T-odd effects in photon-jet production at the Tevatron

0.21 MB·English

by  D. Boer

T-odd effects in photon-jet production at the Tevatron
D. Boer,∗ P.J. Mulders,† and C. Pisano‡
Department of Physics and Astronomy,
Vrije Universiteit Amsterdam,
NL-1081 HV Amsterdam, the Netherlands
(Dated: February 2, 2008)
Theangulardistributioninphoton-jetproductioninpp¯→γjetX isstudiedwithinageneralized
factorization scheme taking into account the transverse momentum of the partons in the initial
hadrons. Within this scheme an anomalously large cos2φ asymmetry observed in the Drell-Yan
8 process could beattributed to theT-odd, spin and transverse momentum dependentparton distri-
0 bution function h⊥q(x,p2). This same function is expected to produce a cos2φ asymmetry in the
1 ⊥
0 photon-jet production cross section. We give the expression for this particular azimuthal asymme-
2 try,whichisestimatedtobesmallerthantheDrell-Yanasymmetrybutstillofconsiderablesizefor
n Tevatron kinematics. This offers a new possibility to study T-odd effects at the Tevatron.
a
J PACSnumbers: 12.38.-t;13.85.Ni;13.88.+e

1
2

]
h I. INTRODUCTION
p
-
p It is well-knownthat the angulardistribution of Drell-Yan leptonpairs displays an anomalouslylargecos2φ asym-
e metry. This was experimentally investigated using π− beams scattering off deuterium and tungsten targets at center
h of mass energies of order 20 GeV [1, 2, 3]. A next-to-leading order (NLO) analysis in perturbative QCD (pQCD)
[
withinthestandardframeworkofcollinearfactorizationfailedtodescribethedata[4]. Morespecifically,theobserved

2 violation of the so-called Lam-Tung relation [5, 6, 7], a relation between two angular asymmetry terms, could not be
v described. The NLO pQCD result is an order of magnitude too small and of opposite sign. This has prompted much
7 theoretical work [4, 8, 9, 10, 11, 12, 13, 14, 15, 16], offering explanations that go beyond the framework of collinear
7
factorization and/or leading twist perturbative QCD.
7
More recently, pd Drell-Yan scattering was studied in a fixed target experiment (√s 40 GeV) at Fermilab [17].
0 ≈
The angular distribution does not display a large cos2φ asymmetry, indicating that the effect that causes the large
.
2 asymmetry in π−N scattering is probably smallfor nonvalence partons. For this reasonone wouldlike to investigate
1 pp¯ scattering, which is expected to be similar to the π−N case (an expectation supported by model calculations
7
[10, 14, 18]). It is an experiment that could be done at the planned GSI-FAIR facility. It can in principle also
0
: be done at Fermilab, although the energy of the collisions is so much higher (√s = 1.96 TeV) that the Drell-Yan
v asymmetry may be quite different in magnitude, possibly much smaller at very high invariant mass Q of the lepton
i
X pair. Nevertheless, it would be interesting to see if NLO pQCD expectations hold at those energies. Recently, such a
study of the angular distribution was done for W-boson production at the Tevatron and a nonzero result compatible
r
a with NLO pQCD [19, 20] was obtained [21]. This may likely be due to the fact that chirality flip effects, such as
the T-odd effect to be discussed here, do not contribute to the cos2φ angular distribution [22, 23]. For neutral
boson production they do contribute howeverand therefore could lead to quite a different result. This remains to be
investigated.
In this paper we consider an asymmetry in the process pp¯ γ jet X that potentially probes the same underlying
→
mechanism and could have certain advantages over Drell-Yan. This photon-jet production process has already been
studied experimentally in the angular integrated case at the Tevatron [24]. Here we will calculate the angular
dependence within the framework as employed in Ref. [16], where transverse momentum and spin dependence of
partons inside hadrons is included1. In that case a nontrivial polarization-dependent quark distribution (denoted by
h⊥q) appears, which offers an explanation for the anomalous angular asymmetry in the Drell-Yan process. The new
1
asymmetryisproportionaltotheanalyzingpoweroftheDrell-Yancos2φasymmetryatthescalesetbythetransverse
momentum of the photon or the jet. The latter asymmetry is expected to decrease with increasing scale [26], but as
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
‡Electronicaddress: [email protected]
1 Photon-jetangular correlationsinppandpp¯collisionshavealsorecentlybeenstudiedinRef.[25]usingthekt-factorizationapproach
applicableatsmallx,wheregluon-gluonscatteringdominates.
2
we willdemonstrate the proportionalityfactorincreases,leadingone to expect a significantasymmetry alsoathigher
energies.
In Section II we discuss the theoretical framework and the expected contributions to the new asymmetry. In
SectionIII westudy the phenomenologyofthis asymmetry,using typicalTevatronkinematicsandcuts. We endwith
a summary of the results and the required measurement.
