Running coupling effects in DIS cross sections for inclusive γ p scattering

Running coupling effects in DIS cross sections for inclusive γ p scattering 1/ 32 Running coupling effects in DIS cross sections for inclusive γ p scattering M.B. Gay Ducati beatriz.gay@ufrgs.br High Energy Physics Phenomenology Group Physics Institute Universidade Federal do Rio Grande do Sul Porto Alegre, Brazil work with E.G. de Oliveira, and J.T. de Santana Amaral

Running coupling effects in DIS cross sections for inclusive γ p scattering 2/ 32 Outline Motivation Deep inelastic scattering (DIS) Dipole formalism: inclusive DIS cross section QCD nonlinear evolution equations (1+1)-dimensional toy model for high energy QCD Results: inclusive cross sections Summary

Running coupling effects in DIS cross sections for inclusive γ p scattering 3/ 32 Introduction High energy QCD evolution nonlinear evolution equations: Pomeron loop equations: generalization of Balitsky JIMWLK hierarchy by including gluon number fluctuations. Mean field approximation (MFA): Balitsky Kovchegov (BK) equation At fixed coupling: The property of geometric scaling, predicted by BK equation, is washed out by the fluctuation effects and replaced by the so called diffusive scaling. Inclusive and diffractive DIS cross sections are expected to show diffusive scaling. Y. Hatta, E. Iancu, C. Marquet, G. Soyez, and D.N. Triantafyllopoulos, Nucl. Phys. A773 (2006) 95. Fluctuation effects have not been observed in the experimental data. M. Kozlov, A. Shoshi and W. Xiang, JHEP 0710 (2007) 020. E. Basso, MBGD, E. G. de Oliveira, J. T. de Santana Amaral, Eur. Phys. J. C58 (2008) 9.

Running coupling effects in DIS cross sections for inclusive γ p scattering 4/ 32 Toy models Pomeron loop equations have a complicated structure and, therefore, are difficult to solve. The properties of the solutions are known only after some approximations and in the asymptotic regime. This difficulty inspired the investigation of simpler models with a smaller number of dimensions. Among them, a (1+1)-dimensional model has shown to mimic high energy QCD with fixed both coupling constant and impact parameter. E Iancu, JT de Santana Amaral, G Soyez, and D Triantafyllopoulos, Nucl.Phys.A 786, 131 (2007) The fixed coupling results show the emergence of the diffusive scaling that washes out the geometric scaling, which appears in the mean field approximation (MFA).

Running coupling effects in DIS cross sections for inclusive γ p scattering 5/ 32 Running coupling and fluctuations The generalization of the model to the running coupling case was done recently. A. Dumitru, E. Iancu, L. Portugal, G. Soyez and D. N. Triantafyllopoulos, JHEP 0708, 062 (2007). The pomeron loop (fluctuation) effects, due to the inclusion of the running of the coupling, are suppressed. This suppression is present up to extremely high values of rapidity Y 200, well beyond the region of interest in QCD phenomenology. Therefore, the running of the coupling restores the geometric scaling behavior of the average dipole scattering amplitude.

Running coupling effects in DIS cross sections for inclusive γ p scattering 6/ 32 Motivation Our purpose is to evaluate inclusive DIS cross sections within the framework of the (1+1)-dimensional model. Four scenarios are possible: MFA with fixed coupling. MFA with running coupling. Stochastic evolution with fixed coupling. Stochastic evolution with running coupling. Identify if the inclusive DIS cross sections calculated with the toy model are subject to the effects of fluctuations. Investigate, for the first time, the consequences of the inclusion of the fluctuations and running coupling effects in the cross sections for inclusive lepton-hadron DIS.

