Glauber Gluons in Soft Collinear Effective Theory and Factorization of Drell-Yan Processes

0.12 MB·English

by  F. Liu

Glauber Gluons in Soft Collinear Effective Theory and
Factorization of Drell-Yan Processes
F. Liu and J.P. Ma
Institute of Theoretical Physics, Academia Sinica, P.O. Box 2735, Beijing 100190, China
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1
Abstract
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Glauber gluons in Drell-Yan processes are soft gluons with the transverse momenta much

n
largerthantheirmomentumcomponentsalongthedirectionsofinitialhadrons. Theirexistence
a
has been a serious challenge in proving the factorization of Drell-Yan processes. The recently
J
proposedsoftcollineareffecttheoryofQCDcanprovideatransparentwaytoshowfactorizations
1
2 for a class of processes, but it does not address the effect of glauber gluons. In this letter we
first confirm the existence of glauber gluons through an example. We then add glauber gluons

] into the effective theory and study their interaction with other particles. In the framework of
h
theeffectivetheorywithglaubergluonsweareabletoshowthatthe effectsofglaubergluonsin
p
- Drell-Yan processes are canceled and the factorization holds in the existence of glauber gluons.
p
OurworkcompletestheprooforargumentoffactorizationofDrell-Yanprocessintheframework
e
h of the soft collinear effective theory.
[

2
v
3
7
QCD factorization theorems enable us to make theoretical predictions from QCD for hard-
9
2 scattering processes, where one can separate long-distance effects from short-distance effects. The
. short-distance effects depend on the details of a process and can be studied with perturbative
2
0 QCD,while long-distance effects can becharacterized by matrix elements of QCDoperators, which
8
depend only on the structure of hadrons and infrared properties of QCD[1]. In general, it is
0
: difficult to prove a factorization theorem,e.g., much effort has been spent to prove factorization for
v
Drell-Yan process[2, 3]. A main obstacle has been the exchange of glauber gluons in initial state
i
X
interactions[4, 3]. The glauber gluons are soft with the transverse momenta much larger than their
r momentum components along the directions of initial hadrons.
a
Recently, the soft collinear effective theory(SCET) has been proposed to study interactions
amongcollinear-andvarioussoftparticlesofQCD[5,6,7]. Theinteractionsareatlong-distanceand
producecollinear- and I.R. divergences in perturbative calculations. Assumingthat SCETincludes
all those particles of QCD, then SCET will re-produce these collinear- and I.R. singularities. It
is interesting to note that with SCET the proofs or arguments of factorization theorems can be
simplified or made in a transparent way. In [6] several examples including Drell-Yan processes
are given. However, the glauber gluons are not included in the proposed SCET[5, 6]. In this
letter we make an attempt to include the glauber gluons into SCET and show that their effect in
the factorization of Drell-Yan process is canceled[6]. Therefore our work completes the proof or
argument of factorization of Drell-Yan processes in the framework of SCET.
BeforeourstudyofSCETbyaddingglaubergluonsandthefactorization ofDrell-Yanprocesses,
we show the existence of glauber gluons in scattering amplitudes. For this we consider the process
q(p ) + q¯(p ) g(k ) + γ∗(q), which contributes to the differential cross section of Drell-Yan
1 2 g
→
1
p2 kg
k
p1
(a) (b)
Figure 1: A diagram which contributes to the process q(p )+q¯(p ) g(k )+γ∗(q) at one loop
1 2 g
→
level. Thebroken lineis acut. Theblack circle denotes theinsertion of theelectromagnetic current
operator.
