GLV - formalism Chen Lin Institute of Particle Physics CCNU, Wuhan - PowerPoint PPT Presentation

GLV - formalism Chen Lin Institute of Particle Physics CCNU, Wuhan October 20, 2016(9421 conference) 1 / 56

Outline 1 Introduction 2 The GLV Model 3 Diagrammatic approach 4 Numerical Results 5 Recursive approach 6 Summary 2 / 56

Introduction In an ultra relativistic heavy ion collision, partons produced from hard collision processes travel through a dense matter previously predicted as Quark-Gluon Plasma (QGP) and losses energy in the surrounding medium. Although collisional energy loss were predicted to be moderate ( dE coll / dx ≪ 1 GeV/fm) 1 , radiative energy loss were expected to be significantly large ( dE rad / dx ≫ few GeV/fm) 2 This radiative energy loss phenomena is called ”Jet Quenching”, and it is one of the signatures for QGP production in RHI-Collisions observed in RHIC(BNL) and LHC(CERN). 1 M.H.Thoma and M. Gyulassy, Nucl. Phys. B 351 (1991) 491 2 J.F. Gunion and G. Bertsch, Phys. Rev. D 25 (1982) 746 3 / 56

Introduction Jet energy loss schemes available in the market 3 : • Modeling of medium • Static scattering centers (BDMPS, Zakharov, GLV, ASW) • Thermally equilibrated, perturbative medium (AMY) • Nuclear medium with short correlation length (Higher Twist) • Resummation schemes • Sum over all possible soft interactions (BDMPS, AMY) • Path integral of hard parton propagation (Zakharov, ASW) • Opacity expansion (GLV) • Evolution scheme (multiple emissions) • Poisson ansatz(BDMPS, GLV, ASW) • Rate equations (AMY) • Modified DGLAP equations (Higher Twist) 3 G.Y. Qin, pres. Jet Quenching in Nuclear Collisions (2010) 4 / 56

Basic idea (radiative energy loss) Radiative energy loss is given as sum over all radiated gluon energy 4 � d ω dN ∆ E rad = d ω ω θ ( E − ω ) (1) The gluon number distribution is proportional to the phase space integral on the radiation amplitude squared dN = 1 � |R rad | 2 (2) d 2 � d R k ⊥ dyd ω color Where it can be extracted from the scattering amplitude of every diagram M rad = M ela i R rad (3) 4 photon radiation is ignored since QED coupling is much less giving rise to negligible cross-section. 5 / 56

G yulassy, L ´ evai, V itev model One begins with the Static Color Screening Yukawa Potential from the GW 5 6 model: 4 πα s V n = 2 πδ ( q 0 q n ) e − i � q n · � x n T an ( R ) ⊗ T an ( n ) , n ) V ( � V ( � q n ) = (4) 2 + µ 2 � q n For small transverse momentum transfer, the elastic cross-section between the jet and the target parton is: q ⊥ ) | 2 d σ el = C R C 2 ( T ) | v ( � (5) d 2 � (2 π ) 2 q ⊥ d A Where the color bookkeeping techniques are described by: Tr ( T a ( R ) T b ( R )) = δ ab C R D R / D A (6) N 2 = c − 1 (7) D A Tr ( T a ( i ) T b ( j )) = δ ab δ ij C 2 ( i ) D i / D A (8) Tr ( T a ( R )) = 0 (9) 5 M. Gyulassy, X. N. Wang, Nucl. Phys. B 420 583 (1994) 6 X. N. Wang, M. Gyulassy, M. Pl¨ umer, Phys. Rev. D 51 3436 (1995) 6 / 56

Feynman rules (example) From the given potential above, one has the Hamiltonian: � � H I ( t ) = dt L = dt ( I − V ) N � � x , t ) T a ( R ) ˆ d 3 � x i ) T a ( i ) φ † ( � x V ( � x − � D ( t ) φ ( � x , t ) (10) i =1 D ( t ) = i ← → ∂ t , A ← → where ˆ ∂ t B = A ( ∂ t B ) − ( ∂ t A ) B . Consider a simple scattering diagram: �� T � �� � � � � p ′ i M = � ( − i ) T exp dt H I ( t ) � p � � � � − T u ( p ′ ) u ( p ) = ( − i )( E p + E p ′ ) × ¯ � x i ) · e i ( p ′ − p ) · � d 4 x � x i T a ( i ) ⊗ T a ( R ) × V ( � x − � (11) i 7 / 56

