text: 10 0.6 #" #" (a) ! ! 0.4 " $# X "# m !" 0.2 0 0 2 4 6 8 Jβ aa 1 FIG. 7. (Color online.) Schematic diagram of the magneti- (b) zation of XY ferromagnets without and with disorder, in the 0.8 presence of a constant magnetic ﬁeld (cid:126)h. The ﬁgure on the left 0.94 0.6 indicates the behavior of m(cid:126) in the presence of (cid:126)h, but when mX 0.92 (cid:15) = 0, while the one on the right is when there is a positive (cid:15). 0.4 0.9 0.2 0.88 6 7 8 the mean ﬁeld equation is given by Eq. (45)), keeping h 0 0 2 4 6 8 strictly positive in both cases. Let us denote the two Jβ Hamiltonians by H and H respectively. We do the h h,(cid:15) comparison by numerical simulations as well as perturba- tively at low temperatures (Sec. below). A perturbation FIG. 8. (Color online.) (a) The X-component of magnetiza- approach, similar to the one in Sec. , can be done at high tion, m , as a function of Jβ. Red crosses represent the case x temperatures also. We refrain from doing it, as the high when the XY model has the applied constant ﬁeld (cid:126)h with temperature behavior in this case is less interesting, in h/J = 0.1, and (a) x = π/3 and (b) x = 0.1. The blue solid view of absence of a phase transition. and the green dashed lines are for the same system but with the additional random ﬁeld of strength (cid:15) = 0.1J and (cid:15) = 0.2J, The length m of the magnetization vector is shrunk in respectively. x is measured in radians. All other quantities the system with the disordered ﬁeld, compared to the or- are dimensionless. dered case. This is seen from numerical simulations (see Figs. 6 (a)), as well as by perturbation techniques at low temperatures. In addition, numerical simulations (shown “order from disorder”. However, such eﬀect is absent in in Fig. 6 (b)) show that the cosine of the phase of the the system when x is small (see Figs. 8 (b) and 9 (top)). magnetization, i.e., cos(φ1), increases at low temperature We also ﬁnd that for a given x, the shift in mx due to in presence of the random ﬁeld. Therefore, the magne- disorder decreases as the ratio J/h increases. tization vector moves towards the X-direction (i.e. the A. Magnetization at low temperature: direction transverse to the applied random ﬁeld). How- Perturbative approach ever, the shift turns out to be zero when x equals to 0, ± π/2. This is also corroborated by perturbative analysis To obtain the behavior of magnetization at low tem- at low temperatures. The schematic diagram in Fig. 7 perature, we will use the implicit function theorem, which shows the low temperature behavior of the length and we now state. Let an equation f(x1, x2) = 0 of(cid:12) two vari- phase of the magnetization with and without disorder, ables x and x be such that f(x , x ) = 0 at (cid:12)(x , x ) = 1 2 1 2 1 2 in the presence of a constant ﬁeld. (x0, x0). x is in general an unknown function of x . But 1 2 2 1 The Y -component, my = m sin φ1, of the magnetiza- we may still understand the character of dx2 , by tion has the same relative behavior as the length m, in dx1 x1=x01 using the fact that (under certain regularity conditions systems described by H and H , i.e., for small (cid:15) > 0 h h,(cid:15) on f near (cid:12)(x0, x0)) (cid:12) (cid:12) it is lower than for (cid:15) = 0. (cid:12) 1 2 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) However, the X-component, m = m cos φ , of the (cid:12) (cid:12) (cid:12) x 1 magnetization, m(cid:126) , behaves in a very interesting way. Its ∂f ∂f dx + 2 = 0. (46) value in the system described by H can be both higher h,(cid:15) ∂x ∂x dx and lower than its value in the system described by Hh 1 (x01,x02) 2 (x01,x02) 1 (x01,x02) depending on the direction of the constant magnetic ﬁeld. The numerical data for the X-component of the magnetic The usual statement of the implicit function theorem ﬁeld, m , are shown for h/J = 0.1, and x = π/3 (Fig. 8 is that when ∂f is nonzero at (x0, x0), we can solve the x ∂x2 1 2 (a)) and x = 0.1 (Fig. 8 (b)). The numerical results for equation f(x , x ) = 0 for x uniquely near this point 1 2 2 x = π/3, where m is signiﬁcantly enhanced in presence and the derivative of the resulting function (x as a func- x 2 of the disorder ﬁeld, signal a random-ﬁeld induced order: tion of x ) at x can then be calculated from the above 1 0 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##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##

text: (a)

text: X

text: m

text: Jβ

text: aa

text: FIG.

text: (Color

text: online.)

text: Schematic

text: diagram

text: of

text: the

text: magneti-

text: (b)

text: zation

text: of

text: XY

text: ferromagnets

text: without

text: and

text: with

text: disorder,

text: in

text: the

text: presence

text: of

text: a

text: constant

text: magnetic

text: ﬁeld

text: The

text: ﬁgure

text: on

text: the

text: left

text: indicates

text: the

text: behavior

text: of

text: in

text: the

text: presence

text: of

text: but

text: when

text: mX

text: while

text: the

text: one

text: on

text: the

text: right

text: is

text: when

text: there

text: is

text: a

text: positive

text: the

text: mean

text: ﬁeld

text: equation

text: is

text: given

text: by

text: Eq.

text: keeping

text: h

text: strictly

text: positive

text: in

text: both

text: cases.

text: Let

text: us

text: denote

text: the

text: two

text: Jβ

text: Hamiltonians

text: by

text: H

text: and

text: H

text: respectively.

