Evaluation of the Nucleon Helicity Flip Form Factor using One and Two Virtual Photons
aa r X i v : . [ h e p - ph ] N ov Evaluation of the Nucleon Helicity Flip Form Factor using One andTwo Virtual Photons
Thorsten Sachs and Patrick Sturm
Institut für Theoretische Physik II,Ruhr-Universität Bochum, D-44780 Bochum, Germany (Dated: November 3, 2018)
Abstract
In this work, we will evaluate the nucleon helicity flip form factor at the limit of large momentumtransfer. Hereby, we will study the exchange of one and two virtual photons separately. For thecalculation of the scattering amplitudes and nucleon transition probability matrix elements, wecombine QCD perturbation theory with an expansion in nucleon distribution amplitudes. Usingthe combination of leading and sub-leading twist nucleon distribution amplitudes, one obtains thedesired form factor. Using this technique, we will obtain a divergent result for the form factor.Nevertheless, the structure of the divergency can be extracted. Finally, we will comment theobtained expressions and discuss the behavior in unpolarized and polarized cross sections. . INTRODUCTION Elastic electron nucleon scattering mediated by the electromagnetic interaction is themost considered process to receive information about the nucleon structure within QCD.Applying the basic one photon exchange approximation, the required nucleon transitionprobability matrix elements are traditionally expressed by the Dirac form factor and thePauli form factor or, equivalently, the magnetic form factor and the electric form factor.For convenience, we use the representation by the magnetic form factor and the Pauli formfactor. Moreover, we concentrate our considerations on the proton form factors h p ( P ′ ) | J emµ (0) | p ( P ) i = ¯ N ( P ′ ) (cid:20) G pM ( Q ) γ µ − F p ( Q ) ( P ′ + P ) µ m N (cid:21) N ( P ) (1) h n ( P ′ ) | J emµ (0) | n ( P ) i = ¯ N ( P ′ ) (cid:20) G nM ( Q ) γ µ − F n ( Q ) ( P ′ + P ) µ m N (cid:21) N ( P ) . (2)At large momentum transfer Q = − q = − ( P ′ − P ) , one just gets the contribution forthe magnetic form factor with the power behavior of Q − . This form factor was measuredin a comprehensive region and calculated with different techniques, basically with the QCDfactorization theorem. Among other form factors, we studied the magnetic form factor in[1]. For further information, we recommend the references in this work.Moving to intermediate values of the momentum transfer, one also gets contributions forthe Pauli form factor with the power behavior of Q − . The different power behavior arisesfrom the helicity flip of the nucleon and so this form factor is also known as helicity flipform factor. Concerning this form factor, experimental data are also available. Moreover,one has discovered a different behavior depending on the type of the experiment.The basic information were taken from unpolarized cross sections. Using the Rosenbluthseparation technique [2], several experiments were performed. Hereby, early experimentsdid not show significant double photon corrections, see [3], [4], [5], [6], [7], [8], [9], [10], orradiative corrections, see [11], [12]. Further experiments were executed in [13], [14], [15], andwith taking into account radiative corrections, see [16], [17]. Moreover, the available datawere fitted in [18]. The consequences of radiative corrections were considered in [19]. Thediscussed technique is useful at low Q , but at larger Q , the contribution of the helicityflip form factor is suppressed by the momentum transfer. However, the electric form factorseems to have the same power behavior as the magnetic form factor. In order to measure thedesired form factor at larger values of Q , one has to study polarized cross sections. During2he last years, the experimental requirements have been created and so various experimentshave been performed. Hereby, one needs a polarized electron beam. From the experimentalperspective, the polarization transfer method, discussed in [20], seems to be in favor. Hereby,one has to measure the polarization of the recoil proton [21], [22], [23], [24], [25], [26], [27],[28]. The alternative is to use polarized proton targets [29], [30]. Concerning these data, theelectric form factor seems to be power suppressed compared to the magnetic form factor.The different measurements were compared in [31], [32], [33].From the theoretical perspective, the calculation of the desired form factor in the large Q region is problematic. Using the QCD