Amplitude Relations Bo Feng based on work with Yi-Jian Du, Rijun Huang, Fein Teng, arXiv:1702.08158, 1703.01269, 1708.04514 The 2nd East Asia Joint workshop, KEK, Japan, Nov 12-17, 2017 Bo Feng Amplitude Relations
Contents Bo Feng Amplitude Relations
I: Motivation Bo Feng Amplitude Relations
Scattering amplitude is one of most important concepts in QFT. It is the bridge connecting experiment data and theoretical prediction Its various properties contain also many important information about the theory, such as Lorentz symmetry, local interaction, unitarity etc. Because this, studying scattering amplitude is also one of main topics in QFT Bo Feng Amplitude Relations
Scattering amplitude is one of most important concepts in QFT. It is the bridge connecting experiment data and theoretical prediction Its various properties contain also many important information about the theory, such as Lorentz symmetry, local interaction, unitarity etc. Because this, studying scattering amplitude is also one of main topics in QFT Bo Feng Amplitude Relations
Scattering amplitude is one of most important concepts in QFT. It is the bridge connecting experiment data and theoretical prediction Its various properties contain also many important information about the theory, such as Lorentz symmetry, local interaction, unitarity etc. Because this, studying scattering amplitude is also one of main topics in QFT Bo Feng Amplitude Relations
Scattering amplitude is one of most important concepts in QFT. It is the bridge connecting experiment data and theoretical prediction Its various properties contain also many important information about the theory, such as Lorentz symmetry, local interaction, unitarity etc. Because this, studying scattering amplitude is also one of main topics in QFT Bo Feng Amplitude Relations
In 1985, by calculating the scattering amplitude of close string theory, Kawai, Lewellen and Tye found an amazing result, the so called KLT relation , which states that: the tree-level scattering amplitude M can be written as � A n ( α ) S [ α | β ] � M n = A n ( β ) α,β where A n , � A n are color ordered scattering amplitudes of Yang-Mills theory, and the S is the momentum kernel. For example A 3 ( 1 , 2 , 3 ) � M 3 ( 1 , 2 , 3 ) = A 3 ( 1 , 2 , 3 ) , A 4 ( 1 , 2 , 3 , 4 ) s 12 � M 4 ( 1 , 2 , 3 , 4 ) = A 4 ( 3 , 4 , 2 , 1 ) [Kawai, Lewellen, Tye; 1985] [Bern, Dixon, Perelstein, Rozowsky; 1999] [Bjerrum-Bohr, Damgaard, Feng, Sondergaard; 2010] Bo Feng Amplitude Relations
Although derived from string very naturally (closed string verses open string), from the point of view of field theory, it is big surprising: Gauge symmetry is symmetry for inner quantities while gravity theory is based on the space-time symmetry, the general equivalence principal for the choice of coordinate. More importantly, the Lagrangian of gauge theory is polynomial with finite number of interaction terms (in fact, only to A 4 term), while the Einstein Lagrangian is highly non-linear and infinite number of interaction terms after perturbative expansion. Bo Feng Amplitude Relations
Although derived from string very naturally (closed string verses open string), from the point of view of field theory, it is big surprising: Gauge symmetry is symmetry for inner quantities while gravity theory is based on the space-time symmetry, the general equivalence principal for the choice of coordinate. More importantly, the Lagrangian of gauge theory is polynomial with finite number of interaction terms (in fact, only to A 4 term), while the Einstein Lagrangian is highly non-linear and infinite number of interaction terms after perturbative expansion. Bo Feng Amplitude Relations
Although derived from string very naturally (closed string verses open string), from the point of view of field theory, it is big surprising: Gauge symmetry is symmetry for inner quantities while gravity theory is based on the space-time symmetry, the general equivalence principal for the choice of coordinate. More importantly, the Lagrangian of gauge theory is polynomial with finite number of interaction terms (in fact, only to A 4 term), while the Einstein Lagrangian is highly non-linear and infinite number of interaction terms after perturbative expansion. Bo Feng Amplitude Relations
Then why there is the KLT relation? The key is the distinction between particles and fields The field carries the tensor representation of Lorentz group, while particle is categorized under the little group. Or in another word, the Lagrangian description uses the field, so is a off-shell description, while the scattering amplitude uses the particle, so is on-shell quantity. The existence of KLT relation is a on-shell property, so it is hard to see by off-shell description. Bo Feng Amplitude Relations
