A simple model for a scalar two-point correlator in the presence of a resonance
AA simple model for a scalar two-point correlator in the presence ofa resonance
Peter C. Bruns
Institut f¨ur Theoretische Physik, Universit¨at Regensburg, D-93040 Regensburg, Germany (Dated: November 14, 2018)
Abstract
We present a simple toy model for a scalar-isoscalar two-point correlator, which can serve as atesting ground for the extraction of resonance parameters from Lattice QCD calculations. We discussin detail how the model correlator behaves when it is restricted to a finite spatial volume, and howthe finite-volume data can be used to reconstruct the spectral function of the correlator in the infinitevolume, which allows to extract properties of the resonance from such data. a r X i v : . [ h e p - l a t ] M a r . INTRODUCTION AND DISCLAIMER In the past years, the extraction of information on hadronic resonances from Lattice QCDdata has become of great interest. Recent lattice studies of this subject can e.g. be found in[1–12], and references therein. In Quantum Field Theory (QFT), resonances are associatedwith poles of scattering or transition amplitudes on unphysical Riemann sheets of the complexenergy surface pertaining to these amplitudes, and all properties of the resonance should refersomehow to these poles in the complex plane [13]. In a finite volume, however, the energyspectrum is discrete, and the correlators and amplitudes are purely real in euclidean time.There are no branch cuts and no Riemann sheets, and it therefore seems nontrivial to relatethe measurements in a finite box to the resonance phenomena observed in (or extracted from)experimental data. Of course, the theory needed to close this gap is already well-developed,see [14–18] for the “classic” articles on the subject, and [19–30] for some recent developments.In the literature just cited, the problem with which we deal here is examined in much moredepth and generality than is attempted in the present article. Here, we focus on a specificsimple field-theoretic toy model - perhaps the simplest one that shows all the features wewant to study on a basic level (resonance dynamics in a finite volume, extraction of datarelated to a form factor in the time-like region in the presence of a resonance), but is stillconsistent with the field-theoretical strictures of unitarity and analyticity. Our first aim is toprovide the practitioner (i.e., people who are concerned with the analysis of realistic latticedata) with an explicit model, as a testing ground for the methods used to deal with realhadronic resonances in Lattice QCD. Our second aim is to present an amenable representationof a non-trivial (momentum-projected) two-point correlator which can be used for pedagogicpurposes, since we have often noticed that the relation between resonance physics andfinite-volume, euclidean-time lattice data is often hard to understand for students in the fieldof Lattice QCD, or even for researchers in related areas. Our third aim is to make sometechnical details of the relevant derivations and calculations available, which are often hard toretrace from the original papers on the subject. Here we try to be as explicit as possible. Wedo not treat discretization effects, and we shall neglect, for the most part, those finite-volumeeffects which are exponentially suppressed with the spatial extent of the finite box, like e.g.finite-volume corrections to masses and coupling constants. The methods to deal with thelatter modifications are very well-known, see e.g. [31–33].2he plan of this article is as follows: In Sec. II, we introduce the general frameworkand the basic properties of the two-point correlator in our model. In Sec. III, we outlinethe construction of the amplitude used to describe the two-body scattering between theelementary, stable particles of our theory (called φ particles here). The model for the correlatorand its behavior in a finite volume are discussed in detail in Sec. IV. In Sec. V, we go througha numerical demonstration of the methods and results obtained in the previous sections, andgive a short conclusion and outlook. 3 I. GENERAL PROPERTIES OF THE SCALAR TWO-POINT CORRELATOR
We consider the two-point correlator of a scalar operator S ( x ), c SS ( x − y ) := (cid:104) | T S ( x ) S ( y ) | (cid:105) , (1)within a model field theory of pseudoscalar φ particles of mass M . We assume that there isan s-wave resonance, called σ here, which occurs in φφ → φφ scattering. For simplicity, theonly asymptotic states for which the operator S ( x ) has non-vanishing matrix elements with thevacuum state are assumed to be the two-particle φφ states, (cid:104) |S ( x ) | φ ( k ) φ ( k ) (cid:105) (cid:54) = 0 (however,in the case where the σ becomes a bound state instead of a resonance, we shall also admit a σ state and (cid:104) |S ( x ) | σ ( k ) (cid:105) (cid:54) = 0). We are, in particular, interested in the value of the matrixelements for a specific prescribed three-momentum p . From now on we let y = (0 , ), anddenote the “momentum-projected” matrix element as˜ c SS ( t, p ) := (cid:90) d x e − i p · x c SS ( t, x ) . (2)According to the general rules of Quantum Field Theory , the correlator can be represented as c SS ( t, x ) = (cid:90) d q (2 π ) e − iqx i M ( q ) , (3)where i M ( q ) is the sum of all Feynman graphs (in the given field theory) with operator inser-tions at two fixed space-time positions x = ( t, x ), t >
