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Journal articles on the topic 'Carrollian holography'

Author: Grafiati
Published: 23 November 2024

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1

Nguyen, Kevin, and Peter West. "Carrollian Conformal Fields and Flat Holography." Universe 9, no. 9 (August 26, 2023): 385. http://dx.doi.org/10.3390/universe9090385.

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The null conformal boundary I of Minkowski spacetime M plays a special role in scattering theory, as it is the locus where massless particle states are most naturally defined. We construct quantum fields on I, which create these massless states from the vacuum and transform covariantly under Poincaré symmetries. Because the latter symmetries act as Carrollian conformal isometries of I, these quantum fields are Carrollian conformal fields. This group theoretic construction is intrinsic to I by contrast to existing treatments in the literature. However, we also show that the standard relativistic massless quantum fields in M, when pulled back to I, provide a realisation of these Carrollian conformal fields. This correspondence between bulk and boundary fields should constitute a basic entry in the dictionary of flat holography. Finally, we show that I provides a natural parametrisation of the massless particles as described by irreducible representations of the Poincaré group and that in an appropriate conjugate basis, they indeed transform like Carrollian conformal fields.
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2

Have, Emil, Kevin Nguyen, Stefan Prohazka, and Jakob Salzer. "Massive carrollian fields at timelike infinity." Journal of High Energy Physics 2024, no. 7 (July 5, 2024). http://dx.doi.org/10.1007/jhep07(2024)054.

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Abstract Motivated by flat space holography, we demonstrate that massive spin-s fields in Minkowski space near timelike infinity are massive carrollian fields on the carrollian counterpart of anti-de Sitter space called Ti. Its isometries form the Poincaré group, and we construct the carrollian spin-s fields using the method of induced representations. We provide a dictionary between massive carrollian fields on Ti and massive fields in Minkowski space, as well as to fields in the conformal primary basis used in celestial holography. We show that the symmetries of the carrollian structure naturally account for the BMS charges underlying the soft graviton theorem. Finally, we initiate a discussion of the correspondence between massive scattering amplitudes and carrollian correlation functions on Ti, and introduce physical definitions of detector operators using a suitable notion of conserved carrollian energy-momentum tensor.
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3

Mason, Lionel, Romain Ruzziconi, and Akshay Yelleshpur Srikant. "Carrollian amplitudes and celestial symmetries." Journal of High Energy Physics 2024, no. 5 (May 2, 2024). http://dx.doi.org/10.1007/jhep05(2024)012.

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Abstract Carrollian holography aims to express gravity in four-dimensional asymptotically flat spacetime in terms of a dual three-dimensional Carrollian CFT living at null infinity. Carrollian amplitudes are massless scattering amplitudes written in terms of asymptotic or null data at $$ \mathcal{I} $$ I . These position space amplitudes at $$ \mathcal{I} $$ I are to be re-interpreted as correlation functions in the putative dual Carrollian CFT. We derive basic results concerning tree-level Carrollian amplitudes yielding dynamical constraints on the holographic dual. We obtain surprisingly compact expressions for n-point MHV gluon and graviton amplitudes in position space at $$ \mathcal{I} $$ I . We discuss the UV/IR behaviours of Carrollian amplitudes and investigate their collinear limit, which allows us to define a notion of Carrollian OPE. By smearing the OPE along the generators of null infinity, we obtain the action of the celestial symmetries — namely, the S algebra for Yang-Mills theory and Lw1+∞ for gravity — on the Carrollian operators. As a consistency check, we systematically relate our results with celestial amplitudes using the link between the two approaches. Finally, we initiate a direct connection between twistor space and Carrollian amplitudes.
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4

Saha, Amartya. "w1+∞ and Carrollian holography." Journal of High Energy Physics 2024, no. 5 (May 13, 2024). http://dx.doi.org/10.1007/jhep05(2024)145.

