Loop Amplitudes in an Extended Gravity Theory

We extend the $S$-matrix of gravity by the addition of the minimal three-point amplitude or equivalently adding $R^3$ terms to the Lagrangian. We demonstrate how Unitarity can be used to simply examine the renormalisability of this theory and determine the $R^4$ counter-terms that arise at one-loop. We find that the combination of $R^4$ terms that arise in the extended theory is complementary to the $R^4$ counter-term associated with supersymmetric Lagrangians.


Introduction
There are different approaches to defining a quantum field theory. One approach is to specify a local Lagrangian density containing fundamental fields and then use the associated Feynman diagrams to determine the scattering amplitudes that contribute to the S-matrix. Alternatively, one can use the on-shell amplitudes to define the theory [1]. In this approach singular structure and symmetries are crucial to constrain and ultimately compute the scattering amplitudes. With enough constraints, higher point amplitudes may be derived from a limited set of lower point functions [2]. In the minimal case all n-point amplitudes may be constructed from only the threepoint amplitudes plus the requirement of factorisation. In [3] such theories were described as fully constructible. Yang-Mills and gravity both have fully constructible tree amplitudes. In [4] we showed that in a theory of gravity deformed by additional three point vertices the leading deformation to the tree amplitudes was also fully constructible.
In this article we examine the one-loop amplitudes for both extended Yang-Mills and gravity and in particular their ultra-violet structure. In agreement with long established results we find that the extended Yang-Mills theory is renormalised [5,6,7,8]. For gravity, we find that additional counter-terms/amplitudes are required. The use of four-dimensional unitarity techniques gives relatively easy access to the ultra-violet structures of these theories.

Structure of the Amplitudes
A one-loop amplitude in a theory of massless particles can be expressed, after a Passarino-Veltman reduction [9], in the form where the I i m are m-point scalar integral functions and the a i etc. are rational coefficients. C is the set of box integral functions with all allowed partitions of the external legs between the corners. For a color-ordered Yang-Mills amplitude the allowed partitions respect the cyclic ordering of the legs. Similarly D and E are the sets of triangle and bubble integral functions. R n is a purely rational term. The integral functions depend upon the number of "massive" legs (or more correctly legs with non-null momenta) so, for example, the triangle functions come in three types : with either one, two or three massive legs. This form is an expansion in terms of the integral functions appearing and so is useful when computing the coefficients from the cut singularities of the amplitude.
Alternately, we can re-express the one-loop amplitude for pure Yang-Mills or gravity in a form which highlights the singular structure of the amplitude, where I n contains the soft-singular Infra-Red (IR) [10] terms of the amplitude and is for a colour ordered gluon amplitude where c Γ = (4π) ǫ Γ 2 (1−ǫ)Γ(1+ǫ)/Γ(1− 2ǫ). For a graviton scattering amplitude [11,12], Within the decomposition of (1) the contributions to I n arise from the box integral functions and the one and two mass triangle integral functions. G n is of the form for gluon scattering amplitudes and for graviton scattering. Within the integral basis decomposition (1), the G n arise from the bubble integral functions. The G n terms contain both the collinear IR singular terms and the UV divergences. The function F n contains the finite transcendental functions. These arise from both the box integral functions and from the three-mass triangle integral function. R n is the remaining finite rational term. The coefficients of the integral functions can be determined using four dimensional generalised unitarity techniques from the on-shell amplitudes. Computing R n from unitarity requires using d = 4 − 2ǫ tree amplitudes.
We will find that the form of the IR singularities is not altered in the extended theories: as might be expected by naive power counting.

