the same in the fragmentation functions and the parton densities. In order to
investigate the uncertainties associated with choice of $\mu_{F}$, several
values of $\mu_{F}$ are considered: $\mu_{F}=2p^{\gamma}_{T}$;
$p^{\gamma}_{T}$; and $p^{\gamma}_{T}/2$. The amplitude $N_{F}$ in Eqs. (51)
and (6.3) is defined in Eq. (6).

Figure 26: The nuclear modification factor $R_{pA}^{\gamma}$ for direct photon
production in minimum bias $p+$Pb collisions at rapidities $\eta_{\gamma}=0$,
$2$, $4$, and $6$ (with the convention that the proton beam moves toward
forward rapidity) obtained from Eq. (6.3) with solutions of the rcBK equation
with different initial nuclear saturation scales. The band labeled CGC-rcBK
includes uncertainties due to the variation of the nuclear saturation scale
and the factorization scale $\mu_{F}$. Similar to Fig. 10, the lines labeled
$N$ are results with a fixed factorization scale, $\mu_{F}=p^{\gamma}_{T}$,
and a fixed saturation scale $Q_{0A}^{2}=NQ_{0p}^{2}$ with
$Q_{0p}^{2}=0.168\,\text{GeV}^{2}/c^{2}$. The range $3<N<7$ is constrained in
Eq. (39). See Ref. [20].

In Figs. 26 and 27, predictions of the direct photon and single inclusive
prompt photon production nuclear modification factors $R_{pA}^{\gamma}(p_{T})$
in minimum bias $p+$Pb collisions at several rapidities. The solutions of the
rcBK equation were obtained using Eq. (6.3) with different initial nuclear
saturation scales. The band labeled CGC-rcBK includes uncertainties due to the
variation of $Q_{0A}^{2}$ within the range given in Eq. (39) with
factorization scales $\mu_{F}=2p_{T}^{\gamma}$, $p_{T}^{\gamma}$, and
$p_{T}^{\gamma}/2$. In Fig. 27, the $\eta_{\gamma}=0$ results include
inclusive prompt photon production calculated in Ref. [135] employing the
Iancu-Itakura-Munier (IIM) saturation model which also provides a good
description of the HERA data [136]. In the IIM method, saturation is
approached from the BFKL region. Therefore the IIM small-$x$ evolution is
different from that obtained with the rcBK equation.

Figure 27: The nuclear modification factor $R_{pA}^{\gamma}$ for single
inclusive prompt photon production in minimum bias $p+$Pb collisions at
$\eta_{\gamma}=0$ and 2 (with the convention that the proton beam moves toward
forward rapidity). The curves are described in Fig. 26. The dashed red line
labeled CGC-IIM was calculated according to the IIM dipole saturation model
[135]. See Ref. [20].

As discussed in Sec. 3.3.3, the results in Figs. 26 and 27 on
$R_{pA}^{\gamma}$, together with those on $R_{pA}^{\rm ch}$ in Fig. 10, can be
used to fix the nuclear saturation scale. Once $Q_{0A}^{2}$ has been
established at one rapidity, the predictions at other rapidities are fixed and
the CGC/saturation dynamics can be tested.

####  6.3.1 Photon-hadron correlations

Photon-hadron azimuthal correlations in $p+A$ and $p+p$ collisions could be an
excellent probe of small-$x$ dynamics [134, 137]. This correlation can be
defined as [134, 137],

$P(\Delta\phi)=\bigg{[}{d\sigma^{p\,A\rightarrow
h(p_{T}^{h})\,\gamma(p_{T}^{\gamma})\,X}[\Delta\phi]\over
d^{2}\vec{b_{T}}\,p_{T}^{h}dp_{T}^{h}\,p_{T}^{\gamma}dp_{T}^{\gamma}\,d\eta_{\gamma}\,d\eta_{h}\,d\phi}\bigg{]}\bigg{[}{d\sigma^{p\,A\rightarrow
h(p_{T}^{h})\,\gamma(p_{T}^{\gamma})\,X}[\Delta\phi=\Delta\phi_{c}]\over
d^{2}\vec{b_{T}}\,p_{T}^{h}dp_{T}^{h}\,p_{T}^{\gamma}dp_{T}^{\gamma}\,d\eta_{\gamma}\,d\eta_{h}\,d\phi}\bigg{]}^{-1}\,\,.$
(52)

The correlation function $P(\Delta\phi)$ is the probability of semi-inclusive
photon-hadron pair production in a certain kinematic region at angle
$\Delta\phi$ relative to production in the same kinematics at fixed reference
angle, $\Delta\phi_{c}=\pi/2$ [134, 137].

Fig. 28 shows the predicted $P(\Delta\phi)$ in minimum bias $p+p$ and $p+$Pb
collisions at 5 TeV for $p^{h}_{T}<p^{\gamma}_{T}$ (left) and
$p^{h}_{T}>p^{\gamma}_{T}$ (right). Given the $p_{T}$ of the produced photon
and hadron, the corresponding correlation can have either a double or single
peak structure. In Ref. [137] it was shown that this feature is related to
saturation physics and is governed by the ratio $p_{T}^{h}/p_{T}^{\gamma}$.
The change from a double to a single peak correlation, depending on the
relative $p_{T}$, is unique to semi-inclusive photon-hadron production: since
the trigger particle in dihadron correlations is always a hadron, it
consistently exhibits a single-peak structure.

Photon-hadron correlations can also be quantified by the coincidence
probability. In contrast to production of a more symmetric final state such as
dihadron production, in photon-hadron production the trigger particle can be
either the prompt photon or the hadron [137]. When the photon is used as the
trigger, the coincidence probability is defined as
$CP_{h}(\Delta\phi)=N^{\text{pair}}_{h}(\Delta\phi)/N_{\gamma}$ where
$N^{\text{pair}}_{h}(\Delta\phi)$ is the photon-hadron yield. The momentum of
the photon trigger, the leading ($L$) particle, is denoted $p^{\gamma}_{T,L}$
while the momentum of the associated ($S$) hadron (typically a $\pi^{0}$) is
denoted $p^{h}_{T,S}$. The azimuthal angle between the photon and the
$\pi^{0}$ is $\Delta\phi$ [137],

$\displaystyle
CP_{h}(\Delta\phi;p^{h}_{T,S},p^{\gamma}_{T,L};\eta_{\gamma},\eta_{h})$
$\displaystyle=$
$\displaystyle\frac{2\pi\int_{p^{\gamma}_{T,L}}dp_{T}^{\gamma}p_{T}^{\gamma}\int_{p^{h}_{T,S}}dp_{T}^{h}p_{T}^{h}\frac{dN^{p\,A\rightarrow
h(p_{T}^{h})\,\gamma(p_{T}^{\gamma})\,X}}{d^{2}\vec{p_{T}}^{\gamma}\,d^{2}\vec{p_{T}}^{h}\,d\eta^{\gamma}\,d\eta^{h}}}{\int_{p^{\gamma}_{T,L}}d^{2}\vec{p_{T}}^{\gamma}\,\frac{dN^{p\,A\rightarrow\gamma(p_{T}^{\gamma})\,X}}{d^{2}\vec{p_{T}}^{\gamma}\,d\eta_{\gamma}}}\,\,.$
(53)

The integrals are carried out within momentum intervals defined by
$p^{\gamma}_{T,L}$ and $p^{h}_{T,S}$. The yields in the above expression are
defined in Eqs. (51) and (6.3). The correlation defined in Eq. (52) can be
considered as a snapshot of the integrand in the coincidence probability
defined in Eq. (53). In the same fashion, one can define a hadron-triggered
coincidence probability [137]. If the $\pi^{0}$ is the trigger particle, the
momenta are instead denoted by $p^{\gamma}_{T,S}$ and $p^{h}_{T,L}$ which then
become the lower limits on the integrals in Eq. (53). The away-side
coincidence probability for the azimuthal correlation of photon-hadron pairs
can also have a double-peak or single-peak structure, depending on the trigger
particle selection and kinematics [137].

Figure 28: The $\gamma-\pi^{0}$ correlation $P(\Delta\phi)$ defined in Eq.
(52) in minimum bias $p+p$ and $p+$Pb collisions at $\sqrt{S}=5$ TeV and
$\eta_{h}=\eta_{\gamma}=3$ (with the convention that the proton beam moves
toward forward rapidity) obtained via the rcBK evolution equation with several
initial saturation scales, $Q^{2}_{0A}=NQ^{2}_{0p}$ with $N=3$, 5, and 7, in
two different transverse momentum regions: $p^{h}_{T}<p^{\gamma}_{T}$ (left)
and $p^{h}_{T}>p^{\gamma}_{T}$ (right). Taken from Ref. [20, 137].

In Fig. 29, predictions of the azimuthal correlations between the produced
prompt photon and hadron, calculated employing the coincidence probability
$CP_{h}$ in $p+$Pb collisions are shown for $\sqrt{s}=0.2$, $5$, and $8.8$
TeV. Equation (6.3) is used in the denominator of Eq. (53). The collinear
divergence was removed from Eq. (53) by introducing the photon fragmentation
function. The numerator is calculated using Eq. (51). An overall normalization
problem may result given that the away-side correlation at $\Delta\phi=\pi/2$
is not sensitive to the collinear singularity. Proper treatment of this
problem requires a full NLO calculation which is currently unavailable.
However, choosing a different photon fragmentation function [68, 129, 131]
will only slightly change the results given the freedom to choose the
fragmentation scale and the rather large uncertainties due to $Q_{0A}^{2}$.
This possible normalization problem is not present in the correlation defined
via Eq. (52).

Figure 29: The $\gamma-\pi^{0}$ coincident probability, $CP_{h}(\Delta\phi)$,
defined in Eq. (53) in minimum-bias $p+A$ collisions at
$\eta_{h}=\eta_{\gamma}=3$ (with the convention that the proton beam moves
toward forward rapidity) for $\sqrt{s}=0.2$, 5 and 8.8 TeV. Taken from Ref.
[20, 137].

Although there are theoretical uncertainties associated with the strength of
the photon-hadron correlations, due to both higher-order corrections and the
less constrained saturation scale $Q_{0A}^{2}$. Nevertheless, the
decorrelation of away-side photon-hadron production increases with energy,
rapidity and density. This decorrelation, together with the appearance of a
double or single-peak structure, are robust predictions of CGC/saturation
effects in the leading-log approximation.

###  6.4 Proton-nucleus dilepton and photon production at the LHC: gluon
saturation and shadowing (R. Baier, F. Gelis, A. H. Mueller and D. Schiff)

Here, a qualitative discussion on saturation and shadowing effects on photon
and dilepton production at very large rapidity in high-energy proton-nucleus
collisions is given. The aim is to describe the various steps of assumptions
leading to the present understanding.

In $\sqrt{s_{{}_{NN}}}=200$ GeV d+Au data on high-$p_{T}$ hadron production at
large rapidity (on the deuteron side) there is a significant suppression of
hadron production compared to the expectation from $p+p$ collisions. This
result suggests that there may be a significant amount of leading-twist gluon
shadowing in the nuclear wavefunction in the region probed by forward hard
scattering at RHIC, with even more expected at the LHC.

