Lee and Weingarten have recently criticized our calculation of quarkonium and
glueball scalars as being "incomplete" and "incorrect". Here we explain the
relation of our calculations to full QCD.Comment: 5 pages,2 epsfigs. Submitted to the Comment section of Phys. Rev. D
  28th April 199

C. Bernard

C. Caso

C.J. Morningstar

E. van Beveren

G.S. Bali

J. Sexton

M. Boglione

M. Golterman

M. Harada

M. R. Pennington

N. Isgur

N.A. Tornqvist

S.R. Sharpe

W. Lee

English

arXiv

Crossref

Comment on “Coupling constant and quark-loop expansion for corrections to the valence approximation”

Boglione, M.

Pennington, M. R.

VU Research Portal

VU Research PortalComment on "Coupling constant and quark-loop expansion for corrections to thevalence approximation"Boglione, M.; Pennington, M. R.published inPhysical Review D2000document versionPublisher's PDF, also known as Version of recordLink to publication in VU Research Portalcitation for published version (APA)Boglione, M., & Pennington, M. R. (2000). Comment on "Coupling constant and quark-loop expansion forcorrections to the valence approximation". Physical Review D, 62(3), 038501.General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal ?Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.E-mail address:vuresearchportal.ub@vu.nlDownload date: 22. May. 2021PHYSICAL REVIEW D, VOLUME 62, 038501Comment on ‘‘Coupling constant and quark-loop expansionfor corrections to the valence approximation’’M. BoglioneVrije Universiteit Amsterdam, De Boelelaan 1081, 1081 HV Amsterdam, The NetherlandsM. R. PenningtonCentre for Particle Theory, Univesity of Durham, Durham DH1 3LE, United Kingdom~Received 28 April 1999; published 6 July 2000!Lee and Weingarten have recently criticized our calculation of quarkonium and glueball scalars as being‘‘incomplete’’ and ‘‘incorrect.’’ Here we explain the relation of our calculations to full QCD.PACS number~s!: 14.40.Cs, 12.39.Mk, 12.40.YxatlfutegethanpDichc-ichfntinathhorasleefaotetiaooleofngd.’’ibu-cor-ndisghtibleamul-sndlorre-ic-mshesecat-tter-sareFortatenlym-Lattice techniques provide an invaluable tool for calculing the properties of hadrons@1#. As a matter of practicanecessity, these calculations involve approximations toQCD. While the spectrum of glueballs has been compuwith increasing precision@2–4#, this is within quenchedQCD. To make contact with experiment requires one tocloser to the full theory by allowing for the creation ofqq̄pairs. Different attempts to do this for light scalars, boquarkonium and glueball, have been made by BoglionePennington~BP! @5# and by Lee and Weingarten~LW! @6#.In a very recent paper, LW have criticized the former attemas beingincompleteand incorrect. We believe their exten-sive discussion is in error in claiming key aspects of QChave been omitted by BP. Let us explain.The BP treatment, like that of Tornqvist@7# and others@8#, is based on a specific approximation to QCD, in whonly hadronic~color singlet! bound states and their interations occur. One begins with the QCD Lagrangian, for whthe only parameters are quark masses and the strength oquark-gluon interaction.LQCD and other scheme-dependeparameters enter on renormalization. One then formallytegrates out the quark and gluon degrees of freedomobtains a Lagrangian involving only hadronic fields witheir