum space)
d2ij = min{E2T,i,E2T,j}[Dh2 + Df2]1/2 (distance between objects)
Calculate min{ d2i , d2ij } for all objects
If d2ij is the smallest, combine objects i and j into a new object
If d2i is the smallest, then object i is a jet
Inclusive in title? Divide by R^2?
JADE algorithm: dij = 2EiEj[1-cos(thetaij)]
invariant mass of 2 particles
2 gluons can be combined into “phantom” jet
kT algorithm: dij2 = 2min{Ei2,Ej2}[1-cos(thetaij)]
minimum relative transverse momentum of two particles
designed to reduce higher order contributions compared to JADE
Longitudinally invariant kT: d2ij = min{E2T,i,E2T,j}[Dh2 + Df2]1/2
Combinations:
ET,ij = ET,i + ET,j
hij = (ET,I * hi + ET,j *hJ)/ ET,ij
fij = (ET,I * fi + ET,j *fJ)/ ET,ij
JADE:
Disadvantages:
Large recombination scheme dependence
kT:
Advantages:
No seed requirements
Same application to cells, hadrons, partons
No overlapping jets
Infrared safe to all orders
Good in ZEUS forward region
Relatively independent of MC recombination scheme
Longitudinally invariant kT:
Infrared and collinear safe to all orders in pQCD
Invariant under longitudinal boosts (along beam axis)
Small experimental and theoretical uncertainities

7. Event Shapes Axis Dependent: Axis Independent: Multi-Jet: Thrust
Broadening
Axis Independent:
C Parameter
Jet Mass
Multi-Jet:
yn_kt: disstance scale at which an event changes from n+1 to n jets using the KT algorithm
Momentum out of plane
Jet Rate

8. Event Plane
Energy flow out of event plane defined by proton direction and thrust major axis
Sensitive to perturbative & non-perturbative contributions
Dijet event:
Perturbative physics takes place in the plane
Non-perturbative physics give rise to out-of-plane momentum

9. Approach to Non-perturbative Calculations
pQCD prediction → measured distribution
Correction factors for non-perturbative (soft) QCD effects
Proposed theory* reduces corrections for any infrared safe event shape variable, F:
Power correction
Independent of any fragmentation assumptions
“Non-perturbative parameter” * – (Dokshitzer, Webber, phys. Lett. B 352(1995)451)
Used to determine the hadronization corrections
Two realms: pQCD and npQCD. pQCD is calculable: uses alpha-s. npQCD is not. Recent theoretical work [2], involving power corrections to event shape variables, provides a new approach to determine the hadronisation corrections compared to monte carlo event generators. The new work involves a phenomenological model based around an effective strong coupling which comes in at low Q. This model conveniently reduces the hadronisation corrections to be of a form containing alpha-s and a new phenomenological variable alpha-0.
Define power correction…..
Why do we care? –> because we can’t calculate features of npQCD hadronization
Future plans involve the fitting of parton level next to leading order curves with power corrections to the mean distributions. This will then yield a value for not only alpha-s but also the new phenomenological variable alpha-0 and allow tests on its universality.
Relate hadronization effects to power corrections. Q dependence of the means of the event shape parameters is presented, from which both the power corrections and the strong coupling constant are determined without any assumption on fragmentation models.

10. Shape Means 2-parameter fit
Simultaneous fit for s and 0
Each shape fit separately
Bins of x and q2
High-x data points are not fit.
Fits use Hessian method for statistical and systematic errors.
All variables with a good 2

11. Mean Parameters Extracted parameters for each shape
Fitted s values consistent to within 5%
Fitted 00.45 to within 10%
Theory errors dominate, except for  axis shapes.

12. Shape Distributions  = 1-T
Fit differential distributions over a limited range.
Bins for which theoretical calculations are expected to be questionable are omitted from fit.
Resummation is applied with DISRESUM.

13. Distribution Parameters
Fits use Hessian method for statistical and systematic errors.
All variables with a good 2
Fits are sensitive to matching method.
0  0.5

14. Jet Rate Event Shape No fits performed up to now.
NLOJET++ gives a good description of the data for higher values of Q.
Waiting on generalized resummation program.

15. 3-jet Event Shape Variable
No fits performed up to now.
First comparison with LO+NLL+PC is shown.
Waiting on generalized resummation program.

16. Jet Substructure
Jet substructure depends mainly on type of primary parton and to lesser extent on the particular hard scattering process
pQCD: gluon initiated jets are broader than quark initiated jets
Jet substructure studied with the jet shape:
(r): fraction of the jet ET inside a cone in the - plane of radius r
Jet dependence:
DIS: none
P: jets broader as jet increases
Etjet dependence:
DIS and p: jets become narrower as ETjet increases
At sufficiently high jet transverse energy, where fragmentation effects become negligible, the jet substructure is expected to be calcculable in pQCD
pQCD predicts that gluon-initiated jets are broader than quark-initiated jets due to the large color charge of the gluon (seen by LEP)
Jets searched with kt in LAB frame
At least one jet with Et>17 and –1 < eta < 2.5
Q^2 < 1 and 0.2 < y < 0.85
Eta dependence:
Comp to QCD
PYTHIA with ISR and FSR
Good desc of data –1..1.5
Data broader than prediction
Thus, quark like –1..0, increasingly gluon like as eta increases
Et dependence

17. Jet Substructure 2
Put something here

18. Multijet Production Jets ordered in ETBRT Jets ordered in LAB Et
Measurement down to low Et
Scale dependence
Good agreement
NLOJET over large range of scales
Eta
Angular distribution
NLOJET over whole eta range
Jets ordered in ETBRT
Jets ordered in LAB

19. Multijet Production 2 R3/2 = trijet/dijet
Dijet: alphas_s^2
Trijet: alpha_s^3
Scale dependence
R3/2 = trijet/dijet
Systematic uncertainties substatially reduced
Very sensitive test of QCD calculation

20. s Values

21. Summary
Mean and differential event shapes have been measured in eP collisions
Power correction fits performed for data for most event shapes
Waiting on resummation terms
Measurement of jet shape and subjet multiplicities demonstrate the validity of the description of the internal structure of jets by pQCD
R3/2 cross section ratio is sensitive to s
Some sensitivity to PDF is also observed