text: 9 − Proof of Theorem 1.1. Start with A0 = A = UUt and apply Lemma 4.1 to ﬁnd σ1 of size m 1 such that A1 = Uσ1Uσt1 is of rank(cid:16)n and(cid:17) satisﬁes (cid:16) (cid:17) − + 1 (14) Tr A−1 1 (cid:54) m −n Tr A−0 1 . m n − Now apply Lemma 4.1 again with U to ﬁnd σ2 of size m 2 such that A2 = U Ut is of σ1 σ2 σ2 rank n and satisﬁes (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) − − + 1 (15) Tr A−2 1 (cid:54) m− −n 1Tr A−1 1 (cid:54) m − n − 1Tr A−0 1 . m n m n − If we continue this procedure, after k steps we ﬁnd σ of size m k such that A = U Ut is of rank n and satisﬁes (cid:16) (cid:17) k (cid:16) (cid:17) k σk σk − + 1 (16) Tr A−k 1 (cid:54) mm− n −n k + 1Tr A−0 1 . (cid:54) − − This holds for any k m n. After k = m n steps, Theorem 1.1 is proved. (cid:3) PIERRE YOUSSEF, UNIVERSITY OF ALBERTA DEPARTMENT OF MATHEMATICAL AND STATISTICAL SCIENCES EDMONTON, AB, CANADA. E-MAIL: PYOUSSEF@UALBERTA.CA ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Proof

text: of

text: Theorem

text: Start

text: with

text: A

text: UUt

text: and

text: apply

text: Lemma

text: to

text: ﬁnd

text: σ1

text: of

text: size

text: m

text: such

text: that

text: is

text: of

label: equation

text: satisﬁes

text: Tr

text: m

text: −n

text: Tr

text: m

text: n

text: Now

text: apply

text: Lemma

text: again

text: with

text: U

text: to

text: ﬁnd

text: σ2

text: of

text: size

text: m

text: such

text: that

text: U

text: Ut

text: is

text: of

text: σ1

text: σ2

text: σ2

text: rank

text: n

text: and

text: satisﬁes

text: Tr

text: m−

text: −n

text: m

text: n

text: m

text: n

text: m

text: n

text: If

text: we

text: continue

text: this

text: procedure,

text: after

text: k

text: steps

text: we

text: ﬁnd

text: σ

text: of

text: size

text: m

text: k

text: such

text: that

text: A

text: U

text: Ut

text: is

text: of

text: rank

text: n

text: and

text: satisﬁes

text: k

text: k

text: σk

text: σk

text: Tr

text: A−k

text: mm−

text: n

text: −n

text: k

text: This

text: holds

text: for

text: any

text: k

text: m

text: n.

text: After

text: k

text: m

text: n

text: steps,

text: Theorem

text: is

text: proved.

text: PIERRE

text: YOUSSEF,

text: UNIVERSITY

text: OF

text: ALBERTA

text: DEPARTMENT

text: OF

text: MATHEMATICAL

text: AND

text: STATISTICAL

text: SCIENCES

text: EDMONTON,

text: AB,

text: CANADA.

text: E-MAIL:

text: PYOUSSEF@UALBERTA.CA

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∗ We report the direct imaging of a novel modulated ﬂux striped domain phase in a nearly twin-free YBCO crystal. These domains arise from instabilities in the vortex structure within a narrow region of tilted magnetic ﬁelds at small angles from the in-plane direction. By comparing the experimental and theoretically derived vortex phase diagrams we infer that the stripe domains emerge from a ﬁrst order phase transition of the vortex structure. The size of domains containing vortices of certain orientations is controlled by the balance between the vortex stray ﬁeld energy and the positive energy of the domain boundaries. Our results conﬁrm the existence of the kinked vortex chain phase in an anisotropic high temperature superconductor and reveal a sharp transition in the state of this phase resulting in regular vortex domains. In type II superconductors (SC) the magnetic ﬁeld pen- to the anisotropy axis rather than to the vortex line. etrates in the form of vortices - magnetic ﬂux tubes each This creates an inversion of the magnetic ﬁeld at some carrying a single ﬂux quantum [1]. Interactions between distance from the vortex core resulting in an attraction vortices and their coupling to crystal defects result in a of the neighboring vortex within the tilt plane [3, 5] rich variety of vortex structures. They include a trian- and formation of dense vortex chains coexisting with the gular or square vortex lattice that can subsequently melt dilute lattice of Abrikosov vortices, as observed in deco- at high temperatures to a vortex liquid, a pinned vortex ration, Hall probe, and electron microscopy experiments glassy state that can sustain high current carrying capac- in layered high-Tc superconductors (see review [6]). ity, and in highly anisotropic superconductors, 2D pan- Formation of patterns including stripe structures is cake vortices that can interact with in-plane Josephson general for a wide variety of condensed matter and col- vortices to create 1D vortex chain states. These states of loidal systems with competing attractive and repulsive vortex matter deﬁne all transport and magnetic proper- interactions, which are usually extended at diﬀerent ties of applied superconductors. length scales [7–9]. In fact, long range attraction and Here we report on an unusual vortex domain struc- short range repulsion of vortices in superconductors with ture that arises following a ﬁrst order phase transition in two order parameters [10] as also in multilayers of dif- the vortex state in nearly untwined YBCO crystal under ferent superconductors [11, 12] can result in formation tilted magnetic ﬁelds. Typically, vortices repulse each of vortex stripes. Experimentally, stripes of vortices in- other when the distance between them is much smaller termittent with Meissner phase were observed in MgB 2 than the penetration depth λ, the scale over which the single crystals [13, 14] and were treated as a possible magnetic ﬁeld of the vortices decays. In this case, the vor- consequence of the two diﬀerent SC condensates in the tices can arrange into a uniformly spaced lattice or form 2-band magnesium diborate. smooth density gradients deﬁned by the critical current, In this work we discovered a new vortex structure of the maximum current that the superconductor can sus- regular stipe domains with alternating ﬂux density, B , z tain before reverting to the normal state. In contrast, at generated by magnetic ﬁelds tilted from the ab-plane in larger distances the interactions between vortices can be a nearly twin-free YBCO platelet crystal. Unlike vor- attractive, resulting in the formation of vortex chains and tex/Meissner domains and vortex chain structures, our bundles (see review [2]). For example, periodic stripe do- vortex domains are created via an instability in the vor- mains and circular bundles of vortices interspersed with tex state [15–22] due to a ﬁrst order phase transition ﬂux free Meissner regions occur in thin niobium disks ∼ characterized by an abrupt jump in the vortex tilt an- with a low Ginzburg-Landau parameter κ = λ/ξ 1, gle [23]. A possibility of such ﬁrst order transition in where ξ is the coherence length setting the size of the the vortex orientation was ﬁrst noticed by Buzdin and vortex core. These vortex domains are similar to the al- Simonov in [24]. During the phase transition, an inter- ternating normal (N) and superconducting Meissner (M) mediate state with domains of diﬀerent vortex phases domains found in the intermediate state of type I SCs (orientations) emerges. We propose that similar to the with a nonzero demagnetizing factor. In the latter, the case of type I SCs and ferromagnets, the regular struc- lamellae M/N domains are known to minimize the mag- ture of domains with alternating density of B reduces z netostatic energy of the normal regions at the expense of the magnetostatic energy of the vortex stray ﬁelds per- the positive energy of the boundaries between the normal pendicular to the surface of the sample. Between the and SC phases. domains with diﬀerent ﬂux orientations vortices gradu- Vortex attraction can also appear in large κ, anisotropic ally rotate forming a domain boundary. Responsible for type II SCs when an applied magnetic ﬁeld is tilted from this rotation supercurrents yield a positive domain wall the anisotropy axis [3–5]. Here, the circulating super- energy which limits the reﬁnement of domains caused by currents that form the vortex tend to ﬂow perpendicular magnetostatics. We suggest that the attractive coupling of tilted vortices into chains along the vortex tilt plane [3–5] and attraction between the neighboring chains with ∗ mutually shifted vortices stabilizes domains with larger ##LTLine##

label: paragraph

text: We

label: abstract

text: report

text: the

text: direct

text: imaging

text: of

text: a

text: novel

text: modulated

text: ﬂux

text: striped

text: domain

text: phase

text: in

text: a

text: nearly

text: twin-free

text: YBCO

text: crystal.

text: These

text: domains

text: arise

text: from

text: instabilities

text: in

text: the

text: vortex

text: structure

text: within

text: a

text: narrow

text: region

text: of

text: tilted

text: magnetic

text: ﬁelds

text: at

text: small

text: angles

text: from

text: the

text: in-plane

text: direction.

text: By

text: comparing

text: the

text: experimental

text: and

text: theoretically

text: derived

text: vortex

text: phase

text: diagrams

text: we

text: infer

text: that

text: the

text: stripe

text: domains

text: emerge

text: from

text: a

text: ﬁrst

text: order

text: phase

text: transition

text: of

text: the

text: vortex

text: structure.

text: The

text: size

text: of

text: domains

text: containing

text: vortices

text: of

text: certain

text: orientations

text: is

text: controlled

text: by

text: the

text: balance

text: between

text: the

text: vortex

text: stray

text: ﬁeld

text: energy

text: and

text: the

text: positive

text: energy

text: of

text: the

text: domain

text: boundaries.

text: Our

text: results

text: conﬁrm

text: the

text: existence

text: of

text: the

text: kinked

text: vortex

text: chain

text: phase

text: in

text: an

text: anisotropic

text: high

text: temperature

text: superconductor

text: and

text: reveal

text: a

text: sharp

text: transition

text: in

text: the

text: state

text: of

text: this

text: phase

text: resulting

text: in

text: regular

text: vortex

text: domains.

text: In

text: type

text: II

text: superconductors

text: (SC)

text: the

text: magnetic

text: ﬁeld

text: pen-

text: to

text: the

text: anisotropy

text: axis

text: rather

text: than

text: to

text: the

text: vortex

text: line.

text: etrates

text: in

text: the

text: form

text: of

text: vortices

text: magnetic

text: ﬂux

text: tubes

text: each

text: This

text: creates

text: an

text: inversion

text: of

text: the

text: magnetic

text: ﬁeld

text: at

text: some

text: carrying

text: a

text: single

text: ﬂux

text: quantum

text: Interactions

text: between

text: distance

text: from

text: the

text: vortex

text: core

text: resulting

text: in

text: an

text: attraction

text: vortices

text: and

text: their

text: coupling

text: to

text: crystal

text: defects

text: result

text: in

text: a

text: of

text: the

text: neighboring

text: vortex

text: within

text: the

text: tilt

text: plane

text: rich

text: variety

text: of

text: vortex

text: structures.

