RUB-TPII-02/99 Hard exclusive electroproduction of two pions and their resonances 1 M.V. Polyakov Petersburg Nuclear Physics Institute, Gatchina, St. Petersburg 188350,Russia and Institut fu¨r Theoretische Physik II, Ruhr–Universit¨at Bochum, D–44780 Bochum, Germany 9 9 9 Abstract 1 We study the hard exclusive production of two pions in the virtual photon fragmentation region n with various invariant masses including the resonance region. The amplitude is expressed in terms a of two-pion light cone distribution amplitudes (2πDA’s). We derive dispersion relations for these J amplitudes,which enablesustofixthemcompletely intermsofππ scattering phasesandafew low- 4 energy subtraction constants determind by the effective chiral lagrangian. Quantitative estimates 1 of the resonance as well ππ background DA’s at low normalization point are made. We also prove certainnewsoftpiontheoremrelatingtwo-pionDA’stotheone-pionDA’s. Crossingrelationsbetween 1 2πDA’sand parton distributions in a pion are discussed. v We demontrate that by studying the shape of the ππ mass spectra (not absolute cross section!) in 5 adiffractiveelectroproduction onecan extract thedeviation ofthemeson (π,ρ,etc.) wavefunctions 1 3 from their asymptotic form 6z(1−z) and hence to get important information about the structure 1 of mesons. We suggest (alternative to S¨oding’s) parametrization of ππ spectra which is suitable for 0 large photon virtuality. 9 9 The aimofthis talk is to outline the approachto two-pionhardelectroproductiondevelopedin[1]. Here / h we summarize the main results, details can be found in [1]. p Hard exclusive electroproduction of mesons is a new kind of hard processes calculable in QCD. In - [2, 3] was shown that a number of exclusive processes of the type: p e h : γ∗(q)+T(p) F(q′)+T′(p′), (1) v L → i X at large Q2 with t = (p p′)2, x = Q2/2(pq) and q′2 = M2 fixed, are amenable to QCD description. − Bj F r The factorization theorem [2] asserts that the amplitude of the process (1) has the following form: a 1 dz dx fT′ (x ,x x ;t,µ)H (Q2x /x ,Q2,z,µ)ΦF(z,µ) Z Z 1 i/T 1 1− Bj ij 1 Bj j Xi,j 0 +power-suppressedcorrections, (2) where fT′ is T T′ skewed parton distribution (for review and referencies see [4]), ΦF(z,µ) is the i/T → j distribution amplitude of the hadronic state F (not necessarily one particle state), and H is a hard ij part computable in pQCD as series in α (Q2). Here we shall study generalproperties of the distribution s amplitudes ΦF(z,µ) when the final hadronic state F is a two pion state (F =ππ). j ∗ The2πDA’swereintroducedrecentlyin[5]inthecontextofQCDdescriptionoftheprocessγ γ 2π. → Inref.[6,1]itwassuggestedthatthe usagesof2πDA’sto describeahardelectroproductionoftwopions leads to universal picture of resonant and non-resonant two pion production. Below we listed main properties of 2πDA’s and shortly discuss their applications to hard diffractive production of two pions off nucleon. 