Renormalization in Open Quantum Field theory I: Scalar field theory

While the notion of open quantum systems is itself old, most of the existing studies deal with quantum mechanical systems rather than quantum field theories. After a brief review of field theoretical/path integral tools currently available to deal with open quantum field theories, we go on to apply these tools to an open version of $\phi^3$ + $\phi^4$ theory in four spacetime dimensions and demonstrate its one loop renormalizability (including the renormalizability of the Lindblad structure).


Introduction and Motivation
Effective field 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 effective field 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 field theories in disguise.
Despite all these successes, there are a variety of phenomena which still resist a clear understanding from the standard effective field 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 effectively behave like an open system because coarse-graining has traced out some degrees of freedom into which the system dissipates. To tackle these systems, one needs to develop a quantum field theory of mixed states where we can trace out degrees of freedom, run on a renormalisation flow and study dualities. This is not a new question. Two of the founders of quantum field theory -Schwinger and Feynman addressed these questions early on and made seminal contributions to the quantum field theories of density matrices. These are the notions of a Schwinger-Keldysh path integral [1,2] and the Feynman-Vernon influence functionals [3,4] -the first addressing how to set up the path-integral for unitary evolution of density matrices by doubling the fields and the second addressing how coarse-graining in a free theory leads to a density matrix path-integral with non-unitary 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], effectively reinvented the Schwinger-Keldysh path integral and proved that the corresponding correlators obey the largest time equation [6,7]. The fourth important advance towards the effective theory of mixed states is the discovery of the quantum master equation by Gorini-Kossakowski-Sudarshan [8] and Lindblad [9]. The quantum master equation prescribes a specific form for the Feynman-Vernon influence functional [3,4] using the constraints that evolution should preserve the trace of the density matrix (trace-preserving) as well as keep the eigenvalues of the density matrix stably non-negative (complete positivity). We will review these ideas and their inter-relations in due turn. Our goal here is to construct a simple relativistic field 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 field with many recent advancements and is of experimental relevance to fields like quantum optics, cold atom physics, non-equilibrium driven systems and quantum information. (See [10][11][12][13][14] for textbook treatments of the subject.) It makes logical sense to test these ideas against relativistic QFTs and how they change under Wilsonian renormalisation. 1 Second, open relativistic QFTs are very relevant by themselves in heavy ion physics and cosmology [17][18][19][20] . Third motivation is to better understand the apparently non-unitary 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 field theories. Hence, it is reasonable to expect that developing the theory of open conformal field 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 field theory : the open version of scalar φ 3 + φ 4 in d = 4 space-time dimensions. One can characterise the effective theory of density matrix of φ 3 + φ 4 theory by a Schwinger-Keldysh (SK) effective action. This action involves the ket field φ R as well as the bra field φ L describing the two side evolution of the density matrix. It takes the form (1.1) 1 We should mention that in the non-relativistic context, various interacting models and their 1-loop renormalisation have already been studied. We will refer the reader to chapter 8 of [15] for textbook examples of 1-loop renormalisation in non-relativistic non-unitary QFTs. The examples include Hohenberg-Halperin classification of dynamics near classical critical points, reaction diffusion models, their critical behavior/scaling and surface growth models including the famous KPZ equations. A more detailed exposition is available in [16].
This is the most general local, power-counting renormalisable, Lorentz invariant and CPT invariant action that could be written down involving φ R and φ L . Note that CPT acts as an anti-linear, anti-unitary symmetry exchanging φ R and φ L and taking i → (−i). It can be easily checked that, under this anti-linear, anti-unitary flip e iS remains invariant provided the couplings appearing in the last line of action {z ∆ , m 2 ∆ , λ ∆ } are real. This action along with a future boundary condition identifying φ R and φ L at future infinity defines the SK effective theory which we will study in this paper.
There are two features of the above action which makes it distinct from the SK effective action of the unitary φ 3 + φ 4 theory. First, there are interaction terms which couple the ket field φ R with the bra field φ L . Such cross couplings necessarily violate unitarity and indicate the breakdown of the usual Cutkosky cutting rules . They are also necessarily a part of 'influence functionals' as defined 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 off all cross couplings between φ R and φ L and set to zero all imaginary couplings in S, we recover the SK effective action of the unitary φ 4 theory : where all couplings are taken to be real. Our aim is to deform φ 4 theory away from this familiar unitary limit and study the theory defined in (1.1) via perturbation theory. The first question one could ask is whether this theory is renormalisable in perturbation theory, i.e., whether, away from unitary limit, the one-loop divergences in this theory can be absorbed into counter terms of the same form. We answer this in affirmative in this work and compute the 1-loop beta functions to be for the mass terms, for the cubic couplings, and (λ 4 + 2σ 4 )(σ 4 + iλ ∆ ) + 3iσ 4 λ ∆ − c.c. (1.5) for the quartic couplings. Note that at 1-loop level we can set z = 1 and z ∆ = 0 since there is no field renormalisation. These equations constitute the central result of this paper. The above set of β functions have a remarkable property which is made evident by deriving the 1-loop renormalisation running of certain combinations of couplings : are preserved under renormalisation! We will prove a non-renormalisation theorem at all 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 Gell-Mann's totalitarian principle [21] that "Everything not forbidden is compulsory" (or as there being new principles in open quantum field theory which forbid some combinations from appearing in perturbation theory). This kind of fine-tuning of couplings which are still protected under renormalisation is a hallmark of open quantum field theories and is a signature of microscopic unitarity [22]. We will now move to briefly describe the significance of the above conditions. We will give three related derivations of the conditions above in this work: 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 satisfied, the structure is robust against perturbative renormalisation.
There is a fourth way of deriving the same conditions, whose deeper significance we will leave for our future work. Say one adds to the above action for the open EFT two Grassmann odd ghost fields g andḡ and demand that the following Grassmann odd symmetry hold for the entire theory : This symmetry then fixes the φ self-couplings to obey equation (1.7). Further, the ghost action is completely fixed to be 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 : first 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 effects.The third question we will comment on but leave out a detailed discussion of, is the modification 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 briefly review the relevant background for our work. This includes the concepts of Schwinger-Keldysh path integrals, their relation to Veltman's cutting rules, Feynman-Vernon influence 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 average-difference basis and we illustrate the great simplification 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 Passarino-Veltman integrals required for open EFT can be found in appendix B and in appendix C.