text: We

text: do

text: the

text: h

text: comparison

text: by

text: numerical

text: simulations

text: as

text: well

text: as

text: perturba-

text: tively

text: at

text: low

text: temperatures

text: (Sec.

text: below).

text: A

text: perturbation

text: FIG.

text: (Color

text: online.)

text: (a)

text: The

text: X-component

text: of

text: magnetiza-

text: approach,

text: similar

text: to

text: the

text: one

text: in

text: Sec.

text: can

text: be

text: done

text: at

text: high

text: tion,

text: m

text: as

text: a

text: function

text: of

text: Jβ.

text: Red

text: crosses

text: represent

text: the

text: case

text: x

text: temperatures

text: also.

text: We

text: refrain

text: from

text: doing

text: it,

text: as

text: the

text: high

text: when

text: the

text: XY

text: model

text: has

text: the

text: applied

text: constant

text: ﬁeld

text: with

text: temperature

text: behavior

text: in

text: this

text: case

text: is

text: less

text: interesting,

text: in

text: h/J

text: and

text: (a)

text: x

text: and

text: (b)

text: x

text: The

text: blue

text: solid

text: view

text: of

text: absence

text: of

text: a

text: phase

text: transition.

text: and

text: the

text: green

text: dashed

text: lines

text: are

text: for

text: the

text: same

text: system

text: but

text: with

text: the

text: additional

text: random

text: ﬁeld

text: of

text: strength

text: and

text: The

text: length

text: m

text: of

text: the

text: magnetization

text: vector

text: is

text: shrunk

text: in

text: respectively.

text: x

text: is

text: measured

text: in

text: radians.

text: All

text: other

text: quantities

text: the

text: system

text: with

text: the

text: disordered

text: ﬁeld,

text: compared

text: to

text: the

text: or-

text: are

text: dimensionless.

text: dered

text: case.

text: This

text: is

text: seen

text: from

text: numerical

text: simulations

text: (see

text: Figs.

text: (a)),

text: as

text: well

text: as

text: by

text: perturbation

text: techniques

text: at

text: low

text: temperatures.

text: In

text: addition,

text: numerical

text: simulations

text: (shown

text: “order

text: from

text: disorder”.

text: However,

text: such

text: eﬀect

text: is

text: absent

text: in

text: in

text: Fig.

text: (b))

text: show

text: that

text: the

text: cosine

text: of

text: the

text: phase

text: of

text: the

text: the

text: system

text: when

text: x

text: is

text: small

text: (see

text: Figs.

text: (b)

text: and

text: (top)).

text: magnetization,

text: i.e.,

text: increases

text: at

text: low

text: temperature

text: We

text: also

text: ﬁnd

text: that

text: for

text: a

text: given

text: x,

text: the

text: shift

text: in

text: mx

text: due

text: to

text: in

text: presence

text: of

text: the

text: random

text: ﬁeld.

text: Therefore,

text: the

text: magne-

text: disorder

text: decreases

text: as

text: the

text: ratio

text: J/h

text: increases.

text: tization

text: vector

text: moves

text: towards

text: the

text: X-direction

text: (i.e.

text: the

text: A.

text: Magnetization

text: at

text: low

text: temperature:

text: direction

text: transverse

text: to

text: the

text: applied

text: random

text: ﬁeld).

text: How-

text: Perturbative

text: approach

text: ever,

text: the

text: shift

text: turns

text: out

text: to

text: be

text: zero

text: when

text: x

text: equals

text: to

text: This

text: is

text: also

text: corroborated

text: by

text: perturbative

text: analysis

text: To

text: obtain

text: the

text: behavior

text: of

text: magnetization

text: at

text: low

text: tem-

text: at

text: low

text: temperatures.

text: The

text: schematic

text: diagram

text: in

text: Fig.

text: perature,

text: we

text: will

text: use

text: the

text: implicit

text: function

text: theorem,

text: which

text: shows

text: the

text: low

text: temperature

text: behavior

text: of

text: the

text: length

text: and

text: we

text: now

text: state.

text: Let

text: an

text: equation

text: two

text: vari-

text: phase

text: of

text: the

text: magnetization

text: with

text: and

text: without

text: disorder,

text: ables

text: x

text: and

text: x

text: be

text: such

text: that

text: f(x

text: x

text: at

text: x

text: in

text: the

text: presence

text: of

text: a

text: constant

text: ﬁeld.

text: x

text: is

text: in

text: general

text: an

text: unknown

text: function

text: of

text: x

text: But

text: The

text: Y

text: -component,

text: my

text: m

text: sin

text: of

text: the

text: magnetiza-

text: we

text: may

text: still

text: understand

text: the

text: character

text: of

text: by

text: tion

text: has

text: the

text: same

text: relative

text: behavior

text: as

text: the

text: length

text: m,

text: in

text: using

text: the

text: fact

text: that

text: (under

text: certain

text: regularity

text: conditions

text: systems

text: described

text: by

text: H

text: and

text: H

text: i.e.,

text: for

text: small

text: h

text: on

text: f

text: near

text: it

text: is

text: lower

text: than

text: for

text: However,

text: the

text: X-component,

text: m

text: m

text: cos

text: φ

text: of

text: the

text: x

text: magnetization,

text: behaves

text: in

text: a

text: very

text: interesting

text: way.

text: Its

text: ∂f

text: ∂f

text: dx

text: value

text: in

text: the

text: system

text: described

text: by

text: H

text: can

text: be

text: both

text: higher

text: ∂x

text: ∂x

text: dx

text: and

text: lower

text: than

text: its

text: value

text: in

text: the

text: system

text: described

text: by

text: Hh

text: depending

text: on

text: the

text: direction

text: of

text: the

text: constant

text: magnetic

text: ﬁeld.

