Inclusive Gluon Production in the QCD Reggeon Field Theory: Pomeron Loops Included

0.28 MB·English

by  Tolga Altinoluk

PreprinttypesetinJHEPstyle-HYPERVERSION
Inclusive Gluon Production in the QCD Reggeon Field Theory:
9
0 Pomeron Loops Included.
0
2

n
a
J

6
1 Tolga Altinoluk1, Alex Kovner1 and Michael Lublinsky2

h] 1Physics Department, University of Connecticut, 2152 Hillside road, Storrs, CT 06269, USA
p 2Physics Department, State University of New York, Stony Brook, NY 11794, USA
-
p
e
h
[ Abstract: We continue the study of hadronic scattering amplitudes at high energy by system-

atically including nonlinear effects of finite partonic density in hadronic wave function as well as
1
v the effects of multiple rescatterings in the scattering process. In this paper we derive expressions
0
for a single inclusive gluon production amplitude and multigluon inclusive production amplitudes
6
5 when the rapidities of all observed gluons are not very different. We show that at leading order
2
these observables exhibit a semiclassical structure. Beyond the semiclassical result, we find that the
.
1
gluon emission has some characteristic features different from the JIMWLK and KLWMIJ limits in
0
9 that the gluons are not emitted independently in rapidity space, but have a correlated component
0
with correlation length (in rapidity space) of order one. We demonstrate the consistency between
:
v
this feature of the multigluon observables and the Hamiltonian of the QCD Reggeon Field Theory
i
X
(H ) derived in the companion paper [1]. We also show that the evolution of these observables
RFT
r
a with total rapidity of the process is generated by HRFT of [1]. We discuss whether this evolution is
equivalent to evolution with H as far as this set of observables is concerned.
JIMWLK
Contents
1. Introduction 1
2. High Energy Evolution as Seen From [1]. 3
2.1 Generalities 3
2.2 Definitions 5
2.3 H 7
RFT
3. The amplitudes Q and the scattering amplitude. 8
4. Inclusive gluon production. 11
4.1 Single gluon inclusive production. 11
4.2 Double and multi gluon inclusive amplitudes. 14
4.3 Evolution with rapidity 18
5. Discussion. 20
A. Appendix - Single gluon inclusive amplitude. 23
1. Introduction
This is the second of two papers devoted to the study of hadronic observables at high energy taking
into account the effects of Pomeron loops. In the first paper [1], we derived the Hamiltonian of
the QCD Reggeon Field Theory [2] which accounts for the effects of nonlinearities in the hadronic
wave function as well as multiple scattering effects in hadronic scattering, and thus fully includes
Pomeron loops. It supercedes the JIMWLK [3],[4],[5],[6] and KLWMIJ [7] evolutions which are
only applicable in parts of the parameter space. This Hamiltonian - H governs the evolution of
RFT
many hadronic observables with energy. In particular it directly gives the evolution of the forward
scattering amplitude. The derivation of [1] was based on the earlier results [8] for the evolution of
the hadronic wave function.
In this paper we extend our analysis to other observables. In particular we derive expressions
for the amplitude of inclusive gluon production at high energy. We consider single gluon production
as well as multi-gluon production when all the emitted gluons are close to each other in rapidity,
so that the rapidity evolution between them does not have to be considered. There has been a
– 1 –
lot of interest in the calculation of inclusive gluon production in the framework of the high energy
evolution. Starting with [9] which calculated single gluon production for DIS in the dipole model,
this observable (as well as multi-gluon production) was considered beyond the dipole model limit in
DIS in [10, 11, 12, 13, 14, 15, 16] and in nucleus-nucleus collisions in [17, 18, 19, 20]. Some, effects of
the Pomeron loops which do not extend to the rapidity of the measured gluons in DIS setting were
studied in [21].
In the present paper we extend these results by including the Pomeron loop effects using the
methods of [1]. Our calculation is applicable to both the DIS situation and the nucleus-nucleus
scattering. We follow the approach and techniques developed in [13, 14]. The extension of these
results to include evolution between the observed gluons is straightforward but we leave it for future
publication.
We note that very recently inclusive gluon production in nucleus-nucleus collision was discussed
in [20]. Our present work has many parallels with [20] even though the methods we use are quite
different. Our results are more complete in the sense that we provide an explicit form of the ob-
servables in terms of the color charge densities of the incoming hadrons (nuclei). It is not necessary
in our approach to solve dynamical equations of motion in order to find the outgoing classical color
fields, as is required by the procedure of [20]. All that is necessary is to average a function of the
incoming charge densities (albeit a complicated one) over the initial probability distribution. The
element of averaging over the initial distributions is also present in the formalism of [20].