1Talk given at conference “Particle and Nuclear Physics with CEBAF at Jefferson Lab”, Dubrovnik, 3-10 November 1998 1 Definition: • Φab(z,ζ,m2ππ) = 41π Z dx−e−2izP+x−outhπa(p1)πb(p2)|ψ¯(x)nˆT ψ(0)|0i(cid:12)x+=x⊥=0, (3) (cid:12) (cid:12) Here, n is a light–like vector (n2 = 0), which we take as n = (1,0,0,1). For any vector, V, the “plus” µ light–conecoordinateisdefinedasV+ (n V)=V0+V3;the“minus”componentasV− =V0 V3. The ≡ · − outgoingpionshavemomentap ,p ,andP p +p isthetotalmomentumofthefinalstate. Finally,T 1 2 1 2 ≡ isaflavormatrix(T =1fortheisosinglet,T =τ3/2fortheisovector2πDA).Thegeneralizeddistribution amplitudes, eq. (3), depend on the following kinematical variables: the quark momentum fraction with respect to the total momentum of the two–pion state, z; the variable ζ p+/P+ characterizing the ≡ 1 distribution of longitudinal momentum between the two pions, and the invariant mass, m2 =P2. ππ The isospin decomposition: • 1 Φab =δabtr(T)ΦI=0+ tr([τa,τb]T)ΦI=1. (4) 2 Symmetries and normalization: From the C-parity one can easily derive the following symmetry • properties of the isoscalar (I =0) and isovector (I =1) parts of 2πDA’s eq. (3): ΦI=0(z,ζ,m2 ) = ΦI=0(1 z,ζ,m2 )=ΦI=0(z,1 ζ,m2 ), ππ − − ππ − ππ ΦI=1(z,ζ,m2 ) = ΦI=1(1 z,ζ,m2 )= ΦI=1(z,1 ζ,m2 ). (5) ππ − ππ − − ππ The isospin one parts of 2πDA’s eq. (3) is normalized as follows: 1 dzΦI=1(z,ζ,m2 )=(2ζ 1)Fe.m.(m2 ), (6) Z ππ − π ππ 0 where Fe.m.(m2 ) is the pion e.m. form factor in the time-like region (Fe.m.(0)=1). π ππ π For the isoscalar 2πDA ΦI=0 we have the following normalization condition: 1 dz(2z 1)ΦI=0(z,ζ,m2 )= 2 Mπζ(1 ζ)FEMT(m2 ), Z − ππ − 2 − π ππ 0 whereMπ isamomentumfractioncarriedbyquarksinapion,FEMT(m2 )isapionformfactorofquark 2 π ππ part of energy momentum tensor normalized by FEMT(0)=1. In ref. [1] this form factor was estimated π in the instanton model of QCD vacuum at low two-pion invariant mass, the result: N m2 FEMT(m2 )=1+ c ππ +... , π ππ 48π2f2 π where f =93 MeV is a pion decay constant. π Double decomposition in conformal and partial waves : Let us decompose 2πDA’s in eigen- • functions of the ERBL [8] evolution equation (Gegenbauer polynomials C3/2(2z 1)) and in partial n − waves of produced pions (Gegenbauer polynomials C1/2(2ζ 1)). Generically the decomposition (for l − both isoscalar and isovector DA’s) has the form: ∞ n+1 Φ(z,ζ,m2 )=6z(1 z) B (m2 )C3/2(2z 1)C1/2(2ζ 1). (7) ππ − nl ππ n − l − nX=0Xl=0 Using symmetry properties Eq. (5) we see that the index n goes over even (odd) and index l goes over odd (even) numbers for isovector (isoscalar) 2πDA’s. Normalization conditions Eq. (6) correspond to BI=1(m2 )=Fe.m.(m2 ). 01 ππ π ππ 2 Relations of 2πDA’s to quark distribution in a pion : By crossing symmetry the 2πDA’s are • related to so-called skewed parton distributions (see review [4]). The latter are defined as: dλ eiλτ πa(p′)T ψ¯f′( λn/2)nˆψf(λn/2) πb(p) = (8) Z 2π h | − | i (cid:8) (cid:9) δabδff′HI=0(τ,ξ,t)+iεabcτfcf′HI=1(τ,ξ,t), whereξ isaskewednessparameterdefinedas: ξ = (n (p′ p))/(n (p′+p)),t=(p′ p)2 andlight-cone vector nµ normalized by (n (p′+p))= 2. By cros−sing·sym−metry w·e can easily expr−ess the moments of · skewed distributions to coefficients B in the expansion (7) (detailed discussion see in [15]): nl 1 N−1n+1 1 1 3 dττN−1HI(τ,ξ,t)= BI (t)ξNC1/2 dx [1 x2] xN−1C3/2(x). (9) Z−1 nX=0 