Heavy Quarks and the Strong Potential

1
Heavy Quarks and the
Strong Potential

• Sally Seidel
• Los Alamos National Laboratory
• 3 May 2006

2
Quantum Chromodynamics (QCD) is the theory of the
strong force. As it binds quarks to form
nucleons and nuclei, the strong force is to a
large degree responsible for the patterns that we
find in nature. QCD has been outstandingly
successful in describing whats observed,
including the evolution and scale dependence of
the coupling as, asymptotic freedom, scaling
violation, jet production rate and shape. But to
fully understand QCD, especially the unique
feature of quark confinement, one needs to be
able to predict the spectrum of bound states that
it permits.
3

• Plan of the talk
• Approaches to calculations of bound quark states,
  and the role of the potential in these
• The role of the Bc meson family in mapping the
  strong potential
• The recent CDF precision measurement of the Bc
  mass
• Bcs at the LHC

4

• Some subtexts of this talk...
• Heres one perspective on why particle physicists
  keep looking for new particles, even though we
  already have 500. Its not stamp collecting! We
  will motivate the hunt for the Bc, the bound
  state of the and c quarks.
• Heres one way that heavy quarks (c, b, t), which
  do not compose the proton or neutron valence and
  may therefore appear to contribute little to the
  structure of the everyday world, can elucidate
  fundamental questions.
• Theres more to life than the Higgs. While the
  Fermilab Tevatrons and CERN Large Hadron
  Colliders programs to search for Higgs and
  Beyond the Standard Model exotics are very rich
  and well motivated, their opportunities for
  probing Standard Model processes are also
  unmatched.

5
A few numbers to remember about scales, masses,
and bound states in QCD...
6

• The coupling as runs with energy. A scale, or
  mass gap, ?QCD, characterizes the boundary
  between the perturbative and non-perturbative
  regimes. The value of ?QCD has been estimated to
  be in the range 200 MeV to 450 MeV.
• Compared to ?QCD, three quarks are light (1-100
  MeV), and three are heavy (1-174 GeV).
• A.C. Benvenuti, et al., Phys. Lett. B 223, 490
  (1989).
  G. Bodwin, et al., PRD 51, 3, 1125
  (1995).

7
The ability to predict the hadron spectrum is a
direct test of our understanding of the
confinement mechanism. QCD alone should be able
to describe the spectroscopy of bound states.
Recent breakthroughs have improved its
precision. Nonetheless it remains technically
challenging (especially for the heavy quarks) as
the theory must naturally describe phenomena at
multiple scales, perturbative and not. Lattice
calculation is difficult because the lattice
spacing must be small compared to 1/mQ but the
grid must be large compared to 1/mQv2, a large
number as the heavy quarks velocity is small.
Thus many approaches have been used to
complement lattice QCD. C.T.H. Davies et al.,
PRL 92, 022001 (2004) and references therein.

8

• Effective Field Theory (EFT) is an alternative to
  the lattice...a quantum field theory in which
  different scales are factorized, leaving adequate
  degrees of freedom to describe phenomena in a
  specific range. Typically an EFT has a potential
  which encodes the effect of degrees of freedom
  that have been integrated out from full QCD.
• EFTs can be classified by the trade-off they
  make between hypotheses required (i.e.,
  factorizations), and precision or range of
  applicability obtained.

9

• Theres been a convergence of results from
• pure QCD (lattice calculation)...While results
  are limited by computational power associated
  with lattice extent (large w.r.t. 1/mv2) and
  granularity (small w.r.t. 1/m), recent
  developments in discretization of light quarks
  (staggered quarks) now permit predictions with
  few-percent precision.
• Non-relativistic QCD...integrates out modes of
  energy and momentum of order mq and describes the
  dynamics of heavy quark pairs at energies much
  smaller than their masses.
• A nice review is given in Heavy Quarkonium
  Physics, Quarkonium Working Group,
  hep-ph/0412158.
• C.W. Bernard et al., Nucl. Phys Proc. Suppl.
  60A, 297 (1998) G.P. Lepage, Nucl. Phys. Proc.
  Suppl. 60A, 267 (1998) C.W. Bernard et al., PRD
  58, 014503 (1998) G.P. Lepage, PRD 59, 074502
  (1999), K. Orginos and D. Toussaint, Nucl. Phys.
  Proc. Suppl. 73, 909 (1999) K. Orginos et al.,
  PRD 60, 054503 (1999) C.W. Bernard et al., PRD
  61, 111502 (2000) K. Orginos and D. Toussaint,
  PRD 59, 014501 (1999).

