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text: HELICOIDAL FLAT SURFACES IN THE 3-SPHERE 15 [18] Palmer, B., and Perdomo, O. M. Rotating drops with helicoidal symmetry. Paciﬁc J. Math. 273, 2 (2015), 413–441. [19] Perdomo, O. M. Helicoidal minimal surfaces in R3. Illinois J. Math. 57, 1 (2013), 87–104. [20] Ripoll, J. B. Uniqueness of minimal rotational surfaces in S3. Amer. J. Math. 111, 4 (1989), 537–547. [21] Spivak, M. A comprehensive introduction to diﬀerential geometry. Vol. IV, second ed. Pub- lish or Perish, Inc., Wilmington, Del., 1979. [22] Suyama, Y. Conformally ﬂat hypersurfaces in Euclidean 4-space II. Osaka Math. J 42 (2005), 573–598. ICMC, Universidade de Sa˜o Paulo, Sa˜o Carlos, Brasil Email address: manfio@icmc.usp.br MAT, Universidade de Bras´ılia, Bras´ılia, Brazil Email address: joaopsantos@unb.br

text: HELICOIDAL

label: paragraph

text: FLAT

text: SURFACES

text: IN

text: THE

text: Palmer,

text: B.,

text: and

text: Perdomo,

text: O.

text: M.

text: Rotating

text: drops

text: with

text: helicoidal

text: symmetry.

text: Paciﬁc

text: J.

text: Math.

text: Perdomo,

text: O.

text: M.

text: Helicoidal

text: minimal

text: surfaces

text: in

text: Illinois

text: J.

text: Math.

text: Ripoll,

text: J.

text: B.

text: Uniqueness

text: of

text: minimal

text: rotational

text: surfaces

text: in

text: Amer.

text: J.

text: Math.

text: Spivak,

text: M.

text: A

text: comprehensive

text: introduction

text: to

text: diﬀerential

text: geometry.

text: Vol.

text: IV,

text: second

text: ed.

text: Pub-

text: lish

text: or

text: Perish,

text: Inc.,

text: Wilmington,

text: Del.,

text: Suyama,

text: Y.

text: Conformally

text: ﬂat

text: hypersurfaces

text: in

text: Euclidean

text: II.

text: Osaka

text: Math.

text: J

text: ICMC,

label: author

text: Universidade

text: de

text: Sa˜o

text: Paulo,

text: Sa˜o

text: Carlos,

text: Brasil

text: Email

text: address:

text: manfio@icmc.usp.br

text: MAT,

text: Universidade

text: de

text: Bras´ılia,

text: Bras´ılia,

text: Brazil

text: Email

text: address:

text: joaopsantos@unb.br


text: HELICOIDAL FLAT SURFACES IN THE 3-SPHERE 7 with − b2 a2 (16) α = − β, a2b2 1 showing that Y (u, v) is invariant by helicoidal motions. Observe that when ± S (cid:3) a = b we have α = 0, i.e., X is a rotational surface in 3. Remark 2. It is important to note that the constant a and b in (12) were considered ∞ in (1, + ) in order to obtain non-zero constant curvatures with its well deﬁned torsions, and then to apply the Bianchi-Spivak construction. This is not a strong restriction since the curvature function κ(t) = t2−1 assumes all values in R \ {0} ∈ ∞ t when t (1, + ). However, by taking a = 1 and b > 1 in (12), a long but straightforward computation give(cid:112)s an unit normal vector ﬁeld 1 N(u, v) = (n , n , n , n ), 1 2 3 4 2(1 +(cid:16) b2) (cid:17) where (cid:16) (cid:17) − v n1(u, v) = b si(cid:16)n u + (cid:17)b + cos(u + bv), v n2(u, v) = b cos u(cid:16)+ b (cid:17)+ sin(u + bv), − v − − n (u, v) = b sin u cos (a bv) , 3 b − − v − − n (u, v) = b cos u sin (u bv) . 4 b Therefore, one shows that this parametrization is also by asymptotic lines where the − − angle function is given by ω(u, v) = 1 b2 v π. Moreover, this is a parametrization b 2 of a Hopf cylinder, since the unit normal vector ﬁeld N makes a constant angle with the Hopf vector ﬁeld (see section 4). We will use the parametrization Y (u, v) given in (14), compose with the stereo- R graphic projection in 3, to visualize some examples with the corresponding con- stants a and b. Figure 1. a = 2 and b = 3. ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: HELICOIDAL

text: FLAT

text: SURFACES

text: IN

text: THE

text: with

label: equation

text: α

text: β,

text: showing

text: that

text: Y

text: (u,

text: v)

text: is

text: invariant

text: by

text: helicoidal

text: motions.

text: Observe

text: that

text: when

text: S

text: a

text: b

text: we

text: have

text: α

text: i.e.,

text: X

text: is

text: a

text: rotational

text: surface

text: in

text: Remark

text: It

text: is

text: important

text: to

text: note

text: that

text: the

text: constant

text: a

text: and

text: b

text: in

text: were

text: considered

text: in

text: in

text: order

text: to

text: obtain

text: non-zero

text: constant

text: curvatures

text: with

text: its

text: well

text: deﬁned

text: torsions,

text: and

text: then

text: to

text: apply

text: the

text: Bianchi-Spivak

text: construction.

text: This

text: is

text: not

text: a

text: strong

text: restriction

text: since

text: the

text: curvature

text: function

text: κ(t)

text: assumes

text: all

text: values

text: in

text: R

text: t

text: when

text: t

text: However,

text: by

text: taking

text: a

text: and

text: b

text: in

text: a

text: long

text: but

text: straightforward

text: computation

text: an

text: unit

text: normal

text: vector

text: ﬁeld

text: N(u,

text: v)

text: (n

text: n

text: n

text: n

text: where

text: v

text: v)

text: b

text: u

text: cos(u

text: bv),

text: v

text: v)

text: b

text: cos

text: b

text: sin(u

text: bv),

text: v

text: n

text: (u,

text: v)

text: b

text: sin

text: u

text: cos

text: (a

text: bv)

text: b

text: v

text: n

text: (u,

text: v)

text: b

text: cos

text: u

text: sin

text: (u

text: bv)

text: b

text: Therefore,

text: one

text: shows

text: that

text: this

text: parametrization

text: is

text: also

text: by

text: asymptotic

text: lines

text: where

text: the

text: angle

text: function

text: is

text: given

text: by

text: ω(u,

text: v)

text: v

text: π.

text: Moreover,

text: this

text: is

text: a

text: parametrization

text: b

text: of

text: a

text: Hopf

text: cylinder,

text: since

text: the

text: unit

text: normal

text: vector

text: ﬁeld

text: N

text: makes

text: a

text: constant

text: angle

text: with

text: the

text: Hopf

text: vector

text: ﬁeld

text: (see

text: section

text: We

text: will

text: use

text: the

text: parametrization

text: Y

text: (u,

text: v)

text: given

text: in

text: compose

text: with

text: the

text: stereo-

text: R

text: graphic

text: projection

text: in

text: to

text: visualize

text: some

text: examples

text: with

text: the

text: corresponding

text: con-

text: stants

text: a

text: and

text: b.

text: Figure

text: a

text: and

text: b

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: 3 A. Phase space analysis It turns out that the physical mechanism underlying the real time control of directed currents we just demon- strated crucially hinges on the mixed phase space struc- (a) (b) ture of our two lattice system. Hence to understand it we perform a systematic analysis of its microscopic compo- sition and analyze its dynamical occupation by the con- sidered particle ensemble. First, to understand the struc- ture of the phase space itself, we take ‘stroboscopic’ snap- shots of particles with diﬀerent initial conditions leading to Poincar´e surfaces of sections (PSOS) which provide a representative overview of the structure of the complete (c) (d) 3D phase space [33]. We also exploit the spatial period- icity of our setup and project the particle position back ∈ to the ﬁrst unit cell x [0, L) of the lattice. The PSOS Figure 3. The position (mod L) and velocity of all the N of the substrate lattice (henceforth referred to as P1) is particles (green) at (a) t = 0.04ttot in case I superposed on symmetric about v = 0 (Fig. C3a black dots) (cid:39)and±contains the PSOS P1(black dots and lines) of the substrate lattice a large central ‘chaotic sea’ 1 between v 1.6. On C (b) t = 0.42t in case III superposed on the PSOS P2(black top of the PSOS, we show the snapshot of the particle tot dots) corresponding to both the substrate and the carrier lat- coordinates, at a given time, used to determine v¯ in case tices and (c) t = 0.95ttot in case IV superposed on the PSOS I of Fig. 2 (green dots in Fig. 3a) illustrating the uniform P3(black dots) corresponding to both the lattices but with − symmetric chaotic diﬀusion of particles through the lat- φ = π. Red solid lines denote the position of the FISCs 2 tice resulting in no transport. The chaotic sea is bounded whereas the red dashed lines indicate the location of the can- C by the ﬁrst invariant spanning curves (FISC; red lines in torus (see text). i , i = 1, 2, 3 denotes the upper, central U,C,L Fig. 3a) which prevents acceleration of our low velocity and lower chaotic layer of P1, P2, and P3 respectively. (d) A | | initial conditions beyond v > 1.6. Contrarily, a parti- zoom into the typical trajectory of a particle initiated at low velocity in the central chaotic sea of the PSOS P2 in Fig. 3b, cle with in|it|i(cid:38)al condition on one of the regular invariant showing the particle’s stickiness to the cantorus. curves at v 1.6 (black lines in Fig. 3a) shows ballistic unidirectional motion through the lattice. Let us now explore how the phase space structure setup, it does not decay but is frozen at its value of the changes in presence of the carrier lattice. Most promi- time of the switch. That is, our atomic ensemble travels nently, the two lattice PSOS (black dots in Fig. 3b), with constant average speed through the symmetrically henceforth referred to as P2, is not mirror symmetric C oscillating lattice. Remarkably, this allows for an intrigu- about the v = 0 axis. Besides the chaotic sea 2 at C ingly simple real time control of the transport velocity: small velocities, it exhibits two additional chaotic layers C (cid:38) Once the desired transport is achieved one simply needs at higher velocities: the upper layer 2 at v 2.2 and C (cid:46) − U to switch oﬀ the carrier lattice. the lower layer 2 at v 2.2. The crucial point now is L that the choice of an appropriate value for γ allows one We now consider a similar case, but switch the carrier C CV to connect 2 , asymmetrically, only with 2 through a lattice subsequently on and oﬀ (case IV in Fig. 2). Once C L ‘cantorus’ structure (red dashed line in Fig. 3b) which is the transport has been frozen for a while we can accel- a hierarchical chain of stable and unstable ﬁxed points, erate it by switching on the carrier lattice again (second C while it remains separated from 2 by a regular invariant green dot on case IV curve). Switching it oﬀ, for a second U curve (red solid line in Fig. 3b). This allows particles to time (second blue dot), freezes also this enhanced trans- C C enter 2 but not 2 . This can be easily seen from the port at a constant strength. Clearly, to achieve a highly L U ﬁxed time snapshot of the particle distribution onto P2 ﬂexible real time control of the transport it would be corresponding to case III denoted by the green dots in desirable to be able to slow it down. This can in fact C Fig. 3b. These particles in 2 still move irregularly but be achieved by switching on the carrier lattice again, L − now only in one direction through the lattice, which is but this time with a phase diﬀerence of φ = π/2 to the origin of the transport we observed in Fig. 2. The fact the substrate (red dot). We see in Fig. 2 that this in- that the transport velocity does not quickly converge to deed slows the transport systematically down. Overall, a constant velocity, but accelerates very slowly, on time we demonstrated a remarkably simple protocol allowing scales of 105 driving periods towards its asymptotic value to enhance, freeze or slowdown the transport of atomic C (case II in Fig. 2) is owed to the cantorus linking 2 and ensembles in two optical lattices on demand. C C 2, which eﬀectively acts as a semi-permeable barrier to L IV. DISCUSSIONS the particles approaching it and slows down the uniform ﬁlling of the accessible parts of the phase space. ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