Running coupling effects in DIS cross sections for inclusive γ p scattering 7/ 32 Dipole frame At smal-x Bj, the γ h process can be described in the so-called dipole frame. In this frame, the virtual photon splits into a quark antiquark (q q) pair. This color dipole interacts with the hadron. Kinematics Invariant mass squared of the system γ h W 2 = (P + q) 2 Photon v Quark Photon virtuality Bjorken-x Rapidity q 2 = (k k ) 2 = Q 2 < 0 x x Bj = High energy limit: W 2, Q2 2P q = Y ln(1/x) 2 Q Q 2 + W 2 x Q2 W 2 0 q µ q µ 1 v Antiquark P µ Hadron

Running coupling effects in DIS cross sections for inclusive γ p scattering 8/ 32 Inclusive cross section In the dipole frame, the DIS cross section of the inclusive γ h scattering can be expressed as: dσ γ Z 1 Z tot d 2 b (Y, X Q2 ) = dv d 2 r ψα(v, γ r; Q) 2 P tot(b,r; Y ). 0 α=l,t ψ γ T/L 2 are the amplitude densities of the q q dissociation of a virtual photon with transversal (T) or longitudinal (L) polarization. x and y are the transverse positions of the quark and antiquark. r = x y is the transverse size of the q q pair. b = (x + y)/2 is its transverse impact parameter. v is the photon longitudinal momentum fraction carried by the quark.

Running coupling effects in DIS cross sections for inclusive γ p scattering 9/ 32 Geometric scaling σ tot γ*p [µb] 10 3 Geometric scaling is a phenomenological feature of DIS. 10 2 First observed in the HERA data on inclusive γ p scattering. Is is expressed as a scaling property of the virtual photon proton cross section: «σ γ p (Y, Q 2 ) = σ γ p Q 2 Qs 2 (Y ) 10 1 ZEUS BPT 97 ZEUS BPC 95 H1 low Q 2 95 ZEUS+H1 high Q 2 94-95 E665 x<0.01 all Q 2 10-1 10-3 10-2 10-1 1 10 10 2 10 3 τ

Running coupling effects in DIS cross sections for inclusive γ p scattering 10/ 32 Dipole picture The dissociation of the virtual photon into the color dipole takes place long before the scattering. The dipole evolves through (small-x Bj) soft gluon radiation until it meets the hadron at the time of the scattering. In the limit N c, a gluon can be effectively replaced with a pointlike quark antiquark pair in a color octet state. Then, a soft gluon emission from a color dipole can be described as the splitting of the original dipole into two new dipoles with a common leg. As the energy increases, the original dipole evolves through successive dipole splittings and becomes an onium, i.e., a collection of dipoles.

Running coupling effects in DIS cross sections for inclusive γ p scattering 11/ 32 Onium hadron scattering The probability for inclusive onium hadron scattering, P tot, refer to the differential cross-section at fixed impact parameter: dσ tot d 2 b (r,b, Y ) = Ptot(b,r; Y ) Ptot(x,y; Y ), The cross section is a priori frame-independent. However, it is most simply evaluated in the frame where the total rapidity of the target is the same as the total rapidity, Y 0 = Y, and the projectile is an elementary dipole: dσ γ tot d 2 b (Y, Q2 ) = Z 1 0 Z dv d 2 r X α=l,t ψ γ α(v, r; Q) 2 2Re T(x,y) Y, T(x,y) Y is the one dipole hadron forward scattering amplitude, (averaged over all the target configurations).

Running coupling effects in DIS cross sections for inclusive γ p scattering 12/ 32 Evolution equations By considering multiple scattering, the resulting evolution of the dipole scattering amplitudes is described by the Balitsky-JIMWLK hierarchy (ᾱ = α sn c/π) Z Y T xy Y = d 2 z K(x,y,z) T xz + T zy T xy T xzt zy Y where K(x,y,z) = ᾱ(x y) 2 /(x z) 2 (z y) 2 In the mean field approximation, this infinite hierarchy reduces to a single closed equation, the Balitsky-Kovchegov (BK) equation Z Y T xy Y = d 2 z K(x,y,z) ˆ T xz Y + T zy Y T xy Y T xz T zy Y When Fourier transformed to momentum space, it belongs to the same universality class of FKPP equation traveling wave solutions geometric scaling

Running coupling effects in DIS cross sections for inclusive γ p scattering 13/ 32 Fluctuations The importance of the gluon (dipole) number fluctuations, not included in the Balitsky s hierarchy, has been recently discovered y 1 y 2 x 1 x 2 v u z New hierarchy: Pomeron Loop Equations: For a projectile with j dipoles: T (j) Y = j ᾱ s T (j) j ᾱ s T (j+1) + j(j 1) 2 ᾱ s α 2 s T (j 1) Complicated transverse plane dependence.

