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String effective actions from sigma-model conformal anomalies
Nuc1ear Phy51c5 8301 (1988) 197-223 N0rth-H011and, Am5terdam
5 7 R 1 N 6 EFFEC71VE A C 7 1 0 N 5 F R 0 M 5 1 6 M A - M 0 D E L C 0 N F 0 R M A L AN0MAL1E5 C. M. HULL* and P. K. 70WN5END DA M7P, 511ver 5treet, Cam6r1d9e C83 9E W, UK Rece1ved 7 5eptem6er 1987
We c0n51der the 519ma m0de1 de5cr161n9 a c105ed 6050n1c 5tr1n9 1n the pre5ence 0f 9rav1t0n, ant1-5ymmetr1c ten50r and d11at0n 6ack9r0und f1e1d5. U51n9 prev10u5 re5u1t5 f0r the metr1c and ant1-5ymmetr1c ten50r tw0-100p 6eta-funct10n5, we der1ve the d11at0n 6eta-funct10n and 06ta1n the fu11 5et 0f c0nd1t10n5 f0r c0nf0rma1 1nvar1ance t0 the 5ame 0rder. We 5h0w that the5e c0nd1t10n5 can 6e der1ved fr0m any 0ne 0f a fam11y 0f 5pace-t1me effect1ve act10n5. After tak1n9 f1e1d redef1n1t10n5 1nt0 acc0unt 0ur act10n a9ree5 w1th that 06ta1ned 6y Met5aev and 75eyt11n.
1. 8eta-funct10n5 and c0nf0rma1 an0ma11e5
A 6050n1c 5tr1n9 pr0Pa9at1n9 1n a d-d1men510na1 5pace-t1me w1th metr1c 91j 1n wh1ch the ant1-5ymmetr1c ten50r 9au9e f1e1d and d11at0n m0de5 0f the 5tr1n9 have vacuum expectat10n va1ue5 61j, ~6, re5pect1ve1y 15 90verned 6y the n0n-11near 519mam0de1 (euc11dean) act10n [1-4]
1 fd2xVr~{~(9,j(¢)y~+6,j(¢)~)0.¢~0~¢j+~a,~(ep)R,~
5 = 2~ra--~
}
(1)
where ¢1 (1 = 0 . . . . . d - 1) are 5pace-t1me c00rd1nate5, x" (/2 = 0,1) are w0r1d-5heet c00rd1nate5 and y~ 15 the w0r1d-5heet metr1c w1th R (~) the c0rre5p0nd1n9 w0r1d5heet curvature 5ca1ar. C0n515tency 0f the 5tr1n9 the0ry re4u1re5 that th15 act10n def1ne a c0nf0rma11y 1nvar1ant 4uantum f1e1d the0ry, and the c0nd1t10n5 f0r c0nf0rma1 1nvar1ance can 6e 1nterpreted a5 ••effect1ve•• f1e1d e4uat10n5 f0r 91j, 6~j and [3, 4]. 7he 519ma-m0de1 can 6e 4uant12ed u51n9 the ver510n 0f the 6ack9r0und f1e1d appr0ach de5cr16ed 1n ref5. [5-7], deve10p1n9 a pertur6at10n the0ry 1n a•, w1th u1tra-v101et d1ver9ence5 re9u1ated 6y d1men510na1 re9u1ar12at10n t09ether w1th the precr1pt10n ~p~p = - y~
(2)
f0r 51mp11fy1n9 pr0duct5 0f e-ten50r5 1n 2 + e d1men510n5. 7he u1tra-v101et d1ver9ence5 are a650r6ed 1nt0 ren0rma112at10n5 0f 9~j, 6~j, • [8] and we def1ne the *Addre55 fr0m 0ct06er 1987: 81ackett La60rat0ry, 1mper1a1 C011e9e, Pr1nce C0n50rt R0ad, L0nd0n 5W7, UK. 0550-3213/88/$03.50•E15ev1er 5c1ence Pu6115her5 8.V. (N0rth-H011and Phy51c5 Pu6115h1n9 D1v1510n)
198
C.M. Hu11,P.K. 70wn5end / 5tr1n9effect1veact10n5
f1-funct10n5 t0 91ve the dependence 0f the ren0rma112ed 5pace-t1me f1e1d5 9~, 64,R~R 0n a ren0rma112at10n ma55-5ca1e/,: 0 f1~ = ~ p, ~ 9 ~ ,
0 f16.= ~ p ~ 61R ,
0 f1~ = - p. - ~ c6R .
(3)
Hencef0rth we 5ha11 dr0p the R 5uperf1x 0n the ren0rma112ed 5pacet1me f1e1d5 a5 the 6are 5pacet1me f1e1d5 w111n0t appear a9a1n 1n th15 paper. 7here 15 50me dependence 0f the f1-funct10n5 0n the 5u6tract10n pre5cr1pt10n, the ch01ce 0f 6ack9r0und/4uantum 5p11t, and the ch01ce 0f wavefunct10n ren0rma112at10n [5-8], 6ut 91ven a def1n1te • 5cheme•• the f1-funct10n5 are c0mp1ete1y determ1ned. 1n the 5cheme u5ed 1n [7] (1.e. 4uant121n9 a5 de5cr16ed 1n 5ect. 9 0f [5], u51n9 m1n1ma1 5u6tract10n, the n0rma1 c00rd1nate expan510n, and mak1n9 n0 wave funct10n ren0rma112at10n 0f the 6ack9r0und 5pace-t1me c00rd1nate) the f1-funct10n5 (1nc1ud1n9 a c0ntr16ut10n fr0m the w0r1d-5heet reparametr12at10n 9h05t5) take the f0rm [4, 8, 9]
f1~= -a•[ R 4 -
H1ktHy + •a•W4] ,
< : -.•[- v%,, + f1¢= 1 ( d - 26) - a•[-•
V2¢ + Y+ ~ 2 ] ,
(4)
3 where H4k = ~3[~62k ] and W4, X4, Y, 2 are p0wer 5er1e5 1n a• wh05e c0eff1c1ent5 are 10ca1 p01yn0m1a15 1n R4k t, H4e, 0~46 and the1r c0var1ant der1vat1ve5. 1t 15 a c0n5e4uence 0f the f0rm 0f the Feynman ru1e5 that 1,V4 and X12 are 1ndependent 0f and that f1~ 15 at m05t 11near 1n ~; we take Y t0 6e ~-1ndependent and 2 t0 6e pr0p0rt10na1 t0 (der1vat1ve5 0f) 4~. We f0und 6y an exp11c1t 2-100p ca1cu1at10n [7] that
W1j = 5(1j)"~- 0(01t),
X1j = 5[1j]-~ 0 ( 0 t ' ) ,
(5)
w1th
54=~1k1,,,~,~
j
"-- m , -
"~k4t,
(6)
where the curvature w1th t0r510n, R (+), 15 def1ned 1n (A.12) 0f append1x A. U51n9 (A.12) and the n0tat10n 0f append1x A we f1nd that 51] can 6e wr1tten a5
3(1j) = 81k•mRj k•m -- ( ~7kH~m1)( ~7k91mj ) ~- 1( ~719k~rn)( ~7j9 k~m) + 2R1k1jHkmnH1mn + 2Hk1mHmn(jR1)nk1 + 2Htp1Hj4kHmkPHm 4~- 2Htp1HfkHk•••••H•
,
5[1j1 = 2(V'kHtm[j) R~1~ - 4(V'kHm41) Hj1PtHm~ + 2( 9r1Hk1j) HkmnH•m, .
(7)
C.M. Hu11,P.K. 70wn5end / 5tr1n9 effect1ve act10n5
199
Recent 1ndependent ca1cu1at10n5 [10-14] a9ree w1th th15 re5u1t, 0r are e4u1va1ent t0 1t after f1e1d redef1n1t10n5 [10,14], and a150 a9ree w1th a re5u1t 6y 805 [15] f0r the 5pec1a1 ca5e 0f a 9r0up man1f01d. 7 0 ca1cu1ate f19 and f16 1t 15 5uff1c1ent t0 c0n51der 0n1y f1at w0r1d-5heet5, 6ut t0 06ta1n f1~ the exten510n t0 curved w0r1d-5heet5 15 needed. U51n9 the techn14ue5 de5cr16ed 1n [5, 6] we f1nd fr0m tw0-100p ca1cu1at10n5 that
Y = aH1jkH 1jk + a~Y + 0(0d•2) , 2 = •(V" 1W2r6)H1ktHJkt + 0(a•).
(8)
7he c0eff1c1ent a and the c0ntr16ut10n Y can 6e f0und fr0m (ted10u5) tw0- and three-100p ca1cu1at10n5, re5pect1ve1y. Rather than ca1cu1ate the5e 4uant1t1e5 d1rect1y, we 5ha11 5h0w 6e10w that they can 6e determ1ned 6y certa1n c0n515tency c0n51derat10n5, w1th the re5u1t that a = - ~ wh11e Y 15 91ven 1n (37). 065erve that there 15 n0 c0ntr16ut10n t0 Y pr0p0rt10na1 t0 the 5pace-t1me curvature 5ca1ar R, 1n a9reement w1th a re5u1t 4u0ted 1n [16]. A1th0u9h there 15 a 2-100p 9raph 91v1n9 5uch a c0ntr16ut10n, 1t 15 cance11ed 6y a 9raph 1nv01v1n9 a 0ne-100p c0unterterm. 7h15 a150 a9ree5 w1th the re5u1t5 0f [18,25], 6ut a5 0ther auth0r5 have c1a1med t0 f1nd a n0n-van15h1n9 c0eff1c1ent f0r th15 term (5ee ref. [4], f0r examp1e) we 91ve deta115 0f the ca1cu1at10n 1n append1x 8. We 5ha11 5ee 6e10w that the a65ence 0f a term pr0p0rt10na1 t0 R 15 a150 re4u1red 6y c0n515tency. 1f the f1-funct10n5 are tr1v1a1 (1.e. 1f they van15h up t0 the am619u1t1e5 1nherent 1n the1r def1n1t10n) then the the0ry 15 r191d 5ca1e 1nvar1ant, 1.e. the 1nte9rated trace an0ma1y van15he5 (f0r c0mpact w0r1d-5heet5). 7he 10ca1 5ca1e, 0r c0nf0rma1, 1nvar1ance needed here re4u1re5 that the trace an0ma1y van15he5 10ca11y, wh1ch re4u1re5 the van15h1n9 0f certa1n ••8-funct10n5•• [5, 6, 16,17] 91ven 6y 85 =
- -• v(,vj
=
. H1,kv
8
= 8 * - •.•v•
,
- .• v
,wjj , (9)
f0r 50me vect0r5 v 1, w 1 c0n5tructed fr0m R1jk1 , H1j k and 0 ~ and the1r der1vat1ve5 and wh1ch can 6e ca1cu1ated pertur6at1ve1y. 80th the f1-funct10n5 and the vect0r5 v~, w1 are 5cheme dependent, 6ut the c0m61nat10n 0ccurr1n9 1n (8) depend5 0n1y 0n the 5u6tract10n pre5cr1pt10n and n0t 0n the nature 0f the 6ack9r0und/4uantum 5p11t 0r wave funct10n ren0rma112at10n5. Further, the dependence 0f the 8-funct10n5 0n the 5u6tract10n 5cheme can 6e a650r6ed 1nt0 a redef1n1t10n 0f the 5pacet1me f1e1d5. 7he effect1ve f1e1d e4uat10n5 f0r the ma551e55 m0de5 0f the 5tr1n9 are, then, that the 8-funct10n5 van15h ( 8 = 0). 1n the next 5ect10n we 5ha11 5h0w, w1th0ut a55um1n9
200
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
the ex15tence 0f an act10n, that the5e 8-funct10n5 5h0u1d 5at15fy certa1n ••c0n515tency c0nd1t10n5••. 1n ref. [7] we c1a1med that the /3-funct10n5 4u0ted a60ve 1ead t0 8-funct10n5 5at15fy1n9 the5e c0nd1t10n5 when • 15 c0n5tant, and 0ne purp05e 0f th15 paper 15 t0 5u65tant1ate th15 c1a1m. Furtherm0re, when we 1nc1ude 4~-dependence, we f1nd that the c0n515tency c0nd1t10n5 and the tw0-100p re5u1t5 a60ve determ1ne/3~ and 8 ~ up t0 0rder a •3, w1th0ut hav1n9 t0 perf0rm three-100p ca1cu1at10n5. 0 u r re5u1t5 f0r the/3 and 8 funct10n5 are 5ummar12ed 1n e45. (41)-(46). 7he va1ue 0f a 51n91e c0eff1c1ent, 4, appear1n9 1n the f1e1d e4uat10n5 8 = 0 rema1n5 undeterm1ned; 1t cann0t 6e f1xed 6y/3-funct10n c0n51derat10n5 a10ne, 6ut w0u1d 6e 6y a tw0-100p c0nf0rma1 an0ma1y ca1cu1at10n (wh1ch we have n0t yet perf0rmed). H0wever, a11 4-dependence can 6e rem0ved t0 th15 0rder 6y a f1e1d redef1n1t10n. We 5ha11 5h0w that, at 1ea5t t0 th15 0rder, the three f1e1d e4uat10n5 8 = 0 can 6e der1ved fr0m an act10n, whatever the va1ue 0f 4. A 5pacet1me effect1ve act10n der1ved fr0m 519ma-m0de1 c0nf0rma1 1nvar1ance ha5 6een 91ven prev10u51y 6y Met5aev and 75eyt11n [10] (and 6y 2an0n [11] f0r c0n5tant d11at0n f1e1d ~; 5ee a150 ref. [14]). 0 u r re5u1t5, 50me 0f wh1ch were rep0rted at the 7r1e5te 1987 5pr1n9 W0rk5h0p 0n 5uper5tr1n95, are 1ar9e1y 1n a9reement w1th the1r5. H0wever, wherea5 1n [10,11,14] 1t wa5 0n1y checked that vary1n9 the act10n 9ave 50me 0f the f1e1d e4uat10n5, we are a61e, w1th 0ur kn0w1ed9e 0f 8 ~, t0 check here that the var1at10n 91ve5 a11 three f1e1d e4uat10n5 8 = 0. Met5aev and 75eyt11n have a150 der1ved an effect1ve act10n fr0m 5tr1n9 5catter1n9 amp11tude5 [10]. We f1nd that th15 a9ree5 w1th the act10n 06ta1ned fr0m 519ma-m0de1 meth0d5 whatever the va1ue 0f 4, wh11e Met5aev and 75eyt11n c1a1m that a9reement 15 0n1y p055161e 1f 4 = ~ [10] (4 15 re1ated t0 the 11 0f [10] 6y 4 = -411).
2. 7he 5tructure 0f the 8-funct10n5
An 1mp0rtant re5tr1ct10n 0n the 5tructure 0f the term5 appear1n9 1n the f1- and 8-funct10n5 can 6e 06ta1ned 6y c0n51der1n9 the Feynman ru1e5. 1f we make the r191d 5ca11n9 91j ~ ~91j ~
61j ~ ~61j ,
~ "~ 0 ,
(10) f0r 50me c0n5tant 9, then 1t can 6e 5h0wn that each 0f the f1- and 8-funct10n5 5ca1e 1n the 5ame way a5 the c0rre5p0nd1n9 f1e1d, e.9. f 1 ~ ~2f19, 8 ~ 8 ~. Further, under the ••H-par1ty•• tran5f0rmat10n H/j~ --~ - H ~ j k
(11)
0ne can 5h0w that f16 and 8 6 a r e 0dd, wherea5 the 0ther funct10n5 are even, e.9. 8~-8~,6ut 8~8~ and 8 ~ 8 ~.
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
201
F r 0 m the 9enera1 f0rm 0f the 8-funct10n5 t0 a11 0rder5 91ven 6y (4.9) and u51n9 the 1dent1t1e5 0f append1x A we 06ta1n
01t
~
6**k1m**
--
(12) 1 a , [ V ~ 8 ~ - 0 ~ 8 ~ . ] =(V~-~J)V~,wj~+ ~J~Vj0~+a~(V~-v~)X,~.
(13)
A5 the f1r5t term 0n the r.h.5. 0f (12) 15 the 9rad1ent 0f a 5ca1ar, th15 mu5t a150 6e true 0f the term 1n cur1y 6racket5 f0r 6ack9r0und5 f0r wh1ch 8 9 = 0, 8 ~ = 0, 1n wh1ch ca5e the ent1re r.h.5. 0f (12) w0u1d 6ec0me ~ P f0r 50me P c0n5tructed fr0m curvature5 etc. 7h15, and the van15h1n9 0f the r.h.5. 0f (13) when 8 ~ = 8 6 = 0, 1mp05e5 5tr1n9ent c0n5tra1nt5, 0r ••c0n515tency c0nd1t10n5••, 0n the f0rm 0f W,j, X~j, v~, w~. F0r examp1e, w1th W,j, X,j a5 91ven 1n (5.6), we 5h0w 6e10w that there d0 1ndeed ex15t vect0r5 d, w 1 5uch that the r19ht-hand 51de 0f (12) 15 the 9rad1ent 0f 50me P when 8~. = 0, 8 ~ = 0, at 1ea5t up t0 term5 0f 0rder a •2. 0 n the 0ther hand, f0r the W,7, X9j 91ven 6y the re5u1t5 0f an ear11er 1nc0rrect 6eta-funct10n ca1cu1at10n [27] there ex15t n0 vect0r5 d, w ~ 5uch that th15 15 true (cf. ref. [23]). N0te that wh11e the5e c0n515tency c0nd1t10n5 are c105e1y re1ated t0 0ne5 that can 6e 06ta1ned 6y demand1n9 that the f1e1d e4uat10n5 8 = 0 6e der1va61e fr0m a 9au9e1nvar1ant act10n [21], we are n0t mak1n9 any a55umpt10n5 a60ut the ex15tence 0f an 1nvar1ant act10n here. 7 0 10we5t 0rder, the m05t 9enera1 f0rm that w and v can take that 15 c0n515tent w1th H-par1ty and 5ca11n9 pr0pert1e5 15 w1= 0 + 0(a•) and v~ = • ~7~ + 0(a•) f0r 50me c0n5tant ~, 50 that the c0n515tency c0nd1t10n5 are 5een t0 6e 5at15f1ed, w1th P = ~ R - - 6 . . 1 j*k, *4 *
.1j,
-~ 1¢ 172(~) - - ~~¢ 2 ( 1 ~ 7 ~ ) ) 2 -+-
0(~) .
