Double Soft Theorems and Shift Symmetry in Nonlinear Sigma Models
DDouble Soft Theorems and Shift Symmetry in Nonlinear SigmaModels
Ian Low
High Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208
Abstract
We show both the leading and subleading double soft theorems of the nonlinear sigma modelfollow from a shift symmetry enforcing Adler’s zero condition in the presence of an unbroken globalsymmetry. They do not depend on the underlying coset
G/H and are universal infrared behaviorsof Nambu-Goldstone bosons. Although nonlinear sigma models contain an infinite number ofinteraction vertices, the double soft limit is determined entirely by a single four-point interaction,together with the existence of Adler’s zeros. a r X i v : . [ h e p - t h ] D ec . INTRODUCTION The study of soft massless particles has a long and rich history [1–6]. The subject gainedrenewed interest following Refs. [7, 8], which stimulated many new analyses [10–13]. Morerecently Ref. [11] proposed new double soft theorems for massless scalars in a variety ofquantum field theories, based on the study of scattering equations [14]. In particular, fornonlinear sigma models (NLSM) with a U ( N ) color (flavor) structure, they presented thefollowing double soft theorems for a color-ordered partial amplitude M (1 , , · · · , n, n + 1 , n + 2) = (cid:0) S (0) + S (1) (cid:1) M (1 , , · · · , n ) + O ( τ ) , (1)where momenta of particles n + 1 and n + 2 are taken soft, p µn +1 = τ p µ and p µn +2 = τ q µ , as τ →
0. The leading and subleading soft factors are S (0) = 12 (cid:18) p n · ( p n +1 − p n +2 ) + p n +1 · p n +2 p n · ( p n +1 + p n +2 ) + p n +1 · p n +2 + p · ( p n +2 − p n +1 ) + p n +2 · p n +1 p · ( p n +2 + p n +1 ) + p n +2 · p n +1 (cid:19) , (2) S (1) = p n +1 ,µ p n +2 ,ν p n · ( p n +1 + p n +2 ) + p n +1 · p n +2 J µνn + p n +2 ,µ p n +1 ,ν p n · ( p n +2 + p n +1 ) + p n +2 · p n +1 J µν , (3)where J µνa is the total angular momentum operator acting on the a th scalar particle J µνa ≡ p µa ∂∂p a,ν − p νa ∂∂p a,µ . (4)The leading double soft factor S (0) contains the famous Adler’s zero [4] when either p µ → q µ →
0. Expanding to the first order in τ , it also reproduces the double soft limitproposed in Ref. [7], which was later studied using BCFW-like recursion relations [9]. Thesubleading double soft factor S (1) was proven with also recursion relations [13].However, it is well-known from the early work of soft pion theorems [4–6] that emission ofsoft pions often are uniquely determined by current algebra, i.e. commutators of vector andaxial currents. Therefore, they are dictated by the Ward identities, which are statements ofthe symmetry in the system. Indeed, many of the recent results concentrate on relating softtheorems to symmetries [8, 12]. This is the viewpoint we wish to pursue. In particular, wewill see that it is possible to derive the double soft theorems for the full scattering amplitudes,i.e. the S -matrix elements, without recourse to color-ordered partial amplitudes. Early studies on the double soft pion emission, using techniques of current algebra, can be found in [5, 6] π a transforming under an unbroken globalsymmetry group H and impose a set of shift symmetries to forbid a scalar mass term, π a → π a + (cid:15) a + · · · , (5)We will see that this condition, together with the requirement of preserving the globalsymmetry H for interaction vertices, is sufficient to prove the double soft theorems in Eq. (1),without recourse to BCFW-like recursion relations. The proof is similar in spirit to thederivation of soft-gluon and soft-graviton theorems using on-shell gauge invariance at tree-level [2, 12].This work is organized as follows. We first clarify the relation between shift symmetryand Adler’s zeros in the next Section, and establish the four-point interaction satisfyingboth the Adler’s zero condition and the unbroken global symmetry group H . Derivation ofthe double soft theorems in NLSM for the full amplitude is presented in Section III. We endwith the Discussions section and also comment on how to recover the double soft theoremfor color-ordered partial amplitudes in Eq. (1) from our results. II. SHIFT SYMMETRY AND ADLER’S ZEROS
