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Pure Spinor Helicity Methods Pure Spinor Helicity Methods Rutger Boels Niels Bohr International Academy, Copenhagen R.B., arXiv:0908.0738 [hep-th] Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 1 / 25 in books? Why you should pay attention, an experiment The greatest common denominator of the audience? Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 2 / 25 Why you should pay attention, an experiment The greatest common denominator of the audience in books? Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 2 / 25 Why you should pay attention, an experiment The greatest common denominator of the audience in books: Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 2 / 25 Why you should pay attention, an experiment The greatest common denominator of the audience in science books? Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 2 / 25 for all legs simultaneously pure spinor helicity methods: precise control over Poincaré and Susy quantum numbers Why you should pay attention subject: calculation of scattering amplitudes in D > 4 with many legs Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 3 / 25 for all legs simultaneously Why you should pay attention subject: calculation of scattering amplitudes in D > 4 with many legs pure spinor helicity methods: precise control over Poincaré and Susy quantum numbers Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 3 / 25 Why you should pay attention subject: calculation of scattering amplitudes in D > 4 with many legs pure spinor helicity methods: precise control over Poincaré and Susy quantum numbers for all legs simultaneously Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 3 / 25 Outline 1 Motivation 2 Covariant representation theory of Poincaré algebra Spin algebra Susy algebra 3 Outlook Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 4 / 25 <=== Zürich Every talk on amplitudes should mention ::: Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 5 / 25 <=== Zürich Every talk on amplitudes should mention ::: Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 5 / 25 one-loop: pure Yang-Mills [Giele, Kunszt, and Melnikov, 08] I uses 6D trees high loops: N = 4, N = 8 [Bern, Dixon, Kosower et. al.], [others] I uses 10D, 11D trees recent quantum leaps in four: what is special about four? string theory (analytic S-matrix type techniques) little is known is there a D > 4 analogue of: MHV amplitudes? (any recent buzzword in D = 4?) pure spinor spaces are higher dimensional twistor spaces Why study amplitudes in higher dimensions? loops in four dimensions dimensional regularization Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 6 / 25 recent quantum leaps in four: what is special about four? string theory (analytic S-matrix type techniques) little is known is there a D > 4 analogue of: MHV amplitudes? (any recent buzzword in D = 4?) pure spinor spaces are higher dimensional twistor spaces Why study amplitudes in higher dimensions? loops in four dimensions dimensional regularization one-loop: pure Yang-Mills [Giele, Kunszt, and Melnikov, 08] I uses 6D trees high loops: N = 4, N = 8 [Bern, Dixon, Kosower et. al.], [others] I uses 10D, 11D trees Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 6 / 25 little is known is there a D > 4 analogue of: MHV amplitudes? (any recent buzzword in D = 4?) pure spinor spaces are higher dimensional twistor spaces Why study amplitudes in higher dimensions? loops in four dimensions dimensional regularization one-loop: pure Yang-Mills [Giele, Kunszt, and Melnikov, 08] I uses 6D trees high loops: N = 4, N = 8 [Bern, Dixon, Kosower et. al.], [others] I uses 10D, 11D trees recent quantum leaps in four: what is special about four? string theory (analytic S-matrix type techniques) Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 6 / 25 is there a D > 4 analogue of: MHV amplitudes? (any recent buzzword in D = 4?) pure spinor spaces are higher dimensional twistor spaces Why study amplitudes in higher dimensions? loops in four dimensions dimensional regularization one-loop: pure Yang-Mills [Giele, Kunszt, and Melnikov, 08] I uses 6D trees high loops: N = 4, N = 8 [Bern, Dixon, Kosower et. al.], [others] I uses 10D, 11D trees recent quantum leaps in four: what is special about four? string theory (analytic S-matrix type techniques) little is known Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 6 / 25 pure spinor spaces are higher dimensional twistor spaces Why study amplitudes in higher dimensions? loops in four dimensions dimensional regularization one-loop: pure Yang-Mills [Giele, Kunszt, and Melnikov, 08] I uses 6D trees high loops: N = 4, N = 8 [Bern, Dixon, Kosower et. al.], [others] I uses 10D, 11D trees recent quantum leaps in four: what is special about four? string theory (analytic S-matrix type techniques) little is known is there a D > 4 analogue of: MHV amplitudes? (any recent buzzword in D = 4?) Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 6 / 25 Why study amplitudes in higher dimensions? loops in four dimensions dimensional regularization one-loop: pure Yang-Mills [Giele, Kunszt, and Melnikov, 08] I uses 6D trees high loops: N = 4, N = 8 [Bern, Dixon, Kosower et. al.], [others] I uses 10D, 11D trees recent quantum leaps in four: what is special about four? string theory (analytic S-matrix type techniques) little is known is there a D > 4 analogue of: MHV amplitudes? (any recent buzzword in D = 4?) pure spinor spaces are higher dimensional twistor spaces Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 6 / 25 Outline 1 Motivation 2 Covariant representation theory of Poincaré algebra Spin algebra Susy algebra 3 Outlook Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 7 / 25 maximal set of commuting operators: kµ, Wµ µ 2 kµk = m , pick spin axis through light-like vector q qµW qµk νΣρσ R = R1 = µ = µνρσ z q 2q · k 2q · k 1;3 9 vectors n1 and n2 such that q; q^; n1; n2 span R , q · ni = 0 massive polarization vectors µ µ µ 1 µ µ µ ((n1 ± in2) · k) µ k mq e± = p n ± in − q e = − 2 1 2 q · k 0 m q · k broken gauge theory in D = 4 ! [R.B., Christian Schwinn, to appear] Warmup: massive vectors in four dimensions higher D massless vectors decompose into massive D = 4 massive spinor helicity: e.g. [Kleiss, Stirling, 85], [Dittmaier, 98] Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 8 / 25 µ 2 kµk = m , pick spin axis through light-like vector q qµW qµk νΣρσ R = R1 = µ = µνρσ z q 2q · k 2q · k 1;3 9 vectors n1 and n2 such that q; q^; n1; n2 span R , q · ni = 0 massive polarization vectors µ µ µ 1 µ µ µ ((n1 ± in2) · k) µ k mq e± = p n ± in − q e = − 2 1 2 q · k 0 m q · k broken gauge theory in D = 4 ! [R.B., Christian Schwinn, to appear] Warmup: massive vectors in four dimensions higher D massless vectors decompose into massive D = 4 massive spinor helicity: e.g. [Kleiss, Stirling, 85], [Dittmaier, 98] maximal set of commuting operators: kµ, Wµ Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 8 / 25 1;3 9 vectors n1 and n2 such that q; q^; n1; n2 span R , q · ni = 0 massive polarization vectors µ µ µ 1 µ µ µ ((n1 ± in2) · k) µ k mq e± = p n ± in − q e = − 2 1 2 q · k 0 m q · k broken gauge theory in D = 4 ! [R.B., Christian Schwinn, to appear] Warmup: massive vectors in four dimensions higher D massless vectors decompose into massive D = 4 massive spinor helicity: e.g. [Kleiss, Stirling, 85], [Dittmaier, 98] maximal set of commuting operators: kµ, Wµ µ 2 kµk = m , pick spin axis through light-like vector q qµW qµk νΣρσ R = R1 = µ = µνρσ z q 2q · k 2q · k Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 8 / 25 massive polarization vectors µ µ µ 1 µ µ µ ((n1 ± in2) · k) µ k mq e± = p n ± in − q e = − 2 1 2 q · k 0 m q · k broken gauge theory in D = 4 ! [R.B., Christian Schwinn, to appear] Warmup: massive vectors in four dimensions higher D massless vectors decompose into massive D = 4 massive spinor helicity: e.g. [Kleiss, Stirling, 85], [Dittmaier, 98] maximal set of commuting operators: kµ, Wµ µ 2 kµk = m , pick spin axis through light-like vector q qµW qµk νΣρσ R = R1 = µ = µνρσ z q 2q · k 2q · k 1;3 9 vectors n1 and n2 such that q; q^; n1; n2 span R , q · ni = 0 Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 8 / 25 broken gauge theory in D = 4 ! [R.B., Christian Schwinn, to appear] Warmup: massive vectors in four dimensions higher D massless vectors decompose into massive D = 4 massive spinor helicity: e.g. [Kleiss, Stirling, 85], [Dittmaier, 98] maximal set of commuting operators: kµ, Wµ µ 2 kµk = m , pick spin axis through light-like vector q qµW qµk νΣρσ R = R1 = µ = µνρσ z q 2q · k 2q · k 1;3 9 vectors n1 and n2 such that q; q^; n1; n2 span R , q · ni = 0 massive polarization vectors µ µ µ 1 µ µ µ ((n1 ± in2) · k) µ k mq e± = p n ± in − q e = − 2 1 2 q · k 0 m q · k Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 8 / 25 broken gauge theory in D = 4 ! [R.B., Christian Schwinn, to appear] Warmup: massive vectors in four dimensions higher D massless vectors decompose into massive D = 4 massive spinor helicity: e.g.
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