Abstract
We propose a method for computing numerically integrals defined via \(i \epsilon \) deformations acting on single-pole singularities. We achieve this without an explicit analytic contour deformation. Our solution is then used to produce precise Monte Carlo estimates of multi-scale multi-loop integrals directly in Minkowski space. We corroborate the validity of our strategy by presenting several examples ranging from one to three loops. When used in connection with four-dimensional regularization techniques, our treatment can be extended to ultraviolet and infrared divergent integrals.
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1 Introduction
The ever-increasing precision of data from particle physics experiments requires a comparable or better level of precision in theoretical predictions, both to establish the parameters of the Standard Model and to search for physics beyond it. To achieve such precision requires the computation of multi-loop amplitudes. A fundamental ingredient of such calculations is the evaluation of master loop integrals (MIs), in terms of which the problem is reduced. This can be performed by analytic, semi-numerical or fully numerical techniques (see [1] for a recent review). Analytic methods are very successful when the class of functions that contribute to the result is known, which usually happens when the number of internal and external masses is limited. However, such a-priori knowledge is not always available, especially when the number of scales increases, so that in these cases one would like to be able to compute MIs numerically, for instance by Monte Carlo (MC) techniques.
In the numerical computation of MIs, an important problem is the appearance of integrable threshold singularities, where single poles are moved away from the real integration domain by the \(i \epsilon \) prescription. These singularities require special treatment, such as a contour deformation into the complex plane [2,3,4,5], or vanishing-width extrapolations methods [6,7,8,9]. Contour deformations are usually controlled by some parameter whose value should be not too small, to guarantee numerical accuracy, and not too large, to avoid crossing branch cuts. In extrapolation methods a series of integrals should be determined that converges to the right value while keeping the computation time low.
This paper explains how integrals defined through the \(i \epsilon \) prescription acting on first-order poles can be evaluated numerically without deforming the integration contour into the complex plane, and how this can be employed to compute MIs appearing in multi-loop calculations. In addition, we demonstrate that this strategy allows one to compute recursively higher-loop functions in terms of lower-loop ones. Other semi-numerical methods relying on one-loop-like objects to build higher loops can be found in [10,11,12]. In [10] a Wick rotation of the loop momentum is needed to avoid singularities. The Feynman parameter space is used in [11], and a contour deformation in [12]. Our method works directly in Minkowski space and avoids contour deformations.
The structure of the paper is as follow. Section 2 details our approach. In Sect. 3 we use it to integrate numerically threshold singularities after an analytic integration over the energy-components of the loop momenta. Section 4 explains how to glue together lower-loop structures to compute numerically certain classes of higher-loop MIs. Finally, in Sects. 5 and 6 we extend our treatment to ultraviolet and infrared divergent configurations regularized via the four-dimensional method of [13].
2 Avoiding contour deformation
In this section we present two methods which avoid contour deformation. The first method uses complex analysis, while the second approach directly works with the original integrand. The two procedures are equivalent, in that they give rise to the same mappings. The numerical results presented in the paper are obtained with method 1 and cross-checked with method 2.
2.1 Method 1
For the sake of clarity, distinct letters (with or without additional subscripts) are used to denote variables ranging in different intervals. In particular, we employ x when \(-1 \le x \le 1\), y if \(0 \le y \le 1\), \(\sigma \) provided \(-\infty< \sigma < \infty \). Finally, \(0 \le \rho \le 1\) stands for a random Monte Carlo (MC) variable.
The core of the procedure is a change of variable such that the \(1/(x+i\epsilon )\) behaviour of the integral
is flattened with \(x \in {\mathbb {R}}\). This is obtained by imposing
where z is a new complex integration variable. In fact, inserting (2) in (1) gives the desired result,
Equation (3) evaluates to \(-i \pi \) along any curve in the z complex plane connecting \(z= 0\) to \(z= 1\) when \(\epsilon \rightarrow 0\). We use this freedom to impose \(x \in {\mathbb {R}}\) by parametrizing
which is real when \(\displaystyle \pi \beta = \ln \frac{\epsilon }{\sin [\pi (1-\alpha )]}\), namely
Therefore
which gives
where \(g_\epsilon \) has been introduced to impose the normalization to 1 also for small but not vanishing values of \(\epsilon \). In summary, after changing variable as in (2), the requirement \(x \in {\mathbb {R}}\) determines the relation between \({\Re }e(z)\) and \({\Im }m(z)\).
Armed with these results, we generalize (1) to an integration over a function
with \(\phi (x)\) sufficiently smooth at \(x= 0\),Footnote 1
Splitting the integration region of (8) into the two sectors with \(x_\alpha < 0\) or \(x_\alpha > 0\) gives
where \(y_\alpha := {\epsilon }/{\tan (\alpha \pi )}\). Equation (11) can be translated to a MC language by looking for the local density g(y) that corresponds to a change of variable \(d \rho = g(y) dy\) reabsorbing the singular behaviour of the integrand of (10),
with \(\int _0^1 dy\, g(y)=1\). By comparing (12) to (11) one determines
The mapping of (13) optimizes the integration over the real part of z (see (7)). This gives stable numerical results when \(\phi (x)\) is such that the \(y_\alpha /\epsilon \) terms in (11) are suppressed. When this is not the case, they generate a large contribution to the variance and, in order to flatten them, the parametrization complementary to (7) is necessary,
which gives
where
Again, (15) is correctly normalized also for a small but not vanishing \(\epsilon \). Comparing (12) to (15) gives now
Multichanneling
Flattening the whole \(1/(x+i\epsilon )\) behaviour of (9) requires a merging of (13) and (16), whose densities we dub \(g_{1} (y)\) and \(g_{2}(y)\). This can be achieved via a multichannel approach with combined density \(g_{\mathrm{c}}(y) := \alpha _1 g_{1}(y)+\alpha _2 g_{2}(y)\) and \( \alpha _1+\alpha _2= 1\),
In (17), \(\rho \) is generated according to the distribution \(g_{1,2}(y)\) with probability \(\alpha _{1,2}\), and the a-priori weights \(\alpha _{1,2}\) can be optimized as described in [14].