(14)
We 5ha11 5ee 6e10w that the c0n515tency c0nd1t10n5 c0nt1nue t0 6e 5at15f1ed at the next 0rder, and we 5ha11 c0mpute the next 0rder c0ntr16ut10n t0 P. When 8 9 = 8 6 = 0 we kn0w that the expre5510n den0ted 6y P 15 a c0n5tant. 0 n the 0ther hand, 1t 15 a c0n5e4uence 0f w0r1d-5heet d1ffe0m0rph15m and c0nf0rma1 Ward 1dent1t1e5 that 1n 6ack9r0und5 f0r wh1ch 8 ~ and 8 ~ van15h t0 a 91ven 0rder 1n a~, 8 e mu5t 6e c0n5tant t0 at 1ea5t that 0rder [4,18, 25], 1.e. ( 8 ~ = 0(a~L), 81~ = 0 ( a ~ c ) } ~ ~18 e = 0 ( a ~ L ) .
(15)
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
202
F r 0 m (4.9), we 5ee that 8 ~ = ~(d - 26) + 0(a•). 7he m05t 9enera1 f0rm that 8 ¢ can take t0 the next 0rder that ha5 the r19ht 5ca11n9 pr0perty can 6e wr1tten a5
( 8 ° / a ~) = ( d -
+ p[ P + ~91J( 89/a,)] + 1~(~7Q[})2
26)/3a• +
f0r 50me c0n5tant5 p, ~,/~, ~-. A5 8 ~ and P mu5t 6e c0n5tant f0r 6ack9r0und5 1n wh1ch 8 9 = 8 6= 0, the c0eff1c1ent5/~ and ~- mu5t van15h, 50 that 8 * = ~ ( d - 26) + pa•[
P + ~91J( 8~/a~)] + 0( a •2)
8=~(d-26)+~0a~[(1+X)R-(13
+~.)H1jkH 1j~
+ 1¢(2 + ~k11y2~ - K2(1[7~312] + 0(a~21 .
(161
7h15 15 t0 6e c0mpared w1th the f0110w1n9 expre5510n f0r 8 ~ wh1ch we 06ta1n u51n9 (4, 8, 9) and the 10we5t 0rder f0rm5 f0r 0, w
8a~=~(d-261+a~[~W2~-1x(W4312-
y]+0(ad).
(171
C0mpar1n9 the ¢6-dependence 0f the5e tw0 expre5510n5 f0r 8 ~, we 5ee that they are c0n515tent 0n1y 1f 0 = 1/x, X = - 1 , and we then 1earn that
Y=
1 - ~
H1jk H 1 j k + 0 ( 0t•).
7hu5 c0n515tency f1xe5 the c0eff1c1ent a 1n (8) and c0nf1rm5 the re5u1t 0f the ca1cu1at10n 0f append1x 8, 1.e. the a65ence 0f a term pr0p0rt10na1 t0 R. 7 0 5ummar12e, kn0w1n9 f19 and f16 ha5 ena61ed u5 t0 determ1ne 8 ~ and f1~ up t0 an unkn0wn c0eff1c1ent r, A 5tra19htf0rward ca1cu1at10n e1ther 0f 8 ~ 0r 0f f1¢ [6] 91ve5 K = 1, 50 the f1na1 re5u1t f0r the 8-funct10n5 t0 10we5t 0rder 1n a• 15
819j= -a•[ R1j- H m H y + X71WfP] + +
80
•3( d -
261-
0(a•2),
+
a•[-• X72~+ ~(W4~) 2 -- 1u 3aa1jk u1jk] x" ] + 0(a,2)
"
(181
We have checked 6y exp11c1t ca1cu1at10n 0f the c0nf0rma1 an0ma1y that th15 15 the c0rrect re5u1t f0r 8 0 when H1jk = 0.
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
203
We n0w w15h t0 extend th15 ana1y515 t0 next 0rder 1n a•. 7 0 th15 end we f1r5t 1ntr0duce the n0tat10n H12 ~- H1mnHjmn~,
H 2 ~ 91jk0'Jk
,
A -= ( d - 26)/3a•,
(19)
and rec0rd the f0110w1n9 1dent1t1e5 5at15f1ed 6y the ten50r 5~j 91ven 1n (6):
( v + - v++) 5<,j) - t/~+~x +~ - v~7 = 2 { < ~ " v~8,~ - 2 ( u ~ ) ~ v~,8~k -/-/7~"w~m v~8r, 0t t
- - H k 1 P ( ~ 7 k H 1m1 ) 8 m9p - -
(VkH1m~)
6 - [ - ~1 .L1k1mD p 1216 V18km ... •k•--mp
+ t1,4A-1~,1-1J~ - 1-11,%~°14~0~% } + 0( ~) , ( v ~j • v,J~)5L~j~
~7
2 { - - 21" *n1k ~ , . 7 . k 8 6 ~ H 1 j H , . k p 1 7 k 8 2 ~ R 6 A~ ( H2)k11V1861k-- ( ~7kH11m ) ( , ~ 7 1 8 9 k ) + Hmff ,¢•7k1-1m• v ~ 1~pt-~- 1--1~ k•m1-1 P1719 -- L1mkP1--1 4L11pR9 -2*~1 **1m ~ k p ** **m1 ~*1 ~ k 4
( 9 2) k1H1 •48 9 4 } 31.-0(0L~) (20)
where 7= 10
1~1jk1 -- 1111jmL1k1 D 4*~1jk1 ** 2 ~* J* ma.1jk1 1(H2"~ ~ ¢H2)1j -- "2~ ] •j•
+ t14 ~41 1421 L1 kmn 6**1jk~" 1m ~" n *~ "
(21)
A5 we have 5een, 01 and w 1 have the f0rm
v,=V1~+a~u1+0(a~2),
w~=a~Y1+0(a~2),
(22)
f0r 50me (a•-1ndependent) vect0r5 u~, Y1. 7hen u51n9 (20)-(22) 1n (12), (13) we f1nd
-
•
2a•[ V"J 17t1uA- 1Wc6Vt1uA + H/k ~7jyk ] = J
1•
1 7 Q 1+ 0(a•2),
Q1 + 0(a•2),
(23)
204
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
where
+a•[-uJ8 9 + R1•"• k V"~8tm 9 - 2(H 2) k1 V~tt8~k- H 1prnH 1km ~7k81p9 - H k f ( VkHt~1)8~ p - ( VkH~1)( X7182m) + 1L[k1mRp 1t6 2.... 1 k1~rnp 4-H
L1mkP1•4 41D6 -- H1Pk(H2)k18~p] 14k ~ ~1 m ~t.~1p
"~-0(0ff2)
"•[-01t[ - u J 8 6 - 2"~11Rk1m ~7k86m -- H1p1HmkP ~7k86m14- 1-1Pmk(V7kL1m1 ~1716
4• (H2)
k117~0 6 • (17kH11m ) 1718 9mk -6 1 R k•mM PR 9 2~1 ~1rn ~ k p
-- HmkPHm14H11P894 -- ( H 2 ) k1H14894] "k- 0(0ff2).
(24)
N0te that 1f u and y were cur1-free, then fr0m (23) the c0n515tency c0nd1t10n5 w0u1d 6e 5at15f1ed. F0r ar61trary ten50r5 5~j there w0u1d n0t 1n 9enera1 6e any vect0r5 u and y 5uch that th15 15 true. 7hat there ex15t 5uch vect0r5 f0r the 51j 91ven 1n (6) 15 a 5tr0n9 check 0n the re5u1t5 0f 0ur f1-funct10n ca1cu1at10n5. (Ear11er 1nc0rrect ca1cu1at10n5 d0 n0t pa55 th15 te5t.) 7he vect0r5 u1, Y1 c0u1d 6e ca1cu1ated d1rect1y and 0ne c0u1d then check whether the re5u1t 15 fu11y c0n515tent w1th (23). 1n pract1ce th15 ca1cu1at10n 15 rather 1nv01ved, 50 that we 5ha11 1n5tead u5e (23), (24) t0 9a1n 1nf0rmat10n a60ut u~ and y9. 7 h e m05t 9enera1 f0rm5 u~ and y~ can take c0n515tent w1th the re4u1red H-par1ty and 5ca11n9 pr0pert1e5 are
u1= V1[k1R + k2 H2 + k 3172~ff" + ~1~[ k5R + k6 H 2
k4(~7(~) 2]
4- k7172(~ "~ k8( ~7(~)2]
+ VJ~P[k9R1j+ k10(H2)12] + knH1jk(W•HJkt ) , y1=k12(~7JH1jk)~7 k~ ,
(25) (26)
f0r 50me c0n5tant5 k n. 1f 8 9 and 8 6 van15h t0 10we5t 0rder, 1.e. 1f 8 9~- 0 + 0(ct•2), 8 6 -~ 0 + 0(ct •2) then Y1 = 0 and 0n1y 5even 0f the term5 1n (25) are 1ndependent; under the5e c1rcum5tance5, 0ne can ea511y check that 1t 15 nece55ary that V~[~uj1-= 0 + 0(a•) f0r (23) t0 6e 5at15f1ed. 7h15 c0nd1t10n rem0ve5 a11 6ut three 0f
C.M. Hu11,P.K. 70wn5end / 5tr1n9effect1veact10n5
205
the 1ndependent term5 1n (25), W1th the re5U1t that U~ Can 6e Wr1tten a5
u1 = U1[ pR + 4H 2 + r( ~7¢)2] + u1,
(27)
f0r 50me c0eff1c1ent5 p, 4, r and 50me vect0r f11 that 5at15f1e5 ~1=0 + 0(a•) whenever 8 9 and 8 6 van15h t0 10we5t 0rder. Hence, u51n9 (27) 1n (23), we c0nc1ude that, when 89 and 8 6 van15h t0 10we5t 0rder, ~71P = 0,
(28)
where P = ~ ( R - - ~ H 2 + 2 V 2 * - - ( 1 1 7 " ~ ) 2)
W2(pR +4H2+r(W(6)2)-2V•1(pR
+ •a•[7+
+4H2+r(W*)2)W1*+A]
+0(a•2),
(29)
wh1ch 15 the exten510n 0f (14) t0 the next 0rder 1n ct•. U51n9 0ur re5u1t5 f0r 8 * t0 0(a•2), we can wr1te 1
--8~=A+P+~9
1
1j
( 8 19 J a )P+ ~ a1A +t 0 ( ~ 2 )
01 t
=A-[-~W2(6+~(~+)2-+H -~-
2]
101t[7-- 101J~20~,j -+ W2( pR + 4H 2 + r( VC6)2) -2W1(PR+4H2+r(W~)2)W1c~+A]+0(a~2),
(30)
where A 15 50me a•-1ndependent expre5510n 0f appr0pr1ate 5ca11n9 we19ht and H-par1ty t0 6e determ1ned. 1f 8 9 = 8 6 = 0, then 8 0 and P are c0n5tant, and hence W1A = 0 + 0(a•). Fr0m (4), (8), (9), (22), (23) we have the a1ternat1ve expre5510n f0r 8 ~ 1 -8°=A
[
3(v0): 1H:]
01 t
x ~ 1j + 1 W 1 ( p R + 4 H 2 + r ( W , ) 2 ) W 1 , + 1 ~ 1 V . , , ] --ct~[Y+~(W1Wj¢~)(H -)
(31)
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
206
and a c0mpar150n 0f th15 w1th (30) y1e1d5 the e4uat10n Y~1-17--1a1J~40 ~1j --4-~ 7 2 ( P R + 4H 2 + r(~7~)2) + 7u1-1V1~
-1v~1(p9+4H2+r(v~)2)v~1~+~(v~,%e~)(H2)~J +~A =0(a~). 8y mean5
(32)
0f the 1dent1ty
( V~1Vj~)( H2) 1j= t( H2)1J[-R1j + (H2)1j - ( 819/a•)] +(1 -t)V1[H~k~(V~Hjk, - (8~t/a~)) ] -(1-t)
[1(V•1(/))V••H2+
(V•1H1k•)(V•;Hjk, - (8~Ja~))]
+0(a•),
(33)
that f0110w5 fr0m (18), f0r ar61trary c0n5tant t, we can wr1te (32) a5
~, ~ j + ~V2(pR + 4H2) + ~t(H2)u(• • JR1j-~H12) [ ~ + ~ 7 ~ ~/,,~ 1--1
9
t
+ •p( V, 4~) V1[ 9k~( 82Ja•)] - •(1 - t ) H~k'~( 86,/a •) } = - ~p (V,~)V~ 7 2 ~ + •(4 + p + ~(1 - t)) V,~ 1Y1H2 -- •p ~1 ~ ~y1 ~,2~ --
2 ~r2(~7~) 2 4- rV1 ~ ¢ 171~ V j4~ + 0(a•).
(34)
065erve that the term 1n 54Uare 6raCket5 1n (34) 15 ~-1ndependent Wh11e the term 1n CUr1y 6raCket5 15 C0n5tant Up t0 term5 0f h19her 0rder 1n a• When 8 9 8 6= 0 + =
0(a•2).
1t 15 c0nven1ent t0 ch005e the parameter t t0 6e 91ven 1n term5 0f 4 6y
p+4+~(1-t)=0
~
t=1+6(4+p),
(35)
50 that the 5ec0nd term 0n the r19ht-hand 51de 0f (34) van15he5. 7hen when 8 9 and 8 6 van15h t0 10we5t 0rder, the 1eft hand 51de 0f (34) 15 1ndependent 0f (/1 (up t0 0(a•)) wh11e the ~-dependence 0f the r19ht hand 51de cann0t 6e e11m1nated u51n9
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
207
8 9 = 8 6 = 0 ( 0 t •2) un1e55 p = r = 0. A5 part1a1 c0nf1rmat10n 0ne can 5h0w that there are a num6er 0f Feynman d1a9ram5 that c0ntr16ute t0 4, wherea5 we have f0und n0ne that c0ntr16ute t0 p 0r r. We c0nc1ude that p = r = 0. C0n51der n0w the ca5e 0f 9enera1 6ack9r0und5 1n wh1ch the 10we5t 0rder f1e1d e4uat10n5 d0 n0t nece55ar11y h01d. A5 p = r = 0 (and ch0051n9 t a5 1n (35)), a11 the 45-dependence 0f (34) mu5t 6e c0nta1ned 1n the cur1y 6racket5. Hence A mu5t 5at15fy A = -f11171d~+ ( 6 4 + 1)(H2)1J(819Ja •) + 34H~k19r~(86t/a~ ) + A ,
(36)
f0r 50me X wh1ch 15 1ndependent 0f • and wh1ch 5at15f1e5 W,.d= 0(a•) when 8 6 = 8 6 = 0(a•2). 8y wr1t1n9 d0wn the m05t 9enera1 f0rm A-can take c0n515tent w1th the5e c0nd1t10n5 and w1th the 5ca11n9 pr0perty (10) we 5ee that 5uch an .4mu5t 6e 0f the f0rm A = c / a •2 f0r 50me c0n5tant c. H0wever, n0 5uch term c0u1d ar15e 1n pertur6at10n the0ry, 50 that we c0nc1ude that A = 0. 7hen u51n9 (35), (36) 1n (34) we f1nd the f0110w1n9 exp11c1t f0rm f0r Y:
= • 1 7 + 1 1j~ • 1 172H2 ~9 ~1j ~4 + 34H~ktW~ WjHJk1 + •(64+ 1 ) ( H 2 ) 1 J ( R 1 j - - ( H 2 ) 1 j ) .
(37)
7 h e n f1e 15 91ven 6y (4.8) and 8 e can 6e f0und fr0m (9) u51n9 the f0rm 0f v 1 91ven a60ve. A1ternat1ve1y, 1n5ert1n9 (37) 1nt0 (31), u51n9 (18) and the 81anch1 1dent1ty f0r H1j k, 91ve5 u5 the expre5510n 18e 01t
=A-[-~
1 V2~, +~(W~)2-~H2]+~a~(7
-101-~v ] + Q + 0(a~2), 2 0 ~1j1
(38) where
Q = ~(H2)1J8~ + 3 4 H 1 ~ [ H J k , 8 ~ - W18~t] -~4~f11W1~,
(39)
wh1ch van15he5 when 8 9 = 8 6 = 0, and
¢)• = ~ + a•4H 2 .
(40)
0 u r re5u1t5 f0r 8 • depend 0n the c0eff1c1ent 4 1n the 5ame way that 0ur 10wer 0rder re5u1t5 u51n9 th15 meth0d depended 0n the c0eff1c1ent •; 1n 60th ca5e5, a ca1cu1at10n 15 needed t0 f1x the c0eff1c1ent.
208
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
Putt1n9 a11 the5e f0rmu1ae t09ether, we f1nd the f0110w1n9f0rm f0r the 8-funct10n5 1n the 4uant12at10n 5cheme 0f [7]:
8 9= a•[-R1j+ (H2)1j1 ~ 1• k1m -1V 2R1kU( H 2 ) k1 + "~)1~ -- ~1a t 2 [[#.1k1m#,~j 14 mnL1k1 ~*tk1m(1**j) ** n
+ •( V1Hk1m)( ~jH 7M ) -- (WkHt,m)(VkHt5) + 2H11pHj4kHmkPHm41 -211fH2,k(H:) kt] + 0(a•3), ~ 6 = 01t( ~7kH1jk ) --
~0[t2[2( ~7k,~ra[j )R1]k•
(41)
m ..}.• 2( X7k",U)(
H2) k,
--4(~71"km[j)H1]knH1mn ]
-1-0(a•3),
(42)
f1v= •(d- 26) + a•(• W2~ + }H 2) - •a•2 [•R1jkt R1jkt- }( U1Hjkt)(97•HJkt) + ~H1j, Hk~H:~Ht~ 1 + ( 6 4 - 1~1-~11jm1-4k1 1~ 21 . . . . ,,•••1jkt-(64-•)(H2)1j(H2)
1j 4V "2H2
+ 24H 1j~ V2H1jk + ( H2) 1j V1~~7j.(/)]"~-0(0~t3) ,
(43)
and f0r the 8-fUnCt10n5
8~= a~{-- R11+ ( H~)1j-- V117jdP 1 t [[1K1k1m1. n n1k1m -- 701 ~ ..}- 2R1ktj( H 2) k1.~ ~at~k1m(1~*j) ")1~ 14 mn14 ~" k1n
+ }(v,1-1~,,.)(9.1-17M) - (v#-1,m1)(v~1-1~5) + 2H1/pHj4kH•nkpHm 4t
-2H, f~j.(H~) ~] + ~v<,~j>} + 0 ( ~ ) ,
•
}0/t[2 1~kH1m[jR1]k1m "~ 2(17kH11j)( H
+a •H1j* -uk
+ a • 117[1Yj1 }
-1-0 ( ~ ) ,
2
)
k1
--
(44)
4( ~71Hkm[j)H1]knH1m n] (45)
209
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
8 ~ = ( d - 26)
2
•
(
3
01¢ [ 11~
1~1jk1
2
-
-- 1151k1 1-1P4mD --Rkt(H2)k1+ 2"~ m "~ ~p4k1
+3(92)1j(H2)1j]}
+ 01'Q +
5rt
rt1 Hjt rrk,,,,,
6"a1jk~a 1rn~a n ~.t
0(0¢'3).
