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Multiloop Feynman integrals - Scholarpedia

# Multiloop Feynman integrals

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Multiloop Feynman integrals appear when quantum-field amplitudes are constructed within perturbation theory. They are integrals over so-called loop momenta. Feynman has invented their graph-theoretical interpretation. (The term Feynman integral' is sometimes used also for path integrals.) Feynman integrals are usually complicated objects even in a one-loop approximation, so that the number of loops equal to two is already considered big.

## Introduction

In perturbation theory, any quantum field model is characterized by a Lagrangian, which is represented as a sum of a free-field part and an interaction part, $$\mathcal{L} = \mathcal{L}_0 + \mathcal{L}_I\ .$$ Amplitudes of the model, e.g. $$S$$-matrix elements and matrix elements of composite operators, are represented as power series in coupling constants. Starting from the $$S$$-matrix represented in terms of the time-ordered exponent of the interaction Lagrangian which is expanded with the application of the Wick theorem, or from Green functions written in terms of a functional integral treated perturbatively, one obtains that, in a fixed perturbation order, the amplitudes are written as finite sums of Feynman integrals which are constructed according to Feynman rules: lines correspond to $$\mathcal{L}_0$$ and vertices are determined by $$\mathcal{L}_I$$ - see (Bogoliubov N, 1983); (Itzykson C, 1980); (Peskin M, 1980); (Zavialov O, 1980).

The basic building block of the Feynman integrals is the propagator that enters the relation $T \phi_i(x_1) \phi_i(x_2) = \;: \phi_i(x_1) \phi_i(x_2): + D_{F,i}(x_1-x_2)\;.$ Here $$D_{F,i}$$ is the Feynman propagator of the field of type $$i\ ,$$ $$T$$ denotes the time-ordered product and the colons denote a normal product of the free fields. The Fourier transforms of the propagators have the form $\tag{1} \tilde{D}_{F,i}(p) \equiv \int {\rm d}^4 x\, e^{i p\cdot x} D_{F,i}(x) = \frac{i Z_i(p)}{(p^2-m_i^2+i 0)^{a_i}}\; ,$

where $${\rm d}^4 x={\rm d} x^0 \,{\rm d} \vec{x}\ ,$$ $$m_i$$ is the corresponding mass, $$Z_i$$ is a polynomial and $$a_i=1$$ or $$2$$ (for the gluon propagator in the general covariant gauge). The scalar product in Minkowski space is $$p\cdot x=p_0 x_0-\vec{p}\cdot\vec{x}=p_0 x_0-p_1 x_1-p_2 x_2-p_3 x_3\ ,$$ with $$p^2=p_0^2-p_1^2-p_2^2-p_3^2\ .$$ For the propagator of the scalar field, one has $$Z=1, a=1\ .$$ The causal $$i 0$$ are usually omitted for brevity. Polynomials associated with vertices of graphs can be taken into account by means of the polynomials $$Z_l\ .$$

For a usual Feynman graph, the denominators are quadratic. Linear denominators usually appear in asymptotic expansions of Feynman integrals within the strategy of expansion by regions (Beneke M, 1998); (Smirnov V, 2002). Such expansions provide a useful link of an initial theory described by some Lagrangian with various effective theories where, indeed, the denominators of propagators can be linear with respect to the external and loop momenta. For example, one encounters the following denominators$p\cdot k\ ,$ with an external momentum $$p$$ on the light cone, $$p^2=0\ ,$$ for the Sudakov limit and with $$p^2\neq 0$$ for the quark propagator of Heavy Quark Effective Theory (HQET). Some non-relativistic propagators appear within threshold expansion and in the effective theory called Non-Relativistic QCD (NRQCD), for example, the denominator $$k_0-\vec{k}^2/(2m)\ .$$

Eventually, one obtains, for any fixed perturbation order, a sum of Feynman amplitudes labelled by Feynman graphs constructed from the given type of vertices and lines. In the commonly accepted physical slang, the graph, the corresponding Feynman amplitude and the corresponding Feynman integral over loop momenta are all often called the diagram'. When dealing with graphs and Feynman integrals one usually does not bother about the mathematical definition of the graph and thinks about something that is built of lines and vertices. Still a graph is an ordered family $${\mathcal{V}, \mathcal{L},\pi_{\pm}}\ ,$$ where $$\mathcal{V}$$ is the set of vertices, $$\mathcal{L}$$ is the set of lines, and $$\pi_{\pm}:\mathcal{L} \to \mathcal{V}$$ are two mappings that correspond the initial and the final vertex of a line. (Mathematicians use the word edge', rather than line'.)

