Comment on ``Success of collinear expansion in the calculation of induced gluon emission''
aa r X i v : . [ h e p - ph ] J un Comment on “Success of collinear expansion in thecalculation of induced gluon emission”
P. Aurenche a , B.G. Zakharov b and H. Zaraket c a LAPTH, Universit´e de Savoie, CNRS, B.P. 110,F-74941 Annecy-le-Vieux Cedex, France b L.D. Landau Institute for Theoretical Physics, GSP-1, 117940,Kosygina Str. 2, 117334 Moscow, Russia c Lebanese University Faculty of Sciences (I),Hadeth-Beirut, Lebanon
Abstract
We show that the arguments against our recent paper on the failure of the collinearexpansion in the calculation of the induced gluon emission raised by X.N. Wang areeither incorrect or irrelevant. . In our recent paper [1] (below referred to as AZZ) we have investigated the relationbetween the light-cone path integral (LCPI) approach [2] (for reviews, see [3, 4, 5]) to theinduced gluon radiation and the higher-twist formalism by Guo, Wang and Zhang (GWZ)[6, 7]. The GWZ approach is based on the Feynman diagram formalism and collinearexpansion. It includes only N = 1 rescattering contribution. The GWZ formalism hasbeen developed for the gluon emission from a fast quark in eA DIS. The gluon spectrumpredicted in [6, 7] contains the logarithmically dependent nucleon gluon density, whichis absent in the LCPI calculations [8]. The AZZ analysis [1] has been motivated by thisdiscrepancy between the GWZ gluon spectrum and the N = 1 contribution to the LCPIspectrum (which in general accounts for arbitrary number of rescatterings).In [1] we have demonstrated that the approximations used in [6, 7] really lead to adisagreement with the LCPI approach [2]. However, contrary to the results of [6, 7] thecorrect use of the collinear expansion gives a zero gluon spectrum. This result is confirmedby the exact calculations of the gluon spectrum within the oscillator approximation in theLCPI [2] and BDMPS approaches [9, 4] which is equivalent to the collinear expansion inmomentum space used in [6, 7]. The nonzero spectrum obtained in [6, 7] is a consequenceof the unjustified neglect of some important terms in the collinear expansion. In [10]Wang has criticized the AZZ analysis [1]. This comment is our reply to Wang’s criticism. As in [1, 6, 7, 10] we consider the induced gluon radiation from a fast massless quarkproduced in eA DIS (as usual q will denote the virtual photon momentum). We takethe virtual photon momentum in the negative z direction, and describe the 0 and 3components of the four-vectors in terms of the light-cone variables y ± = ( y ± y ) / √ h A | ¯ ψ (0) A + ( y ) A + ( y ) ψ ( y ) | A i , and the upper hard parts, H , are calculated perturbatively.1igure 1.In the limit of large struck quark energy all the y + coordinates in the soft part canbe set to zero. Due to conservation of the large p − momenta of fast partons in theFeynman propagators only the Fourier components with p − > y − coordinate)ones. The integrations over the p + momenta of fast partons in the GWZ analysis havebeen performed with the help of the contour integration using the poles of the retardedpropagators. The combinations of different poles leads to the processes with differentlongitudinal momentum transfers (double-hard, hard-soft, and the interferences in theterminology of [6, 7]). The collinear expansion used in [6, 7] corresponds to replacementof the hard part by its second order expansion in the t -channel transverse gluon momentum ~k T 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 . (1)For evaluation of the gluon emission only the second order term in (1) is important.Using the transverse momenta coming from this term and integrating by parts over thetransverse coordinates with the help of the Collins-Soper formula [11] for the gluon densityone can combine the vector potentials in the soft element into the unintegrated gluondensity. This procedure leads to the gluon spectrum in the form of an integral over thefinal gluon transverse momentum with an integrand proportional to the nucleon gluondensity times ∇ k T H | ~k T =0 = 0.In [1] we have demonstrated that the evaluation of the hard parts of the fast partonsin terms of Feynman diagrams in the GWZ formalism [6, 7] is equivalent to that in termsthe transverse Green’s functions used in the LCPI approach [2]. For this reason before thecollinear expansion the hard parts in the GWZ approach should coincide with the N = 1hard parts in the LCPI formalism. The direct comparison performed in [1] shows thatthis is really the case. However, after the collinear expansion the results of [1] and [6, 7]differ. In [1] we have shown that up to the terms suppressed by the small factor R N /L f (hereafter R N is the nucleon radius, L f is the gluon formation length) ∇ k T H | ~k T =0 = 0.The corrections suppressed by the R N /L f are beyond the accuracy of the approximationsof [6, 7]. For this reason under the approximations used in [6, 7] the N = 1 gluonspectrum vanishes. However, according to the GWZ calculations ∇ k T H | ~k T =0 is nonzero.In [10] Wang proceeds to claim that this is the case.2 . In [6, 7] the nonzero second derivative of the hard part at z ≪ z is thefractional gluon momentum) comes from the graph shown in Fig. 1b. The authors use forthe integration variable in the hard part of this graph the transverse momentum of thefinal gluon, ~l T . The ~l T -integrated hard part obtained in [7] (Eq. (15) of [7]) reads (up toan unimportant factor) H ( ~k T ) ∝ Z d~l T ( ~l T − ~k T ) R ( y − , y − , y − ,~l T , ~k T ) , (2)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 ) . (3)Here y − , y − , correspond to the coordinates of the quark interactions with the virtualphoton and t -channel gluons (our z equals 1 − z in [6, 7]). Note that (2) correspondsto the transverse momentum integrated gluon