Table Of ContentComments on the 2nd order bootstrap relation
M.A.Braun
Department of high-energy physics, University of S. Petersburg
9 February 1, 2008
9
9
1
n Abstract. The 2nd order bootstrap relation is discussed in view of the recent critics by F.Fadin,
a
J R.Fiore y A.Papa. It is shown that the strong bootstrap condition and the anzatz to solve it used in
9
our earlier paper are valid at least for the quark part of the next-to-leading contribution.
2
1
v 1 Introduction.
7
4
4
The bootstrap relation plays a key role in the derivation of the BFKL equation up to the 2nd order
1
0 in α . It guarantees that production amplitudes with the gluon quantum number in their t channels
s
9
used for the construction of the absorptive part are indeed given only by a single reggeized gluon
9
/
h exchange and do not contain admixture from two or more reggeized gluon exchanges. The bootstrap
p
relation is known to be satisfied in the lowest order in the coupling constant. Recently the 2nd order
-
p
bootstraprelationwasdiscussedin[1]. Inastrongerformit wasusedin[2]to obtainthe potentialin
e
h thet-channelwiththegluoncolourquantumnumber. Thisapproachwaslatelycritisizedin[3],where
:
v
it was claimed that the strong bootstrap condition used in [2] was not fulfilled and the anzatz used
i
X
to solve it was incorrect. In this note we demonstrate that these objections are totally unfounded.
r
a They are a result of a misinterpretation of the potential used in [2], which is a different quantity as
comparedtothekernelusedin[1,3]. Wealsoderivethe2ndorderbootstraprelationof[1]inasimpler
way and comment on its implication for particle-reggeonscattering amplitudes.
2 Formalism
To introduce the bootstrap relation in a general form it is convenient to use an operator formalism
for the (non-forward) two gluon equation. Let the two gluons with momenta q be described by a
1,2
wave function Ψ(q ,q ). The total momentum q =q +q will always be conserved, so that in future
1 2 1 2
q = q q and the dependence on q and thus q will be suppressed. To facilitate comparison with
2 1 2
−
[1,3] we introduce a metric in which the scalar product of two wave functions is given by
Ψ Ψ dD−2q Ψ∗(q )Ψ(q ), (1)
1 2 1 1 1
h | i≡
Z
where D = 4+2ǫ, ǫ 0, is the dimension used to regularize all the following expressions in the
→
infrared region.
1
2
In this metric, up to order α3, the absorptive part of the scattering amplitude coming from the
s
two-reggeized-gluonexchange is given by
= Φ G(E)Φ (2)
j p t
A h | | i
Here the functions Φ are the so called impact factors which represent the coupling of the external
p,t
particles (t from the target and p from the projectile) to the two exchanged gluons. They are low
energy particle-reggeon scattering amplitudes. They have order α = g2/4π. The Green function
s
G(E) with E =1 j is defined as
−
G(E)=(H E)−1 (3)
−
where H is the two gluon(Hermithean) Hamiltonian. Its explicit expressionin the momentum repre-
sentation is
q H q′ = δD−2(q q′)(ω(q )+ω(q )) V(R)(q ,q′) (4)
h 1| | 1i − 1− 1 1 2 − 1 1
where 1+ω(q ) is the Regge trajectory of the first (second) gluon and V(R) represents the gluonic
1(2)
interaction for the t-channel with the colour quantum number R. In the lowest (1st) order in α one
s
has [4]
ω(1)(q)= aq2 χχ (5)
− h | i
where
χ(q )=1/ q2q2, (6)
1 1 2
q
g2N
a= (7)
2(2π)D−1
and N is the number of colours. As to the gluonic interaction, in the 1st order
ac q2q′2+q2q′2
V(R)(q ,q′)= R 1 2 2 1 q2 (8)
1 1 q2q2q′2q′2 k2 − !
