19 June 1997
PHYSICS
LETTERS 6
Physics Letters B 403 (1997) 139-144
ELSEVIER
x-distribution of deuteron structure function at low-x J.K. Sarma a,1v2,D.K. Choudhury b*3,G.K. Medhi ’ a Electronics Science Department, Gauhati University, Guwahati 781014, Assam, India b Physics Department, Gauhati University, Guwahati 781014, Assam, India c Physics Department, Birjhora Mahavidyalaya, Bongaigaon 783380. Assam, India
Received 25 February 1997
Editor: H. Georgi
Abstract An approximate solution of the Altarelli-Parisi ( AP) equation is presented and the x-distribution of the deuteron structure function is calculated at the low-x limit. The results are compared with the EMC NA 28 experiment data. @ 1997 Published by Elsevier Science B.V.
1. Introduction
u = Jlog(xn/x)
The Altarelli-Parisi (AF’) equations [ 1] are the basic tools to study the Q2-evolution of structure functions. Even though alternative evolution equations [ 21 have been proposed and pursued in recent years to study structure functions especially at low-x, the AP equations have been the basic tools in studying double asymptotic scaling (DAS) [ 31 or extracting the gluon density from the slope of the structure functions at low-x [4]. Based on QCD studies Ball and Forte show [ 31 that evolving a flat input distribution at Q,’ = 1 GeV* with the AP equations leads to a strong rise of F2 at low-x in the region measured by HERA. An interesting feature is that if QCD evolution is the underlying dynamics of the rise, perturbative QCD predicts that at large Q* and small x the structure function exhibits double scaling in the two variables:
0 1997 Published by Elsevier Science B.V. All rights reserved.
PII SO370-2693(97)00483-S
p =
log(%l/~) 9 log(t/to)
with t s log(Q2/A2). This follows from a computation of the asymptotic form of the structure function F2 (x, Q*) at small-x and relies only on the assumption that any increase in F2 (x, Q2) at small x is generated by perturbative QCD evolution. It implies that the AF’ equations have characteristic x-evolution at low x. The present paper reports calculation of x-evolutions for singlet, non-singlet and deuteron structure functions at low x from the same equations. It is based on the approximate solutions of AF’ equation using Taylor expansion at low-x. The formalism was used earlier [5] to the low-x EMC and Fermilab data with reasonable phenomenological s success. It was a natural improvement of an earlier analysis at intermediate x [ 61. In the present paper, in Section 2 we discuss the necessary theory in short. Section 3 gives the result and the discussion.
’ E-mail:
[email protected]. * E-mail:
[email protected]. 3 E-mail:
[email protected]. 0370-2693/97/$17.00
. log(t/r()),
J.K. Sarmu et al/Physics
140
Letters B 403 (1997) 139-144
2. Theory
F;(x/w,t) = F; x+x
Though the theory is discussed earlier [5] yet we have mentioned the essential steps here again for clarity. The AP equation for the singlet structure function has the standard form [ 71
W;(x, f)
- y
at
[{3+41n(l
dw
(1 -w) +2. ---_((I
J.I{w’ + =o, $NF
+
2d2F;(~, r) 8x2 +...
+W*>F;(X/W,t) -2F;(x,f)}
(1 - w)*}G(x/w,
t)dw
I
F;(x/w,t)
(1)
J
+x-&kdFi;,t) . (7)
F;(x,t)
N
Putting (5) and (7) in (2) and (3) and performing u-integrations we get I;(A.,~) =[-(1 +(x(1
I
2
(6)
k=l
f)
=
,
which covers the whole range of u, 0 < u < 1 - x. Neglecting higher order terms, F;( x/w, t) can then be approximated for low x as
where Af = 4/( 33 - 2Nf), Nf being the number of flavour and t = ln(Q2/A2). Defining If(x,
k=l
- .u)}F;(x,t)
I
+xgukdF:(xX,t)
=F;(X,t)
-x)(3+x)]F;(x,t)
-x2)
+2xln(l/x)}‘F’~~.‘)
and
&{(
1 + w2,F,“(x/w,
t) -2Fi(x,
t)} li“(x, t) = Nf
f( 1 - x) (2 - x + %x)G(x, t) [
+(-&x(1
-x)(5-4x+2x2)
+2xln(l/x)}~ Ii(x,
t)
+Nf s x
1,
(9)
where we have used the identity
I =
(8)
{w+ ( 1 -
w)*}G(x/‘w,
t)dw ,
(3)
co uk -=lnl/(l c k
-u).
