Study of double charmonium production around √s = 10.58 GeV with the BaBar detector

Fa oltà diS ienze Matemati he, Fisi hee Naturali

Dottorato di Ri er a in Fisi a

Study of double harmonium

produ tion around

√

s

= 10.58 GeV

with the BaBar dete tor

Dr. Annalisa Ce hi

Advisor:

Prof. Diego Bettoni

Coordinator:

Prof. Filippo Frontera

Referees:

Prof. Antimo Palano

Prof. Dr. Klaus Peters

Introdu tion 5

1 Charmonium physi s 7

1.1 Potential models . . . 7

1.2 Quantum Chromodynami s . . . 12

1.3 Ee tive FieldTheories . . . 14

1.4 Nonrelativisti QCD . . . 16

1.4.1 The NRQCDlagrangian . . . 16

1.5 Experimental studyof harmonium . . . 19

1.5.1 Quarkoniumprodu tion . . . 19

1.5.2 Charmonium spe trum . . . 20

1.5.3 New harmonia . . . 22

2 Double harmonium produ tion 25 2.1 Introdu tion . . . 25

2.2 Cross se tion . . . 26

2.2.1 Color-singletmodel al ulation . . . 27

2.2.2 Cal ulationof the ross se tions . . . 29

2.2.3 Cross se tion for the produ tion of pseudos alar and ve tor doubleheavymesons . . . 32

2.2.4 Light one formalism . . . 33

2.3 Dis repan ybetween theory and experiment . . . 36

2.4 Previous resultson

e

+

_e

−

_{→ J/ψ + X}

. . . 36

2.4.1 X(3940) interpretation . . . 38

2.5 Double

cc

produ tion via

γ

∗

_γ

∗

. . . 38

3 The BABAR experiment 41 3.1 The PEP-II asymmetri ollider . . . 42

3.1.1 Luminosity . . . 44

3.1.2 Ma hine ba kground . . . 45

3.3 Tra kingSystem . . . 48

3.3.1 Sili onVertexDete tor . . . 48

3.3.2 Drift Chamber . . . 48

3.4 Cherenkov dete tor . . . 52

3.5 Ele tromagneti alorimeter . . . 56

3.6 Instrumented uxreturn . . . 59

3.7 Trigger,Data A quisitionand Re onstru tion . . . 65

4 Analysis 67 4.1 Analysis strategy . . . 67

4.2 Data samples and presele tion . . . 67

4.2.1 Datapro essing inBABAR . . . 67

4.2.2 Dataand Monte arlo samples . . . 68

4.2.3 Ba kground . . . 69

4.3 Sele tion strategy . . . 70

4.4 Presele tion . . . 70

4.5 Sele tion uts . . . 72

4.5.1 Numberof hargedtra ks . . . 72

4.5.2 Momentum of the

J/ψ

inthe CM frame . . . 73

4.6 Multivariate Analysis(MVA) . . . 75 4.6.1 Prin ipleof the MVA . . . 75 4.6.2 Appli ation . . . 76 4.6.3 Cal ulationof

N

S

exp

and

N

B

exp

. . . 81 4.6.4 Optimization onsigni an e . . . 84 4.6.5 Summaryonsele tion . . . 85 4.7 Fit to data. . . 87

4.7.1 Fit and toys validation on the signal MC . . . 87

4.7.2 Fit and toys validation on the ba kground . . . 94

4.7.3 Globalt with signal and ba kgroundembedded . . . . 96

4.7.4 Simultaneous t . . . 104

5 Results 109 5.1 Unblind up to 3.8 GeV . . . 109

5.2 Outlook . . . 110

The goal of high energy physi s is to identify the elementary onstituents

of matter and to understand their fundamental intera tions. Over the last

twentyyears,thisendeavorhas beenextraordinarilysu essful. Agauge

the-ory alled Standard Model provides a satisfa tory des ription of the strong,

weak,andele tromagneti intera tionsofalltheknownelementaryparti les.

There are very few dis repan ies between theory and experiment, and most

of them are atthe level of a few standard deviationsor less. However there

are pro esses for whi h experimental results have diered from theoreti al

predi tions by orders of magnitude: soe of these studies are related to the

produ tionof harmonium. This dramati oni t between experimentand

theory presents a unique opportunity to make a signi ant step forward in

our understanding of heavy quarkonium physi s.

Quarkonia play an important role in several high energy experiments.

The diversity, quantity and a ura y of the data still under analysis and

urrently being olle ted inmanyhigh energyexperimentsaroundtheworld

is impressive.

These data ome from experiments of quarkonium formation (BES at

the Beijing Ele tron Positron Collider, E835 at Fermilab, and CLEO atthe

Cornell Ele tronStorage Ring), lean samples of harmonia produ ed in

B-de ays,inphoton-photonfusionandininitialstateradiation,attheB-meson

fa tories (BaBar at PEP-II and Belle at KEKB), in luding the unexpe ted

observation of large amounts of asso iated

(c¯

c)(c¯

c)

produ tion and the

ob-servation of new and possibly exoti s quarkonia states. The CDF and D0

experiments at Fermilab measuring heavy quarkonia prodution from

gluon-gluonfusion in

p¯

p

annihilationsat2TeV; ZEUSandH1,atDESY, studying

harmoniaprodu tioninphoton-gluonfusion;PHENIXandSTAR,atRHIC,

and NA60, at CERN, studying harmonia produ tion, and suppression, in

heavy-ion ollisions [7℄.

This has led tothe dis overy of newstates, new produ tionme hanisms,

newde ays andtransitions, andingeneraltothe olle tionof highstatisti s

expe ted from the BES-III upgraded experiment, while the B fa tories and

the Fermilab Tevatron will ontinue to supply valuable data for few years.

Lateron,newexperimentsatnewfa ilitieswillbe omeoperational(theLHC

experimentsatCERN,PANDAatGSI,hopefullyaSuper-Bfa tory,aLinear

Collider, et .) oering fantasti hallenges and opportunities inthis eld.

Inthisthesistheanalysisondouble harmoniumprodu tionattheenergy

ofthe

Υ

(4S),withtheBABARdataisdo umented. Theaimofthisanalysisis

tounderstandthe me hanismofprodu tionofdouble harmoniumstates fro

e

+

_e

−

annihilation,inparti ularafterthedis repan ieswhi hatthebeginning

of these studiesappeared.

With su essive studies, these dis repan ies have been almost solved.

This analysis was already performed by BABAR ollaboration [28℄, and in

this thesis wewant toupdate that work, with a luminosity early fourtimes

higher (468

f b

−1

. In the mean time, also Belle ollaboration published on

this analysis [16 ℄, obtaining results ompatible with BABAR and ndingout

anew harmonium state, namedX(3940). Weaim here also to onrmthis

state.

This thesis is omposed by ve hapters. Chapter I is an introdu tion

to harmonium spe tros opy, with a des ription of the NRQCD, whi h the

theorethi al framework of this analysi, then the potential models that have

been developed todes ribe the mass spe trum.

The theories related to the double harmonium produ tion me hanisms

are presented inChapter II:in parti ularthe al ulationof the ross se tion

and the di repan ies between theory and experiment.

Then inChapter III the BABAR dete tor isdes ribed.

In Chapter IV there is a des ription on how has been performed the

analysis: theanalysisstrategy,sele tionand utoptimization,andvalidation

of the tare do umented, before the unblind of the interested region.

Charmonium physi s

Until 1974 all the known hadrons were omposed by three quark avours:

theup (

u

),down (

d

)andstrange (

s

). Themassesof thesestateswere rather

small: afew MeV for

u

and

d

, and 100-200 MeV for

s

.

