Renormalization in open quantum field theory. Part I. Scalar field theory

Journal of High Energy Physics, Nov 2017

Abstract 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 ϕ 3 + ϕ 4 theory in four spacetime dimensions and demonstrate its one loop renormalizability (including the renormalizability of the Lindblad structure).

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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 average-di erence basis 5 Running of the coupling constants and physical meaning Action in the average-di 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 Time-like k : computation of divergences B.5.2 Space-like k : computation of divergences B.5.3 Summary of divergences B.6 UV divergences and symmetry factors C Passarino-Veltman diagrams in the average-di erence basis C.1 Passarino-Veltman A type integral in the average-di erence basis C.2 Passarino-Veltman B type integral in the average-di erence basis D Computations in the average-di 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 coarse-graining 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 Schwinger-Keldysh path integral [1, 2] and the Feynman-Vernon in uence functionals [3, 4] | the rst addressing how to set up the path-integral for unitary evolution of density matrices by doubling the elds 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], 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 Gorini-Kossakowski-Sudarshan [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 (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 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, non-equilibrium 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 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 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 space-time 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 + anti-linear, anti-unitary symmetry exchanging R and L and taking i 7! ( i). It can be easily checked that, under this anti-linear, anti-unitary 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 non-relativistic context, various interacting models and their 1-loop renormalisation have already been studied. We will refer the reader to chapter 8 of [17] for textbook examples of 1-loop renormalisation in non-relativistic non-unitary 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 one-loop 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 1-loop 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 1-loop 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 Gell-Mann'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 ne-tuning 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 Schwinger-Keldysh 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 coarse-grained 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 non-unitary, 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 Feynman-Vernon 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 self-couplings 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 Schwinger-Keldysh path integrals, their relation to Veltman's cutting rules, Feynman-Vernon 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 average-di 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 Passarino-Veltman integrals required for open EFT can be found in appendix B and in appendix C. 1.1 Basics of Schwinger-Keldysh theory The Schwinger-Keldysh(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 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 [37] (for example, the correlator used to obtain the Lyapunov exponent involves two time-ordering violations [38]) 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 { 5 { HJEP1(207)4 eld theory including the real time dynamics at nite temperature. It is an essential tool in the non-equilibrium 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 path-integral 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 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 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 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 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 path-integral 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 path-integral (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 future-most (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 Feynman-Vernon [3, 4], we can integrate out the `environment' elds in the Schwinger-Keldysh 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 Feynman-Vernon(FV) coupling in the following) as shown schematically in the gure 1. Here the red-line represents the `environment' elds of Feynman-Vernon 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 on-shell. This also explains why, in usual QFT where we integrate out heavy elds that can never go on-shell in vacuum, no FV coupling or e ective 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 unspeci ed microscopic theory a la Feynman-Vernon. 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 Born-Markov 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 path-integral 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 non-negative. 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 non-unitary (Feynman-Vernon) part of the evolution comes from rest of the terms in r.h.s. The non-unitarity 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 cross-terms 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 Schwinger-Keldysh 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 path-integral 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 Schwinger-Keldysh 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 well-known 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 non-unitary 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 1-loop 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 one-loop 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 Schwinger-Keldysh 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 counter-term method to deal with the divergences. Thus, the open 3 + 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. (−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 counter-term linear in R and L. It is easily demonstrated that the necessary counter-terms 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 counter-terms 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 = space-time 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 counter-term 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 time-like 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 space-like 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 Time-like k : computation of divergences We begin by setting !p = m cosh in the time-like 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 Space-like k : computation of divergences In this subsection, we will consider the space-like case and con rm that the divergence structure is same in the space-like 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 space-like case in (B.22). So, we have in the space-like 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 time-like 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 `quarter-cut' 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 Passarino-Veltman diagrams in the average-di erence basis Let us now take a look at the Passarino-Veltman diagrams in average-di erence 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-di 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 Passarino-Veltman A type integral in the average-di 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 Passarino-Veltman B type integral in the average-di 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 above-mentioned 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 average-di erence basis In section 4, we have already computed the beta functions for the Lindblad violating combinations in the average-di 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. 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Avinash Baidya, Chandan Jana, R. Loganayagam, Arnab Rudra. Renormalization in open quantum field theory. Part I. Scalar field theory, Journal of High Energy Physics, 2017, 204, DOI: 10.1007/JHEP11(2017)204