Graphical rules for SU(N)

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1 M/FP/Prours of Physque Théorque Invrnes n physs nd group theory Grph rues for SU(N) In ths proem, we de wth grph nguge, whh turns out to e very usefu when omputng group ftors n Yng-Ms fed theory onstruted on the guge group SU(N). Ths ppes n prtur to Q, for whh, though N 3, t turns out to e usefu to onsder the extenson to SU(N) for resons whh w eome er n seton Genertors grph representton of the Le ger genertors n the fundment representton t s gven y t. (1) The fundment nes rry rrows to dstngush the two representtons N nd N whh re not equvent for N 3. The grph representton for the genertor (t ) s gven y Sne t s hermtn, (t ). () (t ) One n of ourse, through turn round y π, wrte t (3) thus hermtty (3) mpes tht (4) (5) 1

2 fter settng these rues, gven ordered produt of genertors drwn grphy n e trnsted gery foowng the rrows wrd, s one woud do for r γ mtres. The genertors re normzed onventony through the reton Tr(t t ) 1 δ. (6) 1. erve the Ferz dentty t t 1 (δ δ 1N ) δ δ. (7) [Souton : The ss of the proof s dent to the one used for spnors: I ns t ( 1,, N 1) provde ss for hermtn mtres N N. For gven rtrry hermtn mtrx, one n thus wrte 0 I + t. (8) Usng (6), one gets the expresson of the oeffents pperng n the deomposton (8), n the form 0 Tr N Ths eds to wrte (8) n the form, fousng to the oeffent, et Tr[t. (9).e., for every, Tr N δ +t t [ 1 N δ δ +t t δ δ 0 whh ends the proof fter dentfyng the oeffents of.. Show tht ths dentty reds grphy: 1 1 N (10) [Souton : Ths resut s ovous from the prevous queston. 3. From the deomposton of dentty tng n the tensor produt spe N N, Fnd the grph rues for the snget nd dont proetors. [Souton : The deomposton of dentty tng n the tensor produt spe N N s 1 N + (11) We thus hve P 1 1 N P d (1) (13)

3 4. he tht these proetors gve the pproprte vues for the dmenson of the snget nd dont representtons. [Souton : dm 1 TrP 1 1 N dm d TrP d 1 1. (14) 1 N. (15) few pptons.1 Some oor ftors n fundment representton Let us onsder now few typ oor ftors, whh we w enounter sever tmes. 1. Show tht (t t ) F δ nd ompute F. [Souton : Ths resut s expeted: t t s smr for SU(N), thus from Shur emm t shoud e mutpe of dentty. In order to get ths mutptve ftor F, t s enough to ompute the tre of t t, whh s equ to F N (TrI N). Sne Tr(t t ) 1 δ N 1 (numer of genertors N 1), thus F N 1 N.. erve the sme resut usng the Ferz dentty. gery [Souton :. Grphy. (t t ) t t 1 (δ δ 1N ) δ δ 1 ( N 1 ) δ N 1 N N δ. [Souton : 1 1 N 1 N 1 N N 1 N Thus (t t ) N 1 N δ F δ.e. F (16) 3. erve tht ( t t t ) 1 N t.e. 1 N (17) 3

4 [Souton : Usng Ferz dentty one gets ( t t t ) t t t 1 sne t 0 (genertors re treess). Grphy: ( δ δ 1 ) N δ δ t 1 N t 1 1 N 1 1 N 1 N where we hve used the ft tht Trt 0.. Some oor ftors n dont representton Grph rue n so e gven n order to ompute oor ftor nvovng the dont representton. It requres to rete the dont representton to the fundment one. 1. Prove tht f. (18) [Souton : Ths grph reton rees on the reton Tr([t,t t ) f. (19). erve the foowng dentty: f d f d N δ δ.e. d (0) 4

5 . gery [Souton : The frst proof s purey ger: Tr ( [t, t [t, t ) Tr ( f d t d f e t e) f d f e Tr(t d t e ) 1 fd f d, (1) thus f d f d Tr ( [t, t [t, t ) Tr ( t t t t +t t t t t t t t t t t t ) Tr ( t t t t (t t +t t )t t ) () where we hve used the nvrne of the tre under y permutton, whh fny gves, usng retons (16) nd (17), s we s normztons (6), ( f d f d Tr 1 ) N t t N 1 N t t N Tr(t t ) N δ. (3). Grphy. [Souton : From (18, we hve () (4) Eh of these terms n now e evuted seprtey usng the Ferz dentty: 1 1 N N 1 4N 1 8N ( N 1 8N 1 ) 8N (5) nd 1 1 N N 8N ( 1 8N 1 ). (6) 8N 5

6 where we hve used the ft tht the frst terms n the rght-hnd-sde of the seond ne of the ove equton vnshes sne genertors re treess. Thus, nsertng Eq.(5) nd Eq.(6) nsde Eq.(4) one gets N (7) q.e.d. 3 oor ftors for the n guon produton tree-eve mutguon mptude n e wrtten n n SU(N) Yng-Ms theory s M n [1,, n Tr(t 1 t t n ) m(p 1,ǫ 1 ;p,ǫ ; ;p n,ǫ n ), (8) where 1,, n,p 1,p,,p n nd ǫ 1,ǫ,ǫ n re, respetvey, the oors, moment, nd hetes of the guons, nd the sum s over the nony permuttons of the set [1,, n. Our m s to evute the oor ftors nvoved n the omputton of the squred mtrx eement, verged over oors for the nt stte. Expn why tree-eve mptude ony nvoves snge tres. [Souton : onsder tree-e oor struture, orrespondng to gven Feynmn dgrm nvoved n treeeve mptude nvovng n extern guons. t tree order, the ony owed vertes re 3-guons nd 4-guons ones (qurs re ony nvoved t oop eve). From the oor pont of vew, 4-guons vertex oos e two 3-guons vertes onneted y guon ne, so tht the proem redues to onsder pure tree-e struture ud from 3-guons vertes. It thus oos e (9) where,, nd re themseves tree-e dgrms. Frst, we pss from the dont representton to the fundment one y usng the dentty (18). Ths dentty n e rewrtten s f (30).e. f. (31) For smpty, we now remove the rrows on the fundment ne, tng the onventon tht oop n the fundment representton represent ounterowse oor fows. 6