InNovemberof1974,aremarkablymassiveandnarrowresonan e,named

J, was dis overed [1℄ with a mass of 3.1

GeV/c

2

, de aying to

e

+

_e

−

, in the rea tion

p + Be → e

+

_e

−

_{+ X}

. Simultaneously,the resonan ewasdis overed

[2℄ in the dire t hannel

e

+

_e

−

_{→ hadrons}

(also to

e

+

_e

−

_{, µ}

+

_µ

−

), and was

named the 

ψ

. The dual name

J/ψ

has afterwards persisted.

Withthedis overyofthe

J/ψ

,the existen eofanewquarkavour alled

harm (

c

), with a mass of the order of 1 GeV, as well as the existen e of a

family of states alled harmonia wasdemonstrated.

The

J/ψ

is a member of this family, that is omposed by the bound

states of harm quark and antiquark (

cc

). The harmonium is the most

widely studied heavy quarkonium system, and the goal of this hapter is to

givethe theoreti altoolsne essarytofa ethe quarkonium,andinparti ular

harmonium, physi s.

1.1 Potential models

Whentwoparti lesformaboundstate,the attra tivepotential an be

stud-ied measuring the energy spe trum of the system. In atomi physi s, the

binding energy of the ele tron-nu leus system depends on the orbital

angu-lar momentum(

L

),spin(

S

)and totalangularmomentum(

J = L + S

)state

(negle ting the nu leus angularmomentum

I

). To lassify the energy levels

of the system the spe tros opi notation

n

2S+1

_L

J

is used. A similarpattern of energy levels is present in positronium (the

e

+

_e

−

bound state); this has

Thesame on ept anbeappliedalsotothemesons,whi harethe

quark-antiquark (

qq

) bound states. Also in this ase the spe tros opi notation

n

2S+1

_L

J

forthe lassi ationof the mesonsis used.

The intrinsi parity

P

and harge onjugation

C

of a harmonium state

are relatedto the angular momentum by the relations:

P = (−1)

L+1

,

_{C = (−1)}

L+S

.

Also the

J

P C

notation an be used to lassify the

cc

states.

Quantum Chromodynami s (QCD) is the modern theory of the strong

intera tions. The non perturbative features of QCD prevent the possibility

of des ribing it on the basis of the fundamental theory of the intera tion.

Forthis reasonthenaturalapproa hto harmoniumspe tros opyistobuild

an ee tive potential model des ribing the observed mass spe trum. This

approximation allows to integrate out many fundamental ee ts like gluon

emissionorlightquark pairs andto dealwith anee tive potential whi h is

theresult ofthe

qq

dire t intera tionaswellastheenergy ofthe gluoneld.

This potential should nevertheless reprodu e the two main features of the

boundquarkstates inthetwolimitsof smalland largedistan e: asymptoti

freedom and onnement.

The

cc

system an bedes ribedwith a S hroedingerequation:

HΨ(x) = EΨ(x),

(1.1)

where the hamiltonian forthe

cc

system an be writtenas:

H = H

0

+ H

′

.

(1.2)

H

0

an be expressedasa freeparti lehamiltonian plusa non-relativisti potential

V (r)

:

H

0

= 2m

c

+

p

2

m

c

+ V (r),

(1.3)

where

m

c

is the harm quark mass and

p

itsmomentum.

V (r)

an bebuilt takingintoa ountthe properties ofstrongintera tion

in the limit of small and large distan es. At small distan e the potential

between the quarks for a quark-antiquark pair bound in a olor singlet, is

oulomb-like:

V (r) ∼

4

₃

α

s

_r

(r)

,

where

r

is the distan e between the quarks,

α

s

is the strong oupling onstant and the fa tor 4/3 omes from the group theory of SU(3), related

0

0.1

0.2

0.3

1

10

10

2

µ GeV

α

s

(µ

)

Figure 9.2:

Summary of the values of

( ) at the values of

where they are

Figure 1.1: Summary of the values of

α

s

at the values of

µ

where they are measured[5℄.

The value of the running oupling onstant

α

s

depends on the energy s ale of the intera tion in the way shown in Fig. 1.1 , where is lear the

de rease of

α

s

with in reasing

µ

. At the leading order in the inverse power of

ln(µ

2

_/Λ

2

₎

,

α

s

isdes ribedby:

α

s

(µ) =

4π

β

0

ln(µ

2

/Λ

2

)

,

(1.4)

β

0

= 11 −

2

3

n

f

where

Λ ≃ 0.2 GeV

is the non-perturbative s ale of QCD (the energy

where(1.4)diverges)and

n

f

isthenumberofquarkslighter thanthe energy s ale . It is lear from equation 1.4 that, as the energy s ale of a strong

pro ess de reases and be omes loser to

Λ

,

α

s

in reases and the QCD an not be treated asa perturbative theory.

As a result of (1.4) the oupling

α

s

(µ)

varies logarithmi ally with

µ

, so thatatveryshortdistan es,gluonex hangebe omesweaker. This property,

knownasasymptoti freedom,isresponsibleforthequasi-freebehavior

exhib-ited by quarks inhadronsprobedat very short distan es by deeply inelasti

s attering.

At large distan e, that means at momentum s ales smaller than

Λ ≃

200M eV

the onnement termisdominating. It an bewrittenintheform:

V (r) ∼ kr,

"#$%&' ()* !"#"!$%&'()'*+!",')-"../'+"-+&$.

+,'-.&#/%01'&20.&'3'0 &#5'3#61,'7/88/9#6$*

( !"#$%&'"()1,'0:#6;/&5#11'&20:8#101,'01.1'09#1,1,'0.2'/&5#1.8.6$%;

8.&2/2'61%23':'63#6$/61,' !!'<:' 1.1#/6-.8%'=>6'01&% 1%&'?* " ! @= !? ) #" " #$ #" ! #$ ! %=A& $?

9,'&' "! .63 "" .&' 1,' 0 .8.& .63 1,' -' 1/& /2:/6'610 /7 1,' 6/6;

&'8.1#-#01# :/1'61#.8"=$?(

Figure1.2: PlotoftheQCDpotential(1.1),forquark-gluon oupling

α

s

= 0.20

and

k = 1 Gev/f m

in nature is explained exa tly by the onnement term, be ause it implies

that the energy of a

qq

system in reaseswith the distan e.

By putting together these two behaviors, one an write the Cornell

po-tential, shown inFig. 1.2 [3℄:

V

_{(r) ∼ −}

4

3

α

_s

(r)

r

+ kr,

(1.5)

Withthis potential, the harmonium wave fun tion an beexpressed as:

Ψ(r, θ, φ) = R

nl

(r)Y

l

m

(θ, φ).

(1.6)

This des ription, however, is not enough to reprodu e the mass

dier-en e for harmonium states in the same orbital angular momentum or spin

multiplets.

H

′

inthe equation (1.2 ) in ludesthe spin (

S

) and orbital(

L

) dependent partof thestrongintera tion, explainingthe harmoniumneand hyperne

stru ture [4℄:

H

′

= V

LS

+ V

SS

+ V

tens

.

