OMNÈS REPRESENTATIONS WITH INELASTIC EFFECTS FOR HADRONIC FORM FACTORS

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1 FIELD THEORY, GRAVITATION AND PARTICLE PHYSICS OMNÈS REPRESENTATIONS WITH INELASTIC EFFECTS FOR HADRONIC FORM FACTORS IRINEL CAPRINI National Intitute of Phyic and Nuclear Engineering POB MG 6, Bucharet, R-775 Romania Received December, 4 We derive a generalized Omnè repreentation for the hadronic form factor, which atifie Waton theorem in the elatic region and include the effect of inelatic channel. A an application we dicu the behaviour of the calar form factor of the pion near the KK threhold. The reult are ueful alo for the calculation of the phae produced by the trong final tate interaction in the nonleptonic decay of the K and B meon.. INTRODUCTION The hadronic form factor, defined a matrix element of operator bilinear in the quark field among hadronic tate, play an important role both in perturbative and nonperturbative quantum chromodynamic (QCD). Perturbative QCD predict the behaviour of the form factor only at large momentum tranfer in the pace-like region, where aymptotic freedom hold and the hadronic threhold are abent []. On the other hand, at low energie, chiral perturbation theory (ChPT) provide a ytematic expanion of thee quantitie in power of the momenta and the quark mae. The form factor of the light peudocalar meon were calculated to one-loop in [] and beyond thi order in [3]. The complete evaluation to two loop i given in [4]. Diperion theory provide alo a powerful tool for tudying the form factor and relating their low and the high energy behaviour. In the complete theory of QCD, which include confinement, the form factor are analytic function of real type in the complex energy plane cut along the real axi from the threhold impoed by unitarity to infinity. The mot convenient diperive repreentation i the o-called Omnè repreentation [5], which expree the form factor in term of their phae along the cut. Thi repreentation allow an eay implementation of Waton theorem [6], which tate that in the elatic region the phae of the form factor i equal to the phae of the elatic final tate cattering. Many application of the Omnè repreentation and it mathematical generalization for variou weak and electromagnetic form factor exit in the Rom. Journ. Phy., Vol. 5, No., P. 7 7, Bucharet, 5

2 8 Irinel Caprini literature(ee for intance [7] where diperive and chiral ymmetry contraint on the light meon form factor were derived). The incluion of the inelatic channel in the Omnè repreentation i neceary for calculating quantitie of interet for ChPT like the quadratic radiu of the pion. In particular, the effect of the KK channel on the calar form factor of the pion wa recently a controverial ubject. The problem wa invetigated in the frame of a two-channel generalization of the Omnè repreentation (the ocalled Muhkhelihvili-Omnè (M-O) equation [8]) in [9], [], and more recently in []. The concluion of thee work i that the opening of the KK channel can have important effect on the phae of the calar form factor around GeV, the behaviour depending trongly of the quark tructure of the correponding operator. In Ref. [], on the other hand, it i claimed that the effect of inelaticity i negligible. A thi concluion i baed on a ingle-channel Omnè formalim, it i of interet to include the effect of inelaticity in thi formalim. In the preent paper we addre thi problem and write down a ingle-channel Omnè repreentation which include explicitly the influence of the inelatic channel. We tre that the complete olution i provided only by olving the coupled-channel M-O equation. The formulae which we derive are ueful however ince they offer a rather tranparent picture of the inelatic effect, allowing u to undertand in a qualitative way the different behaviour of the variou form factor. Alo, undertanding the inelatic channel i crucial for predicting the effect of the final tate interaction in nonleptonic weak decay like K ππ or B ππ.. DISPERSIVE REPRESENTATION IN TERMS OF THE PHASE We conider the calar form factor of the pion Γ () =<π( p)( π p ) muuu+ mddd > () and () =<π( p)( π p ) m >, () where = ( p+ p ), u, d, are the quark field and m u, m d, m their current mae. For implicity, we denote generically the above form factor by F(). The function F() i analytic in the -plane cut from the elatic threhold at = 4m π to infinity. The phae δ F ( ) of F() on the cut i defined by the boundary condition iδf ( ) π F ( + iε) = e F ( iε ), > 4m. (3) Thi relation repreent a Riemann boundary value problem [8], with the general olution [5]:

3 3 Omnè repreentation for hadronic form factor 9 ( )d () ()exp δf F = Pn, π (4) ( ) 4mπ where Pn ( ) i a polynomial of degree n. Perturbative QCD predict the aymptotic behaviour [] ( ) ()~ F α,, (5) where α ( ) 4 9ln( = π/ /Λ ) i the QCD running coupling. From the Omnè repreentation (4), thi implie the aymptotic behaviour of the phae δ F ()~( n + ) π+ π,. ln Λ Chiral expanion ugget that the form factor Γ ( ) ha no zero in the complex plane, which mean that n =, the polynomial in (4) reduce to a contant and δγ()~ π+ π,. ln Λ On the other hand, for the form factor (), ChPT indicate a zero cloe to =, which mean that in thi cae n = and the aymptotic behaviour of the phae i δ ()~ π+ π,. ln Λ (6) (7) (8) 3. UNITARITY RELATION AND WATSON THEOREM Along the cut > 4, the form factor F ( ) atifie the unitarity relation m π Im F () =σ () F ()[ f ()] +σ (), (9) in where σ () = 4m π/ and the iocalar S-partial wave parametrized a η iδ () f e =, iσ( ) f () i () in term of the elaticity η and the phae-hift δ. All the function in (9) are evaluated on the upper edge of the cut. Neglecting the mall contribution

4 Irinel Caprini 4 of four pion, the inelatic term σ in( ) can be approximated at low energie by the contribution of the KK channel in K K K πk σ () =θ( 4 m ) σ () F () T (), () where σ 4 K = mk/ i the phae pace, FK ( ) i the kaon calar form factor defined by replacing in Eq. () or () the pion pair by a KK pair, and denote the ππ KK S-wave amplitude. Eq. (9) can be written a or, uing (), a: ( ) T π K F ( + i ε ) iσ() f () F ( i ε ) = iσ in(), () iδ ( ) F ( + iε) η ()e F ( iε ) = iσ (). (3) in In the elatic region < 4 m K, where η = and σ ( ) =, Eq. (3) reduce to in iδ ( ) F ( + iε) = e F ( iε ), < 4m K, (4) which, compared to (3), yield Waton theorem δ () =δ () for < m. Above the KK threhold the phae δ F i no longer equal to the phae hift. F 4 K 4. OMNÈS REPRESENTATIONS WITH INELASTICITY Equation (3) ha the form of an nonhomogeneou Riemann boundary value problem, whoe general olution i given in [8]. Following [6], we look for olution F which atify elatic unitarity and time reveral invariance, which require that the form factor i real-analytic in the cut plane, F ( ) = F ( ). We look for olution of the form F () = GO () (), (5) where O ( ) i an Omnè function defined in term of a certain phae, and G ( ) a reidual function which account for the inelaticity. To atify Waton theorem, the phae of O ( ) mut be equal to δ () below the inelatic threhold, but above it the phae i arbitrary. For every choice of the phae of O ( ) we calculate the remaining function G from the unitarity relation.

5 5 Omnè repreentation for hadronic form factor 4.. OMNÈS FUNCTION DEFINED WITH THE PHASE SHIFT A natural choice i to take the phae of O equal to the phae hift δ along the whole cut up to infinity. So we write the form factor a where F () = G() O() (6) =. (7) ( ) π ( ) () exp O δ d π 4m In order to calculate the reidual function G appearing in (6), we firt notice that by Waton theorem it i real below the inelatic threhold, o it i analytic in the -plane cut only for > 4 m K. From (6) we have Re G ( ) = [ReFcoδ + ImFin δ ], O () Im G ( ) = [ImFcoδ ReFin δ ]. O () (8) On the other hand, by multiplying both ide of (3) with real and imaginary part we have δ and taking the e i ( ) ReFcoδ + Im inδ = Im σine F η ImFcoδ Re in δ = Re eiδ σ in. By comparing with (8) we have F +η iδ (9) η +η iδ Im[ σin e ] Re G ( ) =, O () iδ Re[ σin e ] Im G ( ) =. O () () Therefore, the function G () in the repreentation (6) atifie (up to a polynomial real on the cut) the Omnè repreentation ( ) () exp ψ G = d π () ( ) 4 m K

