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Minimal embedding of vector mesons in the low energy effective chiral lagrangian
Physics Letters B 292 (1992) 377-383 North-Holland
PHYSICS LETTERSB
Minimal embedding of vector mesons in the low energy effective chiral lagrangian E.A. K u r a e v a n d Z.K. Silagadze
Institute of Nuclear Physics, 630 090 Novosibirsk, Russian Federation Received 16 July 1991; revised manuscript received 17 July 1992
It is argued that without requiring gauging of any subgroup of the SUL(3) × SUR(3) chiral symmetryother than U ( 1)~M,the vector meson dominance principle alone is sufficient for including vector mesonsin the low energyeffectivelagrangian.
Pseudoscalar mesons play a somewhat special role among hadrons, being would-be Goldstone bosons associated with the dynamical breaking via quark-antiquark condensation of the SUL(3) × SUR (3) chiral symmetry, which is good enough for QCD because u-, d- and s-quark masses are small compared to the strong interaction scale ~ 1 GeV. So one can expect that in the low energy region it is possible to develop the effective theory of pseudoscalars only and the chiral symmetry of the underlying QCD will to a great extent dictate the properties of such a theory [ 1,2]. The low energy effective action for pseudoscalars containing the minimal possible number of derivatives and correctly reproducing current algebra low energy theorems has the following form [ 3-5 ] (the notations of ref. [ 5 ] are used):
F=F~-~ f dxSp[OuUOUU +]
80~r 2 f S p ( o t s ) ,
M 4
(1)
M 5
where U= exp[ ( 2 i / F , ) ~ ] , F. = 13 5 MeV is the pion decay constant, ot = (0 u U ) U - 1 dxU= (d U)U-~ (it is very useful to use the differential form language [ 5,6 ], and the pseudoscalar meson matrix is
• =
- ~°/~/~+
K°
n/~/g
K°
.
(2)
- 2~/w/-6
The necessity of the second term in ( 1 ) is dictated [3 ] by the requirement of the correct description of QCD chiral anomaly [ 7 ] consequences. Besides, the effective lagrangian without this term will possess a greater symmetry than QCD [ 4 ]. It is clear that ( 1 ) cannot describe all low energy meson physics. First of all it is necessary to include the electromagnetic interactions and this is usually done by replacing ordinary derivatives by covariant ones,
dU--.DU=dU-ieA[Q, U],
( 0o)
(3)
where A =A u dx u is an electromagnetic one-form and
Q=
-I o
•
1
ElsevierSciencePublishers B.V.
3 77
Volume 292, number 3,4
PHYSICS LETTERSB
15 October 1992
Doing so the Wess-Zumino-Witten term ~ fM5 Sp ( a 5) needs some care because we have an integral over a five-dimensional manifold M 5 whose boundary is the Minkowski space-time M 4. As a rule this difficulty is overcome by the use of noetherian trial and error methods for gauging [4,5 ] or by developing some global integration techniques for chiral anomaly [ 8 ]. But the usual method of introducing covariant derivatives can also be applied [ 9 ]. Let us briefly demonstrate this. Under, interchange ( 3 ) differential forms a = (d U) U - ~and fl= U - ~d U= U - ~a U undergo the transformation
ot--.ot-ieAT,
fl~ fl+ ieAS ,
(4)
where T= Q - UQU - ~and S= Q - U- ~QU= - U- ~TU. Taking into account that A 2= 0 and Aa = - aA we get for the Wess-Zumino-Witten term Fwzw=C ~ Sp(a-ieAT)5=C M 5
~ Sp(ots)-5ieC f Sp(a4AT), M 5
(5)
M 5
where C = - i / 8 0 n 2. As to the second term in ( 5 ), it can be rewritten in such a way, that
Sp(a4AT) =ASp [Q(o/4- fl 4) ] = a Sp [ Q d ( a 3 + / ~ 3) ] = - d { A S p [ Q ( a 3 +/~ 3) ]}+ (dA) Sp [Q(o~3q- p 3) ] , where we have used the fact [ 5 ] that even powers of ot and ]~are exact forms: ot an= da2n- t, p2, = _ dp2,- 1. By the use of Stokes' theorem fMS d o = fM4 09, we get the following expression: Fwzw=C ~Sp(aS)+5ieC M 5
f ASp[Q(a3+fl3)]-5ieC f ( d a ) Sp[Q(ot3"kfl3)] , M 4
M 5
Let us further transform the last term Sp [ Q ( a 3 + / / 3 ) ] = S p [ ( Q + UQU -~ ) a 3] = S p [ ( Q + UQU -t ) ( a - i e A T ) 3 ] + ieA Sp[ ( Q + UQU -~ ) ( a Z T - o t T a + TO~ 2 ) ] , hut due to the anticommutativity property of one-forms, Sp ( Q a Q a ) = Sp (flQflQ) = O, and therefore Sp[ ( Q + UQU -1 ) ( ot2T-otTot + Ta 2 ) ] =Sp[ 2Q2( ot2- fl 2) -QaQot + 2QaUQU-lot_ flQflQ ] = 2 d{Sp [Q2(ot +fl) ]} - 2 Sp[Q(dU)Q(dU -1 ) ], the second term is an exact form too, because
Sp[Q(dU)Q(dU-t)] = - d {Sp[a Q U - ~ Q d U - b Q ( d U - ' )QU]}
i f a + b = 1.