Xl=0 nl l (cid:18)ξ(cid:19) Z−1 4 − n If we take the forward limit in this formula, we obtain the relations between moments of quark distribu- tions in a pion and coefficients B : nl 1 MN(π) ≡ Z xN−1(qπ(x)−q¯π(x))=BNI=−11,N(0)AN for odd N, 0 1 M(π) 2 xN−1(q (x)+q¯ (x))=BI=0 (0)A for even N, (10) N ≡ Z π π N−1,N N 0 where A are numerical coefficients (e.g. A = 1,A = 9/5,A = 6/7,A = 5/3, etc.) and q (x) = N 1 2 3 4 π uπ+(x). For example, from eq. (10) we obtain that BI=1(0) = M(π) = 1 what corresponds to nor- 01 1 malization condition (6). Also it is easy to see that BI=0(0) = 5/9M(π) corresponds to normalization 12 2 (7)2. Evolution : The Gegenbauer polynomials C3/2(2z 1) are eigenfunctions of the ERBL [8] evolution n • − equation and hence the coefficients B are renormalized multiplicatively (for even n and odd l): nl α (µ) γn/(2β0) B (m2 ;µ)=B (m2 ;µ ) s , (11) nl ππ nl ππ 0 (cid:18)α (µ )(cid:19) s 0 where β =11 2/3N and the one loop anomalous dimensions are [9]: 0 f − n+1 8 2 1 γ = 1 +4 . n 3(cid:16) − (1+n)(2+n) kX=2 k(cid:17) From the decomposition Eq. (7) and Eq. (11) we can make a simple prediction for the ratio of the ∗ hard P and F waves production amplitudes of pions in the reaction γ p 2πp at asymptotically − − → large virtuality of the incident photon Q2 and fixed m2 : →∞ ππ F wave amplitude 1 − , (12) P wave amplitude ∼ log(Q2)50/(99−6Nf) − or generically for the 2k+1 wave: (2k+1) wave amplitude 1 − . (13) P wave amplitude ∼ log(Q2)γ2k/(2β0) − Soft pion theorems for two-pion distribution amplitudes relate 2πDA’s to distribution ampli- • tude of one pion: 2Toseethisonehastouseadditionallysoftpiontheorem(15) forisoscalar2πDA 3 ΦI=1(z,ζ =1,m2 =0)= ΦI=1(z,ζ =0,m2 =0)=ϕ (z), (14) ππ − ππ π where ϕ (z) is a one pion DA. The analogous theorem for an isoscalar part of the 2πDA’s has the form: π ΦI=0(z,ζ =1,m2 =0)=ΦI=0(z,ζ =0,m2 =0)=0. (15) ππ ππ The soft pion theorem Eq. (14) allows us to relate the coefficients BI=1(m2 ) (see Eq. (7)) and the nl ππ coefficients of expansion of the pion DA in Gegenbauer polynomials ϕ (z)=6z(1 z) 1+ a(π)C3/2(2z 1) . (16) π − (cid:18) n n − (cid:19) n=Xeven The relation has the form: n+1 a(π) = BI=1(0). (17) n nl Xl=1 Dispersion relations and their solution for 2πDA’s: The 2π DA’s are generically complex func- • tions due to the strong interactionof the produced pions. Above the two-pionthreshold m2 =4m2 the ππ π 2πDA’sdevelopthe imaginarypartcorrespondingtothecontributionofon-shellintermediatestates(2π, 4π, etc.). In the regionm2 <16m2 the imaginary part is related to the pion-pion scattering amplitude ππ π by Watson theorem [10]. This relation can be written in the following form (see [1]): ImBI (m2 )=sin[δI(m2 )]eiδlI(m2ππ)BI (m2 )∗ =tan[δI(m2 )]ReBI (m2 ). (18) nl ππ l ππ nl ππ l ππ nl ππ Using Eq. (18) one can write an N-subtracted dispersion relation for the BI (m2 ) nl ππ N−1m2k dk m2N ∞ tanδI(s)ReBI (s) BI (m2 )= ππ BI (0)+ ππ ds l nl . (19) nl ππ Xk=0 k! dm2πkπ nl π Z4m2π sN(s−m2ππ−i0) Solution of such type dispersion relation was found long ago in [11], the solution has the exponential form: N−1m2k dk m2N ∞ δI(s) BI (m2 )=BI (0)exp ππ log BI (0)+ ππ ds l . (20) nl ππ nl (cid:26)kX=1 k! dm2πkπ nl π Z4m2π sN(s−m2ππ−i0)(cid:27) AgreatadvantageofthesolutionEq.