Role of anomalous chromomagnetic interaction in Pomeron and Odderon structures and in gluon distribution
aa r X i v : . [ h e p - ph ] M a r Role of anomalous chromomagnetic interaction inPomeron and Odderon structures and in gluondistribution
Nikolai Kochelev Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research,Dubna, Moscow region, 141980 Russia
Abstract
We calculate the contribution arising from nonperturbative quark-gluon chromomagneticinteraction to the high energy total quark-quark cross section and to gluon distributionsin nucleon. The estimation obtained within the instanton model of QCD vacuum leadsto the conclusion that this type of interaction gives the dominating contribution to thePomeron coupling with the light quarks and to gluon distribution in light hadrons at smallvirtualities of quarks and gluons. We argue that the Odderon, which is the P = C = − [email protected] Introduction
The gluon distribution in nucleon is one of the central quantities in particle physics whichdetermines the high energy cross section values of the huge amount of important processes.In spite of the tremendous achievements in the last years in the measurement of this dis-tribution, full understanding of the dynamics of gluons inside hadrons is absent so far(see reviews [1, 2]). In the Regge theory the behaviour of the gluon distribution functionat small Bjorken x is controlled by the contribution coming from the Pomeron exchangewhich may have so-called ”soft” and ”hard” parts [3]. Usually, the hard Pomeron isassociated with the perturbative BFKL regime [4] and the soft part is assumed to be orig-inated from nonperturbative QCD dynamics [5]. Nonperturbative effects arise from thecomplex structure of QCD vacuum. The instantons are one of the well studied topologicalfluctuations of vacuum gluon fields which might be responsible for many nonperturbativephenomena observed in particle physics (see reviews [6, 7]). Their possible importancein the structure of the Pomeron and gluon distribution was considered in quite differentapproaches [8], [9], [10], [11],[7] for the different approximations to the complicated quark-gluon dynamics in instanton vacuum. In particular, it was shown [12] that instantonslead to the appearance of anomalous chromomagnetic quark-gluon interaction (ACQGI).It was demonstrated that this new type of quark-gluon interaction might be responsiblefor the observed large single-spin asymmetries in various high energy reactions [12, 13].Furthermore, it gives a large contribution to the high energy quark-quark scattering crosssection [14]. The first estimation of the effect of ACQGI on nucleon gluon distributionwas made in [8] and small x behavior g ( x ) ∝ /x corresponding to soft Pomeron wasfound. It was clear from that study that anomalous chromomagnetic interaction shouldalso play an important role in the structure of Pomeron. Indeed, recently the model forsoft Pomeron based on this interaction has been suggested [7].In this paper, we consider the detailed structure of the Pomeron and gluon distributionwith the special attention to the interplay between their perturbative and nonperturbativecomponents. We also discuss the possible manifestation of ACQGI in Odderon exchange. In the general case, the interaction vertex of massive quark with gluon can be written inthe following form: V µ ( k , k , q ) t a = − g s t a [ γ µ F ( k , k , q ) − σ µν q ν M q F ( k , k , q )] , (1)where the form factors F , describe nonlocality of the interaction, k , is the momentumof incoming and outgoing quarks, respectively, and q = k − k , M q is the quark mass,and σ µν = ( γ µ γ ν − γ ν γ µ ) /
