A new look at the nonlinear sigma model 1

Transcription

1 1 Project in collaboration with Jiří Novotný (Prague) and Jaroslav Trnka (Caltech). Main results in [ ] and [ ] K. Kampf Non-linear sigma model 1/20 A new look at the nonlinear sigma model 1 Karol Kampf Charles University, Prague Outline: Overview of low energy QCD Gluon amplitudes non-linear sigma model QCD Montpellier, 1/7/2014

2 Low energy QCD K. Kampf Non-linear sigma model 2/20

3 K. Kampf Non-linear sigma model 2/20 Overview of low energy QCD history: current algebra modern form: ChPT [Weinberg 79], [Gasser,Leutwyler 84-85] present status: ChPT for 2 and 3-flavours up to NNLO for the even sector, and up to NLO for the odd sector (chiral anomaly) extended for intermediate energy region: resonance chiral theory (RChT) many applications: see e.g. some talks at this conference: Soriano, Piandani, Biino, Kolesár, Gonzales-Solis, Roig Garcés, Pham, Knecht,...

4 example: KK,Moussallam: π 0 γγ at two-loop order K. Kampf Non-linear sigma model 3/20

5 K. Kampf Non-linear sigma model 3/20 example: KK,Moussallam: π 0 γγ at two-loop order and this is not even the whole story, one needs also the odd-sector... Now we will keep only: L = F2 4 u µu µ

6 Gluon amplitudes K. Kampf Non-linear sigma model 4/20

7 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 n # diagrams (inc.crossing) # diagrams (col.ordered) calculation tedious, however, some results known [Parke, Taylor 85-86], [Berends, Giele 88] to be extremely simple Is there some better way to calculate? K. Kampf Non-linear sigma model 4/20

8 K. Kampf Non-linear sigma model 5/20 Gluon amplitudes colour ordering stripped amplitude thanks to ordering the only possible poles are: P 2 ij = (p i +p i p j 1 +p j ) 2 Pole structure Weinberg s theorem (one-particle unitarity): on the factorization channel lim M(1,2,...n) = M L (1,2...j,l) P1j 2 0 h l i P 2 1j M R (l,j +1,...n)

9 K. Kampf Non-linear sigma model 6/20 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)

10 K. Kampf Non-linear sigma model 7/20 BCFW relations: factorization channels Cauchy s theorem 1 2πi dz z A(z) = A(0)+ k Res(A,z k ) z k

11 K. Kampf Non-linear sigma model 7/20 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

12 K. Kampf Non-linear sigma model 8/20 BCFW relations P 2 ab (z) = 0 Suppose i (a,...,b) j solution if one and only one i (or j) in (a,a+1,...,b). P 2 ab (z) = (p a +...+p i 1 +p i +zq +p i p b ) 2 = 0 z ab = P2 ab 2(q P ab ) and for allowed helicities it factorizes into two subamplitudes Res(A,z ab ) = s A s L (z ab) i 2(q P ab ) As R(z ab ) 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 )

13 Example: gluon amplitudes # od diagrams for n-body gluon scatterings at tree level n # diagrams (inc.crossing) # diagrams (col.ordered) # BCFW terms Conclusion: it works and it is really fast. Different directions were taken so far, e.g. implementation in multi-jet QCD computations extended to loop level [Bern, Dixon, Kosower 05] BCFW for planar loop integrands N = 4 SYM [Arkani-Hamed, Bourjaily, Cachazo, Caron-Huot, Trnka 10] Amplitudes as volumes of polytopes [Arkani-Hamed, Bourjaily, Cachazo, Hodges, Trnka 10]; Amplituhedron [Arkani-Hamed,Trnka 13] crucial are on-shell objects [Arkani-Hamed et al. 12] perturbative quantum gravity [Witten 03] [Cachazo, Skinner, Mason 12] K. Kampf Non-linear sigma model 9/20

14 Non-linear sigma model K. Kampf Non-linear sigma model 10/20

15 K. Kampf Non-linear sigma model 10/20 Leading order Lagrangian assume general simple compact Lie group G G = SU(N), SO(N), Sp(N),... 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, φ = φ i t i transformation of U: U V R UV 1 L their dynamics given by a Lagrangian (at leading order) L = F2 4 µu µ U 1 where... stands for a trace

16 K. Kampf Non-linear sigma model 11/20 Stripping down group structure in structure constants, we will define n=1 D ab φ ifabc φ c and after some algebra the Lagrangian becomes ( L = φ T ( 1) n ( ) ) 2 2n 2 D 2n 2 (2n)! F φ φ we can combine f abc s into one trace, schematically Tree level on-shell amplitude has a simple group structure M a 1...a n (p 1,...,p n ) = t a σ(1) t a σ(2)...t a σ(n) M(p σ(1),...,p σ(n) ) σ S n/z n

17 K. Kampf Non-linear sigma model 12/20 Properties of stripped amplitudes stripped amplitudes are unique M(p 1,...,p n ) are thus physical we can study different parametrizations [Cronin 67], [Ellis,Renner 70], [Bijnens,KK,Lanz 13], [KK,Novotny,Trnka 13] most familiar: exponential parametrization there we can simply enlarge G to U(N) group φ 0 however decouples results from less restricted U(N) equal to SU(N) 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+ 1+x 2 Cayley : f Caley = 1+x/2 1 x/2

18 K. Kampf Non-linear sigma model 13/20 Explicit example: stripped 4pt amplitude Natural parametrization for diagrammatic calculations: minimal (where off-shell and on-shell stripped vertices are equal) s i,j = P 2 ij 4pt amplitude 2F 2 M(1,2,3,4) = (s 1,2 +s 2,3 )

