Applied Fitting Theory VI. Formulas for Kinematic Fitting

Transcription

1 Alied Fitting heory VI Paul Avery CBX June 9, 1998 Ar. 17, 1999 (rev.) I Introduction Formulas for Kinematic Fitting I intend for this note and the one following it to serve as mathematical references, with eamles, for kinematic fitting as imlemented in the KWFI [1] library. Until the recent advent of silicon tracking, CLEO made little use of kinematic fitting for reasons ranging from ignorance of what it can do, lack of knowledge of eisting software tools and oorly understood charged tracking errors. However, Kalman fitted tracks and silicon tracking rovide better tracking errors, and better understood errors, both of which are imortant for kinematic fitting to aid analyses. he recent availability of high quality osition information has thrust kinematic fitting (as imlemented in the KWFI and VCFI [] libraries) into a far more central role in CLEO analysis, a role which I believe requires etensive documentation to enable hysicists to understand the ower and limitations of kinematic fitting. his note serves as that documentation. It will be constantly udated in my fitting home age * as I accumulate new information and eamles. Most of the algorithms discussed here are imlemented in KWFI. Before I begin, let me define what we are talking about. Kinematic fitting is a mathematical rocedure in which one uses the hysical laws governing a article interaction or decay to imrove the measurements describing the rocess. For eamle, the fact that the three articles coming from a K π π decay must come from a common sace oint can be used to imrove the momentum vectors of the daughter articles, thus imroving the mass resolution of the +. Elicit formulations of this constraint and others based on mass, energy, momentum, etc. are described later. he raw materials for kinematic fitting are charged and neutral tracks obtained, resectively, from measurements made in central tracking chambers and hoton shower detectors. It is assumed that for each track a set of arameters and their covariance matri is available. [3,4] he arameters and their covariance matri contain all the necessary information to aly constraints; one does not need to worry about the detailed hit information. For a somewhat more comlicated eamle, consider the decay sequence * 1

2 B * + π * + + π + K π π Assuming that the neutral ion has been reviously fit, there are several constraints which can be alied: (1) the K π + π mass can be forced to be equal to the mass (1 constraint); () the K and π + from decay intersect at a single sace oint (* 3 = 1 constraint); (3) the π + mass can be constrained to the *+ mass (1 constraint); (4) the, the soft π + from the *+ decay and the π from B decay should intersect at the same sace oint (3* 3 = 3 constraints); and (5) the energies of the final state articles must add u to the beam energy (1 constraint). When the tracks are refit with these 7 constraints using the general algorithm discussed in the Aendi, their arameters will satisfy these constraints and the *+ π mass will be narrower than before. II rack reresentation and notation One of the rewards of writing notes on fitting theory is the number of eole who sto me in the street to shake my hand, telling me breathless stories of how kinematic fitting changed their lives. Whenever I am accosted in this way, the question is invariably raised about why I reresent tracks using the 7-arameter W reresentation, defined as α W = 3, y, z, E,, y, z8, i.e., a 4- momentum and a oint where the 4-momentum is evaluated. Why, my etitioners ersist, do I not use the standard 5-arameter track reresentation used in track fitting? And aren t seven arameters redundant? Well, these are imortant questions, so let me try to elain my reasoning. I designate the 5- arameter reresentation the C format to distinguish it from the W format used here. he C format consists of three variables secifying the article momentum and two variables defining its location in sace (a more recise definition can be found in Aendi I). Most imortantly, it does not secify the location in sace of a article roduction or decay; that would require a sith arameter. he article's initial osition is defined to be at the oint of closest aroach to the reference oint, irresective of its actual origin. For kinematic fitting it is imortant to choose a track reresentation which uses hysically meaningful quantities and is comlete. he first reason why the C format is insufficient for kinematic fitting is that it cannot handle neutral tracks because the curvature is regardless of momentum. One could of course change the reresentation so that the curvature becomes the inverse momentum but that would introduce a different format. Secondly, the W format is much simler to transort in a magnetic field. See Aendi II for details. Finally, the C format does not have enough information to reresent general decays of articles. In kinematic fitting, the energy varies indeendently of the momentum because the mass is in general not constrained. (1)

