SS-matrix bootstrap for resonances
N. Doroud and J. Elias Mir´o
SISSA/ISAS and INFN, I-34136 Trieste, Italy
Abstract
We study the 2 → S -matrix element of a generic, gapped and Lorentz invariant QFTin d = 1 + 1 space time dimensions. We derive an analytical bound on the coupling of theasymptotic states to unstable particles (a.k.a. resonances) and its physical implications. Thisis achieved by exploiting the connection between the S -matrix phase-shift and the roots ofthe S -matrix in the physical sheet. We also develop a numerical framework to recover theanalytical bound as a solution to a numerical optimization problem. This later approach canbe generalized to d = 3 + 1 spacetime dimensions. April 2018 a r X i v : . [ h e p - t h ] S e p ontents S -matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1 The θ -strip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Unstable resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.1 What are unstable resonances? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 A bound on the S -matrix of unstable resonances . . . . . . . . . . . . . . . . . . . . 83.3 Interpretation of the bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Numerical optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.1 A simple S -matrix ansatz for d = 1 + 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2 Towards generalization to higher dimensions . . . . . . . . . . . . . . . . . . . . . . . 175 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18A Nature of the two-particle branch point . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19B Total integrated phase-shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20C Perturbative example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20C.1 The perturbative S -matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21C.2 Resonances, poles and zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Deriving the phenomenological implications of strongly coupled Quantum Field Theories (QFT) ishard. Any new idea or approach to inspect such strongly coupled regime deserves to be scrutinized.The recent progress on the numerical conformal bootstrap [1, 2, 3, 4, 5] – reviving the successful d = 1 + 1 conformal bootstrap [6, 7] – has lead to a revision of the closely related S -matrixbootstrap [8, 9].The old analytical S -matrix bootstrap approach lost momentum with the advent of QCDand due to the difficulties of dealing with the analytic properties of the S -matrix in d = 3 + 1spacetime dimensions. For a compendium of results on the analytic properties of the S -matrix seefor instance [11]. In the present context bootstrap is synonymous to an axiomatic approach, whereout of few physical assumptions one extracts general consequences for physical observables. Forthe S -matrix bootstrap approach, the input assumptions are those of quantum mechanics, specialrelativity and assumptions on the spectrum of particles encoded through analytic properties of the S -matrix elements.Lately, there has been a number of interesting results within the S -matrix bootstrap approach[8, 12, 13]. The key aspects that paved the way for these developments have been to, firstly,identify an interesting and simple enough question that the bootstrap approach can answer and,secondly, the development of a numerical approach to answer the question in general spacetimedimensions. Specifically, ref. [12] found a rigorous analytical upper bound on the coupling betweenasymptotic states of the S -matrix in d = 1 + 1 dimensions. The existence of such upper boundwas expected in higher dimensions and was demonstrated in d = 3 + 1 by means of a numerical See ref. [10] for a testimony. S -matrices in d = 3 + 1 has lots of potential applicationsfor particle physics. For a realistic set up though we would like to study S -matrices that featureunstable resonances. The main purpose of this work is to take the first steps towards developingthis theory. The present work is entirely in d = 1 + 1 and we focus on the 2 → S -matrix elementof the single stable particle of the theory. These simplifying assumptions will allow us to derive anumber of analytical results and intuition that is important before attacking the analogous problemin d = 3 + 1.Section 2 is mostly review and discussion of the analytic properties of the S -matrix. Section 3contains the main result, a bound on the 2 → S -matrix elements that feature unstable resonances.We discuss the interpretation of this bound and the implications for the spectrum of resonances.In section 4 we perform a numerical study that matches the analytical derivations of section 3.Crucially, the numerical approach presented in section 4 admits a generalization to d = 3 + 1dimensions. Finally, we conclude and outline possible directions to develop. S -matrix The main focus of this paper is to study the space of consistent S -matrices in d = 1 + 1 spacetimedimensions. For simplicity we restrict our attention to theories with only a single stable particleof mass m . This is not crucial and the assumption can be relaxed on later studies. We focus onthe elastic 2 → S -matrix element (cid:104) p , p | ˆ S | p , p (cid:105) ≡ S ( p i ) , (2.1)where = (cid:104) p , p | p , p (cid:105) captures the kinematical information. All the interesting physics isencoded in the Lorentz scalar S which is a function of the Mandelstam variable s = ( p + p ) .Note that in two spacetime dimensions the scattering is along a line and thus there is no scatteringangle. Consequently either of the Mandelstam variables t = ( p + p ) or u = ( p + p ) mustvanish, which together with the kinematical constraint s + t + u = 4 m , imply that the function S in Eq. (2.1) is only a function of a single variable s . This function is further constrained bycrossing symmetry S ( s ) = S (4 m − s ) , (2.2) i.e. it is symmetric under the exchange of the s and t channels (or equivalently between the s and u channels). In the rest of this section we will review the analytic properties of S ( s ).Consider the analytical continuation of S ( s ) into the complex s -plane. Generically the function S ( s ) has branch point singularities at the minimal values of s where the process 2 → n iskinematically allowed. For positive s , the lowest such branch point is at s = (2 m ) , the two-particle branch point. Crossing symmetry (2.2) implies the presence of a corresponding branchpoint at s = 0. Generically, in the absence of extra symmetries forbidding particle production,infinitely many branch points are expected on the real line at the minimal values where higher-particle production is kinematically allowed. We have illustrated the branch points at s = 4 m , m (red circles) and the crossing related s = 0 , − m (red squares) in the left plot of Fig. 1. The3hysical S -matrix is obtained in the limit lim (cid:15) → + S ( s + i(cid:15) ) , (2.3)namely by approaching the real line from above without encircling any such branch points. Thephysical s -plane is defined as the trivial analytical continuation of Eq. (2.3) without encircling anybranch point. Pictorially, it consists of the full complex plane minus the cuts on the real axis, seeFig. 1. The two key analytical assumptions on S ( s ) are that all the singularities of the physical s -plane consist only of branch points on the real line; and that along the real axis and below thetwo-particle threshold S ( s ) is a real function. Thus the analytic continuation of S ( s ) satisfies S ∗ ( s ) = S ( s ∗ ) , (2.4)which is often referred to as real analyticity . The last property of S ( s ) follows from unitarity of the full S -matrix, implying the followingconstraint on the 2 → S -matrix element S ( s + i(cid:15) ) S ( s − i(cid:15) ) = f ( s ) with 0 (cid:54) f ( s ) (cid:54) , (2.5)and s > m . Note that we have used real analyticity to write the modulus as | S ( s + i(cid:15) ) | = S ( s + i(cid:15) ) S ( s − i(cid:15) ). Recall that below the inelastic threshold s ∗ (above which 2 → n , with n > f ( s ) = 1 for 4 m < s (cid:54) s ∗ . (2.6)Typically the inelastic threshold is at the three-particle production threshold s ∗ = 9 m or four-particle production threshold s ∗ = 16 m .Unless explicitly stated otherwise, we will refer to S ( s ) as the S -matrix (instead of the → S -matrix element function ). To summarize, the S -matrix is assumed to satisfy crossing (2.2),real-analyticity (2.4), unitarity (2.5) and there are no singularities in the physical s -plane but onlybranch points on the real line associated with the 2 → n ( n (cid:62)
