- Email: [email protected]
Reconstruction of the deep inelastic structure functions from their moments
Volume 74B, number 1, 2
PHYSICS LETTERS
27 March 1978
RECONSTRUCTION OF THE DEEP INELASTIC STRUCTURE FUNCTIONS FROM THEIR MOMENTS
F.J. YNDURAIN
CERN, Geneva,Switzerland and UniversidadAutonoma, Canto Blaneo, MadrM,Spain 1 Received 10 January 1978
A mathematically rigorous and extremely easy to handle method for inverting moments is presented. The method is exemplified by reconstructing the structure function vW~p of deep inelastic scattering from its moments as given by asymptotic freedom.
As is well known, the property of asymptotic freedom [ 1] enjoyed by the colour quark-gluon theory (QCD) of strong interactions allows us, among other things, to calculate the (logarithmic) corrections to naive Bjorken scaling for the deep inelastic scattering structure functions. When comparing theory with experiment, however, a difficulty appears: for the first only gives direct predictions for the moments of the structure functions [2]. For example, ifF(x, Q2) = F2 = vW2, and we define its moments 1
/~n(Q 2) =
fdx F(x, O2)xn,
(1)
0
with the notation of refs. [1] ). A is an energy parameter; experimentally [3-5] A ~ 0.15 to 0.45 GeV. Explicit expressions for the H, P, L, %/3 may be found in ref. [5]. Finally,/~n(Q 2) is supposed to be taken from experiment. Since what we require are file values of F2(x, Q2) at specific values of x, Q2, we are led to a problem of inverting moments. In the existing literature this problem has been tackled in two different ways. MethodL One may invert eqs. (1) and (2) by brute force [2,6]. Replacing eq. (2) by its lowest-order approximation [2], we have 1
then, to a precision O(a2(Q2)), QCD tells us that (for the valence part)
F(x,02) =
fdx F( x
x
02) ~[lOgo2/A2
'
"
"t oTO o/2r'x J (3a)
The kernel T is given by
/ln(Q2 ) = un(Q 2)
+i~
x
3rr + (Hn+ 2 +Pn+2
+Ln+2(Q2))~c(Q 2)
3rr + (Hn+ 2 +Pn+2
+Ln+i(Q2))°lc(O 2)
×/log
Oa/A2~-'~On+ 2/2~°
Here 002 is a reference momentum and aC(O2 ) the effective gluon coupling constant [1] (~c = g 2/4a 1 Permanent address.
68
T(X, x) = ~1
(2)
f
dz x-Az x - l - z
(3b)
--ioo
where A z is the analytic continuation of 70n/230 to complex n -+ z. This method presents a number of serious drawbacks: to obtain T one requires knowledge of all the moments; eqs. (3) are quite unstable, it is impossible to find, other than numerically, the value of T beyond the lowest order approximation. This has led people to use a second type of approac.h [5,7]. Method H. Assume a definite functional form, de-
Volume 74B, number l, 2
PHYSICS LETTERS
pending on certain parameters, for different contributions to F(x, Q2) and fix these parameters by fitting the moments. Now, while this method simplifies the calculations, these still remain complicated and moreever, not only the problem of unstabilities is not solved (just bypassed) but choice of specific functional forms implies potentially dangerous biases * In view of this we think it useful to introduce a third method which possesses the advantage of being at the same time model independent, mathematically rigorous, and extremely simple from a computational point of view. It presents the drawback that it is unable to represent the singularities o f F and, therefore, its behaviour at the endpoints x = 0, 1 for which you have to fall back on previous methods I and II. Our method has been described from a mathematical point of view elsewhere (ref. [8], see also ref. [9] ) and applied successfully to other branches of high-energy [8] or solid state physics [10], but we believe it is the first time it is used for the problem at hand. We will assume that using, say, eq. (2), we have calculated the moments/~0,/al, "", ~N and will then show how to reconstruct F. This we will do in several steps, according to the increasing assumptions we make about this function• To begin with, errors in the /~n shall be neglected; a general discussion about errors will be found at the end of this note.