3 Also, as discussed in the receding aragrah, the osition where a article decays requires three coordinates, one more than secified by the C format. Aendi I describes a rocedure for converting the 5 charged track arameters and their covariance matri to the W format. III General algorithm for kinematic fitting he fitting technique is straightforward and is based on the well-known Lagrange multilier method [3]. It is assumed that the constraint equations can be linearized and summarized in two matrices, which I label and d; secific eressions for and d are given in Section V for constraints most commonly encountered in HEP analyses. Let α reresent the arameters for a set of n tracks (a total of 7n arameters). It has the form of a column vector α = Initially the track arameters have the unconstrained values α, obtained from a track fit, for eamle. he r functions describing the constraints can be written generally as H( α ) =, where H 1H 1, H,H r 6. Eanding around a convenient oint α A yields the linearized equations = H( αa)/ α α αa + H( αa) δα+ d where δα = α α A. hus we see that α α α 1 n (), (3) H1 H1 H 1 α1 α α n H1 αa6 H H H = d = H1αA6 α α α 1 n Hr Hr Hr Hr1αA6 α1 α α n or ij = Hi / α j and di = Hi1αA6. he constraints are incororated using the method of Lagrange multiliers [3] in which the χ is written as a sum of two terms, e.g. χ α 5 = α α V α α + λ δ α + d, (4) where λ is a vector of r unknowns, the Lagrange multiliers. Minimizing the χ with resect to α and λ yields two vector equations which can be solved for the arameters α and their covariance matri: 3

4 Vα α α + λ = (5) δα + d= he latter equation demonstrates clearly that the solution satisfies the constraints. he solution can be written [3] α = α Vα λ λ = V δα + d 3 α 8 α α α α V = V V = V V V V χ = λ V λ = λ 1δα + d6 where δα = α α A. It can be shown that the diagonal elements of the α covariance matri are reduced in size, as eected by intuition. Several things should be noted about the above solution. First, only a single matri must be inverted, the r r matri V. he changes to α caused by the constraints are roagated by matri multilication. Second, the χ is a sum of r distinct terms, one er constraint. However, identifying each of these terms with a articular constraint is only ossible in a loose sense since the contribution of each constraint is correlated with all others through V. Finally, it is useful to comute how far the arameters have to move to satisfy a articular constraint j. he initial distance from satisfaction can be characterized by the quantity δα + d j and the number of standard deviations away from satisfying the constraint is easily calculated to be σ j = δα + d ji i j 4V 9 jj his information can be used to rovide criteria for rejecting background in addition to the overall χ. IV Eamle: Back to back constraint Let's aly the algorithm to a simle case where we want two articles to be back to back. his is equivalent to writing the following constraint equations (this is similar to the constraint discussed in Section V.4): (6) 4

5 + = + y = + = 1 y1 z1 z Initially, the track arameters have the values α α 1 α α 1 = E y z 1 y1 z = α =, where and the initial 7 7 track covariance matrices are denoted V 1 and V. he arameters can be eanded about these values (i.e., α A = α ) giving for and d: = d = y z E y z y1 z1 y z where I have ignored the bending of the track as a function of osition. A clever reader might notice that I have not included the change in energy caused by shifts in momentum for tracks with fied mass (i.e., those found from track fitting or shower finding). However, this constraint is already built in, in the sense that the initial track covariance matri already includes the correlation of energy with momentum. Additional constraints will move the track arameters around in such a way as to reserve the mass *. he matri V is given by V + V V + V V + V V = V = 4 α 9 3V1 8 + V V + V V + V (1) 3 3V + V V + V V + V (7) (8) (9) * It s best to have a coule of beers before thinking seriously about this. 5

