text: 2 distribution q(x) the b coefﬁcients in Eq. 3 are replaced 0,2,2ν by other coefﬁcients c involving integrals of the form 0,2,2ν factor q(x) (see Appendix A). If we approximate the charge 0.08 density of the nucleus as a uniform sphere the resulting form 0.07 (cid:20) (cid:21) factor takes this form: 0.06 9 sin x − 0.05 q(x) = cos x (4) x x 0.04 0.03 As a further simpliﬁcation we can approximate q(x) with a Gaussian function q = e−ax2. The coefﬁcients c can then 0.02 0,2,2ν be expressed analitically as simple functions of Z, a and the 0.01 nuclear radius (Appendix A). Unfortunately also this method 0 faces numerical limitations that we will discuss in Sect. . It 0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.00p0a8th 0le.0n0g0th9 (m0m.0)01 will anyway be used for small scattering angles as a validation for the Monte Carlo approach which we are about to describe Fig. 1. Distribution of the step length for the GEANT4 mixed simulation for in Sect. . 2 GeV/c muons over 1 mm of Lead. III. MONTE CARLO SIMULATION OF MULTIPLE SCATTERING displacement after a certain path length are mainly due to hard collisions, a mixed procedure (with a suitably Multiple Coulomb scattering can be simulated numerically using different approaches: small value of θS) yields spatial and angular distributions that are more accurate than those from a condensed • detailed simulations: each track is simulated at micro- simulation. scopic level as a succession of connected straight seg- ments (free-ﬂights), Changes of direction are determined Since we are interested in rare high-q events a mixed algorithm approach has been preferred. Furthermore in this by sampling the polar deﬂection cos θ from the single scheme the inclusion of nuclear effects can be easily taken scattering distribution, and the azimuthal scattering angle into account when sampling the angular distribution at single- uniformly in [0, 2π]. The length t of the free-ﬂight is sam- − − scattering level. pled from the exponential distribution p(t) = λ 1e t/λ. The simulated angular and spatial displacements are “exact”, i.e. they coincide with those obtained from a IV. THE GEANT4 MONTE CARLO IMPLEMENTATION rigorous solution of the problem (i.e. by mean of the The version of GEANT4 used for this study is 4.9.6.P02 transport equation introduced in the theories of Goudsmit- with the standardSS physics list option to enable the treat- Saudersons [3], [4] and Lewis [5]). Nevertheless detailed ment of single scattering above a certain angular threshold1. simulations are in practice not possible computationally The simulation of single-scattering is dealt with by the routine except for speciﬁc cases involving very low energies and G4WentzelOKandVIxSection.cc. The distribution of thicknesses. the step length used by the GEANT4 simulation is shown • condensed simulations: the global effects of the col- in Fig. 1 for 2 GeV/c muons on 1 mm thick lead target. The lisions during a macroscopic step are simulated us- slope of the exponential distribution corresponds to a value of ing approximations. Usually the theories of Goudsmit- λ = 0.14 µm such that the single-scattering code is invoked (h) Saundersons [3], [4], Molie`re [7], [8] and Lewis [5] are on average about 7000 times for each muon crossing the 1mm used to predict the angular and spatial distributions after thick lead target. each step. At energies where condensed algorithms are The event sampling is done according to the model by required, the great majority of elastic collisions are “soft” Wentzel [9] in the formulation of [10]. The model adopts a collisions with very small deﬂections. Especially the spa- modiﬁed Coulomb potential to take into account the effect of − tial distributions might present signiﬁcant dependencies the screening of the electron cloud: V (r) = (Ze2/r)e r/R. on the choice of the macroscopic step which has to be A slightly modiﬁed version of the Molie`re screening angle (cid:48) carefully studied. χ = ( h¯ )2 is used whose Z and energy dependence is • mixed algorithms: Hard collisions, with scattering angle paarametr2ispeRd as (cfr(cid:32). [10], eq. 23)(cid:18): (cid:19) (cid:114) (cid:33) θ larger than a given value θ , are individually simulated S and soft collisions (with θ < θS) are described by (cid:48) α2a2Z2/3 αZ 2 τ means of a multiple scattering approach. It is clear that, χ = 0 1.13 + 3.76 (5) a p2 β τ + Z2/3 by selecting a conveniently large value for the cutoff angle θ , the number of hard collisions per electron S a being the Bohr radius and τ = E /m. track can be made small enough to allow their detailed 0 − kin (cid:82) If we deﬁne µ = 1(1 cos θ) the differential cross section simulation. The value of the mean free path between the 2 ∼ − (cid:48) − dσ/dµ in presence of screening is proportional to p 2(χ + simulation of hard collisions, λ , is given by λ 1 = a (h) (h) 2πN π σ(θ) sin θdθ. As the ﬂuctuations in the spatial 1“.mac” ﬁle setting: /phys/addPhysics standardSS. θS ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: distribution

text: q(x)

text: the

text: b

text: coefﬁcients

text: in

text: Eq.

text: are

text: replaced

text: by

text: other

text: coefﬁcients

text: c

text: involving

text: integrals

text: of

text: the

text: form

text: factor

text: q(x)

text: (see

text: Appendix

text: A).

text: If

text: we

text: approximate

text: the

text: charge

text: density

text: of

text: the

text: nucleus

text: as

text: a

text: uniform

text: sphere

text: the

text: resulting

text: form

text: factor

text: takes

text: this

text: form:

text: sin

text: x

text: q(x)

text: cos

text: x

text: x

text: x

text: As

text: a

text: further

text: simpliﬁcation

text: we

text: can

text: approximate

text: q(x)

text: with

text: a

text: Gaussian

text: function

text: q

text: The

text: coefﬁcients

text: c

text: can

text: then

text: be

text: expressed

text: analitically

text: as

text: simple

text: functions

text: of

text: Z,

text: a

text: and

text: the

text: nuclear

text: radius

text: (Appendix

text: A).

text: Unfortunately

text: also

text: this

text: method

text: faces

text: numerical

text: limitations

text: that

text: we

text: will

text: discuss

text: in

text: Sect.

text: It

text: will

text: anyway

text: be

text: used

text: for

text: small

text: scattering

text: angles

text: as

text: a

text: validation

text: for

text: the

text: Monte

text: Carlo

text: approach

text: which

text: we

text: are

text: about

text: to

text: describe

text: Fig.

text: Distribution

text: of

text: the

text: step

text: length

text: for

text: the

text: mixed

text: simulation

text: for

text: in

text: Sect.

text: GeV/c

text: muons

text: over

text: mm

text: of

text: Lead.

text: III.

text: MONTE

text: CARLO

text: SIMULATION

text: OF

text: MULTIPLE

text: SCATTERING

text: displacement

text: after

text: a

text: certain

text: path

text: length

text: are

text: mainly

text: due

text: to

text: hard

text: collisions,

text: a

text: mixed

text: procedure

text: (with

text: a

text: suitably

text: Multiple

text: Coulomb

text: scattering

text: can

text: be

text: simulated

text: numerically

text: using

text: different

text: approaches:

text: small

text: value

text: of

text: θS)

text: yields

text: spatial

text: and

text: angular

text: distributions

text: that

text: are

text: more

text: accurate

text: than

text: those

text: from

text: a

text: condensed

text: detailed

text: simulations:

text: each

text: track

text: is

text: simulated

text: at

text: micro-

text: simulation.

text: scopic

text: level

text: as

text: a

text: succession

text: of

text: connected

text: straight

text: seg-

text: ments

text: (free-ﬂights),

text: Changes

text: of

text: direction

text: are

text: determined

text: Since

text: we

text: are

text: interested

text: in

text: rare

text: high-q

text: events

text: a

text: mixed

text: algorithm

text: approach

text: has

text: been

text: preferred.

text: Furthermore

text: in

text: this

text: by

text: sampling

text: the

text: polar

text: deﬂection

text: cos

text: θ

text: from

text: the

text: single

text: scheme

text: the

text: inclusion

text: of

text: nuclear

text: effects

text: can

text: be

text: easily

text: taken

text: scattering

text: distribution,

text: and

text: the

text: azimuthal

text: scattering

text: angle

text: into

text: account

text: when

text: sampling

text: the

text: angular

text: distribution

text: at

text: single-

text: uniformly

text: in

text: The

text: length

text: t

text: of

text: the

text: free-ﬂight

text: is

text: sam-

text: scattering

text: level.

text: pled

text: from

text: the

text: exponential

text: distribution

text: p(t)

text: λ

text: t/λ.

text: The

text: simulated

text: angular

text: and

text: spatial

text: displacements

text: are

text: “exact”,

text: i.e.

text: they

text: coincide

text: with

text: those

text: obtained

text: from

text: a

text: IV.

text: THE

text: MONTE

text: CARLO

text: IMPLEMENTATION

text: rigorous

text: solution

text: of

text: the

text: problem

text: (i.e.

