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Model (in)dependent features of the hard pomeron

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Model (in)dependent features of the hard pomeron
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    a  r   X   i  v  :   h  e  p  -  p   h   /   9   6   1   2   2   3   5  v   1   3   D  e  c   1   9   9   6 Model (In)dependent Features of the Hard Pomeron G. Camici and M. CiafaloniDipartimento di Fisica, Universit´a di Firenzeand INFN Sezione di Firenze Largo E. Fermi 2, 50125 Firenze  DFF 260/11/96 Abstract We discuss the small- x  behaviour of the next-to-leading BFKL equation, de-pending on various smoothing out procedures of the running coupling constant atlow momenta. While scaling violations (with resummed and calculable anomalousdimensions) turn out to be always consistent with the renormalization group, weargue that the nature and the location of the so-called hard Pomeron are dependenton the smoothing out procedure, and thus really on soft hadronic interactions. PACS 12.38.Cy  Interest in the high energy behaviour of Quantum Chromodinamics [1] has recentlyrevived because of the experimental finding [2] of rising structure functions at HERA andhas triggered a number of papers about the next-to-leading (NL) BFKL equation [3] andthe corresponding anomalous dimensions [4, 5]. One of the interesting features of the NL BFKL equation is supposed to be the descrip-tion of running coupling effects, which on one hand raises the question of its consistencywith the renormalization group (R. G.) and on the other hand emphasizes the problemof the singular transverse momentum integration around the Landau pole, sometimesreferred to as the IR renormalon problem [6].The purpose of this note, which is based on a simple treatment of the NL BFKLequation, is to emphasize the distinction between the (leading twist) R. G. features, whichare genuinely perturbative and thus model independent, and the hard Pomeron featureswhich will turn out to be strongly dependent on how the effective running coupling issmoothed out or cut off at low values of   k 2 =  O (Λ 2 ).The problem of the consistency with the R. G. of the BFKL equation with runningcoupling was already analyzed by Collins and Kwiecinski [7] by introducing a sharp cut-off in the transverse momentum integrations. Modifications to the bare Pomeron dueto the running coupling were also analyzed by Lipatov [8], based on some boundaryconditions in the soft region which, in our opinion, are eventually equivalent to settinga cut-off (see below). The cut-off dependence of BFKL type equations has also beeninvestigated [9]. Here we whish actually to point out that different ways of   smoothing out  the running coupling in the large distance, small- k  region yield  different   answers for thenature, location and strength of the (bare) hard Pomeron, while keeping the validity of the leading twist renormalization group factorization, as perhaps to be expected.Let us consider the BFKL equation with running coupling introduced in Refs. [7] and[10] f  A ( t ) =  f  0 A ( t ) + ¯ α s ( t ) ω    dt ′ K  ( t,t ′ ) f  A ( t ′ ) ,  (1)1  where  f  A ( t ) = √  k 2 F  A ( k 2 ) denotes the unintegrated gluon structure function in the hadron A , as a function of   t  = log( k 2 / Λ 2 ), and the BFKL kernel  K  ( t,t ′ ), which possibly containsNL contributions, is supposed to be scale invariant with the spectral representation K  ( t,t ′ ) =    dγ  2 πie ( γ  − 12 )( t − t ′ ) χ ( γ  ) =    + ∞−∞ dσ 2 πe iσ ( t − t ′ ) χ  12 + iσ   (2)which is also assumed to be symmetrical in  t  and  t ′ , so that  χ  12  + iσ   is even in  σ .The form (1) of the NL equation was proved to be valid for the  N  f  -dependent part of the NL kernel in Ref. [5]. But we have also emphasized [11] that the scale of   α s  can bechanged, together with a corresponding change in the scale invariant kernel, so as to leavethe leading twist solution invariant, at NL level accuracy. Therefore, by assuming Eq. (1)we do not emphasize the scale  k 2 as the natural scale for the running coupling 1 , but werather consider a reference form of the BFKL equation which has been widely analyzedpreviously [7-10].We shall also assume that the effective coupling ¯ α s ( t ) =  N  c α s ( t ) π  is smoothed out aroundthe pole at  t  = 0, and that the inhomogeneous term  f  0 ( t ) is peaked at some value  t  = t 0  >  0. We shall often consider in the following the examples α s ( t ) = 1 bt Θ( t − ¯ t ) + 1 b ¯ t Θ(¯ t − t ) ,  (¯ t >  0) ,  (3) f  0 ( t ) = δ  ( t − t 0 ) ,  (4)but our discussion will not be limited to these particular forms.Let us start noticing that if   α s ( t ) ≤ α s ( t M  ) has a maximum at  t  =  t M  , and  χ ( 12 + iσ ) ≥ χ ( 12 ) (as is the case for both the leading and the NL expressions considered so far [3, 5]), the Pomeron singularity  ω  p  has the upper bound ω  p  ≤  ¯ α s ( t M  ) χ  12  .  (5)This follows [7] from general bounds on the norm of   K  , and needs no further expla-nations. We shall show, however, that besides this general result, the properties of thePomeron singularity are very much dependent on the model for  α s ( t ). 1 Actually, the NL calculation[5] suggests that rather  q 2 = ( k − k ′ ) 2 is the natural scale of the runningcoupling. 2  In order to understand this point, we shall discuss the solution to Eq. (1) by using aquasi-local approximation of the kernel  K   valid around the end-point of the  γ  -spectrumin Eq. (2), i.e., K  ( t,t ′ ) ≃ χ  12  1 + a 2 ∂  2 t  + ....  δ  ( t − t ′ ) , a 2 = 12 χ ′′  12  χ  12   .  (6)This equation is based on the simplest polynomial expansion of   χ ( γ  ) around  γ   = 1 / 2,which of course has already been used in the literature [8]. Our purpose here is to obtain,by means of the expansion in Eq. (5) a simple picture of the hard Pomeron properties,and to show that this picture is stable when some higher order polynomial approximationis used.By replacing Eq. (6) into Eq. (1), the latter becomes an inhomogeneous second orderdifferential equation in the  t  variable, of the form α s (¯ t ) α s ( t ) ( f  ( t ) − f  0 ( t )) = ¯ ωω  1 + a 2 ∂  2 t  f  ( t ) ,  (7)  ¯ ω  ≡  ¯ α s (¯ t ) χ (12)  , and its homogeneous part is just a Schroedinger-type equation with a given potential  V  and wave number  k , given by the expressions V  ( t ) = 1 a 2 ω ¯ ω  α s (¯ t ) α s ( t )  − 1  ,  (8) k 2 =  1 −  ω ¯ ω   1 a 2 .  (9)While this potential is always linear for  t >  ¯ t  because of the perturbative behaviour of the running coupling, its form may vary considerably according to how  α s ( t ) is smoothedout in  −∞ < t <  ¯ t  (Fig. 1 (a), (b)) or cut off (Fig. 1 (c)).In the case  α s  is sharply cut-off (Fig. 1 (c)), one has an infinite potential well, whichhas, of course, a discrete spectrum, whose ground state provides the Pomeron pole, inagreement with previous analyses [7-9].3  If instead the smoothed out coupling has a flat behaviour below ¯ t  (Fig. 1 (a)) thespectrum is continuum and the Pomeron is a branch cut singularity with branch point ω  p  = ¯ α s ( −∞ ) χ  12  .  (10)In this case the eigenfunctions are just plane waves for  t  → −∞ , and there appearsto be no reason why their phase-shift should be fixed by some condition [8], contrary tothe case with cut-off.Finally in the intermediate case of Fig. 1(b) there may be an isolated singularity too,depending on the depth of the well.Whatever the relevant case is, the Green’s function of the corresponding Schroedingerequation is easily calculable, and provides the solution of Eq. (7) if we set  f  0 ( t ) = g A δ  ( t − t 0 ). A straightforward analysis shows that for  t > t 0  such solution takes thefactorized form √  k 2 F  A ( k 2 ,Q 20 ) =  f  A ( t,ω ) =  f  R ( t,ω ) g A t 0 f  L ( t 0 ,ω ) ,  t > t 0  = log  Q 20 Λ 2   (11)where  f  R ( f  L ) denote the regular solution of the homogeneous equation for  t  → ∞  ( t  →−∞ ). Their explicit form for  t >  ¯ t  is f  R ( t,ω ) = f  + ( t,ω ) (12) f  L ( t,ω ) = f  − ( t,ω ) + R ( ω ) f  + ( t,ω ) (13)where  R ( ω ) is the reflection coefficient of the well,  f  + ( f  − ) denote the regular (irregular)solutions for  t →∞  in the linear potential, given by the expressions f  ±  ≡   C  ± dγ  √  2 πie ( γ  − 12 ) t − X ( γ  )¯ bω ,  ( t >  ¯ t ) ,  (14) X  ( γ  ) ≡    γ  12 χ ( γ  ′ ) dγ  ′ =  χ  12  γ  −  12  + 16 χ ′′  12  γ  −  12  3 + .... ,  (15)and  C  +  ( C  − ) denote the regular (irregular) contours for the Airy functions (Fig. 2).The reflection coefficient (or  S  -matrix)  R ( ω ) in Eq. (13) is easily found - starting from k 2 = − χ 2 <  0, ( ω >  ¯ ω ) - for the simple model of   α s ( t ) in Eq. (3). In such case the wave4
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