Introduction Radiative habhas can be used as one of the calibrations of the aar electromagnetic calorimeter èemè. Radiative habha events èe, e +! e, e

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1 SLíPUí865 UMSèHEPèè9 October 2 Kinematic Fit for the Radiative habha alibration of aar's Electromagnetic alorimeter æ Johannes M. auer Department of Physics and stronomy, University of Mississippi, University, MS 38677, U.S.. Stanford Linear ccelerator enter, Stanford University, Stanford, 9439, U.S.. bstract For the radiative habha calibration of aar's electromagnetic calorimeter, the measured energy of a photon cluster is being compared with the energy obtained via a kinematic æt involving other quantities from that event. The details of the ætting algorithm are described in this note, together with its derivation and checks that ensure that the ætting routine is working properly. æ Work supported by Department ofenergy contract DEí3í76SF55 and Department ofenergy grant DE-FG2-9ER4622.

2 Introduction Radiative habhas can be used as one of the calibrations of the aar electromagnetic calorimeter èemè. Radiative habha events èe, e +! e, e + æè deposit photons over a large energy range everywhere in the calorimeter. If the momenta of the incoming and outgoing electrons and positrons, as well as the photon's angular position are known, the photon energy can be obtained via a kinematic æt. This æt results in an absolute measurement of the photon energy which then can be compared to the measured photon energy to obtain calibration constants. The radiative habha module is part of aar's Online Prompt Reconstruction èoprè executable. Initial cuts select good electrons, positrons, and photons. Then all possible combinations of triplets èone electron, one positron, one photonè are formed. Each triplet is sent to the ætting routine to calculate its ç 2 est, the ëestimated ç 2 ". The triplet with the lowest ç 2 est is then submitted to the full kinematic æt which returns, among other quantities, the ætted photon energy E fæ and the error matrix of the ætted quantities. The ratio E meas =E fæ is later used to calibrate the calorimeter. Note that no information on the measured photon energy E meas goes into the kinematic æt or ç 2 est. This note is the complete documentation on the algorithm for ætting the radiative habha events for the purpose of calibrating the calorimeter. It describes the whole ætting procedure: the quantities for the kinematic æt and ç 2 est; the derivation and formulas for ç 2 est; the derivation and algorithm for the kinematic æt; tests to check the quality of the kinematic æt. The note details all formulas which go into the computer program so that the program can be checked directly against this document. The derivations contain more details than needed to understand the concept, but the details help to derive, check and recheck all necessary formulas. ctual results of the ætting procedure using real data are not included in this note to keep it a pure code documentation. 2 Deæning the quantities and constraints 2. Measured quantities From the experiment come the following measurements, which shall form the 4-dimensional vector y: P ix, ç y P iy, ç y 2 P iz, ç y 3 msrd momentum in x, y, and z of incoming e, P ix+ ç y 4 P iy+ ç y 5 P iz+ ç y 6 msrd momentum in x, y, and z of incoming e + P ox, ç y 7 P oy, ç y 8 P oz, ç y 9 msrd momentum in x, y, and z of outgoing e, P ox+ ç y P oy+ ç y P oz+ ç y 2 msrd momentum in x, y, and z of outgoing e + ç oæ ç y 3 ç oæ ç y 4 measured ç and ç of the photon The momenta of the incoming electron and positron and their errors are changing run-by-run. The errors of the incoming leptons are given as covariance matrices: V i, = V ixx, V ixy, V ixz, V ixy, V iyy, V iyz, V ixz, V iyz, V izz, V i+ = The errors on P i, and P i+ are assumed to be independent. V ixx+ V ixy+ V ixz+ V ixy+ V iyy+ V iyz+ V ixz+ V iyz+ V izz+ 2

3 The errors of P o, =èp ox, ;P oy, ;P oz, è and P o+ =èp ox+ ;P oy+ ;P oz+ è are also assumed to be independent from each other. They are given in two 3æ3 error matrices: V o, = V oxx, V oxy, V oxz, V oxy, V oyy, V oyz, V oxz, V oyz, V ozz, V o+ = V oxx+ V oxy+ V oxz+ V oxy+ V oyy+ V oyz+ V oxz+ V oyz+ V ozz+ The errors on ç oæ and ç oæ appear in the current analysis without ç-ç-correlations since they were found to be negligibly small, but we still use this 2 æ 2 sub-set of the larger 4 æ 4 error matrix of the Emcluster: è! V V oæ = oççæ V oççæ V oççæ V oççæ ll the errors can be combined in one 4 æ 4 error matrix V all. Its format is like this: V all = V i, V i+ V o, V o+ V oæ = æ æ æ æ 3

