A note on the analytic structure of celestial amplitudes

Celestial amplitudes, obtained by applying Mellin transform and analytic continuation on “ordinary” amplitudes, have interesting properties which may provide useful insights on the underlying theory. Their analytic structures are thus of great interest and need to be better understood. In this paper, we critically examine the analytic structure of celestial amplitudes in a massless low-energy effective field theory. We find that, fixed-order loop contributions, which generate multipoles on the negative β-plane, in general do not provide an accurate description of the analytic structure of celestial amplitudes. By resumming over the leading logarithmic contributions using renormalization group equations (RGEs), we observe much richer analytic structures, which generally contain branch cuts. It is also possible to generate multipoles or shifted single poles if the RGEs satisfy certain relations. Including sub-leading logarithmic contributions is expected to introduce additional corrections to the picture. However, without a new approach, it is difficult to make a general statement since the analytic form of the Mellin transform is challenging to obtain.


Introduction
Celestial amplitudes, reinterpreting scattering amplitudes as correlators in the two dimensional conformal field theory (CFT) [1][2][3], have been extensively studied from various viewpoints, see e.g. refs. [4,5] for overviews of this rapidly growing field. Instead of the usual energy eigenstates, celestial amplitudes consider the scattering of boost eigenstates which have both UV and IR physics involved [6]. With this feature, celestial amplitudes violate the basic Wilsonian decoupling intuition and might provide a new route to probe physics in the UV. Celestial amplitudes also have interesting properties such as providing the correspondence between soft theorem of gauge theory and Ward identities in the celestial CFT [7][8][9][10][11][12][13][14], characterizing the infinite number of non-trivial symmetries of 4-dimensional gauge and gravitational theories in asymptotically flat spacetime [15] that may bring in new perspectives in understanding flat-space quantum field theory and the construction of consistent S-matrix. Recently in ref. [6], celestial amplitudes were applied to a general effective field theory (EFT) with Wilsonian cutoff, and their properties have been studied. Furthermore, with real boost weight β, specific dispersion relations for celestial amplitudes were established in ref. [16], relating the imaginary parts of the celestial amplitudes to their residues at negative even integer values of β. Great efforts are still required to fully understand properties of celestial amplitudes, especially how these properties manifest fundamental principles of quantum field theory.

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The celestial amplitude of a certain process for massless external particles can be obtained by taking the usual amplitude (with energy eigenstates) and performing a Mellin transformation on its energy [3,8,17]. More specifically, a massless 4-point scalar amplitude M is a function of Mandelstam variables s, t, u. With the relation s + t + u = 0, there are only two independent kinematic variables, which can be chosen as the center-of-mass energy ω and an angular variable z, given by 1 with 0 ≤ z ≤ 1. The celestial amplitude A in terms of the boost weight β and the angular variable z is then given by the Mellin transform of the amplitude M with respect to ω, Depending on the form of M(ω, z), the integral only converges in certain regions of β.
One could nevertheless perform analytic continuation on A(β, z) so that it is defined on the entire complex β-plane. It is the analytic structure of A(β, z) on the β-plane that we wish to examine in this paper. For amplitudes with spinning particles, additional factors are needed to characterize the little group scaling, while the celestial amplitudes are still obtained by the same procedure [3,6]. Several important observations were made in ref. [6] about the analytic structure of celestial amplitudes. The analytic structure in the negative (positive) β-plane generally corresponds to the physics in the deep IR (UV). In the deep IR, one could expand the amplitude in terms of ω and obtain an EFT. At tree level, it is simply given by a polynomial of ω, which gives simple poles at negative integers in the β-plane after Mellin transformation. As the residues of these simple poles are given by the corresponding Wilson coefficients, they satisfy various positivity constrains implied from causality and unitarity [18][19][20][21][22][23][24]. On the other hand, the analytic structure in the positive β-plane depends on the nature of the UV theory. Poles at positive integers are expected for a general field theory; while in quantum gravity, the positive β-plane is completely analytic. Loop contributions in general massless EFT can be expressed in terms of a series in log ω, which generates higher order poles in the negative β-plane.
Let us be more specific on the statements for EFTs. A 4-point amplitude in a general massless EFT can be written as where the coefficient c a,b are functions of z (which is not explicitly written). The parameters a and b correspond to the EFT expansion and the loop expansion, respectively. The EFT is only valid in the low energy region. As such, we will only look at the low energy contribution, cutting off the integration at some arbitrary energy ω 0 , assuming it is smaller JHEP08(2022)257 than the scale of the possible UV physics. Without loss of generality, we can set ω 0 = 1 and obtain 2 (1.4) which has a pole at −a of order b + 1 for any given a and b. Therefore, the order of the EFT expansion (parameterized by the power of ω) determines the position of the pole, which are at negative integer values on the β-plane, while the order of the loop correction (parameterized by maximum power of the log ω term) determines the order of the pole. Notice that the above integration has to include physics in the ω → 0 limit to manifest these pole structures. With any finite IR cut-off, taking β → −a gives finite A(β). Unfortunately, the statements above have a critical flaw: as shown in eq. (1.4), for each given a, the contribution to the celestial amplitude is given by a series of poles at β = −a in the form (β+a) b+1 . In the region sufficiently close to β = −a, the terms with higher order poles always dominate, and the perturbative loop expansion breaks down. In this case, the results at a fixed loop order generally would not capture the correct analytic structure. As a naive example, let us consider the Laurent series of the following expression with a small ξ, defined in the region outside the point −a + ξ, (1.5) which can be considered as a loop expansion with ξ being the loop suppression factor (∼ 1/16π 2 ). The left-hand side has a simple pole at −a + ξ, while the right-hand side contains a series of higher order poles at −a. The expansion breaks down in the region |β + a| < |ξ|, and fails to capture the analytic structure in this region! Indeed, this "non-perturbativity" is inherited from the amplitude in eq. (1.3), by noting that the analytic structure in the negative β-plane is dominated by the physics in the deep IR region, ω → 0. It is well known that, the log ω terms become sufficiently large in this region, and higher order terms in log ω are more important and need to be resummed. With resummation, indeed the large log problem is resolved, and the amplitude stays valid even in the ω → 0 limit, unless the couplings become sufficiently large at the IR to reach a confinement scale (e.g. as in QCD). As a first step, we will consider the resummation of leading log contributions using the method of renormalization group (RG), assuming no confinement in the IR. We will show that, at this order, in the massless EFT, the analytic structures of the celestial amplitude are already much richer than the multi-pole structure, and may contain shifted simple poles (as in the example above) or, more generally, branch cuts. A better understanding of these structures could thus provide important insights on the analytic structures of celestial amplitudes in general, and may eventually lead to a deeper understanding of quantum field theory. Of course, it is important to check the effects of sub-leading log contributions, i.e. whether they can generate new analytic structures. As we will discuss later, these contributions bring mathematical challenges in the calculation of celestial amplitudes. New approaches are likely needed to make further progress.

