Shear Viscosity of a Gluon Plasma in Perturbative QCD
aa r X i v : . [ nu c l - t h ] N ov Shear Viscosity of a Gluon Plasma in Perturbative QCD
Jiunn-Wei Chen, Hui Dong, Kazuaki Ohnishi, and Qun Wang Department of Physics and Center for Theoretical Sciences,National Taiwan University, Taipei 10617, Taiwan School of Physics, Shandong University,Shandong 250100, People’s Republic of China Interdisciplinary Center for Theoretical Study and Department of Modern Physics,University of Science and Technology of China,Anhui 230026, People’s Republic of China
Abstract
We calculate the shear viscosity ( η ) to entropy density ( s ) ratio η/s of a gluon plasma in kinetictheory including the gg → gg and gg → ggg processes. Due to the suppressed contribution to η in the gg → gg forward scattering, it is known that the gluon bremsstrahlung gg ↔ ggg processalso contributes at the same order ( O ( α − s )) in perturbative QCD. Using the Gunion-Bertschformula for the gg → ggg matrix element which is valid for the limit of soft bremsstrahlung, wefind that the result is sensitive to whether the same limit is taken for the phase space. Using theexact phase space, the gg ↔ ggg contribution becomes more important to η than gg → gg for α s & × − . Therefore, at α s = 0 . η/s ≃ .
0, between 2.7 obtained by Arnold, Moore andYaffe (AMY) and 0.5 obtained by Xu and Greiner. If the soft bremsstrahlung limit is imposedon the phase space such that the recoil effect from the bremsstrahlung gluon is neglected, thenthe correction from the gg ↔ ggg process is about 10-30% of the total which is close to AMY’sprediction. This shows that the soft bremsstrahlung approximation is not as good as previouslyexpected. I. INTRODUCTION
One of the most surprising discoveries at the Relativistic Heavy Ion Collider (RHIC) isthat the hot and dense matter (believed to be a quark gluon plasma (QGP), see [1, 2, 3, 4]for reviews) formed in collisions appears to be a near-perfect fluid [5, 6, 7, 8, 9, 10, 11, 12,13]. The remanent of the non-central collisions shows collective motion (elliptic flow) witha shear viscosity ( η ) to entropy density ( s ) ratio η/s = 0 . ± . ± . η/s ratio is close to a conjectured minimum bound 1 / π [15], which is motivatedby uncertainty principle [16] and gauge/string duality [17, 18, 19, 20]. Since smaller η/s implies stronger particle interactions, contrary to the conventional picture, the QGPproduced at RHIC tends to be a strongly interacting fluid instead of a weakly interactinggas.However, a recent perturbative QCD calculation of η/s of a gluon plasma by Xu andGreiner (XG) [21] indicates that the gluon elastic scattering gg → gg does not givethe dominant contribution. They found that η/s for the gluon bremsstrahlung process gg ↔ ggg is about 1/7 of that for gg → gg , which means the contribution to the shearviscosity from gg ↔ ggg is 7 times as important as that from gg → gg . This would bring η/s down to 1 / π when strong coupling constant α s ≃ .
6. This implies that the near-perfect QGP might still be described by perturbative QCD and that the conventionalpicture could still be valid. Their conclusion is quite different from an earlier study byArnold, Moore and Yaffe (AMY) [22] (for a recent review, see, e.g., [23]). AMY foundthat gg ↔ ggg only contributes at 10% level for the three flavor quark diffusion constantfor α s < .
