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QCDINTHEREGGELIMIT:FROMGLUONREGGEIZATIONTO PHYSICALAMPLITUDES A.PAPA DipartimentodiFisica,Universita` dellaCalabria, IstitutoNazionalediFisicaNucleare,GruppocollegatodiCosenza, 6 ArcavacatadiRende,I-87036Cosenza,Italy 0 0 ThispaperisabriefsurveyoftheBalitskii-Fadin-Kuraev-Lipatov(BFKL)approachforthedescrip- 2 tionofhardorsemi-hardprocessesintheso-calledReggelimitofperturbativeQCD.Thestarting n pointisafundamental propertyofperturbativeQCD,thegluonReggeization. Thisproperty, com- bined with s-channel unitarity, allows topredict the growth in energy of the amplitude of hard or a J semi-hard processes withexchange of vacuum quantum numbers in thet-channel. Whenalsothe so-calledimpactfactorsofthecollidingparticlesareknown,thennotonlythebehaviorwiththeen- 5 ergy,butthecompleteamplitudecanbedetermined. Thiswasrecentlydone,forthefirsttimewith 2 next-to-leadingorderaccuracyandforaprocesswithcolorlessexternalparticles,inthecaseofthe electroproductionoftwolightvectormesons. 2 v 9 1 Introduction 8 1 1 The BFKL equation1 became very popular in the last years due to the experimental 0 resultsondeepinelasticscatteringofelectronsonprotonsobtainedattheHERAcollider. 6 These results show a power-growth of the gluon density in the proton when the fraction 0 oftheprotonmomentumcarriedbythegluon(i.e. thexBjorkenvariable)decreases. / h 2 Together with the DGLAP evolution equation , the BFKL equation can be used for p - the description of structure functions for the deep inelastic electron-proton scattering at p small values of the x variable. It applies in general to all processes where a “hard” e scale exists (the “hardness” can be supplied either by a large virtuality or by the mass h of heavy quarks) which allows the use of perturbation theory. It is an iterative integral : v equation for the determination of the Green’s function for the elastic diffusion of two i X Reggeized gluons (Section2). TheknowledgeofthisGreen’sfunction isenough forthe determination of the growth in energy of the amplitude of hard or semi-hard processes r a and, for the case of deep-inelastic electron-proton scattering, for the growth in x of the gluondensityintheprotonfordecreasingx. Thekernelofthisequationwasfirstderived intheleading logarithmic approximation (LLA),whichmeansresummation ofallterms of thetype (α lns)n,where α istheQCDcoupling constant and sisthesquare ofthe s s c.m.s. energy. In this approximation total cross sections are predicted to grow at large s with a power of the center-of-mass energy larger than for “soft” hadronic processes (Section3). Unfortunately, in this approximation neither the scale of s nor the argument of the running coupling constant α can be fixed. So, in order to do accurate theoretical pre- s dictions, it was necessary to calculate the radiative corrections to the LLA (Section 4). Thesecorrectionsturnedouttobelargeandwithnegativesignwithrespecttotheleading orderandthishasraisedanongoingdebateonthereliabilityofperturbationtheoryinthis context. Many attempts of improvement have been suggested, but a definite conclusion is still lacking, mainly because of the difficulty to translate any recipe designed to work fortheBFKLGreen’sfunction, whichisanunphysical object, intoadefiniteprediction, checkable byexperiments. The information which is needed together with the BFKL Green’s function in order to build a physical amplitude is given by the so-called “impact factors” of the colliding particles. Atthenext-to-leading orderimpactfactorswerecalculated firstforthecaseof