Thus we have f ( d A ) S p [ Q ( o t 3 + f l 3 ) ] = f ( d A ) S p [ ( Q + U Q U - I ) ( o t - i e A T ) 3] M 5
M S
+ 2 i e f A(dA) d { S p [ a Z ( a + j O ) + a a u - ~ a d U - b a ( d U - ~ ) Q U ] } M5
and, sinceA(dA) (dX) = - d [ A (dA)X] + (dA) (dA)X, we get for (5)
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PHYSICSLETTERSB
15 October i 992
F w z w = C j" Sp(aS)+5ieC ~ A SptQ(ot3+fl3)l M5
M4
-10e2C f A(dA)Sp[QE(ot+fl)+aQU-~QdU-bQ(dU-1)QU] M4
-5ieC f (dA) Sp[ (Q+ UQU-~)(a-ieAT) 3] M 5
+ 10eEC f (dA)(dA) Sp[QE(c~+fl)+aQU-~QdU-bQ(dU-I)QU].
(7)
M5
The last two terms are gauge invariant. For the first one this is obvious because it is expressed through covariant derivatives only, as (4) shows (note that dA = ½Fu~dXUA dx ~ is gauge invariant). As to the last term, its invariance becomes evident if one rewrites it as follows ( a + b= 1 ):
Sp{Q2( ot+ fl) +aQU-~QdU-bQ( dU-~ )QU} =Sp{QE( ot-ieAT+ fl+ ieAS) +aQU-IQ( dU-ieA[ Q,U] ) + bQU-I ( dU-ieA [ Q,U] ) U-~QU} + ieA Sp[ Q2( T - S ) + QU-~Q2U-Q2U-' QU] . In the first term all derivatives have been replaced by covariant ones and the second term equals zero, as is easily checked by substituting T and S. We are interested in a minimal gauge invariant extension of ( 1 ), so one can drop gauge invariant terms in (7). Note also that the remaining part of (7) does not depend on the arbitrary constants a and b, such that a + b = 1, except for the explicitly gauge invariant piece of it. Indeed [ 5 ] Sp [aQU-~Q d U - ba( dU-1)au] = ½Sp [QU-~Q d U - a u a ( dU -~ ) ] + ½( a - b) d [ S p ( Q U - I Q U ) ] and the last term gives a gauge invariant (and parity violating) contribution after integration over M 4, so we drop it. Thus the photon and pseudoscalar meson world at low energies can be described by the effective action
FEM( U,A ) =FN( U,A ) + FWMZW( U,A ) ,
(8)
where
FN(U,A)=~F 2 ~ dxSp[(DuU)(DuU) +1
(9)
M4
and -- -wzw_
EM
~
Sp(ot M5
ie 2
+ ~
5 ) -4-
~
ef A Sp[Q(ot3+fl3) ] M4
~ A(dA) Sp[Q2(ct+fl)+½ QU-'QdU-½ QUQ(dU-')].
(lO)
M 4
It was shown [ 10 ] that (8) correctly reproduces all current algebra low energy theorems [ 11 ]. But even (8) cannot be a complete story for low energy meson physics because pseudoscalar mesons, except pions, are not considerably lighter than other hadrons, so at least vector mesons must be included in a more realistic effective lagrangian [ 12 ]. To include vector (and in general axial-vector) mesons into the game, usually they gauge more widely than the U ( 1 ) subgroup of the SUL (3) × SUR (3) global symmetry group, consider the corresponding gauge bosons 379
Volume 292, number 3,4
PHYSICS LETTERS B
15 October 1992
as vector mesons. However, this procedure can hardly be considered as satisfactory because QCD does not possess a so large local symmetry and it is very unexpected for this symmetry to appear in the low energy limit of QCD. The main idea of this work is that for including vector mesons into the low energy effective lagrangian it is quite sufficient to gauge a U( 1) subgroup (to turn on electromagnetism) and to use a phenomenologically well established principle of vector meson dominance [ 13 1. For the effective action I&( U, A, V) to reproduce the low energy theorems as well as r,, (U, A), let us demand [ 14,151
~DV,exp[ir,,(U,A, VI
exp[irEM(U,A)l=
(11)
(we omit normalization factors in functional integrals). The vector meson dominance suggests the following form for r,,( U, A, V) [ 141:
(12) defined from the p-meson width whereg=g,,,l,h forgp,,
As to the vector meson matrix V, we assume ideal o-o mixing:
PI&- wlfi v=
PK*-
P+ - p@-- w/fi
K*+ K*’
K*O
(13)
-$
Note that ( 11) and ( 12 ) have an infinite number of solutions, exp[ir(U,
V)]= ~DZPexp[i~o(U,Z)]-i.fm$
j dXsp[(2,-vJ(z~-v~)].