text: A.

label: section

text: Phase

text: space

text: analysis

text: It

text: turns

text: out

text: that

text: the

text: physical

text: mechanism

text: underlying

text: the

text: real

text: time

text: control

text: of

text: directed

text: currents

text: we

text: just

text: demon-

text: strated

text: crucially

text: hinges

text: on

text: the

text: mixed

text: phase

text: space

text: struc-

text: (a)

font: default

text: (b)

text: ture

text: of

text: our

text: two

text: lattice

text: system.

text: Hence

text: to

text: understand

text: it

text: we

text: perform

text: a

text: systematic

text: analysis

text: of

text: its

text: microscopic

text: compo-

text: sition

text: and

text: analyze

text: its

text: dynamical

text: occupation

text: by

text: the

text: con-

text: sidered

text: particle

text: ensemble.

text: First,

text: to

text: understand

text: the

text: struc-

text: ture

text: of

text: the

text: phase

text: space

text: itself,

text: we

text: take

text: ‘stroboscopic’

text: snap-

text: shots

text: of

text: particles

text: with

text: diﬀerent

text: initial

text: conditions

text: leading

text: to

text: Poincar´e

text: surfaces

text: of

text: sections

text: (PSOS)

text: which

text: provide

text: a

text: representative

text: overview

text: of

text: the

text: structure

text: of

text: the

text: complete

text: (c)

text: (d)

text: phase

text: space

text: We

text: also

text: exploit

text: the

text: spatial

text: period-

text: icity

text: of

text: our

text: setup

text: and

text: project

text: the

text: particle

text: position

text: back

text: to

text: the

text: ﬁrst

text: unit

text: cell

text: x

text: L)

text: of

text: the

text: lattice.

text: The

text: PSOS

text: Figure

text: The

text: position

text: (mod

text: L)

text: and

text: velocity

text: of

text: all

text: the

text: N

text: of

text: the

text: substrate

text: lattice

text: (henceforth

text: referred

text: to

text: as

text: is

text: particles

text: (green)

text: at

text: (a)

text: t

text: in

text: case

text: I

text: superposed

text: on

text: symmetric

text: about

text: v

text: (Fig.

text: black

text: dots)

text: the

text: PSOS

text: dots

text: and

text: lines)

text: of

text: the

text: substrate

text: lattice

text: a

text: large

text: central

text: ‘chaotic

text: sea’

text: between

text: v

text: On

text: C

text: (b)

text: t

text: in

text: case

text: III

text: superposed

text: on

text: the

text: PSOS

text: top

text: of

text: the

text: PSOS,

text: we

text: show

text: the

text: snapshot

text: of

text: the

text: particle

text: tot

text: dots)

text: corresponding

text: to

text: both

text: the

text: substrate

text: and

text: the

text: carrier

text: lat-

text: coordinates,

text: at

text: a

text: given

text: time,

text: used

text: to

text: determine

text: v¯

text: in

text: case

text: tices

text: and

text: (c)

text: t

text: in

text: case

text: IV

text: superposed

text: on

text: the

text: PSOS

text: I

text: of

text: Fig.

text: (green

text: dots

text: in

text: Fig.

text: illustrating

text: the

text: uniform

text: dots)

text: corresponding

text: to

text: both

text: the

text: lattices

text: but

text: with

text: symmetric

text: chaotic

text: diﬀusion

text: of

text: particles

text: through

text: the

text: lat-

text: φ

text: π.

text: Red

text: solid

text: lines

text: denote

text: the

text: position

text: of

text: the

text: FISCs

text: tice

text: resulting

text: in

text: no

text: transport.

text: The

text: chaotic

text: sea

text: is

text: bounded

text: whereas

text: the

text: red

text: dashed

text: lines

text: indicate

text: the

text: location

text: of

text: the

text: can-

text: C

text: by

text: the

text: ﬁrst

text: invariant

text: spanning

text: curves

text: (FISC;

text: red

text: lines

text: in

text: torus

text: (see

text: text).

text: i

text: i

text: denotes

text: the

text: upper,

text: central

text: U,C,L

text: Fig.

text: which

text: prevents

text: acceleration

text: of

text: our

text: low

text: velocity

text: and

text: lower

text: chaotic

text: layer

text: of

text: and

text: respectively.

text: (d)

text: A

text: initial

text: conditions

text: beyond

text: v

text: Contrarily,

text: a

text: parti-

text: zoom

text: into

text: the

text: typical

text: trajectory

text: of

text: a

text: particle

text: initiated

text: at

text: low

text: velocity

text: in

text: the

text: central

text: chaotic

text: sea

text: of

text: the

text: PSOS

text: in

text: Fig.

text: cle

text: with

text: condition

text: on

text: one

text: of

text: the

text: regular

text: invariant

text: showing

text: the

text: particle’s

text: stickiness

text: to

text: the

text: cantorus.

text: curves

text: at

text: v

text: (black

text: lines

text: in

text: Fig.

text: shows

text: ballistic

text: unidirectional

text: motion

text: through

text: the

text: lattice.

text: Let

text: us

text: now

text: explore

text: how

text: the

text: phase

text: space

text: structure

text: setup,

text: it

text: does

text: not

text: decay

text: but

text: is

text: frozen

text: at

text: its

text: value

text: of

text: the

text: changes

text: in

text: presence

text: of

text: the

text: carrier

text: lattice.

text: Most

text: promi-

text: time

text: of

text: the

text: switch.

text: That

text: is,

text: our

text: atomic

text: ensemble

text: travels

text: nently,

text: the

text: two

text: lattice

text: PSOS

text: (black

text: dots

text: in

text: Fig.

text: with

text: constant

text: average

text: speed

text: through

text: the

text: symmetrically

text: henceforth

text: referred

text: to

text: as

text: is

text: not

text: mirror

text: symmetric

text: C

text: oscillating

text: lattice.

text: Remarkably,

text: this

text: allows

text: for

text: an

text: intrigu-

text: about

text: the

text: v

text: axis.

text: Besides

text: the

text: chaotic

text: sea

text: at

text: C

text: ingly

text: simple

text: real

text: time

text: control

text: of

text: the

text: transport

text: velocity:

text: small

text: velocities,

text: it

text: exhibits

text: two

text: additional

text: chaotic

text: layers

text: C

text: Once

text: the

text: desired

text: transport

text: is

text: achieved

text: one

text: simply

text: needs

text: at

text: higher

text: velocities:

text: the

text: upper

text: layer

text: at

text: v

text: and

text: C

text: U

text: to

text: switch

text: oﬀ

text: the

text: carrier

text: lattice.

text: the

text: lower

text: layer

text: at

text: v

text: The

text: crucial

text: point

text: now

text: is

text: L

text: that

text: the

text: choice

text: of

text: an

text: appropriate

text: value

text: for

text: γ

text: allows

text: one

text: We

text: now

text: consider

text: a

text: similar

text: case,

text: but

text: switch

text: the

text: carrier

text: C

text: CV

text: to

text: connect

text: asymmetrically,

text: only

text: with

text: through

text: a

text: lattice

text: subsequently

text: on

text: and

text: oﬀ

text: (case

text: IV

text: in

text: Fig.