1 2 1 2
q
′
with k = q q (the 1st denominator appears due to our metric (1)) The coefficient c depends
1 − 1 R
on the colour quantum number of the t-channel. For the vacuum channel (R = v) c = 2. For the
v
channel with the gluon colour quantum number (R=g) c =1.
g
In any colour channel the Green function G(E) can be represented via the solutions of the homo-
geneous Schroedinger equation
HΨ =E Ψ . (9)
n n n
These solution may be taken orthonormalized
Ψ Ψ =δ (10)
m n mn
h | i
(of course index n may be continuous). They are assumed to form a complete set
Ψ Ψ =1. (11)
n n
| ih |
n
X
In our metric the last equation means in the momentum space
Ψ (q )Ψ∗(q′)=δD−2(q q′). (12)
n 1 n 1 1− 1
n
X
3
In terms of the eigenfunctions Ψ
n
Ψ Ψ
n n
G(E)= | ih | (13)
E E
n
n −
X
and as a consequence the absorptive part is expressed as
j
A
Φ Ψ Ψ Φ
p n n t
= h | ih | i. (14)
j
A E E
n
n −
X
Note that sometimes it is convenient to consider not the absorptive part of the amplitude for real
particlesbutamplitudesforparticle-reggeonscattering. Onecanevidentlydefinetwosuchamplitudes
Ψ depending on which real particle is taken, from the projectile or target. Formally we define
p,t
Ψ (E)=G(E)Φ . (15)
t t
| i
It satisfies an inhomogeneous Schroedinger equation
(H E)Ψ (E)=Φ . (16)
t t
−
The absorptive part can then be expressed as
= Φ Ψ (E) . (17)
j p t
A h | i
In terms of the eigenfunctions Ψ one can express
n
Ψ Ψ Φ
n n t
Ψ (E)= h | i. (18)
t
E E
n
n −
X
The other function Ψ (E) can be introduced in a similar manner.
p
3 The bootstrap conditions
Now we concentrate on the t-channel with a colour quantum number of the gluon (R = g). The
ideology of the BFKL equation is based on the idea that physical amplitudes in this channel have
the asymptotical behaviour corresponding to an exchange of a single reggeizedgluon. The important
requirement is thus that no contribution should come from the exchange of two or more reggeized
gluons(intheleadingandnext-to-leadingorders). Thisrequirementseemstobeautomaticallyfulfilled
due to the Gribov’s rule of signature conservation. Indeed the two-gluon exchange should have an
overallpositivesignatureandthuscannotcontributetotheamplitudewiththetchannelcorresponding
to the gluon colour quantum number, which has a negative signature. The bootstrap relation is just
a technical tool to implement this requirement.
In order that the amplitude should have the asymptotic behaviour corresponding to a single
reggeized gluon exchange without admixture of any other terms, its absorptive part, considered as a
functionofthe complexangularmomentumj, shouldhavea simplepole atj =1+ω(q)andno other
singularities. From the representation (13 ) we conclude that
First, the spectrum of H in the considered t channel should contain an eigenvalue E = ω(q)
0
−
with the corresponding eigenfunction Ψ
0
(H +ω(q))Ψ =0 (19)
0
4
Second, in the sum over n only the contribution of this eigenfunction Ψ should be present. All
0
other terms should be equal to zero. Strictly speaking, this means that both impact factors Φ
p,t
shouldbe orthogonalto all other eigenfunctions Ψ , n>0. Due to the completeness of the whole set
n
it means that both Φ and Φ should coincide with Ψ , up to a normalization factor.
p t 0
This secondconditionlooksto be verystringent. Note that it shouldbe fulfilled for any projectile
and/ortarget. So, literally understood, it means that coupling of any particle to a pair of reggeonsis
essentiallythesamefunctionofthereggeonicmomenta. Asweshallseebelow,infact,thisconditionis
farweaker,duetothefactthatallourargumentsarelimitedtothetwofirstordersoftheperturbation
theory.
