A general derivation of differential cross section in quark-quark scatterings at fixed impact parameter

0.17 MB·English

by  Shou-wan Chen

A general derivation of differential cross section in quark-quark scatterings at fixed
impact parameter
Shou-wan Chen,1 Jian Deng,1 Jian-hua Gao,2 and Qun Wang1
1Department of Modern Physics and Interdisciplinary Center for Theoretical Study,
University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China
2Department of Physics, Shandong University, Jinan, Shandong 250100, People’s Republic of China
We propose a general derivation of differential cross section in quark-quark scatterings at fixed
impact parameters. The derivation is well defined and free of ambiguity in the conventional one.
Theapproachcanbeappliedtoavarietyofpartonicandhadronicscatteringsinloworhighenergy
particle collisions.
8 PACSnumbers: 25.75.Nq,13.88.+e,12.38.Mh
0
0
2 I. INTRODUCTION

n
a The global polarization effect in quark scatterings in non-central heavy-ion collisions has been predicted due to
J
orbital angular momenta resided in the system as a result of longitudinal flow shear [1, 2, 3, 4]. The effect is defined

5 with respect to the direction orthogonal to the reaction plane determined by the vector of impact-parameter and
1
the beam momentum. The polarization of quarks can be partially carried by hadrons containing these quarks via

hadronization. For example, as a consequence of this global polarization, vector mesons can have spin alignments in
]
h non-centralheavy-ion collisions [5]. Polarizedphotons are recently proposed as a good probe to the polarized quarks
p [6].
- In Ref. [4], the polarization in the quark-quark scatterings has been estimated in a hot medium. The differential
p
e cross-section with respect to the partonic impact parameter is derived by inserting into the differential cross-section
h in mementum space a delta function for transverse momenta, which can in turn be written as an integral over the
[ partonic impact parameter. This derivation has an ambiguity. In this note we will give an alternative and a general

way of deriving the differential cross-section with respect to the partonic impact parameter. We will show that the
1
v differential cross section with respect to impact parameter in our framework is well defined and free of ambiguity.
6
9
2 II. KINEMATICS SETUP
2
.
1 The geometry of a nucleus-nucleus collision at impact parameter b is illustrated in Fig. 1 of Ref. [1]. We assume
0 that two nuclei move along z directions and collide at z =0. The reaction plane is in the y direction. We consider
8 ±
the polarization along the y direction.
0
Let us consider the scattering of quarks with different flavors (P ,λ )+(P ,λ ) (P ,λ )+(P ,λ ) through
: 1 1 2 2 → 3 3 4 4
v the Hard-Thermal-Loop (HTL) resummed gluon propagators, where λ with (i = 1,2,3,4) are spin states and P =
i i
Xi (Ei,piz,piT) are4-momentafor colliding quarks,withlongitudinalandtransversemomentapiz andpiT andenergies
E = p2 +p2 . Note that we treat all quarks massless. We assume the initial momenta are along the z direction,
r i iz iT
a i.e. p = p > 0 and p = p = 0. We will use the shorthand notation p p . The total energy in the
1z 2z 1T 2T T 3T
p − ≡
center-of-mass frame is denoted by √s= (P +P )2.
