Complex Langevin: Correctness criteria, boundary terms and spectrum

The Complex Langevin (CL) method to simulate `complex probabilities', ideally produces expectation values for the observables that converge to a limit equal to the expectation values obtained with the original complex `probability' measure. The situation may be spoiled in two ways: failure to converge and convergence to the wrong limit. It was found long ago that `wrong convergence' is caused by boundary terms; non-convergence may arise from bad spectral properties of the various evolution operators related to the CL process. Here we propose a class of criteria which allow to rule out boundary terms and at the same time bad spectrum. Ruling out boundary terms in the equilibrium distribution arising from a CL simulation implies that the so-called convergence conditions are fulfilled. This in turn has been shown to guarantee that the expectation values of holomorphic observables are given by complex linear combinations of $\exp(-S)$ over various integration cycles. If the spectrum is pathological, however, the CL simulation in general does not reproduce the integral over the desired real cycle.


I. INTRODUCTION
The Complex Langevin (CL) stochastic process [1,2] describes the evolution of a probability distribution on the complexified configuration space, generated by the Fokker-Planck (FP) operator L T .The probability distribution is supposed to converge to an equilibrium distribution which reproduces the averages with respect to a complex measure of a class of holomorphic observables.
The application of the Complex Langevin equation is not without problems, though.Two possibilities of failure have been identified: the process may fail to converge and it may converge to the wrong limit.Failure to converge may occur via exponentially increasing expectation values of certain observables, related to spectrum in the right half complex plane, or it may occur due to spreading of the probability measure, creating slowly decaying 'skirts'.In the latter case expectation values of observables with high powers become mathematically undefined (thus fail to converge in practice).
The conservation of probability under the CL process guarantees a certain trivial stability: expectation values of observables which are bounded on the complexified configuration space will remain bounded under the CL process.This shows that the semigroup generated by FP operator is bounded on an appropriate (Banach) space and exponentially growing modes are not present for bounded observables.But unfortunately this is irrelevant, as we need to consider holomorphic observables, which are not bounded unless they are constant.So for the study of convergence and stability a different mathematical setting is needed.This is done in the next section.
More serious is the problem of wrong convergence.This is typically due to the spreading out of the probability density, leading to slow decay (skirts) of the distribution and the occurrence of boundary terms, as discussed in [24,25].This spreading can be tested by the control variables introduced in the next section.These are used to define a general class of criteria for correctness of the CL results; they are designed to rule out boundary terms but they also rule out instability in the form of exponential or sub-exponential growth in the evolution of observables.The criteria are sufficient, provided certain additional conditions hold; so they are not eliminating all possible failures of the CL method.These limitations as well as open problems are discussed in the last section.
We check some versions of the criteria numerically for a simple model of one variable in Section 3 and for a lattice model in Section 4. By direct numerical determination of the spectrum it is revealed that violation of the criteria does not necessarily imply the presence of 'bad' spectrum.
There is a regime in which failure of the criteria indicate presence of boundary terms and 'wrong convergence' of the CL simulations takes place, yet the the direct determination of the spectrum shows absence of exponentially growing modes.In a simple model we find that the appearance of unwanted spectrum is linked to the Lee-Yang zeroes; the significance of this fact is not yet entirely clear.
A word of caution is in order: the spectrum of a formally defined differential operator depends on the precise definition of the space on which it operates; likewise the relation between the spectrum of an operator and the behavior of the semigroup it generates may be more subtle than we are used to from finite dimensions.Some of these points are addressed in the appendices.
There are, in principle, four possible combinations of boundary terms or no boundary terms, and bad spectrum or no bad spectrum (by bad spectrum we mean eigenvalues of L c , the complex Fokker-Planck operator, with positive real part).We find that, depending on the parameters chosen, bad spectrum and boundary terms may appear together, but boundary terms may also appear without bad spectrum and, of course, there also is also a regime in which neither boundary terms nor bad spectrum occur.So in the presence of boundary terms there is always either wrong convergence or no convergence.Hence the absence of boundary terms is a crucial condition for correctness; the absence of bad spectrum plays only a subsidiary role.Limitations of this statement, concerning situations where the absence of boundary terms, while necessary, is not sufficient for correctness, are discussed in Section V.

