String amplitudes from field-theory amplitudes and vice versa
Song He, Fei Teng, Yong Zhang
TL;DR
This paper develops a method to relate massless string amplitudes to field-theory amplitudes using integration-by-parts reduction and CHY formulas, revealing a double copy structure and deriving new integrands for specific theories.
Contribution
It introduces a generic IBP reduction technique for string correlators and connects string amplitudes to field-theory via CHY formulas, including new integrands for certain gauge theories.
Findings
String amplitudes are the double copy of field-theory amplitudes and specific integrals.
A universal IBP reduction applies to any rational correlator with correct SL(2) weight.
Derived a closed-form CHY integrand for the $(DF)^2+YM+ ext{phi}^3$ theory.
Abstract
We present an integration-by-parts reduction of any massless tree-level string correlator to an equivalence class of logarithmic functions, which can be used to define a field-theory amplitude via a Cachazo-He-Yuan (CHY) formula. The string amplitude is then shown to be the double copy of the field-theory one and a special disk/sphere integral. The construction is generic as it applies to any correlator that is a rational function of correct SL(2) weight. By applying the reduction to open bosonic/heterotic strings, we get a closed-form CHY integrand for the theory.
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String amplitudes from field-theory amplitudes and vice versa
Song He
CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
School of Physical Sciences, University of Chinese Academy of Sciences, No.19A Yuquan Road, Beijing 100049, China
Fei Teng
Department of Physics and Astronomy, Uppsala University, 75108 Uppsala, Sweden
Yong Zhang
CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
School of Physical Sciences, University of Chinese Academy of Sciences, No.19A Yuquan Road, Beijing 100049, China
Abstract
We present an integration-by-parts reduction of any massless tree-level string correlator to an equivalence class of logarithmic functions, which can be used to define a field-theory amplitude via a Cachazo-He-Yuan (CHY) formula. The string amplitude is then shown to be the double copy of the field-theory one and a special disk/sphere integral. The construction is generic as it applies to any correlator that is a rational function of correct SL weight. By applying the reduction to open bosonic/heterotic strings, we get a closed-form CHY integrand for the theory.
††preprint: UUITP-58/18
I Introduction
Recent years have seen enormous advances in understanding structures of scattering amplitudes in quantum field theories (c.f. Arkani-Hamed et al. (2016); Henn and Plefka (2014); Elvang and Huang (2015)), and many crucial insights have originated from string theory. The field-theory limit of the celebrated Kawai-Lewellen-Tye (KLT) relations Kawai et al. (1986) between tree amplitudes in open and closed string theory gives double copy relations linking gauge-theory amplitudes to gravitational ones Bern et al. (1999). Based on a remarkable duality between color and kinematics due to Bern, Carrasco and Johansson (BCJ) Bern et al. (2008), double copy has been extended to quantum level and become the state-of-the-art method for multi-loop calculations in supergravities Bern et al. (2010). Apart from the original KLT relations, string theory has provided constructions of amplitude representation that respects color-kinematics duality at tree and loop level Mafra et al. (2011, 2013a, 2013b); Mafra and Schlotterer (2015); He et al. (2016), and amplitude relations in gauge theory, i.e. BCJ relations Bjerrum-Bohr et al. (2009); Stieberger (2009), and beyond Stieberger and Taylor (2016); Schlotterer (2016).
More surprisingly, it has been realized that tree-level open- and closed-superstring amplitudes themselves can be obtained via a double copy Broedel et al. (2013). By rearranging the worldsheet correlator with massless states in type-I theory Mafra et al. (2013b, a), we can decompose disk amplitudes into field-theory KLT products of super-Yang-Mills (SYM) amplitudes and universal basis of disk integrals called integrals. The former contain the full polarization information but no nontrivial -dependence, while the latter encode expansion and are interpreted as amplitudes in an effective field theory dubbed theory Carrasco et al. (2017a); Mafra and Schlotterer (2017); Carrasco et al. (2017b). Very recently, this has been generalized to bosonic and heterotic strings Huang et al. (2016); Azevedo et al. (2018): now the field-theory amplitudes also contain tachyon poles, and they were conjectured to come from the Lagrangian, with related to its mass parameter Johansson and Nohle (2017). With integrals replaced by certain sphere integrals, we can get closed-string amplitudes via a double copy from these field theories Schlotterer and Stieberger (2013).