1 2
p
III. AMBIGUITY OF DIFFERENTIAL CROSS SECTION WITH RESPECT TO IMPACT
PARAMETER IN CONVENTIONAL TREATMENT
In a conventional derivation of the differential cross section at impact parameter x , one starts with the one in
T
momentum space,
1
dσ = M(P λ ,P λ ,P λ ,P λ )2
1 1 2 2 3 3 4 4
16P P | |
1 2
| · |λ1X,λ2,λ4
d3p d3p
(2π)4δ(4)(P +P P P ) 3 4 , (1)
× 1 2− 3− 4 (2π)32E (2π)32E
3 4
where a sum over the spin states λ ,λ ,λ except λ and an average (with a factor 1/4) over the initial spin states
1 2 4 3
have been taken. One can integrateout p by consuming the delta function δ(3)(p +p p p ) for 3-momentum
4 1 2 3 4
− −
2
conservation, and then carry out the integral over p to remove δ(E +E E E ) for the energy conservation
3z 1 2 3 4
− −
which gives a factor
E E
3 4
dp δ(E +E E E ) = , (2)
3z 1 2 3 4
− − E p(i) E p(i)
Z iX=1,2 3 4z − 4 3z
(cid:12) (cid:12)
(cid:12) (cid:12)
where p(i) =p +p p(i) and p(i) are roots of the energy conserva(cid:12)tion equation an(cid:12)d given by
4z 1z 2z− 3z 3z
E +E 2p2 1/2 p +p
p(1/2) = 1 2 1 T + 1z 2z. (3)
3z ± 2 − p p p p 2
(cid:20) | 1z 2z|− 1z 2z(cid:21)
Then Eq. (1) is simplified as
1 1 d2p
dσ = M(P λ ,P λ )2 T , (4)
64P1 P2 | 3 3 4 4 | E p(i) E p(i) (2π)2
| · |iX=1,2λ1X,λ2,λ4 3 4z − 4 3z
(cid:12) (cid:12)
where we have suppressed the (P1λ1,P2λ2) dependence of the amplitu(cid:12)(cid:12)de. Note that p(cid:12)(cid:12)4T = p3T = pT is implied
− −
in the above expressiondue to momentum conservationand the assumption that the initial state momenta are along
the z-axis, p =p =0. One can rewrite it by inserting a delta function for transverse momenta,
1T 2T
1 d2p d2p′
dσ = T T (2π)2δ(2)(p p′ )
64P P (2π)2(2π)2 T − T
| 1· 2|Z
1
M(P λ ,P λ )M∗(P′λ ,P′λ )
× E p(i) E p(i) 3 3 4 4 3 3 4 4
iX=1,2λ1X,λ2,λ4 3 4z − 4 3z
= 64P1 P d(cid:12)(cid:12)(cid:12)2xT(d22πp)T2(d22πp)′T2(cid:12)(cid:12)(cid:12)ei(pT−p′T)·xT
| 1· 2|Z
1
M(P λ ,P λ )M∗(P′λ ,P′λ ), (5)
× E p(i) E p(i) 3 3 4 4 3 3 4 4
iX=1,2λ1X,λ2,λ4 3 4z − 4 3z
(cid:12) (cid:12)
where P3′ =(E3′,p3z,p′T) and P4′ =(E4′,p4z,−(cid:12)(cid:12)p′T). Then we o(cid:12)(cid:12)btain the differential crosssection at impact parameter
x ,
T
d2σ = 1 d2pT d2p′T ei(pT−p′T)·xT
d2x 64P P (2π)2(2π)2
T 1 2
| · |Z
1
M(P λ ,P λ )M∗(P′λ ,P′λ ). (6)
× E p(i) E p(i) 3 3 4 4 3 3 4 4
iX=1,2λ1X,λ2,λ4 3 4z − 4 3z
(cid:12) (cid:12)
Notethattheaboveexpressionofd2σ/d2xT isn(cid:12)(cid:12)otunique,since(cid:12)(cid:12)onecanmakethereplacementinEq. (5),forexample,
1 1 1
(7)
E p(i) E p(i) → E p(i) E p(i) a1 E′p′(i) E′p′(i) a2
3 4z − 4 3z 3 4z − 4 3z 3 4z − 4 3z
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
with a1+a2 =1, while keep(cid:12)(cid:12)ing the totalcro(cid:12)(cid:12)ss sect(cid:12)(cid:12)ion unchanged.(cid:12)(cid:12)Of(cid:12)(cid:12)course one canm(cid:12)(cid:12)ake many other choices which
conserve the total cross section. In Ref. [4], a = a = 1/2 is used, while Eq. (6) implies a = 1,a = 0. So we see
1 2 1 2
that there is an ambiguity in d2σ/d2x , or in other word, d2σ/d2x is not unique by this definition. The problem
T T
arises when one calculates quantities like the polarization when the integral over x is not made in the whole space
T
(therefore the delta function δ(2)(p p′ ) is not recovered), which would lead to different or inconsistent results.