processes. At one loop level, the amplitude of the process receives a contribution from Fig.1a. We
will use the light-cone coordinate system, in which a vector aµ is expressed as aµ = (a+,a−,~a ) =
⊥
((a0 +a3)/√2,(a0 a3)/√2,a1,a2) and a2 = (a1)2 +(a2)2. We introduce two light cone vectors
− ⊥
n¯µ = (1,0,0,0) and nµ =(0,1,0,0). The momenta are given as:
pµ = (p+,0,0,0), pµ = (0,p−,0,0), kµ = (k+,k−,~k ). (1)
1 1 2 2 g g g g⊥
We consider the contribution in the momentum configuration where the momentum k scales as
g
kµ Q(λ2,1,λ,λ) with Q as a large scale and λ 1. In this configuration the outgoing gluon
g
∼ ≪
is collinear to the initial q¯. For convenience we will take Q = 1 in the following. The leading
contribution from Fig.1 in the limit λ 0 can come from different momentum regions of the
→
loop momentum k with k2 λ2 or with k2 at higher order of λ. These possible regions are: The
∼
collinear regions with k at the order of (1,λ2,λ,λ) or (λ2,1,λ,λ), the soft region with k at the
order of (λ,λ,λ,λ), the ultra-soft region with k at the order of (λ2,λ2,λ2,λ2) and the glauber
region with k at the order of (λ2,λ2,λ,λ). Gluons with momenta in glauber regions are called as
glauber gluons. To show that the glauber gluon gives nonzero contribution to the amplitude, it is
enough to consider the following loop integral involved in the contribution of Fig.1a:
d4k 1 1 1 1
I = i . (2)
(2π)4 · k2+iε · (p k)2+iε · (k k)2 +iε · (k k +p )2+iε
Z 1− g − − g 2
One can perform a power counting in λ for the integral in differen momentum regions to determine
the relative importance of contributions in deferent regions and leading contributions. One finds
that the contributions from the collinear region with k (λ2,1,λ,λ), the ultra-soft region and the
∼
glauber region are at leading order of λ which is (λ−2). The contributions from other regions are
O
at higher order of λ. It is complicated to have a clear separation of leading contributions from the
three regions, because they are all overlapped.
For our purpose we can consider the absorptive part of Fig.1a instead of the total contribution.
We consider the cut diagram of Fig1.a, which is given as Fig.1b. Instead of the integral I we need
to consider the absorptive- or imaginary part of I. The imaginary part is given by:
1 d4k 1 1
ImI = ( 2πi)δ((p k)2) ( 2πi)δ((k k +p )2). (3)
2 (2π)4 · k2+iε · − 1 − · (k k)2 +iε − − g 2
Z g −
From these on-shell conditions and with ~k (λ,λ) one always has k+ λ2 and k− λ2. This
⊥
∼ ∼ ∼
indicates that the absorptive part of I at the leading order of λ is determined only by the glauber
2
region. Performing the power counting for the glauber region one obtains the leading contribution
to ImI as:
1 d4k 1 1
ImI = ( 2πi)δ( 2p+k− k2)
G 2 (2π)4 · k2 +iε · − − 1 − ⊥ · 2k−k+ k2 +iε
Z − ⊥ − g − ⊥
( 2πi)δ( 2k+(k− p−)+2~k ~k k2 +(k p )2)
· − − g − 2 g⊥ · ⊥− ⊥ g − 2
1 d2k 1 1 2 k2
= ⊥ = γ +ln4π ln g⊥ + (λ0),(4)
−8p+1p−2 Z (2π)2k⊥2(~k⊥−~kg⊥)2 8πskg2⊥ "(cid:18)ǫ − (cid:19)− µ2 # O
with s = 2p+p−. In the above we have used dimensional regularization with ǫ = 4 d to regularize
1 2 −
divergences. From the result the integral is divergent. The divergent contribution comes partly
form the region of~k 0 and partly from the region of ~k ~k . The singular contribution from
⊥ ⊥ g⊥
∼ ∼
the region of ~k ~k is because that there are pinched poles in the complex k+-plan for the
⊥ g⊥
∼
integral I. The singularities generated by these pinched poles have brought up a difficulty to prove
the factorization in Drell-Yan processes.
