Dynamical mass generation in BRS invariant theories

Nuc1ear Phy51c58301 (1988) 93-114 N0rth-H011and, Am5terdam

DYNAM1CAL M A 5 5 6 E N E R A 7 1 0 N 1N 8 R 5 1NVAR1AN7 7 H E 0 R 1 E 5 Marc0 E. FA88R1CHE51 Department 0f Phy51c5,Je55e 1v. 8eam5 La60rat0ry 0f Phy51c5, un1ver51ty 0f v1r91n1a, Char10tte5v111e, v1r91n1a22901, u5A

Rece1ved26 May 1987

7he pr061em 0f the 0r191n0f the 9au9e part1c1e•5 ma5515c0n51dered1n the framew0rk 0f the 8R5 5ymmetry. A new appr0ach 15 5u99e5ted where the 9106a1 9au9e 9r0up 5ymmetry 0f the 4uantum the0ry 15 h1dden 6y the c0nden5at10n 0f 60und 5tate5 0f the 9h05t f1e1d51n the pertur6at1ve vacuum, Dynam1ca1ma55 9enerat10n f0r the 9au9e f1e1d5 f0110w5 the 5chw1n9er mechan15m. 1.

1ntr0duct10n

1n th15 w0rk 1 exp10re a new way t0 under5tand the 0r191n 0f the 9au9e part1c1e•5 ma55. 7 h e pr061em 0f where th15 ma55 c0me5 fr0m 15 6y n0 mean5 new and 5evera1 a1ternat1ve 501ut10n5 have a1ready 6een pre5ented 1n the 11terature (f0r a rev1ew, 5ee ref. [1]). A11 the5e f0rmu1at10n5 are 6a5ed 0n the 5chw1n9er mechan15m [2], 1n wh1ch ma55 9enerat10n 15 tr199ered 6y the f0rmat10n 0f a ma551e55 5ca1ar 60und 5tate - the w0u1d-6e 601d5t0ne 6050n. Where the var10u5 f0rmu1at10n5 d1ffer 15 1n the ch01ce 0f the c0n5t1tuent5 0f th15 60und 5tate. 1n 7echn1c010r-11ke the0r1e5 [3,4] the c0n5t1tuent5 are D1rac ferm10n5 (5p1n 1), wherea5 an0ther p05516111ty 15 that the 9au9e part1c1e5 them5e1ve5 - thr0u9h the1r 5e1f-1nteract10n - f0rm ma551e55 5ca1ar5 [5]. 7he 5uper5ymmetr1c ca5e a5 6een ana1y5ed 6y 5h0re [6, 7]. 7 h e pre5ent appr0ach a55ume5 the 5chw1n9er mechan15m, 6ut 15 n0ve1 1n treat1n9 the 9h05t f1e1d5 that appear 1n the 4uantum f0rmu1at10n 0f 9au9e the0r1e5 a5 phy51ca1 part1c1e5 that can f0rm a ma551e55 5ca1ar 60und 5tate. My preference f0r 6a51n9 dynam1ca1 ma55 9enerat10n up0n 9h05t f1e1d5 c0me5 fr0m the fact that - a5 we 5ha11 5ee - ne1ther 9au9e part1c1e5 n0r D1rac ferm10n5 can f0rm a ma551e55 5ca1ar w1th the pr0per 9r0up 5ymmetry t0 1mp1ement the 5chw1n9er mechan15m. 7h15 1eave5 0n1y 9h05t part1c1e5 a5 p055161e cand1date5, 0r e15e re4u1re5 50me ent1re1y new appr0ach 6a5ed 0n new phy51ca1 1dea5. An examp1e 0f the5e new 1dea5 15 the m0de1 ca11ed 7echn1c010r [3, 4], where the ma55 0f the weak-1nteract10n vect0r 6050n5 15 dynam1ca11y 9enerated 1ntr0duc1n9 a new 5uper-5tr0n9 5et 0f ferm10n5 - 1n add1t10n 0f the e5ta6115hed part1c1e5 0f the 5tandard m0de1 [8] - w1th c0n5e4uent pr011ferat10n 0f new part1c1e5. 0550-3213/88/$03.50•E15ev1er 5c1encePu6115her58.V. (N0rth-H011and Phy51c5Pu6115h1n9D1v1510n)

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M. Fa66r1che51 / Dynam1ca1 ma55 9enerat10n

7he 1dea 0f u51n9 the 9h05t f1e1d5 t0 acc0unt f0r an 065erva61e phy51ca1 effect 11ke the 9au9e part1c1e ma55e5 cann0t 6e ju5t1f1ed un1e55 the5e f1e1d5 are c0n51dered a5 rea1 a5 the 9au9e f1e1d5 them5e1ve5. F0rtunate1y, the 9h05t f1e1d can 6e emanc1pated fr0m 1t5 5tatu5 0f a tr1ck t0 9uarantee un1tar1ty t0 6ec0me a rea1 - even 1f d1rect1y un065erva61e- f1e1d u51n9 8R5 5ymmetry [9-11] and the can0n1ca1 4uant12at10n pr0cedure. 7heref0re 1 take the p01nt 0f v1ew 5u99e5ted 1n [12] and c0n51der 8R5 5ymmetry - 1n5tead 0f 10ca1 9au9e 5ymmetry - the fundamenta1 pr1nc1p1e 5hap1n9 part1c1e phy51c5. 8ef0re 91v1n9 the deta115 0f th15 new appr0ach 1n the 60dy 0f the art1c1e, 1t 15 u5efu1 t0 5tre55 what var10u5 0ther m0de15 have 1n c0mm0n w1th the 0ne 1n th15 w0rk. 7he 5chw1n9er mechan15m 15 a the0ry 0f dynam1ca1 ma55 9enerat10n. 7he m0re fam111ar H1995 mechan15m [13], where the n0n-van15h1n9 vacuum expectat10n 0f a 5ca1ar f1e1d 9enerate5 ma55, 5h0u1d 6e th0u9ht 0f a5 an effect1ve the0ry - phen0men01091ca11y u5efu1 6ut c0nceptua11y 1nade4uate. 7he 5chw1n9er m0de1 0f dynam1ca1 ma55 9enerat10n 15 5tr1k1n9 6ecau5e 1t emp10y5 a5 framew0rk a ma551e55 9au9e the0ry ent1re1y 1ack1n9 d1men510na1 parameter 1n 1t5 c1a551ca1 1a9ran91an. H0wever, the 0r191n 0f ma55, and theref0re 0f a 5ca1e, 1n a (f0rma11y) 5ca1e-1nvar1ant 5y5tem, 15 6etter under5t00d after rea1121n9 that the pertur6at1ve treatment 0f the 1nteract10n- 0f any 1nteract10n- ren0rma1121n9 the f1e1d5, exp11c1t1y 6reak5 the c1a551ca1 5ca1e 1nvar1ance 0f the 1a9ran91an. 7heref0re, even 6ef0re any ma55 9enerat10n 0ccur5, the 6reak1n9 0f c1a551ca1 5ca1e 5ymmetry 15 a feature 0f any n0n-tr1v1a1 4uantum f1e1d the0ry. 7h15 6reak1n9 1ead5 eventua11y t0 d1men510na1 tran5mutat10n [14] (where the d1men510n1e55 c0up11n9 c0n5tant 15 traded f0r a d1men510na1 5ca1e) and t0 the p05516111ty 0f dynam1ca1 ma55 9enerat10n. 1t mu5t 6e kept 1n m1nd that 1n a11 the 5cenar105 a60ve 0n1y rat105 0f ma55e5 can 6e mean1n9fu11y def1ned. At th15 5ta9e, even 1f the 1dea 5eem5 a1m05t 1rre515t161e, 1t 15 premature t0 5urrender t0 the temptat10n 0f u51n9 the 5uper5tr1n9 [15] 5ca1e (/1,1~nck= 10-33 cm) t0 p1np01nt the a6501ute 5ca1e 0f the5e ma55e5. 7he 0ut11ne 0f th15 art1c1e 15 a5 f0110w5: 7he 8R5 5ymmetry 15 rev1ewed 1n 5ect5. 2, 3 and 4. 7he aux111ary c0nd1t10n nece55ary 1n the def1n1t10n 0f the H116ert 5pace 0f the the0ry 15 1ntr0duced 1n 5ect. 5 t09ether w1th the c0n5e4uent 4uartet dec0up11n9. 7he re1at10n5h1p 6etween 5ca1e 1nvar1ance and dynam1ca1 ma55 9enerat10n 15 6r1ef1y p01nted 0ut 1n 5ect. 6. 7he 1n5ta6111ty 0f the pertur6at1ve vacuum and the 9au9e part1c1e ma55 are ca1cu1ated, re5pect1ve1y, 1n 5ect5.7 and 8. F1na11y, 5ect. 9 c0nta1n5 a d15cu5510n 0f the re5u1t5.

2. 6au9e 1nvar1ance 1n Yan9-M1115 the0r1e5 Let the 9106a1 9au9e 9r0up 6e def1ned 6y the 9enerat0r5 L a 0f 1t5 L1e a19e6ra. 7he 9au9e f1e1d5 are A~(x) =A~(x)L

a.

