- Email: [email protected]
Meson properties from the ladder Bethe-Salpeter equation
Phy51c5 Letter5 8 266 ( 1991 ) 467-472 N0rth-H011and
P1-1¥51C5 fE77ER5 8
Me50n pr0pert1e5 fr0m the 1adder 8ethe-5a1peter e4uat10n Ken-1ch1 A0k1 Yukawa 1n5t1tutef0r 7he0ret1ca1Phy51c5,Ky0t0 Un1ver51ty,Ky0t0 606, Japan 7a1ch1r0 Ku90 and Mark 6. M1tchard Department 0f Phy51c5,Ky0t0 Un1ver51ty,Ky0t0 606, Japan Rece1ved 20 May 1991
W0rk1n9 1n the 1mpr0ved 1adder appr0x1mat10n t0 QCD, we ca1cu1atethe tw0 p01nt c0rre1at10n funct10n5 f0r the c0mp051te 0perat0r5 ~M~u.Fr0m the5e we extract ma55va1ue5and decay c0n5tant5 f0r the 10we5t1y1n95ca1ar, vect0r and ax1a1vect0r me50n5. C0n51der1n9 the na1vety 0fthe appr0x1mat10n u5ed, and that we have n0 free parameter5 0nce the QCD 5ca1e155et, the re5u1t5are 5urpr151n91y900d.
N0waday5 m a n y h a d r 0 n pr0pert1e5 Can 6e Ca1CU1ated 6y M0nte Car10 51mU1at10n5 0f1att1Ce Q C D . N0nethe1e55 1t 15 5t111de51ra61e t0 deve10p a rea50na61e appr0x1mat10n 5Cheme f0r Ca1CU1at1n9 5UCh 4Uant1t1e5 w1th 1e55 c0mputat10na1 eff0rt. 5uch a 5cheme w0u1d n0t 0n1y 91ve u5 1n519ht 1nt0 the e55ent1a1 dynam1ca1 pr0pert1e5 0f Q C D 1t5e1f 6ut a150 pr0v1de a v1a61e mean5 t0 1nve5t19ate the dynam1c5 0f the var10u5 new m0de15 pr0p05ed 6ey0nd the 5tandard m0de1. 7 h e purp05e 0f th15 1etter 15 t0 5h0w that the 51mp1e 1mpr0ved 1adder appr0x1mat10n 15 actua11y 5urpr151n91y 900d. 1n a prev10u5 p a p e r [ 1 ] we ca1cu1ated the va1ue5 0f5evera1 phy51ca1 4uant1t1e5 c0nnected w1th 5p0ntane0u5 ch1ra1 5ymmetry 6reak1n9 u51n9 the 1mpr0ved 1adder 5chw1n9er-Dy50n and 8ethe-5a1peter e4uat10n5. F1x1n9 f~ = 94 M e V a5 an 1nput we 06ta1ned ( ( ~ ) 16¢v ) 1/3 = 220 + 10 MeV a n d AQcD ~ 500 MeV ~t. 8 0 t h the5e va1ue5 are rather 900d, a1th0u9h 51m11ar num6er5 have 6een 06ta1ned 6y m a n y auth0r5 u51n9 ju5t the 5e1f-ener9y funct10n 27 and the Pa9e15-5t0kar f0rmu1a [2] (f0r a recent reference 5ee f0r examp1e ref. [3 ] ). 1n 0rder t0 check the 1mpr0ved 1adder appr0x1mat10n further, we have n0w extended 0ur w0rk t0 pred1ct the ma55e5 0 f t h e 10we5t 1y1n9 5ca1ar, vect0r and ax1a1 vect0r me50n5, a5 we11 a5 the vect0r and ax1a1 vect0r decay c0n5tant5. Wh115t the 1adder appr0x1mat10n a5 we u5e 1t here 5ay5 n0th1n9 a60ut the phy51c5 0 f c0nf1nement ( a n d theref0re 15 n0t expected t0 91ve r15e t0, f0r 1n5tance, 11near Re99e traject0r1e5), the n0n-re1at1v15t1c 4uark m0de1 1ead5 u5 t0 expect that the c0nf1n1n9 f0rce 15 n0t re1evant t0 the 10we5t 1y1n9 5tate5. 7 h e re5u1t5 0f th15 pre5ent ca1cu1at10n c0nf1rm th15 expectat10n and, t09ether w1th 0ur prev10u5 w0rk, further 5tren9then the hyp0the515 that ch1ra1 5ymmetry 6reak1n9 0ccur5 at 5h0rter d15tance 5ca1e5 than c0nf1nement [4]. 7 h e 6a51c 4uant1ty wh1ch we ca1cu1ate 15 the 6 r e e n funct10n
~6M~8(P; 4) = f d 4 X d 4 r exP [1(4•X+p•r)
] ( 0 [ 7 ~ M ~ ( 0 ) ~ 1 ( X + ~ r ) ~ ( X - - •r)10) ,
( 1)
where ~ M ~ 5h0u1d 6e under5t00d hereafter a5 a c010ur 51n91et 0perat0r w1th def1n1te f1av0ur (f0r examp1e aMd), 1, j are c010ur 1nd1ce5 a n d 0t, f1 are 5p1n0r 1nd1ce5. We ch005e M t0 6e 1, 175, ~u 0r yu~,5, t0 c0up1e t0 the 5ca1ar, P05td0ct0ra1 fe110w0fthe Japan 50c1ety f0r the Pr0m0t10n 0f 5c1ence. ~1 N0te that 0ur AQc015the 1ead1n90rder A ~ =3) wh1ch 15c0mm0n1y kn0wn t0 6e ~ 500 MeV: 1n ref. [ 1] we 1nc0rrect1yc0mpared 0ur re5u1t w1thA M =a) ~ 220 MeV. 0370-2693/91/$ 03.50 • 1991 E15ev1er5c1ence Pu6115her5 8.V. A11r19ht5 re5erved.
467
V01ume266, num6er 3,4
PHY51C5 LE77ER5 8
29 Au9u5t 1991
p5eud05ca1ar, vect0r and ax1a1 vect0r channe15 re5pect1ve1y. C1051n9 the 1e950n th15 vertex we 06ta1n F(42E) = ~31 1 d4P MPc~6m~P(P;4) =1 f d4Xexp(14•X) (01 7~M4/(0) ~M~u(X)10) = 2 1(01~M~18>12
15>
4~ +m~
(2)
7he fact0r 3 1n the f1r5t 11ne c0me5 fr0m c010ur 5U (3) and the f1na1 5um 15 0ver a11 5tate5 wh1ch c0up1e t0 the 0perat0r 4~M~. We w0rk d1rect1y 1n m0mentum 5pace, and 6ecau5e we mu5t W1ck r0tate t0 eva1uate numer1ca11y the 1nte9ra15 wh1ch 0ccur, 0ur re5u1t5 are 1n term5 0f the euc11dean m0mentum 4E- 8y f1tt1n9 t0 e4. (2), we may extract 60th the ma55e5 m5 and the c0up11n95 (01 ~M~15 ). 7he 1mpr0ved 1adder appr0x1mat10n wh1ch we u5e ha5 6een exten51ve1y 5tud1ed e15ewhere. 1t 15 kn0wn t0 pr0duce the c0rrect a5ympt0t1c 6ehav10ur 1n 60th the 5chw1n9er-Dy50n e4uat10n f0r the 4uark 5e1f-ener9y [ 4 ] and the p5eud05ca1ar 601d5t0ne 6050n 8ethe-5a1peter amp11tude [ 5 ], and we refer the reader t0 the 11terature f0r deta115. Here we 0n1y 5ummar12e the appr0x1mat10n a51t app11e5 t0 the ca1cu1at10n 0f 6M. (1) We w0rk 1n the ch1ra111m1t, c0n51der1n9 a m0de1 w1th three f1av0ur5 0f ma551e55 4uark5 each 0ccurr1n9 1n three c010ur5. 7he exten510n t0 n0n-2er0 6are 4uark ma55 (50 a5 t0 ca1cu1ate, f0r examp1e, pr0pert1e5 0f the ka0n) w1116e d15cu55ed 1ater. (11) F0r the 4uark-91u0n vertex we u5e 197u7a, where 7a are the c010ur 5U (3) 9enerat0r5, and 915 the runn1n9 c0up11n9 c0n5tant. After W1ck r0tat10n we 5et 2(p2E) = 3 Q4n( F2) 92(p2) 16(0(p~E2)
1
- 9
1n(p~/A~cD)
+0(,2~p2E)
1 ) 1n(e2/A~cD) .
