Scientific abstract Scattering amplitudes; High energy gravity

Transcription

1 Application No. 681/15 PI Name: Barak Kol Scientific abstract Scattering amplitudes; High energy gravity In recent years it was realized that perturbative gauge theory (and gravity) is far from being well-understood. A myriad of new concepts and tools were found to improve over the standard perturbation theory a la Feynman and often to reduce computation dramatically. However, it is not clear yet how these different concepts fit together optimally. Just like the Lagrangian underlies the standard Feynman rules, some believe that the new concepts hint at an alternative, deep formulation of gauge theory. These issues are the basis for the field known as scattering amplitudes. Clearly, these issues lie at the heart of Quantum Field Theory. They are important not only theoretically but also experimentally as these methods already enabled to compute valuable cross-sections for accelerators including the LHC. The main focus of this proposal is to be the first Israeli group to enter scattering amplitudes, in parallel with A. Sever who recently arrived, to study and review the literature and seek to contribute to the ongoing worldwide effort to address the abovementioned issues by applying our gravitational, geometrical and physical skills. The proposed research group started working in the field in fall 2013 and was already able to find, in the words of a referee, a clearly superior presentation for the basic issue of the color-kinematics decomposition, as well as to find new results regarding the transformation of color structures under permutations. We propose to proceed and perform several concrete studies including searches for a new connection related to shuffle algebras, and for a physical interpretation the appearance of the Eulerian numbers. In addition the group intends to apply its expertise in modern General Relativity (GR) with a stress on connections with High Energy Theory and maintain an effort in the field by following several problems including the Effective Field Theory approach to GR, GR in higher dimensions, the infinite d limit and Extreme Mass Ratio binary motion.

2 Scattering Amplitudes; High Energy Gravitation ISF proposal 2015 Barak Kol Detailed description of the research program This detailed description of the proposal is composed of 3 sections. In the first I describe the current state of affairs for High Energy/Fundamental Interactions physics, both experimentally and theoretically. Assuming close familiarity of the reader with the subject matter, this description will be brief and use a broad brush. In the second section I present the proposed field(s) of research, the main questions to be engaged and their significance. In the last section I describe several concrete research efforts to be pursued and the track record of the proposed PI. 1. Scientific Background Ideally research in theoretical physics should be well-motivated both by experiment and by intrinsic theoretical interest. Here we start with a basic description of the current state of affairs for high energy physics before proceeding to develop the proposed research program. Relevant experiments include The Large Hadron Collider (LHC) The discovery of a 126 GeV Higgs boson was announced in July The accelerator is currently being upgraded to reach 13 TeV center of mass energy (from 8 TeV) and is expected to begin operation in early 2015 [1]. Cosmological observations including CMB polarization. BICEP2 created a stir with the announcement of its results and their proposed interpretation for primordial gravitation waves and inflation [2]. Gravitational wave detectors Advanced LIGO will improve the sensitivity of LIGO by an order of magnitude. It was recently installed and commissioning will continue into 2015 as the project moves towards the first data run [3]. Work is in progress on several other detectors worldwide. Astro-particle experiments and more... 1

3 Relevant theoretical currents and developments include Scattering amplitudes. New methods and concepts to perform calculations in perturbative gauge theory and other theories. To be discussed in more detail in section 2. Integrability of maximally supersymmetric (N=4) gauge theory. "The hydrogen atom of the 21 st century". Quantum Field Theory and Mathematical physics. Bootstrap, supercoformal indices, "physical mathematics". AdS/CMT. The AdS/CFT correspondence inspired gravitational models for Condensed Matter Theory (CMT) systems. Entanglement. Results on entanglement in field theory and its relations with gravity including black hole physics and AdS holography. Modern GR. There is a growth of connections between General Relativity (GR) and High Energy Physics. To be discussed in more detail in section 3. Out of these topics we choose to concentrate on scattering amplitudes, as discussed in the next section, and hence we provide some extended background for it. Background for scattering amplitudes. For some time it is known that the standard perturbation theory based on Feynman diagrams is not the optimal theory for several theories including (non- Abelian) gauge theory and gravity, especially in the presence of a large number of particles and/or small violation of helicity. Here we wish to present a brief and modest survey of the subject s evolution. In the sixties DeWitt observed unexpected cancellations in the 4 particle amplitudes of gravity and Yang Mills. Helicity spinors, whose evolution is described in the review [4], were used in the eighties to obtain the inspiring and simple Parke-Taylor expression for Maximal Helicity Violating (MHV) amplitudes [5]. Progress in the nineties included unitarity and on-shell methods, and ideas from supersymmetry and string theory, see the review [6] and references therein. In 2003 [7] reignited the interest in this field by suggesting an approach involving a string theory in Twistor space. One of the outcomes was a discovery of the recursion relations of [8]. An intriguing color-kinematics duality introduced by Bern-Carrasco-Johansson (BCJ) [9] reduced the number of independent sub-amplitudes to (n-3)! (the coefficients in the relations are kinematics dependent) and this duality was applied to describe gravitational amplitudes as a double copy of gauge theory. See [10] [11] for additional relatively recent work on gravity as the square of gauge theory. 2

