Study of Two-Photon Corrections in the $p\bar{p} --> e+e-$ Process: Hard Rescattering Mechanism

0.33 MB·English

by  Julia Guttmann

Study of Two-Photon Corrections in the pp¯ e+e Process: Hard
−
→
Rescattering Mechanism
Julia Guttmann,1 Nikolai Kivel,1,2,3 and Marc Vanderhaeghen1
1Institut fu¨r Kernphysik, Johannes-Gutenberg Universita¨t, D-55099 Mainz, Germany
1
1 2Helmholtz Institut Mainz, Johannes-Gutenberg Universita¨t, D-55099 Mainz, Germany
0
2 3Petersburg Nuclear Physics Institute, Gatchina, 188350, Russia

n
a (Dated: February 1, 2011)
J

1 Abstract
3

We investigate the two-photon corrections to the process pp¯ e+e− at large momentum trans-
]
h →
p fer, aimed to access the time-like nucleon form factors. We estimate the two-photon corrections
-
p
using a hard rescattering mechanism, which has already been used to calculate the corresponding
e
h
[ corrections to elastic electron-proton scattering. Using different nucleon distribution amplitudes,

1 we findthat the two-photon corrections to the pp¯ e+e− cross sections in themomentum transfer
v →
7
range 5 - 30 GeV2 is below the 1 % level.
6
9
5
. PACS numbers:
1
0
1
1
:
v
i
X
r
a
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I. INTRODUCTION
Electromagnetic form factors (FFs) provide important information on the structure of
the nucleon. Consequently, there has been much effort in their measurement both in the
space-like as well as in the time-like regions.
The space-like electromagnetic FFs, which provide information on spatial distributions
of quarks in the nucleon, can be investigated in elastic electron-proton scattering; for recent
reviews see e.g. Refs. [1–3]. Two experimental methods exist to extract the ratio of electric
(G ) to magnetic (G ) proton FFs. Historically, the first method involves unpolarized
E M
measurements employing the Rosenbluth separation technique, which gives a direct acess to
the space-like FFs through the slope and intercept of the ε-dependence of the cross section
in the one-photon (1γ) exchange approximation :
ε
dσ = (Q2,ε) G2 (Q2)+ G2(Q2) , (1)
C M τ E
h i
where ε and Q2 are the virtual photon polarization parameter and virtuality respectively,
τ = Q2/4m2 , with m the nucleon mass, and where is a known phase space factor. In
N N C
more recent years, polarization experiments using polarized electron beams on polarized
targets or measuring the recoil nucleon polarization, in the elastic e p scattering, provided
−
anotherexperimental waytoaccessG /G . Theratioofpolarizationoftherecoiling proton
E M
perpendicular to its motion (P ) to polarization along its motion (P ) is directly related to
t l
the ratio of electric to magnetic proton FFs
P 2ε G (Q2)
t E
= . (2)
P − τ(1+ε) G (Q2)
l s M
Such polarization experiments have been performed for momentum transfers up to 8.5 GeV2
to date and have shown that the ratio of the electric to magnetic proton FFs is decreasing
with increasing momentum transfer [4–7]. This finding is in contrast to the well known
scaling-behavior of G /G determined by the Rosenbluth separation technique. The dis-
E M
crepancy between data of unpolarized Rosenbluth measurements and polarization experi-
ments has triggered a whole new field studying the influence of two-photon (2γ) exchange
corrections [8–13]; see Ref. [14] for a recent review and references therein. The finding of
those works is that two-photon exchange corrections to the Rosenbluth cross section are a
possible explanation for the discrepancy, whereas the 2γ exchange effects do not impact the
2
polarization transfer extraction of G /G in a significant way. Recently, a first empirical
E M
extraction ofthe three 2γ-exchange amplitudes to elastic electron-protonscattering has been
performed[15]based onmeasurements ofcross sections [16] andpolarizationobservables [17]
at a common value of four-momentum transfer, around Q2 = 2.5 GeV2. It confirms that
a common description of unpolarized measurements and of polarization observables invokes
empirical 2γ-amplitudes with relative magnitude up to about 3 %.
