aa r X i v : . [ h e p - ph ] M a y EVOLUTION EQUATIONS IN QCD AND QED
M. SLAWINSKA
Institute of Nuclear Physics, Polish Academy of Sciences,ul. Radzikowskiego 152, 31-342 Cracow, Poland.
Evolution equations of YFS and DGLAP types in leading order are considered. They arecompared in terms of mathematical properties and solutions. In particular, it is discussed howthe properties of evolution kernels affect solutions. Finally, comparison of solutions obtainednumerically are presented.
Generic QCD evolution equation covering several types of evolution reads: ∂ t D ( t, x ) = D ( t, · ) ⊗ K ( · )( x ) , (1)where the change of a structure function D with a process scale t is given by its convolutionwith the evolution kernel K . I will set the boundary condition D (0 , x ) = δ (1 − x ).Equations of the type (1) appear very often in high energy physics. For the purpose ofthis work, we will consider two simplified cases, originating from QED and QCD radiative pro-cesses. They were chosen due to their mathematical similarities to demonstrate how propertiesof evolution kernels influence their solutions.Let us consider the evolution equation with an additive convolution rule: ∂ t D Y F S ( t, x ) = Z dydu δ ( x − y − u ) K Y F S ( y ) D Y F S ( t, u ) = Z x du K Y F S ( x − u ) D Y F S ( t, u ) (2)with the kernel given by: K Y F S ( u, x ) = Θ( u − x )Θ(1 − u ) 1 u − x − δ ( u − x ) Z xx duu − x . (3)The distribution D Y F S has been calculated by Yennie, Frautschi, and Suura 1 in the contextof exponentiation infrared singularities in QED a .Another example is the evolution equation of DGLAP type 2, describing evolution of gluonicmomentum distribution in the leading order: ∂ t D ( t, x ) = Z dzdu δ ( x − uz ) K DGLAP ( u ) D ( t, z ) ≡ Z x dzz K DGLAP (cid:16) xz (cid:17) D ( t, z ) , (4) a Precisely, we consider distributions obtained from the original one with the substitution x → − x and u → − u .The definition of K Y F S follows from the famous photon spectrum K ( x ) = x − δ ( x ) R dvv he function D ( x ) is by some authors denoted as xD ( x ) 3. The QCD sum rule requires Z dx D ( t, x ) = 1 . (5)The above can be fulfilled if Z dx D (0 , x ) = 1 , (6a) ∂ t Z dx D ( t, x ) = 0 . (6b)The former condition is assured by initial condition, the latter may be obtained from (4): ∂ t Z dxD ( t | x ) = Z dx Z dz Z duδ ( x − uz ) K DGLAP ( z ) D ( t, u )= Z dz K DGLAP ( z ) Z duD ( t, u ) (7)and is fulfilled if Z dz K DGLAP ( z ) = 0 . (8)The condition (8) is assured by the definition K DGLAP = (cid:16) − x (cid:17) + Evolution equations (2) and (4) are governed by similar kernels, with major difference in theconvolution type. To compare them we will introduce regularized kernels and transform additiveevolution into multiplicative. That form would enable us to see that the difference between themlies in the details of kernels definitions.In order to find out interrelations between both evolutions we will work using regularizedkernels. Both kernels can be expressed as a sum of regular (“Θ”) and singular (“ δ ”) parts K ( x ) ≡ K Θ ( x )Θ(1 − x − ε ) − K δ ( x ) δ (1 − x − ε ) . (9)Parameter ε ≪ x -independent) infrared cutoff.We will write both kernels in the form they are used in evolution equations (2) and (4), asfunctions of two variables.In the YFS case: K Y F S ( u, x ) = Θ( u − u − x Θ( u − x − ε ) − δ ( u − x − ε ) ln 1 ε . (10)The DGLAP kernel K DGLAP is usually regularized in the ( x, z ) plane as follows: K DGLAP ( z ) = 11 − z Θ(1 − z − ε ) − δ ( z − − ε ) Z − ε dz − z . (11)The above notation stresses that the kernels enter their evolution equations regularizedin different variables, according to their convolution rules. In the YFS case ε is defined in“additive” variables ( x, u ) and assures x − u >
