Conformal methods for massless Feynman integrals and large methods
We review the large method of calculating high order information on the renormalization group functions in a quantum field theory which is based on conformal integration methods. As an example these techniques are applied to a typical graph contributing to the -function of theory at . The possible future directions for the large methods are discussed in light of the development of more recent techniques such as the Laporta algorithm.
One of the main problems in renormalization theory is the construction of the renormalization group functions. These govern how the parameters of a quantum field theory, such as the coupling constant, depend on scale. In situations where one has to compare with precision data, this ordinarily requires knowing the renormalization group functions to very high orders in a perturbative expansion. The quantum field theories we have in mind are not only the gauge theories of particle physics but also the scalar and fermionic ones which arise in condensed matter problems. These are central in understanding phase transitions. To attain such precision in perturbative expansions means that large numbers of Feynman diagrams have to be determined with the number of graphs increasing with the loop order. Moreover, as the order increases the underlying integrals require more sophisticated methods in order to deduce their value analytically. The widely established methods of computing Feynman graphs will be reported elsewhere in this volume. Here we review an alternative approach which complements explicit perturbative techniques. It does so in such a way that for low loop orders there is overlap but at orders beyond that already known part of the perturbative series can be deduced at all orders within a certain approximation. This is known as the large or large method where is a parameter deriving from a symmetry of the theory such as a Lie group or the number of massless quark flavours, , in Quantum Chromodynamics (QCD). In this method the Feynman graphs are related to those of perturbation theory but because of the nature of the expansion parameter, the powers of the propagators appearing in such graphs are not the canonical value of unity but instead differ from unity by where corresponds to the regularizing parameter of dimensional regularization. In addition beyond leading order in the expansion, the propagator powers will include the anomalous dimensions in addition to the leading or canonical dimension. Therefore, standard perturbative techniques such as integration by parts requires care in its use since one may not be able to actually reduce a graph to a simpler topology. Instead a different technique has had to be refined and developed. It is based on a conformal property of Feynman integrals and we review it here in the context of the large methods. Though it has had some applications in perturbative computations.
The article is organized as follows. We devote the next section to the notation and techniques of computing Feynman graphs using conformal methods in -dimensions. We focus on the general two loop self energy graph in the subsequent section and review the work of Gracey:1 (); Gracey:2 (), upon which this review is mostly based, and others in the methods of evaluating it. These techniques are then applied to a problem in scalar quantum field theory in Section where a graph with internal integrations is evaluated exactly in -dimensions. We conclude in Section with thoughts on the direction in which the technique could be developed next given recent advances in the computation of Feynman graphs using conventional perturbative techniques.
2 Notation and Elementary Techniques
We begin by introducing the notation we will use which will be based on Gracey:1 (); Gracey:2 (). There Feynman graphs were represented in coordinate or configuration space notation. By this we mean that in writing a Feynman integral graphically the integration variables are represented as the vertices. By contrast in momentum space representation the integration variables correspond to the momenta circulating around a loop. So in coordinate space representation propagators are denoted by lines between two fixed points, as illustrated in Figure .
Figure . Coordinate space propagator.
There the power of the propagator is denoted by a number or symbol beside the line. One can map between coordinate and momentum space representation by using a Fourier transform. In the notation of Gracey:1 (); Gracey:2 () we have
where is in coordinate space and is the conjugate momentum. Also for shorthand we set
which is used throughout to avoid the appearance of in the Euler -function. This symbol should not be confused with the mass scale appearing in renormalization group equations. Clearly
which is singular when where is zero or a positive integer. Also vanishes at the negative integers. The elementary identity
follows trivially as does
from the -function identity . With this notation the elementary one loop self energy graph in momentum space is replaced by chain integration in coordinate space representation. This is represented graphically in Figure
Figure . Chain integration.