Step 1: no assumptions, reconstruction in the average. It is well known [8,9] that, in general, point-like reconstruction of F is impossible. Fortunately what experimentalists actually measure is not F(x) +2 at exactly the point x, but (because of uncertainties in v, Q) an average o f F around the point x: this is what is then required of the theoretical prediction. Posed this way the problem ceases to be unstable. It is possible to reconstruct averages o f F with any reasonable weight function; but we will choose a particular set, optimal for the problem at hand [8]. This set
27 March 1978
will be the (normalized) Bernstein polynomials [8,9] :
N-k b ( N , k ) ( x ) - ( N ~ l)! l ~ 0 •
=
x k+l,
(-1) / l!(NZ
k--
(4a)
I)!
k = 0, 1 . . . . . N. The polynomial b(N,k)(x) acts as weight over the point 1
XN, k = f dx b(N,k)(x)x - Nk ++l 2 .' 0
(4b)
so we may define the average o f F around XN, k as 1
F(XN, k) = f 0
dx b(U'k)(x)F(x).
(5a)
Using now eq. (4a), we see that this may be written simply as
N-k ~(XN,k)_ (N~_~.1)! /~0 •
=
(-I)/
I~.(N - k -
l)! I ~ k + l ,
(Sb)
i.e., F is immediately obtained from the moments. The proof that the b (N,k) are suitable weight functions may be found in a number of references [8,9]. Intuitively it is obvious if we realize that they may be rewritten as
b(N,k)(x) =
xk(1 -- x ) N - k f l dx xk(1 -- X) N - k
Therefore the b are positive, normalized to fb = 1 and with a unique maximum near ,3 XN, k" The "focusing power" of b (N,k) is given by the inverse of the dispersion, AN, k,
2x,k = f dx b(U'k)(x)x2 o
-
f dx b(N'~)(x)x 0
(6) = (N - k + 1)/[(N + 2)2(N + 3)].
,1 For example, with the functional form of, e.g.,xVs in ref. [ 71, x V8(x, Q2) = (constant) x ~1(1 - x) r/2, one finks artificially the behaviour ofxV8 at the boundaries, x ~1 , (1 - x) ~2 with the fact that it necessarily is maximum at x= ~1/(~1 + ~2). On the other hand, I am aware that this works both ways, and method II is a good way for pumping in/extracting useful physical information on the different contributions(valence quarks, ocean, gluons) to F(x, Q2). t2 We will drop the Q2 dependence whenever it is irrelevant•
The points at which the average is taken, XN,k, may be varied by varyingN and k to cover the interval ~ta Actually the maximum of b(N,k)(x) lies at Xmax = kiN. This distortion (Xmax ~ XN, k) is indeed necessary since any reasonable weight should not extend beyond the endpoints x = 0, 1. 69
Volume 74B, number 1, 2
PHYSICS LETTERS
27 March 1978
08
0.3 Q2= 11.5 GeV2
Q2=Q~:18 GeV2 0.6
F 0.2
0.1 /Uv &
0.2
dv
/,.-
,
%,
juv&dv 0.2
0.4
0.6
0.B
~
1.0 0
x
A
L
]
I
~t'll-~l
0.2
0.4
0.6
0.8
m
110
x
(a)
(b)
Fig. 1. Comparison of the reconstructed F V with its exact or fitted value. (a) Continuous line: exact FV(x, Q~) (input). Black dots: reconstructed averages. Triangles: point-like reconstruction. Horizontal bars: dispersion of the averages.N = 8. (b) Continuous line: extrapolated FV2(x,Q2) using method II [7]. Small circles: experimental points from ref. [12]. Broken lines: fits to the whole vW2 [7]. Black triangles: reconstructed averages, N = 7; black dots: idem, N = 8; black squares: idem, N = 9. In both cases only the valence quark contributions (Uv, dv) have been taken into account for the reconstruction and for F V itself.
(0, 1) in ~ l / 2 N to ~ l / 3 N bins with precision given by eq. (6). An example of reconstruction of FV(x, Q2) = w~P(u v and d v ) (this means that only the valence quark contribution is taken into account) with N = 7, 8, and 9, the moments given by eq. (2) (as calculated in ref. [5]) is shown in fig. 1 for fixed Q2, varying x, and fig. 2 for fixed x, varying Q2. Once the moments were known, the whole reconstruction took an afternoon with a pocket calculator! With N ~ 50, which any decent computer can handle, reconstruction with dispersion inferior to the experimental one [ 11 ] (~1%) is easily obtained. It may be proved [8] that this is the best you can do without extra assumptions. It is a lot, but with reasonable hypotheses we can still go further.