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The rest of this paper is organized as follows: we begin in section 2 by discussing the general structures of massless EFT amplitudes with resummation of the leading log contributions, focusing on the ω dependence which is most important for the celestial amplitudes. We then discuss in section 3 their corresponding celestial amplitudes below a certain threshold. With the simple case of considering one higher dimensional operator renormalized by a renormalizable operator, general properties of celestial amplitude are presented. The discussions are further extended to the most general case with operators at different dimensions. In section 4, we give an example of a four-scalar scattering amplitude, followed by an interesting case of a four-fermion scattering amplitude that exhibits accidental multipole structures at the leading-log order. Our conclusion is drawn in section 5.

Resummed EFT amplitudes
Resummation of leading log contributions can be efficiently done in EFTs via the RG running of operator coefficients (see [25] for a recent review). Starting from the UV theory, the standard procedure is to integrate out the heavy resonances and match it to the EFT at some matching scale M , which generates a set of operator coefficients. These coefficients are then RG run down to a lower scale of interest µ (e.g. the scale of experimental measurements). If the scale separation is large, the higher order terms in log µ M become non-negligible and need to be resumed. As we argued in the previous section, this resummation must be included in the amplitudes in order to obtain the accurate analytic structures of celestial amplitudes. The inclusion of the resummation in the EFT amplitudes follows from a well known procedure, facilitated by the use of RGEs. In the rest of the section, for completeness and establishing our notation, we review this procedure and derive the result in our context for later use.
Given an EFT Lagrangian, it is straightforward to calculate the amplitude of a process with field theory procedures. However, since our main objective is to Mellin transform an amplitude, it is more illustrative to adopt the on-shell approach and directly parameterize EFTs in terms of amplitudes [26][27][28][29][30][31]. Recently, it was shown that this approach also provides an efficient way of calculating the anomalous dimension matrices and general pattern of possible loop effects in EFTs [32][33][34][35][36]. In a massless EFT, the mapping between higher dimensional operators and the amplitudes is straightforward. We will again use a 4-point massless scalar amplitude for illustration. The generalization to other cases is discussed at the end of this section.
At tree level, the 4-point amplitude can be written as an expansion of the Mandelstam variables (assuming there is no 3-point renormalizable scalar interaction), where we have used the relation s + t + u = 0 to eliminate u. c 0 is a dimensionless coupling while c ) has mass dimension −2 (−4) and corresponds to dimension-6 (8) operator coefficient. In general, each independent kinematic term has an independent coefficient (with different labels on the superscripts), while symmetries may impose additional JHEP08(2022)257 relations among them. Writing M in terms of ω and z, we have (2.2) where for each order in the ω expansion, different kinematic terms are parameterized by different powers of −z. Note that we have chosen the superscripts of the c coefficients to match the power of −z.
As shown in eq. (1.3), loop contributions bring in log ω terms in the amplitudes. Instead of the log expansion in eq. (1.3), we assume the amplitude can be written as where c a are combinations of couplings that enter the loop (which has the same dimension as c (i) a ), and the ". . . " part contains additional terms that are independent of µ at one loop order, which include possible log(z) and log(1 − z) terms from loop kinematics. 3 We have also absorbed factors of 1/16π 2 into γ ij . At this point, eq. (2.3) with eq. (2.4) is simply eq. (1.3) truncated to the order b ≤ 1. However, writing in the form of eq. (2.4) makes it particularly convenient to derive the RGEs. Given that ω and z are kinematic variables (which can vary), for the physical amplitudes to be independent of µ, each c (i) a + γ ijc (j) a log ω µ + . . . term must be separately independent of µ, which gives a one-loop RGE for each c a . (2.5) Note that, the µ dependence inc (j) a can be neglected at the one loop order. These RGEs then capture the log ω dependence of the (re-summed) one loop contribution. In particular, the solution of eq. (2.5), when expanded to one loop order (i.e. not re-summed), is given by where µ 0 is some reference scale, often chosen to be the matching scale M where c 3 For instance, log −t µ 2 = log ω 2 µ 2 + log(z). Note that, in the forward (z → 0) or backward (z → 1) limit, the logarithmic factor log(z) or log(1 − z) diverges and the amplitude may contain IR divergences. Conventionally, we choose arguments of complex numbers to be in the interval (−π, π]. The corresponding branch of the complex logarithm has discontinuities all along the negative real axis.

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which is indeed independent of the renormalization scale µ at the one-loop order. As f a is independent of the renormalization scale order-by-order, we can set µ to specific value in eq. (2.4) and eq. (2.6) to arrive at eq. (2.7). To effectively resum potential large logs from the two separate scales M and ω, we set µ → g(z)ω to minimize the log term in eq. (2.4), and account for large logs in the running of c (i) a from M to ω. 4 With c (i) a (µ) given by the exact solution of the RGE eq. (2.5) instead of the one-loop result eq. (2.6), the amplitude is given by which reproduces eq. (2.7) when expanded to the (not resummed) one-loop order. Note that eq. (2.8) omits the rational loop contributions which are expected to be sub-leading. For a general massless EFT, the rational loop terms will depend on ω polynomially and will contribute to single pole structures of the celestial amplitude. When including nonvanishing mass for the light particles, amplitudes at one loop level also include branch-cuts for producing massive-particles at threshold. The analytical structure of the corresponding celestial amplitude is an interesting topic to be explored for the future. Eq. (2.8) is our master formula for calculating the RG-resumed amplitude.
It is straightforward to apply the above procedure to spinning particles. The amplitude, written in terms of ω and z, contains an additional factor of spinor products from the little group scaling. The detailed derivation of this factor can be found in appendix A and ref. [6]. A 4-point amplitude may also contain massless poles generated by on-shell 3-point amplitudes, though the existence of such 3-point amplitudes are subject to the operator dimensions and little group scalings. 5 In general, one could replace the (−z) i factors in eq. (2.8) with some more general rational functions of z. As long as one chooses a "non-redundant basis" for such z functions to parameterize the f a in eq. (2.4), the above derivation still holds and a RGE can be written down for each c