3. For comparison, XG have η/s ≃ . α s = 0 .
1, while AMY have η/s ≃ . η was computed in [22], the free gluon s is inserted by us for comparison).Both approaches of XG and AMY are based on kinetic theory. However, the mainpoints of differences are: 1) A parton cascade model [24] is used by XG to solve theBoltzmann equation. Since the bosonic nature of gluons is hard to implement in realtime simulations in this model, gluons are treated as a Boltzmann gas (i.e. a classicalgas). For AMY, the Boltzmann equation is solved in a variation method without takingthe Boltzmann gas approximation. 2) The N g ↔ ( N + 1) g processes, N = 2 , , . . . , areapproximated by the effective g ↔ gg splitting in AMY where the two gluons are nearly2ollinear with a small splitting angle, while the gg ↔ ggg process is used in XG wherethe bremsstrahlung gluon is soft but it can have a large splitting angle with its mothergluon. More specifically, in XG, the Gunion-Bertsch formula [25] for the gg → ggg matrix element squared in Eq. (12) is used. This formula is valid for the limit of softbut not necessarily collinear gluon bremsstrahlung. For the phase space, XG uses theexact phase space for the three gluon configurations (called “three-body-like” phase spacein this paper). In principle, if the soft bremsstrahlung limit is a good approximation ofthe gg → ggg process, one should be able to impose the same limit to the phase spaceas well and get approximately the same result. In this limit, the recoil effect from thebremsstrahlung gluon is neglected, and the phase space (for the two hard gluons) is called“two-body-like” here.In this paper, we will perform a third independent calculation for comparison. We willuse the same inputs on the Gunion-Bertsch formula for the gluon scattering amplitudes(modulo a factor 2 in Eq. (12)) with the soft gluon bremsstrahlung approximation as XGbut we will solve the Boltzmann equation variationally as AMY without taking the Boltz-mann gas approximation. We will also test the robustness of the soft gluon bremsstrahlungapproximation by comparing the results with the two- and three-body-like phase space. II. KINETIC THEORY
Using the Kubo formula, η can be calculated through the linearized response functionof a thermal equilibrium state η = − Z −∞ d t ′ Z t ′ −∞ d t Z d x h (cid:2) T ij (0) , T ij ( x , t ) (cid:3) i , (1)where T ij is the spatial part of the off-diagonal energy momentum tensor. In a leadingorder (LO) expansion of the coupling constant, there are an infinite number of diagrams[26]. However, it is proven that the summation of the LO diagrams in a weakly coupled φ theory [26] or in hot QED [27] is equivalent to solving the linearized Boltzmann equationwith temperature-dependent particle masses and scattering amplitudes. The conclusionis expected to hold in weakly coupled systems and can as well be used to compute the LOtransport coefficients in QCD-like theories [22, 28], hadronic gases [29, 30, 31, 32, 33, 34]and weakly coupled scalar field theories [26, 35, 36].The Boltzmann equation of a hot gluon plasma describes the evolution of the color3veraged gluon distribution function f = f ( x, p, t ) ≡ f p ( x ) (a function of space, time andmomentum) as [37, 38, 39, 40, 41] p µ E p ∂ µ f p = C [ f ] , (2)where E p = p for massless gluons. The driving force for the evolution is the particlescattering in the microscopic theory described by the collision term C which is a functionalof f . It is known that, to compute η to LO in the coupling constant α s , we need to include gg → gg and gg ↔ ggg scattering in C [40, 42]. We will show this more explicitly later.In thermal equilibrium, the gluon distribution f p is static, homogeneous and isotropicand hence ∂ µ f p = 0 at every x . This implies C [ f ] = 0 or detailed balance whose solutionis just the Bose-Einstein distribution function f p = 1 / ( e E p /T − V ( x ). Thedeviation from thermal equilibrium can be characterized by the inhomogeneity of V ( x )or the derivative expansions of V ( x ). For simplicity, we work in the comoving frame ofthe fluid element at point x with V = 0 and to the order of first derivatives of V . Thusthe distribution function can be parametrized as f p = f p [1 − χ p (1 + f p )] , (3)where χ p = (cid:2) A ( p ) ∇ · V + B ( p )ˆ p [ i ˆ p j ] ∇ [ i V j ] (cid:3) /T , (4)and where the symmetric traceless combinations ˆ p [ i ˆ p j ] = ˆ p i ˆ p j − δ ij / ∇ [ i V j ] =( ∇ i V j + ∇ j V i ) / − ∇ · V δ ij /
