hepph

DESY

HUBEP

A diquark mo del for containing one heavy

D Eb ert T Feldmann C Kettner and H Reinhardt

1

Institut f urPhysik HumboldtUniversitatzu Berlin Invalidenstrae D Berlin Germany

2

Institut f urTheoretische Physik UniversitatT ubingenAuf der Morgenstelle D T ubingenGermany

June

low energy prop erties of light quark systems on the other Abstract We present a phenomenological ansatz for

hand are dictated by the approximate chiral symmetry of coupling a heavy quark with two light to form a

QCD and its sp ontaneous breaking explaining the emer heavy The heavy quark is treated in the heavy

gence of very light pseudoscalar as Goldstone mass limit and the light quark dynamics is approxi

in the physical sp ectrum The success of this is mated by propagating scalar and axial vector diquarks

due to the fact that the light quark masses q u d s The resulting eective lagrangian which incorp orates

are small compared to the QCD scale m heavy quark and chiral symmetry describ es interactions

q QC D

of heavy baryons with Goldstone b osons in the low en

The interplay b etween chiral and heavy quark sym

ergy region As an application the IsgurWise form fac

metry is thus studied within eective lagrangians for

tors are estimated

strong and weak interactions of with the various

contributing terms restricted by symmetries

The co ecients in front of these terms are in princi

pal free parameters to b e xed by exp eriment but have

also b een related to microscopic quark mo dels for light

Introduction

mesons as well as for heavy mesons containing one

heavy quark Light baryons have b een re

The physics of particles built up with one heavy quark

cently describ ed within quark mo dels as b ound states of

b ecomes much simpler in the heavy quark limit due to

quarks and diquarks resulting in Faddeev type of equa

the subsequent heavy avor and symmetry see eg

tions This pap er is a rst step to obtain an

and references therein These symmetries are realized

analogous description of heavy baryons containing one

as long as the heavy quark mass m is much larger than

Q

heavy quark in the heavy quark limit by making use of

a typical QCD scale m which is the case

Q QC D

the diquark picture Light diquarks are coupled to the

for Q b c Phenomenologically they show up as an

heavy quark by a simple mo del interaction

approximate degeneracy of heavy hadrons diering only

The eective lagrangian for strong interactions of

in the spin of the heavy quark and as a denite scaling

heavy baryons can then b e obtained directly by func

of observables with the heavy quark masses Since the

tional metho ds The introduction of weak heavy quark

heavy quark in the innite mass limit is to b e viewed as

currents also oers the p ossibility of determining the

a static source of color moving with some xed velocity

IsgurWise functions which are the universal form fac

v it is the remaining light degrees of freedom which

tors for heavy quark transitions inside the heavy baryon

determine the hadronic prop erties In Table we present

for a general discussion see for example Weak form

the heavy baryon mass sp ectrum using some estimates

factors and strong coupling constants can then b e used

as given for example in lab eled by isospin I and

to describ e strong and weak semileptonic decays of heavy

strangeness S

baryons like l l

Q Q b c b c

Although Heavy Quark Eective Theory HQET

l

b c

provides the to ol for calculating systematically the p er

turbative QCD corrections one is left with the non

The organization of this pap er is as follows In the

p erturbative features of QCD that lead to the formation

next section we discuss some general prop erties of heavy

of hadrons containing light quarks at low energies The

baryons related to heavy quark and chiral symmetry In

section we will dene our mo del from which the ef

Supp orted by Deutsche Forschungsgemeinschaft under con

fective heavy baryon lagrangian is derived in section

tract Eb

Relations for heavy baryon masses coupling constants

Email feldmannphaphysikhuberlinde

with light mesons and IsgurWise form factors are dis

Supp orted by COSY under contract

1

cussed in some detail The notation is explained later on in the text

D Eb ert et al

Table Heavy baryons masses MeV in parentheses and avor content The mass dierences b etween the rst two columns measure

the deviation from the spin symmetry limit In the avor symmetry limit the b ottom baryon masses should dier from those of the

charmed ones only by a constant equal to m m

c

b

spin partner b ottom baryon notation ij I S

ij

+

0

trivial T ud

c v

b

0 ij +0

T us ds trivial

v c

b

+0

()

