Amplitude Relations Bo Feng based on work with Yi-Jian Du, Rijun - - PowerPoint PPT Presentation
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
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
Bo Feng Amplitude Relations
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 Mn =
An(α)S[α|β] An(β) where An, An are color ordered scattering amplitudes of Yang-Mills theory, and the S is the momentum kernel. For example M3(1, 2, 3) = A3(1, 2, 3) A3(1, 2, 3), M4(1, 2, 3, 4) = A4(1, 2, 3, 4)s12 A4(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,
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,
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,
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
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: An = n
i=1 dzi
Ω(E) I,
[ 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
For the universal part, Ω(E) ≡
′
δ (Ea) = zijzjkzki
δ (Ea) provides the constraints: Scattering equations are defined Ea ≡
2ka · kb za − zb = 0, a = 1, 2, ..., n Only (n − 3) of them are independent by SL(2, C) symmetry
Ea = 0,
Eaza = 0,
Eaz2
a = 0,
Bo Feng Amplitude Relations
Universal part: (n − 3) integrations with (n − 3) delta-functions, so the integration becomes the sum over all solutions of scattering equations
1 det′(Φ)I(z) where det′(Φ) is the Jacobi coming from solving Ea Φab = ∂Ea ∂zb = sab
z2
ab
a = b −
c=a sac z2
ac
a = b ,
Bo Feng Amplitude Relations
Building elements I: four n × n matrices Aab =
za−zb
, for a = b a = b , Bab =
za−zb
, for a = b a = b , Cab =
za−zb
−
c=a ǫa·kc za−zc
for a = b a = b , Xab =
za−zb
, for a = b a = b ,
Bo Feng Amplitude Relations
Building elements II:
[a1a2...an] = (za1 − za2)(za2 − za3)...(zan−1 − zan) (a1a2...an) = (za1 − za2)(za2 − za3)...(zan−1 − zan)(zan − za1) Building elements III: Ψ matrix: A 2n × 2n antisymmetric matrix Ψ: Ψ({ki, ǫi} = A −Ct C B
invariance manifest!
Bo Feng Amplitude Relations
Two most important building blocks of weight two: (I) reduced Pfaffian of Ψ Pf′Ψ = 2(−1)i+j zi − zj PfΨij
ij ,
where 1 ≤ i, j ≤ n and Ψij
ij is the matrix Ψ removing rows i, j and
columns i, j. It is independent of the choice (i, j). (II) color ordered Parker-Taylor factor C(α) = PT(α) = 1 (α(1)...α(n))
Bo Feng Amplitude Relations
One key fact: for all theories described in this frame, CHY-integrand is factorized to two parts: IL IR bi-adjoint scalar Cn(α) Cn(α) Yang-Mills Cn(α) Pf′Ψn Einstein gravity Pf′Ψn Pf′Ψn Born-Infeld (Pf′An)2 Pf′Ψn Non-linear sigma model Cn(α) (Pf′An)2 Yang-Mills-scalar Cn(α) PfXnPf′An Einstein-Maxwell-scalar PfXnPf′An PfXnPf′An Dirac-Born-Infeld (scalar) (Pf′An)2 PfXnPf′An Special Galileon (Pf′An)2 (Pf′An)2
[ Freddy Cachazo, Song He, Ellis Ye Yuan , 2014] [ Freddy Cachazo, Peter Cha, Sebastian Mizera, 2016]
Bo Feng Amplitude Relations
Solving polynomial equations with multiple variables is not easy, but for the CHY case, systematically method has been developed, so we can read out analytic expression straightforward: For the case with only simple pole:
