CONFINEMENT IN MULTI-PARTON SECTORS OF TWO DIMENSIONAL GAUGE - PowerPoint PPT Presentation

CONFINEMENT IN MULTI-PARTON SECTORS OF TWO DIMENSIONAL GAUGE THEORIES Daniele Dorigoni, Gabriele Veneziano, J W 1 How to calculate masses of particles ? • Lattice • Diagonalize Hamiltonian • Light Cone Discretization • QCD equations: coupled Bethe-Salpeter equations on the LC • Simplifications: large N planar diagrams - single traces • less dimensions • even quantum mechanics (but at N → ∞ ) • supersymmetry 1

2 Planar gauge theory in 1+1 dimensions • The history FT on the light cone – C. Thorn (’77) Warm-up: D=1+1, QCD 2 – ’t Hooft (’74) LargeN fermions in funamental irrep − → no multiparton states. YM+with addjoint matter – Klebanov et al. (’93) matter = fermions or scalars ( = reduced Y M 3 ) SY M 2 – Matsumura et al. (’95) D=4 Wilson and Glazek (’93) Hiller et al. (’98) QCD 4 on the light cone – Brodsky et al. (since ’70) 2

One way: Light Cone Discretization 2.1 ∞ n P + = i =1 p + p + i , i > 0 � � n =2 ∞ n K = i =1 r i , K, r i − natural , � � n =2 Cutoff K = ⇒ partitions { r 1 , r 2 , . . . } = ⇒ states |{ r }� = Tr [ a † ( r 1 ) a † ( r 2 ) ...a † ( r p )] | 0 � (1) ⇒ �{ r }| H |{ r ′ }� = |{ r }� = ⇒ E n 3

Second way: integral equations in the continuum 2.2 • Different cutoff – directly in the continuum H | Φ � = M 2 | Φ � (2) ☛ ✟ | Φ � → Φ n ( x 1 , x 2 , . . . , x n ) ↔ ✡ ✠ ☛ ✟ ☛ ✟ ☛ ✟ ☛ ✟ ✏✏ ② PP ❤❤ ② ✭✭ ② ✭✭ ❤❤ PP ✏✏ = + + ✡ ✠ ✡ ✠ ✡ ✠ ✡ ✠ M 2 Φ n ( x 1 . . . x n ) = A ⊗ Φ n + B ⊗ Φ n − 2 + C ⊗ Φ n +2 (3) 4

• Interpretation: proton is invariant against elementary processes • Fundamental: contain DGLAP and BFKL evolution eqns. • Emission and absorption are present (parton recombination) The cutoff: n ≤ n max (4) n max = 2 ’t Hooft equation – exact for QCD 2 (with fundamental fermions) 5

• EQUATIONS ∞ � [ dx ] δ (1 − x 1 − x 2 − . . . x n )Φ n ( x 1 , x 2 , . . . x n ) Tr [ a † ( x 1 ) a † ( x 2 ) . . . a † ( x n )] | 0 � | Φ � = � n =2 EXAMPLE 1: QCD 2 ( fundamental fermions )  1 1  f ( x ) + λ � 1 dy   M 2 f ( x ) = m 2 x + ( y − x ) 2 [ f ( x ) − f ( y )] 1 − x π 0 f ( x ) = Φ 2 ( x, 1 − x ) 6

EXAMPLE 2: SY M 2 restricted to the two-parton sector There are two coupled equations in the bosonic sector  1 1  φ bb ( x ) + λ φ bb ( x )   M 2 φ bb ( x ) = m 2 x + b � 1 − x 2 x (1 − x ) − 2 λ � 1 ( x + y )(2 − x − y ) [ φ bb ( y ) − φ bb ( x )] dy + λ � 1 1 φ ff ( y ) dy � ( y − x ) 2 � π 0 2 π 0 ( y − x ) x (1 − x ) y (1 − y ) x (1 − x ) 4  1 1    φ ff ( x ) M 2 φ ff ( x ) = m 2 x + f 1 − x − 2 λ � 1 [ φ ff ( y ) − φ ff ( x )] dy + λ � 1 1 φ bb ( y ) dy ( y − x ) 2 � π 0 2 π 0 ( x − y ) y (1 − y ) and the single one in the fermionic sector  m 2 x + m 2    φ bf ( x ) + 2 λ φ bf ( x ) f M 2 φ bf ( x ) = b √ x + x   1 − x π − 2 λ � 1 ( x + y ) [ φ bf ( y ) − φ bf ( x )] dy − λ � 1 1 φ bf ( y ) √ xy dy 2 √ xy ( y − x ) 2 π 0 2 π 0 (1 − y − x ) (5) 7