A Feynman graph differs from a graph by distinguishing a subset of vertices which are called external. The external momenta or coordinates on which a Feynman integral depends are associated with the external vertices.

Thus quantities that can be computed perturbatively are written, in any given order of perturbation theory, through a sum over Feynman graphs. For a given graph $$\Gamma\ ,$$ the corresponding Feynman amplitude $\tag{2} G_{\Gamma}(q_1,\ldots,q_{n+1}) = (2\pi)^4\,i \,\delta\left(\sum_i q_i\right) F_{\Gamma}(q_1,\ldots,q_{n})$

can be written in terms of an integral over loop momenta $\tag{3} F_{\Gamma}(q_1,\ldots,q_{n}) = \int{\rm d} ^4 k_1\ldots \int{\rm d} ^4 k_h \prod_{l=1}^L \tilde{D}_{F,l}(r_l) \; ,$

where $${\rm d}^4 k_i={\rm d} k_i^0 \,{\rm d} \vec{k}_i\ ,$$ and a factor with a power of $$2\pi$$ is omitted for brevity. The Feynman integral $$F_{\Gamma}$$ depends on $$n$$ linearly independent external momenta $$q_i=(q_i^0,\vec{q}_i)\ ;$$ the corresponding integrand is a function of $$L$$ internal momenta $$r_l\ ,$$ which are certain linear combinations of the external momenta and $$h=L-V+1$$ chosen loop momenta $$k_i\ ,$$ where $$L,V$$ and $$h$$ are numbers of lines, vertices and (independent) loops, respectively, of the given graph.

One can choose the loop momenta by fixing a tree $$T$$ of the given graph, i.e. a maximal connected subgraph without loops, and correspond a loop momentum to each line not belonging to this tree. (This fixed tree is often called a spanning tree.) Then one has the following explicit formula for the momenta of the lines: $\tag{4} r_l = \sum_{i=1}^h e_{il} k_i + \sum_{i=1}^n d_{il} q_i\;,$

where $$e_{il}=\pm 1$$ if $$l$$ belongs to the $$i$$-th loop and $$e_{il}=0$$ otherwise, $$d_{il}=\pm 1$$ if $$l$$ lies in the tree $$T$$ on the path with the momentum $$q_i$$ and $$d_{il}=0$$ otherwise. The signs in both sums are defined by orientations.

## Divergences

As has been known from early days of quantum field theory, Feynman integrals over loop momenta usually have divergences. This word means that, taken naively, these integrals are ill-defined because the integrals over the loop momenta generally diverge. The ultraviolet (UV) divergences manifest themselves through a divergence of the Feynman integrals at large loop momenta. Consider, for example, the scalar Feynman integral $\tag{5} F_{\Gamma} (q) = \int \tilde{D}_{F}(k) \tilde{D}_{F}(q-k){\rm d}^4 k$

corresponding to the one-loop graph $$\Gamma$$ of Figure 1

where the scalar propagator with the mass $$m$$ is $\tag{6} \tilde{D}_{F}(q) = \frac{i}{q^2-m^2+i 0}\; ,$

Introducing four-dimensional (generalized) spherical coordinates $$k =r \hat{k}$$ in (5), where $$\hat{k}$$ is on the unit (generalized) sphere and is expressed by means of three angles, and counting powers of propagators, one obtains, in the limit of large $$r\ ,$$ the following divergent behaviour$\int_{\Lambda}^{\infty}{\rm d} r\, r^{-1}\ .$

For a general diagram with $$h$$ loops, a similar power counting at large values of the loop momenta gives $$4 h(\Gamma)-1$$ from the Jacobian that arises when one introduces generalized spherical coordinates in the $$(4\times h)$$-dimensional space of $$h$$ loop four-momenta, plus a contribution from the powers of the propagators and the degrees of its polynomials, and leads to an integral $$\int_{\Lambda}^{\infty} {\rm d} r\, r^{\omega-1}\ ,$$ where $\tag{7} \omega = 4 h - 2 L +\sum_l n_l$

is the (UV) degree of divergence of the graph and $$n_l$$ are the degrees of the polynomials $$Z$$ in (1).