(46)
(We have ch05en t0 wr1te/3 • 1n the f0rm (4.8) and 8 • 1n the f0rm (38). 70 pa55 fr0m 0ne f0rm t0 the 0ther re4u1re5 u5e 0f (33).) Fr0m (26), the vect0r Y1 take5 the f0rm Y1 =
k12 ~7k~(96/0t~) •
51m11ar1y, a5 f11 van15he5 w1th 8 9 and 8 6, 1t mu5t take the f0rm f11= ~ V J ~ ( 8 , ~ / a ~ ) + f1V1[9Jk(8~/a~)] + 7(V1~)9k~(8~t/a ~) + 0(a•),
(47)
f0r 50me c0n5tant5 a,/3, y. 51nce y1, f11, and Q van15h w1th the 8-funct10n5 the y~, ~,, and Q dependenee 0f the e4uat10n5 8 = 0 can 6e rem0ved 6y tak1n9 11near c0m61nat10n5 0f them. 7hu5 the c0nd1t10n5 f0r c0nf0rma1 1nvar1ance 0f the 519mam0de1 w1th act10n (1) 6ec0me
R,j - ( H 2 ) , j + V, Vj4) 1 ,[a,1 ~ j k1.,+ + -~0f [1K1k1m1~
2R1kt1(H2)k1+ 2R k1m(1~j) 14 mn1•1k1 a1 n
+
-
+ 2H1~pHj4kHmkPHm41+2H1,pHjP( H2) k~] + 0 ( a •2) = 0, ~7kH1jk -- ( 17"kdP•) HUk +
0t•[( VkH1m[1 ) Rj1k1 m --
( WkH11j )( H 2 ) k1
--2(~7~Hkm[1)Hj]knH1m n]
4-0(0/•2) =0,
2A + V24~~ - (~7~)2 + ~H 2 --01t[ 1-147 ... 1jk1 • R 1 j ( H 2 ) 4~1jk14"1~1jk1-- 1(171Hjk1)(17~HJk1 ) -- 21--91J m .14k1mD +5~1-96~t1jkH11m Hj•.
1j
Hk~n+ ~3(H2~,,.j (H2) 1j] + 0(a•2) = 0 . (48)
210
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
7he 0n1y undeterm1ned 4uant1ty 1n the5e e4uat10n5 15 the va1ue 0f the c0n5tant 4, wh1ch 0n1y appear5 1mp11c1t1ythr0u9h ~/1,(e4. (40)). A11 4-dependence can theref0re 6e rem0ved 6y the f1e1d redef1n1t10n • ~ • - 4a•H 2.
3. Effect1ve act10n5
7he 4ue5t10n 0f whether the e4uat10n5 der1ved 6y demand1n9 519ma-m0de1 c0nf0rma1 1nvar1ance are 06ta1na61e a5 the Eu1er-La9ran9e e4uat10n5 0f a 5pacet1me ••effect1ve act10n•• ha5 6een d15cu55ed 1n a num6er 0f paper5 [10,11,14,19-23]. Here we 5ha11 exh161t a c1a55 0f act10n5 fr0m wh1ch e4. (48) can 6e 50 06ta1ned. We expand the putat1ve effect1ve act10n a5 0~
5 = ~ (a•)"5, = E ( a ~ ) " f d % L , . n=1
(49)
n=1
70 10we5t 0rder 1n a• the act10n 15 [4]
5,--
f da4)Vc9e-e [2A + R -
~ , 2 + (V4~)2] ,
(50)
wh1ch can 6e wr1tten a5
51=
1
e-~[9089.- 28 ~] + 0(a•)
(51)
where we have perf0rmed an 1nte9rat10n 6y part5. 7hat there 5h0u1d 6e a fact0r 0f exp(-4~) can 6e ar9ued 0n 9enera1 9r0und5 [4]. 7he var1at10n 0f 51 y1e1d5 (5ee append1x A)
851 69" = v/~ e-~K°( 84~/a~) + 0(a•), where 4t den0te5 the • vect0r•• and K 0 15 the matr1x
K0
(91j, 61j, ~), +
0
9u
9,,9k,
(52)
8 y den0te5 the • vect0r•• (8~, 8~, 8e),
0
0
-91,]
,0j
/
(53)
2
A5 K 0 15 1nvert161e the Eu1er-La9ran9e e4uat10n5 65/6x0 = 0 are e4u1va1ent t0 the e4uat10n5 8 4" = 0 (at 10we5t 0rder). 1n 0rder that th15 rema1n true t0 the next 0rder
C.M. Hu11,P.K. 70wn5end / 5tr1n9 effect1ve act10n5
211
we re4u1re an 52 5uch that
3"1"(51 + a~52) = Vr9 e - ~ [ K0( 8~/a ~) + a•K1( 8•1•/0t •) + 0 ( a•2)] ,
(54)
f0r 50me matr1x K 1 (wh05e entr1e5 may 1nc1ude •/•-dependent d1fferent1a1 0perat0r5), 6ecau5e the 1nvert16111ty 0f K 0 1mp11e5 the f0rma1 1nvert16111ty 0f K 0 + a•K 1 + •. • a5 a p0wer 5er1e5 1n a•. A5 L mu5t 6e 1nvar1ant under the 5ca11n9 (10), L n can 0n1y c0nta1n term5 5ca11n9 a5 fa -n (wh1ch mu5t a150 a11 have even ••H-par1ty••). Let u5 f1r5t 5upp05e that d = 26; the 9enera1 f0rm 0f 52 1n th15 ca5e 15
e-~{a1R1jktR1)k1+ a2R1jk1H1JmHk1m
52 = f
+a3R1j(H2)1j + a4(W1Hjk1)(9:'~H jk1) + a 5 ( H 2 ) 1 j ( H 2 ) 1j + a6H1jkH11mHJ1nHkmn + a7(H2) 2 + a8R 1j ~71¢6Vjc6 + a9(97~)2R -Jr-a10 971~ 9r1R -{- a11 171dP97j~(92) 1j 4-a12(~7~11})292 4-a13 ~:r1~171H2"-{-a14 ~71~( ~71Hjk1)H{k +a15(~7~) 2 V2~ + a16((W~)2) 2) + 52• ,
(55)
Where
5; =
1
f da~ ~ e - ~ ( ~k189 (99) 1j + )k2( 81Y819j)2[- ~3(8~) 2 +~48~(908~) + ~58~(86) 0) + 0(a•).
(56)
A5 5~ 15 4uadrat1c 1n (10we5t 0rder) 8-funct10n5, 1t5 var1at10n 15 11near 1n 8-funct10n5 and 50 may 6e cance11ed 6y a ch01ce 0f the matr1x K 1 appear1n9 1n (54); (6ecau5e 0f the fact0r 0f a• 1n the K 1 term 1n (54) 0n1y the 10we5t 0rder 8-funct10n5 are re1evant here). 7he parameter5 ~1 -.- k5 are theref0re undeterm1ned, and 1f there 15 any act10n 52 f0r wh1ch (54) 15 5at15f1ed there w111 6e (at 1ea5t) a f1ve-parameter fam11y 0f them (w1th an a550c1ated f1ve-parameter fam11y 0f K 1 matr1ce5). F0r the purp05e 0f determ1n1n9 whether there 15 any 52 5at15fy1n9 (54) we can theref0re 19n0re 52~. 51m11ar1y, any 0ther term 1n 352 pr0p0rt10na1 t0 the 10we5t 0rder 8-funct10n5 can 6e cance11ed 6y a m0d1f1cat10n 0f K 1, 50 that we may free1y u5e the 10we5t 0rder 8 = 0 e45. (1.e. the Eu1er-La9ran9e e4uat10n5 51) t0 51mp11fy 352. 7he fu11 8-funct10n5 0f (44)-(46) may them5e1ve5 6e rep1aced 6y the 51mp1er
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
212
expre5510n5 appear1n9 0n the 1.h.5. 0f the e45. (48) 6ecau5e the d1fference, 6e1n9 pr0p0rt10na1 t0 the 10we5t 0rder 8-funct10n5, can a150 6e a650r6ed 1nt0 K v U51n9 the 1emma5 c011ected 1n append1x A t0 c0mpute the var1at10n 0f 52 we have f0und that (54) 15 5at15f1ed (f0r 50me K1) 1f and 0n1y 1f the a-c0eff1c1ent5 are taken t0 6e a1=~,
a2= -~,
a 7 = {(1 - 6 4 ) ,
a3= -2,
a 8 = 0,
a13 = ~ - 3(1 - 6 4 ) ,
a 9 = 0,
a14 = 2,
a5
a4=0,
3,
a6=
a11 = - 2 ,
a10 = 1(1 - 6 4 ) , a15 = - •(1 - 6 4 ) ,
1 9 , a12= 0,
a16 = 14(1 -- 6 4 ) .
(57) 1f d v~ 26 then 52 mU5t 6e m0d1f1ed t0 1nc1ude the term5
f 6%V~ e-~A{X6(2A + R-•H2+ (X7c6)2) +a17A +a18R
+ a19(~7~)2},
(58)
6ecau5e 8 ~ n0w 1nc1ude5 a A-term. A5 ~k6 15 the C0eff1C1ent 0f a term pr0p0rt10na1 t0 5~, the Var1at10n 0f Wh1Ch 1ead5 t0 a 11near C0m61nat10n 0f 10We5t 0rder 8-fUnCt10n5, 1t W111rema1n Undeterm1ned; a Chan9e 1n ~k6 Can 6e c0mpen5ated 6y a chan9e 1n K 1. 7he new a-c0eff1c1ent5, h0wever, are f1xed t0 6e
a18=
a17 = 10(1 - 6 4 ) ,
a19 =
4(1 - 6 4 ) .
(59)
We have, theref0re, a 51x-parameter fam11y 0f act10n5 at th15 5ta9e, each 0f wh1ch 5at15f1e5 (54) f0r 50me matr1x K1: L11jm14 k1 t" d_~ /22 • f 1 1~ 1~1jk1 1D 52 = J d ~eV$ e - ~4"~1jk1~" --2*~1jk1 . . . . m
-2R11(H2)1j+
11-1 14"1t,,,~ L1ftn • "L1kmn 73 ( H 2 )1j(H 2 ) 1j + ~1jk**
-- 2( H2)1J V1dP17j~ + ~ V 1 ~ V1H2 + 2 V16p( V11-1jk1) U1 jk }
+(1•64)[2(H
+
2• 3 ( V ~ ) 2 + 3 V , 2 ~
3(v
) 2 + 32
3A) 2
-
2 - v2
)
+-~((V~¢32- V•2¢)((R - H 2 + V24~)
(60)
213
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
N0te that the term5 c0nta1n1n9 the fact0r 1 - 64 are pr0p0rt10na1 t0 10wer 0rder f1e1d e4uat10n5. 7h15 mean5 that a11 4-dependence can 6e rem0ved fr0m the effect1ve act10n 6y a f1e1d redef1n1t10n (5ee 6e10w). A5 we 5h0w 1n append1x C, th15 15 a c0n5e4uence 0f the fact that the 4-dependence can 6e rem0ved fr0m the f1e1d e4uat10n5 (48) 6y a f1e1d redef1n1t10n. 7 0 5ummar12e, we have e5ta6115hed that the three f1e1d e4uat10n5 91ven 6y re4u1r1n9 that 8 9, 8 6 and 8 ~ van15h can a11 f0110w fr0m an act10n- prev10u5 auth0r5 d1d n0t check a11 three e4uat10n5. Further, we remark that the c0eff1c1ent5 a , are 1n fact f1xed 6y the 8 9 and 8 6 f1e1d e4uat10n5 a10ne. 7hu5 0n c0mpar1n9 8 ~ = 0 w1th an appr0pr1ate 11near c0m61nat10n 0f the Eu1er-La9ran9e e4uat10n5 there are n0 rema1n1n9 parameter5 that can 6e adju5ted t0 make them a9ree. 7he fact that they d0 a9ree 15 theref0re a very 5tr0n9 check 0n 0ur re5u1t5. 7he f0rm 0f the act10n 5 0f c0ur5e chan9e5 1f we make a f1e1d redef1n1t10n. Any redef1n1t10n 0f f1e1d5 that 15 pr0p0rt10na1 t0 ~51/~1" w111 pr0duce new term5 4uadrat1c 1n (10we5t 0rder) 8-funct10n5 wh1ch can then 6e a650r6ed 6y a chan9e 0f the free parameter5 ~1 ..-~5. 1n add1t10n, a redef1n1t10n 0f the f0rm • ~ • + c0n5tant can 6e a650r6ed 6y re5ca11n9 0/• 0r, 1f d ~ 26, 6y chan91n9 ~6. 7he m05t 9enera1 f0rm that any further redef1n1t10n can take, 1f 1t 15 t0 6e c0n515tent w1th the 5ca11n9 pr0pert1e5 0f 5, 15 t~91j = . 1 ( H 2 ) 1 j + .2V•1 V•j¢ + "3 V•1¢ W j ~ + 9 1 j [ . 4 H 2 + "5 172~ + " 6 A ] ,
661j =
. 7 9 1 j k 17k~ ,
84~ = . 8 H 2 + "9 W24~
(61)
After mak1n9 th15 redef1n1t10n we have the f0110w1n9 15-parameter fam11y 0f act10n5 (depend1n9 0n the 6 k,•5 and the 9 ,•5): 5 = f d % V~-e-~{A [2 +a•(X 3 + 10(1 - 64) + 2 X 6 - d , 6 ) A ] ~- [1 -~ 0/t( - ~4 4- 4(1 - 64) + X 6 - 1 ( d • 4.• [• 31• + 0/t(2X3 .4.• ~k4 -- 31~k6•}• 1 ( d •
2).6)A] R
6 ) , 6 --
2.8 -
,1 -
d.4)A ] H 2
4-[1 "~-0/'(--A 4 + 4(1- 64)+ ~6-"3 -2.9-.:-
•(d-
2). 6- d.5)A](V~) 2
1~ n1jk1 -- 12*•1jk1•" 0 1.11jmHk1 m + (3 + ~1 -- " 1 ) ( H 2 ) 1 j ( H 2 ) + 0/t[[9K1jk11x -~- 111 6~1jk 111 ~ 1mHj1 n Hkmn "~ ~ 1 R 1 j R 1j -k- ~2 R2
1j
214
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
- 2(1 + x 1 - 1 / * 1 ) R , j ( H + ( - 2 X 2 - ~)~4- 1(d• 2)/* 4 - 1*8-
•/*1) RH2
+ (-29(1 - 64) + X2 + -~2~3+ •X4 + ~ ( d - 6)/* 4 + 31/*8+ 1/*1)(H2)2
+
+
+/*,)%
+ (2X2 + 1 ( d - 2 ) / * 5 - / * 9 - •/*2- ~/*3)R(~7~) e + (•(1 - 64) - X1 - 2X= + •X 4 +/*9 + •(d - 2)/*5 )
V1c6V1R
+ ( - 2 - 2X, + X, -/*3 -/*2 +/*7 + / * t ) ( H 2 ) 4 V ~ ~7j~ + ( - 2X2 - }X4 + ~ ( d - 6)/*5 +13/*9 "•}- 6/*21 1 • /*8 -- 1/*1 At- 6/*3
•(d--
2)/*4)92(~)
2
+ ( - •(1 - 64) + • + 3X1 + 2X 2 - 1X 3 - t 6~4 x
+ 2/* 8 - ~(d - 6)/*5 +59/*1 + (d -- 1)/*4 )
•
•/* 39
1Y1dP971H2
U1~P(V~"n,,jk)u1jk
+ 2(1 + X1--)t 5 + •/*2--•/*7--•/*1)
4- (~k 1 -1- ~k2 4- 1~t 3 -- 1~k 4 -- ( d - - 1)/*5-- 2/*9) (1721~) 2
+ ( - •(1 - 64) - 3X1 - •2t 3 + 17k4 +/*9
-
-
1 /*24- 2d/*5-23/* 3 ) ( 1 7 ~ ) 21721~3
1 1 + ( ~ ( 1 - 64) + ~X1 + a)t3 +
1
) ]} + +/*3)((vr¢) 22 (62)
where we have rewr1tten 50me term5 6y 1nte9rat10n 6y part5 and u5e 0f the R1cc1 1dent1ty. 1f any 0f the/*-c0eff1c1ent5 1n th15 act10n are n0n-2er0, the Eu1er-La9ran9e e4uat10n5 w111 n0t 6e expre55161e a5 a matr1x K t1me5 the part1cu1ar e4uat10n5 (48), 6ut they w1116e expre55161e a5 K t1me5 a 51m11ar 5et 0f e4uat10n5 c0rre5p0nd1n9 t0 the 8-funct10n5 06ta1ned 6y mean5 0f a d1fferent ren0rma112at10n pre5cr1pt10n, etc., t0 the 0ne we have u5ed. D1ffer1n9 8-funct10n5 06ta1ned u51n9 d1fferent 5u6tract10n 5cheme5 y1e1d act10n5 that, d1ffer1n9 6y f1e1d redef1n1t10n5, are phy51ca11y e4u1va1ent.
C . M . Hu11, P . K . 7 0 w n 5 e n d /
215
5tr1n9 effect1ve act10n5
Met5aev and 75eyt11n have 5h0wn [10] that the 5tr1n9 5catter1n9 amp11tude5 can 6e 5ummar15ed 6y the act10n
5 = 5 1 + a , f dd6~V/-~e-,~[10 0•jkt 4 * ~ 1 j k 1 a" ,,•,
~0
t41j,~r4k~m
- - 2*•1jk1 . . . .
.•
"~ 6 ** 1 j k * * 1m **
n **
~
( H ) 1j
(u2);J] + 0 ( a •2 )
-
(63)
1n 0rder that the 519ma-m0de1 c0nf0rma1 1nvar1ance c0nd1t10n y1e1d the 5ame 5catter1n9 amp11tude5 1t 15 theref0re nece55ary that there ex15t a ch01ce 0f ~ 5 and ]/~5 5uch that (62) reduce5 t0 (63). W1th the ch01ce 0f X and ]/ c0eff1c1ent5 ]/1 = 2,
]/8 = - 1 + 1(1 - 6 4 ) ,
X 3 = - (1 - 6 4 ) ,
~k4 = - •(1 - 6 4 ) ,
X 6 = - 9(1 - 6 4 ) , (64)
~k1 = ~ 2 = ~k5 = ]/2 = ]13 ~- ]/4 = ]/5 = ]/6 = ]/7 = ]/9 = 0 ,
the act10n (62) 1ndeed reduce5 t0 (63) (even f0r d 4= 26). 1t f0110w5 that the effect1ve act10n 06ta1ned fr0m 519ma-m0de1 c0nf0rma1 1nvar1ance 15 c0n515tent w1th 5tr1n9 5catter1n9 amp11tude5, whatever the va1ue 0f 4. 0 n th15 p01nt we d1ffer fr0m Met5aev and 75eyt11n, wh0 c1a1m that the tw0 act10n5 can 0n1y 6e e4u1va1ent 1f 4 = ~. After a further f1e1d redef1n1t10n, the act10n can 6e wr1tten a5 1
5 = -0/t- f d%V/9 e-~(9'J8~ - 2 8 * ) + 0(a•2).
