The Complex-Mass Scheme and Unitarity in perturbative Quantum Field Theory

We investigate unitarity within the Complex-Mass Scheme, a convenient universal scheme for perturbative calculations involving unstable particles in Quantum Field Theory which guarantees exact gauge invariance. Since this scheme requires to introduce complex masses and complex couplings, the Cutkosky cutting rules, which express perturbative unitarity in theories of stable particles, are no longer valid. We derive corresponding rules for scalar theories with unstable particles based on Veltman's Largest-Time Equation and prove unitarity in this framework.


Introduction
With the discovery of the Higgs boson at the Large Hadron Collider nature again reflects not only the relevance of fundamental principles such as gauge invariance as they are incorporated in theories like the Standard Model (SM), but also that unstable particles play a role as much important as stable ones. The majority of the known fundamental particles are unstable, and in physical observables unstable particles usually play a significant role.
Precision predictions within perturbative Quantum Field Theories (QFT) are still a challenging task, especially when unstable particles are involved. Unstable particles are of nonperturbative nature in the sense that in the usual leading-order (LO) perturbation theory all particles are stable. As a consequence near thresholds or resonances observables even diverge in standard perturbation theory because important contributions are missing. A proper treatment requires the inclusion of finite-width effects via a finite imaginary part in the denominator of the Feynman propagator at least near the poles of unstable particles. This imaginary part results in perturbation theory from a resummation of self-energies.
To date there is no fully established treatment of unstable particles within perturbation theory, although many solutions have been proposed. The problem arises from the need to resum self-energies. However, this resummation, if done carelessly, leads to violation of gauge invariance and gauge independence. Thus, the naive modification of the propagator in the fixedwidth scheme violates Ward identities and the renormalization of the Z-boson mass in the usual on-shell renormalization scheme introduces a gauge dependence as pointed out by Stuart and Sirlin [1][2][3].
For inclusive observables that are dominated by the production of on-shell unstable particles with a small width, finite-width effects can be neglected if the required precision is small compared to the ratio of width and mass of the unstable particles. This so-called narrow-width approximation is, however, insufficient for many applications. A straight-forward gauge-invariant method for the inclusion of the finite width is the factorization scheme introduced in Ref. [4], which consists in the multiplication of the matrix elements with a global resonance factor. However, for more complicated processes it becomes non-trivial to achieve a precision beyond LO. The fermion-loop scheme [5,6] exploits the fact that taking into account only closed fermion loops at the one-loop order allows to perform a gauge-invariant and gauge-independent resummation. By construction this method is restricted to leading-order predictions and to resonances that decay exclusively into fermions. The idea of a gauge-invariant resummation can be carried further by using the background-field method [7][8][9] which allows to perform a Dyson summation without violating Ward identities [10]. While the resummed self-energies still depend on the quantum gauge parameter, this dependence can be fixed by definition, e.g. by using a specific gauge or the prescription of the pinch technique [11]. In practice these methods would require complete NNLO calculations to get NLO accuracy in the region of the resonance. The pole scheme proposed in Ref. [12,13] is based on the fact that both the location of the pole and the residue of the propagator of an unstable particle are gauge-independent. It allows to compute gauge-invariant matrix elements to arbitrary high orders via a Laurent expansion around the complex pole. In practice this method gets quite involved in higher orders, and usually only the leading terms in the Laurent expansion are taken into account, called the leading-pole approximation. Furthermore, effective field theory can be used to describe unstable particles. In the method of Refs. [14,15] non-local gauge-invariant effective operators are introduced that allow the gauge-invariant resummation of self-energies via appropriate choices of free parameters. In the effective-field-theory approach of Refs. [16,17] an expansion in the coupling constant and in the distance from the pole is performed simultaneously. This basically yields a field theoretically elegant way to the pole approximation and can be easily combined with further expansions (see Ref. [18] for a recent application).
The most straightforward and consistent method is in our opinion the Complex-Mass Scheme (CMS) [19][20][21]. It is fully gauge-invariant, valid everywhere in phase space, basically of the same complexity as a calculation for stable particles and applicable to higher orders in perturbation theory [22,23]. Finite widths are introduced by analytically continuing the renormalized mass parameters to appropriate complex values. The introduction of complex parameters immediately raises the question how unitarity is implemented in this scheme. Unitarity is not expected to be violated because the bare Lagrangian is left untouched and only the renormalization procedure is modified as compared to the standard treatment. Moreover, the CMS guarantees exact gauge cancellations through gauge invariance order by order in perturbation theory. Therefore any violation of unitarity should be beyond the order of perturbation theory taken into account completely. It has been shown by Veltman [24] within non-perturbative QFT that unitarity is fulfilled in a theory with unstable particles provided that the unstable particles are excluded from asymptotic states. Since the CMS provides a perturbative description of the full theory it should not violate unitarity, if observables are correctly computed in a valid perturbative regime. Unitarity within the CMS has been touched upon in Ref. [22]. Unitarity in the CMS in a model with a heavy vector boson interacting with a light fermion has been investigated at the one-loop level in Ref. [25].
