One-loop MHV Rules and Pure Yang-Mills
Andreas Brandhuber, Bill Spence, Gabriele Travaglini, Konstantinos, Zoubos
TL;DR
This paper addresses the incompleteness of the standard MHV diagram approach in pure Yang-Mills theory by introducing Lorentz violating counterterms that generate missing rational terms in one-loop amplitudes, aligning with known results.
Contribution
It proposes that Lorentz violating counterterms in a specific regularization scheme produce the missing rational terms in one-loop pure Yang-Mills amplitudes, enhancing the MHV framework.
Findings
Counterterms generate missing rational terms in one-loop amplitudes.
A simple counterterm reproduces all-plus helicity amplitudes.
Results agree with known expressions for one-loop amplitudes.
Abstract
It has been known for some time that the standard MHV diagram formulation of perturbative Yang-Mills theory is incomplete, as it misses rational terms in one-loop scattering amplitudes of pure Yang-Mills. We propose that certain Lorentz violating counterterms, when expressed in the field variables which give rise to standard MHV vertices, produce precisely these missing terms. These counterterms appear when Yang-Mills is treated with a regulator, introduced by Thorn and collaborators, which arises in worldsheet formulations of Yang-Mills theory in the lightcone gauge. As an illustration of our proposal, we show that a simple one-loop, two-point counterterm is the generating function for the infinite sequence of one-loop, all-plus helicity amplitudes in pure Yang-Mills, in complete agreement with known expressions.
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QMUL-PH-07-09
**One-loop MHV Rules and Pure Yang-Mills
**
Andreas Brandhuber, Bill Spence, Gabriele Travaglini and Konstantinos Zoubos111{a.brandhuber, w.j.spence, g.travaglini, k.zoubos}@qmul.ac.uk
*Centre for Research in String Theory
Department of Physics
Queen Mary, University of London
Mile End Road, London, E1 4NS
United Kingdom*
Abstract
It has been known for some time that the standard MHV diagram formulation of perturbative Yang-Mills theory is incomplete, as it misses rational terms in one-loop scattering amplitudes of pure Yang-Mills. We propose that certain Lorentz violating counterterms, when expressed in the field variables which give rise to standard MHV vertices, produce precisely these missing terms. These counterterms appear when Yang-Mills is treated with a regulator, introduced by Thorn and collaborators, which arises in worldsheet formulations of Yang-Mills theory in the lightcone gauge. As an illustration of our proposal, we show that a simple one-loop, two-point counterterm is the generating function for the infinite sequence of one-loop, all-plus helicity amplitudes in pure Yang-Mills, in complete agreement with known expressions.
Contents
1 Introduction
One of the success stories arising from twistor string theory [1] (see [2] for a review) has been the development of new techniques in perturbative quantum field theory. These include recursion relations [3, 4], generalised unitarity [5] and MHV methods (see [6] for a review). One of the key motivations of this work is to provide new approaches to study and derive phenomenologically relevant scattering amplitudes. In particular, this requires that one be able to deal with non-supersymmetric theories, and to include fermions, scalars, and particles with masses. A vital first step is to apply these new methods to pure Yang-Mills (YM) theory, and indeed, some of the first new results inspired by twistor string theory involved pure YM amplitudes at tree- [7, 8, 9, 10, 11, 12, 13, 14] and one-loop [15] level.
A recalcitrant issue in this work is the derivation of rational terms in quantum amplitudes. Unitarity-based techniques [16] and loop MHV methods [17] are successful in obtaining the cut-constructible parts of amplitudes; essentially this is because at some level they are dealing with four-dimensional cuts. In principle performing -dimensional cuts generates all parts of amplitudes [18, 19, 20, 21] as long as only massless particles are involved, however these techniques still appear to be relatively cumbersome. Combinations of recursive techniques and unitarity have led to important progress recently [22, 23, 24, 25, 26, 27, 28, 29, 30, 31], but it would be preferable to have a more powerful prescriptive formulation, particularly keeping in mind that applications to more general situations are sought.
A promising development from this point of view is the Lagrangian approach [32, 33, 34]. Here it has been argued that lightcone Yang-Mills theory, combined with a certain change of field variables, yields a classical action which comprises precisely the MHV vertices. A full Lagrangian description of MHV techniques would in principle give a prescription for applying such methods to diverse theories. The next step in developing this is to understand the quantum corrections in this Lagrangian approach. If one directly uses in a path integral the classical MHV action, containing only purely four-dimensional MHV vertices, then it is immediately clear that this cannot yield all known quantum amplitudes. For example, there is no way to construct one-loop amplitudes where the external gluons all have positive helicities, or where only one gluon has negative helicity, as all MHV vertices contain two negative helicity particles (this issue has been recently discussed in [35]). These amplitudes are particular cases where the entire amplitude consists of rational terms. More generally, it seems clear that the vertices of the classical MHV Lagrangian will not yield the rational parts of amplitudes, but only the cut-constructible terms [15]. Important insights into this question can be obtained from the study of self-dual Yang-Mills theory, which has the same all-plus one-loop amplitude of full YM [36, 37, 38] as its sole quantum correction.111In real Minkowski space, this is in fact its single non-vanishing amplitude. An example, relevant to the discussion in this paper, is given in [35] where it was shown how these amplitudes might be obtained from the Jacobian arising from a Bäcklund-type change of variables which takes the self-dual Yang-Mills theory to a free theory.
A discussion of the full Yang-Mills theory in the lightcone gauge has recently been given by Chakrabarti, Qiu and Thorn (CQT) in [39, 40, 41]. These papers employ an interesting regularisation which, importantly, does not change the dimension of spacetime. For this reason, we find it particularly suitable for setting the scene for the MHV diagram method, which is inherently four-dimensional in current approaches. The regularisation of CQT requires the introduction of certain counterterms, which prove to be rather simple in form. What we will show in this paper is that these simple counterterms provide a very compact and powerful way to represent the rational terms in gauge theory amplitudes; specifically, we will demonstrate that the single two-point counterterm contains all the -point all-plus amplitudes. The way this happens is through the use of the new field variables of [32, 33, 34]. Other counterterms will combine with vertices from the Lagrangian and should generate the rational parts of more general amplitudes. Based on the discussion in this paper, we propose that the counterterms, expressed in the field variables which give rise to standard MHV vertices, in combination with Lagrangian vertices, generate the rational terms previously missing from MHV diagram formulations.
