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Nonvanishing tensor polarization of sea quarks in polarized deuterons
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PHYSICS
LETTERS B
Physics Letters B 398 (1997) 245-251
Nonvanishing tensor polarization of sea quarks in polarized deuterons N.N. Nikolaevaa,bac, W. Schgfer b a ITKP der Universitiit Bonn, NuJ3allee 14-16, D-53115 Bonn, Germany b IKP, KFA Jiilich, D-52425 Jiilich, Germany c L.D. Landau Institute, Kosygina 2, 11 I7 334 Moscow, Russia
Received 4 December 1996; revised manuscript received 20 February 1997 Editor: C. Mahaux
Abstract We show how the dependence of the diffractive nuclear shadowing and of the nuclear excess of pions on the deuteron spin alignment leads to a substantial tensor polarization of sea partons in the deuteron. The corresponding tensor structure function bz ( X, Q*) rises towards small x and we predict the about one per cent tensor asymmetry A2(x, Q*) = b2(x, Q*) /Fw (x, Q2) which by almost two orders in magnitude exceeds the effect evaluated earlier in the impulse approximation. We show that the integral Jo1dx bl (x, Q2) diverges, which implies that the sum rule &r dx bl (x, Q*) = 0 suggested by Close and Kumano does not exist. We comment on the impact of tensor polarization on the determination of the vector spin structure function gld( x, Q*) for the deuteron. @ 1997 Elsevier Science B.V.
The theoretical prejudice supported to a certain extent by experiment is that at high energies which in deep inelastic scattering (DIS) is x -+ 0 total cross sections cease to depend on the beam and target polarizations (hereafter X, Q2 are the DIS variables). For instance, the SLAC El43 data [ 1 ] suggest, and QCD evolution analyses of polarized DIS support [ 21, the vanishing vector polarization of sea quarks in the deuteron. In this paper we show that the tensor polarization of sea quarks in the deuteron is substantial and even rises at x -+ 0. ’ If S is a spin- 1 operator then & = 1 (sj& + Sk&$%32Sik) is the T-even and P-even observable. Conse&tently, for DIS of unpolarized leptons the photoab-
sorption cross sections ygA), crll\) and structure functions F$“’ (x, Q2> = & ( CT$? + @‘) can depend on the spin projection Sn = A onto the target spin quantization axis n: cd*) = co + ff~T#‘Zink+ . . . [461. Hereafter we focus on the y*-target collision axis (the z-axis) chosen for n. The tensor spin structure function b2 can be defined as
bz(x, Q2> = ~[F:+'(x,Q~)+F:-)(x,Q~)-~F,(~)(x,Q~)I = ; xe;x[q:+'(x,Q2) - ‘@‘(x,
1A brief report on these results was presented at the Workshop on Future Physics at HERA [ 31. 0370-2693/97/$17.00
Q2)1 .
By parity conservation
0 1997 Elsevier Science B.V. All rights reserved.
PII SO370-2693(97>00250-5
+q$-)(x,Q2)
f
(1) UC+) = u(-)
and F$+’ = F$ -),
N.N. Nikolaeva,
246
a)
W. SchCfer/Physics
Letters B 398 (1997) 245-251
b)
Fig. 1. Impulse approximation
(a) and Gribov’s inelastic
shadowing
(b),
(c) diagrams
for DE on the deuteron.
&+) = 4$-j. Similarly, UT(A) and aLA) define the transverse, br = 2nbl, and the longitudinal, bL = bp - bT, tensor structure functions. The precursor of the structure function bl has been discussed by Pais in 1967
they differ markedly only when f(* ( y) , f(‘)(y) < 1 [ 81. As a result, in the IA the tensor polarization of quarks A~(x, Q2) = b2(x, Q2)/F2d(x, Q2) is at most several units of 10m4 for x <, 0.5 [ 5,9,10]. Because of
[71. Because vector polarized deuterons are always tensor polarized, in DIS of polarized leptons on a 100 % polarized deuteron the vector spin asymmetry equals Consequently, the determination of AM = gt&$?. the vector spin structure function gtd only is possible if F,* were known:
the sum rule (4)) in the IA A2 (x, Q2) -+ 0 at x -+ 0. Furthermore, the widely discussed sum rule
Q2> = h&*)(x,
gld(X,
(2)
Q2) .