2. In various applications to high energy reactions based onperturbative QCD (pQCD) it is usually assumed that only non-spin flip first term inEq.(1) (Fig.1a) contributes and one can neglect the second term in this equation,Fig.1b,because in the limit of the massless quark this term should be absent due to quark chiralityconservation in massless pQCD. However, it has recently been shown that such assumptionhas no justification in nonperturbative QCD and second term might give in many cases1 ( L ) R ( L ) ( b ) R ( L ) L ( R )I(¯I)( a ) σ µν γ µ qq Figure 1: The quark-gluon coupling: a) perturbative and b) nonperturbative. Symbols R and L denote quark chirality and symbol I ( ¯ I ) denotes instanton (antiinstanton).even a dominant contribution to high energy reactions in comparison with the first one[12, 14].The cornerstone of this phenomenon is the spontaneous chiral symmetry breaking(SCSB) due to the complex topological structure of the QCD vacuum. Indeed, the in-stanton liquid model for QCD vacuum [6, 7] provides the mechanism for such breaking.That mechanism is related to the existence of quark zero modes in the instanton field. Asthe result of SCSB, the light quarks in nonperturbative QCD vacuum have the dynamicalmass, M q . Additionally, t’Hooft quark-quark interaction induced by quark-zero modesleads to the violation of U (1) A symmetry in strong interaction.In high energy reactions one might naively expect the smallness of SCSB effects be-cause of the energy √ s ≫ M q . Indeed, it might be correct for the reactions wherethe dominating contribution comes from quark-exchange diagrams. Within the instantonmodel this type of diagrams is originated from the t’Hooft quark-quark interaction contri-bution. However, instantons also lead to specific quark-gluon chromomagnetic interaction [12] which is presented by the second term in Eq.(1) (Fig.1b). It is evident that this termshould lead to a nonvanishing contribution to high energy reactions because it inducest-channel nonperturbative gluon exchange. The size of the contribution is determined bythe value of anomalous quark chromomagnetic moment (AQCM) µ a = F (0 , , . (2)We should point out that within the instanton model the shape of form factor F ( k , k , q )is fixed: F ( k , k , q ) = µ a Φ q ( | k | ρ/ q ( | k | ρ/ F g ( | q | ρ ) , (3)where Φ q ( z ) = − z ddz ( I ( z ) K ( z ) − I ( z ) K ( z )) ,F g ( z ) = 4 z − K ( z ) (4)are the Fourier-transformed quark zero-mode and instanton fields, respectively, and I ν ( z ), K ν ( z ), are the modified Bessel functions and ρ is the instanton size. The definition of anomalous AQCM used in Eq.(2) differs by a factor of two from the correspondingquantity presented in [12] and [7]. n ( ρ ) innonperturbative QCD vacuum [12]: µ a = − π Z dρn ( ρ ) ρ α s ( ρ ) . (5)The shape of instanton density in the form n ( ρ ) = n c δ ( ρ − ρ c ) , (6)leads to AQCM which is proportional to the packing fraction of instantons f = π n c ρ c invacuum µ a = − πfα s ( ρ c ) . (7)By using the following relation between parameters of the instanton model [15]: f = 34 ( M q ρ c ) , (8)we obtain µ a = − π ( M q ρ c ) α s ( ρ c ) . (9)This formula coincides with the result for AQCM presented in Eq.(7.2) in the paper byDiakonov [7] and shows the direct connection between AQCM and SCSB phenomena.The dimensionless parameter δ = ( M q ρ c ) is one of the main parameters of the instantonmodel. It is proportional to the packing fraction of instantons in QCD vacuum δ ∝ f ≪ ρ − c = 0 . δ MF = 0 .
08 for M q = 170 M eV in the mean field approximation [6] to δ DP = 0 .
