Virtual Next-to-Leading Corrections to the Lipatov Vertex

21 Pages·0.23 MB·English

by  Vittorio Del Duca

EDINBURGH 98/21
MSUHEP-80928
Sept 28, 1998
hep-ph/9810215
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3 Virtual Next-to-Leading Corrections to the Lipatov Vertex
1

2
v
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1 Vittorio Del Duca∗
2
0
1 Particle Physics Theory Group, Dept. of Physics and Astronomy
8
9 University of Edinburgh, Edinburgh EH9 3JZ, Scotland, UK
/
h
p
and
-
p
e
Carl R. Schmidt
h
:
v
i Department of Physics and Astronomy
X
r Michigan State University
a
East Lansing, MI 48824, USA
Abstract
We compute the virtual next-to-leading corrections to the Lipatov vertex in
the helicity-amplitude formalism. These agree with previous results by Fadin and
collaborators, in the conventional dimensional-regularization scheme. We discuss
the choice of reggeization scale in order to minimize its impact on the next-to-
leading-logarithmic corrections to the BFKL equation.
∗
On leave of absence from I.N.F.N., Sezione di Torino, Italy.
1 Introduction
Recently, a long awaited calculation of the next-to-leading-logarithmic (NLL) corrections
to the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation [1]-[3] has been completed [4].
The BFKL program is designed to resum the large logarithms of type ln(sˆ/ tˆ) in semi-
| |
hard strong-interaction processes, which are characterized by two large and disparate
kinematic scales: sˆ, the squared parton center-of-mass energy and tˆ, of the order of the
| |
squared momentum transfer. At leading logarithmic (LL) accuracy there can be large
dependence on the exact choice of the transverse momentum scale in this resummation,
which should be reduced in the NLL calculation. However, the NLL corrections turn
out to be large and negative [4], and when the NLL result is applied to Deep Inelastic
Scattering (DIS) at small x the phenomenological predictions are unreliable [5].
It has been suggested that the bad behavior of the NLL BFKL resummation is due
to the presence of double logarithms of the ratio of two transverse scales, which are
ignored in the resummation [6]. On the other hand, an independent check of the NLL
resummation [4] is not available yet. A partial check has been performed [7]; however, it
relies on the same QCD amplitudes [8, 9, 10, 11, 12] that constitute the building blocks
of the original NLL calculation, and it assumes them to have been computed correctly.
We have undertaken an independent calculation of the QCD amplitudes relevant to the
NLL resummation using the helicity-amplitude formalism [13, 14].
The solution to the LL BFKL equation is an off-shell gluon Green’s function, which
represents the gluon propagator exchanged in the crossed channel of parton-parton scat-
tering. The building blocks of this equation are the Lipatov vertex [1], which summarizes
the emission of a gluon along the propagator, and the LL reggeization term [2], which
summarizes the virtual corrections to the propagator. In NLL approximation, one needs
the reggeization term to the corresponding accuracy [12], plus the real and virtual correc-
tions to the Lipatov vertex. The real corrections to this vertex are given by the emission
of two gluons or of a q¯q pair at comparable rapidities [8, 13], while the virtual corrections
are given by the Lipatov vertex at one-loop accuracy [9, 10, 11]. In this paper we compute
the Lipatov vertex at one-loop accuracy using the helicity-amplitude formalism1.
The outline of this paper is as follows. In section 2 we set up the formalism needed for
extracting the one-loop Lipatov vertex from the five-gluon one-loop helicity amplitudes.
1In order to obtain the one-loop Lipatov vertex, one also needs the helicity-conserving vertices at
one-loop accuracy [9, 14, 15, 16], even though these do not enter directly the calculation of the NLL
corrections to the BFKL kernel.
1
In section 3 we present these helicity amplitudes, as obtained from the work of Bern,
Dixon, and Kosower [17]. In section 4 we consider these amplitudes in the multi-Regge
kinematics in order to extract the Lipatov vertex to O(ǫ0) in the expansion of the space-
time dimension used for regularization of singularities. In section 5 we consider the
amplitudes in the limit that the central emitted gluon becomes soft, so as to extract
the vertex to O(ǫ) in this region. This is necessary to get the correct contribution from
the infrared singularity when integrating over the momentum of this gluon in the squared
amplitude. Finally, in section 6 we discuss the choice of reggeization scale, and we present
our conclusions.
