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The deep-inelastic structure functions at x → 1
Volume 130B, number 3,4
PHYSICS LEI'TERS
20 October 1983
T H E D E E P - I N E L A S T I C S T R U C T U R E F U N C T I O N S A T x -+ 1
E G. DRUKAREV, E.M LEVIN and V.B. ROSENHOUSE Leningrad Nuclear Physws Institute, Lenmgrad. USSR
Received 19 January 1983 Revised manuscript received 26 July 1983
In this letter we discuss the results of our QCD calculations of the structure functions at x ~ 1 While integrating these functions we had to answer a number of questions, which we believe to be of certain physical interest.
l. Are the deep-inelastic processes infrared stable? In other words, are only those distances where the perturbatlve QCD is valid, revolved9 This question is not superfluous. Indeed, the simplest diagram, shown an fig la, contains two quarks . I , close to the mass-
shell. This system is described by the phenomenologlcal function ~ ( A , p±), where PAx (pz) is the longitudinal (transverse) relative m o m e n t u m o f the quarks. The function ~ enters the contribution o f this &agram to the structure function
:~1 Here we consider the two-quark hadrons - the mesons.
D(x' q2) - ( - x2) J
d2k± j 2
\q,
~2=_k 2>0,
J= fdAd2p± (27r)---~
I
\
/
C~
~' \ 2\
p o~
o_
,¢ ( A P i )
i/
\
g
k2<0;
/
\
(1)
oU
/ / %
0_'
a) Pc
P
S ~fl
Pd-
P
k,..,J P-a
P-9 Pal. 2
\ \
Pc
/
/ \
/
13~
Pet Fig. 1. 0 . 0 3 1 - 9 1 6 3 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 03.00 © 1983 North-Holland
223
Volume 130B, number 3,4
PHYSICS LETTERS
Eq. (1) is obtained for the massless quark. One can see that integral (1) diverges at k 2 ~ 0. The confinement mechanism at ~2 ~ X2 (X- 1 is the confinement radms) removes the dwergence. If this situation would have survived during the calculatmns of the total structure functmn, they would have been mvahd since the mare contrlbutmn would have come from small k 2 ~ X2. 2. Can one express the hadron structure functmn through the structure functmn of the quark and the wave function of the hadron quarks~ If there is such formula, it would be useful for phenomenologacal apphcations. 3. What can we say about the hadron let, generated by the quark-spectator? We answered these questions m the framework of the leading loganthmm approximatxon (LIA) m the region: (Xs(q2) (In q2/X2) ln(1 - x) >~ 1 ,
= l" d2pi dA d3p
~(k±,x')
3
X p(p2, a , o)[(l -
x')/ks]
as(k2/(1 -
x')).
(3)
Here the phenomenologmal function ~ describes the two quarks, separated by the large distance 1/X2 whde P gives the exchange by hard gluons with ~2 >> X2 [but #2 < ~:2/( 1 _ x)]. The function 1" was calculated by Brodsky and Lepage [2]. The last factor comes from the exchange by the hard gluon with k 2 = k2/(1 - x) between the two quarks. As usually, we assumed the meson quark transverse momentum to be restricted by the condition This leads to the estimanon ~0~ 1/k±. If one replaces the meson by the photon [ 5 - 7 ] , the quarks transverse momenta become as large as k2 "" q2(l - x) (fig. lc),
%(X2/(1 - x)) ln(1 - x) ~ 1, %(q2) In q2 /X2 ~ 1, (2)
We used the same approach as the authors of refs. [ 1 - 3 ] . There are two possxble mechamsms of the process. In the first the quark obtains the momentum Px (P is the hadron momentum) by exchanging the hard gluon with another quark. It takes place when the quark approach at short distance ~(1 - x)/X 2. At such a distance perturbatlve QCD is vahd. The lowest order QCD term is shown m fig. la. The struck quark c obtains large vlrtuality p2 ~ X2/(I - x). Note that only the special choice of gluon gauge reduces the lowest order QCD contrxbutmn to a single graph [4] So does the axial gauge, which is used in the paper. There is also some probability to find the hadron quark with momentum Px near the mass shell This is the alternative mechanism. Our hypothesis is that this probability is small enough to be neglected. If we consider the next order QCD graphs, only few of them survive while the others cancel each other The exchange by the soft gluons between the meson 224
quarks a and b (fig. la) as well as that between the struck quarks c and c' contribute to the cross section Thus, the struck quark does not interact wxth the others (the impulse approximation). The sum of all the soft gluon exchanges between quarks a and b give us the meson wave function. It IS shown by the diagram of fig. 1b.
p2
~s(X2/(1 -- X)) ln2(1 -- x) ~ 1 ,
%(q2) ln(1 - x ) < 1.