+++0 ij

+++0



S uu fudg dd

c v

c

b

0

0

()

ij

+0

+0

 0

S fsug fdsg

v

c c

b

()

ij

0

 0

S ss

v

c c

b

MeV is the weak decay constant and x Chiral and heavy quark symmetry for heavy

a a

x denotes the light o ctet baryons

a

with a b eing the GellMann matrices

a b ab

In the heavy quark limit it is sucient to deal with the

tr Under global SU SU chi

L R

heavy quark spinors pro jected on the heavy quark veloc

ral transformations these elds transform as

ity v

y y

L U U R

v

im v x

Q

e Qx v Q x

v

which denes the chiral matrix U U L R x as a

nonlinear function of and represents lo cal transforma

where the heavy mass dep endence has b een made ex

tions in SU elds then couple to the vector

V

plicit The heavy quark part of the QCD Lagrangian

and axial vector combinations

then simplies to

i

y y y y

V U V U iU U

L Q iv D G Q O m

HQET v v Q

i

where D G is the covariant derivative with resp ect to

y y y

A U A U

the eld As a consequence physics of strong inter

actions in the heavy quark limit b ecomes indep endent of

The vector eld V transforming as a gauge eld of

the heavy quark spin and avor and conserves the heavy

SU couples via a prop erly dened covariant deri

V

quark velocity Therefore the two light quarks determine

vative The coupling constants to the axial eld A are

the relevant quantum numbers of the heavy baryon The

not restricted by chiral symmetry

coupling of the color avor and spin degrees of freedom

From this it follows that the diquark elds transform

of the two light quarks forming the diquark reads

as

0 0 0 0

ij

C C C C

T j j i j ij ii

UDU U D D U

F F F F

0 0 0 0

ij

T j j i j ij ii

UF U U F F U

and their covariant derivative in SU is dened as

In the following we only consider the color antitriplet

V

part in which together with the color triplet of the

T

D D D iV D iD V

heavy quark forms the physical color singlet baryon

T

state The Pauli principle then requires the diquark elds

D F F iV F iF V

to b e symmetric with resp ect to simultaneous exchange

F D F D F

of spin and avor indices Therefore scalar diquarks

only o ccur in the avor antitriplet and are hence re

Consequently in the heavy mass limit we have two

F

alized by an antisymmetric avor matrix

types of heavy baryons

ud us

D D

i The rst one contains a heavy quark together with the

i

ij

ud ds

A

p

D

D D

two light quarks transforming as the scalar diquark

us ds

ij

D D

D We therefore obtain Dirac spinors for spin

particles

Analogously axialvector diquarks are realized by

the symmetric avor matrix

1

F

Q

Q I

3

2

i

p

ij

A

uu fudg fusg 1

p

T

Q

Q I

v F F F 3

2

p

ij

dd fdsg fudg

1 1

A

Q I Q I p

F

F F F

3 3

2 2

p

ss fusg fdsg

F F F

where v denotes the pro jection on the heavy quark

velocity vT T The two spin orientations of T

To include the interactions with light mesons it is v v v

form the in this case trivial spin symmetry partners

convenient to implement chiral symmetry in the usual

nonlinear realization x exp i xF where F

3

The light o ctet can b e introduced analogously

2

Throughout we consider the avor group SU by the concept of hidden symmetry F

A diquark mo del for baryons containing one heavy quark

ii Coupling the heavy quarks with the light quarks that the pro cess D D violates parity and conse

ij

transforming as the axial vector diquark F one quently the amplitude for T T vanishes in the