text: The

text: numerical

text: data

text: for

text: the

text: X-component

text: of

text: the

text: magnetic

text: The

text: usual

text: statement

text: of

text: the

text: implicit

text: function

text: theorem

text: ﬁeld,

text: m

text: are

text: shown

text: for

text: h/J

text: and

text: x

text: (Fig.

text: is

text: that

text: when

text: ∂f

text: is

text: nonzero

text: at

text: we

text: can

text: solve

text: the

text: x

text: (a))

text: and

text: x

text: (Fig.

text: (b)).

text: The

text: numerical

text: results

text: for

text: equation

text: f(x

text: x

text: for

text: x

text: uniquely

text: near

text: this

text: point

text: x

text: where

text: m

text: is

text: signiﬁcantly

text: enhanced

text: in

text: presence

text: and

text: the

text: derivative

text: of

text: the

text: resulting

text: function

text: (x

text: as

text: a

text: func-

text: x

text: of

text: the

text: disorder

text: ﬁeld,

text: signal

text: a

text: random-ﬁeld

text: induced

text: order:

text: tion

text: of

text: x

text: at

text: x

text: can

text: then

text: be

text: calculated

text: from

text: the

text: above

text: ##LTFigure##

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text: Nuclear Physics A Nuclear Physics A 00 (2020) 1–4 Beam energy dependence of the expansion dynamics in relativistic heavy ion collisions: Indications for the critical end point? Roy A. Lacey Depts. of Chemistry & Physics, Stony Brook University, NY 11974 Abstract The ﬂow harmonic v and the emiss√ion source radii R , R and R ar√e studied for a broad range of centrality selections n − out side long and beam collision energies in Au+Au ( sNN = 7.7 200 GeV) and Pb+Pb ( sNN = 2.76 TeV) collisions at RHIC and the LHC respectively. They validate the acoustic scaling patterns expected f√or hydrodynamic-like expansion over the entire range of beam (cid:104) (cid:105) energies studied. The combined data sets allow estimates for the s dependence of the mean expansion speed c , emission (cid:104) (cid:105) (cid:104) (cid:48)(cid:48)(cid:105) NN (cid:104) (cid:105) s duration ∆τ and the viscous coeﬃcients β th(cid:104)at e(cid:105)ncode the magnitude of the speciﬁc shear viscosity η/s . The estimates indicate initial-state model independ(cid:104)ent val(cid:105)ues of η/s± which ar(cid:104)e large(cid:105)r for the pla±sma produced at 2.76 TeV (LHC) compar√ed to that produced at(cid:104)20(cid:48)(cid:48)0(cid:105) G(cid:104) eV(cid:105) (RH(cid:104)IC)(cid:105)( 4πη/s LHC = 2.2(cid:104) (cid:48)0(cid:48)(cid:105).2 and(cid:104) 4(cid:105)πη/s RHIC = 1.3 0.2). They also sho(cid:104)w a(cid:105)non-monotonic sNN dependence for β , cs and ∆τ , with minima for β and cs , and√a complimentary maximum for ∆τ . These dependencies signal a signiﬁcant change in reaction dynamics in a narrow span of s , which may be linked to reaction trajectories close to NN the critical end point (CEP) in the phase diagram for nuclear matter. Keywords: 1. Introduction Heavy ion collisions provide an important avenue for studying the phase diagram for Quantum Chromodynamics (QCD) [1, 2, 3]. The location of the phase boundaries and the critical end point (CEP), in the plane of temperature vs. baryon chemical potential [(T, µB)-plane], are fundamental “landmarks” of this phase diagram [4]. Lattice QCD ca√lculations suggest that the quark-hadron transition is a crossover at high T and smal√l µB or high collision energies ( s ) [5]. Experimental r√esults from the Relativistic Heavy Ion Collider (RHIC) at s = 200 GeV and the Large NN NN Hadron Collider (LHC) at s = 2.76 TeV, indicate that this transition results in the production of a strongly coupled NN plasma of de-conﬁned quarks and gluons (sQGP) with low speciﬁc shear viscosity η/s, i.e. the ratio of shear vis- cosity η to entropy density s [6]. The validation of this crossover transition, which is a necessary, albeit insuﬃcient, requirement for the existence of the CEP, serves as an important impetus for the ongoing experimental searches. A current strategy for (i) establishing the essential “landmarks” of the phase diagram and (ii) pinning down the thermodynamic and transport properties of each QCD phase, is centered on measurements in energy scans designed to access the broadest possible (T, µB)-domain of the phase diagram. In this proceedings we follow this lead by leveraging the combined measurements for anisotropic ﬂow and HBT radii, recently obtained by PHENIX and STAR in the ﬁrst RHIC Beam Energy Scan (BES-I) and by ATLAS, ALICE and CMS at the LHC. 1Roy.Lacey@Stonybrook.edu 1 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Nuclear

label: paragraph

font: MEUDAV+NimbusSanL-Regu

text: Physics

text: A

text: Nuclear

text: Physics

text: A

text: Beam

label: title

text: energy

text: dependence

text: of

text: the

text: expansion

text: dynamics

text: in

text: relativistic

text: heavy

text: ion

text: collisions:

text: Indications

text: for

text: the

text: critical

text: end

text: point?

text: Roy

label: author

text: A.

text: Lacey

text: Depts.