text: Once

text: C

text: L

text: ‘cantorus’

text: structure

text: (red

text: dashed

text: line

text: in

text: Fig.

text: which

text: is

text: the

text: transport

text: has

text: been

text: frozen

text: for

text: a

text: while

text: we

text: can

text: accel-

text: a

text: hierarchical

text: chain

text: of

text: stable

text: and

text: unstable

text: ﬁxed

text: points,

text: erate

text: it

text: by

text: switching

text: on

text: the

text: carrier

text: lattice

text: again

text: (second

text: C

text: while

text: it

text: remains

text: separated

text: from

text: by

text: a

text: regular

text: invariant

text: green

text: dot

text: on

text: case

text: IV

text: curve).

text: Switching

text: it

text: oﬀ,

text: for

text: a

text: second

text: U

text: curve

text: (red

text: solid

text: line

text: in

text: Fig.

text: This

text: allows

text: particles

text: to

text: time

text: (second

text: blue

text: dot),

text: freezes

text: also

text: this

text: enhanced

text: trans-

text: C

text: C

text: enter

text: but

text: not

text: This

text: can

text: be

text: easily

text: seen

text: from

text: the

text: port

text: at

text: a

text: constant

text: strength.

text: Clearly,

text: to

text: achieve

text: a

text: highly

text: L

text: U

text: ﬁxed

text: time

text: snapshot

text: of

text: the

text: particle

text: distribution

text: onto

text: ﬂexible

text: real

text: time

text: control

text: of

text: the

text: transport

text: it

text: would

text: be

text: corresponding

text: to

text: case

text: III

text: denoted

text: by

text: the

text: green

text: dots

text: in

text: desirable

text: to

text: be

text: able

text: to

text: slow

text: it

text: down.

text: This

text: can

text: in

text: fact

text: C

text: Fig.

text: These

text: particles

text: in

text: still

text: move

text: irregularly

text: but

text: be

text: achieved

text: by

text: switching

text: on

text: the

text: carrier

text: lattice

text: again,

text: L

text: now

text: only

text: in

text: one

text: direction

text: through

text: the

text: lattice,

text: which

text: is

text: but

text: this

text: time

text: with

text: a

text: phase

text: diﬀerence

text: of

text: φ

text: to

text: the

text: origin

text: of

text: the

text: transport

text: we

text: observed

text: in

text: Fig.

text: The

text: fact

text: the

text: substrate

text: (red

text: dot).

text: We

text: see

text: in

text: Fig.

text: that

text: this

text: in-

text: that

text: the

text: transport

text: velocity

text: does

text: not

text: quickly

text: converge

text: to

text: deed

text: slows

text: the

text: transport

text: systematically

text: down.

text: Overall,

text: a

text: constant

text: velocity,

text: but

text: accelerates

text: very

text: slowly,

text: on

text: time

text: we

text: demonstrated

text: a

text: remarkably

text: simple

text: protocol

text: allowing

text: scales

text: of

text: driving

text: periods

text: towards

text: its

text: asymptotic

text: value

text: to

text: enhance,

text: freeze

text: or

text: slowdown

text: the

text: transport

text: of

text: atomic

text: C

text: (case

text: II

text: in

text: Fig.

text: is

text: owed

text: to

text: the

text: cantorus

text: linking

text: and

text: ensembles

text: in

text: two

text: optical

text: lattices

text: on

text: demand.

text: C

text: C

text: which

text: eﬀectively

text: acts

text: as

text: a

text: semi-permeable

text: barrier

text: to

text: L

text: IV.

text: DISCUSSIONS

text: the

text: particles

text: approaching

text: it

text: and

text: slows

text: down

text: the

text: uniform

text: ﬁlling

text: of

text: the

text: accessible

text: parts

text: of

text: the

text: phase

text: space.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∈ ◦ Proof. Let σ K¯ (z) be a fractional linear map such that σ f has coeﬃcients in K. The images ∞ ◦ P u of 0, 1, and under σ f all lie in 1(K). We may therefore apply a further fractional linear u ◦ ◦ ∞ ◦ ◦ transformation τ with K-coeﬃcients so that τ σ f ﬁxes 0, 1, and . That is, τ σ f = f for ◦ u ∈ u v some v. Since τ and σ f have K-coeﬃcients, we know that v K. By uniqueness in the lemma, u (cid:3) we conclude that u = v. ⇒ We are now ready for the proof of the theorem. To prove the implication (1) (2), we take ∈ ∈ ∞ y K and attempt to solve the equation φ(u) = y with u K. If y = , then we may take − ∈ u = 3/2. Otherwise, choose u K¯ such that φ(u) = y. Then the function f has K-rational { ∞ } u critical points 0, 1, , y . By (1), f equivalent to a rational function with K-coeﬃcients. The ∈ u above proposition implies that u K. ⇒ ∈ To prove (2) (1), we start with a rational function f K¯ (z) with K-rational critical points. If f has only two critical points, then each must have multiplicity 2 (by the Riemann-Hurwitz ∞ ∞ formula). Without loss, we assume they are 0 and , and that 0, are ﬁxed by f, so that ∈ − f(z) = az2 for some a K¯ . Evidently a 1f has coeﬃcients in K. Now suppose that f has at least three critical points. Without loss, we may assume that 0, 1, ∞ and are among them. In particular, by the lemma we see that f is equivalent to f for some ∈ ∈ u u K¯ . The remaining critical point is φ(u) K. By (2), both solutions of φ(z) = φ(u) lie in P ∈ 1(K), so that u K. That is, f is equivalent to a rational function with K-coeﬃcients. 3. p-adic fields Our proof of Proposition 1.5 is split into the subcases p = 2, p = 3, and p > 3. Recall that we want to show that Q is φ-perfect for φ(z) = −z2+2z if and only if p = 3. This amounts p 2z+3 P Q ∈ Q to determining whether or not φ(z) = y does or does not have a solution in 1( ) for y . p p Rearranging gives the quadratic equation z2 + 2(1 + y)z + 3y = 0, which has discriminant − − ∆ = 4(y2 y + 1) = (2y 1)2 + 3. (3.1) Q Q Determining if is φ-perfect now amounts to determining whether ∆ is a square in or not ∈ Q p p for every y . p ∈ Z For p = 2, set y = 1 + t with t . Then (3.1) becomes 2 2 ≡ ∆ = 4t2 + 3 3 (mod 4), Q Q which is not a square in . Hence φ(z) = 1 + t has no solution, and is not φ-perfect. (It is 2 2 P Q 2 worth noting that what we have really proved is that the image of 1( ) under φ is disjoint from 2 Z the set 1 + .) 2 2 Q ∈ Q ∈ Z For p = 3, we claim that the discriminant ∆ is a square in for any y . If y , 3 3 3 Z then (3.1) shows that ∆ is a square modulo 3, which is equivalent to ∆ being a square in by (cid:54)∈ Z (cid:48) (cid:48) ∈ Z× 3 Hensel’s lemma. If y , write y = y /3r for some y and r > 0. Then (3.1) shows that − 3 3 ord (∆) = 2r. Multiplying through by 32r gives 3 − ≡ (cid:48) 32r∆ = (2y3r 3r)2 + 32r+1 (2y )2 (mod 3). Z This last quantity is a nonzero square, from which we deduce that 32r∆ is a square in by Hensel’s 3 Q Q lemma. We conclude that ∆ is a square in , and is φ-perfect. 3 3 − Finally, we treat the case p > 3. The resultant of φ(z) = z2+2z is 3, so this rational function may be reduced modulo p to yield a quadratic function φ˜ ∈ F2z(+z3). Note that φ˜(0) = φ˜(−2), so p that φ˜ fails to be injective on P1(F ). As P1(F ) is a ﬁnite set, φ˜ also fails to be surjective. Choose y˜ ∈ F such that φ˜(z) = y˜ has no spolution in Fp , and choose a lift y ∈ Z such that y ≡ y˜ (mod p). p p p 3 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Proof.

text: Let

text: σ

text: K¯

text: (z)

text: be

text: a

text: fractional

text: linear

text: map

text: such

text: that

text: σ

text: f

text: has

text: coeﬃcients

text: in

text: K.

text: The

text: images

text: P

text: u

text: of

text: and

text: under

text: σ

text: f

text: all

text: lie

text: in

text: We

text: may

text: therefore

text: apply

text: a

text: further

text: fractional

text: linear

text: u

text: transformation

text: τ

text: with

text: K-coeﬃcients

text: so

text: that

text: τ

text: σ

text: f

text: ﬁxes

text: and

text: That

text: is,

text: τ

text: σ

text: f

text: f

text: for

text: u

text: u

text: v

text: some

text: v.

text: Since

text: τ

text: and

text: σ

text: f

text: have

text: K-coeﬃcients,

text: we

text: know

text: that

text: v

text: K.

text: By

text: uniqueness

text: in

text: the

text: lemma,

text: u

text: we

text: conclude

text: that

text: u

text: v.

text: We

text: are

text: now

text: ready

text: for

text: the

text: proof

text: of

text: the

text: theorem.

text: To

text: prove

text: the

text: implication

text: we

text: take

text: y

text: K

text: and

text: attempt

text: to

text: solve

text: the

text: equation

text: φ(u)

text: y

text: with

text: u

text: K.

text: If

text: y

text: then

text: we

text: may

text: take

text: u

text: Otherwise,

text: choose

text: u

text: K¯

text: such

text: that

text: φ(u)

text: y.

text: Then

text: the

text: function

text: f

text: has

text: K-rational

text: u

text: critical

text: points

text: y

text: By

text: f

text: equivalent

text: to

text: a

text: rational

text: function

text: with

text: K-coeﬃcients.

