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Heavy Quarks and the Strong Potential
Sally SeidelLos Alamos National Laboratory
3 May 2006
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 what’s observed, including the evolution and scale dependence of the coupling αs, 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.
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
Bc*’s at the LHC
Some subtexts of this talk...
1) Here’s one perspective on why particle physicists keep looking for new particles, even though we already have 500. It’s not stamp collecting! We will motivate the hunt for the Bc, the bound state of the and c quarks.
2) Here’s 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.
3) There’s more to life than the Higgs. While the Fermilab Tevatron’s and CERN Large Hadron Collider’s 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.
b
A few numbers to remember about scales, masses, and bound states in QCD...
The coupling αs runs with energy. A scale, or mass gap, ΛQCD, characterizes the boundary between the perturbative and non-perturbative regimes. The value of ΛQCD has been estimated to be in the range ~200 MeV* to 450 MeV§.
Compared to ΛQCD, 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).
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.
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.
EFT’s can be classified by the trade-off they make between # hypotheses required (i.e., factorizations), and precision or range of applicability obtained.
There’s 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).
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.
EVT )(
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.
Heavy quark bound states are key to elucidating the strong potential.†
b quark mass mb is so much heavier than ΛQCD 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.
Analogizing§ from positronium to “quarkonium”...
For positronium:
•neglecting relativistic corrections, the scale of excitation energies is set by the Rydberg, R = ½μα2
•energy levels given by principal quantum number:
•Virial Thm applies to a spherically symmetric potential, so
Thus for quarkonium:
•photons→gluons and α→αs·color factors
•System is non-relativistic with velocity v ~ αs evaluated at the size of a bound state: v ~ αs(1/r2) where r ~ 1/mv
§ For more details see R.K. Ellis et al., “QCD and Collider Physics,” Cambridge, 1996.
2n
REn
)Order( is so 22 21
2 vETv VkT 2
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.
bb cc
A non-relativistic Coulomb potential would not split 2S
and 1P.
So generalize the potential...
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 σ:
This gives the “Cornell potential”:
Phys. Rev. D 17, 3090 (1978).
r
rrV s )(
3
4)(
rrV )(
2GeV 18.0fm
GeV1
r
E
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)
)/log()( 0rrArV
)/()( 0rrArV
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)
)/1ln(
)/1ln(ln
625
462
)/1ln(
1
75
5321
)/1ln(
1
25
16)01.0(
22
22
2222 r
r
rrrfmrV E
)/1ln(
11
233
12
3
4)(
2222
qqnqV
f
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).
.)( 0)(
0
zAdzig
TrPeW
)(log1
lim)()( 00
WT
rErVT
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).
Each proposed potential function leads to a hypothesized spectrum. For example from Godfrey and Isgur, Phys. Rev. D 32, 189 (1986):
What physical system is best to distinguish among the models?
An excellent laboratory for mapping the strong potential...the Bc system: bound states of one charm and one anti-bottom quark (or their antiparticles):
bc
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 (Δt ~ 10-20-10-23 seconds) by annihilation...
Due to the uncertainty principle, small Δt means resonance widths ΔE are large. Wide states are harder to distinguish from background than narrow ones.
tt
ccbb and
ccbb and
•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 (Δt ~ 10-12 sec) so Bc should be narrow.
Bc{
Bc{
Bc{
}π
}J/ψ
}Bs
}π
}
}Ds
0D
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.
ccbb and
cc
bb
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?...
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, 4.8σ significance)
•mass 6.4 ± 0.39 ± 0.13 GeV/c2
Phys. Rev. Lett. 81, 2432 (1998)
via
•95 ± 12 ± 11 events (D0),
•massDØNote 4539-Conf (2004)
/JBc
/JBc
214.013.0 GeV/c 34.095.5
2.65.54.20
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
/ ,/ JJBc
The definitive mass measurement came from CDF in November 2005 (hep-ex/0505076).
Measurement of the Bc± mass through the decay Bc
± →J/ψ π±
•Analysis relies on the very efficient J/ψ→μ+μ- trigger which provides a high purity data sample.
