Amalgamation of meson-nucleon scattering data

P H Y S I C A L R E V I E W 14 V O L U M E 2 0 , N U M B E R 11 1 D E C E M B E R 1 9 7 9

Amalgamation of meson-nucleon scattering data

R. L. Kelly Lawrence Berkeley Laboratory, Berkeley, California 94720

R. E. Cutkosky Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213

(Received 29 May 1979)

We present a series of numerical and statistical techniques for interpolating and combining ("amalgamating") data from meson-nucleon scattering experiments. These techniques have been extensively applied to .rrp elastic and charge-exchange differential-cross-section and polarization data in the resonance region. The amalgamation is done by fitting a momentum- and angle-dependent interpolating surface to the data over a moderately narrow momentum range, typically - 150 MeV/c, using the interpolating surface to shift data in a narrower central momentum region into fixed angular bins at a predetermined central momentum, and then statistically combining the data in each bin. The fitting procedure takes into account normalization errors, momentum calibration errors, momentum resolution, electromagnetic corrections, threshold structure, and inconsistencies among the data. The full covariance matrix of the amalgamated data is calculated, including contributions of statistical error, systematic error, and interpolation error. Techniques are presented for extracting from the covariance matrix information on the collective statistical fluctuations which correlate the errors of the amalgamated data. These fluctuations are described in terms of "correlation vectors" which facilitate the use of the amalgamated data as input for resonance-region phenomenology.

I. INTRODUCTION

The best modern measurements of two-body meson-nucleon scattering in the resonance region have such high statistical precision that it i s important to take systematic e r r o r s carefully into account when the data a r e used. Some sources of systematic er ror , such a s normalization e r r o r and beam-momentum calibration er ror , a r e rou- tinely monitored and documented by experimentalists, but a r e not always taken into account by data analysts. Otner sources of systematic e r r o r ar i se from unknown experimental biases, and show up only a s discrepancies between the results of overlapping measurements, o r a s discontinuities between nearby measurements. Data analysis itself can introduce systematic biases not present in the original data, e. g . , by the common practice of binning together several measurements made a t slightly different angles and/or momenta.

This paper describes techniques designed to deal with these problems and to produce "amalgamated" differential-cross-section and polarization data in an accurate and economically usable form. The purpose is twofold: first, to resolve many ques- tions about systematic e r r o r s and discrepant data a t an early stage of analysis which i s essentially model independent, and second, to summarize the content of an original data set which may contain hundreds of data extending over a band of momenta by a fixed-momentum data se t which will be smal- le r and more manageable when used in subsequent

stages of analysis. The second purpose i s similar to that of a more commonly employed procedure, which i s to replace the actual data by a set of "Legendre coefficients" or a similar set of parameters. We believe our approach i s superior, because the amalgamated data a r e a more direct and faithful representation of al l the features of the original data. The statistical correlations between the amalg&ated data a r e also smaller and a r e more easily handled in a subsequent analysis than a r e the correlations between expansion coefficients.

The general procedure begins by fitting the available data of a given type in a narrow momentum range with a momentum- and angle-dependent "interpolating surface." The momentum range i s chosen to be narrow enough s o that the interpolating surface can be taken to be quadratic in the laboratory momentum. In practice, this fit often involves many pa.rameters and many constraints. We have developed a fast and accurate fitting procedure using a two-variable orthogonal-polynomi.al technique tailored to the comparatively simple structure of the relevant x2 function. Systematic e r r o r s and systematic discrepancies between different measurements a r e taken into account during the fit of the interpolating surface. Once the surface i s determined, data in a narr6wer central ,momentum range (in practice, about one-third a s wide a s the full range of the fit) where the surface is particularly well determined a r e shifted along the surface to the nearest of a set of closely

20 - 2782 @ 1979 The American Physical Society

20 - A M A L G A M A T I O N O F M E S O N - N U C L E O N S C A T T E R I N G D A T A 2783

spaced preselected angles at a preselected cen- t r a l momentum. The shifted data in each angular "bin" a r e then statistically combined ("amalgamated" ) .

The amalgamated data a r e correlated through their common dependence on the interpolating surface and on the systematic e r r o r s of the original input data. The covariance matrix of the amalgamated data can be calculated directly from the e r r o r s of the input data using the two-variable orthogonal polynomials mentioned above. We have found that the correlation properties of the amalgamated data can be accurately represented in t e rms of collective fluctuations characterized by "correlation vectors." The use of these correlation vectors simplifies the k2 function that one would use in a fit to the amalgamated data, and they can also be used to correct the data for the effect of collective fluctuations.

The techniques described here have been developed in the course of an extensive liN partial- wave analysis,' and have s o far only been applied to 7ip elastic and charge-exchange cross-section and polarization data.2 In most of the following description we use somewhat more general lan- guage appropriate to any elastic or two-body in- elastic meson-nucleon cross-section and polarization data. The generality of the description is somewhat illusory, however, because our methods a r e designed for situations in which there i s a large amount of high-precision data, and this is currently true of only a few meson-nucleon reactions. In principle, the technique could also be extended to deal with other types of data such a s spin-rotation parameter measurements in meson- nucleon scattering o r measurements with the various combinations of polarized targets and polarized beams possible in photoproduction o r pp scattering, but we will not consider such possibilities here.

11. PARAMETRIZATION OF THE INTERPOLATING SURFACE

The purpose of the interpolating surface i s to accurately approximate the true physical values of the measured quantities for which we a r e amalgamating data. We emphasize that this surface i s only an intermediate tool, and is not to be thought of a s the final result. Before considering i t s full energy and angular dependence, let us discuss the angular dependence of the interpolating surface a t fixed energy. The differential c ros s section a t fixed energy, I, i s a sum of squares of real and imaginary parts of invariant amplitudes a l l of which a r e functions of x (=cose) analytic throughout the cut x plane except for singularities

along the rea l axis. The polarization itself is a quotient, but the polarized cross section, IP, i s a bilinear form in the real and imaginary par ts of the invariant amplitudes multiplied by an overall kinematic factor of sine. It i s the quantities I and IP/sin0 for which we actually form interpolating surfaces in the fitting procedure described in the following sections, and their fixed energy behavior can thus be represented by analytic functions of x with singularities a t the same locations a s those of the invariant amplitudes. Specifically, an interpolating surface for meson-nucleon scattering at fixed energy has right- and left-hand cuts starting a t the t- and u-channel thresholds and may have poles and di- poles corresponding to baryon exchange (the same analyticity domain a s that of the invariant amplitude ReB). Fo r an elastic reaction, I and 1P/sin8 also have a singularity at Y = 1 corresponding to Coulomb scattering, but we omit this from the interpolating surface. Coulomb corrections a r e taken into account separately and a r e discussed in Sec. I11 and in Appendix A .

The parametrization we use to represent the angular dependence is a polynomial in the angular variable z of Cutkosky and ~ e o . ~ This variable is defined in te rms of two points on the rea l axis of the x plane denoted a s x+ and -x- which lie to the right and left, respectively, of the physical region (x, > 1). The variable z i s then defined by the requirement that it map the x-plane cut along (x,, w)and (-w, - x-) onto the interior of a unifocal ellipse with the cuts mapped onto the periphery, the interval (-x-, x,) mapped onto the rea l axis, and the points x =*l mapped onto z =&I. This mapping stretches the physical region in the forward and backward peaks while compressing it in the wide-angle region, thus tending to produce flatter structure in z than in x and to thereby reduce the number of te rms required in a polynomial expansion for a good fit to the data. As discussed in Ref. 3, the most rapidly convergent polynomial expansion of a scattering amplitude is a sum of explicit pole te rms and a polynomial in z with x + and -x - located a t the tips of the physical t- and u-channel cuts. We adopt here a simpler and more flexible version of the parametrization in which we omit pole te rms and treat x+ and -x . a s phenomen- ological parameters representing " effective" cut positions a t which the strengths of the right- and left-hand singularities (including poles) f i r s t become appreciable. The reason for omitting explicit pole contributions is twofold, f i r s t because it i s more complicated to include pole contributions in observables than in amplitudeq4 and second because the fitting problem we a r e dealing with here, unlike the problem of determining an amplitude from data, has no continuum ambiguity and the

2784 R . L . K E L L Y A N D R . E . C U T K O S K Y 20 - constraining effect of the known residue of a baryon-exchange pole is consequently l e s s important. The reason f o r not requiring the s t r i c t optimal convergence prescr ipt ion for x, and x- is that it may happen that the main fea tures of the angular dependence of the data a r e controlled by singulari- t i e s which a r e s t ronger and m o r e distant than those which determine truly asymptotic convergence ra tes . In 7iN scattering, fo r example, the right-hand cut begins a t t=4m,2 and is weak there, while the neares t t-channel resonance-exchange poles l ie on unphysical shee t s a t t = (m, i i r , /2 )2 . We have found in pract ice that acceptable f i t s can b e obtained with values of x+ corresponding t o intermediate values of t = (m, - r,)2. F o r the left- hand cut, on the other hand, the s t rong nucleon exchange in n'p elast ic and v-p charge-exchange scat ter ing must b e taken into account by using a value of x. corresponding t o u =. m N 2 , while f o r n-p elast ic scat ter ing we can use al=(m, - A typical example of the mapping f o r nip scat ter ing a t 2 GeV/c is shown in Fig. 1.