(3)
C2 (F) 15 the 5ec0nd Ca51m1r 0f the 4uark repre5entat10n, here 4 f0r the ca5e 0f 5U (3) tr1p1et, A0cD 15 the QCD 5ca1e and ~ ( E> A0cD) 15 an 1nfrared cut 0ff a60ve wh1ch the c0up11n9 run5 acc0rd1n9 t0 the 1ead1n9 0rder ren0rma112at10n 9r0up. 8e10w E the c0up11n9 15 he1d f1xed 1n 0rder t0 av01d any 51n9u1ar1ty. F0r a d15cu5510n 0n the ch01ce 0fm0mentum 5ca1e f0r the ar9ument 0f2 5ee ref. [ 5 ]. (111) We w0rk 1n Landau 9au9e u51n9 Du~(p) = ( - 1 / p 2 ) (9u~-Pu p~/p2) a5 the 91u0n pr0Pa9at0r. (1v) We u5e a 4uark pr0Pa9at0r wh1ch 1nc0rp0rate5 the effect5 0f dynam1ca1 ch1ra1 5ymmetry 6reak1n9, 5v(p) = 1 / [ ~ - 5 ( p 2) ]. 7he 5e1f-ener9y funct10n 5 15 06ta1ned a5 the n0n-2er0 501ut10n 0f the 1mpr0ved 1adder 5chw1n9er-Dy50n e4uat10n w1th runn1n9 c0up11n9 c0n5tant 92•• 92 (max (p ~, k~ ) )
X(p2)=
~
d4 k
C2(F)92~,u5v(k)y~0~,~(k~p) ,
(4)
7he 8ethe-5a1peter e4uat10n f0r 6M 1n the 1mpr0ved 1adder appr0x1mat10n 15 thu5 f
1•m(p; 4) = M +
d4 k ~
C2(F)92yu[--6M(k; 4)]y4D~,~(k--p),
(5)
where the pr0per vertex/~M 15
1•m(P; 4) = 5 ~ (p+ •4) 6M(P; 4 ) 5 ~ (p-- •4) .
(6)
51nCe we are w0rk1n9 1n the C010Ur 51n91et Channe1 the 9r0up the0ry fact0r 15 E7a7a= C2(F). N0t1Ce that f0r 468
V01ume266, num6er 3,4
PHY51C5 LE77ER5 8
29 Au9u5t 1991
the 0ctet channe1 the c0rre5p0nd1n9 fact0r w0u1d 6e k, def1ned 6y ~7a767a=k76. F0r c010ur 5 U ( N ) , k = C2 (F) - • C2 (6) = - 1/2N 50 that th15 channe1 15 repu151ve. F0r 51mp11c1ty we c0n51der the ca5e M = 1 f1r5t. 1n 0rder t0 u5e e4. (5) f0r numer1ca1 ca1cu1at10n, we mu5t c0nvert 1t fr0m a re1at10n 6etween 615p1n0r5 t0 a 5et 0f 5ca1ar e4uat10n5. 70 d0 th15, we expand 6 and F 1n 1nvar1ant amp11tude5 a5
-6M(P; 4) = 5 ( p ; 4) +~P(p; 4) + 4Q(p; 4) + • ( ~ 4 - 4~) 7(p; 4 ) ,
rM(p; 4) =~(p; 4) +~/5(p; 4) + 40• (p; 4) + • (~4-4t~) 7"(p; 4),
(7)
and rewr1te 0ur e4uat10n5 1n term5 0f the5e 4uant1t1e5. We further 51mp11fy the numer1ca1 w0rk 6y ch0051n9 a frame 1n wh1ch the euc11dean vect0r5 (after we have W1ck r0tated ) are 91ven 6y 4 = (4, 0 ), p = ( u, p), k = ( v, k), and 5et x = 1P1, Y= 1k1, 50 that at f1xed 4 the amp11tude5 6ec0me funct10n5 0f 0n1y tw0 var1a61e5, 5(p; 4)= 5(u, x),/5(k; 4) =/5(v, y), etc. When rewr1tten 1n term5 0f the 1nvar1ant amp11tude5, each 0f the tw0 e4uat10n5 e45. (5) and (6) 6ec0me5 f0ur e4uat10n5 re1at1n9 hatted and unhatted amp11tude5. Here f0r rea50n5 0f 5pace we 4u0te 0n1y the f1r5t e4uat10n 1n each ca5e, that f0r 9: 51m11ar e4uat10n5 h01d f0r the 0ther amp11tude5/5, ~ and ~. 7he def1n1t10n 0f the pr0per vertex 91ve5
~= ( •p2+ 142+2+2 - )5+ [ (p2• •4u)2+ + (p2+ 14u)2 - 1p + [ ( 4 u - •42)2+ + (4u+ •42)2• ]Q-42x27,
(8)
etc., where p2 ----X 2 d- U 2, ~ + = ~ ( p 2 + ~42 +•4u), and a11 amp11tude5 are funct10n5 0 f x and u. 51m11ar1ythe 8ethe5a1peter e4uat10n, e4. (5), 1ead5 t0
~ y2dydv49~2 f
9 ( u , x ) = 1+ J
87r3
3
-3 dc°50(k-p)25(v•Y)•
(9)
etc., where p2=U2+X2, k2=v2+y 2, and p.k=uv+xyc050. N0w 0ur ch01ce 0f var1a61e5 a110w5 u5 t0 d0 the an9u1ar 1nte9rat10n51n th15 expre5510n 51nce the c05 0 dependence 15 exp11c1t. We 06ta1n e4uat10n511ke
f ydydv¢-, {(x+y)1+ ( u - v ) 2)
~(u,x)= 1+ j--~y-x ~ m ~ , ~ ( u ~
~ . 5(v,y),
(10)
w1th X = 2 ( m a x ( u 2 + x 2, v2+y 2) ) 91ven 6y e4. (3). E11m1nat1n9 the hatted 4uant1t1e5 we end up w1th a 5et 0f c0up1ed 1nte9ra1 e4uat10n5 0f the f0rm
A(u,x)
(u, x) =
+
dydv8(u,x;v,y)
(v,y) ,
(11)
w1th the 4 × 4 matr1ce5 A ( u, x) and 8 ( u, x; y, v) 6e1n9 91ven 6y e4uat10n511ke e4. ( 8 ) and e4. ( 10 ) re5pect1ve1y. We d15cret12e e4. ( 11 ), and 501ve the re5u1t1n911near e4uat10n numer1ca11y. F0r the vect0r, the e4uat10n5 are a 11tt1e m0re c0mp11cated: we w0rk w1th M=y ~, where 1 15 a 5pace 1ndex, 6ecau5e the fact that 4~= 0 then reduce5 the num6er 0f 1nvar1ant amp11tude5 t0 e19ht, mu1t1p1y1n9 p~, p~, p~, y~, ( ~Y~- Y~ ), ( Y~0- ~Y~), P~(~k~- ~ ) and y tu~Y4 P u4, re5pect1ve1y. We c0nvert e45. ( 6 ) and ( 5 ) t0 5ca1ar e4uat10n5 6y mu1t1p1y1n9 them 6y each 0fp~, ..., Ytu7,Y4pu4~ 1n turn and tak1n9 the trace. 7he a19e6ra wa5 carr1ed 0ut u51n9 the REDUCE a19e6ra1c man1pu1at10n Pr09ram [ 6]. 7he p5eud05ca1ar and ax1a1 vect0r e4uat10n5 re5u1t after 50me tr1v1a1 519n chan9e5 06ta1ned fr0m c0mmut1n9 the extra Y5fact0r. 7he numer1ca1 501ut10n 0fthe 8ethe-5a1peter e4uat10n515 c0mp11cated 6y the fact that the kerne11n e4. ( 11 ) 15 a funct10n 0f f0ur var1a61e5: 1f we d15cret12e each var1a61e 0n N p01nt5 we theref0re have t0 1nvert a matr1x 0f 469
