text: 8 P. R. Kaﬂe et al. the tracer population. Finally, substituting the above mea- Gyr ago events), which have not had enough time to fully sured tracer properties into the Jeans formalism (equation 2) relax suﬀers the worst from this scenario. Also, highl halo we estimate the mass proﬁles of the parent galaxy, which (dominated by the massive accretion events) shows signif- are shown with solid lines in panels (k) and (l). For an easy icant undulations in mass proﬁles at large radius. On the comparison, in panels (k) and (l) we also over-plot the inher- contrary, halos such as halo14 does not contain any domi- ent mass proﬁles of the parent galaxy assuming maximum nant sub-structures and hence, result a well-behaving mass (red dashed-line), minimum (green dashed-line) and aver- proﬁle. Similarly, the halos such as old and lowl, which are age (black dashed-line) values of concentration adopted from expected to be better phase-mixed halos by construction, Fig. 1. The width of the lines in panels (a-j) depict the scat- have comparatively smoother mass proﬁles. ter around the relation obtained from bootstrapping whereas In the end, we like to validate if the erratic mass mea- to avoid further cluttering we only show mean relations in surements at intermittent radii are due to the presence of panels (k) and (l). There are a few trends we note in Fig. 4. sub-structures that are out of equilibrium. The mean radial For example, both the σr and σθ (also σφ, not shown in the velocity vr = 0 is a necessary condition for a system in equi- ﬁgure) attain highest value for small r and vice-versa, which librium (although not a suﬃcient condition). So, |(cid:104)vr(cid:105)|/σr turn-over at r (cid:39) 5 kpc. Also, all of the halos except the can be used to gauge the departure from equilibrium. Note, √ circ have predominantly radial orbits i.e. β > 0 (see panels for N stars in a bin |(cid:104)vr(cid:105)|/(σr/ N) is a better measure of e and f). Furthermore, it can be visually attested that the the statistically signiﬁcance of departure of vr from zero, as logarithmic density distributions ρ(r) of the tracer particles it takes the eﬀect of Poisson noise into account. However, we seem to follow a double power-law with a break at r ∼ 20 choose to study |(cid:104)vr(cid:105)|/σr as it is physically more meaningful kpc. and signiﬁcant. From equation 1 we get, 4.2 Mass reconstruction from tracer population δM/M (cid:39) 2δσr/σr (cid:39) 2vr/σr, (6) For clarity, in Fig. 5 we recast the mass proﬁles of the parent galaxy of 11 ΛCDM (top panels) and the 6 artiﬁ- cial (bottom panels) halos, taken originally from panels (k) assuming that the change in dispersion δσr is of the and (l) of Fig. 4. The magenta bands with solid lines show order of vr. Hence, vr/σr is useful to gauge if a change in the mass proﬁles of the parent galaxy reconstructed from |(cid:104)vr(cid:105)|/σr is enough to explain a corresponding change in the Jeans equation using the spatio-kinematic proﬁles of the mass proﬁle. The magenta dashed lines in all the tiny panels stellar tracer populations. The bands around the lines show of Fig. 5 we show |(cid:104)vr(cid:105)|/σr the proﬁle. Additionally, to check the bootstrapped uncertainties around the mean measure- that a spike in |(cid:104)vr(cid:105)|/σr is statistically signiﬁcant and not ments. Furthermore, the intrinsic mass distributions of the caused by Poisson noise, we show the dispersion with band parent galaxy is shown with the black dashed-line. As men- around |(cid:104)vr(cid:105)|/σr proﬁle using bootstrapping. We ﬁnd that tioned earlier in Section , the intrinsic mass proﬁles of the the erratic spikes seen in the mass proﬁles at many places, artiﬁcial halos are not known as they are constructed from a e.g., halo09 at ∼ 55 kpc, halo02 at ∼ 20 kpc, coincide with mixture of satellites that have been evolved in diﬀerent host spikes in |(cid:104)vr(cid:105)|/σr runs. Moreover, we see that the mass potential. Therefore, in the case of artiﬁcial halos we only reconstructions outside r > 60 kpc generally deteriorate for use the black dashed-lines as a rough guide for the purpose all the halos, which is mainly because of the paucity of mass of a qualitative assessment, which are derived assuming the tracers and dominance of unrelaxed sub-structures in the concentration of the artiﬁcial halos to be an average concen- outskirts. tration of the satellites that make up these halos. The spikes In the ﬁgure the blue solid and dashed lines are the cor- in the mass proﬁles are far more prominent than the uncer- responding reconstructed mass proﬁles of the parent galaxy tainties in the measured values, therefore statistical noise as when the accreted DM particles are used as a dynamical a potential cause for the spikes can be ruled out. We now tracer. The dark matter particles are not directly observable investigate the cause of these spikes. and the discussion of this case is not useful for pragmatic rea- We expect that the mass distributions of the parent sons. However, for the completeness reason we brieﬂy present galaxy monotonically increase as a function of radius. How- and discuss the spatio-kinematic proﬁles and error analysis ever, this is not strictly the case for our estimated mass pro- of this case in the Appendix . For almost all the halos, we ob- ﬁles, again magenta lines shown in Fig. 5. At many places serve that the stellar tracers show comparatively less ridges the mass proﬁles are bumpy (e.g. halo 09 at r ∼ 60 kpc, halo in the reconstructed mass proﬁles compare to the cases when 10 at r ∼ 70 kpc) and in some cases the cumulative mass DM tracers are used. This is because the DM tracer popu- dips (e.g. halo 08 at r > 60 kpc, highl halo at r > 50 kpc), lation have large number of prominent substructures with which is unphysical. Similarly, the mass proﬁles of the highl zero luminosity, and hence zero stellar mass. (cid:38) halo at r 60 kpc and young halo at all radius are also The intermittent noise in the inferred mass distributions bumpy. This discrepant output of our scheme can be under- can be reduced by ﬁtting a smooth parametric models to the stood in conjunction with the radial phase-space diagram velocity dispersions and pressure/density runs that enter the shown in Fig. 2. It is clear from this ﬁgure that at some Jeans analysis. We restrain from doing so mainly because a intermittent radius satellites are just dispersing (say, halo smooth model can not capture the impact of substructures 15 at r ∼ 80 kpc, halo 09 at r ∼ 60 kpc etc). In such ra- in the mass proﬁles. Also, it leads to a natural question of dial shells we will measure biased σr and ρ(r) and hence, what are the good models for the dispersion proﬁles, number the Jeans equation locally fails here as it demands velocity density etc. Therefore, we decided to use the binned data dispersions of stars evenly populating the velocity range not directly. one biased by clumps. The young halo (built from recent <8 A takeaway point from the Fig. 5 is that in overall the MNRAS 000, 1–16 (0000) ##LTLine##

text: P.

text: R.

text: Kaﬂe

text: et

text: al.

text: the

text: tracer

text: population.

text: Finally,

text: substituting

text: the

text: above

text: mea-

text: Gyr

text: ago

text: events),

text: which

text: have

text: not

text: had

text: enough

text: time

text: to

text: fully

text: sured

text: tracer

text: properties

text: into

text: the

text: Jeans

text: formalism

text: (equation

text: relax

text: suﬀers

text: the

text: worst

text: from

text: this

text: scenario.

text: Also,

text: highl

text: halo

text: we

text: estimate

text: the

text: mass

text: proﬁles

text: of

text: the

text: parent

text: galaxy,

text: which

text: (dominated

text: by

text: the

text: massive

text: accretion

text: events)

text: shows

text: signif-

text: are

text: shown

text: with

text: solid

text: lines

text: in

text: panels

text: (k)

text: and

text: (l).

text: For

text: an

text: easy

text: icant

text: undulations

text: in

text: mass

text: proﬁles

text: at

text: large

text: radius.

text: On

text: the

text: comparison,

text: in

text: panels

text: (k)

text: and

text: (l)

text: we

text: also

text: over-plot

text: the

text: inher-

text: contrary,

text: halos

text: such

text: as

text: does

text: not

text: contain

text: any

text: domi-

text: ent

text: mass

text: proﬁles

text: of

text: the

text: parent

text: galaxy

text: assuming

text: maximum

text: nant

text: sub-structures

text: and

text: hence,

text: result

text: a

text: well-behaving

text: mass

text: (red

text: dashed-line),

text: minimum

text: (green

text: dashed-line)

text: and

text: aver-

text: proﬁle.

text: Similarly,

text: the

text: halos

text: such

text: as

text: old

text: and

text: lowl,

text: which

text: are

text: age

text: (black

text: dashed-line)

text: values

text: of

text: concentration

text: adopted

text: from

text: expected

text: to

text: be

text: better

text: phase-mixed

text: halos

text: by

text: construction,

text: Fig.

text: The

text: width

text: of

text: the

text: lines

text: in

text: panels

text: (a-j)

text: depict

text: the

text: scat-

text: have

text: comparatively

text: smoother

text: mass

text: proﬁles.

text: ter

text: around

text: the

text: relation

text: obtained

text: from

text: bootstrapping

text: whereas

text: In

text: the

text: end,

text: we

text: like

text: to

text: validate

text: if

text: the

text: erratic

text: mass

text: mea-

text: to

text: avoid

text: further

text: cluttering

text: we

text: only

text: show

text: mean

text: relations

text: in

text: surements

text: at

text: intermittent

text: radii

text: are

text: due

text: to

text: the

text: presence

text: of

text: panels

text: (k)

text: and

text: (l).

text: There

text: are

text: a

text: few

text: trends

text: we

text: note

text: in

text: Fig.

text: sub-structures

text: that

text: are

text: out

text: of

text: equilibrium.