Celestial amplitude
With the prescription in eq. (2.8) we are now ready to calculate the leading-log resumed amplitude of a given theory and obtain its celestial amplitude. Let us start with the simplest nontrivial case in section 3.1, where an amplitude receives contributions from both a dimensionless coupling and an irrelevant coupling (Wilson coefficient). This case already contains all the essential features of the analytic structures. We then move on to a more complicated case in section 3.2 where a series of Wilson coefficients are considered.

Wilson coefficient running from dimensionless couplings
We consider the following 4-point amplitude with two running couplings λ(µ) and c a (µ) The factor g(z) parameterizes additional z dependence that may come from the loop kinematics. This extra term has been put in the ". . . " part in eq. (2.4) and eq. (2.7). 5 For instance, for particles with spin ≤ 1, the only on-shell 3-point amplitude at the level of dimension-6 operators is the 3-vector one with same helicities. See e.g. refs. [32,34] for more details.

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where for simplicity, we have omitted the possible power-law z dependences, which is not relevant for the following discussion. λ is a dimensionless coupling while c a has mass dimension −a. A typical example of this is the amplitude of a complex scalar φφ → φφ, where the only dimension-6 operator contribution is given by c 2 ω 2 due to the t ↔ u symmetry of the amplitude. Here, we will work in a general framework without referring to any particular model. The one-loop RGEs of λ(µ) and c a (µ) are given by where γ λ and γ c are determined by the particular theory and contain a loop factor (∼ 1/16π 2 ). Note that the form of these RGEs is dedicated by dimensional analysis. In a massless theory, λ(µ) does not receive contributions from c a . One could solve eq. (3.2) first, which gives where we have conveniently fixed the boundary condition at the matching scale, λ M ≡ λ(µ = M ). We will focus on the case with γ λ ≥ 0 so that the dimensionless coupling λ(µ) is finite in the IR region ω → 0 (i.e. no confinement). On the other hand, γ c can be either positive or negative, as the ω a term always dominates over any log divergences to make the EFT contribution well-behaved (i.e. irrelevant) in the IR. Let us first consider a special case in which γ λ → 0, i.e. λ does not run. The solution to eq. (3.3) in this case is given by where we have again fixed the boundary condition at the matching scale M (with c a M ≡ c a (µ = M )). Following the prescription in eq. (2.8), the RG-improved amplitude is given by where for simplicity we have set g(z) = 1, since it generates an overall factor which does not change the analytic structure of celestial amplitudes. The λ M contribution generates a simple pole at β = 0 in the celestial amplitude as in the tree-level case [6]. For the contribution of c a , we have

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where we have naturally chosen to cut off the integration at M as well. 6 The integration is obtained in the region β + a + γ c λ M > 0 where it converges, and analytically continued to the entire β-plane. We see that, the celestial amplitude has a simple pole at β = −a−γ c λ M , which is shifted from the tree-level simple pole at β = −a. This is exactly the case described by eq. (1.5). Without resummation, the series in log ω would generate a series of multipoles at β = −a, which does not capture the correct analytic structure. Note that the shifted-pole observed here for an IR finite amplitude is different from the one related to IR-divergent amplitudes as found in [6]. The case of nonzero γ λ is even more interesting. In this case, the solution to eq.
Note that, eq. (3.8) reduces to eq. (3.5) in the limit γ λ → 0 as expected. The RG-improved amplitude is given by where in the second line we have to simplify the expressions. As g(z) being an order-one factor, we will restrict ourselves to the case of ρ > 0.
Focusing on the c a contribution, the Mellin transformation of the amplitude is given by: where in the second line we have performed changes of variables t = log ω M and then t = (β +α)(ρ−t), and in the last line we used the definition of the incomplete Gamma function, Note again that the integration is obtained in the β > −a region and analytically continued to the entire β-plane. Properties of the incomplete Gamma function [37] lead to a few important observations.
In general, A(β) has a branch cut on (−∞, a]. Furthermore, we find that at β = −a, A(β) is finite (divergent) for r > 1 (r < 1), and logarithmically divergent for r = 1. This dependence on r is related to how fast c a goes to zero or diverges as ω → 0, as shown in eq. (3.8). More details of these results can be found in appendix B.
We also note that, for small non-zero γ λ , (the first line of) eq. (3.9) can be expanded in γ λ which reproduces eq. (3.6) with a series of higher orders terms in log ω, and the celestial amplitude has a series of multipoles at β = −a − γ c λ M . Expanding eq. (3.9) in terms of JHEP08(2022)257 j D l p v N L Z + j I K E P k R 9 J U q N F c / T 6 R k k C p a e C Z z o D o s f r t Z e J f X i / R v u O m P I w T z U K 6 W O Q n A u k I Z W + j I Z e M a j E 1 h F D J z a 2 I j o k k V J t w S i a E r 0 / R / 6 R 9 W r V x 1 b 4 5 r z Q u 8 z i K c A C H c A w 2 1 K A B 1 9 C E F l D w 4 Q G e 4 N m a W I / W i / W 6 a C 1 Y + c w + / I D 1 9 g m C x o 1 X < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " N R g F o B 2 w J k E G 6 C U G n q 6 D / v d 7 p X 0 = " > A A A B 6 X i c d V D J S g N B E K 2 J W 4 x b 1 K O X x i B 4 M f S 4 k M k t 6 M V j F L N A M o S e T k / S p G e h u 0 c I Q / 7 A i w d F v P p H 3 v w b e 5 I R V P R B w e O 9 K q r q e b H g S m P 8 Y R W W l l d W 1 4 r r p Y 3 N r e 2 d 8 u 5 e W 0 W J p K x F I x H J r k c U E z x k L c 2 1 Y N 1 Y M h J 4 g n W 8 y V X m d + 6 Z V D w K 7 / Q 0 Z m 5 A R i H 3 O S X a S L c n Z F C u 4 G r d u a j X b I S r e I 6 M n D n Y d p C d K x X I 0 R y U 3 / v D i C Y B C z U V R K m e j W P t p k R q T g W b l f q J Y j G h E z J i P U N D E j D l p v N L Z + j I K E P k R 9 J U q N F c / T 6 R k k C p a e C Z z o D o s f r t Z e J f X i / R v u O m P I w T z U K 6 W O Q n A u k I Z W + j I Z e M a j E 1 h F D J z a 2 I j o k k V J t w S i a E r 0 / R / 6 R 9 W r V x 1 b 4 5 r z Q u 8 z i K c A C H c A w 2 1 K A B 1 9 C E F l D w 4 Q G e 4 N m a W I / W i / W 6 a C 1 Y + c w + / I D 1 9 g m C x o 1 X < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " N R g F o B 2 w J k E G 6 C U G n q 6 D / v d 7 p X 0 = " > A A A B 6 X i c d V D J S g N B E K 2 J W 4 x b 1 K O X x i B 4 M f S 4 k M k t 6 M V j F L N A M o S e T k / S p G e h u 0 c I Q / 7 A i w d F v P p H 3 v w b e 5 I R V P R B w e O 9 K q r q e b H g S m P 8 Y R W W l l d W 1 4 r r p Y 3 N r e 2 d 8 u 5 e W 0 W J p K x F I x H J r k c U E z x k L c 2 1 Y N 1 Y M h J 4 g n W 8 y V X m d + 6 Z V D w K 7 / Q 0 Z m 5 A R i H 3 O S X a S L c n Z F C u 4 G r d u a j X b I S r e I 6 M n D n Y d p C d K x X I 0 R y U 3 / v D i C Y B C z U V R K m e j W P t p k R q T g W b l f q J Y j G h E z J i P U N D E j D l p v N L Z + j I K E P k R 9 J U q N F c / T 6 R k k C p a e C Z z o D o s f r t Z e J f X i / R v u O m P I w T z U K 6 W O Q n A u k I Z W + j I Z e M a j E 1 h F D J z a 2 I j o k k V J t w S i a E r 0 / R / 6 R 9 W r V x 1 b 4 5 r z Q u 8 z i K c A C H c A w 2 1 K A B 1 9 C E F l D w 4 Q G e 4 N m a W I / W i / W 6 a C 1 Y + c w + / I D 1 9 g m C x o 1 X < / l a t e x i t >