This estimate shows that the Feynman integral is overall UV convergent (no divergences arise from the region where all the loop momenta are large) if the degree of divergence is negative. One says that the Feynman integral has a logarithmic, linear, quadratic, etc. overall divergence when $$\omega=0,1,2,\ldots\ ,$$ respectively. To ensure a complete absence of UV divergences it is necessary to check convergence in various regions where some of the loop momenta become large, i.e. to satisfy the relation $$\omega(\gamma)<0$$ for all the subgraphs $$\gamma$$ of the graph. One calls a subgraph UV divergent if $$\omega(\gamma)\geq 0\ .$$ In fact, it is sufficient to check these inequalities only for one-particle-irreducible (1PI) subgraphs (which cannot be made disconnected by cutting a line). It turns out that these rough estimates are indeed true.

If one turns from momentum space integrals to some other representation of Feynman diagrams, the UV divergences will manifest themselves in other ways. For example, in coordinate space, the Feynman amplitude (for example, (the inverse Fourier transform of (6) in the scalar case) is expressed in terms of a product of the Fourier transforms of propagators $$\prod_{l=1}^L D_{F,l}(x_{l_{\rm i}}-x_{l_{\rm f}})\;$$ integrated over four-coordinates $$x_i$$ corresponding to the internal vertices. Here $$l_{\rm i}$$ and $$l_{\rm f}$$ are the beginning and the end, respectively, of a line $$l\ .$$

The propagators in coordinate space, $\tag{8} D_{F,l}(x) = \frac{1}{(2\pi)^4} \int \tilde{D}_{F,l}(q) e ^{-i x \cdot p} {\rm d} ^4 q \;,$

are singular at small values of coordinates $$x=(x_0,\vec{x})\ .$$ For example the singular behaviour of the scalar propagator is $\tag{9} D_{F}(x) = -\frac{i m}{4\pi^2\sqrt{-x^2+i 0}} K_1 ( i m \sqrt{-x^2+i 0} ) = -\frac{1}{4\pi^2}\frac{1}{x^2-i 0} + O ( m^2 \ln m^2 ) \;,$

where $$K_1$$ is a Bessel function. The leading singularity at $$x=0$$ is given by the value of the coordinate space massless propagator.

Thus, the inverse Fourier transform of the convolution integral (5) equals the square of the coordinate-space scalar propagator which has the singularity $$(x^2-i 0)^{-2}\ .$$ Power-counting shows that this singularity is not locally integrable in four dimensions, and this is the coordinate space manifestation of the UV divergence. In the language of the theory of distributions, this means that, although individual propagators in coordinate space are well-defined distributions their products are usually ill-defined.

The divergences caused by singularities at small loop momenta are called infrared (IR) divergences. First one distinguishes IR divergences that arise at generic values of the external momenta. A typical example of such a divergence is given by the graph where one of the massless lines contains the second power of the corresponding propagator and one obtains a factor $$1/(k^2)^2$$ in the integrand, where $$k$$ is chosen as the momentum of this line. Then, keeping in mind the introduction of generalized spherical coordinates and performing power-counting at small $$k$$ (i.e. when all the components of the four-vector $$k$$ are small), one again encounters a divergent behaviour $$\int_0^{\Lambda} {\rm d} r\, r^{-1}$$ but now at small values of $$r\ .$$ There is a similarity between the properties of IR divergences of this kind and those of UV divergences. One can define, for such off-shell IR divergences, an IR degree of divergence, in a similar way to the UV case. A reasonable choice is provided by the value (Speer E, 1977) $\tag{10} \tilde{\omega}(\gamma) = -\omega(\Gamma/\overline{\gamma}) \equiv \omega(\overline{\gamma}) -\omega(\Gamma)\;,$

where $$\overline{\gamma}\equiv\Gamma\backslash \gamma$$ is the completion of the subgraph $$\gamma$$ in a given graph $$\Gamma$$ and $$\Gamma/\gamma$$ denotes the reduced graph which is obtained from $$\Gamma$$ by reducing every connectivity component of $$\gamma$$ to a point. The absence of off-shell IR divergences is guaranteed if the IR degrees of divergence are negative for all massless subgraphs $$\gamma$$ whose completions $$\overline{\gamma}$$ include all the external vertices in the same connectivity component. The off-shell IR divergences are the worst but they are in fact absent in physically meaningful theories. However, they play an important role in asymptotic expansions of Feynman integrals in momenta and masses (Smirnov V, 2002).

