The quark Regge trajectory at two loops

15 Pages·0.21 MB·English

by  A. V. Bogdan

PreprinttypesetinJHEPstyle-HYPERVERSION
DCTP/02/08, IPPP/02/04, DFTT 03/2002,Budker INP 2002-4 hep-ph/0201240
The quark Regge trajectory at two loops
2
0
0
2 A. V. Bogdan

n
Budker Institute for Nuclear Physics, Novosibirsk State University
a
J 630090 Novosibirsk, Russia

5 E-mail: [email protected]
2
V. Del Duca

1
v Istituto Nazionale di Fisica Nucleare, Sez. di Torino
0 via P. Giuria, 1 - 10125 Torino, Italy
4
E-mail: [email protected]
2
1
0 V. S. Fadin
2
Budker Institute for Nuclear Physics, Novosibirsk State University
0
/ 630090 Novosibirsk, Russia
h
E-mail: [email protected]
p
-
p E. W. N. Glover
e
h Institute for Particle Physics Phenomenology, University of Durham
:
v Durham, DH1 3LE, U.K.
i E-mail: [email protected]
X
r
a
Abstract: The hypothesis of the quark Reggeization is tested by taking the high-energy
limit of the two-loop amplitude for quark-gluon scattering which was recently calculated.
Thelimit is compatible with theReggeization in theleading andthenext-to-leading orders
and allows the determination of the quark trajectory in the two-loop approximation. The
trajectory is presented as an expansion in powers of (D 4) for the space-time dimension
−
D tending to the physical value D = 4.
Keywords: QCD, Jets, NLO and NNLO Computations.
1. Introduction
In the limit of large center of mass energy √s and fixed momentum transfer √ t the
−
most appropriate approach for the description of the scattering amplitudes is given by the
theory of the complex angular momenta or Gribov-Regge theory. One of the remarkable
properties of QCD is the Reggeization of its elementary particles. Unlike QED, where the
electron Reggeizes [1, 2], but the photon remains elementary [3], in QCD both the gluon
[4, 5, 6, 7, 8, 9, 10] and quark [11] Reggeize. We use here the notion “Reggeization” in
the strong sense [12], that means not only the existence of a Reggeon with corresponding
quantum numbers (including signature) and trajectory, but as well the dominance of the
Reggeon contribution to the amplitudes of the processes with these quantum numbers in
each order of perturbation theory. For example, parton-parton scattering amplitudes in
QCD are dominated by gluon exchange in the crossed channel. The Reggeized gluon is a
colour octet state of negative signature in the t-channel, i.e. odd under s u exchange,
↔
and to leading logarithmic (LL) accuracy in ln(s/t ) the virtual radiative corrections to
| |
the parton-parton scattering amplitude with the colour octet state and negative signature
can be obtained, to all orders in α , by the replacement [8]
S
s 1 s jG(t) s jG(t)
− , (1.1)
t → 2 t − t
" #
(cid:18)− (cid:19) (cid:18)− (cid:19)
where j (t) ω(t)+1 is called the Regge trajectory of the gluon.
G
≡
The property of the Reggeization is very important for high energy QCD. The BFKL
equation [7, 8, 9, 10] for the resummation of the leading logarithmic radiative corrections
to scattering amplitudes for processes with gluon exchanges in the t-channel is based on
the gluon Reggeization. The Pomeron, which determines the high energy behaviour of
cross sections, and the Odderon, responsible for the difference of particle and antiparticle
cross sections, appears in QCD as a compound state of two and three Reggeized gluons
respectively. Similarly colorless objects constructed from Reggeized quarks and antiquarks
should also be relevant to the phenomenological description of processes involving the
exchange of quantum numbers.
