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Effective Field Theory and Heavy Quark Physics

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Effective Field Theory and Heavy Quark Physics

Matthias Neubert

Cornell University

Sheldon Fest – May 20-21, 2006

- Have shared excitement of discovery

- Many discoveries:
- Bound states containing heavy quarks
- Oscillations of neutral B mesons
- Flavor-changing transitions between the 3rd and 1st generations (t→d, b→u)
- Penguin-mediated decays (B→Xsγ)
- CP violation in B mixing and decays
- …

- Precision physics:
- CKM elements (|Vcb| at 1.7%, |Vub| at 7%)
- Quark masses (mb at 1.2%)
- Fundamental nonperturbative parameters (Λ, μπ2, Isgur-Wise function, shape function, …)

- Implications for physics beyond the Standard Model
- Scales probed in flavor physics extend far beyond the reach of high-energy colliders (few - 103 TeV)
- New physics must be heavier than we’d like it to be, or it must be hidden very well
- Minimal flavor violation (and extensions thereof)
- In my view, flavor physics is one of the strongest arguments against low-scale SUSY

- Exploit that heavy-quark systems are always characterized by (at least) two largely different scales:
- Systematically separate short-distance physics (perturbative) from long-distance hadronic physics (nonperturbative)

mb» ΛQCD

- Main concept:
- 2 expansion parameters:
and

- 2 expansion parameters:

perturbative

nonperturbative

- Early days:
- Cn at tree-level or one-loop order
- Leading terms and 1/mb corrections

- State of the art:
- Cn at two- or three-loop order
- Include 1/mb2 or even 1/mb3 corrections

- Scale separation
- Lets us compute all perturbative effects
- Often simplifies residual nonperturbative interactions:
- Reduced matrix elements (Isgur-Wise functions, shape functions, heavy-quark parameters)
- Universality of hadronic physics (process independence)
- Applicability of lattice QCD

- Nonrelativistic QCD (NRQCD)
- bound states containing two heavy quarks (quarkonia, Bc, but also B physics on the lattice)

- Heavy-quark effective theory (HQET)
- bound states containing a single heavy quark (B physics at low recoil)

- Soft-collinear effective theory (SCET)
- processes containing soft and energetic partons, including heavy quarks (processes at large recoil)

- Roots of these concepts are rather old
- NRQCD a sibbling of NRQED (studied in 1980s)
- HQET based on symmetry arguments first written down in 1980
- SCET based on eikonal approximation familiar from soft bremsstrahlung in classical electro-dynamics

- Many EFTs for heavy-quark systems are built on concept of symmetries, which simplify the dynamics
- Technically, they result from simplifications of QCD Feynman rules (propagators and vertices) in certain limits, e.g.:

= igstavμ

= igstanμ

[see, e.g., G. Sterman, Nucl. Phys. B 281 (1987) 310]

- Absence of Dirac γμmatrices gives rise to spin symmetries of HQET and SCET
- Independence of hard scales mb, E (after scale separation) can be interpreted as a heavy-flavor symmetry (HQET), or large-recoil symmetry, boost symmetry, scaling (SCET)

- E. Shuryak, Phys. Lett. B 93 (1980) 134:
94 citations

“The masses of light (u,d,s) quarks are rather different, but still there is an approximate SU(3) and a more accurate isotopic SU(2) symmetry on their substitution. The reason for this is that these masses are too small to be important. Something similar occurs for heavy quarks (c,b,…), because their masses are too large on the usual hadronic scale. So, some symmetry for their substitution should exist between 0- and 1- mesons, Σ- and Λ-type baryons etc.”

- E. Shuryak, Nucl. Phys. B 198 (1982) 83:
290 citations

“In this limit hadrons with one heavy quark resemble the hydrogen atom with its fixed center, and many problems of current models of hadronic structure … are made trivial. Mesons made of one very heavy and one light quark are, in some sense, the simplest hadrons in which non-trivial QCD dynamics is essential, so study of them is of great importance. Of course, mesons made out of two very heavy quarks are simpler, but they are next to trivial.”