3. Note that the time derivatives do not appear becausethey can be related to the spatial derivatives by virtue of the conservation of energy mo-mentum tensor. Analogously the deviation of the energy momentum tensor away fromits equilibrium value can be parametrized by the bulk ( ζ ) and shear ( η ) viscosities δT ij = ζ δ ij ∇ · V − η ∇ [ i V j ] . (5)Using the definition in kinetic theory T µν = N g R d p (2 π ) p µ p ν E p f p ( x ), one obtains η = N g β Z d p (2 π ) p E p f p (cid:0) f p (cid:1) B ( p ) , (6)where N g = 16 is the gluon polarization and color degeneracy.4ollowing the standard procedure (see e.g. [26]) and making use of the Boltzmannequation satisfied by B ( p ), Eq.(6) can be recast into η = N g β Z Y i =1 d p i (2 π ) E i | M → | (2 π ) δ ( p + p − p − p )(1 + f )(1 + f ) f f × [ B ij ( p ) + B ij ( p ) − B ij ( p ) − B ij ( p )] + N g β Z Y i =1 d p i (2 π ) E i | M → | (2 π ) δ ( p + p − p − p − p )(1 + f )(1 + f ) f f f × [ B ij ( p ) + B ij ( p ) + B ij ( p ) − B ij ( p ) − B ij ( p )] , (7)where B ij ( p ) ≡ B ( p )(ˆ p i ˆ p j − δ ij ) and M → and M → are amplitudes for gg → gg and gg → ggg processes (or called 22 and 23 processes in this paper), respectively.One useful observation is that the right hand sides of Eqs. (6) and (7) correspond tointegrations over both sides of the Boltzmann equation, or equivalently, a projectionof the Boltzmann equation. It is certainly easier to solve the projected equation thanthe Boltzmann equation itself. However, there would be an infinite number of solutionssatisfying the projected equation, even though the true solution is unique, correspondingto that which gives the largest η (see e.g. [22]). This makes solving η a variationalproblem.To solve for B ( p ), we assume it to be a smooth function which can be expanded in aset of orthogonal polynomials, B ( z ) = z y ∞ X r =0 b r B ( r ) ( z ) , (8)where z = β | p | , B ( r ) ( z ) is a polynomial up to z r and the overall factor z y will be chosenby trial and error to get the fastest convergence [31]. The B ( r ) ( z ) polynomials can beconstructed using the condition Z d p (2 π ) f p (1 + f p ) | p | z y B ( r ) ( z ) B ( s ) ( z ) = T δ rs . (9)One can solve the coefficients b r by equating Eqs. (6) and (7). Then, η is just proportionalto b according to Eqs. (6) and (9). For practical reasons, one uses the approximation B ( z ) = z y P n − r =0 b r B ( r ) ( z ) where n is a finite, positive integer. It can be proved that η isan increasing function of n . Thus, one can systematically approach the true value of η .For y = 1, the series converges rapidly. From n = 2 to 3, η only changes by ∼ | M → | = (12 πα s ) (3 − tu/s − su/t − st/u ) / s = O ( T ). The most singular part of | M → | comes from the colinear region5i.e. either t ≈ u ≈
0) which can be regularized by the Hard-Thermal-Loop (HTL)dressed propagators for gluons. However XG only used the Debye mass m D = (4 πα s ) / T as the regulator just as done in Ref. [44], so for the sake of comparison between AMYand XG, we also use m D as the regulator for soft and collinear divergences in this paper.We will use the HTL gluon propagators in one of our future study. Thus, we consider thenear collinear approximation | M → | ≈ − (12 πα s ) (cid:0) su/t + st/u (cid:1)(cid:12)(cid:12) t ≈ u ≈ . (10)In the center-of-mass (CM) frame, we can use the crossed symmetry between the u - and t -channels and just use two times of the forward angle, t -channel contribution for the sumof the forward ( t -channel) and backward ( u -channel) angle contributions | M → | ≈ − (12 πα s ) su/t (cid:12)(cid:12) t ≈ ≈ (12 πα s ) s ( q T + m D ) (cid:12)(cid:12)(cid:12)(cid:12) q ≈ , (11)where q T is the transverse (with respect to p ) component of q = p − p . Because small q T could also come from large | q | through the u -channel, it is important to note thatwhen using Eq. (11) to calculate the collisional integral, we only pick up the near forwardscattering (around t = q ≈
