Topological Equivalence Theorem and Double-Copy for Chern–Simons Scattering Amplitudes

We study the mechanism of topological mass generation for 3-dimensional Chern–Simons gauge theories and propose a brand-new topological equivalence theorem to connect scattering amplitudes of the physical gauge boson states to that of the transverse states under high-energy expansion. We prove a general energy cancelation mechanism for N-point physical gauge boson amplitudes, which predicts large cancelations of E4 − L → E(4 − L) − N at any L-loop level (L ⩾ 0). We extend the double-copy approach to construct massive graviton amplitudes and to study their structures. We newly uncovered a series of strikingly large energy cancelations E12 → E1 of the tree-level 4-graviton scattering amplitude under high-energy expansion and establish a new correspondence between the 2 energy cancelations in the topologically massive Yang–Mills gauge theory and the topologically massive gravity theory. We further study the scattering amplitudes of Chern–Simons gauge bosons and gravitons in the nonrelativistic limit.


Introduction
The (2+1)-dimensional (3d) Chern-Simons (CS) theories naturally realize gauge-invariant (diffeomo-rphisminvariant) topological mass terms for gauge bosons and gravitons [1].Understanding the underlying mechanism of such topological mass-generations and how it determines the structure the massive gauge boson/graviton scattering amplitudes is important for applying the modern quantum field theories to particle physics and condensed matter physics [1][2] [3].
In this work, we study the dynamics of topological mass-generation for the 3d CS gauge and gravity theories [1].The 3d gauge fields can acquire gauge-invariant topological mass terms à la Chern-Simons [4] and without invoking the conventional Higgs mechanism [5].Adding the 3d CS term will convert the transverse polarization state of massless gauge boson into the physical polarization state of massive gauge boson, which conserves the physical degree of freedom (DoF) of a gauge boson: 1 = 1.For this, we propose a conceptually new Topological Equivalence Theorem (TET) to formulate the topological mass-generation at S-matrix level, which quantitatively connects the scattering amplitudes of the physical polarization states of massive gauge bosons to that of the corresponding transverse gauge bosons.This differs essentially from the conventional equivalence theorem (ET) [6] of the 4d Standard Model and from the Kaluza-Klein (KK) ET for the compactified 5d gauge theories [7][8] [9] and for the compactified 5d General Relativity [10] [11].
We newly develop a general 3d power counting method to count the leading energy dependence of scattering amplitudes in both the topologically massive Yang-Mills (TMYM) theory and topologically massive gravity (TMG).By using the TET identity and power counting method for TMYM theories, we uncover nontrivial energy cancellations among individual diagrams in the treelevel N -gauge boson amplitudes, E 4 → E 4−N , for N⩾ 4.
We will demonstrate that the TET provides a general mechanism to guarantee such highly nontrivial energy cancellations for the 3d massive gauge boson scattering amplitudes as well as the 3d massive graviton scattering amplitudes (through the double-copy construction).
With these, we extend the conventional double-copy approach and construct the massive four-graviton amplitudes of the TMG theory from the corresponding four-gauge boson amplitudes of the TMYM theory.The conventional double-copy method of Bern-Carrasco-Johansson (BCJ) [12] [13] applies to massless gauge/gravity theories and was inspired by the Kawai-Lewellen-Tye (KLT) [14] [15] relation which connects the product of open string amplitudes to that of the closed string at tree level.Some recent works attempted to extend the double-copy method to the 4d massive YM versus Fierz-Pauli-like massive gravity [16] [17] [18], to KK-inspired effective gauge theory with extra global U(1) [19], to the compactified 5d KK gauge/gravity theories [10] [11][20], and to the compactified KK bosonic string theory [21].There are also studies on the double-copy of 3d SUSY CS theroies in the massless limit [22] [23].The recent double-copy studies include the 3d CS gauge theories with or without mat-ter fields and the 3d covariant color-kinematics duality [24][25] [26][27].