(2.1)

M. Fa66r1che51/ Dynam1ca1ma559enerat10n

95

7he f1e1d 5tren9th5 -

=

+ (

x

(2.2)

and the c0var1ant der1vat1ve Dr = 0 r + A~ ×

(2.3)

can 6e def1ned, where (X X y ) a = • 19c~cX6y~, 9 15 the un1ver5a1 9au9e c0up11n9 c0n5tant and c~,~ are the 9r0up 5tructure c0n5tant5. A5 f0r the e1ectr0ma9nef1c f1e1d [16], the pre5ence 0f unphy51ca1 5tate5 make5 1t 1mp055161e t0 def1ne f0r the c1a551ca1 1a9ran91an Lc1 = • ~1~., ~.~, a

(2.4)

the can0n1ca1 m0mentum c0nju9ate t0 A9(x). Acc0rd1n91y, the ham11t0n1an f0rma115m cann0t 6e 1mp1emented. 51nce the can0n1ca1 4uant12at10n pr0cedure h1n9e5 0n the ham11t0n1an can0n1ca1 f1e1d var1a61e5, 1n 0rder t0 pr0ceed w1th 4uant12at10n, the 10ca1 9au9e 5ymmetry mu5t 6e 6r0ken 6y the add1t10n 0f a 9au9e-f1x1n9 term 1n the 1a9ran91an. 7h15 term ha5 the f0110w1n9 f0rm: L~f=

1 ~-a (0-A)2,

(2.5)

where a 15 a 9au9e parameter. 7he 1a9ran91an thu5 def1ned a110w5 f0r a we11-def1ned ham11t0n1an repre5entat10n. 7h15 15 an accepta61e the0ry de5p1te the 1055 0f 10ca1 9au9e 1nvar1ance, 51nce the Ward-7akaha5h1 1dent1t1e5 am0n9 the 6reen funct10n5 (and hence the un1ver5a11ty 0f the c0up11n9 c0n5tant a5 we11 a5 the un1tar1ty 0f the the0ry) f0110w fr0m the 9106a1 8 R 5 5ymmetry. 1n fact, the 1a9ran91an Lc1 + L9f 15 5t1111nVar1ant t0 the 9aU9e tran5f0rmat10n (2.6) 1f the c0nd1t10n

0. (Dha(x)) = 0

(2.7)

15 5at15f1ed. E4. (2.7) 1mp11e5 a 501ut10n 1n wh1ch ha(x) 15 an 0perat0r, 1.e. a dynam1ca1 f1e1d [17]. A term c0rre5p0nd1n9 t0 the k1net1c ener9y 0f 5uch a f1e1d mu5t thu5 6e 1nc1uded 1n the 1a9ran91an. 7he pre5ence 0f new f1e1d5 C(x) pr0p0rt10na1 t0 h(x), ca11ed 9h05t5, 9uarantee5 the un1tar1ty 0f the the0ry [16] 6ecau5e the new 9106a1 tran5f0rmat10n, where the 9au9e funct10n ha(x) 15 a dynam1ca1 f1e1d, 1ead5 t0 the 51avn0v-7ay10r 1dent1t1e5.

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M. Fa66r1che51/ Dynam1ca1ma559enerat10n

H0wever, the n0n-a6e11an 9r0up dynam1c5 1ead5 t0 a c0up11n9 1n e4. (2.7) 6etween the 9h05t5 and the 9au9e f1e1d5. 7h15 5u99e5t5 that, un11ke the e1ectr0ma9net1c ca5e, 1n Yan9-M1115 the0r1e5 9h05t5 f1e1d5 can have dynam1ca1 c0n5e4uence5.

3. 8 R 5 5ymmetry C0ntrary t0 the c1a551ca1 ca5e, where a 51n91e f1e1d, the p0tent1a1 A~(x), 15 nece55ary, f0ur 1ndependent f1e1d5 are p05tu1ated 1n the 4uantum ca5e t0 de5cr16e a pure Yan9-M1115 the0ry. A~,(x), C(x) and C ( x ) have can0n1ca1 d1men510n 0ne, wh11e 8(x), 6e1n9 an aux111ary f1e1d, ha5 d1men510n tw0 and cann0t ac4u1re k1net1c ener9y*. 7he 9h05t5, a1th0u9h 5ca1ar f1e1d5, are ferm10n5**. 7 h e 8 R 5 5ymmetry f0r the5e f0ur f1e1d5 15 def1ned 1n 1t5 9enera1 f0rm 6y the act10n 0f the tw0 9raded 9enerat0r5 Q~ and Q8:

Q8A~ = D~C,

08A~ = D~C,

2Q8C= - ( C X C),

2Q8C= - ( C X C ) ,

Q8C= 8,

0•.8C = 8,

Qa8=0,

Q88=0, 8 +

+ (c ×

= 0,

(3.1)

where an aux111ary f1e1d 8(x) ha5 6een 1ntr0duced*** and the 9r0up 1nd1ce5 0m1tted. 7 h e act10n 0f the5e char9e5 0n a 11near c0m61nat10n 0f f1e1d5 15 def1ned 6y the 9raded Le16n12 ru1e

0 ( X . Y) = ( 0 X ) . Y+ X. ( 0 Y ) ,

(3.2)

where the m1nu5 519n 15 ch05en when a 0dd num6er 0f 9h05t f1e1d5 15 pre5ent, the p1u5 519n 0therw15e. Furtherm0re, the Q8 and Q8 char9e5 are n11p0tent: Q~ = ~2 = 0;

(3.3)

t09ether w1th the 9h05t num6er 0perat0r Qc, def1ned 6y

[1Q¢,C]=C

[1Qc, C ] = - C ,

[10~,A~] =[1Q~,8]=0,

(3.4)

* Any 5uch term w0u1d have can0n1ca1d1men510n9reater than f0ur and 1t w0u1d n0t 6e ren0rma112a61e. ** N0 v101at10n0f the 5p1n-5tat15t1c5the0rem [18] 15h0wever1mp11ed,5ee 5u65ect. 5.1. *** 7h15 aux111aryf1e1d51mp11f1e5the expre5510nf0r the 8R5 tran5f0rmat10n and 1t 15 a550c1ated,6y 1t5 e4uat10n 0f m0t10n, t0 the 5ca1arm0de 0f the 9au9e part1c1e5.

M. Fa66r1che51 / Dynam1ca1 ma55 9enerat10n

97

they 9enerate the 5upera19e6ra

(Q., ~,,} : 0 , [;Qc, ~,1 = - ~ . .

[1Qc, Q81 = Q8

(3.5)

7he tran5f0rmat10n 0f the 9au9e f1e1d5 1n the f1r5t 11ne 0f e4. (3.1) 15 1n the 5ame f0rm a5 the u5ua1 9au9e tran5f0rmat10n

6A~( x ) = D~,2~(x )

(3.6)

6ut the 9au9e funct10n ha(x) 15 n0w a dynam1ca1 f1e1d, name1y the 9h05t. A5 a1ready p01nted 0ut 1n 5ect. 2, th15 15 the 0n1y way t0 ma1nta1n, f0r a n0n-a6e11an 9au9e 9r0up, a re51due 0f the 0r191na1 9au9e 1nvar1ance.

4. 7he can0n1ca1 1a9ran91an 7he m05t 9enera1 8R5-1nvar1ant 10ca1 p01yn0m1a1 1a9ran91an den51ty 5at15fy1n9 certa1n c0nd1t10n5 (can0n1ca1 d1men510n f0ur, 2er0 9h05t num6er, L0rent2 and 9106a11y 9au9e 1nvar1ant) ha5 6een w0rked 0ut 1n ref. [12] a5 L = - - ~,•, ~ .,~ + ~ , ~ + 8 ( a . , 4

+ -~(e× c)}+

( a~e)D~c + , ~ , ( C x c ) = , (4.1)

where, f0r 51mp11c1ty, 1 0m1t the 9r0up 1nd1ce5. 7he aux111ary f1e1d 8 ( x ) can 6e e11m1nated u51n9 1t5 e4uat10n 0f m0t10n

0,A

8(x) . . . .

0~

7(C-X C)

(4.2)

t0 06ta1n

L = Lc1 + L9t + "y( DtF. ) 0~C +

(1-7)(0tF.)D~C + -~7(1- "•1)a(C× C) 2. (4.3)

Ferm10n1c (5p1n-•) matter f1e1d5 can 6e 1nc1uded 0n1y 1n Lc1 1n (4.3). 7he1r can0n1ca1 d1men510n make5 1mp055161e a d1rect c0up11n9 t0 the 9h05t f1e1d5 w1th0ut v101at1n9 the c0nd1t10n5 0n d1men510na11ty and 8R5 1nvar1ance. 7he 8R5-1nvar1ant 1a9ran91an depend5 0n tw0 9au9e parameter5, a and 7. F0r 3• = 1, (4.3) 15 ju5t the Faddeev-P0p0v 1a9ran91an [19]. 7he ch01ce ~, = 0 def1ne5 •t H00ft [20, 21] 9au9e. 7he 11m1t a ~ 0 91ve5 the Landau 9au9e. 1t 15 1mp0rtant t0 n0te that 0n1y f0r ¥ = • can 0ne 1dent1fy C(x) a5 the ant1-part1c1e 0f C(x). 1n th15 ca5e e4. (4.3) re5em61e5 5ca1ar QED and 1 exp101t th15 51m11ar1ty 1n 5ect. 7.