20 October 1983
~0 =
af
d2p± dA p2 A
p(p2, ki ,x')
= c~F(q2(1 - x), k±, x) = ~ { l n [ q 2 ( 1 - x ) / X 2 ] } -c2/~2,
c~=1/137.
(4)
The last equation uses the result obtained In ref. [2]. The structure function of the virtual quark at x 1 in the case (2) has been calculated in ref. [4].
D(x'/x, q2, k2/(1 _ x ')) (1 -- x'/x) 4c2~-1 ( q2 = P(4C2~ ) exp / 4c2 In ~ - A~
(5) -
+4c2ln X2(I ~ x ' ) [ $ J ( ~ ) - ~
+ c2(3 -- 4")'E)~} ,
l~x"
)J
20 October 1983
PHYSICS LETTERS
Volume 130B, number 3,4
from the integration over K2, it should vary in the interval - k 2 >> K2 > X2. It gives
= ~(q2(x' -- X)) -- ~(k2(x ' - x)l(1 - x ' ) ) , AD7= ~(q2(x' - x)) - ~(q2),
(5 con'd)
YI~ is the Euler constant. In eq. (5) we used the usual QCD notations, c 2 = 4/3, ~(k 2) = (1//32) In in k21X 2' , z ; 2 /32 = 1 1 -- ~ n, It lS t h e n u m b e r o f flavours. This struct u r e function is the solution of tile Bethe-Salpeter
eq~aatlon which was obtained by direct summation of the Feynman diagrams We used the method elaborated by Gnbov and Llpatov [8] which was also used by Dokshitzer [9]. Such Bethe-Salpeter equation was written in the paper of Amati et al. [10], where it was used for calculations of the fragmentation functions If we sum all the soft gluon corrections given by eqs. (3), (4),
1 f d K 2 Im[R(K2)/K2] 7r
=
1 +R(-k 2)-R(X 2)
[R(X 2) = 1 ] .
Thus, all the processes of soft gluon radiation by quark d give the same effect as those of the quark with the vmuahty ( - k 2 ) . One can see that these correcnons cancel each other - in the same way as an the case consldered in ref. [2] Hence,
R ( - k 2) = 1, 1
D ( x , q2) = c
2
dx' c - -
Dq ( , 4 1 ( , - X ' ) , q 2 , x ,Ix)
1
f dx' f (d2ki)lk2Dq(IC21(1- x ' ) , q2,x'lx) X V;2(k2/(1 - x), x ' ) .
(6)
We will see that the main contribution to (6) comes from larger k 2 than that to the lowest order term given by eq. (l). Thus, the final state spectator quark d obtains large transverse momentum ~:2 >> X2. Until it had small ~r2 ~ k 2, it could not radiate soft gluons with [~ >> 7t2 , since l2 = al~lq2 /x, a l < 1 - x ',/3l <[3k = - k 2 x / ( 1 - x ,)q2 (we used the Sudakov variables 1 + r = a l p 13lq 2+ l±) The equation for 3k came from the condition Pd = 0 So l 2 <~ ~:2 But for ~2 >> 7t2 it can. Hence, the structure function of the meson is
D(x'q2)=ldK2.~ ~r .,/'~x'f ×Dq((kl2
-
K2)/(1
-x
(8)
× ~o(k2/(1 - x ' ) , x ' ) .
x
d2k± x2
,-) , q 2 , x ,,/ x ),
X S0((k2 - Ic2)/(1 - x ' ) , x ' )
Im[R(g2)/K 2]
(7)
Here K2 is the virtuahty of the quark d. K2 = q2/3k(l --x')/x + k 2, while R (K 2) denotes the radiation of the soft gluons (real and virtual) by the quark d. As we wish to obtain the logarithmic contribution
The integrals (8) can be calculated by the steepest descent method. The main contribution to (8) comes from k 2 ~ X2/(1 - x ) a , a > 0 [4], The equations for a and the saddle point x 0 [4] give a=e/(1-e), (1 - X o ) / ( 1
e=e-~2/2C2=o.034
forn=3,
- x)
= { 4 c 2 [ ~ ( q 2 ) - - ~(~.2/(1 - - X ) ) +
1/J32]} - 1
(9)
Though a is numerically small, the main contribution to (8) comes from ~:2 >> )t2 Thus, the deep inelastic scattering by hadrons at x ~ 1 is infrared stable. Hence. the cross section can be calculated by the perturbative QCD method. This is the answer to the first question, As to the second question, the impulse approximation illustrated by fig ld is valid. The exphclt calculations of integrals (8) by the steepest descent method are presented in ref. [4]. They give for the structure function of the meson with spin S under condition (2)
:1-2 Note that m LLA ~(x) = ~( x ) ,
225
20 October 1983
PHYSICS LETTERS
Volume 130B, number 3,4
0x (
7r132 1/2
~5 -- 127E
/32 )
D,r(x, q 2 ) = f
dk 2 ~ [F(4c2 ~ + 1)(ln k2/X2) c2/~2 ]-1
X [(ln q2/X2)/(ln k2/X2)] ln(1 - x )
(12)
The final integration over k 2 leads to
l-'(~ + 2)
l-x!
(1-x
D, r ~ (ln q2/X2)/ln(1 - x ) ,
[2¢2 ln2(1 - x)
in agreement with the result obtained in ref. [6], but not with that, given by Witten [5]. As to the gluon structure function, note that the gluon can irradiate real gluons before converting into the quark pair (fig. le). The existence o f this channel leads to k 2 ~ X2 and the gluon structure function becomes, as one knows from ref. [6] ,Dg ~ (1 --X)4c2 ~(q2)
+ (3 - 47E)C2 [~(q2) _ ~(X2n/(1 _ x))l) X %2(X2rU(1 - x)) %(1 - x)J2s , ~0 = 1,
4 [ln(1 - x) - 1 ] -2c2/~2 ¢1 = ?
77 = 4c 2 [~(q2) _ ~(k2/(l _ x))] + 2 , e
7= ~
(
f12
)
l+i6e2(1_e)
4e2
-~ln(1-e)"
(10)
Here we omitted also the terms ~e. Eq. (10) gives us not only the double log and main single log terms but also the preexponent factor. The leading double log terms, as(q 2) In q2/X2 ln(1 - x), were given in a number of papers [ 3 - 1 2 ] . The terms as(q2) ln2(1 - x ) were calculated in refs. [2,9,1 1]. Note that due to the large leading values o f k 2 the power of (1 - x) differs from that given in the previous papers. As to the third question, since the virtuality of the quark d can be become as large as ~:2 _ X2/(1 _ x ) a it gives birth to the hadron jet with the mass m 2 X2/(1 - x ) a . So we expect the mass of the hadron jet to increase while l - x diminishes. The situation changes fundamentally if we replace the hadron by the photon. The impulse approximation is still valid The leading contribution 1o the structure function comes now from k 2 ~ k2[(1 - x) q2. In this regionDq(X'/X, q 2 k 2) ~ 1/(x' - x) and the p h o t o n structure function is determined by the size o f the region where we integrated over x ' and In k 2. Since ln(x' - x ) ~ In q2/~.2 In k2/X 2,
D(x'/x, k 2, q2) ~ (x' - x) 4cz~- 1/F(4c2~), = ~ ( q 2 ) _ ~(k2). Eqs. (4), (8) and (11) give us
226
(13)
(11)
We expect the investigation of the structure function at x -~ 1 in the framework o f the impulse approximation to be valid not only in LLA. We think, It can be the basis for phenomenologlcal and Monte Carlo calculations in the description of a large number of processes. We are indebted to Yu.L. Dokshitzer and M.G. Ryskln for fruitful discussions. We also thank V.L Chernyak, A V. Efremov, L.N. Llpatov and A.V. Radushkln for numerous talks in the subject.