(14)
M4 The only restriction on r. ( U,Z) is I’,
(15) Clearly, one gets a minimal embedding of the vector mesons if
rO(
u,z) =rEM
(u,fQ-'Z),
that is
where
dxsp{(a,u-ig[z,,ui)ca~u+-ig(z',u+i)) M4
T~(u,z)= p; j
and (we have symmetrized the Z dZ term) 380
(16)
Volume292, number3,4 i FWZW(U,Z)=-8~7t
PHYSICSLETTERSB
15 October 1992 ig 2
2 f Sp(o/5)+ ] - - ~ 2 f Sp[Z(o~3-l-fl3)]-I- 1 - - ~2 f S p { [ Z d Z - k ( d Z ) Z ] ( o i - k f l ) M 5
M 4
+½ Z U - ~ ( d Z ) ( d U ) + ½ ( d Z ) U - t Z ( d U )
M 4
- ½Z U ( d Z ) ( d U - t )
- ½(dZ)UZ(dU-')}.
Then we have a gaussian integral in (14), but it is calculated approximately [ 15 ] by considering the WessZumino-Witten term as perturbation. F ( U , V ) ~ F W Z W ( U , Z ( ~ P ) ) + F N ( U , Z ( ~ P ) ) - ~rnv l 2
~
d x S p [ ( Z ~ sp) -- Vu)(Z(SP)u-VU) l
,
(17)
M 4
where the Z(usp) saddle point is defined from the normal part. We also expand U relative to q~ and omit vertices which contain more than five particles. Then it is sufficient to know a saddle point with accuracy of terms of order q~z. Expanding Z• -- Z u 2~ ~/x/~, 2 ~ being Gell-Mann matrices, we get
8(rU(u, zu ) - ~, m v2f
a 8Z u
dxSp[(Zu-Vu)(Zu-VU)]
M4
)
--Aab•b(sp)q-Baumo, - - - --u
(18)
z u = z~SP )
where, with the needed accuracy, A~b=--m~,O"b--½g2Sp([2a, q)][2 b, q)]),
Therefore we get for the saddle point g2 7(~,) Vu+ mE ( V u ~ 2 + ~ 2 V u ) _ --u = --
ig
B~u=rnZvV~u - 7 ~ Sp([2~, q)]0uq)) •
292
ig
- ~ v c ~ V u ~ - --mE [q)0u~--(0uq))q)] "
This expression can be rewritten more compactly by the introduction of a covariant derivative Duq~= 0uq)- ig [ Vu, q)], Z(flp) = V u + mi___g_g 2 [Duq), q)]
(19)
Substituting this into (17), we get the effective lagrangian in the above described approximation: (20)
Lerf = L N + L WZW+ L vDM ,
where
LVDM='~"½m2Sp[(Vl.t--g/~laQ)(Vkt-- gA'UQ)], 1
LN= ½Sp[ (Du~) (DU~) ] + -~- (1 - otK)Sp [ q)(Duq)) q)(DU~ ) - q~2(Duq)) (DU~) ] f~
and
381
Volume 292, number 3,4
PHYSICSLETTERSB K-
15 October 1992
~-I-
4-
AA1 :
n-
--
Z5
1 (
5
32
-%÷
c~2 • )
.q+q-/~ AA3 ~- -- 94
A4 : - --,~--- a t
~ ~ 0
~1- o
n'-
~
AA5 : - T
.~.o
_
n ~"
3
(
1. -
3 %
¢x2
)
% (z- 3% )
_
n -°
AA 7
--
- - " ~ ' -
Fig. 1.
t w z w = E . " ~ ( 2 ( 1 - a z + ~a~<)Sp[ ~ ( 0 z ~ ) (0.4)) (0~4)) (0~4)) ] ,t 2 3g 2 ig 4F= Sp[ (0zV.)(0a V . ) 4 ) ] - ~ ( 1 - 3 aK) Sp[ V~(0.4)) (0a4)) (0.4)) ] g2 + ~
Sp{ V.( 0. Va) [4) z (0~4)) - 4)(0.4)) 4)+ ( I - 3 aZ) (O~ 4)) 4) 2 ]
+ (0. V.)V~[ (I - 3 aK) 4) 2(0.4)) --4)(0.4))4)+ (0.4))4) 2 ]
- ( 1 - 6 aK) [ V~4)(O. V~) (0.4))4)+ (0. V.)4)V~4)(O,.4)) ] + ( I - 3 aK) [ V. 4) 2( 0. V~) (0.4)) + (0~,11.)4) 2V~(0.4)) ]
- [v. 4)(0. v~) 4)(0.4)) + (% v.) 4)v~(0.4)) 4)]
}).