Basics of Schwinger-Keldysh theory
The Schwinger-Keldysh(SK) path integrals have been reviewed in [15,20,[23][24][25][26]. Here we will mention some key features : given a unitary QFT and a initial density matrix ρ(t = t i ) = ρ i , we define the SK path integral via Here, U [J] is the unitary evolution operator of the quantum field 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 time-ordering violation. This should be contrasted with the Feynman path-integral which can compute only completely time-ordered correlators.
One could in principle consider the generating functions for correlators with arbitrary number of time-ordering violations [27] (for example, the correlator used to obtain the Lyapunov exponent involves two time-ordering violations [28]) but, in this work, we will limit ourselves to just the usual SK path-integral. The Schwinger-Keldysh path integral gives a convenient way to access the evolution of the most general mixed state in quantum field theory including the real time dynamics at finite temperature. It is an essential tool in the non-equilibrium description of QFTs which is directly defined in Lorentzian signature without any need for analytic continuation from the Euclidean description.
Given an action S[φ, J] of the unitary QFT, we can give a path-integral representation of Z SK [J R , J L ] by introducing a ket field φ R and a bra field φ L : (1.12) The lower limit is the statement that near t = t i the boundary condition for the path-integral is weighed by the initial density matrix ρ i . The upper limit is the statement that the bra and the ket fields should be set equal at far future and summed over in order to correctly reproduce the trace. The factors e iS[φ R ,J R ] and e −iS[φ L ,J L ] correctly reproduce the evolution operators U [J R ] and (U [J L ]) † respectively. If the unitary QFT is in a perturbative regime, the above path integral can be used to set up the Feynman rules [15,23].
1. In a unitary QFT, there are no vertices coupling the bra and the ket fields. The bra vertices are complex conjugates of ket vertices.
2. The ket propagator is time-ordered while the bra propagator is anti-time-ordered. In addition to these, SK boundary conditions also induce a bra-ket propagator which is the on-shell propagator (obtained by putting the exchanged particle on-shell). 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 on-shell.
We will call these rules as Veltman rules after Veltman who re-derived 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 fields 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 non-unitarity. 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 definition of SK path integral in equation (1.11) as reviewed in [24]. We will briefly summarise below the argument for the largest time equation and its relation to SK formalism. We will refer the reader to [29] or [24] for more details.
In the SK path integral, consider the case where the sources obey J R = J L = J(x) beyond a particular point of time t = t f . One can then argue that the path-integral is in fact independent of the source J(x) in the future of t f . This follows from unitarity : the contributions of U [J R ] and U [J L ] † have to cancel each other in Z SK if J R = J L by unitarity.
To convert the above observation into a statement about correlators, we begin by noting that the source J(x) couples to difference operators O R − O L in the SK path integral. If we differentiate the path-integral (1.12) with respect to the common source J(x), it follows that one is basically computing a correlator with the difference operators O R − O L placed in the future of t f . The independence of Z SK on J(x) then implies the vanishing of the correlators with the future-most (or the largest time) operators as difference operators Microscopic unitarity thus requires that correlators of purely difference 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.

Basics of Lindblad theory and Effective theory
Following Feynman-Vernon [3,4], we can integrate out the 'environment' fields in the Schwinger-Keldysh path integral and obtain an effective path integral for the quantum system under question. This inevitably induces a coupling between the bra and ket fields (called Feynman-Vernon(FV) coupling in the following) as shown schematically in the figure 1. Here the red-line represents the 'environment' fields of Feynman-Vernon which couples to the system field via a linear coupling. These 'environment' fields when traced/integrated out induce the unitarity violating FV coupling for the fields describing the open quantum field 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' fields go on-shell. This also explains why, in usual QFT where we integrate out heavy fields that can never go on-shell in vacuum, no FV coupling or effective non-unitarity is induced by Wilsonian RG. 2 We will assume that the open QFT that we are studying in this paper arises from some hitherto unspecified microscopic theory à la Feynman-Vernon.  The FV couplings induced by integrating out environment fields 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 flows from the system to the environment. In this approximation (often termed Born-Markov approximation), one expects a nice local non-unitary EFT and our intent here is to study renormalisation in such an EFT. In the context of open quantum mechanical systems, under a clear separation of timescales, one can derive the Lindblad equation (or the quantum master equation) [8][9][10] for the reduced density matrix of the form Here, H is the Hamiltonian of the system leading to the unitary part of the evolution, whereas the non-unitary (Feynman-Vernon) part of the evolution comes from rest of the terms in RHS. The non-unitarity is captured by a set of operators L α and a set of couplings Γ αβ of the system. It is easily checked that the form above implies i.e., it is trace-preserving. Further, if Γ αβ is a positive matrix, one can show that the above equation describes a dissipative system which keeps the eigenvalues of ρ non-negative. These two properties (along with linearity in ρ) qualify Lindblad form of evolution as a physically sensible dynamics describing an open quantum system. The above equation in Schrödinger picture has an equivalent Heisenberg picture description via an evolution equation for operators : (1.14) Equivalently, one can obtain a path-integral description by adding to the Schwinger-Keldysh action of the system, an influence functional term of the form [26] where we have indicated the way the action should be written in terms of the bra and ket fields in order to correctly reproduce Lindblad dynamics. We note that the Lindblad form of the influence functional has a particular structure which relates the φ R -φ L cross-terms with the imaginary parts of both the φ R action and φ L action. Let us note some important features of the above expression. If we set φ R = φ L in the action above, it vanishes. It is clear that this is exactly the calculation done few lines above in the Schrödinger picture to show that Lindblad evolution is trace-preserving. This is also related to the difference operator decoupling mentioned in the last subsection in the context of Schwinger-Keldysh path integrals. Thus, trace preserving property in the Schrödinger picture becomes difference 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 β to be an identity operator, the Lindblad form then becomes a difference operator, i.e., it can be written as a difference between an operator made of ket fields and the same operator evaluated over the bra fields. This is the form of SK action for a unitary QFT (c.f. equation (1.12)) and it 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 φ R and the pure φ 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 finished this brief review of the necessary ideas, let us turn to the open φ 4 theory whose renormalisation we want to study. We will begin by describing in detail the effective action and the associated Feynman rules in the next section.

