aa r X i v : . [ h e p - ph ] M a y PARTON ENERGY LOSS IN COLLINEAR EXPANSION ∗ B.G. ZAKHAROV
L.D. Landau Institute for Theoretical Physics, GSP-1, 117940,Kosygina Str. 2, 117334 Moscow, Russia
We demonstrate that the N = 1 rescattering contribution to the gluon radiation from a fastmassless quark in eA DIS vanishes in the collinear approximation. It is shown that the nonzero N = 1 gluon spectrum obtained in the higher-twist approach by Guo, Wang and Zhang 3 , There are several approaches to the induced gluon emission from fast partons due to mul-tiple scattering in cold nuclear matter and hot quark-gluon plasma 1 , , , ,
5. The most generalapproach to this phenomenon is the so-called light-cone path integral (LCPI) approach 5 (forreviews, see 6 , , , , N = 1 rescattering and has originally been derived for the gluon emissionfrom a fast quark produced in eA DIS. The analyses 1 , , , , N = 1 gluon spectrum in theLCPI approach 10. But this is not the case. The GWZ gluon spectrum predicted in 3 , , , , We consider the gluon emission from a fast quark produced in eA DIS for the Bjorken variable x B and photon virtuality Q . The transverse momentum integrated distribution for the gq finalstate can be described in terms of the semi-inclusive nuclear hadronic tensor dW µνA /dz (hereafter z = ω/E , where ω is the gluon energy and E is the struck quark energy). The spin effects in therescatterings of fast partons can be neglected. This ensures that the spin structure of dW µνA /dz is the same as for the usual hadronic tensor W µνN in eN DIS. It allows one to describe the ∗ Contributed to 43rd Rencontres de Moriond on QCD and High-Energy Hadronic Interactions, La Thuile,Italy, 8-15 Mar 2008. The work is done in collaboration with P. Aurenche and H. Zaraket df A dz = Z d r n A ( r ) df N ( r ) dz , (1)where df N ( r ) /dz is the in-medium semi-inclusive quark distribution for a nucleon located at r ,and n A ( r ) is the nucleus number density.In the LCPI approach 5 the matrix element of the q → gq ′ in-medium transition is writtenin terms of the wave functions of the initial quark and final quark and gluon in the nucleus colorfield (we omit the color factors and indices) h gq ′ | ˆ S | q i = ig Z dy ¯ ψ q ′ ( y ) γ µ A ∗ µ ( y ) ψ q ( y ) . (2)Each quark wave function in (2) is written as ψ ( y ) = exp( − ip − y + ) √ p − ˆ u λ φ ( y − , ~y T ) , where λ is thequark helicity, ˆ u λ is the Dirac spinor operator, y ± = ( y ± y ) / √
2. The y − dependence of thetransverse wave functions φ is governed by the two-dimensional Schr¨odinger equation i ∂φ ( y − , ~y T ) ∂y − = n [( ~p T − g ~A T ) + m q ]2 µ + gA + o φ ( y − , ~y T ) (3)with µ = p − . The wave function of the emitted gluon can be represented in a similar way. The y − evolution of the transverse wave functions can be described in terms of the Green’s function K forthe Schr¨odinger equation (3). One can show that for the gauges with potential vanishing at largedistances (say, covariant gauges, or Coulomb gauge) one can ignore the transverse potential ~A T .To calculate the N = 1 rescattering contribution we need not use the path integral representationfor the Green’s functions (which is used at the last stage of calculations 5). To obtain the N = 1contribution it is enough to expand K to the second order in the external potential. It allowsone to describe the induced gluon emission in eA DIS in terms of the free Green’s functions K for fast partons and the gluon correlator h A + ( y ) A + ( y ) i in the nucleus. Diagrammaticallyit is represented by a set of diagrams like shown in Fig. 1 in which the horizontal solid linecorresponds to K ( → ) and K ∗ ( ← ), the gluon line shows the gluon correlators, the verticaldashed line shows the transverse density matrices of the final quark and gluon at very large y − .Figure 1.The typical difference in the coordinate y − for the upper and lower γ ∗ qq vertices (which givesthe scale of the quantum nonlocality of the fast quark production) is given by the well knownIoffe length L I = 1 /m N x B . For the nucleon quark distribution L I is the dominating scale in theCollins-Soper formula 11 f N = π R dy − e ix B P + y − h N | ¯ ψ ( − y − / γ + ψ ( y − / | N i . For the gq final2tate the integration over the y − coordinate of the γ ∗ qq vertex is affected by the integrationover the positions of rescatterings and the q → gq splitting. However, for moderate x B when L I ≪ R A one can neglect the effect of rescatterings on the integration over y − . For productionof the final gq states with M gq ≪ Q the restriction on y − from the splitting point can also beignored. Indeed, the typical scale in