Color Randomization of Fast Gluon-Gluon Pairs in the Quark-Gluon Plasma

Abstract

We study the color randomization of two-gluon states produced after splitting of a primary fast gluon in the quark-gluon plasma. We find that for the LHC conditions the color randomization of the gg pairs is rather slow. At jet energies E = 100 and 500 GeV, for typical jet path length in the plasma in central Pb+Pb collisions, the SU(3)-multiplet averaged color Casimir operator of the gg pair differs considerably from its value 2N_c for a fully randomized gg state. Our calculations of the energy dependence for generation of the nearly collinear decuplet gg states, that can lead to the baryon jet fragmentation, show that the contribution of the anomalous decuplet color states to the baryon production should become small at p_T ≳ 10 GeV.

APPENDIX

The contribution to the two-gluon color wave function 〈ab|Ψ〉 of a given irreducible SU(3) multiplet R in the Clebsch-Gordan decomposition (4) can be written as P[R$]_{{cd}}^{{ab}}$〈cd|Ψ〉, where P[R] is the projector operator for the multiplet R given by the general quantum mechanical formula (14). The projectors onto the multiplets 1, 8_A, 8_S, 27, 10, and $\overline {10}$ can be written in terms of the Kronecker deltas, antisymmetric tensor f_abc and symmetric tensor d_abc as

$$P[1]_{{cd}}^{{ab}} = \frac{1}{8}{{\delta }_{{ab}}}{{\delta }_{{cd}}},$$

((40))

$$P[{{8}_{A}}]_{{cd}}^{{ab}} = \frac{1}{3}{{f}_{{abk}}}{{f}_{{kcd}}},$$

((41))

$$P[{{8}_{S}}]_{{cd}}^{{ab}} = \frac{3}{5}{{d}_{{abk}}}{{d}_{{kcd}}},$$

((42))

$$P[27]_{{cd}}^{{ab}} = \frac{1}{2}({{\delta }_{{ac}}}{{\delta }_{{bd}}} + {{\delta }_{{ad}}}{{\delta }_{{bc}}}) - \frac{1}{8}{{\delta }_{{ab}}}{{\delta }_{{cd}}} - \frac{3}{5}{{d}_{{abk}}}{{d}_{{kcd}}},$$

((43))

$$P[10]_{{cd}}^{{ab}} = \frac{1}{4}({{\delta }_{{ac}}}{{\delta }_{{bd}}} - {{\delta }_{{ad}}}{{\delta }_{{bc}}}) - \frac{1}{6}{{f}_{{abk}}}{{f}_{{kcd}}} + \frac{i}{2}Y_{{cd}}^{{ab}},$$

((44))

$$P[\overline {10} ]_{{cd}}^{{ab}} = \frac{1}{4}({{\delta }_{{ac}}}{{\delta }_{{bd}}} - {{\delta }_{{ad}}}{{\delta }_{{bc}}}) - \frac{1}{6}{{f}_{{abk}}}{{f}_{{kcd}}} - \frac{i}{2}Y_{{cd}}^{{ab}},$$

((45))

where

$$Y_{{cd}}^{{ab}} = - \frac{1}{2}({{d}_{{ack}}}{{f}_{{kbd}}} + {{f}_{{ack}}}{{f}_{{kbd}}}).$$

((46))

The two-gluon color wave function for the states 1, 8_S, 27 are symmetric in permutations of gluon color indexes a ↔ b and c ↔ d, and the states 8_A, 10, $\overline {10}$ are antisymmetric. The contribution of the first three terms in the formulas for the decuplet projectors is clearly antisymmetric, the fact that the term $Y_{{cd}}^{{bc}}$ is also antisymmetric for a ↔ b and c ↔ d is evident from the identity

$${{d}_{{ack}}}{{f}_{{kbd}}} + {{f}_{{ack}}}{{d}_{{kbd}}} = {{d}_{{cbk}}}{{f}_{{kda}}} + {{f}_{{cbk}}}{{d}_{{kda}}}.$$

((47))

Calculations of the projectors for 1, 8_S, 27, and 8_A multiplets are trivial, but for 10 and $\overline {10}$ multiplets they are more complicated. The projectors P [10] and P[$\overline {10}$] can be obtained after straightforward (somewhat tedious) calculations with the help of the standard formula (14) using for the decuplet (antidecuplet) states the symmetric spinor tensors Ψ^ijk (Ψ_ijk). For the two-gluon state these tensors can be built using the spinor form of the gluon wave function (g_a$)_{k}^{i}$ = $\frac{1}{{\sqrt 2 }}$(λ_a$)_{k}^{i}$.

