A modern perspective of scattering amplitudes

Transcription

1 1/44 A modern perspective of scattering amplitudes Karol Kampf Charles University, Prague HEP MAD Tana, 25 September 2017 Outline: Introduction & Motivation Effective field theories Scalar sector Spin-1 sector Summary

2 2/44 Introduction Elementary building blocks of nature: + Higgs boson (source: Fermilab Visual Media)

3 3/44 Introduction to understand the basic structure we use scattering experiments basic function: scattering cross section σ: connected with probability for a given process of scattered particles as a function of their energy and momentum + dependence on the angles differential cross-section dσ/dω theoretical prediction based on the quantum field theory basic object: scattering amplitude A dσ/dω A 2 scattering amplitude can be calculated systematically as an expansion of a small parameter using the so-call Feynman diagrams

4 4/44 Motivation Objective of amplitude community: look at familiar objects from different perspective why? two possible answers: 1 technical: we can obtain easier some results 2 conceptual: having completely equivalent reformulation of a given theory can lead to new property discoveries (invisible in traditional formulation) natural description, framework for something new /example: principle of least action for Newton s laws vs. quantum revolution/

5 5/44 Example: gluon amplitudes standard method of calculating n-gluon scattering processes: dominated by pure-gluon interactions in QCD elementary 3pt and 4pt vertices construct all possible Feynman diagrams, e.g.: complicated already for tree level diagrams even for small number of external legs

6 6/44 History: gluon amplitude, tree-level 3pt: 1 diagram, on-shell = 0 4pt: 4 diagrams, can be calculated by hand, differential cross section nice (but intermediate steps complicated) 5pt: calculated in 80, calculation blows up on several pages structure schematically the numerator: single-propagator: (p k ɛ)(ɛ ɛ)(ɛ ɛ), double-propagator: (p i p j )(p k ɛ)(ɛ ɛ)(ɛ ɛ), 6pt: impossible by standard method, but...

7 History: gluon amplitude, tree-level, 6pt SSC approved in 1983 (to be cancelled 10 years later) motivated the following work 7/44

8 8/44 History: gluon amplitude, tree-level, 6pt Parke and Taylor finished the article with:

9 History: gluon amplitude, tree-level, 6pt Parke and Taylor finished the article with: Indeed it was given a year later [Parke, Taylor 86]: A n ( ) = n1 One line formula! The so-called spinor-helicity formalism was introduced (reasonable variables for massless particles) cf. [Mangano,Parke,Xu 87] ij = 2p i p j e iφ ij Is there some better way to calculate? 8/44

10 9/44 The unexpected simplicity is the main motivation for current studies of scattering amplitudes. Target is clear: solution at all (loop) orders. New methods should find correct language. Many ways have been started: toy model ( harmonic oscillator ): N =4 Super Yang Mills Theory, perturbative gravity, pushing trees into loops, applications in phenomenology... But, back to our example (though many statements are quite general)

11 10/44 Example: gluon amplitudes At tree level: colour ordering stripped amplitude M a 1...a n (p 1,... p n ) = σ/z n Tr(t a σ(1)... t a σ(n) )M σ (p 1,..., p n ) M σ (p σ(1),..., p σ(n) ) = M(p 1,..., p n ) M(1, 2,... n) propagators the only poles of M σ thanks to ordering the only possible poles are: P 2 ij = (p i + p i p j 1 + p j ) 2

12 11/44 Pole structure Weinberg s theorem (one particle unitarity): on the factorization channel lim M(1, 2,... n) = M L (1, 2... j, l) i P1j 2 0 P 2 h l 1j M R (l, j + 1,... n)

13 12/44 BCFW relations, preliminaries [Britto, Cachazo, Feng, Witten 05] Reconstruct the amplitude from its poles (in complex plane) shift in two external momenta p i p i + zq, p j p j zq keep p i and p j on-shell, i.e. q 2 = q p i = q p j = 0 amplitude becomes a meromorphic function A(z) only simple poles coming from propagators P ab (z) original function is A(0)

14 13/44 BCFW relations: factorization channels Cauchy s theorem 1 2πi dz z A(z) = A(0) + k Res (A, z k ) z k

15 13/44 BCFW relations: factorization channels Cauchy s theorem 0 = 1 2πi dz z A(z) = A(0) + k Res (A, z k ) z k If A(z) vanishes for z A = A(0) = k Res (A, z k ) z k

