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Spin effects with polarized protons at RHIC
Nuclear PhysicsB361 (1991) 72-92 North-Holland
SPIN EFFECTS WITH POLARIZED PROTONS AT RHIC C. BOURRELY,J.PH. GUILLET* and J. SOFFER Centre de Physique Th~orique, CNRS Luminy, F-13288 Marseille Cedex 9, France
Received 14 January 1991 (Revised 13 March 1991)
We show that the use of RHIC as a polarized pp collider with a high designed luminosity will allow various interesting polarization studies. In particular, as we will see by considering several different reactions, this can help us to pin down the gluon polarized distribution function, to give a better insight for the u- and d-quarks polarization, to make dynamical tests of the Standard Model and to detect the possible existence of subconstituents.
1. Introduction A Relativistic Heavy Ion Collider (RHIC) is now under construction at Brookhaven National Laboratory and a possible realistic use of this machine is to study hadronic interactions with polarized protons with a luminosity of a few 10 32 cm -2 sec -1 and an energy of 50-250 GeV per beam with a polarization of about 70%. Polarization studies at current accelerator energies have been proven to be very useful and the observation, in hadronic collisions, of several puzzling effects like, for example at FNAL, inclusive hyperon production or ~.0 production in the central region, are indeed a real challenge for dynamical models [1]. Moreover, the recent result of the E M C experiment [2] on deep inelastic scattering (DIS) of polarized muons on a polarized target, has stimulated a vast theoretical debate and has generated the production of a huge number of papers [3]. Among the very many basic questions raised by the EMC result are: Is there something wrong with the proton spin? What are the expectations of quark models? What is the exact role of the Adler-Bell-Jackiw anomaly? How large are the perturbative and non-perturbative gluonic contributions to the proton spin? Is there a large fraction of the proton spin carried by the strange quarks? What is the connection with the U(1) problem? Etc. It is fair to say that, at present, the nucleon spin is not yet understood and it requires a special effort on the experimental side in polarized DIS for a more accurate direct determination of the polarized structure functions for both proton and neutron. Polarized pp collisions at supercollider energies *Present address: Theory Division CERN, 1211 Geneva 23, Switzerland. 0550-3213/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
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(LHC, SSC) were studied in detail [4]; one anticipates large single- and doublehelicity asymmetries for various processes which represent significant advantages for signal detection, especially for new physics. In this paper, we wish to argue that polarized pp collisions at RHIC energies can help answering some of the above fundamental issues related to the controversial interpretation of the EMC result, in particular, the magnitude and the sign of the gluon helicity asymmetry AG(x). As we will see, it is also possible to detect parity violating effects related either to the production of the vector gauge bosuns of the Standard Model or to the composite nature of the quarks if the corresponding energy scale is of the order of 1 TeV or so. The paper is organized as follows. In sect. 2 we briefly recall basic kinematics, the definition of the observables and our choices for the various parton distributions. In sect. 3 we will study direct photon production, a process which has a small cross section but whose double helicity asymmetry is very sensitive to AG(x). In sect. 4, we will consider a Standard Model parity violating asymmetry from the production of W ± at large PT which will help the determination of the polarized structure functions for u- and d-quarks. Sect. 5 is devoted to jet studies, and more specifically single-jet production, di-jet production whose cross sections are rather large, and we will show the importance of both the double and the single asymmetries. For this purpose we will give in appendix A all the polarized cross sections for quark-quark scattering, identical or distinct, to lowest order in gluon, y, W, Z exchanges including all interferences contributions. In sect. 6 we will see that compositeness can be tested at RHIC energies from single-jet production. We give our concluding remarks in sect. 7.
2. Basic kinematics, observables and parton distributions
Fundamental interactions at short distances which are probed in pp collisions at high energies, involve hard scattering of quarks and gluons. Let us consider the general hadronic reaction a+b~c+X,
(1)
where c, in the cases we will study below, is either a well-defined particle (hadron or gauge bosun) or a single jet. We will also consider a + b---> c + d + X for double-jet production. In the hard scattering kinematic region, the cross section describing (1) reads, in the QCD parton model, provided factorization holds, as 1
+ b -. c + X) : E 1 + 8,; f
[
+ (i oj')].
U
(2)
74
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The summation runs over all contributing parton configurations, the f(x, Q 2 ) ' s a r e the parton distributions, directly extracted from deep inelastic scattering for quarks and antiquarks, and d~q is the cross section for the interaction of two partons i and j which can be calculated perturbatively, some of which are given in appendix A. If we consider the reaction (1) with both initial hadrons longitudinally polarized, one useful observable is the double helicity hadron asymmetry ALL defined as dora(+)b(+) -- do'a(+)b(_ ) -- do.a(_)b(+ ) + dO'a(_)b(- ) ALL =
dO'a(+)b(+) + dO'a(+)b(- ) + do'a(_)b(+ ) + dO'a(_)b(_ ) '
(3)
which reduces to do.a(+)b(+) -- dtx.a(+)b(_ )
ALL ----
do.a(+)b(+) + dO'a(+)b(_ )
(3')
if parity is conserved because d%(~)b(A,) = d%(_~)b(_X). It is given by
ALLdo" = ~0" 1 +1 6i-"~-f dx~ dx b [Afi(a)( x a, Q2) Af~b)(xb ' Q 2 ) a^.. ^ + (i- j)], ~ L d~.;
(4) assuming the factorization property, where do- is given by eq. (2) and ~ L denotes the corresponding subprocess asymmetry for initial partons i and j. The af(x, Q2)'s are the parton helicity asymmetry defined as
Af(x, 0 2) = / + ( x , Q2) - f _ ( x , O2),
(5)
where f ± are the parton distributions in a polarized hadron with helicity either parallel ( + ) or antiparallel ( - ) to the parent hadron helicity. Recall that the unpolarized parton distributions are f = f++ f_. If only one initial hadron is polarized, say "a", another interesting observable is the sh,gle helicity asymmetry defined as (do-~(_)- do'~(+)) ALd°'= 2 ' (6) a quantity for which one can write an expression similar to eq. (4), provided Aflb)~fj.0,) and ~ L ~ a~', the single helicity subprocess asymmetry. A L is expected to vanish unless some subprocesses involve parity violating interactions, i.e. a~ 4: 0. If both initial beams are polarized, it is also possible to define another parity violating asymmetry, namely A~.v do" = (do.a(-)b(-) -- d°'a(+)b(+)) 2 '
(6')
which for special cases can be about twice as big as A L (see sect. 4). We will then
C. Bourrely et al. / Sph~ effects
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remember that, if at RHIC one achieves the same luminosity with polarized and unpolarized proton beams, it will certainly be more advantageous to measure A~ v rather than A L. Both for polarized and unpolarized parton distributions, we will take our own set in terms of a simple parametrization of their x and Q2 dependences already used in ref. [4]. Concerning the gluon helicity asymmetry AG(x, Q2) there is a standard choice (see e.g. eq. (3.12) in ref. [4]) such that zaG(x)/G(x) is only a few percent up to x -- 0.1. Since the relevant kinematic region stands at small x-values, it results that at small Q2, the integral zaG = fdzaG(x)dx is only 0.2-0.3 and it increases slowly with Q2. This choice means that one expects gluons to carry about 25% of the proton spin and until very recently there were no experimental facts to contradict this reasonable assumption. However after the striking EMC result [2], in particular on the small-x behaviour of the spin-dependent structure function g~(x, Q2), several interpretations and new pictures for the proton spin have been proposed [3]. Among these various possibilities, a suggestion, due to the existence of the anomaly of the axial vector current, leads one to anticipate a ziG(x) much larger than the standard choice, in particular in the small-x region. Therefore, following ref. [5], we will also consider the simple parametrization
AG(x,Q~) =
G(x, Q2o),
x¢
0
x xo,
(7)
which can lead to a much larger value of the integral AG than for the standard choice; the smaller x c the larger ziG. We will take x c ~ 0.2 corresponding to AG ~ 2-3, because this is the value which gives the best agreement with the gt(x, Q2) data. Clearly now gluons give a much too large contribution to the proton spin which ought to be balanced by a large orbital angular momentum. We have no very strong theoretical argument to prefer the standard form or this second choice (see eq. (7)) and, of course, due to the lack of our knowledge other choices are equally possible. These two rather different forms of the gluon helicity asymmetry will allow us to study how much a given reaction is sensitive to AG, whose precise form can only be determined from future data at RHIC. Since we will study in what follows, direct photon and W • production, let us now recall the expression of the differential cross section for producing a gauge boson of mass M with transverse momentum PT and rapidity y. It reads do"
dPTdY = 2PT
,;
fldx
X Xb
× fi(x~,QE)fi(Xb, Q 2 ) - ~ ( s ,^t ,^g O + ( i o j ) ) ,
(8)
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Spin effects
where the explicit M-dependence appears in all relevant kinematic variables,
~=XaXbS, ~ - M 2= --xamTvrse -y and a - M 2= --XbmTVr~.e-y, through the transverse mass m T = (M 2 +p~)X/2 and where (mT/V~-) e y - M2/s
xa(mT/V/s )e-V -- M2/s Xb=
Xa -- (mT/V/~-) e '
'
x0=
1 - (mT/~/~-) e - '
This expression can also be used for single-jet production by making M = 0, and it also holds for direct photon production. For the production of two jets with rapidity y~ and Y2 and with equal and opposite transverse momentum PT, we also recall that the invariant mass distribution reads do2M~ d M = --
LYy
•
×
1
rYmax
dylJy,,,° dY2~'..(1 + S o ) c o s h 2 y
*
tJ
d~o. ) fi(xa,Q2)fi(Xb,Q2)--~(~,t,a)+(i*--~j) ,
(9)
where y* = ~(Y~-Y2) 1 and the jet pair mass is given by M = 2PT cosh y*. Large values of y* correspond to the forward and backward directions at the parton level which should be avoided and this is why it should be restricted to the central region - Y < (y~, Y2) < Y with Y ~ 1. Finally we recall that all the subprocess cross sections and corresponding asymmetries, to complete the calculations of eqs. (8) and (9), are given in ref. [4].