text: The

text: mean

text: radial

text: For

text: example,

text: both

text: the

text: σr

text: and

text: σθ

text: (also

text: σφ,

text: not

text: shown

text: in

text: the

text: velocity

text: vr

text: is

text: a

text: necessary

text: condition

text: for

text: a

text: system

text: in

text: equi-

text: ﬁgure)

text: attain

text: highest

text: value

text: for

text: small

text: r

text: and

text: vice-versa,

text: which

text: librium

text: (although

text: not

text: a

text: suﬃcient

text: condition).

text: So,

text: turn-over

text: at

text: r

text: kpc.

text: Also,

text: all

text: of

text: the

text: halos

text: except

text: the

text: can

text: be

text: used

text: to

text: gauge

text: the

text: departure

text: from

text: equilibrium.

text: Note,

text: circ

text: have

text: predominantly

text: radial

text: orbits

text: i.e.

text: β

text: (see

text: panels

text: for

text: N

text: stars

text: in

text: a

text: bin

text: N)

text: is

text: a

text: better

text: measure

text: of

text: e

text: and

text: f).

text: Furthermore,

text: it

text: can

text: be

text: visually

text: attested

text: that

text: the

text: the

text: statistically

text: signiﬁcance

text: of

text: departure

text: of

text: vr

text: from

text: zero,

text: as

text: logarithmic

text: density

text: distributions

text: ρ(r)

text: of

text: the

text: tracer

text: particles

text: it

text: takes

text: the

text: eﬀect

text: of

text: Poisson

text: noise

text: into

text: account.

text: However,

text: we

text: seem

text: to

text: follow

text: a

text: double

text: power-law

text: with

text: a

text: break

text: at

text: r

text: choose

text: to

text: study

text: as

text: it

text: is

text: physically

text: more

text: meaningful

text: kpc.

text: and

text: signiﬁcant.

text: From

text: equation

text: we

text: get,

text: Mass

text: reconstruction

text: from

text: tracer

text: population

text: δM/M

text: For

text: clarity,

text: in

text: Fig.

text: we

text: recast

text: the

text: mass

text: proﬁles

text: of

text: the

text: parent

text: galaxy

text: of

text: ΛCDM

text: (top

text: panels)

text: and

text: the

text: artiﬁ-

text: cial

text: (bottom

text: panels)

text: halos,

text: taken

text: originally

text: from

text: panels

text: (k)

text: assuming

text: that

text: the

text: change

text: in

text: dispersion

text: δσr

text: is

text: of

text: the

text: and

text: (l)

text: of

text: Fig.

text: The

text: magenta

text: bands

text: with

text: solid

text: lines

text: show

text: order

text: of

text: vr.

text: Hence,

text: vr/σr

text: is

text: useful

text: to

text: gauge

text: if

text: a

text: change

text: in

text: the

text: mass

text: proﬁles

text: of

text: the

text: parent

text: galaxy

text: reconstructed

text: from

text: is

text: enough

text: to

text: explain

text: a

text: corresponding

text: change

text: in

text: the

text: Jeans

text: equation

text: using

text: the

text: spatio-kinematic

text: proﬁles

text: of

text: the

text: mass

text: proﬁle.

text: The

text: magenta

text: dashed

text: lines

text: in

text: all

text: the

text: tiny

text: panels

text: stellar

text: tracer

text: populations.

text: The

text: bands

text: around

text: the

text: lines

text: show

text: of

text: Fig.

text: we

text: show

text: the

text: proﬁle.

text: Additionally,

text: to

text: check

text: the

text: bootstrapped

text: uncertainties

text: around

text: the

text: mean

text: measure-

text: that

text: a

text: spike

text: in

text: is

text: statistically

text: signiﬁcant

text: and

text: not

text: ments.

text: Furthermore,

text: the

text: intrinsic

text: mass

text: distributions

text: of

text: the

text: caused

text: by

text: Poisson

text: noise,

text: we

text: show

text: the

text: dispersion

text: with

text: band

text: parent

text: galaxy

text: is

text: shown

text: with

text: the

text: black

text: dashed-line.

text: As

text: men-

text: around

text: proﬁle

text: using

text: bootstrapping.

text: We

text: ﬁnd

text: that

text: tioned

text: earlier

text: in

text: Section

text: the

text: intrinsic

text: mass

text: proﬁles

text: of

text: the

text: the

text: erratic

text: spikes

text: seen

text: in

text: the

text: mass

text: proﬁles

text: at

text: many

text: places,

text: artiﬁcial

text: halos

text: are

text: not

text: known

text: as

text: they

text: are

text: constructed

text: from

text: a

text: e.g.,

text: at

text: kpc,

text: at

text: kpc,

text: coincide

text: with

text: mixture

text: of

text: satellites

text: that

text: have

text: been

text: evolved

text: in

text: diﬀerent

text: host

text: spikes

text: in

text: runs.

text: Moreover,

text: we

text: see

text: that

text: the

text: mass

text: potential.

text: Therefore,

text: in

text: the

text: case

text: of

text: artiﬁcial

text: halos

text: we

text: only

text: reconstructions

text: outside

text: r

text: kpc

text: generally

text: deteriorate

text: for

text: use

text: the

text: black

text: dashed-lines

text: as

text: a

text: rough

text: guide

text: for

text: the

text: purpose

text: all

text: the

text: halos,

text: which

text: is

text: mainly

text: because

text: of

text: the

text: paucity

text: of

text: mass

text: of

text: a

text: qualitative

text: assessment,

text: which

text: are

text: derived

text: assuming

text: the

text: tracers

text: and

text: dominance

text: of

text: unrelaxed

text: sub-structures

text: in

text: the

text: concentration

text: of

text: the

text: artiﬁcial

text: halos

text: to

text: be

text: an

text: average

text: concen-

text: outskirts.

text: tration

text: of

text: the

text: satellites

text: that

text: make

text: up

text: these

text: halos.

text: The

text: spikes

text: In

text: the

text: ﬁgure

text: the

text: blue

text: solid

text: and

text: dashed

text: lines

text: are

text: the

text: cor-

text: in

text: the

text: mass

text: proﬁles

text: are

text: far

text: more

text: prominent

text: than

text: the

text: uncer-

text: responding

text: reconstructed

text: mass

text: proﬁles

text: of

text: the

text: parent

text: galaxy

text: tainties

text: in

text: the

text: measured

text: values,

text: therefore

text: statistical

text: noise

text: as

text: when

text: the

text: accreted

text: DM

text: particles

text: are

text: used

text: as

text: a

text: dynamical

text: a

text: potential

text: cause

text: for

text: the

text: spikes

text: can

text: be

text: ruled

text: out.

text: We

text: now

text: tracer.

text: The

text: dark

text: matter

text: particles

text: are

text: not

text: directly

text: observable

text: investigate

text: the

text: cause

text: of

text: these

text: spikes.

text: and

text: the

text: discussion

text: of

text: this

text: case

text: is

text: not

text: useful

text: for

text: pragmatic

text: rea-

text: We

text: expect

text: that

text: the

text: mass

text: distributions

text: of

text: the

text: parent

text: sons.

text: However,

text: for

text: the

text: completeness

text: reason

text: we

text: brieﬂy

text: present

text: galaxy

text: monotonically

text: increase

text: as

text: a

text: function

text: of

text: radius.

text: How-

text: and

text: discuss

text: the

text: spatio-kinematic

text: proﬁles

text: and

text: error

text: analysis

text: ever,

text: this

text: is

text: not

text: strictly

text: the

text: case

text: for

text: our

text: estimated

text: mass

text: pro-

text: of

text: this

text: case

text: in

text: the

text: Appendix

text: For

text: almost

text: all

text: the

text: halos,

text: we

text: ob-

text: ﬁles,

text: again

text: magenta

text: lines

text: shown

text: in

text: Fig.

text: At

text: many

text: places

text: serve

text: that

text: the

text: stellar

text: tracers

text: show

text: comparatively

text: less

text: ridges

text: the

text: mass

text: proﬁles

text: are

text: bumpy

text: (e.g.

text: halo

text: at

text: r

text: kpc,

text: halo

text: in

text: the

text: reconstructed

text: mass

text: proﬁles

text: compare

text: to

text: the

text: cases

text: when

text: at

text: r

text: kpc)

text: and

text: in

text: some

text: cases

text: the

text: cumulative

text: mass

text: DM

text: tracers

text: are

text: used.

text: This

text: is

text: because

text: the

text: DM

text: tracer

text: popu-

text: dips

text: (e.g.

text: halo

text: at

text: r

text: kpc,

text: highl

text: halo

text: at

text: r

text: kpc),

text: lation

text: have

text: large

text: number

text: of

text: prominent

text: substructures

text: with

text: which

text: is

text: unphysical.

text: Similarly,

text: the

text: mass

text: proﬁles

text: of

text: the

text: highl

text: zero

text: luminosity,

text: and

text: hence

text: zero

text: stellar

text: mass.

text: halo

text: at

text: r

text: kpc

text: and

text: young

text: halo

text: at

text: all

text: radius

text: are

text: also

text: The

text: intermittent

text: noise

text: in

text: the

text: inferred

text: mass

text: distributions

text: bumpy.

text: This

text: discrepant

text: output

text: of

text: our

text: scheme

text: can

text: be

text: under-

text: can

text: be

text: reduced

text: by

text: ﬁtting

text: a

text: smooth

text: parametric

text: models

text: to

text: the

text: stood

text: in

text: conjunction

text: with

text: the

text: radial

text: phase-space

text: diagram

text: velocity

text: dispersions

text: and

text: pressure/density

text: runs

text: that

text: enter

text: the

text: shown

text: in

text: Fig.