text: of

text: Chemistry

font: CRICBO+rtxi

text: Physics,

text: Stony

text: Brook

text: University,

text: NY

text: Abstract

text: The

label: abstract

text: ﬂow

text: harmonic

text: v

text: and

text: the

text: emiss√ion

text: source

text: radii

text: R

text: R

text: and

text: R

text: ar√e

text: studied

text: for

text: a

text: broad

text: range

text: of

text: centrality

text: selections

text: n

font: WZURKD+txsy

text: out

text: side

text: long

text: and

text: beam

text: collision

text: energies

text: in

text: Au+Au

text: sNN

font: OOTUFF+rtxr

text: GeV)

text: and

text: Pb+Pb

text: sNN

text: TeV)

text: collisions

text: at

text: RHIC

text: and

text: the

text: LHC

text: respectively.

text: They

text: validate

text: the

text: acoustic

text: scaling

text: patterns

text: expected

text: f√or

text: hydrodynamic-like

text: expansion

text: over

text: the

text: entire

text: range

text: of

text: beam

text: energies

text: studied.

text: The

text: combined

text: data

text: sets

text: allow

text: estimates

text: for

text: the

text: s

text: dependence

text: of

text: the

text: mean

text: expansion

text: speed

text: c

text: emission

text: NN

text: s

text: duration

text: ∆τ

text: and

text: the

text: viscous

text: coeﬃcients

text: β

text: the

text: magnitude

text: of

text: the

text: speciﬁc

text: shear

text: viscosity

text: η/s

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text: A

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text: anisotropic

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text: radii,

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text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

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text: Roy A. Lacey / Nuclear Physics A 00 (2020) 1–4 2 √ Figure 1. (Color online) HBT radii vs. R¯ for several mT cuts (as indicated) for (a) Rside, (b) Rout and (c) Rlong for Pb+Pb collis√ions at sNN=2.76 TeV; the data are taken from Ref. [17]. The dashed and dotted curves indicate linear ﬁts to the data (see text). (d) Si vs. 1/ mT; Si are slopes obtained from the respective linear ﬁts to the scaled values of R , R and R , shown in (a), (b) and (c). The dashed, dashed-dot and dotted side out long curves in this panel, represent linear ﬁts. 2. Probes for transport properties and the Critical End Point (CEP) The expansion dynamics of relativistic heavy ion collisions is strongly inﬂuenced by the transport properties of the created medium, as well as the path of the reaction trajectory in the (T, µB)-plane. Such an inﬂuence can manifest as quantiﬁable changes in the magnitude of the space-time extent of the emission source, characterized ∝ by the so-called HB∝T radii Rout, Rside and Rlong; the square o∝f (i) th−e emission source lifetime τ2 R2long, (ii) its geometric size R2 R2 and (iii) the emission duration ∆τ2 (R2 R2 ) [7]. It can also manifest as a signiﬁcant geo side out side m√odulation of the anisotropic ﬂow coeﬃcients vn, depending on the magnitude of η/s. The LHC measurements at sNN = 2.76 TeV, allow investigations of the space-time extent and η/s at high T and small µB; they compliment similar measu√rements fro−m BES-I which allow a systematic study for the µB and T values spanned by the collision energy range sNN = 7.7 200 GeV. Here, it is noteworthy that currently, there are only a few experimental constraints for the (T, µB)-dependence of η/s, especially at the lower beam energies [8]. At the CEP or close to it, anomalies in the dynamic properties of the medium can drive abrupt changes in trans- port coeﬃcients and relaxation rates to give a non-montonic dependence of ηs(T, µB) [6, 9, 10]. An emitting system produced in the vicinity of the CEP is also expected to show a stalling of the expansion speed and a larger emission ∝ − duration manifested as a diﬀerence between Rout and Rside (∆τ2 (R2out R2side)) [11]. Here, the rationale is that, in the vicinity of the CEP, the equation of state (EOS) “softens” considerably and this slows down the speed of expansion (cid:113)(cid:16) (cid:17) and prolongs the emission duration to give Rout > Rside. In prior work [12, 13], we have used the participant eccentricities (εn) and initial transverse size R¯ (1/R¯ = 1/σ2x + 1/σ2y , where σx and σy are the respective root-mean-square widths of the density distributions) obtained with Monte Carlo Glauber (MC-Glauber) simulations, to validated the acoustic nature of of the expansion dynamics [14] in RHIC and LHC collisions. This acoustic property predicts a linear relationship between the expansion time ∝ (t) and the ini(cid:48)ti∝al traverse siz(cid:48)e(cid:48) (∝t R¯), as well as a characteristic linear dependence of ln(vn/εn) on both n2 and 1/R¯, with slopes β (η/s) and β (η/s). We use the latter scaling patterns in conjunction with viscous hydrodynami- (cid:48) (cid:48)(cid:48) (cid:104) (cid:105) cal calculations [15, 16], to calibrate β and β and make estimates of η/s for√the plasma produced in Au+Au and (cid:48)(cid:48) Pb+Pb collisions at 200 GeV and 2.76 TeV respectively. A further study of the s dependence of β , cs and ∆τ is NN then used to search for non-monotonic patterns which could signal the presence of the CEP. 3. Results Representative summaries of the the scaling properties of the HBT radii and v are shown in Figs. 1 and 2 re- n spectively. Figs. 1(a), (b) and (c) validate the expected linear dependence of R , R and R on R¯. They also side out long show the expected decrease in the slope of the respective scaling curves (for R , R and R ) with increasing side out long transverse mass m . The latter conﬁrms the important inﬂuence of the space-momentum correlations which result T from collective expansion. Fig. 1(d) shows that the slopes S , obtained from linear ﬁts (dashed and dotted curves) i 2 ##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## 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text: Roy

text: A.