Then why there is the KLT relation? The key is the distinction between particles and fields The field carries the tensor representation of Lorentz group, while particle is categorized under the little group. Or in another word, the Lagrangian description uses the field, so is a off-shell description, while the scattering amplitude uses the particle, so is on-shell quantity. The existence of KLT relation is a on-shell property, so it is hard to see by off-shell description. Bo Feng Amplitude Relations
Then why there is the KLT relation? The key is the distinction between particles and fields The field carries the tensor representation of Lorentz group, while particle is categorized under the little group. Or in another word, the Lagrangian description uses the field, so is a off-shell description, while the scattering amplitude uses the particle, so is on-shell quantity. The existence of KLT relation is a on-shell property, so it is hard to see by off-shell description. Bo Feng Amplitude Relations
Then why there is the KLT relation? The key is the distinction between particles and fields The field carries the tensor representation of Lorentz group, while particle is categorized under the little group. Or in another word, the Lagrangian description uses the field, so is a off-shell description, while the scattering amplitude uses the particle, so is on-shell quantity. The existence of KLT relation is a on-shell property, so it is hard to see by off-shell description. Bo Feng Amplitude Relations
Then why there is the KLT relation? The key is the distinction between particles and fields The field carries the tensor representation of Lorentz group, while particle is categorized under the little group. Or in another word, the Lagrangian description uses the field, so is a off-shell description, while the scattering amplitude uses the particle, so is on-shell quantity. The existence of KLT relation is a on-shell property, so it is hard to see by off-shell description. Bo Feng Amplitude Relations
A big lesson from the study of scattering amplitudes in recent years is that if we focus on only on-shell quantities, we may Calculate them much fast Get much simpler expression Find deeply hidden relations, such as KLT relation, BCJ relation and the sub-leading order soft theorem of graviton, etc [Bern, Carraso, Johansson, 2008] [Cachazo, Strominger; 2014] Bo Feng Amplitude Relations
A big lesson from the study of scattering amplitudes in recent years is that if we focus on only on-shell quantities, we may Calculate them much fast Get much simpler expression Find deeply hidden relations, such as KLT relation, BCJ relation and the sub-leading order soft theorem of graviton, etc [Bern, Carraso, Johansson, 2008] [Cachazo, Strominger; 2014] Bo Feng Amplitude Relations
A big lesson from the study of scattering amplitudes in recent years is that if we focus on only on-shell quantities, we may Calculate them much fast Get much simpler expression Find deeply hidden relations, such as KLT relation, BCJ relation and the sub-leading order soft theorem of graviton, etc [Bern, Carraso, Johansson, 2008] [Cachazo, Strominger; 2014] Bo Feng Amplitude Relations
A big lesson from the study of scattering amplitudes in recent years is that if we focus on only on-shell quantities, we may Calculate them much fast Get much simpler expression Find deeply hidden relations, such as KLT relation, BCJ relation and the sub-leading order soft theorem of graviton, etc [Bern, Carraso, Johansson, 2008] [Cachazo, Strominger; 2014] Bo Feng Amplitude Relations
With above explanation, the searching of on-shell frame to reach above goals become one of important directions. Some achievements are: String theory Twistor string [Witten, 2003] Grassmanian [Arkani-Hamed, Cachazo, Cheung and Kaplan, 2010] Amplituhedron [Arkani-Hamed and Trnka, 2014] CHY frame [Cachazo, He, and Yuan, 2014] Bo Feng Amplitude Relations
II: General Relation in CHY-frame In this part, we will use one of on-shell frameworks, i.e., the CHY-frame to give a general picture of relations of scattering amplitudes of different theories Bo Feng Amplitude Relations
In 2013, new formula for tree amplitudes of massless theories has been proposed by Cachazo, He and Yuan: � �� n � i = 1 dz i A n = Ω( E ) I , d ω [ Freddy Cachazo, Song He, Ellis Ye Yuan , 2013, 2014] In this frame: Each particle is represented by a puncture in Riemann sphere The expression holds for general D-dimension The box part is universal for all theories The CHY-integrand I determines the particular theory Bo Feng Amplitude Relations
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