The rest of the paper is organised as follows. After giving some background material in section 2, we explicitly derive in section 3 the four point all-plus amplitude from the two-point counterterm of CQT. We follow this by showing that the -point expression, obtained by writing the counterterm in new variables, has precisely the right collinear and soft limits required for it to be the correct all-plus -point amplitude. We present our conclusions in section 4, and our notation and derivations of certain identities have been collected in two appendices.
2 Background
In this section, we first review the classical field redefinition from the lightcone Yang–Mills Lagrangian to the MHV–rules Lagrangian. We then move on to motivate the four–dimensional regularisation scheme we will employ, and argue that it leads directly to the introduction of a certain Lorentz–violating counterterm in the Yang–Mills Lagrangian. We close the section with the remarkable observation that this counterterm provides a simple way to calculate the four–point all-plus one–loop amplitude using only tree–level combinatorics.
2.1 The classical MHV Lagrangian
It seemed clear from the beginning that the MHV diagram approach to Yang-Mills theory must be closely related to lightcone gauge theory. This idea was substantiated by Mansfield [33] (see also [32]). The starting point of [33] is the lightcone gauge-fixed YM Lagrangian for the fields corresponding to the two physical polarisations of the gluon. It was argued convincingly in [33] that a certain canonical change of the field variables re-expresses this lightcone Lagrangian as a theory containing the infinite series of MHV vertices. Some of the arguments in [33] were rather general; these were reviewed in [34], where the change of variables was discussed in more detail, and in particular it was shown how the four- and five-point MHV vertices arise from the change of variables. In this paper we will mainly follow the notation of [34].
The general structure of the lightcone YM Lagrangian, after integrating out unphysical degrees of freedom, is (see appendix A for more details)
[TABLE]
where the gauge condition is with the null vector . Since this Lagrangian contains a vertex, it is not of MHV type. In [33], Mansfield proposed to eliminate this vertex through a suitably chosen field redefinition. Specifically, he performed a canonical change of variables from to new fields , in such a way that
[TABLE]
The remarkable result is that upon inserting this change of variables into the remaining two vertices, the Lagrangian, written in terms of , becomes a sum of MHV vertices,
[TABLE]
The crucial property of Mansfield’s transformation that makes this possible is that, while both and are series expansions in the new fields, has no dependence on the fields while turns out to be linear in . Thus, since the remaining vertices are quadratic in the , the new interaction vertices have the helicity configuration of an MHV amplitude. Mansfield was also able to show that the explicit form of the vertices coincides with the CSW off-shell continuation of the Parke-Taylor formula for the MHV scattering amplitudes, as proposed by [7].
One of the main results of [34] was the derivation of an explicit, closed formula for the expansion of the original fields in terms of the new fields . This was then used to verify that the new vertices are indeed precisely the MHV vertices of [7], at least up to the five-point level. We will now briefly review these results. First, recall that the positive helicity field is a function of the positive helicity field only. It is expanded as follows:
[TABLE]
where . Note that the coordinate is common to all the fields, which is why we have restricted the transformation to the lightcone quantisation surface .
By inserting this expansion into (2.2) and using the requirement that the transformation be canonical, Ettle and Morris succeeded in deriving a very simple expression for the coefficients . After translating to our conventions (see appendix A), they are given by:
[TABLE]
The first few terms in (2.4) are then:
[TABLE]
Similarly, one can write down the expansion of the negative helicity field , which, as discussed above, is linear in , but is an infinite series in the new field . In [34] it was shown that the coefficients in the expansion of are very closely related to those for .222This is perhaps easiest to see [42] by considering that, in the context of SYM, and are part of the same lightcone superfield. The expansion of turns out to be simply
[TABLE]
Thus we see that at each order in the expansion, we need to sum over all possible positions of . Explicitly, the first few terms are:
[TABLE]
Using the above results, it is in principle straightforward to derive the terms that arise on inserting the Mansfield transformation into the two remaining vertices of the theory. For the simplest cases, one can see explicitly that these combine to produce MHV vertices, and some arguments were also given in [33, 34] that this must be true in general.
In supersymmetric theories, the MHV vertices are enough to reproduce complete scattering amplitudes at one loop [43]. However, as we mentioned earlier, for pure YM it is clear that the terms in the MHV Lagrangian (2.3) will not be enough to generate complete quantum amplitudes. For instance, the scattering amplitude with all gluons with positive helicity, which at one loop is finite and given by a rational term, cannot be obtained by only using MHV diagrams, for the simple reason that one cannot draw any diagram contributing to it by only resorting to MHV vertices.333On the other hand, it was shown in [35] that the parity conjugate all-minus amplitude is correctly generated by using MHV diagrams. Another amplitude which cannot be derived within conventional MHV diagrams is the amplitude with only one gluon of negative helicity. Similarly to the all-plus amplitude, this single-minus amplitude vanishes at tree level, and at one loop is given by a finite, rational function of the spinor variables.
The lesson we learn from this is that, in order to apply the MHV method to derive complete amplitudes in pure YM, one should look more closely at the change of variables in the full quantum theory. There are several possible subtleties one should pay careful attention to at the quantum level. First of all, it is possible that the canonical nature of the transformation is not preserved, leading to a non–trivial Jacobian which could provide the missing amplitudes. Another possible source of contributions could come from violations of the equivalence theorem. This theorem states that, although correlation functions of the new fields are in general different from those of the old fields, the scattering amplitudes are actually the same444Modulo a trivial wave-function renormalisation., as long as the new fields are good interpolating fields. These issues were explored in some detail in [35] (see also [34, 42]) where it was shown, for a different (non-canonical) field redefinition, how a careful treatment of these effects can combine to reproduce some of the amplitudes that would seem to be missing at first sight.