The basic parameters of the Reggeons are their trajectories and the interaction ver-
tices. To leading logarithmic accuracy, ω(t) is related to a one-loop transverse-momentum
integration. In dimensional regularization in D =4+2ǫ dimensions this can be written as
µ2 −ǫ
ω(t) = g2 c ω(1) + (g4), (1.2)
S t Γ O S
(cid:18)− (cid:19)
with
2
ω(1) = C , (1.3)
− Aǫ
1 Γ(1 ǫ)Γ2(1+ǫ)
c = − , (1.4)
Γ (4π)2+ǫ Γ(1+2ǫ)
and C = N. The resummation of the real and virtual radiative corrections to parton-
A
parton scattering amplitudes with gluon exchange in the crossed channel is related by
– 1 –
unitarity to the imaginary part of the elastic amplitudes with all possible colour exchanges
inthecrossedchannel. Theradiativecorrectionstotheseamplitudesareresummedthrough
theBFKLequationi.e. atwo-dimensionalintegralequationwhichdescribestheinteraction
of two Reggeized gluons in the crossed channel.
The integral equation is obtained to LL by computing the one-loop corrections to the
gluon exchange in the t channel. They are formed by a real correction: the emission of a
gluonalongtheladder[6]andavirtualcorrection: theone-loopReggetrajectory, Eq.(1.1).
The BFKL equation is then obtained by iterating recursively these one-loop corrections to
all orders in α , to LL accuracy.
s
In recent years, the BFKL equation has been improved to next-to-leading logarithmic
(NLL) accuracy [13, 14, 15]. A necessary ingredient has been the calculation of the two-
loop Reggetrajectory ofthegluon[16,17,18,19]. Recently thegluonReggetrajectory was
re-evaluated [20], in a completely independent way, by taking the high energy limit of the
two-loop amplitudes for parton-parton scattering with gluon exchanges in the t-channel.
The validity of the gluon Reggeization to NLL was confirmed and full agreement with
previous results was found.
So far attention has focussed mainly on processes dominated by Reggeized gluon ex-
change. However, let us consider a scattering process with fermion exchange, namely
quark-gluon scattering, which proceeds via the exchange of a quark in the crossed chan-
nel, and let us take the limit s u. Since in the center-of-mass frame of a two-particle
≫ | |
scattering u = s(1+cosθ)/2, the limit s u corresponds to backward scattering. By
− ≫ | |
crossing symmetry, this is equivalent to quark pair annihilation into two gluons at small
angles (in which case the roles of s and t are exchanged). The contribution from the ex-
change of a colour triplet state and of positive signature i.e. even under s t exchange
↔
Reggeises, so that the virtual radiative corrections to the quark-gluon scattering amplitude
with the colour triplet state and positive signature in the limit s u can be obtained, to
≫ | |
all orders in α and to LL accuracy in ln(s/u), by the replacement [11]
S | |
s 1 s s δT(u) s δT(u)
+ − , (1.5)
u → 2 u u u
" #
r− r− (cid:18)− (cid:19) (cid:18)− (cid:19)
where the quark Regge trajectory is j (u) = δ (u) + 1/2. δ (u) is related to a one-
Q T T
loop transverse-momentum integration, which, for massless quarks and up to replacing the
colour factor C with C , is the same as for the gluon trajectory. Thus in dimensional
A F
regularization δ (u) can be written as,
T
2 µ2 −ǫ
δ (u) = g2C c , (1.6)
T − S F ǫ u Γ
(cid:18)− (cid:19)
with C = (N2 1)/(2N), and c given in Eq. (1.4). This is similar to what happens in
F − Γ
QED, where the electron also Reggeizes [1, 2]. Analogously to the gluon case, an equation
was derived [11] to resum the real and virtual radiative corrections to exchanges with
arbitrary colour state and signature. However, unlike from the BFKL equation for gluon
exchange, that equation has not been solved.
– 2 –
In this paper we explicitly take the high-energy limit of the one-loop [21, 22] and of
the two-loop amplitudes for quark-gluon scattering [23]. This allows us to test the validity
of the quark Reggeization and to calculate the two-loop Regge trajectory of the quark.