From the above discussion we have ImI = ImI (1+ (λ)). To verify this one can work out
G
O
the exact result of the integral I. The exact result can be found in [8]. Through an analytical
continuation and an expansion of the exact result we have from [8]:
1 2 k2
ImI = γ+ln4π ln g⊥ + (k0 ). (5)
8πskg2⊥ "(cid:18)ǫ − (cid:19)− µ2 # O g⊥
Comparing the result in Eq.(4,5) one verifies ImI = ImI (1+ (λ)). One can conclude that the
G
O
exchangeofglaubergluonsdoesgivenonzerocontributionstoscatteringamplitudesatleadingorder
of λ and hence possibly to differential cross sections. The possible contributions due to glauber
gluons are at thesame order of contributions dueto collinear- and ultra-soft gluons. Therefore, one
should include glauber gluons into SCET as an effective theory of QCD to describe interactions
between collinear partons. Although the contributions of glauber gluons in Drell-Yan processes at
twist-2 level are canceled, as shown in [2, 3] and in the current work, but the existence of glauber
gluons has observable effects at twist-3 level. Recent studies of single transverse spin asymmetries
in Drell-Yan- and DIS processes have been shown that these asymmetries are generated through
exchanges of glauber gluons[9]. Hence it is also important to have SCET containing glauber gluons
for a possible study of factorizations of single transverse spin asymmetries.
ToderiveSCETcontaining glaubergluons, wefirstconsiderSCETofonecollinear system,later
we will add another collinear system for our purpose. We will use the space-time representation
of SCET as used in [7]. We choose the collinear system moving along the n¯-direction with the
momentum p at the order
pµ Q(1,λ2,λ,λ), (6)
∼
where Q is a larger scale while λ is small with λ 1. Again we will take Q = 1 in the following.
≪
Interacting particles in such a collinear system can be classified with their momenta at different
orders of λ. They are collinear particles with momenta at the order pµ (1,λ2,λ,λ) , ultra-soft
∼
particles with momenta at the order pµ (λ2,λ2,λ2,λ2), and glauber gluons with momenta at
∼
the order pµ (λ2,λ2,λ,λ). The later have not been incorporated in SCET. Soft particles with
∼
momenta at the order of pµ (λ,λ,λ,λ) will not interact with the above particles because of
∼
momentum conservation. However, their effect will appear in matching operators of full QCD into
3
those of SCET. It has been shown that their effect is canceled in Drell-Yan process. We will not
consider soft particles here.
For the collinear system we decompose the gluon field and quark field of full QCD as:
Aµ = Aµ+Aµ+Aµ , ψ = ξ +η+q. (7)
n¯ g us n¯
IntheaboveA andξ arethecollinear fieldsforcollinear gluonsandcollinear quarks,respectively.
n¯ n¯
η is a small component for collinear quarks which will be integrated out and expressed in terms of
other fields. A and q are the ultra-soft fields for ultra-soft particles. Aµ is the field for glauber
us g
gluons. The space-time dependence of these fields are characterized by the small scale λ in the
following patten. For a collinear field φ which can be the collinear gluon field or the collinear
n¯
quark field we have
∂+φ (x) (1)φ (x), ∂−φ (x) (λ2)φ (x), ∂µφ (x) (λ)φ (x). (8)
n¯ ∼ O n¯ n¯ ∼ O n¯ ⊥ n¯ ∼ O n¯
For a generic ultra-soft field φ and the glauber gluon field we have
us
∂µφ (x) (λ2)φ (x),
us us
∼ O
∂+Aµ(x) (λ2)Aµ(x), ∂−Aµ(x) (λ2)Aµ(x), ∂νAµ(x) (λ)Aµ(x). (9)
g g g g ⊥ g g
∼ O ∼ O ∼ O
By inspecting propagators of particles in different momentum modes we can derive the following
power-scaling:
A+ (1), A− (λ2), Aµ (λ), ξ (λ),
n¯ ∼ O n¯ ∼ O n¯⊥ ∼ O ∼ O
Aµ (λ2), Aµ (λ2), q (λ3). (10)
us g
∼ O ∼ O ∼ O
Using the power-scaling of fields and the scaling of the space-time dependence of the fields one
canexpandtheLagrangianoffullQCDinλwiththeabovefields. Theexpansionisstraightforward.