To reduce the variance when \(\phi (x)\) peaks inside \(-1 \le x \le 1\) in a known way, it is also possible to include an arbitrary numbers of further channels \(g_i(y)\) \((i > 2)\). However, care must be taken due to the fact that the MC weight of (12) includes both the \(f(-y)\) and f(y) contributions. To determine the corresponding density we observe that
which means that if x is randomly chosen in \(-1 \le x \le 1\), the density is \(g_i(-|x|)+ g_i(|x|)\). Hence, the MC weight is \(\big (f(-y)+f(y)\big )/\big (g_i(-y)+g_i(y)\big )\), with \(g_i\) normalized such that
In summary, with \(N_{\mathrm{ch}}\) channels (including \(g_1\) and \(g_2\)), the more general multichannel MC mapping reads
where
with arbitrary (but self-adjustable) weights fulfilling \(\sum _{i=1}^{N_{\mathrm{ch}}} \alpha _i= 1\). In the actual MC used to produce the results presented in this paper we superimpose on \(g_{\mathrm{c}}(y)\) a flat distribution \(g_3(x)= 1/2\) and a channel
which takes care of peaks around \(|x|= 1\).
Principal value integrals
It is often useful to deal with improper integrals, whose behaviour at large values of the integration variables is defined via the Cauchy principal value. The fact that the two symmetric points with respect to \(x=0\) are always considered together, makes the use of (20) very convenient. As a matter of notation, we define
which can be mapped onto the interval \([-1,1]\) by changing variable,
Thus, for instance,
where we understand the symmetric treatment of (20), so that (25) is well defined even when \(\phi (\sigma )\) approaches a constant as \(\sigma \rightarrow \pm \infty \).
Multiple integrals
Equation (20) can be easily extended to n-fold integrals of the type
with \( f( \{ x\})= \phi (\{ x\})/{\prod _{j=1}^{n}(x_j+i \epsilon )}\). Our notation is such that \(\{ x\}= x_1, x_2, \ldots , x_n\) and \(\phi (\{ x\})\) is a smooth function at \(\{ x\}= \{ 0\}\). The result is
where \(~\,d \rho _j= g_{\mathrm{tot}}(y_j) dy_j\) and the numerator stands for a sum over the \(2^n\) terms with positive or negative arguments. For instance, when \(\{ y\}= y_1,y_2\),
Equation (27) can be generalized to more poles per variable, moved away from arbitrary domains \(\in {\mathbb {R}}\), either by partial fractioning the integrand or by splitting the integration region into sub-intervals. However, configurations like that never appear in what follows, so we do not pursue a detailed analysis in this direction.
2.2 Method 2
As an alternative to the above method, one can apply separate changes of variables to flatten the real and imaginary parts of the pole factor(s) in the integrand. Consider the integral
where \(\phi ^\pm (x)=\phi (x)\pm \phi (-x)\). We can write this as
where
Each of these integrals has optimal variance reduction (in the absence of information about \(\phi \)) and is therefore suited to numerical integration as long as \(\phi (x)\) is smooth at \(x = 0\). Note that the \(\epsilon /(x^2+\epsilon ^2)\) and \(x/(x^2+\epsilon ^2)\) behaviours of (29) correspond to the local densities in (13) and (16), respectively.
The method is easily generalised to two variables. Consider
where
with
For MC evaluation, we proceed as follows: for each shot, generate \(x_{1r}, x_{1t}, x_{2r}, x_{2t}\) where
where
uniformly, with \(r_m\) and \(\theta _m\) as in (31). In (34), set \(x_1= x_{1t}\) when the first superscript is 0 and \(x_1= x_{1r}\) when it is 1, and similarly for \(x_2\) according to the second superscript. The weights for the real and imaginary parts are then
The generalisation of (32) to n variables is clear: \(\phi ^{\{k_j\}}\) has superscript \(k_j=1\) in the jth location when there is an \(x_j\) in the integrand, otherwise \(k_j=0\). The symmetrized function \(\Phi \) becomes
where
For MC evaluation, for each shot, generate two points in the n-dimensional hypercube
where again \(0<r_j<r_m\) and \(0<\theta _j<\theta _m\) uniformly. In (39), set \(x_j= x_{jt}\) when \(k_j=0\) and \(x_j= x_{jr}\) when \(k_j=1\). The weight is then
Note that each shot involves 2n random numbers for \(\{x_{jt}\}\) and \(\{x_{jr}\}\), and then \(4^n\) function evaluations at \(x_j=\pm x_{jt}\) and \(\pm x_{jr}\), so the computation time increases rapidly with the number of variables.