The aim of this paper is to study unitarity in scalar field theories in the CMS. In section 2 we shortly review unitarity and the Largest-Time Equation in the case of stable particles and in Section 3 we summarise the Complex-Mass Scheme. In Section 4 we investigate the realization of unitarity in the Complex-Mass Scheme by constructing and exploiting a suitable Largest-Time Equation for unstable particles.

Unitarity and Veltman's Largest-Time Equation for stable particles 2.1 Unitarity
In the language of QFT unitarity means that the S matrix is unitary, i.e. S † S = 1. Separating the non-interacting contributions from S via S =: 1 + iT , one obtains the well-known relation for the transition matrix T . A simple consequence of unitarity is the optical theorem which states that the imaginary part of a forward scattering amplitude T ii is proportional to the total cross section: The connection between (2.1) and the optical theorem (2.2) is established when considering elements of the transition matrix with definite initial and final states, (2.3b) The so-called shadowed region is given by T * while the normal region is given by T and both transition amplitudes are connected by on-shell states which is visualised as a cut (dark hatched line). We denote the equations (2.3) in the following as the unitarity equation. Unitarity is verified by computing the left-hand side of the unitarity equation and comparing it to the righthand side for all possible initial and final states. Direct computation of the left-hand side can be quite involved especially beyond the one-loop level, but with the help of cutting rules, which we introduce in the next section, the problem is theoretically and practically solved.

Veltman's Largest-Time Equation
The LTE can be seen as the analogue to Cutkosky's cutting rules [26], but is straightforward to derive and needs less mathematical tools. The derivation of the LTE can be found, for instance, in Refs. [24,27] and it is based on a decomposition of the Feynman propagator in space-time representation. This decomposition is done, in the case of stable particles, in positive-and negative-time parts in such a way that positive (negative) time is connected to positive (negative) energy flow and vice versa 1 . Let ∆ F (x − y) denote the Feynman propagator in space-time representation then the decomposition is the following: (Decomposition theorem): There exist functions ∆ ± with the properties correspond to positive and negative energy flow, respectively. In Fourier space they take the simple form Given such a decomposition, one can define extended Feynman rules: (The underline operation): Given a Feynman diagram F defined by a set of vertices {x i } and corresponding couplings {g i }, we define new diagrams where one or more of the space-time points x i can be underlined, i.e. x i → x i . This operation shall have the following consequences for propagators connecting the vertices in the original diagram: The energy-flow direction is related to the sign of p 0 , where p 0 is the zeroth component of the four-momentum. We prefer to speak about positive-and negative-time parts instead of positive-and negative-energy parts because of the generalisation to unstable particles introduced in Section 4.1 • i∆ ki is transformed as i∆ ki → −i∆ * ki if two connected space-time points x k , x i are underlined, • any underlined space-time point implies a factor −1 for the corresponding vertex, i.e. if x k → x k , then the corresponding coupling is replaced as ig k → −ig k .
At the level of Feynman diagrams the underline operation is indicated by a circle at the corresponding underlined space-time points. The rules stay the same for couplings with imaginary part, in particular, we stress that the coupling g i is not complex-conjugated for underlined As has been shown by Veltman [24], from these rules the following equation can be derived: 2 (Largest Time Equation): Given a Feynman diagram F defined by a set of vertices {x i } and corresponding couplings {g i }, if the Lagrangian is real and all propagators fulfil the decomposition theorem, then the following equation holds where the sum runs over all possibilities of underlining elements. In total there are 2 N contributions where N is the number of vertices. We note that the LTE holds both for truncated or non-truncated diagrams. The unitarity equation (2.3) is recovered by extracting two contributions from the LTE, namely the one where none of the vertices are underlined and the one where all of them are underlined, i.e. F(x 1 , . . . , x i , . . . , x N ) and F(x 1 , . . . , x i , . . . , x N ). These contributions match iT f i and −iT * if , and the right-hand side of (2.3) can be represented by the remaining terms of the LTE. What then follows is the observation that owing to energy conservation the only nonvanishing terms in the LTE have two well-defined regions, a region with the usual Feynman rules which is always connected to the incoming particles and a region with the "complex-conjugated" Feynman rules (underlined vertices) which is always connected to outgoing particles. The regions are connected to each other by cut propagators ∆ ± , which emerge from propagators where one space-time point is not underlined while the other is. Put in other words: Four-momentum conservation and the given values of external four-momenta forbid certain contributions to the LTE, and the contributions left are the ones where the energy flows from incoming particles to outgoing particles as it is required by the unitarity equation (2.3). This point will be elaborated in more detail later. (Cutkosky's cutting rules): The underline operation together with the LTE are equivalent to Cutkosky's cutting rules, namely that the discontinuity of an amplitude is obtained by replacing propagators in all possible ways by on-shell propagators (2.6), but constrained in such a way that the energy flows from the initial to the final states. For more details, in particular for the derivation of Cutkosky's cutting rules we refer to the original reference [26]. In the following the terminologies Cutkosky's rules, cutting rules and LTE with the usual on-shell cut propagator (2.6) are used as synonyms.