µ
To express the leading order results we introduce the modified derivative ∂ or projection. The
n¯
derivative∂+ onlyacts oncollinear fields,itgives zeroifthederivative acts onultra-soft-orglauber
n¯
gluon fields, i.e.,
∂+φ (x) = ∂+φ (x), ∂+φ (x) = 0, ∂+Aµ = 0. (11)
n¯ n¯ n¯ n¯ us n¯ g
The derivative ∂− acts as an usual derivative on all fields while ∂µ acts only on collinear- or
n¯ n¯⊥
glauber gluon fields, but not on ultra-soft fields. The leading terms of our effective theory can be
written as:
(n¯)(ξ,q,Aµ,Aµ,Aµ )= (Aµ)+ (q,Aµ )+ (ξ ,Aµ,n¯ A ,n¯ A )+ , (12)
n¯ g us g g us us n¯ n¯ n¯ g us
L L L L · · ···
where stand for terms whose contributions to the action S = d4x (n¯) are suppressed by λ.
··· L
In the expansion of the action S one should assign the space-time integration d4x with a correct
R
scaling in λ by noticing that the integration should not change the momentum orders of fields in
different momentum modes.
At leading order glauber gluons can not interact with ultra-soft particles and they also can not
interact with themselves. Hence (A ) is only the kinetic term of the glauber gluons:
g g
L
1
= Tr[∂µAν ∂νAµ][∂ A ∂ A ]. (13)
Lg −2 ⊥ g − ⊥ g ⊥µ gν − ⊥ν gµ
4
It should be noted that only the transverse derivative appears. This fact results in that the
propagator of glauber gluons with the momentum k will be proportional to 1/k2, as already seen
⊥
in Eq.(4). (q,Aµ ) is the part for ultra-soft particles:
us us
L
1
(q,Aµ ) = q¯iγ D q Tr [Dµ ,Dν ] 2, (14)
Lus us · us − 2g2 { us us }
wherethe covariant derivative is defined only with the ultra-soft gluon field, i.e., Dµ = ∂µ+igAµ .
us us
(ξ ,A ,A ,A ) contains collinear particles and their interactions with ultra-soft- and glauber
n¯ n¯ n¯ g us
L
gluons. To express it we introduce the covariant derivative with the modified derivative:
Dµ = ∂µ+igAµ +ignµn¯ (A +A ). (15)
n¯ n¯ n¯ g us
·
With the covariant derivative the part can be expressed as
n¯
L
1
(ξ ,Aµ,n¯ A ,n¯ A ) = ξ¯ in γn¯ D ξ ξ¯ iγ D n γiγ D ξ
Ln¯ n¯ n¯ · g · us n¯ · · n¯ n¯ − n¯ · c⊥2iD+ · · c⊥ n¯
c
1
Tr [Dµ,Dν] 2, (16)
−2g2 { n¯ n¯ }
where the collinear covariant derivative D is defined as Dµ = ∂µ +igAµ. The glauber gluon field
c c n¯
appears in the covariant derivative in the same way as the ultra-soft gluon field. However, the
interaction of collinear gluons with ultra-soft gluons are different than that with glauber gluons
µ
becausetheoperator ∂ does notact on theultra-soft field, buton theglauber field, i.e., ∂ A =
n¯⊥ n¯⊥ g 6
0.
The effective theory at leading order λ is invariant under various gauge transformations, which
are collinear-, ultra-soft- and glauber gauge transformation. The space-time dependence of these
transformations is characterized in the same patten as that of the correspondingfields in Eq. (8,9).
