DOCSLIB.ORG

Loop Amplitudes from MHV Diagrams - Part 1

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Loop Amplitudes from MHV Diagrams - Part 1 Loop Amplitudes from MHV Diagrams - part 1 - Gabriele Travaglini Queen Mary, University of London Brandhuber, Spence, GT hep-th/0407214, hep-th/0510253 Bedford, Brandhuber, Spence, GT hep-th/0410280, hep-th/0412108 Brandhuber, McNamara, Spence, GT in progress 2 HP Workshop, Zürich, September 2006 Outline • Motivations and basic formalism • MHV diagrams (Cachazo, Svrcek,ˇ Witten) - Loop MHV diagrams (Brandhuber, Spence, GT) - Applications: One-loop MHV amplitudes in N=4, N=1 & non-supersymmetric Yang-Mills • Proof at one-loop: Andreas’s talk on Friday Motivations • Simplicity of scattering amplitudes unexplained by textbook Feynman diagrams - Parke-Taylor formula for Maximally Helicity Violating amplitude of gluons (helicities are a permutation of −−++ ....+) • New methods account for this simplicity, and allow for much more efficient calculations • LHC is coming ! Towards simplicity • Colour decomposition (Berends, Giele; Mangano, Parke, Xu; Mangano; Bern, Kosower) • Spinor helicity formalism (Berends, Kleiss, De Causmaecker, Gastmans, Wu; De Causmaecker, Gastmans, Troost, Wu; Kleiss, Stirling; Xu, Zhang, Chang; Gunion, Kunszt) Colour decomposition • Main idea: disentangle colour • At tree level, Yang-Mills interactions are planar tree a a A ( p ,ε ) = Tr(T σ1 T σn) A(σ(p ,ε ),...,σ(p ,ε )) { i i} ∑ ··· 1 1 n n σ Colour-ordered partial amplitude - Include only diagrams with fixed cyclic ordering of gluons - Analytic structure is simpler • At loop level: multi-trace contributions - subleading in 1/N Spinor helicity formalism • Consider a null vector p µ µ µ = ( ,! ) • Define p a a ˙ = p µ σ a a ˙ where σ 1 σ • If p 2 = 0 then det p = 0 ˜ ˜ Hence p aa˙ = λaλa˙ · λ (λ) positive (negative) helicity • spinors a b a˙ b˙ Inner products 12 : = ε ab λ λ , [12] := ε ˙λ˜ λ˜ • ! " 1 2 a˙b 1 2 Parke-Taylor formula 4 + + i j A (1 ...i− ... j− ...n ) = " # MHV 12 23 n1 " #" #···" # ☞ ☞ • Colour decomposition and spinor helicity formalism make the simplicity manifest... • ...but we still have to explain it ! Twistor space • Simple geometrical structure of the amplitudes in twistor space (Witten) MHV Diagrams (Cachazo, Svrcek,ˇ Witten) • String theory on twistor space (Witten) Recursion relations for YM tree amplitudes • (Britto, Cachazo, Feng +Witten) Twistor space (in one slide) • Half Fourier transform on spinor variables ∂ (λ ,λ˜ ) (λ ,µ˜ ) λ˜ i a a˙ −→ a a˙ −→ ∂µ ∂ a,a˙ = 1,2 µ i −→ − ∂λ˜ • Twistor space is a copy of C 4 1 (λ,λ˜ ) (tλ,t− λ˜ ) (λ,µ) (tλ,tµ) −→ ➡ −→ 3 4 ☜ # of supersymmetries Projective twistor space, CP | • Fourier transform can be explicitly carried out for MHV amplitudes ‣ Holomorphicity in the spinor variables • Result ≠ 0 if coordinates of gluons satisfy a˙ aa˙ µ + x λa = 0 x = point in Minkowski space • Line in twistor space (Penrose’s incidence relation) ‣ Maps points in Minkowski space to twistors What about generic