text: The

text: u

text: above

text: proposition

text: implies

text: that

text: u

text: K.

text: To

text: prove

text: we

text: start

text: with

text: a

text: rational

text: function

text: f

text: K¯

text: (z)

text: with

text: K-rational

text: critical

text: points.

text: If

text: f

text: has

text: only

text: two

text: critical

text: points,

text: then

text: each

text: must

text: have

text: multiplicity

text: (by

text: the

text: Riemann-Hurwitz

text: formula).

text: Without

text: loss,

text: we

text: assume

text: they

text: are

text: and

text: and

text: that

text: are

text: ﬁxed

text: by

text: f,

text: so

text: that

text: f(z)

text: for

text: some

text: a

text: K¯

text: Evidently

text: a

text: has

text: coeﬃcients

text: in

text: K.

text: Now

text: suppose

text: that

text: f

text: has

text: at

text: least

text: three

text: critical

text: points.

text: Without

text: loss,

text: we

text: may

text: assume

text: that

text: and

text: are

text: among

text: them.

text: In

text: particular,

text: by

text: the

text: lemma

text: we

text: see

text: that

text: f

text: is

text: equivalent

text: to

text: f

text: for

text: some

text: u

text: u

text: K¯

text: The

text: remaining

text: critical

text: point

text: is

text: φ(u)

text: K.

text: By

text: both

text: solutions

text: of

text: φ(z)

text: φ(u)

text: lie

text: in

text: P

text: so

text: that

text: u

text: K.

text: That

text: is,

text: f

text: is

text: equivalent

text: to

text: a

text: rational

text: function

text: with

text: K-coeﬃcients.

text: p-adic

text: fields

text: Our

text: proof

text: of

text: Proposition

text: is

text: split

text: into

text: the

text: subcases

text: p

text: p

text: and

text: p

text: Recall

text: that

text: we

text: want

text: to

text: show

text: that

text: Q

text: is

text: φ-perfect

text: for

text: φ(z)

text: if

text: and

text: only

text: if

text: p

text: This

text: amounts

text: p

text: P

text: Q

text: Q

text: to

text: determining

text: whether

text: or

text: not

text: φ(z)

text: y

text: does

text: or

text: does

text: not

text: have

text: a

text: solution

text: in

text: for

text: y

text: p

text: p

text: Rearranging

text: gives

text: the

text: quadratic

text: equation

text: y)z

text: which

text: has

text: discriminant

text: y

text: Q

text: Q

text: Determining

text: if

text: is

text: φ-perfect

text: now

text: amounts

text: to

text: determining

text: whether

text: is

text: a

text: square

text: in

text: or

text: not

text: Q

text: p

text: p

text: for

text: every

text: y

text: p

text: Z

text: For

text: p

text: set

text: y

text: t

text: with

text: t

text: Then

text: becomes

text: (mod

text: Q

text: Q

text: which

text: is

text: not

text: a

text: square

text: in

text: Hence

text: φ(z)

text: t

text: has

text: no

text: solution,

text: and

text: is

text: not

text: φ-perfect.

text: (It

text: is

text: P

text: Q

text: worth

text: noting

text: that

text: what

text: we

text: have

text: really

text: proved

text: is

text: that

text: the

text: image

text: of

text: under

text: φ

text: is

text: disjoint

text: from

text: Z

text: the

text: set

text: Q

text: Q

text: Z

text: For

text: p

text: we

text: claim

text: that

text: the

text: discriminant

text: is

text: a

text: square

text: in

text: for

text: any

text: y

text: If

text: y

text: Z

text: then

text: shows

text: that

text: is

text: a

text: square

text: modulo

text: which

text: is

text: equivalent

text: to

text: being

text: a

text: square

text: in

text: by

text: Z

text: Z×

text: Hensel’s

text: lemma.

text: If

text: y

text: write

text: y

text: y

text: for

text: some

text: y

text: and

text: r

text: Then

text: shows

text: that

text: ord

text: Multiplying

text: through

text: by

text: gives

text: (mod

text: Z

text: This

text: last

text: quantity

text: is

text: a

text: nonzero

text: square,

text: from

text: which

text: we

text: deduce

text: that

text: is

text: a

text: square

text: in

text: by

text: Hensel’s

text: Q

text: Q

text: lemma.

text: We

text: conclude

text: that

text: is

text: a

text: square

text: in

text: and

text: is

text: φ-perfect.

text: Finally,

text: we

text: treat

text: the

text: case

text: p

text: The

text: resultant

text: of

text: φ(z)

text: is

text: so

text: this

text: rational

text: function

text: may

text: be

text: reduced

text: modulo

text: p

text: to

text: yield

text: a

text: quadratic

text: function

text: φ˜

text: Note

text: that

text: so

text: p

text: that

text: φ˜

text: fails

text: to

text: be

text: injective

text: on

text: As

text: is

text: a

text: ﬁnite

text: set,

text: φ˜

text: also

text: fails

text: to

text: be

text: surjective.

text: Choose

text: y˜

text: F

text: such

text: that

text: φ˜(z)

text: y˜

text: has

text: no

text: spolution

text: in

text: Fp

text: and

text: choose

text: a

text: lift

text: y

text: Z

text: such

text: that

text: y

text: y˜

text: (mod

text: p).

text: p

text: p

text: p

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Distribution of joint local and total size and of extension for avalanches in the Brownian force ∗ model Mathieu Delorme, Pierre Le Doussal and Kay Jo¨rg Wiese † ‡ The Brownian force model (BFM) is a mean-ﬁeld model for the local velocities during avalanches in elastic interfaces of internal space dimension d, driven in a random medium. It is exactly solvable via a non-linear differential equation. We study avalanches following a kick, i.e. a step in the driving force. We ﬁrst recall the calculation of the distributions of the global size (total swept area) and of the local jump size for an arbitrary kick amplitude. We extend this calculation to the joint density of local and global sizes within a single avalanche, in the limit of an inﬁnitesimal kick. When the interface is driven by a single point we ﬁnd new exponents τ = 5/3 0 and τ = 7/4, depending on whether the force or the displacement is imposed. We show that the extension of a “single avalanche” along one internal direction (i.e. the total length in d = 1) is ﬁnite and we calculate its ∼ − distribution, following either a local or a global kick. In all cases it exhibits a divergence P((cid:96)) (cid:96) 3 at small (cid:96). Most of our results are tested in a numerical simulation in dimension d = 1. I. INTRODUCTION ian random force landscape. Since it can be shown that un- der monotonous forward driving the interface always moves In many physical systems the motion is not smooth, but forward (Middleton’s theorem [18]), the resulting equation of proceeds by avalanches. This jerky motion is correlated over motion for the velocity ﬁeld is Markovian, and amenable to a broad range of space and time scales. Examples are mag- exact methods. netic interfaces, ﬂuid contact lines, crack fronts in fracture, Despite being exactly solvable, the explicit calculation of strike-slip faults in geophysics and many more [1–3]. These avalanche observables in the BFM requires solving a non- systems have been described using the model of an elastic in- linear instanton equation and performing Laplace inversions, terface slowly driven in a random medium. This model is im- which is not always an easy task. Global avalanche properties, portant for avalanches, both conceptually and in applications such as the probability distribution function (PDF) of global [4–6]. The full model of an interface of internal dimension size S, of duration, and of velocity have been obtained for ar- d, in presence of realistic short-ranged disorder is difﬁcult to bitrary driving. Detailed space time properties however are treat analytically, and requires methods such as the Functional more difﬁcult. In Ref. [13] a ﬁnite wave-vector observable Renormalization Group (FRG) [7–14]. was calculated, demonstrating an asymetry in the temporal A simpler version of the model, the so-called Brownian shape. Although the distribution of local avalanche sizes S r force model (BFM) introduced in [10–13] is very interest- has been obtained in some instances, this is not the case for ing in several respects. First it is exactly solvable, and sev- the distribution of the spatial extension (cid:96) of an avalanche, i.e. eral avalanche observables have been calculated, as discussed the range of points which move during an avalanche, an im- below. Second, it was shown [11, 13] to be the appropriate portant observable accessible in experiments. Note that even mean-ﬁeld theory for the space-time statistics of the velocity the fact that an avalanche has a ﬁnite extent, instead of an ﬁeld in a single avalanche for d-dimensional interfaces close exponentially decaying tail in its spatial extension is a non- ≥ to the depinning transition for d d with d = 4 for trivial result, which up to now was only proven for very large uc uc short ranged elasticity and d = 2 for long-ranged elasticity. avalanches in the BFM [19]. uc Remarkably, when considering the dynamics of the center of The aim of this paper is to calculate further observables for mass of the interface, it reproduces the results of the simpler the BFM which contain information about local properties, ABBM model, a toy model for a single degree of freedom such as the joint density of global and local avalanches, and (particle), introduced long ago on a phenomenological basis the distribution of extensions. We consider various protocols, to describe Barkhausen experiments (magnetic noise)[15, 16] where the interface is either driven uniformly in space or at a and much studied since [1, 12, 17]. Last but not least, the single point; in the latter case we identify new critical expo- BFM is the starting point for a calculation of avalanche ob- nents. We study avalanches following a kick, i.e. a step in the servables beyond mean-ﬁeld, using the FRG in a systematic − driving force. expansion in d d [11, 13]. uc The key property which makes the BFM (and the ABBM) The article is structured as follows: In section we recall the model solvable is that the disorder is taken to be a Brown- deﬁnition of the BFM and of the main avalanche observables, together with the general method to obtain them from the in- stanton equation. Section starts by recalling the calculation ∗ of the distributions of the global size (total swept area) S and † of the local jump size S of an avalanche, for an arbitrary kick r ‡ amplitude. In Section we extend this calculation to the joint ##LTLine##

text: Distribution

label: title

text: of

text: joint

text: local

text: and

text: total

text: size

text: and

text: of

text: extension

text: for

text: avalanches

text: in

text: the

text: Brownian

text: force

text: model

text: Mathieu

text: Delorme,

text: Pierre

text: Le

text: Doussal

text: and

text: Kay

text: Jo¨rg

text: Wiese

text: The

label: abstract

text: Brownian

text: force

text: model

text: (BFM)

text: is

text: a

text: mean-ﬁeld

text: model

text: for

text: the

text: local

text: velocities

text: during

text: avalanches

text: in

text: elastic

text: interfaces

text: of

text: internal

text: space

text: dimension

text: d,

text: driven

text: in

text: a

text: random

text: medium.