[ Freddy Cachazo, Song He, Ellis Ye Yuan , 2013,2014] [Baadsgaard, Bjerrum-Bohr, Bourjaily and Damgaard, 2015 ] [Cachazo, Gomez, 2015 ] [Lam, Yao, 2016 ]
For general case with higher order pole
[Baadsgaard, Bjerrum-Bohr, Bourjaily and Damgaard, 2015 ] [Huang, Feng, Luo, Zhu, 2016 ] [Cardona, Feng, Gomez, Huang, 2016 ]
Final conclusion: Any weight two integrand can be decomposed as the sum of PT-factors with kinematic rational coefficients
[ Bjerrum-Bohr, Bourjaily, Damgaard, Feng, 2016]
Bo Feng Amplitude Relations
Solving polynomial equations with multiple variables is not easy, but for the CHY case, systematically method has been developed, so we can read out analytic expression straightforward: For the case with only simple pole:
[ Freddy Cachazo, Song He, Ellis Ye Yuan , 2013,2014] [Baadsgaard, Bjerrum-Bohr, Bourjaily and Damgaard, 2015 ] [Cachazo, Gomez, 2015 ] [Lam, Yao, 2016 ]
For general case with higher order pole
[Baadsgaard, Bjerrum-Bohr, Bourjaily and Damgaard, 2015 ] [Huang, Feng, Luo, Zhu, 2016 ] [Cardona, Feng, Gomez, Huang, 2016 ]
Final conclusion: Any weight two integrand can be decomposed as the sum of PT-factors with kinematic rational coefficients
[ Bjerrum-Bohr, Bourjaily, Damgaard, Feng, 2016]
Bo Feng Amplitude Relations
Solving polynomial equations with multiple variables is not easy, but for the CHY case, systematically method has been developed, so we can read out analytic expression straightforward: For the case with only simple pole:
[ Freddy Cachazo, Song He, Ellis Ye Yuan , 2013,2014] [Baadsgaard, Bjerrum-Bohr, Bourjaily and Damgaard, 2015 ] [Cachazo, Gomez, 2015 ] [Lam, Yao, 2016 ]
For general case with higher order pole
[Baadsgaard, Bjerrum-Bohr, Bourjaily and Damgaard, 2015 ] [Huang, Feng, Luo, Zhu, 2016 ] [Cardona, Feng, Gomez, Huang, 2016 ]
Final conclusion: Any weight two integrand can be decomposed as the sum of PT-factors with kinematic rational coefficients
[ Bjerrum-Bohr, Bourjaily, Damgaard, Feng, 2016]
Bo Feng Amplitude Relations
The generalized KLT relation is the natural consequece of the factorized form of the CHY-integrand: First observation: m[α|β] =
PT(α) 1 det′(Φ) PT(β) = (S[α|β])−1 where the S is KLT kernel and m[α|β] the amplitudes of bi-color-ordered φ3 theory
[ Freddy Cachazo, Song He, Ellis Ye Yuan , 2013] [ Bern, Dixon, Perelstein, Rozowsky, 1998] [ Bjerrum-Bohr, Damgaard, Feng, Sondergaard, 2010]
Bo Feng Amplitude Relations
The generalized KLT relation is the natural consequece of the factorized form of the CHY-integrand: First observation: m[α|β] =
PT(α) 1 det′(Φ) PT(β) = (S[α|β])−1 where the S is KLT kernel and m[α|β] the amplitudes of bi-color-ordered φ3 theory
[ Freddy Cachazo, Song He, Ellis Ye Yuan , 2013] [ Bern, Dixon, Perelstein, Rozowsky, 1998] [ Bjerrum-Bohr, Damgaard, Feng, Sondergaard, 2010]
Bo Feng Amplitude Relations
For general theory with I = IL × IR we derive A =
IL 1 det′(Φ) IR = PT(α) IL det′(Φ) × det′(Φ) PT(α) PT(β) × PT(β) IR det′(Φ) = AL(α) × (φ3)−1 × AR(β) = AL(α) × S[α|β] × AR(β) With both IL = IR = Pf′Ψ we get the original KLT relation.