2) scattering processes. In orderto further elucidate the analytical properties of the S -matrix we will next review a particularlysimple S -matrix. This will also serve as an excuse to introduce the rapidity variable θ which wewill use in the rest of the paper. θ -strip Consider the following classic QFT example in d = 1 + 1: the Sine-Gordon S -matrix element forthe scattering of the lightest breather is given by [15, 16] S SG ( s ) = √ s √ m − s + m (cid:112) m − m √ s √ m − s − m (cid:112) m − m , (2.7) We assume a Z symmetry forbidding a cubic self-interaction of the stable particle. Such cubic self-interactionwould lead to poles at s = m , m . Let us note that in Eq. (2.4) we have assumed that the S -matrix theory is invariant under space parity.The general condition for the two-body S -matrix is Hermitian analyticity S ∗ ij ( s ) = S ji ( s ∗ ) which reduces to realanalyticity only for parity invariant theories [14]. � � �� �� ( � )- �� - ������ ( � ) - � - � - � � � � �� ( θ )- π - � - ����� ( θ ) π Figure 1:
Illustration of the conformal map in Eq. (2.8). The complex s -plane, the left plot, ismapped into the complex θ -strip Im θ ∈ (0 , π ), right plot. We have also depicted the mapping of adashed curve, a dotted curve and a gray grid. where s is the Mandelstam variable. The function S SG ( s ) has a pole at s = m , the mass of thenext-to-lightest breather. The matrix element S SG ( s ) has branch points at s = 0 , m associatedwith the two-particle production threshold. These are square-root branch points and can beresolved by the conformal map s ( θ ) = 4 m cosh ( θ/ . (2.8)Eq. (2.8) maps the entire physical s -plane minus the cuts on the real line into the stripIm θ ∈ (0 , π ) , (2.9)which we will refer to as the physical strip . The transformation is illustrated in Fig. 1. Thetwo-to-two S -matrix element (2.7) in the θ -strip is given by S SG ( θ ) ≡ S α = sinh θ − sinh α sinh θ + sinh α , (2.10)where sinh α = − i m /m (cid:112) − m / (4 m ), and we defined S α ( θ ) for later use. Eq. (2.10) isanalytic at the points θ = 0 and θ = iπ , corresponding to the original branch points of Eq. (2.7) at s = 4 m and s = 0 respectively. The second Riemann sheet of (2.7) reached by traversing a branchcut stemming from the two-particle branch points at s = 0 , m is mapped into Im θ ∈ ( − π, θ = − iπ and θ = iπ are identified; Im θ ∈ [ − π, π ) is the fundamentaldomain of Eq. (2.10), which is periodic under θ ∼ θ + 2 πi .The Sine-Gordon theory is very special as it is an integrable QFT. It follows that thereis no particle production and the full S -matrix factorizes into the product of 2 → S SG ( θ ) is a meromorphic – and thus single valued – function in the θ -strip Im θ ∈ [ − π, π ). For a generic non-integrable QFT however, one has branch points at theinelastic thresholds s = { (3 m ) , (4 m ) , ... } where the matrix elements S → , ... are switched on.Those are mapped into the real line of the θ -strip. As depicted in Fig. 1, they appear both inthe positive and negative real axis of the θ -strip because they can be reached from both Riemannsheets associated to the two-particle branch point. The function S α ( θ ) is commonly called a Coleman-Dalitz-Dyson (CDD) factor.
5s we have seen, the branch point at the two-particle production threshold in the particularexample Eq. (2.7) is two-sheeted. It turns out that this feature is more general and extends tonon-integrable S -matrices, see appendix A for further details. The results of this paper howeverdo not make use of the nature of any of the branch points in the physical s -plane.In d = 1 + 1 dimensions θ has the physical interpretation of being the rapidity difference ofthe incoming particles θ ≡ θ − θ where p i = ( m cosh θ i , m sinh θ i ). The S -matrix literature in d = 1 + 1 dimensions commonly uses this variable. Thus, in the rest of the paper we will considerthe S -matrix as a function S ( θ ) (this is however not crucial and all the results below can bereformulated in the s -plane). For completeness, let us recall that crossing symmetry (2.2) in the θ -strip implies S ( θ ) = S ( iπ − θ ) , (2.11)real analyticity (2.4) reads S ∗ ( θ ) = S ( − θ ∗ ) , (2.12)and unitarity (2.5) in the θ -strip reads S ( θ ) S ( − θ ) = f ( θ ) , (2.13)where 0 (cid:54) f ( θ ) (cid:54) θ . Our goal is to study QFTs with unstable particles or resonances. In this section, we first presentthe operational definition of a resonance before deriving a bound for the S -matrices that featureresonances. Finally, we discuss the interpretation of the bound in the context of a weakly coupledQFT. In perturbation theory unstable particles are often associated with complex poles. These poles lieon higher Riemann sheets that can be reached by traversing the multi-particle branch cuts alongthe real line in the θ -plane. The distinguishing feature of such singularities is that they lead topronounced variations of the phase of the S -matrix evaluated along the real line. Thus, we definea resonance as an abrupt change in the phase of the S -matrixRe 2 δ ( θ ) where 2 iδ ( θ ) ≡ log S ( θ ) , (3.1)without any reference to poles in higher Riemann sheets.Abrupt variations of the phase of the S -matrix typically signal the presence of poles or zerosof the S -matrix in the complex plane and it is up to us to classify such pronounced features of the S -matrix. Of particular interest is when the phase of S ( θ ) abruptly increases by 2 π continuously See appendix C.2 for an explicit example. � � � � � �������� θ � � δ ( θ ) � - ������ ����� - ����� Figure 2: Left:
Section of the complex plane of the phase of Eq. (3.5), the lines δ ( iπ ) ∼ δ ( − iπ )should be indentified. Right: phase-shift of Eq. (3.5) localized around the position of the branchpoints generated by the zeros and poles of S ex ( θ ). and monotonically in θ . Such 2 π phase-shifts stem from the presence of a pair of zeros in the S -matrix S ( θ ) in the physical strip Im θ ∈ (0 , π ). In fact, each zero in the physical θ strip contributeswith an iπ to the total S -matrix phase shift (cid:90) ∞−∞ dθ ∂ θ δ ( θ ) = (cid:88) zeros π . (3.2)Due to crossing symmetry, the zeros θ i of S ( θ ) come in pairs related by crossing S ( θ i ) = S ( iπ − θ i ) =0. In addition, by real analyticity, the roots are also pairwise related by complex conjugation S ∗ ( θ i ) = S ( − θ ∗ i ) = 0. In many physically relevant S -matrices one finds an approximate 2 π changeof the phase in a bounded span θ ∈ [ θ ◦ − γ, θ ◦ + γ ]2∆ δ ≡ (cid:90) θ ◦ + γθ ◦ − γ ∂ θ δ ( θ ) dθ ≈ π , (3.3)where θ ◦ = Re θ i and γ ∼ θ i , the exact choice of the resonance region θ ◦ ± γ is somewhatarbitrary. This is the kind of resonances that we are interested in this paper. For a long-lived unstable particle associated to a long time delay, the 2 π phase-shift is highlylocalized and Im θ i (cid:28)
1. The zeros in the physical strip are accompanied by poles which arehidden behind the multi-particle branch cuts. In terms of the θ variable the S -matrix behaves as e iδ ( θ ) ∼ e iδ θ − θ i θ − θ ∗ i , (3.4)for θ close to | θ i | . The zeros and poles of S ( θ ) result in branch points of δ ( θ ). Pictorially, thisleads to a branch cut that “cuts” the real line. Then, when evaluating 2 δ ( θ ) along the real linethe 2 π phase-shift is a consequence of changing Riemann sheet of the logarithm. See appendix B for the derivation of Eq. (3.2) – this is the relativistic analog of Levinson’s theorem, see forinstance chapter XVII of [17].