text: It

text: is

text: clear

text: from

text: this

text: ﬁgure

text: that

text: at

text: some

text: Jeans

text: analysis.

text: We

text: restrain

text: from

text: doing

text: so

text: mainly

text: because

text: a

text: intermittent

text: radius

text: satellites

text: are

text: just

text: dispersing

text: (say,

text: halo

text: smooth

text: model

text: can

text: not

text: capture

text: the

text: impact

text: of

text: substructures

text: at

text: r

text: kpc,

text: halo

text: at

text: r

text: kpc

text: etc).

text: In

text: such

text: ra-

text: in

text: the

text: mass

text: proﬁles.

text: Also,

text: it

text: leads

text: to

text: a

text: natural

text: question

text: of

text: dial

text: shells

text: we

text: will

text: measure

text: biased

text: σr

text: and

text: ρ(r)

text: and

text: hence,

text: what

text: are

text: the

text: good

text: models

text: for

text: the

text: dispersion

text: proﬁles,

text: number

text: the

text: Jeans

text: equation

text: locally

text: fails

text: here

text: as

text: it

text: demands

text: velocity

text: density

text: etc.

text: Therefore,

text: we

text: decided

text: to

text: use

text: the

text: binned

text: data

text: dispersions

text: of

text: stars

text: evenly

text: populating

text: the

text: velocity

text: range

text: not

text: directly.

text: one

text: biased

text: by

text: clumps.

text: The

text: young

text: halo

text: (built

text: from

text: recent

text: A

text: takeaway

text: point

text: from

text: the

text: Fig.

text: is

text: that

text: in

text: overall

text: the

text: MNRAS

text: ##LTLine##


text: 10 P. R. Kaﬂe et al. 100 1.0 (a) (b) average 75 σmass oﬀset 0.8 50 25 0.6 0 0.4 −25 −50 0.2 −75 −100 0.0 101 102 101 102 r (kpc) r (kpc) 0.035 r/kpc ≥ 10 (c) r/kpc ≥ 10 (d) (per. = 10.6+15.1%) 0.030 (per. = 12.4+15.8%) −14.1 −13.3 1 < r/kpc < 100 1 < r/kpc < 100 0.030 (per. = 10.6+21.4%) (per. = 13.6+22.0%) −20.2 0.025 −24.8 r/kpc < 10 r/kpc < 10 0.025 (per. = 10.9+−2460..74%) (per. = 16.1+−3404..27%) 0.020 0.020 0.015 0.015 0.010 0.010 0.005 0.005 0.000 0.000 −80 −60 −40 −20 0 20 40 60 80 −80 −60 −40 −20 0 20 40 60 80 mass oﬀset(%) mass oﬀset(%) Figure 6. Error analysis of the estimated masses of the parent galaxy using the stellar tracer population of the 11 ΛCDM halos. Panel (a) shows the percentage mass oﬀset and panel (b) shows the random uncertainties in mass measurements, both as a function of distance, where diﬀerent colours represent diﬀerent halos as labelled in Fig.1 and for the aesthetic reason we have not shown the uncertainty bands around these trends. Panels (c) and (d) show distributions of the mass oﬀset for cases where we respectively use standard deviation and root-mean square velocity as a measure for the velocity dispersion. shape of the mass proﬁles of the parent galaxy of the simu- the potential in which they are orbiting. We can estimate lated stellar halos (except for the ﬂagged halos such as highl, the random uncertainty by bootstrapping and this is given young), although bumpier at places where sub-structures lo- by σM. We label the dispersion due to non-equilibrium ef- cally dominate, can be recovered well using the Jeans anal- fects by σnoneq. The dispersion in mass oﬀset can then be ysis. Below we provide a more detail account of the biases w(cid:16)ritten as (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) in our overall mass reconstruction. In Fig. 6 we present the error analysis of the mass mea- σmass oﬀset 2 = σM 2 + σnoneq 2 (7) surements of the parent galaxy using the stellar tracer pop- M M M ulations for all 11 ΛCDM halos. Panel (a) shows the mass oﬀsets (deﬁned in equation 5) in the estimated mass MJeans If we consider the standard error of mean around the av- compared to the intrinsic masses as a function of r. Similarly, erage σmass oﬀset/M relation in Fig. 6 (b), the solid black line in panel (b) we show the fractional uncertainty (σM/M) on coincides within the 2σ conﬁdence interval of the mean rela- the estimated mass measured using bootstrapping as a func- tion in the inner r < 10 kpc. The two relations tracing each tion of r. Diﬀerent colours of the circles or lines in panels (a) other in this regime means that the dispersion in mass oﬀ- and (b) represents diﬀerent halos, the labellings consistent set is dominated by random uncertainty due to Poisson noise with Fig. 1. The average of the coloured lines in panel (b) and can be reduced by increasing the sample size of the trac- is shown with the black dashed-line whereas the black solid ers. Moreover, the large uncertainty in the inner r < 10 kpc line is the dispersion proﬁle of the mass oﬀsets obtained from is of the least concern to us as we are mainly interested in (cid:38) (cid:38) panel (a). measuring mass for r 10 kpc. However, for r 10 kpc, the The mass oﬀset is due to two sources, a) random uncer- dispersion in mass oﬀset is consistently larger than the ran- tainty in the estimator, which is mainly due to Poisson noise dom uncertainty and the diﬀerence keeps on increasing with and b) the tracers not being in dynamical equilibrium with increase of r. This means that dispersion is dominated by MNRAS 000, 1–16 (0000) )%(tesﬀo ssam fdp M/ σ fdp snaeJ snaeJM ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## 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label: paragraph

text: P.

text: R.

text: Kaﬂe

text: et

text: al.

font: default

text: (a)

text: (b)

text: average

text: σmass

text: oﬀset

text: r

text: (kpc)

text: r

text: (kpc)

text: r/kpc

text: (c)

text: r/kpc

text: (d)

text: (per.

text: (per.

text: r/kpc

text: r/kpc

text: (per.

text: (per.

text: r/kpc

text: r/kpc

text: (per.

text: (per.

text: mass

text: oﬀset(%)

text: mass

text: oﬀset(%)

text: Figure

text: Error

text: analysis

text: of

text: the

text: estimated

text: masses

text: of

text: the

text: parent

text: galaxy

text: using

text: the

text: stellar

text: tracer

text: population

text: of

text: the

text: ΛCDM

text: halos.

text: Panel

text: (a)

text: shows

text: the

text: percentage

text: mass

text: oﬀset

text: and

text: panel

text: (b)

text: shows

text: the

text: random

text: uncertainties

text: in

text: mass

text: measurements,

text: both

text: as

text: a

text: function

text: of

text: distance,

text: where

text: diﬀerent

text: colours

text: represent

text: diﬀerent

text: halos

text: as

text: labelled

text: in

text: and

text: for

text: the

text: aesthetic

text: reason

text: we

text: have

text: not

text: shown

text: the

text: uncertainty

text: bands

text: around

text: these

text: trends.

text: Panels

text: (c)

text: and

text: (d)

text: show

text: distributions

text: of

text: the

text: mass

text: oﬀset

text: for

text: cases

text: where

text: we

text: respectively

text: use

text: standard

text: deviation

text: and

text: root-mean

text: square

text: velocity

text: as

text: a

text: measure

text: for

text: the

text: velocity

text: dispersion.

text: shape

text: of

text: the

text: mass

text: proﬁles

text: of

text: the

text: parent

text: galaxy

text: of

text: the

text: simu-

text: the

text: potential

text: in

text: which

text: they

text: are

text: orbiting.

text: We

text: can

text: estimate

text: lated

text: stellar

text: halos

text: (except

text: for

text: the

text: ﬂagged

text: halos

text: such

text: as

text: highl,

text: the

text: random

text: uncertainty

text: by

text: bootstrapping

text: and

text: this

text: is

text: given

text: young),

text: although

text: bumpier

text: at

text: places

text: where

text: sub-structures

text: lo-

text: by

text: σM.

text: We

text: label

text: the

text: dispersion

text: due

text: to

text: non-equilibrium

text: ef-

text: cally

text: dominate,

text: can

text: be

text: recovered

text: well

text: using

text: the

text: Jeans

text: anal-

text: fects

text: by

text: σnoneq.

text: The

text: dispersion

text: in

text: mass

text: oﬀset

text: can

text: then

text: be

text: ysis.

text: Below

text: we

text: provide

text: a

text: more

text: detail

text: account

text: of

text: the

text: biases

text: as

text: in

text: our

text: overall

text: mass

text: reconstruction.

text: In

text: Fig.

text: we

text: present

text: the

text: error

text: analysis

text: of

text: the

text: mass

text: mea-

text: σmass

text: oﬀset

text: σM

text: σnoneq

text: surements

text: of

text: the

text: parent

text: galaxy

text: using

text: the

text: stellar

text: tracer

text: pop-

text: M

text: M

text: M

text: ulations

text: for

text: all

text: ΛCDM

text: halos.

text: Panel

text: (a)

text: shows

text: the

text: mass

text: oﬀsets

text: (deﬁned

text: in

text: equation

text: in

text: the

text: estimated

text: mass

text: MJeans

text: If

text: we

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text: the

text: standard

text: error

text: of

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text: compared

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text: the

text: intrinsic

text: masses

text: as

text: a

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text: r.

text: Similarly,

text: erage

text: σmass

text: oﬀset/M

text: relation

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text: (b),

text: the

text: solid

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text: panel

text: (b)

text: we

text: show

text: the

text: fractional

text: uncertainty

text: (σM/M)

text: on

text: coincides

text: within

text: the

text: 2σ

text: conﬁdence

text: interval

text: of

text: the

text: mean

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text: the

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text: tion

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text: the

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text: r

text: kpc.