Hitherto F:*) , Fi”’ are completely unknown and all the deuteron data have been analyzed under the unwarranted assumption that F,d(x, Q2) for the unpolarized deuteron can be used instead of F:* (x, Q2) in (2). In order to justify this assumption and use the polarized deuteron data for tests of the Bjorken sum rule one needs the direct measurements, and the benchmark theoretical evaluations, of the tensor structure functions of the deuteron. In the impulse approximation (IA) of Fig. la discussed ever since [5] F;“)(x 9Q2) =
x
[
F+Qz)
s
y%) +F2,,(;>Q2)
1
subject to the sum rule for the Doppler-Fermi function
s
%J(“)( y) = I )
where y deuteron deuteron D-wave
(3) smearing
(4)
is a fraction of the lightcone momentum of the carried by the nucleon. For the pure S-wave f (*) (y) = f(‘)(y) and bz(x, Q2) = 0, the admixture makes f(*)(y) # f(‘)(y) but
J
dxbl
(x,
Q2) = 0
(3
holds in the IA. Close and Kumano conjectured that if the sea is unpolarized then this sum rule holds beyond the IA ( [ 111, for earlier discussions see also [12,13]).Forthepositivityconstraintsonbt,2(x,Q2) see [ 141, higher twist effects are discussed in [ 151. Here we report the beyond-the-IA evaluation of the tensor polarization of sea in the deuteron from Gribov’s inelastic shadowing diagrams of Fig. lb and lc [ 161 which lead to a nuclear eclipse effect: crt,t(‘Y*d) = a.t(‘Y*P) +a,,,(‘Y*n) - Aa,h(?‘*d) . In diffractive nuclear shadowing (NSH) the intermediate state X is excited by pomeron exchange (Fig. 1b) , in the second mechanism the state X is excited by pion exchange (Fig. lc) . The both contributions to the eclipse effect are sensitive to the alignment of nucleons in the polarized deuteron. Besides the sea they make gluons too tensor polarized (for an early discussion on gluon densities for spin axis n normal to the collision axis see [6,17,18]). Ever since [ 161 the both contributions to the eclipse effect in the deuteron have been discussed repeatedly [ 19-241, but the dependence on the tensor polarization has not been calculated before. The only exception is Ref. [25], but there the dominant effect of the D-wave has not been considered and the result for bl (x) evaluated in the reggeon exchange model vanishes at x -+ 0, whereas we find bl (x) which rises faster than $, what makes the sum rule integral (5) divergent rather than receiving a finite and small correction, as was discussed in [ 251.
N.N. Nikolaeva, W. Schiifer/Physics
Hereafter we focus on small x and do not consider negligible IA terms. We evaluate diffractive NSH using Gribov’s theory [ 161 extended to DIS in [ 271:
X
J
(4k*)
dM*Sb”)
Letters B 398 (1997) 245-251
where B = Q2/ (Q* + M*) is the Bjorken variable for DIS on pomerons and xp = x/B is a fraction of the nucleon’s momentum taken away by the pomeron, in the target rest frame k, = xpmN. Then diffractive NSH can be cast in the convolution form 1
( y* ---f X)
duD
247
AFCA+ sh
t,-k2
dtdM*
L
’
Q*> = x
(6)
x
Here M is the mass of the state X, k = (kl, momentum of the pomeron,
k, )
+Ad2(xp)FQ,Q*)
is the
, >
+Wqn(P) 1 J&‘WT:‘p
where An!“) (xp) is a nuclear modification pomeron flux functions (i = val, sea) :
,+(4k*) D
=
d3p =4* J C (2%-)3
+ z5(P’)W:!(P) Jz + UO(P)W2(P’) Jz +~(Pb2(P')
2 -3(d*fi)
uo(p’)uo(p)
(d*d) d2klShA)(4k2)e-
X
[3(d*fi)(dS)
Bi ItI .
- (d*d)]
[9(d*B’) (dp) (@fi’) (dfi’) +(d*d)]
,
(7)
1 *A(P) = fi(fiuo(p)(ad) + We (3(@5) (dfi) -(cd))) [26],uc(p),w2(p) aretheSandD-wave functions normalized by the condition SF’ (0) = 1, u are the Pauli spin matrices, P = p/lpi and p is the proton’s momentum in the deuteron, d(A) is the deuteron polarization vector. The polarization dependence of the form factor $,*I (4k*) and the nonvanishing b2 (x, Q*) come only from the S-D-wave interference and D-wave terms cc uo(p)wz(p’) + uc(p’)w2(p> andx WZ(P>W~(P'>. Decompose the diffractive DIS cross section into the pomeron valence and sea structure functions Fip (p) (i = val, sea) and the pomeron fluxes in the proton +; (XP>/XP [29,301,
Because NSH is a leading twist effect [ 27,281 we find a manifestly leading twist bp (x, Q*) . In the practical calculations we use the predictions [29] for Fjp and & (xp) from the color dipole approach to diffractive DIS [28,31] which is known to agree well with the Hl and ZEUS data [ 32,331. Following [ 281 we take for diffraction slopes B,, = Bsp = 6 GeV-* and B Vd = 2B,,,, the results for NSH only weakly depend on Bv,l, B,,, . The pion exchange dominates y*p + nX, contributes substantially to y*p -+ pX at a moderately small xp [ 34,351 and is a well established source of the ii-d asymmetry in DIS on the nucleon [ 36-381. The relevant input is da(y*p + Xn)/dtdM* times the isospin factors. The diagram of Fig. lc can be interpreted as DIS on nuclear excess pions in the deuteron and its contribution to AF, can be cast in the convolution form