Our extended double-copy construction of the 3d massive four-graviton amplitude from the 3d massive fourgauge-boson amplitude at tree level demonstrates strikingly large energy cancellations, E 12 → E 1 , in the highenergy graviton amplitude.With these we establish a new correspondence between the two types of distinctive energy cancellations in the massive gauge-boson amplitudes and the massive graviton amplitudes: E 4 → E 0 in the TMYM theory and E 12 → E 1 in the TMG theory.Finally, for possible applications to the condensed matter system, we further study the scattering amplitudes of Chern-Simons gauge bosons and gravitons in the nonrelativistic limit.

Topological Mass-Generation for Chern-Simons Gauge Theories
The 3d Abelian and non-Abelian CS gauge theories may be called the topologically massive QED (TMQED) and the topologically massive YM (TMYM), respectively.Their Lagrangians take the following forms: where (F µν , A µ ) = (F a µν , A a µ )T a and T a denotes the generator of the SU(N ) gauge group.The matter fields can be further added to the above Lagrangians when needed.We note that in Eq.(1) the gauge bosons acquire a topological mass m = | m| from the CS term, where the ratio s = m/m = ±1 corresponds to their spin [1] [2].Under a general gauge transformation, the action of TMQED theory is invariant up to a trivial surface term.While for the TMYM theory, the change of its action will contribute to a phase factor e i2πwn , where w ∈ Z represents the winding number which follows from the homotopy group Π 3 [SU(N )] ∼ = Z [3] and n is the CS level n = 4π m/g 2 ∈ Z .This ensures the phase factor e i2πwn = 1.
The on-shell gauge field has the plane wave solution x from the equation of motion (EOM), where p µ A a µ = 0 .Thus, the polarization vector obeys the following EOM: Since the CS term does not add any new field, the physical degrees of freedom of each gauge field A a µ is conserved before and after setting m = 0 limit [28][29], i.e., 1 = 1 .The conservation of the physical degrees of freedom of A a µ can be further understood from analyzing the (2+1)d little group [30] [31].
A 3d massive gauge boson in the rest frame has momentum pµ = (m, 0, 0), and its physical polarization vector is solved as can be boosted to ϵ µ P (p) for a general momentum p µ = E(1, βs θ , βc θ ) [31].We find that ϵ µ P (p) can be generally decomposed as: where ϵ µ T (ϵ µ L ) denotes the transverse (longitudinal) polarization vector, β = (1− Ē−2 ) 1/2 , Ē = E/m, and (s θ , c θ ) = (sinθ, cosθ).Hence, using Eq.(3), we can define the on-shell polarization states of the gauge field A a µ : where We note that the 3d massive gauge boson A a µ has 3 possible states in total, including 1 physical polarization state A a P and 2 unphysical polarization states (A a X , A a S ).In contrast, the massless gauge boson contains 1 physical transverse polarization A a T and 2 unphysical polarizations (A a L , A a S ) with ϵ µ L + ϵ µ S ∝ p µ .We observe that adding the CS term for A a µ field dynamically generates a new massive physical state A a P and converts its orthogonal combination A a X into unphysical state, whereas the scalar-polarization state A a S remains unphysical because it appears in the function F a ∝ A a S of the gauge-fixing term: We stress that one cannot naively take massless limit m → 0 for the massive CS theory because it causes the polarization vector ϵ µ L ∝ E/m → ∞ and thus ϵ µ P → ∞ , which makes the physical state A a P → ∞ and thus illdefined.Hence, the current analysis of the dynamics of the massive CS theory is highly nontrivial, from which we will establish a brand-new topological equivalence theorem (TET) in the next section.

Formulation of Topological Equivalence Theorem
The CS action from Eq.( 1) is gauge-invariant, and using the method of Refs.[32][6] we can derive a Slavnov-Taylor-type identity: where F a (p j ) = −ip µ j A a µ and the symbol Φ denotes any other on-shell physical fields after the Lehmann-Symanzik-Zimmermann (LSZ) amputation.Since the function F a contains only one single gauge field A a µ , it is straightforward to amputate each external F a line by the LSZ reduction, where we impose the on-shell condition p 2 j = −m 2 for each external line.From Eq.(3a) and the Energy cancellations for amplitude T [4A a P ] = T c +T s +T t +T u in the 3d TMYM theory, where the contribution of the contact channel is decomposed into three sub-amplitudes according to the color factors, T c = T cs + T ct + T cu .The energy factors are s = s/m 2 = 4 Ē2 and Ē = E/m , whereas for the angular dependence the notations are (s nθ , c nθ ) = (sinnθ, cosnθ).A common overall factor (g 2 /128) in each amplitude is not displayed for simplicity.