II. THEORETICAL FRAMEWORK: CALCULATION OF THE CROSS SECTION
We consider the process
h (P )+h (P ) γ(K )+jet(K )+X, (1)
1 1 2 2 γ j
→
where the four-momentaofthe particlesare givenwithin brackets,andthe photon-jetpairin the finalstate is almost
back-to-back in the plane perpendicular to the direction of the incoming hadrons. To lowest order in pQCD the
reaction is described in terms of the partonic two-to-two subprocesses
q(p )+q¯(p ) γ(K )+g(K ), and q(p )+g(p ) γ(K )+q(K ) . (2)
1 2 γ j 1 2 γ j
→ →
We make a lightcone decomposition of the hadronic momenta in terms of two light-like Sudakov vectors n and n ,
+ −
satisfying n2 =n2 =0 and n n =1:
+ − +· −
M2 M2
Pµ =P+nµ + 1 nµ , and Pµ = 2 nµ +P−nµ . (3)
1 1 + 2P+ − 2 2P− + 2 −
1 2
In general n and n will define the lightcone components of every vector a as a± a n , while perpendicular
+ − ∓
≡ ·
vectors a will always refer to the components of a orthogonal to both incoming hadronic momenta, P and P .
⊥ 1 2
Hence the partonic momenta (p , p ) can be expressed in terms of the lightcone momentum fractions (x , x ) and
1 2 1 2
the intrinsic transverse momenta (p , p ), as follows
1⊥ 2⊥
m2+p2 m2+p2
pµ =x P+nµ + 1 1⊥nµ +pµ , and pµ = 2 2⊥nµ +x P−nµ +pµ . (4)
1 1 1 + 2x P+ − 1⊥ 2 2x P− + 2 2 − 2⊥
1 1 2 2
We denote with s the total energy squared in the hadronic center-of-mass (c.m.) frame, s = (P +P )2 = E2 ,
1 2 c.m.
and with η the pseudo-rapidities of the outgoing particles, i.e. η = ln tan(1θ ) , θ being the polar angles of the
i i − 2 i i
outgoing particles in the same frame. Finally, we introduce the partonic(cid:0)Mandelsta(cid:1)m variables
sˆ=(p +p )2, tˆ=(p K )2, uˆ=(p K )2, (5)
1 2 1 γ 1 j
− −
which satisfy the relations
tˆ 1 uˆ
y = , and =1 y . (6)
−sˆ ≡ eηγ−ηj+1 − sˆ −
FollowingRef.[27]weassumethatatsufficiently highenergiesthe hadroniccrosssectionfactorizesinasoftparton
correlator for each observed hadron and a hard part:
1 d3K d3K
dσh1h2→γjetX = γ j dx d2p dx d2p (2π)4δ4(p +p K K )
2s (2π)32Eγ (2π)32EjZ 1 1⊥ 2 2⊥ 1 2− γ− j
Φ (x ,p ) Φ (x ,p ) H (p ,p ,K ,K )2 , (7)
a 1 1⊥ b 2 2⊥ ab→γc 1 2 γ j
× ⊗ ⊗ | |
aX,b,c
wherethesumrunsoveralltheincomingandoutgoingpartonstakingpartinthesubprocessesin(2). Theconvolutions
indicate the appropriate traces over Dirac indices and H 2 is the hard partonic squared amplitude, obtained from
⊗ | |
the cut diagrams in Figs. 1 and 2 [27]. The parton correlatorsare defined on the lightfront LF (ξ n 0, with n n
−
· ≡ ≡
for parton 1 and n n for parton 2); they describe the hadron parton transitions and can be parameterized
+
≡ →
in terms of transverse momentum dependent (TMD) distribution functions. The quark content of an unpolarized
hadron is described, in the lightcone gauge A n=0 and at leading twist, by the correlator [28]
·
d(ξ P)d2ξ 1 [p/ ,P/]
Φ (x,p ;P)= · ⊥ eip·ξ P ψ(0)ψ(ξ) P = fq(x,p2)P/ +ih⊥q(x,p2) ⊥ , (8)
q ⊥ Z (2π)3 h | | i LF 2(cid:26) 1 ⊥ 1 ⊥ 2M (cid:27)
(cid:5)
3
FIG. 1: Cut diagrams for thesubprocess qq¯→γg.
FIG. 2: Cut diagrams for thesubprocess qg→γq.
where fq(x,p2) is the unpolarized quark distribution, which integrated over p gives the familiar lightcone mo-
1 ⊥ ⊥
mentum distribution fq(x). The time-reversal (T) odd function h⊥q(x,p2) is interpreted as the quark transverse
1 1 ⊥
spin distribution in an unpolarized hadron [28]. Below we will discuss the T-odd nature of this function and its
consequences in more detail.