text: Lacey

text: Nuclear

text: Physics

text: A

label: caption

text: Figure

text: (Color

text: online)

text: HBT

text: radii

text: vs.

text: R¯

text: for

text: several

text: mT

text: cuts

text: (as

text: indicated)

text: for

text: (a)

text: Rside,

text: (b)

text: Rout

text: and

text: (c)

text: Rlong

text: for

text: Pb+Pb

text: collis√ions

text: at

text: TeV;

text: the

text: data

text: are

text: taken

text: from

text: Ref.

text: The

text: dashed

text: and

text: dotted

text: curves

text: indicate

text: linear

text: ﬁts

text: to

text: the

text: data

text: (see

text: text).

text: (d)

text: Si

text: vs.

text: mT;

text: Si

text: are

text: slopes

text: obtained

text: from

text: the

text: respective

text: linear

text: ﬁts

text: to

text: the

text: scaled

text: values

text: of

text: R

text: R

text: and

text: R

text: shown

text: in

text: (a),

text: (b)

text: and

text: (c).

text: The

text: dashed,

text: dashed-dot

text: and

text: dotted

text: side

text: out

text: long

text: curves

text: in

text: this

text: panel,

text: represent

text: linear

text: ﬁts.

text: Probes

text: for

text: transport

text: properties

text: and

text: the

text: Critical

text: End

text: Point

text: (CEP)

text: The

text: expansion

text: dynamics

text: of

text: relativistic

text: heavy

text: ion

text: collisions

text: is

text: strongly

text: inﬂuenced

text: by

text: the

text: transport

text: properties

text: of

text: the

text: created

text: medium,

text: as

text: well

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text: the

text: path

text: of

text: the

text: reaction

text: trajectory

text: in

text: the

text: (T,

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text: Such

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text: magnitude

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text: space-time

text: extent

text: of

text: the

text: emission

text: source,

text: characterized

text: by

text: the

text: so-called

text: HB∝T

text: radii

text: Rout,

text: Rside

text: and

text: Rlong;

text: the

text: square

text: o∝f

text: (i)

text: th−e

text: emission

text: source

text: lifetime

text: τ2

text: (ii)

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text: geometric

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text: (iii)

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text: sNN

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text: allow

text: investigations

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text: space-time

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text: CEP,

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text: equation

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text: considerably

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text: 4 − − Model M /GeV m /GeV N R2 /GeV 2 R2/GeV 2 Υ f T T Gauss-LC 9.460 4.2 0.76 1.91 – Boosted Gaussian 9.460 4.2 0.481 – 0.57 TABLE I: Parameters of the Gauss-LC and Boosted Gaussian models for the Υ wave function. (cid:90) dz ∗ W(r, Q2) = 2πr (Ψ Ψ) . (13) 4π Υ T In Fig. 1 we present our predictions for very low Q2, typical for photoproduction. We obtain that both models predict a peak for small values of r, which is directly associated to the large bottom mass. Moreover, the predicted radius dependence is similar, with the normalization of the Gauss-LC model being smaller than the Boosted Gaussian one. Such diﬀerences have direct implications in the corresponding predictions for the total cross section, as we will demonstrate in the next section. N The non-forward scattering amplitude (x, r, b) contains all information about the target and the strong interaction physics. In the last years, several groups have constructed phenomenological models which satisfy the asymptotic behaviour of the Color Glass Condensate (CGC) formalism [29–31]. In what follows we will use the bCGC model proposed in Ref. [21], which improves the Iancu - Itakura - Munier (IIM) model [32] with the inclusion of the impact parameter dependence in the dipole - proton scattering amplitude. Following [21] we have: (cid:16) (cid:17)  (cid:16) (cid:17)  (cid:2) (cid:3) N N r Qs,p 2 γs+ln(2κ/rλQYs,p) rQ ≤ 2 p(x, r, b) = −0 2 − s,p (14) 1 exp A ln2 (B r Q ) rQ > 2 s,p s,p (cid:48)(cid:48) (cid:48) with Y = ln(1/x) and κ = χ (γ )/χ (γ ), where χ is the LO BFKL characteristic function [33]. The coeﬃcients s s N A and B are determined uniquely from the condition that (x, r, b), and its derivative with respect to rQ , are p s continuous at rQ = 2. In this model, the proton saturation scale Q depends on the impact parameter: s (cid:16) (cid:17) (cid:20) (cid:18) s,p (cid:19)(cid:21) 1 Q ≡ Q (x, b) = x0 λ2 exp − b2 2γs . (15) s,p s,p x 2B CGC The parameter B was adjusted to give a good description of the t-dependence of exclusive J/ψ photoproduction. N CGC The factors , x , λ and γ were taken to be free. Recently the parameters of this model have been updated in Ref. 0 0 s [34] (considering the recently released high precision combined HERA data), being given by γ = 0.6599, B = 5.5 − N × − s CGC GeV 2, = 0.3358, x = 0.00105 10 5 and λ = 0.2063. As demonstrate in Ref. [22], this phenomenological dipole 0 0 describes quite well the HERA data for the exclusive ρ and J/Ψ production. For comparison, in what follows we will N N also use the GBW model [35], which assumes that (x, r, b) = (x, r)S(b) with the forward scattering amplitude being given by Np(x, r) = 1 − e−r2Q2s,p(Y )/4 and Q2s,p(pY ) = (x0/x)λp, with the parameters x0 and λ determined by the ﬁt to the HERA data available in 1999. The parameters of the GBW model have been updated in Ref. [36] considering the ZEUS data available in 2007. In what follows we will use these two sets of parameters in our calculations, with the resulting predictions being denoted GBW and GBW-KSX, respectively. It is important to emphasize that the N GBW model is a model for the forward dipole-target amplitude (x, r), which does not allow us to calculate the p t-dependence of the diﬀerential cross section. Therefore, in the GBW case, we should estimate the total cross section using Eq. (8). III. RESULTS In what follows we will present our predictions for the exclusive Υ photoproduction in photon - proton and proton - proton/nucleus collisions. In order to estimate the cross section we need to specify the exponent λ which determines e R and β. As demonstrated in Ref. [37] the estimate obtained using this approximation for R is strongly dependent g g on the parton distribution used in the calculation. However, this dependence is smaller at large hard scales and small values of x (large energies), which is the case of the Υ production at LHC. Another important aspect is that the incorporation of the skewness correction at small-x in the dipole models still is an open question which deserve more ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: table