v v

ij

gets either spin Dirac elds B or spin heavy quark limit

v

ij

Finally L contains the mo del ansatz for a lo cal cou

RaritaSchwinger elds B These are then sym

v

pling of light diquarks and heavy quarks The transver

metry partners with resp ect to spin rotations of the

sality condition has b een taken into account ex

heavy quark Consequently they can b e combined in

plicitly in the interaction of the heavy quark with the

a multiplet

axial vector diquark by introducing a transversal pro jec

ij

ij ij

tor P g v v The eective couplings G and

p

v B B S

v v v

G have dimension mass G g G g in

ij

tro ducing an inherent cuto scale which we supp ose

B

v

p 0

to b e of the order of GeV Note that b ecause of the

1

Q I Q I

eld transformation this cuto eects the residual

Q I 3 3

3

2

p B 0 C

momentum of the heavy quark k p m v which

B C

Q

p

1

Q I Q I

3 3

Q I A

3

2

is of the order of the light quark momenta

0 p 0

1 1

For an explicitly broken SU m m m

Q

F s u d Q I Q I

3 3

2 2

the parameters in dep end on the various channels

with

characterized by isospin I and strangeness S in Table

The lagrangian can as usual b e rewritten by intro

v S vS S

v v v

x ducing collective baryonic elds T x S

v

v

Heavy avor symmetry further relates the residual mass

y

L tr Q D T hc tr T T

es M M m for dierent heavy avors Q and

v v v v

T T Q

G

similiarly for M M m The transformation of

S S Q

y

the heavy baryon elds and their covariant derivatives

tr S S tr Q F S hc

v v

v v

under chiral symmetry is given by the light quark con

G

tent

where the spin and spin part of the multiplet

T

eld S can b e identied by using the completeness re

T UT U

v v

lation in the heavy mass limit

T

T

U D T U D T T iV T iT V

v v v v v

g v v P P

T

S US U

v v

T

T

vv vv v P P U D S U D S S iV S iS V

v v v v v

v P P

A mo del with heavy quarks and light diquarks

where P pro jects out the transversal spin part

and P the spin part

The mo del we are prop osing treats the two light quarks

After the transformation it is straightforward to

within the heavy baryons as eectively propagating sca

integrate out rst the heavy quark eld and subsequently

lar and axial vector p ointlike diquarks This is sucient

the scalar and axialvector diquarks This leads to

to obtain a dynamics that resp ects chiral and heavy

det M

quark symmetry The mo del lagrangian reads

with

L L L L

T iv T M

v v

L Q iv Q

D

v v

p

y y

tr D D D D M D D

M M M A iv T S

D F v

D

v

p

i h

y

y

M M M A iv S T

D F v

v

tr M F F tr F F

F

S M iv S

p

v v

F

y

L M M tr D A F hc

D F

A D hc

y

where and denote the free propagators of the

D

A F hc tr F

F

scalar and axial vector diquark resp ectively The ab ove

y

determinant gives rise to an eective action log det

L G tr Q D DQ

v v

Tr log It can b e expanded in terms of ordinary

y

g v v F Q Q F G tr

v v

quarkdiquark lo ops resulting in an eective baryon la

grangian This is done in the next section

where we omitted avor and color indices for trans

parency Here L includes the kinetic terms of the heavy

quark and the scalar D and axial vector F diquarks

Lo op expansion and eective lagrangian

with M b eing their eective masses In L their axial

D F

charges enter as eective parameters to describ e the In order to regularize the integrals app earing in the lo op

interaction with soft and mesons Note expansion of we will use the prop ertime metho d

D Eb ert et al

Fig Selfenergy diagram for the baryon eld T Fig Selfenergy diagram for the baryon eld S

v

v

Self energy for S

and rewrite the diquark propagator i DF after a

v

Wickrotation as

Z

The term

2 2

sk M

i

ds e

k M

2

i i Tr S iv S

F

v v

All integrals will then b e given in terms of the incomplete

arising from the lo op expansion of see the Feynman

Gammafunction

diagram in Fig contributes to the self energy part for

Z

the baryon eld S as

v

t

x dt e t

x

S

S S tr g

v v

G

Here the lo op contribution is given by

Self energy for T

v

S

v p

Z

The self energy part for the spin baryon eld T

i

v

d k g

following from is represented by a Feynman diagram

i v k v p i k M

F

in Fig and formally reads

k

T iv T i Tr

v v D

M

F

F F F F

Together with the quadratic term in this contributes

I I I I v p O v p g

to the eective lagrangian

T

tr T T

where

v v

G

i

I M

T

i

where is given in the lowmomentum expansion by

p

T i

v p

M M I

i

i

Z

i

d k

k v k v As b efore the prop er normal and k

i v k v p i k M

p

D

ization of the baryon eld S Z S is p erformed

S

v v

D D

I I v p O v p

F F

by means of Z I I and the residual mass

S

M M m is now connected with the coupling

S S Q

with

G

i

M I

i

F F

I I M Z

S S

p

i

G

M M I

i

i

From this we can read o the renormalization factor

p

D

Strong interactions with light mesons

for the baryon eld T Z T with Z I

v T v T

and the residual mass M M m which is inde

T T Q

p endent of the heavy quark mass and now related to the

The insertions of the axial currents into the diquark

coupling constant G by

propagator generate the related axial couplings of the

heavy baryons to the eld A From the determinant

D

one obtains in the lowmomentum expansion of the

I M Z

T T

G

corresp onding vertex diagrams in Fig

A diquark mo del for baryons containing one heavy quark

Fig Vertex diagram with insertion of the axial current A Fig Vertex diagram with insertion of a weak heavy quark cur

