INVESTIGACiÓN REVISTA MEXICANA VE FíSICA 46 (2) 120--125 ABRIL 2000

Calculation of the QCD chiral coefficients from VMD models

C.A. RamirczEscuela de Física, Un(versidad 'ndustrial de Salllancler

A. A. 678, 8ucaml111lflga, Colombiae~mail: cramjre@lariflo.flet.co

Recibido el 19 de enero de 1999; aceptado el 27 de octubre de 1999

Thc Chiral conserving E4.Chiral cocfficients LI, L2 and £3 are calculated fm three different modcls of Vector Meson Dominance (VMD).It is ruund thatthe resuIt is the same for all three. in agreement with gcnernl considerations shown ¡he equivalence between different VMDIllodels.

KeYlI'ords: QCD; Chiral Lagrangians; VMD

Se calculan los Coeficientes Quiralcs LI• L2 Y £3 para tres modelos diferentes de Dominancia Vectorial (VMD). Se encuentra el mismoresultado parn los tres. en concordancia con cosidernciones generales que muestran la equivalencia entre diferentes modelos de DominanciaVectorial.

lk~cril'lo1"('S: QCD; lagrangianos Quiralcs; dominancia vectorial

PACS: 12.39.Fe; 12.40.Vv

One of lhe mosl popular VMD version is lhe Hidden.SymmelrY (VMD-HS) nne liJo lis Lagrangian is given hy

2. Vector meson dominance (hidden symmetryversion)

(2)

J2,,+-ITo + JJr¡8

/2K"

F2[= f [Tr (X"_\~) +Tr (\ IU + Ul,)] +L.[TrX"X"]'

+ L,[Tr(X"X"W + L3Tr [p;"X,')2], (1)

[ = :: [Tr ID"UI2 + <lTr 12(r" - iV,,)I']

- -21, Tr [F;",(V)F'W(V)], (3)

.'1

have heen proposed and il has heen shown [DI thal there isa more general description and that the different versions areonly particular cases and sOlllctimes they are equivalent. Thepurpose al' this note is to compute the E4 Chiral Symmetryconscrving coefficients for the three different formulations ofVMD and sho\V thal in spite the very differenl appearance lheresults are lhe same for the thcre models.

The Chiral Lagrangian lor SU(3) (Chiral SymmelrY con-serving lerms wi!hout eXlemal currenlS) is [2,3)

wilh (see Appendix for detlnitions and transformations) X ==2Bmq ~ diag.(m;, m;, 2m7,. - m;). U = exp(i7rjFrr).

F, = 94 MeV, X" = 8"UUt and

1. Intrnduction

Chiral Lagrangians [1-31 are a rigorous way lo ohlain QCDpredicliolls in ¡he low energy limil(E « 1 GeV) limil up un-dc(crminaled constanlS. the Chiral Cocfticients. At lhe 1110~

lIlenl il is unfortunalely nol possible lo ohtain those constanlsdircl'lly from QCD even in Latticc Calculations [4] (as mostof lhe hadronic phcnornenology) so they have lO he delcnni-naled experimen!aliy [2J. Al higher energies (E ~ I GeV)lhere is nol a rigorous way to ohlain theoretical QCD pre-dictinns and we havc lo use phenomenological rnodels IikeVeelnr Meson Dominanee (VMD) [6-9], Cons!i!Uenl QuarkModels (QM) [IOJ, EXlended Namhu Jona-Lasinio Models(ENJL) [11J, Non-local Lagtangians (NLL) [12J, Nonrela-liviSlie Quark Models [13J, Sum Rules [14J e!e. NaluraliyIhese Illodcls can he extrapolale lO lhe low energy Iimit topredicI values fOl !he Chital CoefficienlS [15J so Iheir pre-dictions can he compared with lhe experimental values. 11 isalso possihle lo eXlrapolale Chiral Lagrangians predictions.\Virh a given unitarizalion scheme lo (he high energy regionlo oh!ain Ihe resonanees [p(iiO), 10(980) and so onJ [16J.Thus lhe Chiral CoefficienlS and lhe resonance pararnelcrsare decply related: thcy are two differenl ways lo descrihe(he sal1lc low energy physics. 01' particular interest are VMDmodels, or more generally resonance models hecause Iheypcrmit lo parametrize all the experimental results as long as\Ve lake e~lOugh paramelers. It is found. 100 that the Chi~ral eocflicients are saturaled by few resonances, being themos! importanl Ihe Vector [p(iiO)), Sealar [lo(980)J andTensor [h(12iO)J ones 115J (see Tahle 1 for a lis! of Iheknown SU(3) multiplets: Pseudoscalar octel and resonanceolles). The eontrihutions of the Jasl two tend to cancel hc-Iwcen thcl11selves (see Table 11)so lhe Vector Mesan Reso~nances are lhe l11os1important. Several VMD [7,8,2] ll10dels