3. Direct photon production Direct photon production at high PT is a useful probe of the underlying parton-parton interactions and probably one of cleanest reactions to study the perturbative regime. Since the photon originates in the hard scattering subprocess and does not fragment, the cross section is given by eq. (8) with M - - 0 . In the QCD parton model, in the absence of photon bremsstrahlung contributions, direct photons are produced via the q~ annihilation subprocess q~ ~ 7g and the quark-gluon Compton subprocess qg ~ qT. The Compton subprocess has a positive aLL and leads to a contribution to ALL, for ~ ~ 3'X, directly proportional to AG(x, Q2). For the annihilation subprocess one has aLL = - - 1 . First we have calculated the unpolarized cross section for two different values of the production angle 0c.m.= 90 ° and 45 ° corresponding to y = 0 and y = 0.88, respectively. The results for V~--300 and 600 GeV are shown in figs. la, b and clearly the prompt photon yield is more copious at higher energy. We also note that at 45 ° , except at very large PT where the event rates are low, the cross section is not significantly
C. Bourrely et al.
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10 -1
P P "* 7 + a n y t h i n g x/s = 300 G e V
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P P "* 7 + a n y t h i n g ~/s = 600 G e V
10- I
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,o
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1o-5
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,~ 10 -e b
10 -4 10 -7 i 0 -B
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i0 -o
(a)
(b)
10-1o
10-e 0
50
Pr (GeV/c)
100
i
0
50
100
Pr (GeV/c)
Fig. 1. (a) Predicted unpolarized cross sections for ~ - = 300 GeV as a function of PT, for two different values of the c.m. production angle: 0c.m.= 90 ° (dashed curve), 0c.m.= 45 ° (solid curve). (b) Same as (a), but for v~- = 600 GeV.
smaller than at 90° . Direct photon production has been measured in ~p collisions by the UA2 collaboration in this energy range [6] and given the high designed luminosity at RHIC, these cross sections will be measurable up to fairly high PT-Values. Let us now turn to the calculation of ALL. If one uses the standard distribution AG(x, Q2) (eq. (3.12) of ref. [4]), one obtains a very small positive result, say at most (5-10)%. This reflects the dominance of the Compton subprocess and the small magnitude of A Lt. is partly due to the fact that u-quark-gluon and d-quark-gluon lead to opposite sign effects. However if instead, one assumes a large distribution AG like in eq. (7), one gets the results displayed in figs. 2a, b. A L L rises with PT reaching values of the order of 20% or more for 0c.m.= 90 ° and much larger for 0c.m.= 45°. So this shows that the PT dependence of ALL in direct photon production is very sensitive to the gluon polarization and that it should best be measured with photons of rapidity of the order of 1. In fig. 2a at very large PT, ALL is dropping off because in this kinematic region one feels the effect of the annihilation subprocess which gives a negative contribution to ALL. Finally we notice that some of these characteristic features of ALL are also exhibited in the existing predictions for much lower values of v~- corresponding to fixed-target experiments (see refs. [5, 7]).
C. Bourrely et al. / Spin effects
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L
0.6 P P -* 7 + anything
f
\
=
0.4
0.2
(a) 0
50
0
I00
PT (GeV/c)
0.6
I
]
P P ~ 7 + anything ~/s = S00 OeV 0.4
0.2
(bl
0 0
........................ ;..
,
.
.
50
.
. I00
Pr (~eV/e) Fig. 2. (a) Predicted ALL with a large AG for ~ - = 300 G e V as a function of PT, for two different values of the c.m. production angle: 0c.ra.= 90 ° (dashed curve), 0c.m.= 45 ° (solid curve). The small-dashed curve corresponds to a standard AG. (b) Same as (a), but for ~fs = 600 GeV.
4. W + production
Let us now consider the single W + production with a large PT, which is balanced by a hadronic jet. This is similar to the direct photon production we have studied above (see sect. 3) but in this case one should consider the quark-antiquark annihilation subprocess qiqj ~ Wg and the quark gluon Compton subprocess qig--+ qi w with i ~ j . From the decay distribution of the W ± produced at
Bourrely et at / Spineffects
C.
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"x,
~
.
100
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Pr (GeV/c)
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s
150
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x
,
200
Fig. 3. Predicted unpolarized cross sections for ~ = 300 GeV and 600 GeV for y = 0 as a function of PT.
large PT it is possible to discriminate between the scalar and the vector nature of the gluon emitted in the final state [8]. By using this method based on the fact that, unlike the electromagnetic current, the axial current is not conserved, a scalar gluon was excluded at a 2o- level at the CERN collider [9]. Hopefully this will be confirmed by higher-statistics data at RHIC. The differential cross section do-/dp T dy for producing a W with PT and rapidity y is calculated using eq. (8) and we show our results in fig. 3 for y = 0 at two different energies. The production which is more abundant at V~-= 600 GeV is dominated by the Compton diagram and falls out very rapidly at large PT- If the beams are polarized, when the initial partons are carrying a given helicity we recall that for Compton scattering qi(h)g(h) ~ qiW we have d~ ~
rrota s
d [ (h h ) = '
12Xw~2
(1-h)[(1-h)cl([)+Ac2([)]
with cl(F)
= -
(s-M~v)2+
'
c2([)
2(a -M~)2~t~ ,
(10)
C. Bourrely et aL / Spin effects
80 . . . .
0.4
~/s -
i
. . . .
3 0 0 GeV
i
. ~. .....
---"-~S--'--"""""
i
. . . .
- ....................
~-
"
~/s -
: 6 0 0 GeV
p p .-,, W + + jet.
0.2
,#
. . . .
o
-0.2 ~
~
~/s = 600 GeV
~ / s = 3 0 0 GeV
pp
-0.4
0
50
100
P, (Gev/c)
-*W- +jeL
150
200
Fig. 4. Predicted A L in p p ~ W ± + j e t at y = 0 versus PT for ~ - = 3 0 0 GeV (solid curves) and vrS- = 600 GeV (dashed curves).
where x w = sin 0~v is the weak mixing angle. For F:ti(h)g(h)--* ~jW the same formula holds but with h --* - h and A --* - A . The corresponding single helicity asymmetries are d ~ = 1 for polarized quarks and a~. = 1 - c2/c I for polarized gluons. This last quantity being rather small on average, the single hadron helicity asymmetry will be dominated by polarized quarks and is not sensitive to AG. This is clearly shown in our results for A L at y = 0 versus PT displayed in fig. 4, where for W ÷ one sees the trend of Au(x)/u(x) and for W - the trend of Ad(x)/d(x). At fixed PT the effect decreases with increasing energy. Since the W's are left-handed objects, the consideration of A Pv would lead us to predict asymmetries about twice as big as these single helicity asymmetries and this is consistent with some earlier results given in ref. [10].
5. Jet studies 5.1. SINGLE-JET P R O D U C T I O N
The unpolarized cross section for the production of a single jet of rapidity y and transverse momentum P'r is given by eq. (8) with M = 0 but unlike for direct photon production, many pure QCD subprocesses contribute and the event rate is substantially bigger. The main contributions are gluon-gluon scattering which largely dominate at low PT, followed by gluon-quark scattering at medium PT before reaching at very high PT the dominance of quark-quark elastic scattering.
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10 5
104 p p -P jet + anything
lO s
10 s ,-~
10 t
.~, 10-1
~
e~ ~. b
10 -~'
",,
~/s = 600 GeV
1 0 -'a 10 -4
10 -s
"v/s = 300 GeV i0 -e 10 -7
0
50
100
150
200
Pr (GeV/e)
Fig. 5. U n p o l a r i z e d c r o s s s e c t i o n s f o r x / s - = 300 G e V a n d 600 G e V f o r y = 0 a s a f u n c t i o n of PT"
We show in fig. 5 the calculated cross sections at y - - 0 which, as expected, increase with x/S- for a fixed value of PT. These lowest-order QCD predictions are compatible with data obtained by UA1 and UA2 [11], given the large experimental uncertainties. One can hope to reduce the effect of these uncertainties (mainly systematic errors) by taking cross section ratios, so this is the reason why we believe it will be useful to study the behaviour of the asymmetry ALL at RHIC and possibly to detect the effect of a large gluon polarization (for low-energy predictions on ALL see ref. [12]). Since for all the dominant subprocesses the corresponding asymmetries t~L are positive (except for quark-antiquark annihilation), we expect a positive ALL. The results of the calculations for a jet produced at y = 0 are shown in figs. 6a, b at v~ = 300 GeV and 600 GeV versus PT. If one uses the standard gluon distribution A G ( x , Q 2) one obtains a very small ALL for PT < 50 G e V / c which rises at higher PT due to the effect of the quark polarization. However if one uses a large gluon distribution AG(x, Q2) (like eq. (7)) for low PT where gluons dominate, ALL reaches 20% or so. Therefore the measurement of ALL should allow us to discriminate easily between these two possibilities. Since we do not know the sign of AG, we show for illustration in figs. 7a, b the predictions resulting from a large negative AG, by comparison with the standard AG positive. Again we clearly see that in the low-p T region ALL remains positive
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C. B o u r r e l y et al. / S p i n e f f e c t s
p p -* jet + anything ~/s = 300 GeV
0.4
0.2
o
(a) o
50
loo
PT (GeV/c) 0.3
. . . .