The measurements of nucleon FFs at space-like momentum transfers, through elastic
electron-nucleon scattering, are complemented by measurements in the time-like region,
through the crossed processes pp¯ e+e− and e+e− NN¯, which access the vector mesonic
→ →
excitation spectrum of hadrons. The latter process has been measured in recent years at
e+e− facilities, such asDaΦne, CLEO, andBABAR.These measurements have revealed that
the nucleon FFs at time-like momentum transfers are significantly larger than their space-
like counterparts when considering the same magnitude for the virtuality q2 = Q2. In
−
particular, for momentum transfers with magnitude around 10 GeV2, the time-like FFs were
foundtobeenhancedbyafactoroftwo. Newmeasurements areplannedinthenearfutureat
BES-III and at PANDA@FAIR, bringing time-like (positive) q2 values around 20 GeV2 into
reach. Such new measurements will explore the at present still largely uncharted time-like
region in much greater detail and complement our picture of the nucleon.
The time-like FFsare complex quantities due to the interactions of the hadrons in the ini-
tialandfinalstate, respectively. Theirabsolutevaluescanbedeterminedfrommeasurements
of the angular distribution of the unpolarized c.m. cross section in the 1γ-approximation :
1
dσ = (q2) G 2(1+cos2θ)+ G 2sin2θ , (3)
c.m.,1γ M E
C | | τ| |
(cid:20) (cid:21)
whereas the phases are related to polarization observables. Since 2γ-exchange plays a cru-
cial role in the extraction of electromagnetic FFs in the space-like region, investigating its
influence in the time-like region seems to be an obvious task. Even though some theoretical
works have been done [18, 19], there are no comparable calculations so far to estimate the
2γ-exchange corrections for the corresponding time-like processes.
In this work we investigate the 2γ-exchange corrections to the process pp¯ e+e− at
→
large momentum transfer q2. To provide a first estimate of the corrections we consider a
perturbative QCD (pQCD) factorization approach, which has already been used to calculate
the corresponding corrections to elastic electron-proton scattering [13].
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The paper is organized as follows: The general formalism including 2γ-exchange is pre-
sented in Section II. In Section III we estimate the hard 2γ-exchange contribution at large
momentum transfers by relating the 2γ-exchange amplitude to the leading twist nucleon
distributions. The results of the calculation are discussed in Section IV. Some concluding
remarks are given in Section V.
II. GENERAL EXPRESSION OF THE OBSERVABLES INLCUDING 2γ-
EXCHANGE
In order to describe the annihilation of a proton and an antiproton into a lepton pair,
p(p ,λ )+p¯(p ,λ ) l−(k ,h )+l+(k ,h ), (4)
1 N1 2 N2 → 1 1 2 2
where λ , λ , h and h are helicities of the nucleons and leptons respectively, we adopt
N1 N2 1 2
the definitions
p p k k
P = 1 − 2, K = 1 − 2, q2 = (p +p )2, (5)
1 2
2 2
and the Mandelstam variables
s = q2 = (p +p )2, t = (p k )2, u = (p k )2. (6)
1 2 1 2 1 1
− −
The process can be described by two independent kinematical invariants, which we choose
as the variables q2 and t.
The amplitude of the reaction is related by crossing to the corresponding scattering
amplitude for elastic electron-proton scattering. Neglecting the lepton masses, the matrix
element including multi-photon exchange is parameterized by three generalized formfactors.
Several equivalent representations exist. Here we use the representation, which was first
introduced in Ref. [8]. The matrix element can be written in the form
e2 1 1
T = u¯(k ,h)γ v(k , h) v¯(p ,λ ) G˜ γµ F˜ Pµ +F˜ PµK/ u(p ,λ ) ,
q2 2 µ 1 − × 2 N2 M − 2 m 3 m2 1 N1
( (cid:20) N N (cid:21) )
(7)
where G˜ , F˜ and F˜ are complex functions of q2 and t. Neglecting the lepton masses
M 2 3
implicates that the outgoing electron and positron have opposite helicities.
In the following, we also use the generalized form factor
q2
G˜ = G˜ 1 F˜ . (8)
E M − − 4m2 2
N
(cid:16) (cid:17)
4
In the Born approximation G˜ and G˜ reduce to the usual proton FFs and do not depend
M E
˜
on t, while F vanishes. In order to identify the 1γ and 2γ-exchange contributions, we
3
introduce the decompositions
G˜ (q2,t) G (q2)+δG˜ (q2,t),
M M M
≡
G˜ (q2,t) G (q2)+δG˜ (q2,t),
E E E
≡
F˜ (q2,t) δF˜ (q2,t). (9)
3 3
≡
G and G are the time-like proton magnetic and electric FFs and F˜ , δG˜ and δG˜ are
M E 3 M E
amplitudes of order e2, which originate from processes involving the exchange of at least two
photons.