We start with a set of scalars π a furnishing a linear representation of an unbroken globalsymmetry group H , whose group generators T r satisfy the Lie algebra[ T r , T s ] = if rst T t . (6)Under an infinitesimal action of H , the scalars transform as π ( x ) a → π a ( x ) + iα r ( T r ) ab π b ( x ) , (7)where ( T r ) ab is the matrix entry of the generator T i in the particular representation underconsideration. Similar to Ref. [15], we adopt a basis where T r is purely imaginary andanti-symmetric, so that all scalar fields are taken to be real. This is equivalent to writing acomplex scalar in terms of its real and imaginary components.We further assume there is a set of shift symmetries acting on π a , π a → π a + (cid:15) a + · · · , (8)3here · · · contains higher order terms we ignore for now. Assuming the lagrangian is invari-ant under the shift symmetry, L [ π ] → L [ π ] for π a → π a + (cid:15) a , (9)a scalar mass term is forbidden and π a ’s are strictly massless.Without specifying any details of the theory, one can derive the Ward identity corre-sponding to the shift symmetry in the path integral formalism [16], which gives ∂ µ (cid:104) A aµ ( x ) π a ( x ) · · · π a n ( x n ) (cid:105) = i (cid:88) r (cid:104) π a ( x ) · · · π a r − ( x r − ) δ aa r δ (4) ( x − x r ) π a r +1 ( x r +1 ) · · · π a n ( x n ) (cid:105) , (10)where the Noether current corresponding to the shift symmetry is A aµ = δ L δ∂ µ π a . (11)If we take n = 1 in Eq. (10) and Fourier-transform with respect to π a ( x ), the Lehmann-Symanzik-Zimmermann (LSZ) reduction formula then implies ip (cid:104) | ∂ µ A aµ ( x ) | π a ( p ) (cid:105) = if π δ aa e − ip · x , (12)leading to the famous result (cid:104) | A aµ ( x ) | π a ( p ) (cid:105) = if π δ aa p µ e − ip · x . (13)In other words, the Noether current A aµ has a non-vanishing matrix element between vacuumand the one-particle state and, therefore, can create a one-particle pole for π a in the corre-lation functions. The dimensionful parameter f π plays the role of the pion decay constantin low-energy QCD. For n ≥
2, Eq. (10) and the LSZ reduction formula imply the currentconservation p µ (cid:104) f | ˜ A aµ ( p ) | i (cid:105) = 0 , (14)since the right-handed side of Eq. (10) contains only n − n scalar momenta are taken on-shell. As is familiar in low-energy theoremsfor pions, Eq. (13) together with the current conservation imply the single soft limit of thescattering amplitudes of π a ’s vanishes [17] :lim p µ → (cid:104) f + π a ( p ) | i (cid:105) = 0 , (15)4hich is the Adler’s zero condition.The shift symmetry in Eq. (8) implies the effective lagrangian must be derivatively cou-pled, and at lowest order it contains only the kinetic energy, L = 12 ∂ µ π a ∂ µ π a + · · · , (16)where terms neglected are higher dimensional operators containing derivative couplings.Beyond leading order, Ref. [15] proposed extending π a → π a + (cid:15) a to second order, π a → π a + (cid:15) a − f π ( T r ) ab ( T r ) cd π b π c (cid:15) d , (17)which form is dictated by the simple requirement of 1) invariance under the linearly realizedglobal symmetry group H and 2) it reduces to π a i → π a i + (cid:15) a i when all the other scalars π a , a (cid:54) = a i is turned off and set to zero, so as to fulfill the Adler’s zero condition for π a i . The lagrangian invariant under the second order shift symmetry in Eq. (17) is L = 12 ∂ µ π a ∂ µ π a − f π ( T