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Abstract In a 1 + 2D Carrollian conformal field theory, the Ward identities of the two local fields $$ {S}_0^{+} $$ S 0 + and $$ {S}_1^{+} $$ S 1 + , entirely built out of the Carrollian conformal stress-tensor, contain respectively up to the leading and the subleading positive helicity soft graviton theorems in the 1 + 3D asymptotically flat space-time. This work investigates how the subsubleading soft graviton theorem can be encoded into the Ward identity of a Carrollian conformal field $$ {S}_2^{+} $$ S 2 + . The operator product expansion (OPE) $$ {S}_2^{+}{S}_2^{+} $$ S 2 + S 2 + is constructed using general Carrollian conformal symmetry principles and the OPE commutativity property, under the assumption that any time-independent, non-Identity field that is mutually local with $$ {S}_0^{+} $$ S 0 + , $$ {S}_1^{+} $$ S 1 + , $$ {S}_2^{+} $$ S 2 + has positive Carrollian scaling dimension. It is found that, for this OPE to be consistent, another local field $$ {S}_3^{+} $$ S 3 + must automatically exist in the theory. The presence of an infinite tower of local fields $$ {S}_{k\ge 3}^{+} $$ S k ≥ 3 + is then revealed iteratively as a consistency condition for the $$ {S}_2^{+}{S}_{k-1}^{+} $$ S 2 + S k − 1 + OPE. The general $$ {S}_k^{+}{S}_l^{+} $$ S k + S l + OPE is similarly obtained and the symmetry algebra manifest in this OPE is found to be the Kac-Moody algebra of the wedge sub-algebra of w1+∞. The Carrollian time-coordinate plays the central role in this purely holographic construction. The 2D Celestial conformally soft graviton primary $$ {H}^k\left(z,\overline{z}\right) $$ H k z z ¯ is realized to be contained in the Carrollian conformal primary $$ {S}_{1-k}^{+}\left(t,z,\overline{z}\right) $$ S 1 − k + t z z ¯ . Finally, the existence of the infinite tower of fields $$ {S}_k^{+} $$ S k + is shown to be directly related to an infinity of positive helicity soft graviton theorems.
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5

Bagchi, Arjun, Prateksh Dhivakar, and Sudipta Dutta. "AdS Witten diagrams to Carrollian correlators." Journal of High Energy Physics 2023, no. 4 (April 28, 2023). http://dx.doi.org/10.1007/jhep04(2023)135.

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Abstract Carrollian Conformal Field Theories (CFTs) have been proposed as co-dimension one holographic duals to asymptotically flat spacetimes as opposed to Celestial CFTs which are co-dimension two. In this paper, drawing inspiration from Celestial holography, we show by a suitable generalisation of the flat space limit of AdS that keeps track of the previously disregarded null direction, one can reproduce Carrollian CFT correlation functions from AdS Witten diagrams. In particular, considering Witten diagrams in AdS4, we reproduce two and three-point correlation functions for three dimensional Carrollian CFTs in the so-called delta-function branch. Along the way, we construct non-trivial Carrollian three-point functions in the delta-branch by considering a collinear limit. We also obtain a generalised anti-podal matching condition that now depends on the retarded time direction.
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6

Bagchi, Arjun, Prateksh Dhivakar, and Sudipta Dutta. "Holography in flat spacetimes: the case for Carroll." Journal of High Energy Physics 2024, no. 8 (August 20, 2024). http://dx.doi.org/10.1007/jhep08(2024)144.

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Abstract We compare and contrast the two approaches of holography in asymptotically flat spacetimes, viz. the co-dimension two Celestial approach based on the Mellin transformation and the co-dimension one Carrollian approach based on the modified Mellin and elucidate how some of the problems of the Celestial approach can be rectified by the Carrollian one. Considering flat holography as a limit from AdS/CFT makes a co-dimension one dual more plausible, and our previous construction of Carrollian correlations from AdS Witten diagrams is testimony to this. In this paper, we show how to generalize our earlier analysis for operators with spin. We work out a large number of explicit non-trivial examples (twelve) and show matching between the limit of AdS4 Witten diagrams and 3d boundary symmetry considerations, thus making the case for the Carrollian dual even stronger.
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7

Salzer, Jakob. "An embedding space approach to Carrollian CFT correlators for flat space holography." Journal of High Energy Physics 2023, no. 10 (October 13, 2023). http://dx.doi.org/10.1007/jhep10(2023)084.

