Running coupling corrections to inclusive gluon production
aa r X i v : . [ h e p - ph ] J un Running coupling corrections to inclusive gluonproduction
W. A. Horowitz, , Yuri V. Kovchegov Department of Physics, The Ohio State University, Columbus, OH 43210, USA Department of Physics, University of Cape Town, Rondebosch 7701, South AfricaE-mail: [email protected], [email protected]
Abstract.
We calculate running coupling corrections for the lowest-order gluonproduction cross section in high energy hadronic and nuclear scattering using the BLMscale-setting prescription. At leading order there are three powers of fixed coupling;in our final answer, these three couplings are replaced by seven factors of runningcoupling: five in the numerator and two in the denominator, forming a ‘septumvirate’of running couplings, analogous to the ‘triumvirate’ of running couplings found earlierfor the small- x BFKL/BK/JIMWLK evolution equations. It is interesting to note thatthe two running couplings in the denominator of the ‘septumvirate’ run with complex-valued momentum scales, which are complex conjugates of each other, such that theproduction cross section is indeed real. We use our lowest-order result to conjecturehow running coupling corrections may enter the full fixed-coupling k T -factorizationformula for gluon production which includes non-linear small- x evolution.
1. Introduction
This proceedings contribution is based on [1].While an exact analytic formula for gluon production in AA collisions in the ColorGlass Condensate (CGC) framework is still not known, we do know that in pp and pA collisions gluon production at the level of classical gluon fields and leading-ln 1 /x nonlinear quantum evolution is given by the k T -factorization formula [2, 3]: dσd k T dy = 2 α s C F k Z d q φ p ( q , y ) φ A ( k − q , Y − y ) . (1)Here Y is the total rapidity interval of the collision, C F = ( N c − / N c , boldfacevariables denote two-component transverse plane vectors k = ( k , k ), and φ p , φ A arethe unintegrated gluon distributions in the proton and the nucleus, respectively, whichare defined by [2] φ A ( k , y ) = C F α s (2 π ) Z d b d r e − i k · r ∇ r N G ( r , b , y ) (2)and φ p ( k , y ) = C F α s (2 π ) Z d b d r e − i k · r ∇ r n G ( r , b , y ) . (3) unning coupling corrections to inclusive gluon production N G ( r , b , y ) denotes the forward scattering amplitude for a gluondipole of transverse size r with its center located at the impact parameter b scatteringon a target nucleus with total rapidity interval y . N G ( r , b , y ) can, in general, be foundfrom the JIMWLK evolution equation. In the large- N c limit it is related to the quarkdipole forward scattering amplitude on the same nucleus N ( r , b , y ) by N G ( r , b , y ) = 2 N ( r , b , y ) − N ( r , b , y ) , (4)where N ( r , b , y ) can be found from the BK evolution equation. The quantity n G ( r , b , y )from Eq. (3) is also a gluon dipole amplitude, but taken in a dilute regime, where it isfound by solving the linear Balitsky-Fadin-Kuraev-Lipatov (BFKL) evolution equation.Eq. (1) for the gluon production was derived in the fixed coupling approximation.However, the dipole amplitudes N ( r , b , y ), N G ( r , b , y ) and n G ( r , b , y ) are now knownfor the running coupling case due to the completion of the running coupling calculationsfor the BFKL/BK/JIMWLK evolution equations [4, 5]. Using the running-coupling BK(rcBK) equation, the calculational framework presented above, when applied to heavyion collisions by replacing φ p → φ A in Eq. (1), and implemented with a careful inclusionof the nuclear geometry fluctuations, led to the prediction made in [6] of the chargedparticle multiplicity as a function of the collision centrality for the LHC. This predictionwas confirmed by the ALICE data in [7]. However, at the time the prediction [6] wasmade, the scales of the couplings explicitly shown in Eqs. (1), (2), and (3) were notknown and had to be modeled. Our goal here is to fix the scales of those couplings forthe lowest-order gluon production.