r ) ab ( T r ) cd ∂ µ π a π b π c ∂ µ π d . (18)In fact, the full effective lagrangian for the NLSM, which is usually constructed using theformalism of CCWZ [18, 19], can be reproduced to all orders in 1 /f π , without specifying theunderlying coset G/H , if the following ”Closure Condition” is satisfied [15],( T i ) ab ( T i ) cd + ( T i ) ac ( T i ) db + ( T i ) ad ( T i ) bc = 0 . (19)This can be viewed as a consistency condition imposed on the low-energy effective theoryconstructed from the shift symmetry in the infrared. When comparing with the CCWZ for-malism based on a particular coset G/H , the the matrix element ( T i ) ab should be identifiedwith the structure constant of the broken group G in the ultraviolet,( T i ) ab = − if iab = Tr( T i [ X b , X a ]) (20)where T i and X a are generators of H and G/H , respectively. Then the Closure conditionis nothing but the Jacobi identity of the structure constants [15]. Eq. (19) is satisfied The numerical coefficient in the higher order term in Eq. (17) is arbitrary and can be absorbed into thenormalization of f π [15]. With a slight abuse of notation, ( T i ) ab in the left-hand side of Eq. (20) denotes the generator of H in therepresentation under which the scalars π a transform, while T i in the right-hand side of the equation sitsin the adjoint representation of the broken group G . SO ( N ). It is, however, not fulfilled by the fundamentalrepresentation of SU ( N ). In this case one need to enlarge SU ( N ) to SU ( N ) × U (1) thenthe Closure condition is met.At the order of 1 /f π , it is possible to work out the form of the dimension-six operator viaCCWZ to be Tr([ X a , X b ][ X c , X d ]) ∂π a π b π c ∂π d , thereby giving support to the identification( T i ) ab = − if iab . However, the major distinction is the shift symmetry only requires infrareddata on the group generators of the unbroken group H , while CCWZ requires specifying theultraviolet data such as the broken group G .Although the full CCWZ lagrangian for the nonlinear sigma model can be obtainedwithout knowledge of the underlying coset G/H , for the purpose of deriving the double softtheorems we only need the lagrangian, up to the order of 1 /f π , in Eq. (18). III. A DERIVATION OF THE DOUBLE SOFT THEOREMS
The double soft theorems considered in Ref. [11] are for NLSM with a U ( N ) color structureand when the two soft momenta are adjacent to each other in the color-ordered partialamplitudes. We will avoid both assumptions by working instead with the full amplitudes, i.e.the S-matrix elements. If we define M a a ··· a n ( p , p , · · · , p n ) to be the scattering amplitudesof n scalars, it is related to the color-ordered partial amplitudes M σ ( p , · · · , p n ) by M a a ··· a n ( p , p , · · · , p n ) = (cid:88) σ ∈ S n /Z n Tr( T a σ (1) T a σ (2) · · · T a σ ( n ) ) M σ ( p , · · · , p n ) , (21)where the sum is over all permutations of the n indices modulo cyclic permutations. Wealso follow the convention that all momenta are incoming. Moreover, M ( p , · · · , p n ) ≡ M σ ( p , · · · , p n ) for σ = the identity.As was demonstrated in the previous Section, shift symmetry imposes Adler’s zeros onthe amplitudes ∀ i , lim τ → M a a ··· a n ( p , p , · · · , τ p i , · · · , p n ) = 0 , (22)while invariance under the linearly realized global symmetry H requires ∀ r , n (cid:88) i =1 ( T r ) a i b M a ··· a i − ba i +1 ··· a n ( p , p , · · · , p n ) = 0 . (23)6he above constraints can be understood by considering M a ··· a n as a rank- n tensor of theunbroken group H . Using Eq. (7), it transforms under the action of H as M a ··· a n → M a (cid:48) ··· a (cid:48) n = n (cid:88) i =1 (cid:0) iα r ( T r ) a (cid:48) i b (cid:1) M a ··· a i − ba i +1 ··· a n ( p , p , · · · , p n ) . (24)Then Eq. (23) immediately follows by requiring M