amplitudes ? • Scattering amplitudes are supported on curves in twistor space (Witten, 2003) • d = q − 1 + l q = # negative helicity gluons, l = # loops • g ≤ l ‣ Tree MHV: q=2, l=0 ➡ d=1, g=0 (complex line) Amplitude Twistor space structure MHV diagrams ! " ! " M MHV ! ! ! " " ! " M M nMHV ! ! " " ! ! " ! " " ! ! ! nnMHV " M M M " " Why MHV diagrams ? • MHV amplitudes localise on lines in twistor space (Witten) • A line in twistor space corresponds to a point in Minkowski space (Penrose) ➡ • An MHV amplitude can be thought of as a local interaction in space-time ! (Cachazo, Svrcek,ˇ Witten) MHV Rules (tree level) (Cachazo, Svrcek,ˇ Witten) • MHV amplitude ➡ MHV vertex • Off-shell continuation for internal (possibly loop) momenta needed: · Internal momentum is off-shell M M · Need to define spinor λ for an off-shell vector! • Need propagators to connect the MHV vertices Off-shell continuation If L 2 = 0 , we can write • ! Laa˙ = laa˙ + zηaa˙ ‣ η a a˙ := η a η˜ a ˙ is a null reference vector ‣ z = L 2 / 2 ( L η ) is a real number · ‣ l a a˙ : = l a l˜ a˙ is the off-shell continuation, ‣ l L η˜ a˙ (equivalent to CSW’s) a ⇒ aa˙ Internal propagators i Just scalar propagators • P2 + iε • At loop level, the i ε prescription is crucial in correctly determining the integration range Proofs of MHV diagrams Tree level • Covariance (η-independence); singularities (Cachazo, Svrcek,ˇ Witten; Britto, Cachazo, Feng, Witten) • MHV diagrams are a special BCFW recursion relation (Risager) • Lagrangian derivation (Mansfield; Gorsky, Rosly) Loop MHV diagrams (Brandhuber, Spence, GT) • Initial prognosis terribly poor... ‣ Twistor string theory dual to conformal supergravity (not Yang-Mills, not even ordinary supergravity) at the quantum level • ...but we tried anyway ! • Simplest amplitude: 1-loop MHV amplitude in N=4 super Yang-Mills • Computed in 1994 by Bern, Dixon, Dunbar, From TreesKoso tow erLoops, cont’d 1 loop tree A − = A Atree MHV∑× the all order in ! 2-mass easy box function: " " 2me 2 2 c! s − Two-mass easyt bo− x function F (s,t,P ,Q ) = − 2F1 (1, ",1 ",as) + − 2F1 (1, ",1 ",at) −"2 µ2 − − µ2 − − !" # " # P2 " Q2 " − − F 1, ",1 ",aP2 − − F 1, ",1 ",aQ2 − µ2 2 1 − − − µ2 2 1 − − & " # $ % " # $ % 2(pq) with a := P2Q2 st − MHV rules for Loops • Sew d MHV vertices q = # negative helicity gluons, d = q − 1 + l • l = # loops • As at tree level, we use a. CSW off-shell continuation b. Scalar propagators • MHV, 1-loop: d = 2 From Trees to Loops (AB-Spence-Travaglini) • Original prognosis from twistor string theory was negative (Berkovits-Witten), Conformal SUGRA modes spoil duality • Try anyway: • Connect MHV vertices, using the same off-shell continuation as for trees • Chose measure, perform loop integration MHV 1-loop amplitudes in N=4/N=1 SYM (agrees with BDDK) • Calculation from MHV diagrams d M∑ d M Z ! Z m1,m2,h • The sum is over ‣ all possible MHV diagrams ‣ internal particle species (g, f, s) and helicities ‣ different from unitarity-based approach of BDDK • We have to find the