We have made severa l t e s t s which verify that the prescr ipt ion for x, is well matched t o the charac-

FIG. 1. Conformal mapping of the x plane onto the z plane for n +p scattering a t 2 GeV/c. Nearby pole and branch-point singularities in the t and u channels a r e shown, and the "effective" branch points * x , a re indi- cated. The distortion of the physical region is shown by a dashed line which is drawn with equal intervals of 0 .1 in the x plane and with the corresponding mapped intervals in the z plane.

t e r i s t i c s of the data . In part icular , in t e s t s in which we used x, =m (in which c a s e z =x, s o that our expansion is equivalent to the usual one) our f i t s were generally l e s s satisfactory and a l s o required more t e r m s . Choosing x+ to correspond t o t == 4m: a l s o tended t o give l e s s sat isfactory resul ts . The energy dependence of the sur face is handled

more simply because we always fit data over a ra ther narrow range. The energy range is always chosen to be sufficiently narrow s o that quadrat ic interpolation i s sufficient and the surface i s taken t o b e of quadrat ic (or lower) degree in the laboratory momentum 9. We introduce the normalized variable

where @ is a weighted average beam momentum for a l l of the input data. The parameter qo is chosen t o match the amount of momentum-dependent curvature of the sur face required by the data; f o r @ scat ter ing q , is usually about 300 MeV/c. The sur face can now be represented in the f o r m

where f,(z) is a polynomial in z . In principle, the relation between z and r is energy dependent, but we neglect th i s s m a l l effect within the momentum range of a single amalgamation and use a fixed function z ( x ) appropriate to the cen t ra l momentum. In the following discussion we allow K t o be ei ther 0, 1, o r 2, although K = 2 is by f a r the most common c a s e encountered in practice. The higher- o rder coefficients of the polynomials f, a r e con- s t rained to be of comparable magnitude by the "truncation function" of Eq. (3.8). Thus, for K = 2, q0 is the momentum range over which t h e surface develops a la rge amount of angle-dependent curvature.

Threshold s ingular i t ies a r e not introduced into the interpolating sur face itself, but a r e handled in a manner s imilar to the Coulomb corrections. This 1s discussed in Sec. 111 and in Appendix B. The only threshold that has been t reated in detai l s o f a r is the qrz threshold a t 687 MeV/c, but it should a l s o be possible to include the wll threshold a t 1092 MeV/c.

111. DEFINITION OF THE X' FUNCTION

The interpolating sur face is f i t t o experimental data by minimizing the function

where x2 contains the constraints imposed by the


data and + is a "truncation function" (TF) ( ~ e f . 5) which imposes a smooth truncation on the number of parameters used in the fit. In this section we give detailed definitions of x2 and G.

The available world data of a particular type in a narrow momentum range typically consists of several "blocks" of data from different experiments covering various regions of scattering angle a t different momenta. We denote each data block by a Greek subscript, and denote the ith datum of block < as D,,. The inverse-square statistical e r r o r of D,! is called wet and the value of the interpolating surface f(z, ,, v,) a t datum €i is desig- nated by f,!. or the moment we ignore the finite momentum spread of the beam. See Eq. (4.2) for a more precise definition of re, . ] Each data block has an overall normalization e r r o r and a corresponding fitted scale factor. Fo r later convenience, we chose to construct x2 using the recipro- cal of the normalization scale factor A, rather than the scale factor itself. Thus the renormalized datum E i is D,,/x,. As long a s the normalization e r r o r i s small conpared to unity the e r r o r in X, i s the same a s the original normalization e r ro r . Each data block also has a measured beam momentum p, with a calibration e r r o r and a corresponding fitted beam momentum q,. Note that the definition of f,, given above uses the fitted momentum %. Nonanalytic effects (Coulomb-scattering and/or threshold effects) which a r e not allowed for in the interpolating surface a r e taken into account by calculating an explicit correction

term for each datum c m i which includes these effects. The calculation of the correction t e rms is described in Appendices A and B. They a r e to be subtracted from the renormalized input data before these a r e compared with the interpolating surface in the x2 function. Finally, a s discussed in Sec. 11, when dealing with polarization data we multiply by a factor s,,, equal to the corresponding cross- section interpolating surface divided by sinesl , and evaluated a t the fitted momentum q,. The x2 function constructed in this manner is

where s,, = 1 for cross-section data. The ma- t r ices w,', and wf, a r e the inverse covariance.ma- t r ices of the normalizations and beam momenta, respectively, with correlations taken into account by appropriate off-diagonal elements.

A simpler, approximate x2 function obtained from Eq. (3.2) is much more convenient for actual computations. Let

Then the sum over individual data points in X 2 can be rewritten a s

which has the form of an expansion in quantities of order,

I c,, l x (normalization error) , otii = (statistical e r ror ) , (3.5)

There is some arbitrariness in the correction terms in that we can include in them any analytic contributions we like (as long a s these contributions vary slowly enough to be well represented by the interpolating surface), in addition to the specifically nonanalytic effects that they a r e intended to represent. This freedom can be used to keep 1 c,, / small, and it is fairly easy to a r - range that the quantity

/ c,, / /(statistical error) , (3.6)

a r e important and much smaller elsewhere. Thus, for well normalized data @,, is small, and the second and third sums in Eq. (3.4) will be small compared to the first. Furthermore, the summand in the second sum fluctuates in sign so we expect a further reduction by a factor of order (totalnumber of data)lI2 compared to the f irst sum. The weakest point in this line of reasoning occurs for elastic differential-cross-section data a t very small angles, where the Coulomb correction t e rms can in principle become arbitrari ly large. However, it is difficult to make a measurement far into the Coulomb region without encountering backgrounds which also make the statistical e r r o r grow. In explicit checks we have found that even for the most forward available data @,, seldom exceeds

is typically of order unity where nonanalytic effects 0.3. Occasional data points for which O C i > 0.3

2786 R . L . K E L L Y A Y D R . E . C I J T K O S K Y 20 -

can be handled by artificially increasing the s tat is- t ical e r r o r to keep oEi small . Thus, in the r e - mainder of this paper we will use the x2 function

The advantage of this f o r m over Eq. (3 .2 ) is that the correct ion t e r m s can now simply be subtracted f r o m the data before fitting, as in Eq. (3.3) , and d o not enter explicitly into the k2 function itself.

The f r e e p a r a m e t e r s t o be determined by fitting a r e the coefficients of the polynomialsf,(z), the sca le p a r a m e t e r s A,, and the momenta q,. The polynomials f,(z) typically have appreciable coefficients up to o rder 8 o r higher, s o that there is no well-defined s h a r p cutoff point fo r the number of polynomial coefficients retained. We therefore u s e the T F GJ t o impose a smooth truncation on the higher powers of z . This is done by minimizing x2 + 9 , ra ther than x2 alone, where

with the line integral being taken around the unifocal e l l ipse onto which tine t- and u-channel cu t s a r e mapped by the Cutkosky-Deo mapping. The lengths of the semiaxes of the el l ipse a r e typically between two and s ix (depending on the momentum) s o the higher powers of z a r e magnified with r e - spect t o the lower powers on the boundary of the ellipse, and the addition of @ to x2 cu t s off these higher powers smoothly. The region in which the cutoff becomes effective is controlled by adjusting the constant il. The el l ipse shr inks with increasing energy s o that 9 naturally allows the number of effectively f r e e p a r a m e t e r s to increase with increasing energy even if 51 i s held fixed. The part icular weight function used in the integral i s chosen because Chebyshev polynomials a r e orthogonal with respect to this weight and th i s facili- t a tes computation of a s discussed in Sec. IV,

In our applications to np scat ter ing we found that a single value of lom7 (mb/sr)*' for 51 gave generally sat isfactory resu l t s fo r both c r o s s sect ions and polarizations throughout the resonance region. This value w a s a r r ived a t in the usual way, by decreasing h2 until x2 p e r degree of f reedom stop- ped improving. In a few c a s e s where the data w e r e part icular ly s p a r s e and the interpolating sur face was poorly constrained, we used values a s l a rge a s (mb/sr)-'. T h e r e a r e a l s o some data s e t s with pronounced s t ruc ture where 51 can

be decreased fur ther before k2 p e r degree of f ree - dom stops improving. Although we a r e aware that some of th i s s t ruc ture may turn out to be spurious, we have usually attempted to accomo- date it by choosing a conservatively smal l value of Sa, sometimes a s smal l a s lo-' ( m b / ~ r ) - ~ .