Example 3: Y M 2 with addjoint fermionc matter - all parton-number sectors M 2 φ n ( x 1 . . . x n ) = m 2 φ n ( x 1 . . . x n ) x 1 + λ 1 � x 1 + x 2 dyφ n ( y, x 1 + x 2 − y, x 3 . . . x n ) ( x 1 + x 2 ) 2 π 0 + λ dy � x 1 + x 2 ( x 1 − y ) 2 { φ n ( x 1 , x 2 , x 3 . . . x n ) 0 π − φ n ( y, x 1 + x 2 − y, x 3 . . . x n ) }   + λ � x 1 � x 1 − y 1 1 dy dzφ n +2 ( y, z, x 1 − y − z, x 2 . . . x n ) ( y + z ) 2 −   ( x 1 − y ) 2 π 0 0     + λ 1 1 πφ n − 2 ( x 1 + x 2 + x 3 , x 4 . . . x n ) ( x 1 + x 2 ) 2 −   ( x 1 − x 3 ) 2   ± cyclic permutations of ( x 1 . . . x n ) 8

3 Coulomb divergences • IR divergencies (logarithmic) couple different multiplicity sectors • Coulomb divergencies (linear), but they cancel within one multiplicity • Can be done independently for each parton multiplicity p A possibility • − → Solve Coulomb problem first, and then successively add radiation Simplified Hamiltonian, SY M 2 reduced from SY M 4 (Dorigoni), keeping only Coulomb terms = λ � ∞ � k dq H quad q 2 Tr[ A † dk k A k ] (6) C 0 0 π = − g 2 � ∞ � p 1 dq � p 2 dq   H quartic q 2 Tr[ A † p 1 B † q 2 Tr( A † p 2 B † dp 1 dp 2 p 2 B p 2 + q A p 1 − q ] + p 1 B p 1 + q A p 2 − q ) C   2 π 0 0 0 (7) 9

4 Two partons | k, K − k � , k = 1 , .., K − 1 (8) � k | H | k ′ � ⇒ | Φ n � ⇒ Φ n ( k ) FT ⇒ Φ n ( d 12 ) (9) 1.0 0.5 0.5 0.8 0.4 0.4 0.6 0.3 0.3 0.4 0.2 0.2 0.2 0.1 0.1 0.0 0.0 0.0 � 100 � 50 0 50 100 � 100 � 50 0 50 100 � 100 � 50 0 50 100 0.5 0.5 0.5 0.4 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0.0 0.0 0.0 � 100 � 50 0 50 100 � 100 � 50 0 50 100 � 50 0 50 100 Figure 1: ρ n ( d 12 ) , p = 2 , K = 200 , n = 1 , 25 , 50 , 100 , 150 , 199 . 10

5 Three partons - generalization of the ’t Hooft solution to many bodies | k 1 , k 2 , K − k 1 − k 2 � , k 1 = 1 , .., K − 2 , k 2 = 1 , .., K − k 1 − 1 (10) 2 � ⇒ | Φ n � ⇒ Φ n ( k 1 , k 2 ) FT � k 1 , k 2 | H | k ′ 1 , k ′ ⇒ Φ n ( d 13 , d 23 ) (11) 11

Figure 2: ρ 1 ( d 13 , d 23 ) 12

Figure 3: | ρ 10 ( d 13 , d 23 ) 13

Figure 4: ρ 50 ( d 13 , d 23 ) 14

Figure 5: ρ 100 ( d 13 , d 23 ) 15

Figure 6: ρ 200 ( d 13 , d 23 ) 16

Figure 7: ρ 300 ( d 13 , d 23 ) 17

Figure 8: ρ 400 ( d 13 , d 23 ) 18

The highest state Figure 9: ρ 406 ( d 13 , d 23 ) A ”mercedes” configuration 19

”Stringy” plot for two partons 100 80 60 P E 2 Λ 40 20 0 0 50 100 150 200 P �� x 12 � � � x 21 �� Figure 10: Eigenenergies of the, p=2, excited states as a function of the relative separation between two partons, K = 30 , 50 , 100 , 200 . 20