0. The DGLAP kernel is usually regularized“multiplicatively”, such that xz < t D ( t, x ) = Z x (cid:20) Θ(1 − z − ε )1 − z D (cid:16) t, xz (cid:17) z − δ ( z −
1) ln 1 ε D (cid:16) t | xz (cid:17) z (cid:21) dz = Z x (cid:20) Θ(1 − x/u − ε ) u − x D ( t, u ) − δ ( x − u ) ln 1 ε D ( t, x ) (cid:21) du. (12) K DGLAP in (12), however is regularized in ( x, z ) variables, which means that the cutoff is u -dependent. This regularization is given by Θ(1 − x/u − ε ), which transforms into u ≥ x ε . From(12) it is already transparent that different regularization of evolution kernels led to differencesin their “Θ” parts.Since ε is just a parameter set 0 at the end, one can go a step further and regularize DGLAPkernel on the ( x, u ) plane with constant parameter ε ′ : u − x > ε ′ and compare both kernels inthat language. In that case, ε in (12) is x -dependent and should be expressed by x and ε ′ . Itcould be done as follows: Let ε be a (so far not specified) function of x and ε ′ b . One can rewritethe last line of (12) in this new regularization: ∂ t D ( t, x ) = Z x Θ (cid:16) u − x − ε ( x,ε ′ )1 − ε ( x,ε ′ ) x (cid:17) u − x D ( t, u ) − δ ( x − u ) ln 1 ε ( x, ε ′ ) D ( t, x ) du, (13)from which it follows that the requirement of constant cutoff ε ′ leads to a formula ε ′ = ε ( x, ε ′ )1 − ε ( x, ε ′ ) x (14)relating both cutoffs. From (14) we obtain ε ( x, ε ′ ) = ε ′ x + ε ′ ≈ ε ′ x so that DGLAP evolution kernelregularized on the ( x, u ) plane by constant ε ′ has the form: K DGLAP ( x, u ) = Θ( u − x − ε ′ ) 1 u − x − δ ( u − x ) ln xε ′ . (15)Now it is clear that both kernels, although regularized identically and having equal “Θ”parts differ in their singular parts and are therefore different distributions. Another remark concerns normalization of both kernels. It is an intrinsic property of ker-nels, independent of regularization techniques and following from definitions. K DGLAP is nor-malized according to (8), whereas K Y F S inherits its normalization from K ( v ). R K ( v ) dv = R uu − K ( u, x ) dx = 0, whereas the integration range for K Y F S ( u, x ) is [0 , u ]. Therefore Z u K Y F S ( u, x ) dx = Z u K ( v ) dv = a ln u ≤ . (16)As K Y F S ( u, x ) is not normalized according to evolution equation, the normalization of D Y F S is time-dependent (and decreasing). Normalization properties of K DGLAP are imposed by (5),so D DGLAP is normalized by construction. b Redefinition ε → ε ( x ) is a legal operation, as x is just a parameter of the integrand (12). -1 YFS evolutionDGLAP evolutionEvolution time t= 0.200
Comparizon of DGLAP & YFS x -1 YFS evolutionDGLAP evolutionEvolution time t= 0.800
Comparizon of DGLAP & YFS
Figure 1: Numerical results
The graphs in Figure 4 present solutions of (2) and (4) with common initial condition D ( t =0 , x ) = δ (1 − x ),obtained at chosen evolution times.For small evolution times, both distributions are almost identical. The best agreement is inthe region x ≈
1, where both (10) and (15) are very similar. In QCD D ( t | x ≈
1) representscontribution to the total parton density function coming from very few/ very soft emissionsthat do not change the initial momentum of the branching gluon. In YFS evolution this regioncorresponds to y ≈ t one solution may be approximatedby the other.The discrepancies arise for small x , due to differences in singular parts of evolution kernels.They grow larger with increasing t .For t → ∞ , DGLAP solution converges to δ ( x = 0) , and YFS has a non-singular asymptoticsat x → ∼ x at . The right graph in Figure 4 presents comparison of solutions for large evolutiontime. It is visible, that the normalization of D Y F S decreases, whereas the normalization of D DGLAP is constant.
Acknowledgments
The project is partly supported by EU grant MTKD-CT-2004-510126.
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