

 6 Sep 1995

Spin asymmetries in lepton-proton and proton-proton diffractive reactions

S.V.Goloskokov, BLTP, JINR, Dubna, Russia 1

Abstract It is shown that the longitudinal double spin asymmetry All in polarized diffractive Q _Q production depends strongly on the spin structure of the quark-pomeron vertex. Relevant experiments will be possible at HERA with a polarized proton beam.

Diffractive production of high pt jets has been observed experimentally in hadron-hadron collisions [1] and in deep inelastic lepton-proton scattering [2]. Such processes where a proton stays intact or becomes a low mass state are determined at high energies by the pomeron exchange. These reactions with high pt jets can be interpreted as the observation of the partonic structure of a pomeron [3]. In what follows we shall suppose that the observed effects can be predominated by the quark structure of the pomeron [4, 5].

The pomeron is a vacuum t-channel exchange that contributes to high-energy diffractive reactions. There are two well-known models that are used to describe the pomeron contribution. First is the nonperturbative two-gluon exchange model [6, 7], and second is the BFKL pomeron [8] based on the summation of leading perturbative logarithms. Such a "bare" pomeron exchange leads to the mainly imaginary scattering amplitude with a simple quarkpomeron vertex structure

V _qqIP , fl_: (1)

Then spin-flip effects are extremely small and in some sense in experiments [1, 2] the spinaverage distributions inside the pomeron have been studied.

However, the spin structure of the quark-pomeron vertex may be not so simple. Separation of low and high-energy contributions in the pomeron exchange leads to factorization of the quark-proton scattering amplitude into the high-energy spinless "bare" pomeron and lowenergy spin-dependent parts - quark-pomeron and hadron-pomeron vertices [9]. As a result, the pomeron contribution to the quark-proton amplitude looks as follows

T (s; t) = IP (s; t)V _qqIP \Omega V ppIP_ ; (2) where IP is a "bare" pomeron contribution, VqqIP and V ppIP are the quark-pomeron and proton-pomeron vertices, respectively. The quark-pomeron vertex has been calculated perturbatively [9]. It has a form

V _qqIP (k; r) = fl_u0(r) + 2mk_u1(r) + 2k_=ku2(r) + iu3(r)ffl_fffiaekffrfiflaefl5 + imu4(r)oe_ffrff: (3)

1Email: goloskkv@thsun1.jinr.dubna.su

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Here k is a quark momentum, r is a momentum transfer. The quantities ui(r) in (3) are the vertex functions. Note that the structure of the quark-pomeron vertex function (3) is drastically different from the "bare" pomeron. Really, only the term proportional to fl_ corresponds to the standard pomeron vertex (1) which reflects the well-known approximation that the spinless quark-pomeron coupling is like a C = +1 isoscalar photon. The terms u1(r) \Gamma u4(r) lead to the spin-flip in the quark-pomeron vertex in contrast to the term proportional to u0(r). The functions u1(r) \Xi u4(r) at large r2 are not very small [10]. Note that the phenomenological V _qqIP vertex with u0 and u1 terms was proposed in [11].

The proton-pomeron vertex has a form [12]

V _ppIP (p; r) = mp_A(r) + fl_B(r) (4) The quantity A in (4) determines the transverse polarization and B contributes to the longitudinal asymmetry. The ratio m2A=B is about 0.2 [12].

Thus, the pomeron vertices have a complicated spin structure. This should modify different spin asymmetries and lead to new effects in high energy diffractive reactions that can be measured in future spin experiments at HERA, for example.

One of the simplest way to test the quark-pomeron vertex is to study the Q _Q production in diffractive reactions. This sort of reaction has been investigated by different authors for unpolarized particles (see [4, 13, 14], e.g.).

In this report we shall discuss the longitudinal double spin asymmetries in polarized p " p "! p + Q _Q + X and l " p "! l + p + Q _Q diffractive reactions. Such experiments will be possible in future if the proton beam at HERA is polarized. Single transverse spin asymmetry will be discussed in an other talk.

Let us analyse the Q _Q production in diffractive pp scattering (see fig.1). It was shown in our previous estimations [15] that All asymmetry in this case can reach 10 \Xi 12%. Standard kinematical variables look as follows

s = (pi + p)2; t = r2 = (p \Gamma p0)2; xp = pi(p \Gamma p

0)

pip : (5) Here pi and p are initial proton momenta, p0 is a momentum of recoiled hadron, r is a momentum transfer at the pomeron vertex and xp is a part of momenta p carried off by the pomeron. The diagram of Fig.1 is important at small xp that leads to a small invariant mass in the Q _Q system M 2x , _yxps. The diagram with triple pomeron vertex must be considered for large M 2x . Then we shall see more than two high pt jets.