), we deduce the following identity: Using Eqs.(4c) and ( 7), we can reexpress the gauge-fixing function F a (p) = −imA a S as follows: where v a = v µ A a µ .Making the LSZ reduction on Eq.( 6) and combining it with Eq.( 8), we derive the following TET identity for the scattering amplitudes: where Ãa In the above, the residual term T v is suppressed by the factor v µ = O(m/E) ≪ 1 under high energy expansion.The TET identity (9a) states that the A a P -amplitude equals the corresponding A a T -amplitude in the high energy limit.We also observe that the right-hand side (RHS) of Eq.(9a) receives no multiplicative modification factor at loop level, because both A a P and A a T belong to the same gauge field A a µ .This feature differs from the conventional ET [32][33] for the SM Higgs mechanism.
Generalizing the previous power-counting method in 4d theories [34][35] and in 5d theories [10], we derive a new power-counting rule for the 3d CS gauge theories.For a given amplitude, we count the energy-dependence with the power: where (E, E A P , E v ) denote the numbers of the external lines, the external physical states A a P , and the external states with v µ factor, respectively.The V 3 is the number of cubic vertices containing no derivative (which arise from the non-Abelian CS term) and L stands for the number of loops.For the scattering amplitudes of pure gauge bosons (A a P ) with the number of external A a P states E = E A P = N and E v = 0 , we can use Eq.( 10) to deduce its leading individual contributions to be of O(E 4 ) at tree level.For the scattering amplitudes of pure A a T gauge bosons with the number of external . With these, our TET identity (9a) guarantees the energy cancellation in the N -gauge boson (A a P ) scattering amplitude on its left-hand side (LHS): E 4 → E 4−N .This is because on the RHS of Eq.(9a) the pure N -gauge boson A a T -amplitude scales as O(E 4−N ) and the residual term T v (with E v ⩾ 1) scales no more than O(E 3−N ).We can readily generalize this result up to L-loop level and deduce the following energy power cancellations: For the sake of later analysis, we also give the power counting rule on the high-energy leading E-dependence of graviton scattering amplitudes in the TMG theory [31]: where V d3 denotes the number of vertices containing 3 partial derivatives coming from the gravitational CS term in Eq.( 17) and E h P denotes the number of external physical graviton states h P .

Massive Gauge Boson Amplitudes and Energy Cancellations
In this section, we compute explicitly the four gauge boson scattering amplitudes ) in the 3d TMYM theory.They receive contributions from the contact diagram and the pole diagrams via (s, t, u) channels, as shown in the first row of Fig. 1.Using the power counting rule (10), we deduce that the high-energy leading contributions of T [4A a P ] and T [4A a T ] scale like E 4 and E 0 , respectively.Hence, using the TET identity (9a), we would predict the exact energy cancellations at O(E 4 , E 3 , E 2 , E 1 ) in the physical gauge-boson amplitude T [4A a P ], because it should match to the leading energy dependence of T [4A a T ] on the RHS of the TET identity (9a).Then, we compute the full four-point A a P -amplitude at tree level and present it in the following compact form: where the color factors and the explicit expressions of kinematic numerators (N s , N t , N u ) are given in the Supplementary Material [31].We make high energy expansion of the full A a P -amplitude in terms of 1/s or 1/s 0 , where s = s/m 2 , s0 = s 0 /m 2 , and s 0 = s − 4m 2 .Thus, we can explicitly demonstrate the exact energy cancellations at each order of E n (n = 4, 3, 2, 1), which are summarized in Table I.We find that the O(E 4 ) contributions cancel exactly between the contact diagram and the pole diagram in each channel of j = s, t, u .The sum of each O(E n ) contributions (with n = 3, 2, 1) cancels exactly because of the Jacobi identity holds, C s + C t + C u = 0.For comparison, we have performed a parallel analysis of the exact energy cancellations at O(E n ) (with n = 4, 3, 2, 1) under the high energy expansion of 1/s 0 , which are summarized in the Supplementary Material [31].