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M. Fa66r1che51 / Dynam1ca1 ma55 9enerat10n

7he 0vera11 c0n515tency 15 pr0ved 6y the 1ndependence 0f the 5-matr1x 0f the phy51ca1 f1e1d5 fr0m the 9au9e parameter5 "y and a [22]. 5. 7he F0ck 5pace 0f a5ympt0t1c f1e1d5 7he 5tructure 0f the H116ert 5pace 1n wh1ch the 1a9ran91an (4.1) act5 15 5pec1f1ed 6y the 1ntr0duct10n 0f an aux111ary c0nd1t10n that, 5e1ect1n9 the phy51ca1 5tate5, e11m1nate5 the 0ther (unphy51ca1) m0de5. 7h15 H116ert 5pace 15 c0n5tructed a5 a F0ck 5pace 0f appr0pr1ate1y def1ned a5ympt0t1c f1e1d5. 7he5e a5ympt0t1c f1e1d5 are 1501ated 5y5tem5 w1th a c0mp1ete 5et 0f 5tate5 that 06ey free e4uat10n5 0f m0t10n. At th15 p01nt a f1r5t a55umpt10n 15 1ntr0duced. 1 have 1n m1nd a the0ry where the 5ymmetry 0f the 9106a1 9au9e 9r0up 15 eventua11y 5p0ntane0u51y 6r0ken (5ee 5ect. 6) 6y the act10n 0f a 60und 5tate 0f the 0r191na1 f1e1d5. 7heref0re, rather than try1n9 t0 extrap01ate d1rect1y the f1e1d5 1n e4. (4.1), 1 prefer, rever51n9 the u5ua1 pr0cedure, t0 def1ne f1r5t a F0ck 5pace 1n ana109y t0 a H1995-11ke the0ry [13, 23], and hence t0 5h0w that the 1a9ran91an (4.1) 15 a c0n515tent 1nterp01at10n f0r 5uch a5ympt0t1c f1e1d5. 5.1. 7 H E A5YMP7071C F1ELD5

1n the 8R5-1nvar1ant 1a9ran91an den51ty (4.1) the de9ree5 0f freed0m 0f the 9h05t f1e1d5 can 6e 1nte9rated 0ut t0 def1ne an effect1ve 1a9ran91an den51ty. 1 can a55ume that after th15 1nte9rat10n the 0n1y re1evant c0ntr16ut10n 1eft fr0m the 9h05t5 15 the1r ma551e55 60und 5tate, the 601d5t0ne 6050n, that act5 a5 an 1ndependent de9ree 0f freed0m 1n the effect1ve 1a9ran91an. Under the5e hyp0the5e5, 1n the 9au9e 3• = 0 and 0m1tt1n9 the 9r0up 1nd1ce5, the effect1ve 1a9ran91an den51ty c0rre5p0nd1n9 t0 (4.1) 15 Leff = - 1 F4-,~--12~J,~+ ~ m 2 A ~ , A ~ , + m A ~ , 0 7 + ~,0 7 0 ~ , 7 r + ~ a18

2 + 8(0.A).

(5.1)

7h15 15 n0t a der1vat10n, 6ut a 5tatement 0f h0w 0ne can expect the the0ry t0 6ehave, 6y ana109y w1th 51m11ar the0r1e5. E4. (5.1) 15 H1995-11ke w1th0ut the ma551ve H1995 6050n. 7he c0rre5p0nd1n9 e4uat10n5 0f m0t10n are: ( 0"0~ + m 2 ) A , -- - 0 r 8 + rn 0,~r, 0" 0 7 = m 0. A , a8 + 0.a =0.

(5.2)

A1th0u9h e4. (5.2) are n0t dec0up1ed, 1t 15 p055161e t0 d1a90na112e them def1n1n9 tw0 new f1e1d5 1 W~=A, + -~ 0,(-8 + m~), H = ~r - m a f 8 ,

(5.3)

M. Fa66r1che51/ Dynam1ca1ma559enerat10n

99

Where* 0 ~ 0 j = 1. 1 then a55Ume the pre5ence 0f f1Ve a5ympt0t1C f1e1d5**: a ma551Ve 9aU9e f1e1d W~(X), the aUx111ary f1e1d 8a~(X), the tW0 9h05t f1e1d5 Ca~(X) and Ca~(X), and the 601d5t0ne 6050n Ha~(X). ACC0rd1n91y t0 my a55Umpt10n that W~a~(X) and Ha~(X) de5Cr16e a H1995-11ke the0ry, they 06ey the e4Uat10n5 0f m0t10n --a5

(0/t0p-6/9~2)Wa5(X)=0;

~. W a 5 = 0 .

a5

(5.4)

7 h e ma55 1n e4. (5.4) 15 5t111 an Unkn0Wn 4Uant1ty a550C1ated W1th the pre5ence 0f the 10n91tUd1na1 m0de5 1n 1~V~ a~. 7he5e f1e1d5 are C0nneCted t0 the f1e1d5 1n e4. (4.1) thr0U9h the tran5f0rmat10n (5.3). 7heref0re, Wh11e the a5ympt0t1C C0nd1t10n 1n the 5eCt. 7 W111 C0nneCt the f1e1d5 1n e4. (4.1) t0 the 0ne5 1n the effect1Ve 1a9ran91an den51ty 1n e4. (5.1), the H116ert 5paCe 15 C0rreCt1y def1ned U51n9 the deC0Up1ed f1e1d 0f e4. (5.4). 1 Can a150 501Ve e4. (5.4) 1n p1ane WaVe5 and def1ne the re5pect1Ve Creat10n and ann1h11at10n 0perat0r5 f0r each f1e1d. 1n term5 0f the5e, the (ant1-)c0mmutat10n re1at10n5 are

] [6k,6*t] = 0 , [6k,~1*t] = [~19,6tt] = -8~t,

{

}=

(5.5)

where 1 have u5ed the n0tat10n 0f 10wer ca5e f0r W~, 8 and C and the 6 r e e k e4u1va1ent f0r H. ~ t 15 an unkn0wn c0n5tant that need5 n0t 6e 5pec1f1ed f0r what f0110w5. Cur1y 6racket5 5tand f0r ant1-c0mmutat10n, 54uared 6racket5 f0r c0mmutat10n. 7 h e ant1-c0mmutat10n 0f the 9h05t f1e1d5 15 due t0 the1r ferm10n1c character. Even th0u9h they are ant1-c0mmut1n9 f1e1d5 w1th 5p1n 2er0, there 15 n0 v101at10n 0f the 5p1n-5tat15t1c5 the0rem [18] 6ecau5e 0ne 0f the nece55ary hyp0the5e5 0f th15 the0rem 15 that the H116ert 5pace ha5 a p051t1ve-def1n1te metr1c, wherea5 we 1nc1ude 5tate5 0f ne9at1ve n0rm. 5.2. 7HE Aux1L1ARY c0ND1710N AND QUAR7E7 DEc0uPL1N6 7 h e vect0r 5pace ~phy5 0f phy51Ca1 5tate5 mU5t rema1n 1nVar1ant Under the 5-matr1X and the 5Ca1ar pr0dUCt 1n 1t mU5t 6e p051t1Ve-def1n1te. 7 0 CharaCter12e th15 *f 15 a funct10ndef1ned5uch that 0~0~f8 = 8 1f 0 r 0r8 = 0. ** 7he5e are unren0rma112edf1e1d5.

100

M. Fa66r1che51 / Dynam1ca1 ma55 9enerat10n

5u65pace 0f the c0mp1ete vect0r 5pace ~ , Ku90 and 0j1ma [24] def1ne an aux111ary c0nd1t10n u51n9 the 8R5 char9e Q~ ~phy~= {145) ~ 2V•: Q814~) = 0}.

(5.6)

1n the ca5e 0f an a6e11an f1e1d, the c0nd1t10n Q81ff)= 0 reduce5 t0 the fam111ar 6upta-81eu1er [25,26] aux111ary c0nd1t10n 0 . A ( + ) [ ~ ) = 0*, that free5 the phy51ca1 5pace 0f the unwanted 10n91tud1na1 and t1me-11ke 5tate5. 1n 5pace5 w1th 1ndef1n1te n0rm, (X1X)= 0 d0e5 n0t 1mp1y 1X) = 0. Hence 1t 15 p055161e t0 def1ne the 5u65pace

= (1x5 e

h,5:(x1x5 = 0}

(5.7)

that 15 0rth090na1 t0 every vect0r 1n ~phy5"7W0 5tate5 1~) and 1~) + 1X) Cann0t 6e d15t1n9U15hed 6y exper1ment. 7heref0re the H116ert 5paCe J~t~phy5 f0r the C0n5tra1ned f1e1d5 Can 6e def1ned a5 the C0mp1eted 4U0t1ent 5pace ~phy5//3V0. 7h15 mean5 that the 0n1y 5tate5 that are a110Wed t0 C0nta1n Unphy51Ca1part1C1e5 mU5t have a C0m61nat10n 0f f1e1d5 w1th 2er0 n0rm. 7he5e are 9r0Uped 1n the1r 0Wn e4U1Va1enCe C1a55 and d0 n0t affect the phy51C5 0f the 5y5tem. 7he 5Ca1ar pr0dUCt 1n 0~phy~ 15 6y C0n5trUCt10n p051t1Ve def1n1te. 7he 8R5 tran5f0rmat10n pr0pert1e5 0f the a5ympt0t1C f1e1d5 1ntr0dUCed 1n 5U65eCt. 5.1 are Q86k

= Q8Ck

= 0,

Q8Ck = 6 k , Q8~1 = -- 1c k .