text: It

text: is

text: exactly

text: solvable

text: via

text: a

text: non-linear

text: differential

text: equation.

text: We

text: study

text: avalanches

text: following

text: a

text: kick,

text: i.e.

text: a

text: step

text: in

text: the

text: driving

text: force.

text: We

text: ﬁrst

text: recall

text: the

text: calculation

text: of

text: the

text: distributions

text: of

text: the

text: global

text: size

text: (total

text: swept

text: area)

text: and

text: of

text: the

text: local

text: jump

text: size

text: for

text: an

text: arbitrary

text: kick

text: amplitude.

text: We

text: extend

text: this

text: calculation

text: to

text: the

text: joint

text: density

text: of

text: local

text: and

text: global

text: sizes

text: within

text: a

text: single

text: avalanche,

text: in

text: the

text: limit

text: of

text: an

text: inﬁnitesimal

text: kick.

text: When

text: the

text: interface

text: is

text: driven

text: by

text: a

text: single

text: point

text: we

text: ﬁnd

text: new

text: exponents

text: τ

text: and

text: τ

text: depending

text: on

text: whether

text: the

text: force

text: or

text: the

text: displacement

text: is

text: imposed.

text: We

text: show

text: that

text: the

text: extension

text: of

text: a

text: “single

text: avalanche”

text: along

text: one

text: internal

text: direction

text: (i.e.

text: the

text: total

text: length

text: in

text: d

text: is

text: ﬁnite

text: and

text: we

text: calculate

text: its

text: distribution,

text: following

text: either

text: a

text: local

text: or

text: a

text: global

text: kick.

text: In

text: all

text: cases

text: it

text: exhibits

text: a

text: divergence

text: at

text: small

text: Most

text: of

text: our

text: results

text: are

text: tested

text: in

text: a

text: numerical

text: simulation

text: in

text: dimension

text: d

text: I.

text: INTRODUCTION

text: ian

text: random

text: force

text: landscape.

text: Since

text: it

text: can

text: be

text: shown

text: that

text: un-

text: der

text: monotonous

text: forward

text: driving

text: the

text: interface

text: always

text: moves

text: In

text: many

text: physical

text: systems

text: the

text: motion

text: is

text: not

text: smooth,

text: but

text: forward

text: (Middleton’s

text: theorem

text: the

text: resulting

text: equation

text: of

text: proceeds

text: by

text: avalanches.

text: This

text: jerky

text: motion

text: is

text: correlated

text: over

text: motion

text: for

text: the

text: velocity

text: ﬁeld

text: is

text: Markovian,

text: and

text: amenable

text: to

text: a

text: broad

text: range

text: of

text: space

text: and

text: time

text: scales.

text: Examples

text: are

text: mag-

text: exact

text: methods.

text: netic

text: interfaces,

text: ﬂuid

text: contact

text: lines,

text: crack

text: fronts

text: in

text: fracture,

text: Despite

text: being

text: exactly

text: solvable,

text: the

text: explicit

text: calculation

text: of

text: strike-slip

text: faults

text: in

text: geophysics

text: and

text: many

text: more

text: These

text: avalanche

text: observables

text: in

text: the

text: BFM

text: requires

text: solving

text: a

text: non-

text: systems

text: have

text: been

text: described

text: using

text: the

text: model

text: of

text: an

text: elastic

text: in-

text: linear

text: instanton

text: equation

text: and

text: performing

text: Laplace

text: inversions,

text: terface

text: slowly

text: driven

text: in

text: a

text: random

text: medium.

text: This

text: model

text: is

text: im-

text: which

text: is

text: not

text: always

text: an

text: easy

text: task.

text: Global

text: avalanche

text: properties,

text: portant

text: for

text: avalanches,

text: both

text: conceptually

text: and

text: in

text: applications

text: such

text: as

text: the

text: probability

text: distribution

text: function

text: (PDF)

text: of

text: global

text: The

text: full

text: model

text: of

text: an

text: interface

text: of

text: internal

text: dimension

text: size

text: S,

text: of

text: duration,

text: and

text: of

text: velocity

text: have

text: been

text: obtained

text: for

text: ar-

text: d,

text: in

text: presence

text: of

text: realistic

text: short-ranged

text: disorder

text: is

text: difﬁcult

text: to

text: bitrary

text: driving.

text: Detailed

text: space

text: time

text: properties

text: however

text: are

text: treat

text: analytically,

text: and

text: requires

text: methods

text: such

text: as

text: the

text: Functional

text: more

text: difﬁcult.

text: In

text: Ref.

text: a

text: ﬁnite

text: wave-vector

text: observable

text: Renormalization

text: Group

text: (FRG)

text: was

text: calculated,

text: demonstrating

text: an

text: asymetry

text: in

text: the

text: temporal

text: A

text: simpler

text: version

text: of

text: the

text: model,

text: the

text: so-called

text: Brownian

text: shape.

text: Although

text: the

text: distribution

text: of

text: local

text: avalanche

text: sizes

text: S

text: r

text: force

text: model

text: (BFM)

text: introduced

text: in

text: is

text: very

text: interest-

text: has

text: been

text: obtained

text: in

text: some

text: instances,

text: this

text: is

text: not

text: the

text: case

text: for

text: ing

text: in

text: several

text: respects.

text: First

text: it

text: is

text: exactly

text: solvable,

text: and

text: sev-

text: the

text: distribution

text: of

text: the

text: spatial

text: extension

text: of

text: an

text: avalanche,

text: i.e.

text: eral

text: avalanche

text: observables

text: have

text: been

text: calculated,

text: as

text: discussed

text: the

text: range

text: of

text: points

text: which

text: move

text: during

text: an

text: avalanche,

text: an

text: im-

text: below.

text: Second,

text: it

text: was

text: shown

text: to

text: be

text: the

text: appropriate

text: portant

text: observable

text: accessible

text: in

text: experiments.

text: Note

text: that

text: even

text: mean-ﬁeld

text: theory

text: for

text: the

text: space-time

text: statistics

text: of

text: the

text: velocity

text: the

text: fact

text: that

text: an

text: avalanche

text: has

text: a

text: ﬁnite

text: extent,

text: instead

text: of

text: an

text: ﬁeld

text: in

text: a

text: single

text: avalanche

text: for

text: d-dimensional

text: interfaces

text: close

text: exponentially

text: decaying

text: tail

text: in

text: its

text: spatial

text: extension

text: is

text: a

text: non-

text: to

text: the

text: depinning

text: transition

text: for

text: d

text: d

text: with

text: d

text: for

text: trivial

text: result,

text: which

text: up

text: to

text: now

text: was

text: only

text: proven

text: for

text: very

text: large

text: uc

text: uc

text: short

text: ranged

text: elasticity

text: and

text: d

text: for

text: long-ranged

text: elasticity.

text: avalanches

text: in

text: the

text: BFM

text: uc

text: Remarkably,

text: when

text: considering

text: the

text: dynamics

text: of

text: the

text: center

text: of

text: The

text: aim

text: of

text: this

text: paper

text: is

text: to

text: calculate

text: further

text: observables

text: for

text: mass

text: of

text: the

text: interface,

text: it

text: reproduces

text: the

text: results

text: of

text: the

text: simpler

text: the

text: BFM

text: which

text: contain

text: information

text: about

text: local

text: properties,

text: ABBM

text: model,

text: a

text: toy

text: model

text: for

text: a

text: single

text: degree

text: of

text: freedom

text: such

text: as

text: the

text: joint

text: density

text: of

text: global

text: and

text: local

text: avalanches,

text: and

text: (particle),

text: introduced

text: long

text: ago

text: on

text: a

text: phenomenological

text: basis

text: the

text: distribution

text: of

text: extensions.

text: We

text: consider

text: various

text: protocols,

text: to

text: describe

text: Barkhausen

text: experiments

text: (magnetic

text: where

text: the

text: interface

text: is

text: either

text: driven

text: uniformly

text: in

text: space

text: or

text: at

text: a

text: and

text: much

text: studied

text: since

text: Last

text: but

text: not

text: least,

text: the

text: single

text: point;

text: in

text: the

text: latter

text: case

text: we

text: identify

text: new

text: critical

text: expo-

text: BFM

text: is

text: the

text: starting

text: point

text: for

text: a

text: calculation

text: of

text: avalanche

text: ob-

text: nents.

text: We

text: study

text: avalanches

text: following

text: a

text: kick,

text: i.e.

text: a

text: step

text: in

text: the

text: servables

text: beyond

text: mean-ﬁeld,

text: using

text: the

text: FRG

text: in

text: a

text: systematic

text: driving

text: force.