An alternative framework that manifests and extends the double copy in field theories is the Cachazo-He-Yuan (CHY) formulation Cachazo et al. (2014a, b), which expresses tree amplitudes in a large class of massless theories based on scattering equations Cachazo et al. (2014c). Together with loop-level generalizations Adamo et al. (2014); Geyer et al. (2015); Cachazo et al. (2016); Geyer et al. (2016); Geyer and Monteiro (2018), these have led to new double-copy realization of various theories Cachazo et al. (2015a), and one-loop extensions of KLT and amplitude relations He and Schlotterer (2017); He et al. (2018). It is now well established that tree and loop CHY formulas can be derived from worldsheet models known as ambitwistor string theory Mason and Skinner (2014); Casali et al. (2015), where CHY integrand is given by correlators therein. Despite significant progress Siegel (2015); Casali and Tourkine (2016); Azevedo and Jusinskas (2017); Mizera (2018); Bjerrum-Bohr et al. (2014), it is fair to say that the origin of ambitwistor strings, and the relations to strings at finite remain mysterious. A related outstanding question is how such double copies for strings can be understood via ambitwistor strings, especially for bosonic and heterotic strings which would require CHY formulas with nontrivial dependence.
In this letter, we present a double-copy construction for any massless string amplitude in terms of field-theory amplitudes defined by a CHY formula, where the integrand can be directly obtained from the original string correlator. We show algorithmically how to reduce a completely general rational function with correct SL weight (which we refer to as a string correlator), via integration by parts (IBP) to an equivalent class of logarithmic functions, and any of them serves as a CHY half-integrand that gives desired field-theory amplitudes. Once the logarithmic class is reached, the next step is to use scattering equations (SE) to obtain equivalent half-integrands, which are no longer logarithmic but usually take a more compact form and make some properties more manifest. The same holds for closed string correlators where we reduce the holomorphic part by IBP.
As the main application illustrating the power of this method, we perform IBP for all scalar-gluon correlators of compactified open bosonic strings, or equivalently the holomorphic part in heterotic strings, to produce logarithmic functions as CHY half-integrands. By using SE, we further present a remarkably simple formula for the half-integrand to any multiplicities which gives amplitudes for the single-trace sector of the theory. The formula for multitrace amplitudes and pure gluon ones are contained in our result as well, thus it gives CHY representations for all tree amplitudes in this theory.
II CHY integrand from string correlator
The double copy construction at tree-level can be most conveniently expressed by KLT product of color-ordered amplitudes: is defined as
[TABLE]
where the dimensional matrix is the so-called momentum kernel Bjerrum-Bohr et al. (2010) for orderings in a minimal basis. Instead of writing explicitly, it suffices to say that it is the inverse of the matrix formed by double-partial amplitudes in bi-adjoint theory: Cachazo et al. (2014b), for any choice of basis such that is invertible.
A generic massless open-string tree amplitude is given by a disk integral with denoting the ordering for integration domain :
[TABLE]
where , ; one can fix three punctures, e.g. , using SL redundancy, and the product in the Koba-Nielsen factor is over with this fixing. After stripping it off, the (reduced) string correlator is a rational function of ’s which depends on details of vertex operators. The only requirement here is that it has the correct SL weight: with and , it must transform as .
II.1 General claim
Our general claim, as summarized in Fig. 1, is that is the double copy of integrals (disk integral with a Parke-Taylor (PT) factor Broedel et al. (2013))
[TABLE]
with color-ordered amplitudes in a field theory. The latter is defined by a CHY formula, with integrands given by a PT factor as defined above, and a logarithmic function obtained from an IBP reduction of :
[TABLE]
Here the integrals are localized by the delta functions imposing scattering equations Cachazo et al. (2014c, a). Logarithmic functions are defined to have only logarithmic singularities on boundaries of the moduli space of -punctured Riemann spheres; equivalently, such a function can be written as a linear combination of PT factors Mizera (2017); Arkani-Hamed et al. (2017, 2018).
