Implication of CHY formulism

Transcription

1 Zhejiang University, China USTC, November, 2016

2 Contents

3 Part I: Backgrounds

4 In physics, it is always fruitful to study same object from different points of views. For example there are many angles to study scattering amplitudes: Feynman diagrams, String theory, Twistor string theory, On-shell method, Grassmannian method, Amplituhedron, etc. Today, I will discuss one aspect, the CHY-formulism

5 In 2013, new formula for tree amplitudes of massless theories has been proposed by Cachazo, He and Yuan: A n = ( n i=1 dz i) dω Ω(E) I, [ Freddy Cachazo, Song He, Ellis Ye Yuan, 2013, 2014] The box part is universal for all theories The CHY-integrand I determines the particular theory

6 Now we give some explanations for different parts. For the part ( n i=1 dz i) dω Integration variables are z i s, i.e., locations of n external legs in sphere. The formula is invariant under the SL(2, C) transformation z az+b cz+d. The dω is nothing, but the gauge volume and can be written as dω = dzr dzsdz t z rsz st z tr, z ij = z i z j. Dividing dω will reduce integration to (n 3) variables, i.e., three locations can be fixed by SL(2, C) transformation.

7 The second factor Ω(E) δ (E a ) = z ij z jk z ki δ (E a ) a a i,j,k Scattering equations are defined E a b a 2k a k b z a z b = 0, a = 1, 2,..., n Only (n 3) of them are independent by SL(2, C) symmetry E a = 0, E a z a = 0, E a za 2 = 0, a a a

8 Combining these two universal factors: (n 3) integrations with (n 3) delta-functions, so the integration becomes the sum over all solutions of scattering equations z Sol 1 det (Φ) I(z) where det (Φ) is the Jacobi coming from solving E a Φ ab = E a z b = { sab z 2 ab c a sac z 2 ac a b a = b,

9 The third factor, i.e., CHY-integrand I(z), defines a particular theory. Some requirements for physical theories: SL(2, C) invariance require that under the transformation, ( n ) (cz i + d) 4 I(z) (ad bc) 2 I(z). i=1 We will call I having weight four. Should gives only simple pole structure Give the correct factorization limit Give the correct soft limit Respect various symmetries, especially the gauge symmetry

10 Building elements I: A ab = B ab = C ab = X ab = four n n matrices { ka k b z a z b 0 { ɛa ɛ b z a z b { 0, for ɛ a k b z a z b, for c a ɛa kc { 1 z a z b 0 z a z c, for a b a = b, a b a = b, for a b a = b, a b a = b,

11 Building elements II: open chain and closed cycle [a 1 a 2...a n ] = (z a1 z a2 )(z a2 z a3 )...(z an 1 z an ) (a 1 a 2...a n ) = (z a1 z a2 )(z a2 z a3 )...(z an 1 z an )(z an z a1 ) Building elements III: Ψ matrix: A 2n 2n antisymmetric matrix Ψ: ( ) A C t Ψ({k i, ɛ i } = C B

12 Two most important building blocks of weight two: (I) reduced Pfaffian of Ψ Pf Ψ = 2 ( 1)i+j z i z j PfΨ ij ij, where 1 i, j n and Ψ ij 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))

13 Three typical theories: Bi-adjoint φ 3 scalar theory: Carrying two different color groups U(N) U(Ñ) with cubic vertex f abc f xyz φ ax φ by φ cz. The integrand is m φ 3[α β] = PT(α) PT(β) Yang-Mills theory: With the color ordering α I YM = PT(α) Pf Ψ({k, ɛ, z} Gravity theory: It is given by I GR = Pf Ψ({k, ɛ, z} Pf Ψ({k, ɛ, z} [ Freddy Cachazo, Song He, Ellis Ye Yuan, 2013]