Another aim of [35] was to demonstrate how to reproduce one of the above–mentioned rational amplitudes, the one with all–minus helicities, in the MHV formalism. This amplitude is slightly less mysterious than the all–plus amplitude in the sense that one can write down the contributing diagrams using only MHV vertices; however a calculation without a suitable regulator in place would give a vanishing answer, despite the fact that this amplitude is finite. In [35], it was shown, using dimensional regularisation, that the full nonzero result arises from a slight mismatch between four– and ()–dimensional momenta.
It is natural therefore to expect that dimensional regularisation will be helpful also for the problem at hand, which is to recover the rational amplitudes of pure Yang–Mills after the Mansfield transformation. Decomposing the regularised lightcone Lagrangian into a pure four-dimensional part and the remaining –dependent terms, and performing the transformation on the four-dimensional part only, will give rise to several new –dependent terms that can potentially give finite answers when forming loops.
Although this approach shows promise, it is not the one we will make use of in the following. Instead, motivated by the fact that the Mansfield transformation seems to be deeply rooted in four dimensions, we would like to look for a purely four–dimensional regularisation scheme. We now turn to a review of the particular scheme we will use.
2.2 A four–dimensional regulator for lightcone Yang–Mills
In the above we explained why a naïve application of the Mansfield transform leads to puzzles at the quantum level, and discussed possible ways to improve the situation. The conclusion was that, since the missing amplitudes arise from subtle mismatches in regularisation, one should be careful to perform the Mansfield transform on a suitably regularised version of the lightcone Yang–Mills action. Here we will review one approach to the regularisation of lightcone Yang–Mills, which, despite several slightly unusual features, appears to be ideally suited for the problem at hand.
The regularisation we propose to use is inspired by recent work of CQT [39, 40, 41] on Yang–Mills amplitudes in the lightcone worldsheet approach [44, 45]. This is an attempt to understand gauge–string duality which is similar in spirit to ’t Hooft’s original work on the planar limit of gauge theory [46], and aims at improving on early dual model techniques [47, 48]. We recall that one of the main goals in those works is to exhibit the string worldsheet as made up of very large planar diagrams (“fishnets”).
In their recent work, Thorn and collaborators make this statement more precise, using techniques that were unavailable when the original ideas were put forward. It is hoped that, by understanding how to translate a generic Yang–Mills planar diagram to a configuration of fields (with suitable boundary conditions) on the lightcone worldsheet, it will eventually become possible to perform the sum of all these diagrams. This approach to gauge–string duality is thus complementary to that using the AdS/CFT correspondence.
The field content and structure of the worldsheet theory dual to Yang–Mills theory is rather intricate (see e.g. [45]), but for our purposes the details are not important. What is most relevant for us is that one of the principles of this approach is that all quantities on the Yang–Mills side should have a local worldsheet description. This includes the choice of regulator that needs to be introduced when calculating loop diagrams. This requirement led Thorn [49] (see also [50, 51]) to introduce an exponential UV cutoff, which we will discuss in a short while.
Since one of the goals of this programme is to translate an arbitrary planar diagram into worldsheet form (and eventually calculate it), it is an important intermediate goal to understand how to do standard Yang–Mills perturbation theory in “worldsheet–friendly” language. In [39, 40, 41] CQT do exactly that for the simplest case, that of one–loop diagrams in Yang–Mills theory, by analysing how familiar features like renormalisation are affected by the unusual regularisation procedure and other special features of the lightcone worldsheet formalism.
To conclude this brief overview of the lightcone worldsheet formalism, the main point for our current purposes is that it provides motivation and justification for a slightly unusual regularisation of lightcone Yang–Mills, which we will now describe.
Let us momentarily focus on the self–dual part of the lightcone Yang–Mills Lagrangian:
[TABLE]
This action provides one of the representations of self-dual Yang-Mills theory. After transforming to momentum space, we find that the only vertex in the theory is the following (suppressing the gauge index structure):
[TABLE]
As for propagators, following [40], we will use the Schwinger representation:
[TABLE]
In (2.11) is understood to be the appropriate () Wick rotated version of the Minkowski space inner product. For our choice of signature, the latter is
[TABLE]
so that .
We will also make use of the dual or “region momentum” representation, where one assigns a momentum to each region that is bounded by a line in the planar diagram. By convention, the actual momentum of the line is given by the region momentum to its right minus that on its left, as given by the direction of momentum flow555In [40] the flow of momentum is chosen to always match the flow of helicity, but we will not use this convention.. Clearly such a prescription can only be straightforwardly implemented for planar diagrams, which is the case considered in [40]. This is also sufficient for our purposes, since we are calculating the leading single–trace contribution to one–loop scattering amplitudes. Non–planar (multi–trace) contributions can be recovered from suitable permutations of the leading–trace ones (see e.g. [52]).
To demonstrate the use of region momenta, a sample one–loop diagram is pictured in Figure 1.
The “worldsheet–friendly” regulator that CQT employ is simply defined as follows [49]: For a general –loop diagram, with being the loop region momenta, one simply inserts an exponential cutoff factor
[TABLE]
in the loop integrand, where is positive and will be taken to zero at the end of the calculation. This clearly regulates UV divergences (from large transverse momenta), but, as we will see, has some surprising consequences since it will lead to finite values for certain Lorentz–violating processes, which therefore have to be cancelled by the introduction of appropriate counterterms.
Note that has components only along the two transverse directions, hence it breaks explicitly even more Lorentz invariance than the lightcone usually does. This might seem rather unnatural from a field-theoretical point of view, however it is crucial in the lightcone worldsheet approach. Indeed, the lightcone time and (or in practice its dual momentum ) parametrise the worldsheet itself, and are regulated by discretisation; thus, they are necessarily treated very differently from the two transverse momenta which appear as dynamical worldsheet scalars. Fundamentally, this is because of the need to preserve longitudinal () boost invariance (which eventually leads to conservation of discrete ). The fact that the regulator depends on the region momenta rather than the actual ones is a consequence of asking for it to have a local description on the worldsheet.