Step 2: semi-bounded aF/~x, upper and lower bounds for F 2. We assume that there exists a finite constant ,4 L such that either aF(x)/ax + L >~0 or ,4 The case in which L is not constant, hut a known function L(x), is quite similar. 70
--aF(x)/ax + L >~O, then it is possible to obtain very tight upper and lower bounds to F(x) itself. Suppose, e.g., the case is that -aF/ax >~L; and let p = L - aF/ax ~> 0. By partial integration we obtain 1
f dxp(x)x n=npn+l +L/n, n~> 1. 0
(7)
On the other hand 1
L x - F ( x ) = f d x ' O(x')O(x -
x');
(8)
0 to obtain upper and lower bounds for the left-hand side of eq. (8), and hence for F is a standard problem in the theory of moments [9]. What one essentially does is find polynomials p+(x') of degree ~ N - 1 with
p_(x')--
'",
Volume 74B, number 1, 2
PHYSICS LETTERS (o)
0.20 0.18 0.16
_•r:• _
(0.5,Q 2)
fact, in the neighbourhood Ix - XN, kl <~ AN, k, we 3 . may write, with an error ~ A N,k"
F(x) ~ F(XN, k) + (x - XN, k) aF(XN,k)/3X
x =0.5
~"~,~,~
(9)
+ ~(X - XN,k) 2 a 2F(XN, k)/aX 2. Integrating with b(N,k)(x) and using eqs. (4b), (Sa)
0.14 ASF I /
0.12 I
4
o.12
27 March 1978
_
I
6
I
I
8 10 (i 2 (GeV2)
I
12 (b)
,~(o.6,Q~ )
a2d (6) we get2at once F(XN k) ~ F (XN k) -- ½ f 2 k F(XN, k)/ax . We may, to'an accej~tat~le O(A u ~c'J error, replace a2F(XN, t)/ax2 by aZF(XN, k )/ax2 ~The last can be calculated (for k = 2, 3 . . . . . N ) by the same method as/~ before: in fact, integrating by parts, 1
f d x 32F(X)xn = n(n ax 2 0
0.08
F2(06'Q") 0.06
so finally, with F(XN, k) given by eq. (Sb),
MASF l l ~ ASF l / I
l)Un-2,
I
F(XN, k) = F(XN, k) + 6(XN, k),
0.2(GeV2)
Fig. 2. Comparison of the reconstructed F V as a function of Q2. Black points: experimental data from ref. [ 11 ]. Broken fine: fitted F V [5]. Solid lines: (a) reconstructed averages, (b) reconstructed averages and point-like values. N = 8. so then, since p is positive, we may multiply it by this inequality and integrate to obtain the desired bounds in terms of the/1 n. Of course the polynomials p+_ have to be optimized [9,10]. This has been worked out in detail for the densities of states of solids [10] (functions with shapes very similar to F2): no example of application to our case is presented here because, except from a conceptual point of view, such bounds are less useful than the results just discussed or those we will discuss in the following. (The reason is that, due to computational difficulties, it is hard to go b e y o n d N ~ 12 moments.)
Step 3: twice differentiable F, point-like reconstruction * s. Under the assumption that F is twice differentiable except perhaps at the endpoints*6 we can obtain very approximate point-wise reconstruction. In
+s This method is presented here for the first time. ,6 This assumption fails in general for the densities of states of solids (Van Hove singularities), but is very probably valid for our case.
M..
-~
N-k
1 (N+I)!(N
k+l)
× ~ (-1)l(k +/)(k + l/=0 I ! ( N - k - l)! The and sult few
1)
lak+l- 2 "
results for the case F = vW 2 are shown in figs. la 2b. ( F o r fig. 2a 6 is negligible; for fig. l b the rewould be as for fig. 2b.) We see that even with as as ten moments the method is extremely effective.