However, in practice Feynman graphs have more complicated integration points. In other words in coordinate space representation one has more than two lines intersecting at a point. Therefore, more involved integration techniques are required to evaluate the Feynman integrals. One very useful technique is that of uniqueness or conformal integration which was introduced in three dimensions in Gracey:3 (). It has been developed in several ways subsequently and specifically to -dimensions. For example, see Gracey:4 (). We follow Gracey:1 (); Gracey:2 () and use the rule represented in Figure , where is the integration variable,
Figure . Conformal integration when .
which follows when the sum of the exponents of the lines intersecting at the -point vertex add to the spacetime dimension
This is known as the uniqueness condition. By the same token if a graph contains a triangle where the lines comprising the triangle sum to such as
as is the case in Figure , then the unique triangle can be replaced by the vertex on the left side. There are several methods to establish the uniqueness integration rule. If one uses standard text book methods such as Feynman parameters then the integral over can be written as
prior to using, (7). When that condition is set then the hypergeometric function collapses to the geometric series and allows the integration over the Feynman parameter to proceed which results in
Applying the uniqueness condition a second time produces the right hand side of Figure since the hypergeometric function again reduces to the geometric series. This is in such a way that the canonical propagators emerge.
An alternative method is to apply a conformal transformation on the coordinates of the integral, Gracey:2 (). In this approach, which is applicable to any graph in general, one external point is labelled as an origin and given as a coordinate. The other points are denoted by coordinates , and . The conformal transformation changes the integration coordinate as well as the external points through
Thus for two coordinates and undergoing such a transformation we have the lemma
An integration measure also produces contributions to the lines joining to the origin since
Therefore, for the vertex on the left side of Figure this transformation produces the intermediate integral of Figure .
Figure . Vertex of Figure after a conformal transformation with base at where .
To complete the integration requires setting the uniqueness condition (7) which produces a chain integral since the line from to is absent from the graph. To complete the derivation one undoes the original conformal transformations to produce the right hand side of Figure . If one compares the two derivations, the latter is in fact of more practical use. This is because it avoids the use of writing the original integral in terms of Feynman parameters which would become tedious for higher order cases. Also it is simple to implement graphically.
Having recalled the derivation of the uniqueness rule it is straightforward to see that there is a natural extension. In the first derivation there was not a unique way to collapse the hypergeometric function to an elementary type of propagator. Instead this will happen if the sum of the exponents is where is a positive integer. Athough the collapse in this case will not be to the geometric series, it will reduce to simple algebraic functions which are of the propagator type. So, for instance, when we have the result of Figure , Gracey:5 (),
Figure . Conformal integration when .
A similar rule has been constructed and used in Gracey:4 (). We will use Figure later in order to simplify various integrals.
3 Two Loop Self Energy Graph
We can illustrate some of the techniques of conformal integration by considering the massless two loop self energy graph with arbitrary powers, on the propagators. It is illustrated in Figure
Figure . Two loop self energy graph in coordinate space representation.
where we have used the coordinate space representation. Thus the vertices are integrated over rather than the loop momenta. To clarify, the integral of Figure is
where . The structure of this integral has been widely studied and we briefly highlight several properties of relevance. The analysis of Gracey:6 (); Gracey:7 () determined that the symmetry group of the graph was which has elements. Exploiting this the expansion of the integral in with propagator powers of order from unity was determined up to , Gracey:6 (); Gracey:7 (). At it was discovered that the first multi-zeta value occurred, Gracey:7 (). Specifically
where is the Riemann zeta function and in the original notation of Gracey:8 (). Subsequent to this it has been shown that the only numbers which appear in the full series expansion in are mutiple zeta values, Gracey:9 (). While the work of Gracey:6 (); Gracey:7 () illustrated the power of group theory to evaluate master integrals explicitly, using conformal integration allows one to relate two loop self energy integrals by exploiting the masslessness of the original diagrams. This was originally developed in Gracey:1 (); Gracey:2 () and we summarize that here as there appears to be scope nowadays to take this method to three and higher loop order graphs.
The transformations developed in Gracey:2 () fall into several classes. The first is that derived from the elementary use of the Fourier transform. Writing
where and is independent of and corresponds to the value of the integral, then taking the Fourier transform produces an integral which is also the two loop self energy. Though the propagator powers are different. In this sense one can say that the graph is self-dual which is not a property all Feynman graphs have. Thus, Gracey:2 (),
This transformation is known as the momentum representation or MR. It can be easily generalized to other topologies and there is a simple graphical rule for this. Although not immediately apparent from the self energy because of the self-duality, each -valent vertex of the original graph has an associated triangle in the dual graph. For other topologies -valent vertices are mapped to squares and -valent vertices to pentagons with a clear generalization pattern.