The Complex-Mass Scheme
When dealing with gauge theories it is crucial to guarantee gauge invariance which is more involved when unstable particles are present. As pointed out in the introduction, various methods have been developed to describe unstable particles in perturbation theory, but most are only valid near the resonance and lack validity in general phase-space regions. In contrast, the CMS is valid in the full phase space. Its underlying idea is an analytic continuation in the masses of the unstable particles. Being algebraic relations the Ward identities are not violated by such a modification. In practice, the renormalized Lagrangian is rewritten by replacing any appearing mass corresponding to an unstable particle with the complex one in such a way that the bare Lagrangian is not changed. In a way the CMS is just a renormalization scheme with complex renormalization constants.
We sketch the procedure: In the first step renormalized parameters are introduced. Let m 0 denote the bare mass of an unstable particle, then introduce The complex mass µ 2 is attributed to the propagator and resummed while the counter term δµ 2 is treated as a vertex and not resummed. Thus, the propagator in the CMS reads The usual causality prescription [see (2.4)] becomes irrelevant owing to the finite imaginary part of µ 2 = M 2 − iΓM . The procedure implies that the mass counter terms are complex. Since the bare mass is real, the following consistency equation holds Couplings which are purely real in the conventional framework become complex in the CMS if they are related to the masses, which is, for instance, the case for the electroweak mixing angle in the Glashow-Salam-Weinberg theory [21]. We have to employ suited renormalization conditions in order to fix the finite part of the parameters. Usually this is done in the on-shell scheme which is distinguished from others by the fact that the renormalized parameters are equal to physical observables. More concretely, one demands that the renormalized two-point function of a stable particle near its mass p 2 = m 2 is given by the Feynman propagator (2.4). This condition does fix both the mass renormalization and the field renormalization (see e.g. Ref. [28]). The on-shell scheme can be extended to the case of unstable particles and the appropriate renormalization conditions read [21]: Here Σ R denotes the renormalized self-energy of the unstable particle and Σ R is the corresponding renormalized self-energy differentiated with respect to p 2 . The renormalization conditions (3.5) together with the requirement that the bare Lagrangian is real, yielding consistency equations like (3.4), outline a gauge-invariant renormalization procedure. Apart from the validity of Ward identities one must make sure that the renormalization conditions do not introduce a gauge dependence. Given the fact that the complex pole is gauge independent, the renormalization point and the renormalization condition (3.5) are gauge independent. Those proofs were carried out by Stuart [1], Sirlin [2,3,29] and Gambino and Grassi [30]. Even though the renormalization conditions are similar to the ones in the on-shell scheme the difference may be significant as it is the case in the SM for the mass prediction of the W and Z bosons [31]. In view of gauge theories and physical observables, the complex pole is more than a theoretical construct and should be seen as the analogue to the mass for stable particles. For a discussion we refer to Ref. [32].

Unitarity in the CMS
In the CMS the Cutkosky rules are not valid in the sense that their application does not yield the same result as one would get by direct computation of the left-hand side of the unitarity equation (2.3). For instance, the Cutkosky rules require that the discontinuity of the tree-level s-channel diagram vanishes, (4.1) Replacing the stable particle with an unstable one, the direct computation yields In view of the LTE the reason is that the preconditions are not fulfilled and we cannot use the cutting rules for a propagator without having shown that there is a valid decomposition (2.5). As a consequence of the analytical continuation of the S matrix to complex masses algebraic relations are untouched, but operations where complex-conjugation is involved are no longer preserved as it is the case for the unitarity equation. The CMS guarantees gauge invariance, but it is no longer clear how unitarity is implemented. Veltman has shown [24] that for a superrenormalizable theory the S matrix in non-perturbative QFT is unitary on the Hilbert space spanned by only stable particles, Starting from the Källén-Lehmann representation for unstable particles which, in contrast to stable particles, lacks a one-particle pole on the real axis, he showed unitarity by deriving a LTE for dressed propagators. We apply this idea to the CMS in perturbative QFT and derive a corresponding LTE. The comparison to Veltman's unitarity equation (4.3) is done in three steps. In section 4.1 (step one) we construct a decomposition of the CMS propagator of the form (2.5). Then, it follows immediately that we can compute the left-hand side of (4.3) via the LTE as long as we consider interactions with couplings which are either real or have a corresponding complex-conjugated part in the interaction part of the Lagrangian. Otherwise the LTE is still valid, but the amplitude with all space-time points underlined is not any more equal to the complex-conjugated amplitude, and the left-hand side of the unitarity equation can no longer be related to contributions to the LTE. The preconditions for this identification are automatically fulfilled for interaction vertices if the bare Lagrangian is real. The argument fails for the imaginary mass counter term Im δµ 2 because the corresponding counter part resides in the resummed propagator, thus, we have a mismatch in the perturbative order and we have to take care of this contribution differently.
