Higher-loop amplitude monodromy relations in string and gauge theory

Higher-loop amplitude monodromy relations in string and gauge theory

Piotr Tourkine, Pierre Vanhove
DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
Institut de physique théorique, Université Paris Saclay, CNRS, F-91191 Gif-sur-Yvette, France

The monodromy relations in string theory provide a powerful and elegant formalism to understand some of the deepest properties of tree-level field theory amplitudes, like the color-kinematics duality. This duality has been instrumental in tremendous progress on the computations of loop amplitudes in quantum field theory, but a higher-loop generalisation of the monodromy construction was lacking.

In this letter, we extend the monodromy relations to higher loops in open string theory. Our construction, based on a contour deformation argument of the open string diagram integrands, leads to new identities that relate planar and non-planar topologies in string theory. We write one and two-loop monodromy formulæ explicitly at any multiplicity. In the field theory limit, at one-loop we obtain identities that reproduce known results. At two loops, we check our formulæ by unitarity in the case of the four-point super-Yang-Mills amplitude.

preprint: DAMTP-2016-53, IPhT-t16/069

The search for the fundamental properties of the interactions between elementary particles has been the driving force to uncover basic and profound properties of scattering amplitudes in quantum field theory and string theory. In particular, the colour-kinematic duality Bern:2008qj has led to tremendous progress in the evaluation of loop amplitudes in gauge theories Bern:2010ue; Bern:2010tq; Boels:2012ew; Mafra:2014gja; Nohle:2013bfa; Bern:2013yya; Badger:2015lda; Mafra:2011kj; Mafra:2015mja; Chester:2016ojq; Primo:2016omk; Geyer:2015bja; Carrasco:2011hw. One remarkable consequence of this duality is the discovery of unsuspected kinematic relations between tree-level gauge theory amplitudes Bern:2008qj, generated by a few fundamental relations BjerrumBohr:2009rd; Stieberger:2009hq; Feng:2010my; Johansson:2015oia; delaCruz:2015dpa.

The monodromies of the open string disc amplitudes BjerrumBohr:2009rd; Stieberger:2009hq did provide a rationale for the kinematic relations between amplitudes at tree-level in gauge theory. However, while the colour-kinematics duality has been successfully implemented up to the fourth loop order in field theory Bern:2010tq; Boels:2012ew, there is not yet a systematic understanding of its validity to all loop orders. It is therefore natural to seek a higher-loop generalisation of the string theory approach to these kinematic relations.

In this paper we generalise the tree-level monodromy construction to higher-loop open string diagrams (worldsheets with holes). This allows us derive new relation between planar and non-planar topologies of graphs in string theory. The key ingredient in the construction relies on using a representation of the string integrand with a loop momentum integration. This is crucially needed in order to be able to understand zero mode shifts when an external state jumps from one boundary to another. Furthermore, just like at tree-level, the construction does not depend on the precise nature of the scattering amplitude nor the type of theory (bosonic or supersymmetric) considered.

The relations that we obtain in field theory emanate from the leading and first order in the expansion in the inverse string tension . At leading order, we find identities between planar and non-planar amplitudes. At the next order, stringy corrections vanish and we find the loop monodromy relations. They are relations between integrands up to total derivatives, that involve both loop and external momenta. Upon integration, this give relations between amplitude-like integrals with extra powers of loop momentum in the numerator.

At one loop, our string theoretic construction reproduces the field theory relations of Boels:2011tp; Boels:2011mn; Du:2012mt. In observing how the loop momentum factors produce cancellations of internal propagators, we see that BCJ colour-kinematic representations for numerators Bern:2008qj satisfy the monodromy relation at the integrand level. The generality of our construction lead us to conjecture that our monodromies generate all the kinematic relations at any loop order.

We conclude by showing how our construction extends to higher loops in string theory. In particular we write the two-loop string monodromy relations. The field theory limit is subtle to understand in the general case, but we provide a proof of concept with an example in super-Yang-Mills at four-point two-loop, which we check by unitarity. We leave the general field theory relations for future work.