0, and y = (0 , ), and where q = ( q , q )can be interpreted as the four-momentum flowing into the operator insertion at y and out ofthe operator insertion at x . The vertex rule for the operator insertion coupling to φφ is simplygiven by a constant b in our model (and ˜ b in the case of the σ ), so that in the limit where theinteractions are “turned off ” , we have just (cid:104) |S (0) | φ ( k ) φ ( k ) (cid:105) → b and (cid:104) |S (0) | σ ( k ) (cid:105) → ˜ b .For example, the exchange of an “undressed” σ particle (with mass m ) gives a contribution i M σ − ex . = i ˜ b q − m + i(cid:15) ⇒ ˜ c SS ( t, p ) | σ − ex . = ˜ b e − it √ | p | + m (cid:112) | p | + m , (4)where we have used the appropriate i(cid:15) prescription. The relevant Fourier integral can be inferredfrom Eq. (A.18). See e.g. the textbook [34], in particular Chapter 6 and Eqs. (10.4.19,20) therein, and also Chapter 9 of [35]. xchange of a pair of non-interacting φ particles From the exchange of a pair of free pseudoscalar φ particles between the two vertex insertionsresults the contribution i M φφ − ex . = i b (cid:90) d d l (2 π ) d i [( q − l ) − M + i(cid:15) ][ l − M + i(cid:15) ] d → = ib (cid:18) I φφ ( q = 4 M ) + 4 M − q π (cid:90) ∞ M ds (cid:48) σ ( s (cid:48) )( s (cid:48) − M )( s (cid:48) − ( q + i(cid:15) )) (cid:19) =: ib (cid:0) I φφ ( q = 4 M ) + ¯ I φφ ( q ) (cid:1) , σ ( s (cid:48) ) := (cid:114) − M s (cid:48) . (5)The first term in the second line, I φφ ( q = 4 M ), contains a divergent constant for d → ∼ δ ( t ) in ˜ c SS ( t, p ). As we are interested in thebehavior of the correlator for large positive times, this constant term can be dropped. Then,˜ c SS ( t, p ) | φφ − ex . t> = i b (cid:90) ∞ M ds (cid:48) σ ( s (cid:48) )32 π ( s (cid:48) − M ) (cid:90) + ∞−∞ dq (( q ) − ( | p | + 4 M )) e − iq t ( q ) + i(cid:15) − ( | p | + s (cid:48) )= b π (cid:90) ∞ M ds (cid:48) (cid:18) σ ( s (cid:48) )32 π (cid:19) e − i √ s (cid:48) + | p | t (cid:112) s (cid:48) + | p | = b π (cid:90) ∞ ε φφp dE (cid:48) (cid:112) E (cid:48) − ( | p | + 4 M ) (cid:112) E (cid:48) − | p | e − iE (cid:48) t . (6)In the last line, we have introduced ε φφp := (cid:112) | p | + 4 M and a new integration variable E (cid:48) := (cid:112) s (cid:48) + | p | . Integrals of the type of this result are studied in App. A. From those results, we caninfer the behavior of our matrix element contribution for large positive times t (see Eqs. (A.9)and (A.13)-(A.15)) ˜ c SS ( t, p ) (cid:12)(cid:12) φφ − ex . t →∞ −→ b π (cid:113) ε φφp M e − iε φφp t ( it ) . (7)It is reassuring to see that the result is real and positive for euclidean times τ = it .Given that the full amplitude satisfies a dispersive representationIm M ( q ) = − B ∗ ( q ) σ ( q )32 π B ( q ) , M ( q ) = 1 π (cid:90) ∞ M ds (cid:48) Im M ( s (cid:48) ) s (cid:48) − ( q + i(cid:15) ) , (8)with some complex function B ( q ), the above result can be readily generalized,˜ c SS ( t, p ) = 132 π (cid:90) ∞ ε φφp dE (cid:48) (cid:112) E (cid:48) − ( | p | + 4 M ) (cid:112) E (cid:48) − | p | | B ( E (cid:48) ) | e − iE (cid:48) t . (9)Again, given Eq. (8), we find that ˜ c SS ( t, p ) is real and positive for euclidean times τ = it .5 xchange of a pair of non-interacting φ particles - in a finite volume In a cubic box with side length L , employing periodic boundary conditions, the three-momentaare quantized, so that e.g. q = πL N , where N ∈ Z . Accordingly, the momentum integralsnow correspond to discrete summations, (cid:82) d l → (cid:0) πL (cid:1) (cid:80) n ∈ Z with quantized loop momenta.Consider the function I Lφφ ( x ; q ) := 12 (cid:18) πL (cid:19) (cid:90) d l (2 π ) ie − ilx (cid:80) n ∈ Z δ (cid:0) l − πL (cid:0) n + N (cid:1)(cid:1)(cid:104)(cid:0) q − l (cid:1) − M + i(cid:15) (cid:105) (cid:104)(cid:0) q + l (cid:1) − M + i(cid:15) (cid:105) = 12 (cid:90) d l (2 π ) ie − ilx (cid:80) k ∈ Z e i k · ( L l − π N ) (cid:104)(cid:0) q − l (cid:1) − M + i(cid:15) (cid:105) (cid:104)(cid:0) q + l (cid:1) − M + i(cid:15) (cid:105) (10)= − π (cid:88) k ∈ Z (cid:90) + − dz e − iqxz e iπ k · N (2 z − K (cid:16) M ( z ) (cid:112) | x + L k | − t (cid:17) , where x = ( t, x ) and M ( z ) = (cid:113) M − q + q z − i(cid:15) . We have used the Poisson summationformula in the second line, and introduced a Feynman parameter z in the third line. Themodified Bessel function K appears due to the results of App. B. For q > M , M ( z ) is notreal in the whole integration range, and the summations do not necessarily converge there. Letus limit ourselves to q ≤ M for a moment, and let t →
0. For L → ∞ , only the term with k = survives, and so we find I ∞ φφ ( x ; 0) = − π (cid:90) + − dz K ( M | x | ) = − K ( M | x | )16 π = 116 π (cid:18) log (cid:18) M | x | (cid:19) + γ E (cid:19) + O ( | x | ) ,I ∞ φφ ( x ; 0) − I Lφφ ( x ; 0) = 116 π (cid:88) (cid:54) = k ∈ Z K ( M | x + L k | ) , while for q → (2 M, ), we find I ∞ φφ ( x ; (2 M, )) = − π (cid:90) + − dz K (2 M | z || x | ) = 116 π (cid:18) log (cid:18) M | x | (cid:19) + γ E − (cid:19) + O ( | x | ) , so that in the difference lim x → (cid:0) I ∞ φφ ( x ; 0) − I ∞ φφ ( x ; (2 M, )) (cid:1) = (16 π ) − , the regulator | x | − drops out. Starting from the first line of Eq. (10), we use the residue theorem to compute I Lφφ ( x ; q ) − I Lφφ ( x ; 0) x → −→ L (cid:88) n ∈ Z (cid:32) E ( n )+ + E ( n ) − E ( n )+ E ( n ) − ( q + i(cid:15) − (( E ( n )+ + E ( n ) − ) − | q | ) + 14( E ( n )0 ) (cid:33) ,E ( n ) ± = (cid:115)(cid:18) πL (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) N ± (cid:18) n + N (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + M , E ( n )0 = (cid:115)(cid:12)(cid:12)(cid:12)(cid:12) πL n (cid:12)(cid:12)(cid:12)(cid:12) + M , (11)6nd where q ≡ ( q ) − | q | , q = πL N . Combining the above observations, we are in a positionto write down the finite-volume generalization ¯ I fvφφ ( q, L ) of the subtracted (infinite-volume) loopfunction ¯ I φφ ( q ):¯ I fvφφ ( q, L ) := I Lφφ (0; q ) − I Lφφ (0; 0) + (cid:0) I Lφφ (0; 0) − I ∞ φφ (0; 0) (cid:1) + (cid:0) I ∞ φφ (0; 0) − I ∞ φφ (0; (2 M, )) (cid:1) = 116 π − (cid:88) (cid:54) = k ∈ Z K ( M L | k | ) + 1 L (cid:88) n ∈ Z (cid:32) E ( n )+ + E ( n ) − E ( n )+ E ( n ) − ( q + i(cid:15) − (( E ( n )+ + E ( n ) − ) − | q | ) + 1(2 E ( n )0 ) (cid:33) . (12)In this way, we have assured that the subtraction I ∞ φφ (0; (2 M, )) = I φφ ( q = 4 M ) is the samein finite and in infinite volume, so that the “renormalization” does not depend on the volume L . Keeping | N | ∼ | q | fixed, we can write this in a dispersive form as in Eqs. (5) and (6):¯ I fvφφ ( q, L ) = ι q − q π (cid:90) ∞ M ds (cid:48) (cid:80) n ∈ Z (cid:18) π (cid:16) E ( n )+ + E ( n ) − (cid:17) L E ( n )+ E ( n ) − (cid:19) δ (cid:16) s (cid:48) − s (0) n ( q ) (cid:17) s (cid:48) ( s (cid:48) − ( q + i(cid:15) )) ,ι q = 116 π − (cid:88) (cid:54) = k ∈ Z K ( M L | k | ) + (cid:88) n ∈ Z E ( n )0 L ) − π (cid:90) ∞ M ds (cid:48) s (cid:48) (cid:88) n ∈ Z π (cid:16) E ( n )+ + E ( n ) − (cid:17) L E ( n )+ E ( n ) − δ (cid:0) s (cid:48) − s (0) n ( q ) (cid:1) ,s (0) n ( q ) := ( E ( n )+ + E ( n ) − ) − | q | . (13)The subtraction term ι q does not depend on q and therefore drops out in the momentum-projected correlator ˜ c SS ( t > , p ),˜ c fvSS ( t, p ) | φφ − ex . = b π (cid:90) ∞ M ds (cid:48) (cid:88) n ∈ Z π (cid:16) E ( n )+ + E ( n ) − (cid:17) L E ( n )+ E ( n ) − δ (cid:0) s (cid:48) − s (0) n ( p ) (cid:1) e − i √ s (cid:48) + | p | t (cid:112) s (cid:48) + | p | = b L (cid:88) n ∈ Z e − i (cid:16) E ( n )+ + E ( n ) − (cid:17) t (2 E ( n )+ )(2 E ( n ) − ) , (14)with p = πL N , N ∈ Z . It should be clear that Eqs. (6) and (14) just correspond to an insertionof a complete set of free φφ states. Schematically, ˜ c SS ( t, p ) | φφ − ex . is expanded as ∼ (cid:90) d x e − i p · x (cid:90) d q (2 π ) E q (cid:90) d q (2 π ) E q (cid:104) |S ( x ) | φ ( q ) φ ( q ) (cid:105)(cid:104) φ ( q ) φ ( q ) |S (0) | (cid:105) . (15)7 emark on Eq. (12): Note that the limit (cid:15) →
0+ is implicit in Eq. (12). It is obvious that the sum of (real) poleterms cannot be a meaningful approximation to the loop integral for q > M in this limit,which is complex for such q , and has no poles. This is because the integrand has a singularityin the integration range (see e.g. Eq. (5)). In other words, the limits (2 π/L ) → (cid:15) → q comes close to a certain poleposition with index n , and (cid:15) →
0, the corresponding pole cancels out. To this end, we note thefollowing integral representation of the imaginary part of I φφ ( s ), for s > M :1 √ s + i(cid:15) (cid:90) d l (2 π ) (cid:18) s + i(cid:15) − | l | + M ) + | l | + 3 M | l | + M ) (cid:19) = − iσ ( s + i(cid:15) )32 π , (16)which suggests to subtract the corresponding terms in the integrand from ¯ I fvφφ (let us set N = for this demonstration): p (cid:15) ( s, L ) := (cid:18) πL (cid:19) (cid:88) n ∈ Z (cid:32) π E ( n )0 ( s + i(cid:15) − E ( n )0 ) ) + 164 π ( E ( n )0 ) (cid:33) − (cid:18) πL (cid:19) (cid:88) n ∈ Z (cid:32) π √ s + i(cid:15) ( s + i(cid:15) − E ( n )0 ) ) + 18 π ( E ( n )0 ) + 2 M √ s + i(cid:15) ( E ( n )0 ) (cid:33) = (cid:18) πL (cid:19) (cid:88) n ∈ Z ( s + i(cid:15) − M ) E ( n )0 − M √ s + i(cid:15) π ( E ( n )0 ) √ s + i(cid:15) ( √ s + i(cid:15) + 2 E ( n )0 ) . (17)This expression has no pole in s for s > M , so we can take the limit (cid:15) → , L → ∞ to obtainthe integral p (cid:15) ( s, L ) → p ( s, ∞ ) = 1 √ s (cid:90) d l (2 π ) ( s − M ) (cid:112) | l | + M − M √ s (cid:112) | l | + M ) ( √ s + 2 (cid:112) | l | + M )= 116 π ( σ ( s ) artanh( σ ( s )) −
1) = Re ¯ I φφ ( s ) − π . (18)So we have learnt that we can approximate (neglecting in particular exponentially suppressedterms ∼ K ( M L | k | ) in Eq. (12)) for s > M , M L (cid:29) I fvφφ ( s, L ) ≈ Re ¯ I φφ ( s ) + 1 √ sL (cid:88) n ∈ Z (cid:32) s − E ( n )0 ) + ( E ( n )0 ) + 2 M E ( n )0 ) (cid:33) . (19)It is (only) in the sense of Eqs. (16) and (19) that ¯ I fvφφ is a finite-volume “approximation” to¯ I φφ for s > M (for s < M , we have ¯ I fvφφ ≈ ¯ I φφ ).8 II. UNITARY MODEL FOR THE SCATTERING AMPLITUDE
In this section, we describe our model for the σ self-energy, the σφφ vertex function and the φφ scattering amplitude. We split the complete scattering amplitude as T = T p + T np , where T p collects all terms containing the resonance pole, i.e. all s-channel resonance exchangegraphs, while T np contains the non-resonant “background”. Our ansatz for T np is T np ( s ) = 1 α − + ¯ I φφ ( s ) , (20)with a real parameter α and the loop function (see Eq. (5))¯ I φφ ( s ) = − σ ( s )16 π artanh (cid:18) − σ ( s ) (cid:19) , σ ( s ) = (cid:114) − M s . (21)This amplitude has no resonance poles by construction. For α <
0, however, there are “artefact”poles on the negative s -axis of the physical sheet, which have no physical interpretation. Wecan use the background amplitude to construct the σφφ vertex function,Γ( s ) = g − g ¯ I φφ ( s ) T np = g α ¯ I φφ ( s ) , (22)with a real resonance coupling parameter g , and the self-energy of the σ resonanceΣ( s ) = g ¯ I φφ ( s )Γ( s ) . (23)This self-energy obeys the unitarity relation (for real s > M )Im Σ( s ) = Γ ∗ ( s )(Im ¯ I φφ ( s ))Γ( s ) = − Γ ∗ ( s ) (cid:18) σ ( s )32 π (cid:19) Γ( s ) . (24)The pole-part of the scattering amplitude is then constructed as T p ( s ) = − Γ( s ) 1 s − m − Σ( s ) Γ( s ) , (25)and the full scattering amplitude can simply be written as T ( s ) = 1 (cid:16) α + g m − s (cid:17) − + ¯ I φφ ( s ) = 1 (cid:16) α + g m − M (cid:17) − + s − M s − M g /α s − s + ¯ I φφ ( s ) , (26)with s := m + ( g /α ). The mass parameter m can be interpreted as the mass of the“undressed” resonance. The model described above is very simple - it has only one more9arameter than a Breit-Wigner parameterization. Moreover, it is a pure s-wave amplitude,with the s-wave phase shift δ ( s ) given by t ( s ) = 132 π T ( s ) = e iδ ( s ) − iσ ( s ) , s > M . (27)The model amplitude fulfills elastic unitarity exactly, Im t = t ∗ σ ( s ) t for s > M , but allhigher-lying (multi-) particle intermediate states are neglected, so it should be viewed as an ef-fective description valid only below inelastic thresholds. It also does not possess a left-hand cut. • For α > , m > M , the amplitude has a resonance pole on its second Riemann sheet,and is analytic on the cut physical sheet. It is then a solution to the “one-channel Royequation” [36] with one CDD pole ([37], see also Sec. 3 of [38]). • For α > , m < M , there is in addition a bound state at some energy 0 < √ s < M . • For α <