In sections 4.2.1, 4.2.2 (step two), we elaborate the structure of the LTE (2.7) which, in general, has more contributions than the ones actually contributing to the unitarity equation. One then realizes that the CMS has the same structure, up to higher-order corrections, as in the case of stable particles, i.e. the LTE splits into a normal region T and a complex-conjugated region T * , up to terms of higher perturbative order.

Decomposition for the CMS propagator
We start with the construction of a propagator decomposition in the case of the CMS. Since the decomposition is not unique, we list our assumptions which led to it. Basically, one decomposes the propagator in positive-and negative-time parts and adds something to these parts which is zero for positive and negative times, respectively. Working out this idea, one realizes that the allowed transformations have always, in Fourier space, the form of an advanced and retarded propagator ∆ A/R . Hence, our approach consists of defining meromorphic functions ∆ A/R with similar pole structure as in the case of stable particles, and among possible restrictions Occam's Razor suggests The function ∆ A/R must be chosen such that ∆ ± fulfils the decomposition theorem (2.5). The third equation is the condition that the advanced/retarded propagator has only poles in the upper/lower complex plane as it should be. Consequently, we have the same situation as in the case of stable particles, namely θ(±x 0 )FT ∆ A/R (x) = 0, where FT denotes the Fourier transformation. Furthermore, we demand that, similar to the case of stable particles, the retarded propagator turns into the advanced propagator by complex-conjugation and vice versa, which is our last assumption Given these restrictions one can easily derive the unique solutions for ∆ ± . In Fourier space they read: As a first but very important result one verifies that in the limit Γ → 0 + our solutions turn into the stable ones, i.e.
In view of consistency, this means that there is a smooth transition from unstable to stable propagator as the mass M 2 tends below the kinematic limit of instability. On the other hand, this result tells us that in a perturbative expansion in Γ the leading-order "cut" contribution is equal to the cut contribution of a stable particle with the same mass. Apparently, two problems appear: • Given an S matrix, one is usually not allowed to perform an expansion in small Γ, even though perturbation theory predicts O(Γ) = g 2 , where g is the coupling constant. For instance, for the s-channel production of an unstable particle the width is crucial for the finiteness of the result. The question is when we are allowed to do such an expansion or when is it actually necessary since we only want to verify the unitarity equation (2.3).
• At first sight the fact that the cutting rules for the CMS propagator for Γ → 0 + coincide with the ones for stable particles might interfere with Veltman's result, namely that only stable particles appear as asymptotic states in the unitarity equation (4.3).
Both points do not pose any problems, as we show in the upcoming sections.

Kinematic restrictions
The cutting rules are a special subset of the LTE relations where many terms in (2.7) do not contribute because the S matrix underlies physical constraints such as positive energy flow and real masses. These constraints reappear in the LTE amplitudes in form of kinematic factors, e.g. delta and theta functions. The situation is similar for stable and unstable particles, and we consider stable particles first. In our convention the incoming particles are on the left and the outgoing ones on the right. As an example consider the following amplitude in a scalar φ 3 theory: (4.8) If we sum over all possible underlinings the result equals zero except for one contribution which is immediately recognised as a cut (4.9) The example shows that the non-vanishing contributions of the LTE for stable particles split into two separate regions where the normal part (T ) is given by the black dots, while the complexconjugated part (T * ) is given by the white circles. In the following we call this property the cut structure. This means, in particular, that the LTE and simple kinematic arguments lead to the unitarity equation (2.3). Examples of vanishing LTE terms are (4.10) In the first term of (4.10) the cut structure is violated, i.e. there are no two well-defined regions and as a consequence at some vertex in the amplitude the required energy-flow direction is opposed to the physical energy flow which is from the left to the right, and no energy can be transferred to the final state. The second amplitude vanishes because the cutting rules require the intermediate particle to be on-shell which is impossible for stable particles since m 2 < p 2 = (p 1 + p 2 ) 2 . When CMS propagators are involved one would like to have, in particular, the cut structure of surviving LTE terms, but this is a priori not given. For instance, the first term in (4.10) does not vanish when the cut propagator is replaced by the corresponding CMS propagator. These cut-structure violating contributions come from the fact that for the CMS propagator ∆ ± (p, µ) (4.6) there is neither a θ ±p 0 nor a δ p 2 − m 2 but smoothed functions instead. The smoothing does no longer enforce the same strict kinematic constraints as for stable particles. Nevertheless, these contributions are suppressed by at least a factor Γ/M ∼ g 2 , and one obtains the same behaviour for unstable particles in a perturbative sense, meaning that those LTE terms violating the cut structure are always of higher-order in the coupling constant.
Yet, the argument is incomplete since suppressed terms can become relevant as one takes into account higher perturbative orders, i.e. they can be of the same order as higher quantum corrections. The next chapter is devoted to include the imaginary mass counter term in the LTE. We discuss how to simplify LTE relations and we show that the imaginary mass counter term is responsible for the fact that contributions being negligible at a certain perturbative order, stay negligible even if the calculation is extended to higher orders.