References [1] G. Farrar and D. Jackson, Phys Rev. Lett 43 (1979) 246 G. Parisl, Phys. Lett. 84B (1979) 225, V.L Chernyak, Proc. XV LNPI Winter School, Vol. 1 (1980) p 65. [2] S. Brodsky and G.P. Lepage, Phys Lett 87 (1978) 359, Phys. Rev. 22 (1980) 2157. [3] A.V Efremov and A.V Radyushkln, Preprmt E2-87-521 (1981) [4] E.G Drukarev, E,M Levm and V.B. Rosenhouse, Preprint LNPI-764 (1982), Soy. J Nucl Phys. 37 (1983). [5] E. Wxtten, Nucl. Phys. B120 (1977) 189. [6] Yu.L. Dokshltzer, D I. Dyakonov and S.I. Troyan, Phys Rep. 58 (1980) 269 [7] A Chase, Nucl. Phys B189 (1981) 461 [8] V.N. Grabov and L.N. Llpatov, Soy. J Nucl Phys 15 (1972) 438, 675, L N. Llpatov, Soy J Nucl. Phys. 20 (1974) 94. [9] Yu L. Dokshltzer, Zh. Eksp. Teor. Fyz. 71 (1977) 1216. [10] D. Aman, A. Basseto, M. Clafaloni, G. Marchesml and G. Venezlano, Nucl. Phys. B173 (1980) 429 [11] G. Parisl, Phys Lett 90B (1980)295 [12] G CUrCland M. Greco, Phys Lett. 92B (1980) 175, 102B (1981) 280.
PHYSICS LEI'TERS
20 October 1983
T H E D E E P - I N E L A S T I C S T R U C T U R E F U N C T I O N S A T x -+ 1
E G. DRUKAREV, E.M LEVIN and V.B. ROSENHOUSE Leningrad Nuclear Physws Institute, Lenmgrad. USSR
Received 19 January 1983 Revised manuscript received 26 July 1983
In this letter we discuss the results of our QCD calculations of the structure functions at x ~ 1 While integrating these functions we had to answer a number of questions, which we believe to be of certain physical interest.
l. Are the deep-inelastic processes infrared stable? In other words, are only those distances where the perturbatlve QCD is valid, revolved9 This question is not superfluous. Indeed, the simplest diagram, shown an fig la, contains two quarks . I , close to the mass-
shell. This system is described by the phenomenologlcal function ~ ( A , p±), where PAx (pz) is the longitudinal (transverse) relative m o m e n t u m o f the quarks. The function ~ enters the contribution o f this &agram to the structure function
:~1 Here we consider the two-quark hadrons - the mesons.
D(x' q2) - ( - x2) J
d2k± j 2
\q,
~2=_k 2>0,
J= fdAd2p± (27r)---~
I
\
/
C~
~' \ 2\
p o~
o_
,¢ ( A P i )
i/
\
g
k2<0;
/
\
(1)
oU
/ / %
0_'
a) Pc
P
S ~fl
Pd-
P
k,..,J P-a
P-9 Pal. 2
\ \
Pc
/
/ \
/
13~
Pet Fig. 1. 0 . 0 3 1 - 9 1 6 3 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 03.00 © 1983 North-Holland
223
Volume 130B, number 3,4
PHYSICS LETTERS
Eq. (1) is obtained for the massless quark. One can see that integral (1) diverges at k 2 ~ 0. The confinement mechanism at ~2 ~ X2 (X- 1 is the confinement radms) removes the dwergence. If this situation would have survived during the calculatmns of the total structure functmn, they would have been mvahd since the mare contrlbutmn would have come from small k 2 ~ X2. 2. Can one express the hadron structure functmn through the structure functmn of the quark and the wave function of the hadron quarks~ If there is such formula, it would be useful for phenomenologacal apphcations. 3. What can we say about the hadron let, generated by the quark-spectator? We answered these questions m the framework of the leading loganthmm approximatxon (LIA) m the region: (Xs(q2) (In q2/X2) ln(1 - x) >~ 1 ,
= l" d2pi dA d3p
~(k±,x')
3
X p(p2, a , o)[(l -
x')/ks]
as(k2/(1 -
x')).
(3)
Here the phenomenologmal function ~ describes the two quarks, separated by the large distance 1/X2 whde P gives the exchange by hard gluons with ~2 >> X2 [but #2 < ~:2/( 1 _ x)]. The function 1" was calculated by Brodsky and Lepage [2]. The last factor comes from the exchange by the hard gluon with k 2 = k2/(1 - x) between the two quarks. As usually, we assumed the meson quark transverse momentum to be restricted by the condition This leads to the estimanon ~0~ 1/k±. If one replaces the meson by the photon [ 5 - 7 ] , the quarks transverse momenta become as large as k2 "" q2(l - x) (fig. lc),
%(X2/(1 - x)) ln(1 - x) ~ 1, %(q2) In q2 /X2 ~ 1, (2)
We used the same approach as the authors of refs. [ 1 - 3 ] . There are two possxble mechamsms of the process. In the first the quark obtains the momentum Px (P is the hadron momentum) by exchanging the hard gluon with another quark. It takes place when the quark approach at short distance ~(1 - x)/X 2. At such a distance perturbatlve QCD is vahd. The lowest order QCD term is shown m fig. la. The struck quark c obtains large vlrtuality p2 ~ X2/(I - x). Note that only the special choice of gluon gauge reduces the lowest order QCD contrxbutmn to a single graph [4] So does the axial gauge, which is used in the paper. There is also some probability to find the hadron quark with momentum Px near the mass shell This is the alternative mechanism. Our hypothesis is that this probability is small enough to be neglected. If we consider the next order QCD graphs, only few of them survive while the others cancel each other The exchange by the soft gluons between the meson 224
quarks a and b (fig. la) as well as that between the struck quarks c and c' contribute to the cross section Thus, the struck quark does not interact wxth the others (the impulse approximation). The sum of all the soft gluon exchanges between quarks a and b give us the meson wave function. It IS shown by the diagram of fig. 1b.
