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October 9, 2009 Rajendran Raja, MIPP Review 1 The Main Injector Particle Production Experiment (MIPP) at Fermilab –Description of experiment Rajendran Raja Fermilab Review Status of MIPP – E907- Took data till 2006 --18 million events– Analysis status Physics case for MIPP- restart the study of the strong interaction., test scaling laws for inclusive reactions. Obtain nuclear cross sections with 6 beam species for purposes of proton radiography Measure the particle production spectrum with excellent particle id off the NuMI target We designed and built the beamline and the experiment was put together largely from existing hardware that had to be refurbished (to keep costs down). We had to learn to use the TPC, RICH, CKOV, BCKOV. We fabricated the ToF counter and redesigned the RICH front end electronics. All drift chambers were refurbished from existing pieces of apparatus. The hadron calorimeter was re-used. We built the EM calorimeter using a lead absorber.

The Main Injector Particle Production Experiment (MIPP) at … · cross sections of various nuclei including hydrogen. • Using a TPC we measure momenta of ~all charged particles

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Page 1: The Main Injector Particle Production Experiment (MIPP) at … · cross sections of various nuclei including hydrogen. • Using a TPC we measure momenta of ~all charged particles

October 9, 2009 Rajendran Raja, MIPP Review 1

The Main Injector Particle Production Experiment (MIPP) at Fermilab –Description of experiment

Rajendran RajaFermilab

• Review Status of MIPP – E907- Took data till 2006 --18 million events– Analysis status

• Physics case for MIPP- restart the study of the strong interaction., test scaling lawsfor inclusive reactions.

• Obtain nuclear cross sections with 6 beam species for purposes of proton radiography

• Measure the particle production spectrum with excellent particle id off the NuMItarget

• We designed and built the beamline and the experiment was put together largely from existing hardware that had to be refurbished (to keep costs down).

• We had to learn to use the TPC, RICH, CKOV, BCKOV. We fabricated the ToFcounter and redesigned the RICH front end electronics. All drift chambers were refurbished from existing pieces of apparatus. The hadron calorimeter was re-used. We built the EM calorimeter using a lead absorber.

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October 9, 2009 Rajendran Raja, MIPP Review 2

MIPP I–E907-collaboration listY. Fisyak

Brookhaven National LaboratoryR. Winston

EFI, University of ChicagoR.J.Peterson

University of Colorado, Boulder,E.Swallow

Elmhurst College and EFIW.Baker,D.Carey,J.Hylen, C.Johnstone,M.Kostin, H.Meyer, N.Mokhov, A.Para, R.Raja,S. Striganov

Fermi National Accelerator LaboratoryG. Feldman, A.Lebedev, S.Seun

Harvard UniversityP.Hanlet, O.Kamaev,D.Kaplan, H.Rubin,N.Solomey,Y.Torun

Illinois Institute of TechnologyU.Akgun,G.Aydin,F.Duru,E.Gülmez,Y.Gunaydin,Y.Onel, A.Penzo

University of IowaN.Graf, M. Messier,J.Paley

Indiana UniversityP.D.BarnesJr.,E.Hartouni,M.Heffner,J.Klay,D.Lange,R.Soltz, D.Wright

Lawrence Livermore LaboratoryR.L.Abrams,H.R.Gustafson,M.Longo, T.Nigmanov,H-K.Park, D.Rajaram

University of MichiganA.Bujak, L.Gutay,D.E.Miller

Purdue UniversityT.Bergfeld,A.Godley,S.R.Mishra,C.Rosenfeld,K.Wu

University of South CarolinaC.Dukes,L.C.Lu,C.Maternick,K.Nelson,A.Norman

University of Virginia

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October 9, 2009 Rajendran Raja, MIPP Review 3

Brief Description of Experiment• Approved November 2001• Situated in Meson Center 7• Uses 120GeV Main Injector Primary protons to

produce secondary beams of K p from 5 GeV/c to 85 GeV/c to measure particle production cross sections of various nuclei including hydrogen.

• Using a TPC we measure momenta of ~all charged particles produced in the interaction and identify the charged particles in the final state using a combination of dE/dx, ToF, MultiCell Cherenkov and RICH technologies.

• Open Geometry- Lower systematics. TPC gives high statistics. Existing data poor quality.