II. MATHEMATICAL GENERALITIES
To keep the notation simple, we carry out this discussion for one variable.The generalization to many variables requires some care, as discussed in Section IV.

A. Notation
We consider complex measures given by a complex density ρ which is holomorphic and given in terms of an action as Expectation values of holomorphic observables observables O are given by integration over a suitable integration "cycle" (in the terminology of Witten [26]) γ: The CL equation is where dw is the increment of the Wiener process normalized as The evolution of the probability density P on C = R 2 is given by The Fokker-Planck(FP) equation: The evolution of observables is given by transpose which simplifies for holomorphic observables to Formally, these linear evolutions are solved by exponential semigroups, such as exp(tL) etc..

B. A trivial fact
We start with a simple fact.Our probability measures on C ≡ R 2 are given by distributional 'densities' P (x, y; t) on R 2 , i. e. positive distributions (this includes δ distributions and L 1 functions).We denote the standard (total variation) norm of measures on R 2 by ||.|| 1 .For a probability measure P we have Since exp(tL T ) preserves probability, it is a contraction on the space of complex measures, i. e. for any complex density ρ and all t ≥ 0, This means in particular that as an operator on L 1 (R 2 ), L T has no unstable (exponentially growing) modes.More explicitly this can be seen by noting that the Fokker-Planck evolution operator exp(tL T ) has an integral kernel exp(tL T )(x, y; x ′ , y ′ ) ≥ 0 satisfying Mathematically the natural space of observables would be the space of bounded continuous functions, a subspace of L ∞ in C. On this space exp(tL) is really the adjoint (transpose) of exp(tL T ) and so the dual semigroup exp(tL)O is again a contraction (i.e. has ∞ norm ≤ 1).
Unfortunately L ∞ (R 2 ) does not contain any nonconstant holomorphic functions, so clearly the space of observables has to be enlarged; that requires the space of allowed measures to be restricted in such a way that all observables in the enlarged space have well-defined expectation values.