0) to avoid double counting.For the gg → ggg process, we will take the approximation that the bremsstrahlunggluon is very soft (zero rapidity limit) and √ s is much bigger than all transverse momenta.Then the exact result of Ref. [43] reduces to the Gunion-Bertsch formula [25], | M → | ≈ X perm(3 , , (12 πα s ) s ( q T + m D ) πα s q T k T [( q T − k T ) + m D ] (cid:12)(cid:12)(cid:12)(cid:12) q ≈ , (12)where we have inserted the regulator m D as in Ref. [44]. Here k T is the transversecomponent of the bremsstrahlung gluon momentum ( p ) and q T is still the transversecomponent of q = p − p . The three final state gluons are identical particles. Thus, thereare 3! permutations of ( p , p , p ), each gives the same contribution. As explained abovein the 22 case, we need to be careful about using the q T variable. Small q T could meaneither the forward ( t ≈ u ≈
0) scattering. In the convention adoptedfor Eq. (12), one can only pick up the near forward scattering (around t = q ≈
0) butnot the backward scattering otherwise double counting will happen. Our | M → | isderived from the exact result of Ref. [43], where Lorentz invariant Mandelstam variablesare used so there is no this ambiguity, after taking the soft bremsstrahlung limit. Eq.612) is also consistent with the Gunion-Bertsch formula [25] as explicitly demonstratedin App. A. Effectively, the above treatment of collisional integrals leads to a factor 2difference in the gg → ggg contribution to η from that of XG [45] (ours is one half ofXG’s).Naively the gg → gg collision rate is ∝ R dq T | M → | = O ( α s ) and the gg → ggg rate is ∝ R dq T dk T | M → | = O ( α s ) (as will be discussed below, k T has an O ( α s )infrared (IR) cut-off). Thus, the gg → gg process seems more important than gg ↔ ggg .However, this is incorrect. In gg → gg , the amplitude is the largest in the forward andbackward scatterings. But there is no contribution to η in these cases since there is nomomentum redistribution. Mathematically, we have the additional suppression factor[ B ij ( p ) + B ij ( p ) − B ij ( p ) − B ij ( p )] ≃ O ( q T ) in Eq. (7), while no similar suppressionin gg ↔ ggg . Thus, the gg → gg collision rate is proportional to R dq T | M → | q T = O ( α s log α s ), which is of the same order as O ( α s ) of gg ↔ ggg , up to a logarithm [22, 28].This power counting can be used to argue that other processes such as ggg → ggg and gg → gggg (called 33 and 24 processes) are higher order under the assumption that themost important contribution to η comes from the configurations with at most two hardgluons in the initial or the final states. Under this momentum configuration, one observesin Eq. (12) that adding a soft gluon to the 22 process yields a factorizable form for the23 matrix element squared. Schematically, | M | | M | ≃ O ( α s p − T ) , (13)where p T denotes the small momentum scale with p T ≃ O ( q T ) ≃ O ( k T ). Analogously,adding a soft gluon to the 23 process yields | M | | M | ≃ O ( α s p − T ) . (14)Thus, the 33(24) collision rate is smaller than that of 23 by a factor of R dp T | M | / | M | = O ( α s log α s ). This argument can be generalized to other pro-cesses as well. Thus, 22 and 23 are the only processes in the LO under this assumption.The phase space of the 3-gluon state plays an important role in the collisional integralin Eq. (7) for gg ↔ ggg , which is controlled by the delta-functions for energy-momentumconservation. Since we use the Gunion-Bertsch formula, Eq. (12), which is valid for softgluon bremsstrahlung, it is consistent to apply the same condition for energy-momentumconfiguration of the 3-gluon state. This can be done by neglecting the recoil effect due to7he soft gluon bremsstrahlung, i.e. neglecting the momentum of the soft gluon inside thedelta-functions as is done in App. A. Therefore, the phase space for the two near collineargluons in 3-gluon state is 2-body-like. Additionally the exact phase space is 3-body-like ifthe momentum of the soft gluon is kept and treated in equal footing as the other gluonsin the delta-functions. We will see that using the 3-body-like or 2-body-like phase spacemakes a significant difference in the shear viscosity. III. LEADING-LOG RESULT