2 The five-gluon amplitude at high-energy
We are interested in the five-gluon amplitude in the multi-Regge kinematics, which pre-
sumes that the produced gluons are strongly ordered in rapidity and have comparable
transverse momenta:
ya′ y yb′; pa′⊥ k⊥ pb′⊥ . (1)
≫ ≫ | | ≃ | | ≃ | |
In this kinematics the tree-level amplitude for gagb ga′ggb′ scattering may be written
→
[1]
1
Mgtrge→eggg = 2s igfaa′cC−ggν(a0ν)a′(−pa,pa′) t (2)
1
h i
1
× igfcdc′Cνg(0)(qa,qb) t igfbb′c′C−ggν(b0ν)b′(−pb,pb′) ,
2
h i h i
where all external gluons are taken to be outgoing, q are the momenta transferred in the
i
t-channel, i.e. qa = pa pa′ and qb = pb +pb′, and ti qi⊥ 2. The vertices g∗g g,
− − ≃ −| | →
with g∗ an off-shell gluon, are given by [1, 18]
p∗
C−gg+(0)(−pa,pa′) = −1 C−gg+(0)(−pb,pb′) = −pbb′′⊥⊥ , (3)
and the Lipatov vertex g∗g∗ g [18, 19] is
→
q∗ q
Cg(0)(q ,q ) = √2 a⊥ b⊥ , (4)
+ a b
k
⊥
with p = p + ip the complex transverse momentum. The C-vertices transform into
⊥ x y
their complex conjugates under helicity reversal, C∗ ( k ) = C ( k ). The helicity-
{ν} { } {−ν} { }
(0)
flip vertex C is subleading in the high-energy limit. For gluon-quark scattering, g q
++ a b
→
2
ga′gqb′, or quark-quark scattering, qaqb qa′ gqb′, we only need to change the relevant
→
vertices Cgg(0) to Cq¯q(0) and exchange the corresponding structure constants with color
matrices in the fundamental representation [20]. In eq. (2) the mass-shell condition for
the intermediate gluon in the multi-Regge kinematics has been used,
s s
1 2
s = , (5)
k 2
⊥
| |
with s sab, s1 sa′1, s2 s1b′.
≡ ≡ ≡
The virtual radiative corrections to eq. (2) in the LL approximation are obtained, to
all orders in α , by replacing [1, 21]
s
1 1 s α(ti)
i
, (6)
t → t τ
i i (cid:18) (cid:19)
in eq. (2), with α(t) related to the loop transverse-momentum integration
d2k 1
α(t) g2α(1)(t) = α N t ⊥ t = q2 q2 , (7)
≡ s c (2π)2 k2(q k)2 ≃ − ⊥
Z ⊥ − ⊥
and α = g2/4π. The reggeization scale τ is much smaller than any of the s-type invari-
s
ants, τ s,s ,s , and it is of the order of the t-type invariants, τ t ,t . The precise
1 2 1 2
≪ ≃
definition of τ is immaterial to LL accuracy. The infrared divergence in eq. (7) can be reg-
ularized in 4 dimensions with an infrared-cutoff mass. Alternatively, using dimensional
regularization in d = 4 2ǫ dimensions, the integral in eq. (7) is performed in 2 2ǫ
− −
dimensions, yielding
1 µ2 ǫ
α(t) = g2α(1)(t) = 2g2N c , (8)
c Γ
ǫ t!
