Preprints 2022

Classical gravitational spinningspinless scattering at $\mathcal{O}(G^2S^\infty)$
Authors: Rafael Aoude, Kays Haddad, Andreas Helset
Preprint number: UUITP24/22
Making use of the recentlyderived, allspin, oppositehelicity Compton amplitude, we calculate the classical gravitational scattering amplitude for one spinning and one spinless object at $\mathcal{O}(G^2)$ and all orders in spin. By construction, this amplitude exhibits the spin structure that has been conjectured to describe Kerr black holes. This spin structure alone is not enough to fix all deformations of the Compton amplitude by contact terms, but when combined with considerations of the ultrarelativistic limit we can uniquely assign values to the parameters remaining in the eveninspin sector. Once these parameters are determined, much of the spin dependence of the amplitude resums into hypergeometric functions. Finally, we derive the eikonal phase for alignedspin scattering.

Gauge invariance from onshell massive amplitudes and tree unitarity
Authors: Da Liu, Zhewei Yin
Preprint number: UUITP23/22
We study the threeparticle and fourparticle scattering amplitudes for an arbitrary, finite number of massive scalars, spinors and vectors by employing the onshell massive spinor formalism. We consider the most general threeparticle amplitudes with energy growing behavior at most of O(E). This is the special case of the requirement of tree unitarity, which states that the Nparticle scattering amplitudes at treelevel should grow at most as O(E^(4N)) in the high energy hard scattering limit, i.e. at fixed nonzero angles. Then the factorizable parts of the fourparticle amplitudes are calculated by gluing the onshell threeparticle amplitudes together and utilizing the fact that treelevel amplitudes have only simple poles. The contact parts of the fourparticle amplitudes are further determined by tree unitarity, which also puts strong constraints on the possible allowed threeparticle coupling constants and the masses. The derived relations among them converge to the predictions of gauge invariance in the UV theory. This provides a purely onshell understanding of spontaneously broken gauge theories.

Global Structures from the Infrared
Authors: Michele Del Zotto and Iñaki García Etxebarria
Preprint number: UUITP22/22
Abstract: Quantum field theories with identical local dynamics can admit different choices of global structure, leading to different partition functions and spectra of extended operators. Such choices can be reformulated in terms of a topological field theory in one dimension higher, the symmetry TFT. In this paper we show that this TFT can be reconstructed from a careful analysis of the infrared Coulomblike phases. In particular, the TFT matches between the UV and the IR. This provides a purely field theoretical counterpart of several recent results obtained via geometric engineering in various string/M/F theory setups for theories in four and five dimensions that we confirm and extend.

Snowmass White Paper: the DoubleCopy and its Applications
Authors: Tim Adamo, John Joseph M. Carrasco, Mariana CarrilloGonzález, Marco Chiodaroli, Henriette Elvang, Henrik Johansson, Donal O'Connell, Radu Roiban, Oliver Schlotterer
Preprint number: UUITP21/22
The doublecopy is, in essence, a map between scattering amplitudes in a broad variety of familiar field and string theories. In addition to the mathematically rich intrinsic structure, it underlies a multitude of active research directions and has a range of interesting applications in quantum, classical and effective field theories, including broad topics such as string theory, particle physics, astrophysics, and cosmology. This Snowmass White Paper provides a brief introduction to the doublecopy, its applications, current research and future challenges.

The eikonal operator at arbitrary velocities I: the softradiation limit
Authors: Paolo Di Vecchia, Carlo Heissenberg, Rodolfo Russo, Gabriele Veneziano
Preprint: UUITP20/22
Abstract: Observables related to the real part of the gravitational eikonal, such as the deflection angle and time delay, have been found so far to have a smooth postMinkowskian (PM) expansion whose validity extends from the nonrelativistic to the most extreme ultrarelativistic (UR) regime, which smoothly connects with massless particle collisions. To describe gravitational radiation, the eikonal phase has to be promoted to a unitary operator for which we motivate a proposal and start discussing properties in the softradiation limit. A convergent PM expansion is found to only hold below an UR bound (discussed in the GR literature in the seventies) above which a different expansion is instead needed implying, in general, some nonanalyticity in Newton's constant. In this extreme UR regime soft radiative observables receive contributions only from gravitons and are therefore universal. This generalises the pattern discussed in~\cite{DiVecchia:2020ymx} beyond the elastic case.