text: They

text: include

text: a

text: trian-

text: and

text: formation

text: of

text: dense

text: vortex

text: chains

text: coexisting

text: with

text: the

text: gular

text: or

text: square

text: vortex

text: lattice

text: that

text: can

text: subsequently

text: melt

text: dilute

text: lattice

text: of

text: Abrikosov

text: vortices,

text: as

text: observed

text: in

text: deco-

text: at

text: high

text: temperatures

text: to

text: a

text: vortex

text: liquid,

text: a

text: pinned

text: vortex

text: ration,

text: Hall

text: probe,

text: and

text: electron

text: microscopy

text: experiments

text: glassy

text: state

text: that

text: can

text: sustain

text: high

text: current

text: carrying

text: capac-

text: in

text: layered

text: high-Tc

text: superconductors

text: (see

text: review

text: ity,

text: and

text: in

text: highly

text: anisotropic

text: superconductors,

text: pan-

text: Formation

text: of

text: patterns

text: including

text: stripe

text: structures

text: is

text: cake

text: vortices

text: that

text: can

text: interact

text: with

text: in-plane

text: Josephson

text: general

text: for

text: a

text: wide

text: variety

text: of

text: condensed

text: matter

text: and

text: col-

text: vortices

text: to

text: create

text: vortex

text: chain

text: states.

text: These

text: states

text: of

text: loidal

text: systems

text: with

text: competing

text: attractive

text: and

text: repulsive

text: vortex

text: matter

text: deﬁne

text: all

text: transport

text: and

text: magnetic

text: proper-

text: interactions,

text: which

text: are

text: usually

text: extended

text: at

text: diﬀerent

text: ties

text: of

text: applied

text: superconductors.

text: length

text: scales

text: In

text: fact,

text: long

text: range

text: attraction

text: and

text: Here

text: we

text: report

text: on

text: an

text: unusual

text: vortex

text: domain

text: struc-

text: short

text: range

text: repulsion

text: of

text: vortices

text: in

text: superconductors

text: with

text: ture

text: that

text: arises

text: following

text: a

text: ﬁrst

text: order

text: phase

text: transition

text: in

text: two

text: order

text: parameters

text: as

text: also

text: in

text: multilayers

text: of

text: dif-

text: the

text: vortex

text: state

text: in

text: nearly

text: untwined

text: YBCO

text: crystal

text: under

text: ferent

text: superconductors

text: can

text: result

text: in

text: formation

text: tilted

text: magnetic

text: ﬁelds.

text: Typically,

text: vortices

text: repulse

text: each

text: of

text: vortex

text: stripes.

text: Experimentally,

text: stripes

text: of

text: vortices

text: in-

text: other

text: when

text: the

text: distance

text: between

text: them

text: is

text: much

text: smaller

text: termittent

text: with

text: Meissner

text: phase

text: were

text: observed

text: in

text: MgB

text: than

text: the

text: penetration

text: depth

text: λ,

text: the

text: scale

text: over

text: which

text: the

text: single

text: crystals

text: and

text: were

text: treated

text: as

text: a

text: possible

text: magnetic

text: ﬁeld

text: of

text: the

text: vortices

text: decays.

text: In

text: this

text: case,

text: the

text: vor-

text: consequence

text: of

text: the

text: two

text: diﬀerent

text: SC

text: condensates

text: in

text: the

text: tices

text: can

text: arrange

text: into

text: a

text: uniformly

text: spaced

text: lattice

text: or

text: form

text: magnesium

text: diborate.

text: smooth

text: density

text: gradients

text: deﬁned

text: by

text: the

text: critical

text: current,

text: In

text: this

text: work

text: we

text: discovered

text: a

text: new

text: vortex

text: structure

text: of

text: the

text: maximum

text: current

text: that

text: the

text: superconductor

text: can

text: sus-

text: regular

text: stipe

text: domains

text: with

text: alternating

text: ﬂux

text: density,

text: B

text: z

text: tain

text: before

text: reverting

text: to

text: the

text: normal

text: state.

text: In

text: contrast,

text: at

text: generated

text: by

text: magnetic

text: ﬁelds

text: tilted

text: from

text: the

text: ab-plane

text: in

text: larger

text: distances

text: the

text: interactions

text: between

text: vortices

text: can

text: be

text: a

text: nearly

text: twin-free

text: YBCO

text: platelet

text: crystal.

text: Unlike

text: vor-

text: attractive,

text: resulting

text: in

text: the

text: formation

text: of

text: vortex

text: chains

text: and

text: tex/Meissner

text: domains

text: and

text: vortex

text: chain

text: structures,

text: our

text: bundles

text: (see

text: review

text: For

text: example,

text: periodic

text: stripe

text: do-

text: vortex

text: domains

text: are

text: created

text: via

text: an

text: instability

text: in

text: the

text: vor-

text: mains

text: and

text: circular

text: bundles

text: of

text: vortices

text: interspersed

text: with

text: tex

text: state

text: due

text: to

text: a

text: ﬁrst

text: order

text: phase

text: transition

text: ﬂux

text: free

text: Meissner

text: regions

text: occur

text: in

text: thin

text: niobium

text: disks

text: characterized

text: by

text: an

text: abrupt

text: jump

text: in

text: the

text: vortex

text: tilt

text: an-

text: with

text: a

text: low

text: Ginzburg-Landau

text: parameter

text: κ

text: λ/ξ

text: gle

text: A

text: possibility

text: of

text: such

text: ﬁrst

text: order

text: transition

text: in

text: where

text: ξ

text: is

text: the

text: coherence

text: length

text: setting

text: the

text: size

text: of

text: the

text: the

text: vortex

text: orientation

text: was

text: ﬁrst

text: noticed

text: by

text: Buzdin

text: and

text: vortex

text: core.

text: These

text: vortex

text: domains

text: are

text: similar

text: to

text: the

text: al-

text: Simonov

text: in

text: During

text: the

text: phase

text: transition,

text: an

text: inter-

text: ternating

text: normal

text: (N)

text: and

text: superconducting

text: Meissner

text: (M)

text: mediate

text: state

text: with

text: domains

text: of

text: diﬀerent

text: vortex

text: phases

text: domains

text: found

text: in

text: the

text: intermediate

text: state

text: of

text: type

text: I

text: SCs

text: (orientations)

text: emerges.

text: We

text: propose

text: that

text: similar

text: to

text: the

text: with

text: a

text: nonzero

text: demagnetizing

text: factor.

text: In

text: the

text: latter,

text: the

text: case

text: of

text: type

text: I

text: SCs

text: and

text: ferromagnets,

text: the

text: regular

text: struc-

text: lamellae

text: M/N

text: domains

text: are

text: known

text: to

text: minimize

text: the

text: mag-

text: ture

text: of

text: domains

text: with

text: alternating

text: density

text: of

text: B

text: reduces

text: z

text: netostatic

text: energy

text: of

text: the

text: normal

text: regions

text: at

text: the

text: expense

text: of

text: the

text: magnetostatic

text: energy

text: of

text: the

text: vortex

text: stray

text: ﬁelds

text: per-

text: the

text: positive

text: energy

text: of

text: the

text: boundaries

text: between

text: the

text: normal

text: pendicular

text: to

text: the

text: surface

text: of

text: the

text: sample.

text: Between

text: the

text: and

text: SC

text: phases.

text: domains

text: with

text: diﬀerent

text: ﬂux

text: orientations

text: vortices

text: gradu-

text: Vortex

text: attraction

text: can

text: also

text: appear

text: in

text: large

text: κ,

text: anisotropic

text: ally

text: rotate

text: forming

text: a

text: domain

text: boundary.

text: Responsible

text: for

text: type

text: II

text: SCs

text: when

text: an

text: applied

text: magnetic

text: ﬁeld

text: is

text: tilted

text: from

text: this

text: rotation

text: supercurrents

text: yield

text: a

text: positive

text: domain

text: wall

text: the

text: anisotropy

text: axis

text: Here,

text: the

text: circulating

text: super-

text: energy

text: which

text: limits

text: the

text: reﬁnement

text: of

text: domains

text: caused

text: by

text: currents

text: that

text: form

text: the

text: vortex

text: tend

text: to

text: ﬂow

text: perpendicular

text: magnetostatics.

text: We

text: suggest

text: that

text: the

text: attractive

text: coupling

text: of

text: tilted

text: vortices

text: into

text: chains

text: along

text: the

text: vortex

text: tilt

text: plane

text: and

text: attraction

text: between

text: the

text: neighboring

text: chains

text: with

text: mutually

text: shifted

text: vortices

text: stabilizes

text: domains

text: with

text: larger

text: ##LTLine##


text: 3 FIG. 1. (a) Trapped ﬂux (H=0) after ﬁeld cooling from T > T to 60K in H = 1388 Oe. (b) Same after cooling in H = 1388 ◦ c ◦ // Oe tilted by 1.6 . (c) Same after cooling in H = 1388 Oe tilted by 3.5 . (d) Proﬁle of B between two arrows shown in Fig.1b. z (d) Scheme of the stray ﬁelds corresponding to (a). (e) B proﬁle across the sample between arrows shown in (b). (f) Scheme z of the staircase vortices in cuprates. (g) Scheme of the vortex domains. Currents (short arrows) in the domain wall ﬂow along vortices in force free conﬁguration. − FIG. 2. Experimental H H domain existence map. z x vortex attraction as a possible mechanism for stabilizing FIG. 3. Vortex domains at 60K after application and switch- domains with larger Bz. ing oﬀ H⊥∼=210◦0 Oe along∼the s◦hort side tilted from the ab- The vortex phase diagram in tilted ﬁelds is constructed plane by 1.20 (a) and 0.40 (b) using analytic results for vortex-chain energies derived in Ref. [23] (see [31] for details). Our calculations for a bulk sample show abrupt changes in the vortex angle which ponents and on the SC materials parameters [31]. The are mostly driven by the energy of an isolated vortex structure of vortices tilted by a small angle from the in- chain formed in the tilt plane while interactions between plane direction, θ < 1/γ, can be described by the addi- the chains may be neglected. The total energy of a vor- tion of c-axis pancake vortices (kinks) to the Josephson tex chain consists of isolated vortex line and inter-vortex vortex line lattice in purely in-plane ﬁeld. The equilib- coupling contributions, E = Es + Ei . Both terms rium state is obtained by minimizing the sum of E TV TV TV TV depend on the in-plane and c-axis magnetic ﬁeld com- and the energy of interaction of the c-axis ﬂux with ther- ##LTFigure## ##LTFigure## ##LTFigure##

text: FIG.

text: (a)

label: caption

text: Trapped

text: ﬂux

text: after

text: ﬁeld

text: cooling

text: from

text: T

text: T

text: to

text: in

text: H

text: Oe.

text: (b)

text: Same

text: after

text: cooling

text: in

text: H

text: c

text: Oe

text: tilted

text: by

text: (c)

text: Same

text: after

text: cooling

text: in

text: H

text: Oe

text: tilted

text: by

text: (d)

text: Proﬁle

text: of

text: B

text: between

text: two

text: arrows

text: shown

text: in

text: z

text: (d)

text: Scheme

text: of

text: the

text: stray

text: ﬁelds

text: corresponding

text: to

text: (a).

text: (e)

text: B

text: proﬁle

text: across

text: the

text: sample

text: between

text: arrows

text: shown

text: in

text: (b).

text: (f)

text: Scheme

text: z

text: of

text: the

text: staircase

text: vortices

text: in

text: cuprates.

text: (g)

text: Scheme

text: of

text: the

text: vortex

text: domains.