In the leading-log (LL) approximation, one just needs to focus on the small q T contri-bution from the gg → gg process. After performing the small q T expansion to Eq. (7),we obtain ( g = 4 πα s ) η LL ≃ . T g ln(1 /g ) , (15)which agrees with that of [28] very well. Using the entropy density for non-interactinggluons, s = N g π T , we obtain η LL s ≃ . g ln(1 /g ) . (16)This will be used to check our numerical result later. In contrast, we take the Boltzmanngas approximation ( f p = e − E p /T ) used by XG, we get η LL ≃ . T g − ln − (1 /g ) and s = N g π T , which would yield η LL /s ≃ . g − ln − (1 /g ). Thus, the error from takingthe Boltzmann gas approximation for the LL result of η/s is ∼ ∼
70% comesfrom η and ∼
10% comes from s . This suggests that the quantum nature of gluons couldplay an important role on transport coefficients, even though they might not be importantfor thermodynamic quantities. In weak coupling regime, e.g. α s = 10 − , the XG result in[21] gives η /s ≈ . × while the LL result gives η /s ≈ × , which shows a factor4 difference. But the difference from the LL result can be narrowed in Israel-Stewarttheory [46]. IV. TREATMENT OF gg ↔ ggg As mentioned above, both gg → gg and gg ↔ ggg are needed to compute η to theleading order ( O ( α − s )). For the treatment of the 23 process, we consider three cases, (a)8ith the 3-body-like phase space for three gluons and with the LPM effect as the cutofffor the soft gluon; (b) with the 3-body-like phase space and but with m D as the regulatorfor the soft gluon; (c) with the 2-body-like phase space and with m D as the regulator forthe soft gluon.In case (a), the scale of the k T cut-off is set by the Landau-Pomeranchuk-Migdal (LPM)effect, as in Refs. [21] and [22]. Ref. [47] gives an intuitive explanation of the LPM effect:for the bremsstrahlung gluon with transverse momentum | k T | , the mother gluon has atransverse momentum uncertainty ∼ | k T | and a size uncertainty ∼ / | k T | . It takes thebremsstrahlung gluon the formation time t ∼ / | k T | v T ∼ E k / | k T | to fly far enough fromthe mother gluon to be resolved as a radiation. But if the formation time is longer thanthe mean free path l mfp ≈ O ( α − s ), then the radiation is incomplete and it would beresolved as gg → gg instead of gg → ggg . Thus, the resolution scale is set by t ≤ l mfp .This yields the condition | k T | ≥ E k /l mfp ≈ O ( α s ) which is confirmed through carefullyderivations in Ref. [48].Here the mean free path l mfp is given by the collision rate R ≃ /l mfp which sets thescale of the LPM effect is computed via the detailed balance rate. After integration, theBoltzmann equation of Eq.(2) can be written as dndt = n ( R gain − R loss ) , (17)where we have used n = R d p (2 π ) f p . Then the collision rate is the detailed balance rate inthermal equilibrium, R ≡ R equil.gain = R + R + R , (18)where R = N g n Z Y i =1 d p i (2 π ) E i | M | (2 π ) δ ( p + p − p − p ) × f f (cid:0) f (cid:1) (cid:0) f (cid:1) ,R = N g n Z Y i =1 d p i (2 π ) E i | M | (2 π ) δ ( p + p − p − p − p ) × f f (cid:0) f (cid:1) (cid:0) f (cid:1) (cid:0) f (cid:1) ,R = 32 R . (19)Note that our definition of R is the same as that of XG. The phase space for threegluons is 3-body-like in R . And , as mentioned above, only near forward scattering9 α −3 −2 −1 R / T −6 −5 −4 −3 −2 −1
10 1
22, BE +2.5R
23, BE, LPM cutoff R s α
22, Boltzmann, 3 s α
23, Boltzmann, LPM cutoff 3
FIG. 1: (color online) R and R of Eq.(19) shown as functions of α s for BE and Boltzmanngas. (around t = q ≈
0) is included to avoid double counting (see App. A), which gives anadditional factor 1 / R self-consistently since R also depends on R . Our R and R , together with the results for the Boltzmanngas approximation ( f p = e − E p /T , 1 + f p → R , which usesBose-Einstein (BE) statistics, is close to the Boltzmann gas result. Our R , however, getsan enhancement for α s . .