Analogously, for an antiquark,
d(ξ P)d2ξ 1 [p/ ,P/]
Φ¯ (x,p ;P)= · ⊥ e−ip·ξ P ψ(0)ψ(ξ) P = fq¯(x,p2)P/ +ih⊥q¯(x,p2) ⊥ . (9)
q ⊥ −Z (2π)3 h | | i LF 2(cid:26) 1 ⊥ 1 ⊥ 2M (cid:27)
(cid:5)
The gluon correlator in the lightcone gauge is given by [29]
n n d(ξ P)d2ξ
Φµν(x,p ;P) = ρ σ · ⊥ eip·ξ P Tr Fµρ(0)Fνσ(ξ) P
g ⊥ (pn)2Z (2π)3 h | | i LF
· (cid:2) (cid:3) (cid:5)
1 pµpν p2
= gµνfg(x,p2)+ ⊥ ⊥ +gµν ⊥ h⊥g(x,p2) , (10)
2x(cid:26)− ⊥ 1 ⊥ (cid:18) M2 ⊥ 2M2(cid:19) 1 ⊥ (cid:27)
with gµν being a transverse tensor defined as
⊥
gµν =gµν nµnν nµnν . (11)
⊥ − + −− − +
The function fg(x,p2) represents the usual unpolarized gluon distribution, while the T-even function h⊥g(x,p2)
1 ⊥ 1 ⊥
is the distribution of linearly polarized gluons in an unpolarized hadron. We include it here because it potentially
contributes to the observable of interest. However,it will turn out to yield a power-suppressedcontribution.
Intheaboveexpressionsh⊥q appearsastheT-oddpartintheparametrizationofthecorrelatorΦ . Fordistribution
1 q
functions, however, such a T-odd part is intimately connected to the gauge link that appears in the correlator Φ
q
connecting the quark fields. This gauge link is process-dependent and as a result, T-odd functions can appear with
different factors in different processes. The specific factor can be traced back to the color flow in the cut diagrams of
the partonic hard scattering. In situations in which only one T-odd function contributes this has been analyzed in
detailfortheprocessofinterestandrelatedprocesses,cf.e.g.Refs.[30,31]. Itleadstospecificcolorfactorsmultiplying
the T-odd distribution function. In the case of single spin asymmetries (SSA) one T-odd function appears, which
could be the distribution function h⊥q. Taking the appearance in the case of leptoproduction as reference (having a
1
factor +1 for the γ∗q q subprocess), one finds appearance with a different factor in Drell-Yan scattering (factor
1 for qq¯ γ∗ subpro→cess [32, 33, 34]) or yet another factor for the appearance in the SSA in photon-jet production
− →
(factor (N2 +1)/(N2 1) for all contributions shown in Figs 1 and 2 [27]). In the cos2φ asymmetry in Drell-Yan
−
andin photon-jet productionthe situation is different because a product oftwo T-odd functions arises (h⊥qh⊥q¯). As
1 1
we will discuss the factors are in both cases simply +1, which fixes the relative sign of the asymmetries in the two
processes to be +1.
Thefactorinphoton-jetproductionarisesinthefollowingway. Alldiagramshaveonlytwocolorflowcontributions
with relative strength N2 and 1. Color averaging in the initial state gives the usual color factors (N2 1)/N2 for
− −
4
the qq¯ γg process and (N2 1)/N(N2 1) = 1/N for the qg γq process. Including gauge links in the TMD
→ − − →
correlators one schematically has
qq¯ γg : N2 Φ[+((cid:3)†)] Φ[+†((cid:3))] H 2 ∆[−][−†] 1 Φ[−] Φ[−†] H 2 ∆[−][−†], (12)
→ N2 1 q ⊗ q ⊗| | ⊗ g − N2 1 q ⊗ q ⊗| | ⊗ g
− −
qg γq : N2 Φ[−((cid:3))] Φ[+][−†] H 2 ∆[−†] 1 Φ[+] Φ[+][−†] H 2 ∆[−†], (13)
→ N2 1 q ⊗ g ⊗| | ⊗ q − N2 1 q ⊗ g ⊗| | ⊗ q
− −
where the correlators now include gauge links, Φ[+] ψ(0)U[+]ψ(ξ) or Φ[+][−†] F(0)U[+](0,ξ)F(ξ)U[−]† , etc.
g
The gauge links U[±] are the future and past-pointin∼ghones in whichithe path run∼s vhia lightcone , respecitively.