IV. CONSTRUCTION OF ORTHOGONAL POLYNOMIALS

A s a prel iminary t o the discussion of the ful l minimization of x2, we consider h e r e the problem of minimizing x2 with fixed values of the normalization and momeiltunl parameters . This is a l inear least-squares problem which can be solved analytically. We represen t the fitted surface a s

where the functions Tm(z, y) are polynomials in z and 31 and the M + 1 p a r a m e t e r s a,, a r e variable coefficients to h e determined by minimizing x2. It is useful fo r numerous aspec t s of the amalgamation procedure to at tack the problem of determining the coefficients a,, by f i r s t choosing the polynomials T , to diagonalize the a, sec tor d the second-derivative mat r ix of x2. All of the difficulties of the fixed h , and q, minimization problem a r e then contained in the construction of polynom- i a l s T , which satisfy a n appropriate orthogonality condition [ ~ q . (4.10) below], and once these polynomials a r e constructed the determination of the coefficients is trivial. Th is section i s devoted to the formulation of the orthogonality condition and t o the construction of the polynomials which satisfy it .

The t e r m s in X' which a r e bilinear in the coefficients a r e those containingfei2 and those coming f r o m @. The quantity f e i is the average of the fitted surface over the spectrum of the Eth beam. If the beam resolution function is B, (qj we have6

Since f(z, y) i s a t most quadrat ic in y, f c i can be evaluated completely in t e r m s of the average momentum (q),, which we take to be the fitted momentum q,, and the mean squared deviation ((9 - b)s)2)sbE2. Using the decomposition of Eq. (4. 1) we have

where


The TF has been designed to take advantage of the orthogonality property of Chebyshev polynomials on unifocal ellipses. Fo r any ellipse with focii a t z = k 1 we have

where R is the sum of the semiaxes of the ellipse. l To use this relation we represent the polynomials T, a s

where ( L + l ) ( K + 1 ) = 1 W + I. Comparing with Eq. (2.2) we find

Thus, the T F can be expressed a s

where the "truncation matrix" T i s

o,, = QN, . We now have a l l the notation necessary to write

down the coefficient sector of the second-derivative matrix of x2 and the orthogonality condition to be imposed on the T,. This is

When this condition i s satisfied the values of the coefficients which minimize x2 a r e easily found to be

=?U,i~EdEi/sEi.

The actual construction of the orthogonal polynomials can be carried out by a recursion method which is a generalization of the familiar recursion relation

for orthogonal polynomials Q, in a rea l variable x. The lowest-order polynomial i s chosen to be a constant:

Higher-order polynomials a r e generated by ex- pressing them a s a linear combination of a l l lower-order polynomials, plus a linearly independent "leading term" L,:

F o r the f i r s t K polynomials with m > 0 we define L, to introduce higher powers of y . F o r m > K we consecutively introduce higher powers of z in groups of K + 1 linearly independent terms. Spec- if ically,

w h e r e & = m - ( K + l ) . Forexample , i f K = 2 n e w powers of z and y a r e introduced in the following

2 2 2 2 order: 1, y, y , z, zy, zy , z2, z2y, z y , z3 , . . . From Eqs, (4.13) and (4. 14) we immediately find that

Note that unlike the familiar real-variable recursion relation, we cannot truncate the lower side of the sum in (4.13) a t some small value of m - n. This is because the T F is a scalar-product type of integral over complex values of z, and a s a result L, will in general not be orthogonal to any of the polynomials T, with n < m .

We must now solve for the coefficients C, by imposing the orthogonality relation (4.10) on the representation (4.13). Fo r the f i r s t sum in the orthogonality relation we need the representation corresponding to (4.13) for the "evaluated" polynomials T,,. This is

where

For the second sum we will need the analog of Eq. (4.13) for the coefficients Di",. Making a decomposition similar to Eq. (4. 6) for the L,(z, y):

2788 R . L . K E L L Y A N D R . E . C U T K O S K Y -- 20

we obtain

The coefficients A; vanish when (K t 1)1+ k > n?, and by using the definition (4 .17) of the L, and the relation

Z P , ( Z ) = + [ ~ , + ~ ( z ) + ~ , . ~ ( z ) ] , l a 1 (4 .20)

we can express a l l the A coefficients with (K + 1)2 + k 6 m in t e r m s of D coefficients corresponding to lower values of m . These relat ions a r e given below, where we use integers X artd K defined by the decomposition m = (K -t 1)A + K with A 0 and 0 9 K C K :

The procedure for solving the recursion relations is now straightforward. Suppose that we have determined a l l the quantities C,,, , T n S t , and D;, fo r n < m, n' n?. The next stage of the pro- c e s s is to substitute Eqs . (4 .16) and (4 .19) into (4 .10) and impose orthogonality between T m ( z , y ) and a l l T , ( z , y ) with n<m. This gives

which, with the help of (4 .17) and (4 .21) , expres-

s e s the rat ios C,,,/C,, in t e r m s of previously calculated quantities. Imposing the nornlalization condition expressed by Eq. (4 .10) with m = t z we obtain

which is a l so in t e r m s of previously calculated quantities. T,, and DI", can now be calculated f r o m Eqs. (4 .16) and (4 .19) , and one can proceed to index m + 1. The polynomials T,(z, y ) them- selves can ei ther be calculated recursively using the coefficients C, o r direct ly using the coefficients D;.

V. EFFECTS O F INDIVIDUAL COEFFICIENTS AND CONSTRAINTS

In Sec, IV we solved the problem of minimizing x2 when the normalization and momentum param- e t e r s a r e fixed. We now show that the orthogonal- polynomial fo rmal i sm developed there allows u s to make quite specific s tatements about the effect of individual coefficients and constraints on the resulting value of x2 at minimum. The p a r t of x2 which involves the coefficients directly i s

and we can use the resu l t s of Sec. IV [particularly Eqs. (4 .3 ) and (4 .8)-(4. l l ) ] to rewri te this a s

The effect of a n individual coefficient a, on the minimum value of x2, is now c lear . The minimum value is

and the resul t of omitting the ?nth tern1 f rom the s u m (4. 1) i s to increase x2,,,,, by ~2 without changing the values of the remaining coefficients.

To describe the effects of individual constraints we rewri te X" a s

where

20 - A M A L G A M A T I O N O F M E S O N - N U C L E O N S C A T T E R I Y G D A T A 2789

and consider the effect of omitting one of the constraining terms, i. e . , setting one of the we, or w,, to zero. The resulting decrease in x~, , , , , can then be calculated by reminimizing and will generally be greater than the corresponding value of X 2 s t or @ 2 z k at the original minlmum.

Suppose we omit the constraint corresponding to datum qj and denote quantities in which this datum is omitted by primes. Then

x ~ ~ ~ = x ~ ~ - x Z v

A,, = w,, x(~$)' . rn

It is clear from Eqs. ( 5 . 1 2 ) - ( 5 . 15) that the quantity A,, (or A l k ) i s a measure of the "pull" of constraint €? (or l k ) on the fitted parameters E, and on the value of x2,,,,,. The sense in which this 1s true can be made more precise by noting that the orthogonality relation implies that

Thus, it i s natural to identify A,, (or A,,) with the effective number of parameters used in fitting constraint Ei (or I k ) . Equation ( 5 . 1 7 ) can be rewritten in t e rms of the truncation matrix defined in Eq. (4 .9) a s

where

b m = a m - o q j T m q j ,

S m = 6mn - ~ q j ~ m r r j ~ n t l j - Minimizing x ' ~ , we obtain

We identify (M + 1) - Tr7 a s the number of param- e t e r s used in fitting the surface to the data, and T r r a s the number of parameters held fixed by the T F constraint. The quantities k i and A,, a r e found to be quite useful in practice for understand- ing how individual data points and data blocks in- fluence a particular fit and for identifying the posi-

where we have converted to matrix notation. The inverse of S i s

tion and range of the smooth cutoff imposed by the TF.

where VI. ITERATIVE MINIIMIZATION SCHEME

We now consider the problem of minimizing tile full X2 function [ E ~ s . (3 . I ) , (3.7), and (4.8)] with respect to al l of the f ree parameters, including X, and q, a s well a s the polynomial coefficients. We have found that this problem can be efficiently handled by an iterative procedure in which minimization at fixed values of X, and q, a s described above i s alternated with full minimization of a quadratic approximation to X 2 .

Suppose that we have found a set of orthogonal polynomials T , ( z , .v) satisfying ( 4 . 10) and a set of polynomial coefficients a: satisfying ( 4 . 1 1 ) for particular fixed values A: and q! of the normalization and momentum parameters. We now se t

It i s easily verified that the fitted value of the surface corresponding to datum 7j

i s changed by an amount

by the reminimization and that the decrease in the minimum value of x2, i s

x ~ ~ , ~ ~ ~ -X12, ,min = jrZqj/(1 - An,) t ( 5 . 1 3 )

where x2,, i s the value of x2,, a t the original mini- murn.