Including the imaginary mass counter term
A proper description of unstable particles requires resummation of contributions resulting in a finite imaginary part in the propagator. On the other hand, gauge invariance requires that the imaginary counter part to the complex mass enters the Feynman rules. It is not possible [33] to include such a coupling in the LTE relations. However, this is not necessary as we discuss in the following. Consider the insertion of a i(−iΓM ) coupling between two CMS propagators in momentum space, (4.11) This insertion can always be reduced to the usual propagator via simple differentiation with respect to ΓM and, as it becomes clear below, it is important that ΓM is real which is true by construction.
For an arbitrary amplitude F the left-hand side of the unitarity equation (2.3) is computed as follows: Every insertion of i(−iΓM ) can be generated by differentiating specific CMS propagators which we have to identify. Consider all remaining diagrams F τ obtained from F by setting the coupling ΓM to zero, but keeping the resummed counter part in the propagator, then we have On the other hand, instead of setting ΓM to zero we can also remove multiple powers of i(−iΓM )i∆ yielding the same set of diagrams F τ , but allowing us to identify certain propagators in F τ , namely the ones which are located in a chain of i(−iΓM )i∆ insertions and are left over after removing the insertions. Denote the set of those propagators which are linked to diagrams with at least one insertion with Ω := {Ω nτ i τ i }, where τ i identifies the propagator i in the diagram F τ and n τ i represents the maximum number of insertions next to that propagator. The original amplitude F is reconstructed in two steps. First we transform each propagator τ i via Φ Ω : ΓM → ΓM + ω τ i in its denominator and denote the resulting set of diagrams by F τ Φ Ω . Then we differentiate with respect to ω τ i up to n τ i times and set ω τ i to zero in order to generate the needed insertions as in (4.12) and obtain the relation (4.14) By construction F τ Φ Ω is free of imaginary mass counter terms and in the case of a scalar theory, as we consider it here, we can simply commute differentiation with taking the real part. Thus, the result can be expressed by the LTE of F τ Φ Ω and applying the (real) differentiation in (4.14) on it. This representation implies that the cutting rules in T , T * stay the same even if the imaginary mass counter term is included. To see this consider the case where we have a CMS propagator and at least one insertion of i(−iΓM ). Undoing the insertion, one is left with the CMS propagator and after applying the LTE, the propagator i∆(p, µ) either stays the same (∈ T ), transforms into −i∆ * (p, µ) (∈ T * ), or belongs to the cutting region. If the propagator is in the normal region we recover the result (4.12) after applying the differentiation. If the propagator is in the complex-conjugated region, i.e. −i∆ * (p, µ) we can work out the signs and we find Consequently, we recover the usual cutting rules or Feynman rules in the regions T , T * , and needless to say that the result is true for more than just one insertion. The case that the propagator of the unstable particle is on the cut is discussed in the following sections.

Non-resonant contributions of unstable particle propagators
We come back to the question whether contributions, which in the case of stable particles vanish because of kinematic constraints, can actually contribute in the case of unstable particles. To this end we define a simple model we use throughout the rest of the paper. The model consists of an unstable scalar particle φ with mass µ 2 = M 2 − iΓM and a stable scalar particle χ with mass m and the interaction The higher-order contributions O g 2 are of the style of (4.10), i.e. they either violate the cut structure or are further suppressed [owing to s = M 2 and (4.20)], but they have the topology of k

.
As we have demonstrated in Section 4.2.2, we can take into account the imaginary mass counter term via differentiation, and the left-hand side of the unitarity equation (2.3) for k 1 + k 2 reads The first term can never become resonant because it violates the cut structure and the second term is non-resonant as long as the amplitude itself is non-resonant which is true by our assumption s ≈ M 2 . Deriving the leading behaviour of ∆ ± (p, µ) for small Γ/M for p 0 = ± p 2 + M 2 we obtain where f is a smooth function with Taylor expansion in ΓM . The linear dependence in ΓM indicates the suppression and after carrying out the differentiation in (4.19) the linear dependence is eliminated and the resulting order is g 2 × O(Γ 2 ) = g 6 . Since g 4 is the current accuracy of the amplitude (4.17) the contributions from k 1 + k 2 are negligible, which is no coincidence. The argument is easily extended to arbitrary high order. Consider an amplitude up to the order of g n , where n is arbitrary. We compute the LTE according to (4.14) and assume we have a term U ∈ F τ Φ of the order of g m and m ≤ n either violating the cut structure or having at least one non-resonant ∆ ± (p, µ). The order of U is bounded from below as follows where the equality (=) occurs solely for one non-resonant ∆ ± in U. Multiple insertions of i(−iΓM ) result in a systematic elimination of orders as can be easily seen realizing that the differential operator in (4.14) is nothing but the Fourier representation of the translation operator where the series of the exponential function is terminated at the order (ΓM ) The results so far can be summarised as follows: • There exists a decomposition for the CMS propagator (4.6) satisfying the decomposition theorem (2.5), thus, allowing to derive a LTE.