factorization theorem and the basic one photonexchange, the Pauli form factor was studied in [34]. In this work, divergent integrals wereobtained and therefore a cutoff parameter related to an effective size of the nucleon wasintroduced. This form factor was also considered in [35]. Hereby, different models wereconsidered and a cutoff parameter related to an effective mass of the nucleon was discussedwith different logarithmic power behavior.In order to understand the different behavior in the discussed experiments, it has beensuggested that the two photon exchange contribution can cause this situation. Therefore, anadvanced form factor parametrization was developed in [36]. The modified magnetic formfactor was calculated in [37]. The obtained corrections cannot describe the experimentswithout an input from the helicity flip form factor. Therefore, it is necessary to study therequired form factor in one and two photon exchange approximation.Let us apply the technique specified in [1] to evaluate the form factor. We have tocombine QCD perturbation theory with an expansion in nucleon distribution amplitudesagain. Hereby, we have to use the combination of leading and sub-leading twist nucleondistribution amplitudes, studied in [38]. We will start with the one photon exchange and wewill finish with the two photon exchange. Using our technique, we will obtain a divergentresult for the form factor. Nevertheless, the structure of the divergency can be extracted.Concerning the modified helicity flip form factor, we will obtain the dependence on oneadditional variable. Calculating the experimental cross section of the process and usingthe momentum transfer and the scattering angle as variables, one gets a different generalbehavior in the one and two photon exchange approximation. In the first case, the form factordepends on the momentum transfer only. This was already known and so the Rosenbluthseparation technique could be applied. In the second case, the form factor depends on the3omentum transfer and apart from that, it depends on the scattering angle additionally.That means, the Rosenbluth separation technique cannot be used in this case. Moreover,the obtained power behavior of the helicity flip form factor can describe the experimentaldata based on polarized cross sections qualitatively. According to this, we can explain thedifferent behavior in the experiments using unpolarized or polarized cross sections. Finally,we will discuss the required modifications to avoid the divergency. II. ONE PHOTON EXCHANGE APPROXIMATION
Let us start with the presentation of a sample diagram. In the upper part, we see theincoming electron on the left and the outgoing electron on the right. In the lower part, wehave the incoming proton on the left and the outgoing proton on the right. The requiredquark lines denote u , u , d from top to bottom. The designations at the vertices are thecorresponding coordinates and the designations at the lines are the corresponding momenta. l q l ′ u P v P ′ P u P ∆ Λ v P ′ u P ∆ v P ′ Λ P ′ y x y x x x We need the expression for the scattering amplitude. Applying QED Feynman rules, onecan evaluate the leptonic part of the diagram directly M = − i (4 πα em ) Y i =1 Z d y i (2 π ) Z d qq + i u ( l ′ ) γ µ u ( l ) h p ( P ′ ) | J emµ ( y ) | p ( P ) i e iq · ( y − y ) e − iy · ( l − l ′ ) . One can replace h p ( P ′ ) | J emµ ( y ) | p ( P ) i with h p ( P ′ ) | J emµ (0) | p ( P ) i e − iy · ( P − P ′ ) . The y inte-gration leads to q = l − l ′ and the y integration leads to q = P ′ − P , so that4 = − i (4 πα em ) q ¯ u ( l ′ ) γ µ u ( l ) h p ( P ′ ) | J emµ (0) | p ( P ) i . (3)Using Q = − q and P = P + P ′ , we introduce the expansion in nucleon electromagneticform factors M = i (4 πα em ) Q ¯ u ( l ′ ) γ µ u ( l ) ¯ N ( P ′ ) (cid:20) G pM ( Q ) γ µ − F p ( Q ) ¯ P µ m N (cid:21) N ( P ) . (4)The advantage of this expression is the separation of the leptonic and the hadronic part.Consequently, we only need to consider the matrix element h p ( P ′ ) | J emµ (0) | p ( P ) i . This be-havior was already used for the calculation of the magnetic form factor, see [1].Applying the S-matrix expansion including the interaction part of the QCD Lagrangian,one gets the following leading expression for the desired matrix element (4 π ¯ α s ) h p ( P ′ ) | X q e q ¯ ψ q (0) γ µ ψ q (0) T " Y i =1 Z d x i X q i ¯ ψ q i ( x i ) γ α i A α i ( x i ) ψ q i ( x i ) | p ( P ) i . This expansion can be described by 42 Feynman diagrams and Wick