interactions, in an infinite variety of ways, all of whicare determined by the parameters of the underlying theWe then focus on the ten lightest scalar states. The bstates are realized by switching off all their interactionConsequently, their propagators are those of bare particthey are stable. To take this limit, each coupling in thefective Lagrangian of hadronic interactions is multiplied byparameterl i and thesel i are taken to zero. This does nnecessarily correspond to a simple limit of QCD. Neverthless, we plausibly assume that the ten lightest non-interacstates, which result in this limit, are the nine members ofideally mixed quarkonium multiplet and an~orthogonal!glueball. Notice that the namesquarkoniumandglueballarejust a convenient way of referring to the quantum numbersthese states. Individual quark and gluon fields play no rHowever, they are, of course, implicit in the formationhadronic bound states.The Tornqvist@7# and BP treatment is then to switch othe ‘‘ dominant’’ interactions of the light scalars by tuninthe appropriate parametersl i from 0→1 for the couplings of0556-2821/2000/62~3!/038501~2!/$15.00 62 0385-lldtdtthe-ndy.re.s:--ngnf.the bound states to two~or more! pseudoscalars.1 It is byturning on the interactions that the bare states are ‘‘dresseFigure 1 represents the Dyson summation of such contrtions to the inverse propagator. This dressing does notrespond to the creation of a singleqq̄ pair. Multiple pairs andall the gluons~Fig. 1! needed to generate color singlets arespect the chiral limit are implicitly included. Indeed, itwell known @9#, that any picture of pions as simpleqq̄ sys-tems loses contact with the Goldstone nature of the lipseudoscalars, so crucial for describing the world accessto experiment. This important~chiral! limit is embodied inour calculation. The resulting hadronic interactions havedramatic effect on the scalar sector. For instance, thea0 andan f 0 emerge at 980 MeV with largeKK̄ components@10#,even though their bare states are members of an idealtiplet 4–500 MeV/c2 heavier. LW criticize these results anot including the specific gluonic counterterm, Fig. 2, anot explaining why.The explanation is clear: our analysis only includes cosinglet states, both internally and externally as unitarityquires. Colored configurations of whatever kind are implitly included and not readily dissected. If such counterterare relevant to the dressing by pseudoscalar~Goldstone!pairs, they have been included.In spirit, our analysis@5,11# is close to that of Refs.@7,8#.Propagators are dressed by hadron clouds, as in Fig. 1. Tdetermine the right hand cut structure of meson-meson stering amplitudes. However, in the work of Ref.@7#, thiss-channel dynamics is assumed to control the whole scaing amplitude, with left hand cut effects~and crossed-channel exchanges! neglected, even though this violatecrossing symmetry@12,11#. In our treatment@5,11#, particu-larly here where we consider mixing, only propagatorscomputed and no further assumptions are needed.Of course, our analysis does have approximations.instance, the scale of hadronic form factors for a gluish sis assumed to be similar to that of well-establishedqq̄ had-rons. This may not be the case. Moreover, our treatment o1For the glueball, the four pion channel may be particularly iportant.