l a t e x i t s h a 1 _ b a s e 6 4 = " x f r N S j w B b D b 7 r m G t m X a 4 i m t I f W k = " > A A A C F X i c d Z D N S g M x F I U z / l v / R l 2 6 C R b B h Z Y Z U d o u B N G N y w p W i 5 1 a 7 q S Z N j T J D E l G K E N f w o 2 v 4 s a F I m 4 F d 7 6 N m V p B R Q 8 E P s 6 9 N 8 k 9 Y c K Z N p 7 3 7 k x M T k 3 P z M 7 N F x Y W l 5 Z X 3 N W 1 C x 2 n i t A 6 i X m s G i F o y p m k d c M M p 4 1 E U R A h p 5 d h / y S v X 9 5 Q p V k s z 8 0 g o S 0 B X c k i R s B Y q + 3 u K H w Y R A p I F n R B C G i T 4 R c F 3 F 7 T g W E g Y 8 M k v r r e b b t F r + R 5 / o G 3 h 3 P I Z a F a L V c P K t g f O 0 U 0 V q 3 t v g W d m K S C S k M 4 a N 3 0 v c S 0 M l C G E U 6 H h S D V N A H S h y 5 t W p Q g q G 5 l o 6 2 G e M s 6 H R z F y h 5 p 8 M j 9 P p G B 0 H o g Q t s p w P T 0 7 1 p u / l V r p i a q t D I m k 9 R Q S T 4 f i l K O T Y z z i H C H K U o M H 1 g A o p j 9 K y Y 9 s B k Z G 2 T B h v C 1 K f 4 f L v Z K v l f y z / a L R 8 f j O O b Q B t p E 2 8 h H Z X S E T l E N 1 R F B t + g e P a I n 5 8 5 5 c J 6 d l 8 / W C W c 8 s 4 5 + y H n 9 A C 7 f n 3 Y = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " x f r N S j w B b D b 7 r m G t m X a 4 i m t I f W k = " > A A A C F X i c d Z D N S g M x F I U z / l v / R l 2 6 C R b B h Z Y Z U d o u B N G N y w p W i 5 1 a 7 q S Z N j T J D E l G K E N f w o 2 v 4 s a F I m 4 F d 7 6 N m V p B R Q 8 E P s 6 9 N 8 k 9 Y c K Z N p 7 3 7 k x M T k 3 P z M 7 N F x Y W l 5 Z X 3 N W 1 C x 2 n i t A 6 i X m s G i F o y p m k d c M M p 4 1 E U R A h p 5 d h / y S v X 9 5 Q p V k s z 8 0 g o S 0 B X c k i R s B Y q + 3 u K H w Y R A p I F n R B C G i T 4 R c F 3 F 7 T g W E g Y 8 M k v r r e b b t F r + R 5 / o G 3 h 3 P I Z a F a L V c P K t g f O 0 U 0 V q 3 t v g W d m K S C S k M 4 a N 3 0 v c S 0 M l C G E U 6 H h S D V N A H S h y 5 t W p Q g q G 5 l o 6 2 G e M s 6 H R z F y h 5 p 8 M j 9 P p G B 0 H o g Q t s p w P T 0 7 1 p u / l V r p i a q t D I m k 9 R Q S T 4 f i l K O T Y z z i H C H K U o M H 1 g A o p j 9 K y Y 9 s B k Z G 2 T B h v C 1 K f 4 f L v Z K v l f y z / a L R 8 f j O O b Q B t p E 2 8 h H Z X S E T l E N 1 R F B t + g e P a I n 5 8 5 5 c J 6 d l 8 / W C W c 8 s 4 5 + y H n 9 A C 7 f n 3 Y = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " x f r N S j w B b D b 7 r m G t m X a 4 i m t I f W k = " > A A A C F X i c d Z D N S g M x F I U z / l v / R l 2 6 C R b B h Z Y Z U d o u B N G N y w p W i 5 1 a 7 q S Z N j T J D E l G K E N f w o 2 v 4 s a F I m 4 F d 7 6 N m V p B R Q 8 E P s 6 9 N 8 k 9 Y c K Z N p 7 3 7 k x M T k 3 P z M 7 N F x Y W l 5 Z X 3 N W 1 C x 2 n i t A 6 i X m s G i F o y p m k d c M M p 4 1 E U R A h p 5 d h / y S v X 9 5 Q p V k s z 8 0 g o S 0 B X c k i R s B Y q + 3 u K H w Y R A p I F n R B C G i T 4 R c F 3 F 7 T g W E g Y 8 M k v r r e b b t F r + R 5 / o G 3 h 3 P I Z a F a L V c P K t g f O 0 U 0 V q 3 t v g W d m K S C S k M 4 a N 3 0 v c S 0 M l C G E U 6 H h S D V N A H S h y 5 t W p Q g q G 5 l o 6 2 G e M s 6 H R z F y h 5 p 8 M j 9 P p G B 0 H o g Q t s p w P T 0 7 1 p u / l V r p i a q t D I m k 9 R Q S T 4 f i l K O T Y z z i H C H K U o M H 1 g A o p j 9 K y Y 9 s B k Z G 2 T B h v C 1 K f 4 f L v Z K v l f y z / a L R 8 f j O O b Q B t p E 2 8 h H Z X S E T l E N 1 R F B t + g e P a I n 5 8 5 5 c J 6 d l 8 / W C W c 8 s 4 5 + y H n 9 A C 7 f n 3 Y = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " h P + 6 L r U f 2 d 3 t Z a l d q a Q Q v E K M X y w = " > A A A B 2 X i c b Z D N S g M x F I X v 1 L 8 6 V q 1 r N 8 E i u C o z b n Q p u H F Z w b Z C O 5 R M 5 k 4 b m s k M y R 2 h D H 0 B F 2 5 E f C 9 3 v o 3 p z 0 J b D w Q + z k n I v S c u l L Q U B N 9 e b W d 3 b / + g f u g f N f z j k 9 N m o 2 f z 0 g j s i l z l 5 j n m F p X U 2 C V J C p 8 L g z y L F f b j 6 f 0 i 7 7 + g s T L X T z Q r M M r 4 W M t U C k 7 O 6 o y a r a A d L M W 2 I V x D C 9 Y a N b + G S S 7 K D D U J x a 0 d h E F B U c U N S a F w 7 g 9 L i w U X U z 7 G g U P N M 7 R R t R x z z i 6 d k 7 A 0 N + 5 o Y k v 3 9 4 u K Z 9 b O s t j d z D h N 7 G a 2 M P / L B i W l t 1 E l d V E S a r H 6 K C 0 V o 5 w t d m a J N C h I z R x w Y a S b l Y k J N 1 y Q a 8 Z 3 H Y S b G 2 9 D 7 7 o d B u 3 w M Y A 6 n M M F X E E I N 3 A H D 9 C B L g h I 4 B X e v Y n 3 5 n 2 s u q p 5 6 9 L O 4 I + 8 z x 8 4 x I o 4 < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " c z 8 u t z t i y t n R + B Y M Y C X d / R V I M 7 A = " > A A A C C n i c b Z D L S g M x G I X / 8 W 6 9 j W 7 d B E V w o W X G j W 4 E w Y 1 L B d u K n X H 4 J 8 2 0 o U l m S D J C G f o S b n w V N y 4 U 8 Q H c + T a m t Y K 3 A 4 G P c 3 L 7 T 1 o I b m w Q v H t T 0 z O z c / M L i 7 W l 5 Z X V N X 9 9 u W n y U l P W o L n I 9 V W K h g m u W M N y K 9 h V o R n K V L B W 2 j 8 d 5 a 1 b p g 3 P 1 a U d F C y W 2 F U 8 4 x S t s x J / T 5 P j K N N I q 6 i L U m J C h 1 8 U C X d N B 4 e R y i 1 X 5 P p m P / G 3 g 3 o w F v k L 4 Q S 2 Y a L z x H + L O j k t J V O W C j S m H Q a F j S v U l l P B h r W o N K x A 2 s c u a z t U K J m J q / F U Q 7 L j n A 7 J c u 2 W s m T s f j 9 R o T R m I F O 3 U 6 L t m d / Z y P w v a 5 c 2 O 4 o r r o r S M k U / H 8 p K Q W x O R h W R D t e M W j F w g F R z 9 1 d C e + g 6 s q 7 I m i s h / D 3 y X 2 g e 1 M O g H l 4 E s A C b s A W 7 E M I h n M A Z n E M D K N z B A z z B s 3 f v P X o v n 3 V N e Z P e N u C H v N c P C y S d v w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " P u 0 V w v q 2 Y H B U s T e r g g H R 9 K L k 8 3 8 = " > A A A C C n i c d Z D L S g M x G I X / 8 W 6 9 V b d u g i K 4 0 J I R R L s Q B D c u K 1 g V O 7 X 8 k 2 Z q M M k M S U Y o Q 1 / C j a / i x o U i P o A 7 3 8 Z M r a C i B w I f 5 + T 2 n z i T w j p K 3 4 O x 8 Y n J q e m Z 2 c r c / M L i U n V 5 / s y m u W G 8 y V K Z m o s Y L Z d C 8 6 Y T T v K L z H B U s e T n 8 c 1 R m Z / f c m N F q k 9 d P + N t h T 0 t E s H Q e a t T 3 T L k I E o M s i L q o V L Y Y Y M v i q S / p o u D S K d O a H J 5 t d 2 p r t M a p e E u 3 S E l l P J Q r + / V d / d J O H L W Y a R G p / o W d V O W K 6 4 d k 2 h t K 6 S Z a x d o n G C S D y p R b n m G 7 A Z 7 v O V R o + K 2 X Q y n G p A N 7 3 R J k h q / t C N D 9 / u J A p W 1 f R X 7 n Q r d t f 2 d l e Z f W S t 3 y X 6 7 E D r L H d f s 8 6 E k l 8 S l p K y I d I X h z M m + B 2 R G + L 8 S d o 2 + I + e L r P g S v i Y l / 8 P Z T i 2 k t f C E w g y s w h p s Q g h 7 c A j H 0 I A m M L i D B 3 i C 5 +
A + e A x e P u s a C 0 a 9 r c A P B a 8 f V w K d 9 g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " z 0 F 4 1 f L + A n / L 0 p v r 6 j 6 N 6 P 5 u x h w = " > A A A C F X i c d Z D N S g M x F I U z / t b 6 V 3 X p J l g E F 1 o y g m g X g u j G Z Q X b i p 1 a 7 q S Z G k w y Q 5