contractions. Wecan extract the representation of the diagram which we want to study.Let us begin with the determination of the color factor. Therefore, one has to examinethe color structure of the diagram, denoting the color indices with ( a, . . . , i ). We get ¯[ ψ u ( x )] c [ ψ u (0)] a ¯[ ψ u ( x )] g [ ψ u ( x )] d [ t a ] bc [ t a ] de A a α ( x ) A a α ( x ) [ t a ] fg [ t a ] hi A a α ( x ) A a α ( x ) h p ( P ′ ) | [ ¯ ψ u ( x )] b [ ¯ ψ u ( x )] f [ ¯ ψ d ( x )] h | ih | [ ψ u (0)] a [ ψ u ( x )] e [ ψ d ( x )] i | p ( P ) i . Combining all terms and contracting the generators, one gets the color factor C F = 16 ε bfh ε aei δ ca δ gd [ t a ] bc [ t a ] de [ t a ] fg [ t a ] hi δ a a δ a a = 49 . (5)We continue with the evaluation of the Lorentz structure of the diagram, designating theLorentz indices with ( a, . . . , j ). Including C F , we obtain the following expression − (4 π ¯ α s ) e u Y i =1 Z d x i [ γ µ ] ab [ γ α ] cd [ γ α ] ef [ γ α ] gh [ γ α ] ij ¯[ ψ u ( x )] d [ ψ u (0)] a ¯[ ψ u ( x )] h [ ψ u ( x )] e A α ( x ) A α ( x ) A α ( x ) A α ( x ) h p ( P ′ ) | [ ¯ ψ u ( x )] c [ ¯ ψ u ( x )] g [ ¯ ψ d ( x )] i | ih | [ ψ u (0)] b [ ψ u ( x )] f [ ψ d ( x )] j | p ( P ) i . In order to evaluate this expression, we have to apply the representations for the propa-gators and for the projection matrix elements5 π ¯ α s ) e u Y i =1 Z d x i (2 π ) Y j =1 Z d ∆ j ∆ j + i Y k =1 Z d Λ k Λ k + i Z [d u ][d v ] g α α g α α S e − ix · (∆ − Λ − v p ′ ) e − ix · ( − ∆ +Λ + u p ) e − ix · (∆ − Λ − v p ′ ) e − ix · (Λ + u p − v p ′ ) . Computing the integrations, one gets the required momentum conservation constraints ∆ = ( v + v + v ) p ′ − ( u + u ) p ∆ = ( v + v ) p ′ − u p Λ = ( v + v ) p ′ − ( u + u ) p Λ = v p ′ − u p. The component S is the sum of all required structures connected with combinations ofnucleon distribution amplitudes and nucleon spinors. In order to get the desired contribu-tions, one has to combine the twist-3 and twist-4 distribution amplitudes, studied in [38].Furthermore, one has to specify the frame. We prefer to use the light cone decompositiongiven by P µ = p µ + ( m N /Q ) p ′ µ and P ′ µ = p ′ µ + ( m N /Q ) p µ . Using this frame, we can derivethe equation of motion relations and eliminate the small component of the spinor and pro-ceed with the large component only. Let us omit the dependence on the quark momentumfractions. Moreover, we use the standard notation for the spinors.We get the following structures for initial twist-4 and final twist-3 S = ( m N /Q ) ¯ N ( P ′ ) γ α (cid:30) p ′ N ( P ) Tr[ γ µ (cid:30) pγ α (cid:30) ∆ γ α (cid:30) p ′ γ α (cid:30) ∆ ]( V V + A A + V V − A A ) S = ( m N /Q ) ¯ N ( P ′ ) γ α γ (cid:30) p ′ N ( P ) Tr[ γ γ µ (cid:30) pγ α (cid:30) ∆ γ α (cid:30) p ′ γ α (cid:30) ∆ ]( A V + V A − A V + V A ) S = ( m N /
2) ¯ N ( P ′ ) γ α γ λ N ( P ) Tr[ γ µ γ λ γ α (cid:30) ∆ γ α (cid:30) p ′ γ α (cid:30) ∆ ]( − V V + A A ) S = ( m N /
2) ¯ N ( P ′ ) γ α γ γ λ N ( P ) Tr[ γ γ µ γ λ γ α (cid:30) ∆ γ α (cid:30) p ′ γ α (cid:30) ∆ ]( − A V + V A ) S = ( m N ) ¯ N ( P ′ ) γ λ ′ γ α N ( P ) Tr[ γ µ γ α (cid:30) ∆ γ α iσ λ ′ p ′ γ α (cid:30) ∆ ]( T S ) S = ( m N ) ¯ N ( P ′ ) γ λ ′ γ α γ N ( P ) Tr[ γ γ µ γ α (cid:30) ∆ γ α iσ λ ′ p ′ γ α (cid:30) ∆ ]( − T P ) S = (2 m N /Q ) ¯ N ( P ′ ) γ λ ′ γ α N ( P ) Tr[ γ µ iσ pp ′ γ α (cid:30) ∆ γ α iσ λ ′ p ′ γ α (cid:30) ∆ ]( T T − T T + T T ) S = ( m N /Q ) ¯ N ( P ′ ) γ λ ′ γ α γ λ (cid:30) p ′ N ( P ) Tr[ γ µ iσ λp γ α (cid:30) ∆ γ α iσ λ ′ p ′ γ α (cid:30) ∆ ]( T T + 2 T T ) S = ( m N /
2) ¯ N ( P ′ ) γ λ ′ γ α iσ λκ N ( P ) Tr[ γ µ iσ λκ γ α (cid:30) ∆ γ α iσ λ ′ p ′ γ α (cid:30) ∆ ]( T T ) . We get the following structures for initial twist-3 and final twist-46 = ( m N /Q ) ¯ N ( P ′ ) (cid:30) pγ α N ( P ) Tr[ γ µ (cid:30) pγ α (cid:30) ∆ γ α (cid:30) p ′ γ α (cid:30) ∆ ]( V V + A A + V V − A A ) S = ( m N /Q ) ¯ N ( P ′ ) (cid:30) pγ α γ N ( P ) Tr[ γ γ µ (cid:30) pγ α (cid:30) ∆ γ α (cid:30) p ′ γ α (cid:30) ∆ ]( V A + A V + V A − A V ) S = ( m N /
2) ¯ N ( P ′ ) γ λ ′ γ α N ( P ) Tr[ γ µ (cid:30) pγ α (cid:30) ∆ γ α γ λ ′ γ α (cid:30) ∆ ]( − V V + A A ) S = ( m N /
2) ¯ N ( P ′ ) γ λ ′ γ α γ N ( P ) Tr[ γ γ µ (cid:30) pγ α (cid:30) ∆ γ α γ λ ′ γ α (cid:30) ∆ ]( − V A + A V ) S = ( m N ) ¯ N ( P ′ ) γ α γ λ N ( P ) Tr[ γ µ iσ λp γ α (cid:30) ∆ γ α γ α (cid:30) ∆ ]( − S T ) S = ( m N ) ¯ N ( P ′ ) γ α γ γ λ N ( P ) Tr[ γ γ µ iσ λp γ α (cid:30) ∆ γ α γ α (cid:30) ∆ ]( P T ) S = (2 m N /Q ) ¯ N ( P ′ ) γ α γ λ N ( P ) Tr[ γ µ iσ λp γ α (cid:30) ∆ γ α iσ p ′ p γ α (cid:30) ∆ ]( T T − T T + T T ) S = ( m N /Q ) ¯ N ( P ′ ) (cid:30) pγ λ ′ γ α γ λ N ( P ) Tr[ γ µ iσ λp γ α (cid:30) ∆ γ α iσ λ ′ p ′ γ α (cid:30) ∆ ]( T T + 2 T T ) S = ( m N /