V01ume 266, num6er 3,4
PHY51C5 LE77ER5 8
29 Au9u5t 1991
16N 4 e1ement5 (64N 4 f0r the ca5e 0 f t h e vect0r). We are thu5 re5tr1cted t0 rather 5ma11er va1ue5 0 f N t h a n w0u1d 6e 1dea1. We can he1p, h0wever, a 11tt1e 6y u51n9 the kn0wn CP pr0pert1e5 0 f t h e var10u5 amp11tude5 t0 re5tr1ct t0 e4uat10n5 w1th v>•-0 50 that we need 0n1y c0n51der the re910n where a11 var1a61e5 are p051t1ve. 1n1t1a11y we have u5ed N = 18 f0r the 5ca1ar and N = 13 f0r the vect0r. 8ecau5e 0f th15 5ma11 va1ue we mu5t 6e carefu1 1n ch0051n9 0ur d15cret12at10n. We w0rk 1n term5 0f the var1a61e5 U = 1n (u/AQcD), X = 1n (x/AQcD), V=1n(v/A0cD) and Y=1n(y/A0cD), d15cret12ed a5 Un=a+6n w1th n = 1.... , N etc., 6ecau5e the 4uant1t1e5 0f 1ntere5t vary m0re natura11y 0n a 109ar1thm1c 5ca1e. A5 1nput we u5e the va1ue AQcD = 484 MeV 06ta1ned 1n 0ur prev10u5 ca1cu1at10n [ 1 ], and typ1ca11y we take - 3 ~
[8 ( vk• , Y1- ) + 8 ( vk• , Y1+) + 8 ( vk+, Yt- ) + 8 ( vk+, Y1+ ) ] ,
(12)
where vk, vk+, Yt and Y1•+ are the va1ue5 c0rre5p0nd1n9 t0 the d15cret15ed p01nt5 Vk, Vk+ - ~ (3 Vk+ Vk• 1), Y1 and Y1+• - ~ (3Yt+ Y1+1) re5pect1ve1y. 7he th1rd pr061em 15 that 0f re9u1ar12at10n: f0r the 5ca1ar, 0ur vertex funct10n 15 d1ver9ent. We make 1t f1n1te 6y rep1ac1n9 the c0n5tant term M 1n the 8ethe-5a1peter e4uat10n (5) 6y M exp ( - 0tp 2 ), 50 that we actua11y ca1cu1ate the vertex c0nta1n1n9 the 1n5ert10n ~ M exp ( - a p 2) 4/. 7he vect0r vertex funct10n 15 a1ready f1n1te (1n 6 r e e n funct10n5 wh1ch c0nta1n a 51n91e c0n5erved current 1n5ert10n the current 15 n0t ren0rma112ed); h0wever, the 1nte9ra1 1nv01ved 1n c1051n9 the 1e95 0n 6M, e4. (2), 15 a1way5 d1ver9ent, and t0 re9u1ar12e th15 we 1nc1ude a 51m11ar fact0r exp ( - a p 2E). 7he va1ue5 0f a u5ed are 6etween 1 / (2 6 e V ) 2 and 1/ ( 5 6 e V ) 2. 7hu5 what we f1na11y 06ta1n are va1ue5 f0r
F(42) =1 ~ d4Xexp(14•X) (01 ~ M e x p ( - - 0 t , p 2) 9 t ( 0 ) ~ M e x p ( - - a 2 p 2) 4/(X)[0> = ~. ( 0 1 ~ M e x p ( - 0 4 p
15~
2) ¢ ~ 1 5 ) ( 5 1 ~ M e x p ( - a 2 p ~ ) ~u10>
42 + m25
•
(13)
f0r var10u5 0f 42.7yp1ca1 data f0r the p5eud05ca1ar and vect0r ca5e5 15 5h0wn 1n f195. 1a and 16. 7 h e pre5ence 0f the re9u1at0r exp ( - 0tp2) 0f c0ur5e ha5 n0 effect 0n 0ur ma55 determ1nat10n, 51nce ff•Mexp ( - 0tp2E) 9/c0up1e5 t0 any 5tate that ~M9t d0e5. H0wever, t0 ca1cu1ate the c0up11n9 we mu5t u5e the tr1v1a1 1dent1ty
( 01(tM4/15 ) ( 51~M exp( -0tp2E ) 4/10) (01~M~u15) = [ ( 0 1 ~ M e x p ( ~ a p 2
) 4/15) ( 5 1 4 7 M e x p ( • a p 2 )
~/10) ]1/2,
(14)
where the numerat0r 15 06ta1ned fr0m the ca1cu1at10n 0f e4. ( 13 ) w1th t~ = 0, a2 = a and the den0m1nat0r c0me5 fr0m data w1th 04 = a 2 = 0t. We perf0rm tw0 1ndependent ana1y5e5 0f 0ur re5u1t5, f1tt1n9 t0 60th F(42), and the der1vat1ve F• (4~) = dF(42E )/d42 wh1ch we ca1cu1ate numer1ca11y a5 [F(42 + h ) - F(42E - h ) ] /2h u51n9 5u1ta61y 5ma11 h. 80th f1tt1n9 pr0cedure5 were te5ted u51n9 the p5eud05ca1ar data. 7h15 15 c1ear1y 5een t0 have a p01e at 42= 0 (wh1ch a1ready pr0v1de5 a check 0n 0ur w0rk1n9), 6ut 100k1n9 at a 9raph 0 f F ( 4 ~ ) re5tr1cted t0 42 > ( 1 6 e V ) 2 0r 50 the p051t10n 0f the p01e 15 n0 10n9er any m0re 06v10u5 than 15 the 10cat10n 0f the p01e5 1n the 0ther channe15. 80th f1tt1n9 meth0d5 were a61e t0 pred1ct a p01e at 4~ = 0 0n the 6a515 0f the re5tr1cted data w1th an err0r 0f 1e55 than 0.05 470
V01ume 266, num6er 3,4
PHY51C5 LE77ER5 8
29 Au9u5t 1991
7a61e 1
//1 ~ 1~=(2.56~v)
0
42E[MeV2]
× ~1=~3 ~6ey~-~
-~
vect0r ma55,M0 (MeV) ax1a1vect0r ma55,M., (MeV) 5ca1arma55,M,0 (MeV) p decay c0n5tant,fp (MeV) a1 decay c0n5tant,f~, (MeV)
0ur ca1cu1at10n
Exper1menta1 va1ue
710- 830 1110-1420 770- 860 200- 400 270- 720
770• 317] 1260•30 [7] 9 8 3 • 317] 204• 11 [8] 200•20 [9]
[ ~ 1~=(3.56~v~-~
(1500)2 -(800)~ 0
4~ [MeV2] (1270)2
F19. 1. (a) P5eud05ca1ar1nver5epr0pa9at0r. (6) Vect0r 1nver5e pr0Pa9at0r. 1n 60th ca5e5 the cr055e5 mark the ca1cu1ated data p01nt5, wh115t1n (6) the curve5 5h0w the three re50nancef1t5t0 the data.