2
CM*+ Q2) x
c i=val,sea
d&r*
+ dtdM*
X)
=
x $FT*(;.Q*)+
~tot(PP) 4n+&n 16~
c&,(~~)F;~(/3,Q*)e-~‘l~l
Q* 9
(10)
J
- (d*d)]
[3(d*B’)(dP’)
(da) -3(d*fi’)
of the
(8)
!j~$?t,Q*)
1 (11)
where FT is the pion structure function of excess pions equals
,
and the flux
d2kl
248
N.N. Nikolaeva, W. Schiilfer/Physics Letters B 398 (1997) 245-251
An(*)(z) r =-
Z
x
J
6&J %Tz
G;,,rN( z, t>
(25-j2
(t
-
1 ff$-@{ uo(p>uoW(zw
+ u”(p);(p”
[3(d*fi’) (2@‘k)(kd)
i-g>*
w*) - k2)
Table 1 The S-D wave decomposition of the mean multiplicities of excess pions in the deuteron for different polarizations r\ along the y*dcollision axis. The numerical results were obtained for the Bonn and Paris wave functions (AniA’)
-k2(fi’d))
. IO*
Bonn
- (2(kd)(kd*) + uoww*(P>
- (2(kd)(kd*)
-k*)] [Vd*B)
(20%)
(kd)
- k*(H)
-k*)]
+ w*(P)wzw>
Paris
A=0
A=fl
A=0
A=&1
S-wave S-D interference D-wave
0.146 -0.537 0.035
0.999 -0.268 0.085
0.133 -0.676 0.070
0.929 -0.331 0.153
total
-0.356
0.807
-0.466
0.750
2
x [WW (d*fi’) (2(W)(W’) - 3(d*fi’) (2(fi’k)(kd)
- k2(@‘d))
- 3(d*fi) (2(fik)(kd) + 2(kd) (kd*) - k*]
k2(iW’))
- k2(fid))
>
,
(12)
k = (kl, zmN) is the pion momentum (notice the similarity between z and xp), t = -(k: + z*mi)/( lz ), &NN is the rNN coupling and Gr,v,v( z, t) is the ?rNN form factor. At z << 1 relevant to the present analysis the Regge model parameterization Glrm(Z,t)
=exp(-(A~+LYblog(%))(ltl+~~))
The nonrelativistic treatment of the deuteron holds, and binding effects can be neglected, at such a small xp, z ( [ 221 and references therein). The arguments for using the on-mass shell pion structure function are reviewed in [ 381. After the da( y*p -+ nX) will be measured at HERA [ 341, the pion effect will be calculated free of any uncertainties with the off-mass shell behaviour [44]. To have a quick impression on the effect of excess pions notice that at x <, (z) the convolution ( 11) can be approximated by
AF$(x,Q*)
=-($f(fp*)
is
appropriate (for instance, see [ 391) . This form factor is constrained by the pion induced ii-6 asymmetry in the proton, with Ai = 0.3 GeV-*, the Regge slope u$ = 0.9 GeV2 and the Fr from [40] we find the same results as in the phenomenologically successful lightcone approach ( [ 36,37 1, for the review see
[381). Brief comments on bL (x, Q2) are in order. It does not vanish. Evidently, br, from excess pions is found replacing FT in ( 11) by Fz( x, Q2). The experimental data on da; (y* -+ X) / dtdM2 are as yet lacking, we evaluate bL (x, Q*) from NSH using the perturbative QCD approach [ 411 which gives R = bL/bT similar to R = cr~la~ for DIS on protons. We use the Bonn (Table 11 in Ref. [ 421) and Paris [43] wave functions as representatives of the weak and strong D-wave in the deuteron. The deuteron wave functionsin (7) and (12) makeAnLA)(z), An$‘(xp) and An,(,“a(xp) nonvanishing only at xp, z <, (z) N N 0.2, where Rd is the deuteron radius. (&miv-’
+;F+,Q’))(An”‘). The
numbers
of excess
(13) pions
(An:*))
=
l’, $
x An$,$( z ) are cited in Table 1. The pure S- and (negligibly small) D-wave terms give (An(*)) > 0, i.e., the antishadowing effect in A Fsh. The tensor polarization of sea quarks is proportional to (An;*)) - (An;‘)) = 1.16.10-* , whereas NSH for unpolarized deuterons is proportional to 5 ( (2A@f’) + (An(‘))) = 0.42 . lo-* which is strongly reduced by the” S-D wave interference from the pure S-wave contribution. For a comparison, for the number of pions in a nucleon one has (n,)N N 0.2 [36,37]. Table 1 shows the results are stable against the admissible variations of the deuteron wave function. The same is true of the diffractive NSH effects. In Fig. 2 and Fig. 3 we show the contribution from excess pions to AFsh (n, Q*) and bz (x, Q2) calculated from Eqs. (ll), (12) for Q2 = 10 GeV2 and for the
249
N.N. Nikolaeva, W. SchEfer / Physics Letters B 398 (1997) 245-251
0.10 - - - ---
2 a
diffractive NSH effect _ pion excess effect total
0.08 3 -&t 9
I
t
’
“““‘I
’
“““‘I
-
“‘““I
’ ’
- - -’ diffractive NSH effect _ - - -- excess pion effect
0.06 0.04 0.02 0.00 -0.02
0.04
&if
2 a
0.015
0.03
0.010
0.02
9 -m 4
0.01 0.00
1:::: U 1o-5
0.005 0.000 -0.005
lOA
1o-3
1o-2
10-l X
Fig. 2. (a) Nuclear shadowing correction in the deuteron AF,k (x, Q2) for Q2 = 10 GeV2 and its diffractive NSH and pion excess components for the Bonn wave function of the deuteron. (b) The fractional shadowing correction to the deuteron structure function Fzd(X, Q2) for Q2 = 10 GeV2.