text: by

text: mean

text: of

text: the

text: The

text: version

text: of

text: used

text: for

text: this

text: study

text: is

text: transport

text: equation

text: introduced

text: in

text: the

text: theories

text: of

text: Goudsmit-

text: with

text: the

text: standardSS

text: physics

text: list

text: option

text: to

text: enable

text: the

text: treat-

text: Saudersons

text: and

text: Lewis

text: Nevertheless

text: detailed

text: ment

text: of

text: single

text: scattering

text: above

text: a

text: certain

text: angular

text: simulations

text: are

text: in

text: practice

text: not

text: possible

text: computationally

text: The

text: simulation

text: of

text: single-scattering

text: is

text: dealt

text: with

text: by

text: the

text: routine

text: except

text: for

text: speciﬁc

text: cases

text: involving

text: very

text: low

text: energies

text: and

text: The

text: distribution

text: of

text: thicknesses.

text: the

text: step

text: length

text: used

text: by

text: the

text: simulation

text: is

text: shown

text: condensed

text: simulations:

text: the

text: global

text: effects

text: of

text: the

text: col-

text: in

text: Fig.

text: for

text: GeV/c

text: muons

text: on

text: mm

text: thick

text: lead

text: target.

text: The

text: lisions

text: during

text: a

text: macroscopic

text: step

text: are

text: simulated

text: us-

text: slope

text: of

text: the

text: exponential

text: distribution

text: corresponds

text: to

text: a

text: value

text: of

text: ing

text: approximations.

text: Usually

text: the

text: theories

text: of

text: Goudsmit-

text: λ

text: µm

text: such

text: that

text: the

text: single-scattering

text: code

text: is

text: invoked

text: (h)

text: Saundersons

text: Molie`re

text: and

text: Lewis

text: are

text: on

text: average

text: about

text: times

text: for

text: each

text: muon

text: crossing

text: the

text: used

text: to

text: predict

text: the

text: angular

text: and

text: spatial

text: distributions

text: after

text: thick

text: lead

text: target.

text: each

text: step.

text: At

text: energies

text: where

text: condensed

text: algorithms

text: are

text: The

text: event

text: sampling

text: is

text: done

text: according

text: to

text: the

text: model

text: by

text: required,

text: the

text: great

text: majority

text: of

text: elastic

text: collisions

text: are

text: “soft”

text: Wentzel

text: in

text: the

text: formulation

text: of

text: The

text: model

text: adopts

text: a

text: collisions

text: with

text: very

text: small

text: deﬂections.

text: Especially

text: the

text: spa-

text: modiﬁed

text: Coulomb

text: potential

text: to

text: take

text: into

text: account

text: the

text: effect

text: of

text: tial

text: distributions

text: might

text: present

text: signiﬁcant

text: dependencies

text: the

text: screening

text: of

text: the

text: electron

text: cloud:

text: V

text: (r)

text: r/R.

text: on

text: the

text: choice

text: of

text: the

text: macroscopic

text: step

text: which

text: has

text: to

text: be

text: A

text: slightly

text: modiﬁed

text: version

text: of

text: the

text: Molie`re

text: screening

text: angle

text: carefully

text: studied.

text: χ

text: h¯

text: is

text: used

text: whose

text: Z

text: and

text: energy

text: dependence

text: is

text: mixed

text: algorithms:

text: Hard

text: collisions,

text: with

text: scattering

text: angle

text: as

text: eq.

text: θ

text: larger

text: than

text: a

text: given

text: value

text: θ

text: are

text: individually

text: simulated

text: S

text: and

text: soft

text: collisions

text: (with

text: θ

text: θS)

text: are

text: described

text: by

text: αZ

text: τ

text: means

text: of

text: a

text: multiple

text: scattering

text: approach.

text: It

text: is

text: clear

text: that,

text: χ

text: a

text: β

text: τ

text: by

text: selecting

text: a

text: conveniently

text: large

text: value

text: for

text: the

text: cutoff

text: angle

text: θ

text: the

text: number

text: of

text: hard

text: collisions

text: per

text: electron

text: S

text: a

text: being

text: the

text: Bohr

text: radius

text: and

text: τ

text: E

text: /m.

text: track

text: can

text: be

text: made

text: small

text: enough

text: to

text: allow

text: their

text: detailed

text: kin

text: If

text: we

text: deﬁne

text: µ

text: cos

text: θ)

text: the

text: differential

text: cross

text: section

text: simulation.

text: The

text: value

text: of

text: the

text: mean

text: free

text: path

text: between

text: the

text: dσ/dµ

text: in

text: presence

text: of

text: screening

text: is

text: proportional

text: to

text: p

text: simulation

text: of

text: hard

text: collisions,

text: λ

text: is

text: given

text: by

text: λ

text: a

text: (h)

text: (h)

text: π

text: σ(θ)

text: sin

text: θdθ.

text: As

text: the

text: ﬂuctuations

text: in

text: the

text: spatial

text: ﬁle

text: setting:

text: standardSS.

text: θS

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: darm/stnuoc darm/stnuoc 5 G4 S.W. FF G4 S.W. FF 102 G4 dipole FF 103 G4 dipole FF FF = 1 FF = 1 data 11.7 GeV/c data 7.3 GeV/c 102 10 10 1 1 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 q q (mrad) (mrad) Fig. 6. Comparison of the Monte Carlo simulation to the 11.7 GeV/c data-set. Fig. 7. Comparison of the Monte Carlo simulation to the 7.3 GeV/c data-set. Data have been obtained by digitisation from the original paper[14]. Data have been obtained by digitisation from the original paper[14]. parison with the simulation is performed for θ > 5 mrad since target and an upstream Cherenkov counter the effective pion in this region the background contribution can be neglected. contamination was reduced to 4.9×10−6. Emulsion detectors Real data and Monte Carlo events are normalised to each other downstream of the target were also employed to control the above this threshold. The results are shown in Fig. 6 and Fig. 7 pion contamination. for 11.7 GeV/c and 7.3 GeV/c muons respectively. In this case The incoming and outgoing muon directions were deter- the same procedure used for Lead was employed except for mined by using four identical counter hodoscopes made of the fact that the Saxon-Wood parametrisation of the Copper scintillator bars at positions A , A , A and A (Fig. nucleus charge density was used. Experimental data are shown A B C D 3 of [15]). The acceptance of such an arrangement was by the bullets with error bands. The Monte Carlo prediction ◦ between 2 and 14 . Each station was composed of 20 vertical is shown with three options: 1) a point-like nucleus prediction | | scintillators each having dimensions (0.95, 2.54, 15.24) cm, ( F(q) 2 = 1) (gray). 2) the dipole nuclear form factor (red) 3) the smallest dimension being the one in the horizontal plane. the improved form-factor of Eq. 8 (yellow). In both cases the The downstream stations centers were shifted away from the mixed algorithm described in has been used. The band width beam direction by 26.2 cm. The distance along the beam in Monte Carlo histograms represent the statistical uncertainty. between the stations was 2.794 m (AB) and 2.007 m (CD). This meaning of symbols is also preserved in the following. The beam spread in the horizontal and vertical directions The disagreement with the point-like simulation indicates that was determined triggering with a small scintillator (S) located the nuclear density is in this case actually probed. Furthermore along the beam axis in between stations A and A , rotating a better description is obtained by using the improved form- ◦ C D the A and A stations by 90 . The coincidence rate between factor with respect to the dipole one. A B strips in scintillators in A and A is shown in Fig. 6 of B. Lead data with muons (Masek et al.) A B the original paper. This bidimensional rate map was used to The experiment[15] was performed in 1961 proﬁting of generate the incoming direction of muons in the simulation the Bevatron accelerator at the Lawrence radiation labora- accordingly. The vertical spread was generated assuming a ± ◦ tory employing a muon beam with a median momentum of gaussian distribution with a 0.6 r.m.s. This is supported by ± ± ◦ ± ◦ (2.00 0.03) GeV/c and a 3.5% spread. The total number the paper which states a spread of 0.8 and 0.6 in the × of muons incident on the apparatus was 2.5 107. The horizontal and vertical views respectively. A gaussian ﬁt of lead target had a thickness of 14.4 g/cm2 corresponding to the θ distribution obtained from the information contained x 1.268 cm (one half inch). Scattered particles were observed in the hit strip bidimensional distribution (Fig. 6 of [15]) and up to 12◦ (p (cid:39) 400 MeV/c). The muon beam is obtained the detector geometry gives a spread of about 0.6◦ not too far ±T ◦ from a (3.5 0.3) GeV/c pion beam with magnetic selection from the 0.8 quoted in the paper. (see Fig. 3 of the original paper). 2 GeV/c muons are those We have reproduced with a simple simulation both the decaying backward in the π rest frame. This large difference angular spread of the beam and the ﬁnite-solid angle effects in momentum allows a clean π/µ separation. The pion con- introduced by the strip width. This was performed by binning tamination on the target is estimated to be of about 3%. By the true positions according to the detector geometry described using a 107 cm (42”) thick iron absorber downstream of the above. A simple linear trajectory has been assumed in between ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: darm/stnuoc

label: paragraph

font: default

text: darm/stnuoc

text: S.W.