04 from the enhancement factor (1 + f ) which is inverselyproportional to the soft gluon’s bremsstrahlung energy. This enhancement in R makesthe gg ↔ ggg contribution to η smaller in the BE case than the Boltzmann gas. Theenhancement disappears at higher α s where R/T , and hence the IR cut-off, becomesbigger.In case (b) and (c) , we introduce an IR cut-off m D by replacing the 1 /k T factor in Eq.(12) with 1 / ( k T + m D ). Thus, m D not only screens the intermediate states but also theexternal states. This is motivated by demanding the optical theorem to be valid in themedium, even though it need not be the case when the system exchanges particles froma thermal bath. Thus, if the propagator in the loop is screened, then the bremsstrahlunggluon is also screened. This very naive treatment gives | k T | & m D = O ( α s ), which isconsistent with the first treatment in the α s counting.10 . NUMERICAL RESULTS AND DISCUSSIONS We show in Fig. 2 the comparison between η computed with gg → gg alone (denoted as η ) and η computed with gg → gg and gg ↔ ggg (denoted as η ) [49]. In computingthe 23 contribution in case (a) and (b) with the 3-body-like phase space for three gluons,we use different treatments of k T cut-offs: in case (a) we use R = R + 2 . R as thecut-off, where R is self-consistently determined (the blue dashed line in Fig. 1), whilein case (b) we use m D as the regulator. For these two cases we find that adding gg ↔ ggg reduces η by ∼
30% at α s = 10 − where the contribution from gg ↔ ggg is about 1/2of that from gg ↔ gg . The correction is the largest, ∼ α s = 0 .
1. This meansthe gg ↔ ggg contribution is about 3 times that of gg → gg . The behavior shown hereis different from that of XG which shows η /η ∼ / ∼ . gg ↔ ggg contribution is about 7 times as large as gg → gg , for a wide range of α s ( α s = 10 − − . gg ↔ ggg process and the BE statistics versusthe Boltzmann gas approximation used. But we do see the dominance of gg ↔ ggg over gg → gg when α s & × − , as asserted by XG.For case (c) with the 2-body-like phase space for three gluons the effect of the 23process is about 10-30%, which is close to AMY’s result in the whole range of α s . Sinceour result changes dramatically after imposing the soft bremsstrahlung approximation, itmeans this approximation is not as good as previously expected. Thus, it is important togo beyond this approximation to obtain an accurate η .In Fig. 3, η/s as a function of α s is shown for different cases: the LL result η LL /s ofEq. (16), η /s , and η /s for two different k T cut-offs for the 3-body-like phase spaceand that for the 2-body-like phase space. When α s →
0, all these curves should convergeto the LL result. But at α s = 10 − , we have ln(1 /g ) = 2 .
2, which is not large enough todominate the contribution. This is the reason for the deviations of the numerical resultsfrom the LL one in the current range of α s . However, the agreement between the η LL /s and η /s is a good check to our numerical calculations which are carried out by theMonte Carlo method for multi-dimensional integrations. The power of α s dependence ofthese curves are close to ( −
2) as expected. At α s = 0 .
1, with both k T cut-offs for the 3-body-like phase space, the full result η /s ≃ . . α s = 0 . η /s ≃ .
22 and 0.15, respectively, which are larger11 α −3 −2 −1 η / + η +2.5R D D FIG. 2: (color online) η /η shown as a function of α s for the 3-body-like and 2-body-like phase space (PS) of three gluons. There are two different treatments of the cut-off of thebremsstrahlung gluon momentum k T for the 3-body-like phase space. than 0.13 and 0.076 obtained by XG. It is also interesting to note the good agreementusing two different cut-offs for the bremsstrahlung gluon momentum. For the 2-body-likephase space the correction from the 23 process is small and η /s ≈ (70% ∼ η /s ,which is close to AMY’s result.In summary, we have calculated the shear viscosity over entropy density η/s of a gluonplasma in kinetic theory. Due to the suppressed contribution to η in the gg → gg forwardscattering, the gluon bremsstrahlung gg ↔ ggg process also contributes at the same order( O ( α − s )) in perturbative QCD. We find that the gg ↔ ggg contribution becomes moreimportant to η than gg → gg for α s & × − for the 3-body-like phase space for thethree-gluons state. At α s = 0 . η/s ≃ . η/s is about 2 timesas large as that of Xu and Greiner for α s & .