GRV pion structure function [ 401 (for similar parameterizations see also [ 451) . Because the pion effect is small, our estimate of NSH for unpolarized deuterons in Fig. 2 is close to that of Ref. [24] in which only the pure diffractive NSH has been evaluated in the related model for diffractive DIS. A good description of NSH in heavy nuclei in the same approach [46] lends support to our numerical estimates. The results for b2(n, Q2) for the deuteron are shown in Fig. 3. The excess pions play much larger role in bz ( X, Q2) than in nuclear shadowing and dominate b2( x, Q2) at x >, 0.01. At smaller x the diffractive NSH mechanism takes over. In Fig. 2 and Fig. 3 we also show our estimate for bl;(x, Q2). Fig. 3b shows that the tensor polarization of sea quarks in the deuteron A~(x, Q2) is substantial and rises towards small X. Summary and conclusions. Regarding the numerical results, the uncertainties with the wave function of the deuteron are marginal. There are larger uncertainties with the pion and pomeron structure functions, which can be reduced with more data from HERA
X Fig. 3. (a) The tensor structure function of the deuteron b2(x, Q2) at Q2 = 10 GeV2 and its diffractive NSH and pion excess components for the Bonn wave function of the deuteron. (b) The tensor polarization A2( x, Q2) of quarks and antiquarks in the deuteron at Q2 = 10 GeV2. Also shown (by the dotted line) is the longitudinal tensor structure function bL(x, Q2) (in (a)), and the corresponding asymmetry (in (b) )
and do not call into question our finding of the striking spin dependence of DIS which persists, and even gets stronger, at small x. We regard our analysis as a benchmark calculation of the tensor structure function of the deuteron at small X, very similar results must follow from other models if they give similar description of the ii-d asymmetry in the proton and of diffractive DIS. RegFding the measurements of gid in polarized t?‘, ,Gd scattering, our results justify the substitution of the unpolarized deuteron structure function for the more correct but still unknown F.*) in (2) at the present accuracy in the spin asymmetry Aid, but corrections for the tensor polarization will eventually become important for the interpretation of high accuracy DIS off polarized deuterons in terms of the neutron spin structure function gi, and for the precision tests of the Bjorken sum rule. The both diffractive nuclear shadowing and pion ex-
2.50
N.N. Nikolaeva, W. Schiifer/Physics
cess mechanisms give a leading twist bz( n, Q*) . The pion contribution is obviously GLDAP evolving. The sea structure function of the pomeron also obeys the GLDAP evolution [ 3 11. The detailed form of scaling violations in the valence structure function of the pomeron at p -+ 1 remains an open issue [ 30,471, but the contribution from this region of .#?to bp (x, Q2) is marginal. Consequently, QCD evolution of bz( x, Q2) must not be any different from that of F2 ( X, Q2), see also a related discussion in [ 6,181. Furthermore, because of the similar origin of nuclear shadowing and of tensor asymmetry, we can invoke the theoretically [ 27,461 and experimentally (for the review see [ 481) well established weak Q2 dependence of nuclear shadowing. For instance, we expect a similarly large tensor asymmetry in atot(rr>) for real photons ( Q2 = 0). Our finding of the rising tensor polarization of sea quarks in the deuteron invalidates the assumptions invoked in the derivation of the Close-Kumano sum rule (5). Because we find bl (x) = &b:!(x) which rises faster than $, the Close-Kumano integral simply diverges at small x. Finally, Eqs. (9) and ( 12) are fully applicable to calculation of the tensor polarization of gluon densities in the deuteron polarized along the y*d collision axis in terms of gluon structure functions of the pomeron and pion. The results for the gluon structure functions will be presented elsewhere. We are grateful to V.N. Gribov, J. Speth, and B.G. Zakharov for helpful discussions. The useful correspondence with F.E. Close on the sum rule (5) and with A.W. Thomas on the work [23] are acknowledged. N.N.N. thanks Prof. U. MeiBner and D. Schiitte for the hospitality at the Institut f. Theoretische Kernphysik of the University of Bonn. The work of N.N.N. is supported by the DFG grant ME864/13-1. Note added. After the present paper has been submitted for publication, G. Piller informed us of a related analysis of the diffractive shadowing contribution to bl (x) [491.