In these formulas o~K =gEF3/rn 2. Notc that a K = ½ corresponds to the well-known K S R F relation [ 16 ]. W c have checked by explicit calculation that the vector rncsons in (20) do not alter the successful predictions of ( I0 ) in the low cncrgy limit for the n o ~ 2 7, 7 ~ 3 n, 7 ~ 4 ~, 2 7 ~ 3 n, ~ n ~ n n and K K ~ 3 ~ amplitudes. Hcrc wc briefly c o m m e n t on the last case, as the most impressive. The amplitude has the form
382
Volume 292, number 3,4
PHYSICS LETTERS B
15 October 1992
A /t P O" T M(K+(k+)+K-(k-)~n+(q+ )+n-(q-)+n°(qo) )=in-~u,,,,~q+q_qok_ ,
w h e r e f~ =F,#x/~ a n d t h e c o n t r i b u t i o n s to A f r o m d i f f e r e n t F e y n m a n d i a g r a m s in t h e low energy l i m i t are listed in fig. 1. So otK d i s a p p e a r s in the sum. We are grateful to S.I. E i d e l m a n for r e a d i n g t h e m a n u s c r i p t .
References ] 1 ] S. Gasiorowicz and D.A. Geffen, Rev. Mod. Phys. 41 ( 1969 ) 531. [2] S. Weinberg, Physica A 96 (1979) 327; J. Gasser and H. Leutwyler, Ann. Phys. (NY) 158 (1984) 142; Nucl. Phys. B 250 (1985) 465, 517. [ 3 ] J. Wess and B. Zumino, Phys. Lett. B 37 ( 1971 ) 95. [4] E. Witten, Nucl. Phys. B 223 (1983) 422. [5 ]/3. Kaymakcalan, S. Rajeev and J. Schechter, Phys. Rev. D 30 (1984) 594. [6] B. Zumino, Wu Yong-Shi and A. Zee, Nucl. Phys. B 239 (1984) 477. [7] S.L. Adler, Phys. Rev. 177 (1969) 2426; J.S. Bell and R. Jackiw, Nuovo Cimento 60 (1969) 147; W.A. Bardeen, Phys. Rev. 184 (1969) 1848. [8] N.K. Pak and P. Rossi, Nucl. Phys. B 250 (1985) 279. [ 9 ] K.C. Chou, H.Y. Guo, K. Wu and X.C. Song, Phys. Lett. B 134 (1984) 67. [ 10] N.K. Pak and P. Rossi, Phys. Lett. B 148 (1984) 343. [ 11 ] R. Aviv and A. Zee, Phys. Rev. D 5 (1972) 2372. M.V. Terent'ev, Phys. Lett. B 38 (1972) 419. S. Adler, B.W. Lee, S. Treiman and A. Zee, Phys. Rev. D 4 ( 1971 ) 3497. M.V. Terent'ev, Soy. Phys. - Usp. 112 (1974) 37 [in Russian]. [ 12] I. Zahed and G.E. Brown, Phys. Rep. 142 (1987) I; U.-G. Meissner and I. Zahed, Phys. Rev. Left. 56 (1986) 1035; M. Bando, T. Kugo and K. Yamawaki, Prog. Theor. Phys. 73 ( 1985 ) 1541; Phys. Rep. 164 ( 1988 ) 217; G. Ecker et al., Nucl. Phys. B 321 ( 1989 ) 311; Phys. Lett. B 223 ( 1989 ) 425; V. Bernard, N. Kaiser and U.-G. Meissner, Nucl. Phys. B 364 ( 1991 ) 283; J.F. Donoghue, C. Ramirez and G. Valencia, Phys. Rev. D 39 (1989) 1949. [ 13 ] J. Sakurai, Currents and mesons (University of Chicago Press, Chicago, 1969 ). N.M. Kroll, T.D. Lee and B. Zumino, Phys. Rev. 157 (1967) 1376. P.J. O'Donnell, Rev. Mod. Phys. 53 (1981) 673. [ 14 ] Y. Brihaye, N.K. Pak and P. Rossi, Nucl. Phys. B 254 ( 1985 ) 71. [ 15 ] Y. Brihaye, N.K. Pak and P. Rossi, Phys. Lett. B 164 ( 1985 ) 111. [ 16 ] K. Kawarayabashi and M. Suzuki, Phys. Rev. Lett. 16 (1966) 255. Riyazuddin and Fayyazuddin, Phys. Rev. 147 (1966) 1071.