(65)
W1th yet an0ther f1e1d redef1n1t10n and an 1nte9rat10n 6y part5, the act10n 6ec0me5
5=1fd%v/9e-~(91Jf1~-28~)+0(a~2).
(66)
75eyt11n ha5 c1a1med that the act10n 5h0u1d take th15 f0rm t0 a11 0rder5 [19]. 1t 15 p055161e t0 ch005e the parameter5 1n (62) 50 a5 t0 06ta1n a ••pertur6at1ve1y un1tary•• act10n 1n the 5en5e 0f 2w1e6ach [26]. F0r examp1e, 1f d = 26, ch0051n9 ~1 = - 1 ,
X2=1
]/8 = 6/-(1 - 6 4 ) ,
,
X3=
1-
(1-64)
,
~4=
-1-
]/1 = ]/2 = ]/3 = ]/4 = ]/5 = ]/7 = ]/9 = 0 ,
1(1-
64),
X5 = 0 , (67)
216
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
we 06ta1n
-- ,[ 1 n r.1jk1 R 1 j R 1 j + ••R 2 • 1 p 141jmL7k1 -1-0£ [ ~1~1jk11~ -2.~1jk1 . . . . m ..~1L1
1;~1krnn1L1j1L1 1
+ 3 (H2) 2 - 9H ( 7 6 p ) 2 + ~ 7 1 6 P 7 1 H
1j
2 - 9RH1 2
+~R(~7¢)2+ 1(17*)2172~- ~((V¢)2)2]} +0(a~2).
(68)
Append1XA 7he R1emann CUrvatUre 15 91Ven 6y
1 =0 (1} { 1 }{m}--(k•,-••1) R jkt 1, j1 + km 1j
(A.1)
R0Und 6raCket5 den0te 5ymmetr15at10n W1th 5tren9th 0ne wherea5 54uare 6raCket5 den0te ant15ymmetr15at10n w1th 5tren9th 0ne. F0r examp1e,
H1j k =- 3 0[16j~ 1 -- •( 016jk + 0j6k1 + 0k61j ) .
(A.2)
U51n9 the 81anch1 and cyc11c 1dent1t1e5 and 3[~Hjk~1 = 0, wh1ch f0110w5 fr0m the def1n1t10n (A.2), we f1nd the f0110w1n9 u5efu1 1dent1t1e5: R1[k4y
~tRt1jk R
= - ~2R1jk1,
(A.3)
= ~jRk1-
(A.4)
1~7kR1j ,
1~ k 1 m _ 3 k1m 1(k1)m*•j -- ~ R 1 k 1 m R j •
V[1Hj1k, = - ~7[kHt11j , ~7[1Hjk ]~ =
91 17tH1jk,
(A.5)
(A.6) (A.7) (A.8)
~72H1jk = 3W[1VtHjky- 3R1m[1jHk]tm + 3R[/Hjk1t,
(A.9)
C . M . Hu11, P . K . 7 0 w n 5 e n d /
217
5tr1n9 effect1ve act10n5
where the R1cc1 ten50r 15 R1j = Rk1kj •
(A.10)
We def1ne the c0nnect10n and curvature w1th t0r510n 6y jk
+ HJ k~
(1.11)
R(~)~k1---=~0 r ( + ) 1 * -r-( +~ )k1mr ( +Jt) ~1j-w k ~ 1j-=
R j k t +• ~1,Ht11-7-
--
1171Hkj 1 +
(k~1) 1 m - H,~H 1 kj HkmHtj
m
(A.12)
7hen R1• ) = R ( +-)k1kj,
(A.13)
R((1j) • ) = R1j - ( H2)1j
(A.14)
R([1j] + ) = 7- ~7kH1jk
(A.15)
R(+) 1jk1 -- - R (k11j)
(A.16)
R(+) • .4• 2 ( [Uk11 - - - - 3 ~ ++-)H1jk"
(A.17)
70 c0mpute the var1at10n 0f the act10n 1n 5ect. 3 we have u5ed the f0110w1n9 u5efu1 1emma5. C0n51der
5 = fd%- R v6
jkt7Ykt1 ,
(A.18)
where 9 = [det 91j[ and 71jk1 15 a ten50r. 7he var1at10n 0f 5 15
85= f d%Vcd(69~J[-2v~(kvt)6k5 -~
"291j*~k1mn-7k1mn]] + R5k,371 jk•
(A.19)
51m11ar1y, f0r ten50r5 7,j, 7, 7,7k, 55k,
8f d%V~R1f ~j
(A.20)
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
218
(A.21)
+ H~jk 6 7 *j~ , (A.22)
•f dd+~{;k }8Jk1= fdd~1~{~91J[~7k5jk1--~7k5~jk--291j{ 1m}8k1m •
1
k
(A.23) C0n51der the act10n (A.24)
5 = f d a ~ V ~ - e - e ( 6 0 + 61R + 6 2 H 2 + 63(W¢)2). Vary1n9 5 u51n9 (A.19)-(A.23) ee
a5
91ve5
- 61R1 j + 362(H2)1j + 6197197j¢ + (63 - 61)(W ¢ ) ( W j ¢ ) -•91j(61R+62 H 2 + 2 6 1 1 7 2 ¢ + ( 6 3 - 2 6 1 ) ( W ¢ )
e0 85
2 + 6 0 ) , (A.25)
362 ( V7kH1Jk - (97~¢) H1J k ),
(A.26)
86~j e0 85 ---=60 + 61R + 62 H2 + 6312 W2¢ - (W¢)2] .
(A.27)
8¢
1f we ch005e 60=2A,
61 = 1 ,
6 2 -•
- 51,
63
=
1
,
(A.28)
then the act10n (A.24) 6ec0me5 51 (51) and (A.25)-(A.28) y1e1d (52), (53). Append1x
8
Here we 5h0w that there 15 n0 term 1n f1~ pr0p0rt10na1 t0 the R1cc1 5ca1ar R. 7he vert1ce5 we 5ha11 need c0me fr0m the n0rma1 c00rd1nate expan510n 0f 1
1
2 (2rra~)
fd x
+ ... )9,j(,)
0j0A, J,
(8.a)
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
219
90~0~ F19. 1. 7w0-100p d1a9ram c0ntr16ut1n9 a term pr0p0rt10na1 t0 R 1n f1e. 7he d0u61e 11ne 1nd1cate5 the 6ack9r0und f1e1d dependent vertex R1k0~( j 0~,(k 0 ~ 1. 7he wavy 11ne 1nd1cate5 the vertex -h~ 0 ~ ~ 0~J~j(4)) 1nv01v1n9 the tw0-d1men510na1 metr1c pertur6at10n h~,,. 7he d0t5 1nd1cate the p051t10n 0f tw0-d1men510na1 der1vat1ve5 at a vertex.
where (we w0rk 1n euc11dean 5pace)
~,.. = 8.. + h . . ,
~.. = h.. - } ~.h ,
h - ~"~h~..
(8.2)
7he re1evant term5 1n (8.1) are 1
- h " ~ 0 ,..~10 + 1a..a . . . .1:J~..(,/,) ,. 6~ v..¢~ 0.~J~k;,.~j
},
(8.3)
where ~1(x) 15 the n0rma1 c00rd1nate and ep1(x)15 n0w the 6ack9r0und f1e1d (f0r deta115 5ee ref5. [5-7]). We have 1nc1uded a c0var1ant ma55 term t0 re9u1ate 1nfra-red d1ver9ence5. 7he tw0-100p d1a9ram 0f f19. 1 91ve5 the c0ntr16ut10n 0/•
964re
f d•xR( 0~,0.-h~"- ~ h )
(8.4)
t0 the (tw0-d1men510na1) effect1ve act10n. We are u51n9 d1men510na1 re9u1ar12at10n, 50 that --/Lv
•
h 8,,,,-- •eh.
(8.5)
0 t h e r tw0-100p d1a9ram5 that have d1ver9ence5 pr0p0rt10na1 t0 R e1ther 91ve term5 pr0p0rt10na1 t0 m 2 0r c0var1ant12e the re5u1t (8.4), y1e1d1n9 0~p
96rre
f d2x
R
(8.6)
220
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
0
n
F19. 2. 0ne-100p d1a9ram w1th tw0 externa1 4uantum 11ne51ead1n9 t0 the c0unterterm 0f (8.7).
(reca11 that R (v) 15 the tw0-d1men510na15ca1ar curvature). Hence the d1a9ram 0f f19. 1 d0e5 91ve a c0ntr16ut10n t0 f1~ pr0p0rt10na1 t0 R. H0wever, th15 15 cance11ed 6y a 0ne-100p d1a9ram 1nv01v1n9 the c0unterterm 1 f d2x ~ - 7 ~ 3 ~ 247re
3~;R1j (4~)
(8.7)
7h15 c0unterterm 15 needed t0 cance1 the u1tra-v101et d1ver9ence 1n the 0ne-100p d1a9ram 0f f19. 2, wh1ch ha5 tw0 externa1 4uantum 11ne5. A term 0f the f0rm (8.7) 0ccur5 1n the n0rma1 c00rd1nate expan510n 0f the R 4 0 j 0"~J c0unterterm, 6ut n0t w1th the c0rrect c0eff1c1ent. 70 f1nd the r19ht c0eff1c1ent 1t 15 51mp1e5t t0 ca1cu1ate the d1ver9ence 1n f19. 2 d1rect1y, 6ut 1t can a150 6e 06ta1ned fr0m the R u 0,4,18 N~j c0unterterm 1f 0ne take5 1nt0 acc0unt the n0n-11near ren0rma112at10n 0f the 4uantum f1e1d5 that are re4u1red 6y the n0n-11near1ty 0f the 6ack9r0und/4uantum 5p11t [241. 7he 0ne-100p d1a9ram 0f f19. 3 w1th the c0unterterm (8.7) y1e1d5 the c0ntr16ut10n
•
a• fd2xR (0"0~h "~- •Dh)
96~re
(8.8)
t0 the (tw0-d1men510na1) effect1ve act10n. N0t1ce that a1th0u9h 0ne m19ht expect th15
F19. 3. 0ne-100p d1a9ram w1th c0unterterm vertex R1j 3~£:1 8 ~ j c0ntr16ut1n9 a term pr0p0rt10na1 t0 R 1n f1~.
C.M. Hu11,P.K. 70wn5end / 5tr1n9effect1veact10n5
221
d1a9ram t0 c0nta1n 0n1y 1/e 2 p01e5 th15 15 n0t the ca5e, e55ent1a11y 6ecau5e 0f the fact0r 0f e appear1n9 0n the r.h.5. 0f (8.5). A9a1n 0ther 0ne-100p d1a9ram5 w1th c0unterterm5 are e1ther 1rre1evant t0 f1~ 0r 5erve t0 c0var1ant12e (8.8) t0 91ve the c0ntr16ut10n "~ 967re
fd2x9~-R (v)" R
(8.9)
t0 the effect1ve act10n. 7he d1ver9ence5 0f (8.6) and (8.9) cance1, 50 the net c0ntr16ut10n t0 f1~ that 15 pr0p0rt10na1 t0 R 15 2er0, a5 c1a1med. A 51m11ar ca1cu1at10n 5h0w5 that there 15 n0 R c0ntr16ut10n t0 f1~ f0r the (1,1) 0r (1,0) 5uper5ymmetr1c n0n-11near 519ma-m0de15 e1ther.
Append1x C 7 h e c0nd1t10n5 (48) f0r 519ma-m0de1 c0nf0rma1 1nvar1ance 1nv01ve the ar61trary parameter 4 0n1y thr0u9h the 4uant1ty ~ = • + a•4H 2. 7hat 15, 1n term5 0f the new d11at0n f1e1d ~ the5e e4uat10n5 are 4-1ndependent. C1ear1y, 1f the e4uat10n5 1n term5 0f ~ are der1va61e fr0m an act10n, then the act10n can 6e ch05en t0 6e 4-1ndependent (a5 a funct10na1 0f ~ ) . 0 n e can then take ~ ~ (6 6y a f1e1d redef1n1t10n t0 06ta1n a 4-1ndependent act10n f0r the 0r191na1 f1e1d (6. H0w 15 th15 act10n re1ated t0 the 4-dependent act10n f0r • wh05e f1e1d e4uat10n5 are the 0r191na1 4-dependent e4uat10n5 (48)• 7he fact that 4 appear5 0n1y thr0u9h ~ 1n the5e e4uat10n5 d0e5 n0t nece55ar11y mean that 1t appear5 0n1y thr0u9h ~ 1n the act10n, 50 1t 15 n0t even c1ear that the 4-dependence can 6e rem0ved fr0m th15 act10n 6y a f1e1d redef1n1t10n. 7he purp05e 0f th15 append1x 15 t0 5h0w that the 4-dependence can 6e rem0ved 6y a f1e1d redef1n1t10n and that the re5u1t1n9 4-1ndependent act10n 15 e4u1va1ent (after p055161e further f1e1d redef1n1t10n5) w1th that 0ta1ned 6y a d1rect c0n5truct10n 0f a 4-1ndependent act10n 1n term5 0f ~ . 7 h e e4uat10n5 (48) are 91ven a5 a pertur6at10n 5er1e5 1n a~. 1n part1cu1ar ( V ~ ) 2 = ( V ~ ) 2 + 2a•4 V ~ ~7H2 + 0(a,2) and the 0(0ff 2) term can 6e ne91ected at the 0rder 1n a• t0 wh1ch we are w0rk1n9. 7hu5, a1th0u9h a•4H 2 15 n0t nece55ar11y 5ma11, the fact that we w0rk pertur6at1ve1y 1n a• mean5 that we can effect1ve1y 5upp05e that 1t 15. Rather than c0n51der e4uat10n5 and f1e1d redef1n1t10n5 a5 p0wer 5er1e5 1n a~ we theref0re c0n51der 1nf1n1te51ma1 f1e1d redef1n1t10n5 1n exact e4uat10n5. A150, we 5ha11 pre5ent the ar9ument t0 f0110w 1n 9enera11ty 6ecau5e 1t 15 n0t 5pec1f1c t0 the ca5e 1n hand. 7 h e e4uat10n 5a[~6] = 0 f0r a 5et 0f f1e1d5 ~6" w1116e der1va61e fr0m an act10n 1f 5ta, 61--0, 1.e. 1f 5 a - - 5 , a and then 5 15 the act10n. 7he f1e1d e4uat10n 15 theref0re 5 a = 0.
(C.1)
We u5e here the n0tat10n 0f DeW1tt 1n wh1ch the 1ndex a 1a6e15 a11, d15crete 0r
222
C.M. Hu11,P.K. 70wn5end/ 5tr1n9effect1veact10n5
C0nt1nU0U5, Var1a61e5 0n Wh1Ch ~ m a y depend. 1f We m a k e the 1nf1n1te51ma1 f1e1d redef1n1t10n +~.
~ +fa(~)
(C.2)
1n (C.1) we 9et the new e4uat10n 5,~ + 5, ~¢f~ = 0,
(C.3)
(~a6~jrf6, a)(5,6~ - 5,6cfc) =0.
(C.4)
(5 + 5 ~fc),~= 0,
(C.5)
wh1ch 15 06v10u51y e4u1va1ent t0
Ne91ect1n9 0 ( f
2) th15 15
wh1ch 15 the Eu1er-La9ran9e e4uat10n f0r the new act10n 5•[+] = 5 + 5, cfc=5[4~+ f ] .
(c.6)
7h15 15 ju5t the 0r191na1 act10n 1n term5 0f the new f1e1d5 % + f . 7 h e 0n1y n0n-tr1v1a1 p01nt 15 the fact that 0ne mu5t 1ntr0duce the 1nte9rat1n9 fact0r (3a6 + f 6 a) 1n (C.4). App11ed t0 the ca5e 1n h a n d th15 mean5 that the 4 - d e p e n d e n c e 0f the act10n 06ta1ned f r 0 m the 0r191na1 4 - d e p e n d e n t f1e1d e4uat10n5 0f (48) can 6e r e m 0 v e d 6 y the 5h1ft • ~ • - a•4H 2 c0m61ned w1th a 4 - d e p e n d e n t 5h1ft 1n the X-c0eff1c1ent5 t0 t a k e 1nt0 a c c 0 u n t the chan9e 1n the 1nte9rat1n9 fact0r. 0 n e can ver1fy 6 y 1n5pect10n 0f (64) t h a t th15 15 true. W e f0und th15 t0 6e a u5efu1 check 0n the re5u1t5.