−
with
1 Γ(1+ǫ)Γ2(1 ǫ)
c = − . (9)
Γ (4π)2−ǫ Γ(1 2ǫ)
−
In order to go beyond the LL approximation and to compute the one-loop corrections
to the Lipatov vertex, we need a prescription that allows us to disentangle the virtual
corrections to the Lipatov vertex (4) from the corrections to the vertices (3) and the
corrections that reggeize the gluon (6). Such a prescription is supplied by the general
formof the high-energy scattering amplitude, arising froma reggeized gluonin the adjoint
representation of SU(N ) passed in the t - and t -channels [9]. Since only the dispersive
c 1 2
part of the one-loop amplitude contributes to the NLL BFKL kernel, we can use the
3
modified prescription below2. In the helicity basis of eq. (2) this is given by
DispMνaaaν′ad′bνb′νb′νb = 2s igfaa′cDispC−ggνaνa′(−pa,pa′) t1 sτ1 α(t1) (10)
h i 1 (cid:18) (cid:19)
× igfcdc′DispCνg(qa,qb) t1 sτ2 α(t2) igfbb′c′DispC−ggνbνb′(−pb,pb′) ,
h i 2 (cid:18) (cid:19) h i
where
α(t) = g2α(1)(t)+g4α(2)(t)+O(g6)
Cg = Cg(0) +g2Cg(1) +O(g4) (11)
Cgg = Cgg(0) +g2Cgg(1) +O(g4),
are the loop expansions for the reggeized gluon, the Lipatov vertex, and the helicity-
conserving vertex, respectively. In the NLL approximation to the BFKL kernel it is
necessary to compute α(2)(t), Cg(1), and Cgg(1); however, to one loop only Cg(1) and Cgg(1)
appear. Expanding eq. (10) to O(g5) and using eq. (2), we obtain
s s
DispMaa′dbb′ = Mtree 1 + g2 α(1)(t )ln 1 +α(1)(t )ln 2
νaνa′ννb′νb 5 ( " 1 τ 2 τ
+ DispC−ggν(a1ν)a′(−pa,pa′) + DispC−ggν(b1ν)b′(−pb,pb′) + DispCνg(1)(qa,qb) . (12)
gg(0) gg(0) g(0)
C−νaνa′(−pa,pa′) C−νbνb′(−pb,pb′) Cν (qa,qb) #)
Thus, the NLL corrections to Cg(1) can be extracted from the one-loop gg ggg
→
amplitude, by subtracting the one-loop reggeization (8) and the one-loop corrections
to the helicity-conserving vertex. The latter has been computed in the HV and CDR
schemes [9, 14] and in the dimensional-reduction scheme [14] and is given by
DispCgg(1)( p,p′) µ2 ǫ 2 1 τ 32 δ π2
−+ − = c N + ln R +
Cgg(0)( p,p′) Γ( t! " c −ǫ2 ǫ t − 9 − 6 2 !
−+ − − −
5 β β
0 0
+ N , (13)
f
9 − 2ǫ#− 2ǫ)
with β = (11N 2N )/3 and the regularization scheme (RS) parameter
0 c f
−
1 HV or CDR scheme,
δ = (14)
R
( 0 dimensional-reduction scheme.
2The general form of the amplitude given by the exchange of reggeized gluons does not hold for the
absorptivepartofthe one-loopamplitudes,becauseothercolorstructuresoccurinthe high-energylimit.
This has been shown for the four-gluon one-loop amplitude in ref. [14] and it is shown for the five-gluon
one-loop amplitude in appendix C.
4
The last term in eq. (13) is the modified minimal subtraction scheme MS ultraviolet
counterterm. Note that eq. (13) differs from the result in Ref. [14] by the logarithm term,
because in that paper the reggeization scale had been taken to be τ = t.
−
3 The one-loop five-gluon amplitude
The color decomposition of a tree-level multigluon amplitude in a helicity basis is [22]
Mtree = 2n/2gn−2 tr(λdσ(1) λdσ(n))m (p ,ν ;...;p ,ν ), (15)
n ··· n σ(1) σ(1) σ(n) σ(n)
SnX/Zn
where d ,...,d , and ν ,...,ν are respectively the colors and the polarizations of the
1 n 1 n
gluons,theλ’sarethecolormatrices3 inthefundamentalrepresentationofSU(N )andthe
c
sumisover thenoncyclic permutationsS /Z oftheset [1,...,n]. Wetakeallthemomenta
n n
as outgoing, and consider the maximally helicity-violating configurations ( , ,+,...,+)
− −
for which the gauge-invariant subamplitudes, m (p ,ν ;...;p ,ν ), assume the form [22],
n 1 1 n n
p p 4
i j
m ( , ,+,...,+) = h i , (16)
n
− − p p p p p p
1 2 n−1 n n 1
h i···h ih i
where i and j are the gluons of negative helicity. The configurations (+,+, ,..., ) are
− −
then obtained by replacing the pk products with [kp] products. We give the formulae for
h i
these spinor products in appendix A. Using the high-energy limit of the spinor products
(49), the tree-level amplitude for gg ggg scattering may be cast in the form (2).
