Vector symmetry breaking in charmed meson decays

Volume 247, number 2,3 PHYSICS LETTERS B 13 September 1990

Vector symmetry breaking in charmed meson decays

H o w a r d G e o r g i 1,2

Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA

a n d

F u m i y o U c h i y a m a 1,3

Institute of Applied Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan

Received 2 July 1990

We discuss the effects of the breaking of vector symmetry for the two-body hadronic decays of charmed mesons, D + and D °.

1. Introduction

"Vector symmetry" , as discussed in refs. [ 1-3 ], is a realization of the chiral symmetry of QCD in an unfamil iar l imit in which the vector meson masses go to zero along with the Golds tone boson masses. In this l imit , a chiral symmetry is realized in Wigne r - Weyl mode on the part icle states of the theory, with the chiral par tner of the n ident if ied with the longitudinal component of the P. In a previous note [4] , we discussed the appl icat ion of vector symmetry to various heavy meson decays. In this paper, we cal- culate and discuss the leading correct ions to the relations found in ref. [4] (see also ref. [5] ) for the two-body hadronic modes of the charmed mesons. This can be done just as ordinary SU (3) breaking can be incorpora ted into the analysis of part icle decays [6].

We focus on charm decay for several reasons. One is that there is a lot of data [ 7 ]. Many two-body decay modes are seen and measured. Vector symmetry

relates the modes involving light pseudoscalar part i- cles to those involving the light vectors. B meson decay is much less interesting from this point of view, both because the decays into purely light mesons are suppressed by K M angles and because the two-body decays into the pseudoscalar and vector mesons are less impor tan t than mul t ibody final states (or quasi- two-body final states involving heavy resonances) . But the deeper reason is that we believe that we un- ders tand what we are doing in calculating the D decay rates. It seems very plausible that the effective chiral theory descr ipt ion [ 1-3 ] of the interact ions of the light mesons is val id up to the energies involved in D decay, but not much beyond. In principle, al- though we will not discuss it here, we could include all the relevant resonances in this effective theory as mat ter fields, and have a descr ipt ion of the final state interact ions as well.

We do not know how to do this at much higher energies.

Research supported in part by the National Science Founda- tion under Grant #PHY-8714654. 5/90.

2 Research supported in part by the National Science Founda- tion under Grant No. PHY82-17853, supplemented by funds from the National Aeronautics and Space Administration.

3 Supported in part by the Mary lngraham Bunting Institute of Radcliffe College.

2. Notation

We begin by briefly reviewing the notat ion of refs. [ 1 -3 ] . In this paper, we will not discuss the S U ( 3 ) singlet mesons. This simplifies the discussion because we need only discuss the SU (3) symmetr ies of

394 0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)


the theory, and can ignore the U( 1 )'s. We then do not need large-N arguments. We hope to return to these issues in a future publication.

There are two steps in constructing the effective lagrangian.

First, we build the most general lagrangian with a nonlinearly realized

SU(3)L X SU(3)R × SU(3)cL × SU(3)OR

SU(3)L+GL xSU(3)R+GR (1)

in which SU(3)L(R) is the usual chiral SU(3) symmetry of the LH (RH) fermion fields, and SU (3)CL(CR) are gauge symmetries, with gauge fields P L ( R ) - Parity takes L*-~R.

Second, we equate the two gauge fields [ 3 ],

p~=p~ =pU (2)

wherever they appear in the effective lagrangian. The result is an effective lagrangian that describes the pseudoscalar Goldstone bosons, along with massive vector mesons. The identification, (2), breaks the symmetry, ( 1 ), in a well-defined way down to

SU (3)eL × SU (3)o , -~SU (3)c , (3)

SU (3)c is a hidden local symmetry [8] which is the diagonal sum of SU (3) cL and SU (3) cs. In the limit of exact vector symmetry the gauge coupling goes to zero in the vector limit, so that the transverse gauge fields decouple and we are left with independent oc- tets of SU(3)L+GL and SU(3)R+GR Goldstone bosons. The non-zero gauge coupling away from the vector limit produces the nonzero vector boson mass and couplings for the transverse components. In ad- dition to the gauge couplings, this symmetry breaking will generate extra vector symmetry breaking terms in the effective low energy lagrangian. How- ever, these will be suppressed by powers of the vector symmetry breaking parameters, and we need not discuss them here.

We represent the Goldstone bosons in the vector limit by two special unitary matrices,

XL, transforming as a (3, 1, 3, 1 ),

ZL ~LY, L G[ (4)

under the symmetry (where L, R, GL and GR are special unitary matrices representing the symmetry), and

SR, transforming as a ( I, 3, 1, 3 ),

Y,R ~ R~R G~. (5)

Out of these two fields, we can construct the field

U-XLX~ (6)

which transforms as (3, 3, 1, 1 ) and describes the Goldstone bosons of ordinary SU (3) L X SU (3) R/ su(3).