text: Currents

text: (short

text: arrows)

text: in

text: the

text: domain

text: wall

text: ﬂow

text: along

text: vortices

text: in

text: force

text: free

text: conﬁguration.

text: FIG.

text: Experimental

text: H

text: H

text: domain

text: existence

text: map.

text: z

text: x

text: vortex

text: attraction

text: as

text: a

text: possible

text: mechanism

text: for

text: stabilizing

text: FIG.

text: Vortex

text: domains

text: at

text: after

text: application

text: and

text: switch-

text: domains

text: with

text: larger

text: Bz.

text: ing

text: oﬀ

text: Oe

text: along∼the

text: s◦hort

text: side

text: tilted

text: from

text: the

text: ab-

text: The

text: vortex

text: phase

text: diagram

text: in

text: tilted

text: ﬁelds

text: is

text: constructed

text: plane

text: by

text: (a)

text: and

text: (b)

text: using

text: analytic

text: results

text: for

text: vortex-chain

text: energies

text: derived

text: in

text: Ref.

text: (see

text: for

text: details).

text: Our

text: calculations

text: for

text: a

text: bulk

text: sample

text: show

text: abrupt

text: changes

text: in

text: the

text: vortex

text: angle

text: which

text: ponents

text: and

text: on

text: the

text: SC

text: materials

text: parameters

text: The

text: are

text: mostly

text: driven

text: by

text: the

text: energy

text: of

text: an

text: isolated

text: vortex

text: structure

text: of

text: vortices

text: tilted

text: by

text: a

text: small

text: angle

text: from

text: the

text: in-

text: chain

text: formed

text: in

text: the

text: tilt

text: plane

text: while

text: interactions

text: between

text: plane

text: direction,

text: θ

text: can

text: be

text: described

text: by

text: the

text: addi-

text: the

text: chains

text: may

text: be

text: neglected.

text: The

text: total

text: energy

text: of

text: a

text: vor-

text: tion

text: of

text: c-axis

text: pancake

text: vortices

text: (kinks)

text: to

text: the

text: Josephson

text: tex

text: chain

text: consists

text: of

text: isolated

text: vortex

text: line

text: and

text: inter-vortex

text: vortex

text: line

text: lattice

text: in

text: purely

text: in-plane

text: ﬁeld.

text: The

text: equilib-

text: coupling

text: contributions,

text: E

text: Es

text: Ei

text: Both

text: terms

text: rium

text: state

text: is

text: obtained

text: by

text: minimizing

text: the

text: sum

text: of

text: E

text: TV

text: TV

text: TV

text: TV

text: depend

text: on

text: the

text: in-plane

text: and

text: c-axis

text: magnetic

text: ﬁeld

text: com-

text: and

text: the

text: energy

text: of

text: interaction

text: of

text: the

text: c-axis

text: ﬂux

text: with

text: ther-

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: FEYNMAN INTEGRALS AND PERIODS 13 Q for in (3.6) for a graph with external momenta is the same as the one of the associated vacuum graph. We therefore conclude that in the representation of a Feynman graph Γ in (3.11) U and (3.12) the ﬁrst Symanzik polynomial is the determinant of the period matrix of the vacuum graph Γ associated to the graph Γ. 0 F We now turn to the reinterpretation of the second Symanzik polynomial in (3.10) using the world-line methods. F F U Deﬁne ˆ := / one can rewrite this expression in the following way (cid:88) (cid:88) F − n − · (3.19) ˆ = x (m2 iε) + p p G(x , x ; Ω) . i i r s r s ≤ ≤ i=1 1 r,s m where we have introduced the Green funct(cid:18)io(cid:90)n (cid:19) (cid:18)(cid:90) (cid:19) −1 1 xr · − · xr (3.20) G(x , x ; Ω) = d(x , x ) + ω Ω 1 ω , r s 2 r s 2 xs xs where ω = (ω , . . . , ω )T the size l vectors of elementary line elements along the 1 l loops, and d(x , x ) is the distance between the two vertices of coordinates x and r s r x on the graph. This is the expression for the Green function between two points s on the word-line graph constructed in [DS06, section IV]. In section we provide a few examples. One can therefore give an alternative form for the parametric representation of the Feynman graph Γ, by writing splitting the integration over the parameters ≤ ≤ into an integration over the proper-times T with 1 i n of the vacuum graph, ≤ ≤ i 0 and the insertion points x with 1 i m of the extra vertices carrying external i momenta (cid:90) (cid:81) (cid:89) 1 m n0 dT (3.21) IΓD(pi, mi) = D Fˆ3(l−1)+m−lD2 i=1 dxi (dei=t1Ω)D2i . For the case of ϕ3 vertices, at the loop order l, the number of propagators of − vacuum graph is n = 3(l 1) and this formulation leads to a treatment of ﬁeld 0 theory graphs in a string theory manner as in [BBV08a, BBV08b, BBBV08, BG10, Bj10]. The construction applies to any kind of interaction since we never used the details of the valence of the vertices one needs to consider. We provide a few example based on ϕ3 and ϕ4 scalar theories. For instance for the two-loop and three-loop graphs of ﬁgure 2 the period matrix are given by ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: FEYNMAN

text: INTEGRALS

text: AND

text: PERIODS

text: Q

label: list

text: for

text: in

text: for

text: a

text: graph

text: with

text: external

text: momenta

text: is

text: the

text: same

text: as

text: the

text: one

text: of

text: the

text: associated

text: vacuum

text: graph.

text: We

text: therefore

text: conclude

text: that

text: in

text: the

text: representation

text: of

text: a

text: Feynman

text: graph

text: Γ

text: in

text: U

text: and

text: the

text: ﬁrst

text: Symanzik

text: polynomial

text: is

text: the

text: determinant

text: of

text: the

text: period

text: matrix

text: of

text: the

text: vacuum

text: graph

text: Γ

text: associated

text: to

text: the

text: graph

text: Γ.

text: F

text: We

text: now

text: turn

text: to

text: the

text: reinterpretation

text: of

text: the

text: second

text: Symanzik

text: polynomial

text: in

text: using

text: the

text: world-line

text: methods.

text: F

text: F

text: U

text: Deﬁne

text: ˆ

text: one

text: can

text: rewrite

text: this

text: expression

text: in

text: the

text: following

text: way

text: F

text: n

text: ˆ

text: x

text: iε)

text: p

text: p

text: G(x

text: x

text: Ω)

text: i

text: i

text: r

text: s

text: r

text: s

text: r,s

text: m

text: where

text: we

text: have

text: introduced

text: the

text: Green

text: −1

text: xr

text: xr

text: G(x

text: x

text: Ω)

text: d(x

text: x

text: ω

text: Ω

text: ω

text: r

text: s

text: r

text: s

text: xs

text: xs

text: where

text: ω

text: (ω

text: ω

text: )T

text: the

text: size

text: l

text: vectors

text: of

text: elementary

text: line

text: elements

text: along

text: the

text: l

text: loops,

text: and

text: d(x

text: x

text: is

text: the

text: distance

text: between

text: the

text: two

text: vertices

text: of

text: coordinates

text: x

text: and

text: r

text: s

text: r

text: x

text: on

text: the

text: graph.

text: This

text: is

text: the

text: expression

text: for

text: the

text: Green

text: function

text: between

text: two

text: points

text: s

text: on

text: the

text: word-line

text: graph

text: constructed

text: in

text: section

text: IV].

text: In

text: section

text: we

text: provide

text: a

text: few

text: examples.

text: One

text: can

text: therefore

text: give

text: an

text: alternative

text: form

text: for

text: the

text: parametric

text: representation

text: of

text: the

text: Feynman

text: graph

text: Γ,

text: by

text: writing

text: splitting

text: the

text: integration

text: over

text: the

text: parameters

text: into

text: an

text: integration

text: over

text: the

text: proper-times

text: T

text: with

text: i

text: n

text: of

text: the

text: vacuum

text: graph,

text: i

text: and

text: the

text: insertion

text: points

text: x

text: with

text: i

text: m

text: of

text: the

text: extra

text: vertices

text: carrying

text: external

text: i

text: momenta

text: m

text: dT

text: IΓD(pi,

text: mi)

text: D

text: dxi

text: For

text: the

text: case

text: of

text: ϕ3

text: vertices,

text: at

text: the

text: loop

text: order

text: l,

text: the

text: number

text: of

text: propagators

text: of

text: vacuum

text: graph

text: is

text: n

text: and

text: this

text: formulation

text: leads

text: to

text: a

text: treatment

text: of

text: ﬁeld

text: theory

text: graphs

text: in

text: a

text: string

text: theory

text: manner

text: as

text: in

text: The

text: construction

text: applies

text: to

text: any

text: kind

text: of

text: interaction

text: since

text: we

text: never

text: used

text: the

text: details

text: of

text: the

text: valence

text: of

text: the

text: vertices

text: one

text: needs

text: to

text: consider.

text: We

text: provide

text: a

text: few

text: example

text: based

text: on

text: ϕ3

text: and

text: ϕ4

text: scalar

text: theories.

text: For

text: instance

text: for

text: the

text: two-loop

text: and

text: three-loop

text: graphs

text: of

text: ﬁgure

text: the

text: period

text: matrix

text: are

text: given

text: by

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: FEYNMAN INTEGRALS AND PERIODS 3 Amplitudes relations and monodromies 1. Unitarity methods Constructions and computations of quantum ﬁeld theory amplitudes have ex- perienced tremendous progress, leading to powerful methods for evaluating loop amplitudes [BDK96, Bern92, BDDK94, BCF04]. These methods made com- putable many unknown amplitudes and provide an increasing knowledge of gauge theory and gravity amplitudes in various dimensions. These methods are based on the unitarity properties of the scattering ampli- tudes in quantum ﬁeld theory. A quantum ﬁeld theory amplitude is a multivalued function presenting branch cuts associated to particle production. For local and Lorentz invariant quantum ﬁeld theories, the matrix of diﬀusion (cid:80) † S is unitary SS = 1. Therefore the scattering matrix T, deﬁned as S = 1 + iT − † † satisﬁes the relation T T = iTT . The perturbative expansion of the scattering matrix T = ≥ gnA leads to unitarity relation on the perturbative amplitudes n 0 n A . This implies that the imaginary (absorptive) part of the amplitudes A is n n expressible as some phase integral of product of lowest order amplitudes through Cutkosky rules [C60], and dispersion relation are used to reconstruct the full am- plitude. In general the evaluation of the dispersion relations is diﬃcult. Fortunately, at one-loop order, in four dimensions, we know a basis of scalar { } integral functions I speciﬁed by boxes, triangles, bubbles, tadpoles and rational r terms [BDK96, OPP06, EZ07, EKMZ(cid:88)11] − (1.1) A1 loop = c I n r r r where c are rational functions of the kinematics invariants. r An interesting aspect of this construction is that the scalar integral functions have distinctive analytic properties across their branch cuts. For instance the mass- less four-point amplitude can get a contribution from the massless box I (s, t), the 4 one-mass triangles I1m(s) and I1m(t), the massive massive bubbles I (s) and I (t). 3 3 2 2 The ﬁnite part of these functions contain contributions with distinctive discontinu- ities that can be isolated by cuts ∼ − − (1.2) I (s, t) log( s) log( t) 4 ∼ − (1.3) I1m(s) log2( s) 3 ∼ − (1.4) I (s) log( s) . 2 Higher-point one-loop amplitudes have dilogarithm functions entering the ex- − pression of the ﬁnite part, e.g. Li (1 s s /(s s )). Picking a particular → ∞ 2 12 23 34 56 kinematic region s , this function reduces to its branch cut behaviour − 12 ∼ − − − Li (1 s s /(s s )) log( s ) log( s ) + . . . which can be isolated by 2 12 23 34 56 12 23 the cut. It is now enough to look at the discontinuities across the various branch cuts to extract the coeﬃcients c in (1.1). The ambiguity has to be a rational function r of the kinematic invariants. There are various methods to ﬁx this ambiguity that are discussed for instance in [BDK96].