An important fact for our calculations is that the projectors are proportional to the four-gluon color wave functions of the color singlets |$R\bar {R}$〉 built from R and $\bar {R}$ multiplets (16). The fact that the wave function given by (16) describes a color singlet can be checked by calculating the expectation value

$$\langle R\bar {R}{\text{|}}{{T}^{\alpha }}{\text{|}}R\bar {R}\rangle $$

((48))

of the total color generator T^α = $\sum\nolimits_{i = 1}^4 {T_{i}^{\alpha }}$ for four gluons, which should vanish for color singlet states. One can easily show that this is true. Indeed, say, for the contribution from i = 1 we have

$$\langle R\bar {R}{\text{|}}T_{1}^{\alpha }{\text{|}}R\bar {R}\rangle \propto (P[R]_{{cd}}^{{a'b}}){\text{*}}{{f}_{{\alpha a'a}}}P[R]_{{cd}}^{{ab}}.$$

((49))

Since (P[R$]_{{cd}}^{{a' b}}$)* = P[R$]_{{a'b}}^{{cd}}$ and P[R$]_{{cd}}^{{ab}}$P[R$]_{{a'b}}^{{cd}}$) = P[R$]_{{a'b}}^{{ab}}$ ∝ δ_aa' one can see that the left-hand side of (49) ∝ f_aa = 0. The fact that the projector operator (14) is proportional to the color singlet wave function |$R\bar {R}$〉 is not surprising, because the last factor on the right-hand side of (14) is proportional to the wave function of the complex conjugate state 〈cd|$\bar {R}\bar {\nu }$〉 (where the component $\bar {\nu }$ has the “magnetic” quantum numbers opposite to that for ν) with the phase factor similar to that in the Clebsch-Gordan sum over the internal quantum number ν for the color singlet state |$R\bar {R}$〉 [57, 58] built from the states |Rν〉 and |$\bar {R}\bar {\nu }$〉.

The crossing operator U_ts from the s-channel basis to the t-channel one can be calculated with the help of the above formulas for the s-channel projectors and similar formulas for the t-channel basis (that can be obtained by permuting b ↔ c). However, the crossing operation involves also the mixed states |8_A8_S〉 and |8_S8_A〉. It is convenient to use the linear combinations (26), and take the wave functions for the components |8_A8_S〉 and |8_S8_A〉 in the s-channel basis in the form

$$\begin{gathered} \langle abcd{\text{|}}{{8}_{A}}{{8}_{S}}\rangle = \frac{1}{{\sqrt {40} }}{{f}_{{abk}}}{{d}_{{kcd}}}, \\ \langle abcd{\text{|}}{{8}_{S}}{{8}_{A}}\rangle \rangle = \frac{1}{{\sqrt {40} }}{{d}_{{abk}}}{{f}_{{kcd}}}. \\ \end{gathered} $$

((50))

Similar formulas for the t-channel basis are obtained by interchanging b ↔ c. A straightforward calculation gives