I Monodromies on the annulus

One-loop -particle amplitudes in oriented open-string theory are defined on the annulus. They have a gauge group and the following colour decomposition Green:1987mn


The summation over of the external states distributed on the boundaries of the annulus consists of permutations modulo cyclic reordering and reflection symmetry. The quantities and are the external momenta, polarizations and colour matrices in the fundamental representation, respectively. Planar amplitudes are obtained for or with . The color-stripped ordered -gluon amplitude take the following generic form in dimensions


where is the modulus of the annulus and the ’s are the location of the gluons insertions on the string worldsheet – one of them is set to by translation invariance. The loop momentum is defined as the average of the string momentum  D'Hoker:1988ta;


The domain of integration is the union of the ordered sets for and for .

We will show that the kinematical relations at one-loop arise exclusively from shifts in the loop-momentum-dependent part and monodromy properties of the non-zero mode part of the Green’s function in (2)


We refer to the appendix for some properties of the propagators between the same and different boundaries.

The function contains all the theory-dependence of the amplitudes. The crucial point of our analysis is that it does not have any monodromy, therefore the relations that we obtain are fully generic. This function is a product of partition functions, internal momentum lattice of compactification to dimensions, and a prescribed polarisation dependence Green:1987mn; Green:1981ya; Mafra:2012kh; He:2015wgf. The latter is composed of derivatives of the Green’s function. None of these objects have monodromies: that is why the precise form of does not matter for our analysis. This property carries over to higher-loop orders.

i.1 Local and global monodromies

Figure 1: The contour integral (red) vanishes. The two boundaries (black) have opposite orientation.

Let us consider the non-planar amplitude , but where we take the modified integration contour of fig. 1 for . The integrand being holomorphic, in virtue of Cauchy’s theorem, the integral vanishes:


Each separate portion of the integration corresponds to a different ordering and topology. The portions along the vertical sides cancel by periodicity of the one-loop integral (cf. appendix). We are thus left with the contributions from the boundaries and . When exchanging the position of two states on the same boundary, the short distance behaviour of the Green’s function implies


with for a clockwise rotation and for a counter-clockwise rotation. Thus, on the upper part of the contour in figure 1, exchanging the positions of two external states leads to an phase factor multiplying the amplitude


On the lower part of the contour in figure 1, the phases come with the same sign due to an additional sign from in eq. (27). For external states on different boundaries, the Green’s function involves the even function and the ordering does not matter (cf. the appendix).

The main difference with the tree-level case arises from the global monodromy transformation when a state moves from one boundary to the other, . This produces a new phase in the integrand


On non-orientable surfaces the propagator is obtained by appropriate shifts of the Green’s function (4) according the effects of the twist operators Green:1981ya. The local monodromies are the same because they only depend on the short distance behaviour of the propagator, and global monodromies are obtained in an immediate generalisation of our construction.

i.2 Open string relations

We can now collect up all the previous pieces. Paying great care to signs and orientations, according to what was described, the vanishing of the integral along gives the following generic relation111Compared to earlier versions, we correct here a sign mistake in the non-planar phases. Because of this mistake, in fig. 1, we took the cuts of the non-planar vertical contour to be downard cuts, the corrected version has upward cuts. The analysis for the cuts is unchanged. Details on the correct version are given in (Ochirov:2017jby, Appendix B).


where the bracket notation was defined in (8) and we set . In particular, starting from the planar four-point amplitude we find the following formula


We also find, starting from a purely planar amplitude


where now we integrate the vertex operators with ordered position along the contour of fig. 1. The sum is over the shuffle product and the permutation of length , and if in and 1 otherwise. The phase factors with external momenta are the same as at tree-level: the new ingredients here are the insertions of loop-momentum dependent factors inside the integral.

Note that some of our relations involve objects like that seemingly contribute in (1) only if the state 1 is a colour singlet. However, our relations involve colour-stripped objects and are, therefore, valid in full generality. Note also that our relations are valid under the -integration, thus they are not affected by the dilaton tadpole divergence at  Green:1981ya.