CALCULATION OF THE QCD CHIRAL COEFFICIENTS FRO\1 VMD MODELS 121

TABLE 1. Known Resonance Octets and Lhe Psudoscalar one (Chiral Symmclry GoldslOne Bosons). Thcre are several resonances withouta clcar SU(3) partners: 10(400-1200). 10(980). 10(13iO). 10(1500). 10(2020). 10(2060). 10(2200) 1m lhc scalar sectm; and h(12iO).f,(1430). 1;(1525). f,(1565). 12(1640). h(li1O). 12(1810). 12(1950). 12(2010).12(2150). IJ(2020). 12(2300) and 12(2340) lor lhelcnsors.

Octct [= 1 / = 1/2 /=0 NRQM qua. numh.

Scalar 00(980) /,'ó(1430) 10(980)./o(1300)? 13 Po

Sen lar 00(1450) l\ó(1950) lo( 1500)'1. 10(2020)? 23 Po

Pseudoscalar rr 1\ 11. ,]' 11 So

Pscudoscalar rr(1300) /{(1460) ,/(1295). ,¡(li60) 23 So

Pscudoscnlar •• rr(liiO) [,'(1830) ,/(1400 - 14iO). ,/(2225) 33 So

vector p(iiO) [," (892) "'.<1> l' S,

vector . p(1450) K' (1410) ",(1420). ,,(1680) 2351

axial vector ('1(1260) 1\, A (I{, (12iO), 1\, (140Q)) f¡(1285). f¡(1510)? 13p¡

axial vector . '? K,(1650) ? ? 23P.

Tensor 0,(1320) 1\;(1430) f,(12iO).lí(1430)? 13P2

Tensor ? K,(1980) f,(1525). 23 P2

whcrc \~: = gp;¡ and 1n~ = ag2 F;. \Vith a = 2 and!J = !JI' (as iL will he shown lo he the case) wc have theKSRF relation ",2 = 2g~F; [i, 1n An additional lerm(F"Tr [F(V),,,,¡£"] ) can be added bulllp lo irrelcvant no-mixing tcrms can he ahsorvcd in Eq. (3). using the cquation01' Illolion for \~l" 11can he shown that the integral over theveclor t¡ellls can he done at three level hy shifting the ficldsas: \';, --+ \~j - ir!! + &~tL¡tE, + &E,R¡jE,t, so

e" '" ~; [TI' ID"UI2+ oTr 1-2i~;l]1 ? 6

- :j2Tr [F"v( -if"W + O(E ) - terms (4)-.rJp

Using Ihe relation 2r¡w= 2 (D¡lrv - Dvr¡j) = -[AJil Avl-[EJI' El'] wc can show that

F"v( -if,,) = -i (a"f v - avf,,) - i[f,,, f,,]

where ]J = & J11I~- 4m; is Ihe pinn momenta. From theexperimental values [ISJ 111" = (;6S.5:lo 0.6) MeV andr(fI --> 2rr) = (150.;:lo 1.1) MeV we can obtain tbe Cbi-ral Coefficicnls. However these values have to be correctedby a faclor 01' (5i /69)' '" O,S beeausc the YP in Eq. (6) haslo he measurcd in the soft pions region and this is not thecase for the p --+ :lIT decay \vhcre lhe pions are not soft [151.Other resonancc contrihutions and the results of other modelsare SIHl\'o.lnfor comparison in Tahle 11.