Reference5 [1] [2] [3] [4] [5]
C. L0ve1ace, Phy5. Lett. 1358 (1984) 75 E.5. Fradk1n and A.A. 75eyt11n, Phy5. Lett. 1588 (1985) 316; Nuc1. Phy5. 8261 (1985) 1 A. 5en, Phy5. Rev. 32D (1985) 2162; Phy5. Rev. Lett. 55 (1985) 1846 C.6. Ca11an, D. Fr1edan, E. Mart1nec and M.J. Perry, Nuc1. Phy5. 8262 (1985) 593 C.M. Hu11, Lecture5 0n n0n-11near 519ma-m0de15and 5tr1n95, Pr0c. 0f Vanc0uver 5uper F1e1d7he0ry W0rk5h0p, t0 6e pu6115hed [6] C.M. Hu11 and P.K. 70wn5end, Nuc1. Phy5. 8274 (1986) 349, and 1n preparat10n [7] C.M. Hu11 and P.K. 70wn5end, Phy5. Lett. 1918 (1987) 115 [8] D. Fr1edan, Phy5. Rev. Lett. 45 (1980) 1057; Ann. 0f Phy5. 163 (1985) 318 [9] 7. Curtr19ht and C. 2ach05, Phy5. Rev. Lett. 53 (1984) 1799 [10] R.R. Met5aev and A.A. 75eyt11n, Phy5. Lett. 1918 (1987) 354; Nuc1. Phy5. 8293 (1987) 385 [11] D. 2an0n, Phy5. Lett. 1918 (1987) 363 [12] E. 6uada9n1n1 and M. M1ntchev, P15a prepr1nt (rev15ed) (1987) [13] D.R.7. J0ne5, Phy5. Lett. 1928 (1987) 391 [14] 1. Jack and D.A. R055, 50uthampt0n prepr1nt (1987); 1. Jack and 7. J0ne5, Phy5. Lett. 1938 (1987) 449 [15] M. 805, Phy5. Lett. 1898 (1987) 435
C.M. Hu11,P.K. 70wn5end / 5tr1n9 effect1ve act10n5 [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
A.A. 75eyt11n, Phy5. Lett. 1788 (1986) 34; Nuc1. Phy5. 8294 (1987) 383 6. 5h0re, Nuc1. Phy5. 8286 (1987) 349 6. Curc1 and 6. Paffut1, Nuc1. Phy5. 8286 (1987) 399 A.A. 75eyt11n, Phy5. Lett. 1948 (1987) 63 C. Ca11an, 1. K1e6an0v and M.J. Perry, Nuc1. Phy5. 8278 (1986) 78 5.P. de A1w15, Phy5. Rev. D34 (1986) 3760 C. Nune2, Phy5. Lett. 1938 (1987) 195 N.E. Mavr0mat05 and 5.J. P5ychar15, 0xf0rd prepr1nt (1987) P.5. H0we, 6. Papad0p0u105, and K.5. 5te11e, Nuc1. Phy5. 8296 (1988) 26 H. 0560rn, Nuc1. Phy5. 8294 (1987) 595 8. 2w1e6ach, Phy5. Lett. 1568 (1985) 315 8.E. Fr1d11n9 and A.E.M. van de Ven, Nuc1. Phy5. 8268 (1986) 719
223
5 7 R 1 N 6 EFFEC71VE A C 7 1 0 N 5 F R 0 M 5 1 6 M A - M 0 D E L C 0 N F 0 R M A L AN0MAL1E5 C. M. HULL* and P. K. 70WN5END DA M7P, 511ver 5treet, Cam6r1d9e C83 9E W, UK Rece1ved 7 5eptem6er 1987
We c0n51der the 519ma m0de1 de5cr161n9 a c105ed 6050n1c 5tr1n9 1n the pre5ence 0f 9rav1t0n, ant1-5ymmetr1c ten50r and d11at0n 6ack9r0und f1e1d5. U51n9 prev10u5 re5u1t5 f0r the metr1c and ant1-5ymmetr1c ten50r tw0-100p 6eta-funct10n5, we der1ve the d11at0n 6eta-funct10n and 06ta1n the fu11 5et 0f c0nd1t10n5 f0r c0nf0rma1 1nvar1ance t0 the 5ame 0rder. We 5h0w that the5e c0nd1t10n5 can 6e der1ved fr0m any 0ne 0f a fam11y 0f 5pace-t1me effect1ve act10n5. After tak1n9 f1e1d redef1n1t10n5 1nt0 acc0unt 0ur act10n a9ree5 w1th that 06ta1ned 6y Met5aev and 75eyt11n.
1. 8eta-funct10n5 and c0nf0rma1 an0ma11e5
A 6050n1c 5tr1n9 pr0Pa9at1n9 1n a d-d1men510na1 5pace-t1me w1th metr1c 91j 1n wh1ch the ant1-5ymmetr1c ten50r 9au9e f1e1d and d11at0n m0de5 0f the 5tr1n9 have vacuum expectat10n va1ue5 61j, ~6, re5pect1ve1y 15 90verned 6y the n0n-11near 519mam0de1 (euc11dean) act10n [1-4]
1 fd2xVr~{~(9,j(¢)y~+6,j(¢)~)0.¢~0~¢j+~a,~(ep)R,~
5 = 2~ra--~
}
(1)
where ¢1 (1 = 0 . . . . . d - 1) are 5pace-t1me c00rd1nate5, x" (/2 = 0,1) are w0r1d-5heet c00rd1nate5 and y~ 15 the w0r1d-5heet metr1c w1th R (~) the c0rre5p0nd1n9 w0r1d5heet curvature 5ca1ar. C0n515tency 0f the 5tr1n9 the0ry re4u1re5 that th15 act10n def1ne a c0nf0rma11y 1nvar1ant 4uantum f1e1d the0ry, and the c0nd1t10n5 f0r c0nf0rma1 1nvar1ance can 6e 1nterpreted a5 ••effect1ve•• f1e1d e4uat10n5 f0r 91j, 6~j and [3, 4]. 7he 519ma-m0de1 can 6e 4uant12ed u51n9 the ver510n 0f the 6ack9r0und f1e1d appr0ach de5cr16ed 1n ref5. [5-7], deve10p1n9 a pertur6at10n the0ry 1n a•, w1th u1tra-v101et d1ver9ence5 re9u1ated 6y d1men510na1 re9u1ar12at10n t09ether w1th the precr1pt10n ~p~p = - y~
(2)
f0r 51mp11fy1n9 pr0duct5 0f e-ten50r5 1n 2 + e d1men510n5. 7he u1tra-v101et d1ver9ence5 are a650r6ed 1nt0 ren0rma112at10n5 0f 9~j, 6~j, • [8] and we def1ne the *Addre55 fr0m 0ct06er 1987: 81ackett La60rat0ry, 1mper1a1 C011e9e, Pr1nce C0n50rt R0ad, L0nd0n 5W7, UK. 0550-3213/88/$03.50•E15ev1er 5c1ence Pu6115her5 8.V. (N0rth-H011and Phy51c5 Pu6115h1n9 D1v1510n)
198
C.M. Hu11,P.K. 70wn5end / 5tr1n9effect1veact10n5
f1-funct10n5 t0 91ve the dependence 0f the ren0rma112ed 5pace-t1me f1e1d5 9~, 64,R~R 0n a ren0rma112at10n ma55-5ca1e/,: 0 f1~ = ~ p, ~ 9 ~ ,
0 f16.= ~ p ~ 61R ,
0 f1~ = - p. - ~ c6R .
(3)
Hencef0rth we 5ha11 dr0p the R 5uperf1x 0n the ren0rma112ed 5pacet1me f1e1d5 a5 the 6are 5pacet1me f1e1d5 w111n0t appear a9a1n 1n th15 paper. 7here 15 50me dependence 0f the f1-funct10n5 0n the 5u6tract10n pre5cr1pt10n, the ch01ce 0f 6ack9r0und/4uantum 5p11t, and the ch01ce 0f wavefunct10n ren0rma112at10n [5-8], 6ut 91ven a def1n1te • 5cheme•• the f1-funct10n5 are c0mp1ete1y determ1ned. 1n the 5cheme u5ed 1n [7] (1.e. 4uant121n9 a5 de5cr16ed 1n 5ect. 9 0f [5], u51n9 m1n1ma1 5u6tract10n, the n0rma1 c00rd1nate expan510n, and mak1n9 n0 wave funct10n ren0rma112at10n 0f the 6ack9r0und 5pace-t1me c00rd1nate) the f1-funct10n5 (1nc1ud1n9 a c0ntr16ut10n fr0m the w0r1d-5heet reparametr12at10n 9h05t5) take the f0rm [4, 8, 9]
f1~= -a•[ R 4 -
H1ktHy + •a•W4] ,
< : -.•[- v%,, + f1¢= 1 ( d - 26) - a•[-•
V2¢ + Y+ ~ 2 ] ,
(4)
3 where H4k = ~3[~62k ] and W4, X4, Y, 2 are p0wer 5er1e5 1n a• wh05e c0eff1c1ent5 are 10ca1 p01yn0m1a15 1n R4k t, H4e, 0~46 and the1r c0var1ant der1vat1ve5. 1t 15 a c0n5e4uence 0f the f0rm 0f the Feynman ru1e5 that 1,V4 and X12 are 1ndependent 0f and that f1~ 15 at m05t 11near 1n ~; we take Y t0 6e ~-1ndependent and 2 t0 6e pr0p0rt10na1 t0 (der1vat1ve5 0f) 4~. We f0und 6y an exp11c1t 2-100p ca1cu1at10n [7] that
W1j = 5(1j)"~- 0(01t),
X1j = 5[1j]-~ 0 ( 0 t ' ) ,
(5)
w1th
54=~1k1,,,~,~
j
"-- m , -
"~k4t,
(6)
where the curvature w1th t0r510n, R (+), 15 def1ned 1n (A.12) 0f append1x A. U51n9 (A.12) and the n0tat10n 0f append1x A we f1nd that 51] can 6e wr1tten a5
3(1j) = 81k•mRj k•m -- ( ~7kH~m1)( ~7k91mj ) ~- 1( ~719k~rn)( ~7j9 k~m) + 2R1k1jHkmnH1mn + 2Hk1mHmn(jR1)nk1 + 2Htp1Hj4kHmkPHm 4~- 2Htp1HfkHk•••••H•
,
5[1j1 = 2(V'kHtm[j) R~1~ - 4(V'kHm41) Hj1PtHm~ + 2( 9r1Hk1j) HkmnH•m, .
(7)
C.M. Hu11,P.K. 70wn5end / 5tr1n9 effect1ve act10n5
199
Recent 1ndependent ca1cu1at10n5 [10-14] a9ree w1th th15 re5u1t, 0r are e4u1va1ent t0 1t after f1e1d redef1n1t10n5 [10,14], and a150 a9ree w1th a re5u1t 6y 805 [15] f0r the 5pec1a1 ca5e 0f a 9r0up man1f01d. 7 0 ca1cu1ate f19 and f16 1t 15 5uff1c1ent t0 c0n51der 0n1y f1at w0r1d-5heet5, 6ut t0 06ta1n f1~ the exten510n t0 curved w0r1d-5heet5 15 needed. U51n9 the techn14ue5 de5cr16ed 1n [5, 6] we f1nd fr0m tw0-100p ca1cu1at10n5 that
Y = aH1jkH 1jk + a~Y + 0(0d•2) , 2 = •(V" 1W2r6)H1ktHJkt + 0(a•).
(8)
7he c0eff1c1ent a and the c0ntr16ut10n Y can 6e f0und fr0m (ted10u5) tw0- and three-100p ca1cu1at10n5, re5pect1ve1y. Rather than ca1cu1ate the5e 4uant1t1e5 d1rect1y, we 5ha11 5h0w 6e10w that they can 6e determ1ned 6y certa1n c0n515tency c0n51derat10n5, w1th the re5u1t that a = - ~ wh11e Y 15 91ven 1n (37). 065erve that there 15 n0 c0ntr16ut10n t0 Y pr0p0rt10na1 t0 the 5pace-t1me curvature 5ca1ar R, 1n a9reement w1th a re5u1t 4u0ted 1n [16]. A1th0u9h there 15 a 2-100p 9raph 91v1n9 5uch a c0ntr16ut10n, 1t 15 cance11ed 6y a 9raph 1nv01v1n9 a 0ne-100p c0unterterm. 7h15 a150 a9ree5 w1th the re5u1t5 0f [18,25], 6ut a5 0ther auth0r5 have c1a1med t0 f1nd a n0n-van15h1n9 c0eff1c1ent f0r th15 term (5ee ref. [4], f0r examp1e) we 91ve deta115 0f the ca1cu1at10n 1n append1x 8. We 5ha11 5ee 6e10w that the a65ence 0f a term pr0p0rt10na1 t0 R 15 a150 re4u1red 6y c0n515tency. 1f the f1-funct10n5 are tr1v1a1 (1.e. 1f they van15h up t0 the am619u1t1e5 1nherent 1n the1r def1n1t10n) then the the0ry 15 r191d 5ca1e 1nvar1ant, 1.e. the 1nte9rated trace an0ma1y van15he5 (f0r c0mpact w0r1d-5heet5). 7he 10ca1 5ca1e, 0r c0nf0rma1, 1nvar1ance needed here re4u1re5 that the trace an0ma1y van15he5 10ca11y, wh1ch re4u1re5 the van15h1n9 0f certa1n ••8-funct10n5•• [5, 6, 16,17] 91ven 6y 85 =
- -• v(,vj
=
. H1,kv
8
= 8 * - •.•v•
,
- .• v
,wjj , (9)
f0r 50me vect0r5 v 1, w 1 c0n5tructed fr0m R1jk1 , H1j k and 0 ~ and the1r der1vat1ve5 and wh1ch can 6e ca1cu1ated pertur6at1ve1y. 80th the f1-funct10n5 and the vect0r5 v~, w1 are 5cheme dependent, 6ut the c0m61nat10n 0ccurr1n9 1n (8) depend5 0n1y 0n the 5u6tract10n pre5cr1pt10n and n0t 0n the nature 0f the 6ack9r0und/4uantum 5p11t 0r wave funct10n ren0rma112at10n5. Further, the dependence 0f the 8-funct10n5 0n the 5u6tract10n 5cheme can 6e a650r6ed 1nt0 a redef1n1t10n 0f the 5pacet1me f1e1d5. 7he effect1ve f1e1d e4uat10n5 f0r the ma551e55 m0de5 0f the 5tr1n9 are, then, that the 8-funct10n5 van15h ( 8 = 0). 1n the next 5ect10n we 5ha11 5h0w, w1th0ut a55um1n9
200
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
the ex15tence 0f an act10n, that the5e 8-funct10n5 5h0u1d 5at15fy certa1n ••c0n515tency c0nd1t10n5••. 1n ref. [7] we c1a1med that the /3-funct10n5 4u0ted a60ve 1ead t0 8-funct10n5 5at15fy1n9 the5e c0nd1t10n5 when • 15 c0n5tant, and 0ne purp05e 0f th15 paper 15 t0 5u65tant1ate th15 c1a1m. Furtherm0re, when we 1nc1ude 4~-dependence, we f1nd that the c0n515tency c0nd1t10n5 and the tw0-100p re5u1t5 a60ve determ1ne/3~ and 8 ~ up t0 0rder a •3, w1th0ut hav1n9 t0 perf0rm three-100p ca1cu1at10n5. 0 u r re5u1t5 f0r the/3 and 8 funct10n5 are 5ummar12ed 1n e45. (41)-(46). 7he va1ue 0f a 51n91e c0eff1c1ent, 4, appear1n9 1n the f1e1d e4uat10n5 8 = 0 rema1n5 undeterm1ned; 1t cann0t 6e f1xed 6y/3-funct10n c0n51derat10n5 a10ne, 6ut w0u1d 6e 6y a tw0-100p c0nf0rma1 an0ma1y ca1cu1at10n (wh1ch we have n0t yet perf0rmed). H0wever, a11 4-dependence can 6e rem0ved t0 th15 0rder 6y a f1e1d redef1n1t10n. We 5ha11 5h0w that, at 1ea5t t0 th15 0rder, the three f1e1d e4uat10n5 8 = 0 can 6e der1ved fr0m an act10n, whatever the va1ue 0f 4. A 5pacet1me effect1ve act10n der1ved fr0m 519ma-m0de1 c0nf0rma1 1nvar1ance ha5 6een 91ven prev10u51y 6y Met5aev and 75eyt11n [10] (and 6y 2an0n [11] f0r c0n5tant d11at0n f1e1d ~; 5ee a150 ref. [14]). 0 u r re5u1t5, 50me 0f wh1ch were rep0rted at the 7r1e5te 1987 5pr1n9 W0rk5h0p 0n 5uper5tr1n95, are 1ar9e1y 1n a9reement w1th the1r5. H0wever, wherea5 1n [10,11,14] 1t wa5 0n1y checked that vary1n9 the act10n 9ave 50me 0f the f1e1d e4uat10n5, we are a61e, w1th 0ur kn0w1ed9e 0f 8 ~, t0 check here that the var1at10n 91ve5 a11 three f1e1d e4uat10n5 8 = 0. Met5aev and 75eyt11n have a150 der1ved an effect1ve act10n fr0m 5tr1n9 5catter1n9 amp11tude5 [10]. We f1nd that th15 a9ree5 w1th the act10n 06ta1ned fr0m 519ma-m0de1 meth0d5 whatever the va1ue 0f 4, wh11e Met5aev and 75eyt11n c1a1m that a9reement 15 0n1y p055161e 1f 4 = ~ [10] (4 15 re1ated t0 the 11 0f [10] 6y 4 = -411).
2. 7he 5tructure 0f the 8-funct10n5
An 1mp0rtant re5tr1ct10n 0n the 5tructure 0f the term5 appear1n9 1n the f1- and 8-funct10n5 can 6e 06ta1ned 6y c0n51der1n9 the Feynman ru1e5. 1f we make the r191d 5ca11n9 91j ~ ~91j ~
61j ~ ~61j ,
~ "~ 0 ,
(10) f0r 50me c0n5tant 9, then 1t can 6e 5h0wn that each 0f the f1- and 8-funct10n5 5ca1e 1n the 5ame way a5 the c0rre5p0nd1n9 f1e1d, e.9. f 1 ~ ~2f19, 8 ~ 8 ~. Further, under the ••H-par1ty•• tran5f0rmat10n H/j~ --~ - H ~ j k
(11)
0ne can 5h0w that f16 and 8 6 a r e 0dd, wherea5 the 0ther funct10n5 are even, e.9. 8~-8~,6ut 8~8~ and 8 ~ 8 ~.
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
201
F r 0 m the 9enera1 f0rm 0f the 8-funct10n5 t0 a11 0rder5 91ven 6y (4.9) and u51n9 the 1dent1t1e5 0f append1x A we 06ta1n
01t
~
6**k1m**
--
(12) 1 a , [ V ~ 8 ~ - 0 ~ 8 ~ . ] =(V~-~J)V~,wj~+ ~J~Vj0~+a~(V~-v~)X,~.