text: FEYNMAN

text: INTEGRALS

text: AND

text: PERIODS

text: Amplitudes

text: relations

text: and

text: monodromies

label: section

text: Unitarity

text: methods

text: Constructions

text: and

text: computations

text: of

text: quantum

text: ﬁeld

text: theory

text: amplitudes

text: have

text: ex-

text: perienced

text: tremendous

text: progress,

text: leading

text: to

text: powerful

text: methods

text: for

text: evaluating

text: loop

text: amplitudes

text: These

text: methods

text: made

text: com-

text: putable

text: many

text: unknown

text: amplitudes

text: and

text: provide

text: an

text: increasing

text: knowledge

text: of

text: gauge

text: theory

text: and

text: gravity

text: amplitudes

text: in

text: various

text: dimensions.

text: These

text: methods

text: are

text: based

text: on

text: the

text: unitarity

text: properties

text: of

text: the

text: scattering

text: ampli-

text: tudes

text: in

text: quantum

text: ﬁeld

text: theory.

text: A

text: quantum

text: ﬁeld

text: theory

text: amplitude

text: is

text: a

text: multivalued

text: function

text: presenting

text: branch

text: cuts

text: associated

text: to

text: particle

text: production.

text: For

text: local

text: and

text: Lorentz

text: invariant

text: quantum

text: ﬁeld

text: theories,

text: the

text: matrix

text: of

text: diﬀusion

text: S

text: is

text: unitary

text: SS

text: Therefore

text: the

text: scattering

text: matrix

text: T,

text: deﬁned

text: as

text: S

text: iT

text: satisﬁes

text: the

text: relation

text: T

text: T

text: iTT

text: The

text: perturbative

text: expansion

text: of

text: the

text: scattering

text: matrix

text: T

text: gnA

text: leads

text: to

text: unitarity

text: relation

text: on

text: the

text: perturbative

text: amplitudes

text: n

text: n

text: A

text: This

text: implies

text: that

text: the

text: imaginary

text: (absorptive)

text: part

text: of

text: the

text: amplitudes

text: A

text: is

text: n

text: n

text: expressible

text: as

text: some

text: phase

text: integral

text: of

text: product

text: of

text: lowest

text: order

text: amplitudes

text: through

text: Cutkosky

text: rules

text: and

text: dispersion

text: relation

text: are

text: used

text: to

text: reconstruct

text: the

text: full

text: am-

text: plitude.

text: In

text: general

text: the

text: evaluation

text: of

text: the

text: dispersion

text: relations

text: is

text: diﬃcult.

text: Fortunately,

text: at

text: one-loop

text: order,

text: in

text: four

text: dimensions,

text: we

text: know

text: a

text: basis

text: of

text: scalar

text: integral

text: functions

text: I

text: speciﬁed

text: by

text: boxes,

text: triangles,

text: bubbles,

text: tadpoles

text: and

text: rational

text: r

text: terms

text: loop

text: c

text: I

text: n

text: r

text: r

text: r

text: where

text: c

text: are

text: rational

text: functions

text: of

text: the

text: kinematics

text: invariants.

text: r

text: An

text: interesting

text: aspect

text: of

text: this

text: construction

text: is

text: that

text: the

text: scalar

text: integral

text: functions

text: have

text: distinctive

text: analytic

text: properties

text: across

text: their

text: branch

text: cuts.

text: For

text: instance

text: the

text: mass-

text: less

text: four-point

text: amplitude

text: can

text: get

text: a

text: contribution

text: from

text: the

text: massless

text: box

text: I

text: (s,

text: t),

text: the

text: one-mass

text: triangles

text: and

text: the

text: massive

text: massive

text: bubbles

text: I

text: (s)

text: and

text: I

text: (t).

text: The

text: ﬁnite

text: part

text: of

text: these

text: functions

text: contain

text: contributions

text: with

text: distinctive

text: discontinu-

text: ities

text: that

text: can

text: be

text: isolated

text: by

text: cuts

text: I

text: (s,

text: t)

text: log(

text: s)

text: log(

text: t)

text: s)

text: I

text: (s)

text: log(

text: s)

text: Higher-point

text: one-loop

text: amplitudes

text: have

text: dilogarithm

text: functions

text: entering

text: the

text: ex-

text: pression

text: of

text: the

text: ﬁnite

text: part,

text: e.g.

text: Li

text: s

text: s

text: /(s

text: s

text: Picking

text: a

text: particular

text: kinematic

text: region

text: s

text: this

text: function

text: reduces

text: to

text: its

text: branch

text: cut

text: behaviour

text: Li

text: s

text: s

text: /(s

text: s

text: log(

text: s

text: log(

text: s

text: which

text: can

text: be

text: isolated

text: by

text: the

text: cut.

text: It

text: is

text: now

text: enough

text: to

text: look

text: at

text: the

text: discontinuities

text: across

text: the

text: various

text: branch

text: cuts

text: to

text: extract

text: the

text: coeﬃcients

text: c

text: in

text: The

text: ambiguity

text: has

text: to

text: be

text: a

text: rational

text: function

text: r

text: of

text: the

text: kinematic

text: invariants.

text: There

text: are

text: various

text: methods

text: to

text: ﬁx

text: this

text: ambiguity

text: that

text: are

text: discussed

text: for

text: instance

text: in


text: 26 PIERRE VANHOVE (cid:80) In order to determine the diﬀerential equation satisﬁed by the all equal mass banana graphs we provide an alternative expression for the banana integrals. For K2 < ( n m )2(cid:90), one can perform a series expansion i=1 i (cid:89) (cid:80) (cid:80) − n 1 (cid:90) δ(x = 1) dx I2 = (cid:88) n − − i (cid:89) n ∞ − n m(cid:80)2x n x 1 (cid:80)t x [0,+ [n 1 i=1 i i i=1 i i=1 i n δ(x = 1) dx (8.5) = tk n − i . k≥0 [0,+∞[n−1 ( ni=1 m2i xi)k+1( ni=1 xi 1)k+1 i=1 xi Exponentiating the denominators (cid:90) (cid:88) (cid:89) (cid:80) (cid:80) (8.6) tk − − − dudv n dx In2 = k≥0 k!2 [0,+∞[n+1 δ(xn = 1) e u( ni=1 m2ixi) v( ni=1 xi 1) u−kv−k i=1 xii . Using the integral r(cid:90)epresentation for the K Bessel function 0 ∞ √ + − − dx (8.7) e um2x xv x = 2K0(2m uv) 0 one gets (cid:90) (cid:88) (cid:89) − √ − tk n 1 − − (8.8) In2 = 2n 1 k≥0 k!2 i=1 K0(2mi uv) [0,+∞[2 e um2n v ukvkdudv . Now setting uv = (x/2)2, the integral over v gives a K Bessel function with 0 the result (cid:90) (cid:18) (cid:19) (cid:88) (cid:89) √ 2k n − tx 1 (8.9) I2 = 2n 1 K (m x)dx . n [0,+∞[ k≥0 2 k!2 i=1 0 i Using the series expansion of the I(cid:88)B(cid:16)ess(cid:17)el function 0 x 2k 1 (8.10) I (x) = 0 ≥ 2 k!2 k 0 we get the following representation for the banana graph (see [BBBG08] for a previous appearance of this formula at the special values K2 = m2 and all equal masses mi = m) (cid:90) (cid:89) ∞ √ − + n (8.11) I2 = 2n 1 x I ( K2x) K (m x) dx . n 0 0 i 0 i=1 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: PIERRE

text: VANHOVE

text: In

text: order

text: to

text: determine

text: the

text: diﬀerential

text: equation

text: satisﬁed

text: by

text: the

text: all

text: equal

text: mass

text: banana

text: graphs

text: we

text: provide

text: an

text: alternative

text: expression

text: for

text: the

text: banana

text: integrals.

text: For

text: n

text: m

text: one

text: can

text: perform

text: a

text: series

text: expansion

text: i

text: n

text: δ(x

text: dx

text: n

text: i

text: n

text: n

text: n

text: x

text: x

text: [n

text: i

text: i

text: i

text: i

text: n

text: δ(x

text: dx

text: tk

text: n

text: i

text: xi

text: xi

text: Exponentiating

text: the

text: denominators

text: tk

text: dudv

text: n

text: dx

text: δ(xn

text: e

text: u(

text: v(

text: xi

text: u−kv−k

text: xii

text: Using

text: the

text: integral

text: for

text: the

text: K

text: Bessel

text: function

text: dx

text: e

text: xv

text: x

text: uv)

text: one

text: gets

text: tk

text: n

text: uv)

text: e

text: v

text: ukvkdudv

text: Now

text: setting

text: uv

text: the

text: integral

text: over

text: v

text: gives

text: a

text: K

text: Bessel

text: function

text: with

text: the

text: result

text: n

text: tx

text: K

text: (m

text: x)dx

text: n

text: i

text: Using

text: the

text: series

text: expansion

text: of

text: the

text: function

text: x

text: I

text: (x)

text: k

text: we

text: get

text: the

text: following

text: representation

text: for

text: the

text: banana

text: graph

text: (see

text: for

text: a

text: previous

text: appearance

text: of

text: this

text: formula

text: at

text: the

text: special

text: values

text: and

text: all

text: equal

text: masses

text: mi

text: m)

text: n

text: x

text: I

text: K

text: (m

text: x)