In many ways hard photons (or dileptons arising from virtual photons) are a
better probe than high-$p_{T}$ hadrons. Photons are less sensitive to
fragmentation effects while final-state effects are almost absent. Thus, at
$p_{T}\sim 2-3$ GeV/$c$, leading-twist factorization is expected to be
accurate. Hence the nuclear gluon distribution probes $x$ values somewhat
smaller than $10^{-4}$ in $p+$Pb collisions at the LHC.

Increasing the photon rapidity into the forward (proton rapidity) region,
$y>0$, the gluon $x$ probed decreases rapidly, e.g. for virtual photons of
mass $M$ and rapidity $y$ produced in the process $q+g\rightarrow\gamma^{*}$,
$x\simeq(M/\sqrt{s})\exp{(-y)}$. With $\sqrt{s}=5500$ GeV, $M=5.0$
GeV/$c^{2}$, and $y=3.5$, $x=2.7\times 10^{-5}$, so that indeed a fast quark
in the direction of proton rapidity produces virtual photons that probe the
small $x$ gluon distribution in the nucleus.

It is extremely interesting to explore, at least qualitatively, the size of
the suppression one might expect in such reactions. The treatment here follows
Ref. [133] which can be examined for further details. This discussion is based
on a picture derived from the Color Glass Condensate (CGC) effective theory
for the gluon distributioniiiiiiiiiFor recent reviews, see Refs. [138, 139]
and references therein. together with BFKL evolution (see e.g. Ref. [140]) to
reach higher values of $y$.

####  6.4.1 Factorized formula for the inclusive $\gamma^{*}$ cross section

A dimensionless observable, $\sigma(\vec{Q},Y)$, can be written in the
$k_{\bot}$-factorized form

$\sigma(\vec{\mu},Y)=\int\frac{d^{2}q_{T}}{\pi
q^{2}_{T}}\,H(\vec{q}_{T},\vec{\mu})\,\phi_{G}(\vec{q}_{T},Y)\,,$ (54)

where $\mu$ is the hard scale of the reaction, equal to the transverse
momentum, $k_{T}$, of the (massive) photon, i.e.

$\sigma(\vec{k}_{T},Y)\equiv\sigma(\vec{b}_{T},\vec{k}_{T},Y)=\frac{k^{~{}2}_{T}d\sigma}{d^{2}b_{T}d^{2}k_{T}d\ln
z}\,.$ (55)

For simplicity incident quarks are considered instead of protons,
$q+A\rightarrow\gamma^{*}X$, see Fig. 30. Here $k_{T}$ is the transverse
momentum of the $\gamma^{*}$, and $z$ is the longitudinal momentum fraction of
the $\gamma^{*}$ with respect to the incident quark momentum, $z=k_{+}/p_{+}$,
where $p_{+}\rightarrow\infty$. The impact parameter of the $q+A$ collisions
is denoted $\vec{b}$ and $Y=\ln{1/x}$.

bbllx=0,bblly=0,bburx=340,bbury=100, file=Graph.eps,width=11cm

Figure 30: Two gluon exchange graph for quark ($p$) + nucleus ($P$) production
of a real or virtual photon, $\rightarrow~{}\gamma^{(*)}(k)~{}X$.

The leading-twist contribution involves only two exchanged gluons, as shown in
Fig. 30. $H$ is the hard part of the interaction in the $k_{T}$-factorized
form while $\phi_{G}(\vec{q}_{T},Y)/q^{2}_{T}$ is considered to be
proportional to the differential high energy $q+A\rightarrow q+A$ cross
section.

####  6.4.2 Anomalous scaling and shadowing - specific predictions

While there are theoretical uncertainties, there are also robust
characteristic qualitative results. The function $\phi_{G}(k_{T},Y)$ can be
approximated by the scaling function,

$\phi_{G}(k_{T},Y)=\phi_{G}(k_{T}/Q_{s}(\vec{b},Y))\approx\left(k^{2}_{T}/Q^{2}_{s}(\vec{b},Y)\right)^{\lambda_{0}-1},$
(56)

with anomalous dimension $\lambda_{0}\simeq 0.37$ and saturation scale
$Q_{s}$. In $\vec{b}=0$ collisions at fixed $k_{T}$ and $Y$, an anomalous $A$
dependence is predicted, together with shadowing/suppression of the ratio
$R_{pA}$,

$R_{pA}\approx A^{-\lambda_{0}/3}~{}.$ (57)

In order to obtain results at large photon rapidities based on the BFKL
evolution in the presence of saturation, the rapidity dependence of the
saturation scale is chosen to be compatible with phenomenology,

$Q^{2}_{s}(\vec{b}=0,Y=y)=Q^{2}_{s}(Y=0)\exp(\lambda\,y),\,\,~{}~{}\lambda\simeq
0.3\,\,.$ (58)

There is significant suppression expected, $R_{pA}\simeq 0.5$, especially when
the $\gamma^{(*)}$ is produced at forward (in the proton direction)
rapidities, e.g. $y>3$ and $k_{T}>2$ GeV/$c$. More detailed predictions of
forward inclusive prompt photon production and semi-inclusive photon-hadron
correlations in high energy proton-nucleus collisions at the LHC using the CGC
formalism are presented in Refs. [134, 137].

Measurements of photons and dileptons, in addition to hadrons with moderate
transverse momenta but at large rapidities in $p+$Pb collisions at the LHC
will provide important information supporting the saturation picture of high
density gluon dynamics at high energies and small values of $x$.

##  7 Jets

In this section, several results for jet production and modification in media
are presented. Multi-jet production is discussed and predictions for the jet
yields are shown in Sec. 7.1. Cold nuclear matter effects on jet and dijet
production are presented in Sec. 7.2. The angular decorrelation of dijets due
to saturation effects is discussed in Sec. 7.3. Predictions of long-range
near-side azimuthal collimation event with high multiplicity are given in Sec.
7.4. Finally, enhanced transverse momentum broadening in medium is described
in Sec. 7.5.

###  7.1 Multi-jet Production (N. Armesto)

The study of jet production in Pb+Pb collisions at the LHC [141, 142, 143,
144, 145, 146] and their medium modifications, are important for probing the
properties of the hot and dense matter formed in heavy-ion collisions and is
thus a very hot topic in heavy-ion physics. Therefore, studies of (multi-)jet
production in $p+A$ collisions are of great importance as a cold QCD matter
benchmark. Here the jet rates in minimum bias $p+$Pb collisions at the LHC
(4+1.58 TeV per nucleon) are computed using the Monte Carlo code in Refs.
[147, 148, 149] that implements fixed-order perturbative QCD up to next-lo-
leading order. This code produces at most three jets and contains neither
parton cascades nor hadronization corrections.

In the computation, the renormalization and factorization scales have been set
equal and fixed to $\mu=\mu_{F}=\mu_{R}=E_{T}$, where $E_{T}$ is the total
transverse energy in the event. The central set of the NLO MSTW2008 parton
densities (MSTW2008nlo68cl) [114] have been used for the unmodified nucleons.
The nuclear modification of parton densities was examined using the EKS98
[150, 92] and EPS09NLO [48] sets. The standard Hessian method was employed to
estimate the uncertainties coming from nuclear parton densities. The precision
of the computation, limited by CPU time, gives statistical uncertainties
smaller than 10% for the bin with the highest $E_{T}$ in the results shown
here. The anti-$k_{T}$ jet finding algorithm with $R=0.5$ [151] was used. Only
jets with $E_{T}>20$ GeV are considered. The uncertainties due to the choice
of nucleon parton densities, isospin corrections, and scale fixing, together
with the influence of the jet-finding algorithm and the choice of nuclear
targets and collision energies, were discussed elsewhere [152, 153] and are
not considered here.

Figure 31 shows the sum of the 1-, 2- and 3-jet yields, as well as the
individual 2- and 3-jet yields within four pseudorapidity windows (two
central, one backward and one forward) in the lab frame as a function of the
$E_{T}$ of the hardest jet within the acceptance, $E_{T\,_{\rm hardest}}$. The
yields, computed for luminosity ${\cal L}=0.25\times 10^{29}$ cm-2s-1
integrated over a one month ($10^{6}$ s) run (corresponding to an integrated
luminosity of 25 nb-1) [154], are quite large. The yields are given on the
right-hand vertical axes in Fig. 31. High yields are required to study cold
matter effects in high $p_{T}$ multi-jet production. Figure 31 demonstrates
that such studies are feasible. For example, in the backward region,
$-4.75<\eta<-3$, sufficiently high rates for the sum per GeV of 1+2+3-jet
events can be achieved for $E_{T_{\rm hardest}}<50$ GeV. The only regions
where the yields from certain channels are too low for statistically
significant results are those for 3-jet events in the forward region and 2-
and 3-jets in the backward region (only the yields for 2-jet events are
shown). Note that the turnover in the 2-jet yields when $E_{T_{\rm
hardest}}\rightarrow 20$ GeV comes from singularities that appear in the NLO
calculation due to large logarithms because $E_{T_{1}}\sim E_{T_{2}}(\sim
E_{T_{3}})$. Resummation techniques are required to obtain reliable results in
this kinematic region.

Nuclear modifications of the PDFs are very small, maximum ${\cal O}$(20%),
hardly visible in the yields in Fig. 31. Note that the wide uncertainty bands
in the largest $E_{T_{\rm hardest}}$ bins in the forward and backward regions
are due to statistical fluctuations in the Monte Carlo and do not have a
physical origin. The corresponding hot nuclear matter effects in Pb+Pb
collisions are expected to be much larger.

Further work to compare these yields with experimental data would require
consideration of hadronization corrections and the effects of background
subtraction such as in the well-established jet area method [155]. These
should be the subject of future studies.

Figure 31: The 1+2+3- (black), 2- (red) and 3-jet (green) cross sections as a
function of the $E_{T}$ of the hardest jet within the acceptance. Results in
different pseudorapidity windows in the laboratory frame computed for minimum
bias $p+$Pb collisions are shown. The dashed lines are the results without
nuclear modifications of the PDFs. Results with EPS09NLO [48] (solid) and
EKS98 (dotted) [150, 92] are also shown. The bands correspond to uncertainties
computed using the Hessian method for EPS09 [48]. The right vertical axes
gives the scale for the corresponding yields with an integrated luminosity of
25 nb-1. See the text for further details.

###  7.2 Cold matter effects on jet and dijet production (Y. He, B.-W. Zhang
and E. Wang)

Inclusive NLO [156, 157, 158, 159, 160, 161] jet and dijet production are
studied in $p+$Pb collisions. Cold nuclear matter effects are included by
employing the EPS09 [48], DS11 [162, 163] and HKN07 [164] parametrizations of
nuclear parton densities. The numerical results for the inclusive jet
spectrum, the dijet transverse energy and mass spectra, and the dijet triply-
differential cross sections, all using a jet cone size $R=0.4$ in minimum bias
$p+$Pb collisions are shown.

The left-hand side of Fig. 32 illustrates the inclusive jet spectra scaled by
$N_{\rm coll}$ including the nuclear modifications in the central rapidity
region, $|y|<1$. Results for $R_{p{\rm Pb}}$ are also shown. The spectra with
EPS09 and HKN07 show an enhancement in the transverse energy range
$30<E_{T}<200$ GeV. However, the CNM effects are negligible over the entire
region of jet $E_{T}$ with DS11.

