Model-independent constraints on hadronic form factors with above-threshold poles
MModel-independent constraints on hadronic form factors with above-threshold poles
Irinel Caprini
Horia Hulubei National Institute for Physics and Nuclear Engineering,POB MG-6, 077125 Bucharest-Magurele, Romania
Benjam´ın Grinstein
Department of Physics, University of California, San Diego, La Jolla, California 92093, USA
Richard F. Lebed
Department of Physics, Arizona State University, Tempe, Arizona 85287-1504, USA
Model-independent constraints on hadronic form factors, in particular those describing exclusivesemileptonic decays, can be derived from the knowledge of field correlators calculated in perturba-tive QCD, using analyticity and unitarity. The location of poles corresponding to below-thresholdresonances, i.e. , stable states that cannot decay into a pair of hadrons from the crossed channelof the form factor, must be known a priori , and their effect, accounted for through the use ofBlaschke factors, is to reduce the strength of the constraints in the semileptonic region. By con-trast, above-threshold resonances appear as poles on unphysical Riemann sheets, and their presencedoes not affect the original model-independent constraints. We discuss the possibility that theabove-threshold poles can provide indirect information on the form factors on the first Riemannsheet, either through information from their residues or by constraining the discontinuity function.The bounds on form factors can be improved by imposing, in an exact way, the additional informa-tion in the extremal problem. The semileptonic K → π(cid:96)ν and D → π(cid:96)ν decays are considered asillustrations. PACS numbers: 11.55.Fv,13.20.-v,13.30.Ce
I. INTRODUCTION
Since the pioneering works of Meiman [1] and Okubo[2, 3], it has been known that nontrivial constraints onhadronic form factors can be derived from the knowledgeof suitably related field correlators. The method was re-considered in [4] within the modern theory of strong in-teractions, where the correlators relevant for the boundson the K (cid:96) form factors were evaluated in the deep Eu-clidean region by using perturbative QCD.The method exploits unitarity and positivity of thespectral function, and converts a dispersion relation fora correlator of two currents into an integral conditionalong the unitarity cut ( i.e. , above the lowest produc-tion threshold of particles coupled to the currents) forthe modulus-square of the form factors parametrizing therelevant matrix elements in the unitarity sum. From thiscondition and the analyticity properties of the form fac-tors as functions of energy, one can derive, with standardtechniques of complex analysis [5, 6], constraints on thevalues of the form factors and their derivatives at pointsinside the analyticity domain.Many applications of this approach to heavy-quarkform factors describing B → D ( ∗ ) (cid:96)ν semileptonic decays,to heavy-to-light form factors involved in B → π(cid:96)ν or D → π(cid:96)ν decays, or to the light-meson form factors,have been performed in the last 20 years [7–26] (for areview of earlier literature see [27]). A similar formalismhas been applied also to the electromagnetic form factors of the pion [28–31] and proton [32], to the πω form factor[33, 34], and to heavy baryons [35].The presence of singularities below the unitaritythreshold modifies the derivation of the bounds. Themethod can be adapted to include in an exact way thediscontinuity across an unphysical cut below the unitar-ity branch point, present in some cases, related to lighterparticles that can couple to the current [33, 34]. A polesituated below the unitarity threshold, of known posi-tion but unknown residue, can be also accounted for inan optimal way with the technique of Blaschke factors[12, 13]. In such a case, the presence of the pole leads toa weakening of the constraints.Recently, the possible effect of resonances situatedabove the unitarity threshold, close to the physical re-gion, was discussed in [36]. As known from general prin-ciples of quantum field theory [37], the unstable parti-cles are associated to complex poles in the energy plane.Such poles cannot appear on the first Riemann sheet ofthe complex