Errors. (i) First of all we have rounding errors. These are of the order ( N - k)!/[(~(N - k))!] 2 ~ 2 S - k times the relative error in the ~n' This is what limits to ~ 5 0 the number of moments that m a y be used. (ii) Errors in the input I~n(Q2) and in the theoretical expression for t h e / l n should be transmitted stably [8] to the reconstructed F , / ~ However, no useful explicit formula may be given; the best practical procedure [10] is to effect random variations o f the/aA! inside their error bars, and see how this affects F , F. (iii) It is obvious that none o f the methods proposed are able to reproduce singularities; in particular the endpoint behaviour has to be obtained by methods I and II described at the beginning. (iv) The error in eq. (10) is hard to obtain without more assumptions. A reasonable guess, however, is to take i t ~F(XN, k ) [6 (XN, k )/F(XN,k )] 3 / 2 . 71
Volume 74B, number 1, 2
PHYSICS LETTERS
I am grateful to K. Gaemers for useful discussions, and to D. Ross for the same reason and for supplying me with the values of the m o m e n t s as calculated in ref. [5].
[5] [6] [7]
References [8] [1] H.D. Politzer, Phys. Rev. Lett. 30 (1973) 1346; D.J. Gross and F. Wilczek, Phys. Rev. Lett. 30 (1973) 1343; W. Caswell, Phys. Rev. Lett. 33 (1974) 224. [2] D.J. Gross, Phys. Rev. Lett. 32 (1974) 1071. [3] I. Hinchliffe and C.H. Llewellyn Smith, Nucl. Phys. B128 (1977) 93. [4] A. de Rtijula, H. Georgi and H.D. Politzer, Ann. Phys. (NY) 103 (1977) 315;
72
[9] [10] [11] [ 12]
27 March 1978
P.W. Johnson and Wu-ki Tung, Nucl. Phys. B121 (1977) 270. A.J. Buras, E.G. Floratos, D.A. Ross and C.T. Sachrajda, CERN preprint TH 2340 (1977). G. Parisi, Phys. Lett. 43B (1973) 207. A.J. Buras and K.J.F. Gaemers, CERN preprint TH 2322 (1977). C. L6pez and F.J. Yndur~tin, in: Pad~ approximants and their applications, ed. P. Graves-Morris (Academic, New York, 1973), and work quoted therein. J.A. Shohat and J.D. Tamarkin, The problem of moments (Am. Math. Soc., New York, 1943). F. Yndur~in and F.J. Yndur~in, J. Phys. C 8 (1975) 434, and references therein. E.M. Riordan et al., SLAC-PUB-1634 (1976). H. Anderson et al., Measurement of the proton structure function, Proc. Tibilisi Conf. (1976).
PHYSICS LETTERS
27 March 1978
RECONSTRUCTION OF THE DEEP INELASTIC STRUCTURE FUNCTIONS FROM THEIR MOMENTS
F.J. YNDURAIN
CERN, Geneva,Switzerland and UniversidadAutonoma, Canto Blaneo, MadrM,Spain 1 Received 10 January 1978
A mathematically rigorous and extremely easy to handle method for inverting moments is presented. The method is exemplified by reconstructing the structure function vW~p of deep inelastic scattering from its moments as given by asymptotic freedom.
As is well known, the property of asymptotic freedom [ 1] enjoyed by the colour quark-gluon theory (QCD) of strong interactions allows us, among other things, to calculate the (logarithmic) corrections to naive Bjorken scaling for the deep inelastic scattering structure functions. When comparing theory with experiment, however, a difficulty appears: for the first only gives direct predictions for the moments of the structure functions [2]. For example, ifF(x, Q2) = F2 = vW2, and we define its moments 1
/~n(Q 2) =
fdx F(x, O2)xn,
(1)
0
with the notation of refs. [1] ). A is an energy parameter; experimentally [3-5] A ~ 0.15 to 0.45 GeV. Explicit expressions for the H, P, L, %/3 may be found in ref. [5]. Finally,/~n(Q 2) is supposed to be taken from experiment. Since what we require are file values of F2(x, Q2) at specific values of x, Q2, we are led to a problem of inverting moments. In the existing literature this problem has been tackled in two different ways. MethodL One may invert eqs. (1) and (2) by brute force [2,6]. Replacing eq. (2) by its lowest-order approximation [2], we have 1
then, to a precision O(a2(Q2)), QCD tells us that (for the valence part)
F(x,02) =
fdx F( x
x
02) ~[lOgo2/A2
'
"
"t oTO o/2r'x J (3a)
The kernel T is given by
/ln(Q2 ) = un(Q 2)
+i~
x
3rr + (Hn+ 2 +Pn+2
+Ln+2(Q2))~c(Q 2)
3rr + (Hn+ 2 +Pn+2
+Ln+i(Q2))°lc(O 2)
×/log
Oa/A2~-'~On+ 2/2~°
Here 002 is a reference momentum and aC(O2 ) the effective gluon coupling constant [1] (~c = g 2/4a 1 Permanent address.