(5.8)

8eCaU5e 0f the n11p0tenCy 1n e4. (3.3), the 1rredUC161e repre5entat10n5 0f the 8R5 a19e6ra Can 0n1y 6e 9r0Up 51n91et 0r d0U61et. 7he d0U61et repre5entat10n5 C0me a1Way5 1n pa1r5 [24] and are Ca11ed 4Uartet5. E4. (5.8) repre5ent5 0ne 0f the5e 4Uartet5. 7he 5U651d1ary C0nd1t10n rem0Ve5 a11 the f1e1d5 1n the 4Uartet repre5entat10n fr0m the phy51ca1 5pace 6ecau5e they 5pan the 2er0-n0rm 5u65pace 2¢~0. 1n the a65ence 0f 601d5t0ne 6050n5, the 4uartet made 0f the 9h05t5 and the aux111ary f1e1d 15 c0mp1eted 6y the 10n91tud1na1 c0mp0nent5 0f the 9au9e f1e1d5 that are thu5 1eft ma551e55. When, 0n the c0ntrary, 601d5t0ne 6050n5 are pre5ent, they c0mpete w1th the5e 10n91tud1na1 m0de5, expe111n9 them fr0m the dec0up11n9 4uartet. 7heref0re, the 9au9e part1c1e5 c0n5erve the1r 10n91tud1na1 c0mp0nent5 and ac4u1re a dynam1ca1 ma55, a5 1n the examp1e c0n51dered 1n 5u65ect. 5.1. 1n pre5ence 0f 60und 5tate5 0f 9h05t5 and 5p0ntane0u51y 6r0ken 5ymmetry the dec0up11n9 15 ach1eved 6y mean5 0f three 4uartet5 1n5tead 0f ju5t 0ne [27]. * A~+) 5tand5 f0r the p051t1vefre4uencypart 0f A~.

M. Fa66r1ehe51 / Dynam1ca1 ma55 9enerat10n

101

5.3. 8R5 AL6E8RA AND C0NF1NEMEN7 8eCaU5e 0f the CanCe11at10n 0f a11 4Uartet5 fr0m the phy51Ca1 5-matr1X, the 4Uartet deC0Up11n9 1t5e1f ha5 6een 5U99e5ted [24, 28] a5 a p055161e mechan15m 0f c0nf1nement. W1th th15 mechan15m the fact that the the0ry a110w5 0n1y c010r1e55 - that 15, 9au9e 9r0up 1nvar1ant- 60und 5tate5 15 an effect 0f the aux111ary c0nd1t10n (5.6) rather than 0f an 1nf1n1te 61nd1n9 p0tent1a1. Fr0m th15 p01nt 0f v1ew, 4uark5 and 91u0n5 are c0nf1ned 1n51de hadr0n5 1n the 5ame 5en5e that 5ca1ar and 10n91tud1na1 ph0t0n5 1n QED are prevented 6y the 6upta-81eu1er c0nd1t10n fr0m appear1n9 a5 a5ympt0t1c 5tate5. Even th0u9h a d15cu5510n 0f Ku90 and 0j1ma•5 m0de1 15 6ey0nd the 5c0pe 0f th15 w0rk, 0ne 1mp0rtant p01nt ha5 t0 6e c0n51dered. Ku90 and 0j1ma•5 cr1ter10n f0r dec0up11n9 the 91u0n5 (4uark5) 15 the ex15tence 0f a 60und 5tate 6etween a 91u0n (4uark) and a 9h05t. 7h15 c0nd1t10n 15 nece55ary t0 make the 91u0n (4uark) a mem6er 0f a 4uartet and theref0re t0 c0nf1ne 1t. 5u65e4uent1y, N15h1j1ma [29] perfected the Ku90-0j1ma cr1ter10n, 5h0w1n9 that 5uch 60und 5tate5 f0110w fr0m the ex15tence 0f a 60und 5tate 0f tw0 9h05t5. 1f th15 1atter c0nd1t10n were 5uff1c1ent t0 c0nf1ne, there w0u1d 6e a c0ntrad1ct10n w1th the re5u1t 1n the pre5ent w0rk: 1 d0 n0t f1nd any c0nf1nement even when there 15 (5ee 5ect5. 7 and 8) a 60und 5tate 0f 9h05t5. F0rtunate1y, N15h1j1ma•5 c0nd1t10n 15 0n1y nece55ary and n0t 5uff1c1ent [27] and d0e5 n0t app1y 1n the ca5e 0f a 5p0ntane0u51y 6r0ken 5ymmetry, 6ecau5e then the 9au9e 9r0up char9e 15 n0 10n9er we11-def1ned*. 7here 15 thu5 n0 c0ntrad1ct10n 6etween N15h1j1ma•5 re5u1t and my re5u1t, 6ut 51mp1y a d15t1nct10n am0n9 d1fferent pha5e5 1n wh1ch the the0ry can ex15t. A1th0u9h N15h1j1ma•5 w0rk and the pre5ent 0ne treat d1fferent phy51ca1 pr061em5, 1t 15 1mp0rtant t0 n0te that N15h1j1ma [30] (and Ku90 and 0j1ma [28]) ant1c1pated th15 w0rk 1n 5eek1n9 phy51ca1 c0n5e4uence5 0f the 9h05t f1e1d5 wh1ch - a5 n0ted a 6 0 v e - had prev10u51y 6een c0n51dered 0n1y a c0nven1ent tr1ck w1th n0 5uch c0n5e4uence5.

6. A5ympt0t1c freed0m and the 0r191n 0f a 5ca1e 7he 8R5-1nvar1ant 1a9ran91an den51ty def1ned 1n 5ect. 4 15 apparent1y 5ca1e 1nvar1ant 6ecau5e 1t d0e5 n0t c0nta1n any d1men510na1 parameter. H0wever, 6ecau5e 0f the vacuum p01ar12at10n, the c0up11n9 (char9e) 9 depend5 0n the d15tance at wh1ch 1t 15 065erved. 7heref0re, a h1dden 5ca1e A 15 1ntr0duced 1n the 4uantum the0ry 6y the ren0rma112at10n pr0ce55, that ar61trar11y f1xe5 0ne 0f the5e d15tance5. 7he way th15 c0me5 a60ut, and the rea50n why the ar61trary d15tance d0e5 n0t 1ead t0 ar61trary phy51ca1 065erva61e5, 15 de5cr16ed 6y the ren0rma112at10n 9r0up the0ry [16]. 51nce 5ca1e 1nvar1ance 15 6r0ken 6y the ren0rma112at10n pr0cedure, the d1men* 70 have c0nf1nementthe 9au9e 9r0up char9e mu5t 6e we11-def1nedand e4ua1 t0 2er0.

102

M. Fa66r1che51 / Dynam1ca1 ma55 9enerat10n

510n1e55 c0upf1n9 9 15 traded f0r a un1t 0f ma55 (0r 1en9th) - wh1ch 0ne can then u5e t0 expre55 a11 d1men510na1 phy51ca1 4uant1t1e5 (d1men510na1 tran5mutat10n [14]). 6r055 and Neveu [31] p01nted 0ut that 1n a the0ry w1th d1men510na1 tran5mutat10n any phy51ca1 ma55 mu5t 06ey the h0m09ene0u5 ren0rma112at10n 9r0up e4uat10n [16]

[0

0]

A-~--~ + 8(9)-~99 m ( 9 , A ) = 0 ,

(6.1)

where 8 ( 9 ) = A 0 9 / 9 A . E4. (6.1) 51mp1y 5ay5 that the ma55 m(9, A) 15 1nvar1ant under a chan9e 0f A. 7he phy51ca1 ma55 then depend5 0n the c0up11n9 c0n5tant a5

m(9,A)=Aexp(• f9 dx

(6.2)

7h15 15 1n fact the f0rm f0und 1n 5ect. 7. 7he re1at10n5h1p w1th the h19h-ener9y 6ehav10r 0f the the0ry 15 5een 1n the 11m1t 0f van15h1n9 c0up11n9. 1n th15 11m1t e4. (6.2) reduce5 t0 m ( 9 , A) -~ A e x p ( 1 / 8 0 9 2 ) ,

9~ 0

(6.3)

•3+ 0(95). 7he 519n 0f/30 1n the ne19h60rh00d 0f 9 = 0 determ1ne5 a5 8 ( 9 ) ~ ~180~, whether the the0ry 15 1nfrared-5ta61e 0r a5ympt0t1ca11y free. C1ear1y, 0n1y/30 < 0 (a5ympt0t1ca11y free ca5e) make5 e4. (6.3) c0n515tent w1th the rea50na61e re4u1rement that the ma55 90e5 t0 2er0 f0r van15h1n9 c0up11n9. 1n the next tw0 5ect10n5, the c0nc1u510n 0f 6r055 and Neveu that a5ympt0t1ca11y free the0r1e5 c0nta1n n0 ma55 parameter 6ut 9enerate ma55e5 dynam1ca11y 15 5h0wn t0 6e rea112ed w1th the 8R5-1nvar1ant 1a9ran91an 91ven 1n e4. (4.3).

