Renormalization in open quantum field theory. Part I. Scalar field theory
JHE
Renormalization in open quantum eld theory. Part I.
Avinash Baidya 0 1 2 3 4
Chandan Jana 0 1 2 4
R. Loganayagam 0 1 2 4
Arnab Rudra 0 1 2 4
0 Department of Physics, University of California , USA
1 Shivakote , Hesaraghatta Hobli, Bengaluru 560089 , India
2 C.V. Raman Avenue , Bangalore 560012 , India
3 Indian Institute of Science
4 eld theories. After
While the notion of open quantum systems is itself old, most of the existing studies deal with quantum mechanical systems rather than quantum a brief review of eld theoretical/path integral tools currently available to deal with open eld theories, we go on to apply these tools to an open version of 3 + 4 theory in four spacetime dimensions and demonstrate its one loop renormalizability (including the renormalizability of the Lindblad structure).
E ective Field Theories; Renormalization Group

HJEP1(207)4
Scalar
eld theory
quantum
1 Introduction and motivation
2 Introduction to open e ective theory
Lindblad condition from tree level correlators
One loop beta function for m2
One loop beta function for m2
One loop beta function for 3
One loop beta function for 3
One loop beta function for 4
One loop beta function for 4
One loop beta function for
Checking Lindblad condition for mass renormalization
Checking Lindblad condition at the level of cubic couplings
3.10 Checking Lindblad condition for quartic couplings
3.11 Summary of the results
4
Computation in the averagedi erence basis 5 Running of the coupling constants and physical meaning
Action in the averagedi erence basis
One loop computations
Lindblad condition is never violated by perturbative corrections
Linearized analysis around the xed point
Numerical analysis of RG equations
5.2.1
5.2.2
5.2.3
5.2.4
I: Re 4 + 2Re 4 > 0, Im
4 + Im
4 > 0 and Re 3 + Re 3 ? 0
II: Re 4 + 2Re 4 > 0, Im
4 + Im
4 < 0 and Re 3 + Re 3 ? 0
III: Re 4 + 2Re 4 < 0, Im 4 + Im 4 > 0 and Re 3 + Re 3 ? 0
{ i {
Reduction and identities due to largest time equations
B.5 Evaluation of BRP (k)
B.5.1
Timelike k : computation of divergences
B.5.2 Spacelike k : computation of divergences
B.5.3
Summary of divergences
B.6 UV divergences and symmetry factors
C PassarinoVeltman diagrams in the averagedi erence basis
C.1 PassarinoVeltman A type integral in the averagedi erence basis
C.2 PassarinoVeltman B type integral in the averagedi erence basis
D Computations in the averagedi erence basis
D.1 Beta functions for the mass terms
D.1.1
D.1.2
D.1.3
D.1.4
D.2.1
D.2.2
D.2.3
D.2.4
D.2.5
D.3.1
D.3.2
D.3.3
D.3.4
D.3.5
2a vertex
4a vertex
3a d vertex
2a 2d vertex
a 3d vertex
4d vertex
Final
function
D.2 Beta functions for the cubic couplings
Final
function
D.3 Beta functions for the quartic couplings
D.3.6 Final
functions
E Tadpoles
{ ii {
49
49
Introduction and motivation
eld theories are one of the great success stories of theoretical physics. From
our understanding of elementary particles of the standard model to current cosmological
models of evolution of the universe, from the theory of critical phenomena to polymer
physics, the range and success of e ective eld theories is wide and diverse. The concept
and the techniques of renormalisation, in particular have become textbook material and
essential tools in the toolkit of many a theoretical physicist. Over the past few decades,
String theory has further enriched this structure with its system of dualities, including the
shocking suggestion that many theories of quantum gravity are really large N quantum
eld theories in disguise.
Despite all these successes, there are a variety of phenomena which still resist a clear
understanding from the standard e ective eld theory viewpoint. A large class of them
involve dissipation and information loss in evolution. It may be because the systems are
open quantum systems in contact with an environment. Or the system might e ectively
behave like an open system because coarsegraining has traced out some degrees of
freedom into which the system dissipates. To tackle these systems, one needs to develop a
quantum
eld theory of mixed states where we can trace out degrees of freedom, run on a
renormalisation ow and study dualities.
This is not a new question. Two of the founders of quantum eld theory  Schwinger
and Feynman addressed these questions early on and made seminal contributions to the
quantum
eld theories of density matrices. These are the notions of a SchwingerKeldysh
path integral [1, 2] and the FeynmanVernon in uence functionals [3, 4]  the rst
addressing how to set up the pathintegral for unitary evolution of density matrices by doubling
the elds and the second addressing how coarsegraining in a free theory leads to a density
matrix pathintegral with nonunitary evolution.
The third classic result in this direction is by Veltman who, in the quest to give
diagrammatic proofs of Cutkosky's cutting rules [5], e ectively reinvented the
SchwingerKeldysh path integral and proved that the corresponding correlators obey the largest time
equation [6, 7]. The fourth important advance towards the e ective theory of mixed states
is the discovery of the quantum master equation by GoriniKossakowskiSudarshan [8] and
Lindblad [9]. The quantum master equation prescribes a speci c form for the
FeynmanVernon in uence functional [3, 4] using the constraints that evolution should preserve the
trace of the density matrix (tracepreserving) as well as keep the eigenvalues of the density
matrix stably nonnegative (complete positivity). We will review these ideas and their
interrelations in due turn. Our goal here is to construct a simple relativistic eld theory
which elucidates these ideas.
Before we move on to the subject of the paper, let us remind the reader of the broader
motivations which drive this work.
First of all, the theory of open quantum systems
is a
eld with many recent advancements and is of experimental relevance to
elds like
quantum optics, cold atom physics, nonequilibrium driven systems and quantum
information. (See [10{14] for textbook treatments of the subject.) It makes logical sense to
test these ideas against relativistic QFTs [15, 16] and how they change under Wilsonian
{ 1 {
Z
+
+ i
Z
Z
2
2
d h
renormalisation.1 Second, open relativistic QFTs are very relevant by themselves in heavy
ion physics [19{22] and cosmology [23{30]. Third motivation is to better understand the
apparently nonunitary evolution engendered by black holes and to give a quantitative
characterization of the information loss. In particular, AdS/CFT suggests that exterior of
black holes is naturally dual to open conformal eld theories. Hence, it is reasonable to
expect that developing the theory of open conformal eld theories would tell us how to
think about horizons in quantum gravity.