(13)
A5 the f1r5t term 0n the r.h.5. 0f (12) 15 the 9rad1ent 0f a 5ca1ar, th15 mu5t a150 6e true 0f the term 1n cur1y 6racket5 f0r 6ack9r0und5 f0r wh1ch 8 9 = 0, 8 ~ = 0, 1n wh1ch ca5e the ent1re r.h.5. 0f (12) w0u1d 6ec0me ~ P f0r 50me P c0n5tructed fr0m curvature5 etc. 7h15, and the van15h1n9 0f the r.h.5. 0f (13) when 8 ~ = 8 6 = 0, 1mp05e5 5tr1n9ent c0n5tra1nt5, 0r ••c0n515tency c0nd1t10n5••, 0n the f0rm 0f W,j, X~j, v~, w~. F0r examp1e, w1th W,j, X,j a5 91ven 1n (5.6), we 5h0w 6e10w that there d0 1ndeed ex15t vect0r5 d, w 1 5uch that the r19ht-hand 51de 0f (12) 15 the 9rad1ent 0f 50me P when 8~. = 0, 8 ~ = 0, at 1ea5t up t0 term5 0f 0rder a •2. 0 n the 0ther hand, f0r the W,7, X9j 91ven 6y the re5u1t5 0f an ear11er 1nc0rrect 6eta-funct10n ca1cu1at10n [27] there ex15t n0 vect0r5 d, w ~ 5uch that th15 15 true (cf. ref. [23]). N0te that wh11e the5e c0n515tency c0nd1t10n5 are c105e1y re1ated t0 0ne5 that can 6e 06ta1ned 6y demand1n9 that the f1e1d e4uat10n5 8 = 0 6e der1va61e fr0m a 9au9e1nvar1ant act10n [21], we are n0t mak1n9 any a55umpt10n5 a60ut the ex15tence 0f an 1nvar1ant act10n here. 7 0 10we5t 0rder, the m05t 9enera1 f0rm that w and v can take that 15 c0n515tent w1th H-par1ty and 5ca11n9 pr0pert1e5 15 w1= 0 + 0(a•) and v~ = • ~7~ + 0(a•) f0r 50me c0n5tant ~, 50 that the c0n515tency c0nd1t10n5 are 5een t0 6e 5at15f1ed, w1th P = ~ R - - 6 . . 1 j*k, *4 *
.1j,
-~ 1¢ 172(~) - - ~~¢ 2 ( 1 ~ 7 ~ ) ) 2 -+-
0(~) .
(14)
We 5ha11 5ee 6e10w that the c0n515tency c0nd1t10n5 c0nt1nue t0 6e 5at15f1ed at the next 0rder, and we 5ha11 c0mpute the next 0rder c0ntr16ut10n t0 P. When 8 9 = 8 6 = 0 we kn0w that the expre5510n den0ted 6y P 15 a c0n5tant. 0 n the 0ther hand, 1t 15 a c0n5e4uence 0f w0r1d-5heet d1ffe0m0rph15m and c0nf0rma1 Ward 1dent1t1e5 that 1n 6ack9r0und5 f0r wh1ch 8 ~ and 8 ~ van15h t0 a 91ven 0rder 1n a~, 8 e mu5t 6e c0n5tant t0 at 1ea5t that 0rder [4,18, 25], 1.e. ( 8 ~ = 0(a~L), 81~ = 0 ( a ~ c ) } ~ ~18 e = 0 ( a ~ L ) .
(15)
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
202
F r 0 m (4.9), we 5ee that 8 ~ = ~(d - 26) + 0(a•). 7he m05t 9enera1 f0rm that 8 ¢ can take t0 the next 0rder that ha5 the r19ht 5ca11n9 pr0perty can 6e wr1tten a5
( 8 ° / a ~) = ( d -
+ p[ P + ~91J( 89/a,)] + 1~(~7Q[})2
26)/3a• +
f0r 50me c0n5tant5 p, ~,/~, ~-. A5 8 ~ and P mu5t 6e c0n5tant f0r 6ack9r0und5 1n wh1ch 8 9 = 8 6= 0, the c0eff1c1ent5/~ and ~- mu5t van15h, 50 that 8 * = ~ ( d - 26) + pa•[
P + ~91J( 8~/a~)] + 0( a •2)
8=~(d-26)+~0a~[(1+X)R-(13
+~.)H1jkH 1j~
+ 1¢(2 + ~k11y2~ - K2(1[7~312] + 0(a~21 .
(161
7h15 15 t0 6e c0mpared w1th the f0110w1n9 expre5510n f0r 8 ~ wh1ch we 06ta1n u51n9 (4, 8, 9) and the 10we5t 0rder f0rm5 f0r 0, w
8a~=~(d-261+a~[~W2~-1x(W4312-
y]+0(ad).
(171
C0mpar1n9 the ¢6-dependence 0f the5e tw0 expre5510n5 f0r 8 ~, we 5ee that they are c0n515tent 0n1y 1f 0 = 1/x, X = - 1 , and we then 1earn that
Y=
1 - ~
H1jk H 1 j k + 0 ( 0t•).
7hu5 c0n515tency f1xe5 the c0eff1c1ent a 1n (8) and c0nf1rm5 the re5u1t 0f the ca1cu1at10n 0f append1x 8, 1.e. the a65ence 0f a term pr0p0rt10na1 t0 R. 7 0 5ummar12e, kn0w1n9 f19 and f16 ha5 ena61ed u5 t0 determ1ne 8 ~ and f1~ up t0 an unkn0wn c0eff1c1ent r, A 5tra19htf0rward ca1cu1at10n e1ther 0f 8 ~ 0r 0f f1¢ [6] 91ve5 K = 1, 50 the f1na1 re5u1t f0r the 8-funct10n5 t0 10we5t 0rder 1n a• 15
819j= -a•[ R1j- H m H y + X71WfP] + +
80
•3( d -
261-
0(a•2),
+
a•[-• X72~+ ~(W4~) 2 -- 1u 3aa1jk u1jk] x" ] + 0(a,2)
"
(181
We have checked 6y exp11c1t ca1cu1at10n 0f the c0nf0rma1 an0ma1y that th15 15 the c0rrect re5u1t f0r 8 0 when H1jk = 0.
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
203
We n0w w15h t0 extend th15 ana1y515 t0 next 0rder 1n a•. 7 0 th15 end we f1r5t 1ntr0duce the n0tat10n H12 ~- H1mnHjmn~,
H 2 ~ 91jk0'Jk
,
A -= ( d - 26)/3a•,
(19)
and rec0rd the f0110w1n9 1dent1t1e5 5at15f1ed 6y the ten50r 5~j 91ven 1n (6):
( v + - v++) 5<,j) - t/~+~x +~ - v~7 = 2 { < ~ " v~8,~ - 2 ( u ~ ) ~ v~,8~k -/-/7~"w~m v~8r, 0t t
- - H k 1 P ( ~ 7 k H 1m1 ) 8 m9p - -
(VkH1m~)
6 - [ - ~1 .L1k1mD p 1216 V18km ... •k•--mp
+ t1,4A-1~,1-1J~ - 1-11,%~°14~0~% } + 0( ~) , ( v ~j • v,J~)5L~j~
~7
2 { - - 21" *n1k ~ , . 7 . k 8 6 ~ H 1 j H , . k p 1 7 k 8 2 ~ R 6 A~ ( H2)k11V1861k-- ( ~7kH11m ) ( , ~ 7 1 8 9 k ) + Hmff ,¢•7k1-1m• v ~ 1~pt-~- 1--1~ k•m1-1 P1719 -- L1mkP1--1 4L11pR9 -2*~1 **1m ~ k p ** **m1 ~*1 ~ k 4
( 9 2) k1H1 •48 9 4 } 31.-0(0L~) (20)
where 7= 10
1~1jk1 -- 1111jmL1k1 D 4*~1jk1 ** 2 ~* J* ma.1jk1 1(H2"~ ~ ¢H2)1j -- "2~ ] •j•
+ t14 ~41 1421 L1 kmn 6**1jk~" 1m ~" n *~ "
(21)
A5 we have 5een, 01 and w 1 have the f0rm
v,=V1~+a~u1+0(a~2),
w~=a~Y1+0(a~2),
(22)
f0r 50me (a•-1ndependent) vect0r5 u~, Y1. 7hen u51n9 (20)-(22) 1n (12), (13) we f1nd
-
•
2a•[ V"J 17t1uA- 1Wc6Vt1uA + H/k ~7jyk ] = J
1•
1 7 Q 1+ 0(a•2),
Q1 + 0(a•2),
(23)
204
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
where
+a•[-uJ8 9 + R1•"• k V"~8tm 9 - 2(H 2) k1 V~tt8~k- H 1prnH 1km ~7k81p9 - H k f ( VkHt~1)8~ p - ( VkH~1)( X7182m) + 1L[k1mRp 1t6 2.... 1 k1~rnp 4-H
L1mkP1•4 41D6 -- H1Pk(H2)k18~p] 14k ~ ~1 m ~t.~1p
"~-0(0ff2)
"•[-01t[ - u J 8 6 - 2"~11Rk1m ~7k86m -- H1p1HmkP ~7k86m14- 1-1Pmk(V7kL1m1 ~1716
4• (H2)
k117~0 6 • (17kH11m ) 1718 9mk -6 1 R k•mM PR 9 2~1 ~1rn ~ k p
-- HmkPHm14H11P894 -- ( H 2 ) k1H14894] "k- 0(0ff2).
(24)
N0te that 1f u and y were cur1-free, then fr0m (23) the c0n515tency c0nd1t10n5 w0u1d 6e 5at15f1ed. F0r ar61trary ten50r5 5~j there w0u1d n0t 1n 9enera1 6e any vect0r5 u and y 5uch that th15 15 true. 7hat there ex15t 5uch vect0r5 f0r the 51j 91ven 1n (6) 15 a 5tr0n9 check 0n the re5u1t5 0f 0ur f1-funct10n ca1cu1at10n5. (Ear11er 1nc0rrect ca1cu1at10n5 d0 n0t pa55 th15 te5t.) 7he vect0r5 u1, Y1 c0u1d 6e ca1cu1ated d1rect1y and 0ne c0u1d then check whether the re5u1t 15 fu11y c0n515tent w1th (23). 1n pract1ce th15 ca1cu1at10n 15 rather 1nv01ved, 50 that we 5ha11 1n5tead u5e (23), (24) t0 9a1n 1nf0rmat10n a60ut u~ and y9. 7 h e m05t 9enera1 f0rm5 u~ and y~ can take c0n515tent w1th the re4u1red H-par1ty and 5ca11n9 pr0pert1e5 are
u1= V1[k1R + k2 H2 + k 3172~ff" + ~1~[ k5R + k6 H 2
k4(~7(~) 2]
4- k7172(~ "~ k8( ~7(~)2]
+ VJ~P[k9R1j+ k10(H2)12] + knH1jk(W•HJkt ) , y1=k12(~7JH1jk)~7 k~ ,
(25) (26)
f0r 50me c0n5tant5 k n. 1f 8 9 and 8 6 van15h t0 10we5t 0rder, 1.e. 1f 8 9~- 0 + 0(ct•2), 8 6 -~ 0 + 0(ct •2) then Y1 = 0 and 0n1y 5even 0f the term5 1n (25) are 1ndependent; under the5e c1rcum5tance5, 0ne can ea511y check that 1t 15 nece55ary that V~[~uj1-= 0 + 0(a•) f0r (23) t0 6e 5at15f1ed. 7h15 c0nd1t10n rem0ve5 a11 6ut three 0f
C.M. Hu11,P.K. 70wn5end / 5tr1n9effect1veact10n5
205
the 1ndependent term5 1n (25), W1th the re5U1t that U~ Can 6e Wr1tten a5
u1 = U1[ pR + 4H 2 + r( ~7¢)2] + u1,
(27)
f0r 50me c0eff1c1ent5 p, 4, r and 50me vect0r f11 that 5at15f1e5 ~1=0 + 0(a•) whenever 8 9 and 8 6 van15h t0 10we5t 0rder. Hence, u51n9 (27) 1n (23), we c0nc1ude that, when 89 and 8 6 van15h t0 10we5t 0rder, ~71P = 0,
(28)
where P = ~ ( R - - ~ H 2 + 2 V 2 * - - ( 1 1 7 " ~ ) 2)
W2(pR +4H2+r(W(6)2)-2V•1(pR
+ •a•[7+
+4H2+r(W*)2)W1*+A]
+0(a•2),
(29)
wh1ch 15 the exten510n 0f (14) t0 the next 0rder 1n ct•. U51n9 0ur re5u1t5 f0r 8 * t0 0(a•2), we can wr1te 1
--8~=A+P+~9
1
1j
( 8 19 J a )P+ ~ a1A +t 0 ( ~ 2 )
01 t
=A-[-~W2(6+~(~+)2-+H -~-
2]
101t[7-- 101J~20~,j -+ W2( pR + 4H 2 + r( VC6)2) -2W1(PR+4H2+r(W~)2)W1c~+A]+0(a~2),
(30)
where A 15 50me a•-1ndependent expre5510n 0f appr0pr1ate 5ca11n9 we19ht and H-par1ty t0 6e determ1ned. 1f 8 9 = 8 6 = 0, then 8 0 and P are c0n5tant, and hence W1A = 0 + 0(a•). Fr0m (4), (8), (9), (22), (23) we have the a1ternat1ve expre5510n f0r 8 ~ 1 -8°=A
[
3(v0): 1H:]
01 t
x ~ 1j + 1 W 1 ( p R + 4 H 2 + r ( W , ) 2 ) W 1 , + 1 ~ 1 V . , , ] --ct~[Y+~(W1Wj¢~)(H -)
(31)
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
206
and a c0mpar150n 0f th15 w1th (30) y1e1d5 the e4uat10n Y~1-17--1a1J~40 ~1j --4-~ 7 2 ( P R + 4H 2 + r(~7~)2) + 7u1-1V1~
-1v~1(p9+4H2+r(v~)2)v~1~+~(v~,%e~)(H2)~J +~A =0(a~). 8y mean5
(32)
0f the 1dent1ty
( V~1Vj~)( H2) 1j= t( H2)1J[-R1j + (H2)1j - ( 819/a•)] +(1 -t)V1[H~k~(V~Hjk, - (8~t/a~)) ] -(1-t)
[1(V•1(/))V••H2+
(V•1H1k•)(V•;Hjk, - (8~Ja~))]
+0(a•),
(33)
that f0110w5 fr0m (18), f0r ar61trary c0n5tant t, we can wr1te (32) a5
~, ~ j + ~V2(pR + 4H2) + ~t(H2)u(• • JR1j-~H12) [ ~ + ~ 7 ~ ~/,,~ 1--1
9
t
+ •p( V, 4~) V1[ 9k~( 82Ja•)] - •(1 - t ) H~k'~( 86,/a •) } = - ~p (V,~)V~ 7 2 ~ + •(4 + p + ~(1 - t)) V,~ 1Y1H2 -- •p ~1 ~ ~y1 ~,2~ --
2 ~r2(~7~) 2 4- rV1 ~ ¢ 171~ V j4~ + 0(a•).
(34)
065erve that the term 1n 54Uare 6raCket5 1n (34) 15 ~-1ndependent Wh11e the term 1n CUr1y 6raCket5 15 C0n5tant Up t0 term5 0f h19her 0rder 1n a• When 8 9 8 6= 0 + =
0(a•2).
1t 15 c0nven1ent t0 ch005e the parameter t t0 6e 91ven 1n term5 0f 4 6y
p+4+~(1-t)=0
~
t=1+6(4+p),
(35)
50 that the 5ec0nd term 0n the r19ht-hand 51de 0f (34) van15he5. 7hen when 8 9 and 8 6 van15h t0 10we5t 0rder, the 1eft hand 51de 0f (34) 15 1ndependent 0f (/1 (up t0 0(a•)) wh11e the ~-dependence 0f the r19ht hand 51de cann0t 6e e11m1nated u51n9
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
207
8 9 = 8 6 = 0 ( 0 t •2) un1e55 p = r = 0. A5 part1a1 c0nf1rmat10n 0ne can 5h0w that there are a num6er 0f Feynman d1a9ram5 that c0ntr16ute t0 4, wherea5 we have f0und n0ne that c0ntr16ute t0 p 0r r. We c0nc1ude that p = r = 0. C0n51der n0w the ca5e 0f 9enera1 6ack9r0und5 1n wh1ch the 10we5t 0rder f1e1d e4uat10n5 d0 n0t nece55ar11y h01d. A5 p = r = 0 (and ch0051n9 t a5 1n (35)), a11 the 45-dependence 0f (34) mu5t 6e c0nta1ned 1n the cur1y 6racket5. Hence A mu5t 5at15fy A = -f11171d~+ ( 6 4 + 1)(H2)1J(819Ja •) + 34H~k19r~(86t/a~ ) + A ,
(36)
f0r 50me X wh1ch 15 1ndependent 0f • and wh1ch 5at15f1e5 W,.d= 0(a•) when 8 6 = 8 6 = 0(a•2). 8y wr1t1n9 d0wn the m05t 9enera1 f0rm A-can take c0n515tent w1th the5e c0nd1t10n5 and w1th the 5ca11n9 pr0perty (10) we 5ee that 5uch an .4mu5t 6e 0f the f0rm A = c / a •2 f0r 50me c0n5tant c. H0wever, n0 5uch term c0u1d ar15e 1n pertur6at10n the0ry, 50 that we c0nc1ude that A = 0. 7hen u51n9 (35), (36) 1n (34) we f1nd the f0110w1n9 exp11c1t f0rm f0r Y:
= • 1 7 + 1 1j~ • 1 172H2 ~9 ~1j ~4 + 34H~ktW~ WjHJk1 + •(64+ 1 ) ( H 2 ) 1 J ( R 1 j - - ( H 2 ) 1 j ) .
(37)
7 h e n f1e 15 91ven 6y (4.8) and 8 e can 6e f0und fr0m (9) u51n9 the f0rm 0f v 1 91ven a60ve. A1ternat1ve1y, 1n5ert1n9 (37) 1nt0 (31), u51n9 (18) and the 81anch1 1dent1ty f0r H1j k, 91ve5 u5 the expre5510n 18e 01t
=A-[-~
1 V2~, +~(W~)2-~H2]+~a~(7
-101-~v ] + Q + 0(a~2), 2 0 ~1j1
(38) where
Q = ~(H2)1J8~ + 3 4 H 1 ~ [ H J k , 8 ~ - W18~t] -~4~f11W1~,
(39)
wh1ch van15he5 when 8 9 = 8 6 = 0, and
¢)• = ~ + a•4H 2 .
(40)
0 u r re5u1t5 f0r 8 • depend 0n the c0eff1c1ent 4 1n the 5ame way that 0ur 10wer 0rder re5u1t5 u51n9 th15 meth0d depended 0n the c0eff1c1ent •; 1n 60th ca5e5, a ca1cu1at10n 15 needed t0 f1x the c0eff1c1ent.