The other kinds of IR divergences arise when the external momenta considered are on a surface where the Feynman diagram is singular: either on a mass shell or at a threshold. Consider, for example, the massless graph of Figure 2. Let us take $$p_1^2=p_2^2=0$$ and all the masses equal to zero.

The corresponding Feynman integral is $\tag{11} \int \frac{{\rm d}^4k}{(k^2-2 p_1\cdot k) (k^2-2 p_2\cdot k)k^2 } \; .$

At small values of $$k\ ,$$ the integrand behaves like $$1/[(-2 p_1\cdot k) (-2 p_2\cdot k)k^2]\ ,$$ and, with the help of power counting, one sees that there is an on-shell IR divergence due to the region where $$k\to 0$$ (componentwise).

Such IR divergences are local in momentum space, i.e. connected with special points of the loop integration momenta. Collinear divergences arise at lines parallel to certain light-like four-vectors. The same triangle diagram provides a typical example of a collinear divergence. These are divergences at non-zero values of $$k$$ that are collinear with $$p_1$$ or $$p_2$$ and where $$k^2\sim 0\ .$$ This follows from the fact that the product $$1/[(k^2-2p\cdot k) k^2]\ ,$$ where $$p^2=0$$ and $$p\neq 0\ ,$$ generates collinear divergences. To see this let us take residues in the upper complex half plane when integrating this product over $$k_0\ .$$ For example, taking the residue at $$k_0=-|\vec{k}|+i 0$$ leads to an integral containing $$1/(p \cdot k)=1/[p^0 |\vec{k}| (1-\cos\theta)]\ ,$$ where $$\theta$$ is the angle between the spatial components $$\vec{k}$$ and $$\vec{p}\ .$$ Thus, for small $$\theta\ ,$$ one has a divergent integration over angles because of the factor $${\rm d} \cos\theta/(1-\cos\theta)\sim {\rm d} \theta/\theta\ .$$ The second residue generates a similar divergent behaviour – this can be seen by making the change $$k\to p-k\ .$$

Another way to reveal the collinear divergences is to introduce the light-cone coordinates $$k_{\pm} =k_0\pm k_3, \; \underline{k}=(k_1,k_2)\ .$$ If one chooses $$p$$ with the only non-zero component $$p_+\ ,$$ one will see a logarithmic divergence coming from the region $$k_{-}\sim \underline{k}^2\sim 0$$ just by power counting.

These are the main types of divergences of usual Feynman integrals. Various special divergences arise in more general Feynman integrals that can contain linear propagators and appear on the right-hand side of asymptotic expansions in momenta and masses and in associated effective theories. For example, in the Sudakov limit, one encounters divergences that can be classified as UV collinear divergences. Another situation with various non-standard divergences is provided by effective theories, for example, Heavy Quark Effective Theory, NRQCD and potential NRQCD, where special power counting is needed to characterize the divergences.

## Regularization

Feynman integrals over loop momenta are usually divergent, because they can have ultraviolet (UV), infrared (IR), collinear divergences as well as some other kinds of divergences. The standard way of dealing with divergent Feynman integrals is to introduce a regularization. This means that, instead of the original ill-defined Feynman integral, one considers a quantity which depends on a regularization parameter, $$\lambda\ ,$$ and formally tends to the initial, meaningless expression when this parameter takes some limiting value, $$\lambda=\lambda_0\ .$$ This new, regularized, quantity turns out to be well-defined, and the divergence manifests itself as a singularity with respect to the regularization parameter. Experience tells us that this singularity can be of a power or logarithmic type, i.e. $$\ln^n (\lambda-\lambda_0)/(\lambda-\lambda_0)^i\ .$$