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text: r.

text: Diﬀerent

text: colours

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text: the

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text: measuring

text: mass

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text: for

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text: mass

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text: Tractable models of self-sustaining autocatalytic networks Mike Steela and Wim Hordijkb a Biomathematics Research Centre, University of Canterbury, Christchurch, New Zealand Email: mike.steel@canterbury.ac.nz b Institute for Advanced Study, University of Amsterdam, The Netherlands Email: wim@WorldWideWanderings.net Abstract Self-sustaining autocatalytic networks play a central role in living systems, from metabolism at the origin of life, simple RNA networks, and the modern cell, to ecology and cognition. A collectively autocatalytic network that can be sustained from an ambient food set is also referred to more formally as a ‘Reﬂexively Autocatalytic F-generated’ (RAF) set. In this paper, we ﬁrst investigate a simpliﬁed setting for studying RAFs, which are nevertheless rel- evant to real biochemistry and allows for a more exact mathematical analysis based on graph-theoretic concepts. This, in turn, allows for the development of eﬃcient (polynomial-time) algorithms for questions that are computation- ally NP-hard in the general RAF setting. We then show how this simpliﬁed setting for RAF systems leads naturally to a more general notion of RAFs that are ‘generative’ (they can be built up from simpler RAFs) and for which eﬃcient algorithms carry over to this more general setting. Finally, we show how classical RAF theory can be extended to deal with ensembles of catalysts as well as the assignment of rates to reactions according to which catalysts (or combinations of catalysts) are available. Keywords autocatalytic network, directed graph, cycles, closure, reaction rates Preprint submitted to Elsevier today ##LTLine## ##LTLine##

text: Tractable

label: title

text: models

text: of

text: self-sustaining

text: autocatalytic

text: networks

text: Mike

label: author

text: Steela

text: and

text: Wim

text: Hordijkb

text: a

text: Biomathematics

text: Research

text: Centre,

text: University

text: of

text: Canterbury,

text: Christchurch,

text: New

text: Zealand

text: Email:

text: mike.steel@canterbury.ac.nz

text: b

text: Institute

text: for

text: Advanced

text: Study,

text: University

text: of

text: Amsterdam,

text: The

text: Netherlands

text: Email:

text: wim@WorldWideWanderings.net

text: Abstract

text: Self-sustaining

label: abstract

text: autocatalytic

text: networks

text: play

text: a

text: central

text: role

text: in

text: living

text: systems,

text: from

text: metabolism

text: at

text: the

text: origin

text: of

text: life,

text: simple

text: RNA

text: networks,

text: and

text: the

text: modern

text: cell,

text: to

text: ecology

text: and

text: cognition.

text: A

text: collectively

text: autocatalytic

text: network

text: that

text: can

text: be

text: sustained

text: from

text: an

text: ambient

text: food

text: set

text: is

text: also

text: referred

text: to

text: more

text: formally

text: as

text: a

text: ‘Reﬂexively

text: Autocatalytic

text: F-generated’

text: (RAF)

text: set.

text: In

text: this

text: paper,

text: we

text: ﬁrst

text: investigate

text: a

text: simpliﬁed

text: setting

text: for

text: studying

text: RAFs,

text: which

text: are

text: nevertheless

text: rel-

text: evant

text: to

text: real

text: biochemistry

text: and

text: allows

text: for

text: a

text: more

text: exact

text: mathematical

text: analysis

text: based

text: on

text: graph-theoretic

text: concepts.

text: This,

text: in

text: turn,

text: allows

text: for

text: the

text: development

text: of

text: eﬃcient

text: (polynomial-time)

text: algorithms

text: for

text: questions

text: that

text: are

text: computation-

text: ally

text: NP-hard

text: in

text: the

text: general

text: RAF

text: setting.

text: We

text: then

text: show

text: how

text: this

text: simpliﬁed

text: setting

text: for

text: RAF

text: systems

text: leads

text: naturally

text: to

text: a

text: more

text: general

text: notion

text: of

text: RAFs

text: that

text: are

text: ‘generative’

text: (they

text: can

text: be

text: built

text: up

text: from

text: simpler

text: RAFs)

text: and

text: for

text: which

text: eﬃcient

text: algorithms

text: carry

text: over

text: to

text: this

text: more

text: general

text: setting.

text: Finally,

text: we

text: show

text: how

text: classical

text: RAF

text: theory

text: can

text: be

text: extended

text: to

text: deal

text: with

text: ensembles

text: of

text: catalysts

text: as

text: well

text: as

text: the

text: assignment

text: of

text: rates

text: to

text: reactions

text: according

text: to

text: which

text: catalysts

text: (or

text: combinations

text: of

text: catalysts)

text: are

text: available.

text: Keywords

text: autocatalytic

text: network,

text: directed

text: graph,

text: cycles,

text: closure,

text: reaction

text: rates

text: Preprint

text: submitted

text: to

text: Elsevier

text: today

text: ##LTLine##

text: ##LTLine##


text: We ﬁrst considered the special setting of ‘elementary’ systems in which all reactions (or at least those present in the maxRAF) have all their reactants present in the food set. This allows for the structure of the collection of RAFs, irrRAFs, and closed subRAFs to be explicitly described graph-theoretically. As a result, some problems that are computationally intractable in the gen- eral CRS setting turn out to be polynomial-time for an elementary CRS. For example, one can eﬃciently ﬁnd the smallest RAFs in an elementary CRS, which is an NP-hard problem in general (Steel et al., 2013). Also, the number of minimal closed subRAF in an elementary CRS is linear in the size of the set of reactions (for a general CRS, they can be exponential in number). For future work, it may be of interest to determine if there are polynomial-time algorithms that can answer the following questions for an elementary CRS: (i) What is the size of the largest irrRAF? (ii) If inhibition is allowed, then is there a RAF that has no inhibition? The concept of an ‘elementary’ CRS is an all-or-nothing notion. One way to extend the results above could be to deﬁne the notion of ‘level’, whereby a CRS has level k if the length of the longest path from the food set to any reaction product goes through at most k reactions (an elementary CRS thus has level 1). We have not explored this further here but instead, we consider the related alternative notion of a generative RAF. Brieﬂy, a generative RAF allows a RAF to form by eﬀectively enlarging its ‘food set’ with products of reactions, so that each step only requires catalysts that are either present or produced by reactions in the RAF at that stage. Although generative RAFs are more complex than elementary ones, their close connection to elementary RAFs (in a stratiﬁed way) allows for a more tractable analysis than for general RAFs. Moreover, unlike elementary RAFs, no special assumption is required on the underlying CRS; generative RAFs are just a special type of RAF that can be generated in a certain sequential fashion in any CRS. In the ﬁnal section, we considered the impact of rates of RAFs (which need not be generative), and particularly the algorithmic question of ﬁnding a RAF that maximises the rates of its slowest reaction. Not only is this problem solvable in the size of the CRS, but it can also be extended to the slightly more general setting of allowing ‘catalytic ensembles’. The introduction of rates allows for the study of how a population of diﬀerent closed subRAFs might evolve over time, in which primitive subRAF are replaced (out-competed) by eﬃcient ones that rely on new catalysts in place of more primitive ones. We hope to explore this further in future work. 22

text: We

text: ﬁrst

text: considered

text: the

text: special

text: setting

text: of

text: ‘elementary’

text: systems

text: in

text: which

text: all

text: reactions

text: (or

text: at

text: least

text: those

text: present

text: in

text: the

text: maxRAF)

text: have

text: all

text: their

text: reactants

text: present

text: in

text: the

text: food

text: set.

text: This

text: allows

text: for

text: the

text: structure

text: of

text: the

text: collection

text: of

text: RAFs,

text: irrRAFs,

text: and

text: closed

text: subRAFs

text: to

text: be

text: explicitly

text: described

text: graph-theoretically.

text: As

text: a

text: result,

text: some

text: problems

text: that

text: are

text: computationally

text: intractable

text: in

text: the

text: gen-

text: eral

text: CRS

text: setting

text: turn

text: out

text: to

text: be

text: polynomial-time

text: for

text: an

text: elementary

text: CRS.

text: For

text: example,

text: one

text: can

text: eﬃciently

text: ﬁnd

text: the

text: smallest

text: RAFs

text: in

text: an

text: elementary

text: CRS,

text: which

text: is

text: an

text: NP-hard

text: problem

text: in

text: general

text: (Steel

text: et

text: al.,

text: Also,

text: the

text: number

text: of

text: minimal

text: closed

text: subRAF

text: in

text: an

text: elementary

text: CRS

text: is

text: linear

text: in

text: the

text: size

text: of

text: the

text: set

text: of

text: reactions

text: (for

text: a

text: general

text: CRS,

text: they

text: can

text: be

text: exponential

text: in

text: number).

text: For

text: future

text: work,

text: it

text: may

text: be

text: of

text: interest

text: to

text: determine

text: if

text: there

text: are

text: polynomial-time

text: algorithms

text: that

text: can

text: answer

text: the

text: following

text: questions

text: for

text: an

text: elementary

text: CRS:

text: (i)

text: What

text: is

text: the

text: size

text: of

text: the

text: largest

text: irrRAF?

text: (ii)

text: If

text: inhibition

text: is

text: allowed,

text: then

text: is

text: there

text: a

text: RAF

text: that

text: has

text: no

text: inhibition?