14 More theories by: [ Freddy Cachazo, Song He, Ellis Ye Yuan, 2014] [ Freddy Cachazo, Peter Cha, Sebastian Mizera, 2016] Compactifying Generalizing and Squeezing Generalized dimensional reduction Extension from soft limits

15 I L bi-adjoint scalar C n (α) C n (α) Yang-Mills C n (α) Pf Ψ n Einstein gravity Pf Ψ n Pf Ψ n Born-Infeld (Pf A n ) 2 Pf Ψ n Non-linear sigma model C n (α) (Pf A n ) 2 Yang-Mills-scalar C n (α) PfX n Pf A n Einstein-Maxwell-scalar PfX n Pf A n PfX n Pf A n Dirac-Born-Infeld (scalar) (Pf A n ) 2 PfX n Pf A n Special Galileon (Pf A n ) 2 (Pf A n ) 2 I R

16 Part II: Integration

17 Having discussed the two parts, now we combine them to give tree level amplitudes of different theories: Naively, the procedure should be: (1) Finding solution; (2) Evaluate integrand of each solution and sum over. However, solving the scattering equation is not easy: it is equivalent to solve a polynomial equation of degree (n 3)! [ Dolan, Goddard, 2014] Thinking more about the problem: what we really want is not the individual solution, but the sum over selected integrand Thus we should look for method, which has avoided solving scattering equations and implemented selecting information from integrand at same time!

18 Integration rule: [Baadsgaard, Bjerrum-Bohr, Bourjaily and Damgaard, 2015 ] First there is a criteria for the appearance of pole s A = ( i A k i) 2 for a subset A {1, 2,..., n}, i.e., the index χ(a) := L[A] 2( A 1) L[A] be the number ( more accurately it is the difference of number between solid and dashed lines) of lines connecting these nodes inside A and A is the number of nodes. It has nonzero contribution when and only when χ(a) 0 and the pole will be 1 s χ(a)+1 A

19 The Reconstruction of cubic Feynman diagrams: Find all subsets A with χ(a) 0 compatible condition for two subsets A 1, A 2 : they are compatible if one subset is completely contained inside another subset or the intersection of two subsets is empty. Find all maximum compatible combinations, i.e., the combination of subsets with largest number such that each pair in the combination is compatible. For each maximum combination with m subsets, it gives nonzero contribution when and only when m = n 3.

20 Each combination giving nonzero contribution will correspond a (generalized) Feynman diagram with only cubic vertexes Now the key is how to read out expressions of Feynman diagrams? For simple pole, the rule is nothing, but the scalar propagator 1 s A!

21 Example of 6-point { 1, 2}, { 2, 3}, { 4, 5}, { 1, 2, 3} { 1, 2} + { 4, 5} + { 1, 2, 3} { 2, 3} + { 4, 5} + { 1, 2, 3} 1 1 s 12 s 123 s 45 s 23 s 123 s 45

22 Above algorithm works well for m[α β] with two PT -factors [ Freddy Cachazo, Song He, Ellis Ye Yuan, 2013] 1 Example (12345)(13245) with (a 1...a m ) = z a1 a 2...z ama s 23 s 45 1 s 23 s 51

23 However, for other theories, such as YM, GR with Pf Ψ, each individual term contains double pole or triple poles. They are canceled out when and only when summing terms together! Thus we need to find a way to deal with higher poles.

24 What is the Feynman rule for higher order poles? Key: The derivative property of residue of higher order pole makes it quasi-local, i.e., it depends not only the total momentum flow through the propagator, but also momentum configuration at the four corners. Simple pole is completely local. [ Huang, BF, 2016]

25 Feynman rule for single double pole: R I [P A, P B, P C, P D ] = 2P AP C + 2P B P D 2s 2 AB, A B A B D C D C

26 Example: Pole subsets {1, 2, 3}, {1, 2}, {2, 3}, {4, 5}, {5, 6} p 12 p p 3 p 6 2s s 12s p 12p 4 + 2p 3 p 56 2s s 12s p 1p p 23 p 6 2s s 23s p 1p 4 + 2p 23 p 56 2s s 23s 56