measure... From Trees to Loops (AB-Spence-Travaglini) • Original prognosis from twistor string theory was negative (Berkovits-Witten), Conformal SUGRA modes spoil duality • Try anyway: • Connect MHV vertices, using the same off-shell continuation as for trees • Chose measure, perform loop integration • MHVThe 1-loop amplitudes integration in N=4/N=1 SYM measur (agrees with BDDK)e · P L is the momentumdM on the left ! Z m1,m2,h 4 4 d L1 d L2 (4) dM := 2 2 δ (L2 L1 + PL) L1 + iε L2 + iε − Use L = l + z η , and L ( l , z ): • → 4 3 d L dz d l ➡ 2 = L + iε z + isgn (l0η0)ε 2l0 dispersive measure X phase-space measure (Nair measure) Loop integral becomes: (Dispersion integral) X (2-particle LIPS integral) • LIPS integral: ‣ computes the cut of the amplitude ‣ regularise IR divergences: 4−2ε dimensions • Dispersion integral reconstructs the amplitude from cuts The return of the analytic S-matrix ! (1966) Comments • Final result is covariant and agrees with BDDK ‣ Non-trivial cancellation of η-dependence • MHV diagrams work for loops ! • Result incorporates a large number of Feynman diagrams ‣ 1 MHV diagram vs 227,585 Feynman diagrams (7 points) More Comments Dispersion integrals are simplei - no • P2 + iε subtractions needed (van Neerven) • No ghosts • Scalar propagators ‣ MHV diagrams related to lightcone gauge quantisation (more on this later) New form of the 2-mass easy box function 2 2 1 ε ε 2 ε 2 2 2 F(s,t,P ,Q ) = ( s)− + ( t)− ( P )− ( Q )− + B(s,t,P ,Q ) −ε2 − − − − − − ! " B(s,t,P2,Q2) = Li (1 aP2) + Li (1 aQ2) Li (1 as) Li (1 at) 2 − 2 − − 2 − − 2 − From Trees to Loops, cont’d P2 + Q2 s t a = − − P2Q2 st − tree 2 A 2 s = (P + p) t = (P + q)MHV× the all order in ! 2-mass easy box function: - More compact than usual expression " " 2me 2 2 c! s − t − F (s,t,P ,Q ) = − 2F1 (1, ",1 ",as) + − 2F1 (1, ",1 ",at) - Simpler−"2 analµ2 ytic contin− uation− µ2 − − !" # " # P2 " Q2 " − − F 1, ",1 ",aP2 − − F 1, ",1 ",aQ2 − µ2 2 1 − − − µ2 2 1 − − & " # $ % " # $ % 2(pq) with a := P2Q2 st − Further applications • One-loop MHV amplitudes in N=1 super Yang-Mills (Bedford, Brandhuber, Spence, GT; Quigley, Rozali) ‣ Result expressed in terms of finite boxes, and triangles ‣ Agreement with BDDK ‣ No twistor string theory for N=1 Super Yang-Mills... ‣ ...nevertheless MHV diagram method works ! • Cut-constructible part of 1-loop MHV amplitudes in non-supersymmetric Yang-Mills (Bedford, Brandhuber, Spence, GT) ‣ Extends 5-pt and adjacent negative helicity cases of BDK and BDDK ‣ First new result at 1-loop in pure YM ‣ Non-supersymmetric amplitudes are not cut-constructible in 4 dimensions ‣ rational terms are missed by MHV diagrams Next goal: Prove MHV diagrams at one loop • Covariance (η-independence) • Correct singularity structure Andreas Brandhuber’s talk: Friday at 11:25 Summary • MHV diagrams provide a new diagrammatic method to calculate scattering amplitudes at tree and one-loop level in Yang-Mills • Proof for generic one-loop amplitudes: ‣ Feynman Tree Theorem ✂ ‣ Singularity structure .