text: dx

text: n

text: i

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text: 28 PIERRE VANHOVE The diﬀerential operator acting on(cid:88)the integral In2(t) is given by − n 1 dk (9.10) L˜ − = q (t) n 1 k dtk k=0 ≤ ≤ − with the coeﬃcient q (t) polynomials of degree k + 1 for 0 k n 1. The k top degree and lowest-degree polynomial are given by (cid:89) (cid:98) (cid:99) (cid:98) (cid:99) n2 − − (9.11) qn−1(t) = t n2 +η(n) (t (n 2i)2) − i=0 n 1 dq − (t) (9.12) q − (t) = n 1 n 2 2 dt − (9.13) q (t) = t n . 0 ≡ ≡ with η(n) = 0 if n 1 mod 2 and 1 if n 0 mod 2. The inhomogeneous   term S is a constant given by n (cid:90) (cid:88) (cid:16) (cid:17)   (cid:98) (cid:99)  − ∞ n η(n) − 2 x 2k (9.14) S = 2n 1 x q (0) K (x)ndx n k 2 0 0 k=0 − Numerical evaluations for various loop orders give that S = n!. n In section we provide Maple codes for generating the diﬀerential equations for (cid:32) (cid:33) the all equal mass banana in(cid:88)tegrals − n 1 dk − (9.15) q (t) I2(t) = n! . k dtk n k=0 A derivation of the diﬀerential equations for the banana graph can be obtained using Maple and the routines compute Q and rec Q from the paper [BS07] together with the package gfun [SZ94]. For t < n2 the integral converges and(cid:88)we can perform the series expansion (9.16) I2(t) = tk Jk , n n ≥ k 0 where we have introduced the B(cid:90)essel m(cid:16)om(cid:17)ents ∞ 2n + x 2k+1 (9.17) Jk = K (x)n dx . n Γ(k + 1)2 2 0 0 Using the result of [BS07] on the recursion relations satisﬁed by these Bessel − moment c = 22k+1 nΓ(k + 1)2 Jk we deduce that these moments satisfy a n,2k+1 ≥ n recursion relation for k 0 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: PIERRE

text: VANHOVE

text: The

text: diﬀerential

text: operator

text: acting

text: integral

text: is

text: given

text: by

text: n

text: dk

text: L˜

text: q

text: (t)

text: n

text: k

text: dtk

text: with

text: the

text: coeﬃcient

text: q

text: (t)

text: polynomials

text: of

text: degree

text: k

text: for

text: k

text: n

text: The

text: k

text: top

text: degree

text: and

text: lowest-degree

text: polynomial

text: are

text: given

text: by

text: t

text: +η(n)

text: (t

text: (n

text: n

text: dq

text: (t)

text: q

text: (t)

text: n

text: n

text: dt

text: q

text: (t)

text: t

text: n

text: with

text: η(n)

text: if

text: n

text: mod

text: and

text: if

text: n

text: mod

text: The

text: inhomogeneous

text: term

text: S

text: is

text: a

text: constant

text: given

text: by

text: n

text: n

text: η(n)

text: x

text: S

text: x

text: q

text: K

text: (x)ndx

text: n

text: k

text: Numerical

text: evaluations

text: for

text: various

text: loop

text: orders

text: give

text: that

text: S

text: n!.

text: n

text: In

text: section

text: we

text: provide

text: Maple

text: codes

text: for

text: generating

text: the

text: diﬀerential

text: equations

text: for

text: the

text: all

text: equal

text: mass

text: banana

text: n

text: dk

text: q

text: (t)

text: n!

text: k

text: dtk

text: n

text: A

text: derivation

text: of

text: the

text: diﬀerential

text: equations

text: for

text: the

text: banana

text: graph

text: can

text: be

text: obtained

text: using

text: Maple

text: and

text: the

text: routines

text: compute

text: Q

text: and

text: rec

text: Q

text: from

text: the

text: paper

text: together

text: with

text: the

text: package

text: gfun

text: For

text: t

text: the

text: integral

text: converges

text: can

text: perform

text: the

text: series

text: expansion

text: tk

text: Jk

text: n

text: n

text: k

text: where

text: we

text: have

text: introduced

text: the

text: x

text: Jk

text: K

text: (x)n

text: dx

text: n

text: Γ(k

text: Using

text: the

text: result

text: of

text: on

text: the

text: recursion

text: relations

text: satisﬁed

text: by

text: these

text: Bessel

text: moment

text: c

text: nΓ(k

text: Jk

text: we

text: deduce

text: that

text: these

text: moments

text: satisfy

text: a

text: n

text: recursion

text: relation

text: for

text: k

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

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text: ##LTLine##


text: 34 PIERRE VANHOVE [B13] D. Broadhurst, “Multiple Zeta Values and Modular Forms in Quantum Field Theory”, in “Computer Algebra in Quantum Field Theory” p33-72, ed Carsten Schneider Johannes Blu¨mlein (Springer) 2013 [BB09] O. A. Bedoya and N. Berkovits, “Ggi Lectures on the Pure Spinor Formalism of the Superstring,” arXiv:0910.2254 [hep-th]. [Berk01] N. Berkovits, “Covariant Quantization of the Superparticle Using Pure Spinors,” JHEP 0109 (2001) 016 [hep-th/0105050]. [Bern92] Z. Bern, “String Based Perturbative Methods for Gauge Theories,” In *Boulder 1992, Proceedings, Recent directions in particle theory* 471-535. and Calif. Univ. Los Angeles - UCLA-93-TEP-05 (93,rec.May) 66 p. C [hep-ph/9304249]. [BK87] Z. Bern and D. A. Kosower, “A New Approach to One Loop Calculations in Gauge Theories,” Phys. Rev. D 38 (1988) 1888. [BK90] Z. Bern and D. A. Kosower, “Color Decomposition of One Loop Amplitudes in Gauge Theories,” Nucl. Phys. B 362 (1991) 389. [BK91] Z. Bern and D. A. Kosower, “The Computation of Loop Amplitudes in Gauge Theories,” Nucl. Phys. B 379 (1992) 451. [BCJ08] Z. Bern, J. J. M. Carrasco and H. Johansson, “New Relations for Gauge-Theory Ampli- tudes,” Phys. Rev. D 78 (2008) 085011 [arXiv:0805.3993 [hep-ph]]. [BCFIJ07] Z. Bern, J. J. Carrasco, D. Forde, H. Ita and H. Johansson, “Unexpected Cancellations in Gravity Theories,” Phys. Rev. D 77 (2008) 025010 [arXiv:0707.1035 [hep-th]]. [BDDK94] Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, “Fusing Gauge Theory Tree Amplitudes into Loop Amplitudes,” Nucl. Phys. B 435 (1995) 59 [hep-ph/9409265]. [BD91] Z. Bern and D. C. Dunbar, “A Mapping Between Feynman and String Motivated One Loop Rules in Gauge Theories,” Nucl. Phys. B 379 (1992) 562. [BDS93] Z. Bern, D. C. Dunbar and T. Shimada, “String Based Methods in Perturbative Gravity,” Phys. Lett. B 312 (1993) 277 [hep-th/9307001]. [BDK96] Z. Bern, L. J. Dixon and D. A. Kosower, “Progress in One Loop QCD Computations,” Ann. Rev. Nucl. Part. Sci. 46 (1996) 109 [hep-ph/9602280]. [BDK97] Z. Bern, L. J. Dixon and D. A. Kosower, “One Loop Amplitudes for E+ E- to Four Partons,” Nucl. Phys. B 513 (1998) 3 [hep-ph/9708239]. [Beilinson85] A. A. Beilinson, “Higher regulators and values of L-functions”, Journal of Soviet Mathematics 30 (1985), 2036-2070. [BL94] A. Beilinson and A. Levin, “The Elliptic Polylogarithm”, in Motives (ed. Jannsen, U., Kleiman, S,. Serre, J.-P.), Proc. Symp. Pure Math. vol 55, Amer. Math. Soc., (1994), Part 2, 123-190. [BBDFS10a] N. E. J. Bjerrum-Bohr, P. H. Damgaard, B. Feng and T. Sondergaard, “New Iden- tities among Gauge Theory Amplitudes,” Phys. Lett. B 691 (2010) 268 [1006.3214 [hep-th]]. [BBDFS10b] N. E. J. Bjerrum-Bohr, P. H. Damgaard, B. Feng and T. Sondergaard, “Proof of Gravity and Yang-Mills Amplitude Relations,” JHEP 1009 (2010) 067 [1007.3111 [hep-th]]. [BBDSV10] N. E. J. Bjerrum-Bohr, P. H. Damgaard, T. Sondergaard and P. Vanhove, “The Momentum Kernel of Gauge and Gravity Theories,” JHEP 1101 (2011) 001 [arXiv:1010.3933 [hep-th]]. [BBDV09] N. E. J. Bjerrum-Bohr, P. H. Damgaard and P. Vanhove, “Minimal Basis for Gauge Theory Amplitudes,” Phys. Rev. Lett. 103 (2009) 161602 [0907.1425 [hep-th]]. [BBDV13] N. E. JBjerrum-Bohr, J. F. Donoghue and P. Vanhove, “On-Shell Techniques and Universal Results in Quantum Gravity,” arXiv:1309.0804 [hep-th]. [BBFS10] N. E. J. Bjerrum-Bohr, P. H. Damgaard, B. Feng and T. Sondergaard, “Gravity and Yang-Mills Amplitude Relations,” 1005.4367 [hep-th]. [BBV08a] N. E. J. Bjerrum-Bohr and P. Vanhove, “Explicit Cancellation of Triangles in One-Loop Gravity Amplitudes,” JHEP 0804 (2008) 065 [arXiv:0802.0868 [hep-th]]. [BBV08b] N. E. J. Bjerrum-Bohr and P. Vanhove, “Absence of Triangles in Maximal Supergravity Amplitudes,” JHEP 0810 (2008) 006 [arXiv:0805.3682 [hep-th]]. [BG10] J. Bjornsson and M. B. Green, “5 Loops in 24/5 Dimensions,” JHEP 1008 (2010) 132 [arXiv:1004.2692 [hep-th]]. [Bj10] J. Bjornsson, “Multi-Loop Amplitudes in Maximally Supersymmetric Pure Spinor Field Theory,” JHEP 1101 (2011) 002 [arXiv:1009.5906 [hep-th]]. [BEK05] S. Bloch, H. Esnault and D. Kreimer, “On Motives Associated to Graph Polynomials,” Commun. Math. Phys. 267 (2006) 181 [math/0510011 [math.AG]].

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text: Superstring,”

text: [hep-th].

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text: “String

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text: Perturbative

text: Methods

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text: Gauge

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text: In

text: *Boulder

text: Proceedings,

text: Recent

text: directions

text: in

text: particle

text: theory*

text: and

text: Calif.

text: Univ.

text: Los

text: Angeles

text: p.

text: C

text: Z.

text: Bern

text: and

text: D.

text: A.

text: Kosower,

text: “A

text: New

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text: One

text: Loop

text: Calculations

text: in

text: Gauge

text: Theories,”

text: Phys.

text: Rev.

text: D

text: Z.

text: Bern

text: and

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text: A.

text: Kosower,

text: “Color

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text: Theories,”

text: Nucl.

text: Phys.

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text: Nucl.

text: Phys.

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text: Phys.

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text: Z.

text: Bern,

text: D.

text: C.

text: Dunbar

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text: Shimada,

text: “String

text: Based

text: Methods

text: in

text: Perturbative

text: Gravity,”

text: Phys.

text: Lett.

text: B

text: Z.

text: Bern,

text: L.

text: J.

text: Dixon

text: and

text: D.

text: A.

text: Kosower,

text: “Progress

text: in

text: One

text: Loop

text: QCD

text: Computations,”

text: Ann.

text: Rev.

text: Nucl.

text: Part.

text: Sci.

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text: Bern,

text: L.

text: J.

text: Dixon

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text: A.

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text: Partons,”

text: Nucl.

text: Phys.