These transformations are:
Collinear : Aµ Aµ , Aµ Aµ, ξ U ξ , q q,
us us g g n¯ c n¯
→ → → →
i
Aµ U AµU† U ∂µ+ig(Aµ +Aµ),U† ,
n¯ → c n¯ c − g c us g c
h i
Ultra soft: Aµ U AµU† , Aµ U AµU† , ξ U ξ , q U q,
n¯ us n¯ us g us g us n¯ us n¯ us
− → → → →
i
Aµ U Aµ U† U ∂µ,U† ,
us → us us us− g us us
Glauber : Aµ Aµ, Aµ Aµ ,h ξ iU ξ , q q,
g g us us n¯ g n¯
→ → → →
i
Aµ U AµU† U ∂µ+ig(Aµ +Aµ),U† . (17)
n¯ → g n¯ g − g g us g g
h i
Under these transformations the effective Lagrangian is invariant at the leading order of λ, or
the change of the effective Lagrangian is suppressed by λ. One can re-express these transforma-
tions with the modified derivative and neglect higher order effects introduced by the fields in the
transformation so that the effective Lagrangian is exactly invariant. In the above glauber gauge
transformation the glauber field is not changed. However, there is another type of transformations
relatedtotheglaubertransformationdefinedbyU (x˜) =U (x+n¯+x−n),underwhichtheglauber
g⊥ g
field is changed:
i
Aµ U AµU† U ∂µ,U† , Aµ Aµ , q q,
g → g⊥ g g⊥− g g⊥ g⊥ us → us →
h i
Aµ U AµU† +U Aµ ,U† , ξ U ξ . (18)
n¯ → g⊥ n¯ g⊥ g⊥ us g⊥ n¯ → g⊥ n¯
h i
5
Under the above transformation the Lagrangian is invariant. One needs to fix gauges for different
gauge fields and introduces correspondingghost fields. We note that the gauge fixing for ultra-soft-
and glaubergluons can befixedseparately. We will take Feynman gauge for ultra-soft- and glauber
gluons. For collinear gluons we take the covariant gauge for collinear gluons as in [5], where the
covariant derivative should include the glauber gluon field as in Eq.(15).
To study the effects of glauber- and ultra-soft gluons in Drell-Yan process one needs to include
another collinear fields in the n-direction. This can be done by adding the collinear gluon field Aµ
n
and the collinear quark field ξ in the decomposition in Eq.(7) correspondingly and by doing the
n
expansion in λ. At leading order λ the collinear fields in different directions can no interact with
each other. Momentum conservation insures that the collinear fields in different directions can not
bemixed with each other. Then, thesoftcollinear effective theory for two collinear systems moving
in the n-direction and in n¯ direction respectively can be easily written down:
(n¯n) = (Aµ)+ (q,Aµ )+ (ξ ,Aµ,n¯ A ,n¯ A )+ (ξ ,Aµ,n A ,n A )+ . (19)
g g us us n¯ n¯ n¯ g us n n n g us
L L L L · · L · · ···
From the effective theory, the interaction between the collinear fields in different directions is only
through exchanging glauber- and ultra-soft gluons.
Without the presence of glauber gluons the decoupling of ultra-soft gluons can be simply made
through redefinitions of collinear fields, as shown in [5, 6]. In the presence of glauber gluons in
SCET,onecanalsoredefinefieldssothattheredefinedfieldsdonotinteracting withglaubergluons
and ultra-soft gluons. We introduce the following SU(3)-matrices:
0 0
Z (x) = P exp ig dβn¯ A (βn¯ +x) , Z (x) = P exp ig dβn A (βn+x) ,
n¯ g n g
(cid:26)− Z−∞ · (cid:27) (cid:26)− Z−∞ · (cid:27)
0 0
Y (x) = P exp ig dβn¯ A (βn¯ +x) , Y (x) = P exp ig dβn A (βn+x) .