68
T(X, x) = ~1
(2)
f
dz x-Az x - l - z
(3b)
--ioo
where A z is the analytic continuation of 70n/230 to complex n -+ z. This method presents a number of serious drawbacks: to obtain T one requires knowledge of all the moments; eqs. (3) are quite unstable, it is impossible to find, other than numerically, the value of T beyond the lowest order approximation. This has led people to use a second type of approac.h [5,7]. Method H. Assume a definite functional form, de-
Volume 74B, number l, 2
PHYSICS LETTERS
pending on certain parameters, for different contributions to F(x, Q2) and fix these parameters by fitting the moments. Now, while this method simplifies the calculations, these still remain complicated and moreever, not only the problem of unstabilities is not solved (just bypassed) but choice of specific functional forms implies potentially dangerous biases * In view of this we think it useful to introduce a third method which possesses the advantage of being at the same time model independent, mathematically rigorous, and extremely simple from a computational point of view. It presents the drawback that it is unable to represent the singularities o f F and, therefore, its behaviour at the endpoints x = 0, 1 for which you have to fall back on previous methods I and II. Our method has been described from a mathematical point of view elsewhere (ref. [8], see also ref. [9] ) and applied successfully to other branches of high-energy [8] or solid state physics [10], but we believe it is the first time it is used for the problem at hand. We will assume that using, say, eq. (2), we have calculated the moments/~0,/al, "", ~N and will then show how to reconstruct F. This we will do in several steps, according to the increasing assumptions we make about this function• To begin with, errors in the /~n shall be neglected; a general discussion about errors will be found at the end of this note.
Step 1: no assumptions, reconstruction in the average. It is well known [8,9] that, in general, point-like reconstruction of F is impossible. Fortunately what experimentalists actually measure is not F(x) +2 at exactly the point x, but (because of uncertainties in v, Q) an average o f F around the point x: this is what is then required of the theoretical prediction. Posed this way the problem ceases to be unstable. It is possible to reconstruct averages o f F with any reasonable weight function; but we will choose a particular set, optimal for the problem at hand [8]. This set
27 March 1978
will be the (normalized) Bernstein polynomials [8,9] :
N-k b ( N , k ) ( x ) - ( N ~ l)! l ~ 0 •
=
x k+l,
(-1) / l!(NZ
k--
(4a)
I)!
k = 0, 1 . . . . . N. The polynomial b(N,k)(x) acts as weight over the point 1
XN, k = f dx b(N,k)(x)x - Nk ++l 2 .' 0
(4b)
so we may define the average o f F around XN, k as 1
F(XN, k) = f 0
dx b(U'k)(x)F(x).
(5a)
Using now eq. (4a), we see that this may be written simply as
N-k ~(XN,k)_ (N~_~.1)! /~0 •
=
(-I)/
I~.(N - k -
l)! I ~ k + l ,
(Sb)
i.e., F is immediately obtained from the moments. The proof that the b (N,k) are suitable weight functions may be found in a number of references [8,9]. Intuitively it is obvious if we realize that they may be rewritten as
b(N,k)(x) =
xk(1 -- x ) N - k f l dx xk(1 -- X) N - k
Therefore the b are positive, normalized to fb = 1 and with a unique maximum near ,3 XN, k" The "focusing power" of b (N,k) is given by the inverse of the dispersion, AN, k,
2x,k = f dx b(U'k)(x)x2 o
-
f dx b(N'~)(x)x 0
(6) = (N - k + 1)/[(N + 2)2(N + 3)].