plane, and instead are situated on the sec-ond or higher Riemann sheets. The argument used in[36] was based on the remark that a complex pole onthe second sheet close to the real axis produces a localincrease of the modulus of the form factor on the uni-tarity cut. The same increase can be obtained howeverwith a complex singularity of the same position, but sit-uated on the first Riemann sheet. Therefore, in [36] itwas argued that the effect of an above-threshold singu-larity can be mimicked through a complex pole on the a r X i v : . [ h e p - ph ] A ug first Riemann sheet, near the physical region. The lattercan be treated with the standard technique of Blaschkefactors, much like the subthreshold poles. In this way,Ref. [36] estimated the physical effect of the presence ofabove-threshold resonances.In the present paper, we consider the question whetherthe form-factor parametrizations can be improved ifsome knowledge on the above-threshold poles is pro-vided. We start with a brief review of the techniqueof model-independent constraints, presenting in partic-ular the stronger constraints obtained with some addi-tional information outside the semileptonic decay range.In Sec. III we argue first that the presence of an above-threshold pole does not affect the original bounds in thesemileptonic region. Then we investigate whether, fromthe presence of an above-threshold resonance, one canobtain some information on the form factor on the phys-ical sheet and show that, in some cases, the bounds canbe improved by implementing additional information ofthis type. In particular, we find that the most practicalconstraints arise from mapping the effect of the above-threshold resonance to the phase of the form factor alongthe cut. Our conclusions are given in the last section.In a short Appendix, we discuss the connection betweenthe first two Riemann sheets and the canonical variable z used for solving the extremal problem. II. MODEL-INDEPENDENT CONSTRAINTSON HADRONIC FORM FACTORS
We present below, following the review [27] and the re-cent paper [36], the main steps relevant for the derivationof constraints on the form factor parametrizations. As in[36], we concentrate in particular on the form factors rel-evant for the semileptonic decays of pseudoscalar mesons.We consider the heavy-to-light ( Q → q ) vectorlike ( V , A ,or V − A ) quark-transition current J µ ≡ ¯ Q Γ µ q , (1)and the two-point momentum-space Green’s functionΠ µνJ separated into manifestly spin-1 (Π TJ ) and spin-0(Π LJ ) terms:Π µνJ ( q ) ≡ i (cid:90) d x e iqx (cid:10) (cid:12)(cid:12) T J µ ( x ) J † ν (0) (cid:12)(cid:12) (cid:11) = 1 q (cid:0) q µ q ν − q g µν (cid:1) Π TJ ( q ) + q µ q ν q Π LJ ( q ) . (2)The functions Π T,LJ satisfy dispersion relations with pos-itive spectral functions, expressed by unitarity in termsof contributions from a complete set of hadronic states.From the asymptotic behavior predicted by perturbativeQCD, it follows that the dispersion relations require sub-tractions (one for Π LJ and two for Π TJ ). The subtraction constants disappear by taking the derivatives: χ LJ ( q ) ≡ ∂ Π LJ ∂q = 1 π (cid:90) ∞ dt Im Π LJ ( t )( t − q ) ,χ TJ ( q ) ≡ ∂ Π TJ ∂ ( q ) = 1 π (cid:90) ∞ dt Im Π TJ ( t )( t − q ) . (3)Perturbative QCD can be used to compute the functions χ J ( q ) at values of q far from the region where the cur-rent J can produce manifestly nonperturbative effectslike pairs of hadrons. For heavy quarks, Q = c or b , areasonable choice is q = 0, while for Q = s a spacelikevalue, like q = − or q = − , is necessary.The spectral functions Im Π J are evaluated by unitar-ity, inserting into the unitarity sum a complete set ofstates X that couple the current J to the vacuum:Im Π µνJ ( q ) = 12 (cid:88) X (2 π ) δ ( q − p X ) (cid:104) | J µ | X (cid:105) (cid:10) X (cid:12)(cid:12) J † ν (cid:12)(cid:12) (cid:11) . (4)For our purpose, it is enough to take X to be the lightestmeson pair in which one of them (of mass M ) containsa Q quark and the other (of mass m ) contains a ¯ q , anduse the positivity of the higher-mass contributions. Thischoice gives a rigorous lower bound on the spectral func-tions, in terms of the vector or scalar form factors thatparametrize the matrix elements of the