#### Non-perturbative to perturbative QCD via the FFBRST

Eur. Phys. J. C
Non-perturbative to perturbative QCD via the FFBRST
Haresh Raval 0 1
Bhabani Prasad Mandal 0
0 Department of Physics, Institute of Science, Banaras Hindu University , Varanasi 221005 , India
1 Department of Physics, Indian Institute of Technology , Bombay, Mumbai 400076 , India
Recently a new type of quadratic gauge was introduced in QCD in which the degrees of freedom are suggestive of a phase of abelian dominance. In its simplest form it is also free of Gribov ambiguity. However this gauge is not suitable for usual perturbation theory. The finite field dependent BRST (FFBRST) transformation is a method established to interrelate generating functionals for different effective versions of gauge fixed field theories. In this paper we propose a FFBRST transformation suitable for transforming the theory in the new quadratic gauge into the standard Lorenz gauge Faddeev-Popov version of the effective lagrangian. The task is made interesting by the fact that the effective lagrangian is invariant under two different BRST transformations which leads to suitable extension of the previous procedures to accomplish the required result. We are thus able to identify a field redefinition to go from a non-perturbative phase of QCD to perturbative QCD.
1 Introduction
Extensions of the usual Lorenz gauge by including the next
order terms quadratic in gauge fields have been studied in
several contexts [
1–6
]. In [
7,8
] it was shown that a purely
quadratic gauge condition without the linear terms leads to a
suggestive effective lagrangian giving masses to off-diagonal
gluons. The consequences of such a condition to lifting the
Gribov ambiguity were further studied in [
7–9
]. The new type
of quadratic gauge condition is at first introduced as follows,
H a [ Aμ(x )] = Aaμ(x ) Aμa (x ) = f a (x ); for each a
(1)
where f a (x ) is an arbitrary function of x . Several proposals
to establish abelian dominance in the infrared (IR) use what
is called Abelian Projection [
10
]. Such and other algebraic
gauges are usually non-covariant. But as introduced in [
7,8
]
the above gauge is in fact covariant . The gauge prima facie
is an ambiguity free gauge as it is algebraic in nature. Thus,
the quadratic gauge shares the same property of being free of
Gribov copies as the axial gauges nμ Aμa = f a (x ) and the
flow gauge α Aa0 = ∇ · Aa [11] despite being non linear. The
Faddeev–Popov determinant in this gauge is given by
det
δ( Aaμ Aμa )
δ b
= det 2 Aaμ(∂μδab − g f acb Aμc) ,
Therefore, the resulting effective Lagrangian density
contains gauge fixing and ghost terms as follows,
1
LGF + Lghost =− 2ζ
Aaμ Aμa 2 −2
ca Aμa (Dμc)a ,
a
a
where ζ is an arbitrary gauge fixing parameter and (Dμc)a =
∂μca − g f abc Ab cc. Now onwards, we shall drop the
summaμ
tion symbol, but the summation over an index a will be
understood when it appears repeatedly, including when repeated
thrice as in the ghost terms above. In particular,
−ca Aμa (Dμc)a = −ca Aμa ∂μca + g f abcca cc Aμa Ab
μ
(4)
where the summation over indices a, b and c each runs
independently over 1 to N 2 − 1. We should note that ghost
Lagrangian does not have kinetic terms and hence the ghosts
do not propagate in this theory and make no loop
contributions. They act like auxiliary fields, but playing an important
role in the IR. With this understanding, we write the full
effective Lagrangian density in this quadratic gauge as
LQ = − 41 Fμaν F μνa − 2ζ
1
= − 41 Fμaν F μνa + ζ2 F a2 + F a Aaμ Aμa
Aaμ Aμa 2 − 2ca Aμa (Dμc)a
−2ca Aμa (Dμc)a ,
(2)
(3)
(5)
where the field strength Fμaν = ∂μ Aaν (x ) − ∂ν Aaμ(x ) −
g f abc Abμ(x ) Acν (x ) and in the second version the F a are a
set of auxiliary fields called Nakanishi–Lautrup fields [
12
].
As shown in [
9
], the Lagrangian is BRST invariant [
13,14
]
which is essential for the ghost independence of the green
functions and unitarity of the S-matrix. These issues were
studied in Refs. [
7,8
].