We also derive a set of subleading corrections to the double inclusive and multi-gluon ampli-
tudes, which exhibit short range rapidity correlations between the gluons in the final state. These
correlations are formally subleading in α and are therefore not present in the analysis of [20]. In our
s
approach these terms are however under control. Since our derivation of the multi-gluon observables
is closely linked with the derivation of H we are able to show quite generally that the expression
RFT
for H derived in [1] necessarily requires the presence of such short range gluon correlations in
RFT
the multi-gluon observables. We stress that these are not correlations due to the evolution between
the rapidity of the gluons [12, 14], whose perturbative tail we also see in this calculation, but rather
correlations bunching (or anti-bunching) all emitted gluons inside a rapidity interval of order unity.
Our approach allows us to show immediately that the evolution of multi-gluon observables with
respect to ”global” rapidity parameters is given by the RFT Hamiltonian H derived in [1]. By
RFT
”global” rapidity parameters we mean either the total rapidity of the process at fixed gluon rapidity,
or the rapidity difference between the emitted gluons and the projectile; or between the emitted
gluons and the target; but not the rapidity differences between the emitted gluons themselves.
Finally we discuss the question to which extent this energy evolution of the multi-gluon inclusive
observables can be approximated by the JIMWLK evolution, as suggested in [20]. Formally it
appears that H can be replaced by H as long as one considers the evolution of this set
RFT JIMWLK
of observables, with corrections to H being suppressed by powers of α . However we give an
JIMWLK s
example of a situation where such a formal argument fails, and thus advise caution on the subject.
We believe this question merits further investigation.
We start with recapping the notations and some formulae from [1] which we will need for our
derivations. The next section does not contain any new results and is included purely in an attempt
– 2 –
to make the present paper self-contained in terms of notations and definitions. All the formulae
presented in this section are also found in [1]. Reader who is familiar with notations and results of
[1] can skip directly to Section 3.
2. High Energy Evolution as Seen From [1].
2.1 Generalities
We consider a hadronic projectile moving to the right with large energy. We separate the degrees of
freedom of the hadronic wave function into ”valence” - gluons with rapidity greater than some fixed
value, and ”soft” - with rapidities lower than this separation rapidity. The properties of the valence
component of the wave function are characterized by correlators of the color charge density operator
ja(x). The wave function of the hadron can be written as
Ψ = Ω[a,a ,j] v (2.1)
†
| i | i
where a and a are soft gluon creation and annihilation operators. The valence state v has no soft
†
| i
gluons and is therefore annihilated by the soft gluon annihilation operators
a v = 0.
| i
The evolution operator Ω is a unitary operator of the Bogoliubov type
Ω = . (2.2)
CB
Here is a coherent operator that creates the ”classical” Weiszacker-Williams field
C
d2k dη
C = exp i√2 (2π)2 bai(k) 2π [aai(η,k) + a†ia(η,−k)] . (2.3)
(cid:26) Z Z (cid:27)
and is a Bogoliubov type operator responsible for the leading quantum corrections. The rapidity
B p−
variable is defined as η = ln 0 and the creation and annihilation operators are canonical in the
p−
rapidity basis
[aai(η,k),a†jb(ξ,p)] = (2π)3δ(η−ξ)δ2(k−p). (2.4)
The Weiszacker-Williams field ba(x) depends only on transverse coordinates x and is a two
i
dimensional pure gauge field:
1
ba = fabcU bd∂ Udc (2.5)
i −g † i
where fabc are the structure constants of the SU(N) and Uab is an SU(N) group element in the
adjoint representation. The Weiszacker-Williams field is related to the valence color charge density
by
∂ ba(x) = ja(x) (2.6)
i i
The soft gluon operators a(η) are labeled by rapidity variable η as well as transverse coordinates (or
momenta), colorandrotationalindices. TheypopulatetherapidityintervalY equaltotheparameter
– 3 –
of the boost transformation that brings the wave function of the hadron from v to Ψ . We do not
| i | i
denote this cutoff explicitly in most of our formulae and extend the rapidity integration over η to
infinity for all quantities for which the integration converges. The Y dependence is important only
for divergent quantities and eventually this very dependence determines the evolution of physical
observables with rapidity.
