An Instability in the Matrix Solution
of DGLAP Equations
Mehrdad Goshtasbpour and Ali Shafi’i
Dept. of Physics, Shahid Beheshti University, Evin 19834, Tehran, Iran
Abstract. Following, the matrix solution outlined by Ratcliffe [1], and using data points (g 1p n d or)
F2p n d xi Q2i j as the only parameters -thus doing away with the need for a set of free parameters used
in a phenomenological fit to (polarized) parton distributions usually adopted (GRV, MRS, CTEQ,
...)- we arrive at a system of linear equations. Taking the unpolarized case, where there is plenty of
data, and a possible combination of unknowns q 8 Σ, and g, where q 3 xi Q2i1 3F2 p n xi Q2i1 can
be known, there remains three linear equations with three unknowns at every x i , each decomposing
one of the data points F2p xi Q2i j j 1 2 3. The unknowns would be at x i Q2i1 , since the matrix
solution of the nonsinglet and singlet DGLAP evolution equations can be used to determine the
unknowns at x i Q2i j j 2 3 in terms of those at x i Q2i1 .
The determinant of coefficients of these equations is too small to permit a stable solution within
the range of the errors of F2p values. The significance of the problem and possible solutions for it
are discussed.
INTRODUCTION
The simpler nonsinglet evolution equation:
dq x Q2 d ln Q2 α s Q2 2π
1
x
dy
x
P
y
y
q y Q2 (1)
where q represents the usual q3 or q8 the triplet and the octet quark parton distributions and P the respective splitting function, and the more complex ones for singlet
distribution Σ and for g are familiar. The mentioned method by matrix solution consists
primarily of discretizing the Bjorken x variable, via x-bins, and solving the resulting matrix equation exactly as a function of Q2 . Thus, parameterization can use available data
points in x-bins as the only parameters, doing away with free parameters used in other
phenomenological fits to the data.
A numeric code for the matrix solution was prepared and satisfactorily tested, separately for the nonsinglet and the singlet, in comparison to the available parameterized
parton distributions, in particular GRV, as e.g, may be seen in the figures which show
the LO evolution for the nonsinglet.
CP675, Spin 2002: 15th Int'l. Spin Physics Symposium and Workshop on Polarized Electron
Sources and Polarimeters, edited by Y. I. Makdisi, A. U. Luccio, and W. W. MacKay
© 2003 American Institute of Physics 0-7354-0136-5/03/$20.00
299
PROBLEM
Once the program code for evolution of the nonsinglet and singlet, q 3 q8 Σ, and g, at
LO or NLO, is well solved, one is ready to turn to data points (g 1p n d or) F2p n d xi Q2i j .
Now, the problem to solve is the decomposition of each, e.g, F2p xi Q2i j in terms of its
nonsinglet and singlet components, via a system of equations, which turn out to be linear.
After solving these equations, evolution and recomposition into evolved data points is
not a difficult matter.
Using
q3 xi Q2i1 3xi F2 p
n
xi Q2i1 6xi F2p xi Q2i1 F2d xi Q2i1 1
wD (2)
where wD is the probability of D-state for deutron (absorbing its coefficient), q 3 can be
calculated. Thus, eventually, a decomposition can be made to leave us with three linear
equations, each decomposing one of the data points F2p xi Q2i j j 1 2 3. At this stage,
the matrix solution of nonsinglet and singlet DGLAP evolution equations is already used
to allow us to have only three unknowns q 8 Σ, and g at xi Q2i1 .
The determinant of coefficients of these equations is too small, to permit a stable
solution within the range of the errors of F2p values. A closer look indicates that part of
the problem arises due to similarity of two columns of the determinant as q 8 and Σ have
very close values in the large and middle x (or small s x) range where the evolution
begins.
As the difference of q8 and Σ depends on a very small strange distribution s, one would
be tempted to try a change of variables to s and v, where q8 v 2s and Σ v s, while
fixing or giving the small s values. Thus, dropping one equation and one unknown. We
note that this procedure fixes the problem of divergence; as it can be clearly checked
numerically, when, e.g, s values are provided from known GRV distributions, while
working with real data otherwise. Lets see if there is a more fundamental way of
rendering known the value for one of the three variables, namely s. In other words,
reducing the number of variables and equations.
Using the data points F2d xi Q2i1, we have:
q8 6v
and
18 1xi F2d 1
Σ 3v2 9 1xi F2d 1
wD 2 αs3π cg g
(3)
αs 3π cg g
(4)
wD where * means: convolution and cg are defined in the usual way; furthermore, both
equations are meant to be at xi Q2i1 .
Thus, using these replacements, we reduce the number of unknowns to two, v and g at
xi Q2i1 . Note that now we have used an overall of three data points, F2p xi Q2i j j 1 2
and F2d xi Q2i1 , which can not determine all of the four independent original unknowns
q3 q8 Σ, and g, all at xi Q2i1 . Indeed, the two linear equations of decomposition of
F2p xi Q2i j j 1 2 are of the form:
1xi F2p xi Q2i1 16q3 118q8 29Σ αs3π Cg g
300
(5)
1xi F2p xi Q2i2 A3 6q3 A8 18q8 2As9Σ Agαs 3π Cg g
where
fk xi Q2i2 Ak fk xi Q2i1 fk q3 q8 Σ g
(6)
(7)
and Ak , where k 3 8 s g, are known coefficients determined via evolution equations.
Thus, all four variables are kept at xi Q2i1 . Now, using (2), (3), (4), which incidently
are not independent equations, there remains only one independent equation (6) in the
two unknowns v and g at xi Q2i1 , and a second equation via another set of data points
is needed.
It may be seen that the points F2d xi Q2i2 are not in a sense independent of a relatively
simple nonsinglet evolution, and thus provide an exellent check for the solution to the
nonsinglet evolution equation, as the evolution of q 3 defined through (2) together with
F2p xi Q2i2 data points correspond to them.
To create the second equation, what remains are F2p xi Q2i3 . Then the two independent equations are (6) and:
1xi F2p xi Q2i3 B3 6q3 B8 18q8 2Bs9Σ Bgαs 3π Cg g
(8)
These two together with (2) are exactly the equations that can be made for
F2d xi Q2i j j 2 3. Whether the problem can be solved in this manner is not yet
clear.
REFERENCES
1. P. G. Ratcliffe, HEP-PH/0012376
301
FIGURE 1. Our LO Evolution of q 3 of GRV from Q2 75GEV 2 to Q2 5000GEV 2 as compared to
GRV itself
FIGURE 2. Evolution of q 8 , similar to previous figure
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