The main ingredient for what will follow later in this paper is the computation of the one–loop gluon self–energy in the regularisation scheme discussed earlier. This is performed on page 10 of [40], and we will briefly outline it here. This helicity–flipping gluon self–energy, which we denote by , is the only potential self–energy contribution in self–dual Yang–Mills; in full YM we would also have and, by parity invariance, .
There are two contributions to this process, corresponding to the two ways to route helicity in the loop, but they can be easily shown to be equal so we will concentrate on one of them, which is pictured in Figure 2.
In Figure 2, are the outgoing line momenta, is the loop line momentum, and are the region momenta, in terms of which the line momenta are given by
[TABLE]
Remembering to double the result of this diagram, we find the following expression for the self–energy:
[TABLE]
Although we are suppressing the colour structure, the factor of is easy to see by thinking of the double–line representation of this diagram666 For simplicity, we take the gauge group to be .. One of the crucial properties of (2.15) is that the factors of the loop momentum coming from the vertices have cancelled out, hence there are no potential subtleties in the loop integration as . This means that, although for general loop calculations one would have to follow the DLCQ procedure and discretise (as is done for other processes considered in [39, 40, 41]), this issue does not arise at all for this particular integral, and we are free to keep continuous.
To proceed, we convert momenta to region momenta, rewrite propagators in Schwinger representation, and regulate divergences using the regulator (2.13). Employing the unbroken shift symmetry in the region momenta to further set , (2.15) can be recast as:
[TABLE]
Since only appears in the exponential, the integration will lead to a delta function containing , which can be easily integrated and leads to a Gaussian–type integral for . Performing this integral, we obtain (setting , )
[TABLE]
Notice that, had we not regularised using the regulator, we would have obtained zero at this stage. Instead, now we can see that there is a region of the integration (where ) that can lead to a nonzero result. On performing the and integrations, and sending to zero at the end, we obtain the following finite answer:
[TABLE]
Notice that this nonvanishing, finite result violates Lorentz invariance, since it would imply that a single gluon can flip its helicity. Also, it explicitly depends on only the components of the region momenta. Such a term is clearly absent in the tree-level Lagrangian (unlike e.g. the contribution in full Yang–Mills theory), thus it cannot be absorbed through renormalisation – it will have to be explicitly cancelled by a counterterm. This counterterm, which will play a major rôle in the following, is defined in such a way that:
[TABLE]
in other words it will cancel all insertions of , diagram by diagram. Let us note here that, had we been doing dimensional regularisation, all bubble contributions would vanish on their own, so there would be no need to add any counterterms. So this effect is purely due to the “worldsheet–friendly” regulator (2.13).
It is also interesting to observe that in a supersymmetric theory this bubble contribution would vanish777This can in fact be derived from the results of [53], where similar calculations were considered with fermions and scalars in the loop. so this effect is only of relevance to pure Yang–Mills theory.
2.3 The one–loop (++++) amplitude
Now let us look at the all–plus four-point one–loop amplitude in this theory. It is easy to see that it will receive contributions from three types of geometries: boxes, triangles and bubbles. It is a remarkable property888This observation is attributed to Zvi Bern [40]. that the sum of all these geometries adds up to zero. In particular, with a suitable routing of momenta, the integrand itself is zero. Pictorially, we can state this as:
[TABLE]
The coefficients mean that we need to add that number of inequivalent orderings. So we see (and refer to [40] for the explicit calculation) that the sum of all the diagrams that one can construct from the single vertex in our theory, gives a vanishing answer. However, as discussed in the previous section, this is not everything: we need to also include the contribution of the counterterm that we are forced to add in order to preserve Lorentz invariance. Since this counterterm, by design, cancels all the bubble graph contributions, we are left with just the sum of the box and the four triangle diagrams. In pictures,
[TABLE]
where is the known result [54] for the leading–trace part of the four–point all-plus amplitude:
[TABLE]
and the terms in the parentheses clearly cancel among themselves. This leaves the box and triangle diagrams, which are exactly those appearing in the calculation of the parity conjugate amplitude using dimensional regularisation [35], where the bubbles were zero to begin with.
Following [40], we make the obvious, but important for the following, observation that one can change the position of the parentheses:
[TABLE]
where again the terms in the parentheses are zero (by (2.20)). This demonstrates that one can compute the all-plus amplitude just from a tree-level calculation with counterterm insertions (of course, these diagrams are at the same order of the coupling constant as one–loop diagrams because of the counterterm insertion). This remarkable claim is verified in [40], where CQT explicitly calculate the 10 counterterm diagrams and recover the correct result for the four-point amplitude (see pp. 22-23 of [40])999In practice, these authors choose to insert the self-energy result (2.18) in the tree diagrams, so what they compute is minus the all–plus amplitude..
This result, apart from being very appealing in that one does not have to perform any integrals (apart from the original integral that defined the counterterm) so that the calculation reduces to tree–level combinatorics, will also turn out to be a convenient starting point for performing the Mansfield transformation. Specifically, our claim is that the whole series of all-plus amplitudes will arise just from the counterterm action. In the following we will show how this works explicitly for the four-point all-plus case, and then we will argue for the -point case that the corresponding expression derived from the counterterm has all the correct singularities (soft and collinear), giving strong evidence that the result is true in general.
3 The all-plus amplitudes from a counterterm
Having reviewed the relevant new features that arise when doing perturbation theory with the worldsheet–motivated regulator of [49], we now have all the necessary ingredients to perform the Mansfield change of variables on the regulated lightcone Lagrangian. In this section, we will carry out this procedure. We will first regulate lightcone self–dual Yang–Mills, which, as discussed, will require us to introduce an explicit counterterm in the Lagrangian. Then we will perform the Mansfield transformation on the original Lagrangian (converting it to a free theory). We will then show that, upon inserting the change of variables into the counterterm Lagrangian, we recover the all–plus amplitudes as vertices in the theory.