Our paper is organised as follows. In Sect. 2, we discuss the general structure of the
quark-gluon scattering amplitudes, we evaluate them at tree and one-loop level in the limit
s u and decomposethem according to theirreduciblecolour representations exchanged
≫ | |
in the u channel. In Sect. 3 we discuss the Regge ansatz for resumming the LL and NLL
and evaluate the leading and next-to-leading order corrections for the interference of the
tree-amplitude with the Reggeized ansatz in terms of the gluon trajectory and the impact
factors. Sect. 4 is devoted to the analysis of the one- and two-loop Feynman diagram
calculations in the high energy limit and to the extraction of the LL, NLL and NNLL
behaviours. We make a detailed comparison of the two approaches and show that the two
approaches are compatible at the LL and NLL level. This allows the determination of the
quark trajectory to two-loop order which we give as an expansion in powers of (D 4) for
−
the space-time dimension D tending to the physical value D = 4. Finally, our findings are
summarized in Sect. 5.
2. The structure of the quark-gluon scattering amplitude
Let the incoming quark and gluon have momenta p and p respectively, and the outgoing
a b
quark and gluon have momenta p and p respectively.
a′ b′
The most general possible colour decomposition for the amplitude is
M
= 2(TbTb′) +2(Tb′Tb) +δbb′δ , (2.1)
a′a a′a a′a
M A B C
where b, b′ are the colours of the two gluons and a, a′ are the colours of the quarks. The
factor 2 in Eq. (2.1) is due to our choice for the normalization of the fundamental repre-
sentation matrices, i.e. tr(TaTb) = δab/2. The functions , and are colour-stripped
A B C
sub-amplitudes where we have suppressed all dependence on the particle polarisations and
momenta. At lowest order, for general kinematics, only the first two colour structures are
present. They are related by t u exchange, thus in the s u limit only one of them
↔ ≫ | |
contributes. The third colour structure appears first at one-loop order and is symmetric
under u and t exchange.
2.1 Quark-gluon amplitudes at tree and one-loop level
In the high-energy limit s u, quark-gluon scattering (or any other crossing symmetry
≫ | |
related process) is dominated by quark exchange. In this limit, at tree level accuracy only
the configuration for which the outgoing gluon has the same helicity as the incoming quark
will contribute. This is equivalent to say that in the limit s u helicity is conserved
≫ | |
in the s channel. Thus, once the helicity along the quark line is fixed, of the two gluon
helicity configurations allowed at tree level, only one dominates. The only other leading
helicity configuration is the one obtained from this by parity, which flips the helicity of all
the particles in the scattering.
– 3 –
Usingthecolourdecomposition(2.1)inthehelicityformalism[24],wenotethatthereis
only one independent colour-stripped tree sub-amplitude, the Parke-Taylor sub-amplitude
[25]. Using the spinor products in the limit s u (which can be easily derived from the
≫ | |
amplitudes listed in the appendix of Ref. [26] in the limit s t ) in the sub-amplitude for
≫ | |
the leading helicity configuration, we obtain the
2
(0)(p−,p−;p−,p−) = ig2 s pa′⊥ , (2.2)
A a b a′ b′ − Sr−u |pa′⊥|!
where complex transverse coordinates p = px+ipy have been used, and thesupersciptsin
⊥
the argument on the left-hand side label the parton helicities∗. Using Eq. (2.1), the ampli-
tude for quark-gluon scattering q g q g for a generic tree-level helicity configuration
a b a′ b′
→
may be written as,
(0)(pνq,pνg;pνq,pνg′)
M a b a′ b′
= 2i g (Tb) C(0)(pνq,pνg′) s g (Tb′) C(0)(pνg,pνq) , (2.3)
− S a′i qg a b′ u S ia gq b a′
h ir− h i
there the ν’s denote the parton helicities, and we have explicitly enforced helicity conserva-
tion along a massless fermion line. From Eq. (2.2), the tree-level coefficient function C(0)
is,
p
C(0)(p−,p−)= C(0)(p−,p−) = a′⊥ , (2.4)
qg a b′ gq b a′ p
a′
| ⊥|
whilefor s u theunequalhelicity coefficient functionsof typeC(0)( )aresubleading.