rent

p

M i M

D F

The IsgurWise form factors and comparison with

Tr iv S T A hc

v F D

v

Bjorken sum rule constraints

i

i h

0

Due to the heavy spin symmetry matrix elements of

0

A Tr iv S S

F F

v v

0

in the heavy quark weak heavy quark currents Q Q

v v

limit can b e parametrized by means of a few unique form

This results in a renormalization of the axial coupling

factors IsgurWise functions dep ending on the velocity

constants in the eective heavy baryon lagrangian

transfer v v

T A S tr hc

v

v

0

Av v T T

v v

i tr S A v S

v v

0

v hc Gv v S T

v

v

p

with Z Z Z and Z Z The

S S B v v g C v v v v

0

S T S

v v

vertex renormalization constants are calculated as

For one obtains the matrix element of the avour

Z

p

i

symmetry current and consequently the IsgurWise form

M M Z d k

D F

v k i

factors A and B are normalized at zero recoil

A B The form factor G has to vanish

k

identically since there is no coupling to heavy quarks

k M k M M

F D F

which mixes scalar and axial vector diquarks The other

p

DF DF

M M I I

D F form factors are predicted by simply inserting the weak

Z

current into the heavy quark line and calculating the

i

F

I d k Z

onelo op Feynman diagrams see Fig We obtain

k M

F

Z

i

with

d k Av v Z

T

Z

r

v k i v k i

DF

I dx

xM xM

F D

i k M

D

xM xM

F D

r v v

p

ln p

Z

p

r

p

xM xM

DF

D F

dx I

M

F

and

xM xM

F D

v v v B v v g C v v v

Z

iN

C

d k Z

The coupling of the baryons to the light vector elds

S

v k i v k i

V is given by the covariant derivatives In

k k

other words the vector couplings do not suer renormal

g

i k M M ization due to WardIdentities for the SU symme

V

F

try

r

F

B r Z I

S The ab ove results are summarized in the following

eective baryon lagrangian

r

F

C Z I

S

L tr T iv D M T

ef f v T v

S iv D M S tr

v S

The values at the nonrecoil p oint v v are A

v

F

B and C Z I and the three form

S

tr T A S hc

v

v

factors in our mo del are related via the parameter

i tr S A v S

v v indep endent equation

D Eb ert et al

r

lagrangian of strongly interacting heavy baryons where

C B A

r the values of the residual masses M and the axial

T S

couplings have b een related to the input parameters

Phenomenologically it is imp ortant to know the slop e

of the mo del Furthermore we have calculated the three

of these form factors at zero recoil in order to correctly

IsgurWise functions relevant for heavy baryon transi

extrap olate the exp erimental data and to extract jV j

cb

tions and discussed their slop e at the nonrecoil p oint

one of the Standard Mo del parameters entering the CKM

In particular it turns out that the IsgurWise form fac

matrix We obtain

tors of our mo del satisfy the constraints imp osed by the

A

Bjorken sum rules

F

In conclusion our mo del repro duces the phenomeno

B Z I

S

logical structures reecting heavy quark and chiral sym

f g

metry However it would b e desirable to start already

M

F

with an eective mo del of quark avor dynamics like the

for

NambuJonaLasinio mo del which recently has b een

successfully extended to describ e mesons containing a

C B

heavy quark The eective fourquark coupling of the

Although the parameter range of our mo del cannot

NJL mo del includes an attractive interaction in the di

b e restricted suciently by exp erimental data it is worth

quark channel and the diquarks can b e introduced as

comparing our results with the theoretical constraints

collective elds in close analogy to the meson case de

imp osed by Bjorkens sum rules Following the dis

termining their eective masses and couplings in terms

cussion in ref one obtains

of a few parameters

For light baryons it has b een shown that using the

d A

A

quarkdiquark picture one can reduce the

d

three b o dy problem to an eective two b o dy problem

f

the NJL predictions then repro ducing the mass sp ectrum

of the lowest baryon states quite well Gen

jB j B C

eralizing this approach to heavy quark systems would

lead to an analogous equation of light and heavy quarks

d B

C