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Abstract Carrollian conformal field theories (carrollian CFTs) are natural field theories on null infinity of an asymptotically flat spacetime or, more generally, geometries with conformal carrollian structure. Using a basis transformation, gravitational S-matrix elements can be brought into the form of correlators of a carrollian CFT. Therefore, it has been suggested that carrollian CFTs could provide a co-dimension one dual description to gravity in asymptotically flat spacetimes. In this work, we construct an embedding space formalism for three-dimensional carrollian CFTs and use it to determine two- and three-point correlators. These correlators are fixed by the global subgroup, ISO(1, 3), of the carrollian conformal symmetries, i.e., the Bondi-van der Burg-Metzner-Sachs symmetries (BMS). The correlators coincide with well-known two- and three-point scattering amplitudes in Minkowski space written with respect to a basis of asymptotic position states.
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8

Donnay, Laura, Adrien Fiorucci, Yannick Herfray, and Romain Ruzziconi. "Carrollian Perspective on Celestial Holography." Physical Review Letters 129, no. 7 (August 12, 2022). http://dx.doi.org/10.1103/physrevlett.129.071602.

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9

Ciambelli, Luca, Charles Marteau, Anastasios C. Petkou, P. Marios Petropoulos, and Konstantinos Siampos. "Flat holography and Carrollian fluids." Journal of High Energy Physics 2018, no. 7 (July 2018). http://dx.doi.org/10.1007/jhep07(2018)165.

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10

Donnay, Laura, Adrien Fiorucci, Yannick Herfray, and Romain Ruzziconi. "Bridging Carrollian and celestial holography." Physical Review D 107, no. 12 (June 30, 2023). http://dx.doi.org/10.1103/physrevd.107.126027.

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11

Pérez, Alfredo. "Asymptotic symmetries in Carrollian theories of gravity with a negative cosmological constant." Journal of High Energy Physics 2022, no. 9 (September 6, 2022). http://dx.doi.org/10.1007/jhep09(2022)044.

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Abstract Asymptotic symmetries of electric and magnetic Carrollian gravitational theories with a negative cosmological constant Λ are analyzed in 3+1 space-time dimensions. In the magnetic theory, the asymptotic symmetry algebra is given by the conformal Carroll algebra in three dimensions, which is infinite-dimensional and isomorphic to the BMS4 algebra. These results are in full agreement with holographic expectations, providing a new framework for the study of Carrollian holography. On the contrary, in the case of the electric theory, the presence of a negative Λ turns out to be incompatible with a consistent set of asymptotic conditions, that can be traced back to the absence of a sensible ground state configuration. This can be improved if the Carrollian theory obtained from an electric contraction of Euclidean General Relativity is considered. In this case, asymptotic conditions can be constructed with an asymptotic symmetry algebra given by so (1, 4). However, it is shown that the space of spherically symmetric solutions of this theory is degenerate.
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12

Mittal, Nehal, P. Marios Petropoulos, David Rivera-Betancour, and Matthieu Vilatte. "Ehlers, Carroll, charges and dual charges." Journal of High Energy Physics 2023, no. 7 (July 7, 2023). http://dx.doi.org/10.1007/jhep07(2023)065.

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Abstract We unravel the boundary manifestation of Ehlers’ hidden Möbius symmetry present in four-dimensional Ricci-flat spacetimes that enjoy a time-like isometry and are Petrov-algebraic. This is achieved in a designated gauge, shaped in the spirit of flat holography, where the Carrollian three-dimensional nature of the null conformal boundary is manifest and covariantly implemented. The action of the Möbius group is local on the space of Carrollian boundary data, among which the Carrollian Cotton tensor plays a predominent role. The Carrollian and Weyl geometric tools introduced for shaping an appropriate gauge, as well as the boundary conformal group, which is BMS4, allow to define electric/magnetic, leading/subleading towers of charges directly from the boundary Carrollian dynamics and explore their behaviour under the action of the Möbius duality group.
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13

Saha, Amartya. "Carrollian approach to 1 + 3D flat holography." Journal of High Energy Physics 2023, no. 6 (June 12, 2023). http://dx.doi.org/10.1007/jhep06(2023)051.