0, there are unphysical poles on the first Riemann sheet. This is unpleasant,but since these poles could occur at large | s | , such a result could not rule out the presentmodel as a valid description at low energies. Should such a pole, however, occur in thelow-energy region (say, for | s | < M ), the solution should be rejected.Recall that two additional parameters b and ˜ b are introduced, which give the direct couplingof the scalar operator to two free φ fields and to an undressed σ , respectively. IV. UNITARY MODEL FOR THE CORRELATOR
From the simple model for the φφ scattering amplitude described in the previous section, wecan construct a model for the Fourier transform M of the correlator. Let β ( s ) = b (cid:0) − ¯ I φφ ( s ) T np ( s ) (cid:1) , (28)˜ β ( s ) = ˜ b − b ¯ I φφ ( s )Γ( s ) . (29)Then the contributions with ( p ) and without ( np ) resonance pole terms are M p ( s ) = ˜ β ( s ) s − m − Σ( s ) , M np ( s ) = b ¯ I φφ ( s ) β ( s ) , M ( s ) = M p ( s ) + M np ( s ) . (30)10t is then straightforward to show that, for real s = q , M ( s ) = b ¯ I φφ ( s ) + ˜ b (1+ α ¯ I φφ ( s )) − b ¯ I φφ ( s ) gbs − m I φφ ( s ) (cid:16) α + g m − s (cid:17) , M ( m ) = ˜ bg ¯ I φφ ( m ) (cid:16) ˜ b + ¯ I φφ ( m ) (cid:16) α ˜ b − gb (cid:17)(cid:17) , Im M ( s ) = B ∗ ( s ) (cid:0) Im ¯ I φφ ( s ) (cid:1) B ( s ) , B ( s ) = b + ˜ bgm − s I φφ ( s ) (cid:16) α + g m − s (cid:17) . (31)Note that B ( s ) and M ( s ) have the same pole structure as T ( s ) of Eq. (26), so that theconstraints of Eq. (8) are fulfilled for our model amplitude (given that α > , m > M ).Our Eqs. (8) and (9) correspond to a summation over φφ intermediate states: There are no “ σ states” in the case where the σ is a resonance. The resonance effects are all contained in thepole structure of the complex function B ( s ), which is the same as that of T ( s ) (see Eq. (26)), B ( s ) = B ( s ) − B ( s ) ¯ I φφ ( s ) T ( s ) , B ( s ) := b + ˜ bgm − s , (32)so that, from (21), (27) and (32), one can show B ( s ) = | B ( s ) | e iδ ( s ) (“Watson theorem”).In the case where the σ is a bound state, the amplitude M p ( s ) has a pole at s = s σ ! = m σ on the first sheet, which therefore directly appears in the spectrum, just like the “undressed”resonance would (compare Eq. (4)), M p ( s ) = Z σ ˜ β ( s σ ) s − s σ + . . . , Z σ := 11 − Σ (cid:48) ( s σ ) , (33)˜ c SS ( t, p ) | σ b − ex . = Z σ ˜ β ( m σ ) e − it √ | p | + m σ (cid:112) | p | + m σ ( σ b : bound state) . (34)In the resonant case, the pole is located on the second Riemann sheet, at s σ = ( m σ − ( i/ σ ) ,and therefore does not appear in the dispersive representation of the amplitude M (compareEqs. (8), (9)). What can be extracted from the time-dependence of the correlator is only thefunction | B ( s ) | (written as | B ( E (cid:48) ) | in Eq. (9), where p is fixed). With s σ − m − Σ( s σ ) ! = 0 ⇒ ¯ I φφ ( s σ ) = s σ − m g − α ( s σ − m ) (35)(note that ¯ I φφ is to be evaluated on the second sheet here), it is possible to show that (cid:112) Z σ ˜ β ( s σ ) = − lim s → s σ ( s − s σ ) B ( s ) √ Z σ Γ( s ) = − Res s σ B (cid:112) − Res s σ T , (36)which gives us (within our present model) the generalization of the matrix element (cid:104) |S (0) | σ (cid:105) for the case where the σ is a resonance, in terms of the residues of the form factor B and thescattering amplitude T at the resonance pole.11 inite-volume analysis In a finite volume L , the loop function ¯ I φφ in (31) has to be replaced by its finite-volumecounterpart ¯ I fvφφ ( q, L ) (see Eq. (12)), and we deduce that M ( q ) is then given as a series of pole terms, M ( q, L ) = const . + 4 (cid:18) πL (cid:19) ∞ (cid:88) j =0 r j q − s j , s j +1 > s j , (37)where the pole positions are shifted from the values s (0) n (see Eq. (13)) by the interaction, s j ( n ) = s (0) n + O ( α, g ) . In principle, the form Eq. (37) is just given by the discretization ofthe dispersive integral in Eq. (8), which should be a valid approximation to this integral atdistances (cid:29) (2 π/L ) from the positive real s -axis.In the present model, it is easy to see that there are only simple poles: Suppose that there aretwo neighboring pole positions s , of M ( q, L ), s = s + δs > s , and apply the pole conditionsof Eq. (35), ¯ I fvφφ ( s ) − ¯ I fvφφ ( s ) ! = ( s − s ) g ( g − α ( s − m ))( g − α ( s − m )) ⇒ dds ¯ I fvφφ ( s ) δs + O (( δs ) ) ! = δs g ( g − α ( s − m )) + O (( δs ) ) , (38)but the leading term on the right-hand side is non-negative, while the one on the left-hand sideis non-positive (considering Eq. (12) with q → s ), and thus δs cannot become arbitrarily smallas long as L < ∞ , g > s j determines a finite-volume approximationto the (infinite-volume) phase-shift δ ( s ) at s = s j . This is easily seen for toy models as discussedhere, with simple real potentials like V ( s ) = α +( g / ( m − s )), where the phase-shift is explicitlygiven by (see Eqs. (26), (27)) e iδ ( s ) = 32 π (cid:0) V − ( s ) + Re ¯ I φφ ( s ) (cid:1) + iσ ( s )32 π (cid:0) V − ( s ) + Re ¯ I φφ ( s ) (cid:1) − iσ ( s ) ≡ cot δ ( s ) + i cot δ ( s ) − i . (39)Neglecting all finite-volume effects which are exponentially damped with M L , and writing theequation determining the pole positions in the finite volume (see Eq. (35)) as V − ( s j ) + ¯ I fvφφ ( s j ) ! = 0 , (40)we can approximate the expression for the phase shift using Eq. (19), V − ( s j ) + Re ¯ I φφ ( s j ) ≈ − √ s j L (cid:88) n ∈ Z (cid:32) s j − E ( n )0 ) + ( E ( n )0 ) + 2 M E ( n )0 ) (cid:33) , j = 0 , , , . . . (41)12n the continuum limit, the pole positions s i move closer and closer together, and the function M ( q ) acquires a branch cut, and a Riemann sheet structure. The function is then essentiallydetermined by a spectral function, integrated along the unitarity branch cut, as in Eq. (8). Thequestion is whether this spectral function can somehow be (approximately) reconstructed fromthe set of numbers { r i , s i } , in a way similar to the reconstruction of the scattering phase shiftwith the help of the (squared) energy levels s i and Eqs. (39)-(41).From (37) and (2), (3), the momentum-projected correlator in a finite volume