4 2.2 Quantities for the kinematic æt The kinematic æt determines the following numbers: f ix, ç f f iy, ç f 2 f iz, ç f 3 x, y, and z momentum of incoming e, f ix+ ç f 4 f iy+ ç f 5 f iz+ ç f 6 x, y, and z momentum of incoming e + f ox, ç f 7 f oy, ç f 8 f oz, ç f 9 x, y, and z momentum of outgoing e, f ox+ ç f f oy+ ç f f oz+ ç f 2 x, y, and z momentum of outgoing e + ç fæ ç f 3 ç fæ ç f 4 E fæ ç h ç and ç, and energy of the photon ç ç 2 ç 3 four Lagrange multipliers for momentum and energy conservation constraints ç 4 The variables f to f 4 have corresponding measurements. The variable h, the photon energy, is called a ëhidden variable". The vector æ shall be deæned as a 9-element composite of f è4 elementsè, h è elementè, and ç è4 elementsè. 2.3 onstraints We have four constraint equations that have to be satisæed in the kinematic æt: p ix, + p ix+, p x,, p x+, E fæ sin ç fæ cos ç fæ = momentum in x p iy, + p iy+, p y,, p y+, E fæ sin ç fæ sin ç fæ = momentum in y p iz, + p iz+, p z,, p z+, E fæ cos ç fæ = momentum i +, E,, E +, E fæ = energy Here we use, e.g., ç = q p 2 + ix, p2 + iy, p2 + iz, m2 e q f 2 + f f m2 e 3 The estimated ç 2 : ç 2 est This function is calculated for any given electron-positron-gamma triplet to determine which triplet should be used for the kinematic æt. t the end of this subsection, we will have a complete analytical formula for calculating ç 2 est. The formula is based on the diæerence between the initial and ænal momentum, P ç P i, P o. The initial momentum P i is the sum of the momenta of the incoming electron and positron as deæned earlier: P i, and P i+. The measured momenta of the outgoing electron, positron are given by P, and P +. For the outgoing photon, we only have its angles ç æ and ç æ. Using the energy constraint E æ = +, E,, E + we may substitute the unknown photon energy E æ with measured values, and we obtain: P æ ç è +, E,, E + è sin ç æ cos ç æ sin ç æ sin ç æ cos ç æ Of course, n is the normal vector, the direction of the photon. 4 ç E æ n x n y ç E æ n

5 alculating the diæerence to form vector P is easy: P ç P x P y P z = P i, + P i+, P o,, P o+, P oæ In the ideal world, this vector would be exactly zero. For its error matrix V p,we convert V all, the error matrix of y, via a transformation matrix T into V p : V p = T t V all T For the transformation matrix T we have to calculate expressions like j = x; y; z: èe æ n j è = P ix, n j P ix, P i, The transformation matrix is a 3 æ 4 matrix: T = P ix, P iy, P iz, P ix+ P iy+ P iz+,,, ç æ ç æ P ix, P iy, P iz, P ix+ P iy+ P iz+,,, ç æ ç æ P ix, P iy, P iz, P ix+ P iy+ P iz+,,, ç æ ç æ =, P ix, n x, P iy, n x, P iz, n x, P ix+ n x, P iy+ n x, P iz+ n x Px. We note that for Pix,, P ix, n y, P iy, n y, P iz, n y, P ix+ n y, P iy+ n y, P iz+ n y, P ix,, P iy,, P iz,, P ix+, P iy+, P iz+,+ P x, P x, P x, n x n y E, E, E, P y, n x,+ P y, P y, n y E, E, E, P z, P z, n x n y,+ P z, E, E, E,,+ P x+ P x+ P x+ n x n y E + E + E + P y+ n x,+ P y+ P y+ n y E + E + E + P z+ P z+ n x n y,+ P z+ E + E + E +,E æ cos ç æ cos ç æ,e æ cos ç æ sin ç æ E æ sin ç æ E æ sin ç æ sin ç æ,e æ sin ç æ cos ç æ Now wehavev, p =T t V, T, and hence we may calculate ç 2 est: all ç 2 est = P t V, p P 5