208
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
Putt1n9 a11 the5e f0rmu1ae t09ether, we f1nd the f0110w1n9f0rm f0r the 8-funct10n5 1n the 4uant12at10n 5cheme 0f [7]:
8 9= a•[-R1j+ (H2)1j1 ~ 1• k1m -1V 2R1kU( H 2 ) k1 + "~)1~ -- ~1a t 2 [[#.1k1m#,~j 14 mnL1k1 ~*tk1m(1**j) ** n
+ •( V1Hk1m)( ~jH 7M ) -- (WkHt,m)(VkHt5) + 2H11pHj4kHmkPHm41 -211fH2,k(H:) kt] + 0(a•3), ~ 6 = 01t( ~7kH1jk ) --
~0[t2[2( ~7k,~ra[j )R1]k•
(41)
m ..}.• 2( X7k",U)(
H2) k,
--4(~71"km[j)H1]knH1mn ]
-1-0(a•3),
(42)
f1v= •(d- 26) + a•(• W2~ + }H 2) - •a•2 [•R1jkt R1jkt- }( U1Hjkt)(97•HJkt) + ~H1j, Hk~H:~Ht~ 1 + ( 6 4 - 1~1-~11jm1-4k1 1~ 21 . . . . ,,•••1jkt-(64-•)(H2)1j(H2)
1j 4V "2H2
+ 24H 1j~ V2H1jk + ( H2) 1j V1~~7j.(/)]"~-0(0~t3) ,
(43)
and f0r the 8-fUnCt10n5
8~= a~{-- R11+ ( H~)1j-- V117jdP 1 t [[1K1k1m1. n n1k1m -- 701 ~ ..}- 2R1ktj( H 2) k1.~ ~at~k1m(1~*j) ")1~ 14 mn14 ~" k1n
+ }(v,1-1~,,.)(9.1-17M) - (v#-1,m1)(v~1-1~5) + 2H1/pHj4kH•nkpHm 4t
-2H, f~j.(H~) ~] + ~v<,~j>} + 0 ( ~ ) ,
•
}0/t[2 1~kH1m[jR1]k1m "~ 2(17kH11j)( H
+a •H1j* -uk
+ a • 117[1Yj1 }
-1-0 ( ~ ) ,
2
)
k1
--
(44)
4( ~71Hkm[j)H1]knH1m n] (45)
209
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
8 ~ = ( d - 26)
2
•
(
3
01¢ [ 11~
1~1jk1
2
-
-- 1151k1 1-1P4mD --Rkt(H2)k1+ 2"~ m "~ ~p4k1
+3(92)1j(H2)1j]}
+ 01'Q +
5rt
rt1 Hjt rrk,,,,,
6"a1jk~a 1rn~a n ~.t
0(0¢'3).
(46)
(We have ch05en t0 wr1te/3 • 1n the f0rm (4.8) and 8 • 1n the f0rm (38). 70 pa55 fr0m 0ne f0rm t0 the 0ther re4u1re5 u5e 0f (33).) Fr0m (26), the vect0r Y1 take5 the f0rm Y1 =
k12 ~7k~(96/0t~) •
51m11ar1y, a5 f11 van15he5 w1th 8 9 and 8 6, 1t mu5t take the f0rm f11= ~ V J ~ ( 8 , ~ / a ~ ) + f1V1[9Jk(8~/a~)] + 7(V1~)9k~(8~t/a ~) + 0(a•),
(47)
f0r 50me c0n5tant5 a,/3, y. 51nce y1, f11, and Q van15h w1th the 8-funct10n5 the y~, ~,, and Q dependenee 0f the e4uat10n5 8 = 0 can 6e rem0ved 6y tak1n9 11near c0m61nat10n5 0f them. 7hu5 the c0nd1t10n5 f0r c0nf0rma1 1nvar1ance 0f the 519mam0de1 w1th act10n (1) 6ec0me
R,j - ( H 2 ) , j + V, Vj4) 1 ,[a,1 ~ j k1.,+ + -~0f [1K1k1m1~
2R1kt1(H2)k1+ 2R k1m(1~j) 14 mn1•1k1 a1 n
+
-
+ 2H1~pHj4kHmkPHm41+2H1,pHjP( H2) k~] + 0 ( a •2) = 0, ~7kH1jk -- ( 17"kdP•) HUk +
0t•[( VkH1m[1 ) Rj1k1 m --
( WkH11j )( H 2 ) k1
--2(~7~Hkm[1)Hj]knH1m n]
4-0(0/•2) =0,
2A + V24~~ - (~7~)2 + ~H 2 --01t[ 1-147 ... 1jk1 • R 1 j ( H 2 ) 4~1jk14"1~1jk1-- 1(171Hjk1)(17~HJk1 ) -- 21--91J m .14k1mD +5~1-96~t1jkH11m Hj•.
1j
Hk~n+ ~3(H2~,,.j (H2) 1j] + 0(a•2) = 0 . (48)
210
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
7he 0n1y undeterm1ned 4uant1ty 1n the5e e4uat10n5 15 the va1ue 0f the c0n5tant 4, wh1ch 0n1y appear5 1mp11c1t1ythr0u9h ~/1,(e4. (40)). A11 4-dependence can theref0re 6e rem0ved 6y the f1e1d redef1n1t10n • ~ • - 4a•H 2.
3. Effect1ve act10n5
7he 4ue5t10n 0f whether the e4uat10n5 der1ved 6y demand1n9 519ma-m0de1 c0nf0rma1 1nvar1ance are 06ta1na61e a5 the Eu1er-La9ran9e e4uat10n5 0f a 5pacet1me ••effect1ve act10n•• ha5 6een d15cu55ed 1n a num6er 0f paper5 [10,11,14,19-23]. Here we 5ha11 exh161t a c1a55 0f act10n5 fr0m wh1ch e4. (48) can 6e 50 06ta1ned. We expand the putat1ve effect1ve act10n a5 0~
5 = ~ (a•)"5, = E ( a ~ ) " f d % L , . n=1
(49)
n=1
70 10we5t 0rder 1n a• the act10n 15 [4]
5,--
f da4)Vc9e-e [2A + R -
~ , 2 + (V4~)2] ,
(50)
wh1ch can 6e wr1tten a5
51=
1
e-~[9089.- 28 ~] + 0(a•)
(51)
where we have perf0rmed an 1nte9rat10n 6y part5. 7hat there 5h0u1d 6e a fact0r 0f exp(-4~) can 6e ar9ued 0n 9enera1 9r0und5 [4]. 7he var1at10n 0f 51 y1e1d5 (5ee append1x A)
851 69" = v/~ e-~K°( 84~/a~) + 0(a•), where 4t den0te5 the • vect0r•• and K 0 15 the matr1x
K0
(91j, 61j, ~), +
0
9u
9,,9k,
(52)
8 y den0te5 the • vect0r•• (8~, 8~, 8e),
0
0
-91,]
,0j
/
(53)
2
A5 K 0 15 1nvert161e the Eu1er-La9ran9e e4uat10n5 65/6x0 = 0 are e4u1va1ent t0 the e4uat10n5 8 4" = 0 (at 10we5t 0rder). 1n 0rder that th15 rema1n true t0 the next 0rder
C.M. Hu11,P.K. 70wn5end / 5tr1n9 effect1ve act10n5
211
we re4u1re an 52 5uch that
3"1"(51 + a~52) = Vr9 e - ~ [ K0( 8~/a ~) + a•K1( 8•1•/0t •) + 0 ( a•2)] ,
(54)
f0r 50me matr1x K 1 (wh05e entr1e5 may 1nc1ude •/•-dependent d1fferent1a1 0perat0r5), 6ecau5e the 1nvert16111ty 0f K 0 1mp11e5 the f0rma1 1nvert16111ty 0f K 0 + a•K 1 + •. • a5 a p0wer 5er1e5 1n a•. A5 L mu5t 6e 1nvar1ant under the 5ca11n9 (10), L n can 0n1y c0nta1n term5 5ca11n9 a5 fa -n (wh1ch mu5t a150 a11 have even ••H-par1ty••). Let u5 f1r5t 5upp05e that d = 26; the 9enera1 f0rm 0f 52 1n th15 ca5e 15
e-~{a1R1jktR1)k1+ a2R1jk1H1JmHk1m
52 = f
+a3R1j(H2)1j + a4(W1Hjk1)(9:'~H jk1) + a 5 ( H 2 ) 1 j ( H 2 ) 1j + a6H1jkH11mHJ1nHkmn + a7(H2) 2 + a8R 1j ~71¢6Vjc6 + a9(97~)2R -Jr-a10 971~ 9r1R -{- a11 171dP97j~(92) 1j 4-a12(~7~11})292 4-a13 ~:r1~171H2"-{-a14 ~71~( ~71Hjk1)H{k +a15(~7~) 2 V2~ + a16((W~)2) 2) + 52• ,
(55)
Where
5; =
1
f da~ ~ e - ~ ( ~k189 (99) 1j + )k2( 81Y819j)2[- ~3(8~) 2 +~48~(908~) + ~58~(86) 0) + 0(a•).
(56)
A5 5~ 15 4uadrat1c 1n (10we5t 0rder) 8-funct10n5, 1t5 var1at10n 15 11near 1n 8-funct10n5 and 50 may 6e cance11ed 6y a ch01ce 0f the matr1x K 1 appear1n9 1n (54); (6ecau5e 0f the fact0r 0f a• 1n the K 1 term 1n (54) 0n1y the 10we5t 0rder 8-funct10n5 are re1evant here). 7he parameter5 ~1 -.- k5 are theref0re undeterm1ned, and 1f there 15 any act10n 52 f0r wh1ch (54) 15 5at15f1ed there w111 6e (at 1ea5t) a f1ve-parameter fam11y 0f them (w1th an a550c1ated f1ve-parameter fam11y 0f K 1 matr1ce5). F0r the purp05e 0f determ1n1n9 whether there 15 any 52 5at15fy1n9 (54) we can theref0re 19n0re 52~. 51m11ar1y, any 0ther term 1n 352 pr0p0rt10na1 t0 the 10we5t 0rder 8-funct10n5 can 6e cance11ed 6y a m0d1f1cat10n 0f K 1, 50 that we may free1y u5e the 10we5t 0rder 8 = 0 e45. (1.e. the Eu1er-La9ran9e e4uat10n5 51) t0 51mp11fy 352. 7he fu11 8-funct10n5 0f (44)-(46) may them5e1ve5 6e rep1aced 6y the 51mp1er
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
212
expre5510n5 appear1n9 0n the 1.h.5. 0f the e45. (48) 6ecau5e the d1fference, 6e1n9 pr0p0rt10na1 t0 the 10we5t 0rder 8-funct10n5, can a150 6e a650r6ed 1nt0 K v U51n9 the 1emma5 c011ected 1n append1x A t0 c0mpute the var1at10n 0f 52 we have f0und that (54) 15 5at15f1ed (f0r 50me K1) 1f and 0n1y 1f the a-c0eff1c1ent5 are taken t0 6e a1=~,
a2= -~,
a 7 = {(1 - 6 4 ) ,
a3= -2,
a 8 = 0,
a13 = ~ - 3(1 - 6 4 ) ,
a 9 = 0,
a14 = 2,
a5
a4=0,
3,
a6=
a11 = - 2 ,
a10 = 1(1 - 6 4 ) , a15 = - •(1 - 6 4 ) ,
1 9 , a12= 0,
a16 = 14(1 -- 6 4 ) .
(57) 1f d v~ 26 then 52 mU5t 6e m0d1f1ed t0 1nc1ude the term5
f 6%V~ e-~A{X6(2A + R-•H2+ (X7c6)2) +a17A +a18R
+ a19(~7~)2},
(58)
6ecau5e 8 ~ n0w 1nc1ude5 a A-term. A5 ~k6 15 the C0eff1C1ent 0f a term pr0p0rt10na1 t0 5~, the Var1at10n 0f Wh1Ch 1ead5 t0 a 11near C0m61nat10n 0f 10We5t 0rder 8-fUnCt10n5, 1t W111rema1n Undeterm1ned; a Chan9e 1n ~k6 Can 6e c0mpen5ated 6y a chan9e 1n K 1. 7he new a-c0eff1c1ent5, h0wever, are f1xed t0 6e
a18=
a17 = 10(1 - 6 4 ) ,
a19 =
4(1 - 6 4 ) .
(59)
We have, theref0re, a 51x-parameter fam11y 0f act10n5 at th15 5ta9e, each 0f wh1ch 5at15f1e5 (54) f0r 50me matr1x K1: L11jm14 k1 t" d_~ /22 • f 1 1~ 1~1jk1 1D 52 = J d ~eV$ e - ~4"~1jk1~" --2*~1jk1 . . . . m
-2R11(H2)1j+
11-1 14"1t,,,~ L1ftn • "L1kmn 73 ( H 2 )1j(H 2 ) 1j + ~1jk**
-- 2( H2)1J V1dP17j~ + ~ V 1 ~ V1H2 + 2 V16p( V11-1jk1) U1 jk }
+(1•64)[2(H
+
2• 3 ( V ~ ) 2 + 3 V , 2 ~
3(v
) 2 + 32
3A) 2
-
2 - v2
)
+-~((V~¢32- V•2¢)((R - H 2 + V24~)
(60)
213
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
N0te that the term5 c0nta1n1n9 the fact0r 1 - 64 are pr0p0rt10na1 t0 10wer 0rder f1e1d e4uat10n5. 7h15 mean5 that a11 4-dependence can 6e rem0ved fr0m the effect1ve act10n 6y a f1e1d redef1n1t10n (5ee 6e10w). A5 we 5h0w 1n append1x C, th15 15 a c0n5e4uence 0f the fact that the 4-dependence can 6e rem0ved fr0m the f1e1d e4uat10n5 (48) 6y a f1e1d redef1n1t10n. 7 0 5ummar12e, we have e5ta6115hed that the three f1e1d e4uat10n5 91ven 6y re4u1r1n9 that 8 9, 8 6 and 8 ~ van15h can a11 f0110w fr0m an act10n- prev10u5 auth0r5 d1d n0t check a11 three e4uat10n5. Further, we remark that the c0eff1c1ent5 a , are 1n fact f1xed 6y the 8 9 and 8 6 f1e1d e4uat10n5 a10ne. 7hu5 0n c0mpar1n9 8 ~ = 0 w1th an appr0pr1ate 11near c0m61nat10n 0f the Eu1er-La9ran9e e4uat10n5 there are n0 rema1n1n9 parameter5 that can 6e adju5ted t0 make them a9ree. 7he fact that they d0 a9ree 15 theref0re a very 5tr0n9 check 0n 0ur re5u1t5. 7he f0rm 0f the act10n 5 0f c0ur5e chan9e5 1f we make a f1e1d redef1n1t10n. Any redef1n1t10n 0f f1e1d5 that 15 pr0p0rt10na1 t0 ~51/~1" w111 pr0duce new term5 4uadrat1c 1n (10we5t 0rder) 8-funct10n5 wh1ch can then 6e a650r6ed 6y a chan9e 0f the free parameter5 ~1 ..-~5. 1n add1t10n, a redef1n1t10n 0f the f0rm • ~ • + c0n5tant can 6e a650r6ed 6y re5ca11n9 0/• 0r, 1f d ~ 26, 6y chan91n9 ~6. 7he m05t 9enera1 f0rm that any further redef1n1t10n can take, 1f 1t 15 t0 6e c0n515tent w1th the 5ca11n9 pr0pert1e5 0f 5, 15 t~91j = . 1 ( H 2 ) 1 j + .2V•1 V•j¢ + "3 V•1¢ W j ~ + 9 1 j [ . 4 H 2 + "5 172~ + " 6 A ] ,
661j =
. 7 9 1 j k 17k~ ,
84~ = . 8 H 2 + "9 W24~
(61)
After mak1n9 th15 redef1n1t10n we have the f0110w1n9 15-parameter fam11y 0f act10n5 (depend1n9 0n the 6 k,•5 and the 9 ,•5): 5 = f d % V~-e-~{A [2 +a•(X 3 + 10(1 - 64) + 2 X 6 - d , 6 ) A ] ~- [1 -~ 0/t( - ~4 4- 4(1 - 64) + X 6 - 1 ( d • 4.• [• 31• + 0/t(2X3 .4.• ~k4 -- 31~k6•}• 1 ( d •
2).6)A] R
6 ) , 6 --
2.8 -
,1 -
d.4)A ] H 2
4-[1 "~-0/'(--A 4 + 4(1- 64)+ ~6-"3 -2.9-.:-
•(d-
2). 6- d.5)A](V~) 2
1~ n1jk1 -- 12*•1jk1•" 0 1.11jmHk1 m + (3 + ~1 -- " 1 ) ( H 2 ) 1 j ( H 2 ) + 0/t[[9K1jk11x -~- 111 6~1jk 111 ~ 1mHj1 n Hkmn "~ ~ 1 R 1 j R 1j -k- ~2 R2
1j
214
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
- 2(1 + x 1 - 1 / * 1 ) R , j ( H + ( - 2 X 2 - ~)~4- 1(d• 2)/* 4 - 1*8-
•/*1) RH2
+ (-29(1 - 64) + X2 + -~2~3+ •X4 + ~ ( d - 6)/* 4 + 31/*8+ 1/*1)(H2)2
+
+
+/*,)%
+ (2X2 + 1 ( d - 2 ) / * 5 - / * 9 - •/*2- ~/*3)R(~7~) e + (•(1 - 64) - X1 - 2X= + •X 4 +/*9 + •(d - 2)/*5 )
V1c6V1R
+ ( - 2 - 2X, + X, -/*3 -/*2 +/*7 + / * t ) ( H 2 ) 4 V ~ ~7j~ + ( - 2X2 - }X4 + ~ ( d - 6)/*5 +13/*9 "•}- 6/*21 1 • /*8 -- 1/*1 At- 6/*3
•(d--
2)/*4)92(~)
2
+ ( - •(1 - 64) + • + 3X1 + 2X 2 - 1X 3 - t 6~4 x
+ 2/* 8 - ~(d - 6)/*5 +59/*1 + (d -- 1)/*4 )
•
•/* 39
1Y1dP971H2
U1~P(V~"n,,jk)u1jk
+ 2(1 + X1--)t 5 + •/*2--•/*7--•/*1)
4- (~k 1 -1- ~k2 4- 1~t 3 -- 1~k 4 -- ( d - - 1)/*5-- 2/*9) (1721~) 2
+ ( - •(1 - 64) - 3X1 - •2t 3 + 17k4 +/*9
-
-
1 /*24- 2d/*5-23/* 3 ) ( 1 7 ~ ) 21721~3
1 1 + ( ~ ( 1 - 64) + ~X1 + a)t3 +
1
) ]} + +/*3)((vr¢) 22 (62)
where we have rewr1tten 50me term5 6y 1nte9rat10n 6y part5 and u5e 0f the R1cc1 1dent1ty. 1f any 0f the/*-c0eff1c1ent5 1n th15 act10n are n0n-2er0, the Eu1er-La9ran9e e4uat10n5 w111 n0t 6e expre55161e a5 a matr1x K t1me5 the part1cu1ar e4uat10n5 (48), 6ut they w1116e expre55161e a5 K t1me5 a 51m11ar 5et 0f e4uat10n5 c0rre5p0nd1n9 t0 the 8-funct10n5 06ta1ned 6y mean5 0f a d1fferent ren0rma112at10n pre5cr1pt10n, etc., t0 the 0ne we have u5ed. D1ffer1n9 8-funct10n5 06ta1ned u51n9 d1fferent 5u6tract10n 5cheme5 y1e1d act10n5 that, d1ffer1n9 6y f1e1d redef1n1t10n5, are phy51ca11y e4u1va1ent.