6 Irinel Caprini 6 where ψ ( ) i the argument of Re G( ) + iim G( ) : { σ eiδ in } i { σ δ } η Re ψ () = Arctg. +η Im in e () From (6), (7) and () it follow that the phae δ F of the form factor i given by δ () =δ +ψ (). (3) F 4.. OMNÈS FUNCTION DEFINED WITH THE PHASE OF THE PARTIAL WAVE AMPLITUDE An alternative choice i to take the phae of the Omnè function O in (5) equal to the phae δ t of the partial wave amplitude, defined by So, we take where Im f ( ) η coδ δ t () = Arctg Re =. (4) f ( ) η in δ F () = G() O() (5) =. (6) ( ) π ( ) () exp t O δ d π 4m, In the elatic region δ t =δ but above the KK threhold thi equality i no longer valid. In fact, the phae-hift δ ha the peculiarity that it raie rapidly near the KK threhold reaching the value π above (and cloe to) it. From Eq. (4) it folow that when η < and δ pae through π, the phae δ t ha a dip, which become teeper when the elaticity i cloe to. So, above the inelatic threhold the phae δ t of the amplitude i quite different from the phae hift δ. Actually, if δ ( ) =π for < 4 m K, then at thi point phae δ t make a jump by π, t δ ( ) δ ( ) =π, in order to preerve the poitivity of the imaginary part Im f = in δ. We hall aume that parametrization, o that δ reache π only above the KK threhold, a indicate mot experimental δ=δ in the whole elatic region.

7 7 Omnè repreentation for hadronic form factor 3 In order to calculate the reidual function G defined in (5) we notice that for > 4m K the two term in the r.h.. of Eq. (9) are not eparately real, but their um mut be real. Taking the real and the imaginary part of (9) we have: { F f } { F f } Reσ in = ImF Re σ [ ], (7) Im σ in = Im σ ( )[ ]. Inerting in thee relation the repreentation (5) and noting that with the choice (6) the product O [ f ] = O f i real, we obtain Im σin Im G ( ) = σ O () f Reσ / O Im G ( )coδt Re G ( ) =. δ σ in in t f Denoting by ψ ( ) the argument of Re G() + iim G() (8) Im G ( ) ψ () = Arctg, Re G ( ) we write the function G, up to a polynomial, a (9) ( ) () exp ψ G = d π (3) ( ) 4 m K From (5), (6) and (3) it follow that the phae δ F of the form factor i given by δ () =δ +ψ (). (3) F t 5. COMMENTS ON THE SCALAR FORM FACTORS OF THE PION The two approache decribed above are equivalent and mut lead to identical reult. We checked thi equivalence fot the numerical olution of the twochannel M O equation calculated in [] uing the experimental data from [3]. More exactly, we evaluated the right hand ide of the relation (3) and (3) uing a input the correponding quantitie calculated in [] and checked that they lead to identical reult, which moreover coincide with the phae of the form factor obtained in []. The reult are hown in Fig. (reproduced from []), where we indicate the phae hift δ, the phae δ t of the partial wave, and the phae of the form factor Γ and. Below the opening of the inelatic