text: expansion

text: in

text: d

text: d

text: uc

text: The

text: key

text: property

text: which

text: makes

text: the

text: BFM

text: (and

text: the

text: ABBM)

text: The

text: article

text: is

text: structured

text: as

text: follows:

text: In

text: section

text: we

text: recall

text: the

text: model

text: solvable

text: is

text: that

text: the

text: disorder

text: is

text: taken

text: to

text: be

text: a

text: Brown-

text: deﬁnition

text: of

text: the

text: BFM

text: and

text: of

text: the

text: main

text: avalanche

text: observables,

text: together

text: with

text: the

text: general

text: method

text: to

text: obtain

text: them

text: from

text: the

text: in-

text: stanton

text: equation.

text: Section

text: starts

text: by

text: recalling

text: the

text: calculation

text: of

text: the

text: distributions

text: of

text: the

text: global

text: size

text: (total

text: swept

text: area)

text: S

text: and

text: of

text: the

text: local

text: jump

text: size

text: S

text: of

text: an

text: avalanche,

text: for

text: an

text: arbitrary

text: kick

text: r

text: amplitude.

text: In

text: Section

text: we

text: extend

text: this

text: calculation

text: to

text: the

text: joint

text: ##LTLine##


text: 11 We also extended the method to study new driving proto- 1.0 cols relevant to distinct experimental setups. The derived re- sults show new exponents for the small-scale behavior of the 0.8 global avalanche-size distribution following a locally imposed 0.6 displacement, and for the small-scale behavior of the local- size distribution following a localized kick. 0.4 Finally, we presented results for the non-stationary dynam- ics of the BFM, focusing on observables which exist only in 0.2 the position theory, such as the roughness exponent. This ex- plains why both the Larkin roughness and the BFM roughness 0.5 0.5 1.0 1.5 (cid:45) (emerging from the FRG approach), play a role in this model, 0.2 (cid:45) depending on whether the driving is stationary or not. ACKNOWLEDGMENTS 0.4 (cid:45) We thank A. Rosso, A. Kolton and A. Dobrinevski for stim- FIG. 8. Representation of the potential energy V (y) as a function of ulating discussions, PSL for support by Grant No. ANR-10- y, and lines of constant total energy, with E = 0 in red, E > 0 in IDEX-0001-02-PSL, as well as KITP for hospitality and sup- blue and, E < 0 in green. port in part by NSF Grant No. NSF PHY11-25915. Appendix A: Airy functions 1.0 (cid:90) We recall the deﬁnition of the Airy function: 0.5 ∞ Ai(z) := dt eit33 +izt . (A1) −∞ 2π 0.5 0.5 1.0 1.5 (cid:45) (cid:90) ∈ R∗ The following formula is usefulfor a , 0.5 (cid:45) Φ(a, b, c) = dz eaz33 +bz2+(cid:18)cz (cid:19)(A2) 1.0 2iπ (cid:45) C = |a|−1/3e32ab32 −bac Ai | b|2 − c|sg|n(a) . a 4/3 a 1/3 (cid:48) FIG. 9. Phase-space diagram, i.e. trajectories represented with y as It can be obtained from (A1), deforming the contour C, e.g. a function of y. The case E = 0 is in red, E > 0 in blue and to z = − b + iR. and E < 0 in green. We can see that properties of the solution a (periodicity, divergences, etc.) strongly depend on the value of E. Appendix B: General considerations on the instanton equation From ﬁgures 8 and 9, we see that: (i) - there is exactly one positive E = 0 solution y+(x) 1. Sourceless equation (cid:90) ∈ R → deﬁned for all x , up to a shift x x + x . It reads (cid:113) 0 a. Massive case 3/2 dy | | − = x (cid:20) (cid:16) (cid:17) (cid:21) y+(x) y2 2y3 It is useful to start with the simpler sourceless instanton 3 (B2) equation ⇔ 3 3 − x y+(x) = = 1 tanh2 . 1 + cosh x 2 2 (cid:48)(cid:48) − y = y y2 . (B1) (ii) - There is exactly one negative E = 0 (zero energy) (cid:90) − ∈ R∗ solution y (x) deﬁned for all x , namely (cid:113) Here we denote by a prime the derivative with respect to y−(x) dy | | xof. aItpacratnicubleei(notferm(cid:112)praestsed2)aisntahepoctleansstiiaclalVe(qyu)at=ion−oyf2 m+o2tiyo3n, −∞ y2 − 23y3 = x (cid:20) (cid:16) (cid:17) (cid:21) (B3) (cid:48) 3 represented in Fig. 8. Multiplying by y and integrating once, ⇔ − 3 3 − x we obtain y(cid:48) = ± E − V (y), where E is a real integration y (x) = 1 − cosh x = 2 1 coth2 2 . constant equivalent to the total “energy” of the particle. Its (cid:48) (cid:54) phase-space diagram (y, y ) is represented in Fig. 9. (iii) - There are two classes of solutions with E = 0. The ##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##

text: We

text: also

text: extended

text: the

text: method

text: to

text: study

text: new

text: driving

text: proto-

text: cols

text: relevant

text: to

text: distinct

text: experimental

text: setups.

text: The

text: derived

text: re-

text: sults

text: show

text: new

text: exponents

text: for

text: the

text: small-scale

text: behavior

text: of

text: the

text: global

text: avalanche-size

text: distribution

text: following

text: a

text: locally

text: imposed

text: displacement,

text: and

text: for

text: the

text: small-scale

text: behavior

text: of

text: the

text: local-

text: size

text: distribution

text: following

text: a

text: localized

text: kick.

text: Finally,

text: we

text: presented

text: results

text: for

text: the

text: non-stationary

text: dynam-

text: ics

text: of

text: the

text: BFM,

text: focusing

text: on

text: observables

text: which

text: exist

text: only

text: in

text: the

text: position

text: theory,

text: such

text: as

text: the

text: roughness

text: exponent.

text: This

text: ex-

text: plains

text: why

text: both

text: the

text: Larkin

text: roughness

text: and

text: the

text: BFM

text: roughness

text: (emerging

text: from

text: the

text: FRG

text: approach),

text: play

text: a

text: role

text: in

text: this

text: model,

text: depending

text: on

text: whether

text: the

text: driving

text: is

text: stationary

text: or

text: not.

text: ACKNOWLEDGMENTS

text: We

text: thank

text: A.

text: Rosso,

text: A.

text: Kolton

text: and

text: A.

text: Dobrinevski

text: for

text: stim-

text: FIG.

text: Representation

text: of

text: the

text: potential

text: energy

text: V

text: (y)

text: as

text: a

text: function

text: of

text: ulating

text: discussions,

text: PSL

text: for

text: support

text: by

text: Grant

text: No.

text: y,

text: and

text: lines

text: of

text: constant

text: total

text: energy,

text: with

text: E

text: in

text: red,

text: E

text: in

text: as

text: well

text: as

text: KITP

text: for

text: hospitality

text: and

text: sup-

text: blue

text: and,

text: E

text: in

text: green.

text: port

text: in

text: part

text: by

text: NSF

text: Grant

text: No.

text: NSF

text: Appendix

text: A:

text: Airy

text: functions

text: We

text: recall

text: the

text: deﬁnition

text: of

text: the

text: Airy

text: function:

text: Ai(z)

text: dt

text: +izt

text: 2π

text: R∗

text: The

text: following

text: formula

text: is

text: usefulfor

text: a

text: Φ(a,

text: b,

text: c)

text: dz

text: C

text: −bac

text: Ai

text: c|sg|n(a)

text: a

text: a

text: FIG.

text: Phase-space

text: diagram,

text: i.e.

text: trajectories

text: represented

text: with

text: y

text: as

text: It

text: can

text: be

text: obtained

text: from

text: deforming

text: the

text: contour

text: C,

text: e.g.

text: a

text: function

text: of

text: y.

text: The

text: case

text: E

text: is

text: in

text: red,

text: E

text: in

text: blue

text: and

text: to

text: z

text: b

text: iR.

text: and

text: E

text: in

text: green.

text: We

text: can

text: see

text: that

text: properties

text: of

text: the

text: solution

text: a

text: (periodicity,

text: divergences,

text: etc.)

text: strongly

text: depend

text: on

text: the

text: value

text: of

text: E.

text: Appendix

text: B:

text: General

text: considerations

text: on

text: the

text: instanton

text: equation

text: From

text: ﬁgures

text: and

text: we

text: see

text: that:

text: (i)

text: there

text: is

text: exactly

text: one

text: positive

text: E

text: solution

text: y+(x)

text: Sourceless

text: equation

text: R

text: deﬁned

text: for

text: all

text: x

text: up

text: to

text: a

text: shift

text: x

text: x

text: x

text: It

text: reads

text: a.

text: Massive

text: case

text: dy

text: x

text: y+(x)

text: It

text: is

text: useful

text: to

text: start

text: with

text: the

text: simpler

text: sourceless

text: instanton

text: equation

text: x

text: y+(x)

text: cosh

text: x

text: y

text: y

text: (ii)

text: There

text: is

text: exactly

text: one

text: negative

text: E

text: (zero

text: energy)

text: R∗

text: solution

text: y

text: (x)

text: deﬁned

text: for

text: all

text: x

text: namely

text: Here

text: we

text: denote

text: by

text: a

text: prime

text: the

text: derivative

text: with

text: respect

text: to

text: y−(x)

text: dy

text: xof.

text: x

text: represented

text: in

text: Fig.

text: Multiplying

text: by

text: y

text: and

text: integrating

text: once,

text: x

text: we

text: obtain

text: E

text: V

text: (y),

text: where

text: E

text: is

text: a

text: real

text: integration

text: y

text: (x)

text: cosh

text: x

text: constant

text: equivalent

text: to

text: the

text: total

text: “energy”

text: of

text: the

text: particle.