After all the high-energy cancellations, we systematically derive the leading nonzero scattering amplitudes of T [4A a P ] and T [4A a T ] at O(E 0 ) under the 1/s expansion, where (c θ , c 2θ ) = (cosθ, cos2θ).These two amplitudes differ by an amount: which vanishes identically due to the Jacobi identity.Hence, this demonstrates explicitly that the TET (9) holds in the high energy limit.For comparison, we further derive the leading nonzero gauge boson scattering amplitudes at O(E 0 ) under the 1/s 0 expansion: Inspecting Eqs.( 14)-( 15), we find that the two amplitudes of transverse gauge bosons are equal, T 0 [4A a T ] = T ′ 0 [4A a T ], whereas the two amplitudes of physical gauge bosons has , which vanishes identically due to the Jacobi identity.Hence, the leading nonzero amplitudes of O(E 0 ) are universal and independent of the high-energy expansion parameters (either 1/s or 1/s 0 ).
From the above analysis, we have well understood the structure of the four-gauge boson scattering amplitude (13) in the ultraviolet region.We have justified its energy cancellations order by order under the high energy expansion, at each O(E n ) with n = 4, 3, 2, 1, and have proved explicitly the TET (9a) at O(E 0 ).
Next, for possible applications to the condensed matter system and other low energy studies, we further analyze the nonrelativistic limit and make the low energy expansion of the four-point gauge boson scattering amplitudes (13).Thus, we derive the following expanded scattering amplitudes of gauge bosons at the leading order (LO) and next-to-leading order (NLO) of the low energy expansion: We see that under the nonrelativistic expansion at low energies, the LO scattering amplitude T 0 of physical gauge bosons scales as E 0 and the NLO amplitude δT scales as E 2 /m 2 .

Constructing Graviton Scattering Amplitude from Double-Copy
The conventional Einstein gravity in 3d has no physical content [1][36] [37], whereas the topologically massive gravity (TMG) includes the gravitational Chern-Simons (CS) action with which the graviton becomes massive and acquires a physical polarization state h P .The CS action of the TMG theory d 3 xL TMG contains the Lagrangian: where κ = 2/ M Pl is the gravitational coupling constant and M Pl denotes the Planck mass M Pl = 1/(8πG) with G being the Newton constant.
We note that the four-point physical gauge boson scattering amplitude ( 13) is invariant under the generalized gauge transformation: T in the TMYM theory, whereas Feynman diagrams in the second row (blue color) contribute to the fourgraviton scattering process h P h P → h P h P in the TMG theory.Both the gauge boson and graviton scattering processes contain the contributions from the contact diagrams and the (s, t, u) channels.
where the index j = s, t, u and the coefficient ∆ is an arbitrary function of kinematic variables.We find that the numerators {N j } of Eq.( 13) do not manifestly obey the kinematic Jacobi identity, namely, j N j ̸ = 0 .Then, we require the gauge-transformed numerators {N ′ j } to satisfy the Jacobi identity j N ′ j = 0 , with which we determine the coefficient ∆ as follows: With this, we present the full expressions of the gaugetransformed numerators {N ′ j } in Eq.(S23) of the Supplementary Material [31].Thus, from Eq.( 13) we can derive the following gauge-transformed new scattering amplitude: We find that each N ′ j scales as E 3 under high energy expansion, and thus each term in the gauge boson amplitude (13) (with numerators given by {N ′ j }) should scale as E 1 .Using the gauge-transformed numerators {N ′ j }, the individual terms of the amplitude (20) has leading contributions scale as E 1 instead of E 3 .We can verify the exact cancellation of the leading O(E 1 ) contributions by summing up them into the following form: which is proportional to the Jacobi identity and vanishes identically.This cancellation happens in a similar fashion as the last column of Table I, but the sum of all terms of last column of Table I gives a rather different coefficient (containing distinctive angular dependence): We have further verified that by using the amplitude (20) with the gauge-transformed numerators {N ′ j } and making high energy expansion, the nonzero leading contribution to the gauge boson scattering amplitude (13) takes the same form as that of Eq.( 14) at O(E 0 ).This supports our conclusion that the leading nonzero gauge boson anmplitudes at O(E 0 ) are universal, which are independent of the choice of the expansion parameters (such as 1/s or 1/s 0 ) and independent of the basis choice of numerators (N j or N ′ j as connected by the gauge transformations).Next, we use the power counting rule (12) to count the leading high-energy dependence of the four-graviton scattering amplitude at tree level.The four-graviton amplitude receives contributions from the contact diagram and the pole diagrams via (s, t, u) channels, as shown in the second row of Fig. 1.We find that the leading contributions of the individual Feynman diagrams to the physical graviton scattering amplitude scales as E 12 .However, using the extended double-copy approach, we will uncover a series of striking energy-cancellations in the fourgraviton scattering amplitudes, which make the summation of energy-dependent terms cancel all the way from O(E 12 ) down to O(E 1 ).
