# Presentation: 325718

## Combining CDF and D0 Physics Results

- Herbert Greenlee
*Fermilab*

- Fermilab Workshop on Confidence Limits
- March 28, 2000

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## Outline

*W* mass (common systematic error method).
- Top quark mass (correlated error method).
- Leptoquark production limit (counting experiments).

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## Combined W mass (Common Systematic Error Method)

- Assumes that all measurements have the same common error.
- First mass: *m*1  **
- Second mass: *m*2  **
- Common error: ***C*
- Average and error:

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## Combined W mass (cont.)

- Latest Tevatron *W* mass results:
- CDF: *m**W* = 80.433 GeV
- D0:    *m**W* = 80.482 GeV
- Common Error: 0.025 GeV
- Tevatron combined W mass:
*m**W* = 80.454 GeV (common error method)
*m**W* = 80.454 GeV (ignoring correlations)

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## Correlated Error Method

- Assume that measurements are correlated.
- First mass: *m*1  **
- Second mass: *m*2  **
- Correlation Coefficient: ***m****m*******
- Minimize 2:

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## Connection with Common Error Method

- Define “correlated error” ***C**2* = ***1****2*
- Then previous equations reduce to:

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## Calculating Correlation

- Calculate correlation by summing over uncorrelated error categories.
- Usually one assumes ***i** *= 0 or 1.

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## Combining More Than Two Results

- Correlation coefficient ** correlation matrix ***ij*.
- Minimize 2:
- where

- (matrix inverse)

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## Separating Combined Error into Statistical and Systematic Components

- Define the statistical error by propagating errors on the equation for *m*ave.

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## Statistical and Systematic Error (cont.)

- This definition of statistical error is in general different than what one would get by averaging based on statistical errors alone.
- This definition remains sensible as ** 
- Combined statistical error remains less than the combined total error.
- Combined statistical error dominates combined systematic error as ** both approach zero, but systematic approaches zero faster).
- In general, it is not possible to consistently separate the systematic error into components.

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## High Correlation Limit

- If ** > **1/**2 or  > **2/**1 we are in the high correlation region.
- One result enters with negative weight into the average.
- The average mass is not between the two measurements.
- For given **1 and **2, error is maximum at the boundary between high and low correlation regions.
- Weaker experiment has zero weight.
- Central value and error are the same as the stronger experiment.
- In the high correlation region, increasing correlated components of the systematic error of the weaker experiment actually makes the total error smaller.
- If ** = 1, the total error is zero.

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## Slide 12

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## Averaging CDF and D0 Top Quark Mass Results

- Five top quark mass results from both experiments.
- CDF lepton + jets.
- CDF dilepton.
- CDF all jets.
- D0 lepton + jets.
- D0 dilepton.
- Average all five results together simultaneously using correlated error method.
- Reference Fermilab-TM-2084

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## Summary of Top Quark Mass Results and Errors

- Results and systematic errors (100% correlation within each shaded color):
- Correlation matrix and weights:

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## Top Quark Correlation and Weights (cont.)

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## Top Quark Mass Combined Results

- Combined top mass
- With correlation
*m**t* = 174.3 (stat.)  (syst.) GeV/c2
-      = 174.3 GeV/c2
***2*/DOF = 1.92/4
- Without correlation
*m**t* = 174.4 4.2 GeV/ c2

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## Combined First Generation Leptoquark Limit

- Counting experiments:
- Result (cross section or cross section limit) depends on four numbers:
- Number of observed events *n*
- Acceptance **
- Background *b*
- Luminosity L
- Sources of error:
- Counting error (square root of *n*).
- Systematic errors (****, ***b*, **L).
- Correlations (*****b*, ***b*L).

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## Combining Counting Experiments

- Combine events, acceptance and background.
- Propagate errors in sums using standard correlated propagation errors formula.
- Calculate cross section (limit) using combined events, acceptance, and background.
- Detailed procedure described in                    hep-ex/9810015

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## Combined First Generation Leptoquark Limit

- Results:
- CDF
- 0 candidate events.
* ***  < 0.0993 pb.
*M**LQ *> 213.5 GeV/*c*2.
- D0
- 0 candidate events.
* ***  < 0.0663 pb.
*M**LQ *> 225.8 GeV/*c*2.
- Combined
- 0 candidate events.
-  **  < 0.0383 pb.
*M**LQ *> 242.4 GeV/*c*2.

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## Summary

- Three results have been combined between CDF and D0.
- W mass.
- Top quark mass
- First generation leptoquark limit.
- Practical considerations.
- Agreeing on categories of systematic errors.
- Handling high correlation regions.
- Don’t overestimate systematic errors.
- Methods
- Common systematic error method is special case of correlated error method for averaging type of combinations.
- Combined counting method is simple to apply, but is not necessarily optimal.

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