spectrum. Namely this case has beendiscussed in [6, 7] and [1]. In calculating ∇ k T H ( ~k T ) the authors differentiate only thefactor 1 / ( ~l T − ~k T ) . In [1] we have argued that the omitted terms from the factor R areimportant, and after the ~l T integration they almost completely cancel the contributionfrom the 1 / ( ~l T − ~k T ) term. Indeed, the dominating configurations correspond to | y − − y − | ∼ < R N , | y − | ∼ < R N . Neglecting the small corrections suppressed by R N /L f one can putin (3) y − = y − . Then, one can change the variable ~l T → ( ~l T + ~k T ), and the right-hand partof (2) becomes independent of ~k T at all, and one gets ∇ k T H | ~k T =0 = 0. Note that for thetransverse momentum integrated spectrum there is no difference between differentiatingthe integrand of the hard part with respect to ~k T at fixed ~l T or ~l T + ~k T . We emphasize thisfact since in [10] Wang presents the formulas for the fully differential spectrum (in ~l T and z )and says that one should keep the final gluon transverse momentum ~l T fixed in the collinearexpansion. He claims that namely due to ignoring this fact the incorrect conclusion onthe GWZ approach [6, 7] has been done in [1]. However, it is clear misrepresentation ofthe AZZ analysis [1] since in [1] (as in [6, 7]) only the transverse momentum integratedspectrum has been discussed when the above change of the integration variable can safelybe done.In [10] Wang simply ignores the above transparent argument in favor of vanishing ∇ k T H | ~k T =0 . He claims that the contribution from differentiating of the phase factorsentering the hard part “will be linear in ( y − − y − ) /q − or y − /q − which in general aresuppressed by a factor ℓ T r N /q − ...”. and therefore, they cannot cancel the contributionfrom differentiating the 1 / ( ~l T − ~k T ) factor. This statement is clearly wrong. It can easilybe demonstrated calculating ∇ k T H | ~k T =0 for y − = y − , y − = 0. Simple calculation gives ∇ k T H | ~k T =0 ∝ π ∞ Z dl T ( − cos( al T ) l T − a sin( al T ) l T + a cos( al T ) ) , (4)3here a = y − / q − z (1 − z ). The last two terms in the integrand in (4) come from differ-entiating the factor R (according to Wang’s statement these terms should be absent at y − − y − = 0, y − = 0). Introducing the variable τ = al T one obtains ∇ k T H | ~k T =0 ∝ πa ∞ Z dτ − cos( τ ) τ − ∞ Z dτ sin( τ ) τ + ∞ Z dτ cos( τ ) (5)which gives ∇ k T H | ~k T =0 = 0. Indeed after integrating the first term by parts it cancelsthe contribution from the second term in (5). The last integral equals zero. It can beobtained treating this integral as lim δ → R ∞ dx exp( − δx ) cos( x ). In (4), (5) we ignored thekinematical boundaries, and integrated up to infinity. However, accounting for the finitekinematical limit does not change the result. If one introduces a sharp cut-off factor in thegluon emission vertex in the q → qg transition defined in terms of the invariant mass ofthe qg state it gives in terms of the integrand of (4) a sharp cut-off in terms of ( ~l T − ~k T ) .One can easily show that in this case ∇ k T H | ~k T =0 = 0 as well. If one uses a sharp cut-offin terms of the variable ~l T one gets ∇ k T H | ~k T =0 ∼ a sin( a~l T,max ). This strongly oscillating(in y − ) contribution in the sense of evaluation of the gluon spectrum is equivalent to ∇ k T H | ~k T =0 ∼ /~l T,max which can be safely neglected. Thus the above simple analysisdemonstrates that contrary to Wang’s claim the contribution from differentiating thephase factor cancels the the contribution from differentiating 1 / ( ~l T − ~k T ) . We emphasizethat the above arguments concern namely the hard part in the form obtained in [6, 7],and are not related at all to the LCPI approach.In [10] Wang also gives his interpretation of the relation of the GWZ calculations tothe LCPI approach. He gives a “proof” of the fact that the LCPI approach gives the N = 1 spectrum which agrees with the GWZ result. He writes the hard part of [1] interms of the variables ~l T and ~k T (Eq. (23) of [10]) and expands in ~k T only the factor1 / ( ~l T − ~k T ) neglecting the terms which come from the expansion of the phase factor.Of course, this old wrong GWZ prescription leads to the old wrong GWZ result withnonzero gluon spectrum . Note that presenting this “proof” Wang does not pay anyattention to the evident fact that the the collinear expansion in momentum space in theLCPI approach should reproduce the prediction of the oscillator approximation in impactparameter space in which the exact calculations give zero N = 1 spectrum [8, 1]. In summary, we have shown that Wang’s criticism [10] of our recent analysis [1] of therelation between the LCPI [2] and GWZ [6, 7] approaches is unfounded. Using the hardpart exactly in the form of [6, 7] by explicit calculations we have demonstrated that thecollinear expansion gives a vanishing transverse momentum integrated gluon spectrum.It confirms the conclusion of [1] on the falsity of the GWZ calculations [6, 7] predicting anonzero gluon spectrum. Note that the normalization in Wang’s hard part (before the collinear expansion) is incorrect. It isevident since Wang identifies the LCPI hard part with the hard-soft process in GWZ, and claims thatthe double-hard processes are not included in the LCPI-BDMPS-GLV approaches. It is evidently wrongsince the representation of the retarded propagators in terms of the transverse Green’s function obtainedin [1] guarantees that in the LCPI calculations all the combinations of the poles (hard-soft, double-hard,and interferences in the GWZ language) of the propagators are automatically included. cknowledgements The work of BGZ is supported in part by the grant RFBR 06-02-16078-a and the LEAPhysique Th´eorique de la Mati´ere Condes´ee.
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