I R y t C X c O O r u H G h i F v B n W 9 j p r a g o g c C H + f e m + S e M B H c W E I + v I n J q e m Z 2 c J c c X 5 h c W m 5 t L L a M H G q K a v T W M T 6 I g T D B F e s b r k V 7 C L R D G Q o W D O 8 O c n r z V u m D Y / V u e 0 n r C 2 h p 3 j E K V h n d U r b G h 8 G k Q a a B T 2 Q E j p 0 M K Z A u G u 6 M
A h U b L n C l 1 c 7 n V K Z V A j x 9 8 g u z i G X g 2 p 1 v 7 p 3 g P 2 R U 0 Y j 1 T q l 9 6 A b 0 1 Q y Z a k A Y 1 o + S W w 7 A 2 0 5 F W x Q D F L D E q A 3 0 G M t h w o k M + 1 s u N U A b z q n i 6 N Y u 6 M s H r r f J z K Q x v R l 6 D o l 2 G v z u 5 a b f 9 V a q Y 0 O 2 h l X S W q Z o l 8 P R a n A N s Z 5 R L j L N a N W 9 B 0 A 1 d log ω M instead, one then recovers the series in eq. (1.3) with the celestial amplitude having multipoles at β = −a. In both cases, the expanded celestial amplitude could not reproduce the analytic structure of the original one. As explained earlier, this is because the log ω M series diverges in the deep IR ω → 0, which the analytic structures of celestial amplitudes are sensitive to.

