D3-Brane Loop Amplitudes from M5-Brane Tree Amplitudes

We study loop corrections to scattering amplitudes in the world-volume theory of a probe D3-brane, which is described by the supersymmetric Dirac-Born-Infeld theory. We show that the D3-brane loop superamplitudes can be obtained from the tree-level superamplitudes in the world-volume theory of a probe M5-brane (or D5-brane). The M5-brane theory describes self-interactions of an abelian tensor supermultiplet with $(2,0)$ supersymmetry, and the tree-level superamplitudes are given by a twistor formula. We apply the construction to the maximally-helicity-violating (MHV) amplitudes in the D3-brane theory at one-loop order, which are purely rational terms (except for the four-point amplitude). The results are further confirmed by generalised unitarity methods. Through a supersymmetry reduction on the M5-brane tree-level superamplitudes, we also construct one-loop corrections to the non-supersymmetric D3-brane amplitudes, which agree with the known results in the literature.

allow to uniquely fix all the tree-level amplitudes in the theory [18][19][20][21][22][23]; furthermore, the D-brane tree-level amplitudes are an important part of the so-called unifying relations that relate tree amplitudes in a wide range of effective field theories [24].
It is not surprising that, when dimensionally reduced to 4D, these twistor formulations for 6D tree-level amplitudes reproduce the corresponding formulae for the 4D tree-level amplitudes [25]. This paper considers constructing loop corrections to the scattering amplitudes in the 4D theories from the tree-level amplitudes in 6D theories. In particular, we will construct loop corrections to the amplitudes in the D3-brane theory from the M5-brane (or D5-brane) tree-level amplitudes. As a warm up example, we will also consider loop corrections to the scattering amplitudes in 4D N = 4 SYM and N = 8 supergravity from the tree-level amplitudes in 6D SYM and supergravity, respectively. In the case of 4D N = 4 SYM and N = 8 supergravity, the construction reproduces known results in the literature.
With the help of CHY formulation [26,27] of scattering amplitudes, and built on earlier works from ambi-twistor string theory [28,29], it is known that the loop corrections in lower dimensions can be obtained from those in higher dimensions via a peculiar dimensional reduction [30][31][32][33] (see also [34]). In particular, for constructing a n-point one-loop amplitudes in lower dimensions, we begin with a (n+2)-point tree-level amplitudes in the corresponding theory in the higher dimensions, and set n of the external momenta in the lower dimensions whereas the remaining two momenta are taken to be forward and stay in the higher dimensions. The forward momenta play the role of the loop momenta of the one-loop amplitudes in the theory in the lower dimensions. Analogous constructions for the two-loop amplitudes have been pushed forward in [35][36][37], where two pairs of forward momenta become the loop momenta of the two-loop amplitudes. Instead of using the general dimensional CHY formulations, we will apply the 6D twistor formulations in [7][8][9][10][11]14] to construct loop corrections to amplitudes in 4D. The advantage of using the 6D twistor formulae is that they make the supersymmetry manifest, and allow us to conveniently utilise the spinor-helicity formalism, which is a powerful tool for computing scattering amplitudes in 4D theories.
The rest of the paper is organised as follows. Section 2 reviews the 6D scattering equations, and their applications to the tree-level superamplitudes in various 6D supersymmetric theories. Section 3 discusses the construction of loop superamplitudes in 4D theories from the twistor formulae of 6D tree-level superamplitudes with appropriate forward limits. To illustrate the ideas, we take the one-and two-loop four-point amplitudes in N = 4 SYM and N = 8 supergravity as examples. In section 4, we apply the ideas to construct loop corrections to the D3-brane theory. In particular, we focus on the maximally-helicity-violating (MHV) amplitudes at one loop, which are given by contact rational terms. The same re-sults are achieved using generalised unitarity methods. We compute explicitly the four-and six-point amplitudes, and also comment on the structure of the n-point MHV amplitude. In section 5, through a supersymmetric reduction of 6D tree-level superamplitudes in M5-brane theory, we study one-loop corrections to the non-supersymmetric D3-brane amplitudes. We further show that our results are in agreement with those in the reference [38], which were obtained recently by Elvang, Hadjiantonis, Jones and Paranjape using generalised unitarity methods.

6D twistor formulations
In this section, we will briefly review the twistor formulations for tree-level superamplitudes in 6D theories following the references [7][8][9][10]14]. These twistor formulae of 6D tree-level amplitudes will provide the basis for constructing loop corrections to the amplitudes in 4D theories using forward limits.