will be of theform ˜ c fvSS ( t, p ) = 4 (cid:18) πL (cid:19) (cid:88) j r j e − i √ s j + | p | t (cid:112) s j + | p | . (42)Assume that the poles are distributed as s j +1 − s j = 4 (cid:0) πL (cid:1) /ρ ( s j ), with a positive real function ρ ( s ). We can apply a rescaling function to obtain a set of positions ˜ s j with an equidistantdistribution, ˜ s ( s j ) = s + 4 j (cid:0) πL (cid:1) (the prefactors here and in Eqs. (37), (42) have been chosensuch that ρ ( s ) = 1 for the free case, with N = ). In the infinite-volume limit, the sum inEq. (42) tends to the integral˜ c fvSS ( t, p ) → (cid:90) ∞ M d ˜ s r ( s (cid:48) ) e − i √ s (cid:48) + | p | t (cid:112) s (cid:48) + | p | = (cid:90) ∞ M ds (cid:48) ρ ( s (cid:48) ) r ( s (cid:48) ) e − i √ s (cid:48) + | p | t (cid:112) s (cid:48) + | p | , (43)where r ( s (cid:48) ) is a smooth function that interpolates between the values r j . The last equation hasto be compared with Eqs. (6) and (9). With the help of those equations, together with (31)(with ¯ I φφ replaced by ¯ I fvφφ ( q, L ) for the finite volume), and (40), we find (recall that V ( s ) := α + ( g / ( m − s )) ): ρ ( s j ) ! ≈ − (cid:18) σ ( s j )32 π (cid:19) (cid:0) πL (cid:1) V ( s j ) dds (cid:104) V ( s ) ¯ I fvφφ ( s ) (cid:105) s j (cid:12)(cid:12) V ( s j ) ¯ I φφ ( s j ) (cid:12)(cid:12) = − (cid:18) σ ( s j )32 π (cid:19) (cid:0) πL (cid:1) dds (cid:104) ( V ( s )) − + ¯ I fvφφ ( s ) (cid:105) s j (cid:12)(cid:12) ( V ( s j )) − + ¯ I φφ ( s j ) (cid:12)(cid:12) . (44)Note that the last expression is positive because both dds V − and dds ¯ I fvφφ are negative. FromEq. (19), we can express the derivative in the numerator as dds (cid:104) ( V ( s )) − + ¯ I fvφφ ( s ) (cid:105) s j ≈ dds (cid:2) ( V ( s )) − + Re ¯ I φφ ( s ) (cid:3) s j + 12 s j (cid:0) ( V ( s j )) − + Re ¯ I φφ ( s j ) (cid:1) − √ s j L (cid:88) n ∈ Z (cid:16) s j − E ( n )0 ) (cid:17) . (45)13or p (cid:54) = 0, the corresponding energy levels should be used. Inserting (45) in (44), and usingEq. (39) and its derivative w.r.t. s , we find that the spectral function of M (see Eqs. (8), (31))can be reconstructed from the measured matrix elements in the finite volume in the followingsense: (cid:18) σ ( s j )32 π (cid:19) | B ( s j ) | ≈ πρ ( s j )4 (cid:0) πL (cid:1) (cid:34) (cid:18) πL (cid:19) r j (cid:35) , (46)with πρ ( s j )4 (cid:0) πL (cid:1) ≈ dδ ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) s j − sin 2 δ ( s j )4( s j − M ) + 32 π (cid:112) s j − M L (cid:88) n ∈ Z sin δ ( s j ) (cid:16) s j − E ( n )0 ) (cid:17) . (47)The requirement that dδ ( s ) ds − sin 2 δ ( s )4( s − M ) ! > | B ( s j ) | and 4 (cid:0) πL (cid:1) r j in Eq. (46) have the simple interpretation of energy-level densities.To conclude, one can extract resonance properties from numerical finite-volume data as follows: • Determine the (squared) energy levels s i and the coefficients 4 (cid:0) πL (cid:1) r j which determinethe momentum-projected correlator (Eq. (42) evaluated in euclidean time t = − iτ ), • reconstruct the scattering phase shift in the infinite volume via Eq. (39) (with (41)) , • use a parameterization of the scattering amplitude, which satisfies the correct analyticityand unitarity conditions, to make a fit to the phase-shift “data”, and determine the massand width of the resonance (given by the resonance pole position on the second Riemannsheet) by analytic continuation of T ( s ) to the complex s -surface. • Use the energy levels and the scattering phase shift δ ( s ) to compute the density function ρ ( s ) (Eq. (47)). • Obtain | B ( s j ) | from the measured coefficients 4 (cid:0) πL (cid:1) r j via Eq. (46). • Obtain the time-like form factor B ( s ) = | B ( s ) | e iδ ( s ) for s ≥ M . • Employ a parameterization for B ( s ) to analytically continue to the complex plane, asdone for the scattering amplitude T ( s ), and find the resonance decay matrix elementfrom the corresponding residues at the resonance pole (Eq. (36)).We will demonstrate this procedure in the next section.14 . NUMERICAL DEMONSTRATION Our field-theoretical model for the scattering amplitude and the correlator is specified inEqs. (26) and (31). We shall employ units such that the mass of the φ particles is M = 1. Inthese units, the “true values” of the five model parameters will be taken as α = 25 , g = 15 , m = 3 , b = 1 , ˜ b = 23 . (49)For these values, we find that the resonance pole of T ( s ) on the second sheet is located at s σ = (cid:18) m σ − i σ (cid:19) , m σ = 3 . , Γ σ = 0 . , with residuum Res s σ T = − . − . i .Let us assume that the correlator ˜ c SS ( t, ) is measured in a numerical simulation in a finitevolume with M L = 10, with data given in the table below: j s j (cid:0) πL (cid:1) r j We do not attempt any error analysis here - the present section should just serve to demonstratethe plausibility and applicability of our foregoing work. From the values s j , we can evaluatethe phase shift δ ( s ) at the points s = s j in the low-energy region 4 M < s < M . s The plot shows that Eqs. (39), (41) work nicely for
M L = 10. We also see that there is a clearsignature of the resonance in the phase shift. The red curve shows the exact phase shift inthe infinite volume, while the black dots indicate the values computed with our finite-volume15ormulae.We can now make a fit of some parameterization for the scattering amplitude to the phase-shift“data” points. Using our model amplitude of Eq. (26), for example, a fit to the data points in[4 M , M ] returns α fit = 24 . , g fit = 14 . , m ,fit = 3 . , which compared to the “true” values given above is an extremely good result. Analyticallycontinuing to the second Riemann sheet, we find a pole located at m σ,fit = 3 . , Γ σ,fit = 0 . , Res s σ T fit = − . − . i . Of course, one does not know the true form of the scattering amplitude in practice. Let us trya Breit-Wigner-like parameterization (compare (24), (25), (27)), T BW ( s ) := − γ BW s − m BW + iγ BW (cid:16) σ ( s )32 π (cid:17) . (50)This would give m BW = 2 . γ BW = 13 .