To compute the differential cross section of the reaction, we use the center-of-mass (c.m.)
frame, where the momenta of the incoming proton and antiproton have opposite directions.
In this frame the variable t can be related to the c.m.-scattering angle θ between the incident
proton and the outgoing electron. Calculating the cross section up to next order of e2 leads
to the expression
1
dσ = q2 G 2(1+cos2θ)+ G 2sin2θ
c.m. M E
C | | τ| |
"
(cid:0) (cid:1)
+2Re[G δG˜ ∗](1+cos2θ)+ 21Re[G δG˜ ∗]sin2θ
M M E E
τ
+2 Re[G F˜ ∗] 1Re[G F˜ ∗] τ(τ 1)cosθsin2θ , (10)
M 3 E 3
− τ −
#
(cid:16) (cid:17)p
with
q2 e4 τ 1
τ = , (q2) = − . (11)
4m2 C 64π2q2 τ
N r
In the 1γ-exchange approximation, only the first two terms of Eq. (10) contribute to the
cross section and it reduces to the well known formula of the unpolarized cross section:
1
dσ = (q2) G 2(1+cos2θ)+ G 2sin2θ . (12)
c.m.,1γ M E
C | | τ| |
(cid:20) (cid:21)
The other part of Eq. (10) represents the interference of 1γ and 2γ-exchange processes.
In order to determine the imaginary part of the time-like form factors it is necessary to
study polarization observables. An observable which gives acess to the imaginary part of
the electric and magnetic form factor is the single spin asymmetry when either the proton or
antiproton is polarized normal to the scattering plane, which does not require polarization of
5
the leptons inthe final state. Polarization ofthe protonor antiprotonalong or perpendicular
to its motion, but in the scattering plane, in contrast also requires a polarized lepton.
The single spin asymmetry can be defined as
dσ↑ dσ↓
A = − , (13)
y dσ↑ +dσ↓
where dσ↑ (dσ↓) denotes the cross section for an incoming nucleon with positiv (negativ)
perpendicular polarization. In the case of a polarized proton the single spin asymmetry up
to next order in e2 can be obtained as
1
A = C 2sinθ Im[G G∗ ]+Im[G δG˜∗ ]+Im[δG˜ G∗ ] cosθ
y dσ √τ E M E M E M
c.m. (
(cid:16) (cid:17)
+ τ(τ 1) Im[G F˜∗]cos2θ + Im[G F˜∗]sin2θ . (14)
− M 3 E 3
)
p (cid:16) (cid:17)
In contrast to space-like processes the single spin asymmetry A in the time-like region does
y
not vanish in the Born approximation.
III. CALCULATION OF THE 2γ-EXCHANGE CONTRIBUTION AT LARGE q2
To calculate the 2γ-exchange corrections in pp e+e− at large momentum transfers
→
we consider a factorization approach using the concept of hadron distribution amplitudes
(DAs). We follow the experience gained by the space-like process ep ep, for which the
→
amplitudes δG˜ and F˜ were computed [12, 13] at large momentum transfer Q in the form
M 3
of a convolution of a hard kernel H, which can be calculated in QCD perturbation theory,
and the nonperturbative contributions Ψ, which can be related to the DAs of proton and
antiproton, for instance:
δG˜ (Q2,ε) Ψ H(Q2,ε) Ψ, (15)
M
≃ ∗ ∗
where the asterisk denotes the convolutions with respect to the participating quark mo-
mentum fractions. This result represents the leading order contribution with respect to an
expansion in 1/Q. The important feature of such an approach is that the virtualities of both
photons must be large: q2 q2 Q2. The corresponding hard subprocess involves only
− 1 ∼ − 2 ∼
one hard gluon exchange and is therefore suppressed by the strong coupling α . As all
s
∼
spectator quarks are involved in the hard scattering process described by Eq. (15), we shall
refer to it as the hard rescattering contribution.