Feynman rules (cont.) Then one has: u ( p ′ ) u ( p ) i M = ( − i )( E p + E p ′ ) × ¯ � ˜ q ) · e − i � q · � x i T a ( i ) ⊗ T a ( R ) × (2 π ) δ ( E p ′ − E p ) V ( � (12) i Separating the potential and the Dirac spinors, one can see that the Feynman rule for scattering vertex is given as ( − i )( E p + E p ′ ). One can derive the following Feynman rules from the given potential accordingly: − i (2 p 0 − q 0 ) Quark scattering vertex = (13) i / ( p 2 + i ε ) Quark propagator = (14) − ig µν / ( k 2 + i ε ) Gluon propagator = (15) ig s (2 p + k ) µ · ǫ µ T c Emission vertex = (16) 8 / 56

Assumptions and Approximations • Targets are distributed with density: N θ (∆ z j ) ∆ zj � L e ( N ) e − ¯ ρ ( z 1 , · · · , z n ) = (17) Le ( N ) j =1 • The opacity defined by: n = L λ = N σ el ¯ (18) A ⊥ • Energy of jet is high compare to potential screening scale: E + ≈ 2 E ≫ µ (19) • distance between source and scattering center are larger than interaction range: z i − z 0 ≫ 1 (20) µ • One defines the jet with momentum p : M 0 = ie ip · x 0 J ( p ) × 1 (21) 9 / 56

Light-cone kinematics One can define the following light-cone coordinates: � � � k 2 2 ω , � ⊥ k = 2 ω, k ⊥ (22) � � � k ⊥ · � ǫ ⊥ ǫ ( k ) = 0 , ,� ǫ ⊥ (23) ω � 2( E − ω ) , ( � Q ⊥ − � � k ⊥ ) 2 2( E − ω ) , ( � Q ⊥ − � k ⊥ ) 2 p = (24) � � � k 2 ω E − ω + 1) , � ⊥ = 0 , 2 ω ( (25) Q Q ⊥ and their corresponding dot products. We can use the assumption: ( E ≫ ω ≫ Q ) to simplify our calculation. 10 / 56

Gluon tree matrices It is best to work out the matrices below to simplify the calculations follow: Γ α 0 γ ( k ; q 1 ) · ǫ γ ( k ) Γ α ( k ; q 1 ) = Γ 1 = (2 p + k − q 1 ) α Γ α ( k ; q 1 ) Λ 1 = Γ 1 ( ig s t a ) T b (1) ǫ ⊥ · ( � = − 2 g s [2 E � k ⊥ − � q 1 ⊥ ) + ω ( � ǫ ⊥ · � q 1 ⊥ )][ c , b ] T b (1) Γ α 0 µ ( k − q 2 ; q 1 ) g µν Γ ν 0 γ ( k ; q 2 ) · ǫ γ ( k ) Γ α ( k ; q 1 ; q 2 ) = Γ 12 = (2 p + k − q 1 − q 2 ) α Γ α ( k ; q 1 ; q 2 ) Λ 12 = Γ 12 ( ig s t a ) T a 1 (1) T a 2 (2) ǫ ⊥ · ( � = − ig s 4 ω [2 E � k ⊥ − � q 1 ⊥ − � q 2 ⊥ ) + ω� ǫ ⊥ · ( � q 1 ⊥ + 2 � q 2 ⊥ )] × [[ c , a 2 ] , a 1 ] T a 1 (1) T a 2 (2) 11 / 56

Diagrammatic approach � Tr | t 0 · R (0) + t 1 · R (1) + t 2 · R (2) + · · · | 2 dN ∝ (26) � n / 2 where t n ∝ T n � C 2 ( T ) d T a , with Tr ( T odd ) = 0, and Tr ( T even ) = . For a a d A opacity order = n / 2, one has the following opacity expansions: � dN (0) Tr |R (0) | 2 ∝ (27) � Tr |R (0) + t 1 · R (1) + t 2 · R (2) | 2 dN (1) ∝ � C 2 ( T ) d T � � dN (0) + Tr [ R (1)2 + 2 Re ( R (0) † R (2) )] = × (28) d A � Tr |R (0) + t 1 · R (1) + t 2 · R (2) + t 3 · R (3) + t 4 · R (4) | 2 dN (2) ∝ � C 2 ( T ) d T � 2 � dN (1) + Tr [ R (2)2 + 2 Re ( R (1) † R (3) + R (0) † R (4) )] = × (29) d A 12 / 56