,
. . . .
~
. . . .
p p ~ jet + anything ~/s = 600 GeV 0.2
d 0.1
(b) ,
,
0
i
,
I
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,
i
,
i
I
i
100
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i
i
I
150
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h
i
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PT (GeV/c) Fig. 6. (a) Predicted ALL at ~ - = 300 GeV as a function of PT for a standard A G (dashed curve) and a large positive AG (solid curve). (b) Same as (a) but for ~ - = 600 GeV.
because of the term AG(x a) zlG(x b) from gluon-gluon scattering, whereas in the medium-p T region it becomes negative because quark-gluon scattering dominates. In all these predictions at large PT where quark-quark scattering prevails we find, as expected, no sensitivity of ALL to the choice of AG. 5.2. DI-JET P R O D U C T I O N
We consider now the di-jet production reaction pp ~ j e t 1 + j e t 2 + X . In the unpolarized case for the production in the final state of a di-jet of mass M, the
C. Bourrely et al. / Spin effects
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0.2 <1
(a) 0
50
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Pr (ceV/~)
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p p -+ jet + anything ~/s = 600 GeV
0.2
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J 0.1
-0,1
(b) i
,
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I
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i
i
i
I
100
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Pr (GeV/c)
,
,
i
I
150
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i
i
200
Fig. 7. (a) Predicted ALL at ~ - = 300 GcV as a function of PT for a standard AG (dashed curve) and a large negative AG (solid curve). (b) Same as (a) but for 1/s -- 600 GeV.
invariant mass distribution is given by eq. (9). Our predictions are displayed in fig. 8 versus the jet pair mass and we observe that it corresponds to a much larger cross section than in the case of single-jet production and which is growing with increasing energy. Our calculations were done using y = 0.85 and are in reasonable agreement with some earlier UA2 data up to M = 200 G e V / c 2 or so [13], which were also restricted to the pseudorapidity interval 17/I < 0.85. Clearly the underlying dynamics for di-jet is the same as for single-jet production, therefore the general pattern of ALL is expected to be very similar. This is indeed the case
C. Bourrely et
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104 \~, p p -,, jett + jeta + anything 10 a
i0z
101
~ k
i0 -~
~/s= 600GeV
~/s = 300 GeV
i0 -s
lO-a 1 0
50
100
150
200
Jet Pair Mass (GeV/e z) Fig. 8. Predicted invariant mass distribution in di-jet production for x/~ = 300 G e V (solid curve) and = 600 G e V (dashed curve).
as shown in figs. 9a, b, where we see that ALL rises with the jet pair mass to almost 5% for the standard ZIG, whereas it is much larger ( ~ 20%) with the large positive gluon distribution. In fig. 10 we also give for completeness, our prediction for ALL corresponding to a large negative zIG, which turns negative for high mass values, where quark-gluon scattering dominates. 5.3. M A S S S P E C T R O S C O P Y
The intermediate vector bosons W, Z are expected to decay predominantly into quark-antiquark pairs with well-defined branching ratios. Therefore one should observe the existence of these decays in the di-jet mass spectrum by an excess of events over the pure QCD background in the region M ~ Mw, z. Previous ~p collider data [14] have shown some evidence for the W, Z ~ 2 jets signals which is improving with new higher-statistics data [15] and seems to be consistent with the Standard Model expectations. However the W and Z peaks are not resolved and the main difficulty in obtaining a very precise quantitative test arises from the need to detect a signal which is of the order of few percent over the huge QCD background. This requires a very good di-jet mass resolution and a high luminosity.
C. Bourrely et al. / Spin effects 0.25
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~/s
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0.15
2 0,1
0.05
(b) 50
I00 Jet Pair Mass (GeV/e s)
150
200
Fig. 9. (a) Predicted ALL at V~ = 3 0 0 G e V as a function of M for a standard AG (dashed curve) and a large positive AG (solid curve). (b) Same as (a) but for V~ = 600 GeV.
On the theoretical side one should make a detailed study of all the interference effects with pure QCD di-jet production and also, as we will see, polarized beams turn out to be very useful (for some earlier work see ref. [17]). Clearly for a parity violating asymmetry (see eq. (6) or (6')) the pure QCD background will cancel out and we are left with pure electroweak and QCD-electroweak interference terms only. In the case where only one proton beam is polarized, instead of the
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C. Bourrely et at. / Spin effects 0.1
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t
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Jet Pair Mass (GeV/J) Fig. 10. Same as fig.9b, but for a large negative AG.
asymmetry let us consider the cross section difference, dAodo'p(+) do'p(_) dM
dM
dM
'
(11)
where d % ( ± ) / d M are the invariant mass distributions for one initial proton with helicity + or - . The calculations are done using the appropriately modified eq. (9) and the formulae given in appendix A. In eq. (9) we have used Y = 4, allowing the jets to be produced closer to the beam direction, which re-inforces the signal. The results are displayed in figs. lla, b at vr~ = 300 and 600 GeV. We note that both W, Z peaks are visible and even clearly at 600 GeV, with a much sharper shape for the W because of the pure V - A nature of its coupling. With a high-luminosity machine like RHIC, we believe that these predictions can be tested to a high degree of accuracy. A possible way to avoid the problem of systematic errors is to consider the ratio of d % ( ± ) / d M at two different energies. This method can also be used for the search of new gauge bosons decaying into two jets. In the high-mass region, say M > 200 GeV/c 2, where gluon scattering is less important, we expect a better signal-to-background ratio which represents obviously a rather serious advantage. 6. Compositeness Compositeness is one possible approach where one tries to go beyond the Standard Model but, so far, there is no experimental indication for this attractive idea. One way to probe the existence of subconstituents is to study the sensitivity of the unpolarized cross section for some given processes in a specific kinematic
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C. Bourrely et al. / Spin effects 15
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I
80
i
i
i
i
90
Jet Pair Mass
100
(GeV/c s)
I00
l
p p -* jett + jets + anything ~/s = 600 GeV
%, 50 ,.c=
0
{b) =
70
=
=
=
i
i
i
i
I
80
Jet Pair Mass
i
90
i
i
i
100
(GeV/c s)
Fig. 11. (a) Parity violation effect in two jets production from W, Z decay at ~ = 300 GcV. (b) Same as (a) but for ~ = 600 GeV.
region. One usually assumes that the common subconstituents generate some contact interactions whose chirality structure is not known and which will modify the Standard Model amplitude Z sm in a very simple manner. For example the amplitude A~m for qi~ti ~ gg will become
,4; = A ; m 1 +
e 5) ,
(12)
where A is the compositeness energy scale, c and d being two unknown constants.
C
88
Bourrely et aL / Spin effects
The value of A is known to be not too small, say A ~ 1 or 2 T e V and above, and of course for A = oo one recovers the Standard Model situation. According to eq. (12) we expect now a parity violating asymmetry A L whose magnitude will depend on the value of A. For an illustration we have chosen to study it in the case of single-jet production. As we recalled above (see sect. 5) the data on the unpolarized cross section is not accurate enough, due to large systematic errors ( ~ 50%) and therefore any significant enhancement with respect to Standard Model calculations will be difficult to appreciate. On the other hand, the measurement of a
-0.1 x x x xx
#
xx
p p -* j e t + anything
-0.2
~ x
~
~/s = 300 GeV
x
-0.3
(a} -0.4
=
I
0
i
,
,
i
50
I¸
100 P, (GeV/Ù)
0.05
"~
-0.05 I
Illllllll
,,
p p -~ jet + anything
~.,,'.,,~,,
,,
-0.1
~/s = 600 GeV
-0.15
(b) --0.2
,
0
,
,
,
I
50
,
,
,
,
I
100
,
PT (CeV/o)
,
,
,
I
150
,
,
,
,
200
Fig. 12. Predicted A L for ~/s = 300 GeV and y = 0 versus PT, for A = I TeV (dashed curve) and A = 2 TcV (solid curve). (b) Same as (a) but for V~-= 600 GeV.
C. Bourrelyet al. / Spin effects
89
non-zero A L for such a process will be a clear signature for a deviation from the Standard Model. For the various subprocesses which contribute to single-jet production we have used the modified cross sections as in ref. [4] with a left-handed contact interaction i.e. c -- d = 1/x/2. We have checked that the unpolarized cross sections are affected at most by a factor of two. The results for A L are displayed in figs. 12a, b at ~ = 300 GeV and 600 GeV versus P'r for two values of the energy scale A --- 1 TeV and 2 TeV. Therefore we see from these measurements at high PT, that it will be possible to detect the existence of compositeness provided A is not too large. Direct photon production could also be a testing ground reaction but unfortunately in this case, unless A < 1 TeV, A L departures from zero for such high-p T values where the cross section is unobservable.
7. Concluding remarks In this paper we have shown the importance of spin effects in a pp collider with polarized protons in the energy range between fs---300 and 600 GeV. We have given detailed predictions for a selection of reactions, direct photon production, single-jet and di-jet production in some relevant kinematic regions. From the measurement of these double helicity asymmetries it will be possible to pin down the magnitude and the sign of A G ( x ) and to verify directly the interpretation of the EMC result in terms of the axial anomaly. This gluon polarized distribution function can also be extracted by means of other measurable effects as suggested in the literature. These include in polarized deep inelastic scattering, heavy-quark pair production [18] /' p --->f + J/qJ + X and high-p T hadronic jet production [19] f p ~ t ' + j e t ~ + j e t / + X, which would be worth to pursuit at H E R A , provided a polarized proton beam can be realized. Single helicity asymmetries in W + production at large PT are proven to be very useful to improve our knowledge of A u ( x ) and Ad(x), the polarized distribution functions of u- and d-quarks. They can also be used in single-jet production to uncover the existence of compositeness and in di-jet production to detect the hadronic decay modes of standard or new gauge bosons. Finally we have left untouched the subject of transverse polarization, although large effects have been observed at much lower energy values whose theoretical interpretation is far more difficult. We wish to thank G. Bunce and M. Tannenbaum who gave us the motivation for writing this article and for several interesting discussions.