Running coupling effects in DIS cross sections for inclusive γ p scattering 14/ 32 Langevin Equation Approximations: Elementary dipole-dipole amplitude Independence on the impact parameter After Fourier transform to momentum space, one has (ρ i = log(ki 2 /k0)) 2 Y T k = ᾱ sχ( ρ) T k ᾱ s D Y D T (2) k 1,k 2 E = ᾱ sχ( ρ1 ) T (2) k,k + ᾱ s κα 2 s k 2 1δ(k 2 1 k 2 2) T k1, E, D T (2) k 1,k 2 E ᾱ s D T (3) k 1,k 1,k 2 E + (1 2) The hierarchy can be rewritten in the form of the Langevin equation (event-by-event)» q Y T(ρ) = ᾱ χ( ρ)t(ρ) T 2 (ρ) + καst(ρ)η(ρ, 2 Y ), η(ρ, Y ) = 0, η(ρ 1, Y 1)η(ρ 2, Y 2) = 4 δ(ρ1 ρ2)δ(y1 Y2) ᾱ BK equation with a noise term: diffusive approximation stochastic FKPP equation

Running coupling effects in DIS cross sections for inclusive γ p scattering 15/ 32 Consequences of fluctuations The generated front T(ρ) has asymptotic speed smaller than that predicted by the MFA [Brunet e Derrida, 97] vc v c π2 γ cχ (γ c), quando 2 ln 2 (1/αs) 2 αs 1 Different realizations of the same evolution lead to an ensemble of fronts: same shape, shifted from each other along the ρ-axis The front position ρ s ln(q 2 s /k 2 0) is random variable Mean value ρ s (Y ) v c Y Dispersion σ 2 ρ 2 s ρ s 2 ᾱ s DY D: diffusion coefficient

Running coupling effects in DIS cross sections for inclusive γ p scattering 16/ 32 Diffusive scaling The values of ρ s are distributed, in a good approximation, according to a Gaussian probability [Marquet, Soyez e Xiao, 2006] P Y (ρ s) 1» exp (ρs ρs )2, πσ 2 σ 2 The mean amplitude T(ρ, ρ s ) = Z dρ s P Y (ρ s) T(ρ, ρ s) For very high energies and z ρ ρ s γ cσ 2, one has T(z) 1 2 Erfc z σ Dependence on z/σ: diffusive scaling.

Running coupling effects in DIS cross sections for inclusive γ p scattering 17/ 32 Phenomenology γ p inclusive cross section as the convolution, in momentum space, of the amplitude T(k, Y ) and the virtual photon wavefunction (R p is the proton radius): σ γ p (Y, Q 2 ) = α emr 2 pn c Z 0 dk k Z 1 0 dz Ψ(k, z; Q 2 ) 2 T(k, Y ) AGBS model: first parameterization of T(k, Y ) (single event amplitude). Using a Gaussian distribution (P Y (ρ s)) of amplitudes (ρ = log(k 2 /k 2 0)): E D T Y AGBS (ρ, ρ s ) = Z + dρ s P Y (ρ s) T AGBS Y (ρ, ρ s) Fit to ZEUS and H1 (x 0.01, 0.045 Q 2 150 GeV 2 : no evidence of fluctuations χ 2 /n.d.p k 2 0 ( 10 3 ) v c R(GeV 1 ) χ (γ c) D ( 10 3 ) 0.942 1.69 ± 0.16 0.176 ± 0.004 4.83 ± 0.12 6.43 ± 0.29 0 0.942 1.69 ± 0.16 0.176 ± 0.004 4.83 ± 0.12 6.43 ± 0.29 0.0 ± 9.6

Running coupling effects in DIS cross sections for inclusive γ p scattering 18/ 32 (1+1)-dimensional model E Iancu, JT de Santana Amaral, G Soyez, and D Triantafyllopoulos, Nucl.Phys.A 786, 131 (2007) Stochastic particle model in (1+1)-dimensions: Temporal dimension: total rapidity Y. Spatial dimension: position of the particle along an infinite one-dimensional axis x. QCD analogy: Spatial dimension corresponds to the logarithm of the inverse size of a dipole: x log(r 2 0 /r2 ) A hadronic system is specified by the distribution of particles along the one-dimensional axis x. As the rapidity increases, the system of particles changes through the emission of new particles.