text: FF

text: S.W.

text: FF

text: dipole

text: FF

text: dipole

text: FF

text: FF

text: FF

text: data

text: GeV/c

text: data

text: GeV/c

text: q

text: q

text: (mrad)

text: (mrad)

text: Fig.

text: Comparison

text: of

text: the

text: Monte

text: Carlo

text: simulation

text: to

text: the

text: GeV/c

text: data-set.

text: Fig.

text: Comparison

text: of

text: the

text: Monte

text: Carlo

text: simulation

text: to

text: the

text: GeV/c

text: data-set.

text: Data

text: have

text: been

text: obtained

text: by

text: digitisation

text: from

text: the

text: original

text: Data

text: have

text: been

text: obtained

text: by

text: digitisation

text: from

text: the

text: original

text: parison

text: with

text: the

text: simulation

text: is

text: performed

text: for

text: θ

text: mrad

text: since

text: target

text: and

text: an

text: upstream

text: Cherenkov

text: counter

text: the

text: effective

text: pion

text: in

text: this

text: region

text: the

text: background

text: contribution

text: can

text: be

text: neglected.

text: contamination

text: was

text: reduced

text: to

text: Emulsion

text: detectors

text: Real

text: data

text: and

text: Monte

text: Carlo

text: events

text: are

text: normalised

text: to

text: each

text: other

text: downstream

text: of

text: the

text: target

text: were

text: also

text: employed

text: to

text: control

text: the

text: above

text: this

text: threshold.

text: The

text: results

text: are

text: shown

text: in

text: Fig.

text: and

text: Fig.

text: pion

text: contamination.

text: for

text: GeV/c

text: and

text: GeV/c

text: muons

text: respectively.

text: In

text: this

text: case

text: The

text: incoming

text: and

text: outgoing

text: muon

text: directions

text: were

text: deter-

text: the

text: same

text: procedure

text: used

text: for

text: Lead

text: was

text: employed

text: except

text: for

text: mined

text: by

text: using

text: four

text: identical

text: counter

text: hodoscopes

text: made

text: of

text: the

text: fact

text: that

text: the

text: Saxon-Wood

text: parametrisation

text: of

text: the

text: Copper

text: scintillator

text: bars

text: at

text: positions

text: A

text: A

text: A

text: and

text: A

text: (Fig.

text: nucleus

text: charge

text: density

text: was

text: used.

text: Experimental

text: data

text: are

text: shown

text: A

text: B

text: C

text: D

text: of

text: The

text: acceptance

text: of

text: such

text: an

text: arrangement

text: was

text: by

text: the

text: bullets

text: with

text: error

text: bands.

text: The

text: Monte

text: Carlo

text: prediction

text: between

text: and

text: Each

text: station

text: was

text: composed

text: of

text: vertical

text: is

text: shown

text: with

text: three

text: options:

text: a

text: point-like

text: nucleus

text: prediction

text: scintillators

text: each

text: having

text: dimensions

text: cm,

text: F(q)

text: (gray).

text: the

text: dipole

text: nuclear

text: form

text: factor

text: (red)

text: the

text: smallest

text: dimension

text: being

text: the

text: one

text: in

text: the

text: horizontal

text: plane.

text: the

text: improved

text: form-factor

text: of

text: Eq.

text: (yellow).

text: In

text: both

text: cases

text: the

text: The

text: downstream

text: stations

text: centers

text: were

text: shifted

text: away

text: from

text: the

text: mixed

text: algorithm

text: described

text: in

text: has

text: been

text: used.

text: The

text: band

text: width

text: beam

text: direction

text: by

text: cm.

text: The

text: distance

text: along

text: the

text: beam

text: in

text: Monte

text: Carlo

text: histograms

text: represent

text: the

text: statistical

text: uncertainty.

text: between

text: the

text: stations

text: was

text: m

text: (AB)

text: and

text: m

text: (CD).

text: This

text: meaning

text: of

text: symbols

text: is

text: also

text: preserved

text: in

text: the

text: following.

text: The

text: beam

text: spread

text: in

text: the

text: horizontal

text: and

text: vertical

text: directions

text: The

text: disagreement

text: with

text: the

text: point-like

text: simulation

text: indicates

text: that

text: was

text: determined

text: triggering

text: with

text: a

text: small

text: scintillator

text: (S)

text: located

text: the

text: nuclear

text: density

text: is

text: in

text: this

text: case

text: actually

text: probed.

text: Furthermore

text: along

text: the

text: beam

text: axis

text: in

text: between

text: stations

text: A

text: and

text: A

text: rotating

text: a

text: better

text: description

text: is

text: obtained

text: by

text: using

text: the

text: improved

text: form-

text: C

text: D

text: the

text: A

text: and

text: A

text: stations

text: by

text: The

text: coincidence

text: rate

text: between

text: factor

text: with

text: respect

text: to

text: the

text: dipole

text: one.

text: A

text: B

text: strips

text: in

text: scintillators

text: in

text: A

text: and

text: A

text: is

text: shown

text: in

text: Fig.

text: of

text: B.

text: Lead

text: data

text: with

text: muons

text: (Masek

text: et

text: al.)

text: A

text: B

text: the

text: original

text: paper.

text: This

text: bidimensional

text: rate

text: map

text: was

text: used

text: to

text: The

text: was

text: performed

text: in

text: proﬁting

text: of

text: generate

text: the

text: incoming

text: direction

text: of

text: muons

text: in

text: the

text: simulation

text: the

text: Bevatron

text: accelerator

text: at

text: the

text: Lawrence

text: radiation

text: labora-

text: accordingly.

text: The

text: vertical

text: spread

text: was

text: generated

text: assuming

text: a

text: tory

text: employing

text: a

text: muon

text: beam

text: with

text: a

text: median

text: momentum

text: of

text: gaussian

text: distribution

text: with

text: a

text: r.m.s.

text: This

text: is

text: supported

text: by

text: GeV/c

text: and

text: a

text: spread.

text: The

text: total

text: number

text: the

text: paper

text: which

text: states

text: a

text: spread

text: of

text: and

text: in

text: the

text: of

text: muons

text: incident

text: on

text: the

text: apparatus

text: was

text: The

text: horizontal

text: and

text: vertical

text: views

text: respectively.

text: A

text: gaussian

text: ﬁt

text: of

text: lead

text: target

text: had

text: a

text: thickness

text: of

text: corresponding

text: to

text: the

text: θ

text: distribution

text: obtained

text: from

text: the

text: information

text: contained

text: x

text: cm

text: (one

text: half

text: inch).

text: Scattered

text: particles

text: were

text: observed

text: in

text: the

text: hit

text: strip

text: bidimensional

text: distribution

text: (Fig.

text: of

text: and

text: up

text: to

text: (p

text: MeV/c).

text: The

text: muon

text: beam

text: is

text: obtained

text: the

text: detector

text: geometry

text: gives

text: a

text: spread

text: of

text: about

text: not

text: too

text: far

text: ±T

text: from

text: a

text: GeV/c

text: pion

text: beam

text: with

text: magnetic

text: selection

text: from

text: the

text: quoted

text: in

text: the

text: paper.

text: (see

text: Fig.

text: of

text: the

text: original

text: paper).

text: GeV/c

text: muons

text: are

text: those

text: We

text: have

text: reproduced

text: with

text: a

text: simple

text: simulation

text: both

text: the

text: decaying

text: backward

text: in

text: the

text: π

text: rest

text: frame.

text: This

text: large

text: difference

text: angular

text: spread

text: of

text: the

text: beam

text: and

text: the

text: ﬁnite-solid

text: angle

text: effects

text: in

text: momentum

text: allows

text: a

text: clean

text: π/µ

text: separation.

text: The

text: pion

text: con-

text: introduced

text: by

text: the

text: strip

text: width.

text: This

text: was

text: performed

text: by

text: binning

text: tamination

text: on

text: the

text: target

text: is

text: estimated

text: to

text: be

text: of

text: about

text: By

text: the

text: true

text: positions

text: according

text: to

text: the

text: detector

text: geometry

text: described

text: using

text: a

text: cm

text: thick

text: iron

text: absorber

text: downstream

text: of

text: the

text: above.