Appendix A In this appendix we give all the Born cross sections for quark-quark and quark-antiquark scattering where we consider quarks with identical and distinct
90
C. Bourrely et al. / Spin effects
quantum numbers. We calculated these cross sections to lowest order in gluon, y, W and Z exchanges, including all QCD-electroweak interference terms, and assuming that the initial quarks (or antiquarks) have definite helicities a~ and a 2. We take the normalization d t r / d t = (rr/s2)g.iiT,.p where To. denotes the matrix element squared with i and j exchanges. First, let us consider identical quarks, q(al)q(a 2) --+ qq for which we have
T , , = ~a~[~. + A,A=) 7 + u 2
T~v=2a2e
4[
+ (1 - a , a = )
3 tu
(s2 s2 2s2)
(I+A,A2) 7 g + ~ - g + ~ T f f u
+
7
+(1-a,a2)
--~
(A.1)
'
7+~-
,
(A.2)
,= ,= 2j_L)
Tzz = a~[(Ct(1 - a,)(1 - a2) + Ca(1 + a,)(1 + a2) ) t-~z + u'-'~z+ 3 tzU z
+2C~C~(1
- a,a=)
, (A.3)
+
T z , = 2 a z a e ~ [ ( C ~ ( 1 - A,)(1 - aa) + C 2 ( 1 + a,)(1 + A2)) (St7Z + - - $2 uu z
+
- -
tu z +
Suez))
(
+2CRCL(1--Ar~2)
Tzs= ~aza s ( C 2 ( 1 - A x ) ( 1 - A 2 ) + C2(1 +A,)(1 +A2) )
- -
tt z
+
(:;z
- -
uu z
+ --
,
(A.4)
, (A.5)
ut z
s2 T~,g= -~aa~e~(1 + AIA2)~-~,
(A.6)
where tr s, a and a z =t~/sin20w cos20w are the strong, elctromagnetic and Z-boson coupling constants, respectively, and 0 w is the Weinberg angle. The coefficients of the Z coupling to right- and left-handed quarks of charge eq are CR= --eq sin20w,
CL= T ~ - e q s i n 2 O w ,
C. Bourrelyet at. / Spin effects
91
s, t and u are the usual Mandelstam variables and t z - t - M 2,
Uz=u-M
2.
For the line reversed process q(A1)~(A 2) ~ q~, the corresponding T/i are obtained from eqs. (A.1)-(A.6) after making the following substitutions: s ~, u, Az ~ - A 2 (except in the first four expressions (A.1)-(A.4) for the coefficients of t2), 1/u z Sz/(S ~ + r~M~), 1/u~ ~ 1/(s~ + F~Mz), 2 2 where s z --s - Mz2. Second, we consider distinct quarks q(Al)q'(A 2) --* qq' for which we find 2
4 as 7gg= -~ t--T[(1 + AIA2)s 2 + (1 - h l A E ) U 2 ] ,
r =
(A.7)
o~2
27re.e., [ 1+
(A.8)
, = _ _ ~t(c?.cf(1-x,)(1-x2)+c.c.(l~" ~ ,~ T~z
+AI)(1 +
4
,~))~
+(C~C'I~(1-A,)(I+Az)+C~Cf(I+AI)(1-Az))uE],
T~:~' =
2
~za tzt eqeq,[(CLC'L(1
-AI)(I
(A.9)
-A2) + CRC~(1 +Al)(1 +A2))s 2
+ (CLCk(I -At)(1 +A2) + CRCt(I +AI)(I --A2)).2] , (A.10) $2
T~vw=
' = Twz
IVqq,]4(1 - A1)(1 - A 2 ) u 2 ,
~3~ Z lVqQ,12CLC,L(1 -
S2
- , At)(1 - A2) -Uwtz
0~W~ ' Two=
(A.11)
S2
-~ eqeq,lVqq,lZ(1- A , ) ( 1 - Az) -Uwt ,
, 4 rwg= ~aw,~lgqq, I2 (1
(A.12)
(A.13)
S2
-A1)(1 -h2)--
,
(A.14)
Uwt
where a w -- a / s i n z 0 w is the W coupling constant, u w = u - M 2 and Vqq, denotes the qq' element of the weak mixing matrix. For the line reversed reaction q(AI)~'(A2) ~ qFa', the corresponding T/i are obtained from eqs. (A.7)-(A.14) after
C. Bourrely et al. / Spin effects
92
making the following substitutions: s ~ u, A2 __, _A2 ' 1/Uw ~ Sw/(S~ v + F~vMw)2 2 2 2 and 1/U2w .l_>1/(S2w + FwMw). For the crossed channel process q(Al)~(A 2) ~ q'~'
the corresponding T/y are also obtained from eqs. (A.7)-(A.14) after making the following substitutions: s ~, t, A2-->-A 2 (except in the first four expressions (A.7)-(A.10) for the coefficient of u2), 1/t z ~ Sz/(S ~ + F~M~) and 1/t~--, 1/ 2 2 2 (s z + F ~ M ~ ) . F i n a l l y w e m u s t also c o n s i d e r t h e i m p o r t a n t c o n t r i b u t i o n f r o m t h e p r o c e s s e s in w h i c h all f o u r q u a r k s a r e d i s t i n c t i.e. q(At)q'(A 2) -o q"q" for w h i c h we have ~2
T~w--
2 V. , 2
s2
I q¢l ( 1 - A I ) ( 1 - - A 2 ) U 2 w ,
(A.15)
and the crossed channel process q(AI)~"(A 2) --* q"~' for which u2
T'~v=
4 Vq'q"12lVq¢"12(1-Al)(l+A2)s2 w +FwMw2 2 •
(A.16)
Unpolarized cross sections are obtained for A~ = A2 = 0 and we have checked that we recover the results given in ref. [16]. For polarized cross sections, Ranft and Ranft [17] have some incorrect results and leave out some important processes.
References [1] J. Softer, Overview of high-energy physics with polarized particles, invited talk presented at Int. Conf. on Polarization phenomena in nuclear physics, Paris, July 1990, Proc. Editions de Physique 1990, C6, Tome 51, p. 135, ed. A. Boudard and Y. Terrien, and references therein [2] EMC, J. Ashman et al., NucL Phys. B328 (1989) 1 [3] H. Rollnik, Ideas and models for the proton spin, invited talk presented at 9th Int. Syrup. on High-energy spin physics, Bonn, September 1990, to be published in the proceedings [4] C. Bourrely, J. Softer, F.M. Renard and P. Taxil, Phys. Rcp. 177 (1989) 319; See also the errata preprint CPT-87/P.2056 (December 1990) [5] E. Berger and J. Qiu, Phys. Rev. D40 (1989) 778 [6] UA2 Collaboration, R. Ansari et al., Z. Phys. C41 (1988) 395 [7] H.Y. Cheng and S.N. Lai, Phys. Rev. D41 (1990) 91 [8] N. Arteaga-Romero, A. Nicolaidis and J. Silva, Phys. Rev. Lett. 52 (1984) 172 [9] C. Stubenranch, Thesis CE.A-N-2532 (1987) [10] K. Hidaka, Nucl. Phys. B192 (1981) 369 [11] UA2 Collaboration, J.A. Appel et al., Phys. Lett. B160 (1985) 349; UA1 Collaboration, G. Arnison et al., Phys. Lctt. B172 (1986) 461 [12] H.¥. Cheng, S.R. Hwang and S.N. Lai, Phys. Rev. D42 (1990) 2243 [13] UA2 Collaboration, P. Bagnaia et al., Phys. Lett. B138 (1984) 430 [14] UA2 Collaboration, R. Ansari et ai., Phys. Lett. B186 (1987) 452 [15] UA2 Collaboration, J. Alitti et al., preprint CERN-PPE/90-105 (1990) [16] U. Baur, E.W. Glover and A.D. Martin, Phys. Lctt. B232 (1989) 519 [17] F.E. Paige, T.L Trueman and T.N. Tudron, Phys. Rev. D19 (1979) 935; G. R a n t and J. Ranft, Nucl. Phys. B165 (1980) 395 [18] J.Ph. Guillet, Z. Phys. C39 (1988) 75 [19] R.D. Carlitz, J.C. Collins and A.H. Mueller, Phys. Lett. B214 (1988) 229; Z. Kunszt, Phys. Lett. B218 (1989) 243
SPIN EFFECTS WITH POLARIZED PROTONS AT RHIC C. BOURRELY,J.PH. GUILLET* and J. SOFFER Centre de Physique Th~orique, CNRS Luminy, F-13288 Marseille Cedex 9, France
Received 14 January 1991 (Revised 13 March 1991)
We show that the use of RHIC as a polarized pp collider with a high designed luminosity will allow various interesting polarization studies. In particular, as we will see by considering several different reactions, this can help us to pin down the gluon polarized distribution function, to give a better insight for the u- and d-quarks polarization, to make dynamical tests of the Standard Model and to detect the possible existence of subconstituents.