current. Usingthe standard notation t ± ≡ ( M ± m ) , (5)the inequality for the transverse polarization Π TJ can bewritten as 1 πχ TJ ( q ) (cid:90) ∞ t + dt w ( t ) | F ( t ) | ( t − q ) ≤ , (6)where t + is the unitarity threshold, F ( t ) is the vectorform factor, and w ( t ) is a simple, nonnegative function,expressed as a product of phase-space factors dependingupon t + and t − . An analogous expression holds for Π LJ and the scalar form factor.Using the standard dispersion techniques in quantumfield theory [38], one can prove that the semileptonic formfactors are in general analytic functions in the complex t plane, with a unitarity cut along the real axis from t + to ∞ . In some cases, as in B → D(cid:96)ν and B → π(cid:96)ν , theform factors may also exhibit poles situated on the realaxis below the unitarity threshold t + . No analogous polesare present in the form factors relevant in K → π(cid:96)ν and D → π(cid:96)ν decays. All the form factors in semileptonic de-cays satisfy in addition the Schwarz reflection condition,written generically as F ( t ∗ ) = F ∗ ( t ). The form factorsare therefore real on the real t axis below t + , in particu-lar in the semileptonic region 0 ≤ t ≤ t − , where they canbe measured from the decay rates.As shown in the pioneering papers [1–4], one can ob-tain constraints on the form-factor parametrizations inthe semileptonic region, using their analyticity proper-ties and the boundary condition (6). In order to exploitthis condition, it is convenient to map the cut t planeonto the unit disk in the complex z plane defined by theconformal mapping z ≡ ˜ z ( t ; t ) ≡ √ t + − t − √ t + − t √ t + − t + √ t + − t , (7)which maps the cut t complex plane onto the interior ofthe unit disk, such that the branch point t + in mappedonto z = 1 and the two edges of the unitarity cut t ≥ t + map to the boundary | z | = 1. Moreover, z is realfor t ≤ t + . The choice of the free parameter t in (7),which represents the point mapped onto the origin of the z plane, ˜ z ( t ; t ) = 0, will be discussed below.In the variable z , the inequality (6) is written in theequivalent form12 πi (cid:73) C dzz | φ ( z ) F [˜ t ( z ; t )] | ≤ , (8)where ˜ t ( z ; t ) = 4 zt + + t (1 − z ) (1 + z ) (9)is the inverse of (7), and φ ( z ) is an outer function , definedin complex analysis [5] as an analytic function lackingzeros in | z | <
1. In our case, the function φ ( z ) is definedby specifying its modulus | φ ( z ) | = w [˜ t ( z ; t )] | d ˜ z ( t ; t ) /dt | χ T ( q )[˜ t ( z ; t ) − q ] , (10)on the boundary z = e iθ of the unit disk. Then the func-tion for | z | < φ ( z ) = exp (cid:20) π (cid:90) π d θ e iθ + ze iθ − z ln | φ ( e iθ ) | (cid:21) . (11)In particular cases of physical interest, φ ( z ) can be ob-tained in closed form, as a product of simple analyticfunctions (see [27, 36]).From the boundary condition (8), one can derive con-straints on the form factor F ( t ) at points inside the ana-lyticity domain, in particular in the semileptonic region.It is important to emphasize that the use of the outerfunction in (8) ensures the constraints are optimal. As-sume first that the form factor F ( t ) has no singularitiesbelow the unitarity threshold t + , being an analytic func-tion of real type [ F ∗ ( t ) = F ( t ∗ )] in the cut t plane, orequivalently in the unit disk | z | < This definition differs by a minus sign from that adopted in [36]. this is the case for the Kπ or Dπ form factors). Then,expanding as: F ( z ) ≡ F [˜ t ( z ; t )] = 1 φ ( z ) ∞ (cid:88) k =0 a k z k , (12)where the coefficients a k are real, the condition (8) reads: ∞ (cid:88) k =0 a k ≤ . (13)This inequality, which is valid also for any finite sum ofterms, was used in many studies to strongly constrainthe parameters used in the fits to semileptonic data orfor estimating the truncation error [15–17, 19, 21–24].As discussed in several papers, the truncation error isminimized by choosing the parameter t such that thesemileptonic range 0 ≤ t ≤ t − is mapped onto an interval( − z max , z max ) symmetric around the origin in the z plane.This method allowed a high-precision determination ofthe elements V us , V cb , and V ub of the CKM matrix fromexclusive semileptonic decays.The constraints on the Taylor series coefficients a k be-come stronger if some additional information on the formfactor outside the semileptonic range is available. Thegeneral condition involving an arbitrary number of coef-ficients a k and the values of F ( z ) at an arbitrary numberof points inside the unit disk has been derived usingseveral methods and can be found in [27].For the discussion in the next section, it is of interestto give the form of the constraint when one knows thevalues of the form factor F ( z ) at two complex-conjugatepoints, which we denote as z p and z ∗ p , with | z p | <