6 he Lagrange multiliers can be calculated from λ = V1δα + d6, yielding the eressions λ λ λ = V + + V y + y + V 13 z1 + z = V + + V y + y + V 3 z1 + z = V + + V y + y + V 33 z1 + z he udated momentum comonents are calculated using α = α V λ : 1 ( ) 1 ( ) y1 ( ) y1 ( ) z1 ( ) z1 ( ) α = V + V λ V + V λ V + V λ = V + V λ V + V λ V + V λ = V + V λ V + V λ V + V λ Similar equations can be written for the energy and osition comonents by using the aroriate track covariance indices. he χ for the solution is χ y y z1 z = λ δα + d = λ + + λ + + λ + he udated covariance matri can be comuted from Vα = Vα Vα V V α. his gives for the momentum comonents 1V16 3V1 8 3V1 8 1V 6 3V1 8 = "! kl jl $ # ij ij ik kl, 1V6 3V V V V 8 = " ij ij ik kl jl kl! $ #, = " ij ik kl jl kl! $ #, 1V16 3V1 V V 8 where all indices run from 1 to 3. he effect of the fit is to reduce the errors of the individual tracks. Note that the second term in the general formula for V α mies the tracks together so they are correlated after the fit. V Non-verte constraints In this section I comute the elicit form of the and d matrices for constraints commonly encountered in high energy hysics (verte constraints are discussed in the net section). Once these matrices are known the tracks can be kinematically fit using the rocedure described in the revious section. If multile constraints are desired then one just etends the matrices by adding rows to them, one row er constraint. his allows many constraints to be used simultaneously in (11) (1) (13) (14) 6

7 the fit. In all cases the tracks are eanded about their current values 3, y, z, E,, y, z8, so the solution should be seen as changes from these initial values. V.1 Invariant mass constraint he constraint equation which forces a track to have an invariant mass m c is y z c E m =, (15) Eanding about the initial arameters 3, y, z, E,, y, z8, we get for and d: = y z E d = E m y z c (16) V. otal energy constraint he constraint that a track must have a total energy E c can be written E d matrices are trivially comuted to be E c =. he and = 1 (17) d = E E c V.3 otal momentum constraint he constraint that a track must have a total momentum c can be written y z c + + =, (18) Eanding about an initial set of arameters 3, y, z, E,, y, z8, we comute the and d matrices to be = / y / z / d = + + y z c (19) V.4 otal 3-vector constraint his is a set of three constraints that can be written i = () where ci reresents the comonents of the total 3-momentum (i = 1,,3). Comuting and d yields ci 7

8 = d = y z V.5 otal 4-vector constraint his is a set of four constraints that can be written c yc zc µ µ c (1) =, () where µ c reresents the comonents of the total 4-momentum ( µ = 13,,, ). Comuting and d yields = d = y z E E c yc zc c V.6 Particle lies in a lane his constraint, alied to a single track, is useful for cases when one has built a virtual article and wants to further constrain its verte location. It is easier in this case to build the virtual article using a standard verte fit [6] and then aly the additional requirement to the virtual article. he constraint is described by the equation of a lane: η 3 8=, where η is a unit vector normal to the lane and is a oint in the lane. he and d matrices are given by = 3 η ηy ηz8 (4) d=η =η η y η z y z V.7 Particle lies on a line he comments about the usefulness of this constraint from the last subsection aly here. he constraint is described by the equation of a line: = + η s, where is some oint on the line, (3) 8

9 η is a unit vector along the direction of the line and s is the length along the line. Eliminating the distance s we get for the equations 3z z8η / ηz = (5) yy z z η / η = he and d matrices are then = y z 1 1 d = + z η y + z η / η / η z y z If η z is close to zero we can choose either one of the other coordinates as the indeendent variable. V.8 Back to back constraint his is the one constraint which is actually more difficult to eress in our standard W reresentation than in the C reresentation, defined in Aendi I. It eresses the fact that for the decay of a article at rest into two oositely charged articles, the three momentum of the daughter articles is equal and oosite and the two articles must originate at the same oint, a total of 5 conditions. he constraints can be written as follows (this arallels the way they would be eressed in the C reresentation): 1 = φ1 φ = π d1 + d = (7) z1+ z = z z = 1 We can eand these equations in terms of the W track arameters. For simlicity I will assume a solenoidal B field. his yields the following equations eressing the constraints η η y / η / η z z (6) 9