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Abstract The isomorphism between the (extended) BMS4 algebra and the 1 + 2D Carrollian conformal algebra hints towards a co-dimension one formalism of flat holography with the field theory residing on the null-boundary of the asymptotically flat space-time enjoying a 1 + 2D Carrollian conformal symmetry. Motivated by this fact, we study the general symmetry properties of a source-less 1 + 2D Carrollian CFT, adopting a purely field-theoretic approach. After deriving the position-space Ward identities, we show how the 1 + 3D bulk super-translation and the super-rotation memory effects emerge from them, manifested by the presence of a temporal step-function factor in the same. Temporal-Fourier transforming these memory effect equations, we directly reach the bulk null-momentum-space leading and sub-leading soft graviton theorems. Along the way, we construct six Carrollian fields $$ {S}_0^{\pm } $$ S 0 ± , $$ {S}_1^{\pm } $$ S 1 ± , T and $$ \overline{T} $$ T ¯ corresponding to these soft graviton fields and the Celestial stress-tensors, purely in terms of the Carrollian stress-tensor components. The 2D Celestial shadow-relations and the null-state conditions arise as two natural byproducts of these constructions. We then show that those six fields consist of the modes that implement the super-rotations and a subset of the super-translations on the quantum fields. The temporal step-function allows us to relate the operator product expansions (OPEs) with the operator commutation relations via a complex contour integral prescription. We deduce that not all of those six fields can be taken together to form consistent OPEs. So choosing $$ {S}_0^{+} $$ S 0 + , $$ {S}_1^{+} $$ S 1 + and T as the local fields, we form their mutual OPEs using only the OPE-commutativity property, under two general assumptions. The symmetry algebra manifest in these holomorphic-sector OPEs is then shown to be Vir $$ \overset{\wedge }{\ltimes \overline{\textrm{sl}\left(2,{\mathbb{R}}\right)}} $$ ⋉ sl 2 ℝ ¯ ∧ with an abelian ideal.
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14

Stieberger, Stephan, Tomasz R. Taylor, and Bin Zhu. "Carrollian Amplitudes from Strings." Journal of High Energy Physics 2024, no. 4 (April 23, 2024). http://dx.doi.org/10.1007/jhep04(2024)127.

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Abstract Carrollian holography is supposed to describe gravity in four-dimensional asymptotically flat space-time by the three-dimensional Carrollian CFT living at null infinity. We transform superstring scattering amplitudes into the correlation functions of primary fields of Carrollian CFT depending on the three-dimensional coordinates of the celestial sphere and a retarded time coordinate. The power series in the inverse string tension is converted to a whole tower of both UV and IR finite descendants of the underlying field-theoretical Carrollian amplitude. We focus on four-point amplitudes involving gauge bosons and gravitons in type I open superstring theory and in closed heterotic superstring theory at the tree-level. We also discuss the limit of infinite retarded time coordinates, where the string world-sheet becomes celestial.
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15

Bagchi, Arjun, Daniel Grumiller, and Poulami Nandi. "Carrollian superconformal theories and super BMS." Journal of High Energy Physics 2022, no. 5 (May 2022). http://dx.doi.org/10.1007/jhep05(2022)044.

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Abstract We investigate the supersymmetric versions of Bondi-Metzner-Sachs or, equivalently, conformal Carroll symmetry in boundary dimensions d > 3, with applications of flat space holography in mind. We identify the contraction of the relativistic symmetry relevant for our purposes and construct a finite-dimensional Carrollian superconformal algebra (CSA) before proposing an infinite-dimensional lift. We provide the superspace formulation for $$ \mathcal{N} $$ N = 1 CSA and work towards an understanding of the representation theory of the algebra. We conclude with some aspects of $$ \mathcal{N} $$ N > 1 CSA.
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16

Kasikci, Oguzhan, Mehmet Ozkan, Yi Pang, and Utku Zorba. "Carrollian supersymmetry and SYK-like models." Physical Review D 110, no. 2 (July 22, 2024). http://dx.doi.org/10.1103/physrevd.110.l021702.

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This work challenges the conventional notion that in a spacetime dimension higher than one, a supersymmetric Lagrangian invariably consists of purely bosonic terms, purely fermionic terms, as well as boson-fermion mixing terms. By recasting a relativistic Lagrangian in terms of its nonrelativistic and ultrarelativistic sectors, we reveal that an ultrarelativistic (Carrollian) supersymmetric Lagrangian exhibits an exotic feature; that is, it can exist without a purely bosonic contribution. Based on this result, we demonstrate a link between higher-dimensional Carrollian and (0+1)-dimensional quantum mechanical models, yielding higher-order extensions of supersymmetric Sachdev-Ye-Kitaev (SYK) models in which purely bosonic higher-order terms are absent. Given that supersymmetry plays an essential role in improving the quantum behavior and solubility, our findings may lead to interesting applications in non-AdS holography. Published by the American Physical Society 2024
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17

Chen, Bin, and Zezhou Hu. "Bulk reconstruction in flat holography." Journal of High Energy Physics 2024, no. 3 (March 12, 2024). http://dx.doi.org/10.1007/jhep03(2024)064.