16 14/44 BCFW relations P 2 ab (z) = 0 if one and only one i (or j) in (a, a + 1,..., b). Suppose i (a,..., b) j Pab 2 (z) = (p a p i 1 + p i + zq + p i p b ) 2 = = Pab 2 + 2q P abz = 0 solution P 2 ab z ab = 2(q P ab ) P 2 ab (z) = P 2 ab z ab (z z ab ) Thus Res(A, z ab ) = s A s L (z ab) i z ab Pab 2 A s R(z ab ) and for allowed helicities it factorizes into two subamplitudes

17 15/44 BCFW relations Using Cauchy s formula, we have finally as a result A = k,s A s k L (z k) i Pk 2 A s k R (z k ) based on two-line shift (convenient choice: adjacent i,j) recursive formula (down to 3-pt amplitudes) number of terms small = number of factorization channels all ingredients are on shell

18 BCFW Example: gluon amplitudes, 6pt NMHV amplitude 1 shifted momenta: p 1 and p _ + 4 _ _ + 4 _ _ _ 5 3 _ + _ 2_ helicities completely set for the first and third diagram 1 Nomenclature note: A(+,..., +) = 0 and for n > 3 also A(, +,..., +) = 0. Thus maximally helicity-violating (MHV) is A(,, +,... +) and more-negative-helicity amplitudes are next-to-mhv (NMHV), NNMHV, etc. 16/44

19 BCFW Example: gluon amplitudes, 6pt NMHV amplitude 1 shifted momenta: p 1 and p _ + 4 _ _ + 4 _ _ _ 5 3 _ + _ 2_ helicities completely set for the first and third diagram second diagram: no helicity combination possible 1 Nomenclature note: A(+,..., +) = 0 and for n > 3 also A(, +,..., +) = 0. Thus maximally helicity-violating (MHV) is A(,, +,... +) and more-negative-helicity amplitudes are next-to-mhv (NMHV), NNMHV, etc. 16/44

20 17/44 BCFW Example: gluon amplitudes # od diagrams for n-body gluon scatterings at tree level n # diagrams (inc.crossing) # diagrams (col.ordered) # BCFW terms

21 18/44 BCFW Example: proof of MHV ansatz The Parke and Taylor formula A n ( ) = BCFW can be used for an inductive proof n1 for details see standard textbook: Elvang,Huang: Scattering Amplitudes

22 19/44 BCFW recursion relations: problems We have assumed that A(z) 0, for z More generally we have to include a boundary term in Cauchy s theorem. This is intuitively clear: we can formally use the derived BCFW recursion relations to obtain any higher n amplitude starting with the leading interaction. But this does not have to be the correct answer.

23 20/44 BCFW recursion relations: problems example: scalar-qed L = 1 4 F µνf µν Dφ λ φ 4 e e Due to the power-counting the boundary term is proportional to B 2e 2 λ In order to eliminate the boundary term we have to set λ = 2e 2, then the original BCFW works. I.e. we needed some further information (e.g. supersymmetry) to determine the λ piece.

24 Effective field theories 21/44

25 21/44 Effective field theories Now we have infinitely many unfixed λ terms. Schematically L = 1 2 ( φ)2 + λ 4 ( φ) 4 + λ 6 ( φ) Example: 6pt scattering, Feynman diagrams Corresponding amplitude: M 6 = I=poles λ 6 part: not fixed by the pole behaviour. λ λ 6 (...) P I Task: to find a condition in order to link these two terms

26 22/44 Non-linear sigma model [KK, Novotny,Trnka 12 and 13]

27 22/44 Leading order Lagrangian assume general simple compact Lie group G we will build a chiral non-linear sigma model, which will correspond to the spontaneous symmetry breaking (G L G R G V G) G L G R G V consequence of the symmetry breaking: Goldstone bosons ( φ) ( i ) U = exp 2 F φ transformation of U: U V R UV 1 L their dynamics given by a Lagrangian (at leading order) where... stands for a trace L = F 2 4 µu µ U 1

28 23/44 Leading order Lagrangian note we are still general, i.e. the group can be generators t i (φ = φ i t i ) G = SU(N), SO(N), Sp(N),... t a t b = δ ab, [t a, t b ] = i 2f abc t c group structure in structure constants, we will define and rewrite D ab φ if abc φ c L = F 2 4 µu µ U 1 = F 2 4 (U 1 µ U)(U 1 µ U)

29 23/44 Leading order Lagrangian note we are still general, i.e. the group can be generators t i (φ = φ i t i ) G = SU(N), SO(N), Sp(N),... t a t b = δ ab, [t a, t b ] = i 2f abc t c group structure in structure constants, we will define D ab φ and rewrite after some algebra ( L = φ T ( 1) n (2n)! n=1 if abc φ c ( 2 F ) 2n 2 D 2n 2 φ ) φ