→
The color decomposition of one-loop multigluon amplitudes is also known [23]. For
five gluons it is,
M1−loop = (17)
5
25/2g5 tr(λdσ(1)λdσ(2)λdσ(3)λdσ(4)λdσ(5))m (σ(1),σ(2),σ(3),σ(4),σ(5))
5:1

SX5/Z5

+ tr(λdσ(1)λdσ(2))tr(λdσ(3)λdσ(4)λdσ(5))m (σ(1),σ(2);σ(3),σ(4),σ(5)) ,
5:3

S5/XZ2×Z3

where σ(i) is a shorthand for p ,ν in the subamplitudes. The sums are over the
σ(i) σ(i)
permutations of the five color indices, up to cyclic permutations within each trace. The
3Note that eq.(15) differs by the 2n/2 factor from the expression given in ref.[22], because we use the
standard normalization of the λ matrices, tr(λaλb)=δab/2.
5
string-inspired decomposition of the m subamplitudes [17] is
5:1
m = N Ag +(4N N )Af +(N N )As, (18)
5:1 c 5 c − f 5 c − f 5
where Ag, Af, and As get contributions from an N = 4 supersymmetric multiplet, an
5 5 5
N = 1 chiral multiplet, and a complex scalar, respectively. Also, we have
Ax = c m (Vx +Gx) x = g,f,s. (19)
5 Γ 5
For the NLL BFKL vertex we need the five-gluon one-loop subamplitudes only in
the helicity configurations which are nonzero at tree level. We write the functions for
the (1,2,3,4,5) color order for the two relevant helicity configurations below. The func-
tions obtained from the N = 4 multiplet, Vg and Gg, are the same for both helicity
configurations [17],
1 5 µ2 ǫ 5 s s 5 δ
Vg = + ln − j,j+1 ln − j+2,j−2 + π2 R ,
−ǫ2 jX=1 −sj,j+1! jX=1 −sj+1,j+2! −sj−2,j−1! 6 − 3
Gg = 0. (20)
The other functions depend on the helicity configuration. We define
I = [ij] jk [kl] li . (21)
ijkl
h i h i
For the (1−,2−,3+,4+,5+) helicity configuration we have [17],
5 1 µ2 µ2
Vf = ln +ln 2
−2ǫ − 2 " s23! s51!#−
− −
I +I s
Gf = 1234 1245L − 23 (22)
0
2s s s
12 51 (cid:18)− 51(cid:19)
Gf I I (I +I ) s
Gs = + 1234 1245 1234 1245 L − 23
− 3 3s3 s3 2 s
12 51 (cid:18)− 51(cid:19)
I2 s I I
+ 1235 1 35 + 1234 1245 ,
3s2 s s − s 6s2 s s
12 23 51 (cid:18) 12(cid:19) 12 23 51
while for the (1−,2+,3−,4+,5+) helicity configuration, we have
5 1 µ2 µ2
Vf = ln +ln 2
−2ǫ − 2 " s34! s51!#−
− −
I +I s I I s s
Gf = 1325 1342L − 34 + 1324 1342Ls − 23, − 34
− 2s s 0 s s2 s2 1 s s
13 51 (cid:18)− 51(cid:19) 13 51 (cid:18)− 51 − 51(cid:19)
6
I I s s
1325 1352 12 51
+ Ls − , − (23)
s2 s2 1 s s
13 34 (cid:18)− 34 − 34(cid:19)
I2 I2 s s s s
Gs = 1324 1342 2Ls − 23, − 34 +L − 23 +L − 34
−s4 s2 s2 1 s s 1 s 1 s
13 24 51 (cid:20) (cid:18)− 51 − 51(cid:19) (cid:18)− 51(cid:19) (cid:18)− 51(cid:19)(cid:21)
I2 I2 s s s s
1325 1352 2Ls − 12, − 51 +L − 12 +L − 51
−s4 s2 s2 1 s s 1 s 1 s
13 25 34 (cid:20) (cid:18)− 34 − 34(cid:19) (cid:18)− 34(cid:19) (cid:18)− 34(cid:19)(cid:21)
2I3 I s 2I3 I s
+ 1324 1342L − 23 + 1352 1325L − 12
3s4 s s3 2 s 3s4 s s3 2 s
13 24 51 (cid:18)− 51(cid:19) 13 25 34 (cid:18)− 34(cid:19)
1 s I I (I +I ) I3 I I3 I
+ L − 34 1325 1342 1325 1342 [phone] [phone]
3s351 2(cid:18)−s51(cid:19)"− s313 s413s24 s413s25 #
I +I s I2 I2
+ 1325 1342L − 34 + 1325 1342
6s s 0 s 3s4 s s s s
13 51 (cid:18)− 51(cid:19) 13 23 51 34 12