collidingpartons3,4. Amongtheimpactfactorsfortransitionsbetweencolorlessobjects, themostimportant onefrom thephenomenological point ofviewiscertainly theimpact factor for the virtual photon to virtual photon transition, i.e. the γ∗ γ∗ impact factor. Its determination would open the way to predictions of the γ∗γ∗ tot→al cross section and wouldrepresentanecessaryingredientinthestudyoftheγ∗ptotalcrosssection,relevant fordeepinelasticscattering. Itscalculationisrathercomplicatedandonlyafteryear-long 5 effortsitisapproaching completion . Aconsiderable simplificationcanbegainedifoneconsidersinsteadtheimpactfactor for the transition from a virtual photon γ∗ to alight neutral vector meson V = ρ0,ω,φ. Inthiscase,indeed,acloseanalytical expressioncanbeachievedintheNLA,uptocon- 6 tributions suppressed asinverse powersofthephoton virtuality . Theknowledge ofthe γ∗ V impact factor in the NLA allows to determine completely within perturbative QCD→andwithNLAaccuracytheamplitudeofaphysicalprocess,theγ∗γ∗ VV reac- tion7,8(Section5). Thispossibilityisinterestingfirstofallfortheoreticalre→asons,since itcouldbeusedasatest-ground forcomparisons withapproaches differentfromBFKL, suchasDGLAP,andwithpossiblenext-to-leadingorderextensionsofphenomenological models. Moreover,itisusefulfortestingtheimprovementmethodssuggestedatthelevel oftheNLAGreen’sfunctionforsolvingtheproblemoftheinstabilityoftheperturbative series. But it could be interesting also for possible applications to phenomenology. In- deed,thecalculationoftheγ∗ V impactfactoristhefirststeptowardstheapplication → of the BFKLapproach to the description of processes such as the vector meson electro- productionγ∗p Vp,beingcarriedoutattheHERAcollider,andtheproductionoftwo mesons inthe ph→oton collision, γ∗γ∗ VV orγ∗γ VJ/Ψ, which can bestudied at high-energy e+e− andeγ colliders. → → 2 GluonReggeization inperturbativeQCD ThekeyroleinthederivationoftheBFKLequationisplayedbythegluonReggeization. “Reggeization”ofagivenelementaryparticleusuallymeansthattheamplitudeofascat- tering process with exchange of the quantum numbers of that particle in the t-channel goes likesj(t) intheReggelimits t . Thefunction j(t)iscalled “Reggetrajectory” ≫ | | of the given particle and takes the value of the spin of that particle when t is equal to its squared mass. In perturbative QCD, the notion of gluon Reggeization is used in a strongersense. ItmeansnotonlythataReggeonexistswiththequantumnumbersofthe gluonandwithatrajectory j(t) = 1+ω(t)passing through1att = 0,butalsothatthis Reggeon gives the leading contribution in each order of perturbation theory to the amplitude of processes withlarge sandfixed(i.e. notgrowing withs)squared momentum transfer t. Tobedefinite, letusconsider theelastic process A+B A′+B′ withexchange −→ of gluon quantum numbers in the t-channel, i.e. for octet color representation in the t-channel and negative signaturea (see Fig. 1). Gluon Reggeization means that, in the aThenegative(positive)signaturepartofanamplitudeisthepartoftheamplitudewhichisodd(even) undertheexchangeoftheMandelstamvariablessandu. 0 A A A A~ q1 g1 qi q (cid:13)cGicii+1(qi;qi+1) (cid:0)! gi qi+1 gn qn+1 B B0 B B~ Fig.1. (Left)Diagrammaticrepresentationof(A−)A′B′. Thezig-zaglineistheReggeizedgluon,theblackblobsare 8 AB thePPReffectivevertices. Fig.2.