3. Gasser-Leutwyler method

This Illclhod of deal with Vector Mesons f1elds was proposedby Gasser and Lcutwyler in Re!'. 2. The idea is lo use the an-tisyrnmctric tensor field F(F)¡tl' == )..i F¡~vas a fundamentalquantity instead ol" lhe vector lIcld \~I.A 'covariant deriva-livc' for lhe F(\I')JlII-tensor lield CilO be constructed as

where AJI is dcf1ned in the Appcndix. Now we can writc aChiral invariant Lagrangian

wilh

=~F,,,jD-¡ pi"'+F' [''"+O(E6) (9),1 oa,/IV /11" 1 '

("-1. = -~Tr [(D"F",,) (D.,Fv,,)] + ~1Il~Tr(F,,"F''")

+ ;g2"F;Tr [F,,,,e (.\"'.\,\)~]Jn,)

(6)

(5)

( - ¡g"T' ( [ a" J)p-+1.1f -4 1 PJI Jr, Jr

¡ t [ t t]= 4~ (D"U)U, (DvU)U ~

and the linal reslIlt is 115]

~~[ ](," = ~ (Tr.\'".\"')' +2 (Tr.\'"X" )' - 6Tr (.\'".\,,)2 •81Hp

'Fi1 91, 7l"

L¡ = 3" " = -S 1 ,L, =2L¡.L3 = -6Lt•-rr 111'"

To ohtain 9"

(g = !Jp) we have to compute the P --+ 2Jr decaywíd[h. From the Lagrangian wc ohtain ror it

q1. /3f( _. 2) .,,)p---.--,. Jr =--,

GJrl/1.;,(7)

[i ;gpF;1) ["el" " e]• JIV = --- '/\, 4"'14'\1', ,

21/1 f'

( 10)

R,l'. Mex. Fú, ~6 (2) (2lXXl) 12(l-125

122 CARAMIREZ

TABLE 11. Rcsonanccs conlribulion to E4-coefficients. Scalar and Tensor [15] contrihutions are givcn 100. Observe that they are smallerand tcnd lO cancel Oelween Ihemselves. Experimental values and prediclions of olhcr models are shown for comparison. For the No-LocalLagrangian f\1odel (NLL) we take the case of A :;; 1. No!ice (ha! alJ the models are consisten( wilh (he experimental values (unfortunatelyIhe experimental crrors are large).

SOllrce L, L, L3Vcc!orOct. (1.1"'1: ,,(770),clc.) 068 1.36 -4.08Veclor. Ocl. (:~3""1; (1(1450), ctc.) 0.01 (1.02 -0.06Scalar OCL (13])0: (10(980), ctc.) -018 O 0.53Seal,r Sin. (31)0: /0(980)", /0(1370)'!) 0.18 OTensor Del. (13p2: 02(1320). ele) < 0.43 < 0.4, < -1.63Tensor Sin. 03p,: ¡,(1270)'1. /2(1430)'1) <-0.11 < 0.32Tensor Oc!." (231)2: [\"2(1980)?, eIC.) < 0.4 < 0.43 < -1.53Total Resonancc con[rihution 1 x 0.4 2,60100.7 -6.8 x 2.3QM. [101 O.K 1.6 -3ENJL [11 ) 0.K5 1.7 -'¡.2NLL. [121 1.29 2.58 -6.15Exp. [2J 0.9 x 0.3 1.7:f: {l.7 -4.4 x 2,5Exp. (5] [J.4x[J.3 1.35 x [J.3 -3.5:t 1.1

as its gaugc lields. Thc covariant dcrivative of U can he con-structcd as D/IU = D/IU - i[jL¡,U +igU R¡, and we can writelhe Lagrangian fOl Ihis model (S):

where 1('t¡3,/lV = (rJo¡JYtJIJ - g('t,,[j{3J') /2 , and Ll/lV,n{1 =!ln/IUpU" - govD¡JD11 + g¡JvDoD¡J - Y{3I,DnDw Using thc factlhal ¡o.¡J,/lVI1'V,I") = goPY¡J~ \\!c expand lhe propagalor inpowers 01"l/m1, lo have