3.1 Mansfield transformation of
As we saw, the “worldsheet-friendly” regularisation requires us to add a certain counterterm to the lightcone Yang–Mills action, required in order to cancel the Lorentz-violating helicity–flipping gluon selfenergy. As mentioned previously, the calculation of the all–plus amplitude can be tackled purely within the context of self-dual Yang–Mills, which we will focus on from now on. We see that, as a result of this regularisation, the complete action at the quantum level becomes:
[TABLE]
where is the classical Lagrangian for self-dual Yang-Mills introduced in (2.9). Although CQT do not write down a spacetime Lagrangian for , it is easy to see that the following expression would have the right structure:
[TABLE]
This expression depends explicitly on the dual, or region, momenta. In (3.2) we have made use of the simplest way to associate region momenta to fields, which is to assign a region momentum to each index line in double–line notation [46], and thus a momentum , to each of the indices of the gauge field (now slightly extended into a dipole, as would be natural from the worldsheet perspective, where an index is associated to each boundary). Since each line has a natural orientation, the actual momentum of each line can be taken to be the difference of the index momentum of the incoming index line and the outgoing index line. So the momentum of is taken to be . As discussed above, this assignment can only be performed consistently for planar diagrams, which is sufficient for our purposes.
Clearly, the structure of (3.2) is rather unusual. First of all, it depends only on the antiholomorphic () components of the region momenta, and so is clearly not (lightcone) covariant. Even more troubling is the fact that it does not depend only on differences of region momenta, but also on their sums. Since each field thus carries more information than just its momentum, is a non–local object from a four–dimensional point of view (although, as shown in [40], it can be given a perfectly local worldsheet description).
Leaving the above discussion as food for thought, we will now rewrite (3.2) in a more conventional way that is most convenient for inserting into Feynman diagrams,
[TABLE]
In this expression, which can be thought of as the zero–mode or field theory limit of (3.2), all the region momentum dependence is confined to the polynomial factor . This vertex, inserted into tree diagrams, would exactly reproduce the effects of the counterterm pictured in (2.19). Although (3.3) still exhibits some of the apparently undesirable features we discussed above, the calculations in [40] demonstrate that, after summing over all possible insertions of this term, the final result is covariant and correctly reproduces the all–plus amplitudes101010Note that similar–looking treatments using index momenta instead of line momenta for vertices, but which in the end sum up to covariant results have appeared in the context of noncommutative geometry (see e.g. [55]). Although it is possible to write e.g. (3.2) in star–product form, at this stage it is not clear whether that is a useful reformulation.. Therefore, we believe that its problematic properties are really a virtue in disguise, and (as we will see explicitly) they seem to be crucial in obtaining the full series of –point all–plus amplitudes from the Mansfield transformation of a single term.
We are now ready to perform the Mansfield change of variables. In the spirit of the discussion earlier, we will perform the transformation on the classical part of the action only:
[TABLE]
Hence the classical part of the action has been converted to a free theory. Without a regulator, this would be the whole story. However we now see that, within the particular regularisation we are working with, the full Lagrangian contains one extra, one–loop piece, given by in (3.3), which is quadratic in the positive helicity fields . To complete the Mansfield transformation, we will clearly need to expand this term in the new fields , using the Ettle–Morris coefficients (2.4).
Since depends only on the holomorphic fields, we will only need the expansion of in terms of given in (2.4). As a first check that leads to the right kind of structure, note that since depends only on the holomorphic fields, all the new vertices are all–plus. Thus, the full action, when expressed in terms of the fields, takes the schematic form:
[TABLE]
In the next section we will calculate the four–point term and demonstrate that, when restricted on–shell, it reproduces the known form (2.22) for the all–plus amplitude.
3.2 The four–point case
To begin with, we focus on the derivation of the four-point all-plus vertex, whose on-shell version will give us the four-point scattering amplitude. We will thus expand the old fields in the counterterm (3.3) (or (3.2)) up to terms containing four -fields.
When inserting the Ettle–Morris coefficients into (3.3), one has to sum over all possible cyclic orderings with which this can be done. A complication is that now the counterterm itself depends on the ordering. In other words, we need to sum over all the ways of assigning dual momenta to the indices. Schematically, the inequivalent terms that we obtain are:
[TABLE]
where the terms on the first line arise from doing two quadratic substitutions and those on the second from doing one cubic substitution. All the other possibilities are related by cyclicity of the trace. For definiteness, let us now write down what one of these terms means explicitly:111111We suppress the overall factor of until the end of this section. Also, the integrals are implicitly taken to be on the quantisation surface .
[TABLE]
The reason this particular combination of ’s appears here is that, given the ordering we chose, after the Mansfield transformation the counterterm ends up being on leg 4, and its line bounds the regions with momenta and . This is represented pictorially in Figure 3.
Although Figure 3 might suggest that there is a propagator between the counterterm insertion and the location of the original , which has now split into three ’s, this is of course not the case since the whole expression is a vertex at the same point. We have drawn the diagram in this fashion to emphasise which leg the counterterm is located on after the transformation. On the other hand, this vertex is nonlocal (as discussed above, it was nonlocal even in the original variables, but this is now compounded by the Mansfield coefficients, which contain momenta in the denominator), so this notation serves as a useful reminder of that fact.
It is interesting to note that (3.7) is essentially the same expression as the sum of the two channels with the same region momentum dependence that appear in CQT’s calculation of this amplitude using tree–level diagrammatics (compare with Eq. 83 in [40]), which we illustrate in Fig. 4.
Thus we have a picture where one post–Mansfield transform vertex (with ’s) effectively sums two tree–level pre–transformation (with ’s) Feynman diagrams. This is a first indication that our calculation of the all–plus vertex can be mapped, practically one–to–one, to that of the all–plus amplitude on pp. 22-23 of [40].
Another type of contribution to the vertex arises when we transform both of the ’s in . One of the two terms that we find is:
[TABLE]
This contribution can also be mapped to one of the two terms with bubbles on internal lines in CQT.