z 9 F d N r c B l Z F 2 T R h T D e F P 8 P j d 2 K T y r + G S k f H Y / i K K B 1 t I G 2 k I / 2 0 R E 6 R T V U R x T d o Q f 0 h J 6 9 e + / R e / F e v 1 o n v N H M G v o h 7 + 0 T L Z + f c g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " x f r N S j w B b D b 7 r m G t m X a 4 i m t I f W k = " > A A A C F X i c d Z D N S g M x F I U z / l v / R l 2 6 C R b B h Z Y Z U d o u B N G N y w p W i 5 1 a 7 q S Z N j T J D E l G K E N f w o 2 v 4 s a F I m 4 F d 7 6 N m V p B R Q 8 E P s 6 9 N 8 k 9 Y c K Z N p 7 3 7 k x M T k 3 P z M 7 N F x Y W l 5 Z X 3 N W 1 C x 2 n i t A 6 i X m s G i F o y p m k d c M M p 4 1 E U R A h p 5 d h / y S v X 9 5 Q p V k s z 8 0 g o S 0 B X c k i R s B Y q + 3 u K H w Y R A p I F n R B C G i T 4 R c F 3 F 7 T g W E g Y 8 M k v r r e b b t F r + R 5 / o G 3 h 3 P I Z a F a L V c P K t g f O 0 U 0 V q 3 t v g W d m K S C S k M 4 a N 3 0 v c S 0 M l C G E U 6 H h S D V N A H S h y 5 t W p Q g q G 5 l o 6 2 G e M s 6 H R z F y h 5 p 8 M j 9 P p G B 0 H o g Q t s p w P T 0 7 1 p u / l V r p i a q t D I m k 9 R Q S T 4 f i l K O T Y z z i H C H K U o M H 1 g A o p j 9 K y Y 9 s B k Z G 2 T B h v C 1 K f 4 f L v Z K v l f y z / a L R 8 f j O O b Q B t p E 2 8 h H Z X S E T l E N 1 R F B t + g e P a I n 5 8 5 5 c J 6 d l 8 / W C W c 8 s 4 5 + y H n 9 A C 7 f n 3 Y = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " x f r N S j w B b D b 7 r m G t m X a 4 i m t I f W k = " > A A A C F X i c d Z D N S g M x F I U z / l v / R l 2 6 C R b B h Z Y Z U d o u B N G N y w p W i 5 1 a 7 q S Z N j T J D E l G K E N f w o 2 v 4 s a F I m 4 F d 7 6 N m V p B R Q 8 E P s 6 9 N 8 k 9 Y c K Z N p 7 3 7 k x M T k 3 P z M 7 N F x Y W l 5 Z X 3 N W 1 C x 2 n i t A 6 i X m s G i F o y p m k d c M M p 4 1 E U R A h p 5 d h / y S v X 9 5 Q p V k s z 8 0 g o S 0 B X c k i R s B Y q + 3 u K H w Y R A p I F n R B C G i T 4 R c F 3 F 7 T g W E g Y 8 M k v r r e b b t F r + R 5 / o G 3 h 3 P I Z a F a L V c P K t g f O 0 U 0 V q 3 t v g W d m K S C S k M 4 a N 3 0 v c S 0 M l C G E U 6 H h S D V N A H S h y 5 t W p Q g q G 5 l o 6 2 G e M s 6 H R z F y h 5 p 8 M j 9 P p G B 0 H o g Q t s p w P T 0 7 1 p u / l V r p i a q t D I m k 9 R Q S T 4 f i l K O T Y z z i H C H K U o M H 1 g A o p j 9 K y Y 9 s B k Z G 2 T B h v C 1 K f 4 f L v Z K v l f y z / a L R 8 f j O O b Q B t p E 2 8 h H Z X S E T l E N 1 R F B t + g e P a I n 5 8 5 5 c J 6 d l 8 / W C W c 8 s 4 5 + y H n 9 A C 7 f n 3 Y = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " x f r N S j w B b D b 7 r m G t m X a 4 i m t I f W k = " > A A A C F X i c d Z D N S g M x F I U z / l v / R l 2 6 C R b B h Z Y Z U d o u B N G N y w p W i 5 1 a 7 q S Z N j T J D E l G K E N f w o 2 v 4 s a F I m 4 F d 7 6 N m V p B R Q 8 E P s 6 9 N 8 k 9 Y c K Z N p 7 3 7 k x M T k 3 P z M 7 N F x Y W l 5 Z X 3 N W 1 C x 2 n i t A 6 i X m s G i F o y p m k d c M M p 4 1 E U R A h p 5 d h / y S v X 9 5 Q p V k s z 8 0 g o S 0 B X c k i R s B Y q + 3 u K H w Y R A p I F n R B C G i T 4 R c F 3 F 7 T g W E g Y 8 M k v r r e b b t F r + R 5 / o G 3 h 3 P I Z a F a L V c P K t g f O 0 U 0 V q 3 t v g W d m K S C S k M 4 a N 3 0 v c S 0 M l C G E U 6 H h S D V N A H S h y 5 t W p Q g q G 5 l o 6 2 G e M s 6 H R z F y h 5 p 8 M j 9 P p G B 0 H o g Q t s p w P T 0 7 1 p u / l V r p i a q t D I m k 9 R Q S T 4 f i l K O T Y z z i H C H K U o M H 1 g A o p j 9 K y Y 9 s B k Z G 2 T B h v C 1 K f 4 f L v Z K v l f y z / a L R 8 f j O O b Q B t p E 2 8 h H Z X S E T l E N 1 R F B t + g e P a I n 5 8 5 5 c J 6 d l 8 / W C W c 8 s 4 5 + y H n 9 A C 7 f n 3 Y = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " x f r N S j w B b D b 7 r m G t m X a 4 i m t I f W k = " > A A A C F X i c d Z D N S g M x F I U z / l v / R l 2 6 C R b B h Z Y Z U d o u B N G N y w p W i 5 1 a 7 q S Z N j T J D E l G K E N f w o 2 v 4 s a F I m 4 F d 7 6 N m V p B R Q 8 E P s 6 9 N 8 k 9 Y c K Z N p 7 3 7 k x M T k 3 P z M 7 N F x Y W l 5 Z X 3 N W 1 C x 2 n i t A 6 i X m s G i F o y p m k d c M M p 4 1 E U R A h p 5 d h / y S v X 9 5 Q p V k s z 8 0 g o S 0 B X c k i R s B Y q + 3 u K H w Y R A p I F n R B C G i T 4 R c F 3 F 7 T g W E g Y 8 M k v r r e b b t F r + R 5 / o G 3 h 3 P I Z a F a L V c P K t g f O 0 U 0 V q 3 t v g W d m K S C S k M 4 a N 3 0 v c S 0 M l C G E U 6 H h S D V N
We would also like to point out an interesting special case. Note that Γ(s, x k k! if s is a positive integer. Therefore, when r is a nonpositive integer, we have which contains a series of multipoles up to order 1 − r. This is also expected, since in this case the c a contribution in eq. (3.9) can be expanded, and the log series actually terminates at the order −r. For r = 0, we thus have . This is exactly the tree-level result since c a M does not run when r = 0. A negative integer r is generally difficult to obtain, but we find it possible to tune the particle contents and the corresponding quantum numbers for certain theories in a way that negative integer r can be realized at the leadinglog level. In this case, the higher order log contributions accidentally cancel. In section 4 we will show an explicit example of it.
To summarize, we observe the following structures for A(β) on the complex β-plane as shown in figure 1: • r ∈ Z − : poles at −a with pole order up to 1 − r.
• for other values of r: branch cut on (−∞, a].