10 z 1+ y1 + y = y y a y a + a y a = 1 1 / a1+ / a = z1+ z = z1 a y y18 tan a a y 1 1! 1 z + a! " $ # a 3 + y 8 " # $ 1 1 y1 1 1 z y tan a y y # = where = + a y + a, a = BQ, Q i is the charge, and B is the i i i i yi i i i magnetic field strength. Note that the second, third and fifth constraints are eressed at the oint of closest aroach (in the bend lane) to some oint, which I choose to be the origin. he corresonding matrices are obtained by taking the derivatives of these 5 equations with resect to the 14 track arameters, yielding a 5 14 matri. Unfortunately, I don t have the energy to do this at the moment, although the constraints have been imlemented in KWFI. I will ut the derivatives in a later udate to the document. VI Verte constraints VI.1 General solution I will develo some common roerties of verte constraints here before secializing in the rest of this section. Consider a set of n tracks forced to ass through a common oint = 1, y, z6. Since the covariance matri of the verte may be known in advance (what s known as a rior covariance matri), we write the overall χ condition generally as 1 1 F αα V αα V λ Gα E G d (9) where the terms reresent, resectively, the contribution to the χ from the tracks, verte and the constraints. Note that and V reresent the initial verte osition and its covariance matri, while Gα α α A and G A are the deartures of the variables from their eansion oints. For each track i there are two constraint equations, corresonding to the bend and non-bend lanes, resectively. he equations are obtained by eliminating the arc length s from the, y, z equations of motion in Aendi II. For eamle, in a solenoidal B field: i (8) 1

11 a i' yi yi' i i ' i ' yi zi ' zi sin a3 ' ' y8/ a i i i i yi i i where i = i, etc., ai = BQi, Q i is the charge, B is the magnetic field strength and i is the transverse momentum. A generalization of this formula can be derived for B fields oriented along arbitrary directions using the equations in Aendi II: ai = i h i i h i = h h i i sin ai4i i 3i h83 i h 89/ i h a i! where h is a unit vector in the direction of the magnetic field. he E and matrices thus have the simle form E = E E E 1 n = where E i is a 3 matri and i is a 7 matri. E and have this articular block diagonal form because the verte constraints for each track only involve the arameters for that track. he fact that the constraints do not mi tracks greatly simlifies the solution (if the tracks are initially uncorrelated) because the inversion of the n n matri V can be factored into n inversions. In a solenoidal field, the matrices i, E i, and d i for each track are given by E d i i i = i i yi i i i i i sin Ji zi i i zisiryi izisi yizisi ai yi i i i i i i zi i yi zi i i i i yi 1 i i + i zi zi sin Ji ai y + a + a y S R a a y = + S S = 1 y a y where the auiliary quantities J, R, R y, and S are defined as follows: 1 n " $# (3) (31) (3) (33) 11

12 J = a + yy / y ( ) y ( ) y R = + y / S= 1 1J he solution to this roblem is straightforward, but algebraically tedious, and I leave the details to Aendi III. he solution is: δα = δα V λ δ = δ V E λ = V V δ V E λ 1 λ λ = V~ δα + E δ + d= λ + VE δ = V δα + d V~ = V V EV E V where Gα α α A and G A are the deviations of the arameters from their eansion oints. he covariance matrices are V = V + E V E 1 α α α α α α α V = V V V V + V V EV E V V 5 and the χ is given by cov α, =V V EV ~ χ = λ V λ = λ V + λ EV E (37) = λ 1δα + Eδ + d6 he hysical meaning of the covariance matrices can be elained as follows. he verte error matri V is the weighted average of its initial covariance matri and the errors determined from the tracks. he track error matri V α has an initial iece that is decreased by the constraints alied er track and is increased by the wiggle of the verte itself. Note only this last term correlates the tracks with one another. VI. Verte constraint to a fied osition In this case, the verte osition is fied and the solution can be obtained by setting δ =, E = and V =. he solution factors into n ieces, one er track i: (34) (35) (36) 1