interacting with heavy and light diquarks via eective

d

quark exchange while the fundamental diquark prop er

with

ties like the masses and axial couplings will b e

The function A r is obviously compatible

xed by the NJL mo del Although the dynamics is more

with the b ounds in eq Inserting our result for B

involved than in the present simple mo del one exp ects

and C into eq gives

similar symmetry prop erties Investigations along these

lines are in progress

d B

F

Z I

S

d

F

References

C Z I

S

F

M Neub ert Phys Rept

I is p ositive The con which is always true since Z

S

JG KornerM Kramerand D Pirjol ProgPartNuclPhys

straint of eq is also satised as can b e seen in Fig

The result for the IsgurWise form factors A

G Burdman and J F Donoghue Phys Lett B

B and C are plotted in Figs

M B Wise Phys Rev D R

TM Yan HY Cheng CY Cheung GL Lin Y C Lin

Summary and outlo ok

and HL Yu Phys Rev D

P Cho Nucl Phys B ibid B

Focusing on the interplay b etween chiral and heavy

D Eb ert and H Reinhardt Nucl Phys B

quark symmetries in the present pap er we have investi

D Eb ert T Feldmann R Friedrich and H Reinhardt Nucl

gated a phenomenological mo del for the dynamics within

Phys B

a heavy baryon The two light quarks are approximated

W A Bardeen C T Hill Phys Rev D

as eective diquarks and the mo del lagrangian contains

M A Novak M Rho and I Zahed Phys Rev D

their coupling to the heavy quark and to the axial chiral

current as parameters

R T Cahill C D Rob erts and J Prashifka Aust J Phys

Heavy baryon elds have b een introduced by usual

functional integration metho ds such that the diquark

H Reinhardt Phys Lett B

and heavy quark elds could b e integrated out explicitly

D Eb ert and L Kaschluhn Phys Lett B

T Mannel W Rob erts and Z Ryzak Nucl Phys B The lowmomentum expansion then yields an eective

4 (0)

Our notation translates into that of ref via A f

M Bando T Kugo and K Yamawaki Phys Rep

(0) (0)

B g C g

1 2

A diquark mo del for baryons containing one heavy quark

J D Bjorken I Dunietz and J Taron NuclPhys B

A Buck R Alkofer and H Reinhardt Phys Lett B

N Ishii W Bentz and K Yazaki Phys Lett B

ibid B

C Hanhart and S Krewald Phys Lett B

D Eb ert T Feldmann C Kettner H Reinhardt in progress

a

This article was pro cessed by the author using the L T X style le

E

cljour from SpringerVerlag

D Eb ert et al

2 2

Fig The Bjorken b ound f plotted as a function of for three values M

F

Fig The IsgurWise form factor A plotted as a function of

2 2

Fig The IsgurWise form factor B plotted as a function of for three values M F

A diquark mo del for baryons containing one heavy quark

2 2

Fig The IsgurWise form factor C plotted as a function of for three values M F

Figure Captions

Figure Selfenergy diagram for the baryon eld T

v

Figure Selfenergy diagram for the baryon eld S

v

Figure Vertex diagram with insertion of the axial current A

Figure Vertex diagram with insertion of a weak heavy quark current

Figure The Bjorken b ound f plotted as a function of for three

values M

F

Figure The IsgurWise form factor A plotted as a function of

Figure The IsgurWise form factor B plotted as a function of for

three values M

F

Figure The IsgurWise form factor C plotted as a function of for

three values M F

Figures

D

T

T

v

v

Q

v



Figure Selfenergy diagram for the baryon eld T

v

F

S

S

v

v

Q

v



Figure Selfenergy diagram for the baryon eld S v

A

T S T S

v v

v v



Q

v

Figure Vertex diagram with insertion of the axial current A

D F

T S

0

S T

v

0

v

v

v



Figure Vertex diagram with insertion of a weak heavy quark current

f

2

f

2

jM  =13

F

2

f

2

jM  =23

F

2

f

2

jM  =1

F

2 2

Figure The Bjorken b ound f plotted as a function of for three values M

F

A

A

Figure The IsgurWise form factor A plotted as a function of

2

B

2

jM  =13

F

2

B

2

jM  =23

F

2

B

2

jM  =1

F

B

2 2

Figure The IsgurWise form factor B plotted as a function of for three values M

F

C

2

C

2

jM  =13

F

2

C

2

jM  =23

F

2

C

2

jM  =1

F

2 2

Figure The IsgurWise form factor C plotted as a function of for three values M F