We can now tabulate all the terms that we obtain in this way by making the schematic form (3.6) precise. Since the delta–function and trace over parts are the same for all these terms, in Table 1 we just list the rest of the integrand.
To obtain the final form of the vertex, we are now instructed to sum over all these contributions. Thus we can write
[TABLE]
where is given by the following expression:121212For the sake of brevity we omit a subscript in the region momenta appearing in (3.10).
[TABLE]
Comparing this to the expected answer (2.22), we see that the (quadratic) antiholomorphic momentum dependence should arise from the various factors in (3.10). In [40], CQT start from essentially the same expression and demonstrate that it gives the correct result for the all-plus amplitude. Therefore, following practically the same steps as those authors, we can easily see that we obtain the expected answer. However, since we would like to find the full vertex , we will need to keep off–shell information, and so we will choose a slightly different route.
The main complication in bringing (3.10) into a manageable form is clearly the presence of the region momenta. We would like to disentangle their effects as cleanly as possible. Therefore, our derivation will proceed by the following steps:
First, we will show that (3.10) can be manipulated so that the quadratic dependence on region momenta drops out, leaving only terms linear in the region momenta. 2. 2.
Second, we will decompose the resulting expression into a part that depends on the region momenta and one that does not. The –dependent part turns out to have a very simple form, and vanishes on–shell. 3. 3.
Finally, we will show that the –independent part reduces to the known amplitude.
For the first step, we will need the following identity, which is proved in appendix B:
[TABLE]
Also, using the shorthand notation : we note the following very useful identity:
[TABLE]
where and . Noting that, for , , we can use this to rewrite all the region momentum combinations appearing in (3.10) in the following way:
[TABLE]
where we have introduced the notation . We have thus expressed all the quadratic region momentum dependence in terms of the common factor , and, given (3.11), it is clear that this contribution will vanish.131313One could have chosen a different combination of the ’s, but we find the symmetric choice in (3.13) convenient.
After this step, we are left with an expression which is linear in the region momenta. We will now proceed in a similar way, and rewrite all the expressions that contain in terms of a suitably chosen common factor:
[TABLE]
In appendix B we show that the total coefficient of the common factor is
[TABLE]
where is the full covariant momentum squared, and . Thus we see that the complete dependence on the region momenta can be rewritten as follows:
[TABLE]
It is rather satisfying that the region momentum dependence of the vertex takes this simple form, which clearly vanishes when the external legs are on–shell, and thus will not contribute to the all–plus amplitudes.
Having completely disentangled the region momenta from the actual momenta , we will now focus on the terms containing only the latter, which were produced during the decompositions in (3.14). After a few simple manipulations, they can be rewritten as141414We write .
[TABLE]
This expression, together with (3.16) is our proposal for the off–shell four–point all–plus vertex that should be part of the MHV-rules formalism at the quantum level. It would be very interesting to elucidate its structure and bring it into a more compact form. For the moment, however, we will be content to demonstrate that (3.17) is equal on shell to the sought–for amplitude.
To that end, we will follow a similar approach to CQT, and rewrite all the holomorphic spinor brackets in terms of the following three: . To achieve this, we use momentum conservation and a certain cyclic identity (see appendix A) to write
[TABLE]
In a similar way, we can show that
[TABLE]
Collecting all the terms together, and manipulating the resulting expressions, it is straightforward to show that (3.17) simplifies to just
[TABLE]
where we use the notation [34] . Converting to the usual antiholomorphic bracket notation, we rewrite (3.20) as
[TABLE]
Note that so far this expression is completely off shell. We will now show that on shell it reduces to the known result (2.22). In doing this we will keep track of the terms that appear when applying momentum conservation in the form
[TABLE]
These terms are collected in appendix B.
We start by rewriting each of the terms in the last two lines of (3.21) as follows
[TABLE]
We also transform the term using the Schouten identity and also momentum conservation,
[TABLE]
and add up all contributions to the term, which are
[TABLE]
Converting to the spinor bracket, the first of these terms is
[TABLE]
while the remaining terms from (3.23) and (3.24) combine to give
[TABLE]
(where we suppress an overall ) and we see that (3.27) cancels the second term in (3.25), thus showing that (3.26) is the complete on-shell answer. Reintroducing all the prefactors, we thus find that the amplitude is
[TABLE]
Now note that, as discussed in appendix A, in order to convert to the usual Yang–Mills theory normalisation we need to send . We conclude that gives precisely the result (2.22) for the all–plus scattering amplitude.
3.3 The general all–plus amplitude
We have just given an explicit derivation of the four point all-plus amplitude, from the two-point counterterm (3.3). We will argue in the following that this two-point counterterm contains all the all-plus amplitudes.
First, we can see immediately that the counterterm (3.3) has the right kind of structure. Consider the –point all–plus amplitude [56]:
[TABLE]
In terms of spinor brackets this amplitude has terms of the form . A quick look at the Ettle-Morris coefficients shows that, for an –point vertex coming from , they contribute exactly powers of the spinor brackets . Furthermore, there are exactly two powers of coming from the counterterm Lagrangian – one for each power of . Thus the general structure of is appropriate to reproduce (3.29).
Pictorially, we can represent the general –point amplitude, arising from the counterterm in the new variables, as in Figure 5.
Thus we can write this –point all–plus vertex as follows:
[TABLE]
Focusing only on the relevant part of the above expression, and ignoring all coefficients, the general structure we obtain is the following:
[TABLE]
where we have extracted the denominator at the expense of introducing the two missing holomorphic factors and in the numerator. We also made use of the fact that
[TABLE]
applied to the components, to rewrite the two sums in the numerator in terms of the ’s (this gives rise to a minus which we suppress).
It is easy to verify that, for , this sum reproduces the 6 contributions that appeared in the four–point case, and (as we explicitly showed above) combined to give the expected answer. Therefore, we would like to propose that the vertex (3.31) will reduce on–shell to an expression proportional to (3.29). We will not attempt to prove this statement here151515It is perhaps interesting to remark that the proof would involve converting the double sum in (3.31) to the quadruple sum in (3.29)—a state of affairs which has appeared before in a rather different context [20]., but will instead move on to study the general properties of the -point expression (3.30).