text: A

text: simple

text: linear

text: trajectory

text: has

text: been

text: assumed

text: in

text: between

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 8 where u¯ becomes the new virtual control input. Substituting (18) into (14) gives 1 z(q)∆x˙ + z˙(q)∆x = u¯. (19) 2 Proposition 2: The system (19) is passive with respect to the input-output pair (u¯, ∆x). The proof of Proposition 2 shall be similar to that of Proposition 1. (cid:104) (cid:105) Using (7), we can rewrite (18) as (cid:124) (cid:123)(cid:122) (cid:125) − ⊥ − x + x u¯ = z(q)x˙ + J (q) d J (q) q˙. (20) d z 2 z ∗ J ∗ Obviously, if the Jacobian matrix J has full row rank, the virtual control u¯ can be realized by the joint velocity q˙. B. Adaptive Inverse-Jacobian-like Control Let us now start the adaptive controller design based on the passivity enjoyed by the overall kinematic equation. Due to the passivity of the input-output pair (u¯, ∆x), the standard passivity-based design [27] − suggests that the virtual control u¯ = K¯ ∆x with K¯ being a symmetric positive deﬁnite matrix would be qualiﬁed for realizing the image-space tracking, yet, not necessarily give guaranteed performance due to the variation of the depth z(q). To accommodate the varying and uncertain depth, we propose the following virtual control − u¯ = αzˆ(q)∆x (21) where α > 0 is a design constant and zˆ(q) is the estimate of z(q) which is obtained by replacing a in z(q) with its estimate aˆ . The use of the virtual control (21) is inspired by the performance z z guaranteed adaptive control for robot manipulators in [29, Sec. 3.2]. However, it should be emphasized that the virtual control (21) is not the actual control since it does not take into account the dynamic effect of the manipulator. Keeping (21) in mind and based on (20), we deﬁne a joint reference velocity using the estimated Jacobian matrix as ∗ q˙ = Jˆ +zˆ(q)x˙ (22) r r (cid:0) (cid:1) ∗ ∗ ⊥ ∗ where Jˆ is the estimate of J which is obtained by replacing a and a in J with their z z ⊥ ∗ ∗ ∗ ∗ − estimates aˆ and aˆ , respectively, Jˆ + = Jˆ T Jˆ Jˆ T 1 is the standard generalized inverse of z z ∗ − Jˆ (see, e.g., [30]), and x˙ = x˙ α∆x. Differentiating (22) with respect to time yields the joint r d reference acceleration May 27, 2020 DRAFT ##LTLine## ##LTLine##

text: where

text: u¯

text: becomes

text: the

text: new

text: virtual

text: control

text: input.

text: Substituting

text: into

text: gives

text: z(q)∆x˙

text: z˙(q)∆x

text: u¯.

text: Proposition

text: The

text: system

text: is

text: passive

text: with

text: respect

text: to

text: the

text: input-output

text: pair

text: (u¯,

text: ∆x).

text: The

text: proof

text: of

text: Proposition

text: shall

text: be

text: similar

text: to

text: that

text: of

text: Proposition

text: Using

text: we

text: can

text: rewrite

text: as

text: x

text: x

text: u¯

text: z(q)x˙

text: J

text: (q)

text: d

text: J

text: (q)

text: q˙.

text: d

text: z

text: z

text: J

text: Obviously,

text: if

text: the

text: Jacobian

text: matrix

text: J

text: has

text: full

text: row

text: rank,

text: the

text: virtual

text: control

text: u¯

text: can

text: be

text: realized

text: by

text: the

text: joint

text: velocity

text: q˙.

text: B.

text: Adaptive

text: Inverse-Jacobian-like

text: Control

text: Let

text: us

text: now

text: start

text: the

text: adaptive

text: controller

text: design

text: based

text: on

text: the

text: passivity

text: enjoyed

text: by

text: the

text: overall

text: kinematic

text: equation.

text: Due

text: to

text: the

text: passivity

text: of

text: the

text: input-output

text: pair

text: (u¯,

text: ∆x),

text: the

text: standard

text: passivity-based

text: design

text: suggests

text: that

text: the

text: virtual

text: control

text: u¯

text: K¯

text: ∆x

text: with

text: K¯

text: being

text: a

text: symmetric

text: positive

text: deﬁnite

text: matrix

text: would

text: be

text: qualiﬁed

text: for

text: realizing

text: the

text: image-space

text: tracking,

text: yet,

text: not

text: necessarily

text: give

text: guaranteed

text: performance

text: due

text: to

text: the

text: variation

text: of

text: the

text: depth

text: z(q).

text: To

text: accommodate

text: the

text: varying

text: and

text: uncertain

text: depth,

text: we

text: propose

text: the

text: following

text: virtual

text: control

text: u¯

text: αzˆ(q)∆x

text: where

text: α

text: is

text: a

text: design

text: constant

text: and

text: zˆ(q)

text: is

text: the

text: estimate

text: of

text: z(q)

text: which

text: is

text: obtained

text: by

text: replacing

text: a

text: in

text: z(q)

text: with

text: its

text: estimate

text: aˆ

text: The

text: use

text: of

text: the

text: virtual

text: control

text: is

text: inspired

text: by

text: the

text: performance

text: z

text: z

text: guaranteed

text: adaptive

text: control

text: for

text: robot

text: manipulators

text: in

text: Sec.

text: However,

text: it

text: should

text: be

text: emphasized

text: that

text: the

text: virtual

text: control

text: is

text: not

text: the

text: actual

text: control

text: since

text: it

text: does

text: not

text: take

text: into

text: account

text: the

text: dynamic

text: effect

text: of

text: the

text: manipulator.

text: Keeping

text: in

text: mind

text: and

text: based

text: on

text: we

text: deﬁne

text: a

text: joint

text: reference

text: velocity

text: using

text: the

text: estimated

text: Jacobian

text: matrix

text: as

text: q˙

text: Jˆ

text: +zˆ(q)x˙

text: r

text: r

text: where

text: Jˆ

text: is

text: the

text: estimate

text: of

text: J

text: which

text: is

text: obtained

text: by

text: replacing

text: a

text: and

text: a

text: in

text: J

text: with

text: their

text: z

text: z

text: estimates

text: aˆ

text: and

text: aˆ

text: respectively,

text: Jˆ

text: Jˆ

text: T

text: Jˆ

text: Jˆ

text: T

text: is

text: the

text: standard

text: generalized

text: inverse

text: of

text: z

text: z

text: Jˆ

text: (see,

text: e.g.,

text: and

text: x˙

text: x˙

text: α∆x.

text: Differentiating

text: with

text: respect

text: to

text: time

text: yields

text: the

text: joint

text: r

text: d

text: reference

text: acceleration

text: May

text: DRAFT

text: ##LTLine##

text: ##LTLine##


text: eeρ eeρ eeρ eeρ 2 (cid:18) (cid:19) (cid:88) (a) (b) 1 1 1 L[ρ] = −i[Hˆ, ρ] + cˆ ρcˆ† − {cˆ†cˆ , ρ} , (1) 0.5 0.5 k k 2 k k k 0 0 0 10 20 0 10 20 where (cid:126) , and represent diﬀerent relaxation pro- Ωt Ωt = 1 cˆk (c) (d) cesses. For time independent operators the solution is 1 1 formally given by exponentiation, 0.5 0.5 ρ(t) = eLt[ρ(0)]. (2) 0 0 0 10 20 0 10 20 Ωt Ωt If one monitors the time evolution by frequent (contin- FIG. 1. (Color online) Illustration showing the excited state uous) observation the back action conditions the system population as a function of time for a two-level system driven evolution on the measurement outcomes and the evolu- on resonance, and randomly projected at regular intervals ac- cording to measurements in the eigenstate basis. The mea- tion becomes stochastic. surement intervals are diﬀerent in the diﬀerent panels, (a): A realistic approach to continuous or frequent mea- − − − τ = 2.5Ω 1, (b): τ = 1.75Ω 1, (c): τ = 0.75Ω 1 and (d): surements should take the ﬁnite bandwidth and noise − τ = 0.25Ω 1, and show the Quantum Zeno suppression of the properties of the measurement device into acount, see, coherent Rabi oscillations. e.g., [19], but we shall restrict our attention to the ideal case of instantaneous, accurate measurements, repeated at regular intervals . The continuous regime approxi- τ mated by the limit τ → 0 thus assumes a corresponding 1 high bandwidth of detection. [∆S(θ)]2 ≥ , (6) While measurement theory can be formulated more F(θ0) generally, for simplicity we restrict our analysis to projec- tive measurements. Each measurement outcome is thus yields a lower bound of the statistical variance [∆S(θ)]2 an eigenvalue λ of a given operator Λˆ, occurring with the on any unbiased estimate of the unknown quantity θ probability (with the true value θ ). H(cid:2)ere, (cid:3) 0 P(λ) = Tr(Πˆλρ(t)) ≡ ρλλ(t), (3) F(θ) = −E ∂θ2 ln P(D|θ) , (7) where Πˆ = |λ(cid:105) (cid:104)λ| is a projector on the correspond- is the Fisher information which quantiﬁes the depen- ing eigenstaλte. The projection postulate states that the dence of the data outcome probabilities on the unknown conditional state after a measurement performed on the parameter. The Fisher information and the Cramér Rao state at time is [20], bound yield the asymptotic precision of the best possible ρ(t) t estimate, and normally assumes a large number of repe- titions of the experiment (and a corresponding factor N Πˆ ρ(t)Πˆ 1/N in Eq. (6)). In our analysis a self-averaging occurs λ λ (4) ρλ(t) = . due to the repetitive measurements, and we shall ﬁnd P(λ) the corresponding dependence of on and, equiv- F(θ) N alently, on the total time of the probing . A. Fisher information of projective measurement T A data record of projective measurement records N = T/τ outcomes obtained at times t = jτ, D = {λ }N , occurs j j j=1 The acquisition of information about unknown physi- with the probability(cid:89) (cid:0) (cid:1) cal parameters from a measurement record follows Bayes’ rule: Given the stochastic measurement outcome D, the P(D(cid:89)|θ) =(cid:0) Tr |λ (cid:105) (cid:104)λ | eLτ[ρ(t (cid:1))] probability that an unknown parameter has a given value j j j−1 , is given by the corresponding outcome probability con- j θ ditioned on the value θ and its unconditional (prior) prob- = Tr |λj(cid:105) (cid:104)λj| eLτ[|λj−1(cid:105) (cid:104)λj−1|] , (8) abilities, j assuming that the initial state is one of the eigen- P(θ|D) = P(D|θ)P(θ). (5) states, ρ(0) = |λ0(cid:105)(cid:104)λ0|. Since by Eq. (4) the sys- P(D) tem is always projected (reset) in one of the eigen- states upon detection, the probability for an entire data The Cramér Rao bound (CRB) [21], record factors into a product of conditional probabilities ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: eeρ