text: The

text: concept

text: of

text: an

text: ‘elementary’

text: CRS

text: is

text: an

text: all-or-nothing

text: notion.

text: One

text: way

text: to

text: extend

text: the

text: results

text: above

text: could

text: be

text: to

text: deﬁne

text: the

text: notion

text: of

text: ‘level’,

text: whereby

text: a

text: CRS

text: has

text: level

text: k

text: if

text: the

text: length

text: of

text: the

text: longest

text: path

text: from

text: the

text: food

text: set

text: to

text: any

text: reaction

text: product

text: goes

text: through

text: at

text: most

text: k

text: reactions

text: (an

text: elementary

text: CRS

text: thus

text: has

text: level

text: We

text: have

text: not

text: explored

text: this

text: further

text: here

text: but

text: instead,

text: we

text: consider

text: the

text: related

text: alternative

text: notion

text: of

text: a

text: generative

text: RAF.

text: Brieﬂy,

text: a

text: generative

text: RAF

text: allows

text: a

text: RAF

text: to

text: form

text: by

text: eﬀectively

text: enlarging

text: its

text: ‘food

text: set’

text: with

text: products

text: of

text: reactions,

text: so

text: that

text: each

text: step

text: only

text: requires

text: catalysts

text: that

text: are

text: either

text: present

text: or

text: produced

text: by

text: reactions

text: in

text: the

text: RAF

text: at

text: that

text: stage.

text: Although

text: generative

text: RAFs

text: are

text: more

text: complex

text: than

text: elementary

text: ones,

text: their

text: close

text: connection

text: to

text: elementary

text: RAFs

text: (in

text: a

text: stratiﬁed

text: way)

text: allows

text: for

text: a

text: more

text: tractable

text: analysis

text: than

text: for

text: general

text: RAFs.

text: Moreover,

text: unlike

text: elementary

text: RAFs,

text: no

text: special

text: assumption

text: is

text: required

text: on

text: the

text: underlying

text: CRS;

text: generative

text: RAFs

text: are

text: just

text: a

text: special

text: type

text: of

text: RAF

text: that

text: can

text: be

text: generated

text: in

text: a

text: certain

text: sequential

text: fashion

text: in

text: any

text: CRS.

text: In

text: the

text: ﬁnal

text: section,

text: we

text: considered

text: the

text: impact

text: of

text: rates

text: of

text: RAFs

text: (which

text: need

text: not

text: be

text: generative),

text: and

text: particularly

text: the

text: algorithmic

text: question

text: of

text: ﬁnding

text: a

text: RAF

text: that

text: maximises

text: the

text: rates

text: of

text: its

text: slowest

text: reaction.

text: Not

text: only

text: is

text: this

text: problem

text: solvable

text: in

text: the

text: size

text: of

text: the

text: CRS,

text: but

text: it

text: can

text: also

text: be

text: extended

text: to

text: the

text: slightly

text: more

text: general

text: setting

text: of

text: allowing

text: ‘catalytic

text: ensembles’.

text: The

text: introduction

text: of

text: rates

text: allows

text: for

text: the

text: study

text: of

text: how

text: a

text: population

text: of

text: diﬀerent

text: closed

text: subRAFs

text: might

text: evolve

text: over

text: time,

text: in

text: which

text: primitive

text: subRAF

text: are

text: replaced

text: (out-competed)

text: by

text: eﬃcient

text: ones

text: that

text: rely

text: on

text: new

text: catalysts

text: in

text: place

text: of

text: more

text: primitive

text: ones.

text: We

text: hope

text: to

text: explore

text: this

text: further

text: in

text: future

text: work.


text: vertices in R(cid:48) and ending in r. Since R(cid:48) is ﬁnite if we take k > |R(cid:48)| then two vertices on this directed walk must coincide and the resulting sub-walk between this vertex to itself gives a directed cycle in D . Moreover, D |R(cid:48) Q Q contains a directed cycle if and only if this sub-digraph contains a chordless cycle (again, the ‘if’ direction is clear and the ‘only if’ direction follows by the ﬁniteness of R, so shortening each directed cycle by following a chord leads to a sequence of cycles of decreasing length that eventually terminates on a chordless cycle). This establishes Part (i). For Part (ii), a subset R(cid:48) of R has the property that D |R(cid:48) is a chordless Q cycle, which implies (by Part (i)) that R(cid:48) is a RAF. It is also an irrRAF; otherwise, the cycle would have a chord. Conversely, if R(cid:48) has the property that D |R(cid:48) is not a chordless cycle, then either D does not contain a cycle Q Q (in which case it is not a RAF) or it contains a cycle which either has a chord or has other vertices reachable from it, in which case R(cid:48) is not an irrRAF. This establishes the ﬁrst sentence of Part (ii). The arguments for the second and third sentences follow similar lines. For Part (iii), it is clear that the union of one or more cores is a RAF; however, the resulting set of reactions is closed if and only if all reactions that are reachable from that set are also included. For Part (iv), suppose that a core c(cid:48) is reachable from another core c (by deﬁnition, c is not reachable from c(cid:48)). Any closed RAF R(cid:48) that contains both c and c(cid:48) is thus not minimal, since we could delete c(cid:48) and all the reactions that are reachable from c(cid:48) but not from c and obtain a strict subset of R(cid:48) that is also a closed RAF. On the other hand, if R(cid:48) has the property described in Part (iv), then it is a closed RAF by Part (iii) and it is also minimal, since any closed RAF must contain at least one core, alongside all the reactions that are reachable from it. Part (v) follows from Part (iv), since each minimal closed RAF is asso- ciated with exactly one core, and since cores are strongly connected compo- nents of D these cores are vertex-disjoint (i.e. two cores share no reaction). Q Consequently, the number of cores is bounded above by the number of re- actions in the maxRAF of D . Moreover, ﬁnding the strongly connected Q components of any digraph can be done in polynomial time in the size of the digraph (Tarjan, 1972). Each of these strongly connected components can then be tested in polynomial time to determine if it is a core; if so, one can then determine in polynomial time which other vertices are reachable from it. Thus the minimal closed RAFs can be listed in polynomial time in the size of Q. 25

text: vertices

text: in

text: and

text: ending

text: in

text: r.

text: Since

text: is

text: ﬁnite

text: if

text: we

text: take

text: k

text: then

text: two

text: vertices

text: on

text: this

text: directed

text: walk

text: must

text: coincide

text: and

text: the

text: resulting

text: sub-walk

text: between

text: this

text: vertex

text: to

text: itself

text: gives

text: a

text: directed

text: cycle

text: in

text: D

text: Moreover,

text: D

text: Q

text: Q

text: contains

text: a

text: directed

text: cycle

text: if

text: and

text: only

text: if

text: this

text: sub-digraph

text: contains

text: a

text: chordless

text: cycle

text: (again,

text: the

text: ‘if’

text: direction

text: is

text: clear

text: and

text: the

text: ‘only

text: if’

text: direction

text: follows

text: by

text: the

text: ﬁniteness

text: of

text: R,

text: so

text: shortening

text: each

text: directed

text: cycle

text: by

text: following

text: a

text: chord

text: leads

text: to

text: a

text: sequence

text: of

text: cycles

text: of

text: decreasing

text: length

text: that

text: eventually

text: terminates

text: on

text: a

text: chordless

text: cycle).

text: This

text: establishes

text: Part

text: (i).

text: For

text: Part

text: (ii),

text: a

text: subset

text: of

text: R

text: has

text: the

text: property

text: that

text: D

text: is

text: a

text: chordless

text: Q

text: cycle,

text: which

text: implies

text: (by

text: Part

text: (i))

text: that

text: is

text: a

text: RAF.

text: It

text: is

text: also

text: an

text: irrRAF;

text: otherwise,

text: the

text: cycle

text: would

text: have

text: a

text: chord.

text: Conversely,

text: if

text: has

text: the

text: property

text: that

text: D

text: is

text: not

text: a

text: chordless

text: cycle,

text: then

text: either

text: D

text: does

text: not

text: contain

text: a

text: cycle

text: Q

text: Q

text: (in

text: which

text: case

text: it

text: is

text: not

text: a

text: RAF)

text: or

text: it

text: contains

text: a

text: cycle

text: which

text: either

text: has

text: a

text: chord

text: or

text: has

text: other

text: vertices

text: reachable

text: from

text: it,

text: in

text: which

text: case

text: is

text: not

text: an

text: irrRAF.

text: This

text: establishes

text: the

text: ﬁrst

text: sentence

text: of

text: Part

text: (ii).

text: The

text: arguments

text: for

text: the

text: second

text: and

text: third

text: sentences

text: follow

text: similar

text: lines.

text: For

text: Part

text: (iii),

text: it

text: is

text: clear

text: that

text: the

text: union

text: of

text: one

text: or

text: more

text: cores

text: is

text: a

text: RAF;

text: however,

text: the

text: resulting

text: set

text: of

text: reactions

text: is

text: closed

text: if

text: and

text: only

text: if

text: all

text: reactions

text: that

text: are

text: reachable

text: from

text: that

text: set

text: are

text: also

text: included.

text: For

text: Part

text: (iv),

text: suppose

text: that

text: a

text: core

text: is

text: reachable

text: from

text: another

text: core

text: c

text: (by

text: deﬁnition,

text: c

text: is

text: not

text: reachable

text: from

text: Any

text: closed

text: RAF

text: that

text: contains

text: both

text: c

text: and

text: is

text: thus

text: not

text: minimal,

text: since

text: we

text: could

text: delete

text: and

text: all

text: the

text: reactions

text: that

text: are

text: reachable

text: from

text: but

text: not

text: from

text: c

text: and

text: obtain

text: a

text: strict

text: subset

text: of

text: that

text: is

text: also

text: a

text: closed

text: RAF.