Figure 32: The inclusive jet (left) and dijet (right) results in $p$+Pb
collisions at $\sqrt{s}=5$ TeV. The dijet results, presented as a function of
$E_{T2}$ are given for fixed energy $E_{T1}=100$ GeV for jet 1. The rapidity
acceptance for all jets is $|y|<1$. The $E_{T}$ spectra are shown in the upper
part of the plots while the nuclear modification factors are shown in the
lower panels of the figures.

The right-hand side of Fig. 32 displays the rescaled dijet $E_{T}$ spectra as
a function of $E_{T2}$ with jet 1 at a fixed transverse energy of $E_{T1}=100$
GeV employing the same modifications of the parton densities as before. Both
jets are within the central rapidity region, $|y_{1}|<1$ and $|y_{2}|<1$. The
cross sections in the range $10<E_{T}<200$ GeV exhibit a peak near $E_{T}=100$
GeV. Again the spectra with EPS09 and HKN07 are enhanced. On the other hand,
the DS11 set shows very small suppression and enhancement as shown in Fig 32.

Figure 33 gives the rescaled dijet invariant mass $M_{JJ}$ spectra and the
corresponding nuclear modification factors. Here $M_{JJ}$ is defined as
$[(\sum p^{\mu}_{n})^{2}]^{1/2}$ where the sum is over all particles in the
two jets [165]. The maximum rapidity of the two leading jets,
$|y|_{\max}=\max(|y_{1}|,|y_{2}|)$ is defined so that $|y|_{\max}<1$. Jets
with energies greater than 40 GeV in the rapidity range $|y|<2.8$ are
selected. The CNM effects with EPS09 and HKN07 also enhance the $M_{JJ}$
spectra over a wide $M_{JJ}$ region while the effect with DS11 is modest.

Figure 33 also shows the rescaled dijet triply differential cross sections
[166] and their nuclear modifications. The momentum fractions $x_{1}$ and
$x_{2}$ are defined as

$x_{1}=\sum\limits_{i\in{\rm
jet}}\frac{E_{Ti}}{\sqrt{s}}e^{y_{i}}\,\,,~{}~{}x_{2}=\sum\limits_{i\in{\rm
jet}}\frac{E_{Ti}}{\sqrt{s}}e^{-y_{i}}\,\,.$ (59)

where the sums run over all particles in the jets. The results in Fig. 33 are
shown for the rapidity difference between the jets, $y_{*}=0.5(|y_{1}-y_{2}|)$
where $y_{1}$ and $y_{2}$ are the rapidities of the two leading jets. The
results as a function of $x$ show obvious deviations between nuclear
modification factors which implies that the dijet triply-differential cross
sections could be a good observable to distinguish between different shadowing
parameterizations [167].

Figure 33: (Left) The dijet invariant mass spectra in $p+$Pb collisions.
(Right) The dijet triply differential cross sections in $p+$Pb collisions. The
nuclear modification factors, $R_{p{\rm Pb}}$, are also shown.

###  7.3 Angular decorrelation of dijets as a signature of gluon saturation
(K. Kutak and S. Sapeta)

Here predictions for the emergence of saturation [168] effects on dijet
production in $p+$Pb scattering at the LHC. The results are based on a study
performed in Refs. [169, 170]. The saturation scale characterizes formation of
a dense system of partons. There is growing evidence that the phenomenon of
gluon saturation indeed occurs [171, 172]. This calculation employs a high-
energy factorization formalism which accounts for both the high energy scale
of the scattering and the hard momentum scale $p_{T}$ provided by produced
hard system [173]. Dijets separated by large rapidity gaps [174] are
considered here. More specifically, the focus is on a case where one jet is
measured in the central rapidity region of the detectors while the other is at
large rapidity, see Fig. 34. Such final states probe the parton density in one
of the hadrons at low momentum fraction $x$ while the other is at relatively
large $x$. Predictions of azimuthal decorrelations of dijets production in
$p+$Pb relative to $p+p$ collisions are presented.

Figure 34: Jet production in the forward (assuming the proton moves toward
positive rapidity) region in hadron-hadron collisions.

Assuming, without loss of generality, that $x_{1}\simeq 1$ and $x_{2}\ll 1$
(if the proton moves in the direction of forward rapidity), the cross section
takes the form

$\displaystyle\frac{d\sigma}{dy_{1}dy_{2}dp_{T1}dp_{T2}d\Delta\phi}$
$\displaystyle=$
$\displaystyle\sum_{a,c,d}\frac{p_{T1}p_{T2}}{8\pi^{2}(x_{1}x_{2}s)^{2}}$ (60)
$\displaystyle\mbox{}\times{\cal M}_{ag\to
cd}x_{1}f_{a/A}(x_{1},\mu^{2})\,\phi_{g/B}(x_{2},k^{2}_{T})\frac{1}{1+\delta_{cd}}\,\,,$

where

$k_{T}^{2}=p_{T1}^{2}+p_{T2}^{2}+2p_{T1}p_{T2}\cos\Delta\phi\,\,,$ (61)

and $\Delta\phi$ is the azimuthal distance between the outgoing partons. The
squared matrix element ${\cal M}_{ag\to cd}$ includes $2\to 2$ process with
one off-shell initial-state gluon and three on-shell partons $a$, $c$, and
$d$. The following partonic subprocesses contribute to the production of the
dijet system: $qg\to qg$; $gg\to q\bar{q}$; and $gg\to gg$ [175]. The off-
shell gluon in Eq. (60) is described by the unintegrated gluon density
$\phi_{g/B}(x_{2},k^{2}_{T})$, a solution of a nonlinear evolution equation
[176, 177] that depends on the longitudinal momentum fraction $x_{2}$, and the
transverse momentum of the off-shell gluon, $k_{T}$. On the side of the on-
shell parton, probed at high proton $x_{1}$, the collinear parton density
$f_{a/A}(x_{1},\mu^{2})$ is appropriate.

Figure 35: Ratio of differential cross sections for central-forward dijet
production at $\sqrt{s}=5\text{ TeV}$ as functions of azimuthal distance
between the jets $\Delta\phi$ for three different jet $p_{T}$ cuts. On the
left, the selected rapidities correspond to the CMS detector while, on the
right, the rapidity range appropriate for ALICE is shown. The calculations
have been made with the convention that the proton beam moves toward forward
rapidity.

Figure 35 shows the ratios of the differential cross sections for central-
forward dijet production in $p+$Pb relative to $p+p$ as a function of the
azimuthal distance between the jets, $\Delta\phi$. In the region
$\Delta\phi\sim\pi$, the gluon density is probed at small $k_{T}$, where it is
strongly suppressed due to nonlinear effects. The ratio is a signal of
saturation that is sensitive to the enhancement of saturation effects afforded
by the larger $A$ of the Pb nucleus.

The left-hand side of Fig. 35 shows the results for various jet $p_{T}$ cuts:
$p_{T\,{\rm jet}}>15$, 25, and 35 GeV/$c$ while the jet rapidities are
restricted to positive values within the coverage of the central and forward
CMS detectors. On the right-hand side of Fig. 35, lower $p_{T}$ cuts,
$p_{T}>10$ and 20 GeV/$c$, are used and the rapidity is restricted to positive
values within the central and larger rapidity range of the ALICE detector. The
results show that, when the two jets are back-to-back, the cross section ratio
is significantly smaller than one. This is a consequence of stronger gluon
saturation in the Pb nucleus than in a proton. Thus the unintegrated gluon
distribution in the region of small and medium $k_{T}$ is more suppressed in
Pb than in protons, as shown in Fig. 35. In addition to the effects shown
here, the region near $\Delta\phi\simeq\pi$ is sensitive to Sudakov effects
which further suppress the cross section. Therefore, refinement along those
lines could be envisaged in the future. However, the ratio shown here is less
sensitive to these corrections. It is worth emphasizing that the suppression
due to saturation predicted in Fig. 35 is strong and extends over a large
enough $\Delta\phi$ range for experimental observation, even if the region
near $\Delta\phi=\pi$ could be further refined.

###  7.4 Predictions for long-range near-side azimuthal collimation in high
multiplicity p+Pb collisions (K. Dusling and R. Venugopalan)

Rapidity-separated dihadron correlation measurements can provide valuable
insight into the gluon dynamics of the nuclear wavefunction. For example, it
was recently realized [172] that a novel “near-side” azimuthal collimation in
high multiplicity $p+p$ events [178] is a consequence of nonlinear gluon
interactions and was found to be in excellent agreement with computations in
the Color Glass Condensate (CGC) effective field theory [179, 180].

The situation in $p+$Pb collisions is more attractive. The degree of near-side
collimation is highly sensitive to the fine structure of the unintegrated
gluon distribution. It can be shown that the collimated yield is enhanced by
$\left(Q_{s}^{\rm Pb}/Q_{s}^{p}\right)^{2}$ in asymmetric collisions. Here
$Q_{s}^{\rm Pb}$ is the saturation scale probed in the lead nucleus and can be
estimated to scale with the number of participants, $Q_{s}^{2,\rm Pb}\approx
N_{\rm part}Q_{s}^{2,p}$, where $Q_{s}^{p}$ is the saturation scale for the
proton. Therefore, for high multiplicity events where many nucleons in the
lead nucleus participate, a significant near-side collimation is observed
[181]. Similar to the $p+p$ data, calculations within the CGC effective theory
are able to explain the data [182].

This last study combined the physics of saturation and BFKL dynamics in order
to understand the near-side collimation and the recoiling away-side jet. Both
are required for a full understanding of the azimuthal structure of dihadron
correlations. The per-trigger yields shown in Fig. 36 are well described by
this framework for the symmetric $p_{T}^{\rm trig}\sim p_{T}^{\rm assoc}$
windows where data are currently available. Predictions for the asymmetric
$p_{T}^{\rm trig}\neq p_{T}^{\rm assoc}$ windows are also shown.

Figure 36: Correlated yield $1/N_{\rm trig}d^{2}N/d\Delta\phi$ after ZYAM
(zero yield at minimum, used to remove the $\Delta\Phi$-independent pedestal)
as a function of $|\Delta\phi|$ integrated over $2\leq|\Delta\eta|\leq 4$ for
the most central multiplicity bin $N_{\rm trk}^{\rm offline}\geq 110$. The CMS
data [181] have currently only been provided for the diagonal components,
$p_{T}^{\rm trig}\sim p_{T}^{\rm assoc}$, of the correlation matrix. The
theory curves are the result of adding the BFKL contribution responsible for
the away-side jet and the “dipole”-like Glasma contribution. The solid black
curve is the result for $Q_{0p}^{2}=0.504$ GeV2 on $N_{\rm part}^{\rm Pb}=14$
and the dashed green is for $Q_{0p}^{2}=0.336$ GeV2 on $N_{\rm part}^{\rm
Pb}=16$.