An obvious way of regularizing UV-divergent Feynman integrals is to introduce a cut-off at large values of the loop momenta. Another well-known regularization procedure is the Pauli–Villars regularization (Pauli W, 1949), which is described by the replacement $\frac{1}{p^2-m^2} \to \frac{1}{p^2-m^2}-\frac{1}{p^2-M^2}$ and its generalizations. For finite values of the regularization parameter $$M\ ,$$ this procedure clearly improves the UV asymptotics of the integrand. Here the limiting value of the regularization parameter is $$M=\infty\ .$$

If the integer powers $$a_l$$ in the propagators are replaced by general complex numbers $$\lambda_l$$ one obtains an analytically regularized (Speer E, 1968) Feynman integral where the divergences of the diagram are encoded in the poles of this regularized quantity with respect to the analytic regularization parameters $$\lambda_l\ .$$ Consider, for example, the analytically regularized Feynman integral $\tag{12} F_{\Gamma} (q,\lambda_1,\lambda_2) = \int \frac{{\rm d}^4 k}{(k^{2} -m^2+i 0)^{\lambda_1} [(q-k)^{2}-m^2 + i 0]^{\lambda_2}}$

corresponding to the one-loop graph $$\Gamma$$ of Figure 1 Power counting at large values of the loop momentum reveals the divergent behaviour $$\int_{\Lambda}^{\infty}{\rm d} r\, r^{\lambda_1+\lambda_2-3},$$ which results in a pole $$1/(\lambda_1+\lambda_2-2)$$ at the limiting values of the regularization parameters $$\lambda_l=1\ .$$

## Dimensional regularization and parametric representations of Feynman integrals

A very important type of regularization successfully applied in practice is dimensional regularization, where the regularization parameter is the space-time dimension considered as a general complex number - see (Bollini C, 1972); 't Hooft G, 1977); Breitenlohner P, 1977). In particular, it is compatible with gauge invariance. One way to introduce dimensional regularization is of algebraic character. It is based on certain axioms for integration in a space with non-integer dimension. After evaluating a Feynman integral according to the algebraic rules, one arrives at some concrete function of these parameters but, before integration, one is dealing with an abstract algebraic object.

An analytic way to introduce dimensional regularization is based on a representation of Feynman integrals as integrals over alpha (Schwinger) parameters. In the case of $$d$$ integer dimensions (e.g., for the physical value $$d=4$$) and in the case of scalar propagators the alpha representation has the form: $\tag{13} F_{\Gamma}(q_1,\ldots,q_n;d) = e^{i \pi [h(1-d/2)-L]/2}\pi^{h d/2} \int_0^\infty {\rm d}\alpha_1 \ldots\int_0^\infty{\rm d}\alpha_L \mathcal{U}^{-d/2} e ^{i \mathcal{V}/\mathcal{U}-i \sum m_l^2 \alpha_l},$

where $$\mathcal{U}$$ and $$\mathcal{V}$$ are the well-known functions $\tag{14} \mathcal{U} = \sum_{T\in T^1} \prod_{l{\not\!\, \in} T} \alpha_l,$

$\tag{15} \mathcal{V}= \sum_{T\in T^2} \prod_{l{\not\!\, \in} T} \alpha_l \left( q^T\right)^2 .$

In (14), the sum runs over trees of the given graph, and, in (15), over 2-trees, i.e. subgraphs that do not involve loops and consist of two connectivity components; $$\pm q^T$$ is the sum of the external momenta that flow into one of the connectivity components of the 2-tree $$T\ .$$ (It does not matter which component is taken because of the conservation law for the external momenta.) The products of the alpha parameters involved are taken over the lines that do not belong to the given tree $$T\ .$$ The functions $$\mathcal{U}$$ and $$\mathcal{V}$$ are homogeneous functions of the alpha parameters with the homogeneity degrees $$h$$ and $$h+1\ ,$$ respectively.