In this work, we take a modest step towards answering these questions by setting up
and studying the simplest looking open quantum
eld theory: the open version of scalar
3 + 4 in d = 4 spacetime dimensions. One can characterise the e ective theory of density
1 m2 2R + 33! 3R + 44! 4R + 23! 2R L + 34! 3R L
2
i
d x
1 m2? 2L + 33?! 3L + 44?! 4L + 23?! 2L R + 34?! 3L R
2
i
(1.1)
R L +
antilinear, antiunitary symmetry exchanging
R and L and taking i 7! ( i). It can be
easily checked that, under this antilinear, antiunitary ip eiS remains invariant provided
the couplings appearing in the last line of action fz ; m2 ;
with a future boundary condition identifying
e ective theory which we will study in this paper.
g are real. This action along
R and
L at future in nity de nes the SK
There are two features of the above action which makes it distinct from the SK e ective
action of the unitary
3 + 4 theory. First, there are interaction terms which couple the
ket eld
R with the bra eld
L. Such cross couplings necessarily violate unitarity and
indicate the breakdown of the usual Cutkosky cutting rules. They are also necessarily a
part of `in uence functionals' as de ned by Feynman and Vernon and are generated only
when a part of the system is traced out [3, 4]. A more obvious way the above action
violates unitarity is due to the fact that S is not purely real. If we turn o
all cross
couplings between
R and
L and set to zero all imaginary couplings in S, we recover the
SK e ective action of the unitary
4 theory:
S ;Unitary =
+
Z
Z
2
d x
1 m2 2R + 33! 3R + 44! 4Ri
2
d x
1 m2 2L + 33! 3L + 44! 4Li
2
(1.2)
1We should mention that in the nonrelativistic context, various interacting models and their 1loop
renormalisation have already been studied. We will refer the reader to chapter 8 of [17] for textbook
examples of 1loop renormalisation in nonrelativistic nonunitary QFTs. The examples include
Hohenbergwhere all couplings are taken to be real. Our aim is to deform
4 theory away from this
familiar unitary limit and study the theory de ned in (1.1) via perturbation theory.
The rst question one could ask is whether this theory is renormalisable in perturbation
theory, i.e., whether, away from unitary limit, the oneloop divergences in this theory can
be absorbed into counter terms of the same form. We answer this in a rmative in this
work and compute the 1loop beta functions to be
dm2
d ln
dm2
d ln
=
=
1
for the mass terms,
for the cubic couplings, and
functions have a remarkable property which is made evident by
deriving the 1loop renormalisation running of certain combinations of couplings:
These equations show that the conditions
Im z z = 0 ; Im m2
m2 = 0 ; Im 3 +3 Im 3 = 0 ; Im
orders in perturbation theory to prove that the above conditions are never corrected at any
order in loops. One can think of this as violating GellMann's totalitarian principle [31]
that \Everything not forbidden is compulsory" (or as there being new principles in open
eld theory which forbid some combinations from appearing in perturbation
theory). This kind of netuning of couplings which are still protected under renormalisation
is a hallmark of open quantum
eld theories and is a signature of microscopic unitarity [32].
We will now move to brie y describe the signi cance of the above conditions. We will
give three related derivations of the conditions above in this work:
1. In the SchwingerKeldysh formalism, the microscopic unitarity demands that di
erence operators (i.e., operators of the form OR
OL) have trivial correlators. This, as
a statement about correlation functions, should hold even in the coarsegrained open
e ective eld theory. The decoupling of di erence operators then naturally lead to
the conditions above.
2. Relatedly, while the open EFT is nonunitary, one can demand that a certain weaker
version of Veltman's largest time equation be obeyed. This then leads to the
condiHJEP1(207)4
tions above.
tions above.
3. The trace preserving and the complete positivity of the evolution demands that the
FeynmanVernon in uence functional be of the Lindblad form. Insisting that the
dynamics of the open EFT be of the Lindblad form naturally leads to the
condiSg =
Z
where
Thus, a certain weak form of unitarity still holds in the open EFT and is explicitly realized
by the conditions above. And once these conditions are satis ed, the structure is robust
against perturbative renormalisation.
There is a fourth way of deriving the same conditions, whose deeper signi cance we will
leave for our future work. Say one adds to the above action for the open EFT two
Grassmann odd ghost elds g and g and demand that the following Grassmann odd symmetry
hold for the entire theory:
R =
L = g + g ;
g = ( R
L) ;
g =
( R
L) :
(1.8)
This symmetry then xes the
selfcouplings to obey equation (1.7). Further, the ghost
action is completely xed to be
R L +Y4? 2L
) gg
1
2!
1
3
1
3
1
3
zg = Re z ;
mg2 = Re m2 ;
Y3 = (Re 3 + Re 3) + (Im 3
Im 3) ;
Y4 = (Re 4 + Re 4) + (Im 4 +
) ;
Y
= (Re 4 + 4 Re 4)
i
4
4
i
{ 4 {
(1.9)
(1.10)
If the boundary conditions/initial states are chosen such that the ghosts do not propagate,
our computations of the beta functions still hold. We will leave a detailed examination of
these issues to the future work. We will also not address in this work various other crucial
questions on the derivation of a open EFT: rst is the problem of infrared divergences in
the unitary theory which need to be tackled correctly to yield a sensible open EFT. Second
is the related question of the appropriate initial states and dealing with various transient
e ects.The third question we will comment on but leave out a detailed discussion of, is the
modi cation of the cutting rules in the open EFT. We hope to return to these questions
in the future.
Organization of the paper.
The rest of the paper is organized as follows. In the
rest of the introduction, we will very brie y review the relevant background for our work.
This includes the concepts of SchwingerKeldysh path integrals, their relation to Veltman's
cutting rules, FeynmanVernon in uence functionals for open EFTs and the Lindblad form
for the evolution. The readers who are familiar with these concepts are encouraged to skim
through these subsections in order to familiarize themselves with our notation.
In section 2 we will write down the action for the open EFT and set up the propagators
and Feynman rules. We will also discuss the conditions under which the evolution density
matrix of the theory is of Lindblad form. In section 3 we compute the one loop beta
function for various coupling constants. The result of the section is summarized in 3.11.
In section 4, we rewrite the theory in averagedi erence basis and we illustrate the great
simpli cation that happens in this basis. The details of the computation in this basis can
be found in appendix D. In section 4.3, we present a proof that the Lindblad condition is
never violated under perturbative corrections. Section 6 consists of the conclusion of our
analysis and various future directions. Appendix A describes some of our notations and
conventions. Computation of the various one loop PassarinoVeltman integrals required
for open EFT can be found in appendix B and in appendix C.
1.1
Basics of SchwingerKeldysh theory
The SchwingerKeldysh(SK) path integrals have been reviewed in [17, 19, 33{36]. Here
we will mention some key features: given a unitary QFT and a initial density matrix
(t = ti) = i, we de ne the SK path integral via
ZSK[JR; JL]
TrnU [JR] i (U [JL])y
o
(1.11)
Here, U [J ] is the unitary evolution operator of the quantum
eld theory deformed by
sources J for some operators of the theory. This path integral is a generator of all correlation
functions with at most one timeordering violation. This should be contrasted with the
Feynman pathintegral which can compute only completely timeordered correlators.
One could in principle consider the generating functions for correlators with arbitrary
number of timeordering violations [37] (for example, the correlator used to obtain the
Lyapunov exponent involves two timeordering violations [38]) but, in this work, we will
limit ourselves to just the usual SK pathintegral. The SchwingerKeldysh path integral
gives a convenient way to access the evolution of the most general mixed state in quantum
{ 5 {
HJEP1(207)4
eld theory including the real time dynamics at nite temperature. It is an essential tool in
the nonequilibrium description of QFTs which is directly de ned in Lorentzian signature
without any need for analytic continuation from the Euclidean description.