6 What is the meaning of this ç 2? We can say that the 4 input variables are used to measure P, and ç 2 est tells us the deviation of the measured P from the expected P, which is zero. 4 The kinematic æt For the derivation of the kinematic æt algorithm, we follow the description of Louis Lyons, page 5, 52 ëë. 4. The ç 2 -Function The real ç 2 -function can be written down in the following way: ç 2 = èf, mè t V, all èf, mè + ç ëp xi, + p xi+, p xo,, p xo+, E æ sin ç æ cos ç æ ë + ç 2 ëp yi, + p yi+, p yo,, p yo+, E æ sin ç æ sin ç æ ë + ç 3 ëp zi, + p zi+, p zo,, p zo+, E æ cos ç æ ë + ç 4 ë +, E o,, E o+, E æ ë The constraint equations are here included via Lagrange multipliers. To minimize this ç 2, we could use a standard package like MINUIT, but standard packages are always slower than specially adapted code. Since the ç 2 -minimization is being done millions of times, it pays oæ to write special code for the minimization. In addition, MINUIT is not supported in aar's Online Prompt Reconstruction. 4.2 Derivation of kinematic æt algorithm t the minimum of ç 2, its ærst derivatives are to be zero. equations: Lyons uses for this the following ç 2 æ i = for i = to 4 ç 2 h ç 2 The three equation sets can be written as: y = here h = E æ = æ 5 ç k = here ç = æ 6 etc. 2 G èf, mè+d t ç = E t ç = = y The factor 2 in front ofv, is missing in Lyons' book ëë. We could easily remove this factor from our formulas all by re-deæning the Lagrange multipliers in the ç 2 -function with a factor 2. This would not change the æt result or errors, as long as the subsequent calculations were carried out consistently. 6

7 where G is the 4 æ 4 inverse error matrix of the measurements which we also call V, D = =æ ::: =æ 4 2 =æ ::: 2 =æ 4 3 =æ ::: 3 =æ 4 4 =æ ::: 4 =æ 4 and E = =æ 5 2 =æ 5 3 =æ 5 4 =æ 5 We now expand the constraint equations around f and h, and we obtain for the four equations k with k = to 4: k ç èè k + We may rewrite this into: 4X i= èè f i 4X i= èè f i èf i, m i è+ èè h all. èf i, f èè i è+ èè h èh, hèè è= èè èh, hèè è=, èè k + Now we collect everything, use the deænitions for M, Y, and Z, with and we see: M = 2G Dt E t D E Y = R = èè, Dèf, mè = f, m h, h ç 4X i= èè f i Z = èf èè i,r èf ;h è 2 èf ;h è 3 èf ;h è,dèf,mè; 4 èf ;h è MY =Z, m èè i è This is the equation we have to solve. Since the constraint equations = contain non-linear functions like sin ç æ, Eq. èè is only an approximation, and we have to iterate as described in the next section. 4.3 Recipe for the kinematic æt algorithm The matrix M and the vector Z are functions of the measurements and their error matrices as well as of the parameters æ. The vector Y is, as mentioned above, and can be calculated with: Y = f, m h, h ç Y = M, Z Here is the iteration: Initially, we will use for the æt quantities f = m, i.e. the measured quantities. For h = h, we calculate the photon energy via simple energy conservation. These 7

8 together with the measured quantities allow us to calculate M and Z. We multiply the inverse of M with Z and obtain Y. This result will then give us a better set of f and h, which we again use to calculate M and Z, and then a better Y. nd we continue until our constraint equations are suæciently fulælled and the quantities f and h are stable. It might happen that the iteration does not converge at the minimum, but wanders oæ into unphysical numbers. In that case, it would be good to have a certain boundary box around the point. If the step would make the point lie outside the box, then the program would change the step so that the point would be back inside. It might be good to implement this, although the radiative habha ætting does not seem to need this part of the algorithm. 4.4 Details of matrices and vectors used in the kinematic æt We deæne the following variables: E i = 8 é é: E, E + for i =,2,3 èp ix,, p iy,, p iz, è for i =4,5,6 èp ix+, p iy+, p iz+ è for i =7,8,9 èp x,,p y,,p z, è for i =,, 2 èp x+, p y+, p z+ è s i = è for i ç 6 èpix,, p iy,, p iz,, p ix+, p iy+, p iz+ è, for ié6 èp x,,p y,,p z,,p x+,p y+,p z+ è For 4 æ 4 matrix D we need the following expressions: Row j = to 3, columns i = to 2: j æ i = è Row j =, column i = 3: Row j =, column i = 4: Row j = 2, column i = 3: Row j = 2, column i = 4: Row j = 3, column i = 3: s i if i = j or i = j +3ori=j+6ori=j+9 else æ 3 =,E fæ cos ç fæ cos ç fæ =,æ 5 cos æ 3 cos æ 4 æ 4 = E fæ sin ç fæ sin ç fæ = æ 5 sin æ 3 sin æ 4 2 æ 3 =,E fæ cos ç fæ sin ç fæ =,æ 5 cos æ 3 sin æ 4 2 æ 4 =,E fæ sin ç fæ cos ç fæ =,æ 5 sin æ 3 cos æ 4 3 æ 3 = E fæ sin ç fæ = æ 5 sin æ 3 8