Radiation reaction for spinning blackhole scattering
Authors: Francesco Alessio, Paolo Di Vecchia
Preprint: UUITP19/22
Abstract: Starting from the leading soft term of the 5point amplitude, involving a graviton and two Kerr black holes, that factorises into the product of the elastic amplitude without the graviton and the leading soft factor, we compute the infrared divergent contribution to the imaginary part of the twoloop eikonal. Then, using analyticity and crossing symmetry, we determine the radiative contribution to the real part of the twoloop eikonal and from it the radiative part of the deflection angle for spins aligned to the orbital angular momentum, the loss of angular momentum and the zero frequency limit of the energy spectrum for any spin and for any spin orientation. For spin one we find perfect agreement with recent results obtained with the supersymmetric worldline formalism.

The SAGEX Review on Scattering Amplitudes Chapter 2: An Invitation to ColorKinematics Duality and the Double Copy
Authors: Zvi Bern, John Joseph Carrasco, Marco Chiodaroli, Henrik Johansson, Radu Roiban
Preprint: UUITP18/22
Abstract: Advances in scattering amplitudes have exposed previouslyhidden colorkinematics and doublecopy structures in theories ranging from gauge and gravity theories to effective field theories such as chiral perturbation theory and the BornInfeld model. These novel structures both simplify higherorder calculations and pose tantalizing questions related to a unified framework underlying relativistic quantum theories. This introductory minireview article invites further exploration of these topics. After a brief introduction to colorkinematics duality and the double copy as they emerge at tree and looplevel in gauge and gravity theories, we present two distinct examples: 1) an introduction to the web of doublecopyconstructible theories, and 2) a discussion on the application of the double copy to calculations relevant to gravitationalwave physics.

2Group Symmetries and MTheory
Authors: Michele Del Zotto, Iñaki García Etxebarria, Sakura SchäferNameki
Preprint: UUITP17/22
Quantum Field Theories engineered in Mtheory can have 2group symmetries, mixing 0form and 1form symmetry backgrounds in nontrivial ways. In this paper we develop methods for determining the 2group structure from the boundary geometry of the Mtheory background. We illustrate these methods in the case of 5d theories arising from Mtheory on ordinary and generalised toric CalabiYau cones, including cases in which the resulting theory is nonLagrangian. Our results confirm and elucidate previous results on 2groups from geometric engineering.

Angular momentum of zerofrequency gravitons
Authors: Paolo Di Vecchia, Carlo Heissenberg, Rodolfo Russo
Preprint: UUITP16/22
By following closely Weinberg's soft theorem, which captures the $1/\omega$ pole contribution to the amplitude for soft graviton emissions ($\omega<\Lambda$) on top of an arbitrary background hard process, we calculate the expectation value of the graviton's angular momentum operator for arbitrary collisions dressed with soft radiation. We find that the result becomes independent of the cutoff $\Lambda$ on the graviton's frequency, effectively localizing at $\omega=0$. In this way, our result captures the contribution to the angular momentum that comes from the zerofrequency modes. Like the soft theorem, our formula has an exact dependence on the kinematics of the hard particles and is only a function of their momenta.
As an example, we discuss in some detail the case of the $2 \to 2$ scattering of spinless particles in General Relativity and ${\cal N}=8$ supergravity. 
Exact TT deformation of twodimensional Maxwell theory
Authors: Luca Griguolo, Rodolfo Panerai, Jacopo Papalini, Domenico Seminara
Preprint: UUITP15/22
TTdeformed twodimensional quantum Maxwell theory on the torus is examined, taking into account nonperturbative effects in the deformation parameter μ. We study the deformed partition function solving the relevant flow equation at the level of individual flux sectors. Summing exactly the “instanton” series, we obtain a welldefined expression for the partition function at arbitrary μ. For μ > 0, the quantum spectrum of the theory experiences a truncation, the partition function reducing to a sum over a finite set of positiveenergy states. For μ < 0 instead, the appearance of nonperturbative contributions in μ drastically modifies the structure of the partition function, regularizing its naive divergences through instantonlike subtractions. For each flux sector, we show that the semiclassical contribution is dominated by the deformed classical action. The theory is observed to undergo infiniteorder phase transitions for certain values of μ, associated with the vanishing of Polyakovloop correlators.