8 4 Irinel Caprini 8 channel all the phae depicted are equal. Above the KK threhold, δ Γ ha a pronounced dip and then follow cloely the phae δ t of the cattering amplitude, taying below the phae hift δ by approximately π. Thi behaviour of the form factor Γ i confirmed by the experimental data on the central production of pion pair in pp colliion [7]. In the notation of the previou ection, thi mean that for the form factor Γ the additional phae ψ from (3) i negative and approache rapidly the value π, while the additional phae ψ from (3) i cloe to zero. On the other hand, the phae of the form factor follow cloely the phae hift alo above the KK threhold, which mean that the additional phae ψ from (3) i mall, while the phae ψ from (3) i large and cloe to π. Fig.. The phae δ Γ of the pion form factor Γ() calculated from the twochannel M-O equation [] (olid line). The dahed and dotted line decribe the phae hift δ and the phae δ t of the partial wave amplitude, repectively. The dah-dotted line depict the phae δ of the form factor (). Thi behaviour, obtained numerically in [], can be undertood qualitatively uing the expreion derived in Section 4. For illutration, we conider the repreentation given in ubection 4., where the Omnè function i expreed in term of the phae hift. A we mentioned, the experimental data on ππ cattering [3] indicate that the phae hift δ () raie very rapidly and reache the value π jut above the inelatic KK threhold. Moreover, jut above thi threhold the elaticity η ha a harp decreae, indicating a very trong inelaticity, and then rather quickly approache again the elatic value η =.

9 9 Omnè repreentation for hadronic form factor 5 From the expreion () it follow that, if the elaticity η () i cloe to, the additional phae ψ appearing in (3) i equal to modulo ±π. Denoting by δ σin the phae of the complex quantity σ in = σ in exp( iδ σin ) and omitting the irrelevant poitive factor, we notice from (8) that the phae ψ depend on the ign of the quantitie G σ in Re ~ in( δ +δ )/( η ) (3) G σ in Im ~ co( δ +δ ). (33) In the vicinity of the threhold the phae δ σ in i uppreed by the phae pace. Since, a we mentioned above, the phae hift δ i cloe to π, the quantity co( δ σ in +δ ), which determine the ign of the imaginary part of G, i negative. Moreover, jut above the threhold, where δ i till le than π, the real part ReG defined in (3) i poitive. Thi mean that the point aociated to the complex quantity G i ituated in the fourth quadrant of the trigonometric circle, and ψ <. From (3) it follow therefore that the inelaticity ha the effect of lowering the phae of the form factor. The evolution of the phae ψ at higher energie depend on the ign of the real part ReG. If in( δ +δ ) > (34) σ in then ReG > and the point aociated to G remain in the fourth quadrant. Hence, when the inelaticity η approache again the value, the phae ψ tend to through negative value, and the phae of the form factor approache δ from below. But if in( δ +δ ) <, (35) σ in then the point aociated to G enter the third quadrant, and ψ π when η become cloe to. The deciive role i played therefore by the phae of the inelatic term σ in ~ FK ( ) T πk ( ) defined in (). Thi quantity can be undertood by noticing that the two-channel unitarity equation given in [9] are atified by the following expreion: F () = c() T () + c() T () ππ πk F () = c () T () + c () T (), K πk KK (36)