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text: Beilinson,

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text: Mathematics

text: A.

text: Beilinson

text: and

text: A.

text: Levin,

text: “The

text: Elliptic

text: Polylogarithm”,

text: in

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text: (ed.

text: Jannsen,

text: U.,

text: Kleiman,

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text: Serre,

text: J.-P.),

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text: P.

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text: Damgaard,

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text: Iden-

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text: B

text: [hep-th]].

text: N.

text: E.

text: J.

text: Bjerrum-Bohr,

text: P.

text: H.

text: Damgaard,

text: B.

text: Feng

text: and

text: T.

text: Sondergaard,

text: “Proof

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text: and

text: Yang-Mills

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text: Relations,”

text: JHEP

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text: E.

text: J.

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text: P.

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text: Vanhove,

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text: 7 10 E vac Δ Δ = 0 Al M|BN BN Cu E conduction band Pt Ag W W c Au M M|BN Φ E E d) (eV) 1 DD Mc(BdN) (*d) EF Φ*p Φnp v g physisorption (N 1 2 3 valence band B M 0.1 D Metal h-BN E Δ vac Δ BN M|BN conduction band 0.01 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4 d (Å) WM WM|BN Φn Eg chemisorption | E FIG. 5. (Color online) Potential steps ∆M|BN(d) at metal h- F Φ* Φp BN interfaces as in Fig. 4 plotted on a logarithmic scale for p 2 3 valence band the metals on which h-BN is physisorbed. The solid line gives 1 the ﬁt function of Eq. 5; the dashed line gives the function ∆ (d) used for graphene on these metals in Ref. 30. c FIG. 6. (Color online) Top: schematic illustration of the en- | ergy levels at a metal h-BN interface for three layers of h-BN ˚A−1. 2 when the ﬁrst layer is physisorbed. Evac and EF are the vac- uum level and the Fermi level, respectively. W , W | , and This ﬁt function is shown in Fig. 5 along with the data M M BN ∆ | are the work function of the clean metal surface, that points, which shows that the ﬁt is remarkably good. An M BN | of the metal h-BN structure, and the interface potential step, exception seems to be Al for d < 3.0 ˚A, but this is to respectively. E , E , and E are the top of the valence band v c g be expected as there regime chemisorption sets in. The of h-BN, the bottom of the conduction band, and the band deviation of the function from a simple exponential is gap, respectively. Φ and Φ are the Schottky barrier heights p n most prominent for d > 4.0 ˚A. The reason for this is that for holes and electrons, respectively. In the physisorbed case ∗ | − | ≈ the dipole resulting from exchange repulsion decreases Φp = Ev EF Φp. Bottom: if the ﬁrst layer of h-BN rapidly to zero at large distances. What remains is a is chemisorbed on the metal (resulting in a distorted h-BN small dipole that results from van der Waals interactions layer) a small additional potential step ∆BN of order 0.1 eV between the metal and h-BN56 that does not go expo- develop∗s between the ﬁrst and the second h-BN layer. In gen- eral, Φ and Φ can diﬀer by much more, by electron volts nentially to zero as a function of d, but rather as a power p p (see Tables I and II). law dα. In the present case, the numbers become too small to pinpoint the exact value of α. A function of the form given in Eq. (5) was used step ∆ (d) remained that could be ﬁtted with the same by Khomyakov et al. to describe the potential step c functional form as Eq. (5). The function ∆ (d) extracted encountered in the physisorption of graphene on metal c in Refs. 30 and 32 is also shown in Fig. 5. For distances substrates30,32. The bonding of h-BN to the metals stud- in the typical physisorption regime 2.8 < d < 4.0 ˚A, this ied in the present paper is very similar. A comparison | function is very close to that obtained for metal h-BN of the equilibrium binding separations and energies in interfaces, indicating that in their adsorption on metal Table I to those for graphene on metals in Ref. 30 gives surfaces, h-BN and graphene behave very similarly. the same classiﬁcation into chemisorption and physisorp- C. Energy level alignment tion for both graphene and h-BN. The h-BN layer forms a slightly stronger bond with the metals than graphene A schematic illustration of the Schottky barrier formed does. In the case of graphene the analysis of the poten- | at a metal h-BN interface is shown in Fig. 6 (top). We tial step is complicated by the electron transfer between the metal and graphene30,32,58. In the case of h-BN, this deﬁne the p-type Schottky barrier height as cannot (easily) happen because it is a large band gap in- sulator. After modeling the eﬀect of electron transfer in − − | Φ = E W + ∆ | (d) ∆ . (6) the metal graphene case, a contribution to the potential p v M M BN BN Here W is the work function of the clean metal sur- M face, ∆ | (d) is the interface potential step discussed 2 T±he 9±5% co±nﬁdence intervals of the parameters a0,1,2 are M BN 291, 184, 29−, respectively. The −decay parameter γ varies in the previous section, ∆BN is a small potential step be- between 1.82 ˚A 1 for Ag and 1.92 ˚A 1 for Pt. The value 1.85 tween the ﬁrst and second h-BN layer (discussed below) − ˚A 1 gives a reasonable ﬁt for all metals. and E is the position of the top of the valence band of v ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: E

text: vac

text: Δ

text: Δ

text: Al

text: M|BN

text: BN

text: Cu

text: E

text: conduction

text: band

text: Pt

text: Ag

text: W

text: W

text: c

text: Au

text: M

text: M|BN

text: Φ

text: E

text: E

text: d)

text: (eV)

text: DD

text: Mc(BdN)

text: (*d)

text: EF

text: Φ*p

text: Φnp

text: v

text: g

text: physisorption

text: (N

text: valence

text: band

text: B

text: M

text: D

text: Metal

text: h-BN

text: E

text: Δ

text: vac

text: Δ

text: BN

text: M|BN

text: conduction

text: band

text: d

text: (Å)

text: WM

text: WM|BN

text: Φn

text: Eg

text: chemisorption

text: E

text: FIG.

text: (Color

text: online)

text: Potential

text: steps

text: ∆M|BN(d)

text: at

text: metal

text: h-

text: F

text: Φ*

text: Φp

text: BN

text: interfaces

text: as

text: in

text: Fig.

text: plotted

text: on

text: a

text: logarithmic

text: scale

text: for

text: p

text: valence

text: band

text: the

text: metals

text: on

text: which

text: h-BN

text: is

text: physisorbed.

text: The

text: solid

text: line

text: gives

text: the

text: ﬁt

text: function

text: of

text: Eq.

text: the

text: dashed

text: line

text: gives

text: the

text: function

text: (d)

text: used

text: for

text: graphene

text: on

text: these

text: metals

text: in

text: Ref.

text: c

text: FIG.

text: (Color

text: online)

text: Top:

text: schematic

text: illustration

text: of

text: the

text: en-

text: ergy

text: levels

text: at

text: a

text: metal

text: h-BN

text: interface

text: for

text: three

text: layers

text: of

text: h-BN

text: when

text: the

text: ﬁrst

text: layer

text: is

text: physisorbed.

text: Evac

text: and

text: EF

text: are

text: the

text: vac-

text: uum

text: level

text: and

text: the

text: Fermi

text: level,

text: respectively.

text: W

text: W

text: and

text: This

text: ﬁt

text: function

text: is

text: shown

text: in

text: Fig.

text: along

text: with

text: the

text: data

text: M

text: M

text: BN

text: are

text: the

text: work

text: function

text: of

text: the

text: clean

text: metal

text: surface,

text: that

text: points,

text: which

text: shows

text: that

text: the

text: ﬁt

text: is

text: remarkably

text: good.

text: An

text: M

text: BN

text: of

text: the

text: metal

text: h-BN

text: structure,

text: and

text: the

text: interface

text: potential

text: step,

text: exception

text: seems

text: to

text: be

text: Al

text: for

text: d

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text: but

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text: respectively.

text: E

text: E

text: and

text: E

text: are

text: the

text: top

text: of

text: the

text: valence

text: band

text: v

text: c

text: g

text: be

text: expected

text: as

text: there

text: regime

text: chemisorption

text: sets

text: in.

text: The

text: of

text: h-BN,

text: the

text: bottom

text: of

text: the

text: conduction

text: band,

text: and

text: the

text: band

text: deviation

text: of

text: the

text: function

text: from

text: a

text: simple

text: exponential

text: is

text: gap,

text: respectively.

text: Φ

text: and

text: Φ

text: are

text: the

text: Schottky

text: barrier

text: heights

text: p

text: n

text: most

text: prominent

text: for

text: d

text: ˚A.

text: The

text: reason

text: for

text: this

text: is

text: that

text: for

text: holes

text: and

text: electrons,

text: respectively.

text: In

text: the

text: physisorbed

text: case

text: the

text: dipole

text: resulting

text: from

text: exchange

text: repulsion

text: decreases

text: Φp

text: Ev

text: EF

text: Φp.

text: Bottom:

text: if

text: the

text: ﬁrst

text: layer

text: of

text: h-BN

text: rapidly

text: to

text: zero

text: at

text: large

text: distances.

text: What

text: remains

text: is

text: a

text: is

text: chemisorbed

text: on

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text: h-BN

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text: Waals

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text: additional

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text: develop∗s

text: between

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text: and

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text: In

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text: Φ

text: can

text: diﬀer

text: by

text: much

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text: by

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text: volts

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text: law

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text: al.

text: to

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text: form

text: as

text: Eq.

text: The

text: function

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text: extracted

text: encountered

text: in

text: the

text: physisorption

text: of

text: graphene

text: on

text: metal

text: c

text: in

text: Refs.

text: and

text: is

text: also

text: shown

text: in

text: Fig.

text: For

text: distances

text: The

text: bonding

text: of

text: h-BN

text: to

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text: in

text: the

text: typical

text: physisorption

text: regime

text: d

text: ˚A,

text: this

text: ied

text: in

text: the

text: present

text: paper

text: is

text: very

text: similar.

text: A

text: comparison

text: function

text: is

text: very

text: close

text: to

text: that

text: obtained

text: for

text: metal

text: h-BN

text: of

text: the

text: equilibrium

text: binding

text: separations

text: and

text: energies

text: in

text: interfaces,

text: indicating

text: that

text: in

text: their

text: adsorption

text: on

text: metal

text: Table

text: I

text: to

text: those

text: for

text: graphene

text: on

text: metals

text: in

text: Ref.

text: gives

text: surfaces,

text: h-BN

text: and

text: graphene

text: behave

text: very

text: similarly.

text: the

text: same

text: classiﬁcation

text: into

text: chemisorption

text: and

text: physisorp-

text: C.

text: Energy

text: level

text: alignment

text: tion

text: for

text: both

text: graphene

text: and

text: h-BN.

text: The

text: h-BN

text: layer

text: forms

text: a

text: slightly

text: stronger

text: bond

text: with

text: the

text: metals

text: than

text: graphene

text: A

text: schematic

text: illustration

text: of

text: the

text: Schottky

text: barrier

text: formed

text: does.

text: In

text: the

text: case

text: of

text: graphene

text: the

text: analysis

text: of

text: the

text: poten-

text: at

text: a

text: metal

text: h-BN

text: interface

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text: shown

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text: Fig.

text: (top).