n¯ us n us
− · − ·
(cid:26) Z−∞ (cid:27) (cid:26) Z−∞ (cid:27)
(20)
With these matrices we redefine all collinear fields as:
i
Aµ = Z Y A˜µY† Z† Z ∂µ +ig nµA−,Z† , ξ = Z Y χ ,
n¯ n¯ n¯ n¯ n¯ n¯ − g n¯ ⊥ s us n¯ n¯ n¯ n¯ n¯
h i h i
i
Aµ = Z Y A˜µY† Z† Z ∂µ +ig n¯µA+,Z† , ξ = Z Y χ . (21)
n n n n n n− g n ⊥ s us n n n n n
h i h i
With the re-defined fields the Lagrangian containing the interactions with glauber gluons at the
leading order of λ becomes
(ξ ,Aµ,n¯ A ,n¯ A )= (χ ,A˜µ,0,0), (ξ ,Aµ,n A ,n A ) = (χ ,A˜µ,0,0). (22)
n¯ n¯ n¯ g us n¯ n¯ n¯ n n n g us n n n
L · · L L · · L
This result tells that the redefined collinear fields will not interact with glauber gluons and ultra-
soft gluons. It should be noted that there is no difference between S-matrix elements calculated
with the original Lagrangian and those calculated with the Lagrangian of redefined fields. For our
redefinition of fields this can also be realized with path-integral of SCET by noting the fact that
the Jacobian’s of changing field variables are unit.
Now we are in position to discuss the factorization for Drell-Yan process. The process is given
as
h +h ℓ+ +ℓ− +X, (23)
n¯ n
→
6
whereonehadronh movesinthen-directionwhileanotherh movesinthen¯-direction. Atleading
n n¯
order of QED the differential cross-section is determined by the hadronic tensor:
Wµν(q) = d4xe−iq·x h h Jµ(x)Jν(0)h h , Jµ = ψ¯γµψ. (24)
n¯ n n n¯
h | | i
Z
To show the factorization one needs first to match the operator Jµ(x)Jν(0) to the operators of the
soft collinear effective theory at the leading order in λ. In the matching, the interactions between
collinear particles which are not presented in SCET are assembled by different Wilson lines:
0 0
W (x) = P exp ig dαn A (αn+x) , W (x) = P exp ig dαn¯ A (αn¯ +x) .
n¯ n¯ n n
− · − ·
(cid:26) Z−∞ (cid:27) (cid:26) Z−∞ (cid:27)
(25)
At leading order of λ the operators consists of collinear fields only and their relative space-time
dependence can only be of x−n for φ fields and of x+n of φ . Taking the operators consisting
n¯ n
only of quark fields in SCET as example, the matching is:
Wµν(q) = dk+dk−Cµν(q,k+n¯,k−n) dx−dy+exp ix−k+ iy+k−
1 2 ij 1 2 − 1 − 2
Z Z (cid:16) (cid:17)
h h (ξ¯ W )(x−n)Γ(i)(W†ξ )(0) (ξ¯ W )(y+n¯)Γ(j)(W†ξ )(0) h h + (,26)
n¯ n n¯ n¯ n¯ n¯ n¯ n n n n n n n¯
·h | | i ···
h ih i
where stand for other possible operators at leading order and terms at higher orders in λ which
···
will be neglected. The leading order is at λ4. For simplicity we will discuss the factorization for
the above term in detail. Those terms containing other possible operators can be handled in the
same way. In the above we have already used the properties
n¯ γξ = γ−ξ = 0, n γξ = γ+ξ = 0 (27)
n¯ n¯ n n
· ·
to decouple the Dirac indices. Hence the Γ -matrices are given by
n¯
Γ(1),Γ(2),Γ(3) = (n γ,n γγ ,n γγν,). (28)
n¯ n¯ n¯ 5 ⊥
· · ·
(cid:16) (cid:17)
(i) (i)
Γ is defined by replacing n with n¯ in Γ .
n n¯
The functions C can be calculated with perturbative theory. If one uses perturbative theory
to calculate the hadronic tensor by replacing the hadrons with parton states, one will have I.R.-
and collinear singularities. These singularities are reproduced by the matrix elements of SCET
operators through construction of SCET, e.g., those in the right hand side of the above equation.