###  7.5 Enhancement of transverse momentum broadening (H. Xing, Z.-B. Kang,
I. Vitev and E. Wang)

Both initial-state and final-state multiple scattering lead to acoplanarity,
or momentum imbalance of the two leading final-state particles. To quantify
this effect, the transverse momentum imbalance $\vec{q}_{T}$ is defined as:

$\displaystyle\vec{q}_{T}=\vec{p}_{T1}+\vec{p}_{T2},$ (62)

with the average squared transverse momentum imbalance

$\displaystyle\langle q_{T}^{2}\rangle=\left(\int
d^{2}\vec{q}_{T}q_{T}^{2}\frac{d\sigma}{d\mathcal{PS}\,d^{2}\vec{q}_{T}}\right)\left(\frac{d\sigma}{d\mathcal{PS}}\right)^{-1}\,\,.$
(63)

Here, $d\sigma/d\mathcal{PS}$ is the differential cross section with the
appropriate phase space factor, determined separately for each process. For
example, in $p+A$ collisions, $d\mathcal{PS}=dy_{1}dy_{2}dp_{T}^{2}$ for
photon+jet production and $d\mathcal{PS}=dy_{1}dy_{2}dp_{T1}dp_{T2}$ for
photon+hadron production.

The enhancement of the transverse momentum imbalance (or nuclear broadening)
in $h+A$ ($h=p$, $\gamma^{*}$) collisions relative to $h+p$ collisions can be
quantified by the difference:

$\displaystyle\Delta\langle q_{T}^{2}\rangle=\langle
q_{T}^{2}\rangle_{hA}-\langle q_{T}^{2}\rangle_{hp}\,\,.$ (64)

The broadening $\Delta\langle q_{T}^{2}\rangle$ is a result of multiple quark
and gluon scattering and is a direct probe of the properties of the nuclear
medium. Both initial-state and final-state multiple parton interactions are
taken into account to calculate the nuclear broadening $\Delta\langle
q_{T}^{2}\rangle$ for photon+jet (photon+hadron) and heavy quark (heavy meson)
pair production in $p+A$ collisions.

The nuclear broadening $\Delta\langle q_{T}^{2}\rangle$ can be calculated in
perturbative QCD. A specific method based on double parton scattering has been
discussed in detail in Refs. [183, 80]. The derivation discussed here closely
follows Ref. [80]. The leading contribution to the nuclear broadening comes
from double scattering: either in the initial-state or the final-state. The
contributions from these diagrams in the covariant gauge can be calculated to
obtain the following expression for the nuclear broadening of photon+jet
production in $p+A$ collisions:

$\displaystyle\Delta\langle q_{T}^{2}\rangle$ $\displaystyle=$
$\displaystyle\left(\frac{8\pi^{2}\alpha_{s}}{N_{c}^{2}-1}\right)$ (65)
$\displaystyle\mbox{}\times\frac{\sum_{a,b}f_{a/p}(x^{\prime})\left[T_{b/A}^{(I)}(x)H^{I}_{ab\to\gamma
d}(\hat{s},\hat{t},\hat{u})+T_{b/A}^{(F)}(x)H^{F}_{ab\to\gamma
d}(\hat{s},\hat{t},\hat{u})\right]}{\sum_{a,b}f_{a/p}(x^{\prime})f_{b/A}(x)H^{U}_{ab\to\gamma
d}(\hat{s},\hat{t},\hat{u})}\,\,.$

Here $T_{b/A}^{(I)}(x)=T_{q/A}^{(I)}(x)$ (or $T_{g/A}^{(I)}(x)$) are twist-4
quark-gluon (or gluon-gluon) correlation functions associated with initial-
state multiple scattering, defined as [183, 80]

$\displaystyle T_{q/A}^{(I)}(x)$ $\displaystyle=$
$\displaystyle\int\frac{dy^{-}}{2\pi}\,e^{ixp^{+}y^{-}}\int\frac{dy_{1}^{-}dy_{2}^{-}}{2\pi}\,\theta(y^{-}-y_{1}^{-})\,\theta(-y_{2}^{-})$
(66) $\displaystyle\mbox{}\times\frac{1}{2}\,\langle p_{A}|F_{\alpha}^{\
+}(y_{2}^{-})\bar{\psi}_{q}(0)\gamma^{+}\psi_{q}(y^{-})F^{+\alpha}(y_{1}^{-})|p_{A}\rangle\,\,,$
$\displaystyle T_{g/A}^{(I)}(x)$ $\displaystyle=$
$\displaystyle\int\frac{dy^{-}}{2\pi}\,e^{ixp^{+}y^{-}}\int\frac{dy_{1}^{-}dy_{2}^{-}}{2\pi}\,\theta(y^{-}-y_{1}^{-})\,\theta(-y_{2}^{-})$
(67) $\displaystyle\mbox{}\times\frac{1}{xp^{+}}\,\langle
p_{A}|F_{\alpha}^{~{}+}(y_{2}^{-})F^{\sigma+}(0)F^{+}_{~{}\sigma}(y^{-})F^{+\alpha}(y_{1}^{-})|p_{A}\rangle\,\,.$

The corresponding twist-4 correlation functions associated with final-state
multiple scatter are $T_{q/A}^{(F)}(x)$ and $T_{g/A}^{(F)}(x)$. They are
defined as in Eqs. (66) and (67), except the $\theta$-functions are replaced
such that [183, 80],

$\displaystyle\theta(y^{-}-y_{1}^{-})\,\theta(-y_{2}^{-})\to\theta(y_{1}^{-}-y^{-})\,\theta(y_{2}^{-})\,\,.$
(68)

The broadening in photon + hadron production is calculated similarly.

Nuclear broadening is evaluated in a formalism where multiple scattering
contributes to the cross section via higher-twist matrix elements in the
nuclear state. This framework follows a well-established QCD factorization
formalism for particle production in $p+A$ collisions and has previously been
used to describe cold nuclear matter effects such as energy loss, dynamical
shadowing and broadening. This work differs from more generic parton
broadening phenomenology in because the color and kinematic structures of the
hard part are evaluated exactly. In particular, the nuclear enhancement of the
transverse momentum imbalance is studied in dijet and photon+jet production.

Figure 37: Nuclear broadening, $\Delta\langle q_{T}^{2}\rangle$, is shown for
dijet (left) and photon+jet (right) production at specified rapidities in
$p+A$ collisions as a function of $N_{\rm coll}$. Results are shown for
$y_{1}=y_{2}=2$ in $\sqrt{s}=5$ TeV $p+$Pb collisions and $y_{1}=y_{2}=1$ at
$\sqrt{s}=200$ GeV d+Au collisions. (The calculations have been made with the
convention that the proton beam moves toward forward rapidity.) The jet
transverse momentum is integrated over $30<p_{T}<40$ GeV/$c$ at $\sqrt{s}=5$
TeV and $15<p_{T}<25$ GeV/$c$ at $\sqrt{s}=200$ GeV. The red line shows the
result for RHIC kinematics with scattering parameter $\xi^{2}=0.12$ GeV2 while
the yellow band represents the variation of $\xi^{2}$ in the LHC kinematics.

The results are shown in Fig. 37. The line gives the baseline $\Delta\langle
q_{T}^{2}\rangle$ determined from RHIC as a function of $N_{\rm coll}$. The
band is the result in the LHC kinematics for a plausible range of the
scattering parameter $\xi^{2}$, defined in Eq. (29). The band is broader for
dijet production and $\Delta\langle q_{T}^{2}\rangle$ is also larger. There is
also a somewhat larger deviation of the bottom edge of the band from the RHIC
result for dijet production.

##  8 Gauge Bosons

Gauge boson production is discussed in this section. Section 8.1 makes
predictions for $W$ and $Z^{0}$ $p_{T}$ and rapidity distributions in $p+p$
and $p+$Pb collisions. The $W^{\pm}$ charge asymmetry as a function of the
decay lepton rapidity is also shown. Section 8.2 describes a calculation of
the resummed $Z^{0}$ $p_{T}$ distribution while transverse broadening of
vector boson production is discussed in Sec. 8.3.

###  8.1 $W$ and $Z$ production and $W^{\pm}$ charge asymmetry (P. Ru, E.
Wang, B.-W. Zhang and W.-N. Zhang)

Production of the gauge bosons, $W^{+},W^{-}$ and $Z^{0}$, is discussed. The
transverse momentum and the rapidity distributions in min-bias $p+$Pb
collisions at $\sqrt{s}=5$ TeV are calculated at next-to-leading order and
next-to-next-to-leading order. The fiducial cross sections $\sigma_{\rm fid}$
are calculated in the fiducial phase space for vector boson production used by
ATLAS Collaboration [184, 185] in both $p+$Pb and $p+p$ collisions.

The fiducial $Z^{0}$ cross section is the inclusive cross section $p+\,{\rm
Pb}\to Z^{0}/\gamma^{*}+X$ multiplied by the branching ratio for
$Z^{0}/\gamma^{*}\to l^{+}l^{-}$ within the fiducial acceptance. Here $X$
denotes the underlying event and the recoil system. The fiducial acceptance is
assumed to be the same as that defined by ATLAS in $p+p$ collisions [184] with
the following cuts on the lepton transverse momentum and pseudorapidity, and
the dilepton invariant mass: $p_{T}^{l}>20$ GeV/$c$; $|\eta^{l}|<2.4$; and
$66<m_{ll}<116$ GeV/$c^{2}$.

The fiducial $W^{\pm}$ cross sections are the inclusive cross sections
$p+\,{\rm Pb}\to W^{\pm}+X$ multiplied by the branching ratios for $W^{\pm}\to
l\nu$ within the fiducial acceptance. Following the ATLAS definition [185],
the acceptance cuts on lepton and neutrino transverse momentum and
pseudorapidity as well as the $W$ transverse mass are: $p_{T}^{l}>20$ GeV/$c$;
$|\eta_{l,\nu}|<2.4$; $p_{T}^{\nu}>25$ GeV/$c$; and
$m_{T}=\sqrt{2p_{T}^{l}p_{T}^{\nu}(1-\cos(\phi^{l}-\phi^{\nu}))}>40$
GeV/$c^{2}$.

Results are simulated to NLO ($\mathcal{O}(\alpha_{s})$) and NNLO
($\mathcal{O}(\alpha_{s}^{2})$) in the total cross section employing
$\mathtt{DYNNLO}$ for Drell-Yan-like production in hadron-hadron collisions
[186]. The MSTWNLO and MSTWNNLO proton parton densities are used, along with
the EPS09 [48] and DSSZ [163] shadowing parameterizations.

Tables 8.1 and 8.1 shows the fiducial cross sections, normalized to their per
nucleon values for better comparison. The NLO and NNLO values are shown for
comparison, both for $p+$Pb and $p+p$ collisions at the same energy. The ratio
between the NNLO and NLO cross section, giving some indication of the
theoretical uncertainty and the convergence of the perturbative expansion for
gauge boson production, is $\sim 1.02$ showing that the higher order
corrections are small. When the NLO and NNLO results are compared in $p+$Pb
collisions with the EPS09 shadowing parameterization, a similar correction is
found. The difference between the EPS09 and DSSZ parameterizations, both
calculated at NLO, is also quite small, on the order of 1%. There is a slight
decrease in the total fiducial $Z^{0}$ cross section. There is a larger
decrease for $W^{+}$ and an enhancement in the fiducial cross section in
$p+$Pb collisions. This is less an effect of shadowing than it is of isospin
since $u\overline{d}\to W^{+}$ and $\overline{u}d\to W^{-}$ and there are more
$d$ quarks in the lead nucleus, causing the enhanced cross section.