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Abstract In this note, we discuss the bulk reconstruction of massless free fields in flat space from the highest-weight representation of boundary Carrollian conformal field theory (CCFT). We expand the bulk field as a sum of infinite descendants of a primary state defined in the boundary CCFT, and discuss the Lorentz invariant bulk-boundary propagator in detail for the BMS3/CCFT2 case. In our calculation, it is necessary to introduce a nonzero mass at the beginning and take it as vanishing at the end. The framework we proposed has the potential to probe local bulk physics from the boundary CCFT.
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18

Figueroa-O’Farrill, José, Emil Have, Stefan Prohazka, and Jakob Salzer. "Carrollian and celestial spaces at infinity." Journal of High Energy Physics 2022, no. 9 (September 1, 2022). http://dx.doi.org/10.1007/jhep09(2022)007.

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Abstract We show that the geometry of the asymptotic infinities of Minkowski spacetime (in d + 1 dimensions) is captured by homogeneous spaces of the Poincaré group: the blow-ups of spatial (Spi) and timelike (Ti) infinities in the sense of Ashtekar-Hansen and a novel space Ni fibering over $$ \mathcal{I} $$ I . We embed these spaces à la Penrose-Rindler into a pseudo-euclidean space of signature (d + 1, 2) as orbits of the same Poincaré subgroup of O(d + 1, 2). We describe the corresponding Klein pairs and determine their Poincaré-invariant structures: a carrollian structure on Ti, a pseudo-carrollian structure on Spi and a “doubly-carrollian” structure on Ni. We give additional geometric characterisations of these spaces as grassmannians of affine hyperplanes in Minkowski spacetime: Spi is the (double cover of the) grassmannian of affine lorentzian hyperplanes; Ti is the grassmannian of affine spacelike hyperplanes and Ni fibers over the grassmannian of affine null planes, which is $$ \mathcal{I} $$ I . We exhibit Ni as the fibred product of $$ \mathcal{I} $$ I and the lightcone over the celestial sphere. We also show that Ni is the total space of the bundle of scales of the conformal carrollian structure on $$ \mathcal{I} $$ I and show that the symmetry algebra of its doubly-carrollian structure is isomorphic to the symmetry algebra of the conformal carrollian structure on $$ \mathcal{I} $$ I ; that is, the BMS algebra. We show how to reconstruct Minkowski spacetime from any of its asymptotic geometries, by establishing that points in Minkowski spacetime parametrise certain lightcone cuts in the asymptotic geometries. We include an appendix comparing with (A)dS and observe that the de Sitter groups have no homogeneous spaces which could play the rôle that the celestial sphere plays in flat space holography.
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19

Bhattacharyya, Arpan, and Poulami Nandi. "Circuit complexity for Carrollian Conformal (BMS) field theories." Journal of High Energy Physics 2023, no. 7 (July 12, 2023). http://dx.doi.org/10.1007/jhep07(2023)105.

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Abstract We systematically explore the construction of Nielsen’s circuit complexity to a non-Lorentzian field theory keeping in mind its connection with flat holography. We consider a 2d boundary field theory dual to 3d asymptotically flat spacetimes with infinite-dimensional BMS3 as the asymptotic symmetry algebra. We compute the circuit complexity functional in two distinct ways. For the Virasoro group, the complexity functional resembles the geometric action on its co-adjoint orbit. Using the limiting approach on the relativistic results, we show that it is possible to write BMS complexity in terms of the geometric action on BMS co-adjoint orbit. However, the limiting approach fails to capture essential information about the conserved currents generating BMS supertranslations. Hence, we refine our analysis using the intrinsic approach. Here, we use only the symmetry transformations and group product laws of BMS to write the complexity functional. The refined analysis shows a richer structure than only the geometric action. Lastly, we extremize and solve the equations of motion (for a simple solution) in terms of the group paths and connect our results with available literature.
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20

Bekaert, Xavier, Andrea Campoleoni, and Simon Pekar. "Holographic Carrollian conformal scalars." Journal of High Energy Physics 2024, no. 5 (May 22, 2024). http://dx.doi.org/10.1007/jhep05(2024)242.