Hence the perturbative functions are free from I.R.- and collinear divergence. At this stage the
long-distance- and short-distance effects are factorized, but the factorization of Drell-Yan process
is not complete. The factorization is completely achieved if one can show that the matrix element
of the two-hadron state in Eq.(26) can be factorized into a product of two matrix elements of
one-hadron state. In Eq.(26) the operator (ξ¯ W )(x−n)Γ(i)(W†ξ )(0) can emit ultra-soft- and
n¯ n¯ n¯ n¯ n¯
glauber gluons to interact with the state hnhand the emitted gluons cani also be absorbed by the
operator (ξ¯ W )(y+n¯)Γ(j)(W†ξ )(0) . This prevents us at first look from factorizing the matrix
n n n n n
element ohf two-hadron state in Eq.(26i) as a product of two matrix elements of one-hadron state.
For the factorization one needs to show that the total effects of gluon-emissions are canceled.
7
Before discussingthe factorization, we discuss a general feature of theλ-expansion in the space-
time representation. We consider a product of collinear fields φ , denoted as Φ , multiplied with
n¯ n¯
any ultra-soft field φ . The integral of the product has the following property:
us
dβΦ (βn+x)φ (βn+x) = dβΦ (βn+x) φ (x) 1+ (λ2) . (29)
n¯ us n¯ us
O
Z (cid:20)Z (cid:21) n o
Thisequationcanbederivedbyexpansiontheultra-softfieldasφ (βn+x) = φ (x)+β∂+φ (x)+
us us us
. The derivative ∂+φ (x) is at the order of λ2, while the integration variable β is related to the
us
···
collinear direction and is taken as at the order of λ0. Therefore we have the above equation. Now
we use the redefined collinear fields in Eq.(21) to express the matrix elements in Eq.(26). Under
the re-definition the Wilson W and W become by using the property
n n¯
W (x) = V (x)W˜ (x)V†(x) 1+ (λ2) , V (x) = Z (x)Y (x),
n n n n n n n
O
W (x) = V (x)W˜ (x)V†(x)n1+ (λ2)o, V (x) = Z (x)Y (x) (30)
n¯ n¯ n¯ n n¯ n¯ n¯
O
n o
where W˜ and W˜ are defined by replacing A and A in Eq.(25) with A˜ and A˜ , respectively.
n n¯ n n¯ n n¯
The hadronic matrix element with the redefined fields become:
h h (ξ¯ W )(x−n)Γ(i)(W†ξ )(0) (ξ¯ W )(y+n¯)Γ(j)(W†ξ )(0) h h
n¯ n n¯ n¯ n¯ n¯ n¯ n n n n n n n¯
h | | i
= h hh (χ¯ W˜ ) (x−n)Γ(i)(W˜ †χ )i(h0) V†(x−n)V (0) V†(y+n¯i)V (0)
n¯ n n¯ n¯ i n¯ n¯ n¯ j n¯ n¯ n n
h | ij kl
h i h i
(χ¯ W˜ ) (y+n¯)Γ(j)(W˜ †χ ) (0)h h 1+ (λ2) , (31)
n n k n n n l n n¯
| i O
h i
where ijlk are color indices. With the definition of V (x−n) and V (y+n¯) one knows that the
n¯ n
x−-dependence in V (x−n) and the y+-dependence in V (y+n¯) are characterized by λ2. Since the
n¯ n
space-time separation x− and y+ will be integrated over in the hadronic tensor, hence we can have
the following approximation under the integrations:
V†(x−n)V (0) V†(y+n¯)V (0) = δ δ + (λ2). (32)
n¯ n¯ n n ij lk
ij kl O
h i h i
Using the fact that redefined collinear fields donot interact with glauber- and ultra-soft gluons and
also that the collinear fields in different directions do not interact, we can re-write Eq.(31) under
the integrations in Eq.(26) with Eq.(32) as:
h h (ξ¯ W )(x−n)Γ(i)(W†ξ )(0) (ξ¯ W )(y+n¯)Γ(j)(W†ξ )(0) h h
n¯ n n¯ n¯ n¯ n¯ n¯ n n n n n n n¯
h | | i
= h h(χ¯ W˜ )(x−n)Γ(i)(W˜ †χ )(0)ihh h (χ¯ W˜ )(y+n¯)Γ(j)(W˜i†χ )(0)h
n¯ n¯ n¯ n¯ n¯ n¯ n¯ n n n n n n n
h | | i h | i
h ih i
1+ (λ2) . (33)
· O
n o