text: On

text: the

text: other

text: hand,

text: if

text: has

text: the

text: property

text: described

text: in

text: Part

text: (iv),

text: then

text: it

text: is

text: a

text: closed

text: RAF

text: by

text: Part

text: (iii)

text: and

text: it

text: is

text: also

text: minimal,

text: since

text: any

text: closed

text: RAF

text: must

text: contain

text: at

text: least

text: one

text: core,

text: alongside

text: all

text: the

text: reactions

text: that

text: are

text: reachable

text: from

text: it.

text: Part

text: (v)

text: follows

text: from

text: Part

text: (iv),

text: since

text: each

text: minimal

text: closed

text: RAF

text: is

text: asso-

text: ciated

text: with

text: exactly

text: one

text: core,

text: and

text: since

text: cores

text: are

text: strongly

text: connected

text: compo-

text: nents

text: of

text: D

text: these

text: cores

text: are

text: vertex-disjoint

text: (i.e.

text: two

text: cores

text: share

text: no

text: reaction).

text: Q

text: Consequently,

text: the

text: number

text: of

text: cores

text: is

text: bounded

text: above

text: by

text: the

text: number

text: of

text: re-

text: actions

text: in

text: the

text: maxRAF

text: of

text: D

text: Moreover,

text: ﬁnding

text: the

text: strongly

text: connected

text: Q

text: components

text: of

text: any

text: digraph

text: can

text: be

text: done

text: in

text: polynomial

text: time

text: in

text: the

text: size

text: of

text: the

text: digraph

text: (Tarjan,

text: Each

text: of

text: these

text: strongly

text: connected

text: components

text: can

text: then

text: be

text: tested

text: in

text: polynomial

text: time

text: to

text: determine

text: if

text: it

text: is

text: a

text: core;

text: if

text: so,

text: one

text: can

text: then

text: determine

text: in

text: polynomial

text: time

text: which

text: other

text: vertices

text: are

text: reachable

text: from

text: it.

text: Thus

text: the

text: minimal

text: closed

text: RAFs

text: can

text: be

text: listed

text: in

text: polynomial

text: time

text: in

text: the

text: size

text: of

text: Q.


text: Part (vi) follows from Part (v) since Q contains a closed subRAF if and only if it contains a minimal closed subRAF. (cid:50) 8.2. Proof of Theorem 2 For Part (i), the proof of the ﬁrst claim is by induction on k. For the base case k = 1, if R (cid:54)= ∅, then R is the maxRAF of Q|F and so is a closed 1 1 RAF of Q. Now suppose that R is a generative RAF. In this case, R k−1 k is the maxRAF of Q|F ∪ π(R ) and so R is a closed RAF of this CRS. k−1 k For the second part, observe that if (R(cid:48), i ≥ 1) is a generative RAF sequence i for a generative RAF R(cid:48), then, by induction on k, R(cid:48) ⊆ R , since R is k k k constructed by taking the maxRAF of R , so it contains any other closed k−1 RAF of R . k−1 For Part (ii), observe that R(Q(cid:48)) is, by deﬁnition, a subset of R(cid:48). Since R(Q(cid:48)) is generative (by Part (i)), the ‘if’ direction is established. Conversely, suppose that R(cid:48) is generative for Q. Then R(cid:48) is also generative for Q(cid:48). Since R(Q(cid:48)) ⊆ R(cid:48), and since R(Q(cid:48)) is the maximal generative RAF for Q, it follows that R(Q(cid:48)) = R(cid:48). For Part (iii), the proof of the ﬁrst claim follows from the fact that the maxRAF can be computed in polynomial time in the size of the CRS (Hordijk and Steel, 2004); the second claim now follows from Part (ii). For Part (iv), the algorithm ﬁrst constructs the generative RAF sequence R , R , . . . to obtain R(Q(cid:48)) = R(cid:48) (where the last equality is from Part (ii)). 1 2 One then sequentially applies Part (vi) of Theorem 1 to determine whether there is a closed RAF strictly within R , and if not whether there is within 1 R , and so on. If no such closed (and strict subset) RAF is detected then 2 R(cid:48) contains no other generative RAF as a strict subset, otherwise any such subset found is a generative RAF that lies strictly inside R(cid:48). (cid:50) 8.3. Proof of Theorem 3 Let L = {f(x, r) : (x, r) ∈ C}, and let M = max L. Consider the CRS Q(cid:48) = (X(cid:48), R∗, C∗, F ) obtained from Q by ﬁrst deleting any uncatalysed reaction and then replacing each reaction r that is catalysed by (say) k ≥ 1 molecule types with k distinct copies of this reaction r , . . . , r , each of which 1 k is catalysed by a diﬀerent one of the k molecule types. Thus each reaction r in R∗ is catalysed by exactly one molecule type, which we will denote as x(r). For the associated catalysis set C∗ = {(x(r), r) : r ∈ R∗}, let f(cid:48) be the rate function induced by f (i.e. if r ∈ R is replaced by r , . . . , r ∈ R∗ then 1 k f(cid:48)(x(r ), r ) := f(x(r), r)). For each (cid:96) ∈ L let: i i R∗ = {r ∈ R∗ : f(cid:48)(x(r), r) ≥ (cid:96)}. (cid:96) 26 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Part

text: (vi)

text: follows

text: from

text: Part

text: (v)

text: since

text: Q

text: contains

text: a

text: closed

text: subRAF

text: if

text: and

text: only

text: if

text: it

text: contains

text: a

text: minimal

text: closed

text: subRAF.

label: section

text: Proof

text: of

text: Theorem

text: For

text: Part

text: (i),

text: the

text: proof

text: of

text: the

text: ﬁrst

text: claim

text: is

text: by

text: induction

text: on

text: k.

text: For

text: the

text: base

text: case

text: k

text: if

text: R

text: then

text: R

text: is

text: the

text: maxRAF

text: of

text: Q|F

text: and

text: so

text: is

text: a

text: closed

text: RAF

text: of

text: Q.

text: Now

text: suppose

text: that

text: R

text: is

text: a

text: generative

text: RAF.

text: In

text: this

text: case,

text: R

text: k

text: is

text: the

text: maxRAF

text: of

text: Q|F

text: π(R

text: and

text: so

text: R

text: is

text: a

text: closed

text: RAF

text: of

text: this

text: CRS.

text: k

text: For

text: the

text: second

text: part,

text: observe

text: that

text: if

text: i

text: is

text: a

text: generative

text: RAF

text: sequence

text: i

text: for

text: a

text: generative

text: RAF

text: then,

text: by

text: induction

text: on

text: k,

text: R

text: since

text: R

text: is

text: k

text: k

text: k

text: constructed

text: by

text: taking

text: the

text: maxRAF

text: of

text: R

text: so

text: it

text: contains

text: any

text: other

text: closed

text: RAF

text: of

text: R

text: For

text: Part

text: (ii),

text: observe

text: that

text: is,

text: by

text: deﬁnition,

text: a

text: subset

text: of

text: Since

text: is

text: generative

text: (by

text: Part

text: (i)),

text: the

text: ‘if’

text: direction

text: is

text: established.

text: Conversely,

text: suppose

text: that

text: is

text: generative

text: for

text: Q.

text: Then

text: is

text: also

text: generative

text: for

text: Since

text: and

text: since

text: is

text: the

text: maximal

text: generative

text: RAF

text: for

text: Q,

text: it

text: follows

text: that

text: For

text: Part

text: (iii),

text: the

text: proof

text: of

text: the

text: ﬁrst

text: claim

text: follows

text: from

text: the

text: fact

text: that

text: the

text: maxRAF

text: can

text: be

text: computed

text: in

text: polynomial

text: time

text: in

text: the

text: size

text: of

text: the

text: CRS

text: (Hordijk

text: and

text: Steel,

text: the

text: second

text: claim

text: now

text: follows

text: from

text: Part

text: (ii).

text: For

text: Part

text: (iv),

text: the

text: algorithm

text: ﬁrst

text: constructs

text: the

text: generative

text: RAF

text: sequence

text: R

text: R

text: to

text: obtain

text: (where

text: the

text: last

text: equality

text: is

text: from

text: Part

text: (ii)).

text: One

text: then

text: sequentially

text: applies

text: Part

text: (vi)

text: of

text: Theorem

text: to

text: determine

text: whether

text: there

text: is

text: a

text: closed

text: RAF

text: strictly

text: within

text: R

text: and

text: if

text: not

text: whether

text: there

text: is

text: within

text: R

text: and

text: so

text: on.

text: If

text: no

text: such

text: closed

text: (and

text: strict

text: subset)

text: RAF

text: is

text: detected

text: then

text: contains

text: no

text: other

text: generative

text: RAF

text: as

text: a

text: strict

text: subset,

text: otherwise

text: any

text: such

text: subset

text: found

text: is

text: a

text: generative

text: RAF

text: that

text: lies

text: strictly

text: inside

text: Proof

text: of

text: Theorem

text: Let

text: L

text: {f(x,

text: r)

text: (x,

text: r)

text: C},

text: and

text: let

text: M

text: max

text: L.

text: Consider

text: the

text: CRS

text: R∗,

text: C∗,

text: F

text: obtained

text: from

text: Q

text: by

text: ﬁrst

text: deleting

text: any

text: uncatalysed

text: reaction

text: and

text: then

text: replacing

text: each

text: reaction

text: r

text: that

text: is

text: catalysed

text: by

text: (say)

text: k

text: molecule

text: types

text: with

text: k

text: distinct

text: copies

text: of

text: this

text: reaction

text: r

text: r

text: each

text: of

text: which

text: k

text: is

text: catalysed

text: by

text: a

text: diﬀerent

text: one

text: of

text: the

text: k

text: molecule

text: types.