383
PHYSICS LETTERSB
Minimal embedding of vector mesons in the low energy effective chiral lagrangian E.A. K u r a e v a n d Z.K. Silagadze
Institute of Nuclear Physics, 630 090 Novosibirsk, Russian Federation Received 16 July 1991; revised manuscript received 17 July 1992
It is argued that without requiring gauging of any subgroup of the SUL(3) × SUR(3) chiral symmetryother than U ( 1)~M,the vector meson dominance principle alone is sufficient for including vector mesonsin the low energyeffectivelagrangian.
Pseudoscalar mesons play a somewhat special role among hadrons, being would-be Goldstone bosons associated with the dynamical breaking via quark-antiquark condensation of the SUL(3) × SUR (3) chiral symmetry, which is good enough for QCD because u-, d- and s-quark masses are small compared to the strong interaction scale ~ 1 GeV. So one can expect that in the low energy region it is possible to develop the effective theory of pseudoscalars only and the chiral symmetry of the underlying QCD will to a great extent dictate the properties of such a theory [ 1,2]. The low energy effective action for pseudoscalars containing the minimal possible number of derivatives and correctly reproducing current algebra low energy theorems has the following form [ 3-5 ] (the notations of ref. [ 5 ] are used):
F=F~-~ f dxSp[OuUOUU +]
80~r 2 f S p ( o t s ) ,
M 4
(1)
M 5
where U= exp[ ( 2 i / F , ) ~ ] , F. = 13 5 MeV is the pion decay constant, ot = (0 u U ) U - 1 dxU= (d U)U-~ (it is very useful to use the differential form language [ 5,6 ], and the pseudoscalar meson matrix is
• =
- ~°/~/~+
K°
n/~/g
K°
.
(2)
- 2~/w/-6
The necessity of the second term in ( 1 ) is dictated [3 ] by the requirement of the correct description of QCD chiral anomaly [ 7 ] consequences. Besides, the effective lagrangian without this term will possess a greater symmetry than QCD [ 4 ]. It is clear that ( 1 ) cannot describe all low energy meson physics. First of all it is necessary to include the electromagnetic interactions and this is usually done by replacing ordinary derivatives by covariant ones,
dU--.DU=dU-ieA[Q, U],
( 0o)
(3)
where A =A u dx u is an electromagnetic one-form and
Q=
-I o
•
1
ElsevierSciencePublishers B.V.
3 77
Volume 292, number 3,4
PHYSICS LETTERSB
15 October 1992
Doing so the Wess-Zumino-Witten term ~ fM5 Sp ( a 5) needs some care because we have an integral over a five-dimensional manifold M 5 whose boundary is the Minkowski space-time M 4. As a rule this difficulty is overcome by the use of noetherian trial and error methods for gauging [4,5 ] or by developing some global integration techniques for chiral anomaly [ 8 ]. But the usual method of introducing covariant derivatives can also be applied [ 9 ]. Let us briefly demonstrate this. Under, interchange ( 3 ) differential forms a = (d U) U - ~and fl= U - ~d U= U - ~a U undergo the transformation
ot--.ot-ieAT,
fl~ fl+ ieAS ,
(4)
where T= Q - UQU - ~and S= Q - U- ~QU= - U- ~TU. Taking into account that A 2= 0 and Aa = - aA we get for the Wess-Zumino-Witten term Fwzw=C ~ Sp(a-ieAT)5=C M 5
~ Sp(ots)-5ieC f Sp(a4AT), M 5
(5)
M 5
where C = - i / 8 0 n 2. As to the second term in ( 5 ), it can be rewritten in such a way, that
Sp(a4AT) =ASp [Q(o/4- fl 4) ] = a Sp [ Q d ( a 3 + / ~ 3) ] = - d { A S p [ Q ( a 3 +/~ 3) ]}+ (dA) Sp [Q(o~3q- p 3) ] , where we have used the fact [ 5 ] that even powers of ot and ]~are exact forms: ot an= da2n- t, p2, = _ dp2,- 1. By the use of Stokes' theorem fMS d o = fM4 09, we get the following expression: Fwzw=C ~Sp(aS)+5ieC M 5
f ASp[Q(a3+fl3)]-5ieC f ( d a ) Sp[Q(ot3"kfl3)] , M 4
M 5
Let us further transform the last term Sp [ Q ( a 3 + / / 3 ) ] = S p [ ( Q + UQU -~ ) a 3] = S p [ ( Q + UQU -t ) ( a - i e A T ) 3 ] + ieA Sp[ ( Q + UQU -~ ) ( a Z T - o t T a + TO~ 2 ) ] , hut due to the anticommutativity property of one-forms, Sp ( Q a Q a ) = Sp (flQflQ) = O, and therefore Sp[ ( Q + UQU -1 ) ( ot2T-otTot + Ta 2 ) ] =Sp[ 2Q2( ot2- fl 2) -QaQot + 2QaUQU-lot_ flQflQ ] = 2 d{Sp [Q2(ot +fl) ]} - 2 Sp[Q(dU)Q(dU -1 ) ], the second term is an exact form too, because
Sp[Q(dU)Q(dU-t)] = - d {Sp[a Q U - ~ Q d U - b Q ( d U - ' )QU]}
i f a + b = 1.