Load more

Recommended publications
  • Beyond Feynman Diagrams Lecture 2
    Beyond Feynman Diagrams Lecture 2 Lance Dixon Academic Training Lectures CERN April 24-26, 2013 Modern methods for trees 1. Color organization (briefly) 2. Spinor variables 3. Simple examples 4. Factorization properties 5. BCFW (on-shell) recursion relations Beyond Feynman Diagrams Lecture 2 April 25, 2013 2 How to organize gauge theory amplitudes • Avoid tangled algebra of color and Lorentz indices generated by Feynman rules structure constants • Take advantage of physical properties of amplitudes • Basic tools: - dual (trace-based) color decompositions - spinor helicity formalism Beyond Feynman Diagrams Lecture 2 April 25, 2013 3 Color Standard color factor for a QCD graph has lots of structure constants contracted in various orders; for example: Write every n-gluon tree graph color factor as a sum of traces of matrices T a in the fundamental (defining) representation of SU(Nc): + all non-cyclic permutations Use definition: + normalization: Beyond Feynman Diagrams Lecture 2 April 25, 2013 4 Double-line picture (’t Hooft) • In limit of large number of colors Nc, a gluon is always a combination of a color and a different anti-color. • Gluon tree amplitudes dressed by lines carrying color indices, 1,2,3,…,Nc. • Leads to color ordering of the external gluons. • Cross section, summed over colors of all external gluons = S |color-ordered amplitudes|2 • Can still use this picture at Nc=3. • Color-ordered amplitudes are still the building blocks. • Corrections to the color-summed cross section, can be handled 2 exactly, but are suppressed by 1/ Nc Beyond Feynman Diagrams Lecture 2 April 25, 2013 5 Trace-based (dual) color decomposition For n-gluon tree amplitudes, the color decomposition is momenta helicities color color-ordered subamplitude only depends on momenta.
    [Show full text]
  • Maximally Helicity Violating Amplitudes
    Maximally Helicity Violating amplitudes Andrew Lifson ID:22635416 Supervised by Assoc. Prof. Peter Skands School of Physics & Astronomy A thesis submitted in partial fulfilment for the degree of Bachelor of Science Advanced with Honours November 13, 2015 Abstract An important aspect in making quantum chromodynamics (QCD) predictions at the Large Hadron Collider (LHC) is the speed of the Monte Carlo (MC) event generators which are used to calculate them. MC event generators simulate a pure QCD collision at the LHC by first using fixed-order perturbation theory to calculate the hard (i.e. energetic) 2 ! n scatter, and then adding radiative corrections to this with a parton shower approximation. The 2 ! n scatter is traditionally calculated by summing Feynman diagrams, however the number of Feynman diagrams has a stronger than factorial growth with the number of final-state particles, hence this method quickly becomes infeasible. We can simplify this calculation by considering helicity amplitudes, in which each particle has its spin either aligned or anti-aligned with its direction of motion i.e. each particle has a specific helicity. In particular, it is well-known that the helicity amplitude for the maximally helicity violating (MHV) configuration is remarkably simple to calculate. In this thesis we describe the physics of the MHV amplitude, how we could use it within a MC event generator, and describe a program we wrote called VinciaMHV which calculates the MHV amplitude for the process qg ! q + ng for n = 1; 2; 3; 4. We tested the speed of VinciaMHV against MadGraph4 and found that VinciaMHV calculates the MHV amplitude significantly faster, especially for high particle multiplicities.