• The LTE does not allow to include the imaginary mass counter term directly, but it can be introduced via (4.14).
• We have shown that cut-structure-violating terms as well as all cuts of non-resonant propagators can always be neglected no matter at which order in the coupling constant the violation takes place. Thus, only correctly cut LTE amplitudes have to be taken into account which is required by unitarity.
Further, from the stated results it follows immediately that unitarity is fulfilled automatically if there are no resonant ∆ ± (p, µ). The missing piece which has yet to be investigated is when ∆ ± (p, µ) becomes resonant. This happens when internal momenta are integrated out, or usually when certain phase-space integrations are carried out.

Resonant contributions of unstable particle propagators
In this section we finally discuss what happens when ∆ ± (p, µ) is resonant. Then, the corresponding terms in the LTE usually are no longer negligible as for instance the second term of (4.10) for an unstable s-channel particle and s ≈ M 2 . Naively interpreting the unitarity equation (2.3) would lead us to the conclusion that only the sums of all diagrams on both sides of the unitarity equation coincide, though, in the case of stable particles diagrams can be separated according to their topology and perturbative order. Perturbative unitarity then follows from the fact that the coupling can be chosen arbitrarily meaning that we can, in principle, distinguish between orders by varying the coupling. This argument fails when the theory is renormalized according to the CMS. The distinction of perturbative orders does no longer work because of resummation, and we actually have to consider sums of diagrams, but the occurrence of non-trivial dependencies between topologically different Feynman diagrams can be excluded at least in scalar theories.
Before proceeding we mention one problem which inevitably complicates the forthcoming discussion: When speaking of a perturbative order one usually refers to the expansion in the coupling constant. For the CMS there is no such perturbative order because there is no welldefined order in g owing to resummation. The CMS propagator 1/(s − µ 2 ) contributes to the perturbative order, but has no Taylor expansion near the resonance, i.e. for s ≈ M 2 . Since ΓM is assumed to be of order g 2 M 2 , the change in behaviour occurs near the resonance. Strictly speaking, we cannot investigate the CMS for perturbative unitarity because we cannot assign a perturbative order to amplitudes in the framework of the CMS. However, we can speak of relative orders in the sense that if the left-hand side of the unitarity equation has a phase-spacedependent order then also the right-hand side does, and the order of the quotient is independent of phase space. Our strategy is therefore to identify the diagrams not only by their topology, but also by the perturbative order in certain phase-space regions where it is well-defined, e.g. for s − M 2 < ΓM or s − M 2 ΓM . At this point we stress that we are not trying to carry out an expansion in ΓM , we only identify the diagrams necessary for unitarity to hold. The only expansion done is in the propagators ∆ ± connecting the regions T , T * , thus, the unitarity equation is considered with finite-width effects.
We start again with the example of the s-channel production of an unstable particle. For the resonant case one must perform a Laurent expansion to capture the leading behaviour, The LTE at LO reads for where in the last step we made use of (4.23) and the identity The LTE does not know about the diagrammatic significance of ΓM which we have to determine and plug in by hand, and, as discussed before, ΓM must be computed at least by the one-loop renormalization conditions (3.5). We denote Σ 1 R the renormalized (according to the CMS) oneloop self-energy of the unstable particle and Σ 1 the corresponding unrenormalised one. The self-energies are related to the counterterm δµ 2 by the renormalization conditions (3.5), and the consistency equation (3.4) links δµ 2 and ΓM , so we obtain a relation between ΓM and Σ 1 ΓM = Im δµ 2 = − Re iΣ 1 (p 2 ) p 2 =µ 2 . (4.26) In the next step we assume that the analytic continuation of the self-energy behaves well enough at p 2 ≈ M 2 , i.e. we suppose that which can be obtained formally by performing an expansion in p 2 , but sometimes a Taylor expansion is not possible as it is the case when infrared singularities appear. Then, one usually has logarithmic corrections, but they do not bother us as long as the limit g → 0 exists. In the next step we make use of the assumption (4.27) and find for ΓM : This equation expresses what is known from the on-shell scheme, i.e. the width is the cut through loops and can be interpreted as the decay width. At one-loop order the widths of the CMS and the on-shell scheme coincide, but this is no longer true at higher orders and we are not able to argue this way in the general case. Nevertheless, let us make use of this result to demonstrate unitarity at one-loop order. At this order we can directly apply the cutting rules to Σ 1 since there are no intermediate unstable particles (in our model) (0.1) . (4.29) This result together with (4.24) and (4.26) yields exactly what is requested by unitarity, namely (4.30) One can verify, that no double counting occurs and that the one-loop amplitude k 3 in (4.17) is cancelled by the mass counter term k 2 as it is required by the renormalization condition (3.5). The Laurent expansion does not make sense when the momentum p is an internal loop or an integrated phase-space momentum. It turns out that not only the argument, but also the result is the same. As discussed in Section 4.1 the solutions for the decomposition ∆ ± (p, µ) for small widths are given by the cutting rules (4.7). As Veltman has shown, one does not expect unstable particles in asymptotic states, and this dilemma is resolved at LO as follows. Similarly to the identity (4.25), we have in distributional sense that where we used (4.26) and (4.29) in the last step. This shows that the LO resonant ∆ ± can be interpreted as a higher-order cut amplitude which is what is required by Veltman's unitarity equation. For instance, consider the one-loop self-energy of the stable particle denoted asΣ 1 .