Given an action S[ ; J ] of the unitary QFT, we can give a pathintegral representation
of ZSK[JR; JL] by introducing a ket eld
R and a bra eld L:
Z R(t=1)= L(t=1)
i( R; L)
[d R][d L] eiS[ R;JR] iS[ L;JL]
(1.12)
The lower limit is the statement that near t = ti the boundary condition for the
pathintegral is weighed by the initial density matrix i. The upper limit is the statement that
the bra and the ket elds should be set equal at far future and summed over in order to
correctly reproduce the trace. The factors eiS[ R;JR] and e iS[ L;JL] correctly reproduce the
evolution operators U [JR] and (U [JL])y respectively.
If the unitary QFT is in a perturbative regime, the above path integral can be used to
set up the Feynman rules [17, 33].
1. In a unitary QFT, there are no vertices coupling the bra and the ket elds. The bra
vertices are complex conjugates of ket vertices.
2. The ket propagator is timeordered while the bra propagator is antitimeordered. In
addition to these, SK boundary conditions also induce a braket propagator which
is the onshell propagator (obtained by putting the exchanged particle onshell). We
will term these propagators as cut propagators. The terminology here is borrowed
from the discussion of Cutkosky cutting rules where one thinks of the dividing lines
between the bra and ket parts of the diagram as a `cut' of the diagram where particles
go onshell.
We will call these rules as Veltman rules after Veltman who rederived these rules in his
study of unitarity [6, 7]. To reiterate, a fundamental feature of Veltman rules is the fact
that in a unitary theory, bra and ket elds talk only via cut propagators but not via cut
vertices. As we will see in the following, this ceases to be true in an open QFT where, as
Feynman and Vernon [3, 4] showed, there are novel cut vertices which signal nonunitarity.
One of the fundamental features of the Veltman rules is a statement called the largest
time equation which is fundamental to Veltman's approach to proving perturbative
unitarity and cutting rules. The largest time equation is a direct consequence of the de nition of
SK path integral in equation (1.11) as reviewed in [34]. We will brie y summarise below
the argument for the largest time equation and its relation to SK formalism. We will refer
the reader to [39] or [34] for more details.
In the SK path integral, consider the case where the sources obey JR = JL = J (x)
beyond a particular point of time t = tf . One can then argue that the pathintegral is in
fact independent of the source J (x) in the future of tf . This follows from unitarity: the
contributions of U [JR] and U [JL]y have to cancel each other in ZSK if JR = JL by unitarity.
To convert the above observation into a statement about correlators, we begin by noting
that the source J (x) couples to di erence operators OR
OL in the SK path integral. If we
{ 6 {
cut propagator of Env. field
×
=
×
×
Unitary QFT
di erentiate the pathintegral (1.12) with respect to the common source J (x), it follows that
one is basically computing a correlator with the di erence operators OR
OL placed in the
with the futuremost (or the largest time) operators as di erence operators OR
Microscopic unitarity thus requires that correlators of purely di erence operators are
trivial and any macroscopic open EFT should faithfully reproduce this condition. One of
the main motivations of this work is to understand how these conditions get renormalized
and the relation of these conditions to the Lindbladian form studied in open quantum
system context.
1.2
Basics of Lindblad theory and e ective theory
Following FeynmanVernon [3, 4], we can integrate out the `environment' elds in the
SchwingerKeldysh path integral and obtain an e ective path integral for the quantum
system under question.
This inevitably induces a coupling between the bra and ket
elds (called FeynmanVernon(FV) coupling in the following) as shown schematically in
the
gure 1.
Here the redline represents the `environment'
elds of FeynmanVernon
which couples to the system
eld via a linear coupling. These `environment' elds when
traced/integrated out induce the unitarity violating FV coupling for the elds describing
the open quantum
eld theory.
Note that the propagator that induces FV coupling is necessarily a cut propagator of
the environment which means that the FV coupling is only induced in the regime where
the `environment'
elds go onshell.
This also explains why, in usual QFT where we
integrate out heavy elds that can never go onshell in vacuum, no FV coupling or e ective
nonunitarity is induced by Wilsonian RG.2 We will assume that the open QFT that we
are studying in this paper arises from some hitherto unspeci ed microscopic theory a la
FeynmanVernon.
The FV couplings induced by integrating out environment elds need not always be
local. A local description for the resultant open QFT is often accomplished by working
with a limit where the time scales in the environment are assumed to be very fast compared
to the rate at which the information
ows from the system to the environment. In this
approximation (often termed BornMarkov approximation), one expects a nice local
nonunitary EFT and our intent here is to study renormalisation in such an EFT. In the context
2Note that this is true about dilute states which are near vacuum state. A counterexample is at
of open quantum mechanical systems, under a clear separation of timescales, one can derive
the Lindblad equation (or the quantum master equation) [8{10] for the reduced density
matrix of the form
i~
d
dt
Equivalently, one can obtain a pathintegral description by adding to the
SchwingerKeldysh action of the system, an in uence functional term of the form [36]
SF V = i
is a positive matrix, one can show that the above
equation describes a dissipative system which keeps the eigenvalues of
nonnegative.
These two properties (along with linearity in ) qualify Lindblad form of evolution as a
physically sensible dynamics describing an open quantum system. The above equation
in Schrodinger picture has an equivalent Heisenberg picture description via an evolution
equation for operators:
dt
i~ dA = [A; H] + i X
Ly AL
Ly L A
1
2 ALy L
:
(1.14)
Here, H is the Hamiltonian of the system leading to the unitary part of the evolution,
whereas the nonunitary (FeynmanVernon) part of the evolution comes from rest of the
terms in r.h.s. The nonunitarity is captured by a set of operators L and a set of couplings
HJEP1(207)4
of the system. It is easily checked that the form above implies
where we have indicated the way the action should be written in terms of the bra and ket
elds in order to correctly reproduce Lindblad dynamics. We note that the Lindblad form
of the in uence functional has a particular structure which relates the
R L crossterms
with the imaginary parts of both the
R action and
L action.
Let us note some important features of the above expression. If we set
is also related to the di erence operator decoupling mentioned in the last subsection in
the context of SchwingerKeldysh path integrals. Thus, trace preserving property in the
Schrodinger picture becomes di erence operator decoupling at the level of SK path integral
for the EFT.
We also note that if we take one of the Lindblad operators say L to be an identity
operator, the Lindblad form then becomes a di erence operator, i.e., it can be written as
a di erence between an operator made of ket elds and the same operator evaluated over
the bra elds. This is the form of SK action for a unitary QFT (cf. equation (1.12)) and it
{ 8 {
merely shifts the system action. But when both Lindblad operators are not identity, one
gets various cross terms and associated imaginary contributions to the pure
R and the
pure L action. Thus, once the cross couplings are determined, one can use the Lindblad
form to determine all imaginary couplings. This is the route we will take to write down
the Lindblad conditions like the ones in equation (1.7).
Having
nished this brief review of the necessary ideas, let us turn to the open
4
theory whose renormalisation we want to study. We will begin by describing in detail the
e ective action and the associated Feynman rules in the next section.