88, and a pole with m σ,BW = 2 . σ,BW = 0 . T ( s ) of Eq. (26) and the values α fit , g fit , m ,fit , we can now evaluatethe numbers ρ ( s j ) with the help of Eq. (47) : j ρ ( s j ) 1.011 0.884 1.557 1.055 2.106 0.878 0.579 The lowest level with j = 0 is excluded here because δ ( s ) is only defined for s ≥ M . Now weare in a position to compare the outcome for the right-hand side of Eq. (46) (numerical results)with the “true” values on the left-hand side, given by our model for | B ( s j ) | and the “true”parameters of Eq. (49): j ρ ( s ) → In the last row of the previous table, we also give the outcome if ρ ( s ) is set to 1, which wouldbe the interaction-free limit of this function. We see that this would be a bad approximationif the interaction is enhanced as e.g. in the resonance region.16e point out that the expression in Eq. (47) should be used instead of the original definition ρ ∼ (cid:0) πL (cid:1) / ∆ s stemming from the “brute force” discretization of the s (cid:48) -integral, because someinformation on the interaction in the infinite-volume limit (derived only from the spectrum)has already been implemented in (47). We demonstrate this in the plot below. s Ρ The red curve shows the function according to Eq. (47) (but note that it is not the graph of ρ , which is only defined at the s j ). The black and the blue points mark the values ρ ( s j ) =4 (cid:0) πL (cid:1) / ( s j +1 − s j ) and ρ ( s j ) = 4 (cid:0) πL (cid:1) / ( s j − s j − ), respectively, which should lead to thesame limit function ρ ( s ). One observes that the uncertainty involved in the discretization ofthe s (cid:48) -integral is quite large in the region where the phase shift rapidly varies from one energyeigenvalue to the next: there, the energy discretization cannot resolve the rapid variation ofthe final-state interaction in the infinite volume.From our numerical results for δ ( s j ) and | B ( s j ) | , we can now infer the complex values of theform-factor B ( s j ). These results are collected in the following tables. j B ( s j ) (“num.”) 2 .
806 + 1 . i .
951 + 1 . i .
048 + 3 . i .
041 + 5 . ij B ( s j ) (“num.”) − .
091 + 4 . i − .
264 + 1 . i − .
271 + 0 . i Now we have to use some parameterization of the form-factor in order to be able to performan analytic continuation to the complex plane. We will again simply use our model amplitude(Eq. (31)), though in practice one will usually have to resort to some effective parameterizations,17hich might entail a considerable uncertainty. Fitting to the values in the above table (withour results for α fit , g fit , m ,fit ) yields b fit = 0 . , ˜ b fit = 0 . , very close to our “true” values. The continuation of our model amplitude to the second Riemannsheet (where ¯ I φφ ( s ) → ¯ I φφ ( s ) − iσ ( s )16 π ) yields (compare Eq. (36))Res s σ B = − . − . i ⇒ − Res s σ B (cid:112) − Res s σ T = 0 .
609 + 0 . i , while the “true” value is found (continuing our model with the true parameters to the pole onthe second sheet) to be (cid:112) Z σ ˜ β ( s σ ) = (1 .
051 + 0 . i ) · (0 .
583 + 0 . i ) = 0 .
612 + 0 . i . This is the value of the matrix element for the “ σ decay constant” associated with the operator S ( x ). Apparently, the same procedure can be applied for other operators with different matrixelements for φφ and σ (with the same level density function ρ ( s ), which depends only onthe scattering phase and can thus be determined from the finite-volume spectrum alone).While the model discussed here is too simple to draw any general conclusions about realisticapplications, we still believe that the main features of the physical problem in question cannicely be demonstrated in this framework. We hope that it has become clear that thereis nothing mysterious to the relation between physical resonances and finite-volume latticedata in the resonant channel (or at least, it is not more mysterious than the concept ofthe complex energy plane). Additional problems appearing in realistic applications are thepossibility of multiple open decay channels, possibly with more than two particles in thefinal states, discretization (lattice spacing) effects, and the use of smearing for source andsink operators. For the discussion of these more involved quaestions, we refer the readerto the literature cited in the introduction, and to future work along the lines of the present study. Acknowledgments
I thank Maxim Mai, Andreas Sch¨afer and Philipp Wein for discussions on the manuscript. Thiswork was supported by the Deutsche Forschungsgemeinschaft SFB/Transregio 55.18 ppendix A: Some useful integrals
For real t (cid:54) = 0, b >
0, we find the result (cid:90) ∞ b dq cos qtq (cid:112) q − b = sgn( t ) π b (cid:16) J ( bt ) (cid:16) bt π H ( bt ) − (cid:17) + btJ ( bt ) (cid:16) − π H ( bt ) (cid:17)(cid:17) − π b = π b (cid:18) b | t | F (cid:18)
12 ; 32 , −
14 ( bt ) (cid:19) − (cid:19) . (A.1)where J n and H m are the Bessel J and Struve H functions. Taking derivatives with respect to t , we obtain results for more integrals (which appear to be diverging on first sight), (cid:90) ∞ b dq sin ( qt ) (cid:112) q − b = − b π J ( bt ) b | t | , (A.2) (cid:90) ∞ b dq q cos ( qt ) (cid:112) q − b = − b π J ( bt ) b | t | . (A.3)For real t (which we always assume here), we also find a result in terms of Bessel’s Y, (cid:90) ∞ b dq cos ( qt ) (cid:112) q − b = b π Y ( b | t | ) b | t | , (A.4)and together with Eq. (A.2) (cid:90) ∞ b dq e iqt (cid:112) q − b = b π Y ( b | t | ) − i sgn( t ) J ( b | t | )) b | t | . (A.5)A generalization of Eq. (A.1) is h n ( t, b ) := (cid:90) ∞ b dq cos qtq n +1 (cid:112) q − b = | t | n − sin ( nπ ) Γ(1 − n ) F (cid:18) −
12 ; n, n + 12 ; −
14 ( bt ) (cid:19) + b − n √ π (cid:0) n − (cid:1) Γ( n + 1) F (cid:18) − n ; 12 , − n ; −
14 ( bt ) (cid:19) . (A.6)Note that the limits lim n → h n ( t, b ), lim n → h n ( t, b ) etc. exist. For n →
0, one recovers (A.1). Theasymptotic expansions of the Bessel and Struve functions are well-known, and we find that, forlarge bt (cid:29) h ( t, b ) → − (cid:114) π b t / sin (cid:16) bt + π (cid:17) , and since ∂ ∂t h n +1 ( t, b ) = − h n ( t, b ), the asymptotic form of the h n must be in general h n ( t, b ) → − b n (cid:114) π b t / sin (cid:16) bt + π (cid:17) , (A.7)neglecting all suppressed powers of t − . Note that there can not be an additional polynomialpart because, for t > h n ( t, b ) is bounded by its value as t → | h n ( t, b ) | < lim t → h n ( t, b ) = b − n √ π n ! (cid:12)(cid:12)(cid:12)(cid:12) Γ (cid:18) n − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) .