Theexplicitformof isnotimportanthere,butitsaction onthegluoncreation andannihilation
B
operators is linear
βα = BaαB† = Θαβ[j] aβ + Φαβ[j] a†β, βα† = Ba†αB† = Θ∗αβ[j] a†β + Φ∗αβ[j]aβ (2.7)
It turns out to be useful to define separately the matrix
N(p,k) = Θ(p,k) Φ(p, k), (2.8)
− −
where we have indicated explicitly only the transverse momentum dependences. The transformation
matrices Θ, Φ and N depend on the valence color charge density and where calculated in [1]. The
explicit form of Θ and Φ is not important for us in this paper. The expression for N is
2i 2i
N(p,η;k,ξ) = p (1 2l)k p (1 2L) k (2.9)
e (ξ η) eξ η iǫh | − | i−h | − D2e (ξ η) ∂2eξ η +iǫ| i
− − − − − ∂2 − − − D2 −
where D is the transverse covariant derivative in the background filed b, and the longitudinal pro-
jectors l and L are defined as
ed
∂ ∂ 1
lab i jδab; Lab Dae(b) Ddb(b) (2.10)
ij ≡ ∂2 ij ≡ i D2(b) j
(cid:20) (cid:21)
The vacuum of the operator β as calculated in [1] is
0 β = 0 = e14Trln(1−Λ†Λ) e−12a†Λa† 0 (2.11)
| i B| i | i
with
Λ = Θ 1Φ. (2.12)
−
Eqs.(2.1,2.3,2.11) characterize the wave function of a hadron with rapidity Y. This hadron,
which we call the ”projectile” scatters on another hadron- the ”target”. The target is specified
by a distribution of color fields αa. Calculation of many interesting observables in the scattering
process, including multi-gluon inclusive amplitudes, can be represented as the double average over
the projectile and the target wave functions of some operator Oˆ
O¯ = DαWT[α] O¯; O¯ v Oˆ[j]v = DjWP[j]O[j]. (2.13)
T
h i ≡ h | | i
Z Z
Here WP[j] is a weight functionals representing the probability distributions of the projectile color
charge j, while WT[α] is the same for the target fields α. The rapidity evolution of this observable
is given by H (2.32) in the following sense [4, 1]
RFT
d δ
O¯ = Dj WP[j] H j, O[j]. (2.14)
RFT
dY − δj
Z (cid:20) (cid:21)
– 4 –
or equivalently
d δ
WP[j] = H j, WP[j]. (2.15)
dY Y − RFT δj Y
(cid:20) (cid:21)
The Hamiltonian H is formally defined as the expectation value of the unitary operator Rˆ in
RFT a
the hadronic wave function Ψ
| i
δ d
H j, = 0Ω [j,a,a ]Rˆ Ω[j,a,a ]0 . (2.16)
RFT † † a † Y=0
δj −dY h | | i|
(cid:20) (cid:21)
with
Rˆa e Rxjsaoft(x) δjaδ(x) (2.17)
≡
and
dη
ja (x) = g a b(η,x) Ta ac(η,x) (2.18)
soft 2π † bc
Z
The operator Rˆ shifts the valence color charge density by the charge density of the soft gluons. This
a
reflects the fact that in the boosted wave function not only valence, but also soft gluons participate
in the scattering, and thus the effective color charge density must include both j and j .
soft
H was calculated in [1]. To summarize the results of [1] we have to introduce (alas - at the
RFT
risk of being repetitive) some additional notations.
2.2 Definitions
First, since the color charge density operators do not commute, we need to keep track of the ordering
in eq.(2.16). As a result we have to define the right and left color charge density operators J and
R
J . The labels refer to their order in the calculation of the matrix element of O and the Hamiltonian
L
H . The operator Ω in eq.(2.16) has to be understood as a function of the right charge operator
RFT
J , while the operator Ω is a function of J . This is important, since observable O generically
R † L
depends on the color charge density, and in the calculation of its matrix element in Ψ it always
| i
appears sandwiched between Ω and Ω. Defining J and J specifies the relative ordering between
† R L
the color charge operators in the wave function and those in the observable. As shown in [22], the
rightandleftchargedensitiesarerelatedbytherotation withunitarymatrixR(heretistheordering
variable [7])
1 δ ab
Rab(x) = exp g dt Tc . (2.19)
P { δjc(x,t)
(cid:20) Z0 (cid:21)
The matrix R has the meaning of a scattering matrix of the gluon of the target on the projectile[22].