text: eeρ

text: eeρ

text: eeρ

label: equation

text: (a)

text: (b)

text: L[ρ]

text: −i[Hˆ,

text: ρ]

text: cˆ

text: ρcˆ†

text: {cˆ†cˆ

text: ρ}

text: k

text: k

text: k

text: k

text: k

text: where

text: and

text: represent

text: diﬀerent

text: relaxation

text: pro-

text: Ωt

text: Ωt

text: cˆk

text: (c)

text: (d)

text: cesses.

text: For

text: time

text: independent

text: operators

text: the

text: solution

text: is

text: formally

text: given

text: by

text: exponentiation,

text: ρ(t)

text: Ωt

text: Ωt

text: If

text: one

text: monitors

text: the

text: time

text: evolution

text: by

text: frequent

text: (contin-

text: FIG.

text: (Color

text: online)

text: Illustration

text: showing

text: the

text: excited

text: state

text: uous)

text: observation

text: the

text: back

text: action

text: conditions

text: the

text: system

text: population

text: as

text: a

text: function

text: of

text: time

text: for

text: a

text: two-level

text: system

text: driven

text: evolution

text: on

text: the

text: measurement

text: outcomes

text: and

text: the

text: evolu-

text: on

text: resonance,

text: and

text: randomly

text: projected

text: at

text: regular

text: intervals

text: ac-

text: cording

text: to

text: measurements

text: in

text: the

text: eigenstate

text: basis.

text: The

text: mea-

text: tion

text: becomes

text: stochastic.

text: surement

text: intervals

text: are

text: diﬀerent

text: in

text: the

text: diﬀerent

text: panels,

text: (a):

text: A

text: realistic

text: approach

text: to

text: continuous

text: or

text: frequent

text: mea-

text: τ

text: (b):

text: τ

text: (c):

text: τ

text: and

text: (d):

text: surements

text: should

text: take

text: the

text: ﬁnite

text: bandwidth

text: and

text: noise

text: τ

text: and

text: show

text: the

text: Quantum

text: Zeno

text: suppression

text: of

text: the

text: properties

text: of

text: the

text: measurement

text: device

text: into

text: acount,

text: see,

text: coherent

text: Rabi

text: oscillations.

text: e.g.,

text: but

text: we

text: shall

text: restrict

text: our

text: attention

text: to

text: the

text: ideal

text: case

text: of

text: instantaneous,

text: accurate

text: measurements,

text: repeated

text: at

text: regular

text: intervals

text: The

text: continuous

text: regime

text: approxi-

text: τ

text: mated

text: by

text: the

text: limit

text: τ

text: thus

text: assumes

text: a

text: corresponding

text: high

text: bandwidth

text: of

text: detection.

text: While

text: measurement

text: theory

text: can

text: be

text: formulated

text: more

text: generally,

text: for

text: simplicity

text: we

text: restrict

text: our

text: analysis

text: to

text: projec-

text: tive

text: measurements.

text: Each

text: measurement

text: outcome

text: is

text: thus

text: yields

text: a

text: lower

text: bound

text: of

text: the

text: statistical

text: variance

text: an

text: eigenvalue

text: λ

text: of

text: a

text: given

text: operator

text: Λˆ,

text: occurring

text: with

text: the

text: on

text: any

text: unbiased

text: estimate

text: of

text: the

text: unknown

text: quantity

text: θ

text: probability

text: (with

text: the

text: true

text: value

text: θ

text: P(λ)

text: Tr(Πˆλρ(t))

text: ρλλ(t),

text: F(θ)

text: −E

text: ∂θ2

text: ln

text: P(D|θ)

text: where

text: Πˆ

text: is

text: a

text: projector

text: on

text: the

text: correspond-

text: is

text: the

text: Fisher

text: information

text: which

text: quantiﬁes

text: the

text: depen-

text: ing

text: eigenstaλte.

text: The

text: projection

text: postulate

text: states

text: that

text: the

text: dence

text: of

text: the

text: data

text: outcome

text: probabilities

text: on

text: the

text: unknown

text: conditional

text: state

text: after

text: a

text: measurement

text: performed

text: on

text: the

text: parameter.

text: The

text: Fisher

text: information

text: and

text: the

text: Cramér

text: Rao

text: state

text: at

text: time

text: is

text: bound

text: yield

text: the

text: asymptotic

text: precision

text: of

text: the

text: best

text: possible

text: ρ(t)

text: t

text: estimate,

text: and

text: normally

text: assumes

text: a

text: large

text: number

text: of

text: repe-

text: titions

text: of

text: the

text: experiment

text: (and

text: a

text: corresponding

text: factor

text: N

text: Πˆ

text: ρ(t)Πˆ

text: in

text: Eq.

text: In

text: our

text: analysis

text: a

text: self-averaging

text: occurs

text: λ

text: λ

text: ρλ(t)

text: due

text: to

text: the

text: repetitive

text: measurements,

text: and

text: we

text: shall

text: ﬁnd

text: P(λ)

text: the

text: corresponding

text: dependence

text: of

text: on

text: and,

text: equiv-

text: F(θ)

text: N

text: alently,

text: on

text: the

text: total

text: time

text: of

text: the

text: probing

text: A.

text: Fisher

text: information

text: of

text: projective

text: measurement

text: T

text: A

text: data

text: record

text: of

text: projective

text: measurement

text: records

text: N

text: T/τ

text: outcomes

text: obtained

text: at

text: times

text: t

text: jτ,

text: D

text: {λ

text: }N

text: occurs

text: j

text: j

text: The

text: acquisition

text: of

text: information

text: about

text: unknown

text: physi-

text: with

text: the

text: cal

text: parameters

text: from

text: a

text: measurement

text: record

text: follows

text: Bayes’

text: rule:

text: Given

text: the

text: stochastic

text: measurement

text: outcome

text: D,

text: the

text: Tr

text: |λ

text: eLτ[ρ(t

text: probability

text: that

text: an

text: unknown

text: parameter

text: has

text: a

text: given

text: value

text: j

text: j

text: is

text: given

text: by

text: the

text: corresponding

text: outcome

text: probability

text: con-

text: j

text: θ

text: ditioned

text: on

text: the

text: value

text: θ

text: and

text: its

text: unconditional

text: (prior)

text: prob-

text: Tr

text: abilities,

text: j

text: assuming

text: that

text: the

text: initial

text: state

text: is

text: one

text: of

text: the

text: eigen-

text: P(θ|D)

text: P(D|θ)P(θ).

text: states,

text: Since

text: by

text: Eq.