(1.7) The various terms of intera tion are des ribedin the following:

ˆ spin-orbit (

V

LS

): spin-orbit for es between quarks are present for both ve tor and s alar intera tions, but in dierent form. We nd for

quarks ofequal mass

m

c

:

V

LS

= (L · S)(3

dV

V

dr

−

dV

S

dr

)/(2m

2

c

r)

(1.8)

where

V

S

and

V

V

are the s alar and ve tor omponents of the non-relativisti potential V(r). This term splits the states with the same

orbitalangularmomentum dependingonthe

(L · S)

expe tation value

(ne stru ture);

ˆ spin-spin (

V

SS

): the hyperne ele tromagneti intera tion between a proton and an ele tron leads to a 1420 MHz level splitting between

singlet and triplet states of atomi hydrogen. In light-quark systems,

a similar spin-spin for e due to single-gluon-ex hange between quarks

generates the splittings between the masses of the pion and the

ρ

res-onan e, the nu leon and the

∆

resonan e, the

Σ

and the

Λ

hyperons,

and so on. The spin-spin intera tion is of the form:

V

SS

=

2(S

1

· S

2

)

3m

2

c

∇

2

_V

V

(r)

(1.9)

and the expe tation valuefor

S

1

· S

2

is +1/4for

S

=1and -3/4 for

S

=0;

ˆ tensor (

V

T

): the tensor potential, in analogy with ele trodynami s, ontains the tensor ee ts of the ve tor potential:

V

T

=

S

12

12m

2

c

(

1

3

dV

V

dr

−

d

2

_V

V

dr

2

),

S

12

= 2 [3(S · ˆr)(S · ˆr) − S

2

].

where

S

12

has nonzero matrix elementsonly for

L 6= 0

.

Evenifthe QCDtheory allowstodes ribeonthebasis thefoundamental

theoryofthe intera tions,asexplainedinthenextse tion,othersuggestions

for the fun tional form of the binding potential V(r) exist, but they are

essentially oin identwith thevaluesfrom(1.1)inthe regionfrom0.1to1.0

fm, the dimension s ale of the

cc

system, and lead to similar results.

Another possibility topredi tthe harmoniummass spe trumis to

on a dis rete Eu lidean spa e-time grid. Indeed, QCD has been very

su - essful inpredi ting phenomena involving largemomentum transfer. In this

regime the oupling onstant is small and perturbation theory be omes a

reliabletool.

On the other hand, at the s ale of the hadroni world,

µ

≤ 1GeV

, the

oupling onstant is of order unity and perturbative methods fail. In this

domain latti e QCD provides a non-perturbative tool for al ulating the

hadroni spe trum and the matrix elements of any operator within these

hadroni states from rst prin iples. Sin e no new parameters oreld

vari-ables are introdu ed in this dis retization, LQCD retains the fundamental

hara ter of QCD.

The eld theory fundamental prin iples and the path integral an be

used to al ulate on a omputer the properties of the strong intera tion,

with Monte Carlo integration of the Eu lidean path integral. The value of

thelatti espa ing,usually denotedwith

a

, an bede ideddependingonthe

spe i problem that has to be solved.

The only tunable input parameters in these simulations are the strong

oupling onstantandthe baremassesofthequarks. Our beliefisthatthese

parametersare pres ribedby someyetmorefundamental underlying theory,

however,within the ontext of the standard model they have to be xed in

terms of anequal numberof experimental quantities.

1.2 Quantum Chromodynami s

Quantum Chromodynami sisa quantumeld theory obtained fromthe full

StandardModel(SM)bysettingthe weakandele tromagneti oupling

on-stantstozeroandfreezingthes alardoublettoitsva uumexpe tationvalue.

What remains is a Yang-Mills (YM) theory with lo al gauge group SU(3)

( olour) ve torially oupled to six Dira elds (quarks) of dierent masses

(avours). The ve tor elds in the YM Lagrangian (gluons) live in the

ad-joint representation and transform like onne tions under the lo al gauge

group whereas the quark elds live in the fundamental representation and

transform ovariantly. The QCD Lagrangian is

L

QCD

= −

1

4

F

a

µν

F

a µν

+

X

{q}

¯

q(iγ

µ

_D

µ

− m

q

)q

(1.10)

where

{q} = u, d, s, c, b, t,

F

_µν

a

= ∂

µ

A

a

ν

− ∂

ν

A

a

µ

+ gf

abc

A

b

µ

A

c

ν

,

D

µ

= ∂

µ

− iT

a

A

a

µ

and

f

abc

aretheSU(3)stru ture onstantsand

T

a

formabasisofthe

founda-mental representation of the SU(3) algebra. When oupled to

ele tromag-netism, gluons behave as neutral parti les whereas

u

,

c

and

t

quarks have

harges +2/3 and

d

,

s

and

b

quarks have harges -1/3.

The mainproperties ofQCD,whi hhavebeenpartiallyillustrated inthe

previous se tions,are the following:

ˆ It is Poin aré, parity, time reversal and hen e harge onjugation

in-variant. ItisinadditioninvariantunderU(1)

6

whi himpliesindividual

avour onservation.

ˆ Being a non-Abelian gauge theory, the physi al spe trum onsists of

olour singlet states only. The simplest of these states have the

quan-tum numbers of quark-antiquark pairs (mesons) or of three quarks

(baryons) although other possibilities are not ex luded.

ˆ TheQCDee tive oupling onstant

α

s

(q)

de reasesasthemomentum transfer s ale

q

in reases (asymptoti freedom) [8 , 9℄, as also already

explainedinse tion1.1. Thisallowstomakeperturbative al ulations

in

α

s

athigh energies.

ˆ At low energies it develops an intrinsi s ale (mass gap), usually

re-ferred as

Λ

QCD

, whi h provides themain ontributiontothe masses of most light hadrons. At s ales

q ∼ Λ

QCD

, α

s

(q) ∼ 1

and perturbation theory annot be used. Investigations must be arried out using

non-perturbative te hniques, the bestestablished of whi h islatti e QCD.

Quarks are onventionally divided intolight

m

q

l

+

_l

−

_Λ

QCD

(q = u, d, s)

and heavy

m

Q

≫ Λ

QCD

(Q = c, b, t)

: 1

m

u

= 1.5 − 3.3 MeV, m

u

= 3.5 − 6.0 MeV, m

s

= 70 − 130 MeV,

m

c

= 1.27

+0.07

−0.11

M eV, m

b

= 4.20

+0.17

−0.07

GeV, m

t

= 171.2 ± 2.1 GeV

ˆ Iflightquarkmassesarenegle ted,theU(1)

3

avour onservation

sym-metryoftheQCDLagrangianinthisse torisenlargedtoaU(3)

N

U(3)

1

group. The axial U(1) subgroup is expli itly broken by quantum

ef-fe ts(axial anomaly). Theve tor U(1)subgroupprovideslightavour

onservation. The remainingSU(3)

N

SU(3) subgroup,knownas hiral

symmetry group, turns out to be spontaneously broken down to the

diagonal SU(3) (avour symmetry). This produ es eight Goldstone

bosons, whi h, upon taking into a ount the expli it breaking of the

symmetry due to the non-zero quark masses, a quire masses that are

mu hsmaller than

Λ

QCD

.

ˆ Hadrons ontainingheavyquarkshavemassesoftheorderof

m

Q

rather than of the order

Λ

QCD

. They enjoy parti ular kinemati al features that allow for spe i theoreti al treatments.

1.3 Ee tive Field Theories

From the point of view of QCD the des ription of hadrons ontaining two

heavy quarks is a rather hallenging problem, whi h adds to the

ompli a-tionsof thebound stateineld theorythose omingfromanonperturbative

low-energy dynami s. A proper relativisti treatment of the bound state

basedon the Bethe-Salpeter equation [10 ℄ has proveddi ult. Perturbative

al ulationshaveturnedout unpra ti alathigherorderandthe methodhas

been abandonedinre entQCD al ulations. Moreover, theentanglementof

allenergymodes inafullyrelativisti treatmentismoreanobsta lethanan

advantageforthefa torizationofphysi alquantitiesintohigh-energy

pertur-bativeandlowenergynonperturbative ontributions. Partialsemirelativisti

redu tions and models have been often adopted to over ome these

di ul-ties at the pri e to introdu e un ontrolled approximations and lose onta t

withQCD.Thefully relativisti dynami s an, inprin iple,betreated

with-out approximations inlatti e gaugetheories. This isin perspe tive the best

founded and most promising approa h,as already said inse tion 1.1 .