A set of less obvious transformations can be deduced from the uniqueness condition. First, we define the shorthand notation, Gracey:2 (),
and illustrate the technique for one case. If one considers the central propagator it can be replaced by a chain integral. Although there are an infinite number of ways of doing this one can choose the exponents of the chain so that the top vertex is unique. In other words
where is the intermediate integration point. As the vertex of Figure is now unique the conformal integration rule can be used to rewrite the integral. This results in, Gracey:2 (),
In the notation of Gracey:2 () this transformation is known as . It is elementary to see that there are five other such transformations which are denoted by , , , and . The syntax is that when an arrow points in a general upwards direction it is a transformation on the vertex and by contrast in a downwards direction it relates to the vertex. The propagator which one replaces by a chain to make the vertex unique is in correspondence with the direction of the arrow. While these six transformations operate on the internal vertices there are two which act on each of the external vertices. One can complete the uniqueness of one of these by realizing that the integral itself is a propagator with power as indicated in (16), Gracey:2 (). For example, if the right external point is chosen as the base integration vertex then the appending propagator has power . This produces
and this is denoted by . The corresponding transformation on the left external point is called .
The final set of transformations are based on the conformal transformations (11) and (12) together with the effect they have on the two vertex measures, Gracey:2 (). One can choose either of the external vertices as the origin of the transformation. Once decided the result of the conformal transformation is that all propagators joining to the origin have their powers changed to the difference of and the sum of the exponents at the point at the other end of that propagator. This means all points including those not directly connected to the base point in the first place. For the two loop self energy there are no such points but for higher loop graphs this will be the case. We will give an example of this in Section . Thus the conformal left transformation is, Gracey:2 (),
where there is no -function factor and this is denoted by CL in contrast to CR which is the transformation based on the right external vertex as the origin of the conformal transformation. The full set of transformations and the result of applying each to the graph of Figure are summarized in a Table in Gracey:2 (). However, as brief examples of the transformations the integral of (15) is related as follows
Though the latter follows from a simple rotation of the integral as well.
Aside from the transformations there are other techniques which allow one to evaluate the two loop self energy and higher order graphs. Perhaps the most exploited is that of integration by parts which was introduced for (16) in Gracey:10 (). It determined that the first term in the expansion of was and has also been used in other applications, Gracey:2 (). Indeed more recently the technique has been developed by Laporta in Gracey:11 () to produce an algorithm which relates all integrals in a Feynman graph to a base set of master integrals. These can then be evaluated by direct methods to complete the overall computation. In the coordinate space representation we use here the basic rule is given in Figure
Figure . Integration by parts in coordinate space representation.
where the or on a line indicates that the power of that propagator is increased or decreased by unity. For example, with this, Gracey:10 (),
which can be expanded in powers of . Clearly the series can only involve rationals and . Indeed the rule can also be applied to more general cases. In Gracey:2 () it was shown that
for arbitrary and . However, not all graphs can be integrated by parts. An example of such a case is for non-unit , and . Another example is (15), Gracey:7 (), whose expansion has a non-Riemann zeta value at some point in the expansion. Indeed this is perhaps an indication of an obstruction to integrability.
While integration by parts allows one to reduce the powers of various propagators
Figure . Reduction formula for two loop self energy based on the generalized transformation.
by unity within a Feynman diagram it is not the only method to achieve this. A modification of the uniqueness method can be used to derive rules similar to Figure . Specifically if one chooses the exponents of the propagators to be then one finds the extension given in Figure . Using this rule and repeating the analysis of the transformations on the two loop self energy graph provides relations specific to this topology, Gracey:5 (). For instance, extending to have the upper vertex exponents summing to gives the relation in Figure where the or on the right side indicates that the exponent of that line is increased or decreased by unity. In Figure provided and then the powers of the respective propagators can be reduced by unity. However, this restriction is a drawback if one wishes to reduce graphs which have unit exponents. Instead it is possible to extend the method which produced the relation of Figure . For instance, rather than begin with the general two loop self energy and applying the generalized uniqueness rule, one can use one of the transformations of Gracey:2 () and then apply a rule like that of Figure before applying the transformation inverse to the original one. In this way one can build up a suite of relations.