text: the

text: sys-

text: P(D)

text: tem

text: is

text: always

text: projected

text: (reset)

text: in

text: one

text: of

text: the

text: eigen-

text: states

text: upon

text: detection,

text: the

text: probability

text: for

text: an

text: entire

text: data

text: The

text: Cramér

text: Rao

text: bound

text: (CRB)

text: record

text: factors

text: into

text: a

text: product

text: of

text: conditional

text: probabilities

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Proof. Begin with an edge e = (v, vs ). In addition to e, the ver- − m tex vs has 2M 1 other incident edges. One is the type m edge m (vs , vs2 ), while the others have one of the type k forms (vs , vs s ) m m− ∈ { } \ { } ∈ Xm m k or (vs s 1, vs ), where k 1, . . . , M m . Let y + satisfy m k m e y(vs ) = 1, and let y , respectively y denote the restriction of y to m k m the subtrees with root vertex vs and initial edges of type k, respec- m tively m. As a consequence of Corollary 4.2, for a ﬁxed value of k the two functions y agree as functions of the distance from vs on the k m two edges of type k incident on vs . m Let z denote the restriction of y to the edge e. The continuity m and derivative vertex conditions at vs relate z to the restrictions m m y , y . Using local edge coordinates which identify vs with l for k m m m the edge e, and which identify vs with 0 for the o(cid:88)ther incident edges, m the initial data for z at vs is m m (cid:48) (cid:48) (cid:48) z (l , λ) = 1, z (l , λ) = y (0, λ) + 2 y (0, λ). m m m m m k (cid:54) k=m (cid:88) The function z (x, λ) may be written as m − − (cid:48) (cid:48) − z (x, λ) = C (l x, λ) [y (0, λ) + 2 y (0, λ)]S (l x, λ). m m m m k m m (cid:54) k=m (cid:88) Evaluation at x = 0 gives − (cid:48) (cid:48) z (0, λ) = C (l , λ) [y (0, λ) + 2 y (0, λ)]S (l , λ). m m m m (cid:88) k m m (cid:54) k=m (cid:48) − (cid:48) (cid:48) (cid:48) (cid:48) z (0, λ) = C (l , λ) + [y (0, λ) + 2 y (0, λ)]S (l , λ). m m m m k m m (cid:54) k=m As noted above, on [0, l ] the function z (x, λ) is a scalar multiple m m of y (x, λ), so that z (x, λ)/z (0, λ) = y (x, λ). In particular, m m m m (cid:48) (cid:48) z (0, λ) y (0, λ) = (cid:80)m m z (0, λ) (cid:80)m − (cid:48) (cid:48) (cid:48) (cid:48) C (l , λ) + [y (0, λ) + 2 (cid:54) y (0, λ)]S (l , λ) = m m − (cid:48) m k=m (cid:48) k m m . C (l , λ) [y (0, λ) + 2 (cid:54) y (0, λ)]S (l , λ) m m m k=m k m m This can be rewritten as 17 ##LTLine## ##LTLine##

text: Proof.

text: Begin

text: with

text: an

text: edge

text: e

text: (v,

text: vs

text: In

text: addition

text: to

text: e,

text: the

text: ver-

text: m

text: tex

text: vs

text: has

text: other

text: incident

text: edges.

text: One

text: is

text: the

text: type

text: m

text: edge

text: m

text: (vs

text: while

text: the

text: others

text: have

text: one

text: of

text: the

text: type

text: k

text: forms

text: (vs

text: vs

text: s

text: m

text: m−

text: Xm

text: m

text: k

text: or

text: (vs

text: s

text: vs

text: where

text: k

text: M

text: m

text: Let

text: y

text: satisfy

text: m

text: k

text: m

text: e

text: y(vs

text: and

text: let

text: y

text: respectively

text: y

text: denote

text: the

text: restriction

text: of

text: y

text: to

text: m

text: k

text: m

text: the

text: subtrees

text: with

text: root

text: vertex

text: vs

text: and

text: initial

text: edges

text: of

text: type

text: k,

text: respec-

text: m

text: tively

text: m.

text: As

text: a

text: consequence

text: of

text: Corollary

text: for

text: a

text: ﬁxed

text: value

text: of

text: k

text: the

text: two

text: functions

text: y

text: agree

text: as

text: functions

text: of

text: the

text: distance

text: from

text: vs

text: on

text: the

text: k

text: m

text: two

text: edges

text: of

text: type

text: k

text: incident

text: on

text: vs

text: m

text: Let

text: z

text: denote

text: the

text: restriction

text: of

text: y

text: to

text: the

text: edge

text: e.

text: The

text: continuity

text: m

text: and

text: derivative

text: vertex

text: conditions

text: at

text: vs

text: relate

text: z

text: to

text: the

text: restrictions

text: m

text: m

text: y

text: y

text: Using

text: local

text: edge

text: coordinates

text: which

text: identify

text: vs

text: with

text: l

text: for

text: k

text: m

text: m

text: m

text: the

text: edge

text: e,

text: and

text: which

text: identify

text: vs

text: with

text: for

text: the

text: incident

text: edges,

text: m

text: the

text: initial

text: data

text: for

text: z

text: at

text: vs

text: is

text: m

text: m

text: z

text: (l

text: λ)

text: z

text: (l

text: λ)

text: y

text: λ)

text: y

text: λ).

text: m

text: m

text: m

text: m

text: m

text: k

text: k=m

text: The

text: function

text: z

text: (x,

text: λ)

text: may

text: be

text: written

text: as

text: m

text: z

text: (x,

text: λ)

text: C

text: (l

text: x,

text: λ)

text: [y

text: λ)

text: y

text: λ)]S

text: (l

text: x,

text: λ).

text: m

text: m

text: m

text: m

text: k

text: m

text: m

text: k=m

text: Evaluation

text: at

text: x

text: gives

text: z

text: λ)

text: C

text: (l

text: λ)

text: [y

text: λ)

text: y

text: λ)]S

text: (l

text: λ).

text: m

text: m

text: m

text: m

text: k

text: m

text: m

text: k=m

text: z

text: λ)

text: C

text: (l

text: λ)

text: [y

text: λ)

text: y

text: λ)]S

text: (l

text: λ).

text: m

text: m

text: m

text: m

text: k

text: m

text: m

text: k=m

text: As

text: noted

text: above,

text: on

text: l

text: the

text: function

text: z

text: (x,

text: λ)

text: is

text: a

text: scalar

text: multiple

text: m

text: m

text: of

text: y

text: (x,

text: λ),

text: so

text: that

text: z

text: (x,

text: λ)/z

text: λ)

text: y

text: (x,

text: λ).

text: In

text: particular,

text: m

text: m

text: m

text: m

text: z

text: λ)

text: y

text: λ)

text: m

text: z

text: λ)

text: C

text: (l

text: λ)

text: [y

text: λ)

text: y

text: λ)]S

text: (l

text: λ)

text: m

text: m

text: m

text: k=m

text: k

text: m

text: m

text: C

text: (l

text: λ)

text: [y

text: λ)

text: y

text: λ)]S

text: (l

text: λ)

text: m

text: m

text: m

text: k=m

text: k

text: m

text: m

text: This

text: can

text: be

text: rewritten

text: as

text: ##LTLine##

text: ##LTLine##


text: (cid:16) (cid:17) − ξ2 − (λ) 1 ξ (λ)2 = m 1 S ξ + 1, (4.16) m S − ξ − (λ) m m m 1 m 1 or by using the quadratic formula, (cid:16) (cid:17) (cid:104)(cid:16) (cid:17) (cid:105) − − − ξ2 − (λ) 1 ξ2 − (λ) 1 2 1/2 2ξ (λ) m 1 S = m 1 S2 + 4 . (4.17) m S − ξ − (λ) m S − ξ − (λ) m m 1 m 1 m 1 m 1 The variables ξ , . . . , ξ can be successively eliminated from the M 2 ﬁrst equation. Starting with k = M and continuing up the list of indices, the ﬁrst equation can be written as a polynomial equation for ξ with coeﬃcients which are polynomials in ξ , . . . , ξ − and the k 1 k 1 entire functions S (l , λ). Repeated use of the substitution (4.16), m k followed by clearing of the denominators, reduces the ﬁrst equation to degree one in ξ . These substitutions result in an equivalent system k ∈ (cid:91) of equations as long as λ / Z , where 0 { } Z = S (l , λ) = 0 . (4.18) 0 m m m − Now solve for 2ξ , subtract [ξ2 − (λ) 1]S /[S − ξ − (λ)], use k m 1 m m 1 m 1 the substitution (4.17), and square both sides. Since squaring is a two-to-one map, it will not change the dimension of the set of solu- tions. After applying these substitutions, the variable ξ has been k eliminated, and after clearing the denominators, the modiﬁed j = 1 equation is a polynomial in ξ (λ), . . . , ξ − (λ) with entire coeﬃcients. 1 k 1 The substitution process provides a common bound N for the degrees of the polynomials p . m Suppose the ﬁnal version of the ﬁrst equation does not have pos- − itive degree for ξ . Deﬁne Q = F (ξ , λ) F (ξ , λ) for m = − 1 m m m m+1 m+1 1, . . . , M 1. The system of equations for ξ , . . . , ξ is then the system 1 M Q = 0, . . . , Q − = 0, 1 M 1 where − ∂Q 1 1 − ∂Q 1 1 − m m = [ 1], = [ 1], ∂ξ S ξ2 (λ) ∂ξ S ξ2 (λ) m m m m+1 m+1 m+1 23 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ξ2

text: (λ)

text: ξ

text: m

text: S

text: ξ

text: m

text: S

text: ξ

text: (λ)

text: m

text: m

text: m

text: m

text: or

text: by

text: using

text: the

text: quadratic

text: formula,

text: ξ2

text: (λ)

text: ξ2

text: (λ)

text: 2ξ

text: (λ)

text: m

text: S

text: m

text: m

text: S

text: ξ

text: (λ)

text: m

text: S

text: ξ

text: (λ)

text: m

text: m

text: m

text: m

text: m

text: The

text: variables

text: ξ

text: ξ

text: can

text: be

text: successively

text: eliminated

text: from

text: the

text: M

text: ﬁrst

text: equation.