(Right)DiagrammaticrepresentationoftheproductionamplitudeAA˜B˜+nintheLLA. AB Reggekinematical regions u ,tfixed(i.e. notgrowingwiths),theamplitude ≃ − → ∞ ofthisprocesstakestheform A′B′ s j(t) s j(t) (cid:16)A−8(cid:17)AB = ΓcA′A "(cid:18)−−t(cid:19) −(cid:18)−t(cid:19) # ΓcB′B . (1) Here c is a color index and Γc are the particle-particle-Reggeon (PPR) vertices, not P′P 9 depending on s. This form of the amplitude has been proved rigorously to all orders ofperturbation theoryintheLLA.Inthisapproximation theReggeized gluontrajectory, 10 j(t) 1+ω(t),enterswith1-loop accuracy ,andwehave ≡ g2t N dD−2k g2NΓ(1 ǫ)Γ2(ǫ) ω(1)(t) = ⊥ = − ( q2)ǫ . (2π)D−1 2 k2(q k)2 − (4π)D/2 Γ(2ǫ) − ⊥ Z ⊥ − ⊥ Here D = 4 + 2ǫ has been introduced in order to regularize the infrared divergences and the integration is performed in the space transverse to the momenta of the initial collidingparticlesb. IntheNLAtheform(1)hasbeencheckedinthefirstthreeordersof 11 perturbation theory andisonlyassumedtobevalidtoallorders. 3 BFKLintheLLAandthe“bootstrap” Amplitudes with quantum numbers in the t-channel different from the gluon ones are obtained intheBFKLapproach bymeansofunitarity relations, thuscalling forinelastic amplitudes. In the LLA, the main contributions to the unitarity relations from inelastic amplitudes come from the multi-Regge kinematics, i.e. when rapidities of the produced particles are strongly ordered and their transverse momenta do not grow with s. In the bInthefollowing,sincethetransversecomponentofanymomentumisobviouslyspace-like,thenotation p2⊥ =−p~2willbealsoused. pA pA0 (cid:8)A0A q1 q1(cid:0)q q1 q1(cid:0)q q1 q1(cid:0)q q10 q10(cid:0)q G G = q1 q1(cid:0)q+ G q2 q2(cid:0)q q2 q2(cid:0)q q2 q2(cid:0)q (cid:8)B0B pB pB0 Fig.3.(Left)DiagrammaticrepresentationofAA′B′(foradefinitecolorgrouprepresentation)asderivedfroms-channel AB unitarity. TheovalsaretheimpactfactorsoftheparticlesAandB,thecircleistheGreen’sfunctionfortheReggeon- Reggeonscattering. Fig.4.(Right)SchematicrepresentationoftheBFKLequationintheLLA. multi-Regge kinematics, the real partc of the production amplitudes takes a simple fac- torized form,duetogluonReggeization, AA˜B˜+n = 2sΓc1 n γPi (q ,q ) si ωi 1 1 sn+1 ωn+1Γcn+1, (2) AB AÃ i=1 cici+1 i i+1 (cid:18)sR(cid:19) ti!tn+1(cid:18) sR (cid:19) B˜B Y where s is an energy scale, irrelevant in the LLA, γPi (q ,q ) is the (non-local) R cici+1 i i+1 effective vertex for the production of the particles P with momenta k = q q in i i i i+1 − the collisions of Reggeons with momenta q and q and color indices c and c , i i+1 i i+1 q p ,q p ,s = (k +k )2,k p−,k p andω standsforω(t ), 0 ≡ A n+1 ≡ − B i i−1 i 0 ≡ A˜ n+1 ≡ B˜ i i with t = q2. In the LLA, P can be only the state of a single gluon (see Fig. 2). By i i i using s-channel unitarity and theprevious expression fortheproduction amplitudes, the amplitude of the elastic scattering process A+B A′ +B′ at high energies can be −→ writtenas AA′B′ = is dD−2q1 dD−2q2 δ+i∞ dω Φ(R,ν)(~q ;~q;s ) AB (2π)D−1 Z ~q12~q1′2 Z ~q22~q2′2 Zδ−i∞ sin(πω) R,ν A′A 1 0 X s ω s ω − τ G(R)(~q ,~q ,~q)Φ(R,ν)( ~q ; ~q;s ) . (3) × s − s ω 1 2 B′B − 2 − 0 (cid:20)(cid:18) 0 (cid:19) (cid:18) 0(cid:19) (cid:21) Hereandbelowq′ q q,q q isthemomentumtransfer intheprocess, thesumis i ≡ i− ∼ ⊥ over the irreducible representations R of the color group obtained in the product of two adjoint representations and over the states ν of these representations, τ is the signature equal to +1( 1) for symmetric (antisymmetric) representations and s is an artificial 0 − energyscale,whichdisappearsinthefullexpressionoftheamplitudeateachfixedorder cTheimaginarypartgivesanext-to-next-to-leadingcontributionintheunitarityrelations. (R,ν) (R,ν) of approximation. Φ and Φ are the so-called impact factors in the t-channel A′A B′B color state (R,ν). The first of them is related to the convolution of the PPR effective vertices Γ and Γ , the second to the convolution of Γ and Γ d. G(R) is the AÃ A′A˜ B˜B B′B˜ ω Mellin transform of the Green’s functions for Reggeon-Reggeon scattering (see Fig. 3). (R) Thedependence fromsisdeterminedbyG ,whichobeystheequation(seeFig.4) ω ωG(R)(~q ,q~ ,q~) = ~q2~q′2δ(D−2)(~q ~q ) ω 1 2 1 1 1− 2 dD−2q + r K(R)(~q ,~q ;~q)G(R)(~q ,~q ;~q), (4) ~q2(~q ~q)2 1 r ω r 2 Z r r − whoseintegral kernel, K(R)(~q ,q~ ;~q)= [ω( ~q2)+ω( ~q′2)]δ(D−2)(~q ~q )+K(R)(~q ,~q ;~q) , (5) 1 2 − 1 − 1 1− 2 r 1 2 is composed by a “virtual” part, related to the gluon trajectory, and by a “real” part, (R) K , related to particle production in Reggeon-Reggeon collisions. In the LLA, the r “virtual”partofthekerneltakescontributionfromthegluonReggetrajectorywith1-loop accuracy, ω(1),whilethe“real”parttakescontribution fromtheproduction ofonegluon (B) intheReggeon-Reggeon collisionatBornlevel,K . TheBFKLequationisgivenby RRG Eq.(4)whent = 0andforsingletcolorrepresentationinthet-channel,otherwiseEq.(4) iscalledthe“generalized” BFKLequation. Therepresentation (3)oftheelasticamplitude, A+B A′+B′,derivedfroms- −→ channelunitarity, forthepartwithgluonquantumnumbersinthet-channel(R = 8,τ = 1), must reproduce the representation (1) with one Reggeized gluon exchange in the − t-channel, with LLA accuracy. This consistency is called “bootstrap” and was checked 1 in the LLA already in Ref. . Subsequently, a rigorous proof of the gluon Reggeization 9 intheLLAwasconstructed . The part of the representation (3) with vacuum quantum numbers in the t-channel (R = 0, τ = +1) for the case of zero momentum transfer is relevant for the total cross section ofthescattering ofparticlesAandB,viatheopticaltheorem: m AB d2~q Φ (~q ,s ) d2~q Φ ( ~q ,s ) σ (s)= I sAAB = 1 A 1 0 2 B − 2 0 (6) AB s 2π ~q 2 2π ~q 2 Z 1 Z 2 δ+i∞ dω s ω G (~q ,~q ) , ω 1 2 × 2πi s Z (cid:18) 0(cid:19) δ−i∞ whereDhasbeenputequalto4,assumingthatthestateAandB arecolorless andhave 13 therefore goodinfraredbehavior ,and (0) G (~q ,~q ) ω 1 2 G (~q ,~q ) . ω 1 2 ≡ ~q2~q2 1 2 dFortheprecisedefinitionofimpactfactors,aswellasoftheBFKLkernelK(R) whichappearsbelow, 12 seeRef. . ThisGreen’sfunction satisfiestheequation ωG (~q ,q~ )= δ2(~q ~q )+ d2~qK(~q ,~q)G (~q,~q ), (7) ω 1 2 1 2 1 ω 2 − Z where K(0)(~q ,~q ) 1 2 K(~q ,~q ) . (8) 1 2 ≡ ~q2~q2 1 2 The eigenfunctions and the corresponding eigenvalues of the LLA singlet kernel are 14 known(see,forinstance, Ref. ): N α d2~q K(~q ,q~ )(~q2)γ−1 = c sχ(γ)(~q2)γ−1 , 2 1 2 2 π 1 Z with Γ′(γ) χ(γ) =2ψ(1) ψ(γ) ψ(1 γ), ψ(γ) . − − − ≡ Γ(γ) The set of functions (~q2)γ−1 with γ = 1/2+iν, < ν < forms a complete set, 2 −∞ ∞ sothat δ+i∞ dω +∞ dν σ (s) = (9) AB δ−Zi∞ 2πi Z−∞ 2π2[ω− Ncπαsχ(1/2+iν)] d2~q d2~q s ω 1Φ (~q ,s ) 2Φ ( ~q ,s ) (~q2)−iν−3/2(~q2)+iν−3/2 . × 2π A 1 0 2π B − 2 0 s 1 2 Z Z (cid:18) 0(cid:19) The maximal value of χ(γ) on the integration contour is χ(1/2) = 4ln2, which corre- sponds to the maximal eigenvalue of the kernel ωB = (α N /π)4ln2, where the sub- P s c script“P”standsfor“Pomeron”. ItistheneasytoseethatEq.(9)leads,bysaddlepoint evaluation oftheν-integration, tothefollowingresult: sωPB σLLA . (10) ∼ √lns For α = 0.15 one gets ωB 0.40, much larger than the corresponding value for the s P ≃ crosssectionofsofthadronicprocesses( 0.08),butinroughagreementwiththepower- ≃ growth in x observed for the gluon density in the proton at small x and large virtuality Q2. The relation (10) shows that unitarity is violated, since the cross section overcomes the Froissart-Martin bound. This is obvious since, in the LLA, only a definite set of intermediate states, aswehaveseen, contributes tothes-channel unitarity relation. This means that the BFKL approach cannot be applied at asymptotically high energies. In order to identify the applicability region of the BFKLapproach, it is necessary to know the scale of sand the argument of the running coupling constant, which are not fixedin theLLA. 