Letters B 398 (1997) 245-251 [2] M. Ansehnino, (1995) 1;
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[lo]
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cm ELSEVIER
17 April 1997
PHYSICS
LETTERS B
Physics Letters B 398 (1997) 245-251
Nonvanishing tensor polarization of sea quarks in polarized deuterons N.N. Nikolaevaa,bac, W. Schgfer b a ITKP der Universitiit Bonn, NuJ3allee 14-16, D-53115 Bonn, Germany b IKP, KFA Jiilich, D-52425 Jiilich, Germany c L.D. Landau Institute, Kosygina 2, 11 I7 334 Moscow, Russia
Received 4 December 1996; revised manuscript received 20 February 1997 Editor: C. Mahaux
Abstract We show how the dependence of the diffractive nuclear shadowing and of the nuclear excess of pions on the deuteron spin alignment leads to a substantial tensor polarization of sea partons in the deuteron. The corresponding tensor structure function bz ( X, Q*) rises towards small x and we predict the about one per cent tensor asymmetry A2(x, Q*) = b2(x, Q*) /Fw (x, Q2) which by almost two orders in magnitude exceeds the effect evaluated earlier in the impulse approximation. We show that the integral Jo1dx bl (x, Q2) diverges, which implies that the sum rule &r dx bl (x, Q*) = 0 suggested by Close and Kumano does not exist. We comment on the impact of tensor polarization on the determination of the vector spin structure function gld( x, Q*) for the deuteron. @ 1997 Elsevier Science B.V.
The theoretical prejudice supported to a certain extent by experiment is that at high energies which in deep inelastic scattering (DIS) is x -+ 0 total cross sections cease to depend on the beam and target polarizations (hereafter X, Q2 are the DIS variables). For instance, the SLAC El43 data [ 1 ] suggest, and QCD evolution analyses of polarized DIS support [ 21, the vanishing vector polarization of sea quarks in the deuteron. In this paper we show that the tensor polarization of sea quarks in the deuteron is substantial and even rises at x -+ 0. ’ If S is a spin- 1 operator then & = 1 (sj& + Sk&$%32Sik) is the T-even and P-even observable. Conse&tently, for DIS of unpolarized leptons the photoab-
sorption cross sections ygA), crll\) and structure functions F$“’ (x, Q2> = & ( CT$? + @‘) can depend on the spin projection Sn = A onto the target spin quantization axis n: cd*) = co + ff~T#‘Zink+ . . . [461. Hereafter we focus on the y*-target collision axis (the z-axis) chosen for n. The tensor spin structure function b2 can be defined as
bz(x, Q2> = ~[F:+'(x,Q~)+F:-)(x,Q~)-~F,(~)(x,Q~)I = ; xe;x[q:+'(x,Q2) - ‘@‘(x,
1A brief report on these results was presented at the Workshop on Future Physics at HERA [ 31. 0370-2693/97/$17.00
Q2)1 .
By parity conservation
0 1997 Elsevier Science B.V. All rights reserved.
PII SO370-2693(97>00250-5
+q$-)(x,Q2)
f
(1) UC+) = u(-)
and F$+’ = F$ -),
N.N. Nikolaeva,
246
a)
W. SchCfer/Physics
Letters B 398 (1997) 245-251
b)
Fig. 1. Impulse approximation
(a) and Gribov’s inelastic
shadowing
(b),
(c) diagrams
for DE on the deuteron.
&+) = 4$-j. Similarly, UT(A) and aLA) define the transverse, br = 2nbl, and the longitudinal, bL = bp - bT, tensor structure functions. The precursor of the structure function bl has been discussed by Pais in 1967
they differ markedly only when f(* ( y) , f(‘)(y) < 1 [ 81. As a result, in the IA the tensor polarization of quarks A~(x, Q2) = b2(x, Q2)/F2d(x, Q2) is at most several units of 10m4 for x <, 0.5 [ 5,9,10]. Because of
[71. Because vector polarized deuterons are always tensor polarized, in DIS of polarized leptons on a 100 % polarized deuteron the vector spin asymmetry equals Consequently, the determination of AM = gt&$?. the vector spin structure function gtd only is possible if F,* were known:
the sum rule (4)) in the IA A2 (x, Q2) -+ 0 at x -+ 0. Furthermore, the widely discussed sum rule
Q2> = h&*)(x,
gld(X,
(2)
Q2) .