If we omit the TF constraint corresponding to w,, a similar calculation gives equations analo- gous to ( 5 . 12) and ( 5 . 1 3 ) :

;L =A:+ ah, , (6. 1)

and expand X2 to second order in 6am, 6&, and 64, . Define the parameter and derivative vectors

2790 R . L . K E L L Y A N D R . E . C U T K O S K Y

and second-derivative mat r ices

Using a ~ ~ / a a , ( ~ = 0 and

+a2x2/aa,aan (, =. 6,,

d=(:2), a=(::), ==(I1( Dlz) . (6. 5) DT2 D22

Minimization with respec t to 6 gives (6.3)

6 = ~ " d (6.6)

and we can take advantage of the special f o r m of d and D to find that

the second-order expansion of x2 now becomes F o r reference we give below explicit expressions

x2 = ~ ~ ~ - 2 d ~ 6 + €iTD6+O(b3), fo r the derivat ives that appear in d and D, where (6'4)

p r imes onfEi and s., denote differentiation with where respect to q,

1 ax2 if€ i ix = C r ( & f e i t - s e i d E i ) + C w:.(+ I ) ,

1 ax2 w, ix, f ..sf. - ~ = ~ ~ ( f ~ ~ - ~ ) ( ~ € f € ~ - ~ € ~ ~ ~ ~ ) + ~ ~ ~ € ~ ( ~ ~ - f i ~ ) , z? 3%

1 a2x2 -__ =C . 2 aa,ax, sGi T""zb,fei - s,id,i),

1 a2x2 2sLiTmi) - s E i A i tki - U) s ' . T . + A ~ / ; ~ T , ~ ] ,

SE i S, i

I a2x2 W E if, i 2

2 a h a k - \ v (FT~) +w:.?

1 a2x2

2fi'i~:i +fEi ~ e " i + 2fii(si i)l) + &jfii - M Y ] } + W:q

S~ i Se i Se i

F o r most purposes t e r m s proportional to A, f,, - seidei a r e sufficiently smal l to be safely neglected in the above expressions f o r the second derivatives. This has no effect on the final minimum, which occurs a t d2 = 0, and does not degrade the convergence ra te of the i terat ive procedure signi- ficantly. In part icular , i t is never necessary to compute the second derivat ives xi and s:'~ because they a r e contained in a t e r m proportional to \ f,, - seidei.

Our basic i terat ion scheme is to find a s e t of polynomials and coefficients a t fixed values of A, and q,, then shift X, and q, according to Eq. (6.7), find new polynomials and coefficients, e tc . However, it is well known that the type of multi- dimensional Newton-Raphson approximation which led to Eq. (6. 7) can have se r ious instability problems, and we mus t modify th i s scheme somewhat to avoid these difficulties. At the outset of a minimization we s t a r t f r o m initial values of k = 1 and

I

q, = p , and hold q, fixed, iterating with the a, and A, p a r a m e t e r s only until a s table solution is found. The q, var iables a r e then released and the full i terat ive procedure is followed. The initial minimization a t fixed q, is necessary because the momentum derivatives of x2 a r e poorly known during the initial i terat ive s teps, and large, unstable, highly correlated shif ts of the normalization and momentum variables away f rom their input values can occur i f the full i terat ive scheme is applied a t the outset. After each calculation of 6 i t is useful to check that the adjusted p a r a m e t e r s actually give a decrease in x2. This is done by evaluating x2 approximately to fourth o rder in 6 and comparing the resul t with the previous value. If it is found that x2 has actually increased, we replace 6 by Pb where the sca le factor P is chosen to minimize x2. The fourth-order evaluation of x2 resu l t s in a cubic equation for P which can be solved analytically. Sometimes x2 will appear to decrease when


6 is chosen, but because the approximate fourth- order evaluation i s insufficiently accurate, it will be found that X' has actually increased when an exact evaluation i s made with new polynomials and coefficients in the next iterative step. In this case we multiply 6 by a factor of 0.3 and try again. Failures requiring the scale factor P or the factor of 0.3 a r e often associated with unstable behavior of the interpolating surface rather than A, and q,, because the latter a r e directly constrained by wr, and XI:,. We can therefore often correct this behavior and move closer to the minimum by temporarily holding the interpolating surface fixed a s we shift X, and q,, i. e . , by replacing Eqs. (6.8) and (6.7) with 6 , = 0 and 6 , This replacement is also useful in the initial iteration when A, f i r s t departs from unity and in the f i r s t step in which q , i s allowed to depart from f i e . With these safeguards against instability the iterative procedure usually converges in somewhat less than 40 full steps, i. e . , somewhat less than 40 reevaluations of the polynomials and their coefficients.

VII. ERROR ADJUSTMENT

The x2 confidence levels of f i ts obtained a s described in the previous sections a r e often very small. This is due to unknown experimental biases and e r r o r s in some of the data, and these effects will propagate into the amalgamated data unless they a r e explicitly removed. The nature of the problem can be clearly seen in histograms of the data point and data-block confidence-level distributions calculated on the assumption of Gaussian e r ro r s . Examples a r e shown in Fig. 1 of Ref. 2. Instead of being flat, the distributions a r e peaked at low confidence levels. These peaks a r e nearly always present though their heights and widths vary with momentum, The data block confidence level distribution i s usually even more sharply peaked than that of the data points, indicating a fairly even scattering of bad data among the different blocks.

We deal with this problem by doing the x2 minimization in two passes. After the f i r s t pass e r r o r ba r s of data in the low-confidence-level peak a r e stretched a s described below, and the data is then refit. After the second fit the stretching i s done again, but a t this stage the low-confidence-level peak has essentially disappeared so the effect i s minor. The stretching algorithm is defined in te rms of

where N, i s the number of data points (including normalizations and momenta) and

is the effective number of degrees of freedom. (N, i s the number of normalization and momentum parameters contributing to x?) The quantities jiZei and similarly defined quantities for the normalizations and momenta a r e expected to be dis- tributed approximately in a x2 distribution for one degree of freedom if the e r r o r s a r e truly Gaus- sian. The e r r o r eEi of datum Ei 1s stretched according to the algorithm

e, unchanged if ', i < 6; ,

and a similar procedure is applied to the normalization and momentum convariance matrices. Thus, stretching begins when g Z e i exceeds 6 2 , and becomes extreme when ji$i exceeds 6,*; 60 and 6, a r e chosen to lie near the edge and the middle of the low-confidence-level peak, respectively. Typical values a r e 6, = 2 and 6,- 3. About 10% of the e r r o r s a r e usually adjusted by this algorithm, and only about half of these a r e stretched by a factor of more than 1.5.

Provision i s also made for simultaneous stretching of a l l the e r r o r ba r s in data blocks that remain poorly fit after the above procedure i s carried out, but this i s seldom necessary and the overall stretching factor i s seldom larger than about 1.2.

VIII. INTERPOLATION, ERROR PROPAGATION, AND hhl ALGAMATION

The covariance matrix of the shifted data is obtained by calculating their response to fluctuations in the input data. These fluctuations a r e represented in te rms of a statistical model of input data in which the data actually used a r e considered to be a single sample point in a space of Gaussian random variables whose mean values a r e the true physical values of the measured quantities. The x2 function corresponding to a general sample point in this space in the same approximation a s that of Eq. (3.7) is

The general sample point is here represented by the quantities

2792 R . L . K E L L Y A N D R . E . C U T K O S K Y - 20

which have the part icular values d, ,, 1, p,, and 0, respectively, in the actual fit. h he D,, in Eq, (8.2) should not be confused with the Dei in Eqs. (3 .2 ) and (3.3). ] The quantities with supersc r ip t 0 in (8.2) represen t the mean values of the random variables which a r e assumed to be equal to the t r u e physical values of the relevant quantities. The inverse covariance mat r ices of D,,, A,, P,, and A, a r e taken to be TiEi (diagonal), $in, GI:,, and T ~ , respectively, where the mat r ices k, G', and k2 a r e the original mat r ices w, wl, and w2 a s modified by e r r o r - b a r s t retching and 7 is the truncation mat r ix defined in Eqs. (4.9). Inclusion of the quantities A, with covariance mat r ix r-' in the space of random var iab les allows for fluctuations of the appropriate sca le in the n pyzovi values of the coefficients.

The error-propagat ion calculation does not take into account the effect of fluctuations in the input cross-sect ion data on the shifted polarization data through the factor s, , . We neglect this effect because the cross-sect ion data a r e generally con- siderably more prec i se than the polarization data. T e s t s have been made to check that the effect is in fact negligible. We a l s o neglect fluctuations in the adjusted inverse covariance mat r ices k, iil', and 6' and in thc truncation matr ix T .