4 A search for the underlying geometry, and especially that underlying dual superconformal invariance of N = 4, led to the Grassmannian, the space of k planes in n dimensional space, where k is related to the number of negative helicity gluons and n is the total number of external gluons. This was successively refined into the positive Grassmannian [12], a subset of the Grassmannian which generalizes the simplex, and most recently into the Amplituhedron [13], which is a larger object which can be triangulated by positive Grassmannians. A simple formula for gravitational tree-level MHV amplitudes was presented [14]. The Gross- Mende scattering equations were found to play a central role and allowed Cachazo-He-Yuan (CHY) [15] to obtain a remarkable closed formula for all tree level amplitudes in both (pure) gauge theory and gravity, without reference to helicity violation or helicity spinors. Excellent and relatively recent reviews include [16] [17] [18]. 2. Objectives and expected significance In this section we describe the reasons which led us to recently enter the field of scattering amplitudes. Some outstanding questions. Currently we have several exciting new concepts and techniques which pertain to scattering amplitudes including the helicity spinors, unitarity and on-shell methods, BCFW recursion relations, the BCJ color-kinematics duality, the CHY formula and finally the Amplituhedron, as described in section 1. These developments suggest numerous questions, including What is the physical interpretation of the space of variables for the Gross-Mende scattering equations, and that of the equations themselves? These variables are valued in the complex plane, and in the Gross-Mende context they are interpreted as points on the string worldsheet. A formulation of Yang-Mills amplitudes in terms of string theory in ambitwistor space suggests a possible similar interpretation. It would be nice if an even more direct physical interpretation could be found. Can the CHY formula be further simplified? In specific cases such as MHV it is known to be equivalent to the simpler Parke-Taylor expression. It is not clear when such simplifications are possible. What is the relation between the CHY formula and the formulation in terms of the Amplituhedron? The Gross-Mende scattering equations make no appearance in the Amplituhedron formulation, and vice versa, the Amplituhedron does not appear manifestly in the CHY formula. Yet, the two describe the same amplitudes, and hence could be expected to be linked. Does the CHY formula incorporate already all the insights from other known principles including unitarity cuts, the BCFW recursion relations, and the BCJ color-kinematics duality? More generally, it is known that standard Feynman diagrams can be improved upon, and several such improved concepts and techniques are known. Yet, there are no clear rules as to which technique is to be used in any given case, and neither was shown to be optimal. Hence it is clear 3

5 to me that the optimal procedure to compute scattering amplitudes in gauge theory and gravity is not fully known yet. Moreover, conceptually it is not clear how the various new concepts would all fit into a single unifying framework. Just like the Lagrangian and the action formulation underlie the standard perturbation theory à la Feynman, some believe that such a unifying framework will be associated with an alternative deep formulation of gauge theory, which would be the ultimate goal of studying scattering amplitudes. Expected Significance. Scattering amplitudes lie at the very core of Quantum Field Theory (QFT), both experimentally through accelerator experiments and theoretically. 60 years after the advent of QFT, and after all the effort invested over the years in the computation of Feynman diagrams, it is very surprising to find that some key concepts remain hidden. The current situation of the field indicates a clear potential for progress which will change both the way we compute amplitudes for accelerator experiments and possibly the way we think about gauge theory and other theories including GR. Hence the expected significance cannot be overestimated. Based on this potential we chose to direct the group towards entering this field last year (fall 2013). We worked on the decomposition into color and kinematics, which is the first step in the theory, by studying color structures and were able to obtain new insights and new results at tree level and 1-loop, including the role of the shuffle and split operation and a characterization of color structure transformation under permutations [19] [20] [21]. Closely related and important work on color structures appeared in [22] [23] [24]. We intend to continue to study and review the literature and seek to contribute to the ongoing effort to address the above-mentioned issues. I believe that our experience with gravity, field theory and more generally with abstract geometry prepares us well for the task, as demonstrated by our track record of theoretical innovation (below) and by our early results. I am encouraged by the anonymous and kind referee who found our presentation in [19] to be clearly superior and a well-known mathematician for his interest [25]. At the same time we intend to continue and apply the available expertise of the group in modern GR, especially in connection with high energy physics, and maintain our efforts in the field. We defer the description of objectives of this part to the next section. 3. Methodology This section describes specific problems that I intend for the group to engage with during the next couple of years. We start with scattering amplitudes. Shuffle algebras and amplitudes. Shuffle algebras are known to apply to iterated integrals and polylogarithms which appear in the evaluation of Feynman diagrams, see for example the lectures [26] and the talk [27]. Our work [19] identified this structure in a different context within scattering amplitudes, namely for color structures. We speculate that this may not be a coincidence, and we plan to study the iterated integrals and to examine whether a direct connection with color structures may exist, and its possible impact on the computation of subamplitudes. Eulerian numbers. The Eulerian numbers are defined in a combinatorial context through a certain partition of the set of permutations. In the context of scattering amplitudes they were 4