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Abstract We provide holographic realisations in Minkowski spacetime of a free conformal Carrollian scalar field living at null infinity. To this end, we first show that the electric and magnetic limits of a relativistic conformal scalar are equivalent and we study the representation of the Carroll, Poincaré and BMS algebras that is realised on the resulting solution space. We then realise it as a quotient of the solution space of a free massless scalar in Minkowski spacetime with unusual falloff, in full analogy with the interpretation of Dirac’s singleton as a shortened scalar in Anti de Sitter spacetime.
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21

Banerjee, Aritra, Rudranil Basu, Arpan Bhattacharyya, and Nilachal Chakrabarti. "Symmetry resolution in non-Lorentzian field theories." Journal of High Energy Physics 2024, no. 6 (June 19, 2024). http://dx.doi.org/10.1007/jhep06(2024)121.

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Abstract Starting from the computation of Symmetry Resolved Entanglement Entropy (SREE) for boosted intervals in a two dimensional Conformal Field Theory, we compute the same in various non-Lorentzian limits, viz, Galilean and Carrollian Conformal Field Theory in same number of dimensions. We approach the problem both from a limiting perspective and by using intrinsic symmetries of respective non-Lorentzian conformal algebras. In particular, we calculate the leading order terms, logarithmic terms, and the $$ \mathcal{O}(1) $$ O 1 terms and explicitly show exact compliance with equipartition of entanglement, even in the non-Lorentzian system. Keeping in mind the holographic origin of SREE for the Carrollian limit, we further compute SREE for BMS3-Kac-Moody, which couples a U(1) × U(1) theory with bulk gravity.
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22

Ciambelli, Luca, Charles Marteau, P. Marios Petropoulos, and Romain Ruzziconi. "Gauges in three-dimensional gravity and holographic fluids." Journal of High Energy Physics 2020, no. 11 (November 2020). http://dx.doi.org/10.1007/jhep11(2020)092.

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Abstract Solutions to Einstein’s vacuum equations in three dimensions are locally maximally symmetric. They are distinguished by their global properties and their investigation often requires a choice of gauge. Although analyses of this sort have been performed abundantly, several relevant questions remain. These questions include the interplay between the standard Bondi gauge and the Eddington-Finkelstein type of gauge used in the fluid/gravity holographic reconstruction of these spacetimes, as well as the Fefferman-Graham gauge, when available i.e. in anti de Sitter. The goal of the present work is to set up a thorough dictionary for the available descriptions with emphasis on the relativistic or Carrollian holographic fluids, which portray the bulk from the boundary in anti-de Sitter or flat instances. A complete presentation of residual diffeomorphisms with a preliminary study of their algebra accompanies the situations addressed here.
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23

Campoleoni, Andrea, Luca Ciambelli, Arnaud Delfante, Charles Marteau, P. Marios Petropoulos, and Romain Ruzziconi. "Holographic Lorentz and Carroll frames." Journal of High Energy Physics 2022, no. 12 (December 1, 2022). http://dx.doi.org/10.1007/jhep12(2022)007.

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Abstract Relaxing the Bondi gauge, the solution space of three-dimensional gravity in the metric formulation has been shown to contain an additional free function that promotes the boundary metric to a Lorentz or Carroll frame, in asymptotically AdS or flat spacetimes. We pursue this analysis and show that the solution space also admits a finite symplectic structure, obtained taking advantage of the built-in ambiguities. The smoothness of the flat limit of the AdS symplectic structure selects a prescription in which the holographic anomaly appears in the boundary Lorentz symmetry, that rotates the frame. This anomaly turns out to be cohomologically equivalent to the standard holographic Weyl anomaly and survives in the flat limit, thus predicting the existence of quantum anomalies in conformal Carrollian field theories. We also revisit these results in the Chern-Simons formulation, where the prescription for the symplectic structure admitting a smooth flat limit follows from the variational principle, and we compute the charge algebra in the boundary conformal gauge.
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24

Grumiller, Daniel, and Max Riegler. "Carrollian c functions and flat space holographic RG flows in BMS3/CCFT2." Physical Review D 108, no. 12 (December 7, 2023). http://dx.doi.org/10.1103/physrevd.108.126008.

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