    [Show full text]
  • MHV Amplitudes in N=4 SUSY Yang-Mills Theory and Quantum Geometry of the Momentum Space
    Outline Introduction and tree MHV amplitudes BDS conjecture and amplitude - Wilson loop correspondence c=1 string example and fermionic representation of amplitudes Quantization of the moduli space On the fermionic representation of the loop MHV amplitudes Towards the stringy interpretation of the loop MHV amplitudes. MHV amplitudes in N=4 SUSY Yang-Mills theory and quantum geometry of the momentum space Alexander Gorsky ITEP GGI, Florence - 2008 (to appear) Alexander Gorsky MHV amplitudes in N=4 SUSY Yang-Mills theory and quantum geometry of the momentum space Outline Introduction and tree MHV amplitudes BDS conjecture and amplitude - Wilson loop correspondence c=1 string example and fermionic representation of amplitudes Quantization of the moduli space On the fermionic representation of the loop MHV amplitudes Towards the stringy interpretation of the loop MHV amplitudes. Introduction and tree MHV amplitudes BDS conjecture and amplitude - Wilson loop correspondence c=1 string example and fermionic representation of amplitudes Quantization of the moduli space On the fermionic representation of the loop MHV amplitudes Towards the stringy interpretation of the loop MHV amplitudes. Alexander Gorsky MHV amplitudes in N=4 SUSY Yang-Mills theory and quantum geometry of the momentum space Outline Introduction and tree MHV amplitudes BDS conjecture and amplitude - Wilson loop correspondence c=1 string example and fermionic representation of amplitudes Quantization of the moduli space On the fermionic representation of the loop MHV amplitudes
    [Show full text]
  • Unitarity Methods for Scattering Amplitudes in Perturbative Gauge Theories
    Unitarity Methods for Scattering Amplitudes in Perturbative Gauge Theories by MASSIMILIANO VINCON A report presented for the examination for the transfer of status to the degree of Doctor of Philosophy of the University of London Thesis Supervisor DR ANDREAS BRANDHUBER Centre for Research in String Theory Department of Physics Queen Mary University of London Mile End Road London E1 4NS United Kingdom The strong do what they want and the weak suffer as they must.1 1Thucydides. Declaration I hereby declare that the material presented in this thesis is a representation of my own personal work unless otherwise stated and is a result of collaborations with Andreas Brandhuber and Gabriele Travaglini. Massimiliano Vincon Abstract In this thesis, we discuss novel techniques used to compute perturbative scattering am- plitudes in gauge theories. In particular, we focus on the use of complex momenta and generalised unitarity as efficient and elegant tools, as opposed to standard textbook Feynman rules, to compute one-loop amplitudes in gauge theories. We consider scatter- ing amplitudes in non-supersymmetric and supersymmetric Yang-Mills theories (SYM). After introducing some of the required mathematical machinery, we give concrete exam- ples as to how generalised unitarity works in practice and we correctly reproduce the n- gluon one-loop MHV amplitude in non-supersymmetric Yang-Mills theory with a scalar running in the loop, thus confirming an earlier result obtained using MHV diagrams: no independent check of this result had been carried out yet. Non-supersymmetric scat- tering amplitudes are the most difficult to compute and are of great importance as they are part of background processes at the LHC.
    [Show full text]
  • Thesis Presented for the Degree of Doctor of Philosophy
    Durham E-Theses Hadronic productions of a Higgs boson in association with two jets at next-to-leading order WILLIAMS, CIARAN How to cite: WILLIAMS, CIARAN (2010) Hadronic productions of a Higgs boson in association with two jets at next-to-leading order , Durham theses, Durham University. Available at Durham E-Theses Online: http://etheses.dur.ac.uk/414/ Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in Durham E-Theses • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full Durham E-Theses policy for further details. Academic Support Oce, Durham University, University Oce, Old Elvet, Durham DH1 3HP e-mail: [email protected] Tel: +44 0191 334 6107 http://etheses.dur.ac.uk 2 Hadronic production of a Higgs boson in association with two jets at next-to-leading order Ciaran Williams A Thesis presented for the degree of Doctor of Philosophy Institute for Particle Physics Phenomenology Department of Physics University of Durham England August 2010 Dedicated to My Parents Hadronic production of a Higgs boson in association with two jets at next-to-leading order Ciaran Williams Submitted for the degree of Doctor of Philosophy August 2010 Abstract We present the calculation of hadronic production of a Higgs boson in association with two jets at next-to-leading order in perturbation theory.