Computing the LTE and making use of the result (4.31) yields up to higher orders: This can be generalized and proves unitarity as long as the leading behaviour of ∆ ± is sufficient.
Having discussed the leading contributions, it is still unclear how unitarity can be verified at higher orders which, especially by direct computation, is far more involved and probably not even possible in general. In the following, we focus on general properties of the LTE in order to prove perturbative unitarity within the CMS in general.
For LTEs of higher-order amplitudes the approximation (4.31) is not sufficient, and more terms in the expansion of ∆ ± in Γ/M must be taken into account. However, with the expansion the diagrammatic interpretation gets lost, and it is difficult to compare the result to the righthand side of the unitarity equation. Motivated by Veltman's approach, namely to derive LTEs for dressed propagators, we try to identify the LTE of higher-order two-point functions as higherorder cut two-point functions without using (4.31), i.e. we aim at defining cuts that fulfil Provided that (4.33) is true, if the LTE of an arbitrary amplitude can be rearranged in such a way that the cut region is given by LTEs of two-point functions instead of the usual ∆ ± , then unitarity follows immediately since we have the correct cut-structure (as shown in Section 4.2.3) and valid cuts (4.33) (to be defined). Thus, the problem is reduced to the problem of studying LTEs of two-point functions and this is necessary because the CMS mixes loop orders and only the LTE of the whole two-point function can yield well-defined cuts, which we show below. The idea is the following: For a specific amplitude we consider all diagrams up to a certain order in the CMS. Applying the LTE yields contributions with the correct cut structure. In contributions where unstable propagators are cut, these are iteratively replaced by cuts of the full propagator upon including the needed higher-order contributions. That this replacement is valid can be justified as follows. Consider a LTE contributionF with a ∆ ± (p, µ) originating from a specific CMS propagator ∆(p, µ) somewhere in a diagram F. If the diagram is of highest considered order, we can simply use (4.31) to replace ∆(p, µ) by a cut through stable particles. If the diagram F is not of highest order, there are diagrams which have the same structure as F but more self-energy insertions next to that propagator. Among the contributions to the LTE of these higher-order diagrams are terms which have the same structure asF, but where instead of the ∆ ± LTE components of two-point functions appear (originating from the self-energy insertions), and collecting all these terms we retrieveF with a nested LTE of a two-point function.
We first give a simple example for a LTE of an amplitude with nested two-point functions. We show that the nested two-point functions reappear as a LTE of the two-point functions, i.e. we can identify cut two-point functions. Consider For the purpose of demonstration, assume values of s off the resonance which is less complicated (the case s ≈ M 2 is taken care of as described in (4.41) below). Computing the LTE of F yields, up to higher orders, represents the LTE of the nested two-point function. Underlined endpoints in two-point functions do not come with couplings and the underlining rules must be extended. We define the LTE of two-point functions by pretending there were couplings (ig) 2 at the end-points allowing us to make use of the usual underlining rules. After removing the endpoint couplings (dividing by g 2 ), the difference between the two-point function with and without couplings is a sign, (ig) 2 /g 2 = −1, which is the reason why we have to take 2 Re instead of −2 Re. We arrive at i.e. the cut through the two-point function is expressed by a cut of a two-point function of lower order and the cut representation is consistent with (4.33). After isolating cut two-point functions the normal region T and the complex-conjugated region T * are given by and notice that for this way of identifying terms in the LTE external states need not to be on-shell. Having shown how to rearrange LTEs of matrix elements in order to identify nested LTEs of two-point functions, we elaborate the meaning of (4.33). One point which in our current framework turns out to be a real problem beyond one loop is the diagrammatic significance of ΓM . At some point in our calculation, in particular when ∆ ± is resonant, we have to plug in the expression for ΓM obtained from the renormalization condition. This problem can be circumvented by making use of resummed results, i.e. instead of using the usual perturbative expansion we represent two-point functions by their fully resummed equivalent deliberately taking into account non-significant (higher) perturbative orders. This eliminates some of the partial resummation of the CMS and is sufficient for a diagrammatic interpretation. Returning to the statement that, in contrast to the on-shell scheme, ΓM does not represent well-defined cutcontributions, one realizes that 2 Re iΣ R := 2 Re [iΣ R − ΓM ] (4.39) does, which can be understood as follows. We express the resummed two-point function as and computing the LTE of this expression we obtain where the cut self-energies are defined by (4.39) and normal and complex-conjugated self-energies are just the usual ones renormalized according to the CMS. This definition of cuts is in agreement with Veltman's unitarity equation (4.3) which has yet to be shown. In (4.39), (4.40), (4.41a) and (4.41b), i.e. for dressed propagators there is no partial resummation. Starting from the CMS, we can construct dressed propagators by resumming renormalized self-energies. After full resummation the iΓM in the propagator cancels with the iΓM of iΣ R and all explicit iΓM expressions disappear which is the reason why (4.39) represents well-defined cuts. In this limit unitarity is not violated as has been shown by Veltman [24] and it is left to understand that nothing goes wrong when going from full resummation to the CMS. This is formally shown as follows: We only need to study cut two-point functions, i.e. we need to show that the right-hand side of (4.41a) is equal to (4.41b). Assume the left-hand side of (4.41a) is given at n loop order, then equation (4.41a) tells us that the LTE of an n loop two-point function can be computed by LTEs of n loop self-energies, where we actually mean the self-energies (4.39). As in our example (4.37), these self-energy LTEs are iterated LTEs of one-to (n − 1)-loop two-point functions. Thus, one makes the induction hypothesis that LTEs of two-point functions (4.41) represent well-defined cuts at n − 1 loops. Expanding the two-point function on the left-hand side of (4.41a) at n loops the statement (4.41b) follows from the induction hypothesis, the start of the induction being given by (4.29). Thus, cuts are defined iteratively as in the example (4.37) and each cut through an unstable particle, possibly belonging to higher-order contributions and after collecting all terms like in (4.32) results in a nested cut two-point function (4.42) which itself can have cut unstable particles.