a ··· a n = M a (cid:48) ··· a (cid:48) n . Alternatively, it canbe derived from the Ward identity associated with the H invariance as well as the LSZreduction [9].The last ingredient we need for the derivation is the Feynman rule for the 4-point vertexin the effective lagrangian in Eq. (18), written using the shorthand notation s ij ≡ ( p i + p j ) , iV a a a a ( p , p , p , p ) == (cid:88) σ ∈ cycl f ( T i ) a a σ (2) ( T i ) a σ (3) a σ (4) ( s σ (4) − s σ (2) σ (4) + s σ (2) σ (3) − s σ (3) ) , (25)where we only sum over cyclic permutations of { , , } .Before considering the double soft limit, it is instructive to consider the single soft limitleading to the Adler’s zero. The single soft amplitude receives contributions from 1) the”pole diagram” where the soft leg is attached to one of the external hard legs and 2) the gutdiagram where the soft leg is attached to an internal line [17]. When the soft momentumis taken to zero, p µ = τ p µ and τ →
0, the pole diagram could potentially develop a softsingularity, because the propagator immediately following the soft leg can now go on-shellas τ →
0, 1( k + τ p ) − m = 12 k · p τ = O (cid:18) τ (cid:19) . (26)In theories with a shift symmetry the massless scalar must be derivatively coupled, whichmeans each coupling carries a positive power of momentum and could potentially cancelthe soft singularity, thereby yielding a finite contribution. For Nambu-Goldstone bosons,however, no cubic couplings exist. Then the Adler’s zero condition implies the gut diagramsmust vanish in the limit τ → For massless scalars with shift symmetry the cubic coupling must contain derivative. However all kinematicinvariants formed by three light-like momenta vanish. IG. 1:
An example of the pole diagram in double soft limit, where both soft legs are attached tothe same external hard leg. of diagrams where a pole in the propagator could potentially develop, while everything elsebelongs to the gut diagram which has no soft singularity. Thus the scattering amplitude of n + 2 scalars can be written as M a a ··· a n +2 ( p , p , · · · , p n +2 ) = N a a ··· a n +2 ( p , p , · · · , p n +2 )+ n (cid:88) i =1 (cid:102) M a i a n +1 a n +2 b ( p i , p n +1 , p n +2 , q i ) 1 q i (cid:102) M a ··· a i − ba i +1 ··· a n ( p , · · · , − q i , · · · p n ) , (27)where N a a ··· a n +2 represents the contribution from the gut diagrams while the pole diagramfactorizes into product of two semi-on-shell amplitudes, (cid:102) M , defined as scattering amplitudeswith one of the momenta taken off-shell. Momentum conservation implies q i = − ( p i + p n +1 + p n +2 ) and its associated propagator in the pole diagram becomes on-shell when p n +1 and p n +2 become soft simultaneously. The four-point semi-on-shell amplitude can be obtainedfrom the Feynman rule in Eq. (25), after using the notation T abcd = ( T r ) ab ( T r ) cd , (cid:102) M a i a n +1 a n +2 b ( p i , p n +1 , p n +2 , q i ) = 13 f π (cid:2) T a i a n +1 a n +2 b (cid:0) s ( n +1)( n +2) − s i ( n +2) (cid:1) + T a i a n +2 ba n +1 (cid:0) s ( n +1) i − s ( n +1)( n +2) (cid:1) + T a i ba n +1 a n +2 (cid:0) s ( n +2) i − s ( n +1) i (cid:1)(cid:3) , (28)We are interested in the limit p µn +1 → τ p µn +1 and p µn +2 → τ p µn +2 become soft as τ → p µn +1 → N a a ··· a n +2 ( p , · · · , p n , , p n +2 )= 13 f π n (cid:88) i =1 (cid:0) T a i a n +1 a n +2 b − T a i ba n +1 a n +2 (cid:1) (cid:102) M a ··· a i − ba i +1 ··· a n ( p , · · · , p i + p n +2 , · · · p n ) . (29)8f we further take p n +2 → N a a ··· a n +2 ( p , · · · , p n , , f π n (cid:88) i =1 (cid:0) T a i a n +1 a n +2 b − T a i ba n +1 a n +2 (cid:1) M a ··· a i − ba i +1 ··· a n ( p , · · · , p i , · · · p n )= 13 f π n (cid:88) i =1 T a i a n +1 a n +2 b M a ··· a i − ba i +1 ··· a n ( p , · · · , p i , · · · p n ) . (30)where