2.3 Choosing \(\epsilon \)
Here we perform a study of the value of \(\epsilon \) to be used in practice. More specifically, we compare the numerical and analytic determinations of the three-fold test integral
whose behaviour at \(x_j \sim 0\) mimics a typical multi-dimensional environment. The result of this comparison is given in Fig. 1, where the solid (dashed) line represents the real (imaginary) part of (43). Bullets and squares with errors are the MC predictions for \({\Re }e\left[ T(\epsilon )\right] \) and \({\Im }m\left[ T(\epsilon )\right] \), respectively. To quantify the effect of a nonzero \(\epsilon \) on the MC estimate \(Q_\mathrm{MC}(\epsilon )\pm \varDelta Q(\epsilon )\) of a known quantity Q, it is convenient to introduce the estimators
Requiring the \(\epsilon \ne 0\) bias on \(Q_{\mathrm{MC}}(\epsilon )\) to be of the order of the maximum between \(\epsilon \) and the relative MC error gives the condition
Table 1 reports the R value of the entries of Fig. 1 and their MC accuracy defined as
From Fig. 1 and Table 1 we infer that a range \(10^{-8} \le \epsilon \le 10^{-6}\) is adequate to achieve MC estimates accurate at the level of three parts in \(10^5\). Since the results presented in this paper are never more accurate than this, we set, for definiteness, \(\epsilon = 10^{-7}\). However, the last row of Table 1 shows that numerically stable predictions are produced also with a smaller \(\epsilon \) and a larger MC statistics. From this, we deduce that the \(\epsilon \ne 0\) bias can be reduced to be negligible in most practical applications, and that the accuracy of our method is driven by the MC error.
3 A semi-numerical integration algorithm for MIs
In this section we illustrate how the approach of Sect. 2 can be successfully applied to produce stable and precise semi-numerical MC estimates of loop MIs. This is achieved in two steps. Firstly, we integrate analytically over the energy components of the loop momenta, which is always doable by means of the Cauchy integral theorem. In addition, depending on the case at hand, some of the loop angular integrals can also be performed analytically. In this way, integral representations of MIs can be easily obtained. Secondly, we give up any attempt towards a fully analytic integration, which may be difficult, and integrate numerically over the left-over loop components. The integrand to be evaluated is usually plagued by threshold singularities. Single poles migrate towards the real integration domain for some kinematic configurations, so that a blind numerical integration over denominators deformed by the Feynman \(i \epsilon \) prescription gives large errors. However, this is precisely the situation for which our approach is designed. We mitigate these problems by retaining a finite small value of \(\epsilon \), flattening the real and imaginary parts of pole contributions, and applying multichannel mappings.Footnote 2 In what follows we illustrate the performance of this strategy by means of two examples.
3.1 A one-loop example
Consider the three-point function of Fig. 2 in the case \(m_0=m_1=m_2= m\), \(p^2_1=p^2_2=0\) and timelike P. Rescaling all momenta by m
gives
with
One splits
with \(\displaystyle R^2_2 := \rho ^2+ \frac{\tau }{4}+\sqrt{\tau }\rho c_\theta +1\). Thus
where \(\displaystyle R^\pm _2 := \sqrt{({\sqrt{\tau }}/{2}\pm \rho )^2+1}\). The cut of the logarithms with \(+ i \epsilon \) (\(- i \epsilon \)) is in the lower (upper) t complex half-plane, so that the integration over t in (49) is trivial once one rewrites
with \(\displaystyle R^2_0 := \rho ^2+1\). The results is
where
and
When \(\sqrt{\tau } > 2\), the first integrand of (52) develops a pole at \(r= i \epsilon \) that migrates towards the integration region in the limit \(\epsilon \rightarrow 0\). Treating this with the strategy of Sect. 2 gives the results presented in Table 2.
3.2 A two-loop example
We study the two-loop self-energy scalar diagram of Fig. 3.
For a timelike \(P/m= (\sqrt{\tau },{\mathbf {0}})\) and \(m_1= m\) it reads
where
and
with \(\mu _0:= m^2_0/m^2\). Integrating over the angular variables gives
The integration over \(t_1\) and \(t_2\) is trivial and produces
where
and \(A_i := r_i+\lambda _i \sqrt{\tau }/2\). Threshold singularities at \(r_{1,2}= i \epsilon \) are present when \(\lambda _{1,2}= +1\) if \(\sqrt{\tau } > 2\). When \(m_0= 0\), a two-dimensional implementation of the method of Sect. 2 gives the results reported in Table 3. Larger MC errors correspond to smaller values of \(\rho \). However, we observe that when \(\mu _0 \ne 0\) this effect is mitigated. For instance, a \(10^{9}\) MC-point estimate with \(\rho = .1\) gives
4 Gluing together lower-loop structures
Here we show how higher-loop integrals can be expressed in terms of lower-loop building blocks. Throughout this section dimensionful quantities are rescaled by an arbitrary mass m, so that loop momenta are written as in (54) and, in particular,
Furthermore, we define
and study cases up to a \(P := p_1+p_2 \rightarrow p_3+p_4\) kinematics of the form
Rescaled propagators belonging to the loop momentum q are denoted by
In addition, we define
andFootnote 3
The essence of the procedure is to use \(\sigma _0\) and \(\sigma _1\) as integration variables of the method of Sect. 2. This is achieved by multiplying the integrand by
where
This gives rise to the appearance of the following three functionals,
where \(j=2,3\). Assuming \(J_1\) independent of any angular variable and \(J_2\) (\(J_3\)) independent of \(\theta \) (\(\phi \)) allows one to compute the functionals once for all. \(\Phi _1\) reads
As for \(\Phi _2\), one has
with
Note that (66) and (67) have been analytically continued to configurations with any sign of \(\tau \), \(\tau _1\), \(\tau _2\), \(\lambda _{12}\). Finally
in which
4.1 One-loop examples
To elucidate the procedure, we first consider gluing tree-level structures to compute the three-point function of Fig. 2 – with arbitrary kinematics and masses – and the box diagram of Fig. 4 with \(p^2_i= 0\) and \(m_0=m_1=m_2=m_3=m\), which we dub C and \(D_0\), respectively. Equations (63), (67) and (69) produce
In (71) the integration over the azimuth angle \(\phi \) is trivial,
while the integral over \(c_\theta \) in (72) has to be dealt with numerically using the method of Sect. 2, which gives
where
Improving the numerical accuracy
When inserted in (71) and (72), Eqs. (73) and (74) could potentially produce inaccurate results when strong cancellations are expected among different integration regions. This happens if
Note that case (a) is relevant to C but not to \(D_0\), since
while (b) applies to both C and \(D_0\) due to the common \(1/\tau \) prefactor. In the following paragraphs we illustrate how numerical inaccuracies caused by the configurations (a) and (b) in (76) can be circumvented.
