Z. Phys. C - Particles and Fields 30, 151-156 (1986) ~r Physik C

and �9 Springer-Verlag 1986

Charmed Baryons in Quantum Chromodynamics

V.M. Belyaev and B.Yu. Blok

Institute of Theoretical and Experimental Physics, B. Cheremushkinskaya 25, SU-117259 Moscow, USSR

Received 20 February 1985; in revised form 12 July 1985

Abstract. The masses of charmed baryons with spin 1/2 (A~ +, 2;+ +) and their residues into quark currents are calculated on the basis of QCD sum rules method. The obtained values of masses are in good agreement with experiment. Arguments are given in favour of existence of resonance X~ + § with negative parity and mass close to 2, 6 GeV.

1. Introduction

QCD sum rule method, firstly suggested by Shifman et al. [1] for calculation of mesonic masses and generalised for the case of baryons in [2-4] enables one to determine model-independently, within the framework of Quantum Chromodynamics itself, masses of big number of hadrons. In the present paper we continue the investigation of QCD sum rules for baryonic masses and obtain sum rules for masses of charmed baryons with spin i/2, which have one charmed c-quark.

Until now only the lowest states in antitriplet-A ~+, and in sextet-,S~ + § were observed. A strange baryon A § with quantum numbers of antitriplet has been dis- covered recently. The best established baryon is A ~+ [5, 6]. The existence of 27 + + and A + needs further verification.

Since 1976, when charmed baryons were firstly observed, various experiments have been performed for their search and investigation of their properties. One may hope to observe a big number of charmed baryons in recently built new machines, where the corresponding experiments are held. So, the question of charmed baryons mass spectrum is of big import- ance now.

QCD sum rules were firstly applied to charmed baryons in [7]. However only the limit of infinitely heavy charmed c-quark was discussed, anomalous dimensions of operators and gluon condensate con- tribution was not taken into account.

The purpose of the present work is to obtain the

values of A + and X + + masses. Here we take into account the finiteness of c-quark mass, anomalous dimensions of operators and gluon condensate contribution. As a result, we derive relativistic sum rules for A + and X + +

2. Description of the Method

Following the standard QCD sum rule method procedure, described in [1,2], we calculate the polarisation operator

H~1(q) = i S eiqX ( T {rh(X), t~j(0) } )od4 x (1)

where i, j-spin indexes of the current t/with quantum numbers of the discussed baryons.

The polarisation operator is calculated in the region q2 ~ _ 1 GeV 2, where, on one side, we can use perturba- tion theory, but on other side, there is a big contribu- tion from nonperturbative effects, closely related to non-trivial structure of QCD vacuum. Those effects can be taken into account with the help of operator expansion:

T {q(x), F/(0)} = ~ C.(xZ)O.(O) (2) t l

where O,-operators and C,(x2)--the correspond- ing coefficient functions.

Performing the calculation of the left, "theoretical", part of the sum rules, we take into account operators with dimension d < 6 (Fig. 1 a-h; double lines corres- pond to heavy quark propagator): the unit operator I, quark condensate ~ (uu or aTd), operators G 2, ~G"uv(2~/2)a,r p and products of four quark operators f fF ~b f i f th : the contribution of the latter is determined with the help of factorization gypotesis [1].

The anomalous dimensions of baryonic currents are the same, as for baryonic currents with only light quarks, because we work in the region lower, than the thereshold of c-quark creation:y~, = - 2 / 9 . We also

152 V.M. Belyaev

( x ~ "j (,~j)

Q b c

",,...__.1/ d e f

g h

Fig. la-b. The Feynman graphs for polarization operator. Double lines correspond to c-quark propagators

" .~ I 1 a~_= =.,~ ,x q(x) ~10) qlx) F:l(O)

o b

Fig. 2a and k The graphs for c-quark propagators in the gluon field

assume:

4n as(q2) - 9 In ( - qZ/A 2) (3)

where A = 100 MeV. Operator expansion normalisation point is/~ = 0, 5 GeV.