10 6 Irinel Caprini where Tππ = f, T π K and T KK denote the S-wave projection of the ππ ππ, ππ KK and KK KK amplitude, repectively, and the function c () and c( ) are real for > 4m π. If the coefficient c and c are poitive, the relation (36) imply, by the parallelogram rule for vector addition, that the phae of the pion form factor F π i larger than the phae of T ππ and maller than the phae of T πk, while the phae of the kaon form factor F K i larger than the phae of T KK and maller than the phae of T πk. We recall that by unitarity the phae of the nondiagonal amplitude T πk i the um of the phae hift of the diagonal element [9]. The experimental data [3] [5] indicate that the phae hift δ K of the KK KK tranition i negative. Uing a relation imilar to () (with δ replaced by δ K ), we obtain for the phae of T KK poitive value in the econd quadrant. Let u conider firt the form factor Γ ( ) defined in (). The coefficient c and c take value conitent with the aymptotic condition (7). The explicit calculation with data from [3] indicate that Γ ()~ c () Tππ(), which implie that δγ ~ δ t. The Omnè formalim i conitent with thi reult: inerting Γ K()~ c () TπK() in the expreion (), it follow that the δ σ in i cloe to and the relevant quantity in Eq. (3) i in( δ +δ ) ~ in δ. Since above the σ in KK threhold δ become rapidly greater than π, the inequality (35) hold, which mean that the difference δγ δ tend to π, a hown in Fig.. Thi reult i quite table with repect to the parametrization of the unitary S-matrix. Indeed, even if the firt term in (36) i not dominant and the phae of σ in i negative, the correction i not very large, and till lead to in( δ σ in +δ ) <, due to the large value of δ. In the cae of the form factor defined in (), the aymptotic condition (8) elect a different pattern for the coefficient c and c, which may be negative. It follow that δ σ in i a large negative phae, o that the um δ σ in +δ become le than π and the inequality (34) hold. Therefore, when the elaticity η approache, the difference δ δ tend to. 6. CONCLUSIONS In the preent paper we derived ingle channel Omnè repreentation for the hadronic form factor, which include explicitly the effect of the inelatic

11 Omnè repreentation for hadronic form factor 7 channel in the unitarity um. A we dicued in Section 4, the reult provide a qualitative undertanding of the calar form factor of the pion in the vicinity of the inelatic KK threhold. We mention that the reult are ueful alo for including the effect of final tate recattering in the nonleptonic decay like K ππ and B ππ, which are of interet for the CP-violation parameter in the Standard Model. Diperion relation and Omnè repreentation for the amplitude of thee decay, including the effect of initial and final tate interaction, were derived in [8], [9]. The function G decribing the inelatic channel in K ππ decay wa expanded in a power erie baed on a conformal mapping [9]. In the cae of B nonleptonic decay, the additional phae () produced by the final tate interaction can be evaluated in term of the weak decay amplitude into intermediate peudocalar and vector meon, uing Regge theory for the trong recattering amplitude []. Acknowledgment. Thi work wa upported by a grant of the Romanian Academy, under the contract /4. REFERENCES. G. P. Lepage and S. J. Brodky, Phy. Rev. D, 57 (98).. J. Gaer and H. Leutwyler, Nucl. Phy. B 5, 57 (985). 3. J. Gaer and U. G. Meiner, Nucl. Phy. B 357, 9 (99). 4. J. Bijnen, G. Colangelo and P. Talavera, JHEP 985, 4 (998). 5. R. Omnè, Nuovo Cim. 8, 36 (958). 6. K. M. Waton, Phy. Rev. 95, 8 (954). 7. I. Caprini, Eur. Phy. J. C 3, 47 (). 8. N. I. Mukhelihvili, Singular Integral Equation, Noordhoff-Groningen (953). 9. J. F. Donoghue, J. Gaer and H. Leutwyler, Nucl. Phy. B 343, 34 (99).. B. Mouallam, Eur. Phy. J. C 4, ().. B. Ananthanarayan, I. Caprini, G. Colangelo, J. Gaer and H. Leutwyler, Phy. Lett. B 6, 8 (4).. F. J. Yndurain, Phy. Lett. B 578, 99 (4) [Erratum-ibid. B 586, 439 (4)]. 3. B. Hyam et al., Nucl. Phy. B64, 34 (973). 4. D. Cohen et al., Phy. Rev. D, 595 (98). 5. W. Wetzel et al., Nucl. Phy. B 5, 8 (976). 6. T. N. Pham and T. N. Truong, Phy. Rev. D 6, 896 (977). 7. D. Morgan and M. R. Pennington, Phy. Rev. D 58, 3853 (998). 8. I. Caprini, L. Micu and C. Bourrely, Eur. Phy. J. C, 45 (). 9. C. Bourrely, I. Caprini and L. Micu, Eur. Phy. J. C 7, 439 (3).. J. Donoghue, Phy. Rev. Lett., 77, 78 (996).