Introduction to Open effective theory
Let us begin by writing down the action for the most general open quantum field 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, [24]) and SK boundary conditions, is given by

Lindblad condition
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 field 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 φ 3 + φ 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 We note that CPT constrains this action to vanish when φ R = φ L . As a result, there are no conditions on these real couplings from the Lindblad structure.

Imaginary Quadratic terms of the action
The imaginary part of the quadratic terms is given by The Lindblad condition is given by

Imaginary Cubic coupling
Now we compute the imaginary part of the cubic terms in the action The Lindblad condition is given by

Imaginary Quartic coupling
The imaginary part of action at the level quartic coupling is given by The Lindblad condition at for the quartic couplings is given by

Exact propagators
The ket field φ R and the bra field φ L in SK path-integral satisfy the following boundary condition Owing to this boundary condition and the mixing term between φ R and φ L fields, the kinetic matrix derived from the action (2.1) is given by where the ε prescription implements Schwinger-Keldysh boundary conditions. We define the kinetic matrix K by Its inverse (viz., the propagator) can be written as where, Please note that when the Lindblad conditions (2.4) are satisfied, we have z = 1 . (2.14) 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 off-diagonal entries, i.e., The corresponding property in the unitary quantum field theory is the well-known relation between the various correlators in the Keldysh formalism [15]. This can equivalently be reformulated as the vanishing of two point function of two difference correlators : In this work, we will work in the limit where the non-unitary couplings Im[m 2 ] and m 2 ∆ are considered as perturbations to Re[m 2 ], and similarly, Im[z 2 ] and z 2 ∆ are considered small compared to Re[z 2 ]. Further, since 1-loop correction to the propagators do not generate field renormalisation we can also set z = 1. In this limit, the propagators in equation (2.12) reduced to those given by figure 2.

Feynman rules
In this paper henceforth, we will set z = z ∆ = 1 (which is not renormalised at one-loop in d=4 dimensions). We will treat all other parameters in our action except the real part of m 2 (i.e., Re(m 2 )) perturbatively. This includes λ 3 , σ 3 , λ 4 , σ 4 and λ ∆ , as well as Im m 2 and m 2 ∆ . The propagators of φ fields are given below. We have used solid blue and dotted blue lines for φ R (ket fields) and φ L (bra fields) fields respectively. Note that in the cut propagators P and M the energy is restricted to flow from the ket field to the bra field.
We will now set up the Veltman rules for the vertices to compute SK correlators in the open Factor

Lindblad condition from tree level correlators
In a unitary Schwinger-Keldysh theory, the correlator of difference operators vanishes to all order in perturbation theory. This is equivalent to Veltman's largest time equation (see for example [24]). One could ask whether this statement continues to hold true in the non-unitary 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 difference operator two point functions. We can extend this to higher point functions simply. Consider the tree level correlator of three difference operators the correlator of four difference operators is given by The correlators of the three and the four difference 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 difference operators. One can, in fact, show the following statement [22]: consider an open EFT, which is obtained by tracing out some subset of fields in an underlying unitary theory. Then, the unitarity of the underlying theory implies that the open EFT satisfies the Lindblad condition.

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 φ 3 + φ 4 theory. The main aim in this section will be to demonstrate the following three claims : 1. Despite the novel UV divergences that occur in the open φ 3 + φ 4 theory, one can use a simple extension of the standard counter-term method to deal with the divergences. Thus, the open φ 3 + φ 4 theory is one-loop 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 non-unitary 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 one-loop renormalisation are proportional to the Lindblad conditions.
We shall provide an all-order 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.

One loop beta function for m 2
We will now begin a discussion of various loop diagrams. The simplest is perhaps the tadpole diagrams which can be cancelled by a counter-term linear in φ R and φ L . It is easily demonstrated that the necessary counter-terms do not violate the Lindblad condition (See appendixE).
Let us compute the one loop beta function for m 2 . We shall consider all the one loop Feynman diagrams that contributes to the process φ R → φ R . One can verify that there are mainly two types of diagrams -one class of diagrams due to the cubic couplings, as depicted in figure 4, and the other class of diagrams due to quartic couplings, depicted in figure 5.
The sum of the contribution from all the Feynman diagrams is given by Using the results in (B.79a)-(B.79d), we can see that the contribution is divergent and one needs to add one loop counter-terms δm 2 , in the MS scheme, to absorb the divergences.
Using the standard methods of quantum field theory, one can then compute the one loop beta function as

Figure 4:
One Loop corrections to m 2 due to cubic couplings   Now, we will compute the one loop beta function for m 2 ∆ . As in the case of m 2 , there will again be two classes of diagrams. The diagrams due to cubic and quartic couplings are as shown in figure 7 and in figure 8 respectively. The sum over all the contributions is given by Some of these one loop contributions are divergent and one needs to add one loop counter-terms. The m 2 ∆ counter-term in MS scheme is given by and the beta function for m 2 ∆ is given by

Checking Lindblad condition for mass renormalization
From equation (3.3), we find that the beta function for Im m 2 is given by Now, using equation (3.7) and equation (3.6), one gets the beta function for (Im Figure 9: Diagrammatic representation of the Ten 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.