Extrapolation 1: in K → ∞ 0.5 0.4 0.3 E 2 Λ x 0.2 0.1 0.0 0 50 100 150 200 250 300 P x � P �� x 12 � � � x 21 �� Figure 11: 21

Extrapolation 2: in a = 2 π P → 0 0.50 0.48 E 2 0.46 Λ x 0.44 0.42 0.000 0.005 0.010 0.015 0.020 a 2 �Π x Figure 12: 22

Families of states with three partons 10 10 10 5 5 5 0 0 0 � 5 � 5 � 5 � 10 � 10 � 10 � 10 � 5 0 5 10 � 10 � 5 0 5 10 � 10 � 5 0 5 10 10 10 10 5 5 5 0 0 0 � 5 � 5 � 5 � 10 � 10 � 10 � 10 � 5 0 5 10 � 10 � 5 0 5 10 � 10 � 5 0 5 10 Figure 13: Contour plots of ρ n ( d 13 , d 23 ) , as partons are moved further away. Series A : n = 10 , 19 , 28 , 41 , 54 , 72 , 4 ≤ l = | d 12 | + | d 23 | + | d 31 | ≤ 14 . The minimal distance between partons = 1. 23

5 5 5 0 0 0 � 5 � 5 � 5 � 5 0 5 � 5 0 5 � 5 0 5 5 5 0 0 � 5 � 5 � 5 0 5 � 5 0 5 Figure 14: Series B. As above but on the Dalitz plot. Now diquarks are allowed, d min = 0 24

all series 30 25 20 P E 3 15 Λ 10 5 0 0 20 40 60 80 P �� x 12 � � � x 23 � � � x 31 �� Figure 15: ρ 406 ( d 13 , d 23 ) 25

”Stringy” plot for three partons one series � K � 40,60,80,100 100 80 60 P E 3 Λ 40 20 0 0 50 100 150 200 250 300 P x Figure 16: Eigenenergies of the, p=3, excited states as a function of the combined length of strings stretching between three partons. = ⇒ String tensions extracted from E 2 ( l ) and E 3 ( l ) seem to be consistent. 26

Four partons Figure 17: Structure of eigenstates with four partons. Contour plots in three relative distances ( d 14 , d 24 , d 34 ) for states no. 1,9,35,60,100,165 spanning the whole range of states for K = 12 , r max = 165 . 27

K � 20 20 15 P E 4 10 Λ 5 0 0 20 40 60 80 100 P �� x 12 � � � x 23 � � � x 34 � � � x 41 �� Figure 18: Eigenenergies of the four parton states vs. the combined string length (all series). 28

6 Inclusive distributions 6.1 Number of pairs at distance ∆ p − 1 d p − 1 � i =1 δ (∆ − d ip ) | ψ r ( � � ∆ p − 1 ) | 2 , D r (∆) = ∆ p − 1 (12) � 29

2.0 1.0 1.0 0.8 0.8 1.5 0.6 0.6 1.0 0.4 0.4 0.5 0.2 0.2 0.0 0.0 0.0 � 10 � 5 0 5 10 � 10 � 5 0 5 10 � 10 � 5 0 5 10 1.0 1.0 1.0 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.0 � 10 � 5 0 5 10 � 10 � 5 0 5 10 � 10 � 5 0 5 10 0.7 0.7 0.7 0.6 0.6 0.6 0.5 0.5 0.5 0.4 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0.0 0.0 0.0 � 10 � 5 0 5 10 � 10 � 5 0 5 10 � 10 � 5 0 5 10 0.7 0.7 0.7 0.6 0.6 0.6 0.5 0.5 0.5 0.4 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0.0 0.0 0.0 � 10 � 5 0 5 10 � 10 � 5 0 5 10 � 10 � 5 0 5 10 Figure 19: Inclusive parton densities for four partons and for lower states r = 1 , 4 , 5 , 6 , 9 , 12 , 13 , 14 , 15 , 20 , 26 , 29 , K = 27 , r max = 2600 . 30