Following [8], we firstly calculate all diagrams in x-representation. For the propagator of free massive quark we have:

SC(x) = im2 K l ( m ~ ) m22 4n 2 x / ~ 2 4n2x 2 K2(m x ~ 2 ) �9

(4)

It is easy to obtain the following propagators of massive quark in gluon field in Foch-Schwinger gauge Aux u = O:

= G~x ~ m K l ( m x / - x 23 (:~a.~ + apa~) &(x) (

- 2 i m K o ( r n ~ ) a v ~ . ~ (5)

and

Sodx ) - - 32 . 26 -

+ ix2mK2(mx/~-~)

+ 6 i ~ K l ( m ~ x Z ) } (6)

and B.Yu. Blok: Charmed Baryons in Quantum Chromodynamics

where a, b are colour indexes, apx = i/2 (TpVx - V~Tp). It is necessary to remark, that a part of propagator

S~2 (6):

(~s G2 )~ (.)~ -~- ~ ) 32.26 (7)

corresponds to the condensate of heavy c-quarks, whose contribution we take into account during the calculation of gluon condensate contribution. The corresponding term for light quarks is already accumulated in the following expression:

(: u"~(x), ~(0): )o = - ~ 6 ~ (:aU:)o +-.-. (8)

The nontrivial question arises to choose numerical value of c-quark mass. The majority of authors (see, for example, [1,9, 10]) do not use the physical mass of c-quark, which corresponds to the pole of quark propagator, but the mass in the euclidean point of

2 defined as: normalisation: p2 = _ me,

B(-m~) mc - A ( - rn~) (9)

where A(p2)p+B(p 2) is the numerator of the propagator and Landau gauge is used. It can be shown that in this care mc ~ 1, 26 GeV, and the magnitude of ~,-corrections for charmonium sum rules is minimal. But in [11] the arguments were given, that such defini- tion of mass is no thing more, than a convenient method for taking into account ~,-corrections in charmonium sum rules: firstly, the calculation of spiral amplitudes in charmonium decay ~b--*37 gives mc=1.35_0.05 GeV, secondly, the value of mass obtained in [1,9,10] is gauge dependent. The remarkable compensation of a~-corrections to heavy quark loop occurs only in Landau gauge, where they almost completely go into the renormalisation of mass. At last, a big value of a~-corrections to the mesonic loop is connected with coulomb effects, which change the wave function of c-quark, but have no attitude to mass renormalisation.

So, we conclude, that physical mass of c-quark is known up to accuracy of 10%, and the mass in the euclidean point of normalisation, has uncertainty, because ~-corrections are different for different sum rules. So, because we don't take into account ~- corrections, we must vary c-quark mass near the physical point of normalisation mc = 1.35 GeV, looking for the best agreement of various sum rules. This value of mass can slightly (~ 10%) differ from the real physical mass, and from the euclidean mass, corresponding to charmonium sum rules. Thus the c-quark mass is now a new parameter, which lets to improve the agreement of the sum rules. We also take into account the anomalous dimension of those mass: Ym = - 4/9.

Let us consider the right hand side, or, phenomeno-

V.M.: Belyaev and B.Yu. Blok: Charmed Baryons in Quantum Chromodynamics

logical part of the sum rules. Two different structure arise: the first has an odd number of 7-matrices-t), the second has no y-matrices and corresponds to the unit matrix I. We will exactly take into account in these structures the contribution of the lowest baryonic state (in the approximation of infinitely narrow resonances). The contribution of the rest heavier states will be approximated by the continuum, which begins from some threshold so and is determined by quark loop contribution (Fig. 1). For a function at each structure in the polarisation operator we have the following dispersion relation:

1 ~ Imf ( s )ds . Q2 : _ q2 s@)=;! 7z ' (lO)

or, after Borel transformation [1]:

(Q2),+1 / d "~" f n ( M 2 ) = B f ( Q 2 ) = l i m ~ { - - / f(QZ) (11)

Q2/n = M 2

we get:

fB(M2) = l~z i e-~/~t~Imf(s)ds (12)

where Imf(s)--the spectral density. Further, the continuum contribution will be always transfered to the left part of sum rule. Because of the presence of integral exponents in f/~(M2), it will be useful to use the following equation:

m 2 oo ( c r 9(s) e-S/M~ ds (13)

where

g(s) - (mZ~)k- ~ O(S -- m~) i (s ' - z)"-'dz (t4) ( n - 1)! ~ z~

0-function shows that the threshold in (13) is equal to c-quark mass. The consideration of the continuum contribution means the change of the upper integration limit in (13).