6 e V 2. N0nethe1e55, 6ecau5e 0f the ar61trary nature 0f 0ur f1tt1n9, we re9ard 0ur numer1ca1 va1ue5 a5 900d 0n1y when 60th f1t5 91ve 51m11ar re5u1t5. A5 a further check, we c0nf1rm that the ma55e5 pred1cted are 1ndependent 0f the parameter5 a u5ed 1n the exp0nent1a1 re9u1at0r5. F0r the f1t t0 F we f1nd va1ue5 0f r, a n d m ~ wh1ch m1n1m12e the funct10n 2 F1
2
•
1n th15 ca5e we f1nd emp1r1ca11y that the 6e5t f1t 15 06ta1ned 6y f1tt1n9 three re50nance5 [that 15 tak1n9 n = 3 1n e4. (15) ] t0 the data: 0ne w1th a very 1ar9e ma55 a n d ne9at1ve re51due 1n 0rder t0 m0de1 the effect 0f the cut0ff exp ( - a p ) 2, 0ne 0f wh1ch 91ve5 the ma55 a n d c0up11n9 we are 1ntere5ted 1n a n d a th1rd wh1ch ref1ect5 the pre5ence 0f re5t 0 f t h e 5pectrum up t0 the 5ca1e m 2~ 1/0t. F0r the der1vat1ve we m1n1m12e
~ .~/F~(4~)] . [(F• (42)+ ~=, ~ (42E+m~)2][
(16)
Here the 54uare 1n the den0m1nat0r5 0f the tr1a1 funct10n mean5 that the 1ar9e ma55 ne9at1ve re51due term re4u1red 1n the f1t t0 F 6ec0me51rre1evant1y 5ma11, a n d a tw0 re50nance f1t ( m = 2 ) 15 5uff1c1ent. U51n9 the va1ue AQcD = 484 MeV wh1ch we prev10u51y 06ta1ned 6y f1x1n9f~ = 94 MeV #2, we 06ta1n the re5u1t5 5h0wn 1n ta61e 1, where a11 the 4uant1t1e5 are 1n MeV. 0 u r def1n1t10n5 0f the decay c0n5tant5 are a5 f0110w5:
(0]u7ud(x) 1P) = e x p ( -14.x) f~M0eu,
<0[
ayu75d(x)1a, ) = e x p ( -14.x) f~,M,,~ u ,
where 4 and E are the m 0 m e n t u m a n d p01ar12at10n 0f the part1c1e. A5 0ur re5u1t5 we 51mp1y 91ve the ran9e 0f va1ue5 06ta1ned a5 the cut0ff parameter5 a~ a n d 0L2 vary 0ver the ran9e5 0 ~ < a 1 ~ 1 / ( 2 6 e V ) 2 and 1 / ( 5 6 e V ) 2< a2 ~< 1 / ( 2 6eV)2. 7 h e a55umpt10n that 0ur re5u1t5 can 6e m0de1ed 6y a tw0 0r three re50nance f1t 15 ade4uate t0 e5t1mate the ma55e5 we are 1ntere5ted 1n t0 w1th1n 1e55 than 20%, 1ar9e1y 1ndependent 0f the ch01ce 0 f t h e cut0ffparameter5 a~ and a2 1n e4. (13). H0wever, th15 a55umpt10n c1ear1y cau5e5 50me 0 f t h e c0nt1nuum 5pectrum t0 6e 1nc1uded 1n the re51due 0f the part1c1e wh1ch we w15h t0 ca1cu1ate: thu5 the e5t1mate5 0f the5e
~2 1n fact we 5h0u1d 5tr1ct1yhave u5ed the va1ue0ff~ 1n the ch1ra111m1t,wh1ch 1550me 5% 5ma11erthan the 065erved va1ue. H0wever, 91venthe appr0x1mate nature 0f0ur numer1ca1w0rk 5uch a 5ma11d1fference151rre1evanthere. 471
v01ume 266, num6er 3,4
PHY51C5 LE77ER5 8
29 Au9u5t 1991
re51due5 vary 9reat1y, and 50 d0 the va1ue5 we 91ve f0r the decay c0n5tant5. 7 h e pr061em 15 cau5ed 6y the need t0 extrap01ate 0ur data 1nt0 the t1me11ke re910n where the p01e5 are 10cated. We are n0w w0rk1n9 t0 extend 0ur pr0cedure t0 ca1cu1ate d1rect1y w1th t1me11ke 42, a n d we expect t0 6e a61e t0 f1x the va1ue5 0 f t h e var10u5 re51due5 much m0re accurate1y u51n9 th15 appr0ach. 7 h e auth0r5 w0u1d 11ke t0 thank M. 8 a n d 0 , K. H19a5h1j1ma, J. K0da1ra, U . - 6 . Me18ner, 7. M u t a a n d K. Yamawak1 f0r d15cu5510n5. M . 6 . M . ackn0w1ed9e5 the Japan 50c1ety f0r the Pr0m0t10n 0 f 5c1ence f0r the fe110w5h1p, a n d thank5 the M1n15try 0f Educat10n, 5c1ence a n d Cu1ture f0r the 6rant-1n-A1d f0r Enc0ura9ement 0 f F0re19n Y0un9 5c1ent15t ( # 01795056). 7.K. 15 5upp0rted 1n part 6y the 6rant-1n-A1d f0r C00perat1ve Re5earch ( # 02302020) and the 6rant-1n-A1d f0r 5c1ent1f1c Re5earch ( # 02640225) fr0m the M1n15try 0 f Educat10n, 5c1ence and Cu1ture.
Reference5 [ 1] K.-1.A0k1,M. 8and0, 7. Ku90, M.6. M1tchard and H. Nakatan1, Pr09.7he0r. Phy5. 84 (1990) 683. [2] H. Pa9e15and 5.5t0kar, Phy5. Rev. D 20 (1979) 2947. [3] A. 8arducc1, R. Ca5a16u0n1, 5. De Curt15, D. D0m1n1c1and R. 6att0, Phy5. Rev. D 38 (1988) 238. [4] K. H19a5h1j1ma,Phy5. Rev. D 29 (1984) 1228. [ 5 ] K.-1.A0k1,M. 8and0, 7. Ku90 and M.6. M1tchard, Ky0t0 prepr1nt KUN5 1035 (1990). [6] A.C. Hearn, Reduce 3.3 U5er•5 Manua1 (1987 ). [ 7 ] Part1c1eData 6r0up, 6.P. Y05t et a1., Rev1ew 0f part1c1e pr0pert1e5, Phy5. Lett. 8 204 ( 1988 ) 1. [8] J. 6a55er and H. Leutwy1er,Phy5. Rep. 87 (1982) 77. [9] N. 159ur, C. M0rn1n95tar and C. Reader, Phy5. Rev. 39 (1989) 1357.