The form of the second term of the expression (4)
appearing in the ghost lagrangian contains ghost bilinears
multiplying terms quadratic in gauge fields. Hence if the
nonpropagating ghosts are assumed to be frozen they amount
to a non-zero mass matrix for the gluons. To strengthen this
connection it is necessary to assume that the vacuum
corresponds to ghost condensation. This was achieved through
introducing a Lorenz gauge fixing term for one of the
diagonal gluons, in addition to the purely quadratic terms of Eq.
(1). This gauge fixing gives the propagator to the
corresponding ghost field. Using this ghost propagator, one can give
nontrivial vacuum values to bilinears ca cc within the
framework Coleman-Weinberg mechanism as described in [
7,8
].
We shall revisit the point in the next section also.
The resulting mass matrix for the gluons has N (N − 1)
non-zero eigenvalues only and thus has nullity N − 1. Thus,
the N (N − 1) off-diagonal gluons acquire masses and the
rest N − 1 diagonal gluons remain massless. The massive
off-diagonal gluons are presumed to provide evidence of
Abelian dominance, which is a signature of quark
confinement. This and other phenomena that emerge in this gauge,
such as the avoidance of Gribov ambiguity were studied
explicitly in [
7–9
]. Quark confinement and Gribov
ambiguity are important non-perturbative issues. And this gauge
therefore proves to be important in studying non-perturbative
regime of QCD.
The finite field dependent BRST (FFBRST)
transformation was introduced for first time in Ref. [
6
] by
integrating infinitesimal usual BRST transformations. Such FFBRST
transformations have exactly the same form as the
infinitesimal ones, with the difference that the infinitesimal global
anti-commuting parameter is replaced by an anti-commuting
but finite parameter dependent on space time fields, but with
no explicit dependence on space time coordinates. The
meaning of “finite anti-commuting parameter” is that if we
calculate the Green’s functions for such parameters between
vacuum and a state with gauge and ghost fields we get finite
values as opposed to infinitesimal values. Being finite in
nature FFBRST transformation does not leave the path
integral measure invariant even though other properties of usual
infinitesimal BRST transformation are intact. Thus the
generating functional to a BRST invariant theory is not invariant
under FFBRST. Jacobian of such finite transformation
provides a non-trivial factor which depends on FFBRST
parameter.
Due to this non-trivial Jacobian FFBRST transformations
are simultaneously field redefinitions as well as BRST
transformations on the fields being redefined. They are thus
capable of connecting generating functionals of two different
BRST invariant theories and have been used to study
different gauge field theoretic models with various effective actions
[
15–28
]. In this paper we construct an appropriate FFBRST
transformation to establish the connection at the level of
generating functionals between the recently introduced quadratic
gauge with substantial implications in the non-perturbative
QCD [
7–9
] and the familiar Lorenz gauge which is
suitable to describe the perturbative QCD. This is novel
connection since previous connections were either between two
gauges suitable for only perturbative sector e.g., connection
between Lorenz and axial gauges or they had no such unique
field theoretic meaning attached to them. We should here
mention that the same FFBRST however does not
explicitly connect the vacuum in the quadratic gauge with which
non-Perturbative phase of QCD is associated to the vacuum
in Lorenz gauge to which perturbative phase of QCD
corresponds. To understand this, we first discuss the vacua of both
the theories.
As discussed above, the vacuum in the quadratic gauge
is provided by the SU (N ) symmetric ghost condensation
of bilinears [
7,8
] which is non perturbative in nature since
the non perturbative confining phase corresponds to this
vacuum [
7,8
] and it arises at Lagrangian level only as is
clear from Eq. (4). Such a vacuum does not exist in the
theory of Lorenz gauge. The vacuum in the Lorenz gauge
is provided by the mix condensate of gluon and ghost,
F μν Fμν + cc arising from the following sorts of loops
(Fig. 1) [
29
].
Nature of this vacuum is perturbative as it arises out of
loops and, over and above it formal perturbation theory of
QCD is built. Contribution of these loops is however trivial
at the tree level O(g0) since it vanishes in the dimensional
regularization scheme as the gluon propagator is p12 .