7s an illustration, consider the following function S ex ( θ ) = S α ( θ ) S − α ∗ ( θ ) , (3.5)where Im α > S α ( θ ) was defined in Eq. (2.10). The function S ex ( θ ) is a realistic S -matrixelement because it satisfies the unitary equation, crossing symmetry and it is real analytic. S ex ( θ ) has zeros at θ = α, − α ∗ and poles at complex conjugate points as required by unitarity S ex ( θ ) S ex ( − θ ) = 1. The left plot in Fig. 2 shows a section of the fundamental domain of thecomplex plane of Im δ ( θ ) that includes the zeros and poles of S ( θ ). We have depicted branchcuts connecting the zeros on the physical strip with the poles on the lower stip θ ∈ ( − π, S -matrix is evaluated. On the righthand side we have ploted Re 2 δ ( θ ) on a segment along the real line. As 2 δ ( θ ) goes through theregion θ ≈ Re α , i.e. near the location of the branch points, the function 2 iδ = log S is evaluatedin a higher Riemann sheet and the imaginary part is shifted by 2 π . In section 3.3 we discuss aperturbative QFT with the same qualitative picture as the S -matrix in Eq. (3.5).To close up this section, let us insist that in general we will not refer to complex poles of the S -matrix. Instead, we focus on the zeros of S ( θ ) in the physical θ -strip (or physical s -plane), whichare in a one-to-one correspondence with each π contribution to the total phase-shift (3.2). Thispicture avoids the need to discuss the nature of the branch points of S ( θ ) on the real line and theanalytical continuation of the function S ( θ ) around such branch points which requires a case bycase analysis. Instead, it only requires the trivial analytical continuation of S ( θ ) into the physicalsheet which, by definition, is always available. Note also that the operational definition of unstableresonance that we are employing is physically meaningful because the phase-shift is experimentallyaccessible (it can also be extracted from lattice Monte Carlo simulations [20]). S -matrix of unstable resonances The two-dimensional S -matrix can be written as follows S ( θ ) = (cid:89) j S α j ( θ ) exp (cid:18) − (cid:90) + ∞−∞ dθ (cid:48) πi log f ( θ (cid:48) )sinh( θ − θ (cid:48) + i(cid:15) ) (cid:19) for Im( θ ) ∈ [0 , π ) (3.6)where (cid:15) is an arbitrarily small positive parameter and S α j ( θ ) denotes a CDD factor defined in(2.10): S α ( θ ) = sinh θ − sinh α sinh θ + sinh α . The set { α j } parametrizes the position of the zeros and poles of S ( θ ) in the physical strip. Aswritten in Eq. (3.6) the set { α j } may contain repeated elements in order to account for the correctorder of the poles and zeros of S ( θ ). The function S α ( θ ) saturates unitarity S α ( θ ) S α ( − θ ) = 1along the entire real line. The function f ( θ ) parametrizes the amount of inelasticity, see (2.13).Eq. (3.6) will play a crucial role in our discussion below so let us review its derivation. Note however that a generic product of CDD factors leads to a finite volume spectrum E i ( R ) with branch pointsingularities at finite volume [18, 19]. In fact, for our particular physical set up with a single stable particle (cid:81) j S α j ( θ ) has no poles in the physicalstrip but only zeros. .2.1 Discussion of Eq. (3.6) Let us define φ ( θ ) ≡ i∂ θ δ ( θ ) and consider the following dispersion relation φ ( θ ) = (cid:73) ∂ C θ dθ (cid:48) πi φ ( θ (cid:48) )sinh( θ (cid:48) − θ ) , (3.7)where ∂ C θ is a closed contour encircling a region C θ where φ ( θ ) is regular, i.e. such that S ( θ ) isholomorphic and does not vanish in C θ . Next we apply Cauchy’s theorem and blow the contour inEq. (3.7) to the boundary of the physical strip. In doing so, we must subtract the zeros of S ( θ ) inthe physical strip φ ( θ ) = (cid:88) j (cid:18) α j − θ ) + 1sinh( iπ − α j − θ ) (cid:19) + (cid:90) ∞−∞ dθ (cid:48) πi φ ( θ (cid:48) )sinh( θ (cid:48) − θ ) + (cid:90) −∞ + iπ ∞ + iπ dθ (cid:48) πi φ ( θ (cid:48) )sinh( θ (cid:48) − θ ) , (3.8)where the θ = 0 , iπ lines are approached from above and below, respectively. In Eq. (3.8), zeroscome in pairs related by crossing S ( θ ) = S ( iπ − θ ) and we have dropped the contribution from thecontour arcs at infinity. This can be justified by assuming that S ( θ ) is polynomially bounded. Crossing symmetry implies φ ( θ + iπ ) = φ ( − θ ). Therefore, the last integral in Eq. (3.8) can bewritten as (cid:82) ∞−∞ dθ (cid:48) πi φ ( − θ (cid:48) ) / sinh( θ (cid:48) − θ ) and we are led to φ ( θ ) = (cid:88) j (cid:18) α j − θ ) + 1sinh( iπ − α j − θ ) (cid:19) − (cid:90) ∞−∞ dθ (cid:48) πi ∂ θ (cid:48) log f ( θ (cid:48) )sinh( θ (cid:48) − θ ) . (3.9)where we have used φ ( θ ) + φ ( − θ ) = ∂ θ log f ( θ ), by Eq. (2.13). Finally, integrating by parts withrespect to θ (cid:48) the integral in Eq. (3.9) and using S ( θ ) = exp (cid:82) dθφ ( θ ) we are led to Eq. (3.6).A key point of Eq. (3.6) is that the roots { α j } of the S -matrix are factored out. The factor inEq. (2.10) shows that each zero α j in the physical strip has an accompanying pole located at − α j in the unphysical strip Im θ ∈ ( − π, α j of the S -matrix in Eq. (3.6) has a pole at − α j . Eq. (3.6) only applies in thephysical strip. In order to analytically continue Eq. (3.6) into the unphysical strip we need to knowthe nature of the branch point singularities on the real line. If the two-particle threshold branchpoint at s = 4 m is a square-root singularity (in the Mandelstam s -plane), then the conformal map s = 4 m cosh ( θ/