9 Row j = 3, column i = 4: Row j = 4, columns i = to 2: Row j = 4, column i = 3 and 4: 3 æ 4 = 4 æ i = s i æ i E i 4 æ 3 = 4 æ 4 = The 4 æ matrix E is: E =, sin ç fæ cos ç fæ,sin ç fæ sin ç fæ,cos ç fæ, =,sin æ 3 cos æ 4, sin æ 3 sin æ 4, cos æ 3, 4.5 The error matrix of the æt The second partial derivatives of ç 2 appear in the error matrix of the æt parameters: è!, H = 2 æ i æ j So in our case, H is a 9 æ 9 matrix. The detailed expressions for the second derivatives of ç 2 will be given in the following section. 4.6 Tests for goodness of æt fter completing the iteration on the kinematic æt, one wants to make sure that all quantities are indeed correct. esides the obvious tests that the constraint equations are satisæed, one can check that indeed a minimum was reached. For this, one may wiggle each ænal value æ to æ 4 and recalculate ç 2. In our case we have in the ç 2 -function the terms with the Lagrange multipliers. Just recalculating the ç 2 function will not lead to correct results, since the found vector æ is a minimum only when also requiring the constraints. So one has to redo the æt while forcing the selected element of æ to the oæ-minimum value. This wiggling allows us to map out the minimum, and it also tells us whether the æt error returned for that parameter is reasonable. If we æxe fæ to be æç æ æt way from the real æt result, then the ç 2 should rise by in either direction. When mapping out this rise, one will see the shape of a parabola. When the formulas are complicated andèor one is far away from the minimum, the parabola will be distorted. In our case, we can indeed calculate the æt error for E fæ, but if this would be impossible, one can ænd the æt error by mapping out the minimum with the above described re-ætting with æxed values. The æç-error is then deæned to be where ç 2 is unit above the minimum. s mentioned, this function may be distorted when far away from the minimum. complicated ç 2 -function might even distort the æç-area. In this case, one can take the minimum and two 9

10 points very close to it, æt a parabola through these three points, and take the sigma from that parabola as the error. The same process also works for the hidden parameter èætted photon energyè, and we deænitely have to re-æt since the ætted photon energy only appears in the constraints, where the Lagrange multipliers would inæuence the outcome. Here is how we have to modify the formulas for re-ætting: 4.6. Re-ætting with æxed E fæ We want to redo the æt with the photon energy æxed to E æx = E fæ + æ. To the ç 2 -function, we add the term + X èe æ, E æx è 2 where X is a large number compared to the original ç 2. If we now minimize this new ç 2 -function, the additional term adds a large penalty toany deviation of E æ from E æx. Going through the derivation again, we ænd the following places that have to be changed in the code: æ First partial derivative ç2 for i = 5 ëfor i = kë has the additional term ë+2xèe æi æ, E æx è". æ No change to second partial derivatives. æ Matrix M has the additional term ë+2x" at è5,5è. This means that the è5,5è-element of M is no longer zero. æ Vector Z has an additional term at position 5: Z =,2Xèh, E æx è,r. These are all necessary changes. The iteration should converge again, but this time always result in E æ = E æx for suæciently large X Re-ætting with æxed f k Let us now wiggle one of the measurement variables æ to æ 4. When æxing f k to f k = f k æx,we add the term + X èf k, f k æx è 2 to the ç 2 -function. gain, X is a large number compared to the original ç 2. The following changes have to be made in the formulas of the algorithm: æ The ærst partial derivative ç 2 =æ i gets for i = k the additional term ë+2xèf k, f k æx è". æ gain no change to second partial derivatives. æ Matrix M gets at position èk; kè the additional term ë+2x". æ Vector Z has at position k the entry ë,2xèm k, f k æx è".