The ((cid:3)) contributions indicate the presence of a 1Tr(U[+]U[−]†) term. Following Ref. [31], the c±or∞relator Φ (and
N q
similarly Φ ) can be split into two parts, Φ[+((cid:3)†)] = Φ[+] +δΦ[+((cid:3)†)], where the non-universal part δΦ vanishes
q q q q q
upon p -integration or p -weighting. In what follows we will omit these non-universal parts, but in principle they
T T
could contribute (and potentially spoil factorization as recently discussed in Refs [35, 36, 37]) when one considers
cross sections that are differential in measured transverse momenta. Ideally one should consider performing the
appropriate weighting to remove their possible contributions altogether. We, however, keep the expressions for the
cross sections differential because the weighting is often difficult in experimental data analyses and the relation with
the Drell-Yan expressions is more straightforward. After weighting one would be left only with the TMD functions
Φ[±] and Φ[±†], in which the gauge link determines the sign with which T-odd functions are multiplied, giving in qq¯
q q
scatteringforh⊥q apositivesignfromΦ[+] andanegativesignfromΦ[−]. Ifnofactorsarisefromtheothercorrelators
1 q q
one sees that in SSA involving only one T-odd function h⊥q, this function appears in photon-jet production with the
1
abovementioned factor (N2+1)/(N2 1) but otherwise the normal partonic cross section. This factor and similar
−
ones for other T-odd functions were used in Ref. [27]. In the present case of a product of two T-odd functions, we
obtainfrombothcombinationsofcorrelatorsΦ[+] Φ[+†] andΦ[−] Φ[−†] inEq.(12)nowapositivesign. Hence,the
q ⊗ q q ⊗ q
contribution h⊥qh⊥q¯ appears with just a factor +1, justifying the use of the parametrizations in Eqs (8) and (9) in
1 1
combinationwith the normalpartonichardscatteringamplitudes squared. Forthe contributioninvolvingthe T-even
gluon function h⊥g no process dependent color factors need to be considered.
1
The issue of factorization is left as an open question and the same applies to resummation. In Refs [38, 39]
the resummation for the cos2φ asymmetry is addressed and found to be unclear beyond the leading logarithmic
approximation. A similar situation could apply to the asymmetry in the photon-jet production case, but we will not
address this here. Our present goal is to point out how the contribution of h⊥qh⊥q¯enters the asymmetry expression
1 1
in leading order and to give an estimate of its expected magnitude.
Inordertoderiveanexpressionforthecrosssectionintermsofpartondistributions,weinserttheparametrizations
(8)-(10) of the TMD quark, antiquark and gluon correlators into (7). Furthermore, utilizing the decompositions of
the parton momenta in (4), the δ-function in (7) can be rewritten as
2 1 1
δ4(p1+p2 k1 k2) = δ x1 ( Kγ⊥ eηγ+Kj⊥ eηj) δ x2 ( Kγ⊥ e−ηγ+Kj⊥ e−ηj)