2
Introduction to open e ective theory
Let us begin by writing down the action for the most general open quantum
eld theory,
consisting of a real scalar which can interact via cubic and quartic interactions, given
in (2.1). The most general action, taking into account CPT symmetry(See for example, [34])
and SK boundary conditions, is given by
HJEP1(207)4
S =
Z
Z
Z
+
+ i
Z
ddx
ddx
ddx
1
2
2
1
2
1
2
m2 2R +
m2? 2L +
44! 4R +
Imposing CPT and demanding that the action (2.1) should be of the Lindblad form, we get
four constraints among the coupling constants  one for eld renormalisation, one for the
mass, one for the cubic coupling and one for quartic coupling terms. We begin by tabulating
all the power counting renormalisable Lindblad terms in the 3 + 4 theory in table 1. Also
tabulated are the conditions resulting from insisting that our action be of Lindblad form
(we call these the Lindblad conditions). We will now consider various parts of the action
in turn and rewrite them in a way that the Lindblad conditions become manifest.
Real terms of the action.
The real part of the action is given by
Re [S] =
Z
Z
Z
ddx
ddx
ddx
2
1
2
Re 4
Re 3
4!
3!
h
4
R
( 3R
4L +
3L) +
Re 4
3!
Re 3
2!
2i +
1
2
2
R
R L
R L( R
Re m2
2
R
2
L
2
L
L)
We note that CPT constrains this action to vanish when
no conditions on these real couplings from the Lindblad structure.
Im z
Im m2
4
4
4
Im
Im
Im
Im 3
Im 3
m2
1
2
Im 3 +
Im 3 = 0
3!
2!
) Im
3 + 3Im
3 = 0
Lindblad couplings
Im z = z
Im m2 = m2
Im
Imaginary quadratic terms of the action.
The imaginary part of the quadratic terms
ddx
ddx
ddx
2
z
Im m2
2R + 2
L
R L
=
L)
2
+
ddx (z
Im m2
R L
The Lindblad condition is given by
z
= Im [z] ;
m2 = Im [m2] ;
Imaginary cubic coupling.
Now we compute the imaginary part of the cubic terms in
the action
Im [S3] =
=
Z
Z
ddx
ddx
3!
3!
Im 3
( R
Im 3 3R +
Im 3 3L +
3!
L)( 2R
Im 3 2
2!
2L)+
The Lindblad condition is given by
ddx
ddx
ddx
1
4!
1
4!
2!2!
The imaginary part of action at the level quartic coupling
The Lindblad condition at for the quartic couplings is given by
L in SK pathintegral satisfy the following boundary
Owing to this boundary condition and the mixing term between
R and L elds, the
kinetic matrix derived from the action (2.1) is given by
K =
i(z k2 + m2
z k2 + m2
i")
2 "
(k0)
z k2 + m2
2 "
( k )
0 !
i(z? k2 + (m2)? + i")
where the " prescription implements SchwingerKeldysh boundary conditions. We de ne
the kinetic matrix K by
1
2
iS 3
( R ( k) L ( k)) K
R(k) !
L(k)
2!2!
)Im
= 2 Im
4 + 4Im
where,
z
1+
= z 1
+z 1
Its inverse (viz., the propagator) can be written as
h R( k) R(k)i h R( k) L(k)i !
h L( k) R(k)i h L( k) L(k)i
i
Re[zk2+m2] i"
2
(Re[zk2 +m2]) (k0)
( i)
Re[zk2 +m2] i"
Re[zk2 +m2]+i"
Im[zk2 +m2] z k2 +m2
z k2 +m2
Im[zk2 +m2]
!
2
i
(Re[zk2 +m2]) ( k0) !
Re[zk2+im2]+i"
Propagator R
Propagator P
Propagator M Propagator L
p2+m−i2−iε
2πδ+(p2 + m2)
2πδ−(p2 + m2)
p2+mi2+iε
p
p
p
Please note that when the Lindblad conditions (2.4) are satis ed, we have
Further, it can be easily checked that in this limit, the sum of diagonal entries in the
propagator matrix is equal to the sum of o diagonal entries, i.e.,
K
1
RR +
K
1
LL =
K
1
RL +
K
1
LR
The corresponding property in the unitary quantum eld theory is the wellknown relation
between the various correlators in the Keldysh formalism [17]. This can equivalently be
reformulated as the vanishing of two point function of two di erence correlators:
K
1
R L;R L = 0:
In this work, we will work in the limit where the nonunitary couplings Im[m2] and m2 are
considered as perturbations to Re[m2], and similarly, Im[z2] and z
2 are considered small
compared to Re[z2]. Further, since 1loop correction to the propagators do not generate
eld renormalisation we can also set z = 1. In this limit, the propagators in equation (2.12)
reduced to those given by gure 2.
2.3
Feynman rules
In this paper henceforth, we will set z = z
= 1 (which is not renormalised at oneloop
in d=4 dimensions). We will treat all other parameters in our action except the real part
of m2(i.e., Re(m2)) perturbatively. This includes 3
, 3
, 4
, 4 and
, as well as Im m2
and m2 .
The propagators of
elds are given below. We have used solid blue and dotted
blue lines for
R(ket
elds) and
L(bra
elds)
elds respectively. Note that in the cut
propagators P and M the energy is restricted to ow from the ket eld to the bra eld.
We will now set up the Veltman rules for the vertices to compute SK correlators in
the open 3 + 4 theory (see table 2 and gure 3).
(2.14)
(2.15)
(2.16)
2R L
R 2L
3R L
R 3L
2 2
R L
( i 3)(2 )d (P p)
(i ?3)(2 )d (P p)
( i 3)(2 )d (P p)
(i 3?)(2 )d (P p)
( i 4)(2 )d (P p)
(i ?4)(2 )d (P p)
( i 4)(2 )d (P p)
(i 4?)(2 )d (P p)
(
) (2 )d (P p)
iλ?3
−iλ4
iσ4?
−iσ3
iλ?4
−λΔ
In a unitary SchwingerKeldysh theory, the correlator of di erence operators vanishes to
all order in perturbation theory. This is equivalent to Veltman's largest time equation (see
for example [34]). One could ask whether this statement continues to hold true in the
nonunitary theory. We have already remarked during our discussion of propagators around
equation (2.12) that the quadratic Lindblad conditions are equivalent to the vanishing
of di erence operator two point functions. We can extend this to higher point functions
simply. Consider the tree level correlator of three di erence operators
h( R(p1)
=
i( 3
L(p1))( R(p2)
3
)
the correlator of four di erence operators is given by
h( R(p1)
=
i( 4
L(p1))( R(p2)
4
)
4i( 4
L(p2))( R(p3)
L(p3))( R(p4)
L(p4))i
4
)
6
= 2(Im 4 + 4 Im 4
3
)
The correlators of the three and the four di erence operators are precisely given by the
Lindblad violating couplings. This implies that at tree level, the Lindblad conditions are
the same as the vanishing of correlators of di erence operators.
One can, in fact, show the following statement [32]: consider an open EFT,
which is obtained by tracing out some subset of elds in an underlying unitary theory.
Then, the unitarity of the underlying theory implies that the open EFT satis es the
(2.17)
(2.18)
Lindblad condition.