• First Physics run- 18 million events 2005. Ended Feb 2006– Analyzing data

• Physics Case for MIPP-I

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Goal of MIPP-Program • Measure Neutrino Flux <=> Meson Cross-Section (Xf, Pt)

» Primary ±/K±/K0 in proton - Target collision in open Geometry (Single-arm spectrometer measurements lose correlation)

» Measure meson production cross section where the beam is ±/K±/p± at different momenta (5<Pbeam<85) using a suite of nuclear targets, thin and thick, that compose neutrino beam elements (Target-holder, Horns, collimators, decay-pipe, Air/He) i.e. provide a matrix of measurements that will be directly used in the Meson-Transport -Flux in NuMI(MINOS/NOvA), DUSEL, Atmospheric, Neutrino-factory,

• Empirically gird the hadron-cascade simulation: MIPP uses =>targets spanning periodic table (H-->U); =>thin (1% Lambda) to thick (2.5% Lambda); =>Beam (p/Pi+-/K+-) from 5<Pbeam<120 GeV

• Study of the Strong Interaction

Page 5: The Main Injector Particle Production Experiment (MIPP) at … · cross sections of various nuclei including hydrogen. • Using a TPC we measure momenta of ~all charged particles

October 9, 2009 Rajendran Raja, MIPP Review 5

We have a theory of the strong interaction—in theory

• Why study non-perturbative QCD? Answer:- We do not know how to calculate a single cross section in non-perturbative QCD! This is >99% of the total QCD cross section. Perturbative QCD has made impressive progress. But it relies on structure functions for its calculations, which are non-perturbative and derived from data.

• All “established” scaling laws (Feynman scaling, KNO scaling, rapidity plateaus) are violated. Regge theory is a framework. Predictions violatable by adding more trajectories.

• Most existing data are of low statistics with poor particle id and high systematics(single arm spectrometers)

• Most models tuned on a variety of old data on single particle inclusives. They manage to get the longitudinal profiles of hadronic showers correct. Transverse profiles requires understanding and modelling of particle correlations (i.e full non-perturbativeQCD dynamics). Data not available. One is at the mercy of a large number of models that are not consistent with each other. A lot of manpower is invested in “tuning and validating” models, that are inadequate to answer the detailed questions precision experiments are asking today.

• Technology has improved since the bubble chamber and today we are in a position to acquire high quality production data that can completely revolutionize our ability to predict hadronic showers. This will greatly increase our understanding of systematicsin a series of experiments, fixed target neutrino experiments being a significant beneficiary.

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October 9, 2009 Rajendran Raja, MIPP Review 6

General scaling law of particle fragmentation

• States that the ratio of a semi-inclusive cross section to an inclusive cross section

• where M2,s and t are the Mandelstam variables for the missing mass squared, CMS energy squared and the momentum transfer squared between the particles a and c. PRD18(1978)204.

• Using EHS data, we have tested and verified the law in 12 reactions (DPF92) but only at fixed s.

• MIPP will in principle test this in 36 reactions. MIPP upgrade can extend these scaling relation tests to two particle inclusive reactions which requires more statistics.

f a b c Xf a b c X

f M s tf M s t

Msubset subsetsubset

( )( )

( , , )( , , )

( )

2

22

Page 7: The Main Injector Particle Production Experiment (MIPP) at … · cross sections of various nuclei including hydrogen. • Using a TPC we measure momenta of ~all charged particles

October 9, 2009 Rajendran Raja, MIPP Review 7

Scaling Law-EHS results

Page 8: The Main Injector Particle Production Experiment (MIPP) at … · cross sections of various nuclei including hydrogen. • Using a TPC we measure momenta of ~all charged particles

October 9, 2009 Rajendran Raja, MIPP Review 8

NuMI target hadroproduction-Sparsity of data

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October 9, 2009 Rajendran Raja, MIPP Review 9

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October 9, 2009 Rajendran Raja, MIPP Review 10

Page 11: The Main Injector Particle Production Experiment (MIPP) at … · cross sections of various nuclei including hydrogen. • Using a TPC we measure momenta of ~all charged particles

October 9, 2009 Rajendran Raja, MIPP Review 11

MIPP Secondary BeamInstalled in 2003. Excellent performance. Ran it successfully in MIPP from 5-

85 GeV/c secondaries and 120 GeV/c primary protons. Excellent particle ID capabilities using 2 Beam Cerenkovs. For low momenta (<~10 GeV/c) ToF is used for pid. Design principles and lessons learned used in M-test upgrade.