text: Starting

text: with

text: k

text: M

text: and

text: continuing

text: up

text: the

text: list

text: of

text: indices,

text: the

text: ﬁrst

text: equation

text: can

text: be

text: written

text: as

text: a

text: polynomial

text: equation

text: for

text: ξ

text: with

text: coeﬃcients

text: which

text: are

text: polynomials

text: in

text: ξ

text: ξ

text: and

text: the

text: k

text: k

text: entire

text: functions

text: S

text: (l

text: λ).

text: Repeated

text: use

text: of

text: the

text: substitution

text: m

text: k

text: followed

text: by

text: clearing

text: of

text: the

text: denominators,

text: reduces

text: the

text: ﬁrst

text: equation

text: to

text: degree

text: one

text: in

text: ξ

text: These

text: substitutions

text: result

text: in

text: an

text: equivalent

text: system

text: k

text: of

text: equations

text: as

text: long

text: as

text: λ

text: Z

text: where

text: Z

text: S

text: (l

text: λ)

text: m

text: m

text: m

text: Now

text: solve

text: for

text: 2ξ

text: subtract

text: (λ)

text: /[S

text: ξ

text: (λ)],

text: use

text: k

text: m

text: m

text: m

text: m

text: the

text: substitution

text: and

text: square

text: both

text: sides.

text: Since

text: squaring

text: is

text: a

text: two-to-one

text: map,

text: it

text: will

text: not

text: change

text: the

text: dimension

text: of

text: the

text: set

text: of

text: solu-

text: tions.

text: After

text: applying

text: these

text: substitutions,

text: the

text: variable

text: ξ

text: has

text: been

text: k

text: eliminated,

text: and

text: after

text: clearing

text: the

text: denominators,

text: the

text: modiﬁed

text: j

text: equation

text: is

text: a

text: polynomial

text: in

text: ξ

text: (λ),

text: ξ

text: (λ)

text: with

text: entire

text: coeﬃcients.

text: k

text: The

text: substitution

text: process

text: provides

text: a

text: common

text: bound

text: N

text: for

text: the

text: degrees

text: of

text: the

text: polynomials

text: p

text: m

text: Suppose

text: the

text: ﬁnal

text: version

text: of

text: the

text: ﬁrst

text: equation

text: does

text: not

text: have

text: pos-

text: itive

text: degree

text: for

text: ξ

text: Deﬁne

text: Q

text: F

text: (ξ

text: λ)

text: F

text: (ξ

text: λ)

text: for

text: m

text: m

text: m

text: m

text: M

text: The

text: system

text: of

text: equations

text: for

text: ξ

text: ξ

text: is

text: then

text: the

text: system

text: M

text: Q

text: Q

text: M

text: where

text: ∂Q

text: ∂Q

text: m

text: m

text: ∂ξ

text: S

text: ξ2

text: (λ)

text: ∂ξ

text: S

text: ξ2

text: (λ)

text: m

text: m

text: m

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: − − shows µ+(σ) µ (σ) = δ (σ) = 0. m m m Now suppose δ (σ) = 0 in open neighborhood of σ. Theorem 4.12 m notes that R (x, t, λ) extends holomophically to a neighborhood of ∈ m σ. If g L2(Γ) with support in a ﬁnite set of edges, the function (cid:104) (cid:105) R (λ)f, g also extends holomorphically as a single valued function m ⊂ in an interval (α, β) containing σ. If [a, b] (α, β), then for any ∈ f L2(e) (cid:104)1 (cid:105) [P + P ]f, g = 0. [a,b] (a,b) 2 The set of g with with support in a ﬁnite set of edges is dense in L2(Γ), ∈ so 1[P +P ]f = 0 for all f L2(e). By linearity [P +P ]h = 2 [a,b] ∈(a,b) [a,b] (a,b) 0 for any h L2(Γ) with support in a ﬁnite set of edges, and since the projections are bounded we conclude that P + P = 0 and (α, β) [a,b] (a,b) is in the resolvent set [14, p. 357]. ∈ ∞ \ Corollary 4.15. Assume σ [0, ) Z and for m = 1, . . . , M ± (cid:54) ± 0 suppose µ (σ) = 1. Then σ is in the resolvent set of ∆+q if and only m if µ+(σ) is real valued in open neighborhood of σ for m = 1, . . . , M. m Proof. If µ+(σ) is real valued, then the symmetry µ (λ) = µ (λ) m − m m established in Lemma 3.7 means µ (σ) has the same real value. The m (cid:54) same symmetry also implies that δ (σ) = 0 if µ+(σ) is not real valued. m m ≥ ≥ Theorem 4.16. For q 0 and M 2 the spectrum of ∆ + q has a strictly positive lower bound. (cid:90) Proof. Since (cid:104) (cid:105) (cid:104) (cid:105) | | (∆ + q)f, f = ∆f, f + q f 2, T M it suﬃces to verify the result when q = 0. Consider the case when the edge lengths l are all equal to 1. Then m the system (4.8) reduces to − − (2M 1)µ2(λ) 2MC(1, λ)µ(λ) + 1 = 0. (cid:112) The quadratic formula gives ± − − 2MC(1, λ) 4M2C2(1, λ) 4(2M 1) µ(λ) = − . 2(2M 1) 27 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: shows

text: µ+(σ)

text: µ

text: (σ)

text: δ

text: (σ)

text: m

text: m

text: m

text: Now

text: suppose

text: δ

text: (σ)

text: in

text: open

text: neighborhood

text: of

text: σ.

text: Theorem

text: m

text: notes

text: that

text: R

text: (x,

text: t,

text: λ)

text: extends

text: holomophically

text: to

text: a

text: neighborhood

text: of

text: m

text: σ.

text: If

text: g

text: with

text: support

text: in

text: a

text: ﬁnite

text: set

text: of

text: edges,

text: the

text: function

text: R

text: (λ)f,

text: g

text: also

text: extends

text: holomorphically

text: as

text: a

text: single

text: valued

text: function

text: m

text: in

text: an

text: interval

text: (α,

text: β)

text: containing

text: σ.

text: If

text: [a,

text: b]

text: (α,

text: β),

text: then

text: for

text: any

text: f

text: [P

text: P

text: ]f,

text: g

text: [a,b]

text: (a,b)

text: The

text: set

text: of

text: g

text: with

text: with

text: support

text: in

text: a

text: ﬁnite

text: set

text: of

text: edges

text: is

text: dense

text: in

text: so

text: +P

text: ]f

text: for

text: all

text: f

text: By

text: linearity

text: [P

text: +P

text: ]h

text: [a,b]

text: ∈(a,b)

text: [a,b]

text: (a,b)

text: for

text: any

text: h

text: with

text: support

text: in

text: a

text: ﬁnite

text: set

text: of

text: edges,

text: and

text: since

text: the

text: projections

text: are

text: bounded

text: we

text: conclude

text: that

text: P

text: P

text: and

text: (α,

text: β)

text: [a,b]

text: (a,b)

text: is

text: in

text: the

text: resolvent

text: set

text: p.

text: Corollary

text: Assume

text: σ

text: Z

text: and

text: for

text: m

text: M

text: suppose

text: µ

text: (σ)

text: Then

text: σ

text: is

text: in

text: the

text: resolvent

text: set

text: of

text: ∆+q

text: if

text: and

text: only

text: m

text: if

text: µ+(σ)

text: is

text: real

text: valued

text: in

text: open

text: neighborhood

text: of

text: σ

text: for

text: m

text: M.

text: m

text: Proof.

text: If

text: µ+(σ)

text: is

text: real

text: valued,

text: then

text: the

text: symmetry

text: µ

text: (λ)

text: µ

text: (λ)

text: m

text: m

text: m

text: established

text: in

text: Lemma

text: means

text: µ

text: (σ)

text: has

text: the

text: same

text: real

text: value.

text: The

text: m

text: same

text: symmetry

text: also

text: implies

text: that

text: δ

text: (σ)

text: if

text: µ+(σ)

text: is

text: not

text: real

text: valued.

text: m

text: m

text: Theorem

text: For

text: q

text: and

text: M

text: the

text: spectrum

text: of

text: q

text: has

text: a

text: strictly

text: positive

text: lower

text: bound.

text: Proof.