1. Introduction A Relativistic Heavy Ion Collider (RHIC) is now under construction at Brookhaven National Laboratory and a possible realistic use of this machine is to study hadronic interactions with polarized protons with a luminosity of a few 10 32 cm -2 sec -1 and an energy of 50-250 GeV per beam with a polarization of about 70%. Polarization studies at current accelerator energies have been proven to be very useful and the observation, in hadronic collisions, of several puzzling effects like, for example at FNAL, inclusive hyperon production or ~.0 production in the central region, are indeed a real challenge for dynamical models [1]. Moreover, the recent result of the E M C experiment [2] on deep inelastic scattering (DIS) of polarized muons on a polarized target, has stimulated a vast theoretical debate and has generated the production of a huge number of papers [3]. Among the very many basic questions raised by the EMC result are: Is there something wrong with the proton spin? What are the expectations of quark models? What is the exact role of the Adler-Bell-Jackiw anomaly? How large are the perturbative and non-perturbative gluonic contributions to the proton spin? Is there a large fraction of the proton spin carried by the strange quarks? What is the connection with the U(1) problem? Etc. It is fair to say that, at present, the nucleon spin is not yet understood and it requires a special effort on the experimental side in polarized DIS for a more accurate direct determination of the polarized structure functions for both proton and neutron. Polarized pp collisions at supercollider energies *Present address: Theory Division CERN, 1211 Geneva 23, Switzerland. 0550-3213/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
C. Bourrely et aL / Spin effects
73
(LHC, SSC) were studied in detail [4]; one anticipates large single- and doublehelicity asymmetries for various processes which represent significant advantages for signal detection, especially for new physics. In this paper, we wish to argue that polarized pp collisions at RHIC energies can help answering some of the above fundamental issues related to the controversial interpretation of the EMC result, in particular, the magnitude and the sign of the gluon helicity asymmetry AG(x). As we will see, it is also possible to detect parity violating effects related either to the production of the vector gauge bosuns of the Standard Model or to the composite nature of the quarks if the corresponding energy scale is of the order of 1 TeV or so. The paper is organized as follows. In sect. 2 we briefly recall basic kinematics, the definition of the observables and our choices for the various parton distributions. In sect. 3 we will study direct photon production, a process which has a small cross section but whose double helicity asymmetry is very sensitive to AG(x). In sect. 4, we will consider a Standard Model parity violating asymmetry from the production of W ± at large PT which will help the determination of the polarized structure functions for u- and d-quarks. Sect. 5 is devoted to jet studies, and more specifically single-jet production, di-jet production whose cross sections are rather large, and we will show the importance of both the double and the single asymmetries. For this purpose we will give in appendix A all the polarized cross sections for quark-quark scattering, identical or distinct, to lowest order in gluon, y, W, Z exchanges including all interferences contributions. In sect. 6 we will see that compositeness can be tested at RHIC energies from single-jet production. We give our concluding remarks in sect. 7.
2. Basic kinematics, observables and parton distributions
Fundamental interactions at short distances which are probed in pp collisions at high energies, involve hard scattering of quarks and gluons. Let us consider the general hadronic reaction a+b~c+X,
(1)
where c, in the cases we will study below, is either a well-defined particle (hadron or gauge bosun) or a single jet. We will also consider a + b---> c + d + X for double-jet production. In the hard scattering kinematic region, the cross section describing (1) reads, in the QCD parton model, provided factorization holds, as 1
+ b -. c + X) : E 1 + 8,; f
[
+ (i oj')].
U
(2)
74
C. Bourrely et al. / Spflz effects
The summation runs over all contributing parton configurations, the f(x, Q 2 ) ' s a r e the parton distributions, directly extracted from deep inelastic scattering for quarks and antiquarks, and d~q is the cross section for the interaction of two partons i and j which can be calculated perturbatively, some of which are given in appendix A. If we consider the reaction (1) with both initial hadrons longitudinally polarized, one useful observable is the double helicity hadron asymmetry ALL defined as dora(+)b(+) -- do'a(+)b(_ ) -- do.a(_)b(+ ) + dO'a(_)b(- ) ALL =
dO'a(+)b(+) + dO'a(+)b(- ) + do'a(_)b(+ ) + dO'a(_)b(_ ) '
(3)
which reduces to do.a(+)b(+) -- dtx.a(+)b(_ )
ALL ----
do.a(+)b(+) + dO'a(+)b(_ )
(3')
if parity is conserved because d%(~)b(A,) = d%(_~)b(_X). It is given by
ALLdo" = ~0" 1 +1 6i-"~-f dx~ dx b [Afi(a)( x a, Q2) Af~b)(xb ' Q 2 ) a^.. ^ + (i- j)], ~ L d~.;
(4) assuming the factorization property, where do- is given by eq. (2) and ~ L denotes the corresponding subprocess asymmetry for initial partons i and j. The af(x, Q2)'s are the parton helicity asymmetry defined as
Af(x, 0 2) = / + ( x , Q2) - f _ ( x , O2),
(5)
where f ± are the parton distributions in a polarized hadron with helicity either parallel ( + ) or antiparallel ( - ) to the parent hadron helicity. Recall that the unpolarized parton distributions are f = f++ f_. If only one initial hadron is polarized, say "a", another interesting observable is the sh,gle helicity asymmetry defined as (do-~(_)- do'~(+)) ALd°'= 2 ' (6) a quantity for which one can write an expression similar to eq. (4), provided Aflb)~fj.0,) and ~ L ~ a~', the single helicity subprocess asymmetry. A L is expected to vanish unless some subprocesses involve parity violating interactions, i.e. a~ 4: 0. If both initial beams are polarized, it is also possible to define another parity violating asymmetry, namely A~.v do" = (do.a(-)b(-) -- d°'a(+)b(+)) 2 '
(6')
which for special cases can be about twice as big as A L (see sect. 4). We will then
C. Bourrely et al. / Sph~ effects
75
remember that, if at RHIC one achieves the same luminosity with polarized and unpolarized proton beams, it will certainly be more advantageous to measure A~ v rather than A L. Both for polarized and unpolarized parton distributions, we will take our own set in terms of a simple parametrization of their x and Q2 dependences already used in ref. [4]. Concerning the gluon helicity asymmetry AG(x, Q2) there is a standard choice (see e.g. eq. (3.12) in ref. [4]) such that zaG(x)/G(x) is only a few percent up to x -- 0.1. Since the relevant kinematic region stands at small x-values, it results that at small Q2, the integral zaG = fdzaG(x)dx is only 0.2-0.3 and it increases slowly with Q2. This choice means that one expects gluons to carry about 25% of the proton spin and until very recently there were no experimental facts to contradict this reasonable assumption. However after the striking EMC result [2], in particular on the small-x behaviour of the spin-dependent structure function g~(x, Q2), several interpretations and new pictures for the proton spin have been proposed [3]. Among these various possibilities, a suggestion, due to the existence of the anomaly of the axial vector current, leads one to anticipate a ziG(x) much larger than the standard choice, in particular in the small-x region. Therefore, following ref. [5], we will also consider the simple parametrization
AG(x,Q~) =
G(x, Q2o),
x¢
0
x xo,
(7)
which can lead to a much larger value of the integral AG than for the standard choice; the smaller x c the larger ziG. We will take x c ~ 0.2 corresponding to AG ~ 2-3, because this is the value which gives the best agreement with the gt(x, Q2) data. Clearly now gluons give a much too large contribution to the proton spin which ought to be balanced by a large orbital angular momentum. We have no very strong theoretical argument to prefer the standard form or this second choice (see eq. (7)) and, of course, due to the lack of our knowledge other choices are equally possible. These two rather different forms of the gluon helicity asymmetry will allow us to study how much a given reaction is sensitive to AG, whose precise form can only be determined from future data at RHIC. Since we will study in what follows, direct photon and W • production, let us now recall the expression of the differential cross section for producing a gauge boson of mass M with transverse momentum PT and rapidity y. It reads do"
dPTdY = 2PT
,;
fldx
X Xb
× fi(x~,QE)fi(Xb, Q 2 ) - ~ ( s ,^t ,^g O + ( i o j ) ) ,
(8)
C.Bourrelyet al. /
76
Spin effects
where the explicit M-dependence appears in all relevant kinematic variables,
~=XaXbS, ~ - M 2= --xamTvrse -y and a - M 2= --XbmTVr~.e-y, through the transverse mass m T = (M 2 +p~)X/2 and where (mT/V~-) e y - M2/s
xa(mT/V/s )e-V -- M2/s Xb=
Xa -- (mT/V/~-) e '
'
x0=
1 - (mT/~/~-) e - '
This expression can also be used for single-jet production by making M = 0, and it also holds for direct photon production. For the production of two jets with rapidity y~ and Y2 and with equal and opposite transverse momentum PT, we also recall that the invariant mass distribution reads do2M~ d M = --
LYy
•
×
1
rYmax
dylJy,,,° dY2~'..(1 + S o ) c o s h 2 y
*
tJ
d~o. ) fi(xa,Q2)fi(Xb,Q2)--~(~,t,a)+(i*--~j) ,
(9)
where y* = ~(Y~-Y2) 1 and the jet pair mass is given by M = 2PT cosh y*. Large values of y* correspond to the forward and backward directions at the parton level which should be avoided and this is why it should be restricted to the central region - Y < (y~, Y2) < Y with Y ~ 1. Finally we recall that all the subprocess cross sections and corresponding asymmetries, to complete the calculations of eqs. (8) and (9), are given in ref. [4].