− − s (cid:18) −√s | | | | (cid:19) (cid:18) −√s | | | | (cid:19)
δ2(p +p K K ) , (14)
× 1⊥ 2⊥− γ⊥− j⊥
with corrections of order (1/s2). After integration over x and x , which fixes the parton momentum fractions by
1 2
the first two δ-functions oOn the r.h.s. of (14), the resulting hadronic cross section consists of two contributions, i.e.
dσh1h2→γjetX dσ[fq,g] dσ[h⊥q]
= 1 + 1 , (15)
dη d2K dη d2K d2q dη d2K dη d2K d2q dη d2K dη d2K d2q
γ γ⊥ j j⊥ ⊥ γ γ⊥ j j⊥ ⊥ γ γ⊥ j j⊥ ⊥
withq K +K . Weareinterestedineventsinwhichthephotonandjetareapproximatelyback-to-backinthe
transve⊥rs≡e plaγn⊥e, thejr⊥efore q K , K . The cross section dσ[fq,g] depends on the unpolarized (anti)quark
| ⊥|≪| γ⊥| | j⊥| 1
5
and gluon distributions fq,g, while dσ[h⊥q] depends on the (anti)quark function h⊥q. More explicitly,
1 1 1
dσ[fq,g] αα
1 = s δ2(q K K ) e2 d2p d2p δ2(p +p q )
dηγd2Kγ⊥dηjd2Kj⊥d2q⊥ sK2γ⊥ ⊥− γ⊥− j⊥ Xq qZ 1⊥ 2⊥ 1⊥ 2⊥− ⊥
1
(1 y)(1+y2)fq(x ,p2 )fg(x ,p2 )
×(cid:26)N − 1 1 1⊥ 1 2 2⊥
1 N2 1
+ y(1+(1 y)2)fq(x ,p2 )fg(x ,p2 )+ − (y2+(1 y)2)
N − 1 2 2⊥ 1 1 1⊥ N2 (cid:20) −
(fq(x ,p2 )fq¯(x ,p2 )+fq(x ,p2 )fq¯(x ,p2 )) , (16)
× 1 1 1⊥ 1 2 2⊥ 1 2 2⊥ 1 1 1⊥ (cid:21)(cid:27)
and
dσ[h⊥q] 2αα
1 = s δ2(q K K ) e2 d2p d2p δ2(p +p q )
dηγd2Kγ⊥dηjd2Kj⊥d2q⊥ −sK2γ⊥ ⊥− γ⊥− j⊥ Xq qZ 1⊥ 2⊥ 1⊥ 2⊥− ⊥
y(1 y) (K p )(K p )+(K p )(K p )
− (p p )+ γ⊥· 1⊥ j⊥· 2⊥ γ⊥· 2⊥ j⊥· 1⊥
× M M (cid:18) 1⊥· 2⊥ K2 (cid:19)
1 2 γ⊥
N2 1
− (h⊥q(x ,p2 )h⊥q¯(x ,p2 )+h⊥q(x ,p2 )h⊥q¯(x ,p2 )) , (17)
× N2 1 1 1⊥ 1 2 2⊥ 1 2 2⊥ 1 1 1⊥ (cid:27)
where the sums always run over quarks and antiquarks. Power-suppressedterms of the order (1/(K4 s)), such as
O γ⊥
the ones proportional to the gluon distribution function h⊥g, are neglected throughout this paper. If we define the
1
function
1
(x ,x ,q2) e2 d2p d2p δ2(p +p q ) 2(hˆ p )(hˆ p ) (p p )
H 1 2 ⊥ ≡ M M qZ 1⊥ 2⊥ 1⊥ 2⊥− ⊥ (cid:18) ⊥· 1⊥ ⊥· 2⊥ − 1⊥· 2⊥ (cid:19)
1 2 Xq
(h⊥q(x ,p2 )h⊥q¯(x ,p2 )+h⊥q(x ,p2 )h⊥q¯(x ,p2 )), (18)
× 1 1 1⊥ 1 2 2⊥ 1 2 2⊥ 1 1 1⊥
with hˆ q /q , and denote with φ , φ andφ the azimuthal angles,in the hadronic center-of-massframe, of the
≡ ⊥ | ⊥| γ j ⊥
outgoing photon, jet and vector q respectively, then (17) can be rewritten as
⊥
dσ[h⊥q] 2αα N2 1
1 = s y(1 y) − δ2(q K K )
dηγd2Kγ⊥dηjd2Kj⊥d2q⊥ −sK2γ⊥ − N2 ⊥− γ⊥− j⊥
2 K{l Km} q{lqm}
γ⊥ j⊥ ⊥ ⊥ δlm (x ,x ,q2)
× 2K2 (cid:18) q2 − (cid:19)H 1 2 ⊥
l,Xm=1 γ⊥ ⊥
2αα N2 1
= s y(1 y) − δ2(q K K )
−sK2 − N2 ⊥− γ⊥− j⊥
γ⊥
cos(2φ φ φ ) (x ,x ,q2). (19)
× ⊥− γ − j H 1 2 ⊥
The approximation K K has been used in the derivation of the second equation in (19). Furthermore, the
γ⊥ j⊥
| |≈| |
same approximation allows us to derive the following relations, starting from the definition of q in terms of K
⊥ γ⊥
and K ,
j⊥
q
| ⊥| cosφ = cosφ +cosφ ,
K ⊥ γ j
γ⊥
| |
q
| ⊥| sinφ = sinφ +sinφ . (20)
K ⊥ γ j
γ⊥
| |
Since q K , (20) implies cosφ cosφ and sinφ sinφ , so that
| ⊥|≪| γ⊥| j ≈− γ j ≈− γ
cos(2φ φ φ ) cos2(φ φ ) cos2(φ φ ), (21)