One loop beta function for λ 3
Now we will compute the one loop beta function for various cubic couplings. The Passarino-Veltman C and D integrals will have no contribution to the one loop β function for the cubic (and quartic) couplings, since they are UV finite 3 . Hence, we shall not consider those Feynman diagrams in our analysis. We begin with the beta function computation of λ 3 . The diagrams for one of the channels are depicted in figure 9. The other two channels are obtained by interchanging 1 ←→ 3 and 2 ←→ 3. The sum over the all the Feynman diagrams is given by Using the results in (B.79a)-(B.79d), we see that the one loop contributions are divergent and we need to add one loop counter-terms δλ 3 to cancel the divergences.
Following the standard methods of quantum field theory, we compute the one loop beta function to be 3.5 One loop beta function for σ 3 As described in the previous subsection, we will only consider PV B type diagrams for two of the channels are depicted in figure 10 and 11. The remaining channel is obtained by interchanging 1 ←→ 2 in the diagrams in figure 11. The sum over all the contributions is given by Here iσ 3 is the tree level contribution. The term iM 1 (k 3 ) denotes the sum over Feynman diagrams in figure 10 whereas iM 2 (k 2 ) denotes the sum over Feynman diagrams in figure 11. The contribution iM 2 (k 1 ) is obtained by interchanging 1 ↔ 2 in figure 11. The expression for iM 1 (k 3 ) is given by be UV finite. We will leave the detailed computation including these finite contributions to future work.
is cancelled by the following counter-term Figure 11: Diagrammatic representation of the Ten The expression for iM 2 (k 2 ) is given by The divergent contribution from iM 2 (k 2 ) (and from iM 2 (k 1 )) are cancelled by the following counter-term Hence the total one loop beta function for σ 3 is given by

Checking Lindblad condition at the level of cubic couplings
From equation (3.11), we obtain the beta function for Im λ 3 as and the beta function of Im σ 3 can be computed from the imaginary part of equation (3.17). We obtain Adding these two equations we get Again, one can see that the one loop beta function for the Lindblad violating cubic coupling is zero when there is no Lindblad violating coupling in the tree level Lagrangian.

One loop beta function for λ 4
Now we proceed to compute the one loop beta function for the quartic couplings. We will only consider the bubble diagrams since the triangle and box diagrams are finite. Let us consider all the one loop Feynman diagrams that contribute to the process Figure 13: Diagrammatic representation of the one loop counter-term for λ 4 diagrams are depicted in figure 12. The sum over all the Feynman diagrams is given by Using the results in (B.79a)-(B.79d), it's easy to see that the contribution is divergent and we need to add a one loop counter-term δλ 4 to cancel the divergences; Using the standard methods of quantum field theory, we can compute the one loop beta function to be By setting σ 4 = λ ∆ = 0 we recover the standard result of unitary φ 4 theory.

One loop beta function for σ 4
Again, only the Passarino-Veltman B type diagrams contribute to the one loop beta function for σ 4 . All the B type diagrams are depicted in figure 14. The sum over all of them is given by The one loop counter-term for σ 4 is given by and the one loop beta function is found to be 3.9 One loop beta function for λ ∆ The Passarino-Veltman B type contributions for s-channel and t-channel is being shown in figure  15 and figure 16 respectively. u-channels diagrams are obtained by interchanging 1 ↔ 2 in figure  16. The sum over all the contributions is given as The one-loop divergence can removed by adding the following counter-term and one loop beta function for λ ∆ is given by This equation, along with (3.7) and (3.8), implies that if one starts with a Lindblad theory then one loop renormalization preserves the Lindblad condition.

Summary of the results
We started with the most general Lagrangian of a mixed system described by a scalar field with cubic and quartic coupling in (1.

Computation in the average-difference basis
In section 3, we had computed the one loop beta functions for various couplings of an open φ 3 + φ 4 theory. In particular, by looking at the Lindblad violating couplings, we found that the Lindblad condition is preserved under one loop renormalization. In this section, we will rewrite the perturbation theory in a different basis where this fact is manifest. We would also like to prove that the preservation of Lindblad conditions hold to arbitrary perturbative order. The proof that we present here is very much inspired by a correponding argument in the context of cutting rules in a unitary theory and uses a version of Feynman tree theorem. The basis we shift to is often termed the Keldysh basis. It is made of the average and difference of bra and ket fields. This basis has an advantage that the difference operator decouplings are more manifest in this basis while it obscures the cutting rule interpretation of various diagrams involved. While the unitary vertices are mixed up with the Feynman-Vernon couplings in this basis, the computations do greatly simplify owing to lesser number of divergent diagrams and vanishing of difference-difference propagator. Our discussion here would necessarily be brief, since the details are straightforward and similar to the computation in the previous section. For a more detailed presentation, we refer the reader to appendix D

Action in the average-difference basis
We define φ d and φ a such that where the subscripts d and a denote 'difference' and 'average' respectively. The Lagrangian in this basis is given by The Feynman rules in this basis are given in figure 33. Note that the terms in the last two lines of the Lagrangian involves only the average fields φ a . The coefficients of the purely average couplings are exactly the Lindblad violating couplings. This is expected for the following reason : since φ d vanishes when φ R = φ L , the terms that can contribute to the imaginary part of the action, in that limit, are the pure φ a vertices. Since all Linblad terms vanish in this limit, it follows that pure φ a vertices should be Lindblad violating. In addition, we observe that in the open φ 3 + φ 4 theory, all Lindblad violating couplings are of pure average type. This clear separation of the Lindblad violating couplings is the most salient aspect of this basis, making it easy to trace their renormalisation. The propagators in this basis are given by [15,23,24] a : Please note that we will use a different color for propagators in the average-difference basis. Also, we shall be using results presented in appendix C. In this basis, only the tadpole A a diverges

One loop computations
As mentioned before, the computation greatly simplifies in this basis. All the computations in average-difference basis can be found in appendix D. Here we shall demonstrate only a few examples. For instance, let us compute the beta function of one of the Lindblad violating terms, (Im m 2 − m 2 ∆ ). In figure 18, we have considered all the divergent diagrams (i.e., the diagrams involving A a , B af and B ab ) that contribute to the process φ d → φ d . The total contribution to the process is given by Figure 18: Renormalization of the Lindblad violating mass term in the average-difference basis Hence, the one loop beta function for the Lindblad violating mass term is given by We had obtained the same result (equation (3.8)) in the other basis. Notice that the beta function of the Lindblad violating term can easily be computed just by computing one process in this basis.
Similarly, one can calculate the beta function of the Lindblad violating term (Im λ 3 +3 Im σ 3 ). Divergent diagrams for one particular channel is depicted in figure 19. There are two more channels. The total contribution is given as  Figure 19: Renormalization of the Lindblad violating cubic coupling in the average-difference basis 2(Im λ 3 + 3 Im σ 3 ) Following the standard procedures, we can very easily compute the one loop beta function and it is given by Now, let us we compute the beta function of Lindblad violating term (Im λ 4 + 4 Im σ 4 − 3λ ∆ ) by computing the process a vertex, which is depicted in figure 20. The total contribution is given by Hence, the one loop beta function is The usefulness of average-difference basis is quite evident from these three calculations. The complete computation in average-difference basis can be found in appendix D.