Whilst the explicit calculation for the four point case was rather involved as we saw earlier, the study of the general properties of the –point amplitudes proves much simpler. In particular, we will show that the collinear and soft limits of the expressions proposed for the –point case can be very easily shown to be correct. Let us start by introducing some simplifying notation. One can write the change of variables for the field as
[TABLE]
where
[TABLE]
and generally
[TABLE]
(for simplicity, we are dropping inconsequential constant factors in this discussion). This notation is similar to that of [34]. Integrations and the insertion of suitable delta functions are understood, and can be illustrated by comparing the short-hand expressions above with the full equations given earlier. It will prove convenient to define
[TABLE]
We will use the expression in the following, where the dot in the first placemark in the means that one substitutes in that place the negative of the sum of the other momenta. Then the result which we have proved above for the four point amplitude can be expressed as
[TABLE]
or very simply
[TABLE]
It is clear that the general conjecture that all the –point all plus amplitudes are generated from the two-point counterterm (3.3) translates into the proposal that the -point all-plus amplitude is given by
[TABLE]
Let us now show that the expression on the right-hand side of (3.39) has precisely the same soft and collinear limits as the known amplitude on the left-hand side.
Collinear limits
Under the collinear limit
[TABLE]
the -point amplitude behaves as
[TABLE]
where we relabel after the limit is taken (the and factors involve momenta rather than spinors, which is why the -dependent factor is , rather than the conventional ).
Consider the behaviour of the right-hand side of (3.39) under the limit (3.40). The first point is that if the indices lie on different ’s, then there are no poles generated in this collinear limit. This is clear from the explicit expressions for the ’s in (3.35). Thus we may ignore any terms of this type. It is then immediate from the explicit forms of the ’s that
[TABLE]
for any , with (the first index in never contributes in a collinear limit, as one can see from the conjecture (3.39)). Thus we see that the expressions have the right sort of collinear behaviour. It is straightforward to see that the coefficients in (3.39) also get relabelled correctly in the collinear limit; they are not explicitly involved as they refer to pairs of momenta attached to different fields, and as we saw, these do not contribute.
It is then immediate to see that the summation over the products of ’s in (3.39) reduces correctly in the collinear limit to the required summation over products of ’s with one fewer leg in total. Hence the proposal (3.39) for the amplitude has precisely the same collinear limits as the physical amplitude.
Soft limits
We also find that there is a simple derivation of the soft limits of the expression in (3.39). In the soft limit
[TABLE]
the -point amplitude behaves as
[TABLE]
where we assume cyclic ordering as usual, so that, for example, . The soft function is given in terms of the momentum brackets by
[TABLE]
The functions have a simple behaviour under soft limits. One has immediately that in the soft limit ,
[TABLE]
for (with ). For the soft limits corresponding to the case missing in the above, we need the results
[TABLE]
which follow from the definitions of the ’s, and
[TABLE]
which follows from the cyclic identity . Finally, from relabelling the ’s we have in the soft limit that . Then it follows that in the soft limit
[TABLE]
as required.
Again, it is then easy to see that the summation over the products of ’s in (3.39) reduces correctly in the soft limit to the required summation over products of ’s with one fewer leg in total. Hence the proposal (3.39) for the amplitude has precisely the same soft limits as the physical amplitude.
4 Discussion
Whilst new, twistor-inspired methods for calculating amplitudes in gauge theory have led to much progress, the lack of a systematic action-based formulation which incorporates these new ideas has been an impediment to further developments. MHV diagrams have the two advantages of being closely allied to the twistor picture, as well as providing an explicit realisation of the dispersion and phase space integrals fundamental to unitarity-based methods. However, without an action formalism, standard MHV methods have so far been mainly restricted to massless theories at one-loop level, and to the cut-constructible parts of amplitudes.
The advent of a classical MHV Lagrangian for gauge theory, derived from lightcone YM theory [32, 33, 34], provides the basis for transcending these limitations. In order for this to be realised, it is necessary to describe the quantum MHV theory. What we have done in this paper is to investigate this quantum theory. Using the regularisation methods of [39, 40, 41], we have provided arguments that the simplest one-loop counterterm in the quantum MHV theory – a two point vertex – provides an extraordinarily concise generating function for the infinite sequence of one-loop, all-plus helicity amplitudes in YM theory. We showed this by explicit calculation for the four-point case, and then proved that the soft and collinear limits of the conjectured -point amplitude precisely matched those of the correct answer.
We would like to emphasise that the simplicity of our approach — which reduced the calculations of the loop amplitudes we considered to tree–level algebraic manipulations— is largely due to the four–dimensional nature of the regularisation scheme we employed. By staying in four dimensions, we preserve the appealing features of the inherently four–dimensional field redefinition of [32, 33].
Based upon this result, it is very natural to conjecture that the full quantum YM theory is correctly described by this quantum MHV Lagrangian. The correct ingredients appear to be present. For example, in the approach of [39, 40, 41] there arise one-loop counterterms with helicities . We studied the counterterm in this paper, arguing that when expressed in the variables this generates the full set of all-plus amplitudes. Transforming the counterterm to variables will generate an infinite sequence of single–minus vertices. There will be other contributions to single-minus vertices from combinations of all-plus vertices and MHV vertices. It would be surprising if the combined contributions of these did not lead to the correct YM single-minus expressions. Certainly all of these have the correct powers of spinor brackets for this to be the case.
Transforming the and counterterms to variables will lead to new contributions to MHV vertices161616In the MHV case there are additional counterterms noted in [41] which may also need to be taken into account in future discussions.. The MHV vertices from the classical MHV Lagrangian only generate the cut-constructible parts of YM loop amplitudes, such as the one-loop MHV amplitude. These new contributions might be expected to lead to the missing, rational parts. This would also potentially explain why in [57] the combination of all-plus vertices with MHV tree vertices did not yield the correct single-minus amplitudes – these additional MHV contributions are missing.