The idea that string amplitude can be written as a double copy has been realized in various theories Mafra et al. (2013a, b); Huang et al. (2016); Azevedo et al. (2018) and expected to hold for generic . For example, by using intersection theory, we can obtain from by computing intersection numbers 111See Sebastian Mizera’s talk at Amplitude 2018. In this work, we will give a concrete and streamlined IBP algorithm that reduces a generic explicitly to a class of logarithmic functions; the algorithm lands on a nonminimal basis of factors, making further simplifications accessible.
We actually have an equivalence class of logarithmic functions denoted as : any gives the same string integral as that of . As we will show shortly, gives exactly needed for the double copy, via CHY formula with a PT factor; any in the class gives the same result since they are also in an equivalent class on the support of SE. While the explicit form of gets more complicated as grows, we emphasize that it can usually be simplified greatly by use of SE (see Fig. 1). The manipulation from to is in a sense the reversed one as going from to : while being non-logarithmic, usually allows an all-multiplicity expression that makes other properties, such as gauge invariance, manifest. The primary example of this double copy is “type-I SYM” Mafra et al. (2013b, a), where the class of logarithmic functions from type-I correlator can be used to produce SYM amplitudes. It is now well known that the CHY half-integrand for external gluons, which takes a remarkably simple form 222We refer the readers to the original paper Cachazo et al. (2014a) for the definition of the matrix and more details on CHY formulas., is equivalent to on the support of SE.
Last but not least, one can generalize the double copy to closed-string amplitudes by replacing integrals by certain sphere integrals. As conjectured in Schlotterer and Stieberger (2013); Stieberger (2014); Stieberger and Taylor (2014) and proven in Schlotterer and Schnetz (2018); Brown and Dupont (2018), the latter can be obtained as the single-valued (sv) projection Schnetz (2014); Brown (2014) of open-string amplitudes, and e.g., we have “type-II sv(type-I) SYM”. All of our discussions hold for such double copies where IBP is applied to the holomorphic part of closed-string correlator (with the antiholomorphic part untouched): schematically we write Azevedo et al. (2018) , with the CHY half-integrand for obtained from the holomorphic correlator, .
II.2 Derivation: IBP reduction and double copy
We can conveniently represent any rational functions involving only ’s by a graph: we denote each in the denominator (numerator) by a solid (dashed) line directed from node to . With the gauge fixing, e.g. , all the labeled trees are logarithmic functions since we can write them as a sum of PT factors Gao et al. (2017).
The derivation of our general claim consists of two steps. We first give a method for reducing any string correlator to logarithmic functions via IBP relations. Next, we show that such logarithmic functions are precisely the CHY integrands for those field theories that enter the string-amplitude double copy mentioned above.
IBP reduction
We start with a simple example , which, after we gauge fix and subtract the total derivative , reduces to a logarithmic function, in this case a PT factor after restoring the SL covariance:
[TABLE]
where and . This example represents the simplest scenario where only a single subcycle and a labeled line are left after gauge fixing . The next simplest case is represented by tadpole graphs, as depicted in Fig. 2a, in which a single labeled line connects to the subcycle. One can reduce them to a sum of labeled trees by recursively using the IBP relation Schlotterer (2016):
[TABLE]
where and are obtained by matching . This relation holds when is multiplied by a function independent of to . The total derivative of is not used to derive this relation, and we identify it as the node shared by the subcycle and the tail. For example, Fig. 2a can be processed as:
[TABLE]
Notice that the tail always gets shortened, such that we can turn tadpole graphs into a sum of labeled trees by repeating the IBP relation (II.2).
By connecting more labeled lines to a tadpole, we get multibranch graphs, as shown in Fig. 2b. If we remove a subgraph including part of the subcycle, the rest of the graph turns into a labeled tree, which can be written as a sum of labeled lines through algebraic operations Gao et al. (2017). For example, we can view the chain in Fig. 2b as a labeled tree rooted on . Then partial fraction identities lead to
[TABLE]
In this way, we can turn all the multibranch graphs into tadpoles, and consequently labeled trees via IBP.