the semi-on-shell n -point amplitude (cid:102) M a ··· a n now becomes the on-shell amplitude M a ··· a n , since all external momenta are now on-shell as both p n +1 and p n +2 are taken tozero. Notice the term containing T a i ba n +1 a n +2 = ( T r ) a i b ( T r ) a n +1 a n +2 vanishes due to theconstraints in Eq. (23), which arises from the unbroken global symmetry H . Similarly,letting p µn +2 → N a a ··· a n +2 ( p , · · · , p n , p n +1 , − f π n (cid:88) i =1 (cid:0) T a i a n +2 ba n +1 − T a i ba n +1 a n +2 (cid:1) (cid:102) M a ··· a i − ba i +1 ··· a n ( p , · · · , p i + p n +1 , · · · p n ) . (31)Again letting p n +2 → H group as well as the Closure condition in Eq. (19).Next we will take both p n +1 and p n +2 soft simultaneously. To compare with the proposeddouble soft theorems in Ref. [11], we will keep the propagator in 1 /q i in tact withoutexpanding in τ ,1 q i = 1( p i + τ p n +1 + τ p n +2 ) → τ ( s i ( n +1) + s i ( n +2) ) + τ s ( n +1)( n +2) . (32)On the other hand, we will only keep terms up to O ( τ ) in the semi-on-shell amplitudes.The contribution from the gut diagram gives, after expanding in power series in τ , N a ··· a n +2 ( p , · · · , p n +1 , p n +2 ) = N a ··· a n +2 ( p , · · · , p n , , τ p µn +1 ∂∂ ¯ p µn +1 (cid:12)(cid:12)(cid:12)(cid:12) ¯ p µn +1 =0 N a ··· a n +2 ( p , · · · , p n , ¯ p n +1 , τ p µn +2 ∂∂ ¯ p µn +2 (cid:12)(cid:12)(cid:12)(cid:12) ¯ p µn +2 =0 N a ··· a n +2 ( p , · · · , p n , , ¯ p n +2 ) . (33)After plugging in the conditions on N a ··· a n +2 in Eqs. (29) and (31), which are obtained from9equiring Adler’s zeros in the amplitudes, we arrive at N a ··· a n +2 ( p , · · · , p n +1 , p n +2 ) = 13 f π n (cid:88) i =1 T a i a n +1 a n +2 b M a ··· b ··· a n + τ f π n (cid:88) i =1 (cid:0) T a i a n +1 a n +2 b − T a i ba n +1 a n +2 (cid:1) p µn +2 ∂∂p µi M a ··· b ··· a n − τ f π n (cid:88) i =1 (cid:0) T a i a n +2 ba n +1 − T a i ba n +1 a n +2 (cid:1) p µn +1 ∂∂p µi M a ··· b ··· a n , (34)where again the semi-on-shell amplitudes (cid:102) M have been replaced by the n -point on-shellamplitudes M a ··· b ··· a n ≡ M a ··· b ··· a n ( p , · · · , p i , · · · , p n ), as all external momenta become on-shell after the expansion in τ .Similarly the n -point semi-on-shell amplitude in the contribution from the pole diagramin Eq. (27) can be expanded up to O ( τ ), (cid:102) M a ··· b ··· a n ( p , · · · , p i + p n +1 + p n +2 , · · · p n ) = M a ··· b ··· a n +( p n +1 + p n +2 ) µ ∂∂p µi M a ··· b ··· a n , (35)while the four-point semi-on-shell amplitude can be expanded in τ explicitly using Eq. (28).Putting everything together, we obtain the following double soft theorems: M a ··· a n +2 ( p , · · · , p n +2 ) = (cid:0) S (0) + S (1)sym + S (1)asym (cid:1) M a ··· b ··· a n ( p , p , · · · , p n ) , (36)where S (0) = n (cid:88) i =1 f ( T r ) a i b ( T r ) a n +1 a n +2 p i · ( p n +2 − p n +1 ) p i · ( p n +1 + p n +2 ) + p n +1 · p n +2 , (37) S (1)asym = n (cid:88) i =1 f ( T r ) a i b ( T r ) a n +1 a n +2 p νn +1 p µn +2 p i · ( p n +1 + p n +2 ) + p n +1 · p n +2 J µνi , (38) S (1)sym = n (cid:88) i =1 f (cid:2) ( T r ) a i a n +1 ( T r ) a n +2 b + ( T r ) a i a n +2 ( T r ) a n +1 b (cid:3) × p n +1 · p n +2 p i · ( p n +1 + p n +2 ) + p n +1 · p n +2 . (39)The angular momentum operator J µνi for the i th particle is defined in Eq. (4). IV. DISCUSSIONS
Having derived the leading and subleading double soft theorems in NLSM, we concludewith several comments and discussions: 10