text: Thus

text: each

text: reaction

text: r

text: in

text: R∗

text: is

text: catalysed

text: by

text: exactly

text: one

text: molecule

text: type,

text: which

text: we

text: will

text: denote

text: as

text: x(r).

text: For

text: the

text: associated

text: catalysis

text: set

text: C∗

text: {(x(r),

text: r)

text: r

text: R∗},

text: let

text: be

text: the

text: rate

text: function

text: induced

text: by

text: f

text: (i.e.

text: if

text: r

text: R

text: is

text: replaced

text: by

text: r

text: r

text: R∗

text: then

text: k

text: r

text: f(x(r),

text: r)).

text: For

text: each

text: L

text: let:

text: i

text: i

text: R∗

text: {r

text: R∗

text: r)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: S S 2 3 r r 9 5 r r r 2 1 3 S S 1 2 r r 4 7 S S r6 3 1 r 8 D D∗ Q Q Figure 2: The directed graph DQ for an elementary CRS Q (adapted from an experi- mental system of Ashkenasy et al. (2004)) that has three strongly connected components (S , S , S ), of which S and S are cores. The associated (acyclic) condensation digraph ∗1 2 3 1 2 DQ is shown on the right. The unique minimal closed RAF is S ∪ S ; the other closed 2 3 RAF is the full set itself, namely S ∪ S ∪ S . The reactions subsets S , S ∪ S , S ∪ S , 1 2 3 1 1 3 1 2 S ∪ S , and S are all RAFs but not closed RAFs. A computer-based search ﬁnds 305 1 3 2 RAFs altogether. There are six chordless cycles in this CRS, which correspond to the six irrRAFs: {r }, {r }, {r }, {r , r }, {r , r } and {r , r }. Note that this representation of 2 5 8 1 4 4 7 3 7 the CRS is in terms of the molecules produced by reactions that have reactants in the food set. However, each reaction produces a single (and unique) product so we can identify the product with the reaction in this example. Remarks: • Parts (ii)–(vi) of Theorem 1 hold even when Q is not elementary, pro- vided that Q(cid:48) = (X, R(cid:48), C, F ) is elementary where R(cid:48) is the maxRAF of Q. • Although cores share no reactions in common, it is quite possible for minimal closed RAFs to share reactions in common. • The last sentence of Part (ii) implies that the size of the smallest RAF is equal to the length of the shortest directed cycle in D and this can Q be found in polynomial time in |Q| (by a depth-ﬁrst-search or network ﬂow techniques). This is in contrast to the problem of ﬁnding the size of a smallest RAF in a general CRS, which has been shown to be NP-hard in Steel et al. (2013). 8 ##LTFigure## ##LTLine## ##LTLine##

text: S

text: S

text: r

text: r

text: r

text: r

text: r

text: S

text: S

text: r

text: r

text: S

text: S

text: r

text: D

text: D∗

text: Q

text: Q

text: Figure

text: The

label: caption

text: directed

text: graph

text: DQ

text: for

text: an

text: elementary

text: CRS

text: Q

text: (adapted

text: from

text: an

text: experi-

text: mental

text: system

text: of

text: Ashkenasy

text: et

text: al.

text: that

text: has

text: three

text: strongly

text: connected

text: components

text: (S

text: S

text: S

text: of

text: which

text: S

text: and

text: S

text: are

text: cores.

text: The

text: associated

text: (acyclic)

text: condensation

text: digraph

text: ∗1

text: DQ

text: is

text: shown

text: on

text: the

text: right.

text: The

text: unique

text: minimal

text: closed

text: RAF

text: is

text: S

text: S

text: the

text: other

text: closed

text: RAF

text: is

text: the

text: full

text: set

text: itself,

text: namely

text: S

text: S

text: S

text: The

text: reactions

text: subsets

text: S

text: S

text: S

text: S

text: S

text: S

text: S

text: and

text: S

text: are

text: all

text: RAFs

text: but

text: not

text: closed

text: RAFs.

text: A

text: computer-based

text: search

text: ﬁnds

text: RAFs

text: altogether.

text: There

text: are

text: six

text: chordless

text: cycles

text: in

text: this

text: CRS,

text: which

text: correspond

text: to

text: the

text: six

text: irrRAFs:

text: {r

text: {r

text: {r

text: {r

text: r

text: {r

text: r

text: and

text: {r

text: r

text: Note

text: that

text: this

text: representation

text: of

text: the

text: CRS

text: is

text: in

text: terms

text: of

text: the

text: molecules

text: produced

text: by

text: reactions

text: that

text: have

text: reactants

text: in

text: the

text: food

text: set.

text: However,

text: each

text: reaction

text: produces

text: a

text: single

text: (and

text: unique)

text: product

text: so

text: we

text: can

text: identify

text: the

text: product

text: with

text: the

text: reaction

text: in

text: this

text: example.

text: Remarks:

text: Parts

text: (ii)–(vi)

text: of

text: Theorem

text: hold

text: even

text: when

text: Q

text: is

text: not

text: elementary,

text: pro-

text: vided

text: that

text: (X,

text: C,

text: F

text: is

text: elementary

text: where

text: is

text: the

text: maxRAF

text: of

text: Q.

text: Although

text: cores

text: share

text: no

text: reactions

text: in

text: common,

text: it

text: is

text: quite

text: possible

text: for

text: minimal

text: closed

text: RAFs

text: to

text: share

text: reactions

text: in

text: common.

text: The

text: last

text: sentence

text: of

text: Part

text: (ii)

text: implies

text: that

text: the

text: size

text: of

text: the

text: smallest

text: RAF

text: is

text: equal

text: to

text: the

text: length

text: of

text: the

text: shortest

text: directed

text: cycle

text: in

text: D

text: and

text: this

text: can

text: Q

text: be

text: found

text: in

text: polynomial

text: time

text: in

text: |Q|

text: (by

text: a

text: depth-ﬁrst-search

text: or

text: network

text: ﬂow

text: techniques).

text: This

text: is

text: in

text: contrast

text: to

text: the

text: problem

text: of

text: ﬁnding

text: the

text: size

text: of

text: a

text: smallest

text: RAF

text: in

text: a

text: general

text: CRS,

text: which

text: has

text: been

text: shown

text: to

text: be

text: NP-hard

text: in

text: Steel

text: et

text: al.

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##


text: 2 analyticity properties. To make a very simple situa- tion, let us examine the connected part of the following k p p FIG. 1. Example of vacuum bubble. k′ = p − k II. A PARADOX AND ITS RESOLUTION IN (a) (b) LIGHT-FRONT PERTURBATION THEORY FIG. 2. The two orderings in x+ for the Green function of two A. Light-front perturbation theory; conventions φ2 operators in free-ﬁeld theory. Only graph (a) is non-zero when the external plus-momentum p+ is positive. We use light-front coordinates where components of − a Lorentz vector are deﬁned as x = (x+, x , x ), and T where, given a choice of Cartesian coordinates, x+ = momentum-space Green function of composite operators √ − √ (x0 + x3)/ 2, x = (x0 − x3)/ 2, and x denotes the in the theory(cid:90)of a free scalar ﬁeld of mass m: T remaining transverse coordinates. Graphs in light-front perturbation theory — e.g., [1] · — are speciﬁed with particular orderings in the space- Π(p2) = d2x eip x(cid:104)0|T 1φ2(x)1φ2(0)|0(cid:105) . (2) 2 2 conn. time coordinate x+ for the vertices. There are integrals over the values of momentum components k+ and k j j,T for the lines, constrained by conservation of the plus and The factors of 1/2 are to get a standard normaliza- transverse components. The values of k+ are restricted tion convention, and the use of the connected part means to physical positive values k+ > 0 for forward moving that from each instance of the operator φ2 is subtracted j momenta. The integrand has an “energy denominator its vacuum expectation value. To make the calculations factor” of the following form for each intermediate state maximally simple, while still exhibiting the principles at I stake, we work in 1 + 1 space-time dimensions, without any transverse dimensions. A standard textbook prop- (cid:80) erty is that Π(p2) is an analytic function whose only sin- i gularity is a branch point at the threshold point p2 = , (1) − k2 + m2 4m2. If the second φ2 operator were at a general po- P − ∈ j,T j + i(cid:15) ext j I 2k+ sition y, we would change x in the exponent to x − y j without change in Π(p2), by translation invariance. The use of time-ordering of the operators in Eq. (2) where the sum is over particles in the intermediate might appear to suggest the use of equal-time quantiza- state, corresponding to lines in the graph. Each de- tion. However, it is known that the deﬁnition is fully nominator is the diﬀerence between the external minus- covariant. When x is space-like, the operators commute, component of momentum and the on-shell plus-component and then the ordering is irrelevant. When x is time-like, of momentum for the intermediate state. In graphs, we it is frame-independent as to which position is future- use the convention that x+ increases from left to right, (cid:80) most, and then the corresponding operator is deﬁned to and correspondingly the ﬂow of positive k+ is from left j be on the left. The situation with light-like separation to right. Note that plus momentum is conserved, so that is governed by the usual rules for analyticity and for the Pe+xt = j∈I kj+. distributional nature of the ﬁelds. With the basic rules for light-front perturbation the- A Green function of composite operators is used in- ory, vacuum bubbles like Fig. 1 are zero. This is because stead of an actual S-matrix element to give maximum the external momentum is zero. Hence it has zero plus simplicity. In QCD, we often use such Green functions component: Pe+xt = 0, and there are no possible interme- to formulate the strong-interaction part of a scattering diate states: All the lines have positive plus momentum with electroweak particles, with the composite operators from left to right, and these can never sum to the zero corresponding to the coupling between an electroweak external plus momentum. The vanishing of the vacuum ﬁeld and QCD ﬁelds. bubbles is what leads to the statement of the triviality of Given that free-ﬁeld theory is used, there are exactly the vacuum. two x+-ordered graphs for Π(p2), as in Fig. 2. B. A paradox When the external momentum has a positive plus com- ponent, i.e., p+ > 0, the rules for x+-ordered perturba- In this section, I show how the basic rules for light- tion theory give a single allowed ordering in x+, symbol- front perturbation theory applied at vanishing external ized in Fig. 2(a). The value of the graph, including its plus momentum lead to an inconsistency with standard symmetry factor 1/2, is ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: analyticity

text: properties.