- N. Isgur and M.B. Wise, Phys. Lett. B 237 (1990) 527:
1387 citations

“The new symmetries allow us to obtain absolutely normalized model-independent predictions in the heavy-quark limit of all the form factors for the Q1→Q2 induced weak pseudoscalar to pseudoscalar and pseudoscalar to vector transitions in terms of a single universal function ξ(t) with ξ(0)=1.”

Symmetry relations and normalization of Isgur-Wise function can be used to extract |Vcb|

[MN, Phys. Lett. B 264 (1991) 455]

- Heavy-quark symmetries have many important ramifications in phenomenology
- In particular:
- Exclusive |Vcb| determination (spin-symmetry relations among different B→D* form factors)
- Form-factor relations for heavy-to-light transitions such as B→π,ρ and D→ π,ρ (flavor symmetry), which may help for exclusive |Vub| determination

- In spectroscopy, find relations between level spacings in bottom and charm systems
- E.g.:
- mB*2-mB2≈ 0.49 GeV2 vs. mD*2-mD2≈ 0.55 GeV2
- mB2*2-mB12≈ mD2*2-mD12 ≈ 0.17 GeV2
- mBs-mB≈ mDs-mD ≈ 0.10 GeV
- and many more …

- QCD factorization approach achieves scale separation at leading power in 1/mb for a large class of exclusive hadronic B decays (alternative: pQCD approach)
- Decay amplitudes predicted model-independently, including their normalization and strong-interaction phases

[Beneke, Buchalla, MN, Sachrajda (1999); Keum, Li, Sanda (2000)]

- In language of effective field theory, this is accomplished by matching QCD amplitudes onto matrix elements in SCET

[Bauer, Pirjol, Stewart (2000)]

QCD → SCETI → SCETII (HQET)

Extraction of γ:

B→PV modes have smaller penguins than B→PP modes

Smaller uncertainties when γ is extracted from time-dependent rates in B→πρ decays

Result:

B→πρ

Old data

New data

B→ ππ

Old data

γ= (62 ± 8)o

New data

[Beneke, MN (2003)]

At leading power in ΛQCD/mb, the B→Xsγ decay rate factorizes into a convolution of three objects:

[MN (1993); Bigi, Shifman, Uraltsev (1993); Korchemsky, Sterman (1994)]

- Master formula for the partial rate:

Γ ~ H(μh) * U(μh,μi) * J(μi) * U(μi,μ0) * M(μ0)

QCD → SCET → RG evolution → HQET → RG evolution → Local OPE

μh ~ mb

μi ~√mbΔ

μ0 ~ Δ=mb-2E0

Perturbation theory

Hadronic physics

- Master formula for the partial rate:

[MN (2004)]

μh ~ mb

μi ~√mbΔ

μ0 ~ Δ=mb-2E0

Perturbation theory

Hadronic physics

- After resummation, largest uncertainties are probed by variations of the low scale μ0~Δ≈1 GeV:

NLO

[Becher, MN (in preparation)]

NNLO

F(1.8 GeV) / default

μ0/Δ

- Generic problem in QCD:
- Resummation for processes with multiple scales
- Interplay of soft and collinear emissions → Sudakov double logarithms
- Jet physics: MX2 « Q2
- Soft: low momentum pμ→0
- Collinear: p || pX with p2→0

- Examples: DIS, fragmentation, Drell-Yan, Higgs production, event shapes, inclusive B decays, …

MX

- QCD factorization formula for DIS (x→1):
- Most transparent to derive this in SCET: need hard-collinear,anti-collinear, and soft-collinear modes (called “soft” in the literature)
- Resum threshold logarithms by solving RGEs of SCET in momentum space

[Manohar (2003); Becher, MN, Pecjak (in preparation)]

- Exact momentum-space formula for resummed structure function F2(x,Q2):
- Vastly simpler than existing results
- Free of spurious Landau poles

[Becher, MN (2006)]

- We love heavy-quark physics because …
- it probes a sector of particle physics hiding great mysteries
- over the past 25 years, it has offered us lots of real data
- it has been characterized by close collaborations between experimenters and theorists
- theoretical work is crucial to experimental success

- Look forward to continuing these studies in LHC era, when particle physics moves back to Europe
- Sheldon will continue to be part on it …

LHCb week in Barcelona (September 2005)