T − T
IV. DIFFERENTIAL CROSS SECTION AT FIXED IMPACT PARAMETER
In order to solve the ambiguity in the previous section, we will derive in this section the cross sections of parton-
partonscatterings at fixed impact parameterin a generalapproach. To this end, we need to introduce particle states
labeled by transverse positions and longitudinal momenta, which we call states in the mixed representation. They
3
are connected with states in momentum space by Fourier transform in transverse sector. We express in the mixed
representation the final states in the scatterings as
A d2p
p ,λ ,x = T 3Teip3T·x3T p ,λ ,
| 3z 3 3Ti (2π)2 | 3 3i
Z
A d2p
p ,λ ,x = T 4Teip4T·x4T p ,λ , (8)
| 4z 4 4Ti (2π)2 | 4 4i
Z
where A is the area in the transverse plane. The S-matrix element from the initial to final states then reads
T
S = p ,λ ,x ;p ,λ ,x S p ,λ ,p ,λ
fi 3z 3 3T 4z 4 4T 1 1 2 2
h | | i
A d2p A d2p
= T 3T T 4Te−ip3T·x3Te−ip4T·x4T(2π)4δ(4)(P +P P P )
(2π)2 (2π)2 1 2− 3− 4
Z
1 1 1 1
M(P λ ,P λ ), (9)
3 3 4 4
×√2E V √2E V √2E V √2E V
1 2 3 4
where the final state momenta areP =(E ,p ,p ) andP =(E ,p ,p ). The squaredmatrix element becomes
3 3 3z 3T 4 4 4z 4T
S 2 = ATd2p3T ATd2p4T ATd2p′3T ATd2p′4Te−i(p3T−p′3T)·x3Te−i(p4T−p′4T)·x4T
| fi| (2π)2 (2π)2 (2π)2 (2π)2
Z
(2π)8δ(4)(P +P P P )δ(4)(P +P P′ P′)
× 1 2− 3− 4 1 2− 3− 4
1 1
M(P λ ,P λ )M∗(P′λ ,P′λ ), (10)
×V416E E E E E′E′ 3 3 4 4 3 3 4 4
1 2 3 4 3 4
where P3′ = (E3′,p3z,p′3T) and P4′ = (E4′p,p4z,p′4T). Note that the longitudinal momenta of P3 and P3′ are the same,
so are P and P′. The transverse parts of the delta-functions ensure p = p =p and p′ = p′ =p′ . We
can integ4rate ou4t p′ and p to get rid of the four delta functions in th3eTtran−sve4Trse secTtor and3aTrrive−at4T T
T T
S 2 = A4 d2p d2p′ e−i(pT−p′T)·(x3T−x4T)δ(0,z)(P +P P P )δ(0,z)(P +P P′ P′)
| fi| T T T 1 2− 3− 4 1 2− 3− 4
Z
1 1
M(P λ ,P λ )M∗(P′λ ,P′λ ), (11)
×V416E E E E E′E′ 3 3 4 4 3 3 4 4
1 2 3 4 3 4
whereδ(0,z) denote thedelta funcptions forthe energyandthez componentofmomenta. The differentialcrosssection
is then
d2x V Ldp Ldp
dσ = T 3z 4z S 2
fi
A 4v τ 2π 2π | |
T rel λ1X,λ2,λ4Z
= d2x 1 1 d2p dp d2p′ dp e−i(p3T−p′3T)·xT
Tτ 64(2π)3 T 3z T 4z
Z
1
δ(0,z)(P +P P P )δ(E +E E′ E′)
×v E E E E E′E′ 1 2− 3− 4 1 2− 3− 4
rel 1 2 3 4 3 4
× Mp(P3λ3,P4λ4)M∗(P3′λ3,P4′λ4), (12)
λ1X,λ2,λ4
wherewehavedefinedx x x . Inthe firstlinewehaveusedthatthedifferentialcrosssectionisproportional
T 3T 4T
to the fraction d2x /A w≡ith A−= d2x . Here L is the length along the z direction and v = P P /(E E )
T T T T rel 1 2 1 2
| · |
the relative velocity of incident partons. We also used V =A L and 2πδ(z)(0)= L (because two delta functions for
R T
z-momenta are identical). Note that τ is the time period of the scattering. It is obvious to see d2σ/d2x > 0 from
T
Eq. (12).