Baryonic masses calculation is based on the following demands to the sum rules:

1. The continuum contribution is less than 30%. 2. The contribution of higher (d > 6) power correc-

tions is less than 30%. 3. Good approximation of functions In I~ and In 12

by straight lines (with minimal dispersion), and sufficiently wide region, where this approximation is implemented, e.g. where sum rules work: A M 2 > 400 MeV 2. Here 11, /z-functions in structures with odd and even number of ?-matrices respectively.

4. In the region, where both sum rules are fulfilled simultaneously they must agree:

11/12 = M R = const (15)

and the slopes of corresponding straight lines

153

must be equal. The continuum threshold and c-quark mass are two variables, and their best choice corres- ponds to the best agreement of the sum rules. The detailes of the determination of So and baryon mass see in [3].

The error in the calculation of baryonic masses is due, mainly, to the following facts:

1. Dispersion in the slopes of straight lines. 2. Uncertainty in the continuum threshold. 3. Uncertainty in condensate values. The absolute

error in the mass value A MR is the same, as in the nucleon case [2, 3], but the relative error is twice less (nearly 10%).

3. Choice of the Currents

The most general form of quark current with quantum numbers of A~ + and without covariant derivatives is:

T = eabc { a( ua Cdb)c c + b/u" C 7 5 db)? s c C

q- c(uaTu?scdb)?u?sCC} = ath + bl12 + c1~3 (16)

where C is the charge conjugation matrice. The direct calculation of the diagram of Fig. la

shows, that it's contribution to the structure with even number of ?-matrices for correlators (01T{ql, 01} 10) and (0[T{t/s,O3}10) has negative sign, opposite to the sign of the contribution to the correlator <0[ T{//2 ,tl2 }10). Resonances with different parties give the contributions with different signs to this structure, hence, demanding the dominance of the state JP = 1/2 + in the sum rules, we see, that in our calcula- tions for antitriplet we must use the current

rlT= ~abc(uaC? sdb)? 5cC. (17)

Namely, the bare loop contribution is propor- tional to b 2 - a 2 - c 2, therefore the contribution of resonancer with negative parity is minimal when a = c = 0 .

The same current was used in I-7] where nonrelati- vistic sum rule for A ~+ was used.

Notice, that operators ff~ and gsffG"~a~(2"/2)~b do not contribute to the sum rules for current (17).

The general current without derivatives with quantum numbers of X + + has the following form:

rls = eabc { a(u~Ccb)u ~ + b(uaC? scb)? suC}, (18)

The some arguments as in the antitriplet case, show that the state de= 1/2 + dominates in sum rules, if a = - b = 1. So we use the following current

tls = I?'abc { (blaCcb)uC--(blaC ? sCb)7 5uC }. (19)

For this current the contribution of operators q~ff and gs~Pau~G~(2/2)"~, in to the sum rules also vanishes.

The choice of currents in QCD sum rules was discuss in detail in [12].

The variation of a, b, c only increases the contribution of negative parity resonances. So the

154

accuracy of results does not increase when we use currents different from (17) and (19).