text: Its

text: phase-space

text: diagram

text: (y,

text: y

text: is

text: represented

text: in

text: Fig.

text: (iii)

text: There

text: are

text: two

text: classes

text: of

text: solutions

text: with

text: E

text: The

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

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text: 7 → ∞ IV. DRIVING AT A POINT: AVALANCHE SIZES To impose the displacement, we consider the limit m2 . In that limit u = w , and the local size of the avalanche x=0,t 0,t Here we brieﬂy study avalanche sizes for an interface S is equal to δw . 0 0 driven only in a small region of space, e.g. at a point. There While the local size S is ﬁxed by the driving, we can calcu- 0 are two main cases: late the distribution of global sizes. It is obtained in Appendix • the local force on the point is imposed, which in our using an instanton equation with a Dirac mass term. It can framework means to consider a local kick δw = be mapped onto the same instanton equation as studied for the x δw δ(x). In the massless setting it amounts to use joint PDF of local and global sizes. The Laplace-transform of f = δf δ(x), the result for the PDF is given in Eq. (E6). Its small-driving x limit, i.e. the density, is • the displacement u of one point of the interface is √ x=0,t imposed. 3 ∼ − ρ(S) = S τloc.driv. (40) Γ(1/4)S7/4 As we now see this leads to different universality classes and exponents. with a distinct exponent 7 A. Imposed local force τloc.driv. = 4 . (41) V. DISTRIBUTION OF AVALANCHE EXTENSIONS Consider an avalanche following a local kick at x = 0, i.e. δw = δw δ(x). x 0 In this section we study the distribution of avalanche ex- In the BFM the distribution of the global size of an avalanche (cid:82) tensions. In the BFM they can be calculated analytically. We does not depend on whether the kick is local in space or not. start by recalling standard scaling arguments. One still obtains [13] the global-size distribution as given in A. Scaling arguments for the distribution of extensions Eq. (16) with δwˆ = δw = δw . x x 0 The distribution of the local size at the point of the kick is more interesting. The calculation is performed in Appendix As mentioned in the last section, we expect that the global . For simplicity we restrict to d = 1, the general case can be size S and the extension (cid:96) of avalanches are related by the − obtained as above by inserting factors of Ld 1. The full result scaling relation fliomr itthδewPD→F, P0δwit0(sSim0)p,liiﬁsegsi.veNnotiinng(CP7) an(Sd i)s (cid:39)bulδkwy. ρI(nSth)e, S ∼ (cid:96)d+ζ (42) 0 (cid:16) δw0 0 (cid:17) 0 0 the corresponding local-size density becomes (cid:28) in the region of small avalanches S S (in dimensionfull m − 1 (cid:48) units). From the deﬁnition of the avalanche-size exponent ρ(S ) = Ai 31/3S2/3 . (36) 0 31/3S5/3 0 ∼ − 0 P(S) S τ (43) At small S , or equivalently in the massless limit at ﬁxed 0 and using the change of variables P(S)dS = P((cid:96))d(cid:96) we ﬁnd δf = m2δw , it diverges as 0 0 ∼ − − − P((cid:96)) (cid:96) κ with κ = 1 + (τ 1)(d + ζ) . (44) (cid:39) S 5/3 ∼ − ρ(S ) 0 S τ0,loc.driv. . (37) 0 S0(cid:28)1 32/3Γ(1/3) 0 Using the value for τ from the NF relation (35) we obtain This leads to a new avalanche exponent − 2 τ = 2 . (45) d + ζ 5 τ = . (38) 0,loc.driv. 3 For SR elasticity, this yields The cutoff at small size is given by the driv≡ing, S0 ∼ δw03/2. κ = d + ζ − 1 . (46) At large S the PDF is cut by the scale S 1 and decays as 0 m − The prediction for the BFM is that ζ = 4 d and τ = − √ BFM BFM (cid:39) √S 3/2 − 3/2, which leads to ρ(S0) S0(cid:29)1 2 0π31/4 e 2S0/ 3 . (39) κ = 3 (47) BFM B. Imposed displacement at a point in all dimensions. We will now check this prediction from the scaling relations with exact calculations on the BFM model in We analyze the problem in the massless case. To impose the displacement at point x = 0 we replace in the equation d = 1. → B. Instanton equation for two local sizes of motion (1) and (3), m2 m2δ(x). Hence there is no global mass, but a local one to drive the interface at a point. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: IV.

text: DRIVING

text: AT

text: A

text: POINT:

text: AVALANCHE

text: SIZES

text: To

text: impose

text: the

text: displacement,

text: we

text: consider

text: the

text: limit

text: In

text: that

text: limit

text: u

text: w

text: and

text: the

text: local

text: size

text: of

text: the

text: avalanche

text: Here

text: we

text: brieﬂy

text: study

text: avalanche

text: sizes

text: for

text: an

text: interface

text: S

text: is

text: equal

text: to

text: δw

text: driven

text: only

text: in

text: a

text: small

text: region

text: of

text: space,

text: e.g.

text: at

text: a

text: point.

text: There

text: While

text: the

text: local

text: size

text: S

text: is

text: ﬁxed

text: by

text: the

text: driving,

text: we

text: can

text: calcu-

text: are

text: two

text: main

text: cases:

text: late

text: the

text: distribution

text: of

text: global

text: sizes.

text: It

text: is

text: obtained

text: in

text: Appendix

label: list

text: the

text: local

text: force

text: on

text: the

text: point

text: is

text: imposed,

text: which

text: in

text: our

text: using

text: an

text: instanton

text: equation

text: with

text: a

text: Dirac

text: mass

text: term.

text: It

text: can

text: framework

text: means

text: to

text: consider

text: a

text: local

text: kick

text: δw

text: be

text: mapped

text: onto

text: the

text: same

text: instanton

text: equation

text: as

text: studied

text: for

text: the

text: x

text: δw

text: δ(x).

text: In

text: the

text: massless

text: setting

text: it

text: amounts

text: to

text: use

text: joint

text: PDF

text: of

text: local

text: and

text: global

text: sizes.

text: The

text: Laplace-transform

text: of

text: f

text: δf

text: δ(x),

text: the

text: result

text: for

text: the

text: PDF

text: is

text: given

text: in

text: Eq.

text: Its

text: small-driving

text: x

text: limit,

text: i.e.

text: the

text: density,

text: is

text: the

text: displacement

text: u

text: of

text: one

text: point

text: of

text: the

text: interface

text: is

text: imposed.

text: ρ(S)

text: S

text: τloc.driv.

text: As

text: we

text: now

text: see

text: this

text: leads

text: to

text: different

text: universality

text: classes

text: and

text: exponents.

text: with

text: a

text: distinct

text: exponent

text: A.

text: Imposed

text: local

text: force

text: τloc.driv.

text: V.

text: DISTRIBUTION

text: OF

text: AVALANCHE

text: EXTENSIONS

text: Consider

text: an

text: avalanche

text: following

text: a

text: local

text: kick

text: at

text: x

text: i.e.

text: δw

text: δw

text: δ(x).

text: x

text: In

text: this

text: section

text: we

text: study

text: the

text: distribution

text: of

text: avalanche

text: ex-

text: In

text: the

text: BFM

text: the

text: distribution

text: of

text: the

text: global

text: size

text: of

text: an

text: avalanche

text: tensions.

text: In

text: the

text: BFM

text: they

text: can

text: be

text: calculated

text: analytically.

text: We

text: does

text: not

text: depend

text: on

text: whether

text: the

text: kick

text: is

text: local

text: in

text: space

text: or

text: not.

text: start

text: by

text: recalling

text: standard

text: scaling

text: arguments.

text: One

text: still

text: obtains

text: the

text: global-size

text: distribution

text: as

text: given

text: in

text: A.

text: Scaling

text: arguments

text: for

text: the

text: distribution

text: of

text: extensions

text: Eq.

text: with

text: δwˆ

text: δw

text: δw

text: x

text: x

text: The

text: distribution

text: of

text: the

text: local

text: size

text: at

text: the

text: point

text: of

text: the

text: kick

text: is

text: more

text: interesting.

text: The

text: calculation

text: is

text: performed

text: in

text: Appendix

text: As

text: mentioned

text: in

text: the

text: last

text: section,

text: we

text: expect

text: that

text: the

text: global

text: For

text: simplicity

text: we

text: restrict

text: to

text: d

text: the

text: general

text: case

text: can

text: be

text: size

text: S

text: and

text: the

text: extension

text: of

text: avalanches

text: are

text: related

text: by

text: the

text: obtained

text: as

text: above

text: by

text: inserting

text: factors

text: of

text: Ld

text: The

text: full

text: result

text: scaling

text: relation

text: fliomr

text: itthδewPD→F,

text: an(Sd

text: i)s

text: ρI(nSth)e,

text: S

text: the

text: corresponding

text: local-size

text: density

text: becomes

text: in

text: the

text: region

text: of

text: small

text: avalanches

text: S

text: S

text: (in

text: dimensionfull

text: m

text: units).