integrating over the splitting points is given by the gluonformation length L f ∼ E/M gq which is much bigger than L I at M gq ≪ Q . This is valid forboth the vacuum DGLAP and the induced gluon emission. Also, at L I ≪ L f one can take forthe lower limit of the integration over the splitting points for the upper and lower parts of thediagrams in Fig. 1 the position of the struck nucleon. Then the quark production and gluonemission become independent and the df N ( r ) /dz can be approximated by the factorized form df N ( r ) dz ≈ f N dP ( r ) dz , (4)where dP/dz is the induced gluon spectrum described by the right parts of the diagrams evalu-ated neglecting the quantum nonlocality of the fast quark production.Due to confinement the typical separation of the arguments in the gluon correlators is of theorder of the nucleon radius, R N . It allows one to replace the fast parton propagators betweenthe gluon fields in the graphs like Fig. 1b by δ functions in impact parameter space. Thisapproximation is valid for parton energy ≫ /R N . It follows from the Schr¨odinger diffusionrelation for the parton transverse motion ρ ∼ L/E . Also, the smallness of the fast partondiffusion radius at the longitudinal scale ∼ R N allows one to replace in other transverse Green’sfunctions the y − coordinates by the mean values of the arguments of the vector potentials in thegluon correlators. This approximation corresponds to a picture with rescatterings of fast partonson zero thickness scattering centers (nucleons). The inequality ω ≫ /R N for the emitted gluonis equivalent to R N ≪ L f . For this reason, in the picture of thin nucleons the contribution of thegraphs like Fig. 1c,d with gluon correlators connecting the initial quark and final quark or gluoncan be neglected since they are suppressed by the small factor R N /L f . These approximationshave been used in the original formulation of the LCPI approach 5 (the BDMPS 1 and GLV2 approaches use them as well). Note that (similarly to the case of the quark-gluon plasma12) each gluon correlator appears only in the form of an integral over ∆ y − = y − − y − and at y +2 = y +1 . One can easily show that this ensures gauge invariance of the result (to leading orderin α s ).In 3 , eA DIS is described by the diagrams like shown in Fig. 2.Figure 2.The lower soft part is expressed in terms of the matrix element h A | ¯ ψ (0) A + ( y ) A + ( y ) ψ ( y ) | A i ,and the upper hard parts are calculated perturbatively. Due to conservation of the large p − momenta of fast partons in the Feynman propagators only the Fourier components with p − > y − coordinate) ones. One can show that the Feynman diagram treatment of 3 , K ( ~y T, , y − | ~y T, , y − ) = i Z dp + d~p T (2 π ) exp [ − ip + ( y − − y − ) + i~p T ( ~y T, − ~y T, )] p + − ~p T + m p − + i G r ( y − y ) = 14 π Z ∞ dp − p − e − ip − ( y +2 − y +1 ) "X λ ˆ u λ ¯ˆ u λ K ( ~y T, , y − | ~y T, , y − )+ iγ − δ ( y − − y − ) δ ( ~y T, − ~y T, ) i . (6)Here ˆ u λ and ¯ˆ u λ act on the variables with indices 2 and 1, respectively. The last term in (6) isthe so-called contact term. It does not propagate in y − and can be omitted in calculating thenuclear final-state interaction effects for fast partons. Using (6) and a similar representation forthe gluon propagator the hard parts in the higher-twist method can be represented in termsof the transverse Green’s functions as it is done in the LCPI treatment. We emphasize thatthe description of the hard parts in terms of the transverse Green’s functions automaticallyincludes all the processes in the GWZ approach (hard-soft, double-hard, and interferences inthe terminology of 3 , The calculation of the diagrams like shown in Fig. 1 is simplified by noting that the freetransverse Green’s function can be written as K ( ~y T, , y − | ~y T, , y − ) = θ ( y − − y − ) X p T φ p T ( ~y T, , y − ) φ ∗ p T ( ~y T, , y − ) , (7)where φ p T ( ~y T , y − ) is the plane wave solution to the Schr¨odinger equation for A µ = 0 with thetransverse momentum ~p T . It allows one to represent the upper and lower parts of the diagramsshown in Fig. 1 in the form R dy − d~y T φ ∗ q ′ ( ~y T , y − ) φ ∗ g ( ~y T , y − ) φ q ( ~y T , y − ) where the outgoing andincoming wave functions have the form of the plane waves with sharp changes of the transversemomenta at the points of interactions with the external gluon fields. This method has previouslybeen used in 13 for investigation of the role of the finite kinematical boundaries. All the hardparts evaluated with the help of the plane waves agree with that obtained in Refs. 3 , N = 1 spectrum can bewritten as 10 , dP ( r ) dz = ∞ Z r dξn A ( ~r T , r + ξ ) dσ ( z, ξ ) dz . (8)Here dσ ( z, ξ ) /dz is the cross section of gluon emission from the fast quark produced at distance ξ from the scattering nucleon. At z ≪ , dσ ( z, ξ ) dz = 2 α s [1 + (1 − z ) ] z Z d~p T d~k T ~k T xdG ( k T , x ) d~k T H ( ~p T , ~k T , z, ξ ) , (9) H ( ~p T , ~k T , z, ξ ) = " ~p T − ( ~p T − ~k T ) ~p T ~p T ( ~p T − ~k T ) · " − cos i~p T ξ Ez (1 − z ) ! . (10)Here the limit x → dG ( k T , x ) /d~k T is the unintegrated nucleon gluon density in theCollins-Soper form 11, which at x ≪ dG ( k T , x ) d~k T = N c − x π α s C F Z d~ρ exp ( − i~k T ~ρ ) ∇ σ ( ρ ) , (11)4here σ ( ρ ) is the well known dipole cross section.The collinear expansion corresponds to replacement of the hard part H by its second orderexpansion in ~k T (we suppress all the arguments except for ~k T for clarity) H ( ~k T ) ≈ H ( ~k T = 0) + ∂H∂k αT (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ~k T =0 k αT + ∂ H∂k αT ∂k βT (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ~k T =0 · k αT k βT . (12)Only the second order term in (12) is important, which gives to a logarithmic accuracy dσ ( z, ξ ) /dz ∝ R d~p T xG ( p T , x ) ∇ k T H ( ~p T , ~k T , z, ξ ) | ~k T =0 . But from (10) one can easily obtain ∇ k T H | ~k T =0 = 0.It is also seen from averaging of the hard part over the azimuthal angle of ~k T which gives h H ( ~p T , ~k T , z, ξ ) i ∝ θ ( k T − p T ). Thus, contrary to the expected dominance of the region k T ∼ < p T only the region k T > p T contributes to the gluon emission, and formal use of the collinearexpansion gives completely wrong result with zero gluon spectrum.The zero N = 1 gluon spectrum in the collinear approximation agrees with prediction ofthe harmonic oscillator approximation in the BDMPS 1 and LCPI 5 approaches. The oscillatorapproximation in 1 , σ ( ρ ) = C ρ . This parametrization is equivalent to the approximation of the vector potential bythe linear expansion A + ( y − , ~y T + ~ρ ) ≈ A + ( y − , ~y T ) + ~ρ ∇ y T A + ( y − , ~y T ) which can be traced tothe collinear expansion in momentum space. The first term in the expansion in the density ofthe spectrum in the oscillator approximation in the BDMPS and LCPI approaches correspondsto N = 2, and the term with N = 1 rescattering is absent 10. In terms of the representation (9)absence of the N = 1 contribution in the oscillator approximation is a consequence of the factthat in this case dG/d~k T ∝ δ ( ~k T ) (as one sees from (11)). The vanishing N = 1 spectrum is in a clear contradiction with the nonzero result of 3 ,
4. Thisdiscrepancy is strange enough since Eqs. (8)-(10) are completely equivalent to the formulationof 3 , ,
4. This puzzle has a simple solution. In 3 , z ≪ ~l T . The ~l T -integrated hard part obtained in 4 (Eq. 15 of 4) reads (up to an unimportantfactor) H ( ~k T ) ∝ Z d~l T ( ~l T − ~k T ) R ( y − , y − , y − ,~l T , ~k T ) , (13)where R ( y − , y − , y − ,~l T , ~k T ) = 12 exp " i y − ( ~l T − ~k T ) − (1 − z )( y − − y − )( ~k T − ~l T ~k T )2 q − z (1 − z ) × " − exp i ( y − − y − )( ~l T − ~k T ) q − z (1 − z ) ! · " − exp − i y − ( ~l T − ~k T ) q − z (1 − z ) ! (14)is an analog of the last factor in the square brackets in (10) for y − = 0, y − = y − ( y − , y − , correspond to the quark interactions with the virtual photon and t -channel gluons, our z equals1 − z in 3 , ∇ k T H ( ~k T ) the authors differentiate only the factor 1 / ( ~l T − ~k T ) .However, the omitted terms from the factor R are important. After the ~l T integration theyalmost completely cancel the contribution from the 1 / ( ~l T − ~k T ) term. Indeed, after putting y − = y − and changing the variable ~l T → ( ~l T + ~k T ) the right-hand part of (13) does not depend on ~k T atall. We emphasize that even without the change of the variable (13) leads to ∇ k T H | ~k T =0 = 0 ifone performs differentiating correctly which is not done in 3 ,
4. The difference between y − and y − ∇ k T H | ~k T =0 suppressed by the small factor ∼ ( R N /L f ) .Such contributions may be viewed as zero since they are beyond predictive accuracy of theapproximations used in 3 , a . . In summary, we have demonstrated that the collinear expansion fails in the case of gluonemission from a fast massless quark produced in eA DIS. In this approximation the N = 1rescattering contribution to the gluon spectrum vanishes. The nonzero gluon spectrum obtainedin 3 , , eA DIS and jet quenching in AA collisions do not make sense. Acknowledgments
This work is supported in part by the grant RFBR 06-02-16078-a and the LEA PhysiqueTh´eorique de la Mati´ere Condes´ee.
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