7. 1n5ta6111ty 0f the vacuum

E4. (4.3) def1ne5 a c1a55 0f 1a9ran91an5 that depend 0n tw0 9au9e parameter5, a and ~,. 7he 5-matr1x 0f the phy51ca1 f1e1d5 15 1ndependent 0f the ch01ce 0f the5e parameter5 6ut the appr0x1mat10n5 u5ed 1n the ca1cu1at10n 6e10w depend 0n a and ~/. 7he ch01ce y = • make5 e4. (4.3) re5em61e the 1a9ran91an 0f 5ca1ar QED. 7h15 ch01ce 15 the 0n1y 0ne f0r wh1ch the 8R5-1nvar1ant 1a9ran91an 15 herm1t1an. 8ef0re exp101t1n9 th15 he1pfu1 51m11ar1ty t0 5ca1ar QED, a d1fference 5h0u1d 6e p01nted 0ut: 7he 1a9ran91an (4.3) ha5 n0 1nteract10n 4uadrat1c 1n 60th 9au9e f1e1d5 and 9h05t5. 7h15 1mphe5 the a65ence 0f rad1at1ve ma55 9enerat10n f0r the 9h05t5 6ecau5e 1t 15 ju5t 5uch a 4uadrat1c 1nteract10n that dr1ve5 d1men510na1 tran5mutat10n [14]. 1n 0ther w0rd5, e4. (4.3) 15 the the0ry 0f a tru1y ma551e55 5ca1ar part1c1e; wherea5 5ca1ar QED, at 4uantum 1eve1, 15 a1way5 the the0ry 0f a ma551ve 5ca1ar f1e1d

M. Fa66r1che51 / Dynam1ca1 ma55 9enerat10n

103

(C01eman-We1n6er9 mechan15m [14]). 7heref0re the ex15tence 0f a 60und 5tate 1n the 8R5-1nvar1ant the0ry 5h0u1d 1ead t0 1n5ta6111ty 0f the pertur6at1ve vacuum 6ecau5e, a5 a1ready p01nted 0ut 1n the prev10u5 5u65ect10n, the ma551e55ne55 0f the c0n5t1tuent5 make5 a c0nden5at10n 0f the5e 60und 5tate5 1n the 9r0und 5tate ener9et1ca11y fav0ra61e. An ana1090u5 c0nden5at10n ha5 6een exh161ted 1n re1at1v15t1c 4uantum mechan1c5 [32] a5 the effect 0f a 5upercr1t1ca1 61nd1n9 p0tent1a1. A 5ca1e 15 9enerated 6ecau5e the 5tr0n9 p0tent1a1 ha5 t0 6e cut 0ff t0 def1ne a 5e1f-adj01nt ham11t0n1an. 1n 4uantum f1e1d the0ry the c0rre5p0nd1n9 5ca1e 15 9enerated a5 f0110w5: 7he 9au9e-1ndependence 0f e4. (4.1) make5 1t p055161e t0 tran5f0rm the 1a9ran91an t0 the u5ua1 Faddeev-P0p0v f0rm tak1n9 3• = 0. 1n th15 f0rm, 1t 15 ea5y t0 5h0w the a5ympt0t1c freed0m 0f the the0ry. 7hu5, 1n the way 5h0wn 1n 5ect. 6, the ren0rma112at10n 9r0up parameter A 5et5 a p055161e cut 0ff. 7he runn1n9 c0up11n9 c0n5tant 9(4) reache5 1n A the upper 11m1t 0f the re910n 0f m0menta 1n wh1ch the 1n5ta6111ty ex15t5 and 9(A) 15 theref0re 1t5 cr1t1ca1 va1ue, 6e10w wh1ch n0 60und 5tate 15 expected. 7he ex15tence 0f a cr1t1ca1 c0up11n9 d0e5 n0t 1mp1y a 1ar9e va1ue f0r the c0up11n9 1t5e1f. 7hu5 1t may 6e rea50na61e t0 91ve the c0up11n9 a va1ue 5ma11 en0u9h t0 perm1t pertur6at1ve appr0x1mat10n t0 the 8ethe-5a1peter kerne1. Hav1n9 exp1a1ned the feature5 1 9enera11y expect 0f the 1a9ran91an den51ty 1n e4. (4.3), 1 a55ume a 5upercr1t1ca1 c0up11n9 9 > 9(A) and 501ve the 8ethe-5a1peter e4uat10n f0r the 9h05t 60und 5tate. 7he kerne1 15 truncated t0 the 1adder appr0x1mat10n, and 1 ch005e the Feynman 9au9e (a = 1). A def1n1te ch01ce 0f 9au9e 15 nece55ary 6ecau5e the 1adder appr0x1mat10n 15 n0t 9au9e 1nvar1ant. 7.1. 6 R 0 U P

PR0PER71E5 0F 7HE 80UND 57A7E

7he nature 0f the c0nden5ate 15 def1ned 6y the pr0pert1e5 0f the 8ethe-5a1peter 60und 5tate wave funct10n ,/,a6(4, p ) =

f d4x e,4.~(017Ca(x/2)C6(~x/2)1P)"

(7.1)

7he 9enera112ed Pau11 pr1nc1p1e re4u1re5 that 4,a6 carry the ant1-5ymmetr1c adj01nt repre5entat10n 0f the 9au9e 9r0up; that 15

~a6(4, p) ~xfa6cX~(4~p).

(7.2)

70 5ee th15, c0n51der a permutat10n 0f the tw0 9h05t part1c1e5 1n th15 60und 5tate. 7he effect 0f 5uch an exchan9e 15 de5cr16ed 1n e4. (7.3) 6e10w 6y the chan9e5 1n the f0ur c0n5erved 4uant1t1e5 L, 5, 1 and C9 - re5pect1ve1y an9u1ar m0mentum, 5p1n, 9au9e 9r0up 5ymmetry and the 9h05t-char9e par1ty ( - 1) L+5+51+521C~ = • 1,

(7.3)

104

M. Fa66r1che51 / Dynam1ca1 ma55 9enerat10n

where the - 1 0n the r19ht-51de-hand c0me5 a60ut 6ecau5e the 9h05t5 ant1c0mmute and the 51 are the1r 5p1n5. A 5ca1ar 60und 5tate 0f 5ca1ar5 ha5 L = 5 = 0 and 51 = 0. Furtherm0re, C9 can 6e c0n515tent1y a55umed even 51nce, un11ke e1ectr1c char9e par1ty 1n QED, the Yan9-M1115 f1e1d ha5 even 9h05t-char9e par1ty. 7heref0re 1 = - 1 , name1y the 9au9e 9r0up fact0r mu5t 6e ant1-5ymmetr1c. 7he 10we5t ant15ymmetr1c repre5entat10n 15 the adj01nt 0ne and theref0re the 5ca1ar CC 60und 5tate mu5t carry 1t. 1t f0110w5 that 1f 5uch a 60und 5tate ex15t5 the 9au9e 9r0up 5ymmetry 0f the vacuum w1116e 6r0ken 6ecau5e 0f the c0nden5at10n 0f 5uch 60und 5tate5, and the 5ymmetry 0f the 1a9ran91an h1dden. 7h15 w0u1d n0t 6e the ca5e f0r 60und 5tate5 0f the 9au9e part1c1e5 them5e1ve5 [33] 0r f0r 60und 5tate5 0f D1rac ferm10n5. F0r e1ther 0f the5e the c0nden5at10n, 1f 1t happen5 at a11, pr0ceed5 1n the 1nvar1ant (5ymmetr1c) channe1 0f the 9au9e 9r0up and n0 5p0ntane0u5 6reak1n9 0f the 9au9e 5ymmetry 15 expected 6ecau5e the vacuum - even after a c0nden5at10n 0f the5e 60und 5tate5 - w0u1d rema1n 5ymmetr1c. 7heref0re, 1 w11119n0re the c0ntr16ut10n 0f the 9au9e part1c1e and D1rac ferm10n 5ect0r5 t0 the 8ethe-5a1peter e4uat10n 6e10w. 1t 15 1mp0rtant t0 rea112e h0w cruc1a11y the ex15tence 0f a 60und 5tate 1n the adj01nt channe1 h1n9e5 0n the tw0 pr0pert1e5 0f hav1n9 5p1n 2er0 and dea11n9 w1th ferm10n1c c0n5t1tuent5. N0 k1nd 0f part1c1e can have the adj01nt channe1 5tr0n9er than the 1nvar1ant 0ne, and 0n1y f0r a 5ca1ar ferm10n 15 the c0nden5at10n 1n the adj01nt channe1 p055161e 6ecau5e the 5ymmetry 0f the wave funct10n ru1e5 0ut the ex15tence 0f a 9r0up 51n91et 60und 5tate.

7.2. 50LU710N 0F 7HE 8E7HE-5ALPE7ER EQUA710N 7 h e 8ethe-5a1peter e4Uat10n f0r the 60Und 5tate 0f the 9h05t f1e1d5 1n e4. (4.3) 15 (0m1tt1n9 the 9r0Up 1nd1Ce5 and W1Ck-r0tat1n9 a11 m0menta) (1p2 + p 2 + p . p ) ( • p 2 + p 2 • p . p ) X ( p , p )

N92 f°A~42d42d~2 ( (p+4)2•p2 - 2} X(4, P),

8(2~r)4

(p•

4)2

(7.4)

where the f1r5t term 1n the cur1y 6racket ar15e5 fr0m the 9au9e part1c1e exchan9e and the 5ec0nd term fr0m the CC 4-p01nt 1nteract10n. Wh11e the 0ne-9au9e-part1c1e exchan9e 15 attract1ve 1n the adj01nt channe1, the c0ntact (4-p01nt) 1nteract10n 15 repu151ve 1n the 5ame channe1. 7he c0nd1t10n that an attract1ve 4-p01nt vertex w0u1d make the ener9y 0f the 5y5tem un60unded 6e10w can 6e u5ed t0 11m1t the phy51ca11y adm155161e va1ue5 0f the 9au9e parameter a t0 a >1 0. E4. (7.4) 15 a Fredh01m 1nte9ra1 e4uat10n when A < ~ . Expand1n9 1n 6e9en6auer p01yn0m1a15 c9~ [34], the 1nte9rat10n 0f the an9u1ar part 1n e4. (7.4) can 6e

M. Fa66r1che51/ Dynam1ca1ma559enerat10n

105

perf0rmed, 1eav1n9 an 1nf1n1te 5et 0f c0up1ed e4uat10n5 [(p2 + 1p2) 2 -

•p2p2]60(p2 ) • •p2P262(p2 )