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• also...
• Perturbative non-relativistic QCD...further
  integrates out phenomena at scale of momentum
  transfer (mv) relative to scale of kinetic energy
  (mv2).
• Phenomenological potential models...which often
  begin with
• so are most applicable to the heaviest, least
  relativistic, bound states.

11
So a reasonable experimental goal is to map the
strong potential. We know that the detailed
shape of a potential determines the energies at
which its states are bound.
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• Heavy quark bound states are key to elucidating
  the strong potential.
• b quark mass mb is so much heavier than ?QCD that
  a perturbative expansion in 1/mb is well
  motivated.
• The heavy quark and antiquark relative velocities
  v are much less than c, permitting a
  non-relativistic treatment... then in analogy
  with QED, expect splittings between states with
  the same quantum numbers to be of size mv2 and
  hyperfine splittings of size mv4.
• E. Eichten and C. Quigg, hep-ph/9402210.

13

• Analogizing from positronium to quarkonium...
• For positronium
• neglecting relativistic corrections, the scale of
  excitation energies is set by the Rydberg, R
  ½µa2
• energy levels given by principal quantum number
• Virial Thm applies to a
  spherically symmetric potential, so
• Thus for quarkonium
• photons?gluons and a?ascolor factors
• System is non-relativistic with velocity v as
  evaluated at the size of a bound state v
  as(1/r2) where r 1/mv
• For more details see R.K. Ellis et al., QCD
  and Collider Physics, Cambridge, 1996.

14

• However...
• while bound states of light quarks can be modeled
  by a perturbed Coulombic spectrum, the spectrum
  of and states (which being heavy probe
  closer to the non-perturbative regime) is known
  to be not Coulombic.

A non-relativistic Coulomb potential would not
split 2S and 1P. So generalize the potential...
15
At short distances, lowest order perturbation
theory gives a Coulomb-like potential for
one-gluon exchange but this does not include
confinement. Experimentally, production
typically occurs at an energy scale 1 GeV
(typical hadron mass) at a separation of 1 fm
(typical hadron size). So at long distances,
one-gluon exchange can be replaced by bunched
color flux tubes with linear energy density
s This gives the Cornell potential Phys.
Rev. D 17, 3090 (1978).
16

• Spin-independent features of spectroscopy
  have been shown to be described by this form.
  Other proposed spin-independent potentials tuned
  to match charmonium and bottomonium spectra
  include the
• Martin potential
• Logarithmic potential, produces mass-independent
  level spacings
• Phys. Lett. B 93, 338 (1980).
  
  Phys. Lett. B
  71, 153 (1977)

17

• Richardson potential, which assumes one-gluon
  exchange but explicitly incorporates the scale
• Buchmüller-Tye potential, which includes 2-loop
  running at small distances and interpolation
  between the limits of large and small r
• Phys. Lett. B 82, 272 (1979)
  
  Phys. Rev. D 24, 132 (1981)

18

• The QCD-inspired spin-dependent potential has
  been written down
• Transform QCD Lagrangian ? NRQCD Lagrangian.
• Write down the gauge-invariant Green
  function G(T) in the path integral
  representation. Insert a complete set of
  eigenstates with eigenenergies En.
• Make a Wick rotation and using the Feynman-Kac
  formula obtain the ground state energy
  E0(G(-iT)).
• For an infinitely heavy quark, G is a product
  involving a static Wilson loop
• Heavy quark kinetic energy ? 0, leaving potential
  
• N. Brambilla and A. Vairo, hep-ph/9904330
  
  B. Thacker and G. Lepage,
  PRD 43, 196 (1991).

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• To order 1/m2, the result is
• but the resulting full spectrum has not been
  calculated.
• Phys. Rev. D 63, 014023 (2001) Phys. Rev. D 63,
  054007 (2001) Phys. Rev. D 67, 034018 (2003)
  Phys. Rev. Lett. 88, 012003 (2002).

20

• Each proposed potential function leads to a
  hypothesized spectrum. For example from Godfrey
  and Isgur, Phys. Rev. D 32, 189 (1986)

21
What physical system is best to distinguish among
the models?
22
An excellent laboratory for mapping the strong
potential...the Bc system bound states of one
charm and one anti-bottom quark (or their
antiparticles)
23

• What makes Bc a good laboratory for comparing
  data to theory on the shape of the strong
  potential?
• modeling the binding of a two-body ( ) system
  is easier than modelling three bodies (qqq)---so
  start with a meson!
• The heavier the better, to suppress relativistic
  effects---but cannot form, because top
  quarks decay before binding.
• bind but decay rapidly (?t
  10-20-10-23 seconds) by annihilation...
• Due to the uncertainty principle, small ?t means
  resonance widths ?E
  are large. Wide states are harder to distinguish
  from background than narrow ones.