Therefore, the vacuum in the Lorenz gauge, F μν Fμν + cc is
trivial i.e., F μν Fμν + cc = 0 at tree level. Such vacuum
composed of mix condensate does not exist in the theory of
the quadratic gauge as the two point gluon and ghost
functions are formally absent in the theory. Thus, we can see that
confining phase of theory in the quadratic gauge and free
gluon phase in the Lorenz gauge belong to vacua of entirely
different nature. It is clear that the FFBRST technique based
on the parameter constructed in Eq. (18) explicitly does
not connect two vacua discussed above,
SU (N ) symmetric cc (in the quadratic gauge)
F F B RST
F μν Fμν + cc (in the Lorenz gauge).
This conclusion is not at all surprising as the FFBRST is
designed to connect two theories and not the quantities
derivable from the theory regardless of the fact that theories belong
to the same or two different vacua.
2 Connecting two different regimes
As discussed in the introduction, the main non perturbative
result of the quadratic gauge was established with the help
of additional gauge fixing for one of the gluons. The
presence of this additional gauge fixing does reintroduce the
Gribov ambiguity for this component but this is the price to be
paid for an explicit demonstration of effective masses for the
off diagonal gluons. Hence, our aim here is to connect the
generating functional corresponding to the effective actions
in quadratic gauge with additional Lorenz gauge fixing for
one of the gluons to that in the usual Lorenz gauge through
the technique of the FFBRST transformation. To do so, we
write the effective action of the quadratic gauge with
additional gauge fixing for one of the gluons which is as
follows
where stands for ∂μ∂μ and a set of additional fields
G3, d3, d3 correspond to the additional Lorenz gauge of the
diagonal gluon A3 and the ghosts d3, d3 are treated as SU (3)
singlets.
As a first approach it is easy to see that the action in Eq. (6)
is invariant under the following nilpotent BRST
transformation
(7)
(8)
δω
g
As for the previous transformations, We observe that the
second last of Eq. (8) is inhomogeneous wave equation for BRST
differential δd3 with a simple form of the right hand side
acting as the source, which surely admits a local solution for
δd3(x ) for given fields. We see that now this set of
transformations has become applicable for FFBRST technique as
δG3, δ d3 are non zero. Further, the last three
transformawhere δω infinitesimal, anticommuting and global
parameter. We note that the second last transformation the Eq. (7)
can in general be solved for δd3 explicitly. This set of
transformations differs from the usual BRST transformation in the
composite form of the last of the Eq. (7), which can always
be defined with certainty locally in the same spirit of first
five transformations since all fields and their derivatives by
construction are well defined at every spacetime point x , it
is not the differential equation in the δd3 and the is an
invertible operator (similar concept has appeared in the case
of Lagrangian in the literature, see for example Ref. [
30
]).
However, these transformations are not useful for FFBRST
technique since the transformations of G3 and d3 are trivial.
Therefore, we need to introduce a new set of BRST
transformations under which the action (6) is also invariant. This
clearly shows that the passage from BRST to FFBRST
transformations is non trivial. This conclusion has never been
obvious from earlier works. The transformations are as
follows
δcd = δ2ω f dbccbcc
δcd = δgω F d
δω
g
tions in Eq. (8) are not nilpotent, but they satisfy in their
compact form the following higher degree closed algebra
δ2 (δd3) d3
δ2 (δ3 d3) d3
The remaining one can be easily derived from one of these
algebras. Thus, we are prompted to restore the nilpotency
and simplify the algebra, to the extent possible. To do so, we
express the effective action (6) in terms of a new auxiliary
field B3,
transformation having been made nilpotent through the
introduction of an auxiliary field. We shall next achieve the stated
connection using these unusual transformations (11) in the
FFBRST technique.