6D scattering equations
The scattering amplitudes of massless particles in 6D are described in terms of the 6D spinor-helicity formalism [39], which expresses the 6D massless momentum as where A, B = 1, 2, 3, 4 are spinor indices of Lorentz group Spin (1,5). Here we have used the short-hand notations where a, b andâ,b are little-group indices. For a massless particle in 6D, the little group is Spin(4) ∼ SU(2) L × SU(2) R , so a, b in the above equation are the indices of SU(2) L with a, b = 1, 2, andâ,b =1,2 refer to SU(2) R .
The kinematics of massless particles in 6D can be nicely described by the 6D scattering equations, either via rational maps [7][8][9] or equivalently the polarised scattering equations [10]. We will utilise the polarised scattering equations in our discussions, 2 which take the 2 All the discussions also apply to the formalism based on rational maps, see the reference [14] for a detailed discussion of rational maps and their equivalence to the polarised scattering equations via a symplectic Grassmannian.
following form where we mod out the SL(2, C) symmetries acting on world-sheet coordinates σ i as well as on the coordinates u i . The rational functions λ Aa (σ) are given by Here λ Ab j are the 6D helicity spinors that we introduced in (1), and the constraints in (3) implies the momentum conservation, i.e.
It is convenient to choose the little-group spinors ǫ i that enter in the constraints v i ǫ i = 1 to be ǫ i,a = (0, 1). For such a choice, the delta-function constraints v i ǫ i = 1 are solved by v i,a = (1, v i ).
It is worth noticing that (3) can be recast into a matrix form, where Λ A is a 2n-dimensional vector encoding the external helicity spinors, and V is a n × 2n matrix that follows from (3) with σ ij := σ i − σ j , and Ω is the symplectic metric Importantly, under the condition v i ǫ i = 1, the matrix V obeys the following symplectic condition [14], Therefore the space of the matrices V forms the symplectic Grassmannian [40,41]. Finally, one may define a different set of 6D scattering equations with the helicity spinorsλâ i,A , which is needed for determiningṽ i ,ũ i . The variablesṽ i ,ũ i are required for constructing superamplitudes of the non-chiral theories such as 6D SYM, D5-brane theory and supergravity theories.

6D tree-level superamplitudes
To describe the on-shell supersymmetry in 6D, we further introduce Grassmann variables, η I i,a andηĨ i,â , with I = 1, 2, · · · , N andĨ = 1, 2, · · · , N for a theory with (N , N ) supersymmetry, and i is the particle label. In particular, the supercharges are defined by and similarlyqĨ We note the supercharges obey the correct supersymmetry algebra, {q A,I i ,q B J,i } = δ I J p AB i , and similar algebra relations forqĨ A,i ,q A,i,Ĩ . The superamplitudes should be annihilated by the total supercharges, namely Q A,I The on-shell spectrum of a supersymmetric theory can be packaged into an on-shell superfield with the help of the Grassmann variables, η I i,a andηĨ i,â , and the on-shell superamplitudes are functions of helicity spinors as well as the Grassmann variables. Let us begin with the 6D SYM with (1, 1) supersymmetry. The on-shell spectrum of the theory is given by where, for instance, φ 11 is one of four scalars of the theory, and A aâ is the 6D gluon. Similar to the CHY construction of scattering amplitudes [26,27], the tree-level amplitudes of a generic 6D theory take the following form in the twistor formulations In the above formula, the measure dµ 6D n imposes the 6D scattering equations which are given in (3) or (5), and I L and I R specify the dynamics of the theory, which will be called as left and right integrand. In the case of 6D (1, 1) SYM, the twistor formula of the tree-level superamplitudes is given as where the left and right integrands take the following form Here η,η are the Grassmann version of Λ A in (6), η = {η 1,1 , η 2,1 , . . . , η n,1 , η 1,2 , η 2,2 , . . . , η n,2 } , η = {η 1,1 ,η 2,1 , . . . ,η n,1 ,η 1,2 ,η 2,2 , . . . ,η n,2 } , and the Grassmann delta functions in I (1,1) L imply the conservation of supercharges of 6D (1, 1) supersymmetry.
The n × n matrix H n has the following entries [10] Here, just as ǫ i,a , we can chooseǫ i,â = (0, 1). Note that H ii is independent of the choice of little-group index a, namely it is a Lorentz scalar. The reduced determinant det ′ H in I where H [ij] [kl] means that we remove the i-th and j-th columns as well as the k-th and l-th rows, and the result is independent of the choices of i, j and k, l. Note that the conjugate variables such asṽ i ,ũ i appeared in the integrands are determined by scattering equations in (10).
Let us turn to the M5-brane theory and D5-brane theory. The D5-brane theory has (1, 1) supersymmetry and contains the same spectrum as SYM, that is given in (13). The M5-brane theory is a chiral theory with (2, 0) supersymmetry, and the on-shell superfield is a tensor multiplet, which is given as with I = 1, 2, and B ab is the on-shell 6D self-dual tensor. For the D5-brane theory and the M5-brane theory, only the even-multiplicity amplitudes are non-trivial, and the oddmultiplicity ones vanish identically. The left integrands for the M5-brane theory and the D5-brane theory of the twistor formulations are in fact the same, where S n , which is only defined for even n, is an n × n matrix with entries given as with p i , p j being the 6D momenta. The reduced Pfaffian of S n , Pf ′ S n , is defined as where (S n ) kl kl is an (n−2) × (n−2) matrix with the k-th and l-th rows and columns of S n removed, and the result is independent of the choice of k, l. The right integrand of the D5-brane superamplitudes is given by whereas the right integrand of the M5-brane superamplitudes takes a simpler form, The fermionic delta functions in I D5 R lead to the (1, 1) supersymmetry for the D5-brane theory, and the fermionic delta functions in I M5 R encode (2, 0) supersymmetry for the M5brane theory. The object U n that appears in the above formulae is an n × n anti-symmetric matrix with entries given by and one may define the conjugate matrixŨ n in a similar fashion usingũ i . Note that Pf ′ S n is related to det ′ H n through the following identity which is true under the support of 6D scattering equations. In summary, the tree-level superamplitudes in M5-brane theory is given by with I M5 L and I M5 R given in (23) and (27), respectively. A similar formula can be written down for the D5-brane superamplitudes. From the twistor formulae, we can obtain explicit superamplitudes, for instance, the four-point amplitude of M5-brane theory is simply where recall that Q 4 is the supercharge, and it is defined as Q A,I