3
One loop beta function
In this section, we compute the beta function for all the mass terms and coupling constants
that appear in the action of the open
to demonstrate the following three claims:
3 +
4 theory. The main aim in this section will be
1. Despite the novel UV divergences that occur in the open 3 + 4 theory, one can use a
simple extension of the standard counterterm method to deal with the divergences.
Thus, the open 3 + 4 theory is oneloop renormalisable.
2. Once these UV divergences are countered, the standard derivation of beta functions
and RG running also goes through, except for the fact that one has to now also
renormalise the nonunitary couplings.
3. We will also demonstrate that the running of a certain combinations of the couplings,
the ones which given by the Lindblad conditions (equation (2.4), equation (2.6) and
equation (2.8) respectively), under oneloop renormalisation are proportional to the
Lindblad conditions.
We shall provide an allorder proof in the next section that the Lindblad conditions are
never violated under perturbative corrections. Here we shall use the notations and results
presented in appendix B.
(−iλ3)2 BRR(k)
2
(iσ3∗)2 BLL(k)
(−iσ3)2BLR(k)
(iσ3∗)(−iλ3) BP M (k)
(iσ3∗)(−iλ3) BM P (k)
(−iσ3)2BP P (k)
HJEP1(207)4
2
(−iλ3)(−iσ3)BP R(k)
(−iλ3)(−iσ3)BM R(k)
(iσ3∗)(−iσ3)BP L(k)
(iσ3∗)(−iσ3)BM L(k)
One loop beta function for m2
We will now begin a discussion of various loop diagrams. The simplest is perhaps the
tadpole diagrams which can be cancelled by a counterterm linear in
R and
L. It is
easily demonstrated that the necessary counterterms do not violate the Lindblad condition
(See appendix E).
Let us compute the one loop beta function for m2. We shall consider all the one loop
Feynman diagrams that contributes to the process
R !
R. One can verify that there
are mainly two types of diagrams  one class of diagrams due to the cubic couplings, as
depicted in
gure 4, and the other class of diagrams due to quartic couplings, depicted in
gure 5 (the corresponding counter term is shown in gure 6).
2
(−iλ4) AR
+
+
im2
( i 3)
2
2
2
2
(−λΔ) AL
1
2
−iδm2
×
×
2
The sum of the contribution from all the Feynman diagrams is given by
One loop beta function for m2
Now, we will compute the one loop beta function for m2 . As in the case of m2, there will
again be two classes of diagrams. The diagrams due to cubic and quartic couplings are
Using the results in (B.79a){(B.79d), we can see that the contribution is divergent and one
needs to add one loop counterterms m2, in the MS scheme, to absorb the divergences.
m2
MS
=
1
1
(4 )
(4 )
2
h
2 ( 3)
Using the standard methods of quantum
eld theory, one can then compute the one loop
beta function as
m2 =
1 h
(4 )2 ( 3)
2
If set 3 = 4 =
spacetime dimensions.
( 3?)2 + 2
3 3 + j 3j
2
+ ( 4
i
+ 2 4) Re m2 i
= 0, then we get back the standard results of 3 + 4 theory in d = 4
as shown in gure 7 and in gure 8 respectively. The sum over all the contributions is
Some of these one loop contributions are divergent and one needs to add one loop
counterterms. The m2 counterterm in MS scheme is given by
4(Re 3 + Re 3)Im 3 + (2
2Im 4)(Re m2)i
m2
MS
=
1 h
1
d
4
1
2
+ ( E
and the beta function for m2 is given by
m2 =
1
Checking Lindblad condition for mass renormalization
From equation (3.3), we nd that the beta function for Im m2 is given by
d (Im m2)
dln
1 h
=
(4 )2 2(Re 3 + Re 3)(Im 3 + Im 3)
+ (Im 4 + 2 Im 4
)(Re m2)i
Now, using equation (3.7) and equation (3.6), one gets the beta function for (Im m2
2
(Im m2 m2 ) =
(4 )2 [(Im 3 +3 Im 3)(Re 3 +Re 3)+(Im 4 +4Im 4 3
)(Re m2)]
equation (3.8) shows that the one loop beta function for Lindblad violating mass terms
vanish in the absence of Lindblad violating cubic (equation (2.6)) and quartic coupling
(equation (2.8)) at the tree level.
3.4
One loop beta function for
3
Now we will compute the one loop beta function for various cubic couplings. The
PassarinoVeltman C and D integrals will have no contribution to the one loop
function for the
(3.4)
(3.5)
(3.6)
(3.7)
m2 )
(3.8)
A further reduction is possible using largest time equations and their concomitant
cutting rules:
BRR(k) + BRL(k) = BRP (k) + BRM (k)
BLR(k) + BLL(k) = BLP (k) + BLM (k)
BP R(k) + BP L(k) = BP P (k) + BMM (k)
BMR(k) + BML(k) = BMP (k) + BRM (k)
From applying these identities, we can conclude that Re[BRP (k) + BRP ( k)] and
BRP ( k)] is symmetric under m $ m exchange. Further, the real part(viz.,
the cut) of BRP (k) integral is given by
BRP (k)+BRP (k) = Re BR+L (k)
k
2
(B.54)
(B.55)
(B.56)
(B.57)
(B.58)
(B.59)
These conditions in turn lead to the identities:
=
Vol Sd 2 "
1+2
+
k
2
(4 )
2
2
+
m)2 +k2 + ( k0)
( k2)
BRP (k) + BLP (k) = BMR(k) + BML(k)
BRP (k) + BLM (k) = BMR(k) + BP L(k)
BRP (k) + BP R(k) = BRM (k) + BMR(k)
BP R(k) + BP L(k) = BRM (k) + BLM (k)
BP R(k) + BML(k) = BRM (k) + BLP (k)
BLP (k) + BP L(k) = BLM (k) + BML(k)
BRP (k) + BML(k) = BRP (k) + [BRP (k)]m$m
BRP (k)m$m = BRP ( k)m$m
From, these identities we get
which in turn obey
BR+L(k)
BR+R(k) = BRP ( k)
BR+R(k) + BR+L(k) = BRP (k) + BRP ( k) = BRP (k) + BRM (k)
Another implication is
Re BR+R(k) =
Vol Sd 2 "
+
m2
k2
m2 2# d2 3
1
2
(k0)
( k0)
(m
m)2
k
2
( k2)
k
2
(4 )2
2
The following combination is symmetric under m $ m as well as k $
k exchange.
BRP (k) + BRP (k) + BP M (k)
=
Vol S
( k2)
k
2
d
2
2
(B.60)
The basic integral BRP (k) has the following form on the timelike case:
=
4 d Vol(Sd 2) Z 1
m
(!p2
d!p !2
p
(!p2
d!p 2M (!p
m2) d 2 3
(M + !p)2
i"
m2) d 2 3
!p)
i"
where, we have de ned
with
The same integral in the spacelike case takes the form
!
p
4 d Z
2Q
q
jj
m2
m2
m2
2M
m2 + Q2
2Q
:
BRP (k) =
dd 1
q
1
1
(2 )d 1i 2!q qjj + qjj
i"
Our aim in this subsection is to evaluate these integrals and extract out the appropriate
divergences.