Yang-Mills Case
Before looking at gravity theories, we consider the case of gluon scattering in pure Yang-Mills. We will consider a color ordered formalism and examine the color ordered partial amplitudes which have cyclic symmetry. The full amplitude can be reconstructed by multiplying by the color factors and summing over permutations. We also restrict to the leading in N c color ordered partial amplitude for which the one-loop leading in color amplitudes have a factor of N c .
We define the theory from the fundamental three point amplitudes, where h i is the helicity of the i-th leg. For gluons the helicity is ±1. Three point amplitudes vanish for real momenta but may be non-zero if we allow complex momenta. If we express the momenta in spinor variables p αα = λ αλα then amplitudes become functions of the bilinears a b = ǫ αβ λ α a λ β b and [a b] = −ǫαβλα aλβ b . Since, for a three point amplitude of massless particles we must have s ab = a b [b a] = 0, there are two possibilities: Consequently we can build three point amplitudes either from i j or [i j] but not both. Under scaling λ i −→ t i λ i ,λ i −→ t −1 iλ i the amplitude must scale as t −2h i . Finally requiring that the amplitude vanishes for real momenta leads to the following unique non-zero three-point gluon amplitudes, The first two amplitudes are the well known MHV ("Maximally Helicity Violating") and MHV amplitudes. They form the top elements in a multiplet involving all the helicities of N=4 super-Yang-Mills. The parameter α is dimensionful with mass dimension minus two, ie. α = c/M 2 where c is dimensionless and M is some mass scale 1 and the α expansion can be considered as an expansion in inverse powers of the mass M.
In this letter we consider expanding the theory by including the α-vertices in addition to the MHV vertices. The amplitudes in this theory can then be expanded as a power series in α, where A α 0 n is the usual Yang-Mills amplitude. From a Lagrangian field theory viewpoint we are extending the theory by where α ′ = −αg/3 [13]. Having defined the three-point vertices, factorisation can be used [14,4] to obtain the leading four-point tree amplitudes: (These expressions match those obtained from Feynman diagram calculations [15], or color-kinematics duality [13] or scattering equations [16]) The combinations K ++++ and K −+++ carry all the necessary spinor weight of the amplitude with |K ++++ | = |K −+++ | = 1 for real momenta. The factor K ++++ has manifest cyclic symmetry but is also fully crossing symmetric since Similarly K −+++ has manifest flip-symmetry but is also invariant under exchange of the positive helicity legs, 2 ↔ 3 etc. These factors can be written in many ways, e.g.
For the pure Yang-Mills theory, all tree amplitudes can be constructed using factorisation from the three-point trees [17]: i.e. the theory is "constructible" using the definition of [3]. The leading deformed amplitudes A α n also can also be constructed in this way, however amplitudes beyond this leading deformation are not constructible purely from factorisation. This will be pursued further in the context of gravity theories later.
We now wish to determine the one-loop amplitudes in the extended theory to leading order in α. Unitarity methods have proven very efficient in determining one-loop amplitudes using the on-shell tree amplitudes.
Working in four dimensions the simplicity of the four-dimensional trees greatly simplifies the calculation of the coefficients of the integral functions, but does not allow us to compute the finite rational terms R. Since we are mainly interested in the UV singularities, which come with an accompanying ln(µ 2 /s), four dimensional two-particle cuts suffice [18].