The leading term in the chiral theory that depends on the Goldstone boson fields will involve two derivatives:

-~f 2 [tr (DUSLDuS[) + tr(DU-rRDuS~)], (7)

where, after we equate the gauge the L and R fields as in (2),

DU~V'L(R) = O'U~L(R) - - igZL(R)P~" ( 8 )

In a notation similar to that of refs. [ 1-3], we define

L"=-iX[D"SL, R u_= - i~'~DU~Y'R, (9)

which transform like

L"~GLL"G~, R'--,GRR"G~. (lO)

In unitary gauge, we can write these simply in terms of the p fields and the Goldstone boson fields, ~:

Ta ( 0 ~ a _mppU ) +..., L"= TaL~--~ 7-

To R"= TaR~..~ ~ ( - ~Xa -mpp~) +... , (1 1 )

where Ta for a = 1, ..., 8 are the Gell-Mann matrices.

3. Decay rates

The low energy weak interactions that produce D decays will appear in the effective theory as small terms with specific transformation properties under the symmetry, (1), determined by the transformation properties of the corresponding terms in the low- energy weak hamiltonian. The key fact is that these terms are constructed primarily out of left-handed quarks, which are singlets under SU(3)R. This fact led to the result of ref. [4], that the amplitudes for

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two pseudoscalars, one pseudoscalar and one vector, and two vectors are all equal in the vector limit, in the usual sense of the vector limit that the pseudoscalar is related to the longitudinal component of the vector.

In a theory with vector symmetry [ 1-3 ], the leading term in an effective lagrangian contributing to two-body hadronic D decay has the form

~" 20¢jab ~ j L ~ L b , , ( 12 )

where ~ f o r j = u , d or s, is the D meson field for the state cj. The coefficient, C~j~b= C~jb ~ could involve derivatives acting on the ~ field, but they are irrelevant to the on-shell amplitude, so we treat c~ as a constant. As we showed in ref. [41, this term gives equal contributions to the decays into pseudosealars and the longitudinal components of the corresponding vector mesons.

In this note, we will describe the leading correc- tions to the relations discussed in ref. [ 4] due to the breaking of vector symmetry. These effects will be of two kinds. Firstly, there will be effects of the transverse components of the vector mesons, both in (12), and in higher order terms explicitly involving the transverse vector fields, such as

g f 2~yab ( D. ~j)pS'~L~,~ ( 13 )

and

l g2 f 2Gb~jpU~pb,,~, (14)

where pU~=&p~ &pU_g[pU, p,,] is the p field strength. Secondly, there will be vector symmetry breaking effects proportional to the "quark constitu- ent mass",/z in the notation of refs. [ 1-3 ]. We will show that such terms give rise to contributions to the effective lagrangian o f the form

f 2fljat,~jLUaRbu, ( 15 )

in which the ~Ra, the Goldstone bosons of the SU (3) R appear. Note that unlike a , the fl and 7 coefficients are not symmetric in a ~ b .

Now the amplitudes for the decays look like

( Ttl~Tt2b l H w [Uj)

= -- (Pl P2) (O~jab--flj, b--fljb,~), (16)

(Ttlap2b IHw [Uj)

= -- imo(ple2)(ajab-- f l jab+fl jba--m~yjba) , (17)

(P,,Pzb IHw IDj)

= m~(el E2) (ajab +fljab + fljb, - m27ja;, - mZb;,a)

-m~[ (p, p2) ( ~, e2)- (p, ~) (p~E, ) ]

X (Tjab + Yjba + 6jab). (18)

There are a number of things to notice about these amplitudes. The factors of mp that appear in ( 17 ) and (18) come from the form of ( 11 ), (13) and (14), and we have used the relation m~=gf. There is a po- tential uncertainty here because this relation is wrong by a factor of about x/2, at least if we use tree-level parameters [2,3 ]. However, this uncertainty can be soaked up into the definition of 7. The factors of m 2 and m 2 in ( 17 ) and ( 18 ) come from the kinematics. We have written the amplitude in (18) as a sum of two terms, because the interference between them is small, suppressed by a factor of 2 2 m l m 2 / ( p l p 2 ) 2. To the order we are working (remember that the vector limit is an expansion in powers of the vector meson masses), we can ignore the second term. Only the first term contains the product of a with fl or y, which is the leading correction to the zeroth order results. The fi term, (14), involving two transverse p fields, makes no contribution to this order, and we will not discuss it further.