2) ¯ N ( P ′ ) iσ λ ′ κ ′ γ α γ λ N ( P ) Tr[ γ µ iσ λp γ α (cid:30) ∆ γ α iσ λ ′ κ ′ γ α (cid:30) ∆ ]( T T ) . Computing every structure, one always gets the dependence on ¯ N ( P ′ ) ¯ P µ N ( P ) as pre-dicted. One can simplify the expression by exchanging u ↔ v for contributions of initialtwist-3 and final twist-4. Consequently, we obtain a representation depending on initialtwist-4 and final twist-3 only.Let us now present the result of the discussed diagram depending on the integration overthe quark momentum fractions. When we compare with the separated hadronic part of (4),we can extract the contribution to the desired form factor. In order to get the completeresult, one also needs the contributions of the other diagrams designated by C , F p ( Q ) = − (4 π ¯ α s ) e u m N Q Z [d u ] u ( u + u ) [d v ] v ( v + v ) D + C . (6)The component D is the sum of the remaining twist combinations of distribution ampli-tudes connected with multiple quark momentum fractions D = [ V V + A A ](2( u + u )( v + v )) D = [ V V − A A ]( v − ( u + u )( v + v )) D = [ V A − A V ]( v + ( u + u )( v + v )) D = [ T S − T P ](+2 v − u + u ) v ) D = [ T T + T T ]( − v − u + u ) v ) . Finally, we must insert the nucleon distribution amplitudes. Unfortunately, the corre-sponding integration is divergent. This divergency arises from endpoint singularities. Thatmeans, the integrals get divergent when a quark has no momentum or the full momentum of7he nucleon. In order to analyze the structure of the divergency, one can introduce a cutoffparameter Ω . Therefore, one has to respect that in case of infinite momentum transfer theintegration must go from zero to one for every quark momentum fraction. According to this,we always integrate from Ω /Q to − Ω /Q , keeping in mind that the introduced parameterhas the same dimension as the momentum transfer.Computing the modified integration, we can extract the structure of the divergency. Thegeneral behavior does not depend on the chosen polynomial expansion of the distributionamplitudes. Moreover, this behavior is identical for all other required diagrams as well.Consequently, we can generally express the power behavior of the helicity flip form factordepending on the cutoff parameter F p ( Q ) ∝ Q − ln ( Q / Ω) . (7)We derived the expected power behavior of Q − and we obtained a double logarithmicdivergency in the case of Ω → . This behavior is in agreement with [34]. III. TWO PHOTON EXCHANGE APPROXIMATION
Let us begin with the discussion about an important behavior of this situation. In theone photon case, the inversion of the lepton direction delivers the same contribution tothe scattering amplitude. This statement is not true in the two photon case, because theinversion of the lepton direction produces another diagram. Therefore, one has to distinguishbetween the box diagram and the cross diagram. The corresponding contributions to thescattering amplitude must be calculated separately.We start with the presentation of the box diagram. In the upper part, we see the incomingelectron on the left and the outgoing electron on the right. In the lower part, we have theincoming proton on the left and the outgoing proton on the right. The required quark linesdenote u , u , d from top to bottom. The designations at the vertices are the correspondingcoordinates and the designations at the lines are the corresponding momenta.8 Γ q q l ′ u P v P ′ P u P v P ′ u P ∆ v P ′ Λ P ′ y y x y y x We need the expression for the scattering amplitude. Applying QED Feynman rules,one can evaluate the leptonic part of the diagram directly. Moreover, one can neglect theelectron mass in the propagator M B = − i (4 πα em ) Y i =1 Z d y i (2 π ) Y j =1 Z d q j q j + i Z d ΓΓ + i u ( l ′ ) γ µ (cid:30) Γ γ µ u ( l ) h p ( P ′ ) | J emµ ( y ) J emµ ( y ) | p ( P ) i e iq · ( y − y ) e iq · ( y − y ) e − i Γ · ( y − y ) e − iy · l e iy · l ′ . The integration over y leads to Γ = l − q and the integration over y leads to Γ = l ′ + q .Combining them, one gets the representation