Our goal is now to calculate the covariance ma- t r ix of the fitted p a r a m e t e r s that i s implied by th i s prescr ipt ion for the s tat is t ical nature of the input data. We denote the variable p a r a m e t e r s a s

use polynomials which satisfy a n orthogonality condition appropriate to the c a s e of mean-value

(8' 2, input data:

where

0 2 GEi =$, E(A:)2/(~Ei)

and T L i is the quantity in Eq. (4.4) evaluated a t 0

ye =ye = (4: - 4)/q0. The conditions f o r a minimum of x2, satisfied by a t , A:, and q! & r e

n ~ = ~ G e > h l D P i m e t , +C mn AO n )

e t se0i n

Using these definitions and minimum conditions we now expand x', to second order about i t s mean value minimum, i. e . , to second order in the quantities 6D,,, 6 4 , 6P,, EA,, 6am, EA,, and &iq,. T o simplify the notation we a l s o define the following quantities (where Y and s Lake t.he values 1 and 2):

where the quantities with supersc r ip t z e r o a r e the %. ;LOO .

6Km - 7,6A, +ELF T,,,i 6DEi values taken a t the minimum of X2, fo r the c a s e n e i S e i

of mean value input data. In the following we a l s o The resul t of the second-order expansion is then


All t e rms in X contain factors proportional to the deviation of the mean value input data from the corresponding fitted quantities. Assuming that the parametrization has been appropriately chosen, we expect these factors to be f i r s t order in magnitude and to fluctuate in sign. This results in a suppression of the linear t e rms in Z relative to the second-order part of X2, which i s positive definite, and a further suppression of the quadratic t e rms in H which will actually be of third-order magnitude. On this basis we neglect Z in the following calculations.

Our next step i s to obtain the fitted values of 6a,, €4, and 69, in t e rms of 6DE ,, &A6, 6P,, and 6A,. Minimizing the expression (8.7) for x2, (with Z neglected) gives the following relations:

We can now use relations ( 8 , l l ) to express the fluctuations in the shifted data in t e rms of the fluctuations in the input data. A shifted datum D,,, which is originally the ith data point of data block E and i s shifted to central bin b a t the central momentum q , i s defined to be

where sb and c, a r e defined similarly to s,, and c, i, rb, is unity for cross-section data and i s (sin'J,)/(sin'J,,) for polarization data, and

where the barred quantities indicate values a t Thus, renormalized cross-section data a r e shifted minimum. Converting to matrix notation these parallel to the fitted surface while the deviations equations become of renormalized polarization data from the fitted

6Z= 6 ~ - fibit polarization a r e modulated by sine. To exhibit the fluctuating part of the shifted datum we rewrite

S26z=6G-/3T6Z. (8.10) it as

The solution i s

6~ = ~ " 6 k ,

6E= ~ " 6 6 , where

6k=6K-~Q"bG,

66 = 6G - pT6K, f bO = z a i T m b .

m

A = 1 - pa-'PT, (8.12)

Expanding D,,, - D:,, we find that the fluctuating part of Db, , is, to f i r s t order, B = ~ - P * P .

D O . For later use we note the relation F2,t 6-

~ . ~ = y ( 6 ~ , , - ~ 6& - qp.)

A"= 1 + p ~ - ' p ~ (8.13) (8.19)

which may be obtained a s follows: +T (2 - *) 6zm. A-'=A-'(A +pa-lpT) = 1 + ~ - l ( p a - ~ ~ ) ~ - l p ~

Following the same argument that we used to neg- = 1 +A"(AP)B"P~ = 1 + P B - ' ~ ~ . (8.14) lect E, we expect that the approximation

R . L . K E L L Y A N D R . E . C U T K O S K Y

is accura te to f i r s t order . We can therefore re - place D;,/A: in (8 .19 ) and obtain

where

" . Equations (8 .20) and (8 .11) express A,,, a s a

l inear function of the fluctuations in the input data (6Z,6zs,) =-(PB"),,, . which have the prescr ibed covariances

The las t th ree en t r ies of Eqs. (8 .26) make up the

( 8 , 2 2 ) covariance matr ix of the fitted parameters . We denote this matr ix a s a whole a s

(6Am6An) = 72. u =

Using these we can calculate the covariance of two shifted data:

The following a r e useful intermediate s teps in the calculation:

( 6 ~ , i 6 ~ ~ ) = ~ " € ~ , O , i / s , O , ,

( OG,) = 6,,FS, ,/s,0, , ( 6K,6Kn) = g m n , ( 8 .24 )

(6G, 6G,) =a, ,, , (6Km6G,,) =P,,, ,

U is proportional to the inverse second-derivative matr ix of X2, with t e r m s contained in Z neglected. T o make an explicit comparison with the second derivat ives displayed in Eqs. (6.9), we introduce a supersc r ip t 0 on the mat r ices Dl,, DZ2 , an,d D which denotes ( 1 ) evaluation a t the mean value minimum, (2) neglect of t e r m s proportional t o f , o i - s : , ~ : ~ , and (3) replacement of a l l input weight mat r ices by their s t retched ver - sions, i. e . , w,, --&,,, etc. Then it is easily verified that

( 6 k m 6kn) =A,,,,, , (8 .25) The covariance mat r ix of the shifted data can

( 6 ~ y s 6 ~ m ) = B e , s v , now be obtained direct ly f r o m Eqs. (8 .20) and (6km6C,,J = - ( P s ~ - ~ B ) , , , , (8 .26 ) . The resul t is

Although the fo rmal derivation of (8 .29 ) has been quantities, i t is of course impossible t o use these facilitated by expanding about a point correspond- values in a numerical evaluation of V. In prac- - ing to the t rue physical values of the relevant tice, we make the replacements aL4a,, k:-h,,


q:-&, T;, -T,,,, where the barred quantities a r e the values a t minimum for the particular fit under consideration. These replacements a re no less accurate than the various other f irst-order approximations involved in the derivation of (8.29).

The f i r s t te rm of V i s primarily due to the er- r o r s of the original data while the remaining terms represent e r r o r s of interpolation, renormalization, and momentum shifting. These latter e r r o r s a r e generally somewhat smaller than, but comparable to, those of the original data.

The final step of our procedure i s the construction of the amalgamated data and their covariance matrix. The amalgamated datum in bin b is a linear combination of the shifted data in that bin

with normalized coefficients

The covariance matrix of the amalgamated data is

We choose the coefficients Ybei to minimize the variance of D, subject to the normalization constraint (8.31), i. e. , we require

where I* i s a Lagrange multiplier. This yields

where wb is the inverse of the submatrix of V per- taining to bin b.

IX. CORRELATION VECTORS

The amalgamated data D, obtained in Sec. VIII a r e intended to be useful a s precise input data for fitting programs, such a s partial-wave-analysis programs. They a r e more complicated than " raw" experimental data, however, because they have highly correlated e r r o r s a s expressed by their covariance matrix C,, . The correlations ar i se through the mutual dependence of the amalgamated data on the interpolating surface and on the systematic e r r o r s of the original input data. Most of the e r r o r correlation corresponds to collective fluctuations with rather smooth angular variation, although more complicated correlations also occur. Although the matrix C,, contains complete information on the e r r o r correlations, including

their collective aspects, this information i s not expressed in a particularly transparent way. In this section we show how to extract from C,, a simple, quantitative description of the collective fluctuations. One result of this will be the ability to perform a fit to the amalgamated data with a x2 function which involves only single sums over the data points, rather than a double sum over a l l the matrix elements of C, , . A more important result will be the ability to extract from a particular fit, fitted amplitudes for the collective fluctuations. These amplitudes can be used to perform collective adjustments to the data, in a direct generalization of the common procedure of re- normalizing data using a fitted scale factor.