A nonrelativisti treatment of the heavyquarkoniumdynami s, whi h is

suggested by the large mass of the heavyquarks, has lear advantages. The

velo ityofthequarksintheboundstateprovidesasmallparameterinwhi h

thedynami als alesmaybehierar hi allyorderedandthe QCDamplitudes

systemati ally expanded. Fa torization formulas be ome easier to a hieve.

Aprioriwedonotknowifanonrelativisti des riptionwillworkwellenough

for all heavy quarkonium systems in nature: for instan e, the harm quark

may not be heavyenough. The fa t that most of the theoreti al predi tions

are based on su h a nonrelativisti assumption and the su ess of most of

We may, however, also take advantage of the existen e of a hierar hy

of s ales by substituting QCD with simpler but equivalent Ee tive Field

Theories (EFTs). EFTshavebe omein reasinglypopularinparti lephysi s

during the last de ades.

They provide a realization of Wilson renormalization group ideas [11℄

and fullyexploit the properties oflo alquantum eld theories. AnEFT isa

quantum eld theory with the following properties:

a) it ontainsthe relevantdegreesoffreedom todes ribephenomenathat

o ur in ertainlimited rangeof energies and momenta;

b) it ontainsanintrinsi energys ale

Λ

thatsetsthelimitofappli ability

of the EFT.

TheLagrangianofanEFTisorganizedinoperatorsofin reasingdimension,

hen e, an EFT is in general non-renormalizable in the usual sense. In spite

of this, it an be made nite to any nite order in

1/Λ

by renormalizing

(mat hing) the onstants (mat hing oe ients)infrontof the operatorsin

the Lagrangian untilthat order. This meansthat one needs more

renormal-ization onditions when the order in

1/Λ

is in reased. However, even if the

only way of xing the onstants would be by means of experimental data,

this would redu ebut not spoilthe predi tive powerof the EFT.If the data

are abundant, the onstants an be t on e for ever and used later on to

make predi tions onnew experiments.

The prototype of EFT for heavy quarks is the Heavy Quark Ee tive

Theory (HQET), whi h is the EFT of QCD suitable to des ribe systems

with onlyone heavyquark [12 , 13 ℄. These systems are hara terizedby two

energy s ales:

m

and

Λ

QCD

. HQET is obtained by integrating out the s ale

m

and built as asystemati expansionin powersof

Λ

QCD

/m

.

As dis ussedabove, bound states made of two heavy quarks are

hara -terized by more s ales. Integrating out only the s ale

m

, whi h for heavy

quarks an be done perturbatively, leads to an EFT, Nonrelativisti QCD

(NRQCD) [14, 15℄, that still ontains the lower s ales as dynami al degrees

of freedom. Disentangling the remaining s ales is relevant both te hni ally,

sin e it enables perturbative al ulations otherwise quite ompli ate, and

more fundamentally, sin e it allows to fa torize nonperturbative

ontribu-tions intothe expe tation values ormatrixelements offewoperators. These

may be eventually evaluated on the latti e, extra ted from the data or

al- ulatedin QCDva uummodels.

In the next se tion we will give a briefgeneral introdu tion to NRQCD,

1.4 Nonrelativisti QCD

Aparti ularlyelegantapproa hforseparatingrelativisti fromnonrelativisti

s alesistore asttheanalysisintermsofnonrelativisti quantum

hromody-nami s(NRQCD)[15℄,anee tiveeldtheorydesignedpre iselytoseparate

therelativisti physi sofannihilation(whi hinvolvesmomenta

p ∼ M

)from

thenonrelativisti physi sofquarkoniumstru ture(whi hinvolves

p ∼ Mv

).

NRQCD onsists of a nonrelativisti S hroedinger eld theory for the

heavyquarkand antiquarkthat is oupled tothe usualrelativisti eld

the-ory for light quarks and gluons. The theory is made pre isely equivalent to

full QCD through the addition of lo al intera tions that systemati ally

in- orporaterelativisti orre tionsthrough anygivenorderintheheavy-quark

velo ity

v

. It is an ee tive eld theory, with a nite ultraviolet uto of

order

M

thatex ludesrelativisti states (statesthatare poorlydes ribedby

nonrelativisti dynami s). A heavy quark in the meson an u tuate into

a relativisti state, but these u tuations are ne essarily short-lived. This

meansthat the ee tsof the ex ludedrelativisti states anbemimi kedby

lo al intera tions and an, therefore,be in orporated into NRQCD through

renormalizations of its innitely many oupling onstants. Thus,

nonrela-tivisti physi s is orre tly des ribed by the nonperturbative dynami s of

NRQCD, while all relativisti ee ts are absorbed into oupling onstants

that an be omputed asperturbationseries in

α

s

(M).

The main advantage oered by NRQCD overordinary QCD inthis

on-text is that it is easier to separate ontributions of dierent orders in

v

in

NRQCD. Thus, we are able not only to organize al ulations to all orders

in

α

s

, but also to elaborate systemati ally the relativisti orre tions to the onventional formulas.

1.4.1 The NRQCD lagrangian

Themost importantenergy s alesforthe stru ture andspe trumofaheavy

quarkonium system are

M v

and

M v

2

, where

M

is the mass of the heavy

quark

Q

and

vl

+

_l

−

₁

isitsaveragevelo ityinthemesonrestframe. Momenta

of order

M

play only a minor role in the omplex binding dynami s of the

system. We an take advantage of this fa t in our analysis of heavyquark

mesonsby modifyingQCD intwosteps.

We start with full QCD, in whi h the heavy quarks are des ribed by

4- omponent Dira spinor elds. In the rst step, we introdu e an ultraviolet

momentum uto that is of order

M

. This uto expli itly ex ludes

rela-tivisti heavyquarksfromthetheory,aswellasgluonsandlightquarkswith

sin e the importantnonperturbative physi sinvolvesmomenta of order

M v

orless. Of ourse,therelativisti stateswearedis ardingdohavesomeee t

on the low energy physi s of the theory. However, any intera tion involving

relativisti intermediate states is approximately lo al, sin e the

intermedi-ate states are ne essarily highly virtual and so annot propagate over long

distan es. Thus, generalizing standard renormalization pro edures, we

sys-temati ally ompensate for the removal of relativisti states by adding new

lo al intera tionstothelagrangian. Toleadingorderin

1/Λ

or, equivalently,

1/M

, these new intera tions are identi al in form to intera tions already present in the theory, and so the net ee t is simply to shift bare masses

and harges. Beyond leading order in

1/M

, one mustextend the lagrangian

to in lude nonrenormalizable intera tions that orre t the low energy

dy-nami sorder-by-order in

1/M

. In this uto formulationof QCD,allee ts

that arise from relativisti states, and only these ee ts, are in orporated

into renormalizations of the oupling onstants of the extended lagrangian.

Thus, in the uto theory, relativisti and nonrelativisti ontributions are

automati ally separated. This separation is the basis for an analysis of the

annihilationde ays of heavyquarkonia.