Thus we have f ( d A ) S p [ Q ( o t 3 + f l 3 ) ] = f ( d A ) S p [ ( Q + U Q U - I ) ( o t - i e A T ) 3] M 5
M S
+ 2 i e f A(dA) d { S p [ a Z ( a + j O ) + a a u - ~ a d U - b a ( d U - ~ ) Q U ] } M5
and, sinceA(dA) (dX) = - d [ A (dA)X] + (dA) (dA)X, we get for (5)
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PHYSICSLETTERSB
15 October i 992
F w z w = C j" Sp(aS)+5ieC ~ A SptQ(ot3+fl3)l M5
M4
-10e2C f A(dA)Sp[QE(ot+fl)+aQU-~QdU-bQ(dU-1)QU] M4
-5ieC f (dA) Sp[ (Q+ UQU-~)(a-ieAT) 3] M 5
+ 10eEC f (dA)(dA) Sp[QE(c~+fl)+aQU-~QdU-bQ(dU-I)QU].
(7)
M5
The last two terms are gauge invariant. For the first one this is obvious because it is expressed through covariant derivatives only, as (4) shows (note that dA = ½Fu~dXUA dx ~ is gauge invariant). As to the last term, its invariance becomes evident if one rewrites it as follows ( a + b= 1 ):
Sp{Q2( ot+ fl) +aQU-~QdU-bQ( dU-~ )QU} =Sp{QE( ot-ieAT+ fl+ ieAS) +aQU-IQ( dU-ieA[ Q,U] ) + bQU-I ( dU-ieA [ Q,U] ) U-~QU} + ieA Sp[ Q2( T - S ) + QU-~Q2U-Q2U-' QU] . In the first term all derivatives have been replaced by covariant ones and the second term equals zero, as is easily checked by substituting T and S. We are interested in a minimal gauge invariant extension of ( 1 ), so one can drop gauge invariant terms in (7). Note also that the remaining part of (7) does not depend on the arbitrary constants a and b, such that a + b = 1, except for the explicitly gauge invariant piece of it. Indeed [ 5 ] Sp [aQU-~Q d U - ba( dU-1)au] = ½Sp [QU-~Q d U - a u a ( dU -~ ) ] + ½( a - b) d [ S p ( Q U - I Q U ) ] and the last term gives a gauge invariant (and parity violating) contribution after integration over M 4, so we drop it. Thus the photon and pseudoscalar meson world at low energies can be described by the effective action
FEM( U,A ) =FN( U,A ) + FWMZW( U,A ) ,
(8)
where
FN(U,A)=~F 2 ~ dxSp[(DuU)(DuU) +1
(9)
M4
and -- -wzw_
EM
~
Sp(ot M5
ie 2
+ ~
5 ) -4-
~
ef A Sp[Q(ot3+fl3) ] M4
~ A(dA) Sp[Q2(ct+fl)+½ QU-'QdU-½ QUQ(dU-')].
(lO)
M 4
It was shown [ 10 ] that (8) correctly reproduces all current algebra low energy theorems [ 11 ]. But even (8) cannot be a complete story for low energy meson physics because pseudoscalar mesons, except pions, are not considerably lighter than other hadrons, so at least vector mesons must be included in a more realistic effective lagrangian [ 12 ]. To include vector (and in general axial-vector) mesons into the game, usually they gauge more widely than the U ( 1 ) subgroup of the SUL (3) × SUR (3) global symmetry group, consider the corresponding gauge bosons 379
Volume 292, number 3,4
PHYSICS LETTERS B
15 October 1992
as vector mesons. However, this procedure can hardly be considered as satisfactory because QCD does not possess a so large local symmetry and it is very unexpected for this symmetry to appear in the low energy limit of QCD. The main idea of this work is that for including vector mesons into the low energy effective lagrangian it is quite sufficient to gauge a U( 1) subgroup (to turn on electromagnetism) and to use a phenomenologically well established principle of vector meson dominance [ 13 1. For the effective action I&( U, A, V) to reproduce the low energy theorems as well as r,, (U, A), let us demand [ 14,151
~DV,exp[ir,,(U,A, VI
exp[irEM(U,A)l=
(11)
(we omit normalization factors in functional integrals). The vector meson dominance suggests the following form for r,,( U, A, V) [ 141:
(12) defined from the p-meson width whereg=g,,,l,h forgp,,
As to the vector meson matrix V, we assume ideal o-o mixing:
PI&- wlfi v=
PK*-
P+ - p@-- w/fi
K*+ K*’
K*O
(13)
-$
Note that ( 11) and ( 12 ) have an infinite number of solutions, exp[ir(U,
V)]= ~DZPexp[i~o(U,Z)]-i.fm$
j dXsp[(2,-vJ(z~-v~)].