472
P1-1¥51C5 fE77ER5 8
Me50n pr0pert1e5 fr0m the 1adder 8ethe-5a1peter e4uat10n Ken-1ch1 A0k1 Yukawa 1n5t1tutef0r 7he0ret1ca1Phy51c5,Ky0t0 Un1ver51ty,Ky0t0 606, Japan 7a1ch1r0 Ku90 and Mark 6. M1tchard Department 0f Phy51c5,Ky0t0 Un1ver51ty,Ky0t0 606, Japan Rece1ved 20 May 1991
W0rk1n9 1n the 1mpr0ved 1adder appr0x1mat10n t0 QCD, we ca1cu1atethe tw0 p01nt c0rre1at10n funct10n5 f0r the c0mp051te 0perat0r5 ~M~u.Fr0m the5e we extract ma55va1ue5and decay c0n5tant5 f0r the 10we5t1y1n95ca1ar, vect0r and ax1a1vect0r me50n5. C0n51der1n9 the na1vety 0fthe appr0x1mat10n u5ed, and that we have n0 free parameter5 0nce the QCD 5ca1e155et, the re5u1t5are 5urpr151n91y900d.
N0waday5 m a n y h a d r 0 n pr0pert1e5 Can 6e Ca1CU1ated 6y M0nte Car10 51mU1at10n5 0f1att1Ce Q C D . N0nethe1e55 1t 15 5t111de51ra61e t0 deve10p a rea50na61e appr0x1mat10n 5Cheme f0r Ca1CU1at1n9 5UCh 4Uant1t1e5 w1th 1e55 c0mputat10na1 eff0rt. 5uch a 5cheme w0u1d n0t 0n1y 91ve u5 1n519ht 1nt0 the e55ent1a1 dynam1ca1 pr0pert1e5 0f Q C D 1t5e1f 6ut a150 pr0v1de a v1a61e mean5 t0 1nve5t19ate the dynam1c5 0f the var10u5 new m0de15 pr0p05ed 6ey0nd the 5tandard m0de1. 7 h e purp05e 0f th15 1etter 15 t0 5h0w that the 51mp1e 1mpr0ved 1adder appr0x1mat10n 15 actua11y 5urpr151n91y 900d. 1n a prev10u5 p a p e r [ 1 ] we ca1cu1ated the va1ue5 0f5evera1 phy51ca1 4uant1t1e5 c0nnected w1th 5p0ntane0u5 ch1ra1 5ymmetry 6reak1n9 u51n9 the 1mpr0ved 1adder 5chw1n9er-Dy50n and 8ethe-5a1peter e4uat10n5. F1x1n9 f~ = 94 M e V a5 an 1nput we 06ta1ned ( ( ~ ) 16¢v ) 1/3 = 220 + 10 MeV a n d AQcD ~ 500 MeV ~t. 8 0 t h the5e va1ue5 are rather 900d, a1th0u9h 51m11ar num6er5 have 6een 06ta1ned 6y m a n y auth0r5 u51n9 ju5t the 5e1f-ener9y funct10n 27 and the Pa9e15-5t0kar f0rmu1a [2] (f0r a recent reference 5ee f0r examp1e ref. [3 ] ). 1n 0rder t0 check the 1mpr0ved 1adder appr0x1mat10n further, we have n0w extended 0ur w0rk t0 pred1ct the ma55e5 0 f t h e 10we5t 1y1n9 5ca1ar, vect0r and ax1a1 vect0r me50n5, a5 we11 a5 the vect0r and ax1a1 vect0r decay c0n5tant5. Wh115t the 1adder appr0x1mat10n a5 we u5e 1t here 5ay5 n0th1n9 a60ut the phy51c5 0 f c0nf1nement ( a n d theref0re 15 n0t expected t0 91ve r15e t0, f0r 1n5tance, 11near Re99e traject0r1e5), the n0n-re1at1v15t1c 4uark m0de1 1ead5 u5 t0 expect that the c0nf1n1n9 f0rce 15 n0t re1evant t0 the 10we5t 1y1n9 5tate5. 7 h e re5u1t5 0f th15 pre5ent ca1cu1at10n c0nf1rm th15 expectat10n and, t09ether w1th 0ur prev10u5 w0rk, further 5tren9then the hyp0the515 that ch1ra1 5ymmetry 6reak1n9 0ccur5 at 5h0rter d15tance 5ca1e5 than c0nf1nement [4]. 7 h e 6a51c 4uant1ty wh1ch we ca1cu1ate 15 the 6 r e e n funct10n
~6M~8(P; 4) = f d 4 X d 4 r exP [1(4•X+p•r)
] ( 0 [ 7 ~ M ~ ( 0 ) ~ 1 ( X + ~ r ) ~ ( X - - •r)10) ,
( 1)
where ~ M ~ 5h0u1d 6e under5t00d hereafter a5 a c010ur 51n91et 0perat0r w1th def1n1te f1av0ur (f0r examp1e aMd), 1, j are c010ur 1nd1ce5 a n d 0t, f1 are 5p1n0r 1nd1ce5. We ch005e M t0 6e 1, 175, ~u 0r yu~,5, t0 c0up1e t0 the 5ca1ar, P05td0ct0ra1 fe110w0fthe Japan 50c1ety f0r the Pr0m0t10n 0f 5c1ence. ~1 N0te that 0ur AQc015the 1ead1n90rder A ~ =3) wh1ch 15c0mm0n1y kn0wn t0 6e ~ 500 MeV: 1n ref. [ 1] we 1nc0rrect1yc0mpared 0ur re5u1t w1thA M =a) ~ 220 MeV. 0370-2693/91/$ 03.50 • 1991 E15ev1er5c1ence Pu6115her5 8.V. A11r19ht5 re5erved.
467
V01ume266, num6er 3,4
PHY51C5 LE77ER5 8
29 Au9u5t 1991
p5eud05ca1ar, vect0r and ax1a1 vect0r channe15 re5pect1ve1y. C1051n9 the 1e950n th15 vertex we 06ta1n F(42E) = ~31 1 d4P MPc~6m~P(P;4) =1 f d4Xexp(14•X) (01 7~M4/(0) ~M~u(X)10) = 2 1(01~M~18>12
15>
4~ +m~
(2)
7he fact0r 3 1n the f1r5t 11ne c0me5 fr0m c010ur 5U (3) and the f1na1 5um 15 0ver a11 5tate5 wh1ch c0up1e t0 the 0perat0r 4~M~. We w0rk d1rect1y 1n m0mentum 5pace, and 6ecau5e we mu5t W1ck r0tate t0 eva1uate numer1ca11y the 1nte9ra15 wh1ch 0ccur, 0ur re5u1t5 are 1n term5 0f the euc11dean m0mentum 4E- 8y f1tt1n9 t0 e4. (2), we may extract 60th the ma55e5 m5 and the c0up11n95 (01 ~M~15 ). 7he 1mpr0ved 1adder appr0x1mat10n wh1ch we u5e ha5 6een exten51ve1y 5tud1ed e15ewhere. 1t 15 kn0wn t0 pr0duce the c0rrect a5ympt0t1c 6ehav10ur 1n 60th the 5chw1n9er-Dy50n e4uat10n f0r the 4uark 5e1f-ener9y [ 4 ] and the p5eud05ca1ar 601d5t0ne 6050n 8ethe-5a1peter amp11tude [ 5 ], and we refer the reader t0 the 11terature f0r deta115. Here we 0n1y 5ummar12e the appr0x1mat10n a51t app11e5 t0 the ca1cu1at10n 0f 6M. (1) We w0rk 1n the ch1ra111m1t, c0n51der1n9 a m0de1 w1th three f1av0ur5 0f ma551e55 4uark5 each 0ccurr1n9 1n three c010ur5. 7he exten510n t0 n0n-2er0 6are 4uark ma55 (50 a5 t0 ca1cu1ate, f0r examp1e, pr0pert1e5 0f the ka0n) w1116e d15cu55ed 1ater. (11) F0r the 4uark-91u0n vertex we u5e 197u7a, where 7a are the c010ur 5U (3) 9enerat0r5, and 915 the runn1n9 c0up11n9 c0n5tant. After W1ck r0tat10n we 5et 2(p2E) = 3 Q4n( F2) 92(p2) 16(0(p~E2)
1
- 9
1n(p~/A~cD)
+0(,2~p2E)
1 ) 1n(e2/A~cD) .