24

• The strong and electromagnetic forces conserve
  flavor, so the two flavors (b and c) of the Bc
  cannot annihilate via them. Bc must decay weakly
• Weak decays intrinsically take longer (?t 10-12
  sec) so Bc should be narrow.

p
J/?
Bc
Bs
Bc
p
Bc

Ds
25
We expect the energy levels of Bc and its excited
states to lie in the same range as the known
bound states of . Those
particles have splittings much smaller than their
quark masses... implying bound quark
velocities vcharm 0.5 in vbottom 0.3 in
...so the bound states are approximately
non-relativistic. Bc should be likewise
non-relativistic, simplifying the form that can
be used to describe its potential.
26
The Bc has a high mass...about 6 GeV...so it can
only be produced at the highest energy colliders.
Precision measurements of it and its excited
states Bc should in principal provide a map of
the strong potential. And the data?...
27

• Only about 130 events containing a ground-state
  Bc have been observed. The first observations
  were made at the Fermilab Tevatron through
  semileptonic decays
• via
• events (CDF, 4.8s significance)
• mass 6.4 0.39 0.13 GeV/c2
• Phys. Rev. Lett. 81, 2432 (1998)
• via
• 95 12 11 events (D0),
• mass
• DØNote 4539-Conf (2004)

28
The presence of the neutrino prevented full
reconstruction of events, leading to a relatively
large uncertainty on the Bc mass. A precision
mass measurement requires full reconstruction of
the decay, for example
29
The definitive mass measurement came from CDF in
November 2005 (hep-ex/0505076).
30

• Measurement of the Bc mass through the decay
  Bc ?J/? p
• Analysis relies on the very efficient J/??µµ-
  trigger which provides a high purity data sample.
• 360 pb-1 in at

• Silicon microstrip tracker (L00SVXISL) in 1.4
  T axial field
• Open-cell wire drift chamber (COT)
• muon chambers (CMUCMX) to ? lt 1.0

31

• Muon selection
• Require candidate tracks match in COT and CMU or
  CMX
• Select µµ- pairs with pT gt 1.5(2.0) GeV/c in
  CMU(CMX) to form J/? candidate with mass
  2.7 lt M(µµ-) lt
  4.0 GeV/c2.

32

• Reconstructing Bc?J/?p offline
• every track has r-f measurement on 3 SVX layers
• reconfirm COT - CMU/CMX track match
• 3.042 lt M(µµ-) lt 3.152 GeV/c2
• assign pion ID to every other charged track with
  pT gt 400 MeV/c
• Constrain M(µµ-) to world average for J/?, 3.096
  GeV/c2
• Fit J/? and p to common 3D vertex save all
  combinations for which fit converges
• Form primary vertex from remaining tracks

33

• The remaining cuts were selected in blind
  analysis mode to avoid bias. Data in the search
  mass window 5.6 lt M(Bc) lt
  7.2 GeV/c2
  were temporarily hidden by substituting
  a known 3-track invariant mass value. Window
  width is 2s about Bc mass obtained from CDF
  semileptonic search and 100x wider than expected
  mass resolution of 14 MeV/c2.
• Using Monte Carlo, vary cuts to maximize
• The 1.5 selects signals 3s above background
  fluctuation.
• G. Punzi, PHYSTAT2003 and arXivphysics/0308063.

34

• Cuts were developed for variables in fully
  simulated Monte Carlo events with
• Bc mass 6.4 GeV/c2
• Bc lifetime 0.46 ps
• theoretical pT spectrum, checked by a harder
  spectrum
• C.-H. Chang et al., hep-ph/0309120.
• A. V. Berezhnoy et al., Z. Phys A 356, 79 (1996).

35

• Variables used in selection
• 3-track 3-d vertex fit ?2 lt 9 (4 d.o.f.)
• pion contribution to the fit ?2p lt 2.6
• impact parameter of the Bc candidate lt 65 µm in
  r-f
• (ct)max lt 750 µm where t is Bc proper decay time
• pion pT gt 1.8 GeV/c
• 3-d angle between Bc candidate and vector from
  primary to secondary vertex lt 0.4 rad
• significance of the projected decay length of the
  Bc onto its transverse momentum direction,
  Lxy/s(Lxy) gt 4.4.
• 390 candidates remain.

36

• Validate cuts (minus ct) on control data sample
  B ? J/? K
• reconstruct 2378 57 B, with correct mass
  (5279.0 0.3 MeV/c2) including B ? J/? p
  contribution
• mass resolution 11.5 0.3 MeV/c2

37

• Predict Bc events from
• B yield
• trigger and recon efficiency in range 0.35-0.85,
  depending on Bc lifetime and pT spectrum
• 
  measured by CDF semileptonically
• theoretical calculations of
  BR(Bc ?J/?p)/ BR(Bc
  ?J/?l?)