For this purpose, we extend the conventional massless double-copy method [12][13] to the case of TMYM theories.Applying the correspondence of the extended colorkinematics duality C j → N ′ j to the gauge-transformed four-point massive gauge boson amplitude (20), we construct the scattering amplitude of massive gravitons with physical polarization, M[h P h P → h P h P ] ≡ M[4h P ], as follows: where we have made the gauge-gravity coupling conversion g → κ/4 .We stress that, as a key point, the above double-copy construction must be applied directly to the full gauge-boson amplitude (20) without high energy expansion.Substituting the numerators (N ′ s , N ′ t , N ′ u ) [31] into Eq.(23), we derive the following exact tree-level scattering amplitude of massive gravitons: where in the numerator the (Q j , Qj ) are polynomial functions of the dimensionless Mandelstam valiable s = s/m 2 (= s0 + 4): The above massive graviton scattering amplitude can be also reexpressed in terms of the Mandelstam variable s 0 ( = s − 4m 2 ) which does not contain any massdependence, as shown in Sec. 4 of the Supplementary Material [31].
Then, we expand Eq.( 24) by the high energy expansion of 1/s and derive the four-graviton scattering amplitude at the leading order: which has the distinctive scaling of O(mE).We can also make the high energy expansion of 1/s 0 (with s 0 = s − 4m 2 ) and the LO graviton amplitude takes the same form as Eq.( 26) except the replacement s 1/2 → s 1/2 0 .From the LO graviton amplitude (26), we can derive its s-partial wave amplitude a 0 as follows: where we have added an angular cut on the scattering angle (δ ⩽ θ ⩽π −δ ) to remove the collinear divergences of the integral.We see that the above partial wave amplitudes have good high energy behaviors and remain finite in the high energy limit E→ ∞.Imposing the unitarity conditions |Re(a 0 )| < 1/2 and |Im(a 0 )| < 1 [38][39], we deduce the following constraints: which can be readily obeyed.This shows that the 3d TMG exhibits good ultraviolet (UV) behavior, unlike the conventional 3d Fierz-Pauli-type of massive gravity models.
We note that the leading individual terms of the numerators (N j , N ′ j ) scale as (E 5 , E 3 ) respectively [31], where the gauge transformation (18) causes the energy cancellations of E 5 → E 3 in each new numerator N ′ j .This has an important impact on the energy dependence of the double-copied graviton amplitude (23).Namely, in each channel, the amplitude N ′ 2 j /(s j −m 2 ) contains leading energy dependence behaving as E 4 , rather than E 8 from N 2 j /(s j − m 2 ) .In comparison with the leading energy-dependence of each individual contribution of the tree-level four-graviton amplitude which scales as E 12 by the direct power counting of individual Feynman diagrams, our double-copy construction (23) demonstrates that in each channel the graviton scattering amplitude could have the leading energy dependence of E 4 at most.Hence, the double-copy construction guarantees a series of large energy cancellations in the original fourgraviton scattering amplitude, E 12 → E 4 , which brings the leading E-dependence down by a large power factor of E 8 = E 4×2 .