2Γ = ( l + l ′ ) + ( q − q ) . We obtain M B = − i (4 πα em ) Y i =1 Z d y i (2 π ) Y j =1 Z d q j q j + i ¯ u ( l ′ ) γ µ (cid:30) Γ γ µ u ( l ) h p ( P ′ ) | J emµ ( y ) J emµ ( y ) | p ( P ) i e − iy · q e − iy · q . (8)We finish with the presentation of the cross diagram. In the upper part, we see theincoming electron on the right and the outgoing electron on the left. In the lower part, wehave the incoming proton on the left and the outgoing proton on the right. The requiredquark lines denote u , u , d from top to bottom. The designations at the vertices are thecorresponding coordinates and the designations at the lines are the corresponding momenta.9 ′ Γ q q lu P v P ′ P u P v P ′ u P ∆ v P ′ Λ P ′ y y x y y x We need the expression for the scattering amplitude. Applying QED Feynman rules, onecan evaluate the leptonic part of the diagram directly. Furthermore, one can neglect theelectron mass in the propagator M C = − i (4 πα em ) Y i =1 Z d y i (2 π ) Y j =1 Z d q j q j + i Z d ΓΓ + i u ( l ′ ) γ µ (cid:30) Γ γ µ u ( l ) h p ( P ′ ) | J emµ ( y ) J emµ ( y ) | p ( P ) i e iq · ( y − y ) e iq · ( y − y ) e − i Γ · ( y − y ) e − iy · l e iy · l ′ . The integration over y leads to Γ = l − q and the integration over y leads to Γ = l ′ + q .Combining them, one gets the representation
2Γ = ( l + l ′ ) + ( q − q ) . We obtain M C = − i (4 πα em ) Y i =1 Z d y i (2 π ) Y j =1 Z d q j q j + i ¯ u ( l ′ ) γ µ (cid:30) Γ γ µ u ( l ) h p ( P ′ ) | J emµ ( y ) J emµ ( y ) | p ( P ) i e − iy · q e − iy · q . (9)The overall result for the scattering amplitude is given by M = M B + M C . Let usnow introduce the expansion in nucleon electromagnetic form factors. Unfortunately, theleptonic and the hadronic part are not separated in this case. Nevertheless, one can show theexistence of a separated representation for M . Whereas the basic expression just dependsone the nucleon momenta, the modified expression also depends on the lepton momenta. Thederivation can be taken from [36]. Therefore, we can use Q = − q and assume q = l − l ′ together with q = P ′ − P . Furthermore, we need P = P + P ′ and L = l + l ′ . One gets10 = i (4 πα em ) Q ¯ u ( l ′ ) γ µ u ( l ) ¯ N ( P ′ ) (cid:20) ˜ G pM γ µ − ˜ F p ¯ P µ m N + ˜ F p ¯ (cid:30) L ¯ P µ m N (cid:21) N ( P ) . (10)All form factors depend on Q and one additional variable. Therefore, we choose thedimensionless quantity ω defined by ω = 4( ¯ P · ¯ L ) /Q . At large Q , one gets the boundarycondition ω ≥ . In principle, one can generally expand every form factor as ˜ F = F + δF ,where F is the single photon exchange contribution and δF is the multi photon exchangecontribution. We do not use this decomposition because we consider the one and two photonexchange separately. The leading form factors are considered in [37].Let us now study the matrix element h p ( P ′ ) | J emµ ( y ) J emµ ( y ) | p ( P ) i . The evaluation ofthis matrix element must be combined with the other terms in (8) and (9) to derive a resultfor the form factor. Applying the S-matrix expansion including the interaction part of theQCD Lagrangian, one gets the following leading expression for this matrix element − (4 π ¯ α s )2 h p ( P ′ ) | Y j =2 X q j e q j ¯ ψ q j ( y j ) γ µ j ψ q j ( y j ) T " Y i =1 Z d x i X q i ¯ ψ q i ( x i ) γ α i A α i ( x i ) ψ q i ( x i ) | p ( P ) i . This expansion can be described by 12 Feynman diagrams and Wick contractions. Wecan extract the representation of the diagram which we want to study.Let us begin with the determination of the color factor. Therefore, one has to examinethe color structure of the diagram. We denote the color indices with ( a, . . . , f ). We get ¯[ ψ u ( x )] d [ ψ u ( y )] a [ t a ] cd [ t a ] ef A a α ( x ) A a α ( x ) h p ( P ′ ) | [ ¯ ψ u ( y )] b [ ¯ ψ u ( x )] c [ ¯ ψ d ( x )] e | ih | [ ψ u ( y )] b [ ψ u ( y )] a [ ψ d ( x )] f | p ( P ) i . Combining all terms and contracting the generators, one gets the color factor C F = 16 ε bce ε baf δ da [ t a ] cd [ t a ] ef δ a a = − . (11)We continue with the evaluation of the Lorentz structure of the diagram. Therefore, wedesignate the Lorentz indices with ( a, . . . , h ). Hereby, we have to distinguish between thebox and cross contribution. Nevertheless, we have to include C F in both representations.Keeping in mind
2Γ = ( l + l ′ ) + ( q − q ) , one gets the expression for (8) M B = i (4 π ¯ α s )(4 πα em ) e u Y i =1 Z d x i Y j =1 Z d y j (2 π ) Y k =1 Z d q k q k + i ¯ u ( l ′ ) γ µ (cid:30) Γ γ µ u ( l )[ γ µ ] ab [ γ µ ] cd [ γ α ] ef [ γ α ] gh ¯[ ψ u ( x )] f [ ψ u ( y )] a A α ( x ) A α ( x ) e − iy · q e − iy · q h p ( P ′ ) | [ ¯ ψ u ( y )] c [ ¯ ψ u ( x )] e [ ¯ ψ d ( x )] g | ih | [ ψ u ( y )] d [ ψ u ( y )] b [ ψ d ( x )] h | p ( P ) i .