2) resolves the singularity and we can analytically continue Eq. (3.6) provided weavoid other possible branch points on the real line. Then we may conclude that S ( θ ) has poles inthe unphysical strip at the same positions as the factors S α j ( θ ). But again, this conclusion is notstrictly necessary.Even if we can analytically continue the function Eq. (3.6) across the branch points on the realline, the poles of S α j ( θ ) may be canceled by the exponential factor ∼ e (cid:82) log f/ sinh in Eq. (3.6), whichat the same time can generate poles in unphysical sheets reached by traversing higher particleproduction branch cuts not related to the two-particle branch point. We need this kind of technical assumption to prove Eq. (3.6). However, as discussed in section 3.2.3 below, thisassumption is not crucial for the bound on the S -matrix that we are about to derive. .2.2 The bound Consider a generic point θ = ˜ θ + it , with ˜ θ ∈ R and t ∈ (0 , π ) in the physical strip. Then, theabsolute value of the S -matrix is given by | S ( θ ) | = (cid:89) j | S α j ( θ ) | exp (cid:32) sin t (cid:90) + ∞−∞ dθ (cid:48) π cosh(˜ θ − θ (cid:48) ) | sinh( θ − θ (cid:48) ) | log f ( θ (cid:48) ) (cid:33) , (3.10)where we have used Re[ i sinh ∗ ( θ − θ (cid:48) )] = sin t cosh(˜ θ − θ (cid:48) ). Note that log f ( θ ) (cid:54) (cid:54) f ( θ (cid:48) ) (cid:54) | S ( θ ) | (cid:54) (cid:89) j | S α j ( θ ) | , (3.11)for θ in the physical strip. Eq. (3.10) applies in the whole physical θ -strip and in particular itimplies | S (cid:48) ( α i ) | (cid:54) | S (cid:48) α i ( α i ) | (cid:89) j (cid:54) = i | S α j ( α i ) | , (3.12)at the position of each zero α i in the physical strip.We shall see in section 3.3 that there is a direct relation between | S (cid:48) ( α i ) | and the parametercontrolling the perturbative expansion, i.e. the dimensionless coupling constant. More precisely,we will show that S (cid:48) ( α i ) is proportional to the ratio of the expansion parameter and the square ofthe width of the resonance. Thus, for a fixed width and with an abuse of language, we will call S (cid:48) ( α i ) : coupling to the resonance α i , (3.13)without reference to the actual underlying coupling constants of the possible Lagrangian descrip-tion. The derivation of (3.11) presented above explicitly accounts for inelasticity f ( θ ) and knowledge of f ( θ ) results in a stronger bound than (3.11). Note, however, that the bound can be obtained in amore general setting as follows. Consider the nowhere-vanishing function h ( θ ) = S ( θ ) / (cid:89) j S α j ( θ ) , (3.14)where the product in the denominator runs over all zeros of S ( θ ) with the appropriate order. Byconstruction, h ( θ ) is a holomorphic function in the physical strip and is bounded on the boundariesIm θ b = 0 , π , since | S α k ( θ b ) | = 1 and | S ( θ b ) | (cid:54)
1. Therefore, by the Hadamard three-lines theorem, | h ( θ ) | is bounded in the physical strip by its value at the boundary and we are led to Eq. (3.11). Inrefs. [21, 13] a similar argument is used to bound the residue of the poles of S ( θ ) on the θ ∈ [0 , iπ )segment in the physical strip which are associated with stable particles.10he simple derivation of (3.11) given above does not require the S -matrix to admit a represen-tation of the form (3.6). As an example of such an S -matrix of broad interest, consider S g ( θ ) = e g √ cosh ( θ/ √ − cosh ( θ/ ≡ e ig sinh θ , (3.15)where g (cid:62)
0. The S -matrix (3.15) does not admit a representation of the form (3.6) with finitelymany factors of S α . In fact one can check the the above S -matrix can be obtained in the limitwhere we have an infinite product of S α factors [19]: e ig sinh θ = lim n →∞ ( − n n (cid:89) j =1 S α ( θ ) (3.16)where S α is given by (2.10) with sinh α = 2 in/g . The S -matrix S g ( θ ) has infinitely many phase-shifts of the type in Eq. (3.3) that can be interpreted as resonant particles [22]. This is easilyexplained from the infinite-product representation above: each S α factor accounts for a pair of(simple) zeros in the physical strip giving rise to a phase shift of 2 π . In the limit n → ∞ in (3.16)we end up with an infinite number of coincident zeros at θ = ∞ + iπ/
2, and the total (integrated)phase shift is infinite. The S -matrix in (3.15) can be viewed as an integrable deformation[19, 22, 25] corresponding to the upward flow generated by certain irrelevant operators (in theRG sense). Moreover, a special limit of (3.15), namely lim g → S g , appears in the context of theeffective string description of Yang-Mills flux tubes [26].Eq. (3.11) implies that an S -matrix with at least n zeros at { α j } has a magnitude less than orequal to (cid:81) nj =1 | S α j ( θ ) | , with | S α ( θ ) | (cid:54) θ in the physical strip. Therefore, we do not necessarilyneed to know the spectrum of unstable resonances up to arbitrarily high energy in order to obtain ameaningful bound. Additional knowledge of UV resonances makes the bound more stringent. Thisobservation is key for the bounds in (3.11-3.12) to be sensible from an effective low energy physicsstanding point where we do not necessarily want to commit to a particularly detailed spectrum ofUV resonances beyond a certain energy cutoff. To illustrate this point we have plotted | S (cid:48) ( θ ) | as afunction of x ≡ Re θ for the S-matrix S = S θ S − θ ∗ (black solid line) in the left plot in Fig. 3. Anyother theory which features resonances at { θ , − θ ∗ } has a coupling | S (cid:48) ( θ ) | which falls below thisline. For comparison, the same plot depicts | S (cid:48) ( θ ) | for the S-matrix S = S θ S − θ ∗ S θ S − θ ∗ featuringa second pair of resonances at { θ , − θ ∗ } . The dotted line depicts | S (cid:48) ( θ ) | for θ = 4 + iπ/ θ = 6 + iπ/
9. As can be seen from the plots the closer the resonancesare to each other the stricter the bound on | S (cid:48) ( θ ) | gets. Thus, the effects of possible further heavyresonances | θ − θ j | (cid:29) | S (cid:48) ( θ ) | , the larger the minimum mass gap with the nearest resonance θ . Namely, themass gap increases monotonically as | S (cid:48) ( θ ) | increases. This is illustrated in the right plot of Fig. 3for S = S θ S − θ ∗ S θ S − θ ∗ with θ = 3 + iπ/ θ = iπ/ S (cid:48) ( θ ) because below a critical coupling there is no bound on the mass gapfor a system with only two resonances { θ , θ } . Further assumptions on the spectrum of possiblehigher mass resonances would lead to stricter bounds on the separation Re θ − Re θ . Higher order zeros can be factorized by point-splitting, S nα = (cid:81) nj =1 S α + j(cid:15) , where (cid:15) ∼ e − n . Other examples with infinitely many resonances are the elliptic (doubly periodic) S -matrix such as ref.[23, 24]. �� ��� ��� ��� ��� ��� ��� ������������������������ � | ∂ � ( � + � π / � ) | ������ ��������� �� θ � = � + � π / � ��� ��� ��� ��� �������� �������� �� ������ ����� ��������� | ∂ � ( θ � )| � � � � � � � � � θ � � � �� � � - � � - � � � � � � � � � �� � � � ���� � � ������ ��������� �� θ � = � + � π / � Figure 3:
In the left plot, maximal coupling to the lowest mass resonance at θ = x + iπ/ S -matrix with a single resonance (solid black), and an S -matrix with a second resonance at θ = 6+ iπ/ θ = 4 + iπ/ θ asa function of the coupling to the θ resonance for an S -matrix with two resonances. Consider an effective action describing the low energy dynamics of two massive scalar fields witha cubic interaction in two dimensions, S = (cid:90) d x (cid:20) (cid:0) ∂ µ π∂ µ π − m π + ∂ µ σ∂ µ σ − M σ (cid:1) − λ σπ − . . . (cid:21) , (3.17)where · · · denote further interactions of the fields that stabilize the potential at large field valuesbut whose coupling constant is much smaller than λ/m and are therefore inconsequential for thediscussion below. S -matrix Due to the cubic vertex in (3.17), for
M > m the particle excitations of σ are unstable and candecay to lighter particles. This instability manifests itself as a resonance in the ππ → ππ scatteringwhich can be analyzed with perturbation theory. The ππ → ππ component of the S -matrix isgiven by S = S ( s ) = (cid:18) i M ( s )2 √ s √ s − m (cid:19) (3.18)where the identity is the inner product of two particle states = (cid:104) p , p | p , p (cid:105) . The amplitude M is given by the sum of the σ -exchange diagrams in the s, t and u -channel: i M ( s ) = + + . (3.19)Up to higher order loop corrections and non-perturbative effects M is given by M ( s ) = λ M + λ πm − λ s − M − Π( s ) − λ m − s − M − Π(4 m − s ) , (3.20) See appendix C for details. s ) is the (amputated) two-point function given byΠ( s ) = λ π tanh − (cid:112) ss − m (cid:112) s ( s − m ) . (3.21)As discussed in section 2, we find that Eq. (3.18) is crossing symmetric S ( s ) = S (4 m − s ), it isanalytic in the domain 0 < | s ± i(cid:15) | < m and real along the real line in that domain. At thetwo-particle production threshold s = 4 m there is a square-root branch point and by crossingsymmetry we find another one at s = 0.It is convenient to resolve the square-root singularities at s = 0 , m by means of the conformalmap in Eq. (2.8). Henceforth we work in units such that m = 1. As a function of θ , the S -matrixin (3.18) is single-valued and given by S ( θ ) = 1+ iλ θ (cid:34) M + λ π −
14 cosh (cid:0) θ (cid:1) − M − Π( θ ) + 14 sinh (cid:0) θ (cid:1) + M + Π( iπ − θ ) (cid:35) , (3.22)where the amputated two-point function is given byΠ( θ ) = λ π θ − iπ sinh θ , (3.23)for θ in the fundamental domain θ ∈ ( − iπ, iπ ]. For
M > S -matrix in Eq. (3.22) has four poles and four zeros. The two s -channel poles arelocated at θ ± p = ± (cid:18) θ ◦ + λ θ ◦ π sinh θ ◦ (cid:19) − iλ
16 sinh θ ◦ , (3.24)where θ ◦ = 2 cosh − (cid:0) M (cid:1) , and the two zeros are located at θ ± z = ± (cid:18) θ ◦ + λ θ ◦ π sinh θ ◦ (cid:19) + iλ
16 sinh θ ◦ , (3.25)in the physical strip. Note that θ ± z = − θ ∓ p , as required by unitarity. The location of the remainingpoles and zeros follow from crossing symmetry, i.e. S ( θ ) = S ( iπ − θ ), and are located at θ c ± p/z = iπ − θ ± p/z . (3.26)Let us remark that the zeros { θ ± z , θ c ± z } lie above the real axis corresponding to the physical sheetwhile the poles { θ ± p , θ c ± p } lie below the real axis corresponding to the second Riemann sheet of the s = 0 , m branch points. Thus, the picture is qualitatively similar to the one depicted in Fig. 2. Eq. (3.23) is valid in the fundamental domain θ ∈ ( − iπ, iπ ]. However, it can be extended to the entire complex θ -plane by means of the identity θ − iπ = log (cid:16) sinh θ/ − cosh θ/ θ/ θ/ (cid:17) , valid in the fundamental domain, where the r.h.s.is periodic in the imaginary axis direction with period θ ∼ θ + 2 πi . .3.3 Bound In order to better understand the scope of the bound (3.12) lets consider it in the context of theperturbative example. Near the root at θ = θ i one of the denominators in the perturbative S -matrix (3.22) is of order λ , see appendix C for details. Consequently the dominant contributionto S (cid:48) ( θ i ) goes like λ − . For θ i = θ ± z given by (3.25) we find S (cid:48) ( θ i ) = − iλ M ( M −
4) + O ( λ ) , (3.27)thus Eq. (3.12) bounds ∼ M /λ (in units of m = 1). The width of the resonance associated to θ i is given by Γ = − Im Π( θ ◦ ) /M = λ / (4 M √ M −
4) (3.28)and hence S (cid:48) ( θ i ) can be expressed as | S (cid:48) ( θ i ) | = λ M Γ . (3.29)Note that at this order in perturbation theory the effective perturbative parameter is given by λ eff = λM (as can be seen in (3.22)). In light of this observation, Eq. (3.29) can be interpretedas follows: for a resonance particle σ with a fixed width Γ, the strength of the coupling between σ and the stable particles π is constrained to obey bound (3.12). This indicates that for fixedwidth, | S (cid:48) ( θ i ) | is directly related to the coupling parameter of the stable particles to the resonanceparticle θ i and can therefore be (loosely) referred to as the coupling. We remark that the bound issaturated at this order of perturbation theory, and that higher order corrections set | S (cid:48) ( θ i ) | withinthe bound due to particle production S → > The results presented in the previous section do not admit a straightforward generalization tohigher dimensions where the analytical approach proves cumbersome. For this reason here wepresent an alternative approach utilizing numerical methods which can be generalized to higherdimensions. This approach can be summarized in two key steps. First, we construct an ansatzfor the S -matrix which encodes analyticity and crossing symmetry as well as the location of theunstable resonances. The free parameters of the ansatz include the resonance coupling parameters.The second step is to maximize these coupling parameters under the constraint of unitarity S ( θ ) S ( − θ ) (cid:54) → S -matrix element admits a simple expansion which we can exploitto build our ansatz. Although this ansatz does not generalize to higher dimensions it serves asa simple framework to demonstrate how the numerical approach outlined above is implemented.Thus we first present the numerical approach using this ansatz before presenting a more generalansatz which can readily be generalized to higher dimensions.14 � - � � � ���������� θ - ����� - ��� - ��� ��� ��� - ������ ρ - ���� - Figure 4:
Illustration of the conformal map in Eq. (4.1) with β = iπ . S -matrix ansatz for d = 1 + 1Let us denote by S { θ j } ans our ansatz for the 2 → S -matrix element. This ansatz is labelled by thelocation of its roots in the θ -strip { θ j } . To encode holomorphicity of the function S { θ j } ans in a domain D θ of the θ -strip we consider a conformal map ρ from D θ into the unit disk. The S -matrix, viewedas a function of ρ , is therefore holomorphic inside the unit disk. A holomorphic function insidethe unit disk is analytic and therefore admits an absolutely convergent Taylor expansion insidethe disk. Thus S { θ j } ans ( ρ ) can be defined via its Taylor expansion inside the unit disk which makesholomorphicity of S { θ j } ans ( ρ ( θ )) in D θ manifest.To make crossing symmetry S { θ j } ans ( θ ) = S { θ j } ans ( iπ − θ ) manifest we require the map ρ ( θ ) to satisfy ρ ( θ ) = ρ ( iπ − θ ). Such a map can be viewed as a biholomorphic map between the fundamentaldomain Im θ ∈ (0 , π/
2) and the unit disk. A conformal map with such properties is given by ρ ( θ ) = sinh θ − i sinh θ + i . (4.1)As illustrated in Fig. 4, under the map Eq. (4.1) the crossing symmetric point θ = iπ/ ρ θ i = ρ ( θ i ), the function g ( ρ ) ≡ S { θ j } ans ( ρ ) / (cid:89) j ( ρ − ρ θ j ) (4.2)is nowhere vanishing and holomorphic inside the unit disk and therefore admits an absolutelyconvergent Taylor expansion g ( ρ ) = z (1 + ∞ (cid:88) n =1 c n ρ n ) , (4.3)where the overall factor is given by z ≡ g (0) > S -matrix element S { θ j } ans ( ρ ) = z (cid:89) j ( ρ − ρ θ j ) (cid:0) ∞ (cid:88) n =1 c n ρ n (cid:1) , (4.4)which is holomorphic, crossing symmetric and encodes the location of the zeros ρ θ j . The parameters { z, c i } are constrained by unitary of the S -matrix Eq. (4.5) but are otherwise arbitrary realparameters.Our next step is to maximize z over the space of the expansion coefficients { c n } in Eq. (4.4)under the constraint of unitarity. The unitarity bound along the real line in the θ -strip translatesto an analogous bound along the boundary of the unit disk parameterized as ρ = e iφ , S { θ j } ans ( e iφ ) S { θ j } ans ( e − iφ ) (cid:54) φ ∈ [0 , π ] . (4.5)Maximizing over z is tantamount to maximizing over the coupling to the resonances ∂ θ k S { θ j } ans ( ρ θ k ),with the location of the roots of S { θ j } ans held fixed. All couplings are maximized simultaneously, asfollows from (3.12). In order to set up the numerical code to maximize over z , the series in (4.4) has to be truncated.Therefore in the numerical code we maximize z in the truncated ansatz S { θ j } M ( ρ ) = z (cid:89) j ( ρ − ρ θ j ) (cid:0) M (cid:88) n =1 c n ρ n (cid:1) . (4.6)In addition, the constraint (4.5) is imposed in a large but finite number of points on the unit circle,namely it is evaluated at K points φ ∈ { , π/K, π/K, . . . , π } . (4.7)The only approximation made in this implementation is in the truncation of the series in (4.6).Convergence as M is increased is fast and even keeping the first few terms leads to precise results.As an example, consider { θ j } = { i, − i } , (4.8)in Eq. (4.6). We maximize (4.6) over z under the unitary constraint for the set of zeros in (4.8) andwe get the white lines depicted in Fig. 5. To obtain such result we increased M and K until we gota convergent result, the plots shown are for { M, K } = { , } . The white lines are super-imposedover black thicker lines. These correspond to the S -matrix theory S ex of (3.5) with α → iS ex ( θ ) = S i ( θ ) S − i ( θ ) , (4.9)in the left plot, while in the right plot we compare with the phase-shift of S ex . The numericalresults have been obtained with
Mathematica ’s function
FindMaximum . The computation is cheap, Recall that S ex saturates unitarity at all physical energies. In d = 3 + 1, only the trivial S -matrix saturatesunitarity at all energies, since any non-trivial S -matrix has a finite amount of particle production. From thisperspective, wether or not we can identify a Lagrangian model leading to S ex is somewhat anecdotic and special to d = 1 + 1 physics. ��� - ������������ � �� � ( � � ϕ ) � � � � ( � � ϕ ) ��� ��� ��� ��� ��� ��� ����� - � �� - � ϕ � �� � - � � � ������� � � δ ( θ ) � � � � ��� - � �� - � θ δ �� � - δ � � Figure 5:
In the left plot solid black lines depict the imaginary part, real part and absolute value of S th ( e iφ ). Superimposed we show the imaginary part (dashed white), the real part (doted white) andabsolute value (solid white) of S ans ( e iφ ). In the lower left plot, sharing the same horizontal axis, weshow the real part (dotted) and the imaginary part (dashed) of S th ( e iφ ) − S ans ( e iφ ). Finally the plotto the right is a comparison of the phase of S ans (white) and S th (black) in the θ strip. taking O (1) min. of time and ∼ S -matrices with manymore resonances is also feasible and leads to equally good results. It is convenient to make crossing symmetry explicit by extending S into a symmetric functionof two Mandelstam variables S ( s, t ) = S ( t, s ). An ansatz of this form is much more suitable forgeneralization to higher dimensions. In 1 + 1 dimensions the physical S -matrix is obtained byconstraining S ( s, t ) to the plane s + t = 4 m . This function is analytical in s and t up to thebranch points on the real line.Next, we build an ansatz S { ω i } ans − ( s, t ) encoding analyticity, crossing, and the location of theresonances. Similar to what we did in section 4.1 we encode analyticity in each variable s and t by conformally mapping the domain of holomorphicity into the unit disk and subsequently definethe function as a Taylor series in the poly-disk. Such a conformal map is provided by ω ( s ) = √ m − √ m − s √ m + √ m − s . (4.10)As we did before, we factor out the zeros of S { ω i } ans − ( s, t ), and expand the nowhere vanishing partin a (convergent) double-expansion whilst fulfilling all the physical assumptions, S { ω i } ans − ( s, t ) = z (cid:89) i [ ω i − ω ( s )] [ ω i − ω ( t )] (cid:16) ∞ (cid:88) m,n =1 c n,m ω ( s ) n ω ( t ) m (cid:17) . (4.11)Here c m,n are symmetric and real. Eq. (4.11) can be equivalently written in the corresponding θ s , θ t -strips. The existence of such a double expansion Eq. (4.11) is easy to show in two dimensions. To17his end note that the map (4.1), with θ = 2 cosh − ( √ s m ), has the following convergent expansion ρ ( s ) = (cid:2) ω ( s ) + ω (4 m − s ) (cid:3) ∞ (cid:88) n =0 ( − n ω (4 m − s ) n ω ( s ) n . This, together with (4.4), results in the double expansion (4.11) with t = 4 m − s .Evaluating S { ω i } ans − ( s, t ) at t = 4 m − s , and maximizing over z under the unitarity constraintwe obtain comparable results to the ones showed in Fig. 5. Eq. (4.11) admits