⊥ γ j ⊥ γ ⊥ j
− − ≈− − ≈− −
and (19) takes the form
dσ[h⊥q] 2αα N2 1
1 = s y(1 y) − δ2(q K K )cos2(φ φ ) (x ,x ,q2) .
dηγd2Kγ⊥dηjd2Kj⊥d2q⊥ sK2γ⊥ − N2 ⊥− γ⊥− j⊥ ⊥− j H 1 2 ⊥
(22)
6
Substituting (16)and(22)into(15), andintegratingoverK (alternatively,integratingoverK wouldleadtothe
j⊥ γ⊥
same equations as presented below, but with the replacement K K ), we obtain
γ⊥ j⊥
↔
dσh1h2→γjetX dσh1h2→γjetX
= d2K
dηγdηjd2Kγ⊥d2q⊥ Z j⊥ dηγd2Kγ⊥dηjd2Kj⊥d2q⊥
αα 1 1
= s (1 y)(1+y2) (x ,x ,q2)+ y(1+(1 y)2) ˜(x ,x ,q2)
sK2 (cid:26)N − G 1 2 ⊥ N − G 1 2 ⊥
γ⊥
N2 1
+ − (y2+(1 y)2) (x ,x ,q2)+ 2y(1 y) (x ,x ,q2) cos2(φ φ )
N2 (cid:20) − F 1 2 ⊥ − H 1 2 ⊥ ⊥− γ (cid:21)(cid:27)
(23)
where, in analogy to (18), the following convolutions of distribution functions have been utilized:
(x ,x ,q2) e2 d2p d2p δ2(p +p q )(fq(x ,p2 )fq¯(x ,p2 )+fq(x ,p2 )fq¯(x ,p2 )),
F 1 2 ⊥ ≡ qZ 1⊥ 2⊥ 1⊥ 2⊥− ⊥ 1 1 1⊥ 1 2 2⊥ 1 2 2⊥ 1 1 1⊥
Xq
(x ,x ,q2) e2 d2p d2p δ2(p +p q )fq(x ,p2 )fg(x ,p2 ),
G 1 2 ⊥ ≡ qZ 1⊥ 2⊥ 1⊥ 2⊥− ⊥ 1 1 1⊥ 1 2 2⊥
Xq
˜(x ,x ,q2) e2 d2p d2p δ2(p +p q )fq(x ,p2 )fg(x ,p2 ). (24)
G 1 2 ⊥ ≡ qZ 1⊥ 2⊥ 1⊥ 2⊥− ⊥ 1 2 2⊥ 1 1 1⊥
Xq
Alternatively, equation (23) can be rewritten as
dσh1h2→γjetX 1 dσh1h2→γjetX
= 1+ (y,x ,x ,q2) cos2(φ φ ) , (25)
dηγdηjd2Kγ⊥d2q⊥ π2 dηγdηjdK2γ⊥dq2⊥ (cid:18) A 1 2 ⊥ ⊥− γ (cid:19)
with
dσh1h2→γjetX 1 dσh1h2→γjetX
= dφ dφ d2K
dηγdηjdK2γ⊥dq2⊥ 4Z ⊥ γ j⊥ dηγd2Kγ⊥dηjd2Kj⊥d2q⊥
π2αα 1 1
= s (1 y)(1+y2) (x ,x ,q2)+ y(1+(1 y)2) ˜(x ,x ,q2)
sK2 (cid:26)N − G 1 2 ⊥ N − G 1 2 ⊥
γ⊥
N2 1
+ − (y2+(1 y)2) (x ,x ,q2) (26)
N2 − F 1 2 ⊥ (cid:27)
and the azimuthal asymmetry
(y,x ,x ,q2) = ν(x ,x ,q2)R(y,x ,x ,q2), (27)
A 1 2 ⊥ 1 2 ⊥ 1 2 ⊥
where
2 (x ,x ,q2)
ν(x ,x ,q2) = H 1 2 ⊥ (28)
1 2 ⊥ (x ,x ,q2)
F 1 2 ⊥
containsthedependenceonh⊥q andisidenticaltotheazimuthalasymmetryexpressionthatappearsintheDrell-Yan
1
process [16], with the scale Q equal to K . The ratio
γ⊥
| |
π2αα dσh1h2→γjetX −1
R = s y(1 y) (x ,x ,q2)
sK2 − F 1 2 ⊥ (cid:18)dη dη dK2 dq2 (cid:19)
γ⊥ γ j γ⊥ ⊥
N2y(1 y) (x ,x ,q2)
= − F 1 2 ⊥
N(1 y)(1+y2) (x ,x ,q2)+Ny(1+(1 y)2) ˜(x ,x ,q2)+(N2 1)(y2+(1 y)2) (x ,x ,q2)
− G 1 2 ⊥ − G 1 2 ⊥ − − F 1 2 ⊥
(29)
only depends on the T-even distribution functions fq,g(x,p2).
1 ⊥
7
0000....8888
x = 0.21, x = 0.23
RRRR 1 2
0000....7777 x = 0.13, x = 0.14
1 2
x = 0.06, x = 0.07
1 2
0000....6666 x = 0.04, x = 0.05
1 2
0000....5555
222200000000 GGGGeeeeVVVV ==== |||| KKKKγγγγ −−−−|||| ||||
0000....4444
111122220000
0000....3333
66665555
0000....2222
33336666
0000....1111
0000
0000 0000....2222 0000....4444 0000....6666 0000....8888 1111
yyyy
FIG.3: TheratioRdefinedin(29)asafunctionofy,calculated accordingto(35)fordifferentvaluesofx1,x2,|Kγ⊥|typical
of the Tevatron experiments [40].
III. PHENOMENOLOGY: THE AZIMUTHAL ASYMMETRY
The process pp¯ γjetX is currently being analyzed by the DØ Collaboration at the Tevatron collider [24, 40].