Bo Feng Amplitude Relations
Expansion: If we define coefficient c(α) =
β S[α|β] × AR(β) we have
A =
AL(α) × S[α|β] × AR(β) =
c(α)AL(α) Similar we have A =
with c(β) =
α AL(α) × S[α|β]
Bo Feng Amplitude Relations
Expansion: If we define coefficient c(α) =
β S[α|β] × AR(β) we have
A =
AL(α) × S[α|β] × AR(β) =
c(α)AL(α) Similar we have A =
with c(β) =
α AL(α) × S[α|β]
Bo Feng Amplitude Relations
Expansion: If we define coefficient c(α) =
β S[α|β] × AR(β) we have
A =
AL(α) × S[α|β] × AR(β) =
c(α)AL(α) Similar we have A =
with c(β) =
α AL(α) × S[α|β]
Bo Feng Amplitude Relations
We must emphasize that the expansion of amplitudes of one theory by another theory is hard to guess from the Lagrangian, but is very natural from CHY frame!
Bo Feng Amplitude Relations
In this part, we will discuss one example, i.e., expanding the amplitudes of Einstein-Yang-Mill theory into color-ordered amplitudes of pure Yang-Mills theory. We will start from the single trace part of EYM theory, then to mutli-trace part.
Bo Feng Amplitude Relations
First form of single trace part: The Recursive Expansion Let us start with a few examples: With one graviton AEYM
n,1 (1, . . . , n; p) =
(ǫp · Yp)AYM
n+1(1, {2, . . . , n − 1} ✁ {p}, n) ,
where the summation ✁ is over all shuffles σ ✁ σ, i.e., all permutation sets of σ ∪ σ while preserving the ordering of each σ, σ. At the left, p is graviton while at the right p becomes effectively a gluon, we call it as "turning a graviton to gluons"
Bo Feng Amplitude Relations
First form of single trace part: The Recursive Expansion Let us start with a few examples: With one graviton AEYM
n,1 (1, . . . , n; p) =
(ǫp · Yp)AYM
n+1(1, {2, . . . , n − 1} ✁ {p}, n) ,
where the summation ✁ is over all shuffles σ ✁ σ, i.e., all permutation sets of σ ∪ σ while preserving the ordering of each σ, σ. At the left, p is graviton while at the right p becomes effectively a gluon, we call it as "turning a graviton to gluons"
Bo Feng Amplitude Relations
First form of single trace part: The Recursive Expansion Let us start with a few examples: With one graviton AEYM
n,1 (1, . . . , n; p) =
(ǫp · Yp)AYM
n+1(1, {2, . . . , n − 1} ✁ {p}, n) ,
where the summation ✁ is over all shuffles σ ✁ σ, i.e., all permutation sets of σ ∪ σ while preserving the ordering of each σ, σ. At the left, p is graviton while at the right p becomes effectively a gluon, we call it as "turning a graviton to gluons"
Bo Feng Amplitude Relations
With two gravitons: AEYM
n,2 (1, 2, . . . , n; p, q)
=
(ǫp · Yp)AEYM
n+1,1(1, {2, . . . , n − 1} ✁ {p}, n; q)
+
(ǫp · Fq · Xq)AYM
n+2(1, {2, . . . , n − 1} ✁ {q, p}, n)
Feature:
Expansion is done recursively, i.e., each time just turning a graviton to gluon Manifest gauge invariant with field strength F µν
q
:= qµǫν
q − ǫµ qqν
The treatment of p, q is not symmetric. Different choices of the first elements give different, but equivalent expressions
Bo Feng Amplitude Relations
With two gravitons: AEYM
n,2 (1, 2, . . . , n; p, q)
=
(ǫp · Yp)AEYM
n+1,1(1, {2, . . . , n − 1} ✁ {p}, n; q)
+
(ǫp · Fq · Xq)AYM
n+2(1, {2, . . . , n − 1} ✁ {q, p}, n)
Feature:
Expansion is done recursively, i.e., each time just turning a graviton to gluon Manifest gauge invariant with field strength F µν
q
:= qµǫν
q − ǫµ qqν
The treatment of p, q is not symmetric. Different choices of the first elements give different, but equivalent expressions
Bo Feng Amplitude Relations
With two gravitons: AEYM
n,2 (1, 2, . . . , n; p, q)
=
(ǫp · Yp)AEYM
n+1,1(1, {2, . . . , n − 1} ✁ {p}, n; q)
+
(ǫp · Fq · Xq)AYM