As discussed, for example in [23], it can be taken as the basic degree of freedom of the Reggeon Field
Theory. We can then write [22]
δ
gJa(x) = gtr R(x)Ta ; (2.20)
R − δR (x)
(cid:26) † (cid:27)
δ
gJa(x) = gtr TaR(x) ;
L − δR (x)
(cid:26) † (cid:27)
Ja(x) = [R(x)J (x)]a.
L R
– 5 –
Accordingly we define two classical fields, b and b :
R L
1 1
baRi = −g fabcU†bd[JR]∂iUdc[JR]≡ −g fabcUR†bd∂iURdc;
1 1
baLi = −g fabcU†bd[JL]∂iUdc[JL]≡ −g fabcUL†bd∂iULdc (2.21)
The matrices Θ, Φ and Λ also ”bifurcate” into Right and Left quantities.
Λ Λ[J ] = Θ 1[J ]Φ[J ]; Λ Λ[J ] = Θ 1[J ]Φ[J ]. (2.22)
R R − R R L L − L L
≡ ≡
It is thus clear that for the purposeof the calculation of the average of any observable the proper
ordering of the factors of the charge density is equivalent to the substitution
Ω† → Ω†L = BL† CL†; Ω → ΩR = CRBR (2.23)
where the subscript L (R) indicates that the respective operator depends on J (J ). Due to the
L R
presence of the operator Rˆ in eq.(2.16) (and in other similar averages) it turns out that the Left
a
quantities arealwaysrotatedbythematrixR. Itisthusconvenienttointroducethebarredquantities
Ω¯ ¯ ¯ ; ¯ Rˆ , ¯ Rˆ Rˆ . (2.24)
L L L L a† L L a† L a
≡ C B B ≡ B C ≡ C
and accordingly
¯ba R abbb ; Λ¯ R Λ R; N¯ = N R (2.25)
Li † Li L † L L L
≡ ≡
Finally we introduce another set of matrices
K = (ΘΘ¯ ΦΦ¯ ) ; E = (ΦΘ¯T ΘΦ¯T) ; (2.26)
αβ † † αβ αβ αβ
− −
Ξ = K 1E. (2.27)
−
which satisfy
KET EKT = 0, KK EE = 1; K K ETE = 1; K E ETK = 0; (2.28)
† † † ∗ † ∗
− − − −
1 1
= K K ; = KT K . (2.29)
† ∗
1 ΞΞ 1 Ξ Ξ
† †
− −
Above we have omitted the indices L and R. All the barred quantities depend on J while the
L
unbarred ones - on J . We will stick to this practice in the following, whenever it does not lead to
R
confusion.
Thematrices K,E andΞaretheanalogsoftheBogoliubov transformeq.(2.7) fortransformation
between the operators β ( defined by eq.(2.7) with Θ , etc.) and β¯ ( defined by eq.(2.7) with Θ¯ ,
R R L R
etc.). The matrices K and E, which we will need in the following have been calculated in [1]
K(p,η;k,ξ) = ihp|2(1−l−LR)R†(l−LL)+(1−l−LR)(∆¯ −∆†R)R†(1−l−LL)|ki
i 1
+ p (1 2l)R (1 2l)k i p (1 2L ) R (1 2L )k
1 eξ−η +iǫh | − † − | i− h | − R 1 DR2 eξ η iǫ † − L | i
− − D¯2 − −
E(p,η;−k,ξ) = ihp|2(1−l−LR)R†(l−LL)+(1−l−LR)(∆¯ −∆†R)R†(1−l−LL)|ki
i 1
+ p (1 2l)R (1 2l)k i p (1 2L ) R (1 2L )k (2.30)
1+eξ−ηh | − † − | i− h | − R 1+ DR2 eξ η † − L | i
D¯2 −
– 6 –
with
D¯ R D R; ∆¯ = R ∆ R. (2.31)
† L † L
≡
In eq. (2.30) ∆ is an operator in the transverse space, which we do not specify, since it does not
appear in the following, D and D are transverse covariant derivatives in the background fields b
R L R
and b respectively, and longitudinal projectors L are defined as in eq.(2.10). In eq.(2.30) all D
L R(L) R
are ordered to the left of D .