Substituting this result and expressing the redefined fields with the original fields, we complete the
factorization for Eq. (26):
Wµν(q) = dk1+dk2−Ciµjν(q,k1+n¯,k2−n)· dx−e−ix−k1+hhn¯|(ξ¯n¯Wn¯)(x−n)Γ(n¯i)(Wn¯†ξn¯)(0)|hn¯i
Z (cid:20)Z (cid:21)
dy+e−iy+k2− hn (ξ¯nWn)(y+n¯)Γ(nj)(Wn†ξn)(0)hn 1+ (λ2) + . (34)
· h | | i O ···
(cid:20)Z (cid:21)h i
8
In the matching in Eq.(26) there are contributions from color-octet operators at leading order like
(ξ¯ W )(x−n)Γ(i)Ta(W†ξ )(0) (ξ¯ W )(y+n¯)Γ(j)Ta(W†ξ )(0) , (35)
n¯ n¯ n¯ n¯ n¯ n n n n n
h ih i
represented by . With the above procedure and the fact that the hadrons are color-less, one can
···
show that these color-octet operators do not contribute at leading order of λ.
In the above we have shown that the effects of glauber gluons are completely canceled at the
leading order of λ in the same way of the cancelation of ultra-soft gluons. The above result is
invariant at leading order of λ under various gauge transformation. We note that the Wilson line
W and W under the transformations given in Eq. (17) are transformed as:
n n¯
Collinear : W U W 1+ (λ2) , W U W 1+ (λ2) .
n n n n¯ n¯ n¯
→ O → O
Ultra soft : W U W (cid:16)U† 1+ (cid:17)(λ2) , W U(cid:16)W U† 1(cid:17)+ (λ2) .
n us n us n¯ us n¯ us
− → O → O
Glauber : W U W U† 1(cid:16)+ (λ2) ,(cid:17) W U W U† 1+(cid:16) (λ2) . (cid:17) (36)
n g n g n¯ g n¯ g
→ O → O
(cid:16) (cid:17) (cid:16) (cid:17)
At the leading order of λ there are other three operators appearing in the matching in Eq.(26).
They are:
ξ¯ ξ G+µG+ν, ξ¯ ξ G−µG−ν, G−µG−νG+µ′G+ν′, (37)
n n n¯ n¯ n¯ n¯ n n n n n¯ n¯
where we have suppressed the color structure, spin structures, space-time dependence and the
Wilson line of collinear gluon in different directions. Thecollinear field strength tensors are defined
withcorrespondingcollinear gluonfields,theindicesµ,ν,µ′ andν′ aretransverse. Thefactorization
or the cancelation of effects through exchanges of glauber- and ultra-soft gluons can be shown in a
similar way as in the above. With the factorization one can define for each hadron various parton
distributions and write the final result for the hadronic tensor in a compact form. The detailed
results can be found somewhere in the literature.
To summarize: In this letter we have studied the effects of glauber gluons in the framework of
SCET, which had not been considered in the originally proposed SCET. We have confirmed the
existence of glauber gluons through an example. By adding the glauber gluons into SCET we have
found that the glauber gluons only interact with collinear particles. Furthermore, we have shown
in the framework of SCET with glauber gluons that the effects of glauber gluons are canceled in
Drell-Yan process, i.e., the factorization in the present of glauber gluons still holds. Therefore our
work completes the proof or argument of factorization of Drell-Yan process in the framework of
SCET. Our result is in the agreement with the traditional proof of Drell-Yan process where the
existence of glauber gluon has been a challenging difficulty. However the proof here is given in a
more transparent way.
Acknowledgments
WethankProf. M.YuandDr. F.Fengformanyinterestingdiscussions. Thisworkissupported
by National Nature Science Foundation of P.R. China(No. 10721063,10575126 and 10975169).
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