Lindblad condition is never violated by perturbative corrections
In this section, we will give an all order perturbative argument for why Lindblad conditions are not violated to arbitrary order in perturbation theory. Consider the action in the averagedifference basis given in (4.2). From this expression we note that all the Lindblad violating couplings of open φ 3 + φ 4 theory appear as the coupling constants for the pure average vertices. Our argument below can be easily extended to any open QFT which has the property that all Lindblad violating vertices are pure average vertices. Note that the converse is always true in an open EFT : any pure average vertex is necessarily Lindblad violating (since it contributes to the action even in the φ R = φ L limit). Now we want to show that if we start from the open φ 3 + φ 4 theory, then the Lindblad condition(s) are never violated under perturbative corrections using the fact that they are all of pure average type. In other words, by assuming that there is no pure average vertex at tree level, and that there is no difference-difference propagator, we would like to show that such a vertex/propagator can never be generated under loop corrections. We will prove it in three steps.
We will begin with an • Assumption: At tree level, one has no pure-average vertex and no pure-difference propagator. All Lindblad violating couplings are assumed to be pure average vertices and hence are taken to vanish at tree level.
• Statement 1 : Say we assume that there is no pure-average 1PI 2 point vertex generated at g loop. Then, it implies that there is no pure-difference 1 PI propagator generated at g

loop.
Proof : According to our starting assumption, there is no tree level pure-difference propagator. Such a propagator can then only be generated by a Feynman diagram of type depicted in figure 21.
dif-dif propagator ≡ pure average two point vertex • Statement 2 : If there is no pure-average 1PI vertex at g loops, there is no such vertex at g + 1 loops. This statement via induction, then implies that pure-average 1PI vertices are never generated at any loop order. By our previous statement, this implies then that pure-difference 1 PI propagators are also never generated at any loop order. In order to prove this, we first prove -Statement 2a: Consider a Feynman diagram contributing to a pure-average 1PI vertex. There must be at least one vertex (internal or external) such that the following is true : there exists a closed path completely made of b-type propagators which begins and ends in that vertex (it may or may not pass through external vertices). Proof : Since we are considering a diagram contributing to a pure-average 1PI vertex, all the external propagators at every external vertex are of a-type. By our assumption, there is no pure-average 1PI vertex. Thus, there should be at least one d-type line leaving at a given external vertex. Since there is no pure-difference propagator, this d-type line necessarily converts itself into an a-type line : thus the propagator is of average-difference b-type with the arrow leaving the external vertex. This propagator thus ends as an a-type line either in the vertex that one began with, or another external vertex or an internal vertex.
In the first case, we have obtained the desired result : there exists a closed path completely made of b-type propagators which begins and ends in that vertex.
In the second case, we note that the external vertex has an external a-type leg, and the b-type propagator which went from the starting vertex also ends with an a-type leg on the second vertex. Since there are no pure-average vertices, there should necessarily be a d-type leg which is going out of the second vertex. The d-type leg can again only be a part of an b-type propagator since there is no pure-difference propagator. A similar argument also applies to the third case of an internal vertex. We can now follow the b-type propagators and repeat the argument again. This process would terminate (since we are looking at a finite graph) and we would return to some vertex on the path for second time, closing the loop.
where, ω p = ( p 2 + m 2 ) 1 2 and G R (x − y) denotes the retarted propagator. We will now use the result that a closed loop of retarted propagators is identically zero. This statement is a part of the Feynman tree theorem [30]. A closed loop of retarted propagators can be written as where, k i denotes the external momenta. Since all the poles in a retarded propagator are below the real p 0 axis. So, one can close the contour from above, picking no residues and, as a result, the integral vanishes. (QED) • Statement 2a and 2b imply that if there is no pure average vertex operator or a pure difference propagator at g loop then there will be no such vertex/propagator at g + 1 loop. From this we conclude, via induction, that if there is no Lindblad violating coupling at tree level, such a coupling is never generated by perturbative corrections.(QED) This then concludes our argument in the average-difference basis that the Lindblad violating couplings are never generated in loops. The readers familiar with cutting rule arguments ala Veltman in unitary theories would recognise the style of the above argument. The proof that difference operators decouple at arbitrary loops in a unitary theory, or equivalently the proof that Keldysh causal structure is preserved under loop corrections for a unitary theory bear a close resemblance to the proof above. The surprise here is that the argument goes through even without assuming unitarity. We also note the perturbative nature of the above argument, since it invokes the fact that the graphs at any given loop order are finite. It would be interesting to try and give a non-perturbative proof of the statement of this section.
With this formal proof in hand, in next section, we will now turn to a preliminary study of the RG running in our open EFT. The interesting question is to map out behaviour novel to open EFTs which cannot be found in unitary QFTs.

Running of the coupling constants and physical meaning
In this section, we will perform an analysis of the running of couplings from our 1-loop beta functions. Given the many couplings involved in the the RG equations in (1.3),(1.4) and (1.5), we will begin with a judicious rewriting of our equations. Once the Lindblad conditions are imposed, we obtain the following count for the couplings : 1. 5 quartic couplings + 1 lindblad condition (1.7) =⇒ 4 independent quartic couplings 2. 4 cubic couplings+ 1 lindblad condition (1.7) =⇒ 3 independent cubic couplings 3. 3 mass terms + 1 lindblad condition (1.7) =⇒ 2 independent mass terms Our RG equations for these 9 independent variables can then be recast into the following convenient form : Note the simple structure of the 9 coupled differential equations given above. We have ordered them such that the jth equation depends only on the variables appearing in the first j To analyse the nature of this fixed point, we turn to the first equation which drives them all. It can be written as This implies that depending on the sign of the initial value, Re λ 4 + 2Re σ 4 either increases or decreases as we go to higher energy scales. As we will see, this sign controls whether the theory is UV free or IR free. We recognise in the RG equation for Re λ 4 + 2Re σ 4 the usual φ 4 coupling RG equation with Re λ 4 + 2Re σ 4 serving as an effective φ 4 coupling. The asymptotically free regime and the negative beta function corresponds to this effective φ 4 coupling turning negative and is hence akin to the theory studied by Symanzik [31]. 4 We will begin by performing a linearised analysis around the fixed point mentioned above and follow it up with a more detailed numerical analysis.