In fact, we further discover a series of striking energy cancellations of E 4 → E 1 in the full graviton scattering amplitude (23), which rely on the sum of all three kinematic channels.We summarize these exact Ecancellations in Table II.We note that an S-matrix element S with E external states and L loops has massdimension D S = 3 − E/2 in the 3d spacetime [31].Thus, the four-point graviton scattering amplitude M[4h P ] in 3d has mass-dimension 1, and contains the gravity coupling κ 2 of mass-dimension −1.Hence, we can express the graviton scattering amplitude M[4h P ] = κ 2 M[4h P ], where M[4h P ] has mass-dimension 2 and is determined by the two dimensionful parameters (E, m).Thus we can deduce its scaling behavior M[4h P ] ∝ m n 1 E n 2 with n 1 +n 2 = 2 , under the high energy expansion.For the energy terms of E n 2 with n 2 = 4, 3, 2, we deduce its corresponding mass-power factor n 1 = −2, −1, 0 , respectively.This means that in the massless limit m→ 0 , the physical graviton amplitude M[4h P ] would go to infinity (for n 2 ⩾ 3) or remains constant (for n 2 = 2 ).However, we observe that in the massless limit m→ 0 , the 3d graviton field becomes unphysical and has no physical degrees of freedom [37].Hence the scattering amplitude M[4h P ] should vanish since the physical graviton h P no longer exists in this limit.This means that the m n 1 E n 2 terms with n 1 = −2, −1, 0 should vanish and the physical scattering amplitude M[4h P ] has to start with the leading behavior of m 1 E 1 , just as the behavior shown in Eq.( 26).This is why the energy cancellations should hold at each order of (E 4 , E Exact energy cancellations at each order of (E 4 , E 3 , E 2 ) in our double-copied four-graviton scattering amplitude (23).A common overall factor (κ 2 m 2 /2048) in each entry is not displayed for simplicity.
structure of the massive graviton amplitude (24).
Finally, for possible applications to the condensed matter system and other low energy studies, we analyze the nonrelativistic limit and make the low energy expansion of the double-copied four-graviton scattering amplitude (24).Thus, we derive the following LO and NLO scattering amplitudes of massive gravitons under the low energy expansion: ) where s ≪ m 2 .It shows that under nonrelativistic expansion, the LO graviton amplitude M 0 [4h P ] scales as E 0 m 2 and the NLO graviton amplitude δM[4h P ] behaves as E 2 m 0 .

Conclusions and Discussions
Studying the mechanism of topological mass-generations and its impact on the structure the massive gaugeboson/graviton scattering amplitudes in the 3d Chern-Simons theories is important for applying the modern quantum field theories to particle physics and condensed matter physics [1][2] [3].In this Letter, we systematically studied the high energy behaviors of the gauge-boson/graviton scattering amplitudes in the topologically massive Yang-Mills (TMYM) theory and the topologically massive gravity (TMG) theory [1].We found that making the high energy expansion uncovers large energy cancellations E 4 → E 4−N for each N -point massive gauge boson scattering amplitude.These energy cancellations are ensured by the topological equivalence theorem (TET) identity (9) as we newly proposed in Section 3.This is highly nontrivial because naively taking the massless limit would cause the (physical, longitudinal) polarization vectors in Eq.(3) diverge, (ϵ µ P , ϵ µ L ) → ∞, and thus make the physical state of the topologically massive gauge boson A a P ill-defined.The nontrivial and consistent approach is to take the high energy expansion for a fixed nonzero gauge boson mass m ̸ = 0 and prove the large energy cancellations by using the TET identity (9), as we demonstrated in Section 3.Moreover, we further extended the conventional massless double-copy approach to the present massive TMYM and TMG theories.We constructed the massive four-graviton scattering amplitude and uncovered its structure as in Eqs.( 23)-( 26) and Table II.