©2000 The American Physical Society01-1totenetwifte-wnngallnde-ga-n-onriti-U-o-sunntri-byCOMMENTS PHYSICAL REVIEW D 62 038501incorporates interactions with two pseudoscalars and,lesser extent, with multipion channels. It is these that demine both the sign and magnitude of the mass shifts geated. For the quarkonium states, the dressing by the lightpseudoscalar channels always produces a downward shmass. The size of these shifts of between 100 and 500 M~depending on flavor! is set phenomenologically@10# by theK0* (1430). A much smaller shift of 10–25 MeV for the precursor glueball is set by the strength of the glueball to tpseudoscalar coupling calculated on the lattice by Sextoeal. @13#. The suppression of the couplings of the resulti‘‘dressed’’ hadron to two pseudoscalars happens@11# irre-spective of the exact mass of the bare glueball@2–4#. TheFIG. 1. A pictorial representation of mixing between quarknium and glueball physical states through common meson-mechannels, included in the Dyson summation of the scalar bostate propagator.o03850ar-r-oinVoinclusion of more channels, likerr andK* K *̄ , may well beimportant in dressing this state and may alter the rather smmass shifts we found for that sector in both magnitude asign. Of course, only physically accessible hadronic intermdiate states contribute to the imaginary part of the propator, Fig. 1. Unopen channels contribute only to the real~ordispersive! part and result in renormalizations of the udressed parameters.By including in our calculation key aspects of the hadrworld, in the way described here and in@5#, we believe wemust have approached closer to full QCD—despite the ccism of Lee and Weingarten.The authors ackowledge partial support from the ETMR Program EuroDAFNE, Contract No. CT98-0169.ondFIG. 2. Feynman diagram representing the counterterm cobution to the glueball-quarkonium mixing amplitude as givenLee and Weingarten@6#.n,,@1# M. Teper, ‘‘Glueball masses and other physical propertiesSU~N! gauge theories inD5311: a review of lattice resultsfor theorists,’’ hep-ph/9812187.@2# UKQCD Collaboration, G.S. Baliet al., Phys. Lett. B389, 378~1993!.@3# J. Sextonet al., Phys. Rev. Lett.75, 4563~1995!.@4# C.J. Morningstar and M. Peardon, inLattice 96, Proceedingsof the Workshop, St. Louis, MO, 1996@Nucl. Phys. B~Proc.Suppl.! 53, 917 ~1997!#; Phys. Rev. D56, 4043 ~1997!; 60,034509~1999!.@5# M. Boglione and M.R. Pennington, Phys. Rev. Lett.79, 1998~1997!.@6# W. Lee and D. Weingarten, Phys. Rev. D59, 094508~1999!.@7# N.A. Tornqvist, Acta Phys. Pol. B16, 503 ~1985!; Z. Phys. C68, 647 ~1995!.@8# E. van Beverenet al., Z. Phys. C30, 615 ~1986!.f @9# C. Bernard and M. Golterman, Phys. Rev. D46, 853 ~1992!;S.R. Sharpe, Nucl. Phys. B~Proc. Suppl.! 30, 213 ~1993!; M.Golterman, Acta Phys. Pol. B25, 1731~1994!.@10# Particle Data Group, C. Casoet al., Eur. Phys. J. C3, 1~1998!.@11# M. Boglione, Ph.D. thesis submitted to the University of TuriItaly, 1997; M. Boglione and M.R. Pennington~in prepara-tion!.@12# N. Isgur and J. Speth, Phys. Rev. Lett.77, 2332~1996!; N.A.Tornqvist and M. Roos,ibid. 77, 2333~1996!; M. Harada, F.Sannino, and J. Schechter,ibid. 78, 1603 ~1997!; N.A. Torn-qvist and M. Roos,ibid. 78, 1604~1997!.@13# J. Sextonet al., in Lattice 95, Proceedings of the WorkshopMelbourne, Australia, 1995@Nucl. Phys. B~Proc. Suppl.! 42,128 ~1995!#.1-2