text: Model

text: M

text: /GeV

text: m

text: /GeV

text: N

text: /GeV

text: Υ

text: f

text: T

text: T

text: Gauss-LC

text: Boosted

text: Gaussian

text: TABLE

text: I:

text: Parameters

text: of

text: the

text: Gauss-LC

text: and

text: Boosted

text: Gaussian

text: models

text: for

text: the

text: Υ

text: wave

text: function.

label: equation

text: dz

text: W(r,

text: (Ψ

text: Ψ)

text: 4π

text: Υ

text: T

text: In

text: Fig.

text: we

text: present

text: our

text: predictions

text: for

text: very

text: low

text: typical

text: for

text: photoproduction.

text: We

text: obtain

text: that

text: both

text: models

text: predict

text: a

text: peak

text: for

text: small

text: values

text: of

text: r,

text: which

text: is

text: directly

text: associated

text: to

text: the

text: large

text: bottom

text: mass.

text: Moreover,

text: the

text: predicted

text: radius

text: dependence

text: is

text: similar,

text: with

text: the

text: normalization

text: of

text: the

text: Gauss-LC

text: model

text: being

text: smaller

text: than

text: the

text: Boosted

text: Gaussian

text: one.

text: Such

text: diﬀerences

text: have

text: direct

text: implications

text: in

text: the

text: corresponding

text: predictions

text: for

text: the

text: total

text: cross

text: section,

text: as

text: we

text: will

text: demonstrate

text: in

text: the

text: next

text: section.

text: N

text: The

text: non-forward

text: scattering

text: amplitude

text: (x,

text: r,

text: b)

text: contains

text: all

text: information

text: about

text: the

text: target

text: and

text: the

text: strong

text: interaction

text: physics.

text: In

text: the

text: last

text: years,

text: several

text: groups

text: have

text: constructed

text: phenomenological

text: models

text: which

text: satisfy

text: the

text: asymptotic

text: behaviour

text: of

text: the

text: Color

text: Glass

text: Condensate

text: (CGC)

text: formalism

text: In

text: what

text: follows

text: we

text: will

text: use

text: the

text: bCGC

text: model

text: proposed

text: in

text: Ref.

text: which

text: improves

text: the

text: Iancu

text: Itakura

text: Munier

text: (IIM)

text: model

text: with

text: the

text: inclusion

text: of

text: the

text: impact

text: parameter

text: dependence

text: in

text: the

text: dipole

text: proton

text: scattering

text: amplitude.

text: Following

text: we

text: have:

text: N

text: N

text: r

text: Qs,p

text: rQ

text: p(x,

text: r,

text: b)

text: −0

text: s,p

text: exp

text: A

text: (B

text: r

text: Q

text: rQ

text: s,p

text: s,p

text: with

text: Y

text: and

text: κ

text: χ

text: (γ

text: )/χ

text: (γ

text: where

text: χ

text: is

text: the

text: LO

text: BFKL

text: characteristic

text: function

text: The

text: coeﬃcients

text: s

text: s

text: N

text: A

text: and

text: B

text: are

text: determined

text: uniquely

text: from

text: the

text: condition

text: that

text: (x,

text: r,

text: b),

text: and

text: its

text: derivative

text: with

text: respect

text: to

text: rQ

text: are

text: p

text: s

text: continuous

text: at

text: rQ

text: In

text: this

text: model,

text: the

text: proton

text: saturation

text: scale

text: Q

text: depends

text: on

text: the

text: impact

text: parameter:

text: s

text: s,p

text: Q

text: Q

text: (x,

text: b)

text: λ2

text: exp

text: s,p

text: s,p

text: x

text: CGC

text: The

text: parameter

text: B

text: was

text: adjusted

text: to

text: give

text: a

text: good

text: description

text: of

text: the

text: t-dependence

text: of

text: exclusive

text: J/ψ

text: photoproduction.

text: N

text: CGC

text: The

text: factors

text: x

text: λ

text: and

text: γ

text: were

text: taken

text: to

text: be

text: free.

text: Recently

text: the

text: parameters

text: of

text: this

text: model

text: have

text: been

text: updated

text: in

text: Ref.

text: s

text: (considering

text: the

text: recently

text: released

text: high

text: precision

text: combined

text: HERA

text: data),

text: being

text: given

text: by

text: γ

text: B

text: N

text: s

text: CGC

text: GeV

text: x

text: and

text: λ

text: As

text: demonstrate

text: in

text: Ref.

text: this

text: phenomenological

text: dipole

text: describes

text: quite

text: well

text: the

text: HERA

text: data

text: for

text: the

text: exclusive

text: ρ

text: and

text: J/Ψ

text: production.