Hitherto F:*) , Fi”’ are completely unknown and all the deuteron data have been analyzed under the unwarranted assumption that F,d(x, Q2) for the unpolarized deuteron can be used instead of F:* (x, Q2) in (2). In order to justify this assumption and use the polarized deuteron data for tests of the Bjorken sum rule one needs the direct measurements, and the benchmark theoretical evaluations, of the tensor structure functions of the deuteron. In the impulse approximation (IA) of Fig. la discussed ever since [5] F;“)(x 9Q2) =
x
[
F+Qz)
s
y%) +F2,,(;>Q2)
1
subject to the sum rule for the Doppler-Fermi function
s
%J(“)( y) = I )
where y deuteron deuteron D-wave
(3) smearing
(4)
is a fraction of the lightcone momentum of the carried by the nucleon. For the pure S-wave f (*) (y) = f(‘)(y) and bz(x, Q2) = 0, the admixture makes f(*)(y) # f(‘)(y) but
J
dxbl
(x,
Q2) = 0
(3
holds in the IA. Close and Kumano conjectured that if the sea is unpolarized then this sum rule holds beyond the IA ( [ 111, for earlier discussions see also [12,13]).Forthepositivityconstraintsonbt,2(x,Q2) see [ 141, higher twist effects are discussed in [ 151. Here we report the beyond-the-IA evaluation of the tensor polarization of sea in the deuteron from Gribov’s inelastic shadowing diagrams of Fig. lb and lc [ 161 which lead to a nuclear eclipse effect: crt,t(‘Y*d) = a.t(‘Y*P) +a,,,(‘Y*n) - Aa,h(?‘*d) . In diffractive nuclear shadowing (NSH) the intermediate state X is excited by pomeron exchange (Fig. 1b) , in the second mechanism the state X is excited by pion exchange (Fig. lc) . The both contributions to the eclipse effect are sensitive to the alignment of nucleons in the polarized deuteron. Besides the sea they make gluons too tensor polarized (for an early discussion on gluon densities for spin axis n normal to the collision axis see [6,17,18]). Ever since [ 161 the both contributions to the eclipse effect in the deuteron have been discussed repeatedly [ 19-241, but the dependence on the tensor polarization has not been calculated before. The only exception is Ref. [25], but there the dominant effect of the D-wave has not been considered and the result for bl (x) evaluated in the reggeon exchange model vanishes at x -+ 0, whereas we find bl (x) which rises faster than $, what makes the sum rule integral (5) divergent rather than receiving a finite and small correction, as was discussed in [ 251.
N.N. Nikolaeva, W. Schiifer/Physics
Hereafter we focus on small x and do not consider negligible IA terms. We evaluate diffractive NSH using Gribov’s theory [ 161 extended to DIS in [ 271:
X
J
(4k*)
dM*Sb”)
Letters B 398 (1997) 245-251
where B = Q2/ (Q* + M*) is the Bjorken variable for DIS on pomerons and xp = x/B is a fraction of the nucleon’s momentum taken away by the pomeron, in the target rest frame k, = xpmN. Then diffractive NSH can be cast in the convolution form 1
( y* ---f X)
duD
247
AFCA+ sh
t,-k2
dtdM*
L
’
Q*> = x
(6)
x
Here M is the mass of the state X, k = (kl, momentum of the pomeron,
k, )
+Ad2(xp)FQ,Q*)
is the
, >
+Wqn(P) 1 J&‘WT:‘p
where An!“) (xp) is a nuclear modification pomeron flux functions (i = val, sea) :
,+(4k*) D
=
d3p =4* J C (2%-)3
+ z5(P’)W:!(P) Jz + UO(P)W2(P’) Jz +~(Pb2(P')
2 -3(d*fi)
uo(p’)uo(p)
(d*d) d2klShA)(4k2)e-
X
[3(d*fi)(dS)
Bi ItI .
- (d*d)]
[9(d*B’) (dp) (@fi’) (dfi’) +(d*d)]
,
(7)
1 *A(P) = fi(fiuo(p)(ad) + We (3(@5) (dfi) -(cd))) [26],uc(p),w2(p) aretheSandD-wave functions normalized by the condition SF’ (0) = 1, u are the Pauli spin matrices, P = p/lpi and p is the proton’s momentum in the deuteron, d(A) is the deuteron polarization vector. The polarization dependence of the form factor $,*I (4k*) and the nonvanishing b2 (x, Q*) come only from the S-D-wave interference and D-wave terms cc uo(p)wz(p’) + uc(p’)w2(p> andx WZ(P>W~(P'>. Decompose the diffractive DIS cross section into the pomeron valence and sea structure functions Fip (p) (i = val, sea) and the pomeron fluxes in the proton +; (XP>/XP [29,301,
Because NSH is a leading twist effect [ 27,281 we find a manifestly leading twist bp (x, Q*) . In the practical calculations we use the predictions [29] for Fjp and & (xp) from the color dipole approach to diffractive DIS [28,31] which is known to agree well with the Hl and ZEUS data [ 32,331. Following [ 281 we take for diffraction slopes B,, = Bsp = 6 GeV-* and B Vd = 2B,,,, the results for NSH only weakly depend on Bv,l, B,,, . The pion exchange dominates y*p + nX, contributes substantially to y*p -+ pX at a moderately small xp [ 34,351 and is a well established source of the ii-d asymmetry in DIS on the nucleon [ 36-381. The relevant input is da(y*p + Xn)/dtdM* times the isospin factors. The diagram of Fig. lc can be interpreted as DIS on nuclear excess pions in the deuteron and its contribution to AF, can be cast in the convolution form