6 conjectured [28] and proven inductively [29] to count the number of solutions for certain relevant equations. I expect that the combinatorial definition is directly related to their appearance in scattering amplitudes and may enable insight and a non-inductive proof, which I intend to seek. Integration by parts. This is the name of a certain well-known recursive technique to compute multi-loop Feynman integrals [30] [31], and hence it is related to scattering amplitudes, even if somewhat indirectly. I believe that currently we do not know which integrals can be reduced using this technique and what would the number of master integrals be. I hope to engage with these questions. Continuation. I may work on open problem identified in our earlier scattering amplitudes papers on color structures and the role of the (2 ) representation of, and in particular to apply the newly determined symmetries of color structures to the computation of sub-amplitudes. To follow. In addition I intend to continue and follow the literature on scattering amplitudes including the CHY formulation, the Amplituhedron, the Hodges formulation, BCJ duality and more Modern GR The specific objectives in this direction are: Compose a review of the Effective Field Theory (EFT) approach to GR. Several groups in the world worked on this subject in the last decade, and our group made some significant contributions. I would like to write a review of these developments. Continuation. I may work on open problem identified in our earlier GR papers including higher relativistic anomalies in the two-body problem [32], renormalization and numeric implementation of equivalent periodic sources in EMR [33] and higher terms in the black hole effective action [34]. To follow. I intend to keep following key topics in modern GR including the following fields where I already made contributions: The EFT approach to GR, the post-newtonian approximation [35] [36], GR in higher dimensions [37] [38] [39] [40], the infinite dimension limit [41] [42] [43], and the Extreme Mass Ratio approximation Track record of proposed PI Clearly the most important research resource is the PI himself (BK). A description of the main scientific achievements and their recognition follows. Contributions to the Effective Field Theory (EFT) approach to GR, including the definition and basic theory of Non-Relativistic Gravitational (NRG) fields. Following Goldberger and Rothstein BK worked on the EFT approach to GR, a theory which applies ideas from quantum field theories to a wide range of problems in GR (and elsewhere) involving a scale hierarchy, including the post-newtonian two-body problem. He (together with his Ph.D. student, Smolkin) introduced a decomposition of the metric field which is natural and useful for the stationary as well as the Newtonian limit, and applied it to caged black holes (BHs within a compact dimension) and to the binary motion [44] [45]. These fields were compared by Gilmore and Ross 5

7 [46] against other possibilities and were chosen for reproducing the two-body effective action within the EFT method to order 2PN. Later they were used for 3PN as well [47]. In 2008 he won the fifth prize in the Gravity Essay Contest for identifying the BH effective degrees of freedom at low frequencies [48]. Predicting the phase diagram of the black-hole black-string transition. In 2001 a leading researcher initiated a debate regarding the end-state of the Gregory-Laflamme black string instability. On the following year BK predicted the end-state and the full phase diagram and argued for it by novel qualitative methods including Morse theory and topology change [49]. At the time many other possibilities were present in the literature, but BK's predictions were finally confirmed by numerical solutions [50] [51]. In 2007 BK organized in Jerusalem an international workshop on this topic. He organized another international workshop in 2012 titled 40 years of black hole thermodynamics. Definition and basic theory of (p,q) webs for supersymmetric field theories and more. In 1997 (p,q) 5-brane configurations arose in the realization of 5d super-symmetric field theories. BK (together with Aharony and Hanany) abstracted and defined (p,q) webs (a self-coined term) [52]. These webs were applied to both 5-brane webs and string webs in 5d field theories as well as later (by Bergman and BK) to 4d N=4 SUSY. In addition he foresaw mathematical applications for describing both Calabi-Yau manifolds and certain complex polynomials. Leung and Vafa [53] realized the first prediction through the theory of Toric Geometry. The second prediction became to be realized by the theory of Tropical Geometry which applied webs to answer a basic mathematical problem, that of counting the number of curves (of prescribed degree) passing through prescribed points. Surprisingly the counting works not only for complex numbers but also for real numbers. BK was invited to participate and speak at several meetings on tropical geometry: at MSRI Berkeley in 2009, in Ein-Gedi Israel in spring 2013 and a meeting in Oberwolfach to be held in spring Finding the local form of the conformal manifold for a super-conformal field theory. Obtaining the conformal manifold (super-conformal moduli space) for super-conformal field theories is a natural and basic problem. In BK stated and argued for the role of D-terms of the global group in this moduli space [54], and wrote down the formula for its local form in a draft [55]. This formula was later confirmed and rigorously proven by Green-Komargodsky- Seiberg-Tachikawa-Wecht [56]. Group leading track record. Since 2002 BK leads the research in his group consisting on average of some 4 Ph.D. students. He supervised six students to Ph.D. graduation (Sorkin, Gorbonos, Asnin, Smolkin, Levi and Hadar) and one to M.Sc. (Haddad). So far four of these proceeded to post-doc positions abroad, one to a position in the high-tech industry and one to a Ph.D. program in Barcelona. 6