19 K. Kampf Non-linear sigma model 13/20 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

20 K. Kampf Non-linear sigma model 13/20 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

21 Explicit example: stripped 10pt amplitude K. Kampf Non-linear sigma model 13/20

22 Recursion relations K. Kampf Non-linear sigma model 14/20

23 K. Kampf Non-linear sigma model 14/20 Semi-on-shell amplitudes Definition J a,a 1,...,a n n (p 1,...,p n ) = 0 φ a (0) π a 1 (p 1 )...π an (p n ) off-shellness in p 2 n+1 0 p n+1 = n j=1 In analogy with vertex and on-shell amplitudes we can define flavour-stripped J n (p 1,...,p n ). Then M(p 1,p 2,...,p n+1 ) = lim p 2 n+1 0 p 2 n+1j n (p 1,p 2,...,p n ) p j Normalization J 1 (p) = 1

24 Berends-Giele recursive relations used for gluons [Berends,Giele 88] 1 1 j 1 j = Σ m,{jk} j 2 n j m 1 +1 j m = n J(1,2,...,n) = i n p 2 n+1 m=2 n m+1 j 1 =1 n m+2 j 2 =j 1 +1 n m+(m 1) j m 1 =j m 2 +1 iv m+1 (p(1,j 1 ),p(j 1 +1,j 2 ),...p(j m 1 +1,n), p(1,n)) J(1,...,j 1 )J(j 1 +1,...,j 2 ) J(j m 1 +1,...,n). K. Kampf Non-linear sigma model 15/20

25 K. Kampf Non-linear sigma model 16/20 # of diagrams: geometry suggested by [Susskind, Frye 70] Number of ways to divide polygon (n-gon) with non-crossing diagonals so that the resulting objects have number of edges equal to number of legs of allowed vertices.

26 K. Kampf Non-linear sigma model 16/20 # of diagrams: geometry suggested by [Susskind, Frye 70] Number of ways to divide polygon (n-gon) with non-crossing diagonals so that the resulting objects have number of edges equal to number of legs of allowed vertices. Example: QCD with only gluon vertices, 5-gluon fusion (n.b. # is 10)

27 K. Kampf Non-linear sigma model 16/20 # of diagrams: geometry suggested by [Susskind, Frye 70] Number of ways to divide polygon (n-gon) with non-crossing diagonals so that the resulting objects have number of edges equal to number of legs of allowed vertices. Example: QCD with only gluon vertices, 5-gluon fusion (n.b. # is 10)

28 K. Kampf Non-linear sigma model 16/20 # of diagrams: geometry suggested by [Susskind, Frye 70] Number of ways to divide polygon (n-gon) with non-crossing diagonals so that the resulting objects have number of edges equal to number of legs of allowed vertices. Another example: 8-pion fusion (n.b. # is 21) x 4 x 8 x 8 x 1

29 BCFW-like reconstruction K. Kampf Non-linear sigma model 17/20

30 K. Kampf Non-linear sigma model 17/20 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 can assume the (k + 1)-times subtracted Cauchy formula which leads to the desired generalization 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

31 K. Kampf Non-linear sigma model 17/20 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 can assume the (k + 1)-times subtracted Cauchy formula which leads to the desired generalization 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

32 K. Kampf Non-linear sigma model 18/20 Semi-on-shell amplitude M due to the derivative couplings the standard BCFW shift leads to A(z) z for z we need two subtracted formula and two values A(z 1 ) and A(z 2 ) This is difficult to obtain. Way out: different continuation + give up on-shellness We had A(p 1,p 2,...,p n+1 ) = lim p 2 n+1 0 p 2 n+1j n (1,...,n)

33 Semi-on-shell amplitude M due to the derivative couplings the standard BCFW shift leads to A(z) z for z we need two subtracted formula and two values A(z 1 ) and A(z 2 ) This is difficult to obtain. Way out: different continuation + give up on-shellness Now we define M n (p 1,p 2,...,p n+1 ) = p 2 n+1j n (1,...,n) and the complex deformation as (all even-line shift) p i p i (z) : p 2i+1 (z) = p 2i+1 and p 2i (z) = zp 2i i.e. M n (z) M 2n+1 (p 1,zp 2,p 3,...zp 2n,p 2n+1 ) For its properties we can use directly the properties of J. K. Kampf Non-linear sigma model 18/20

34 K. Kampf Non-linear sigma model 19/20 Semi-on-shell amplitude M: properties M(z) is meromorphic function of z with single poles. physical amplitude for z = 1 and on-shell amplitude using scaling properties for J A(p 1,...,p 2n+2 ) = lim p 2 2n+2 0 M n (1) 1) lim M n (z) = (p 1 +p p 2n+1 ) 2 z 0 (2F 2 ) n p2 (2F 2 ) n 2) M n (z) = O(z 0 ) as z once subtracted reconstruction formula M n (z) = p2 (2F 2 ) n + P Res(M n,z P ) z z P z z P

35 Conclusion K. Kampf Non-linear sigma model 20/20

36 K. Kampf Non-linear sigma model 20/20 Conclusion Main message: BCFW reconstruction can be used for a broader class than was expected before. We have demonstrated its applicability for the SU(N) non-linear sigma model, which is a non-renormalizable model with infinite number of interaction vertices. We gain: faster way how to calculate amplitudes (so far implemented for tree-level diagrams) and new view on these quantities. This can be used to obtain better understanding of its properties (used for proofs of so-called Adler zero(s) as conjectured in N. Arkani-Hamed, F. Cachazo and J. Kaplan: 08). Thank you for your attention.