The soft theorems for the NLSM are written entirely using infrared data: ( T i ) ab isthe generator of the unbroken global symmetry group in the IR, without referenceto the broken group in the UV. This reflects the fact that Nambu-Goldstone bosonsinterpolate the different degenerate vacua and their interactions encode the structureof the vacua. • We have only used the explicit form of the four-point interaction, together with theexistence of Adler’s zeros, to derive the soft theorems. This observation is fairly generaland should apply to other types of theories, including gluons and gravitons. In thesecases the cubic interaction will also enter, when each soft leg is connected to a differentexternal hard leg via three-point interactions. • We can uplift the soft theorems to the formalism of CCWZ by using the identificationin Eq. (20). Then the leading soft factor S (0) agrees with the well-known double soft-pion theorem [5–7], which is determined by the commutator of the two soft indices[ X a n +1 , X a n +2 ]. While the leading order soft factor S (0) is anti-symmetric in a n +1 and a n +2 , the next-to-leading soft factor contains both an anti-symmetric component S (1)asym and a symmetric component S (1)sym . • We derived the soft theorems for the full amplitudes, which include cases when thetwo soft legs are adjacent to each other, or when there is a hard leg sandwiched by thetwo soft legs. This can be seen by applying Eq. (20) to express the group-theoreticfactor in the soft factors in color-ordered form,( T r ) ab ( T r ) cd = Tr (cid:0) [ X a , X b ][ X c , X d ] (cid:1) (40)= Tr (cid:0) X a X b X c X d (cid:1) − Tr (cid:0) X a X b X d X c (cid:1) +Tr (cid:0) X b X a X d X c (cid:1) − Tr (cid:0) X a X a X d X c (cid:1) , (41)from which we see terms that are anti-symmetric in a n +1 and a n +2 arises from diagramswhere the two soft legs are adjacent to each other. The leading soft factor S (0) receivescontribution only from this class of diagrams. Diagrams where there is a hard legsandwiched between the soft legs are next-to-leading order in the soft expansion andonly contribute to S (1)sym . There is no contribution at this order in τ from other typesof configurations, consistent with the finding of Ref. [13] using recursion relations.11 Using the color decomposition in Eq. (40), we can also recover the result for color-ordered partial amplitudes in Eq. (1). More specifically, the partial amplitude M (1 , · · · , n + 2) receives contributions from diagrams similar to Fig. 1, where the( n +1) th and ( n +2) th legs are attached only to either the 1 st hard leg in the color-orderof { b ( n + 1)( n + 2)1 } , or the n th hard leg, in the color-order of { bn ( n + 1)( n + 2) } .Here the index b represents the ”color” of the off-shell leg carrying the momentum q i = − ( p n +1 + p n +2 + p i ). Singling out these two contributions in Eq. (37–39) repro-duces Eq. (1) exactly.Last but not least, it seems plausible that the same approach of zooming in on four-point (and three-point) couplings would allow one to derive the other double soft theoremsproposed in Ref. [11]. One obvious question is then the connection to other approachesfor deriving the soft theorems, which involve either the recursion relation or the scatteringequations. It would be interesting to clarify this connection in the future. In addition, itwould also be interesting to extend the present analysis to cases including spontaneouslybroken spacetime symmetry. Acknowledgments
The author would like to thank Cliff Cheung and Yu-tin Huang for comments on themanuscript. This work is supported in part by the U.S. Department of Energy under Con-tracts No. DE-AC02-06CH11357 and No. de-sc0010143. [1] F. E. Low, Phys. Rev. , 1428 (1954); M. Gell-Mann and M. L. Goldberger, Phys. Rev. ,1433 (1954).[2] F. E. Low, Phys. Rev. , 974 (1958); T. H. Burnett and N. M. Kroll, Phys. Rev. Lett. ,86 (1968).[3] S. Weinberg, Phys. Rev. , B1049 (1964); S. Weinberg, Phys. Rev. , B516 (1965);D. J. Gross and R. Jackiw, Phys. Rev. , 1287 (1968); R. Jackiw, Phys. Rev. , 1623(1968).[4] S. L. Adler, Phys. Rev. , B1022 (1965).