1 1Dfl,1,/,,~= -.) lnd.¡w - -'1 D.OJ~.IW +.. (11)

mp mI'

. - F;T'(D U)( U)t 1T. ( L' F,R ')L - 4"" 1 /1 DI¡ - ¡ 1 F¡w + ¡IV

" -M-T,( 2 2) fl .{ " t}+ 2" 1 L,. + 11" + 2'Tt L ,UR"U , ( 13)

The Palh integral on F'¡IV is donc in lhe usual way to 9h-lain

[('fr = _Ji DniJ,¡1V Ji + 0(E6)o{3 • ¡'V ,

= __ 1_ Ji I''"'o~ Ji + 0(E6)2'/1/~ o{3 ,lnp

= e;" + 0(E6), ( 12)

Thc conslants Fw, JI, Y and ñ are thc unrenonnalizcd con-stants and their relalions with lhe physical ones will be oh-taincd later. In this model lhe Weinherg rclation m~ ::::2m~is nol neccssarily valid and in lhis way it is possible lo solveconsislency problellls [191 1'01' Ihe ease 01' iJ i O, The La-grangian can he rcwrittcn as

• - r li L" ,iR" 1T, (F(L)2 F(R)')L - LO + . /1 i + '/1 .¡ - 4" 1 ¡IV + ' IJV.,

,n- (., ") ( 1)+ -Tr L- + 11- + flTr L UR"U2 1I JI 1I ,This is Ihe same expression 01' Ihe VMD-HS case 01' Eq. (6).howcvcr wc havc to show that the [J" is the same in hOlh ver-sions. The p -)- 27T dccay Lagrangian is given hy lhe lastterm 01' Eq. (9). The decay width can be obtained (Ihe calcu-lation is ditTcrcnt, hecause lhe fundamental veclor ficld is (hetensor) in agrcement with Eq. (7). Thereforc lhe result for lheChiral Cocfficienls is lhe same.

- .)r _ F; T. (,. ,-,>1)1..-0 - 4 1 ."JI"\. ,

- .),i .gF;T('i}')11I = -1 2 r '" 1I ,

Ji - .gF; Tt, (Ai \" )l' - 1 2 .• J' ,

.) ? qJ.t'l.m- =: Al- + '--" ,

2

wherc mp ami HIn are Ihe physicalmasscs. Thc Effeclivc La-grangian can hc ohtaincd hy doing the integral on the Gauge

4. VMD in Ihe Yang-MiIIs version

Finally \Ve want 10 consider lhe Yang-Mills or gaugedmodcl (S). In Ihis version 01' VMD Ihe Chiral Symmetry ispromotcd lo hc a local une, wilh the resonance fields [p(770)ami ([1(1270). or Ljl = t(p¡, + (lj,), RI, = t(PJI - (//

1)1

m~ = 11~2- B (14)

Re" Mex, Fís. 46 (2) (2(XXJ)t 20-125

CALCULATlON OFTHE QCD CHIRAL COEFFICIENTS FROM VMD MODELS 123

Thc E1-Lagrangian is

wherc the physical pion-dccay constant is

F2 = t2 (1 _ y2 t2/m2)1f Jr 1f e¡.

(24)

(23)

(22)

(21 )

Y (Jt2 ( y' fr2)--.!'... - _._'- 1 - --'m2 - m2p2 21n'}.'

(J 'e¡ 1f e¡

L ( J) Y t; ( y2 t;))[FJlv --2 '}.= -t---:!2 1 - -2 2 ~ ~v,in 1He¡ 111e¡

FI/( 1 J) yt;( y2t;)y¡IV -7jDn R = 1'2 '}. 1 - -2' 2 IIV.