text: To

text: make

text: a

text: very

text: simple

text: situa-

text: tion,

text: let

text: us

text: examine

text: the

text: connected

text: part

text: of

text: the

text: following

text: k

text: p

text: p

text: FIG.

text: Example

text: of

text: vacuum

text: bubble.

text: k′

text: p

text: k

text: II.

text: A

text: PARADOX

text: AND

text: ITS

text: RESOLUTION

text: IN

text: (a)

text: (b)

text: LIGHT-FRONT

text: PERTURBATION

text: THEORY

text: FIG.

text: The

text: two

text: orderings

text: in

text: x+

text: for

text: the

text: Green

text: function

text: of

text: two

text: A.

text: Light-front

text: perturbation

text: theory;

text: conventions

text: φ2

text: operators

text: in

text: free-ﬁeld

text: theory.

text: Only

text: graph

text: (a)

text: is

text: non-zero

text: when

text: the

text: external

text: plus-momentum

text: p+

text: is

text: positive.

text: We

text: use

text: light-front

text: coordinates

text: where

text: components

text: of

text: a

text: Lorentz

text: vector

text: are

text: deﬁned

text: as

text: x

text: (x+,

text: x

text: x

text: and

text: T

text: where,

text: given

text: a

text: choice

text: of

text: Cartesian

text: coordinates,

text: x+

text: momentum-space

text: Green

text: function

text: of

text: composite

text: operators

text: x

text: and

text: x

text: denotes

text: the

text: in

text: the

text: a

text: free

text: scalar

text: ﬁeld

text: of

text: mass

text: m:

text: T

text: remaining

text: transverse

text: coordinates.

text: Graphs

text: in

text: light-front

text: perturbation

text: theory

text: e.g.,

text: are

text: speciﬁed

text: with

text: particular

text: orderings

text: in

text: the

text: space-

text: eip

text: conn.

text: time

text: coordinate

text: x+

text: for

text: the

text: vertices.

text: There

text: are

text: integrals

text: over

text: the

text: values

text: of

text: momentum

text: components

text: k+

text: and

text: k

text: j

text: j,T

text: for

text: the

text: lines,

text: constrained

text: by

text: conservation

text: of

text: the

text: plus

text: and

text: The

text: factors

text: of

text: are

text: to

text: get

text: a

text: standard

text: normaliza-

text: transverse

text: components.

text: The

text: values

text: of

text: k+

text: are

text: restricted

text: tion

text: convention,

text: and

text: the

text: use

text: of

text: the

text: connected

text: part

text: means

text: to

text: physical

text: positive

text: values

text: k+

text: for

text: forward

text: moving

text: that

text: from

text: each

text: instance

text: of

text: the

text: operator

text: φ2

text: is

text: subtracted

text: j

text: momenta.

text: The

text: integrand

text: has

text: an

text: “energy

text: denominator

text: its

text: vacuum

text: expectation

text: value.

text: To

text: make

text: the

text: calculations

text: factor”

text: of

text: the

text: following

text: form

text: for

text: each

text: intermediate

text: state

text: maximally

text: simple,

text: while

text: still

text: exhibiting

text: the

text: principles

text: at

text: I

text: stake,

text: we

text: work

text: in

text: space-time

text: dimensions,

text: without

text: any

text: transverse

text: dimensions.

text: A

text: standard

text: textbook

text: prop-

text: erty

text: is

text: that

text: is

text: an

text: analytic

text: function

text: whose

text: only

text: sin-

text: i

text: gularity

text: is

text: a

text: branch

text: point

text: at

text: the

text: threshold

text: point

text: If

text: the

text: second

text: φ2

text: operator

text: were

text: at

text: a

text: general

text: po-

text: P

text: j,T

text: j

text: ext

text: j

text: I

text: sition

text: y,

text: we

text: would

text: change

text: x

text: in

text: the

text: exponent

text: to

text: x

text: y

text: j

text: without

text: change

text: in

text: by

text: translation

text: invariance.

text: The

text: use

text: of

text: time-ordering

text: of

text: the

text: operators

text: in

text: Eq.

text: where

text: the

text: sum

text: is

text: over

text: particles

text: in

text: the

text: intermediate

text: might

text: appear

text: to

text: suggest

text: the

text: use

text: of

text: equal-time

text: quantiza-

text: state,

text: corresponding

text: to

text: lines

text: in

text: the

text: graph.

text: Each

text: de-

text: tion.

text: However,

text: it

text: is

text: known

text: that

text: the

text: deﬁnition

text: is

text: fully

text: nominator

text: is

text: the

text: diﬀerence

text: between

text: the

text: external

text: minus-

text: covariant.

text: When

text: x

text: is

text: space-like,

text: the

text: operators

text: commute,

text: component

text: of

text: momentum

text: and

text: the

text: on-shell

text: plus-component

text: and

text: then

text: the

text: ordering

text: is

text: irrelevant.

text: When

text: x

text: is

text: time-like,

text: of

text: momentum

text: for

text: the

text: intermediate

text: state.

text: In

text: graphs,

text: we

text: it

text: is

text: frame-independent

text: as

text: to

text: which

text: position

text: is

text: future-

text: use

text: the

text: convention

text: that

text: x+

text: increases

text: from

text: left

text: to

text: right,

text: most,

text: and

text: then

text: the

text: corresponding

text: operator

text: is

text: deﬁned

text: to

text: and

text: correspondingly

text: the

text: ﬂow

text: of

text: positive

text: k+

text: is

text: from

text: left

text: j

text: be

text: on

text: the

text: left.

text: The

text: situation

text: with

text: light-like

text: separation

text: to

text: right.

text: Note

text: that

text: plus

text: momentum

text: is

text: conserved,

text: so

text: that

text: is

text: governed

text: by

text: the

text: usual

text: rules

text: for

text: analyticity

text: and

text: for

text: the

text: Pe+xt

text: j∈I

text: kj+.

text: distributional

text: nature

text: of

text: the

text: ﬁelds.

text: With

text: the

text: basic

text: rules

text: for

text: light-front

text: perturbation

text: the-

text: A

text: Green

text: function

text: of

text: composite

text: operators

text: is

text: used

text: in-

text: ory,

text: vacuum

text: bubbles

text: like

text: Fig.

text: are

text: zero.

text: This

text: is

text: because

text: stead

text: of

text: an

text: actual

text: S-matrix

text: element

text: to

text: give

text: maximum

text: the

text: external

text: momentum

text: is

text: zero.

text: Hence

text: it

text: has

text: zero

text: plus

text: simplicity.

text: In

text: QCD,

text: we

text: often

text: use

text: such

text: Green

text: functions

text: component:

text: Pe+xt

text: and

text: there

text: are

text: no

text: possible

text: interme-

text: to

text: formulate

text: the

text: strong-interaction

text: part

text: of

text: a

text: scattering

text: diate

text: states:

text: All

text: the

text: lines

text: have

text: positive

text: plus

text: momentum

text: with

text: electroweak

text: particles,

text: with

text: the

text: composite

text: operators

text: from

text: left

text: to

text: right,

text: and

text: these

text: can

text: never

text: sum

text: to

text: the

text: zero

text: corresponding

text: to

text: the

text: coupling

text: between

text: an

text: electroweak

text: external

text: plus

text: momentum.

text: The

text: vanishing

text: of

text: the

text: vacuum

text: ﬁeld

text: and

text: QCD

text: ﬁelds.

text: bubbles

text: is

text: what

text: leads

text: to

text: the

text: statement

text: of

text: the

text: triviality

text: of

text: Given

text: that

text: free-ﬁeld

text: theory

text: is

text: used,

text: there

text: are

text: exactly

text: the

text: vacuum.

text: two

text: x+-ordered

text: graphs

text: for

text: as

text: in

text: Fig.

text: B.

text: A

text: paradox

text: When

text: the

text: external

text: momentum

text: has

text: a

text: positive

text: plus

text: com-

text: ponent,

text: i.e.,

text: p+

text: the

text: rules

text: for

text: x+-ordered

text: perturba-

text: In

text: this

text: section,

text: I

text: show

text: how

text: the

text: basic

text: rules

text: for

text: light-

text: tion

text: theory

text: give

text: a

text: single

text: allowed

text: ordering

text: in

text: x+,

text: symbol-

text: front

text: perturbation

text: theory

text: applied

text: at

text: vanishing

text: external

text: ized

text: in

text: Fig.

text: The

text: value

text: of

text: the

text: graph,

text: including

text: its

text: plus

text: momentum

text: lead

text: to

text: an

text: inconsistency

text: with

text: standard

text: symmetry

text: factor

text: is

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

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