C. Conditions on the space of probability measures
To get a nontrivial space of holomorphic observables, we have to consider weighted spaces of measures and observables.We define these spaces in terms of a strictly positive weight function σ(x, y), growing at infinity, using the norms By a slight abuse of notation we denote the space of complex measures ρ on C with ||ρ|| 1,σ < ∞ by Of course the space of observables also has to be chosen in such a way that the evolution operator L c and the semigroup exp(tL c ) leave it invariant; this means that we cannot limit ourselves to a finite dimensional subspace, such as polynomials up to a given order.
The choice of the weight σ is dictated by the class of observables we want to consider: it has to be chosen such that they lie in L ∞ σ −1 .In other words: σ has to grow at least as fast as all the observables we want to consider.We should also choose σ so that it does not grow too much faster than the observables in our chosen space, because we do not want it to cause a 'false alarm' about slow decay.
For the noncompact case of R d the usual set of observables consists of all polynomials, so we need σ to grow faster than any power at infinity.Possible choices are or In the following we use the second choice σ ≡ σ 2 with α = 1.
For the case of a compact real configuration space, such as U (1), the natural space of observables is spanned by the exponentials exp(inz), which grow exponentially in the imaginary direction.So in this case we need a stronger than exponential decay in the measure; we may choose for instance or gauge models.
It is not automatically true that exp(tL T ) is a contraction from L 1 σ to itself; we have to make an assumption quantifying the necessary decay of the measure evolving under the CL process: with a constant C σ independent of t.Written out, (17) says The weight function plays the role of a non-holomorphic control variable, which is required to have a well-defined and bounded expectation value under the probability measure P evolving according to the Fokker-Planck equation.
If the CL process is ergodic, the choice of the initial distribution P (x, y; 0) does not matter and we may for instance choose P (x, y; 0) = δ(x)δ(y) which is convenient for numerical checks.
Eq. ( 17) or (18 imply so expectation values of observables will be bounded in time; i. e. Assumption A implies that there are no exponentially growing modes showing up. Assumption A is a criterion for correctness: Since it guarantees strong enough decay on P (x, y; t), it implies the absence of boundary terms for the observables in L ∞ 1/σ .Condition A has the following relation to the 'drift criterion' of [27]: the latter can be interpreted as a special case; it requires the existence of an α > 0 such that Assumption A is satified for the This shows that for any polynomial S of higher than second degree, the drift criterion is stronger than the versions ( 13) and ( 14).In compact cases, the drift grows exponentially in the noncompact directions, so the control variable σ d,α is again stronger than ( 15) and ( 16).It is in fact stronger than necessary, i. e. it might signal incorrectness of certain CL results when they are in fact correct.
This has been found to actually occur in some cases of the one-link U (1) model [29] but it might also happen for polynomial models.
Under Assumption A the left hand side of ( 21)is bounded uniformly in t, so the right hand side is uniformly bounded as well i. e. there are no unstable modes of L showing up.
In more detail the argument goes as follows: by assumption choosing now this bounds the left hand side of (22) by If there is no boundary term, i. e. ( 21) holds, we thus find which shows the absence of unlimited growth (exponentially or otherwise) in O(x + iy; t).
Restricting L to the subspace of holomorphic functions in L ∞ σ −1 , it can be replaced by L c due to the Cauchy-Riemann equations, so under our assumption, L c as well can have no unstable modes in this space.L. L. Salcedo [30] pointed out that a simple quartic model sheds light on some problems of the CL method.The model is defined by the action which we investigate for λ ≥ 0, m 2 > 0 and complex h (below we denote the imaginary part of h with h I ), corresponding to the complex density (Note that a lattice version of this model was studied in [31] using the Complex Langevin equation.) As remarked by Salcedo, given λ > 0, m 2 ≥ 0, the partition function Z(h) = ρ(x)dx vanishes for certain values purely imaginary of h (so-called Lee-Yang zeroes), leading to divergent expectation values of x n , whereas the CL equation does not show anything special at these values; so clearly the CL results cannot be correct.This fact is borne out by numerical studies, which show deviations between the exact results and the numbers produced by CL, becoming most dramatic near the Lee-Yang zeroes.Here we want to point out that these deviations are linked to massive failures of assumption A for Im h larger than some value h c , so Eq. ( 25) does not hold there, we expect boundary terms to occur and we cannot use the criterion to rule out spectrum of L c in the right half plane.In the next subsection the spectrum of L c is directly determined numerically with the result that the appearance of unstable modes, while not coinciding with the appearance of boundary terms, is actually linked to the Lee-Yang zeroes (see also [31]).
To give a definite meaning to the spectrum of L c , we consider it as an operator in the Hilbert space The spectrum is then the same as the spectrum of considered as an operator on (Maybe it would be more natural to continue working in the Banach space defined in the previous section, but for spectral considerations Hilbert spaces are more convenient.) We are interested in complex 'magnetic fields' h, so H is a Schrödinger operator with a complex potential.
Let's now consider purely imaginary h: the spectrum of hermitian part (H + H † )/2 now reaches down to −h 2 I /4 < 0; in fact is an eigenvector of (H + H † )/2 with eigenvalue −h 2 /4.This shows that exp(tL c ) is not a contractive semigroup on H S .But the numerics presented in the next subsection suggests that nevertheless remains bounded for all t > 0, provided h I is small enough.ψ 0 = exp(−S(x)/2) is an eigenvector of −H with eigenvalue 0 and presumably exp(tL c )φ converges to a multiple of ψ 0 for all φ ∈ H S in this case (see discussion in Appendix A.)