13 α = α V λ λ i i α i i i = V δα + d i i i i i i i α = 4 i i 9 α = α i i αi i i i αi χ = λiv i λ i = λ i iδαi + di V V V V V V V he tracks remain uncorrelated after the fit because each track is fit searately to the fied oint. VI.3 Verte constraint to an unknown osition In this case the verte osition must be determined from the constraints. he simlest aroach is to assign large values to V, the initial verte covariance matri, and aly the method from the first subsection. his method of assigning a large rior covariance matri to a set of arameters is discussed in more detail in ref. [3]. Note that when using this technique one has an effective number of degrees of freedom equal to n 3 because the fit causes the three verte arameters to move an insignificant number of standard deviations. VII Verte constraint with the verte osition satisfying other conditions Suose one wanted to determine a verte whose osition satisfies some other constraint. For + eamle, in ee collisions the vertical sread of the beam may be much smaller than the resolution so that the verte effectively lies in a horizontal lane (1 constraint), or the verte might lie on the trajectory of another article ( constraints). he fit can be carried out in basically two ways. One method is to incororate the etra information directly in the verte fit. his leads to a somewhat more comlicated but solvable roblem, ecet that several verte fitting routines need to be written, one for each variation. A better method is to do the fit in two stes, first determining the verte using the algorithm from Section VI and then alying the additional constraints. his two ste rocedure is mathematically identical to alying all the constraints simultaneously, as I have roven elsewhere [3]. he final χ is obtained by adding together the χ values for the two stes. Note, however, that the inut tracks are not udated by the two ste rocedure. o udate the track arameters using the etra information, we would have to udate the track covariance matrices after the initial verte constraint, a rocess that would introduce correlations between the tracks. his is not articularly difficult to work out, but it does not seem necessary at this time to kee track of the udated inut tracks. (38) 13

14 VIII Miscellaneous verte maniulations VIII.1 Averaging two vertices Suose we have two estimates of a verte, each with a different covariance matri. How do we combine them using the full information? he best value, which I call z, can be found by turning the roblem into a χ minimization: χ ave = z V z + zy Vy zy (39) he best z is the one that minimizes this χ. A straightforward rocedure gives the solution z= V + V 1 4 y 9 4V + Vy y9 = + V3V + Vy8 1y6 1y 6 3V V 8 1y 6 1 with χ ave = + y. Note that this solution is the generalization of the wellknown average of two quantities having different errors (it is trivial to generalize to any number of vertices). Moreover, the above formulation only requires the inverse of a single matri, 3V + Vy8 1, so V and V y can individually be singular. he averaging rocedure offers an interesting way to factor the verte fitting roblem. Suose you have n tracks whose common verte you are trying to find. he tracks can be broken u into two disjoint sets with n 1 and n tracks, resectively, and searate vertices v 1 and v can be found. he overall verte osition is just the weighted average of the two vertices and the total χ is equal to χ1 + χ + χ ave. his result rovides a useful way of testing verte fitting algorithms. As described in the last Section, however, the averaging rocedure rovides no way to udate the inut tracks going into the verte. VIII. esting for equality of two vertices It is useful occasionally to test whether two vertices are identical. he roblem can be cast into a constraint roblem, and the χ determined in this way turns out to be the same as in the revious subsection, as a moment's reflection tells us it should. If the vertices are denoted and y, having covariance matrices V and V y, resectively, the χ for equality is given by 5 5 χ equal (4) = y V + Vy y (41) he value of χ equal can be converted to a confidence level using the fact that there are three degrees of freedom. 14