= ~Mr2f d42 4260(42)(3p4[91(42, p2) + 9•1(42, p2)] -[p2+42+p2]90(42, p2)}

(7.5)

f0r n = 0, and [(p2

+ 1p2) 2 -

•p2p2]60(p2 ) • •p2p2[6n+ 2 + 6n•2 ]

~k~ 2 - 2(n + 1) f d42

426"(42)(3p4[9"-1(42• p2) + 9,+x(42, p2)] -[p2+42+p2]9,(a2, p2)}, (7.6)

f0r n >1 2. 7he c0eff1c1ent5 6, are def1ned 6y the expan510n 0~

X(4, P) = ~ 6,,(4 2, P2)(9,a(c05f1)

(7.7)

n=0

(f1 15 the an91e 6etween 4 and P), and 1 u5ed 1

(p•4)2

E9m4(2~p2C )1c(°59(0~)m

9,,(42, p2)• 0(42-p2) 0(p2-42) 42 (p/4)m+ p2 (4/p)m, c05(,) -

4•p 141 1P1

(7.8)

Under the a55umpt10n P2/A2 << 1, the e4uat10n5 1n (7.5) dec0up1e, reduc1n9 the pr061em t0 the 501ut10n 0f the 51n91e, n = 0 e4uat10n

90(x) = Xf01dy K( x, y )90(y ) , where

(7.9)

x =p2/A2, y = 42/A2, 90=pf~ + p 4 60 • • = N92/(16~r)2, a = P2/4A2, and

K(x, y) = ~

a2VF~~ {2(1+y)•4a}[0(x-y) x

+ 0 ( yy- x ) ] . (7.10)

M.Fa66r1che51/ Dynam1ca1ma559enerat10n

106

7he kerne1 1n e4. (7.10) 15 5ymmetr1c (6y 1n5pect10n), 54uare-1nte9ra61e 6ecau5e (f0r a 5ma11 6ut n0t 2er0)

f01f011K(x,y)[2dx6y=3[109(1/a2)]2; and p051t1ve a5

a ~

0

(7.11)

a5 can 6e pr0ved 6y wr1t1n9

2f01dxf(x) f01dyf(y)(x +y)[ 0(x ~-y) 1 x

+ 0(yy- x) ]

= 4f01dyf(y){ f9 dxf(x)(1+ Y ) } > 0,

(7.12)

where f(x) = V~4~(x)/~ x 2 15 an ar61trary funct10n. 70 pr0ve the 1ne4ua11ty 1n e4. (7.12) 1 5h0w that 60th term5 1n the 6racket 0f e4. (7.12) are p051t1ve def1n1te: 1et

F( y ) = fy1d xf(x )/x 50 that

f(y) = -yF•(y)

(7.13)

and F(1) = 0. 7hu5, the f1r5t term can 6e rewr1tten

4f016yf(Y) f0y6xf(x)= --2f016y ~-~-~y{fV16Xf(X)}2=2( 1f0 6Xf(X)} >0.2 (7.14) 51m11ar1y, f0r the 5ec0nd term

4f01dyf(Y)Yfy1dxf(x)x - -4f01dyy2F•(y)F(y ) =4f01dYyF2(y ) > 0. (7.15) 80th term5 are thu5 p051t1ve 1f f:9 0; acc0rd1n91y the kerne1 15 p051t1ve. 7he 5ma11e5t e19enva1ue 15 theref0re 60th p051t1ve (a5 a ~ 0) and 60unded a60ve and 6e10w [35]: [A4j-1/4 ~ ~k0 ~ A~2/A4 ~

(7.16)

where

A2n = f1dXK2n(X, X) .'o

K2, = f01dtK,(x, t)K,(t, y),

/4=K(x,y). (7.17)

M. Fa66r1che51 / Dynam1ca1m055 9enerat10n

0m1tt1n9 h19her 0rder term5, 2 ) t 0 - 109(4A2/p2) + 0

(1)

1092(4A2/p2 ) ,

107

(7.18)

and the de51red re5u1t read5 p 2 = 4A2exp ( _ (16¢r)2/2N92).

(7.19)

Even th0u9h p2 15 m0n0t0n1c 1n the c0up11n9 92, the 501ut10n 15 pre5ent f0r any 92 > 0.7h15 c0rr060rate5 the 1adder appr0x1mat10n a5 10n9 a5 the cr1t1ca1 9(A) 15 5ma11 en0u9h. 7he 60und 5tate thu5 f0und refer5 t0 a ne9at1ve ma55 6ecau5e e4. (7.19) 15 1n the euc11dean m0mentum-5pace. 1n 0ther w0rd5, the pertur6at1ve vacuum, fr0m where th15 ma55 15 mea5ured, cann0t 6e a 5ta61e m1n1mum 0f the 5y5tem ener9y 6ecau5e the 501ut10n (7.19) 1mp11e5 t h a t - un1e55 h19her-0rder term5 can c0n5p1re t0 pr0duce effect1ve1y repu151ve CC f0rce5- a c0nden5ate 0f 60und 5tate5 (7.1) 10wer5 the ener9y 0f the 5y5tem. 51nce the 60und 5tate5 (7.1) carry the 9r0up adj01nt repre5entat10n, the1r c0nden5at10n h1de5 the 9106a1 9au9e 9r0up 5ymmetry. 7he t19ht-61nd1n9 11m1t 0f e4. (7.4) 15 a 2er0 ma55 60und 5tate that repre5ent5 the 5et 0f 601d5t0ne 6050n5 a550c1ated w1th the h1d1n9 0f 5uch 9106a1 5ymmetry and they a110w f0r the 5chw1n9er mechan15m (5ee next 5ect10n) t0 pr0ceed. 1 n0te h0wever that 1n the 2er0-ma55 11m1t A---, m e4. (7.4) 15 n0 10n9er 0f Fredh01m type; the 1nfrared 51n9u1ar1ty re5u1t1n9 fr0m A--* m pr0duce5 a c0nt1nu0u5 5pectrum. H0wever, an exact 501ut10n 0f e4. (7.4) 1n th15 11m1t 15 kn0wn [36] and th15 make5 p055161e the ca1cu1at10n 0f the dynam1ca1 ma55e5 1n 5ect. 8. 7.3, 7HE F1ELD5 1N 7HE NEW VACUUM

8ecau5e 0f the vacuum 1n5ta6111ty, the 9h05t f1e1d5 1n the 1nterp01at1n9 re910n ac4u1re a dynam1ca1 ma55 91ven 6y p2/4 1n e4. (7.19). 7h15 ma55 9enerat10n can 6e under5t00d a5 1ntr0duc1n9 a ma55 term 1n e4. (7.4) and 1mp051n9 the 5ta6111ty c0nd1t10n p2>~ 0. 7he 9h05t ma55e5 are thu5 the 5ta611121n9 mechan15m. 7he5e ma55e5 are an effect 0f the 5upercr1t1ca1 1nteract10n and van15h f0r the a5ympt0t1c f1e1d5. 7he 10we5t va1ue 0f m 5uch that P 2 • rn 2 >1 0 15 the va1ue 0f the dynam1ca11y 9enerated 9h05t f1e1d ma55. At the 5ame t1me a ma551e55 5ca1ar 5tate, the 601d5t0ne 6050n, appear5. 7h15 501ut10n 15 the t19ht-61nd1n9 11m1t 0f the 8ethe-5a1peter e4uat10n and 1t5 pre5ence 15 a c0n5e4uence 0f 601d5t0ne•5 the0rem [37]. 7he a5ympt0t1c c0nd1t10n5 are thu5 that the ma551ve 9h05t part1c1e5 90 1nt0 the ma551e55 f1e1d5 Ca5(x)

c ( x ) --,. 2Uc

5(x),

C( x ) ~ 2~/:Ca~( x ) ;

(7.20)

108

M. Fa66r1che51 / Dynam1ca1 ma55 9enerat10n

the c0mp051te 60und 5tate 90e5 1nt0 the 601d5t0ne 6050n f1e1d ~a~(x) 11m 7C( x + ~ ) C ( x - ~ ) --* 21/297a5( x ),

(7.21)

~ --*0

where a11 three 11m1t5 are taken f0r x 0 ~ - m and 7 1n e4. (7.21) 15 the t1me 0rder1n9 0perat0r. 7he 2 c0n5tant5 are ren0rma112at10n fact0r5 that rem1nd u5 that the tran5f0rmat10n t0 the a5ympt0t1c f1e1d5 15 n0t can0n1ca1. A5 p01nted 0ut 1n 5u65ect. 5.1, the5e f1e1d5 mu5t f1r5t 6e r0tated t0 the dec0up1ed W~a5 and H a~ 6ef0re the phy51c5 0f the 5y5tem can 6e apprec1ated. 7he pr061em 0f the va1ue 0f the ma55 ac4u1red 6y the 9au9e part1c1e5 15 d15cu55ed 1n the next 5ect10n. 8. 7 h e 9au9e part1c1e•5 ma55 8.1. 5 C H W 1 N 6 E R MECHAN15M

1n 5ect. 7, 1 5h0wed that the 8R5-1nvar1ant 1a9ran91an a110w5 the f0rmat10n 0f 60und 5tate5 0f the 9h05t f1e1d5. 7heref0re the pertur6at1ve vacuum 0f the 1nteract1n9 f1e1d5 15 n0t 5ta61e w1th re5pect t0 f0rmat10n 0f a c0nden5ate that 6reak5 the 9106a1 9au9e 5ymmetry. 7he 5tructure 0f the H116ert 5pace 15 thu5 n0n-tr1v1a1 and 1t ha5 6een ana1y5ed 1n 5ect. 5. 1n Landau 9au9e, the 8ethe-5a1peter e4uat10n f0r the ma551e55 w0u1d-6e 601d5t0ne 6050n 15 ju5t the e4uat10n f0r a t19ht 60und 5tate 1n 5ca1ar QED w1th n0 4-p01nt 1nteract10n. 7h15 pr061em ha5 6een 501ved 6y Fe1dman and Fu1t0n [36]; the1r 501ut10n a150 5h0w5 the ex15tence 0f a ma551e55 60und 5tate 0f the 5ca1ar part1c1e5. 7 h e fact that 1n Landau 9au9e an e x a c t - a16e1t 1n 1adder appr0x1mat10n 0n1y - 501ut10n f0r the 8ethe-5a1peter wave funct10n 0f the 601d5t0ne 6050n5 15 kn0wn, make5 p055161e t0 ca1cu1ate the dynam1ca1 9au9e-part1c1e ma55e5 created 6y the 5chw1n9er mechan15m. 1n th15 mechan15m the der1vat1ve c0up11n9 0f the w0u1d-6e 601d5t0ne 6050n t0 the (vect0r) 9au9e 6050n v1a A~ 0~ra 1nduce5 a p01e term 1n the 9au9e part1c1e•5 p01ar12at10n ten50r

17~( k ) = { 9~,~k2- k~,k~} Ha6( k2).