Running coupling effects in DIS cross sections for inclusive γ p scattering 19/ 32 Particle systems A hadronic system is described by the probability P(n(x), Y ) of each configuration n(x). Two systems of particles interact: the left mover the projectile, and the right mover the target. The projectile rapidity is Y Y 0 and the target rapidity is Y 0. For the purpose of the calculation of inclusive cross sections, Y 0 = Y and all the evolution is in the target. The projectile is a single particle at x = log(r 2 0/r 2 ), where r is the virtual photon dipole size.

Running coupling effects in DIS cross sections for inclusive γ p scattering 20/ 32 S-matrix The S matrix is a priori frame independent: d S dy 0 = 0 The S-matrix of the scattering of two given configurations is:»z S[n, m] = exp dx Rdx Ln(x R)m(x L) ln σ(x R x L). σ(x R x L) = 1 τ(x R x L) is the S matrix for the scattering of two elementary particles of logarithmic sizes x R and x L. The elementary particle-particle scattering amplitude τ(x y) is chosen as τ(x y) = α(x)α(y) exp( x y ) α(x)α(y)k(x, y), that is analogous to the corresponding (approximate) expression in QCD. The average S-matrix is given by Z S Y = DnDm P R[n(x R), Y Y 0]P L[m(x L), Y 0]S[n(x R), m(x L)].

Running coupling effects in DIS cross sections for inclusive γ p scattering 21/ 32 Deposite rate An evolution step corresponds to a small increment dy. Only one extra particle is emitted in each step. The quantity f z[n(x)] is the deposit rate density. f z[n(x)]dzdy is the probability that an extra particle with size in the interval (z, z + dz) will be emitted, given that the initial configuration was n(x). In the toy model, the deposit rate is given by the following expression: f z[n(x)] = Tz[n(x)]. α(z) Where»Z T z[n(x)] = 1 exp dx n(x) ln σ(z x).

Running coupling effects in DIS cross sections for inclusive γ p scattering 22/ 32 Particle evolution As one particle is emitted, the final configuration consists in the same particles as the initial configuration plus an additional particle. P({n}, Y ) evolution: Z Z dp[n(x), Y ] = f z[n(x) δ xz]p[n(x) δ xz, Y ] dy z A generic observable O has evolution given by: Z O Y = f z[n(x)] {O[n(x) δ xz] O[n(x)]} Y Y. z z f z[n(x)] P[n(x), Y ]. The evolution of specific observables (for example the particle densities and the scattering amplitudes) can be obtained from this equation.

Running coupling effects in DIS cross sections for inclusive γ p scattering 23/ 32 Evolution of amplitudes The evolution equation for the amplitude of the scattering between a projectile which consists of a single particle of a given logarithmic size x and a generic target is given by: T x Y = αx Z z K xz T z(1 T x), This is the first equation of an infinite hierarchy. In the RHS, there is the T matrix T xt z for the scattering of a projectile made with two particles. Analogous to the first equation of the hierarchy of pomeron loop equations (extended to running coupling).

Running coupling effects in DIS cross sections for inclusive γ p scattering 24/ 32 Evolution of amplitudes Only in the second equation of the hierarchy the term corresponding to the gluon recombination appears: Z Z T xt y = α x K xz T zt y(1 T x) + α y K yz T zt x(1 T y) Y z z Z + α xα y α zk xzk yz T z(1 T x)(1 T y). z In the mean field approximation (MFA), the whole hierarchy reduces to a single closed equation, which is obtained by making TT = T T : Z T x Y = αx K xz [ T z T z T x ]. z Analogous to the BK equation.

Running coupling effects in DIS cross sections for inclusive γ p scattering 25/ 32 Coupling In the case of fixed coupling, we set α s = 0.2. In the case of running coupling, α s = 1/(βx). However, at negative values of x this coupling must be frozen to the value α 0 (chosen here to be 0.7.) Therefore, the running coupling used is: with c = 0.1. α s = 1 βc ln(e x/c + e 1/(α 0βc) ), The initial conditions must be chosen in a way that they are already saturated in the frozen region. Therefore, the dynamics is restricted to the region where α s = 1/(βx).

Running coupling effects in DIS cross sections for inclusive γ p scattering 26/ 32 Numerical analysis To calculate the cross sections using the (1+1)-dimensional model, numerical analysis are employed. The x-axis is discretized in sites of size 1/8. The initial condition is set to be of 20 particles in each site for x < 6 and no particles with x > 6. The evolution in Y will drive this front to higher values of x.