5] R. F. Dashen and M. Weinstein, Phys. Rev. , 1261 (1969).[6] S. Weinberg, Phys. Rev. Lett. , 879 (1966); S. Weinberg, Phys. Rev. D , 674 (1970).[7] N. Arkani-Hamed, F. Cachazo and J. Kaplan, JHEP , 016 (2010) [arXiv:0808.1446 [hep-th]].[8] A. Strominger, JHEP , 152 (2014) [arXiv:1312.2229 [hep-th]]; T. He, V. Lysov, P. Mi-tra and A. Strominger, JHEP , 151 (2015) [arXiv:1401.7026 [hep-th]]; F. Cachazo andA. Strominger, arXiv:1404.4091 [hep-th]; D. Kapec, V. Lysov, S. Pasterski and A. Strominger,JHEP , 058 (2014) [arXiv:1406.3312 [hep-th]].[9] K. Kampf, J. Novotny and J. Trnka, JHEP , 032 (2013) [arXiv:1304.3048 [hep-th]].[10] E. Casali, JHEP , 077 (2014) [arXiv:1404.5551 [hep-th]]; B. U. W. Schwab andA. Volovich, Phys. Rev. Lett. , no. 10, 101601 (2014) [arXiv:1404.7749 [hep-th]]; Z. Bern,S. Davies and J. Nohle, Phys. Rev. D , no. 8, 085015 (2014) [arXiv:1405.1015 [hep-th]]; S. He, Y. t. Huang and C. Wen, JHEP , 115 (2014) [arXiv:1405.1410 [hep-th]];A. J. Larkoski, Phys. Rev. D , no. 8, 087701 (2014) [arXiv:1405.2346 [hep-th]]; F. Cac-hazo and E. Y. Yuan, arXiv:1405.3413 [hep-th]; N. Afkhami-Jeddi, arXiv:1405.3533 [hep-th]; T. Adamo, E. Casali and D. Skinner, Class. Quant. Grav. , no. 22, 225008 (2014)[arXiv:1405.5122 [hep-th]]; Y. Geyer, A. E. Lipstein and L. Mason, Class. Quant. Grav. ,no. 5, 055003 (2015) [arXiv:1406.1462 [hep-th]]; B. U. W. Schwab, JHEP , 062 (2014)doi:10.1007/JHEP08(2014)062 [arXiv:1406.4172 [hep-th]]; M. Bianchi, S. He, Y. t. Huang andC. Wen, Phys. Rev. D , no. 6, 065022 (2015) [arXiv:1406.5155 [hep-th]]; J. Broedel, M. deLeeuw, J. Plefka and M. Rosso, Phys. Rev. D , no. 6, 065024 (2014) [arXiv:1406.6574[hep-th]]; C. D. White, Phys. Lett. B , 216 (2014) [arXiv:1406.7184 [hep-th]]; M. Zlot-nikov, JHEP , 148 (2014) [arXiv:1407.5936 [hep-th]]; C. Kalousios and F. Rojas, JHEP , 107 (2015) [arXiv:1407.5982 [hep-th]]; Y. J. Du, B. Feng, C. H. Fu and Y. Wang,JHEP , 090 (2014) [arXiv:1408.4179 [hep-th]]; H. Luo, P. Mastrolia and W. J. Tor-res Bobadilla, Phys. Rev. D , no. 6, 065018 (2015) [arXiv:1411.1669 [hep-th]]; J. Broedel,M. de Leeuw, J. Plefka and M. Rosso, Phys. Lett. B , 293 (2015) [arXiv:1411.2230 [hep-th]]; B. U. W. Schwab, JHEP , 140 (2015) [arXiv:1411.6661 [hep-th]]; W. M. Chen,Y. t. Huang and C. Wen, Phys. Rev. Lett. , no. 2, 021603 (2015) [arXiv:1412.1809 [hep-th]]; W. M. Chen, Y. t. Huang