text: We

text: tial

text: step

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text: the

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text: the

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text: the

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text: the

text: p-type

text: Schottky

text: barrier

text: height

text: as

text: cannot

text: (easily)

text: happen

text: because

text: it

text: is

text: a

text: large

text: band

text: gap

text: in-

text: sulator.

text: After

text: modeling

text: the

text: eﬀect

text: of

text: electron

text: transfer

text: in

text: Φ

text: E

text: W

text: (d)

text: the

text: metal

text: graphene

text: case,

text: a

text: contribution

text: to

text: the

text: potential

text: p

text: v

text: M

text: M

text: BN

text: BN

text: Here

text: W

text: is

text: the

text: work

text: function

text: of

text: the

text: clean

text: metal

text: sur-

text: M

text: face,

text: (d)

text: is

text: the

text: interface

text: potential

text: step

text: discussed

text: T±he

text: co±nﬁdence

text: intervals

text: of

text: the

text: parameters

text: are

text: M

text: BN

text: respectively.

text: The

text: −decay

text: parameter

text: γ

text: varies

text: in

text: the

text: previous

text: section,

text: ∆BN

text: is

text: a

text: small

text: potential

text: step

text: be-

text: between

text: ˚A

text: for

text: Ag

text: and

text: ˚A

text: for

text: Pt.

text: The

text: value

text: tween

text: the

text: ﬁrst

text: and

text: second

text: h-BN

text: layer

text: (discussed

text: below)

text: ˚A

text: gives

text: a

text: reasonable

text: ﬁt

text: for

text: all

text: metals.

text: and

text: E

text: is

text: the

text: position

text: of

text: the

text: top

text: of

text: the

text: valence

text: band

text: of

text: v

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text: 5 where h is the holonomy going from the source Since the Pachner moves that we are considering are e(0),π(0) of the edge e to the reference point π(0), and the ﬂuxes only reﬁnement moves, they cannot remove the refer- have to be inverted if their orientation does not agree ence frame π(0) of an integrated ﬂux observable. These with the one outgoing from π(0) that deﬁnes the general ﬂux observables are cylindrically consistent. Indeed, one orientation of the path. In the Abelian case the deﬁnition can understand heuristically that the Pachner moves add (11) reduces simply to the sum of the ﬂuxes, without the only pieces of ﬂat geometry, for which the Gauss con- adjoint action. straints are satisﬁed. Thus the vector associated to the ∗ This integrated ﬂux observable corresponds geometri- path π does not change under a Pachner move. ∗ cally to the displacement vector between the nodes π (0) In order to show this explicitly, we have once again to ∗ and π (1) of the triangulation. Under the action of a specify the replacement rules and argue that the observ- gauge transformation, it transforms by the adjoint ac- ables commute with the Pachner moves. tion of the group in the frame in which the individual For the 1–3 Pachner move, the invariance of the inte- ∗ ﬂuxes are transported, i.e. the source vertex π(0). grated ﬂuxes is immediate since the edges e of the trian- ∗ The composition of these ﬂux observables is deﬁned gulation deﬁning the path π cannot be aﬀected by the in the standard way as follows. Suppose that X ∗ is move. However, in order to show this explicitly, one has ∗ π1 a ﬂux observable deﬁned along a path π and X ∗ is to keep track of the new holonomies that are introduced ∗ 1 π2 deﬁned along π such that π (1) = π (0). Transporting by the 1–3 move. Our replacement rule is such that the 2 1 2 the ﬂuxes in the same frame in order to add them, we new path in the ribbon graph after the 1–3 move has to have the composition rule start from the same source π(0), end at the same ter- − minal point π(1), go along the same ﬂux variables (or X ∗ ◦ X ∗ = X ∗ + h 1X ∗h , (12) π2 π1 π1 π1 π2 π1 their inverse), and transport these ﬂuxes using the new where h is the holonomy from π (0) to π (0). holonomies. π1 1 2 ∗ It is convenient to introduce the ribbon picture [25], Concerning the 2–2 move, if the path π does not go which simultaneously represents the graph Γ and its along the internal edge of the triangulation that is being ∗ dual Γ as depicted in ﬁgure 4. A ribbon can be as- ﬂipped by the move, the invariance of the integrated ﬂux signed holonomy and ﬂux data with the composition rule follows from the same reasoning as in the 1–3 case dis- I (g , 0) ◦ ( , X ) = (g , X ). Ribbons can then be com- cussed above. If the path (or a portion of it) corresponds 1 1 1 1∗ posed following a path (cid:0)π . This deﬁnes the integrat(cid:1)ed to the edge that is being ﬂipped, one has to use the re- ﬂuxes through placement rule depicted on ﬁgure 5. This replacement − − − rule is unique as long as one follows the arrows of the (g , X 1) ◦ (g , X ) = g g , X ∗ = X + g 1X 1g , 2 2 1 1 2 1 π 1 1 2 1 ribbon representation. (13) Finally, let us conclude with an important remark con- which is a semi–direct product structure. cerning the reference frame to which the individual ﬂuxes are transported. It can happen that this reference frame π X1 g1 g2 X2−1 coincides with a vertex of Γ that is moved by the 1–3 g1 g2 or 2–2 move. This is illustrated for example on ﬁgure 2, where the vertex v is replaced by the three vertices A π∗ A Γ∗ v , v , and v . Therefore, if the reference frame of the B C D path deﬁning X ∗ coincides with the vertex v , there is π A g3 Γ g3 an ambiguity in the choice of placement of the reference frame after the Pachner move. While this is not problem- FIG. 4: Replacement of a portion of triangulation (dashed atic for the holonomy observables because of their gauge triangle) and its dual by a ribbon graph that contains both invariance, it aﬀects the integrated ﬂuxes since they are the holonomies and the ﬂuxes. Our convention is such that gauge–covariant. Since the construction of the measure the ribbons have a clockwise orientation, and that the pairs in the next section will involve a choice of a maximal (holonomy, ﬂux) have positive orientation. A path deﬁning tree with root r, we can choose to deﬁne the integrated an integrated ﬂux X ∗ is represented with thick lines. π ﬂuxes transported to the root. We discussed how the root behaves under Pachner moves in section 3. Gauge–invariant observables can be easily obtained by Finally, one can use the fact that the Poisson bracket taking the Lie algebra trace over two integrated ﬂuxes between the simplicial ﬂuxes (7) and the holonomies re- transported to a common reference frame, or by group ∗ produce the Poisson structure on T G to derive the Pois- averaging combinations of exponentiated ﬂuxes as in the son(cid:8)action of th(cid:9)e in(cid:16)te(cid:104)grated ﬂux (11) on a(cid:105) fu(cid:17)nction ψ(ge). next section. Details on the Poisson algebra of these If e (cid:54)⊂ π, this action is vanishing. If e ⊂ π, it is given by ﬂuxes and holonomies will be discussed in [18]. − i X , ψ(g ) = h 1 Rh ψ (g ). (14) π e π(0),e(0) π(0),e(0) e ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: h

text: is

text: the

text: holonomy

text: going

text: from

text: the

text: source

text: Since

text: the

text: Pachner

text: moves

text: that

text: we

text: are

text: considering

text: are

text: of

text: the

text: edge

text: e

text: to

text: the

text: reference

text: point

text: and

text: the

text: ﬂuxes

text: only

text: reﬁnement

text: moves,

text: they

text: cannot

text: remove

text: the

text: refer-

text: have

text: to

text: be

text: inverted

text: if

text: their

text: orientation

text: does

text: not

text: agree

text: ence

text: frame

text: of

text: an

text: integrated

text: ﬂux

text: observable.

text: These

text: with

text: the

text: one

text: outgoing

text: from

text: that

text: deﬁnes

text: the

text: general

text: ﬂux

text: observables

text: are

text: cylindrically

text: consistent.

text: Indeed,

text: one

text: orientation

text: of

text: the

text: path.

text: In

text: the

text: Abelian

text: case

text: the

text: deﬁnition

text: can

text: understand

text: heuristically

text: that

text: the

text: Pachner

text: moves

text: add

text: reduces

text: simply

text: to

text: the

text: sum

text: of

text: the

text: ﬂuxes,

text: without

text: the

text: only

text: pieces

text: of

text: ﬂat

text: geometry,

text: for

text: which

text: the

text: Gauss

text: con-

text: adjoint

text: action.

text: straints

text: are

text: satisﬁed.

text: Thus

text: the

text: vector

text: associated

text: to

text: the

text: This

text: integrated

text: ﬂux

text: observable

text: corresponds

text: geometri-

text: path

text: π

text: does

text: not

text: change

text: under

text: a

text: Pachner

text: move.

text: cally

text: to

text: the

text: displacement

text: vector

text: between

text: the

text: nodes

text: π

text: In

text: order

text: to

text: show

text: this

text: explicitly,

text: we

text: have

text: once

text: again

text: to

text: and

text: π

text: of

text: the

text: triangulation.

text: Under

text: the

text: action

text: of

text: a

text: specify

text: the

text: replacement

text: rules

text: and

text: argue

text: that

text: the

text: observ-

text: gauge

text: transformation,

text: it

text: transforms

text: by

text: the

text: adjoint

text: ac-

text: ables

text: commute

text: with

text: the

text: Pachner

text: moves.

text: tion

text: of

text: the

text: group

text: in

text: the

text: frame

text: in

text: which

text: the

text: individual

text: For

text: the

text: Pachner

text: move,

text: the

text: invariance

text: of

text: the

text: inte-

text: ﬂuxes

text: are

text: transported,

text: i.e.

text: the

text: source

text: vertex

text: grated

text: ﬂuxes

text: is

text: immediate

text: since

text: the

text: edges

text: e

text: of

text: the

text: trian-

text: The

text: composition

text: of

text: these

text: ﬂux

text: observables

text: is

text: deﬁned

text: gulation

text: deﬁning

text: the

text: path

text: π

text: cannot

text: be

text: aﬀected

text: by

text: the

text: in

text: the

text: standard

text: way

text: as

text: follows.

text: Suppose

text: that

text: X

text: is

text: move.

text: However,

text: in

text: order

text: to

text: show

text: this

text: explicitly,

text: one

text: has

text: π1

text: a

text: ﬂux

text: observable

text: deﬁned

text: along

text: a

text: path

text: π

text: and

text: X

text: is

text: to

text: keep

text: track

text: of

text: the

text: new

text: holonomies

text: that

text: are

text: introduced

text: π2

text: deﬁned

text: along

text: π

text: such

text: that

text: π

text: π

text: Transporting

text: by

text: the

text: move.

text: Our

text: replacement

text: rule

text: is

text: such

text: that

text: the

text: the

text: ﬂuxes

text: in

text: the

text: same

text: frame

text: in

text: order

text: to

text: add

text: them,

text: we

text: new

text: path

text: in

text: the

text: ribbon

text: graph

text: after

text: the

text: move

text: has

text: to

text: have

text: the

text: composition

text: rule

text: start

text: from

text: the

text: same

text: source

text: end

text: at

text: the

text: same

text: ter-

text: minal

text: point

text: go

text: along

text: the

text: same

text: ﬂux

text: variables

text: (or

text: X

text: X

text: X

text: h

text: ∗h

text: π2

text: π1

text: π1

text: π1

text: π2

text: π1

text: their

text: inverse),

text: and

text: transport

text: these

text: ﬂuxes

text: using

text: the

text: new

text: where

text: h

text: is

text: the

text: holonomy

text: from

text: π

text: to

text: π

text: holonomies.