The most generic graph with no in the numerator or connected subcycles has the form of Fig. 2c, consisting of a product of tadpole and multibranch graphs, and a single labeled tree 333If there are more than one labeled trees, there must also be connected subcycles as a result of the SL weight constraint.. Following the above prescription, we can turn a tadpole or multibranch subgraph into labeled trees planted on the other connected components, reducing the number of subcycles by one. This algorithm allows us to recursively eliminate all the subcycles, and the final result contains only labeled trees, thus the function we obtain is logarithmic.
As we will see shortly, the above discussion applies to open bosonic and heterotic strings. We defer the treatment of generic rational functions involving numerators and connected subcycles to Appendix A.
String double copy
Now we prove that given by the obtained above is indeed the one in the double copy . As a logarithmic function, we can write , but the coefficients are not unique since the PT factors are not linearly independent. As shown for open strings Mafra et al. (2013a); Broedel et al. (2013) and CHY integrals Cachazo et al. (2014b), they satisfy BCJ relations by IBP and SE equivalence moves respectively. Consequently, any of them can be expressed in terms of the dimensional minimal basis:
[TABLE]
By using Eq. (2) and (3), we have
[TABLE]
for any . The sum in the bracket is a field-theory amplitude defined by CHY formula: recalling that , thus we have
[TABLE]
which gives exactly as in Eq. (4). Note that Eq. (9) makes it manifest that satisfies BCJ relations, which is necessary for a consistent KLT double copy. One can also define by the LHS, which can be rewritten in a form that respects color/kinematics duality with BCJ numerators given by linear combinations of ’s. However, we find it extremely useful to turn to CHY formula on the RHS: from our explicit result , we can use SE to simplify it to some , for which it is possible to obtain a closed form.
One can apply the IBP reduction to type-I correlators and obtain logarithmic functions, where the coefficients give the BCJ numerators for SYM Mafra et al. (2013b). For gluons, the standard CHY half-integrand can also be reduced via SE to , giving equivalent BCJ numerators Du and Teng (2017). Remarkably, these BCJ numerators are free of any tachyon poles (though they appear in intermediate steps as in above examples), and unlike , both and are homogenous in , all of which are of course guaranteed by the worldsheet supersymmetry. Next we will turn to bosonic/heterotic strings, where BCJ numerators and contains tachyon poles and nontrivial dependence. We will see that our two-step method again gives a remarkably simple formula for .
III New formulas for bosonic and heterotic strings
Let’s first review correlators for (compactified) open bosonic strings and heterotic strings. Since the former is identical to the holomorphic part of the latter, we will focus on the heterotic-string case and everything carries over to the open-string case. By a slight abuse of notation, we denote closed-string measure as , and the single-trace tree amplitude of gravitons and gluons is given by Schlotterer (2016)
[TABLE]
where denotes the gluon color ordering. The holomorphic function contains half of the graviton vertex operator correlation function, taking the form
[TABLE]
where we sum over permutations in , each of which is written as a product of disjoint cycles. The cycle factor is given by , and for . The antiholomorphic function is the type-I correlator that encodes vertex operators of gluons and the other half of the gravitons.
For the single-trace integrand, we can always gauge fix the gluon such that we only need to break the -cycles in by IBP. The one-graviton integrand is automatically in the logarithmic form since contains no -cycles, i.e. . For two gravitons, we have , which has only one -cycle . We can break it by the IBP
[TABLE]
which holds when does not involve . This leads to the logarithmic function
[TABLE]
with the SL covariance restored. As another example, we present the IBP of the three-graviton integrand in Appendix B. Similar calculation can also be found in Schlotterer (2016) for a slightly different purpose. Although the complexity of grows quickly with more gravitons, it is still well controlled under our algorithm. Moreover, as the second step, we can greatly simplify by using SE and obtain a closed-form for the most generic single-trace sector.
Similar treatment can in principle be applied to multitrace and pure graviton sectors. We give several such examples in Appendix E and F.