C . M . Hu11, P . K . 7 0 w n 5 e n d /
215
5tr1n9 effect1ve act10n5
Met5aev and 75eyt11n have 5h0wn [10] that the 5tr1n9 5catter1n9 amp11tude5 can 6e 5ummar15ed 6y the act10n
5 = 5 1 + a , f dd6~V/-~e-,~[10 0•jkt 4 * ~ 1 j k 1 a" ,,•,
~0
t41j,~r4k~m
- - 2*•1jk1 . . . .
.•
"~ 6 ** 1 j k * * 1m **
n **
~
( H ) 1j
(u2);J] + 0 ( a •2 )
-
(63)
1n 0rder that the 519ma-m0de1 c0nf0rma1 1nvar1ance c0nd1t10n y1e1d the 5ame 5catter1n9 amp11tude5 1t 15 theref0re nece55ary that there ex15t a ch01ce 0f ~ 5 and ]/~5 5uch that (62) reduce5 t0 (63). W1th the ch01ce 0f X and ]/ c0eff1c1ent5 ]/1 = 2,
]/8 = - 1 + 1(1 - 6 4 ) ,
X 3 = - (1 - 6 4 ) ,
~k4 = - •(1 - 6 4 ) ,
X 6 = - 9(1 - 6 4 ) , (64)
~k1 = ~ 2 = ~k5 = ]/2 = ]13 ~- ]/4 = ]/5 = ]/6 = ]/7 = ]/9 = 0 ,
the act10n (62) 1ndeed reduce5 t0 (63) (even f0r d 4= 26). 1t f0110w5 that the effect1ve act10n 06ta1ned fr0m 519ma-m0de1 c0nf0rma1 1nvar1ance 15 c0n515tent w1th 5tr1n9 5catter1n9 amp11tude5, whatever the va1ue 0f 4. 0 n th15 p01nt we d1ffer fr0m Met5aev and 75eyt11n, wh0 c1a1m that the tw0 act10n5 can 0n1y 6e e4u1va1ent 1f 4 = ~. After a further f1e1d redef1n1t10n, the act10n can 6e wr1tten a5 1
5 = -0/t- f d%V/9 e-~(9'J8~ - 2 8 * ) + 0(a•2).
(65)
W1th yet an0ther f1e1d redef1n1t10n and an 1nte9rat10n 6y part5, the act10n 6ec0me5
5=1fd%v/9e-~(91Jf1~-28~)+0(a~2).
(66)
75eyt11n ha5 c1a1med that the act10n 5h0u1d take th15 f0rm t0 a11 0rder5 [19]. 1t 15 p055161e t0 ch005e the parameter5 1n (62) 50 a5 t0 06ta1n a ••pertur6at1ve1y un1tary•• act10n 1n the 5en5e 0f 2w1e6ach [26]. F0r examp1e, 1f d = 26, ch0051n9 ~1 = - 1 ,
X2=1
]/8 = 6/-(1 - 6 4 ) ,
,
X3=
1-
(1-64)
,
~4=
-1-
]/1 = ]/2 = ]/3 = ]/4 = ]/5 = ]/7 = ]/9 = 0 ,
1(1-
64),
X5 = 0 , (67)
216
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
we 06ta1n
-- ,[ 1 n r.1jk1 R 1 j R 1 j + ••R 2 • 1 p 141jmL7k1 -1-0£ [ ~1~1jk11~ -2.~1jk1 . . . . m ..~1L1
1;~1krnn1L1j1L1 1
+ 3 (H2) 2 - 9H ( 7 6 p ) 2 + ~ 7 1 6 P 7 1 H
1j
2 - 9RH1 2
+~R(~7¢)2+ 1(17*)2172~- ~((V¢)2)2]} +0(a~2).
(68)
Append1XA 7he R1emann CUrvatUre 15 91Ven 6y
1 =0 (1} { 1 }{m}--(k•,-••1) R jkt 1, j1 + km 1j
(A.1)
R0Und 6raCket5 den0te 5ymmetr15at10n W1th 5tren9th 0ne wherea5 54uare 6raCket5 den0te ant15ymmetr15at10n w1th 5tren9th 0ne. F0r examp1e,
H1j k =- 3 0[16j~ 1 -- •( 016jk + 0j6k1 + 0k61j ) .
(A.2)
U51n9 the 81anch1 and cyc11c 1dent1t1e5 and 3[~Hjk~1 = 0, wh1ch f0110w5 fr0m the def1n1t10n (A.2), we f1nd the f0110w1n9 u5efu1 1dent1t1e5: R1[k4y
~tRt1jk R
= - ~2R1jk1,
(A.3)
= ~jRk1-
(A.4)
1~7kR1j ,
1~ k 1 m _ 3 k1m 1(k1)m*•j -- ~ R 1 k 1 m R j •
V[1Hj1k, = - ~7[kHt11j , ~7[1Hjk ]~ =
91 17tH1jk,
(A.5)
(A.6) (A.7) (A.8)
~72H1jk = 3W[1VtHjky- 3R1m[1jHk]tm + 3R[/Hjk1t,
(A.9)
C . M . Hu11, P . K . 7 0 w n 5 e n d /
217
5tr1n9 effect1ve act10n5
where the R1cc1 ten50r 15 R1j = Rk1kj •
(A.10)
We def1ne the c0nnect10n and curvature w1th t0r510n 6y jk
+ HJ k~
(1.11)
R(~)~k1---=~0 r ( + ) 1 * -r-( +~ )k1mr ( +Jt) ~1j-w k ~ 1j-=
R j k t +• ~1,Ht11-7-
--
1171Hkj 1 +
(k~1) 1 m - H,~H 1 kj HkmHtj
m
(A.12)
7hen R1• ) = R ( +-)k1kj,
(A.13)
R((1j) • ) = R1j - ( H2)1j
(A.14)
R([1j] + ) = 7- ~7kH1jk
(A.15)
R(+) 1jk1 -- - R (k11j)
(A.16)
R(+) • .4• 2 ( [Uk11 - - - - 3 ~ ++-)H1jk"
(A.17)
70 c0mpute the var1at10n 0f the act10n 1n 5ect. 3 we have u5ed the f0110w1n9 u5efu1 1emma5. C0n51der
5 = fd%- R v6
jkt7Ykt1 ,
(A.18)
where 9 = [det 91j[ and 71jk1 15 a ten50r. 7he var1at10n 0f 5 15
85= f d%Vcd(69~J[-2v~(kvt)6k5 -~
"291j*~k1mn-7k1mn]] + R5k,371 jk•
(A.19)
51m11ar1y, f0r ten50r5 7,j, 7, 7,7k, 55k,
8f d%V~R1f ~j
(A.20)
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
218
(A.21)
+ H~jk 6 7 *j~ , (A.22)
•f dd+~{;k }8Jk1= fdd~1~{~91J[~7k5jk1--~7k5~jk--291j{ 1m}8k1m •
1
k
(A.23) C0n51der the act10n (A.24)
5 = f d a ~ V ~ - e - e ( 6 0 + 61R + 6 2 H 2 + 63(W¢)2). Vary1n9 5 u51n9 (A.19)-(A.23) ee
a5
91ve5
- 61R1 j + 362(H2)1j + 6197197j¢ + (63 - 61)(W ¢ ) ( W j ¢ ) -•91j(61R+62 H 2 + 2 6 1 1 7 2 ¢ + ( 6 3 - 2 6 1 ) ( W ¢ )
e0 85
2 + 6 0 ) , (A.25)
362 ( V7kH1Jk - (97~¢) H1J k ),
(A.26)
86~j e0 85 ---=60 + 61R + 62 H2 + 6312 W2¢ - (W¢)2] .
(A.27)
8¢
1f we ch005e 60=2A,
61 = 1 ,
6 2 -•
- 51,
63
=
1
,
(A.28)
then the act10n (A.24) 6ec0me5 51 (51) and (A.25)-(A.28) y1e1d (52), (53). Append1x
8
Here we 5h0w that there 15 n0 term 1n f1~ pr0p0rt10na1 t0 the R1cc1 5ca1ar R. 7he vert1ce5 we 5ha11 need c0me fr0m the n0rma1 c00rd1nate expan510n 0f 1
1
2 (2rra~)
fd x
+ ... )9,j(,)
0j0A, J,
(8.a)
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
219
90~0~ F19. 1. 7w0-100p d1a9ram c0ntr16ut1n9 a term pr0p0rt10na1 t0 R 1n f1e. 7he d0u61e 11ne 1nd1cate5 the 6ack9r0und f1e1d dependent vertex R1k0~( j 0~,(k 0 ~ 1. 7he wavy 11ne 1nd1cate5 the vertex -h~ 0 ~ ~ 0~J~j(4)) 1nv01v1n9 the tw0-d1men510na1 metr1c pertur6at10n h~,,. 7he d0t5 1nd1cate the p051t10n 0f tw0-d1men510na1 der1vat1ve5 at a vertex.
where (we w0rk 1n euc11dean 5pace)
~,.. = 8.. + h . . ,
~.. = h.. - } ~.h ,
h - ~"~h~..
(8.2)
7he re1evant term5 1n (8.1) are 1
- h " ~ 0 ,..~10 + 1a..a . . . .1:J~..(,/,) ,. 6~ v..¢~ 0.~J~k;,.~j
},
(8.3)
where ~1(x) 15 the n0rma1 c00rd1nate and ep1(x)15 n0w the 6ack9r0und f1e1d (f0r deta115 5ee ref5. [5-7]). We have 1nc1uded a c0var1ant ma55 term t0 re9u1ate 1nfra-red d1ver9ence5. 7he tw0-100p d1a9ram 0f f19. 1 91ve5 the c0ntr16ut10n 0/•
964re
f d•xR( 0~,0.-h~"- ~ h )
(8.4)
t0 the (tw0-d1men510na1) effect1ve act10n. We are u51n9 d1men510na1 re9u1ar12at10n, 50 that --/Lv
•
h 8,,,,-- •eh.
(8.5)
0 t h e r tw0-100p d1a9ram5 that have d1ver9ence5 pr0p0rt10na1 t0 R e1ther 91ve term5 pr0p0rt10na1 t0 m 2 0r c0var1ant12e the re5u1t (8.4), y1e1d1n9 0~p
96rre
f d2x
R
(8.6)
220
C.M. Hu11, P.K. 70wn5end / 5tr1n9 effect1ve act10n5
0
n
F19. 2. 0ne-100p d1a9ram w1th tw0 externa1 4uantum 11ne51ead1n9 t0 the c0unterterm 0f (8.7).
(reca11 that R (v) 15 the tw0-d1men510na15ca1ar curvature). Hence the d1a9ram 0f f19. 1 d0e5 91ve a c0ntr16ut10n t0 f1~ pr0p0rt10na1 t0 R. H0wever, th15 15 cance11ed 6y a 0ne-100p d1a9ram 1nv01v1n9 the c0unterterm 1 f d2x ~ - 7 ~ 3 ~ 247re
3~;R1j (4~)
(8.7)
7h15 c0unterterm 15 needed t0 cance1 the u1tra-v101et d1ver9ence 1n the 0ne-100p d1a9ram 0f f19. 2, wh1ch ha5 tw0 externa1 4uantum 11ne5. A term 0f the f0rm (8.7) 0ccur5 1n the n0rma1 c00rd1nate expan510n 0f the R 4 0 j 0"~J c0unterterm, 6ut n0t w1th the c0rrect c0eff1c1ent. 70 f1nd the r19ht c0eff1c1ent 1t 15 51mp1e5t t0 ca1cu1ate the d1ver9ence 1n f19. 2 d1rect1y, 6ut 1t can a150 6e 06ta1ned fr0m the R u 0,4,18 N~j c0unterterm 1f 0ne take5 1nt0 acc0unt the n0n-11near ren0rma112at10n 0f the 4uantum f1e1d5 that are re4u1red 6y the n0n-11near1ty 0f the 6ack9r0und/4uantum 5p11t [241. 7he 0ne-100p d1a9ram 0f f19. 3 w1th the c0unterterm (8.7) y1e1d5 the c0ntr16ut10n
•
a• fd2xR (0"0~h "~- •Dh)
96~re
(8.8)
t0 the (tw0-d1men510na1) effect1ve act10n. N0t1ce that a1th0u9h 0ne m19ht expect th15
F19. 3. 0ne-100p d1a9ram w1th c0unterterm vertex R1j 3~£:1 8 ~ j c0ntr16ut1n9 a term pr0p0rt10na1 t0 R 1n f1~.
C.M. Hu11,P.K. 70wn5end / 5tr1n9effect1veact10n5
221
d1a9ram t0 c0nta1n 0n1y 1/e 2 p01e5 th15 15 n0t the ca5e, e55ent1a11y 6ecau5e 0f the fact0r 0f e appear1n9 0n the r.h.5. 0f (8.5). A9a1n 0ther 0ne-100p d1a9ram5 w1th c0unterterm5 are e1ther 1rre1evant t0 f1~ 0r 5erve t0 c0var1ant12e (8.8) t0 91ve the c0ntr16ut10n "~ 967re
fd2x9~-R (v)" R
(8.9)
t0 the effect1ve act10n. 7he d1ver9ence5 0f (8.6) and (8.9) cance1, 50 the net c0ntr16ut10n t0 f1~ that 15 pr0p0rt10na1 t0 R 15 2er0, a5 c1a1med. A 51m11ar ca1cu1at10n 5h0w5 that there 15 n0 R c0ntr16ut10n t0 f1~ f0r the (1,1) 0r (1,0) 5uper5ymmetr1c n0n-11near 519ma-m0de15 e1ther.
Append1x C 7 h e c0nd1t10n5 (48) f0r 519ma-m0de1 c0nf0rma1 1nvar1ance 1nv01ve the ar61trary parameter 4 0n1y thr0u9h the 4uant1ty ~ = • + a•4H 2. 7hat 15, 1n term5 0f the new d11at0n f1e1d ~ the5e e4uat10n5 are 4-1ndependent. C1ear1y, 1f the e4uat10n5 1n term5 0f ~ are der1va61e fr0m an act10n, then the act10n can 6e ch05en t0 6e 4-1ndependent (a5 a funct10na1 0f ~ ) . 0 n e can then take ~ ~ (6 6y a f1e1d redef1n1t10n t0 06ta1n a 4-1ndependent act10n f0r the 0r191na1 f1e1d (6. H0w 15 th15 act10n re1ated t0 the 4-dependent act10n f0r • wh05e f1e1d e4uat10n5 are the 0r191na1 4-dependent e4uat10n5 (48)• 7he fact that 4 appear5 0n1y thr0u9h ~ 1n the5e e4uat10n5 d0e5 n0t nece55ar11y mean that 1t appear5 0n1y thr0u9h ~ 1n the act10n, 50 1t 15 n0t even c1ear that the 4-dependence can 6e rem0ved fr0m th15 act10n 6y a f1e1d redef1n1t10n. 7he purp05e 0f th15 append1x 15 t0 5h0w that the 4-dependence can 6e rem0ved 6y a f1e1d redef1n1t10n and that the re5u1t1n9 4-1ndependent act10n 15 e4u1va1ent (after p055161e further f1e1d redef1n1t10n5) w1th that 0ta1ned 6y a d1rect c0n5truct10n 0f a 4-1ndependent act10n 1n term5 0f ~ . 7 h e e4uat10n5 (48) are 91ven a5 a pertur6at10n 5er1e5 1n a~. 1n part1cu1ar ( V ~ ) 2 = ( V ~ ) 2 + 2a•4 V ~ ~7H2 + 0(a,2) and the 0(0ff 2) term can 6e ne91ected at the 0rder 1n a• t0 wh1ch we are w0rk1n9. 7hu5, a1th0u9h a•4H 2 15 n0t nece55ar11y 5ma11, the fact that we w0rk pertur6at1ve1y 1n a• mean5 that we can effect1ve1y 5upp05e that 1t 15. Rather than c0n51der e4uat10n5 and f1e1d redef1n1t10n5 a5 p0wer 5er1e5 1n a~ we theref0re c0n51der 1nf1n1te51ma1 f1e1d redef1n1t10n5 1n exact e4uat10n5. A150, we 5ha11 pre5ent the ar9ument t0 f0110w 1n 9enera11ty 6ecau5e 1t 15 n0t 5pec1f1c t0 the ca5e 1n hand. 7 h e e4uat10n 5a[~6] = 0 f0r a 5et 0f f1e1d5 ~6" w1116e der1va61e fr0m an act10n 1f 5ta, 61--0, 1.e. 1f 5 a - - 5 , a and then 5 15 the act10n. 7he f1e1d e4uat10n 15 theref0re 5 a = 0.
(C.1)
We u5e here the n0tat10n 0f DeW1tt 1n wh1ch the 1ndex a 1a6e15 a11, d15crete 0r
222
C.M. Hu11,P.K. 70wn5end/ 5tr1n9effect1veact10n5
C0nt1nU0U5, Var1a61e5 0n Wh1Ch ~ m a y depend. 1f We m a k e the 1nf1n1te51ma1 f1e1d redef1n1t10n +~.
~ +fa(~)
(C.2)
1n (C.1) we 9et the new e4uat10n 5,~ + 5, ~¢f~ = 0,
(C.3)
(~a6~jrf6, a)(5,6~ - 5,6cfc) =0.
(C.4)
(5 + 5 ~fc),~= 0,
(C.5)
wh1ch 15 06v10u51y e4u1va1ent t0
Ne91ect1n9 0 ( f
2) th15 15
wh1ch 15 the Eu1er-La9ran9e e4uat10n f0r the new act10n 5•[+] = 5 + 5, cfc=5[4~+ f ] .
(c.6)
7h15 15 ju5t the 0r191na1 act10n 1n term5 0f the new f1e1d5 % + f . 7 h e 0n1y n0n-tr1v1a1 p01nt 15 the fact that 0ne mu5t 1ntr0duce the 1nte9rat1n9 fact0r (3a6 + f 6 a) 1n (C.4). App11ed t0 the ca5e 1n h a n d th15 mean5 that the 4 - d e p e n d e n c e 0f the act10n 06ta1ned f r 0 m the 0r191na1 4 - d e p e n d e n t f1e1d e4uat10n5 0f (48) can 6e r e m 0 v e d 6 y the 5h1ft • ~ • - a•4H 2 c0m61ned w1th a 4 - d e p e n d e n t 5h1ft 1n the X-c0eff1c1ent5 t0 t a k e 1nt0 a c c 0 u n t the chan9e 1n the 1nte9rat1n9 fact0r. 0 n e can ver1fy 6 y 1n5pect10n 0f (64) t h a t th15 15 true. W e f0und th15 t0 6e a u5efu1 check 0n the re5u1t5.
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