15 IX Building virtual articles A virtual article is made by combining a set of measured tracks into a single one. Common eamles in high energy hysics are the π, K s and Λ, all of which decay into two secific kinds of articles and have had techniques develoed esecially for finding them. We would like to generalize this rocess so that, for eamle, one could combine the daughter roducts in K π + π and make the arent. Once constructed with a set of arameters and a covariance matri, the virtual article indistinguishable from any other measured article and can be used in further kinematic fits or used to build still other articles. he rocess of creating virtual article offers an alternative way of kinematically fitting a decay sequence. Consider the cascade decay described at the beginning of this note: B * + π * + + π + K π π he full fit roceeds as follows. First one builds the from the K π + π tracks and calculates a χ. hen the is forced to satisfy a mass constraint, generating another χ. his rocess is reeated until the B is constructed; the total χ for the sequence is the sum of the individual χ s. he values for the track arameters and the total χ are eactly what would have been obtained if the entire sequence had been fit simultaneously. his is a secial case of a theorem I roved in Ref. [3], which says that the results of a fit do not deend on whether constraints are used sequentially or simultaneously. he mathematical details of how virtual articles are constructed are described in ref. [6]. (4) 15

16 Aendi I Converting charged track information to the W format he C reresentation is used in charged track fitting in CLEO and several other HEP eeriments. he 5 arameters αc =(, c φ, d, λ, z ), measured relative to the reference oint 1r, yr, zr6, describe the entire shae of the heli, where c =1/ R (R is the radius of curvature), φ is the φ of the track momentum vector at the oint of closest aroach to the reference oint, d is the signed distance of closest aroach to the reference oint in the y lane, λ = cot θ, where θ is the olar angle measured from the +z ais, and z is the z osition of the track at the oint of closest aroach to the reference oint in the y lane. I assume that the B field is oriented along the +z direction. he seven W arameters at the oint of closest aroach to the reference oint can then be calculated in terms of the five canonical arameters, as shown below: a = cosφ c a = sinφ c a = λ c E = ( a/ c) 1+ λ + m = r d y= yr + d z = z + z r sinφ cosφ (43) where a = Bq, B is the magnetic field in esla and q is the charge in units of e. he values of,,, E 3 y z 8 defined here are used by default in CLEO analyses. he 7 7 covariance matri for the W arameters can be comuted by taking the differentials of the above equations: 16

17 δ = δcyδφ c y δy = δc+ δφ c z δz = δc+ δλ c λ δe δ δλ Ec c = + E y δ = δd yδφ y δy=+ δd + δφ δz = δz V W can be comuted from V C using the above formulas and roagation of errors. For eamle, y 1VW6 = 1VC6 + 1VC6 + V c c 1 y1 C6 y y 1VW6 = 1VC6 + V V 1 11 c c z 1VW6 = 1VC6 1VC6 y 1V C6 c c C 6 1 y 1 C 6 hese follows from the definition VW 11 cov, = δ δ, etc. (44) (45) 17

18 Aendi II Motion of a charged article in a magnetic field It can easily be shown by solving the Lorentz force equation d/ dt = qv B that a charged article traces out a heli in sace, with the ais of the heli lying along the magnetic field direction. For a solenoidal field aligned along the z direction, as encountered in most colliding beam eeriments, articles move in circles in the r φ lane with a radius of curvature R= / a, where a = Bq, B is the magnetic field strength in esla, q is the charge and is the momentum transverse to the z ais. he equations of motion as a function of the ath length s are then: = cos ρs ysin ρs y = ycos ρs+ sin ρs z = z y = + sin ρs 1cos ρs5 a a (47) y y= y + sin ρs+ 1cos ρs5 a a z z = z + s where ρ = a/,, y, z 1 6 is a known oint on the heli and, y, z is its 3-1,, 6 to momentum there. hey are functions of s, the arc length measured from the oint y z 1, y, z6 (the radius of curvature is R a terms of osition: = a y y = / ). Eliminating s we can eress momentum in y = y + a = z z (48) When the B field is not aligned along the z ais, it is useful to write the equations in a rotationally invariant form. We define a right-handed coordinate system based on these vectors: 3 h 8 h h h (49) where h is a unit vector ointing along the direction of the magnetic field. By identifying the comonents from the solenoidal case with the general case, the momentum and osition 18