2
CM*+ Q2) x
c i=val,sea
d&r*
+ dtdM*
X)
=
x $FT*(;.Q*)+
~tot(PP) 4n+&n 16~
c&,(~~)F;~(/3,Q*)e-~‘l~l
Q* 9
(10)
J
- (d*d)]
[3(d*B’)(dP’)
(da) -3(d*fi’)
of the
(8)
!j~$?t,Q*)
1 (11)
where FT is the pion structure function of excess pions equals
,
and the flux
d2kl
248
N.N. Nikolaeva, W. Schiilfer/Physics Letters B 398 (1997) 245-251
An(*)(z) r =-
Z
x
J
6&J %Tz
G;,,rN( z, t>
(25-j2
(t
-
1 ff$-@{ uo(p>uoW(zw
+ u”(p);(p”
[3(d*fi’) (2@‘k)(kd)
i-g>*
w*) - k2)
Table 1 The S-D wave decomposition of the mean multiplicities of excess pions in the deuteron for different polarizations r\ along the y*dcollision axis. The numerical results were obtained for the Bonn and Paris wave functions (AniA’)
-k2(fi’d))
. IO*
Bonn
- (2(kd)(kd*) + uoww*(P>
- (2(kd)(kd*)
-k*)] [Vd*B)
(20%)
(kd)
- k*(H)
-k*)]
+ w*(P)wzw>
Paris
A=0
A=fl
A=0
A=&1
S-wave S-D interference D-wave
0.146 -0.537 0.035
0.999 -0.268 0.085
0.133 -0.676 0.070
0.929 -0.331 0.153
total
-0.356
0.807
-0.466
0.750
2
x [WW (d*fi’) (2(W)(W’) - 3(d*fi’) (2(fi’k)(kd)
- k2(@‘d))
- 3(d*fi) (2(fik)(kd) + 2(kd) (kd*) - k*]
k2(iW’))
- k2(fid))
>
,
(12)
k = (kl, zmN) is the pion momentum (notice the similarity between z and xp), t = -(k: + z*mi)/( lz ), &NN is the rNN coupling and Gr,v,v( z, t) is the ?rNN form factor. At z << 1 relevant to the present analysis the Regge model parameterization Glrm(Z,t)
=exp(-(A~+LYblog(%))(ltl+~~))
The nonrelativistic treatment of the deuteron holds, and binding effects can be neglected, at such a small xp, z ( [ 221 and references therein). The arguments for using the on-mass shell pion structure function are reviewed in [ 381. After the da( y*p -+ nX) will be measured at HERA [ 341, the pion effect will be calculated free of any uncertainties with the off-mass shell behaviour [44]. To have a quick impression on the effect of excess pions notice that at x <, (z) the convolution ( 11) can be approximated by
AF$(x,Q*)
=-($f(fp*)
is
appropriate (for instance, see [ 391) . This form factor is constrained by the pion induced ii-6 asymmetry in the proton, with Ai = 0.3 GeV-*, the Regge slope u$ = 0.9 GeV2 and the Fr from [40] we find the same results as in the phenomenologically successful lightcone approach ( [ 36,37 1, for the review see
[381). Brief comments on bL (x, Q2) are in order. It does not vanish. Evidently, br, from excess pions is found replacing FT in ( 11) by Fz( x, Q2). The experimental data on da; (y* -+ X) / dtdM2 are as yet lacking, we evaluate bL (x, Q*) from NSH using the perturbative QCD approach [ 411 which gives R = bL/bT similar to R = cr~la~ for DIS on protons. We use the Bonn (Table 11 in Ref. [ 421) and Paris [43] wave functions as representatives of the weak and strong D-wave in the deuteron. The deuteron wave functionsin (7) and (12) makeAnLA)(z), An$‘(xp) and An,(,“a(xp) nonvanishing only at xp, z <, (z) N N 0.2, where Rd is the deuteron radius. (&miv-’
+;F+,Q’))(An”‘). The
numbers
of excess
(13) pions
(An:*))
=
l’, $
x An$,$( z ) are cited in Table 1. The pure S- and (negligibly small) D-wave terms give (An(*)) > 0, i.e., the antishadowing effect in A Fsh. The tensor polarization of sea quarks is proportional to (An;*)) - (An;‘)) = 1.16.10-* , whereas NSH for unpolarized deuterons is proportional to 5 ( (2A@f’) + (An(‘))) = 0.42 . lo-* which is strongly reduced by the” S-D wave interference from the pure S-wave contribution. For a comparison, for the number of pions in a nucleon one has (n,)N N 0.2 [36,37]. Table 1 shows the results are stable against the admissible variations of the deuteron wave function. The same is true of the diffractive NSH effects. In Fig. 2 and Fig. 3 we show the contribution from excess pions to AFsh (n, Q*) and bz (x, Q2) calculated from Eqs. (ll), (12) for Q2 = 10 GeV2 and for the
249
N.N. Nikolaeva, W. SchEfer / Physics Letters B 398 (1997) 245-251
0.10 - - - ---
2 a
diffractive NSH effect _ pion excess effect total
0.08 3 -&t 9
I
t
’
“““‘I
’
“““‘I
-
“‘““I
’ ’
- - -’ diffractive NSH effect _ - - -- excess pion effect
0.06 0.04 0.02 0.00 -0.02
0.04
&if
2 a
0.015
0.03
0.010
0.02
9 -m 4
0.01 0.00
1:::: U 1o-5
0.005 0.000 -0.005
lOA
1o-3
1o-2
10-l X
Fig. 2. (a) Nuclear shadowing correction in the deuteron AF,k (x, Q2) for Q2 = 10 GeV2 and its diffractive NSH and pion excess components for the Bonn wave function of the deuteron. (b) The fractional shadowing correction to the deuteron structure function Fzd(X, Q2) for Q2 = 10 GeV2.