D3-brane massive tree-level amplitudes
After a dimensional reduction to 4D, the M5-brane superfield defined in (22) reduces to the D3-brane superfield, which is identical to the superfield of N = 4 SYM, given as (in the non-chiral form), where we have identified {η 1 ,η1} of 6D (1, 1) supersymmetry as η I − with I = 1, 2, and {η 2 ,η2} as ηĨ + withĨ = 1, 2. The supercharges are given by So A 11 is identified as the minus-helcity photon (or gluon, in the case of SYM) A − , A 22 as the plus-helcity photon (or gluon) A + etc. The tree-level amplitudes in D3-brane theory are obtained through a dimension reduction by setting all the external momenta of the M5-brane amplitudes in 4D [7]. The dimension reduction procedure leads to the twistor formulations for the tree-level superamplitudes in D3-brane theory [25].
Here we are interested in tree-level superamplitudes in D3-brane theory with massive states, since they are relevant for constructing loop corrections that we will consider in the section 4.2 using generalised unitarity methods. The massive tree-level amplitudes in D3-brane theory can also be obtained from the M5-brane tree-level amplitudes by a careful dimension reduction. In particular, the masses may be viewed as extra dimensional momenta as KaluzaKlein modes. The D3-brane superfield with massive sates is straightforwardly obtained from (22) where a = 1, 2 are the little group indices of massive particles in 4D, for instance A ab is the 4D massive vector. The superamplitudes in the D3-brane theory with both massless and massive states are obtained from (30) by setting the 6D helicity spinors as for the massless states, and for the massive states with complexified masses given by µ j = λ 1 j λ 2 j andμ j = λ1 jλ2 j . 3 We will mostly consider the case with two massive states (say, they are particles i and j), in which case µ i = −µ j = µ andμ i = −μ j =μ, because the extra dimensional momenta should be conserved. For instance, at four points, from (31), we find the superamplitude with two massive states and two massless are given by where particles 1 and 2 are massive, and 3 and 4 are massless. From the superamplitude, we can also obtain component amplitudes, for instance, where φ 1 ,φ 2 are massive scalars with mass µ.
For higher-point amplitudes, we can construct the superamplitude by writing down an ansatz consisting of factorisation terms, which are obtained from lower-point amplitudes, as well as some possible contact terms, and then compare the ansatz with the twistor formula (30) to determine any unfixed parameters. For instance, at six points, the factorisation terms of the tree-level amplitude with two massive states are shown in Fig.(1), where each diagram takes the form of We have used the four-point superamplitude given in (31) with the understanding that kinematics are projected to 4D as described in the above paragraphs. The Grassmann integration is to sum over the intermediate states, and "Perm" represents summing over all the independent permutations (we will use the same notation in the later sections). It is straightforward to check that the factorisation terms as shown in Fig.(1) agree with (30), it therefore implies that no contact term exists at six points. This is consistent with the known fact that a six-point supersymmetric contact term with six derivatives is not allowed 3 The same procedure has been applied to 6D SYM amplitudes to obtain massive amplitudes in 4D N = 4 SYM on the Coulomb branch [8]. Figure 1: The vertices, Q L and Q R , represent four-point superamplitudes δ 8 (Q L ) and δ 8 (Q R ). They are glued together by an on-shell propagator K. The curvy lines denote massless particles in 4D, while the solid line indicates 4D massive particles. In the above diagram, we choose leg-1 and leg-6 to be massive states in 4D.
by (2, 0) (and (1, 0)) supersymmetry [42]. This is also in agreement with what was found in [38] for the all-plus and single-minus amplitudes (with two additional massive scalars). 4 Similar computation applies to higher-point amplitudes. In particular, consider the eightpoint superamplitudes, for which the factorisation terms schematically take the form of We find in general contact terms are required to match with the twistor formula (30), except for the special cases with the helicity configurations being all-plus and single-minus photons (with two additional massive scalars), which agrees with the results of [38]. However, for more general helicity configurations, we do require contact terms. For instance, the amplitude of two-minus photons, with the factorisation terms given in (40) (after projecting to the corresponding component amplitude), we find the following contact term is required to match with twistor formula (30), where φ 7 ,φ 8 are massive scalars with mass µ. We will come back to this term when we consider its contributions to loop corrections.