•360 pb-1 in at TeV 96.1spp
•Silicon microstrip tracker (“L00+SVX+ISL”) in 1.4 T axial field:
•Open-cell wire drift chamber (“COT”)
•muon chambers (“CMU+CMX”) to |η| < 1.0
B
12 )GeV/c( 0015.0/)( TT pp
•Muon selection
•Require candidate tracks match in COT and CMU or CMX
•Select μ+μ- pairs with pT > 1.5(2.0) GeV/c in CMU(CMX) to form J/ψ candidate with mass
2.7 < M(μ+μ-) < 4.0 GeV/c2.
•Reconstructing Bc±→J/ψπ± offline
•every track has r-φ measurement on ≥3 SVX layers
•reconfirm COT - CMU/CMX track match
•3.042 < M(μ+μ-) < 3.152 GeV/c2
•assign pion ID to every other charged track with pT > 400 MeV/c
•Constrain M(μ+μ-) to world average for J/ψ, 3.096 GeV/c2
•Fit J/ψ and π to common 3D vertex; save all combinations for which fit converges
•Form primary vertex from remaining tracks
•The remaining cuts were selected in “blind analysis” mode to avoid bias. Data in the search mass window
5.6 < M(Bc) < 7.2 GeV/c2 were temporarily hidden by substituting a known 3-track invariant mass value. Window width is ±2σ 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 ≥3σ above background fluctuation.
*G. Punzi, PHYSTAT2003 and arXiv:physics/0308063.
. 5.1 av
F
B
S
•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).
•Variables used in selection:
•3-track 3-d vertex fit χ2 < 9 (4 d.o.f.)
•pion contribution to the fit χ2π < 2.6
•impact parameter of the Bc candidate < 65 μm in r-φ
•(ct)max < 750 μm where t is Bc proper decay time
•pion pT > 1.8 GeV/c
•3-d angle between Bc candidate and vector from primary to secondary vertex < 0.4 rad
•significance of the projected decay length of the Bc onto its transverse momentum direction, Lxy/σ(Lxy) > 4.4.
•390 candidates remain.
•Validate cuts (minus ct) on control data sample B± → J/ψ K±
•reconstruct 2378 ± 57 B±, with correct mass (5279.0 ± 0.3 MeV/c2) including B± → J/ψ π± contribution
•mass resolution 11.5 ± 0.3 MeV/c2
•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/ψπ±)/ BR(Bc± →J/ψℓ±ν)
)(/)( uc BBRBBR
•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 Gaussian+linear bkg, Gaussian width linear in mass from 13 to 19 GeV/c2. Fit parameters are #Signal, #Background, and Bkg slope.
•For the data, measure
av
F
B
S
5.1maxmax
• Predict the Σmax distribution for the null hypothesis. Use Monte Carlo background sample: linear (combinatoric) + physical (inclusive Bc→J/ψX with BR’s from theory†).
• Σmax 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).
• Scrutinize events in the signal region. Discover 2 classes of unacceptable fitted tracks used as pions:
(1) insufficient COT hits for good mass resolution, so incompatible with presumed narrow Gaussian
(2)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.
•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.
To confirm the broad peak’s 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 χ2 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.
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)
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.
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 Tevatron’s 2 TeV) beginning 2007.
And they really mean 2007!... LHC Construction and Installation Schedule
Ready June 2007
The new CERN control room...
The ATLAS and CMS Experiments will be there.
A conservative estimate† predicts 10,000 events could be fully reconstructed from one year’s ATLAS data at luminosity 1033 cm-2s-1 (integrated, 10 fb-1), assuming
•σ( ) = 500 μb → 5 x 1012 pairs
•trigger muon pT > 6 GeV/c and |η| < 1.6
•Prob(b →Bc(*)) ≈ 10-3
•BR(Bc → J/ψπ; J/ψ → μμ) ≈ 10-4
•Combined detection efficiency ~ 1%
†F. Albiol et al., ATLAS Note ATL-PHYS-94-058 (1994).
/JBc
bb
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) + ππ from Bc
*(2S) → Bc*(1S) + ππ
.
#C. Quigg, Proc. Snowmass 1993, 439.
cc BB * cc BB *
First steps: Seeing the 1S, 2S and 2P levels at LHC...
1) reconstruct the hadronic decays
2) detect Bc+γ(455 MeV)
3) detect γ(353 MeV), γ(382 MeV), γ(397 MeV) in coincidence with Bc
*→Bc+γ(72 MeV)
This could be enough to definitively specify the strong potential.