30 24/44 Stripping down Now we want to build up the interaction vertex We need to learn how to connect structure constants together t a t b = δ ab [t a, t b ] = i 2f abc t c f abc = i 2 [t a, t b ]t c f abc t c = i 2 [t a, t b ] i.e. we can combine f abc s into one trace, schematically and thus the Feynman rule for the interaction vertices can be written as V a 1 a 2...an n (p 1, p 2,..., p n ) = t aσ(1) t aσ(2)... t aσ(n) V n (p 1,..., p n ) σ S n/z n

31 25/44 Stripping and ordering Up to now general group: we didn t need any property of f abc or t i. From now on: we will simplify the problem setting G = SU(N). Simplification due to the completeness relation: Xt a t a Y = XY 1 N X Y N 2 1 a=1 double trace has to cancel out two vertices are connected via a propagator (δ ab ) ordering of t a i in the final single trace is conserved The tree graphs built form the stripped vertices and propagators are decorated with cyclically ordered external momenta.

32 26/44 G = SU(N) U(N) stripped amplitudes and vertices are unique M(p 1,..., p n ) are thus physical we can study different parametrizations [Cronin 67],.., [Bijnens,KK,Lanz 12] already mentioned exponential parametrization there we can simply enlarge G to U(N) group ( ) i 2 U = exp F N φ0 Û, Û SU(N) φ 0 however decouples: L (2) = 1 2 φ0 φ 0 + F 2 4 µû µ Û 1 results from less restricted U(N) equal to SU(N)

33 G = U(N) different parametrizations 2 General form of the parametrization U(φ) f(x) f(x) = a k x k, f( x)f(x) = 1 k=0 exponential : f exp = e x minimal : f min = x x 2 Cayley f Caley = 1+x/2 1 x/2 2 For details see Appendix A in KK,Novotny,Trnka 13. For original literature see Gursey60,Cronin 67, Ellis,Renner 70 27/44

34 G = U(N) different parametrizations 2 General form of the parametrization U(φ) f(x) f(x) = a k x k, f( x)f(x) = 1 k=0 exponential : f exp = e x minimal : f min = x x 2 w 2k+1,n = Cayley f Caley = 1+x/2 1 x/2 ( w k,n = ( 1)k 1 2n+2 ) 1+δ kn (2n+2)! k+1 w k,n = ( 1)k 1+δ kn 1 2 2n The stripped Feynman rules can be written ( ) 2 n n 2n+2 V 2n+2 (s i,j ) = ( 1) n F 2 w k,n where s i,j P (i, j) 2. k=0 ( 1 ( 1)n k )( 2 n k δ 2k+1,n k+1 n k i=1 s i,i+k 2 For details see Appendix A in KK,Novotny,Trnka 13. For original literature see Gursey60,Cronin 67, Ellis,Renner 70 27/44 )

35 28/44 Explicit example: stripped 4pt amplitude Natural parametrization for diagrammatic calculations: minimal w min 2k,n = 0 Thus off-shell and on-shell stripped vertices are equal. 4pt amplitude 2F 2 M(1, 2, 3, 4) = (s 1,2 + s 2,3 )

36 29/44 Explicit example: stripped 6pt amplitude 4F 4 M(1, 2, 3, 4, 5, 6) = = (s 1,2 + s 2,3 )(s 1,4 + s 4,5 ) s 1,3 + (s 1,4 + s 2,5 )(s 2,3 + s 3,4 ) s 2,4 + (s 1,2 + s 2,5 )(s 3,4 + s 4,5 ) s 3,5 (s 1,2 + s 1,4 + s 2,3 + s 2,5 + s 3,4 + s 4,5 ) This can be rewritten as 4F 4 M(1, 2, 3, 4, 5, 6) = 1 (s 1,2 + s 2,3 )(s 1,4 + s 4,5 ) s 1,2 + cycl, 2 s 1,3

37 30/44 Explicit example: stripped 8pt amplitude 8F 6 M(1, 2, 3, 4, 5, 6, 7) = = 1 (s 1,2 + s 2,3 )(s 1,4 + s 4,7 )(s 5,6 + s 6,7 ) (s 1,2 + s 2,3 )(s 1,4 + s 4,5 )(s 6,7 + s 7,8 ) 2 s 1,3 s 5,7 s 1,3 s 6,8 + (s 1,2 + s 2,3 )(s 4,5 + s 4,7 + s 5,6 + s 5,8 + s 6,7 + s 7,8 ) 2s 1,2 1 s 1,3 2 s 1,4 + cycl