8 References [1] "CERN announces LHC restart schedule," [Online]. Available: [Accessed 12 October 2014]. [2] P. A. R. Ade et. al. [BICEP2 collaboration], "Detection of B-Mode Polarization at Degree Angular Scales by BICEP2," Phys. Rev. Lett., vol. 112, p , [3] "Advanced LIGO News," [Online]. Available: [Accessed 12 October 2014]. [4] M. L. Mangano and S. J. Parke, "Multiparton amplitudes in gauge theories," Phys. Rept., vol. 200, p. 301, [5] S. J. Parke and T. R. Taylor, "An Amplitude for n Gluon Scattering," Phys. Rev. Lett., vol. 56, p. 2459, [6] Z. Bern, L. J. Dixon and D. A. Kosower, "Progress in one loop QCD computations," Ann.Rev. Nucl. Part. Sci., vol. 46, p. 109, [7] E. Witten, "Perturbative gauge theory as a string theory in twistor space," Commun. Math. Phys., vol. 252, p. 189, [8] R. Britto, F. Cachazo, B. Feng and E. Witten, "Direct proof of tree-level recursion relation in Yang-Mills theory," Phys. Rev. Lett., vol. 94, p , [9] Z. Bern, J. M. Carrasco and H. Johansson, "New Relations for Gauge-Theory Amplitudes," Phys. Rev. D, vol. 78, p , [10] N. E. J. Bjerrum-Bohr, P. H. Damgaard, B. Feng and T. Sondergaard, "Gravity and Yang-Mills Amplitude Relations," Phys. Rev. D, vol. 82, p , [11] N. E. J. Bjerrum-Bohr, P. H. Daamgard, R. Monteiro and D. O'Connell, "Algebras for Amplitudes," JHEP, vol. 1206, p. 061, [12] N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, A. B. Goncharov, A. Postnikov and J. Trnka, "Scattering Amplitudes and the Positive Grassmannian," arxiv: [13] N. Arkani-Hamed and J. Trnka, "The Amplituhedron," arxiv: [14] A. Hodges, "A simple formula for gravitational MHV amplitudes," arxiv: [15] F. Cachazo, S. He and E. Y. Yuan, "Scattering of Massless Particles in Arbitrary Dimension," arxiv: [16] L. J. Dixon, "Scattering amplitudes: the most perfect microscopic structures in the universe," J. Phys. A, vol. 44, p , [17] H. Elvang and Y.-t. Huang, "Scattering amplitudes," arxiv: [hep-th]. [18] L. J. Dixon, "A brief introduction to modern amplitude methods," arxiv: [hep-ph]. [19] B. Kol and R. Shir, "Color structures and permutations," accepted for publication by JHEP, arxiv: [hep-th]. 7