(3)
C2 (F) 15 the 5ec0nd Ca51m1r 0f the 4uark repre5entat10n, here 4 f0r the ca5e 0f 5U (3) tr1p1et, A0cD 15 the QCD 5ca1e and ~ ( E> A0cD) 15 an 1nfrared cut 0ff a60ve wh1ch the c0up11n9 run5 acc0rd1n9 t0 the 1ead1n9 0rder ren0rma112at10n 9r0up. 8e10w E the c0up11n9 15 he1d f1xed 1n 0rder t0 av01d any 51n9u1ar1ty. F0r a d15cu5510n 0n the ch01ce 0fm0mentum 5ca1e f0r the ar9ument 0f2 5ee ref. [ 5 ]. (111) We w0rk 1n Landau 9au9e u51n9 Du~(p) = ( - 1 / p 2 ) (9u~-Pu p~/p2) a5 the 91u0n pr0Pa9at0r. (1v) We u5e a 4uark pr0Pa9at0r wh1ch 1nc0rp0rate5 the effect5 0f dynam1ca1 ch1ra1 5ymmetry 6reak1n9, 5v(p) = 1 / [ ~ - 5 ( p 2) ]. 7he 5e1f-ener9y funct10n 5 15 06ta1ned a5 the n0n-2er0 501ut10n 0f the 1mpr0ved 1adder 5chw1n9er-Dy50n e4uat10n w1th runn1n9 c0up11n9 c0n5tant 92•• 92 (max (p ~, k~ ) )
X(p2)=
~
d4 k
C2(F)92~,u5v(k)y~0~,~(k~p) ,
(4)
7he 8ethe-5a1peter e4uat10n f0r 6M 1n the 1mpr0ved 1adder appr0x1mat10n 15 thu5 f
1•m(p; 4) = M +
d4 k ~
C2(F)92yu[--6M(k; 4)]y4D~,~(k--p),
(5)
where the pr0per vertex/~M 15
1•m(P; 4) = 5 ~ (p+ •4) 6M(P; 4 ) 5 ~ (p-- •4) .
(6)
51nCe we are w0rk1n9 1n the C010Ur 51n91et Channe1 the 9r0up the0ry fact0r 15 E7a7a= C2(F). N0t1Ce that f0r 468
V01ume266, num6er 3,4
PHY51C5 LE77ER5 8
29 Au9u5t 1991
the 0ctet channe1 the c0rre5p0nd1n9 fact0r w0u1d 6e k, def1ned 6y ~7a767a=k76. F0r c010ur 5 U ( N ) , k = C2 (F) - • C2 (6) = - 1/2N 50 that th15 channe1 15 repu151ve. F0r 51mp11c1ty we c0n51der the ca5e M = 1 f1r5t. 1n 0rder t0 u5e e4. (5) f0r numer1ca1 ca1cu1at10n, we mu5t c0nvert 1t fr0m a re1at10n 6etween 615p1n0r5 t0 a 5et 0f 5ca1ar e4uat10n5. 70 d0 th15, we expand 6 and F 1n 1nvar1ant amp11tude5 a5
-6M(P; 4) = 5 ( p ; 4) +~P(p; 4) + 4Q(p; 4) + • ( ~ 4 - 4~) 7(p; 4 ) ,
rM(p; 4) =~(p; 4) +~/5(p; 4) + 40• (p; 4) + • (~4-4t~) 7"(p; 4),
(7)
and rewr1te 0ur e4uat10n5 1n term5 0f the5e 4uant1t1e5. We further 51mp11fy the numer1ca1 w0rk 6y ch0051n9 a frame 1n wh1ch the euc11dean vect0r5 (after we have W1ck r0tated ) are 91ven 6y 4 = (4, 0 ), p = ( u, p), k = ( v, k), and 5et x = 1P1, Y= 1k1, 50 that at f1xed 4 the amp11tude5 6ec0me funct10n5 0f 0n1y tw0 var1a61e5, 5(p; 4)= 5(u, x),/5(k; 4) =/5(v, y), etc. When rewr1tten 1n term5 0f the 1nvar1ant amp11tude5, each 0f the tw0 e4uat10n5 e45. (5) and (6) 6ec0me5 f0ur e4uat10n5 re1at1n9 hatted and unhatted amp11tude5. Here f0r rea50n5 0f 5pace we 4u0te 0n1y the f1r5t e4uat10n 1n each ca5e, that f0r 9: 51m11ar e4uat10n5 h01d f0r the 0ther amp11tude5/5, ~ and ~. 7he def1n1t10n 0f the pr0per vertex 91ve5
~= ( •p2+ 142+2+2 - )5+ [ (p2• •4u)2+ + (p2+ 14u)2 - 1p + [ ( 4 u - •42)2+ + (4u+ •42)2• ]Q-42x27,
(8)
etc., where p2 ----X 2 d- U 2, ~ + = ~ ( p 2 + ~42 +•4u), and a11 amp11tude5 are funct10n5 0 f x and u. 51m11ar1ythe 8ethe5a1peter e4uat10n, e4. (5), 1ead5 t0
~ y2dydv49~2 f
9 ( u , x ) = 1+ J
87r3
3
-3 dc°50(k-p)25(v•Y)•
(9)
etc., where p2=U2+X2, k2=v2+y 2, and p.k=uv+xyc050. N0w 0ur ch01ce 0f var1a61e5 a110w5 u5 t0 d0 the an9u1ar 1nte9rat10n51n th15 expre5510n 51nce the c05 0 dependence 15 exp11c1t. We 06ta1n e4uat10n511ke
f ydydv¢-, {(x+y)1+ ( u - v ) 2)
~(u,x)= 1+ j--~y-x ~ m ~ , ~ ( u ~
~ . 5(v,y),
(10)
w1th X = 2 ( m a x ( u 2 + x 2, v2+y 2) ) 91ven 6y e4. (3). E11m1nat1n9 the hatted 4uant1t1e5 we end up w1th a 5et 0f c0up1ed 1nte9ra1 e4uat10n5 0f the f0rm
A(u,x)
(u, x) =
+
dydv8(u,x;v,y)
(v,y) ,
(11)
w1th the 4 × 4 matr1ce5 A ( u, x) and 8 ( u, x; y, v) 6e1n9 91ven 6y e4uat10n511ke e4. ( 8 ) and e4. ( 10 ) re5pect1ve1y. We d15cret12e e4. ( 11 ), and 501ve the re5u1t1n911near e4uat10n numer1ca11y. F0r the vect0r, the e4uat10n5 are a 11tt1e m0re c0mp11cated: we w0rk w1th M=y ~, where 1 15 a 5pace 1ndex, 6ecau5e the fact that 4~= 0 then reduce5 the num6er 0f 1nvar1ant amp11tude5 t0 e19ht, mu1t1p1y1n9 p~, p~, p~, y~, ( ~Y~- Y~ ), ( Y~0- ~Y~), P~(~k~- ~ ) and y tu~Y4 P u4, re5pect1ve1y. We c0nvert e45. ( 6 ) and ( 5 ) t0 5ca1ar e4uat10n5 6y mu1t1p1y1n9 them 6y each 0fp~, ..., Ytu7,Y4pu4~ 1n turn and tak1n9 the trace. 7he a19e6ra wa5 carr1ed 0ut u51n9 the REDUCE a19e6ra1c man1pu1at10n Pr09ram [ 6]. 7he p5eud05ca1ar and ax1a1 vect0r e4uat10n5 re5u1t after 50me tr1v1a1 519n chan9e5 06ta1ned fr0m c0mmut1n9 the extra Y5fact0r. 7he numer1ca1 501ut10n 0fthe 8ethe-5a1peter e4uat10n515 c0mp11cated 6y the fact that the kerne11n e4. ( 11 ) 15 a funct10n 0f f0ur var1a61e5: 1f we d15cret12e each var1a61e 0n N p01nt5 we theref0re have t0 1nvert a matr1x 0f 469