Self-Quenching The scattering matrix: M (0) iJ ( p + k ) e i ( p + k ) · x 0 ( ig s )(2 p + k ) µ ǫ µ i ∆( p + k ) c = rad ǫ ⊥ · � iJ ( p + k ) e i ( p + k ) · x 0 ( − 2 g s ) E − ω � k ⊥ = c � E k 2 ⊥ M (0) el i R (0) = (30) rad The radiation amplitude squared: 1 1 |R (0) Tr |R (0) � � rad | 2 rad | 2 = d R d R i j � 2 1 � E − ω = 16 πα s C R (31) � E k 2 ⊥ 13 / 56

Absorption (optional) In the QGP heat bath, the jet parton can either emit or absorb a gluon, one will take into account the Bose enhancement and absorption factor k | ) = ( e | � k | / T − 1) − 1 in the phase space integration. 7 N ( | � � d Φ = d 3 | � 1 + N ( | � : if k 0 = | � k | 1 k | ) k | for emission (32) : if k 0 = | � N ( | � (2 π ) 3 2 | � k | ) k | for absorption k | dN = 1 Tr |R (0) | 2 d Φ (33) d R Then: � d | � � 2 dN (0) 2 C R α s k ⊥ | � E − ω 1 = | � dyd ω π E k ⊥ | � � (1 + N ( | � k | )) δ ( ω − | � k | ) + N ( | � k | ) δ ( ω + | � × k | ) (34) 7 E. Wang, X.N. Wang, Phys. Rev. Lett. 87 , 142301 (2001) 14 / 56

Divergence and virtual correction Virtual processes: � d | � � E 2 − | � � dN (0) k | 2 dyd ω = − 2 α s C R k ⊥ | 2 [1 + 2 N ] δ ( ω ) (35) | � E 2 π k ⊥ | Then, the gluon spectrum is: dN (0) + dN (0) dN (0) 1 2 = dyd ω dyd ω � d | � �� E − ω � 2 2 α s C R k | (1 + N ) δ ( ω − | � = k | ) | � π E k | � 2 � E − ω N δ ( ω + | � + k | ) E � 2  E 2 − | � � k | 2 − (1 + 2 N ) δ ( ω ) (36)  E 2 15 / 56

Energy Loss Using the gluon number spectrum, one can then calculate the energy loss: dyd ω dN (0) � ∆ E (0) rad = dyd ω ωθ ( E − ω ) (37) � Note that d ωδ ( ω ) ω = 0, which means that the virtual gluon does not contribute to the total energy loss. Then: � d | � 2 α s C R k ⊥ | � ∆ E (0) dx [(1 − x ) 2 θ (1 − x ) = E rad | � π k ⊥ | − 4 xN θ (1 − x ) − (1 + x ) 2 N θ ( x − 1)] (38) 16 / 56

Energy loss (Analysis) We can write the energy loss in three terms: ∆ E (0) rad = ∆ E (0) a rad − ∆ E (0) b rad − ∆ E (0) c (39) rad Note that ∆ E (0) a is the energy loss from emission at T = 0. ∆ E (0) b and rad rad ∆ E (0) c are the energy absorption at finite temperature. looking at the rad ratio of ∆ E (0) b rad / ∆ E (0) a rad = 12 T / E , which means that if E < 12 T , anti-self-quenching happens. E.g. at SPS( T = 150 MeV ), jet with E < 1 . 8 GeV will absorb energy instead of quenching. ∆ E (0) c is negligible rad for E ≫ T and becomes significant when E ≪ T . However, the zeroth order self-quenching calculations over-estimates the energy loss at high-energy collisions and thus will not be use in the future, but demonstrates how to systematically calculate jet-quenching. 17 / 56