3. Direct photon production Direct photon production at high PT is a useful probe of the underlying parton-parton interactions and probably one of cleanest reactions to study the perturbative regime. Since the photon originates in the hard scattering subprocess and does not fragment, the cross section is given by eq. (8) with M - - 0 . In the QCD parton model, in the absence of photon bremsstrahlung contributions, direct photons are produced via the q~ annihilation subprocess q~ ~ 7g and the quark-gluon Compton subprocess qg ~ qT. The Compton subprocess has a positive aLL and leads to a contribution to ALL, for ~ ~ 3'X, directly proportional to AG(x, Q2). For the annihilation subprocess one has aLL = - - 1 . First we have calculated the unpolarized cross section for two different values of the production angle 0c.m.= 90 ° and 45 ° corresponding to y = 0 and y = 0.88, respectively. The results for V~--300 and 600 GeV are shown in figs. la, b and clearly the prompt photon yield is more copious at higher energy. We also note that at 45 ° , except at very large PT where the event rates are low, the cross section is not significantly
C. Bourrely et al.
/ Spineffects
77 i
10 -1
P P "* 7 + a n y t h i n g x/s = 300 G e V
.
.
.
.
i
P P "* 7 + a n y t h i n g ~/s = 600 G e V
10- I
10 -z 10 -s
"u
'U
10-2
10 - 4
,o
.~,
1o-5
10 -3
,~ 10 -e b
10 -4 10 -7 i 0 -B
io-~ I
i0 -o
(a)
(b)
10-1o
10-e 0
50
Pr (GeV/c)
100
i
0
50
100
Pr (GeV/c)
Fig. 1. (a) Predicted unpolarized cross sections for ~ - = 300 GeV as a function of PT, for two different values of the c.m. production angle: 0c.m.= 90 ° (dashed curve), 0c.m.= 45 ° (solid curve). (b) Same as (a), but for v~- = 600 GeV.
smaller than at 90° . Direct photon production has been measured in ~p collisions by the UA2 collaboration in this energy range [6] and given the high designed luminosity at RHIC, these cross sections will be measurable up to fairly high PT-Values. Let us now turn to the calculation of ALL. If one uses the standard distribution AG(x, Q2) (eq. (3.12) of ref. [4]), one obtains a very small positive result, say at most (5-10)%. This reflects the dominance of the Compton subprocess and the small magnitude of A Lt. is partly due to the fact that u-quark-gluon and d-quark-gluon lead to opposite sign effects. However if instead, one assumes a large distribution AG like in eq. (7), one gets the results displayed in figs. 2a, b. A L L rises with PT reaching values of the order of 20% or more for 0c.m.= 90 ° and much larger for 0c.m.= 45°. So this shows that the PT dependence of ALL in direct photon production is very sensitive to the gluon polarization and that it should best be measured with photons of rapidity of the order of 1. In fig. 2a at very large PT, ALL is dropping off because in this kinematic region one feels the effect of the annihilation subprocess which gives a negative contribution to ALL. Finally we notice that some of these characteristic features of ALL are also exhibited in the existing predictions for much lower values of v~- corresponding to fixed-target experiments (see refs. [5, 7]).
C. Bourrely et al. / Spin effects
78
L
0.6 P P -* 7 + anything
f
\
=
0.4
0.2
(a) 0
50
0
I00
PT (GeV/c)
0.6
I
]
P P ~ 7 + anything ~/s = S00 OeV 0.4
0.2
(bl
0 0
........................ ;..
,
.
.
50
.
. I00
Pr (~eV/e) Fig. 2. (a) Predicted ALL with a large AG for ~ - = 300 G e V as a function of PT, for two different values of the c.m. production angle: 0c.ra.= 90 ° (dashed curve), 0c.m.= 45 ° (solid curve). The small-dashed curve corresponds to a standard AG. (b) Same as (a), but for ~fs = 600 GeV.
4. W + production
Let us now consider the single W + production with a large PT, which is balanced by a hadronic jet. This is similar to the direct photon production we have studied above (see sect. 3) but in this case one should consider the quark-antiquark annihilation subprocess qiqj ~ Wg and the quark gluon Compton subprocess qig--+ qi w with i ~ j . From the decay distribution of the W ± produced at
Bourrely et at / Spineffects
C.
i0 -t
.
.
.
.
,
.
.
.
.
,
.
.
.
.
~
79 . . . .
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+ jet
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~ ~/s- BOOGeV
"l~b10-7 ~/s=3OOGe~
xxx~'xxxxx~'x xx,,xxxx
I0-8 10-o
.
0
.
.
,
.
50
.
.
,
"x,
~
.
100
.
.
Pr (GeV/c)
.
s
150
,
,
x
,
200
Fig. 3. Predicted unpolarized cross sections for ~ = 300 GeV and 600 GeV for y = 0 as a function of PT.
large PT it is possible to discriminate between the scalar and the vector nature of the gluon emitted in the final state [8]. By using this method based on the fact that, unlike the electromagnetic current, the axial current is not conserved, a scalar gluon was excluded at a 2o- level at the CERN collider [9]. Hopefully this will be confirmed by higher-statistics data at RHIC. The differential cross section do-/dp T dy for producing a W with PT and rapidity y is calculated using eq. (8) and we show our results in fig. 3 for y = 0 at two different energies. The production which is more abundant at V~-= 600 GeV is dominated by the Compton diagram and falls out very rapidly at large PT- If the beams are polarized, when the initial partons are carrying a given helicity we recall that for Compton scattering qi(h)g(h) ~ qiW we have d~ ~
rrota s
d [ (h h ) = '
12Xw~2
(1-h)[(1-h)cl([)+Ac2([)]
with cl(F)
= -
(s-M~v)2+
'
c2([)
2(a -M~)2~t~ ,
(10)
C. Bourrely et aL / Spin effects
80 . . . .
0.4
~/s -
i
. . . .
3 0 0 GeV
i
. ~. .....
---"-~S--'--"""""
i
. . . .
- ....................
~-
"
~/s -
: 6 0 0 GeV
p p .-,, W + + jet.
0.2
,#
. . . .
o
-0.2 ~
~
~/s = 600 GeV
~ / s = 3 0 0 GeV
pp
-0.4
0
50
100
P, (Gev/c)
-*W- +jeL
150
200
Fig. 4. Predicted A L in p p ~ W ± + j e t at y = 0 versus PT for ~ - = 3 0 0 GeV (solid curves) and vrS- = 600 GeV (dashed curves).
where x w = sin 0~v is the weak mixing angle. For F:ti(h)g(h)--* ~jW the same formula holds but with h --* - h and A --* - A . The corresponding single helicity asymmetries are d ~ = 1 for polarized quarks and a~. = 1 - c2/c I for polarized gluons. This last quantity being rather small on average, the single hadron helicity asymmetry will be dominated by polarized quarks and is not sensitive to AG. This is clearly shown in our results for A L at y = 0 versus PT displayed in fig. 4, where for W ÷ one sees the trend of Au(x)/u(x) and for W - the trend of Ad(x)/d(x). At fixed PT the effect decreases with increasing energy. Since the W's are left-handed objects, the consideration of A Pv would lead us to predict asymmetries about twice as big as these single helicity asymmetries and this is consistent with some earlier results given in ref. [10].
5. Jet studies 5.1. SINGLE-JET P R O D U C T I O N
The unpolarized cross section for the production of a single jet of rapidity y and transverse momentum P'r is given by eq. (8) with M = 0 but unlike for direct photon production, many pure QCD subprocesses contribute and the event rate is substantially bigger. The main contributions are gluon-gluon scattering which largely dominate at low PT, followed by gluon-quark scattering at medium PT before reaching at very high PT the dominance of quark-quark elastic scattering.
C
Bourrely et al. / Spin effects
81
10 5
104 p p -P jet + anything
lO s
10 s ,-~
10 t
.~, 10-1
~
e~ ~. b
10 -~'
",,
~/s = 600 GeV
1 0 -'a 10 -4
10 -s
"v/s = 300 GeV i0 -e 10 -7
0
50
100
150
200
Pr (GeV/e)
Fig. 5. U n p o l a r i z e d c r o s s s e c t i o n s f o r x / s - = 300 G e V a n d 600 G e V f o r y = 0 a s a f u n c t i o n of PT"
We show in fig. 5 the calculated cross sections at y - - 0 which, as expected, increase with x/S- for a fixed value of PT. These lowest-order QCD predictions are compatible with data obtained by UA1 and UA2 [11], given the large experimental uncertainties. One can hope to reduce the effect of these uncertainties (mainly systematic errors) by taking cross section ratios, so this is the reason why we believe it will be useful to study the behaviour of the asymmetry ALL at RHIC and possibly to detect the effect of a large gluon polarization (for low-energy predictions on ALL see ref. [12]). Since for all the dominant subprocesses the corresponding asymmetries t~L are positive (except for quark-antiquark annihilation), we expect a positive ALL. The results of the calculations for a jet produced at y = 0 are shown in figs. 6a, b at v~ = 300 GeV and 600 GeV versus PT. If one uses the standard gluon distribution A G ( x , Q 2) one obtains a very small ALL for PT < 50 G e V / c which rises at higher PT due to the effect of the quark polarization. However if one uses a large gluon distribution AG(x, Q2) (like eq. (7)) for low PT where gluons dominate, ALL reaches 20% or so. Therefore the measurement of ALL should allow us to discriminate easily between these two possibilities. Since we do not know the sign of AG, we show for illustration in figs. 7a, b the predictions resulting from a large negative AG, by comparison with the standard AG positive. Again we clearly see that in the low-p T region ALL remains positive
82
C. B o u r r e l y et al. / S p i n e f f e c t s
p p -* jet + anything ~/s = 300 GeV
0.4
0.2
o
(a) o
50
loo
PT (GeV/c) 0.3
. . . .
,
. . . .