(14)
M4 The only restriction on r. ( U,Z) is I’,
(15) Clearly, one gets a minimal embedding of the vector mesons if
rO(
u,z) =rEM
(u,fQ-'Z),
that is
where
dxsp{(a,u-ig[z,,ui)ca~u+-ig(z',u+i)) M4
T~(u,z)= p; j
and (we have symmetrized the Z dZ term) 380
(16)
Volume292, number3,4 i FWZW(U,Z)=-8~7t
PHYSICSLETTERSB
15 October 1992 ig 2
2 f Sp(o/5)+ ] - - ~ 2 f Sp[Z(o~3-l-fl3)]-I- 1 - - ~2 f S p { [ Z d Z - k ( d Z ) Z ] ( o i - k f l ) M 5
M 4
+½ Z U - ~ ( d Z ) ( d U ) + ½ ( d Z ) U - t Z ( d U )
M 4
- ½Z U ( d Z ) ( d U - t )
- ½(dZ)UZ(dU-')}.
Then we have a gaussian integral in (14), but it is calculated approximately [ 15 ] by considering the WessZumino-Witten term as perturbation. F ( U , V ) ~ F W Z W ( U , Z ( ~ P ) ) + F N ( U , Z ( ~ P ) ) - ~rnv l 2
~
d x S p [ ( Z ~ sp) -- Vu)(Z(SP)u-VU) l
,
(17)
M 4
where the Z(usp) saddle point is defined from the normal part. We also expand U relative to q~ and omit vertices which contain more than five particles. Then it is sufficient to know a saddle point with accuracy of terms of order q~z. Expanding Z• -- Z u 2~ ~/x/~, 2 ~ being Gell-Mann matrices, we get
8(rU(u, zu ) - ~, m v2f
a 8Z u
dxSp[(Zu-Vu)(Zu-VU)]
M4
)
--Aab•b(sp)q-Baumo, - - - --u
(18)
z u = z~SP )
where, with the needed accuracy, A~b=--m~,O"b--½g2Sp([2a, q)][2 b, q)]),
Therefore we get for the saddle point g2 7(~,) Vu+ mE ( V u ~ 2 + ~ 2 V u ) _ --u = --
ig
B~u=rnZvV~u - 7 ~ Sp([2~, q)]0uq)) •
292
ig
- ~ v c ~ V u ~ - --mE [q)0u~--(0uq))q)] "
This expression can be rewritten more compactly by the introduction of a covariant derivative Duq~= 0uq)- ig [ Vu, q)], Z(flp) = V u + mi___g_g 2 [Duq), q)]
(19)
Substituting this into (17), we get the effective lagrangian in the above described approximation: (20)
Lerf = L N + L WZW+ L vDM ,
where
LVDM='~"½m2Sp[(Vl.t--g/~laQ)(Vkt-- gA'UQ)], 1
LN= ½Sp[ (Du~) (DU~) ] + -~- (1 - otK)Sp [ q)(Duq)) q)(DU~ ) - q~2(Duq)) (DU~) ] f~
and
381
Volume 292, number 3,4
PHYSICSLETTERSB K-
15 October 1992
~-I-
4-
AA1 :
n-
--
Z5
1 (
5
32
-%÷
c~2 • )
.q+q-/~ AA3 ~- -- 94
A4 : - --,~--- a t
~ ~ 0
~1- o
n'-
~
AA5 : - T
.~.o
_
n ~"
3
(
1. -
3 %
¢x2
)
% (z- 3% )
_
n -°
AA 7
--
- - " ~ ' -
Fig. 1.
t w z w = E . " ~ ( 2 ( 1 - a z + ~a~<)Sp[ ~ ( 0 z ~ ) (0.4)) (0~4)) (0~4)) ] ,t 2 3g 2 ig 4F= Sp[ (0zV.)(0a V . ) 4 ) ] - ~ ( 1 - 3 aK) Sp[ V~(0.4)) (0a4)) (0.4)) ] g2 + ~
Sp{ V.( 0. Va) [4) z (0~4)) - 4)(0.4)) 4)+ ( I - 3 aZ) (O~ 4)) 4) 2 ]
+ (0. V.)V~[ (I - 3 aK) 4) 2(0.4)) --4)(0.4))4)+ (0.4))4) 2 ]
- ( 1 - 6 aK) [ V~4)(O. V~) (0.4))4)+ (0. V.)4)V~4)(O,.4)) ] + ( I - 3 aK) [ V. 4) 2( 0. V~) (0.4)) + (0~,11.)4) 2V~(0.4)) ]
- [v. 4)(0. v~) 4)(0.4)) + (% v.) 4)v~(0.4)) 4)]
}).