Linearized analysis around the fixed point
In this section, we study linearized beta functions around the fixed points and find the eigenvalues and eigenvectors of the beta function matrix. Consider small deviations around the fixed points where, we have assumed The linearized beta functions for Re λ 4 + 2Re σ 4 , Im λ 4 + Im σ 4 and Re λ 3 + Re σ 3 are zero. This suggests that 1 , 2 and 3 remain constant (i.e., they are marginal couplings at the fixed point).
The rest of the linearized beta functions about the fixed point can be written as in terms of the RG time t ≡ ln µ (4π) 2 . Here we have defined the coupling constant matrix G as and the beta function matrix B is given by The six eigenvalues of the matrix B are -0, 0, 0, 1 , 2 1 , 3 1 . The corresponding eigenvectors are given by: Eigenvalues of the matrix B suggest that three out of the six coupling combinations are marginal at the fixed point. The asymptotic behavior of the rest of the variables depend only on the sign of 1 or Re λ 4 + 2Re σ 4 . A positive 1 would mean that the couplings become relevant in UV, whereas a negative 1 would mean that the couplings are relevant in IR. This conforms to the intuition we presented in the beginning of this section : the coupling Re λ 4 + 2Re σ 4 runs like the quartic coupling of an ordinary φ 4 theory : the theory is IR free for positive value of this combination whereas it is UV free (asymptotically free) for negative value of this combination. This coupling then drives all other couplings to be either IR free or asymptotically free. Let us now extend our analysis beyond the linearised regime around tthe fixed points, given by We will begin by re-examining eqn.(5.1) to gain more qualitative insight on the nature of running in this theory: 1. We will begin with the statement that, depending on the sign of the initial value, Re λ 4 + 2Re σ 4 either increases or decreases as we go to higher energy scales . Thus, we can have two distinct scenarios 2. The second equation depends upon the sign of (Re λ 4 + 2Re σ 4 ) as well as on the sign of the initial value of Im λ 4 + Im σ 4 . For instance, keeping a positive Re λ 4 + 2Re σ 4 and a negative initial value of Im λ 4 + Im σ 4 would result in a decreasing behavior as shown in figure 23 and figure 25. Thus, we have two further sub-cases, depending upon the sign of Im λ 4 + Im σ 4 .
3. The third and fourth equation implies that the evolution of Re λ 4 and Im λ 4 − 4 Im σ 4 depends only on the values of Re λ 4 + 2 Re σ 4 and Im λ 4 + Im σ 4 , given the assumption that the imaginary couplings are small compared to the real ones.

The fifth equation is similar to the second equation.
Hence, there will again be two subcases.
5. It's easy to verify that, with the help of similar reasonings, the remaining equations will not provide us with further sub-cases.
We found that the key conclusion remains unchanged for Re λ 3 + Re σ 3 ≷ 0. So we will always be considering the case Re λ 3 + Re σ 3 ≷ 0 together. Thus, we conclude that we can broadly have 8 cases in total and they basically correspond to the two sides of either of these three fixed points: each fixed point will provide two cases and we have 2 3 cases altogether.
With this insight, we will proceed to a more detailed numerical analysis.

Numerical analysis of RG equations
In this subsection, we continue our analysis of the various possible cases in the RG evolution equations. It is useful to have a rough criteria to check the validity of our analysis and as to when the analysis can be interpreted physically. We will perform this analysis only for the Lindblad theory, where the coupling constants obey the Lindblad conditions. We shall always work in a regime where the imaginary couplings are smaller compared to the real ones (since this is the regime where our beta functions were derived). Moreover, we will demand the following bounds which seem to be reasonable from the point of microsocopic unitarity [22]. We will deem the couplings which do not satisfy this bound as unphysical in the following. The initial conditions are chosen keeping these physical bounds in consideration and we shall analyse the dynamics corresponding to all the possible behaviors.

I:
Re λ 4 + 2Re σ 4 > 0, Im λ 4 + Im σ 4 > 0 and Re λ 3 + Re σ 3 ≷ 0 The first equation in (5.1) tells us that the sign of Re λ 4 + 2Re σ 4 will remain positive in this regime. In particular, Re λ 4 + 2Re σ 4 evolves in the same way as λ Unitary 5 . Now, from the second equation, one can see that Im λ 4 +Im σ 4 will keep increasing if it starts at a positive initial value, but at a slower rate compared to Re λ 4 + 2Re σ 4 . Similarly, from the third and fourth equation, one can see that, keeping in mind the assumptions, both Re λ 4 and Im λ 4 − 4Im σ 4 will increase in the way as shown in figure 22. Note here that Im λ 4 − 4Im σ 4 rises faster than Im λ 4 + Im σ 4 and thus, it results in a continuously increasing Im λ 4 and a decreasing Im σ 4 as shown in the second diagram in figure 22. Also, the increase of Im λ 4 is faster than the decrease of Im σ 4 and thus, under the RG flow, λ ∆ becomes positive, which is unphysical. Evolution of the remaining cubic couplings and mass terms variables will not affect the evolution Im λ 4 and Im σ 4 . So, both the sub-cases due to different signs of Re λ 3 + Re σ 3 would have a positive λ ∆ and thus, these two cases can be deemed as unphysical.

II:
Re λ 4 + 2Re σ 4 > 0, Im λ 4 + Im σ 4 < 0 and Re λ 3 + Re σ 3 ≷ 0 The evolution of each variable for this case is depicted in figure 23. We observe that the couplings do not violate the physical conditions throughout. One can see that the couplings become stronger in the UV and attain a Landau pole.