The pomeron will be a leading contribution when the energy sp in the Qp system is sufficiently large. One can obtain

sp = (k + p)2 * jtj + M

2Q

xp : (6)

2

p p' pp i

r Figure 1: The diffractive Q _Q production in pp reaction. So, sp is sufficiently large when jtj is a few GeV 2 and xp , 0:1 \Xi 0:2 . For such jtj the perturbative QCD can be used to calculate the spin structure of quark-pomeron vertex.

We shall calculate the longitudinal double spin asymmetry determined by the relation

All = \Delta oeoe = oe(

!() \Gamma oe(!))

oe(!)) + oe(!() : (7) In the longitudinal asymmetry case only the fl_B(r) term at the proton-pomeron vertex (4) contributes.

In calculations we shall use the hypothesis that the pomeron couples to a single quark (or an antiquark) for simplicity. The integration over the all Q _Q phase space will be performed. For a planar loop we find

oe(\Delta oe) = F (IP ) cx

p

Z 1

y0 dyg(y)(\Delta g(y)) Z

syxp=4 0

d2k?N oe(\Delta oe)(xp; k2?; ui; jtj)q

1 \Gamma 4k2?=syxp(k2? + M 2Q)2 : (8)

Here g(\Delta g) are the gluon spin-average and spin structure function of the proton, k? is a transverse part of the momentum in the loop, MQ is a quark mass, N oe(\Delta oe) is a trace over the quark loop. In (8) F (IP ) is a function which contains "bare" pomeron contribution IP (s; t) and pomeron-proton vertices V ppIP (see (2)). This, function is the same for oe and \Delta oe in (7) and cancels at All. Similar forms were found for the nonplanar loop.

We find that oe / 1=x2p at small xp. This behaviour is associated usually with the pomeron flux factor for ffIP (0) = 1. However, \Delta oe is proportional to ffl_*fffirfi::: / xpp. Thus, additional xp appears and we find that \Delta oe / 1=xp at small xp.

In calculations of the integrals (8) the off-mass-shell behaviour of the pomeron structure functions ui has been considered. It was found that the following forms can be used as a good approximation for all functions at not small jtj

ui(k2?; jtj) = jtjjtj + k2

? u

i(0; jtj): (9)

As a result, the convergence of integrals (8) over k? is improved.

We use the simple form of the u0(r) vertex function

u0(r) = _

20

_20 + jtj; r

2 = jtj;

3

with _0 , 1Gev introduced in [16]. The functions u1(r) \Xi u4(r) at jtj ? 1GeV 2 were calculated in perturbative QCD [10].

The main contribution to \Delta oe is proportional to the first moment of \Delta g

\Delta g = Z

1

0 dy\Delta g(y); (10) which is unknown now. However, the magnitude of \Delta g can be large, \Delta g , 3. This large magnitude of \Delta g is important in the explanation of the proton spin [17].

In calculation of oe we use the simple form of the gluon structure function

g(y) = Ry (1 \Gamma y)5; R = 3: This form corresponds to the pomeron with ffIP (0) = 1. Just the same approximation for the pomeron exchange has been used in calculations. The analysis can be made for the pomeron with ffIP (0) = 1 + ffi and more complicated structure functions but it does not change the results drastically.

The resulting asymmetry depends on the ratio

Cg = \Delta gR : (11) If (10) is fulfilled, Cg , 1. This magnitude will be used in what follows.

For a standard form of the pomeron vertex (1) we find

All = \Gamma

2xp(ln jtjM2

Q \Gamma 3)

ln jtjM2

Q (2 ln

sxp

4jtj + ln

jtj M2Q )

: (12)

For the pomeron vertex (3) the axial-like term V _(k; r) / u3(r)ffl_fffiaekffqfiflaefl5 is extremely important in asymmetry. The formula for asymmetry is more complicated in this case.

Our predictions for All asymmetry at ps = 40GeV and xp = 0:2 for the standard quarkpomeron vertex (fl_ term is taken into account in (3)) and the spin-dependent quark-pomeron vertex (3) are shown in Fig.2 for light quarks and in Fig.3 for the heavy (C) quark. It is easy to see that the obtained asymmetry strongly depends on the structure of the quark-pomeron vertex. For the spin-dependent quark-pomeron vertex, All asymmetry is smaller by factor 2 because oe in (7) is larger in this case. This is connected with the contribution of other ui structures.

As it was mentioned above, the asymmetry in pp polarized diffractive reactions depends on an unknown \Delta g spin gluon structure function of the proton. To obtain more explicit results, let us study Q _Q diffractive production in lepton-proton reaction. At small xp it is determined by a diagram similar to that shown in Fig.1 with the lepton and photon instead of the gluon structure function of proton in the upper part of Fig.1.

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Fig.2 Fig.3 Figure 2: The All asymmetry of light quarks production. Solid line -for standard; dot-dashed line -for spin-dependent quark-pomeron vertex.