    [Show full text]
  • Studies in Field Theories: Mhv Vertices, Twistor Space, Recursion Relations and Chiral Rings
    STUDIES IN FIELD THEORIES: MHV VERTICES, TWISTOR SPACE, RECURSION RELATIONS AND CHIRAL RINGS Peter Svr·cek Advisor: Edward Witten A DISSERTATION SUBMITTED TO THE FACULTY OF PRINCETON UNIVERSITY IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY RECOMMENDED FOR ACCEPTANCE BY THE DEPARTMENT OF PHYSICS April 2005 Copyright °c 2005 by Peter Svr·cek All rights reserved. ii Abstract In this thesis we study di®erent aspects of four dimensional ¯eld theories. In the ¯rst chapter we give introduction and overview of the thesis. In the second chapter we review the connection between perturbative Yang-Mills and twistor string the- ory. Inspired by this, we propose a new way of constructing Yang-Mills scattering amplitudes from Feynman graphs in which the vertices are o®-shell continuations of the tree level MHV amplitudes. The MHV diagrams lead to simple formulas for tree-level amplitudes. We then give a heuristic derivation of the diagrams from twistor string theory. In the third chapter, we explore the twistor structure of scattering amplitudes in theories for which a twistor string theory analogous to the one for N = 4 gauge theory has not yet been proposed. We study the di®erential equations of one-loop amplitudes of gluons in gauge theories with reduced supersymmetry and of tree level and one-loop amplitudes of gravitons in general relativity and supergravity. We ¯nd that the scattering amplitudes localize in twistor space on algebraic curves that are surprisingly similar to the N = 4 Yang-Mills case. In the next chapter we propose tree-level recursion relations for scattering am- plitudes of gravitons.
    [Show full text]
  • Unitarity and On-Shell Recursion Methods for Scattering Amplitudes
    FACULTY OF SCIENCE UNIVERSITY OF COPENHAGEN PhD thesis Kasper Risager Larsen Unitarity and On-Shell Recursion Methods for Scattering Amplitudes Academic advisor: Poul Henrik Damgaard Submitted: 23/11/07 Unitarity and On-Shell Recursion Methods for Scattering Amplitudes by Kasper Risager Submitted for the degree of Ph.D. in Physics from the Niels Bohr Institute, Faculty of Science, University of Copenhagen. November 2007 Abstract This thesis describes some of the recent (and some less recent) developments in calculational techniques for scattering amplitudes in quantum field theory. The focus is on on-shell recursion relations in complex momenta and on the use of unitarity methods for loop calculations. In particular, on-shell recursion is related to the MHV rules for computing tree-level gauge amplitudes and used to extend the MHV rules to graviton scattering. Combinations of unitarity cut techniques and recursion is used to argue for the “No-Triangle Hypothesis” in N = 8 supergravity which is related to its UV behaviour. Finally, combinations of unitarity and recursion is used to demonstrate the full calculation of a one- loop amplitude involving a Higgs particle and four gluons in the limit of large top mass. Contents 1 Introduction 5 1.1ScatteringAmplitudesinTheoryandExperiment.......... 5 1.1.1 BackgroundProcessesattheLHC.............. 6 1.1.2 ApproachingQuantumGravity?............... 7 1.2ScatteringAmplitudesintheComplexPlane............ 7 1.2.1 The Unitary and Analytic S-Matrix............. 8 1.2.2 GeneralizingfromLoopstoTrees............... 10 1.2.3 FullAmplitudesfromTrees.................. 11 1.3AboutThisThesis........................... 12 1.3.1 Omissions............................ 12 1.3.2 Outline............................. 12 1.4ConclusionandOutlook........................ 13 1.4.1 ACompetitorintheCalculationRace............ 13 1.4.2 UnderstandingQuantumFieldTheories........... 14 2 Preliminaries 15 2.1ColourOrdering...........................