Now we briefly outline the procedure for the passage
from the BRST transformations to the FFBRST
transformations. We start with making the infinitesimal global
parameter δω field dependent by introducing a numerical parameter
κ (0 ≤ κ ≤ 1) and making all the fields κ dependent such
that φ (x , κ = 0) = φ (x ) and φ (x , κ = 1) = φ (x ), the
transformed field. The symbol φ generically describes all the
fields A, c, c, F, d3, d3, B3, G3. The BRST transformation
in Eq. (11) is then written as
dφ = δb[φ (x , κ)] (φ (x , κ)) dκ
(13)
where is a finite field dependent anti-commuting
parameter and δb[φ (x , κ)] is the form of the transformation for
the corresponding field as in Eq. (11). The FFBRST is then
constructed by integrating Eq. (13) from κ = 0 to κ = 1 as
[
6
]
φ ≡ φ (x , κ = 1) = φ (x , κ = 0) + δb[φ (0)] [φ (x )] (14)
where [φ (x )] = 01 dκ [φ (x , κ)]. Like usual BRST
transformation, FFBRST transformation leaves the effective
action in Eq. (10) invariant. However, since the
transformation parameter is field dependent in nature, FFBRST
transformation does not leave the path integral measure, Dφ invariant
and produces a non-trivial Jacobian factor J . This J can
further be cast as a local functional of fields, ei SJ (where the
SJ is the action representing the Jacobian factor J ) if the
following condition is met [
6
]
Dφ (x , κ)
1 d J d SJ
J dκ − i dκ
ei(SJ +Sef f ) = 0.
(15)
(16)
(17)
The last three transformation rules satisfy the following
algebra
Thus the procedure for FFBRST may be summarised as (i)
calculate the infinitesimal change in Jacobian, 1J ddκJ dκ using
We see that first two rules of Eq. (12) are nilpotent and the
last one is ‘almost’ nilpotent now. It is interesting to compare
Eqs. (9) and (12). The second of the Eq. (12) implies the
following
δ2d3 = δnd3 for all n ≥ 2.
This is the unusual example of the BRST transformation with
idempotent algebra. Thus, we see that the introduction of
B3 has made a substantial difference in the algebra of the
transformations, with the novel feature of the algebra of the
for infinitesimal BRST transformation, + or − sign is for
Bosonic or Fermion nature of the field φ respectively (ii)
make an ansatz for SJ , (iii) then prove the Eq. (15) for this
ansatz and finally (iv) replace J (κ) by ei SJ in the generating
functional
W =
Dφ (x )ei Sef f (φ) =
Dφ (x , κ) J (κ)ei Sef f (φ(x,κ)).
Setting κ = 1, this would then provide the new effective
action Se f f = SJ + Se f f .
Now we proceed to construct a FFBRST transformation
with an appropriate parameter to connect the generating
functionals in the quadratic gauge with additional Lorenz gauge
for the diagonal gluon A3 and the Lorenz gauge. We construct
the finite field dependent parameter as
The γ1, γ2 and γ3 are constant parameters and 2 = 0.
Group index a is summed over. This FFBRST
transformation is particularly different among others [
15–28
] due to the
unique form of transformations (11) and by the fact that the
field dependent parameter in Eq. (18) contains two ghosts
c, d with two different transformation properties unlike
others where there is only one ghost. We now calculate the
change in the Jacobian 1J ddκJ due to the FFBRST with the
parameter in Eq. (18), under which the measure changes
Dφ (κ) → J (κ)Dφ (κ) as
1 d J 1
J dk = − g
−
δ
δ
− δ d3
i
= g
δ
δ Aaμ
d4x
f abccbcc
2 δca
Dμc a
− δδca F a
∂ Aμ3 + B3
δ
− δG3
d3
d4x 2γ2ca Dμc a Aμa + γ3ca ∂μ Dμc a
− F a γ2 Aaμ Aμa + γ3∂μ Aμa
− γ1G3 ∂μ Aμ3 + B3
+ γ1d3 d3
+ δ∂δμ Aa ∂μ Dμc a
μ
Since 1J ddkJ does not contain terms with as
multiplicative factor, the κ dependence in SJ (κ) is multiplicative [
6
].