The total vector boson production cross sections in the fiducial phase space,
$\sigma_{\rm fid}$, in units of nb in $p+p$ collisions. The results at NLO and
NNLO are compared.  $p+p$ $\sigma_{\rm fid}$ (nb) Decay channel MSTWNNLO
MSTWNLO $Z\to e^{+}e^{-}$ 0.339 0.332 $W^{+}\to e^{+}\nu$ 2.35 2.30 $W^{-}\to
e^{-}\nu$ 1.47 1.44

The total vector boson production cross sections per nucleon in the fiducial
phase space, $\sigma_{\rm fid}/\langle N_{\rm bin}\rangle$ in $p+$Pb
collisions. Columns 2 and 3 compare the results at NLO and NNLO calculated
with EPS09 while columns 3 and 4 compare the EPS09 and DSSZ shadowing
parameterizations at NLO.  $p+$Pb $\sigma_{\rm fid}/\langle N_{\rm
coll}\rangle$ (nb) MSTWNNLO MSTWNLO MSTWNLO Decay channel EPS09 EPS09 DSSZ
$Z\to e^{+}e^{-}$ 0.338 0.328 0.329 $W^{+}\to e^{+}\nu$ 2.06 2.02 2.05
$W^{-}\to e^{-}\nu$ 1.54 1.52 1.53

Figure 38 shows the $Z^{0}$ $p_{T}$ distributions in $p+$Pb collisions at both
NLO and NNLO. The NNLO result is somewhat higher and not as smooth as the NLO
calculations which appear independent of the choice of shadowing
parameterization. Differences between the results with EPS09 and DSSZ, which
can be attributed to shadowing effects rather than isospin, are only apparent
when the ratio $R_{p\,{\rm Pb}}(p_{T})$ is formed. The slight decrease in per
nucleon yield in $p+$Pb relative to $p+p$ seen in the total cross sections in
Tables 8.1 and 8.1 are due to the lowest $p_{T}$ bin, $p_{T}<20$ GeV/$c$. At
higher $p_{T}$, the ratio increases above unity. However, the effect is not
significantly larger than 5% over the entire $p_{T}$ range.

Figure 38: (Left) Normalized $Z^{0}$ differential cross section
$(1/\sigma_{\rm fid})(d\sigma_{\rm fid}/dp_{T})$. (Right) The suppression
factor $R_{pA}(p_{T})$ in 20 GeV/$c$ $p_{T}$ bins.

The individual $W^{+}$ and $W^{-}$ $p_{T}$ distributions are shown at NLO and
NNLO on the left-hand side of Fig. 39. Again, the NLO results are smoother
than the NNLO calculations. The difference in the overall cross sections are
clearly observable: in the fiducial range of the calculations, the $W^{+}$
cross section is $\approx 35$% greater than the $W^{-}$ cross section. The
ratio of $p+$Pb to $p+p$ is shown on the right-hand side of Fig. 39. The
$W^{-}$ ratio is larger than unity and increasing strongly with $p_{T}$. The
effect is due to the greater abundance of $d$ quarks in the Pb nucleus (126
neutrons vs. 82 protons). The valence $d$ quark distributions in the neutrons,
equivalent to the valuence $u$ quark distributions in the protons, have a
larger density at relatively high $x$, causing the observed increase.
Conversely, the lower density of valence $u$ quarks in the Pb nucleus causes
$R_{p{\rm Pb}}<1$ over the entire $p_{T}$ range. The sum of the two charged
gauge bosons shows a trend very similar to that of the $Z^{0}$ in Fig. 38,
revealing a result closer to the true shadowing effect. Even though the
$W^{+}$ cross section is greater than that of the $W^{-}$, the isospin effect
on the $W^{-}$ is large enough to make the ratio larger than unity at high
$p_{T}$.

Figure 39: (Left) The normalized $W=W^{+}+W^{-}$ differential cross section
$(1/\sigma_{\rm fid})(d\sigma_{\rm fid}/dp_{T})$. (Right) The suppression
factor $R_{pA}(p_{T})$ for $W$ production given in 20 GeV/$c$ $p_{T}$ bins.

Figures 40 and 41 show the $Z^{0}$, $W^{+}$ and $W^{-}$ rapidity
distributions, normalized per nucleon. For comparison the $p+p$ and $p+$Pb
distributions are shown both at NLO and NNLO. Aside from numerical
fluctuations at NNLO, the order of the calculation makes little difference in
either the shape or the magnitude of the rapidity distributions. The $p+p$
distributions are all symmetric around $y=0$ while the $p+$Pb distributions
are peaked in the direction of the Pb nucleus.

Figure 40: The $Z^{0}$ rapidity distribution $(1/\langle N_{\rm
coll}\rangle)(d\sigma_{\rm fid}/dy)$. Results are shown for both $p+$Pb and
$p+p$ collisions in the center-of-mass frame for both systems.

There are significant differences between the $W^{+}$ and $W^{-}$
distributions even for $p+p$ collisions. The $W^{+}$ distribution is
considerably broader with peaks away from $y=0$, at $|y|\sim 1.7$ due to the
larger average momentum fraction $x$ of the valence $u$ quarks in the proton.
The greater density of valence $u$ quarks in the proton leads to the larger
overall cross section. The isospin effect tends to make the $W^{+}$ rapidity
distribution more symmetric since the valence $u$ quark distribution in the
neutron, equivalent to the valence $d$ distribution in the proton, has a
smaller average $x$ and lower density which reduces the cross section while
removing the peaks away from midrapidity. The $W^{-}$ distribution in $p+p$
collisions is both smaller and narrower than the $W^{+}$. This distribution
shows the strongest isospin effect with rapidity.

Figure 41: The $W^{+}$ (left) and $W^{-}$ (right) rapidity distributions
$(1/\langle N_{\rm coll}\rangle)(d\sigma_{\rm fid}/dy)$. Results are shown for
both $p+$Pb and $p+p$ collisions in the center-of-mass frame for both systems.

These differences are reflected in the $W^{\pm}$ charge asymmetry, defined as
$(N_{W^{+}}-N_{W^{-}})/(N_{W^{+}}+N_{W^{-}})$ and shown as a function of the
decay lepton pseudorapidity. The results are given in Fig. 42. The $p+p$
asymmetry is symmetric around $y=0$ with a strong dip at midrapidity. The
origin is clear from the individual $W^{+}$ and $W^{-}$ rapidity
distributions. There is, however, a strong forward/backward asymmetry in
$p+$Pb collisions. In the direction of the proton beam, the asymmetry follows
that of the $p+p$ result. It falls off and becomes negative in the direction
of the lead beam. This trend is independent of the order of the calculation
and the shadowing parameterization used.

Figure 42: The $W^{\pm}$ charge asymmetry,
$(N_{W^{+}}-N_{W^{-}})/(N_{W^{+}}+N_{W^{-}})$, as a function of the charged
lepton pseudorapidity in both $p+p$ and $p+$Pb collisions in the center-of-
mass frame for both systems.

###  8.2 Nuclear modification of the transverse momentum spectrum of $Z^{0}$
production (Z.-B. Kang and J.-W. Qiu)

The Collins-Soper-Sterman formalism [187] is used to calculate $Z^{0}$
production over the full $p_{T}$ range at the LHC,

$\displaystyle\frac{d\sigma_{A+B\rightarrow Z^{0}+X}}{dy\,dp_{T}^{2}}$
$\displaystyle=$ $\displaystyle\frac{1}{(2\pi)^{2}}\int
d^{2}b\,e^{i\vec{p}_{T}\cdot\vec{b}}\,\tilde{W}(b,M_{Z},x_{1},x_{2})+Y(p_{T},M_{Z},x_{1},x_{2})\,.$
(69)

The $\tilde{W}$ term gives the dominant contribution when $p_{T}\ll M_{Z}$
while the $Y$ term is perturbatively calculable, see Ref. [188], allowing a
smooth transition from the resummed low $p_{T}$ region to $p_{T}\sim M_{Z}$
where the fixed-order perturbative QCD calculations work well. In Eq. (69),
$x_{1}=e^{y}\,M_{Z}/\sqrt{s}$ and $x_{2}=e^{-y}\,M_{Z}/\sqrt{s}$ while
$\tilde{W}$ is given by [188]

$\tilde{W}(b,M_{Z},x_{1},x_{2})=\left\\{\begin{array}[]{ll}\tilde{W}^{\rm
P}(b,M_{Z},x_{1},x_{2})&\quad\mbox{$b\leq b_{\rm max}$}\\\ \tilde{W}^{\rm
P}(b_{\rm max},M_{Z},x_{1},x_{2})\,\tilde{F}^{\rm
NP}(b,M_{Z},x_{1},x_{2};b_{\rm max})&\quad\mbox{$b>b_{\rm
max}$}\end{array}\right.$ (70)

where $b_{\rm max}\sim 1/$(few GeV) is a parameter that specifies the region
in which $\tilde{W}^{\rm P}$ is perturbatively valid, and $\tilde{F}^{\rm NP}$
is a nonperturbative function determining the large $b$ behavior of
$\tilde{W}$ and is defined below. In Eq. (70), $\tilde{W}^{\rm
P}(b,M_{Z},x_{1},x_{2})$ includes all powers of large perturbative logarithms
resummed from $\ln(1/b^{2})$ to $\ln(M_{Z}^{2})$ [187]

$\tilde{W}^{\rm P}(b,M_{Z},x_{1},x_{2})={\rm e}^{-S(b,M_{Z})}\,\tilde{W}^{\rm
P}(b,c/b,x_{1},x_{2})\,,$ (71)

where $c$ is a constant of order one [187, 188], and

$S(b,M_{Z})=\int_{c^{2}/b^{2}}^{M_{Z}^{2}}\,\frac{d{\mu}^{2}}{{\mu}^{2}}\left[\ln\left(\frac{M_{Z}^{2}}{{\mu}^{2}}\right)A(\alpha_{s}({\mu}))+B(\alpha_{s}({\mu}))\right],$
(72)

with perturbatively-calculated coefficients $A(\alpha_{s})$ and
$B(\alpha_{s})$ given in Ref. [188] and references therein. The perturbative
factor in Eq. (71), $\tilde{W}^{\rm P}(b,c/b,x_{1},x_{2})$, has no large
logarithms. It is expressed as

$\tilde{W}^{\rm
P}(b,c/b,x_{1},x_{2})=\sigma_{0}\sum_{i=q,\bar{q}}f_{i/A}(x_{1},\mu=c/b)\,f_{\bar{i}/B}(x_{2},\mu=c/b)\,$
(73)

where $\sigma_{0}$ is the leading order $q\overline{q}\rightarrow Z^{0}$
partonic cross section [188]. The functions $f_{i/A}$ and $f_{\bar{i}/B}$ are
the modified parton distributions given by [187]

$f_{i/A}(x_{1},\mu)=\sum_{a}\int_{x_{1}}^{1}\frac{d\xi}{\xi}\,C_{i/a}(x_{1}/\xi,\mu)\,\phi_{a/A}(\xi,\mu)$
(74)

where $\sum_{a}$ is over $a=q,\bar{q},g$, $\phi_{a/A}(\xi,\mu)$ are the normal
proton or effective nuclear parton distribution functions (PDFs) and
$C_{i/a}=\sum_{n=0}C_{i/a}^{(n)}(\alpha_{s}/\pi)^{n}$ are perturbatively
calculable coefficient functions for finding a parton $i$ from a parton $a$,
given in Ref. [188].