III:
Re λ 4 + 2Re σ 4 < 0, Im λ 4 + Im σ 4 > 0 and Re This is a case where the couplings are relevant in IR and remain within the physical bounds throughout as can be observed in figure 24. In this case, Re λ 4 + 2Re σ 4 becomes asymptotically free as can be seen from the first equation in (5.1). The second and fifth equation, meanwhile, tells us that Im λ 4 + Im σ 4 and Re λ 3 + Re σ 3 would go to zero as we go to higher energies. This would also mean that Im λ 4 − 4 Im σ 4 becomes constant as Im λ 4 + Im σ 4 goes to zero. Im λ 4 and Im σ 4 become constant at higher energies and λ ∆ attains a fixed point. With similar reasonings, one can predict the behavior of other couplings. Re λ 4 + 2Re σ 4 Im λ 4 + Im σ 4 Re λ 3 + Re σ 3 One can observe from figure 25 that this case can be deemed as unphysical as λ ∆ attains a positive value. It basically comes about due to the sign of Im λ 4 + Im σ 4 as can be seen from the second equation in (5.1).

Conclusion and Future directions
In this work, we considered a simple φ 3 + φ 4 toy model of an open quantum field theory in which the renormalisation and the running of couplings could be studied. By enumerating all power-counting renormalisable terms, we demonstrate that the theory is 1-loop renormalisable whereby all UV divergences can be absorbed into appropriate counter-terms. This is in analogy with the standard result for a unitary QFT. The novelty lies in the non-unitary Feynman-Vernon couplings and the corresponding UV divergences which result in a β function for such non-unitary couplings. One of the main results of our paper is that these beta functions surprisingly protect a particular fine-tuning of couplings which is associated with demanding that the non-unitary evolution be that of Lindblad form. We end with an all loop argument on why this protection should extend to any order on perturbation theory. The work described in this article has various natural extensions -to large N models, to theories with fermions and theories with gauge fields. Given our experience with supersymmetric field theories, open versions of supersymmetric theories may well provide an exactly solvable model of an open QFT where one can study non-perturbative physics as well as dualities. We hope to return to these issues in the immediate future.
With this work, we hope to have convinced the reader of the charms of hitherto unexplored world of open quantum field theories. In many aspects, they closely mimic the familiar paradigm of unitary quantum field theories but yet deviate from them in interesting ways. Very basic conceptual issues like renormalisation or anomalies or non-pertubative physics (as that of instantons) are yet ill-understood.     International Centre for Theoretical Sciences, Tata Institute of Fundamental Research (ICTS-TIFR), Bengaluru. A.R. would like to thank ICTS-TIFR, Bengaluru for hospitality during the initial stages of this work. A. would like to thank support from Kishore Vaigyanik Protsahana Yojana (KVPY) funded by the Department of Science and Technology, Government of India. A. would also like to thank Indian Institute of Science (IISc) and ICTS-TIFR for the hospitality provided during this work. RL would also like to acknowledge his debt to all those who have generously supported and encouraged the pursuit of science in India.

A Notations and Conventions
A.1 Most commonly used acronyms • The loop integrals are named as the following. We start from the left(bottom) vertex and move in the counter-clockwise direction for s-channel(t, u channel) diagrams(s).

A.2 Conventions for Feynman integrals
Since more general diagrams can appear in this context we will introduce a suitable notation. Following the standard notation [33,34], we will use A for tadpole diagrams and B for bubble diagrams. In addition, we will use R, L, P, M as subscripts to denote the corresponding propagators as present in the diagrams. 2.

Figure 26: PV One loop A type integrals in SK theory
We are using slightly different normalization from [34] for Passarino-Veltman integrals. The relation between our integrals and the integrals in [34] is given below We also note that Passarino-Veltman definitions use mostly negative metric η µν = diag (1, 1, 1, 1) , while in this work we use mostly positive metric η µν = diag (1, 1, 1, 1). This fact has to be taken into account while comparing our expressions in terms of momentum-square against the standard expressions in discussions of PV integrals.
In SK theory there are four A type integrals. They are given by Here In SK theory of one-scalar, there are ten B type integrals (compared to one B type integral in ordinary QFT of a single scalar). For the sake of generality, we will evaluate the most general scalar B-type integrals with unequal masses, m andm, that can occur in an open EFT perturbation theory. These are 16 in number and they are defined as: Figure 27: PV one loop B type integrals in SK theory. The momentum and mass corresponding to the lower propagator is denoted by p µ and m respectively, whereas the momentum and mass corresponding to the upper propagator are q µ andm respectively. The momenta p and q are taken to flow anti-clockwise in the loop.
In the following appendix, we will evaluate these integrals and their divergences.

B Evaluating Passarino-Veltman Loop Integrals for open φ 3 + φ 4 theory
In this section, we describe in some detail the loop integrals that appear in the perturbation theory of open φ 3 +φ 4 theory. While some of the integrals are familiar from usual QFT textbooks and a few other integrals occur in discussions of cutting rules, as far as the authors are aware, the majority of the integrals described in this section are not analyzed elsewhere. Hence, these integrals are described in some detail with a special focus on the new kind of features that occur when we try to do integrals in the real time (most of the integrals in this section do not admit Wick rotation because of their unusual i prescriptions) .

B.1 Passarino-Veltman A type integrals
There are four A type PV integrals : A R , A L , A P , A M . They satisfy the following relations Using this result and the relations (B.1)-(B.2), we can determine A L and A P . A L , A P and A M are given by Let us now compute this integral in terms of these kinematic variables. We have which is the two-body phase space as advertised.
Restoring the kinematic constraints, we get The integral can then be expanded near d = 4 to get Replacing k µ by −k µ we get These two expressions can be added to get When m =m, we can write and B.3 Integrals B P P (k) and B M M (k) We now turn to the 'cross-cut' integrals B P P (k) and B M M (k) which do not occur in the usual discussions of cutting rules in a unitary theory. They are loop integrals peculiar to open QFTs with their own characteristic kinematic behavior.

B.3.1 Time-like k µ
We will begin by examining B P P (k) for time-like k µ . In the rest frame of k µ i.e., k µ = (M, 0), we can do similar manipulations as in the previous subsection to get where we have used with p being the appropriate momentum which solves the kinematics (see below). For M > 0, i.e., k µ being future time-like, we recognize the integral as the one describing the phase space for a deep in-elastic scattering process :m with momentum p strikes against the target M at rest converting it into the particle m traveling with momentum p. The kinematics is solved by which is sensible for M < m −m.Thus, in this kinematic regime we get For M < 0, i.e., k µ being past time-like, we recognize the integral as the one describing the phase space for the two body decay ofm into a particle of mass |M | and m. The kinematics is solved by which is sensible form > |M | + m.Thus, in this kinematic regime we get Thus, we conclude that for time-like k µ , Note that B P P (k) = B P P (−k) could have been directly deduced from the integral form.