Loops from trees
This section will describe the general ingredients for constructing loop amplitudes from tree-level amplitudes in higher dimensions using the twistor formulations. We would like to emphasise that our construction makes the supersymmetry manifest, and utilises the powerful spinor helicity formalism. This is due to the fact that, instead of the generaldimensional CHY formulae, we will apply the 6D twistor formulations that we reviewed in the previous section, from which we will construct the loop corrections to 4D superamplitudes.
To illustrate the idea, we will study the well-known one-and two-loop superamplitudes in N = 4 SYM and N = 8 supergravity using our construction. The loop corrections to the D3-brane superamplitudes (as well as non-supersymmetric D3-brane amplitudes) will be studied in the following sections.

Loop corrections from higher-dimensional tree amplitudes
It was argued [32,33] that with an appropriate forward limit, the CHY scattering equations for the tree-level amplitudes give rise to the one-loop scattering equations that were originally obtained from the ambi-twistor theory [28,29]. To construct the loop corrections to n-point scattering amplitudes in lower-dimensional theories, we start with the (n+2)-point treelevel amplitudes in corresponding theory in the higher dimensions. We then set n of the external momenta in the lower dimensions of the loop amplitudes that we are interested in, whereas the remaining two momenta, which still stay in the higher dimensions, are taken to be forward. This pair of the forward momenta play the role of the loop momenta of the one-loop amplitudes in the lower dimensions. Analogously, with a careful analysis, the twoloop corrections to n-point amplitudes can be obtained from the (n+4) tree-level in higher dimensions with two pairs of forward momenta that become the loop momenta of the loop amplitudes [35][36][37].
Explicitly, the forward limit procedure in our construction works as the following: To compute an n-point one-loop amplitude in 4D, we set the kinematics (the spinor-helicity variables) of the (n+2)-point tree-level amplitude in 6D being parametrised as which become the massless external kinematics of the one-loop amplitude in 4D. We also use k αα i = λ α iλα i to denote massless 4D momenta for external particles. As for the remaining two helicity spinors of the forward-limit particles, we set them to be , namely the 6D momenta p AB n+1 , p AB n+2 are forward and the 4D part is identified as the loop momentum ℓ with ℓ 2 = 0. Furthermore, we should identify the Grassmann variables η I i ,ηĨ i (for i = 1, 2, · · · , n) with the Grassmann variables of lower-dimensional supersymmetric theories (as we will see explicitly in examples in the following sections). As for the forward-limit pair, we set such that the supercharges q I n+1 = −q I n+2 andqĨ n+1 = −qĨ n+2 . This is required for the conservation of supercharges of external particles, namely n i=1 q I i = n i=1qĨ i = 0. With this set up, the twistor formulation for the one-loop amplitude is then given as where we have denoted the n-point one-loop amplitude in 4D as A denotes the 6D (n+2)-point tree-level amplitude with the special forward kinematics given in (42) and (43), as well as (44) for the Grassmann variables. The Grassmann integral is to sum over all the internal states for the pair of the particles that are taken to be forward. In the last step of (45), we have expressed the 6D tree-level amplitudes in the twistor formulations, with the measure dµ 6D n+2 given in (5), and the precise form of the integrand I L and I R depends on the theory we are considering. As discovered in [30] (see also [43]), this formulation typically leads to loop integrands with linear propagators, and we will see more examples of these linear propagators in the following sections. 5