1. Sincethe functions satisfy the Schwarz reflection property, onehas F ( z ∗ p ) = F ∗ ( z p ). Using, as in [27], the techniqueof Lagrange multipliers for imposing the additional con-straints at z p and z ∗ p , a straightforward calculation givesthe inequality K − (cid:88) k =0 a k ≤ − F ( z p , ξ ) , (14)where F is defined as F ( z p , ξ ) = 2(1 − | z p | ) | − z p | | z p | K ( z p − z ∗ p ) (15) × (cid:34) Re (cid:32) ξ z ∗ Kp − z ∗ p (cid:33) − | ξ | | z p | K − | z p | (cid:35) , in terms of the point z p and the complex quantity ξ = φ ( z p ) F ( z p ) − K − (cid:88) k =0 a k z kp . (16) In complex analysis, if instead of the L norm (8) the boundarycondition is expressed by means of the L ∞ norm, the problem isknown as a combined Schur-Carath´eodory and Pick-Nevanlinnainterpolation problem [5, 6]. The inequality (14) defines an allowed domain for first K coefficients a k in terms of the input complex value F ( z p )entering the variable ξ . One can check from (15) that thefunction F is positive for | z p | < a k and F ( z p ). Therefore, the domain defined by (14) issmaller than that given by the condition K − (cid:88) k =0 a k ≤ F ( z p ) improves the constraints on the parameters in thesemileptonic region. We note, however, that the improve-ment is small if the point z p is close to the boundary ofthe unit disk, since F is small for | z p | close to 1.Another additional piece of information that can im-prove the constraints is knowledge of the phase of theform factor along a part of the unitarity cut. In somecases, as for the pion electromagnetic form factor or the K (cid:96) form factors, the phase is related by Fermi-Watsontheorem [39, 40] to the phase shift of the correspondingelastic scattering amplitude, which is known with preci-sion, for instance from the solution of Roy equations [41].In the present context (as discussed in the next section),it is of interest to note that one can approximately obtainthe phase on a part on the cut using the mass and widthof a nearby resonance.Using this information as an additional constraintleads to a modified optimization problem, solved for thefirst time for the K (cid:96) form factors in [42]. Several general-izations have been discussed more recently in [20, 21, 27].For completeness, we give below the constraint on thefirst K coefficients a k when the phase arg F ( t ) is knownon the region 0 ≤ t ≤ t in (for the derivation, see Sec. 4of the review [27]).We denote by ζ in ≡ ˜ z ( t in ; t ) = e iθ in (18)the image on the unit circle in the z plane of the point t in + i(cid:15) situated on the upper edge of the cut [the point t in − i(cid:15) being mapped onto exp( − iθ in )]. Then the domainallowed for the coefficients a k is given by K − (cid:88) k =0 a k + 1 π K − (cid:88) k =0 a k θ in (cid:90) − θ in d θ λ ( θ ) sin [ kθ − Φ( θ )] ≤ , (19)where Φ( θ ) = arg[ F ( e iθ )] + arg[ φ ( e iθ )] , (20)and λ ( θ ) is the solution of the integral equation K − (cid:88) k =0 a k sin[ kθ − Φ( θ )] = λ ( θ ) − π θ in (cid:90) − θ in d θ (cid:48) λ ( θ (cid:48) ) K Φ ( θ, θ (cid:48) ) , (21) for θ ∈ ( − θ in , θ in ), where the kernel is defined as K Φ ( θ, θ (cid:48) ) ≡ sin[( K − / θ − θ (cid:48) ) − Φ( θ ) + Φ( θ (cid:48) )]sin[( θ − θ (cid:48) ) / . (22)The inequality (19) describes an allowed domain for a k that is smaller than the original domain (17), which rep-resents the improvement introduced by knowledge of thephase on a part of the unitarity cut.In the above derivations, the crucial role was playedby the fact that the form factor is analytic in the cut t plane. As discussed above, the form factors relevantfor K → π(cid:96)ν and D → π(cid:96)ν