(8.1)

7h15 p01e pr0duce5 a ma55 f0r the 9au9e vect0r 6050n5 6ecau5e 1t 5h1ft5 the 51n9u1ar1ty 1n the1r pr0pa9at0r (1n Landau 9au9e)

9~ - k~k J k 2

= ( - 1 ) k2[ 0 + n06(k2)]

(8.2)

fr0m k 2 = 0 t0 k 2 = ( m 2 ) a 6 , where (m2)a6 15 the re51due at 2er0 0f the 5e1f-ener9y

M. Fa66r1che51/ Dynam1ca1ma559enerat10n

109

~-P.4

1

~-P - 4 v

~(P1 F19. 1. 7he p01e term 1n the 9au9e part1c1e•5 p01ar12at10n ten50r. 7he da5hed 11ne5 repre5ent 9h05t part1c1e5; the 9h05t5 f0rm a 2er0-ma55 60und 5tate (d0tted 11ne)w1th the 9au9e 9r0up 4uantum num6er5 0f the 9au9e 6050n.

51n9u1ar1ty 1~a6(0) =

m2)a6 k2

(8.3)

7he ma55 matr1x 15 de9enerate, (m2)a6 = m2~a6, 6ecau5e the 9r0up 1551mp1e. 7he ma55 can 6e wr1tten (5ee f19. 1) m 2= ~ ( 0 ) 0 ( 0 ) ,

(8.4)

where, 6y L0rent21nvar1ance, ~(p2)p~

9N 4 fP~,6(~P + (2•n")

4)X,(4)6(•P-4)d44,

(8.5)

the 6(p)•56e1n9 the 9h05t pr0pa9at0r5 and Xp(4) the 8ethe-5a1peter wave funct10n. 1 have avera9ed 0ut the 9106a19au9e 9r0up 1nd1ce5 t091ve the fact0r N f0r the L1e 9r0up 5U(N). • have a150 r0uted the 1nterna1 m0menta 1n 5uch a way that 0n1y the va1ue 0f • f0r 2er0 m0mentum 15 needed. Let then X = Xe( P = 0), 2 = 92/m2 and everyth1n96e 1n un1t50f ener9y 5uch that m = 1. Fr0m e4. (8.5) 4 ( 0 ) = 4(2~r) 2 9 N f000 (22d~)

"

(8.6)

Fe1dman and Fu1t0n [36] 91ve the c105ed f0rm expre5510n f0r the 5pace-t1me wave funct10n 0f the 601d5t0ne 6050n

X(2) = 6AX2(2 + 1)~-2F(a, 6; c; - 2 ) ,

(8.7)

where X = 92/16~r2, F(a, 6; ¢; 2) 156au55•5 hyper9e0metr1c funct10n, A 15 the

M. Fa66r1che51/ Dynam1ca1ma559enerat10n

110

n0rma112at10n C0n5tant t0 6e determ1ned, and a(•) = • -- 1(1 -- 24h) 1/2, a(~k) = 22-- •(1 -- 24X) 1 / 2 - (1 -- 6X) 1/2, 6 ( h ) = 22-- •(1 -- 24~k) 1/2 + (1 -- 6X) 1/2,

c(x)=3.

(8.8)

7he c0eff1c1ent5 a, a and 6 are funct10n5 0f the c0up11n9 and the1r rea1 part5 are 60unded 6y 0 < R e ( a ) ~< •;

0 < R e ( a ) ~< 3;

3 ~< R e ( 6 ) ~< ~ + 1v/3-.

(8.9)

7he5e 60und5 make the 1nte9ra15 6e10w we11-def1ned. 7he 1nte9ra1 repre5entat10n 0f the hyper9e0metr1c funct10n [34] y1e1d5

2 • ( 0 ) = 6NAX - 9

F(c) r(6)r(c-

6)

Jofldtlb-X(1t)c-6-1r°¢d2J0

22(1 + 2 ) a - 4 (1 + t2) a

(8.10)

that can 6e 1nte9rated* t0

6NA~ 2 F(a + 1 - a ) F ( 1 - a + 2a + 6) • (0)

9

r(2a

. + 4 -

)r(1 -

+

(8.11)

+ 6)

E4. (8.11) 15 1n ener9y un1t5 w1th m = 1 . 7 0 1ntr0duce 6ack the u5ua1 un1t5 1 need an expre5510n f0r the dynam1ca1 ma55 0f the 9h05t5. 1n u51n9 the re5u1t 0f 5ect. 7 1n a d1fferent 9au9e 1 make the unpr0ved a55umpt10n that 5uch a re5u1t 15 0n1y weak1y 9au9e dependent. 51nce 1 am here 0n1y 1ntere5ted 1n a 4ua11tat1ve p1cture, th15 a55umpt10n, wh11e 1t mu5t 6e 60rne 1n m1nd, 5eem5 accepta61e (6ut 1 c0me 6ack t0 th15 pr061em 1n the f1na1 d15cu5510n). 1 theref0re have M

2 ~

~ (0) ~ (0)A2eXp( -- 8 / N h )

(8.12)

1n a9reement W1th e4. (6.3). 8.2. 7HE DYNAM1CAL MA55 7 h e expre5510n 1n e4. (8.12) 15 the f1na1 re5U1t. 7he pr0dUCt 0f the F fUnCt10n5 Can 6e expre55ed a5 a C0m61nat10n 0f hyper6011C fUnCt10n5. * 1 have u5ed 3.197(5) and 7.512(3)1n ref. [38].

M. Fa66r1che51 / Dynam1ca1 ma55 9enerat10n

111

70 determ1ne the n0rma112at10n c0n5tant A, 1 wr1te the char9e carr1ed 6y the 60und 5tate

Q~ = f d2xj~(x).

(8.13)

7he F0ur1er tran5f0rm 0f the current 15

j0(p) = 2p0f~,~X6(p)5(c(p)

(8.14)

Q = 2 f dp°p°e-1P°x°fa5cX6(p°)7(c(p° )

(8.15)

50 that

where p = 0.51nce Qa 15 1ndependent 0f t1me, at x ° = 0, 1 06ta1n the c0nd1t10n

=

6x

•

(8.16)

where

1(~)=f01dUU6-1(1--U)2-6f01dVV~-1(1--V)2-7~f0°°d2

22(2 + 1) "+a-4 (1 + u2)a(1 + v2) ~ (8.17)

ha5 t0 6e ca1cu1ated numer1ca11y. 7he ma55 1ncrea5e5 w1th c0up11n9 9 and van15he5 f0r 9 = 0 (free the0ry) a5 0ne w0u1d expect. H0wever, the c0up11n9 mu5t 6e 5ma11 en0u9h t0 9uarantee the c0n515tency 0f the ca1cu1at10n. 7he ren0rma112at10n 9r0up parameter A 5et5 the 5ca1e f0r the d1men510na1 ma55. 7heref0re, numer1ca1 va1ue5 f0r the ma55 need a prec15e va1ue 0f A and 0f the c0rre5p0nd1n9 c0up11n9 9. 7here are n0 data 1n the weak-1nteract10n ca5e. 1n QCD A 15 kn0wn 6ut 0n1y w1th 109ar1thm1c prec1510n. 1n the pre5ent ca5e, th15 1ack 0f prec1510n 15 n0t a 5er10u5 11m1tat10n 6ecau5e 0f the crude appr0x1mat10n a1ready u5ed 1n ca1cu1at1n9 e4. (8.12). E4. (8.12) can 6e wr1tten, f0r N = 3, a5

v2=9(x)v(x)

(8.18)

where

9A2~k

[ (8.19)

M. Fa66r1che51 / Dynam1ca1 ma55 9enerat10n

112

15 a very 5en51t1ve funct10n 0f the c0up11n9 ~, wherea5

v(X) =

F(a + 1 - a ) r ( 1 F(2a+ 4 - a ) r ( 1

- a + 2a + 6) - a +

a

+ 6)

F*(a + 1 - a)F*(1 - a + 2a + 6)

×F*(2a+4-a)F*(1-a+a+6)

1 x 1()----~

(8.20)