33 for M q = 345 MeV within Diakonov-Petrov model (DP) [16]. For the strongcoupling constant at the scale of instanton average size [6],[7] α s ( ρ c ) ≈ . , (10)we obtain the following values for AQCM: µ MFa ≈ − . , µ DPa ≈ − . δ in Eq.(9) for AQCM. Indeed, this formula can be rewritten in thefollowing form: µ a = − S δ, (12)where S = 2 π/α s ( ρ ) is the Euclidean instanton action. The typical value of this actionis very large [6, 7] S ≈ ÷
15 (13)and leads to the compensation of the δ smallness effect on AQCM.3ithin the instanton model approach the first term in Eq.(1) is related to the nonzeromode contribution to quark propagator in the instanton field. The nonzero modes con-tribution to quark propagator can be approximated with high accuracy by perturbativepropagator [6]. Due to zero mode dominance for the light quarks, [6], we can expect thatfor the light quarks this sort of contribution should be suppressed in comparison with thesecond term in Eq.(1). However, for heavy quark the first term should dominate becausethere are no zero modes for heavy quark in the instanton field. Furthermore, instantoninduced form factors in the chromomagnetic part of interaction suppress the contributionof the second term for highly virtual quark and/or gluon. Therefore, form factor in thefirst term in Eq.(1) might be chosen in the form F ( k , k , q ) = Θ( | k | − µ )Θ( | k | − µ )Θ( | q | − µ ) , (14)where µ is the factorization scale between perturbative and nonperturbative regimes. Inour estimation below we will use µ ≈ /ρ c ≈ . Let us estimate the contribution of the vertex, Eq.(1), to the total high energy quark-quark scattering cross section. The leading diagrams contributing to the non-spin flipamplitude of q − q scattering are shown in Fig.2 and for colorless t-channel exchangepresents the model of the Pomeron. The imaginary part of the total forward scatteringamplitude gives the total quark-quark cross-section. ( b ) ( c ) g g g ( a ) I g g g ¯I ¯I I¯II Figure 2: The fine Pomeron structure in the model with perturbative interaction andnonperturbative ACQGI: a) perturbative contribution, b) interference perturbative andnonperturbative vertices, c) nonperturbative contribution. The symbol I ( ¯ I ) denotesinstanton (antiinstanton).So, in our model Pomeron includes the pure perturbative exchange (Fig.2a), nonper-turbative (Fig.2c) diagrams and the mixed graph (Fig.2b).By using the relation, Eq.(9), the total contribution to quark-quark cross section forthe quarks with small virtualities is σ total = σ pert + σ mix + σ nonpert , (15)where σ i = Z ∞ q min dσ i ( t ) dt dq , (16)4 σ ( t ) pert dt = 8 πα s ( q )9 q dσ ( t ) mix dt = α s ( q ) π | µ a | ρ c F g ( | q | ρ c )3 q dσ ( t ) nonpert dt = π µ a ρ c F g ( | q | ρ c )32 , (17)where q = − t and q min ≈ /ρ c for perturbative and mixed contributions and q min = 0for pure nonperturbative (Fig.2c) contribution. m - a s (mb) Figure 3: The contibution to the total quark-quark cross section as a function of AQCM:perturbative (dashed line) , mixed (dotted line), nonperturbative (dashed-dotted line)and their sum (solid line).For the strong coupling constant, the following parametrization was used for the case N f = 3: α s ( q ) = 4 π q + m g ) / Λ QCD ) , (18)where Λ QCD = 0 .
280 GeV and the value m g = 0 .
88 GeV was fixed from the requirement α s ( q = 1 /ρ c ) ≈ π/ α s ( q ) → constant as q → . < | µ a | < . µ a = − M q = 280 MeV. This mass is in agreement withrecent result of analysis of dressed-quark propagator within DSE approach involving the5attice-QCD data from [21]. We will adopt this value in our estimations below. For thatset of parameters the total quark-quark cross section σ totalqq = 3 .
05 mb is the sum of thefollowing partial cross sections: σ pertqq = 0 . mb, σ mixqq = 1 . mb, σ nonpertqq = 1 . mb, (20)and it is not far away from ”experimental” quark constituent model value σ expqq ≈ σ inP P ( ¯ P ) = 36 mb in the energy range where they are approximately constant.One may expect also an additional contribution to the total cross section arises from themultigluon and multiquark emission induced by the quark-gluon-instanton vertex. It willbring our esimation to the experimental value. It follows from Eq.(20) that the contribu-tion to the quark-quark cross section due to non-perturbative chromagnetic quark-gluoninteraction is about 80% and the contribution from pure perturbative exchange is about20% and quite small. Therefore, within our model the dynamics of soft Pomeron is de-termined not by the γ µ -like quark-gluon vertex (Fig.1a) as in most conventional modelsfor the Pomeron, but by the σ µν vertex pictured in Fig.1b. The widely assuming state-ment is that the difference in the dynamics of soft and hard Pomerons comes from thedifference in their dependence on such kinematic variables as total energy and transfermomenta. From our point of view, the main source of difference between two exchangesarises from a completely different spin structure of quark-gluon interaction inside thePomeron exchange.In our above estimation above only simplest contributions to the Pomeron exchangepresented in Fig.2 was considered. Due to pure spin one t-channel exchange they lead tocross section independent of the energy. Therefore the effective Pomeron intercept α P = 1in this approximation. It is well known that the experimental data show that the valueof soft Pomeron intercept α P (0) ≈ .