L
2.3 H
RFT
The result of [1] for H to leading order in α in the concise matrix notation is
RFT s
d
H = b(x) ¯b(x) N¯ (η,x;ξ,y)K 1(ξ,y;λ,z)N(λ,z;ζ,u) b(u) ¯b(u)
RFT † −
dY − −
( Zx,y,z,u Zη,ξ,λζ
(cid:2) (cid:3) (cid:2) (cid:3)
1
+ Tr ln (KK ) . (2.32)
† Y=0
4 |
)
The Tr ln term in this expression is subleading in α and will not be considered any further. While
s
b is independent of rapidity, both N and K depend on rapidity differences only. Introducing
N (x,y) N(x,η,y,ξ) = [1 l L] (2.33)
⊥ ≡ Zη−ξ − −
which will reappear in the next section and performing the rapidity integrations the result for the
first term in eq.(2.32) can be recast in the form
1
H = [b R b ] (1 l L )
RFT R † L L
π − − −
1
(1 2l)R (1 2l) + (1 2L )R (1 2L ) − (1 l L ) [b R b ].(2.34)
† R † L R R † L
× − − − − − − −
h i
Reinstating explicitly transverse coordinate dependences this reads
1 1 (x z) (x z)
H = [bb (x)R ba(x) ba (x)] δ 2 − i − j
RFT 8π3 Ri † − Li ij(x z)2 − (x z)4
Zx,y,z,z¯ (cid:20) − − (cid:21)
1 (y z¯) (y z¯)
× δac + [UL†(x)UL(z)]ac K˜−j1kcd(z,z¯) δkl(y z¯)2 − 2 −(y k z¯)4− l
h i ⊥ (cid:20) − − (cid:21)
× δde + [UR†(z¯)UR(y)]de [beRl(y) − R†ef(y)bfLk(y)] (2.35)
h i
with
1 1 (x z) (x z) 1 (z y) (z y)
K˜ ab(x,y) = δ 2 − i − k δ 2 − k − j
⊥ij 2π2 Zz(cid:20) ik(x−z)2 − (x−z)4 (cid:21) (cid:20) kj(z−y)2 − (z−y)4 (cid:21)
ab
× R†ab(z) + UR†(x)UR(z)R†(z)UL†(z)UL(y) (2.36)
(cid:26) (cid:27)
h i
With all the definitions in place we can now proceed to calculation of the multi-gluon spectrum.
– 7 –
3. The amplitudes Q and the scattering amplitude.
We willstartby rederivingtheresults of [1]ina slightly moregeneral way whichlater will allow usto
calculate additional observables. First, we remind the reader the general formalism of [23], which we
will use here. The matrix element eq.(2.16) pertinent to the calculation of H can be represented,
RFT
using the optical theorem as
1
01 Ω Rˆ Ω 0 = 0(1 Ω Rˆ Ω) (1 Ω Rˆ Ω)0 . (3.1)
h | − † a | i 2h | − † a† − † a | i
This assumes that the amplitude is real, which in our case can be verified directly by examining
eq.(2.34). Introducing the complete basis of intermediate states and defining the amplitudes
Q (x ,η ) = x ,η ;...;x ,η (1 Ω Rˆ Ω)0 = n Ω¯ Ω 0 (3.2)
n i i 1 1 n n † A †
h | − | i h | | i
we can write
1 d ∞
H = Πn [dx dη ] Q (x ,η )Q (x η ) . (3.3)
RFT 2 dY i=1 i i †n i i n i i |Y=0
n=1 Z
X
As explained in [23], the amplitudes Q has the meaning of emission amplitudes of n gluons in one
n
stepofevolution. TheknowledgeofQ inprincipleallowsonetoconstructalsoexclusiveobservables,
n
for example constraining the number of particles produced in the final state. In the next section we
will use Q to calculate single and double inclusive gluon production.
n
Our aim is to write the expression for Q in the normal ordered form, so that the amplitudes
n
can be easily read off.