Comment on "Coupling constant and quark-loop expansion for corrections to the valence approximation"

DSpace at VU

PHYSICAL REVIEW D, VOLUME 62, 038501Comment on ‘‘Coupling constant and quark-loop expansion
for corrections to the valence approximation’’
M. Boglione
Vrije Universiteit Amsterdam, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands
M. R. Pennington
Centre for Particle Theory, Univesity of Durham, Durham DH1 3LE, United Kingdom
~Received 28 April 1999; published 6 July 2000!
Lee and Weingarten have recently criticized our calculation of quarkonium and glueball scalars as being
‘‘incomplete’’ and ‘‘incorrect.’’ Here we explain the relation of our calculations to full QCD.
PACS number~s!: 14.40.Cs, 12.39.Mk, 12.40.YxLattice techniques provide an invaluable tool for calculat-
ing the properties of hadrons @1#. As a matter of practical
necessity, these calculations involve approximations to full
QCD. While the spectrum of glueballs has been computed
with increasing precision @2–4#, this is within quenched
QCD. To make contact with experiment requires one to get
closer to the full theory by allowing for the creation of qq¯
pairs. Different attempts to do this for light scalars, both
quarkonium and glueball, have been made by Boglione and
Pennington ~BP! @5# and by Lee and Weingarten ~LW! @6#.
In a very recent paper, LW have criticized the former attempt
as being incomplete and incorrect. We believe their exten-
sive discussion is in error in claiming key aspects of QCD
have been omitted by BP. Let us explain.
The BP treatment, like that of Tornqvist @7# and others
@8#, is based on a specific approximation to QCD, in which
only hadronic ~color singlet! bound states and their interac-
tions occur. One begins with the QCD Lagrangian, for which
the only parameters are quark masses and the strength of the
quark-gluon interaction. LQCD and other scheme-dependent
parameters enter on renormalization. One then formally in-
tegrates out the quark and gluon degrees of freedom and
obtains a Lagrangian involving only hadronic fields with
their interactions, in an infinite variety of ways, all of which
are determined by the parameters of the underlying theory.
We then focus on the ten lightest scalar states. The bare
states are realized by switching off all their interactions.
Consequently, their propagators are those of bare particles:
they are stable. To take this limit, each coupling in the ef-
fective Lagrangian of hadronic interactions is multiplied by a
parameter l i and these l i are taken to zero. This does not
necessarily correspond to a simple limit of QCD. Neverthe-
less, we plausibly assume that the ten lightest non-interacting
states, which result in this limit, are the nine members of an
ideally mixed quarkonium multiplet and an ~orthogonal!
glueball. Notice that the names quarkonium and glueball are
just a convenient way of referring to the quantum numbers of
these states. Individual quark and gluon fields play no role.
However, they are, of course, implicit in the formation of
hadronic bound states.
The Tornqvist @7# and BP treatment is then to switch on
the ‘‘ dominant’’ interactions of the light scalars by tuning
the appropriate parameters l i from 0→1 for the couplings of0556-2821/2000/62~3!/038501~2!/$15.00 62 0385the bound states to two ~or more! pseudoscalars.1 It is by
turning on the interactions that the bare states are ‘‘dressed.’’
Figure 1 represents the Dyson summation of such contribu-
tions to the inverse propagator. This dressing does not cor-
respond to the creation of a single qq¯ pair. Multiple pairs and
all the gluons ~Fig. 1! needed to generate color singlets and
respect the chiral limit are implicitly included. Indeed, it is
well known @9#, that any picture of pions as simple qq¯ sys-
tems loses contact with the Goldstone nature of the light
pseudoscalars, so crucial for describing the world accessible
to experiment. This important ~chiral! limit is embodied in
our calculation. The resulting hadronic interactions have a
dramatic effect on the scalar sector. For instance, the a0 and
an f 0 emerge at 980 MeV with large KK¯ components @10#,
even though their bare states are members of an ideal mul-
tiplet 4–500 MeV/c2 heavier. LW criticize these results as
not including the specific gluonic counterterm, Fig. 2, and
not explaining why.
The explanation is clear: our analysis only includes color
singlet states, both internally and externally as unitarity re-
quires. Colored configurations of whatever kind are implic-