4 BFKLintheNLA In the NLA, the Regge form of the elastic amplitude (1) and of the production amplitudes (2),impliedbygluonReggeization, hasbeenchecked onlyinthefirstthreeorders 11 of perturbation theory . In order to derive the BFKL equation in the NLA, gluon Reggeization is assumed to be valid to all orders of perturbation theory. It becomes im- portant, therefore, tocheckthevalidityofthisassumption. In the NLA it is necessary to include into the unitarity relations contributions which differ from those inthe LLAby having one additional powerof α or one powerless in s lns. The first set of corrections is realized by performing, only in one place, one of the following replacements intheproduction amplitudes (2)entering thes-channelunitarity relation: ω(1) ω(2) , Γc(Born) Γc(1-loop) , γGi(Born) γGi(1-loop) . −→ P′P −→ P′P cici+1 −→ cici+1 The second set ofcorrections consists in allowing the production in the s-channel inter- mediatestateofonepairofparticleswithrapiditiesofthesameorderofmagnitude,both inthecentralorinthefragmentationregion(quasi-multi-Reggekinematics). Thisimplies onereplacement amongthefollowingonesintheproduction amplitudes (2)entering the s-channel unitarity relation: Γc(Born) Γc(Born) , γGi(Born) γQQ(Born) , P′P −→ {f}P cici+1 −→ cici+1 γGi(Born) γGG(Born) . cici+1 −→ cici+1 Here Γ stands for the production of a state containing an extra particle in the frag- {f}P mentation region of the particle P in the scattering off the Reggeon, γQQ(Born) and cici+1 γGG(Born) aretheeffective vertices fortheproduction ofaquarkanti-quark pairandofa cici+1 two-gluon pair,respectively, inthecollision oftwoReggeons. The detailed program of next-to-leading corrections to the BFKL equation was for- 15 mulated in Ref. and was carried out over a period of several years (for an exhaustive 14 review, see Ref. ). It turns out that also in the NLAthe amplitude for the high energy elastic process A+B A′ +B′ can be represented as in Eq. (3) and in Fig. 3. The −→ Green’s function obeys an equation with the same form as Eq. (4), witha kernel having the same structure as inEq. (5). Here the “virtual” part ofthe kernel takes also the contribution from the gluon trajectory at 2-loop accuracy, ω(2)11, while the “real” part of the kernel takes the additional contribution from one-gluon production in the Reggeon- Reggeon collisions at 1-loop order, K(1) 16,17,18,19, from quark anti-quark pair pro- RRG (B) 20 21 duction atBornlevel,K (Ref. fortheforwardcase, Ref. forthenon-forward RRQQ (B) 22 case)andfromtwo-gluonpairproduction atBornlevel,K (Ref. fortheforward RRGG 23 24 case,Ref. forthenon-forward, octetcase,Ref. forthenon-forward, singlet case). Theconsistency between the representation (3) of the elastic amplitude, A+B A′ +B′, derived from s-channel unitarity, for the part with gluon quantum number−s→in the t-channel (R = 8, τ = 1), and the representation (1) with one Reggeized gluon − exchange in the t-channel (“bootstrap”) is of crucial importance in the NLA.In this approximation, indeed, gluonReggeization wasonlyassumedinorder toderivetheBFKL equation. Moreover, the check of the bootstrap is also a (partial) check of the correct- ness of calculations which were performed mostly by one research group. In the NLA, 12 the bootstrap leads totwoconditions tobefulfilled ,oneonthe NLAoctetkernel, the otherontheNLAoctetimpactfactors, whichhavebothbeenverified21,25,3. 