2. Running coupling corrections: strategy
To include running coupling corrections we follow the BLM scale-setting procedure [8],which is known to be correct at least at the leading order in 1 /N c . One first needsto resum the contribution of all quark bubble corrections giving powers of α µ N f ,with N f the number of quark flavors and α µ the physical coupling at some arbitraryrenormalization scale µ . We then complete N f to the full beta-function by replacing N f → − π β (5)in the obtained expression. Here β = 11 N c − N f π (6)is the one-loop QCD beta-function. After this, the powers of α µ β should combine intophysical running couplings α s ( Q ) = α µ α µ β ln Q µ (7)at various momentum scales Q which would follow from this calculation. We use theM S renormalization scheme.As was originally argued in [9], including running coupling corrections into thediagrams of Fig. 1 below assuming that only a gluon can be produced in the final unning coupling corrections to inclusive gluon production µ . Following [9] we rectify the problem by redefining the gluon production cross-section to include production of collinear gluon–gluon and quark–anti-quark pairs withthe invariant mass lower than some collinear IR cutoff Λ coll . This new observable iscompletely µ -independent and expressible in terms of the running coupling constants.
3. Running coupling corrections to LO gluon production
Gluon production at the lowest order in the coupling is shown in Fig. 1 in the A + = 0light-cone gauge. At this order the unintegrated gluon distribution is φ ( k , y ) = α s C F π k . (8)such that Eq. (1) reduces to dσd k T dy = 2 α s C F π k Z d q q ( k − q ) . (9)Our goal is to set the scales for the three couplings in Eq. (9). qk − q k q qk k A B C aci ′ p , ip , j j ′ σ σ ′ σ σ ′ acb Figure 1.
Diagrams contributing to the lowest-order gluon production in quark-quarkscattering at high energy in the A + = 0 light-cone gauge. The final result for the lowest-order gluon production cross section with the runningcoupling corrections included is [1] dσd k T dy = 2 C F π α s (Λ coll ) k Z d q q ( k − q ) α s ( q ) α s (( k − q ) ) α s ( Q ) α s ( Q ∗ ) (10)with the momentum scale Q being a complicated function of q and k − q given in [1].An interesting feature is that the scale Q is complex-valued! The cross-section (10)is, of course, real, as it contains a complex-valued coupling constant multiplied by itsconjugate, α s ( Q ) α s ( Q ∗ ). Eq. (10) clearly looks like the fixed-coupling cross-section(9) with three factors of fixed-coupling replaced by the seven running couplings: wechose to refer to this structure as the septumvirate of couplings [1].An important feature of the scale Q is that in the q → q → k ) limitln Q µ MS = ln k µ MS + 12 (11) unning coupling corrections to inclusive gluon production dσd k T dy ≈ C F π α s (Λ coll ) α s ( k ) α s ( Q s )( k ) ln k Q s . (12)
4. Ansatz for the running coupling corrections in the k T -factorizationformula Using the lowest-order expression (10) we would like to conjecture the following running-coupling generalization of Eq. (1): dσd k T dy = 2 C F π k Z d q φ p ( q , y ) φ A ( k − q , Y − y ) α s (Λ coll ) α s ( Q ) α s ( Q ∗ ) (13)with the new (rescaled) distribution functions defined by (cf. Eqs. (2) and (3)) φ A ( k , y ) = C F (2 π ) Z d b d r e − i k · r ∇ r N G ( r , b , y ) (14)and φ p ( k , y ) = C F (2 π ) Z d b d r e − i k · r ∇ r n G ( r , b , y ) . (15)One may use this ansatz, along with N G obtained from the rcBK evolution to furtherimprove the existing CGC phenomenology for particle multiplicity along with itscentrality and rapidity dependence at RHIC and LHC. Acknowledgments
This research is sponsored in part by the U.S. Department of Energy under Grant No.de-sc0004286.
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