(10)
k=l
one can recast ( 1) as
Using (8) and (9) in (4) we obtain
aF;(x, t>
- +[{3+41n(l
-x)}Fl(x,t)
dr
+1f(x,t)
+fi(x,t)]
=o.
(4)
Let us introduce the variable u = 1 -w and note that
aF;(x, r) Af - 7 A(x)F;(x,t) dt [ cW;(x, t) + B(x) + C(X)G(X, dX
00
c
Uk.
(5)
k=O
The series (5) is convergent for In/ < 1. Since x < w < 1, so 0 < u < 1 - x and hence the convergence criterion is satisfied. Using (5) we can rewrite,
+
D(x)
dG(X, . _ dX
t>
t> I
(11)
=o,
where A(x)
=3+41n(l
-x)
-(I
-x)(3-x),
(12)
J.K. Sauna et al./Physics
B(x)
= x( 1 - x2> + 2xln( l/x)
C(x)
=$f(l
D(x)
=-$Vf.(l
,
-x)(2-x+2x2),
(13)
The general solution of (20) is
(14)
F(u,V)
=0,
(22)
where F is an arbitrary function
-x)(5-4x+2x2)
+ 2xln( l/x).
(15)
In order to solve ( 11)) we need to relate the singlet distribution Fi(x, t) with the gluon distribution G( x, t). For small x and high Q2, the gluon is expected to be more dominant than the sea. For lower Q’ (Q2 cv A2), however, there is no such clear cut distinction between the two. For simplicity, we therefore, assume, G(x, I) = KF;(x,
141
Letters B 403 (1997) 139-144
t) ,
(16)
where K is a parameter to be determined from experiments. But the possibility of the breakdown of relation ( 16) also can not be ruled out. Then from Eq. ( 11) we get
u(x,t,
and
F;) = C,
and V( x, t, F;) = C2
(23)
form a solution of the equations dt
dx P(x,t,F,S) Solving
dF;
= Q(x,r,F2)
= R(x,t,
F2)
(24)
(24) one obtains
u(x, t, F;) = tX”(x)
(25)
and V(x, t, F;) = F,(x,
t)Y”(x)
,
(26)
where
c?Fi’(x, t) dt L(x,K)F,(x,t)
+M(x,K)
=0,
aF;(x, t) ax
1
X”(x) =exp
Y”(x) = exp
Llx, K) = A(x) M(x, K) = B(x)
+ KC(x)
,
+ KD(x)
(18) .
(19)
The general solution of (17) can now be obtained by recasting it in the standard form
Jww]
and
(17)
where
[ l/A f
(27)
1.
[I
L(x)/M(x)dx
Thus the structure function Fi( x, t) (22) with u and V given by (25) and tively. It thus has no unique solution. possibility is that a linear combination to satisfy (20) so that
(28) has to satisfy (26), respecThe simplest of u and V is
(29)
A,yu + B,V = 0.
X3(x> s 1
Putting the values of u and V in (29) we obtain =
(20)
R(x,t,F;),
F;(x,t)
=-$1.
where P(x, t, F;) = AfM(x, Q(x,t,F;) and R(x,r,F,)
.
(30)
Defining
K) ,
=--I,
[ Y”(x)
(21)
F;(x,ro)
=
-2. s
to.
X”(x)
-
[ YS(x)
1
(31)
one then has = -AfL(x,K)F,(x,t).
F,s(x, r) = F;l‘(x, to) . (t/to)
.