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October 9, 2009 Rajendran Raja, MIPP Review 12

Beam line details• Design borrowed from muon collider final focus.• 95 meters from primary to secondary target. Low momentum

Pions survive.• We went thru 7 designs before settling on this.• Dispersion at collimator.• Beam has to be parallel at the experimental target. • Final design chosen due to Carol Johnstone.• Beam particle id using two differential beam Ckovs.• Gave this design to M-test to avoid test beam community

from wanting to use MIPP.• They have a slightly longer version of this design that bends

horizontally. • With upgraded power supplies that regulate at low current

and Hall probes , it is possible to increase momentum range from ~1 GeV/c - 85 GeV/c for secondaries of both charges. Charged kaons from ~3GeV/c (decay).

• Removing Cu primary target and adding pinhole collimator, we brought in low intensity beam of 120 geV protons for NuMI.

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October 9, 2009 Rajendran Raja, MIPP Review 13

Jolly Green Giant

TPC

Cerenkov

Time of Flight

Rosie

RICH

Chambers

Neutron Calorimeter

MIPPMain Injector Particle Production Experiment (FNAL-E907)

Jolly Green Giant

TPC

Cerenkov

Time of Flight

Rosie

RICH

Chambers

Neutron Calorimeter

EM shower detector

MIPPMain Injector Particle Production Experiment (FNAL-E907)

Vertical cut plane

Page 14: The Main Injector Particle Production Experiment (MIPP) at … · cross sections of various nuclei including hydrogen. • Using a TPC we measure momenta of ~all charged particles

October 9, 2009 Rajendran Raja, MIPP Review 14

MC7 during Installation –Had to install struts supporting magnets in M-Bottom; Clean out Hyper Cp; Redo the building since it was not big enough. No crane coverage-Access doors carefully planned. Good insulation and AC.

Roof is non-canonical; does not leak.

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October 9, 2009 Rajendran Raja, MIPP Review 15

TPC

Page 16: The Main Injector Particle Production Experiment (MIPP) at … · cross sections of various nuclei including hydrogen. • Using a TPC we measure momenta of ~all charged particles

October 9, 2009 Rajendran Raja, MIPP Review 16

Beam Cherenkovs

)3

Density (mlb/ft0 10 20 30 40 50 60 70

Fractio

n of tr

iggers

-310

-210

-110

out×Upstrm Beam Cherenkov in out×Upstrm Beam Cherenkov in

)3

Density (mlb/ft0 10 20 30 40 50 60 70

Fractio

n of tr

iggers

-310

-210

-110

out×Dwnstrm Beam Cherenkov in out×Dwnstrm Beam Cherenkov in

)3

Density (mlb/ft0 10 20 30 40 50 60

Frac

tion o

f trigg

ers

-410

-310

-210

-110

1

out×Upper Beam Cherenkov in

)3

Density (mlb/ft0 10 20 30 40 50 60

Fra

ctio

n o

f tr

igg

ers

-410

-310

-210

-110

1

out×Lower Beam Cherenkov in

+40 GeV/c

-40 GeV/c

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October 9, 2009 Rajendran Raja, MIPP Review 17

RICH rings pattern recognized

Page 18: The Main Injector Particle Production Experiment (MIPP) at … · cross sections of various nuclei including hydrogen. • Using a TPC we measure momenta of ~all charged particles

October 9, 2009 Rajendran Raja, MIPP Review 18

Comparing Beam Cherenkov to RICH for +40 GeV beam triggers-No additional cuts!

Reconstructed ring radius (cm)18 20 22 24 26 28 30

richProtonEntries 987

Mean 21.91

RMS 0.5726

Reconstructed ring radius (cm)18 20 22 24 26 28 30

0

50

100

150

200

250

300

350

400

450richProton

Entries 987

Mean 21.91

RMS 0.5726

Distribution of RICH Ring Radii with Proton Trigger

Pro

tons

, 99.

8%P

roto

ns, 9

9.8%

Pro

tons

, 99.