Loop corrections to SYM and supergravity superamplitudes
To illustrate the ideas, we consider the one-and two-loop amplitudes in 4D N = 4 SYM and N = 8 supergravity from our construction. Under the dimensional reduction, the 6D SYM superfield (13) reduces to the superfield of 4D N = 4 SYM (in the non-chiral form) as shown in (32). For the pair of particles that we take to be forward (i.e. the particles (n+1) and (n+2)), we set the Grassmann variables as According to the procedure discussed in the previous section, the one-loop superamplitudes in N = 4 SYM are then given by with the left and right integrands given by where the cyclic sum is to sum over all the cyclic permutation of α. Again, it should be understood that the kinematics in the above formula are taken to be: n of the helicity spinors are in 4D, and two remain in 6D and being forward, whereas the Grassmann variables are identified according to (32) and (46).
As an example, we construct the one-loop corrections to the four-point superamplitude in 4D N = 4 SYM. Following the procedure discussed previously, we begin with the six-point tree-level superamplitude of 6D (1, 1) SYM, and dimensionally reduce four of the external kinematics (including the fermionic ones) to 4D N = 4 SYM, whereas the remaining two of them are in 6D and are taken to be forward, as illustrated in Fig.(2). We find that the one-loop integrand of the four-point amplitude in 4D N = 4 SYM is given by where the supercharges Q 4 ,Q 4 are defined in (33). The box integral I box (k 1 , k 2 , k 4 ) is defined as where the integral is in the linear propagator representation and we also sum over four cyclic permutations. The result is in agreement with [30]. Importantly, as we show in Appendix A, the linear-propagator representation (50) is equivalent to the standard box integral with quadratic propagators, namelỹ  (42)), while the particle with the solid line still remains in 6D kinematics. The red line represents the forward-limit particle pair with 6D kinematics, which is understood as loop momentum.
Similar construction applies to the one-loop corrections to the four-point superamplitude in 4D N = 8 supergravity. For the N = 8 supergravity loop amplitudes, we begin with the six-point tree-level superamplitude of (2, 2) supergravity. The prescription then leads to the following result M where Q 4 ,Q 4 are defined in (33), but now with I = 1, 2, 3, 4 andĨ = 1, 2, 3, 4 for the N = 8 supersymmetry. We also sum over all the permutations due to the permutation symmetry of gravity amplitudes, and (52) indeed reproduces the known result in [30].
The construction may be generalised to higher loops, especially the two-loop corrections [35][36][37]. For describing two-loop corrections, we will need (n+4)-point tree amplitudes as input, and set two pairs of particles in forward limit (p AB n+1 , p AB n+2 are forward, so as p AB n+3 , p AB n+4 ), and require their supercharges to cancel among each other. To be explicit, the second pair of forward-limit particles (n+3) and (n+4) should obey the same relation in (43) and (44) as the first particle pair (n+1) and (n+2) do. With the similar setup as one-loop amplitude, we can write down the twistor formulation of two-loop amplitude as where ℓ 1 and ℓ 2 are the momenta of the two forward-limit particles, (n+1) and (n+3), which are identified as loop momenta with ℓ 2 1 = 0 and ℓ 2 2 = 0. We denote the two-loop n-point amplitude in 4D as A 4D,n . The internal states of all forward-limit particles are summed over by performing Grassmann integral.
We have checked that (54) gives correct two-loop four-point amplitude; for instance, the planar part computed by using the integrand in (56) is shown to be equal to where the planar two-loop boxes, I planar α 1 α 2 , α 3 α 4 are defined in the Appendix.A of [35], see equation (A5). We have also checked the agreement between our formula and the known result for the non-planar sector. Similarly, the four-point two-loop superamplitude in 4D N = 8 supergravity can be obtained by considering the eight-point tree-level amplitude of the 6D (2, 2) supergravity, which is expressed in the twistor formulation as given in (21). We have verified that the result of this construction is in the agreement with the known result [35,49].