text: π1

text: It

text: is

text: convenient

text: to

text: introduce

text: the

text: ribbon

text: picture

text: Concerning

text: the

text: move,

text: if

text: the

text: path

text: π

text: does

text: not

text: go

text: which

text: simultaneously

text: represents

text: the

text: graph

text: Γ

text: and

text: its

text: along

text: the

text: internal

text: edge

text: of

text: the

text: triangulation

text: that

text: is

text: being

text: dual

text: Γ

text: as

text: depicted

text: in

text: ﬁgure

text: A

text: ribbon

text: can

text: be

text: as-

text: ﬂipped

text: by

text: the

text: move,

text: the

text: invariance

text: of

text: the

text: integrated

text: ﬂux

text: signed

text: holonomy

text: and

text: ﬂux

text: data

text: with

text: the

text: composition

text: rule

text: follows

text: from

text: the

text: same

text: reasoning

text: as

text: in

text: the

text: case

text: dis-

text: I

text: (g

text: X

text: (g

text: X

text: Ribbons

text: can

text: then

text: be

text: com-

text: cussed

text: above.

text: If

text: the

text: path

text: (or

text: a

text: portion

text: of

text: it)

text: corresponds

text: 1∗

text: posed

text: following

text: a

text: path

text: This

text: deﬁnes

text: the

text: to

text: the

text: edge

text: that

text: is

text: being

text: ﬂipped,

text: one

text: has

text: to

text: use

text: the

text: re-

text: ﬂuxes

text: through

text: placement

text: rule

text: depicted

text: on

text: ﬁgure

text: This

text: replacement

text: rule

text: is

text: unique

text: as

text: long

text: as

text: one

text: follows

text: the

text: arrows

text: of

text: the

text: (g

text: X

text: (g

text: X

text: g

text: g

text: X

text: X

text: g

text: π

text: ribbon

text: representation.

text: Finally,

text: let

text: us

text: conclude

text: with

text: an

text: important

text: remark

text: con-

text: which

text: is

text: a

text: semi–direct

text: product

text: structure.

text: cerning

text: the

text: reference

text: frame

text: to

text: which

text: the

text: individual

text: ﬂuxes

text: are

text: transported.

text: It

text: can

text: happen

text: that

text: this

text: reference

text: frame

text: π

text: coincides

text: with

text: a

text: vertex

text: of

text: Γ

text: that

text: is

text: moved

text: by

text: the

text: or

text: move.

text: This

text: is

text: illustrated

text: for

text: example

text: on

text: ﬁgure

text: where

text: the

text: vertex

text: v

text: is

text: replaced

text: by

text: the

text: three

text: vertices

text: A

text: π∗

text: A

text: Γ∗

text: v

text: v

text: and

text: v

text: Therefore,

text: if

text: the

text: reference

text: frame

text: of

text: the

text: B

text: C

text: D

text: path

text: deﬁning

text: X

text: coincides

text: with

text: the

text: vertex

text: v

text: there

text: is

text: π

text: A

text: Γ

text: an

text: ambiguity

text: in

text: the

text: choice

text: of

text: placement

text: of

text: the

text: reference

text: frame

text: after

text: the

text: Pachner

text: move.

text: While

text: this

text: is

text: not

text: problem-

text: FIG.

text: Replacement

text: of

text: a

text: portion

text: of

text: triangulation

text: (dashed

text: atic

text: for

text: the

text: holonomy

text: observables

text: because

text: of

text: their

text: gauge

text: triangle)

text: and

text: its

text: dual

text: by

text: a

text: ribbon

text: graph

text: that

text: contains

text: both

text: invariance,

text: it

text: aﬀects

text: the

text: integrated

text: ﬂuxes

text: since

text: they

text: are

text: the

text: holonomies

text: and

text: the

text: ﬂuxes.

text: Our

text: convention

text: is

text: such

text: that

text: gauge–covariant.

text: Since

text: the

text: construction

text: of

text: the

text: measure

text: the

text: ribbons

text: have

text: a

text: clockwise

text: orientation,

text: and

text: that

text: the

text: pairs

text: in

text: the

text: next

text: section

text: will

text: involve

text: a

text: choice

text: of

text: a

text: maximal

text: (holonomy,

text: ﬂux)

text: have

text: positive

text: orientation.

text: A

text: path

text: deﬁning

text: tree

text: with

text: root

text: r,

text: we

text: can

text: choose

text: to

text: deﬁne

text: the

text: integrated

text: an

text: integrated

text: ﬂux

text: X

text: is

text: represented

text: with

text: thick

text: lines.

text: π

text: ﬂuxes

text: transported

text: to

text: the

text: root.

text: We

text: discussed

text: how

text: the

text: root

text: behaves

text: under

text: Pachner

text: moves

text: in

text: section

text: Gauge–invariant

text: observables

text: can

text: be

text: easily

text: obtained

text: by

text: Finally,

text: one

text: can

text: use

text: the

text: fact

text: that

text: the

text: Poisson

text: bracket

text: taking

text: the

text: Lie

text: algebra

text: trace

text: over

text: two

text: integrated

text: ﬂuxes

text: between

text: the

text: simplicial

text: ﬂuxes

text: and

text: the

text: holonomies

text: re-

text: transported

text: to

text: a

text: common

text: reference

text: frame,

text: or

text: by

text: group

text: produce

text: the

text: Poisson

text: structure

text: on

text: T

text: G

text: to

text: derive

text: the

text: Pois-

text: averaging

text: combinations

text: of

text: exponentiated

text: ﬂuxes

text: as

text: in

text: the

text: of

text: ﬂux

text: on

text: ψ(ge).

text: next

text: section.

text: Details

text: on

text: the

text: Poisson

text: algebra

text: of

text: these

text: If

text: e

text: π,

text: this

text: action

text: is

text: vanishing.

text: If

text: e

text: π,

text: it

text: is

text: given

text: by

text: ﬂuxes

text: and

text: holonomies

text: will

text: be

text: discussed

text: in

text: i

text: X

text: ψ(g

text: h

text: Rh

text: ψ

text: (g

text: π

text: e

text: e

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:90) (cid:88) n (cid:107) − (cid:107) ≤ | | h2 (DxkI Ak)ed0(x)/hϕh,Ω 2L2(Ω;C2) hC √ ϕh,Ω(x) 2dx, (4.12) k=1 B0( hC) (cid:90) (cid:90) then the Poincar´e estimate gives | | ≤ − | | e2d0(x)/h ϕh,Ω(x) 2dx h 1C √ ϕh,Ω(x) 2dx. (4.13) Ω B0( hC) Let χ a smooth cut-oﬀ function satisfying ≤ ∈ χ(x) = 1 if d (x) (S + η )/2 and χ(x) = 0 if x / Ω. 0 (cid:88) 0 0 Then the function − − − ϕh,θ(x) = eiθ(x ω)χ(x ω)ϕh,Ω(x ω) ∈ ω Γ (cid:27) is Γ-periodic and satisﬁes − − (P h(x, h(D θ)) Eh)ϕh,θ(x) = rh,θ and (cid:107) (cid:107) ≤ −Ω (cid:107) (cid:107) (4.14) rh,θ L2(Wn;C2) Ce (S0+η0)/(2h) ϕh,θ L2(Wn;C2) and (cid:90) | (cid:88) < rh,θ ϕh,θ > W C = L2( n; 2) − − − eiθ(ρ ω) ([P h(x, hD); χ]ϕh,Ω)(x ω).(χϕh,Ω)(x ρ)dx ∈ Wn ω,ρ Γ0 { ± ± } (cid:88) with Γ = 0, β , . . . , β and 0 1 n − n − − [P h(x, hD); χ] = 2h2i ∂ χ(D I A ) h2∆χI. x x k k k (cid:12) k=1 (cid:12) (cid:12) (cid:12) So (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 1 | ≤ − (cid:107) (cid:107) < rh,θ ϕh,θ > W C Ce S0/h. (4.15) ϕh,θ 2 W C L2( n; 2) L2( n; 2) The proof comes easily from (4.12) and (4.13) as in [Ou] or in [He]. Using the same argument of [He-Sj-1] as in the proof of (4.10), we get that | − − | ≤ − Eh,θ Eh τh,θ Ce (S0+η0)/h) , (4.16) Ω 1 | with τh,θ = (cid:107) (cid:107) < rh,θ ϕh,θ > W C . ϕh,θ 2 W C L2( n; 2) L2( n; 2) (cid:3) Theorem 4.2 follows from (4.15) and (4.16) 11 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: n

text: (DxkI

text: hC

text: ϕh,Ω(x)

text: hC)

text: then

text: the

text: Poincar´e

text: estimate

text: gives

text: ϕh,Ω(x)

text: h

text: ϕh,Ω(x)

text: Ω

text: hC)

text: Let

text: χ

text: a

text: smooth

text: cut-oﬀ

text: function

text: satisfying

text: χ(x)

text: if

text: d

text: (x)

text: (S

text: η

text: and

text: χ(x)

text: if

text: x

text: Ω.

text: Then

text: the

text: function

text: ϕh,θ(x)

text: eiθ(x

text: ω)χ(x

text: ω)ϕh,Ω(x

text: ω)

text: ω

text: Γ

text: is

text: Γ-periodic

text: and

text: satisﬁes

text: (P

text: h(x,

text: h(D

text: θ))

text: Eh)ϕh,θ(x)

text: rh,θ

text: and

text: −Ω

text: rh,θ

text: Ce

text: ϕh,θ

text: and

text: rh,θ

text: ϕh,θ

text: W

text: C

text: n;

text: eiθ(ρ

text: ω)

text: ([P

text: h(x,

text: hD);

text: χ]ϕh,Ω)(x

text: ω).(χϕh,Ω)(x

text: ρ)dx

text: Wn

text: ω,ρ

text: Γ0

text: with

text: Γ

text: β

text: β

text: and

text: n

text: n

text: [P

text: h(x,

text: hD);

text: χ]

text: χ(D

text: I

text: A

text: x

text: x

text: k

text: k

text: k

text: So

text: rh,θ

text: ϕh,θ

text: W

text: C

text: Ce

text: ϕh,θ

text: W

text: C

text: n;

text: n;

text: The

text: proof

text: comes

text: easily

text: from

text: and

text: as

text: in

text: [Ou]

text: or

text: in

text: [He].

text: Using

text: the

text: same

text: argument

text: of

text: as

text: in

text: the

text: proof

text: of

text: we

text: get

text: that

text: Eh,θ

text: Eh

text: τh,θ

text: Ce

text: Ω

text: with

text: τh,θ

text: rh,θ

text: ϕh,θ

text: W

text: C

text: ϕh,θ

text: W

text: C

text: n;

text: n;

text: Theorem

text: follows

text: from

text: and

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##