11 4.6.3 onædence Level If all errors of the measurements are nicely described by Gaussian distributions, and if all events are what we think they are, i.e., èin our caseè radiative habhas, then the ç 2 values of the æts should be distributed like the ç 2 -distribution for n = 3 è3 because out æt is a 3-constraint ætè. Instead of looking at the ç 2 distributions directly, it is easier to map the ç 2 to a æat distribution with values between and. This value is then called the conædence level è.l.è of the event. If the ç 2 is really distributed as it should be, the conædence level will have a æat distribution. So we are looking for two things in the.l. distribution: èè Most of the region should have a æat distribution. If not, the errors used in the æt might be too large or too small. If the errors are underestimated, the ç 2 will be larger than expected, and the conædence level distribution will be tilted downward èwhen going from to è. Vice-versa, if the errors are overestimated, the.l. distribution will be tilted upward. More information on the validity of errors might be obtained from the ëpull" distributions described later. è2è peak at zero indicates events that do not fulæll the kinematics of radiative habhas at all. They will result in very large ç 2 è=very small.l., close to zeroè. These events can come from backgrounds or misidentiæed tracks. What can we do? We can improve our selection criteria. Or we can cut out all events belonging to that peak, taking only those events that are part of the æat distribution. cut on the conædence level is, of course, equivalent to a cut on ç The ëpull" For each measured variable, one can plot the so-called ëpull" ë2ë or ënormalized stretch values" ë3ë ë4ë: meas, æt pull p = p çmeas, ç æt The minus sign in the square root comes from the strong correlation between the measured and the ætted quantity, and ëstill puzzles many users" ë2ë. If all measured errors were estimated correctly and the conditions for the æt were satisæed èe.g., the event was really a radiative habha eventè, then the pull quantity will be distributed like a Gaussian centered at with ç =. If an error is for example overestimated, the pull quantity will have a more narrow distribution. In this case, the conædence level should also be aæected, displaying a tilt in its distribution. To check whether a systematic increase or decrease of one or more errors would improve the pull andèor the conædence level distributions, one can redo the whole analysis with increased or decreased errors. Perhaps one can ænd a set of corrections that create nice pull distributions and a nice conædence level distribution. If the errors are really not correct, one should talk with the colleagues who are responsible for the errors. However, abnormal pull quantities might not be always created by incorrect errors. Systematically shifted measurements could also cause such symptoms. 5 ç 2 -Function First Derivatives For this set of equations, we will use the following notation: L i = 8 é é: ç ç 2 ç 3 for i =,4,7, èp ix,, p ix+, p x,, p x+ è for i =2,5,8, èp iy,, p iy+, p y,, p y+ è for i =3,6,9,2 èp iz,, p iz+, p z,, p z+ è

12 Now we calculate the ærst partial derivatives of the ç 2 -function, i.e., the 9 equations ç 2 =æ i. For i = to 2: ç 4X 2 =2 V, all æ èf f i ij j, m j è+s i L i +s i ç 4 i E i For i = 3: ç 2 For i = 4: For i = 5: For i = 6: For i = 7: For i = 8: For i = 9: æ i = 2 ç 2 = 2 4X j= 4X j= æ i = 2 ç 2 j= V, all ij èf j, m j è, ç E æ cos ç æ cos ç æ, ç 2 E æ cos ç æ sin ç æ + ç 3 E æ sin ç æ V, all ij èæ j, m j è,æ 6 æ 5 cos æ 3 cos æ 4, æ 7 æ 5 cos æ 3 sin æ 4 + æ 8 æ 5 sin æ 3 = 2 4X j= 4X j= V, all ij èf j, m j è+ç E æ sin ç æ sin ç æ, ç 2 E æ sin ç æ cos ç æ V, all ij èæ j, m j è+æ 6 æ 5 sin æ 3 sin æ 4, æ 7 æ 5 sin æ 3 cos æ 4 æ i =,ç sin ç æ cos ç æ, ç 2 sin ç æ sin ç æ, ç 3 cos ç æ, ç 4 =,æ 6 sin æ 3 cos æ 4, æ 7 sin æ 3 sin æ 4, æ 8 cos æ 3, æ 9 ç 2 æ i = p xi, + p xi+, p x,, p x+, E æ sin ç æ cos ç æ = æ + æ 4, æ 7, æ, æ 5 sin æ 3 cos æ 4 ç 2 æ i = p yi, + p yi+, p y,, p y+, E æ sin ç æ sin ç æ = æ 2 + æ 5, æ 8, æ, æ 5 sin æ 3 sin æ 4 ç 2 æ i = p zi, + p zi+, p z,, p z+, E æ cos ç æ = æ 3 + æ 6, æ 9, æ 2, æ 5 cos æ 3 ç 2 æ i = +, E,, E +, E æ = +, E,, E +, æ 5 2