a generalizationfrom d = 1 + 1 to d = 3 + 1 spacetime dimensions as was done in [13]. In d = 3 + 1 there is ananalogous definition of resonance presented in section 3.1, in terms of phase-shifts and roots of thecomponents of the S -matrix in the partial wave decomposition. In a forthcoming publication weplan to study the space of S -matrices in d = 3 + 1 that feature unstable resonances. In this work we have found a bound on the coupling of asymptotic states to unstable resonances,Eq. (3.12), which is saturated in the limit of maximal elasticity of the 2 → S -matrix element.The bound for each resonance is improved as the number of resonances is increased, or the gapbetween the resonances is decreased. Therefore (3.12) can be interpreted as setting a minimalmass gap between the resonances as a function of the coupling to the resonances. In section 4we have recovered the analytical results of section 3 as a numerical solution. This consists innumerically maximizing the coupling to the resonances of the S -matrix ansatz (4.6) or (4.11)under the constraint of unitarity.There are a number of interesting directions left to be developed. For instance, in d = 1 + 1spacetime dimensions, generalizing our results to systems involving many non-trivial 2 → S -matrix elements is of interest and could prove instructive for more involved problems in higherdimensions. This would also facilitate making contact with integrable models such as ref. [27]which have a known Lagrangian description and feature unstable particles. Another interestingdirection in d = 1 + 1 is to constrain unstable resonances by studying the crossing symmetryconstraints on four-point functions in the boundary of AdS in d = 1 + 1 and subsequently takingthe flat space limit [8].Perhaps the most promising direction to pursue, from the particle physics point of view, is togeneralize the results obtained here to d = 3 + 1. In section 4 we have explained a possible routetowards such generalization. This avenue promises many applications to particle physics and modelbuilding beyond the Standard Model. It would also be interesting to study unstable particles ofhigher spin in the 2 → S -matrix element which can be achieved via incorporating thecorresponding Legendre polynomials in the S -matrix ansatz.A naive generalization of the bound (3.12) to d = 3 + 1 suggests that we should find a maximalvalue of | S (cid:48) | for unstable resonances which is saturated in the limit where the amount of particleproduction is minimized. Furthermore, we anticipate an interesting interplay between the maximalvalue of | S (cid:48) | and the number of resonances allowed below a given energy. For instance, in analogy18 igure 6: Analytic continuation of Eq. (2.5) through the branch cut. to the result of section 3.2.3, in d = 3 + 1 we expect that given the value S (cid:48) ( θ i ) of the lightestresonance there is a lower bound on the mass of the next-to-lowest lying resonance. Acknowledgements
We thank G. Mussardo, S. Rychkov, B. van Rees and G. Villadoro for the useful discussions. We arealso grateful to J. Penedones, M. Serone and L. Vitale for the useful discussions and comments onthe draft. N. D. is supported by the PRIN project “Non-perturbative Aspects Of Gauge TheoriesAnd Strings”.
A Nature of the two-particle branch point
Eq. (2.5) can be used to argue that the two-particle threshold branch point is a square-root sin-gularity, see for instance [28]. The argument goes as follows. Consider the analytical continuationof S ( s + i(cid:15) ) into the second Riemann sheet by following a full anti-clockwise rotation around s = 4 m , see Fig. 6. Lets call such analytically continued function G ( s ). Then, under suchanalytical continuation, the unitary equation becomes S ( s − i(cid:15) ) G ( s − i(cid:15) ) = f ( s ) , (A.1)where we made use of continuity S ( s − i(cid:15) ) = G ( s + i(cid:15) ) (and assumed that f ( s ) has no branchpoints). Then, by taking the ratio between Eq. (2.5) and Eq. (A.1) one obtains G ( s − i(cid:15) ) = S ( s + i(cid:15) ) . (A.2)The latter equation would imply that rotating around the two-particle branch point twice the S -matrix is invariant. Therefore, if the S -matrix has a branch point and if the branch point isan algebraic singularity then it must be a square-root type singularity. A similar result can beobtained in d = 3 + 1 dimensions [11]. Note that this argument alone can not exclude the possibility of two-sheeted essential singularities. Total integrated phase-shift
The function S ( θ ) is meromorphic inside the physical strip Im θ ∈ (0 , π ). The so called argumentprinciple implies (cid:73) ∂P dθ πi ∂ θ (2 iδ ( θ )) = N z − N p . (B.1)where the closed contour integral encircles some region P in the physical strip and N z and N p arethe number of zeros and poles in that region, weighted by their order. In our particular physicalset up P is inside the physical θ -strip and thus there are no poles ( N p = 0) due to stable particles.Now, using crossing symmetry we have ∂ θ δ ( iπ + θ ) = − ∂ θ δ ( − θ ) and therefore the total phase-shift can be written as a contour integral (cid:90) ∞−∞ dθ ∂ θ δ ( θ ) = 12 (cid:73) C dθ ∂ θ δ ( θ ) , (B.2)with C is a contour encircling the whole physical strip. Then, by Cauchy residue theorem, (3.2)follow from (B.2). For simplicity we have assumed that S ( θ ) asymptotes to a constant at θ → ±∞ so that the contribution from the segments at infinity vanishes – this assumption can be relaxed. C Perturbative example
In this appendix we provide further details of the perturbative QFT discussed in section 3.3. TheFeynman rules for the theory in (3.17) are= ik − m + i(cid:15) , = ik − M + i(cid:15) , = − iλ , (C.1)where the plain line denotes the propagator for π and the dashed line denotes the propagator for σ . First order of business is to compute the loop corrections to propagation of σ which is capturedby the (amputated) diagram − i Π( k ) = (C.2)The loop integral can be carried out explicitly and yieldsΠ( k ) = i − iλ ) (cid:90) d q (2 π ) iq − m + i(cid:15) i ( k − q ) − m + i(cid:15) = λ π tanh − (cid:113) k k − m √ k √ k − m , (C.3)where the limit (cid:15) → k < m the loop correction Π( k ) is real. Above the two-particle thresholdΠ( k ) has a non-vanishing imaginary partIm Π( k ) = − λ (cid:112) k ( k − m ) θ ( k − m ) . (C.4) See for instance Theorem 4.1 of [29].