There is finally a third bootstrap condition which is a relation for the impact factor. With only
the state Ψ contributing we get from (13)
0
Φ Ψ Ψ Φ
p 0 0 t
= h | ih | i (20)
j
A j 1 ω(q)
− −
Integratingthisoverjwithapropersignaturefactorwefindthecorrespondingamplitudeasafunction
of s
q2 ω(q) Φ Ψ Ψ Φ s ω(q) s ω(q)
p 0 0 t
(s)= s h | ih | i − + (21)
A − s sinπω(q) q2 q2
(cid:18) 0(cid:19) "(cid:18) (cid:19) (cid:18) (cid:19) #
This should be compared with the standard form of the contribution of a Reggeized gluon, with a
scale given by q2,
s s ω(q) s ω(q)
(s)= Γ (q)Γ (q) − + (22)
A − p t q2 "(cid:18)q2 (cid:19) (cid:18)q2(cid:19) #
Here Γ’s represent the coupling of the projectile or target particles to the exchanged reggeized gluon
(”particle-particle-Reggeon vertices”, in the terminology of [1]). Comparing (21) and (22) we find a
relation which should be satisfied both for the projectile and target
q2 ω(q)/2 ΦΨ
0
Γ(q)=q h | i (23)
(cid:18)s0(cid:19) |sinπω(q)|
In fact the impact factor Φ itself involves the vertex Γp(in the lowest order it is completely expressed
through it). So Eq. (23) is a non-trivial condition to be satisfied for the coupling of any particle to
the reggeized gluon. It is the third bootstrap condition.
4 Leading and next-to-leading orders
It is well known that the bootstrap conditions described in the preceding section are fulfilled in
the leading order (LO) [5]. Indeed the homogeneous LO Schroedinger equation has an eigenvalue
E(1) = ω(1)(q) with the corresponding eigenfunction Ψ(0) = cχ (Eq. (6), c = 1/ χχ is the
0 − 0 h | i
normalization factor): p
H(1)χ= ω(1)(q)χ (24)
−
(upper indeces always mark the order in α ). One easily checks that (24) is true with the explicit
s
expressions (4)– (8) for the Hamiltonian and χ. Due to the trivial form of the latter Eq. (24) is in
fact a simple relation between the gluon trajectory and the gluonic interaction integrated over one of
its momenta.
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It can also be trivially shown that in the lowest order the impact factors for any external particle
reduce to χ as required. Indeed (see[1]) in the LO for any particle
√N
Φ= i Γχ. (25)
− 2
From this one gets
√N √πω(1)
Ψ(0) Φ = i Γ χχ = Γ (26)
h 0 | i − 2 h | i q
p
wherewehaveused(5). Puttingthisin(23)wefindthatalsothethirdbootstrapconditionissatisfied
in the LO, when we can neglect the power factor and take sinπω πω
≃
Now we pass to the next-to-leading order (NLO). Again we have to fulfil three different require-
ments. First, the spectrum of the Hamiltonian should include the gluon trajectory also in the NLO,
Eq. (19) We present the corresponding eigenfunction as
Ψ =cχ+Ψ(1) (27)
0 0
where χ is the known LO eigenfunction and the second term is the (unknown) NLO contribution.
Separating perturbation orders in the Hamiltonian and in the eigenvalue E = ω(q) and taking the
0
−
NLO we get an equation for Ψ(1)
0
(H(1)+ω(1)(q))Ψ(1) = c(H(2)+ω(2)(q))χ (28)
0 −
For this equation to be soluble, the inhomogeneous term should be orthogonal to the solution of the
corresponding homogeneous equation, that is, to χ, recall Eq. (24). So we get a condition
χH(2)+ω(2)(q)χ =0 (29)
h | | i
With χ a known function, this equation is in fact a relation between the gluonic NLO interaction in
the colour channel R = g and the NLO Regge trajectory, integrated over the initial and final gluon
relative momenta. This relation was first obtained in [1] from different arguments. Note that in
contrast to the Eq. (19), valid for any q and q , relation (29) is only an identity in q. So its contents
1
is much weaker than that of (19). For this reason we shall call it a weak bootstrap contition, leaving
the definition of the strong bootstrap condition for the equations (24) in the LO and (19) in the LO
and NLO.
Nowwecometothesecondcondition,whichisthatalsointheNLOthereshouldbenocontribution
to the absorptive part other than from the gluonic Regge pole. However this condition is trivially
staisfied in the NLO. Indeed matrix elements
Φ Ψ , n=0
p.t n
h | i 6
with eigenfunctions different from Ψ are all zero in the lowest order. So they have at least order α .