13 6 ç 2 -Function Second Derivatives For i = to 2 and j = to 2: = 2V, æ j æ + all s Ei,fi ij iç 2 =E i 4 E 2 i i = 2 V, all ij, s iç 4 fi fj E 3 i = 2 V, all ij = 2 V, all ij, s iæ 9 æ 2 i,e 2 i E 3 i = 2 V, all ij, s iæ 9 æi æj E 3 i else if i = j if E i = E j by deænition For i = to 2 and j =3to4: For i = to 2 and j = 5: For i = to 2 and j =6to8: =2V, all ij = = s i if L i = L j by deænition = else For i = to 2 and j = 9: For i = 3 and j = 3: = s i f i E i = s i æ i E i = 2V, æ j æ + all ç ij E æ sin ç æ cos ç æ + ç 2 E æ sin ç æ sin ç æ + ç 3 E æ cos ç æ i For i = 3 and j = 4: For i = 3 and j = 5: = 2 V, all ij + æ 6æ 5 sin æ 3 cos æ 4 + æ 7 æ 5 sin æ 3 sin æ 4 + æ 8 æ 5 cos æ 3 = 2V, æ j æ + all ç ij E æ cos ç æ sin ç æ, ç 2 E æ cos ç æ cos ç æ i For i = 3 and j = 6: = 2 V, all ij + æ 6æ 5 cos æ 3 sin æ 4, æ 7 æ 5 cos æ 3 cos æ 4 =,ç cos ç æ cos ç æ, ç 2 cos ç æ sin ç æ + ç 3 sin ç æ =,æ 6 cos æ 3 cos æ 4, æ 7 cos æ 3 sin æ 4 + æ 8 sin æ 3 =,E æ cos ç æ cos ç æ =,æ 5 cos æ 3 cos æ 4 3

14 For i = 3 and j = 7: =,E æ cos ç æ sin ç æ =,æ 5 cos æ 3 sin æ 4 For i = 3 and j = 8: For i = 3 and j = 9: For i = 4 and j = 4: = E æ sin ç æ = æ 5 sin æ 3 = For i = 4 and j = 5: For i = 4 and j = 6: = 2V, æ j æ + all ç ij E æ sin ç æ cos ç æ + ç 2 E æ sin ç æ sin ç æ i = 2 V, all ij + æ 6æ 5 sin æ 3 cos æ 4 + æ 7 æ 5 sin æ 3 sin æ 4 = ç sin ç æ sin ç æ, ç 2 sin ç æ cos ç æ = æ 6 sin æ 3 sin æ 4, æ 7 sin æ 3 cos æ 4 = E æ sin ç æ sin ç æ = æ 5 sin æ 3 sin æ 4 For i = 4 and j = 7: For i = 4 and j = 8 and 9: For i = 5 and j = 5: For i = 5 and j = 6: æjæi =,E æ sin ç æ cos ç æ =,æ 5 sin æ 3 cos æ 4 = = =,sin ç æ cos ç æ =, sin æ 3 cos æ 4 For i = 5 and j = 7: =,sin ç æ sin ç æ =, sin æ 3 sin æ 4 4

15 For i = 5 and j = 8: For i = 5 and j = 9: For i =6to9andj=6to9: References =,cos ç æ =, cos æ 3 =, = ëë Lyons, L., Statistics for nuclear and particle physicists èambridge Univ. Press, 986è. ë2ë Eadie, W. T. et al., Statistical methods in experimental physics ènorth-holland, msterdam, 986è. ë3ë V. lobel, Least squares methods, p. I 27, in ock, R. K. et al. èeds.è, Formulae and methods in experimental data evaluation with emphasis on high energy physics. èeuropean Phys. Soc., 984è ë4ë Roe,. P., Probability and statistics in experimental physics, èspringer-verlag, 992è. cknowledgments I thank ill Dunwoodie èslè for his patience with my numerous questions. His help and expertise were invaluable in understanding all issues on the estimated ç 2 and the kinematic æt. 5