Supersymmetric D3-brane amplitudes at one loop
In this section, we consider one-loop corrections to the superamplitudes in D3-brane theory using forward limits of higher-dimensional tree-level amplitudes, following the general prescription of the previous sections. The results will be further confirmed using generalised unitarity methods. The higher-dimensional amplitudes that are relevant for constructing the loop corrections to D3-brane amplitudes are the tree-level amplitudes in M5-brane theory 6 .
Because the construction makes the supersymmetry manifest, the amplitudes with all-plus and single-minus helicity configurations manifestly vanish. So the first non-trivial helicity configurations are those with two minus photons, namely the MHV amplitudes.
It is known that the non-trivial tree-level amplitudes in the D3-brane theory are helicity conserving [17]. So the MHV amplitudes vanish at tree level (except for the four-point case), and they do not have non-trivial four-dimensional cuts at one-loop order. Therefore the oneloop MHV amplitudes (with more than four points) can only be rational terms. To extract the rational terms, it is necessary to consider the loop momenta in general d dimensions. We will treat the extra-dimensional loop momenta as masses, therefore, it is equivalent to consider 4D loop momenta with massive states running in the loop.
In our construction, this is set up by separating the massless 6D momenta (which will be taken to be forward) into ℓ 4D and two extra dimensions, such that when the loop momentum ℓ 4D is put on-shell, we have ℓ 2 4D = −µ 2 . With such convention, we write a 6D momentum as its 4D component (ℓ 4D := ℓ) with extra dimensions, that will be called as p 4 and p 5 , which correspond to the components of the fifth and the sixth dimension, respectively. The 6D momentum is explicitly written as then the massless condition is Since we have identified the momentum of a forward-limit particle as the loop momentum as shown in (43), µ andμ can also be expressed in terms of λ A n+1,a : We now have a massive particle in the loop with a loop momentum,l = ℓ+µ withl 2 = ℓ 2 +µ 2 .
Also, the linear propagators are unchanged since µ is in the extra dimension, so we havẽ for a 4D external momentum k i .
With this setup, the one-loop D3-brane amplitudes can be obtained from the tree-level M5-brane amplitude by a similar construction we outlined in previous sections, where d 4 η n+1 = dη 1 n+1,1 dη 2 n+1,1 dη 1 n+1,2 dη 2 n+1,2 , and Grassmann integration is to sum over the superstates of the forward-limit particles. Again in the above formula, it should be understood that it is the tree-level forward-limit amplitude on the right-hand side that gives the one-loop integrand. This tree-level M5-brane amplitude is expressed in the twistor formulation as where the left and right integrands I M5 L and I M5 R are given in (23), and (27), respectively.

One-loop corrections to D3-brane superamplitudes
Let us begin with the one-loop correction to a four-point amplitude, which is very similar to the case of four-point one-loop amplitudes of N = 4 SYM in the previous section. We will show that the one-loop amplitude receives correction from the bubble diagram in the where A (0) D3 ,4 is the tree-level amplitude of the D3-brane theory, and it is given by Note that it is the supersymmetrisation of the higher-derivative term F 4 . As we have seen in the previous section, in this construction the loop integrands are typically in the linear propagator representation. For the four-point case we consider here, the bubble integral is defined by .
(67) Recalll = ℓ + µ, so thatl 2 = ℓ 2 + µ 2 andl · (k 1 + k 2 ) = ℓ · (k 1 + k 2 ). As shown in Appendix A, (67) is equivalent to the standard bubble integral with quadratic propagators The result (65) is in agreement with the result in the reference [50] that was originally obtained using unitarity cuts [51,52], which is in the quadratic propagator form (68), and only massless loop propagators were considered. The bubble integral (67) or (68) is UV divergent in 4D, therefore the result (65) leads to a UV counter term for the D3-brane effective Lagrangian, which is of the form d 4 F 4 (and its supersymmetric completion), or equivalently in momentum space it is given as The above four-point superamplitude is expressed in a non-chiral form, where the superfield is given by (32). For describing the MHV superamplitudes at higher points which we will study shortly, it is more convenient to use the chiral version. The chiral version is obtained by a Grassmann Fourier transform. For instance, the chiral superfield is obtained from non-chiral superfield given in (32) through, After combining η I − and ξĨ and denoting them as η A with A = 1, 2, 3, 4, which transform under SU(4) R-symmetry, we obtain the superfield in a chiral form, and the supercharges take the form In the chiral representation, the four-point tree-level D3-brane superamplitude is given by [42] A (0) where [12] 34 2 is the Jacobian factor from the Grassmann Fourier transform, and importantly it is permutation invariant and 34 2 in the denominator is not a pole. When expressed in the chiral form, the four-point D3-brane superamplitude at one loop takes the following form where we define Q αA n = n i=1 q αA i . We will continue to utilise the chiral representation for the discussion on higher-point D3-brane superamplitudes.
The one-loop corrections to higher-point superamplitudes can be obtained similarly from the general formula (63). To illustrate the idea, we will consider the six-point MHV amplitude. Using the formula (63) with n = 6, we find the MHV amplitude is proportional to an overall supercharge, δ 8 (Q 6 ), as required by N = 4 supersymmetry. Explicitly, the one-loop superamplitude is constructed by taking forward the eight-point tree-level superamplitude as shown in the Fig.(4). Note that the contact term of eight-point amplitude does not contribute. We find that the integrand for the one-loop six-point MHV amplitude takes a very similar form as the four-point result given in (74). It is given as The integral I triangle is the scalar triangle integral in the linear propagator representation, which takes the following form This result is verified by (63) for n = 6 by solving numerically the scattering equations in formula with the forward kinematics. There are three terms with linear propagators in the above equation, each of them can be understood as assigning the forward pair of legs in different places; for example, the first term in (75) is shown in the Fig.(4). In principle, the one-loop corrections to higher-point amplitudes can be obtained in a similar fashion; however, solving scattering equations with higher-point kinematics becomes more and more difficult. We hope to develop better numerical and analytical methods to handle this issue, which we leave as a further research direction.