As for case (a), a preliminary analysis is in order to understand the mechanism that makes C finite despite (7778).Footnote 4 We define
in terms of which
Now the \(\sigma _0\) integral is convergent by power counting and \(\lambda \) in (61) behaves as
where \(\alpha \) and \(\beta \) are constants. Replacing \(\lambda \) with \({\tilde{\lambda }}\) in (73) produces an integrand in which all branch points and poles are located in the lower \(\sigma _0\) complex half-plane. As a result, the integral over \(\sigma _0\) approaches zero when \(\sigma _{10} \rightarrow \pm \infty \), so that the \(\Lambda \rightarrow \infty \) limit exists. This same reasoning allows one to construct a class of vanishing integrals defined as
with
where \({\tilde{q}}\) is the asymptotic \(|\sigma _{10}| \rightarrow \infty \) limit of q,
Again, all cuts and poles lie in the lower \(\sigma _0\) complex half-plane, so that
Now \(\alpha \) and \(\beta \) can be set to obtain a local cancellation of the problematic large \(\sigma _{10}\) configurations.Footnote 5 An explicit calculation with \(\tau _1= \tau _2= 0\) gives
When \(\tau _i \ne 0\) the accuracy of (71) is improved by the non-vanishing external masses, so that (85) is relevant to this case as well. In summary, the formula
produces numerically stable results with \(\alpha \) and \(\beta \) given in (85) when the same sequence of \(\sigma _0\) and \(\sigma _1\) values are used in both C and \({\tilde{C}}\).
It turns out that the configurations of type (b) of C are also cured by the subtraction in (86). Thus, we are only left with the discussion of the case (b) for \(D_0\). In the \(\tau \rightarrow 0\) region it is convenient to give up the exact formula and use, instead, a few terms of a Taylor expansion in \(\tau \) and \(\chi \) obtained with the method given in Appendix A,
By doing that, it is easy to find a value of \(\tau \) below which the exact result is well approximated by (87), and above which (72) is accurate.
Results
Here we present the numerical outcome of a MC based on (71), (72), (86) and (87).
The results for \(C(P^2,0,0,m,m,m)\) are shown in Figs. 5 and 6. In the latter, the relative difference between the MC and the analytic (AN) non-zero results of the former is plotted in terms of
where R and I refer to the real and imaginary parts, respectively. With the given MC statistics, the analytic result is reproduced by (86) within a few parts in \(10^4\),Footnote 6 which is an accuracy comparable to the one of Table 2, but obtained with 50 times more points. The main reason for this difference is that the one-dimensional representation (52) is now replaced by the twofold integration (71). On the other hand, the gluing algorithm is highly modular and can be extended to more complex situations, as we will see in the next subsection.
Table 4 displays our results for \(D_0\) as a function of \(\tau \) and the scattering angle \( -{\chi }/{\tau }= \left( 1+\cos \theta _{13} \right) /2 \). We use the MC evaluation of (72) above \(\tau = 1\) and the Taylor expansion of (87) when \(\tau < 1\). In the former case, an accuracy of a few parts in \(10^3\) is reached with \(10^9\) MC points.
4.2 Two and three-loop examples
Two-loop self-energy
The two-loop diagram of Fig. 3 can be easily obtained by gluing together a one-loop triangle and a tree-level structure. For instance, with \(m_0=m_1=0\) one has
Equation (89) suffers the inaccuracies of type (a) of (76). To cure this, we use the same strategy described in Sect. 4.1. First, we consider an explicit representation of the triangle as a function of \(\sigma _0\) and \(\sigma _{10}\) in (79),
with \(\lambda _0\) in (75),
and
From this, it is easy to determine a \(|\sigma _{10}| \rightarrow \infty \) asymptotic approximant of (90) that gives zero upon integration over \(\sigma _{0,1}\). The results readsFootnote 7
where \({\tilde{\lambda }}_0:= (\sigma _{10}-\tau )^2-4 \tau \), \(\Lambda _c\) is arbitrary and \(\tilde{F}(\sigma _0,\sigma _{10})\) is constructed by replacing in (91) the \(u_j, v_j\) with their asymptotic counterparts \({\tilde{u}}_j, {\tilde{v}}_j\) defined as
with
Finally, in the same spirit as (86), we rewrite
where we understand a local subtraction of the large \(|\sigma _{10}|\) configurations.
In Table 5 we present our numerical estimates based on (95) with \(\Lambda _c= 1/2\). An accuracy of the order of \(10^{-3}\) is achieved with \(10^9\) MC points. We also studied the stability of (95) at small values of \(\tau \). For instance, when \(\rho = 100\) (\(\tau =0.04\)) we obtain
Note that determining \({\tilde{S}}_2(\Lambda _c)\) requires an analytic knowledge of the integrand. When this is not possible, an alternative approach is to cut away the problematic configurations in a controlled manner. For instance, discarding in (89) integration points with
is expected to produce an error of \({{\mathcal {O}}}(\Lambda ^2)\). Indeed we checked that, with \(\Lambda = 0.01\), the fraction of the integral discarded by (97) is always below the errors reported in Table 5.