n+2(1, {2, . . . , n − 1} ✁ {q, p}, n)
Feature:
Expansion is done recursively, i.e., each time just turning a graviton to gluon Manifest gauge invariant with field strength F µν
q
:= qµǫν
q − ǫµ qqν
The treatment of p, q is not symmetric. Different choices of the first elements give different, but equivalent expressions
Bo Feng Amplitude Relations
With two gravitons: AEYM
n,2 (1, 2, . . . , n; p, q)
=
(ǫp · Yp)AEYM
n+1,1(1, {2, . . . , n − 1} ✁ {p}, n; q)
+
(ǫp · Fq · Xq)AYM
n+2(1, {2, . . . , n − 1} ✁ {q, p}, n)
Feature:
Expansion is done recursively, i.e., each time just turning a graviton to gluon Manifest gauge invariant with field strength F µν
q
:= qµǫν
q − ǫµ qqν
The treatment of p, q is not symmetric. Different choices of the first elements give different, but equivalent expressions
Bo Feng Amplitude Relations
With three gravitons: AEYM
n,3 (1, . . . , n; p, q, r)
=
(ǫp · Yp)AEYM
n+1,2(1, {2, . . . , n − 1} ✁ {p}, n; q, r)
+
(ǫp · Fq · Yq)AEYM
n+2,1(1, {2, . . . , n − 1} ✁ {q, p}, n; r)
+
(ǫp · Fr · Yr)AEYM
n+2,1(1, {2, . . . , n − 1} ✁ {r, p}, n; q)
+
(ǫp · Fq · Fr · Yr)AYM
n+3(1, {2, . . . , n − 1} ✁ {r, q, p}, n)
+
(ǫp · Fr · Fq · Yq)AYM
n+3(1, {2, . . . , n − 1} ✁ {q, r, p}, n) .
Bo Feng Amplitude Relations
Some remarks: The recursive expansion is first obtained from gauge invariance principle for each graviton. Then it is re-derived in CHY-frame. We can recursively expand to pure Yang-Mill amplitude. Then there are three ways to write down the final expansion: the ordered splitting form, the KK-basis form and the tree diagram representation of KK-basis form. Although it is simple example, starting from it, other expansions can be done straightforwardly
Bo Feng Amplitude Relations
Some remarks: The recursive expansion is first obtained from gauge invariance principle for each graviton. Then it is re-derived in CHY-frame. We can recursively expand to pure Yang-Mill amplitude. Then there are three ways to write down the final expansion: the ordered splitting form, the KK-basis form and the tree diagram representation of KK-basis form. Although it is simple example, starting from it, other expansions can be done straightforwardly
Bo Feng Amplitude Relations
Some remarks: The recursive expansion is first obtained from gauge invariance principle for each graviton. Then it is re-derived in CHY-frame. We can recursively expand to pure Yang-Mill amplitude. Then there are three ways to write down the final expansion: the ordered splitting form, the KK-basis form and the tree diagram representation of KK-basis form. Although it is simple example, starting from it, other expansions can be done straightforwardly
Bo Feng Amplitude Relations
For generic case m-trace EYM amplitude with |h| gravitons: First write down the recursive expansion of single trace EYM amplitude with |H| = m − 1 + |h| gravitons If in the expansion, the graviton is still graviton, we just replace it by the trace of gluons A(1, 2 . . . r H) − → A(1, 2 . . . r |t t t1 |t t t2 | . . . |t t tm−1 h) If in the expansion, the graviton has been turned into gluons, we do replacement the graviton by the KK-expansion of the trace of gluons: K K K(Tr Tr Tr s, a, b) = {Kt
t t1 a1,b1, Kt t t2 a2,b2 . . . Kt t ts as,bs} ,
and modify corresponding kinematic factor (Fha)µν → −(kbj)µ(kaj)ν The key is turning a graviton into the trace
Bo Feng Amplitude Relations