Figure . Another reduction formula for two loop self energy.
One such useful relation is illustrated in Figure which is derived in several stages. The first is to construct a relation similar to that of Figure by first applying to the graph of Figure and then undoing it by applying the rule of Figure to the same external vertex. This produces a relation where increases by unity in each of the three resulting graphs. The second stage is to apply this rule to the graph of Figure after a CR transformation has been enacted. To complete the derivation the final step is to undo with another CR transformation. Thus the value of each graph on the right hand side of Figure is one less than that of the graph on the left side. This reduction has coefficients on the right hand side which are non-singular for unit propagators. Other rules can be derived by this method and a fuller set are recorded in Appendix B of Gracey:12 (). It is worth noting that similar rules based on the generalized uniqueness where developed in Gracey:4 ().
4 QFT Application
Having discussed the general techniques for determining massless Feynman integrals using conformal methods, we illustrate their usefulness in a practical problem in a quantum field theory. Specifically we focus on the determination of the critical exponents at a phase transition in various models in the large expansion. The background which we describe here is based on a series of articles, Gracey:1 (); Gracey:2 (); Gracey:13 (), where exponents were determined in -dimensions at and . The fact that -dimensional results are computable means that information on the renormalization group functions can be deduced in various spacetime dimensions. This is due to a special feature of critical point field theories and that is that at a non-trivial fixed point of the renormalization group flow the critical exponents correspond to the associated renormalization group function at that fixed point. Thus information on the renormalization group functions is encoded in these exponents. Moreover, at a fixed point several quantum field theories can lie in the same universality class despite having different structures. This is invariably as a consequence of a common interaction in the Lagrangian. Thus the same exponents can be used to access the structure of the renormalization group functions of two different theories. Further, as the spacetime dimension is not used as a regulator, information on the exponents can be deduced simultaneously in several different dimensions such as three and four. For more background to the use of the renormalization group equation at near criticality in quantum field theories see, for example, Gracey:14 ().
For the application of the conformal methods we consider here we concentrate on the nonlinear model which is critically equivalent in -dimensions to theory. For the latter theory the Lagrangian is
where is the coupling constant and . Introducing an auxiliary field equates this Lagrangian to
At criticality it is the interaction which drives the dynamics and thus it is straightforward to see that in this formulation the Lagrangian interaction is the same as that of the nonlinear model when the fields are constrained to lie on an -dimensional sphere. The constraint would have a final term linear in rather than a quadratic one together with a different coupling constant. This essentially is the origin of both field theories being in the same universality class. The linear or quadratic terms in at criticality serve effectively to define the structure of the propagators. In coordinate space representation these are, Gracey:1 (); Gracey:2 (),
where and are -independent amplitudes and and are the scaling dimensions of the fields. The latter comprise two parts. The first is the canonical dimension and the other is the anomalous dimension. Here
where is the anomalous dimension of and is the vertex anomalous dimension. The former is related to the renormalization group function which is also termed the anomalous dimension, , by
where is the value of the coupling constant at the critical point,
and is the wave function renormalization constant. (In (31) we have temporarily used to denote the standard renormalization group scale that underlies any renormalization group equation.) To determine the values of the exponents to a particular order in requires solving the skeleton Schwinger-Dyson equation for the -point functions at the same order. We do not discuss that formalism here, which can be found in Gracey:1 (); Gracey:2 (), as our focus is rather on the evaluation of the Feynman graphs contributing to these equations. Though we should say that the presence of the non-zero anomalous dimensions in the propagators means that in -point functions there are no self energy corrections on any internal propagator as otherwise there would be double counting. So the number of graphs to consider is smaller than the corresponding perturbative case.