III.1 The single-trace integrand
Now we present the single-trace CHY half-integrand , resulted from our two-step reduction of string integrand (10):
[TABLE]
where is given by the cycle expansion
[TABLE]
For length one and two cycles, we have and , where is given by , and is the linearized field strength. For longer cycles (), we have , where is recursively generated from through
[TABLE]
with for . In this equation, the summation is over all cyclic partitions (CP) of a cycle into words, where the length of each word is at least two; denotes the first element of , which is the label in that succeeds the word . For example, for , we have four CPs with only one word, , , , , and two more with two words, and ; the sum on the RHS of Eq. (16) thus include terms like , and , etc. Given a word and the next label , the function is defined as
[TABLE]
where , and in particular . The bracket stands for the index antisymmetrization, for example, . The summation range ensures that the right hand side of Eq. (16) only contains those ’s with shorter cycles, since agrees with only when . In Appendix C, we give examples of up to five gluons.
Note that the closed-formula for is still a conjecture, since it is generalized from explicit calculations at low points. However, as will be discussed in great detail in He et al. , the formula has passed various nontrivial consistency checks. On the support of SE, we have checked numerically up to eleven points that Eq. (14) agrees with the logarithmic form resulted from the IBP of the string integrand (10). We have also checked up to eight points that the gauge amplitude produced by Eq. (14) agrees with that calculated from the Feynman diagrams of with massless external particles. Further consistency checks are provided in Appendix D.
IV Conclusion and discussion
In this work we have presented an IBP reduction method for correlators of any massless string amplitude, , to a class of logarithmic functions, . The latter can be used as CHY half-integrand (with the other half given by PT) to define field-theory amplitudes, and their double copy with disk/sphere integrals gives the original string amplitude. Although generally takes a complicated form, one can use SE to simplify it and even get a closed form to all multiplicities. We have demonstrated this two-step method for open (compactified) bosonic/heterotic strings, and obtained remarkably simple formulas for single-trace and certain multi-trace sectors of theory. It can bring new insight into this theory, as well as conformal gravity which is obtained from a double copy with YM Johansson and Nohle (2017); Johansson et al. (2018).
Our discussion is completely generic since it only requires to be a rational function with correct SL weight. It can thus be applied beyond those well-studied conventional string theories, for example, to the dual model proposed in Ref. Baadsgaard et al. (2016). An important open question concerns the application of IBP reduction to string correlators with massive states, and it would be fascinating to see if similar structures exist there.
On the other hand, the CHY half-integrands obtained in this work contain explicit dependence. It will be interesting to understand how they can emerge from certain model of ambitwistor string Mason and Skinner (2014); Casali et al. (2015); Geyer and Monteiro (2018). Some attempts have been made along this direction, and give the correct three-point results Azevedo and Jusinskas (2017). Last but not least, it is highly desirable to extend certain aspects of our construction to loop-level string amplitudes/correlators, and see how they are related to field-theory amplitudes/ambitwistor string correlators.
Acknowledgements.
It is our pleasure to thank Thales Azevedo, Henrik Johansson, Sebastian Mizera and Oliver Schlotterer for inspiring discussions and helpful comments on our draft. S.H.’s research is supported in part by the Thousand Young Talents program, the Key Research Program of Frontier Sciences of CAS under Grant No. QYZDB-SSW-SYS014 and Peng Huanwu center under Grant No. 11747601. F.T. is supported by the Knut and Alice Wallenberg Foundation under grant KAW 2013.0235, and the Ragnar Söderberg Foundation under grant S1/16.
Appendix A More on IBP
In Sec. II.2, we have proved that those gauge-fixed string correlators without numerators or connected subcycles can be reduced into logarithmic functions via IBP. Now we complete the proof for generic rational function with correct SL weight by showing that we can always reduce them to the former case. We start with an example . We can absorb the numerator into the Koba-Nielsen factor as
[TABLE]
Now we can gauge fix and perform the IBP as in Eq. (5), while the Mandelstam variables in are shifted as and . We then restore the gauge covariance and release the cross ratio from :
[TABLE]
The product in the bracket can be rearranged into a sum of rational functions without in the numerators:
[TABLE]
Although the second term is not a logarithmic function, we have reduced the problem to the one already solved in Eq. (5): by using this IBP relation again, we obtain the final logarithmic function:
[TABLE]
which involves exotic tachyon poles.