Three-loop self-energy
Our next example is the scalar three-loop self-energy of Fig. 7, which is attained by gluing together the two triangles
by means of \(\Phi _1\),
Now the integrand vanishes fast enough at large \(\sigma _{0,1}\), so that no subtraction is needed. Our results for the case \(\{\mu _k\}= \{1,1,0,0,0,0,0,0\}\) are shown in Fig. 8, where we compare them with digitized curves obtained from Fig. 4 of [10]. The agreement is good, but our points tend to overshoot the lines at large \(\tau \). However, the quality of the plot in Ref. [10] is poor there and the digitization may be misleading. Thus, we cross-checked internally our high-energy results by comparing the outcome of two independent MCs based on method 1 and 2 of Sect. 2, respectively. We did not find any systematic difference in the range \(7< \tau < 50\). Finally, we observe that dealing with arbitrary mass configurations poses no difficulties whatsoever. For example, when \(\{\mu _k\}= \{1,1,2,3,4,5,6,7\}\), using OneLOop [16] to evaluate the \(C_{L,R}\) triangles gives, at \(\tau = 10\) and with \(10^{9}\) points,
Planar two-loop vertex
Consider now the two-loop scalar vertex of Fig. 9. Gluing the one-loop triangle on the left
by means of \(\Phi _2\) gives
This representation holds true with any sign of \(\tau , \tau _1, \tau _2\) and for any choice of internal masses.Footnote 8 In addition, it does not require subtracting large \(\sigma _{0,1}\) configurations. In Table 6 we collect a few results obtained by computing the triangle with OneLOop. The last row refers to the Standard-Model-like \(Z\rightarrow \nu {\bar{\nu }}\) case with \(m_0=m_1= m_e\), \(m_2=m_3=m_4=M_W\), \(m_5= 0\), \(P^2= M^2_Z\) and \(p^2_{1,2}= 0\). This shows that the MC error is under control also for configurations with large mass gaps. Note that \(V_2(\{\mu _k\})\) can also be obtained by gluing together the box on the right \(D_R := D(\tau _1,\tau _2,\sigma _3+\mu _3,\sigma _4+\mu _4,\tau ,\sigma ^{34}_b,\mu _1,\mu _2,\mu _0,\mu _5)\) and the tree-level decay on the left. When \(k^2\) is greater than zero one has
The loop momentum of the \(m_3\) line of Fig. 9 flows through just one propagator of \(D_R\). Because of that, a “technical” cut \(\sqrt{\tau ^2/(\tau ^2+(\sigma _3-\sigma _4)^2)} < \Lambda \) is required to damp the inaccurate large \(\sigma _{3,4}\) behaviour of (101). By power counting, the discarded contribution is of \({{\mathcal {O}}}(\Lambda ^4)\). With \(\Lambda = 0.05\) and \(10^9\) MC points (101) reproduces the numbers of Table 6 at the percent level.
Non-planar two-loop vertex
The same reasoning leading to (101) allows one to compute the non-planar two-loop vertex of Fig. 10,
where
Now the loop momentum of the \(m_3\) line flows through two propagators of \(D^{np}\). This provides an additional damping factor in (102) with respect to (101), so that large \(\sigma _{3,4}\) configurations do not lead to numerical inaccuracies and no technical cut is required. A few numerical results are collected in Table 7.
Planar and non-planar double box
The planar and non-planar two-loop double boxes are depicted in Fig. 11a, b, respectively. They read
where
are the one-loop boxes on the left and right sides of Fig. 11a, b, respectively. In Figs. 12 and 13 we present a comparison between our estimates and the results presented in [7] for the case
in the region of \(\tau \) where \(|\cos \theta _{13}| \le 1\).Footnote 9 The agreement is very good. As benchmark values, we list in Table 8 the MC entries of Figs. 12 and 13, together with their statistical errors.
More complex structures
In all cases presented so far we could perform an easy analytic integration over at least one angular variable of the loop momentum \(\omega \). When the number of loops and legs increases, integrating analytically over \(c_\theta \) and/or \(\phi \) is not trivial any more, so that the number of numerical integrations required by the gluing procedure reaches its maximum value, i.e. four. In this paragraph we give a few examples of the gluing approach in these more complex situations. In particular, we use the multichannel approach of Sect. 2.1 to study the scalar triple box \(B_3\) and two-loop pentabox \(E_2\) of Figs. 14 and 15, respectively.
As for \(B_3\), inserting (63) into the integrand gives
where
are the one-loop boxes on the left and right sides of Fig. 14. The change of variable
produces
When \(\tau = 50\), \(\chi = -4\), \(\tau _i= \mu _0= \mu _1= 1\), \(\mu _{\tiny 2\div 9}= 0\), (109) gives, with \(10^9\) MC shots and using OneLOop to evaluate \(D_{L,R}\),
Note that it is not difficult to deal with non-planar configurations. For instance, the diagram obtained from Fig. 14 by interchanging the vertices \(v_1 \leftrightarrow v_2\) and \(v_3 \leftrightarrow v_4\) is as in (106), but with
Finally, the pentabox depicted in Fig. 15 reads
where \(E(\omega )\) is the one-loop pentagon on the right side, which depends upon ten independent invariants – built from the momenta \(q\!=\! m \omega , \,p_{3\div 5}\) – and the five masses \(m_{3\div 7}\). The presence of the \(1/(\sigma _2+i \epsilon )\) propagator forces one to trade the integral over \(c_\theta \) for an integration over \(\sigma _2\) to be dealt with the method of Sect. 2. This, together with the change of variable of (108), gives
where
To provide a benchmark value, we have chosen to evaluate (113) with \(\mu _{0\div 2}= 1\), \(\mu _{3\div 7}= 0\) at a particular \(p_1+p_2 \rightarrow p_3+p_4+p_5\) phase-space point satisfying \(\tau = 50\), \(\tau _1= \tau _2=0\) and \(\tau _{3 \div 5}= 1\), namely
with \(r\!=\! \sqrt{193/450}\), \(c_{34}\!=\! -159/193\) and \(s_{34}\!=\! \sqrt{1-c^2_{34}}\). We employed CutTools [18] to reduce \(E(\omega _n)\) to one-loop boxes. Computing the latter with \(\mathtt {OneLOop}\) gives, with \(10^9\) MC shots,
Once again, non-planar configurations are obtained without extra effort. For instance, if the vertex v of Fig. 15 is moved to the \(m_3\) line, (112) still holds by simply modifying the one-loop pentagon \(E(\omega )\) accordingly.