4. Polarisation Operators

The direct calculation of diagrams of Fig. 1 leads to the following expressions for Borel transformed functions at different structures in the polarisation operator:

The sextet case: Structure 4, x = rn~/M2:

(2=) 4 e -*/M= Im H~ (s)ds

/c o

= 3M6(Es(x)- 2 E4(x) + E3(x) )

+ a2/6e -x + b/48MZ(EE(X) - E3(x)) (20)

structure 1

( 2 2 ) 4 ~- ~ M~ j e- / Im H~

TC 0

= 3rncm6(E4(x)- 2E 3 (x) + E2 (x)) M 4

+ aZ/3mc e-~ + b/48--(3Ez(x)-4E3(x)) mc

+ b/32mcMZ(2E~(x) - 3Ez(x)). (21)

The antitriplet case: structure

(2n)4 S e -~/~t~ Im H~ (s)ds 7~ 0

= 3 M 6 ( E 5 ( x ) - 2 E 4 ( x ) + E3(x))

a 2 bM 2 + ~-e -~ + ~ - ( E 3 ( x ) + 2Ez(x)) (22)

structure 1

(2n) 4 e-~/M~ Im 1I~ (s)ds

7~ 0

= 3mcm6(E4(x) - 2E 3 (x) + E2 (x))

me a2 x b M 4 + - - ~ - e - + ~ -~ -c (3E2(x) - 4Ea(x))

b~C M2 E2(x) (23) 32

Here the following values and designations for conden- sates are used:

(/Vu )o = ( aYd)o ~ - (0, 24) 3 GeV 3

a = - (2n)2 (tVU)o =0 , 55 GeV 3

0, OeV, \ n / 0

b=(2n)2( ~G2)o" (24,

The formulae used in the calculations are given in the Appendix.

V.M. Belyaev and B.Yu. Blok: Charmed Baryons in Quantum Chromodynamics

5. Sum Rules for Antitriplet

Taking into account the continuum contribution and anomalous dimensions of the operators, we can obtain from (22), (23) the following sum rules for A +"

Structure 4"

M 6 ~ L-4/9 { (EI (x) - EI (y) )x2 ( 6 + 4x + X---~ )

a2L4/9 -x bM 2L_4/9~ -x e + e (5 x)

6- 192

= fiE e M~~ (25) Ae "

Structure 1:

mcM6L-4/9 {e-X(1 + ~x +X---~)-e-'(l + y+-3x

~t

+ �89 + 9xy ) - 9xZ)+ 3 x(x + y + 1)ln(x/y)t

-- me M 6 L- 4-/9 (E 1 ( X ) - - E 1 (Y))" x'(3 + 3 x + x2/2)

a2m L4"/9 e-X 4-L -'*/9bM4 le-X(1 +2x) + 6 c - 48mc

_ e-Y(1 +x+y+3xln(x /y ) ) - (El (x) -El (y) )

�9 x(2x + 3 ) - ~ M Z L - 4 / 9 ( e - x - e -y

__~2 A/f "9-M2 /M2 (26) - - x ( E l ( x ) - E l ( y ) ) ) _ p A c , , . A c ~ ~ ,

Here mc = mco(L')-4/9, where mco is the mass of charmed quark in the normalisation point p2 = _ me o2

L - - ~(/~2) " L '-~(m2~ (27) e~(m2) ' e~(m z)

y=so/M 2, Mao--mass of A +, and fl~c=(Zn)4fl~o, where flZ--a square of residue of A + into the current t / r :

(0 lq r [A + ) = fly(p); (~ - mc)V(p) = 0 (28)

v(p)--the Dirac spinor. The analysis of the sum rules (25), (26) shows

(Fig. 3), that the sum rule with the even number of v-matrices works well when 1, 4 < M 2 < 1, 8 GeV 2. (because when M z < 1, 4 GeV z higher power correc- tions are too large, and when M 2 > 1, 8 GeV 2 continuum contribution is too large). The sum rule for the structure with the odd number of v-matrices works well, when 1, 8 < M 2 < 2, 2 GeV 2.

The continuum threshold is found to be about s o ~ 10 GeV 2. The change of c-quark mass gives us relatively small variation of the baryon mass, determined from each of the sum rules (25), (26)

V.M.: Belyaev and B.Yu. Blok: Charmed Baryons in Quantum Chromodynamics

I2(OeV 7 } I.dGeV 6)

/, / /

. /

t / / /

0.06 - / / / / 0.1 - //

004-

0.02-

0 i i - / i =- 1 2 1 2

o Ma(GeV 2 ) b M2{ GeV z )

Finally, we have for antitriplet:

MA~ = 2.3 + 0.1 GeV, s o ~ 10 GeV 2

~ 2 = 0.8 -t- 0.1 GeV 6.