Further evidence for the conjecture that the quantum MHV Lagrangian is equivalent to quantum YM theory would be welcome. One could start with seeking explicit proofs of the above proposals. One can also investigate beyond massless one-loop gauge theory – an advantage of the Lagrangian approach is that the inclusion of masses, and of fermions and scalars, is in principle clear. There are other issues raised by this work. It is plausible that the potential quantum versions of the twistor space formulations of gauge theory [58, 59, 60] are most likely to be allied to the quantum theory discussed here – one simple reason for believing this is that the regularisation employed here keeps one in four dimensions. Perhaps there are simple twistor space analogues of the counterterms discussed above.
Finally, although for our purposes the lightcone worldsheet approach to perturbative gauge theory provided simply the motivation for a particular choice of regularisation scheme, we believe that it would be fruitful to further explore possible connections between that framework and the twistor string programme.
Addendum: We would like to thank Paul Mansfield and Tim Morris for having informed us that they have recently been pursuing research related to that presented in this paper. Their work, which is complementary to ours in that it employs dimensional regularisation, has now appeared in [61].
Acknowledgements
It is a pleasure to thank Paul Heslop, Gregory Korchemsky, Paul Mansfield, Tim Morris and Adele Nasti for discussions. We would like to thank PPARC for support under the Rolling Grant PP/D507323/1 and the Special Programme Grant PP/C50426X/1. The work of GT is supported by an EPSRC Advanced Fellowship EP/C544242/1 and by an EPSRC Standard Research Grant EP/C544250/1.
Appendix A Notation
Lightcone conventions
Here we summarise our lightcone conventions. We start off by introducing lightcone coordinates
[TABLE]
We also have , , and so on. The scalar product between two vectors and is written as
[TABLE]
We choose as our lightcone time coordinate, therefore the lightcone gauge used in this paper is defined by
[TABLE]
This condition can be written as , where is a constant null vector, chosen to have components (hence , ).
To any four-vector we associate the bispinor defined by
[TABLE]
We also define holomorphic and anti-holomorphic spinors as
[TABLE]
from which it follows that
[TABLE]
This is of course consistent with the on-shell condition . Furthermore, comparing (A.4) and (A.6) and choosing as specified earlier, we see that a generic off-shell vector can be decomposed as
[TABLE]
where
[TABLE]
(A.7) and (A.8) are the familiar decompositions of off-shell vectors in the MHV literature [62, 17, 63, 15].
The off-shell holomorphic spinor product is defined as:
[TABLE]
whereas for the antiholomorphic spinors we define
[TABLE]
In these conventions, one finds
[TABLE]
or, in the case where and are on shell, . In the standard QCD literature conventions it is customary to define ; this can be obtained by simply re-defining the inner product of two anti-holomorphic spinors, , to be the negative of the right hand side of (A.10).
Useful identities
The form (A.9) is very convenient for deriving identities for that also involve the components. For instance, one has:
[TABLE]
It is also easy to see how to apply momentum conservation, take say , and substitute
[TABLE]
Then we have
[TABLE]
We have also used the momentum bracket notation from [34]
[TABLE]
Lightcone Yang–Mills action
Here we give the form of the lightcone Yang–Mills action that we use in this paper. As discussed in more detail in [35], starting from the YM Lagrangian , imposing the lightcone gauge (A.3), and integrating out the component which appears quadratically, the final lightcone theory contains only the two physical components and [64, 65, 66], which we associate with positive and negative helicity respectively. The Lagrangian takes the simple form (2.1)
[TABLE]
with
[TABLE]
Note that, in agreement with CQT, we have used the normalisation . In order to convert to the usual conventions for Yang–Mills theory, we therefore need to rescale .
Relation to the notation of CQT
To compare our notation to that of [39, 40, 41], note that we employ outgoing momenta instead of incoming, therefore the all–plus amplitudes in these works would be all–minus from our perspective, and should thus be conjugated when comparing. Also, our time evolution coordinate is taken to be rather that , which (among other changes) implies that of CQT becomes . Our metric is also taken to have opposite signature to that in CQT. Finally, CQT define momentum brackets and , which are just our and brackets respectively.
Appendix B Details on the four–point calculation
In this appendix we prove two results that were used in section 3, namely equations (3.11) and (3.15). To make the expressions more compact, instead of momentum brackets we use the following notation:
[TABLE]
The variables satisfy the simple relation:
[TABLE]
while momentum conservation is applied as
[TABLE]
Also, to minimise clutter, in this appendix we use the notation .
Proof of the quadratic identity
In order to show (3.11), it is convenient to divide out by the factor (which is there anyway in (3.10)) in order to bring it to the form
[TABLE]
Expanding out the two last terms in (B.4) as
[TABLE]
we apply momentum conservation on each of the four components of the first term of (B.5), in the following way:
[TABLE]
Clearly these transformations have been chosen to cancel the first four terms in (B.4). Collecting the remaining terms, we obtain
[TABLE]
thus showing (3.11).
Proof of the linear identity
We will now outline the proof ot the linear (in region momenta) identity (3.15). Converting it to the notation used in the appendix, and performing simple manipulations, we find (suppressing the overall factor):
[TABLE]
Similarly to the previous case, we will rewrite the second line in the final expression in such a way that we completely cancel all the terms in the first line. To do that we use
[TABLE]
and
[TABLE]
What remains after substituting these is
[TABLE]
Now we collect various terms together to rewrite as
[TABLE]
Clearly the term on the last line vanishes by momentum conservation. We now restore all labels to write the final result as
[TABLE]
where we used that (and similarly for ), and where . Using momentum conservation on both terms, we rewrite them as
[TABLE]
For each momentum we have that , therefore we can rewrite the above as
[TABLE]
The term vanishes, hence, noticing also that , we conclude that
[TABLE]
Off-shell terms in the four-point case
For completeness, we also give the form of the off-shell terms that arose in the manipulations leading to (3.26).
Using the notation they are :
[TABLE]
This expression, together with in (3.16), should be added to (3.26) in order to recover a fully off-shell four–point vertex.
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