decays do not have sub-threshold singularities, while the form factors involvedin B → D ( ∗ ) (cid:96)ν and B → π(cid:96)ν decays have subthresholdpoles, corresponding to particles stable with respect tostrong decays into ¯ BD and ¯ Bπ , respectively.As remarked for the first time in [12, 13], it is possibleto derive constraints on the form factor even if the residueof the pole is not known. Denoting by z p the position ofthe pole in the z -variable, the inclusion of the pole canbe done in an optimal way with respect to the condition(8) by using a so-called Blaschke factor [5]: B ( z ; z p ) ≡ z − z p − zz ∗ p , (23)which is a function analytic in | z | ≤ z = z p and has modulus unity for z on the unit circle: | B ( ζ ; z p ) | = 1 , ζ = e iθ . (24)By using (24), one obtains from (8), with no loss of in-formation, the equivalent condition12 πi (cid:73) C dzz | B ( z ; z p ) φ ( z ) F [˜ t ( z ; t )] | ≤ . (25)Taking into account that the product B ( z ; z p ) F ( z ) is an-alytic in | z | <
1, we write the most general parametriza-tion of the form factor as F ( z ) = 1 B ( z ; z p ) φ ( z ) ∞ (cid:88) k =0 a k z k , (26)where the coefficients a k still satisfy (13).Since by the maximum modulus principle | B ( z ; z p ) | < | z | <
1, the constraints in the semileptonic regionderived from (26) are weaker than those valid when nosubthreshold poles are present.
III. ABOVE-THRESHOLD POLES
The possible effect of an above-threshold resonance wasinvestigated in [36], starting with the remark that a polein the form factor at the same position as the resonancepole, but situated on the first Riemann sheet, creates aBreit-Wigner lineshape indistinguishable from that cre-ated by a physical second-sheet pole equally near theunitarity cut. Therefore, the effect of a second-sheet polewas simulated by a pole situated on the first sheet. InAppendix A we give for completeness the positions in the z plane of a second-sheet pole, z II p , and its counterpart onthe first sheet, z I p , for some particular form factors. Thetreatment of the fake pole at z p ≡ z I p by the techniqueof Blaschke factors, as shown in the previous section, ledto the conclusion that an above-threshold resonance hasthe effect of weakening the unitarity bounds. The ef-fect was found to be small in the case of the Kπ and Dπ vector form factors. However, since any informationon the modulus of the form factor on the cut is coveredby the rigorous condition (6), which is the main ingre-dient of the formalism, one can see that accounting forthe fake pole is not necessary. Thus, the presence of anabove-threshold pole does not affect the bounds in thesemileptonic region.On the other hand, it is known that a pole of the scat-tering amplitude as a function of c.m. energy squaredon a higher Riemann sheet can produce in some cases(such as elastic 2 → S -matrix element at thecorresponding point on the first sheet. This property isuseful in practice: In [43], the mass and width of the σ scalar resonance were found by performing the analyticcontinuation of the Roy equations for ππ scattering intothe first sheet of the complex plane and looking for thezeros of the S matrix.One might ask whether a similar property exists forform factors. In order to answer this question, we con-sider in more detail the analytic continuation to the sec-ond Riemann sheet. According to the general dispersiveapproach in field theory [38], it is useful to consider, alongwith a given form factor F ( t ), the corresponding ampli-tude (of definite angular momentum and isospin) of theelastic scattering of two hadrons of masses M and m . Wereview below some well-known facts about these quanti-ties that are useful for our purpose.Denoting by f ( t ) the relevant partial wave of the in-variant elastic amplitude, elastic unitarity is expressedas Im f ( t ) = ρ ( t ) f ( t ) f ∗ ( t ) , t + ≤ t ≤ t in , (27)where ρ ( t ) = (cid:112) (1 − t + /t )(1 − t − /t ) is the dimensionlessphase space. This relation is valid in the elastic region,below the opening of the first inelastic threshold t in . Un-less otherwise specified, by real t above the threshold t + ,we mean the value t + i(cid:15) , on the upper edge of the cut.Equation (27) has the well-known solution [38] f ( t ) = e iδ ( t ) sin δ ( t ) ρ ( t ) , t + ≤ t ≤ t in , (28)in terms of the phase shift δ ( t ).The relation (27) provides also the route for analyticcontinuation to the second Riemann sheet. Using the Schwarz reflection property f ∗ ( t ) = f ( t ∗ ), we write (27)as f ( t + i(cid:15) ) − f ( t − i(cid:15) ) = 2 iρ ( t ) f ( t + i(cid:15) ) f ( t − i(cid:15) ) . (29)The amplitude f II ( t ) on the second sheet is defined bygluing the lower edge of the cut in the first sheet to theupper edge on the cut in the second sheet, i.e. , by requir-ing f II ( t + i(cid:15) ) = f ( t − i(cid:15) ). Understanding all quantitieswithout a superscript as defined on the first Riemannsheet, we write Eq. (29) as f II ( t ) = f ( t )1 + 2 iρ ( t ) f ( t ) . (30)The S matrix is defined on the first sheet as S ( t ) = 1 + 2 iρ ( t ) f ( t ) , (31)and on the second sheet as S II ( t ) = 1 − iρ ( t ) f II ( t ) . (32)Using the definition (30) of f II ( t ), one obtains: S II ( t ) = 1 S ( t ) . (33)From this relation it follows that the poles of f II ( t ) [andof S II ( t )] correspond to zeros of S ( t ) on the first sheet,the property mentioned at the beginning of this section.Turning now to form factors, elastic unitarity impliesthe relation [38]Im F ( t ) = ρ ( t ) F ∗ ( t ) f ( t ) , (34)valid for t in the elastic region, t + ≤ t ≤ t in .A first consequence of (34) is the well-known Fermi-Watson theorem [39, 40]: Since the right-hand side isknown to be real, the phase of the form factor must beequal to the phase shift of the amplitude (28):arg[ F ( t )] = arg[ f ( t )] = δ ( t ) , t + ≤ t ≤ t in . (35)Moreover, by defining, in analogy to f II ( t ), F II ( t + i(cid:15) ) ≡ F ( t − i(cid:15) ) , (36)one obtains from (34): F II ( t ) = F ( t )1 + 2 iρ ( t ) f ( t ) = F ( t ) S ( t ) . (37)Assuming that F ( t ) does not vanish at the zero of S ( t ), F II ( t ) has a pole at that position. So, the second-sheetpoles of the form factor and the S -matrix element havethe same position, a known universality property of thepoles in S -matrix theory. The relation (37) shows alsothat the analytic structure of the function F II ( t ) is morecomplicated that that of F ( t ): besides the unitarity cut,it has the same branch points as S ( t ), in particular thoselying on the left-hand cut produced by crossed-channelexchanges [38].We show now that it is possible to express the value of F ( t ) on the first sheet, at the value of t corresponding tothe second-sheet pole position, in terms of the residuesof the poles of the form factor and the amplitude on thesecond sheet. From (33) and (37) one has: F ( t ) = F II ( t ) S ( t ) = F II ( t ) S II ( t ) . (38)Denoting by t p one of the pole positions on the secondsheet, in the vicinity of the pole one can write f II ( t ) = r f t − t p + g ( t ) , (39)and F II ( t ) = r F t − t p + h ( t ) , (40)where the functions g and h are regular at t = t p . Usingthese expressions and (32) in (38) and taking the limit t → t p gives F ( t p ) = i ρ ( t p ) r F r f . (41)From the Schwarz principle, F ( t ∗ p ) = F ∗ ( t p ), the valueof F at t ∗ p (still on the first sheet) is the complex con-jugate of the expression (41). As shown in the previoussection, this additional condition on the first sheet can beincluded exactly in the Meiman-Okubo problem, leadingto an improvement of the bounds in the semileptonic re-gion. The relation (14) gives the allowed domain of thecoefficients a k in terms of this additional information. Itcan be viewed therefore as a new sum rule relating theresidues of the above-threshold poles on the second Rie-mann sheet to the parameters describing the semileptonicdecays.In practice, if the ratio r F /r f is not known, one canreverse the argument and use (14) as a constraint on theresidues, in terms of the coefficients a k determined fromfits to semileptonic decay data. However, the correlationis expected to be small, due to the fact that, as shownin Appendix A, in cases of interest the point z p ≡ z I p is close to the boundary | z | = 1. Therefore, the valueof the new sum rule in this case is of more formal thanphenomenological significance.Of more practical value turns out to be another con-sequence of unitarity that is valid on the