27 Feynman rule for single triple pole R II [P A, P B, P C, P D ] = (2P AP C )(2P A P D ) 4sAB 3 + (2P B P C )(2P B P D ) + (2P C P A )(2P C P B ) + (2P D P A )(2P D P B ) (P2 A P2 B )2 + (P 2 C P2 D )2 4s 3 AB 4s 3 AB + 2 (PA 2 + P2 B )(P2 C + P2 D ) 9 4sAB 3. A B P A P B D C P D P C

28 Pole subsets {1, 2}, {3, 4}, {5, 6}, {3, 4, 5}, {3, 4, 6}, {3, 5, 6}, {4, 5, 6}

29 s 15 s12 2 s s 23s 15 34s 56 s12 3 s s 24s 15 34s 56 s12 3 s + 1 s 16 34s 56 s12 2 s 56 s12 2 s s 16s 23 34s 56 s12 3 s 34s 56 s 16s 24 s 3 12 s 34s 56 + s 13s 23 s 3 12 s 56s s 16s 26 s 3 12 s 34s s 15s 25 s 3 12 s 34s s 14s 24 s 3 12 s 56s 356.

30 Feynman rule for duplex-double pole: R III [P A, P B, P E, P C, P D ] = (2P AP D )(2P B P C ) (2P A P C )(2P B P D ) sab 2 s2 CD (P2 E )(2P AP D + 2P B P C 2P A P C 2P B P D ) 4sAB 2. s2 CD A B P A P B E P E D C P D P C

31 Feynman rule for triplex-double pole: p 1 p 2 p 5 p 6 p 3 p 4 p 3 p 4 p 1 p 2 p 5 p p 6 p 5 p 4 p 3 p 2 p p 1 p 2 P A P B 5 4 p 6 p 3 P F P C p 5 p 4 P E P D

32 Although Feynman rule method is very convenient for practical calculation, deriving rule for higher order poles is not systematic. A systematic way is to use the Monodromy identity or Cross ratio identity [ Bjerrum-Bohr, Bourjaily, Damgaard, Feng, 2016] [ Cardona, Feng, Gomez, Huang, 2016]

33 Scattering equations can be rewritten as 1 = b a,q,p s ab s aq z aq z bp z ab z qp Using it, one can derive (Λ Λ = N ) 1 = 1 s Λ i Λ/{j} β Λ/{α} s iβ z ij z βα z iβ z jα The key: the numerator z ij z βα reduce the χ(λ) by one

34 Claim: Any weight two integrand can be decomposed as the sum of PT-factors with kinematic coefficients [ Bjerrum-Bohr, Bourjaily, Damgaard, Feng, 2016]

35 We need to use two kinds of identities: The cross ratio identities hold only on the support of scattering equation The identity holds algebraically, i.e., the Schouten identity [ab] [ca][bc] = [ad] [ca][dc] + [db] [cd][bc] which leads to following open up relation = ( 1) A 2 σ (A 1 A R 2 )

36 Now we outline the proof: Any term in integrand can be written as I L = with cross ratios defined by 1 (α(1)...α(n)) F(R ijkl) R ijkl = [ij][kl] [ik][jl] Now we reduce cross ratio factor one by one.

37 First step: using the factor [ij] to open up the closed cycle and we get two possible configurations

38 For the second one, we open up the small closed cycle to get the one big cycle. Done!

39 For the first one, we need to use the cross ratio identity

40 Part III: Implications

41 Having established necessary tools, now we can discuss how to get very nontrivial relations among tree-level scattering amplitudes of various theories from the point of view of CHY formulism.