38

• Expect 10-50 Bc events
• Scan search region in 10 MeV/c2 intervals with a
  sliding window from -100 MeV/c2 to 200 MeV/c2
  about each nominal peak. Asymmetric window
  minimizes contributions from partially
  reconstructed decays (see below). There are 131
  possible such windows.
• For each window fit Gaussianlinear bkg, Gaussian
  width linear in mass from 13 to 19 GeV/c2. Fit
  parameters are Signal, Background, and Bkg
  slope.
• For the data, measure

39

• Predict the Smax distribution for the null
  hypothesis. Use Monte Carlo background sample
  linear (combinatoric) physical (inclusive
  Bc?J/?X with BRs from theory).
• Smax near 6290 MeV/c2 with 19 6 events.
  Probability that this is a random fluctuation
  0.17.
• V.V. Kiselev, Phys. Atom.Nucl. 67, 1559 (2004).

40

• Scrutinize events in the signal region. Discover
  2 classes of unacceptable fitted tracks used as
  pions
• insufficient COT hits for good mass resolution,
  so incompatible with presumed narrow Gaussian
• poor SVX resolution in z-direction
• These are found to contribute 10 of signal but
  40 of the combinatorial bkg. Remove these
  events. 220 candidates remain. Demand good
  silicon z information on the pion and at least
  one muon.

41

• Perform an unbinned likelihood fit over the full
  mass range.
• 14.6 4.6 events are observed (probability of
  random fluctuation 0.012)
• Mass 6285.7 5.3 1.2 MeV/c2 (0.08
  uncertainty).
• The broad low-mass enhancement is real but
  partially reconstructed Bc decays.

42

• To confirm the broad peaks identity, note
• physical bkg pions (left side band LSB) should
  have small impact parameter dxy, but combinatoric
  pions (right side band RSB) should not.
  Strategy
• Relax cuts on impact parameter of the Bc
  candidate and on ?2 of the 3-d vertex fit, to
  make a signal in the dxy distribution rise above
  the combinatorics. Plot dxy.
• Subtract LSB-RSB of dxy.

• Result the curve for Bu data also describes well
  the pattern in the Bc.
• Low dxy excess (224 59 events) consistent with
  MC.

43

• Systematics
• measurement of track parameters (0.3 MeV/c2)
• momentum scale (0.6 MeV/c2)
• These are evaluated from the B control data.
• possible differences between B and Bc pT spectra
  (0.5 MeV/c2)
• fitting uncertainties knowledge of background
  shape and signal width (0.9 MeV/c2)

44
This precision measurement of the Bc mass
provides the baseline against which models of the
strong potential can be calibrated. But to map
the shape of the potential, we need to know what
other stationary states it supports, and we need
precision mass measurements on them. So we need
the excited states Bc too.
45
To see the excited states in substantial numbers,
we probably need more energy. The Large Hadron
Collider will provide proton-proton collisions at
center-of-mass energy 14 TeV (compare Tevatrons
2 TeV) beginning 2007.
46
And they really mean 2007!...
LHC
Construction and Installation Schedule
Ready June 2007
47
The new CERN control room...
48
The ATLAS and CMS Experiments will be there.
49

• A conservative estimate predicts 10,000
  events could be fully reconstructed
  from one years ATLAS data at luminosity 1033
  cm-2s-1 (integrated, 10 fb-1), assuming
• s( ) 500 µb ? 5 x 1012 pairs
• trigger muon pT gt 6 GeV/c and ? lt 1.6
• Prob(b ?Bc()) 10-3
• BR(Bc ? J/?p J/? ? µµ) 10-4
• Combined detection efficiency 1
• F. Albiol et al., ATLAS Note ATL-PHYS-94-058
  (1994).

50
A comparable number of Bc should be produced in
approximately 15 narrow bc states predicted to
lie below the BD flavor threshold (7.14
GeV). These are reconstructed through and
with electromagnetic decays expected to dominate
for all but the 2S levels. The challenge
efficient detection of a 72 MeV M1-photon in
coincidence with an observed Bc decay. This is
needed to distinguish Bc(2S) ? Bc(1S) pp from
Bc(2S) ? Bc(1S) pp . C. Quigg, Proc.
Snowmass 1993, 439.
51

• First steps Seeing the 1S, 2S and 2P levels at
  LHC...
• reconstruct the hadronic decays
• detect Bc?(455 MeV)
• detect ?(353 MeV), ?(382 MeV), ?(397 MeV) in
  coincidence with Bc?Bc?(72 MeV)
• This could be enough to definitively specify the
  strong potential.