08 [22]. In spite of the fact that empirically softPomeron intercept close to one, its deviation from one leads to visible energy dependenceof the total and diffractive cross sections and to a large subleading contributions at veryhigh energies. Some of diagrams which provide such subleading contributions in our
I I I II I II ¯ I ¯ I ¯ I ¯ I ¯ I ¯ I ¯ I ¯ Ia ) b ) c )¯ I I
Figure 4: The example of the diagrams which give the contribution to energy-dependentpart of Pomeron exchange.model are presented in Fig.4. It is evident that at low energy such contributions should6e suppressed by even powers of small packing fraction of instantons in QCD vacuum, f n < / n , n = 2 , ... . However, due to their logarithmic growth with increasing of energy theymight give the dominant contribution at very large energy. The calculation of thesecontributions is beyond of this paper and will be the subject of the separate publication[23]. Within the conventional approach, the Odderon P=C=-1 partner of Pomeron, originatesfrom three gluon exchange (Fig.4a) with non-spin-flip perturbative-like quark-gluon vertex[24], [25],[26], [27],[28]. The experimental support of the existence of such exchange comesfrom high energy ISR data on the difference in the dip structure around | t |≈ . t [19]. ( b ) ( c )( a ) I ¯I ¯III¯I I ¯II Figure 5: The structure of Odderon exchange: a) non spin-flip perturbative three gluonexchange, b) and c) nonperturbative spin-flip contributions.According to our model, the perturbative part of the Odderon, Fig.5a, in the regionof momentum transfer | t | / ≤ /ρ c is expected to be much smaller in comparison withthe nonperturbative part presented by the graphs, Fig.5b and Fig.5c . ( b )( a ) ¯II I Figure 6: The example of the diagrams which give the contribution to spin-flip componentof Pomeron.It is clear that the first diagram gives rise to the non spin-flip amplitude of quark-quark scattering, the diagram in Fig.5b leads to single spin-flip and the diagram in Fig.6c The detailed calculation will be published elsewhere. n = < λ i λ i | λ f λ f > (see e.g. [20]), where n = 1 , ..., λ i , ( f , ) are helicities of initial (final) quarks, respectively, one can seethat the graph in Fig.5a gives the contribution to the Φ and Φ amplitudes, diagram inFig.5b contributes to the Φ amplitude, and Fig.5c gives rise to the Φ amplitude. Ourconjecture is that the spin-flip amplitude dominates in Odderon exchange. Therefore, onemight expect that Odderon should strongly interfere with the spin-flip part of Pomeron.Some of the diagrams which give the rise to the spin-flip part of the Pomeron are presentedin Fig.6.We would like to mention that in [29], [30] and [31] an alternative mechanism for thespin-flip component of Pomeron and Odderon [29], was discussed. This mechanism isbased on the existence of the quark-diquark component in the nucleon wave function. It was shown above that the Pomeron structure is rather complicated. It includes per-turbative, ”hard”, and nonperturbative, ”soft”, parts and their interference, ”soft-hard”part.