p2
~s(X2/(1 -- X)) ln2(1 -- x) ~ 1 ,
%(q2) ln(1 - x ) < 1.
20 October 1983
~0 =
af
d2p± dA p2 A
p(p2, ki ,x')
= c~F(q2(1 - x), k±, x) = ~ { l n [ q 2 ( 1 - x ) / X 2 ] } -c2/~2,
c~=1/137.
(4)
The last equation uses the result obtained In ref. [2]. The structure function of the virtual quark at x 1 in the case (2) has been calculated in ref. [4].
D(x'/x, q2, k2/(1 _ x ')) (1 -- x'/x) 4c2~-1 ( q2 = P(4C2~ ) exp / 4c2 In ~ - A~
(5) -
+4c2ln X2(I ~ x ' ) [ $ J ( ~ ) - ~
+ c2(3 -- 4")'E)~} ,
l~x"
)J
20 October 1983
PHYSICS LETTERS
Volume 130B, number 3,4
from the integration over K2, it should vary in the interval - k 2 >> K2 > X2. It gives
= ~(q2(x' -- X)) -- ~(k2(x ' - x)l(1 - x ' ) ) , AD7= ~(q2(x' - x)) - ~(q2),
(5 con'd)
YI~ is the Euler constant. In eq. (5) we used the usual QCD notations, c 2 = 4/3, ~(k 2) = (1//32) In in k21X 2' , z ; 2 /32 = 1 1 -- ~ n, It lS t h e n u m b e r o f flavours. This struct u r e function is the solution of tile Bethe-Salpeter
eq~aatlon which was obtained by direct summation of the Feynman diagrams We used the method elaborated by Gnbov and Llpatov [8] which was also used by Dokshitzer [9]. Such Bethe-Salpeter equation was written in the paper of Amati et al. [10], where it was used for calculations of the fragmentation functions If we sum all the soft gluon corrections given by eqs. (3), (4),
1 f d K 2 Im[R(K2)/K2] 7r
=
1 +R(-k 2)-R(X 2)
[R(X 2) = 1 ] .
Thus, all the processes of soft gluon radiation by quark d give the same effect as those of the quark with the vmuahty ( - k 2 ) . One can see that these correcnons cancel each other - in the same way as an the case consldered in ref. [2] Hence,
R ( - k 2) = 1, 1
D ( x , q2) = c
2
dx' c - -
Dq ( , 4 1 ( , - X ' ) , q 2 , x ,Ix)
1
f dx' f (d2ki)lk2Dq(IC21(1- x ' ) , q2,x'lx) X V;2(k2/(1 - x), x ' ) .
(6)
We will see that the main contribution to (6) comes from larger k 2 than that to the lowest order term given by eq. (l). Thus, the final state spectator quark d obtains large transverse momentum ~:2 >> X2. Until it had small ~r2 ~ k 2, it could not radiate soft gluons with [~ >> 7t2 , since l2 = al~lq2 /x, a l < 1 - x ',/3l <[3k = - k 2 x / ( 1 - x ,)q2 (we used the Sudakov variables 1 + r = a l p 13lq 2+ l±) The equation for 3k came from the condition Pd = 0 So l 2 <~ ~:2 But for ~2 >> 7t2 it can. Hence, the structure function of the meson is
D(x'q2)=ldK2.~ ~r .,/'~x'f ×Dq((kl2
-
K2)/(1
-x
(8)
× ~o(k2/(1 - x ' ) , x ' ) .