Noting that
since there are no non-vanishing four dimensional cuts for these amplitudes, the non-vanishing α 1 amplitudes are the all-plus and the single minus.
Calculating the s-channel two-particle cut for the all-plus amplitude, as shown in fig.1, gives Figure 1: The two-particle cut of the all-plus amplitude.
where we also have a configuration where the α vertex is on the right hand side. Manipulating eq. (16) using we have Now so Now, rather than perform the integration over the cut momenta we recognise this as the cut of a covariant integral which we can evaluate: where we only keep terms with a s-channel cut in the resultant integral [19,20]. Consequently we replace C 2 by a triangle integral with linear numerator where I 2 (s) is the scalar bubble integral function and I 3 (s) is the one-mass scalar triangle integral function which only depend on the kinematic variable s, we obtain Doubling this to account for inserting the F 3 operator on the opposite side of the cut and combining with the t-channel cut leads to the amplitude where we have used Note that there are no box functions in this amplitude and the triangle functions generate the soft part of the IR term. The overall coefficient of ǫ −1 is Figure 2: The bubbles for single minus which is proportional to the tree and so the one-loop UV infinity leads to a renormalisation of the αF 3 term.
We also compute the one-loop contribution to the single minus amplitude A (1) (1 − , 2 + , 3 + , 4 + ). There are three non-zero configurations as shown in fig. 2. The first of these is This can be rearranged to which is the two particle cut of a quadratic box. Replacing this by a covariant integral and only keeping terms with a s-channel cut we obtain The other terms are a little more complex (requiring a quartic box integral) and yield Combining these gives The box functions combine with the triangle functions to generate the IR singular terms and the finite transcendental function, where and we find no corrections to the IR structure. We also have the UV terms: the coefficient of ǫ −1 in these terms is which matches the singularity found for the A(1 + 2 + 3 + 4 + ) case.
Equations (26) and (34) are the singularities in the bare amplitudes. The amplitudes contain universal collinear IR singularities [10]: where β 0 = 11N c /3. We determine the UV divergence by first subtracting these from the bare singularity. When renormalising the theory there must be a simultaneous renormalisation of g 2 and α. The renormalisation of g 2 is unaltered since it is determined by the α 0 amplitudes.
so that (reinserting g and N c ) This value of 7N c /3 matches previous calculations of the anomalous dimension of Yang-Mills extended by the F 3 operator [5, 6, 7, 8].

Extended Gravity Amplitudes
We now consider extending gravity by additional three point vertices. For gravitons with h = ±2 the possible three point amplitudes are 2 From a Lagrangian viewpoint we are considering the theory The non-vanishing four-point α 0 tree is These four-point amplitudes due to a R 3 term have been computed using field theory methods long ago [21] but can also be obtained from the threepoint amplitudes by factorisation [14,4] . These expressions also appear as the UV infinite pieces of both two-loop gravity in four dimensions [22,23] and one-loop gravity in six dimensions [24]. We evaluate the one-loop amplitude for the all-plus amplitude via the two particle cuts Arranging the trees carefully we obtain where both integrals yield the same resultant integral function. After doubling this to account for inserting the R 3 operator on the opposite side of the cut, this can be rewritten as The triangle functions correctly generate the IR terms and so the the amplitude can be expressed as with F 4 = 0 and The coefficient of ǫ −1 is (49) In Einstein gravity the IR singularity is of the form s ln(s)/ǫ [11, 12] and the additional vertex will not affect this by power counting. Therefore the rational ǫ −1 singularities in eq. (49) represent the ultra-violet divergence. Unlike the Yang-Mills case this is not a renormalisation of the cubic vertex but must be cancelled by the addition of a four-point vertex produced by a higher-dimension local operator.
As a consistency check we also consider the single minus amplitude. The s-channel bubble in this case has three configurations: These give contributions to the coefficients of the bubble integral functions. We find giving the overall coefficient of I 2 (s) to be Extracting the UV divergence we find and so this amplitude has no UV divergence. In summary, the UV infinities for the four-point one-loop amplitudes are, (re-inserting the appropriate factors) Unlike the Yang-Mills case, the UV infinity is not removed by a renormalisation of the three-point vertex but requires the addition of a four-point vertex which acts as a counterterm. In the following section we examine the counterterm that is required.

R 4 operators
From [25] the general R 4 counterterm in arbitrary dimension is where and the combination vanishes on-shell in any dimension due to it being proportional to the Euler form In D = 4 these R 4 tensors reduce to two independent tensors. One of these is the square of the "Bel-Robinson" tensor [26] which was shown to be consistent with supersymmetry and thus became a candidate counterterm for supergravity theories [27]. In higher dimensions this tensor extends to a two-parameter set [28].
Computing with the general counterterm [24,29] where we see explicitly the two independent choices of tensors expressed in amplitudes. In general a tensor could be expressed as where R 4 + is the counterterm consistent with supersymmetry and R 4 − is orthogonal to it, in the sense that it yields M c (1 − , 2 − , 3 + , 4 + ) = 0. A general counterterm would be a combination of the two.
Clearly the order α 1 theory is made UV finite by the addition of the orthogonal counterterm, R 4 − . This could be realised, for example, by choosing