I2 I2 I2 I2 I I
+ 1324 1342 + 1325 1352 1342 1325 ,
3s4 s s s s 3s4 s s s s − 6s2 s s
13 23 24 34 51 13 25 12 34 51 13 34 51
with the functions L ,L ,L ,Ls defined in Appendix B. For both the helicity configura-
0 1 2 1
tions above, the functions Vs and Vf are related by,
Vf 2
Vs = + . (24)
− 3 9
In the expansion in ǫ, eq. (20-24) are valid to O(ǫ0). The amplitude (17) defined in terms
of eq. (18-24) is MS regulated. Using eq. (19, 20, 24), we can write the m subamplitude
5:1
(18) as the sum of a universal piece, which is the same for both helicity configurations,
and a non-universal piece, which depends on the helicity configuration,
m = mu +mnu , (25)
5:1 5:1 5:1
with
mu = c m N Vg, (26)
5:1 Γ 5 c
2
mnu = c m β Vf +(4N N )Gf +(N N ) Gs + .
5:1 Γ 5 0 c − f c − f 9
(cid:20) (cid:18) (cid:19)(cid:21)
In addition,
1
m (4,5;1,2,3) = m (σ(1),σ(2);σ(3),σ(4),5) , (27)
5:3 5:1
N
c COPX(1,2,3)
4
where only the N -independent, unrenormalized contributions to m are included [17]
f 5:1
(1,2,3)
and COP denotes the subset of permutations of S that leave the ordering of (1,2,3)
4 4
unchanged up to a cyclic permutation [23].
7
4 The one-loop corrections to the Lipatov vertex
To obtain the next-to-leading logarithmic corrections to the Lipatov g∗g∗ g vertex,
→
we need the amplitude M1−loop(B−,A−,A′+,k+,B′+) in the high-energy limit. We must
5
consider each of the color orderings in eq. (17) and expand the subamplitudes (18) in
powers of t/s, retaining only the leading power, which yields the leading and next-to-
leading terms in ln(s/t). In fact, at NLL we only need to keep the dispersive parts of the
subamplitudes m and m . By direct inspection of eq. (19-24) it is straightforward to
5:1 5:3
show that if a given color ordering of m is suppressed by a power of t˜/s˜ at tree-level,
5
where t˜= t , t , k 2 and s˜= s, s , s , then the corresponding color ordering of m will
1 2 ⊥ 1 2 5:1
| |
also be suppressed at one-loop.
For the Mtree amplitude in the multi-Regge kinematics, the leading color orderings
n
are obtained by untwisting the color flow, in a such a way to obtain a double-sided color-
flow diagram, and by retaining only the color-flow diagrams which exhibit strong rapidity
orderings of the gluons on both sides of the diagram, without regard for the relative
rapidity ordering between the two sides [18, 24]. Easy combinatorics then show that
there are 2n−2 such color orderings, and because of the reflection and cyclic symmetries
of the subamplitudes only 2n−3 need to be determined, e.g. all the ones which have at
least (n 2)/2 gluons on one side of the color-flow diagram. For the Mtree amplitude,
− 5
and thus for m , we have eight leading color orderings, out of which four need to be
5:1
determined, and we can choose them to be (A−,A′+,k+,B′+,B−), (A−,A′+,k+,B−,B′+),
(A−,A′+,B′+,B−,k+) and (A−,k+,B′+,B−,A′+). The other four leading subamplitudes
are then obtained by taking the ones above in reverse order, which yields an overall minus
sign.