GRV pion structure function [ 401 (for similar parameterizations see also [ 451) . Because the pion effect is small, our estimate of NSH for unpolarized deuterons in Fig. 2 is close to that of Ref. [24] in which only the pure diffractive NSH has been evaluated in the related model for diffractive DIS. A good description of NSH in heavy nuclei in the same approach [46] lends support to our numerical estimates. The results for b2(n, Q2) for the deuteron are shown in Fig. 3. The excess pions play much larger role in bz ( X, Q2) than in nuclear shadowing and dominate b2( x, Q2) at x >, 0.01. At smaller x the diffractive NSH mechanism takes over. In Fig. 2 and Fig. 3 we also show our estimate for bl;(x, Q2). Fig. 3b shows that the tensor polarization of sea quarks in the deuteron A~(x, Q2) is substantial and rises towards small X. Summary and conclusions. Regarding the numerical results, the uncertainties with the wave function of the deuteron are marginal. There are larger uncertainties with the pion and pomeron structure functions, which can be reduced with more data from HERA
X Fig. 3. (a) The tensor structure function of the deuteron b2(x, Q2) at Q2 = 10 GeV2 and its diffractive NSH and pion excess components for the Bonn wave function of the deuteron. (b) The tensor polarization A2( x, Q2) of quarks and antiquarks in the deuteron at Q2 = 10 GeV2. Also shown (by the dotted line) is the longitudinal tensor structure function bL(x, Q2) (in (a)), and the corresponding asymmetry (in (b) )
and do not call into question our finding of the striking spin dependence of DIS which persists, and even gets stronger, at small x. We regard our analysis as a benchmark calculation of the tensor structure function of the deuteron at small X, very similar results must follow from other models if they give similar description of the ii-d asymmetry in the proton and of diffractive DIS. RegFding the measurements of gid in polarized t?‘, ,Gd scattering, our results justify the substitution of the unpolarized deuteron structure function for the more correct but still unknown F.*) in (2) at the present accuracy in the spin asymmetry Aid, but corrections for the tensor polarization will eventually become important for the interpretation of high accuracy DIS off polarized deuterons in terms of the neutron spin structure function gi, and for the precision tests of the Bjorken sum rule. The both diffractive nuclear shadowing and pion ex-
2.50
N.N. Nikolaeva, W. Schiifer/Physics
cess mechanisms give a leading twist bz( n, Q*) . The pion contribution is obviously GLDAP evolving. The sea structure function of the pomeron also obeys the GLDAP evolution [ 3 11. The detailed form of scaling violations in the valence structure function of the pomeron at p -+ 1 remains an open issue [ 30,471, but the contribution from this region of .#?to bp (x, Q2) is marginal. Consequently, QCD evolution of bz( x, Q2) must not be any different from that of F2 ( X, Q2), see also a related discussion in [ 6,181. Furthermore, because of the similar origin of nuclear shadowing and of tensor asymmetry, we can invoke the theoretically [ 27,461 and experimentally (for the review see [ 481) well established weak Q2 dependence of nuclear shadowing. For instance, we expect a similarly large tensor asymmetry in atot(rr>) for real photons ( Q2 = 0). Our finding of the rising tensor polarization of sea quarks in the deuteron invalidates the assumptions invoked in the derivation of the Close-Kumano sum rule (5). Because we find bl (x) = &b:!(x) which rises faster than $, the Close-Kumano integral simply diverges at small x. Finally, Eqs. (9) and ( 12) are fully applicable to calculation of the tensor polarization of gluon densities in the deuteron polarized along the y*d collision axis in terms of gluon structure functions of the pomeron and pion. The results for the gluon structure functions will be presented elsewhere. We are grateful to V.N. Gribov, J. Speth, and B.G. Zakharov for helpful discussions. The useful correspondence with F.E. Close on the sum rule (5) and with A.W. Thomas on the work [23] are acknowledged. N.N.N. thanks Prof. U. MeiBner and D. Schiitte for the hospitality at the Institut f. Theoretische Kernphysik of the University of Bonn. The work of N.N.N. is supported by the DFG grant ME864/13-1. Note added. After the present paper has been submitted for publication, G. Piller informed us of a related analysis of the diffractive shadowing contribution to bl (x) [491.
Letters B 398 (1997) 245-251 [2] M. Ansehnino, (1995) 1;
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[lo]
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