Fig.3a and b. The sum rule behaviour for A~. In this and in the next graphs the following notations are used: solid lines correspond to the r.h.s, of the sum rules, dashed l ines-- to the 1.h.s. of the sum rules, graphs a - - t o the sum rules for structure q, graphs h - - fo r structure I. The following values of the mass, residue and continuum threshold are chosen: MA~ : 2.3 GeV, ~ 2 = 0.8 GeV 6, s o = 10 GeV 2

0,3-

02-

0.1

0

11 (GeV 's )

/ ' / " 0.5- 0J.- 0.3- 0.2-

~ y 0.1 -

i - - 0

1 2 M2(GeV2l b

I2(GeV 7)

M 2(GeV z )

Fig. 4a and b. The sum rule behaviour for Z'~. The masses, residues of baryons and the continuum threshold are:

Mr~ = 2.4 GeV, Mx, ~ = 2.6 GeV, j ~ = 3.6 GeV 6, ~ , = I G e V 6, s 0 = 1 0 G e V 2

separately. The best choice of m~o following from the condition of the best fit of straight line In I~ and In I e is found to be m~o = 1.35 + 0.05 GeV. This agrees well with the results of [11]. In this case the mass ofA~ + is equal to 2.3 + 0.1 GeV. When M 2 ~ 1.8 GeV the ratio I2/I~=MR,,~l.8 GeV. This slight contradiction between M R and MA~ is due to the following facts: we do not take into account G-corrections; 2) both sum rules work simultaneously in a very small region; 3) the uncertainty in M R value is more than in MA~ because the slope of In IL2 enables us to determine M 2 , thus the error in MA~ decreases. Notice that independent variation of m~o in both sum rules which imitates the possible contribution, of ~, corrections to those sum rules allows us to come to an excellent agreement of both sum rules: M R ~ 2.2 GeV when M 2

1.8 GeV 2, and MAo = 2.3 __+ 0.1 GeV. Here m~o = 1.3 GeV for the structure I, m~o = 1.4 GeV for the structure p. The proof of this, however, requires the calculation of es corrections.

155

(29)

6. Sum Rules for Sextet

Considering the sum rules for the sextet case we see that if we take into account in the right-hand pheno- menological side of the sum rule only one resonance 22 + +, then the slopes ofln 11 and In I z are nearly equal: Mx~ ,-~ 2.5 GeV, but the ratio 11/12 = M z ,,, 1.4 GeV is unexpectedly small and falls down slowly with M 2 increasing. In order to explain this contradiction one should suggest the existence of resonance 22* which is created by the same current as 22+ § but has negative parity and a mass close to the mass of 22+ +. This resonance gives a negative contribution into the right- hand side of the sum rule for the structure I. As a result, the ratio

12/11 . f12 _ 3 2

but not MR; here 31, ]32 are the residues of these resonances into the current q~, M R is the mass of these resonances.

Taking into account the continuum contribution and anomalous dimensions of the operators we get from (20), (21) the following sum rules for the sextet: The structure 0:

M6 L - 4 / 9 { (E 1 ( x ) - E 1 (y))x2(6 + 4 x + x2/2)

+ e- X ( 1 - 3 x - 7 x2-X~ ) - e - r ( 1 - ( 4 x - y)

- (4 x y - y 2/2) - x4/2y - 6 x 2 In (x/y)) } + aZ/6L4/9e- ~ + b/96M2L4/9(e- x(1 + x)

- - e Y(1 + xZ/y)-- (E 1 (x) - - E 1 (y))x(2 + x)) = ~ e x p ( - z 2 Mso/M )

+ ]~g. exp ( - Mg./M2). (30)

The structure I:

�88 + 52x +X---~)

- e-r(1 + y + ~x + 1(y2 + 9xy - 9x 2)