Openstring integrals with multiple unintegrated punctures at genus one
Authors: André Kaderli and Carlos Rodriguez
Preprint number: UUITP14/22
We study integrals appearing in intermediate steps of oneloop openstring amplitudes, with multiple unintegrated punctures on the $A$cycle of a torus. We construct a vector of such integrals which closes after taking a total differential with respect to the $N$ unintegrated punctures and the modular parameter $\tau$. These integrals are found to satisfy the elliptic KnizhnikZamolodchikovBernard (KZB) equations, and can be written as a power series in $\alpha$'  the string length squared  in terms of elliptic multiple polylogarithms (eMPLs). In the $N$puncture case, the KZB equation reveals a representation of $B_{1,N}$, the braid group of $N$ strands on a torus, acting on its solutions. We write the simplest of these braid group elements  the braiding one puncture around another  and obtain generating functions of analytic continuations of eMPLs. The KZB equations in the socalled universal case is written in terms of the genusone DrinfeldKohno algebra $\mathfrak{t}_{1,N} \rtimes \mathfrak{d}$, a graded algebra. Our construction determines matrix representations of various dimensions for several generators of this algebra which respect its grading up to commuting terms.

Snowmass White Paper: String Perturbation Theory
Authors: Nathan Berkovits, Eric D'Hoker, Michael B. Green, Henrik Johansson, Oliver Schlotterer
Preprint number: UUITP13/22
Abstract: The purpose of this White Paper is to review recent progress towards elucidating and evaluating string amplitudes, relating them to quantum field theory amplitudes, applying their predictions to string dualities, exploring their connection with gravitational physics, and deepening our understanding of their mathematical structure. We also present a selection of targets for future research.

Searching for Kerr in the 2PM amplitude
Authors: Rafael Aoude, Kays Haddad, Andreas Helset
Preprint number: UUITP12/22
Abstract: The classical scattering of spinning objects is well described by the spinorhelicity formalism for heavy particles. Using these variables, we derive spuriouspolefree, oppositehelicity Compton amplitudes (factorizing on physical poles to the minimal, allspin threepoint amplitudes of ref. [1]) in the classical limit for QED, QCD, and gravity. The cured amplitudes are subject to deformations by contact terms, the vast majority of whose contributions we can fix by imposing a relation between spin structuresmotivated by lower spin multipoles of black hole scatteringat the second postMinkowskian (2PM) order. For QED and gravity, this leaves a modest number of unfixed coefficients parametrizing contactterm deformations, while the QCD amplitude is uniquely determined. Our gravitational Compton amplitude allows us to push the stateoftheart of spinning2PM scattering to any order in the spin vectors of both objects; we present results here and in the auxiliary file 2PMSpin8Aux.nb up to eighth order in the spin vectors. Interestingly, despite leftover coefficients in the Compton amplitude, imposing the aforementioned relation between spin structures uniquely fixes some higherspin parts of the 2PM amplitude.