≫ | | ±∓
Squaring and summing Eq. (2.3) over helicity and colour, we obtain,
s 128 s
(0) 2 = 8 C2 N g4 = g4 . (2.5)
|M | F c S u 3 S u
hel,col − −
X
in agreement with the s u limit of the squared tree quark-gluon amplitudes.
≫ | |
InRef.[22],thecoefficients inthecolourdecomposition(2.1)oftheone-loopamplitude
for the quark-gluon scattering have been calculated in the ’t Hooft-Veltman and in the
dimensional reduction infrared schemes. Using the results of Ref. [22], the sub-amplitude
of Eq. (2.2) and the tree amplitude of Eq. (2.3), we can write the unrenormalised one-loop
amplitude in the helicity configuration of Eq. (2.2), in the ’t Hooft-Veltman scheme and in
the limit s u as,
≫ | |
(1)(p−,p−;p−,p−)=
M a b a′ b′
2C s 2(C +C ) 3C
g˜2(u) (0)(p−,p−;p−,p−) F ln A F + F +(π2 7)C
S M a b a′ b′ ǫ u − ǫ2 ǫ − F
(cid:20) − (cid:18)− (cid:19) (cid:21)
2 s
+ g˜S2(u)δbb′δa′a A(0)(p−a,p−b ;p−a′,p−b′) ǫ ln −t +O(ǫ). (2.6)
(cid:18)− (cid:19)
∗Notethat,contrarytotheconventionsofthehelicityformalismwhereallparticlesaretakenasoutgoing,
here we label momenta and helicities as in thephysical scattering.
– 4 –
to leading accuracy in s/u. Note that on the right hand side the first term, which is
proportional to the tree amplitude Eq. (2.3), is real. The second term contains the new
colour structure introduced at the one loop level and is proportional to the tree sub-
amplitude (2.2). It is purely imaginary, since to this accuracy ln( s/ t) iπ, where
− − ≃ −
we used the usual prescription ln( s) = ln(s) iπ, for s > 0. In Eq. (2.6) and further in
− −
our paper we have rescaled the coupling as
µ2 −ǫ
g˜2(u) = g2c . (2.7)
S S Γ u
(cid:18)− (cid:19)
In the one-loop amplitude helicity is not conserved in the s channel, thus the unequal
helicity coefficient functions of type C(1)( ) are no longer subleading. In fact, from
±∓
Ref. [22] we obtain
g4 s
(1)(p−,p∓;p−,p±) = 2i S (C C ) (TbTb′) , (2.8)
M a b a′ b′ (4π)2 A− F u a′a
r−
which is finite and can be written in the form of Eq. (2.3) by replacing the one of the
helicity conserving coefficient functions of type C(0) with the helicity violating coefficient
function of type C(1) where
C(1)(p−,p+) = C(1)(p+,p−)= gS2 (C C )|pa′⊥|. (2.9)
qg a b′ gq b a′ −(4π)2 A− F pa′
⊥
Note that in the calculation of a production rate the coefficient function of type C(1)( )
±∓
appear only in the square of a one-loop amplitude.
Taking the interference of the one loop amplitude (2.6) with the tree amplitude, (2.3),
and summing over helicity and colour of initial and final states, we obtain,
(1) (0)∗ = (0) 2 g˜2(u) (2.10)
M M |M | S
heXl,col(cid:16) (cid:17) hXel,col
2C s 2(C +C ) 3C iπ 1
F ln A F + F +(π2 7)C + (ǫ).