Our example captures the essence of the general idea. In any SL weight two integrand , if appear in the numerator, and must also appear in the denominator in, say, and . We can remove the numerator by absorbing the cross ratio into the Koba-Nielsen factor, schematically shown as
[TABLE]
By repeating this operation, we obtain an integrand that contains only disjoint subcycles. After gauge fixing, we can then follow Sec. II.2 and reduce it to a sum of PT factors via IBP relations, like the first row of Eq. (A).
Next, we release a cross ratio, say, from the deformed Koba-Nielsen factor . For each PT factor, we first use KK relations to move next to , and then gauge fix . The resultant integrand only contains a tadpole and a labeled line:
[TABLE]
which can be turned into a sum of PT factors via IBP according to Sec. II.2. This process is captured by Eq. (A) and (21) in our example. By recursively releasing cross ratios from the deformed Koba-Nielsen factor, we will eventually land on a logarithmic function together with the original string measure, which completes our constructive proof.
Appendix B Single trace integrand with three gravitons
In this section, we derive the three-graviton CHY integrand from IBP. This calculation is similar to the one in Ref. Schlotterer (2016), but we present the result in a different form. We start with , given by Eq. (11):
[TABLE]
and we work with the gauge fix .
Our first step is to separate out the length-two and three -cycles contained in and combine them with the second row of Eq. (B):
[TABLE]
where we have defined and
[TABLE]
In Eq. (B), the two rows above the dashed line do not contain any -cycles, as one can check straightforwardly. The first three terms below the dashed line of Eq. (B) contain only length-two -cycles, while the last two terms contain only length-three cycles. All the length-two cycles can be broken by Eq. (12) and the following relation:
[TABLE]
which holds when its coefficient does not contain . We note that this IBP generates new length-three cycles.
Next, we can break all the length-three cycles by the IBP relation
[TABLE]
which holds when its coefficient does not contain either or . The final result is
[TABLE]
After being multiplied by a PT factor, Eq. (B) is just the logarithmic form integrand for the three-graviton case.
Since is also a CHY integrand, we can use SE to rewrite it into a more compact form. For example, the summation in the last row of Eq. (B) can be reduced to
[TABLE]
One can easily show that on the support of SE, Eq. (B) agrees with the given in the next section. For generic cases, the operation resembles the inverse one as shown in Cardona et al. (2016); Bjerrum-Bohr et al. (2016); Huang et al. (2017).
Appendix C Single-trace examples
Now we provide a few examples for our generic single-trace formula given in Sec. III.1. The integrand involving two gravitons is given by , where
[TABLE]
which agrees with the logarithmic function (13) after some use of SE. For three gravitons, Eq. (15) gives
[TABLE]
where, according to Eq. (16), is given by
[TABLE]
In the four-graviton integrand , the only piece that has not appeared in or is in the length-four cycle factor , which is given by
[TABLE]
The first example of index antisymmetrization for function as shown in Eq. (17) is
[TABLE]
It is contained in
[TABLE]
that appears in the length-five cycle factor .
Appendix D Consistency checks
Last but not least, we list some all-multiplicity checks. From Eq. (16) and (17), one can see that satisfies the same cyclic and reflection property as . Gauge invariance is manifest since is built from ’s and is gauge invariant on the support of SE. Moreover, in the limit , we have if , such that factorizes as , recovering the universal single soft behavior Weinberg (1965) after the CHY integration. Furthermore, as , all the tachyon poles drop out, and by induction it is easy to see that all the -factors vanish as . Consequently, reduces to the cycle expansion of a Pfaffian Lam and Yao (2016), reproducing the single-trace YM-scalar integrand Cachazo et al. (2015b). On the other hand, the ’s vanish as when , leaving only ’s in , which lands exactly on the integrand Azevedo and Engelund (2017).