The lagrangian for NRQCDis:

L

N RQCD

= L

light

+ L

heavy

+ δL

(1.11)

The gluons and the

n

f

avors of light quarks are des ribed by the fully relativisti lagrangian

L

light

= −

1

2

trG

µν

G

µν

₊

X

¯

qiDq

(1.12)

where

G

µ

is the gluon eld-strength tensor expressed in the form of an SU(3) matrix, and

q

is the Dira spinor eld for a light quark. The

gauge- ovariant derivative is

D

µ

_{+ igA}

µ

, where

A

µ

_{= (φ, A)}

is the SU(3)

matrix-valued gaugeeldand

g

is theQCD oupling onstant. Thesum in(1.12)is

overthe

n

f

avorsof light quarks.

The heavy quarksand antiquarks are des ribed by the term

L

heavy

= ψ

†



iD

t

+

D

2

2M



ψ + χ

†



iD

t

−

D

2

2M



χ,

(1.13)

where

ψ

is the Pauli spinor eld that annihilates a heavy quark,

χ

is

the Pauli spinor eld that reates a heavy antiquark, and

D

t

and

D

are the time and spa e omponentsof the gauge- ovariant derivative

D

µ

. Color

L

light

+L

heavy

des ribesordinaryQCD oupledtoaS hroedingereldtheory for the heavyquarks and antiquarks.

Therelativisti ee tsoffullQCDarereprodu edthroughthe orre tion

term

δL

in the lagrangian

L

N RQCD

[14℄.

In parti ular the orre tionterms most important forheavyquarkonium

are bilinearin the quark eld orantiquark eld:

L

bilinear

=

c

1

8M

3

ψ

†

_(D

2

₎

2

ψ − χ

†

(D

2

)

2

χ

+

c

2

8M

2

ψ

†

(D · gE − gE · D)ψ − χ

†

(D · gE − gE · D)χ

+

c

3

8M

2

ψ

†

(iD × gE − gE × iD)ψ − χ

†

(iD × gE − gE × iD)χ

+

c

4

2M

ψ

†

(gB · σ)ψ − χ

†

(gB · σ)χ

,

(1.14) where

E

i

_{= G}

0i

and

B

i

=

1

2

ǫ

ijk

_G

jk

arethe ele tri andmagneti omponents

of the gluoneld strength tensor

G

µν

. By harge onjugation symmetry, for

every termin(1.14)involving

ψ

,there isa orrespondingterm involving the

antiquarkeld

χ

,withthe same oe ient

c

i

,up toasign. Theoperatorsin (1.14 )mustberegularized,andtheythereforedependontheultraviolet uto

orrenormalizations ale

Λ

of NRQCD.The oe ients

c

i

(Λ)

alsodepend on

Λ

insu h away as to an el the

Λ

-dependen e of the operators.

Noti e that

L

bilinear

doesn't ontain mixed two-fermionoperators involv-ing

χ

†

and

ψ

(or

ψ

†

and

χ

), orrespondingtotheannihilation(orthe reation)

of a

Q ¯

Q

pair. Indeed su h terms are ex luded from the lagrangian as part

of the denition of NRQCD: if su h an operator annihilates a

Q ¯

Q

pair, it

would, by energy onservation, have to reate gluons (or light quarks) with

energiesof order

M

. The amplitude forannihilation of a

Q ¯

Q

pair into su h

highenergygluons annotbedes ribeda uratelyinanonrelativisti theory

su has NRQCD.

The oe ients

c

i

must be tuned as fun tions of the oupling onstant

α

s

, the heavy-quark mass parameter infull QCD, and the ultraviolet uto

Λ

of NRQCD, sothat physi al observables are the same asin fullQCD. Inprin iple,innitelymanytermsarerequiredintheNRQCDlagrangian

inordertoreprodu e fullQCD,but inpra ti e onlyanite numberofthese

isneededforpre ision toanygivenorder inthetypi alheavy-quark velo ity

1.5 Experimental study of harmonium

1.5.1 Quarkonium produ tion

Quarkonia an be produ ed in several ways, whi h rea h dierent states

within the spe trum. The rst three listed here are mere reversals of

Q ¯

Q

de ay pro esses and are sket hed inFig. 1.3 a), b), and ).

In ele tron-positron olliders, the rea tion results in

e

+

_e

−

_{→ γ}

∗

_{→ Q ¯}

_Q

states that an ouple to a virtual photon, namely

n

3

_S

1

su h as

J/ψ

and

Υ

with a tiny admixture of

n

3

_D

1

. Dire t resonan e formation oers the advantageoflargeprodu tionrates,givinga esstobran hingfra tionseven

as small as

10

−5

, as well as higher a ura y in the measurements of masses

and widths..

Two-photon ollisions allow dire t reation of

J = 0, 2

states, e.g.

η

[c,b]

,

χ

[c,b][0,2]

. While theyare readilyavailableat

e

+

_e

−

ma hines,theysuerfrom

small produ tionrates. Still theyprovidean important ontribution in that

they an be used fordis overy purposes.

Hadron ma hines, being able to form any quarkonia state in prin iple

by annihilation of the

p¯

p

pair into gluons, ontinue to ontribute mostly to

the study of produ tion of harmonia. This environment suers from large

ba kground;thereby one has to fo us onex lusive de ays.

Twomore s enarios: downward transitions within the system providean

importantroute tootherwise not rea hable states. AnyB-fa tory has a ess

to

cc

states through weak de ays of the

b

quark, with the two pro esses

sket hed inFig. 1.3 d) and e).

An important ba kground for the rea tion

e

+

_e

−

_{→ Q ¯}

_{Q → X}

, or more

expli itly,

e

+

_e

−

_{→ γ}

∗

_{→ Q ¯}

_{Q → X}

, is the ase in whi h no intermediate

Q ¯

Q

resonan e isformed. The presen e ofthis hanneladdsto themeasured ross-se tion both dire tly and by interferen e, whi h an bea sizeable

on-tribution. Inmostmeasurements,this ontribution isnottakenintoa ount

orsubtra ted. This ba kgroundneeds tobe either measured,by running o

the relevant resonan e, or al ulated. In measurements of the ross-se tion

as fun tionof energy (s ans),the non-resonant produ tion an be expli itly

taken intoa ount when tting the line shape.

In addition tothat, the produ tion of double harmonium in

e

+

_e

−

anni-hilations has re ently been observed at the B fa tories. The produ tion of

double harmoniumin

e

+

_e

−

annihilationwasdis overedbytheBelle

ollabo-rationfromdata olle tedatthe

Υ (4S)

resonan e ata enter-of-massenergy

s =

p(10.6)GeV

by studying the re oil momentum spe trum of the

J/ψ

in

e

+

_e

−

_{→ J/ψ}

+ X [16 ℄. The measured ross se tion for double harmonium

predi tion of NRQCD in the non-relativisti limit. This large dis repan y

was rather puzzling. This way to produ e harmonium is the main topi of

thisthesis,andthenext hapterwillbededi atedtothedouble harmonium

produ tionphysi s.

1.5.2 Charmonium spe trum

The spe trum of harmonium states is shown in Fig. 1.4 . The potential

model, des ribed in se tion 1.1 , an explain with the spin-spin intera tion

term (

V

SS

) the splitting among spin singlet and triplet states like

J/ψ

and

η

c

, and with the spin-orbit intera tion (

V

LS

) the splitting among states like

χ

c 0,1,2

.

The harmoniumspe trum onsistsofeightnarrowstatesbelowtheopen

harm threshold (3.73 GeV) and several tens of states abovethe threshold.

All eight states below

D ¯

D

threshold are wellestablished, but whereas

thetriplet statesare measuredwith verygooda ura y,the same annotbe

said for the singlet states.

The

η

c

wasdis overedalmostthirtyyearsagoandmanymeasurementsof itsmass andtotal widthexist. Despite the largevariety ofavailabledata on

it,the pre isedetermination of its mass and width is still an openproblem.