•.• lHe¡ me¡

"-1 .'-.)2-,ff y- F. ( [1'F; ) (V2)

[,4 = -8 1. 1 - -2 '}. Tr .•'l. IJV •nle¡ 1l1e¡

The eoemeient is re1ated to the (1 --+ 2rr deeay eonstant YP byIhe cquation, as will be show

And finally we obtain C4lT

It can be shown

- -¡ 1I 2 2L = LO+ J"Li +m L" + /,(L,,, R,,),

2i' r hi R" + m n.R2LO = LO + ji i ""2 ¡tI

- 1 - 2 ( 1- )£~fT = Lo - -4 ') (J;J + /4 - -2 2 J/I, RJ¡ •m. rl1

= L~ +J~'Ri,,+RDR1R+/,( - 2,:,2 J", R,,), (16)

whcre in arder lo do the remaining R¡I integral wc wrote C~lf

in the givcn form. with

I 1 2LO = [o - 4~JiJI'm

P" = hi" _ ~Tr(J"U>,iUt)R 2m2 '

Ji, = Ji, + B Tr (>,iUR"Ut) ,

/,(L,,, R,,) = -¡n. (F(L);,v + F(R);,v). (t5)

The integral aver LI, can he done al tree level and lo arderfOllr dcrivatives lo obtain

Fields. Let us start with the L" field. We ean write the La-grangian as

In ihis case new El-lerms are ohtaincd and Frr beco mesrcnormalizcd. The E2 total contribution is

Howevcr we slill have lo provc Eg. (24). The Lagrangian forthe P -+ 2'lT decay can be writtcn as

with ,JI' = i[JF-;XII, 11."= _igF-;YIJ. Now we can in-tegratc out the (l JI ficltls. To do that we have lo replace"" --+ - (J" - ",,) /2m~. Thus we get for the tensor fields.kccping only the tcrms that contri hu te to the decay

(25)

1[((1--+ 271") = 2Tr(J"U' + h"R")

- ¡Tr [F(Ln);w + F(Rn);w]

+ /1)2 n. (L' + R2) + BTr (L UR"Ut) (26)2 JI ¡I JI

" ¡!JF;F(L, R);", = :¡:-¡-.,

'-m;:¡

x{p¡w(X1w+}/lW) =F g2~7PIW[XII+Y¡¡,xv+yV]}. (27)4m;;

Thus we obtain again the same rcsult of Eq. (6)

F2L = 2.n. (X" yt) +Cff + 0(E6

){'Ir 4 .• 1I 4 .

(20)I

( 19)

( 18)

(17)

-2Ji" = _iyF• (I+~.)n. (>,iy,'.),R 2 1n2

__ I_J2 = _y2t;Tr(X )[,,j)4m2 '1' 81n2 JI' '

2 - 4 2l,i Di) Ji _ Y F. mp T. (V"X-t)

--, /lIt H' R - -------z-z 1 ~'\. 11'4 /' 8 In Tna

F') ., -') F?C.ff = ---"-(1 _ y.-F;)n. ()[" )[1) = ---"-T (X"XI)l.1 2'" 11 ,f 1 JI

'"i l/I'e¡ 'i

Doing the HJt integral wc obtain

_ r 1 2 1 iJ' ij Ji RC..(f - L.-O - -4 .)JiJ1 - -4JR DR Rm-

(J"1 )+/, -2m2'-2DRJR".

Aftcr sorne computation we can find that

Then

1 ( fr2)'L((I--+ 2rr) = :¡Tr[(I,,(J" + "")] - it ¡m~ Tr((I,w[X" + Y", X" + Y"J)

B ( . F"') . -., ? .,',+ -.- - '2Y ; Tr [(I"ut(X" + Y")U - (I"U(X" + y")ut] = ';J

FF7(1+ -n, - ~y-~; m~)Tr((I,,[D"<p,<pJ),

'i HIe¡ 'i ; m,;; 2 '111(1(28)

Re!'. Mex. Fis. 46 (2) (2000) 120-125

124 C.A. RAMIREZ

TAnLE 111. Chiral Transformations. and definitions of the relevan! Ficlds. Z",., :; D¡,Z" - a" Z¡.&' "fZ¡,

hcm

u,

rO'

A"

Z,.

F,•••(Z)

f~v

SU(3)L X SU(3)R Tranf.