Running coupling effects in DIS cross sections for inclusive γ p scattering 27/ 32 Numerical analysis In the mean field approximation, the evolution in Y is done by the means of a fixed-step ODE solver provided by the Gnu Scientific Library. In the stochastic evolution, the system of particles is simulated with 10 4 events. At each Y step, first the step size is randomly calculated from a exponential decay distribution with mean equal to the inverse of total probability of emission of a new particle. Secondly, the site of the new particle is randomly assigned considering the deposit rate of each site. The physical scattering amplitude is the average over the events of each individual scattering amplitude.

Running coupling effects in DIS cross sections for inclusive γ p scattering 28/ 32 Saturation scale The saturation scale of each event is identified with the front position in the axis x. The front position (x s) is given by the expression: Z x s = x s Y=0 + T(x, Y ). x> x s Y=0 From our initial conditions, x s Y=0 = 6. Like r 2 = r 2 0 exp ( x), the saturation scale is defined to be: Qs 2 = 1 expx s. r0 2 In the case of multiple events, the dispersion of fronts is given by: σ 2 = x 2 s x s 2.

Running coupling effects in DIS cross sections for inclusive γ p scattering 29/ 32 Results Results The cross section with integration in x is given by: dσ γ tot d 2 b (Y, Q2 ) = πr2 0 2 Z 1 0 dv Z dx e x X α=l,t ψ γ α(v, x; Q) 2 P tot(x; Y ), Is our results, we use r 0 = 1 GeV 1.

Running coupling effects in DIS cross sections for inclusive γ p scattering 30/ 32 Results: fixed coupling 10 2 FC and MFA Inclusive cross sections with fixed coupling as a function of Q 2 / Q 2 s. Top: MFA geometric scaling. As Y increases, also does the GS window. In the GS window, cross sections at different Y have the same shape. Bottom: fluctuations no geometric scaling. As Y increases, the shape of the cross section changes. Fluctuations are important in the fixed coupling case, breaking geometric scaling. dσ γ tot d 2 b (Y,Q2 ) dσ γ tot d 2 b (Y,Q2 ) 10 3 10 4 10 2 10 3 10 4 Y = 0 Y = 10 Y = 20 Y = 30 Y = 40 Y = 50 Y = 60 Y = 70 Y = 80 Y = 90 Y = 100 1 10 2 10 4 Q 2 / Q s 2 Y = 0 Y = 10 Y = 20 Y = 30 Y = 40 Y = 50 Y = 60 Y = 70 Y = 80 Y = 90 Y = 100 FC and fluctuations 1 10 2 10 4 Q 2 / Q s 2

Running coupling effects in DIS cross sections for inclusive γ p scattering 31/ 32 Results: running coupling 10 2 RC and MFA Inclusive cross sections with running coupling as a function of Q 2 / Q 2 s. Top: MFA geometric scaling. Bottom: fluctuations geometric scaling too. As Y increases, also does the GS window. dσ γ tot d 2 b (Y,Q2 ) 10 3 10 4 10 2 Y = 0 Y = 20 Y = 40 Y = 60 Y = 80 Y = 100 Y = 120 Y = 140 Y = 160 Y = 180 Y = 200 1 10 2 10 4 Q 2 / Q s 2 RC and fluctuations The running of the coupling suppresses the fluctuation effects. Geometric scaling behavior is restored. dσ γ tot d 2 b (Y,Q2 ) 10 3 10 4 Y = 0 Y = 20 Y = 40 Y = 60 Y = 80 Y = 100 Y = 120 Y = 140 Y = 160 Y = 180 Y = 200 1 10 2 10 4 Q 2 / Q s 2

Running coupling effects in DIS cross sections for inclusive γ p scattering 32/ 32 Summary We evaluated inclusive DIS cross sections using a (1+1)-dimensional model for high energy QCD to describe the evolution of dipole scattering amplitudes Geometric scaling behavior was seen in the mean field approximation, both with running and fixed coupling. In the stochastic evolution with fixed coupling, the geometric scaling is washed out, giving rise to diffusive scaling. In the stochastic evolution with running coupling, the geometric scaling is recovered. The inclusion of running coupling makes the mean field approximation enough to describe the DIS cross sections.