and C. Wen, JHEP , 150 (2015) [arXiv:1412.1811 [hep-th]]; A. J. Larkoski, D. Neill and I. W. Stewart, JHEP , 077 (2015) [arXiv:1412.3108 hep-th]]; A. Sabio Vera and M. A. Vazquez-Mozo, JHEP , 070 (2015) [arXiv:1412.3699[hep-th]]; C. Cheung, K. Kampf, J. Novotny and J. Trnka, Phys. Rev. Lett. , no.22, 221602 (2015) [arXiv:1412.4095 [hep-th]]; P. Di Vecchia, R. Marotta and M. Mojaza,JHEP , 137 (2015) [arXiv:1502.05258 [hep-th]]; A. E. Lipstein, JHEP , 166 (2015)[arXiv:1504.01364 [hep-th]]; T. Adamo and E. Casali, Phys. Rev. D , no. 12, 125022 (2015)[arXiv:1504.02304 [hep-th]]; T. Klose, T. McLoughlin, D. Nandan, J. Plefka and G. Travaglini,JHEP , 135 (2015) [arXiv:1504.05558 [hep-th]]; A. Volovich, C. Wen and M. Zlot-nikov, JHEP , 095 (2015) [arXiv:1504.05559 [hep-th]]; M. Bianchi and A. L. Guerri-eri, JHEP , 164 (2015) [arXiv:1505.05854 [hep-th]]; L. V. Bork and A. I. Onishchenko,arXiv:1506.07551 [hep-th]; P. Di Vecchia, R. Marotta and M. Mojaza, arXiv:1507.00938 [hep-th]; A. L. Guerrieri, arXiv:1507.08829 [hep-th]; S. Chin, S. Lee and Y. Yun, JHEP , 088(2015) [arXiv:1508.07975 [hep-th]]; A. Strominger, arXiv:1509.00543 [hep-th]; P. Di Vecchia,R. Marotta and M. Mojaza, arXiv:1511.04921 [hep-th]; A. Brandhuber, E. Hughes, B. Spenceand G. Travaglini, arXiv:1511.06716 [hep-th]; T. T. Dumitrescu, T. He, P. Mitra and A. Stro-minger, arXiv:1511.07429 [hep-th].[11] F. Cachazo, S. He and E. Y. Yuan, Phys. Rev. D , no. 6, 065030 (2015) [arXiv:1503.04816[hep-th]].[12] Z. Bern, S. Davies, P. Di Vecchia and J. Nohle, Phys. Rev. D , no. 8, 084035 (2014)[arXiv:1406.6987 [hep-th]].[13] Y. J. Du and H. Luo, JHEP , 058 (2015) [arXiv:1505.04411 [hep-th]].[14] F. Cachazo, S. He and E. Y. Yuan, JHEP , 149 (2015) [arXiv:1412.3479 [hep-th]].[15] I. Low, Phys. Rev. D , no. 10, 105017 (2015) [arXiv:1412.2145 [hep-th]].[16] See, for example, M. E. Peskin and D. V. Schroeder, “An Introduction to quantum fieldtheory,” Reading, USA: Addison-Wesley (1995) 842 p.[17] For an excellent review, see Chapter 3 in S. Coleman, “Aspects of Symmetry,” CambridgeUniversity Press (1985).[18] S. R. Coleman, J. Wess and B. Zumino, “Structure Of Phenomenological Lagrangians. 1,”Phys. Rev. (1969) 2239.[19] C. G. . Callan, S. R. Coleman, J. Wess and B. Zumino, “Structure Of PhenomenologicalLagrangians. 2,” Phys. Rev. (1969) 2247.(1969) 2247.