TheParti leDataGroup(PDG)[5℄valueofthemassis2980.3

±

1.2MeV/

c

2

:

the erroronthe

η

c

mass isstill aslarge as1.2 MeV/

c

2

, tobe omparedwith

few tens of KeV/

c

2

for the

J/ψ

and

ψ

′

and few hundreds of KeV/

c

2

for the

χ

c 0,1,2

. The situation is even worse for the total width: the PDG average

F

F

4

4

4

4

H



H



H



H



H



H



4

4

4

4

H



H



( )

4

4

S

S

4

4

S

S

E

T

( )

4

4

V

T

:



( )

a)

b)

c)

d)

e)

–

Figure1.3: Heavyquarkoniaprodu tiondiagrams. Produ tion(left)andtheir

orrespondingde ay (right)pro esses: a)

e

+

e

−

_{→ γ}

∗

_{→ Q ¯}

_Q

; b)

γγ → Q ¯

Q

; )

p¯

p →

gluons

→ Q ¯

Q

; d) Quarkoniumde-ex itation byemission oftwo pions; e) reating harmoniumfroma Bmeson.

Figure 1.4: The harmoniumspe trum.

is 26.7

±

3.0 MeV. The most re ent measurements have shown that the

η

c

width is larger than was previously believed, with values whi h are di ult

to a omodate in quark models. This situation points to the need for new

high-pre ision measurements of the

η

c

parameters.

Therstexperimentaleviden eofthe

η

c

(2S)

wasreportedbythe Crystal Ball Collaboration [17℄, but this nding was not onrmed in subsequent

sear hes in

p¯

p

or

e

+

_e

−

experiments. The

η

c

(2S) was nally dis overed by the Belle ollaboration [18 ℄ in the hadroni de ay of the B meson

B →

K + η

c

(2s) → K + (K

s

K

−

π

+

)

with amass whi hwasin ompatiblewiththe

Crystal Ball andidate. The Bellendingwasthen onrmed by CLEOand

BaBar [19, 20℄, whi h observed this state in twophoton fusion. The PDG

value of the mass is 3637

±

4 MeV/

c

2

, and the width isonly measured with

ana ura y of 50%: 14

±

7 MeV/

c

2

.

The

1

_P

1

stateof harmonium(

h

c

)isofparti ularimportan einthe deter-mination of the spin-dependent omponentof the

qq

onnement potential.

The

h

c

has been observed by CLEO [21℄ in the rea tion

ψ(2S) → π

0

_h

c

→

(γγ)(γη

c

)

with amass of3524.4

±

0.6

±

0.4 MeV/

c

2

than 5

σ

.

Parti le

n

2S+1

_L

J

J

P C

Mass(MeV) Width(MeV)

η

c

1

1

S

0

0

−+

2980.3

±

1.2 26.7

±

3.0

J/ψ

1

3

S

1

1

−−

3096.916

±

0.011 (93.2

±

2.1)

×

10

−3

χ

c0

1

3

P

0

0

++

3414.75

±

0.31 10.2

±

0.7

χ

c1

1

3

P

1

1

++

3510.66

±

0.07 0.89

±

0.05

χ

c2

1

1

P

2

2

++

3556.20

±

0.09 2.03

±

0.12

η

c

(2S)

2

1

S

0

0

−+

3637

±

4 14

±

7

ψ(2S)

2

3

_S

1

1

−−

3686.09

±

0.04 0.317

±

0.009

Table1.1: Quantumnumbers,massesandwidthofthe harmoniumstateswith

massbelowtheopen harmprodu tionthresholdfrom PDG.[5 ℄.

The region above

D ¯

D

threshold is ri h in interesting new physi s. In

thisregion, losetothe

D ¯

D

threshold, oneexpe tstondthefour1Dstates.

Oftheseonlythe

1

3

_D

1

,identiedwiththe

ψ

(3770)resonan eanddis overed bythe MarkI ollaboration[22℄,has beenestablished. Itisawideresonan e

(

Γ(ψ(3770) = 27.3 ± 1.0MeV/c

2

), whi h de ays predominantly to

D ¯

D

. The

J

= 2 states (

1

1

_D

2

and

1

3

_D

2

) are predi ted to be narrow, be ause parity onservation forbids their de ay to

D ¯

D

. In addition to the

D

states, the

radial ex itations of the

S

and

P

states are predi ted to o ur above the

open harm threshold. None of these stateshave been positivelyidentied.

Some of the features of harmonium states are summarized in table 1.1.

In the next se tion, there will be more details on the experimental on the

energyregionabove

D ¯

D

threshold,inparti ularforthosenewstatesre ently,

mainly dis overed at the B-fa tories.

1.5.3 New harmonia

A lot of new states have re ently been dis overed (X, Y, Z mesons),mainly

inthe hadroni de aysof the B meson: thesenew states are asso iatedwith

harmoniumbe ausetheyde aypredominantlyinto harmoniumstatessu h

as the

J/ψ

or the

ψ

′

, but their interpretation is far from obvious. In this

se tion,abriefsummary ofthe experimental dataand the possible

interpre-tationis presented.

Curiously,three harmonium-likestateswereobservedwithsimilarmasses

near 3.94 GeV/

c

2

, but in quite dierent pro esses, as summarized in table

1.2 [26 ℄.

The harmoniumlike state X(3940) has been observed by Belle in the

double harmonium produ tioninthe pro ess

e

+

_e

−

_{→ J/ψ D ¯}

_D

∗

inthe mass

State

J

P C

Mass(MeV/

c

2

)

Γ

(MeV/

c

2

) De ay Produ tion Collaboration

X(3940)

?

?+

3942

+7

−6

±

8 37

+26

−28

±

8

D ¯

D

∗

_e

+

_e

−

_{→ J/ψ X(3940)}

Belle X(4160)

?

?+

4156

+25

−20

±

15 139

+11

−61

±

21

D

∗

_D

¯

∗

_e

+

_e

−

_{→ J/ψ X(4160)}

Belle Y(3940)

?

?+

3943

±

11

±

13 87

±

22

±

26

ωJ/ψ

B → KY (3940)

Belle Y(3940)

?

?+

3914.6

+3.8

−3.4

±

2.0 34

+12

−8

±

5

ωJ/ψ

B → KY (3940)

BaBar Z(3930)

2

++

3929

±

5

±

2 29

±

10

±

2

γγ → Z(3940)

Belle Table1.2: MeasuredparametersoftheXYZ(3940)states.

for the BABAR ollaboration. X(3940) state is tentatively identied with

η

c

(3S)

. In addition Belle found a new harmoniumlike state, X(4160), in thepro esses

e

+

_e

−

_{→ J/ψ X(4160)}

de ayinginto

D

∗

_D

¯

∗

withasigni an eof

5.1

σ

[23 ℄. BoththeX(3940)andtheX(4160)de aytoopen harmnalstates

andtherefore ouldbeattributedto

3

1

_S

0

and

4

1

_S

0

onventional harmonium states. However, the problem with this assignment is that potential models

predi tmassesfortheselevelstobesigni antlyhigherthanthosemeasured

for the X(3940) and X(4160).

TheY(3940)statewasobservedbyBelleasanear-thresholdenhan ement

in the

ωJ/ψ

invariant mass distribution for ex lusive

B → KωJ/ψ

de ays

with astatisti al signi an e of8.1

σ

[24 ℄. AlsoBABAR found an

ωJ/ψ

mass

enhan ement at

∼

3.915 GeV/

c

2

in the de ays

B → K

0,+

_{→ ωJ/ψ}

[25 ℄ and

onrmed the Belle result. The Y(3940) mass is two standard deviations

lower than the Z(3930) mass, and three standard deviations lower than for

the X(3940); the width agrees with the Z(3930) and X(3940) values.