We have thus shown that the kinematic relations (9) relate planar and non-planar open string topologies, which normally have independent colour structures. This is the one-loop generalisation of the string theory fundamental monodromies that generates all amplitude relations at tree-level in string theory BjerrumBohr:2009rd; Stieberger:2009hq. Thus, we conjecture our one-loop relations (9), written for all the permutations of the external states, generate all the one-loop oriented open string theory relations. Let us now turn to the consequences in field theory.

Ii Field theory relations

Gauge theory amplitudes are extracted from string theory ones in the standard way. We send and keep fixed the quantity that becomes the Schwinger proper-time in field theory. We also set , with . The Green’s function of eq. (4) reduces to the sum of the field theory worldline propagator and a stringy correction


(for details see appendix).222In bosonic open string one would need to keep to the terms of the order because of the Tachyon. At leading order in , open string amplitudes reduce to the usual parametric representation of the dimensional regulated gauge theory amplitudes Green:1982sw; Bern:1992cz.333See also Bern:1990cu; Bern:1990ux; Bern:1991aq for equivalent closed string methods All the monodromy phase factors reduce to and from (11) we recover the well-known photon decoupling relations between non-planar and planar amplitudes Bern:1994zx, with ,


This is an important consistency check on our relations.

At the first order in we get contributions from expansion of the phase factors but as well potential ones from the massive stringy mode coming from . The analysis of the appendix of Green:1999pv shows that this contributes to next order in , which, importantly, allow us to neglect it here. Therefore, the field theory limit of (9) gives a new identity


These relations are the one-loop equivalent of the fundamental monodromy identities Feng:2010my; Johansson:2015oia; delaCruz:2015dpa that generates all the amplitude relations at tree-level.

In particular, using (13), we obtain the relation between planar gauge theory integrands with linear power of loop momentum


These are the relations derived in Boels:2011tp; Boels:2011mn; Du:2012mt: this constitutes an additional check on our formulæ.

Let us now analyse the effect of the linear momentum factors at the level of the graphs. At this point we pick any representation of the integrand in terms of cubic graphs only and the field theory limit defines the loop momentum as the internal momentum following immediately the leg .444This is checked by matching with usual definition of the Schwinger proper times. We then rewrite the loop momentum factors as differences of propagators. Hence, each individual graph with numerator produces two graphs with one fewer propagator, e.g. {fmffile}pent-red


Then, there always exist another graph that will produce one of the two reduced graphs as well, with a different numerator . In the previous example, it would be the pentagon for the massive box with corner. Finally, reduced graphs also arise directly from string theory, when vertex operators collide Bern:1992cz. In (15), these always appear in such combinations of two graphs, say and ; {fmffile}box-box


The color ordered 3-point vertex is antisymmetric, so and the terms cancel. We then realize that the graphs entering the monodromy relations can be organised by triplets of Jacobi numerators times denominator. In a BCJ representation, all these triplets vanish identically and eq. (14) is satisfied at the integrand level. Thus, any BCJ representation satisfies these monodromy relations, but the converse is not true.

Iii Toward Higher-loop relations

Higher-loop oriented open string diagrams are worldsheets with holes, one for each loop.555 We do not consider string diagrams with handles in this work. They lead to non-planar corrections Berkovits:2009aw. Just like at one loop, we consider the integral of the position of a string state on a contractible closed contour that follows the interior boundary of the diagram (cf. for instance fig. 2). The integral vanishes without insertion of closed string operator in the interior of the diagram. This constitutes the essence of the monodromy relations at higher-loop.

Figure 2: Two-loop integrand monodromy. Integration over the red contour vanishes. Given the definition of the loop momentum in eq. (18), parallel integrations along cancel only up to a shift in the loop momentum.

Because the exchange of two external states on the same boundary depends only on the local behaviour of the Green’s function, we have the same local monodromy transformation as at tree-level.