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where r a = γ a /γ λ , λ(t = 0) = 1, c 2 (t = 0) = 1, and A 4 , A 6 are some constants fixed by the boundary conditions of c 4 and c 6 . For simplicity, we have also fixed the g(z) in eq. (2.8) to be one. Several interesting observations can be made for the A(β) in eq. (3.21). First, it can be shown that, in the limit γ λ → 0, A(β) still reduces to a series of shifted simple poles, following the derivation in the previous section. The poles are shifted by factors depending on only γ a 's (γ λ , γ 2 , γ 4 , . . .) which are the anomalous dimension matrices involving λ, not depending on the ones from only dimensional Wilson coefficients (γ 4 , γ 6 , . . .). Second, for nonzero γ λ , the analytic structure of A(β) also only depends on γ a 's, as a result of the structure of the running couplings in eq. (3.16). The analytical structure of A(β) is thus sensitive to specific contributions to the RGEs. This pattern should be general if we include only one dimensionless couplings, e.g. λ, y or g. When including multiple dimensionless couplings, the analytical structure of A(β) is difficult to obtain and may have entangled dependence on various anomalous dimensions. Generally, eq. (3.21) contains branch cuts on the negative β axis starting from β = 0, −2 , −4 . . ., unless the corresponding parameters (r 2 , r 4 , 2r 2 − 1, r 6 , . . . ) are non-positive integers, as discussed in the previous section.

λφ 4
For simplicity, we will focus on the Higgs sector with the dimensionless coupling λ|H † H| 2 in the Standard Model EFT (SMEFT) and consider the four scalar scattering amplitude A(H 2 αβ H †2 αβ ) with α, β,α,β being the indices of SU(2) L . This scattering amplitude receives contributions from the following two dimension-6 operators in the unbroken phase: with the corresponding Wilson coefficients C H and C HD , respectively. The anomalous dimension matrix for SMEFT has been computed in [38][39][40][41][42][43] with the following RGEs: where we also assume massless particles so that the λ is only renormalzied by itself and is solved as: with λ M = λ(µ = M ). Solving eq. (4.4), we obtain Their contribution to A(H 2 αβ H †2 αβ ) are given by: where the renormalization scale is set to the collision energy of particles H 2 αβ as µ = ω. As A(H 2 αβ H †2 αβ ) receives contributions from the dimension six operator with r = 1(C H ) or r = 1/2(C HD ), its celestial amplitude diverges at β = −2 and has branch cut on β ∈ (−∞, −2].