× ǫ u − ǫ2 ǫ − F − ǫ C O
(cid:20) − (cid:18)− (cid:19) F(cid:21)
2.2 The colour structure in the u-channel
In order to gain more insight in how the amplitudes in the colour decomposition (2.1) are
relatedtotheexchangeofparticularcolourstates,wedecomposethequark-gluonscattering
amplitudesaccordingtotheirreduciblecolourrepresentations, 3 8= 3 ¯6 15,exchanged
⊗ ⊕ ⊕
in the u channel,
= (Pbb′) (χ) , (2.11)
a′a χ
M M
χ
X
where we recall that a and a′ are quark colour indices and b and b′ are gluon colour indices,
and χ = 3, ¯6, 15. M are colour-stripped coefficients and (Pbb′) (χ) are the colour
χ a′a
projectors,
1
(Pbb′) (3) = (TbTb′) ,
a′a a′a
C
F
– 5 –
1 1
(Pbb′) (¯6) = δbb′δ (TbTb′) (Tb′Tb) ,
a′a a′a a′a a′a
2 − N 1 −
−
1 1
(Pbb′) (15) = δbb′δ (TbTb′) +(Tb′Tb) , (2.12)
a′a a′a a′a a′a
2 − N +1
which fulfill the usual property of projectors,
(Pbb′) (χ) (Pb′b′′) (χ′) = δ (Pbb′′) (χ). (2.13)
a′a aa′′ χχ′ a′a′′
We find that
1
= 2C + ,
M3 FA− NB C
= + ,
¯6
M −B C
= + . (2.14)
15
M B C
2.3 Signature of the exchanged state
In addition to having a particular colour, exchanged Reggeons must have a particular sig-
nature. Inother wordsthey areeither even (positive signature)or odd(negative signature)
under the exchange of s and t. We therefore define the amplitudes of specific signature to
be,
1
± = ( (s t)). (2.15)
Mχ 2 Mχ±Mχ ↔
Note that only the colour triplet positive signature exchange is expected to Reggeize. This
is the contribution that will generate the LL and NLL behaviour of the amplitude. The
otherpositivesignaturecontributionsandallofthenegativesignaturecontributionsarenot
given by simple poles. In the LL approximation they can be obtained using the equation
for amplitudes with fermion exchange derived in [11], but beyond the LL it is not known
how to analyse these structures. Note also that (s t) is in fact the amplitude for
χ
M ↔
quark-antiquark annihilation into gluons.
As mentioned earlier, the colour structure δbb′δ occurs first at one-loop, while the
a′a
Born contribution to (Tb′Tb) vanishes in the Regge limit. Therefore, in the Regge limit,
a′a
(0)±
many of the amplitudes vanish in the Born approximation. In fact, from Eqs. (2.2)
χ
M
and (2.3) we obtain,
(0)− (0)+ (0)− (0)+ (0)−
= = = = = 0, (2.16)
M3 M¯6 M¯6 M15 M15
(0)+
and only is non-zero,
M3
(0)+ = 2C (0). (2.17)
M3 FA
From Eq. (2.6), for the one-loop coefficient of the positive signature triplet, we have
(1)+ = g˜2(u) (0)+
M3 S M3
C s s 2(C +C ) 3C
F ln +ln − A F + F +(π2 7)C + (ǫ),
× ǫ u u − ǫ2 ǫ − F O
(cid:26)− (cid:20) (cid:18)− (cid:19) (cid:18)− (cid:19)(cid:21) (cid:27)
while for the higher-dimensional representations we have,
(1)+ (1)+
= = 0. (2.18)
M¯6 M15
– 6 –
Thus, the colour representations ¯6 and 15 are not present in the positive signature, at
one-loop level and to leading power accuracy in s/u. In fact, in the LL approximation [11],
+ = + = 0 (2.19)
M¯6 M15
to all orders.
For the one-loop coefficients of negative signature, we obtain,
1 s s 1 s
(1)− = g˜2(u) (0)+ C ln ln − + ln − + (ǫ),
M3 S M3 ǫ − F u − u C t O
(cid:26) (cid:20) (cid:18)− (cid:19) (cid:18)− (cid:19)(cid:21) F (cid:18)− (cid:19)(cid:27)
1 1 s
(1)− = (1)− = g˜2(u) (0)+ ln − + (ǫ). (2.20)
M¯6 M15 S M3 ǫ C t O
F (cid:18)− (cid:19)
Thus in the one-loop amplitude of negative signature all the three representations ex-
changed in the u channel contribute, to leading power accuracy in s/u.