~
. . . .
p p ~ jet + anything ~/s = 600 GeV 0.2
d 0.1
(b) ,
,
0
i
,
I
50
,
i
,
i
I
i
100
,
i
i
I
150
i
h
i
,
200
PT (GeV/c) Fig. 6. (a) Predicted ALL at ~ - = 300 GeV as a function of PT for a standard A G (dashed curve) and a large positive AG (solid curve). (b) Same as (a) but for ~ - = 600 GeV.
because of the term AG(x a) zlG(x b) from gluon-gluon scattering, whereas in the medium-p T region it becomes negative because quark-gluon scattering dominates. In all these predictions at large PT where quark-quark scattering prevails we find, as expected, no sensitivity of ALL to the choice of AG. 5.2. DI-JET P R O D U C T I O N
We consider now the di-jet production reaction pp ~ j e t 1 + j e t 2 + X . In the unpolarized case for the production in the final state of a di-jet of mass M, the
C. Bourrely et al. / Spin effects
83
I
0.4
0.2 <1
(a) 0
50
100
Pr (ceV/~)
0,3
J
p p -+ jet + anything ~/s = 600 GeV
0.2
t
J 0.1
-0,1
(b) i
,
,
,
I
50
i
i
i
i
I
100
i
Pr (GeV/c)
,
,
i
I
150
i
i
i
200
Fig. 7. (a) Predicted ALL at ~ - = 300 GcV as a function of PT for a standard AG (dashed curve) and a large negative AG (solid curve). (b) Same as (a) but for 1/s -- 600 GeV.
invariant mass distribution is given by eq. (9). Our predictions are displayed in fig. 8 versus the jet pair mass and we observe that it corresponds to a much larger cross section than in the case of single-jet production and which is growing with increasing energy. Our calculations were done using y = 0.85 and are in reasonable agreement with some earlier UA2 data up to M = 200 G e V / c 2 or so [13], which were also restricted to the pseudorapidity interval 17/I < 0.85. Clearly the underlying dynamics for di-jet is the same as for single-jet production, therefore the general pattern of ALL is expected to be very similar. This is indeed the case
C. Bourrely et
84
al. / Spin effects
104 \~, p p -,, jett + jeta + anything 10 a
i0z
101
~ k
i0 -~
~/s= 600GeV
~/s = 300 GeV
i0 -s
lO-a 1 0
50
100
150
200
Jet Pair Mass (GeV/e z) Fig. 8. Predicted invariant mass distribution in di-jet production for x/~ = 300 G e V (solid curve) and = 600 G e V (dashed curve).
as shown in figs. 9a, b, where we see that ALL rises with the jet pair mass to almost 5% for the standard ZIG, whereas it is much larger ( ~ 20%) with the large positive gluon distribution. In fig. 10 we also give for completeness, our prediction for ALL corresponding to a large negative zIG, which turns negative for high mass values, where quark-gluon scattering dominates. 5.3. M A S S S P E C T R O S C O P Y
The intermediate vector bosons W, Z are expected to decay predominantly into quark-antiquark pairs with well-defined branching ratios. Therefore one should observe the existence of these decays in the di-jet mass spectrum by an excess of events over the pure QCD background in the region M ~ Mw, z. Previous ~p collider data [14] have shown some evidence for the W, Z ~ 2 jets signals which is improving with new higher-statistics data [15] and seems to be consistent with the Standard Model expectations. However the W and Z peaks are not resolved and the main difficulty in obtaining a very precise quantitative test arises from the need to detect a signal which is of the order of few percent over the huge QCD background. This requires a very good di-jet mass resolution and a high luminosity.
C. Bourrely et al. / Spin effects 0.25
I
85
I
0.2
0.15
2 0.1
~/s
=
.........
3 0 0 GeV
.
0.05
(a) 0
. . . .
, ' - - ' ] ' - -
0 0.25
I
,
i
*
50 Jet Pair Mass .
.
.
.
I
.
.
.
.
,
I
,
i
,
,
100 (GeV/c 2) I
.
.
.
.
150
}
.
.
.
.
I
.
.
.
.
p p ~ jet I + jet z + anything 0.2
~/s = 600 GeV
0.15
2 0,1
0.05
(b) 50
I00 Jet Pair Mass (GeV/e s)
150
200
Fig. 9. (a) Predicted ALL at V~ = 3 0 0 G e V as a function of M for a standard AG (dashed curve) and a large positive AG (solid curve). (b) Same as (a) but for V~ = 600 GeV.
On the theoretical side one should make a detailed study of all the interference effects with pure QCD di-jet production and also, as we will see, polarized beams turn out to be very useful (for some earlier work see ref. [17]). Clearly for a parity violating asymmetry (see eq. (6) or (6')) the pure QCD background will cancel out and we are left with pure electroweak and QCD-electroweak interference terms only. In the case where only one proton beam is polarized, instead of the
86
C. Bourrely et at. / Spin effects 0.1
.
.
.
.
I
.
.
.
.
t
.
I
,
.
.
.
I
.
I
,
.
.
.
0.05
2 /s = 600 -0.05
-0.1
. 0
.
.
.
t
,
,
,
,
50
,
,
,
100
150
,
,
, 200
Jet Pair Mass (GeV/J) Fig. 10. Same as fig.9b, but for a large negative AG.
asymmetry let us consider the cross section difference, dAodo'p(+) do'p(_) dM
dM
dM
'
(11)
where d % ( ± ) / d M are the invariant mass distributions for one initial proton with helicity + or - . The calculations are done using the appropriately modified eq. (9) and the formulae given in appendix A. In eq. (9) we have used Y = 4, allowing the jets to be produced closer to the beam direction, which re-inforces the signal. The results are displayed in figs. lla, b at vr~ = 300 and 600 GeV. We note that both W, Z peaks are visible and even clearly at 600 GeV, with a much sharper shape for the W because of the pure V - A nature of its coupling. With a high-luminosity machine like RHIC, we believe that these predictions can be tested to a high degree of accuracy. A possible way to avoid the problem of systematic errors is to consider the ratio of d % ( ± ) / d M at two different energies. This method can also be used for the search of new gauge bosons decaying into two jets. In the high-mass region, say M > 200 GeV/c 2, where gluon scattering is less important, we expect a better signal-to-background ratio which represents obviously a rather serious advantage. 6. Compositeness Compositeness is one possible approach where one tries to go beyond the Standard Model but, so far, there is no experimental indication for this attractive idea. One way to probe the existence of subconstituents is to study the sensitivity of the unpolarized cross section for some given processes in a specific kinematic
87
C. Bourrely et al. / Spin effects 15
i
i
p p -* jet, + jet~ + anything ~/s = 300 GeV
10
d ,o c~
(a) -5
=
t
i
70
I
i
i
i
i
I
80
i
i
i
i
90
Jet Pair Mass
100
(GeV/c s)
I00
l
p p -* jett + jets + anything ~/s = 600 GeV
%, 50 ,.c=
0
{b) =
70
=
=
=
i
i
i
i
I
80
Jet Pair Mass
i
90
i
i
i
100
(GeV/c s)
Fig. 11. (a) Parity violation effect in two jets production from W, Z decay at ~ = 300 GcV. (b) Same as (a) but for ~ = 600 GeV.
region. One usually assumes that the common subconstituents generate some contact interactions whose chirality structure is not known and which will modify the Standard Model amplitude Z sm in a very simple manner. For example the amplitude A~m for qi~ti ~ gg will become
,4; = A ; m 1 +
e 5) ,
(12)
where A is the compositeness energy scale, c and d being two unknown constants.
C
88
Bourrely et aL / Spin effects
The value of A is known to be not too small, say A ~ 1 or 2 T e V and above, and of course for A = oo one recovers the Standard Model situation. According to eq. (12) we expect now a parity violating asymmetry A L whose magnitude will depend on the value of A. For an illustration we have chosen to study it in the case of single-jet production. As we recalled above (see sect. 5) the data on the unpolarized cross section is not accurate enough, due to large systematic errors ( ~ 50%) and therefore any significant enhancement with respect to Standard Model calculations will be difficult to appreciate. On the other hand, the measurement of a
-0.1 x x x xx
#
xx
p p -* j e t + anything
-0.2
~ x
~
~/s = 300 GeV
x
-0.3
(a} -0.4
=
I
0
i
,
,
i
50
I¸
100 P, (GeV/Ù)
0.05
"~
-0.05 I
Illllllll
,,
p p -~ jet + anything
~.,,'.,,~,,
,,
-0.1
~/s = 600 GeV
-0.15
(b) --0.2
,
0
,
,
,
I
50
,
,
,
,
I
100
,
PT (CeV/o)
,
,
,
I
150
,
,
,
,
200
Fig. 12. Predicted A L for ~/s = 300 GeV and y = 0 versus PT, for A = I TeV (dashed curve) and A = 2 TcV (solid curve). (b) Same as (a) but for V~-= 600 GeV.
C. Bourrelyet al. / Spin effects
89
non-zero A L for such a process will be a clear signature for a deviation from the Standard Model. For the various subprocesses which contribute to single-jet production we have used the modified cross sections as in ref. [4] with a left-handed contact interaction i.e. c -- d = 1/x/2. We have checked that the unpolarized cross sections are affected at most by a factor of two. The results for A L are displayed in figs. 12a, b at ~ = 300 GeV and 600 GeV versus P'r for two values of the energy scale A --- 1 TeV and 2 TeV. Therefore we see from these measurements at high PT, that it will be possible to detect the existence of compositeness provided A is not too large. Direct photon production could also be a testing ground reaction but unfortunately in this case, unless A < 1 TeV, A L departures from zero for such high-p T values where the cross section is unobservable.