    [Show full text]
  • A Simpler Prescription for MHV Graviton Tree Amplitudes in Superstring Theory
    A simpler prescription for MHV graviton tree amplitudes in superstring theory Tiago Ursulino [email protected] Instituto de F´ısica Te´orica, Universidade Estadual Paulista (IFT-Unesp) S~aoPaulo, SP, Brazil September 2013 Abstract We extend the Berkovits-Maldacena prescription for MHV amplitudes of the open superstring to the closed superstring, showing that in the α0 = 0 limit it reduces to the result of supergravity found recently by Hodges. We also verify that this prescription calculates the correct superstring tree level MHV amplitude for 4 gravitons including α0 corrections. arXiv:1308.6456v2 [hep-th] 5 Nov 2013 Keywords: Scattering amplitudes, superstrings and heterotic strings, extended su- persymmetry Contents 1 Introduction 1 2 MHV amplitudes in super-Yang-Mills and supergravity 3 2.1 Helicity spinors . 3 2.2 Supersymmetric extensions of YM and gravity . 6 2.3 MHV amplitudes for gluons . 8 2.4 MHV amplitudes for gravitons . 9 2.4.1 An overview of the formulas so far; symmetry and efficiency . 9 3 MHV amplitudes in superstring theory 14 3.1 Berkovits-Maldacena prescription . 14 3.2 The analogous prescription for closed strings . 16 3.2.1 Scattering with n=4 gravitons . 17 3.2.2 The α0=0 limit, for arbitrary n . 19 4 Conclusions 26 A Comparing our MHV scattering amplitude for n=4 gravitons with usual superstring theory 27 1 Introduction Berkovits and Maldacena [1] have proposed 5 years ago a prescription to calculate MHV amplitudes for gluons at tree level inside the context of string theory, in a 4 dimensional spacetime. Their
    [Show full text]
  • R4 Counterterm and E7(7) Symmetry in Maximal Supergravity
    SLAC–PUB–13848 AEI-2009-110 ITP-UH-18/09 arXiv:0911.5704 4 counterterm and E symmetry in maximal supergravity R 7(7) Johannes Br¨odela,b and Lance J. Dixonc a Max-Planck-Institut f¨ur Gravitationsphysik Albert-Einstein-Institut, Golm, Germany b Institut f¨ur Theoretische Physik Leibniz Universit¨at Hannover, Germany c SLAC National Accelerator Laboratory Stanford University Stanford, CA 94309, USA Abstract The coefficient of a potential 4 counterterm in = 8 supergravity has been shown previously R N to vanish in an explicit three-loop calculation. The 4 term respects = 8 supersymmetry; R N hence this result poses the question of whether another symmetry could be responsible for the cancellation of the three-loop divergence. In this article we investigate possible restrictions from the coset symmetry E7(7)/SU(8), exploring the limits as a single scalar becomes soft, as well as a double-soft scalar limit relation derived recently by Arkani-Hamed et al. We implement these relations for the matrix elements of the 4 term that occurs in the low-energy expansion of closed- R string tree-level amplitudes. We find that the matrix elements of 4 that we investigated all obey R the double-soft scalar limit relation, including certain non-maximally-helicity-violating six-point amplitudes. However, the single-soft limit does not vanish for this latter set of amplitudes, which suggests that the E symmetry is broken by the 4 term. 7(7) R Work supported in part by US Department of Energy contract DE-AC02-76SF00515. Contents 1 Introduction 2 2 Coset structure, hidden symmetry and double-soft limit 6 3 String theory corrections to field theory amplitudes 8 3.1 Tree-level amplitudes in = 4 SYM and =8Supergravity.
    [Show full text]
  • Non-Renormalization Theorems Without Supersymmetry
    CALT-2015-024 Non-renormalization Theorems without Supersymmetry Clifford Cheung and Chia-Hsien Shen Walter Burke Institute for Theoretical Physics California Institute of Technology, Pasadena, CA 91125∗ (Dated: May 11, 2015) We derive a new class of one-loop non-renormalization theorems that strongly constrain the running of higher dimension operators in a general four-dimensional quantum field theory. Our logic follows from unitarity: cuts of one-loop amplitudes are products of tree amplitudes, so if the latter vanish then so too will the associated divergences. Finiteness is then ensured by simple selection rules that zero out tree amplitudes for certain helicity configurations. For each operator we define holomorphic and anti-holomorphic weights, (w, w) = (n − h, n + h), where n and h are the number and sum over helicities of the particles created by that operator. We argue that an operator Oi can only be renormalized by an operator Oj if wi ≥ wj and wi ≥ wj , absent non-holomorphic Yukawa couplings. These results explain and generalize the surprising cancellations discovered in the renormalization of dimension six operators in the standard model. Since our claims rely on unitarity and helicity rather than an explicit symmetry, they apply quite generally. INTRODUCTION where n(A) and h(A) are the number and sum over helic- ities of the external states. Since A is physical, its weight Technical naturalness dictates that all operators not is field reparameterization and gauge independent. The forbidden by symmetry are compulsory—and thus gen- weights of an operator O are then invariantly defined by erated by renormalization. Softened ultraviolet diver- minimizing over all amplitudes involving that operator: gences are in turn a telltale sign of underlying symmetry.