Figure 3: The All asymmetry of heavy (C) quarks production. Solid line -for standard; dot-dashed line -for spin-dependent quark-pomeron vertex.

The standard set of kinematical variables looks as follows [2]

s = (pl + p)2; Q2 = \Gamma q2; t = (p \Gamma p0)2 y = pqp

lp ; x =

Q2 2pq ; fi =

Q2 2q(p \Gamma p0) ; xp =

q(p \Gamma p0)

qp ; (13)

where pl; p0l and p; p0 are initial and final lepton and proton momenta, respectively, q = pl \Gamma p0l.

The asymmetry is determined by formula (7). For a planar loop we find

oe(\Delta oe) = F (IP ) cx

p

Z syxp=4

0

d2k?N oe(\Delta oe)(xp; k2?; ui; jtj)q

1 \Gamma 4k2?=syxp(k2? + M 2Q)2 (14)

The main contributions to All asymmetry in the discussed region are determined by the u0 and u3 structures in (3). For a standard form of the pomeron vertex (1) we have

All = \Gamma xP y

Q2(2 \Gamma y)(ln( jtjm2

Q ) \Gamma 3)

[2jtj(1 \Gamma y) ln( Q

2

jtjfi ) + Q2y(2 \Gamma y) ln(

jtj m2Q )]

: (15)

The formulae for oe and \Delta oe for the pomeron vertex (3) can be found in [18]. Note that \Delta oe is proportional to Q2. As a result the asymmetry must increase with Q2 (15).

Our predictions for All asymmetry for energy ps = 300GeV estimated from perturbative vertex functions for y = 0:5 and xp = 0:2 for the standard quark pomeron vertex and the

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Fig.4 Fig.5 Figure 4: The jtj dependence of All asymmetry of light and heavy (C) quarks production at fixed Q2 = 10GeV 2. Solid line -for standard; dot-dashed line -for spin-dependent quarkpomeron vertex.

Figure 5: The Q2 dependence of All asymmetry of light and heavy (C) quarks production at fixed jtj = 3GeV 2. Solid line -for standard; dot-dashed line -for spin-dependent quarkpomeron vertex.

spin-dependent quark pomeron are shown in Fig. 4,5. In fig.4 the jtj dependence of All for fixed Q2 = 10GeV 2 (fi = 0:25) is shown. In fig.5 one can see the Q2 dependence of All for fixed jtj = 3GeV 2. The obtained asymmetry is not small and strongly depends on the spin structure of the quark-pomeron vertex. Asymmetry decreases with jtj growing and increase with Q2 growing.

The estimations show that total integrated cross section of light quark production in lp reaction is about 0:2 \Xi 0:1 nb [4, 14]. Our calculation shows that the cross section for C quark production has to be smaller by factor 3 \Xi 10.

The obtained asymmetries in lepton-proton and proton-proton reactions have some common properties

ffl The asymmetry for heavy quark production is sufficiently large and positive. ffl Asymmetry is opposite in sign for light and heavy quarks. This is determined by a

factor like (ln(jtj=M 2Q) \Gamma 3) that appears in the asymmetry. It is different in sign for MQ , 0:005GeV and MQ , 1:5GeV for the investigated momenta transfer.

ffl Asymmetry decreases with energy only logarithmically All , 1=ln(sxp=(4jtj).

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ffl Asymmetry is equal to zero at xp = 0. So, it is better to study it at xp = 0:1 \Xi 0:2 ffl Obtained asymmetry strongly depends on the structure of the quark-pomeron vertex. ffl The axial-like term V _(k; r) / u3(r)ffl_fffiaekffqfiflaefl5 is extremely important in asymmetry.

This term is proportional to momenta transfer r (r2 = jtj) and can be measured only for the pomeron contribution with nonzero momenta transfer. Then the detection of a final proton is extremely important, otherwise, the integration over jtj can be performed (see [2] e.g.).

The discussed spin-dependent contributions to the quark-pomeron and hadron-pomeron vertex functions modify different spin asymmetries and lead to new effects in high energy diffractive reactions which can be measured in spin experiments at future accelerators.

To summarize, we have presented in this letter the perturbative QCD analysis of the longitudinal double spin asymmetry in diffractive 2-jet production in lp and pp processes. The model prediction shows that the ALL asymmetry can be studied and the information about the spin structure of the quark-pomeron vertex can be extracted. It should be emphasized that the obtained spin effects are completely determined at fixed momenta transfer by the large-distance contributions in quark (gluon) loops. So, they have a nonperturbative character. The investigation of spin effects in diffractive reactions is an important test of the spin sector of QCD at large distances.

This work was supported in part by the Russian Fond of Fundamental Research, Grant 94-02-04616 and Heisenberg-Landau Grant.

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