Beyond Cubic Vertices
The non-extended theory of graviton scattering ( and of gluons) is constructible: that is the entire S-matrix can be generated by demanding that the amplitudes are factorisable [30]. In practice the factorisation can be excited by the BCFW shift. In the extended theory the leading deformation of the S-matrix is also constructible [4] albeit by using alternative shifts [31,32]. However at order α 2 , if we consider M α 2 4:tree (1 − , 2 − , 3 + , 4 + ) there is a single factorisation as shown in fig. 3. The amplitude has the correct factorisation for any choice of β. This ambiguity means we also have to specify the four-point amplitude to determine the S-matrix.
In the diagrammar approach this ambiguity arises due to the existence of a polynomial function with the correct symmetries and spinor and momentum weight. From a field theory perspective, additional counterterms can contribute to this amplitude. Specifically, we could deform the theory via and the four-point amplitude is only specified once C α and C β are determined. From a constructibility viewpoint, defining the theory from its four-point amplitudes is much less constrained than using the three-point amplitudes because momenta constraints do not limit the vertex to be constructed only from λ i orλ i but can involve both (or more likely momenta invariants). Specifically we could introduce a fundamental amplitude From a Lagrangian perspective this would be implemented by a D 2 R 4 operator giving non-vanishing M(1 − , 2 − , 3 + , 4 + ) but vanishing M(1 + , 2 + , 3 + , 4 + ) and M(1 − , 2 + , 3 + , 4 + ). As the all-plus and single-minus amplitudes vanish in a supersymmetric theory, any operator compatible with supersymmetry that generates a non-vanishing four-point MHV amplitude will suffice. A specific choice of this is [33] t 10 t 8 ∂ 2 R 4 = t a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 a 10 where t a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 a 10 10 = δ a 1 a 4 δ a 2 a 5 δ a 3 a 6 δ a 7 a 9 δ a 8 a 10 − 4δ a 1 a 4 δ a 2 a 10 δ a 3 a 5 δ a 6 a 7 δ a 8 a 9 (67) and, where possible, this should be anti-symmetrised with respect to the pairs of indices a 2 ↔ a 3 etc. and symmetrised with respect to pairs of couples of indices (a 2 a 3 ) ↔ (b 1 b 2 ) [33].
Although constructability from three-point vertices is an attractive concept, unfortunately we find the theory is not completely specified by the three-point vertex.

Conclusions
We have studied the S-matrix of extended Yang-Mills and gravity using a diagrammar approach in which the theory is defined by its on-shell amplitudes. If we wish to extend either pure Yang-Mills or gravity by the addition of a three-point interaction there is an essentially unique choice. This choice leads to a theory in which the leading deformation is constructible from three-point amplitudes although higher order deformations require further information to fix the amplitudes. In this letter we have studied the oneloop corrections to these theories and demonstrated how Unitarity can be used to simply examine the renormalisability. For Yang-Mills the one-loop UV infinities renormalise the three-point vertex at leading order. For gravity however the UV infinities must be cancelled by four-point amplitudes arising from a different source. Extending the S-matrix of gravity by the addition of the minimal three-point amplitude is equivalent to adding R 3 terms to the Lagrangian. From a Lagrangian view point this is then renormalised at oneloop by R 4 counterterms. For the leading deformations we find that these counterterms are the combination that is orthogonal to the R 4 counterterm associated with supersymmetric Lagrangians.

Acknowledgements
This work was supported by STFC grant ST/L000369/1. GRJ was supported by STFC grant ST/M503848/1. JHG was supported by the College of Science (CoS) Doctoral Training Centre (DTC) at Swansea University.