8%

Reconstructed ring radius (cm)20 22 24 26 28 30

richKaonEntries 1084

Mean 27.19

RMS 1.134

Reconstructed ring radius (cm)20 22 24 26 28 30

0

50

100

150

200

250

300

350

400

450richKaon

Entries 1084

Mean 27.19

RMS 1.134

Distribution of RICH Ring Radii with Kaon TriggerP

roto

ns,

3.7

%P

roto

ns,

3.7

%

Ka

on

s, 9

3.6

%

Pio

ns,

2.2

%P

ion

s, 2

.2%

Reconstructed ring radius (cm)20 22 24 26 28 30

richPionEntries 1214

Mean 29.11

RMS 0.341

Reconstructed ring radius (cm)20 22 24 26 28 30

0

100

200

300

400

500

600

700

800

richPionEntries 1214

Mean 29.11

RMS 0.341

Distribution of RICH Ring Radii with Pion Trigger

Pio

ns,

99.8

%P

ions,

99.8

%

Reconstructed ring radius (cm)20 22 24 26 28 30 32

Reconstructed ring radius (cm)20 22 24 26 28 30 32

0

200

400

600

800

1000

1200

Distribution of RICH Ring Radii in Beam

Pro

tons

, 54.

3%

Kao

ns, 3

.7%

Pio

ns, 4

1.4%

Pro

tons

, 54.

3%P

roto

ns, 5

4.3%

Kao

ns, 3

.7%

Pio

ns, 4

1.4%

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October 9, 2009 Rajendran Raja, MIPP Review 19

MIPP (FNAL E907) Target: NuMIRun: 15007SubRun: 0Event: 160 Sat Jul 16 200511:22:30.687398 *** Trigger ***BeamWord: 0080Bits: 80D7

z (cm)

-80 -60 -40 -20 0 20 40 60 80

x (cm)

-40-20

020

40

y (

’cm

’)

-40

-20

0

20

40

z (cm)-80 -60 -40 -20 0 20 40 60 80

x (

cm)

-40

-20

0

20

40

x (cm)-40 -20 0 20 40

y (

’cm

’)

-40

-20

0

20

40

z (cm)-80 -60 -40 -20 0 20 40 60 80

y (

’cm

’)

-40

-20

0

20

40

MIPP (FNAL E907) Target: NuMIRun: 15007SubRun: 0Event: 160 Sat Jul 16 200511:22:30.687398 *** Trigger ***BeamWord: 0080Bits: 80D7

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MIPP (FNAL E907) Target: NuMIRun: 15007SubRun: 0Event: 160 Sat Jul 16 200511:22:30.687398 *** Trigger ***BeamWord: 0080Bits: 80D7

MIPP (FNAL E907) Target: NuMIRun: 15007SubRun: 0Event: 160 Sat Jul 16 200511:22:30.687398 *** Trigger ***BeamWord: 0080Bits: 80D7

MIPP (FNAL E907) Target: NuMIRun: 15007SubRun: 0Event: 160 Sat Jul 16 200511:22:30.687398 *** Trigger ***BeamWord: 0080Bits: 80D7

MIPP (FNAL E907) Target: NuMIRun: 15007SubRun: 0Event: 160 Sat Jul 16 200511:22:30.687398 *** Trigger ***BeamWord: 0080Bits: 80D7

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MIPP (FNAL E907) Target: NuMIRun: 15007SubRun: 0Event: 160 Sat Jul 16 200511:22:30.687398 *** Trigger ***BeamWord: 0080Bits: 80D7

MIPP (FNAL E907) Target: NuMIRun: 15007SubRun: 0Event: 160 Sat Jul 16 200511:22:30.687398 *** Trigger ***BeamWord: 0080Bits: 80D7

MIPP (FNAL E907) Target: NuMIRun: 15007SubRun: 0Event: 160 Sat Jul 16 200511:22:30.687398 *** Trigger ***BeamWord: 0080Bits: 80D7

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MIPP (FNAL E907) Target: NuMIRun: 15007SubRun: 0Event: 160 Sat Jul 16 200511:22:30.687398 *** Trigger ***BeamWord: 0080Bits: 80D7

MIPP (FNAL E907) Target: NuMIRun: 15007SubRun: 0Event: 160 Sat Jul 16 200511:22:30.687398 *** Trigger ***BeamWord: 0080Bits: 80D7

MIPP (FNAL E907) Target: NuMIRun: 15007SubRun: 0Event: 160 Sat Jul 16 200511:22:30.687398 *** Trigger ***BeamWord: 0080Bits: 80D7

MIPP (FNAL E907) Target: NuMIRun: 15007SubRun: 0Event: 160 Sat Jul 16 200511:22:30.687398 *** Trigger ***BeamWord: 0080Bits: 80D7