A key point is that the double-copy construction must be applied to the exact gauge boson amplitude (20) without high energy expansion.From these, we discovered a series of strikingly large energy cancellations in the fourpoint massive graviton scattering amplitude at tree level: for the 3d TMG theory.Our analysis has newly established a striking correspondence between the two types of distinctive energy cancellations of four-point massive scattering amplitudes: E 4 → E 0 in the TMYM theory and E 12 → E 1 in the TMG theory.In Eq.( 30), the exact energy cancellations in the four-graviton scattering amplitude by a large power of E 11 are even much more severe than the energy cancellations E 10 → E 2 in the massive four-longitudinal KK graviton scattering amplitudes of the compactified 5d gravity theory as found by explicit calculations [40][41] and by the KK double-copy construction [10][11].Our discovery of the striking energy cancellations of E 12 → E 1 newly demonstrates that the massive graviton scattering amplitudes in the 3d TMG theory have much better UV behavior than the naive expectation based on the conventional power counting of Feynman diagrams.This also encourages us to further establish the renormalizability of the TMG theory by extending our massive double-copy approach up to loop levels.For the possible applications to the condensed matter system and other low energy studies, we further presented the nonrelativistic scattering amplitudes of the massive gauge bosons in Eq.( 16) and of the massive gravitons in Eq. (29).A substantial extension of the main content of this Letter is presented in our companion long paper [42] (where the nonrelativistic scattering amplitudes are not shown).
where the structure constant appears in the commutator [T a , T b ] = iC abc T c , with T a denoting the generator of the gauge group SU(N ).

S3. Power Counting Method for 3d Chern-Simons Theories
Consider a scattering S-matrix element S having E external states and L loops (L ⩾ 0).Thus, the amplitude S has a mass-dimension [42]: where the number of external states E = E B + E F with E B and E F being the numbers of external bosonic and fermionic states, respectively.We denote the number of vertices of type-j as V j , where each vertex V j includes d j derivatives, b j bosonic lines, and f j fermionic lines.Then, the total mass-dimension of the energy-independent coupling constant in the amplitude S is given by For each Feynman diagram contributing to the amplitude S , we denote the number of the internal lines as I = I B +I F with I B ( I F ) being the number of the internal bosonic (fermionic) lines.Thus, we have the following general relations: where V = j V j is the total number of vertices in a given Feynman diagram.Hence, from Eqs.(S14)-(S16), we derive the leading energy-power dependence of the amplitude S as follows: Then, we note that the following relations must be obeyed: where V d denotes the number of all cubic vertices including one partial derivative and V 3 denotes the number of cubic vertices without partial derivative [10].With these, we can derive the following power counting rule on the leading energy dependence: where E A P denotes the number of external physical gauge bosons A a P (= ϵ µ P A a µ ) and E v represents the number of external gauge boson states contracted with the factor v µ .
For the topologically massive gravity (TMG) theory, we note that the leading graviton self-interaction vertex comes from the CS term which always contains 3 partial derivatives.Thus, for a given graviton vertex of this kind we have d j = 3 and f j = 0 in Eq.(S17), which lead to j V j d j = 3V d3 and V = V d3 in such leading diagrams, where V d3 is the number of vertices including 3 partial derivatives.Hence, the leading energy contribution to the pure graviton scattering amplitude in 3d spacetime is given by the Feynman diagrams including the CS graviton vertices with 3 derivatives, and thus is determined as follows: where E h P denotes the number of external physical graviton states h P (= ϵ µν P h µν ) and V d3 represents the number of vertices including 3 partial derivatives.For the tree-level diagrams, we have L = 0 and V d3 = E h P − 2 .Hence, we can further express the leading energy-power dependence (S20) as follows:

FIG. 1 .
FIG.1.Feynman diagrams in the first row (red color) contribute to the four-gauge boson scattering processesA a P A b P →A c P A d P and A a T A b T →A c T A dT in the TMYM theory, whereas Feynman diagrams in the second row (blue color) contribute to the fourgraviton scattering process h P h P → h P h P in the TMG theory.Both the gauge boson and graviton scattering processes contain the contributions from the contact diagrams and the (s, t, u) channels.
3 , E 2 ), in accord with what we have discovered in Table II by the explicit analysis of the energy