The non-perturbative function $\tilde{F}^{\rm NP}$ in Eq. (70) has the form,

$\displaystyle F^{\rm NP}(b,M_{Z},x_{1},x_{2};b_{\rm max})$ $\displaystyle=$
$\displaystyle\exp\Bigg{\\{}-\ln\left(\frac{M_{Z}^{2}b_{\rm
max}^{2}}{c^{2}}\right)\left[g_{1}\left((b^{2})^{\alpha}-(b_{\rm
max}^{2})^{\alpha}\right)\right.$ (75) $\displaystyle+$
$\displaystyle\left.g_{2}\left(b^{2}-b_{\rm
max}^{2}\right)\right]-\bar{g}_{2}\left(b^{2}-b_{\rm
max}^{2}\right)\Bigg{\\}}\,.$

where the explicit logarithmic dependence, $\ln(M_{Z}^{2}\,b_{\rm
max}^{2}/c^{2})$, was derived by solving the Collins-Soper equation [187]. The
${g}_{2}$ term is a result of adding a general power correction to the
renormalization group equation while the $\bar{g}_{2}$ term represents the
size of the intrinsic transverse momentum of active partons [188].

The coefficients of the two terms proportional to $b^{2}$ in Eq. (75) can be
combined [189],

$\displaystyle G_{2}=\ln\left(\frac{M^{2}b_{\rm
max}^{2}}{c^{2}}\right)\,g_{2}+\bar{g}_{2}\,,$ (76)

to sum the dynamical and intrinsic power corrections. By requiring the first
and second derivatives of $\tilde{W}$ to be continuous at $b=b_{\rm max}$, the
parameters $\alpha$ and $g_{1}$ in Eq. (75) can be uniquely fixed, leaving
only one parameter, $G_{2}$, sensitive to the power corrections and other
nonperturbative effects. Taking $\bar{g}_{2}=0.25\pm 0.05$ GeV2 and
$g_{2}=0.01\pm 0.005$ GeV2, $G_{2}^{pp}=0.324$ GeV2. Predictions employing Eq.
(69) are consistent with all $p+\overline{p}$ and $p+p$ data from the Tevatron
and the LHC [188].

The EPS09 NLO parameterization [48] is used to account for the leading-twist
nuclear effects on the parton densities in $p+A$ collisions. Following the
method proposed in Ref. [189], the nuclear-size-enhanced multiple scattering
effects are accounted for by choosing $g_{2}\to g_{2}A^{1/3}$. Then for
$Z^{0}$ production, $G_{2}^{p{\rm Pb}}=0.689{\rm~{}GeV}^{2}$.

In Fig. 43, the predictions for $Z^{0}$ production are shown. The cross
section including resummation in Eq. (69) is evaluated employing the CTEQ6M
parton densities at factorization scale
$\mu=M_{T}/2=0.5\sqrt{M_{Z}^{2}+p_{T}^{2}}$. The $Y$ term is calculated at NLO
in $\alpha_{s}$ [188].

Figure 43:  $Z^{0}$ boson production in $p+p$ and $p+$Pb collisions at
$\sqrt{s}=5$ TeV and $y=0$ [190].

The upper panel shows the $Z^{0}$ production cross section as a function of
$p_{T}$. The black dashed curve is the $p+p$ baseline while the red solid
curve shows the minimum bias $p+$Pb result. The blue dotted curve is the
minimum bias $p+$Pb result without the $A^{1/3}$ enhancement of $g_{2}$ so
that the nuclear-size-enhanced dynamical power corrections from multiple
scattering are absent. The lower panel presents the nuclear modification
factor $R_{pA}$.

In Fig. 43, the red solid curves are almost indistinguishable from the dotted
curves. Thus, power corrections are not important for $Z^{0}$ production at
LHC energies. Therefore, $Z^{0}$ production in $p+A$ collisions is an ideal
probe of the modification of the parton densities in nuclei as well as of the
high energy “isospin” effect.

The $Z^{0}$ cross section at $y=0$ is dominated by gluon-initiated
subprocesses for $p_{T}>20$ GeV/$c$. That is, $R_{pA}$ is an excellent
observable to study nuclear modifications of the gluon distribution,
heretofore effectively unknown, especially at the values of $x$ and $\mu$
probed by $Z^{0}$ production. At factorization scale $\mu=M_{Z}$, the EPS09
gluon shadowing factor [48] is less than unity (shadowing) for $x<0.005$ and
greater than unity (antishadowing) over a sufficiently large range,
$0.005<x<0.2$, as shown in Fig. 44.

Figure 44:  The EPS09 shadowing ratio at scale $Q=M_{Z}$:
$R_{i}^{A}=f_{i/A}(x,Q^{2})/f_{i/p}(x,Q^{2})$. The red solid curve shows the
gluon ratio while the blue dashed curve shows the valence $u$-quark ratio.

As indicated in the lower half of Fig. 43, the nuclear modification factor
$R_{pA}$ for $Z^{0}$ production is suppressed at $p_{T}<10$ GeV/$c$ and
enhanced at high $p_{T}$, up to $p_{T}\sim 70$ GeV/$c$. The low $p_{T}$
suppression is a result of low $x$ shadowing in EPS09. However, the strong
enhancement over such a large $p_{T}$ range was unexpected. After a careful
examination of the kinematics, it was found that, at $y=0$, $x\sim
M_{T}/\sqrt{s}\sim 0.013$, already in the EPS09 antishadowing region. Thus,
the clear enhancement of $R_{pA}$ in the large $p_{T}$ region in Fig. 43 can
be explained by the broad EPS09 antishadowing region at the $Z^{0}$ scale. The
large $p_{T}$ enhancement of $R_{pA}$ vanishes for other shadowing
parameterizations with smaller gluon antishadowing. Therefore, the measurement
of $R_{pA}$ for $Z^{0}$ production in $p+$Pb collisions at the LHC provides a
clean and unique test of gluon antishadowing as proposed in the EPS09
parameterization. Furthermore, $R_{pA}$ is a direct measurement of nuclear
gluon distribution since the cross section is dominated by gluon-initiated
subprocesses for $p_{T}>20$ GeV/$c$.

###  8.3 Transverse momentum broadening of vector boson production (Z.-B. Kang
and J.-W. Qiu)

Finally, transverse momentum broadening of inclusive vector boson production,
$A(p_{1})+B(p_{2})\to V[{\rm J/}\psi,\,\Upsilon,\,Z^{0}](q)+X$ at the LHC is
discussed. The average squared transverse momentum of vector boson production
is

$\displaystyle\langle q_{T}^{2}\rangle(y)_{AB}\equiv\int
dq_{T}^{2}\,q_{T}^{2}\,\frac{d\sigma_{AB\to V}}{dy\,dq_{T}^{2}}\left[\int
dq_{T}^{2}\,\frac{d\sigma_{AB\to V}}{dy\,dq_{T}^{2}}\right]^{-1}\,.$ (77)

The transverse momentum broadening in $p+$Pb collisions is defined as

$\displaystyle\Delta\langle q_{T}^{2}\rangle_{p\,{\rm Pb}}(y)\equiv\langle
q_{T}^{2}\rangle(y)_{p\,{\rm Pb}}-\langle q_{T}^{2}\rangle(y)_{pp}\,.$ (78)

Following the derivation in Refs. [183, 130], the first nonvanishing
contribution to the transverse momentum broadening of heavy quarkonium
production is

$\displaystyle\Delta\langle q_{T}^{2}\rangle_{\rm HQ}^{\rm
CEM}=\left(\frac{8\pi^{2}\alpha_{s}}{N_{c}^{2}-1}\,\lambda^{2}\,A^{1/3}\right)\frac{(C_{F}+C_{A})\,\sigma_{q\bar{q}}+2\,C_{A}\,\sigma_{gg}+\Delta\sigma_{gg}}{\sigma_{q\bar{q}}+\sigma_{gg}}$
(79)

where the superscript “CEM” indicates that heavy quarkonium production is
evaluated in the Color Evaporation Model (CEM). A similar result was derived
in the NRQCD approach [183]. The $\sigma_{q\bar{q}}$ and $\sigma_{gg}$
partonic cross sections are contributions from quark-antiquark and gluon-gluon
subprocesses, respectively [183]. The $\Delta\sigma_{gg}$ term is a small,
color-suppressed correction to the $gg$ subprocess derived in Ref. [130]. In
the region where the $gg$ subprocess dominates heavy quarkonium production,
$\sigma_{gg}\gg\sigma_{q\bar{q}},\,\Delta\sigma_{gg}$, heavy quarkonium
broadening is further simplified as [183]

$\displaystyle\Delta\langle q_{T}^{2}\rangle_{\rm HQ}^{\rm CEM}\approx
2\,C_{A}\left(\frac{8\pi^{2}\alpha_{s}}{N_{c}^{2}-1}\,\lambda^{2}\,A^{1/3}\right)\,.$
(80)

In Fig. 45, the predictions of transverse momentum broadening of Drell-Yan
type vector boson production in $p+$Pb collisions at the LHC at $y=0$ are
shown as a function of $N_{\rm coll}$. To determine the effective dependence
on $N_{\rm coll}$, the $A^{1/3}$ in Eq. (79) is replaced by $A^{1/3}N_{\rm
coll}(b)/N_{\rm coll}(b_{\rm min\,bias})$. In $p+$Pb collisions at the LHC, a
Glauber-model calculation with $\sigma_{\rm NN}^{\rm in}=70$ mb at
$\sqrt{s}=5$ TeV gives $N_{\rm coll}(b_{\rm min\,bias})\sim 7$. In addition to
heavy quarkonium production, the broadening of $W/Z^{0}$ production,
calculated using the formalism derived in Ref. [183], is also shown in Fig.
45. The dramatic difference in the magnitude of the broadening between heavy
quarkonium and $W/Z^{0}$ production in Fig. 45 should be a signature QCD
prediction.

Figure 45: The transverse momentum broadening of vector boson production in
$p+$Pb collisions at $y=0$, shown as a function of $N_{\rm coll}$. The
$\Upsilon$ (red solid), $J/\psi$ (red dashed), $W^{\pm}$ (black solid), and
$Z^{0}$ (black dashed) results are given [190].

