Post on 20-Oct-2021
transcript
Twisted Boundary ConditionsJonathan Flynn
Tsukuba LQCD&PP 15 Dec 2004 1/17
Contents
θ-BC: twisted boundary conditions
TChPT: twisted chiral perturbation theory
Partial twisting
Applications?
Tsukuba LQCD&PP 15 Dec 2004 2/17
Some recent work
N-N phase shifts PF Bedaque nucl-th/0402051
Pseudoscalar mesondispersion relation(quenched)
GM de Divitiis,R Petronzio andN Tantalo
hep-lat/0405002
θ-BC and two-particlestates
GM de Divitiisand N Tantalo
hep-lat/0409154
ChPT analysis CT Sachrajdaand G Villadoro
hep-lat/0411033
Tsukuba LQCD&PP 15 Dec 2004 3/17
Boundary Conditions
PBC: lattice momenta quantised
pi =2π
Lni
lowest non-zero momentum is quite large, big gaps
non-periodic or twisted spatial boundary conditions:allow continuously variable offset in the comb of allowedthree-momenta
Tsukuba LQCD&PP 15 Dec 2004 4/17
Twisted BC in QCD
Lq = q̄(x)( /D +M)q(x)
observables should besingle-valued: OK ifaction is single-valued ona torus
⇒ field satisfies
ψ(x + eiL) = Uiψ(x)
for i = 1, 2, 3, where Ui
is a symmetry of theaction
for general diagonal M,Ui should be diagonal(CSA of U(3))
Ui = exp(iΘi)
allowed momenta:
pi =2πni
L+θi
L
Tsukuba LQCD&PP 15 Dec 2004 5/17
Twisted BC: 2
change variable:
q̃(x) = e−iΘ·xq(x) (Θ0 = 0)
q̃ satisfies PBC
Lagrangian:
Lq = ¯̃q(x)( /̃D +M)q̃(x)
with
D̃µ = Dµ + iBµ, Bi = Θi/L, B0 = 0
Tsukuba LQCD&PP 15 Dec 2004 6/17
Twisted BC: 3
propagator encodes shift: S(x, y)→ S̃(x, y)
S̃(x) = 〈q̃(x)¯̃q(0)〉 =∫
dk0
2π
1
L3
∑
k
eik·x
i(/k + /B) +M
sum over k = 2πn/L
momentum in denominator is shifted by θ/L
Tsukuba LQCD&PP 15 Dec 2004 7/17
Twisted BC on the Lattice
change of variable modifies the lattice covariantderivatives:
∇Θµψ(x) = eiΘµ/LUµ(x)ψ(x + µ̂) − ψ(x)
∇Θ ∗µ ψ(x) = ψ(x) − e−iΘµ/LU†µ(x − µ̂)ψ(x − µ̂)
Tsukuba LQCD&PP 15 Dec 2004 8/17
Twisted BC on the Lattice
change of variable modifies the lattice covariantderivatives:
∇Θµψ(x) = eiΘµ/LUµ(x)ψ(x + µ̂) − ψ(x)
∇Θ ∗µ ψ(x) = ψ(x) − e−iΘµ/LU†µ(x − µ̂)ψ(x − µ̂)
inverting the modified operator encodes the momentumshift Θ/L in the calculated propagator
Tsukuba LQCD&PP 15 Dec 2004 8/17
Twisted BC on the Lattice
change of variable modifies the lattice covariantderivatives:
∇Θµψ(x) = eiΘµ/LUµ(x)ψ(x + µ̂) − ψ(x)
∇Θ ∗µ ψ(x) = ψ(x) − e−iΘµ/LU†µ(x − µ̂)ψ(x − µ̂)
inverting the modified operator encodes the momentumshift Θ/L in the calculated propagator
hadron momentum shifted by sum of quark shifts
dDPT: quenched study of pseudoscalar mesondispersion relation
SV: ChPT analysis
Tsukuba LQCD&PP 15 Dec 2004 8/17
SV Chiral PT Analysis
Exponential suppression of finite-volume correctionsfrom θ-BC for quantities without FSI (masses, decayconstants, semileptonic FF’s)
Tsukuba LQCD&PP 15 Dec 2004 9/17
SV Chiral PT Analysis
Exponential suppression of finite-volume correctionsfrom θ-BC for quantities without FSI (masses, decayconstants, semileptonic FF’s)
Not possible in general to extract matrix elements usingθ-BC for amplitudes involving FSI (eg K→ ππ)
Tsukuba LQCD&PP 15 Dec 2004 9/17
SV Chiral PT Analysis
Exponential suppression of finite-volume correctionsfrom θ-BC for quantities without FSI (masses, decayconstants, semileptonic FF’s)
Not possible in general to extract matrix elements usingθ-BC for amplitudes involving FSI (eg K→ ππ)
The above remain true for ‘partial twisting’: θ-BC forvalence, PBC for sea
Tsukuba LQCD&PP 15 Dec 2004 9/17
SV Chiral PT Analysis
Exponential suppression of finite-volume correctionsfrom θ-BC for quantities without FSI (masses, decayconstants, semileptonic FF’s)
Not possible in general to extract matrix elements usingθ-BC for amplitudes involving FSI (eg K→ ππ)
The above remain true for ‘partial twisting’: θ-BC forvalence, PBC for sea
They construct effective Lagrangian in presence of θ-BC.
