Towards One-Loop MHV Techniques

Transcription

1 Towards One-Loop MHV Techniques Carola F. Berger Snowmass - August 22, 2005 Carola F. Berger, SLAC Snowmass - August 22, 2005

2 Towards One-Loop MHV Techniques Carola F. Berger Snowmass - August 22, 2005 Carola F. Berger, SLAC Snowmass - August 22, 2005

3 Tree Level Techniques Parke-Taylor (1986) Twistor space Witten (2003) ij n1 δ ( i k i) ij =ū (k i )u + (k j ) d λe iµ λ u + (k i )=(λ i ) α ; u (k i )=( λ i ) α In general: curve in twistor space of genus g l and of degree d = q 1 + l Carola F. Berger, SLAC Snowmass - August 22,

4 Tree Level Techniques Parke-Taylor (1986) Twistor space Witten (2003) ij n1 δ ( i k i) ij =ū (k i )u + (k j ) d λe iµ λ u + (k i )=(λ i ) α ; u (k i )=( λ i ) α In general: curve in twistor space of genus g l and of degree d = q 1 + l MHV rules Cachazo, Svrcek, Witten (2004) Off-shell Parke-Taylor amplitudes connected by scalar propagator More efficient than Feynman rules for QCD (and SUSY) tree amplitudes. Lagrangian connection? Carola F. Berger, SLAC Snowmass - August 22,

5 Tree Level Techniques contd. On-shell recursion relations Britto, Cachazo, Feng, and Witten ( ) A n = r,s 1 A λ L P 2 λ=± r,s A λ R (1) Proof: Shift gluons i, j p i (z) = λ i ( λ i z λ j ) p j (z) = (λ j + zλ i ) λ j (2) amplitude A n (z) = A n (p 1,..., p i (z),..., p j (z),...) (3) Carola F. Berger, SLAC Snowmass - August 22,

6 Tree Level Techniques contd. Amplitude is on-shell for all z, momentum is conserved. Internal momentum 1/P 2 kl 1/P 2 kl (z) A n (z) has only simple poles. Physical scattering amplitude A(0) 0 = 1 2πi C dz z A(z) If A(z ) 0 A(0) = poles α Res z=zα A(z) z = r,s λ=± A λ L(z = z α ) 1 Pr,s 2 A λ R (z = z α) (4) Carola F. Berger, SLAC Snowmass - August 22,

7 One Loop - Supersymmetric Amplitudes Supersymmetric amplitudes - cut constructible dlips( l 1, l 2 )A tree ( l 1, m 1,..., m 2, l 2 )A tree ( l 2, m 2 + 1,..., m 1 1, l 1 ) (5) legs in D = 4 dimensions (integral in D = 4 2ε dimensions) also, generalized cuts reduction to box, triangle, bubble integrals with coefficients obtainable from tree amplitudes Carola F. Berger, SLAC Snowmass - August 22,

8 One Loop - QCD Amplitudes SUSY decomposition reduction to scalar loop in SUSY amplitudes rational terms intimately linked to (poly)logarithmic contributions Carola F. Berger, SLAC Snowmass - August 22,

9 One Loop - QCD Amplitudes SUSY decomposition reduction to scalar loop in SUSY amplitudes rational terms intimately linked to (poly)logarithmic contributions N = 0-2 approaches Generalized cuts (poly)logarithmic terms; rational terms require evaluation to higher order in ε (Bedford, Brandhuber, Spence, Travaglini ( ) ) Carola F. Berger, SLAC Snowmass - August 22,

10 One Loop - QCD Amplitudes SUSY decomposition reduction to scalar loop in SUSY amplitudes rational terms intimately linked to (poly)logarithmic contributions N = 0-2 approaches Generalized cuts (poly)logarithmic terms; rational terms require evaluation to higher order in ε (Bedford, Brandhuber, Spence, Travaglini ( ) ) Recursion relations (Bern, Dixon, Kosower ( ) + in progress, Forde, CFB) Carola F. Berger, SLAC Snowmass - August 22,

11 New features at one loop branch cuts One Loop - QCD Amplitudes contd. Carola F. Berger, SLAC Snowmass - August 22,

12 New features at one loop branch cuts One Loop - QCD Amplitudes contd. double poles: multiparticle factorization simple poles collinear factorization one-loop splitting amplitudes for (+ + +) and ( ) helicity configurations [ab] ab 2 Carola F. Berger, SLAC Snowmass - August 22,