The coupling constant at the critical point is denoted by and is defined as a nontrivial zero of the -function, . As we are working in -dimensions such a non-trivial zero exists in our theories since away from the spacetime dimension where the theory is renormalizable the coupling constant becomes dimensionful. Hence the first term of the -dimensional -function depends on . Moreover, will depend on the parameters of the theory which in our case here is . Thus . Similarly and . These can all be expanded in powers of where is large in such a way that the coefficients of are -dependent. Thence if one expresses these coefficients in powers of where for theory or for the nonlinear model, then one can deduce the coefficients in the corresponding renormalization group equation to all orders in perturbation theory at that order in . In this respect it is important to note that in the large expansion or do not play the role of a regulator as they would do in conventional perturbation theory.
Instead to see the origin of where a regulator is required one should consider the simple two loop contribution to the self energy graph given in Figure in coordinate space representation.
Figure . Two loop self energy for .
To use conformal methods one has to check the sum of the exponents at a vertex in coordinate space representation. From (29) one can see that
However, from the structure of the renormalization group equation at criticality the anomalous dimensions and begin as . More, specifically
Thus at leading order in the basic vertex is unique, Gracey:2 (). Hence at this order one can integrate at either of the vertices and produce the first contribution to the integral which is . The second integration is a simple chain and naively gives . This is clearly ill-defined due to the zeroes and singularities deriving from the -function. However, this graph was chosen to illustrate the fact that the graph and indeed the theory requires a regularization in this critical point formulation. The method developed in Gracey:1 (); Gracey:2 () was to use analytic regularization which is introduced by shifting the vertex anomalous dimension by an infinitesimal amount, , via
In some respect one is in effect performing a perturbative expansion in the vertex anomalous dimension, Gracey:1 (); Gracey:2 (). Consequently even at leading order the graph of Figure no longer has a unique vertex due to a non-zero . Therefore, to determine the graph to the finite part in requires the addition and subtraction of the graphs of Figure , Gracey:2 ().
Figure . Subtracted graphs for computation of two loop self energy.
These two graphs have been chosen in such a way that their singularity structure in exactly matches that of Figure , Gracey:2 (). Clearly they represent simple chain integrals which can be determined as where the singularity is clearly regularized. To complete the evaluation introduces another technique, which we will use later, to extract a finite term of a graph. This is a temporary regularization, Gracey:2 (). If one subtracts the graphs of Figure from that of Figure , the combination is finite with respect to which is therefore not required and can be set to zero. Thus one can complete the first integration at the upper vertex of each graph. (Without a regularization the point where one integrates in each graph has to be the same and thence the order of integration is important.) This produces for each graph. However, each of the three subsequent chain integrals has a singular exponent, . To circumvent this the lower two propagators of all three graphs are temporarily regularized by where is arbitrary. Thus the three graphs give
where for and not equal to zero or a negative integer and is derivative of the logarithm of the -function.
A more involved example which uses many of the techniques of the previous section occurs in the computation of the correction to the -function in theory. The relevant critical exponent is which is related through the critical renormalization group equation to the -function slope at criticality. In this case it has the form
and the explicit forms for are deduced from the part of the Schwinger-Dyson equations corresponding to corrections to scaling. In other words the propagators of (28) are extended to
In principle other corrections can appear here corresponding to other exponents such as that for the -function of the nonlinear model but one tends to focus on one calculation at a time. The effect of the corrections is that to deduce within the Schwinger-Dyson formalism all Feynman diagrams with one correction insertion on a propagator contribute at each particular order in . While the expression for appeared in Gracey:15 () the explicit evaluation of the contributing graphs has not been detailed. Thus we discuss one such diagram here as the approach can be readily adapted to the other graphs. It is given in Figure . To see that it is each closed loop of fields contributes a factor of and each propagator is . This is due to the fact that the amplitude is , Gracey:1 (); Gracey:2 (). As there are four of the former and five of the latter then this gives overall which is one factor of more than the previous order graph of Figure . Finally, another factor of derives from the actual Schwinger Dyson formalism used to determine . The double line on one propagator in Figure denotes the correction.
Figure . Particular graph contributing to the theory -function at .