text: From

text: the

text: deﬁnition

text: of

text: the

text: avalanche-size

text: exponent

text: ρ(S

text: Ai

text: P(S)

text: S

text: τ

text: At

text: small

text: S

text: or

text: equivalently

text: in

text: the

text: massless

text: limit

text: at

text: ﬁxed

text: and

text: using

text: the

text: change

text: of

text: variables

text: P(S)dS

text: we

text: ﬁnd

text: δf

text: it

text: diverges

text: as

text: κ

text: with

text: κ

text: (τ

text: ζ)

text: S

text: ρ(S

text: S

text: Using

text: the

text: value

text: for

text: τ

text: from

text: the

text: NF

text: relation

text: we

text: obtain

text: This

text: leads

text: to

text: a

text: new

text: avalanche

text: exponent

text: τ

text: d

text: ζ

text: τ

text: For

text: SR

text: elasticity,

text: this

text: yields

text: The

text: cutoff

text: at

text: small

text: size

text: is

text: given

text: by

text: the

text: driv≡ing,

text: κ

text: d

text: ζ

text: At

text: large

text: S

text: the

text: PDF

text: is

text: cut

text: by

text: the

text: scale

text: S

text: and

text: decays

text: as

text: m

text: The

text: prediction

text: for

text: the

text: BFM

text: is

text: that

text: ζ

text: d

text: and

text: τ

text: BFM

text: BFM

text: √S

text: which

text: leads

text: to

text: e

text: κ

text: BFM

text: B.

text: Imposed

text: displacement

text: at

text: a

text: point

text: in

text: all

text: dimensions.

text: We

text: will

text: now

text: check

text: this

text: prediction

text: from

text: the

text: scaling

text: relations

text: with

text: exact

text: calculations

text: on

text: the

text: BFM

text: model

text: in

text: We

text: analyze

text: the

text: problem

text: in

text: the

text: massless

text: case.

text: To

text: impose

text: the

text: displacement

text: at

text: point

text: x

text: we

text: replace

text: in

text: the

text: equation

text: d

text: B.

text: Instanton

text: equation

text: for

text: two

text: local

text: sizes

text: of

text: motion

text: and

text: Hence

text: there

text: is

text: no

text: global

text: mass,

text: but

text: a

text: local

text: one

text: to

text: drive

text: the

text: interface

text: at

text: a

text: point.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 Our second result is that it is possible to shift variables of integration in the 1EH action within the path integral to eliminate this mixed propagator. We are then left with a relatively simple set of Feynman rules; there are now just two propagating ﬁelds (that do not mix) and three vertices. This is an improvement over the situation that occurs in the 2EH action where there is one propagating ﬁeld and an inﬁnite number of vertices with an arbitrary number of external ﬁelds. We then demonstrate the utility of our result by computing the two-point function to one-loop order using an arbitrary gauge ﬁxing parameter. In the limiting case in which this parameter equals one, we reproduce the result of Ref. [4]. The ﬁrst order formalism has also been used for doing loop calculations in gravity in Ref. [5], though in the models considered there it is not clear if the ﬁrst and second order formalisms are equivalent. We begin by considering the 1YM action. II. THE FIRST ORDER YANG-MILLS ACTION (cid:0) (cid:1) It is evident that the 1YM Lagrangian L −1 − 1 = Fa ∂µAaν ∂νAaµ + g(cid:15)abcAbµAcν + Fa Faµν (2.1) 1Y M 2 µν 4 µν is classically equivalent to the 2YM Lagrangian (cid:0) (cid:1) L = −1 ∂ Aa − ∂ Aa + g(cid:15)abcAb Ac 2 (2.2) 2Y M 4 µ ν ν µ µ ν L L as upon substitution of the equation of motion for Fa that follows from (2.1) back into , follows. µν 1Y M 2Y M The 1YM and 2YM Lagrangians have the gauge invariance δFa = g(cid:15)abcFb θc (2.3a) µν µν δAa = ∂ θa + g(cid:15)abcAb θc; (2.3b) µ µ µ (cid:90) (cid:90) L we are led to the path integral for 1Y M D D L L Z = Aa Fa ∆ (A) exp i ddx ( + ) (2.3c) µ µν FP 1Y M gf L where ∆ (A) is the Faddeev-Popov determinant associated with the gauge ﬁxing Lagrangian . (More than one FP gf gauge ﬁxing may occur [6–8].) The ﬁeld Aa (but not Fa ) interacts with other “matter” ﬁelds. µ µν If in Eq. (2.3c) we perform the shift (cid:0) (cid:1) → − Fa Fa + ∂ Aa ∂ Aa + g(cid:15)abcAb Ac (2.4) µν µν µ ν ν µ µ ν (cid:90) (cid:90) (cid:20) (cid:21) then we ﬁnd that D D 1 L L Z = Aa Fa ∆ (A) exp i ddx Fa Faµν + + . (2.5) µ µν FP 4 µν 2Y M gf L the integral over Fa decouples and the usual generating functional associated with is recovered with its three- µν 2Y M (cid:104) (cid:105) (cid:104) (cid:105) point and four-point vertices. (In its unshifted form, Eq. (2.3c) results in the three propagators AA , FF and (cid:104) (cid:105) (cid:104) (cid:105) AF and the vertex FAA [2, 3].) We can also make the shift (cid:0) (cid:1) → − Fa Fa + ∂ Aa ∂ Aa (2.6) µν µν µ ν ν µ ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Our

text: second

text: result

text: is

text: that

text: it

text: is

text: possible

text: to

text: shift

text: variables

text: of

text: integration

text: in

text: the

text: action

text: within

text: the

text: path

text: integral

text: to

text: eliminate

text: this

text: mixed

text: propagator.

text: We

text: are

text: then

text: left

text: with

text: a

text: relatively

text: simple

text: set

text: of

text: Feynman

text: rules;

text: there

text: are

text: now

text: just

text: two

text: propagating

text: ﬁelds

text: (that

text: do

text: not

text: mix)

text: and

text: three

text: vertices.

text: This

text: is

text: an

text: improvement

text: over

text: the

text: situation

text: that

text: occurs

text: in

text: the

text: action

text: where

text: there

text: is

text: one

text: propagating

text: ﬁeld

text: and

text: an

text: inﬁnite

text: number

text: of

text: vertices

text: with

text: an

text: arbitrary

text: number

text: of

text: external

text: ﬁelds.

text: We

text: then

text: demonstrate

text: the

text: utility

text: of

text: our

text: result

text: by

text: computing

text: the

text: two-point

text: function

text: to

text: one-loop

text: order

text: using

text: an

text: arbitrary

text: gauge

text: ﬁxing

text: parameter.

text: In

text: the

text: limiting

text: case

text: in

text: which

text: this

text: parameter

text: equals

text: one,

text: we

text: reproduce

text: the

text: result

text: of

text: Ref.

text: The

text: ﬁrst

text: order

text: formalism

text: has

text: also

text: been

text: used

text: for

text: doing

text: loop

text: calculations

text: in

text: gravity

text: in

text: Ref.

text: though

text: in

text: the

text: models

text: considered

text: there

text: it

text: is

text: not

text: clear

text: if

text: the

text: ﬁrst

text: and

text: second

text: order

text: formalisms

text: are

text: equivalent.

text: We

text: begin

text: by

text: considering

text: the

text: action.

text: II.

text: THE

text: FIRST

text: ORDER

text: YANG-MILLS

text: ACTION

text: It

text: is

text: evident

text: that

text: the

text: Lagrangian

text: L

text: −1

text: Fa

text: ∂µAaν

text: ∂νAaµ

text: Fa

text: Faµν

text: M

text: µν

text: µν

text: is

text: classically

text: equivalent

text: to

text: the

text: Lagrangian

text: L

text: −1

text: Aa

text: Aa

text: Ac

text: M

text: µ

text: ν

text: ν

text: µ

text: µ

text: ν

text: L

text: L

text: as

text: upon

text: substitution

text: of

text: the

text: equation

text: of

text: motion

text: for

text: Fa

text: that

text: follows

text: from

text: back

text: into

text: follows.

text: µν

text: M

text: M

text: The

text: and

text: Lagrangians

text: have

text: the

text: gauge

text: invariance

text: δFa

text: θc

text: µν

text: µν

text: δAa

text: θa

text: θc;

text: µ

text: µ

text: µ

text: L

text: we

text: are

text: led

text: to

text: the

text: path

text: integral

text: for

text: M

text: D

text: D

text: L

text: L

text: Z

text: Aa

text: Fa

text: (A)

text: exp

text: i

text: ddx

text: µ

text: µν

text: FP

text: M

text: gf

text: L

text: where

text: (A)

text: is

text: the

text: Faddeev-Popov

text: determinant

text: associated

text: with

text: the

text: gauge

text: ﬁxing

text: Lagrangian

text: (More

text: than

text: one

text: FP

text: gf

text: gauge

text: ﬁxing

text: may

text: occur

text: The

text: ﬁeld

text: Aa

text: (but

text: not

text: Fa

text: interacts

text: with

text: other

text: “matter”

text: ﬁelds.

text: µ

text: µν

text: If

text: in

text: Eq.

text: we

text: perform

text: the

text: shift

text: Fa

text: Fa

text: Aa

text: Aa

text: Ac

text: µν

text: µν

text: µ

text: ν

text: ν

text: µ

text: µ

text: ν

text: then

text: we

text: ﬁnd

text: that

text: D

text: D

text: L

text: L

text: Z

text: Aa

text: Fa

text: (A)

text: exp

text: i

text: ddx

text: Fa

text: Faµν

text: µ

text: µν

text: FP

text: µν

text: M

text: gf

text: L

text: the

text: integral

text: over

text: Fa

text: decouples

text: and

text: the

text: usual

text: generating

text: functional

text: associated

text: with

text: is

text: recovered

text: with

text: its

text: three-

text: µν

text: M

text: point

text: and

text: four-point

text: vertices.

text: (In

text: its

text: unshifted

text: form,

text: Eq.

text: results

text: in

text: the

text: three

text: propagators

text: AA

text: FF

text: and

text: AF

text: and

text: the

text: vertex

text: FAA

text: We

text: can

text: also

text: make

text: the

text: shift

text: Fa

text: Fa

text: Aa

text: Aa

text: µν

text: µν

text: µ

text: ν

text: ν

text: µ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##