6
This mechanism can be simply generalized to the crossing channel pp e+e− at large
→
momentum transfer q2 > 0. A typical diagram of the leading pQCD contribution to the 2γ-
exchange correction to theannihilation amplitude is illustrated in Fig.1. The simple analysis
p
e+
d u u
γ∗
γ∗
−
e
p
FIG. 1: Diagram for pp¯ e+e− including the exchange of two hard photons
→
allows to conclude that a description of the corresponding hard time-like subprocess can be
obtained directly from the space-like one using the crossing symmetry. Therefore we obtain
that the leading order asymptotic behavior of the time-like 2γ-exchange amplitudes can be
represented by the same form as Eq. (15) for the space-like process.
In the case of q2,t m2 the momenta of proton and antiproton in the c.m.-frame can
≫ N
be expressed by two light-like vectors n, n¯:
n¯
p √s , n¯ = (1,0,0,1),
1
≃ 2
n
p √s , n = (1,0,0, 1). (16)
2
≃ 2 −
The lepton momenta are defined as
√s √s
kµ = η¯ nµ +η n¯µ +kµ,
1 2 2 ⊥
√s √s
kµ = η nµ +η¯ n¯µ kµ, (17)
2 2 2 − ⊥
where, at large momentum transfer, η, η¯ and k can be determined from
⊥
t u
η , η¯= 1 η , k2 ηη¯s, (18)
≃ −s − ≃ −s ⊥ ≃
7
with the obvious restriction 0 < η < 1. Note that the kinematic variable η can be expressed
in terms the e− c.m. angle θ as :
1
η (1+cosθ) (19)
≃ 2
The proton matrix element at leading twist level is described by twist-three nucleon DAs
as:
4 0 εijku Wi[λ n]u Wj[λ n]d Wk[λ n] p = Dx e−ip1+(Pxiλi)Ψ (x ), (20)
α 1 β 2 σ 3 1 i αβσ i
Z
(cid:10) (cid:12) (cid:12) (cid:11)
with meas(cid:12)ure given by Dx = dx dx dx δ(1 (cid:12) x x x ), and where
i 1 2 3 1 2 3
− − −
0
q W[x] q (x)Pexp ig dt (n A)(x+tn) . (21)
α α
≡ ·
(cid:26) Z−∞ (cid:27)
Following Ref. [23], the function Ψ (x ) can be expressed as :
αβσ i
Ψ (x ) = V(x ) p [1n/¯ C] γ N+ +A(x ) p [1n¯/γ C] N+
αβσ i i 1+ 2 αβ 5 σ i 1+ 2 5 αβ σ
(cid:2)+T(x(cid:3)) p [1n/¯γ C] γ⊥γ N(cid:2)+ ,(cid:3) (22)
i 1+ 2 ⊥ αβ 5 σ
where N+ n¯/n/N represents the large component of the nucleon (cid:2)spinor, C(cid:3)is charge con-
≡ 4
jugation matrix: C−1γ C = γT, and the scalar functions A, V, T stand for the nucleon
µ − µ
DAs.
The hard rescattering contribution to the time-like 2γ-exchange amplitudes δG˜ , and
M
s/m2 F˜ canbeobtainedfromtheresultsforelasticep-scattering[13]usingcrossing relations
N 3
for the hard perturbative subprocess. In our kinematics, expressed by Eqs. (16, 17, 18), this
leads to the following substitution:
Q2 q2 iε, ζ η, (23)
→ − − →
whereζ isthekinematicalparameterintroducedinRef.[13]. Wethenobtainforthetime-like
2γ-exchange amplitudes :
α α 2π 2 d[y ] d[x ] 4(2η 1)x y Φ(y ,x )
δG˜ (q2,η) = em s i i − 2 2 i i , (24)
M
− q4 3 y y y¯ x x x¯ [x η¯+y η x y ][x η+y η¯ x y ]
(cid:18) (cid:19) Z 1 2 2 1 2 2 2 2 − 2 2 2 2 − 2 2
s α α 2π 2 d[y ] d[x ] 2(x y¯ +x¯ y )Φ(y ,x )
F˜ (q2,η) = em s i i 2 2 2 2 i i , (25)
m2 3 q4 3 y y y¯ x x x¯ [x η¯+y η x y ][x η+y η¯ x y ]
N (cid:18) (cid:19) Z 1 2 2 1 2 2 2 2 − 2 2 2 2 − 2 2
where Φ denotes the specific combination of the nucleon distribution amplitudes:
Φ(y ,x ) = Q 2 [(V′ +A′)(V +A)+4T′T](3,2,1)
i i u
+Q Q [(V′ +A′)(V +A)+4T′T](1,2,3)+2Q Q [V′V +A′A](1,3,2), (26)
u d u d
8
andthenumbersinthebracketsdefinetheorderofthemomentumfractionsinthearguments
of the DAs: V′V(3,2,1) V′(y ,y ,y )V(x ,x ,x ). We also introduced the quark charges
3 2 1 3 2 1
≡
Q = +2/3, Q = 1/3, the fine structure coupling α = e2/(4π), and the QCD coupling
u d em
−
α .