19e turn to a much more complicated integral, g ( t ; a, b ) := (cid:90) ∞ b dq (cid:112) q − b (cid:112) q − a cos qt , for b > , ≤ a < b . (A.8)It is obviously even in t and a . The integral does not exist for t = 0. However, taking the limit t →
0+ along the real line, it can be checked that it tends to g ( t → a, b ) → − bE (cid:16) a b (cid:17) ,where E ( k ) = π F ( , − ; 1; k ) is the well-known elliptic integral. The expression for g ( t ; 0 , b )is given in (A.1). Since q ≥ b > a , we can expand the square-root in the denominator of theintegrand of (A.8) and integrate term by term, using (A.6): g ( t ; a, b ) = (cid:90) ∞ b dq cos qt ∞ (cid:88) n =0 Γ (cid:0) n + (cid:1) Γ (cid:0) (cid:1) Γ( n + 1) (cid:112) q − b q n +1 a n = ∞ (cid:88) n =0 Γ (cid:0) n + (cid:1) Γ (cid:0) (cid:1) Γ( n + 1) h n ( t, b ) a n . According to Eq. (A.8), a term linear in | t | can only be generated from h and h . Carefullytaking the limits n → n →
1, one finds g ( t ; a, b ) = − bE ( a /b ) + π ( b − a ) | t | + O ( | t | ).Reordering this series, it can also be shown that lim a →± b g ( t ; a, b ) = − sin( bt ) /t for a ∈ ] − b, b [.Using Eq. (A.7), we deduce that the asymptotic form for large t is given by g ( t ; a, b ) t →∞ −→ − (cid:114) π b b √ b − a sin (cid:0) bt + π (cid:1) t / . (A.9)Integrating Eq. (A.4) over t , mathematica gives the result (cid:90) ∞ b dq sin ( qt ) q (cid:112) q − b = π b G , , bt , (cid:12)(cid:12)(cid:12)(cid:12) , − − , ; − , , (A.10)in terms of the (generalized) Meijer-G function. More generally, similar to Eq. (A.6)˜ h n ( t, b ) := (cid:90) ∞ b dq sin qtq n +1 (cid:112) q − b = t n − cos ( nπ ) Γ(1 − n ) F (cid:18) −
12 ; n, n + 12 ; −
14 ( bt ) (cid:19) + b − n t √ π n − n + ) F (cid:18) − n ; 32 , − n ; −
14 ( bt ) (cid:19) . (A.11)Again, one can check that the limits lim n → ˜ h n ( t, b ), lim n → ˜ h n ( t, b ) etc. exist. From the knownasymptotic behavior of the hypergeometric functions, we can infer that, for large t , h n ( t, b ) → b n (cid:114) π b t / cos (cid:16) bt + π (cid:17) , (A.12)and therefore, similar to Eq. (A.9), (cid:90) ∞ b dq (cid:112) q − b (cid:112) q − a sin qt t →∞ −→ (cid:114) π b b √ b − a cos (cid:0) bt + π (cid:1) t / . (A.13)20aking repeated time derivatives of the above results, one also shows (cid:90) ∞ b dq (cid:112) q − b (cid:112) q − a q n e − iqt t →∞ −→ (cid:114) π b b n +1 √ b − a e − ibt ( it ) / , (A.14)and by taking derivatives w.r.t. a , (cid:90) ∞ b dq (cid:112) q − b (cid:112) q − a q n e − iqt t →∞ −→ (cid:114) π b b n +1 √ b − a e − ibt ( it ) / , etc . (A.15)In addition to the Fourier integrals discussed above, it is useful to know (cid:90) ∞ dq cos qt ( q + M ) n = √ π t n − (2 √ M ) n − Γ( n ) K − n ( √ M t ) , n > . (A.16)The modified Bessel functions of half-integer degree can be expressed through exponentialfunctions. For example, one has K ( z ) = K − ( z ) = √ πe − z / √ z and thus, for example (cid:90) ∞ dq cos qtq + M = π √ M e −√ M | t | , (A.17)for real M (cid:54) = 0. It is straightforward to see that such formulae can be analytically continuedto imaginary M , at the cost of the introduction of an i(cid:15) -prescription. For real z , we find fromthe theorem of residues (cid:90) ∞ dq cos qtq − z ± i(cid:15) = ∓ iπ e ∓ i | zt | | z | . (A.18)Similarly, decomposing the integrand into partial fractions, (cid:90) ∞ dq cos qt ( q − z + i(cid:15) )( q − z + i(cid:15) ) = iπz − z (cid:18) e − i | z t | | z | − e − i | z t | | z | (cid:19) , z (cid:54) = z , (A.19) (cid:90) ∞ dq cos qt ( q − z + i(cid:15) ) ( q − z + i(cid:15) ) = iπ ( z − z ) (cid:18) (3 z − z − i | z t | ( z − z )) e − i | z t | | z | − e − i | z t | | z | (cid:19) . ppendix B: A Fourier integral in d = Consider the integral I M ( x − y ) := (cid:90) d l (2 π ) ie − il · ( x − y ) l − M + i(cid:15) . The integral diverges if the space-time distance four-vector x − y approaches zero. First, wechoose a time-like distance here, and set x − y = ( t, ), with some t >
0. We perform the l -integration by the method of residues, closing the contour in the lower complex l -plane: I M ( x − y ) = (cid:90) d l (2 π ) (cid:90) + ∞−∞ dl π ie − il t (cid:104) l − (cid:16)(cid:112) | l | + M − i(cid:15) (cid:17)(cid:105) (cid:104) l − (cid:16) − (cid:112) | l | + M + i(cid:15) (cid:17)(cid:105) = 14 π (cid:90) ∞ p dp (cid:112) p + M − i(cid:15) e − i √ p + M t (B.1)= M π (cid:90) ∞ dφ sinh φ e − ( i ( M − i(cid:15) ) t ) cosh φ = M π (cid:18) K ( iM t ) iM t (cid:19) = M K ( iM t )4 π it . We have used the substitution pM = sinh φ , and an integral formula for the modified Besselfunctions of the second kind, K ν ( z ), which obey2 νz K ν ( z ) = K ν +1 ( z ) − K ν − ( z ) , (B.2) dK ν ( z ) dz = −