42 The first highly nontrivial result: the understanding of bi-color-ordered amplitude of φ 3 theory: m[α β] = PT(α) z Sol = (S[α β]) 1 1 det (Φ) PT(β) [ Freddy Cachazo, Song He, Ellis Ye Yuan, 2013] [ Bern, Dixon, Perelstein, Rozowsky, 1998] [ Bjerrum-Bohr, Damgaard, Feng, Sondergaard, 2010]

43 Second important result: KLT relation [ Kawai, Lewellen, Tye, 1986] GR = Pf Ψ z Sol = PT(α) Pf Ψ det (Φ) 1 det (Φ) Pf Ψ det (Φ) PT(α) PT(β) PT(β) Pf Ψ det (Φ) = YM L (φ 3 ) 1 YM R = YM L (α) S[α β] YM R (β) [ Bern, Dixon, Perelstein, Rozowsky, 1998] [ Bjerrum-Bohr, Damgaard, Feng, Sondergaard, 2010]

44 In fact, KLT relation is just one result of recent Color-Kinematic Duality form of amplitudes: c i n i A YM = D i cubic diagrams such that if c i + c j + c k = 0 by Jacobi relation of Lie algebra, then we have n i + n j + n k = 0 Propose: If we can write the YM-amplitude in above form, the corresponding gravity amplitude can be obtained immediately as A YM = cubic diagrams at both tree-level or loop-levels. ñ i n i D i [ Bern, Carrasco, Johansson, 2008]

45 CHY-formula provides a natural framework to understand. The key is following observation of integrand: I = I L I R = Double copy structure A = z Sol I L 1 det (Φ) I R = PT(α) I L det (Φ) det (Φ) PT(α) PT(β) PT(β) I R det (Φ) = A L (α) S[α β] A R (β)

46 The explicit construction of BCJ-numerator n i : For I = PT(α)I R, if we can rewrite I R = α S n 2 n 1αn (1α(2)α(3)...α(n 1)n) then the n 1αn s provide the basis for BCJ-numerator n i. [ Freddy Cachazo, Song He, Ellis Ye Yuan, 2013] For YM-theory, the I R = Pf Ψ. By our claim, it can be written as α S n 1 ñ 1αn (1α(2)α(3)...α(n 1)α(n)) Now we can use the KK-relation to rewrite 1 (1α(2)α(3)...α(n 1)α(n)) = ( ) β 1 (1β(2)β(3)...β(n 1)n) β S n 2

47 Expanding amplitudes by amplitudes: I L I R Yang-Mills C n (α) Pf Ψ n Einstein gravity Pf Ψ n Pf Ψ n Einstein-YM PT r (α) PfΨ S Pf Ψ n Born-Infeld (Pf A n ) 2 Pf Ψ n 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]

48 From the table: Yang-Mills-scalar C n (α) PfX n Pf A n Einstein-Maxwell-scalar PfX n Pf A n PfX n Pf A n Dirac-Born-Infeld (scalar) (Pf A n ) 2 PfX n Pf A n we see that amplitudes of Einstein-Maxwell-scalar and Dirac-Born-Infeld can be expanded by color ordered Yang-Mills-scalar amplitudes. I L I R

49 I L I R Non-linear sigma model C n (α) (Pf A n ) 2 Special Galileon (Pf A n ) 2 (Pf A n ) 2 Amplitudes of Special Galileon theory can be expanded by amplitudes of Non-linear sigma model

50 Further generalization: All amplitudes in CHY-formulism can be expanded by m[α β] of bi-adjoint φ 3 theories Reasons: (1) m[α β] provides the scalar cubic Feynman diagrams; (2) CHY-formulism provides a method to address m[α β] with kinematic factors. = in some sense m[α β] is the Feynman diagram in CHY-formulism!

51 A few last remarks: There are many aspects we have not touched, such as the soft behaviors of various theories Although we have focused only on tree-level, important progress has been achieved for loop-level [ Geyer, Mason, Monteiro, Tourkine, 2015, 2016] There are still many problems to be understood, such as including fermions, scalar etc in arbitrary dimension

52 Thanks a lot for listening!!!