19 comonents can be written (these can be derived directly from the equations of motion under the Lorentz force) = h h cos ρs h sin ρs+ h h h h h h r= r sin ρs 1 cos ρs + h s a a = a r r h (5) 19

20 Aendi III erivation of the Verte Constraint Solution We write the overall χ condition for a verte constraint as α 1 5 χ = α α V α α + V + λ δ α + E δ + d (51) where the terms reresent, resectively, the contribution to the χ from the tracks, the verte and the verte constraint. Note that and V reresent the initial verte osition and its covariance matri, while, E and d are defined in Section VI. Note that Gα α α A and G A are the deviations of the track and verte coordinates about their eansion oints. o cast the χ equation into a familiar form, we write it the using etended matrices combining information from the verte and the tracks: V E1 1 ~ α ~ α = α ~ = 1 V1 E V = α V E giving the new equation n n n n χ α 4 9 (5) 1 = α ~ α ~ ~ ~ ~ + δ ~ ~ α α λ V α + d (53) he minimization of this χ with resect to all the variables has the standard solution [3]: δ ~ δ ~ ~ α = α Vα ~ λ ~ λ = V δ ~ ~ α + d 4 9 ~ ~ V~ = Vα ~ 4 9 (54) ~ ~ Vα ~ = Vα ~ Vα ~ V~ Vα ~ ~ χ = λ V δ ~ ~ λ = λ α + d 4 9 where G ~ ~ ~ α α α A. he verte and track art of the solution can be searated out to give: δα = δα V λ δ = δ V E λ α λ λ = V~ δα + E δ + d= λ + VE δ = V δα + d ~ = α V V EV E (55)

21 with covariance matrices α α α α α V = V V V~ V V = V V E V~ EV 5 cov α, =V V~ EV ~ ~ he only non-trivial art of the calculation involves the calculation of V~ = V~ 4 α 9, which is a n n matri. his is where the articular form of the matri comes into lay. We write ~ ~ V~ = V~ = EV E + V We invert this using the Woodbury theorem, A+ UV = A A U VA U VA (56) 4 α 9 3 α 8 (57) , where A is an invertible matri. U and V should be selected to make the matri in arentheses have small dimensions. Choosing U = EV and V = E yields the inverse V~ = V V EV 1+ E V EV E V 1 1 = V V E V + E V E E V he quantity in arentheses is of size 3 3. V is the block diagonal matri V = V 1 V (58) V = V 4 9 (59) V n i i i i and each V i is a matri. hus the roblem reduces to inverting n matrices and one 3 3 matri and doing a lot of matri multilication. he full solution reduces to: δα = δα V λ with covariance matrices δ = δ V E λ = V V δ V E λ α 1 λ λ = V~ δα + E δ + d= λ + VE δ = V δα + d V~ = V V EV E V (6) 1

22 V = V + E V E 1 1 α α α α α α V = V V V V + V V EV E V V 5 and the χ is given by cov α, =V V EV χ α = V~ = + 3V EVE 8 = λ 1δα + Eδ + d6 λ λ λ λ (61) (6) References 1. Paul Avery, ata Analysis and Kinematic Fitting with the KWFI Library, CBX 98, or Bill Ford, he VCFI Verte Fitting Program, CSN Paul Avery, Alied Fitting heory I: General Least Squares heory, CBX 91 7, 4. Paul Avery, Alied Fitting heory IV: Formulas for rack Fitting, CBX 9 45, 5. Paul Avery, Alied Fitting heory V: rack Fitting Using the Kalman Filter, CBX 9 39, 6. Paul Avery, Alied Fitting heory VII: Building Virtual Particles, CBX 97,