We can evaluate Eq. (12) as follows. First we intergrateout p to removeδ(p +p p p ). The remaining
4z 1z 2z 3z 4z
− −
two delta functions enforce energy conservation,which canbe removedby carryingout integralsoverthe magnitudes
of transverse momenta p′ and p . We end up with
T T
d2σ = 1 1 2πdϕ 2πdϕ′ pm3zaxdp e−ipT(cosϕ−cosϕ′)xT E3E4
d2xT τ 64(2π)3|P1·P2|Z0 Z0 Zpm3zin 3z (E3+E4)2
M(P λ ,P λ )M∗(P′λ ,P′λ ), (13)
× 3 3 4 4 3 3 4 4
λ1X,λ2,λ4
4
where ϕ and ϕ′ are azimuthal angles of p and p′ relative to the direction of x respectively. The magnitude of
T T T
transverse momentum satisfying the energy conservation for P and P′ is proved to be the same and given by
3,4 3,4
(E +E )2+p2 (p +p p )2 2 1/2
p = 1 2 3z − 1z 2z− 3z p2 . (14)
T ((cid:20) 2(E1+E2) (cid:21) − 3z)
The energies are
E =E′ = p2 +p2,
3 3 3z T
q
E =E′ = (p +p p )2+p2. (15)
4 4 1z 2z − 3z T
q
The integral p is in the range [pmin,pmax] where
3z 3z 3z
(E +E )2 (p +p )2 p +p −1
pmax = 1 2 − 1z 2z 1 1z 2z ,
3z 2(E +E ) − E +E
1 2 (cid:20) 1 2 (cid:21)
(E +E )2 (p +p )2 p +p −1
pmin = 1 2 − 1z 2z 1+ 1z 2z . (16)
3z − 2(E +E ) E +E
1 2 (cid:20) 1 2 (cid:21)
The amplitude square is given by
M(P3λ3,P4λ4)M∗(P3′λ3,P4′λ4) = g4cqqE1E2 E3E4E3′E4′J1µµ′J2νν′∆µν(P1−P3)∆∗µ′ν′(P1−P3′) (17)
Here g is the quark-gluon coupling constant and α p= g2/(4π). The color factor for qq scatterings is c =
s qq
(1/N2)(δabδab/4) = 2/9, where 1/N2 is from the average over the initial state colors. Jµµ′ and Jνν′ are tensors
c c 1 2
only dependent on momentum directions,
Jµµ′ 1 Tr[u(P′,λ )u(P ,λ )γµP γσγµ′]
1 ≡ E E E′ 3 3 3 3 1σ
1 3 3
Jνν′ p1 Tr[u(P′,λ )u(P ,λ )γνP γσγν′] (18)
2 ≡ E E E′ 4 4 4 4 2σ
2 4 4 Xλ4
p
where u(P ,λ ) denotes the spinor for the parton i with the spin state λ along the reference direction n e ,
i i i y
and u = u†(P ,λ )γ is its conjugate. ∆ is the HTL resummed gluon propagator [7, 8, 9]. Here we≡only
i,λi i i 0 µν
take the magnetic gluon exchange into account which invloves the magnetic mass µ is introduced to regulate the
m
divergence arising from the soft gluon exchange. The magnetic mass µ is proportional to the temperature T,
m
µ = 0.255(N /2)1/2g2T. The numerical result from Eq. (13) is given in Fig. 1 for collisions in the center-of-mass
m c
system of the colliding quarks. The polarized part is much less than unpolarized one. Both the unpolarized and
polarized differential cross sections show an oscillation feature.