A. Checking Assumption A
We choose the weight function this allows to consider observables in the (Banach) space defined by the norm which contains all polynomials in z = x + iy and which also lie in the Hilbert space H S (28).
In Fig. 1 we show the expectation values σ 2 t and σ d,α t (α = 0.1, 0.5, 1.0) under the CL process for short Langevin times t, for the parameters λ = 1, m 2 = 0.1, h I = 1.0.We see a sharp increase of both quantities starting above t = 1.2; σ d,1.0 t takes off a little earlier than the other expectation values; this is not surprising considering the stronger growth for α = 1.0.So for these parameters, Assumption A is violated with all choices of the weight function.We also conclude from Fig. 1 that for the parameters chosen and t < 1.2 there are no visible boundary terms.This means that via integration by parts P (x, y; t)O(x + iy)dx dy = P (x, y; 0)O(x + iy; t)dx dy .
Notice that increasing t up to 1.2 the data tend towards the correct equilibrium values, but before they can reach them, boundary terms appear and drive the results away from the correct ones (for z 2 ) or make it impossible to determine them due to huge fluctuations (for z 4 ).In other models [24,25] it was found that for suitable choice of parameters, there is a 'plateau' in t, i. e.
an interval in t in which the CL results were consistent with the correct ones, before the boundary terms appeared.For the model at hand, this situation also occurs for smaller values of h I .
The distribution appears for t < 1.2 to decay faster than 1/σ 2 and 1/σ d,α , for larger t the decay becomes slower, probably power-like, leading to boundary terms and failure of CL.
It is possible to understand why around t = 1.2 the character of the CL process changes and a skirt begins to show up: since we have no noise in the imaginary (y) direction, motion in this direction cannot be faster than that determined by the deterministic equation The flow pattern of the drift is such that |K y | has the largest downward size for x = 0, in fact the solution of (37 will reach −∞ after a finite time t c .Since we are starting the process at the origin, for a time t < t c , in the presence of noise in the x direction, no value lower than y 0 (t) can be reached; so P (x, y; t) is supported in a strip y 0 (t) < y < 0 and it is well localized in x, so no skirt can arise.t c is determined from (37) by for h I = 1 this gives In Fig. 2  We should remark it does not make sense to expect correctness for low powers and failure for the higher ones, because that would mean failure of the 'consistency conditions' linking different powers (see [32]).This again shows that it is necessary to work in a space of observables invariant under L c and exp(tL c ).
We should also note that with our choice of purely imaginary h, the exact values of z n for n even are purely real, whereas for n odd they are purely imaginary.As for the CL values, due to the symmetry x → −x of the drift force K, this is also true as long as we have convergence.
We conclude from Fig. 2 that for λ = 1, h I ≤ 0.25 there are no visible boundary terms; the values of Re z n agree within the errors with the exact ones (the same is true for the imaginary parts) and L T as well as L c have no spectrum in the right hand plane.
It should be noted that the first Lee Yang zero for our choice λ = 1, m 2 = 0.1 is at h 1 ≈ 2.52i, so the blowup of σ both for finite t (Fig. 1) and for the equilibrium in Fig. 2 happens for |h I | much smaller than |h 1 |.This is because the form of Assumption A we used is sufficient but not actually necessary to rule out "unstable modes".In the next subsection it is shown by direct numerics that spectrum in the right half plane only appears for |h It is also noteworthy that apparently it does not make much difference whether one chooses σ 2 or one of the σ d,α as control variables as long as α is not too large: the blowup happens pretty much in the same place for the different α values, both in t and in h.
But the simulations, together with the results of the next subsection, also make manifest that spectrum in the left hand plane alone does not guarantee correctness because it does not rule out boundary terms.The massive failure of expectation values z 2 and z 4 long before the first Lee-Yang zero shows this clearly.We also look at boundary terms for the observables z and z 2 ; as explained in [24,25]), they are obtained as In Fig. 3 we show the plot of the boundary terms of these observables.As one observes, the first boundary term of z seems to be consistent with zero (in the infinite cutoff limit) for all magnetic fields, however the second boundary term is nonzero above h I > 0.5.For the observable z 2 , already the first boundary term shows nonzero values above h I > 0.5.This confirms the assessment made above using the control variables σ 2 and σ d,α .

B. Direct determination of the spectrum
In this section we investigate the spectrum of the L c operator for the model ( 26), given by For the numerical investigation we used several bases e (a) i : where e n is the monomial basis, and Φ n (z) are the eigenfunctions of the Hamiltonian of the corresponding harmonic oscillator.We truncate these bases using the first N basis vectors, and calculate the spectrum of the resulting N × N (in general complex) matrix using the QR algorithm with explicit shifts [33].The numerical diagonalization requires the usage of high precision numbers, e.g. at N = 1024 we use floating point numbers with a mantissa of 1024 bits to avoid the appearance of spurious eigenvalues due to precision loss.
To transform L c into the bases given above one writes e.g.
where A = 0 gives the monomial base e (1) and A = 1 gives the base e (2) .One than calculates Thus L c in this basis is given by First we investigated the convergence of the spectrum as the truncation is improved.At h = 0, the spectrum of L c is known to consist of non-positive real eigenvalues.In Fig. 4 we show the spectrum of L c for zero magnetic field for different truncations in the e (2) basis.We observe that non-real eigenvalues appear for the truncated matrices, however, as the truncation is improved, more and more eigenvalues appear on the real axis, and the converged eigenvalues are all real (and non-positive).In principle the spectrum of the L c operator in other bases should converge to the same eigenvalues, provided the bases are related by a bounded linear map with a bounded inverse (for more general basis changes this may fail, see Appendix A).The convergence rate (with increasing truncation), however, may be basis dependent even then.From the bases mentioned above, e (2) shows by far the fastest convergence rate.
Next we investigate the number of positive real-part eigenvalues (which make the L c evolution of some observables unstable) as a function of the magnetic field h.At zero real part of the magnetic field, as Im h is increased, eigenvalues appear with positive real parts.As observed in Fig. 5, the number of such eigenvalues increases by one precisely at the Lee-Yang zeroes of the theory.For higher magnetic field magnitudes, higher truncation of the L c operator has to be used for convergence, as observed in Fig. 5.In Fig. 6 we show the number of eigenvalues with positive real part as a function of the complex h parameter.Note that by the Lee-Yang theorem, which applies here [34,35], zeroes only occur for purely imaginary h.