(20)isthatitgivesthe2πDA’sinawiderangeofenergiesinterms ofknownππ phaseshiftsandafew subtractionconstants(usuallytwoissufficient). The keyobservation is that these constants (non-perturbative input) are related to the low-energy behaviour of the 2πDA’s at m2 0. In the low energy region they can be computed using the effective chiral lagrangian. ππ → Amplitudes of ππ resonances production: We give here an example how ρ meson distribution • amplitude can be expressed in terms of 2πDA. General formula for DA’s of resonances with any spin can be found in [1]. In [1] we obtained the following expression for the coefficients of the expansion of ρ meson DA in Gegenbauer polynomials ϕ (z)=6z(1 z) 1+ a(ρ)C3/2(2z 1) , ρ − (cid:18) n n − (cid:19) n=Xeven in terms of only subtraction constants entering Eq. (20) N−1 a(ρ) =BI=1(0)exp c(n1)m2k , (21) n n1 k ρ (cid:0)Xk=1 (cid:1) 4 where the subtraction constants c(nl) can be expressed in terms of BI=1(m2 ) at low energies k nl ππ 1 dk ck(nl) = k!dm2πkπ[log Bnl(m2ππ)−log Bl−1l(m2ππ)](cid:12)(cid:12)(cid:12)m2 =0 . (22) (cid:12) ππ (cid:12) The normalization constants f can be computed as: ρ √2Γ ImFe.m.(m2) ρ π ρ f = . (23) ρ g ρππ Here we quote the result for the chirally even ρ meson DA extracted from 2πDA using Eqs. (21,23) and values of low-energy subtraction constants calculated using effective chiral lagrangian ( see details and results of these calculations in refs. [6, 1]) : ϕ (z)=6z(1 z) 1 0.14C3/2(2z 1) 0.01C3/2(2z 1)+... , (24) ρ − − 2 − − 4 − (cid:2) (cid:3) withnormalizationconstantf =190MeVaccordingtoEq.(23)inagoodagreementwith experimental ρ value f = 195 7 MeV [14]. The ρ meson DA’s were the subject of the QCD sum rules calculations ρ ± [12,13],ourresultEq.(24)isinaqualitativedisagreementwiththeresultsofQCDsumrulecalculations, (ρ) the sign of a obtained here is opposite to the QCD sum rules predictions a = 0.18 0.1 [12] and 2 2 ± a(ρ) = 0.08 0.02, a(ρ) = 0.08 0.03 [13] (these results refer to normalization point µ = 1 GeV). Let 2 ± 4 − ± us note that actually our sign of a(2ρ) follows from: 1) soft pion theorem (14) 2) the relation of BN−1,N to momentsofquarkdistributioninapion(10)3)relationa(ρ) =BI=1(0)exp(c(21)m2)(seeeq.(21))and 2 21 1 ρ 4) the fact that a(π) 0. Combining all this we can state that: 2 ≈ 7 sign(a(ρ))=sign(BI=1(0))=sign(a(π) BI=1(0))=sign(a(π) M(π))= sign(M(π)). (25) 2 21 2 − 23 2 − 6 3 − 3 Here in the last equality we assumed that a(π) < 7M(π) what is satisfied in the model and seems 2 6 3 phenomenologically. QCD parametrization of two pion spectrum in diffractive two pion production: Thedepen- • dence of the two-pion hard production amplitude on the m factorizes into the factor: ππ 1 dz A ΦI=1(z,ζ,m2 ;Q¯2), (26) ∝Z z(1 z) ππ 0 − for the two pions in the isovector state, and 1 (2z 1) A dz − ΦI=0(z,ζ,m2 ;Q¯2), (27) ∝Z z(1 z) ππ 0 − for the pions in the isoscalarstate (e.g.for the π0π0 production). In the aboveequations we showedalso the dependence of the 2πDA’s on the scale of the process Q¯2, which is governedby the ERBL evolution equation [8]. For the hard exclusive reactions off nucleon at small x (see e.g. recent measurements [7]) the Bj production of two pions in the isoscalar channel is strongly suppressed relative to the isovector channel because the former is mediated by C-parity odd exchange. At asymptotically large Q2 one can use the asymptotic form of isovector 2πDA: lim ΦI=1(z,ζ,m2 ;Q¯2)=6Fe.m.