~l"u~k~L\~k{ ~ ~L~~~ = Ev~Ekl/J' -t :EVllJ,:E~

r "r "t +" ° "t/' -t .•..•v ¡""'''V •....•v ,.......v\:1: -t :EVY:i:~~

.\,. --T ELXJ.'Et

}~.-t ERX¡,:Ek

-t EL.\¡,,,El

-t EUXI'IJE~1

~1.[.4,. - (~~O,.~ve]Et=:EvAJ.E~ + aIJEv:E~

~II[B,. + eE~O"Ev(]~k=EvB"Et, + Eval-'E~

~zZI.Ez +i(o¡.&Ez)E1EzF¡.v(Z)EiEv f~"Et,

Dcllnicions

u = ('xp(i1T/F.).1T = 1T',"; T. = >.,/2. Tr (T'Tl) = J;j /2\ ::= 2Bdiag.(mu, I/)d, m~) ::= diag.(m"., m"., 2m/{ - tu".)

u = ((",. = i ((to,.( - O,.a') = ie(O,.U)e

r,. = 4 (~, 0,.(+ o,.(e)H =(',e r(\'~ = r,"x,. = (O,.U)ut = (A,. + B,.)e. X,\ = -X"

l~.= UI(O"U) = UI x"u. l~!= -l~.)\"" == 01'.\" - D"X" = [X¡" .\'""j}~'" == DI' }~" - DI' }~, = _ut X¡,,,UA,. = 0,.(('

~z=I'XJl(iT.Oz). Z=L,R,F, Z,.=Z~T'

F",,(Z) = O,.Z" - O"Z,. + igziZ,,, Z,,]. Z = L, R, l',.4

f~"= eF(L)''''( r (F(R)""(I

wherc we have L1scd lhe cquation 01'mol ion for p. al trce levelD1l (l/u, + 111~f!l' = Oand droppcd lhe terms proportionalto lhepinn mass. As \Ve can sec lhcre are threc contributions lo lhep --+ 2rr dccay. They corrcspond lo lhe threc Feynman dia-grams possihlc contrihutions: fJ ---+ Jr -71", P -+ (al -t 7í) - JT

and p -t (al -t 11') - (al ---+ 71"). Using lhe expression for Bfmm Eq. (14) we get

igF,'2¡n'2 ( 19'2F,2)L((I-+ 21T)= 4F'~ -E,- I-;¡-f Tr((I,.[D"q,,<,&])~ .; m;¡ _ ma

Appcndix ,\: Misccllancous rclations

For any A, BE SU(N) we have Iha!

N1._1

L Tr(YA) . Tr(.\;LJ)i=l

22Tr (..lB) - N Tr(A) . Tr(LJ),

- 19"'r. ( [D''''' "'1)= - 1 Pll 'P. 'P ..¡

To ohlain Eq. (24) as we wanled lo prove.

Ackllowlcd¡:mcnt~

(29)N2_1 ?L Tr(.\; AY LJ)= 2Tr(A) . Tr(LJ) - j~ Tr(AB), (30)i=l

Tr(ALJAB) = -2Tr(A' LJ')

+ ~(TrA')(TrLJ2) + (TrALJ)2,This rescarch was supportcd in part by lhe DIF de Cienciasof lhe Universidad Industrial dc Santander (Colombia).

1. S. Wcinhcrg. I'hysica 1\96 (1979) 327; Phys. Re\'. 166 (l96X)156X; S. Colcmall. 1. Wcss. nnd B. Zumino, Ph)'S. Re\'. 177(1969) 2239; 177 (1969) 2247: 1. Cronin, Phys. R('\~ 161(1967) I-1X3; H. Gcorgi, Weak Illfemctions afl(J Modem Par-ride I'h)'Jics. tBcnjamin. California, USA, 19R5).

whcre lhe laSl one is v"lid only 1m any A. BE SU(3).

2. 1. Gasscr alld 11. Lctltwylcr,Allfl. Ph~.s. (N.Y) lSR (1984) 142;Nucl. Phys. lUSO (1<)gS) 495; J. Gasscr, Nucl. Phys. B279(19X7) 65.

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Re\'. Mi'x. F{<¡.4fi (2) (2000) 120--125

CALCULATlONUFTHE QCll CHIRALCOEFFICIENTS FROMVMll MOllELS 125

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