$${{\left( {\begin{array}{*{20}{c}} {{\text{|}}11\rangle } \\ {{\text{|}}{{8}_{A}}{{8}_{A}}\rangle } \\ {{\text{|}}{{8}_{S}}{{8}_{S}}\rangle } \\ {{\text{|}}2727\rangle } \\ {{\text{|}}10\overline {10} \rangle } \\ {{\text{|}}\overline {10} 10\rangle } \\ {{\text{|}}{{{({{8}_{A}}{{8}_{S}})}}_{ + }}\rangle } \\ {{\text{|}}{{{({{8}_{A}}{{8}_{S}})}}_{ - }}\rangle } \end{array}} \right)}_{t}} = \left( {\begin{array}{*{20}{c}} {\frac{1}{8}}&{\frac{1}{{\sqrt 8 }}}&{\frac{1}{{\sqrt 8 }}}&{\frac{{3\sqrt 3 }}{8}}&{\frac{{\sqrt 5 }}{{4\sqrt 2 }}}&{\frac{{\sqrt 5 }}{{4\sqrt 2 }}}&0&0 \\ {\frac{1}{{\sqrt 8 }}}&{\frac{1}{2}}&{\frac{1}{2}}&{ - \frac{{\sqrt 3 }}{{2\sqrt 2 }}}&0&0&0&0 \\ {\frac{1}{{\sqrt 8 }}}&{\frac{1}{2}}&{ - \frac{3}{{10}}}&{\frac{{3\sqrt 3 }}{{10\sqrt 2 }}}&{ - \frac{1}{{\sqrt 5 }}}&{ - \frac{1}{{\sqrt 5 }}}&0&0 \\ {\frac{{3\sqrt 3 }}{8}}&{ - \frac{{\sqrt 3 }}{{2\sqrt 2 }}}&{\frac{{3\sqrt 3 }}{{10\sqrt 2 }}}&{\frac{7}{{40}}}&{ - \frac{{\sqrt 3 }}{{4\sqrt {10} }}}&{ - \frac{{\sqrt 3 }}{{4\sqrt {10} }}}&0&0 \\ {\frac{{\sqrt 5 }}{{4\sqrt 2 }}}&0&{ - \frac{1}{{\sqrt 5 }}}&{ - \frac{{\sqrt 3 }}{{4\sqrt {10} }}}&{\frac{1}{4}}&{\frac{1}{4}}&{ - \frac{1}{{\sqrt 2 }}}&0 \\ {\frac{{\sqrt 5 }}{{4\sqrt 2 }}}&0&{ - \frac{1}{{\sqrt 5 }}}&{ - \frac{{\sqrt 3 }}{{4\sqrt {10} }}}&{\frac{1}{4}}&{\frac{1}{4}}&{\frac{1}{{\sqrt 2 }}}&0 \\ 0&0&0&0&{ - \frac{1}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}}&0&0 \\ 0&0&0&0&0&0&0&{ - 1} \end{array}} \right) \times {{\left( {\begin{array}{*{20}{c}} {{\text{|}}11\rangle } \\ {{\text{|}}{{8}_{A}}{{8}_{A}}\rangle } \\ {{\text{|}}{{8}_{S}}{{8}_{S}}\rangle } \\ {{\text{|}}2727\rangle } \\ {{\text{|}}10\overline {10} \rangle } \\ {{\text{|}}\overline {10} 10\rangle } \\ {{\text{|}}{{{({{8}_{A}}{{8}_{S}})}}_{ + }}\rangle } \\ {{\text{|}}{{{({{8}_{A}}{{8}_{S}})}}_{ - }}\rangle } \end{array}} \right)}_{s}}$$

((51))

As one can see the crossing matrix is real and symmetrical.

A straightforward calculation of the 6 × 6 diffraction matrix in the s-channel basis with the help of the above formulas for the projectors gives (the order of states is the same as in (10))

$$\begin{gathered} \hat {\sigma }(\rho ) \\ = {{\sigma }_{8}}(\rho ) \times \left( {\begin{array}{*{20}{c}} 2&{ - \frac{1}{{\sqrt 2 }}}&0&0&0&0 \\ { - \frac{1}{{\sqrt 2 }}}&{\frac{3}{2}}&{ - \frac{1}{2}}&{ - \frac{1}{{\sqrt 6 }}}&0&0 \\ 0&{ - \frac{1}{2}}&{\frac{3}{2}}&0&{ - \frac{1}{{\sqrt 5 }}}&{ - \frac{1}{{\sqrt 5 }}} \\ 0&{ - \frac{1}{{\sqrt 6 }}}&0&{\frac{3}{2}}&{ - \frac{2}{{\sqrt {30} }}}&{ - \frac{2}{{\sqrt {30} }}} \\ 0&0&{ - \frac{1}{{\sqrt 5 }}}&{ - \frac{2}{{\sqrt {30} }}}&1&0 \\ 0&0&{ - \frac{1}{{\sqrt 5 }}}&{ - \frac{2}{{\sqrt {30} }}}&0&1 \end{array}} \right), \\ \end{gathered} $$

((52))

where σ₈(ρ) is the dipole cross section for a color singlet two-gluon system of the size ρ.