text: For

text: comparison,

text: in

text: what

text: follows

text: we

text: will

text: N

text: N

text: also

text: use

text: the

text: GBW

text: model

text: which

text: assumes

text: that

text: (x,

text: r,

text: b)

text: (x,

text: r)S(b)

text: with

text: the

text: forward

text: scattering

text: amplitude

text: being

text: given

text: by

text: Np(x,

text: r)

text: and

text: with

text: the

text: parameters

text: and

text: λ

text: determined

text: by

text: the

text: ﬁt

text: to

text: the

text: HERA

text: data

text: available

text: in

text: The

text: parameters

text: of

text: the

text: GBW

text: model

text: have

text: been

text: updated

text: in

text: Ref.

text: considering

text: the

text: ZEUS

text: data

text: available

text: in

text: In

text: what

text: follows

text: we

text: will

text: use

text: these

text: two

text: sets

text: of

text: parameters

text: in

text: our

text: calculations,

text: with

text: the

text: resulting

text: predictions

text: being

text: denoted

text: GBW

text: and

text: GBW-KSX,

text: respectively.

text: It

text: is

text: important

text: to

text: emphasize

text: that

text: the

text: N

text: GBW

text: model

text: is

text: a

text: model

text: for

text: the

text: forward

text: dipole-target

text: amplitude

text: (x,

text: r),

text: which

text: does

text: not

text: allow

text: us

text: to

text: calculate

text: the

text: p

text: t-dependence

text: of

text: the

text: diﬀerential

text: cross

text: section.

text: Therefore,

text: in

text: the

text: GBW

text: case,

text: we

text: should

text: estimate

text: the

text: total

text: cross

text: section

text: using

text: Eq.

text: III.

text: RESULTS

text: In

text: what

text: follows

text: we

text: will

text: present

text: our

text: predictions

text: for

text: the

text: exclusive

text: Υ

text: photoproduction

text: in

text: photon

text: proton

text: and

text: proton

text: proton/nucleus

text: collisions.

text: In

text: order

text: to

text: estimate

text: the

text: cross

text: section

text: we

text: need

text: to

text: specify

text: the

text: exponent

text: λ

text: which

text: determines

text: e

text: R

text: and

text: β.

text: As

text: demonstrated

text: in

text: Ref.

text: the

text: estimate

text: obtained

text: using

text: this

text: approximation

text: for

text: R

text: is

text: strongly

text: dependent

text: g

text: g

text: on

text: the

text: parton

text: distribution

text: used

text: in

text: the

text: calculation.

text: However,

text: this

text: dependence

text: is

text: smaller

text: at

text: large

text: hard

text: scales

text: and

text: small

text: values

text: of

text: x

text: (large

text: energies),

text: which

text: is

text: the

text: case

text: of

text: the

text: Υ

text: production

text: at

text: LHC.