Before embarking on a general discussion of fluctuation-affected data, we will consider a particular simple example by way of introduction. The example i s a se t of data f ib with independent "statistical" e r r o r s i e , for each data point and an overall normalization e r r o r of +n. More pre- cisely, we represent t ) , by a statistical model in which

where the random variables X and db have means 1 and z b , respectively, and have the following covariance matrix:

where

6 h a X - 1 ,

6db = db - zb . Expanding Bb to f i r s t order in 6X and 6db we obtain

The covariance matrix of the normalization-error- affected data i s

ebc = ( 6 Z b 6 4 ) = 6,,eb2 + n2zbzc . (9.5)

Now suppose that we wish to approximate the covariance matrix Cb, of some actual amalgamated data Db by a parametrization of the type obtained in Eq. (9.5). We need to choose values for d,, e,, and n. There i s no unique way to do this, but after testing several approaches we have settled on the following method. Fo r zb we use the fitted value corresponding to bin b,

The e r r o r eb is chosen by requiring that the dia-

2796 R . L . K E L L Y A N D R . E . C U T K O S K Y 20 -

gonal e lements of C and C be equal. Th is gives eb in t e r m s of n:

eb2 = Cbb -- n2Zb2 . (9.7)

Finally, to determine n itself we define the resi- dual correlat ion matr ix

t r y to devise a way to fit the "normalized" variables db and the normalization variable X simul- taneously, even though we only have data on the combination Adb. The obvious procedure i s to represent the normalized variables by

Bb/x=db - db6x (9.16)

and to introduce an auxiliary normalization param- e t e r to represen t A. F o r l a te r convenience we will actually use a normalization parameter 5 which represen ts 6X/n. Since the normalized variables have independent e r r o r s &eb and 6X/n has unit e r r o r we a r e led to guess that the following x 2 function is appropriate:

and the s u m of s q u a r e s of i t s off-diagonal e lements

n is chosen to minimize r, giving

Note that nothing in Eqs. (9.7) and (9.10) guaran- t ees that eB2 > 0 and n2 > 0. This depends on whe- ther the amalgamated data really do have the s ta- t is t ical character of normalization-error-affected data s o that the parametr izat ion embodied in C is adequate to provide a good approximation t o C .

Once we have determined a n approximate e r r o r mat r ix we may consider using it in a fit to the amalgamated data. We would then do the f i t by minimizing the approximate x2 function

where

Minimizing \jr2 with respect to 5 gives

This shows two things. F i r s t , our guessed x2 function 5' is indeed cor rec t because once 5 is eliminated, $2,,, is identical in f o r m to i2 which was explicitly constructed f rom the cor rec t e r r o r mat r ix fo r the model data. Second, when the fit is compfeted and Fb has been determined by minimizing q2,, , , we a r e able to construct explicitly the fitted value of A, which i s 1 + n 2 . We can then return to the original data and construct re - normalized data - - - -

(5 I ) ) mr 5 Db/( l + nZ) = D, - nd,Z (9.20)

where Fb is the value of the fitting function a t bin b. The inverse.of C is

where

So z2 reduces to in which the effect of normalization fluctuation h a s been suppressed. In a n application to_actual amalgamated data with C approximated by C we inter- p re t where

a s renormalized data. The renormalization procedure is especially useful in a highly constrained fit with ji2 entering a s one contribution to a total X" function which includes contributions f rom many other data besides D,. T o the extent that the normalization of Fb is overconstrained by the other data, the fitted value of in such a case may provide a particularly accurate measure of the normalization of D,.

The above formalism is not only illustrative,

Thus ? ' h a s the property of involving only single s u m s over the data points.

It is of interest to consider the ra ther peculiar looking x2 function of Eq. (9.14) fur ther and to interpret it in t e r m s of our s tat is t ical model of normalization-error-affected data. Returning to the model, suppose we have a s e t of data f i b and the associated covariance matr ix C, which we want to fit with s o m e fitting function F,. Let u s


but actually provides a useful f i r s t approximation to the description of the correlated e r r o r struc- t,ure of amalgamated data. In most cases a significant amount of correlation does a r i s e from uncertainties in overall normalization. However, this parametrization employs only one f ree parameter n2 to describe a l l the off-diagonal elements of C, and it is not sufficiently precise for general use. We next develop a more flexible parametrization which allows the detailed structure of the specific fluctuations affecting a particular data se t to be extracted from its covariance matrix.

Consider a statistical model of fluctuation-affected data f i b specified in t e rms of random variables $ and 5, a s

where

The "correlation vectors" K" describe the profiles of 1V statistically independent fluctuations whose amplitudes a r e given by the random variables 5,. We will require that the correlation vectors be linearly independent, and will impose this condition by requiring them to satisfy the following orthogonality relation:

The particular form of the orthogonality relation is chosen for later convenience. The choice involves no significant loss of generality because, in practice, the orthogonality conditions always represent a small number of relations between a large number of free parameters. The covariance matrix of the fluctuation-affected data i s

If we now wish to approximate an actual covariance matrix C using this parametrization we must choose values for a large number of free param- e t e r s (the diagonal e r r o r s eb and the elements of N orthogonal correlation vectors). We have found that a practical way to do this is to require equality of the diagonal elements of C and C, and to minimize F, defined a s in Eq. (9.91, by varying the correlation vectors one a t a time. This constrained minimization procedure i s described in Ap-

pendix C. An indication of the accuracy of the approximation of C by i s given by the final value of 2r/No(No - I), where N o is the number of occu- pied bins. This i s the mean square value of bp,, and i t should be small compared to unity. How- ever, i t is important to realize that although we have found minimization of I? to be a particularly stable and simple w?y to fit C with the parametrization embodied in C, the final value of r itself i s a rather arbitrary measure of the accuracy of this approximation. Other quantitative measures a r e given below, and these a r e also used in asses- sing the accuracy of approximation. In our applications we have obtained adequate accuracy with one or two correlation vectors. The best accuracy is usually attained by choosing M = 2, but the addition of a secoild vector does not always lead to significant improvement so we sometimes choose N = 1. Occasionally it even happens that no improvement i s possible over the simple normalization-error parametrization, so we use N -; 1 with K' given by Eq. ( ~ 1 6 ) .

The approximate X 2 function for a fit to the ainal- gamated data with the approximated e r r o r matrix i s of the same form a s Eq. (9. ll), but with

where

So i2 reduces to

where

The particularly simple form of 2" i s a result of our choice of the orthogonality relation for the correlation vectors. We interpret %2 by returning to the statistical model of Eqs. (9.22) and (9.23). To fit the random variables d, and 5, simultane- ously, using only data on the combination given by B5, we introduce auxiliary parameters 5, to represent the fluctuation amplitudes t, and construct the quantities

to represent the "unfluctuated" variables d,. We a r e thus led to the x 2 function

2798 R. L . K E L L Y A N D R. 6 . C I J T K O S M Y 20 -

where

Minimization with respec t to En gives

A s with Eys. (9.19) these resu l t s vindicate our choice of q2 and provide u s with explicit formulas fo r the fitted values of the fluctuation amplitudes. In an application to a~na lgamated data with C approximated by C we interpret

a s adjusted data i n which the effects of fluctuations have been suppressed. The adjusted data will be part icular ly accura te when F,, and hence m z,, a r e overconstrained by other data besides Db. The col!ective fluctuation amplitudes Z, represen t pr imari ly the effects of the normalization and momentum calibration uncertaint ies of the original data. The fitted normalizations and momenta of different data s e t s a r e correlated, but they a r e not completely determined. Uncertainties in the interpolating sur face a l s o contribute to the collective fluctuations.

We close this section by considering measures of the accuracy of approximation of C by other than the r m s value of 6pbc. Consider the x2 function for a s e t of amalgamated data Db with respec t to their own mean values IS,:

Now since

((Db - f5,)(oc - B,)) = CbC

by definition, we have

(x ') = T~c"C = N o .

Using the relation (valid f o r Gaussian s tat is t ics)

we find s imilar ly that

Thus, we reproduce the well-known resu l t s that the mean and central var iance of x2 a r e No and 2No, respectively. To judge the accuracy of (? fo r pract ical applications we calculate the mean and central variance of

f o r comparison with the ideal values of No and 2No. Using Eqs. (9.36) and (9.38) one finds that

It is important to monitor these two quantities in practice, because it is quite possible to achieve a s m a l l value of 2 r / ~ ' , ( N ~ - 1) accompanied by bad values of the mean and central variance. This can usually be cor rec ted by stopping the minimization of r somewhat shor t of a n absolute minimum. Typical values of these quantities that we obtain in applications with N = 2 a r e l e s s than 0. 1 for-the r m s value of 6p,,, o r d e r unity fo r the bias T ~ C - ' C - No, and 2, ONo t o 2. 4No f o r the central variance.

X. CONCLUSIONS

We have presented techniques fo r amalgamating data f r o m two-body meson-nucleon scat ter ing experiments . The techniques take account of s ta t is- tical experimental e r r o r s , known systematic experimental e r r o r s , unknown experimental b iases which appear a s inconsistencies between overlapping o r neighboring data se t s , and e r r o r s of interpolation. The resulting amalgamated data a r e highly correlated. We a r e able to paramet r ize the correlat ions in terrns of collective fluctuations using one o r two correlat ion vectors .

The techniques described herein have been applied to a collection of a l l existing ?p elast ic and a"p charge-exchange differential-cross-section and polarization data between 349 and 2055 Mev/c. We have produced amalgamated data for these react ions a t 35 momenta f r o m 429 to 1995 Mev/c, in angular bins of 3 " spacing. Some of these re - su l t s a r e described in Ref. 2. We believe that these data will be part icular ly useful f o r partial- wave analysis and other resonance-region phenomenology. A description of our own partial-wave analysis using these data is given in Ref. 1. The data will shortly be made available to interested users . The computer p rogram developed f o r this project runs on the LBL CDC-7600, and could be modified f o r use with data on various two-body reactions. The program will a l s o be supplied to interested u s e r s . Inquir ies should be directed to one of the authors.