,1 For example, with the functional form of, e.g.,xVs in ref. [ 71, x V8(x, Q2) = (constant) x ~1(1 - x) r/2, one finks artificially the behaviour ofxV8 at the boundaries, x ~1 , (1 - x) ~2 with the fact that it necessarily is maximum at x= ~1/(~1 + ~2). On the other hand, I am aware that this works both ways, and method II is a good way for pumping in/extracting useful physical information on the different contributions(valence quarks, ocean, gluons) to F(x, Q2). t2 We will drop the Q2 dependence whenever it is irrelevant•
The points at which the average is taken, XN,k, may be varied by varyingN and k to cover the interval ~ta Actually the maximum of b(N,k)(x) lies at Xmax = kiN. This distortion (Xmax ~ XN, k) is indeed necessary since any reasonable weight should not extend beyond the endpoints x = 0, 1. 69
Volume 74B, number 1, 2
PHYSICS LETTERS
27 March 1978
08
0.3 Q2= 11.5 GeV2
Q2=Q~:18 GeV2 0.6
F 0.2
0.1 /Uv &
0.2
dv
/,.-
,
%,
juv&dv 0.2
0.4
0.6
0.B
~
1.0 0
x
A
L
]
I
~t'll-~l
0.2
0.4
0.6
0.8
m
110
x
(a)
(b)
Fig. 1. Comparison of the reconstructed F V with its exact or fitted value. (a) Continuous line: exact FV(x, Q~) (input). Black dots: reconstructed averages. Triangles: point-like reconstruction. Horizontal bars: dispersion of the averages.N = 8. (b) Continuous line: extrapolated FV2(x,Q2) using method II [7]. Small circles: experimental points from ref. [12]. Broken lines: fits to the whole vW2 [7]. Black triangles: reconstructed averages, N = 7; black dots: idem, N = 8; black squares: idem, N = 9. In both cases only the valence quark contributions (Uv, dv) have been taken into account for the reconstruction and for F V itself.
(0, 1) in ~ l / 2 N to ~ l / 3 N bins with precision given by eq. (6). An example of reconstruction of FV(x, Q2) = w~P(u v and d v ) (this means that only the valence quark contribution is taken into account) with N = 7, 8, and 9, the moments given by eq. (2) (as calculated in ref. [5]) is shown in fig. 1 for fixed Q2, varying x, and fig. 2 for fixed x, varying Q2. Once the moments were known, the whole reconstruction took an afternoon with a pocket calculator! With N ~ 50, which any decent computer can handle, reconstruction with dispersion inferior to the experimental one [ 11 ] (~1%) is easily obtained. It may be proved [8] that this is the best you can do without extra assumptions. It is a lot, but with reasonable hypotheses we can still go further.
Step 2: semi-bounded aF/~x, upper and lower bounds for F 2. We assume that there exists a finite constant ,4 L such that either aF(x)/ax + L >~0 or ,4 The case in which L is not constant, hut a known function L(x), is quite similar. 70
--aF(x)/ax + L >~O, then it is possible to obtain very tight upper and lower bounds to F(x) itself. Suppose, e.g., the case is that -aF/ax >~L; and let p = L - aF/ax ~> 0. By partial integration we obtain 1
f dxp(x)x n=npn+l +L/n, n~> 1. 0
(7)
On the other hand 1
L x - F ( x ) = f d x ' O(x')O(x -
x');
(8)
0 to obtain upper and lower bounds for the left-hand side of eq. (8), and hence for F is a standard problem in the theory of moments [9]. What one essentially does is find polynomials p+(x') of degree ~ N - 1 with
p_(x')--
'",
Volume 74B, number 1, 2
PHYSICS LETTERS (o)
0.20 0.18 0.16
_•r:• _
(0.5,Q 2)
fact, in the neighbourhood Ix - XN, kl <~ AN, k, we 3 . may write, with an error ~ A N,k"
F(x) ~ F(XN, k) + (x - XN, k) aF(XN,k)/3X
x =0.5
~"~,~,~
(9)
+ ~(X - XN,k) 2 a 2F(XN, k)/aX 2. Integrating with b(N,k)(x) and using eqs. (4b), (Sa)
0.14 ASF I /
0.12 I
4
o.12
27 March 1978
_
I
6
I
I
8 10 (i 2 (GeV2)
I
12 (b)
,~(o.6,Q~ )
a2d (6) we get2at once F(XN k) ~ F (XN k) -- ½ f 2 k F(XN, k)/ax . We may, to'an accej~tat~le O(A u ~c'J error, replace a2F(XN, t)/ax2 by aZF(XN, k )/ax2 ~The last can be calculated (for k = 2, 3 . . . . . N ) by the same method as/~ before: in fact, integrating by parts, 1
f d x 32F(X)xn = n(n ax 2 0
0.08
F2(06'Q") 0.06
so finally, with F(XN, k) given by eq. (Sb),
MASF l l ~ ASF l / I
l)Un-2,
I
F(XN, k) = F(XN, k) + 6(XN, k),
0.2(GeV2)
Fig. 2. Comparison of the reconstructed F V as a function of Q2. Black points: experimental data from ref. [ 11 ]. Broken fine: fitted F V [5]. Solid lines: (a) reconstructed averages, (b) reconstructed averages and point-like values. N = 8. so then, since p is positive, we may multiply it by this inequality and integrate to obtain the desired bounds in terms of the/1 n. Of course the polynomials p+_ have to be optimized [9,10]. This has been worked out in detail for the densities of states of solids [10] (functions with shapes very similar to F2): no example of application to our case is presented here because, except from a conceptual point of view, such bounds are less useful than the results just discussed or those we will discuss in the following. (The reason is that, due to computational difficulties, it is hard to go b e y o n d N ~ 12 moments.)