ThersisanalternativewaytodotheintegralsinEq. (12). Onecanintergrateoutp toremoveδ(p +p p p )
4z 1z 2z 3z 4z
− −
and then p to remove δ(E +E E E ) as in Eq. (2). We finally obtain the differential cross section with
3z 1 2 3 4
− −
respect to the impact parameter,
d2σ = 1 1 d2p d2p′ e−i(pT−p′T)·xTδ(E +E E′ E′)
d2x τ 64(2π)3 T T 3 4− 3− 4
T Z
1 E E
3 4
×iX=1,2λ1X,λ2,λ4 |P1·P2| E3E4E3′E4′ E3p4(iz)−E4p3(iz)
M(P λ ,P λ )M∗(Pp′λ ,P′λ ), (cid:12) (cid:12) (19)
× 3 3 4 4 3 3 4 4 (cid:12)(cid:12) (cid:12)(cid:12)
Note that all final state energies are functions of p and p′ . The root p(i) is given by Eq. (3) and p(i) is given
T T 3z 4z
by p(i) = p +p p(i). We now rewrite the remaining delta-function into an integral over a time t which is the
4z 1z 2z − 3z
conjugate variable of the uncertainty of the final state energies arising from the specified transverse positions in the
final state,
τ/2
2πδ(E +E E′ E′) dtei(E3+E4−E3′−E4′)t, (20)
3 4− 3− 4 ≈
Z−τ/2
5
Figure1: (color online) Thepolarized andunpolarized differentialcrosssections. Theexact resultsfrom Eq. (13)isinthered
solid line (unpolarized) and blue dashed line (polarized part multiplied by a factor of 10). The sum of the unpolarized part
d2(σu(0p)ol+σu(1p)ol)/d2xT from Eqs. (29,32,34) is in the red dotted line, while that of the polarized part d2(σp(0o)l+σp(1o)l)/d2xT
(multiplied by a factor of 10) is in the blue dash-dotted line. The parameters are chosen to be √s=20 GeV, T =200 MeV,
τ =1.1 fm and αs =1. In theperturbation approach to obtain d2(σ(0)+σ(1))/d2xT we use pctut=3 GeV.
0.20
0.15
0.10
Σ2d€€€€€€€€€€€€ 2dxT 0.05
0.00
-0.05
0.2 0.4 0.6 0.8 1.0
x HfmL
T
where we have assumed T . Inserting the above into Eq. (19), we obtain
→∞
d2σ = 1 d2p d2p′ e−i(pT−p′T)·xT
d2x 64(2π)4 P P T T
T | 1· 2|Z D
1 E E
3 4
×iX=1,2λ1X,λ2,λ4 E3E4E3′E4′ E3p4(iz)−E4p3(iz)
M(P λ ,P λp)M∗(P′λ ,P(cid:12)′λ ) , (cid:12) (21)
× 3 3 4 4 3 3 (cid:12)(cid:12)4 4 it (cid:12)(cid:12)
where the time average is defined by
h···it
1 τ/2 dt( )ei(E3+E4−E3′−E4′)t. (22)
h···it ≡ τ ···
Z−τ/2
At small transverse momenta in the center-of-mass system where p p′ √s, we have (E +E E′ E′)
T ∼ T ≪ 3 4− 3− 4 ∼
p2/√s p′2/√s. If p2 t/√s6p2τ/√s 1 or τ √s/p2, we can expand the phase factor in Eq. (21) as
T ∼ T T| | T ≪ ≪ T
ei(E3+E4−E3′−E4′)t 1+i(E +E E′ E′)t 1(E +E E′ E′)2t2+ . (23)
≈ 3 4− 3− 4 − 2 3 4− 3− 4 ···
To the leading order, we have E3 ≈E3′, E4 ≈E4′ and ei(E3+E4−E3′−E4′)t ≈1, Eq. (21) becomes
d2σ(0) = 1 d2p d2p′ e−i(pT−p′T)·xT 1
d2x 64(2π)4 P P T T (i) (i)
T | 1· 2|Z iX=1,2λ1X,λ2,λ4 E3p4z −E4p3z
M(P λ ,P λ )M∗(P′λ ,P′λ ). (cid:12) (cid:12) (24)
× 3 3 4 4 3 3 4 4 (cid:12)(cid:12) (cid:12)(cid:12)
It is interesting to see the above is unique to the leading order since
1 1 1
. (25)