A M A L G A M A T I O N O F M E S O N - N U C L E O N S C A T T E R I N G D A T A 2799

ACKNOWLEDGMENTS normalize the rea l part of e-2iu3fN to enforce

Our applications of amalgamation techniques to vp scattering data were made in a fruitful col- laboration with David Hodgkinson and the late J ames Sandusky. We a r e grateful to the many experimentalists who supplied data with well- documented discussions of systematic e r r o r s for use in this project. In the formative stages of this analysis we benefited from numerous discussions with experimentalists concerning the basic con- cepts of what we have come to call"ama1gama- tion. " We a r e particularly indebted to Owen Chamberlain and Herbert Steiner in this regard. One of u s (RLK) is pleased to acknowledge the hospitality of the Max-Planck-Institut fiir Physik und Astrophysik, Miinchen during a visit when part of this paper was written. This work was supported by the Division of High Energy Physics of the U. S. Department of Energy under Contracts Nos. W-7405-ENG-48 and EY-76-02-3066, and by the Elementary Particle Physics Program of the U.S. National Science Foundation under Grant No. PHY76-21097.

APPENDIX A: ELECTROMAGNETIC CORRECTIONS

The electromagnetic (em) part of the correction c,{ (defined in Sec. 111) for an elastic differential- cross-section datum consists of pure Coulomb scattering and Coulomb-nuclear interference contributions. That is, if we write the scattering amplitudes a s sums of em and nuclear par t s

f =fern + f ~ , (All &?=gem + g ~ ,

the corresponding correction t e rm i s

Ifem12 + Igem12 + 2Re(femfN*) +2Re(g,mgN*). (A21

We use the amplitudes of Tromborg e t al.' for f,, and gem, calculated in the manner described in Sec. I1 C of the following paper. '

General expressions for the nuclear amplitudes a r e given in Eqs. (2. 32) of the following paper, but in applications to 7ip data amalgamation we have modified these somewhat. F i rs t , rather than multiplying each partial wave by the appropriate %

Coulomb phase factor, we multiply by an average overall phase factor eZt03, where u3 is the non- relativistic Coulomb phase shift for F waves. We have verified that a:, i s a reasonable average of C J s (see Ref. 1) for D, F, G , and H waves in the relevant energy range. S and P waves a r e omitted from the average because the Coulomb phase shift is small compared to typical statistical e r r o r s in the phases of these waves. Second, because we a r e most interested in accurately reproducing the single-photon-exchange pole contribution, we re-

agreement with forward dispersion-relation deter- minations of the rea l part at 0". We also ignore the energy dependence of the empirical partial waves used in c ~ n s t r u c t i n g f ~ and gN and use fixed values obtained by linear interpolation to the central momentum q,. This i s necessary because of the er ra t ic energy dependence of empirical partial-wave amplitudes. The e r r o r introduced in the em corrections a t momenta different from q, by this choice of amplitudes i s a smooth function of momentum which i s compensated by the interpolating surface. The parametrization of

fN and gN i s thus

where Ref, (0 ") i s the dispersion-relation calculation for the forward rea l part evaluated a t P,, and f,, g, a r e the standard partial-wave sums for fN,

gN evaluated a t q, without Coulomb phase factors. The identification of Re fD(Oo) with the "Coulomb- phase-free" version off, i s discussed in Sec. IID of the following paper. In our applications to 7ip scattering we have used the dispersion-relation predictions of Engelmann and end rick' and the partial-wave amplitudes of Ayed.

em corrections and the threshold corrections discussed in Appendix B have been applied to differential-cross-section data only. These corrections a r e unimportant for polarization data because of their lower statistical precision. Since the corrections a r e made before fitting they must be made a t the measured momenta p, rather than the fitted momenta q,. This i s acceptable for em corrections which show little energy variation over ranges corresponding to typical momentum calibration e r ro r s . This point is more delicate for threshold corrections, and i s discussed further in Appendix B.

With em effects removed in the manner described, the interpolating surface a t 0" represents the forward nuclear differential c ross section. We have included in our cross-section data s e t s (both elastic and charge-exchange) predfctions (with e r ro r s ) for this quantity obtained by using the total c ros s sections and the optical theorem, along with the dispersion-theory predictions for the rea l par t s of the amplitudes a s calculated by Engelmann and Hendrick. These predictions help to determine the shape of the interpolating surface near the forward direction, and also help to determine the normalization parameters of the other data sets . However, we do not include these 0" predictions in forming the amalgamated data by the method described in Sec. VIII.

2600 R . L . K E L L Y A N D R . E . C U T K O S I C Y

APPENDIX R: CORRECTIONS FOR THRESHOLD STRUCTUKE

We begln with a review of the effect of an inelas- t ic threshold on a communicating open channel. Consider the S matr ix fo r N two-body channels with definite values of a l l conserved quantum numbers J, P, etc. We examine the behavior of S when the f i r s t N - 1 channels a.re open, and the Nth is a n S-wave channel near threshold. Let q be the c. m. niomentum in the Nth channel s o that Snra " J? (bf < N) and S,v, .- 1 q near q = 0. The general f o r m of S to lowest nontrivial order in q is

P ,

where So and S, a r e symmetr ic (N - I ) x (A' - 1) matr ices , I3 is a n (N- 1)-component vector, and A is a sca la r . We require that S be unitary above threshold (q 0) and that S,, + Slq be unitary below threshold (y = i 1 q 1). After some algebra one finds that this leads to the following relat ions among the p a r a m e t e r s of S:

In part icular , this implies that fo r M < N,

Eliminating B , we obtain

which i s the basic relation between a n opening production channel and the corresponding elast ic channel. The t e r m :(s,,)~ produces a square-root cusp in SjWM because q a (s - st,) lI2. F o r N = 2 the S matrix reduces to

Equation ( ~ 5 ) displays the s imple relation between the elast ic phase shift a t threshold and the phase of the production amplitude which is charac te r i s - t i c of the two-channel case.

We now specialize to ?r-p scatteririg near the threshold a t 1488 MeV (687 MeV/c), and let S be the IJP = +$- S matr ix. It is assumed that the above description in t e r m s of N two-body channels is adequate f o r our purposes, although multibody channels account fo r nearly a l l of the inelasticity a t th i s threshold. The production c r o s s section

near threshold is

where q, and q, a r e the c. m. momenta in the n-p and Vn channels, respectiveiy, and

T h i s determines 1 B,, 1 in t e r m s of the slope of the production c r o s s section a t threshold. The measured value of a( is 2 1 . 2 1 1 . 8 ~ b / ( M e v / c ) . l1

Using ( ~ 3 ) and (B7) we find that the elast ic T- mat r ix element near threshold is

where ff is the phase of B,, . Bhandari and chaoi2 have determined ff to be 4 1 ° i 6" by fitting the backward v-p elast ic differential-cross-section data of Debenhani ef a1. lS The ?rN Sll amplitude is fair ly elast ic near the 7hz threshold (V -0. 91, s o it is not surpris ing that a is consistent with the threshold value of the elast ic phase shift 6 -39".

Consider now the problem of amalgamating n-p elast ic o r charge-exchange differential-cross- section data in a range of laboratory momenta pz >PI, >PI , which includes 687 MeV/c. We will construct correct ion te rms , contributions to cei, which represent the interference between the cusp t e r m in Eq. (B8) and the regular par t of the f amplitude a t threshold. AS discussed in Sec. 111, we a r e f r e e to modify Eq. (B8) by adding analytic t e r m s which can be fit by the interpolating s u r - face, e. g . , we can add a quadrat ic polynomial in p,,. This freedom can be used to control the magnitude of the correct ion t e r m s away f r o m the immediate vicinity of threshold. Thus, we para - metr ize T a s

where T , contains the cusp contribution, TQ is a quadrat ic approximation to T,, and To is a con- s tant to be determined. It is convenient to introduce a new variable

which v a r i e s f r o m - 1 to 1 a s p , v a r i e s f rom p1 to p2. We choose T, to be proportional to (x - x,)lt2 and to be normalized to a g r e e with the square-root singularity in Eq. (B8). This gives

where the square root is positive fo r x,, < x < 1 and positive imaginary for - 1 < x < xth. The con- s tant D is


F o r TQ we construct a quadratic function of x which approximates (x - xth)ll%n the range 1 % I < 1:

where P, i s the lth Legendre polynomial and

Note that h, is in general complex. TQ is now given by

TQ = i ~ e ~ ~ ~ h(x) . (B15)

The constant T o is determined by linear interpolation of the cusp-f ree quantity 2' - (T, - TQ) to threshold, using the empirical Sll partial-wave amplitudes of Ref. 10 for T.