Step 3: twice differentiable F, point-like reconstruction * s. Under the assumption that F is twice differentiable except perhaps at the endpoints*6 we can obtain very approximate point-wise reconstruction. In
+s This method is presented here for the first time. ,6 This assumption fails in general for the densities of states of solids (Van Hove singularities), but is very probably valid for our case.
M..
-~
N-k
1 (N+I)!(N
k+l)
× ~ (-1)l(k +/)(k + l/=0 I ! ( N - k - l)! The and sult few
1)
lak+l- 2 "
results for the case F = vW 2 are shown in figs. la 2b. ( F o r fig. 2a 6 is negligible; for fig. l b the rewould be as for fig. 2b.) We see that even with as as ten moments the method is extremely effective.
Errors. (i) First of all we have rounding errors. These are of the order ( N - k)!/[(~(N - k))!] 2 ~ 2 S - k times the relative error in the ~n' This is what limits to ~ 5 0 the number of moments that m a y be used. (ii) Errors in the input I~n(Q2) and in the theoretical expression for t h e / l n should be transmitted stably [8] to the reconstructed F , / ~ However, no useful explicit formula may be given; the best practical procedure [10] is to effect random variations o f the/aA! inside their error bars, and see how this affects F , F. (iii) It is obvious that none o f the methods proposed are able to reproduce singularities; in particular the endpoint behaviour has to be obtained by methods I and II described at the beginning. (iv) The error in eq. (10) is hard to obtain without more assumptions. A reasonable guess, however, is to take i t ~F(XN, k ) [6 (XN, k )/F(XN,k )] 3 / 2 . 71
Volume 74B, number 1, 2
PHYSICS LETTERS
I am grateful to K. Gaemers for useful discussions, and to D. Ross for the same reason and for supplying me with the values of the m o m e n t s as calculated in ref. [5].
[5] [6] [7]
References [8] [1] H.D. Politzer, Phys. Rev. Lett. 30 (1973) 1346; D.J. Gross and F. Wilczek, Phys. Rev. Lett. 30 (1973) 1343; W. Caswell, Phys. Rev. Lett. 33 (1974) 224. [2] D.J. Gross, Phys. Rev. Lett. 32 (1974) 1071. [3] I. Hinchliffe and C.H. Llewellyn Smith, Nucl. Phys. B128 (1977) 93. [4] A. de Rtijula, H. Georgi and H.D. Politzer, Ann. Phys. (NY) 103 (1977) 315;
72
[9] [10] [11] [ 12]
27 March 1978
P.W. Johnson and Wu-ki Tung, Nucl. Phys. B121 (1977) 270. A.J. Buras, E.G. Floratos, D.A. Ross and C.T. Sachrajda, CERN preprint TH 2340 (1977). G. Parisi, Phys. Lett. 43B (1973) 207. A.J. Buras and K.J.F. Gaemers, CERN preprint TH 2322 (1977). C. L6pez and F.J. Yndur~tin, in: Pad~ approximants and their applications, ed. P. Graves-Morris (Academic, New York, 1973), and work quoted therein. J.A. Shohat and J.D. Tamarkin, The problem of moments (Am. Math. Soc., New York, 1943). F. Yndur~in and F.J. Yndur~in, J. Phys. C 8 (1975) 434, and references therein. E.M. Riordan et al., SLAC-PUB-1634 (1976). H. Anderson et al., Measurement of the proton structure function, Proc. Tibilisi Conf. (1976).