B.5.1
Timelike k : computation of divergences
We begin by setting !p = m cosh in the timelike case to get
BRP (k) =
4 d Vol S
d 2 md 3 Z 1
4M (2 )d 1 i
where we have de ned
=
Vol S
d 2
m
M
=
p
m
m2
2
4
m2
cosh
d
i d=2 1
i"
:
m2
2M m
cosh
(B.61)
(B.62)
(B.63)
(B.64)
Thus, we have reduced our analysis to the integral
F ( )
0 i d=2 1
cosh
(B.65)
This integral can then analyzed in detail to study the analytic structure of this integral.
But, for our purposes, it is su cient to extract the divergences.
For our computation of
function, we need to extract out the divergent part of these
integrals. Focusing on the large !p contribution, we can approximate BRP (k) by
m2
4
2
d!p !
2
p
2
m2 d2
16m
3M
3 d
2
+
p
1
2M !p4
M 2 +m2 m2
4M 2!p5
+O !p 6
4 d M 2 +m2 m2
2
+: : :
(B.66)
16m
3M
+
M 2 +m2 m2
2M 2
2
4m2
4
2e E
1
2
+: : :
(B.67)
4 d Z
2Q
4 d Vol(Sd 2) Z 1
1
(2 )d 1i 2!q qjj + qjj
i"
=
4 d
d!q? !q?(!q2
?
d!p 2M (!p
m2) d2 3
!p) i"
(!p2
1
m2
q
jj
Near d = 4, this gives
i
(4 )2
so that
given by
where, we have de ned
B.5.2
Spacelike k : computation of divergences
In this subsection, we will consider the spacelike case and con rm that the divergence
structure is same in the spacelike case. Let us rst get the real part of BRP (k), which is
m2) d2 4 Z 1
1
1
dqjj q!q2 + qj2j qjj + qjj
?
1
(B.68)
(B.69)
i"
Let us rst get the real part of BRP (k), which is given by
d!q? !q? (!q2
1
qjj + qjj
d!q? !q? (!q2
?
i"
?
real part
4 d
4 d
Z 1
The above equation can easily be seen to be equal to BP P (k) for the spacelike case
in (B.22). So, we have in the spacelike case
Let us now get the imaginary part of BRP (k), which is also the divergent part. It is
=
i
+ ln
4
m2
2e E
1
+ : : : : (B.72)
+ ln
4
m2
2e E
1
+ : : : :
(B.73)
We see that these divergences are same as the timelike case.
B.5.3
Summary of divergences
We now summarize the divergences in various integrals. We start with
and
=
i
(4 )
2
k
2
2k2
div BR+L
div BR+R
MS
MS
=
=
i
i
(4 )
(4 )
2
2
m2
m2
k2
2
d
4
d
2
2
d
4
+ ln
4
1
+ ln
1
4 e E
1
4 e E
+ ln
4 e E
dqjj q!q2 + q2
jj
1
?
2 i (qjj + qjj)
MS
MS
MS
MS
i
i
2
4
+ ln
d
4
4
4
+ ln
+ ln
2
2
1
+ ln
4 e E
1
1
4 e E
4 e E
1
4 e E
When m = m, we can thus summarize the divergence of `quartercut' integrals as
div[BRP ]
div[BLP ]
MS
MS
= div[BP R]
= div[BP L]
MS
MS
= div[BRM ]
= div[BMR]
= div[BLM ]
= div[BML]
MS
MS
This along with
MS
MS
=
=
h 2
(4 )2
h 2
4
i
(4 )2
h 2
d
+ ln
4
+ ln
1
4 e E
1
4 e E
summarizes all the divergences needed in this work.
B.6
UV divergences and symmetry factors
In this subsection, we will collect the UV divergences of various B type diagrams for the
convenience of the reader.
div[BRR(k)]
div[BLR(k)]
div[BP P (k)]
div[BP R(k)]
MS
MS
MS
MS
=
= 0
= 0
=
1
2
i
2
(4 )2 d
4
+ ln
1
4 e E
div[BRR(k)]
MS
Further, we have
MS
MS
MS
= div[BLL]
?
= div[BP M ]
= div[BMR]
= div[BP L]
= div[BML]
= div[BMP ]
= div[BP P ]
MS
MS
MS
MS
= 0
=
1
2
MS
We also give below the symmetry factors of the corresponding diagrams in gure 28. The
divergences given above along with the symmetry factors provides a quick way to write
down appropriate
functions for the open QFT. In the ensuing gure 29 and gure 30, we
tabulate a set of useful diagrammatic identities which relate the various SK loop integrals.
(B.76)
(B.77)
(B.78)
(B.79a)
(B.79b)
(B.79c)
(B.79d)
(B.80a)
(B.80b)
(B.80c)
×
1
×
× + ×
× + 2 ×
×
BRR(k)
BLL(k)
2BLR(k)
=
×
× + ×
× + 2 ×
×
BP M (k)
BM P (k)
2BM M (k)
× + ×
× = ×
× + ×
BRR(k)
BLR(k)
BP R(k)
BMR(k)
× + ×
× = ×
× + ×
BP M (k)
BP P (k)
BP R(k)
BP L(k)
× + ×
× = ×
× + ×
BMP (k)
BMM (k)
BMR(k)
BML(k)
× + ×
× = ×
× + ×
BLL(k)
BLR(k)
BP L(k)
BML(k)
×
×
×
×
Complex
Conjugation
Aa
Ab = Af = 0
PassarinoVeltman diagrams in the averagedi erence basis
Let us now take a look at the PassarinoVeltman diagrams in averagedi erence basis.
It's worth remembering here that only three out of the four propagators, in this basis,
are nonvanishing: the `d' propagator vanishes. This means that we have lesser number
of nonvanishing diagrams in this basis. As a matter of fact, some of the nonvanishing
diagrams (in averagedi erence basis) do not diverge. All these facts add up to give only
a few divergent one loop diagrams  only one A type and two B type integrals. Thus,
computations for the beta functions greatly simpli es in this basis. We will not try to
evaluate the PV integrals from scratch.
We will express the integrals in the
averagedi erence basis in terms of the integrals in the R L basis and then, use the results from
the previous sections to determine the former.
C.1
PassarinoVeltman A type integral in the averagedi erence basis
There are two A type PV integrals in this basis: Aa and Ab = Af (see gure 31). Using
the relations in equation (4.3) and equation (B.3){(B.4), it's easy to check that we get
1
2
(4 )2 (d
4)
+ ln
4
m2
2e E
1 m2
1
2
Aa =
(AR + AL) =
Ab = AR
Af = AR
AM = 0
AP = 0
(C.1)
(C.2)
HJEP1(207)4
C.2
PassarinoVeltman B type integral in the averagedi erence basis
There are six PV B type integrals in this basis: Baa, Baf , Bab, Bbf , Bfb and Bff = Bbb.
The corresponding Feynman diagrams can be found in gure 32.