Accidental multipole structures
In this subsection, we consider an explicit example where the celestial amplitude generated by an EFT exhibits accidental multipole structures, as mentioned in the previous section. For the sake of convenience, we restrict ourselves to dimension-6 operators in the SMEFT, but with strong coupling and the weak couplings turned off in the following discussions. In particular, we consider the four-fermion scattering amplitudes A(L α Q aβ uȧe), which receives contributions from the two operators with the corresponding Wilson coefficients C (1) lequ and C lequ , respectively [44]. We will fix the hypercharges that fermions carry to be the SM values, and modify them later on to achieve the desired accidental multipole structure. The running of the Wilson coefficients are then given by [41][42][43] µ d dµ C (1) lequ = g 2 1 16π 2 − 6(y 2 e + y e y u − y e y q + y q y u )C (1) lequ − 24(y q + y u )(2y e − y q + y u )C +2(y 2 e − y e y q + y e y u − 2y 2 q + 5y q y u − 2y 2 u )C where in the SM, y q = 1 6 , y e = −1, (with n G the number of generations). To solve the differential equations, we can rotate to a basis in which the running of the dimension-six operators is diagonal and can be expressed as:

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where P is the invertible matrix to diagonalize the anomalous dimension matrix. The (ratios of) anomalous dimensions turn out to be γ c1 /γ g = −0.485 . . . and γ c3 /γ g = 0.226 . . ., which are irrational numbers. The RGEs can be easily solved, and the solutions follow the pattern in eq. (3.8): The four fermion amplitude focusing on the positive-helicity configurations can then be calculated as: lequ (µ) [14][23] + C (1) lequ (µ) [14][23] with the renormalization scale set to µ = ω. The corresponding celestial amplitude can then be seen to follow our previous discussions, and its analytic structure depends on the two important ratios, γ c1 /γ g and γ c3 /γ g . Both ratios will lead to branch-cut structures in the complex β plane running from −∞ to −2. The celestial amplitude A(β) is also divergent (both real and imaginary parts) at β = −2. As γ c1 < 0, C 1 (ω) is divergent in the small ω region and dominates the contributions to the analytical structure of the celestial amplitudes.
Let us now change the quantum numbers of the fermions in order to obtain the accidental multipole structures. Following the discussion in the previous section, it is desirable to tune the quantum numbers such that γ c1 /γ g and γ c3 /γ g are either zero or negative integers. A consistent quantum field theory coupled to gravity may only have fermions with rational hypercharges [45][46][47], and we will stick to this case. Note that, the hypercharges also needs to satisfy the relation y l + y q − y e − y u = 0 to have hypercharge conservation for the two operators, which are already implemented in eq. (4.10). Consider first the case of γ i = 0 (i = c1, c3), there is no solution except for the trivial one with y e = y q = y u = y l = 0. The single pole structure only shows up trivially and preserves to higher loop orders. For an Abelian gauge theory, γ g cannot be zero for non-vanishing γ i and the shifted-pole structure cannot be achieved at one loop level (or higher-order levels). Note that if we instead consider non-Abelian gauge theories, it is possible to have γ g = 0 for the one-loop beta function and observe shifted-pole structure for the four-fermion celestial amplitudes. We further consider the possibility of γ i /γ g being a negative integer. For simplicity, we require y q + y u = 0 to avoid operator mixing at one-loop level. We are able to find solution for γ c3 /γ g = −1 (y e = −y u , y d = y u , y l = y u , n G = 1, n c = 3) for the running of C (3) lequ . This solution gives γ c1 /γ g = 3/5. The celestial amplitude for A(L α Q aβ uȧe) thus has both JHEP08(2022)257 pole structure of order 2 at β = −2 and branch cut structure for β < −2. With these hypercharges, however, the U(1) 3 Y anomaly cancellation requires existences of heavy chiral fermions charged under the U(1) Y gauge group. Other negative integer values for γ i /γ g can not be achieved unless we choose smaller values of n c , e.g. to be one. Larger n G increases γ g which results in even smaller |γ i /γ g | < 1, evading solutions of negative integer values of γ i /γ g .
We note here again that the multipole structures in the celestial amplitude are due to the termination of leading log contribution (to C (3) lequ ) at finite loop order. This feature should be accidental, and would not hold, for instance, once the resummation of sub-leading log terms or contributions from more than one dimensionless couplings are included. Due to the difficulties of solving the RGEs and performing the corresponding Mellin transformations, in general we are not able to do analytical calculations for higher loop-order contributions. However, we do note that in certain SU(N ) non-Abelian gauge theories with coupling g, γ g = 0 at the one-loop order, while with higher order loop contributions the theory can flow to an interacting fixed point in the IR [48][49][50][51] for certain numbers of fermion flavors charged. At this fixed point, the gauge coupling g does not run, and the celestial amplitude generated by the higher dimensional operators have shifted-pole structures (as in eq. (3.7)) instead of branch cuts.

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tributions capture at least qualitatively all possible analytical structures in the negative β-plane, or if new structures can be generated by higher-order contributions. To include the next-leading log contributions, two-loop RGEs are needed, which can be difficult to solve analytically. Furthermore, a complicated function of ω can also be difficult to Mellin transform analytically. In massive theories, the ω dependences in the amplitudes are also generally more complicated. The investigation of the analytic structures of celestial amplitudes in these more complicated cases turn out to be challenging, and may require novel tools and strategies.
While the analytic structures of momentum space amplitudes have clear physical meanings for either poles or branch cuts, the implications of those in celestial amplitudes are far less clear at the moment. It is also difficult to write down dispersion relations in the β-plane (the ones in ref. [16] are in the ω-plane), as the behavior of the contour at infinity is not well understood. A better understanding of the physics information encoded in the celestial amplitudes may also tell us what analytic structures are expected in general. where the finite piece comes from the fact that the absolute value of the summation over k is smaller than α × e |β+a|/κ with α > 0. As Γ(x) is finite for x = 0, −1, −2, . . ., the only singularity possible in the above celestial amplitude is at β = −a region. Notice that due to the (β + a) r−1 term, the celestial amplitude also has branch cut for β + a < 0 if r − 1 is non-integer. We then have: The above equations shows that the branch cut vanishes when r is nonpositive interger r ∈ Z − . Notice that when r = 0, the real part has single pole at β = −a as expected.
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