5 UV divergences
We deal with ultraviolet divergent integrals following the FDR approach of [13], in which the divergent configurations are extracted from the original integrand by partial fractioning.Footnote 10 The resulting expressions are integrable in four dimensions and nicely match the algorithm of Sect. 2. We illustrate our procedure by means of the scalar two-point function of Fig. 16,
where
with \(M^2_1(q) := m^2_1+2 (q \cdot P)-P^2\) and \({\bar{q}}^2 := q^2-\mu ^2+i\epsilon \).
By partial fractioningFootnote 11
This gives, by definition of FDR integration,
where \(\mu _{\scriptscriptstyle \mathrm R}\) is the finite renormalization scale. It is convenient to take the limit \({\mu \rightarrow 0}\) directly at the integrand level and substitute \(\mu \) with \(\mu _{\scriptscriptstyle \mathrm R}\) only in the logarithms. This is achieved by rewriting
where \(D_i := {\bar{D}}_i+\mu ^2\). By doing so, \(\mu ^2\) is dropped everywhere, except in the logarithmically UV divergent part of the vacuum, where it is replaced by \(\mu _{\scriptscriptstyle \mathrm R}^2\). In this way, no \(\mu \rightarrow 0\) limit is required, so that (120) is a good starting point for a numerical treatment. In what follows we describe how the methods of Sects. 3 and 4 can be adapted to deal with (120). More complex multi-loop UV divergent configurations can be treated likewise.
5.1 Integrating over the loop energy component
We take \(m_1=m_0=m\) for simplicity. A rescaling as in (4748) produces
where
with
The Cauchy integral theorem allows one to compute
Inserting this in (121) and using \(r= R_0-\sqrt{\tau }/2\) as a new integration variable gives
where
If \(\sqrt{\tau } > 2\) the pole at \(r= i \epsilon \) migrates towards the integration contour when \(\epsilon \rightarrow 0\). Treating this with our numerical approach produces the results collected in Table 9.
Integrals with a polynomial degree of divergence can be treated in exactly the same way. As an example, Appendix B details the case of the one-point function
5.2 Gluing substructures
The gluing approach of Sect. 4 can be easily extended to (116). Inserting (63) in (120) gives
with
and \(\nu \) defined in (123). The presence of a double pole is an obstacle to a direct numerical treatment of (128). In fact, our algorithm is designed to deal with single poles only. However, we observe that, if \(\nu \) has a finite imaginary part, the singularity never approaches the real axis. In particular, \(B_{\scriptscriptstyle \mathrm FDR}(-i\nu )\) is better suited than \(B_{\scriptscriptstyle \mathrm FDR}(\nu )\) to be evaluated numerically if \(\nu \in {\Re }\). Besides, the connection between the two can be derived by differentiating (120),
which gives
\(B_{\scriptscriptstyle \mathrm FDR}(-i\nu ^\prime )\) still suffers from numerical inaccuracies of type (76) (a). To cure this, we locally subtract from it an approximant, \( \tilde{B}_{\scriptscriptstyle \mathrm FDR}(-i\nu ^\prime ) = 0\), constructed in such a way that, after changing variables as in (79), all cuts and poles lie in the lower \(\sigma _0 \) complex half-plane. This is obtained by replacing in (128) \(\lambda ^{\frac{1}{2}}\theta (\lambda ) \rightarrow \sqrt{\lambda ^2-4\tau i \epsilon }\, \theta (\lambda +4\tau \sigma _0)\). In summary, we rewrite
In Table 10 we present our estimates for \(B_{\scriptscriptstyle \mathrm FDR}(1/4)\) with \(\mu _0=\mu _1=1\) obtained by means of Eqs. (131) and (132). The figures match the results of Table 9, although with larger errors. However, we point out that the gluing method is more flexible when it comes to generic kinematics. For instance, with \(\tau = -10\), \(\mu _0=1\), \(\mu _1=4\), one obtains, with \(10^8\) MC points,
to be compared to the analytic value \(-i\, 27.220\).
6 IR divergences
We deal with IR divergent integrals by means of the FDR approach of [19], where a small mass \(\mu \), added to judiciously chosen propagators, is used as a regulator of both infrared and collinear divergences. In this section, we illustrate how this allows one to combine virtual and real contributions prior to integration. After that, our method can be used to evaluate numerically loop integrals where the IR configurations are locally subtracted. We study, in particular, the IR divergent scalar triangle
that appears in a \(P \rightarrow p_1+p_2\) decay with \(p_i^2= 0\) and \(s:= P^2\). However, our findings can be generalized to more complex environments.