(32)
J.K. Sarma et al./Physics
142
which gives the t-evolution of singlet structure function Fi (x, t) . Again defining
Letters B 403 (1997) 139-144
FFS(x,t)
= FFS(xo,t)
[J
‘jl/AfB(x)
x exp F(xo,
t) =
XV(x)
-$.
I.
Y”(x)
s
I
(33)
[$i$
=F2(xo,t)
(38) which give the t and x-evolutions of non-singlet structure function FFs. The F2 deuteron and proton structure functions measured in deep inelastic electro-production can be written in terms of singlet and non-singlet quark distribution functions as
1
a Y’(X) 1
[
,
-10
*=1”
one then has
Fi(x,t)
- A(x)/B(x)}dx]
I=xo
so that F; = IFS 9 2 x
1
x exp
-- L(x) M(x)
AfM(x)
& I
X0
and
1
(34)
which gives the x-evolution of Fi (x, t). On the other hand, the AP equation for the nonsinglet structure function
F”2 = LFNS + hF;,
at
-
+[{3+4ln(l
Using (32) and (34) in (39) we will get the t and x-evolutions of the deuteron structure function at low x as
+2
s
- 2F,Ns(x, t)}
=o
t) = F;(n,
to) . (t/to)
(41)
and
-~)}F~~(x,t)
F;(x,
I
dw p{(l (l-w)
(40)
18 2
F:(L dFNS(x, t)
(39)
to) = F$(xo, t)
+ w2)F,NS(x/w,t) xexp
1
[I
‘{l/A,M(x)
-L(*)lM(r)jdx]
(42)
-'iO
using the input functions (35) F;(x&)
can be written as
and
c?FF’( x, f)
F;(xo,
at AFRO
= $F,s(x, to)
+ B(x)
aFpS (x, t) dx
Similarly using (32) and (37) in (40) we have the t-evolution of the proton structure function at low x as
]
=o
t) = $F;s(xo, t) .
(36)
F;(x,t)
= F;(x,to)
(t/to)
(43)
which is free from the additional assumption (16). Using the same procedure as for the singlet equation, Eq. (36) yields
using the input functions
FFS(x,t)
But the x-evolution of the proton structure function like those of the deuteron structure function is not possible by this methodology; because to extract the
and
= F;‘(x,to)
.(t/to)
(37)
F;(x,to)
= $F~S(~,to)
+ $F;(x,to).
J.K. Sarma et al./Physics Letters B 403 (1997) 139-144
0.4 0.2 -
10-z
***b
l _*
l-T
10‘2
-
10-l
k*
10'2
1F'
10'
lo-'
X
Fig. I. Nucleon structure function, F2(D) obtained by EMC NA 2X from deuteron as a function of x for different intervals of Q? (in GeV’ ) Statistical errors are indicated by bars; systematic errors are shown by the bands beneath. In addition to the marked errors there is an overall normalization error of 7%. Here solid lines are our results (J?q. (42) for Nf = 4 and K N 10-1012. Input data points are given by arrow heads. x-evolution of the proton structure function we are to put ( 34) and (38) in (40). But as the functions inside the integral sign of Eqs. (34) and (38) are different, we need to separate the input functions Ft( x0, t) and FFs (x0, t) from the data points to extract the xevolution of the proton structure function, which is not possible.
3. Results and discussion In our earlier analyses 151 we observed the excellent phenomenological success of the r-evolutions of deuteron and proton structure functions. Here we analyse the x-evolutions of the deuteron structure function. For a quantitative analysis we evaluate the integrals that occurred in (42) for Nf = 4 and present the results in Fig. 1 (solid lines) for EMC NA 28 deuteron data [ 81 in the K z 1O-1O’2 range. Input data points indicated by arrow heads are taken from experiments.