Correction terms a r e now constructed from the f amplitude:

where f o is the partial-wave sum for f with the Sll amplitude replaced by the quantity T o constructed above, and with the other partial waves evaluated a t thresho1.d by simple linear interpolation. K i s the isospin factor for the reaction under consideration; $ for n-p --a-p and - a / 3 for a'p ' n O n . The correction term i s

This must be evaluated a t ( c o s ~ ) , ~ and integrated over the momentum spectrum of data block € before being added to c, i . The momentum averaging is particularly important when p, i s close to 687 ~ e ~ / c . If the spectral shape is a polynomral in p,, and we neglect the weak momentum dependence of fO/4n, the integral can be evaluated exactly by Gaussian integration in the variable / (x - x,,) over appropriate subranges. For most n-p data we have used a rectangular momentum spectrum for this integration. Fo r the data of Debenhanl et a1. 13'14 we use a triangular shape appropriate to the conditions of that experiment.

We finally consider the problems associated with making the above corrections a t the measured momenta p, rather than the fitted momenta q,. This is not really justified for threshold effects, and a better procedure would be to adjust the correction terms iteratively a s the % a r e being determined s o that they end up being evaluated a t

the iitted momenta. In our present applications, however, we have followed the simpler procedure for two reasons. F i rs t , much of the existing r-p data near 687 MeV/c a r e taken a t too widely spaced momenta and/or a r e insufficiently precise for cusp effects to be clearly present. The com- plication of evaluating cei a t q, rather thar, p, i s unwarranted for these data. Second, the data of Debenham et al., which does display prominent cusp effects, has a momentum spectrum with a 1.2% full width a t half maximum and a momentum calibration e r r o r of *O. 1%. Thus, the difference betweenp, and q, i s completely washed out by the momentum bite integration. Similar, though less extreme, mismatches between momentum bite and calibration e r r o r a r e present in the other existing high-precision data se ts near the Vn threshold.

APPENDIX C: DETERMINATION OF THE CORRELATION VECTORS

We consider the problem posed in Sec. IX of approximating a given covariance matrix Cbc by the parametrization given in Eq. (9.25) subject to the orthogonality constraint of Eq. (9.24). Our approach will be to require equality of the diagonal elements of C and 5: and to minimize F, defined a s in Eq. (9.91, by varying the correlation vectors one a t a time. To describe the procedure we introduce the notation

db = c b b l e b 2 ,

u ; = g / G b ,

Now suppose we want to iterate vector 1, holding the rest fixed. We introduce the matrix a:

which has,known matrix elements given by p,, and the fixed vectors. It is also useful to simplify the orthogonality conditions by replacing the vector elements vt with new independent variables,

where the latter equality follows from the requirement that d ~ , ~ = 0. The problem is now to vary xb so a s to minimize

R . L . K E L L Y A N D R . E . C U T K O S K Y

where

subject to the constraints

etrization we have obtained adequate accuracy by always choosing N - 1 o r N=2. In ei ther c a s e we

(C5) begin with a single correlat ion vector appropriate to normalization fluctuations

K: = nf , / s , , (C16)

x u n - 0 f o r ? z > l . C b b - (C6) where n2 is given by Eq. (9. 101, and then vary K,' to minimize I?. If AT=2 i t is important to con-

We do this by a n i terat ive Newton-Raphson mini- s t ruc t a fairly good guess f o r the second vector mization of the quantity before beginning the Newton-Kaphson iteration.

To describe this, l e t a b e defined a s in Eq. (C2) , 5 2 = r + 2 x K, (x~u") , ( c 7 ) except that I;' is now the variable vector with 21'

n> 1 held fixed. We calculate the quantities

where the K , a r e Lagrange mult ipl iers and we a r e using vector notation for the s u m over bins. De- Sbc = abdadc(l - O b d ) ( f - 6 c d )

note the derivative vector and second-derivative d

matrix of l? a t x, =x: by =v:v: (v:)2(1 - 6hd) ( l - d c l ) + 0 ( 6 p ) ,

and expand SZ to second o r d e r about x::

n= 0, - 2 (a' - K ~ ( V ~ ) ' ) 6x + (6x)'P(dx), (C9)

and our initial guess f o r u2 is

where where B i s the value of b for which t, is a maxi- 6 x = x - x " . ( c 1 0 ) mum. The second t e r m in Eq. (C18) approximate-

ly orthogonalizes 1)' to v'. Now we go through a n Minimization of Si de te rmines the increment in x

i teration in which we al ternately s c a l e v 2 by the to be factor

The s tar t ing vector x0 is assumed to sat isfy the which minimizes r and then replace v 2 by v2 - av ' , orthogonality conditions, s o the Lagrange multi- where p l ie r s a r e determined by requiring that

Qnm= ( V " ) ~ P - ~ Z I ~ . ((213)

Thus,

K n = C g-'n,(vm)t~-l f f , n31

(C14)

where Q is the submatr ix of Q obtained by deleting the f i r s t row and column. The s tep f r o m x0 to x resu l t s in a decrease in r given by

r - r0 = CY+P-~CY - C QArn/cn~, . n m

(C15)

In applications of the correlation-vector param-

is chosen to orthogonalize v1 and v 2 to f i r s t o rder , During this i terat ion l imits must be imposed on the s ize of each element of v2 to ensure that a,, > (v;)'. The p r o c e s s converges when a becomes vanishingly small , and the full minimization procedure with a l l e lements of v2 varying in- dependently can then begin. Alternative variation of 2)' and v2 is continued until convergence is achieved. Instabilities of the Newton-Raphson method sometimes occur, but they can usually be avoided by temporari ly switching to a s imple s teepest descent minimization.

20 --.

A h l A E C $ M 4 T l O N O F M E S O N - N U C L E O N S C A T ' T F R I N G D A T A 2803

'I{. E. Cutkosky et al., iollowing papers, Phys. Rev. !3 z, 2804 (1979); 0, 2839 (1979); Phys. Hev. Lett . 37, 645 (1976);iand in Proceedings of' t h e Topical Con- -- .fefe.r.enc~ on Baryon Resonances, edited by R. 'r. Ross and D. H. Saxon (Gxford TJniv. P r e s s , London, 1976), p. 48.

2 ~ . P. Hodgkinson et a1 ., in Proceedings of the Topi- cczl Conference on Baryon Resonances, edited by E . T. Ross and D. H. Saxon (Gxford Univ., P r e s s , London, 1976), p. 41.

'R. E . Cutkosky and B. B. Deo, Phys. Rev. 174, 1859 (1968).

4 ~ . E. Cutkosky arid B. B. Deo, Phys. Rev. Lett. 20, 1272 (1968).

"The truncation function i s roCerred to a s a "conver gcncc tes t function" in Refs . 3 and 4 arid in many otllcr publications. We prefer fne new terminology a s being m o r e cicscriptive.

%n principle, the integration i i i Eq. (4.2) should be done at fixed label-aiory angle ra ther than fixed ccnter-of- m a s s angle. F o r rrN sca l te r iug in the resonance region this distinction i s insignificant f o r typical momeiitum bites (+2% o r bet ter) and angular reso!utions (10.01 o r worse in cos'd). A m o r e signi- f icant question is: how to ex l racr tile r m s width of the

beam b , f r o m a quoted mornentum hi te , The momeu- tun1 bite i s usually only vaguely defined in publications. As a ru le we in te rpre t it a s relerir lg t,o the half width a t half maximum, and multiply hy a fac tor of 0.60 to es t imate b e . This is the c o r r e c t factor for a t rap- ezoidal m o ~ l ~ e n t u m distribution with flat-lop to base ra t io of 0.6. F o r comparison, !he c o r r e c t fac tors a r e 0.58 and 0.66 for flat-Lop to bzse ra t ios of 1 . 0 and 0.3, rrspectively.

"l'o avoit? any possible confusion, we note that P , i s used in Ihis pakxr to designate the l th Chebyshev polynorriial. The riotation T, is reserved for the two- variable polynomials in z and y

'5. Trornborg et ai ., Phys. Rev. I) 15, 725 (1977). ''Y. R. Engelnrann and R . E . Hendrick, Phys. Rev. D s , 2891 (1977) and private communication. "k. Ayed, University of Paris-Sud thes i s , Report No.

CEA-N-1921, 1976 (unpublished). "D. M. Binnie et a l . , Phys. Rev. D j , 2789 (1973). '%. Bhandari and Y.-A. Chao, Phys . Rev. D 2, 192

(197'7). '%. C. &benham et a2 ., Phys. Rev. D 12, 2545 (1975). " ~ e have used only the a-p elast ic data f rom Ref. 13.

Tile charge-exchange data appear to suf fe r f rom nor- m?iiz:iiiiol; problems.

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Amalgamation of meson-nucleon scattering data