2Γ = ( l + l ′ ) + ( q − q ) , one gets the expression for (9) M C = i (4 π ¯ α s )(4 πα em ) e u Y i =1 Z d x i Y j =1 Z d y j (2 π ) Y k =1 Z d q k q k + i ¯ u ( l ′ ) γ µ (cid:30) Γ γ µ u ( l )[ γ µ ] ab [ γ µ ] cd [ γ α ] ef [ γ α ] gh ¯[ ψ u ( x )] f [ ψ u ( y )] a A α ( x ) A α ( x ) e − iy · q e − iy · q h p ( P ′ ) | [ ¯ ψ u ( y )] c [ ¯ ψ u ( x )] e [ ¯ ψ d ( x )] g | ih | [ ψ u ( y )] d [ ψ u ( y )] b [ ψ d ( x )] h | p ( P ) i . In order to evaluate these expressions, we have to apply the representations for thepropagators and for the projection matrix elements.We present the expression for (8) with
2Γ = ( l + l ′ ) + ( q − q ) at first M B = − i (4 π ¯ α s )(4 πα em ) e u Y i =1 Z d x i (2 π ) Y j =1 Z d y j (2 π ) Y k =1 Z d q k q k + i Z d ∆∆ + i Z d ΛΛ + i ¯ u ( l ′ ) γ µ (cid:30) Γ γ µ u ( l ) Z [d u ][d v ] g α α S e − iy · ( q + u p − v p ′ ) e − iy · ( q − ∆+ u p ) e − ix · (∆ − Λ − v p ′ ) e − ix · (Λ+ u p − v p ′ ) . We present the expression for (9) with
2Γ = ( l + l ′ ) + ( q − q ) at last M C = − i (4 π ¯ α s )(4 πα em ) e u Y i =1 Z d x i (2 π ) Y j =1 Z d y j (2 π ) Y k =1 Z d q k q k + i Z d ∆∆ + i Z d ΛΛ + i ¯ u ( l ′ ) γ µ (cid:30) Γ γ µ u ( l ) Z [d u ][d v ] g α α S e − iy · ( q + u p − v p ′ ) e − iy · ( q − ∆+ u p ) e − ix · (∆ − Λ − v p ′ ) e − ix · (Λ+ u p − v p ′ ) . The appearing integrations are identical in both cases. After computation of these in-tegrations, one gets the required momentum conservation constraints. We notice that thephoton momenta do not depend on ω consequentially q = v p ′ − u p ∆ = ( v + v ) p ′ − u pq = ( v + v ) p ′ − ( u + u ) p Λ = v p ′ − u p. The component S is the sum of all required structures connected with combinations ofnucleon distribution amplitudes and nucleon spinors. In order to get the desired contribu-tions, one has to combine the twist-3 and twist-4 distribution amplitudes, studied in [38].Furthermore, one has to specify the frame. We prefer to use the light cone decompositiongiven by P µ = p µ + ( m N /Q ) p ′ µ and P ′ µ = p ′ µ + ( m N /Q ) p µ . Using this frame, we can derive12he equation of motion relations and eliminate the small component of the spinor and pro-ceed with the large component only. Let us omit the dependence on the quark momentumfractions. Moreover, we use the standard notation for the spinors.We get the following structures for initial twist-4 and final twist-3 S = ( m N /Q ) ¯ N ( P ′ ) γ α (cid:30) p ′ N ( P ) Tr[ γ µ (cid:30) pγ µ (cid:30) ∆ γ α (cid:30) p ′ ]( V V + A A + V V − A A ) S = ( m N /Q ) ¯ N ( P ′ ) γ α γ (cid:30) p ′ N ( P ) Tr[ γ γ µ (cid:30) pγ µ (cid:30) ∆ γ α (cid:30) p ′ ]( A V + V A − A V + V A ) S = ( m N /
2) ¯ N ( P ′ ) γ α γ λ N ( P ) Tr[ γ µ γ λ γ µ (cid:30) ∆ γ α (cid:30) p ′ ]( − V V + A A ) S = ( m N /
2) ¯ N ( P ′ ) γ α γ γ λ N ( P ) Tr[ γ γ µ γ λ γ µ (cid:30) ∆ γ α (cid:30) p ′ ]( − A V + V A ) S = ( m N ) ¯ N ( P ′ ) γ λ ′ γ α N ( P ) Tr[ γ µ γ µ (cid:30) ∆ γ α iσ λ ′ p ′ ]( T S ) S = ( m N ) ¯ N ( P ′ ) γ λ ′ γ α γ N ( P ) Tr[ γ γ µ γ µ (cid:30) ∆ γ α iσ λ ′ p ′ ]( − T P ) S = (2 m N /Q ) ¯ N ( P ′ ) γ λ ′ γ α N ( P ) Tr[ γ µ iσ pp ′ γ µ (cid:30) ∆ γ α iσ λ ′ p ′ ]( T T − T T + T T ) S = ( m N /Q ) ¯ N ( P ′ ) γ λ ′ γ α γ λ (cid:30) p ′ N ( P ) Tr[ γ µ iσ λp γ µ (cid:30) ∆ γ α iσ λ ′ p ′ ]( T T + 2 T T ) S = ( m N /
2) ¯ N ( P ′ ) γ λ ′ γ α iσ λκ N ( P ) Tr[ γ µ iσ λκ γ µ (cid:30) ∆ γ α iσ λ ′ p ′ ]( T T ) . We get the following structures for initial twist-3 and final twist-4 S = ( m N /Q ) ¯ N ( P ′ ) (cid:30) pγ α N ( P ) Tr[ γ µ (cid:30) pγ µ (cid:30) ∆ γ α (cid:30) p ′ ]( V V + A A + V V − A A ) S = ( m N /Q ) ¯ N ( P ′ ) (cid:30) pγ α γ N ( P ) Tr[ γ γ µ (cid:30) pγ µ (cid:30) ∆ γ α (cid:30) p ′ ]( V A + A V + V A − A V ) S = ( m N /
2) ¯ N ( P ′ ) γ λ ′ γ α N ( P ) Tr[ γ µ (cid:30) pγ µ (cid:30) ∆ γ α γ λ ′ ]( − V V + A A ) S = ( m N /
2) ¯ N ( P ′ ) γ λ ′ γ α γ N ( P ) Tr[ γ γ µ (cid:30) pγ µ (cid:30) ∆ γ α γ λ ′ ]( − V A + A V ) S = ( m N ) ¯ N ( P ′ ) γ α γ λ N ( P ) Tr[ γ µ iσ λp γ µ (cid:30) ∆ γ α ]( − S T ) S = ( m N ) ¯ N ( P ′ ) γ α γ γ λ N ( P ) Tr[ γ γ µ iσ λp γ µ (cid:30) ∆ γ α ]( P T ) S = (2 m N /Q ) ¯ N ( P ′ ) γ α γ λ N ( P ) Tr[ γ µ iσ λp γ µ (cid:30) ∆ γ α iσ p ′ p ]( T T − T T + T T ) S = ( m N /Q ) ¯ N ( P ′ ) (cid:30) pγ λ ′ γ α γ λ N ( P ) Tr[ γ µ iσ λp γ µ (cid:30) ∆ γ α iσ λ ′ p ′ ]( T T + 2 T T ) S = ( m N /