143
It is seen that our integrals are almost independent of the K-values particularly in the x < 0.1 range. These results conform well to the data especially for Q2 < 2 GeV2; but for Q2 > 2 GeV2, F.f grows faster as x decreases. This is a possible indication of the breakdown of (16) at high-Q2. A clearer testing of our result is actually the relation (38) which is free from the additional assumption ( 16). But non-singlet data is not sufficiently available at low x to test our result - Eq. (38). Generally the x-distributions of structure functions are assumed at a fixed low Q2 = Q,” value by various experimental and theoretical constraints and there is no universal agreement among these different assumptions. Then the values of structure functions at higher Q2 values are calculated from evolution equations. But here we present a method to calculate the xdistribution of the deuteron structure function for any value of Q2. By knowing the value of the structure function at a fixed value of x = x0, we can evaluate it for other values of x in the low-x region. This is a possible alternative to the various other phenomenological x-distributions discussed in the literature. Traditionally the AP equations provide a means of calculating the manner in which the parton distributions change at fixed x as Q2 varies. This change comes about because of the various types of parton branching emission processes and the x-distributions are modified as the initial momentum is shared among the various daughter partons. However the exact rate of modifications of x-distributions at fixed Q2 cannot be obtained from the AP equations since it depends not only on the initial x but also on the rates of change ofparton distributions with respect to x, d”F( x)/dx” (II = 1 to co), upto infinite order. Physically this implies that at high x, the parton has a large momentum fraction at its disposal and as a result radiates partons (including gluons) in innumerable ways, some of them involving complicated QCD mechanisms. However for low x, many of the radiation processes will cease to occur due to momentum constraints and the x-evolutions get simplified. It is then possible to visualise a situation in which the modification of the x-distribution simply depends on its initial value and its first derivative. In this simplified situation, the AP equations give information on the shapes of the x-distribution as demonstrated in this paper. Our result also indicates that the shapes of the x-distributions of all the structure func-
144
J.K. Sanna et al./Physics Letters B 403 (1997) 139-144
at low x which are some combinations of nonsinglet and singlet structure functions, are the same for all values of Q*. This is observed in all data including the HERA data. tions
References I I] G. Altarelli and G. Parisi, Nucl. Phys. B 12 ( 1977) 298; G. Altarelli, Phys. Rep. 81 ( 1981) 1; V.N. Gribov and L.N. Lipatov, Sov. J. Nucl. Phys. 20 ( 197.5) 94; Y.L. Dokshitzer, Sov. Phys. JETP 46 ( 1977) 641. [ 21 Y.Y. Balitsky and L.N. Lipatov, Sov. J. Nucl. Phys. 28 ( 1978) 822; E.A. Kuraev, L.N. Lipatov and Fadin, Sov. Phys. JETP 45 (1977) 199; L.V. Gribov, E.M. Zevin and M.G. Ryskin, Phys. Rep. 100 (1983) 1; Nucl. Phys. B 188 (1981) 555.
131 R.D. Ball and S. Forte, Phys. Lett. B 335 (1994) 77; B 336
141
151
I61
171 181
(1994) 77; A. De Rujula, S.L. Glashow, H.D. Politzer, S.B. Treiman, E Wilczek and A. Zee, Phys. Rev. D 10 (1974) 1649; A. Zee, F. Wilczek and S.B. Treiman, Phys. Rev. D 10 ( 1974) 2881. K. Prytz, Phys. Lett. B 311 ( 1993) 286; 332 ( 1994) 393; Kalpana Bora and D.K. Choudhury, Phys. Lett. B 354 ( 1995) 151; A.V. Kotokov and G. Parente, Phys. Lett. B 379 ( 1996) 195. D.K. Choudhury and J.K. Sarma, Pramana - J. Phys. 38 (1992) 481; 39 (1992) 273; J.K. Sarma and B. Das, Phys. Len. B 304 ( 1993) 323. D.K. Choudhury and A. Saikia, Pramana - J. Phys. 29 ( 1987) 385; 33 (1989) 359; 34 (1990) 85. L.F. Abbott, W.B. Atwood and R.M. Bamett, Phys. Rev. D 22 (1980) 582. EMC, NA 28; M. Ameodo et al., Nucl. Phys. B 333 ( 1990) 1; Phys. L&t. B 21 I (1988) 493.