38 Explicit example: stripped 10pt amplitude 31/44

39 32/44 Generalization of reconstruction formula: subtractions [Benincasa, Conde 11] [Feng et al. 11] [KK,Novotny,Trnka 12] In introduction: A(z) 0 for z Suppose some deformation of the external momenta p k p k (z) so that A(z) z k for z we have to generalize and use the (k + 1)-times subtracted Cauchy formula A(z) = n i=1 Res (A; z i ) z z i k+1 j=1 z a j k+1 + z i a j j=1 A(a j ) k+1 l=1,l j z a l a j a l

40 32/44 Generalization of reconstruction formula: subtractions [Benincasa, Conde 11] [Feng et al. 11] [KK,Novotny,Trnka 12] In introduction: A(z) 0 for z Suppose some deformation of the external momenta p k p k (z) so that A(z) z k for z we have to generalize and use the (k + 1)-times subtracted Cauchy formula A(z) = n i=1 Res (A; z i ) z z i k+1 j=1 z a j z i a j if for a i we have A(a i ) = 0

41 33/44 BCFW-like reconstruction of non-linear sigma model for the first time we have used the amplitude methods for EFT number of diagrams much lower can be used for a proof of double Adler zero [conjectured in N. Arkani-Hamed, F. Cachazo and J. Kaplan,: What is the Simplest Quantum Field Theory? 08] drawback: tailor-made for one special model, in special limit (two derivatives, massless case, tree-level) done using the off-shell object : this is against the amplitude idea Lesson learnt: low energy limit, the Adler zero, plays a role of gauge symmetry

42 Scalar effective field theories 34/44

43 Natural classification Soft limit of one external leg of the tree-level amplitude Interaction term A(tp 1, p 2,..., p n ) = O(t σ ), as tp 1 0 Then another natural parameter is e.g. L = m φ n ρ = m 2 n 2 L = m φ 4 + m φ 6 so these two diagrams can mix: p 2m 2 p m ρ = ρ, i.e. rho is fixed 34/44

44 35/44 We want to focus on a non-trivial case for: L = m φ n : m < σn or σ > (n 2)ρ + 2 n

45 36/44 First case: ρ = 0 (i.e. two derivatives) Schematically for single scalar case L = 1 2 ( φ)2 + i λ i 4( 2 φ 4 ) + i λ i 6( 2 φ 6 ) +... similarly for multi-flavour (φ i : φ 1, φ 2,...). non-trivial case σ = 1 Outcome: single scalar: free theory multiple scalars (with flavour-ordering): non-linear sigma model n.b. it represents a generalization of [Susskind, Frye 70], [Ellis, Renner 70]

46 37/44 Second case: ρ = 1, σ = 2 (double soft limit) 1. focus on the lowest combination and fix the form: L int = c 2 ( φ φ) 2 + c 3 ( φ φ) 3 condition: c 3 = 4c 4 2

47 37/44 Second case: ρ = 1, σ = 2 (double soft limit) 1. focus on the lowest combination and fix the form: L int = c 2 ( φ φ) 2 + c 3 ( φ φ) 3 condition: c 3 = 4c find the symmetry φ φ b ρ x ρ + b ρ ρ φ φ (again up to 6pt so far)

48 Second case: ρ = 1, σ = 2 (double soft limit) 1. focus on the lowest combination and fix the form: L int = c 2 ( φ φ) 2 + c 3 ( φ φ) 3 condition: c 3 = 4c find the symmetry φ φ b ρ x ρ + b ρ ρ φ φ (again up to 6pt so far) 3. ansatz of the form c n ( φ φ) n + c n+1 ( φ φ) n φ φ 4. in order to cancel: 2(n + 1)c n+1 = (2n 1)c n i.e. c 1 = 1 2 c 2 = 1 8, c 3 = 1 16, c 4 = 5 128,... 37/44

49 38/44 Second case: ρ = 1, σ = 2 (double soft limit) 4. in order to cancel: 2(n + 1)c n+1 = (2n 1)c n i.e. c 1 = 1 2 c 2 = 1 8, c 3 = 1 16, c 4 = 5 128,... solution: L = 1 ( φ φ) This theory known as a scalar part of the Dirac-Born-Infeld [1934] DBI action Scalar field can be seen as a fluctuation of a 4-dim brane in five-dim Minkowski space Á Remark: soft limit and symmetry are equivalent

50 39/44 Third case: ρ = 2, σ = 2 (double soft limit) Similarly to previous case we will arrive to a unique solution: the Galileon Lagrangian d+1 L = d n φl der n 1 L der n = ε µ 1...µ d ε ν 1...ν d n µi νi φ i=1 n=1 d j=n+1 It possesses the Galilean shift symmetry φ φ + a + b µ x µ (leads to EoM of second-order in field derivatives) η µj ν j = (d n)! det { ν i νj φ } n i,j=1.