In these formulas o~K =gEF3/rn 2. Notc that a K = ½ corresponds to the well-known K S R F relation [ 16 ]. W c have checked by explicit calculation that the vector rncsons in (20) do not alter the successful predictions of ( I0 ) in the low cncrgy limit for the n o ~ 2 7, 7 ~ 3 n, 7 ~ 4 ~, 2 7 ~ 3 n, ~ n ~ n n and K K ~ 3 ~ amplitudes. Hcrc wc briefly c o m m e n t on the last case, as the most impressive. The amplitude has the form
382
Volume 292, number 3,4
PHYSICS LETTERS B
15 October 1992
A /t P O" T M(K+(k+)+K-(k-)~n+(q+ )+n-(q-)+n°(qo) )=in-~u,,,,~q+q_qok_ ,
w h e r e f~ =F,#x/~ a n d t h e c o n t r i b u t i o n s to A f r o m d i f f e r e n t F e y n m a n d i a g r a m s in t h e low energy l i m i t are listed in fig. 1. So otK d i s a p p e a r s in the sum. We are grateful to S.I. E i d e l m a n for r e a d i n g t h e m a n u s c r i p t .
References ] 1 ] S. Gasiorowicz and D.A. Geffen, Rev. Mod. Phys. 41 ( 1969 ) 531. [2] S. Weinberg, Physica A 96 (1979) 327; J. Gasser and H. Leutwyler, Ann. Phys. (NY) 158 (1984) 142; Nucl. Phys. B 250 (1985) 465, 517. [ 3 ] J. Wess and B. Zumino, Phys. Lett. B 37 ( 1971 ) 95. [4] E. Witten, Nucl. Phys. B 223 (1983) 422. [5 ]/3. Kaymakcalan, S. Rajeev and J. Schechter, Phys. Rev. D 30 (1984) 594. [6] B. Zumino, Wu Yong-Shi and A. Zee, Nucl. Phys. B 239 (1984) 477. [7] S.L. Adler, Phys. Rev. 177 (1969) 2426; J.S. Bell and R. Jackiw, Nuovo Cimento 60 (1969) 147; W.A. Bardeen, Phys. Rev. 184 (1969) 1848. [8] N.K. Pak and P. Rossi, Nucl. Phys. B 250 (1985) 279. [ 9 ] K.C. Chou, H.Y. Guo, K. Wu and X.C. Song, Phys. Lett. B 134 (1984) 67. [ 10] N.K. Pak and P. Rossi, Phys. Lett. B 148 (1984) 343. [ 11 ] R. Aviv and A. Zee, Phys. Rev. D 5 (1972) 2372. M.V. Terent'ev, Phys. Lett. B 38 (1972) 419. S. Adler, B.W. Lee, S. Treiman and A. Zee, Phys. Rev. D 4 ( 1971 ) 3497. M.V. Terent'ev, Soy. Phys. - Usp. 112 (1974) 37 [in Russian]. [ 12] I. Zahed and G.E. Brown, Phys. Rep. 142 (1987) I; U.-G. Meissner and I. Zahed, Phys. Rev. Left. 56 (1986) 1035; M. Bando, T. Kugo and K. Yamawaki, Prog. Theor. Phys. 73 ( 1985 ) 1541; Phys. Rep. 164 ( 1988 ) 217; G. Ecker et al., Nucl. Phys. B 321 ( 1989 ) 311; Phys. Lett. B 223 ( 1989 ) 425; V. Bernard, N. Kaiser and U.-G. Meissner, Nucl. Phys. B 364 ( 1991 ) 283; J.F. Donoghue, C. Ramirez and G. Valencia, Phys. Rev. D 39 (1989) 1949. [ 13 ] J. Sakurai, Currents and mesons (University of Chicago Press, Chicago, 1969 ). N.M. Kroll, T.D. Lee and B. Zumino, Phys. Rev. 157 (1967) 1376. P.J. O'Donnell, Rev. Mod. Phys. 53 (1981) 673. [ 14 ] Y. Brihaye, N.K. Pak and P. Rossi, Nucl. Phys. B 254 ( 1985 ) 71. [ 15 ] Y. Brihaye, N.K. Pak and P. Rossi, Phys. Lett. B 164 ( 1985 ) 111. [ 16 ] K. Kawarayabashi and M. Suzuki, Phys. Rev. Lett. 16 (1966) 255. Riyazuddin and Fayyazuddin, Phys. Rev. 147 (1966) 1071.
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