0 s
The representation (13) contains products of two such matrix elements. So the part of (13) which
comesfromthe states otherthanΨ isatleasttwoorderssmallerthanthe leadingterm(ofthe order
0
α2). Thereforeit does not contribute in the NLO at all. Thus in the NLO, irrespective ofthe formof
s
the impact factors, there is no contribution to the absorptive part of the amplitude from states other
than a single reggeized gluon.
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It is remarkable that this is only true for the absorptive part of the scattering amplitude for
real particles. If one considers the particle- reggeon amplitudes instead, then admixture of other
intermediate states, apart from a single reggeized gluon, seems possible. Indeed take Ψ (E) as an
t
example. One immediately sees from Eq. (18) that terms with n = 0 will generally give a nonzero
6
contribution in the NLO unless the impact factor coincides with the eigenfunction Ψ also in the
0
the NLO (which seems improbable for every possible externalparticle). This contributiondisappears
whenonetakesthe productin (17)due to orthogonalityoftheeigenfunctionwithn=0to allothers.
However in the particle-reggeon amplitude Ψ (E) itself the NLO contribution from higher gluonic
t
states may be present.
This circumstance makes one think about the presence of such states in the reggeon-reggeon
amplitudes which enter the unitarity relation for the absorptive part of the initial amplitude. Should
such states be present, it would invalidate the derivation of the BFKL equation in the NLO.
The form which takes the third bootstrap condition (23) in the NLO was obtained in [1]. As far
as we know it has not been checked for any particle so far.
5 Solution of the strong bootstrap condition
In this section we are going to demonstrate that the anzatz proposed in [6] and used in [2] actually
solves the strong bootstrap relation (19) in the LO and NLO orders. The ansatz introduces a single
function η(q) through each both the potential and the gluon Regge trajectory are expressed. In [2]
we used an unsymmetric potential W defined as follows
η(q ) η(q ) 1 η(q)
′ 1 2
W(q ,q )= + (30)
1 1 η(q′) η(q′) η(k) − η(q′)η(q′)
(cid:18) 1 2 (cid:19) 1 2
The trajectory is expressed via η(q) as
η(q)
ω(q)= dD−2q (31)
1
− η(q )η(q )
Z 1 2
This anzatz guarantees fulfilment of the bootstrap relation in the form
dD−2q′W(q ,q′)=ω(q) ω(q ) ω(q ) (32)
1 1 1 − 1 − 2
Z
In the LO one has
η(0)(q)=q2/a (33)
(see Eq. (7)).
TopasstothesymmetricpotentialV onehasfirstseparatefromW theintegrationmeasurefactor
1/(η(q′)η(q′)) and then symmetrize substituting this factor for 1/ η(q )η(q )η(q′)η(q′). This gives
1 2 1 2 1 2
the desired relation p
η(q )η(q )
′ 1 2 ′
W(q ,q )= V(q ,q ) (34)
1 1 sη(q1′)η(q2′) 1 1
The bootstrap relation (32) then transforms into
1 1
dD−2q′V(q ,q′) =(ω(q) ω(q ) ω(q )) (35)
1 1 1 η(q′)η(q′) − 1 − 2 η(q )η(q )
Z 1 2 1 2
p p
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This is nothing but the equation (19) for the gluon Regge pole solved by
1
Ψ (q )=c (36)
0 1
η(q )η(q )
1 2
So our anzatz just solves the strong bootstrap relaption.
Tocompareourresultswith[3]wehavefinallytorelateoursymmetricpotentialV totheirreducible
kernel K used in [1,3]. The latter is defined in respect to the metric with a factor q2q2 in the
r 1 2
denominator. Therefore identification with V requires passing to our metric. We then find
′
K (q ,q )
V(q ,q′)= r 1 1 (37)
1 1
q2q2q′2q′2
1 2 1 2
q
Combining (34) and (37) we find the final relation of the irreducuble kernel K of [1,3] and the
r
unsymmetric potential W of [2]
η(q′)η(q′)
K (q ,q′)= q2q2q′2q′2 1 2 W(q ,q′) (38)
r 1 1 1 2 1 2 sη(q1)η(q2) 1 1
q
This relationhasnot, inallprobability,beentakeninto accountin[3]when discussingourresults. In
the following section we shall show that with the form of η(q) matched to the quark contribution to
the gluon Regge trajectory one obtains the correct form of the quark contribution to the kernel K .