Baa =
Baf =
Bab =
1
4
1
2
1
2
(BRR + BRL + BLR + BLL)
(BRR
(BRR
Baa
Bbf
div[Baf ]
div[Bab] MS
div[Bfb] MS
div[Bbf ]
div[Bff ]
MS
MS
MS
MS
= 0
1
2
1
2
Bab
Bf b
×
Baf
Bf f = Bbb = 0
×
To compute the divergences for the abovementioned integrals, we use the results given in
equations (B.79a){(B.80c) in section B.6. So, we have
i
2(4 )2 d
2(4 )2 d
2
2
4
4
+ ln
+ ln
1
1
4 e E
4 e E
(C.3)
= div[Bbb] MS
= 0
We shall use these results for the computations in section 4 and in the next section.
D
Computations in the averagedi erence basis
In section 4, we have already computed the beta functions for the Lindblad violating
combinations in the averagedi erence basis and we found that it matches with our computations
in the
R L basis. For the sake of completion, we calculate the beta function for rest of
the mass terms and the rest of the coupling constants in this basis. This computation
enables one to verify the beta functions computed in
R L basis. We shall start o by
providing the set of Feynman rules in this basis.
The propagators in this basis are given in equation (4.3). The vertex factors in this
basis are given by
Beta functions for the mass terms
In order to compute the beta functions of the masses Re m2; Im m2 and m2 , we need
to compute three di erent correlators. As usual, we omit all the nite terms which are
irrelevant for beta function computation.
We have chosen the following three correlators:
D.1.1
2a vertex
First we consider a !
a via 2a vertex. There are three divergent one loop contribution,
as depicted in the rst row of the gure 34. The total contribution is given by
2(Im m2
m2 ) + (2 diagrams)
2(Im
d
Re m2
3)
2
4
d
+ ln
2
4
1
4 e E
+ ln
1
4 e E
where, the rst term is from tree level contribution and the rest are from loop level. Thus,
from equation (D.1), the beta function for (Im m2
m2 ) is given by
(D.1)
(D.2)
d
dln
Next we consider a !
a via a d vertex. It has same divergent diagrams as that of (D.1),
but with di erent vertex factors. The corresponding Feynman diagrams are depicted in
(4 )2 (Im 3 +3 Im 3)(Re 3 + Re 3)+
(4 )2 (Im 4 +4 Im 4 3
Re m2
×
2(Im m2 − m2Δ)
×
(−i)Re m2
×
12 (Im m2 + m2Δ)
2(Im λ3 + 3 Im σ3)
(−i)(Re λ3 + Re σ3)
12 (Im λ3 − Im σ3)
HJEP1(207)4
1
1
2
−4i (Re λ3 − 3 Re σ3)
3
2
3
2
3
4
1
2
×
i
1
1
2
3
4
2
2(Im λ4 + 4 Im σ4 − 3λΔ)
(−i)(Re λ4 + 2 Re σ4)
12 (Im λ4 + λΔ)
1
2
×
3
4
(−4i) (Re λ4 − 2 Re σ4)
18 (Im λ4 − 4 Im σ4 − 3λΔ)
the second row of the gure 34. The total contribution is given by
( i)Re m2 + ( i)(Re 3 + Re 3) ( i)(Re 3 + Re 3) 2(4 )2 d
4
+ ln 4 e E
1
+ 2(Im 3 + 3 Im 3)
+ ( i)(Re 4 + 2 Re 4) R2(e4 m)22 d
2
12 (Im 3
Im 3) 2(4 )2 d
4
+ ln 4 e E
i
1
2
4
+ ln 4 e E
1
(D.3)
1
2
(Im m2 + m2 ) + (2 diagrams)
(Im
3
Im
Re m2
d
2
4
3)
2
+ ln
+ ln
1
4 e E
1
4 e E
The Feynman diagrams for a !
a via
2d vertex are depicted in the third row of the
gure 34. The tree level and one loop contributions are given by
Im 3)(Re 3 + Re 3) +
Re m2
(4 )2 (Im 4 +
) (D.6)
(Im
4 +
1
2
2
dln
(Im m2 + m2 ) =
(4 )2 (Im 3
D.1.4
Final
function
tions (D.2), (D.4) and (D.6)
dm2
dln
dm2
dln
(4 )2 Im
From this equation we obtain the beta function for (Im m2 + m2 )
The beta functions of m2 (= Re m2 + i Im m2) and m2 can be obtained from the
equaAgain, the rst term is the tree level and the rest are the one loop contributions. Thus,
from (D.3) the beta function for (Re m2) is as follows
dln
Re m2 =
1
(4 )2 (Re 3 + Re 3)
2
1
Re m2
(4 )2
(Re 4 + 2 Re 4)
D.1.3
2d vertex
3)(Im
Im
3)
( 3 + 3)( 3 + 3 + 2i Im
3) +
i
i
i
(D.4)
(D.5)
(D.7)
(D.8)
+(3 channels) 2(Im 3 +3 Im 3) ( i)(Re 4 +2 Re 4)
+(3 channels) 2(Im 4 +4 Im 4 3
) ( i)(Re 3 + Re 3)
i
2
i
3a vertex
2(Im 3 +3 Im 3)
D.2
Beta functions for the cubic couplings
We have four cubic coupling constants and the corresponding vertices are 3
a
, 2a d
, a 2d,
3d and we need to compute four correlators. In each case, we will keep only the divergent
The tree level and one loop Feynman diagram for d !
the rst row of the gure 35. These contributions are given by
d d via 3a vertex is depicted in
φa
d
dln
(4 )2 (Im
3
(4 )2 (Im
3
1
2
(4 )2 (Im
(4 )2 (Im
3 + 3 Im
3
(Im
3) =
3)(Re 4 + 2 Re 4)
2
a d vertex
Now we compute d !
( i)(Re 3 + Re 3)
a a via 2a d vertex. The relevant tree level and one loop Feynman
diagrams are shown in the second row of the gure 35 and these contributions are given by
+(3 diagrams) ( i)(Re 3 + Re 3) ( i)(Re 4 +2 Re 4)
+(1 diagram) 2(Im 4 +4 Im 4 3
(Im 3 Im 3)
1
2
+(2 diagram)
(Im 4 +
) 2(Im 3 +3 Im 3)
2(4 )2
i
2(4 )2
+ln
2
d 4
2
d 4
1
4 e E
+ln
+ln
1
1
4 e E
4 e E
(D.10)
This implies that the beta function for (Re 3 + Re 3) is given by
dln
(Re 3 + Re 3) =
(4 )2 (Re 3 + Re 3)(Re 4 + 2 Re 4)
4 +
)(Im
3 + 3 Im
Im
3)
(D.11)
3
3)
dln
d
dln
2
1
1
2
3
The contribution (upto one loop) to a !