Data on the cross s→ection, differential in η , η and K2 , have been taken at √s=1.96 TeV, and were considered in
γ j j⊥
a preliminary study [40] with the following kinematic cuts:
K >30 GeV, 1<η <1 (central region)
γ⊥ γ
| | −
K >15 GeV, 0.8 η 0.8 (central), 1.5< η <2.5 (forward); (30)
j⊥ j j
| | − ≤ ≤ | |
the transverse momentum imbalance between the photon and the jet being constrained by the relation
q <12.5+0.36 K (GeV) . (31)
| ⊥| ×| γ⊥|
Suchangularintegratedmeasurementsareonlysensitivetothetransversemomentumintegratedpartondistributions
fq,g(x)= dp2 fq,g(x,p2) , (32)
1 Z ⊥ 1 ⊥
as can be seen from the leading order expression of the cross section,
dσpp¯→γjetX παα 1 1
= s e2 (1 y)(1+y2)fq(x )fg(x )+ y(1+(1 y)2)fq(x )fg(x )
dη dη dK2 sK2 q(cid:26)N − 1 1 1 2 N − 1 2 1 1
γ j γ⊥ γ⊥ Xq
N2 1
+2 − (y2+(1 y)2)fq(x )fq(x ) , (33)
N2 − 1 1 1 2 (cid:27)
obtained after integrating (26), together with the definitions in (24), over q2. Here we have used that the antiquark
⊥
contribution in the antiproton equals the quark contribution inside a proton. A study of the angular dependent
crosssectionin (25)will providevaluableinformationonthe TMDdistribution functionh⊥q(x,p2), ifthe azimuthal
1 ⊥
asymmetry turnsouttobesufficientlysizeableintheavailablekinematicregion. Modelcalculations[10,18]applied
A
to the pp¯Drell-Yan process have shown that the quantity ν in (27) is of the order of 30% or higher for q of a few
| ⊥|
8
GeV and Q values of (1 10) GeV. Therefore, a study of the order of magnitude of as a function of x , x and
1 2
q2 requires an estimaOte of−the ratio R defined in (29). This will be obtained as follows.A
⊥
First of all, the unknown TMD distribution functions appearing in (29) are evaluated assuming a factorization of
their transverse momentum dependence (see, for example, [41, 42]), that is
fq,g(xp2)=fq,g(x) (p2) , (34)
1 , ⊥ 1 T ⊥
with fq,g(x) being the usual unpolarized parton distributions and (p2) being a generic function, taken to be the
same f1or all partons and often chosen to be Gaussian. The q2-depTende⊥nce of R then drops out and (29) takes the
⊥
form
2N2y(1 y) e2fq(x )fq(x )
R= − q q 1 1 1 2 .
e2 N(1 y)(1+y2)fq(x )fg(x )+Ny(1+(1 Py)2)fq(x )fg(x )+2(N2 1)(y2+(1 y)2)fq(x )fq(x )
q q − 1 1 1 2 − 1 2 1 1 − − 1 1 1 2
P (cid:8) (35)(cid:9)
We consideronly lightquarks,i.e. the sumin (35) runs overq =u,u¯,d, d¯,s, s¯, andwe use the leading orderGRV98
set [43] for the parton distributions, at the scale µ2 =K2 .
γ⊥
Our results for R as a function of y are shown in Fig. 3 at some fixed values of the variables x , x and K ,
1 2 γ⊥
| |
typical of the Tevatron experiments [40]. The values of x and x considered correspond to their average when both
1 2
the photon and the jet are in the central rapidity region, where η η 0 and x x . In this case y 0.5,
j γ 1 2
≈ ≈ ≈ ≈
where R turns out to be largest. Evidently, R increases as x and x increase, due to the small contribution, in the
1 2
denominator, of the gluon distributions fg(x) in the valence region.
1
Hence, we see that the asymmetry is a product of a large Drell-Yan asymmetry term ν and a factor R that is
A
estimated to be in the 10%-50% range for Tevatron kinematics. This leads us to conclude that an asymmetry in
the order of 5%-15% is possible in the central region. This could allow a study of the distribution function h⊥Aq in
1
pp¯ γjetX at the Tevatron.
→
IV. SUMMARY AND CONCLUSIONS
In this paper we have calculated the cross section of the process pp¯ γjetX within a generalized factorization
→
scheme,takingintoaccountthetransversemomentumofthepartonsintheinitialprotonandantiproton. Inparticular,
wehavestudiedthecontributionfromtheT-odd,spinandtransversemomentumdependentpartondistributionh⊥q,
1
which leads to an azimuthal asymmetry similar to the one observed in the Drell-Yan process. Based on the fact that
the latter asymmetry is large in π−N scattering and therefore most likely also in pp¯collisions, we have obtained an
estimate for the analogous asymmetry in photon-jet production. The latter asymmetry is expected to be a factor of
2-10smallerfortypicalTevatronkinematics,whichmaystillbesufficientlylargetobemeasurable. Thiswouldoffera
newpossibilityofmeasuringT-oddeffects usingthishighenergycollider. Asimilarmeasurementcouldbe performed
at pp colliders as well, however, there one expects a significantly smaller contribution due to the absence of valence
antiquarks.
The asymmetry measurement itself requires the reconstruction of both the length and the direction of the photon
andjetmomentatransversetothe beam. The angularasymmetryis thenacos2φasymmetry,withthe angleφgiven
by the difference between the angle of either one of the two transverse momenta, K or K , and the angle of
j⊥ γ⊥
their sum, q = K +K . Realizing the fact that the uncertainty in the latter angle φ may be rather large
⊥ j⊥ γ⊥ ⊥
when relatively small q values are considered, it may, as an initial step, be convenient to integrate the angular
| ⊥|
distribution over four quadrants which can then be added and subtracted in the appropriate way to gain statistics.
The asymmetry is found to be largest in the central rapidity region, where η η 0.
j γ
≈ ≈
Acknowledgments
We would like to thank Cedran Bomhof for useful comments and discussions. This researchis part of the research
programof the “Stichting voor Fundamenteel Onderzoek der Materie (FOM)”, which is financially supported by the
“Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)”.