26 Another set ofbootstrap conditions, proposed in Ref. , wasintroduced. Theywere called “strong” bootstrap conditions, since their fulfillment implies that of the bootstrap conditions considered so far. They constrain the form of the octet impact factors and of theoctetBFKLkernel. TheirfulfillmenthasbeenverifiedinRefs.27,28. Recentlyishas been understood that these strong bootstrap conditions are a subset of those which arise fromtherequestofconsistencywiths-channelunitarityoftheinelasticamplitudesinthe 29 Regge form which enter the BFKLapproach . Their fulfillment amounts to prove the 29 gluonReggeization intheNLA . Asfor the exchange of vacuum quantum numbers in the t-channel, the NLA correc- tionstotheBFKLkernelleadtoalargecorrectiontotheBFKLPomeronintercept30,31. 14 Indeed, onegetsnow N α (~q2) α (~q2)N d2~q K(~q ,~q )(~q2)γ−1 = c s 1 χ(γ)+ s 1 cχ(1)(γ) (~q2)γ−1 , 2 1 2 2 π π ! 1 Z whereχ(1)(γ)isaknownfunction14. Weobservethatnowtheargumentofα isnotun- s definedasintheLLAcase. Therelativecorrectionr(γ)definedasχ(1)(γ) = r(γ)χ(γ) − inthepointγ = 1/2+iν turnsouttobeverylarge30,31, 1 n n f f r 6.46+0.05 +0.96 2 ≃ N N3 (cid:18) (cid:19) c c andleadstoaNLAPomeronintercept ω = ωB(1 2.4ωB) , ωB = 4ln2Nα (~q2)/π . P P − P P s A lot of papers have been devoted to the problem of this large correction (see, for in- 32 stance, ). AsanticipatedintheIntroduction, theunderstanding ofthewaytotreatthese large corrections can be helped by considering the full NLA amplitude of hard QCD physical processes, instead of limiting the attention to the unphysical BFKL Green’s function. The construction of a physical NLA amplitude in the BFKL approach, how- ever, calls for the determination of the NLA impact factors of the colorless particles involved in the process. This determination can be achieved by means of perturbation theory only in case a hard enough scale exists, such as a large photon virtuality or the massofaheavyquark. 5 Theinclusion ofimpactfactors: theamplitudefor theelectroproduction of two lightvectormesons Recently, the impact factor for the transition from a virtual photon with longitudinal polarization to a light vector meson with longitudinal polarization was calculated with 6 NLA accuracy . It can be used, together with the NLA BFKL Green’s function to build with NLA accuracy the amplitude for the production of two light vector mesons (V = ρ0,ω,φ) in the collision of two virtual photons. In the kinematics s Q2 ≫ 1,2 ≫ Λ2 ,whereQ2 arethephotonvirtualities,otherhelicityamplitudesareindeedpower QCD 1,2 suppressed, with a suppression factor m /Q and the light vector meson mass can V 1,2 ∼ beputequaltozero. Aswehaveseenbefore, theforwardamplitude maybepresented asfollows δ+i∞ s d2~q d2~q dω s ω 1 2 m ( )= Φ (~q ,s ) Φ ( ~q ,s ) G (~q ,~q ). I s A (2π)2 ~q 2 1 1 0 ~q 2 2 − 2 0 2πi s ω 1 2 Z 1 Z 2 Z (cid:18) 0(cid:19) δ−i∞ (11) This representation for the amplitude is valid with NLA accuracy. Here Φ (~q ,s ) 1 1 0 and Φ ( ~q ,s ) are the impact factors describing the transitions γ∗(p) V(p ) and 2 2 0 1 γ∗(p′) − V(p ), respectively. The Green’s function in (11) obeys the→BFKL equa- 2 → tion(7). Thescales isartificialandmustdisappear inthefullexpression fortheampli- 0 tude ateach fixedorder ofapproximatione. Using the result for the meson NLAimpact 6 factorsuchcancellationwasdemonstratedexplicitlyinRef. fortheprocessinquestion. 