(m2 )z(1 z)(2ζ 1), (28) Q¯2→∞ ππ π ππ − − 5 where Fe.m.(m2 ) is the pion e.m. form factor in the time-like region. Therefore the shape of π+π− π ππ mass spectrum in the hard electroproduction process at small x and asymptotically large Q2 should Bj be determined completely by the pion e.m. form factor in time-like region: lim A eiδ11(m2ππ) Fe.m.(m2 )(2ζ 1). (29) Q2→∞ ∝ | π ππ | − Deviationoftheπ+π− massspectrumfromitsasymptoticformEq.(29)(“skewing”)canbeparametrized at small x and large Q¯2 in the form: Bj A∝eiδ11(m2ππ)|Fπe.m.(m2ππ)|h1+B21(0;µ0)exp{c(121)m2ππ}(cid:18)ααss((Qµ¯02))(cid:19)50/(99−6Nf)i(2ζ−1)+ eiδ31(m2ππ)B23(0;µ0)exp{b23m2ππ+R31(m2ππ)}(cid:18)ααs((Qµ¯2))(cid:19)50/(99−6Nf)C31/2(2ζ−1) s 0 +higher powers of 1/log(Q¯2), (30) here m4 ∞ δI(s) d RI(m2 )=Re ππ ds l and b = log B (m2 ) . (31) l ππ π Z4m2π s2(s−m2ππ−i0) nl dm2ππ nl ππ (cid:12)(cid:12)(cid:12)m2ππ=0 (cid:12) (cid:12) We see that the deviation of the ππ invariant mass spectrum from its asymptotic form Eq. (29) in this approximationischaracterizedbyafewlow-energyconstants(B (0),B (0),c(21),b ),otherquantities- 21 23 1 23 the pion e.m. form factor and the ππ phase shifts–are known from low-energyexperiments. In principle, using the parametrization(31)one canextractthe values ofthese low-energyparametersfromthe shape of ππ spectrum (not absolute cross section!) in diffractive production experiments. Knowing them one can obtain the deviation of the π meson DA (see Eq. (17)) a(π) =B (0)+B (0), 2 21 23 and the ρ meson DA (see Eq. (21)) a(ρ) =B (0)exp(c(21)m2), 2 21 1 ρ from their asymptotic form 6z(1 z), as well as the normalization constants for the DA of isovector − resonances of spin three. Inanalysisofexperimentsontwopiondiffractiveproductionoffnucleon(seee.g. [7])thenon-resonant backgroundisdescribedbySo¨dingparametrization[16],whichtakesintoaccountrescatteringofproduced pions on final nucleon. Let us note however that in a case of hard (Q2 ) diffractive production the →∞ final state interaction of pions with residual nucleon is suppressed by powers of 1/Q2 relative to the leading twist amplitude. Here we proposed alternative leading-twist parametrization(31) describing the so-called “skewing” of two pion spectrum. Acknowledgments IamgratefultoD.Diakonov,M.Diehl,L.Frankfurt,K.Goeke,P.Pobylitsa,A.Radyushkin,A.Scha¨fer, M. Strikman, O. Teryaev and C. Weiss for inspiring discussions. This work has been supported in part by the BMFB (Bonn), Deutsche Forschungsgemeinschaft(Bonn) and by COSY (Ju¨lich). References [1] M.V. Polyakov,hep-ph/9809483. [2] J. Collins,L. Frankfurt, and M. Strikman, Phys. Rev. D56 (1997) 2982. 6 [3] A. V. Radyushkin, Phys. Rev. D56 (1997) 5524. 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