3. Regge Theory Interpretation
Let’s choose, for definiteness, the QCD Compton scattering process
q(p )+g(p ) q(p )+g(p ). (3.1)
a b a′ b′
→
We will usephysical polarizations of gluons, sothat their polarization vectors satisfy e(p )
b
·
p = e(p ) p = 0 and e(p ) p = e(p ) p = 0. Then, for massless quarks, the
b b b′ b′ b′ b′ b
· · ·
contribution of the Reggeized quark to the amplitude for colour triplet exchange with even
signature can be presented [11] as,
11 s δT(u) s δT(u)
+ = Γ − − + Γ , (3.2)
R3 QG q 2 u u GQ
⊥ " #
6 (cid:18)− (cid:19) (cid:18)− (cid:19)
where q is the transverse to the (p ,p ) plane part of the momentum transfer q = p p ,
⊥ a b a′ b
−
u = q2 = q2 in the Regge limit, Γ and Γ are the Reggeon vertices for the G Q and
⊥ QG GQ →
Q G transitions and δ determines the quark Regge trajectory. Note that for massless
T
→
quarks δ depends only on q2, so that instead of two complex conjugate trajectories with
T ⊥
opposite parities for massive quarks wehave asingle one. We assumethat thequark Regge
trajectory has the perturbative expansion,
δ (u) = g˜2(u) δ(1) +g˜4(u) δ(2) + (g˜6(u)). (3.3)
T S T S T O S
At leading order [11] and in D = 4+2ǫ dimensions,
2C
δ(1) = F. (3.4)
T − ǫ
This is related to the analogous one-loop gluon trajectory by
C
δ(1) = Fω(1).
T C
A
– 7 –
The general structure of the Reggeon vertices is determined by relativistic invariance
and colour symmetry. For the quark-gluon vertices the general structure is
e(p ) q q
ΓQG = −gSu¯(pa′)Tb 6e(pb)(1+δe(u))+ bq·2 6 ⊥δq(u) ,
(cid:20) ⊥ (cid:21)
e∗(p ) q q
ΓGQ = −gS 6e∗(pb′)(1+δe(u))+ b′q2· 6 ⊥δq(u) Tb′u(pa) . (3.5)
(cid:20) ⊥ (cid:21)
Note that it is straightforward to relate this general vertex structure to the scattering of
particular particle helicities, Section 2.1. The functions δ and δ represent the radiative
e q
corrections to the Born vertices. They are functions of q2 in the massless case and have
⊥
the perturbative expansion
δ (u) = g˜2(u) δ(1) +g˜4(u) δ(2) + (g˜6),
e S e S e O S
δ (u) = g˜2(u) δ(1) +g˜4(u) δ(2) + (g˜6). (3.6)
q S q S q O S
In the one-loop approximation the corrections were obtained in [27] and have the form
C 1 3(1 ǫ) 1 ǫ
δ(1) = ω(1) F − +ψ(1)+ψ(1 ǫ) 2ψ(1+ǫ) + ,
e 2C ǫ − 2(1+2ǫ) − − 2ǫ − 2(1+2ǫ)
(cid:20) A (cid:18) (cid:19) (cid:21)
(3.7)
ǫ 1
δ(1) = ω(1) 1+ . (3.8)
q 2(1+2ǫ) N2
(cid:18) (cid:19)
Note,thatwhereasδ(1) hasasoft(1/ǫ2)singularitythatmustbecancelledbyrealradiation,
e
(1)
δ is finite as ǫ 0, since the corresponding spin structure is absent at leading order.