Functions Beyond Multiple Polylogarithms for Precision Collider Physics
Authors: Jacob Bourjaily, Johannes Broedel, Ekta Chaubey, Claude Duhr, Hjalte Frellesvig, Martijn Hidding, Robin Marzucca, Andrew McLeod, Marcus Spradlin, Lorenzo Tancredi, Cristian Vergu, Matthias Volk, Anastasia Volovich, Matt von Hippel, Stefan Weinzierl, Matthias Wilhelm, Chi Zhang.
Preprint number: UUITP11/22
Abstract: Feynman diagrams constitute one of the essential ingredients for making precision predictions for collider experiments. Yet, while the simplest Feynman diagrams can be evaluated in terms of multiple polylogarithms  whose properties as special functions are well understood  more complex diagrams often involve integrals over complicated algebraic manifolds. Such diagrams already contribute at NNLO to the selfenergy of the electron, $t \bar{t}$ production, $\gamma \gamma$ production, and Higgs decay, and appear at two loops in the planar limit of maximally supersymmetric YangMills theory. This makes the study of these more complicated types of integrals of phenomenological as well as conceptual importance. In this white paper contribution to the Snowmass community planning exercise, we provide an overview of the state of research on Feynman diagrams that involve special functions beyond multiple polylogarithms, and highlight a number of research directions that constitute essential avenues for future investigation.

2d Sigma Models and Geometry
Authors: Ulf Lindström
Preprint number: UUITP10/22
Supersymmetric nonlinear sigma models have target spaces that carry interesting geometry. The geometry is richer the more supersymmetries the model has. The study of models with two dimensional world sheets is particularly rewarding since they allow for torsionful geometries. In this review I describe and exemplify the relation of $2d$ supersymmetry to Riemannian, complex, bihermitian, $(p,q)$ hermitean, Kähler, hyperkähler, generalised geometry and more.

Localizing nonlinear $N=(2,2)$ sigma model on $S^2$
Authors: Victor Alekseev, Guido Festuccia, Victor Mishnyakov, Nicolai Terziev and Maxim Zabzine
Preprint number: UUITP09/22
We present a systematic study of ${\cal N}=(2,2)$ supersymmetric nonlinear sigma models on $S^2$ with the target being a Kähler manifold. We discuss their reformulation in terms of cohomological field theory. In the cohomological formulation we use a novel version of 2D selfduality which involves a $U(1)$ action on $S^2$. In addition to the generic model we discuss the theory with target space equivariance corresponding to a supersymmetric sigma model coupled to a nondynamical supersymmetric background gauge multiplet. We discuss the localization locus and perform a oneloop calculation around the constant maps. We argue that the theory can be reduced to some exotic model over the moduli space of holomorphic disks.

Spinorhelicity formalism for massive and massless amplitudes in five dimensions
Authors: Marco Chiodaroli, Murat Günaydin, Henrik Johansson and Radu Roiban
Preprint Number: UUITP08/22
Fivedimensional gauge and gravity theories are known to exhibit striking properties. D=5 is the lowest dimension where massive tensor states appear naturally, providing a testing ground for perturbative insights into sixdimensional tensor theories. Fivedimensional supergravities are highly constrained and admit elegant geometric and algebraic formulations, with global symmetries manifest at the Lagrangian level. In this paper, we take a step towards the systematic investigation of amplitudes in five dimensions, and present a fivedimensional version of the spinorhelicity formalism, applicable to massless, massive and supersymmetric states. We give explicit representations for onshell spinor and polarization variables such that the littlegroup symmetry and gauge redundancy are manifest. Massive selfdual tensor states are discussed in some detail, as well as all the onshell supermultiplets that can appear in mattercoupled gauge and supergravity theories. As a byproduct of considering supersymmetry in the presence of central charge, we obtain massless tendimensional MajoranaWeyl spinors as products of fivedimensional massive spinors.

Uses of Killing and KillingYano Tensors
Authors: Ulf Lindström and Özgür Sarıoğlu
Preprint Number: UUITP07/22
In this contribution we have collected some facts about Killing and KillingYano tensors that we feel are of general interest for researchers working on problems that rely on differential geometry. We also include some of our recent studies pertaining to currents, charges and (super)invariants for particles and tensionless strings.