The dimensionally regularized Feynman integral corresponding to a given graph $$\Gamma$$ can be defined by means of (13), where the quantity $$d$$ is considered as a complex number. This is a function of kinematical invariants $$q_i \cdot q_j$$ constructed from the external momenta and contained in the function $$\mathcal{V}\ .$$ The external momenta $$q_i$$ as well as the metric tensor $$g_{\mu\nu}$$ are treated as elements of an algebra of covariants, where one has in particular, $$g_{\mu}^{\mu}=d\ .$$ This algebra also includes the $$\gamma$$-matrices with anticommutation relations $$\gamma_\mu\gamma_\nu+\gamma_\nu\gamma_\mu= 2 g_{\mu\nu}$$ so that $$\gamma^\mu\gamma_\mu=d\ .$$ The definition of the anti-symmetric tensor $$\varepsilon_{\kappa\mu\nu\lambda}$$ in this algebra is more subtle.

Thus the dimensionally regularized Feynman integrals are defined as linear combinations of tensor monomials in the external momenta and other algebraic objects with coefficients that are functions of the scalar products $$q_i \cdot q_j$$ and are given by $$\alpha$$-parametric integrals.

Besides alpha parameters, the closely related Feynman parameters are often used. For a product of two propagators raised to general powers, one writes down the following relation: $\tag{16} \frac{1}{(m_1^2-p_1^2)^{\lambda_1} (m_2^2-p_2^2)^{\lambda_2}} = \frac{\Gamma(\lambda_1+\lambda_2)}{\Gamma(\lambda_1) \Gamma(\lambda_2)} \int_0^1 \frac{{\rm d} \xi\, \xi^{\lambda_1-1}(1-\xi)^{\lambda_2-1}}{ [(m_1^2-p_1^2) \xi + (m_2^2-p_2^2)(1-\xi)]^{\lambda_1+\lambda_2}}$

which is usually applied to a pair of appropriately chosen propagators.

If an obvious generalization of (16) is applied to an arbitrary number of propagators and performs the integration over all loop momenta one will arrive at a parametric representation which can also be obtained from (13) by making the change of variables $$\alpha_l=\eta \alpha'_l\ ,$$ with $$\sum \alpha'_l=1\ ,$$ performing the integration over $$\eta$$ from $$0$$ to $$\infty$$ explicitly, one obtains $\tag{17} F_{\Gamma}(q_1,\ldots,q_n;d) =(-1)^L \left(i\pi^{d/2} \right)^h \Gamma(L-h d/2) \int_0^\infty {\rm d}\alpha_1 \ldots\int_0^\infty{\rm d}\alpha_L \, \delta\left( \sum \alpha_l-1\right) \frac{ \mathcal{U}^{a-(h+1) d/2}} { \left(-\mathcal{V} +\mathcal{U}\sum m^2_l \alpha_l\right)^{a-h d/2}} \; .$

A folklore Cheng–Wu theorem says that the same formula (17) holds with the delta function $\tag{18} \delta\left( \sum_{l\in \nu} \alpha_l-1\right)\;,$

where $$\nu$$ is an arbitrary subset of the lines $$1,\ldots,L\ ,$$ when the integration over the rest of the $$\alpha$$-variables, i.e. for $$l\overline{\in}\nu\ ,$$ is extended to the integration from zero to infinity.

The parametric representations (13) and (17) are used for various purposes, in particular, to prove theorems on renormalization Hepp K, 1966); (Bergère M, 1974); [#BM|Breitenlohner P, 1977)]]; (Zavialov O, 1980), on asymptotic expansions of Feynman integrals in momenta and masses (Smirnov V, 2002), to evaluate Feynman integrals analytically and numerically (Smirnov V, 2004).

Although a regularization makes it possible to deal with divergent Feynman integrals, it does not actually remove divergences, because this operation is of an auxiliary character so that sooner or later it will be necessary to switch off the regularization. To provide UV finiteness of physical observables evaluated through Feynman diagrams, another operation, called renormalization, is used (Bogoliubov N, 1983); (Itzykson C, 1980); (Peskin M, 1980); (Zavialov O, 1980). This operation is described, at the Lagrangian level, as a redefinition of the bare parameters of a given Lagrangian by inserting counterterms. The renormalization at the diagrammatic level is called $$R$$-operation. It removes the UV divergence from individual Feynman integrals.

The IR and collinear divergences are also removed but in another way. After the contribution of radiative corrections represented in terms of Feynman integrals is summed up with the contribution of real radiation, these divergences are cancelled with similar divergences present in this second part of perturbative corrections, so that physical results for amplitudes or cross-sections turn out to be finite.