s
In general, the time-like amplitudes are complex functions. At tree level, the expressions
of Eqs. (24) and (25) do not contain an imaginary part explicitly. This can be simply
understood : the s-channel cut requires the on-shell photons (see e.g. diagram in Fig.1)
but at large q2 their propagators are highly virtual and hence the tree amplitudes are real.
Therefore we can obtain nontrivial imaginary contributions only from the loop corrections.
In particular, computing leading logarithms associated with the renormalization of DAs and
QCD coupling α one obtains imaginary contributions generated by time-like logarithms:
s
ln[ q2 iε] = ln[q2] iπ. Such effects can be easily accounted for by using the well known
− − −
formula for the analytic continuation of α [20]:
s
α (q2)
α ( q2) = s +..., (27)
s
− 1 iβ α (q2)/4
0 s
−
where β = 11 2/3n is the first term of the β-function. Eq. (27) includes resummed
0 f
−
large corrections β α which can be important at intermediate energies where α is not
0 s s
∼
too small. Similarly, solving the renormalization group equation, one obtains an imaginary
part originating from the evolution of DAs. However, the resulting imaginary contributions
provide quite small numerical effects for the regions of q2 which we are going to discuss
below, see e.g. Ref. [21].
As can be seen from Eqs. (24, 25) the leading behavior of the amplitudes δG˜ and
M
s/m2 F˜ goes as 1/q4, whereas δF˜ is suppressed in the large momentum transfer limit,
N 3 2
since it behaves as 1/q6. We may expect that at intermediate energies 5 10 GeV2 the
∼ −
effective scaledefining theapplicability oftheperturbativeexpansion isalready largeenough
in order to apply the present formalism. In what follows, we assume that the scale of the
strong coupling in Eqs. (24, 25) is of order µ2 0.6 q2.
R ≃
To evaluate the convolution integrals given in Eqs. (24, 25), we need to consider a model
description for the twist-3 DAs. In Ref. [22] the asymptotic behavior of the DAs and their
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f r r
N − +
(10−3 GeV2)
COZ [24] 5.0 0.5 4.0 1.5 1.1 0.3
± ± ±
BLW [25] 5.0 0.5 1.37 0.35
±
QCDSF [26] 3.23 1.06 0.33
TABLE I: Parameters entering the proton DA (at µ = 1 GeV) for three parameterizations (COZ,
BLW, and the lattice evaluation from QCDSF) used in this work.
first conformal moments can be found as
V(x ) 120x x x f [1+r (1 3x )],
i 1 2 3 N + 3
≃ −
A(x ) 120x x x f r (x x ),
i 1 2 3 N − 2 1
≃ −
1
T(x ) 120x x x f 1+ (r r )(1 3x ) , (28)
i 1 2 3 N − + 3
≃ 2 − −
(cid:20) (cid:21)
where the DAs depend on the three parameters f , r and r . For our calculations, we
N + −
consider two phenomenological models for the DAs, which have been discussed in the litera-
ture : COZ [24] and BLW [25], as well as one description based on lattice QCD calculations
[26]. The corresponding parameters are presented in Table I. One notices that the param-
eters r und r in the BLW model and in the lattice calculations are nearly comparable,
+ −
whereas the overall normalization f is about 2/3 smaller for the lattice DA as compared
N
with the description of the BLW model. In contrast to the BLW model and lattice calcu-
lations, the parameters r and r are about 3 times larger in the COZ description of the
+ −
nucleon DAs. Below, we will provide calculations using the first two models, COZ and BLW.
The results following from the lattice calculations can easily be approximated by scaling the
BLW results. All parameters from the Table I have been evolved with leading logarithmic
accuracy.
Using the parametrization of Eq. (28), the convolution integrals can be computed and
10