For generic case m-trace EYM amplitude with |h| gravitons: First write down the recursive expansion of single trace EYM amplitude with |H| = m − 1 + |h| gravitons If in the expansion, the graviton is still graviton, we just replace it by the trace of gluons A(1, 2 . . . r H) − → A(1, 2 . . . r |t t t1 |t t t2 | . . . |t t tm−1 h) If in the expansion, the graviton has been turned into gluons, we do replacement the graviton by the KK-expansion of the trace of gluons: K K K(Tr Tr Tr s, a, b) = {Kt
t t1 a1,b1, Kt t t2 a2,b2 . . . Kt t ts as,bs} ,
and modify corresponding kinematic factor (Fha)µν → −(kbj)µ(kaj)ν The key is turning a graviton into the trace
Bo Feng Amplitude Relations
For generic case m-trace EYM amplitude with |h| gravitons: First write down the recursive expansion of single trace EYM amplitude with |H| = m − 1 + |h| gravitons If in the expansion, the graviton is still graviton, we just replace it by the trace of gluons A(1, 2 . . . r H) − → A(1, 2 . . . r |t t t1 |t t t2 | . . . |t t tm−1 h) If in the expansion, the graviton has been turned into gluons, we do replacement the graviton by the KK-expansion of the trace of gluons: K K K(Tr Tr Tr s, a, b) = {Kt
t t1 a1,b1, Kt t t2 a2,b2 . . . Kt t ts as,bs} ,
and modify corresponding kinematic factor (Fha)µν → −(kbj)µ(kaj)ν The key is turning a graviton into the trace
Bo Feng Amplitude Relations
For generic case m-trace EYM amplitude with |h| gravitons: First write down the recursive expansion of single trace EYM amplitude with |H| = m − 1 + |h| gravitons If in the expansion, the graviton is still graviton, we just replace it by the trace of gluons A(1, 2 . . . r H) − → A(1, 2 . . . r |t t t1 |t t t2 | . . . |t t tm−1 h) If in the expansion, the graviton has been turned into gluons, we do replacement the graviton by the KK-expansion of the trace of gluons: K K K(Tr Tr Tr s, a, b) = {Kt
t t1 a1,b1, Kt t t2 a2,b2 . . . Kt t ts as,bs} ,
and modify corresponding kinematic factor (Fha)µν → −(kbj)µ(kaj)ν The key is turning a graviton into the trace
Bo Feng Amplitude Relations
For generic case m-trace EYM amplitude with |h| gravitons: First write down the recursive expansion of single trace EYM amplitude with |H| = m − 1 + |h| gravitons If in the expansion, the graviton is still graviton, we just replace it by the trace of gluons A(1, 2 . . . r H) − → A(1, 2 . . . r |t t t1 |t t t2 | . . . |t t tm−1 h) If in the expansion, the graviton has been turned into gluons, we do replacement the graviton by the KK-expansion of the trace of gluons: K K K(Tr Tr Tr s, a, b) = {Kt
t t1 a1,b1, Kt t t2 a2,b2 . . . Kt t ts as,bs} ,
and modify corresponding kinematic factor (Fha)µν → −(kbj)µ(kaj)ν The key is turning a graviton into the trace
Bo Feng Amplitude Relations
Besides on above expansion of EYM theory, using the CHY-frame, we can find other relations among different theories by a few operations (Dimension reduction, Squeezing, Generalized Dimension reduction):
[ Yuan’s talk, 2017]
Bo Feng Amplitude Relations
More examples of expansions I: IL IR Yang-Mills Cn(α) Pf′Ψn Einstein gravity Pf′Ψn Pf′Ψn Einstein-YM PTr(α) PfΨS Pf′Ψn Born-Infeld (Pf′An)2 Pf′Ψn Thus we see that amplitudes of Einstein, Einstein-YM, Born-Infeld can be expanded by the color-ordered YM amplitudes.
[ Stieberger, Taylor, 2016] [ Nandan, Plefka, Schlotterer, Wen, 2016] [ de la Cruz, Kniss, Weinzierl , 2016]
Bo Feng Amplitude Relations
More examples of expansions II: IL IR Yang-Mills-scalar Cn(α) PfXnPf′An Einstein-Maxwell-scalar PfXnPf′An PfXnPf′An Dirac-Born-Infeld (scalar) (Pf′An)2 PfXnPf′An we see that amplitudes of Einstein-Maxwell-scalar and Dirac-Born-Infeld can be expanded by color ordered Yang-Mills-scalar amplitudes.