text: Since

text: q)f,

text: f

text: ∆f,

text: f

text: q

text: f

text: T

text: M

text: it

text: suﬃces

text: to

text: verify

text: the

text: result

text: when

text: q

text: Consider

text: the

text: case

text: when

text: the

text: edge

text: lengths

text: l

text: are

text: all

text: equal

text: to

text: Then

text: m

text: the

text: system

text: reduces

text: to

text: λ)µ(λ)

text: The

text: quadratic

text: formula

text: gives

text: λ)

text: λ)

text: µ(λ)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: T Figure 1: A rescaled graph 2 F to ι in . Since this word can be reduced to the empty word, it M − − must have adjacent symbols s s 1 or s 1s . This means the ver- m m m m T tices w , . . . , w are not distinct, so no such cycle exists. Since is 1 K T M connected by Proposition 3.2 and has no cycles, is a tree. M T 3.3 Abelian subgroups and multipliers for M ∈ T Each edge e = (v, vs) is a bridge. With v = v− and vs = v , ± M T + the subgraphs Γ described above will be subtrees of , denoted by T ± e X± M . The vector spaces are as in Lemma 2.1. e (cid:15) T ∈ C\ ∞ Lemma 3.4. Suppose e = (v, vs) is an edge of and λ [0, ). ± X± ± M If y is a nontrivial element of , then y is nowhere vanishing on T ± e . e ± ∈ T ± ± Proof. Suppose y (x ) = 0 for some x . First notice that y 0 0 e T must then vanish identically on the subtree consisting of points T ± 0 x of with the property that paths from x to v must include 1 e 1 x . Otherwise, a nonnegative self-adjoint operator ∆ + q could be 0 9 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: T

label: caption

text: Figure

text: A

text: rescaled

text: graph

text: F

text: to

text: ι

text: in

text: Since

text: this

text: word

text: can

text: be

text: reduced

text: to

text: the

text: empty

text: word,

text: it

text: M

text: must

text: have

text: adjacent

text: symbols

text: s

text: s

text: or

text: s

text: This

text: means

text: the

text: ver-

text: m

text: m

text: m

text: m

text: T

text: tices

text: w

text: w

text: are

text: not

text: distinct,

text: so

text: no

text: such

text: cycle

text: exists.

text: Since

text: is

text: K

text: T

text: M

text: connected

text: by

text: Proposition

text: and

text: has

text: no

text: cycles,

text: is

text: a

text: tree.

text: M

text: T

label: section

text: Abelian

text: subgroups

text: and

text: multipliers

text: for

text: M

text: T

text: Each

text: edge

text: e

text: (v,

text: vs)

text: is

text: a

text: bridge.

text: With

text: v

text: v−

text: and

text: vs

text: v

text: M

text: T

text: the

text: subgraphs

text: Γ

text: described

text: above

text: will

text: be

text: subtrees

text: of

text: denoted

text: by

text: T

text: e

text: X±

text: M

text: The

text: vector

text: spaces

text: are

text: as

text: in

text: Lemma

text: e

text: T

text: C\

text: Lemma

text: Suppose

text: e

text: (v,

text: vs)

text: is

text: an

text: edge

text: of

text: and

text: λ

text: X±

text: M

text: If

text: y

text: is

text: a

text: nontrivial

text: element

text: of

text: then

text: y

text: is

text: nowhere

text: vanishing

text: on

text: T

text: e

text: e

text: T

text: Proof.

text: Suppose

text: y

text: (x

text: for

text: some

text: x

text: First

text: notice

text: that

text: y

text: e

text: T

text: must

text: then

text: vanish

text: identically

text: on

text: the

text: subtree

text: consisting

text: of

text: points

text: T

text: x

text: of

text: with

text: the

text: property

text: that

text: paths

text: from

text: x

text: to

text: v

text: must

text: include

text: e

text: x

text: Otherwise,

text: a

text: nonnegative

text: self-adjoint

text: operator

text: q

text: could

text: be

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: with just two of the us required in a correlator. These operators are invariant under the discrete symmetry that changes the sign of Ψ, χ and the ghosts. Imposing invariance under this symmetry will exclude mixing between the ingredients of these operators thought of as parts of Sl and others that might be part of Sr. We will refer to this as GSO symmetry. The correlators of these currents lead to the reduced Pfaﬃans of CHY: (cid:104) (cid:105) (cid:48) 1 u u v . . . v = Pf (M) = − Pf(M ) , 1 2 3 n 12 σ σ   1 2 × × where M is the skew 2n 2n matrix with n n block decomposition        −  · · A CT k k (cid:15) (cid:15) i j i j M = , A = , B = , ij ij σ σ ij ij C B and · (cid:15) k (cid:54) − · j C = , i = j, C = (cid:15) P (σ ) , ij ii i i σ ij and M is M with the ﬁrst two rows and columns removed. 12 3.3 Comb system S . CS The comb system [24] was introduced as a way of obtaining colour factors as se- quences of contractions of structure constants rather than as colour ordered traces. In general, such contractions can be generated from trivalent diagrams with the structure constants fabc of some Lie algebra at the vertices and contractions δab along the inter- nal edges. It is well known that these are linearly dependent as a consequence of the − Kleiss-Kuijf relations with a basis being given by ‘combs’, with n 2 vertices lined up in a row [32, 33] and end points given by 1 and n: → · · · fa1a2b1fb1a3b2 fbn−3an−1an. The comb system has the remarkable property that, in conjunction with worldsheet supersymmetry, only these combs arise from correlators and not the multitrace terms that arise from an ordinary current algebra. This system arises from an action for ∈ ⊗ matter ﬁelds ρ, ρ˜, q, y g(cid:90) K1/2 i.e., worldsheet spinors taking values in the Lie (cid:16) (cid:17) algebra g of some gauge group. The worldsheet action is · · · ¯ ¯ 1 S = ρ˜ ∂ρ + q ∂y + χρ [ρ, ρ˜] + [q, y] CS 2 – 10 – ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: with

text: just

text: two

text: of

text: the

text: us

text: required

text: in

text: a

text: correlator.

text: These

text: operators

text: are

text: invariant

text: under

text: the

text: discrete

text: symmetry

text: that

text: changes

text: the

text: sign

text: of

text: Ψ,

text: χ

text: and

text: the

text: ghosts.

text: Imposing

text: invariance

text: under

text: this

text: symmetry

text: will

text: exclude

text: mixing

text: between

text: the

text: ingredients

text: of

text: these

text: operators

text: thought

text: of

text: as

text: parts

text: of

text: Sl

text: and

text: others

text: that

text: might

text: be

text: part

text: of

text: Sr.

text: We

text: will

text: refer

text: to

text: this

text: as

text: GSO

text: symmetry.

text: The

text: correlators

text: of

text: these

text: currents

text: lead

text: to

text: the

text: reduced

text: Pfaﬃans

text: of

text: CHY:

text: u

text: u

text: v

text: v

text: Pf

text: (M)

text: Pf(M

text: n

text: σ

text: σ

text: where

text: M

text: is

text: the

text: skew

text: matrix

text: with

text: n

text: n

text: block

text: decomposition

text: A

text: CT

text: k

text: k

text: i

text: j

text: i

text: j

text: M

text: A

text: B

text: ij

text: ij

text: σ

text: σ

text: ij

text: ij

text: C

text: B

text: and

text: k

text: j

text: C

text: i

text: j,

text: C

text: P

text: (σ

text: ij

text: ii

text: i

text: i

text: σ

text: ij

text: and

text: M

text: is

text: M

text: with

text: the

text: ﬁrst

text: two

text: rows

text: and

text: columns

text: removed.

text: Comb

text: system

text: S

text: CS

text: The

text: comb

text: system

text: was

text: introduced

text: as

text: a

text: way

text: of

text: obtaining

text: colour

text: factors

text: as

text: se-

text: quences

text: of

text: contractions

text: of

text: structure

text: constants

text: rather

text: than

text: as

text: colour

text: ordered

text: traces.

text: In

text: general,

text: such

text: contractions

text: can

text: be

text: generated

text: from

text: trivalent

text: diagrams

text: with

text: the

text: structure

text: constants

text: fabc

text: of

text: some

text: Lie

text: algebra

text: at

text: the

text: vertices

text: and

text: contractions

text: δab

text: along

text: the

text: inter-

text: nal

text: edges.

text: It

text: is

text: well

text: known

text: that

text: these

text: are

text: linearly

text: dependent

text: as

text: a

text: consequence

text: of

text: the

text: Kleiss-Kuijf

text: relations

text: with

text: a

text: basis

text: being

text: given

text: by

text: ‘combs’,

text: with

text: n

text: vertices

text: lined

text: up

text: in

text: a

text: row

text: and

text: end

text: points

text: given

text: by

text: and

text: n:

text: The

text: comb

text: system

text: has

text: the

text: remarkable

text: property

text: that,

text: in

text: conjunction

text: with

text: worldsheet

text: supersymmetry,

text: only

text: these

text: combs

text: arise

text: from

text: correlators

text: and

text: not

text: the

text: multitrace

text: terms

text: that

text: arise

text: from

text: an

text: ordinary

text: current

text: algebra.

text: This

text: system

text: arises

text: from

text: an

text: action

text: for

text: matter

text: ﬁelds

text: ρ,

text: ρ˜,

text: q,

text: y

text: i.e.,

text: worldsheet

text: spinors

text: taking

text: values

text: in

text: the

text: Lie

text: algebra

text: g

text: of

text: some

text: gauge

text: group.

text: The

text: worldsheet

text: action

text: is

text: S

text: ρ˜

text: ∂ρ

text: q

text: ∂y

text: χρ

text: [ρ,

text: ρ˜]

text: [q,

text: y]

text: CS

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##