Tensionless Strings and Killing(Yano) Tensors
Authors: Ulf Lindström and Özgür Sarıoğlu
Preprint Number: UUITP06/22
We construct invariants for bosonic and spinning tensionless (null) strings in back grounds that carry Killing or KillingYano tensors of mixed type. This is facilitated by the close relation of these strings to point particles. We apply the construction to the Minkowski and to the KerrNewman background. 
A tale of tails through generalized unitarity
Authors: Alex Edison, Michèle Levi
Preprint Number: UUITP05/22
We introduce a novel framework to study highorder gravitational effects on a binary from the scattering of its emitted gravitational radiation. Here we focus on the radiationreaction due to the background of the binary’s gravitational potential, namely on the socalled tail effects. We start from the effective field theory of a binary composite particle, and through multiloop and generalized unitarity methods, we derive the effective action of the dynamical multipoles, the energy spectrum, and the observable flux due to these effects. We proceed through the third subleading such radiationreaction effect – at the fourloop level and the seventh order in postNewtonian gravity – shedding new light on the higherorder effects, and completing the state of the art.

Oneloop amplitudes in EinsteinYangMills from forward limits
Authors: Franziska Porkert, Oliver Schlotterer
Preprint number: UUITP04/22
We present a method to compute the integrands of oneloop EinsteinYangMills amplitudes for any number of external gauge and gravity multiplets. Our construction relies on the doublecopy structure of EinsteinYangMills as (super)YangMills with the socalled YM+phi^3 theory  pure YangMills coupled to biadjoint scalars  which we implement via oneloop CachazoHeYuan formulae. The YM+phi^3 building blocks are obtained from forward limits of treelevel input in external gluons and scalars, and we give the composition rules for any number of traces and orders in the couplings g and kappa. One the one hand, we spell out supersymmetry and dimensionagnostic relations that reduce loop integrands of EinsteinYangMills to those of pure gauge theories. On the other hand, we present fourpoint results for maximal and halfmaximal supersymmetry where all supersymmetry cancellations are exposed. In the halfmaximal case, we determine sixdimensional anomalies due to chiral hypermultiplets in the loop.

Gravitational waves in dark bubble cosmology
Authors: Ulf Danielsson, Daniel Panizo and Rob Tielemans
Preprint number: UUITP03/22
Abstract: In this paper we construct the 5D uplift of 4D gravitational waves in de Sitter cosmology for the brane world scenario based on a nucleated bubble in AdS5. This makes it possible to generalize the connection between the dark bubbles and Vilenkin's quantum cosmology to include gravitational perturbations. We also use the uplift to explain the interpretation of the apparently negative energy contributions in the 4D Einstein equations, which distinguish the dark bubble scenario from RandallSundrum.

Vanishing Yukawa Couplings and the Geometry of String Theory Models
Authors: L. B. Anderson, J. Gray, M. Larfors, M. Magill
Preprint number: UUITP02/22
Abstract: We provide an overview of recent work which aims to understand patterns of vanishing Yukawa couplings that arise in models of particle physics derived from string theory. These patterns are seemingly linked to a plethora of different geometrical structures and our understanding of the subject has yet to be consolidated in a unified framework. This short note is based upon a talk that was given by one of the authors at the Nankai Symposium on Mathematical Dialogues. Therefore it is aimed at a mathematical audience of mixed academic background.

Higher Symmetries of 5d Orbifold SCFTs
Authors: Michele Del Zotto, Jonathan J. Heckman, Shani Nadir Meynet, Robert Moscrop, Hao Y. Zhang
Preprint number: UUITP01/22
Abstract: We determine the higher symmetries of 5d SCFTs engineered from Mtheory on a C^{3}/Γ background for Γ a finite subgroup of SU(3). This resolves a longstanding question as to how to extract this data when the resulting singularity is nontoric (when Γ is nonabelian) and/or not isolated (when the action of Γ has fixed loci). The BPS states of the theory are encoded in a 1D quiver quantum mechanics gauge theory which determines the possible 1form and 2form symmetries. We also show that this same data can also be extracted by a direct computation of the corresponding defect group associated with the orbifold singularity. Both methods agree, and these computations do not rely on the existence of a resolution of the singularity. We also observe that when the geometry faithfully captures the global 0form symmetry, the abelianization of Γ detects a 2group structure (when present). As such, this establishes that all of this data is indeed intrinsic to the superconformal fixed point rather than being an emergent property of an IR gauge theory phase.