6 Theimpactfactorsareknownasanexpansion inα (seeRef. ) s 4πe f Φ (~q) =α D C(0)(~q 2)+α¯ C(1)(~q 2) , D = q V N2 1, (12) 1,2 s 1,2 1,2 s 1,2 1,2 −N Q c − c 1,2 h i q where f is the meson dimensional coupling constant (f 200MeV) and e should V ρ q ≈ bereplaced bye/√2,e/(3√2)and e/3forthecaseofρ0,ω andφmesonproduction, − respectively. Using the known impact factors and the NLA Green’s function, spectrally decom- 8 posedonthebasisoftheLLAkerneleigenfunctions, onegets +∞ m ( ) s s α¯s(µR)χ(ν) I s A = dν α2(µ )c (ν)c (ν) D D (2π)2 s s R 1 2 1 2 Z (cid:18) 0(cid:19) −∞ (1) (1) c (ν) c (ν) 1+α¯ (µ ) 1 + 2 (13) s R ×" c1(ν) c2(ν) ! s β 10 dln(c1(ν)) +α¯2(µ )ln χ¯(ν)+ 0 χ(ν) χ(ν)+ +i c2(ν) +2ln(µ2) , s R s  8N − 3 dν R  (cid:18) 0(cid:19) c    (1) where c (ν) (c (ν)) are the coefficient of the expansion of the LLA (NLA) impact 1,2 1,2 factors forthetwophotons inthebasis formedbytheeigenfunctions oftheLLAkernel. eForunderstandingthereasonswhythisscalewasintroducedandentersalsothedefinitionoftheimpact factors,seeRefs.14,12. 8 All the other functions of ν in (13) are known (see Ref. ). It is possible to write this amplitude intheformofaseries, as Q Q m 1 1 2 I sA = α (µ )2 (14) D D s (2π)2 s R 1 2 ∞ s n s n−1 b + α¯ (µ )nb ln +d (s ,µ )ln , 0 s R n n 0 R × s s (cid:20) n=1 (cid:18) (cid:18) 0(cid:19) (cid:18) 0(cid:19) (cid:19)(cid:21) X wheretheb andd coefficientscanbeeasilydeterminedbycomparison with(13). One n n shouldstressthatbothrepresentations oftheamplitude(14)and(13)areequivalentwith NLA accuracy, since they differ only by next-to-NLA (NNLA) terms. It is easily seen from Eq. (14) that the amplitude is independent in the NLA from the choice of energy andstrongcoupling scales. Indeed, withtherequired accuracy, α¯ (µ )β µ2 α¯ (µ ) = α¯ (µ ) 1 s 0 0 ln R (15) s R s 0 − 4Nc µ20!! andtherefore termsα¯nlnn−1slns andα¯nlnn−1slnµ cancelin(14). s 0 s R Heresomenumerical resultsarepresented fortheamplitude giveninEq.(14)forthe Q = Q Qkinematics,i.e. inthe“pure”BFKLregime. Theotherinterestingregime, 1 2 ≡ Q Q or vice-versa, where collinear effects could come heavily into the game, will 1 2 ≫ notbeconsideredhere. InthenumericalanalysispresentedbelowtheseriesintheR.H.S. ofEq.(14)hasbeentruncatedton = 20,afterhavingverifiedthatthisproceduregivesa verygoodapproximation oftheinfinitesumforY 10. Theb andd coefficients forn = 5ands =≤Q2 = µ2 can becalculated numeri- n n f 0 R callyandturntobethefollowingones: b = 17.0664 0 b = 34.5920 b = 40.7609 b = 33.0618 b = 20.7467 1 2 3 4 b = 10.5698 b = 4.54792 b = 1.69128 b = 0.554475 5 6 7 8 (16) d = 3.71087 d = 11.3057 d = 23.3879 d = 39.1123 1 2 3 4 − − − − d = 59.207 d = 83.0365 d = 111.151 d = 143.06 . 5 6 7 8 − − − − These numbers make visible the effect of the NLA corrections: the d coefficients are n negative and increasingly large in absolute values asthe perturbative order increases. In this situation the optimization of perturbative expansion, in our case the choice of the renormalization scaleµ andoftheenergyscales ,becomesanimportantissue. Below R 0 33 theprinciple ofminimalsensitivity (PMS) isadopted. Usually PMSisused tofixthe value of the renormalization scale for the strong coupling. Here this principle is used in a broader sense, requiring the minimal sensitivity of the predictions to the change of both the renormalization and the energy scales, µ and s . More precisely, we replace R 0 in(14)ln(s/s )withY Y ,whereY = ln(s/Q2)andY = ln(s /Q2),andstudythe 0 0 0 0 − dependence oftheamplitudeonY . 0 Ithasbeenfoundthat,forseveralfixedvaluesofQ2andn ,atanyvalueoftheenergy f Y there are wide regions of values of the parameters Y and µ where the amplitude is 0 R