unitarity cutbelow the first inelastic threshold. By dividing both sidesof (29) by the product f ∗ f , one has1 f ∗ ( t ) − f ( t ) = 2 iρ ( t ) , (42)which implies Im (cid:20) f ( t ) (cid:21) = − ρ ( t ) . (43) FIG. 1: Phase of the Dπ scalar form factor as a function ofthe c.m. energy E = √ t . The solution of this equation is f ( t ) = 1 ψ ( t ) − iρ ( t ) , (44)where the undetermined function ψ ( t ) is real on the elas-tic part of the unitarity cut, t + ≤ t ≤ t in . If a narrowresonance of mass M and width Γ is present, this func-tion can be parametrized as ψ ( t ) ∼ M − tM Γ , (45)up to factors holomorphic in a region t + < t < t in , where t in denotes the first inelastic threshold. By including allthese factors in an energy-dependent Γ( t ), we can write,with a good approximation, the phase of the form factorin a limited energy region above the threshold as:arg[ F ( t )] = arctan (cid:20) M Γ( t ) M − t (cid:21) . (46)This relation can be generalized to the case where over-lapping resonances occur. In such a case, it is a well-known feature of S -matrix theory that simply summingBreit-Wigner resonances does not preserve unitarity, andthe proper treatment would require allowing Γ not only tobe dependent on energy, but also a matrix-valued quan-tity over the various channels.In Fig. 1 we show the phase δ of the scalar Dπ form factor obtained from (46), using the standard Breit-Wigner expression Γ( t ) = Γ ρ ( t ) /ρ ( M ), with the mass M = 2 .
351 GeV and width Γ = 0 .
230 GeV of the D ∗ res-onance [44]. We can assume that this value of the phaseis a good approximation in the elastic region, below theopening of inelastic channels.As discussed in the previous section, this additionalinformation leads to a stronger constraint in the semilep-tonic region, given by Eq. (19). This constraint can beeasily derived by solving the integral equation (21) forthe function λ ( θ ) and using this solution in (19). Forillustration, we present below the result of this analy-sis for the scalar Dπ form factor. We take the value χ LV ( q = 0) = 0 .
016 from Ref. [25] and the outer func-tion from Refs. [25, 27]: φ ( z ) = √ (cid:113) χ LV (0) √ π m D − m π m D + m π (1 − z )(1 + z ) / × (1 − zz − ) / (1 + z − ) / , (47)where we take for simplicity t = 0 in (7) and use thenotation z − ≡ ˜ z ( t − ; 0).Taking for illustration K = 5, we obtain the alloweddomain for the coefficients a k , k ≤ . a + 1 . a + 1 . a + 1 . a + 2 . a + 0 . a a − . a a − . a a − . a a − . a a − . a a − . a a + 0 . a a + 0 . a a + 2 . a a ≤ . (48)In this calculation, we assume that the phase is givenup to the first inelastic threshold t in = (2 .
42 GeV) dueto the Dη channel. The results are actually quite stableagainst the variation of t in around this value.It is easy to see that the constraint (48) is stronger thanthe standard condition (17). In a typical application tosemileptonic processes, the lowest coefficients a k are de-termined from fits of the data, and the aim is to set abound on the next coefficient, which gives an estimate ofthe truncation error. In practical applications (see for in-stance [24]), the optimal values of the parameters are usu-ally small, far from saturating the upper bound (17). Tosimulate such a situation, we take, for instance, the inputvalues a = 0 . , a = 0 . , a = 0 .
07 and a = 0 .
05, forwhich the left hand side of (17) is 0.024. With this input,we obtain the constraint | a | ≤ .
99 from the standardinequality (17), and the smaller range − . ≤ a ≤ . δF ( t − ) at the end t − (corresponding to z − ) of the semileptonic region. Fromthe parametrization (12), one can write this error as: δF ( t − ) ≈ | a | z − | φ ( z − ) | . (49)Using the above limits on a and the values z − = 0 . φ ( z − ) = 0 .
176 in our case, we obtain from (49) theuncertainties δF ( t − ) ≈ .
063 using the standard con-straint (17) and δF ( t − ) ≈ .
043 using the improved con-straint (48), which amounts to an improvement by about30%. Similar results are obtained for a large class ofinput values for the lowest coefficients. One can use also the optimal value of t discussed inSec. II, for which the semileptonic region is mapped ontoa symmetric range in the z plane. From Eq. (A7), weobtain in our case t = 1 .