C.1 The perturbative S -matrix Incorporating the loop correction to the propagator (C.3), the quantum corrected propagator for σ is given by G ( k ) = = ik − M − Π( k ) + i(cid:15) (C.5)With this propagator we can readily compute the ππ → ππ component of the S -matrix. Thecontributing diagram in the s -channel is= − iλ s − M − Π( s ) + i(cid:15) = − λ G ( s ) (C.6)where s = ( p + p ) . In the centre of mass frame we have p = p = ( p , p ) and p = p = ( p , − p ).Thus t = ( p − p ) = 4 m − s and u = ( p − p ) = 0. The corresponding amplitude in the t -channel amounts to − λ G ( t ) = − λ G (4 m − s ) and the u -channel diagram yields a contributionequal to − λ G ( u ) = − λ G (0). The one-loop amplitude – up to higher order corrections – isgiven by i M ( s ) = iλ M + λ πm − iλ s − M − Π( s ) − iλ m − s − M − Π(4 m − s ) . (C.7)Then, the ππ → ππ component of the S -matrix is given by S = S ( s ) = (cid:32) i M ( s )2 (cid:112) s ( s − m ) (cid:33) (C.8)where the identity is the inner product of two particle states = (cid:104) p , p | p , p (cid:105) , which is given by = (2 π ) E E (cid:0) δ ( p − p ) δ ( p − p ) + δ ( p − p ) δ ( p − p ) (cid:1) , (C.9)and the extra factor multiplying M ( s ) arises from the identity (2 π ) δ (2) ( p + p − p − p ) = / (2 √ s √ s − m ).A few remarks regarding S ( s ) defined in (C.8) are in order. As we will demonstrate below, theabove S -matrix has complex poles hidden behind a branch cut stemming from the two-particlethreshold. These poles correspond to the exchange of an on-shell unstable σ particle which amountsto a resonance in the scattering process whose width is determined byΓ = − Im Π( M ) /M ren (C.10) Namely, vertex corrections and box diagrams. S ( s ) stemming from suchpoles. Alternatively we can extract the same information about the resonance from the zeros of S ( s ) as they too give rise to a branch cut in log S ( s ) and thus result in a phase shift. Moreoverthe zeros are in a sense more fundamental as they are not hidden behind any branch cuts and wedo not need to analytically continue the S -matrix through branch cuts to study them. Below weanalyze the resonance in our perturbative example first by studying the poles and later throughthe zeros. C.2 Resonances, poles and zeros
As discussed in section 3.1, resonances are associated with branch cuts in the s -plane for thefunction log S ( s ) which typically connect a pair of pole and zero of the S -matrix the location andresidue of which determine the features of the resonance. Here we study the zeros and poles of theperturbative S -matrix (C.8). C.2.1 Poles in s -plane Poles associated to unstable resonances are not visible on the complex s -plane and are hiddenbehind multi-particle branch cuts of the S -matrix. To illustrate this in our perturbative examplenote that (C.8) has square-root branch points at s = 0 and at s = 4 m . We take the branch cutsto stretch along the real axis from −∞ to 0 and from 4 m to + ∞ . The location of the poles of S ( s ) due to the exchange of a σ in the s -channel are determined by s − M − Π( s ) = 0 . (C.11)Now consider the following ansatz for the solution s ∗ = 4 m + s ◦ e iϕ +2 πin (C.12)where s ◦ is taken to be positive, ϕ ∈ [0 , π ) and n = 0 , √ s − m . Plugging our ansatz into the equation (C.11) and takingthe real part we obtain s ◦ cos ϕ + 4 m − M − ReΠ( s ∗ ) = 0 . (C.13)This implies s ∗ = M + O ( λ ) and since 4 m − M − ReΠ( M ) < ϕ ∈ (0 , π/ ∪ (3 π/ , π ). Therefore, the imaginary part of (C.11) simplifies to s ◦ sin ϕ + λ − n cos( ϕ/ M √ s ◦ = 0 . (C.14)where we have used (C.4) and have omitted higher order terms using that sin ϕ = O ( λ ). It isevident that the equation has no solution for n = 0, i.e. in the physical sheet. To find a solutionwe have to take n = 1 which as can be seen from (C.12) corresponds to the analytic continuationof the function √ s − m into the second sheet. Here we have assumed M (cid:62) m . For M < m we find a pole on the real axis and below the two-particlethreshold corresponding to production of a stable particle of mass M . s = 4 m we now switch to the θ -variable, introduced in section 2.1.Recall that θ = θ ( s ) maps the two-sheeted Riemann surface associated to Eq. (C.8) into the strip θ ∈ [ − iπ, iπ ). C.2.2 Poles and zeros of the S -matrix The S -matrix on the θ -strip, in units m = 1, was given in (3.22): S ( θ ) = 1+ iλ θ (cid:34) M + λ π −
14 cosh (cid:0) θ (cid:1) − M − Π( θ ) + 14 sinh (cid:0) θ (cid:1) + M + Π( iπ − θ ) (cid:35) . (C.15)We remind the reader that the above expressions are perturbative results valid only up to order λ . We are interested in finding poles and zeros of the S -matrix in the fundamental domain ofcomplex θ . To this end we can use a series expansion for small λ . The poles arise when one ofthe denominators vanishes. Since S = 1 + O ( λ ), the zeros lie in the regions where one of thedenominators is of order λ . Thus we can look for location of zeros and poles in parallel. Thelocation of the zeros and poles arising from the s -channel contribution are determined by theequation 4 sinh θ (cid:0) ( θ/ − M − Π( θ ) (cid:1) = iaλ (C.16)where a = 0 for poles and a = 1 for zeros.Note that the t and u -channel contributions to (C.16) appear at order λ along with contri-butions from other diagrams we have not considered. These contributions only affect the positionof the pole and the zero at order λ . We are interested in poles and zeros near s ◦ = M or θ ±◦ = ± − (cid:0) M (cid:1) . We therefore look for perturbative solutions of the form θ ± σ = θ ±◦ + λ θ ±∗ (C.17)Plugging this ansatz into (C.16) we find8 θ ±∗ sinh θ ±◦ − θ ±◦ − iπ π = ia (C.18)and therefore θ ±∗ = 18 sinh θ ±◦ (cid:18) θ ±◦ π + i ( a −
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