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October 9, 2009 Rajendran Raja, MIPP Review 20

TPC Reconstructed tracks MIPP (FNAL E907) Target: BerylliumRun: 12719SubRun: 0Event: 9 Mon Feb 28 200503:18:40.377278 *** Trigger ***BeamWord: 0400Bits: C44F

z (cm)

-80 -60 -40 -20 0 20 40 60 80

x (cm)

-40-20

020

40

y (

’cm

’)

-40

-20

0

20

40

z (cm)

-80 -60 -40 -20 0 20 40 60 80

x (cm)

-40-20

020

40

y (

’cm

’)

-40

-20

0

20

40

z (cm)-80 -60 -40 -20 0 20 40 60 80

x (

cm)

-40

-20

0

20

40

track 7track 3track 10track 8track 11

track 6

track 2

track 1

track 21

track 22

X

x (cm)-40 -20 0 20 40

y (

’cm

’)

-40

-20

0

20

40

z (cm)-80 -60 -40 -20 0 20 40 60 80

y (

’cm

’)

-40

-20

0

20

40

track 7track 3

track 10track 8

track 11

track 6track 2

track 1

track 21

track 22

X

MIPP (FNAL E907) Target: BerylliumRun: 12719SubRun: 0Event: 9 Mon Feb 28 200503:18:40.377278 *** Trigger ***BeamWord: 0400Bits: C44F

MIPP (FNAL E907) Target: BerylliumRun: 12719SubRun: 0Event: 9 Mon Feb 28 200503:18:40.377278 *** Trigger ***BeamWord: 0400Bits: C44F

MIPP (FNAL E907) Target: BerylliumRun: 12719SubRun: 0Event: 9 Mon Feb 28 200503:18:40.377278 *** Trigger ***BeamWord: 0400Bits: C44F

MIPP (FNAL E907) Target: BerylliumRun: 12719SubRun: 0Event: 9 Mon Feb 28 200503:18:40.377278 *** Trigger ***BeamWord: 0400Bits: C44F

MIPP (FNAL E907) Target: BerylliumRun: 12719SubRun: 0Event: 9 Mon Feb 28 200503:18:40.377278 *** Trigger ***BeamWord: 0400Bits: C44F

MIPP (FNAL E907) Target: BerylliumRun: 12719SubRun: 0Event: 9 Mon Feb 28 200503:18:40.377278 *** Trigger ***BeamWord: 0400Bits: C44F

MIPP (FNAL E907) Target: BerylliumRun: 12719SubRun: 0Event: 9 Mon Feb 28 200503:18:40.377278 *** Trigger ***BeamWord: 0400Bits: C44F

MIPP (FNAL E907) Target: BerylliumRun: 12719SubRun: 0Event: 9 Mon Feb 28 200503:18:40.377278 *** Trigger ***BeamWord: 0400Bits: C44F

MIPP (FNAL E907) Target: BerylliumRun: 12719SubRun: 0Event: 9 Mon Feb 28 200503:18:40.377278 *** Trigger ***BeamWord: 0400Bits: C44F

MIPP (FNAL E907) Target: BerylliumRun: 12719SubRun: 0Event: 9 Mon Feb 28 200503:18:40.377278 *** Trigger ***BeamWord: 0400Bits: C44F

MIPP (FNAL E907) Target: BerylliumRun: 12719SubRun: 0Event: 9 Mon Feb 28 200503:18:40.377278 *** Trigger ***BeamWord: 0400Bits: C44F

MIPP (FNAL E907) Target: BerylliumRun: 12719SubRun: 0Event: 9 Mon Feb 28 200503:18:40.377278 *** Trigger ***BeamWord: 0400Bits: C44F

MIPP (FNAL E907) Target: BerylliumRun: 12719SubRun: 0Event: 9 Mon Feb 28 200503:18:40.377278 *** Trigger ***BeamWord: 0400Bits: C44F

MIPP (FNAL E907) Target: BerylliumRun: 12719SubRun: 0Event: 9 Mon Feb 28 200503:18:40.377278 *** Trigger ***BeamWord: 0400Bits: C44F

MIPP (FNAL E907) Target: BerylliumRun: 12719SubRun: 0Event: 9 Mon Feb 28 200503:18:40.377278 *** Trigger ***BeamWord: 0400Bits: C44F