7 The dgrm (9) thus reds (3) Let us ppy Ferz dentty to the guon whh onnets the two fundment oops. Ths eds to 1 1 N + 1 N (33) + 1 N + 1 N (34) so tht +.(35) Thus, usng reursve proof endng when rehng the sheves of the tree, t s er tht the whoe struture n e redued to snge tre of the produt of the n genertors n the fundment representton, the vrous terms eng the tre Tr(t 1 t t n ) nd ther non-y permuttons. 3.1 prtur se We frst onsder the oor ftor ssoted to Tr(t 1 t t n ). We denote the orrespondng oor ftor s n 1 (N t t n ) 1 1) Tr(t1 (N 1) T n. (36) [Souton : s premnry step, et us study the struture of Tr(t 1 t t n ). It reds [Tr(t 1 t t n ) Tr(t 1 t t n ) Tr(t n t t 1 ) (37) where we hve used the ft tht t re hermtn. Thus, Tr(t 1 t t n ) Tr(t 1 t t n ) Tr(t n t t 1 ) (38) 7

8 whh reds grphy 1 1 n Tr(t 1 t t n ) 3 4 n1 3 (39) the seond equty eng trv y n ovous deformton. 1. Show tht n 4 [Souton : Ovousy, T 1 0, (40) T N 1 4 T 3 (N )(N 1) 8N, (41). (4) T 1 0. Next, from Eq. (15), T 1 N 1 4. Fny, T N F 1 1 N ( N 1 4N 1 ) N 1, 4N so tht. Justfy the ft tht T 3 (N )(N 1) 8N. T n ( ) n N (43) for N ( t Hooft mt). [Souton : One n ppy the Ferz dentty s we dd ove for T nd T 3. In the rge N mt, ony the frst term n Eq. (10) remns. Eh guon thus ontrutes wth ftor of N/, edng to ( ) n N T n. 8

9 3. Prove tht T n+1 N n F 1 N T n. (44) [Souton : The proof rees on the use of the Ferz dentty (10), whh gves n+1 n 3 1 n 3 1 N n Now 1 1 n 3 F n1 3 n F, 4 4 thus edng to Eq. (44). 4. Sove the reton (44) nd show tht [Souton : Introdung the reurson (44) reds wth K N F 1N. Ths s soved y T n N 1 N n 1 () n [ 1(1N ) n1. (45) n (N)n N T n, n+1 K n + n, n K 1Kn1 1K,.e. Thus n (N 1) N (1(1N ) n1 ). T n Nn () n n N 1 1 [ 1(1N N n () n ) n1, whh stsfes, s expeted, T n ( ) N n. 9

10 3. The pnr mt 1. For few numer of guons, evute the other oor ftors ourng when squrng the vrous mtrx eements nvoved n Eq. (8). For smpty, we now repe SU(N) y U(N), nd we restrt ourseves to the N mt. In 1974 (Nuer Physs 7 (1974) ), t Hooft proved tht one n me smpe evuton of the sng of gven dgrm n ths U(N) Yng-Ms theory y smpe nvestgton of the two-dmenson surfe n one-to-one orrespondene wth ths dgrm. The phys motvton to do so s to ntrodue new expnson prmeter, owng for systemt ssfton of dgrms, esdes the usu oupng g nvoved n perturtve methods. We w not gve here the deted Feynmn rues for the U(N) Yng-Ms theory. The ony thng whh we need s the ft tht qurs (ntqur) ve n the fundment representton N ( N) of U(N), whe the guon re n the dont representton. The vrous propgtor thus rry oor ndes n ordne wth these representtons, whe the vertes nvove the genertors n fundment (for the qur-guon vertex) nd dont representtons (for 3-guons, 4-guons vertes nd guon-ghost vertex). Ths s ustrted n Fg. 1 (for smpty we do not onsder 4-guons nd ghosts vertes). Fgure 1: t Hooft representton for U(N) guge theory. To me mppng etween the oor struture of gven Feynmn dgrm nd two-dmenson surfe, one shoud tth tte surfes to eh oor ndex oop. Ths eds to g surfe, wth edges formed y the qur nes. Ths surfe s n gener mutpy onneted: t ontns worm hoes (for exmpe euse of n ntern qur oop). The surfe n e osed y tthng tte surfes to the qur oops seprtey.. Fnd the surfe to e drwn for the oor strutures nvestgted n seton Fnd the surfes orrespondng to the other oor ftors ourng n Eq. (8), for the se of n nd n 3 guons. n you dentfy the dfferene etween the type of surfes whh re nvoved? 4. Justfy the ft tht the strutures nvestgted n seton 3.1 re the domnnt one for gven vue of n. Wht s the topoogy of the orrespondng surfes? 10

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