Like at one loop, the global monodromy of moving the external state 1 from one boundary to another boundary by crossing the cycle leads to the factor . The loop momenta are the zero-modes of the string momenta  D'Hoker:1988ta. The string integrand depends on them through the factor:


Importantly, the integration path between and in (18) depends on a homology class. This implies that this expression has an intrinsic multivaluedness, corresponding to the freedom of shifting the loop momentum by external momenta when punctures cross through the cycles.666 Doing the Gaussian integration reduces to the standard expression of the string propagator, which is single valued on the surface.. Choosing one for each of these contours induces a choice of cuts on the worldsheet along given cycles that renders the expression single-valued. Our choice to make the cycle join at some common point also removes the loop momentum shifting ambiguity and give globally defined loop momenta.

A two-loop example.

The generalisation of (9) gives the two-loop integrated relations777Compared to earlier versions, we corrected a sign in the non-planar phases. Higher-loop phases are related to the ones at one-loop by the factorisation limit of the string amplitude.


At four points we get


where etc. are planar two-loop amplitude integrand, and are the two non-planar amplitude integrands with the external state 1 on the -cycle with , as fig. 2. The field theory limit of that relation, at leading order in , leads to


where are the leading colour field theory single trace amplitudes, and with our choice of orientation of the cycles is the double trace field theory amplitude. We recover the relation obtained by unitarity method in Feng:2011fja. For SYM, the graphs are essentially scalar planar and non-planar double boxes Bern:1997nh, and this relation is easily verified by inspection, thanks to the antisymmetry of the three-point vertex. At order , we conjecture that the field theory limit yields;


These relations are not reducible to KK-like colour relations, like these of Naculich:2011ep, just like at tree-level where BCJ kinematic relation go beyond KK ones. An extension of the one-loop argument Tourkine:2012ip indicates that the massive string corrections to the field theory limit of the propagator does not contribute at the first order in . A detailed verification of this kind of identities will be provided somewhere else, but we give below a motivation by considering the two-particle discontinuity in the case of SYM. The two-particle -channel cut of the two-loop amplitude is the sum of two contributions, with one-loop and tree-level amplitudes, and  Bern:1998ug, respectively:


where and are the on-shell cut loop momenta. The -channel two-particle cut of (22) gives a first contribution


where and are the cut momenta. This expression vanishes thanks to the monodromy relation between the four-point tree amplitudes in the parenthesis Bern:2008qj; BjerrumBohr:2009rd; Stieberger:2009hq. The second contribution is


where is the one-loop loop momentum and and are the cut momenta. This expression vanishes thanks to the four-point one-loop monodromy relation (15) in the parenthesis. We believe that this approach has the advantage of fixing some ambiguities in the definition of loop momentum in quantum field theory. And the implications of the monodromy relations at higher-loop in maximally supersymmetric Yang-Mills, by applying our construction to the world-line formalism of Dai:2006vj, will be studied elsewhere.

Finally, we note that our construction should applies to both the bosonic or supersymmetric string, as far as the difficulties concerning the integration of the supermoduli Witten:2012bh can be put aside.


We would like to thank Lance Dixon for discussions and Tim Adamo, Bo Feng, Michael B. Green, Ricardo Monteiro, Alexandre Ochirov, Arnab Rudra for useful comments on the manuscript.

The research of PV has received funding the ANR grant reference QST 12 BS05 003 01, and the CNRS grants PICS number 6430. PV is partially supported by a fellowship funded by the French Government at Churchill College, Cambridge. The work of PT is supported by STFC grant ST/L000385/1. The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme “Gravity, Twistors and Amplitudes” where work on this paper was undertaken. This work was supported by EPSRC grant no EP/K032208/1.


Appendix A Planar and non-planar Green function

The Green function between two external states on the same boundary of the annulus is given by with


and between two external states on the different boundaries of the annulus is given by thanks to the relation between the functions under the shift




The periodicity around the loop follows from


and an appropriate redefinition of the loop momentum.

The string theory correction to the field theory propagator in (12) is


is the contribution of massive string modes propagating between two external states on the same boundary and on different boundaries.


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