text: Another

text: important

text: aspect

text: is

text: that

text: the

text: incorporation

text: of

text: the

text: skewness

text: correction

text: at

text: small-x

text: in

text: the

text: dipole

text: models

text: still

text: is

text: an

text: open

text: question

text: which

text: deserve

text: more

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 8 Fx = Dbx− − Dbx + Da−x − Dax + Dc−x − Dcx− Qa+ = (Bia,j)2 ,Qa− = (Bia,j−+1)2 , i,j i,j 1 i,j i 1,j i,j i 1,j i,j 1 i,j 4J+ s i,j 4J s − − i,j i,j +R+ (y y ) + R+− (y − y − ) +Ri+,j−1(i,yji,j−i+11−,j yi+1i,,jj−+11) + Ri−i−11,,jj−1i(y1i−,j+1,1j − yii,j−1,1j) Qbi,+j = (4BJib+,j)s2 ,Qbi,−j = (B4ibJ+−1,js)2 , − − − − i,j i,j +R − (y y − ) + R − (y − y ), i 1,j i,j+1 i 1,j+1 i,j 1 i+1,j 1 i+1,j (Bc )2 − (Bc )2 Fy = Dby− − Dby + Da−y − Day + Dc−y − Dcy− Qci,+j = 4Ji+,j s ,Qci,j = 4Ji−,j s . (A12) i,j − i,j 1 i−,j i 1−,j i,j i 1,j − i,j 1 i,j i,j R+ (x x ) R+− (x − x − ) i,j i+1,j i,j+1 i 1,j i 1,j+1 i 1,j − − − − − R+ − (x − x − ) R − − (x − x − ) Boundary conditions are applied as holonomic con- i,j 1 i,j 1 i+1,j 1 i 1,j 1 i 1,j i,j 1 − − − − − − straints. For example, periodic boundary in x can be Ri−1,j(xi,j+1 xi−1,j+1) Ri,j−1(xi+1,j−1 xi+1,j), realized with xNx,j = x1,j + Lx, and rigid wall with (A9) x = 0, x = L . N and L stand for the resolution 1,j Nx,j x x x and domain size in x respectively. where afxundeﬁned [1]ifx1 fnum [1]1ﬁrstoftwo fx [1]secondoftwo noop [0]secondoftworef[1]@startlink1@hrefhref[1]1@endlinkanitize@url (cid:104) (cid:0) (cid:1) [0]startlink[1]endlink[0]rl [1]href 1bib@innerbibempty W. A. Newcomb, noop Nuclear Fusion Supplement 2, R+ = p(cid:0)+i,j (cid:1)+ 1 (cid:0) (cid:1)Ba 2(cid:105)Ia+ 451 (1962). i,j (J+ )γs 4(J+ )2s i,j i,j i,j i,j P. J. Morrison and J. M. Greene, noop Physical Review + Bb 2 Ib+ + Bc(cid:104) (cid:0)2 Ic+ ,(cid:1) Letters 45, 790 (1980). i,j i,j i,j i,j − D. D. Holm, J. E. Marsden, and T. S. Ratiu, noop Ad- R− = p−(cid:0)i,j +(cid:1) −1 (cid:0) B(cid:1)a 2(cid:105)Ia− vances in Mathematics 137, 1 (1998). i,j (J )γs 4(J )2s i,j+1 i,j i,j i,j J.-G. Liu and W.-C. Wang, noop Journal of Computa- + Bb 2 Ib− + Bc 2 Ic− , (A10) tional Physics 174, 12 (2001). i+1,j i,j i,j i,j T. A. Gardiner and J. M. Stone, noop Journal of Com- putational Physics 205, 509 (2005). and J. E. Marsden and M. West, noop Acta Numerica 2001 10, 357 (2001). H. Qin and X. Guan, noop Physical Review Letters 100, − − − Dax =Qb+(x x ) + Qb − (x x − ) 035006 (2008). i,j i,j i,j+1 i+1,j i,j 1 i,j i+1,j 1 − − − − − Qc+(x x ) Qc − (x x − ), A. Stern, Y. Tong, M. Desbrun, and J. E. Marsden, noop i,j i,j+1 i,j i,j 1 i+1,j i+1,j 1 − − − arXiv preprint arXiv:0707.4470 (2007). Dbx =Qa+(x x ) + Qa− (x x − ) i,j i,j i+1,j i,j+1 i 1,j i,j i 1,j+1 − − − − − J. Squire, H. Qin, and W. M. Tang, noop Physics of Qc+(x x ) Qc− (x x − ), i,j i+1,j i,j i 1,j i,j+1 i 1,j+1 Plasmas 19, 084501 (2012). − − − Dcx =Qa+(x x ) + Qa (x x ) i,j i,j i,j+1 i,j i,j i+1,j+1 i+1,j D. Pavlov, P. Mullen, Y. Tong, E. Kanso, J. E. Marsden, − − − − − Qb+(x x ) Qb (x x ), and M. Desbrun, noop Physica D: Nonlinear Phenomena i,j i+1,j i,j i,j i+1,j+1 i,j+1 − − − 240, 443 (2011). Day =Qb+(y y ) + Qb − (y y − ) i,j i,j i,j+1 i+1,j i,j 1 i,j i+1,j 1 − − − − − M. Kraus, noop arXiv preprint arXiv:1307.5665 (2013). Qc+(y y ) Qc − (y y − ), i,j i,j+1 i,j i,j 1 i+1,j i+1,j 1 − − − E. S. Gawlik, P. Mullen, D. Pavlov, J. E. Marsden, and Dby =Qa+(y y ) + Qa− (y y − ) i,j i,j i+1,j i,j+1 i 1,j i,j i 1,j+1 M. Desbrun, noop Physica D: Nonlinear Phenomena − − − − − Qc+(y y ) Qc− (y y − ), 240, 1724 (2011). i,j i+1,j i,j i 1,j i,j+1 i 1,j+1 − − − Dcy =Qa+(y y ) + Qa (y y ) M. Desbrun, A. N. Hirani, M. Leok, and J. E. Marsden, i,j i,j i,j+1 i,j i,j i+1,j+1 i+1,j − − − − − noop arXiv preprint math/0508341 (2005). Qb+(y y ) Qb (y y ), i,j i+1,j i,j i,j i+1,j+1 i,j+1 (A11) A. N. Hirani, Discrete exterior calculus, noop Ph.D. the- sis, California Institute of Technology (2003). D. Longcope and H. Strauss, noop Physics of Fluids B: where Plasma Physics 5, 2858 (1993). ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Fx

text: Dbx−

text: Dbx

text: Da−x

text: Dax

text: Dc−x

text: Dcx−

text: Qa+

text: ,Qa−

text: i,j

text: i,j

text: i,j

text: i

text: i,j

text: i

text: i,j

text: i,j

text: s

text: i,j

text: s

text: i,j

text: i,j

text: +R+

text: (y

text: y

text: R+−

text: (y

text: y

text: Qbi,+j

text: ,Qbi,−j

text: i,j

text: i,j

text: +R

text: (y

text: y

text: R

text: (y

text: y

text: i

text: i

text: i,j

text: (Bc

text: (Bc

text: Fy

text: Dby−

text: Dby

text: Da−y

text: Day

text: Dc−y

text: Dcy−

text: Qci,+j

text: s

text: ,Qci,j

text: s

text: i,j

text: i,j

text: i−,j

text: i

text: i,j

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text: i,j

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text: i,j

text: R+

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text: x

text: R+−

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text: i

text: i

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text: x

text: R

text: (x

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text: Boundary

text: conditions

text: are

text: applied

text: as

text: holonomic

text: con-

text: i,j

text: i,j

text: i

text: i

text: i,j

text: straints.

text: For

text: example,

text: periodic

text: boundary

text: in

text: x

text: can

text: be

text: realized

text: with

text: xNx,j

text: Lx,

text: and

text: rigid

text: wall

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text: x

text: x

text: L

text: N

text: and

text: L

text: stand

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text: the

text: resolution

text: Nx,j

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text: x

text: x

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text: where

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text: noop

text: Nuclear

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text: i,j

text: i,j

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text: J.

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text: noop

text: Physical

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text: and

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