##  9 Acknowledgments

The research of J. L. Albacete is supported by a fellowship from the Théorie
LHC France initiative funded by the IN2P3. The work of N. Armesto was
supported by the European Research Council grant HotLHC ERC-2001-StG-279579;
by Ministerio de Ciencia e Innovación of Spain grants FPA2008-01177,
FPA2009-06867-E and Consolider-Ingenio 2010 CPAN CSD2007-00042; by Xunta de
Galicia grant PGIDIT10PXIB 206017PR; and by FEDER. G. G. Barnafóldi was
partially supported by the János Bolyai Research Scholarship of the Hungarian
Academy of Sciences. G. G. Barnafóldi, M. Gyulassy, and P. Levai also
acknowledge Hungarian grants OTKA PD73596, NK77816, NK106119, NIH
TET_10-1_2011-0061 and ZA-15/2009. J. Barrette and V. Topor Pop are supported
by the Natural Sciences and Engineering Research Council of Canada. S. De is
grateful to the Department of Atomic Energy of India for financial support. A.
Dumitru is supported by the DOE Office of Nuclear Physics through Grant No.
DE-FG02-09ER41620 and by The City University of New York through the PSC-CUNY
Research Award Program, grant 65041-0043. K. Dusling is supported by the US
Department of Energy under DOE Contract No. DE-FG02-03ER41260. K. J. Eskola is
supported by the Academy of Finland, Project 133005. R. J. Fries would like to
acknowledge support by NSF CAREER Award PHY-0847538 and by the JET
Collaboration and DOE grant DE-FG02-10ER41682. H. Fujii and Y. Nara are
supported in part by Grant-in-Aid for Scientific Research (B) 22340064. F.
Gelis is supported by the Agence Nationale de la Recherche project
11-BS04-015-01. M. Gyulassy is supported by the Division of Nuclear Science,
U.S. Department of Energy, under Contract No. DE-AC03-76SF00098 and DE-
FG02-93ER-40764 (associated with the JET Topical Collaboration Project). I.
Helenius is supported by the Magnus Ehrnrooth Foundation. The work of B. Z.
Kopeliovich was partially supported by Fondecyt (Chile) grant No. 1090291. The
work of K. Kutak and S. Sapeta was partially supported by the Foundation for
Polish Science with the grant Homing Plus/2010-2/6. The work of A. H. Mueller
is supported in part by the US Department of Energy under contract No. DE-
FG02-92ER-40699. The work of J. Nemchik was supported by grants VZ MŠMT
6840770039 and LA 08015 (Ministry of Education of the Czech Republic). M.
Petrovici is supported by the Romanian Authority for Scientific Research,
CNCS-UEFIS-CDI project number PN-II-ID-2011-3-0368. The work of A. H. Rezaeian
was partially supported by Fondecyt (Chile) grant No. 1110781. R. Venugopalan
was supported by US Department of Energy under DOE Contract No. DE-
AC02-98CH10886. The work of R. Vogt was performed under the auspices of the
U.S. Department of Energy by Lawrence Livermore National Laboratory under
Contract DE-AC52-07NA27344 and within the framework of the JET Collaboration.
The work of X.-N. Wang was performed under the auspices of the U.S. Department
of Energy under Contract No. DE-AC02-05CH11231, by the National Natural
Science Foundation of China under grant No. 11221504, and within the framework
of the JET Collaboration. I. Vitev is supported by the US Department of
Energy, Office of Science, under Contract No. DE-AC52-06NA25396 and by the
LDRD program at LANL.

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# Geometrical dependence in Casimir-Polder repulsion

John Joseph Marchetta<EMAIL_ADDRESS>Department of Physics, Southern
Illinois University–Carbondale, Carbondale, Illinois 62901, USA  Prachi
Parashar<EMAIL_ADDRESS>John A. Logan College, Carterville,
Illinois 62918, USA  K. V. Shajesh<EMAIL_ADDRESS>Department of Physics,
Southern Illinois University–Carbondale, Carbondale, Illinois 62901, USA

###### Abstract

Repulsion, induced from quantum vacuum fluctuations, for an anisotropically
polarizable atom on the symmetry axis of an anisotropically polarizable
annular disc is studied. There exists two torsion free points on each side of
the annular disc, where the interaction energy is orientation independent. The
position of second of the two torsion free points, on either side of the disc,
is shown to determine the orientation dependence of the atom at distances far
from the plate, revealing a geometrical dependence. In the ring limit of the
annular disc, new repulsion emerges.

The Casimir effect is an umbrella term referring to interactions induced by
quantum fluctuations in the electromagnetic field. The associated forces
become dominant in the nanoscale, for neutral configurations. The related van
der Waals and London dispersion forces have to do with the domain in which the
materials interact weakly at short distances van der Waals (1873); Eisenschitz
and London (1930); London (1930); Hettema (2000). Casimir and Polder Casimir
and Polder (1948) extended London dispersion force to include retardation
effects at large distances using fourth order perturbation theory. Casimir
then computed the attraction between two neutral conducting plates, which was
inaccessible to perturbation theory, by evaluating the electromagnetic zero
point energy outside and inside the plates with only the difference
contributing to a finite inward pressure Casimir (1948). Thus, Casimir’s
result suggested that the zero point energy could have measurable
consequences.

Repulsive forces from the Casimir effect counter the intuition given for its
usual attractive nature. Nevertheless, the first repulsive forces were found
early on in three-body van der Waals interactions by Axilrod and Teller
Axilrod and Teller (1943) and Muto Muto (1943), and by Craig and Power Craig
and Power (1969a, b) in Casimir-Polder interactions between anisotropically
polarizable atoms. The counterintuitive nature of repulsion induced from
quantum fluctuations explains why it was not until a decade ago that a
repulsive effect was realized between conductors in the (strong and retarded)
nonperturbative Casimir regime. This came by in Ref. Levin  _et al._ (2010),
where it was argued on physical grounds that the interaction energy between an
elongated conductor and a perfectly conducting metal sheet with a circular
aperture could have a local minima. They showed that the Casimir force could
become repulsive when the needle got sufficiently close to the aperture. It
was suggested that the anisotropy in the geometry of a highly conducting
object corresponds to an effective anisotropic permittivity and the interplay
of these anisotropic permittivities leads to non-monotonic interaction
energies that causes repulsion McCauley  _et al._ (2011); Shajesh and Schaden
(2012). Only few attempts have been made to understand the result in Ref.
Levin  _et al._ (2010) analytically. In Ref.Eberlein and Zietal (2011), using
the method of inversion from electrostatics, it was shown that even in the van
der Waals (weak and non-retarded) limit, similar configurations lead to
repulsion. In Ref. Abrantes  _et al._ (2018), the method of inversion was
again used to show that in the (weak and non-retarded) van der Waals regime an
anisotropically polarizable atom placed along the symmetry axis of a toroid
could experience repulsion. An advance was made in the retarded regime when an
analytic formula for the (weak and retarded) Casimir-Polder energy between
dielectric bodies was derived in Refs. Shajesh and Schaden (2012) and Milton
_et al._ (2012a), demonstrating that repulsion is possible in configurations
with anisotropically polarizable atoms and anisotropic dielectric materials.
In spite of the above successes in the weak scenario, analytic derivation in
the strong regime of a closed-form expression for the interaction energy
between a polarizable atom and a highly conducting plate with an aperture
still remains elusive Milton  _et al._ (2011, 2012b); Shajesh  _et al._
(2017).

Repulsion in Casimir effect also appears if the interacting bodies possess
certain electric and magnetic properties Boyer (1974), or if the medium’s
conductivity is greater than that of the materials’ Dzyaloshinskii  _et al._
(1961). However, in the current level of our understanding, apparently, these
repulsions have different origins from those discussed in this paper: that of
interaction energies between anisotropically polarizable atoms and
anisotropically polarizable dielectrics.

Figure 1: Point atom of polarizability ${\bm{\alpha}}$ above a dielectric ring
of polarizability ${\bm{\sigma}}$. The atom is on the symmetry axis of the
ring.

In this Letter, we analyze the interaction, including the orientation
dependence, of an anisotropically polarizable atom above the center of an
anisotropically polarizable ring depicted in Fig. 1 using the methods of Refs.
Shajesh and Schaden (2012); Milton  _et al._ (2012a). A full exposition of the
interaction for each polarization basis of the ring and the atom is available
in Ref. Marchetta  _et al._ (2020), which serves as supplementary material for
this article. Casimir-Polder repulsion for an atom interacting with the ring
demonstrates a richer dependence on polarizability than the corresponding
interaction with a plate with a circular aperture in Ref. Shajesh and Schaden
(2012). In Refs. Shajesh and Schaden (2012); Milton  _et al._ (2012a) it was
shown that the atom gets repelled at short distances only when its orientation
deviates no more than a critical angle away from the axis perpendicular to the
plate. An orientation independent point, where the energy is independent of
the orientation of the atom and thus feels no torsion there, was found on both
sides of the plate. It was established that the energy is minimized when its
orientation is parallel to that of the plate for distances less than the
torsion free point, and perpendicular to the plate for larger distances.

We show that, for certain polarizabilities of the ring in Fig. 1, see Ref.
Marchetta  _et al._ (2020) for details, there exist two torsion-free heights
on each side of the ring, in contrast to a single torsion free point on each
side for a plate with circular aperture. We show that for specific
orientations the atom could feel a repulsive force when it is near the center
of the ring, and, in addition, at intermediate distances from the center of
the ring, for example see Fig. 2. To understand this additional repulsive
region, and the additional torsion-free point, we analyze the configuration
involving an atom interacting with an anisotropically polarizable annular
disc. Keeping the inner radius of the annular disc fixed and varying the outer
radius, we can smoothly deform the disc into a ring, or a plate of infinite
extent with a circular aperture. We find upper bounds on the outer radius of
the disc that allows for repulsion at intermediate distances. The second
torsion-free height moves farther away as the outer radius of the disc
increases. Interestingly, even though one torsion-free point moves to infinity
in the plate limit for certain polarizabilities, it does not move afar
indefinitely for all the polarizabilities.

In this Letter we focus our attention to the case when the ring is polarizable
in the direction of the symmetry axis of the ring. For an extensive analysis
we refer to Ref. Marchetta  _et al._ (2020). The discussion in this article
and repulsive Casimir effect in general is expected to be of considerable
interest in nanotechnology, because it could assist in reducing ‘stiction’
among the components within nano devices. As a simpleminded application of a
configuration studied here, we entertain the plausibility that the interaction
between a ring and atom could serve as a prototype for a rotaxane. We discuss
an idealized cycle that could be implemented when designing such molecular
machines.

The Casimir-Polder interaction energy between dilute dielectric bodies is
given in terms of the electric susceptibilities ${\bm{\chi}}_{i}$, $i=1,2,$ of
the bodies as Marchetta  _et al._ (2020)

$E_{12}=-\frac{\hbar
c}{2}\int_{-\infty}^{\infty}\frac{d\zeta}{2\pi}\,\text{Tr}\Big{[}{\bf\Gamma}_{0}\cdot{\bm{\chi}}_{1}\cdot{\bf\Gamma}_{0}\cdot{\bm{\chi}}_{2}\Big{]},$
(1)

where the trace Tr is over both the coordinate and matrix indices. The
polarizable atom in Fig. 1 is suitably described in terms of the
polarizability tensor ${\bm{\alpha}}(\omega)$ of the atom,
${\bm{\chi}}_{1}({\bf r})={\bm{\alpha}}(\omega)\delta^{(3)}({\bf r}-{\bf
r}_{0})$, where ${\bf r}_{0}$ is the position of the atom. The ring,
similarly, is described in terms of the polarizability per unit length
${\bm{\sigma}}(\omega)$ of the ring, ${\bm{\chi}}_{2}({\bf
r})={\bm{\sigma}}(\omega)\delta(\rho-a)\delta(z)$, where $a$ is the radius of
the ring. The polarizabilities are dependent on frequency $\omega$ and the
interaction energy is expressed as an integral over the imaginary frequency
$\omega=i\zeta$. The free Green dyadic ${\bm{\Gamma}}_{0}$ in Eq. (1) is given
by

${\bf\Gamma}_{0}({\bf
r};i\zeta)=\frac{e^{-|\zeta|r}}{4\pi\,r^{3}}\Big{[}-u(|\zeta|r)\,{\bf
1}+v(|\zeta|r)\,\hat{\bf r}\hat{\bf r}\Big{]}$ (2)

with $u(x)=1+x+x^{2}$ and $v(x)=3+3x+x^{2}$. For large distances between the
bodies, the interaction energy in Eq. (1) only gets contributions from very
small frequencies. In the Casimir-Polder limit, the polarizabilities can be
approximated by their static values, $\omega=i\zeta=0$, and then the
$\zeta$-integration can be completed easily.