Bo Feng Amplitude Relations
More examples of expansions III: IL IR Non-linear sigma model Cn(α) (Pf′An)2 Special Galileon (Pf′An)2 (Pf′An)2 Amplitudes of Special Galileon theory can be expanded by amplitudes of Non-linear sigma model
Bo Feng Amplitude Relations
Further generalization: All amplitudes in CHY-formulism can be expanded by m[α|β] of bi-adjoint φ3 theories Why?
m[α|β] provides the scalar cubic Feynman diagrams. Any Feynman diagram is based on it. CHY-formulism provides a method to address m[α|β] with kinematic factors. in some sense m[α|β] is the on-shell Feynman diagram in CHY-formulism!
Bo Feng Amplitude Relations
Further generalization: All amplitudes in CHY-formulism can be expanded by m[α|β] of bi-adjoint φ3 theories Why?
m[α|β] provides the scalar cubic Feynman diagrams. Any Feynman diagram is based on it. CHY-formulism provides a method to address m[α|β] with kinematic factors. in some sense m[α|β] is the on-shell Feynman diagram in CHY-formulism!
Bo Feng Amplitude Relations
Further generalization: All amplitudes in CHY-formulism can be expanded by m[α|β] of bi-adjoint φ3 theories Why?
m[α|β] provides the scalar cubic Feynman diagrams. Any Feynman diagram is based on it. CHY-formulism provides a method to address m[α|β] with kinematic factors. in some sense m[α|β] is the on-shell Feynman diagram in CHY-formulism!
Bo Feng Amplitude Relations
Further generalization: All amplitudes in CHY-formulism can be expanded by m[α|β] of bi-adjoint φ3 theories Why?
m[α|β] provides the scalar cubic Feynman diagrams. Any Feynman diagram is based on it. CHY-formulism provides a method to address m[α|β] with kinematic factors. in some sense m[α|β] is the on-shell Feynman diagram in CHY-formulism!
Bo Feng Amplitude Relations
Further generalization: All amplitudes in CHY-formulism can be expanded by m[α|β] of bi-adjoint φ3 theories Why?
m[α|β] provides the scalar cubic Feynman diagrams. Any Feynman diagram is based on it. CHY-formulism provides a method to address m[α|β] with kinematic factors. in some sense m[α|β] is the on-shell Feynman diagram in CHY-formulism!
Bo Feng Amplitude Relations
Bo Feng Amplitude Relations
The revolution in scattering amplitudes in recent years has pushed forward some important concepts, i.e., such as
Various relations discussed in this talk is hard to understand from off-shell Lagrangian formula, but is easy to see from some on-shell frames These things hint that there maybe different descriptions
not completely figured out Although there are huge progresses we have made, there are still more waiting us to discover and to understand!
Bo Feng Amplitude Relations
The revolution in scattering amplitudes in recent years has pushed forward some important concepts, i.e., such as
Various relations discussed in this talk is hard to understand from off-shell Lagrangian formula, but is easy to see from some on-shell frames These things hint that there maybe different descriptions
not completely figured out Although there are huge progresses we have made, there are still more waiting us to discover and to understand!
Bo Feng Amplitude Relations
The revolution in scattering amplitudes in recent years has pushed forward some important concepts, i.e., such as
Various relations discussed in this talk is hard to understand from off-shell Lagrangian formula, but is easy to see from some on-shell frames These things hint that there maybe different descriptions
not completely figured out Although there are huge progresses we have made, there are still more waiting us to discover and to understand!
Bo Feng Amplitude Relations
The revolution in scattering amplitudes in recent years has pushed forward some important concepts, i.e., such as
Various relations discussed in this talk is hard to understand from off-shell Lagrangian formula, but is easy to see from some on-shell frames These things hint that there maybe different descriptions
not completely figured out Although there are huge progresses we have made, there are still more waiting us to discover and to understand!
Bo Feng Amplitude Relations
Bo Feng Amplitude Relations