51 40/44 Surprise: ρ = 2, σ = 3 (enhanced soft limit) general galileon: three parameters (in 4D) only two relevant (due to dualities [KK, Novotny 14])

52 40/44 Surprise: ρ = 2, σ = 3 (enhanced soft limit) general galileon: three parameters (in 4D) only two relevant (due to dualities [KK, Novotny 14]) we demanded O(p 3 ) behaviour we have verified: possible up to very high-pt order suggested new theory: special galileon [Cheung,KK,Novotny,Trnka ]

53 40/44 Surprise: ρ = 2, σ = 3 (enhanced soft limit) general galileon: three parameters (in 4D) only two relevant (due to dualities [KK, Novotny 14]) we demanded O(p 3 ) behaviour we have verified: possible up to very high-pt order suggested new theory: special galileon [Cheung,KK,Novotny,Trnka ] symmetry explanation: hidden symmetry [K. Hinterbichler and A. Joyce ] φ φ + s µν x µ x ν 12λ 4 s µν µ φ ν φ

54 41/44 New recursion for effective theories [Cheung, KK, Novotny, Shen, Trnka 2015]

55 41/44 The high energy behaviour forbids a naive Cauchy formula A(z) 0 for z Can we instead use the soft limit directly?

56 41/44 The high energy behaviour forbids a naive Cauchy formula A(z) 0 for z Can we instead use the soft limit directly? yes! The standard BCFW not applicable, we propose a special shift: p i p i (1 za i ) on all external legs This leads to a modified Cauchy formula dz A(z) z Π i (1 a i z) σ = 0 note there are no poles at z = 1/a i (by construction). Now we can continue in analogy with BCFW

57 42/44 A Periodic Table of Effective Field Theories [Cheung, KK, Novotny, Shen, Trnka 2016]

58 Classification of EFTs We use the set of four parameters: (ρ, σ, v, d) 3 2 trivial soft behavior Gal sgal 1 P(X) WZW DBI forbidden 0 NLSM /44

59 43/44 Photon scatterings [work in progress]

60 43/44 Very well known: effective field theory of light-by-light scattering: Euler-Heisenberg Then we can study 6-pt scattering directly as: And so on for higher orders. Systematic study is in under way...

61 44/44 Summary we have studied many trees [ Ny hazo tokana tsy mba ala ] we have provided a new look at effective field theories, i.e. non-renormalizable theories one can employ very similar tools as for renormalizable theories focused first on properties of scalar effective theories first very preliminary results for spin-1 technical benefit: can calculate differently scattering amplitude (BCFW-like relations) other benefits: we can see or prove new properties (e.g. double Adler zero) classification of relevant theories why it can be very useful: e.g. these scalar theories relevant for Beyond Standard Gravity

62 44/44 Summary we have studied many trees [ Ny hazo tokana tsy mba ala ] we have provided a new look at effective field theories, i.e. non-renormalizable theories one can employ very similar tools as for renormalizable theories focused first on properties of scalar effective theories first very preliminary results for spin-1 technical benefit: can calculate differently scattering amplitude (BCFW-like relations) other benefits: we can see or prove new properties (e.g. double Adler zero) classification of relevant theories why it can be very useful: e.g. these scalar theories relevant for Beyond Standard Gravity see next

63 Motivation II Our universe is composed of 5% normal matter 25% dark matter 70% dark energy why dark energy? we know that space is expanding and natural question: how this is changing, or naively how this is decreasing big surprise in 1998: space is accelerating (!) nobel priset /44

64 45/44 Motivation II General relativity: R µν 1 2 Rg µν = 1 m P T µν

65 45/44 Motivation II General relativity: after 1998 R µν 1 2 Rg µν 1 m P T µν

66 45/44 Motivation II General relativity: after 1998 Two possible solutions 1 dark energy R µν 1 2 Rg µν 1 m P T µν 2 modification of GR R µν 1 2 Rg µν = 1 m P (T µν m 2 P Λg µν ) R µν 1 2 Rg µν + Λg µν = 1 m P T µν

67 45/44 Motivation II possible modification of GR massive graviton V (r) M m 2 P 1 r e mr massless graviton: two degrees of freedom massive graviton: five degrees, one can see them as 1 graviton (massless), 1 photon and 1 scalar We have to understand all possible theories of spin-1 and spin-0