Here we consider ${\bm{\alpha}}(0)=\alpha_{1}\hat{\bf e}_{1}\hat{\bf e}_{1}$
and ${\bm{\sigma}}(0)=\sigma_{z}\hat{\bf z}\hat{\bf z}$, where $\hat{\bf z}$
is chosen to be in the direction of the symmetry axis of the ring and
$\hat{\bf e}_{1}$ represents the orientation of the polarizability of the atom
as described in Fig. 1. The interaction energy for the atom and ring is found
to be Marchetta  _et al._ (2020)

$\displaystyle E=-\frac{\hbar
c\alpha_{1}\sigma_{z}}{64\pi}\frac{a}{(a^{2}+h^{2})^{\frac{11}{2}}}\Big{[}(26a^{4}+3h^{2}a^{2}+40h^{4})$
(3) $\displaystyle+(26a^{4}-123h^{2}a^{2}+40h^{4})\cos 2\theta\Big{]},$

where the angle $\theta$ is the deviation of the atom’s orientation from the
symmetry axis of the ring in Fig. 1, while $h$ and $a$ denote the height of
the atom above the ring and the radius of the ring, respectively.

Figure 2: The interaction energy between an atom with polarizability
${\bm{\alpha}}=\alpha_{1}\,\hat{\bf e}_{1}\hat{\bf e}_{1}$ and a ring of
polarizability ${\bm{\sigma}}=\sigma_{z}\,\hat{\bm{z}}\hat{\bm{z}}$ is
illustrated in bottom left corner of figure. The energy in Eq. (3) is plotted
as function of height $h$ for different orientations of $\hat{\bf e}_{1}$ with
respect to the symmetry axis of the ring. The first region of repulsion occurs
for $|h|<0.47\,a$ and $60.88^{\circ}<\theta<119.12^{\circ}$. A second region
of repulsion is not visible, however, the inset provides a zoomed in view of
this non-monotonicity, where the switch from attraction to repulsion occurs.
Repulsive force experienced by the atom in the direction of the symmetry axis
will be a manifestation of negative slopes in these plots.

In Fig. 2, we plot the interaction energy of Eq. (3) as a function of height
$h$ for various orientations $\theta$. The energy is minimized at $h=0$ for
$\theta=0^{\circ}$, suggesting that for $h=0$ the atom tries to align its
direction of polarizability with that of the ring. The heights for which the
interaction energy is orientation independent and torsion-free is determined
when the coefficient of $\cos(2\theta)$ in Eq. (3) is zero, which is at
${h}=\pm 0.48\,a$ and $h=\pm\,1.69a$. Upon crossing a torsion-free point, the
orientation that minimizes the energy changes by $90^{\circ}$ as shown in the
inset of Fig 2. Repulsive forces on the atom requires the parameters to
satisfy the inequality

$\frac{h}{a}<\pm\sqrt{\frac{(19+181\cos{2\theta})\pm\sqrt{D}}{80(1+\cos{2\theta})}},$
(4)

where

$D=-6039-11682\cos 2\theta+20601\cos^{2}2\theta.$ (5)

Repulsion occurs when $60.88^{\circ}<\theta<119.12^{\circ}$ for short
distances and has the maximum extent in height for $|h|<0.47\,a$ at
$\theta=90^{\circ}$. When $|\theta|<13.27^{\circ}$, repulsion emerges for a
range of intermediate distances. This range is of maximum extent when
$\theta=0^{\circ}$, for $1.24\,a<|h|<1.41\,a$. The repulsive force at these
intermediate distances, though small in magnitude, might lead to interesting
applications.

We consider an annular disk to understand the differences between the ring and
an aperatured plate. The annular disc is described by ${\bm{\chi}}_{2}({\bf
r})={\bm{\lambda}}(\omega)\theta(b-\rho)\theta(\rho-a)\delta(z)$, where
${\bm{\lambda}}(\omega)$ is the polarizability per unit area of the annular
disc. Here $b$ is the outer radius of the disc and $a$ is the inner radius. We
consider the case ${\bm{\lambda}}(0)=\lambda_{z}\hat{\bf z}\hat{\bf z}$. The
limit as $b\to\infty$ corresponds to a plate with a circular aperture of
radius $a$. Taking the delicate limit of $b\to a$ with the surface
polarizability $\lambda_{z}\to\infty$, such that $\sigma_{z}=\lambda(b-a)$ is
kept fixed, constructs a ring Marchetta  _et al._ (2020). The energy of the
annular disc is computed to be Marchetta  _et al._ (2020)

$\displaystyle E$ $\displaystyle=$ $\displaystyle-\frac{\hbar
c\alpha_{1}\lambda_{z}}{64\pi}\frac{1}{5}\frac{(-1)}{(\rho^{2}+h^{2})^{\frac{9}{2}}}\Big{[}(26\rho^{4}+17h^{2}\rho^{2}+26h^{4})$
(6) $\displaystyle\hskip 28.45274pt+(26\rho^{4}-73h^{2}\rho^{2}+6h^{4})\cos
2\theta\Big{]}\bigg{|}^{\rho=b}_{\rho=a}.$

Unlike the case of ring, in Eq. (4), the repulsive regions can not be captured
in a closed-form expression for the case of a disc, but can be easily
determined numerically. The intermediate region of repulsion, as described in
the zoomed in view of Fig. 2 for a ring, vanishes when the width of the
annulus is above the critical values given in Table 1 for the specific
orientations of the atom.

Orientation |  Criteria for second region of repulsion
---|---
$\theta=0^{\circ}$ | $a<b<1.6505a$
$\theta=6^{\circ}$ | $a<b<1.559a$
$\theta=9^{\circ}$ | $a<b<1.4379a$
$\theta=12^{\circ}$ | $a<b<1.2323a$
Table 1: This table lists criteria for the region of repulsion for
intermediate distances repulsion to occur between an anisotropically
polarizable atom and an annular disc with polarizability in the $z$-direction.
The second column lists the upper bound on the outer radius $b$ of the disc
for orientation angles $\theta$ of the atom.

The four orientation independent heights, $\pm h_{1}$ and $\pm h_{2}$, for the
case of disc are determined by

$\frac{(26b^{4}-73b^{2}h^{2}+6h^{4})}{(b^{2}+h^{2})^{\frac{9}{2}}}-\frac{(26a^{4}-73a^{2}h^{2}+6h^{4})}{(a^{2}+h^{2})^{\frac{9}{2}}}=0,$
(7)

with the solutions above the disc satisfying

$0.48a\xleftarrow[\text{ring}]{a\leftarrow
b}|h_{1}|\xrightarrow[\text{plate}]{b\to\infty}0.60a$ (8a) and
$1.69a\xleftarrow[\text{ring}]{a\leftarrow
b}|h_{2}|\xrightarrow[\text{plate}]{b\to\infty}3.44a.$ (8b)

A difference in the systematics of the features in the energy occurs between
axial versus radial polarizability of the disc Marchetta  _et al._ (2020). The
four orientation independent heights for the case of radial polarizability of
the disc are given by

$0.36a\xleftarrow[\text{ring}]{a\leftarrow
b}|h_{1}|\xrightarrow[\text{plate}]{b\to\infty}0.44a$ (9a) and
$3.45a\xleftarrow[\text{ring}]{a\leftarrow
b}|h_{2}|\xrightarrow[\text{plate}]{b\to\infty}\infty.$ (9b)

Unlike the case of radial polarizability above, in the case of axial
polarizability the two orientation independent points, two on each side of the
disc, exist even in the limit when the disc becomes a plate. In this scenario,
the second orientation independent height increases to a finite value in the
plate limit. This implies that the orientation preference of the atom at $h=0$
and $h\to\infty$ is the same for axial polarizability. This should be
contrasted with that of radially polarizable disc in which case the
orientation preference of the polarizability of the atom at $h=0$ is
orthogonal to that at $h\to\infty$.

Figure 3: Difference in interaction energy $\Delta E$ between $\theta=0$ and
$\theta=\pi/2$ plotted as a function of height $h$. The intersection of the
energy curves with abscissa represent orientation independent points, marked
$h_{1}$ and $h_{2}$. The orientation dependence at each height is shown as
double-arrowed vectors adjacent to the horizontal axis. Observe that the
orientation preferences for the plate is different for radial and axial
polarizabilities.

We summarize our results in Fig. 3, which highlights the orientation
independent torsion free points and the domains of repulsion. For large
distances from the aperture the atom tends to orient its polarizability
perpendicular to the polarizability of the plate when the plate is polarizable
in the radial direction, while it tends to orient parallel when the plate is
polarizable in the axial direction. This feature at $h\to\infty$ is completely
determined by the the existence of the second torsion free point. The
repulsion felt at intermediate distances is about six orders of magnitude
weaker than the repulsion at short distances which still lacks experimental
evidence. It is not clear if the repulsions found for intermediate region of
heights will persist when the conductivity of the plate is larger.

Figure 4: The bottom left part of the figure displays a prototype of a Casimir
machine using an atom and a ring. The interaction energy of the configuration
as a function of height $h$ for $\theta=0$ and $\theta=90^{\circ}$ are
plotted. The points $A$, $B$, $C$, and $D$, on the plots are used to construct
a cyclic path.

As a simpleminded application of the configuration discussed here, we propose
a prototype of Casimir machine along the lines of a rotaxane Anelli  _et al._
(1991), rendering its applicability to molecular transportation, molecular
switching, and nanorecording. By requiring the atom to be fixed on an axis and
allowing the ring to move with center on the axis, we consider a cyclic path
involving the points $A\to B\to C\to D\to A$ in Fig. 4. The atom starts at the
center of the ring while orientated parallel to the polarizability of the
ring, denoted by $A$. We put work into the system to rotate the atom by
$90^{\circ}$ so that the energy is maximized at point $B$. The ring is then
repelled from the atom to $h=0.47a$. By expending a small amount of energy the
ring adjusts its position to the torsion free point at $h_{e}=0.48a$ labeled
in the diagram by $C$. Position $D$ represents the ring at $h_{e}$ but with
the atom rotated, without cost of internal energy, to $\theta=0$. It is
energetically favorable for the atom to then return to the center of ring and
complete the cycle.

In summary, the interaction between an anisotropically polarizable dielectric
annulus and an anisotropically polarizable atom reveals tight constraints on
their relative geometrical orientations in their directions of polarizability
for repulsion. We have shown that the interaction energies are characterized
by torsion free points and domains of repulsion. The intuition gained from
this study will guide us in finding a complete understanding of Casimir
repulsion.

JJM acknowledges support from the REACH Program at Southern Illinois
University–Carbondale. PP and KVS remember Martin Schaden for collaborative
assistance.

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