Generalised unitarity methods
In this section, we will construct the one-loop amplitudes through the d-dimensional generalised unitary methods [53]. Again the extra dimensional loop momenta will be viewed as the masses of the internal propagating particles. We will find the results agree with those computed in the previous section using the twistor formulations. Of course, they are in different representations: one in the linear propagator representation, the other in the standard quadratic propagator representation.
Let us first begin with the four-point case. The one-loop amplitude only receives contribution from the bubble diagram, which can be formed by gluing two four-point superamplitudes as shown in the Fig.(5). Explicitly, it is given by where the explicit form of the four-point superamplitudes are given by Figure 5: The four-point bubble diagram is formed by gluing two tree-level four-point superamplitudes. We identify K 1 as loop momentum (K 1 = ℓ), then the on-shell conditions (K 2 i = 0) can be read as ℓ 2 = 0, and (ℓ + k 1 + k 2 ) 2 = 0.
Note for the external states, they are massless, therefore From (77), we deduce that the one-loop correction to the MHV amplitude is given by In the above formula we have summed over the permutations, and after linearise the propagators as we show in Appendix A, the result agrees with the one obtained from the twistor formula given in (74).
We now consider the six-point amplitude. Due to the fact that there is no six-point contact term, for the MHV amplitude, only the triangle diagram is non-trivial, for which we glue three four-point superamplitudes as shown in the Fig.(6). We pair up external leg-(1, 2), (3,4), and (5,6) in three different corners, and the internal massive lines are denoted as K 1 , K 2 , and K 3 . The suprsymmetric gluing result of four-point superamplitudes gives Explicit evaluation of the above formula leads to the following result for the one-loop correction to the six-point MHV amplitude Here we have chosen the loop momentum to be K 3 (K 3 =l), and we have also summed over permutations to obtain the complete answer. Again, as explained in Appendix A, (82) is in agreement with (75), which we obtained from the twistor formulations.  Figure 6: Legs-(1,2), (3,4), and (5,6) are glued in three different corners with on-shell propagators, K 1 , K 2 and K 3 . When K 3 is identified with the loop momentum, the diagram gives the contribution as (82).
As for a general n-point MHV amplitude, it is easy to see that it contains a (n/2)-gon integrand which takes the same form as the four-and six-point amplitudes we have computed, namely, However, as we know that for general helicity configurations, the tree-level amplitudes (with two massive states) at higher points require contact terms, (e.g. (41) for the eight-point case), it implies that besides the (n/2)-gon topology, the one-loop amplitude in general also receives contributions from the integrands with lower-gon topologies. For instance, for the eight-point MHV amplitude, a bubble diagram will also contribute due to the contact term shown in (41) (as well as its supersymmetric completion). We leave the computation of the one-loop corrections to the n-point MHV amplitude as a future research direction.

Rational terms of MHV D3-brane amplitudes at one loop
In the previous sections, we obtained the one-loop integrand for the MHV amplitude of the D3-brane theory, either using the forward limit of the twistor formulations or the generalised unitary methods. The one-loop integral can be performed explicitly using the dimension shifting formula [54], see for instance the Appendix C of [38]. Explicitly, for a m-gon scalar integral, we have where we have taken D = 4 − 2ǫ and considered the small ǫ limit. We therefore obtain the one-loop correction to the six-point MHV amplitude of the D3-brane theory, which is given by a contact rational term, It is straightforward to see that the above result is the unique answer that has the right power counting and correct little-group scaling, and further require the answer do not possess poles.
Indeed, we do not allow MHV amplitudes to have poles since the theory has no three-point amplitudes. Similarly, at n points, the unique answer that is consistent with the power counting and little-group scaling takes the following form The above result is also in agreement with (83) after performing the integral using (84).
However, as we commented that the (n/2)-gon integral (83) is only a part of the full answer for n > 6, therefore the overall coefficient of (86) has to be determined by explicit computations after including all the lower-topology integrands.
Finally, due to the fact that the scalars of D3-brane theory are Goldstone bosons of spontaneously breaking of translation and Lorentz rotation, the corresponding amplitudes with these scalars should obey enhanced soft behaviour [18,19], where k 1 is the momentum of one of the scalars. The enhanced soft behaviour was further argued to be valid when the loop corrections are taken into account [55]. To study the soft behaviour of the scalar fields, we consider the rational term with the helicity configuration (φ 1 ,φ 2 , 3 − , 4 + , · · · , n + ), where φ 1 ,φ 2 are scalars. From (85), we see that the rational term is then proportional to It is easy to see that each term in the permutation goes as O(k 2 1 ) (or O(k 2 2 )) in the soft limit k 1 → 0 (or k 2 → 0), which is consistent with the enhanced soft behaviour of the D3-brane theory.