E p(i) E p(i) ≈ E p(i) E p(i) a1 E′p′(i) E′p′(i) a2
3 4z − 4 3z 3 4z − 4 3z 3 4z − 4 3z
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
The next-to-leading order no(cid:12)n-vanishing con(cid:12)tribu(cid:12)tion comes from(cid:12) th(cid:12)e term t2 in E(cid:12)q. (23) since the linear term is
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) ∼ (cid:12)
odd in t whose integral gives zero in the range [ τ/2,τ/2]. It reads
−
d2σ(1) = τ2 d2p d2p′ e−i(pT−p′T)·xT
d2x −1536(2π)4P P T T
T 1 2
| · |Z
6
(E +E E′ E′)2 E E
3 4− 3− 4 3 4
×iX=1,2λ1X,λ2,λ4 E3E4E3′E4′ E3p4(iz)−E4p3(iz)
×M(P3λ3,P4λ4)Mp∗(P3′λ3,P4′λ4). (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) (26)
For a simple case in the center-of-mass frame of two colliding quarks, where p = p , p = p and E =
1z 2z 3z 4z 1
E =E =E =√s/2. To the leading order, we have and E′ =E′ √s/2, then Eq. (−24) is evaluat−ed as
2 3 4 3 4 ≈
d2σ(0) α2s d2p d2p′ e−i(pT−p′T)·xT
d2x ≈ 36π2 T T
T Z
1 1 p p′
1+iλ n n T − T
×p2 +µ2 p′2 +µ2 3 1· × √s
T m T m (cid:26) (cid:20) (cid:21)(cid:27)
α2 2 dA(x )
= s A2(x )+ λ n (n x )A(x ) T
T 3 1 T T
9 √s · × dx
(cid:20) T (cid:21)
d2σu(0p)ol d2σp(0o)l b
= +λ , (27)
d2x 3 d2x
T T
where we have taken only the magnetic gluon exchange into account and the magnetic mass µ is introduced to
m
regulate the divergence arising from the soft gluon exchange. We denoted n =e as the direction of p and used
1 z 1
1 pcTut p J (p x )
A(x ) = d2p e±ipT·xT =2π dp T 0 T T , (28)
T T p2 +µ2 T p2 +µ2
Z T m Z0 T m
where J is the Bessel function and pcut is the cutoff to regulate the ultraviolet divergence. The polarized and
0 T
unpolarized differential cross sections can be obtained
d2σ(0) α2
upol = sA2(x ),
d2x 9 T
T
d2σ(0) 2α2 dA(x )
pol = sn (n x )A(x ) T . (29)
d2x 9√s · 1× T T dx
T T
Note that the above result is just Eqs. (40,41) in Ref. [4]. b
The next-to-leading order contribution is evaluated as
d2σ(1) τ2α2s d2p d2p′ e−i(pT−p′T)·xT(p2 p′2)2
d2x ≈ −216π2s T T T − T
T Z
1 1 p p′
1+iλ n n T − T
×p2 +µ2 p′2 +µ2 3 1· × √s
T m T m (cid:26) (cid:20) (cid:21)(cid:27)
d2σ(1) d2σ(1)
upol pol
= +λ , (30)
d2x 3 d2x
T T
where we have used
2 (p2 p′2)2
(E +E E′ E′)2 = √s 2 s/4 p2 +p′2) T − T . (31)
3 4− 3− 4 − − T T ≈ s/4
(cid:20) q (cid:21)
The unpolarized part reads
d2σu(1p)ol = τ2α2s d2p d2p′ e−i(pT−p′T)·xT(p4 +p′4 2p2p′2)
d2x −216π2s T T T T − T T
T Z
1 1 τ2α2
= s(A A A2), (32)
×p2 +µ2 p′2 +µ2 − 27s 0 1− 2
T m T m
where we used
pcTut pni
A (x ) dp T J (p x ). (33)
i T ≡ Tp2 +µ2 0 T T
Z0 T m
7
Figure 2: The polarized and unpolarized differential cross sections in the leading order from Eq. (29) (left panel) and the
next-to-leadingorder from Eqs. (32,34) (right panel). The parameters are set to thesame valuesas in Fig. 1.