The Z(3930) state was found by Belle in two-photon ollisions

γγ →

D ¯

D

with a mass

∼

3.930 GeV/ 2. The produ tion rate and the angular distributioninthe

γγ

enter-of-massframefavortheinterpretationofZ(3930)

as the

χ

c2

(2P) harmoniumstate.

For all other new states (X(3872), Y(4260), Y(4320) and so on) the

in-terpretationisnotatall lear,withspe ulations rangingfromthe missing

cc

Double harmonium produ tion

2.1 Introdu tion

The ex lusive produ tion of a pair of double heavy mesons with -quarks

in

e

+

_e

−

annihilation has attra ted onsiderable attention in the last years.

In fa t, at the beginning of these studies, the ross se tion of the pro ess

e

+

_e

−

_{→ J/ψ η}

c

,whi hwasmeasured inthe experimentsonBABARand Belle dete tors at the energy

√

s

= 10.6 GeV, resultedto be

σ(e

+

e

−

_{→ J/ψ + η}

c

) × B(η

c

→≥ 2 charged) =

(

25.6 ± 2.8 ± 3.4 [

27

]

17.6 ± 2.8

+1.5

−2.1

[

28

]

(2.1)

and led to a dis repan y with the theoreti al al ulation in the framework

of nonrelativisti QCD (NRQCD) by an order of magnitude. This

on lu-sion is basedon al ulationsin whi h the relativemomentaof heavyquarks

and bound state ee ts inthe produ tionamplitude were not takeninto

a - ount. A set of al ulations was performed to improve the nonrelativisti

approximation for the pro ess.

Inparti ular,relativisti orre tionstothe rossse tion

σ(e

+

_e

−

_{→ J/ψ +}

η

c

)

were onsideredina olorsingletmodelinreferen e[29 ℄usingthemethods of NRQCD[14℄. A synthesis of this method will be done inse tion 2.2.1 .

Anotherattempttotakeintoa ounttherelativisti orre tionswasdone

intheframeworkofthelight- oneformalism [30,31℄,des ribedhereinse tion

2.2.4 . With this formalism the dis repan y between experiment and theory

an be eliminated ompletely by onsidering the intrinsi motion of heavy

quarks formingthe doubly heavymesons.

In addition, perturbative orre tions of order

α

s

to the produ tion am-plitudewere al ulated inreferen e [32℄, where Zhang, Gao and Chao ould

On a ount of dierent values of relativisti orre tions obtained in

ref-eren es [29 , 30 , 31 ℄ and the importan eof a relativisti onsideration of the

pro ess

e

+

_e

−

_{→ J/ψ + η}

c

in solving the doubly heavy meson produ tion problem, Ebert and Martynenko [33 ℄ have performed a new investigation of

relativisti and bound state ee ts. This investigation is based on the

rela-tivisti quarkmodelwhi hprovidesthesolutioninmanytasksofheavyquark

physi s. In [34, 35℄ theyhave demonstrated how the originalamplitude,

de-s ribing the physi al pro ess, must be transformed in order to preserve the

relativisti plusboundstate orre tions onne tedwiththeone-parti lewave

fun tionsand the wave fun tionof a two-parti le bound state.

In parti ular, in paper [33 ℄ they extend the method to the ase of the

produ tion of a pair (

P + V

) of double heavy mesons ontaining quarks of

dierentavours

b

and

c

. They onsidertheinternalmotionofheavyquarks

in both produ ed pseudos alar

P

and ve tor

V

mesons, and the results of

the ross-se tion will be presented in se tion2.2.3 .

Two more se tions are in this hapter: se tion 2.4 where a synthesis

of the re ent results on the analysis

e

+

_e

−

_{→ γ}

∗

_{→ J/ψ}

+ X is done and

the possible interpretations of the state X(3940), whi h is expe ted to be

seen in the double harmonium produ tion pro ess via one virtual photon,

are illustrated. This last se tion is parti ularly interesting for this analysis,

whi hhasinitsaimsalsoto onrmthisstate,also seeninBelleinthe re oil

spe trum. Finallyinse tion2.5wewillbrieydes ribethetheory on erning

thedouble harmoniumprodu tionwithtwovirtual photonsinvolvedinthe

pro ess (

e

+

_e

−

_{→ γ}

∗

_γ

∗

_{→ cccc}

).

2.2 Cross se tion

If harmoniumisthe onlyhadronintheinitialornalstate,the olor-singlet

model should be a urate up to orre tions that are higher order in

v

. The

simplestexamplesofsu hpro esses areele tromagneti annihilationde ays,

su has

J/ψ → e

+

_e

−

and

η

c

→ γγ

,and ex lusiveele tromagneti produ tion pro esses,su h as

γγ → η

c

.

Another pro ess for whi h the olor-singlet model should be a urate is

e

+

_e

−

annihilation intoexa tly two harmonia. There are nohadrons in the

initial state, and the absen e of additional hadrons in the nal state an

beguaranteed experimentallyby the monoenergeti natureofa 2-bodynal

state. For many harmonia,the NRQCDmatrix element an be determined

from the ele tromagneti annihilation de ay rate of either the harmonium

state itself or of another state related to it by spin symmetry. Cross

suppressed by powersof

v

2

without any unknown phenomenologi al fa tors.

One problem with

e

+

_e

−

annihilation into ex lusive double harmonium

isthat the rossse tions arevery smallatenergieslarge enoughtotrustthe

predi tions of perturbative QCD. A naive estimate of the ross se tion for

J/ψ + η

c

inunits of the ross se tion for

µ

+

_µ

−

is:

R[J/ψ + η

c

] ∼ α

2

s



m

c

v

E

beam



6

.

(2.2)

The2powersof

α

s

arethefewestrequiredtoprodu ea

cc + cc

nalstate. There is a fa tor of (

m

c

v

3

) asso iated with the wavefun tion at the origin

for ea h harmonium. These fa tors in the numerator are ompensated by

fa tors of the beam energy

E

beam

in the denominatorto get a dimensionless ratio.

As an example, onsider

e

+

_e

−

annihilation with enter-of-mass energy

2

E

beam

= 10.6 GeV. If we set

v

2

_≈

0.3,

α

s

≈

0.2, and

m

c

≈

1.4 GeV, we get the naive estimate

R[J/ψ + η

c

] ≈

4

×

10

−7

. This should be ompared

to the total ratio

R[hadrons] ≈

3.6 for all hadroni nal states [36 ℄. The

de ay of the

J/ψ

into the easily dete table

e

+

_e

−

or

µ

+

_µ

−

modes suppresses

the observable ross se tionby anotherorder of magnitude.

Fortunately,theeraofhigh-luminosityBfa torieshasmadethe

measure-mentofsu hsmall rossse tionsfeasible. BraatenandLee[29 ℄ al ulatedthe

ross se tions for ex lusive double- harmoniumprodu tion via

e

+

_e

−

annihi-lation intoa virtual photon. This pro ess produ es only harmonium states

with opposite harge onjugation. The ross se tionsfor harmonium states

with thesame harge onjugation, whi h pro eed through

e

+

_e

−

annihilation

into two virtual photons[37, 38℄ will be illustrated inse tion 2.5 .

2.2.1 Color-singlet model al ulation

In this se tion, the ross se tions for

e

+

_e

−

annihilation through a virtual

photonintoadouble- harmoniumnalstate

H

1

+ H

2

are al ulatedbyusing the olor-singletmodel. The olor-singletmodel(CSM) anbeobtainedfrom

the NRQCD fa torization formula by dropping all of the olour-o tet terms

and all but one of the olour-singletterms. The term that is retained isthe

one in whi h the quantum numbers of the

Q ¯

Q

pair are the same as those of

the quarkonium.