4. Grouptheory

The weak hamiltonian transforms under the un- broken chiral symmetry (which we will abbreviate just as SU (3)j_ X SU ( 3)6, rather than reproducing all of ( 1 ) ) like a ( 15, 1 ) + (6, 1 ). In a tensor notation, we can represent these two pieces of Hw by tensors with the following structure:

[15]f,,=[15l,kn;, [15]~;=0, [81k'---[61 k;, (19)

where all the indices are SU(3 )e indices. The amplitudes a, fl and 7 must be constructed from the tensors of (19), the invariant tensors of SU(3)LXSU(3)R , and spurions which describe the breaking of vector symmetry. The important spurion is the quark con- stituent mass, It, which transforms as (3, 3) + (3, 3 ), which means that it behaves like a tensor with one upper SU(3)L index and one lower SU(3)R index, or vice versa. The symmetry breaking occurs because /zf is actually set proportional to the identity matrix,

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/.t) =/.za). (20)

The a amplitude survives in the symmetry limit, and so contains no factors oflt. It has the form

k l ajab=AI TajTbm[15]~ +A2{Ta, Tb}~[15]~

+A3ejkm{Ta, Tb}~[6] 'm, (21)

where Aj are unknown reduced matrix elements. The fl amplitude does contain a factor of/~. It has

the form

m T k l n T m k I - ]no. fljab=BiTa,~ b, pl~[15]lm"l-B2~-oml anTb jPk[6 (22)

An interesting feature of (22) is that t he j index must go on the Gell-Mann matrix for the vector meson state b. In particular, if the vector meson Pb, does not contain the same antiquark as the original D state, then fljab vanishes.

The ~ amplitude, like a , contains no factors of~t. It is already suppressed because it involves a transverse p field. This shows up in (17) and (18) as the ex- plicit factors of m 2 or m 2 in front of ~,. Thus addi- tional factors o f p would be even higher order in the symmetry breaking. This term in the effective lagrangian is obtained by building the corresponding term with PL, and then equating gauge fields. Thus the group theoretical structure is similar to that o f the a term, involving only SU (3) L indices. Indeed the only difference between a and y is that there are addi- tional contributions because y is not constrained to be symmetric in a ~ b . Unfortunately, there are several unknowns:

Yjab = Cl k l T a j T b m [ 1 5 ] ~ + C 2 { T a , Tb}/k[ 1 5 ]~

+ C 3 { j k m { T a , T b } k [ 6 ] l m " k C 4 [ T a , Tb lk[ 15]~

+ C s ~ j k m [ T a , T b ] l k [ 6 ] lm

-t- C6~jmn{~krsT~ar T~s + e ( kln) } [ 15 ]'~

-bC7{ejk, T~amTlb n + P ( j m n ) } [~]mn (23)

5. Discussion

At this point, we can simply use the amplitudes, ( 16) - ( 18 ), and the coefficients, (21 ) - (23) , to fit to the observed branching ratios, and other properties of the decays. This is an important task, but it is hard

to avoid getting lost in the multidimensional param- eter space. Instead, we will derive relations among the decays that follow from the symmetry structure, making use of the fact that fl and 7 are supposed to be small compared to a if we are near the vector limit.

To lowest nontrivial order in vector symmetry breaking, the partial decay rates from ( 16 ) - ( 18 ) are proportional to the following:

oc [Pl I (P, P2)ZlOejab--fljab--fljba I 2

[P~ [ (P~p2)2{ lOljab lZ-- 2 Re[ a~ab(fljab + fljba) ] }, (24)

F(Dj--+~laP2b )

oc m ~ Ip, [ (pl E2 ) 2 l Ogjab -- fljab + fljba -- m 2 7jba I z

~m~ IPl I (Pl e2) 2

X{ [Ogjab 12--2 R e [ OLTab( fljab -- fljba"l'- m2~jba) ] }, (25)

F(Dj--+PlaP2b ) ~ m 4 IPl I (El {2) 2

X [Ogjab'~-fljab'~-fljba--m217jab--m~Yjba [2

~ m 4 IPl [((-l(-2)2{[ajab [2

- 2 Re [ a~'~b ( --fljab--fljba "[-m2~jab "~m2yjba)]} . (26)

We expect, from ( 24 ) - ( 26 ),

F(Dj--+~la~2b )

= [ p l [ ( p l p 2 ) 2 l . / # ( D j ~ l a ~ 2 b ) [ 2 , (27)

F(Dj--+l~laP2b )

=m2lp, I(Pl~2)zl+#(Uj-+~,apzb)l 2, (28)

F(Dj--+ pla p2b )

= m 4 IP, I (El ~-2)2],at[(Uj-+PlaP2b)]2, (29)

where

],/l/[(Uj_+~la~2 b ) ]2+ ] j{ (Uj_+PlaP2 b ) [2

= I~¢(Dj--,~,ap2b)12+ I~¢(Dj~l~P2a)12. (30) To compare (30) with data, one should do a com-

plete analysis of the data, using ( 27 ) - ( 29 ). We have not done this. However, below we show some tanta- lyzing results, obtained by combining data from ref. [7 ] with an analysis of resonance structure in four-

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Table 1 Data on D decays. Data on decays with two vector mesons in the final state is from ref. [ 9 ]. The rest of the data is from ref. [ 7 ].