1, largely due to the factor 2 difference incollisional integrals for the gg ↔ ggg process and the Bose-Einstein statistics versus theBoltzmann gas approximation used. We have observed that using m D as the regulator fortransverse momentum of the soft bremsstrahlung gluon agrees well with that using therate as the cut-off for the LPM effect in η for the current range of α s . In dealing with the12 α −3 −2 −1 / s η −1
10 110 +2.5R D D FIG. 3: (color online) η/s versus α s for (a) the leading-log result in Eq. (15), (b) the result ofthe 22 process only, the full result with 22+23 processes for the 3-body-like phase space (PS) ofthree gluons where the k T cut-off is set by (c) m D or (d) the LPM effect, and (e) the full resultwith 22+23 processes for the 2-body-like phase space (PS).
23 process it is consistent to implement the soft gluon condition in the energy-momentumconfiguration of the three-gluons state that there is one soft gluon, which results in the 2-body-like phase space for the three-gluons state, since we use the Gunion-Bertsch formulafor the 23 matrix element which is valid only for soft gluon bremsstrahlung. In thiscase we obtain results close to AMY’s. To test which is the correct description for thephase space of three gluons in the 23 process, or in other words, to test if the Gunion-Bertsch formula is still valid for general 3-body-like momentum configurations, a furtherand comprehensive study with the exact matrix element is needed.Acknowledgement: JWC and QW thank Zhe Xu for clarifying the definition of colli-sional rates used in the LPM effect and for many helpful discussions. The authors alsothank Carsten Greiner and Guy Moore for helpful comments. JWC and KO are supportedby the NSC and NCTS of R.O.C.. QW is supported in part by the ’100 talents’ projectof Chinese Academy of Sciences (CAS) and by the National Natural Science Foundationof China (NSFC) under the grants 10675109 and 10735040. HD is supported by NSFCunder the grant 10847149. 13
PPENDIX A: THE CROSS SECTION FOR 23 FROM GUNION-BERTSCHFORMULA
In the center-of-mass frame of 1 and 2, the cross section is written by, σ = 12 s Z Y i =3 d k i (2 π ) E i | M | (2 π ) δ ( k + k − k − k − k )= 27 π α s Z d k q T + m D ) δ ( E + E − E − E ) Z d k T dy q T k T [( q T − k T ) + m D ]= 27 π α s Z d q T q T + m D ) Z d k T dy q T k T [( q T − k T ) + m D ] . (A1)Since we use the Gunion-Bertsch formula for soft gluon bremsstrahlung, we assume the5th gluon is soft, so we made the approximation in the second equality of Eq. (A1), δ ( k + k − k − k − k ) ≈ δ ( k + k − k − k ) , (A2)which means the phase space is dominated by the 22 process. We also used E = E = E = E = √ s/ Z d k δ ( E + E − E − E ) = 12 Z d qδ ( E − E )= 12 Z d q T dq z δ ( E − q ( E + q z ) + q T )= Z d q T E p E − q T ≈ Z d q T (A3)where k = k + q . Note that a factor of 2 is given from the two roots for q z in theequation E = p ( E + q z ) + q T , i.e. q z = − E ± p E − q T which correspond to forwardand backward solution q z = −√ s, t = − s, q T = 0. Eq. (A1) is 2 times as largeas that derived in Ref. [44]. One has to choose the forward scattering and get the factor1/2, Z forward d k δ ( E + E − E − E ) = 12 Z d q T . (A4)Then the differential cross section from Eq. (A1) becomes dσ d q T d k T dy = 272 π α s q T + m D ) q T k T [( q T − k T ) + m D ] , (A5)which reproduces the result in [44]. [1] M. Gyulassy and L. McLerran, Nucl. Phys. A , 30 (2005) [arXiv:nucl-th/0405013].
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20% (4 πη/s = 27 ± α s . This error, although significant, is still consistent with an O ( α s ) error.) error.