15 r0u9h1y c0n5tant 1n the ran9e 0f c0up11n95 c0n51dered. 7he exper1menta1 QCD c0up11n9 c0n5tant* aM5 -- 0.2 91ve5 •t ~ 1- F0r AM5 --- 1 6 e V the 91u0n ma55 turn5 0ut t0 6e very 5ma11. 7h15 15 rea55ur1n9 51nce at th15 ener9y re91me, the 9au9e-6050n ma55 5h0u1d 6e a very 5ma11 c0rrect10n t0 the a5ympt0t1ca11y free the0ry. Extrap01at1n9 t0 •t = •, 1 06ta1n M-~ 10 MeV. 7h15 5u65tant1ate5 the p05516111ty 0f mea5ura61e ma55 f0r a va1ue 0f the c0up11n9 c0n5tant 5t111 1n51de the pertur6at1ve d0ma1n. M0re ma551ve 91u0n5 can pre5uma61y ex15t 6ut 0n1y 0ut51de the d0ma1n 0f my appr0x1mat10n5. 9. D15cu5510n

A1th0u9h var10u5 k1nd5 0f 1n5ta6111ty t0ward c0nden5ate-f0rmat10n are 6e11eved t0 character12e any 4uantum 9au9e the0ry [40, 41], 1n th15 w0rk 1 91ve ar9ument5 that 5u99e5t the pre5ence 0f a new 1n5ta6111ty. 7he 9h05t 5ect0r under90e5 a c0nden5at10n that carr1e5 the adj01nt repre5entat10n 0f the 9au9e 9r0up and theref0re the vacuum d0e5 n0t 5hare the 5ymmetry 0f the 1a9ran91an. 7he 601d5t0ne the0rem 5h0w5 that ma551e55 5ca1ar 5tate5 are pre5ent w1th the 4uantum num6er5 0f the 6r0ken-9r0up 9enerat0r5. 7he 5chw1n9er mechan15m can thu5 pr0ceed and an e1e9ant way t0 de5cr16e 1t 15 0ffered 6y the 4uartet dec0up11n9 0f Ku90 and 0j1ma. Even 1f the 9au9e 9r0up 5ymmetry 15 h1dden, the 9au9e part1c1e5 ma551ve and the the0ry de5cr16e5 the5e ma551ve exc1tat10n5 rather than the expected ma551e55 0ne5, the p05516111ty 0f rep1ac1n9 the H1995 f1e1d5 a1t09ether w1th 5uch a m0de1 15 far fr0m 06v10u5. A5 any 0ther dynam1ca1 mechan15m, 1f true 1t w0u1d r1d u5 0f the H1995. H0wever, 1 am aware 0f hav1n9 0n1y p01nted 0ut t0 a p05516111ty rather than pr0ved a v1a61e a1ternat1ve t0 the H1995 mechan15m. 7he numer1ca1 re5u1t 1n 5ect. 8 15 5u99e5t1ve 6ut the appr0x1mat10n5 u5ed mu5t 6e 60rne 1n m1nd. 7he m05t 5er10u5 11m1tat10n 15 the 1adder appr0x1mat10n f0r the 8ethe-5a1peter e4uat10n. Unf0rtunate1y 1 d0 n0t 5ee any a1ternat1ve t0 th15 appr0x1mat10n 6ecau5e even the next 0rder f0r the kerne1 15 t00 d1ff1cu1t t0 501ve 1n c105ed f0rm. 1 f0und very 1ntere5t1n9 h0w far 1t 15 p055161e t0 90 u51n9 e55ent1a11y pertur6at1ve meth0d5 1n f1nd1n9 a c1ear1y 501ut10n. 7h15 15 an effect 0f the 51mp11c1ty 0f the 9h05t 1nteract10n c0m61ned w1th the ma551e55ne55 0f the w0u1d-6e

n0n-pertur6at1ve

* 1 h a v e u5ed a n a p p r 0 x 1 m a t e va1ue f r 0 m the re5u1t [39] 0f a 5 = 0.19 •+ 0.06 a n d AM5 = ,,. (~ 88 +1"360.746eV 1n e + e react10n d a t a .

M. Fa66r1che51 / Dynam1ca1ma55 9enerat10n

113

601d5t0ne 6050n5. 7he5e tW0 feature5 make the 8ethe-5a1peter e4Uat10n tracta61e. At the 5ame t1me, a 5er10U5 pr061em rema1n5: the 9aU9e dependence 0f the 501Ut10n. Wh11e the herm1t1C1ty C0nd1t10n 5eem5 5tr0n9 en0U9h t0 f1X the Va1Ue 0f ~, the 0ther parameter a 15 ar61trary. 7he 5tUdy 0f th15 9aU9e dependence 15 d1ff1CU1t6eCaU5e 1t re4U1re5 a 6etter appr0x1mat10n than the 1adder 0ne. EVen th0U9h, 1n 5eCt. 7 and 5eCt. 8, the ex15tenCe 0f a 501Ut10n 15, re5pect1Ve1y, pr0Ved 1n Feynman and 1n Landau 9aU9e (at 1ea5t 1n the ma551e55-60Und 5tate 11m1t), further 5tUdy 0f th15 pr061em 15 5t111 nece55ary. 7he p05516111ty 0f the ex15tence 0f Va1Ue5 0f et f0r Wh1Ch the 5ymmetry 6reak1n9 501Ut10n d15appear5 15 part1CU1ar1y W0rr150me. 51nCe the pre5ent examp1e 0f a 5ChW1n9er mechan15m 5eem5 t0 W0rk 1rre5peCt1Ve 0f the 9aU9e 9r0Up, the phy51Ca1 app11Cat10n5 are n0t 11m1ted t0 the Ca5e 0f Weak 1nteraCt10n5 Where there 15 a rea50na61e C0nf1denCe 1n the ma551Vene55 0f the VeCt0r 6050n5. 1n QCD, 0n the C0ntrary, th15 C0nf1denCe 15 m1551n9. My Va1Ue5 f0r the 91U0n ma55 5h0W that a 1ar9e ran9e 0f ener9y 15 p055161e. EXtrap01at1n9 t0 n0n-pertur6at1ve c0up11n95 (1ar9er than 1) the5e ma55e5 can 6e a5 1ar9e a5 hundred5 0f 6eV•5. 7 h e 1nterp1ay 6etween ma55e5 and c0nf1nement 15, h0wever, far fr0m 6e1n9 c1ear. At th15 5ta9e, the rea1 te5t f0r the the0ry 15 1n weak-1nteract10n phy51c5 6ecau5e, f0r th15 ca5e, the vect0r 6050n ma55e5 are kn0wn. 7he va1ue 0f the ren0rma112at10n 9r0up 5ca1e parameter A, 0n the 0ther hand, 15 unkn0wn. 7he rea50n f0r th15 1ack 0f 1nf0rmat10n 15 the 11tt1e 1ntere5t 0f 5uch a 4uant1ty 1n the 5tandard m0de1 [8]. 1f A c0u1d 6e determ1ned - even w1th1n 109ar1thm1c prec1510n - tw0 5cenar105 are p055161e: e1ther the phy51ca1 ma55e5 are determ1na61e f0r c0up11n95 1n51de the pertur6at1ve re91me 0r they cann0t 6e determ1ned 1n a 5e1f-c0n515tent way u51n9 the pre5ent appr0ach 6ecau5e the c0up11n95 are t00 1ar9e. 1n the 5ec0nd ca5e, the u5efu1ne55 0f my w0rk w0u1d 0n1y 6e c0nceptua1 un1e55 a fu11y n0n-pertur6at1ve 501ut10n can 6e f0und. 51nce 1 d0 n0t have much c0nf1dence 1n the p05516111ty 0f f1nd1n9 5uch a n0n-pertur6at1ve 501ut10n 1n c105ed f0rm* 1 9reat1y fav0ur the f1r5t 5cenar10. F1na11y, 1 have t0 5tre55 that 1 c0mp1ete1y 1eft 0ut fr0m th15 w0rk 0ne pr061em: 7 h e 0r191n 0f the D1rac ferm10n (4uark5 and 1ept0n5) ma55e5. 7h15 15 an 1mp0rtant 4ue5t10n 6ut 1t d0e5 n0t 5eem t0 6e an5wera61e 1n term 0f 9h05t f1e1d 1nteract10n5 6ecau5e, a5 p01nted 0ut 1n 5ect. 4, there 15 n0 c0up11n9 6etween D1rac part1c1e5 and the 9h05t5. H0wever, 0n a pure1y 5pecu1at1ve 1eve1, 1t 15 w0rthwh11e t0 ment10n that 5uch a c0up11n9 - w1th the c0n5e4uent p055161e ma55 9enerat10n - can 6e re1ated t0 a 6reak1n9 0f part 0f the 8R5 5ymmetry [43]. 51nce 1 c0u1d n0t w0rk 0ut any def1n1te pred1ct10n fr0m 5uch a m0de1, 1 dec1ded t0 0m1t 1t fr0m th15 w0rk. 7 h e pr00f 0f the p051t1v1ty 0f the kerne1 1n 5eCt. 7 Wa5 f0Und 6y Pr0fe550r JU11an V. N061e. 1 W0U1d a150 11ke t0 thank h1m f0r many fru1tfU1 d15CU5510n5. 1 thank * Except perhap5 0n the 1att1ce[42],u51n9c0mputer 51mu1at10n5.

114

M. Fa66r1che51 / Dynam1ca1ma55 9enerat10n

Pr0fe550r H.-J. We6er f0r p01nt1n9 0ut ref. [29] t0 me. 7h15 w0rk wa5 5upp0rted 1n part 6y the U5 Nat10na1 5c1ence F0undat10n. Reference5 [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43]

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