q
→
There is no ansatz for any of the odd signature exchanges or for the even signature
¯6 and 15 exchanges. These contributions do not correspond to simple poles and cannot
be described by an ansatz of the form of Eq. (3.2). As mentioned earlier, in the LL
approximation + and + arezerotoallorders. TheLLnegativesignaturecontributions
M¯6 M15
through to (D 4) can be obtained from the equation derived in [11] and are given by
O −
1 1 C s
− = (0)+g˜2(u) C + iπ+g˜2(u) F(2iπln +π2)+ (g˜4(u)) ,
M3 M3 S F C ǫ − S ǫ u O S
(cid:18) F(cid:19) (cid:18) (cid:18)| |(cid:19) (cid:19)
(3.9)
1 C +2C 1 s
− = (0)+g˜2(u) iπ+g˜2(u) A F − (2iπln +π2)+ (g˜4(u)) ,
M¯6 M3 S C ǫ − S 2ǫ u O S
F (cid:18) (cid:18)| |(cid:19) (cid:19)
(3.10)
1 C +2C +1 s
− = (0)+g˜2(u) iπ+g˜2(u) A F (2iπln +π2)+ (g˜4(u)) .
M15 M3 S C ǫ − S 2ǫ u O S
F (cid:18) (cid:18)| |(cid:19) (cid:19)
(3.11)
Note that at the one-loop level Eqs. (2.19), (3.2)-(3.8) and (3.9)-(3.11) give all of the
contributions to the scattering amplitude that survive in the Regge limit. They are in
agreement withEqs.(2.6), (2.8), (2.18), (2.18) and(2.20) obtained fromexact calculations.
– 8 –
3.1 Projection by tree-level amplitude
Let us denote the projection of the tree amplitude (0) summed over spins and colours
M
on a generic amplitude as,
M
(0) = (0)† . (3.12)
hM |Mi M M
spin,col
X
For the Reggeized amplitude we obtain the projection normalised by the square of the
Born amplitude to be,
(0) +
hM |R3i (3.13)
(0) (0)
hM |M i
1+exp−iπδT(u) (1+δ (u))δ (u) δ2(u)
= exp(δ (u)L) (1+δ (u))2 + e q + q ,
T e
2 1+ǫ 4(1+ǫ)2
(cid:0) (cid:1) " #
(0)+
where L = ln(s/ u). Note that coincides with the Born amplitude so that setting
− R3
δ = δ = δ = 0 produces unity.
e T q
Writing + as a perturbative series,
R3
+ = g˜2n(u) (n)+, (3.14)
R3 S R3
n
X
and expanding Eq. (3.13) to first order gives the one-loop contribution to the Reggeized
(1)+
amplitude such that
R3
(0) (1)+ (1)
δ π
hM |R3 i = δ(1)L+2δ(1) + q i δ(1), (3.15)
(0) (0) T e 1+ǫ − 2 T
hM |M i
that agrees with the results of [21, 22]. For the two-loop contribution we obtain
hM(0)|R(32)+i = (δ(1)T)2 L2+ δ(2) + 2δ(1) + δq(1) δ(1) L
(0) (0) 2 T e 1+ǫ T
" ! #
hM M i
π2 δ(1)δ(1) (δ(1))2 δ(2)
(δ(1))2+(δ(1))2+ e q + q +2δ(2) + q
− 4 T e 1+ǫ 4(1+ǫ)2 e 1+ǫ
(1) (1)
π δ δ
i (δ(1))2L+ q T +2δ(1)δ(1) +δ(2) . (3.16)
− 2 T 1+ǫ e T T
!
Note that the Reggeized form for positive signature colour triplet exchange does not
saturate the possible contributions to the cross section in the Regge limit. There is also
a contribution from the negative signature exchange. This negative signature contribution
appearsfirstintheimaginarypartofthe(normalized)projectionontheone-loopamplitude
and also in the logarithmic imaginary part and non-logarithmic real parts of the two-loop
amplitude.
Therefore,ifthestatementaboutthequarkReggeization isvalidinthenext-to-leading
logarithmic order, the logarithmic terms in the right-hand side of Eqs. (3.15) and (3.16)
must coincide with corresponding terms for the total amplitude, which can be found using
the results of [23] in the appropriate limit.
– 9 –