B.3.2 Space-like k µ
We will next study B P P (k) when k µ is space-like. We set k µ = {0, Q = √ k 2 , 0 d−2 } where we take Q > 0 without loss of generality. We have where, in the penultimate step we have definedω p⊥ ≡ p 2 ⊥ + (Q − p || ) 2 +m 2 . In the last step, we have used with the definitions The rest of the integral is (d − 2) dimensional transverse phase space with the transverse mass given by We thus get Restoring the kinematical constraints, we get Putting together the various kinematical regimes, we obtain Expanding around d = 4 we get where we have used B M M (k) = B P P (−k) = B P P (k). Taking m =m, the second factor vanishes and we obtain

B.4 Reduction of divergent integrals to B RP (k)
We now turn to the 'quarter-cut' integrals B RP (k), B RM (k), B LP (k) and B LM (k) which also do not occur in the usual discussions of cutting rules in a unitary theory. They are also loop integrals peculiar to open QFTs. However, unlike the integrals considered in the last section, they do not evaluate to on-shell phase space for various processes. This off-shell nature means that they exhibit UV divergences and hence are crucial to the issue of renormalizability of open QFTs. When the open QFT is renormalisable, these diagrams contribute to β functions of an open QFT. As before, we will evaluate this integrals in various kinematic regimes and then put together the answers at the end. We will consider the integral This is the characteristic integral which leads to UV divergences in open QFT. Before analyzing this integral further, we will show that the other divergent integrals can be reduced to this integral. We start with The integrals with the subscripts exchanged can be obtained by exchanging m andm (thus exchanging p µ and q µ ) and reversing k µ . For example, B P R (k) = B RP (−k)| m↔m and similarly for other integrals : It is convenient to define the following combination of integrals : As we will see in next subsection, using these combinations, the rest of the divergent integrals can also be reduced to B RP (k).
where we have used the definition given in equation(B.35). Next, we turn to We then look at We want to now write the answer of the contour integral with a definite ε prescription. An examination of the sign of resulting ε's shows that the form depends now on the sign of M as well as m −m. A careful examination of ε's give Transcribing it into B RP integrals, we obtain It follows that We will now turn to the case of space-like k µ to prove similar relations in that case.

B.4.2 Space-like k µ : reduction of divergent integrals
We will study B RP (k) when k µ is space-like. We set k µ = {0, Q = √ k 2 , 0 d−2 } where we can take Q > 0 without loss of generality.
where, we have defined Now, we move on to calculating B RL (k), given by Now, we take p = −q in the first integral and p = k + q for the second integral to write Similarly, Let us now do the B RR (k) integral for space-like k µ : Now, we take p || = −q || in the first integral and p || = Q + q || in the second, to get Here, in the penultimate step, we have done some variable redefinitions to obtain an answer similar to the time-like case. We can finally compute

B.4.3 Summary of divergent integrals
We can now put together various cases and write This is apart from the other divergent integrals : We note that all these integrals can be written in terms of B RP (k) as advertised.

B.4.4 Reduction and identities due to largest time equations
These conditions in turn lead to the identities : From, these identities we get which in turn obey The same integral in the space-like case takes the form where, we have defined Our aim in this subsection is to evaluate these integrals and extract out the appropriate divergences.

B.5.1 Time-like k µ : computation of divergences
We begin by settingω p =m cosh η in the time-like case to get Thus, we have reduced our analysis to the integral This integral can then analyzed in detail to study the analytic structure of this integral. But, for our purposes, it is sufficient to extract the divergences. For our computation of β function, we need to extract out the divergent part of these integrals. Focusing on the largeω p contribution, we can approximate B RP (k) by

(B.66)
Near d = 4, this gives In this subsection, we will consider the space-like case and confirm that the divergence structure is same in the space-like case. Let us first get the real part of B RP (k), which is given by where, we have defined Let us first get the real part of B RP (k), which is given by The above equation can easily be seen to be equal to B P P (k) for the space-like case in (B.22). So, we have in the space-like case Let us now get the imaginary part of B RP (k), which is also the divergent part. It is given by when, m =m, we get We see that these divergences are same as the time-like case.

B.5.3 Summary of divergences
We now summarize the divergences in various integrals. We start with When m =m, we can thus summarize the divergence of 'quarter-cut' integrals as 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.
We also give below the symmetry factors of the corresponding diagrams in figure 28. The divergences given above along with the symmetry factors provides a quick way to write down appropriate β functions for the open QFT. In the ensuing figure 29 and figure 30, we tabulate a set of useful diagrammatic identities which relate the various SK loop integrals. Figure 28: Symmetry factors for all the ten one loop integrals

C Passarino-Veltman diagrams in the average-difference basis
Let us now take a look at the Passarino-Veltman diagrams in average-difference basis. It's worth remembering here that only three out of the four propagators, in this basis, are non-vanishing: the 'd' propagator vanishes. This means that we have lesser number of non-vanishing diagrams in this basis. As a matter of fact, some of the non-vanishing diagrams (in average-difference 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 simplifies in this basis. We will not try to evaluate the PV integrals from scratch. We will express the integrals in the average-difference basis in terms of the integrals in the φ R -φ L basis and then, use the results from the previous sections to determine the former.

D Computations in the average-difference basis
In section 4, we have already computed the beta functions for the Lindblad violating combinations in the average-difference basis and we found that it matches with our computations in the φ R -φ 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 φ R -φ L basis. We shall start off by providing the set of Feynman rules in this basis.
The propagators in this basis are given in equation

D.2 Beta functions for the cubic couplings
We have four cubic coupling constants and the corresponding vertices are φ 3 a , φ 2 a φ d , φ a φ 2 d , φ 3 d and we need to compute four correlators. In each case, we will keep only the divergent parts as before.
So, the counter-terms obey Lindblad condition if there is no Lindblad violating cubic couplings.