itly included and not readily dissected. If such counterterms
are relevant to the dressing by pseudoscalar ~Goldstone!
pairs, they have been included.
In spirit, our analysis @5,11# is close to that of Refs. @7,8#.
Propagators are dressed by hadron clouds, as in Fig. 1. These
determine the right hand cut structure of meson-meson scat-
tering amplitudes. However, in the work of Ref. @7#, this
s-channel dynamics is assumed to control the whole scatter-
ing amplitude, with left hand cut effects ~and crossed-
channel exchanges! neglected, even though this violates
crossing symmetry @12,11#. In our treatment @5,11#, particu-
larly here where we consider mixing, only propagators are
computed and no further assumptions are needed.
Of course, our analysis does have approximations. For
instance, the scale of hadronic form factors for a gluish state
is assumed to be similar to that of well-established qq¯ had-
rons. This may not be the case. Moreover, our treatment only
1For the glueball, the four pion channel may be particularly im-
portant.©2000 The American Physical Society01-1
COMMENTS PHYSICAL REVIEW D 62 038501incorporates interactions with two pseudoscalars and, to a
lesser extent, with multipion channels. It is these that deter-
mine both the sign and magnitude of the mass shifts gener-
ated. For the quarkonium states, the dressing by the light two
pseudoscalar channels always produces a downward shift in
mass. The size of these shifts of between 100 and 500 MeV
~depending on flavor! is set phenomenologically @10# by the
K0*(1430). A much smaller shift of 10–25 MeV for the pre-
cursor glueball is set by the strength of the glueball to two
pseudoscalar coupling calculated on the lattice by Sexton et
al. @13#. The suppression of the couplings of the resulting
‘‘dressed’’ hadron to two pseudoscalars happens @11# irre-
spective of the exact mass of the bare glueball @2–4#. The
FIG. 1. A pictorial representation of mixing between quarko-
nium and glueball physical states through common meson-meson
channels, included in the Dyson summation of the scalar bound
state propagator.03850inclusion of more channels, like rr and K*K*¯ , may well be
important in dressing this state and may alter the rather small
mass shifts we found for that sector in both magnitude and
sign. Of course, only physically accessible hadronic interme-
diate states contribute to the imaginary part of the propaga-
tor, Fig. 1. Unopen channels contribute only to the real ~or
dispersive! part and result in renormalizations of the un-
dressed parameters.
By including in our calculation key aspects of the hadron
world, in the way described here and in @5#, we believe we
must have approached closer to full QCD—despite the criti-
cism of Lee and Weingarten.
The authors ackowledge partial support from the EU-
TMR Program EuroDAFNE, Contract No. CT98-0169.
FIG. 2. Feynman diagram representing the counterterm contri-
bution to the glueball-quarkonium mixing amplitude as given by
Lee and Weingarten @6#.@1# M. Teper, ‘‘Glueball masses and other physical properties of
SU~N! gauge theories in D5311: a review of lattice results
for theorists,’’ hep-ph/9812187.
@2# UKQCD Collaboration, G.S. Bali et al., Phys. Lett. B 389, 378
~1993!.
@3# J. Sexton et al., Phys. Rev. Lett. 75, 4563 ~1995!.
@4# C.J. Morningstar and M. Peardon, in Lattice 96, Proceedings
of the Workshop, St. Louis, MO, 1996 @Nucl. Phys. B ~Proc.
Suppl.! 53, 917 ~1997!#; Phys. Rev. D 56, 4043 ~1997!; 60,
034509 ~1999!.
@5# M. Boglione and M.R. Pennington, Phys. Rev. Lett. 79, 1998
~1997!.
@6# W. Lee and D. Weingarten, Phys. Rev. D 59, 094508 ~1999!.
@7# N.A. Tornqvist, Acta Phys. Pol. B 16, 503 ~1985!; Z. Phys. C
68, 647 ~1995!.
@8# E. van Beveren et al., Z. Phys. C 30, 615 ~1986!.@9# C. Bernard and M. Golterman, Phys. Rev. D 46, 853 ~1992!;
S.R. Sharpe, Nucl. Phys. B ~Proc. Suppl.! 30, 213 ~1993!; M.
Golterman, Acta Phys. Pol. B 25, 1731 ~1994!.
@10# Particle Data Group, C. Caso et al., Eur. Phys. J. C 3, 1
~1998!.
@11# M. Boglione, Ph.D. thesis submitted to the University of Turin,
Italy, 1997; M. Boglione and M.R. Pennington ~in prepara-
tion!.
@12# N. Isgur and J. Speth, Phys. Rev. Lett. 77, 2332 ~1996!; N.A.
Tornqvist and M. Roos, ibid. 77, 2333 ~1996!; M. Harada, F.
Sannino, and J. Schechter, ibid. 78, 1603 ~1997!; N.A. Torn-
qvist and M. Roos, ibid. 78, 1604 ~1997!.
@13# J. Sexton et al., in Lattice 95, Proceedings of the Workshop,
Melbourne, Australia, 1995 @Nucl. Phys. B ~Proc. Suppl.! 42,
128 ~1995!#.1-2