For positive values of the invariants s we use the prescription ln( s ) = ln(s ) iπ.
ij ij ij
− −
Thus, in the multi-Regge kinematics and at NLL the dispersive part of the universal
piece (20) becomes, using the spinor products (49),
1 µ2 ǫ µ2 ǫ µ2 ǫ
DispVg = 2 +2 +
−ǫ2 " t1! t2! k⊥ 2! #
− − | |
2 µ2 ǫ µ2 ǫ 1 t δ 4
+ y + y ln2 1 R + π2, (28)
1 2
ǫ " t1! t2! #− 2 t2 − 3 3
− −
where we have written the leading logarithms in terms of the physical rapidity intervals
y = ln(s /√ t k ) and y = ln(s /√ t k ). The non-universal piece (26) becomes,
1 1 1 ⊥ 2 2 2 ⊥
− | | − | |
after rewriting all the phases in terms of q q∗ and some algebraic manipulation,
a⊥ b⊥
mnu σ(B−),σ(A−),σ(A′+),σ(k+),σ(B′+)
5:1
(cid:16) (cid:17)
8
= m σ(B−),σ(A−),σ(A′+),σ(k+),σ(B′+) c
5 Γ
(cid:16)3β β µ2 ǫ µ2 ǫ 64(cid:17) 10
0 0
+ N + N
c f
× (−2 ǫ − 2ǫ " t1! t2! #− 9 9
− −
β L (t /t )
0 (t +t +2q q∗ ) 0 1 2 (29)
− 2 1 2 a⊥ b⊥ t
2
N N L (t /t ) q q∗
+ c − f k 2 [2t t +(t +t +2 k 2)q q∗ ] 2 1 2 a⊥ b⊥ ,
3 | ⊥| "− 1 2 1 2 | ⊥| a⊥ b⊥ t32 − 2t1t2 #)
wherethefirst termistheMSultravioletcounterterm, andwherethepermutations σ span
the eight leading color orderings. Combining eq. (28) and (29), we see that the dispersive
parts of the leading m subamplitudes are all proportional to the corresponding tree-
5:1
level subamplitudes. Therefore, by the tree-level U(1) decoupling equations [23, 25] the
m subamplitudes vanish,
5:3
Dispm σ(B−),σ(A−),σ(A′+),σ(k+),σ(B′+) = 0+O(t/s). (30)
5:3
(cid:16) (cid:17)
Thus, we conclude that the dispersive part of the one-loop five-gluon amplitude is simply
proportional to the tree amplitude to leading power in t/s. Combining eq. (28) and (29),
it is given to O(ǫ0) by
DispM1−loop(A−,A′+,k+,B′+,B−) = Mtree(A−,A′+,k+,B′+,B−)g2c
5 5 Γ
1 µ2 ǫ µ2 ǫ µ2 ǫ
N 2 +2 +
× ( c "−ǫ2 " t1! t2! k⊥ 2! #
− − | |
2 µ2 ǫ µ2 ǫ 1 t δ 4
+ y + y ln2 1 R + π2
1 2
ǫ " t1! t2! #− 2 t2 − 3 3 #
− −
3β β µ2 ǫ µ2 ǫ 64 10
0 0
+ N + N (31)
c f
−2 ǫ − 2ǫ " t1! t2! #− 9 9
− −
β L (t /t )
0 (t +t +2q q∗ ) 0 1 2
− 2 1 2 a⊥ b⊥ t
2
N N L (t /t ) q q∗
+ c − f k 2 [2t t +(t +t +2 k 2)q q∗ ] 2 1 2 a⊥ b⊥ .
3 | ⊥| "− 1 2 1 2 | ⊥| a⊥ b⊥ t32 − 2t1t2 #)
Note that only the real part of this amplitude contributes to the NLL corrections to the
BFKL equation. It can easily be obtained using Re(q q∗ ) = (t +t + k 2)/2.
a⊥ b⊥ − 1 2 | ⊥|
Using eq. (8, 12, 13) and eq. (31), we can extract the NLL corrections to the Lipatov
vertex to O(ǫ0)
DispCg(1)(q ,q ) 1 µ2 ǫ τ 1 µ2 ǫ τ
ν a b = c N ln + ln
Cνg(0)(qa,qb) Γ( c"ǫ −t1! |k⊥|2 ǫ −t2! |k⊥|2
9