r
6 The quark contribution to the trajectory and potential
We are going to show that, first, the quark contribution to the gluon Regge trajectory has the form
implied by our anzatz (31). We shall determine the partof η(q) coming from this contribution. Then
using (30) and (38)) we shall find the correspondingpart of the irreducible kernel K and compare it
r
with the results of [3].
The part of the gluon trajectory which comes from quark is given by the expression [3]
dD−2q
ω(2) =2bq2 1(q2ǫ q2ǫ q2ǫ), (39)
Q q2q2 − 1 − 2
Z 1 2
where
g4NN Γ(1 ǫ)Γ2(2+ǫ)
F
b= − . (40)
(2π)D−1(4π)2+ǫǫΓ(4+2ǫ)
On the other hand, presenting
η(q)=η(0)(q)(1+ξ(q)) (41)
we have in the NLO from (31)
dD−2q
ω(2)(q)= aq2 1(ξ(q) ξ(q ) ξ(q )) (42)
− q2q2 − 1 − 2
Z 1 2
As we observe, the form of (39) follows this pattern. Comparing (39) and (42) we identify
2b
ξ (q)= q2ǫ (43)
Q
− a
Now we pass to the irreducible kernel K . From (30) and (38) we express it via ξ(q) as
r
1 1 q2q′2
K(2)(q ,q′)= a q2q2q′2q′2 1 2 (ξ(q )+ξ(q′) ξ(q′) ξ(q ) 2ξ(k))+
r 1 1 2 1 2 1 2 k2sq′2q2 1 2 − 1 − 2 −
q h 1 2
8
1 q2q′2 q2
2 1 (ξ(q )+ξ(q′) ξ(q ) ξ(q′) 2ξ(k)) (2ξ(q) ξ(q ) ξ(q ) ξ(q′) ξ(q′))
k2sq2′2q12 2 1 − 1 − 2 − − q12q22q1′2q2′2 − 1 − 2 − 1 − 2 i
q (44)
Putting here the found ξ(q), Eq. (43), we obtain the quark contribution
q2q′2
KQ(q ,q′)= b 1 2 (q2ǫ+q′2ǫ q2ǫ q′2ǫ 2k2ǫ)+
r 1 1 − k2 1 2 − 2 − 1 −
h
q2q′2
2 1 (q2ǫ+q′2ǫ q2ǫ q′2ǫ 2k2ǫ) q2(2q2ǫ q2ǫ q2ǫ q′2ǫ q′2ǫ) (45)
k2 2 1 − 1 − 2 − − − 1 − 2 − 1 − 2
i
Comparing this expression with the one found in [3] (Eq. (47) of that paper) we observe that they
areidentical. This meansthatourbootstrapconditionandthe anzatzto solveit arevalidatleastfor
the quark contribution in the NLO.
7 Conclusions
Comparing our bootstrap results [2] for the gluonic interaction in the gluon colour channel with the
directcalculations of its quark partin [3] we find a complete agreement. This leavesa certaindose of
optimism as to the total potential calculated in [2] being the correct one.
8 Acknowledgements
The author expresses his deep gratitude to Profs. Bo Anderson and G.Gustafson for their hospitality
during his stay at Lund University, where these comments were written.
9 References
1. V.S.Fadin and R.Fiore, hep-ph/9807472.
2. M.A.Braun and G.P.Vacca, hep-ph/9810454.
3. V.S.Fadin, R.Fiore and A.Papa, hep-ph/9812456.
4. V.S.Fadin, E.A.Kuraev and L.N.Lipatov, Phys. Lett. B60 (1975) 50.
5. L.N.Lipatov, Yad. Fiz. 23 (1976) 642.
6. M.A.Braun, Phys. Lett. B345 (1995) 155; B348 (1995) 190.