2) ¯ N ( P ′ ) iσ λ ′ κ ′ γ α γ λ N ( P ) Tr[ γ µ iσ λp γ µ (cid:30) ∆ γ α iσ λ ′ κ ′ ]( T T ) . Computing every structure, we get the dependence on multiple combinations of leptonand nucleon spinors. Nevertheless, it is possible to express all these combinations as functionsof the desired component ¯ u ( l ′ ) γ µ u ( l ) ¯ N ( P ′ ) ¯ P µ N ( P ) only. Therefore, we have to use that thecombination ¯ u ( l ′ ) γ µ γ u ( l ) ¯ N ( P ′ ) ¯ P µ γ N ( P ) does not contribute. Using u + u + u = 1 and v + v + v = 1 , one gets convenient representations for all components. We notice that in13he obtained result for M , the leptonic and the hadronic part are separated now. One canapply the same twist exchange as in the previous case.Let us now present the result of the discussed diagram connection depending on theintegration over the quark momentum fractions. When we compare with (10), we canextract the contribution to the desired form factor. In order to get the complete result, onealso needs the contributions of the other diagrams designated by C , ˜ F p ( ω, Q ) = (4 π ¯ α s )(4 πα em ) e u m N Q Z [d u ] u u ( u + u ) [d v ] v v ( v + v ) ( ω/ (( u ( v + v ) + ( u + u ) v ) − ( u − v ) ω )) D + C . (12)The component D is the sum of the remaining twist combinations of distribution ampli-tudes connected with multiple quark momentum fractions D = [ V V + A A ](2 u ( u + u )( v + v ) + 2 v ( v + v ) ) D = [ V V − A A ](4 u v ( v + v ) − u ( u + u )( v + v ) − v ( v + v ) ) D = [ V A − A V ](4 u v ( v + v ) + u ( u + u )( v + v ) + v ( v + v ) ) D = [ T S − T P ](+8 u v ( v + v ) − u ( u + u ) v − v v ( v + v )) D = [ T T + T T ]( − u v ( v + v ) − u ( u + u ) v − v v ( v + v )) . Finally, we must insert the nucleon distribution amplitudes again. Unfortunately, thecorresponding integration is divergent. This behavior is similar to the one photon exchange,but now we get another singularity. This divergency just appears at the limit ω = 1 andit is also an endpoint singularity. In order to analyze the structure of the divergency, weintroduce an analogous cutoff parameter Ω as applied in the previous case.Computing the modified integration, we can extract the structure of the divergency. Thegeneral behavior does not depend on the chosen polynomial expansion of the distributionamplitudes. Moreover, this behavior is identical for all other required diagrams as well.Consequently, we can generally express the power behavior of the helicity flip form factordepending on the cutoff parameter ˜ F p ( Q ) ∝ Q − ln ( Q / Ω) . (13)We derived the expected power behavior of Q − and we obtained a double logarithmicdivergency in the case of Ω → . This behavior is in agreement with [34].14 V. CONCLUSION AND OUTLOOK
We want to emphasize that the result has the same power behavior for the helicity flipform factor in the one and two photon exchange approximation. According to this, one canstudy these contributions simultaneously. Concerning the modified helicity flip form factorof the two photon exchange, we obtained the dependence on one additional variable whichcan be related to the scattering angle of the experimental cross section. This behavior causesproblems for the interpretation of unpolarized cross sections. Furthermore, we realize thatthe obtained power behavior of the helicity flip form factor can describe the experimentaldata based on polarized cross sections qualitatively. These conclusions can explain thedifferent behavior in the experiments using unpolarized or polarized cross sections.Finally, we have to discuss the required modifications to avoid the divergency. Theappearing double logarithmic singularities indicate the existence of not included soft contri-butions. This is a consequence of the factorization approach where possible contributionsfrom remaining soft spectator quarks are considered as power suppressed. Meanwhile, thereare evidences that those contributions cannot be neglected. Using a soft effective theory,the behavior of soft contributions is discussed in [39]. In this work, it has been pointed outthat the discussed soft contributions have the same power behavior as the factorized contri-butions and so they must be taken into account. Unfortunately, the required techniques toget all possible soft contributions are still in development.The studies about various nucleon form factors in multi photon exchange approximationincluding factorizable and non-factorizable contributions are an interesting topic which re-quires further investigations. Meanwhile, also comprehensive reviews were written, see [40]and [41] to get an overview about the obtained achievement.We want to thank Dr. N. Kivel for useful discussions and Prof. Dr. M. V. Polyakov forenabling this work.The work has been supported by BMBF grant 06BO9012. [1] T. Sachs and P. Sturm, arXiv:1111.0463 [hep-ph] (2011).[2] M. N. Rosenbluth, Phys. Rev. , 615 (1950).