Non-supersymmetric D3-brane amplitudes at one loop
The loop corrections to scattering amplitudes in 4D lower-supersymmetric theories or nonsupersymmetric theories can be obtained by a supersymmetry reduction on 6D tree-level superamplitudes such that only relevant states (instead of the full super multiplets) run in the loops. Using this idea, we will study loop corrections to the amplitudes in the nonsupersymmetric Born-Infeld (BI) theory. In particular, for the BI theory, we project the external states to be photons and the internal particles to be a pair of massive vectors.
In the following sections, we will consider one-loop amplitudes with all-plus external photons (the Self-Dual sector) and single-minus external photons (the Next-to-Self-Dual sector). They vanish identically in the supersymmetric theory, and for the non-supersymmetric theory, they are purely rational terms at one-loop order. Therefore, just as in the case of one-loop MHV amplitudes in the supersymmetric theory, to extract the rational terms we require the internal particles propagating in the loop to be massive. We find that the results from the forward-limit construction agrees with those in [38], which were computed originally using the generalised unitarity methods [51,52].

Self-Dual sector
We begin with the amplitudes in the Self-Dual (SD) sector, namely the amplitudes with allplus helicity configuration. We perform a supersymmetric reduction by choosing all external legs to be plus-helicity photons and the forward-limit particles to be a pair of massive vectors or massive scalars 7 . With such construction, the one-loop n-point amplitude in the SD sector is given by where d 2 η i,a = 1 2 ǫ IJ dη I i,a dη J i,a . This choice of Grassmann variables integration projects all the external legs (i.e. particles 1 to n) to be plus-helicity photons, and it also sets the internal particles n+1 and n+2 to be scalars.
In the case of n = 4, carrying out the Grassmann integral in (89) and solving the scattering equations, we find that the one-loop correction to the four-point amplitude in the SD sector is given by where the bubble integral I bubble (k 1 , k 2 ) is given in (67). The above expression agrees with the loop integrand in (4.1) of [38], except for the propagators being linearised. Similar computation applies to higher-point cases. Let us consider the six-point case here, that is given by (89) with n = 6, from which we find which agrees with the loop integrand in (4.4) of [38]. Again our result is in the linear propagator representation, but it is equivalent to that of [38] as shown in Appendix A.
We note that, at least for the cases we studied here, the one-loop integrands of amplitudes in the SD sector of non-supersymmetric D3-brane theory take a very similar form as those of the MHV amplitudes in the supersymmetric D3-brane theory. In fact, they are related to each other by exchanging the factor (µ 2 ) 2 in amplitudes of the SD sector with δ 8 (Q) in the supersymmetric amplitudes.

Next-to-Self-Dual sector
The computation for the one-loop amplitudes in the Next-to-Self-Dual (NSD) sector is very similar. They are the amplitudes with single-minus helicity configuration, so we only need to change the choice of Grassmann variables from all-plus to single-minus. Applying the similar construction as (89), we have We assign particle-n to be a minus-helicity photon and the rest of the particles to be plushelicity photons. The forward-limit particles n+1 and n+2 are again chosen to be scalars.
The results are in the agreement with (4.17) and (4.19) of [38], after translating the quadratic propagators into the linear ones as discussed in Appendix A.
Finally, we comment that the above supersymmetric reduction procedure is very general, and it can be applied to other non-supersymmetric theories. For instance, we have checked explicitly that the procedure reproduces well-known results of some rational terms in pure Yang-Mills theory [54]. They are obtained from the supersymmetric reduction of the amplitude in 6D SYM with the forward limit as we described.
The shift of the loop momentum in the second term results into the linear-propagator representation of the bubble integral where ≃ denotes equivalence upon integration.
Similar computation applies to the integrals with non-trivial numerators such as those given in (93) and (95). In particular, one can show that (93) is equivalent tõ and under the loop integration (95) is equivalent tõ