7. Concluding remarks In this paper we have shown the importance of spin effects in a pp collider with polarized protons in the energy range between fs---300 and 600 GeV. We have given detailed predictions for a selection of reactions, direct photon production, single-jet and di-jet production in some relevant kinematic regions. From the measurement of these double helicity asymmetries it will be possible to pin down the magnitude and the sign of A G ( x ) and to verify directly the interpretation of the EMC result in terms of the axial anomaly. This gluon polarized distribution function can also be extracted by means of other measurable effects as suggested in the literature. These include in polarized deep inelastic scattering, heavy-quark pair production [18] /' p --->f + J/qJ + X and high-p T hadronic jet production [19] f p ~ t ' + j e t ~ + j e t / + X, which would be worth to pursuit at H E R A , provided a polarized proton beam can be realized. Single helicity asymmetries in W + production at large PT are proven to be very useful to improve our knowledge of A u ( x ) and Ad(x), the polarized distribution functions of u- and d-quarks. They can also be used in single-jet production to uncover the existence of compositeness and in di-jet production to detect the hadronic decay modes of standard or new gauge bosons. Finally we have left untouched the subject of transverse polarization, although large effects have been observed at much lower energy values whose theoretical interpretation is far more difficult. We wish to thank G. Bunce and M. Tannenbaum who gave us the motivation for writing this article and for several interesting discussions.
Appendix A In this appendix we give all the Born cross sections for quark-quark and quark-antiquark scattering where we consider quarks with identical and distinct
90
C. Bourrely et al. / Spin effects
quantum numbers. We calculated these cross sections to lowest order in gluon, y, W and Z exchanges, including all QCD-electroweak interference terms, and assuming that the initial quarks (or antiquarks) have definite helicities a~ and a 2. We take the normalization d t r / d t = (rr/s2)g.iiT,.p where To. denotes the matrix element squared with i and j exchanges. First, let us consider identical quarks, q(al)q(a 2) --+ qq for which we have
T , , = ~a~[~. + A,A=) 7 + u 2
T~v=2a2e
4[
+ (1 - a , a = )
3 tu
(s2 s2 2s2)
(I+A,A2) 7 g + ~ - g + ~ T f f u
+
7
+(1-a,a2)
--~
(A.1)
'
7+~-
,
(A.2)
,= ,= 2j_L)
Tzz = a~[(Ct(1 - a,)(1 - a2) + Ca(1 + a,)(1 + a2) ) t-~z + u'-'~z+ 3 tzU z
+2C~C~(1
- a,a=)
, (A.3)
+
T z , = 2 a z a e ~ [ ( C ~ ( 1 - A,)(1 - aa) + C 2 ( 1 + a,)(1 + A2)) (St7Z + - - $2 uu z
+
- -
tu z +
Suez))
(
+2CRCL(1--Ar~2)
Tzs= ~aza s ( C 2 ( 1 - A x ) ( 1 - A 2 ) + C2(1 +A,)(1 +A2) )
- -
tt z
+
(:;z
- -
uu z
+ --
,
(A.4)
, (A.5)
ut z
s2 T~,g= -~aa~e~(1 + AIA2)~-~,
(A.6)
where tr s, a and a z =t~/sin20w cos20w are the strong, elctromagnetic and Z-boson coupling constants, respectively, and 0 w is the Weinberg angle. The coefficients of the Z coupling to right- and left-handed quarks of charge eq are CR= --eq sin20w,
CL= T ~ - e q s i n 2 O w ,
C. Bourrelyet at. / Spin effects
91
s, t and u are the usual Mandelstam variables and t z - t - M 2,
Uz=u-M
2.
For the line reversed process q(A1)~(A 2) ~ q~, the corresponding T/i are obtained from eqs. (A.1)-(A.6) after making the following substitutions: s ~, u, Az ~ - A 2 (except in the first four expressions (A.1)-(A.4) for the coefficients of t2), 1/u z Sz/(S ~ + r~M~), 1/u~ ~ 1/(s~ + F~Mz), 2 2 where s z --s - Mz2. Second, we consider distinct quarks q(Al)q'(A 2) --* qq' for which we find 2
4 as 7gg= -~ t--T[(1 + AIA2)s 2 + (1 - h l A E ) U 2 ] ,
r =
(A.7)
o~2
27re.e., [ 1+
(A.8)
, = _ _ ~t(c?.cf(1-x,)(1-x2)+c.c.(l~" ~ ,~ T~z
+AI)(1 +
4
,~))~
+(C~C'I~(1-A,)(I+Az)+C~Cf(I+AI)(1-Az))uE],
T~:~' =
2
~za tzt eqeq,[(CLC'L(1
-AI)(I
(A.9)
-A2) + CRC~(1 +Al)(1 +A2))s 2
+ (CLCk(I -At)(1 +A2) + CRCt(I +AI)(I --A2)).2] , (A.10) $2
T~vw=
' = Twz
IVqq,]4(1 - A1)(1 - A 2 ) u 2 ,
~3~ Z lVqQ,12CLC,L(1 -
S2
- , At)(1 - A2) -Uwtz
0~W~ ' Two=
(A.11)
S2
-~ eqeq,lVqq,lZ(1- A , ) ( 1 - Az) -Uwt ,
, 4 rwg= ~aw,~lgqq, I2 (1
(A.12)
(A.13)
S2
-A1)(1 -h2)--
,
(A.14)
Uwt
where a w -- a / s i n z 0 w is the W coupling constant, u w = u - M 2 and Vqq, denotes the qq' element of the weak mixing matrix. For the line reversed reaction q(AI)~'(A2) ~ qFa', the corresponding T/i are obtained from eqs. (A.7)-(A.14) after
C. Bourrely et al. / Spin effects
92
making the following substitutions: s ~ u, A2 __, _A2 ' 1/Uw ~ Sw/(S~ v + F~vMw)2 2 2 2 and 1/U2w .l_>1/(S2w + FwMw). For the crossed channel process q(Al)~(A 2) ~ q'~'
the corresponding T/y are also obtained from eqs. (A.7)-(A.14) after making the following substitutions: s ~, t, A2-->-A 2 (except in the first four expressions (A.7)-(A.10) for the coefficient of u2), 1/t z ~ Sz/(S ~ + F~M~) and 1/t~--, 1/ 2 2 2 (s z + F ~ M ~ ) . F i n a l l y w e m u s t also c o n s i d e r t h e i m p o r t a n t c o n t r i b u t i o n f r o m t h e p r o c e s s e s in w h i c h all f o u r q u a r k s a r e d i s t i n c t i.e. q(At)q'(A 2) -o q"q" for w h i c h we have ~2
T~w--
2 V. , 2
s2
I q¢l ( 1 - A I ) ( 1 - - A 2 ) U 2 w ,
(A.15)
and the crossed channel process q(AI)~"(A 2) --* q"~' for which u2
T'~v=
4 Vq'q"12lVq¢"12(1-Al)(l+A2)s2 w +FwMw2 2 •
(A.16)
Unpolarized cross sections are obtained for A~ = A2 = 0 and we have checked that we recover the results given in ref. [16]. For polarized cross sections, Ranft and Ranft [17] have some incorrect results and leave out some important processes.
References [1] J. Softer, Overview of high-energy physics with polarized particles, invited talk presented at Int. Conf. on Polarization phenomena in nuclear physics, Paris, July 1990, Proc. Editions de Physique 1990, C6, Tome 51, p. 135, ed. A. Boudard and Y. Terrien, and references therein [2] EMC, J. Ashman et al., NucL Phys. B328 (1989) 1 [3] H. Rollnik, Ideas and models for the proton spin, invited talk presented at 9th Int. Syrup. on High-energy spin physics, Bonn, September 1990, to be published in the proceedings [4] C. Bourrely, J. Softer, F.M. Renard and P. Taxil, Phys. Rcp. 177 (1989) 319; See also the errata preprint CPT-87/P.2056 (December 1990) [5] E. Berger and J. Qiu, Phys. Rev. D40 (1989) 778 [6] UA2 Collaboration, R. Ansari et al., Z. Phys. C41 (1988) 395 [7] H.Y. Cheng and S.N. Lai, Phys. Rev. D41 (1990) 91 [8] N. Arteaga-Romero, A. Nicolaidis and J. Silva, Phys. Rev. Lett. 52 (1984) 172 [9] C. Stubenranch, Thesis CE.A-N-2532 (1987) [10] K. Hidaka, Nucl. Phys. B192 (1981) 369 [11] UA2 Collaboration, J.A. Appel et al., Phys. Lett. B160 (1985) 349; UA1 Collaboration, G. Arnison et al., Phys. Lctt. B172 (1986) 461 [12] H.¥. Cheng, S.R. Hwang and S.N. Lai, Phys. Rev. D42 (1990) 2243 [13] UA2 Collaboration, P. Bagnaia et al., Phys. Lett. B138 (1984) 430 [14] UA2 Collaboration, R. Ansari et ai., Phys. Lett. B186 (1987) 452 [15] UA2 Collaboration, J. Alitti et al., preprint CERN-PPE/90-105 (1990) [16] U. Baur, E.W. Glover and A.D. Martin, Phys. Lctt. B232 (1989) 519 [17] F.E. Paige, T.L Trueman and T.N. Tudron, Phys. Rev. D19 (1979) 935; G. R a n t and J. Ranft, Nucl. Phys. B165 (1980) 395 [18] J.Ph. Guillet, Z. Phys. C39 (1988) 75 [19] R.D. Carlitz, J.C. Collins and A.H. Mueller, Phys. Lett. B214 (1988) 229; Z. Kunszt, Phys. Lett. B218 (1989) 243