    [Show full text]
  • Refining MHV Diagrams
    Refining MHV Diagrams by Andreas Brandhuber Queen Mary, University of London Munich, June 19, 2007 hep-th/0612007 AB-Spence-Travaglini arxiv:0704.0245 [hep-th] AB-Spence-Travaglini-Zoubos Outline • Motivation • MHV Diagrams • Tree MHV Diagrams (Cachazo-Svrcek-Witten) • Loop MHV Diagrams (AB-Spence-Travaglini) • One-loop Amplitudes in Pure Yang-Mills from MHV diagram • The all-minus helicity amplitudes • MHV diagrams from LC Lagrangian (Gorsky-Rosly, Mansfield, Morris-Ettle) • Completion of MHV diagram method • all-plus helicity amplitude from Lorentz violating counterterm • non-canonical, “holomophic” field transformation • Conclusions Goal • The goal of this talk is to address two questions: I. Do MHV Diagrams (a la CSW) provide a new, complete, perturbative expansion of Yang-Mills? II. And if not can we find a suitable completion of the MHV diagram method? Motivations • Simplicity of scattering amplitudes unexplained by textbook Feynman methods: e.g. Park-Taylor formula for MHV tree amplitude <--+++...+> • New ideas from twistor string theory (initiated by Witten 2003) help to “explain” this simplicity • Novel, practical tools to calculate physically relevant amplitudes ➡ LHC is coming • New reorganisations of conventional perturbation theory Spinor helicity Amplitudes formalism ColourSpinor decomposition helicity formalism • Main idea:A =disentangleA( λi, colourλi,;hi ) Scattering p µ null vector Amplitudes { } • At tree lep µ v el n, Yullang-Mills vector interactions are planar •Color Ordering:• µ µ DefineColour p a-or a ˙ = der p µ edσ
    [Show full text]
  • Conformal Properties of All-Plus Scattering Amplitudes Konforme Eigenschaften Von Streuamplituden
    Conformal Properties of All-Plus Scattering Amplitudes Konforme Eigenschaften von Streuamplituden Submitted by Edward Wang Bachelor Thesis Faculty of Physics Ludwig-Maximilians-Universit¨atM¨unchen Supervisor Prof. Dr. Johannes Henn August 22, 2019 Abstract Symmetry properties of scattering amplitudes often lead to simplifications in their computation and in their final expression. Gluonic scattering amplitudes in pure Yang-Mills theory in the all-plus helicity configuration exhibit remarkable signs of conformal symmetry. At one-loop order they are found to be conformally invariant, but this observation still lacks an explanation. In the present thesis we analyze the conformal properties of all-plus gluon scattering amplitudes at one-loop order and present an attempt to rewrite them in terms of manifestly conformally invariant objects. This might lead to a better understanding of the origin of this symmetry for these amplitudes. i Acknowledgements I would like to thank Prof. Henn for introducing me to this exciting topic and for the fruitful discussions. I would like to thank Dr. Kai Yan for patiently answering all my questions. I would like to express my deep gratitude to Simone Zoia who accompanied me in every step of my research project and who very thoughtfully guided me through all challenges faced throughout this project. I would like to thank my family and friends for supporting me in my professional and academic decisions, especially my brother William, who always encouraged me and accompanied me throughout my studies. I also want to thank my friend Sven Jandura for proofreading this thesis. ii Contents 1 Introduction 1 2 Tools 3 2.1 Yang-Mills Theory .
    [Show full text]