IV. THE XY MODEL
In this section we test the proposed control variables of the correctness criterion for the three dimensional XY model defined by the action where x represents the space-time coordinate on a 2+1 dimensional cubic lattice, x + ν is the neighboring lattice point of x in the direction ν, and µ is the chemical potential.At µ > 0 the action is in general complex, resulting in a sign problem hindering Monte Carlo simulations of the theory.This model has been previously investigated using the CLE in [36], and its boundary terms in [25].The sign problem of this model can be solved using the worldline formulation [37].
To investigate the boundary terms in the XY model, we first defined the norm where the field configurations satisfying N IM < C for some real C enclose the real manifold in a bounded domain.We than investigate the observable  48) as a function of β, for various α values, at µ = 0.1, measured on an 8 3 lattice.We used the exponent γ = 1.2 as indicated.
• Absence of boundary terms in the equilibrium measure of CL ensures that the "convergence conditions" (CC) [32] and the Schwinger-Dyson equations (SDE) are satisfied.This remains true also in the presence of a kernel.
• As shown in [40], the SDE imply that the expectation values with the equilibrium measure are given by a complex linear combination of the integrations over inequivalent integration cycles.
If there are several inequivalent integration cycles, each of them will represent a zero mode of L T c (here we have to consider L T c as an operator acting on a space of linear functionals on the space of observables).Integration cycles connect different zeroes of ρ(z) = exp(−S(z), which may be finite or infinite, or they wind around compact directions of the configuration space.
The real (physical) integration cycle is not always reproduced by CL, but the introduction of a kernel may remedy this.
• In some cases the existence of inequivalent integration cycles is also accompanied by nonergodicity of the CL process, i.e. the existence of different equilibrium distributions depending on the starting point.This will mean that the real Fokker-Planck operator L T has more that one zero mode.But there are also examples where the CL process is ergodic, yet the eigenvalue 0 of L T c is degenerate (see for instance the example in [40]).
Examples of the "mixing" of several integration cycles compromising the correctness of the CL simulations are plentiful, see for instance [39,40], as well as in Appendix B of [6], where for certain kernels the spectrum of L c is no longer on the left hand side of the complex plane (which is signalled correctly by the criterion developed in this paper), the boundary terms seem to vanish, and in fact the results can be expressed as a complex linear combination of the integration cycles [40].
The special case of non-ergodicity occurs in simple models with zeroes in the complex density ρ in [39].If ρ has a finite zero on the original real integration cycle, there are typically at least two inequivalent integration cycles, starting at that zero and going to infinity in different directions.
More zeroes lead to supplementary cycles connecting 2 zeroes.Such cycles also occur or in the case of compact models, connecting two finite zeroes.In fact, here it is easy to see that the eigenvalue 0 of L T c is degenerate: we may multiply ψ 0 = exp(−S) by the characteristic function of an interval between two zeroes (one of which may be at infinity), thereby producing a new eigenfunction of L T c with eigenvalue 0. But ergodicity may also fail in simple quartic models without finite zeroes, for instance for where numerics strongly suggests that there are two different equilibrium distributions [29].Another example of apparent nonergodicity is found in Appendix B of [6], in that case involving CL with a constant kernel.
In a non-ergodic situation, in particular in the presence of zeroes of ρ , it may depend on the starting point of CL which integration cycle or which linear combination of cycles is represented.
For lattice models it is of course quite difficult to determine all the possible integration cycles as well as the linear combination of them representing the original problem.
To summarize, we have located the main problems of the Complex Langevin method: First, insufficient decay of the probability distribution generated by the process, which leads to boundary terms and spoils the averages.Second, degeneracy of the zero mode of L T c , which is related to inequivalent integration cycles of the theory (this includes ergodicity problems).The first problem has been thoroughly studied both in simple models and in lattice simulations of realistic models.
It can be tested for using an on-line measurement, sometimes even correction of the CL results can be performed [24,25].In this paper we have proposed some diagnostic observables which signal the first as well as the second problem.
These problems (especially the second one) need further investigation, probably with the introduction of (field dependent) kernels.If a kernel has the effect of forcing the equilibrium distribution But as remarked in the appendix of [41], this solution is not unique, even in the limit β → 0.
A second solution for β = 0 is This solution is for t > 0 holomorphic in z ∈ C, but it has an essential singularity at t = 0. On closer inspection it is seen that as soon as |Imz| > |Rez|, the solution blows up as t → 0, so it does not solve the initial value problem everywhere.It solves it only in the wedge |Im z| < |Re z|.
The solution (B5) is correctly represented by the real Langevin process on the positive or negative real half-axis.There is no boundary term and no unstable mode.
But if we want to simulate the one-pole model on a line parallel to the real axis, only the first solution can be used.The CL process then produces a superposition of the positive and negative half-lines (the coefficients actually depend on the starting point of the process), as shown in [40]; the CL process is not ergodic.There is an unstable mode present, CL produces an incorrect result and there is a boundary term at the origin term signaling incorrectness [41].
(left panel) we present equilibrium values for λ = 1, m 2 = 0.1, varying h I from 0 to 1.We show expectation values of σ 2 , as well as σ d,α (α = 0.1, 0.5, 1.0); in the right panel we show Re z 2 and Re z 4 as well as |z| 2 and |z| 4 .We see that σ 2 blows up at h I = 0.25, while σ d,α blow up at already at h I = 0.2; z 4 starts deviating from the exact value around h I = 0.3, whereas z 2 starts showing already considerable deviation from the exact value starting at h I = 0.4.At h I = 0.3 the observable z 4 starts to show increasing errors, while the control variables show huge values log( σ 2 ) ≃ O(10 − 100).