This implies that the fields in the ansatz for the SJ can be
taken to be κ independent. With this fact in mind, we make
the following ansatz for the SJ to compensate the Jacobian
contribution of FFBRST transformation
1
SJ [φ, κ] = g
d4x α1(κ)Fa Aaμ Aμa + 2α2(κ)ca Aμa Dμc a
+ α3(κ)Fa∂μ Aμa + α4(κ)ca∂μ Dμc a
+ α5(κ)G3 ∂μ A3μ + B3 + α6(κ)d3 d3
In order to satisfy the condition in Eq. (15), the following
equation must be obeyed
Dφ[x, κ] Fa Aaμ Aμa(−γ2 − α˙1) + 2ca Aμa(Dμc)a(γ2 − α˙2)
+Fa∂μ Aμa(−γ3 − α˙3) + ca∂μ(Dμc)a(γ3 − α˙4)
+(−γ1 − α˙5)G3(∂μ A3μ + B3)
+(γ1 − α˙6)d3 d3 ei(Sef f +SJ ) = 0,
which gives the following relation among parameters
α˙1 = −α˙2 = −γ2
α˙3 = −α˙4 = −γ3
α˙5 = −α˙6 = −γ1.
The Eqs. (23) have the obvious solutions
α1 = −α2 = −γ2κ; α3 = −α4 = −γ3κ,
α5 = −α6 = −γ1κ
We choose the arbitrary parameters γ1 = 1, γ2 = 1, γ3 = −1
in Eq. (24). Thus, the additional Jacobian contribution at
κ = 1 is
SJ =
d4x
− F a Aaμ Aμa + 2ca Aμa Dμc a
+F a ∂μ Aμa − ca ∂μ Dμc a
−G3 ∂μ A3μ + B3
+ d3 d3 .
Adding this Jacobian contribution, SJ to the Se f f in Eq. (10)
we obtain at κ = 1 the Lorenz gauge as follows
(19)
Se f f + SJ =
d4x
− 21ξ (B3)2 + ζ2 F a2 + F a ∂μ Aμa
−ca ∂μ(Dμc)a
= SL
Here the term ξ1 (B3)2 is redundant which can be put to zero
by using EOM for B3. Now, we may further apply second
FFBRST such that ζ → ζ in the same Lorenz gauge by well
known methods [
6
]. We may summarize this symbolically as
the conversion from one theory to another,
Ze f f =
DφeiSef f F F B RST
−→
Dφ (κ)ei(Sef f +SJ )
(20)
=
Dφ ei SL = Z L ,
(21)
(22)
(23)
(24)
(25)
(26)
(27)
where α j (κ), j = 1, . . . , 6, are arbitrary functions with
initial condition αi (κ = 0) = 0 while the fields themselves are
κ independent. We calculate,
Thus, we have connected two theories with two different
regimes of applicability. This is a connection also between
theories with and without propagating ghosts.
3 Conclusion
The spirit of BRST invariance was to establish the unitarity
of the S-matrix in gauge theories whose gauge fixed versions
contain ghost degrees of freedom. This technique was
substantially extended in the FFBRST approach to permit field
redefinitions transforming the effective action with one
possible gauge fixing to that of another. In some of the recent
earlier work the interesting features of a purely quadratic gauge
condition without the usual Lorenz condition have been
studied and shown to lead to several interesting properties of the
non-perturbative QCD vacuum in the IR limit. At first site
the effective degrees of freedom entering here, the off
diagonal gluons with masses, appear unrelated to those entering
the perturbation theory calculations and which are
compatible with the elegant UV properties of Yang–Mills theories.
In this paper we have resorted to the FFBRST technique to
establish a direct formal connection between the two varieties
of the QCD effective lagrangians. Several technical
difficulties are encountered in this process and it has required us to
make suitable extensions to the FFBRST method. In
particular a new auxiliary field is required to ensure nilpotency of the
modified BRST transformations. The resulting field
redefinitions which connect the degrees of freedom capturing the
IR behaviour of QCD vacuum with those of the UV version
suitable to perturbative computations need to be studied
further. Also, the extensions of the FFBRST technique proposed
here can be put to use for other similar problems.
Acknowledgements We acknowledge the fruitful discussion with
Prof. Urjit A. Yajnik. This work is partially supported by Department
of Science and Technology, Govt. of India under National Postdoctoral
Fellowship scheme with File No. ‘PDF/2017/000066’. One of us (BPM)
acknowledges the support from Physics Department, IIT-Bombay for a
visitation during which the work was initiated.
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