3] D. Yount and J. Pine, Phys. Rev. , 1842 (1962).[4] A. Browman, F. Liu, and C. Schaerf, Phys. Rev. , B1079 (1965).[5] R. L. Anderson et al. , Phys. Rev. Lett. , 407 (1966).[6] W. Bartel et al. , Phys. Lett. B 25 , 242 (1967).[7] R. L. Anderson et al. , Phys. Rev. , 1336 (1968).[8] B. Bouquet et al. , Phys. Lett.
B 26 , 178 (1968).[9] J. Mar et al. , Phys. Rev. Lett. , 482 (1968).[10] L. Camilleri et al. , Phys. Rev. Lett. , 149 (1969).[11] Y. Tsai, Phys. Rev. , 1898 (1961).[12] L. W. Mo and Y. Tsai, Rev. Mod. Phys. , 205 (1969).[13] J. Litt et al. , Phys. Lett. B 31 , 40 (1970).[14] L. E. Price et al. , Phys. Rev.
D 4 , 45 (1971).[15] W. Bartel et al. , Nucl. Phys.
B 58 , 429 (1973).[16] R. C. Walker et al. , Phys. Rev.
D 49 , 5671 (1994).[17] L. Andivahis et al. , Phys. Rev.
D 50 , 5491 (1994).[18] P. E. Bosted et al. , Phys. Rev.
C 51 , 409 (1995).[19] J. Arrington, Phys. Rev.
C 69 , 022201 (2004).[20] R. G. Arnold, C. E. Carlson, and F. Gross, Phys. Rev.
C 23 , 363 (1981).[21] B. D. Milbrath et al. , Phys. Rev.
Lett. 80 , 452 (1998).[22] M. K. Jones et al. , Phys. Rev. Lett. , 1398 (2000).[23] O. Gayou et al. , Phys. Rev. C 64 , 038202 (2001).[24] O. Gayou et al. , Phys. Rev. Lett. , 092301 (2002).[25] V. Punjabi et al. , Phys. Rev. C 71 , 055202 (2005).[26] G. MacLachlan et al. , Nucl. Phys.
A 764 , 261 (2006).[27] G. Ron et al. , Phys. Rev. Lett. , 202002 (2007).[28] A. J. R. Puckett et al. , Phys. Rev. Lett. , 242301 (2010).[29] M. K. Jones et al. , Phys. Rev. C 74 , 035201 (2006).[30] C. B. Crawford et al. , Phys. Rev. Lett. , 052301 (2007).[31] J. Arrington, Phys. Rev. C 68 , 034325 (2003).[32] M. E. Christy et al. , Phys. Rev.
C 70 , 015206 (2004).[33] I. A. Qattan et al. , Phys. Rev. Lett. , 142301 (2005).
34] A. V. Belitsky, X. Ji, and F. Yuan, Phys. Rev. Lett. , 092003 (2003).[35] S. J. Brodsky, J. R. Hiller, D. S. Hwang, and V. A. Karmanov, Phys. Rev. D 69 , 076001(2004).[36] P. A. M. Guichon and M. Vanderhaeghen, Phys. Rev. Lett. , 142303 (2003).[37] N. Kivel and M. Vanderhaeghen, Phys. Rev. Lett. , 092004 (2009).[38] V. Braun, R. J. Fries, N. Mahnke, and E. Stein, Nucl. Phys. B 589 , 381 (2000).[39] N. Kivel and M. Vanderhaeghen, Phys. Rev.
D 83 , 093005 (2011).[40] C. F. Perdrisat, V. Punjabi, and M. Vanderhaeghen, Prog. Part. Nucl. Phys. , 694 (2007).[41] J. Arrington, P. G. Blunden, and W. Melnitchouk, Prog. Part. Nucl. Phys. , 782 (2011)., 782 (2011).