Charge onjugation symmetryrequiresone ofthe harmoniatobea

C =

−

state and the other to be a

C = +

state. The

C = −

states with narrow widths are the

J

P C

_{= 1}

−−

states

J/ψ

and

ψ(2S)

, the

1

+−

state

h

c

,and the yet-to-be-dis overed

2

−−

The

C = +

states with narrowwidths are the

0

−+

states

η

c

and

η

c

(2S)

, the

J

++

states

χ

cJ

(1P),

J = 0, 1, 2

, and the yet-to-be-dis overed

2

−+

state

η

c2

(1D).The resultswill be express intermsof the ratio

R[H

1

+ H

2

]

dened by

R[H

1

+ H

2

] =

σ[e

+

_e

−

_{→ H}

1

+ H

2

]

σ[e

+

_e

−

_{→ µ}

+

_µ

−

_]

(2.3) In the text, only the results for

R

summed over heli ity states will be

given. Theseresults mayfa ilitatethe use ofpartialwaveanalysis toresolve

theexperimental double- harmoniumsignalinto ontributions fromthe

var-ious harmonium states.

When the

e

+

_e

−

beam energy

E

beam

ismu h largerthan the harm quark mass

m

c

, the relative sizes of the various double- harmonium ross se tions aregovernedlargelybythenumberofkinemati suppressionfa tors

r

2

,where

the variable

r

is denedby

r

2

=

4m

2

c

E

2

beam

.

(2.4)

If we set

m

c

= 1.4 GeV and

E

beam

= 5.3 GeV, the value of this small parameter is

r

2

= 0.28. The asymptoti behavior of the ratio

R[H

1

+ H

2

]

as

r →

0 an be determined from the heli ity sele tion rules for ex lusive

pro essesinperturbativeQCD.Forea hofthe

cc

pairsinthenalstate,there

isasuppressionfa torof

r

2

duetothe largemomentumtransfer requiredfor

the

c

and

c

¯

to emerge with small relative momentum. Thus, at any order

in

α

s

, the ratio

R[H

1

+ H

2

]

must de rease at least as fast as

r

4

as

r → 0

.

Howeveritmayde rease morerapidly dependingonthe heli itystatesofthe

two hadrons. There is of ourse a onstraint on the possible heli ities from

angularmomentum onservation:

|λ

1

− λ

2

|

=0 or1.

The asymptoti behavior of the ratio

R[H

1

(λ

1

) + H

2

(λ

2

)]

depends on the heli ities

λ

1

and

λ

2

. The heli ity sele tion rules imply that the slowest asymptoti de rease

R ∼ r

4

an o ur only if the sum of the heli ities of

the hadrons is onserved. Sin e there are no hadrons in the initial state,

hadron heli ity onservation requires

λ

1

+ λ

2

= 0. The only heli ity state that satises both this onstraint and the onstraint of angular momentum

onservationis(

λ

1

, λ

2

)=(0, 0). Foreveryunitof heli ity by whi h thisrule isviolated, there is afurther suppression fa tor of

r

2

.

So, the resulting estimatefor the ratio R atleading order in

α

s

is

R

QCD

[H

1

(λ

1

) + H

2

(λ

2

)] ∼ α

2

s

(v

2

)

3+L

1

+L

2

(r

2

)

2+|λ

1

+λ

2

|

.

(2.5)

The fa tor of

v

3+2L

for a harmoniumstate with orbitalangular

ourse be further suppression fa tors of

r

2

that arise from the simple

stru -ture of the leading-order diagrams for

e

+

_e

−

_{→ cc + cc}

inFig. 2.1, but these

suppression fa tors are unlikely to persist tohigher orders in

α

s

. TheQEDdiagrams for

e

+

_e

−

_{→ cc(}

3

_S

1

) + cc

inFig. 2.2 give ontributions

to

R[J/ψ + H

2

]

that s ale in a dierent way with

r

. This ase is a tually

interesting for the analysis do umented in this thesis. As

r →

0, there is a

ontribution to the ross se tion from these diagrams into the ross se tion

for

γ + H

2

and the fragmentation fun tionfor

γ → J/ψ

. Thisfragmentation

pro essprodu es

J/ψ

ina

λ

J/ψ

=

±

1heli ity state. Thehard-s attering part of the pro ess produ es only one

cc

pair with small relative momentum, so

there is one fewer fa tor of

r

2

relative to equation2.5 . The ross se tion for

γ + H

1

is still subje t tothe heli ity sele tion rules of perturbative QCD,so

the pure QED ontribution tothe ratio Rhas the behavior

R

QED

[J/ψ (±1) + H

2

(λ

2

)] ∼ α

2

(v

2

)

3+L

2

(r

2

)

1+|λ

2

|

.

(2.6)

There may also be interferen e terms between the QCD and QED

on-tributions whoses aling behavioris intermediatebetween equations2.5 and

2.6 .

2.2.2 Cal ulation of the ross se tions

In this se tion, the ross se tions for ex lusive double- harmonium

produ -tion in

e

+

_e

−

annihilationattheB fa tories ispresented, and partially

al u-lated.

The resultsinse tion2.2.1were expressedintermsof theratioRdened

inequation 2.3. The orresponding ross se tions are:

σ[H

1

+ H

2

] =

4πα

2

3s

R[H

1

+ H

2

]

(2.7)

The ratios R depend on a number of inputs: the oupling onstants

α

s

and

α

, the harmquark mass

m

c

, and the NRQCD matrix elements

hO

1

i

.

The value of the QCD oupling onstant

α

s

depends onthe hoi e ofthe s ale

µ

. In the QCDdiagrams of Fig. 2.1 , the invariant mass ofthe gluon is

ps/2

. Wetherefore hoosethe s aletobe

µ

=5.3GeV.Theresulting value of the QCD oupling onstant is

α

s

(µ)

= 0.21.

The numeri al value forthe pole mass

m

c

ofthe harm quarkisunstable under perturbative orre tions, so it must be treated with are. Sin e the

expressionsfor the ele tromagneti annihilation de ay rates in ludethe

per-turbative orre tion of order

α

s

the appropriate hoi e for the harm quark mass

m

c

in these expressions is the pole mass with orre tions of order

α

s

in luded. It an be expressed as

P

₁

P

₂

(a)

(b)

(c)

(d)

Figure 2.1: QCD diagrams that an ontribute to the olor-singlet pro ess

γ

∗

_{→ cc + cc}

→

P

₂

P

₁

(b)

(a)

Figure 2.2: QED diagrams that an ontribute to the olor-singlet pro ess

γ

∗

_{→ cc(}

3

S

1

) + cc

m

c

= ¯

m

c

( ¯

m

c

)



1 +

4

3

α

s

π



.

(2.8)

Taking the running mass of the harm quark to be

m

¯

c

( ¯

m

c

)

= 1.2

±

0.2 GeV, the NLO pole mass is

m

c

= 1.4

±

0.2 GeV.

The Braaten-Lee predi tions for the double harmonium ross se tions

without relativisti orre tions are givenin table 2.1 . 1

TheBraaten-Leepredi tionsforthedouble harmonium rossse tionsfor

the S-wave states (

η

c

, η

c

(2S), J/ψ , ψ(2S)

) in luding the leading relativisti orre tion are obtained by multiplyingthe values intable 2.1 by the fa tor:

1

Onlyvalues interestingfor this analysis have been reported. For all al ulations see