Ω¯ Ω 0 = ¯ ¯ 0 = ¯ ei√2(¯b b)(a+a†) ¯ ¯ 0 . (3.4)
† † † † − †
| i B C CB| i B BB B| i
where
dη
√2(¯b−b)(a+a†) ≡ √2 d2x ¯bai(x)−bai(x) 2π aai(x,η)+aai†(x,η) . (3.5)
Z h iZ h i
For thefirstfactor, usingthefactthat ¯istheBogoliubov operatorwiththeaction similartoeq.(2.7)
B
with Θ, Φ Θ¯, Φ¯ we have
→
¯ ei√2(¯b b)(a+a†) ¯ = ei√2(¯b b)(N¯†a+N¯Ta†) = e (¯b b)N¯†N¯(¯b b)ei√2(¯b b)N¯Ta†ei√2(¯b b)N¯†a. (3.6)
† − − − − − − −
B B
The state ¯ 0 is the vacuum of the operator
†
B B| i
β˜ = ¯ a ¯ = Ka + Ea ; (3.7)
† † †
B B BB
with K and E defined in eq.(2.26). Comparing eq. (3.7) with eq. (2.7), we can use eq. (2.11) with
the substitution Λ Ξ to write
→
¯† 0 = e14Trln(1−Ξ†Ξ)e−12a†Ξa† 0 . (3.8)
B B| i | i
Now using the fact that the annihilation operator can be represented as a= δ we can write
δa†
ei√2(¯b−b)N¯†ae−12a†Ξa† 0 = e−21(a†+i√2(¯b−b)N¯†) Ξ (a†+i√2(¯b−b)N¯†) 0 . (3.9)
| i | i
– 8 –
And so, finally the normal ordered form is
Ω¯†Ω 0 = e14Trln(1−Ξ†Ξ)e−(¯b−b)[N¯†N¯−N¯†ΞN¯∗](¯b−b) ei√2(¯b−b)[N¯T−N¯†Ξ]a† e−21a†Ξa† 0 . (3.10)
| i | i
Using the relations eqs.(2.28, 2.27) and the definition of Ξ in terms of Θ,Φ,Θ¯,Φ¯ it is straightforward
to show
N = KN¯ EN¯ ; N¯ = K N +ETN . (3.11)
∗ † ∗
−
Thus we can rewrite eq.(3.10) as
Ω¯†Ω 0 = e−41Trln(K†K) e−(¯b−b)N¯†K−1N(¯b−b) ei√2a†K−1N(¯b−b) e−12a†Ξa† 0 . (3.12)
| i | i
Taking the overlap of eq.(3.12) with the vacuum ( 0 Ω¯ Ω 0 ) we reproducethe result of [1] eq.(2.32),
†
h | | i
while the last two factors in eq.(3.12) allow us to calculate Q in a straightforward manner.
n
Scattering amplitude once again.
Before moving on to discussion of inclusive gluon spectrum, we rederive the expression for the scat-
tering amplitude and for the RFT Hamiltonian using eq.(3.3). This will explicitly establish the
equivalence of the two approaches and will also serve as a consistency check on our expressions for
the amplitudes Q . According to eq.(3.3) we need to extract the term linear in Y from the sum
n
of squares of the amplitudes Q integrated over the transverse coordinates and rapidities. The am-
n
plitudes Q are obtained by expanding the exponent in eq.(3.12) to appropriate order. In general
n
therefore the calculation will involve expressions of the type
0 [(¯b b)N K 1a]m [aΞ a]k [a K 1N (¯b b)]l [a Ξa ]n 0 . (3.13)
† †− † † − † †
h | − − | i
Herem, n, l, k aresomeinteger powerstobesummedover. For thepurposeofthepresentdiscussion
we will think of the matrix Ξ(ξ η) as short range in rapidity space. Although this is not really the
−
case, as discussed in detail in [1], the nonlocal in rapidity terms have to be subtracted from the final
result, since they lead to higher powers of Y. Recall that the field ¯b b does not depend on rapidity.
−
Thus after contracting all a’s with a ’s in eq.(3.13) we will generate expressions of two types and
†
their products (for simplicity of notation we drop in this schematic discussion the factors of NK 1
−
which accompany powers of (¯b b))
−
(¯b b) [Ξ Ξ]k(η,ξ)(¯b b); Tr[Ξ Ξ]l. (3.14)
† †
− −
Zηξ
Itiseasy toseethateach oneofthesetermsisoforderY. Thusweonlyneedtoconsidercontractions
in eq.(3.13) that lead to appearance of only one such ”connected” term, rather than product of two
or more - the ”disconnected” graphs . This has an immediate consequence that only terms of order
(¯b b)2 have to be kept. Higher powers of b necessarily lead to disconnected graphs and thus
−
necessarily to higher powers of Y. On the other hand, the terms which do not contain any powers
of b are suppressed by a power of coupling constant α . Thus we are lead to the conclusion that the
s
only terms that contribute to H are connected contractions of order (¯b b)2. With this in mind
RFT
−
the calculation becomes straightforward.
– 9 –