+ 3x(x + y + 1)In(x/y))- (El(x)

- El(y))x(3 + 3x + x2/2) + ~mcL4/9e -~

bM 4 + ~ L - 4 / 9 ( e - X ( 1 + 2x) - e-r(1 + y + x

q43/T/c

+ 3 x In (x/y)) - (E~ ( x ) - El(y))x

bmc A / I 2 1 _ 4 / 9 ( [ K , -(2x + 3) )_ ~ - . . . . t ,~l (x)

- E1 (y))(3 x + 2) - 3 e - x + e - r(3 + 2 In (x/y))

= ~2oMsoe-M~:M~ -- ]~, Mx, e - ~ m~, (31)

156

Here M : , / 3 : are the mass and residue of this resonance, fl =/3 (2z02. From these sum rules we get (see Fig. 3):

Mr~ = 2.4 + 0.1 GeV, Mz . ~ 2.6 GeV

s o ~ 10 GeV 2. (32)

The residues f12. and f12 are I_+0.7GeV 6 and 3.6 _+ 2.4 GeV 6, respectively.

7. Conclusion

Thus, we have determined the mass spectrum of spin 1/2 charmed baryons basing on relativistic QCD sum rules. We have found that

M A o = 2 . 3 _ 0 . 1 G e V s 0 a ~ 1 0 G e V 2

Mz~ = 2.4 + 0.1 GeV Soz ~ ~ 10GeV 2. (33)

The results obtained are in a good agreement with the experimental data MAo = 2.28 GeV, Mso = 2.42 GeV [51.

Besides, we predict the existence of resonance 27* with negative parity and the same rest quantum numbers, as 22 + +" M : .-~ 2.6 GeV.

Acknowledgement. The authors thank T.M. Alley, A.V. Smilga and M.A. Shifman for useful discussions.

Appendix

We give here the derivation of the formulae useful for the Borel transformation of the functions at the struc- tures in the polarisation operator.

To calculate it we must know the Borel transforms of Fourier transformations of the functions

K~(mx/~). K~(mx/~)~ x (,#:;)"

where K~(z) is the MacDonald Function which arises in the propagator of a massive quark.

Let us proceed from the Minkovsky space to eu- clidean one. We have

ileiqXK,(mx/Tx2)d4x ( jeT:

= Ie- x,K (m' *)d4xl

v 7~2 ~ D n+v-Se-Q2/4p2

1 / /FI - V __ m2 "O ~ , 1 v;~p2)dp

where

l l / n - -V , > ,1_v;5)

V.M. Belyaev and B.Yu. Blok: Charmed Baryons in Quantum Chromodynamics

1 = 2 5 x " - ' - 1 (1 - x2) v- 1 e=2/(4(, - x2lo2 dx

0

and x 1 are euclidean space coordinates. For derivation of the last formula we used an integral representation for the MacDonald function:

K/m ~/7~, t V ~ ' l ' = r ( v + l / 2 ) ~ . . --~x/Tm oJdtc~ + tzy+,/2"

Using the formula

BM2e-QZ/4P 2 = 6 ( 1 / M 2 -- 1 /4p 2)

we concluded that the Borel transforms have the form

--f~xZ]t --f~SX2~n-~ ire2 mn+v_ 2

�9 O ( ~ , 1 - v;m2/M 2 )

(A,I)

if it stands in the structure I,

Kv(m x / ~ ) ~2 (_xf__x2), ' 2 , _ 4 m M "+v-4

"r (T'n--v 1--v;m2/M z)" (A.2)

if it stands in the structure q. Here

o9 1 !e-'Zta-'-~(t - 1)'-~dt, (A.3) e) =

It is easy to see that in the cases we are interested in, (A.3) can be expressed through integral exponents; if 13 __< 2 and ~ is an integer, c~ > 1.

O(0~,/3;Z)= 1 [~) -~-0 ~7~ x k E Ck- l(--1) a - l - k E ~ + l - f l - k ( Z )

~3 E,(z) = ~ t-"e-Z'&.

1

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