12 ( K ν − ( z ) + K ν +1 ( z )) . (B.3)Note that the result for I M ( x − y ) = I M ( t ) is real for t = − iτ (euclidean time τ ), and real M .Let us also look at the general case of Eq. (B.1), for t := x − y , r := | x − y | . Again, weperform the l -integration by the method of residues, closing the contour in the lower complex l -plane (assuming Re t > I M ( x − y ) = (cid:90) ∞ | l | d | l | (2 π ) π (cid:112) | l | + M (cid:18) e i | l | r − e − i | l | r i | l | r (cid:19) e − it √ | l | + M . We employ the substitutions | l | /M = sinh φ , t = s cosh ξ , r = s sinh ξ ( ⇒ s = √ t − r ), andmake use of the fact that the resulting integrand is even in φ , I M ( x − y ) = M π is (cid:90) + ∞−∞ dφ sinh φ sinh ξ (cid:0) e iMs sinh φ sinh ξ − e − iMs sinh φ sinh ξ (cid:1) e − iMs cosh φ cosh ξ . (B.4) See e.g. [40], pp. 367, 377. We have corrected for an unusual sign convention factor ( − ν in the definitionof the K ν ( z ) used in that book, so that our signs agree with those used by mathematica ® . φ cosh ξ ± sinh φ sinh ξ = cosh ( φ ± ξ ), and substituting φ (cid:48) = φ ∓ ξ in the firstand second term, respectively, we find I M ( x − y ) = M π is (cid:90) + ∞−∞ dφ sinh φ sinh ξ (cid:0) e − iMs cosh( φ − ξ ) − e − iMs cosh( φ + ξ ) (cid:1) (B.5)= M π is (cid:90) + ∞−∞ dφ (cid:48) sinh ( φ (cid:48) + ξ ) − sinh ( φ (cid:48) − ξ )sinh ξ e − iMs cosh φ (cid:48) = M π is (cid:90) + ∞−∞ dφ (cid:48) cosh φ (cid:48) e − iMs cosh φ (cid:48) ( B. = M π is K ( iM s ) = K (cid:0) iM √ t − r (cid:1) π i √ t − r . We have again used the fact that the integrand is even under φ (cid:48) ↔ − φ (cid:48) , and partial integrationusing ddφ φ = − sinh φ cosh φ , (cid:90) ∞ dφ z sinh φ e − z cosh φ = (cid:90) ∞ dφ cosh φ e − z cosh φ = K ( z ) . This proves the general version of Eq. (B.1), which could of course also be deduced from thespecial case above and Lorentz invariance of the integrand.23
1] X. Feng, K. Jansen and D. B. Renner, Phys. Rev. D (2011) 094505 [arXiv:1011.5288 [hep-lat]].[2] C. B. Lang, D. Mohler, S. Prelovsek and M. Vidmar, Phys. Rev. D (2011) no.5, 054503Erratum: [Phys. Rev. D (2014) no.5, 059903] [arXiv:1105.5636 [hep-lat]].[3] M. G¨ockeler, R. Horsley, M. Lage, U.-G. Meißner, P. E. L. Rakow, A. Rusetsky, G. Schierholzand J. M. Zanotti, Phys. Rev. D (2012) 094513 [arXiv:1206.4141 [hep-lat]].[4] D. Mohler, PoS LATTICE (2012) 003 [arXiv:1211.6163 [hep-lat]].[5] J. J. Dudek et al. [Hadron Spectrum Collaboration], Phys. Rev. D (2013) no.3, 034505 Erra-tum: [Phys. Rev. D (2014) no.9, 099902] [arXiv:1212.0830 [hep-ph]].[6] C. B. Lang, L. Leskovec, D. Mohler and S. Prelovsek, JHEP (2015) 089 [arXiv:1503.05363[hep-lat]].[7] D. J. Wilson, R. A. Briceno, J. J. Dudek, R. G. Edwards and C. E. Thomas, Phys. Rev. D (2015) no.9, 094502 [arXiv:1507.02599 [hep-ph]].[8] G. S. Bali et al. [RQCD Collaboration], Phys. Rev. D (2016) no.5, 054509 [arXiv:1512.08678[hep-lat]].[9] D. Guo, A. Alexandru, R. Molina and M. D¨oring, Phys. Rev. D (2016) no.3, 034501[arXiv:1605.03993 [hep-lat]].[10] R. A. Briceno, J. J. Dudek, R. G. Edwards and D. J. Wilson, Phys. Rev. Lett. (2017) no.2,022002 [arXiv:1607.05900 [hep-ph]].[11] J. J. Wu, H. Kamano, T.-S. H. Lee, D. B. Leinweber and A. W. Thomas, arXiv:1611.05970[hep-lat].[12] V. M. Braun et al. , arXiv:1612.02955 [hep-lat]. Accepted for publication in JHEP.[13] J. Gegelia and S. Scherer, Eur. Phys. J. A (2010) 425 [arXiv:0910.4280 [hep-ph]].[14] L. Maiani and M. Testa, Phys. Lett. B (1990) 585.[15] M. L¨uscher, Nucl. Phys. B (1991) 531.[16] M. L¨uscher, Nucl. Phys. B (1991) 237.[17] U.-J. Wiese, Nucl. Phys. Proc. Suppl. (1989) 609.[18] L. Lellouch and M. L¨uscher, Commun. Math. Phys. (2001) 31 [hep-lat/0003023].[19] U.-G. Meißner, K. Polejaeva and A. Rusetsky, Nucl. Phys. B (2011) 1 [arXiv:1007.0860[hep-lat]].
20] V. Bernard, M. Lage, U.-G. Meißner and A. Rusetsky, JHEP (2011) 019 [arXiv:1010.6018[hep-lat]].[21] M. D¨oring, U.-G. Meißner, E. Oset and A. Rusetsky, Eur. Phys. J. A (2011) 139[arXiv:1107.3988 [hep-lat]].[22] H. X. Chen and E. Oset, Phys. Rev. D (2013) no.1, 016014 [arXiv:1202.2787 [hep-lat]].[23] M. T. Hansen and S. R. Sharpe, Phys. Rev. D (2012) 016007 [arXiv:1204.0826 [hep-lat]].[24] V. Bernard, D. Hoja, U.-G. Meißner and A. Rusetsky, JHEP (2012) 023 [arXiv:1205.4642[hep-lat]].[25] R. A. Briceno, M. T. Hansen and A. Walker-Loud, Phys. Rev. D (2015) no.3, 034501[arXiv:1406.5965 [hep-lat]].[26] D. R. Bolton, R. A. Briceno and D. J. Wilson, Phys. Lett. B (2016) 50 [arXiv:1507.07928[hep-ph]].[27] R. A. Briceno and M. T. Hansen, Phys. Rev. D (2016) no.1, 013008 [arXiv:1509.08507 [hep-lat]].[28] D. Agadjanov, M. D¨oring, M. Mai, U.-G. Meißner and A. Rusetsky, JHEP (2016) 043[arXiv:1603.07205 [hep-lat]].[29] Z. H. Guo, L. Liu, U.-G. Meißner, J. A. Oller and A. Rusetsky, Phys. Rev. D (2017) no.5,054004 [arXiv:1609.08096 [hep-ph]].[30] L. Wang, Z. Fu and H. Chen, arXiv:1702.08337 [hep-lat].[31] M. L¨uscher, Commun. Math. Phys. (1986) 177.[32] P. Hasenfratz and H. Leutwyler, Nucl. Phys. B (1990) 241.[33] G. Colangelo, S. D¨urr and C. Haefeli, Nucl. Phys. B (2005) 136 [hep-lat/0503014].[34] S. Weinberg, “The Quantum Theory of Fields. Vol. 1: Foundations”, Cambridge (1995).[35] C. Itzykson and J. B. Zuber, “Quantum Field Theory”, New York, USA: McGraw-Hill (1980).[36] J. Gasser and G. Wanders, Eur. Phys. J. C (1999) 159 [hep-ph/9903443].[37] L. Castillejo, R. H. Dalitz and F. J. Dyson, Phys. Rev. (1956) 453.[38] O. Brander, Commun. Math. Phys. (1975) 97.[39] E. P. Wigner, Phys. Rev. (1955) 145.[40] E. T. Whittaker and G. N. Watson, “A Course of Modern Analysis”, Cambridge University Press(1996).(1955) 145.[40] E. T. Whittaker and G. N. Watson, “A Course of Modern Analysis”, Cambridge University Press(1996).