MIPP (FNAL E907) Target: BerylliumRun: 12719SubRun: 0Event: 9 Mon Feb 28 200503:18:40.377278 *** Trigger ***BeamWord: 0400Bits: C44F

MIPP (FNAL E907) Target: BerylliumRun: 12719SubRun: 0Event: 9 Mon Feb 28 200503:18:40.377278 *** Trigger ***BeamWord: 0400Bits: C44F

MIPP (FNAL E907) Target: BerylliumRun: 12719SubRun: 0Event: 9 Mon Feb 28 200503:18:40.377278 *** Trigger ***BeamWord: 0400Bits: C44F

MIPP (FNAL E907) Target: BerylliumRun: 12719SubRun: 0Event: 9 Mon Feb 28 200503:18:40.377278 *** Trigger ***BeamWord: 0400Bits: C44F

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October 9, 2009 Rajendran Raja, MIPP Review 21

Data Taken In current runData Summary

27 February 2006 Acquired Data by Target and Beam Energy

Number of events, x 106 Target E

Z Element Trigger Mix 5 20 35 40 55 60 65 85 120

Total

Empty1 Normal 0.10 0.14 0.52 0.25 1.01K Mass2 No Int. 5.48 0.50 7.39 0.96 14.330

Empty LH1 Normal 0.30 0.61 0.311 LH Normal 0.21 1.94 1.98 1.73

7.08

p only 1.084 Be

Normal 0.10 0.561.75

C Mixed 0.21C 2% Mixed 0.39 0.26 0.47

1.336

NuMI p only 1.78 1.7813 Al Normal 0.10 0.10

p only 1.0583 Bi

Normal 0.52 1.262.83

92 U Normal 1.18 1.18Total 0.21 2.73 0.86 5.48 0.50 13.97 0.96 2.04 4.63 31.38

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October 9, 2009 Rajendran Raja, MIPP Review 22

ReconstructedProton-Carbonat 120 GeV/cEvent

TPC X

TPC Y Global Y

Global X

RICH

Wire Chambers EM Cal

Hadron CalTO

F W

all

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October 9, 2009 Rajendran Raja, MIPP Review 23

Acceptances and resolutions• Full MC Geant3 based. Use known tracks and

match them to found tracks. MC display

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MC Event Display

October 9, 2009 Rajendran Raja, MIPP Review 24

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October 9, 2009 Rajendran Raja, MIPP Review 25

Acceptances

Page 26: The Main Injector Particle Production Experiment (MIPP) at … · cross sections of various nuclei including hydrogen. • Using a TPC we measure momenta of ~all charged particles

October 9, 2009 Rajendran Raja, MIPP Review 26

Feynman x acceptance

Page 27: The Main Injector Particle Production Experiment (MIPP) at … · cross sections of various nuclei including hydrogen. • Using a TPC we measure momenta of ~all charged particles

October 9, 2009 Rajendran Raja, MIPP Review 27

MIPP Momentum resolution

Track Momentum GeV/c

0 10 20 30 40 50 60 70 80 90 100

dp

/p

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

HARP

MIPP

HARP MIPP Resolution Comparison

Page 28: The Main Injector Particle Production Experiment (MIPP) at … · cross sections of various nuclei including hydrogen. • Using a TPC we measure momenta of ~all charged particles

October 9, 2009 Rajendran Raja, MIPP Review 28

Analysis Status• We have published NIM articles.• General NIM article on experiment in

preparation.• NIM article on Calorimeter published• NIM article on Charged Kaon mass

measurement using RICH rings (a new technique) submitted.

• NUMI target hadroproduction analysis very close to completion.

• Calorimeter neutron production cross section analyses 90% complete.

Page 29: The Main Injector Particle Production Experiment (MIPP) at … · cross sections of various nuclei including hydrogen. • Using a TPC we measure momenta of ~all charged particles

October 9, 2009 Rajendran Raja, MIPP Review 29

Scaling Law

Continuing on to physical t values, one gets

Essentially, it states that semi-inclusive cross sections are not all independent but are connected by these relations.

)(),,()(

)(),,()(22

22

MDtsMFXabc

MDtsMFXabc

sXs

X

)()(),,()(),,(

)()( 2

22

22

MMDtsMFMDtsMF

XabcXabc

subX

Xsub sub

)()()( 2M

XcabfXcabf

subsub