a a via a 2d vertex is given by
1
2
(Im
3
Im
3)
+(3 diagrams) ( i)(Re 3 + Re 3)
(Im
4 +
+(2 diagrams) ( i)(Re 4 +2 Re 4)
(Im
( 4i) (Re 4 2 Re 4) 2(Im
3 +3 Im
3)
1
2
1
2
)
Im
3)
3
+ln
4 e E
1
4 e E
+ln
1
4 e E
(D.12)
The corresponding Feynman diagram can be found in the third row of the gure 35. Hence
the beta function for (Im
3
Im
3) is as follows
(Im
3
Im
3) =
(4 )2 (Re 3 + Re 3)(Im
4 +
)
(4 )2 (Re 4 + 2 Re 4)(Im
(4 )2 (Re 4
2 Re 4)(Im
3 + 3 Im
3)
3
Im
3)
(D.13)
of 3
, 3
dln
d 3 =
dln
d 3 =
(4 )2
3 h
1 h
(4 )2 ( 4 + 2 4)( 3
4( 3 + 3) + 4( 3 + 3) + i
( 3
3
i
3) + 4( 3 + 2 3 + 3 3)
i
( 3 + 2 3 + 3 3)
3d vertex
4
i (Re 3 3 Re 3)
The tree level and the one loop Feynman diagrams for a !
a a via 3d vertex is depicted
in the fourth row of the gure 35 and the contribution from these diagrams are given by
1
2
+(3 channels) ( i)(Re 3 + Re 3)
(Im
4 +
(Im
3
Im
3)
( 4i) (Re 4 2 Re 4)
+ln
1
4 e E
2
4 e E
+ln
2
1
which leads to the following beta function for (Re 3
3 Re 3)
(Re 3
3 Re 3) =
(4 )2 (Re 3 + Re 3)(Re 4
2 Re 4)
3
(4 )2 (Im
4 +
)(Im
3
Im
3)
D.2.5
Final function
From the equations (D.9), (D.11), (D.13) and (D.15) we can compute the beta functions
(D.14)
(D.15)
i
(D.16)
1
1
1
2
3
2
3
2
3
2
1
2
2
2
2
1
1
1
φa
φaφa
Beta functions for the quartic couplings
In this subsection we compute the beta functions for the quartic couplings in the
averagedi erence basis. There are ve di erent quartic coupling constants: these are the coupling
constants multiplying the operators 4a
, 3a d
, 2a 2d
, a 3d
, 4d. For all these couplings there
are two distinct divergent diagrams (similar to the cubic coupling constants). As in the
last subsection, we keep only the divergent terms from the one loop contributions.
D.3.1
4a vertex
We start by computing d d !
in the rst row of the gure 36 and the corresponding contributions are given by
d d via 4a vertex. The Feynman diagrams are depicted
2(Im
4 + 4 Im
4
3
)
2(Im
4 + 4 Im
4
( i)(Re 4 + 2 Re 4)
2(4 )2
2
d
4
+ ln
3
1
4 e E
)
(D.17)
dln
D.3.2
dln
dln
D.3.3
3
a d vertex
2a 2d vertex
Thus the beta function for (Im
) is given by
6
) =
)(Re 4 + 2Re 4) (D.18)
Next we compute d d !
a d via 3a d vertex, which is depicted in the second row of
the gure 36. The tree level and one loop contributions from these Feynman diagrams are
(Re 4 + 2 Re 4) + (3 channels)
2(Im
(Im
4 +
2(4 )2
2
+ ln
( i)(Re 4 + 2 Re 4)
+ ln
1
4 e E
)
1
4 e E
we can determine the beta function for (Re 4 + 2 Re 4)
(Re 4 + 2 Re 4) =
(43)2 h(Re 4 + 2 Re 4)
2
(Im 4 + 4 Im 4 3
)(Im 4 +
(D.19)
)
(D.20)
(D.21)
D.3.4
Here we determine the tree level and one loop contribution to a a !
The Feynman diagrams can be found in the fourth row in gure 36 and the corresponding
a a via a 3d vertex.
The tree level and one loop Feynman diagrams for a a !
depicted in the third row of the gure 36, contributes as follows
a a via 2a 2d vertex, which is
1
2
(Im
4 +
+(3 channels)
+(2 channels)
dln
1
2
1
2
(Im
(Im
4 +
4 +
( i)
4
1 h
(4 )2 5(Im
(Re 4 2 Re 4)
From this equation we evaluate the beta function for (Im
) ( i)(Re
+2 Re 4)
) ( i)(Re 4 +2 Re 4)
i
4 +
d 4
),
2
d 4
2
d 4
+ln
+ln
+ln
1
1
4 e E
4 e E
1
4 e E
(Im
4 +
) =
4 +
)(Re 4 + 2 Re 4)
i
(D.22)
contributions are
( i)
4
(Re 4
2 Re 4)
1
2
( i)
4
(Im
4 +
(Im
4 +
+ (3 channels)
(Re 4
2 Re 4)
1
2
+ ln
1
4 e E
( i)(Re 4 + 2 Re 4)
4
+ ln
1
4 e E
(D.23)
The beta function for (Re 4
2 Re 4) is given by
dln
D.3.5
(Re 4 2 Re 4) =
4d vertex
)
The last computation of this section is a a !
a a via 4d vertex, which is depicted in the
fth row in gure 36. The tree level and one loop contribution to this process is given by
1
8
(Im
4
4
3
+ (2 diagrams)(3 channels)
(Im
( i)
4
(Re 4
2 Re 4)
1
2
2(4 )2
4 +
2
4
+ ln
1
4 e E
this equation determines the the beta function for (Im 4 4 Im 4 3
) and it is given by
6
(4 )2 (Im
(Im
4
4
3
) =
4 +
(Re 4
2 Re 4)
(D.26)
D.3.6
Final functions
From the ve equations  (D.18), (D.20), (D.22), (D.24) and (D.26) we can determine the
beta functions of 4 (= Re 4 + i Im
4), 4 (= Re 4 + i Im
4) and
and they are
dln
given by
dln
d 4 =
d 4 =
dln
dln
(4 )2
(4 )2
1
1
i
(D.27)
3 h 24 + 2 4( 4 + i
3 h 42 + ( 4 + 4)( 4
) + 2 i
) + 2 i
h ( 4 + i
h
( 4 + 2 4)( 4 + i
(4 )2
4( 4
c:c:i
2 42 + 5i 4
) + 2( 4)2 + 5i 4
i
( 4 + 4 + 4 + i
i
vertex
2
1
2
1
2
1
2
1
2
4
3
4
3
4
3
4
3
4
4
4
4
4
4
{ 77 {
φaφa
2
(−iλ3) AR
2
(iσ3?) AL
HJEP1(207)4
(E.2)
iM = +
iRe m2
(4 )2
3
2
2
E
Tadpoles
In this appendix we compute various one loop tadpoles in this theory.
R 1 loop tadpole.
The Feynman diagrams contributing to one loop tadpole of R is
being drawn in gure 37. The sum of the contribution from all the Feynman diagrams is
Now using result from appendix B we get the following divergent contribution
( i 3) AR +
(i 3?) AL + ( i 3)AM
2
3? + 2 3
2
2
d
4
+ ln
1
2e E
This can be removed a counterterm of the form
iRe m2
(4 )2
Re 3 + Re 3 + i(Im 3 + 3 Im 3)
d
4
+ ln
4
1
2e E
Checking lindblad condition.
Now we want to check whether the counterterms that
were added to remove the tadpoles satisfy the Lindblad condition or not.
i
h
?i =
1
(4 )2 [Im 3 + 3 Im 3
]
2
d
4
+ ln
1
2e E
Re m2
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