x
d2k± x2
,-) , q 2 , x ,,/ x ),
X S0((k2 - Ic2)/(1 - x ' ) , x ' )
Im[R(g2)/K 2]
(7)
Here K2 is the virtuahty of the quark d. K2 = q2/3k(l --x')/x + k 2, while R (K 2) denotes the radiation of the soft gluons (real and virtual) by the quark d. As we wish to obtain the logarithmic contribution
The integrals (8) can be calculated by the steepest descent method. The main contribution to (8) comes from k 2 ~ X2/(1 - x ) a , a > 0 [4], The equations for a and the saddle point x 0 [4] give a=e/(1-e), (1 - X o ) / ( 1
e=e-~2/2C2=o.034
forn=3,
- x)
= { 4 c 2 [ ~ ( q 2 ) - - ~(~.2/(1 - - X ) ) +
1/J32]} - 1
(9)
Though a is numerically small, the main contribution to (8) comes from ~:2 >> )t2 Thus, the deep inelastic scattering by hadrons at x ~ 1 is infrared stable. Hence. the cross section can be calculated by the perturbative QCD method. This is the answer to the first question, As to the second question, the impulse approximation illustrated by fig ld is valid. The exphclt calculations of integrals (8) by the steepest descent method are presented in ref. [4]. They give for the structure function of the meson with spin S under condition (2)
:1-2 Note that m LLA ~(x) = ~( x ) ,
225
20 October 1983
PHYSICS LETTERS
Volume 130B, number 3,4
0x (
7r132 1/2
~5 -- 127E
/32 )
D,r(x, q 2 ) = f
dk 2 ~ [F(4c2 ~ + 1)(ln k2/X2) c2/~2 ]-1
X [(ln q2/X2)/(ln k2/X2)] ln(1 - x )
(12)
The final integration over k 2 leads to
l-'(~ + 2)
l-x!
(1-x
D, r ~ (ln q2/X2)/ln(1 - x ) ,
[2¢2 ln2(1 - x)
in agreement with the result obtained in ref. [6], but not with that, given by Witten [5]. As to the gluon structure function, note that the gluon can irradiate real gluons before converting into the quark pair (fig. le). The existence o f this channel leads to k 2 ~ X2 and the gluon structure function becomes, as one knows from ref. [6] ,Dg ~ (1 --X)4c2 ~(q2)
+ (3 - 47E)C2 [~(q2) _ ~(X2n/(1 _ x))l) X %2(X2rU(1 - x)) %(1 - x)J2s , ~0 = 1,
4 [ln(1 - x) - 1 ] -2c2/~2 ¢1 = ?
77 = 4c 2 [~(q2) _ ~(k2/(l _ x))] + 2 , e
7= ~
(
f12
)
l+i6e2(1_e)
4e2
-~ln(1-e)"
(10)
Here we omitted also the terms ~e. Eq. (10) gives us not only the double log and main single log terms but also the preexponent factor. The leading double log terms, as(q 2) In q2/X2 ln(1 - x), were given in a number of papers [ 3 - 1 2 ] . The terms as(q2) ln2(1 - x ) were calculated in refs. [2,9,1 1]. Note that due to the large leading values o f k 2 the power of (1 - x) differs from that given in the previous papers. As to the third question, since the virtuality of the quark d can be become as large as ~:2 _ X2/(1 _ x ) a it gives birth to the hadron jet with the mass m 2 X2/(1 - x ) a . So we expect the mass of the hadron jet to increase while l - x diminishes. The situation changes fundamentally if we replace the hadron by the photon. The impulse approximation is still valid The leading contribution 1o the structure function comes now from k 2 ~ k2[(1 - x) q2. In this regionDq(X'/X, q 2 k 2) ~ 1/(x' - x) and the p h o t o n structure function is determined by the size o f the region where we integrated over x ' and In k 2. Since ln(x' - x ) ~ In q2/~.2 In k2/X 2,
D(x'/x, k 2, q2) ~ (x' - x) 4cz~- 1/F(4c2~), = ~ ( q 2 ) _ ~(k2). Eqs. (4), (8) and (11) give us
226
(13)
(11)
We expect the investigation of the structure function at x -~ 1 in the framework o f the impulse approximation to be valid not only in LLA. We think, It can be the basis for phenomenologlcal and Monte Carlo calculations in the description of a large number of processes. We are indebted to Yu.L. Dokshitzer and M.G. Ryskln for fruitful discussions. We also thank V.L Chernyak, A V. Efremov, L.N. Llpatov and A.V. Radushkln for numerous talks in the subject.
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