I I ¯I g g g g g gq q q ( a ) ( b ) ( c ) Figure 7: The diagrams contributing to nucleon gluon distribution: a) ”hard”-perturbative, b) ”hard-soft” interference perturbative and nonperturbative exchanges,c) ”soft”-nonperturbative part.Therefore, this structure should also manifest itself in the structure of gluon distri-bution in nucleon. One of the ways to show it is in the use of a DGLAP-like approach[32],[33] with the modified quark splitting function P Gq according with the vertex, Eq.(1)[8]. The diagrams giving the contribution to nucleon gluon distribution in our model arepresented in Fig.7.At present, unintegrated gluon distibution is widely used in different applications (see,for example, [34, 35]). To calculate this distribution, we use the convolution model formula f ( x, k ⊥ ) = N q k ⊥ Z x dyy P Gq ( x/y, k ⊥ ) q V ( y ) , (21)where N q = 3 is the number of valence quarks in nucleon, q V is the valence quark distri-bution function in nucleon, P Gq is the quark splitting function as defined in [33], and weneglect possible intrinsic momentum dependence of q V related to the confinement scale.8 k (GeV ) ^ f(x,k ) ^ Figure 8: The unintegrated gluon distribu-tion at x = 10 − : solid (dashed) line is total(perturbative) contribution. -3 -2 -1 xg(x) x Figure 9: Perturbative (dashed line) and to-tal (solid line) contributions to gluon dis-tribution at Q = 1 GeV in comparisonwith some of the phenomenological fits: dot-ted line is ALEKHIN02LO set and dashed-dotted line is MSTW2008LO fit [36].The splitting function for the general vertex Eq.(1) is given by the formula P Gq ( z, k ⊥ ) = C F z (1 − z ) k ⊥ π ( k ⊥ + M q z ) X λ T r n (ˆ k C + M q ) U µ ( t )(ˆ k A + M q ) ¯ U ρ ( t ) o ǫ µ ( λ ) ǫ ∗ ρ ( λ ) , (22)where U µ ( t ) = V µ (0 , , t ), k A ( k C ) is momentum of initial (final) quark, t = q = ( k A − k C ) ,¯ U = γ U † γ and λ is gluon helicity. In the infinite momentum frame k A = ( P, P + M q P , ~ ⊥ ) k C = ((1 − z ) P + k ⊥ + M q − z ) P , (1 − z ) P, − ~k ⊥ ) q = ( zP − k ⊥ + M q z − z ) P , zP, ~k ⊥ ) , (23)the result for splitting function is P Gq ( z, k ⊥ ) = C F k ⊥ πz ( k ⊥ + M q z ) × [( q α s ( | t | )Θ( | t | − Λ ) + q α s (1 /ρ c ) µ a F g ( | t | )) z + 2((1 − z ) α s ( | t | )Θ( | t | − Λ ) + α s (1 /ρ c ) µ a k ⊥ M q F g ( | t | ))] , (24)where | t | = ( k ⊥ + M q z ) / (1 − z ) is the gluon virtuality in Fig.7.The integrated distribution is given by g ( x, Q ) = Z Q dk ⊥ k ⊥ f ( x, k ⊥ ) , (25)9or estimation we use a simple form for valence quark distribution q V ( x ) = 1 .
09 (1 − x ) √ x (26)with the normalization Z q V ( x ) dx = 1 . (27)The result of calculation of unintegrated gluon distribution at x = 10 − is presented inFig.8 as a function of k ⊥ . The result for integrated gluon distibution at small Q = 1 GeV is pictured in Fig.8. It is evident that the nonperturbative contributiondominates in both unintegrated and integrated gluon distributions. For the large Q perturbative contribution starts to dominate due to its stronger Q dependence. Such adifference in the Q dependence is directly related to the difference in the k ⊥ behaviorbetween perturbative and nonperturbative contributions coming from the spin-non-flipand spin-flip part of the general quark-gluon vertex, Eq.(1). In Fig.9 we also presentthe comparison of our result with some available phenomenological parametrizations. Bytaken into account the uncertainties in the extraction of gluon distribution from the dataand our simple parametrization for valence quark distribution we may say that agreementis rather good. In summary, we suggest a new approach to the Pomeron and Odderon structures and gluondistribution in hadrons. It is based on the modified quark-gluon vertex which includes thenon-perturbative spin-flip part related to anomalous chromomagnetic interaction. It isshown that this interaction gives the main contribution to the Pomeron coupling to smallvirtuality light quarks and to the gluon distribution in nucleon. Our conjecture is that theorigin of the difference between ”soft” and ”hard” Pomerons is related to the differencein the spin structure of quark-gluon interaction governing these effective exchanges. Wegive the arguments in favor of the spin-flip dominance in Odderon exchange.
The author is very grateful to I. O. Cherednikov, A.E. Dorokhov, S. V. Goloskokov,E.A. Kuraev, N.N. Nikolaev and L.N. Lipatov for useful discussions. This work was sup-ported in part by RFBR grant 10-02-00368-a, by Belarus-JINR grant, and by Heisenberg-Landau program.
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