V01ume 266, num6er 3,4
PHY51C5 LE77ER5 8
29 Au9u5t 1991
16N 4 e1ement5 (64N 4 f0r the ca5e 0 f t h e vect0r). We are thu5 re5tr1cted t0 rather 5ma11er va1ue5 0 f N t h a n w0u1d 6e 1dea1. We can he1p, h0wever, a 11tt1e 6y u51n9 the kn0wn CP pr0pert1e5 0 f t h e var10u5 amp11tude5 t0 re5tr1ct t0 e4uat10n5 w1th v>•-0 50 that we need 0n1y c0n51der the re910n where a11 var1a61e5 are p051t1ve. 1n1t1a11y we have u5ed N = 18 f0r the 5ca1ar and N = 13 f0r the vect0r. 8ecau5e 0f th15 5ma11 va1ue we mu5t 6e carefu1 1n ch0051n9 0ur d15cret12at10n. We w0rk 1n term5 0f the var1a61e5 U = 1n (u/AQcD), X = 1n (x/AQcD), V=1n(v/A0cD) and Y=1n(y/A0cD), d15cret12ed a5 Un=a+6n w1th n = 1.... , N etc., 6ecau5e the 4uant1t1e5 0f 1ntere5t vary m0re natura11y 0n a 109ar1thm1c 5ca1e. A5 1nput we u5e the va1ue AQcD = 484 MeV 06ta1ned 1n 0ur prev10u5 ca1cu1at10n [ 1 ], and typ1ca11y we take - 3 ~
[8 ( vk• , Y1- ) + 8 ( vk• , Y1+) + 8 ( vk+, Yt- ) + 8 ( vk+, Y1+ ) ] ,
(12)
where vk, vk+, Yt and Y1•+ are the va1ue5 c0rre5p0nd1n9 t0 the d15cret15ed p01nt5 Vk, Vk+ - ~ (3 Vk+ Vk• 1), Y1 and Y1+• - ~ (3Yt+ Y1+1) re5pect1ve1y. 7he th1rd pr061em 15 that 0f re9u1ar12at10n: f0r the 5ca1ar, 0ur vertex funct10n 15 d1ver9ent. We make 1t f1n1te 6y rep1ac1n9 the c0n5tant term M 1n the 8ethe-5a1peter e4uat10n (5) 6y M exp ( - 0tp 2 ), 50 that we actua11y ca1cu1ate the vertex c0nta1n1n9 the 1n5ert10n ~ M exp ( - a p 2) 4/. 7he vect0r vertex funct10n 15 a1ready f1n1te (1n 6 r e e n funct10n5 wh1ch c0nta1n a 51n91e c0n5erved current 1n5ert10n the current 15 n0t ren0rma112ed); h0wever, the 1nte9ra1 1nv01ved 1n c1051n9 the 1e95 0n 6M, e4. (2), 15 a1way5 d1ver9ent, and t0 re9u1ar12e th15 we 1nc1ude a 51m11ar fact0r exp ( - a p 2E). 7he va1ue5 0f a u5ed are 6etween 1 / (2 6 e V ) 2 and 1/ ( 5 6 e V ) 2. 7hu5 what we f1na11y 06ta1n are va1ue5 f0r
F(42) =1 ~ d4Xexp(14•X) (01 ~ M e x p ( - - 0 t , p 2) 9 t ( 0 ) ~ M e x p ( - - a 2 p 2) 4/(X)[0> = ~. ( 0 1 ~ M e x p ( - 0 4 p
15~
2) ¢ ~ 1 5 ) ( 5 1 ~ M e x p ( - a 2 p ~ ) ~u10>
42 + m25
•
(13)
f0r var10u5 0f 42.7yp1ca1 data f0r the p5eud05ca1ar and vect0r ca5e5 15 5h0wn 1n f195. 1a and 16. 7 h e pre5ence 0f the re9u1at0r exp ( - 0tp2) 0f c0ur5e ha5 n0 effect 0n 0ur ma55 determ1nat10n, 51nce ff•Mexp ( - 0tp2E) 9/c0up1e5 t0 any 5tate that ~M9t d0e5. H0wever, t0 ca1cu1ate the c0up11n9 we mu5t u5e the tr1v1a1 1dent1ty
( 01(tM4/15 ) ( 51~M exp( -0tp2E ) 4/10) (01~M~u15) = [ ( 0 1 ~ M e x p ( ~ a p 2
) 4/15) ( 5 1 4 7 M e x p ( • a p 2 )
~/10) ]1/2,
(14)
where the numerat0r 15 06ta1ned fr0m the ca1cu1at10n 0f e4. ( 13 ) w1th t~ = 0, a2 = a and the den0m1nat0r c0me5 fr0m data w1th 04 = a 2 = 0t. We perf0rm tw0 1ndependent ana1y5e5 0f 0ur re5u1t5, f1tt1n9 t0 60th F(42), and the der1vat1ve F• (4~) = dF(42E )/d42 wh1ch we ca1cu1ate numer1ca11y a5 [F(42 + h ) - F(42E - h ) ] /2h u51n9 5u1ta61y 5ma11 h. 80th f1tt1n9 pr0cedure5 were te5ted u51n9 the p5eud05ca1ar data. 7h15 15 c1ear1y 5een t0 have a p01e at 42= 0 (wh1ch a1ready pr0v1de5 a check 0n 0ur w0rk1n9), 6ut 100k1n9 at a 9raph 0 f F ( 4 ~ ) re5tr1cted t0 42 > ( 1 6 e V ) 2 0r 50 the p051t10n 0f the p01e 15 n0 10n9er any m0re 06v10u5 than 15 the 10cat10n 0f the p01e5 1n the 0ther channe15. 80th f1tt1n9 meth0d5 were a61e t0 pred1ct a p01e at 4~ = 0 0n the 6a515 0f the re5tr1cted data w1th an err0r 0f 1e55 than 0.05 470
V01ume 266, num6er 3,4
PHY51C5 LE77ER5 8
29 Au9u5t 1991
7a61e 1
//1 ~ 1~=(2.56~v)
0
42E[MeV2]
× ~1=~3 ~6ey~-~
-~
vect0r ma55,M0 (MeV) ax1a1vect0r ma55,M., (MeV) 5ca1arma55,M,0 (MeV) p decay c0n5tant,fp (MeV) a1 decay c0n5tant,f~, (MeV)
0ur ca1cu1at10n
Exper1menta1 va1ue
710- 830 1110-1420 770- 860 200- 400 270- 720
770• 317] 1260•30 [7] 9 8 3 • 317] 204• 11 [8] 200•20 [9]
[ ~ 1~=(3.56~v~-~
(1500)2 -(800)~ 0
4~ [MeV2] (1270)2
F19. 1. (a) P5eud05ca1ar1nver5epr0pa9at0r. (6) Vect0r 1nver5e pr0Pa9at0r. 1n 60th ca5e5 the cr055e5 mark the ca1cu1ated data p01nt5, wh115t1n (6) the curve5 5h0w the three re50nancef1t5t0 the data.