FIG. 3 .
FIG. 3. The boundary terms in the quartic model (26) for m 2 = 0.1, λ = 1 and Re h = 0: the imaginary parts of the first and second boundary terms of the observable z and the real part of the first boundary term of the observable z 2 is shown as a function of the cutoff C = |z| 2 , for various h I = Im h values, as indicated.

FIG. 4 .
FIG. 4. The spectrum of the L c operator at λ = 1, m 2 = 0.1, h = 0. On the right panel, a part of the spectrum closer to zero is shown.

FIG. 5 . 3 FIG. 6 .
FIG.5.The number of eigenvalues of L c with positive real part as a function of the imaginary part of magnetic field h, at Re h = 0, for different truncations in the basis e(2) .The Lee-Yang zeroes of the theory are indicated by vertical lines.Right: zoom in around the second Lee-Yang zero.The parameters used are λ = 1, m 2 = 0.1.

1 FIG. 7 .
FIG. 7. The boundary terms of the action density in the XY model as a function of β.
[28]dicates insufficient deacy of the probability distribution, i. e. the presence of skirts.It should not be confused with 'runaways', i. e. breakdown of the simulation after a finite time; this problem was eliminated in all cases encountered by the use of adaptive step size[28].The Langevin operator L is the formal transpose of L T .It is the true transpose if and only if there are no boundary terms.Since Assumption A guarantees the absence of boundary terms for observables in the appropriate space L ∞ 1/σ , we have indeed P (x, y; t)O(x, y)dx dy = P (x, y; 0)O(x, y; t)dx dy , FIG. 2. Equilibrium CL results.Left: σ 2 t and σ α t , α = 0.1, 0.5, 1.0 vs. h I = Im h, at Re h = 0. Right: Re z 2 and Re z 4 as well as |z| 2 and |z| 4 vs. h I = Im h, at Re h = 0; red lines: exact results.