Tsukuba LQCD&PP 15 Dec 2004 9/17
Twisted ChPT
Twisted BC:
Σ(x + eiL) = UiΣ(x)U†i
Redefine fields:
Σ̃(x) = e−iΘ·x/LΣ(x)eiΘ·x/L
Tsukuba LQCD&PP 15 Dec 2004 10/17
Twisted ChPT
Twisted BC:
Σ(x + eiL) = UiΣ(x)U†i
Redefine fields:
Σ̃(x) = e−iΘ·x/LΣ(x)eiΘ·x/L
to get
LChPT =f 2
8〈D̃µΣ̃†D̃µΣ̃〉 −
f 2
8〈Σ̃χ† + χΣ̃†〉
whereD̃µΣ̃ = ∂µ + i[Bµ, Σ̃]
Tsukuba LQCD&PP 15 Dec 2004 10/17
Twisted ChPT: 2
Standard ChPT Lagrangian coupled to vector field Bµ.
Effect of twist on mesons found from [Bi, Σ̃]:
Tsukuba LQCD&PP 15 Dec 2004 11/17
Twisted ChPT: 2
Standard ChPT Lagrangian coupled to vector field Bµ.
Effect of twist on mesons found from [Bi, Σ̃]:
neutral mesons:
[Bi, π0] = 0 −→ no shift
Tsukuba LQCD&PP 15 Dec 2004 11/17
Twisted ChPT: 2
Standard ChPT Lagrangian coupled to vector field Bµ.
Effect of twist on mesons found from [Bi, Σ̃]:
neutral mesons:
[Bi, π0] = 0 −→ no shift
charged mesons shifted by difference of the twists of thetwo valence quarks
[Bi, π±] = ±(θui − θdi)
Lπ±
Tsukuba LQCD&PP 15 Dec 2004 11/17
Twisted ChPT: 2
Standard ChPT Lagrangian coupled to vector field Bµ.
Effect of twist on mesons found from [Bi, Σ̃]:
neutral mesons:
[Bi, π0] = 0 −→ no shift
charged mesons shifted by difference of the twists of thetwo valence quarks
[Bi, π±] = ±(θui − θdi)
Lπ±
allowed values of meson momenta shifted in externalstates and in propagators
Tsukuba LQCD&PP 15 Dec 2004 11/17
dDPT Quenched Study
fixed volume L3T withL = 3.2r0 ≈ 1.6 fm
163 × 32
243 × 48
323 × 64
O(a) improved
4 quark masses
invert for each of
|θ| = 0,√
3, 2√
3, 3√
3
calculate pseudoscalarmeson correlator withone quark twisted
Expected mesonmomentum
|p| = |θ|L=
0.000 GeV0.217 GeV0.433 GeV0.650 GeV
cf. 2π/L ≈ 0.785 GeV
Tsukuba LQCD&PP 15 Dec 2004 12/17
dDPT: 2
extract effective energies
interpolate to fixedphysical quark masses
extrapolate a→ 0 atfixed quark mass, fixed L
Tsukuba LQCD&PP 15 Dec 2004 13/17
dDPT: 2
extract effective energies
interpolate to fixedphysical quark masses
extrapolate a→ 0 atfixed quark mass, fixed L
relativistic dispersionrelation
E2i j =M2
i j + |θ|2/L2
is well satisfied
Tsukuba LQCD&PP 15 Dec 2004 13/17
dDPT: 3
Dispersion relation test
2
3
4
5
6
7
8
9
0 0.5 1 1.5 2 2.5 3
(r0
E)2
(r0 |θ|/L)2
Tsukuba LQCD&PP 15 Dec 2004 14/17
Partial Twisting
ChPT analysis above was in full QCD
Do you have to twist the sea quarks by the sameamount as the valence quarks?
Tsukuba LQCD&PP 15 Dec 2004 15/17
Partial Twisting
ChPT analysis above was in full QCD
Do you have to twist the sea quarks by the sameamount as the valence quarks?
SV say ‘Not always’
extend analysis to a partially twisted chiral Lagrangiancorresponding to Nv +Ns quarks with Nv ghostquarks
for processes with at most one hadron in externalstates and where shift does not introduce cuts in thecorrelator
Tsukuba LQCD&PP 15 Dec 2004 15/17
Partial Twisting: 2
Example from SV: fK± for untwisted d and s quarks:
fK(L) − fK(∞)
fK(∞)
= −m2π
f 2π
e−mπL
(2πmπL)3/2
94 u untwisted(1
2
∑
i cosθi +34
)
u fully twisted(∑
i cosθi − 34
)
u partially twisted
Tsukuba LQCD&PP 15 Dec 2004 16/17
Applications?
Numerical tests with twisted valence quarks onuntwisted sea quarks: meson dispersion relations,meson decay constants with different meson momenta
Tsukuba LQCD&PP 15 Dec 2004 17/17
Applications?
Numerical tests with twisted valence quarks onuntwisted sea quarks: meson dispersion relations,meson decay constants with different meson momenta
Heavy-to-light semileptonic decays (eg: D→ πlν)
at fixed quark masses: map out full q2 range
chiral extrapolation: generate points at fixedE = v · pπ for different light quarks and avoid fittingto a model FF
Tsukuba LQCD&PP 15 Dec 2004 17/17