6 e V 2. N0nethe1e55, 6ecau5e 0f the ar61trary nature 0f 0ur f1tt1n9, we re9ard 0ur numer1ca1 va1ue5 a5 900d 0n1y when 60th f1t5 91ve 51m11ar re5u1t5. A5 a further check, we c0nf1rm that the ma55e5 pred1cted are 1ndependent 0f the parameter5 a u5ed 1n the exp0nent1a1 re9u1at0r5. F0r the f1t t0 F we f1nd va1ue5 0f r, a n d m ~ wh1ch m1n1m12e the funct10n 2 F1
2
•
1n th15 ca5e we f1nd emp1r1ca11y that the 6e5t f1t 15 06ta1ned 6y f1tt1n9 three re50nance5 [that 15 tak1n9 n = 3 1n e4. (15) ] t0 the data: 0ne w1th a very 1ar9e ma55 a n d ne9at1ve re51due 1n 0rder t0 m0de1 the effect 0f the cut0ff exp ( - a p ) 2, 0ne 0f wh1ch 91ve5 the ma55 a n d c0up11n9 we are 1ntere5ted 1n a n d a th1rd wh1ch ref1ect5 the pre5ence 0f re5t 0 f t h e 5pectrum up t0 the 5ca1e m 2~ 1/0t. F0r the der1vat1ve we m1n1m12e
~ .~/F~(4~)] . [(F• (42)+ ~=, ~ (42E+m~)2][
(16)
Here the 54uare 1n the den0m1nat0r5 0f the tr1a1 funct10n mean5 that the 1ar9e ma55 ne9at1ve re51due term re4u1red 1n the f1t t0 F 6ec0me51rre1evant1y 5ma11, a n d a tw0 re50nance f1t ( m = 2 ) 15 5uff1c1ent. U51n9 the va1ue AQcD = 484 MeV wh1ch we prev10u51y 06ta1ned 6y f1x1n9f~ = 94 MeV #2, we 06ta1n the re5u1t5 5h0wn 1n ta61e 1, where a11 the 4uant1t1e5 are 1n MeV. 0 u r def1n1t10n5 0f the decay c0n5tant5 are a5 f0110w5:
(0]u7ud(x) 1P) = e x p ( -14.x) f~M0eu,
<0[
ayu75d(x)1a, ) = e x p ( -14.x) f~,M,,~ u ,
where 4 and E are the m 0 m e n t u m a n d p01ar12at10n 0f the part1c1e. A5 0ur re5u1t5 we 51mp1y 91ve the ran9e 0f va1ue5 06ta1ned a5 the cut0ff parameter5 a~ a n d 0L2 vary 0ver the ran9e5 0 ~ < a 1 ~ 1 / ( 2 6 e V ) 2 and 1 / ( 5 6 e V ) 2< a2 ~< 1 / ( 2 6eV)2. 7 h e a55umpt10n that 0ur re5u1t5 can 6e m0de1ed 6y a tw0 0r three re50nance f1t 15 ade4uate t0 e5t1mate the ma55e5 we are 1ntere5ted 1n t0 w1th1n 1e55 than 20%, 1ar9e1y 1ndependent 0f the ch01ce 0 f t h e cut0ffparameter5 a~ and a2 1n e4. (13). H0wever, th15 a55umpt10n c1ear1y cau5e5 50me 0 f t h e c0nt1nuum 5pectrum t0 6e 1nc1uded 1n the re51due 0f the part1c1e wh1ch we w15h t0 ca1cu1ate: thu5 the e5t1mate5 0f the5e
~2 1n fact we 5h0u1d 5tr1ct1yhave u5ed the va1ue0ff~ 1n the ch1ra111m1t,wh1ch 1550me 5% 5ma11erthan the 065erved va1ue. H0wever, 91venthe appr0x1mate nature 0f0ur numer1ca1w0rk 5uch a 5ma11d1fference151rre1evanthere. 471
v01ume 266, num6er 3,4
PHY51C5 LE77ER5 8
29 Au9u5t 1991
re51due5 vary 9reat1y, and 50 d0 the va1ue5 we 91ve f0r the decay c0n5tant5. 7 h e pr061em 15 cau5ed 6y the need t0 extrap01ate 0ur data 1nt0 the t1me11ke re910n where the p01e5 are 10cated. We are n0w w0rk1n9 t0 extend 0ur pr0cedure t0 ca1cu1ate d1rect1y w1th t1me11ke 42, a n d we expect t0 6e a61e t0 f1x the va1ue5 0 f t h e var10u5 re51due5 much m0re accurate1y u51n9 th15 appr0ach. 7 h e auth0r5 w0u1d 11ke t0 thank M. 8 a n d 0 , K. H19a5h1j1ma, J. K0da1ra, U . - 6 . Me18ner, 7. M u t a a n d K. Yamawak1 f0r d15cu5510n5. M . 6 . M . ackn0w1ed9e5 the Japan 50c1ety f0r the Pr0m0t10n 0 f 5c1ence f0r the fe110w5h1p, a n d thank5 the M1n15try 0f Educat10n, 5c1ence a n d Cu1ture f0r the 6rant-1n-A1d f0r Enc0ura9ement 0 f F0re19n Y0un9 5c1ent15t ( # 01795056). 7.K. 15 5upp0rted 1n part 6y the 6rant-1n-A1d f0r C00perat1ve Re5earch ( # 02302020) and the 6rant-1n-A1d f0r 5c1ent1f1c Re5earch ( # 02640225) fr0m the M1n15try 0 f Educat10n, 5c1ence and Cu1ture.
Reference5 [ 1] K.-1.A0k1,M. 8and0, 7. Ku90, M.6. M1tchard and H. Nakatan1, Pr09.7he0r. Phy5. 84 (1990) 683. [2] H. Pa9e15and 5.5t0kar, Phy5. Rev. D 20 (1979) 2947. [3] A. 8arducc1, R. Ca5a16u0n1, 5. De Curt15, D. D0m1n1c1and R. 6att0, Phy5. Rev. D 38 (1988) 238. [4] K. H19a5h1j1ma,Phy5. Rev. D 29 (1984) 1228. [ 5 ] K.-1.A0k1,M. 8and0, 7. Ku90 and M.6. M1tchard, Ky0t0 prepr1nt KUN5 1035 (1990). [6] A.C. Hearn, Reduce 3.3 U5er•5 Manua1 (1987 ). [ 7 ] Part1c1eData 6r0up, 6.P. Y05t et a1., Rev1ew 0f part1c1e pr0pert1e5, Phy5. Lett. 8 204 ( 1988 ) 1. [8] J. 6a55er and H. Leutwy1er,Phy5. Rep. 87 (1982) 77. [9] N. 159ur, C. M0rn1n95tar and C. Reader, Phy5. Rev. 39 (1989) 1357.
472