Let us illustrate the procedure at the example of our toy theory. Consider the two-point function of an unstable particle at two-loop order R,φ is the according to the CMS renormalized two-loop self-energy of the unstable particle φ, and Σ 2(1) R,φ and Σ 2(2) R,φ denote the one-loop and two-loop renormalized contributions to the two-loop self-energy, respectively. At this point we have to demand that the perturbation series is valid in the sense that the two-point function is well approximated by where the two-point function on the right-hand side is defined as the resummed propagator for which the self-energy is evaluated and renormalized at two-loop order. Then, we can compute the LTE with the help of (4.41a) and up to higher orders we have Notice that the imaginary part of δµ 2 dropped out and only the real part is left over renormalising the one-and two-loop self-energies. In what follows we could have argued with the induction hypothesis, but we work out this example and we compute the LTE of Σ which can be generalized to show that all two-loop self-energies in this theory yield, up to higher orders, well-defined cut self-energies. Having dealt with the two-loop self-energies, if we combine the one-and two-loop result in (4.45), we have shown (4.41b) at two-loop order. In case of nested unstable two-point functions we would have had to use the induction hypothesis.
Our results can be summarised as follows: • In the beginning we demonstrated how to compute LTEs of arbitrary amplitudes leading us to the result that the cut structure is guaranteed in a perturbative sense which is the basis for unitarity. One is left to check that ∆ ± yields well-defined cuts.
• From the solutions of ∆ ± for CMS propagators we concluded that unitarity holds off resonances. The leading behaviour of resonant ∆ ± can be interpreted as the cut one-loop two-point function [see equation (4.31)].
• The leading approximation of ∆ ± is not enough beyond one-loop and higher-order corrections need to be included. The LTEs of different loop orders do not separately represent valid cuts in the sense of Veltman's definition and they must be considered simultaneously because owing to the partial resummation the diagrams are connected to each other and cancellations take place. Whenever there is a cut through unstable particle lines we identify cuts by iteratively including appropriate higher-order contributions resulting in nested LTEs of two-point functions. For the LTE of the two-point functions up to a given order a valid cut interpretation can be assigned which is consistent with the interpretation of Veltman, i.e. only lines of stable particles are cut.

Conclusions
The Complex-Mass Scheme provides a straightforward method to consistently implement unstable particles in perturbative calculations. Formally, the procedure is an analytic continuation of matrix elements to complex masses with appropriate renormalization condition.
In the Complex-Mass Scheme the Cutkosky cutting rules can no longer be used to verify unitarity, and it is not clear how perturbative unitarity is implemented. Following Veltman, we derived a Largest-Time Equation within the Complex-Mass Scheme which could then be used to obtain a diagrammatic representation for the imaginary part of scattering amplitudes, also when unstable particles are present.
Our derivation of the Largest-Time Equation is based on the decomposition theorem and we showed that an appropriate decomposition can be achieved for the Complex-Mass Scheme propagator. As a result, one finds that the would-be cut propagators ∆ ± (p, µ) of unstable particles are smoothed versions of the stable ones. In case of stable particles the Largest-Time Equation coincides with the Cutkosky cutting rules, but including unstable particles leads to additional contributions which can be interpreted as contributions where the energy flow is backward. Performing an expansion solely of would-be cut propagators ∆ ± (p, µ) of unstable particles in Γ/M does indeed yield cutting rules where unstable resonant ∆ ± (p, µ) can be replaced by higher-order cuts through stable particles only. In this way, we recover the perturbative statement of Veltman's result in the Complex-Mass Scheme, namely that a QFT is unitary up to higher orders only if unstable particles are excluded from asymptotic states.