The most important check is to show that the integrand given by (14) has the correct massless factorization properties for the amplitudes He et al. . Following Cachazo et al. (2014b), and PT’s split correspondingly in the factorization limit , thus we only need to consider the behavior of . The scalar channel, which separates the gluons into two sets, is simple: the leading order gives a simple pole , and factorizes nicely since all ’s that contain gluons in both sets only contribute to higher orders. In other words, the cycle expansion form shown in Eq. (15) trivializes the scalar factorization channel.
The gluon channel is more interesting. For simplicity, we consider the special one that separates the gluons and the scalars. The leading order is now a double pole whose coefficient is proportional to evaluated on the -point kinematics, while the subleading order provides the simple pole . The leading order must vanish, and indeed we have checked up to eight points that
[TABLE]
When , this statement is simply that the matrix has corank two for -point kinematics. Remarkably, inherits this corank-two property for finite ! One can verify this statement by showing that the coefficient of each tachyon pole structure of vanishes on the support of the SEs, as we will present in He et al. .
Appendix E Towards multi-trace integrands
The generalization of the single-trace string integrand to color traces is straightforward:
[TABLE]
where and . Following the same prescription outlined in Sec. III, we can in principle get the CHY integrand for multitrace sectors.
Interestingly, we can also get multitrace by taking certain coefficients of the single-trace integrand . As the most general formula will be given in He et al. , here we simply present it for some cases. To start with, the double-trace integrand can be obtained by the prescription
[TABLE]
where denotes another gluon trace (in addition to ) and denotes the new set of gravitons. The simplest one is the pure-gluon double trace case already given in Schlotterer (2016); Azevedo et al. (2018): . We also give a closed-form expression for double trace with one additional graviton, denoted by label :
[TABLE]
where we first sum over all pairs of labels , and for each choice we read out words and by rewriting the cycle in the form ; then the sum is over words with endpoints and , with shuffle of and in between. On the support of SEs, Eq. (39) is invariant under the relabeling of the traces, although this property is not explicit.
Furthermore, we can extract a third gluon trace from the double-trace integrand by the same process of Eq. (38). Remarkably, the pure-gluon triple-trace result also takes a compact form:
[TABLE]
At the leading order of , both Eq. (39) and (E) reproduce the YM-scalar theory in the corresponding multitrace sectors Cachazo et al. (2015a); Chiodaroli et al. (2015, 2017).
We have checked that Eq. (39) and (E) agree with IBP result starting from Eq. (34) on the support of SE. We can get generic multitrace integrands by repeating Eq. (38), which has been numerically checked up to eleven points and five traces with IBP results.
We provide two examples for Eq. (39) and (E), both of which are at six points:
[TABLE]
[TABLE]
Appendix F Pure gluon integrand
It turns out to be convenient to write pure integrand as
[TABLE]
where the second term vanishes at . Moreover, by taking in , we should reproduce the half-integrand for a single insertion of vertex He and Zhang (2017). This instructs us to use a similar cycle expansion for :
[TABLE]
where counts the number of cycles with length at least two. The bracket stands for certain “averaging” over cycles with length at least two, which will be spelled out shortly, while length-one cycles factor out as:
[TABLE]
The longest cycles involved in have length .
At four points, the amplitude is explicitly given in Azevedo et al. (2018), which can be produced by
[TABLE]
where the last three terms are given by the relabeling of
[TABLE]
When going to higher points, the single soft limit fixes all the terms involving length-one cycles, while the requirement of no double pole at massless factorization channels puts strong constraints on the terms without length-one cycles. Actually, we find that up to seven points these constraints lead to a unique answer to the ansatz (44), which involves a modified version of :
[TABLE]
where depends on , the number of partitions. For , we always have , such that the only relevant ’s are and . In the following, we give explicitly the building blocks of up to seven points.
Five points:
Besides Eq. (47), we also need
[TABLE]
The overall normalization here is different from that in Eq. (47) because of the different normalizations adopted for and longer cycles.
Six points:
In addition to Eq. (47) and (49), there are three more building blocks:
[TABLE]
Seven points:
There are again three new building blocks emerge at :
[TABLE]
Up to seven points, we find that simply averages over and . We leave the study of generic cases to future works.
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