[1] S.Falciano et al. [NA10 Collaboration], Z. Phys. C 31 (1986) 513.
9
[2] M. Guanziroli et al. [NA10 Collaboration], Z. Phys. C 37 (1988) 545.
[3] J. S. Conway et al.,Phys. Rev.D 39 (1989) 92.
[4] A.Brandenburg, O.Nachtmann and E. Mirkes, Z. Phys. C 60 (1993) 697 .
[5] C. S. Lam and W. K.Tung, Phys. Rev.D 18 (1978) 2447.
[6] C. S. Lam and W. K.Tung, Phys. Rev.D 21 (1980) 2712.
[7] E. N.Argyres and C. S.Lam, Phys.Rev.D 26 (1982) 114.
[8] A.Brandenburg, S.J. Brodsky,V. V.Khoze and D. Muller, Phys.Rev.Lett. 73 (1994) 939.
[9] K.J. Eskola, P.Hoyer, M. Vanttinenand R.Vogt, Phys. Lett.B 333 (1994) 526.
[10] D.Boer, S. J. Brodsky and D. S.Hwang, Phys.Rev.D 67 (2003) 054003.
[11] Z. Lu and B. Q. Ma, Phys.Rev.D 70 (2004) 094044.
[12] D.Boer, A. Brandenburg,O. Nachtmann and A. Utermann,Eur. Phys.J. C 40 (2005) 55.
[13] Z. Lu and B. Q. Ma, Phys.Lett. B 615 (2005) 200.
[14] L. P.Gamberg and G. R. Goldstein, Phys.Lett. B 650 (2007) 362.
[15] A.Brandenburg, A.Ringwald and A.Utermann, Nucl.Phys. B 754 (2006) 107.
[16] D.Boer, Phys. Rev.D 60 (1999) 014012.
[17] L. Y.Zhu et al. [FNAL-E866/NuSea Collaboration], Phys. Rev.Lett. 99 (2007) 082301.
[18] V.Barone, Z. Lu and B. Q. Ma, Eur. Phys. J. C 49 (2007) 967.
[19] E. Mirkes, Nucl. Phys. B 387 (1992) 3.
[20] E. Mirkes and J. Ohnemus,Phys.Rev.D 51 (1995) 4891.
[21] D.E. Acosta et al. [CDF Collaboration], Phys. Rev.D 73 (2006) 052002 .
[22] C. Bourrely and J. Soffer, Nucl. Phys.B 423 (1994) 329.
[23] D.Boer, Phys. Rev.D 62 (2000) 094029.
[24] A.Kumar [DØ Collaboration], arXiv:0710.0415 [hep-ex].
[25] T. Pietrycki and A. Szczurek,Phys.Rev. D 76 (2007) 034003.
[26] D.Boer, Nucl. Phys.B 603 (2001) 195 .
[27] A.Bacchetta, C. Bomhof, U.D’Alesio, P. J. Mulders and F. Murgia, Phys. Rev.Lett. 99 (2007) 212002.
[28] D.Boer and P. J. Mulders, Phys.Rev. D 57 (1998) 5780.
[29] P.J. Mulders and J. Rodrigues, Phys. Rev.D 63 (2001) 094021.
[30] C. J. Bomhof and P.J. Mulders, JHEP 0702 (2007) 029.
[31] C. J. Bomhof and P.J. Mulders, arXiv:0709.1390 [hep-ph],to bepubl.Nucl. Phys.B.
[32] J. C. Collins, Phys. Lett. B 536 (2002) 43.
[33] A.V. Belitsky, X.Ji and F. Yuan,Nucl.Phys. B 656 (2003) 165.
[34] D.Boer, P. J. Mulders and F. Pijlman, Nucl. Phys. B 667 (2003) 201.
[35] J. Collins and J. W. Qiu, Phys.Rev.D 75 (2007) 114014.
[36] W. Vogelsang and F. Yuan,Phys.Rev.D 76 (2007) 094013.
[37] J. Collins, arXiv:0708.4410 [hep-ph].
[38] D.Boer and W. Vogelsang, Phys. Rev.D 74 (2006) 014004.
[39] E. L. Berger, J. W. Qiu and R.A. Rodriguez-Pedraza, Phys.Rev.D 76 (2007) 074006.
[40] O.Atramentov,talkgivenattheXVInternationalWorkshoponDeep-InelasticScatteringandRelatedSubjects(DIS2007),
Munich,Germany, 16-20 April2007.
[41] P.J. Mulders and R. D.Tangerman, Nucl. Phys.B 461 (1996) 197 [Erratum-ibid. B 484 (1997) 538].
[42] U.D’Alesio and F. Murgia, Phys. Rev.D 70 (2004) 074009.
[43] M. Glu¨ck, E. Reyaand A. Vogt, Eur.Phys. J. C 5 (1998) 461.