9 [20] B. Kol and R. Shir, "1-loop color structures and sunny diagrams," arxiv: [hep-th]. [21] B. Kol and R. Shir, "Perturbative gauge theory and 2+2=4," arxiv: [hep-th]. [22] S. G. Naculich, "All-loop group-theory constraints for color-ordered SU(N) gauge-theory amplitudes," Phys. Lett. B, vol. 707, p. 191, [23] A. C. Edison and S. G. Naculich, "Symmetric-group decomposition of SU(N) group-theory constraints on four-, five-, and six-point color-ordered amplitudes," JHEP, vol. 1209, p. 069, [24] C. Reuschle and S. Weinzierl, "Decomposition of one-loop QCD amplitudes into primitive amplitudes based on shuffle relations," Phys. Rev. D, vol. 88, p , [25] J. Stasheff, private communication. [26] S. Weinzierl, "Feynman Graphs," arxiv: [hep-ph]. [27] J. Henn, "Mathematical methods for scattering amplitudes," in Amplitudes 2014, Paris. [28] M. Spradlin and A. Volovich, "From Twistor String Theory To Recursion Relations," Phys. Rev. D, vol. 80, p , [29] F. Cachazo, S. He and E. Y. Yuan, "Scattering in Three Dimensions from Rational Maps," JHEP, vol. 1310, p. 141, [30] K. G. Chetyrkin and F. V. Tkachov, "Integration by Parts: The Algorithm to Calculate beta Functions in 4 Loops," Nucl. Phys. B, vol. 192, p. 159, [31] V. A. Smirnov, Feynman integral calculus, Berlin: Springer, [32] B. Kol and B. Nabet, "Leading anomalies, the drift Hamiltonian and the two-body system," arxiv: [gr-qc], submitted to Phys. Rev. D. [33] B. Kol, "Self-force from equivalent periodic sources," arxiv: [gr-qc]. [34] B. Kol and M. Smolkin, "Black hole strereotyping: induced gravito-electric polarization," JHEP, vol. 1202, p. 10, [35] L. Blanchet, "Gravitational radiation from post-newtonian sources and inspiralling compact binaries," Living Rev. Rel., vol. 9, p. 4, [36] G. Schaefer, "Post-Newtonian methods: Analytic results on the binary problem," Fundam. Theor. Phys., vol. 162, p. 167, [37] B. Kol, "The Phase transition between caged black holes and black strings: A Review," Phys. Rept., vol. 422, p. 119, [38] R. Emparan, "Blackfolds," in Black Holes in Higher Dimensions, Cambridge University Press, [39] N. A. Obers, "Black Holes in Higher-Dimensional Gravity," Lect. Notes Phys., vol. 769, p. 211, [40] R. Gregory, "The Gregory-Laflamme instability," in Black holes in higher dimensions, Cambridge University Press, [41] V. Asnin, D. Gorbonos, S. Hadar, B. Kol, M. Levi and U. Miyamoto, "High and Low 8

10 Dimensions in The Black Hole Negative Mode," Class. Quant. Grav., vol. 24, p. 5527, [42] R. Emaparan, R. Suzuki and K. Tanabe, "The large D limit of General Relativity," JHEP, vol. 1306, p. 009, [43] N. E. J. Bjerrum-Bohr, "Quantum gravity at a large number of dimensions," Nuc. Phys. B, vol. 684, p. 209, [44] B. Kol and M. Smolkin, "Classical Effective Field Theory and Caged Black Holes," Phys. Rev. D, vol. 77, p , [45] B. Kol and M. Smolkin, "Non-Relativistic Gravitation: From Newton to Einstein and Back," Class. Quant. Grav., vol. 25, p , [46] J. B. Gilmore and A. Ross, "Effective field theory calculation of second post-newtonian binary dynamics," Phys. Rev. D, vol. 78, p , [47] S. Foffa and R. Sturani, "Effective field theory calculation of conservative binary dynamics at third post-newtonian order," Phys. Rev. D, vol. 84, p , [48] B. Kol, "The Delocalized Effective Degrees of Freedom of a Black Hole at Low Frequencies," Gen. Rel. Grav., vol. 40, p. 2061, [49] B. Kol, "Topology change in general relativity, and the black hole black string transition," JHEP, vol. 0510, p. 049, [50] H. Kudoh and T. Wiseman, "Connecting black holes and black strings," Phys. Rev. Lett., vol. 94, p , [51] L. Lehner and F. Pretorius, "Black strings, low viscosity fluids and violation of cosmic censorship," Phys. Rev. Lett., vol. 105, p , [52] O. Aharony, A. Hanany and B. Kol, "Webs of (p,q) five-branes, five-dimensional field theories and grid diagrams," JHEP, vol. 9801, p. 002, [53] C. N. Leung and C. Vafa, "Branes and toric geometry," Adv. Theor. Math. Phys., vol. 2, p. 91, [54] B. Kol, "On conformal deformations," JHEP, vol. 0209, p. 046, [55] B. Kol, "On Conformal Deformations II," arxiv: [hep-th]. [56] D. Green, Z. Komargodski, N. Seiberg, Y. Tachikawa and B. Wecht, "Exactly Marginal Deformations and Global Symmetries," JHEP, vol. 1006, p. 106,