Our strategy is based on combining together cut-diagrams that are individually divergent, but whose sum is finite. We use a scalar massless \(g \varphi ^3\) theory defined through the Feynman rules of Fig. 17,
where we have introduced propagators with positive and negative values of the energy and the momentum component along the x direction. The cuts contributing to \(\varphi ^{*} \rightarrow \varphi \varphi (\varphi )\) are listed in Fig. 18 where, to make contact with (134),
The diagrams are organized in pairs sharing collinear singularities. For instance, in \(\mathrm D_{a}\) the energy component of propagator 1 is the sum of those of propagators 2 and 3. Thus, propagators 1 and 3 never pinch in the \(q^0_{1}\) complex plane, and propagator 3 can only become collinear to 4. Likewise, in \(\mathrm D_{b}\) the sign of the momentum components along x only allows particles 3 and 4 to become collinear to 5. In both cases, we regulate the singular splitting by including a small mass \(\mu \) in propagators 3 and 4, leaving 5 massless.Footnote 12 In summary, \(\mathrm D_{a}+D_{b}\) is free of collinear divergences, and the same happens for \(\mathrm D_{c}+D_{d}\). In addition, \(\mathrm D_{a}+D_{b}+D_{c}+D_{d}\) is also free of infrared singularities. A similar reasoning applies to \(\mathrm D_{e}+D_{f}+D_{g}+D_{h}\), but with an opposite sign of the energy component of \(q_1\). The previous analysis shows that the three-particle cuts \(\mathrm D_{b}\) and \(\mathrm D_{d}\) can be used as local countertems for \(\mathrm D_{a}+D_{c}\).Footnote 13 This requires common reference frames. One can employ two different routings for \(\mathrm D_{a}+D_{b}\) and \(\mathrm D_{c}+D_{c}\). However, they must coincide in the limit \(q_1 \rightarrow 0\) to guarantee the cancellation of the soft behaviour of \(\mathrm D_{a}+D_{b}+D_{c}+D_{d}\). In particular, when computing \(\mathrm D_{a,b}\) we assign a momentum \(q_2\) to propagator 4, from left to right, and choose
On the other hand, we calculate \(\mathrm D_{c,d}\) with \(q_2\) assigned to propagator 5 and
The result of the computation is reported in Appendix C in terms of integrals over
It is convenient to further split \(\mathrm D_{a,c}= D^{s}_{a,c}+D^{u}_{a,c}\), where the superscripts \(^{\mathrm{s,u}}\) refer to the subtracted and unsubtracted regions, which correspond to the integration intervals \(\sqrt{\eta }< R_1 < 1/2\) and \(1/2< R_1 < \infty \), respectively. In fact, \(\mathrm D_{b,d}\) contribute in the subtracted region only, and \(\mathrm D^{u}_{a}+D^{u}_{c}\) is free of IR singularities,
An analytic calculation [19] shows that \(\sum _\mathrm{j=a,b,c,d} D_j= 0\). Hence, one must have
In Table 11 we display our numerical estimate of K based on Eqs. (C.22), (C.29) and (C.34). The correct result is precisely approached and the MC error does not grow when decreasing \(\eta \), which is an indication that the local cancellation works as expected. Finally, we point out that the outlined strategy can be turned into a fully exclusive local subtraction algorithm by introducing suitable phase-space mappings, as described in [20].
7 Conclusion and outlook
We have presented a flexible method for the numerical treatment of loop integrals in four-dimensional Minkowski space, without the need of explicit contour deformation. This is achieved by exploiting the \(i\epsilon \) prescription with a small finite value of \(\epsilon \) and making changes of variables to reduce the variance of both the real and imaginary parts of the integrand. We propose a semi-numerical approach, in which an analytic integration over loop time-components is followed by multichannel Monte Carlo integration. In some cases, further integrations can be performed before the final numerical step. The method lends itself readily to the evaluation of complex multi-loop structures by gluing together simpler substructures. It also deals easily with processes involving many different external and propagator mass scales, where analytical results are difficult to obtain.
In practice, we find that \(10^9\) Monte Carlo shots with \(\epsilon \sim 10^{-7}\) (in terms of some relevant mass scale) can yield relative precision of the order of \(10^{-4}\) for one-loop diagrams and \(10^{-3}\) for two- and three-loops obtained by gluing together analytical results for one-loop substructures. As for the performance of our algorithms, we report in Table 12 the time to produce \(10^6\) MC shots with method 1 for a few representative cases. It ranges from a few tenths of a second to more than a minute. Method 2 gives somewhat slower timings.
We have focused on scalar integrals without any structure in the numerator, but we expect that the treatment of loop tensors should follow the same guidelines described in this paper. In particular, the approach of Sect. 3, in which an analytic integration is performed over the loop time-component, should work as it stands. As for the gluing method of Sect. 4, adding structures in the numerator could potentially lead to worse behaviour that needs to be corrected by local subtractions of large loop configurations, as done in Eqs. (86), (95), (132), or by the technical cuts described in (97) and (101). We leave a detailed study of this subject for further investigation.
We have sketched out how our method can be extended to UV and IR divergent configurations. Again, a deeper investigation is left for the future.
In summary, we believe that a numerical treatment of virtual corrections in four dimensions, of the type we have proposed, could be very beneficial in the computation of complicated multi-leg multi-scale amplitudes. More specifically, we think that the direction to go would be to integrate directly the amplitude as a whole, rather than the separate MIs. This could mitigate some of the large gauge cancellations among individual contributions, if common loop momentum routings are chosen for classes of diagrams. In addition, Monte Carlo integration of the loops and over the phase-space of real emissions can be merged, potentially stabilising and speeding up the calculation.
Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors’ comment: This is a theoretical paper with no external associated data.]
Notes
From now on, we omit \(\lim _{\epsilon \rightarrow 0}\) and consider \(\epsilon \) as an infinitesimal parameter.
The use of threshold counterterms [15] could further improve the precision of our approach.
\(\sigma ^{ij}_{a,b,c,d}\) are the invariants \((q-p_{1,2,3,4})^2/m^2\) computed at values of t and \(\rho \) satisfying the conditions \( t^2-\rho ^2= \sigma _i+\mu _i\) and \((t-\sqrt{\tau })^2-\rho ^2= \sigma _j+\mu _j\).