97 GeV and z − = 0 . z − , the error estimated from (49) is muchsmaller, but the constraints on the coefficient a are sim-ilar to those reported above. In this case too, the im-provement brought by the incorporation of the phase δ turns out to be quite important. IV. SUMMARY AND CONCLUSIONS
In this paper we have continued the discussion ofthe effect of above-threshold singularities on model-independent form-factor parametrizations, initiated inRef. [36]. We emphasized the fact that the presence ofabove-threshold poles does not affect the strength of theoriginal model-independent constraints. By exploitingthe connection between the first and the second Riemannsheets of a generic semileptonic form factor, we have de-rived a relation between the value of the form factor onthe first Riemann sheet at the point t p that is the image ofthe location of the resonance pole on the unphysical (sec-ond) Riemann sheet, and the residues of the form factorand of the related elastic scattering amplitude. Using thisexpression in the combined constraint (14) involving thecoefficients a n of a Taylor series expansion in the variable z and the values of the form factor at the two complex-conjugate points, we derived a new sum rule relating theparametrization in the semileptonic region to the residuesof the second-sheet poles of the form factor F and thecorresponding elastic scattering amplitude f . We arguedhowever that the effect of this additional information inimproving the model-independent constraints is expectedto be small. Finally, we showed that from the mass andwidth of a narrow resonance, one can approximately ob-tain the phase of the form factor on a limited part of theunitarity cut. By including this additional information inthe extremal problem, one obtains stronger constraints,given in (19), on the form-factor parametrization in thesemileptonic region. This second method appears to beof more immediate utility in phenomenological applica-tions. Acknowledgments
I.C. acknowledges support from the Ministry of Re-search and Innovation, Contract PN 16420101/2016.B.G. was supported by the U.S. Department of Energyunder Grant de-sc0009919. R.F.L. was supported bythe U.S. National Science Foundation under Grant No.1403891.
Appendix A: Uniformization of the two-sheetRiemann surface by z mapping In this Appendix we discuss the connection betweenthe canonical variable z in Eq. (7) used for solving theextremal problem in Sec. II and the Riemann structureof the elastic cut of the semileptonic form factor F ( t ).We first note that (7) can be written as z ≡ ˜ z ( t ; t ) = √ t + − t + ik ( t ) √ t + − t − ik ( t ) , (A1)in terms of the function k ( t ) = (cid:112) t − t + . (A2)We recall that the first Riemann sheet is defined byarg( t − t + ) ∈ (0 , π ), while the second sheet is definedby arg( t − t + ) ∈ (2 π, π ). It follows that the first Rie-mann sheet corresponds to arg k ( t ) ∈ (0 , π ), which im-plies k I ( t ) >
0, and the second Riemann sheet corre-sponds to arg k ( t ) ∈ ( π, π ), which implies k I ( t ) < k I ( t ) is the imaginary part of k ( t ). Denoting by k R ( t ) the real part of k ( t ), we obtain from (A1): | z | = [ √ t + − t − k I ( t )] + k R ( t )[ √ t + − t + k I ( t )] + k R ( t ) . (A3)From this relation it follows that k I ( t ) > ⇒ | z | < ,k I ( t ) < ⇒ | z | > . (A4)Therefore, the first Riemann sheet of the t plane, where k I ( t ) >
0, is mapped inside the unit circle in the z plane,while the second sheet, where k I ( t ) <
0, is mapped out-side the unit circle. In standard terminology, the variable(7) achieves the uniformization of the Riemann surfaceof the elastic cut, i.e. , it maps the two Riemann sheetsonto a single plane.For the discussion in Sec. III, it is useful to have arelation between the images in the z plane of the pole onthe second sheet, and of the corresponding complex point situated on the first sheet. This relation follows from thesymmetry property˜ t ( z ; t ) = ˜ t ( z − ; t ) , (A5)satisfied by (9), which shows that the images in the z plane of the first-sheet and second-sheet points corre-sponding to the same complex t value are inverse to eachother, z I p = 1 z II p . (A6)For a numerical illustration, we take for definiteness t = t + (cid:20) − (cid:114) − t − t + (cid:21) , (A7)to achieve a symmetric semileptonic range ( − z max , z max ),as discussed in Sec. II. 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