Decay Branching ratio Corrected

D+--,I~*°n+ 1 7 +0.8 7 6 +3.6 • - - 0 . 8 • - - 3 . 6

--* K°P + 6.6 + 1:67 20.6+-~:o 3 --,l~°n + 2 Q+o.4 4 ~+o.6

• ° - - 0 . 4 • J - - 0 . 6

-~ l~.*°p + 4.8 _+ 1.2 + 1.4 29.8 + 7.4 -+ 8.7

D°- ,K*-n + 4 Q+o.7 21 ~+3.1 , o - - 0 . 7 . - / - - 3 . 1

--K p+ 8.2+I: 2 25.6+_33:7 __,K-n+ a 77+0.37 6 n+O.6

J " - - 0 . 3 2 " v - - 0 . 5

--,K*-p + 6.2+2.3+2 38.8_ + 14.4_+ 12.5

--, I~*°n ° 2.1 +o:~ 9.4+~:W - + 0 . 3 1 ~ + 0 . 9 --'K°p ° 0.47_o~1 .-'-0.7

. g o n o 2.8_+~: 3 4.5_+~:~ --,K*°p ° 1.9 _+ 0.3 _+ 0.7 11.9 _+ 1.9 _+ 4.4

some extent , these d i f ferences can be absorbed into a

d i f ferent va lue o f g r i n ( 2 7 ) - (29 ) , but clearly, vec to r

s y m m e t r y is badly broken. This makes our per turba-

t ive t r e a t m e n t o f the cor rec t ions dangerous. N e v e r -

theless, the cor rec t ions seem to p rese rve the relat ion,

( 30 ) , w i th in the errors.

A c k n o w l e d g e m e n t

F.U. wishes to thank to M. Suzuki for discussions.

H .G. wishes to thank D. Hi t l in , R. Mor r i son and M.

Whi there l l for d iscuss ions and D.F. D e J o n g h for sup-

plying a copy o f ref. [ 9 ]. Par t o f this work was done

at the Ins t i tu te for Theore t i ca l Physics in Santa Bar-

bara. H .G. is grateful to the s taf f o f the I T P for its

hospi ta l i ty and help.

body decays f r o m ref. [9] . R a t h e r than c o m p a r i n g

( 2 7 ) - ( 2 9 ) direct ly, we discuss only the b ranch ing

ratios, correct ing the quo ted branching ratios with the

k inemat ic factors impl ic i t in ( 27 ) - ( 29 ), wi th g f = m o.

The results for the three we l l -measured groups o f de-

cays are shown in table 1. Specifically, to ob ta in the

cor rec ted b ranch ing rat ios, we have d i v i d e d by a

phase-space fac tor and by the k inema t i c fac tor f rom

( 2 7 ) - (29 ) (bo th n o r m a l i z e d to 1 for massless final

state par t ic les ) .

Evident ly , the d i f ferences be tween the cor rec ted

b ranch ing ra t ios o f m o d e s that should have equal

b ranch ing rat io in the vec to r l imi t are ra ther large• To

R e f e r e n c e s

[ 1 ] H. Georgi, Phys. Rev. Lett. 63 (1989) 1917; Nucl. Phys. B 331 (1990) 311.

[2] A. Falk and M. Luke, preprint HUTP-89/A057. 13 ] P. Cho, preprint HUTP-90/A016. [4] H. Georgi and F. Uchiyama, Phys. Len. B 238 (1990) 395. [ 5 ] T. Mannel, Phys. Lett. B 244 (1990) 502. [6] M. Muraskin and S.L Glashow, Phys. Rev. 132 (1963) 482. [7] Particle Data Group, G.P. Yost el al., Review of particle

physics, Phys. Lett. B 204 (1988) 1. [8] See M. Bando, T. Kugo and K. Yamawaki, Phys. Rep. 164

(1988) 217-314. [9] D.F. DeJongh, Resonant substructure in K.nnn decays of D

mesons, Cal Tech Thesis (May 1990).

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Vector symmetry breaking in charmed meson decays