13 New features at one loop branch cuts One Loop - QCD Amplitudes contd. double poles: multiparticle factorization simple poles collinear factorization one-loop splitting amplitudes for (+ + +) and ( ) helicity configurations [ab] ab 2 boundary terms A(z ) 0 Carola F. Berger, SLAC Snowmass - August 22,

14 New features at one loop branch cuts One Loop - QCD Amplitudes contd. double poles: multiparticle factorization simple poles collinear factorization one-loop splitting amplitudes for (+ + +) and ( ) helicity configurations [ab] ab 2 boundary terms A(z ) 0 spurious (unphysical) singularities as s 1 s 2 ( ) ln s1 s 2 (s 1 s 2 ) 2 Carola F. Berger, SLAC Snowmass - August 22,

15 One Loop - QCD Amplitudes contd. Solutions merge unitarity with recursion A n (0) = dz z Disc C n(z) + poles Res C n(z) z + poles Res C n(z) z (6) where C n from cuts, R n rational pieces - can be constructed via recursion relations. Carola F. Berger, SLAC Snowmass - August 22,

16 One Loop - QCD Amplitudes contd. Solutions merge unitarity with recursion A n (0) = dz z Disc C n(z) + poles Res C n(z) z + poles Res C n(z) z (6) where C n from cuts, R n rational pieces - can be constructed via recursion relations. double shifts to avoid boundary terms (j < l < n) λ j λ j z λ l z nj lj λ n λ l λ l + zλ j λ n λ n + z nj lj λ j (7) Carola F. Berger, SLAC Snowmass - August 22,

17 One Loop - QCD Amplitudes contd. double poles - from soft factors A L (ˆk,..., P ij, l)s(ˆk, P ij, l) 1 Pij 2 S(m, P ij, ˆn)A R (ˆn, P ij,..., r) (8) S(a, s +, b) = ab as sb independent of helicities of a, b. S(a, s, b) = [ab] [as][sb] (9) Carola F. Berger, SLAC Snowmass - August 22,

18 One Loop - QCD Amplitudes contd. double poles - from soft factors A L (ˆk,..., P ij, l)s(ˆk, P ij, l) 1 Pij 2 S(m, P ij, ˆn)A R (ˆn, P ij,..., r) (8) S(a, s +, b) = ab as sb independent of helicities of a, b. S(a, s, b) = [ab] [as][sb] (9) spurious singularities L 1 (r) = L 0(r)+1 1 r, L 0 (r) = ln r 1 r. Introduction of extra rational terms! avoid double counting. ( ) ( ) ln s1 s 2 L s1 1 (s 1 s 2 ) 2 s 2 s 2 2 (10) need to subtract overlap terms to Carola F. Berger, SLAC Snowmass - August 22,

19 References Conventional techniques: L. J. Dixon, TASI 1995, hep-ph/ Z. Bern, L. J. Dixon and D. A. Kosower, Ann. Rev. Nucl. Part. Sci. 46, 109 (1996) [hep-ph/ ]. Twistor space: E. Witten, Commun. Math. Phys. 252, 189 (2004) [hep-th/ ]. CSW: F. Cachazo, P. Svrcek and E. Witten, JHEP 0409, 006 (2004) [hep-th/ ]. On-shell recursion (BCF): R. Britto, F. Cachazo and B. Feng, Nucl. Phys. B 715, 499 (2005) [hepth/ ]. R. Britto, F. Cachazo, B. Feng and E. Witten, Phys. Rev. Lett. 94, (2005) [hep-th/ ]. Carola F. Berger, SLAC Snowmass - August 22,

20 References One-loop unitarity approach: A. Brandhuber, B. Spence and G. Travaglini, Nucl. Phys. B 706, 150 (2005) [hep-th/ ]. J. Bedford, A. Brandhuber, B. Spence and G. Travaglini, Nucl. Phys. B 706, 100 (2005) [hep-th/ ]; Nucl. Phys. B 712, 59 (2005) [hep-th/ ]; and hep-th/ Recursive approach at one loop: Z. Bern, L. J. Dixon and D. A. Kosower, Phys. Rev. D 71, (2005) [hep-th/ ]; hep-ph/ ; hep-ph/ Carola F. Berger, SLAC Snowmass - August 22,