The principal value integrations in (71) are not sufficient to regularize the large \(\sigma _{0,1}\) behaviour. In fact, \(\Phi ^{(\mu _0,\mu _1)}_2[1]\) approaches two different constants when \(\sigma _{0,1} \rightarrow \infty \) or \(\sigma _{0,1} \rightarrow -\infty \), so that no cancellation is possible.
Additionally, \(\Lambda _0\) can be used to control when such a local subtraction has to be performed.
Similar results are obtained when \(\tau < 0\). For instance, with \(\tau = -10^3\) Eq. (86) gives \(m^2 C= 1(1)\times 10^{-6} -i\,0.23566(5)\), to be compared to the analytic value \(-i\, 0.23561\).
The inclusion of complex masses is in principle possible, although a dedicated study is needed in this case to assess the numerical accuracy, especially for small width-to-mass ratios.
An analytic continuation to nonphysical configurations is possible, although we did not try it.
At one loop, FDR is equivalent to the \(\overline{\mathrm{MS}}\) scheme of Dimensional Regularization.
The UV divergent piece is dubbed vacuum and written between square brackets by convention.
Note that adding \(\mu ^2\) also to 1 and/or 2 does not change the asymptotic \(\mu \rightarrow 0\) limit of the result. This is used, for instance, in (135).
An analogous procedure holds for the last four cuts of Fig. 18.
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Acknowledgements
The work of RP is supported by the SRA grant PID2019-106087GB-C21 (10.13039/501100011033), by the Junta de Andalucía grants A-FQM-467-UGR18 and P18-FR-4314 (FEDER), and by the COST Action CA16201 PARTICLEFACE. The work of BW was partially supported by STFC HEP consolidated grants ST/P000681/1 and ST/T000694/1.
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Appendices
Appendix A: Taylor expansions
The Taylor expansions for integrals like (71) and (72) can be obtained using the general result
where
which can be established by induction.
For (71), we first make a shift of variable \(q\rightarrow q+p_2\). Then for \(\tau _1= \tau _2=0\), \(\mu _1= \mu _2=1\) we have
Applying (A.1), and noting that terms with \(l\ne k\) vanish as they would involve \(\tau _1\) or \(\tau _2\),
where \(\tau =2p_1\cdot p_2/m^2\). For \(\tau _1,\tau _2\ne 0\), we have instead of (A.3)
This expansion can be extended to general masses by the substitutions
For the expansion of (72) we have
Note that the number of factors of \(q^\mu \) must be even, \(j+k+n=2K\). Then
where \(N=j+k+l+4\). Substituting (A.2), we obtain (87).
Appendix B: The one-point FDR integral
We compute the one-loop integral
with \({\bar{D}}_0\) given in Eq. (117). Extracting the vacuum produces the expansion
hence
By taking the \({\mu \rightarrow 0}\) limit at the integrand level and replacing \(\mu \) with \(\mu _{\scriptscriptstyle \mathrm R}\) in the logarithmic divergent vacuum one obtains
Choosing now \(m=m_0\) gives
where
with \(R_0\) and \(R_\nu \) defined in (123). One computes
which gives
where
Note that there is no pole in this case. In Table 13 we report a comparison between a numerical implementation of (B.16) and the analytic result
Appendix C: The IR integrals
Here we obtain onefold integral representations for the cut-diagrams \(\mathrm{D_{a,b,c,d}}\) of Sect. 6 and prove (139).
The diagram \(\mathrm D_b\):
Choosing the momenta as in (136) gives
with
Note that a harmless \(\mu ^2\) has been added to propagator 1 and that the Heaviside function forces propagator 5 to have a positive component of the momentum along x. Using the three Dirac delta functions one arrives at
with \(R_{1,2}\) in (138), \(R^+ := R_1+R_2\) and
Integrating analytically over \(R_2\) produces logarithms with boundaries determined by the three Heaviside functions. The result reads
The diagram \({\mathrm{D}}_a\):
We split \({\mathrm{D}}_a\) into two components, \({\mathrm{D}}_a=D^+_a+D^-_a\), with positive and negative values of \(q_1\) along x. Choosing the momenta as in (136) produces
with the same \(\sigma _{1\div 5}\) of (C.19). Using the two delta functions gives
where
and
\({{\mathrm{D}}^-_a}\) is obtained from (C.24) by replacing
Hence
Computing \(I_1\) with the Cauchy integral theorem gives
Note the appearance of the same denominator structures of (C.20). An integration over \(R_2\) produces
where
The diagram \({\mathrm{D}}_c\):
It is the complex conjugate of (C.29).
The diagram \({\mathrm{D}}_d\):
Inserting a harmless \(\mu ^2\) in propagator 5 and choosing the momenta as in (137) gives
where
Using the delta functions produces
with \(c_\theta \) as in (C.21). Integrating over \(R_2\) gives
where
and
The combination \({\mathrm{D}}^{u}_{a}+D^{u}_{c}\):
This is obtained from (C.29) and its complex conjugate by replacing the integration region \(\sqrt{\eta }< R_1 < \infty \) by \(1/2< R_1 < \infty \). Upon doing so, \(\eta \) can be set to 0 and \(R^\pm _2= R_1 \pm 1/2\). Introducing \(x= 2 R_1\) gives
from which (139) follows.
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Pittau, R., Webber, B. Direct numerical evaluation of multi-loop integrals without contour deformation. Eur. Phys. J. C 82, 55 (2022). https://doi.org/10.1140/epjc/s10052-022-10008-6
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DOI: https://doi.org/10.1140/epjc/s10052-022-10008-6