0.01
0.20
Unpolarized 0.00
0.15
Polarized
ΣHL20d€€€€€€€€€€€€€€€€ 2dxT0.10 ΣHL21d€€€€€€€€€€€€€€€€ 2dxT--00..0021 Unpolarized
0.05
Polarized
-0.03
0.00
0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0
x HfmL x HfmL
T T
For i=0,1,2 with n =1,5,3. The polarized part turns out to be
i
d2σ(1) τ2α2 d
pol = s n (n x ) (A A A2). (34)
d2x −27s3/2 · 1× T dx 0 1− 2
T T
ThenumericalresultsofEqs. (29,32,34)areshowninFig. 2.bWeseethatboththeleadingandnext-to-leadingparts
are damped out above 0.4 fm. The next-to-leading contributions of the unpolarized and polarized parts are about
1/6 of the leading counterparts. Both the unpolarized and polarized differential cross sections show an oscillation
feature. Thesumsoftheleadingandnext-to-leadingcontributionsareshownasthereddotted(unpolarized)andblue
dash-dottedline (polarized) in Fig. 1. We already mentioned that a cutoff in transversemomentum pcut is needed to
t
regulate the integrals in the leading and next-to-leading differential cross sections. The cross section results depend
on pcut in the perturbation. The time scale τ is also a quantity to be determined. We can find the range of the τ by
t
equating the exact result d2σ/d2x from Eq. (13) and the sum d2(σ(0) +σ(1))/d2x in the perturbation approach
T T
from Eqs. (29,32,34) at a specified value of pcut.
t
V. SUMMARY AND DISCUSSION
We have proposed a general approach to the differential cross section with respect to impact parameters, which is
well-defined and free of ambiguity existing in the conventional approach. The main difference of our approach from
the conventional one is that (1) in the conventional approach the transfer of small transverse momenta is implied,
while the general approach is valid for all transverse momenta; (2) there are two independent delta functions in the
general approach for energy conservation in the cross section formula, which arises from fixing impact parameters in
the final state partons making the total final state energy be uncertain. While in the conventional approach the two
delta functions for energy conservation are identical turning the second one to be the infinite interaction time τ.
As a simple illustration of our formalism, we evaluated the polarized and unpolarized differential cross sections at
small angle quark-quark scatterings in the center-of- mass system of the colliding quarks. To smoothly connect the
general approach and the conventional one, we propose an expansion in terms of ∆E =E E 1/τ with E and
f i i
− ∼
E theinitialandfinalstateenergiesincollisions. The leadingordercontributionreproducedtheconventionalresult,
f
i.e. that of Ref. [4].
The general formulation in this paper can also be applied to many other parton-parton scatterings in heavy ion
collisions or even proton-proton collisions [10].
Acknowledgement. The authors thank Zuo-tang Liang and Xin-nian Wang for insightful discussions. Q.W.
acknowledges supports in part by the startup grant from University of Science and Technology of China (USTC) in
association with 100-talent project of Chinese Academy of Sciences (CAS) and by NSFC grant No. 10675109.
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