The presence of such a correction means that the graph is -finite. Moreover, since we only want the value as a function of rather than and we can replace the exponents of the lines by their canonical values. If one was computing then the anomalous dimensions of each exponent would need to be retained at . The benefit of this restriction here is that of the ten vertices eight are unique. There are ten integrations to do over the vertices rather than the six of the loops as we are in coordinate space representation. Given this high degree of uniqueness the graph can be reduced rather quickly to one with fewer integrations. To do this one can use a variety of the rules we had earlier aside from uniqueness such as conformal transformation, unique triangle, insertion at an internal or external vertex. Ultimately one produces the graph of Figure .
Figure . Reduced integral of Figure .
This graph cannot be reduced any further since there are no unique vertices or triangles. Though various vertices or triangles are one unit from uniqueness. Moreover, integration by parts cannot be used since at some point one produces an unregularized exponent, such as or , or a zero in a denominator factor. In some sense this graph could be regarded as a master integral since it arises in several of the other graphs contributing to the Schwinger-Dyson equation. Moreover, it is worth noting that in strictly four dimensions the propagators of the graph would all have unit exponents. As an aside if an interested reader has been applying the conformal techniques to reduce the diagram and obtains similar exponents but distributed differently around the diagram then it will be related to that of Figure by applying the transformations discussed for the master -loop self energy. We note here that if a conformal transformation is applied to the graph of Figure with the left internal point as the CL base, then that would introduce a new line from the top right internal vertex to the base. This illustrates comments made earlier.
To proceed further and reduce the graph to a known function of requires an integration by parts but this requires modifying the integral first. Though before this can be achieved safely one has to introduce a temporary regularization to handle hidden singularities at a later stage of the computation. This technique has been applied by others, Gracey:4 (); Gracey:16 (). For our case we have choosen the regularization of Figure . How one chooses the temporary regularization is not unique. However, it is chosen here so that after application of the integration by parts rule of Figure the resulting four graphs have either unique vertices or triangles which are -dependent and which regularize any singularity after subsequent integration. For the integration by parts we use the top left
Figure . Temporary regularization of previous graph to reduce it to two loop basic graphs.
internal vertex of Figure with the line joining the quartic vertex as the reference line of the rule of Figure . This produces the four graphs of Figures -.
Figure . First graph after integration by parts.
All but the third have at least one unique vertex while that has a unique triangle. In our earlier notation the first two graphs of Figure and are
As both of these are -finite and have no -singular coefficients, one can set to zero in each. The final evaluation is by a two loop reduction formula similar to those of Figures and .
Figure . Second graph after integration by parts.
For the remaining two graphs of Figures and one has to treat them together due to the singular propagator exponents as will be evident. After integrating the respective unique triangle and vertex they combine to produce
Figure . Third graph after integration by parts.
As the external coefficient includes a factor of then the quantity inside the square brackets needs to be evaluated to .
Figure . Fourth graph after integration by parts.
This is not possible exactly for both integrals. (It is for the first.) Instead since one only needs the part itself one can achieve this by evaluating the integral
From the two -loop graphs we are interested in the term of this integral clearly corresponds to the piece we require. Moreover, it can be evaluated exactly using as it then reduces to an integral to which one can apply a -loop recurrence relation similar to that of Figure . The final expression for the graph of Figure is
Setting reproduces the established leading order value for the wheel of three spokes, Gracey:17 (), which provides a useful check. Finally, all the other contributing graphs are evaluated in a similar way and the full expression for , after using the Schwinger Dyson formalism, is given in Gracey:15 ().
5 Future Directions
We close the article by discussing several directions in which this approach could move. First, the extension of scalar field theories to non-abelian gauge theories has been considered in Gracey:18 (); Gracey:19 (); Gracey:20 (); Gracey:21 (); Gracey:22 () for various applications where information is needed on the renormalization group functions of operators in deep inelastic scattering and the -function. That approach is based on the observations of Gracey:23 () using the number of quark flavours, , as the expansion parameter. Rather than use the full QCD Lagrangian one exploits the critical point equivalence with the non-abelian Thirring model, Gracey:23 (),